The Sommerfeld Model: Ground State and Heat Capacity

The previous lesson showed that treating the conduction electrons as a classical gas fails: the heat capacity is a hundred times too small, and the culprit is the exclusion principle. Arnold Sommerfeld's 1927 fix keeps Drude's picture of free electrons in a box but counts their states quantum-mechanically and fills them according to Fermi–Dirac statistics. Everything measurable about the electron gas — its energy, pressure, compressibility, and its tiny heat capacity — follows from one object: the density of states $g (E)$ , the number of single-electron levels per unit energy.

Quantizing the electron gas in a box

Take $N$ electrons in a cube of side $L$ and volume $V = L^{3}$ , ignoring both the electron–electron interaction and the periodic potential of the ions. Each electron obeys the free-particle Schrödinger equation

- \frac{ℏ ^{2}}{2 m} \nabla^{2} ψ_{k} (r) = E ψ_{k} (r),

whose solutions are plane waves $ψ_{k} (r) = V^{- 1/2} e^{i k \cdot r}$ with energy

E (k) = \frac{ℏ ^{2} k ^{2}}{2 m} .

To count states cleanly we impose Born–von Kármán periodic boundary conditions, $ψ (r + L \overset{x}{^}) = ψ (r)$ and likewise for $\overset{y}{^}, \overset{z}{^}$ . This wraps the box onto a torus; it changes nothing about the bulk but makes the allowed wavevectors a discrete grid,

k_{x} = \frac{2 π}{L} n_{x}, k_{y} = \frac{2 π}{L} n_{y}, k_{z} = \frac{2 π}{L} n_{z}, n_{x}, n_{y}, n_{z} \in Z .

Each allowed $k$ occupies a volume $(2 π / L)^{3} = 8 π^{3} / V$ of $k$ -space, so the number of $k$ -points per unit volume of $k$ -space is $V /8 π^{3}$ . Because each spatial state holds two electrons (spin up and down), the number of electron states in a $k$ -space region of volume $Ω_{k}$ is

d N = 2 \cdot \frac{V}{8 π ^{3}} Ω_{k} = \frac{V}{4 π ^{3}} Ω_{k} .

The Fermi sphere

In the ground state the electrons fill the lowest-energy levels, and since $E$ depends only on $∣ k ∣$ , the occupied region is a sphere in $k$ -space — the Fermi sphere — of radius $k_{F}$ , the Fermi wavevector. Its surface is the Fermi surface; for free electrons it is a perfect sphere.

In the ground state the allowed k-points fill a sphere of radius k_F. Each grid cell has volume (2 pi / L) cubed and holds two electrons; the sphere surface is the Fermi surface, and every occupied state lies inside it.

Setting the total occupied volume $\frac{4}{3} π k_{F}^{3}$ equal to the count of $N$ electrons through $d N$ above,

N = \frac{V}{4 π ^{3}} \cdot \frac{4}{3} π k_{F}^{3} = \frac{V k _{F}^{3}}{3 π ^{2}},

so the Fermi wavevector is fixed entirely by the electron density $n = N / V$ :

k_{F} = (3 π^{2} n)^{1/3} .

The Fermi energy is the energy of the topmost filled level and the Fermi velocity is the speed of an electron on the Fermi surface:

E_{F} = \frac{ℏ ^{2} k _{F}^{2}}{2 m} = \frac{ℏ ^{2}}{2 m} (3 π^{2} n)^{2/3}, v_{F} = \frac{ℏ k _{F}}{m} .

For a monovalent metal like sodium or copper, $n \sim 1 0^{29} m^{- 3}$ gives $k_{F} \sim 1.4 \overset{˚}{A}^{- 1}$ (comparable to a reciprocal-lattice vector, a fact that returns in band theory), $E_{F} \sim 3$ – $7 eV$ , and $v_{F} \sim 1 0^{6} m/s$ , about $1%$ of the speed of light. The corresponding Fermi temperature $T_{F} = E_{F} / k_{B}$ runs from $30, 000$ to $80, 000 K$ , so at any real temperature $T / T_{F} ≲ 0.01$ and the gas is deeply degenerate.

The density of states

Most calculations need not the individual levels but their number per unit energy. Counting states inside a sphere of radius $k (E) = 2 m E /ℏ$ ,

N (E) = \frac{V}{4 π ^{3}} \cdot \frac{4}{3} π k^{3} = \frac{V}{3 π ^{2}} (\frac{2 m E}{ℏ ^{2}})^{3/2},

and differentiating gives the density of states $g (E) = d N / d E$ :

g (E) = \frac{V}{2 π ^{2}} (\frac{2 m}{ℏ ^{2}})^{3/2} E = \frac{3}{2} \frac{N}{E _{F}} \frac{E}{E _{F}} .

The characteristic square-root growth is a three-dimensional signature: in $d$ dimensions $g (E) \propto E^{d /2 - 1}$ , so a two-dimensional gas has a constant density of states and a one-dimensional gas diverges as $E^{- 1/2}$ at each band bottom. A tidy relation follows by evaluating at the Fermi energy:

g (E_{F}) = \frac{3 N}{2 E _{F}} = \frac{3 n}{2 E _{F}} V .

The three-dimensional density of states g(E) grows as the square root of energy. At T = 0 all levels below E_F are full (shaded). At finite T the Fermi-Dirac factor rounds the step over a width of a few k_B T, promoting electrons from just below E_F to just above it.

