MP464: Solid State Physics Problem Sheet

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1 MP464: Solid State Physics Problem Sheet 1 Write down primitive lattice vectors for the -dimensional rectangular lattice, with sides a and b in the x and y-directions respectively, and a face-centred rectangular lattice, with a conventional cell of the same dimensions. Determine what happens to the primitive lattice vectors in each case under reflections in the x and y-axes. The results are different because the space groups are different. A two dimensional lattice has primitive lattice vectors a 1 and a. A different choice of primitive lattice vectors a 1 and a can always be written as a linear combination of the first choice, a 1 = α 11 a 1 + α 1 a a = α 1 a 1 + α a a1 a α11 α 1 α 1 α a = 1 a, with integer co-efficients α 11, α 1, α 1 and α. Show that this requires that α 11 α α 1 α 1 = ±1 hint: all primitve cells have the same volume. Which of the following are legitimate primitive lattice pairs i ii iii a 1 = a 1 + a, a = a ; a 1 = a 1 a, a = a 1 + a ; a 1 = a 1 + 5a, a = a 1 + 3a? A three dimensional lattice has primitive lattice vectors a 1, a and a 3. Is a primitive set? a 1 = 3a 1 + a + a 3 a = 3a 1 + a + 5a 3 a 3 = a 1 + a + 3a 3 3 A two-dimensional centred rectangular lattice has a conventional cell with sides a and b. Show that the special case b = a corresponds to a square lattice.

4 A two-dimensional centred rectangular lattice has a conventional cell with sides a and b. Show that the special case b = 3a corresponds to a hexagonal lattice.

2 4 A two-dimensional centred rectangular lattice has a conventional cell with sides a and b. Show that the special case b = 3a corresponds to a hexagonal lattice. Identify the -d lattice type, the symmetries of the lattice and and the symmetries of the tiling below, found in the Alhambara palace in Andalusia, Spain. Which symmetries of the lattice are absent in the tiling? 8 5 Derive the ratio c a = 3 quoted in the lectures for the hexagonal close packed HCP structure hint: find the height of a tetrahedron with a base consisting of an equilateral triangle of side a. Why is the HCP structure not a Bravais lattice? 6 Find primitive lattice vectors for the 3-dimensional hexagonal lattice and calculate the volume of a primitive cell in terms of a and b. 7 Calculate the angle between the primitive lattice vectors given in the lectures for the BCC lattice. 8 Show that a FCC lattice with an extra point in the centre of a conventional cell, i.e. a chimera of a FCC and a BCC lattice, is not a Bravais lattice. 9 Show that a BCC lattice with conventional cell spacing a is equivalent to a facecentred orthorhombic lattice with b = c = a hint: rotate the BCC lattice through 45 about a conventional cell edge. 10 Why is there no face-centred tetragonal lattice in the Bravais list of 3-dimensional lattices? 11 Derive the following packing fractions: i -dimenional hexagonal lattice: ii Diamond: 3π 16. π 3 ;

3 1 Calculate the packing fraction for a face centred cubic crystal with a monatomic basis consisting of spherical atoms. Given that lead crystallises in a face centred cubic structure with a monatomic basis, calculate the density of lead. Note: Lead has atomic mass 07. and spherical atoms with radius m. 13 The lattice plane defined by any three Miller indices h, k, l is always a - dimensional Bravais lattice. Identify which -dimensional Bravais lattice is associated with the 111, 110 and 100 planes of a simple cubic lattice. 14 Show that the reciprocal of the BCC lattice is a FCC lattice and calculate the volume of a primitive cell of the reciprocal lattice in terms of the lattice constant a of a conventional cell of the direct lattice. Show that in general the reciprocal of a reciprocal lattice is the original direct lattice. 15 Sketch the first three Brilouin zones for a -dimensional hexagonal lattice. 16 Calculate the structure factor for a FCC lattice. 17 A crystal consisting of simple cubic lattice, with lattice spacing a and primitive lattice vectors a 1 = aˆx, a = aŷ and a 3 = aẑ, has a diatomic basis of two identical atoms, one at the origin and one at a ˆx + ŷ + ẑ. In the lectures it was shown that the structure factor eliminates all Bragg reflections associated with reciprocal lattice vectors G hkl with h + k + l an odd integer. The same crystal can be viewed as a body centred cubic lattice with a monatomic basis. All reciprocal lattice vectors then give Bragg peaks. How are these two pictures reconciled? 18 A two-dimensional square lattice has lattice spacing a. How many Bragg peaks will be present for incoming X-rays with wavelength a λ a? 19 The Lennard-Jones potential for Neon is { σ 1 σ } 6 Ur = 4ǫ r r with ǫ = J and σ = m. Calculate the ratio of the cohesive energies for Neon in BCC and FCC structures. Note: in the notation used in the lectures the lattice sums for a BCC lattice are A 1 = and A 6 = For any physical system exhibiting oscillatory behaviour of the form e ikx ωt, the phase of the oscillation is the same for points satisfying x t = ω K. The velocity v P = ω K is called the phase velocity. The phase velocity is only physically relevant if the dispersion relation is such that ωk is a linear function of K, more generally it is the group velocity,

