By the end of this lecture, you should be able to:
- explain why only electrons near the Fermi energy contribute appreciably to thermal properties at ordinary temperatures;
- use the Fermi-Dirac distribution to describe the free electron gas at finite temperature;
- derive and interpret the linear-in-\(T\) electronic heat capacity of a metal;
- relate electrical conductivity to the shifted Fermi sphere and to electron scattering;
- derive the electronic thermal conductivity and the Wiedemann-Franz law;
- explain the origin and limitations of the thermopower of a free electron gas.
11.1 Why Electrons Matter for Thermal Properties
In an insulator, heat is mainly stored and transported by lattice vibrations. In a metal, there is an additional channel: the delocalized conduction electrons. These electrons can store energy, carry charge, and transport heat.
The central point is that conduction electrons in a metal are not a classical gas. They are fermions and obey the Pauli principle. As a result, most electrons are blocked from participating in low-temperature thermal processes because the nearby states are already occupied.
A useful analogy is a completely filled lecture hall. If almost every seat is occupied, only people close to the few empty seats can move. Likewise, in a metal at ordinary temperatures, only electrons in a narrow energy window around the Fermi energy can change their state.
This lecture therefore uses the free electron gas as a minimal model for electronic thermal properties. It neglects the periodic crystal potential and electron-electron interactions, but it correctly captures the most important quantum-statistical effect: Fermi-Dirac occupation.
11.2 Free Electron Gas at Low and Finite Temperature
11.2.1 Ground-State Scales
For free electrons, the energy is
\[ E(k)=\frac{\hbar^2 k^2}{2m_{\mathrm e}}. \]
At \(T=0\), the Pauli principle fills all one-electron states from the bottom of the band up to the Fermi energy \(E_F\). In three dimensions the occupied states form a sphere in \(k\)-space, the Fermi sphere. Its radius is the Fermi wave vector
\[ k_F=(3\pi^2 n)^{1/3}. \]
The associated Fermi energy, Fermi temperature, and Fermi velocity are
\[ E_F=\frac{\hbar^2 k_F^2}{2m_{\mathrm e}}, \qquad T_F=\frac{E_F}{k_B}, \qquad v_F=\frac{\hbar k_F}{m_{\mathrm e}}. \]
For typical metals,
\[ E_F \sim \text{a few eV}, \qquad T_F \sim 10^4 \dots 10^5\ \mathrm{K}. \]
Thus, even room temperature corresponds to
\[ T \ll T_F. \]
This inequality is the reason why metals at ordinary temperatures are strongly degenerate Fermi systems rather than classical gases.
11.2.2 Density of States Near the Fermi Energy
For a three-dimensional free electron gas, the density of states including both spin directions is
\[ D(E) = \frac{V}{2\pi^2} \left( \frac{2m_{\mathrm e}}{\hbar^2} \right)^{3/2} E^{1/2}. \]
The value at the Fermi energy can be written in the compact form
\[ D(E_F)=\frac{3}{2}\frac{N}{E_F}. \]
The density of states near \(E_F\) controls the low-temperature thermal response, because only states close to \(E_F\) can be thermally redistributed.
11.2.3 Fermi-Dirac Distribution
At finite temperature, the occupation probability of a one-electron state with energy \(E\) is
\[ f(E) = \frac{1}{\exp\left(\frac{E-\mu}{k_B T}\right)+1}. \]
Here \(\mu\) is the chemical potential. The function \(f(E)\) tells us the probability that a state of energy \(E\) is occupied.
At \(T=0\),
\[ f(E) = \begin{cases} 1, & E\le E_F,\\ 0, & E>E_F. \end{cases} \]
At \(T>0\), the sharp edge is smeared over an energy interval of order \(k_B T\) around \(\mu\). Since \(k_B T\ll E_F\) in metals, only a small fraction of electrons participates in the thermal redistribution.
11.2.4 Chemical Potential at Low Temperature
Because the number of electrons is fixed, the chemical potential is determined by
\[ N = \int_0^\infty D(E) f(E)\,dE. \]
At \(T=0\),
\[ \mu(0)=E_F. \]
At low but finite temperature, the Sommerfeld expansion gives
\[ \mu(T) = E_F \left[ 1 - \frac{\pi^2}{12} \left( \frac{T}{T_F} \right)^2 \right]. \]
The correction is of order \((T/T_F)^2\). For ordinary metals near room temperature, this correction is very small, so one usually sets
\[ \mu \simeq E_F. \]
For a smooth function \(g(E)\) and \(k_B T\ll \mu\),
\[ \int_0^\infty g(E) f(E),dE \simeq \int_0^\mu g(E),dE + \frac{\pi^2}{6} (k_B T)^2 \left. \frac{dg}{dE} \right|_{E=\mu} +\cdots . \]
The expansion works because \(f(E)\) differs appreciably from a step function only in a narrow energy interval of width \(\sim k_B T\) around \(E=\mu\).
