Ashcroft Mermin eng · Problem 3

Inhomogeneous Semiconductors

Physics — Statistical Mechanics · phonon-corrections-electronic-energy

Problem

Phonon Corrections to the Electronic Energy In the limit ωD→0\omega_D \to 0, the correction (26.27) to the electronic energy reduces to that of the Hartree-Fock approximation as modified by Thomas-Fermi screening (Chapter 17, page 344): εkTF=εk0−∫k′<kFdk′(2π)34πe2∣k−k′∣2+k02.(26.56)\varepsilon_{\mathbf{k}}^{TF} = \varepsilon_{\mathbf{k}}^0 - \int_{k'<k_F} \frac{d\mathbf{k}'}{(2\pi)^3} \frac{4\pi e^2}{|\mathbf{k}-\mathbf{k}'|^2 + k_0^2}. \tag{26.56} When the phonon frequencies are not considered vanishingly small, (26.27) differs appreciably from (26.56) only for those values of the integration variable k′\mathbf{k}' for which εk′\varepsilon_{\mathbf{k}'} is within O(ℏωD)O(\hbar\omega_D) of εk\varepsilon_{\mathbf{k}}. Since ℏωD\hbar\omega_D is small compared with εF\varepsilon_F, the region of k′\mathbf{k}' for which the correction is appreciable is a shell about the surface εk′=εk\varepsilon_{\mathbf{k}'} = \varepsilon_{\mathbf{k}} that is thin (on the scale of the dimensions of the zone). We may exploit this fact to simplify the correction term by writing the integral over k′\mathbf{k}' as an integral over energy ε′\varepsilon', and an integral over the constant-energy surfaces εk′=ε′\varepsilon_{\mathbf{k}'} = \varepsilon'. As ε′\varepsilon' varies, the variation of the term in (εk−ε′)2(\varepsilon_{\mathbf{k}} - \varepsilon')^2 in the denominator of (26.27) is very important, since the denominator vanishes within this range. However, the remaining ε′\varepsilon' dependence of the integrand (due to the fact that k′\mathbf{k}' is constrained to a surface of energy ε′\varepsilon') leads to very little variation as ε′\varepsilon' varies within O(ℏωD)O(\hbar\omega_D) of εF\varepsilon_F. It is thus a good approximation to replace the k′\mathbf{k}' integrations over the surfaces εk′=ε′\varepsilon_{\mathbf{k}'} = \varepsilon' by integrations over the single surface εk′=εk\varepsilon_{\mathbf{k}'} = \varepsilon_{\mathbf{k}}. After this replacement, the only ε′\varepsilon' dependence left comes from the explicit term in ε′\varepsilon' in the denominator. The integral over ε′\varepsilon' is then easily performed. (a) Show that under this approximation, εk=εkTF−∫εk′=εkdS′8π3∣∂ε/∂k∣4πe2∣k−k′∣2+k02×12ℏω(k−k′)ln⁡∣εF−εk−ℏω(k−k′)εF−εk+ℏω(k−k′)∣.(26.57)\begin{aligned} \varepsilon_{\mathbf{k}} = \varepsilon_{\mathbf{k}}^{TF} &- \int_{\varepsilon_{\mathbf{k}'}=\varepsilon_{\mathbf{k}}} \frac{dS'}{8\pi^3|\partial\varepsilon/\partial\mathbf{k}|} \frac{4\pi e^2}{|\mathbf{k}-\mathbf{k}'|^2+k_0^2} \nonumber\\ &\times \frac{1}{2}\hbar\omega(\mathbf{k}-\mathbf{k}') \ln\left|\frac{\varepsilon_F - \varepsilon_{\mathbf{k}} - \hbar\omega(\mathbf{k}-\mathbf{k}')}{\varepsilon_F - \varepsilon_{\mathbf{k}} + \hbar\omega(\mathbf{k}-\mathbf{k}')}\right|. \tag{26.57} \end{aligned} (b) Show that (26.57) immediately implies that the phonon-corrected Fermi surface εk=εF\varepsilon_{\mathbf{k}} = \varepsilon_F is identical to the uncorrected Fermi surface, εkTF=εF\varepsilon_{\mathbf{k}}^{TF} = \varepsilon_F. (c) Show that when εk\varepsilon_{\mathbf{k}} is several times ℏωD\hbar\omega_D from εF\varepsilon_F, the phonon correction is smaller than the Thomas-Fermi correction by O(ℏωD/εF)(ℏωD/[εk−εF])O(\hbar\omega_D/\varepsilon_F)(\hbar\omega_D/[\varepsilon_{\mathbf{k}} - \varepsilon_F]). (d) Show that when εk−εF\varepsilon_{\mathbf{k}} - \varepsilon_F is small compared with ℏωD\hbar\omega_D, (26.57) reduces to (26.28) and (26.29).

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