Weak localization effects indisordered graphene

Thesis Subtitle

Hemant Kumar

Department of PhysicsAalto University

Finland

Contents

1 Introduction 11.1 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Transport measurements in graphene . . . . . . . . . . . . . . . . 21.3 Weak localization . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Device Fabrication 42.1 Cleaning of substrate . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Graphene Exfoliation . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Raman spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Resist coating . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.2 Electron Beam Exposure . . . . . . . . . . . . . . . . . . 9

2.5 Metalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Lift-Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Measurement set-up 113.1 Measurement scheme . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Cryogenic apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Wiring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Experimental part 144.1 Sample geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Electrical characterization of the device . . . . . . . . . . . . . . 14

4.2.1 Dirac curve . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.2 Carrier mobility . . . . . . . . . . . . . . . . . . . . . . . 164.2.3 Resistance variation with temperature . . . . . . . . . . . 164.2.4 Dirac curve under weak magnetic fields . . . . . . . . . . 17

A Fabrication recipe 19

1

Abstract

Graphene a perfect 2-D material made of carbon atoms two-dimensional elec-tron gas lying on the surface. Due to the low spin-orbit coupling of carbonatoms, graphene is an ideal candidate for the spin bus in spin transistors. Thefluctuation of residual charge densities due to its lattice defects and trappedcharge in gate oxide is main concern towards the realization of the spin bus ingraphene. The weak localization in graphene is a potential tool to characterizesuch fluctuations. Here in this thesis, I probed the weak localization effectsin graphene on SiO2. For this, I fabricated the micrometer-sized graphene de-vice and measured the conductance fluctuation in dilution refrigerator in thepresence of magnetic field.

Chapter 1

Introduction

In 2003, Andre Geim along with his colleague Kostya Novoselov took a blockof graphite and some Scotch tape. They repeatedly stuck and peeled back theScotch tape until they managed to get down to few atomic sheets of carbonwhich produced a material that is an excellent conductor of heat and electricityand has an extreme mechanical strength about 100 times stronger than thestrongest steel. In 2010, for his work in graphene, they won Nobel prize inphysics.

The purpose of this chapter is to give a brief review on the electronic prop-erties of graphenes mainly from a theoretical point of view. Later on, I will alsopresent transport studies at a weak magnetic field on monolayer graphene, witha particular emphasis on weak localization and backscattering of electrons.

1.1 Band structure

Graphene, a planar atomic sheet of carbon atoms arranged together in a denselypacked hexagonal honeycomb lattice with two atoms A and B per unit cell asshown in Fig. 1.1. The stability of graphene, a 2-dimensional structure is dueto its tightly packed carbon atoms in which each carbon atom having an atomicnumber of six and has an electronic configuration of 1s22s22p2. The valenceelectron of each carbon atoms in graphene is sp2 hybridized.

In graphene valence and conduction band form a conically shaped structurewhich meets at the K and K′ points of the Brillouin zone and creates the zeroband gap (Fig. 1.1). The point, where the conduction and the valence bandtouches each other is called the Dirac point. The low energy properties ofgraphene in the vicinity of Dirac points (K, K′) satisfy the following equation:

E(k) = ±h× vf√k2x + k2y

where a + sign corresponds to the conduction band and a - sign to the valenceband and vf is the Fermi velocity, vf = 1× 106 m/s.

1

Figure 1.1: Left to right: The honeycomb lattice of graphene in which a unitcell is represented by a two carbon atoms denoted by A and B, the conduction(upper) band and the valence (lower) band touches each other in six points ofa hexagon, the K and K′ points and linear dispersion at low energies.

