Boosting the efficiency of ab initio electron-phonon coupling calculations through dual interpolation
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Boosting the efficiency of ab initio electron-phonon coupling calculations through dual
interpolation
Anderson S. Chaves,1, 2 Alex Antonelli,2 Daniel T. Larson,3 and Efthimios Kaxiras3, 1
1 John
A. Paulson School of Engineering and Applied Sciences,
Harvard University, Cambridge, Massachusetts, 02138, USA
2 Gleb Wataghin Institute of Physics and Center for Computing in Engineering & Sciences,
University of Campinas, PO Box 13083-859, Campinas, SP, Brazil
3 Department of Physics, Harvard University, Cambridge, Massachusetts, 02138, USA
(Dated: July 14, 2020)
The coupling between electrons and phonons in solids plays a central role in describing many
arXiv:2006.16954v2 [cond-mat.mtrl-sci] 10 Jul 2020
phenomena, including superconductivity and thermoelecric transport. Calculations of this coupling
are exceedingly demanding as they necessitate integrations over both the electron and phonon
momenta, both of which span the Brillouin zone of the crystal, independently. We present here an ab
initio method for efficiently calculating electron-phonon mediated transport properties by dramatically
accelerating the computation of the double integrals with a dual interpolation technique that combines
maximally localized Wannier functions with symmetry-adapted plane waves. The performance gain
in relation to the current state-of-the-art Wannier-Fourier interpolation is approximately 2ns × M,
where ns is the number of crystal symmetry operations and M, a number in the range 5 − 60, governs
the expansion in star functions. We demonstrate with several examples how our method performs
some ab initio calculations involving electron-phonon interactions.
PACS numbers: 71.15.Nc,36.40.-c,72.80.Ga
The electron-boson coupling is ubiquitous in phys- phonon momenta, both spanning the entire Brillouin
ical phenomena through the whole spectrum of the Zone (BZ), independently. This double integration re-
physics of solids. In particular, electron-phonon (el-ph) quires very fine sampling of the electron and phonon
coupling plays a fundamental role in the renormaliza- wavevectors to achieve numerical convergence, which
tion of electronic and vibrational energy scales, thus represents the bulk of the computational burden. The
determining the coupling itself, with important con- application of crystal symmetry properties for the full
sequences for transport properties[1–4]. Conventional integration of TP, which depend directly on the el-ph
superconductivity is a case in point, where the interac- matrix elements, is not allowed. Even if the wavevector
tions between electrons and phonons give rise to Cooper k in the el-ph matrix element lies within the symmetry-
pairing.[5] Other examples include the temperature de- reduced portion (irreducible wedge) of the BZ, the trans-
pendence of electronic conductivity and thermoelectric ferred momenta k + q spread out in the whole zone
transport properties,[6, 7] as well as phonon-assisted because q belongs to an uniform mesh. Dense sampling
optical absorption in indirect-gap semiconductors.[8] In- of the BZ is prohibitive, the reason being the connec-
terest in thermoelectrics has increased rapidly in recent tion between transferred momenta with equally dense
years, partly due to the expectation of discovering higher k-point meshes.
