SPINTRONICS: A FIRST-PRINCIPLES STUDY

Volodymyr Karpan

Samenstelling promotiecommissie:

Prof. dr. ?. ? Universiteit Twente, voorzitterProf. dr. P. J. Kelly Universiteit Twente, promotorProf. dr. P. H. Dederichs Institut fur Festkorperforschung, JulichProf. dr. C. Filippi Universiteit Twente, Universiteit LeidenProf. dr. J. L. Herek Universiteit TwenteProf. dr. B. Koopmans Technische Universiteit EindhovenProf. dr. W. L. Vos Universiteit TwenteDr. M. Zwierzycki Instytut Fizyki Molekularnej, Poznan

This work was supported financially by the “NanoNed” project.

The work described in this thesis was carried out at the“Computational Material Sci-ence”(CMS) group, Faculty of Science and Technology (TNW), University of Twente.

Spintrontronics: fist-principles study.V.M. Karpan,ISBN: ??-???-????-?Thesis Universiteit Twente, Enschede.Copyright c© V.M. Karpan, 2008Printed by ??????, Enschede

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University of Twente (UT)

SPINTRONICS: A FIRST-PRINCIPLES STUDY

PROEFSCHRIFT

ter verkrijging vande graad van doctor aan de Universiteit Twente,

op gezag van de rector magnificus,prof. dr. W. H. M. Zijm,

volgens besluit van het College voor promotiesin het openbaar te verdedigen

op vrijdag 20 juni 2008 om 13.15 uur

door

Volodymyr M. Karpan

geboren op 16 September 1977te Horodenka, Oekraine

Computational Material Science (CMS)

Dit proefschrift is goedgekeurd door:

Prof.dr. P. J. Kelly promotor

to Yulya, my parents

and teachers

Contents

1 Introduction 11.1 Spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Spin injection into semiconductors . . . . . . . . . . . . . . . . 6

1.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . 81.2.2 TB-MTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.3 Wave function matching . . . . . . . . . . . . . . . . . . . . . 121.2.4 Landauer-Buttiker approach . . . . . . . . . . . . . . . . . . . 14

1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Spin injection from Fe into InAs 192.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 FeInAs revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Fe|BL|InAs system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.1 Ideal Fe|Au|InAs junction . . . . . . . . . . . . . . . . . . . . . 272.4.2 Disordered Fe|Au|InAs junction . . . . . . . . . . . . . . . . . . 30

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Influence of Roughness and Disorder on Tunneling Magnetoresis-tance 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Methods, Models and Technical Details . . . . . . . . . . . . . . . . . 383.3 Barrier profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 Ideal Fe|vacuum|Fe MTJ. Transport calculations . . . . . . . . . . . . 42

3.4.1 Parallel configuration: Majority channel . . . . . . . . . . . . . 433.4.2 Parallel configuration: Minority channel . . . . . . . . . . . . . 44

vii

viii Contents

3.5 Effect of interfacial roughness on the TMR . . . . . . . . . . . . . . . 493.5.1 Majority transmission . . . . . . . . . . . . . . . . . . . . . . . 523.5.2 Minority transmission . . . . . . . . . . . . . . . . . . . . . . . 533.5.3 Antiparallel spin alignment . . . . . . . . . . . . . . . . . . . . 55

3.6 Substitutional disorder: FexCo1−x electrodes . . . . . . . . . . . . . . 553.7 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 573.8 Appendix 1: The k-point sampling . . . . . . . . . . . . . . . . . . . . 593.9 Appendix 2: Configurational averaging and the size of the scattering

region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 Recovering Julliere model: ab-initio study 674.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Early models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3 Ab-initio results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.1 Factorization of conductance . . . . . . . . . . . . . . . . . . . 734.3.2 Factorization of conductance in case of finite bias . . . . . . . . 76

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Graphene and Graphite as Perfect Spin Filters 835.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3 Geometry and electronic structure of TM|Grn|TM . . . . . . . . . . . 88

5.3.1 Graphite and graphene . . . . . . . . . . . . . . . . . . . . . . 885.3.2 Graphene on Ni(111) substrate . . . . . . . . . . . . . . . . . . 905.3.3 Ni|Grn|Ni(111) junction . . . . . . . . . . . . . . . . . . . . . . 91

5.4 Electron transport through a FM|Grn|FM junction . . . . . . . . . . . 945.4.1 Specular interface . . . . . . . . . . . . . . . . . . . . . . . . . 945.4.2 Ni|Cum|Grn|Cum|Ni (111) . . . . . . . . . . . . . . . . . . . . 975.4.3 Effect of disorder . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 100Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 A new material system for highly planar electronics 105Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Chapter 1

Introduction

In this thesis, spin-polarized transport through multilayered systems is studied. Ageneral introduction to the eld of spintronics is presented in this chapter. Prob-lems and progress in the main research areas of spintronics: magnetoresistance andspin injection into nonmagnetic materials are discussed. Then the major details ofthe theoretical approach, namely the density functional theory, linear-mun tin or-bitals method, mode-matching scheme and nally the LandauerBüttiker transportformalism are introduced.

1.1 Spintronics

The central object of study in this thesis is the handling of spin degree of freedomof an electron in electronic transport. Just as a planet orbiting the sun and spinningabout its own axis possesses both orbital and spin angular momenta, so does anelectron orbiting a nucleus. Since the electron has no radial extent and cannot be“turning” about its axis, this is purely an analogy. The spin is an intrinsic propertyof the electron and has a constant value. However, just as the planets can rotateclockwise or anticlockwise, the electron spin can also be considered to be clockwiseor anticlockwise: there are two different kinds of spin according to whether the pro-jection of the spin onto a given quantization axis is ±~ 1

2 , which we term spin“up”andspin “down”. The relative number of electrons with spin-up and spin-down is veryimportant for the magnetic properties of a chosen material. Nonmagnetic materialsare characterized by the same number of electrons with the same properties in bothspin channels. As for the magnetic materials there is an imbalance in the density ofstates for spin up and spin down electrons as schematically illustrated in Fig. 1.1(a,b).Depending on the number of electrons below the Fermi level in each spin channel,the channel with more (fewer) states below the Fermi level is termed the majority(minority) spin channel. For nonmagnetic materials both channels are identical. Thespin of an electron is an intrinsic angular momentum s which is directly coupled to itsmagnetic moment m by the relation m = −gsµB(s/~) (where µB = e~/(2me) is theBohr magneton and gs the Lande factor which is roughly equal to two). Spin was first

1

2 Introduction

measured in the famous experiment of Stern and Gerlach in 1921. Afterwards, itstheoretical explanation was developed by Kronig who, however, decided not to pub-lish these results because calculated linear speed on the surface of electron was largerthan the speed of light. Later, in 1925 appeared theoretical study of spin by Uhlen-beck and Goudsmit [1]. In 1927 Pauli introduced spin into a quantum mechanics. Asignificant contribution to understanding spin was made by Dirac who, by combiningquantum mechanics and special relativity, developed a theoretical framework fromwhich the magnetic moment and “spin” of electrons followed automatically. Despitethe central role played by electron spin in many areas of condensed matter physics, itbarely figured in the mainstream of charge-based electronics. The situation changedwith the discovery of oscillatory interlayer exchange coupling in Fe|Cr and Co|Cumultilayers by Grunberg et al. [2] and later by Parkin et al. [3–5]. This led to thenear simultaneous discovery of the giant magnetoresistance effect (GMR) by two ex-perimental research groups led by A. Fert [6] in Paris and P. Grunberg [7] in Julich.The GMR effect is a milestone in condensed matter physics for which both Fert andGrunberg were awarded the Nobel Prize in Physics in 2007. From the study and ap-plication of this effect a new branch of a solid state physics has emerged, which is nowcalled spintronics or spin electronics. This field refers to study of the role played byelectron spin in the processes studied by the solid state physics, and possible devicesthat specifically exploit spin properties instead of or in addition to charge degrees offreedom. Current efforts in this field involve several major directions from which wewould like to discuss spin transport phenomena in metal and semiconductor-baseddevices. The purpose of the former one is perfecting the existing magnetoresistance(MR) - based technology by either developing new materials with larger spin po-larization of electrons or making improvements or variations in the existing devicesthat allow for better spin filtering. The second field, focuses on finding novel waysof generating and utilizing spin-polarized currents. These include investigation ofspin transport in semiconductors and looking for ways in which semiconductors canfunction as spin polarizers and spin valves. In the following subsections we are goingto discuss major issues that stand in these two fields in more detail.

1.1.1 Magnetoresistance

The most important measurable characteristics in this field is the electric resistance.The effect which is going to be discussed lies in a possibility to change the resistance ofa device such that we can detect it. Here we study only metal-based devices thereforewe focus on a ferromagnetic metal (FM). It is well-known that the electrical resistanceof a FM can be changed in three different ways. The simplest way is to influencethe electrical resistance by changing temperature. Resistance grows as a function oftemperature due to the increase of scattering on thermally activated ions. This, infact, serves as a criterion for distinguishing metals from semiconductors (SC) whereelectrical resistance decreases as a function of temperature due to the increase in thenumber of free electrons. We can also control electrical resistance of a FM by changingits magnetization by applying an external magnetic field at fixed temperature. Thethird way is associated with the direction of magnetization relative to the current

1.1. Spintronics 3

CuEF

transmissionmajority

1

2

3

Co

minority

EF

majority

Fermi surface projections

(a)

(b)

(c) (d)

(e) (f)

(g)

(h)minority

10

Figure 1.1: Schematic illustration of spin densities of states in Cu (a) and Co(b). Fermi surface projection of Cu majority (c) and (d) and Co for majority (e)and minority (f) spin channels. Spin polarization of current in Cu|Co junction ismade clear with majority (g) and minority (h) transmissions.

(also temperature dependent). This effect is called the anisotropic magnetoresistance(AMR) and it was discovered by W. Thompson in 1857 [8]. This effect proved tobe very important due to its application in magnetic recording. Here, we would liketo discuss the successors of the AMR effect - giant and tunneling magnetoresistanceeffects.

The GMR is a quantum mechanical effect which is observed in a system con-sisting from alternating ferromagnetic (FM) and nonmagnetic metal (NM) layers(FM|NM|FM junction). Depending on the relative orientation of the magnetizationsin the magnetic layers, the device resistance changes from small (magnetizations par-allel) to large (magnetizations antiparallel). An external magnetic field is appliedto change the relative orientation of magnetization directions. Examples of systemswhere the GMR effect is observed include various material combinations from whichwe focus on Co|Cu|Co(111) junction to go through the explanation of the GMR effectin more details. As it will be shown below the interface between FM and NM (Co|Cu)plays a crucial role in spin-dependent transport. The chosen junction contains twosuch Co|Cu interfaces, therefore, for the sake of simplicity we start by discussingthe spin-transport through a single interface. The bias voltage is applied in sucha way that electrical current flows perpendicular to the interface in the so-calledcurrent perpendicular to the plane (CPP) geometry. We also assume perfect peri-odicity in the plane parallel to Co|Cu interface. Next, we assume that the electriccurrent driven through the Co|Cu junction consists of two independent componentsfor majority and minority channels. This approximation is called Mott’s “two currentmodel” and it work best at low temperatures where the spin relaxation length, λsf(how far an electron can travel before it loses its spin information), is much larger

4 Introduction

than the elastic mean free path, `e (the average distance which an electron travelsbefore it is elastically scattered at a defect) [9–11]. The total resistance of the Co|Cujunction can then be represented within the two-current resistor scheme as a sum oftwo parallel resistors standing for two spin channels. Most important is that theseresistors are characterized by different resistance and the current injected from Cointo Cu is spin-polarized. For qualitative demonstration of this effect we use resultsof our numerical method which is described in the next section. These results areshown in Fig. 1.1. In Fig. 1.1(c-f) we show projections of the Fermi surfaces of Coand Cu on a plane perpendicular to [111] direction. As can be seen, the Fermi surfaceprojection (FSP) of Cu and Co majority have a lot in common, so as the proper-ties of electrons in these channels. Therefore, the transmission of majority electronfrom Co to Cu endures almost no scattering and the resistance is small. On thecontrary, the FSP of Cu and Co minority are quite different and, therefore, strongscattering of electrons is expected. Henceforth, due to spin-dependent mismatch ofthe electronic properties of these two materials the spin filtering of a current injectedfrom Co into Cu at the Co|Cu interface occurs. Experimental measurements of thespin polarization of a current injected into a nonmagnetic material are rather com-plicated. This problem can be circumvented by adding another FM layer as shownin Fig. 1.2. We assume that thickness of a NM layer is such that in the absence ofan external magnetic field the magnetizations of FM leads are oriented antiparallel.Such relative orientation, however, can be switched to parallel if an external magneticfield of a certain strength along one of the magnetization direction is applied. Thetotal resistance of magnetic junction can then be calculated using the two-currentresistor model as it is illustrated in Fig. 1.2. According to this model, the conduc-tance of a junction in case of antiparallel orientation is GAP = 2/(Rmaj + Rmin)and GP = (Rmaj + Rmin)/(2RmajRmin) in the case of parallel ones. Usually, GP islarger than GAP . Therefore, due to a difference between Rmaj and Rmin a nonzeromagnetoresistance GMR= (GP −GAP)/GAP appears.

Tunneling magnetoresistance (TMR) effect is observed in a magnetic tunnel junc-tion (MTJ) which is a junction illustrated in Fig. 1.2 modified by replacing a non-magnetic metallic spacer layer with a nonmagnetic insulator (I). In MTJs the conduc-tance depends on the relative orientation of FM electrodes magnetizations just likedescribed above. In FM|I|FM the electron transfer mechanism has tunneling charac-ter. Spin-effect appears due to spin-filtering of the electron with one spin orientationat FM|I interface. The magnetoresistance effect observed in CPP-MTJs is usuallylarger than that of the CPP-GMR which makes it valuable for practical applications.It worth mentioning that the TMR effect was discovered before the GMR effect in1975 by M. Julliere [12]. In his pioneering experiments Julliere was able to measurenonzero TMR in Fe|Ge-oxide|Co system. Unfortunately, these results were difficultto reproduce1 and it took about 20 years for experimentalists to make system wherethe TMR effect was found to be reproducible. Successful room temperature mag-netic tunneling transport measurements in CoFe|AlOx|Co junctions were performedby Moodera [13] in 1995. These results have strongly accelerated study of spin-

1as mentioned by A.Fert in his Nobel lecture

1.1. Spintronics 5

maj

FM NM FM

parallel configuration (H≠0)

min

maj

min

FM NM FM

antiparallel configuration (H=0)

maj

min

min

maj

RminRmaj

Rmin Rmaj

maj

min maj

minRmaj

Rmin Rmin

Rmajmaj

min

maj

min

Figure 1.2: Schematic layout of the two states of a spin-valve structure andelectron scattering in a magnetic|nonmagnetic multilayer. In antiparallel alignment(b) both type of electrons are subject to strong scattering while in the parallelorientation (a) only one type of electrons encounters a high resistant magneticlayer. The two current model shows that the parallel alignment (c) between theferromagnetic layers results in a lower resistance than the antiparallel alignment(d).

transport due to its possible application to magnetic sensors and magnetic memory.On one hand aluminum oxide has proven to be the most suitable and commonly usedinsulator barrier in industry. MTJ based on this insulator are now routinely fabri-cated with well reproducible characteristics. With a proper choice of materials and anoptimized junction preparation the TMR ratio can reach up to 70% at room temper-ature [14]. On the other hand, scientific research in this area is now focused on studyof MTJ with a crystalline MgO tunnel barrier. First experiments on MgO-basedtunnel junctions have reported large values of TMR exceeding 200%. These were theTMR measured in Fe|MgO|Fe(001) by Parkin et al. [15] and FeCo|MgO|FeCo(001)Yuasa et al. [16] CPP junctions. While the observed magnetoresistance effect wascertainly larger than that for MTJ with amorphous barriers, it was still two orders ofmagnitude smaller than that predicted theoretically [17, 18]. This discrepancy willbe addressed in Chapter 3. Though the questions regarding some very importantissues in this area remain unsolved, research to date has focused on achieving evenhigher TMR values at room temperature. New records for room temperature TMRare regularly renewed. The current record stands at 500% for FeCoB|MgO|FeCoBjunction with amorphous FeCoB leads [19]. For more details on the GMR and TMReffects we refer to the reviews [20–26].

The MR effects just discussed can be used to make magnetic sensors and magnetic

6 Introduction

gateFM

source drain

insulatorinsulator

semiconductor

FM

Vg = 0

gateFM

source drain

insulator

FM

Vg >> 0

Figure 1.3: Datta-Das spin transistor. Without the gate voltage the electronsfrom FM source pass through the channel and are detected on by another FMelectrode. With the gate voltage on, the electric field causes spins to process dueto the magnetic interaction. The electrons with their spin misaligned with drainmagnetization direction can not pass through the circuit.

memory. The variation of electric resistance is used to detect small changes in mag-netic fields by magnetic sensors. Advances of these effects have already been used,the magnetoelectronic devices are inside the vast majority of computers and laptops.Most modern hard drives use a GMR spin valve, a device that reads information offthe individual disks that make up a hard drive. Such magnetic reading head can bemade small which enabled a thousand fold increase in the storage capacity of diskdrives since it was introduced in 1998. A couple of years ago TMR reading heads wereintroduced by Seagate for laptop and desktop drives. Nowadays, the TMR head ismature technology for hard disk drives. Different application of MR effects could bea new type of computer memory known as MRAM (Magnetoresistive Random AccessMemory). In 2006, the company Freescale began to offer 1MB MRAM devices.

1.1.2 Spin injection into semiconductors

Since the 1970s conventional electronic microprocessors have operated with packets ofelectronic charge propagating along semiconductor channels that are getting smallerall the time. This progress of microelectronics is often summarized in Moore’s Lawaccording to which microprocessors will double in power every 18 months as moretransistors can be fitted in a chip. Although this trend will continue for the next fewyears, the silicon technology is reaching the end of the road map as the size of individ-ual bits approaches the dimension of atoms. The existing magnetoelectronic deviceswork successfully as switches or valves but they can not be used to amplify electricsignals. However, possible incorporation of spin degree of freedom in semiconductorelectronics promises new logic devices with enhanced functionality. Therefore, spinrelaxation and spin transport in semiconductors are of fundamental research interestnot only for being basic solid state physics issues, but also for the potential thesephenomena have in electronic technology.

There are many basic questions pertaining to combining semiconductors with

1.1. Spintronics 7

magnetic metals. For example, whether placing a doped semiconductor in contactwith magnetic metal would impede spin transport across the interface is far from well-understood [27]. There were several schemes proposed for spintronic devices based onthe metal-oxide-semiconductor technology, practical realization of which is consideredas important steps towards spin-based semiconductor electronics. Here we would liketo discuss the field effect spin-transistor proposed in 1989 by S. Datta and B. Das [28].Such a device consists of a FM injector and a FM detector of spin-polarized currentand a SC which provides a channel for two-dimensional electron transport betweenthem. This transistor is schematically shown in Fig. 1.3. As in a conventional fieldeffect transistor there is a third electrode (gate) that generates an electric field tomodulate the current in the transport channel. The magnetizations of injector anddetector are assumed to be parallel. In case of zero gate voltage, every electronemitted from the injector with its spin oriented along the magnetization direction itshould be possible to enter the detector in the absence of any changes to spins duringtransport. For nonzero gate voltage the gate electrode produces an electric field thatcauses the spins to precess. The electron current is then modulated as the electronswith their spin parallel to the magnetization pass to the detector. If the orientationis antiparallel there is no current as shown in right panel in Fig. 1.3. Despite years ofeffort this effect has not been convincingly demonstrated experimentally. Problemswhich impede realization of such device can be divided into three groups: efficientinjection of spin-polarized current from FM into SC, transfer of electrons through SCwithout losing their spin and eventually detection of such current by FM electrode[29]. Already the realization of efficient spin injection appeared to be rather difficult.On one hand, the significant polarization of electrical conductivity in a real FM resultsin a spin-polarized electrical current. By choosing a FM and a SC such that to achievean efficient spin filtering at the FM|SC interface, the most straightforward way of spin-injection from a FM into a SC would be the formation of an ohmic contact betweenthem. However, it was shown from the experiments that efficiency of spin injectionvia ohmic contact into a SC is small. The problem appears due to the conductivitymismatch [30] between FM and SC. The effectiveness of the spin injection dependson the ratio between spin-dependent FM conductivity σFM and spin-independent SCconductivity σSC and a substantial spin injection occurs if σFM ≈ σSC . However, fortypical FM and SC σFM >> σSC the spin injection efficiency is small. This problemcan be resolved by making use of an additional tunneling barrier [31] at the FM|SCinterface see Fig. 1.3 or by means of a Schottky barrier formed at FM|SC interface likein case of Fe|GaAs junction. Another possible solution is to compensate the resistancewith the spin-dependent FM|SC interface resistance [32–34] in junctions like Fe|InAswhich is studied in more detail in Chapter. 2. As an alternative approach one can usediluted magnetic semiconductors (DMS) as the spin injector. DMS, however, havelimited practical application due to low (of the order of 100 K) Curie temperature[35, 36]. One can also use half-metallic ferromagnets [37] as the 100% spin-polarizedferromagnetic injectors, although these are challenging materials with which to work.

8 Introduction

1.2 Numerical Method

From the discussion in the previous subsection it is made obvious that effects like MRof a magnetic junction or efficiency of spin injection from FM into SC are the mani-festations of spin-dependent transport in complicated multilayered structures and itsstudy is our primary target. Firstly, just mentioned effects are due to the spin filter-ing at FM|NM or FM|I or FM|SC interfaces, therefore, accurate description of theelectronic and structural properties of such interfaces as well as their bulk materialsare very important for accurate description of spin-dependent transport. Secondly, itis also important that we use method which would require no free parameters and bematerial specific. Therefore, we have chosen first-principles approach for our needs.

In this section we discuss our theoretical method in more detail. It can be dividedinto two parts: (i) the selfconsistant calculations of the potential and correspondingelectronic band structure and (ii) transport calculations. In the first part we usetight-binding linear muffin-tin orbitals method in order to find self-consistent solu-tion of Kohn-Sham equation in local density approximation. Then, in the secondpart we use mode matching scheme to calculate transmission probability amplitudeswhich enter the Landauer-Buttiker formalism to calculate the conductance in linear-response regime.

1.2.1 Density Functional Theory

It is known from the quantum mechanics that the expectation values of observablesfor a quantum mechanical system can be obtained once its wave function Ψ, whichis the solution of the Schrodinger equation

HΨ(r1, r2, ..., rN ) = EΨ(r1, r2, ..., rN ), (1.1)

is known. In order to solve Eq. (1.1) one has to specify the Hamiltonian H. Werestrict ourself to consideration of the electronic properties. The nuclear degrees offreedom are taken into account in the form of an external potential acting on theelectrons, therefore, the wave function is a function of the electronic coordinatesonly. This approach is called the Born-Oppenheimer or adiabatic approximation andit simplifies the problem by taking into account the fact that nucleus are usuallymuch heavier than the electrons. Therefore, the Hamiltonian of a system with Nelectrons can be written as following

H = − ~2

2m

∑i

∇2ri

+ Vext(ri) +12

∑i 6=j

e2

|ri − rj |, (1.2)

where the first term is the kinetic energy operator, second term describes electron-ionand ion-ion interactions and the third one stands for the electron-electron interaction.

Analytical solutions of Eqs. (1.1), (1.2) are only possible for a very simple systems.Therefore, many numerical methods for solving the Schrodinger equation have beendeveloped. The method which we will use here is local density approximation ofdensity functional theory (LDA-DFT). It is one of the most popular and successful

1.2. Numerical Method 9

parameter-free, material-specific approaches in quantum mechanics. As such theLDA-DFT provides us with an accurate description of the electronic ground stateproperties for wide range of itinerant electron many-particle systems. This approachwas first formulated by Hohenberg and Kohn [38] in the form of two theorems. Thefirst theorem states that the energy of the ground state is a unique functional of theelectron density

E[ρ(r)] = F [ρ(r)] +∫Vext(r)ρ(r)d3r. (1.3)

The second theorem states that the density functional reaches its minimum at theexact ground state density ρGS(r) and the total energy of the ground state can bewritten as

EGS = F [ρGS(r)] +∫Vext(r)ρGS(r)d3r. (1.4)

However, the DFT approach in the form of Eq. (1.4) is of a little practical use sincethe functional F is not known. The next step was made by Kohn and Sham [39] whoproposed to replace the complex system of many interacting particles in an externalpotential Vext(r) by a system of noninteracting particles in an effective potentialVeff(r). In this case the density of electrons is the sum over densities of a system ofnoninteracting particles, described by single-particle wave functions ψi,

ρ(r) =N∑i=1

|ψi(r)|2, (1.5)

which can be obtained by solving a set of equations for N noninteracting particles

[− ~2

2m∇2 + Veff(r)]ψi(r) = εiψi(r), (1.6)

where the effective potential is

Veff(r) = Vext(r) +∫

ρ(r)| r− r′ |

d3r +δExc[ρ]δρ(r)

. (1.7)

Eqs. (1.5) and (1.6) are called the Kohn-Sham equations. They allow us an exacttreatment of the many-body system in the ground state and have to be solved self-consistently. The last term of Eq. (1.7) Exc = Ex +Ec is the sum of the exchange andcorrelation energies. Exchange-correlation energy is, in principle, the only unknownterm in the Kohn-Sham equations. In order to determine it an approximation hasto be made. Kohn and Sham suggested to use homogeneous electron gas of electrondensity ρ to define exchange-correlation energy per electron εxc(ρ). The approachwhich is called the local density approximation (LDA) for the exchange-correlationfunctional Exc of inhomogeneous systems with density ρ(r) is then assumes that thelocal exchange-correlation energy εxc(ρ(r)) is equal to εxc(ρ) of a model system withρ ≡ ρ(r)

ELDAxc =

∫ρ(r)εxc(ρ(r))dr, (1.8)

10 Introduction

There are few schemes available for parameterizing of the exchange-correlation energywithin the LDA. The most frequently employed parameterizations are due to vonBarth and Hedin [40], Ceperley and Alder [41] as parameterized by Perdew andZunger [42] and Vosko, Wilk and Nusair [43].

