By the end of this lecture, you should be able to:
- explain why the classical Drude model fails for the electronic heat capacity of metals;
- derive the linear-in-\(T\) electronic heat capacity in the Sommerfeld model and interpret the Sommerfeld coefficient \(\gamma\);
- combine the electronic and phononic low-temperature contributions into \(C = \gamma T + \beta T^{3}\);
- distinguish energy current from heat current and write the coupled transport equations for charge and heat;
- derive the Sommerfeld form of the Wiedemann–Franz law and summarize the Seebeck, Peltier, and Thomson effects;
- identify the main thermal limitations of the free-electron model and motivate the transition to Bloch electrons.
11.1 Roadmap
- Thermal questions that the Drude model can and cannot answer
- Quantum statistics and the Sommerfeld correction to thermodynamics
- Electronic heat capacity and low-temperature separation of \(C(T)\)
- Heat currents, thermal conductivity, and the Wiedemann–Franz law
- Thermoelectric phenomena and the thermal failures of the free-electron model
| Symbol | Meaning |
|---|---|
| \(n_e\) | conduction-electron density |
| \(\tau\) | relaxation time |
| \(\epsilon_F\) | Fermi energy |
| \(T_F\) | Fermi temperature, defined by \(\epsilon_F = k_B T_F\) |
| \(\rho(\epsilon)\) | electronic density of states per unit volume |
| \(\rho(\epsilon_F)\) | density of states at the Fermi energy |
| \(c_{el}\) | electronic heat capacity per unit volume |
| \(C_{el}\) | molar electronic heat capacity |
| \(\gamma\) | Sommerfeld coefficient |
| \(\mathbf{j}\) | electric current density |
| \(\mathbf{j}_Q\) | heat-current density |
| \(\sigma\) | electrical conductivity |
| \(\kappa_e\) | electronic thermal conductivity |
| \(L_0\) | Sommerfeld Lorenz number |
| \(S\) | Seebeck coefficient |
| \(\Pi\) | Peltier coefficient |
| \(\mu_T\) | Thomson coefficient |
| \(v_F\) | Fermi velocity |
11.2 Electronic Contributions to Thermal Properties
In Lecture 10, the thermal properties of solids were discussed from the perspective of the lattice: phonons give the dominant contribution over a broad temperature range, and at low temperature the Debye result implies a \(T^{3}\) law for the phonon heat capacity. Today we add the electronic channel. In metals, conduction electrons also store and transport thermal energy, but in a way that is qualitatively different from the classical gas picture. In this lecture we will understand this second contribution resulting in the standard low-temperature form of the heat capacity \[ C(T) = \gamma T + \beta T^{3} . \] Here \(\gamma\) is the electronic coefficient and \(\beta\) is the phonon coefficient. Both coefficients are material dependent.
The key question of this lecture is therefore not whether electrons contribute to thermal properties, but why their contribution is so much smaller than the classical expectation and why it is nevertheless crucial at low temperature.
11.3 Drude Model
11.3.1 Drude in a Nutshell
The Drude model treats conduction electrons as a classical gas of charge \(-e\) and mass \(m_e\), with collisions summarized by a relaxation time \(\tau\). In a static electric field, the drift velocity is \[ \mathbf{v}_{dr} = - \frac{e\tau}{m_e}\mathbf{E}. \] Here \(\mathbf{v}_{dr}\) is the average drift velocity, \(\tau\) is the mean time between collisions, and \(\mathbf{E}\) is the electric field. The associated current density is \[ \mathbf{j} = - e n_e \mathbf{v}_{dr} = \sigma \mathbf{E} , , \] so that \[ \sigma = \frac{n_e e^{2}\tau}{m_e} , . \] Here \(\sigma\) is the electrical conductivity and \(n_e\) is the conduction-electron density.
The Drude model is a phenomenological starting point for understanding the electronic contribution to thermal conductivity. The same mobile electrons that carry charge are also expected to carry energy.
11.3.2 Classical Electronic Heat Capacity
In the classical picture, equipartition gives an average energy of \[ \frac{3}{2}k_B T \] per electron. The resulting electronic heat capacity per unit volume is therefore \[ c_{el}^{cl} = \frac{3}{2} n_e k_B , \] Here \(c_{el}^{cl}\) is the classical electronic heat capacity per unit volume. This result is temperature independent.
This is the first major thermal failure of the Drude model. Experimentally, the electronic contribution is not a temperature-independent constant. Instead, it is much smaller and becomes visible mainly at low temperature, where the phonon contribution has already fallen strongly.