figure-of-merit materials, boosted by the nanotechnol- Specialized numerical techniques have been devel-
ogy revolution. [9] A major goal of theory has been oped to address this problem. One attempt to simplify
to predict thermoelectric transport properties directly the brute-force integration is based on pre-screening of
from atomistic-scale calculations without any adjustable subsets of the reciprocal space, such as relevant con-
or empirical parameters,[4, 6, 7] particularly combining duction pockets within the neighborhood of band ex-
density functional theory (DFT) and many-body pertur- trema that significantly contribute to the integral. Al-
bation theory. Despite great advances, such calculations ternatively, interpolation schemes, such as linear[12] or
remain very demanding and still pose a challenge, even Wannier-based[1, 13] ones, have been developed to im-
for simple crystalline bulk systems. prove convergence. In particular, the interpolation of
A well-established approach, using a first-principles the el-ph matrix elements on the basis of Wannier func-
description of el-ph coupling, relies on solving the el- tions introduced by Giustino, Cohen, and Louie,[13] has
ph matrix elements through density-functional pertur- proven very successful in calculating properties with
bation theory (DFPT)[11]. The el-ph matrix elements, more favorable scaling than using directly the DFPT ap-
g(k, k + q) = (hk + q| δq,β V KS |ki)uc , correspond to the proach. In this Letter, we present a novel method for
electronic scattering calculated from the variations of the computation of el-ph mediated TP which uses two
the Kohn-Sham (KS) potential due to phonon pertur- interpolations: the first one is the usual Wannier-Fourier
bations with wavevector q and branch index β within (W-F) interpolation, followed by a second one based
the unit cell (uc). To obtain transport properties (TP), on symmetry-adapted plane-waves (PW). Our method
| g(k, k + q)|2 must be integrated over the electron and leads to an efficient sampling of extremely fine, homo-
2
8
10 280
(a) (b) 3
2.4
Lorenz function, Λ (10-8WΩ/K2)
Thermal conductivity, κ (W/mK)
Λn
Electrical Conductivity, σ (S/m)
Seebeck Coefficient, S (µV/K)
140 Sp
107 2.4
2.2
0
Λp
1.8
106 2 -140 Sn
σn 1.2
5 1.8 -280 κn
10
σp
0.6
-420 κp
4
1.6
10
0
0.25 0.5 1 2 4 0.25 0.5 1 2 4
Carrier Concentration (1020 cm-3) Carrier Concentration (1020 cm-3)
Figure 1. Thermoelectric transport properties TP for p- and n-type doped Si polycrystals: (a) Electrical conductivity, σ, (red)
compared with experimental values[10] (filled black symbols) and Lorenz function, Λ, (blue); (b) Seebeck coefficient, S, (red)
compared with experimental data[10] (filled black symbols) and thermal conductivity due to the carriers, κ, (blue) calculated
from the relaxation times due to scatterings by el-ph coupling and ionized impurities (see text for details).
geneous k and q grids, with a significant decrease in Wigner-Seitz (WS) supercell with Born-von-Kármán
computational cost compared to W-F calculations with (BvK) periodic boundary conditions, Uk0 (uq0 ) is a di-
a single interpolation. agonalizer matrix over k0 (q0 ) indices from Wannier to
To illustrate the capability of our method, we con- Bloch representations for electrons (phonons) and the
sidered realistic properties of solids (see SM for more el-ph matrix elements in the Wannier representation are
details). Fig. 1 shows the calculated thermoelectric TP given by
for Si polycrystals using our dual interpolation method
1
to calculate the relaxation time within the Boltzmann g(Re , R p ) =
Np ∑ e−i(k·Re +q·R p ) U†k+q g(k, q)Uk u−q 1 ,
transport equation (BTE). We calculated the electrical k,q
conductivity, σ, Seebeck coefficient, S, Lorenz function, (2)
Λ, and thermal conductivity due to the carriers, κ, as Uk is a unitary matrix corresponding to the rotation
functions of carrier concentration. Our results for n of the corresponding electronic states from Bloch to
and p-doped Si polycrystals agree reasonably well with Wannier representations within the gauge of maximally
available experimental TP. For these calculations we also localized Wannier functions (MLWF),[14] and uq is a uni-
included the scattering by ionized impurities, within the tary rotation matrix from Bloch to MLWF for phonons.
Brooks-Herring theory (see SM), considering in all cal- The strength of the W-F interpolation method is the fact
culations a fixed unitary ratio between impurities and that one only needs to perform calculations over the
carrier concentrations, which are the only input param- initial coarse k, q meshes, and then can use Eq. (1) to
eter along with the crystal structure. Fig. 2 shows the determine g(k0 , q0 ) on finer k0 , q0 meshes. For this, we
results of phonon-assisted optical absorption for Si at neglect the matrix elements outside the WS supercell
296K and 78K as a function of the photon energies. Our generated from the initial coarse BZ mesh.
calculations, based on our dual interpolation method The accuracy of W-F calculations strongly depends
and the theory developed by Hall, Bardeen and Blatt (see on the spatial localization of g(Re , R p ) within Eq. (1). A
[8]), are in good agreement with experimental results. more detailed analysis suggests g(Re , R p ) should decay
Before presenting our method we review the basic in the variable Re at least with the rapidity of MLWFs.
concept and analyze the advantages and drawbacks of For Re = 0, g(0, R p ) decays with R p due to the screened
W-F interpolation. Using W-F interpolation, the el-ph Coulomb interaction of the dipole potential generated by
matrix elements g(k, q) can be calculated on coarse k, q atomic displacement. Thus the localization of g(Re , R p )
meshes and then interpolated onto much finer k0 , q0 depends strongly on the dielectric properties of the sys-
meshes through simple matrix multiplication.[13] The tem. In particular, Friedel oscilations[16] (|R p |−3 ) and
matrix elements on the fine mesh are given by quadrupole behavior[17] (|R p |−4 ) are intimately related
to the screening properties of metals and nonpolar semi-
1 0 0
g(k0 , q0 ) =
Ne ∑ ei(k ·Re +q ·R p ) Uk0 +q0 g(Re , R p )U†k0 uq0 , conductors, respectively.
