Electron–phonon (e–ph) interaction plays a crucial role in determining many physical properties of the materials, such as the superconducting transition temperature, the relaxation time and mean free path of hot carriers, the temperature dependence of the electronic structure, and the formation of the vibrational polaritons. In the past two decades, the calculations of e–ph properties from first-principles has become possible. In particular, the renormalization of electronic structures due to e–ph interaction can be evaluated, providing greater insight into the quantum zero-point motion effect and the temperature dependence behavior. In this perspective, we briefly overview the basic theory, outline the computational challenges, and describe the recent progress in this field, as well as future directions and opportunities of the e–ph coupling calculations.

In recent years, researchers have shown an increased interest in electron–phonon (e–ph) interactions, which are at the heart of accurately predicting many chemical and physical properties;1 for example, they elucidate the phonon mechanism responsible for superconductivity,2–4 they induce the distortion of the crystal lattice for the excess carrier,5–10 and they determine the carrier lifetime11–15 and mobility in materials.16–28 In addition, e–ph interactions enable a significant increase in catalytic reactivity.29,30 They also contribute to the temperature dependence of electronic structure properties of materials31–40 and can make the phonon-induced band structure renormalization even at zero K because of the quantum zero-point motion of the nuclei.31,32,41 As a result, e–ph interactions have been a “hot topic” in condensed matter theory for almost a century, but only during the recent two decades, the accurate first principles assessment of these effects has become possible, which enables a quantitative understanding of the above multiple phenomena and motivates further research.

However, due to the complexity and heavy computational cost of the non-empirical e–ph interaction, first principles calculations are not an easy task. The elementary theoretical framework needs to be constructed to treat electron–electron and electron–phonon interaction on the same quantum mechanical level. At the same time, efficient computational techniques that dramatically reduce the computational cost of first principles calculations of e–ph coupling needs to be developed, which can improve the predictive power of e–ph interaction calculations. In other words, continued efforts are needed to improve the computational accuracy and efficiency of the e–ph interaction, as well as extend them to more complex materials.

For the first principle e–ph renormalization, two methods exist: perturbative formulations based on the density functional perturbation theory42–47 and frozen-phonon method based on the finite-difference approach,48 both can enable the calculation of the zero-point renormalization and temperature dependence of the band structure for a variety of materials.31,32,41,49–56 The key ingredient of the first principles calculation of the e–ph induced band structure renormalization is the second-order derivative of the eigenvalues with respect to the atomic displacements. This derivative splits into two leading terms—the Fan term and the Debye–Waller term. The Fan term corresponds to the first-order phonon-induced perturbation of the electronic energy up to the second order in the ionic displacement. The Debye–Waller term is the second-order phonon-induced perturbation of the Hamiltonian, which is difficult to be evaluated.54 The rigid-ion approximation (RIA) is adopted in the Allen, Heine, and Cardona (AHC) theory to simplify the calculation, but the exact full Debye–Waller term need to include both the RIA term and the non-RIA term.54 On the other hand, the frozen-phonon approach can naturally capture the full Debye–Waller term, which has been found to be small in solid31,32,54,57 but is non-negligible in molecule systems.41 Furthermore, the frozen-phonon approach can extend the calculation beyond the semilocal density functional theory (DFT) and consider the many-body corrections.32 

For the perturbative method, the computational challenges are associated with the calculation of the e–ph matrix elements, which is computationally expensive when a large number of electron and phonon wave vectors in the Brillouin zone are needed. On the other hand, the finite-difference method requires a large supercell to model the explicit motion of the nuclei in the system, so the scaling of computational cost restricts the simulation size. Recent methodological developments have tackled these challenges; for example, the Wannier-interpolation method has been introduced in the study of e–ph interaction4 and has been successfully employed in a variety of applications with the perturbative method. The methodological development on the finite difference side, such as nondiagonal supercells method58 and one-shot method,59 can dramatically reduce the computational cost of first-principles calculation of the e–ph interaction. Thanks to these developments, the calculation of the e–ph from first principles has become prosperous in the past few years and will play a prominent role in forthcoming years. The aim of this perspective is to describe the current state of the art and to discuss the remaining challenges for further research.

The remainder of this perspective is organized as follows: in Sec. II, we present challenges in the theoretical description of the e–ph interaction and its computational formalism. We first introduce the perturbative approach within Green’s function framework, then we derivate the Allen–Heine–Cardona theory to describe e–ph renormalization in condensed matter physics. We also describe the finite difference approach and some recent developments in this direction, which can greatly reduce the computational cost of the calculations. In Sec. III, we review the calculations of the e–ph interactions, which include the insulators and semiconductors; furthermore, we analyze the trends of e–ph interactions, under which circumstances accounting for e–ph interactions become essential for accurate computational bandgap predictions. In Sec. IV, we describe the current challenges and future opportunities in the e–ph interaction calculation.