1.2.2 TB-MTO

The eigenvalues of Kohn-Sham equations Eq. (1.5), (1.6) for free atom that corre-spond bound states form a discrete set of energy levels called atomic orbitals. Ifmany atoms are brought together to form a solid, their atomic orbitals form bands ofenergies due to a large number of orbitals and small energy difference between them.Therefore, in solid state physics, the electronic band structure (or simply band struc-ture) of a solid describes ranges of energy that an electron is “forbidden” or “allowed”to have. Moreover, it greatly determines an important characteristic - transport ofelectrons.

There are different kinds of approaches to calculate the band structure. Histori-cally, among the first ones was cell method proposed by Wigner and Seitz in 1933. Inthis method a crystal is divided into Wigner-Seitz cells inside of which the atomic po-tential has spherical symmetry. Due to complicated boundary conditions, in atomicsphere approximation (ASA) it was proposed to replace the Wigner-Seitz cells withspheres of the same volume. In 1937 Sleter proposed to use non-overlapping atom-centered “muffin-tin” spheres with the spherically symmetric potential inside themand the constant potential in the remaining interstitial part. The corresponding wavefunctions - muffin-tin orbitals (MTO) form a flexible, minimal basis set leading tohighly efficient computational schemes for solving the Kohn-Sham equations [44–46].

In order to build the MTOs of a solid we start by considering a single atomicsphere. Then the ASA is recovered (i) by taking the kinetic energy in the interstitialregion to be zero and (ii) by expanding the muffin-tin spheres so that they fill allspace. As it was already mentioned above the atomic potential inside an atomicsphere of radius S has spherical symmetry therefore its wave function can be foundby solving the radial Schrodinger equation, while outside the sphere the wave functionis solution of the Laplace equation ∇2Ψ = 0. An energy dependent orbital can bewritten as

ΦL(E, r) = ilYL(r)

ul(r, E) if r 0 S;[Dl+l+1

2l+1 ( rS )l + l−Dl

2l+1 ( rS )−l−1]ul(S,E) if r > S,

(1.9)

where L stands for both l and m quantum numbers, ul is solution of the radialSchrodinger equation and YL the spherical harmonics. Dl = Su′l(E,S)/ul(E,S) isthe logarithmic derivative of ul(E,S) at r ≡ S. By subtracting from the partial wave,both inside and outside the atomic sphere, the part Dl+l+1

2l+1 ( rS )l which is irregular atinfinity a new orbital is formed which is regular, continuous and differentiable in all

1.2. Numerical Method 11

space

φL(E, r) = ilYL(r)

2l+1l−Dl

ul(r,E)ul(S,E) −

Pl(E)2(2l+1) ( rS )l if r 0 S;

( rS )−l−1 if r > S,

(1.10)

where

Pl(E) = 2(2l + 1)Dl(E) + l + 1Dl(E)− l

. (1.11)

However, one should remember that Eq.(1.10) is not the solution of the Schrodingerequation inside the atomic sphere.

Our next step would be to consider a crystal. In such a case we center an atomicsphere on every atom. We also take into account the fact that inside every atomicsphere we have “tails” of the MTOs coming from the rest of the atomic spheres.Therefore, for the linear combination of MTOs centered on different atoms (sites)

Ψ(E, r) =∑R,L

φL(E, rR)CRL, (1.12)

together with expansion of these tails around different sites R′ which is given by

ilYL(rR)(rRS

)−l−1 = −∑L′

(rR′

S)l′ 12(2l′ + 1)

il′Y ′L(rR′)SR′L′,RL (1.13)

the partial solution is recovered if we require that∑R′,L′

[PRL(E)δRR′δLL′ − SRL,R′L′ ]CR′L′ = 0. (1.14)

where SR′L′,RL are the expansion coefficients which form a matrix of structure con-stants and rR ≡ r −R and rR ≡ |r −R|. Eq. (1.14) is called the tail-cancellationcondition which is, in fact, the energy dependent KKR equation. The tail-cancellationequation contains all the information about the geometry in the structure constantsSRL,R′L′ and the atomic potentials in the potential function PRL(ε). By introducinga set of “screening” constants α

Pα(E) = P (E) + P 0(E)αP 0(E) = P 0(E)(1− αP 0(E)

)−1, (1.15)

andSα = S + SαSα = S (1− αS)−1

, (1.16)

a short range (“tight-binding”) MTO (TB-MTO) can be defined allowing us to con-sider only first and second nearest neighbors in the case of close-packed structure.In the following we use the notation α for the set of parameters which minimize thehopping. Note, that the form of tail-cancellation condition remains unchanged whenrewritten with the screening transformation. A big disadvantage of Eq. (1.14) is thatit contains energy dependence in the potential function Pα(E) which complicates so-lution of Eq. (1.14). This problem can be solved using energy-independent linearized

12 Introduction

MTO (LMTO). In our work we use the TB-LMTO surface Green function (SGF)method which was developed to study the electronic structure of interfaces and lay-ered systems. When combined with the coherent-potential approximation (CPA), itallows self-consistent calculation of the electronic structure, charge and spin densitiesof layered materials with disorder [47]. This approach satisfies our requirements ofbeing able to treat complex electronic structures efficiently. Keep in mind, that fortransport studies we only need to know the potential function at the Fermi energy.

1.2.3 Wave function matching

In this subsection we summarize our approach to solving the scattering problem.This problem is formulated for an infinite system consisting of the region of interest(an interface, junction etc.) sandwiched between two semi-infinite ideal leads. Byreplacing the semi-infinite leads by appropriate energy dependent boundary condi-tions, it can be reduced to finite size. This can be achieved by the wave-functionmatching (WFM) method for calculating the transmission and reflection matricesdue to Ando [48]. In the next we will give a very simple introduction to this methodand full details of the formalism and of the implementations can be found in [49–51].

To keep matters simple we assume that there exists two-dimensional transla-tional symmetry in the plane perpendicular to the transport directions so that statesare characterized by a lateral wave vector k‖ in the corresponding two-dimensionalBrillouin zone. Furthermore we assume nearest neighbour interactions only. Thescreened KKR equation (1.14) written in its components has a form of infinite chainof equations with I (a composite layer index) running from −∞ to ∞

−Sk‖I,I−1CI−1 +

(PαI,I(E)− Sk‖

I,I

)CI − S

k‖I,I+1CI+1 = 0, (1.17)

whereS

k‖I,J =

∑T∈TI,J

Sα(T)eik‖T, (1.18)

is the Bloch summation of the screened structure constant matrix over the set ofvectors TI,J that describes hopping from lattice site in the I-th layer to lattice sitesin layer J . In the following explicit reference to k‖, E and α is omitted to keep thenotation simple. CI describes the amplitude of atomic orbital on the I-th latticesite. The boundary conditions are determined if Eq. (1.17) is solved in finite leads.For finite leads the matrix elements in equation of motion (1.17) does not depend onlayer index I which in combination with Bloch-Floquet theorem (the wave functionsof adjacent cells of a periodic system can be related via constant amplitude/phasefactor λ) transforms Eq. (1.17) into a generalized linear eigenvalue problem for λ. Bycalculating the wavevectors and velocities of the lead modes, the right- and left-goingmodes together with evanescent modes can be found for a fixed energy (Fermi levelin most cases) and k‖.

Having obtained the modes of the ideal leads, we match them to the scatteringregion. We assume that the scattering region is localized in space: 1 ≤ I ≤ N . For

1.2. Numerical Method 13

I = 0 Eq. (1.17) gives:

−S0,−1C−1 + (P0,0 − S0,0)C0 − S0,1C1 = 0, (1.19)

Assuming that the general solution in the left lead consists of the right- and left-goingpropagating modes, that is incoming and reflected waves with amplitudes A and B,respectively, we can write the amplitude at the I = −1 site as

C−1 = Aeik(−a) +Be−ik(−a) ≡ Aλ−1L,+ +Bλ−1

L,− = Aλ−1L,+ + (C0 −A)λ−1

L,− (1.20)

where the last step follows because for I = 0 we have C0 = A+B. In the scatteringproblem we assume that we have set the incoming wave so A is fixed. We can nowuse Eq. (1.20) to eliminate C−1 in Eq. (1.19), rewriting it as

(P0,0 − S0,0 − S0,−1λ−1L,−1)C0 − S0,1C1 = AS0,−1(λ−1

L,+ − λ−1L,−), (1.21)

which truncates the infinite chain of equations (1.17) from the left. For I = N + 1,Eq. (1.17) gives:

−SN+1,NCN + (PN+1,N+1 − SN+1,N+1)CN+1 − SN+1,N+2CN+2 = 0, (1.22)

In the right lead we assume there is only a transmitted, outgoing wave. Then,CN+2 = CN+1λR,+,

−SN+1,NCN + (PN+1 − SN+1,N+1 − βRλR,+)CN+1 = 0, (1.23)

and the infinite chain of equations (1.17) from the right is also truncated. UsingEqs. (1.21) and (1.23), Eq. (1.17) can be written in matrix form

(P I− S′) Ψ = q, (1.24)

where Ψ is a finite dimensional vector that contains the coefficients CI in the scat-tering region plus those on the two boundary atoms, i.e. I = 0, . . . , N + 1. q is a“source” vector of length N + 2, whose elements are zero, except the for first onewhich is

q0 = AS0,−1

(λ−1L,+ − λ

−1L,−

). (1.25)

S′ is a finite (N +2)× (N +2) structure constants matrix and P is a diagonal matrixof potential functions characterizing the AS potentials. All the matrix elements ofS′ are identical to those of the original structure constant matrix, except for the firstand the last diagonal elements, which are

S′0,0 = S0,0 + ΣL(E), S′N+1,N+1 = SN+1,N+1 + ΣR(E), (1.26)

withΣL(E) = S0,−1λ

−1L,− and ΣR(E) = SN+1,N+2λR,+. (1.27)

In Green function language ΣL/R(E) are called the self-energies of the left and rightleads. They contain all the information concerning the coupling of the scatteringregion to the leads. The self-energies are complex and energy dependent.

14 Introduction

Thus, we have replaced an infinite dimensional problem, Eq. (1.17), by a finite di-mensional one, Eq. (1.24). Once Eq. (1.24) is solved, all that remains is to extract thetransmission coefficient amplitude given by the amplitude of the wave function in theright lead normalized to the amplitude of the incoming wave, and (flux) normalizedwith velocities [48, 50]

t =

√vR/aRvL/aL

CN+1

A, (1.28)

where vL and vR are velocities of propagating states and aL and aR are latticeconstants in transport direction in left and right leads respectively.

Using WFM, simple scattering problems with tight-binding Hamiltonians can besolved analytically in a straightforward way, as illustrated in [50, 52].

1.2.4 Landauer-Buttiker approach

In order to reach our main purpose - quantitative study of transport phenomena ininhomogeneous materials we have to find a way to calculate conductance. In themacroscopic conductor the conductance is described by Ohm’s law

G = σA

L, (1.29)

where A is the conductor cross section area, L is the length of the conductor, andσ is its conductivity. From Eq. (1.29) it is clear that if conductor gets smaller andsmaller A → 0 and/or L → 0 the conductivity of such system is not defined. Thisis due to the fact that Ohm’s law relies on the assumption that the conductivity isa macroscopically defined quantity which is homogeneous within the conductor. Onatomic scale this is not valid anymore and Ohm’s law in its classical definition (1.29)breaks down.

On the other hand the Landauer-Buttiker scattering formalism is an establishedmethod for describing transport if the dimensions of the conductor is intermediate be-tween microscopic and macroscopic. This range is called mesoscopic and spans fromtens to hundreds of nanometers. On such a scale conductor does not exhibit the ohmicbehavior because its size is smaller than three characteristic length scales[53]: (i) thede Broglie wavelength, (ii) mean free path, and (iii) the phase-relaxation length.Landauer and Buttiker formulated the problem of electronic transport in terms ofscattering matrices where the transmission matrix element tm,n is the probabilityamplitude that a state |n〉 in the left-hand lead incident on the scattering regionfrom the left is scattered into a state |m〉 in the right-hand lead. The conductanceG = dI/dV |V=0 in the linear response regime is given by

GLB =e2

h

∑n,m

|tm,n|2 = Tr[tt†]. (1.30)

The Landauer-Buttiker approach is intuitively very appealing because transportthrough nanostructures is naturally described in terms of the scattering of electron

1.3. Thesis outline 15

waves that is transmission and reflection. Usually, explicit calculation of the scat-tering states is avoided by making use of the invariance properties of the trace inEq. (1.30) to calculate the conductance directly from Green functions expressed insome convenient localized orbital representation [54]. However, we want to makecontact with a large body of theoretical literature on mesoscopic physics [55, 56]which requires knowing the full microscopic transmission and reflection matrices andto make explicit use of the scattering states to analyze our numerical results. Ourrequirement of physical transparency is satisfied by choosing a computational schemewhich yields the full scattering matrix and not just the conductance.

1.3 Thesis outline

The aim of this thesis is to study the issues for the development of new spintronicdevices and to provide an theoretical explanation for experimental observations andto predict new effects. The method which we are going to use for these purposes wasbriefly described in the previous section.

In the next chapters the results of our study of spin-dependent transport in awide range of layered junctions are presented.

In Chapter 2 we present detailed study of spin injection into SC in Fe|InAs andFe|Au|InAs junctions. In such ideal systems large spin-dependent interface resis-tance has been shown to solve the conductivity mismatch problem. However, in-terface disorder reduces total interface resistance significantly [32]. From our studywe have found that decrease of polarization of injected from Fe into InAs currentis proportional to interface disorder. To anticipate intermixing of FM and SC (in-terface disorder) we propose to use a buffer layer made of Au. We demonstratedecrease in sensitivity of spin-dependent interface resistance to the interface disorderin Fe|Au|InAs systems.

In Chapters 3 and 4 we study transport in semirealistic model tunneling junc-tion. In particular we show that the resonant tunneling plays an important rolein FeCo|vacuum|FeCo ideal magnetic tunnel junction. Due to this contribution theminority parallel conductance dominates among all the rest of the conductance com-ponents in parallel and antiparallel configurations which leads to very large valuesof the TMR. In Chapter 3 we study the effect of interface roughness and magneticalloy disorder in the leads on the tunneling magnetoresistance and compare it withthat of ideal case. Observation of new relation between the antiparallel componentof conductance and parallel ones has lead us to better understanding of Julliere’smodel. The results of this study are presented in Chapter 4.

In Chapters 5 and 6 we propose completely new family of systems for spintronicdevices. Based upon observation of mismatch between the Fermi surface projection ofgraphite and FM (Ni and Co in both fcc and hcp cases) minority and good matchingbetween the Fermi surface projection of graphite and FM majority together with thefact that their lattice constants are almost identical we predict perfect spin filteringat graphite(graphene)|FM(111) interface. We predict 100% (maximum) pessimisticmagnetoresistance in FM|graphite|FM(111) system. We also observe weak sensitivity

16 Introduction

of the MR to the interface roughness and alloy disorder. Furthermore, based on thefact that filtering occurs on one interface we study system like FM|graphite|NM(111)to demonstrate that perfect spin-injection (100% spin polarization of injected current)into NM takes place. We hope that low temperature Andreev reflection experimentswould allow to prove this experimentally. From our first principles calculations wepredict that two- dimensional hexagonal-BN and BC2N are direct gap insulator andsemiconductor respectively. These results allow us to introduce new family of highlyplanar devices.

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BIBLIOGRAPHY 17

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18 Introduction

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Chapter 2

Spin injection from Fe intoInAs

It has been proposed that the interface resistance of a defect-free (001) interfacebetween bcc Fe and zinc blende semiconductors (such as GaAs, InAs or ZnSe) can beso large and spin-dependent [1, 2] that direct spin injection should be possible. In [2]however, it was pointed out that even a modest amount of interface disorder would besucient to destroy the spin-dependence. In this chapter, we extend our earlier ab-initio study to much lower concentrations of disorder. We show that the minority-spintransmission is very sensitive to local interface geometry and the factors governingthe polarization quenching are identied in the dilute limit where a single Fe atomoccupies an interface In or As site. In principle, the equilibrium geometry of thesecongurations could be determined by total energy minimization. However, becauseinterface structures are frequently kinetically determined, there is no guarantee thatthe lowest energy congurations will dominate the interface transmission behaviour.Rather than attempting such a computationally very demanding total-energy study,we attempt to circumvent the problem entirely by inserting a buer layer (BL) be-tween Fe and InAs to prevent Fe minority-spin states coupling to the semiconductordirectly while still preserving the transmission spin-polarization. We identify a can-didate BL and demonstrate by explicit calculation that it preserves the transmissionpolarization of the ideal epitaxial structure remarkably well. Disorder at the Fe|BLinterface are shown to have small eect on the large transmission polarization. How-ever, we have found that spin polarization depends sensitively on the disorder atnonmagnetic BL|InAs interface. We expect the BL to work similarly for Fe|GaAsand Fe|MgO interfaces.

2.1 Introduction

Achieving efficient injection of a spin-polarized current into a semiconductor is a nec-essary condition for realizing“spintronic”devices which combine traditional semiconductor-

19

20 Spin injection from Fe into InAs

based electronics with control over spin degrees of freedom. Thanks to the robustness(high Curie temperature, TC) of their magnetism, elemental metallic ferromagnetssuch as Fe, Co and Ni should be ideal sources of polarized electrons. Unfortunatelydevices based upon metals suffer from the “conductivity mismatch” problem [3]. Thevery large, spin-independent resistivity of the semiconductor dominates the poten-tial drop throughout the device and the spin-dependent contribution of the metallicpolarizer is by comparison negligible. One way around this problem is to use mag-netic semiconductors but at the moment the Curie temperatures of these materialsare still too low for practical applications. Another solution is to introduce a largespin-dependent interface resistance. To date, spin injection has been demonstratedin systems where such a resistance originates either from an intrinsic Schottky bar-rier (SB) (Fe|GaAs [4] and Fe|GaAlAs [5–7]) or from an additional insulating layer[8]. The record room temperature polarization for injection from metallic electrodescurrently stands at 32% using the latter technique [9]. However, such barriers havedrawbacks. They limit the maximum current which can be passed through the inter-face and can lead to large thermal dissipation at the interface. An ideal system forrealizing spin-injection would be an Ohmic contact between a metallic ferromagnetand a semiconductor. One possible candidate is the Fe|InAs system. Unlike Fe andGaAs which are almost perfectly lattice matched, there is a large (5.4%) lattice mis-match between Fe and InAs. Nevertheless, it has been demonstrated that Fe filmscan be grown epitaxially on top of InAs [10, 11]. In spite of the absence of a Schot-tky barrier, it has been pointed out [2] that the interface resistance (IR) resultingfrom differences in the electronic structures is quite sizable and its spin dependencesufficiently large to overcome the conductivity mismatch problem. Unfortunatelythe spin-asymmetry of the transmission through a Fe|InAs interface was found to bequenched by disorder.

In this chapter we present a more detailed theoretical study of the Fe|InAs(001)system focussing on interfaces with low concentrations of substitutional impurities.We study the properties of a Fe|BL|InAs system with an additional buffer layer (BL)or bilayer introduced between Fe and InAs. The chapter is organized as follows. Inthe next section we describe the method used to calculate the interface resistance.Our new results on spin injection from Fe into InAs are given in Section 2.3 wherewe explicitly demonstrate that the quenching of spin injection is proportional to theinterface disorder. In Section 2.4 we study the effect of a buffer layer introducedto reduce the sensitivity of spin polarization to interface disorder. We study theeffect of disorder at both interfaces formed with the buffer layer and finish with someconclusions.

2.2 Method

The method we use is essentially the same as that employed in the earlier shortstudy by Zwierzycki et al. [2] and discussed in more detail in [12]. In the firststep, a potential profile for the Fe|InAs interface is calculated self-consistently withinthe local spin density approximation (LSDA) of density-functional theory (DFT).

2.2. Method 21

This was done using the layer TB-LMTO (tight-binding linearized muffin-tin orbital)[13] surface Green’s function (SGF) method [14] in the atomic-sphere approximation(ASA) [15]. Throughout this study the exchange and correlation potentials we use arethose calculated by Ceperley and Alder [16] and parameterized by Perdew and Zunger[17]. The lattice mismatch between Fe (aFe = 2.866A ) and InAs (aInAs = 6.058A )was taken into account by tetragonally distorting iron so that the in-plane latticeconstants of the two materials match. The height of the Fe unit cell was reduced inorder to preserve its volume. The interlayer separation at the interface was calculatedusing the condition of local space-filling with atomic sphere radii kept at their bulkvalues. A similar procedure was used on introducing buffer layers.

Since the atomic spheres approximation (ASA) [15] for the potential works verywell for close-packed solids, we adopt the usual procedure [18] of introducing addi-tional “empty spheres” at the tetrahedral interstitial positions in the zinc blende (zb)structure, i.e. atomic spheres without nuclear charge, effectively converting the opendiamond structure into a close-packed one where every sphere has eightfold coordi-nation. Besides In and As atoms at (0, 0, 0) and (1

4 ,14 ,

14 ), the unit cell of InAs then

contains two types of empty spheres, E1 and E2 at (12 ,

12 ,

12 ) and ( 3

4 ,34 ,

34 ) positions,

respectively, all in units of InAs lattice constant. For simplicity, equal sphere sizesare used for In, As, E1 and E2.

A well-documented deficiency of DFT is the failure of the Kohn-Sham eigenvaluespectrum to reproduce the single particle gap of even weakly correlated semiconduc-tors and insulators. This is systematically and seriously underestimated in the LDA[19]. In particular, the experimental band gap of InAs is 0.42 eV [20] but no gap ora much smaller gap is found in LDA calculations. In our study, the gap in InAs wasopened using a“scissor operator”correction applied as follows: an attractive constantterm was added to the potential inside the As atomic sphere and the Kohn-Shamequations iterated to self-consistency. This procedure was carried out for differentvalues of the constant until a value was found which reproduced the experimentalband gap. Once the correct gap was obtained, the InAs and Fe electronic structureswere lined up by applying a constant shift to the InAs potential until the bottom ofthe InAs conduction band (EC) and the Fermi energy (EF ) were aligned. In mostcalculations we chose EF − EC = 0.02 eV, corresponding to a doping concentrationof about 1017cm−3. The resulting band structure is plotted in Fig. 2.4(d) in the kzdirection for k‖ = 0, i.e., the Γ−X direction for bulk InAs or the point Γ in the two-dimensional BZ (2D BZ). Since there are two Fe atoms in 2D unit cell, its 2D BZ isfolded down and additional states, marked with dashed lines in Fig. 2.4(a,b), appearat Γ. These states come from the corners of the the unfolded 2D BZ and correspondto the P-N direction in the original 3D BZ.

In the second step, the self-consistent ASA potentials are used to calculate en-ergy dependent transmission matrices using a TB-MTO wave-function matching [21](WFM) scheme [12, 22]. The conductance (in units of e2/h) is given by the Landauer-Buttiker formula

Gσ =∑µ,ν,k||

|tµν(k||)|2, (2.1)

22 Spin injection from Fe into InAs

where tµν(k||) is a transmission coefficient and ν and µ denote incoming and trans-mitted Bloch waves, respectively. The resistance R = 1/G, calculated as the inverseof the conductance given by Eq. 2.1, cannot be interpreted as an interface resistancebecause if this procedure is applied to an “interface” between two identical materials,R does not vanish. The reason is that the conductance (per unit area) of perfectcrystalline materials is finite [23]. To calculate interface resistances (IR) we must usean expression derived by Schep et al. which corrects for the Sharvin conductance inthe limiting case of two identical materials [24]:

RFe/InAs =h

e2

[1∑|tµν |2

− 12

(1NFe

+1

NInAs

)](2.2)

The first term is the inverse of the Landauer-Buttiker conductance and NFe(InAs) isthe Sharvin conductance (in units of e2/h) of Fe (InAs). The integration over the2D BZ is performed with a sampling density corresponding to 106 k-points for the1 × 1 interface unit cell. Disorder is modeled using lateral supercells [12, 22] withthe potentials calculated using a version of the coherent potential approximation[14, 25] (CPA) generalized to treat disorder which is only homogeneous within alayer. Where necessary, a large number of disorder configurations (usually more than10) was considered in order to estimate an error bar. To study low concentrationsof disorder in this way, large supercells are required. In the present study, lateralsupercells containing 32 In (or As) or 64 Fe atoms in each atomic layer were used.

Little is known about the microscopic structure of Fe|InAs interfaces. In partic-ular, we are not aware of structure relaxation calculations for this system. Howeverthe results obtained for the closely related Fe|GaAs interface [26, 27] suggest thatAs-termination is energetically favorable for GaAs. Though experimental results [11]suggest the formation of an FeAs alloy at the Fe|InAs(001) interface, we will examineboth terminations and model interface disorder as one layer of FexAs1−x (FexIn1−x)with Fe substituting interfacial As (In) atoms in case of As (In) terminated InAsinterface.