Re ,R p Despite the advantages of the W-F interpolation
(1) and its more favorable scaling, there are still some
where Re and R p are primitive lattice vectors of the drawbacks. The method is computationally intensive
3
105 interpolation over the whole k-space on a finer grid.
296K Such an interpolation problem may present severe dif-
104 ficulties if one uses an inadequate basis set. We show
Absorption Coefficient, α (cm-1)
78K below that such an interpolation over the entire BZ, com-
3
10 bined with periodic or other boundary conditions, can
be properly constructed from a basis set possessing the
102 appropriate crystal symmetry.
1
10
100
As the full symmetry of the crystal’s reciprocal space
is contained in the function f (k̄l ), it is natural to use
10-1 symmetry-adapted PW or star functions, Υm (k0 ), as a
basis set to Fourier expand f [19]:
10-2
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 M
Photon Energy (eV) f˜(k0 ) = ∑ am Υm (k0 ) , (3)
m =1
Figure 2. Phonon-assisted optical absorption, α, of Si at
296K and 78K as a function of photon energies. Our results
where Υm (k0 ) = n1s ∑{υ} exp(i (υRm ) · k0 ) , with the
(blue and magenta) are in good agreement with corresponding
experimental results (black)[15]. sum running over all ns point group symmetry oper-
ations {υ} on the direct lattice translations, Rm . Star
functions obey orthogonality relations involving BZ
summations[19], are totally symmetric under all point-
when many k/q points are needed to achieve con-
group operations, and are ordered such that the mag-
verged values for TP. The main computational oper-
nitude of Rm is nondecreasing as m increases, defining
ations are the simple matrix multiplications shown in
each star function to a given shell of lattice vectors. By
Eq. 1, with a computational complexity of O(n3 ) for taking into account the symmetry, it is expected that
classical computation, where n is the matrix size in- this expansion would converge much faster than using
0 q0
volved. For final dense grids with N kf (N f ) k0 (q0 ) a regular Fourier expansion. Following the approach
points, the number of floating-point operations reach proposed by Shankland-Koelling-Wood[20, 21], we take
0 q0 the number of star functions in the expansion, M, to be
≈ N kf × N f n3 . To reduce the computational cost of greater than the number of data points (M > nk̄ ). We
Wannier-based calculations, some strategies have been then require the fit function, f˜, to pass through the data
adopted, including double grid schemes[6] or (quasi) points and use the extra freedom from additional basis
Monte-Carlo (MC) integrations.[18] In the former, only functions to minimize a spline-like roughness functional
the bandstructure and phonon dispersion are calcu- in order to suppress oscillations between data points,
lated over an ultrafine grid (N f × N f × N f ), while el-ph resulting in a well behaved function throughout the BZ.
matrix elements are computed over a moderate grid
(Nel − ph × Nel − ph × Nel − ph , with Nel − ph = s × N f and
s = 1/2, 1/3) and extrapolated to the ultrafine grid as- We adopt the spline-like roughness functional defined
suming that the el-ph coupling function is smooth. The by Pickett, Krakauer and Allen [22],
drawbacks here consist of the extrapolation which is
often fraught with risk, and the very modest gain fac- M
tor of the method. On the other hand, by using (quasi) Π= ∑ | a m |2 ρ ( R m ) (4)
MC integration, one has to test very dense sets of ran- m =2
dom (or quasi-random) k or q-points, which is a serious 2 2
drawback.