The challenges in the methodological development of the e–ph interaction are mainly twofold: First, how to build a theoretical framework to treat the electron–electron and electron–phonon interaction on the same quantum mechanical level? Second, how to compute the e–ph interaction efficiently from the first principles? In the following, we first introduce the perturbative formulations approach based on the many-body Green’s function framework, and we also derivate the Allen–Heine–Cardona (AHC) theory. Then, we introduce the finite difference method for practical calculations.

Within the adiabatic approximation, the Kohn–Sham single particle states ψn and their eigenenergies ϵn can be calculated by solving a Kohn–Sham (KS) single particle equation,
ĥKSψn=ϵnψn,
(1)
where ĥKS refers to the Kohn–Sham Hamiltonian. The electron states ψn can be calculated with density functional theory (DFT).60 Similarly, the phonon eigenstates are obtained from a dynamical equation,
jβΦiα,jβujβ,λ=Miωλ2uiα,λ,
(2)
where i and j refer to different atoms, α and β refer to the three coordinates, Mi is the atomic mass of atom i, λ is the index of the vibrational mode, u,λ is the eigen displacement vector of atom i in direction α associated with the vibrational mode λ, and Φ is the Hessian (force constant) matrix. The above equation can be further written as an eigenvalue equation,
jβΦiα,jβMiMjejβ,λ=ωλ2eiα,λ,
(3)
where ejβ,λ=Mjujβ,λ is the solution of the above eigenvalue equation. The Hessian matrix and the phonon frequencies ωλ can be obtained with density functional perturbation theory (DFPT).61–63 
In 1970, Hedin and Lundqvist64 published a set of exact equations to describe both the electron–electron and electron–phonon interaction simultaneously in Green’s function framework. In this scheme, the screened Coulomb interaction W can be decomposed into two contributions: electronic (Wee) and ionic screening (Weph), as shown in Fig. 1. The self-energy that connects a system of non-interacting electrons with the fully interacting system then also takes on a separable form,
Σ=Σe-e+Σe-ph.
(4)
Here, Σe-e includes the electron–electron interaction, whereas the interaction with the nuclei is confined to Σeph. The full solution of these Green’s function equations64 is not tractable right now, which implies that approximations have to be used.
FIG. 1.

(a) The diagrams for the contribution of the electron–electron interaction. (b) Diagrammatic representation of the Fan and Debye–Waller terms for the electron–phonon interaction.

FIG. 1.

(a) The diagrams for the contribution of the electron–electron interaction. (b) Diagrammatic representation of the Fan and Debye–Waller terms for the electron–phonon interaction.

Close modal

In the description of the electron–electron interaction, Hedin’s GW approximation65 for the self-energy, as shown schematically in Fig. 1(a), has been the method of choice for the calculation of quasiparticle band structures of solids, and for a variety of materials, good agreement with measured photoemission spectra has been achieved.64,66–68 Moreover, the screened Coulomb interaction We−e includes surface polarization effects, also known as image effects,69 that renormalize the energy levels of adsorbed molecules compared to their gas phase values. Due to their computational expense, GW calculations have mostly been performed for bulk systems and small clusters and molecules,66–68 but the number of calculations for a complex system is increasing.

The development of first principles methods for the electron–phonon interaction has been trailing that of the electron–electron interaction. Efficient approaches to compute the electron–phonon interaction by, e.g., means of density-functional perturbation theory have become available only fairly recently.61,62,70 They enabled the first principles calculation of band structure renormalizations due to electron–phonon coupling for real materials,31,32,49,71 which was previously done using the empirical pseudopotential method43,72 or on the level of models.73 The e–ph renormalization for the electronic states is shown in Fig. 1(b). The Fan term corresponds to the first-order phonon-induced perturbation of the electronic energy up to the second order and is similar to the GW self-energy, which is shown in Fig. 1(a). The Debye–Waller term is the second-order phonon-induced perturbation treated within perturbation theory. Both the Fan and the Debye–Waller self-energies can be expressed in terms of the electron–phonon coupling matrix elements,64 as will be illustrated later in this section.

