Fröhlich electron-phonon vertex from first principles
FFr¨ohlich electron-phonon vertex from first principles
Carla Verdi and Feliciano Giustino ∗ Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
We develop a method for calculating the electron-phonon vertex in polar semiconductors andinsulators from first principles. The present formalism generalizes the Fr¨ohlich vertex to the caseof anisotropic materials and multiple phonon branches, and can be used either as a post-processingcorrection to standard electron-phonon calculations, or in conjunction with ab initio interpolationbased on maximally localized Wannier functions. We demonstrate this formalism by investigatingthe electron-phonon interactions in anatase TiO , and show that the polar vertex significantlyreduces the electron lifetimes and enhances the anisotropy of the coupling. The present workenables ab initio calculations of carrier mobilities, lifetimes, mass enhancement, and pairing inpolar materials. PACS numbers: 71.38.-k, 63.20.dk
The electron-phonon interaction (EPI) is a cornerstoneof condensed matter physics, and plays important rolesin a diverse array of phenomena. Recent years have wit-nessed a surge of interest in ab initio calculations of EPIs,leading to new techniques and many innovative applica-tions in the case of metals and non-polar semiconduc-tors [1–12]. In contrast to this fast-paced progress, in thecase of polar semiconductors and insulators the study ofEPIs from first principles has not gone very far, owingto the prohibitive computational costs of EPI calcula-tions for polar materials. For example, a fully ab initio calculation of the carrier mobility of a polar semiconduc-tor has not been performed yet, while such calculationshave recently been reported for non-polar semiconduc-tors such as silicon [13] and graphene [14]. Given thefast-growing technological importance of polar semicon-ductors, from light-emitting devices to transparent elec-tronics, solar cells and photocatalysts [15–17], developingaccurate and efficient computational methods for study-ing EPIs in these systems is of primary importance.At variance with metals and non-polar semiconduc-tors, in polar materials two or more atoms in the unitcell carry nonzero Born effective charge tensors [18]. Asa consequence, the fluctuations of the ionic positions cor-responding to longitudinal optical (LO) phonons at longwavelength generate macroscopic electric fields which cancouple strongly to electrons and holes, leading to theso-called Fr¨ohlich interaction [19]. Up to now this in-teraction has not been taken into account in ab initio calculations of EPIs; the two key obstacles towards adescription of Fr¨ohlich coupling from first principles are(i) the Fr¨ohlich coupling was designed to describe sim-ple isotropic systems with one LO phonon, and (ii) theelectron-phonon vertex diverges for q →
0, where q is thephonon wavevector. The first obstacle relates to the fun-damental question on how to define the Fr¨ohlich couplingin the most general way. The second obstacle rendersfirst-principles calculations extremely demanding, sincea correct description of the singularity requires a veryfine sampling of the Brillouin zone. In this work we address the challenges (i) and (ii) aboveby developing a general formalism for first-principles cal-culations of the Fr¨ohlich vertex. Our strategy consists inseparating the short-range and the long-range contribu-tions to the electron-phonon matrix elements, and iden-tifying the Fr¨ohlich coupling with the long-range com-ponent. We translate our formalism into a powerfulcomputational scheme, whereby the short-range compo-nent is calculated using state-of-the-art Wannier-Fourierelectron-phonon interpolation [20], and the singular cou-pling is calculated using the Born effective charges andthe high-frequency dielectric permittivity tensor. As afirst demonstration of this approach we calculate carrierlifetimes in anatase TiO .The Fr¨ohlich model [19] describes the interaction ofan electron in a parabolic band with a dispersionless LOphonon of frequency ω LO . The electron is in an isotropicdielectric medium with static and high-frequency per-mittivities (cid:15) and (cid:15) ∞ , respectively. In this model theelectron-phonon coupling matrix element takes the form: g q = i | q | (cid:20) e πε πN Ω ¯ hω LO (cid:18) (cid:15) ∞ − (cid:15) (cid:19)(cid:21) , (1)where q is the phonon wavevector, Ω the unit cell vol-ume, N the number of unit cells in the Born-von K´arm´ansupercell, and e , ε , and ¯ h are the electron charge, vac-uum permittivity, and reduced Planck constant, respec-tively. Equation (1) shows that the Fr¨ohlich coupling g q diverges at long wavelengths, q →
