PHY380 Solid State Physics

Professor Maurice Skolnick, Dr Dmitry Krizhanovskii and Professor David Lidzey

Syllabus 1. The distinction between insulators, semiconductors and metals. The periodic table.

Quantitative aspects.

2. Basic crystal structures. The crystalline forms of carbon.

3. Density of states, Fermi-Dirac statistics. Free electron model.

4. Electrical transport. Resistivity and scattering mechanisms in metals. Temperature dependence.

5. The nearly free electron model. The periodic lattice, Bragg diffraction, Brillouin zones.

6. Prediction of metallic, insulating behaviour: periodic potential and tight-binding descriptions.

7. Real metals, shapes of Fermi surfaces.

8. Soft x-ray emission.

1 http://www.sheffield.ac.uk/physics/teaching/phy380

8. Effective mass. Electrons and holes.

9. Optical absorption in semiconductors. Excitons. Comparison with metals.

10. Doping, donors and acceptors in semiconductors. Hydrogenic model.

11. Semiconductor statistics. Temperature dependence.

12. Temperature dependence of carrier concentration and mobility. Compensation. Scattering mechanisms.

13. Hall effect, cyclotron resonance. Landau levels in magnetic field.

14. Plasma reflectivity in metals and semiconductors.

15. Magnetism (6 lectures)

The Nobel Prizes 2009 and 2010

2

PHY380: Some General Points Recommended Textbooks Solid State Physics, J R Hook and H Hall, Wiley 2nd edition Introduction to Solid State Physics, C Kittel, Wiley 7th edition The Solid State, H M Rosenberg Oxford 1989 All the contents of the course, to a reasonable level, can be found in Hook and Hall. Kittel has wider coverage, and is somewhat more advanced. Ashcroft and Mermin is a more advanced, rigorous textbook, with rigorous proofs.

3

Relation to Previous Courses This course amalgamates much of the previous PHY330 and the magnetism section of PHY331. Assessment The course will be assessed by an end of semester exam (85%) and two home-works (15%) in the middle and towards the end of the semester respectively (1 November, 13 December deadlines) Prerequisite PHY250, 251, Solids (L R Wilson) Lecture Notes The notes provide an overview of the main points, and all important figures. Many more details will be given during lectures. Students thus need to take detailed notes during lectures to supplement the hand-outs.

4

Overall Aims Electrons in solids: determine electrical and optical properties Crystal lattice: bands, band gaps, electronic properties metals, semiconductors and insulators Underpin large parts of modern technology: computer chips, light emitting diodes, lasers, magnets, power transmission etc, etc Nanosize structures important modern development The next slides gives some examples: there are many more

5

6

32nm transistors. Intel web site

Telecommunications laser: Oclaro

25nm

Multi- colour LED strip light

Electronics, computing

Lighting, displays Telecommunications, internet

Integrated circuit

http://www.aztex.biz/tag/integrated-circuits/

Data storage (cd, dvd, blu-ray)

Other major, modern-day applications from condensed matter physics: Magnetic materials hard disks, data storage Superconductors magnets, storage ring at e.g. CERN, magnetic levitation Liquid crystal displays Solar cells Mobile communications, satellite communications

7

Research in Semiconductor Physics There is a highly active research group in the department in the field of semiconductor physics Opportunities for projects (3rd and 4th year), and PhDs See http://ldsd.group.shef.ac.uk/ for more details, or see me for more details

8

Topic 1: Metals, semiconductor and insulators overview and crystal lattices Range of electron densities Metals: Typical metal (sodium), electron density n=2.6x1028m-3 Insulators (e.g. diamond): electron density very small (Eg ~ 5.6eV, ~5000K >>kBT at 300K) Semiconductors: electron density controllable, and is temperature dependent, in range ~1016m-3 to ~1025m-3 Conductivity is proportional to electron density

9

Importance of bands and band gaps Determine electron density and hence optical and electronic

properties

Understanding of origin will be important part of first 7 lectures

Bands and band gaps arise for interaction of electrons with periodic crystal lattice

Three schematic diagrams illustrating differences in bands, gaps and their filling in metals, semiconductors and insulators will be given in the lecture (these are important, simple starting point for course)

10

I

II IV

Note also: Transition metals Noble metals 11

With relation to previous slide: Group 1: alkali metals, partially filled bands Group II: alkaline earths Group IV: semiconductors, insulators, filled bands + transition metals, noble metals