If we consider the electron–electron interaction on the level of the GW approximation, the frequency-dependent Green’s function reads
Gnk(ω)=(ωΣnkGW(ω)ΣnkFan(ω)ΣnkDW)1,
(5)
and the imaginary part defines the spectral function,
Ank(ω)=1π|IGnk(ω)|.
(6)
After the electron–phonon coupling matrix elements have been calculated, the Fan and Debye–Waller self-energies can be computed as follows:49,
ΣnkFan(ω)=nqλ|gnnkqλ|2NqB(ωqλ)+1fnkqωϵnkqωqλi0++B(ωqλ)+fnkqωϵnkq+ωqλi0+,
(7)
ΣnkDW=qλ1Nqnn12|g̃nnkqλ|2ϵnkϵnk[2B(ωqλ)+1].
(8)
Here, fnk are the Fermi–Dirac occupation factors, and B(ωqλ) are the Bose–Einstein occupation factors. gnnkqλ is the e–ph matrix element, which describes the transition amplitude from electronic state ψn(k) with band index n and momentum k to a Bloch state ψn(k + q) with band index n′ and momentum k + q mediated by a phonon with mode index λ and momentum q,
gnnkqλ=12ωqλψn(k+q)ĥksuλ(q)ψn(k),
(9)
where the electron states ψn(k) and ψn(k + q) are calculated with the DFT method, and the phonon dispersions ωqλ are obtained with the DFPT method.61–63 The Debye–Waller electron–phonon matrix elements g̃nnkqλ can be written as
|g̃nnkqλ|2=12ωqλiα,iαtiα,iαλ(q)hnn,iα*(k)hnn,iα(k),
(10)
tiα,iαλ(q)=uiα,λ*(q)uiα,λ(q)+uiα,λ*(q)uiα,λ(q),
(11)
hnn,iα(k)=λuiα,λ1(0)ω0λ1/2gnnk0λ,
(12)
where u,λ(q) is a mass-scaled phonon eigenvector e,λ(q), i.e., uiα,λ(q)=Mi1/2eiα,λ(q), and the atomic index and the Cartesian directions are labeled by i and α, respectively.
In this way, the renormalization effects due to both the electron–electron and the electron–phonon interaction are included in the spectrum. However, this equation has never been solved before. Instead, the problem has been simplified by, e.g., Giustino et al.49 and Cannuccia and Marini50 by replacing the quasiparticle energies (i.e., the effect of the GW self-energy) in Eq. (5) with DFT Kohn–Sham eigenvalues,
Gnk(ω)=(ωϵnkksΣnkFan(ω)ΣnkDW)1.
(13)
Equation (13) is conceptually simpler and computationally more tractable and has been solved for trans-polyacetylene, and the renormalization of the electronic band structure has been shown in Fig. 2(a). When the e–ph interaction is weak, Eq. (13) has one solution, and the spectral function has one peak, this is shown schematically for trans-polyacetylene in Fig. 2(b). However, if the electron–phonon interaction is strong, Eq. (13) may have more than one solution.50,73 The electronic states are then not only renormalized but also split into several polaronic states as shown in Fig. 2(c). In certain cases, this might even wash out the main peak to a point where the quasiparticle picture breaks down.50 We refer to the resulting quasiparticles as polarons whose energies are obtained as poles of Eq. (13),
ϵnk(T)=ϵnkks+ZnkΣnkFan(ω)+ΣnkDW,
(14)
with
Znk=(1RΣnkFan(ω)ω)1.
(15)
FIG. 2.

(a) Effect of electron–phonon interaction on the band structure of trans-polyacetylene. Bidimensional representation of the probability amplitude. The intensities of peaks are associated with a colored scale, from white to black. The solid black lines are the KS valence bands(From50). (b) Schematic diagram to show the influence of electron–phonon interaction on the spectral function when electron–phonon interaction is small. (c) Schematic diagram to show the influence of the electron–phonon interaction on the spectral function when electron–phonon interaction is large.

FIG. 2.

(a) Effect of electron–phonon interaction on the band structure of trans-polyacetylene. Bidimensional representation of the probability amplitude. The intensities of peaks are associated with a colored scale, from white to black. The solid black lines are the KS valence bands(From50). (b) Schematic diagram to show the influence of electron–phonon interaction on the spectral function when electron–phonon interaction is small. (c) Schematic diagram to show the influence of the electron–phonon interaction on the spectral function when electron–phonon interaction is large.

Close modal
In the limit Znk equals 1, we obtain the Allen–Heine–Cardona (AHC) theory,74 in which the spectral function of the polaron is given by a delta function. The effect of the electron–phonon interaction is then to renormalize every electronic state into a polaronic state shifted in energy by non-adiabatic Fan term and DW term, which is called non-adiabatic AHC theory,
ΔEAHCna=ΣnkFan-na(ϵnkks)+ΣnkDW,
(16)
ΣnkFan-na(ϵnkks)=nqλ|gnnkqλ|2NqB(ωqλ)+1fnkqϵnkksϵnkqωqλiδ0++B(ωqλ)+fnkqϵnkksϵnkq+ωqλiδ0+.
(17)
Furthermore, in the adiabatic approximation, the phonon frequencies ωqλ are simply dropped, which is considered small with respect to eigenenergy differences in the denominator of Eq. (17), and the adiabatic Fan term can be written as
ΣnkFan-ad=nqλ|gnnkqλ|2Nq2B(ωqλ)+1ϵnkksϵnkq,
(18)
and the adiabatic AHC equation,
ΔEAHCad=ΣnkFan-ad(ϵnkks)+ΣnkDW.
(19)

It should be noted that such an adiabatic AHC approach will diverge at band edges for infrared-active materials.31,75 This is because, in the adiabatic approach, the electrons are supposed to have time to adjust to the change of potential; however, in the polar materials, the “fast” zone-center LO phonons cannot be followed by the “slow” electron, so the adiabatic approach could breakdown.31,76 The divergence of the adiabatic AHC approach can be avoided by using a non-adiabatic equation [Eq. (17)] or using (generalized) Fröhlich mode.75,77 It should be noted that the recently proposed GWPT (GW perturbation theory)78,79 approach could directly compute the changes in the electronic self-energy due to perturbing phonons within the GW framework, and it constructs the e–ph matrix element at the GW level, so it can calculate the electron–phonon (e–ph) interactions with the full inclusion of the GW nonlocal, energy-dependent self-energy effects. Its relation with Eq. (5), which treats electron–electron and electron–phonon interaction simultaneously in Green’s function framework, is still an open question and needs further investigation.

