Quantum Condensed Matter Physics

Quantum Condensed Matter Physics Part II Lent/Easter 2012 Lent term notes 1 This handout complements your notes and the standard text books. The handout is not the course! May 7, 2012 1 based on a comprehensive handout by P B Littlewood 2 Contents 1 Fermi and Bose gases 7 1.1 Free electron gas in three-dimensions . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Fermi surface, and density of states . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Thermal properties of the electron gas . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Lattice dynamics and phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Lattice specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Electrodynamics of metals 17 2.1 Scattering and electrical conductivity of metals . . . . . . . . . . . . . . . . . . 17 2.2 High frequency response – optical conductivity of metals . . . . . . . . . . . . . 22 2.3 Screening and Thomas-Fermi theory . . . . . . . . . . . . . . . . . . . . . . . . 25 3 The variety of condensed matter 31 3.1 Types of solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 The binding of crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Complex matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Periodic Solids and Diffraction 41 4.1 The description of periodic solids . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 The reciprocal lattice and diffraction . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Diffraction conditions and Brillouin zones . . . . . . . . . . . . . . . . . . . . . . 44 5 Electronic structure 1 47 5.1 Tight binding: Linear combination of atomic orbitals . . . . . . . . . . . . . . . 47 5.2 Diatomic molecule revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Linear chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.4 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.5 Linear chain revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.6 LCAO method in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 4 CONTENTS 5.7 Periodic boundary conditions and counting states in 3 dimensions . . . . . . . . 52 6 Electronic structure 2 55 6.1 Nearly free electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.2 Plane wave expansion of the wavefunction . . . . . . . . . . . . . . . . . . . . . 55 6.3 The Schrödinger equation in momentum space . . . . . . . . . . . . . . . . . . . 56 6.4 One-dimensional chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.5 Pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7 Band theory of metals and insulators 61 7.1 Bands and Brillouin zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.2 Metals and insulators in band theory . . . . . . . . . . . . . . . . . . . . . . . . 62 7.3 Examples of band structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8 Experimental probes of the band structure 69 8.1 Quantum oscillations – de Haas van Alphen effect . . . . . . . . . . . . . . . . . 69 8.2 Optical transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.3 Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.4 Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 9 Semiclassical dynamics 81 9.1 Wavepackets and equations of motion . . . . . . . . . . . . . . . . . . . . . . . . 81 9.2 Electrons and holes in semiconductors . . . . . . . . . . . . . . . . . . . . . . . 84 CONTENTS 5 Preface Books There are many good books on solid state and condensed matter physics, but the subject is rich and diverse enough that each of these contains both much more and much less than the topics covered in this course. The two classic textbooks are Kittel, and Ashcroft and Mermin. These are both at the correct level of the course, and have the virtue of clear exposition, many examples, and lots of experimental data. Slightly more concise, though in places a little more formal is Ziman. Grosso and Parravicini has a somewhat wider coverage of material, but much of it goes well beyond the level of detail required for this course. Marder is at about the right level (though again with more detail than we shall need), and has a nice blend of quantum properties with statistical and classical properties. A well illustrated modern treatment of most topics in this course is also given by Ibach and Lüth. OUP have recently issued a series of short texts in condensed matter physics. They are more detailed than needed for this course, but are quite accessible and excellent for reference. The most relevant for this course is Singleton. • C.Kittel, Introduction to Solid State Physics, 7th edition, Wiley, NY, 1996. • N.W.Ashcroft and N.D.Mermin, Solid State Physics, Holt-Saunders International Editions, 1976. • J.M.Ziman, Principles of the Theory of Solids, CUP, Cambridge, 1972. • H. Ibach and H. Lüth, Solid State Physics, Springer 1995. • J. Singleton, Band Theory and the Electronic Properties of Solids, OUP 2001. • M.P. Marder, Condensed Matter Physics, Wiley, NY, 2000. Covers both quantum matter and mechanical properties. • G.Grosso and G.P.Parravicini, Solid State Physics, AP, NY, 2000. A wide coverage of material, very bandstructure oriented, very detailed. • A very good book, though with a focus on statistical and “soft” condensed matter that makes it not so relevant for this course, is P.M.Chaikin and T.Lubensky, Principles of Condensed Matter Physics, CUP, Cambridge, 1995. These notes Treat these notes with caution. If you are looking for text book quality, then you should look at text books. Polished and optimised treatments of the topics covered here have been published in a number of excellent books, listed above. For much of its duration the course follows the book by Singleton, and where it does not, the books by Ashcroft&Mermin, as well as Kittel, give essential support. Reading up in text books is not only useful revision of the lecture material, it also serves to give important background information and context, which is difficult to provide 6 CONTENTS within the scope of the lectures. Now that you made it into the second half of the third year, you should really give that a try! What is the purpose of these notes, then? (i) They may help you to fill in some gaps or make corrections in your personal lecture notes, when the lecture moved too quickly to keep accurate notes. (ii) They may contain some information which was only superficially touched on during the lectures but not explicitly written down. (iii) They may contain some alternative approaches, which were not given in the lectures but which may be interesting to know about, as understanding comes from combining and reconciling many approaches to the same topic. The notes have a strictly complementary function. Do not attempt to learn condensed matter physics from these notes alone. In many places, treatments given in the lectures were much simpler and more direct than what you will find in the notes. The lecture overheads together with your personal notes from the lectures form the backbone of this course – and where they do not suffice, text books and these notes as background reading material may help. The notes are not to be memorised for examination: often they include detailed derivations that are there to satisfy the curious, for completeness, and for background. The lectures will be presented using more qualitative and physical descriptions. In a few places, and particularly where they discuss material that is not easy to find collected in textbooks, the notes are much more lengthy. Material which is explicitly non-examinable is placed in small type; but in general, no detailed derivations will be required for examination. You may find it worthwhile, however, to work through some of this at least once. Chapter 1 The Fermi and Bose gases 1.1 Free electron gas in three-dimensions Consider a free electron gas, confined to a three-dimensional box of side L. The free particle Schrodinger equation is h̄2 − 2m ∂2 ∂2 ∂2 + + ψ(r) = ǫψ(r) ∂x2 ∂y 2 ∂y 2 ! (1.1) which has the following eigenstates: ψk (r) = N sin(kx x) sin(ky y) sin(kz z) with energy ǫk = h̄2 |k|2 2m (1.2) (1.3) Owing to the restriction to the box (0 < x < L, 0 < y < L, 0 < z < L)) the allowed values of k are discrete. π (1.4) k = (nx , ny , nz ) L with nx , ny , nz positive integers. It is more convenient to introduce wavefunctions that satisfy periodic boundary conditions, namely ψ(x + L, y, z) = ψ(x, y, z) (1.5) and similarly for y and z directions. These are of the form of a plane wave ψk (r) = exp(ik · r) (1.6) where the eigen-energies are identical to Eq. (1.4) but the restriction on momentum being k= 2π (nx , ny , nz ) L (1.7) with nx , ny , nz positive or negative integers. Together with the spin quantum number m, the components of k are the good quantum numbers of the problem. 7 8 CHAPTER 1. FERMI AND BOSE GASES 1.2 Fermi surface, and density of states In the ground state at zero temperature, the fermi gas can then be represented by filling up all the low energy states up to a maximum energy ǫF (the fermi energy) corresponding to a sphere of radius the fermi momentum kF in k − space. Each triplet of quantum numbers kx , ky , kz accounts for two states (spin degeneracy) and occupies a volume (2π/L)3 . The total number of occupied states inside the fermi sphere is N =2· 4/3πkF3 (2π/L)3 (1.8) so that the fermi wave-vector is written in term of the electron density n = N/V as kF = (3π 2 n)1/3 (1.9) We are often interested in the density of states, g(E), which is the number of states per unit energy range. Calculate it by determining how many states are enclosed by a thin shell of energy width dE, viz. g(E)dE = 2 · Volume of shell in k − space 4πk 2 dk =2· Volume of k − space per state (2π)3 /V hence , (1.10) 1 V V m 2mE 2 2 dk g(E) = 2 . (1.11) 4πk = (2π)3 dE π 2 h̄2 h̄2 The factor of 2 is for spin degeneracy. Often, the density of states is given per unit volume, so the factor of V disappears.  1.3 Thermal properties of the electron gas The occupancy of states in thermal equilibrium in a fermi system is governed by the fermi distribution 1 (1.12) f (E) = (E−µ)/k T B e +1 where the chemical potential µ can be identified (at zero temperature) with the fermi energy EF of the previous section. The number density of particles is n = N/V 2 X 1 X f (Ei ) = f (ǫk ) V i V k 1 Z dkf (ǫk ) = 3 Z4π = = dE g(E)f (E) (1.13) The internal energy density u = U/V can then be written in the same fashion: u= Z dE Eg(E)f (E) (1.14) 9 1.3. THERMAL PROPERTIES OF THE ELECTRON GAS Eq. (1.14) will be used to derive the electronic specific heat cv = ∂u/∂T |v at constant volume. The estimation is made much simpler by realising that in almost all cases of interest, the energy scale set by temperature kB T (≈ 0.025 eV at room temperature) is much less than the fermi energy EF (a few eV in most metals). From Eq. (1.14) ∂f (E) (1.15) ∂T Notice that the fermi function is very nearly a step-function, so that the temperature-derivative is a function that is sharply-peaked for energies near the chemical potential. The contribution to the specific heat then comes only from states within kB T of the chemical potential and is much less than the 3/2kB per particle from classical distinguishable particles. From such an argument, one guesses that the specific heat per unit volume is of order cv = Z dE Eg(E) cv ≈ N kB T kB V EF (1.16) Doing the algebra is a little tricky, because it is important to keep the number density to stay fixed (Eq. (1.13)) — which requires the chemical potential to shift (a little) with temperature since the density of states is not constant. A careful calculation is given by Ashcroft and Mermin. But to the extent that we can take the density of states to be a constant, we can remove the factors g(E) from inside the integrals. Notice that with the change of variable x = (E − µ)/kB T   ex 1 dµ x df (1.17) × = x + dT (e + 1)2 T kB T dT The number of particles is conserved, so we can write Z dn ∂f (E) = 0 = g(EF ) dE dT ∂T (1.18) which on using Eq. (1.17) becomes ∞   1 dµ x ex . × + 0 = g(EF )kB T dx x (e + 1)2 T kB T dT −∞ Z The limits can be safely extended to infinity: the factor approximation dµ/dT = 0. ex (ex +1)2 To the same level of accuracy, we have Z ∂f (E) cv = g(EF ) dE E ∂T Z ∞ dx (µ + kB T x) = g(EF )kB T = 2 g(EF )kB T 2 = Z −∞ ∞ dx −∞ (1.19) is even, and hence at this level of x ex x 2 (e + 1) T x2 ex (ex + 1)2 π 2 k T g(EF ) 3 B (1.20) The last result is best understood when rewritten as cv = π 2 kB T nkB 2 EF confirming the simple argument given earlier and providing a numerical prefactor. (1.21) 10 CHAPTER 1. FERMI AND BOSE GASES The calculation given here is just the leading order of an expansion in powers of (kB T /EF )2 . To next order, one finds that the chemical potential is indeed temperature-dependent:   1 πkB T 2 4 µ = EF 1 − ( ) + O(kB T /EF ) (1.22) 3 2EF but this shift is small in a dense metal at room temperature, and may usually be neglected. 1.4 Lattice dynamics and phonons One-dimensional monatomic chain Our model consists of identical atoms connected by springs, shown in Fig. 1.1 Figure 1.1: A one-dimensional linear chain. The atoms are shown in their equally spaced equilibrium conditions in the top row, and with a periodic distortion below. The bottom figure plots the displacements un as arrows, and the curve shows how this is a sine-wave of period 6a, in this case. In equilibrium, the atoms are uniformly spaced at a distance a, and we now look for oscillations about the equilibrium position. We assume the crystal is harmonic, so that the spring restoring force is linearly dependent upon the extension. Then if we take the displacement of the nth atom (which is at the point rn = na) to be un , its equation of motion is m ∂ 2 un = K(un+1 − un ) + K(un−1 − un ) ∂t2 (1.23) We guess that the solution is a wave, of the form un (t) = uo cos(qrn − ω(q)t) (1.24) Here the wavelength of the wave is λ = 2π/q, and the period is T = 2π/ω(q); to check that this is a solution, and to determine the frequency we substitute in the equation of motion. To do this is left as an exercise, and a few lines of algebra will show that the solution Eq. (1.24) exists provided that qa mω 2 (q) = 2K(1 − cos(qa)) = 4K sin2 ( ) (1.25) 2 so that qa ω(q) = 2(K/m)1/2 sin( ) (1.26) 2 1.4. LATTICE DYNAMICS AND PHONONS 11 Figure 1.2: Dispersion relation between frequency and wavevector for a one-dimensional monatomic chain Eq. (1.25) is called a dispersion relation — the relation between the frequency of the mode and its wavevector, or equivalently the relationship between the wavelength and the period. The wavevector q is inversely related to the wavelength; note that for long wavelength modes (i.e. q → 0), the relationship is linear, viz ω(q) = (K/m)1/2 (qa) (1.27) which is the same as for a wire with tension Ka and density m/a. In the long wavelength limit, we have compressive sound waves that travel with a velocity v = a(K/m)1/2 . Because this kind of wave behaves like a sound wave, it is generally called an acoustic mode. The dispersion is not linear for larger values of q, and is in fact periodic (Fig. 1.2). The periodicity can easily be understood by reference to Eq. (1.24). Suppose we choose q = 2π/a. Note then that 2π qrn = × na = 2πn (1.28) a so that all the atoms displace together, just as if q = 0. In general it is straightforward to show that if one replaces q by q + integer × 2π/a, then the displacements are unchanged – so we may simplify our discussion by using only q vectors in the range π π − ≤q≤ . (1.29) a a This is called the first Brillouin zone. One-dimensional diatomic chain The monatomic chain contains only acoustic modes, but the phonon spectrum becomes more complex if there are more atoms per unit cell. As an illustration, we look at the diatomic chain. For simplicity, we use again a phenomenological model of balls and springs, but now with two different atoms in the unit cell, two different masses and two different spring constants (see Fig. 1.3). We can now write down two equations of motion, one for each type of atom: ∂ 2 unA = K(unB − unA ) + K ′ (un−1,B − unA ) ∂t2 ∂ 2 unB mB = K ′ (un+1A − unB ) + K(un,A − unB ) 2 ∂t mA (1.30) 12 CHAPTER 1. FERMI AND BOSE GASES Figure 1.3: Diatomic chain The solution of this is a little more complicated than before, but we can now intuitively see that there ought to be a new type of phonon mode by considering a particular limit of the parameters. Suppose the two atoms are quite strongly bound together in pairs, as sketched in the figure above: then we might expect that K ≫ K ′ , and to a first approximation the pairs can be treated as independent molecules. (We will also simplify the analysis by taking mA = mB = m.) Then every molecule will have a vibrational mode where the two atoms oscillate out of phase with each other with a frequency ωo2 = 2K/m . (1.31) The corresponding coodinate which undergoes this oscillation is uopt (q = 0) = uA − uB (1.32) where I have explicitly remarked that this is at q = 0 if each molecule undergoes the oscillation in phase with the next. We can of course make a wavelike solution by choosing the correct phase relationship from one unit cell to the next — as sketched in Fig. 1.4, but if K ′ ≪ K this will hardly change the restoring force at all, and so the frequency of this so-called optical phonon mode will be almost independent of q. Figure 1.4: Dispersion of the optical and acoustic phonon branches in a diatomic chain, and a schematic picture of the atomic displacements in the optical mode at q=0 There are now two branches of the dispersion curve, along one of which the frequency vanishes linearly with wavevector, and where the other mode has a finite frequency as q → 0(see Fig. 1.5). The name “optical” arises because at these long wavelengths the optical phonons can interact (either by absorption, or scattering) with light, and are therefore prominent features in the absorption and Raman spectra of solids in the infrared spectrum. Phonons in three-dimensional solids The descriptions above are not too hard to generalise to three- dimensional solids, although the algebra gets overloaded with suffices. 