Sommerfeld expansion

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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 \beta is a large quantity, the integral can be expanded[1][2] in terms of \beta as

\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)^2H^\prime(\mu) + O \left(\frac{1}{\beta\mu}\right)^4

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

Application to the free electron model[edit]

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 H(\varepsilon). For these integrals we can then identify \beta as the inverse temperature and \mu as the chemical potential. Therefore, the Sommerfeld expansion is valid for large \beta (low temperature) systems.

Derivation to second order in temperature[edit]

We seek an expansion that is second order in temperature, i.e., to \tau^2, where \beta^{-1}=\tau=k_BT is the product of temperature and Boltzmann's constant. Begin with a change variables to \tau x=\varepsilon -\mu:

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, I=I_1+I_2, and rewrite I_1 using the change of variables x\rightarrow-x:

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}\,.


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 I_1,

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

to obtain:

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 -\tau \mathrm{d}x =  \mathrm{d}\varepsilon in the first term of I_1. Combine I=I_1+I_2 to obtain:

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 \tau is sufficiently small and H(\varepsilon) is sufficiently smooth:

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

to obtain,

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:

\int_0^{\infty} \frac{x\mathrm{d}x}{e^{x} + 1}=\frac{\pi^2}{12}.

Hence,

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)\,

Notes[edit]

  1. ^ Ashcroft & Mermin 1976, p. 760.
  2. ^ http://www.physik.uni-regensburg.de/forschung/fabian/pages/mainframes/teaching/teaching_files/files%20of%20mf_statistical_physics/Sommerfeld.pdf
  3. ^ http://www.sosmath.com/tables/integral/integ38/integ38.html

References[edit]