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Temperature Dependent Electron-Phonon Scattering and Electron Mobility in

SrTiO$_{3}$ Perovskite from First Principles

Preprint · June 2018

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 Temperature Dependent Electron-Phonon Scattering and Electron Mobility
in SrTiO3 Perovskite from First Principles

Jin-Jian Zhou,1 Olle Hellman,1, 2 and Marco Bernardi1, ∗

1
Department of Applied Physics and Materials Science,
California Institute of Technology, Pasadena, California 91125, USA
2
Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA

Structural phase transitions and soft phonon modes pose a longstanding challenge to comput-
ing electron-phonon (e-ph) interactions in strongly anharmonic crystals. Here we develop a first-
principles approach to compute e-ph scattering and charge transport in materials with anhar-
arXiv:1806.05775v1 [cond-mat.mtrl-sci] 15 Jun 2018

monic lattice dynamics. Our approach employs renormalized phonons to compute the temperature-
dependent e-ph coupling for all phonon modes, including the soft modes associated with ferroelec-
tricity and phase transitions. We show that the electron mobility in cubic SrTiO3 is controlled by
scattering with longitudinal optical phonons at room temperature and with ferroelectric soft phonons
below 200 K. Our calculations can accurately predict the temperature dependence of the electron
mobility between 150−300 K, and reveal the origin of the T −3 dependence of the electron mobility in
SrTiO3 . Our approach enables first-principles calculations of e-ph interactions and charge transport
in broad classes of crystals with phase transitions and strongly anharmonic phonons.

Strontium titanate (SrTiO3 ) is a prototypical per- and related materials with phase transitions and strong
ovskite oxide that has attracted interest due to its in- anharmonicity. Computing from first principles the scat-
triguing physical properties and technological applica- tering between electrons and soft phonons as a function
tions [1, 2]. Similar to other perovskites, SrTiO3 exhibits of temperature remains an open challenge of broad rele-
structural phase transitions (it is cubic above, and tetrag- vance to materials physics.
onal below 105 K) with associated soft phonon modes In this Letter, we develop an ab initio approach to
that change their frequency with temperature [3–5]. This compute the e-ph coupling as a function of temperature
strongly anharmonic lattice dynamics is found broadly in strongly anharmonic crystals. We apply it to com-
in materials of technological interest − among others, pute the phonon dispersions and the temperature depen-
metal-halide perovskites, oxides and chalcogenides. The dence of the electron mobility in cubic SrTiO3 , obtain-
complex interplay between electronic and lattice degrees ing results in excellent agreement with experiment. Our
of freedom makes it challenging to microscopically un- method allows us to quantify the contribution of different
derstand electron-phonon (e-ph) interactions and charge acoustic, optical and soft modes to e-ph scattering and
transport in these materials. transport. We find that both the AFD and the ferro-
Despite extensive studies, the charge conduction mech- electric soft modes couple strongly with electronic states
anisms in SrTiO3 are still debated [6–9]. The elec- near the conduction band edge. We show that the T −3
tron mobility in cubic SrTiO3 exhibits a roughly T −3 dependence of the mobility is due to an interplay be-
temperature dependence above 150 K [9, 10], where tween the LO and the ferroelectric soft phonons, which
carrier transport is typically limited by e-ph scatter- dominate e-ph scattering at temperatures above and be-
ing. However, it is still controversial whether the tem- low 200 K, respectively, while the AFD soft mode has a
perature dependence is due to scattering of electrons negligible contribution due to a lack of scattering phase
with longitudinal optical (LO) phonons, ferroelectric space. Our work provides a practical ab initio approach
soft phonons [6, 7], or soft phonons associated with to study e-ph coupling and charge transport in materials
the cubic-to-tetragonal antiferrodistortive (AFD) phase with anharmonic phonons, of which SrTiO3 and related
transition [8, 11]. While the arguments supporting each perovskite oxides are a paradigmatic case.
mechanism are based on phenomenological models, mi- The key ingredients for computing e-ph scattering
croscopic insight and quantitative analysis from first- and charge transport are the e-ph matrix elements
principles calculations are still missing. gmnν (k, q), which quantify the probability amplitude to
Recently developed ab initio calculations of e-ph scatter from an initial Bloch state |nki (with band n
coupling and phonon-limited carrier mobility [12–16] and crystal momentum k) to a final state |mk + qi by
are based on density functional perturbation theory emitting or absorbing a phonon with wavevector q, mode
(DFPT) [17], which cannot handle strongly anharmonic index ν, energy ~ωνq and displacement eigenvector eνq ,
lattice dynamics. Since DFPT predicts imaginary fre-
quencies for the soft modes and lacks thermal effects, the s
~ X eκα
typical workflow of ab initio e-ph and charge transport gmnν (k, q) = √ νq hmk + q |∂qκα V | nki ,
calculations [12–14] cannot be applied to cubic SrTiO3 2ωνq κα Mκ
(1)
 2