1.4. LATTICE DYNAMICS AND PHONONS 13 Figure 1.5: Pattern of atomic displacements for an acoustic and an optical phonon of the same wavevector. Rather than a one-dimensional wavevector k corresponding to the direction of the 1D chain, there is now a three-dimensional dispersion relation ω(k), describing waves propagating in different directions. Also, there are not just compressional waves, but also transverse, or shear waves, that have a different dispersion from the longitudinal (compressional) waves. (These exist in a crystal in any dimension, including our 1D chain, where they can be visualised with displacements perpendicular to the chain direction.) Quite generally, for each atom in the unit cell, one expects to find three branches of phonons (two transverse, and one longitudinal); always there are three acoustic branches, so a solid that has m atoms in its unit cell will have 3(m − 1) optical modes. And again, each optical modes will be separated into two transverse branches and one longitudinal branch.1 Density of states Just as for the electron gas problem we need to write down the density of states for phonons. First, we need to count how many modes we have and understand their distribution in momentum space. In the 1D monatomic chain containing N atoms (assume N very large), there are just N degrees of freedom (for the longitudinal vibration) and therefore N modes. This tells us (and we can see explicitly by looking at boundary conditions for an N-particle chain) that the allowed k-points are discrete, viz kn = N N N 2π n ; n = (− , − + 1, ..., ] , L 2 2 2 (1.33) so that k runs from −π/a to π/a, with a = N/L, the lattice constant. Notice this is the same spacing of k-states for the electron problem, and the only difference is that because the atoms are discrete, there is a maximum momentum (on the Brillouin zone boundary) allowed by counting degrees of freedom. By extension, in three dimensions, each branch of the phonon spectrum still contains N states in total, but now N = L3 /Ωcell with Ωcell the volume of the unit cell, and L3 = V the volume of the crystal. The volume associated with each allowed k-point is then (2π)3 ∆k = L3 1 (1.34) The separation between longitudinal and transverse is only rigorously true along lines of symmetry in k-space. 14 CHAPTER 1. FERMI AND BOSE GASES Figure 1.6: Comparison of Debye density of states (a) with that of a real material (b). There are 3 acoustic branches, and 3(m − 1) optical branches. It is convenient to start with a simple description of the optical branch(es), the Einstein model, which approximates the branch as having a completely flat dispersion ω(k) = ω0 . In that case, the density of states in frequency is simply DE (ω) = N δ(ω − ω0 ) . (1.35) We have a different results for the acoustic modes, which disperse linearly with momentum as ω → 0. Using a dispersion ω = vk, and following the earlier argument used for electrons, we get the Debye model 4πk 2 dk V ω2 DD (ω) = (1.36) = 2 3 . (2π/L)3 dω 2π v Of course this result cannot apply once the dispersion curves towards, the zone boundary, and there must be an upper limit to the spectrum. In the Debye model, we cut off the spectrum at a frequency ωD , which is determined so that the total number of states (N ) is correctly counted, i.e. by choosing Z ωD 0 dωDD (ω) = N (1.37) which yields 6π 2 v 3 N . (1.38) V Notice that this corresponds to replacing the correct cutoff in momentum space (determined by intersecting Brillouin zone planes) with a sphere of radius 3 ωD = kD = ωD /v . 1.5 (1.39) Lattice specific heat Phonons obey Bose-Einstein statistics, but their number is not conserved and so the chemical potential is zero, leading to the Planck distribution n(ω) = 1 . exp(h̄ω/kB T ) − 1 (1.40) 15 1.5. LATTICE SPECIFIC HEAT The internal energy is U= Z For the Einstein model UE = and the heat capacity CV = ∂U ∂T ! dωD(ω)n(ω)h̄ω N h̄ωo eh̄ωo /kB T − 1 = N kb V h̄ωo kB T !2 (1.41) (1.42) eh̄ωo /kB T . (eh̄ωo /kB T − 1)2 (1.43) At low temperatures, this grows as exp−h̄ωo /kB T and is very small, but it saturates at a value of N kB (the Dulong and Petit law) above the characteristic temperature θE = h̄ωo /kB .2 At low temperature, the contribution of optical modes is small, and the Debye spectrum is appropriate. This gives Z ωD h̄ω V ω2 . (1.44) UD = dω 2 3 h̄ω/k T B 2π v e −1 0 Power counting shows that the internal energy then scales with temperature as T 4 and the specific heat as T 3 at low temperatures. The explicit formula can be obtained as CV = 9N kB  T θD 3 Z θ /T D 0 dx x4 e x , (ex − 1)2 (1.45) where the Debye temperature is θD = h̄ω/kB . We have multiplied by 3 to account for the three acoustic branches. 2 This is per branch of the spectrum, so gets multiplied by 3 in three dimensions 16 CHAPTER 1. FERMI AND BOSE GASES Chapter 2 Electrodynamics of metals 2.1 Scattering and electrical conductivity of metals Starting from minimal assumptions, we will model electrical conduction in metals as a result of acceleration of electrons, on the one hand, balanced by scattering processes, on the other. A crude approach might start from an equation of motion for the individual electrons: v̇ = mq E − v/τ , where q and m are the charge and mass of an electron, and τ denotes a relaxation time, during which an electron would tend to lose its velocity in the absence of an external force. This starting point leads to the same results as the route described below, but it also leads to a serious conceptual problem: in reality, the individual electrons do not relax into zero velocity states. Rather, most of them are always in high wavenumber states, which correspond to velocities of up to about 1% of the speed of light. 2.1.1 Hydrodynamic formulation The alternative hydrodynamic formulation discussed here consists in considering fields, which vary slowly with position and time and arise from coarse-graining (averaging over small lengthand time-scales) the microscopic electronic coordinates. We need three such fields: • ρ(r, t), the charge density = qn(r, t). • v(r, t), the average (or drift) velocity of the electron system. And from the above two: • j(r, t) = ρv, the current density or flow field. The drift velocity v arises from averaging the actual electron velocities over an (unspecified) volume, which has to be smaller than the wavelength of a spatially varying applied field. The current density has to obey the continuity equation ρ̇ = −∇j = −∇ (ρv) 17 (2.1) 18 CHAPTER 2. ELECTRODYNAMICS OF METALS In the limit of long wavelengths, we can neglect this build-up of charge density and assume ρ̇ = 0, i.e. an incompressible fluid. This gives for the rate of change of the current density: d j ≃ ρv̇ dt (2.2) The drift velocity relates to the total momentum of the electron system via v = Npm , where N is the number of electrons in the system and m is the electron mass. The momentum, in turn, changes in the presence of an applied electric or magnetic field. If there were no collisions, which can remove momentum from the electron system, then we would have ṗ = N f (t) = N mq (E + v × B) 1 , where again N is the total number of electrons, f is the external force acting on an individual electron, and q is the charge of an electron. E and B are the applied electric and magnetic field. Relaxation time Collisions, or scattering, will introduce a further term, which represents the decay of the electron momentum in the absence of an external force. Note that electron-electron collisions do not give rise to a decay of momentum in any obvious way: they would appear to conserve the momentum of the electron system. It turns out that at a more advanced level of analysis, they do contribute to the relaxation of momentum, but let us for the moment neglect this contribution. The electron momentum decays, then, because of collisions of the electrons against lattice imperfections, such as impurities, dislocations etc., and – in the wider sense – lattice distortions caused by lattice vibrations. We could model the influence of electron scattering events by making two very simplifying assumptions: • Electron collisions randomise an electron’s momentum, so that – on average – the contribution of an electron to the total momentum is 0 after a collision. • The probability for a collision to occur, P , is characterised by a single relaxation time τ : P (collision in [t, t + dt]) = dt/τ . 1 The connection between the quantum mechanical description of the electrons and this classical result can be made by treating the electrons as wave-packets. The energy ǫ(k) is the frequency associated with the phase rotation of the wavefunction, ψk e−iǫ(k)t/h̄ , but for the motion of a wave packet in a dispersive band, we should use the group velocity, dω/dk, or as a vector ṙ = vg = h̄−1 ∇k ǫ(k) , (2.3) where r is the position of the wavepacket. The momentum of the wave-packet is p = h̄k. A force F causes work to be done on the electron at a rate dǫ dk dǫk = = F vg dt dk dt (2.4) which leads to the key relation h̄ . dk = F = −e(E + v ∧ B) = −e(E + h̄−1 ∇k ǫ(k) ∧ B) dt (2.5) 2.1. SCATTERING AND ELECTRICAL CONDUCTIVITY OF METALS 19 Figure 2.1: The net current from the fermi sea at rest is zero, but a shift by a small momentum leads to a net imbalance. From these assumptions, we find that the probability that a particular electron has not scattered in the time interval [t, t + dt] is 1 − dt/τ . As only the electrons which have not scattered contribute to the total momentum (the momentum of the others randomises to zero on average), this gives a total momentum after time t + dt: p(t + dt) = (1 − dt/τ )(p(t) + N f (t)dt) + ... (2.6) This gives rise to a differential equation for the momentum: ! 1 d p = N f (t) + dt τ (2.7) If we wanted to relate this result to the k-space representation of the electron system as a filled (Fermi) volume in k-space, we could recast this equation in terms of the rate of change of wavevector, as p = h̄k: each electron assumes, on average, an additional contribution to its wavevector, δk = p/(N h̄), which follows the differential equation ! 1 d + h̄δk = f (t) dt τ (2.8) Almost invariably, δk ≪ kF , so the shift in the Fermi sea is very small — and the net current is entirely produced by the small imbalance near the Fermi surface. Just as for the specific heat and paramagnetic suseptibility, it is only the properties of electrons near the fermi surface that determine the conductivity of a metal. 2.1.2 Electrical conductivity We can extract the drift velocity v from the total momentum p by dividing by particle number N and particle mass m∗ . 2 Now, we express the equation of motion for the current density via j = ρv as: ! d ρ 1 j = ∗ f (t) (2.9) + dt τ m 2 We use the more general m∗ for the particle mass, rather than the more specific electron mass me , because in solids, the effective charge carrier mass can differ dramatically from me 20 CHAPTER 2. ELECTRODYNAMICS OF METALS Replacing f (t) by the Lorentz force and setting the charge q equal to −e, where e is the magnitude of the charge of an electron, we obtain: ! d ρe 1 j = − ∗ (E + v × B) + dt τ m (2.10) Let us first examine the case B = 0, E 6= 0: the response of the electron system to a timevarying applied electric field. The amplitude of this response will depend on the frequency at which the field is varying. The simplest way to extract this frequency dependence is to insert a trial solution of the form j = jω e−iωt and a time-varying field of the form E = Eω e−iωt . This transforms the above differential equation into an algebraic equation: jω = − 1 ρe Eω m∗ 1/τ − iω (2.11) Expressing ρ as −en, where n is the electron number density, and using the definition of the conductivity j = σE, we obtain: σω = 1 ne2 , ∗ m 1/τ − iω (2.12) where the prefactor ne2 /m∗ = ωp2 is the square of the plasma frequency (see below). In the low frequency limit, this expression for the conductivity tends to the DC conductivity σ0 = ne2 τ m∗ (2.13) The conductivity can also be written in terms of the mobility µ = eτ /m∗ σ = neµ = ne2 τ m∗ (2.14) Thermal conductivity of metals Particles with velocity v, mean free path ℓ and specific heat C are expected to yield a thermal conductivity K = Cvℓ/3. For a free fermi gas, we get the correct answer from this formula by using the electronic specific heat, the characteristic carrier velocity vF , and the mean free path for carriers on the fermi surface ℓ = vF τ . Hence, using the relationship between the Fermi velocity and the Fermi energy EF = m∗ vF2 /2, Kel = 2 2 π 2 nkB π 2 nkB Tτ T · vF · vF τ = 2 EF 3m∗ (2.15) Almost invariably, the electronic thermal conductivity is bigger than that due to the lattice. K and σ are of course closely related, being both proportional to the scattering time and the density, as is natural. The ratio π2 K = σT 3  kB e 2 is expected to be a constant, independent of material parameters. Wiedemann-Franz law, which works strikingly well for simple metals. (2.16) This proportionality is the 2.1. SCATTERING AND ELECTRICAL CONDUCTIVITY OF METALS 21 Figure 2.2: The upper figure shows the geometry of a Hall bar, with the current flowing uniformly in the x-direction, and the magnetic field in z. The lower figure shows the steady state electron flow (arrows) in a section normal to ẑ. When a voltage Ex is first applied, and Ey is not yet established, the electrons will deflect and move in the (downward) y-direction. The y-surfaces of the crystal then become charged, producing the field Ey which exactly cancels the Lorentz force −evx B. 2.1.3 Transport in magnetic fields Returning to Eq. (2.10), we now study the electrical transport in a transverse magnetic field, i.e. a static magnetic field B in the ẑ direction, and static currents and electrical fields in the x − y plane. The equations of motion for charge carriers with charge q (= −e in the case of electrons, but we will later see that there can also be positively charged carriers called holes) are now 3 .   q2n (Ex + vy B) m∗ q2n = (Ey − Bvx ) m∗ q2n Ez = m∗ ∂t + τ −1 jx =   ∂t + τ −1 jy   ∂t + τ −1 jz (2.17) In steady state , we set the time derivatives ∂t = d/dt = 0, and get the three components of the current density jx jy jz qτ = qn Ex + βvy m∗   qτ = qn Ey − βvx m∗ qτ = qn ∗ Ez m with the dimensionless parameter β = frequency and the scattering rate. 3  qB τ m∗  (2.18) = ωc τ = µB the product of the cyclotron Again, we define e to be the magnitude of the charge of an electron. 22 CHAPTER 2. ELECTRODYNAMICS OF METALS Hall effect If a current j = qv is flowing in a magnetic field B, Eq. (2.10) leads to a force on the carrier qv × B, normal to the direction of flow. Since there is no flow in the normal direction, there must exist a counterbalancing electric field E = −v × B. This is the Hall effect. Consider the rod-shaped geometry of Fig. 2.2. The current is forced by geometry to flow only in the x-direction, and so jy = 0, vy = 0 which gives vx = qτ Ex , m∗ (2.19) and Ey = βEx (2.20) It turns out that for high mobility materials, and large magnetic fields, it is not hard to reach large values of |β| ≫ 1, so that the electric fields are largely normal to the electrical currents. The Hall coefficient is defined by RH = 1 Ey = jx B nq (2.21) Notice that it is negative for electrons, but importantly is independent of the effective mass, and grows with decreasing carrier density. For holes of charge +e the sign is positive. The Hall effect is an important diagnostic for the density and type of carriers transporting the electrical current in a semiconductor. The simple picture of parabolic bands works quite well for alkali metals, where the predicted Hall coefficient is within a few percent of the expected value for a parabolic free electron band. But Be, Al, and In all have positive Hall coefficients accounted for by a band-structure with hole pockets that dominates the Hall effect. 