2.3 FeInAs revisited

It was shown in Ref. [2] that while a perfect Fe|InAs (001) interface acts as a veryeffective spin-filter, the spin-dependence of the interface transmission (or of the cor-responding IR) is very sensitive to interface disorder. In particular, even the smallestamount of disorder considered at the time, corresponding to 1 in 8 In (or As) atomssubstituted by Fe, was capable of quenching the polarization almost completely. Itis natural to ask how much disorder can be tolerated while still maintaining an ac-ceptable polarization. To answer this question we revisit the problem using largerlateral supercells that allow us to study concentrations of disorder 4 times lower thanpreviously [2]. The results are summarized in Fig. 2.1, where in the top (bottom)left panels we show majority and minority conductances as a function of the number

2.3. FeInAs revisited 23

Figure 2.1: Conductance (on the left) and interface resistance (on the right)for As- (upper panels) and In-terminated (lower panels) Fe|InAs interfaces as afunction of the fraction of interfacial In atoms substituted by Fe for majority (N)and minority (H) spins. The green and blue lines connect the values averaged overconfigurations. Corresponding polarizations are shown in the insets.

of As (In) atoms substituted by Fe for As (In)-terminated interfaces. The symbolsdenote the values calculated for various randomly generated configurations of dis-order and the lines connect the average values. The corresponding IRs are shownin the panels on the right in Fig. 2.1 where the average polarizations are shown asinsets. Starting with perfect interfaces, we see spin-polarizations approaching 100%for both interface terminations, with the majority channel dominating over a muchless conducting minority channel. As explained in Refs. [1, 2], this is the result ofa symmetry-related selection rule. For perfect k||-preserving interfaces, transmissionoccurs only in the small area around the center of the 2D BZ (Γ point) correspondingto the occupied states at the bottom of the InAs conduction band (see Fig. 2.4). AtΓ, these states have the full ∆zb

1 symmetry of the C2v group. The very similar ∆1

states (of the C4v symmetry group) in the Fe majority band can transmit into InAsvery efficiently with probability approaching unity. In the Fe minority spin channel

24 Spin injection from Fe into InAs

−3 −2 −1 0 1 2 30

0.00004

0.00008

In termination

−3 −2 −1 0 1 2 30

0.00005

0.0001

0.00015

displacement of Fe impurity potential (eV)

cond

ucta

nce

(e2 /h

)

As termination

Figure 2.2: Conductance for As- (left panel) and In-terminated (right panel)Fe|InAs interfaces as a function of single Fe impurity potential shift in a supercellof 32 atoms for for majority (N) and minority (H) spins.

there are ∆2′ states which are formally compatible with the InAs ∆zb1 symmetry

states. However the differences in spatial distribution and orbital composition ofthese states (in-plane dxy for ∆2′ versus predominantly s and pz for ∆zb

1 ) reduces thetransmission probability at Γ to the 10−2 range. As seen in Fig. 2.1 the introductionof disorder in the form of Fe substituting some of the interfacial In or As atoms, hasa quite small effect in the majority channel. In the minority channel, the picturechanges much more dramatically where disorder breaks the symmetry and overridesthe selection rule which prevented transmission of minority spin states. This leads tothe opening of new channels for transmission through the interface and the averageconductance increases roughly in proportion to the fraction of substituted atoms, asseen in Fig. 2.1 for both As and In terminated Fe|InAs interfaces. The polarization,originally close to 100% and positive, decreases to zero at a disorder concentration ofabout 5/32 and becomes negative when the fraction of substituted atoms is increased.The increase of transmission in the minority channel occurs via diffusive scattering 1

which dominates the transmission even for the lowest concentration (1/32) shown inFig. 2.1 and accounts for 80% and 93% of the total value for As- and In-termination,respectively.

It might, however, be misleading to consider only average values of the trans-mission since some configurations of disorder presumably have a lower energy thanothers and may be present at the interface with a correspondingly higher probability.Fig. 2.1 exhibits a very substantial spread of values for microscopically

1By diffusive we mean scattering into the states with different k‖.

2.3. FeInAs revisited 25

−10% −5% 0% 5% 10%

0.9

1

1.1

1.2

1.3

spacial displacement of Fe impurity

norm

aliz

ed in

terf

ace

resi

stan

ce

Figure 2.3: Normalized interface resistance of As-terminated and Fe|InAs inter-faces as a function of a single Fe impurity displacement in 4 × 4 lateral supercellsin units of Fe|InAs interface distance for majority (N) and minority (H) spins.

different configurations of disorder, especially in the minority channel. In fact thedistribution of values for majority and minority channels overlaps for concentrationsabove 2/32. Interestingly, there seems to be a positive correlation between the valueof the minority conductance and the degree of clustering of the Fe impurities.

The spread in conductances in Figs. 2.1 for different configurations of disorderindicates a strong sensitivity of the Fe|InAs interface transmission to fine details ofthe interface structure. If we were to allow the atoms to relax to their minimumenergy configurations, it is conceivable that we would observe sizeable changes in theconductance. Since a geometry optimization for a system of this size is not possible[26, 27], we instead illustrate the sensitivity with model calculations in the spirit ofa tight-binding Hamiltonian. We focus on the dilute limit of a single subsitutionalFe atom on one of 32 interface In (or As) sites.

In the first of two model studies, we added, non self-consistently, a constantattractive or repulsive potential shift inside the atomic sphere of the single Fe im-purity. The effect on the minority and majority conductances is shown in Fig. 2.2as a function of the potential shift. The shift is seen to have only a limited effectin the majority channel. For minority spins however there are large resonant-liketransmission peaks which can be moved in and out of resonance by the shift. Theeffect is especially large for As where it can lead to almost complete quenching ofthe polarization. Such features were not observed when the same potential shift wasapplied to one of the interface In (or As) atoms in the absence of Fe impurities. The

26 Spin injection from Fe into InAs

potential shifts applied in Fig. 2.2 are of course entirely arbitrary. One can howeverimagine that they correspond qualitatively to changes in the position of Fe d statewhich might result from changes in the local geometry and chemical coordination ofan Fe impurity if it were allowed to relax.

The sensitivity to the structure is further illustrated in Fig. 2.3 where we show theeffect on the interface resistance of diplacing a single Fe impurity along the (001) axis,perpendicular to the interface. The atomic sphere potential is not recalculated self-consistently for each displacement but is displaced rigidly. Negative displacementscorrespond to displacements towards the Fe lead and positive ones in the directionof the InAs lead. Displacements are given in units of the Fe-InAs interfacial distancewhich is about 1.4A in our calculations. The values of IR are given relative totheir zero-displacement values. While the majority channel is not changed at all, theminority one varies by as much as 40% for the range of displacement studied.

In view of the great sensitivity of the transmission through an Fe|InAs interfaceto the details of its structure and composition, it is difficult to make any detailedinterpretation of experimental observations without having much more informationabout the interfaces which were studied, or to make useful predictions if the interfacestructure and composition cannot be controlled experimentally. Though it is, in prin-ciple, possible to theoretically determine relaxed interface geometries by total energyminimization, in practice an extensive study of systems of this size are prohibitivelyexpensive. Even if it were possible, the relevance of the results of such calculationsto real experiments is doubtful because real interfaces are frequently metastable andtheir structures depend on the growth conditions, annealing etc. Even though Fig. 2.1shows that spin-injection is in theory possible, it may be impossible to approach thislimit in practice if Fe reacts with InAs during growth.

The resonant features of Fig. 2.2 indicate that the effectiveness with which Feimpurities facilitate forward scattering in the minority channel quenching the polar-ization, is determined in large part by the local electronic structure of the impurityand the way it bonds with the InAs host. This is turn suggests that the spin-injectionproperties of the system could be improved if Fe with its open d-shell were preventedfrom reacting with the InAs substrate during growth. In the following sections westudy the properties of systems where such reactions are prevented by introducing abuffer layer (BL) of non-magnetic metal at the Fe|InAs interface.

2.4 Fe|BL|InAs system

A suitable buffer layer would be a nonmagnetic metal whose lattice constant matchedthose of both the ferromagnet and the semiconductor to permit epitaxial growth. Itshould not act as a barrier to transmission of ∆1 states from Fe into InAs. The lattercondition is met by e.g. noble metals with their Fermi level-crossing bands possessing∆1 symmetry along Γ – X direction. In the following we present calculations for aAu buffer layer. The lattice constants of fcc gold and bcc Fe are known to matchvery well and near perfect interfaces can be prepared experimentally. It has also beenshown that, despite the lattice mismatch, crystalline Au can be grown on top of InAs

2.4. Fe|BL|InAs system 27

‘

‘

Fe bct MAJ Fe bct MIN Au fct InAs

Figure 2.4: Band structure of Fe bct majority and minority, Au fct and InAsin (001) growth direction. Downfolded bands are marked with green dashed lines.Only unfolded bands are labeled with the names of their irreducible representations.

[28]. In the following, we assume a thin layer of face-centred tetragonal Au whosein-plane lattice constant has been adjusted to match that of InAs. The out-of-planelattice constant is chosen to preserve the volume of the bulk cubic unit cell withaAu = 4.078A.

2.4.1 Ideal Fe|Au|InAs junction

We first present results for a junction with specular Fe|Au and Au|InAs interfaces.In Fig. 2.5 the conductance is shown as a function of the distance between the Fermienergy and the bottom of the InAs conduction band, EF − EC , for a junction witha five monolayer (ML) thick Au buffer layer. The Au layer should not have to be sothin that pinholes might occur during growth. It should not be so thick that spin-flipscattering becomes appreciable. Results for both In- and As-terminations are shownin the figure. We see that the polarization of injected current is higher than 90%for a wide range of values of EF − EC . We can understand these results in terms ofthe band structures shown in Fig. 2.4. In the center of the 2D BZ, the ∆1 states ofAu couple effectively with the states of the same symmetry in majority spin Fe andwith ∆zb

1 in InAs thus preserving the high transmission of the majority channel. Thefiltering of minority carriers on the other hand is even more effective than for pureFe|InAs. Unlike the ∆zb

1 states in InAs, the ∆1 states of Au are strictly orthogonalto ∆2′ in Fe which are thus forced to tunnel through the buffer layer to reach the

28 Spin injection from Fe into InAs

0.05 0.15 0.250

0.002

0.004

cond

ucta

nce

(e2 /h

)

0.1 0.2

96

98

100

EF−E

c, eV

pola

rizat

ion,

(%

)

0.05 0.15 0.250

0.002

0.004

EF−E

c (eV)

0.1 0.2

96

98

100

EF−E

c, eV

pola

rizat

ion,

(%

)Figure 2.5: Conductance (in units of e2/h) through ideal Fe|Au5|InAs junctionfor As-terminated (left panel) and In-terminated (right panel) Au|InAs interfacesas a function of concentration of dopants in InAs for majority (N) and minority(H) spins. Corresponding polarizations are shown on the insets.

semiconductor. These arguments only hold strictly for the states at Γ where distinctsymmetries are well defined. As EF − EC increases and a larger area of the 2D BZaround Γ becomes occupied with InAs conduction band states, the effectiveness ofthe above selection rule decreases leading to a gradual reduction of the polarization.

The results shown in Fig. 2.5 were obtained for a single thickness of buffer layer,5 ML of Au. Multiple reflections occur in ideal layered structure and can sometimeslead to substantial thickness-dependence of the transmission. Therefore, before pro-ceeding to study the effect of interface disorder, we should study how the resultspresented in Fig. 2.5 are modified by size effects. As just discussed, the situationin the majority channel is simple. The Au ∆1 state matches the Γ states in Fe andInAs very well and is therefore only very weakly confined within the buffer layer.We expect the modulation of the majority spin transmission probabilities to be veryweak and, for finite values of EF −EC , to be washed out in course of the 2D BZ in-tegration. Note that confinement occurring outside the central area of 2D BZ, wherethere are no states in InAs, will not affect transport properties. The minority spinchannel, where Au ∆1 states encounter the “symmetry gap” in Fe (see Fig. 2.4), ismore complicated.

Fig. 2.6 shows the ∆1-symmetry local density of states (LDOS) on the interfaceFe atoms at k|| = Γ as a function of energy for a large range of Au thickness. A smallimaginary part (z = 0.0001 Ry) was added to the energy in order to make the DOSplots smoother (transport calculations are performed for real energies). The top andbottom panels of the figure show the minority spin ∆1-symmetry bulk DOS for

2.4. Fe|BL|InAs system 29

-1 -0.5 0 0.5 1

E-EF, (eV)

4

0DOS InAs

Fenum

ber

of

Au m

onola

yers

25

20

15

10

5

30

DOS

0

10

1

0

2

3

4

-1

-2

-3

-4

Figure 2.6: Local density of minority spin states with ∆1 symmetry (on alogarithmic colour scale) for interface Fe atoms at the Γ point as a function of theenergy and the thickness of the Au buffer layer. The minority spin local DOS for∆1-symmetry states is shown on a linear scale for bulk Fe (upper panel) and InAs(lower panel).

tetragonal Fe and cubic InAs, respectively. The symmetry gap in Fe for minorityspin states with ∆1-symmetry can be seen in the vanishing DOS in the energy range-1 to +0.2 eV with respect to the Fermi energy in the top panel of the figure. The0.42 eV InAs gap is clearly visible between -0.44 and -0.02 eV in the bottom panel.For junctions with more than 11 MLs of Au, the potentials were prepared non-self-consistently by repeating the potential of the central Au layer. For energies fallingwithin the band gap of InAs, Au minority-spin ∆1 states are perfectly confinedinside the buffer layer leading to the formation of quantum well (QW) states, welldocumented in the literature [29]. These are visible in Fig. 2.6 as sharp, well quantizedresonances. Below and above the InAs gap, Au states are able to spill into thesemiconductor by coupling to the InAs ∆zb

1 band. The sharp resonances are thenreplaced by a relatively weakly modulated DOS on the interface Fe atoms. At about0.2 eV above the Fermi energy, the Fe ∆1 symmetry gap ends, the Au minority-spin∆1 states are not confined on either side and changing the thickness of Au does

30 Spin injection from Fe into InAs

not lead to any significant modulation of the interface Fe DOS. For the energieswe are primarily interested in - low in the conduction band of InAs and in the Fesymmetry gap - the LDOS modulation visible on the logarithmic scale used in Fig. 2.6is rather weak on an absolute scale and unlikely to substantially alter the transmissionfrom the poorly matched ∆2′ states of Fe. As we move away from Γ and have tointegrate over a larger region of reciprocal space about Γ, the confinement at theFe|Au interface decreases, the quantization becomes weaker (the maxima occurringat different energies) and the net effect decreases. To test the above picture explicitly,we calculated the conductance as a function of EF − EC for a range of buffer layerthicknesses (1 – 10 ML) and found no signs of ∆1 QW effects in either spin-channel.

In the minority channel the downfolded “corner” states (dashed lines close tozero energy in Fig. 2.4(c) are strongly confined within the Au layer leading to theformation of QW states with a symmetry different to the ∆1 symmetry discussed inthe context of Fig. 2.6. The confinement follows from a combination of the orbitalcharacter of these states (which is free-electron-like px, py) and the in-plane Blochphase coming from downfolding the states at the “corner” of the Au surface BZ ontothe origin Γ of the smaller, common interface BZ which is determined by the materialwith the largest lattice constant InAs. This Bloch phase factor dictates that eventhough the orbital composition of the Au states is qualitatively similar to that ofthe four ∆5 states in the Fe minority band, the coupling between them is identicallyzero at Γ and remains weak around it. If the Au “corner” states are brought downto the Fermi level, e.g. by altering the amount of the tetragonal distortion of Au,then resonant transport can occur, allowing minority ∆5-derived states in Fe to betransmitted into InAs for some values of k‖. The effect can lead to a substantialreduction of the polarization but occurs only for specific combinations of EC − EFand thickness of the Au buffer layer. Consequently it should be possible to avoid itin experiment, especially as it is unlikely to survive the presence of disorder.

2.4.2 Disordered Fe|Au|InAs junction

The next step is to study the effect of interface disorder on the transmission througha Fe|Au|InAs junction. We fix the Au thickness at 5 ML and start by introducingdisorder at the magnetic Fe|Au interface while keeping the Au|InAs interface clean.We consider substitutional disorder with Fe impurities replacing Au on some interfacesites. The calculated spin-dependent resistances are shown in Fig. 2.7 as a functionof the fraction of the substituted Au atoms. Note that while the maximum fractionconsidered here (14/64) is formally the same as in Fig. 2.3, the actual areal densityof impurities is twice as large here because there are twice as many Au atoms perunit area which can be substituted with Fe as there are In (or As) sites. We see thesame main trends as we saw in Fig. 2.7: for both terminations the majority chan-nel is almost unaffected by disorder while the minority resistances are substantiallyreduced. Comparing Figs. 2.3 and 2.7, we note, however, that the minority resis-tances are almost an order of magnitude larger than in the original Fe|InAs system.Consequently, average polarizations do not vanish as they did previously but

2.4. Fe|BL|InAs system 31

104

105

106

107

108

0 2/64 6/64 10/64 14/64

inte

rfac

e re

sist

ance

(fΩ

m2 )

As termination

0

50

100

2/64 6/64 10/64 14/64

pola

rizat

ion

(%)

104

105

106

107

108

0 2/64 6/64 10/64 14/64

In termination

fraction of substituted Au atoms

0

50

100

2/64 6/64 10/64 14/64

pola

rizat

ion

(%)

Figure 2.7: Interface resistance of As- (left panel) and In- (right panel) termi-nated Fe|Au|InAs junctions as a function of the fraction of interfacial Au atomssubstituted by Fe for majority (N) and minority (H) spins.

remain positive. However, the configuration spread in the minority channel remainssubstantial and, for In termination, overlap with the majority values occurs.

As a further test of the stability of these results, calculations similar to thoseof Fig. 2.3 were performed by displacing a single Fe impurity perpendicular to theinterface. The change in interface resistance is shown in Fig. 2.8 as a function of thedisplacement of one Fe impurity (1/64) given as a percentage of the Fe–Au distance(1.57 A). Negative displacements are towards the Fe electrode, positive ones towardsAu. The calculations were performed non-self-consistently and the resistances in theplot are normalized to their zero displacement values. Comparing Figs. 2.3 and 2.8we see that the relative change in the minority channel is about 4 times smaller inthe present case.

Turning now to disorder at the non-magnetic Au|InAs interface, we consider twoscenarios for substitutional disorder: (i) with In (As) atoms substituting Au in thebuffer layer, for which results are shown in the left-hand panels of Fig. 2.9 and (ii)with Au atoms substituting In (As), for which results are shown in the right-handpanels of Fig. 2.9. The upper panels are for As termination, the lower panels for Intermination. As before, the results for the majority spin case are quite insensitive todisorder and do not need to be discussed in any detail.

In scenario (i), the In (As) impurities can occupy either of two inequivalent setsof sites as there are two Au atoms for each In (As) interface atom. The two inequiv-alent sites are considered separately and the results are shown as filled and emptysymbols for the minority spin case. Mixed configurations were not considered. Theconfiguration averaged results, shown as continuous and dashed lines, scarcely differ

32 Spin injection from Fe into InAs

−10% −5% 0% 5% 10%0.9

0.95

1

1.05

spacial displacement of Fe impurity

norm

aliz

ed in

terf

ace

resi

stan

ce

Figure 2.8: Normalized interface resistances of As-terminated Fe|Au5|InAs junc-tion as a function of a single Fe impurity displacement in units of Fe|Au interfacedistance for majority (N) and minority (H) spins.

for As termination (upper left) but are very different for In termination (lower left).For both terminations, the configuration averaged resistances for the majority andminority channels are well separated. The configuration spread for As-termination isexceptionally large for the sites depicted with open triangles and dashed line yieldingzero polarization for some higher concentration configurations. Clearly, it would beimportant to establish the relative stability of the two different sites by determiningthe difference in their energies. Results for the second scenario, with Au substitut-ing In or As interface atoms, are shown on the right-hand side of Fig. 2.9. Theminority spin resistance is seen to be quenched much more rapidly by disorder atthe non-magnetic interface than by disorder at the magnetic interface, with the av-erage polarization dropping quickly to zero and becoming negative. For all but thelowest concentrations of disorder considered, some minority-spin configurations haveresistances which are lower than the majority spin resistance making it important toknow whether such configurations are energetically favoured or not.

Comparison of Figs. 2.7 and 2.9 with Fig. 2.1 shows that overall, introducing agold buffer layer leads to a higher minority channel interface resistance and a higherpolarization for a given level of interface disorder. However, the results are alsosensitive to the details of the interface disorder: the chemical nature of a substitu-tion, its local atomic structure and how a given concentration of impurity atoms isconfigured. In many cases the spread of resistance values in the minority channelis very large, with resistances for some configurations overlapping with that of themajority channel, even when average values remain well separated. Surprisingly, the

2.4. Fe|BL|InAs system 33

Figure 2.9: Majority- (N, 4) and minority- (H,O) spin resistance of a Fe|Au|InAsjunction for In- and As-terminated interface in case of Au|InAs interface disorderwith In(As) substituting Au atoms. Filled and empty symbols denote the configu-rations where As (In) impurities occupy only one of the two inequivalent Au sitesshown on the left top(bottom) panels. Majority (N) and minority (H) spins resis-tance of Fe|Au|InAs junction for In- and As-terminated in case of Au|InAs interfacedisorder with Au substituting As(In) atoms shown on the right top(bottom) panels.

polarization seems to be more sensitive to the disorder at the non-magnetic Au|InAsinterface. This is unfortunate because while it is probably possible to prepare nearlyideal Fe|Au interfaces [30], the same is less likely for the non-magnetic Au|InAs in-terface. The situation is also likely to deteriorate if there is disorder present at bothinterfaces simultaneously.

34 Spin injection from Fe into InAs

2.5 Conclusions

In this chapter an earlier study [2] of the effect of interface disorder on the transmis-sion properties of Fe|InAs interfaces has been extended. In particular, it has beenshown that for small concentrations of substitutional disorder, with Fe substitutinginterface In or As atoms, a proportionality between the minority spin conductanceand the concentration of disorder is observed. Interface disorder results in quenchingof the spin polarization of current injected from Fe into InAs. It has been possible todemonstrate that in the dilute limit of a single interface impurity, the interface trans-mission is very sensitive to the details of the interface atomic structure by applying ashift to the potential and/or position of the Fe impurity atom. This suggests tryingto improve the spin-injection efficiency by preventing Fe from directly interactingwith InAs by introducing a buffer layer which is reasonably lattice-matched to bothFe and InAs. A suitable candidate is Au which is unlikely to mix with Fe [30]. Auis a noble metal which at the Fermi level has only states with ∆1 symmetry alongtheΓ – X direction. We showed that a Au buffer layer increases the efficiency of spininjection in an ideal Fe|Au|InAs system compared to an ideal Fe|InAs interface. Athin metal layer like Au can form quantum well states for energies and k‖ vectorsfor which the transmission into Fe and InAs is poor. These states, however, playan important role only for clean interfaces and they are quenced by small amountsof interface disorder which destroy the translational symmetry and conservation ofcrystal momentum parallel to the interface.

The most important part of these work concerns the effect of interface disorderon the polarization of current injected into InAs in the Fe|Au|InAs junction. Theeffect was again studied using larger lateral supercells to model disorder. We foundthat disorder at the Fe|Au interface reduces the polarization but to a smaller degreethan for Fe|InAs. Unfortunately, disorder on the non-magnetic Au|InAs interface isshown to have much stronger impact and can quench the polarization or change itssign.

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[2] M. Zwierzycki, K. Xia, P. J. Kelly, G. E. W. Bauer, and I. Turek, Phys. Rev. B67, 092401 (2003).

[3] G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. van Wees,Phys. Rev. B 62, R4790 (2000).

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BIBLIOGRAPHY 35

[7] T. J. Zega, A. T. Nahbicki, S. C. Erwin, I. Zutic, and G. Kioseoglou, Phys. Rev.Lett. 96, 196101 (2006).

[8] A. Fert and H. Jaffres, Phys. Rev. B 64, 184420 (2001).

[9] X. Jiang et al., Phys. Rev. Lett. 94, 056601 (2005).

[10] Y. B. Xu, E. T. M. Kernohan, M. Tselepi, J. A. C. Bland, and S. Holmes, Appl.Phys. Lett. 73, 399 (1998).

[11] L. Ruppel et al., Phys. Rev. B 66, 245307 (2002).

[12] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and G. E. W. Bauer, Phys.Rev. B 73, 064420 (2006).

[13] O. K. Andersen, O. Jepsen, and D. Glotzel, Canonical description of the bandstructures of metals in, in Highlights of Condensed Matter Theory, edited byF. Bassani, F. Fumi, and M. P. Tosi, International School of Physics ‘EnricoFermi’, Varenna, Italy” pp. 59–176, North-Holland, Amsterdam, 1985.

[14] I. Turek, V. Drchal, J. Kudrnovsky, M. Sob, and P. Weinberger, Elec-tronic Structure of Disordered Alloys, Surfaces and Interfaces (Kluwer, Boston-London-Dordrecht, 1997).

[15] O. K. Andersen, Phys. Rev. B 12, 3060 (1975).

[16] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).

[17] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).

[18] D. Glotzel, B. Segall, and O. K. Andersen, Sol. State Comm. 36, 403 (1980).

[19] R. M. Martin, Electronic Structure: Basic Theory And Practical Methods (Cam-bridge University Press, Cambridge, United Kingdom, 2004), pp. 42–44.

[20] S. M. Sze, Physics of Semiconductor Devices, second ed. (John Wiley & Sons,New York, United States of America, 1981).

[21] T. Ando, Phys. Rev. B 44, 8017 (1991).

[22] K. Xia et al., Phys. Rev. B 63, 064407 (2001).

[23] Y. V. Sharvin, Zh. Eksp. Teor. Fiz. 48, 984 (1965), [Sov. Phys. JETP 21, 655(1965)].

[24] K. M. Schep, J. B. A. N. van Hoof, P. J. Kelly, G. E. W. Bauer, and J. E.Inglesfield, Phys. Rev. B 56, 10805 (1997).

[25] P. Soven, Phys. Rev. 156, 809 (1967).

[26] S. Mirbt, B. Sanyal, C. Isheden, and B. Johansson, Phys. Rev. B 67, 155421(2003).

36 Spin injection from Fe into InAs

[27] S. C. Erwin, S.-H. Lee, and M. Scheffler, Phys. Rev. B 65, 205422 (2002).

[28] C. Ohler, C. Daniels, A. Forster, and H. Luth, J. Vac. Sci. Technol. B 15, 702(1997).

[29] M. Milun, P. Pervan, and D. P. Woordruff, Rep. Prog. Phys. 65, 99 (2002).

[30] F. J. A. den Broeder, D. Kuiper, A. P. van de Mosselaer, and W. Hoving, Phys.Rev. Lett. 60, 2769 (1988).

Chapter 3

Influence of Roughness andDisorder on TunnelingMagnetoresistance

In this chapter we present a systematic, quantitative study of the effect of inter-face roughness and lead disorder on the transport properties of Fe|vacuum|Fe andFeCo|vacuum|FeCo MTJs. From parameter-free, electronic structure based calcu-lations we find that surface roughness has very strong effect on the spin-polarizedtransport. The effect of lead disorder is weaker but still sufficient to suppress thehuge TMR predicted for ideal systems.