0. This singular-ity poses a challenge to ab initio calculations of EPIs inpolar materials.In general the vertex describing electron-one phononinteractions can be expressed via the coupling matrix el-ement g mnν ( k , q ) = (cid:104) ψ m k + q | ∆ q ν V | ψ n k (cid:105) . This quan-tity has the meaning of probability amplitude for thescattering between the initial electronic state | ψ n k (cid:105) andthe final state | ψ m k + q (cid:105) via the perturbation ∆ q ν V dueto a phonon with crystal momentum q , branch ν andfrequency ω q ν . The matrix elements g mnν ( k , q ) can becalculated starting from density functional perturbation a r X i v : . [ c ond - m a t . m t r l - s c i ] O c t theory [21], and have been employed to investigate manyproperties involving EPIs, for example the electron ve-locity renormalization [22] and lifetimes [23, 24], phononsoftening [3] and lifetimes [25, 26], phonon-assisted ab-sorption [27, 28], critical temperature in conventional su-perconductors [10, 29, 30], and resistivity [13, 14]. Thekey ingredient of all these calculations is the evaluationof g mnν ( k , q ) on extremely dense Brillouin zone grids,which is computationally prohibitive. This difficultyhas been overcome with the development of an ab ini-tio interpolation strategy for the electron-phonon vertex[20, 31] based on maximally-localized Wannier functions[32]. The approach of Ref. 20 relies on the spatial lo-calization of the scattering potential and of the electronwavefunctions when expressed in a real-space Wannierrepresentation.While the method of Ref. 20 was successfully appliedto metals and non-polar semiconductors [1, 6, 12, 14, 23,28, 33, 34], the same strategy breaks down in the case ofpolar materials. In fact the singularity in Eq. (1) impliesthat the scattering potential is long-ranged in real space,hence the Fr¨ohlich vertex is not amenable to Wannier-Fourier interpolation. By the same token less refinedlinear interpolation strategies are equally inadequate.In order to deal with the polar singularity we separatethe short- ( S ) and long-range ( L ) contributions to thematrix element: g mnν ( k , q ) = g S mnν ( k , q ) + g L mnν ( k , q ) . (2)If all contributions leading to the long-wavelength diver-gence are collected inside g L , then the short-range com-ponent will be regular and amenable to Wannier-Fourierinterpolation. This strategy is analogous to the calcula-tion of LO-TO splittings in polar materials by separatingthe analytical and non-analytical parts of the dynamicalmatrix [35].We now derive an expression for g L starting from thefollowing ansatz : the macroscopic electric field generatedby the nuclei and experienced by the electrons can be ob-tained by associating an electric point dipole p = e Z ∗ · u to each atom, where Z ∗ = Z ∗ αβ is the Born effective charge tensor and u = u α is the displacement fromequilibrium [here and in the following Greek indices in-dicate Cartesian coordinates, and we use the notations( B · c ) α = (cid:80) β B αβ c β , a · B · c = (cid:80) αβ a α B αβ c β ]. Thisnotion draws from the very definition of Born chargesas the sources of the macroscopic polarization [18]. Ouransatz amounts to following similar steps as in the origi-nal work of Fr¨ohlich [19], although we are replacing ionicpoint charges by Born effective charge tensors. A moreformal theory of polar electron-phonon coupling can bedeveloped by starting from a many-body approach [36],and using the analytical properties of the dielectric ma-trix [37]. Since in polar insulators the atomic oscillationsaround equilibrium take place over timescales which aremuch longer than the electronic response time, we canassume following Fr¨ohlich [19] that the electrostatic po-tential generated by the dipole p is screened by the high-frequency (electronic) permittivity. This choice corre-sponds to assuming the adiabatic approximation. In themost