1.2 Transport measurements in graphene

The basic graphene field-effect transistor (FET) consist of a graphene film withtwo metal electrodes placed on a highly doped Si substrate as shown in Fig.1.1 (a). The carrier density induced due to back gate voltage can be calculatedaccording to the following equation:

n =εoεrVGte

= αVg

where εo is the permittivity of free space (εo = 8.854 × 10−12 F/m), εr is therelative permittivity of SiO2 (εr = 3.9) t is the thickness of SiO2 layer (inour case t = 282 nm) and e is the electron charge (e = 1.602× 10−19 C). Aftersubstituting the values the proportionality coefficient α is found to be 7.64×1010

cm−2V −1.The conductivity of a graphene strongly depends on the position of the

Fermi level. In an undoped graphene, the fermi level lies at the Dirac pointwith a completely filled valence band and the empty conduction band. Thedensity of states and hence carrier concentration vanishes at the Dirac point andbecomes smaller than the charged impurity which results in the breakage of thesystem into puddles of electrons and holes. The presence of charge impuritiesin the form of puddles of electron and hole induce a density distribution inthe graphene sample which is responsible for the finite minimum conductivity.The finite minimum conductivity can also be induced by several mechanismssuch as corrugations in the graphene sheet which shift the position of the Diracpoint, molecular doping, external gate voltage, thermal fluctuations and crystaldefects.

The dependence of the resistivity on gate voltage is shown in Fig. 1.2 (b).In doped graphene fermi level is shifted from the Dirac point, depending onthe type of doping concentration. When the fermi level lies below or above theDirac point, the conduction is usually carried out by holes in the valence bandor electrons in the conduction band respectively. In doped graphene, the carrier

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(a) Graphene FET (b) Dirac cone

Figure 1.2: Schematic diagram of a graphene field effect transistor with Si (P++)act as a back gate and conductance is measured between source and drainthrough graphene and Dirac curve

Figure 1.3: An electron trajectory in a diffusive system which is not in thestraight line due to random scattering by charged impurities. The phase of theelectron waves is same as the two paths are identical and their interference willbe constructive in nature.

density at various gate voltages can be obtained from the relation:

n = α(Vg − VDirac)

where α = 7.64 × 1010cm−2V −1 from equation 1.1 and VDirac (Dirac voltage)is the gate voltage for which the maximum resistivity is observed.

1.3 Weak localization

Weak localization is a well-known Quantum interference effects which arise atlow temperatures. Weak localization is usually due to the backscattering andconstructive interference between two paths of the electron along closed loops,traveling in opposite directions as shown in Fig. 1.3. This constructive interfer-ence increased the resistance of the sample, which is known as Weak localizationeffects.

3

Chapter 2

Device Fabrication

The most basics fabrication of a graphene based device consists of follow-ing steps: cleaning of the substrate, exfoliation of graphene, identification ofgraphene using Raman spectroscopy, e-beam Lithography, metallization andlift-off. In the final stage, in order to connect the device to measurement setup,the chip is bonded to a chip carrier. In this chapter, I will explain aforemen-tioned steps in details.

2.1 Cleaning of substrate

Graphene, a one-atom-thick sheet of carbon atoms need a substrate over which itcan be stabilized. Without an underlying substrate, a two-dimensional crystalcan not be thermally stable and it will wrinkle/crumbled. In general, a flatsubstrate is preferred for a long mean free path in ballistic samples. The mostprevalent substrate is SiO2/Si. The thickness of the SiO2 has to be carefullyselected in order to have a good optical contrast. Oxide thickness of 90 nm and282 nm is two such candidate for optical characterization of the graphene, 90nm shows better contrast, see Fig. 2.1.

I choose 282 nm oxide Si P++ for our device fabrication where P-doped Siacts as a back gate for our device. Thicker oxide allows us to use larger gatevoltage over the graphene without the electrical breakdown of the SiO2.

The substrate size at the low-temperature laboratory is of 5 mm×5 mm andit has pre-patterned Ti/Au markers. Si chips usually contain organic residuesand Si/SiO2 debris. For organic residue, I washed the substrate in Acetone thenrinse in IPA which is then followed by oxygen plasma. For heavier debris likeSiO2 and Si, chips are sonicated in Acetone.

