PHY380 Solid State Physics

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

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 2 nd edition Introduction to Solid State Physics, C Kittel, Wiley 7 th 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 homeworks (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

Electronics, computing Integrated circuit http://www.aztex.biz/tag/integrated -circuits/ 25nm 32nm transistors. Intel web site Lighting, displays Telecommunications, internet Data storage (cd, dvd, blu-ray) Multi- colour LED strip light Telecommunications laser: Oclaro 6

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 (3 rd and 4 th 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=2.6x10 28 m -3 Insulators (e.g. diamond): electron density very small (E g ~ 5.6eV, ~5000K >>k B T at 300K) Semiconductors: electron density controllable, and is temperature dependent, in range ~10 16 m -3 to ~10 25 m -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 (a) Space lattice (b) Basis, containing two different ions Space Lattice plus basis This figure and slide 14 not covered in lecture here for extra (useful) information Space lattice plus basis (Fig Kittel) Lattice translation vector T = u 1 a 1 + u 2 a 2 + u 3 a 3 (c) Crystal structure a 1, a 2, a 3 lattice constants (spacings of atoms) Position vector r' = r +T 13

Space lattices in two dimensions Primitive (unit) cell defined by translation vectors 3D 14

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 a 1, a 2, a 3 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 ( ) planes, [ ] directions (covered in 2 nd year) 18

Diamond The Crystalline Forms of Carbon Graphite Carbon nanotube http://diahttp://www.theage.com.au http://physics.berkeley.edu/research/lanzara/ http://www.azonano.com/ Buckyball C 60 Graphene 2010 Nobel Prize to Geim and Novoselov http://diahttp://www.theage.com.au http://en.wikipedia.org/wiki/graphene 19

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 20

Comparison of two crystalline forms of carbon 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 21

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 22

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 f ( E) = exp[ 1 ( E E )] kt F / For T 0, f(e) = 1 for E < E F f(e) = 0 for E > E F ~k B T f(e) E/k B in units of 10 4 K 24

Free Electron Theory Leads to condition for allowed k-values next two pages 25

Periodic boundary conditions (box, side L) to count states ψ ( x + L, y, z) = ψ ( x, y, z) ψ k ( r) = e ik. r = e i( k x x+ k y y+ k z z) is travelling wave solution provided that k x = 0, ±2π/L, ±4π/L... 2πn/L, where n is a positive or negative integer Proof: 2πn expik x ( x + L) = expi ( x + L L) 2πnx = expi expi2πn L i2πnx = exp = L expik x x cos 2πn + isin 2πn = 1+ 0 = 1 26

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 k x = 0, ±2π/L, ±4π/L... 2πn/L In one dimension, one allowed value of k for range of k of 2π/L 27

Substituting into Schrödinger equation gives r ik k e r. ) ( = ψ 2 2 2 2 2 2 2 ) ( 2 k m k k k m E z y x k = + + = Parabolic dispersion of free particle with mass m Corresponds to with, 2 2 k p m p E = = 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

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 (2π/L) 3 Volume of sphere in k-space up to energy E, wavevector k is 4 3 3 πk F Then calculate number of available states from E = 0 to E F, and hence derive expression for density of states + the Pauli exclusion principle 30

Number of states, Fermi wavevector and Fermi energy E F = 2 2m ( 2 3π n) 2 3 31

Values of T F, k F, E F, v F for sodium and their significance (37000K, 0.96x10 10 m -1, 3.2eV, 1.07x10 6 m/sec) 32

Topic 2 summary 1. Electrons are Fermions and obey Fermi-Dirac statistics and the Pauli exclusion principle 2. States up to E F filled, above E F empty 3. Form of the density of states proportional to E 1/2 4. Expressions and quantitative values for E F, k F, v F (these are important!) 33

Topic 3: Conductivity Drude theory of conductivity based on free electron model Ion cores ignored, periodic lattice ignored, effective mass Zero frequency approximation, Ohm s Law Displacement of Fermi sphere by electric field and scattering processes Phonon and defect scattering, Matthiesen s rule 34

Newton s 2 nd Law d v m dt dk dt = e = e Include scattering m d v dt v + τ τ scattering time ( E + v x B) ( E + v x B) = e ( E + v x B) Deduce velocity Define mobility Deduce current density, conductivity and Ohm s Law j = nev eτ µ = m 2 ne τ σ = m d.c conditions, B = 0 mv τ = ee 35

Fermi sea of electrons in applied electric field, and scattering processes For derivation of displacement in k- space see next slide 36

Motion of electrons in electric field and scattering: change in wavevector Alternatively: v D δk eτe = m mv = = eeτ 5 x 10 8 smaller than k F So displacement of Fermi sea by electric field is very small Scattering counters acceleration of electrons by electric field 37

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 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 38

For phonons (conservation of energy and wavevector): k E el i el i + k + ω ph ph = = k E el f el f 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 39

Combination of two types of scattering Phonon scattering is temperature dependent Scattering by imperfections is temperature independent Matthiesen s rule (additive combination of contributions from phonon and defect scattering) 40

