By the end of this lecture, you should be able to:
- explain why lattice vibrations modulate the electronic potential and therefore scatter Bloch electrons;
- derive the linearized form of the electron–phonon perturbation from small ionic displacements;
- state the crystal-momentum and energy selection rules for phonon absorption and emission;
- explain the deformation-potential picture for long-wavelength longitudinal acoustic phonons;
- connect electron–phonon scattering to electrical resistivity, electronic thermal conductivity, and finite phonon lifetimes;
- interpret phonon broadening and softening as observable consequences of electron–phonon coupling.
12.1 Roadmap
- From the static lattice of Lecture 11 to a vibrating lattice
- Linearized electron–phonon coupling and the scattering picture
- Deformation potential as a continuum, long-wavelength limit
- Temperature dependence of scattering and transport consequences
- Phonon lifetime, linewidth, and the spectroscopic signature of coupling
| Symbol | Meaning |
|---|---|
| \(\mathbf{R}_{l}^{0}\) | equilibrium position of ion \(l\) |
| \(\mathbf{u}_{l}\) | displacement of ion \(l\) from \(\mathbf{R}_{l}^{0}\) |
| \(U_{0}(\mathbf{r})\) | static lattice-periodic electron–ion potential |
| \(\delta U(\mathbf{r})\) | change in potential caused by lattice vibrations |
| \(\psi_{\mathbf{k}}(\mathbf{r})\) | Bloch state of crystal momentum \(\mathbf{k}\) |
| \(\mathbf{q}\) | phonon wave vector |
| \(\lambda\) | phonon branch or polarization index |
| \(\omega_{\lambda}(\mathbf{q})\) | phonon angular frequency |
| \(\mathbf{e}_{\lambda}(\mathbf{q})\) | phonon polarization vector |
| \(\mathbf{G}\) | reciprocal-lattice vector |
| \(g_{\lambda}(\mathbf{k},\mathbf{q})\) | electron–phonon matrix element |
| \(N_{\mathbf{q}\lambda}\) | Bose occupation number of phonons |
| \(\tau_{el\text{–}ph}\) | electron lifetime limited by electron–phonon scattering |
| \(\rho(T)\) | electrical resistivity |
| \(\kappa_{e}\) | electronic thermal conductivity |
| \(\Delta(\mathbf{r})\) | local dilatation, \(\delta V/V\) |
| \(C\) | deformation-potential constant |
| \(\Gamma_{\lambda}(\mathbf{q})\) | phonon decay width in energy units |
12.2 From Electronic Contributions to a Vibrating Lattice
In Lecture 11, electrons contributed to thermal transport through quantities such as \(C_{el}\), \(v_F\), and a relaxation time \(\tau\). What was still missing was a microscopic origin of that relaxation time. In a real crystal, ions are not fixed: they vibrate about their equilibrium positions, and these vibrations modulate the periodic potential felt by the electrons. That modulation is the microscopic origin of electron–phonon coupling.
The starting point is therefore to separate the static lattice potential from the vibrational correction: \[ U_{el\text{–}ion}(\mathbf{r};{\mathbf{R}_{l}}) = U_{0}(\mathbf{r}) + \delta U(\mathbf{r}) . \] Here \(U_{el\text{–}ion}(\mathbf{r};{\mathbf{R}_{l}})\) is the full electron–ion potential, \(U_{0}(\mathbf{r})\) is the equilibrium lattice-periodic part obtained when \(\mathbf{R}_{l}=\mathbf{R}_{l}^{0}\), and \(\delta U(\mathbf{r})\) is the perturbation caused by the ionic displacements.
The separation is justified because electrons adjust much faster than the ions move. This can be formulated as an adiabatic or Born–Oppenheimer decoupling: the electrons see an almost instantaneous ionic configuration, while the ions move in an averaged electronic background. For our purposes, this means that the Bloch states of the static crystal remain the natural starting basis, and the vibrational correction can be treated as a perturbation.