4

(a) 282 nm (b) 90 nm

Figure 2.1: An optical image of graphene placed on a silicon substrate coveredwith a 282 nm and 90 nm silicon dioxide

2.2 Graphene Exfoliation

There are two approaches for preparing a graphene flake:

1. Scotch Tape Method in which graphene is extracted from an already ex-isting graphite crystal.

2. Chemical vapor deposition (CVD) in which the graphene layer can begrown directly on a silicon substrate.

Scotch Tape Method provides graphene of very high quality and purity due tothe low complexity, however, the size of the obtained flakes is too poor typicallywith a lateral size of 10 µm - 20 µm.

I used ”Scotch Tape Method” for exfoliation of graphene. This technique isbased on the Micro-Mechanical Exfoliation (MME) in which a chunk of graphitecalled graphenium and a piece of scotch tape is used. By placing the adhesivetape on a graphenium crystals, multiple-layer graphene gets attached to thetape. The multiple-layer graphene is then cleaved into various flakes of few layersof graphene by repeated peeling of the scotch tape. The best part in the adhesivetape is placed on the silicon wafer and hard pressed for two minutes. The scotchtape and the wafer are placed on a 100 C hot plate, which makes removal ofthe tape easier. The scotch tape is peeled off from the silicon substrate, leavingbehind graphene and graphite as shown in the Fig. 2.2.

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Figure 2.2: An optical image of a large exfoliated graphene after heating bothscotch tape and the wafer at 100 C.

Figure 2.3: Raman spectra of mono-layer graphene fitted with a convolution ofGaussian and Lorentzian function.

2.3 Raman spectroscopy

Raman spectroscopy is a spectroscopic technique based on inelastic scattering ofmonochromatic laser light. In an Inelastic scattering the frequency of photonsin monochromatic light changes upon interaction with a sample. The thickness,disorder, doping, strain and thermal conductivity of graphene can be learnedfrom the Raman spectrum. The position and shape of the two major Ramanbands G and 2D in graphene is one of the most important pieces of informationfor physics of graphene and can be used for determining the number of layers.Fig. 2.3 shows the G and 2D peak of the monolayer graphene.

The G band - The G band is a sharp band that appears around 1587 cm−1

in the Raman spectrum of graphene. The band is an in-plane vibrational modeinvolving the sp2 hybridized carbon atoms that comprise the graphene sheet.It is extremely sensitive to strain, doping, and temperature. The width and thefrequency of the G band can be used to monitor the doping level in graphene.The position of the G band shifts to lower energy as the layer thicknessincreases representing the softening of the bonds between carbon atoms.

6

0 0.5 1 1.5 2 2.5 3 3.5 41572

1574

1576

1578

1580

1582

1584

1586

1588

1590

layers

G p

eak

posi

tion

(1/c

m)

G peak position vs number of layers

0 0.5 1 1.5 2 2.5 3 3.5 42660

2665

2670

2675

2680

2685

2690

2695

layers

2D p

eak

(1/c

m)

2D peak position vs number of layers

0 0.5 1 1.5 2 2.5 3 3.5 45

10

15

20

25

layers

G p

eak

FW

HM

(1/

cm)

G Peak FWHM vs number of layers

0 0.5 1 1.5 2 2.5 3 3.5 4

30

40

50

60

70

layers

2D p

eak

FW

HM

(1/

cm)

2D Peak FWHM vs number of layers

Figure 2.4: Comparison of Raman G and 2D peaks of the monolayer, bilayer,trilayer graphene. The substrate (Si with 282 nm SiO2) and laser wavelength(633 nm) are the same for all samples.

The D band - The D band is known as the disorder band or the defect band andis due to lattice motion away from the center of the Brillouin zone. It appearsbetween 1270 and 1450 cm−1, indicates defects or edges in the graphene sample.The D band is typically weak in high-quality graphene. The intensity of the Dband is directly proportional to the level of defects in the sample.