Rm
with ρ( Rm ) = 1 − c1 Rmin + c2 ( RRmin
m 6
) , where
Our method proceeds as follows. For clarity, we de-
scribe the procedure for doing the partial q0 integration Rm = |Rm |, Rmin is the magnitude of the smallest
first, but one can easily switch the order of integration. nonzero lattice vector, and c1 = c2 = 3/4. Such a func-
We begin by computing g(k, q) over coarse k and q tional is more physically appealing than the original
meshes. Next, using W-F interpolation, we determine functional proposed by Shankland-Koelling-Wood, in
g over a finer q0 mesh and perform the partial integra- the sense that departures of f˜ is minimized from its
tion at each of the nk̄ irreducible k-points, k̄l , corre- mean value, a1 , instead of zero. The main problem in
sponding to a moderate regular k-mesh (kr ). Thus we the expansion by star functions in Eq. (3) is the determi-
obtain a function f (k̄l ) ∝ BZ | g(k̄l + q0 , k̄l )|2 dq0 con-
R
nation of the Fourier coefficients, am . Thus, a Lagrange
taining first-principles el-ph coupling properties defined multiplier method can be used toward this goal, once
at selected high-symmetry points in the corresponding the problem has been reduced to minimizing Π sub-
k-space. Given f (k̄l ), the next goal is to find a smooth ject to the constraints, f˜(k̄l ) = f (k̄l ), in relation to am .4
Consequently, the result of this minimization is following point group operations of crystal symmetry.
The corresponding translation vectors can be given as
n −1
(
ρ( Rm )−1 ∑l =k̄ 1 λ∗l Υ∗m (k̄l ) − Υ∗m (k̄nk̄ ) , m > 1, R = u1 a1 + u2 a2 + u3 a3 , in which a1 , a2 , a3 are related
am = M to the crystal’s direct primitive vectors. Such points
f (k̄nk̄ ) − ∑m =2 am Υm ( k̄nk̄ ), m = 1,
are generated inside a sphere with a radius defined as
(5) 0
R = 3 · nk̄ · ns · M · Ω/4π, in which Ω is the volume
p3
in which the Lagrange multipliers, λ∗l , can be evaluated
from of the unit cell. Consequently, R0 determines the full ex-
tension of the real space and can be properly increased,
nk̄ −1 for example, by increasing M, the number of star func-
f (k̄ p ) − f (k̄nk̄ ) = ∑ H pl λ∗l , (6) tions per k-point. In order to capture crystal anisotropy,
l =1 the extension of the real space can be determined for
each crystal direction, defining spheres for each crys-
with tallographic axis with √ the maximum radius given by
M Υm (k̄ p ) − Υm (k̄nk̄ ) Υ∗m (k̄l ) − Υ∗m (k̄nk̄ )
Rmax (t) = I NT ( R0 · bt · bt ) + 1, where bt are the re-
H pl = ∑ , spective reciprocal primitive vectors, with t = {1, 2, 3},
m =2
ρ( Rm ) and I NT ( x ) takes the largest integer number that does
(7) not exceed the magnitude of x.
a positive-definite symmetric matrix that can be deter- The star functions are ordered in such a way that the
mined once for a given crystal problem and can be easily magnitude of Rm is nondecreasing as m increases. Thus,
crafted numerically. a 3D array containing all vectors are sorted considering
Once the Fourier coefficients are determined, a repre- their concentric radius, r, from the sphere center defined
sentation of f˜ is generated, which can be written more for each axis, and provided that all vectors, R, have
clearly as a linear mapping of the W-F data, different star functions, m. The magnitude of each Rm
vector is defined through the metric tensor formalism.
nk̄ −1
For all Rm in the Bravais lattice, the reciprocal lattice
f˜(k0 ) = ∑ J (k̄l , k0 )[ f (k̄l ) − f (k̄nk̄ )] , (8) is characterized by a set of wavevectors k, such that,
l =1
e2πik·Rm = 1. Given Rm and k in the same direction,
where J is the interpolation formula given by the magnitude of the vector k in the reciprocal space
is given by |k| = (k1 u1 + k2 u2 + k3 u3 )/r = (nint (1) +
nk̄ −1 M
[Υ∗m (k̄ p ) − Υ∗m (k̄nk̄ )]Υm (k0 ) nint (2) + nint (3))/r, where nint (t) = 1, 2, ..., k max (t) are
J (k̄l , k0 ) = ∑ ∑ ρ( Rm )H pl
, (9) integer numbers with k max (t) = 2Rmax (t) + 1. To de-
p =1 m termine all k vectors from Rm , a 3D FFT is performed.