2.2 High frequency response – optical conductivity of metals To evaluate the response of the electron gas in a metal to a time-varying field, we treat the electrons as a charged fluid, and displace them relative to the fixed ionic background (jellium again) by an amount u(r), which is essentially the time integral of the drift velocity v: u = uq ei(q·r−ωt) uo k q (2.22) This is a longitudinal wave — displacement parallel to the wavevector. This distinction is worthwhile because a transverse wave uo ⊥ q would not induce internal electric fields, since in that case ∇ · E ∝ q · E = 0 The longitudinal displacement of the charge induces a polarisation due to the conduction electrons of Pc = ρu , (2.23) 2.2. HIGH FREQUENCY RESPONSE – OPTICAL CONDUCTIVITY OF METALS 23 If we include also the polarisation of the ionic cores by atomic processes, we find that the total polarisation is P = ρu + ǫ0 χ∞ E , (2.24) where χ∞ relates the atomic polarisation to the electric field: P∞ = ǫ0 χ∞ E. Note, before we go on, the key equations relating the relevant electric fields and the current density: ǫ0 E ǫ0 χE ǫ0 ǫE j = = = = D−P P D σE (2.25) Since the time derivative of u is the drift velocity v, and the current density is j = ρv, we find for the rate of change of P: 4 Ṗ = j + ǫ0 χ∞ Ė (2.26) Fourier transforming this equation, e.g. by inserting P = Pω e−iωt , j = jω e−iωt , E = Eω e−iωt , and using Eqn. 2.25 we obtain: jω = σω Eω = −iωǫ0 (χω − χ∞ )Eω ⇒ σω = −iωǫ0 (ǫω − ǫ∞ ) , (2.27) where ǫ∞ is the permittivity which the material would have, if the contributions of the conduction electrons were not counted. If the atomic polarisability is zero, then ǫ∞ = 1 and σω = −iωǫ0 (ǫω − 1). We can also turn this result around and solve for ǫω : ǫω = i Earlier (Eqn. 2.12), we found σω = σω + ǫ∞ ǫ0 ω ne2 1 . m∗ 1/τ −iω ǫ(ω) = ǫ∞ − (2.28) Inserting this into Eqn. 2.28 gives ωp2 ω 2 + iω/τ , (2.29) where ωp is the Plasma frequency: ωp2 ne2 = ǫ0 m∗ (2.30) At frequencies larger than 1/τ the conductivity rapidly falls off: ℜσ(ω) = 4 σ(0) . 1 + ω2τ 2 (2.31) An alternative approach, which avoids introducing the field u(r, t), could consider ∇Pc = −ρc and ρ̇c = −∇j to find Ṗc = j, and then add in the polarisation due to atomic polarisability, to give the key equation 2.26 24 CHAPTER 2. ELECTRODYNAMICS OF METALS Figure 2.3: Generic diagram of an inelastic scattering experiment. The incident particle in this case an electron is scattered to a final state of different energy and momentum. By comparing the incident and scattered spectra, one deduces the energy loss spectrum of the internal collective excitations in the medium. For high energy electrons - typically used in an EELS experiment - the momentum loss (q − k) is small. Consequently, the dielectric function near q = 0 can be extracted directly from measurements of the ac conductivity σ(ω) by optical absorption of light. A comparison between optical measurements and electron energy loss measurements is shown in Fig. 2.4. Since D is generated (by Poisson’s law) by the external potential, and E is generated by the screened potential, another way of writing the dielectric function (see also Eq. (2.45)) is Vtot (q, ω) = Vext (q, ω) ǫ(q, ω) (2.32) Defined this way, we have already estimated a formula for the static dielectric function in Eq. (2.47). Plasmons. Eq. (2.29) describes the response of a damped harmonic oscillator with reso√ nance frequency ωp / ǫ∞ . In metals, ωp >> τ −1 and usually ǫ∞ ≃ 1. At resonance, ǫω → 0, which implies (because D = ǫ0 ǫω E) a finite amplitude of oscillation for P despite zero forcing field D. These are modes of free oscillation: solutions of the form uo eiωp t with high resonance frequency in the eV energy range – a massive mode. This classical discussion does not generate any dispersion for the plasmon, i.e. the plasma frequency is found to be q-independent. It turns out that the classical theory is exact for q → 0, but there are quantum transitions that are entirely missed at short wavevectors. Since ǫ measures the charge response of a solid, then plasmons are generated by any charged probe. The classic experiment to observe plasmons is Electron Energy Loss Spectroscopy (EELS), where a high energy electron is sent into the sample and the energy loss monitored. As in any driven oscillator, energy is dissipated at or near the resonant frequency (with the width in frequency depending on the damping of the oscillator). An EELS spectrum will therefore have a peak near the plasma frequency. 2.3. SCREENING AND THOMAS-FERMI THEORY 25 Figure 2.4: . Electron energy loss spectrum for Ge and Si (dashed lines) compared to values of Im(1/ǫ) extracted directly from measurements of the optical conductivity. [From H.R.Philipp and H.Ehrenreich, Physical Review 129, 1550 (1963) 2.3 Screening and Thomas-Fermi theory One of the most important characteristics of the metallic state is the phenomenon of screening. If we insert a positive test charge into a metal, it attracts a cloud of electrons around it, so that at large distances away from the test charge the potential is perfectly screened - there is zero electric field inside the metal. Notice that this is quite different from a dielectric, where the form of the electrostatic potential is unchanged but the magnitude is reduced by the dielectric constant ǫ. Screening involves a length-scale: a perturbing potential is not screened perfectly at very short distances. Why not? In a classical picture, one might imagine that the conduction electrons simply redistribute in such a way as to cancel any perturbing potential perfectly. This would require precise localisation of the electrons, however, which in quantum mechanics would incur too high a penalty in kinetic energy. Just as in the hydrogen atom the electron cannot sit right on top of the proton, a balance is reached in metals between minimising potential and kinetic energy. This leads to number density building up in the vicinity of a perturbing potential, which will screen the potential over a short but finite distance. Response to an external potential The aim of this calculation is to estimate the response of a free electron gas to a perturbing potential. The perturbing potential could be caused by charges outside the metal, but it could also be due to extra charges placed inside the metal. 26 CHAPTER 2. ELECTRODYNAMICS OF METALS We begin with the free electron gas in a metal, without an externally applied perturbing potential. The electrostatic potential in the metal, V0 (r) is connected to the charge distribution ρ0 (r) via ρ0 (r) ∇2 V0 (r) = − (2.33) ǫ0 In the simplest model of a metal, we consider the positive background charge to be smeared out homogeneously throughout the metal. The electron gas moves on top of this positive background. This is the plasma or ’Jellium’ model for a metal. ρ0 = 0 everywhere in this case.5 In the presence of a perturbing potential Vext (r), the electron charge density ρ(r) will redistribute, ρ(r) = ρ0 (r) + δρ(r), causing a correction to the potential V (r) = V0 (r) + δV (r): ∇2 δV (r) = − δρ(r) ǫ0 (2.34) In order to make progress, we need to link the charge density redistribution δρ to the applied potential Vext . For long-wavelength perturbations, it is plausible that in a region surrounding the position r the perturbing potential effectively just shifts the free electron energy levels, which is equivalent to assuming a spatially varying Fermi energy. This is the essence of the Thomas Fermi approximation. Thomas-Fermi approximation The Thomas-Fermi theory of screening starts with the Hartree approximation to the Schrödinger equation. The Hartree approximation is to replace the many-body pairwise interaction between the electrons by a set of interactions between a single electron and the charge density made up from all the other electrons, i.e. by a one-body potential for the ith electron ′ ′ 2 e2 X Z e Z ′ ρ(r ) ′ |ψj (r )| dr dr = , Ucoul (r) = − 4πǫ0 |r − r′ | 4πǫ0 j6=i |r − r′ | (2.35) where the summation is over all the occupied states ψj . This clearly takes into account the averaged effect of the Coulomb repulsion due to all the other electrons. This introduces enormous simplicity, because instead of needing to solve an N-body problem, we have a (selfconsistent) one-body problem. It contains a lot of important physics, and turns out to be an approximation that is good provided the electron density is high enough. We will discuss better theories later, in the special topic of the electron gas. We shall treat the case of “jellium”, where the ionic potential is spread out uniformly to neutralise the electron liquid. Note: the average charge density is therefore always zero! The metal is neutral. An external potential will, however, cause a redistribution of charge, leading to local accumulation of positive or negative charge, which will tend to screen the external potential. The net effect will be that the total potential seen by an individual electron in the Schrödinger equation is less than the external potential. We wish to calculate the charge density induced by such an external potential ρind ([Vext (r)]). 5 The correction to the charge density, δρ, does not include those charges (ρext ) which may have been placed inside the metal to set up the perturbing potential. They obey ∇2 Vext = −ρext /ǫ0 . 27 2.3. SCREENING AND THOMAS-FERMI THEORY δ n(r) µ EF (r) −e Vtot (r) −e Vext (r) Figure 2.5: Thomas-Fermi approximation Jellium. The potential in the problem is the total potential (external plus induced, Vtot = Vext + δV ) produced by the added charge and by the non-uniform screening cloud (see Fig. 2.5) − h̄2 2 ∇ ψ(r) + (−e)(δV (r) + Vext (r))ψ(r) = Eψ(r) . 2m (2.36) Slowly varying potential. Assume that the induced potential is slowly varying enough that the energy eigenvalues of Eq. (2.36) are still indexed by momentum, but just shifted by the potential as a function of position: E(k, r) = E0 (k) − eVtot (r)) , (2.37) 2 2 where E0 (k) follows the free electron, parabolic dispersion h̄2mk . This only makes sense in terms of wavepackets, but provided the potential varies slowly enough on the scale of the Fermi wavelength 2π/kF , this approximation is reasonable. Constant chemical potential. Keeping the electron states filled up to a constant energy µ requires that we adjust the local Fermi energy EF (r) (as measured from the bottom of the band) such that6 . µ = EF (r) − eVtot (r) , (2.38) Local density approximation. We assume that EF just depends on the local electron number density n via the density of states per unit volume gV (E): Z EF gV (E)dE = n . (2.39) This means that a small shift in the Fermi energy, δEF gives rise to a change in the number density δn = gV (EF )δEF . The Fermi energy shift, in turn, is linked (via Eqn. 2.38) to Vtot as δEF = e(δV + Vext ), from which we obtain δn = egV (EF )(δV + Vext ) . 6 One is often sloppy about using EF and µ interchangeably; here is a place to take care (2.40) 28 CHAPTER 2. ELECTRODYNAMICS OF METALS Linearised Thomas-Fermi. When the added potential Vext is small, the induced number density δn is small, and therefore the number density n cannot differ very much from the density n0 of the system without the potential (n = n0 + δn). We may then express Eqn.2.34 in a linearised form with respect to the perturbing potential: ∇2 δV (r) = e2 gV (EF ) (δV (r) + Vext (r)) ǫ0 (2.41) Density response. This is solved by Fourier transforms, for instance by assuming an oscillatory perturbing potential Vext = Vext (q)eiqr and a resulting oscillatory induced potential δV = δV (q)eiqr : δV (q) = − e2 gV (EF )/ǫ0 qT2 F V (q) = − Vext (q) , ext q 2 + e2 gV (EF )/ǫ0 q 2 + qT2 F (2.42) 1 where we have collected e2 gV (EF )/ǫ0 into the Thomas Fermi wave vector qT F = (e2 gV (EF )/ǫ0 ) 2 , which for the free electron gas is calculated as qT2 F = 4 kF 2.95 −1 2 1 me2 k = = ( √ Å ) . F π 2 ǫ0 h̄2 π aB rs (2.43) 2 ǫ0 ≃ 0.53 Å is the Bohr radius and rS is the Wigner-Seitz radius, defined by Here, aB = 4πh̄ me2 3 −1 (4π/3)rS = n . For the induced number density we obtain: nind (q) = ǫ0 q 2 Vext (q) , e [1 + q 2 /qT2 F ] (2.44) Dielectric permittivity. In general, this phenomenon is incorporated into electromagnetic theory through the generalised wavevector dependent dielectric function ǫ(q). The dielectric function relates the electric displacement D to the electric field E, in the form ǫ0 ǫ(q)E(q) = D(q). While the gradient of the total potential Vtot = V0 + δV + Vext = δV + Vext (V0 = 0 for Jellium) gives the E− field, the gradient of the externally applied potential Vext gives the displacement field D. As E and D are related via the relative permittivity, ǫ, the potentials from which they derive are also connected by ǫ: Using Eq. 2.42 we find Vext (q) = ǫ(q) (δV (q) + Vext (q)) (2.45) q2 Vtot (q) = Vext (q) 2 , q + qT2 F (2.46) and for ǫ(q): ǫT F (q) = 1 + qT2 F q2 . (2.47) Screening. ǫT F ∝ q −2 at small q (long distances), so the long range part of the Coulomb potential (also ∝ 1/q 2 ) is exactly cancelled. In real space, if vext = Q/r is Coulombic (long 2.3. SCREENING AND THOMAS-FERMI THEORY 29 range), V (r) = (Q/r)e−qT F r is a short-range Yukawa, or screened potential7 . In a typical metal, rs is in the range 2 − 6, and so potentials are screened over a distance comparable to the interparticle spacing; the electron gas is highly effective in shielding external charges. 7 This form is originally due to P.Debye and E.Hückel, Zeitschrift für Physik 24, 185, (1923) and was derived for the theory of electrolytes; it appears also in particle theory under the name of the Yukawa potential; the physics in all cases is identical 30 CHAPTER 2. ELECTRODYNAMICS OF METALS Chapter 3 The variety of condensed matter 3.1 Types of solids What holds a solid together? Cohesion is ultimately produced by the electrostatic interaction between the nuclei and the electrons, but depending on the particular atomic structure the types of solids can be very different. 