Electron-phonon interaction

The electron-phonon interaction couples electronic states to the atomic displacements represented by phonons. A lattice vibration changes the potential seen by an electron; its quantized displacement permits phonon absorption and emission. This interaction affects electrical resistance, electronic thermal conductivity, lattice dynamics and pairing in conventional superconductors.

For electron energy transfer $\Delta$ and phonon energy ${\Omega }_q=\hslash {\omega }_q$, the symmetrized pair amplitude is $V_{\mathrm{eff}}=2|M_q{|}^2{\Omega }_q/({\Delta }^2-{\Omega }_q^2)$. It is attractive for $|\Delta |<{\Omega }_q$. This interaction is retarded, and screened Coulomb repulsion must also be considered.

A second-quantized electron-phonon Hamiltonian with free electrons, harmonic phonons and terms linear in phonon displacement, $M_q{c}_{k+q,\sigma }^{†}{c}_{k,\sigma }(b_q+b_{-q}^{†})$. The name is used for both general linear electron-phonon models and the more specific long-range coupling to polar optical modes; the intended dispersion and matrix element must be stated.

An anti-Hermitian generator $S$ is chosen so $[S,H_0]=-H_{e\text{-}ph}$. Expanding $e^{S}H{e}^{-S}$ leaves $H_0+[S,H_{e\text{-}ph}]/2$ to second order. Eliminating virtual phonons produces an effective electron-electron interaction, plus energy shifts. Resonant denominators and higher orders limit this perturbative elimination.

A feature in phonon dispersion caused by a nonanalytic wavevector dependence of electronic screening. In a quasi-one-dimensional conductor, enhanced response near $2k_F$ can soften a lattice mode and signal an approaching Peierls transition.

A Peierls transition, named for Rudolf Peierls, is a metal-insulator transition caused by a Peierls distortion of an initially metallic chain.^[ref] A modulation at $2k_F$ mixes states near opposite Fermi points and can open a band gap at the Fermi energy. At half filling with one atomic orbital per chemical atom, this modulation doubles the original lattice period and alternates strong and weak bonds. In a weak-coupling zero-temperature description, the Peierls energy gain is proportional to $-({\delta }^2/t)\ln (t/|\delta |)$ whereas the elastic cost is proportional to ${\delta }^2$, where $t>0$ is the undistorted hopping strength and $\delta$ its small alternating change. The logarithm permits a distortion at arbitrarily weak coupling in the ideal model. Finite temperature, finite size, interchain coupling and thermal fluctuations matter for a real transition; strictly one-dimensional reasoning alone does not establish finite-temperature long-range order.

The experiment follows phonon peaks as both momentum and energy transfer are varied. In inelastic neutron scattering, $\mathbf{q}={\mathbf{k}}_i-{\mathbf{k}}_f$ locates the lattice modulation and $\hslash \omega =E_i-E_f$ measures the mode energy, so a softened mode can be distinguished from static diffraction alone. Measurements on deuterated TTF-TCNQ between $60$ and $300\mathrm{K}$ found a growing Kohn anomaly near $2k_F$ below about $150\mathrm{K}$ as the $54\mathrm{K}$ transition was approached.^[ref]

A Peierls distortion modulates the atomic positions and chemical bonds of a one-dimensional chain with a partly filled energy band. In the half-filled example, alternating short and long bonds double the primitive unit cell. The distortion is the structural change; the associated Peierls transition is the change in electronic conduction.^[ref]

For spin-degenerate electrons at atomic spacing $a$, alternate the hopping amplitudes $-(t+\delta )$ and $-(t-\delta )$, with $t>0$ and $|\delta |<t$. The doubled primitive unit cell gives an off-diagonal Bloch Hamiltonian entry $-(t+\delta )-(t-\delta )e^{-2ika}$. Its two eigenvalues are the displayed energy bands, for $|k|\le \pi /(2a)$. The band gap at the edge of the Brillouin zone is $4|\delta |$. At half filling there are two electrons per doubled primitive unit cell, so the lower band is full and the upper empty. This explicit model realizes the period-doubling form of a Peierls distortion.