12

Crystal Lattices

The nature of the crystal lattice, and the number of electrons in the outer shell determine the conduction properties of most elements Periodic arrangement of atoms Space lattice plus basis (Fig Kittel)

Lattice translation vector T = u1a1 + u2a2 + u3a3 a1, a2, a3 lattice constants (spacings of atoms) Position vector r' = r +T

(c) Crystal structure

(a) Space lattice

(b) Basis, containing two different ions

13

Space Lattice plus basis This figure and slide 14 not covered in lecture here for extra (useful) information

Space lattices in two dimensions Primitive (unit) cell defined by translation vectors

14

3D

Cubic lattices

Lattice points defined by translation vectors Also note diamond is fcc space lattice Primitive basis: 2 atoms for each point of lattice (Kittel page 19)

15

Primitive (unit) cell: Parallelipiped defined by axes a1, a2, a3 sc, bcc and fcc lattices, lattice points per cell and per unit volume Simple cubic: 1 lattice point per unit cell bcc: 2 lattice points per unit cell fcc: 4 lattice points per unit cell Number of lattice points per unit volume?

16

Periodic table and crystal structures

17

Planes and directions

18

( ) planes, [ ] directions (covered in 2nd year)

http://diahttp://www.theage.com.au http://physics.berkeley.edu/research/lanzara/

http://en.wikipedia.org/wiki/Graphene

Diamond Graphite

Buckyball C60

http://diahttp://www.theage.com.au

Graphene

Carbon nanotube

http://www.azonano.com/

2010 Nobel Prize to Geim and Novoselov

The Crystalline Forms of Carbon

19

20

2010 Nobel Prize for Physics A Geim and K Novoselov Graphene, single sheet of carbon atoms: high electron motilities, electrons with new properties, very strong, electronics and sensor applications potentially

21

Key properties of diamond Cubic (diamond) crystal lattice (see slide 19) Very hard, high strength, insulator, chemically inert, very high thermal conductivity, optically transparent Key properties of graphene Hexagonal crystal lattice (see slides 19, 20), two dimensional plane Very strong, metallic but conductivity can be controlled, unique linear dispersion relations (E v k ), very high thermal conductivity, adsorbate properties

Comparison of two crystalline forms of carbon

22

Topic 1 summary 1. Distinctions between metals, semiconductors and insulators, in

particular the widely differing electron densities

2. Impact on everyday life

3. Importance of band gaps, and filling of bands, in controlling these properties

4. Periodic lattice gives rise to bands, band gaps

5. The crystal structures of carbon

Topic 2: Free Electron Model

This is the simplest theory of conduction in metals, based on a non-interacting gas of electrons (which obey Fermi Dirac statistics). It ignores the presence of the crystal lattice. It explains some basic properties, but fails to account for many others e.g. which elements are metallic, the colour of metals, electrons and holes etc, for which we need band theory. Based on the free electron Fermi gas Electrons are Fermions which obey Fermi-Dirac statistics (and the Pauli exclusion principle)

23

Fermi-Dirac distribution function

( ) kTEEEf F /]exp[1)(

=

For T 0, f(E) = 1 for E < EF f(E) = 0 for E > EF

E/kB in units of 104 K

f(E)

~kBT

24

25

Leads to condition for allowed k-values next two pages

Free Electron Theory

Periodic boundary conditions (box, side L) to count states

),,(),,( zyxzyLx =+

is travelling wave solution )(.)( zkykxkirikk zyxeer

++==

provided that kx = 0, 2/L, 4/L .... 2n/L, where n is a positive or negative integer Proof:

1012sin2cos =+=+ nin

)(2exp)(exp LxL

niLxikx +=+

niLnxi 2exp2exp=

xikLnxi

xexp2exp ==

26

27

Counting of States (important, needed to evaluate e.g. the density of states, Fermi energy and other key properties)

Allowed values of k are thus kx = 0, 2/L, 4/L .... 2n/L In one dimension, one allowed value of k for range of k of 2/L