general case of anisotropic solid this will be givenby the tensor (cid:15) ∞ = (cid:15) ∞ ,αβ . By solving the anisotropicPoisson’s equation with the dipole p placed at the originof the reference frame we find: V L ( r ) = i πN Ω e πε p · (cid:88) q (cid:88) G (cid:54) = − q ( q + G ) e i ( q + G ) · r ( q + G ) · (cid:15) ∞ · ( q + G ) , (3)where G indicates a reciprocal lattice vector, and thewavevectors q belong to a regular grid of N points in theBrillouin zone. This result is derived in the SupplementalMaterial [38]. Now we consider that one contribution inthe form of Eq. (3) arises from each atom κ in the position τ κ R = τ κ + R , where R denotes a lattice vector. For agiven phonon with wavevector q belonging to the branch ν the atomic displacement pattern is given by ∆ τ ( q ν ) κ R =(¯ h/ N M κ ω q ν ) e i q · R e κν ( q ). In this expression e κν ( q )represents a vibrational eigenmode normalized within theunit cell. If we make the replacement p → e Z ∗ κ ∆ τ ( q ν ) κ R inside Eq. (3), we obtain our main result for the matrixelement g L : g L mnν ( k , q ) = i π Ω e πε (cid:88) κ (cid:18) ¯ h N M κ ω q ν (cid:19) (cid:88) G (cid:54) = − q ( q + G ) · Z ∗ κ · e κν ( q )( q + G ) · (cid:15) ∞ · ( q + G ) (cid:104) ψ m k + q | e i ( q + G ) · r | ψ n k (cid:105) , (4)where the bracket is to be evaluated within the Born-vonK´arm´an supercell. Details about this derivation are pro-vided in the Supplemental Material [38]. If we considerthe more restrictive situation of an isotropic dielectric,we find that Eq. (4) reduces correctly to the Fr¨ohlichvertex in Eq. (1). This result can be obtained by using the relation between the Born charges and the static andhigh-frequency permittivities [39], and by invoking theLyddane-Sachs-Teller relations [40]. Since g L reduces tothe Fr¨ohlich limit under the assumptions used in the orig-inal work [19], Eq. (4) represents the generalization of theFr¨ohlich vertex for ab initio calculations. We note that FIG. 1. (a) Calculated phonon dispersions in anatase TiO along high-symmetry lines in the Brillouin zone. The LO phononsdiscussed in the main text are highlighted by arrows. (b) Calculated electron-phonon matrix elements, with (blue solid lines)and without (red dashed lines) the polar coupling g L from Eq. (4). The calculations using Wannier-Fourier interpolations(lines) are compared to direct DFPT calculations at each wavevector (filled discs). Here we show the gauge-invariant traceof | g | over degenerate states. In the calculation of g mnν ( k , q ) we set the initial electronic state | ψ n k (cid:105) to the bottom of theconduction band at Γ, the final electronic state | ψ m k + q (cid:105) to the bottom of the conduction band, and the phonon branch to bethe highest (LO) optical mode. (c) Spherical average of the electron-phonon matrix elements, 4 πq | g | , with (blue) and without(red) the polar coupling, and using the simple Fr¨ohlich model in Eq. (1) (gray). any choice of the polar electron-phonon coupling whichdid not have exactly the same limit as Eq. (4) for q → phonon self-energy and hence phonon fre-quencies without the correct LO-TO splitting. In fact, inthe long wavelength limit our ansatz leads precisely to theelectron-phonon vertex of Ref. 36. The advantage of ourformulation is that the physics behind the polar singular-ity becomes transparent. One interesting property of ourpolar vertex in Eq. (4) over Eq. (1) is that it naturallytakes into account the periodicity of the lattice, whichis not the case for the original Fr¨ohlich vertex. Further-more we do not need to make any assumptions on whichLO mode should be considered, since our formalism in-corporates seamlessly the coupling to all modes, and thecoupling strength is automatically suppressed whenever Z ∗ κ · e κν ( q ) is transverse to q + G .Taken together Eqs. (2) and (4) provide a practicalrecipe for calculating EPIs in polar materials. In thesimplest approach one could perform calculations involv-ing only the polar coupling g L , in order to determine themagnitude of these effects. In this case the phase factors (cid:104) ψ m k + q | e i ( q + G ) · r | ψ n k (cid:105) can safely be replaced by their q + G → δ mn , and the ensuing calculations of g L become trivial post-processing operations. An exam-ple of this calculation