11.3.3 Drude Heat Conduction and the Classical Wiedemann–Franz Law
In the kinetic picture, the electronic thermal conductivity is written as \[ \kappa_e^{cl} = \frac{1}{3}\ell v c_{el}^{cl} , \] where \(\ell = v\tau\) is the mean free path, \(v\) is the characteristic particle speed, and \(c_{el}^{cl}\) is the electronic heat capacity per unit volume. Substituting the classical expressions gives \[ \kappa_e^{cl} = \frac{3}{2}\frac{n_e \tau}{m_e} k_B^{2} T . \]
Combining this with the Drude conductivity yields the following ratio of thermal and electrical conductivity
Here \(L_D\) is the classical Lorenz number predicted by the Drude model.
The structure of this result is suggestive: the ratio is independent of \(n_e\) and \(\tau\). The same scattering processes affect charge and heat transport in a similar way. However, the numerical value is not correct. The classical treatment gets the order of magnitude only accidentally.
11.4 Sommerfeld Model
11.4.1 Fermi–Dirac Distribution
The Sommerfeld model keeps the free-electron picture in real space but replaces classical statistics by Fermi–Dirac statistics. The equilibrium occupation is \[ f_{0}(\epsilon) = \frac{1}{e^{(\epsilon-\mu)/(k_B T)} + 1} , \] where \(f_0(\epsilon)\) is the probability that a one-electron state of energy \(\epsilon\) is occupied, and \(\mu\) is the chemical potential.
At low temperature, \(\mu\) remains very close to the Fermi energy: \[ \mu(T) = \epsilon_F \left[ 1 - \frac{\pi^{2}}{12}\left(\frac{k_B T}{\epsilon_F}\right)^{2} + \cdots \right] , . \] Equivalently, with \(\epsilon_F = k_B T_F\), \[ \mu(T) = \epsilon_F \left[ 1 - \frac{\pi^{2}}{12}\left(\frac{T}{T_F}\right)^{2} + \cdots \right] . \] Here \(\epsilon_F\) is the Fermi energy and \(T_F\) is the Fermi temperature.
The important physical consequence is that thermal excitations are confined to a narrow energy window of width of order \(k_B T\) around \(\epsilon_F\).
11.4.2 Why Only a Small Fraction of Electrons Matter
This is the central difference from the classical gas. In a metal, almost all states below \(\epsilon_F\) are already occupied, and the Pauli principle blocks further occupation. As a result, electrons deep inside the Fermi sea are thermally inert. Only electrons within a shell of thickness \(\sim k_B T\) around the Fermi surface can be rearranged by temperature.
A useful sanity check is therefore: \[ \frac{\text{thermally active electrons}}{\text{all conduction electrons}} \sim \frac{k_B T}{\epsilon_F} = \frac{T}{T_F} \ll 1 . \] Since \(T_F\) is typically much larger than laboratory temperatures, only a tiny fraction of all conduction electrons contributes to the low-temperature heat capacity.
This one observation explains why the classical value is far too large.
11.5 Electronic Heat Capacity in the Sommerfeld Model
11.5.1 Main Result
Using the Sommerfeld expansion, one obtains for the electronic heat capacity per unit volume \[ c_{el} = \frac{\pi^{2}}{3}\rho(\epsilon_F) k_B^{2} T . \] Here \(\rho(\epsilon_F)\) is the density of states at the Fermi energy.
For free electrons, one may rewrite this as \[ c_{el} = \frac{\pi^{2}}{2} n_e k_B \frac{k_B T}{\epsilon_F} . \]
This is the basic thermal result of the Sommerfeld model. It differs qualitatively from the Drude prediction in two ways:
- it is linear in \(T\) rather than constant;
- it is smaller than the classical value by a factor of order \(k_B T/\epsilon_F\).
11.5.2 Molar Form and the Sommerfeld Coefficient
Experimentally, one often writes the molar electronic heat capacity as \[ C_{el} = \gamma T , \] where \(\gamma\) is the Sommerfeld coefficient.
Thus \(\gamma\) measures the low-energy electronic density of states. In the free-electron model, a larger \(\rho(\epsilon_F)\) means a larger low-temperature electronic heat capacity. In real materials, deviations of \(\gamma\) from the free-electron estimate are therefore an indication that the free-electron picture is incomplete.
11.5.3 Interpretation
There are two complementary ways to read the result \[ c_{el} \propto \rho(\epsilon_F) T . \]
First, thermodynamics at low temperature is controlled by the density of available states right at the Fermi level. Second, the linear factor in \(T\) reflects the thermal width of the shell around the Fermi surface in which occupations can change.