Fill the lower energy band of the dimerized tight-binding chain with both spin states. Take zero on-site energy and measure ground-state energy per doubled primitive unit cell. State counting in a Bloch band and $q=ka$ giveEquivalently, for $|\delta |\le t$, $\mathcal{E}(\delta )=-(8t/\pi )E(\sqrt{1-{\delta }^2/t^2})$, where $E(k)$ is the complete elliptic integral of the second kind in the modulus convention.

For fixed $t>0$ and $|\delta |/t\to 0$, the change isThe logarithm comes from states near the edge of the Brillouin zone. Put $u=\pi /2-q$. To leading order the square root there is $\sqrt{t^2{u}^2+{\delta }^2}$. For a fixed small cutoff $\eta >0$, direct integration givesThe rest of the band, and the higher-order corrections near its edge, contribute only $O({\delta }^2/t)$; multiplying by $-8/\pi$ gives the displayed asymptotic expansion. Here the remainder uses Big O notation. The negative logarithmic term exceeds any fixed quadratic elastic cost $K{\delta }^2$, $K>0$, for sufficiently small nonzero $\delta$. This explains the zero-temperature instability to a Peierls distortion in the ideal chain model.

In electron-phonon scattering, an electron changes its energy by $\pm \hslash {\omega }_{\mathbf{q}}$ while its crystal momentum changes by a phonon wavevector modulo a reciprocal lattice vector. Charge-transport relaxation weights a scattering angle $\theta$ by $1-\cos \theta$. Small-angle events can therefore relax heat efficiently while relaxing electrical current weakly. At temperatures above the phonon excitation scale, scattering becomes approximately elastic in the electron thermal shell; residual elastic disorder often dominates at very low temperature.

For free-electron recoil and a linear acoustic branch, $E_e(q)={\hslash }^2{q}^2/(2m_e)$ and $E_{\mathrm{ph}}(q)=\hslash c_{s}q$. Their ratio is $q\lambda$, so equality occurs at $q=1/\lambda$. This is a momentum/localization crossover scale, formally half a sound-speed analogue of the Reduced Compton wavelength. It does not replace the speed of light in electron relativity or determine a mean free path. A Fermi-surface electron emitting a phonon has fractional radial loss $\delta \simeq q/(2\lambda k_F^2)$ when the transferred energy is small.

For a spherical Fermi surface of radius $k_F$ lying inside a zone with shortest reciprocal lattice vector magnitude $G>2k_F$, a translated neighbouring Fermi sphere is separated from it by $q_U=G-2k_F$. An electron-phonon Umklapp scattering process satisfying $\mathbf{k}+\mathbf{q}={\mathbf{k}}^{\prime }+\mathbf{G}$ therefore needs a phonon wavevector of at least $q_U$. For an approximately linear acoustic dispersion its activation scale isAt $T\ll T_U$ these phonons are sparsely occupied; once they become thermally available, momentum-relaxing scattering can grow rapidly. Normal small-angle scattering and residual disorder need not vanish below this scale, so it is a crossover estimate rather than a sharp universal conductivity threshold.

For acoustic modes ${\omega }_q\simeq c_{s}q$, thermally active wavevectors have $q_T\sim k_{B}T/(\hslash c_s)$. When $q_T\ll k_F$, a nearly elastic event on a spherical Fermi surface has $1-\cos \theta =q^2/(2k_F^2)$, suppressing momentum relaxation. In the clean three-dimensional acoustic model, the thermal relaxation rate scales as $T^3$ while the electrical transport rate gains two extra powers and scales as $T^5$. With electronic heat capacity proportional to $T$, this gives $K_e\propto T^{-2}$ and electrical resistivity proportional to $T^5$ in the phonon-limited regime. Residual elastic scattering cuts off these behaviours as temperature falls further.^[ref]