is shown in Fig. S2 in the Supple-mental Material [38]. A more refined strategy consists ofcomputing the complete matrix elements g = g L + g S byexploiting Wannier-Fourier interpolation. In this case weneed to perform the following steps: (i) evaluate the com-plete matrix elements g on coarse Brillouin zone grids;(ii) subtract g L so as to obtain the short-ranged partof the matrix element, g S ; (iii) apply Wannier-Fourierelectron-phonon interpolation to the short-range matrixelement, following Ref. 20; (iv) add up the short-rangepart and the long-range part at arbitrary k and q points after interpolation. This strategy enables the calculationof millions of electron-phonon matrix elements for polarmaterials with ab initio accuracy, and at the computa-tional cost of a standard calculation of phonon disper-sions. In order to correctly capture the electronic phaseswe stress that this procedure requires the interpola-tion of the overlaps in Eq. (4). This is achieved by us-ing the rotation matrices U nm k appearing in the def-inition of maximally localized Wannier functions [32]: | w m R (cid:105) = (cid:80) n k e − i k · R U nm k | ψ n k (cid:105) . Using this definitionand by considering small q + G we obtain the result: (cid:104) ψ m k + q | e i ( q + G ) · r | ψ n k (cid:105) = (cid:104) U k + q U † k (cid:105) mn . (5)The matrices U k are known at every point of the coarsegrid from the calculation of maximally localized Wannierfunctions, and can be obtained at all other points via theinterpolation of the electron Hamiltonian [41].In order to demonstrate our approach we consider theelectron-phonon coupling in a prototypical polar semi-conductor, anatase TiO . Very recently Fr¨ohlich physicshas been studied in this material by means of angle-resolved photoelectron spectroscopy [42]. Figure 1(a)shows the phonon dispersion relations of anatase TiO calculated using Quantum ESPRESSO [43] [44]. The po-lar coupling is manifest in the LO-TO splitting whichcan be seen around the Γ point for the infrared-active E u and A modes (the LO modes are highlighted byarrows in the plot). In Fig. 1(b) we compare represen-tative electron-phonon matrix elements calculated fromDFPT with the result of our polar Wannier-Fourier in-terpolation, along high-symmetry lines in the Brillouinzone. Our interpolated matrix elements were obtainedby performing explicit DFPT calculations on a coarse4 × × FIG. 2. (a) Diagram of the Migdal self-energy Σ used to calculate electron lifetimes. The straight and wiggly lines representthe electron and phonon Green’s functions, respectively. The circles are the electron-phonon matrix elements. (d) Close-up ofthe conduction bands of anatase TiO near the band bottom at Γ, taken as the zero of the energy. (b), (e) Electron linewidthsarising from electron-phonon scattering, ImΣ, along the Γ Z and the Γ X lines respectively, near the band bottom. The energiesof the LO phonons shown in Fig. 1(a) are indicated by vertical dashed lines. (c), (f) Electron lifetimes from (b) and (e). In(b), (e) and (c), (f) the blue solid lines are computed using the complete vertex, while the red dashed lines are obtained usingstandard Wannier interpolation [31]. that our method perfectly reproduces the polar singular-ities at Γ, and the behavior of the matrix element any-where in the Brillouin zone. For comparison in the samefigure we also show the short-range part of the matrix el-ement | g S | (red line) which is clearly non-singular near Γ.We stress that a standard interpolation strategy without taking into account the polar coupling completely failsin reproducing the correct behavior (see Fig. S1 in theSupplemental Material [38]). In order to quantify the im-portance of the polar divergence, in Fig.1(c) we considerthe quantity 4 πq | g | ( q = | q | ). This quantity is ubiqui-tous in electron-phonon calculations since most physicalproperties involve Brillouin zone integrations containing | g mnν ( k , q ) | d q [45]. The short-range component (red)severely underestimates the complete coupling strength,even after the singularity has been lifted by the volume el-ement prefactor 4 πq . Similarly, when using the Fr¨ohlichmodel in Eq. (1), the coupling is significantly overesti-mated (gray). This demonstrates that the incorporationof the long-range