Condensed Matter Physics J. Ellis (10 Lectures)

Condensed Matter PhysicsJ. Ellis (10 Lectures) Periodic Systems: Overview of crystal structures, the reciprocal lattice. Phonons: Phonons as normal modes – classical and quantum picture. 1D monatomic chain, 1D diatomic chain, examples of phonons in 3D. Debye theory of heat capacity, thermal conductivity of insulators. Electrons in solids: Free electron model: Fermi-Dirac statistics, concept of Fermi level, electronic contribution to heat capacity. Bulk modulus of a nearly free electron metal. Electrical and thermal conductivity. Wiedemann-Franz law. Hall effect. Nearly free electron model: Derivation of band structure by considering effect of periodic lattice on 1-D free electron model. Bloch’s theorem. Concept of effective mass. The difference between conductors, semiconductors and insulators explained by considering the band gap in 2D. Hole and electron conduction. Doping of semiconductors, p and n types, pn junctions – diodes, LEDs and solar cells. Books In general the course follows the treatment in Solid State Physics, J.R. Hook and H.E. Hall (2nd edition, Wiley, 1991). Introduction to Solid State Physics, Charles Kittel (8th edition, Wiley, 2005) is highly recommended. (need not be the latest edition) Another book, generally available in College libraries and may usefully be consulted is The Solid State, Rosenberg H M (3rd edn OUP 1988) Webpage http://www-sp.phy.cam.ac.uk/~je102/

Condensed Matter Physics: Periodic Structures • Course deals with crystalline materials – can be extended later to amorphous materials. • Crystalline structure characterised by set of lattice points – each in equivalent environment, but not necessarily at the position of an atom. • Each lattice point will have associated with it one or more atoms - the ‘basis’. e.g. NaCl • (Mathematically, the lattice would be represented by an array of delta functions, and the crystal described by a convolution of the lattice with a function that described e.g. the electron density associated with the basis.) Structure Lattice Basis = *

Condensed Matter Physics: Unit Cells • Lattice described by a unit cell – which may have one lattice point per unit cell (a ‘primitive’ unit cell) or more than one (‘non-primitive). e.g. for cubic: • Primitive Cubic • Face Centred Cubic (fcc) • Body Centred Cubic (bcc) • How many lattice points per unit cell? Either count those at corners and face centres with weight 1/8 and ½ respectively, or move whole cell so that no lattice points are on the sides/corners, and count lattice points inside the cell. Non primitive unit cell, 4 lattice points per unit cell Primitive unit cell, 1 lattice point per unit cell Non primitive, 2 lattice points per unit cell

Bravais Lattices In 3D there are 14 different lattices – know as ‘Bravais’ lattices. P=primitive I= body centred F=Face centred on all faces A,B,C = centred on a single face Need to remember the P,I, and F forms of the cubic unit cells

c b a Directions • Unit cells characterised by the 3 ‘lattice vectors’ (a,b, and c) that define their edges. e.g. for a face centred cubic (fcc) lattice • Directions given in terms of basis vectors – a direction: would be writen as [u,v,w]. • In a cubic lattice are all related by symmetry. They are together denoted by . Unit Cell Lattice Vectors Non primitive Primitive c b a

z z (111) (120) y y c/l x x b/l b/2 a/l a/l Planes • The notation describing a set of uniformly spaced planes within a crystal is defined as follows: • Assume one of the planes passes through the origin • Look at where the next plane cuts the three axes that are defined by the three lattice vectors. • If the plane cuts the three axes at a/h , b/k , c/l , then the set of planes is described by the Miller indices (h,k,l), and {h,k,l} indicates all planes related to (h,k,l) by symmetry. • If h,k, or l is zero it indicates that the plane is parallel to the respective axis. e.g. (showing only the plane next to one that contains the origin)

Fourier Transforms and The Reciprocal Lattice • 1D periodic functions • A 1D periodic function, f(x)=f(a+x), can be represented as a Fourier series: The wave vectors used, kh are a uniformly separated set of points in 1D wave vector (k) space. • To illustrate how a 3D Fourier series is built up consider the orthorhombic case (a≠b≠ c, 90° between axes) • In 2D, the coefficients Ch vary with y: • But the function is periodic in y, so represent Ch(y) as a Fourier series: • In 3D the Chkvary with z: • And again since the function is periodic in z, Chk (z) can be written as a Fourier series: • Hence the 3D series is: • k vectors, (kh,kk,kl), needed for the Fourier transform form a lattice in 3D reciprocal space known as the RECIPROCAL LATTICE.