Substituting into Schrdinger equation gives rikk er.)( =

22

2222

2)(

2k

mkkk

mE zyxk

=++=

Parabolic dispersion of free particle with mass m Corresponds to with ,

2

2

kpm

pE ==

p is termed the crystal momentum, and k the wavevector

Dispersion Relation

28

Density of States The Fermi energy and Fermi surface Key properties of metals

29

30

Need to determine number of states in k-space up to a given energy (the Fermi energy) One allowed wavevector in volume element of k-space of (2/L)3 Volume of sphere in k-space up to energy E, wavevector k is Then calculate number of available states from E = 0 to EF, and hence derive expression for density of states

3

34

Fk

+ the Pauli exclusion principle

Number of states, Fermi wavevector and Fermi energy

31

( ) 3222 32

nm

EF

=

Values of TF, kF, EF, vF for sodium and their significance (37000K, 0.96x1010 m-1, 3.2eV, 1.07x106m/sec)

32

33

Topic 2 summary 1. Electrons are Fermions and obey Fermi-Dirac statistics and

the Pauli exclusion principle

2. States up to EF filled, above EF empty

3. Form of the density of states proportional to E1/2

4. Expressions and quantitative values for EF, kF, vF (these are important!)

Topic 3: Conductivity Drude theory of conductivity based on free electron model

Ion cores ignored, periodic lattice ignored, effective mass

Zero frequency approximation, Ohms Law

Displacement of Fermi sphere by electric field and

scattering processes

Phonon and defect scattering, Matthiesens rule

34

Deduce velocity Define mobility Deduce current density, conductivity and Ohms Law

mnemenej

2

v

=

=

=

35

Newtons 2nd Law Include scattering scattering time

d.c conditions , B = 0

( )

( )BEedtdk

BEedtdm

xv

xvv

+=

+=

( )BEedtdm xvvv +=

+

Eem =v

Fermi sea of electrons in applied electric field, and scattering processes

For derivation of displacement in k-space see next slide

36

So displacement of Fermi sea by electric field is very small Scattering counters acceleration of electrons by electric field

Motion of electrons in electric field and scattering: change in wavevector

Alternatively:

37

eEmvk

mEevD

==

=

5 x 108 smaller than kF

For metals two scattering mechanisms are important 1. Lattice scattering - phonons

2. Imperfections (defects)

impurity atoms, vacancies, lattice defects

Scattering collisions which are important are those which relax momentum gained from E-field Scattering must be across Fermi sea i.e. large k, small E

38

Phonon scattering Fermi energy ~ 3 eV

Phonons have maximum energy

~50 meV

Scattering must be to an empty state

Thus only electrons close to Fermi surface can be scattered

Must conserve energy and momentum

Collisions which relax momentum gained in applied electric field lead to resistance

Must be across Fermi sea: Large k small E

39

Situation is similar for defect scattering However, in this case collisions are elastic, but still with large

momentum change as for phonons

It is again scattering with large k which is effective in leading to resistance (as for phonon scattering)

For phonons scattering is inelastic, but energy change is negligible

For phonons (conservation of energy and wavevector):

elfph

eli

elfph

eli

EE

kkk

=+

=+

Combination of two types of scattering Phonon scattering is temperature dependent Scattering by imperfections is temperature independent Matthiesens rule (additive combination of contributions from phonon and defect scattering)

40

Scattering of electrons is not by ions Instead by impurities and defects Electrons propagate freely in periodic structure (see Bragg scattering later) Mean free path lB> 1m or more lB >> interatomic spacing, so collisions not with ions

41

Additional point (important)

Topic 3: Summary Theory of conductivity based on free electron model

Ion cores ignored, periodic lattice ignored. Electrons treated with

effective mass

Displacement of Fermi sphere by electric field and scattering processes

Phonon and defect scattering. Contributions are additive. Matthiesens rule

Scattering processes which relax momentum across the Fermi sea are the important ones (in opposite direction to acceleration by field)

Scattering is not by the ions of the lattice. 42

43

Topic 4: Electrons in periodic lattice, nearly free electron model

Many experimental observations are not explained by free electron theory, including: 1. Existence of bands, band gaps 2. Existence of non-metals 3. Effective mass 4. Colours of metals 5. High frequency conductivity 6. Existence of holes 7. Nature of the Hall effect The periodic lattice is all important in explaining these and other phenomena

44

Periodic lattice gives rise to Bragg diffraction of electron waves

45

o

aak

nn

90

2sin22

=

===

for waves travelling down 1D chain

Therefore k=n/a Bragg condition for 1D chain Electron wave is scattered by 2/a (= G) (reciprocal lattice vector)

1D chain

46

Origin of band gap from Bragg diffraction (following Kittel, chapter 7, 7th edition)

Continued next 2 slides

47

See diagram next slide

With lower and higher energy respectively

Two solutions with different energy at same wavelength (and hence wavevector). Leads to band gap.