coupling using Eq. (2) and (4) is essen-tial for ab initio calculations of EPIs in polar materials.For completeness Figs. S3 and S4 in the SupplementalMaterial show that similar conclusions apply to two otherprototypical polar compounds, GaN and LiF [38].As a first example of application of our method wecalculate the lifetimes of conduction electrons in anataseTiO arising from the EPI. We determine the lifetimes τ n k from the imaginary part of the Migdal electron self-energy Σ n k [Fig. 2(a)], using τ n k = ¯ h/ n k [46]. Weconsider electrons in the conduction band along the Γ Z and the Γ X high-symmetry lines, with energies near theband bottom [Fig. 2(d)]; we evaluate the self-energy using 512,000 inequivalent phonon wavevectors (80 × × A u LOphonon at Γ. We can also resolve a weaker coupling witha threshold of 40 meV, associated with a E u LO phonon.Therefore, for electronic states close to the bottom of theconduction band we find that the coupling is dominatedby the A u and E u LO modes around 100 meV (at 20 K),in agreement with the experimental findings of Ref. [42].We note that in this case electron-electron interactionsdo not contribute to the linewidths (within the G W approximation) since the electron energy is below theenergy thresholds for electron-hole pair generation (thefundamental gap) and for plasmon emission. Turning tothe lifetimes, we see in Fig. 2(c) and (f) that these are es-sentially infinite below the phonon emission theshold, butthey are reduced to ∼ Z to X . This is relatedto the anisotropy of the band structure, and to the factthat the coupling is strongest for small phonon wavevec-tors [38]. For comparison we also show in Fig. 2 calcu-lations performed with the standard Wannier interpola-tion technique [31], which fails to correctly reproduce thematrix elements. The lifetimes are incorrectly enhancedby up to an order of magnitude; at the same time theanisotropy is mostly washed out [red curves in Fig. 2(c)and (f)]. This comparison demonstrates the importanceof the polar singularity in the correct calculation of elec-tron lifetimes in TiO . We expect to find similar effectsin related properties, such as polaron binding energy andcarrier mobilities.In conclusion, we introduced a method for studyingelectron-phonon interactions in polar semiconductors andinsulators. Our method generalizes the Fr¨ohlich theoryvia a consistent description of short-range and long-rangecontributions to the coupling strength. The present for-malism can be employed either for complementing first-principles calculations at no extra cost, or for perform-ing efficient and accurate ab initio calculations usingWannier-Fourier interpolation. We expect that our ap-proach will enable the calculation of many properties be-yond the reach of current methods, including tempera-ture dependent mobilities in polar semiconductors, dy-namics of photoexcited carriers, and superconductivityin doped oxides.This work was supported by the LeverhulmeTrust (Grant RL-2012-001) and the UK Engineeringand Physical Sciences Research Council (Grant No.EP/J009857/1). This work used the ARCHER UK Na-tional Supercomputing Service via the AMSEC Leader-ship project, and the Advanced Research Computing fa-cility of the University of Oxford. Note added. — Recently a related study of polar electron-phonon couplings was reported [47]. ∗ [email protected][1] F. Giustino, M. L. Cohen, and S. G. Louie, Nature (Lon-don) , 975 (2008).[2] G. 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Carla Verdi and Feliciano Giustino
Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
POINT DIPOLE IN ANISOTROPIC DIELECTRICMEDIUM
Here we derive Eq. (3) of the main text. Let us con-sider a point charge Q placed at the position τ . We alsoinclude a uniform compensating background − Q/N