The methods described above have recently been used to investigate the electron–phonon interaction in carbon based materials. For example, based on the adiabatic AHC theory [using the adiabatic Fan term shown in Eq. (18)], Giustino et al.49 have calculated the electron–phonon renormalization of the bandgap in diamond, and they found that the Fan and DW terms are of comparable magnitude. Cannuccia and Marini50 interpret this in terms of polaronic states that they obtained by solving Eq. (13), in good agreement with optical absorption measurements. They also computed the band structure of the polymer trans-polyacetylene including the electron–phonon renormalization (see Fig. 2) and observed strong polaronic effects and the break down of the quasiparticle picture in certain parts of the band structure. Further work on the diamond was reported by Poncé et al.,31 who use both the non-adiabatic [Eq. (17)] and the adiabatic [Eq. (18)] AHC theory to investigate the e–ph zero-point renormalization and the temperature dependence of the direct and indirect bandgaps, as shown in Fig. 3. The converged non-adiabatic zero-point renormalization for the direct/indirect bandgap of the diamond is −415.8/−329.79 meV, which underestimates the experimental ones by 6.8%/9.4%. The calculated slope at high temperatures is −0.504 meV/K for the direct bandgap and is −0.435 meV/K for the indirect bandgap, which also underestimates the experimental ones by 16% and 19%. Such underestimations are due to neglecting the anharmonicity, non-rigid-ion terms, and many-body effects.

FIG. 3.

Temperature dependence of the diamond gaps using the non-adiabatic temperature dependence on a 75 × 75 × 75 q-grid with Lorentzian extrapolation to vanishing imaginary parameter δ. The slopes at high temperatures are −0.504 meV/K for the direct gap of the diamond and −0.435 meV/K for the indirect one. The experimental points from Clark et al.80 and Logothetidis et al.81 (using first or second-derivative line-shape analysis) are shifted so that the lowest temperature point matches the theoretical line. Reproduced with permission from Poncé et al., J. Chem. Phys. 143, 102813 (2015). Copyright 2015 AIP Publishing LLC.

FIG. 3.

Temperature dependence of the diamond gaps using the non-adiabatic temperature dependence on a 75 × 75 × 75 q-grid with Lorentzian extrapolation to vanishing imaginary parameter δ. The slopes at high temperatures are −0.504 meV/K for the direct gap of the diamond and −0.435 meV/K for the indirect one. The experimental points from Clark et al.80 and Logothetidis et al.81 (using first or second-derivative line-shape analysis) are shifted so that the lowest temperature point matches the theoretical line. Reproduced with permission from Poncé et al., J. Chem. Phys. 143, 102813 (2015). Copyright 2015 AIP Publishing LLC.

Close modal

In order to address the dynamical and anharmonic effects on the e–ph renormalization of the electronic structure, Antonius et al.57 investigated diamond, BN, LiF, and MgO, and they found that using a dynamical scheme to compute the frequency-dependent self-energy [Eq. (13)] can reduce zero-point renormalization (ZPR) by as much as 40%. In addition, the anharmonic effects reduce the ZPR of certain states by as much as 60% compared with the static AHC theory. Furthermore, it has been conjectured that other materials containing atoms of the second row of the Periodic Table (e.g., B, C, N, O, F) have also exhibited large electron–phonon effects,45 for example, in SiC,55 GaN,75 and ZnO.8 

Despite the fact that the electron–phonon interaction can lead to considerable corrections to the electronic levels even at zero temperature,74 it has been mostly neglected in the past due to the associated computational challenges. These include, for instance, the fine sampling of the Brillouin zone,4 the slow convergence of the electron–phonon self-energies with respect to unoccupied states,4,41 and the non-rigid-ion-approximation elements in the Debye–Waller term.41,54 As a result, considerable attention has, instead, been devoted to the development of computational techniques for the electron–phonon matrix elements gnnkqλ. They can be computed directly in any electronic structure code using a supercell approach82,83 or using the much more elegant density-functional perturbation theory.4,84,85 However, it has been demonstrated that a fine sampling of the Brillouin zone is often needed to achieve numerical convergence.11,86 Giustino et al.4 have developed an interpolation technique based on Wannier functions to significantly reduce the computational cost. The spatial locality of the Wannier functions is exploited to achieve this in a computationally efficient manner. The basic idea of their method is to use Wannier functions as a compact, intermediate representation of real-space in a forward and backward Fourier transformation of the electron–phonon coupling matrix element from a coarse Brillouin-zone sampling to a dense one.