3.2 The binding of crystals Inert gases The inert gases have filled electron shells and large ionisation energies. Consequently, the electronic configuration in the solid is close to that of separated atoms. Since the atoms are neutral, the interaction between them is weak, and the leading attractive force at large distances comes from the van der Waals interaction, which gives an attractive potential proportional to 1/R6 . This form can be loosely derived by thinking of an atom as an oscillator, with the electron cloud fluctuating around the nucleus as if on a spring. The centre of the motion lies on top of the atom, but if the cloud is displaced, there will be a small dipole induced, say p1 . Such displacements happen as a result of zero-point motion of the electron cloud in the potential of the nucleus. A distance R away from the atom there is now an induced electric field ∝ p1 /R3 . A second atom placed at this point will then have a dipole induced by the electric field of the first: p2 ∝ αp1 /R3 , where α is the atomic polarizability. The second dipole induces an electric field at the first, which is now E1 ∝ p2 /R3 ∝ αp1 /R6 . (3.1) The energy of the system is then changed by an amount D E ∆U = h−p1 · E1 i ∝ −α p21 /R6 . (3.2) Notice that it depends on the expectation value of the square of the dipole moment < p21 >, which is non-zero, and not the square of the expectation value < p1 >2 , which would be zero. 31 32 CHAPTER 3. THE VARIETY OF CONDENSED MATTER Figure 3.1: Two dipoles represent model atoms that are arranged along a line, with the positive charges (+e) fixed at the positions 0, R, and the negative charges (-e) at the points x1 , R + x2 . If the atoms move together so that the electron charge distributions begin to overlap, repulsive forces come into play. While there is of course a contribution from the direct electrostatic repulsion of the electrons, more important is the Pauli exclusion principle that prevents two electrons having their quantum numbers equal. The effect of Pauli exclusion can be seen by an extreme example, of overlapping two Hydrogen atoms entirely, with the electrons for simplicity assumed to be in the same spin state. In this case, while two separated atoms may be both in the 1S ground state, the combined molecule must have a configuration 1s2s, and thus is higher by the promotion energy. Calculations of the repulsive interaction are complex but the answer is clearly short-ranged. They are often modelled empirically by an exponential form e−R/Ro , or a power law with a large power. A commonly used empirical form to fit experimental data on inert gases is the Lennard-Jones potential A B U (R) = − 6 + 12 (3.3) R R with A and B atomic constants obtained from gas-phase data. With the exception of He, the rare gases from close-packed (face-centered cubic) solids with a small cohesive energy, and low melting temperatures. Helium is special because zero-point motion of these light atoms is substantial enough that they do not solidify at zero pressure down to the absolute zero of temperature. The quantum fluids 3 He and 4 He have a number of extraordinary properties, including superfluidity. Ionic Crystals Given the stability of the electronic configurations of a rare gas, atoms that are close to a filled shell will have a tendency to lose or gain electrons to fill the shell. • The energy for the reaction M − > M + + e− in the gas phase is called the ionization energy I. • The energy for the reaction X + e− − > X − in the gas phase is called the electron affinity A. • The cohesion of an ionic molecule can overcome the energy cost I + A by the electrostatic attraction, e2 /R 33 3.2. THE BINDING OF CRYSTALS • In a solid, the electrostatic interaction energy for a diatomic crystal1 is Uelectrostatic = 1 XX Uij 2 i j (3.4) where Uij = ±q 2 /Rij is the sum of all Coulomb forces between ions. If the system is on a regular lattice of lattice constant R, then we write the sum Uelectrostatic = − 1 αM q 2 2 R (3.5) where αM is a dimensionless constant that depends only on the crystal structure. • The evaluation of αM is tricky, because the sum converges slowly. Three common crystal structures are N aCl (αM = 1.7476), CsCl (1.7627), and cubic ZnS or Zincblende (1.6381). • To the attractive Madelung term must be added the repulsive short range force, and we now have the added caveat that ions have different sizes, explaining why N aCl has the rocksalt structure, despite the better electrostatic energy of the CsCl structure. Covalent crystals The covalent bond is the electron pair or single bond of chemistry. Model Hydrogen. Two overlapping atomic orbitals on identical neighbouring atoms will hybridise. Because the Hamiltonian must be symmetric about a point centered between the ions then the eigenstates must have either even or odd parity about this center. If we have a simple system of two one electron atoms - model hydrogen - which can be approximated by a basis of atomic states φ(r − R) (assumed real) centered on the nucleus R, then two states of even and odd parity are ψ± (r) = φ(r − Ra ) ± φ(r − Rb ) (3.6) ψ+ has a substantial probability density between the atoms, where ψ− has a node. Consequently, for an attractive potential E+ < E− , and the lower (bonding) state will be filled with two electrons of opposite spin. The antibonding state ψ− is separated by an energy gap Eg = E− − E+ and will be unfilled. The cohesive energy is then approximately equal to the gap Eg 2 Covalent semiconductors. If we have only s-electrons, we clearly make molecules first, and then a weakly bound molecular solid, as in H2 . Using p, d, orbitals, we may however make directed bonds, with the classic case being the sp3 hybrid orbitals of C, Si, and Ge. These are constructed by hybrid orbitals s + px + py + pz + 3 other equivalent combinations, to make new orbitals that point in the four tetrahedral directions: (111), (1̄1̄1), (1̄11̄), (11̄1̄). These directed orbitals make bonds with neighbours in these tetrahedral directions, with each atom donating one electron. The open tetrahedral network is the familiar diamond structure of C, Si and Ge. 1 Beware the factor of 1/2, which avoids doublePcounting the interaction energy. P The energy of a single ion i due to interaction with all the other ions is Ui = j6=i Uij ; the total energy is 12 i Ui 2 Actually twice (two electrons) half the gap, if we assume that E± = Eatom ± 12 Eg 34 CHAPTER 3. THE VARIETY OF CONDENSED MATTER Figure 3.2: Tetrahedral bonding in the diamond structure. The zincblende structure is the same but with two different atoms per unit cell Ionic semiconductors. In GaAs and cubic ZnS the total electron number from the pair of atoms satisfies the ”octet” rule, and they have the identical tetrahedral arrangement of diamond, but with the atoms alternating. This is called the zinclende structure. The cohesion in these crystals is now part ionic and part covalent. There is another locally tetrahedral arrangement called wurtzite which has a hexagonal lattice, favoured for more ionic systems. With increasing ionic components to the bonding, the structures change to reflect the ionicity: group IV Ge (diamond), III-V GaAs (Zincblende), II-VI ZnS (zincblende or wurtzite), II-VI CdSe (wurtzite), and I-VII N aCl (rocksalt). Metals Metals are generally characterised by a high electrical conductivity, arising because the electrons are relatively free to propagate through the solid. Close packing. Simple metals (e.g. alkalis like Na, and s-p bonded metals such as Mg and Al) usually are highly coordinated (i.e. fcc or hcp - 12 nearest neighbours, sometimes bcc - 8 nearest neigbours), since the proximity of many neighbouring atoms facilitates hopping between neighbours. Remember that the fermi energy of a free electron gas (i.e. the average kinetic energy per particle) is proportional to kF2 ∝ a−2 ∝ n2/3 (here a is the lattice constant and n the density; the average coulomb interaction of an electron in a solid with all the other electrons and the other ions is proportional to a−1 ∝ n1/3 . Thus the higher the density, the larger the kinetic Va (r) Vb (r) Ea Eb Figure 3.3: A simple model of a diatomic molecule. The atomic hamiltonian is Hi = T + Vi (r), with T the kinetic energy −h̄2 ∇2 /2m and Vi the potential. We keep just one energy level on each atom. 3.3. COMPLEX MATTER 35 energy relative to the potential energy, and the more itinerant the electrons.3 By having a high coordination number, one can have relatively large distances between neigbours - minimising the kinetic energy cost - in comparison to a loose-packed structure of the same density. Screening. Early schooling teaches one that a metal is an equipotential (i.e. no electric fields). We shall see later that this physics in fact extends down to scales of the screening length λ ≈ 0.1nm, i.e. about the atomic spacing (though it depends on density) - so that the effective interaction energy between two atoms in a metal is not Z 2 /R (Z the charge, R the separation), but Z 2 e−R/λ /R and the cohesion is weak. Trends across the periodic table. As an s-p shell is filled (e.g. Na,Mg,Al,Si) the ion core potential seen by the electrons grows. This makes the density of the metal tend to increase. Eventually, the preference on the right-hand side of the periodic table is for covalent semiconductor (Si, S) or insulating molecular (P, Cl) structures because the energy is lowered by making tightly bound directed bonds. Transition metals. Transition metals and their compounds involve both the outer s-p electrons as well as inner d-electrons in the binding. The d-electrons are more localised and often are spin-polarised in the 3d shell when they have a strong atomic character (magnetism will be discussed later in the course). For 4d and 5d transition metals, the d-orbitals are more strongly overlapping from atom to atom and this produces the high binding energy of metals like W (melting point 3700 K) in comparison to alkali metals like Cs (melting point 300 K). 3.3 Complex matter Simple metals, semiconductors, and insulators formed of the elements or binary compounds like GaAs are only the beginning of the study of materials. Periodic solids include limitless possibilities of chemical arrangements of atoms in compounds. Materials per se, are not perhaps so interesting to the physicist, but the remarkable feature of condensed matter is the wealth of physical properties that can be explored through novel arrangements of atoms. Many new materials, often with special physical properties, are discovered each year. Even for the element carbon, surely a familiar one, the fullerenes (e.g. C60 ) and nanotubes (rolled up graphitic sheets) are recent discoveries. Transition metal oxides have been another rich source of discoveries (e.g. high temperature superconductors based on La2 CuO4 , and ferromagnetic metals based on LaM nO3 ). f −shell electron metals sometimes produce remarkable electronic properties, with the electrons within them behaving as if their mass is 1000 times larger than the free electron mass. Such quantum fluid ground states (metals, exotic superconductors, and superfluids) are now a rich source of research activity. The study of artificial meta-materials becins in one sense with doped semiconductors (and especially layered heterostructures grown by molecular beam epitaxy or MBE), but this subject is expanding rapidly due to an influx of new tools in nanomanipulation and biological materials. Many materials are of course not crystalline and therefore not periodic. The physical description of complex and soft matter requires a separate course. 3 Note the contrast to classical matter, where solids are stabilised at higher density, and gases/liquids at lower density. 36 CHAPTER 3. THE VARIETY OF CONDENSED MATTER Glasses If one takes a high temperature liquid (e.g. of a metal) and quenches it rapidly, one obtains a frozen structure that typically retains the structure of the high-temperature liquid. Meltquenched alloys of ferromagnets are often prepared this way because it produces isotropic magnetic properties. For most materials the amorphous phase is considerably higher in energy than the crystalline, so the system has to be frozen rapidly, far from its equilibrium configuration. A few materials make glassy states readily, and the most common example is vitreous silica (SiO2 ). Crystalline forms of silica exist many of them!) and all are network structures where each Si bonds to four oxygen neighbours (approximately tetrahedrally) and each O is bonded to two Si atoms. Since the O2− ion is nearly isotropic, the orientation of one tetrahedral group respect to a neighbouring group about the connecting Si − O − Si bond is not fixed, and this allows for many possible crystalline structures, but especially for the entropic stabilisation of the glass phase. Whatever the arrangement of atoms, all the electrons are used up in the bonding, so glass is indeed a good insulator. The characteristic feature of a strong or network glass is that on cooling the material becomes increasingly viscous, often following the Vogel-Fulcher law, C η ∝ e T −T0 (3.7) implying a divergence of the viscosity η at a temperature T0 . Once η reaches about 1012 Pa s, it is no longer possible to follow the equilibrium behaviour. Consequently, debates still rage about whether or not the glass transition is a “true” phase transition, or indeed whether or no the temperature T0 has physical meaning. Polymers The classic polymers are based on carbon, relying on its remarkable ability to adopt a variety of local chemical configurations. Polyethylene is built from repeating units of CH2 , and more complex polymers are constructed out of more complex subunits. Because the chains are long, and easily deformed or entangled, most polymers are glassy in character, and therefore their physical properties are largely dominated by entropic considerations. The elasticity of rubber is produced by the decrease in entropy upon stretching, not by the energetic cost of stretching the atomic bonds. Many simple polymers are naturally insulating (e.g. the alkanes) or semiconducting, but it is sometimes possible to ”dope” these systems so that there are electronic states. They become interesting for a number of reasons in technology and fundamental science. Because a simple polymer chain can often be modelled as a one-dimensional wire, they provide a laboratory for the often unusual properties of one-dimensional electronic systems. Because the tools of organic chemistry allow one to modify the physical properties of polymers in a wide range of ways (for example, by adding different side chains to the backbones), one can attempt to tune the electronic and optic properties of heterogeneous polymer structures to make complex devices (solar cells, light-emitting diodes, transistors) using a very different medium from inorganic semiconductors. Liquid crystals Like a single atom, polymers are isotropic, because they are very long. Shorter rod-shaped molecules however have an obvious orientational axis, and when combined together to make a 3.3. COMPLEX MATTER 37 liquid crystal one can construct matter whose properties are intermediate between liquid and solid. Nematics. An array of rods whose centres are arranged randomly has no long-range positional order (just like a liquid), but if the rods are oriented parallel to each other has long-range orientational order, like a molecular crystal. This is a nematic liquid crystal. The direction in space of the orientational order is a vector n̂ called the director. The refractive index of the material will now be different for light polarized parallel and perpendicular to the director. Cholesterics. It turns out if the molecule is chiral then the director need not point always in the same direction, and in a cholesteric liquid crystal the direction of n̂ twists slowly in a helix along an axis that is perpendicular to it. Usually the pitch of the twist is much longer than size of the rod, is a strong function of temperature, and frequently close to the wavelength of visible light. Smectics. Smectics additionally have long-range positional order along one direction, usually to be thought of as having layers of molecules. So called Smectic A has the director parallel to the planes, whereas in Smectic C the director is no longer perpendicular (and may indeed rotate as a function of position). In Smectic B the molecules in the plane have a crystalline arrangement, but different layers fall out of registry. This is a kind of quasi-2D solid. Quasicrystals As a last piece of exotica, the classic group theory of crystal structures proves the impossibility of building a Bravais lattice with five-fold symmetry. Nature is unaware of this, and a series of metallic alloys have been found that indeed have crystals with axes of three, five, and tenfold symmetry. These materials are in fact physical representations of a mathematical problem introduced by Penrose of tiling of a plane with (e.g.) two rhombus shaped tiles that have corner angles of 2π/10 and 2π/5. A complete tiling of the plane is possible, though the structure is not a periodic lattice (it never repeats). 38 CHAPTER 3. THE VARIETY OF CONDENSED MATTER Figure 3.4: Liquid crystal structures Schematic representation of the position and orientation of anisotropic molecules in: (a) the isotropic phase; (b) the nematic phase; (c) the smectic-A phase; and (d) the smectic-C phase. [From Chaikin and Lubensky] 3.3. COMPLEX MATTER 39 Figure 3.5: Scanning tunnelling microscope image of a 10 nm2 quasicrystal of AlP dM n with a Penrose tiling overlaid. [Ledieu et al Phys.Rev.B 66, 184207 (2002)] 40 CHAPTER 3. THE VARIETY OF CONDENSED MATTER Chapter 4 Periodic solids and diffraction 4.1 The description of periodic solids An ideal crystal is constructed from the infinite repetitition of identical structural units in space. The repeating structure is called the lattice, and the group of atoms which is repeated is called the basis. The basis may be as simple as a single atom, or as complicated as a polymer or protein molecule. This section discusses briefly some important definitions and concepts. For a more complete description with examples, see any of the textbooks recommended in the introduction. Lattice. The lattice is defined by three fundamental (called primitive ) translation vectors ai , i = 1, 2, 3. The atomic arrangement looks the same from equivalent points in the unit cell: r′ = r + X i ni ai ∀ integer ni . (4.1) Primitive unit cell. The primitive unit cell is the parallelipiped formed by the primitive translation vectors ai , and an arbitrary lattice translation operation can be written as T= X ni a i (4.2) i There are many ways of choosing a primitive unit cell, but the lattice so formed is called a Bravais lattice. Wigner-Seitz cell A most convenient primitive unit cell to use is the Wigner-Seitz cell, constructed as follows: Draw lines to connect a given lattice point to all of its near neighbours. Then draw planes normal to each of these lines from the midpoints of the lines. The smallest volume enclosed in this way is the Wigner-Seitz primitive unit cell. Point group. The are other symmetry operations that can be performed on a lattice, for example rotations and reflections. The collection of symmetry operations, which applied about a lattice point, map the lattice onto itself is the lattice point group. This includes reflections and rotations; for example a 2D square lattice is invariant under reflections about the x and y axes, as well as through axes at an angle of π/4 to the x and y axes, and rotations through any multiple of π/2. Remember that adding a basis to a primitive lattice may destroy some of the point group symmetry operations. There are five distinct lattice types in two dimensions, and 14 in three dimensions. 