3.1 Introduction

Spintronics is a rapidly developing branch of a modern condensed matter physics. Itaims at exploiting the quantum spin state of the electrons as well as their charge.Control over additional degree of freedom leads to observation of new effects. Mostimportant for practical applications is the magnetoresistance - dependence of resis-tance of a system on external magnetic field. Here we concentrate on the tunnel-ing magnetoresistance (TMR) which refers to the dependence of the resistance ofa FM1|I|FM2 (ferromagnet|insulator|ferromagnet) magnetic tunnel junction (MTJ)on the relative orientation of the magnetization directions of the ferromagnetic elec-trodes when these are changed from being antiparallel (AP) to parallel (P): TMR =(RAP − RP )/RP ≡ (GP − GAP )/GAP . Since the discovery of large values of TMRin MTJs based upon ultrathin layers of amorphous Al2O3 as insulator, [1] a consid-erable effort has been devoted to exploiting the effect in sensors and as the basis fornon-volatile memory elements. Understanding TMR has been complicated by thedifficulty of experimentally characterizing FM|I interfaces. The chemical composi-tion of the interface has been shown [2] to have a strong influence on the magnitudeand polarization of the TMR and knowledge of the interface structure is a neces-

37

38 Inuence of Roughness and Disorder on TMR

sary preliminary to analyzing MTJs theoretically. In the absence of detailed struc-tural models of the junctions and the materials-specific electronic structures whichcould be calculated with such models, the effect was interpreted in terms of electrodeconduction-electron spin polarizations Pi, using a model suggested by Julliere [3] inwhich the TMR = 2P1P2/(1 − P1P2). A great deal of discussion has focused onthe factors contributing to the quantity [4] P but the use of amorphous oxide asbarrier material made impossible a detailed theoretical study with which to confrontexperiment [5, 6].

The situation changed quite drastically with the recent observation of large valuesof TMR at room temperature in FeCo|MgO|FeCo MTJs in which the MgO tunnelbarrier was mono- [7, 8] or poly-crystalline [9]. This work was motivated in partby the prediction [10, 11] by materials-specific transport calculations of huge TMRvalues for ideal Fe|MgO|Fe structures. This new development lends fresh urgencyto the need to understand the factors governing the sign and magnitude of TMRbecause the largest observed value of 353% at low temperature, [8] is still well belowthe ab-initio predicted values of order 10,000% for the relevant thicknesses of MgO[10]. Some effort has been devoted to explaining the discrepancy in terms of interfacerelaxation [12] or the formation of a layer of FeO at the interface [13, 14] but the roleof interface disorder has only been speculated upon.

In this chapter, which is a follow-up to our earlier study, [15] we use first principleselectronic structure calculations to study the effect of roughness and lead disorderon TMR in MTJs with a vacuum barrier and Fe or Fe1−xCox alloy electrodes. Avacuum barrier was chosen due to its simplicity and because there are many studiesof spin-dependent vacuum tunneling in its own right [16–20].

The chapter is organized as follows: in the next section we discuss theoreticalapproach used to obtain results presented in remaining sections. In Sec. 3.3 we brieflydiscuss the electronic structure of the ideal MTJ concentrating on the potential profileof VTB. Sec. 3.4 is devoted to the detailed study of the transport properties of idealFe|vacuum|Fe junction. In the next two sections we discuss the results obtained forMTJ with disorder in the form of surface roughness (Sec. 3.5) and substitutionaldisorder in the leads (Sec. 3.6). We finish with the conclusions in the last section.Technical details concerning the convergence of our calculations are relegated to thetwo appendices.

3.2 Methods, Models and Technical Details

We study transport properties of the MTJs in the linear-response regime within theframework of the local spin-density approximation (LSDA) of the density functionaltheory (DFT) in a two-step procedure. In the first step self-consistent potentials,charge- and spin-densities are determined for a system (Fig. 3.1) consisting of twosemi-infinite ideal leads and the scattering region (SR). The scattering is restrictedto the latter part of the system which encompasses vacuum tunneling barrier (VTB)

3.2. Methods, Models and Technical Details 39

VCA VCAFe Co

Co Fe

Left lead Right leadScattering region

Fe Fe

Left lead Right leadScattering region

CPA CPA

N­2

N

N­1

N

roughness

disorder

Figure 3.1: Top figure represents Fe|vacuum|Fe MTJ with interface roughness(surface coverage). Grey circles stand for Fe atoms. SR in this figure is given byvacuum barrier and 4MLs of Fe on each side of it. Interface roughness is mod-elled as an incomplete Fe layers. Bottom figure represents alloy lead disorder inFeCo|vacuum|FeCo MTJ. Grey and black circles in the SR stand for Fe and Coatoms respectively. In this case FeCo atoms in the leads are treated in the virtualcrystal approximation (VCA). These atoms are represented with brown circles.

and several monolayers (MLs) of metal immediately next to it. The assumed in-planetranslational invariance preserves k|| as good quantum number. The self consistentcalculations are performed using surface Green’s function [21] implementation ofthe tight-binding linear muffin-tin orbitals (TB-LMTO) method [22]. Throughoutthe paper we use spd basis set and exchange-correlation potential is parametrizedaccording to von Barth and Hedin [23]. During the calculations the potentials in thescattering region are allowed to relax. The potentials of the leads are kept at theirpreviously calculated bulk values. Atomic spheres approximation (ASA) is used withthe sites of the fixed bcc lattice occupied with either atomic or empty spheres (ES),the latter used to model VTB. In the case of Fe only MTJs (Fig. 3.1, top panel) anexperimental lattice constant aFe = 2.866A is used. (001) is the transport direction.

In the second step, the self-consistent potentials are used to calculate the elementsof the scattering matrix (transmission and reflection coefficients) using a TB-MTOimplementation [24] of wave-function matching (WFM) scheme due to Ando [25].With these the conductance can be calculated using Landauer-Buttiker formula [26].This involves integration over two-dimensional Brillouin zone (2D BZ). Convergence

40 Inuence of Roughness and Disorder on TMR

20 40 60 80 1001.6

1.8

2

2.2

2.4sp

in m

omen

t, µ B

Co concentration (%)Fe Co

CPA

VCA

Figure 3.2: Bulk magnetic moments of FeCo bcc alloy as the function of alloyconcentration calculated in the VCA (solid line) and CPA (dashed line) approxi-mations.

of the results with respect to sampling density is discussed in Appendix 3.8.The disorder is modelled using lateral supercells with typical size of 100 atoms

(10x10). The convergence with respect to supercell size is discussed in Appendix 3.9.Two different kinds of disorder, shown in panels of Fig. 3.1, are considered.

The first one is the roughness at Fe|vacuum interface (Fig. 3.1, top panel). Inthis case the topmost layers of either electrode is covered with additional Fe atoms,occupying the sites of underlying bcc lattice. The potentials inside atomic and emptyspheres are calculated using the layered version [21] of a coherent potential approxi-mation (CPA) [27] with the surface layer treated formally as substitutional FexES1−xalloy. The spheres are then randomly distributed in the lateral supercell with theconcentration of Fe corresponding to the assumed surface coverage x.

The second case studied is the junction with electrodes composed of the randomFe1−xCox alloy (Fig. 3.1, bottom panel). It is known from the experiments that Coforms a substitutional random alloy with Fe (Fe1−xCox) which assumes bcc structurein the concentration range from x = 0.0 to x w 0.7 [28]. For higher Co concentrationsa phase transition to the hcp structure occurs. For the sake of simplicity we assumehere a bcc crystal structure for all concentrations. The FexCo1−x alloy lattice constantis calculated from Vegard’s law whereby

aFexCo1−x = (xaFe + (1− x)aCo (3.1)

Here aCo = 2.817A [29] is a lattice constant of bcc Co determined assuming thatthe volumes of bcc and hcp cells are identical. Like previously the atomic potentials

3.3. Barrier prole 41

inside SR are calculated using CPA and the atomic spheres distributed randomlyin lateral supercells. Note that we do not introduce additional roughness at theFM|vacuum interface. Using this approach it is possible to model substitutionaldisorder in the part of the electrodes falling inside the scattering region. However,doing the same for the leads would require the use of supercells also in the transportdirection as the WFM scheme is based on the assumption that the leads possessthe full translational symmetry. It would considerably increase the computationalcosts and is in practice not feasible. Instead we describe the leads using virtualcrystal approximation (VCA) where the alloy is replaced by homogeneous materialconsisting of ”atoms” with fractional atomic numbers calculated using equation akinto Eq. (3.1). Given their closeness in the periodic table and similarity of their bcclattice constants one expects VCA to work reasonably for the alloy consisting ofFe and Co. We test it explicitly by comparing the magnetic moments of the bulkFexCo1−x alloy calculated using VCA and CPA. An excellent agreement is foundbetween VCA and CPA values for the whole range of concentrations as shown inFig. 3.2. Furthermore the values agree well with other experimental and theoreticalresults [30]. Note that only spin magnetic moment is calculated with the orbitalcontribution being neglected. Treating the scattering region and leads differentlymeans that the fictitious interface is introduced at the boundaries between theseparts of the system (Fig. 3.1, bottom panel). However its presence is of no practicalconsequence as the resistance of the whole structure is strongly dominated by thetunneling barrier. This point is further discussed in Appendix 3.9.

All conductances in the paper are given in e2/h units and normalized to the areaof the in-plane 2D unit cell, equal to a2

FexCo1−x.

3.3 Barrier profile

The usual point of reference when discussing TMR is the textbook problem of freeelectron tunneling through a rectangular potential barrier. Before proceeding it isuseful to establish to what extent our VTB conforms to this simple model.

In order to determine the potential profile of VTB we start with the self consistentcalculations for ideal Fe|vacuum|Fe junction. Then for each “layer” of vacuum we usecorresponding ES to populate the sites of bulk bcc structure and perform band-structure calculations. The position of the bottom of resulting parabolic band withrespect to the Fermi energy of the system is taken to be the barrier height for a givenlayer. The results of this procedure are shown in Fig. 3.3 for two different thicknessesof VTB. It can be seen that for a sufficiently thick barrier (20 MLs equals 28A) theshape is indeed approximately rectangular. The work function extracted from the20 MLs plot equals to 4.75 eV which agrees reasonably with the experimental valueof 4.5 eV [31]. We note that for MTJs with barrier thickness exceeding 10 MLs thecalculated conductances are smaller than the numerical accuracy [32]. Because ofthat in the following we shall restrict our calculations to barriers of 10 MLs or less.An example of the potential profile for thinner (8 MLs) VTB is shown in the inset ofFig. 3.3

42 Inuence of Roughness and Disorder on TMR

4 6 8 10 12 14 16 180

1

2

3

4

5p

ote

ntia

l b

arr

ier

(eV

)

layer number

FeVac20Fe

2 3 4 5 6 70

1

2

3

4

5 FeVac8Fe

Figure 3.3: Potential profile of vacuum barrier for Fe|vacuum|Fe MTJs withbarrier width of 20 MLs with respect to Fermi energy. Inset: profile of the potentialprofile for the barrier of 8 MLs.

3.4 Ideal Fe|vacuum|Fe MTJ. Transport calculations

Before embarking on a study of roughness and disorder effects on the TMR it isimportant to understand what factors determine the transport properties of an idealMTJ. We start by calculating conductance of ideal Fe|vacuum|Fe system with VTBof increasing thickness. The results for majority and minority spin channels in par-allel configuration of the magnetic moments, GmajP and GminP , are shown in Fig. 3.4together with a single spin conductance for antiparallel configuration, GσAP . It isinstructive to compare these results with the predictions of a simple model of freeelectrons tunneling through a rectangular barrier. In this model the decay of the con-ductance is asymptotically described by by a simple exponential factor exp(−2dκ)where d is the thickness of the barrier and κ =

√(2m/~2)Vb with Vb being the height

of the barrier.All three curves in Fig. 3.4 reach exponential limit for VTB thicker than 8 MLs

with the same decay rate (i.e. the slope of the curve). Extracting the barrier heightfrom the slopes of the curves yields Vb ≈ 4.7 eV in good agreement with the valueextracted from the potential profile of Fig. 3.3.

3.4. Ideal Fe|vacuum|Fe MTJ. Transport calculations 43

2 4 6 8 1010

−12

10−10

10−8

10−6

10−4

10−2

100

thickness of vacuum

cond

ucta

nce

(e2 /h

)

MIN

MAJ

AP

2 4 6 8 1010

1

102

103

104

105

TMR (%)

Figure 3.4: Conductances GminP (H), GmajP (N), and GσAP (×) of an idealFe|vacuum|Fe MTJ as a function of the barrier thickness (measured in units oflayers of a bcc lattice). The corresponding values of TMR are shown in the inset.

The TMR, shown in the inset of Fig. 3.4, increases with the thickness of VTBand eventually saturates at about 20,000%. This number is comparable to the TMRthose calculated from the first-principles for ideal Fe|MgO|Fe MTJ [10, 11].

Even though all three curves reach the same asymptotic limit of exponentialattenuation, their behavior is very different for thinner barriers. This is the conse-quence of different underlying mechanisms of transmission. We will discuss these inthe following subsections concentrating on majority and minority channel in parallelconfiguration.

3.4.1 Parallel configuration: Majority channel

Majority conductance for parallel configuration exhibit the simplest behavior of thethree shown in Fig. 3.4. For all but the thinnest barriers the curve follows a simpleexponential decay. Accordingly we expect the tunneling to qualitatively follow thepredictions of the free electron model. This is confirmed by examination of the k‖-resolved transmission shown in the right hand panel of Fig. 3.5. The transmission hasa maximum in the center of 2D BZ (Γ point) and decays rapidly once we move awayfrom the center. As the thickness of VTB increases the maximum value decreasesexponentially and the transmission becomes increasingly more concentrated aroundΓ in accordance with free electron model.

Nearly spherical symmetry of the transmission plot might seem surprising whencompared with the projection of the majority Fermi surface (FSP) shown in the left

44 Inuence of Roughness and Disorder on TMR

6MLs

-12

-10

-8

-141

2

3

4 FSP

Figure 3.5: Fe majority Fermi surface projection (FSP) on plane perpendicularto [001] direction (on the left) and k‖ resolved majority transmission for VTB of 6MLs shown on a logarithmic scale (right panel). The results are shown in full first2D BZ.

hand panel of Fig. 3.5. The latter is multiply sheeted and far from free electronsphere. This apparent contradiction can be resolved by noting that the transmissionis dominated by the contribution from a single sheet of the Fe’s FS which possessesrelatively simple geometry and is visible as a rectangular cyan shape in the back-ground of the FSP plot. Thanks to their high symmetry, ∆1 at Γ, and mostly sporbital character these states are able to couple most effectively to the simple decayingmodes in the vacuum layer.

3.4.2 Parallel configuration: Minority channel

The dependence of minority conductance on VTB thickness (Fig. 3.4) is quite dif-ferent from that seen for majority spins. For thicknesses greater than 3 MLs theminority channel dominates the conductance giving rise to large and negative po-larization [33] of the current in the parallel configuration. The differences are evenmore apparent for k‖-resolved transmission shown in Fig. 3.6 for varying thicknessof VTB. The symmetry of the plots clearly deviates from spherical symmetry of tun-neling in the majority channel. The most important feature however is that for allbut thinnest barriers the transmission comes predominantly from the small areas ofvery high (comparable or equal to unity) transmission. One set of such ”hot spots”or resonances is located immediately next to the Γ point. In the following we willbe referring to these as A type resonances. Another set of very narrowly definedresonances is located further away from the center of the 2D BZ. The position ofthese B type resonances is marked with arrows in Fig. 3.6. The dominance of ”hotspots” is more apparent when a linear scale is used. An example is shown in Fig. 3.7

3.4. Ideal Fe|vacuum|Fe MTJ. Transport calculations 45

2MLs

6MLs

4MLs

8MLs

-15

-10

-5

0

B

Figure 3.6: The k‖ resolved minority transmission for VTB of 2, 4, 6 and 8 MLsshown on a logarithmic scale. For the latter the positions of B resonance are shownwith arrows. The results are shown in full first 2D BZ.

for VTB of 8 MLs.Resonant peaks in k‖ transmission has been found in first principles calculations

of tunneling through both vacuum [34–36] and other barriers [10–12]. The resonancesarise when there exists a surface state on the surfaces of metallic electrodes. Suchstate can indeed be found for Fe(001) surface [16]. Its existence is made possible bythe so called ”symmetry gap” present in Fe’s band structure. Along Γ−H directionin 3D BZ, corresponding to Γ point in 2D BZ, the two ∆1 bands are separated by agap of about 3 to 4 eV depending on the spin orientation. The other bands present inthis range of energies possess distinctly different symmetries. Consequently a surfacestate of ∆1 symmetry can exists at Γ point provided that its energy falls withinthe gap. The surface states exist for both majority and minority spins. They arehowever positioned very differently with respect to the Fermi level which reflects thedifferences in the positions of ∆1 symmetry gaps in the bulk band structures. Themajority channel surface state is located about 2 eV beneath the Fermi level and

46 Inuence of Roughness and Disorder on TMR

Kx (π/aFe) Ky (π/aFe)

-1

1 -1

1

0.2

0.4

0.6

0.8

1.0tr

ansm

issi

on

A

B

0

Figure 3.7: The k‖ resolved transmission for Fe|vacuum|Fe MTJ with barrierwidth of 8 MLs shown using linear scale. A and B labels stand for two resonancesclose to Γ and far from the center of the 2D BZ correspondingly.

consequently does not influence transmission. The minority surface state on theother hand is located close to the Fermi level and it is closely associated with the“hot spots” visible in Figs. 3.6 and 3.7.

The existence of surface state can be demonstrated by explicit calculation of den-sity of states (DOS). However it is difficult to distinguish its features in the plotsof the full or even layer-resolved DOS (LDOS). Instead we calculate for minoritychannel k‖-, layer- and orbital-resolved DOS. Setting k‖ = (0, 0) and summing overthe orbitals compatible with ∆1 symmetry (s,pz and d3z2−r2) we then obtain, forthe surface layer, the results shown in Fig. 3.8 [37]. There are no ∆1 states in theminority band structure of the bulk Fe for the energy range shown in the figure.Indeed the same procedure repeated for the layers away from the interface showsthat the peaks visible in Fig. 3.8 vanish over the distance of few monolayers from theFe|vacuum interface (see Ref. [38] for details). The surface state consists predomi-nantly of d3z2−r2 orbitals extending into the vacuum. For VTB of limited thicknessthe states at both Fe surfaces can interact through the vacuum forming bonding-antibonding pairs. This leads to the splitting of the peak visible clearly in Fig. 3.8for 5 MLs curve. As the thickness of VTB increases the splitting becomes weakerand the two peaks merge into one as seen for 8 MLs curve in the plot.

The surface state located exactly at Γ is orthogonal to states in the bulk and thusdoes not contribute to the coherent tunneling studied in this paper. However, as wemove away from the center of 2D BZ the surface state evolves into resonant states of

3.4. Ideal Fe|vacuum|Fe MTJ. Transport calculations 47

−0.2 −0.1 0 0.1 0.20

20

40

60

80

5 ML

8 ML

E−EF [eV]

LDO

S [s

tate

s/(e

V a

tom

spi

n)]

Figure 3.8: ∆1 contribution to the minority LDOS of the interfacial Fe atom fork‖ = Γ in the Fe|vacuum|Fe MTJ for 5 MLs (solid line) and 8 MLs (dashed line)vacuum.

lower symmetry capable of coupling to the states in the bulk. These states pos-sess weak but non-negligible in-plane dispersion (i.e. k‖ dependence). Just as thesurface state the resonant states can interact through the barrier forming bonding-antibonding pairs. Whenever either of the pair aligns with the Fermi level a resonanttunneling can take place. There are two parameters that characterize the resonanttunneling: the surface resonance half-width ΓR and bonding-antibonding splitting ∆or equivalently the resonance life time tR = ~/ΓR and hopping time tH = ~π/∆.The resonant transmission without attenuation (T = 1) is observed if ∆ ≥ ΓR [34].

In order to map out the positions of resonant states in 2D BZ we have plottedthe minority k‖-resolved LDOS for atomic layer at the interface and in the bulk ofFe. These are shown in Fig. 3.9 for VTB of 8 MLs together with the projected Fermisurface. The energy has been fixed at the Fermi level. The interfacial LDOS is vividlydifferent from that of the bulk layer with characteristic sharply peaked structures.The amplitudes of the peaks drop rapidly as me move way from Fe|vacuum interface.What is more the presence of the peaks coincide with the “hot spots” of tunnelingtransmission shown in Fig. 3.6.

With the increasing thickness of the barrier the bonding-antibonding splittingdecreases and as a result the position of the resonances in the 2D BZ changes as well.However, as long as the splitting is large enough for the resonance condition to be

48 Inuence of Roughness and Disorder on TMR

-5

-4

-3

bulk

interfacesubinterface

1

2

3

4FSP

B

Figure 3.9: Minority spin FSP on the plane perpendicular to [001] direction (topleft). The LDOS (logarithmic scale) as a function of the k‖ for Fe atom in bulk (topright) at the subinterface (bottom left) and interface (bottom right). The LDOScalculations were performed for energy equal to the Fermi level. The positions ofB resonance are shown with arrows in the right-bottom panel.

fulfilled, the maximum transmission remains equal to one. This is illustrated inFig. 3.10 where the maxima of k‖-resolved transmission for A and B resonances areshown separately. The maximum transmission for majority channel and for a singlespin transmission in AP configuration are shown as well for comparison. We see thatthe maximum for A type resonances start decreasing only for VTBs thicker than 7MLs. This coincides with the onset of the exponential decay for the total minoritychannel conductance shown in Fig. 3.4. The correlation might seem surprising giventhat the maximum for B type resonances remains equal to one for even thicker bar-riers. However, in practice the contribution to the total conductance comes entirelyfrom A type resonances. The reason for this is that B type peaks are extremelylocalized in the momentum space. This is illustrated in Fig. 3.11 where we plot across-section through the A and B type peaks in transmission. We see that the widthsof the peaks differ by as much as three orders of magnitude. Thus the regions of

3.5. Eect of interfacial roughness on the TMR 49

2 3 4 5 6 7 8 910

−10

10−5

100

thickness of vacuum

max

. tra

nsm

issi

on MAJ AP MIN A

MIN B

Figure 3.10: Maximum transmissions for majority (N) and minority (H) spinchannels and one spin channel in antiparallel configuration (×) are shown as afunction of barrier width.

high transmission associated with B type resonances form effectively a set of measurezero and do not contribute to the integral over 2D BZ.

In the AP configuration surface state appears in minority spin channel on only oneof Fe|vacuum interfaces. However, as seen in Fig. 3.10, for sufficiently thin barriereven a single resonant state can give rise to significant transmission. Just as forparallel configuration there exists a rough correspondence between the features ofthe curves in Figs. 3.10 and 3.4.

3.5 Effect of interfacial roughness on the TMR

An ideal tunnel junction considered in the previous section is impossible to realize inpractice; there will always be some finite amount of disorder in the form of surfaceroughness, islands, dislocations etc. In this section we will consider Fe|vacuum|Fejunction with disorder introduced in the form of incomplete (rough) surface layers,modeled by occupying, at random, a fraction of the lattice sites of the topmost layerwith Fe atoms. A sketch of the system is shown in the top panel of Fig. 3.1. Threedifferent mechanisms can influence the transmission through the MTJ with rough Fesurfaces. Firstly, introduction of the disorder breaks the point group symmetry ofthe system and is bound to destroy the surface state together with the associatedresonant states. This in turn should result in the huge decrease of minority conduc-tance as the resonant tunneling is mostly responsible for the high values seen for idealstructures. Secondly, in case of the weak disorder the surface state might possibly

50 Inuence of Roughness and Disorder on TMR

0.0 0.02850

0.1

0.2

0.3

trans

mis

sion

ky

kx=0A

0.242234 0.2422660

0.2

0.4

0.6

0.8

1

trans

mis

sion

ky

kx=0.578194B

Figure 3.11: Profiles of peaks in minority transmission probability of A (top)and B (bottom) resonances for VTB of 8 MLs.

survive but it can be shifted with respect to the Fermi level and the associated reso-nances broadened. That latter effect might in some cases lead to an increase of theconductance depending on the details like the original energy of the resonant stateand the amplitude of bonding-antibonding splitting. Thirdly, introducing randomcoverage implies that in some areas the local thickness of VTB will be smaller thanfor original ideal junction. In view of steep exponential dependence of tunneling onthickness (Fig. 3.4) this can lead to increase of conductances. This effect is likelyto influence the majority channel with its simple free-electron like tunneling. Thesebroad predictions are supported by results of calculations for 50% surface coverageshown in Fig. 3.12. 10x10 lateral supercells were used and the results were averagedover 20 configurations of disorder. The disorder was generated by randomly extract-ing half of Fe atoms from one of Fe surfaces and depositing them (also randomly) onthe other surface thus keeping the average VTB thickness unchanged. As expectedthe minority channel exhibits the most spectacular changes in comparison to theconductances of ideal system also shown in Fig. 3.12 for comparison. The values ofthe minority conductances decrease by up to four orders in magnitude and are nowactually smaller than the majority ones. The conductance of majority channel is

3.5. Eect of interfacial roughness on the TMR 51

2 4 6 8 1010

−12

10−10

10−8

10−6

10−4

10−2

100

thickness of vacuum

co

nd

ucta

nce

(e

2/h

)

MIN

MAJ

AP

2 4 6 8 1010

1

102

103

104

105

TMR (%)

Figure 3.12: Conductances GminP (H), GmajP (N), and GσAP (×) of anFe|vacuum|Fe MTJ as a function of the barrier thickness (measured in units oflayers of a bcc lattice). The dashed lines are for ideal junctions. The solid lines areconfiguration-averaged conductances for rough junctions prepared by removing, atrandom, half of the Fe atoms from one surface and depositing them, at random, onthe other surface. The results are normalized to the 1× 1 surface unit cell used forthe ideal case. The large symbols refer to data points which also appear in Fig. 3.4.

moderately increased and the antiparallel case falls in between the two parallel con-ductances. The values of TMR (see the inset in Fig. 3.12) are greatly reduced andare now in the range of 10 to 100% in qualitative agreement with experiments [7–9].

All three conductances for disordered system shown in Fig. 3.12 decay exponen-tially with the thickness of the barrier already for very thin barriers. This, togetherwith the huge drop of minority conductance, suggests that disorder in the form of50% coverage is sufficient to kill the surface resonant effects in the transmission.

In order to better understand the effect of the disorder it is useful to follow theevolution of the conductance with the increasing surface coverage. This is shown inFig. 3.13 together with the TMR in the inset. The VTB thickness has been fixed to8 MLs and the roughness is created by depositing additional Fe atoms on surfacesof both electrodes. This means that 50% point in Fig. 3.13 corresponds to 7MLs inFig. 3.12. Equivalent points in both figures have been marked with large symbols. Aswe can see there are qualitative differences in the dependence of the conductances onsurface coverage. In the following subsections we will discuss these in more details.