The above approaches based on AHC theory are computationally efficient but require analytic derivatives of the Hamiltonian. These are available for standard DFT approaches, such as the local density approximation (LDA), but not for the more advanced hybrid functionals or the GW method. However, the self-interaction error of LDA can lead to a significant underestimation of the e–ph interaction when compared to more accurate GW calculations. It remains to be clarified whether the underestimation of LDA can be corrected by using hybrid DFT approaches that yield an improved description of the electronic band structure. In the following, we will describe the finite-difference methods, which can be used in conjunction with advanced electronic structure methods that go beyond the semilocal DFT.

In this section, we present the computational approach for electron–phonon renormalization with the finite-difference method. For a detailed review of the finite-difference method in the study of e–ph interactions, we refer the reader to Ref. 48.

Within the adiabatic harmonic approximation, the zero-point renormalization for the electric eigenstates can be written with the eigen displacement (u,λ) as31,32
δϵn=λ4ωλiα,jβuiα,λ2ϵnRiαRjβujβ,λ.
(20)
By applying zλ=Mλ1/2iαRiαuiα,λ to Eq. (20), we can write the zero-point renormalization in a more concise form as
δϵn=λ4ωλMλ2ϵnzλ2,
(21)
where zλ is the collective atomic displacement along the vibrational eigenmode λ. Mλ is a normalization factor of mode λ, defined as Mλ=(iαuiα,λ2)1.
The above zero-point renormalization for electronic states is for the cluster case. It can be extended to the condensed matter solid case by considering Brillouin integration and also the polar correction,
δεn(k)=δεnA(k)+δεnP(k),
(22)
i.e., the adiabatic δεnA(k) term and polar correction δεnP(k) term, in which n is the electronic band index, and k is the reciprocal-space point of the electron. The adiabatic contribution δεnA(k) is given by the integral over the Brillouin-zone with volume ΩBZ,
δεnA(k)=λdqΩBZ4ωλ(q)2εi(k)δλq2.
(23)
Here, ωλ(q) is the phonon frequency, λ is the phonon mode index, and q is the reciprocal-space point of the phonon. In the finite-difference formalism,32,76,87,88 the response of the electronic states to individual phonon modes ωλ(q) is computed via small displacements (δλq) from equilibrium along the respective normalized mass-weighted phonon eigenvector eλ(q). In addition to the above adiabatic term, the polar correction term to the ZPR is also important. For strongly polar materials, the supercell adiabatic approaches are problematic.75 In order to prevent the divergence, we need to know the real physical mechanism behind this divergence. As clearly explained by Miglio et al.,75 the original physical mechanism is the “impossibility for the electron to follow the phonon dynamics,” so using the non-adiabatic AHC approach31,75 or the (generalized) Fröhlich model75,77 could solve this diverged problem.89 Shang et al.76 use the Fröhlich correction method to give the missing part of the contribution to the ZPR of the electronic structure for 82 octet binary materials in both the zincblende and the rocksalt structures. In this formalism, the singular Fröhlich interaction at the band edges is treated as
δεnP(k)=±αnωLOπtan1qFqi.
(24)
Here, ωLO is the highest longitudinal optical frequency, qF is the upper limit of such Fröhlich integral,77 and αi is the Fröhlich coupling constant defined for band i.

There are three advantages for the finite-difference method: first, it can include the full Debye–Waller term to get rid of rigid-ion approximation; second, it has the ability to use arbitrary electronic band structure calculation method; and third, the finite-difference method can also enable exploring the anharmonic effects that go beyond the AHC theory. Capaz et al.87 adopted finite-difference and tight-binding methods to investigate the temperature dependence of the bandgap for semiconducting single-wall carbon nanotubes. Laflamme et al.90 investigate the e–ph interactions in C60 molecules with hybrid functionals and found that such functional could strongly increase the e–ph coupling parameter by 40%. Han and Bester51,91 used the finite-difference method and first-principles density functional theory approach to study the e–ph coupling and observed a large e–ph effect on the electronic properties of carbon and silicon nanoclusters. Antonius et al.32 studied the temperature dependence of the direct bandgap of diamond with many-body perturbation theory using the GW approximation, as shown in Fig. 4, and they found the perturbative approaches based on LDA underestimate the zero-point renormalization (ZPR) by around 200 meV; while GW correction could increase the ZPR by 42% (from −437 meV by LDA to −622 meV by GW). In addition, the high-temperature slope is also increased by more than 50% (from −0.42 meV/K by LDA to −0.67 meV/K by GW) with GW correction and achieves good agreement with experimental data. Monserrat92 confirmed that the impact of the many-body corrections (GW approximation) is as large as 50% in diamond, silicon, and TiO2, while it is weaker within 5% in LiF and MgO. Similar calculations using such an approach have also been reported in investigating the e–ph renormalization of the band structure in the perovskite CsSnI3,93 in hybrid perovskite CH3NH3PbI,34 and in metal halide perovskite metal halide perovskites belonging to the class ABX3 (A = Rb, Cs; B = Ge, Sn, Pb; and X = F, Cl, Br, I).94 

FIG. 4.

