Sommerfeld expansion

Integrals of this type appear frequently when calculating electronic properties, like the heat capacity, 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.

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,

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

We can obtain higher order terms in the Sommerfeld expansion by use of a generating function for moments of the Fermi distribution. This is given by

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

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 ${\displaystyle \int _{0}^{\infty }{\frac {d\epsilon }{2\pi }}\sinh(\epsilon \tau /\pi ){\frac {1}{e^{\beta \epsilon }-1}}={\frac {1}{4\tau }}\left\{1-{\frac {\tau T}{\tan \tau T}}\right\},\quad 0<\tau T<\pi .}$

• Sommerfeld, A. (1928). "Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik". Zeitschrift für Physik. 47 (1–2): 1–3. Bibcode:1928ZPhy...47....1S. doi:10.1007/BF01391052. S2CID 116911592.
• Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Thomson Learning. p. 760. ISBN 978-0-03-083993-1.