52 Inuence of Roughness and Disorder on TMR

0 10 20 30 40 5010

−8

10−7

10−6

10−5

10−4

cond

ucta

nce

(e2 /h

)

surface coverage (%)

surface coverage (%)MIN

AP

MAJ

0 20 40

101

103

105

TM

R (

%)

Figure 3.13: Configuration-averaged conductances GminP (H), GmajP (N), andGσAP (×) of an Fe|vacuum|Fe MTJ with 8 MLs barrier width as a function of thesurface coverage, normalized to a 1 × 1 surface unit cell. The dashed line denotesGσAP predicted from (3.2) and the dash-dotted line gives the majority conductancecalculated using Eq. (3.2). Inset: TMR as a function of the surface coverage. Thedashed line is the value predicted using Julliere’s expression and a calculated DOSpolarization of 55%. Large symbols refer to the similarly marked data points in theprevious figure.

3.5.1 Majority transmission

Majority channel for parallel configuration exhibits the most trivial behavior of thethree conductances shown in Fig. 3.13. Conductance increases as the function ofsurface coverage due to reduction of an effective barrier thickness. A simple ’weak-link’ model can be used to approximate these results. According to this model atransmission depends locally only on the thickness of the barrier. Conductance ofthe rough MTJ with 8 MLs barrier is then described as weighted average over theconductances of ideal systems with VTB of 6,7 and 8 MLs:

GmajP (x) = P6(x)Gmaj6 + P7(x)Gmaj7 + P8(x)Gmaj8 (3.2)

where GmajP (n) is the majority conductance of ideal junction with VTB of n MLsand the weights are defined as: P6(x) = x2, P7(x) = 2x(1− x) and P8(x) = (1− x)2.The results calculated using Eq. (3.2) are shown as dash-dot line in Fig. 3.13. Thevalues predicted for intermediate coverages differ from the results of first principlescalculations. However the overall trend and the values at extreme points of the plotare well reproduced.

3.5. Eect of interfacial roughness on the TMR 53

Figure 3.14: Minority conductance as a function of surface coverage in case of5, 8 and 10 MLs VTBs. Note that results for 5 and 10 MLs are multiplied by 100.Results for 5 MLs are shown using the linear scale on the right hand side of theplot.

3.5.2 Minority transmission

The minority conductance exhibits fairly complex dependence on the amount of sur-face coverage. For coverages of up to 5% we see a rapid decrease followed by a plateauof approximately constant (on logarithmic scale) values extending to about 20% ofcoverage. For even larger coverages the conductance value decreases again, reaches aminimum around 35% and afterwards start to increase. In the last part the minorityand majority curves are practically parallel. This suggests that the same mechanism,namely the decrease of the average thickness of the barrier, is responsible for theobserved growth. Similarity to the majority channel indicates also that the reso-nant effects, dominating transmission through an ideal system, have been entirelydestroyed at this point.

A huge decrease of conductance is visible already for coverages as small as 1%. Itis unlikely that surface state and associated resonances are destroyed by disorder soweak. Instead, a second mechanism discussed in the beginning of this section mightbe a culprit. Changes to a local electronic structure, in particular broadening of theresonances, can either decrease or increase the transmission depending on the detailslike the position of the peak with respect to the Fermi energy and the strength ofbonding-antibonding splitting. Both possibilities are demonstrated in Fig. 3.14 wherewe have plotted the minority conductance calculated for VTB of 5,8, and 10 MLs.The behavior of 8 and 10 MLs curves is qualitatively very similar, although the values

54 Inuence of Roughness and Disorder on TMR

-7.6

-5.25

-11

-6

specular contribution

diffusive contribution

20% 50%35%

(a) (b) (c)

(d) (e) (f)

Figure 3.15: Specular (top panels) and diffusive (bottom panels) contributions oftransmission calculated for 20% (a,d), 35% (b,e) and 50% (c,f) of surface coverage(logarithmic scales). Calculations are performed in 20x20 SC. The single k‖-pointin supercell 2D BZ was chosen to be close to the Γ point. The results are averagedover 40 disordered configurations.

of course differ very substantially. For 5 MLs however the introduction of weakdisorder results in a modest increase of the conductance. Note that 5 MLs resultsare shown using linear scale given on the right hand side of the plot.

The fact that minority conductance, while substantially reduced, remains farlarger than the majority one for coverages of up to 20% suggests that transmission inthis region of the curve is still assisted by the resonant states on Fe surface. A possiblescenario is that the random disorder removes the mirror symmetry of the junction andeffectively “detunes” resonant states. In this situation the perfect resonant tunnelingwithout attenuation is no longer possible. However even the single resonant statecan contribute significantly to the transmission as we have seen in Sec. 3.4 for APconfiguration.

The conductance of MTJ in the presence of the disorder can be factorized into

3.6. Substitutional disorder: FexCo1−x electrodes 55

the specular and diffusive contribution:

G = Gspec +Gdiff =e2

h

∑k‖

T (k‖,k‖) +e2

h

∑k‖ 6=k′‖

T (k‖,k′‖) (3.3)

where each of the transmissions is summed over the incoming and outgoing statesT (k‖,k′‖) =

∑µν |tµν(k‖,k′‖)|

2. As the disorder present in the system increases thediffusive term in Eq. (3.3) start playing increasingly more important role. Usinga channel decomposition technique (see Ref. [24] for details) we can separate thetwo terms. In Fig. 3.15 we show the k‖-resolved specular [T spec(k‖) = T (k‖,k‖)]ad diffusive [T diff (k‖) =

∑k′‖(6=k‖)

T (k‖,k′‖)] transmissions for varying coverages.The calculations were performed using 20x20 supercells with a single k‖-point in thedownfolded BZ. This is equivalent to 400 k‖-point sampling of the original BZ. Inall three cases the diffusive contribution outweighs the specular one with most of thecontribution coming from the area around the center of 2D BZ. It is only for thehighest concentration that the transmission becomes more evenly spread over the2D BZ.

3.5.3 Antiparallel spin alignment

The AP conductance assumes values intermediate between GmajP and GminP . Inter-estingly it can be approximated by a simple formula

GσAP =√GmajP GminP . (3.4)

As we see in Fig. 3.13, where the results of Eq. (3.4) are shown using dashed line,perfect agreement sets in as soon as the surface resonance contribution is killed byroughness [39]. Somewhat similar relation has been demonstrated before for the idealMTJs [40]. It is easy to show that as long as Eq. (3.4) holds the TMR is positive.

3.6 Substitutional disorder: FexCo1−x electrodes

In this section we study the transport properties of the MTJ with the electrodescomposed of random Fe1−xCox alloy. As discussed in Sec. 3.2 the calculations areperformed for the structure shown schematically in the bottom panel of Fig. 3.1. Theuniform potential in the leads is calculated using VCA. The substitutional disorderis introduced within the scattering region with Fe and Co atoms distributed over thelateral supercell. The potentials for these are calculated using CPA. We denote suchstructure as VCA|CPA|vacuum|CPA|VCA. Four MLs of disordered alloy are used onboth sides of VTB with the thickness of the latter fixed to 8 MLs. See Appendix 3.9for the discussion of the associated convergence issues.

The results of transport calculations for VCA|CPA|vacuum|CPA|VCA structureare shown in Fig. 3.16 using dashed lines. The overall trend seen in the figure is thedecrease of the conductance with the increasing concentration of Co. The only

56 Inuence of Roughness and Disorder on TMR

0 20 40 60 80 100

10−8

10−7

10−6

10−5

cond

ucta

nce

(e2 /h

)

TMR

(%)

Co concentration (%)

Co concentration (%)

MIN

AP

MAJ

0 40 80102

103

104

Figure 3.16: GminP (H), GmajP (N), and GσAP (×) for anFe1−xCox|vacuum|Fe1−xCox MTJ with 8 MLs barrier width as a function ofx, the concentration of Co atoms, calculated in the virtual crystal (VCA: solidlines) and CPA/supercell (SC: dashed lines) approximations. Conductances areconfiguration averaged and normalized to a 1× 1 surface unit cell. Inset: TMR inVCA () and CPA/SC (•) approximations.

exception is the minority channel exhibiting an increase in a limit of Co concentrationsover 80% that is when a limit of pure Co electrode is approached. Similarly TMRdecreases with the increasing concentration of Co down to the 100% range and picksup again only for the largest concentrations. Overall the effect of substitutionaldisorder is similar to that of surface roughness discussed in the preceding section.The most spectacular changes take place in the minority channel and lead to thereduction of TMR ratio. However, no change of the polarization sign is observed.

The interpretation of numerical results is difficult in this case as the effects ofsubstitutional disorder and the changes to the electronic structure are intertwined.In order to shed some light on the importance of the latter we have performed trans-port calculations also for ideal MTJ with the whole electrodes described using VCA.The results are shown as solid lines in Fig. 3.16 and empty symbols in the inset.The agreement between both sets of results is nearly perfect for majority and APcurves. For minority channel we see substantial differences for concentrations up to10%. Above that both conductances assume comparable values up until 80% Coconcentration where they start to diverge again. The two TMR curves reflect thisbehavior. As there is no symmetry breaking for VCA|vacuum|VCA structure the

3.7. Discussion and Conclusions 57

E-EF, eV

LDO

S [s

tate

s/(e

vat

om

spin

)]

Figure 3.17: The LDOS with ∆1 symmetry for different concentrations x ofFexCo1−x alloy shown as a function of energy.

evolution of the conductance in this case must be related to the changes in under-lying electronic structure. The transport for VCA|vacuum|VCA junction is initiallydominated by the resonant tunneling in the minority channel. However as the Coconcentration increases the surface state is gradually removed from the vicinity ofFermi level. This is illustrated in Fig. 3.17 where we have shown the ∆1 contributionto the LDOS at Γ and the surface layer of the electrode. Once the contribution ofthe resonant states is switched off, the conductances for both structures become verysimilar. For VCA|CPA|vacuum|CPA|VCA MTJ the process is decrease of the mi-nority conductance is sped up by the symmetry breaking and the presence of diffusescattering

The antiparallel conductance is once again well described by Eq. (3.4).

3.7 Discussion and Conclusions

In this chapter the transport properties of FexCo1−x|vacuum|FexCo1−x tunnel junc-tions using the first principles WFM method have been studied. Starting with ideal

58 Inuence of Roughness and Disorder on TMR

Fe|vacuum|Fe system it has been found that the minority channel in parallel con-figuration is dominated by resonant tunneling also found in other first-principlescalculations [10, 11, 41]. This effect is responsible for the predicted TMR ratio of20,000%. The resonances take the form of “hot spots” that is the regions of 2D BZwhere the tunneling takes place with little or no attenuation. Two kind of such reso-nances have been found. Type A spots are located close to the center of 2D BZ andare closely related to the surface state of ∆1 symmetry which form on Fe(001) sur-face. The other set, type B, consists of points further removed from the center of thezone. This latter type resonances do not contribute meaningfully to the integratedconductance because of their extreme localization in the momentum space.

However, ideal junction will never be fabricated. In fact there are experimentalevidences for presence of interfacial roughness and lead disorder etc in the real MTJs[7]. Disorder (surface roughness or substitutional disorder in the leads) has beenfound to decrease the huge TMR predicted for ideal system to order of magnitudeof the experimental values [7, 9]. The interface roughness has two main competingeffects: i) it increases the transmission due to decrease of effective barrier width andii) by breaking the point group symmetry of the system it destroys the resonantand surface states thus decreasing the minority conductance and consequently TMR.The latter effect is most important for MTJs studied in the present paper where theresonant effects dominate the tunneling for ideal system. Even the small amountof disorder (roughness) is sufficient to substantially degrade the transmission in theminority channel even though the resonant states are likely not removed at this point.In the limit of strong disorder the minority conductance exhibits the same thicknessdependence as the majority one which indicates that the resonance effects are entirelyremoved in this limit.

The effect of FexCo1−x alloy leads was studied using both VCA (ideal system)and CPA/supercell (substitutional disorder) approach. It has been found that theeffect of disorder in this case is generally similar, although less spectacular, to thatof surface roughness. Interestingly, the changes of the electronic structure caused byalloying Fe with Co are in themselves enough to effectively remove the surface andassociated resonant states from the vicinity of the Fermi level. In this regime thereis little difference between the results of VCA and CPA/supercell calculations.

Extremely high TMR ratios predicted by other groups for ideal Fe|Mg0|Fe [10–12, 41] overestimate by far the values found in the experiment [7, 9]. The results forclean and disordered MTJs suggest that these experiments are still in the disorderlimited regime. It should be noted however that direct application of our results toMgO based junctions can be disputed. Unlike for our systems, the tunneling throughideal MgO is not dominated by minority channel. Instead, the conduction is largestin the majority channel thanks to good matching of ∆1 band states in Fe to theevanescent solutions within MgO energy gap. Therefore, even though the resonanceshas been predicted for the minority conductance their destruction by disorder mightnot have equally potent effect as for our Fe|vacuum|Fe.

Another interesting aspect of these results is that for the disordered junctionsthe conductance of antiparallel configuration can be successfully approximated usingsimple formula of Eq. (3.4). This suggests that the tunneling through a general non-

3.8. Appendix 1: The k-point sampling 59

MAJ

1.321

1.322

1.323

×10−8

Q 40060010002600

cond

ucta

nce

(e2 /h

)

1/Q2

MIN

1.5

1.8

2.1

×10−4

0 2×10−6 4×10−6 6×10−6

MAJ

1.6

1.65

×10−8

Q6080100240

cond

ucta

nce

(e2 /h

)

1/Q2

MIN1.0

1.2

1.4

×10−5

0 1×10−4 2×10−4 3×10−4

Figure 3.18: Conductance of ideal Fe|vacuum|Fe MTJ (left hand panel) andFe|vacuum|Fe MTJ with 50% roughness in 10 × 10 lateral supercell (right handpanel) for majority and minority spins for 8 MLs barrier width plotted as a functionof the normalized area element in the Brillouin zone summation 1/Q2, where Q isthe number of intervals along the reciprocal lattice vector.

symmetric junction can be expressed in a factorized form reminiscent of the Julliereformula [3].

3.8 Appendix 1: The k-point sampling

Calculation of the conductance involves the summation over a set of k‖ points that isintegration over the 2D BZ. Naturally, a question arises about the number of pointsnecessary to achieve sufficient accuracy. In view of the qualitative differences in thenature of transmission through ideal and disordered system we will address these tocases separately.

The majority and minority parallel conductances in case of ideal Fe|vacuum|FeMTJ (left hand panel) and Fe|vacuum|Fe MTJ with 50% roughness in 10×10 lateralsupercell (right hand panel) with 8 MLs of vacuum barrier are shown in Fig. 3.18 asthe function of the normalized area element in the Brillouin zone summation 1/Q2,where Q is the number of divisions of the reciprocal lattice vectors. As it can beseen both conductances in case of ideal MTJ are essentially converged for Q over600. Note that resolving of the B type resonances would require Q > 107. Howeverthanks to their extreme localization these resonances can be safely ignored as theydo not contribute to the total integrated conductance. The same procedure has beenapplied to a system with surface roughness at 50% coverage modelled using 10x10

60 Inuence of Roughness and Disorder on TMR

MAJ

2.2

2.7

x10−6

4 6 8 10 12 14 16 18 20

10−7

10−6

cond

ucta

nce

(e2 /h

)

size of the supercell

MIN

Figure 3.19: Conductance of Fe|vacuum|Fe MTJ with 50% roughness in a√H ×

√H lateral SC for majority and minority spins for 8 MLs barrier width

plotted as a function of√H. The results are given for different randomly generated

configurations of disorder.

supercells. We see that both conductances, especially minority, converge much fasterand the values are stabilized already for Q > 80. Better convergence is caused bythe destruction of the localized resonances in the minority channel. The other factorinvolved is that in downfolded BZ of the supercell the sampling for a given value ofQ corresponds to 100 times (for 10x10 supercell) as many points in the original BZ.

3.9 Appendix 2: Configurational averaging and thesize of the scattering region

Modeling of the disorder by means of lateral supercells becomes exact for infinitelylarge supercells. Of course all practical calculations have to be performed using thecells of the finite size. It is therefore important to estimate the inaccuracies introducedby the finite size of the cells. In order to estimate these, the conductance calculationsfor disordered (surface roughness) Fe|vacuum|Fe junction using 20 (40 for smallersystems) different configurations of disorder for superlattices of increasing size (

√H)

have been performed. The results, shown in Fig. 3.19, demonstrate relatively slow

3.9. Appendix 2 61

0 10 20 30 40 5010

−8

10−7

10−6

10−5

10−4

co

nd

ucta

nce

(e

2/h

)

surface coverage (%)

MIN

AP

MAJ

0 10 20 30 40 50

−0.1

0

0.1

Figure 3.20: Conductances GminP (H), GmajP (N), and GσAP (×) of anFe|vacuum|Fe MTJ with 8 MLs barrier width as a function of the surface coverage,normalized to a 1 × 1 surface unit cell are shown for all calculated configurationsof disorder at the interface layers. Inset: The difference in conductance betweentwo antiparallel spin channels normalized to the full AP conductance shown as afunction of surface coverage.

10 30 50 70 9010

−9

10−8

10−7

10−6

10−5

co

nd

ucta

nce

[e

2/h

]

Co concentration (%)

MIN

AP

MAJ

0 0.5 1−0.02

0

0.02

0.04

Figure 3.21: Conductances GminP (H), GmajP (N), and GσAP (×) of anFe1−xCox|vacuum|Fe1−xCox MTJ with 8 ML barrier width as a function of x, theconcentration of Co atoms, calculated in the CPA/supercell approximation. Re-sults are shown for all calculated configurations of disorder. Inset: The differencein conductance between two antiparallel spin channels normalized to the full APconductance.

62 Inuence of Roughness and Disorder on TMR

4.2

4.22

4.24

R!A

(10−

8 "m

2 )

MAJ

4 6 8 10 12

3.5

4

4.5

5

number of disordered Fe0.9Co0.1 layers

R!A

(10−

10 "

m2 )

MIN

Figure 3.22: Resistance of Fe0.9Co0.1|vacuum|Fe0.9Co0.1 MTJ as a function ofnumber of disordered FeCo monolayers on each side of the vacuum tunneling barrierin the scattering region.

convergence in minority channel in comparison to the metallic interfaces [24]. Thisreflect the fact that the tunneling transmission probabilities are hugely sensitive (ex-ponential dependence) to the local variation in the thickness of VTB. The averagedvalues on the other hand are well converged already for

√H = 10 case. Increasing

the size of the supercell makes the calculations considerably more expensive. Asthere is little to be gained from using supercells larger than 10x10, this size for allthe calculations performed in the main body of the text has been chosen.

In order to verify that above mentioned convergence is not significantly differentalso for coverages smaller than 50% we show in Fig. 3.20 the non-averaged results cor-responding to Fig. 3.13. The lines connect the averaged values, same as in Fig. 3.13.As we can see the spread of values can be considerable but the spin channels remainwell separated and the general trends are well reproduced by averaged values. Thedisorder described using the supercells of finite size and a single configuration of scat-terers breaks the mirror symmetry of the ideal junction and allows the conductancesof the two spin channels in AP configuration to diverge. However a truly random

3.9. Appendix 2 63

20 40 60 800

50

100

150

resi

stiv

ity (

µΩ×c

m)

Co concentration (%)

MAJ

MIN

Figure 3.23: Resistivity of disordered Fe1−xCox alloy as a function of Co con-centration x for both majority and minority spin channels.

disorder is expected to restore the symmetry and brings the two spin channels intoagreement. Therefore the agreement between configuration-averaged AP conduc-tances serves as a useful metrics for the validity of our approach to disorder modeling.In the inset of Fig. 3.20 we show the normalized difference of averaged antiparallelconductances ε defined as:

ε = 2GmajAP −GminAP

GmajAP +GminAP

(3.5)

where the GσAP is configurationally averaged AP conductance. As we can see thevalues of ε remain modest all throughout the range of coverages. Similar conclusionscan be drawn for alloy disorder case as it is demonstrated by Fig. 3.21.

Another problem arises in calculations for the junctions with FexCo1−x electrodeswhere the substitutional disorder is restricted only to the part of the electrodes con-tained within the scattering region (Fig. 3.1, bottom panel). Simple VCA approx-imation is used for the remaining parts of electrodes. This approach makes thecalculations feasible but can potentially also be a source of inaccuracies. In Fig. 3.22we show resistances calculated for different configurations of substitutional disorderin Fe0.9Co0.1 alloy and MTJs with 4 to 12 MLs included in the scattering region onboth sides of the barrier. The spread of values is relatively large and convergenceslow, especially in the minority channel. The average values, on the other hand,remain essentially the same even though we are increasing the size of the scatteringregion. The reason for this is that the resistance of MTJ is dominated by the tunnel-ing through the barrier. Indeed using the values of resistivity for FexCo1−x, plottedin Fig 3.23, we conclude that the resistance of several layers of an alloy is insignificantcompared to the resistances given in Fig. 3.22 for the whole MTJ by many ordersof magnitude. The numbers in Fig. 3.23 were determined by the calculation of the

64 Inuence of Roughness and Disorder on TMR

inverse of the conductance for the FexCo1−x slab of varying thickness and extractingthe resistivities from the slope of the curve (Ohm’s low). All the calculations in themain body of the text were performed with just 4 MLs of disordered alloy on bothsides of VTB. As we have seen in Sec. 3.6 it is enough to break the symmetry anddestroy the resonant states in the minority channel. This is further confirmed inFig. 3.21 which corresponds to the Fig. 3.16 in the main text. As we can see thespread of non-averaged values is insignificant on the scale of the plot. The normalizeddifferences of the AP conductances, shown in the inset, are also in good agreement.

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66 Inuence of Roughness and Disorder on TMR

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[37] The WFM method does not require the introduction of complex energies whencalculating the Green function matrix. However it is convenient to add imag-inary part when calculating DOS in order to smooth out the plots. All theDOS calculations in the current manuscript were performed with imaginary partη = 0.034eV .

[38] P. A. Khomyakov, G. Brocks, V. Karpan, M. Zwierzycki, and P. J. Kelly, Phys.Rev. B 72, 035450 (2005).

[39] For ideal MTJs, the resonant tunneling contribution will be quenched for suffi-ciently thick barriers so Eq. (3.2) can also be expected to hold in that situation.

[40] K. D. Belashchenko et al., Phys. Rev. B 69, 174408 (2004).

[41] K. D. Belashchenko, J. Velev, and E. Y. Tsymbal, Phys. Rev. B 72, 140404(2005).

[42] I. I. Mazin, Phys. Rev. Lett. 83, 1427 (1999).

[43] J. M. MacLaren, X.-G. Zhang, and W. H. Butler, Phys. Rev. B 56, 11827 (1997).

Chapter 4

Recovering Julliere model:ab-initio study

Model proposed by Jullière is the most cited model in the context of tunneling mag-netoresistance (TMR) study which relates the TMR to another experimentally mea-sured quantity - tunnelling spin polarization (TSP). Despite numerous experimentalevidences that support this model, the physical basis of tunnelling density of states(TDOS) that determines the TSP is not generally dened, and our understanding ofthe Jullière model is not complete.Based upon the results of ab-initio study of semirealistic Fe|vacuum|Fe magnetictunnel junction we recover this model and dene the TDOS in terms of measurablequantities. With our results we complement the previous studies of the model andgeneralize it for the nite bias case. As an illustration, we study the case of biasdependence of TMR in asymmetric magnetic tunnel junctions (MTJ).

4.1 Introduction

Study of magnetoresistance (MR) effects attracts much attention due to its possibleapplication in making mesoscopic scale magnetic sensors and magnetic random-accessmemory elements. Here we focus on the tunnelling magnetoresistance (TMR) effectwhich is observed in magnetic tunnel junctions (MTJ) that consist of two ferromag-netic leads (FM) and a thin film of insulator (I) sandwiched between them. Theelectrical resistance of a FM|I|FM junction depends on the relative orientation ofmagnetization direction in the leads which changes from antiparallel (AP) to parallel(P) if an external magnetic field is applied along one of the magnetization directions.Therefore, a nonzero TMR = (GP −GAP )/GAP = (RP −RAP )/RP where GP (RP )and GAP (RAP ) are the total conductances (resistances) of a junction in the P andAP orientations respectively, can be measured.

Despite significant progress achieved in making the MTJs with high room- tem-perature TMR [1], some of the problems like understanding of the Julliere model [2],

67

68 Recovering Jullière model: ab-initio study

has not been resolved yet. This model is the earliest and the most frequently usedone to study the TMR. The Julliere model is based upon two approximations. Firstit is assumed that the spin of electrons is conserved during the tunneling. This as-sumption is also known as Mott’s two-current approximation [3, 4]. Then, we assumethat conductances GP and GAP in each spin channel σ =maj, min are proportionalto the product of spin-dependent tunnelling densities of state (TDOS) ρσi in left andright FM leads i = L,R

GP = GmajP +Gmin

P ∝ ρmajL ρmaj

R + ρminL ρmin

R ,

GAP = G1AP +G2

AP ∝ ρmajL ρmin

R + ρminL ρmaj

R .

(4.1)

Then, TMR = (2PLPR)/(1 − PLPR), where PL and PR are the tunneling spin po-larizations (TSP) of left and right leads given by Pi = (ρmaj

i − ρmini )/(ρmaj

i + ρmini ).

Note, that if FM leads are identical ρσ = ρσL = ρσR the GAP is then given by

GAP = 2√GmajP Gmin

P ∝ 2ρmajρmin. (4.2)

From Eqs. (4.1) and (4.2) it follows that a MTJ can be separated into left and rightparts whose electronic properties are characterized by some TDOS then the P andAP conductances in each spin channel can be factorized in terms of correspondingspin-dependent TDOS. Therefore, in order to demonstrate that the works in practiceone have to define and measure the TDOS to predict the TMR and compare it withactually measured TMR. It appeared that experimentally it is possible to measurethe TSP. In such experiments the TSP is detected by a superconductor (S) in FM|I|Sjunctions [5]. By now, numerous experiments [6–8] have proven that the TMR valuespredicted from the Julliere model and those actually measured in FM|I|FM junctionsare in good agreement. Despite big success of such an admittedly simple model, theproblem of defining the physical basis of the TDOS remains a hot topic. Julliereused a classical definition of the TDOS due the Meservey and Tedrow [9] accordingto which the TDOS that defines the TSP is given by the density of states (DOS)in the FM leads. This is also partly supported by the recent experimental resultspresented in Ref. [10] where the TSP was shown to depend on the s-character DOS inFM leads. However, another experiments relate the TSP to the electronic propertiesof a tunnel barrier [11] or electronic properties of FM|I interface [7, 12]. As a resultthe straightforward definition of TDOS in terms of measurable quantities is stillmissingand one can have only an intuitive understanding of physical basis of theTDOS.