Temperature dependence of the direct bandgap 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. Reprinted with permission from Antonius et al., Phys. Rev. Lett. 112, 215501 (2014). Copyright 2014 The American Physical Society.

FIG. 4.

Temperature dependence of the direct bandgap 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. Reprinted with permission from Antonius et al., Phys. Rev. Lett. 112, 215501 (2014). Copyright 2014 The American Physical Society.

Close modal

One of the disadvantages of the finite-difference method is the need to build the large supercell for q-point at small fractional coordinates (m1/n1, m2/n2, m3/n3), due to the fact that, in the standard finite-difference calculation, the diagonal supercells contained n1 × n2 × n3 primitive cells need to be built; in order to address this problem, Lloyd–Williams and Monserrat58 have shown that the smallest nondiagonal supercell only contains a number of primitive cells equal to the least common multiple of n1, n2, and n3. So for a uniform N × N × N q-grid, the size of the supercell can be dramatically reduced from N3 to N primitive cells, which could reduce the computational cost by orders of magnitude.

Besides the supercell size problem, the computational cost of the sum over many individual configurations for each phonon mode in Eq. (23) is also very heavy. In order to address this problem, Monserrat92,95 has proposed the thermal line method by replacing the sum over individual phonon modes by one or a small number of representative configurations for the mean value. Zacharias and Giustino59 extended this approach to phonon-assisted optical absorption using only one-shot calculation with a suitable choice of an “optimum” distorted atomic configuration by displacing the atoms from equilibrium by an amount,
Δτiα=λ(1)(λ1)eiα,λMi12ωλ.
(25)
Furthermore, Patrick and Giustino96 used an importance sampling Monte Carlo scheme to calculate the phonon-assisted optical absorption of diamondoids. Monserrat95 used Monte Carlo sampling over thermal lines to estimate accurate vibrational averages. Chen et al.97 proposed a non-uniform strategy to perform Brillouin zone integration that provides a significant computational advantage compared to the usual approach based on regular uniform grids, which has accelerated the finite-difference approach by factors of 6–7.

The first principles calculation could shed light on the impact of the e–ph interactions on the electronic structure calculation, which could answer the following questions: What is the nature of the quantum mechanical states or levels? Are they purely of electronic character? How strong is the e–ph renormalization of electronic states (polarons)? What influence do the e–ph interactions have on the electronic structure compared to the electron–electron interaction? We will briefly introduce some of these investigations in this section.

Electron–phonon interactions affect the electronic properties from 0 K to high temperature. The temperature dependence of the electronic structure has been investigated for a large group of materials, which generally show the normal behavior that the fundamental band-gap energy decreases with temperature (Varshni effect98), while others (e.g., CuGaS2 and AgGaS2,99 copper halides100) are considered to be abnormal because the opposite is observed (inverse Varshni effect99); such “anomalous” effects of temperature on the electronic band structures have been attributed to the presence of noble metals with d electrons.99 In fact, the frontier orbitals of d-electron solids can impart a variety of exotic electronic properties. In a joint experiment-theory study, Shang et al.101 show that e–ph interaction can affect the d-orbital derived bands in the opposite way from sp-orbital derived bands; by using first principles calculation, it was also demonstrated that the anisotropic environment of a crystal field, and specifically, the crystal field splitting of the d-bands, introduces new factors that modify e–ph interaction in previously unexpected ways and achieve quantitative agreement with two-photon photoemission study of temperature dependent electronic band energies of TiO2. Thus, how a crystal field and e–ph interaction affects the d-bands of solids provides a new way to analyze and control their electronic properties. As shown in Fig. 5, we show that the e–ph interaction affects the O 2p orbital derived valence bands of the two materials in the same way, and the Ti 3d and Si 2p orbitals derived conduction bands in the opposite way. In both cases, the valence bands maximums (VBMs) monotonically increase in energy with temperature in our calculations. By contrast, the temperature dependences of conduction bands minimums (CBMs) of TiO2 and SiO2 have the opposite signs, which was attributed to their dominant orbital characters. The opposite temperature dependences of the TiO2 and SiO2 CBMs can be explained by how the e–ph coupling affects the energies of the dominant atomic orbitals. In the quantitative first-principles approach, we define the e–ph factors as fλ=2ωλMλ2ϵnzλ2, the e–ph contribution from each mode can be evaluated, and the summed over all modes can determine the slope of the temperature dependence of each band, as shown in Fig. 5(c), raising of the CBM energy with temperature in TiO2 occurs because the number of modes with a positive e–ph factors is larger than ones with a negative one. By contrast, the decrease of the CBM energy with temperature in SiO2 occurs because all of the e–ph factors values are negative. Similarly, using the first principles approach, Villegas et al.71 also discussed the anomalous temperature dependence of the bandgap of black phosphorous. It should be noted that, in addition to the contribution from atomic motion, there is an important contribution from the thermal expansion/contraction of the lattice that is caused by the anharmonic terms in the lattice potential, which can be evaluated with quasiharmonic approximation,102 and any study looking at temperature dependent bands should also evaluate the effects of thermal expansion, as have been done in the above investigations.71,101

FIG. 5.

