Electron-phonon interactions from first principles

This article reviews the theory of electron-phonon interactions in solids from the point of view of ab initio calculations. While the electron-phonon interaction has been studied for almost a century, predictive nonempirical calculations have become feasible only during the past two decades. Today it is possible to calculate from first principles many materials properties related to the electron-phonon interaction, including the critical temperature of conventional superconductors, the carrier mobility in semiconductors, the temperature dependence of optical spectra in direct and indirect-gap semiconductors, the relaxation rates of photoexcited carriers, the electron mass renormalization in angle-resolved photoelectron spectra, and the nonadiabatic corrections to phonon dispersion relations. In this article a review of the theoretical and computational framework underlying modern electron-phonon calculations from first principles as well as landmark investigations of the electron-phonon interaction in real materials is given. The first part of the article summarizes the elementary theory of electron-phonon interactions and their calculations based on density-functional theory. The second part discusses a general field-theoretic formulation of the electron-phonon problem and establishes the connection with practical first-principles calculations. The third part reviews a number of recent investigations of electron-phonon interactions in the areas of vibrational spectroscopy, photoelectron spectroscopy, optical spectroscopy, transport, and superconductivity.

Figure 1

Diagrammatic representation of the phonon Green’s function and self-energy. (a) Dyson equation for the phonon propagator, Eq. (135). The thick wavy line represents the fully interacting, nonadiabatic propagator; the thin wavy line is the adiabatic propagator; the disk is the nonadiabatic self-energy. (b) Lowest-order diagrammatic expansion of the phonon self-energy in terms of the bare electron-phonon vertices and the RPA electronic polarization. The small dots are the bare electron-phonon coupling functions, and the thin lines are the noninteracting (for example, Kohn-Sham) electron Green’s functions. This diagram is the simplest possible term which begins and ends with a phonon line. (c) Nonperturbative representation of the phonon self-energy in terms of the bare coupling, the dressed coupling (large gray disk), the fully interacting electron Green’s functions (thick lines), and the vertex $\mathrm{\Gamma }$ from Eq. (83). This diagram was proposed by [215] and describes the first line of Eq. (141). (d) Schematic representation of the relation between the dressed electron-phonon coupling $g$ and the bare coupling $g^{\mathrm{b}}$, from Eq. (144). [390] reports a similar diagram, although with the bare coupling function on the far right; the difference stems from the present choice of using the irreducible polarization $P=GG\mathrm{\Gamma }$ instead of the reducible polarization employed by Vogl.

Figure 2

Diagrammatic representation of the electron Green’s function and electron-phonon self-energy. (a) Dyson equation for the electron Green’s function, Eq. (151). The thick straight line represents the fully dressed electron propagator, the thin straight line is the propagator calculated at clamped nuclei, and the disk is the electron-phonon self-energy. (b) Decomposition of the electron-phonon self-energy into Fan-Migdal self-energy, Eq. (153), Debye-Waller contribution, Eq. (154), and the remainder given by Eq. (155). (c) Fan-Migdal electron-phonon self-energy expressed in terms of the dressed electron-phonon coupling function (dark gray disk as in Fig. 1), the fully interacting electron Green’s functions (thick straight line), the fully interacting phonon propagator (thick wavy line), and the vertex $\mathrm{\Gamma }$ from Eq. (83). (d) Debye-Waller contribution resulting from the fully interacting phonon propagator (thick wavy line) and the matrix element in Eq. (40) (hatched disk). (e) Correction to Hedin’s $GW$ self-energy arising from the modification of the electronic structure induced by the electron-phonon interaction. $W_{\mathrm{e}}$ is the screened Coulomb interaction of Eq. (94) (bold dashed double line). $W_{\mathrm{e}}^{\mathrm{cn}}$ is the screened Coulomb interaction evaluated at clamped nuclei (thin dashed double line). ${\mathrm{\Gamma }}^{\mathrm{cn}}$ is the vertex of Eq. (83), but evaluated at clamped nuclei.