12.3 Linearized Electron–Phonon Coupling
12.3.1 Small Displacements and the Perturbing Potential
For small ionic displacements, the potential may be expanded to first order: \[ \delta U(\mathbf{r}) = \sum_{l} \mathbf{u}_{l}\cdot \left. \frac{\partial U_{el\text{–}ion}(\mathbf{r};{\mathbf{R}_{m}})}{\partial \mathbf{R}_{l}} \right|_{{\mathbf{R}_{m}^{0}}} . \] Here \(\mathbf{u}_{l}=\mathbf{R}_{l}-\mathbf{R}_{l}^{0}\) is the displacement of ion \(l\), and the derivative is evaluated at the equilibrium ionic positions \({\mathbf{R}_{m}^{0}}\).
This is the central microscopic statement: the coupling is linear in displacement to leading order. The approximation is reliable because thermal vibrations are small compared with interatomic distances over most of the harmonic regime.
A useful sanity check:
- if all \(\mathbf{u}_{l}=0\), then \(\delta U(\mathbf{r})=0\) and we recover the static crystal;
- the larger the displacement field, the larger the perturbation acting on the electrons.
12.3.2 Normal-Mode Form of the Lattice Vibration
To make the perturbation physically transparent, write a normal mode of lattice vibration as \[ \mathbf{u}_{l}(t) = \mathbf{e}_{\lambda}(\mathbf{q})\ u_{\lambda\mathbf{q}}\ e^{i(\mathbf{q}\cdot \mathbf{R}_{l}^{0}-\omega_{\lambda}(\mathbf{q})t)} +\text{c.c.} \] Here \(\mathbf{q}\) is the phonon wave vector, \(\lambda\) labels the branch or polarization, \(\mathbf{e}_{\lambda}(\mathbf{q})\) is the polarization vector, \(u_{\lambda\mathbf{q}}\) is the mode amplitude, and c.c. denotes the complex conjugate required to make the displacement real.
Substituting this mode into the linearized perturbation shows that a single phonon mode acts as a traveling modulation of the crystal potential. That modulation can scatter an electron from one Bloch state into another.
12.3.3 Matrix Element and Crystal-Momentum Selection Rule
The transition amplitude between Bloch states \(\psi_{\mathbf{k}}\) and \(\psi_{\mathbf{k}'}\) is \[ g_{\lambda}(\mathbf{k},\mathbf{q};\mathbf{k}') = \langle \psi_{\mathbf{k}'} \vert \delta U_{\lambda\mathbf{q}} \vert \psi_{\mathbf{k}} \rangle . \] Here \(\delta U_{\lambda\mathbf{q}}\) is the part of the perturbing potential associated with the mode \((\mathbf{q},\lambda)\), and \(g_{\lambda}(\mathbf{k},\mathbf{q};\mathbf{k}')\) is the corresponding electron–phonon matrix element.
Because the perturbation carries the lattice phase factor \(e^{i\mathbf{q}\cdot \mathbf{R}_{l}^{0}}\), the sum over lattice sites contains \[ \sum_{l}e^{i(\mathbf{k}-\mathbf{k}'+\mathbf{q})\cdot \mathbf{R}_{l}^{0}} . \] This sum vanishes unless \[ \mathbf{k}'=\mathbf{k}+\mathbf{q}+\mathbf{G} . \] Here \(\mathbf{G}\) is a reciprocal-lattice vector.
This is the crystal-momentum selection rule for electron–phonon scattering.
- If \(\mathbf{G}=0\), the process is normal.
- If \(\mathbf{G}\neq 0\), the process is umklapp.
The accompanying energy condition is \[ \varepsilon_{\mathbf{k}'} = \varepsilon_{\mathbf{k}} \pm \hbar\omega_{\lambda}(\mathbf{q}) . \] Here \(\varepsilon_{\mathbf{k}}\) and \(\varepsilon_{\mathbf{k}'}\) are the electron energies before and after scattering, and the plus or minus sign corresponds to phonon absorption or emission, respectively.
12.3.4 Interpretation
The important conceptual point is not merely that electrons scatter, but that the vibrating lattice breaks the exact translational symmetry of the static lattice. Once the vibration is resolved into phonon modes, however, the missing momentum is carried by the phonon wave vector \(\mathbf{q}\), modulo a reciprocal-lattice vector.
