First-principles computations of phonon-limited carrier mobilities in semiconductors have recently gained popularity. Such calculations are indeed crucial for the discovery and development of new functional materials. In state-of-the-art approaches, Fourier-based interpolation schemes are used to obtain the electron-phonon matrix elements on the very dense wavevector grids needed to converge carrier lifetimes and mobilities. In polar semiconductors, the long-range electrostatic interactions lead to a divergence of the matrix elements, rendering their interpolation unstable. For this reason, ab initio methods have been recently developed to model the non-analytical behavior of the matrix elements for q→0 [1]. Most of the studies performed so far have focused on this Fröhlich divergence generated by dynamical dipoles. However, additional non-analytical terms are present in the q→0 limit [2]. In this work, we analyze the role played by the dynamical quadrupoles and show that an accurate interpolation is obtained only when both dipolar and quadrupolar fields are taken into account. We discuss their impact on the accuracy and on the convergence of carrier mobilities both in polar and non-polar semiconductors. [1] Phys. Rev. Lett. 115, 176401 (2015) [2] Phys. Rev. B 13, 694 (1976) https://arxiv.org/abs/2002.00628 https://arxiv.org/abs/2002.00630

1. Phonon-limited carrier mobility
in semiconductors : importance of
the dynamical quadrupoles
Guillaume Brunin1,2, Henrique Miranda1, Matteo Giantomassi1,
Miquel Royo3, Massimiliano Stengel3, Matthieu Verstraete4
Xavier Gonze1, Gian-Marco Rignanese1,2, and Geoffroy Hautier1
[1] UCLouvain (Belgium) / [2] FNRS (Belgium) / [3] ICMAB (Spain) / [4] ULiège (Belgium)
APS March Meeting
Denver (CO), USA, 5 March 2020

2. In semiconductors, the intrinsic mobility at high T
is governed by electron-phonon interactions
[Yu & Cardona, Fundamentals of Semiconductors]

3. • In the relaxation-time approximation of the linearized Boltzmann formalism, the electron
mobility is given by:
• The electron lifetimes are given by:
Theories have been developed to describe carrier transport
ne electron concentration
velocity operator
Ω unit cell volume
ΩBZ Brillouin zone volume
vnk
nνq Bose-Einstein occupation function
fmk+q Fermi-Dirac occupation function
electron energy for state nk
phonon energy for state νq
εnk
ωνq
with
with
μe,αβ =
−1
Ωne
∑
n∈CB
∫
dk
ΩBZ
vnk,αvnk,βτnk
∂f
∂ε
εnk
τ−1
nk = 2π
∑
m,ν
∫BZ
dq
ΩBZ
|gmnν(k, q)|2
× [(nνq + fmk+q)δ(εnk − εmk+q + ωνq) +
(nνq + 1 − fmk+q)δ(εnk − εmk+q − ωνq)],

4. Electron-phonon interactions can be calculated from DFPT
• The electron-phonon coupling matrix elements are obtained as:
where is the Bloch wave function for state nk
is the 1st-order variation of the KS potential V
due to a phonon mode at wavevector q and branch index ν
ψnk
ΔνqV
gmnν(k, q) = ⟨ψmk+q ΔνqV ψnk⟩
ΔνqV = eiq⋅r 1
2ωνq
∑
κα
eκα,ν(q)
Mκ
Vκα,q(r)
phonon energy for state νq
phonon eigenvector for the atom κ
Mκ mass of the atom κ
lattice-periodic scattering potential from DFPT
ωνq
eκα,ν(q)
Vκα,q(r)
with

5. • Following the approach of Verdi and Giustino [Phys. Rev. Lett. 115, 176401 (2015)]
we can write:
• The short range part is smooth in q-space and can be interpolated:
The scattering potential can be separated into
short range ( ) and long range ( ) contributions𝒮 ℒ
Vκα,q(r) = V 𝒮
κα,q(r) + Vℒ
κα,q(r)
gmnν(k, q) = g 𝒮
mnν(k, q) + gℒ
mnν(k, q)
Fourier interpolation
• Wannier
• Atomic orbitals
• Plane waves
DFPT
multipole-effects
Perturbo [Zhou et al., arXiv:2002.02045]
[Poncé et al., Comput. Phys. Commun. 209, 116 (2016)]
[Brunin et al., arXiv:2002.00630]