The 2D band - The 2D band is referred to as an overtone of the D band andit is the result of a two-phonon lattice vibrational process. The 2D band is astrong band in graphene and it appears at approximately 2700 cm−1. Theposition and shape of the 2D band depend on the excitation frequency of thelaser and can be used for the determining the number of layers in graphene.

The peak positions and full-width half maximum (FWHMs) of G and 2D peakshave been plotted as a function of the number of layers as shown in Fig. 2.4.The peak intensity ratio of the 2D and G band can also be used to identifythe monolayer graphene. The ratio I2D/IG of these bands for a high-qualitysingle-layer graphene will be seen to be equal to 2 as shown in Fig. 2.5.

The Raman spectrum curve is fitted with a convolution of a Lorentzian anda Gaussian function to get a more accurate value for the full width at halfmaximum (FWHM). The G peak doesnt give sufficient information to distin-guish between different thicknesses, on the other hand, the 2D peak positionor FWHM, provides a definite method for identifying monolayer graphene. InFig. 2.4, the positions and FWHMs of G and 2D peaks have been plotted as afunction of the number of layers.

7

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

I2D

/IG

Mon

olay

er

I2D

/IG

for Monolayer graphene

Figure 2.5: Intensity ratio of the 2D and G band for identification of a monolayergraphene.

Figure 2.6: A schematic flowchart to fabricate a field effect transistor (FET)

2.4 Lithography

In the low-temperature laboratory, I used electron beam lithography to fabricatethe electrical leads to the graphene. The substrate is spin coated and using afocused beam of electrons the pattern is exposed to form a negative mask. Theexposed resist is subsequently developed using a specific developer to createa negative mask. A metal layer is then evaporated using e-beam evaporatorwhich metallized the fabricated leads. The remaining resist and excess metalare removed using a solvent in the ”lift-off” process. A schematic diagram asshown in Fig. 2.6, describes the process of making an electrical lead to thegraphene.

8

2.4.1 Resist coating

A thin layer of the resist of uniform thickness, to cover the substrate for lithog-raphy process is achieved by using resist coating. The substrate is usually bakedat 150C for 10 minutes before spinning to remove water which may cause poorresist adhesion. I used PMMA and a copolymer - 3 of methyl methacrylate andmethacrylic acid (MMA-MAA) e-beam resist with a minimum baking temper-ature of 180C and 150 respectively for 5 minutes. I used a double-layer resistbecause it provides a greater gain in resolution as compared to a thick singlelayer resist.

The resist is then pipetted onto the substrate while the device is placed ona spin coater which has a vacuum that holds the device being spun. Duringthe rotation the resist thickness decreases until a balance between adhesion andcentrifugal force is reached. The thickness of a particular resist is determinedby the spinning speed and resist concentration. The resulting resist coating hassome bulging at the edges and a uniform thickness at the center.

2.4.2 Electron Beam Exposure

In electron beam lithography (EBL), the resist is exposed to the desired elec-trode pattern to fabricate the metallization mask. I made the pattern usingcomputer-aided design (CAD) software. The design CAD file is then convertedinto a format that is readable by lithography system. The lithography systemsends commands to a scan generator which rasters focussed electron beam overthe resist. The movement of the e-beam is controlled by applying discrete volt-age steps to deflection coils of electron column. A beam blanker is used whenthe beam shifts to different parts of the pattern. A high-resolution pattern isobtained by optimizing the total pixel dose, so that pattern did not get overex-posed or underexposed. The optimum pixel dose is determined experimentallyto a dummy chip and the process is called dose test.

After exposure to the e-beam, the device is developed using a developer, forPMMA and copolymer, I used methyl isobutyl ketone (MIBK). I diluted theMIBK with isopropanol to achieve a high resolution. The development time for2-layer resist in 1:3 MIBK : IPA is 7-10 second and depends on the thicknessof resist. The development process is usually stopped by rinsing the device inpure IPA.

