Sommerfeld expansion

A Sommerfeld expansion is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics. Physically, the integrals represent statistical averages using the Fermi–Dirac distribution.

When the inverse temperature \( {\displaystyle \beta } \) is a large quantity, the integral can be expanded[1][2] in terms of \( {\displaystyle \beta }\) as

\( {\displaystyle \int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1}}\,\mathrm {d} \varepsilon =\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +{\frac {\pi ^{2}}{6}}\left({\frac {1}{\beta }}\right)^{2}H^{\prime }(\mu )+O\left({\frac {1}{\beta \mu }}\right)^{4}} \)

where \( {\displaystyle H^{\prime }(\mu )}\) is used to denote the derivative of \( {\displaystyle H(\varepsilon )}\) evaluated at \( {\displaystyle \varepsilon =\mu }\) and where the \( {\displaystyle O(x^{n})}\) notation refers to limiting behavior of order \( {\displaystyle x^{n}}\) . The expansion is only valid if \( {\displaystyle H(\varepsilon )}\) vanishes as \( {\displaystyle \varepsilon \rightarrow -\infty } \) and goes no faster than polynomially in \( {\displaystyle \varepsilon }\) as \( {\displaystyle \varepsilon \rightarrow \infty } .\)

Application to the free electron model

Integrals of this type appear frequently when calculating electronic properties in the free electron model of solids. In these calculations the above integral expresses the expected value of the quantity \( {\displaystyle H(\varepsilon )}\) . For these integrals we can then identify \( {\displaystyle \beta }\) as the inverse temperature and \( {\displaystyle \mu } \) as the chemical potential. Therefore, the Sommerfeld expansion is valid for large \( {\displaystyle \beta }\) (low temperature) systems.
Derivation to second order in temperature

We seek an expansion that is second order in temperature, i.e., to \( {\displaystyle \tau ^{2}} \) , where \( {\displaystyle \beta ^{-1}=\tau =k_{B}T} \) is the product of temperature and Boltzmann's constant. Begin with a change variables to \( {\displaystyle \tau x=\varepsilon -\mu } :

\( {\displaystyle I=\int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1}}\,\mathrm {d} \varepsilon =\tau \int _{-\infty }^{\infty }{\frac {H(\mu +\tau x)}{e^{x}+1}}\,\mathrm {d} x\,,} \)

Divide the range of integration, \( {\displaystyle I=I_{1}+I_{2}}\) , and rewrite \( {\displaystyle I_{1}}\) using the change of variables \( {\displaystyle x\rightarrow -x} \) :

\( {\displaystyle I=\underbrace {\tau \int _{-\infty }^{0}{\frac {H(\mu +\tau x)}{e^{x}+1}}\,\mathrm {d} x} _{I_{1}}+\underbrace {\tau \int _{0}^{\infty }{\frac {H(\mu +\tau x)}{e^{x}+1}}\,\mathrm {d} x} _{I_{2}}\,.} \)

\( {\displaystyle I_{1}=\tau \int _{-\infty }^{0}{\frac {H(\mu +\tau x)}{e^{x}+1}}\,\mathrm {d} x=\tau \int _{0}^{\infty }{\frac {H(\mu -\tau x)}{e^{-x}+1}}\,\mathrm {d} x\,} \)

Next, employ an algebraic 'trick' on the denominator of \( {\displaystyle I_{1}}\) ,

\( {\displaystyle {\frac {1}{e^{-x}+1}}=1-{\frac {1}{e^{x}+1}}\,,} \)

to obtain:

\( {\displaystyle I_{1}=\tau \int _{0}^{\infty }H(\mu -\tau x)\,\mathrm {d} x-\tau \int _{0}^{\infty }{\frac {H(\mu -\tau x)}{e^{x}+1}}\,\mathrm {d} x\,} \)

Return to the original variables with \({\displaystyle -\tau \mathrm {d} x=\mathrm {d} \varepsilon } \) in the first term of \( {\displaystyle I_{1}} . Combine \({\displaystyle I=I_{1}+I_{2}}\) to obtain:

\( {\displaystyle I=\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +\tau \int _{0}^{\infty }{\frac {H(\mu +\tau x)-H(\mu -\tau x)}{e^{x}+1}}\,\mathrm {d} x\,} \)

The numerator in the second term can be expressed as an approximation to the first derivative, provided \( {\displaystyle \tau }\) is sufficiently small and \( {\displaystyle H(\varepsilon )} is sufficiently smooth:

\( {\displaystyle \Delta H=H(\mu +\tau x)-H(\mu -\tau x)\approx 2\tau xH'(\mu )+\cdots \,,} \)

to obtain,

\( {\displaystyle I=\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +2\tau ^{2}H'(\mu )\int _{0}^{\infty }{\frac {x\mathrm {d} x}{e^{x}+1}}\,} \)

The definite integral is known[3] to be:

\( {\displaystyle \int _{0}^{\infty }{\frac {x\mathrm {d} x}{e^{x}+1}}={\frac {\pi ^{2}}{12}}} .