11.3 Electronic Heat Capacity
The electronic heat capacity is the first major thermal property for which the classical Drude picture fails.
11.3.1 Classical Expectation and Its Failure
In a classical gas, equipartition assigns an energy of order \(k_B T\) to each thermally active degree of freedom. The corresponding classical Drude estimate for the electronic heat capacity is
\[ C_V^{\mathrm{class}} = 3Nk_B. \]
The precise prefactor depends on the counting convention, but the important point is independent of this detail: the classical result is temperature independent and far too large.
Experimentally, the electronic heat capacity of metals is much smaller and is approximately proportional to \(T\) at low temperature. This failure is one of the clearest signs that the conduction electrons must be treated as a degenerate Fermi gas.
11.3.2 Internal Energy of the Electron Gas
The internal energy is obtained by summing the energy of each one-electron state, weighted by its occupation probability:
\[ U = \sum_{\mathbf k,\sigma} E(k) f(E_k). \]
Using the density of states, this becomes
\[ U = \int_0^\infty E D(E) f(E)\,dE. \]
For a three-dimensional free electron gas,
\[ U = \frac{V}{2\pi^2} \left( \frac{2m_{\mathrm e}}{\hbar^2} \right)^{3/2} \int_0^\infty \frac{E^{3/2}} {\exp\left(\frac{E-\mu}{k_B T}\right)+1} \,dE. \]
This integral is evaluated at low temperature using the Sommerfeld expansion.
11.3.3 Sommerfeld Result for the Heat Capacity
Applying the Sommerfeld expansion gives
\[ U = U(0) + \frac{\pi^2}{6} (k_B T)^2 D(E_F). \]
Therefore,
\[ C_V = \left( \frac{\partial U}{\partial T} \right)_V = \frac{\pi^2}{3} k_B^2 T D(E_F). \]
Dividing by the volume gives the heat capacity per unit volume,
\[ c_V = \frac{C_V}{V} = \frac{\pi^2}{3} k_B^2 \frac{D(E_F)}{V} T. \]
Using
\[ D(E_F)=\frac{3}{2}\frac{N}{E_F}, \]
we obtain
\[ c_V = \frac{\pi^2}{2} \frac{n k_B^2}{E_F} T = \frac{\pi^2}{2} n k_B \frac{T}{T_F}. \]
The standard form is
\[ c_V=\gamma T, \]
with the Sommerfeld coefficient
\[ \gamma = \frac{\pi^2}{3} k_B^2 \frac{D(E_F)}{V} = \frac{\pi^2}{2} \frac{n k_B^2}{E_F}. \]
11.3.4 Total Low-Temperature Heat Capacity of a Metal
A real metal contains both electrons and lattice vibrations. At low temperature, the leading contributions are
\[ C_p = \gamma T + A T^3. \]
The linear term is electronic. The cubic term is the Debye phonon contribution.
Dividing by \(T\) gives
\[ \frac{C_p}{T} = \gamma + A T^2. \]
Thus a plot of \(C_p/T\) versus \(T^2\) should be a straight line at sufficiently low temperature. The intercept gives \(\gamma\), and the slope gives the lattice coefficient \(A\).
11.3.5 What the Sommerfeld Coefficient Measures
The Sommerfeld coefficient is proportional to the electronic density of states at the Fermi energy:
\[ \gamma \propto D(E_F). \]
Therefore, heat-capacity measurements provide a direct low-temperature probe of the electronic states near \(E_F\).
For simple metals, especially alkali metals, the measured \(\gamma\) is reasonably close to the free-electron estimate. For transition metals and heavy-fermion compounds, it can be much larger. This indicates that the free-electron mass and the free-electron density of states are no longer adequate. In a more complete theory, one introduces band masses and interaction-renormalized effective masses.
11.4 Electrical Conductivity of the Degenerate Electron Gas
Electrical conductivity is not itself a thermal property, but it is needed to understand electronic heat transport and the Wiedemann-Franz law.