Despite the complexity of this problem, the theoretical studies [13–19] have madeprogress in gaining an insight into the Julliere model. Results of these studies relatedto our work will be discussed in the next section. Then, in Section 4.3 we presentour own findings based on ab-initio approach. We introduce new, equivalent to theJulliere expression (4.1), factorization procedure, which allows us studying both zeroand finite bias cases. Finally, we finish with our conclusions in Section 4.4.

4.2. Early models 69

d

Vb

Figure 4.1: Schematic representation of free-electron tunneling through an idealrectangular barrier.

4.2 Early models

In several decades since the Julliere model was proposed, a number of theoreticalapproaches were proposed to study the spin-dependent transport in a MTJ. One ofthe first was the free electron approach where the spin-dependent transport of freeelectrons propagating along the tunnel junction with an ideal rectangular barrier ofheight Vb and width d is studied [13]. Such model is schematically shown in Fig. 4.1.The exchange splitting is included by considering different electron potentials for twospin channels Vmaj and Vmin. Tunneling conductance in case of transverse periodicityin a junction is given by

Gσ =e2

h

∑k‖

Tσ(k‖), (4.3)

where k‖ is electron transverse momentum and Tσ(k‖) is the spin-dependent trans-mission coefficient. Matching the wave functions and their derivatives at the left andright interfaces between the FM leads and barrier the Tσ(k‖) can be found. ForVmaj < EF , Vmin < EF , EF < Vb and in the limit of thick tunnel barriers (large d)Tσ(k‖) is reduced to a simple expression

Tσ(k‖) =16kσLk

σRκ

2

(kσ2L + κ2)(kσ2

R + κ2)exp(−2κd), (4.4)

where kσi =√

(2m/~2)(EF − Vσ)− k2‖ is z component of a wavevector (normal to

the interface) of the electrons at the Fermi level in the left and right leads andκ =

√(2m/~2)(Vb − EF )− k2

‖ describes the exponential decay of the electron trans-mission coefficient inside the tunnel barrier. Taking into account that magnetization

70 Recovering Jullière model: ab-initio study

directions of left and right electodes can be P or AP this equation can be rewrittenin more suitable form

Tσσ′(k‖) = TσL (k‖)exp(−2κd)Tσ

′

R (k‖), (4.5)

where TσL (k‖) (Tσ′

R (k‖)) are the spin-dependent transmission coefficients calculatedat the left (right) FM|I interface

Tσi (k‖) =4kσi κ

(kσ2i + κ2)

. (4.6)

In Eq. (4.5) σ = σ′ and σ 6= σ′ for P and AP magnetization orientations respectively.To calculate the conductance one have to sum over the all k‖ contributions in thetwo-dimensional Brillouin zone (2D BZ) according to Eq. (4.9). This, however, isnot necessary in the limit of thick barriers, because due to exponential decay in κ allthe terms except for the k‖ = 0 can be safely neglected. Then the total P and APconductances are

GP =e2

hexp(−2κd)

[TmajL (0)TmajR (0) + TminL (0)TminR (0)

],

(4.7)

GAP =e2

hexp(−2κd)

[TmajL (0)TminR (0) + TminL (0)TmajR (0)

].

From comparison of Eqs. (4.8) and (4.1) we conclude that the role of TDOS ρσifrom the Julliere model is now played by the transmission coefficient Tσi (0) ande2exp(−2κd)/h is the proportionality coefficient which is missing in Eq. (4.1). Slon-czewski’s TSP can be written as

PSi =Tmaji (0)− Tmini (0)Tmaji (0) + Tmini (0)

= Piκ2 − kmaji kmini

κ2 + kmaji kmini

, (4.8)

where

Pi =kmaji − kmini

kmaji + kmini

, (4.9)

In the contrast to the classical definition of TSP due to Meservey and Tedrow, theTSP given by Eq. (4.8) depends on the electronic properties of not only the FM leadsgiven by Pi but also a tunnel barrier. Therefore PSi is also known as generalized TSP.For free electrons, in the limit of large d and small Vb, the functional dependence ofthe TMR on the TSP is well described by Eq. (4.8) [13, 17]. Eqs. (4.6) is veryimportant as it demonstrates that the transmission coefficients are factorized intoindependent terms that correspond to left and right leads and a tunnel barrier. Incomparison with the Julliere model which was formulated from general arguments,the practical application of free electron approach is, unfortunately, limited.

The factorization of transmission can also be recovered from the general argu-ments [18]. Similarly to above mentioned transverse periodicity is assumed in a MTJ

4.3. Ab-initio results 71

and an electron is scattered on left and right FM|I interfaces. The multiple scatteringcontributions can be neglected assuming large thickness of a tunnel barrier. Thenthe spin dependent transmission coefficient that connect states in left and right FMleads for every k‖ of the 2D BZ is

Tσσ′

L,R(k‖) = TσL (k‖)Tm(k‖)Tσ′

R (k‖) (4.10)

where TσL (k‖)(Tσ′

R (k‖)) describes incoming (outgoing) state in left (right) FM leadand Tm(k‖) stands for spin-independent decay inside a tunnel barrier. For thicktunnel barriers the summation over k‖ is again reduced to a leading term k‖ = 0,and the conductance of an electron through a MTJ can be factorized in a form similarto Eq. (4.5).

Contrary to the free-electron model, for realistic MTJ the exact analytic expres-sions for spin-dependent transmission probabilities Tσi and for the spin-independentdecay inside the barrier can not be derived. Here, we suggest to use ab-initio approachto calculate them. On one hand it has been demonstrated from ab-initio study ofmaterial-specific ideal MTJ that the factorization of transmission coefficient given byEq.(4.10) does not work [18, 19]. On the other hand, for MTJs with different kindsof disorder the factorization and therefore the Julliere model has been shown to workwell using the tight-binding calculations [14, 15] or Bardeen’s transfer-Hamiltonianapproach [16]. Therefore, we conclude that the disorder which should diminish themultiple-scattering contribution is very important and has to be taken into account.Keeping in mind that it is not possible to make a perfect MTJ we expect our resultsfor disordered junctions to be more reliable compared to an ideal MTJ case.

4.3 Ab-initio results

In order to test the validity of factorization procedure or equivalently the Jullieremodel we study spin-dependent transport of electrons using ab-initio approach ina semirealistic MTJ. In our method we first solve self-consistently the Kohn-Shamequations using the tight-binding linear muffin-tin orbital [20] surface Green’s func-tion method [21]. Disordered systems are treated using the layered coherent potentialapproximation [21, 22]. The atomic spheres potentials serve as an input to the sec-ond step where the transmission matrix entering Landauer’s transport formalism[23] is calculated using a tight-binding muffin-tin orbital implementation [24, 25] ofa wave-function matching scheme due to Ando [26]. Disorder is modeled in largesupercells, containing 10×10 atoms per monoatomic layer (ML), by distributing theself-consistent atomic potentials at random with an appropriate concentration for asmany configurations as are required. The method allows us to calculate“exact”valuesof conductances GP , GAP and therefore TMR. We can also calculate atom-resolvedlocal densities of states (LDOS) in order to estimate the TMR using the classicaldefinition of TSP due to Meservey [9].

We proceed by focusing on study of spin-dependent transport in ideal and disor-dered Fe|vacuum|Fe MTJs which have been studied before in Chapter. 3 and Ref. [27].There are also many studies of spin-dependent vacuum tunneling in its

72 Recovering Jullière model: ab-initio study

0 20 40 60 80 100

10−7

10−6

10−5

10−4

10−3

cond

ucta

nce

(e2 /h

)

surface coverage (%)

MIN

MAJ

AP

0 20 40 60 80 1000

200

400

600

TM

R (

%)

Figure 4.2: Configuration-averaged conductances GminP (H), GmajP (N), and GσAP(×) of an Fe|vacuum|Fe MTJ as a function of the surface coverage, normalized to a1×1 surface unit cell. The barrier width for the MTJ with clean surface for x = 0%(x = 100%) is 8 MLs (6 MLs). The dash-dotted line denotes GσAP predicted fromEq. (4.11) and the dashed line denotes GσAP predicted from Eq. (4.12). Inset: TMRas a function of the surface coverage.

own right [28–32]. A disorder in MTJ can be taken into account in different waysand we do it in the next way. Vacuum barrier is characterized by an absence of adisorder inside the barrier, therefore, the disorder at the Fe|vacuum surface (surfaceroughness) or in the Fe leads can be studied. We focus on study of surface roughnessin Fe|vacuum|Fe MTJ which as we have shown in Chapter. 3 has stronger effecton the TMR than alloy lead disorder. The effect of surface disorder on the TMR isillustrated in Fig. 4.2. In this figure the dependences of GσP and GσAP of conductancesfor each spin channel as a function of surface disorder are shown. Surface coveragex characterizes disorder and corresponds to a concentration of Fe impurity atomson Fe surface. Note, that in this figure x is the same for left and right surfacesx = xL = xR, therefore, we call such a MTJ symmetric. Starting (ending) point inthis figure x = 0% (x = 100%) corresponds to ideal Fe|vacuum|Fe MTJ with 8MLs

4.3. Ab-initio results 73

(6MLs) of vacuum thickness 1.Increase of GmajP in Fig. 4.2 is attributed to narrowing of tunnel barrier as the

surface coverage increases. More complicated behavior exhibited by GminP is dueto the large and roughness-sensitive surface resonant tunneling contribution to theconductance in this spin channel [33]. For ideal or weakly disordered surfaces theresonant tunneling contribution to the total tunneling transmission is crucial. Thesurface disorder quenches this contribution and therefore reduces the GminP . If thecontribution from the resonance tunneling becomes negligible, GminP increases as afunction of disorder2 due to decrease of the tunnel barrier. The corresponding TMRis shown in the inset. As it is seen the TMR varies from the huge values to the ordersof experimentally observed ones. Hence, the surface disorder closes a gap betweenthe experimentally measured and much larger theoretically predicted TMR values[34, 35]. More technical details together with the deeper analysis of spin-dependenttransport in Fe|vacuum|Fe MTJ one can find in Ref. [27] or Chapter. 3.

As it was already mentioned classical interpretation of the Julliere model is basedupon assumption that TDOS is defined by DOS in FM leads [9]. Before we proceed,let us check this assumption. From Fig. 4.2 it is obvious that significant variation ofthe TMR can not be attributed solely to the invariant LDOS in FM leads. Perhaps,LDOS at the surface should be taken into account? We have studied LDOS calculatedfor different atoms as well as their possible combinations (not shown here) but none ofthem could be identified with the TDOS which would explain the results in Fig. 4.2.

4.3.1 Factorization of conductance

The validity of above mentioned factorization of transmission is checked by comparingthe calculated and the factorized AP conductance given by

GσAP =∑k‖

√TmajP (k‖)TminP (k‖), (4.11)

where TσP (k‖) are the k‖-resolved transmission coefficients through the junction forspin channel σ for P orientation. Factorized GσAP is shown in Fig. 4.2 with dash-dotted line respectively. As it is seen from the figure the agreement between factorizedand calculated AP conductances is very good in the whole range of surface coveragesexcept for ideal junction with 6MLs of vacuum (x=100%). To explain this discrepancy

we study the relation of calculated TAPσ(k‖) and factorized√TmajP (k‖)TminP (k‖)

transmission coefficients as a function of k‖ in the two-dimensional Brillouin zone(2D BZ) as it is shown in Fig. 4.3. From this figure it is clear that the factorizationworks well in most of the k‖-points except for the areas where the surface resonantstates are located. For these points the transmission coefficient in these areas can notbe factorized because the surface resonant tunneling is not observed in separated FM|Isystem. Therefore, Eq. 4.11 fails in case of significant nonfactorized transmissioncontribution which appears due to inseparable effects like the surface states

1thickness of a vacuum ML is equal to a half lattice constant of Fe aFe/2= 1.433 A2x=50% corresponds to the most disordered surface

74 Recovering Jullière model: ab-initio study

0.8

0.9

1.0

1.1

1.26 ML

Figure 4.3: Factorization of transmission Eq. (4.11) is tested by calculating

a relation TAP (k‖)/√TmajP (k‖)TminP (k‖) for every k‖ of the 2D BZ for ideal

Fe|vacuum|Fe MTJ with 6MLs of vacuum.

resonances in our case. As it was already mentioned in Chapter. 3 this contribu-tion from such inseparable effect reduces as a function of barrier thickness, therefore,Eq. (4.11) gives better estimation of GσAP for ideal junction with larger barrier thick-ness as it is demonstrated by Fig. 4.2 for 8MLs of vacuum barrier.

In order to demonstrate that factorization Eq. 4.11 works for any given barrierthickness the contribution from the surface resonance states should be diminished.For this purpose the surface roughness is taken into account. The factorized APconductance in such a case exhibits good agreement with the calculated results asis illustrated by Fig. 4.2. Nearly perfect agreement which sets in the range of x =12 . . . 96% is attributed to almost complete elimination of the surface resonance dueto roughness.

Compared with an ideal junction case, the analysis of transmission probabilityTσ(k‖ is more complicated in case of roughness due to the downfolding of 2D BZ.In such supercell calculations every downfolded k‖-point contains 100 k‖ of unfolded2D BZ. Another disadvantage appears from comparison of the Julliere expressionfor symmetric junctions Eq. (4.2) with Eq. (4.11). In case of a thin tunnel barrierone have to carry out a sum over all k‖3 and the TSP is defined in terms of Tσ(k‖which in general can not be measured experimentally. Therefore, we suggest using

3In the next we consider only the disordered MTJ therefore all k‖ thereafter correspond to k‖of downfolded 2D BZ

4.3. Ab-initio results 75

an alternative factorization of conductance given by

GσAP =√GmajP

√GminP =

√∑k‖

TmajP (k‖)√∑

k′‖

TminP (k′‖). (4.12)

which does not contain such disadvantages. Strictly speaking, a difference betweentwo factorization procedures (Eqs. (4.11) and (4.12)) lies in the order in which thesummation over the k‖ is carried out. In contrast to Eqs. (4.11), the factorizationof conductance takes into account a contribution from the transmissions coefficientsthat connect states in k‖ in the left lead of MTJ with states in all k′‖ 6= k‖ on theright side.

However, due to roughness the Tσ(k‖) is almost uniformly distributed over theBZ which in turn provides us with a good agreement between these two factorizationprocedures as it is demonstrated by Fig. 4.2. n this figure the factorized GσAP is givenby dashed line. A difference between AP conductance calculated from Eqs. (4.11)and (4.12) becomes significant for slightly disordered and ideal junctions where thedistribution of TminP (k′‖) over the 2D BZ is not uniform due to the surface resonancestates. One should also keep in mind that Eqs. (4.11) and (4.12) become identical inthe limit of large supercell sizes when it is enough to 2D BZ is enough to calculatethe transmission in a single k′‖ = 0 point to calculate the conductance.

Practical use of Eq. (4.12) appears from its comparison with Julliere model givenby Eq. (4.2). As it is seen the role of ρσi is now played by the measurable quantity√Gσi and the TSP can now be defined as

Pi =

√Gmaji −

√Gmini√

Gmaji +√Gmini

. (4.13)

Using Eq. (4.12) it can be easily shown that TMR of a symmetrical MTJ takes onlynonnegative values as demonstrated in the inset in Fig. 4.2.

Generalization of factorization Eq. (4.12) for junctions with different disorderat the left and right surfaces (asymmetric MTJ) is possible by means of followingexpression

Gσσ′(xL, xR) =

√GσP (xL)Gσ′P (xR) (4.14)

where xL and xR stand for two surface coverages at left and right Fe|vacuum surfacesand GσP (xi) is the σ spin component of total P conductance of a symmetric MTJ withsurface coverage xi. Eq. (4.14) together with conductances calculated for symmetricaljunctions allows us to calculate GσP and GσAP and eventually the TMR of variousasymmetrical Fe|vacuum|Fe MTJ. In order to to check the quality of this procedurewe benchmark the TMR values calculated by means of Eq. (4.14) against the exactTMR as it is demonstrated by Fig. 4.4. The asymmetric junctions are constructedby fixing disorder on left surface at xL = 5% or xL = 12% and varying disorder onthe right Fe|vacuum surface. As it can be seen almost perfect agreement betweenpredicted and exact TMR values sets for surface coverages that are large enough

76 Recovering Jullière model: ab-initio study

xL=12% xRxL=5% xR

Figure 4.4: Configuration-averaged TMR of asymmetric Fe|vacuum|Fe MTJwith 8 MLs barrier width, normalized to a 1 × 1 surface unit cell as a function ofthe surface coverage xR, while surface coverage of left surface is fixed at xL = 5%(on the right) and xL = 12% (on the left). Red solid lines stand for the results ofour transport calculations while the black dashed lines show the factorized values,obtained from Eq. (4.14).

to destroy the surface resonance state. Therefore, the agreement demonstrated forxL = 12% is better than that of xL = 5%. Note, that in this figure the resultsare averaged over many disorder configurations. Contrary to symmetric MTJ, TMRof asymmetric junctions can be negative if the asymmetry of Fe|vacuum|Fe junctiongiven by a difference between xL and xR is large enough. The TMR becomes negativeif the TSP for left and right sides of a junction have opposite signs, which is exactlywhat is demonstrated by Fig. (4.4). Fixed TSP of left part is negative, and wechange the TSP of the right part by varying the surface coverage xR. Once thesign of right TSP becomes positive (around x = 24%) the TMR becomes negative.These conclusions are also in qualitative agreement with the experimental findings:negative TMR has been measured in asymmetric MTJ [11, 36, 37].

4.3.2 Factorization of conductance in case of finite bias

In the original formulation of Julliere model there is no explicit reference to theenergy dependence. Therefore, we assume that use of factorization procedure forany given energy should also be possible. The energy dependence of conductancepaves a way for important characteristics - bias dependence of the TMR. The TMRas a function of finite bias V has been investigated in numerous experiments andtheoretical studies starting from the Julliere [2]. Here, we focus on theoretical study

4.3. Ab-initio results 77

−0.2 −0.1 0 0.1 0.2−30

−20

−10

0

10

TM

R (

%)

bias voltage (V)

−0.2 0 0.2

−40

0

40

xR=28%

xL=12%

TS

P (

%)

bias voltage (V)

Figure 4.5: Configurationally-averaged over 7 configurations TMR as the func-tion of bias voltage is shown for asymmetrical MTJ with 8 MLs of vacuum andwith xL = 12% and xR = 28%. Red solid line stands for exact calculation. Blackdashed line is for TMR calculated using factorization approach.

of asymmetric MTJs which have been shown to manifest a peculiar sensitivity of signand value of a TMR to the direction and strength of applied bias [11, 36, 37].

First we calculate the bias dependence of the TMR. The details of procedure areas following. We begin by applying a rigid ±V/2 shift to the atomic potentials inthe leads and neglect the charge rearrangement. Such non-self-consistent procedurewhen applied to MTJ was shown to provide with a reasonable description at smallbias voltages compared with the self-consistent results [38]. The conductance as afunction of applied finite bias V is obtained by integrating over the energy intervalgiven by µL = EF + V/2 and µR = EF − V/2,

Gσσ′(xL, xR, V ) =

1µL − µR

∫ µL

µR

Gσσ′(xL, xR, ε)dε (4.15)

where µL and µR are the chemical potentials of left and right leads correspondinglyand Gσσ

′(xL, xR, ε) is P or AP conductance in a general case of asymmetric junc-

tion averaged over many configurations. In our numerical scheme the integrationis replaced by integral sum with energy step ∆ε = 0.02eV for biases higher then

78 Recovering Jullière model: ab-initio study

±0.04eV 4. Hereafter, positive bias corresponds to µL > µR and µL < µR to the caseof negative bias. From Eq. (4.15) it becomes obvious that the number of channelsparticipating in electron transport increases and the total transmission probabilitychanges if finite bias voltage V is applied. The results of such calculations are shownin Fig. 4.5 with a solid line. In this figure we demonstrate bias dependence of TMRfor asymmetric junction with xL = 12% and xR = 28% and fixed thickness vac-uum of 8 MLs. As it can be seen from this figure the negative TMR at zero bias ischanged significantly if finite bias is applied. For positive biases the TMR becomespositive which is qualitatively similar to the experimental findings in asymmetricFe|GaAs|Fe(001) MTJ [36].

From the point of view of computational costs the finite bias calculations fordisordered systems are rather expensive. In order to reduce these costs we proposeto use the factorization procedure which in case of finite positive bias is illustrated byFig. 4.6. First we calculate GmajP and GminP of two symmetric junctions with differentsurface disorder x = xL and x = xR in the range of energies εL = [EF − V,EF ] andεR = [EF , EF + V ] with a step ∆ε. Then we factorize the conductance

Gσσ′(xL, xR, ε) =

√Gσ(xL, ε− V/2)Gσ′(xR, ε+ V/2) (4.16)

to calculate the energy dependent spin components of P and AP conductances ofasymmetric junction. Note, that the AP conductance for both spin channel are fac-torized into spin components of the P conductance. Afterwards, we sum over theenergies to calculate the bias dependent P and AP conductances. We repeat thisprocedure for a number of different configurations to averaged conductances overthe disorder. Finally, we calculate the TMR. The results of such energy dependentfactorization procedure are shown with dashed line in Fig. 4.5. As it is seen the fac-torized and calculated TMR values are in good agreement. Discrepancy between thefactorized and exactly calculated TMR appears due to errors of numerical integrationand a limited number of configurations studied here.

Factorization of conductance can again be used to explain the behavior of theTMR as a function of finite bias. In the inset in Fig. 4.5 we show the TSP for leftand right parts of the MTJ as a functions of bias voltage. As it is seen from theseresults the change of TMR sign depends on the signs of the TSPs. Similar argumentswere used for zero bias. These conclusions also agree with other studies [39].

In a view of above mentioned it is important to discuss the computational effectfrom the energy-dependent factorization procedure. Assuming that we are interestedin calculating bias dependence of TMR in symmetric and asymmetric MTJ with ndifferent surface roughness, the total amount of exact calculations would be n(n−1)/2per spin per P or AP orientation and per configuration. At the same time with thefactorization procedure one needs to perform only n such calculations per spin perconfiguration. Moreover, the AP conductance is factorized into the components ofP one. Therefore, in a limit of large n, the number of calculations needed to beperformed for calculating bias dependence of the TMR in the MTJs with n differentsurface roughness is n times smaller than that of exact procedure.

4for smaller biases ∆ε = 0.01eV

4.3. Ab-initio results 79

YES

NO

YES

NO

YES

NO NO

YES integrate

End

σσ' σσ'L R L R

1G (x ,x ,V) = G (x ,x ,ε) dεL

RL R

μ

μμ μ− ∫

Start

x = xL

εL = EF - V

majP L LG (x ,ε )minP L LG (x ,ε )

εL > EF

choose correct x

x = xR

εR = EF

majP R RG (x ,ε )minP R RG (x ,ε )

εR< EF+ V

xL - left surface coveragexR - right surface coverageV - bias voltageε = ε L + V/2 = εR - V/2Δε – energy step

calculate calculate

εL = εL+ Δ ε εR = εR+ Δ ε

factorizeσσ' σ σ'

L R P L P RG (x ,x ,ε) = G (x ,ε V/2)G (x ,ε V/2)− +

Figure 4.6: Flowchart that represens factorization Julliere-like procedure ex-ploited here to calculate bias dependence of TMR. Three stages of procedure aremade clear. First we calculate energy dependent conductances of symmetrical sys-tems Gσ(xL, ε+V/2) and Gσ

′(xR, ε−V/2). Then using factorized expression (4.16)

we calculate P and AP energy dependent conductances of asymmetric junctionGσσ

′(xL, xR, ε). Finally we calculate bias dependent conductances using Eq. (4.15)

and TMR.

80 Recovering Jullière model: ab-initio study

4.4 Conclusions

In conclusion, using ab initio approach we have studied the Julliere model. In thismodel it is assumed that a MTJ can be separated into left and right independentparts that are characterized by some TDOS. The total conductance in each spinchannel through the junction is then factorized in terms of TDOS for left and rightparts. Therefore the model or equivalently the factorization of conductance works ifthere is no significant contribution from the inseparable processes like quantum wellstates, surface localized resonances etc. to total conductance. Polarization of TDOS(TSP) can be measured experimentally and the validity of the Julliere model hasbeen demonstrated from numerous experiments [6–8]. Despite plenty of experimentalresults, the physical basis of the TDOS is unclear.

Spin-dependent transport of electrons in a MTJ has been investigated in a numberof theretical studies. From the ab initio calculations for ideal material-specific MTJsit has been demonstrated that tunnel transmission does not factorize due to presenceof surface resonant tunneling [18, 19]. However, the factorization of transmissionwas shown to work in disordered MTJ using the tight-binding approximation [14, 15]and Bardeen’s transfer-Hamiltonian method [16]. The role played by disorder isimportant because it destroys the surface resonance states and the Julliere model isrecovered.

Taking this into account we have studied transport in Fe|vacuum|Fe MTJ withinterface roughness. Conductances estimated by the factorization of transmission co-efficients for every Tσ(k‖) and therefore Julliere model were found in good agreementswith those calculated exactly. Disagreement in case of ideal system is attributed tothe surface resonance states whose contribution to the total conductance can notbe factorized. Taking into account that Tσ(k‖) is not an observable quantity whichlimits the field of practical application of factorization procedure we suggest usingnew factorization procedure that operates with conductances that can be measuredexperimentally. We have demonstrated that the conductances estimated from bothfactorization procedures agree with each other due to almost uniform distribution oftransmission Tσ(k‖) over two-dimensional Brillouin zone in case of disorder.

