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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 anticipateassigning 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 usedin crystallography. [6]

At room temperature, iron selenide (FeSe) has a primitive tetragonal crystal lat-tice with two Fe and two Se atoms per unit cell. The fractional coordinates are

Fe : 0, 0, 0 ; 12 ,

12 , 0

Se : 12 , 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 allscrew 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 hk0Bragg reflections, where n is an integer.

(d) Give the fractional coordinates of a point of inversion symmetry, and state thelocation of a 4 symmetry axis. [11]

Diffraction measurements show that FeSe undergoes a small distortion to an or-thorhombic structure on cooling below 90 K. At the onset of the transition, the tetrag-onal 110 reflection splits into two peaks of equal intensity, whereas the tetragonal 200reflection does not split. Describe the structural distortion in the z = 0 plane. [4]

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2. Describe the principles of inelastic neutron scattering for the measurement ofphonon dispersion relations. Include in your description an outline of an experimentalarrangement, and explain how the energy and wave vector of the phonons are obtainedfrom the observations. [8]

The intermetallic compound YNi2B2C has a body-centred tetragonal lattice withconventional 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 latticefor −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 parallelto the c axis? [8]

A neutron spectrometer records the scattering from a crystal of YNi2B2C as afunction of neutron energy transfer E at a fixed scattering vector Q = (0.2, 0, 8) inreciprocal lattice units. The spectrum recorded at a temperature of 20 K contains apeak at E = 7 meV due to the excitation of an acoustic phonon. The intrinsic widthof the peak is found to be 0.1 meV. Estimate the lattice thermal conductivity alongthe a axis of YNi2B2C, and compare your estimate with the experimental value for thetotal thermal conductivity of 39 W m−1 K−1 at 20 K. Account for any discrepancy, andsuggest a likely origin for the broadening given that YNi2B2C is a superconductor. [9]

[ The volume heat capacity of YNi2B2C at 20 K is 37× 103 J K−1 m−3. ]

[ The superconducting transition temperature of YNi2B2C is 15 K. ]

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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 numbersL, S and J for the orbital, spin and total angular momentum of an isolated ion of Eu2+

(4f7). [7]

The magnetic free energy of the cubic ferromagnet europium oxide (EuO) may bewritten in terms of the magnetisation M = (Mx,My,Mz) as

F = F0 + a(T − Tc)M2 + bM4 + c (M2

xM2y +M2

xM2z +M2

yM2z ) ,

where the constants a, b, c, F0 and Tc are independent of temperature T , and a > 0,b > 0 and 2b+ c > 0 .

(a) Show that there is no spontaneous magnetisation for T > Tc.

(b) Below Tc, the spontaneous magnetisation is along one of the cubic high-symmetrydirections. Deduce the direction for (i) c > 0 and (ii) c < 0.

(c) Obtain an expression for the temperature dependence of the spontaneous mag-netisation for c < 0. [12]

The magnetic properties of EuO are only weakly anisotropic. Explain why this isso, and given that EuO is not a metal, suggest an appropriate form for a spin Hamilto-nian to describe its ferromagnetic behaviour. [6]

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4. Describe the physical origin of the exchange interaction in magnetic systems, andexplain why it often leads to an energy of the form −J S1 · S2. Describe the type ofexchange 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 cbetween neighbouring spins. The magnetic interactions are described by the XY spinHamiltonian

H = −∑n

[J1(SxnS

xn+1 + SynS

yn+1) + J2(SxnS

xn+2 + SynS

yn+2)

],

where n runs over all sites in the chain, and J1 and J2 are exchange constants.

Two possible ordered magnetic ground states are (i) ferromagnetic alignment,with mean-field energy per spin EF , and (ii) antiferromagnetic alignment, with mean-field energy per spin EAF . Show that EF = −S2 (J1 +J2) and EAF = S2(J1−J2), whereS is the spin value. [4]

Give a physical explanation for why a negative J2 increases the energy of boththe ferromagnetic and antiferromagnetic ground states. [4]

An alternative magnetic ground state is a spin helix, in which the spin componentson the nth site are given by

Sxn = S cos(nqc) and Syn = S sin(nqc) .