(a) CBM orbitals of TiO2 and SiO2 are plotted within their unit cells (red balls-O ions, gray balls-Ti ions, blue balls-Si ions, and yellow-CBM orbitals). (b) Calculated VBM- and CBM-state energy shifts for TiO2 and SiO2. The VBM shifts are caused by the O-2p orbitals and are similar for TiO2 and SiO2; for CBMs, the Ti-3d and Si-2s orbitals shift oppositely. Each type of orbital has a characteristic temperature-dependent behavior. (c) The e-p factor for phonons of increasing frequencies for the CBM states of TiO2 (gray) and SiO2 (blue), which cause their opposite shifts. Reprinted with permission from Shang, et al., Phys. Rev. Res. 1, 033153 (2019). Copyright 2019 The American Physical Society.

FIG. 5.

(a) CBM orbitals of TiO2 and SiO2 are plotted within their unit cells (red balls-O ions, gray balls-Ti ions, blue balls-Si ions, and yellow-CBM orbitals). (b) Calculated VBM- and CBM-state energy shifts for TiO2 and SiO2. The VBM shifts are caused by the O-2p orbitals and are similar for TiO2 and SiO2; for CBMs, the Ti-3d and Si-2s orbitals shift oppositely. Each type of orbital has a characteristic temperature-dependent behavior. (c) The e-p factor for phonons of increasing frequencies for the CBM states of TiO2 (gray) and SiO2 (blue), which cause their opposite shifts. Reprinted with permission from Shang, et al., Phys. Rev. Res. 1, 033153 (2019). Copyright 2019 The American Physical Society.

Close modal

The e–ph interaction can also induce topological phase transitions. Monserrat and Vanderbilt103 investigate the temperature dependence of the band structure for the Bi2Se3 family of topological insulators and find that these materials can turn into normal insulators at high temperatures. Similarly, using the first-principle approach, Antonius and Louie104 studied the topological phase diagram of BiTl(S1−δSeδ)2 as a function of doping and temperature. The e–ph interaction renormalized the band structure at a finite temperature and induced the transition between the topological insulators and the conventional insulators. The e–ph interaction promotes the BiTlS2 from a normal insulator to a topological insulator at high temperatures; on the contrary, in BiTlSe2, the topological phase exists at low temperatures and turns into a normal insulator at high temperatures. As shown in Fig. 6, the temperature dependence of the valence bands maximum (VBM) and the conduction bands minimum (CBM) for various intermediate doping between BiTlS2 and BiTlSe2 is given. In BiTlS2 and for low doping (δ ≲ 0.3), the bandgap decreases as the temperature increases, inducing the topological phase above a critical temperature. With high doping (δ ≳ 0.55) and in BiTlSe2, the bandgap closes with increasing temperature, and the system is changed from a topological phase to a trivial phase above a critical temperature.

FIG. 6.

Temperature dependence of the top of the valence bands and the bottom of the conduction bands for different doping between BiTlS2 and BiTlSe2. Reprinted figure with permission from G. Antonius and S. G. Louie, Phys. Rev. Lett. 117, 246401 (2016). Copyright 2016 The American Physical Society.

FIG. 6.

Temperature dependence of the top of the valence bands and the bottom of the conduction bands for different doping between BiTlS2 and BiTlSe2. Reprinted figure with permission from G. Antonius and S. G. Louie, Phys. Rev. Lett. 117, 246401 (2016). Copyright 2016 The American Physical Society.

Close modal

In the above studies, the first principles assessment of the e–ph interactions has become possible, which allows us to clarify how the nuclear motion determines the dependence of the electronic structure on temperature. However, apart from some empirical rule, little is yet known in this regard about the trends in chemical and structural space, i.e., under which circumstances accounting for e–ph interactions become essential for accurate computational band structure predictions. Cardona and Thewalt105 found that the zero-point renormalization (ZPR) “should be proportional to M−1/2 for monatomic crystals” (M is the isotopic mass), they adopt such a rule to obtain the ZPR from the isotopic mass derivatives of the gaps, which is in good agreement with the values obtained from the linear extrapolation method for 17 materials (Table III in Ref. 105). Shanget al.76 reported the e–ph renormalization of the electronic structure for 82 octet binary materials in both the zincblende (ZB) and rocksalt (RS) structures, which demonstrated that the ZPR value is proportional to the product of the inverse fourth root of the masses (δεiA(k)1ωλ1(MIMJ)1/21(MIMJ)1/4), confirmed the mass relation for monatomic crystals, and also extended the mass rule to non-monatomic crystals, as shown in Fig. 7. Furthermore, it was found that ZPR is not mainly governed by an inverse square root of the mass, but other characteristics of the materials, such as the crystal structure, electronic structure, as well as polar correction, are also essential for ZPR. Shang and Yang88 also systemically studied the ZPR of the HOMO–LUMO gap for 32 molecules and find that the ZPR does not only relate to the atomic masses but quite relates to the electronic structure properties of the molecules.