Sanity check:
- if \(\rho(\epsilon_F)\) is larger, more low-energy excitations are available, so \(c_{el}\) should increase;
- if \(T \to 0\), no thermal smearing remains, so \(c_{el}\to 0\) linearly.
11.6 Low-Temperature Separation
At low temperature, the heat capacity of a metal has two leading contributions:
- an electronic term linear in \(T\);
- a phonon term cubic in \(T\).
Therefore \[ C(T) = \gamma T + \beta T^{3} , \] where \(\gamma\) is the Sommerfeld coefficient and \(\beta\) is the coefficient of the Debye \(T^{3}\) law established previously for phonons.
A useful rearrangement of this equation is \[ \frac{C(T)}{T} = \gamma + \beta T^{2} . \]
This form is convenient because a plot of \(C/T\) versus \(T^{2}\) should be linear at sufficiently low temperature. The intercept gives \(\gamma\), while the slope gives \(\beta\).
In this formula both lattice thermodynamics and electronic thermodynamics come together:
- the \(T^{3}\) term represents the phonon density of states near \(\omega = 0\);
- the \(T\) term represents the electronic density of states near \(\epsilon_F\).
11.7 Heat Currents in an Electron Gas
11.7.1 Heat Current Is Not Energy Current
For transport, one must distinguish energy flow from heat flow. The heat-current density is \[ \mathbf{j}_Q = \mathbf{j}_E - \mu\ \mathbf{j}_n , \] where \(\mathbf{j}_E\) is the energy-current density, \(\mathbf{j}_n\) is the particle-current density, and \(\mu\) is the chemical potential.
This subtraction is essential. When particles move, they carry not only thermal energy but also the chemical work associated with moving particles between reservoirs. Heat current is the part of the energy current that remains after removing this convective contribution.
11.7.2 Coupled Charge and Heat Transport
In linear response, the electric and heat currents are \[ \mathbf{j} = K_0\left(\mathbf{E} + \frac{\nabla \mu}{e}\right) - K_1\left(-\frac{\nabla T}{T}\right) , \tag{11.1}\]
\[ \mathbf{j}_Q = K_1\left(\mathbf{E} + \frac{\nabla \mu}{e}\right) + K_2\left(-\frac{\nabla T}{T}\right) . \tag{11.2}\]
Here \(K_0\), \(K_1\), and \(K_2\) are transport integrals defined by
\[ K_n = e^{-n} \int d\epsilon\ \left(-\frac{\partial f_0}{\partial \epsilon}\right) \sigma(\epsilon)(\epsilon-\mu)^n . \] Using the Sommerfeld expansion, we can simplify the transport coefficients to
\[ \begin{align} K_0 &= \sigma(\mu) + \left.\frac{\pi^2}{6}(k_BT)^2\frac{d^2 \sigma(\epsilon)}{d\epsilon^2}\right|_{\epsilon=\mu}, \\ K_1 &= \left.\frac{\pi^2}{3e}(k_BT)^2\frac{d\sigma(\epsilon)}{d\epsilon}\right|_{\epsilon=\mu}, \\ K_2 &= \frac{\pi^2}{3e^2}(k_BT)^2\sigma(\mu), \end{align} \]
where the coefficient \(K_0\) is the electrical conductivity to lowest order.
The same coefficient \(K_1\) appears in both equations. This is not accidental. It is the linear-response signature that charge transport and heat transport are coupled irreversible processes.
11.7.3 Pure Thermal Conduction
When no electric current flows, we can cancel the electric field in Equation 11.1 and Equation 11.2 and obtain the thermal conductivity \[ \kappa_e = \frac{K_2}{T} - \frac{K_1^{2}}{K_0 T} . \]
At low temperature, the second term is suppressed by powers of \(k_B T/\epsilon_F\), and for the Sommerfeld electron gas one finds \[ \kappa_e = \frac{\pi^{2}}{9} k_B^{2} T , \rho(\epsilon_F) v_F^{2}\tau , . \] Using the heat-capacity formula, this becomes \[ \kappa_e = \frac{1}{3} c_{el} v_F^{2}\tau , . \] Here \(v_F\) is the Fermi velocity.
This has the same kinetic form as in the classical gas, but the physics has changed completely:
- the relevant velocity is now \(v_F\), not the thermal velocity;
- the temperature dependence enters through \(c_{el}\), not through the speed.
That is the decisive quantum correction.