41 42 CHAPTER 4. PERIODIC SOLIDS AND DIFFRACTION Figure 4.1: . The Wigner-Seitz cell for the BCC and FCC lattices Space group. The translational symmetry and the point group symmetries are subgroups of the full symmetry of the lattice which is the space group. Every operation in the space group consists of a rotation, reflection, or inversion followed by a translation. However, the space group is not necessarily just the sum of the translational symmetries and the point symmetries, because there can be space group symmetries that are the sum of a proper rotation and a translation, neither of which are independently symmetries of the lattice. The number of possible lattices is large. In three dimensions there are 32 distinct point groups, and 230 possible lattices with bases. Two of the important lattices that we shall meet later are the body-centred and face-centred cubic lattices, shown in Fig. 4.1. Index system for crystal planes If you know the coordinates of three points (not collinear), this defines a plane. Suppose you chose each point to lie along a different crystal axis, the plane is then specified by giving the coordinates of the points as xa1 + ya2 + za3 (4.3) Of course the triad (xyz) need not be integers. However, one can always find a plane parallel to this one by finding a set of three integers (hkl) where x/h = y/k = z/l. (hkl) is called the index of the plane. When we want to refer to a set of planes that are equivalent by symmetry, we will use a notation of curly brackets: so {100} for a cubic crystal denotes the six equivalent symmetry planes (100), (010), (001), (1̄00), (01̄0), (001̄), with the overbar used to denote negation. 4.2 The reciprocal lattice and diffraction The reciprocal lattice as a concept arises from the theory of the scattering of waves by crystals. You should be familiar with the diffraction of light by a 2-dimensional periodic object - a diffraction grating. Here an incident plane wave is diffracted into a set of different directions in a Fraunhofer pattern. An infinite periodic structure produces outgoing waves at particular angles, which are determined by the periodicity of the grating. What we discuss now is the generalisation to scattering by a three-dimensional periodic lattice. First calculate the scattering of a single atom (or more generally the basis that forms the unit cell) by an incoming plane wave, which should be familiar from elementary quantum 43 4.2. THE RECIPROCAL LATTICE AND DIFFRACTION Figure 4.2: Illustration of Bragg scattering from a crystal mechanics. An incoming plane wave of wavevector ko is incident on a potential centred at the point R. At large distances the scattered wave take the form of a circular wave. (See figure Fig. 4.2) The total field (here taken as a scalar) is then ψ ∝ eiko ·(r−R) + cf (r̂) eiko |r−R| |r − R| (4.4) All the details of the scattering is in the form factor f (r̂) which is a function of the scattering angle, the arrangement and type of atom, etc. The total scattered intensity is just set by c and we will assume it is small (for this reason we do not consider multiple scattering by the crystal) For sufficiently large distance from the scatterer, we can write r·R ko |r − R| ≈ ko r − ko r Define the scattered wavevector r k = ko r and the momentum transfer q = ko − k (4.5) (4.6) (4.7) we then have for the waveform ψ∝e iko ·r " eiq·R 1 + cf (r̂) r # . (4.8) Now sum over all the identical sites in the lattice, and the final formula is ψ∝e iko ·r " 1+c X i eiq·Ri fi (r̂) r # . (4.9) Away from the forward scattering direction, the incoming beam does not contribute, and we need only look at the summation term. We are adding together terms with different phases q · Ri , and these will lead to a cancellation unless the Bragg condition is satisfied q · R = 2πm (4.10) for all R in the lattice, and with m an integer (that depends on R). The special values of q ≡ G that satisfy this requirement lie on a lattice, which is called the reciprocal lattice. 1 One can check that the following prescription for the reciprocal lattice will satisfy the Bragg condition. The primitive vectors bi of the reciprocal lattice are given by a2 ∧ a3 and cyclic permutations . (4.11) b1 = 2π a1 · a2 ∧ a3 1 We can be sure that they are on a lattice, because if we have found any two vectors that satisfy Eq. (4.10), then their sum also satisfies the Bragg condition. 44 CHAPTER 4. PERIODIC SOLIDS AND DIFFRACTION 4.3 Diffraction conditions and Brillouin zones For elastic scattering, there are two conditions relating incident and outgoing momenta. Conservation of energy requires that the magnitudes of ko and k are equal, and the Bragg condition requires their difference to be a reciprocal lattice vector k − ko = G. The combination of the two can be rewritten as G G k· = ( )2 . (4.12) 2 2 Eq. (4.12) defines a plane constructed perpendicular to the vector G and intersecting this vector at its midpoint. The set of all such planes defines those incident wavevectors that satisfy the conditions for diffraction (see Fig. 4.3). Figure 4.3: Ewald construction. The points are the reciprocal lattice of the crystal. k0 is the incident wavevector, with the origin chosen so that it terminates on a reciprocal lattice point. A sphere of radius |k0 | is drawn about the origin, and a diffracted beam will be formed if this sphere intersects any other point in the reciprocal lattice. The angle θ is the Bragg angle of Eq. (4.14) This condition is familiar as Bragg’s Law. The condition Eq. (4.12) may also be written as 2π π sin θ = λ d (4.13) where λ = 2π/k, θ is the angle between the incident beam and the crystal planes perpendicular to G, and d is the separation between the plane and the origin. 4.3. DIFFRACTION CONDITIONS AND BRILLOUIN ZONES 45 Since the indices that define an actual crystal plane may contain a common factor n, whereas the definition used earlier for a set of planes removed it, we should generalise Eq. (4.13) to define d to be the spacing between adjacent parallel planes with indices h/n, k/n, l/n. Then we have 2d sin θ = nλ (4.14) which is the conventional statement of Bragg’s Law. To recap: • The set of planes that satisfy the Bragg condition can be constructed by finding those planes that are perpendicular bisectors of every reciprocal lattice vector G. A wave whose wavevector drawn from the origin terminates in any of these planes satisifies the condition for elastic diffraction. • The planes divide reciprocal space up into cells. The one closest to the origin is called the first Brillouin zone. The nth Brillouin zone consists of all the fragments exterior to the (n − 1)th plane (measured from the origin) but interior to the nth plane. • The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. This will play an important role in the discussion of electronic states in a periodic potential. • The volume of each Brillouin zone (adding up the fragments) is equal to the volume of the primitive unit cell of the reciprocal lattice, which is (2π)3 /Ωcell where Ωcell is the volume of the primitive unit cell of the crystal. 46 CHAPTER 4. PERIODIC SOLIDS AND DIFFRACTION Chapter 5 Electronic structure from local orbitals 5.1 Tight binding: Linear combination of atomic orbitals Perhaps the most natural view of a solid is to think about it as a collection of interacting atoms, and to build up the wavefunctions in the solid from the wavefunctions of the individual atoms. This is the linear combination of atomic orbital (LCAO) or tight-binding method. 5.2 Diatomic molecule revisited Remember our modelling of a covalently bonded diatomic molecule, where we worked with a highly restricted basis on one orbital per atom. For identical atoms, the full Hamiltonian consists of H = T + Va + Vb (5.1) with T the kinetic energy and Va , Vb the (identical potentials) on the two atoms. The basis set consists of two states |a > and |b > that satisfy T + Va |a > = E0 |a > T + Vb |b > = E0 |b > (5.2) (5.3) so that E0 is the eigenenergy of the atomic state, and we look for solutions |ψ >= α|a > +β|b > (5.4) We solve this in the usual way: Project H|ψ >= E|ψ > onto < a| and < 2 to get the simultaneous equations ! ! α Ẽ0 − E t =0 (5.5) β t∗ Ẽ0 − E neglecting the overlap elements < a|b >. Here Ẽ0 = Haa = ha |T + Va + Vb | ai = Ea + ha |Vb | ai 47 (5.6) 48 CHAPTER 5. ELECTRONIC STRUCTURE 1 is a shift of the atomic energy by the crystal field of the other atom(s). The more interesting term is the hopping matrix element that couples the atomic states together:1 t = Hab = ha |T + Va + Vb | bi (5.7) For t < 0, the new eigenstates are 1 |ψi = √ [|1i ∓ |2i] E = Ẽ0 ± |t| 2 (5.8) The lower energy (bonding) state has electron density higher between atoms. The higher energy (antibonding) state has node between atoms. 5.3 Linear chain Now let us generalised this to a ring of N identical atoms (i.e. a chain with periodic boundary conditions: X |ψi = ci |ii (5.9) i which now generates the matrix equations           2 E0 − E t 0 t E0 − E t 0 t E0 − E ... ... ... 0 0 0 t 0 0 ... 0 t ... 0 0 ... 0 0 ... ... ... ... E0 − E t ... t E0 − E           c1 c2 c3 ...          cN −1  (5.10) cN We have dropped the tilde on the diagonal elements — this is anyway just a parameter. The solutions are (check by substitution) (m) cj E (m) 5.4 jm 1 = √ exp 2πi N N   2πm = Eo + 2t cos N   (5.11) m = 0, 1, ..., N − 1 (5.12) Bloch’s theorem The periodicity of the solutions of the 1D ring is a simple example of a very general property of eigenstates in an infinite periodic potential. The Hamiltonian for a particle in a periodic potential h i Hψ(r) = −h̄2 ∇2 /2m + U (r) ψ(r) = Eψ(r), 1 (5.13) Note the sign of t depends on the symmetry of the orbitals: for s-states, with an attractive potential Vi < 0, then t is negative; but for px states t is positive for atoms aligned along x. 2 The t’s in the corners make this matrix symmetric under translations: it is called a circulant 49 5.4. BLOCH’S THEOREM Figure 5.1: Eigenvalues of the 1D chain Eq. (5.11) are confined to a band in energy centred on the (shifted) atomic energy level Ẽ0 . If N is very large, the energies form a continuous band and are periodic in m. Then we replace the index m by the continuous crystal momentum k = 2πm/N a, with a the lattice constant. So we could label the states more symmetrically by keeping a range −N/2 + 1 < m < N/2 (or −π/a < k < π/a); this is called the first Brillouin zone. where U (r + R) = U (r) for all R in a Bravais lattice. Bloch’s theorem states that they have the form ψnk (r) = eik·r unk (r) (5.14) unk (r + R) = unk (r) (5.15) ψnk (r + R) = eik·R ψnk (r) (5.16) where or, alternatively, that Notice that while the potential is periodic, the wave function consists of a plane wave times a periodic function. n is an index, call the band index, and we shall see the physical meaning of both n and k in a moment. Proof of Bloch’s theorem Here we sketch a proof of Bloch’s theorem, and we shall give a somewhat more intuitive (but longer) one later. First, let us define a translation operator TR , which when operating on any function, shifts the argument by a lattice vector R : TR f (r) = f (r + R) (5.17) It is straightforward3 to then show that TR commutes with the Hamiltonian: TR H = HTR (5.18) TR TR′ = TR′ TR = TR+R′ ; (5.19) Furthermore the translation operators commute with themselves. 3 Operate with the translation operator on Hψ and use the periodic symmetry of the potential 50 CHAPTER 5. ELECTRONIC STRUCTURE 1 We may now use a fundamental theorem of quantum mechanics; two commuting operators can be chosen to have the same eigenstates, so Hψ = Eψ TR ψ = c(R)ψ (5.20) Applying the results of Eq. (5.18) and Eq. (5.19), we see that the eigenvalues of T must satisfy c(R)c(R′ ) = c(R + R′ ) (5.21) Now let ai be three primitive vectors of the lattice, and write c(ai ) = e2πixi (5.22) which is just a definition of the xi , but we have chosen this form because the boundary conditions will in the end force the xi to be real. Since a general Bravais lattice vector can be written as R = n1 a1 + n2 a2 + n3 a3 , we can then use the rule of Eq. (5.21) to show that c(R) = c(a1 )n1 c(a2 )n2 c(a3 )n3 = e2πi(x1 n1 +x2 n2 +x3 n3 ) (5.23) which is precisely of the form c(R) = eik·R when k = x1 b 1 + x2 b 2 + x3 b 3 (5.24) and the bi are reciprocal lattice vectors that satisfy bi · aj = 2πδij . This is precisely Bloch’s theorem in the form given in Eq. (5.14). 5.5 Linear chain revisited We can use Bloch’s theorem to revisit the linear chain problem. If we want to make up a wave-function using only one-orbital per unit cell we now know that it must be of the form 1 X ikRj φ(r − Rj ) e ψk (r) = √ N j (5.25) and by comparison to the result of explicit calculation in Eq. 5.11 we can connect the notation: Rj = ja ; k = 2π m ; a = lattice constant Na (5.26) Translational symmetry restricts the phase relationship from site to site. With one orbital per unit cell there is then no choice in the wavefunction. Now evaluate the energy E(k) = hψk |H| ψk i 1 X −ikRj hφj |H| φm i eikRm e = N j,m (5.27) (5.28) and then writing Rm = Rj + Rn E(k) = = 1 X ikRn × hφj |H| φj+n i e N j,n 1 X X ikRn × hφ0 |H| φn i e N j n = t(eika + e−ika ) = 2t cos ka (5.29) (5.30) (5.31) 51 5.6. LCAO METHOD IN GENERAL k has to be restricted to a window to avoid double counting. The values of k are discrete, but very close together, spaced by ∆k = 2π/L, where L = N a. The range of k must cover kmax − kmin = 2π/a to give N states. Frequently, its convenient to choose the range −π/a < k < π/a, the first Brillouin zone. 5.6 LCAO method in general It is now fairly clear how to extend this method to higher dimensions, and to multiple orbitals per atom, and to multiple unit cells, with wavefunctions of the form ψm,k (r) = norb X 1 X 1 N2 n=1 R eik·R c(m) n (k)φn (r − R) . (5.32) • R are the lattice vectors. φn is the nth orbital in each unit cell. • m is the band index:if we have norb basis functions, we will have m = 1, ..., norb bands. • k is a three-dimensional vector, now restricted to a three-dimensional Brillouin zone (the Wigner-Seitz cell of the reciprocal lattice) (m) • cn (k) is then the matrix of coefficients to be determined by diagonalising the Hamiltonian. The dimension of the matrix concerned is norb × norb . In practice, how to do this: • Write down a Bloch state made up out of a single orbital in each unit cell. 1 X ik·R e φn (r − R) . φn,k (r) = 1 N2 R (5.33) • Bloch states of different k are orthogonal 1 X ik·R−ik′ ·R′ hφm,k′ |φn,k i = e hφm (r − R′ )|φn (r − R)i N ′ (5.34) R,R = = 1 X i(k−k′ )·R ik′ ·R′′ e e hφm (r − R − R′′ )|φn (r − R)i N R,R′′ X ′ ′′ δ(k − k′ ) eik ·R < φm (r − R′′ )|φn (r) > (5.35) (5.36) R′′ ≈ δ(k − k′ )δm,n X R′′ ′ ′′ eik ·R δR′′ ,0 = δ(k − k′ )δm,n (5.37) where the very last line can, in principle, be made exact by choosing a basis of states that are orthogonal in real space (and then known as Wannier functions). • This forms a basis to solve the problem, i.e. the eigenstates are now known to be linear combinations X ψm,k (r) = c(m) (5.38) n (k)φn,k (r) n • The eigenstates and eigenvalues are determined by the diagonalisation of the matrix Hm,n (k), which has elements Hm,n (k) = = hφm,k |H| φn,k i 1 X −ik·R′ e hφm (r − R′ ) |H| φn (r − R)i eik·R N ′ (5.39) (5.40) R,R = = 1 X X −ik·R′′ e hφm (r − R − R′′ ) |H| φn (r − R)i N ′′ R R X ′′ e−ik·R hφm (r − R′′ ) |H| φn (r)i R′′ (5.41) (5.42) 52 CHAPTER 5. ELECTRONIC STRUCTURE 1 • Notice that the phase factor involves k · R, where R are lattice vectors, not distances between atoms. 