FIG. 7.

Relationship between the ZPR of the direct bandgap (ZB structure) and the mass of the constituting atoms. Reprinted figure with permission from Shang et al., J. Phys. Chem. C 125, 6479–6485 (2021). Copyright 2021 The American Chemical Society.

FIG. 7.

Relationship between the ZPR of the direct bandgap (ZB structure) and the mass of the constituting atoms. Reprinted figure with permission from Shang et al., J. Phys. Chem. C 125, 6479–6485 (2021). Copyright 2021 The American Chemical Society.

Close modal

Electron–phonon interactions can also result in the formation of polarons, which are formed by the excess carrier, that induce a distortion of the crystal lattice and in turn promotes localization, so the polaronic corrections to the band renormalization also need to be carefully assessed. In recent work, a unified approach to polarons and ZPR has been presented using a self-consistent many-body Green’s function theory.106 This approach reduces to the AHC theory when a large polaron exists, and it reduces to the ab initio polaron equations107 when a small polaron exists. The study for LiF shows that the polaron localization effects could dominate over the standard AHC theory.106 

It is now understood that e–ph interactions play an important role in predicting many chemical and physical properties, and first principles calculations of e–ph interactions have provided quantitative and deep insights in many areas of condensed matter physics from metals to semiconductors, from carrier mobility to polaron formation, and from optical properties to topological phase transitions. In this perspective, we discussed the theoretical formalism for describing the e–ph interaction in both the perturbative approach and the nonperturbative adiabatic approach, and we also reviewed recent calculations in many materials. On the one hand, the perturbative method can include the dynamical effect by moving beyond the adiabatic approximation; on the other hand, the finite-difference approach enabled the e–ph calculation to move beyond semilocal DFT and the harmonic approximation.

A further application for e–ph interaction will include the interplay between e–ph and many complex physical and chemical phenomena; for example, the chemical reactions can be influenced by inherent dissipative processes, which involve energy transfer between the conduction electrons and the ionic motion. In photocatalysis, it would be a great breakthrough if one can couple electronic excitation with the vibration mode, which corresponds to a desired reaction coordinate, which could offer great promise for a significant increase of reaction selectivity since the e–ph interaction can make excitation energy efficiently transfer to the corresponding reaction pathway. In addition, for the e–ph interaction in the magnetic systems, there are still also many open questions.

Besides the application, the development of efficient computational methods is also demanding. In the ab initio e–ph calculation, the key technical challenge is getting the e–ph coupling matrix elements for different electronic states and phonon modes. In the reciprocal-space formalism, the calculation of each e–ph matrix element requires an individual density functional perturbation theory (DFPT) calculation, which can become computationally expensive quite rapidly, for example, in the calculation of charge carrier dynamics,108 since the e–ph matrix elements at a large number of electron and phonon wave vectors in the Brillouin zone are needed, the e–ph calculation can become a computational bottleneck. In order to overcome this problem, various interpolation techniques have been proposed in the previous work: Giustino et al.4 suggested to Fourier-transform the reciprocal-space electron–phonon coupling elements to real-space using Wannier function (WF) interpolation, which has been successfully applied in recent e–ph calculations.1,12 However, the generation of WF becomes challenging for structurally complex systems with large supercells, in which the required initial guess for constructing the WFs is not apparent. In order to address this problem, the direct Fourier interpolation of the perturbed potential approach73,109 has been taken by the ABINIT software,109 which can interpolate the response potential at an arbitrary reciprocal point using the inverse Fourier transformation. Agapito et al.13 presented an interpolation method by employing the atomic orbital (AO) wave function, which removes the trial-and-error steps needed to build the WF and is ideal for e–ph calculation in complex materials. Since the AO-based interpolation for e–ph matrix elements works, one may wonder whether one can directly use the AO basis set to represent the e–ph matrix element in real space, just following the way of using the AO basis set to represent the electronic Hamiltonian110–112 and the force constants;63 furthermore, one could even reformulate the e–ph interaction in a fully localized representation based on WF or AO without going back to Bloch space by Fourier interpolation, as also suggested in Ref. 113. In addition, we can also go beyond AHC by adding the e–ph self-energy into the self-consistency GW framework to see the influence of electron–electron and electron–phonon on the same footing.

Finally, we expect that the first principles e–ph interaction calculations will be applied in various complex systems and gain deep insight into the interplay between e–ph and complex physical phenomena.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 22003073, T2222026, and 22288201). H.S. would like to thank Professor Matthias Scheffler, Professor Yi Luo, Professor Patrick Rinke, Dr. Gabriel Antonius, and Professor Samuel Poncé for inspiring discussions.

The authors have no conflicts to disclose.

Honghui Shang: Data curation (lead); Formal analysis (lead); Methodology (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Jinlong Yang: Conceptualization (lead), Supervision (lead), Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

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