where ∂qκα V ≡ p eiqRp ∂pκα V and ∂pκα V is the varia-

P
tion of the Kohn-Sham potential for a unit displacement
of atom κ (with mass Mκ and located in the unit cell at
Rp ) in the Cartesian direction α. TDEP 200K 300K
80

Phonon energy (meV)

To compute the e-ph coupling at finite temperature DFPT Exp.
in anharmonic crystals, we use in Eq. (1) temperature-
dependent renormalized phonon energies ω̃νq (T ) and
eigenvectors ẽνq (T ) that include anharmonic effects and
are obtained with the temperature-dependent effective 40
potential (TDEP) method [18]. TDEP extracts the ef-
fective interatomic force constants (IFCs) that best de-
scribe the anharmonic Born-Oppenheimer potential en-
ergy surface at a given temperature. For comparison, we 0
also compute harmonic phonons and e-ph coupling using
DFPT. For the TDEP calculations, we prepare 4 × 4 × 4 𝚪 X M 𝚪 R X
(320 atom) supercells with thermal displacements cor-
responding to a given temperature T , perform density FIG. 1. Phonon dispersions of cubic SrTiO3 , computed with
functional theory (DFT) calculations (see below) on the TDEP at 200 K (teal lines) and 300 K (red lines), and for
comparison with DFPT (gray dashed line). Experimental re-
supercells to collect atomic displacements and forces, and
sults at room temperature (from Refs. [34, 35]) are shown
extract the effective force constants at each temperature with open circles.
by least-squares fitting [18]. This process is repeated it-
eratively until convergence [19]. Note that in polar ma-
terials the IFCs contain a long-range contribution due to
the dipole-dipole interactions [20, 21]. We develop a new with temperature. Figure 1 also shows that the phonon
method to accurately include the long-range contribution dispersions obtained with TDEP at 300 K are in excellent
in the IFCs; the method is outlined in the Supplemental agreement with experiment [34, 35]. The TDEP phonon
Material [22] and will be detailed elsewhere. dispersions at 200 K and 300 K match closely, except for
The e-ph matrix elements gmnν (k, q) are com- the lowest-energy ferroelectric soft mode at Γ and the soft
puted, both using harmonic (DFPT) and anharmonic AFD mode at R, for which energy renormalization due to
(TDEP) phonons, with our in-house developed per- anharmonic interactions is significant. Although DFPT
turbo code [23], which is also employed to efficiently is inaccurate for the soft modes, it generates reasonable
compute the e-ph scattering rates [13] and the elec- dispersions for high-energy phonons (above ∼30 meV)
tron mobility using an iterative solution of the linearized that are consistent with TDEP results. The correction
Boltzmann transport equation (BTE) [24]. Briefly, we scheme introduced in this work [22] allows us to accu-
perform DFT calculations on SrTiO3 within the Perdew- rately account for the long-range contribution to the IFCs
Burke-Ernzerhof generalized gradient approximation [25] and to obtain accurate LO mode dispersions near Γ. By
using the Quantum Espresso package [26]. Fully rel- contrast, recent work [38, 39] using the mixed-space ap-
ativistic norm-conserving pseudopotentials that include proach [40] shows unusual oscillations along Γ−R and
the spin-orbit coupling (SOC) [27, 28] are employed, to- Γ−M in the highest LO mode dispersion, which are an
gether with the experimental lattice constant of 3.9 Å [29] artifact.
and a plane-wave kinetic energy cutoff of 85 Ry. Wannier To quantitatively study the coupling strength between
interpolation [30] in combination with the polar correc- electrons and different phonon modes, we analyze the
tion [31, 32] is employed to evaluate the e-ph matrix ele- absolute value of the e-ph matrix elements in Eq. (1),
ments on very fine Brillouin zone grids. We adopt coarse |gmnν (k, q)|. We choose k = 0 (the Γ point) as the ini-
8 × 8 × 8 q-point grids for DFPT calculations, and Wan- tial electron momentum, and compute the square root
nier functions for the Ti-t2g orbitals are constructed from of the gauge-invariant trace of |g|2 over the three lowest
Bloch states on a coarse 8 × 8 × 8 k-point grid using the conduction bands, for phonon wavevectors q along a high
Wannier90 code [33]. Fine grids with up to 1253 k-points symmetry Brillouin zone path. Results are given for both
are used to converge the mobility. anharmonic phonons computed at 200 K with TDEP
Figure 1 compares phonon dispersions computed with and for harmonic phonons from DFPT for comparison.
DFPT with those obtained using TDEP at 200 K and The mode-resolved e-ph coupling strengths, |gν (q)|, are
300 K. The DFPT result exhibits unstable soft phonon shown in Fig. 2(a) and mapped with a color scale on the
modes with negative energies both at the zone center (Γ phonon dispersions in Fig. 2(b) for better visualization.
point) and corners (R and M points), consistent with We find that the two highest-energy LO modes, labeled
previous work [36, 37]. In the TDEP result, the soft LO-1 and LO-2 in Fig. 2(a,b), exhibit the strongest cou-
phonons are stable, and their energy shifts continuously pling with electrons; for these modes, |gν (q)| diverges
 3

