2753 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS TRINITY TERM 2011 Wednesday, 22 June, 9.30 am 12.30 pm Answer four questions. Start the answer to each question in a fresh book. A list of physical constants and conversion factors accompanies this paper. The numbers in the margin indicate the weight that the Examiners anticipate assigning to each part of the question. Do NOT turn over until told that you may do so. 1
1. Explain what is meant by the terms Landau levels and filling factor in connection with a two-dimensional electron gas (2DEG) in a perpendicular magnetic field. [5] 0 Graphene is made from a single layer of carbon atoms. The valence and conduction bands form conical valleys which meet as shown in the diagram. There are two such valleys per Brillouin zone. The electronic dispersion is given by E(k) = ±v F h k, where k is the electron wavevector relative to the axis of the cone and v F is the Fermi velocity. (a) In an applied magnetic field of flux density B the energy of the jth Landau level in graphene is given by E j = v F 2e hbj, where the spin splitting has been neglected. Show that when the chemical potential is half-way between Landau levels N s = 4eB h (j + 1 2 ) and 1 ρ xy = 4e2 h (j + 1 2 ), where N s is the number of carriers per unit area and ρ xy is the transverse Hall resistivity. [10] (b) Sketch the expected form of ρ 1 xy (N s) for N s in the range 5 10 16 to 5 10 16 m 2 and B = 14T. [5] (c) Giventhatv F 10 6 ms 1 ingraphene, suggestwhythequantumhall effectcanbe observed at room temperature in graphene but only at much lower temperatures in traditional 2DEG systems. [5] 2753 2
2. Explaining your assumptions, show how the tight-binding model can be used to obtain the following expression for the electronic dispersion in a crystal with a single isotropic orbital per lattice point, E(k) = Ẽ0 T 0t(T)exp(ik T), (1) where the T are lattice vectors. Give expressions for Ẽ0 and t(t). [10] For the case of a square lattice with nearest-neighbour and next-nearest-neighbour orbital overlap only, show that equation (1) may be written E(k) = Ẽ0 2t 1 (cosk x a+cosk y a) 4t 2 cosk x acosk y a, (2) where a is the lattice parameter. [5] The copper-oxide superconductors can be modelled as two-dimensional metals on a square lattice. The electronic dispersion in one copper-oxide superconductor can be approximated by equation (2) with parameters Ẽ0 = 0.24eV, t 1 = 0.18eV and t 2 = 0.08eV, and with the Fermi energy at E = 0. Calculate the Fermi wavevectors along the reciprocal-space paths (i) (0,π/a) (π/a,π/a), (ii) (0,0) (π/a,π/a), and (iii) (0,π/a) (π/a,0). Hence, sketch the Fermi surface in the first Brillouin zone. [6] a a In another copper-oxide superconductor, it is suggested that an electronic instability creates an additional potential whose periodicity is described by a doubled unit cell, as shown by the dotted square in the figure. Determine the modified Brillouin zone scheme and sketch the expected Fermi surface for the doubled unit cell. [4] 2753 3 [Turn over]
3. Explain what is meant by ferroelectricity. State the symmetry requirement for a crystal to be ferroelectric. [4] Bariumtitanate(BaTiO 3 )becomesferroelectricbelowt C = 395K,andundergoes the following sequence of structural phase transitions on cooling: Pm3m 395K P4mm 270K Cm2m 180K R3m. The different phases are related by small distortions of the lattice. The table below gives the directions of the unit cell axes for the three ferroelectric phases in terms of the Cartesian axes of the cubic Pm3m phase. P4mm [100] [010] [001] Cm2m [110] [110] [001] R3m [101] [110] [011] For each of the three ferroelectric phases, (a) explain the meaning of the space group symbol and draw a diagram showing the point group symmetries; [9] (b) deduce the direction of the ferroelectric polarization relative to the cubic axes, explaining your reasoning; [6] Explain how you might detect the structural change from Pm3m to P4mm using x-ray powder diffraction data. When a single crystal of BaTiO 3 is cooled through T C no net ferroelectric polarization is observed. Explain why not, and suggest an experiment that could measure the ferroelectric polarization. [6] 2753 4