By factorizing conductances we have been able to predict the TMR for junctionswith different roughness at left and right interfaces xL and xR correspondingly (asym-metric MTJ) by using the results for symmetrical junctions xL = xR. Negative TMRhas been shown in cases if difference (asymmetry) between xL and xR increases. Sim-ilarly to the Julliere model, the negative sing has been attributed to opposite signsof TSP for left and right parts of junction. Further, the application of factorizationapproach has been extended to the energy dependent case which has allowed us tostudy bias dependence of the TMR. We have demonstrated good agreement betweenexactly calculated and factorized TMR as a function of applied bias. Change of theTMR sing as a function of bias has been shown to correlate again with the change ofsing of the TSP.

BIBLIOGRAPHY 81

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[14] E. Y. Tsymbal and D. G. Pettifor, Phys. Rev. B 58, 432 (1998).

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82 Recovering Jullière model: ab-initio study

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Chapter 5

Graphene and Graphite asPerfect Spin Filters

Based upon the observations (i) that their in-plane lattice constants match almostperfectly and (ii) that their electronic structures overlap in reciprocal space for onespin direction only, perfect spin ltering for interfaces between graphite and (111)fcc or (0001) hcp Ni or Co is predicted. The spin ltering is quite insensitive tolattice mismatch, roughness and disorder. The formation of a chemical bond betweengraphite and the open d-shell transition metals that might complicate or even preventspin injection into a single graphene sheet can be simply prevented by dusting Ni orCo with one or a few monolayers of Cu while still preserving the ideal spin injectionproperty.

5.1 Introduction

We recently predicted a perfect spin filtering effect for ultra-thin films of graphitesandwiched between two ferromagnetic leads [1]. This prediction emerged from theinterface between two rapidly developing branches of condensed matter physics: mag-netoelectronics and graphene electronics [2, 3]. Magnetoelectronics exploits the ad-ditional degree of freedom presented by the intrinsic spin and associated magneticmoment of electrons while graphene electronics is based upon the unique electronicproperties of two-dimensional graphene sheets. Based on the giant magnetoresistanceeffect discovered twenty years ago [4, 5], magnetoelectronics was rapidly applied tomaking improved read head sensors for hard disk recording and is a promising tech-nology for a new type of magnetic storage device, a magnetic random access mem-ory. The giant magnetoresistance (GMR) effect is based on the spin dependence ofthe transmission through interfaces between normal and ferromagnetic metals (FM).The effect is largest when the current passes through each interface in a current-perpendicular-to-plane (CPP) measuring configuration but the absolute resistance

83

84 Graphene and Graphite as Perfect Spin Filters

of metallic junctions is too small for practical applications and the current-in-plane(CIP) configuration with a much smaller MR is used in practice. Replacing thenon-magnetic metal spacer with a semiconductor [6] or insulator (I) [7, 8] results inspin-dependent tunneling and much larger resistances are obtained with FM|I|FMmagnetic tunnel junctions (MTJs). Substantial progress has been made in increasingthe MR effect by improving experimental techniques, and by replacing the amorphousAl2O3 insulator with crystalline MgO [9, 10]. Though there is a relatively large lat-tice mismatch of 3.8% between Fe and MgO, the tunneling magnetoresistance (TMR)in Fe|MgO|Fe junctions has been reported to reach values as high as 180% at roomtemperature [11]. Low temperature values as high as 1010% have been reported forFeCoB|MgO|FeCoB MTJs [12, 13]. The sensitivity of TMR (and spin injection) todetails of interface structure [14, 15] makes it difficult to close the quantitative gapbetween theory and experiment so it is important for our understanding of TMRto be able to prepare interfaces where disorder does not dominate the spin filteringproperties. This remains a challenge due to the high reactivity of the open-shelltransition metal (TM) ferromagnets Fe, Co, and Ni with typical semiconductors andinsulators.

With this in mind, we wish to draw attention to a quite different material sys-tem in which a thin graphite film is sandwiched between two ferromagnetic leads.Graphite is the ground state of carbon and as one of the most important elementalmaterials, its electronic structure has been studied in considerable detail. It consistsof weakly interacting sheets of strongly bonded carbon atoms. Because of the weakinteraction between these “graphene” or “monolayer graphite” sheets, the electronicstructure of graphite is usually discussed in two steps: first, in terms of the electronicstructure of a single sp2-bonded sheet, followed by consideration of the interactionbetween sheets [16–18]. From these early, and many subsequent studies, it is knownthat graphene is a “zero-gap semiconductor” or a semimetal in which the Fermi sur-face is a point at the “K” point in two-dimensional reciprocal space. The physicalproperties associated with this peculiar electronic structure have been studied theo-retically in considerable detail, in particular in the context of carbon nanotubes whichcan be considered as rolled-up graphene sheets [19]. With the very recent discovery

Graphene Co Ni Cuaexpt

fcc (A) [20] 3.544 3.524 3.615aexpt

hex (A) 2.46 2.506 2.492 2.556aLDA

hex (A) 2.45 2.42 2.42 2.49d0 (A) 2.04 2.03 3.18

Table 5.1: Lattice constants of Co, Ni, Cu, and graphene, ahex ≡ afcc/√

2.Equilibrium separation d0 for a single graphene sheet on top of the graphite(0001),Co, Ni or Cu(111) surfaces as calculated within the framework of the DFT-LSDAusing the in-plane lattice constant ahex = 2.46 A.

5.1. Introduction 85

Co hcpNi fcc

Gr

(a)

MAJ

(b)

Cu

MIN

1 2 3 4K

MΛ

Co fcc Ni hcp(c) (e) (g)

(d) (f) (h)

(i) (j)

Figure 5.1: Fcc Fermi surface (FS) projections onto a plane perpendicular to the(111) direction: Ni fcc majority (a) and minority (b) spins; Co fcc majority (c) andminority (d) spins; Ni hcp majority (e) and minority (f) spins; Co hcp majority (g)and minority (h) spins; Cu fcc (i). For graphene and graphite, surfaces of constantenergy are centred on the K point (j). The number of FS sheets is given by thecolour bar.

and development of an exceptionally simple procedure for preparing single and mul-tiple graphene sheets, micromechanical cleavage [21], it has became possible to probethese predictions experimentally. Single sheets of graphene turn out to have a veryhigh mobility [22] that manifests itself in a variety of spectacular transport phe-nomena such as a minimum conductivity, anomalous quantum Hall effect (QHE)[23, 24], bipolar supercurrent [25] and room-temperature QHE [26]. Spin injectioninto graphene using ferromagnetic electrodes has already been realized [27, 28]. Thelow atomic number of carbon implying weak spin-orbit interaction should trans-late into very long intrinsic spin-flip scattering lengths, a very desirable propertyin the field of spin electronics or “spintronics”, which aims to combine traditionalsemiconductor-based electronics with control over spin degrees of freedom. However,the room temperature MR effect of ∼ 10% observed in lateral, current-in-plane (CIP)graphene-based devices with soft permalloy leads is still rather small [27].

86 Graphene and Graphite as Perfect Spin Filters

Instead of a CIP geometry, we consider a CPP TM|Gr|TM junction, where TMis a close-packed surface of fcc or hcp Ni or Co and Gr is graphite (or n sheets ofgraphene, Grn). We argue that such a junction should work as a perfect spin fil-ter. The essence of the argument is given in Table 5.1 and Fig. 5.1. According toTable 5.1, the surface lattice constants of (111) Ni, Co and Cu match the in-planelattice constants of graphene and graphite almost perfectly. The lattice mismatchof 1.3% at the Ni(111)|Gr interface is, in fact, one of the smallest for the magneticjunctions that have been studied so far. This small lattice mismatch suggests thatthe TM|Gr|TM junction might be realized experimentally, for example, using thechemical vapor deposition technique [29–31]. Assuming perfect lattice matching atthe TM|Gr interface, it is possible to directly compare the Fermi surface (FS) pro-jection of graphite with the FS projections of Cu fcc (111), Ni and Co in both fcc(111) and hcp (0001) directions, see Fig. 5.1.

The Fermi surface of graphene is a point at the high-symmetry K point in recip-rocal space. The Fermi surfaces of graphite and doped graphene are centred on thispoint and close to it. Figure 5.1 shows that there are no majority spin states for Niand Co around the K point whereas minority spin states exist (almost) everywherein the surface BZ. Only the minority spin channel should then contribute to trans-mission from a close-packed TM surface into graphite. In a TM|Gr|TM junction,electrons in other regions of reciprocal space on the left electrode will have to tunnelthrough graphite to reach the right electrode. If the graphite film is taken thickenough to suppress tunneling, majority spin conductance will be quenched and onlyminority spin conductance through the graphite will survive i.e. perfect spin filteringwill occur.

In this chapter, we wish to study quantitatively the effectiveness of this spin fil-tering: how it depends on the thickness of the graphite film, the geometry of theclean metal-graphite interface, interface roughness and disorder, and lattice mis-match. While we will be mainly concerned with the CPP geometry, we will alsocomment on the applicability of some of our conclusions to the CIP geometry. Thechapter is organized as follows. In Sec. 5.2 a brief description of the computationalmethod together with the most important details of the calculations is given. InSec. 5.3 the electronic and structural properties of bulk materials like Ni, Co, graphiteand graphene, that are used for constructing TM|Grn|TM junctions, using the tight-binding linear muffin tin orbitals (TB-MTO) within the atomic-sphere approximation(ASA) are studied. These results are benchmarked against plane-wave pseudopoten-tial calculations. Section 5.4 contains the results of spin-dependent electron transportcalculations for specular interfaces (ideal junction) as well as for junctions with in-terface roughness and alloy disorder. Finally, the conclusions are drawn in Sec. 5.5.

5.2 Computational Method

The starting point for our study is an atomic structure calculated by minimizing thetotal energy within the local spin density approximation (LSDA) of density functional

5.2. Computational Method 87

theory (DFT). This was done using a plane-wave pseudopotential (PWP) methodbased upon projector augmented wave (PAW) pseudopotentials [32] as implementedin the VASP program [33–35]. The interaction between graphite and the TM surfaceis modelled using a repeated slab geometry of six metal layers with a graphene sheeton top and a vacuum thickness of ∼ 12 A. A dipole correction is applied to avoidinteractions between periodic images of the slab [36]. The surface Brillouin zone(SBZ) was sampled with a 36 × 36 k-point grid and the SBZ integrals carried outwith the tetrahedron integration scheme. A plane wave kinetic energy cutoff of 400 eVwas used. The plane-wave pseudopotential method is used to calculate energy bandstructures, charge transfer, binding energies and work functions for single TM|Grinterfaces [1, 37]. The equilibrium distances d0 between the graphene sheet and theTM surfaces are summarized in Table 5.1.

The equilibrium geometries are used as input for a self-consistent TB-LMTO [38]calculation for the TM|Grn|TM junction. The Kohn-Sham potentials are used to cal-culate spin-dependent transmission probabilities through the TM|Grn|TM junctionusing the wave-function matching scheme [39–41]. To do this, the junction is dividedinto three parts: the scattering region sandwiched between semi-infinite left and rightleads, all of which are split into layers that are periodic in the lateral direction. Thesemi-infinite leads are assumed to be ideal periodic crystals in which the electronstates (modes) are wave functions with Bloch translational symmetry. Accordingto the Landauer-Buttiker formalism of transport, the conductance for minority andmajority spin channels can be calculated by summing up all the probabilities fortransmitting an electron from the electron modes in the left lead through the junc-tion into electron modes in the right leads [40, 42, 43].

We can study the effect of various types of disorder on the transmission using thesame formalism and computer codes by modelling the disorder within large lateralsupercells [39, 40] and averaging over many configurations of disorder generated bychoosing positions of impurity atoms or imperfections randomly. Three types of dis-order are studied: interface roughness, interface alloying and lattice mismatch. Inthe first two cases, averaging is performed over a minimum of ten configurations ofdisorder. To model interface roughness, some surface atoms are removed (replacedby empty spheres with nuclear charges that are zero in the ASA) and the ASA poten-tials are calculated self-consistently using a layer version [44] of the coherent potentialapproximation (CPA) [45]. The effect of interface alloying which might occur if de-position of a thin layer of copper on Ni or Co (“dusting”) leads to intermixing ismodelled in a similar fashion. Thirdly, the small lattice mismatch between graphiteand TM is modelled by “cutting and pasting” AS potentials from self-consistent cal-culations for TM|Grn|TM junctions with two different in-plane lattice constants. Thetwo systems are then combined using a supercell of the size determined by the latticemismatch. The lateral BZ of the supercell is sampled with a 24 × 24 k-point gridfor the self-consistent TB-ASA-LMTO calculations. To converge the conductance,denser grids containing 800×800, 20×20 and 8×8 k-points are used for 1×1 (idealjunction), 5× 5 and 20× 20 supercells, respectively.

88 Graphene and Graphite as Perfect Spin Filters

5.3 Geometry and electronic structure of TM|Grn|TM

In this section we describe in more detail how the electronic structure of TM|Grn|TMjunctions for TM=Cu, Ni or Co is calculated. These close-packed metals can be grownwith ABC stacking in the (111) direction (fcc), or with AB stacking in the (0001)direction (hcp). We neglect the small lattice mismatch of 1.3%, 1.9% and 3.9% forthe Ni|Gr, Co|Gr, and Cu|Gr interfaces, respectively, and assume the junction in-plane lattice constant to be equal to that of graphite, aGr = 2.46 A. In the atomicspheres approximation (ASA), the atomic sphere radii of Ni, Co and Cu are thenrTM = 2.574 a.u. The ASA works well for transition metals like Co, Ni or Cu whichhave close-packed structures. For materials like graphite which has a very openstructure with an in-plane lattice constant aGr = 2.46 A, and an out-of-plane latticeconstant cGr = 6.7 A, the unmodified ASA is not sufficient. Fortunately, a reasonabledescription of the crystal potential can be obtained by packing the interstitial spacewith empty spheres [47]. This procedure should satisfy the following criteria: (i) thetotal volume of all atomic spheres has to be equal to the volume of the entire system(space filling), and the (ii) overlap between the atomic spheres should be as small aspossible.

5.3.1 Graphite and graphene

To see how this procedure works in practice, we benchmark the TB-MTO-ASA bandstructure of graphite against the “exact” band structure calculated with the PWPmethod. To preserve the space group symmetry which is D4

6h (P63/mmc) for graphite[46], the positions of the atomic spheres are chosen at Wyckoff positions. There are

Model Atom Wyckoff position radiusposition parameters (a.u.)

I C1 2b 1.556C2 2c 1.556E1 2a 1.454E2 2d 1.61E3 4f z=0.5 1.454E4 24l x=1/3, y=0, z=0.376 1.454

II C1 2b 1.56C2 2c 1.56E 4f z=0.398 2.18

Table 5.2: Atomic parameters including five types of empty spheres: E, E1, E2,E3,E4, (with nuclear charge Z=0); atomic positions according to Wyckoff notationsof space group D4

6h (P63/mmc) Ref.[46]; position parameters and atomic sphereradii.

5.3. Geometry and electronic structure of TM|Grn|TM 89

Model I Model II

top view

side view

Figure 5.2: Top and side perspective views (top and bottom panels) of graphitewhere the potential is represented in the atomic spheres approximation using addi-tional, empty atomic spheres. Model I (left) contains 32 empty spheres in a unit cellcontaining 4 carbon atoms (red spheres). Model II (right), contains just 4 emptyspheres. For model I, gray, green, blue and yellow spheres display the positions ofE1, E2, E3 and E4 empty spheres respectively. For model II, there is just one typeof empty sphere (green).

twelve different Wyckoff positions consistent with D46h symmetry, so the best choice

of empty spheres is not immediately obvious. We construct two models that describethe band structure close to the Fermi energy well compared to the PWP results;this is what is most relevant for studying transport in the linear response regime.Model I with 32 empty spheres per unit cell and model II with only 4 empty spheresper unit cell both preserve the symmetry of graphite within the ASA. The crystalstructures of graphite packed with empty spheres according to these two models isshown schematically in Fig. 5.2 1. The Wyckoff labels, atomic sphere coordinatesand radii are given in Table 5.2. Figure 5.3 shows the band structure of graphiteobtained with the TB-MTO-ASA for models I and II compared to the “exact” PWPband structure. Both models are seen to describe the graphite π bands around the

1note that not all the empty spheres in a unit cell are shown in Fig. 5.2

90 Graphene and Graphite as Perfect Spin Filters

Λ

E−

EF (

eV)

−2

−1

0

1

2

K T M

Model I

Λ K T M

Model II

Figure 5.3: Band structure of graphite for model I (on the left), and model II(on the right). Green dots and black lines correspond to band structures calculatedusing the PWP and TB-MTO-ASA methods, respectively.

Fermi energy very well. Model I provides a very good description of the bandswithin ±2 eV of the Fermi energy, while the smaller basis model II is quite goodwithin ±1 eV. At the cost of including many more empty spheres, model I providesa better description of the crystal potential between the graphene planes than modelII. For this reason we use model I to study the transport properties of ideal junctions,junctions with interface roughness and alloy disorder. To be able to handle the large20 × 20 lateral supercells needed to model lattice mismatch of 5% at the TM|Grinterface, we use model II.

5.3.2 Graphene on Ni(111) substrate

The next step is to put a monolayer of graphite (graphene) on top of the Ni(111)substrate at a distance d0 from the metal surface. From our studies of the energeticsof graphene on TM(111), we found [1, 37] that the lowest energy configuration (with3m symmetry) for TM=Ni or Co corresponds to an “AC” configuration in which onecarbon atom is positioned on top of a surface TM atom (an “A” site) while the thesecond carbon atom is situated above a third layer TM atom (a “C” site), whereA and C refer to the ABC stacking of fcc close-packed planes, see Fig. 5.5. Thisis in agreement with another recent first-principles calculations [48] as well as withexperiments [30, 31] for graphene on the Ni(111) surface. The electronic structure ofa single graphene sheet will depend on d0 and the details of such graphene-metallicsubstrate contacts can be expected to play an important role in current-in-plane(CIP) devices [27, 28]. For the less strongly bound BC configuration of Gr on Ni,the equilibrium separation is rather large d0 ∼ 3.3 A and the characteristic band

5.3. Geometry and electronic structure of TM|Grn|TM 91

Λ K M− 2

− 1

0

1

2

E (

eV

)

Λ K M

MIN

− 2

− 1

0

1

2

E (

eV

)BC AC

MA

J

Figure 5.4: The results of PWP calculations of majority (top panels) and minority(bottom panels) spin band structures (green) of a single graphene layer absorbedupon (both sides of) a 13 layer (111) Ni slab for a BC configuration with d0 = 3.3A, and an AC configuration with d0 = 2.0 A. The bands replotted in black usingthe carbon pz character as a weighting factor are superimposed. The Fermi energyis indicated by the horizontal dashed line.

structure of an isolated graphene sheet is clearly recognizable; see Fig. 5.4. For thelowest energy AC configuration, the interaction between the graphene sheet and Nisurface is much stronger, a gap is opened in the graphene derived pz bands and thereare no graphene states at the K-point in reciprocal space at the Fermi energy forthe minority spin channel. This may prevent efficient spin injection into graphene inlateral, CIP devices [27].

5.3.3 Ni|Grn|Ni(111) junction

The transmission of electrons through a TM|Grn|TM junction will obviously dependon the geometry of the metal-graphite contacts. Rather than carrying out a totalenergy minimization explicitly for every different value of n, we assume that the weakinteraction between graphene sheets will not influence the stronger TM|Gr interac-tion and construct the junction using the “AC” configuration and the equilibriumseparation d0 = 2.03 A for each interface, as shown in Fig. 5.5. The interstitial spaceat the TM|Gr interfaces is filled with empty spheres using a procedure analogous tothat described for bulk graphite [49].

92 Graphene and Graphite as Perfect Spin Filters

ACA

d0

B A B C

d0

C/2

(a) (b)

C1 C2

A

d0

d0C1 C2

C1 C2

Figure 5.5: “AC” model of TM|Grn|TM structure for (a) even and (b) oddnumber of graphene sheets. Carbon atoms are represented by small red spheres, TMatoms by larger gray spheres. The configuration shown in (a) is a c1c1 configurationwith the carbon atom labelled c1 above an“A”site surface layer TM atom of the topand the bottom electrodes. The other carbon atom, c2, is above a third layer TMatom on a “C” site. An equivalent c2c2 configuration in which the c2 atoms are ontop of TM atoms can be realized by rotating the top and bottom electrodes by 180

about a vertical axis through the second layer “B” sites; this effectively interchangesc1 and c2. Two other equivalent configurations c1c2 and c2c1 can be realized in ananalogous fashion by rotating either the top or the bottom electrode through 180.For two sheets of graphene stacked as in graphite, a c2c2 configuration is sketchedin (b). Interlayer distance is indicated as d0 and c/2 is the distance separating twoneighbouing graphene sheets.

Because the two carbon atoms c1 and c2 in the graphene unit cell are equivalent,either of them can be positioned above a surface Ni atom on an A site with the otheron the C site in an “AC” configuration, without changing the total energy. Sincethis can be done for each TM|Gr interface separately, four different configurationsof the TM|Gr|TM junction can be constructed by rotating one or both electrodesthrough 180 about a vertical axis through the second layer B sites which interchangeselectrode A and C sites in Fig. 5.5. We label these four different configurations c1c1,c1c2, c2c1 and c2c2 in terms of the carbon atoms which are bonded to A site TMatoms. For more than one graphene sheet, the second sheet breaks the symmetrybetween the c1 and c2 atoms. While we have not checked this explicitly, we expectthe corresponding energy difference to be small and will neglect it.

5.3. Geometry and electronic structure of TM|Grn|TM 93

−2

−1

0

1

2 (a)

E−E

F (eV

)

MA

J

PWP

(b)

TB−LMTO−ASA

−2

−1

0

1

2 (c)

Λ K

E−E

F (eV

)

MIN

T M

(d)

Λ K T M

Figure 5.6: Lateral band structures of the Ni|Gr|Ni ideal junction with a mono-layer of graphite and 6 monolayers of (111) Ni that on both sides in case of “AC”model c1c2 configuration for majority (top panels) and minority (bottom panels)spin channels. The PWP calculations are shown on the left (green dotted lines)and the TB-LMTO-ASA results are on the right (black solid lines).

In Figure 5.4 we saw that the graphene π states interacted strongly with the nickelsurface in the minimum energy “AC” configuration. The interaction with the metalsubstrate made the c1 and c2 carbon atoms inequivalent and led to the opening of anenergy gap in the graphene π bands. Having constructed an interface geometry, wecan examine the band structure of the Ni|graphene|Ni (111) junction as a functionof k‖, the two dimensional Bloch vector. This is shown in Fig. 5.6 for a single sheetof graphene sandwiched between two slabs of Ni each of which is 6 monolayers thick.The bands on the left-hand side were calculated using the plane-wave pseudopotential(PWP) method and those on the right with the TB-LMTO-ASA. Assuming thePWP as a benchmark, we see that the Ni-related bands are described well by theTB-LMTO-ASA - as might be expected since the ASA is known to work well for close-packed solids. The second thing we see is that there is no gap in the graphene π bands.This is because the c1c2 configuration used in the calculation has inversion symmetryand the symmetry between the two carbon atoms is restored; see Fig. 5.5(a). Thethird point to be made is that the charge transfer from graphene (work function: 4.5

94 Graphene and Graphite as Perfect Spin Filters

eV) to Ni (work function: 5.5 eV) and strong chemisorption leads to the formation ofa potential step at the interface and a significant shift of the graphene π bands withrespect to the Fermi level which is pinned at that of bulk Ni. We find similar resultsfor Co|Gr|Co(111) and Co|Gr|Co(0001) junctions. There is a difference between theposition of the graphene π-derived bands, most noticably at the K point, in the PWPand TB-LMTO-ASA band structures shown in Fig. 5.6 for both spin channels. Itappears that the interface dipole is not adequately described by the ASA. Since fromthe point of view of describing transmission of electrons through this junction, theelectronic band structure is the most important measure of the quality of our basis,description of the potential etc, this discrepancy is disturbing and will certainlyhave quantiative consequences. However, our most important conclusions will bequalitative and will not depend on this aspect of the electronic structure.

5.4 Electron transport through a FM|Grn|FM junc-tion

Using the method and geometries outlined above, we proceed to study the spin-dependent transmission through ideal Ni|Gr|Ni junctions in the CPP geometry as afunction of the thickness of the graphite spacer layer. We then discuss how interfaceroughness, alloy disorder and graphite-metal lattice mismatch affect the spin-filteringproperties of the junctions using large lateral supercells to model the various typesof disorder. Similar results are found for all the TMs shown in Fig. 5.1.

5.4.1 Specular interface

The spin-dependent transmission through Ni|Grn|Ni (111) junctions is shown inFig. 5.7 for parallel (P) and antiparallel (AP) orientations of the magnetizationin the nickel leads, in the form of the conductances GσP and GσAP with σ = min,maj. All the conductance values are averaged over the four interface configurationsof the Ni|Grn|Ni junction which are consistent with AC confiurations of the Ni|Gr(111) interface. Gmaj

P and GσAP are strongly attenuated, while GminP saturates to an

n-independent value. The magnetoresistance (MR) defined as

MR =RAP −RP

RAP× 100% ≡ GP −GAP

GP× 100%, (5.1)

rapidly approaches its maximum possible value of 100%, as shown in the right insetin Fig. 5.7. This pessimistic definition of MR is more convenient here because GAP

vanishes for large n. It is usually the optimistic version, that approaches 1012 %in our calculations but does not saturate, that is quoted [9, 10, 50, 51]. The leftinset in Fig. 5.7 shows how the conductance depends on the particular configurationof the junction. The minority spin conductance in the parallel configuration, whichdominates the magnetoresistance behaviour, is highest for the c1c1 configuration with

5.4. Electron transport through a FM|Grn|FM junction 95

Figure 5.7: Conductances GminP (O), Gmaj

P (M), and GσAP (×) averaged over thefour configurations c1c1, c1c2, c2c1 and c2c2 of a Ni|Grn|Ni junction as a function ofthe number of graphene monolayers n for ideal junctions. Right inset: magnetore-sistance MR as a function of n. Left inset: minority parallel conductance Gmin

P (O)given for four different configurations. The points which are circled and connectedwith a dashed line are the values which were shown in Ref. [1].

a value of GminP ∼ 10−2G0. This is approximately an order of magnitude larger than

GminP for the c2c1, c1c2 and c2c2 configurations. The c1c2 and c2c1 configurations

are equivalent so the corresponding values of GminP should be identical. The small

differences which can be seen in the figure are an indication of the overall accuracy ofthe numerical calculation. The points which are circled and connected with a dashedline are the values which were shown in Ref. [1].