(a) 103 surements from Ref. [10]. The temperature dependence

LO-1 TDEP of our computed mobility is in excellent agreement with
LO-2 DFPT
|g| (meV) experiment. By fitting the data at 150−300 K with a
T −n power law, we get n ≈ 3.09 for the experimental
102 data and n ≈ 3.12 for our computed mobility, namely
an error in the exponent within 1%. Both the RT
approximation and the iterative BTE solution exhibit a
𝚪 X M 𝚪 R X T −3 temperature dependence of the mobility, though the
(b)
log(|g|) iterative solution mobilities are roughly 15% higher than
in the RT approximation. Our results clearly show that
Phonon energy (meV)

LO-1 >10
80 the T −3 dependence of the mobility can be explained
8 through the e-ph scattering alone.
6
There is a subtle interplay between the e-ph scattering
LO-2
40
mechanisms regulating the temperature dependence of
ferroelectric 4 the electron mobility. We analyze the e-ph scattering
soft
rates for each phonon mode and their contribution to
<2
transport, highlighting the role of the soft modes. We
0 LA focus on the four modes labeled in Fig. 2(b) − the two
𝚪 X M 𝚪 R X LO modes, the longitudinal acoustic (LA) mode and the
ferroelectric soft mode near Γ − that are most relevant
FIG. 2. (a) Absolute value of the e-ph matrix elements com- for transport at 150−300 K. Figure 3(b) shows the
puted with anharmonic phonons from TDEP (teal dots) and mode-resolved scattering rates at three representative
harmonic phonons from DFPT (red dots). (b) Phonon dis- temperatures. Also shown in Fig. 3(b) is the integrand
persions overlaid with a log-scale color map of |gν (q)|.
in Eq. (2) at each temperature, which quantifies how
much electronic states at a given energy contribute to
as 1/q for q approaching Γ due to the Fröhlich inter- transport [13]. Between 150−300 K, the integrand is
action [41]. Notably, both the ferroelectric soft mode non-zero only within ∼100 meV of the conduction band
near Γ and the AFD soft mode at R couple strongly minimum (CBM), so that the e-ph scattering rates in
with electrons. While for the LO phonons the DFPT that energy range can accurately quantify which phonon
and TDEP results are in agreement, the coupling be- modes limit the mobility.
tween electrons and soft modes exhibits an unphysical At 300 K, the two LO modes dominate e-ph scattering,
divergence in DFPT, whereas using TDEP anharmonic exhibiting scattering rates an order of magnitude higher
phonons gives a physical, finite value of |g|. The strong than any other mode. The LO-mode scattering rate has
