New method for solving Boltzmann's equation for electrons in metals
New method for solving Boltzmann's equation for electrons in metals
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DOI：
10.1103/physrevb.17.3725
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发表时间：
1978-05
影响因子：
3.7
通讯作者：
P. B. Allen
中科院分区：
文献类型：
--
作者：
P. B. Allen
引入了一组新的基函数，它由费米面调和函数FJ（k）和能量（ε− μ）k B T中的多项式σ n（ε）的乘积组成.前者在费米曲面上是标准正交的，而后者则是权函数为− <$f <$ε的标准正交的。根据这一套的确切的半经典玻尔兹曼方程采取了特别简单的形式，给出了一个矩阵方程，它可能会被截断在低阶高精度。与变分方法的联系是简单的。在1× 1矩阵处截断给出了通常的变分解，其中假设φ k与电导率的ν kx成正比，与热导率的（ε− μ）ν kx成正比。在声子散射的情况下，给出了散射算符的矩阵元QJn，J′ n′的显式方程，并给出了弱各向异性情况下精确的ρ的微扰公式.矩阵元是谱函数α 2（±，J，J′）F（Ω）上的简单积分，它推广了超导理论中的电子-声子谱函数α 2 F（Ω）.类比玻尔兹曼理论和Eliashberg理论之间的超导体的Tc描述。明确了高温电阻与s波或p波转变温度之间的密切关系。
A new set of basis functions is introduced, consisting of products of Fermi-surface harmonics F J (k) and polynomials σ n (ε) in the energy (ε− μ) k B T. The former are orthonormal on the Fermi surface, and the latter are orthonormal with weight function−∂ f∂ ε. In terms of this set the exact semiclassical Boltzmann equation takes a particularly simple form, giving a matrix equation which can probably be truncated at low order to high accuracy. The connection with variational methods is simple. Truncating at a 1× 1 matrix gives the usual variational solution where φ k is assumed proportional to ν kx for electrical conductivity and (ε− μ) ν kx for thermal conductivity. Explicit equations are given for the matrix elements Q J n, J′ n′ of the scattering operator for the case of phonon scattering, and a perturbation formula for ρ is given which is accurate for weak anisotropy. The matrix elements are simple integrals over spectral functions α 2 (±, J, J′) F (Ω) which generalize the electron-phonon spectral function α 2 F (Ω) used in superconductivity theory. Analogies are described between Boltzmann theory and Eliashberg theory for T c of superconductors. The intimate relations between high-temperature resistance and the s-or p-wave transition temperature are made explicit.