4 v g = dωk dk, which is physically relevant. Calculate the group and phase velocities for the dispersion relation derived in the lectures for a one-dimensional crystal with a basis consisting of two ions with masses M 1 and M and identical spring constants. Show that the group velocity is zero for K at a Brillouin zone boundary. 1 In the lectures the dispersion relation for a one-dimensional crystal, with a monatomic basis of atoms with mass M and lattice spacing a, was derived using a simple model using Hooke s law with spring constants C, K a ωk = ω 0 sin, where ω 0 = C/M. Show that this dispersion relation gives a density of states Dω = N π 1 ω 0 ω, where N is the number of lattice points in the crystal. This diverges at the zone boundary as K π a and ω ω 0. A divergence like this in the dispersion relation is not uncommon and is called a van Hove singularity. A one-dimensional lattice with lattice spacing a has a diatomic basis consisting of two ions with identical masses m and equilibrium positions na and na+d, with 0 < d < a/. Assume that each ion only interacts with its nearest neighbours on either side and model the force using Hooke s law with spring constant D on the left and C on the right of the ions at equilibrium positions na. Write down Newton s second law for the displacement u n t away from equilibrium of ions at equilibrium positions na and the displacement v n t away from equilibrium of ions at equilibrium positions na + d. Using plane-wave solutions of the from u n = ǫ 1 e ikna ωt, v n = ǫ e ikna ωt derive the coupled equations mω C + D ǫ 1 + C + De ika ǫ + = 0 C + De ika ǫ 1 + mω C + D ǫ = 0 for ǫ 1 and ǫ. Derive the dispersion relation with ω = C + D m ± 1 m C + D + CD coska ǫ = C + DeiKa C + De ika ǫ 1. Show that the group velocity of both modes vanishes at a Brillouin zone boundary.

5 3 Consider a monatomic one-dimensional crystal of unit cell dimension a in which atoms separated by pa interact with a force constant C p. Show that the dispersion relation is ω K = 4 p K a C p sin. M Show that, in the long-wavelength small K limit, p ω K = a M p p C p 1 K, provided the sum is finite. 4 Show that the heat capacity for a gas of phonons in the Debye approximation, without assuming low temperature, is C V = 9Nk B T Θ D 3 xd 0 x 4 dx e x 1. is 5 The thermal energy of a collection of harmonic oscillators with frequency ω K ωmax UT, V = h ω < n > + 1 Dωdω. 0 In the lectures the zero point energy of the oscillators hω was ignored in the calculation of the specific heat, as it is independent of temperature, but does it depend on the volume? Including this term in the Debye approximation show that the thermal energy, assuming 3 acoustic modes, is UV, T = π V k B T 4 10v 3 h 3 4 3N + 3 hv 4 3 π V 1 3, were v is the speed of sound. For a monatomic one-dimensional crystal, with atoms of mass M and spring constant C C, we saw that v = a. Express v in terms of V to find the how U above depends on M V. 6 If an electron gas has number density n then nearest neighbour electrons are typically a distance 1/n 1 3 apart. Determine the ratio of the Coulomb energy due to electrostatic repulsion between two electrons to the their Fermi energy in such a gas. Calculate this ratio for electrons in a monovalent metal with a typical lattice spacing a = m. Show that the Coulomb energy becomes less important as the lattice spacing decreases, the more dense the gas the less important Coulomb repulsion is and the better the ideal gas approximation gets!