11.4.1 Definition
The electrical conductivity \(\sigma\) relates the electrical current density \(\mathbf J_{\mathrm e}\) to the electric field:
\[ \mathbf J_{\mathrm e} = \sigma \mathbf E = -\sigma \nabla \varphi_{\mathrm{el}}. \]
Here \(\varphi_{\mathrm{el}}\) is the electric potential.
11.4.2 Drude Relaxation Picture
In the Drude model, the field accelerates electrons, while scattering relaxes the drift velocity. The equation of motion is
\[ m_{\mathrm e}\frac{d\mathbf v}{dt} = -e\mathbf E = m_{\mathrm e}\frac{\mathbf v_D}{\tau}. \]
Here \(\mathbf v_D\) is the drift velocity and \(\tau\) is the mean relaxation time. In the stationary state,
\[ \mathbf v_D = -\frac{e\tau}{m_{\mathrm e}}\mathbf E. \]
The current density is then
\[ \mathbf J_{\mathrm e} = -en\mathbf v_D = \frac{ne^2\tau}{m_{\mathrm e}}\mathbf E, \]
so
\[ \sigma = \frac{ne^2\tau}{m_{\mathrm e}}. \]
The same algebra survives in the Sommerfeld model, but the interpretation changes.
11.4.3 Sommerfeld Picture: Shifted Fermi Sphere
In thermal equilibrium, every occupied state with wave vector \(\mathbf k\) is balanced by an occupied state with wave vector \(-\mathbf k\). The average wave vector is therefore zero, and no current flows.
In an electric field, the occupied Fermi sphere is displaced by a small amount \(\delta\mathbf k\). The current density becomes
\[ \mathbf J_{\mathrm e} = -en \frac{\hbar}{m_{\mathrm e}} \delta\mathbf k. \]
The field changes the average wave vector according to
\[ \left. \frac{\partial \langle \mathbf k\rangle}{\partial t} \right|_{\mathrm{force}} = -\frac{e\mathbf E}{\hbar}. \]
Scattering relaxes the displaced distribution back toward equilibrium:
\[ \left. \frac{\partial \langle \mathbf k\rangle}{\partial t} \right|_{\mathrm{scatt}} = -\frac{\delta\mathbf k}{\tau}. \]
In the stationary state,
\[ -\frac{e\mathbf E}{\hbar} - \frac{\delta\mathbf k}{\tau} = 0, \]
so
\[ \delta\mathbf k = -\frac{e\tau}{\hbar}\mathbf E. \]
Substitution gives Ohm’s law again:
\[ \mathbf J_{\mathrm e} = \frac{ne^2\tau}{m_{\mathrm e}}\mathbf E, \]
and therefore
\[ \sigma = \frac{ne^2\tau}{m_{\mathrm e}} = \frac{ne^2\ell}{m_{\mathrm e} v_F}. \]
The mean free path is
\[ \ell=v_F\tau. \]
The velocity entering \(\ell\) is the Fermi velocity, not the small drift velocity.
11.4.4 Temperature Dependence of Electrical Resistivity
Since
\[ \sigma=\frac{ne^2\tau}{m_{\mathrm e}}, \]
the temperature dependence of the resistivity
\[ \rho=\frac{1}{\sigma} \]
comes mainly from the temperature dependence of the relaxation time \(\tau\).
Important scattering mechanisms in simple metals are:
- scattering by phonons;
- scattering by defects and impurities;
- scattering by sample surfaces.
If several scattering processes act independently, the empirical Matthiessen rule states that the scattering rates add:
\[ \frac{1}{\tau} = \frac{1}{\tau_1} + \frac{1}{\tau_2} + \frac{1}{\tau_3} +\cdots . \]
Equivalently, the resistivity is often written as a residual part plus a phonon part,
\[ \rho(T) = \rho_0+\rho_{\mathrm{ph}}(T). \]
The residual resistivity
\[ \rho_0=\mathrm{const} \]
comes from defects, impurities, and surfaces. It remains finite as \(T\to 0\) in a real sample.
For electron-phonon scattering,
\[ \rho_{\mathrm{ph}}\propto T \qquad (T\gg \Theta_D), \]
because the thermal phonon population is approximately proportional to \(T\) at high temperature.
At low temperature,
\[ \rho_{\mathrm{ph}}\propto T^5 \qquad (T\ll \Theta_D). \]
This \(T^5\) behavior combines two effects: the number of thermally excited acoustic phonons is strongly reduced, and low-temperature phonons mostly cause small-angle scattering, which is inefficient at relaxing electrical current.