2.5 Metalization

Physical vapor deposition techniques are used to metallized the resist maskcreated by e-beam lithography. The masked substrate is placed in a vacuumchamber and the metal is evaporated from a crucible. During deposition thepressure in the crucible range from 10−5 mbar to less than 10−9 mbar so thatthe particles from the source do not collide with the gas molecules.

The material which sits in a water-cooled copper hearth is heated usinga focused electron beam. The e-beam is generated by heating a filament to

9

produce electrons which are then accelerated using high voltages (5 kV to 20kV). The electron beam is focussed onto the copper hearth using magnetic andelectric fields.

The evaporated metal condensed on the masked substrate where the thick-nesses of metal range from a few nanometers to thousands of nanometers witha deposition rate of 0.1 nm/s to 10 nm/s. A quartz oscillator which sits in thevacuum monitored the thickness of the evaporated material. The propertiesof the evaporated metal depend on vacuum level, deposition rate, and processresidues. For low contact resistance to graphene, low vacuum pressure, and lowdeposition rates are required.

An addition layer of different metal generally consists of 5 nm of Cr or Tiis used to improve adhesion to the substrate followed by the main electrodematerial. I used Gold of thickness 50nm for all superconducting contacts tographene.

2.6 Lift-Off

The process of removing the resist and excess metal after metallizing the con-tacts is commonly referred to as lift-off. In this step, the resist and excess metalare dissolved in an organic solvent. For PMMA and copolymer, I used acetone.The lift-off can be accelerated by heating the acetone at 60 C and using asyringe to spray acetone on the resist. After detachment of residual metal, thedevice is washed with IPA and subsequently dried using pressurized nitrogen.

2.7 Bonding

It is a micro-welding technique for electrical interconnection of the device tothe PCB part of the sample holder using electrically insulating low-temperaturevarnish. The electrical contacts between the bonding pads on the substrate andaluminum wire are achieved using ultrasonic bonding. The size of the on-chipbonding pads influences the performance of the device. I used bonding padsof size 200 µm × 200 µm to avoid parasitic capacitances which can tune theelectronic property of the device.

For details fabrication recipe, please look into supplemental information:Appendix A.

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Chapter 3

Measurement set-up

In electronics transport measurements, the fabricated device needs to be pro-tected from the outside world through cooling and shielding, while keeping themconnected to the macroscopic measurement setup. In this chapter, measurementscheme, cryogenic setups used for cooling and the wiring will be addressed.

3.1 Measurement scheme

The measurement scheme for electrical characterisation of the device is pre-sented in Fig. 3.1. I performed all the measurements using current biasedlock-in scheme. The phase lock-in AC transport measurements have a bettersignal to noise ratio in comparison to dc measurement.

A reference sinusoidal signal of fixed amplitude and frequency from lock-in(Stanford Research System SR830) is fed to sample through a high resistancefor current bias measurement. The signal from the sample is filtered to removethe high-frequency components and amplified using current to voltage pream-plifier (SR 570). The output of the preamplifier is measured with respect to thereference signal in lock-in.

The back gate of the sample is connected to a Keithley voltage source. Anexternal current carrying coil is used to produce a low magnetic field in therange between -7 to 7 milli tesla (mT).

The controlled voltage from lock-in is sent to a current source which is thenfed into the coil. The current in the coil generates an induced magnetic fieldwhich is given by the following expression:

B =µoNiR

2

2(R2 + z2)1/2

where R is the radius of the current carrying loop, i is the current, N is thenumber of turns, z is the distance to axis of the loop from the centre and µo

is the permmeability of free space (µo = 12.57 × 10−7 T-m/A). A computer

11

Figure 3.1: Scheme of the measurement setup.

records the amplitude and phase of the detected signal, the back gate voltageand the current delivered to the electromagnet.