Ω sothat the supercell is neutral overall (the supercell consistsof N unit cells). The medium is an anisotropic dielec-tric with the permittivity tensor (cid:15) = (cid:15) αβ . The potentialgenerated by this point charge and the background isobtained as the solution of the Poisson equation [1]:( ∇· (cid:15) ·∇ ) φ ( r ; τ ) = − ε X T (cid:20) Q δ ( r − τ − T ) − QN Ω (cid:21) , (S1)where T are the lattice vectors of the Born-von K´arm´ansupercell. The electrostatic potential φ is periodic in thesupercell, therefore it can be expanded in planewaves as: φ ( r ; τ ) = P q , G e i ( q + G ) · r φ ( q + G ; τ ), where the q vec-tors belong to a regular grid of N points in the Brillouinzone, and the G vectors belong to the reciprocal lattice ofthe crystal unit cell. By replacing this expansion insideEq. (S1) we obtain, apart from a constant: φ ( r ; τ ) = 4 π Ω Q πε N X q X G = − q e i ( q + G ) · ( r − τ ) ( q + G ) · (cid:15) · ( q + G ) . (S2) The electrostatic potential generated by a dipole p = Q u centered at the origin of the reference frame is simplygiven by φ dip ( r ) = lim u → φ ( r ; u ) − φ ( r ; 0): φ dip ( r ) = − i π Ω 14 πε N X q X G = − q p · ( q + G ) e i ( q + G ) · r ( q + G ) · (cid:15) · ( q + G ) , (S3)which is the same as Eq. (3) of the main text after setting V L = − eφ . LONG-RANGE COMPONENT OF THEELECTRON-PHONON MATRIX ELEMENT
Here we comment on the derivation of Eq. (4) in themain text. The perturbation potential entering the ma-trix element g L is obtained by replacing the dipole p inEq. (S3) with e Z ∗ κ · ∆ τ ( q ν ) κ R , for each atom κ in the position τ κ R = τ κ + R . Here R is a lattice vector within the Born-von K´arm´an supercell. As stated in the main text thedisplacement of each atom in the vibrational eigenmode( q , ν ) is given by ∆ τ ( q ν ) κ R = (¯ h/ N M κ ω q ν ) e i q · R e κν ( q ).By combining these relations with Eq. (S3) we obtainimmediately: V L q ν ( r ) = i π Ω e πε X κ (cid:18) ¯ h N M κ ω q ν (cid:19) X G = − q ( q + G ) · Z ∗ κ · e κν ( q ) e i ( q + G ) · ( r − τ κ ) ( q + G ) · (cid:15) · ( q + G ) . (S4)Here we made use of the completeness relation: P R e i ( q − q ) · R = N δ qq , with δ the Kronecker symbol. The matrix elements is calculated as h ψ m k + q | V L q ν ( r ) | ψ n k i : g L mnν ( k , q ) = i π Ω e πε X κ (cid:18) ¯ h N M κ ω q ν (cid:19) X G = − q ( q + G ) · Z ∗ κ · e κν ( q )( q + G ) · (cid:15) · ( q + G ) h ψ m k + q | e i ( q + G ) · ( r − τ κ ) | ψ n k i . (S5)As usual the braket is to be evaluated within the Born-von K´arm´an supercell. Equation (4) in the main text is obtained from this expression by taking the limit e − i ( q + G ) · τ κ → g L given in Eq. (S5) couldbe altered to include only one reciprocal lattice vectorin the summation, namely ˜ G q such that | q + ˜ G q | =min G | q + G | . This would correspond to retaining onlythe residue of the pole in reciprocal space. In practicethis choice is not convenient since it leads to a Fr¨ohlichvertex with discontinuous q -derivatives at the Brillouinzone boundaries. Instead, in order to minimize the num-ber of G vectors in Eq. (S5) while preserving a smooth(i.e. infinitely differentiable) Fr¨ohlich matrix element, itis sufficient to truncate the sum to the smallest shell of G vectors which surrounds q . In those cases where one isinterested in q vectors within the first Brillouin zone, theabove choice corresponds to retaining only the handful ofreciprocal lattice vectors which define the first zone. ANISOTROPY OF THE ELECTRONLINEWIDTHS
The electron linewidths are obtained from the imagi-nary part of the self-energy Σ arising from the electron-phonon coupling, calculated within the Migdal approxi-mation [2]:Σ n k = X mν q | g mnν ( k , q ) | (cid:20) n q ν + f m k + q (cid:15) n k − (cid:15) m k + q + ¯ hω q ν − iη + n q ν + 1 − f m k + q (cid:15) n k − (cid:15) m k + q − ¯ hω q ν − iη (cid:21) . (S6)Here f m k + q and n q ν are the Fermi-Dirac and theBose-Einstein occupation numbers, respectively, (cid:15) n k and (cid:15) m k + q are electron energies, ¯ hω q ν is the energy of aphonon with wavevector q and polarization ν , and η is asmall broadening (10 meV). Figures 2(b) and (e) of the main text show the calcu-lated linewidths of conduction electrons in anatase TiO near the band bottom at Γ. These linewidths are highlyanisotropic, varying by a factor of three when goingfrom the Z point to X . This feature is related to theanisotropy of the band structure, and to the fact thatthe coupling is strongest for small phonon wavevectors.In fact, after we take the limit of vanishing temperaturein Eq. (S6), the imaginary part of the self-energy yields afactor δ ( (cid:15) n k − (cid:15) m k + q − ¯ hω q ν ) that signifies the energy con-servation in the scattering process. In the direction Γ X ,the large curvature