coupling between electrons and soft modes is essential to a two-plateaux structure as a function of energy [13, 14];
understanding electron dynamics in SrTiO3 . the low-energy plateaux, which mainly contributes to
The temperature-dependent e-ph matrix elements are transport, corresponds to LO phonon absorption, a
employed to compute the mobility using both the relax- thermally activates process with a rate proportional
ation time (RT) approximation [13] and an iterative so- to the LO phonon occupation, NLO ≈ e−~ωLO /kT . As
lution of the BTE [24] that goes beyond the RT approx- the temperature is reduced, the contribution from LO
imation. Briefly, we compute the e-ph scattering rates mode scattering thus drops exponentially and at lower
(and their inverse, the RTs τnk ) from the imaginary part temperatures transport is dominated by scattering with
of the lowest-order e-ph self-energy [42]. The mobility is low-energy phonons, including acoustic and soft modes.
computed as At 200 K, scattering from the ferroelectric soft mode
  is significant. Its rate is larger than the scattering rate
∂f X α
Z
2e β with the LO-1 mode, and it is second only to scattering
µαβ (T ) = dE − Fnk (T )vnk δ(E − εnk ),
nc Vuc ∂E with the LO-2 mode. At 150 K, ferroelectric soft mode
nk
(2) scattering is dominant in the energy range of interest for
where εnk and vnk are the electron energy and velocity, transport, with a smaller contribution from scattering by
respectively, α and β are Cartesian directions, f is the LA phonons. Our results show that, while LO phonon
Fermi-Dirac distribution, Vuc is the unit cell volume and scattering limits the mobility at room temperature, the
nc is the electron concentration. Fnk is computed as ferroelectric soft phonons play a crucial role at lower
τnk vnk in the RT approximation, or obtained by solving temperatures, dominating over other scattering mecha-
the BTE iteratively [24]. nisms near 150 K. The T −3 mobility dependence is due
Figure 3(a) shows our calculated electron mobility as to the combined effects of the LO mode and ferroelectric
a function of temperature (obtained using the iterative soft mode scattering. Although the coupling between
BTE solution) and compares it with experimental mea- electrons and the AFD soft mode at R is also strong [see
 4

(arb. unit)
(a) (b) 0.05 0.25 0.9

300 K 200 K 150 K

Int.
exp
exp(ax)*x**(bx)
0 0 0
0 50 100 150 0 50 100 150
computed50
0 100 150

Electron mobility (cm2/Vs) n = 3.12 exp(at)*x**(bt)

LO-1 LO-1 LO-1
1022 n 2 22 2

Scattering Rate (THz)

(THz)

Scattering Rate (THz)

