A11046W1 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 2015 Wednesday, 17 June, 2.30 pm 5.45 pm 15 minutes reading time 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 the meaning of the terms screw axis, glide plane and point symmetry used in crystallography. [6] At room temperature, iron selenide (FeSe) has a primitive tetragonal crystal lattice with two Fe and two Se atoms per unit cell. The fractional coordinates are 1 Fe : 0, 0, 0 ; 2, 1 2, 0 1 Se : 2, 0, η ; 0, 1 2, η (0 < η < 0.5). Sketch the crystal structure in projection down the z axis onto the z = 0 plane. [4] (a) With the aid of a separate diagram of the unit cell, specify the locations of all screw axes, if any. (b) What is the translation associated with the z = 0 glide plane? (c) Show that the glide symmetry in (b) leads to the condition h + k = 2n on the hk0 Bragg reflections, where n is an integer. (d) Give the fractional coordinates of a point of inversion symmetry, and state the location of a 4 symmetry axis. [11] Diffraction measurements show that FeSe undergoes a small distortion to an orthorhombic structure on cooling below 90 K. At the onset of the transition, the tetragonal 110 reflection splits into two peaks of equal intensity, whereas the tetragonal 200 reflection does not split. Describe the structural distortion in the z = 0 plane. [4] A11046W1 2
2. Describe the principles of inelastic neutron scattering for the measurement of phonon dispersion relations. Include in your description an outline of an experimental arrangement, and explain how the energy and wave vector of the phonons are obtained from the observations. [8] The intermetallic compound YNi 2 B 2 C has a body-centred tetragonal lattice with conventional unit cell parameters a = 0.353 nm and c = 1.05 nm. (a) On a scale of 1 cm = 5 nm 1, draw the (h, 0, l) section of the reciprocal lattice for 2 < h < 2 and 2 < l < 2. (b) Sketch the first Brillouin zone on your drawing. (c) Explain why the phonon mode frequencies at the reciprocal space positions (1, 0, 0) and (0, 0, 1) are the same. (d) Why are there only two acoustic mode frequencies for phonons propagating parallel to the c axis? [8] A neutron spectrometer records the scattering from a crystal of YNi 2 B 2 C as a function of neutron energy transfer E at a fixed scattering vector Q = (0.2, 0, 8) in reciprocal lattice units. The spectrum recorded at a temperature of 20 K contains a peak at E = 7 mev due to the excitation of an acoustic phonon. The intrinsic width of the peak is found to be 0.1 mev. Estimate the lattice thermal conductivity along the a axis of YNi 2 B 2 C, and compare your estimate with the experimental value for the total thermal conductivity of 39 W m 1 K 1 at 20 K. Account for any discrepancy, and suggest a likely origin for the broadening given that YNi 2 B 2 C is a superconductor. [9] [ The volume heat capacity of YNi 2 B 2 C at 20 K is 37 10 3 J K 1 m 3. ] [ The superconducting transition temperature of YNi 2 B 2 C is 15 K. ] A11046W1 3 [Turn over]
3. State Hund s rules for determining the magnetic ground state of an isolated ion. Briefly describe the underlying physics. Determine the ground state quantum numbers L, S and J for the orbital, spin and total angular momentum of an isolated ion of Eu 2+ (4f 7 ). [7] The magnetic free energy of the cubic ferromagnet europium oxide (EuO) may be written in terms of the magnetisation M = (M x, M y, M z ) as F = F 0 + a(t T c ) M 2 + b M 4 + c (M 2 xm 2 y + M 2 xm 2 z + M 2 y M 2 z ), where the constants a, b, c, F 0 and T c are independent of temperature T, and a > 0, b > 0 and 2b + c > 0. (a) Show that there is no spontaneous magnetisation for T > T c. (b) Below T c, the spontaneous magnetisation is along one of the cubic high-symmetry directions. Deduce the direction for (i) c > 0 and (ii) c < 0. (c) Obtain an expression for the temperature dependence of the spontaneous magnetisation for c < 0. [12] The magnetic properties of EuO are only weakly anisotropic. Explain why this is so, and given that EuO is not a metal, suggest an appropriate form for a spin Hamiltonian to describe its ferromagnetic behaviour. [6] A11046W1 4
4. Describe the physical origin of the exchange interaction in magnetic systems, and explain why it often leads to an energy of the form J S 1 S 2. Describe the type of exchange interaction that can be important in an antiferromagnetic oxide. [7] An infinite linear chain of identical spins lies along the z axis with a separation c between neighbouring spins. The magnetic interactions are described by the XY spin Hamiltonian H = n [ J1 (S x ns x n+1 + S y ns y n+1 ) + J 2(S x ns x n+2 + S y ns y n+2 ) ], where n runs over all sites in the chain, and J 1 and J 2 are exchange constants. Two possible ordered magnetic ground states are (i) ferromagnetic alignment, with mean-field energy per spin E F, and (ii) antiferromagnetic alignment, with meanfield energy per spin E AF. Show that E F = S 2 (J 1 + J 2 ) and E AF = S 2 (J 1 J 2 ), where S is the spin value. [4] Give a physical explanation for why a negative J 2 increases the energy of both the ferromagnetic and antiferromagnetic ground states. [4] An alternative magnetic ground state is a spin helix, in which the spin components on the n th site are given by S x n = S cos(nqc) and S y n = S sin(nqc). Obtain E H, the mean-field energy per spin for the spin helix, and show that E H is a minimum when cos(qc) = α J 1 /J 2 and J 2 < 0, where α is a constant to be determined. Hence, show that the spin helix is always the most stable ground state when J 2 < 0 and J 2 > α J 1. [10] A11046W1 5 [Turn over]