The Reciprocal Lattice: the General Case • For a periodic function in 3D with a lattice described by lattice vectors a, b and c, all the wavevectors you need in 3D k space for a 3D Fourier transform representation are: (Always use primitive unit cells.) • The set of G vectors given by all possible integer values of h,k, and l is known as the reciprocal lattice. The G vectors are know as reciprocal lattice vectors. • A periodic function f(r) can then be expressed as the 3D Fourier series. • Since the dot product of a lattice vector (ua+vb+wc) with a reciprocal lattice vector Ghkl is 2p(uh+vk+wl) – an integer multiple of 2p - if you move by a lattice vector the phase of the exponentials remains unchanged giving the same value for f(r) and the correct periodicity in real space.

The Reciprocal Lattice. An Orthorhombic Example • Orthorhombic: a≠b≠ c, 90° between axes, a, b, c form a right handed set. • Reciprocal lattice vectors: • View structure down ‘c’ axis: • If the angles between the a,b, and c axes are not 90° then a axis in real space will not necessarily be parallel to the A axis in reciprocal space. (See hexagonal example later.) Reciprocal Space Lattice Real Space Lattice b B=2p/b a A=2p/a

The Reciprocal Lattice and Miller Index Planes • The first plane (after the plane going through the origin) with a Miller index (h, k, l) goes through the points: • The normal to this plane is parallel to the cross product of two vectors in this plane, and hence to Ghkl : • For a plane wave, wavevector Ghkl, the difference in phase between a point on the plane that goes through the origin, and a point in the plane shown in the diagram above is: (The phase difference between two points separation r is k.r. Ghkl is perpendicular to the planes and so any vector, r joining a pair of points, one in each plane, will do.) • Thus the set of planes with Miller indices (h,k,l) are perpendicular Ghkl, and the phase of a wave, wavevector Ghkl changes by 2p between one plane and the next. The set of planes have the same spacing, therefore, as wavefronts of the wave with wavevector Ghkl z c/l Ghkl y b/k x a/h

Reciprocal Lattice and Miller Index Planes: Orthorhombic Example • Examples of Miller indices and G vectors Reciprocal Space 3rd index undefined as we are looking in 2D Real Space (01•) planes G01• b B a A G000 Reciprocal Space Real Space (02•) planes G02• b B a A

Reciprocal Lattice: Orthorhombic and Hexagonal Examples Reciprocal Space Real Space (12•) planes G12• b B a A Reciprocal Space Real Space (10•) planes G000 B b A a G10• Note:[1] G vector perpendicular to planes, and of length inversely proportional to the plane spacing. [2] If the lines (=planes in 3D) drawn in the figure were ‘wave crests’ then the wavevector of that wave would be the associated G vector

X-ray and Neutron Diffraction • The diffraction of x-rays and neutrons from a solid is used to study structure. • The phase of a wave changes by k.r over distance r • The condition for diffraction from a crystal relates to the scattering wave vectorks, which is the difference: ks = kf - ki between the wavevectors of the outgoing (kf) and incoming (ki) beams. • If the scattering wavevector is equal to a reciprocal lattice vector then since the product of a lattice vector with a reciprocal lattice vector is an integer multiple of 2p, all equivalent points within the crystal (e.g. all identically located atoms) will scatter in phase and give a strong outgoing beam. i.e. the diffraction condition is: ks = Ghkl and: kf = ki + Ghkl phase difference (kf –ki).r=ks.r Scattering objects Extra phase ki.r outgoing wave wavevector kf incoming wave wavevector ki r Extra phase kf.r

kf G ki X-ray and Neutron Diffraction: Energy Conservation • Conservation of energy requires that the incoming and outgoing wavevectors (once the scattering particle is free of the crystal) must be of equal magnitude. • Condition for diffraction neatly represented by Ewald’s sphere construction: both ki and kf must lie on the surface of a sphere, and be separated by a G vector. • It is clear that it is quite possible that for a particular incident condition there is no diffraction from a crystal – the Ewald construction is quite specific on ki and kf . • Diffraction from powders overcomes this by having a large number of crystals in different orientations. Ewald’s Construction