Figure 3

Temperature-dependent band gap and lifetimes in an idealized semiconductor or insulator. (a) Temperature dependence of the band gap according to Eq. (168) (thick solid blue line). The straight thin black line is the asymptotic expansion at high temperature; this line intercepts the vertical axis at the band gap calculated with clamped nuclei $E_{\mathrm{g}}^{\mathrm{cn}}$. The difference between the latter value and the band gap at $T=0$ including the EPI gives the zero-point renormalization $\mathrm{\Delta }{E}_{\mathrm{g}}^{\mathrm{ZP}}$. (b) Temperature dependence of the electron linewidth (solid blue line) and lifetimes (dashed red line) using the same model as in (a). The zero-point broadening is ${\mathrm{\Gamma }}_{n\mathbf{k}}^{\mathrm{ZP}}$. This simplified trend is valid only when the electron energy is at least one phonon energy away from a band extremum, so that both phonon emission and phonon absorption processes are allowed. The parameters of the model are $E_{\mathrm{g}}^{\mathrm{cn}}=1\mathrm{eV}$, $\mathrm{\Delta }{E}_{\mathrm{g}}^{\mathrm{ZP}}=100\mathrm{meV}$, $\hslash {\omega }_0=100\mathrm{meV}$, and ${\mathrm{\Gamma }}_{n\mathbf{k}}^{\mathrm{ZP}}=50\mathrm{meV}$. These values are representative of common semiconductors.

Figure 4

Two-dimensional maps of the electron spectral function $A(\mathbf{k},\omega )$ for electrons coupled to a dispersionless phonon of frequency ${\omega }_0$. The noninteracting bands are given by $ϵ(\mathbf{k})=-{ϵ}_{\mathrm{F}}+{\hslash }^2|\mathbf{k}{|}^2/2m^{*}$, and the Fermi level coincides with the top of the energy window. The matrix element is $|g{|}^2=\hslash {\omega }_0/N_{\mathrm{F}}$ when the electron energies differ by less than the cutoff ${ϵ}_{\max }$ and zero otherwise ($N_{\mathrm{F}}$ is the density of states at the Fermi level). (a) Spectral function for the case ${ϵ}_{\mathrm{F}}=10\hslash {\omega }_0$ (white on blue, black), noninteracting band structure (solid line, yellow, light gray), and fully interacting band structure within the Brillouin-Wigner perturbation theory (solid line, red, dark gray). (b) Spectral function for the case ${ϵ}_{\mathrm{F}}=2\hslash {\omega }_0$. The model parameters are $m^{*}=0.1m_{\mathrm{e}}$, $\hslash {\omega }_0=100\mathrm{meV}$, $\eta =20\mathrm{meV}$, and ${ϵ}_c=5\mathrm{eV}$. For clarity the calculated spectral functions are cut off at the value $3{\mathrm{eV}}^{-1}$ and normalized. The self-energy is shifted by a constant so as to have ${\mathrm{\Sigma }}^{\mathrm{ep}}(0)=0$. This correction guarantees the fulfillment of Luttinger’s theorem about the volume enclosed by the Fermi surface ([263]).

Figure 5

Spatial decay of the electron-phonon matrix elements of diamond in the Wannier representation: (a) $\max |g_{mn\kappa \alpha }({\mathbf{R}}_p,0)|$ vs $|{\mathbf{R}}_p|$, and (b) $\max |g_{mn\kappa \alpha }(0,{\mathbf{R}}_{p^{\prime }})|$ vs $|{\mathbf{R}}_{p^{\prime }}|$. The maximum values are taken over all subscript indices, and the data are normalized to the largest value. The insets show the same quantities in logarithmic scale. The calculations were performed using the local density approximation to DFT. From [153].

Figure 6

Comparison between Wannier-interpolated electron-phonon matrix elements and explicit DFPT calculations for diamond. The interpolated matrix elements were calculated starting from a coarse $4^3$ Brillouin-zone grid (dotted line, black), a $6^3$ grid (dashed line, blue), and a $8^3$ grid (solid line, red). The dots indicate explicit DFPT calculations. In this example $|n\mathbf{k}⟩$ is set to the valence band top at $\mathrm{\Gamma }$; $|m\mathbf{k}+\mathbf{q}⟩$ spans ${\mathrm{\Lambda }}_3$, ${\mathrm{\Delta }}_5$, and ${\mathrm{\Sigma }}_2$ bands, and the phonon is set to the highest optical branch. From [153].

Figure 7

Wannier interpolation of electron-phonon matrix elements for anatase ${\mathrm{TiO}}_2$. The initial state $|n\mathbf{k}⟩$ is set to the bottom of the conduction band at $\mathrm{\Gamma }$, the final state $|m\mathbf{k}+\mathbf{q}⟩$ spans the bottom of the conduction band along high-symmetry lines, and the phonon is the highest LO mode. The dots correspond to explicit DFPT calculations. The red dashed line is the short-range component of the matrix elements $g^{\mathcal{S}}$. The solid blue curve represents the matrix elements $g^{\mathcal{S}}+g^{\mathcal{L}}$, as obtained from the modified Wannier interpolation of Sec. 6a3. The interpolation was performed starting from a coarse $4\times 4\times 4$ unshifted grid. From [387].