The ground-state energy and pressure

Integrating $E g (E)$ over the filled sphere gives the total energy at $T = 0$ :

U_{0} = 0 \int E_{F} E g (E) d E = \frac{3}{2} \frac{N}{E _{F}^{3/2}} 0 \int E_{F} E^{3/2} d E = \frac{3}{5} N E_{F},

so the average energy per electron is $\frac{3}{5} E_{F}$ , not the classical $\frac{3}{2} k_{B} T$ . Even at absolute zero the gas carries an enormous zero-point energy because the exclusion principle forces electrons into high- $k$ states. This energy depends on volume through $E_{F} \propto n^{2/3} \propto V^{- 2/3}$ , so it exerts a degeneracy pressure

P = - \frac{\partial U _{0}}{\partial V} = \frac{2}{3} \frac{U _{0}}{V} = \frac{2}{5} n E_{F},

of order $1 0^{10} Pa$ ( $1 0^{5} atm$ ) in a typical metal. It is this Fermi pressure, resisted by the electrostatic binding to the ions, that sets the compressibility (bulk modulus) of a metal; the free-electron estimate lands within a factor of a few of the measured values for the alkali metals.

Finite temperature: the Sommerfeld expansion

At $T > 0$ the occupation of a level of energy $E$ is the Fermi–Dirac function

f (E) = \frac{1}{e ^{(E - μ) / k_{B} T} + 1},

with $μ$ the chemical potential, fixed by holding $N$ constant. The function drops from $1$ to $0$ over a window of a few $k_{B} T$ centred on $μ$ ; at $T = 0$ it is a step and $μ = E_{F}$ . Any thermodynamic average has the form $\int H (E) g (E) f (E) d E$ , and because $- d f / d E$ is a sharp spike at $μ$ of width $k_{B} T$ , such integrals are dominated by the immediate neighbourhood of the Fermi surface. Expanding a smooth function $H (E)$ about $μ$ and integrating term by term gives the Sommerfeld expansion,

- \infty \int \infty H (E) f (E) d E = - \infty \int μ H (E) d E + \frac{π ^{2}}{6} (k_{B} T)^{2} H^{'} (μ) + \frac{7 π ^{4}}{360} (k_{B} T)^{4} H^{'''} (μ) + \dots .

The expansion parameter is $(k_{B} T / E_{F})^{2} \sim 1 0^{- 4}$ , so the leading correction is already superb. Applying it to the number constraint fixes how $μ$ drifts below $E_{F}$ as the gas warms,

μ (T) = E_{F} [1 - \frac{π ^{2}}{12} (\frac{k _{B} T}{E _{F}})^{2} + \dots],

a shift of parts in $1 0^{4}$ at room temperature, so $μ \approx E_{F}$ is an excellent approximation.

The Fermi-Dirac occupation f(E) at T = 0 (sharp step at E_F, solid) and at finite T (rounded, dashed). Its derivative minus df/dE is a spike of width a few k_B T centred on E_F: only electrons in that shell change their occupation and can absorb heat.

The electronic heat capacity

The heat capacity is $C_{V} = (\partial U / \partial T)_{V}$ . A quick physical estimate gets the answer up to a number: only the fraction $\sim k_{B} T / E_{F}$ of electrons lying within $k_{B} T$ of the Fermi surface can be thermally excited, and each gains energy $\sim k_{B} T$ , so

U (T) - U_{0} \sim N \frac{k _{B} T}{E _{F}} k_{B} T, C_{V} \sim N k_{B} \frac{k _{B} T}{E _{F}} = N k_{B} \frac{T}{T _{F}} .

The tiny factor $T / T_{F}$ supplies the suppression that repairs Drude's failure. The Sommerfeld expansion supplies the exact coefficient. Applying it to $U = \int E g (E) f (E) d E$ with the number constraint gives

C_{V} = \frac{π ^{2}}{3} g (E_{F}) k_{B}^{2} T \equiv γ T, γ = \frac{π ^{2}}{3} k_{B}^{2} g (E_{F}) = \frac{π ^{2}}{2} \frac{N k _{B}}{T _{F}} .

The linear coefficient $γ$ (the Sommerfeld coefficient) is directly proportional to the density of states at the Fermi surface — a fact that makes low-temperature heat-capacity measurement one of the cleanest probes of $g (E_{F})$ in any metal.

Separating electrons from phonons

At low temperature the total heat capacity of a metal is the sum of the electron term and the Debye phonon term, which varies as $T^{3}$ :

C_{V} = γ T + β T^{3} .

Dividing by $T$ linearizes the data:

\frac{C _{V}}{T} = γ + β T^{2} .

A plot of $C_{V} / T$ against $T^{2}$ is therefore a straight line whose intercept is the electronic $γ$ and whose slope is the phonon $β$ (from which the Debye temperature follows). This is the standard experimental separation of the two contributions, and it works because the electron and phonon terms carry different powers of $T$ .

Plotting C over T against T squared gives a straight line. The intercept is the electronic Sommerfeld coefficient gamma; the slope is the phonon coefficient beta, from which the Debye temperature follows. Two metals with different densities of states at E_F show different intercepts.

Two loose ends point ahead. First, measured values of $γ$ agree with the free-electron prediction only to within factors of order unity: in transition metals the discrepancy is large, and it is customary to absorb it into a thermal effective mass $m^{*}$ through $γ \propto m^{*}$ , a first hint that the periodic lattice reshapes $g (E_{F})$ — the subject of band theory. Second, the same thermal shell that carries the heat also carries the electric and thermal current; the next lesson puts the Fermi surface in motion under applied fields and derives the Wiedemann–Franz law and the Hall effect from it.