5.7 Periodic boundary conditions and counting states in 3 dimensions We saw that the spacing between k-points in 1D was 2π/L, where L is the linear dimension of the crystal. • This generalises to 3 dimensions: the volume associated with each k is ∆k = (2π)3 V (5.43) with V the volume of the crystal. • Within each primitive unit cell of the reciprocal lattice there are now precisely N allowed values of k, (N being the number of unit cells in the crystal). • In practice N is so big that the bands are continuous functions of k and we only need to remember density of states to count. • The bandstructure is periodic in the reciprocal lattice En (k + G) = En (k) for any reciprocal lattice vector G. It is sometimes useful to plot the bands in repeated zones, but remember that these states are just being relabelled and are not physically different. • Since electrons are fermions, each k-point can now be occupied by two electrons (double degeneracy for spin). So if we have a system which contains one electron per unit cell (e.g. a lattice of hydrogen atoms), half the states will be filled in the first Brillouin zone. Two electrons per unit cell fills a Brillouin zone’s worth of k states. Periodic boundary conditions and volume per k-point A formal proof of the number of allowed k-points uses Bloch’s theorem, and follows from the imposition of periodic boundary conditions: ψ(r + Ni ai ) = ψ(r) (5.44) where Ni are integers, with the number of primitive unit cells in the crystal being N = N1 N2 N3 , and ai primitive lattice vectors. Applying Bloch’s theorem, we have immediately that eiNi k·ai = 1, (5.45) so that the general form for the allowed Bloch wavevectors is k= 3 X mi i Ni bi , for mi integral. (5.46) with bi primitive reciprocal lattice vectors. Thus the volume of allowed k-space per allowed k-point is just b1 b2 b3 1 ∆k = · ∧ = b1 · b2 ∧ b3 . (5.47) N1 N2 N3 N 5.7. PERIODIC BOUNDARY CONDITIONS AND COUNTING STATES IN 3 DIMENSIONS53 Since b1 · b2 ∧ b3 = (2π)3 N/V is the volume of the unit cell of the reciprocal lattice (V is the volume of the crystal), Eq. (5.47) shows that the number of allowed wavevectors in the primitive unit cell is equal to the number of lattice sites in the crystal. We may thus rewrite Eq. (5.47) ∆k = (2π)3 V (5.48) 54 CHAPTER 5. ELECTRONIC STRUCTURE 1 Chapter 6 Electronic structure from plane waves 6.1 Nearly free electrons The tight binding method is clearly the appropriate starting point for a theory when the atomic potential is very strong, and the hopping probability for an electron to move from site-to-site is small. Here we explore the other limit, where instead the lattice potential is assumed to be weak, and the kinetic energy is the most important term. 1 6.2 Plane wave expansion of the wavefunction Bloch’s theorem suggests that we should be able to expand the electron wavefunction in terms of plane waves: X ck−G |k − Gi; , (6.1) |ψk i = G where the sum runs over all reciprocal lattice vectors G. Where does this come from? Bloch’s theorem states that ψk (r) is a product of a plane wave eikr and a function ck (r) with the periodicity of the lattice. We can Fourier-expand the periodic function as a sum over all P P reciprocal lattice vectors, G ck+G eiGr , which can be relabeled to G ck−G e−iGr . This gives P P for ψk (r) = hr|ψk i = G ck−G ei(k−G)r = G ck−G hr|k − Gi. If you are unconvinced, you can test compliance with Bloch’s theorem by checking that Eq. (5.16) applies: writing T̂R for translation by a lattice vector, we find T̂R |ψk i = X G ei(k−G)R ck−G |k − Gi = eikR |ψk i , (6.2) as e−iGR = 1 for any reciprocal lattice vector G. In this form, the electron wavefunction appears as a superposition of harmonics, whose wavevectors are related by reciprocal lattice vectors G. 1 There are lengthy descriptions of this approach in all the textbooks. A nice treatment can be found, for example, in the book by Singleton. The discussion below starts with a particularly convenient form of the electron wavefunction, which is constrained by Bloch’s theorem. This makes the derivation of the key results quicker, but you may still benefit from reading the more expansive discussions in the books. 55 56 CHAPTER 6. ELECTRONIC STRUCTURE 2 6.3 The Schrödinger equation in momentum space Writing the Hamiltonian as Ĥ = Ĥ0 + U , where H0 gives the kinetic energy and U is the periodic potential of the lattice, we are looking for the eigenvalues Ek in Ĥ|ψk i = Ek |ψk i . (6.3) Left multiply with a plane wave state hk|: hk|Ĥ|ψk i = Ek ck = hk|Ĥ0 |ki + X G hk|U (r)|k − Gick−G (6.4) We can identify hk|U (r)|k − Gi as the Fourier component UG of the periodic potential, where 1 Z NZ −iG·r UG = dr e−iG·r U (r) , (6.5) dr e U (r) = V V unit cell and conversely, U (r) = X UG eiG·r . (6.6) G ∗ Since the potential is real UG = U−G . We obtain the key equation:  (0)  E k − E k ck + (0) where the kinetic energy Ek = X UG ck−G = 0 , (6.7) G h̄2 2 k . 2m It is often convenient to rewrite q = k + K, where K is a reciprocal lattice vector chosen so that q lies in the first Brillouin zone, when Eq. (6.7) is just 2 " 6.4 X h̄2 UG−K cq−G = 0 (q − K)2 − E cq−K + 2m G # ! (6.8) One-dimensional chain Let’s get some insight from a simple model of a one-dimensional chain, but now simplifying the atomic potential so it just contains the leading Fourier components, i.e U (x) = 2U0 cos 2 2πx a (6.9) In order to get this explicit form, you have to relabel the set G → G + K in the second summation - but G is just a dummy label. 57 6.4. ONE-DIMENSIONAL CHAIN Figure 6.1: Band structure in the “empty lattice”, where U = 0, but we pretend to keep Bloch’s theorem, so we have multiple bands in each Brillouin zone. Turning on the potential U0 splits the degeneracies and opens up band gaps Then the secular determinant runs just down the tri-diagonal3 ... ... ... (k + 4π )2 − E U0 a 2π 2 U0 U0 (k + a ) − E U0 k2 − E U0 2π 2 U0 (k − a ) − E U0 4π 2 U0 (k − a ) − E ... ... ... (6.10) The physical interpretation of this is that an incident plane wave with wavevector k can be scattered by the potential into a state of k ± 2π/a. Multiple scattering then mixes these terms together. If U0 is small, we should be able to treat it perturbatively, remembering to take care of degeneracies. Of course if U0 = 0, we get the free electron eigenvalues (m) E0 (k) = (k − 2πm/a)2 , m = ..., −2, −1, 0, 1, 2, ... (6.11) which are now repeated, offset parabolas. Remember Bloch’s theorem tells us that k as a crystal momentum is conserved only within the first Brillouin zone (see Fig. 6.1). Now suppose U0 is turned on, but is very small. It will be important only for those momenta when two free electron states are nearly degenerate, for example, m=0,1 are degenerate when k = π/a. Near that point, we can simplify the band structure to the 2x2 matrix (k − 2π 2 ) a U0 −E U0 k2 − E ! c1 c0 ! (6.12) The solution of the determinantal leads to a quadratic equation: 1q 2 1 (k − (k − 2π/a)2 )2 + 4U02 E ± (k) = (k 2 + (k − 2π/a)2 ) ± 2 2 (6.13) Exactly at k = π/a, the energy levels are 0 E ± (π/a) = Eπ/a ± |U0 |, 3 We set h̄2 /2m = 1 for a moment (6.14) 58 CHAPTER 6. ELECTRONIC STRUCTURE 2 2p/a + y ya Figure 6.2: Energy bands in one dimension, and amplitudes of wavefunctions at the band edges and if we choose the potential to be attractive U0 < 0, the wavefunctions are (aside from normalisation) ψ − (π/a) = cos(πx/a) , ψ + (π/a) = sin(πx/a) . (6.15) The wavefunctions are plotted, along with the potential, in Fig. 6.2. Figure 6.3: Diatomic chain of atoms. Note that if the potentials on the two atoms are identical, and δ = 0, the chain converts to a monatomic chain of period a/2 6.5 Pseudopotential The NFE method and the tight-binding method are not accurate methods of electronic structure determination; nevertheless both of them exhibit the basic principles. They are commonly used to write down simple models for bands, with their parameters fit to more sophisticated calculations, or to experiment. It turns out that band gaps in semiconductors are usually fairly small, and the true dispersion can be modelled by scattering from a few Fourier components of the lattice potential. The reason is that the relevant scattering potential for valence band electrons is however MUCH smaller than the full atomic potential ze2 /r of an electron interacting with a nucleus of charge z. The effective potential for scattering of the valence electrons by the atomic cores is a weak pseudopotential. When we consider the band structure of a typical solid, we are concerned only with the valence electrons, and not with those tightly bound in the core, which remain nearly atomic. If we solve the full Schrödinger equation with the real Coulomb potential, we expect to calculate 59 6.5. PSEUDOPOTENTIAL not just the valence electronic states, but also the atomic like core states. A pseudopotential reproduces the valence states as the lowest eigenstates of the problem and neglects the core states. Figure 6.4: Pseudopotential: The true potential V (r) has a wavefunction for the valence electrons that oscillates rapidly near the core. The pseudopotential Vs (r) has a wavefunction Φs (r) that is smooth near the core, but approximates the true wavefunction far from the core region. A weak pseudopotential acting on a smooth pseudo-wavefunction gives nearly the same energy eigenvalues for the valence electrons as the full atomic potential does acting on real wavefunctions. Away from the atomic cores, the pseudopotential matches the true potential, and the pseudo-wavefunction approximates the true one. A formal derivation of how this works can be given using the method of orthogonalised plane waves. The atomic states are well described by the Bloch functions fnk of the LCAO or tight-binding scheme Eq. (5.32). Higher states, which extend well beyond the atoms will not necessarily be of this kind, but they must be orthogonal to the core levels. This suggests that we should use as a basis 4 |χk >= |k > − X n βn |fnk > , (6.16) where |k > is a plane wave, and the coefficients βn (k) are chosen to make the states χ orthogonal to the core states |fnk >. The states in Eq. (6.16) are orthogonalised plane waves (OPW); away from the core, they are plane wave like, but in the vicinity of the core they oscillate rapidly so as to be orthogonal to the core levels. We can now use the OPW’s as basis states for the diagonalisation in the same way that we used plane waves in the NFE, viz X |ψk >= αk−G |χk−G > . (6.17) G This turns out to converge very rapidly, with very few coefficients, and only a few reciprocal lattice vectors are included in the sum. The following discussion explains why. Suppose we have solved our problem exactly and determined the coefficients α. Now consider the 4 We use Dirac’s notation of bra and ket, where |k > represents the plane wave state exp(ik · r), and < R φ1 |T |φ2 > represents the matrix element dr φ∗1 (r)T (r)φ2 (r) of the operator T . 60 CHAPTER 6. ELECTRONIC STRUCTURE 2 sum of plane waves familiar from the plane-wave expansion, but using the same coefficients, i.e. X αk−G |k − G > , |φk >= (6.18) G and then 5 it is easily shown that X |ψ >= |φ > − n < fn |φ > |fn > . Then substitute into the Schrodinger equation H|ψ >= E|ψ >, which gives us X H|φ > + (E − En ) < fn |φ > |fn >= E|φ > (6.19) (6.20) n We may look upon this as a new Schrödinger equation with a pseudopotential defined by the operator X Vs |φ >= U |φ > + (E − En ) < fn |φ > |fn > (6.21) n which may be written as a non-local operator in space Z (Vs − U )φ(r) = VR (r, r′ )φ(r′ ) dr′ , where VR (r, r′ ) = X n (E − En )fn (r)fn∗ (r′ ) . (6.22) (6.23) The pseudopotential acts on the smooth pseudo-wavefunctions |φ >, whereas the bare Hamiltonian acts on the highly oscillating wavefunctions |ψ >. One can see in Eq. (6.21) that there is cancellation between the two terms. The bare potential is large and attractive, especially near the atomic core at r ≈ 0; the second term VR is positive, and this cancellation reduces the total value of Vs especially near the core. Away from the core, the pseudopotential approaches the bare potential. 5 Saving more notation by dropping the index k Chapter 7 Bandstructure of real materials 7.1 Bands and Brillouin zones In the last chapter, we noticed that we get band gaps forming by interference of degenerate forward- and backward going plane waves, which then mix to make standing waves. Brillouin zones. What is the condition that we get a gap in a three-dimensional band structure? A gap will arise from the splitting of a degeneracy due to scattering from some fourier component of the lattice potential, i.e. that E0 (k) = E0 (k − G) (7.1) which means (for a given G) to find the value of k such that |k|2 = |k − G|2 . Equivalently, this is 2 G G = k· (7.2) 2 2 which is satisfied by any vector lying in a plane perpendicular to, and bisecting G. This is, by definition, the boundary of a Brillouin zone; it is also the Bragg scattering condition, not at all coincidentally.1 Electronic bands. We found that the energy eigenstates formed discrete bands En (k), which are continuous functions of the momentum k and are additionally labelled by a band index n. The bands are periodic: En (k + G) = En (k). Bloch’s theorem again. The eigenstates are of the form given by Bloch’s theorem ψnk (r) = eik·r unk (r) (7.3) where u(r) is periodic on the lattice. Notice that if we make the substitution k → k + G, Eq. (7.3) continues to hold. This tells us that k can always be chosen inside the first Brillouin zone for convenience, although it is occasionally useful to plot the bands in an extended or repeated zone scheme as in Fig. 6.2. Crystal momentum. The quantity h̄k is the crystal momentum, and enters conservation laws for scattering processes. For example, if an electron absorbs the momentum of a phonon of 1 Notice that the Bragg condition applies to both the incoming and outgoing waves in the original discussion in Chapter 4, just with a relabelling of G → −G 61 62 CHAPTER 7. BAND THEORY OF METALS AND INSULATORS wavevector q, the final state will have a Bloch wavevector k′ = k + q + G, where G is whatever reciprocal lattice vector necessary to keep k′ inside the Brillouin zone. Physical momentum can always be transferred to the lattice in arbitrary units of h̄G. Notice that depending on the energy conservation, processes can thus lead to transitions between bands. Counting states. In a big system, the allowed k-points are discrete but very closely spaced. Each occupies a volume (2π)3 ∆k = (7.4) V with V the volume of the crystal. Thus within each primitive unit cell or Brillouin zone of the reciprocal lattice there are now precisely N allowed values of k, (N being the number of unit cells in the crystal). Even number rule. Allowing for spin, two electrons per real space unit cell fills a Brillouin zone’s worth of k states. 