10 µ⇠T 10
102 10
10 10
102

Rate(THz)
LO-2 LO-2 LO-2
n = 3.09

Scatteringrates
1
10
101 10
1
101 1011
10

Scattering
1011
10
Soft
Soft
This work Soft
Cain et al. LA
0 00
10
100 10
10 LA 1000
10 LA
1000
10
150 200 250 300 00 50
50 100
100 150
150 00 50
50 100
100 150
150 00 50
50 100
100 150
150
Temperature (K) E - ECBM (meV)
EE- -EE CBM (meV)
CBM (meV) EE -- E
ECBM (meV)
CBM (meV) E - ECBM (meV)

FIG. 3. (a) Computed electron mobility as a function of temperature (blue circles), compared with experimental values
(black squares) taken from Ref. [10]. (b) Mode-resolved e-ph scattering rates, as a function of conduction band energy, at
temperatures of 300 K, 200 K and 150 K (one per panel from left to right). The scattering rates are given for the two LO
modes, the ferroelectric soft mode and the LA mode labeled in Fig. 2(b). The integrand in Eq. (2), which shows how much
electronic states at a given energy contribute to transport, is also plotted at each temperature. The zero of the energy axis is
the CBM.

Fig. 2(b)], the AFD soft mode does not scatter electrons lowest order is needed to more accurately compute the
appreciably due to a lack of scattering phase space near mobility in the polaron transport regime. One expects
the CBM. Our accurate treatment of soft phonons and that including higher-order e-ph interactions would
their temperature-dependent e-ph scattering is crucial suppress the e-ph RTs and lower the computed mobility
to gain these new microscopic insights. towards the experimental value. While developing an ab
As seen in Fig. 3(a), the computed mobility is almost initio theory of polaron transport will be the subject of
an order of magnitude higher than experiments. We future work, it is clear that the temperature dependence
maintain that our results are accurate within the of the mobility agrees well with experiment within our
band-like picture of transport and the lowest-order of lowest-order approach.
perturbation theory in the e-ph interaction. Note that In summary, we developed a first-principles approach
our calculations include SOC effects and a bandstructure to compute e-ph interactions and charge transport in
with accurate electron effective masses [22], use phonons materials with phase transitions and anharmonic lattice
in excellent agreement with experiments (including the dynamics. Accurately treating the soft modes reveals
soft modes), treat scattering from all phonon modes on the origin of the T −3 temperature dependence of the
the same footing, are carefully converged using ultra-fine electron mobility in cubic SrTiO3 , which we show to be
grids, and employ an accurate iterative solution of BTE due to the combined e-ph scattering from LO and soft
to obtain the mobility. We found previous work that ferroelectric modes. Our work paves the way to studying
reported ab initio calculations of the room temperature charge carrier dynamics in broad classes of materials
electron mobility of SrTiO3 in good agreement with ex- with anharmonic phonons, including perovskite oxides,
periment [37]. In that work, the mobility was computed metal-halide perovskites and chalcogenides.
within the RT approximation, without including SOC
effects or soft modes, and by including only LO phonon This work was supported by the Joint Center
scattering with an approximate treatment. Our results for Artificial Photosynthesis, a DOE Energy Innovation
show that accurate calculations within ab initio band Hub, supported through the Office of Science of the U.S.
theory and the lowest-order e-ph interaction significantly Department of Energy under Award No. de-sc0004993.
overestimate the electron mobility in SrTiO3 . M.B. acknowledges support by the National Science
We argue that the discrepancy between the measured Foundation under Grant No. ACI-1642443, which
and the ab initio band-like mobility is due to polaron provided for basic theory and electron-phonon code
effects [43], which are known to occur in oxide crystals development. O.H. acknowledge support from the
like SrTiO3 with mobilities of less than ∼10 cm2 /V s at EFRI-2DARE program of the National Science Founda-
room temperature [44]. The presence of a large polaron tion, Award No. 1433467. This research used resources
in SrTiO3 is well-known experimentally [45, 46]. A of the National Energy Research Scientific Computing
theory that includes strong e-ph interactions beyond the Center, a DOE Office of Science User Facility supported
 5

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