This is why a static Bloch state has only a finite lifetime in a vibrating crystal: it is not an exact eigenstate of the full, time-dependent problem.
12.4 Deformation Potential
12.4.1 Continuum Limit for Long-Wavelength Acoustic Phonons
For long-wavelength longitudinal acoustic phonons, the lattice may be treated as an elastic continuum. The relevant scalar field is the local dilatation \[ \Delta(\mathbf{r}) \equiv \frac{\delta V}{V} = \nabla\cdot\mathbf{u}(\mathbf{r}) . \] Here \(\mathbf{u}(\mathbf{r})\) is the continuum displacement field, \(\delta V/V\) is the local relative volume change, and \(\Delta(\mathbf{r})\) measures local compression or expansion.
The deformation-potential approximation states that this local compression shifts the electronic potential by \[ U_{def}(\mathbf{r})=C\ \Delta(\mathbf{r}) . \] Here \(C\) is the deformation-potential constant.
This is a very useful simplification because it converts a complicated microscopic potential change into a local scalar coupling to compression.
12.4.2 Free-Electron Estimate of the Coupling Strength
In a free-electron picture, the Fermi energy scales with density as \[ \varepsilon_{F}\propto n^{2/3}\propto V^{-2/3} . \] Here \(n\) is the electron density and \(V\) is the local volume. Therefore a small local volume change gives \[ \delta \varepsilon_{F} = -\frac{2}{3}\varepsilon_{F}\frac{\delta V}{V} = -\frac{2}{3}\varepsilon_{F}\Delta(\mathbf{r}) . \] Here \(\delta \varepsilon_{F}\) is the local shift of the Fermi energy.
Because the chemical potential must remain uniform in equilibrium, the electronic potential must shift in the opposite direction: \[ U_{def}(\mathbf{r}) = \frac{2}{3}\varepsilon_{F}\Delta(\mathbf{r}) . \] Hence \[ C=\frac{2}{3}\varepsilon_{F} . \] Here \(\varepsilon_{F}\) is the Fermi energy of the electron gas.
This result is conceptually important even if the numerical prefactor is only approximate. It shows that the deformation potential is not arbitrary: it is tied to how compression changes the electronic energy scale.
12.4.3 Physical Scope of the Approximation
The deformation-potential picture is most natural for long-wavelength longitudinal acoustic phonons.
- Long wavelength means the crystal can be viewed locally as a continuum.
- Longitudinal means the displacement has a nonzero divergence.
- Acoustic means the distortion corresponds to compression and expansion rather than internal relative motion within the basis.
For short wavelengths, optical phonons, or strongly ionic systems, a more detailed microscopic description is needed.
12.5 Electron–Phonon Scattering, Resistivity, and Thermal Transport
12.5.1 Scattering Probability
Once the lattice vibration is resolved into phonon modes, the scattering probability has the standard golden-rule structure \[ W_{\mathbf{k}\rightarrow\mathbf{k}'} = \frac{2\pi}{\hbar} |g_{\lambda}(\mathbf{k},\mathbf{q};\mathbf{k}')|^{2} \Big[ (N_{\mathbf{q}\lambda}+1) \delta(\varepsilon_{\mathbf{k}'}-\varepsilon_{\mathbf{k}}+\hbar\omega_{\lambda}(\mathbf{q})) + N_{\mathbf{q}\lambda} \delta(\varepsilon_{\mathbf{k}'}-\varepsilon_{\mathbf{k}}-\hbar\omega_{\lambda}(\mathbf{q})) \Big] . \]
Here \(W_{\mathbf{k}\rightarrow\mathbf{k}'}\) is the transition probability per unit time, \(N_{\mathbf{q}\lambda}\) is the phonon occupation number, and the two delta functions enforce energy conservation for emission and absorption, respectively. The final state must also satisfy \(\mathbf{k}'=\mathbf{k}\pm\mathbf{q}+\mathbf{G}\).
Two immediate consequences follow:
- the scattering strength grows with the phonon population;
- even in a perfect crystal, thermal vibrations alone give electrons a finite lifetime.