Origin of band gap from Bragg diffraction

Bragg diffraction leads to band gaps, since cos2(x/a), sin2(x/a) charge distributions at k=n/a

Two solutions at same wavelength (k-vector)

Energy gaps occur when waves have wavelength which is in synchronism with the lattice

48

49

As noted earlier, at Bragg condition electron wave is scattered by k = 2/a (= G) (reciprocal lattice vector) Lattice potential (Fourier components) mixes waves at these points in dispersion in unperturbed band-structure (in (a) above), giving rise to gaps in (b)

50

Group velocity dkdE

dkdvg

1==

is zero at zone boundary, corresponds to standing wave

Continuing last slide

51

To summarise Topic 4 Bragg diffraction defines edge of Brillouin zone.

Group velocity at Bragg condition (at zone boundary)

is zero

Bragg diffraction, and hence band gaps, occurs for

waves (k-values) in synchronism with lattice periodicity

General condition for Bragg diffraction,

G is reciprocal lattice vector

Gk =

52

Nobel Prize in Physics 2009; Strong relevance to Solid State Physics Charles K Kao, Optical fibres, Basis of internet data transmission

Combines semiconductor laser sources, modulators, detectors, knowledge of optical absorption mechanisms in solids

53

Willard S. Boyle and George E. Smith, Charge Coupled Device Detectors

Digital imaging device in cameras, fax machines, scanners, telescopes and many other types of modern instrumentation. Based on silicon integrated circuit technology and field effect transistors

Readout of information from each pixel

54

Key Points of Topics 1-4 1. Existence of bands and band gaps vital to explain key properties of

electrons in solids 2. Band region of allowed electron states in E(k) space 3. Band gap - region of forbidden states, no allowed states 4. Explains distinction between metals, semiconductors and insulators 5. Fermi-Dirac distribution function. States filled up to Fermi wavevector 6. Behaviour of Fermi sphere under applied electric field, small

perturbation 7. Scattering mechanisms. Scattering is not by ions of lattice. 8. Bragg scattering gives rise to band gaps 9. Bragg condition defines k-vectors at which Bragg scattering occurs 10.Treatment of k-vectors for which waves in synchronism with lattice

provides insight into origin of band gaps 11.General condition for Bragg diffraction 12. Outer shell electrons provide dominant contribution to conduction (see

periodic table)

Gk =

55

Atomic configuration of sodium: 1s2, 2s2, 2p6, 3s1 n = 1, 2 shells tightly bound and give rise to lower energy valence bands. Do not contribute to conduction. 3s electron is weakly bound and leads to conduction. Half filled band

56

Topic 5: Introduction to Brillouin zones, half-filled and filled bands

Number of states in a band Monovalent atoms metallic Insulators: can only occur for even number of valence electrons

Group II elements, nevertheless are metallic. Concept of overlapping bands

57

Each unit cell contributes one value of k to each Brillouin zone, and hence to each band

Including spin, 2N states per band

If one atom per unit cell (monovalent), then band half filled alkali, noble metals

Insulators can only occur for even number of valence electrons per primitive cell (e.g. C, Si, Ge, which are 4 valent, plus have 2 atoms per primitive cell)

Group II elements could be insulators, but bands overlap, so metals, but relatively poor metals (also see Hall effect where there is hole conduction)

Counting of states and filling of bands

Periodic boundary conditions (following from pages 26, 27) k = 0, 2/L, 4/L .... 2n/L L is length of chain of atoms, n is an integer If N is number of atoms, the lattice constant a is equal to L/N Total number of states between /a is N More strictly, N is number of primitive unit cells in chain

58

Conduction in half-filled and filled bands

I

II IV

Note also: Transition metals Noble metals 59

Alkali metals and noble metals have one outer shell electron: partially filled band and hence metal Group IV: semiconductors, insulators, 4 outer shell filled bands Group II: even number of outer shell electrons, but overlapping bands. Hence metallic.

60

61

How bands can overlap

And thus overlapping bands i.e. energy in second band less than that in first

in

Ec < Eb for mkEg 2

22