4. Explain what is meant by the terms Wigner Seitz construction and Brillouin zone. What are the advantages of using the Wigner Seitz construction to describe Brillouin zones of high-symmetry crystals? [5] Explain how neutron inelastic scattering can be used to measure phonon dispersion relations. Include a brief description of one type of neutron scattering spectrometer that could be used for the measurements. What are the main difficulties in using X-ray inelastic scattering to measure phonon dispersion relations? [8] Copper (Cu) has a cubic F lattice. The Brillouin zone for a cubic F lattice is shown below (left) with labels for certain symmetry points. The diagram on the right shows the measured phonon frequencies of Cu as a function of wavevector along different directions in reciprocal space. Γ X K Γ L a* c* b* Frequency (THz) 8 7 6 5 4 3 2 (00ξ) L T T 1 T 2 (ξξ0) L (ξξξ) L T 1 0 0 0.2 0.4 0.6 0.8 1 (000) ξ (001) (110) 0.8 0.6 0.4 ξ 0.2 0 0.1 0.2 0.3 0.4 0.5 (000) ξ (½½½) (a) Draw a diagram of the plane in reciprocal space containing the reciprocal lattice vectors (00ξ) and (ξξ0). Draw the boundary of the Brillouin zone centred on (000) and label the neighbouring reciprocal lattice points which have non-zero structure factors. (b) Explain why the positions (001) and (110) in reciprocal space are equivalent. (c) Explain why there are three distinct branches in the phonon dispersion diagram along Γ K, but only two distinct branches along Γ X and Γ L. (d) Assuming nearest-neighbour interactions only, calculate the ratio of the velocities of sound for longitudinal (L) and transverse (T) phonons with wavevectors (00ξ). Compare your answer with the data.. [12] 2753 5 [Turn over]
5. Show that the density of states per unit volume in the conduction band of a three-dimensional (3D) semiconductor is given by g 3D (E) = α E E g, where E is the electron energy and E g is the energy at the bottom of the conduction band. Express α in terms of the electron effective mass m e and h. [6] Sketch a typical absorption spectrum for optical transitions near the direct gap of (a) a bulk semiconductor, and (b) semiconductor quantum wells in which the carriers are confined in 2D, 1D, and 0D. [4] The figure below shows absorption spectra at a temperature of 2K for CdTe semiconductor quantum dots of different average diameter d, embedded in a glass matrix. Absorbance (arb. units) 1.0 0.8 0.6 0.4 0.2 0.0 d=10.2nm d=8.6nm d=6.6nm d=5.2nm d=5.0nm 1.6 1.8 2.0 2.2 2.4 2.6 Photon Energy (ev) Derive an expression for the confinement energy of an electron or hole inside a quantum dot. Assume the dot is a cube with side d and has infinite potential barriers on all sides. Explain qualitatively why the spectra shift in energy with dot diameter and suggest why the observed absorption peaks are broadened. [8] Use the derived expression to extract an estimate of each sample s average dot diameter from the observed absorption peaks. Compare your results with the diameters determined from small-angle X-ray scattering measurements, given in the figure. Give reasons for any discrepancies observed. [7] [The electron and hole effective masses near the band edge of CdTe are m e = 0.096m e and m h = 0.4m e respectively. The band gap energy of CdTe at a temperature of 2K is E g =1.606eV.] 2753 6
6. (a) State Hund s rules for determining the magnetic ground state of an isolated ion and comment qualitatively on the underlying physics. Determine the ground state quantum numbers L, S and J for the orbital, spin and total angular momentum for an isolated ion of Co 2+ (3d 7 ) and Co 3+ (3d 6 ). [7] (b) Discuss the magnetic ground states of Co 2+ and Co 3+ ions located at the centre of (i) a regular octahedron, and (ii) a regular tetrahedron of O 2 ions. Consider separately the cases of weak and strong crystal fields. Theinsulator Co 3 O 4 contains Co 3+ ions in octahedral coordination and Co 2+ ions in tetrahedral coordination. Account for the experimental finding that Co 3+ ions carry no magnetic moment, while Co 2+ ions have a moment of 3µ B per