The Interaction of Electrons and Lattice Vibrations

Abstract

The concept of the “particle” as an entity by itself makes sense only if its life time in a given state is fairly long even with the interactions.

Problems

• 4.1 According to the equation

$$ K = \frac{1}{3}\sum\limits_{m} {C_{m} \bar{V}_{m} \lambda_{m} ,} $$

the specific heat C_m can play an important role in determining the thermal conductivity K. (The sum over m means a sum over the modes m carrying the energy.) The total specific heat of a metal at low temperature can be represented by the equation

$$ C_{v} = AT^{3} + BT, $$

where A and B are constants. Explain where the two terms come from.

• 4.2 Look at Figs. 4.7 and 4.9 for the thermal conductivity of metals and insulators. Match the temperature dependences with the “explanations.” For (3) and (6) you will have to decide which figure works for an explanation.

(1) T:

(a) Boundary scattering of phonons ${\text{K}} = C\bar{V}\lambda /3$, and $\bar{V},\lambda$ approximately constant

(2) T²:

(b) Electron–phonon interactions at low temperature changes cold to hot electrons and vice versa

(3) constant:

(c) C_v ∝ T

(4) T³:

(d) T > θ_D, you know ρ from Bloch (see Problem 4.4), and use the Wiedemann–Franz law

(5) Tⁿe^β/T:

(e) C and $\bar{V} \cong {\text{ constant}}$. The mean squared displacement of the ions is proportional to T and is also inversely proportional to the mean free path of phonons. This is high-temperature umklapp

(6) T⁻¹:

(f) Umklapp processes at not too high temperatures

• 4.3 Calculate the thermal conductivity of a good metal at high temperature using the Boltzmann equation and the relaxation-time approximation. Combine your result with (4.160) to derive the law of Wiedemann and Franz.

• 4.4 From Bloch’s result (4.146) show that σ is proportional to T⁻¹ at high temperatures and that σ is proportional to T⁻⁵ at low temperatures. Many solids show a constant residual resistivity at low temperatures (Matthiessen’s rule). Can you suggest a reason for this?

• 4.5 Feynman [4.7, p. 226], while discussing the polaron, evaluates the integral

$$ I = \int {\frac{{{\text{d}}\varvec{q}}}{{q^{2} f\left( \varvec{q} \right)}},} $$

[compare (4.112)] where

$$ d\varvec{q} = {\text{d}}q_{x} \;{\text{d}}q_{y} \;{\text{d}}q_{z} , $$

and

$$ f\left( \varvec{q} \right) = \frac{{\hbar^{2} }}{2m}\left( {2\varvec{k} \cdot \varvec{q} - q^{2} } \right) - \hbar \omega_{\text{L}} , $$

by using the identity:

$$ \frac{1}{{K_{1} K_{2} }} = \int\limits_{0}^{1} {\frac{{{\text{d}}x}}{{[K_{1} x + K_{2} \left( {1 - x} \right)^{2} ]}}} . $$

• a. Prove this identity

• b. Then show the integral is proportional to

  $$ \frac{1}{k}\sin^{ - 1} \frac{{K_{3} k}}{\sqrt 2 }, $$

  and evaluate K₃.

• c. Finally, show the desired result:

$$ E_{{\varvec{k},0}} = - \alpha_{\text{c}} \hbar \omega_{\text{L}} + \frac{{\hbar^{2} k^{2} }}{{2m^{ * * } }}, $$

where

$$ m^{ * * } = \frac{{m^{ * } }}{{1 - \frac{{\alpha_{c} }}{6}}}, $$

and m* is the ordinary effective mass.