To demonstrate that spin-filtering occurs due to high transmission of minority spinelectrons around the K point, we plot the majority- and minority-spin transmissionfor the P configuration as a function of k‖ for two graphite films of different thicknessin Figure 5.8. A single sheet of graphene (a monolayer of graphite) is essentiallytransparent with a conductance of order G0 in both spin channels. In the minorityspin channel, the transmission is very low or vanishes close to Γ and along the highsymmetry Γ-K line, close to K, in spite of there being one or more sheets of Fermisurface in these regions of reciprocal space. This is a clear indication of the

96 Graphene and Graphite as Perfect Spin Filters

MAJ

MIN

1 ML 5 ML

(a)

(b) Г K-H

r

(e)

(c)

(d)

0.0 0.2 0.4 0.6 0.8 1.0

MIN at K-point

KГ.

Figure 5.8: Transmission as a function of the transverse crystal momentum k‖in the two dimensional interface BZ for a c1c2 configuration of an ideal Ni|Grn|Ni(111) junction in a parallel state. (a) and (b) are for a single graphene sheet, n = 1;(c) and (d) are for n = 5; (e) shows the minority spin transmission in a small area,r = 0.057 (2π/aGr) around the K point for 5 ML of graphite on an enlarged scale.

importance of matrix element effects: selection rules resulting from the incompati-bility of wave functions on either side of the interface [40].

For thicker graphite, the majority transmission must be zero around Γ and aroundthe K point because there are no states there in the Ni leads. The only contributionto the conductance comes from tunneling through graphite in regions of the 2D-BZwhere there are Ni states and the gap between graphite bonding and antibonding πstates is small. This occurs close to the M point; see Fig. 5.3 and Ref. [52]. Becausethe gap decreases going from M to K, the transmission increases in this direction. Atthe edge of the Fermi surface projection, the velocity of the Bloch electrons in theleads is zero so that the maximum transmission occurs just on the M side of theseedges.

The total minority transmission consist of two contributions. On the one handthere is a tunneling contribution from throughout the 2D-BZ which, depending onthe particular k‖ point, is determined by the gap in graphite as well as by the

5.4. Electron transport through a FM|Grn|FM junction 97

compatibility of the symmetries, at that point, of the wave functions in Ni and ingraphite. On the other hand there is a large transmission from the neighbourhoodof the K point coming from the Bloch states there in graphite. Once these havecoupled to available states in Ni, this contribution doesn’t change as more layers ofgraphite are added. Perfect spin-filtering (100% magnetoresistance) occurs when thetunneling contributions are essentially quenched compared to the minority spin Kpoint contribution. For four MLs of graphite the polarization is within a percentof 100% and for five MLs it is for all intents and purposes complete. The onlydiscernible transmission in Figs. 5.8(c-e) is found close to the K point. Magnificationof this region in Fig. 5.8(e) shows a certain amount of structure in the transmission.This can be explained in terms of the multiple sheets of Ni minority spin Fermi surfacein the vicinity of K (Fig. 5.1) and the small but finite dispersion of the graphite bandsperpendicular to the basal plane [52]. The transmission is seen to have the threefoldsymmetry of the junction.

The spin-filtering does not depend on details of how graphite is bonded to the fer-romagnetic leads as long as the translational symmetry parallel to the metal-graphiteinterfaces is preserved. We have verified this by performing explicit calculations (re-sults not shown here) for junctions in the“AB”and“BC”configurations with differentmetal-graphite separations d.

5.4.2 Ni|Cum|Grn|Cum|Ni (111)

In Section 5.3, we saw that the electronic structure of a sheet of graphene dependsstrongly on its separation from the underlying TM substrate. For Co and Ni, equi-librium separations of the order of 2.0 A were calculated for the lowest energy ACconfiguration (see Table 5.1), the interaction was strong and the characteristic lineardispersion of the graphene electronic structure was destroyed, Fig. 5.4. For a sepa-ration of 3.3 A, the small residual interaction does not destroy the linear dispersion.Unlike Co and Ni, Cu interacts only weakly with graphene, there is only a smallenergy difference between the “asymmetric” AC configuration with d0 = 3.3 A andthe slightly more weakly bound “symmetric” BC configuration with d0 = 3.4 A, andbonding to Cu preserves the characteristic graphene electronic structure, opening uponly a very small gap of about 10 meV at the Dirac point [53].

Should it be desirable to avoid forming a strong bond between graphite and theTM electrode, then it should be a simple matter of depositing one or a few layers ofCu on e.g. Ni. Such a thin layer of Cu will adopt the in-plane lattice constant of Niand graphite will bind to it weakly so that the electronic structure of the first layerof graphite will be only weakly perturbed. Because Cu oxidizes less readily than Nior Co, it may be used as a protective layer. Cu has no states at or around the Kpoint for either spin channel (Fig. 5.1) so it will simply attenuate the conductanceof the minority spin channel at the K point. This is demonstrated in Fig 5.9 wherethe magnetoresistance of a Ni|Cum|Grn|Cum|Ni junction is shown as a function ofthe number m of layers of Cu when there are 5 MLs and 7 MLs of graphite. TheMR decreases as the thickness of Cu is increased, reducing the transmission of the Kpoint channel. The decrease of MR can be compensated by increasing the thickness

98 Graphene and Graphite as Perfect Spin Filters

Ni|Cu|Gr|Cu|Ni

Figure 5.9: Magnetoresistance as a function of a number of Cu monolayers onboth left and right Ni leads in case of 5 ML (dashed line) and 7 ML (solid line) ofgraphene.

of graphite. These conclusions are consistent with the qualitative conclusions drawnabove in connection with Fig 5.1(i).

Although the linear dispersion of the graphene bands is essentially unchangedby adsorption on Cu, application of an in-plane bias will destroy the translationalsymmetry parallel to the interface upon which our considerations have been based.The finite lateral size of a Ni|Cu electrode will also break the translational symmetryin a CIP measuring configuration and edge effects may destroy the spin-injectionproperties. We will return to this issue in the last chapter.

5.4.3 Effect of disorder

Lattice MismatchSofar, we have assumed TM and graphite lattices which are commensurate in-plane.In practice there is a lattice mismatch with graphite of 1.3% for Ni, 1.9% for Co and3.9% for Cu which immediately poses the question of how this will affect the perfectspin-filtering. While lattice mismatch between lattices with lattice constants a1 anda2 can in principle be treated by using n1 units of lattice 1 and n2 units of lattice2 with n1a1 = n2a2, in practice we cannot perform calculations for systems with nmuch larger than 20 which limits us to treating a large lattice mismatch of 5%. Toput an upper limit on the effect of a 1.3-1.9% lattice mismatch, we performed

5.4. Electron transport through a FM|Grn|FM junction 99

Ni|Gr|Ni

Figure 5.10: Magnetoresistance as a function of n for: (circles) ideal junctions;(diamonds) Ni|Grn|Cu50Ni50|Ni junctions where the surface layer is a disorderedalloy; (squares) Ni|Grn|Ni junctions where the top layer of one of the electrodes isrough with only half of the top layer sites occupied. The supercell conductancesare normalized to the 1 × 1 surface unit cell used for the ideal case. For therough surface layer, the error bars indicate the spread of MR obtained for differentconfigurations. Inset: schematic representation of Ni|Grn|Ni junction with alloydisorder (roughness) at the right Ni|Gr interface. Ni atoms are given by large grayballs while Cu (missing) atoms in the case of alloy disorder (roughness) are givenby large black balls. Positions of carbon atoms are represented by small grey balls.

calculations for a Ni|Gr5|Ni junction matching 19 × 19 unit cells of Ni in-plane to20 × 20 unit cells of graphite. The effect of this 5% lattice mismatch was to reducethe (pessimistic) magnetoresistance from 100% to 90% (or ∼ 900% in the optimisticdefinition). We conclude that the actual Ni|Gr mismatch of 1.3% should not be aserious limiting factor in practice.

Incommensurability is not the only factor that might reduce the magnetoresis-tance. Preparing atomically perfect interfaces is not possible and raises the questionof how sensitive the perfect spin-filtering will be to interface roughness or disorder.Our studies of spin injection in Chapter 2 [15] and TMR in Chapter 3 [14] suggestthey may be very important and can even dominate the spin transport properties.Interface RoughnessWe envisage a procedure for making Ni|Grn|Ni junctions in which thin graphite lay-ers are prepared by micromechanical cleavage of bulk graphite onto a SiO2 covered

100 Graphene and Graphite as Perfect Spin Filters

Si wafer [54] into which TM (Ni or Co) electrodes have been embedded. We assumethat the (111) electrodes can be prepared in ultrahigh vacuum and characterized onan atomic scale and that the surfaces are flat and defect free. Layers of grapheneare peeled away until the desired value of n is reached. Assuming it will be possibleto realize one essentially perfect interface, we have studied the effect of roughness atthe second interface, assuming it is prepared by evaporation or some similar method.The graphite is assumed to be atomically perfect and all of the roughness occursin the metal interface layer. We model this roughness as in Ref. [14] by removinga certain percentage of the top layer atoms. The atomic sphere potentials are cal-culated using the layer version [44] of the coherent potential approximation (CPA)[45]. The CPA-AS potentials are then distributed at random with the appropriateconcentration in 5× 5 lateral supercells and the transmission is calculated in a CPPgeometry for a number of such randomly generated configurations. The effect onthe magnetoresistance of removing half a monolayer of Ni is shown in Fig. 5.10 asa function of the number of graphite layers. 50% roughness at one interface is seento reduce the 100% magnetoresistance to about 70% pessimistic (230% optimistic).Interface DisorderThe last type of disorder we consider is a layer of interface alloy. We imagine thatdepositing a layer of Cu on Ni to prevent graphite bonding to the Ni has led to alayer of Ni and Cu mixing. In a worst case scenario, we assume all of the disorder isin the surface layer and assume this to be a Ni50Cu50 random alloy. The potentialsare once again calculated self-consistently using the layer CPA and the transmissioncalculated as for roughness. The effect on a monolayer of CuNi alloy is to reduce theMR to 90% (900% in the optimistic definition) for a thick graphite film, as shown inFig. 5.10.

These results indicate that the momentum transfer induced by the scatteringdue to imperfections is insufficient to bridge the large gap about the K point inthe majority spin FS projections. Alternatively, it may be possible to prepare twoseparate, near-perfect TM|Gr interfaces and join them using a method analogousto vacuum bonding [55]. Graphite has a large c-axis resistivity [56]. If one of theTM|Gr interfaces is ideal and the graphite layer is sufficiently thick, then it shouldbe possible to achieve 100% spin accumulation in a high resistivity material makingit suitable for injecting spins into semiconductors [57]. Because carbon is so light,spin-flip scattering arising from spin-orbit interaction should be negligible.

5.5 Discussion and Conclusions

Motivated by the recent progress in preparing and manipulating discrete, essentiallyatomically perfect graphene layers, we have used parameter-free, materials specificelectronic structure calculations to explore the spin transport properties of a novelTM|Grn|TM system. Perfect spin-filtering is predicted for ideal TM|Grn|TM junc-tions with TM = Co or Ni in both fcc and hcp crystal structures. The spin filteringstems from a combination of almost perfect matching of Gr and TM lattices and

5.5. Discussion and Conclusions 101

unique features of their electronic band structures. Graphite films have occupiedstates at the Fermi level only around the K-point in the first (interface) BZ. Close-packed surfaces of fcc and hcp Ni and Co have only minority spin states in the vicinityof the K point, at the Fermi energy. For a modest number of layers of graphite, trans-port from one electrode to the other can only occur via the graphite state at or aroundthe K point. Thus, perfect spin filtering occurs if in-plane translational symmetry ispreserved.

The electronic structure calculations presented here have shown that there is astrong hybridization of carbon π orbitals with Ni (Co) surfaces which may suppressspin-injection for both CIP and CPP geometries due to a substrate-induced bandgapin graphene. It has been suggested to use TM|Cum|Grn|Cum|TM junctions where nobangap opening exists in graphene because the Cu substrate binds graphene weakly.Moreover, using a few monolayers of Cu(111) on the top of Ni(111) electrodes doesnot degrade the MR for a sufficiently thick graphite film. The TM|Cum|graphenelayered structure can be made by intercalation of Cu underneath a graphene sheeton the Ni(111) surface [58]. The use of Cu might also prevent rapid oxidation ofthe Ni(Co)(111) surfaces, which could be important for making practical devices.Explicit studies of rough or disorder interfaces indicate that the spin-filtering effectwill not be critically sensitive to these types of breaking of translational symmetry.

As already discussed in connection with Fig. 5.1, the graphite film in a TM|Grn|TMjunction acts as a tunneling barrier (similar to magnetic tunnel junctions) for ma-jority spin electrons whereas it is conducting for minority spin electrons. The minor-ity electron can then transmit via the graphite states similarly to the GMR effect.However, the TM|Grn|TM junction has several important advantages if comparedwith a conventional TMR junction. Firstly, it has three times smaller lateral lat-tice mismatch compared to the 3.8% for commonly-used Fe|MgO|Fe(001) junction.This should reduce some of the strain and amount of defects that otherwise limitthe thickness and degrade the efficiency of spin injection. Secondly, record break-ing values of MR in the TM|Grn|TM junction are predicted. The spin polarizationwill approach 100% for an ideal junction with n > 3 graphite monolayers, and onlyreduce to 70-90% for junctions with extremely large disorder and roughness at theTM|Grn interfaces. Thirdly, this spin-filtering effect should be less sensitive to thetemperature assuming sufficiently thick graphite interlayer. This conclusion is basedupon several observations. The first one is the highest tunneling rate of majoritystates at M point demonstrated by Fig. 5.8(a). And the second is the large energygap (almost 1 eV) between the Fermi level and graphite band at this point shown inFig. 5.3. Typical thermal broadening is, therefore, too small to allow high transmis-sion of the majority electrons in TM|Grn|TM system which could otherwise decreasethe spin polarization.

In conclusion, we propose a new class of lattice-matched junctions, TM|Grn|TM,that exhibit exceptionally high magnetoresistance effect which is robust to the inter-face disorder and roughness, and insensitive to the temperature. All that make thesejunctions highly attractive for possible applications in spintronic devices.

102 Graphene and Graphite as Perfect Spin Filters

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Chapter 6

A new material system forhighly planar electronics

Lattice matched materials have gured prominently in the discovery of new physicaleects at interfaces. Because the in-plane lattice constants of graphite, hexagonalboron nitride (h-BN) and of the close-packed surfaces of Co, Ni and Cu match almostperfectly, it should be possible using micromechanical cleavage [1] to prepare idealinterfaces between materials which are respectively, a semimetal, a wide band gapsemiconductor, ferromagnetic and nonmagnetic metals. This observation motivatedus to investigate theoretically a range of novel layered devices. Using parameter-freeenergy minimization and electronic transport calculations, we show how h-BN canbe combined with the perfect spin ltering property of Ni|graphite and Co|graphiteinterfaces [2] to make perfect tunnel junctions or ideal spin injectors with any desiredresistance-area product. Noting that monolayers of h-BN and the structurally relatedBC2N are direct gap semiconductors, we propose a novel system of planar electronicand light emitting devices.

Progress in increasing the capacity of magnetic hard disk drives depends on find-ing materials with large magnetoresistance (MR) ratios and suitable resistance-area(RA) products for use as read-head sensors. Present read-head technology is basedupon magnetic tunnel junctions (MTJ) with amorphous Al2O3 barriers and tun-neling magnetoresistance (TMR) ratios of 20-70%. The next generation of readheads requiring a larger TMR effect will be based upon “giant TMR” MTJs withMgO tunnel barriers and room temperature TMR ratios of 200-500% [3]. Unlessstated otherwise, we use, the so-called optimistic definition for magnetoresistance,MR = (RAP − RP )/RP ≡ (GP − GAP )/GAP . Reducing the area A of a junctionwith a fixed tunnelling resistance per unit area (RA = constant) will increase theresistance R. To reduce RA, the insulating oxide barrier can be made thinner. Cur-rently tunnel junctions are so thin (tMgO ∼ 1.0 nm or only 4-5 atomic layers thick)

105

106 A new material system for highly planar electronics

1 3 5 7

10−12

10−10

10−8

10−6

10−4

number of graphene monolayers

co

nd

ucta

nce

(e

2/h

)

MIN

MAJ

AP

1 3 5 7

25

50

75

100

MR

(%

)

Figure 6.1: Conductances GminP (O), Gmaj

P (M), and GσAP (×) of a Ni|BN|Grn|Nijunction as a function of the number of sheets of graphene n. Results for 2 sheetsof BN are shown. Inset: magnetoresistance as a function of n for ideal junctions.Here, the pessimistic definition, MR = (RAP −RP )/RAP ≡ (GP −GAP )/GP whichdoes not diverge when GAP is zero, is used.

that further reduction will introduce pinholes. If A is to be reduced even further, itwill become necessary to find systems with equally high or higher TMR ratios andlower RA products.

We recently showed [2] that a very few atomic layers of graphite sandwichedbetween close-packed Ni or Co electrodes should have an infinite magnetoresistancebecause neither of these transition metal ferromagnets (FM) has majority spin statesin the vicinity of the K point of reciprocal space where the graphite Fermi surface islocated. As a consequence, the majority spin conductance is attenuated exponentiallywhen the number of graphene sheets (Gr) is increased. There are Ni and Co minorityspin states at the K point and once they couple to the Bloch states in graphite, theyare not attenuated when its thickness is increased. A FM|Grn|FM magnetic tunneljunctions has a very low RA product (∼ 0.5 Ωµm2 ) which depends very weakly onthe number n of graphene sheets. This intriguing behaviour depends upon the happycoincidence that the K points of all three materials coincide which in turn resultsfrom a near perfect matching of the in-plane lattice parameters of graphite and closepacked surfaces of Ni and Co. Though this prediction of perfect

107

1 3 5 7 9

10−10

10−8

10−6

10−4

10−2

100

number of BN monolayers

co

nd

ucta

nce

(e

2/h

)

1 3 5 7 90

20

40

60

opt. T

MR

(%

)

1 3 5 7 9−50

−30

−10

10

pola

risation (

%)

Figure 6.2: Conductances GminP (O), Gmaj

P (M), and GσAP (×) of a Ni|BNn|Nijunction as a function of the number of BN layers n for ideal junctions. Insets:optimistic magnetoresistance as a function of n for ideal junction (on the right) andpolarization of the parallel conductance P = (Gmaj

P −GminP )/(Gmaj

P +GminP ).

spin filtering has yet to be experimentally confirmed, the physical principles uponwhich it is based are very well established and robust. Detailed calculations show aremarkable insensitivity to interface roughness, disorder and lattice mismatch.

For other applications it is desirable to have a large RA product. Spin electronicsor “spintronics” aims to introduce into conventional semiconductor-based electronicsthe additional spin degree of freedom used to such good effect in metal-based “mag-netoelectronics” . Attempts to inject spins directly into semiconductors encounter aso-called “conductivity mismatch” problem: the difference in spin-up and spin-downresistivity in conventional ferromagnetic metals is negligible compared to the verymuch larger spin-independent resistivity of semiconductors [4]. This problem can beresolved by injecting spins through a spin-dependent tunnel [5] or Schottky barrier[6] but up till now the spin polarizations achieved at room temperature are far fromcomplete. Here we show how the RA product of a FM|Grn|FM junction with FM =Ni or Co, can be made arbitrarily large without reducing the polarization, by insert-ing m sheets of h-BN to make an FM|BNm|Grn|FM(111) MTJ. Hexagonal BN is alarge band gap semiconductor with an indirect gap of 6 eV[7]. More importantly, ithas the same honeycomb structure as graphene, almost the same lattice parameter,

108 A new material system for highly planar electronics

AlGr

Figure 6.3: The most stable configuration of graphene on (111) surfaces of thefcc non-magnetic metals Al, Ag, Au, Pd and Pt with one carbon atom on top ofa metal atom (“A” site), and the second carbon on a hollow site (“C” site). Theinterface unit cell contains 8 carbon and 3 metal atom.

and can be prepared in monolayer form by micromechanical cleavage [1]. Insertingtwo layers of h-BN increases the RA product by more than three yields of magnitudewithout any deterioration in the polarization (Fig.6.1). By adjusting the numberof layers of h-BN, the RA product can be varied essentially arbitrarily. Relaxedstructures were determined ab-initio by energy minimization [2, 8] and conductanceswere also calculated from first-principles [9].

The large magnetoresistance of Fe|MgO|Fe MTJs [10–13] is attributed to thecrystallinity of the MgO tunnel barrier. Since h-BN is crystalline and its in-planelattice constant matches those of (111) Ni and Co to better than a percent, weinvestigated the magnetoresistance of FM|BNm|FM(111) MTJs with m sheets of h-BN. The results are shown in Fig. 6.2 for Ni electrodes. It can be seen that in thewide barrier limit the magnetoresistance vanishes. The small MR found for thinbarriers can be traced to the existence of a surface state in the minority channel onNi(111). As the barrier width increases, the contribution from this surface state isquenched. MgO is a cubic material with a conduction band minimum at the Γ-pointthat is much lower in energy than at other high symmetry points. States at the Fermienergy of the metal electrode which match the s-like symmetry of this conductionband minimum are attenuated much more slowly in MgO than states with othersymmetries. In the case of Fe, there is a state with this orbital character at theFermi energy for majority spin but not for minority spin. By contrast, the bottomof the conduction band (top of the valence band) of h-BN at the K, Γ, M, H and

109

5 7 910

−12

10−10

10−8

10−6

10−4

number of graphene monolayers

co

nd

ucta

nce

(e

2/h

) MINORITY

MAJORITY

Figure 6.4: Conductances GminP (O), Gmaj

P (M) of a Ni|Grn|Al junction as afunction of the number of graphene layers n for ideal junctions.

L (respectively, K, Γ, M, H, A and L) high symmetry points in reciprocal spacehave very similar energies [7] so that there is no preferential tunnelling of stateswith a particular orbital character which might translate as in the Fe-MgO case intopreference for a particular spin channel. Perfect lattice matching is not sufficientto obtain a large magnetoresistance and h-BN must be used in combination withgraphite to obtain a large TMR and large RA product.

The perfect spin filtering properties of graphite on close-packed surfaces of Nior Co means that this hybrid system, which we denote FM(111)|Grn, behaves asa half-metallic material and can be used to inject a 100% spin-polarized currentinto nonmagnetic materials. As an example, we consider spin injection into metallicaluminium where till now the most successful means of injecting spin has been byusing an aluminium oxide tunnel barrier [14].

The lattice constants of the face-centred cubic non-magnetic metals (NM) Al, Ag,Au, Pd and Pt are such that a 2× 2 unit cell of graphene containing 8 carbon atomsmatches a

√3×√

3 surface unit cell of the (111) non-magnetic metal almost perfectly.The most stable configuration of a graphene|NM(111) interface is determined withoutintroducing free parameters by minimizing the density functional theory (DFT) totalenergy within the local density approximation. The lowest energy“AC”configurationillustrated in Fig. 6.3 has one carbon atom above a NM atom (the surface “A” sites)

110 A new material system for highly planar electronics

while the other is above a third layer Al “C” site. A and C refer to the conventionalABC stacking of the layers in an fcc crystal. The equilibrium interlayer distance atthe interface is calculated to be d = 3.41 for aluminium. For the other non-magneticmetals Ag, Au, Pd, and Pt, equilibrium geometries and bonding energies can be foundelsewhere [15]. Once an interface geometry has been determined, the conductancecan be calculated. The results are shown in Fig. 6.4 for Ni|Grn|Al(111) as a functionof n. This figure demonstrates the saturation of the minority spin injection and rapidexponential attenuation of the majority spin injection resulting in 100% polarizationof the injected carriers.

In principle, FM(111)|Grn could be used to inject spins into a doped semicon-ductor. In practice, it might be necessary to include some layers of h-BN in orderto match the impedance of the semiconductor (or to apply a finite bias, though thiswould go beyond the linear response regime we have implicitly assumed so far). Toperform detailed atomistic calculations it is necessary to consider a semiconductorwith the same in-plane lattice parameter as graphite - or which can be matched usinga reasonably sized supercell. The layered BCxN alloy is in this respect a very attrac-tive system because it is intermediate between graphite and h-BN and indeed, h-BNbelongs to this system with x = 0. Because its bandgap is 6 eV, considerable efforthas been devoted to the possibility of using h-BN as a source of ultraviolet light [16].Unfortunately, it would appear that h-BN is an indirect band gap material [7], as isBC2N [17].

Our calculations of the electronic structure of BCxN indicate however, that mono-layers of h-BN or BC2N are direct band gap materials. Since calculations for agraphite|BN|graphite sandwich yield a transmission of order 0.1, a monolayer is tootransparent to sustain an applied voltage. To inject carriers into a monolayer of h-BN,the electrodes would have to be in the plane of the monolayer of h-BN as sketchedin the upper panel of Fig 6.5. A well-known difficulty with this configuration is thatelectrons and holes injected from either side can pass through the semiconductorwithout recombining to emit light. This can be solved by constructing a quantumwell in which the electrons and holes are trapped until they recombine. In this casethe obvious material to use for the quantum well is BC2N with a band gap of order2 eV [17, 18]. A sketch of the quantum well structure is given in the lower panel ofFig 6.5. These configurations are two dimensional analogues of BxCyNz nanotubeheterojunctions [19].

Acknowledgments: This work is supported by “NanoNed”, a nanotechnology pro-gramme of the Dutch Ministry of Economic Affairs. It is part of the research programsof “Chemische Wetenschappen” (CW) and “Stichting voor Fundamenteel Onderzoekder Materie” (FOM) and the use of supercomputer facilities was sponsored by the“Stichting Nationale Computer Faciliteiten” (NCF), all financially supported by the“Nederlandse Organisatie voor Wetenschappelijk Onderzoek” (NWO).

111Gr|h-BN|Gr LED

Grh-BN

Gr

EF

Gr|h-BN|BC2N|h-BN|Gr Quantum Well

Gr h-BNBC2N

h-BN Gr

EF

Figure 6.5: Top: a two dimensional graphene|BN|graphene tunnel junction. Bot-tom: a two dimensional graphene|BN|BC2N|BN|graphene quantum well.

112 A new material system for highly planar electronics

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