11.8 Wiedemann–Franz Law in the Sommerfeld Model
Combining the Sommerfeld expressions for \(\kappa_e\) and \(\sigma\) yields \[ \kappa_e = \frac{\pi^{2}}{3}\left(\frac{k_B}{e}\right)^{2} T \sigma , . \] Equivalently, \[ \frac{\kappa_e}{\sigma T} = L_0 \qquad \text{with} \qquad L_0 = \frac{\pi^{2}}{3}\left(\frac{k_B}{e}\right)^{2} . \] Numerically, \[ L_0 \approx 2.45 \times 10^{-8}\ \mathrm{V^{2}K^{-2}} . \]
This is the Sommerfeld form of the Wiedemann–Franz law. It improves the Drude result by replacing the incorrect classical prefactor \(\frac{3}{2}\) with \(\frac{\pi^{2}}{3}\).
11.8.1 Interpretation and Sanity Check
A useful interpretation is that the same quasifree carriers transport both charge and heat. If scattering is described by a single relaxation time and only states near the Fermi surface matter, the same microscopic ingredients appear in both conductivities and cancel in the ratio.
A sanity check is immediate:
- if scattering becomes stronger, both \(\sigma\) and \(\kappa_e\) decrease;
- if the same carriers and the same \(\tau\) control both, their ratio should be more robust than either conductivity separately.
That is exactly what the Lorenz number expresses.
11.9 Thermoelectric Phenomena
11.9.1 Seebeck Effect
If a temperature gradient is applied under open-circuit conditions, an electric field develops. The Seebeck coefficient is defined by \[ \mathbf{E} + \frac{\nabla \mu}{e} = S \nabla T \qquad (\mathbf{j}=0) . \] Here \(S\) is the Seebeck coefficient, also called the thermopower.
In the transport notation above, \[ S = - \frac{1}{T}\frac{K_1}{K_0} . \tag{11.3}\]
The Sommerfeld expansion then gives the Cutler–Mott formula \[ S= -\frac{\pi^{2}k_B^{2}T}{3e} \left. \frac{d\ln \sigma(\epsilon)}{d\epsilon} \right|_{\epsilon=\epsilon_F} . \]
In the simplest free-electron approximation, this reduces to \[ S \approx - \frac{\pi^{2}}{3}\frac{k_B^{2}T}{e\epsilon_F} . \]
The negative sign is natural in the free-electron picture: thermal diffusion moves electrons from hot to cold, building up negative charge on the cold side. At room temperature this has a value of order a few \(\mu\mathrm{V/K}\).
11.9.2 Relative Thermopower
For a thermocouple made of materials \(a\) and \(b\), the measurable thermoelectromotive force is \[ \mathcal{E}_{ab}(T,T_0) = \int_{T_0}^{T}dT\ \bigl(S_a - S_b\bigr) . \] Thus the relative Seebeck coefficient is \[ S_{ab} = S_a - S_b . \]
11.9.3 Peltier Effect
When an electric current passes through a junction of two materials at common temperature, heat is absorbed at one contact and released at the other. The Peltier coefficient is defined by \[ \Pi = \left.\frac{j_Q}{j}\right|_{\nabla T = 0} . \] Using the defintion of the Seebeck coefficient in Equation 11.3, we obtain \[ \Pi = ST . \]
Therefore we can express heat exchanged at a junction between two metals as \[ \frac{dQ}{dt} = (\Pi_a - \Pi_b) j = (S_a - S_b)Tj . \]
11.9.4 Thomson Effect
If a current flows through a single material in the presence of a temperature gradient, there is reversible heat release or absorption in addition to Joule heating.
Writing the constitutive relations as \[ \mathbf{E} = \frac{1}{\sigma}\mathbf{j} + S \nabla T , \qquad \mathbf{j}_Q = \Pi \mathbf{j} - \kappa_e \nabla T , \] one finds a reversible contribution proportional to \(\mathbf{j}\cdot\nabla T\).
Defining the Thomson coefficient \(\mu_T\) by \[ \left(\frac{\partial q}{\partial t}\right)_{rev} =-\mu_T \mathbf{j}\cdot\nabla T , \] the second Kelvin relation is \[ \mu_T = T \frac{dS}{dT} = T \frac{d}{dT}\left(\frac{\Pi}{T}\right) . \]
Together, the Seebeck, Peltier, and Thomson effects show that electric and heat currents are not independent transport channels.
11.10 Where the Free-Electron Model Fails for Thermal Properties
11.10.1 Failure 1: The Drude Electronic Heat Capacity Is Wrong
The classical prediction \[ c_{el}^{cl} = \frac{3}{2} n_e k_B \] is qualitatively incorrect. The measured electronic contribution is much smaller and linear in \(T\), not constant.
11.10.2 Failure 2: The Sommerfeld Model Fixes the Scaling but Not All Materials
The Sommerfeld model correctly predicts that the electronic heat capacity is proportional to \(\rho(\epsilon_F)T\), and it works reasonably well for simple metals. However, for many materials the density of states inferred from heat-capacity data differs substantially from the free-electron estimate. The measured Sommerfeld coefficient can be much larger than the ideal-electron prediction.