Strong Scattering of Waves in Crystals • Neutrons (provided the sample is typically thinner than 1cm) and x-rays are scattered at most once as they pass through a crystal. • If you try to send a beam of electrons through a crystal it is very strongly scattered – the mean free path depends on energy, but takes a minimum at 50-100eV of about 6Å in a typical metal. • If you imagine starting a beam of electrons inside a crystal with a particular wave vector k, it will quickly be scattered into a set of waves travelling with wavevectors k + Ghkl . • There will then be more scattering, but now, since diffraction simply adds a G vector to the intial wave vector, it will be from one of these new set of waves to another – indeed some may be scattered back into the original wave with wave vector k. • After a while a sort of equilibrium is reached with the rate of scattering out of a particular set of waves equalling the rate of scattering into it. Once this has happened, no further effect of the scattering can be seen, and this explains why despite the large scattering cross sections, as we shall see later, electrons can behave as if they move through a crystal unimpeded. • We shall see later that because electrons moving through a crystal with a certain wave vector (k) can in fact have some of their ‘probability amplitude’ in whole set of associated waves (wavevectors k + Ghkl ), one may have to allow for this by making a correction to the ‘effective mass’ that they seem to have.

Condensed Matter Physics: Phonons • Aims: • Lattice vibrations ‘normal modes’/’phonons’ • Establish concepts by considering modes of a 1-dimensional, harmonic chain, both monatomic and diatomic. • Examples of phonons in a 3D lattice. • Debye theory of heat capacity • 3kB/atom at ‘high’ temperatures. aT3 at low temperatures. • What are high/low temperatures, concept of Debye temperature. • Thermal Conductivity of insulating crystals  aT3 at low temperatures.  aT-1 at high temperatures. • Strong effect of defects and specimen dimensions at low temperatures.

Atomic Motion in a Lattice • In a solid, the motion of every atom is coupled to that of its neighbours – so cannot describe motion atom by atom – use ‘normal mode’ approach instead. • The motion of a ‘harmonic’ system (objects connected by ‘Hook’s law’ springs), can be described as a sum of independent ‘normal modes’ in which the coordinates all oscillate at same frequency and maintain fixed ratios to each other. • Motion of atoms must be described quantum mechanically, but we will use the results that: • The displacement patterns of the classical normal modes are the same as the ratios of the coordinates in the quantum mechanical ones. • The energy of the quantum mechanical modes is expressed in terms of the frequency (w) of the classical mode: • In a solid these quantised normal modes are called phonons.

k k k m m x2 x1 Normal Modes - Classical View: 2 Coordinate Example • Mode 1: Mode 2: • General solution. 1/√2 for normalisation

Normal Modes - Quantum View: 2 Coordinate Example • The frequencies and amplitude ratios are the same as for the classical case, but the energy is quantized Each term has only one variable but their sum is constant, so each must be constant giving two Independent equations

Lattice vibrations • 1-D harmonic chain • Take identical masses, m, separation a connected by springs (spring constant, a): • This is a model limited to “nearest-neighbour” interactions. Equation of motion for the nth atom is: • We have N coupled equations (for N atoms). • Take cyclic boundary conditions – N+1th atom equivalent to first (will be discussed later). • All masses equivalent – so the normal mode solutions must reflect this symmetry and all have the same amplitude (u0) and phase relation to their neighbours, i.e.

-p p Phased Lattice Vibrations: Frequency of Modes • Look for normal mode solutions • Each coordinate has time dependence: • Substitute into equation of motion: • Phase, d, only has unique meaning for a range of 2p: makes most sense to consider w as a function of d over the range –p to p, giving:

Lattice Vibrations: Nature of Modes • Can write the amplitude of the nth atom as:` • Can write the phase difference between successive atoms in terms of a wavevector, conventionally written as q for phonons: • Now it is clear that the modes are waves travelling along the chain of atoms: • The dispersion relation for these waves is: • Since the phase, d, only has unique meaning for the range –p/2 to p/2, q only has a unique value over the range: • Energy stored in mode is i.e. a ground state of energy ħw/2 plus: n phonons each of energy ħw. • Momentum of a phonon turns out to be ħq . • Velocity = w/q(If you can see the wave move you must have formed a wavepacket, so velocity is group velocity) na is the distance x along the chain