ion. [8] (c) KCoF 3 has a simple cubic crystal lattice with spacing a = 0.4 nm and Co 2+ ions located at 0,0,0. Below 135K, the magnetic moments of the Co 2+ ions order antiferromagnetically such that each moment aligns antiparallel to each of its six nearest neighbours. Show that this magnetic structure can be represented by a cubic unit cell with spacing a = 2a. A neutron diffraction experiment is performed on a powder sample of KCoF 3. Show that the selection rule for magnetic Bragg scattering at wavevector Q = (h,k,l)π/a is that h, k, and l are odd integers. Hence, explain how you could detect the antiferromagnetic order by neutron diffraction. [6] [The intensity of magnetic scattering of neutrons depends on a structure factor F = j µ j exp(iq r j ), where the sum extends over all sites j in the magnetic unit cell, with position vectors r j and ordered moments µ j.] (d) Elemental Co is a ferromagnet. The saturated moment is 1.7µ B per cobalt atom. Explain this observation and discuss qualitatively how you would expect this value to change upon application of high pressure. [4] 2753 7 [Turn over]
7. Briefly describe how interactions lead to a transition to spontaneous magnetic order in a ferromagnet below the Curie temperature using a mean-field model. What is a suitable quantity to define as an order parameter for the transition to magnetic order in (i) a ferromagnet, and (ii) an antiferromagnet? [6] A square-lattice anisotropic magnet is described by the Hamiltonian H = 1 2 ij ( )] [J SiS z j z +J Si x Sj x +S y i Sy j +gµ B B i whereinthefirstsumiandj arenearest-neighboursites, andthelasttermisthezeeman energy in an external magnetic field B applied along the z-axis. When J > J the ground state at B = 0 is a collinear antiferromagnet in which neighbouring spins point alternately along the positive and negative z-direction. Calculate the mean-field energy per spinof this state. Determine the mean-field ground states at B = 0 for (i) J > J, (ii) J > J, (iii) J > J. [5] Assuming J > J > 0, consider an alternative magnetic structure in which the moments have their z components aligned ferromagnetically along the field and the components in the xy plane ordered antiferromagnetically, as illustrated in the figure. Show that the mean-field energy of this (spin-flop) state is S z i, E SF (θ) = gµ B BSsinθ+ n 2 (J +J )S 2 f(θ) n 2 J S 2 (per spin), where n is the number of nearest neighbours and f(θ) is a function to be determined. Find how the equilibrium angle θ varies with B, and show that the spin-flop state becomes energetically more favourable than the collinear antiferromagnetic state above a critical field B 1 to be determined. [9] M M S z θ θ SPIN FLOP 0 B 1 B 2 B Account for the shape of the magnetization curve plotted in the figure, and determine the values of J and J for the square-lattice antiferromagnet K 2 CoF 4, for which B 1 = 80T, B 2 = 130T, g = 2.1 and S = 3/2. [5] 2753 8
8. Describe briefly the BCS theory of superconductivity, mentioning the experimental evidence which shows that electrons are paired in the superconducting state. [5] Starting from the fact that superconducting pairs have zero momentum in the absence of a magnetic field, justify the form of the London equation µ 0 λ 2 ( J)+B = 0. Find an expression for λ in terms of the density n, charge q and mass m of the superconducting pairs. Hence, estimate λ for a typical elemental superconductor. [5] What condition on the penetration depth must be satisfied if a material is to behave as a type-ii superconductor? Derive an expression for a, the spacing between vortices in a type-ii superconductor exhibiting a triangular flux-line lattice, as a function of the applied magnetic field. [5] Neutrons of wavelength 1 nm are used to study the flux-line lattice by diffraction. Calculate a and the lowest two angles of diffraction for a type-ii superconductor in a magnetic field of flux density 0.5 T. What difficulties would be encountered with this method for studying the flux-line lattice of YBa 2 Cu 3 O 7, for which λ = 150nm? [10] [The magnetic flux quantum Φ 0 = h/2e = 2.07 10 15 Wb.] 2753 9 [LAST PAGE]