Solid State I Tutorial Question Sheet 1

1. Dimension-independent forms of Ohm’s law are crucial in parameterising the properties of metals. Given a single crystal of dimensions as sketched below, you pass a current, I, and obtain a resistance by measuring a voltage, V, between contacts 1 and 2. How do you convert to the corresponding resistivity, ρ, assuming homogeneous current paths through the sample? How does ρ relate to the conductivity, σ?

I

L12

1 2

L

t

w

2. The alkali metal sodium has one conduction electron per atom, and an average atomic separation of 3.35 Å. Its resistivity at 273 K is 4.2 µΩcm. Make a classical estimate of the scattering, or relaxtion time, τ.

The Hall effect played an important role in understanding the subtleties of real metals. By considering the Lorentz force -ev × B on electrons moving in a magnetic field B, show that in steady state, a transverse field electric field Ey results from applying an electric field Ex and a magnetic field Bz.

x

yz

Ex jx

The Hall coefficient RH = Ey/jxBz is like a transverse resistivity describing this process. Show that in the Drude model, RH = -1/ne, where n is the carrier concentration and e the electron charge. What is the magnetoresistance, ρ(Bz)? How could one account for a positive Hall coefficient in the Drude model?

3. Much of this longer question should be basically revision. If you struggle with it, note that its contents are covered in all the standard text books.

4. Calculate TF, kF and vF for potassium, using data from the standard text books. Other systems exist in nature which can be treated (at various levels of approximation) as quantum Fermi gases. One is liquid 3He, which has a density of 81 kg m-3. Estimate TF, kF and vF in this case as well.

5. Calculate the density of states per unit energy g(ε) for the quantum Fermi gas in one/two dimensions, and compare with the three-dimensional result derived in lectures. Express your answers per unit length/area.

6. Summary and revision of the Drude and Sommerfeld models.

Write down the definition and units for ρ, σ, n, vdrift, vrandom(definition for Drude and Sommerfeld models), τ, l, κ, c, k, εF and TF. This exercise is much more valuable if you make use of thought about the units to enhance your understanding of the quantities concerned and their inter-relationship!

7. X-ray diffraction is performed using the setup shown below, with the pattern imaged on a planar detector a long way behind the sample.

Incident x-rays

Scattered x-rays

Sample

DetectorWhat form will the pattern take for a) a single crystal illuminated by monochromaticx-rays, b) a single crystal illuminated by white x-rays (Laue setup) c) a powder sample illuminated by monochromatic x-rays and d) a powder sample illuminated by white x-rays. Make rough sketches, and describe your reasoning.

8. Given this periodic array, identify a) the lattice b) the basis c) the Wigner-Seitz unit cell d) another primitive unit cell.

9. Derive and sketch the reciprocal lattice for an orthorhombic lattice (a,b,c orthogonal, a ≠ b ≠ c), and draw in the first Brillouin zone.

10. Take only the a* b* plane of the lattice from question 9, and treat it as a two-dimensional rectangular reciprocal lattice. Draw the first two Brillouin zones. Then take only the a* axis, treating it as a one-dimensional linear reciprocal lattice, and draw the first three zones.

Tutorial sheet 3

11. Check the self-consistency of the ‘filled band is insulating’ picture discussed in lectures. Try to ignore the fact that you can’t make a Fermi surface in a filled band, and instead construct particles from wave packets centres on states at π/a. What is the group velocity? What is the character of the wave-functions here? Are either of these consistent with a metallic state?

12. Recall the following from lectures: ‘… So insulators, which cannot carry a current, must contain electrons too. In a metal they must be free to move, and in an insulator they must be stuck. Iasked my tutor why this was so - and he told me that it was not understood. It was good to know the limits of knowledge at the time’ Quote from Sir Nevill Mott, writing about his time as a student in Cambridge, around 1925. Armed with our understanding of the effects of the periodic potential, we can now account for diamond being an insulator. Give a critique of Mott’s statement that the electrons in an insulator must be stuck.