3.2 Cryogenic apparatus

The quantum behavior in fabricated graphene device required a minimum tem-perature of 20 mK or below. A dilution refrigerator is a most convenient wayto provide continuous cooling to temperatures as low as 2 mK. The coolingpower is generated by the mixing of the helium-3 (He-3) ’lighter concentratedphase’ and helium-4 (He-4) ’heavier dilute phase’ in the mixing chamber. In acontinuous cooling system helium-3 (He-3) must be extracted from the heavierdilute phase and returned into the lighter concentrated phase. This is achievedby distilling of helium-3 (He-3) in a still which is internally connected by thepumping lines to the heavier diluted phase in the mixing chamber. A resistiveheater in the still is used to maximize the evaporation rate of helium-3(He-3).

These pumping lines are attached to the heat exchanger which carry awayheat from the mixing chamber where the experiment is thermally isolated. Theminimum achievable temperature was reached in a piece of rhodium metal usinga slightly different technique (nuclear de-magnetism), which was cooled to 100pK. This record is achieved in the low-temperature laboratory, Helsinki whereI did my internship.

3.3 Wiring

Electronic transport measurements at cryogenic temperatures require wires thatconnect room-temperature electronics to the sample. However, transmission ofsignals through these measurement lines is affected by electromagnetic radia-tion from the outside. A strong filtering is usually required to suppress high-frequency noise. These lines are filtered using multiple filtering techniques to

12

create a high attenuation over a wide frequency range.

Manganin twisted pairs - Manganin twisted pairs, an alloy made up of 83%copper, 13% manganese, and 4% nickel. It is used to connect 4 K cold plate tothe outside world thus minimizing heat transfer between two points and tominimize electromagnetic noise pick-up. The resistivity of twisted pairs isaround 60 /m along with a distributed capacitance of 330pF/m which providesa large attenuation at high frequencies with a cut-off frequency around 1 GHz.

Thermocoax - All measurement lines from 4.2K to cold plate usethermocoax cables, which additionally filter the high frequencies components.At mixing chamber plate, the additional filtering is added by integratingcopper powder filter and RC filter inside the shielded box to eliminateelectromagnetic noise. The overall, filter of the dc wiring is in the range of 10kHz. Copper powder filter also provides thermal equilibrium of the electricleads from high-temperature parts of a cryostat. These signals are furtherfiltered at different stages using RLC filter which prevents the sample from thehigh voltage spikes. A lock-in amplifier of high sensitivity is used to detectsthe signals from the sample.

Shielded sample stage - In transport measurements, the device andmeasurement electronics were placed inside of the electromagnetically shieldedbox. This box is usually made of copper or brass. The cover is sealed makingthe box vacuum and radiation tight.

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Chapter 4

Experimental part

4.1 Sample geometries

In transport measurements, several sample geometries can be used to investigatethe intrinsic behavior of graphene. In simplest two terminal measurements, thetwo leads are used for both as current leads and for voltage measurement. Thecurrent from the current source results in a voltage drop in the graphene andcontacts, giving additional resistance. A voltmeter measures the voltage dropbetween the sample electrodes, as shown in Fig. 4.1 (a). The measured voltage:

V ≈ I(Rsample + 2rcontact)

Finally, with the four-terminal configuration, one can exclude the wire impedancefrom the measurement. The current is fed through the electrode connected tothe graphene sample and the voltage drop is measured from the central sectionof the sample as shown in Fig. 4.1 (b). The measured voltage:

V = (I − i)Rsample − i(2rcontact + 2rlead)

V ≈ IRsample

4.2 Electrical characterization of the device

4.2.1 Dirac curve

In the experiments, I investigated two graphene field effect transistor of differentchannel length, for which I measured the resistivity versus charge density usingfour-terminal configuration. Before cooling down the sample to mK tempera-ture, I scanned the graphene resistance as a function of the charge density (n)at room temperature as shown in the Fig. 4.2.