Hence,

\( {\displaystyle I=\int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1}}\,\mathrm {d} \varepsilon \approx \int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +{\frac {\pi ^{2}}{6\beta ^{2}}}H'(\mu )\,} \)

Generating functions

A generating function for moments of the Fermi distribution function is

\( {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}e^{\tau \epsilon /2\pi }\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{\tau }}\left\{{\frac {({\frac {\tau T}{2}})}{\sin({\frac {\tau T}{2}})}}e^{\tau \mu /2\pi }-1\right\},\quad 0<\tau T/2\pi <1.} \)

Here \( {\displaystyle k_{\rm {B}}T=\beta ^{-1}}\) and Heaviside step function \( {\displaystyle -\theta (-\epsilon )}\) subtracts the divergent zero-temperature contribution. Expanding in powers of \( {\displaystyle \tau }\) gives, for example [4]

\( {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}=\left({\frac {\mu }{2\pi }}\right),} \)
\( {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}\left({\frac {\epsilon }{2\pi }}\right)\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{2!}}\left({\frac {\mu }{2\pi }}\right)^{2}+{\frac {T^{2}}{4!}},} \)
\( {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{2!}}\left({\frac {\epsilon }{2\pi }}\right)^{2}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{3!}}\left({\frac {\mu }{2\pi }}\right)^{3}+\left({\frac {\mu }{2\pi }}\right){\frac {T^{2}}{4!}},} \)
\( {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{3!}}\left({\frac {\epsilon }{2\pi }}\right)^{3}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{4!}}\left({\frac {\mu }{2\pi }}\right)^{4}+{\frac {1}{2!}}\left({\frac {\mu }{2\pi }}\right)^{2}{\frac {T^{2}}{4!}}+{\frac {7}{8}}{\frac {T^{4}}{6!}},} \)
\( {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{4!}}\left({\frac {\epsilon }{2\pi }}\right)^{4}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{5!}}\left({\frac {\mu }{2\pi }}\right)^{5}+{\frac {1}{3!}}\left({\frac {\mu }{2\pi }}\right)^{3}{\frac {T^{2}}{4!}}+\left({\frac {\mu }{2\pi }}\right){\frac {7}{8}}{\frac {T^{4}}{6!}},}\)
\( {\displaystyle \int _{-\infty }^{\infty }{\frac {d\epsilon }{2\pi }}{\frac {1}{5!}}\left({\frac {\epsilon }{2\pi }}\right)^{5}\left\{{\frac {1}{1+e^{\beta (\epsilon -\mu )}}}-\theta (-\epsilon )\right\}={\frac {1}{6!}}\left({\frac {\mu }{2\pi }}\right)^{6}+{\frac {1}{4!}}\left({\frac {\mu }{2\pi }}\right)^{4}{\frac {T^{2}}{4!}}+{\frac {1}{2!}}\left({\frac {\mu }{2\pi }}\right)^{2}{\frac {7}{8}}{\frac {T^{4}}{6!}}+{\frac {31}{24}}{\frac {T^{6}}{8!}}.} \)

A similar generating function for the odd moments of the Bose function is \( {\displaystyle \int _{0}^{\infty }{\frac {d\epsilon }{2\pi }}\sinh(\epsilon \tau /\pi ){\frac {1}{e^{\beta \epsilon }-1}}={\frac {1}{4\tau }}\left\{1-{\frac {\tau T}{\tan \tau T}}\right\},\quad 0<\tau T<\pi }\)

Notes

Ashcroft & Mermin 1976, p. 760.
Fabian, J. "Sommerfeld's expansion" (PDF). Universitaet Regensburg. Retrieved 2016-02-08.
"Definite integrals containing exponential functions". SOS Math. Retrieved 2016-02-08.

R. Loganayagam, P. Surówka (2012). "Anomaly/Transport in an Ideal Weyl gas". JHEP. 04: 2012:97. arXiv:1201.2812. Bibcode:2012JHEP...04..097L. doi:10.1007/JHEP04(2012)097.

References

Sommerfeld, A. (1928). "Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik". Zeitschrift für Physik 47: 1–3. Bibcode:1928ZPhy...47....1S. doi:10.1007/BF01391052.
Ashcroft, Neil W.; Mermin, N. David (1976). "Solid State Physics". Thomson Learning: 760. ISBN 978-0-03-083993-1.

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