of the band makes it possible to haveelectronic transitions with small momentum transfer. Inthe direction Γ Z the small curvature requires electronictransitions with larger momentum transfer in order toconserve energy. As the polar matrix element goes as1 /q , the different curvatures lead to enhanced linewidthsfor electrons with momentum along Γ X , as compared toelectrons with momentum along Γ Z . This effect is notcorrectly captured by using a standard Wannier-Fourierinterpolation [red lines in Fig. 2(b) and (e) in the maintext]. In particular, the linewidths are systematically un-derestimated in the standard interpolation, and the erroris larger for the linewidths along Γ X , as they involve cou-pling to phonons with small wavevectors. [1] L. D. Landau and E. M. Lifshits, Electrodynamics of Con-tinuous Media (Pergamon, Oxford, 1960).[2] G. Grimvall,
The Electron-Phonon Interaction in Metals (North-Holland, New York, 1981).[3] F. Giustino, M. L. Cohen, and S. G. Louie, Phys. Rev. B , 165108 (2007). FIG. S1. (a) Calculated electron-phonon matrix elements using our polar Wannier-Fourier interpolation of Eq. (2) in themain text (blue solid lines) and the standard interpolation [3] (red dashed lines). The calculations using Wannier-Fourierinterpolations (lines) are compared to direct DFPT calculations at each wavevector (filled discs) as in Fig. 1(b) in the maintext. (b) Spherical average of the electron-phonon matrix elements, 4 πq | g | , with the polar Wannier-Fourier interpolation (blueline) and with the standard interpolation (red line). The standard interpolation corresponds to applying the Wannier-Fourierinterpolation technique of Ref. [3] to the entire electron-phonon vertex g on the l.h.s. of Eq. (2). The matrix element g fromdensity-functional perturbation theory goes as 1 / | q | for q →
0, and vanishes at q = 0 due to the periodic boundary conditions.As a result the Wannier-Fourier method effectively tries to interpolate a strongly fluctuating function, leading to a spurious dipat q = 0 and unphysical behavior elsewhere in the Brillouin zone. This problem is solved by separating the vertex into long-and short-range components, and performing Wannier-Fourier interpolation only on the short-range part g S .FIG. S2. (a), (c) Electron linewidths in anatase TiO arising from electron-phonon scattering, ImΣ, along the Γ Z and the Γ X lines respectively, near the band bottom [as in Fig. 2(b) and (e) of the main text]. The energies of the LO phonons shown inFig. 1(a) of the main text are indicated by vertical dashed lines. (b), (d) Electron lifetimes from (a) and (c). In (a), (c) and(b), (d) the blue solid lines are computed using the complete electron-phonon vertex g in Eq. (2), while the black dashed linesare obtained by using only the long-range part of the electron-phonon vertex, g L in Eq. (2). The two approaches yield verysimilar results in this case. The agreement between calculations performed using g or g L implies that the electron lifetimes inanatase TiO are dominated by polar interactions with optical phonons. FIG. S3. (a) Calculated phonon dispersions in w-GaN along high-symmetry lines in the Brillouin zone. The phonons areinterpolated from a 6 × × × × | g | over degenerate states. In the calculation of g mnν ( k , q ) we set the initial electronic state | ψ n k i to the top of thevalence band at Γ, the final electronic state | ψ m k + q i to the top of the valence band, and the phonon branch to be the highest(LO) optical mode. (c) Spherical average of the electron-phonon matrix elements, 4 πq | g | , with the present method (blue)and with the standard interpolation (red). These calculations were performed using the lattice parameter a =3.158 ˚A.FIG. S4. (a) Calculated phonon dispersions in rocksalt LiF along high-symmetry lines in the Brillouin zone. The phonons areinterpolated from a 6 × × × × | g | over degenerate states, we set the initial electronic state to the top of the valence band at Γ, thefinal electronic state to the top of the valence band, and the phonon branch to be the highest (LO) optical mode. (c) Sphericalaverage of the electron-phonon matrix elements, 4 πq | g | , with the present method (blue) and with the standard interpolation(red). These calculations were performed using the lattice parameter aa