The position of the Dirac point, where maximum resistivity is observed de-pends on initial doping concentration. Usually, because of the initial doping,

14

(a) (b)

Figure 4.1: (a) Two-terminal configurations and (b) four-terminal configurationused for transport measurements.

−8 −6 −4 −2 0 2 4 6 8

x 1011

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

Charge density (n)

Res

istiv

ity (

Ω)

Resistivity vs Charge density at room temperature (292.3495K)

(a) Graphene resistivity as a function ofcharge density at room temperature.

0 0.5 1 1.5 2 2.5 3

x 1012

1000

2000

3000

4000

5000

6000

7000

charge density (n)

Res

istiv

ity (

Ω)

Resistivity vs Charge density at 630 mK while cooling down

(b) D1

−4 −3 −2 −1 0 1 2

x 1012

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

5

Charge density (n)

Res

ista

nce

(Ω)

Resistance vs Charge density at 1K

(c) D2

Figure 4.2: Dirac curve for device (a) D1 having scaling factor w/l = 0.187 (b)D2 having scaling factor w/l = 1.15.

the position of the Dirac point shifts from the ideal curve. The initial doping ingraphene can be calculated by measuring the FWHM of a fitted 2-dimensionalgaussian curve to resistivity versus charge density plot as shown in Fig. 4.2 (b,c). The initial doping is calculated to be for device D1: n = 1.6 × 1012 cm−2

and for device D2: 3.2857 × 1012 cm−2. The scaling factor from resistance tosheet resistance for device (D1) is w/l = 0.187, for device (D2) is w/l = 1.15.

The obtained resistivity versus gate voltage curves was very broad this canbe explained by thermodynamics stability of the two-dimensional (2D) mate-rial. The stability of graphene is due to the presence of large three-dimensionalbulk structure beneath which cause microscopic corrugations in graphene. As

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−2.5 −2 −1.5 −1 −0.5 0 0.5 1

x 1012

0

1

2

3

4

5

6

7

8x 10

−4

Charge density

Ele

ctric

al c

ondu

ctiv

ity

Conductivity vs Charge density

Figure 4.3: Conductivity of a graphene measured as a function of charge density.The ratio of the slope of the linear fits indicates that in this sample mobility ofelectrons is not equal to the mobility of holes.

a result, the position of the Dirac point is not constant for the whole graphenesheet which causes the broadening of the density of states.

4.2.2 Carrier mobility

The mobility of electrons/holes is affected by the strength of intrinsic or extrin-sic scattering mechanisms. Intrinsic scattering is caused due to scattering ofphonons while extrinsic includes scattering due to defects, impurities or adsor-bates. The extrinsic scattering caused by impurities implies the purity of thesample and can be increased with technological improvement.

In undoped graphene, the strength of the intrinsic scattering is same for bothelectrons or holes and does not depend on charge hence the electron mobility µe

is equal to the hole mobility µh. The mobility of the electron and hole can beverified experimentally by measuring the electrical conductivity as a function ofgate voltage as shown in Fig. 4.3.

σ(Vg) = ni(Vg)eµi

where ni(Vg) is the concentration of charge carriers electrons or holes calculatedin chapter 1. The plot between electrical conductivity and gate voltage showsa linear dependence and the ratio of the slope of lines gives the mobility, here| µe/µh | ≈ 0.8. Thus, the ratio between mobilities of electrons and holes is notuniversal and depends on the quality and quantity of impurity in graphene.

4.2.3 Resistance variation with temperature

I measured the resistance of monolayer graphene using the field effect transistorconfiguration as a function of temperature at a constant gate voltage (Vg = -12.6 V) near the charge neutrality point. At higher temperature (above around150 K) resistance increases with temperature due to scattering from high energyphonons. But below around 400 mK resistance often decreases as increase intemperature as shown in Fig. 4.4. This behavior suggests that resistivity atcharge neutrality point is usually governed by charged impurities puddles ofelectrons and holes.