Additional point (important) 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 l B > 1µm or more l B >> interatomic spacing, so collisions not with ions 41

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. Matthiesen s 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

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 43

Periodic lattice gives rise to Bragg diffraction of electron waves 44

1D chain 2π nλ = n = k o θ = 90 2asinθ = 2a for waves travelling down 1D chain Therefore k=±nπ/a Bragg condition for 1D chain Electron wave is scattered by 2π/a (= G) (reciprocal lattice vector) 45

Origin of band gap from Bragg diffraction (following Kittel, chapter 7, 7 th edition) Continued next 2 slides 46

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. 47

Origin of band gap from Bragg diffraction Bragg diffraction leads to band gaps, since cos 2 (πx/a), sin 2 (πx/a) charge distributions at k=±nπ/a Two solutions at same wavelength (k-vector) Energy gaps occur when waves have wavelength which is in synchronism with the lattice 48

As noted earlier, at Bragg condition electron wave is scattered by k = 2π/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) 49

Continuing last slide Group velocity v g d = ω = dk 1 de dk is zero at zone boundary, corresponds to standing wave 50

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, k = G G is reciprocal lattice vector 51

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 52

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 53

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 k = G 12. Outer shell electrons provide dominant contribution to conduction (see periodic table) 54

Atomic configuration of sodium: 1s 2, 2s 2, 2p 6, 3s 1 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 55

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 56

Counting of states and filling of bands Periodic boundary conditions (following from pages 26, 27) k = 0, ±2π/L, ±4π/L... 2πn/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 ±π/a is N More strictly, N is number of primitive unit cells in chain 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) 57

Conduction in half-filled and filled bands 58

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

How bands can overlap in : E c can be less than E b for: E c < E b for 2 k E g < 2m 2 And thus overlapping bands i.e. energy in second band less than that in first 61

Overlapping bands: energy of state in second band lower than in first Consequence: some of states in second band filled before uppermost states in first Leads to two partially filled bands Electrons and holes anomalous Hall coefficient 62

Summary, Topic 5 Total number of states in 1D chain, using periodic boundary conditions = N, where N is number of atoms. Given by total number of allowed k- values. Each unit cell contributes one value of k to each Brillouin zone, and hence to each band. Including spin gives 2N states per band Monovalent atoms with one atom per unit cell (alkali and noble metals), band half filled, expect metallic Insulators: can only occur for even number of valence electrons per primitive cell e.g. C, Si, Ge 4 valence electrons plus 2 atoms per primitive cell But group II elements, the alkaline earths (metals) have even number of electrons, expected to be insulators, but are metallic. Overlapping bands. Can only occur in 2 and 3D. Simple proof for 2D. 63

Summary of Bragg diffraction, Brillouin zones 1. Bragg condition defines edges of Brillouin zones 2. For one dimension, simple proof of condition k = ±π/a (page 45) 3. In general k = G 4. Can also be understood in terms of mixing of particular values of k by Fourier components of periodic lattice potential (page 49) (Kittel pages 34-36 for rigorous treatment) 5. Dependence of band gaps on atomic number, differing roles of inner and outer shells. 64

Topic 6: Construction and Properties of Brillouin Zones Use generalised Bragg condition to construct Brillouin Zones Definition and properties of Brillouin Zones Consequences for Fermi surfaces Different zone schemes Essential steps to understand shapes of Fermi surfaces of real metals (and hence conduction properties) 65

Bragg Diffraction: 66

In 1D, rederivation of Bragg s Law G = 2π/a in 1D 67

Geometrical constructions to obtain Brillouin Zones Also see next slide Hook and Hall (p334) of G 1 Perpendicular bisectors 68

Construction of Brillouin Zones for Square Lattice 69

Definition of Brillouin zones 70

1. Generalised Bragg condition 2k. G = G 2 2. defines boundaries of Brillouin zones. k lies on perpendicular bisector of G. 3. Construction of 1 st, 2 nd, 3 rd zones 4. If Fermi surface is sufficiently large that it crosses Brillouin zone boundaries, then shape of Fermi surface will be strongly modified. 71

Reduced zone scheme Translation vector Hook and Hall (p116-118) Also see Hook and Hall p39 for physical discussion By reciprocal lattice translation (2π/a), can translate points in higher zones into first zone 72

Repeated, reduced and extended zone schemes Rely on reciprocal lattice translations One Brillouin zone one band in extended zone scheme 73

Shapes of Fermi surface resulting from Brillouin Zone structure Superimpose Fermi circle on Brillouin Zones using ( k + G) ψ (k) ψ = Additional mechanism for occurrence of partially filled bands Complicated shapes of Fermi surfaces 74