The Meaning of phonon wavevector q. • The wavevector q gives the phase shift between successive unit cells. • q is defined on the range where G =2p/a is the smallest reciprocal lattice vector. • q has no meaning between lattice points, so is equivalent to q+G. • Remember: a wave vector that is a reciprocal lattice vector gives a phase shift of 2np between two points separated by a lattice vector. • Free space is uniform, so a phase shift along a wave given by f=kr works for any r. In a crystal, space is not uniform - equivalent points are separated by a lattice vector, and f=kr only has meaning if r is a lattice vector. q+G q Amplitude Phase shift between lattice points is meaningless n

Phonon dispersion • Dispersion curves • w versus q gives the wave dispersion • Key points • The periodicity in q (reciprocal space) is a consequence of the periodicity of the lattice in real space. Thus the phonon at some wavevector, say, q1 is the same as that at q1+nG, for all integers n, where G=2p/a (a reciprocal lattice vector). • In the long wavelength limit (q→0) we expect the “atomic character” of the chain to be unimportant.

Limiting behaviour • Long wavelength limit • dispersion formula • leads to the continuumresult (see IB waves course) • These are conventional sound-waves. • Short wavelength limit • “Atomic character” is evident as the wavelength approaches atomic dimensions q→p/a. l=2a is the shortest, possible wavelength. • Here we have a standing wave w/q=0 q→0 Continuum result Y - Young’s modulus r - density

Momentum of a Phonon: ħq • Need to extend our concept of momentum to something that works for phonons – a so called ‘crystal momentum’. • If, for example, a neutron hits a crystal and creates a phonon, we want a definition of phonon momentum such that momentum will be conserved in the scattering/phonon creation process. • For a static lattice we simply have diffraction, and to get a large scattered intensity all the scatterers have to scatter in phase, i.e. (kf - ki).r = 2np (r is a lattice vector: separation of identical atoms) and for this to be true kf - ki= G. • If the lattice is now distorted by a phonon, the way each atom scatters will be modified by an extra phase term, q.r, so, if the scattered amplitudes are all to add up, the scattering wavevector will have to give an extra phase difference between lattice positions of q.r. i.e: (kf - ki).r = 2np+ q.r and kf - ki = G + q . • This means that on scattering the crystal changes momentum by ħ(G + q). ħG is the momentum transfer due to diffraction from the lattice causing the whole crystal to recoil, and so it is sensible to define the momentum of the phonon as ħq – after scattering either you have created a phonon momentum -ħq or you have annihilated one of momentum ħq. • But you say, you can’t just define momentum anyhow you like – surely it is something that exists and we have to measure it. Not at all. Momentum and energy entered physics as constructions created to make the maths of doing physics easy. Consider potential energy – to what measurable ‘real’ quantity can you add an arbitrary offset and everything is still ok? Why is energy conserved ?– because we carefully define all forms of energy so that it is, at least that is how the idea started.

Momentum of a Phonon: ħq,but is it reasonable? (non examinable) • The problem is that if you really do have a infinite uniform wave in an infinite lattice, there are as many atoms going forwards as backwards and it carries no momentum. • The total (classical) momentum is carried somehow by all the atoms in the crystal – what we are trying to do is divide it notionally between momentum carried by the phonon and that associated with motion of the centre of mass – so its complicated. • However, if you make a wavepacket out of a small spread of wavevectors, then if you give the wavepacket an energy ħw and add up the momentum associated with all the vibrating particles they don’t quite cancel out, but do indeed give ħq. • Effectively, after the neutron scatters the whole crystal starts to move, carrying momentum ħG, and inside the crystal is a wavepacket of vibrations travelling through the crystal that carries a net extra momentum of ħq. • If you are considering scattering of a neutron, you are not considering an infinite crystal, so one can reconcile normal momentum with crystal momentum. • To understand the infinite case (non quantum mechanically) – you have to take the limit of the wavepacket going to infinite length – which is approaching infinitiy in a different way from saying that we have uniform oscillations throughout the crystal and let the crystal size go to infinity, so you get a different result for the momentum when you go to the limit in a different way. • If an inifinte lattice has to supply ħG or ħq of momentum it does not change the state of motion of the lattice (i.e. the lattice does not start to move) because it has infinite mass.