In practice, k max (t) defines the number of data points
which transforms one k-mesh into another one, that is, on each dimension and should be carefully taken as
k̄l → k0 . This is the main result of our approach, which the product of small primes in order to improve the
allows great computational savings by transforming the efficiency of FFT.
W-F data obtained over the k-mesh of irreducible points In fact, the FFT computational complexity is
(k̄l ) into a homogeneous dense grid (k0 ) that is larger O( N log N ), where N corresponds to the number of
than the regular grid (kr ) that generates such irreducible data points related to the product of FFT dimensions,
points. One important point to stress is that J does 3
namely
√ N√ = k max√ (1) × k max (2) × k max (3) ≈ 8R0 ·
not depend on data, but it is completely defined by the
b1 · b1 · b2 · b2 · b3 · b3 ≈ 6/π (nk̄ × ns × M ). Con-
lattice, namely the set of irreducible sampling (k̄l ), the
sequently, the number of floating-point operations by
number of star functions (M), and the form of spline- q0
like roughness functional (Π). In practice, in order to using our approach is ≈ N f × nk̄ n3 + 6/π (nk̄ × ns ×
get a denser mesh, we rely on a Fast Fourier Transform M) ln (6/π (nk̄ × ns × M)). The first term comes from
(FFT) from the real space to the reciprocal space in order the first W-F interpolation by using nk̄ irreducible points,
to compute the expansion given in Eq. (3). We take while the second one comes from the symmetry-adapted
advantage of the BvK periodic boundary conditions to PW interpolation. The gain in performance by using our
increase the real space by the expansion factor, M, as method in comparison with single W-F calculation, to
will be explained below, to get proportionally a new get approximately the same final homogeneous grid,
homogeneous k0 -mesh finer than the original one. As a can be given by ≈ 2(ns × M), assuming N kf ≈ N
0
result we get the full integration over very fine k0 and 0 q0
q0 meshes in order to calculate transport properties, that and N kf = N f . Clearly, high symmetry systems al-
is, TP ∝ ∑k0 f˜(k0 ) . low greater computational savings, however the factor
Our implementation for the second interpolation is M, typically ranging from 5 − 60 enables remarkably
based on modifications and adaptations of some subrou- significant performance gain even for low symmetry
tines of the BoltzTraP[26] code. Lattice points and their systems.
respective star functions are generated in the real space In order to test our implementation, we carried out TP5 Figure 3. (a) Scattering rates for Si at 300 K calculated with the dual interpolation method (green dots), DFT with linear interpolation (black[23] and light blue[4] lines and black squares[12]), W-F interpolation using EPW (purple triangles[24] and light orange circles), and tight-binding calculations (red[25] line). The W-F calculations used (30)3 k/(60)3 q ((100)3 k/(40)3 q) meshes in calculations represented by the purple triangles (orange circles). (b) Walltime required to perform calculations for the electron self-energy due to el-ph coupling in the Fan-Migdal approximation, for different grid sizes. Empty magenta triangles correspond to the direct calculation using EPW over homogeneous grids. Empty (filled) green squares, empty (filled) red circles and empty (filled) blue diamonds correspond to the calculations using T-EPW (only second PW interpolation), starting from 256, 1661 and 5216 irreducible k points, respectively (see text for details). calculations for silicon. We computed the imaginary part on W-F interpolations (EPW) grows almost exponen- of the electron self-energy in the Fan-Migdal approxima- tially with increasing k0 /q0 density. By applying our tion, Im Σ, which gives the relaxation time due to e-ph method a drastic reduction in the computational time scattering, and consequently, thermoelectric TP using is obtained, which is generally greater than two orders the BTE, as shown in Fig. 1. Additionally, we also stud- of magnitude, as can be observed from the curves for ied phonon-assisted optical absorption for Si, as shown the total time of T-EPW. These curves also demonstrate in Fig. 2. More details about these calculations can a roughly exponential growth with k0 /q0 density, which be found in the Supplemental Material (SM). Since the is due to increasing the grid size in the first W-F in- first step in computing the double BZ integrals is based terpolation, independent of the number of initial irre- on a W-F interpolation, our implementation has been ducible points k̄l . In these test calculations we consid- built on top of the Electron-Phonon Wannier (EPW) [27] ered nk̄ = 256, 1661, 5216, leading to regular meshes of code, which is contained in the Quantum Espresso pack- kr = (20)3 , (40)3 , (50)3 . The plateaus in the computa- age [28]. We have modified the EPW code in order tional time for T-EPW are due to increasing the value of to include the second PW interpolation, as described M from 5 to 60, while keeping fixed the grid of first W-F above, which we call Turbo-EPW (T-EPW). In Fig. 3(a) interpolation ((100)3 q0 points). As shown in Fig. 3(b), we show Im Σ for Si at 300 K calculated using T-EPW using PW interpolation one can achieve much denser in comparison with other approaches, namely, previous grids by increasing the value of M with negligible in- DFT with linear interpolation[4, 12, 23], W-F interpola- crease in computational time. As shown in the SM, the tion with EPW[24], and tight-binding calculations[25]. accompanying error due to the increase of M to generate Our results, calculated using (100)3 k0 /(100)3 q0 grids, denser grids diminishes by increasing nk̄ ; a solution that are in good agreement with other W-F calculations us- can also be used to minimize errors from possible kink ing the EPW code directly on (30)3 k0 /(60)3 q0 [24] and structures derived from band crossings, which leads to (100)3 k0 /(40)3 q0 grids, but with significantly reduced a Gibbs ringing in Fourier series analysis. computational time. In summary, our method can be used to calculate effi- In order to estimate the performance gain of our ap- ciently el-ph-based TP. The computational performance proach, in Fig. 3(b) we show the computational time gain is remarkable, being ≈ 2(ns × M) faster than state- required to finalize the calculation of Im Σ for differ- of-the-art EPW calculations without loosing accuracy. ent k0 /q0 grids, all using the same computational hard- It should be emphasized that this novel approach can ware. The time required for calculations based only also be used as an efficient and stable numerical tool in
6
order to calculate ubiquitous double BZ integrals, and
potentially extending to many further applications, for
instance phonon-assisted nonlinear optical properties,
superconducting critical temperature and its related ther-
modynamic properties and electron-plasmon coupling
TP from first-principles. Moreover, this method may
allow previously impractical calculations and can serve
as a starting point to explore the effects of the vertex
corrections to the Migdal approximation as well as to ad-
dress the e-ph coupling in complex systems with many
atoms in the unit cell. This last capability would be
useful for the discovery of efficient materials for energy
applications, such as high-performance thermoelectrics.
Appendix A Details of the calculations for Si
First, we compute the self-consistent potential and Figure 4. Convergence analysis of the scattering rate of Si at
Kohn-Sham states on a 12 × 12 × 12 Monkhorst-Pack 300K due to el-ph coupling in the Fan-Migdal approximation
as a function of the electron energy. The calculation has been
k-point grid using DFT and lattice-dynamical properties
performed on different q meshes, namely, (20)3 (green crosses),
with DFPT[11] on a 3 × 3 × 3 q-point grid, as imple-
(60)3 (blue squares), (100)3 (red dots) q-meshes in the first W-
mented in the Quantum Espresso distribution[28] us-
F interpolation, while keeping fixed the number of irreducible
ing the Perdew-Burke-Ernzerhoff exchange-correlation k points in the second plane-waves interpolation, leading to
functional[29]. We used a full-relativistic norm- (100)3 k-mesh.
conserving optimized Vanderbilt pseudopotential[30].
The unit cell consists of Si in the diamond structure
with an experimental lattice parameter of 5.43 Å. The
e-ph matrix elements are first computed on coarse grids, k-mesh, is within ≈ ±6%. For the remaining, the differ-
then they are determined in the significantly finer ence between the data points is about the same, within
grids using both Wannier-Fourier (W-F) interpolation ≈ ±3%.
only, through EPW code and our dual interpolation
method, Turbo-EPW (T-EPW). Maximally localized Wan-
nier functions[14] for the wannierization procedure are Appendix C Calculation of electron self-energy and
obtained from Wannier90.[31] Thus, Bloch-to-Wannier thermoelectric properties
rotation matrices and then Wannier-to-Bloch diagonal-
izer matrices are used to interpolate el-ph matrix ele-
The expression for the imaginary part of electronic
ments.
self-energy due to el-ph coupling in the Fan-Migdal
approximation can be derived from quantum field
theory[1] and it is expressed as
Appendix B Convergence analysis
dq
Z
Σ00n,k (ω, T ) = π ∑ |g (k, q)|2
Fig. 4 shows how the scattering rate aproaches con- m,β BZ
Ω BZ mn,β
vergence by increasing the mesh size of W-F interpo- "
lation, from (20)3 to (100)3 q points, while keeping ×
nqβ ( T ) + f mk+q δ(ω − (emk+q − e F ) + ωqβ )
the number of irreducible points fixed at 5216 k points.