Summing over all allowed final states gives an inverse lifetime \[ \frac{1}{\tau_{el\text{–}ph}(\mathbf{k})} = \sum_{\mathbf{k}',\lambda} W_{\mathbf{k}\rightarrow\mathbf{k}'} . \] Here \(\tau_{el\text{–}ph}(\mathbf{k})\) is the lifetime of an electron in Bloch state \(\mathbf{k}\) against phonon scattering.
12.5.2 Temperature Dependence
There are two limiting trends for metals:
- at sufficiently high temperature, the lifetime scales approximately as \(\tau_{el\text{–}ph}\propto T^{-1}\);
- at very low temperature, the lifetime grows more rapidly as the thermal phonon population collapses, with \(\tau_{el\text{–}ph}\propto T^{-3}\) for the single-particle lifetime in metals.
The high-temperature trend is easy to understand: when many phonon modes are thermally occupied, the Bose factor increases roughly linearly with \(T\), so the scattering rate increases and the lifetime decreases.
The low-temperature trend is also physically intuitive: only long-wavelength phonons remain populated, the available phase space shrinks strongly, and scattering is suppressed.
12.5.3 Resistivity Versus Temperature
Because electron–phonon scattering relaxes electronic momentum, it is one of the main microscopic origins of the resistivity of metals. At the qualitative level relevant for the present lecture:
- once phonons are well populated, the electron–phonon contribution to \(\rho(T)\) grows roughly linearly with temperature;
- as \(T\to 0\), the phonon population vanishes and the electron–phonon contribution to the resistivity falls rapidly toward zero.
This directly connects to Lecture 11: the relaxation time in transport is not just a phenomenological parameter but can be traced to scattering from thermally excited lattice vibrations.
12.5.4 Consequence for Electronic Thermal Conductivity
From Lecture 11 we know that the electronic thermal conductivity may be written schematically as \[ \kappa_{e}\sim \frac{1}{3}C_{el}v_{F}^{2}\tau , \] where \(C_{el}\) is the electronic heat capacity, \(v_{F}\) is the Fermi velocity, and \(\tau\) is the transport relaxation time.
Electron–phonon coupling now gives a microscopic meaning to \(\tau\): when phonons become more abundant, electron scattering is stronger, the mean free path is shorter, and \(\kappa_{e}\) is reduced. Thus electron–phonon coupling matters for thermal properties in two distinct ways:
- it limits charge transport and therefore electrical conductivity;
- it limits heat transport by shortening the lifetime of the same electronic carriers.
This is the microscopic complement to the Wiedemann–Franz discussion of Lecture 11.
12.6 Phonon Lifetime and Linewidth
12.6.1 Phonons Also Acquire a Finite Lifetime
Electron–phonon coupling does not only scatter electrons. It also allows a phonon to create an electron–hole pair, or the reverse process, so phonons themselves are no longer infinitely long-lived. In perturbation theory, this appears as a complex correction to the phonon energy: \[ \hbar\widetilde{\omega}_{\lambda}(\mathbf{q}) = \hbar\omega_{\lambda}(\mathbf{q}) + \Delta E_{\lambda}(\mathbf{q}) - i\Gamma_{\lambda}(\mathbf{q}) . \] Here \(\widetilde{\omega}_{\lambda}(\mathbf{q})\) is the renormalized phonon frequency, \(\Delta E_{\lambda}(\mathbf{q})\) is the real energy shift, and \(\Gamma_{\lambda}(\mathbf{q})\) is the decay width induced by the interaction.
The corresponding phonon lifetime is \[ \tau_{\lambda}(\mathbf{q}) = \frac{\hbar}{2\Gamma_{\lambda}(\mathbf{q})} . \] Here \(\tau_{\lambda}(\mathbf{q})\) is the phonon lifetime.
This equation is one of the most important thermal statements of this lecture: once phonons interact with electrons, they acquire a finite linewidth and therefore a finite lifetime.
12.6.2 Spectroscopic Meaning
A finite phonon lifetime shows up experimentally as a broadened phonon line in spectroscopy.
- In Raman or infrared spectroscopy, broader peaks indicate faster decay.