11.4.5 Residual Resistance Ratio
The residual resistance ratio is defined as
\[ \mathrm{RRR} = \frac{\rho(300\ \mathrm K)}{\rho_0}. \]
Large values of \(\mathrm{RRR}\) indicate very pure metals with small residual resistivity. In alloys or strongly disordered metals, impurity scattering dominates and \(\mathrm{RRR}\) can be close to one.
11.5 Electronic Thermal Conductivity
11.5.1 Definition
The electronic thermal conductivity \(\kappa\) relates the heat-current density \(\mathbf J_h\) to the temperature gradient:
\[ \mathbf J_h = -\kappa \nabla T. \]
The minus sign means that heat flows from hot to cold.
11.5.2 Kinetic Estimate
In analogy with phonon heat conduction, the electronic thermal conductivity can be written as
\[ \kappa = \frac{1}{3} c_V v^2 \tau. \]
For a degenerate electron gas, the relevant speed is the Fermi velocity,
\[ v=v_F. \]
Using
\[ c_V = \frac{\pi^2}{2} n k_B \frac{T}{T_F} \]
and
\[ v_F^2 = \frac{2E_F}{m_{\mathrm e}} = \frac{2k_B T_F}{m_{\mathrm e}}, \]
we obtain
\[ \kappa = \frac{1}{3} \left( \frac{\pi^2}{2} n k_B \frac{T}{T_F} \right) \left( \frac{2k_B T_F}{m_{\mathrm e}} \right) \tau. \]
Therefore,
\[ \kappa = \frac{\pi^2}{3} \frac{n k_B^2 \tau}{m_{\mathrm e}} T. \]
11.5.3 Wiedemann-Franz Law
The electrical conductivity is
\[ \sigma = \frac{ne^2\tau}{m_{\mathrm e}}. \]
The electronic thermal conductivity is
\[ \kappa = \frac{\pi^2}{3} \frac{n k_B^2\tau}{m_{\mathrm e}} T. \]
Taking the ratio eliminates \(n\), \(m_{\mathrm e}\), and \(\tau\):
\[ \frac{\kappa}{\sigma} = \frac{\pi^2}{3} \left( \frac{k_B}{e} \right)^2 T. \]
This is the Wiedemann-Franz law. Equivalently,
\[ L_0 \equiv \frac{\kappa}{\sigma T} = \frac{\pi^2}{3} \left( \frac{k_B}{e} \right)^2. \]
The Lorenz number is
\[ L_0 = 2.44\times 10^{-8}\ \mathrm{W\,\Omega\,K^{-2}}. \]
The law works best when the same scattering processes limit both electrical and thermal transport.
11.5.4 Temperature Dependence of \(\kappa\)
Using
\[ \kappa=L_0\sigma T=\frac{L_0T}{\rho}, \]
we can infer the qualitative temperature dependence from \(\rho(T)\).
At very low temperature, residual scattering dominates:
\[ \rho\simeq \rho_0, \qquad \kappa\propto T. \]
At low temperatures where electron-phonon scattering gives \(\rho\propto T^5\),
\[ \kappa\propto T^{-4}. \]
At high temperatures where \(\rho\propto T\),
\[ \kappa\simeq \mathrm{const}. \]
Thus, pure metals often show a maximum in \(\kappa(T)\). The maximum is sharper and occurs at lower temperature in very clean samples. In alloys, impurity scattering suppresses the maximum.
11.6 Thermopower
Thermopower describes how a temperature gradient generates an electric field in an open circuit.
11.6.1 Physical Origin
Consider a metal whose left side is hotter than its right side. Electrons near the hot side have, on average, more thermal energy than electrons near the cold side. They diffuse from hot to cold. Since electrons carry negative charge, negative charge accumulates near the cold side and positive charge is left behind near the hot side. This charge separation creates an electric field that opposes further diffusion.
In the stationary open-circuit state, the diffusion tendency and the electric-field-driven drift cancel.