1st Brillouin Zone • Periodicity: All the physically distinguishable modes lie within a single span of 2p/a. • First Brillouin Zone (BZ) • we chose the range of q to liewithin |q| p/a. This is the 1st BZ. • Number of modes must equal the number of atoms, N, in the chain and for finite N the allowed q values are discrete, separation 2p/Na (see ‘waves in a box’ later). • To Summarise: Each mode (at particular q) is a quantised, simple-harmonic oscillator, E= ħw(n+1/2). Phonons have particle character – bosons: each mode can have any number of phonons in it with: Energy=ħw, Mom.= ħq, Velocity = w/q. The unique modes lie within the first B.Z.. 1st Brillouin zone (shaded)

Measurement of Phonons • Basic principle: • Need a probe with a momentum and energy comparable to that of the phonons e.g. thermal energy neutrons for bulk, and He atoms at surfaces. X-rays can have correct wavelength, but the energy is so high it is hard to resolve the small changes induced by phonon interactions. (At l=1Å, energy is 12.4keV – typical phonon energies are up to 40meV) • Particle hits the lattice and creates/annihilates phonons. • Illuminate sample with a monochromatic beam – incident wavevector ki • Energy analyse scattered signal – peaks in signal correspond to single phonon creation/annihilation occurring at a particular kf. • Use of conservation laws • Energy of phonon (+ = creation, - = annihilation): • Crystal momentum conservation for phonon creation: • Crystal momentum conservation for phonon annihilation:

Rotating Disk Chopper Measurement of Phonons II • To measure energy of probe can use time of flight techniques – e.g. helium atom scattering (HAS): • Time flight of individual atoms through apparatus – to determine energy transfer on scattering. HAS ToF data for Cu(100) surface Single phonon creation peaks Relative Intensity Elastic peak Phys. Rev. B 48, 4917, (1993) Energy Transfer/meV

Diatomic lattice • Technically a lattice with a basis • proceeding as before. Equations of motion are:Trial solutions:substituting giveshomogeneous equations require determinant to be zero giving a quadratic equation for w2. mA mB Two solutions for eachq

Acoustic and Optic modes • Solutions • q→0: • Optic mode (higher frequency) • Acoustic mode (lower frequency) Effective massm w=(2a/mB) w=(2a/mA) Periodic: all distinguishable modes lie in |q|<p/2a

Displacement patterns • Displacements shown as transverse to ease visualisation. • Acoustic modes: Neighbouring atoms in phase • Optical modes: Neighbouring atoms out of phase • Zone-boundary modes • q=p/2a; l=2p/q=4a (standing waves) • Higher energy mode – only light atoms move • Lower energy mode – only heavier atoms move

Origin ofoptic and acoustic branches • Effect of periodicity • The modes of the diatomic chain can be seen to arise from those of a monatomic chain. Diagrammatically: Monatomic chain, period a period in q is p/a for diatomic chain Modes with q out- side new BZ period ‘backfolded’ into new BZ by adding ± G=p/a Acoustic and optical modes Energy of optical and acoustic modes split if alternating masses different

Diatomic chain:summary • Acoustic modes: • correspond to sound-waves in the long-wavelength limit. Hence the name. w→0 as q→0 • Optical modes: • In the long-wavelength limit, optical modes interact strongly with electromagnetic radiation in polar crystals. Hence the name. • Strong optical absorption is observed. (Photons annihilated, phonons created.) w→finite value as q→0 • Optical modes arise from folding back the dispersion curve as the lattice periodicity is doubled (halved in q-space). • Zone boundary: • All modes are standing waves at the zone boundary, w/q= 0: a necessary consequence of the lattice periodicity. • In a diatomic chain, the frequency-gap between the acoustic and optical branches depends on the mass difference. In the limit of identical masses the gap tends to zero.

Phonons in 3-D crystals: Monatomic lattice • Example: Neon, an f.c.c. solid: • Inelastic neutron scattering results in different crystallographic directions • Many features are explained by our 1-D model: • Dispersion is sinusoidal (nearest neighbour. interactions). • All modes are acoustic (monatomic system) Phys. Rev. B 11, 1681, (1975)

Neon:a monatomic, f.c.c. solid • Notes: (continued) • There are two distinct types of mode: • Longitudinal (L), with displacements parallel to the propagation direction, • These generally have higher energy • Transverse (T), with displacements perpendicular to the propagation direction • These generally have lower energy • They are often degenerate in high symmetry directions (not along (0)) • Minor point (demonstrating that real systems are subtle and interesting, but also complicated): • L mode along (0) has 2 Fourier components, suggesting next-n.n. interactions (see Q 3, sheet 1). In fact there are only n.n. interactions • The effect is due to thefcc structure. Nearest-neighbour interactions from atom, A (in plane I) join toatom C (in plane II) and to atom B (in plane III) thuslinking nearest- and next-nearest-planes.

