Sommerfeld ExpansionEdit
Sommerfeld expansion is a staple technique in quantum statistical mechanics that provides a precise way to approximate integrals weighted by the Fermi-Dirac distribution at low temperatures. It is especially valuable for systems of degenerate fermions, such as electrons in metals, where the occupation function behaves like a sharp step near the chemical potential. The method expresses thermodynamic quantities as a smooth baseline plus a series of temperature-dependent corrections, organized in even powers of the small parameter k_B T.
Named in honor of Arnold Sommerfeld, the expansion exploits the fact that at temperatures well below the Fermi temperature, the dominant physics is governed by states near the chemical potential μ. The resulting asymptotic series involves derivatives of a smooth accompanying function g(ε) evaluated at μ. This yields compact formulas that connect microscopic details of the energy dependence of g to macroscopic observables such as energy, entropy, and specific heat. In practice, the technique underpins the classic result that the electronic contribution to the specific heat of metals varies linearly with temperature at low T, a hallmark of degenerate fermion behavior.
Historical context and foundations
The Sommerfeld expansion emerged from efforts to refine the free-electron model of metals by incorporating quantum statistics more accurately. Early work in quantum statistics showed that the behavior of electrons at low temperatures could be captured by expanding around the Fermi surface. The method owes its name to the work of Arnold Sommerfeld in the 1920s and 1930s, who used it to derive concise expressions for thermodynamic quantities of degenerate electron gases. The technique complemented the broader framework of statistical mechanics and quantum statistical mechanics, and it remains a standard tool within condensed matter physics and low-temperature physics.
The core idea is to evaluate integrals of the form ∫ dε g(ε) f(ε), where f(ε) is the Fermi-Dirac distribution, by expanding in powers of the small parameter T/T_F, with T_F the Fermi temperature. The result is an asymptotic series that isolates a zero-temperature contribution and systematic finite-temperature corrections. In this sense, the Sommerfeld expansion formalizes the intuition that, for a degenerate electron gas, the physics is controlled by a narrow energy window near μ.
Mathematical framework
Consider an integral of the form I(T) = ∫ dε g(ε) f(ε; μ, T), where f is the Fermi-Dirac distribution f(ε) = [e^( (ε−μ)/(k_B T) ) + 1]^{-1}, and g(ε) is a smooth function of energy that encodes the system’s density of states and other spectral properties. For temperatures much smaller than the characteristic electronic energy scales, I(T) admits an expansion in even powers of k_B T:
I(T) ≈ ∫_{-∞}^{μ} g(ε) dε + (π^2/6) (k_B T)^2 g'(μ) + (7π^4/360) (k_B T)^4 g'''(μ) + …
Key ideas: - The leading term is the zero-temperature contribution, obtained by integrating up to the chemical potential. - The first finite-temperature correction is proportional to T^2 and involves the first derivative g'(μ). - Higher-order corrections involve higher derivatives of g at μ and higher powers of T. - The expansion is valid when μ lies well within a region where g(ε) is smooth and when T is small compared with the relevant energy scales, such as the Fermi energy E_F.
A central application is to the grand canonical potential or related thermodynamic quantities. For example, the electronic energy, entropy, and the specific heat can be extracted by applying the expansion to the appropriate g(ε), often with g(ε) proportional to the density of states or to related spectral functions. In the canonical case of a three-dimensional free electron gas, one finds the classic result for the electronic specific heat C_v ∝ T with a coefficient proportional to the density of states at μ, N(μ).
For convenience, the expansion is often presented with μ approximately equal to the Fermi energy at T = 0, μ ≈ E_F, and expressed in terms of the density of states g(μ) and its derivatives. In more complicated band structures or dimensionalities, the same structure holds but with the appropriate derivatives evaluated at the actual μ.
Applications in condensed matter physics
Electronic specific heat and thermodynamics of metals: The Sommerfeld expansion explains why the electronic contribution to the specific heat of metals grows linearly with temperature at low T, with a coefficient proportional to the density of states at the Fermi level. This connection is encapsulated in expressions like C_v ≈ (π^2/3) k_B^2 T g(μ), where g(μ) is the density of states at the chemical potential. See Fermi-Dirac distribution and density of states for context.
Internal energy and chemical potential corrections: The expansion yields systematic corrections to U(T) and μ(T) in terms of g and its derivatives. These corrections are essential when modeling metals with nontrivial band structure or when high-precision thermodynamics is required.
Extensions to different dimensions and band structures: While the textbook 3D free-electron gas is the archetype, the Sommerfeld method generalizes to two-dimensional electron gases, one-dimensional systems, and multi-band metals, with the basic pattern of temperature corrections preserved but with the appropriate g(ε) and its derivatives evaluated at the appropriate μ.
Connections to finite-temperature transport and response: Since many transport coefficients depend on the energy dependence of the DOS and on the distribution of states near μ, the Sommerfeld expansion informs approximate analytic expressions for quantities like thermopower and low-temperature conductivities in clean, degenerate metals.
Educational value in quantum statistics: The expansion provides a clear, tangible link between microscopic spectral properties and macroscopic thermodynamics, making it a standard teaching tool in courses on quantum statistical mechanics and condensed matter physics.
Limitations and generalizations
Validity and regime of applicability: The expansion assumes T ≪ T_F and smooth behavior of g(ε) near μ. It is an asymptotic series, and truncation at a given order introduces controlled, small errors that scale with higher powers of T. In systems with sharp features in g(ε) near μ, such as Van Hove singularities, the standard expansion may require modification or alternative techniques.
Electron-electron interactions: The basic form presumes non-interacting or weakly interacting electrons, sometimes framed within Fermi-liquid theory. In strongly correlated or highly disordered systems, the simple Sommerfeld picture can fail or require substantial revision, and one must account for many-body effects that alter the energy dependence of g(ε) or introduce new low-energy scales.
Magnetic fields and other couplings: External fields, spin splitting, or coupling to lattice vibrations (phonons) can modify the relevant spectral functions and chemical potential in nontrivial ways. The expansion can be adapted to these situations, but the expressions become more model dependent.
Generalizations to grand potentials and finite temperatures: The same methodology underpins expansions of the grand potential and related thermodynamic functionals, with g(ε) replaced by the appropriate spectral weight. It remains a flexible tool in the broader study of finite-temperature quantum systems.