7.2 Metals and insulators in band theory The last point is critical to the distinction that band theory makes between a metal and an insulator. A (non-magnetic) system with an even number of electrons per unit cell may be an insulator. In all other cases, the fermi energy must lie in a band and the material will be predicted2 to be a metal. Metallicity may also be the case even if the two-electron rule holds, if different bands overlap in energy so that the counting is satified by two or more partially filled bands. Notation The bandstructure En (k) defines a function in three-dimensions which is difficult to visualise. Conventionally, what is plotted are cuts through this function along particular directions in k-space. Also, a shorthand is used for directions in k-space and points on the zone boundary, which you will often see in band structures. • Γ = (0, 0, 0) is the zone centre. • X is the point on the zone boundary in the (100) direction; Y in the (010) direction; Z in the (001) direction. Except if these directions are equivalent by symmetry (e.g. cubic) they are all called ”X”. • L is the zone boundary point in the (111) direction. • K in the (110) direction. • You will also often see particular bands labelled either along lines or at points by greek or latin capital letters with a subscript. These notations label the group representation of the state (symmetry) and we won’t discuss them further here. 2 Band theory may fail in the case of strongly correlated systems where the Coulomb repulsion between electrons is larger than the bandwidth, producing a Mott insulator 63 7.2. METALS AND INSULATORS IN BAND THEORY 1D 2D 3D Figure 7.1: Density of states in one (top curve), two (middle curve) and three (lower curve) dimensions Density of states We have dealt earlier with the density of states of a free electron band in 1. The maxima Emax and minima Emin of all bands must have a locally quadratic dispersion with respect to momenta measured from the minima or maxima. Hence the density of states (in 3D) near the minima will be the same g(E > ∼ V m∗ Emin ) = 2 2 π h̄ 2m∗ (E − Emin ) h̄2 !1 2 . (7.5) as before, with now however the replacement of the bare mass by an effective mass m∗ = (m∗x m∗y m∗z )1/3 averaging the curvature of the bands in the three directions 3 . A similar form 1 must apply near the band maxima, but with now g(E) ∝ (Emax − E) 2 . Notice that the flatter the band, the larger the effective mass, and the larger the density of states4 . Since every band is a surface it will have saddle points (in two dimensions or greater) which are points where the bands are flat but the curvature is of opposite signs in different directions. Examples of the generic behaviour of the density of states in one, two and three dimensions are shown in Fig. 7.1. The saddle points give rise to cusps in the density of states in 3D, and a logarithmic singularity in 2D. For any form of E(k), the density of states is g(E) = X n 3 gn (E) = X Z dk δ(E − En (k)) , 4π 3 n Since the energy E(k) is a quadratic form about the minimum, the effective masses are defined by ∂ 2 E(k) 2 ∂kα kmin 4 along the principal axes α of the ellipsoid of energy. The functional forms are different in one and two dimensions. (7.6) h̄2 m∗ α = 64 CHAPTER 7. BAND THEORY OF METALS AND INSULATORS Figure 7.2: Surface of constant energy Because of the δ-function in Eq. (7.6), the momentum integral is actually over a surface in k-space Sn which depends on the energy E; Sn (EF ) is the Fermi surface. We can separate the integral in k into a two-dimensional surface integral along a contour of constant energy, and an integral perpendicular to this surface dk⊥ (see Fig. 7.2). Thus Z Z dS gn (E) = dk⊥ (k) δ(E − En (k)) 3 S (E) 4π Z n 1 dS , (7.7) = 3 |∇ E (k)| 4π ⊥ n Sn (E) where ∇⊥ En (k) is the derivative of the energy in the normal direction.5 Notice the appearance of the gradient term in the denominator of Eq. (7.7), which must vanish at the edges of the band, and also at saddle points, which exist generically in two and three dimensional bands. Maxima, minima, and saddle points are all generically described by dispersion (measured relative to the critical point) of h̄2 2 h̄2 2 h̄2 2 kx ± ky ± k (7.8) E(k) = E0 ± 2mx 2my 2mz z If all the signs in Eq. (7.8) are positive, this is a band minimum; if all negative, this is a band maximum; when the signs are mixed there is a saddle point. In the vicinity of each of these critical points, also called van Hove singularities, the density of states (or its derivative) is singular. In two dimensions, a saddle point gives rise to a logarithmically singular density of states, whereas in three dimensions there is a discontinuity in the derivative. 7.3 Examples of band structures Metals If there are not an even number of electrons per unit cell, the chemical potential must lie in a band, and there will be no energy gap. Because there are low-lying electronic excitations, the system is a metal. The fermi surface is the surface in momentum space that separates the 5 We are making use of the standard relation δ(f (x) − f (x0 )) = δ(x − x0 )/|f ′ (x0 )| 7.3. EXAMPLES OF BAND STRUCTURES 65 Figure 7.3: Band structure of Al (solid line) compared to the free electron parabolas (dotted line). Calculations from Stumpf and Scheffler, cited by Marder. Figure 7.4: Band structure of Cu metal [from G.A.Burdick, Phys. Rev.129,138 (1963)], cited by Grosso and Parravicini filled from the empty states. In a simple metal like N a (3s1 - 1 valence electron) or Al ( 3s2 p1 - 3 valence electrons) this is nearly a sphere like the free electron gas. In other cases (e.g. Cu, 4s3d10 ) the sphere extends in some directions to meet the Brillouin zone boundary surface. There can be situations where several bands are cut by the fermi energy, and the topology of fermi surfaces is sometime complicated. Semimetals Even if there are the right number of electrons to fill bands and make a semiconductor, the bands may still overlap. Consequently, the chemical potential will intersect more than one band, making a pocket of electrons in one band and removing a pocket of electrons from the band below (which as we shall see later, are sometimes called holes). This accounts for the metallicity of Ca and M g (which have two electrons per unit cell), and also As, Sb and Bi. 66 CHAPTER 7. BAND THEORY OF METALS AND INSULATORS Figure 7.5: Fermi surface of Cu The latter, despite being group V elements, have crystal structures that contain 2 atoms per unit cell and therefore 10 valence electrons. We have previously alluded to graphite, which is a special kind of semimetal. We noted that a graphene sheet has conduction and valence bands that touch at special points on the zone boundary. Over all except these points, the band structure has a gap - thus graphene is more correctly titled a zero-gap semiconductor. Semiconductors and insulators If there are an even number of electrons per unit cell, then it is possible (if the bands don’t overlap) for the occupied states all to lie in a set of filled bands, with an energy gap to the empty states. In this case the system will be a semiconductor or insulator. Such is the case for the group IV elements C, Si and Ge, as well as the important III-V compounds such as GaAs and AlAs. These elements and compounds in fact have 2 atoms per unit cell (diamond or zincblende structure) and have a total of 8 valence electrons per unit cell — 4 filled bands. The band structures of Si, Ge, and GaAs are shown in Fig. 7.6 and Fig. 7.7. The maximum of the valence bands of all the materials is at Γ. Si and Ge are both indirect gap materials, because the conduction bands have minima either in the (100) direction (Si) or the (111) direction (Ge). 7.3. EXAMPLES OF BAND STRUCTURES 67 Figure 7.6: Pseudopotential band structure of Si and Ge [M.L.Cohen and T.K.Bergstresser Phys.Rev141, 789 (1966)]. The energies of the optical transitions are taken from experiment. Figure 7.7: (1966)] Band structure of GaAs [M.L.Cohen and T.K.Bergstresser Phys.Rev141, 789 68 CHAPTER 7. BAND THEORY OF METALS AND INSULATORS Figure 7.8: The valence charge density for Ge, GaAs, and ZnSe from an early pseudopotential calculation, plotted along a surface in a 110 plane that contains the two atoms of the unit cell. Note the (pseudo-)charge density shifting from the centre of the bond in Ge to be almost entirely ionic in ZnSe. [M.L.Cohen, Science 179, 1189 (1973)] Chapter 8 Experimental probes of the band structure 8.1 Quantum oscillations – de Haas van Alphen effect In high magnetic fields B > 1 T and in pure samples, many material properties have been found to oscillate as a function of applied magnetic field. The form of these quantum oscillations, in particular the frequency of oscillation, can be used to infer the shape of the Fermi surface and other key electronic properties. 8.1.1 Size of cyclotron orbits A full quantum mechanical treatment of the motion of electrons in a strong magnetic field is problematic. When the lattice potential can be neglected, for free electrons, the Schrödinger equation can be solved directly. For real materials, however, the lattice potential is essential to the band structure and cannot be neglected. In this case, progress can be made with a semiclassical treatment which makes use of the Bohr-Sommerfeld quantisation condition (see, e.g. Kittel ch. 9):   I 1 pdr = n + h (8.1) 2 Here, p is the canonical momentum, conjugate to the position r. The canonical momentum can be written as the sum of the kinetic (or mv-) momentum, mv, and the field momentum, qA. Particles with charge q moving in a strong magnetic field B are forced into an orbit by the Lorentz force: mv̇ = q ṙ × B. This relation connects the components of velocity and position of the particle in the plane perpendicular to B and can be integrated: mv⊥ = qr × B, where r is measured from the centre of the orbit. This allows us to write p = mv + qA = q(r × B + A), and using flux), we obtain: I pdr = q I r × Bdr + qΦ = −qΦ , 69 H Adr = Φ (the magnetic (8.2) 70 CHAPTER 8. EXPERIMENTAL PROBES OF THE BAND STRUCTURE because r × Bdr = −B r × dr = −2BAr , where Ar is the real space area enclosed by the orbit’s projection onto the plane perpendicular to B. H H We arrive at the conclusion that the flux threading the real space orbit is quantised: 1 h Φn = A(n) r Bn = (n + ) 2 e (8.3) Can we relate the motion of the electron in real space to the accompanying motion in kspace? From our earlier result for the relation between momentum and position, mv⊥ = h̄k⊥ = qr × B, we find that the k-space orbit has the same shape as the real space orbit, but is turned . This means that the area enclosed by the k-space orbit Ak by 90 degrees and stretched by Bq h̄ is Ak =  2 e h̄ B 2 Ar , (8.4) where the electron charge e has been inserted for the more general q. Combining this result with Eqn. 8.3, we find Ak = 8.1.2 1 2πe B(n + ) h̄ 2 (8.5) Density of states oscillations In a magnetic field, the allowed k-states no longer form a regular lattice in reciprocal space, as k is no longer a good quantum number. All the k-states in the vicinity of a k-orbit superimpose to form the orbital motion of the electrons. The electrons now ’live’ on a set of cylinders, the Landau tubes, with quantised cross-sectional areas. These cylinders, whose cross-sectional area expands with increasing field B, cut through the zero-field Fermi surface of the metal. What effect will this have on the B-dependence of the density of states at the Fermi level, g(EF )? Considering a particular slice ⊥ B through the Fermi surface with area Ak , this will now only contribute to g(EF ), if its area coincides with the area of one of the Landau tubes. As B increases, one Landau tube after the other will 2πe (n + 12 ). Consequently, the contribution of this satisfy this condition, at field values B1n = h̄A k slice to g(EF ) oscillates with a period ∆  1 B  = 1 Bn+1 − 1 2πe 1 = Bn h̄ Ak (8.6) This is the Onsager relation, which links the period of quantum oscillations to the crosssectional area of the Fermi surface. There remains one important consideration: in reality, we can only measure quantum oscillations associated with extremal orbits. These arise, where a Landau tube can touch, rather than cut through, the Fermi surface. At such regions of the Fermi surface, there are many closelying orbits with nearly identical cross-section, which causes the corresponding density of states oscillations to add coherently. For the rest of the Fermi surface, the oscillations attributed to each orbit have different period and they add incoherently, which wipes out the effect. 8.1. QUANTUM OSCILLATIONS – DE HAAS VAN ALPHEN EFFECT 71 The number of points collected by each orbit F I Figure 8.1: Quantisation of k-space orbits in high magnetic fields. In 2D, the grid of allowed states in zero field collapses onto rings spaced according to the Onsager relation (upper panel). In 3D, these rings extrude to cylinders - the Landau tubes. Quantum oscillations will be detected for extremal Fermi surface cross-sections, as successive Landau tubes push through the Fermi surface with increasing magnetic field B. 8.1.3 Experimental observation of quantum oscillations Many observable properties depend directly on the density of states at the Fermi level, and many of these have been used to detect quantum oscillations. The classic example is the magnetic susceptibility χ, which according to simple theory is proportional to g(EF ). Measurements of χ(B) at low temperature exhibit oscillations which, when plotted versus (1/B) allow the determination of extremal Fermi surface cross-sections. This is called the de Haas-van Alphen effect. Similar oscillations can be observed in measurements of electrical resistivity (’Shubnikov-de Haas’), of the magnetisation, of the sample length and of the entropy – which can be picked up by measuring the temperature oscillations of a thermally isolated sample. Generally, these experiments require • High purity samples: the electronic mean free path must be long enough to allow the 72 CHAPTER 8. EXPERIMENTAL PROBES OF THE BAND STRUCTURE Amplitude [a.u.] 0.1 2f Signal [a.u.] 0.05 100 10 1 0.1 0 5 10 15 DHvA Frequency [kT] 20 0 17 −0.05 10 18 12 14 Magnetic Field [T] 16 18 Figure 8.2: De Haas-van Alphen oscillations observed in the transition metal oxide CaVO3 [Inoue, Bergemann, Hase, Julian, Phys. Rev. Lett. 88, 236403 (2002)]. electrons to complete roughly one cyclotron orbit before scattering. • High magnetic field: high magnetic fields make the cyclotron orbits tighter, which equally helps to fulfil the mean free path condition. • Low temperature: The density of states oscillations are smeared out, when the Fermi surface itself is smeared by thermal broadening of the Fermi-Dirac distribution. Typically, experiments are carried out below 1 K for transition metal compounds, and below 100 mK for heavy fermion compounds (see Appendix). 8.2 Optical transitions The band structure provides the excitation spectrum of the solid. The ground state of the system involves filling states up to the fermi energy, but we can also excite the system in different ways. One of the simplest is the absorption of a photon, which can be visualised as an excitation of an electron from an occupied state into an empty state, leaving behind a ”hole” in the valence band. See Fig. 8.3. The minimum gap in a semiconductor is the energy difference between the highest occupied state and the lowest unoccupied state, and this is the threshold for optical absorption (neglecting excitonic physics, see later). In some semiconductors, the maximum valence band state and the minimum in the conduction band occur at the same momentum - in such a direct gap system, direct optical excitation is allowed at the minimum gap, and an important example is GaAs. Si and Ge are example of indirect gap materials, because the conduction band minimum is toward the edge of the zone boundary. The minimum energy transition is at large momentum, and therefore cannot be accomplished by direct absorption of a photon. The lowest energy transition is instead a phonon-mediated transition where the energy is provided by the photon and the momentum provided by the phonon. This is much less efficient than direct optical absorption. 8.2. OPTICAL TRANSITIONS 73 Figure 8.3: Direct absorption by light is a nearly vertical transition since the wavevector of a photon with energy of order a semiconductor gap is much smaller than the typical momentum of an electron. (a) In a direct gap semiconductor, such as GaAs, the lowest energy available states for hole and electron are at the same momentum, and the optical threshold is at the vertical energy gap. (b) IN an indirect gap material (e.g. Si or Ge), the minimum energy excitation of electron and hole pair connects state of different momenta - and a phonon of momentum q must be excited concurrently with the photon. Luminescence is the inverse process of recombination of an electron-hole pair to emit light. It comes about if electrons and holes are injected into a semiconductor (perhaps electrically, as in a light-emitting diode). Obviously, this process will not be efficient in an indirect gap semiconductor but is more so in a direct gap material. This simple fact explains why GaAs and other III-V compounds are the basis of most practical opto-electronics in use today, whereas Si is the workhorse of electrical devices. 74 CHAPTER 8. EXPERIMENTAL PROBES OF THE BAND STRUCTURE Figure 8.4: The interband absorption spectrum of Si has a threshold at the indirect gap Eg ≈ 1.1 eV which involves a phonon and is very weak. The energies E1 and E2 correspond to critical points where the conduction and valence bands are vertically parallel to one another; absorption is direct (more efficient) and also enhanced by the enhanced joint density of electron and hole states. [E.D.Palik, Handbook of the optical constants of solids, AP, 1985] . 