Phonons in 3-D crystals: Diatomic lattice • Example: NaCl, has sodium chloride structure! • Two interpenetrating f.c.c. lattices • Main points: • The 1-D model gives several insights, as before. There are: • Optical and acoustic modes (labels O and A); • Longitudinal and transverse modes (L and T). • Dispersion along () is simplest and most like our 1-D model • () planes contain, alternately, Na atoms and Cl atoms (other directions have Na and Cl mixed) Phys. Rev. 178 1496, (1969)

NaCl phonons • Notes, continued… • Note the energy scale. The highest energy optical modes are ~8 THz (i.e. approximately 30 meV). Higher phonon energies than in Neon. The strong, polar bonds in the alkali halides are stronger and stiffer than the weak, van-der-Waals bonding in Neon. • Minor point: • Modes with same symmetry cannot cross, hence the avoided crossing between acoustic and optical modes in (00) and (0) directions. • Ignore the detail for present purposes

Conservation Laws and Symmetry • Lagrangian Mechanics • Newton 2 normally considered in Cartesian coordinates: ‘F=ma’ • Can generalise to non-Cartesian coordinates, but now write equations of motion in terms of derivatives of the Lagrangian: L=K.E. – P.E. . Field of ‘analytical dynamics’ based on this idea. • Conservation Laws • A key result of analytical dynamics is Noether’s theorem – for every symmetry in the Lagrangian (i.e. in the system), there is an associated conservation law. e.g. it turns out that: • If the system’s behaviour is independent of the time you set it going, energy is conserved. • If the system’s behaviour is independent of where it is in space , momentum is conserved. • If the system’s behaviour is independent of the its angular orientation, angular momentum is conserved. • In a crystal – space is no longer uniformbut has a new symmetry – its periodic, so the law of conservation of momentum is replaced by a new law – the conservation of ‘crystal momentum’ in which momentum is conserved to within a factor of ħG. E.g. • Diffraction: wavevector allowed to change by factors of G • Phonon creation: • Adding a G vector to a phonon’s wavevector does not change its properties, but its crystal momentum changes by ħG

A mathematical aside (for interest - non examinable). • L=K.E.-P.E (e.g. S.H.O. ) • Equations of motion: Euler-Lagrange equations: SHO: • Conjugate momentum: (SHO: ) • If L is independent of qi: • Energy conservation? Under many circumstances the Hamiltonian H (defined as ) is the energy, and since , if L does not have explicit time dependence , H and hence energy is conserved. Noether’s theorem

The Use of Conservation Laws • What do conservation laws tell you? • Conservation laws tell you what is allowed to happen – it is not possible to have an outcome of an event that violates a valid conservation law. • Unless conservation laws permit only one outcome, they do not tell you what will actually happen, nor how fast it will happen. e.g. conservation of crystal momentum inside a periodic solid tells you what possible outgoing momenta a diffracted particle may have, ( ) but they do not tell you how intense the outgoing beams will be.

Thermal Properties of Insulating Crystals: Heat Capacity • Thermal energy is stored in the phonons • Need to know how much energy is stored in each mode. • Need to know how many phonon modes there are. • Need to sum the thermal energy over all modes • Heat capacity is then the derivative of the thermal energy. • Energy stored in a phonon ‘normal mode’ • Each mode has an energy E=ħw(n+ ½) where n is the number of phonons in the mode. • The factor of ½ is the ‘zero point’ energy – it cannot be removed. Since thermal energies are taken to be zero in the ground state, it will be ignored in this treatment. • It the solid is in thermal contact with some fixed temperature ‘reservoir’ then the probability of the mode having n phonons relative to the chance of it having none is given by a Boltzmann factor: Pn= exp(-n ħw/kBT)

Energy/normal mode, continued • Calculate average energy stored in a particular normal mode (ith ) by averaging over all possible values of n (0 to ∞). Planck’s formula for a single oscillator