6 7 In the Sommerfeld expansion we encounter the integral x e x e x + 1 dx. Show that this evaluates to π 3. Hint: you may find it useful first to show that x e x dx = 4 xe x dx, then expand e x e x 1 = 1+e x n=0 1n e nx, swap the summation and integration and change the integration variable from x to y = nx, then finally use the fact that this?. n=1 1 n n = π 1 can you prove 8 Derive the expression for the first two non-zero terms in a Sommerfeld expansion for the thermal energy of a free electron gas given in the lectures UT = 3 5 Nε F + π 4 Nε kb T F +... ε F Note: you may assume the definite integral 0 x π e x +1dx = 1. 9 An electron moves in crossed electric and magnetic fields: a constant electric E = Eˆx in the x-direction and a constant magnetic field B = Bẑ in the z-direction. Using the Lorentz force law, F = ee+v B where v is the electron s velocty, write down the equations of motion for the electron. Solve for the motion when the electron no component of velocity in the z-direction and its initial motion is purely in the x-y plane. Show that the electron s velocity can be decomposed into a rotating component superimposed on a constant drift in the y-direction. 30 In the lectures the x and y components of the electron velocity in a Hall experiment were shown to be related by j y v x = τe m E x ω c τv y, v y = τe m E y + ω c τv x, where ω c = eb m is the cyclotron frequency and τ the scattering time. Show that the components of the current density j = env are related to the electric field by jx σ 0 1 ωc τ Ex =, 1 + ωcτ ω c τ 1 E y where σ 0 = ne τ m is the Drude conductivity. The matrix σ = σ 0 1 ωc τ 1 + ωcτ ω c τ 1

7 is called the conductivity tensor and its inverse ρ = σ 1 the resistivity tensor, relating E to j Ex ρxx ρ = xy jx. Show that where R H = 1 ne E y is the Hall co-efficient. ρ yx ρ yy 1 σ ρ = 0 R H B R H B 31 The vector potential for a uniform magnetic field in the z-direction can be chosen to be A = Byˆx, where B is a constant, so that B = A = Bẑ. The Hamiltonian for a free electron, without spin, moving in this field is Show that 1 σ 0 j y, H = h m y + 1 z + i h m x eyb. ψx = e ik xx+k z z fy is an eigenfunction of this Hamiltonian, with eigenvalue E, if fy satisfies h d f [ h m dy + kz m + 1 ] mω cy y 0 fy = Efy, where ω c = eb m is the cyclotron frequency and y 0 depends on k x as y 0 = hk x mωc. This is the Schrödinger equation for a simple harmonic oscillator in one-dimension with frequency ω c and energy levels E n = n + 1 h k z hω c +. m These energy levels are called Landau levels. How are the energy levels changed when electron spin is included in the calculation? 3 Sketch the band structure for a FCC lattice in the free electron approximation. 33 Derive the dispersion relation quoted in the lectures, Ek = h m k + π π a a k ± π π am a a k + π h from the sub-matrix of the central equation from band theory, E 0 k Ek Ũ 1 Ũ 1 E 0 k π, a Ek Ũ 1,

8 where E 0 k = h k m is the unperturbed energy of a free electron. In the lectures the eigenfunctions were calculated assuming that Ũ1 was real. What are the eigenfunctions when Ũ1 is complex? 34 Calculate the dispersion relation and the band gap arising from the potential Ux = Ũn cos πnx a in the central equation, where Ũn is real. 35 A free particle with mass m moving in one-dimension has a Hamiltonian H 0 = h m with eigenvalues E 0 k = h k m, and eigenfunctions which are plane waves, ψ kx = e ikx. In the theory of quantum mechanics a result from time-independent perturbation theory states that, when the Hamiltonian is perturbed to H 0 +Ux, the eigenvalues are modified to Ek = E 0 k + Ek where d dx, Ek = ψ xuxψxdx dx. ψ xψx An electron moving in a one-dimensional crystal, with lattice spacing a and N lattice sites, experiences a periodic potential Ux = Ũ1 cos πx a due to the lattice ions, where Ũ 1 is a positive constant. Show that Ek = 0 for plane waves ψx = e ikx but for standing waves, { 1 ψ ± x = e ikx + e ikx = coskx 1 e ikx e ikx = sinkx, a gap opens up at k = π a with π E ± = a ±Ũ1 4. Give a physical reason why sin kπ a has lower energy than cos kπ a. i

MP464: Solid State Physics Problem Sheet

MP464: Solid State Physics Problem Sheet 1) Write down primitive lattice vectors for the -dimensional rectangular lattice, with sides a and b in the x and y-directions respectively, and a face-centred