8.3 Photoemission The most direct way to measure the electron spectral function directly is by photoemission, although this is a difficult experiment to do with high resolution. In a photoemission experiment, photons are incident on a solid, and cause transitions from occupied states to plane wave-like states well above the vacuum energy; the excited electron leaves the crystal and is collected in a detector that analyses both its energy and momentum.1 The photon carries very little momentum, so the momentum of the final electron parallel to the surface is the same as the initial state in the solid, while of course the perpendicular component of the momentum is not conserved. Photoemission data is therefore most easy to interpret when there is little dispersion of the electronic bands perpendicular to the surface, as occurs in anisotropic layered materials. It is fortunate that there are many interesting materials (including high-temperature superoconductors) in this class. If one analyses both the energy and the momentum of the outgoing electron, (this is Angle Resolved Photo-Emission Spectroscopy, or ARPES) one can determine the band structure directly. Integrating over all angles gives a spectrum that is proportional to the total density of states. 1 For a detailed discussion of photoemission experiments, see Z.X.Shen and D.S.Dessau, Physics Reports, 253, 1-162 (1995) 8.4. TUNNELLING 75 Figure 8.5: Schematics of a photoemission experiment. The optical transitions are nearly vertical, so the electrons are excited from the valence bands to high energy excited states (above the vacuum energy necessary to escape from the crystal) with the same crystal momentum. In this case the two transitions that can be excited by a single frequency will yield a double peak in the kinetic energy distribution of the escaped electrons. When the excited electrons escape through the surface of the crystal, their momentum perpendicular to the surface will be changed. If the surface is smooth enough, the momentum of the electron parallel to the surface is conserved, so the angle of the detector can be used to scan kk The ideal schematic for interpreting an ARPES experiment would then be as shown in Fig. 8.5. An example of real data is shown in Fig. 8.7. Photoemission can give information only about occupied states. The technique of inverse photoemission involves inserting an electron of known energy into a sample and measuring the ejected photon. Since the added electron must go into unoccupied state, this spectroscopy allows one to map out unoccupied bands, providing information complementary to photoemission. 8.4 Tunnelling Tunnelling spectroscopies (injecting or removing electrons) through a barrier have now evolved to be very important probes of materials. The principle here is that a potential barrier allows one to maintain a probe (usually a simple metal) at an electrical bias different from the chemical potential of the material. Thus the current passed through the barrier comes from a nonequilibrium injection (tunnelling) through the barrier. A model for a simple metal tunnelling into a more complex material is shown in Fig. 8.8. With the metal and sample maintained at different electrical potentials separated by a bias eV , 76 CHAPTER 8. EXPERIMENTAL PROBES OF THE BAND STRUCTURE Figure 8.6: Idealised results from a photoemission experiment. A peak is observed at the band energy in each spectrum, but disappears when the band crosses the Fermi energy then the current through the junction can be estimated to be of the form I∝ Z µ µ+eV gL (ω)gR (ω)T (ω) (8.7) where T is the transmission through the barrier for an electron of energy ω and gL and gR are the densities of states.2 If the barrier is very high so that T is not a strong function of energy, and if the density of states in the contact/probe is approximately constant, then the energy-dependence comes entirely from the density of states inside the material. Notice then that the differential conductivity is proportional to the density of states (see Fig. 8.8): dI/dV ∝ g(µ + eV ) . (8.8) It is difficult to maintain very large biases, so most experiments are limited to probing electronic structure within a volt or so of the fermi energy. Tunnel junctions are sometimes fabricated by deposition of a thin insulating layer followed by a metal contact. The technique of scanning tunnelling microscopy (STM) uses a small tip, with vacuum as the surface barrier. Because the tunnel probability is an exponential function of the barrier thickness, this scheme provides high (close to atomic, in some cases) spatial resolution, even though the tip radius will be nm or larger. By hooking this up to a piezoelectric drive in a feedback loop, it has proved possible to provide not only I − V characteristics at a single point, but also spatial maps of the surface. Scanned probe spectroscopies have advanced to become extraordinary tools at the nanoscale. As well as STM, it is possible to measure forces near a surface (atomic force microscopy, AFM ), which is particularly useful for insulating samples. It has proven possible to manipulate individual atoms, to measure the magnetism of a single spin, and with small single-electron transistors to study to motion of single electron charges in the material. 2 Strictly this formula applies when the tunnelling process does not conserve momentum parallel to the interface, i.e. if the surface is rough or disorded. 8.4. TUNNELLING 77 Figure 8.7: . Photoemission spectra on the two dimensional layered metal Sr2 RuO4 . The bands are nearly two-dimensional in character, so the interpretation of the photoemission data is straightforward – different angles (see Fig. 8.5 )correspond to different in-plane momenta. The upper panels show energy scans for different angles that correspond to changing the inplane momentum in the direction from the centre of the Brillouin zone Γ towards the centre of the zone face M and the corner X. Several bands cross the Fermi energy, with different velocities, and sharpen as their energies approach EF . The left hand lower panel plots the positions of the peaks as a function of momentum at the fermi energy, to be compared with the band structure calculation of the fermi surface(s) on the lower right. [Experiment from Damascelli et al, PRL; theory from Mazin et al PRL 79, 733 (1997)] 78 CHAPTER 8. EXPERIMENTAL PROBES OF THE BAND STRUCTURE Figure 8.8: Schematic description of tunnelling between two materials maintained at a relative bias eV . The current is approximately given by the integrated area between the two chemical potentials (provided the matrix element for tunnelling is taken constant.) If the density of states of the contact (or probe, labelled 1 in the figure )is also slowly varying, then the differential conductance dI/dV is proportional to the density of states of the material itself, at the bias eV above the chemical potential µ2 . 8.4. TUNNELLING 79 Figure 8.9: Differential conductance of a tunnel junction between superconducting Pb and metallic Mg reveals the gap in the density of states of superconducting lead. [I. Giaever, Nobel Prize Lecture, 1973] 80 CHAPTER 8. EXPERIMENTAL PROBES OF THE BAND STRUCTURE Figure 8.10: An array of Fe atoms arranged in a corral on the surface of Cu traps a surface electron state whose density can be imaged by STM. M.F. Crommie, C.P. Lutz, D.M. Eigler, E.J. Heller. Surface Review and Letters 2 (1), 127-137 (1995). Chapter 9 Semiclassical model of electron dynamics 9.1 Wavepackets and equations of motion We now want to discuss the dynamics of electrons in energy bands. Because the band structure is dispersive, we should treat particles as wave-packets. The band energy ǫ(k) is the frequency associated with the phase rotation of the wavefunction, ψk e−iǫ(k)t/h̄ , but for the motion of a wave packet in a dispersive band, we should use the group velocity, dω/dk, or as a vector ṙ = vg = h̄−1 ∇k ǫ(k) , (9.1) where r is the position of the wavepacket. All the effects of the interaction with the lattice are contained in the dispersion ǫ(k). If a force F is applied to a particle, the rate of doing work on the particle is dǫ dk dǫk = = F vg dt dk dt (9.2) which leads to the key relation h̄ dk = F = −e(E + v ∧ B) = −e(E + h̄−1 ∇k ǫ(k) ∧ B) dt (9.3) where we have introduced electric E and magnetic B fields. The effect of an electric field is to shift the crystal momentum in the direction of the field, whereas the effect of a magnetic field is conservative - the motion in k-space is normal to the gradient of the energy. Thus a magnetic field causes an electron to move on a line of constant energy, in a plane perpendicular to the magnetic field. This property is the basis of magnetic techniques to measure the fermi surface of metals. Bloch oscillations Suppose we have a one-dimensional electron band, such as shown in in Fig. 9.1. The group velocity is also shown — note that it reaches maximum size about half way to the zone boundary, 81 82 CHAPTER 9. SEMICLASSICAL DYNAMICS Figure 9.1: Band energy E(k) (solid line) and group velocity v(k) dashed line in a simple 1D band. A wavepacket progressing its crystal momentum according to Eq. (9.4) accelerates as k increases from zero, and then slows and reverses direction as k approaches the zone boundary. and then decreases to zero at the zone boundary. If an electron in this band were subject to a constant electric field, we get eEt k(t) = k(0) − , (9.4) h̄ so that the wave packet of electrons oscillates up and down the energy surface. It we start from the minimum of the band, then the group velocity grows linearly in time as for a free electron accelerating (though with a mass different from the free electron mass). However, on approaching the zone boundary, the group velocity slows - the acceleration of the particle is opposite to the applied force. What is actually happening is buried within the semiclassical model via the dispersion ǫ(k): as the wavepacket approaches the Brillouin zone boundary, real momentum (not crystal momentum k) is transferred to the lattice, so that on reaching the zone boundary the particle is Bragg-reflected. Thus a DC electric field may be used - in principle - to generate an AC electrical current. All attempts to observe these Bloch oscillations in conventional solids has so far failed. The reason is that in practice it is impossible to have wavepackets reach such large values of momentum as π/a due to scattering from impurities and phonons in the solid. We will incorporate scattering processes in the theory in a moment. It turns out however, that one can make artificial periodic potentials in a semiconductor superlattice. The details of this process will be discussed later, but for our purposes the net effect is to produce a square well potential that is periodic with a periodicity that can be much longer than the atom spacing. The corresponding momentum at the zone boundary is now much smaller, so the wavepacket does not have to be excited to such high velocities. The signature of the Bloch oscillations is microwave radiation produced by the oscillating charge at a frequency that is proportional to the DC electrical field. 83 9.1. WAVEPACKETS AND EQUATIONS OF MOTION Figure 9.2: Schematic diagram of energy versus space of the conduction and electron bands in a periodic heterostructure lattice. The tilting is produced by the applied electric field. The levels shown form what is called a Wannier-Stark ladder for electron wavepackets made by excitation from the valence band in one quantum well, either vertically (n=0) or to neighbouring (n = ±1) or next-neighbouring n = ±2 wells of the electron lattice. In the experiment, electrons (and holes) are excited optically by a short-pulse laser whose frequency is just above the band gap of the semiconductor (i.e. a few ×1015 Hz). The electrical radiation (on a time scale of picoseconds) is monitored as a function of time and for different DC electrical biases, shown on the left panel. The spectral content is then determined by taking a fourier transform of the wavepackets (right panel); at large negative voltages one sees a peak at a frequency that increases with increasing bias. The device is not symmetric, and therefore has an offset voltage of about -2.4 V before the Bloch oscillation regime is reached. [From Waschke et al. Physical Review Letters 70, 3319 (1993).] Approximations and justification for the semiclassical model A full justification of the semiclassical model is not straightforward and we will not go into that here. [See Kittel, Appendix E, and for a more formal treatment J. Zak, Physical Review 168 686 (1968)] • Note that at least the semiclassical picture takes note of the fact that the Bloch states are stationary eigenstates of the full periodic potential of the lattice, and so there are no collisions with the ions. • We must be actually describing the motion of a wavepacket X ψn (r, t) = g(k − k′ )ψnk′ (r, t) exp [−iǫn (k′ )t/h̄] where g(k) → 0 if |k| > ∆k (9.5) k′ The wavepacket is described by a function g(k) that is sharply peaked, of width ∆k, say. Clearly ∆k ≪ 1/a, with a the lattice constant (otherwise the packet will disperse strongly). • The size of the packet in real space is therefore ∆R ≈ 1/∆k. Consequently the semiclassical model can only be used to describe the response to fields that vary slowly in space, on a scale much larger than the lattice constant. 84 CHAPTER 9. SEMICLASSICAL DYNAMICS • n, the band index, is assumed to be a good quantum number. Clearly if the lattice potential were tiny, we would expect to return to free electrons, and be able to accelerate particles to high energies and make transitions between bands. Rather naturally the constraint is that the characteristic field energies be small in comparison to the band gap Egap : they are in fact 2 eEa ≪ Egap /EF (9.6) with EF the characteristic fermi energy, or overall bandwidth. The electric fields in a metal rarely exceed 1 V m− 1, when the LHS of this inequality is about 10− 10 eV; not in danger. • The corresponding constraint on magnetic fields is 2 h̄ωc ≪ Egap /EF (9.7) with ωc = eB/mc the cyclotron frequency. This corresponds to about 10−2 eV in a field of 1 Tesla, so that strong magnetic fields indeed may cause transitions between bands, a process of magnetic breakdown. • The last condition is that of course that the frequency of the fields must be much smaller than the transition energies between levels, i.e. h̄ω ≪ Egap 9.2 Electrons and holes in semiconductors An immediate consequence of this picture is that filled bands are inert. If all the electrons states in a zone are occupied, then the total current is got by integrating the group velocity over the whole zone; but the group velocity is the gradient of a periodic function; so this integral yields zero. Indeed all insulating solid elements have either even valence, or a lattice containing an even number of atoms in the basis, and therefore filled bands. It is of interest to consider what happens to a filled band with one electron removed. This can be created by absorption of a photon whose energy exceeds the energy gap of a semiconductor, to make a transition of an electron from the valence band into the conduction band [See Fig. 9.3]. The removal of an electron from a filled band leaves a hole , which in fact can be viewed as a fermionic particle with distinct properties. Hole momentum. kh = −ke (9.8) ǫh (kh ) = −ǫe (ke ) (9.9) This can be seen from the optical absorption experiment. The light produces a (nearly) vertical transition and gives no momentum to the electron hole pair. Since the initial state is a filled band with total momentum zero, Eq. (9.8) follows. Hole energy. This sign is needed because (measuring energies from the top of the band) removing an electron of lower energy requires more work. Hole velocity. A combination of the first two rules then gives vh = h̄−1 ∇kh ǫh (kh ) = h̄−1 ∇ke ǫe (ke ) = ve (9.10) Effective mass. The dispersion at the bottom (top) of the bands is parabolic, and therefore can be approximated as h̄2 k 2 (9.11) ǫ = ǫ0 + 2m∗ 9.2. ELECTRONS AND HOLES IN SEMICONDUCTORS 85 Figure 9.3: Absorption of a photon creates an electron-hole pair, with an energy ǫe +ǫh +2Egap but adds negligible momentum to the system. Hence the hole momentum is the negative of the momentum of the empty electronic state, and its energy is positive (measured conventionally from the top of the band). defining an effective mass m∗ . We have m∗h = −m∗e (9.12) so the hole mass is positive at the top of the electron band. Hole charge. The effective charge of a hole is positive, as can be seen by taking the equation of motion for the electron h̄ dke = −e(E + ve ∧ B) dt (9.13) and making the replacement ke → −kh and ve → vh , giving h̄ dkh = e(E + vh ∧ B) dt (9.14) The same result comes from noticing that the current carried by the hole evh must be the same as the (missing) current (not) carried by the empty electron state.