6. In polar semiconductors, the long-range is
dominated by the diverging Fröhlich potential
• Remove Frölich divergence from on the coarse (DFPT) mesh
• Interpolate the short-range
matrix elements
• Add back to
all the points of the fine
mesh
gmnν(k, q)
g 𝒮
mnν(k, q)
gℒ(F)
mnν (k, q)
[Poncé et al., Comput. Phys. Commun. 209, 116 (2016)]
gℒ(F)
mnν (k, q) = i
4π
Ω ∑
κ
1
NqMκωνq
×
∑
G≠−q
(q + G) ⋅ Z*κ ⋅ eκν(q)
(q + G) ⋅ ϵ∞ ⋅ (q + G)
× ⟨ψ ei(q+G)⋅(r−τκ)
ψ⟩
the effective charge tensor
the dielectric tensor
Z*κ
ϵ∞
with
{

7. This approach was adopted in most calculations so far…
• However, there is something puzzling when we consider the convergence of the electron
mobility of semiconductors w.r.t. the initial q-mesh used for the DFPT calculations: it is really
slow!

8. In fact, the scattering potential is missing something…
• In Si, the Born effective charges are zero and the imaginary part of the potential ( ) does not
diverge for q → 0.
• In GaAs, the Fröhlich-like model ( ) correctly describes the divergence of the imaginary part
of the potential close to q → 0.
• In both materials, the real part of the potential ( ) presents a discontinuity
for q → 0 which is not captured by the Fröhlich-like model ( ).
ℜ ¯VDFPT
ℑ ¯VDFPT
ℜ ¯Vℒ(F)
ℑ ¯Vℒ(F)
Average potential over the unit cell for selected atomic perturbations

9. The Fourier interpolation introduces unphysical oscillations
for q → 0 which are reflected in the matrix elements gmnν
• Silicon case: 9 × 9 × 9 q-point grid
• Already observed by Agapito and Bernardi [Phys. Rev. B 97, 235146 (2018)] .
DFPT FI

10. • Following the work of Vogl [Phys. Rev. B 13, 694 (1976)], we distinguish
the long-range Fröhlich and quadrupole contributions
• Similar approach (no term) recently used by Jhalani et al. [arXiv:2002.08351]
Vℒ(F)
κα,q (r) Vℒ(Q)
κα,q (r)
ℰ
To describe the discontinuities, we need to go
beyond the Fröhlich model including quadrupole interactions
Vℒ
κα,q(r) =
4π
Ω ∑
G≠−q
i(qβ + Gβ)Z*κα,β
(qδ + Gδ)ϵ∞
δδ′(qδ′ + Gδ′)
ei(qη+Gη)(rη−τκη)
+
4π
Ω ∑
G≠−q
(qβ + Gβ)Z*κα,βvHxc,ℰγ(r)(qγ + Gγ)
(qδ + Gδ)ϵ∞
δδ′(qδ′ + Gδ′)
ei(qη+Gη)(rη−τκη)
+
4π
Ω ∑
G≠−q
1
2
(qβ + Gβ)Qβγ
κα(qγ + Gγ)
(qδ + Gδ)ϵ∞
δδ′(qδ′ + Gδ′)
ei(qη+Gη)(rη−τκη)
Vℒ(F)
κα,q (r)
Vℒ(Q)
κα,q (r)
the dynamical quadrupole tensor
the change of the Hxc potential w.r.t. the electric field
in Cartesian coordinates
Qκα
vHxc,ℰγ(r) ℰ
with
{

11. Including the quadrupole solves the discrepancies w.r.t. DFPT
• The discontinuity in the real part of the potential ( ) for q → 0 is now captured by the
quadrupole term ( ) for both Si and GaAs
ℜ ¯VDFPT
ℑ ¯VDFPT
ℜ ¯Vℒ(F)
ℑ ¯Vℒ(F)
Average potential over the unit cell for selected atomic perturbations
ℜ( ¯Vℒ(F)
+ ¯Vℒ(Q)
)
ℑ( ¯Vℒ(F)
+ ¯Vℒ(Q)
)

12. Including the quadrupole solves the discrepancies w.r.t. DFPT
• Silicon case: 9 × 9 × 9 q-point grid
DFPT FI FI+Q

13. The electron mobility converge faster…
• The error is around 10% in Si and goes up to 30% in GaAs where, due to the small effective
mass, most of the scattering channels for electrons close to the band edge involve small
momentum transfer.
• In the polar systems investigated so far, we observed that gives the most important
contribution compared to the term (in GaAs, ignoring this term changes the electron mobility
by 0.1%).
Qκα
ℰ
10%
30%

14. Conclusion
• Important physics is missing in current electron-phonon computations
▶ One should include terms up to quadrupoles in the long-range model of the electron-phonon
matrix elements
▶ Without this term, the error on the mobility can be as big as 30%
• Papers with more details:
▶ https://arxiv.org/abs/2002.00628
▶ https://arxiv.org/abs/2002.00630
• Developments available in ABINIT v9, soon to be released