13. We began the course by saying that carbon exists in all three possible low temperature states of matter (metal, insulator and superconductor). Solving the Schrödinger equation for non-interacting electrons in a periodic potential has showed us how diamond can be an insulator, even though carbon contains weakly bound outer electrons. In the picture that we have developed, could potassium, nitrogen or indeed lanthanum copper oxide, La2CuO4, ever be insulators?

14. Diamond and indium antimonide (InSb) are both insulators, with band gaps (between the highest energy filled band and lowest energy empty band) of 5.5 eV and 0.15 eV respectively. Are both materials transparent to visible light? If not, why not?

15. A one-dimensional metal sees such a weak periodic potential that it is the pure sinusoid V(x) = V1cos(2πx/a). Will there be a band gap at ± 2π/a if the bands are drawn in the extended zone scheme?

1e21

BE /E

E −ω+ω=ε ω Tkh

hh

16. The average energy of a quantum mechanical oscillator of frequency ωE at temperature T is:

(equation 50 from the lecture notes). As we saw, Einstein modelled the vibrations of a solid as 3N of these oscillators. Show that for kBT >> ωE, .TkB≅ε

∑ ∫ ωω

−ω+ω=

∞

ωbranches 0

/ d)(1e2

1/B

gVE TkTOT h

hh

17. In lectures we wrote down the exact expression for the totalvibrational energy per unit volume of a solid:

And then applied the Debye approximation to obtain the specific heat. Show that the exact expression can be used to derive the same result in the low temperature limit (hint: Hook and Hall do the algebra on p.58,

but it will be ‘good for you’ to practise it yourselves. There is an obvious reason for the exact agreement between the results of the two treatments. What is it?

18. An important step to understanding the low temperature transport properties of solids is to realise that the number of excited phonons varies as T3 for T << ΘD. Show that this is so. 19. Two 2D metals, A and B, have Fermi surfaces shown in blue, sketched in the first Brillouin zone. Which will have the broadest temperature range in which the resistivity varies as T5, and why?

A B

20. Practice with manipulating Fourier transforms. Sketch the Fourier space representation of the transforms F(s) of:

a) cos2xb) cos2πx/ac) cosx + 3cos3xd) 1 + sin2xe) sin2xsinx (make use of the convolution theorem)

21. The specific heat of the oxide metal Sr2RuO4 is shown below. What are the electronic term, γ, and the Debye temperature, ΘD, if we assume that its structure is tetragonal with primitive cell dimensions a = b = 3.8 Å and c = 6.3 Å?

80

40

0 100 200T2 (K2)

c/T

(mJ/

mol

K2 )

22. By considering the dispersion relations for electrons, x-rays and neutrons, give a critique of which is likely to be the best probe for measurement of phonon dispersion relations by inelastic scattering.

23. Will the existence of optic phonon branches affect conclusions that we drew about the phonon heat capacity (cv ~ T3 for T << ΘD and T independent for T >> ΘD)?

24. Convolution and finite size effects in diffraction. Using your knowledge of the convolution theorem, sketch the diffraction pattern that you would expect to see from an array of of atoms ten atoms long compared to one of infinite length. The lattice spacing in both cases is a.25. KCl crystallises in the face-centred cubic structure with a diatomic basis at (0,0,0) and (1/2,0,0) (referenced to the cubic lattice vectors a,b and c). K and Cl have essentially identical atomic scattering factors. What simpler atomic structure will the x-ray pattern suggest to exist?

Tutorial sheet 6

26. a) Show that for a 3D material with a spherical Fermi surface and a k independent scattering time τ, where as usual the mean free path l = vFτ.

lh

2F

2

ke∝σ

lh

2F

2

231 keπ

=σb) The exact answer is . Using the example of a

simple cubic lattice of side a and a monatomic basis, show that this special case reduces to the Drude formula .

*

2

mne τ=σ

c) Show how expression (83) from the notes allows the density ofstates g(ε) to be expressed in a form in which it is proportional to m*kF, and how the exact expression

22F*

hπkm is equivalent to the

expression obtained in question 5.

27. The well-known high temperature superconductor YBa2Cu3O7 is a layered structure containing 2D sheets of CuO2 and 1D chains of CuO. Assuming a) that each gives a half-filled parabolic band, b) that these bands are independent, and c) that the material has 2D square symmetry, sketch its Fermi surface.