Heat Capacity at High Temperatures • Low temperature (kBT<<ħw) limit of energy: • High temperature limit of energy (1>>ħw/ kBT) • How many phonon modes? • If a crystal contains N atoms, you need 3N coordinates to describe position of all N atoms and so there will be 3N normal modes. • Thermal behaviour of whole crystal at high temperatures: • Since each mode stores kBT of energy at ‘high’ temperatures, and there are 3N modes, then the total energy stored at high temperatures is 3NkBT and the heat capacity for kBT >>ħwi is 3NkB. (Basis of Dulong and Petit’s law, 1819 – heat capactiy/atomic weight constant.)

Debye Theory: The Aim • Thermal behaviour of whole crystal at intermediate and low temperatures. • At ‘non-high’ temperatures, Ei depends on ħwi so we need a way of summing the contribution all the modes: • The first step is to convert the sum to an integral: where g(w)=dN/dw is the ‘density of states and g(w)dw gives the number of phonon states dNwith energies lying between w and w+dw. • The actual g(w) is complicated – the Debye theory of heat capacity works by producing a simplified model for g(w) so that the integral for Etotal can be performed.

Boundary Conditions and Models for g(w): Permitted k values. • Reflecting B.C. • Reflecting boundary conditions give standing wave states. • At boundary may have node (photons, electrons) or antinode (phonons). • Cyclic boundary conditions • N+1th atom equivalent to 1st. • Travelling wave solutions. • 1D you can wrap into a circle, but cyclic b.c.harder to justify in 2 and 3D. • Can consider repeating your block on N atoms with identical units to fill infinite space and requiring all blocks to have identical atomic displacement patterns. Ok classically, but hard to get the quantum mechanics correct. • ‘Infinite’ extent: • As soon as you use the modes derived you make a wavepacket of some sort with zero amplitude at infinity, which fits any b.c. at infinity, but this method does not give density of states. Cyclic B.C. Reflecting B.C. n=3 n=3 n=2 n=2 n=1 n=1 A

Debye Model: g(w) for ‘Waves in a Box’ • For small values of k (long wavelengths) phonons look like sound waves – with a linear dispersion relation w=vsk where vs is a mean speed of sound (see discussion later). • The Debye model assumes this is true for all wavelengths – not just long ones – i.e. it ignores the structure in g(w) due to the atomic nature of the material. • g(w) is calculated by assuming that the crystal is a rectangular box of side lengths A,B,C. We use reflecting b.c., though cyclic b.c. give same results • In each dimension the there must be a whole number of half wavelengths across the box so as to fit the boundary conditions, i.e. in each direction A=nl/2 and k=np/A. The total wavevector of the phonon must be: • Volume per state in k space is p3/(ABC) i.e. p3/V where V is the volume of the box. • (k not q is used in this derivation because the idea of waves in a box applies to many problems in physics – including black body radiation and the free electron model of a solid, and by convention k is used in these derivation.)

g(w) for ‘Waves in a Box’ II All states in the shell have same |k| • First, work out g(k) from no. of states, dN, that have a wavevector of magnitude between k and k +dk.These states lie within the positive octant of a spherical shell of radius k and thickness dk. (k +ve for standing waves.) • For each phonon mode there are two transverse and one longitudinal polarisation, i.e. 3 modes per point in k space. • For cyclic BC, states 2x as far apart –vol/state = 8p3/V but require full shell, not just +ve octant, – net result same g(k). • Can show (Wigner) results independent of shape of box. States uniformly distributed in k-space 1 state “occupies a volume” (p3/ABC) Volume of shell 3 Polarisations/k state Vol. of one state

No. of phononsin dw at w Energy per phonon(Planck formula) Internal Energy in Debye Model • Heat capacity follows from differentiating the internal energy (as usual). • For the present we will ignore the zero point motion. • Need to make sure you integrate over the correct number of modes – use the fact that if there are N atoms in the crystal (volume V) then there are 3N modes. Debye suggested simply stopping the integral (at the ‘Debye frequency’, wD) once 3N modes have been covered, i.e. • Internal energy • Hence: (We can now see that the appropriate mean velocity is where vL and vT are the longitudinal and transverse sound wave velocities)

Condensed Matter Physics J. Ellis (10 Lectures)