Figure 8

Frequency of the Raman $G$ band of graphene vs carrier concentration. The black filled disks are from Raman measurements of gated graphene on a silicon substrate at 295 K. The thick horizontal dashed line (red) shows the variation of the $E_{2g}$ mode frequency with doping, within the adiabatic approximation. The solid blue line shows the variation of the frequency calculated by including nonadiabatic frequency renormalization. From [326].

Figure 9

Comparison between measured (${\omega }_{\exp }$) and calculated (${\omega }_{\text{th}}$) vibrational frequencies of the $E_{2g}$ mode of graphite intercalation compounds. Open symbols are adiabatic DFPT calculations, and filled symbols are calculations including the nonadiabatic corrections. The line corresponds to ${\omega }_{\text{th}}={\omega }_{\exp }$. From [348].

Figure 10

Phonon dispersion relations of ${\mathrm{CaC}}_6$ calculated using Wannier interpolation. The dashed lines (black) and the solid lines (red) represent the standard adiabatic calculation and the nonadiabatic phonon dispersions, respectively. From [69].

Figure 11

Calculated Fan-Migdal self-energy of the surface state at the Be(0001) surface. (a) Imaginary part of the self-energy, obtained from Eq. (188). The dashed (black) line is the self-energy evaluated using the DFT-LDA bands; the solid lines (color, gray scale) correspond to the self-energy calculated by taking into account the renormalization of the DFT band structure by the electron-phonon interaction. (b) Real part of the self-energy, using the same color, gray scale code as in (a). The inset in (a) compares the renormalized band structure (color) with the “bare” DFT band (black dashed line). The inset of (b) shows the renormalization of the band velocity induced by the electron-phonon interaction. From [121].

Figure 12

(a), (b) Calculated real part of the Fan-Migdal self-energy in pristine and $n$-doped graphene, respectively (solid black lines). The doping level is $4\times {10}^{13}{\mathrm{cm}}^{-2}$. The dashed lines correspond to a simplified analytical model where particle-hole symmetry is assumed. (c), (d) Electron band velocity renormalization resulting from the self-energies in (a) and (b). All calculations in (a)–(d) were performed using DFT-LDA. From [308]. (e) Calculated spectral function of $n$-doped graphene for one of the branches of the Dirac cone. ${\omega }_{\text{ph}}$ indicates the characteristic phonon energy leading to the photoemission kink, and $E_{\mathrm{D}}$ denotes the energy of the Dirac point. The calculations include $GW$ quasiparticle corrections. From [311].

Figure 13

Temperature dependence of the direct band gap of diamond calculated using the Allen-Heine theory. The upper curve shows the results obtained within DFPT at the LDA level. The lower curve was obtained via $GW$ calculations in the frozen-phonon approach. Triangles are experimental data. The zero-point renormalization calculated by including $GW$ quasiparticle corrections is 628 meV. From [28].

Figure 14

Phonon-assisted optical absorption in silicon: comparison between first-principles calculations (solid lines, orange) and experiment (circles, blue). The calculations were performed using the theory of [178], as given by Eq. (200). Spectra calculated at different temperatures were shifted horizontally so as to match the experimental onsets. From [302].

Figure 15

Electron relaxation times in GaAs resulting from electron-phonon scattering. (a) Calculated relaxation times as a function of electron energy with respect to the conduction band bottom. The color code (gray shades) of the data points identifies the valley where each electronic state belongs. (b) Schematic representation of the conduction band valleys in GaAs. From [50].

Figure 16

Temperature-dependent mobility of $n$-type silicon. The solid line (red) indicates the mobility calculated using the linearized Boltzmann transport equation, Eq. (202), and the dashed line (blue) corresponds to the energy relaxation time approximation. The triangles and diamonds are experimental data points. From [253].

Figure 17

(a) Energy distribution of the superconducting gap function of ${\mathrm{MgB}}_2$ as a function of temperature, calculated using the anisotropic Migdal-Eliashberg theory. The gap vanishes at the critical temperature (in this calculation $T_{\mathrm{c}}=50\mathrm{K}$). Two distinct superconducting gaps can be seen at each temperature. (b) Density of electronic states in the superconducting state of ${\mathrm{MgB}}_2$ at various temperatures calculated within the Migdal-Eliashberg theory. From [270].

Figure 18

Superconducting order parameter $\chi (\mathbf{r},{\mathbf{r}}^{\prime })$ in real space, calculated for (a) ${\mathrm{MgB}}_2$ and (b) hole-doped graphane. The plots show a top view (top) and a side view (bottom) of the hexagonal layers in each case. The variable $\mathbf{s}=\mathbf{r}-{\mathbf{r}}^{\prime }$ is the relative coordinate in the order parameter, while the center-of-mass coordinate is placed in the middle of a B-B bond or a C-C bond. From [257].