Summary: Topic 6 Generalised Bragg Condition: 2k. G = G 2 Brillouin zone boundaries defined by intersection of k with perpendicular bisectors of reciprocal lattice vectors G Reciprocal lattice vector in 1D G = 2πx/a Generalise to 3D First Brillouin Zone is the set of points in reciprocal space that can be reached from origin without crossing any Bragg plane Generalise to 2 nd, nth zones All Brillouin Zones have the same volume ψ ( k + G) = ψ ( k) Basis of reduced, repeated and extended zones. Approximate proof and consequences. 75

Topic 7: Fermi Surfaces in Metals, Their Forms and Their Measurement Topic 6 has introduced effect of periodic potential and of Brillouin zones on shapes of Fermi surface Topic 7 is concerned with the shapes of Fermi surfaces in real metals, and the role of the crystal lattice potential and its periodicity Will discuss silicon band structure in next lecture - download 76

Band must intersect Brillouin Zone boundary at right angles (2D picture, also holds in 3D) (as for band at zone boundary in 1D) 77

Real Fermi surfaces e.g. copper, silver, gold Fermi surface in copper Repeated zone scheme fcc lattice in real space bcc lattice in reciprocal space Belly, neck and dogs bone orbits Distortion of Fermi surfaces by periodic potential at Brillouin zone boundaries (as in 2D on previous slide) Alkali metals e.g. Na, K Fermi surface lies inside 1 st Brillouin zone, and is only very slightly distorted 78

Origin of neck orbits: Energy of band lowered as it approaches zone boundary So states at higher k may be populated Thus spherical Fermi surface distorted Dog s Bone Hole-like constant energy surface: easily visualised in extended zone scheme Other ways to produce holes?? 79

Intermediate summary: 1. Periodic potential produces gaps at zone boundary 2. Fermi surface intersects zone boundary at right angles 3. Crystal potential rounds out sharp corners in Fermi surface 4. Total volume enclosed by Fermi surface depends only on electron density independent of details of potential 80

Soft X-ray emission Method to measure conduction electron distribution in solids 1. Only outer shell electrons contribute 2. All inner shells are filled, and play no role e.g. In Na 1s, 2s, 2p shells filled (p55) 3. Can measure energy distribution of conduction electrons by soft x-ray emission 4. Use high energy electron bombardment to create hole in one of inner shells 5. Conduction electron falls into hole. X-ray photon emitted 6. Distribution of emitted x-rays gives measure of conduction electron distribution Related topics due to conduction electrons: plasmons, plasma reflectivity see later in course. Determine Fermi energy from plasma frequency. 81

Soft x-ray emission spectrum 0 E F 82

Summary Topic 7 1. Periodic potential produces gaps at boundaries 2. Fermi surface must intersect zone boundary at right angles 3. Crystal potential rounds out sharp corners in Fermi surface 4. Total volume enclosed by Fermi surface depends only on number of electrons why. Independent of potential 5. All Brillouin zones have same volume 6. Copper, silver, gold, belly, neck and dog s bone orbits 7. Alkali metals much simpler 8. Soft X-Ray emission measures electron distribution of occupied bands in solids. Complementary to conductivity, Hall effect, plasma reflectivity 83

Topic 8: Tight Binding Model 1. Levels sharp in isolated atoms 2. When atoms brought together, Pauli principle does not allow energies of electrons on different atoms to be the same. 3. For N atoms, bands formed to accommodate 2N electrons band contains 2N states 4. Tight binding since electrons assumed to be associated initially with individual atoms 5. Shape of different bands different, since orbital leading to different bands are different and have different overlap 84

Why? See Rosenberg book 85

The band structure of silicon as a real example Atomic levels broaden into bands The band at 0 ev is the valence band The next band to higher energy is the conduction band Derive from outermost electron states in atomic Si

Summary: Topic 8 1. Tight binding model is alternative approach to understand band formation (intuitive approach starting from atomic orbitals) 2. Degeneracy of levels lifted due to wavefunction overlap 3. Predicts 2N states per band as does periodic potential model 87

Topic 9: Effective Mass, Electrons and Holes We have shown in previous topics that electrons and holes are not scattered by the ions of the crystal lattice (except at the Bragg condition) However the ions and the periodic potential do lead to a measurable change in the properties of the charge carriers: they lead to effective masses which are not equal to the free electron mass We also introduce the concept of holes in this topic 88

Derivation of expression for effective mass At zone boundary v g =?? 89

m*/m e Variation of effective mass with E and k k 90

See diagram on previous page 91

Electrons and holes in electric field 92

Examples holes in semiconductors Partially filled bands in metals: group II elements See Hall effect, cyclotron resonance to determine sign of charge carriers Also note large range of effective masses 93

Pictorial representation of motion of empty states (holes) in electric field Supplement Filled band: no current Remove one electron Current is minus that carried by one electron i.e. -(-e)v = +ev 94

Summary Topic 9 1. Derivation of expression for effective mass for charge carrier in energy band. 2. Variation of m * with k across Brillouin zone 3. Concept of holes, positive mass, positive charge particle. 4. Empty electron state in otherwise filled band 5. Charge transport by electrons and holes 6. Large range of effective masses m 2 d E dk * 2 = 2 95