- In inelastic neutron or x-ray scattering, the linewidth likewise encodes the inverse lifetime.
- The real part of the interaction shifts the phonon frequency, while the imaginary part broadens the line.
Thus phonon linewidth is not merely a spectroscopic detail: it is a direct observable of electron–phonon coupling.
12.6.3 Short Interpretation
The same interaction that makes electrons poor ballistic carriers also makes phonons poor eternal normal modes. Electron–phonon coupling therefore renormalizes both subsystems simultaneously:
- electrons acquire finite lifetimes and energy shifts;
- phonons acquire finite lifetimes and energy shifts.
That mutual renormalization is the microscopic core of the chapter.
Optional: Kohn Anomalies and Soft Phonons
At special wave vectors, especially around \(q\approx 2k_{F}\), the electron–phonon correction to the phonon energy can become unusually large. In three dimensions this produces a kink or anomaly in the phonon dispersion, known as a Kohn anomaly. In one-dimensional or strongly anisotropic systems the effect is much stronger and may drive a soft phonon and ultimately a lattice instability of Peierls type.
For the present lecture, the essential message is simpler: strong electron–phonon coupling may reveal itself not only by a linewidth but also by a measurable frequency shift and, in extreme cases, by a soft mode.
12.7 Take-Home Messages
- Electron–phonon coupling originates from the modulation of the electron–ion potential by lattice vibrations.
- To leading order, the coupling is linear in the ionic displacement field.
- A phonon scatters an electron between Bloch states while conserving crystal momentum up to a reciprocal-lattice vector.
- Normal and umklapp processes are distinguished by whether such a reciprocal-lattice shift is needed.
- The deformation-potential picture is the simplest description of long-wavelength longitudinal acoustic phonons.
- Electron–phonon scattering is a major microscopic source of the temperature-dependent resistivity of metals.
- The same interaction limits electronic thermal conductivity by shortening the electronic mean free path.
- Phonons also acquire finite lifetimes, so electron–phonon coupling appears experimentally as phonon broadening and frequency shifts.
12.8 Problem Set
Linearized coupling. Starting from the full potential \(U_{el\text{–}ion}(\mathbf{r};{\mathbf{R}_{l}})\), show that the first-order correction caused by small displacements \(\mathbf{u}_{l}\) is \[ \delta U(\mathbf{r}) = \sum_{l} \mathbf{u}_{l}\cdot \left. \frac{\partial U_{el\text{–}ion}}{\partial \mathbf{R}_{l}} \right|_{{\mathbf{R}_{m}^{0}}} . \] Explain physically why the zeroth-order term is just the static lattice potential.
Crystal-momentum selection rule. Assume the scattering matrix element contains the lattice sum \[ \sum_{l}e^{i(\mathbf{k}-\mathbf{k}'+\mathbf{q})\cdot \mathbf{R}_{l}^{0}} . \] Show that it vanishes unless \(\mathbf{k}'=\mathbf{k}+\mathbf{q}+\mathbf{G}\). What is the difference between a normal process and an umklapp process?
Deformation potential. Use the free-electron scaling \(\varepsilon_{F}\propto V^{-2/3}\) to derive \[ \delta\varepsilon_{F} = -\frac{2}{3}\varepsilon_{F}\frac{\delta V}{V} . \] Why does this imply the deformation-potential estimate \[ U_{def}(\mathbf{r})=\frac{2}{3}\varepsilon_{F}\Delta(\mathbf{r})\ ? \]
Temperature trends in transport. Give a qualitative explanation for why the electron–phonon contribution to the resistivity is roughly linear in temperature at high \(T\) but falls rapidly as \(T\to 0\).
Phonon lifetime and linewidth. Starting from \[ \hbar\widetilde{\omega}_{\lambda}(\mathbf{q}) = \hbar\omega_{\lambda}(\mathbf{q}) + \Delta E_{\lambda}(\mathbf{q}) - i\Gamma_{\lambda}(\mathbf{q}) , \] show that a finite \(\Gamma_{\lambda}(\mathbf{q})\) implies a finite lifetime. Why does this lead to a broadened phonon line in spectroscopy?