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Figure 4.4: Resistance of a monolayer graphene as a function of temperature.

−60 −50 −40 −30 −20 −10 0 100

5

10

15x 10

4

Gate Voltage (V)

Res

ista

nce

(Ω)

R vs Vg at 1K

7 mT0 mT−7 mT

Figure 4.5: Dirac curves at different values of magnetic field. The Dirac curveplotted with red is at 0 mT and shows maximum resistance as compared toother two plots with blue and green which is due to weak localization effects.

4.2.4 Dirac curve under weak magnetic fields

The effects of weak localization and backscattering of electrons in graphenecan be experimentally studied by applying a weak perpendicular magnetic fieldwhich randomizes the phase between the two electron waves and thus destroysthe interference which leads to decrease in resistance as seen from the Fig. 4.4.

I sequentially performed voltage sweeps and measured the resistance for 3different values of magnetic field: 0 mT, 7 mT and -7 mT. The Dirac curve for7 mT and -7 mT overlap each other, as expected because they are measured forthe same absolute value of magnetic feld.

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Appendix A

Fabrication recipe

In this appendix, I will explain the recipe for fabricating a device having narrowchannels using e-beam lithography in the clean room of the Aalto University.

Step 1: Substrate Cleaning procedures -I used the following procedure to clean the silicon substrate:

1. I washed the substrate by rinsing in acetone to remove the organicresidue followed by sonication in a hot (75 C - 80 C) bath ofdichloroethane (DCE) for 10 minutes.

2. I dissolved the acetone and DCE in IPA for 5 minutes and then blowwith dry N2.

3. For removing the bulk of organic contamination, I clean the device withO2 plasma for 5 minutes.

Step 2: PMMA - MMA spin procedures -I used Polymethyl methacrylate (PMMA) and a co-polymer - 3 of methylmethacrylate (MMA) as an e-beam resist. The following procedure that I usedfor achieving a uniform thickness of resist:

1. I placed the cleaned substrate onto spinner chunk and turn on thevacuum pump. Before placing the substrate I set the desired spin rateand time using a spin monitor, for both PMMA and MMA I used 4000RPM for 60 seconds. At the same time, I preheat the hot plate to atemperature of 150 C for further steps.

2. I baked the substrate onto the hot plate which was preheated in step 1,for PMMA and MMA I baked it to a temperature of 150 C and 180 Crespectively.

Step 2: E - beam lithography procedures -I used the following procedure for achieving small features with high resolutionfor the fabricated device as shown in Fig. A.1.

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Figure A.1: An optical image of a large exfoliated graphene after heating bothscotch tape and the wafer at 100 C.

1. I exposed the desired areas with a 20 kV electron beam using a scanningelectron microscope with an area does of 200 µC/cm2 (does depends onpattern geometry and size). For small features of size, less than 1micrometer I used the lowest probe current to avoid overspill and forlarge features (bonding pads) I used the maximum value of probecurrent.

2. I developed the pattern for 15 seconds in 1:3 fresh mixture of MIBK :IPA. The development is stopped by putting the substrate in pure IPAfor 1 minute.

3. After exposure to e-beam onto the substrate, I sputter the desired metalon the wafer. I used 5 nm of Ti or Cr as a sticking layer to improve theadhesion to the substrate followed by a 50 nm of gold, main electrodematerial.

4. I removed the excess metal and resist by soaking the device in hotacetone (60 C) for two hours. After detachment of residual metal, Iwashed it with isopropanol.

This recipe was tested for different values of electron dose on a dummy chip asshown in Fig. A.2.

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(a) Underexposed pattern (b) Overexposed pattern

(c) Ideal pattern

Figure A.2: Dose test on a dummy chip.

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