The second interpolation by star functions leads into #
a converged grid with (100)3 k points. Fig. 5 shows +[nqβ ( T ) + 1 − f mk+q ]δ(ω − (emk+q − eF ) − ωqβ ) ,
the difference in scattering rates calculated by different
approaches, namely, different parameters in the sec- (10)
ond plane-waves interpolation (PWI), over equivalent
meshes. The analysis shows that the accompanying er- where nqβ ( T ) and f mk+q are the Bose-Einstein and the
ror due to the increase in M decreases by enlarging the Fermi-Dirac distributions, Ω BZ is the BZ volume, m
number of irreducible points, nk̄ (see main text). More- and n are the corresponding electronic states, while
over, the difference between scattering rates calculated β represents the phonon branch, emk+q are the elec-
over (100)3 q/ 1661 k points expanded by using M = 5 tronic eigenenergies of the state mk + q and ωqβ are the
to reach (100)3 k-mesh and data from (100)3 q/ 256 corresponding eigenfrequencies with wavevector q and
k points expanded by using M = 30 to reach (100)3 phonon branch β. Basically, from the first W-F interpola-7
perimental conditions of zero temperature gradient
(∇ T = 0) and zero electric current, the kinetic co-
efficient tensors can be identified with the electrical
conductivity tensor, σ = Λ(0) , the Seebeck coeffi-
cient tensor, S = (eT )−1 Λ(1) /Λ(0) , and the charge
carrier contribution to thermal conductivity tensor,
−1
κ e = ( e 2 T ) −1 Λ (1) · Λ (0) · Λ (1) − Λ (2) . Conse-
quently, once we have τn,k from Eq. (11), we can com-
pute the electrical conductivity, the Seebeck coefficient,
Lorenz function and the charge carrier contribution
to thermal conductivity from the solution of Eq. (13)
and Eq. (12). Note that the both bandstructure and
phonon dispersion have also been interpolated by the
method presented in the main text over the same grid
as electron self-energy. Indeed, we have implemented
these equations on top of the BoltzTraP code,[26] from
which we can obtain these transport properties directly
Figure 5. Difference between scattering rates computed by
from first principles.
using different parameters in the second plane-waves interpo-
lation, over equivalent meshes. Green crosses correspond to
the difference between scattering rates calculated over (100)3
Appendix D Scattering by ionized impurities
q/ 1661 k points expanded by using M = 5 to reach (100)3 k-
mesh and data from (100)3 q/ 256 k points expanded by using
M = 30 to reach (100)3 k-mesh. Blue squares correspond to For the calculation of thermoelectric transport proper-
the difference between scattering rates calculated over (100)3 ties of n- and p-type Si polycrystals, we have also consid-
q/ 5216 k points expanded by using M = 10 to reach (180)3 ered the scattering by ionized impurities. Such scattering
k-mesh and data from (100)3 q/ 1661 k points expanded by
has been treated theoretically by Brooks and Herring
using M = 30 to reach (180)3 k-mesh. Red dots correspond to
(B-H)[32, 33] by considering a screened Coulomb po-
the difference between scattering rates calculated over (100)3
tential, the Born approximation for the evaluation of
q/ 5216 k points expanded by using M = 8 to reach (160)3
transition probabilities and neglecting perturbation ef-
k-mesh and data from (100)3 q/ 1661 k points expanded by
fects of the impurities on the electron energy levels and
wave functions. In the B-H theory the electron is scat-
using M = 25 to reach (160)3 k-mesh.
tered independently by dilute concentrations of ionized
centers randomly distributed in the semiconductor.