5. A metal has a simple spherical Fermi surface and an isotropic, energy-independent effective mass. A magnetic field B is applied parallel to z. Explain the principles needed to derive the following equation: { dv m dt + v } = ee ev B, τ where m is the effective mass, τ 1 is the scattering rate, and the other symbols have their usual meaning. [2] For circularly polarized light of angular frequency ω, where E ± = (E x ±ie y ) exp(iωt) with E x = E y, show that the resulting current J ± = (J x ± ij y ) exp(iωt) is related to E by the classical dynamic conductivity given by σ ± = σ 0 1 + i(ω ± ω c )τ, where σ 0 = ne 2 τ/m, n is the density of electrons, and ω c = eb/m. [6] Explain what the consequences of this relation are for the optical properties of conducting materials in the limits of (i) a high density of electrons, zero magnetic field; (ii) a low density of electrons, high magnetic fields; (iii) a high density of electrons, high magnetic fields. [9] Cyclotron resonance is commonly used as a technique to measure the effective mass of electrons in both semiconductors and metals. Explain why the experimental arrangement is different for the two types of materials and sketch the experimental arrangement and typical results for the two cases. [8] A11046W1 6
6. For the electronic dispersion in a crystal with a single orbital per lattice point, the tight-binding model can be used to obtain an expression of the form E(k) = E 0 T 0 t(t) exp(ik T), where T represents the lattice vectors. Explain the meaning of the terms E 0 and t(t), and for the case of a rectangular lattice with interactions from the nearest neighbour atoms in the x and y directions only, show that the equation above leads to the expression E(k) = E 0 2t x cos(k x a) 2t y cos(k y b), where the lattice parameter a < b, and t x and t y must be appropriately defined. Sketch the first Brillouin Zone, labelling the Γ, X, Y, and M-points. For the case where t x = 2t y, sketch the dispersion relation along the path Γ M X Γ Y M Γ. [8] A monovalent metal crystallizes with the above crystal structure. What will be the approximate value of the Fermi energy and the shape of the Fermi surface for this material? Where would you expect additional energy gaps to open up? A threedimensional (3D) tetragonal crystal of the same metal crystallizes with a = b < c and t x = t y = 2t z, where the 3D notation makes the z-axis equivalent to the y-axis in the two-dimensional (2D) case. Sketch the approximate shape of the Fermi surface. What techniques could you use in order to test out your predictions? [10] When a magnetic field, B, is applied to this metal along the z direction, the low temperature resistivity is found to show two series of oscillations with different frequencies in 1/B. Explain what is the origin of these oscillations. Use the dispersion relation above to estimate the ratio of the two frequencies. [7] A11046W1 7 [Turn over]
7. Excitons are typically thought of as members of two different families, Frenkel and Wannier-Mott excitons. Compare and contrast the properties of the two families and give examples of materials which belong to each family, explaining why you have chosen them. [6] Transmission through a thin layer of a new bulk semiconducting material is used to deduce its absorption spectrum, as shown in the figure below at 4.2 K and 77 K. Explain what causes the characteristic behaviour observed, which curve corresponds to which temperature, and deduce as much as you can about the properties of the material. With the additional information that the electron and hole effective masses are both 0.2 m e (where m e is the standard mass of the electron), what further information can you deduce? [9] Absorption coefficient, 10 6 m -1 1 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 Photon energy, ev A quantum well is to be constructed from the new material by surrounding a 5 nm thick layer with a thick layer of a second material with a band gap of 2.3 ev. Estimate the energy levels for the electrons and holes in the quantum well and explain how the absorption spectrum will be changed, including a sketch of what you think it may look like at 4.2 K in the region 1.6 2.4 ev. [10] A11046W1 8
8. Explain the origin of the Ginzberg-Landau equation below: J = i qh q2 (ψ ψ ψ ψ ) ψ 2 A. 2m m Using this equation, show that the magnetic flux Φ through a superconducting ring is quantised as Φ = nφ0, where n is an integer and Φ0 = h/2e. [9] Give a brief description of how you might observe flux quantisation in a typical superconducting ring, as shown in Figure (a) above. [4] For a superconducting ring containing two identical Josephson junctions (as shown in Figure (b) above), show that the current I across the device is given by πφ I = 2 IC cos sin δ, Φ0 where δ is a phase difference that must be defined. δ1 and δ2 are the phase differences across the individual junctions, and IC is a constant. [7] Show that at low current I IC, this device behaves as a flux tunable inductor, with inductance Φ0. Leff = 4πIC cos πφ Φ0 [5] A11046W1 9 [LAST PAGE]