Obtain EH , the mean-field energy per spin for the spin helix, and show that EH is aminimum when cos(qc) = −αJ1/J2 and J2 < 0, where α is a constant to be determined.Hence, show that the spin helix is always the most stable ground state when J2 < 0and |J2| > α |J1|. [10]

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5. A metal has a simple spherical Fermi surface and an isotropic, energy-independenteffective mass. A magnetic field B is applied parallel to z. Explain the principles neededto derive the following equation:

m∗{

dv

dt+

v

τ

}= −eE− ev ×B ,

where m∗ is the effective mass, τ−1 is the scattering rate, and the other symbols havetheir usual meaning. [2]

For circularly polarized light of angular frequency ω, where E± = (Ex±iEy) exp(iωt)with |Ex| = |Ey|, show that the resulting current J± = (Jx ± iJy) exp(iωt) is related toE by the classical dynamic conductivity given by

σ± =σ0

1 + i(ω ± ωc)τ,

where σ0 = ne2τ/m∗ , n is the density of electrons, and ωc = eB/m∗ . [6]

Explain what the consequences of this relation are for the optical properties ofconducting 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 effectivemass of electrons in both semiconductors and metals. Explain why the experimentalarrangement is different for the two types of materials and sketch the experimentalarrangement and typical results for the two cases. [8]

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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) = E0 −∑T6=0

t(T) exp(ik ·T) ,

where T represents the lattice vectors. Explain the meaning of the terms E0 and t(T),and for the case of a rectangular lattice with interactions from the nearest neighbouratoms in the x and y directions only, show that the equation above leads to the expres-sion

E(k) = E0 − 2tx cos(kxa)− 2ty cos(kyb) ,

where the lattice parameter a < b , and tx and ty must be appropriately defined. Sketchthe first Brillouin Zone, labelling the Γ, X, Y, and M-points. For the case where tx = 2ty ,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 bethe approximate value of the Fermi energy and the shape of the Fermi surface forthis material? Where would you expect additional energy gaps to open up? A three-dimensional (3D) tetragonal crystal of the same metal crystallizes with a = b < c andtx = ty = 2tz , where the 3D notation makes the z-axis equivalent to the y-axis in thetwo-dimensional (2D) case. Sketch the approximate shape of the Fermi surface. Whattechniques 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, thelow temperature resistivity is found to show two series of oscillations with differentfrequencies in 1/B. Explain what is the origin of these oscillations. Use the dispersionrelation above to estimate the ratio of the two frequencies. [7]

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7. Excitons are typically thought of as members of two different families, Frenkeland Wannier-Mott excitons. Compare and contrast the properties of the two familiesand give examples of materials which belong to each family, explaining why you havechosen them. [6]

Transmission through a thin layer of a new bulk semiconducting material is usedto 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 towhich 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 both0.2 me (where me is the standard mass of the electron), what further information canyou deduce? [9]

1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64

1

Abs

orpt

ion

coef

ficie

nt, 1

06 m-1

Photon energy, eV

A quantum well is to be constructed from the new material by surrounding a 5 nmthick layer with a thick layer of a second material with a band gap of 2.3 eV. Estimatethe energy levels for the electrons and holes in the quantum well and explain how theabsorption spectrum will be changed, including a sketch of what you think it may looklike at 4.2 K in the region 1.6–2.4 eV. [10]

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8. Explain the origin of the Ginzberg-Landau equation below:

J = −iqh̄

2m(ψ∗∇ψ − ψ∇ψ∗)− q2

m|ψ|2A .

Using this equation, show that the magnetic flux Φ through a superconducting ring isquantised 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 typicalsuperconducting ring, as shown in Figure (a) above. [4]

For a superconducting ring containing two identical Josephson junctions (as shownin Figure (b) above), show that the current I across the device is given by

I = 2 IC cos

(πΦ

Φ0

)sin δ ,

where δ is a phase difference that must be defined. δ1 and δ2 are the phase differencesacross 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

Leff =Φ0

4πIC cos(πΦΦ0

) .

[5]

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