11.6.2 Definition
The thermopower, or Seebeck coefficient, is defined by
\[ \mathbf E = S \nabla T. \]
11.6.3 Free-Electron Estimate
In the simple free-electron argument, a temperature gradient produces a diffusion velocity
\[ \mathbf v_{\mathrm{diff}} = -\frac{\tau}{6} \frac{d v^2}{dT} \nabla T. \]
The electric field produces a drift velocity
\[ \mathbf v_{\mathrm{drift}} = -\frac{e\tau}{m_{\mathrm e}}\mathbf E. \]
In the stationary open-circuit state,
\[ \mathbf v_{\mathrm{diff}} + \mathbf v_{\mathrm{drift}} = 0. \]
This gives
\[ \frac{1}{3} \frac{d}{dT} \left( \frac{mv^2}{2} \right) \nabla T + e\mathbf E = 0. \]
The derivative of the average electronic energy per particle is the heat capacity per particle,
\[ \frac{d}{dT} \left( \frac{mv^2}{2} \right) = \frac{c_V}{n}. \]
Therefore,
\[ \mathbf E = -\frac{1}{3ne} c_V \nabla T. \]
Using \(\mathbf E=S\nabla T\), we obtain
\[ S = -\frac{1}{3ne}c_V. \]
With the Sommerfeld result for \(c_V\),
\[ c_V = \frac{\pi^2}{2} n k_B \frac{T}{T_F}, \]
the thermopower becomes
\[ S = -\frac{\pi^2}{6} \frac{k_B}{e} \frac{T}{T_F}. \]
This gives a magnitude of order
\[ |S|\sim 1\ \mu\mathrm{V/K} \]
for simple metals near room temperature.
11.6.4 Why the Free-Electron Thermopower Often Fails
The free-electron model predicts a small negative thermopower. Many real metals have a thermopower with the opposite sign or with a magnitude not captured by this estimate.
The reason is that thermopower is especially sensitive to details near the Fermi energy: band structure, effective masses, and the energy dependence of scattering. These details are absent from the free-electron model and require the Bloch-electron description.
- Metals contain an electronic thermal channel in addition to the phonon channel.
- The conduction electrons in ordinary metals form a degenerate Fermi gas.
- Only electrons close to the Fermi energy can change occupation at low temperature.
- The electronic heat capacity is linear in temperature and is governed by the density of states at the Fermi energy.
- Low-temperature heat-capacity data can separate electronic and phonon contributions.
- Electrical resistivity is controlled primarily by scattering from phonons, defects, impurities, and surfaces.
- Electronic thermal conductivity is large in clean metals because the same mobile electrons carry charge and heat.
- The Wiedemann-Franz law follows when the same scattering time controls electrical and thermal transport.
- Thermopower is small in the free-electron model and is highly sensitive to physics beyond that model.
Problem Set
Chemical Potential at Low Temperature
For a three-dimensional free electron gas,
\[ \mu(T) = E_F \left[ 1 - \frac{\pi^2}{12} \left( \frac{T}{T_F} \right)^2 \right]. \]
Explain why \(\mu(T)\simeq E_F\) is an excellent approximation for ordinary metals near room temperature.
Electronic Heat Capacity
Starting from
\[ U = U(0) + \frac{\pi^2}{6} (k_B T)^2 D(E_F), \]
derive
\[ C_V = \frac{\pi^2}{3} k_B^2 T D(E_F). \]
Then use
\[ D(E_F)=\frac{3}{2}\frac{N}{E_F} \]
to show that
\[ c_V = \frac{\pi^2}{2} \frac{n k_B^2}{E_F}T. \]
Low-Temperature Separation of Electron and Phonon Heat Capacity
At low temperature, the measured heat capacity of a metal is
\[ C_p = \gamma T + A T^3. \]
Show how a plot of \(C_p/T\) versus \(T^2\) can be used to extract \(\gamma\) and \(A\).
Electrical Conductivity from the Shifted Fermi Sphere
In the Sommerfeld picture, an applied electric field shifts the Fermi sphere by
\[ \delta\mathbf k = -\frac{e\tau}{\hbar}\mathbf E. \]
Using
\[ \mathbf J_{\mathrm e} = -en\frac{\hbar}{m_{\mathrm e}}\delta\mathbf k, \]
derive
\[ \sigma = \frac{ne^2\tau}{m_{\mathrm e}}. \]
Wiedemann-Franz Law
Use
\[ \sigma = \frac{ne^2\tau}{m_{\mathrm e}} \]
and
\[ \kappa = \frac{\pi^2}{3} \frac{n k_B^2\tau}{m_{\mathrm e}}T \]
to derive the Wiedemann-Franz law and identify the Lorenz number.
Thermopower of the Free Electron Gas
The free-electron estimate for the thermopower is
\[ S = -\frac{\pi^2}{6} \frac{k_B}{e} \frac{T}{T_F}. \]
Explain the physical origin of the negative sign and why this expression is expected to be small in ordinary metals.