The per-unit-time transition probability for the scatter-
tion we can get f (k̄) = Σ00n,k̄ (ω, T ), over the irreducible ing of charge carriers by ionized impurities can be given
points, which will be interpolated throughout the whole in the plane-wave approximation as
BZ by star functions, resulting in Σ00n,k0 (ω, T ) over denser
k0 grids. 2π Ni
W (k|k0 ) =
Σ00 is directly related to the scattering rate, that is, h̄ V
inversely proportional to the relaxation time Z 2
U (r) exp i (k − k0 ; r) dr δ(e(k0 ) − e(k)) ,
(14)
1
= 2Σ00n,k (ω = 0, T ) , (11)
τn,k where U (r) is the scattering potential and Ni is the ion-
ized impurity concentration.
which enters in kinetic transport equations. Indeed, the
kinetic coefficient tensors can be expressed through A long-range Coulomb field, U (r ) = eφ(r ) = ±e2 /ζr,
with potential φ at a point r of the crystal is created
! by the presence of positive (donor) or negative (accep-
∂ f (0) (µ; e, T )
Z
(α) 2
Λ (µ; T ) = e Ξ(e, µ, T )(e − µ) −
α
de , tor) impurity ions, within a medium with dieletric con-
∂e stant ζ. The straightforward application of this field in
(12) Eq. (14) leads into a logarithmic divergence, and hence, a
where µ is the chemical potential and Ξ(e, µ, T ) is the screened Coulomb potential has to be considered. From
transport distribution kernel given by the B-H theory the potential can be expressed in a more
rigorous form as φ(r ) = ±e/ζr (exp{(−r/r0 )}), where
dk
Z
Ξ(e, µ, T ) = ∑ vn,k ⊗ vn,k τn,k (µ, T )δ(e − en,k ) 3 , r0 is the radius of ion field screening defined by
n 8π
(13) 4πe2 ∂ f0
Z
−2
r0 (k) = − g(e)de , (15)
with vn,k being the electron velocity. From both ex- ζ0 ∂e(k)8
where f (0) (e) is the equilibrium electron distribution with
function, ζ 0 is the static dielectric constant, and g(e)
is the density of states, calculated numerically on an
vim (k) gmj,β (k, q)
energy grid with spacing de sampled over Nk k-points S1 (kq) = ∑ em,k − eik − h̄ω + iΓm,k , (20)
m
dk 1 δ(e − en,k )
Z
g(e) = ∑ δ(e − en,k ) 8π3 =
ΩNk ∑ de
,
n n,k
gim,β (k, q)vmj (k + q)
(16) S2 (kq) = ∑ em,k+q − eik ± h̄ωβq + iΓm,k+q , (21)
where Ω is the volume of the unit cell. m
Within the relaxation time approximation for the Boltz-
mann transport equations, the relaxation time for the and
scattering of the charge carriers by ionized impurities
can be expressed as 1 1
P = (n βq + ± )( f − f j,k+q ) . (22)
2 2 ik
h̄ζ 0 2 ∂e(k)
τimp (k) = k2 (17)
4
2πe Ni Fimp (k) ∂k In these equations, ω is the photon frequency, c is the
speed of light, nr is the refractive index (for silicon we
where used nr = 3.4), and λ is the photon polarization. S1 and
S2 are the two possible ways for the indirect absorption
η
Fimp (k) = ln(1 + η ) − , (18) process, while P is related to the carrier and phonon
1+η statistics. For calculations of Si, the DFT band gap and
all conduction bands have been shifted up by 0.7eV to
is the screening function with η = (2kr0 )2 . Here, we simulate experimental gap. Our calculations have been
∂e(k)
interpolated ∂k within Eq. (17) by using the plane- carried out over (60)3 q/(40)3 k meshes. It took ≈ 36
waves interpolation (see the main text) and used Math- minutes to perform the calculation by using our dual
iessen’s rule to consider both the scattering by ionized interpolation method on 8 CPU cores at the Odyssey
impurities and phonons. cluster (Harvard University).
Acknowledgments
Appendix E Calculation of phonon-assisted optical
absorption
The authors thank the Harvard FAS Research Comput-
To calculate the phonon-assisted absorption coeffi- ing facility and the Brazilian CCJDR-IFGW-UNICAMP
cient, we use the Fermi’s golden rule expression[8, 34]: for computational resources. A.S.C. and A.A. grate-
fully acknowledge financial support from the Brazil-
4π 2 e2 1 1 ian agency FAPESP under Grants No.2015/26434-2,
α(ω ) = 2
ωcnr (ω ) Ω Nk Nq ∑ |λ · (S1 + S2 )|2 (19) No.2016/23891-6, No.2017/26105-4, No.2018/01274-0
βijkq and No.2019/26088-8. A.S.C. also acknowledges the
× Pδ(e j,k+q − ei,k − h̄ω − ±h̄ω βq ) , kind hospitality of SEAS-Harvard University.
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