This means that the relevant low-energy electronic states are not described adequately by a simple free-electron band with bare mass \(m_e\).
11.10.3 Failure 3: Wiedemann–Franz Is Not Universally Valid
The Sommerfeld Lorenz number is a substantial improvement over the Drude model, especially near room temperature. However, the Wiedemann–Franz law is not satisfied over the full temperature range of real metals. It works around room temperature and again asymptotically at very low temperature, but it fails in an intermediate temperature window.
The physical message is that a single, energy-independent relaxation time is too crude once different scattering mechanisms contribute with different temperature dependences.
11.10.4 Failure 4: The Seebeck Coefficient Is Often Wrong in Sign or Magnitude
The simplest free-electron model predicts a negative thermopower of small magnitude. In reality, many metals have positive Seebeck coefficients, and semiconductors can show values orders of magnitude larger.
This failure is especially important thermally because the Seebeck coefficient is directly sensitive to the energy dependence of transport near the Fermi level. The sign and magnitude depend on details that the free-electron model suppresses: band structure, the periodic lattice potential, and the energy dependence of scattering.
11.11 Outlook on the Next Lecture: From Free Electrons to Bloch Electrons
This lecture has reached the limit of what a free-electron picture can do for thermal physics. It explains why the electronic heat capacity is linear in \(T\), why \(\gamma\) probes low-energy electronic states, and why electrical and thermal transport are coupled. It also shows where the approximation fails:
- the density of states is often not free-electron-like;
- effective masses are renormalized;
- thermopower is highly sensitive to band-structure details;
- deviations from the Wiedemann–Franz law reflect nontrivial scattering physics.
The next logical step is therefore to place electrons into the periodic potential of the crystal. In the Bloch-electron picture, the free-electron quantities \(\epsilon(\mathbf{k})\), \(v_F\), and \(\rho(\epsilon_F)\) are replaced by band-dependent quantities. This is the natural framework for understanding why some metals remain simple, why others do not, and why thermal coefficients are often fingerprints of the band structure.
11.12 Take-Home Messages
- The Drude model captures the idea of mobile charge carriers but fails badly for the electronic heat capacity because it uses classical statistics.
- In a degenerate electron gas, only states within a narrow thermal window around the Fermi energy contribute to low-temperature thermodynamics.
- The Sommerfeld model predicts an electronic heat capacity linear in temperature, summarized experimentally by the Sommerfeld coefficient.
- In a metal at low temperature, the total heat capacity is the sum of a linear electronic term and a cubic phonon term.
- Heat current is not the same as energy current; charge and heat transport are coupled irreversible processes.
- The Sommerfeld model yields the improved Wiedemann–Franz law with the universal Lorenz number \(L_0\).
- Seebeck, Peltier, and Thomson effects are different manifestations of the same coupled charge–heat transport physics.
- The remaining failures of the free-electron model point directly toward band structure and Bloch electrons.
11.13 Problem Set
Classical versus quantum electronic heat capacity. Compare the classical Drude result for the electronic heat capacity with the Sommerfeld result. Why does the classical model overestimate the electronic contribution so strongly?
Low-temperature separation. Starting from the electronic contribution \(C_{el}=\gamma T\) and the phonon contribution \(C_{ph}=\beta T^{3}\), show that \(C/T\) is linear in \(T^{2}\). Explain how \(\gamma\) and \(\beta\) can be extracted from low-temperature data.
Wiedemann–Franz law. Use \[ \sigma = \frac{1}{3}e^{2}\rho(\epsilon_F)v_F^{2}\tau \] and \[ \kappa_e = \frac{1}{3}c_{el}v_F^{2}\tau \] together with \[ c_{el} = \frac{\pi^{2}}{3}\rho(\epsilon_F)k_B^{2}T \] to derive the Sommerfeld form of the Wiedemann–Franz law.
Thermopower sign in the simplest model. The Cutler–Mott formula states that \[ S = - \frac{\pi^{2}k_B^{2}T}{3e} \left. \frac{d\ln \sigma(\epsilon)}{d\epsilon} \right|_{\epsilon=\epsilon_F} . \] Explain why the simplest free-electron model predicts a negative Seebeck coefficient. Then state one reason why real metals may show a positive value instead.
Heat current versus energy current. Why is the heat current not equal to the energy current? Give a brief physical explanation of the term subtracted in \[ \mathbf{j}_Q = \mathbf{j}_E - \mu \mathbf{j}_n . \]