Supplemental Material for:
Respective Roles of Electron-Phonon and Electron-Electron Interactions
in the Transport and Quasiparticle Properties of SrVO
3
David J. Abramovitch,
1, 2
Jernej Mravlje,
3, 4
Jin-Jian Zhou,
5
Antoine Georges,
6, 2
and Marco Bernardi
1,
∗
1
Department of Applied Physics and Materials Science, and Department of Physics,
California Institute of Technology, Pasadena, California 91125
2
Center for Computational Quantum Physics, Flatiron Institute,
162 5th Avenue, New York, New York 10010, USA
3
Joˇzef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
4
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
5
School of Physics, Beijing Institute of Technology, Beijing, 100081, China
6
Coll`ege de France, Paris, France and CCQ-Flatiron Institute, New York, NY, USA
I. COMPUTATIONAL DETAILS
A. DFT, DFPT, and Transport Calculations
We calculate the electronic structure, lattice vibrations, and electron-phonon (
e
-ph) perturbation potentials using
DFT and DFPT with the
Quantum Espresso
package [1]. In all calculations, we use the experimental lattice
parameter of 3.842
̊
A [2, 3], together with a 12
×
12
×
12 electronic
k
-point grid and a 6
×
6
×
6
q
-point phonon
grid, a plane-wave kinetic energy cutoff of 100 Rydberg energy, and the PBEsol exchange-correlation functional [4].
We project the electronic structure onto the three t
2
g
orbitals using Wannier90 [5]. We use the
Perturbo
code to
compute and interpolate the
e
-ph coupling matrix elements and to calculate the
e
-ph self-energy, spectral function,
and transport properties [6]. The Green-Kubo based transport calculations use a 80
3
k
-point grid with 200,000
q
-
points randomly sampled in the Brillouin zone. For Boltzmann transport equation calculations, we use a 120
3
k
-point
grid above 100 K, and a 200
3
k
-point grid below 100 K, together with a 40
3
q
-point grid.
B. DMFT Calculations
We calculate the electron-electron scattering contribution to the self energy within DFT+DMFT as implemented
in TRIQS package [7] using a first principles extension [8] and Pad ́e analytical continuation [7]. The DFT calculations
are performed with Wien2k [9] using the experimental lattice parameter of 3.842
̊
A and the PBE exchange-correlation
functional. The local orbitals for the DMFT calculations are constructed within the energy window [
−
1
.
5
,
1
.
9] eV
covering the t
2
g
bands, which are evaluated on a 50
3
k
-point grid. We use interaction parameters
U
= 4
.
5 eV and
J
= 0
.
675 eV in the Kanamori convention. The impurity problem is solved for each temperature using the continuous-
time quantum Monte Carlo hybridization expansion solver [7, 10, 11] with 200 Monte Carlo steps per measurement
cycle and 2
.
56
×
10
9
total cycles and a sufficiently long thermalization of 4
×
10
6
Monte Carlo cycles.
C. Fermi Surface Averaged
e
-ph Coupling
The average
e
-ph coupling on the Fermi surface,
|
g
ν
(
q
)
|
shown in Fig. 1(c) in main text, is calculated as
|
g
ν
(
q
)
|
=
s
1
N
(
ε
F
)
X
mn
k
|
g
νmn
(
k
,
q
)
|
2
×
δ
(
ε
n
k
−
ε
F
)
(1)
where
ε
n
k
is the band energy,
ε
F
is the Fermi energy,
N
(
ε
F
) is the electronic density of states at the Fermi energy,
and
g
νmn
(
k
,
q
) is the
e
-ph coupling matrix element for electron bands
n
and
m
, electron momenta
k
and
k
+
q
, and
phonon mode
ν
and momentum
q
[6].
∗
bmarco@caltech.edu
2
II. HIGHER-ENERGY BEHAVIOR OF THE SPECTRAL FUNCTION AND SELF-ENERGIES
FIG. S1. Spectral functions and self-energies of SVO as in Fig. 2 of main text, but shown in a larger energy window from the
Fermi surface. Here, the kinks at
∼
60 meV can be seen more clearly as they contrast the dispersion of the rest of the quasiparticle
spectrum. In addition, the
e
-
e
contribution to the imaginary self-energy becomes larger than the
e
-ph contribution at higher
energies. Based on the Kramers-Kronig relations, this explains why the
e
-
e
interactions dominate band renormalization.
3
III. EFFECTS OF HIGHER-ORDER ELECTRON-PHONON INTERACTIONS AND VERTEX
CORRECTIONS ON THE RESISTIVITY
First, we examine the use of the lowest-order approximation to compute the
e
-ph self-energy. While this approxima-
tion is typically accurate for metals, SVO has a smaller bandwidth, greater phonon energy, and greater
e
-ph coupling
than many conventional metals, and thus higher-order contributions may play a greater role [12, 13]. We compute the
e
-ph limited resistivity using a cumulant resummation method [14] capable of treating delocalized (large) polarons due
to intermediate-strength
e
-ph interactions. The results are plotted in Fig. S2. We find that the cumulant resummation
increases the resistivity by only
∼
13% at 290 K and a somewhat larger amount at lower temperatures, indicating
that polarons play a negligible role in SVO. While a more detailed study of polarons in strongly correlated metals
such as SVO is beyond the scope of this paper, this calculation suggests that using the lowest-order
e
-ph self-energy
is sufficient to describe
e
-ph interactions in SVO.
Second, we examine our neglect of vertex corrections in the Green-Kubo calculations [15, 16]. Vertex corrections
improve the description of backscattering and the momentum dependence of
e
-ph scattering, which is particularly
important at low temperature, where transport is dominated by small-momentum acoustic phonons [15]. While vertex
corrections are more difficult to include in Green’s function based methods, they can be included straightforwardly
in the semiclassical limit by solving the full Boltzmann transport equation (BTE). Conversely, the relaxation time
approximation (RTA) of the BTE neglects vertex corrections, and thus we can assess the role of vertex corrections by
comparing the resistivity obtained from the RTA and the full solution of the BTE. The resistivity calculated with these
two methods is plotted in Fig. S2. We find that including vertex corrections (full BTE) increases the resistivity only
by a small amount (less than
∼
10% in the temperature range studied here) and makes the temperature dependence
slightly more pronounced. Our calculations show that optical phonons dominate
e
-ph scattering above 100 K (see SM
section IV), further justifying neglecting vertex corrections in this temperature range.
FIG. S2. The
e
-ph limited resistivity as a function of temperature, computed using different approximations. Comparing the
lowest-order (Fan-Migdal) and cumulant approximations shows the effect of higher-order
e
-ph interactions, while comparing
the full BTE and RTA shows the effect of vertex corrections in the semiclassical limit. The results show that both effects are
negligible in SVO.
4
IV. TEMPERATURE DEPENDENT CONTRIBUTION OF DIFFERENT PHONON MODES TO
ELECTRON-PHONON SCATTERING
A. Origin of the Temperature Dependence
We examine the
e
-ph limited resistivity in SVO. In the 50
−
200 K temperature range, we find that all phonon
momenta contribute to the resistivity, although some optical phonons with energies between 15
−
90 meV and strong
e
-ph coupling are frozen out. This represents an intermediate regime between the low-temperature limit, with acoustic
phonon-limited
T
5
resistivity, and the high-temperature limit with all phonon modes populated and
T
-linear resistivity.
In this regime, where the momentum dependence of
e
-ph scattering is not critical, we can use the RTA to approximate
the resistivity [6]:
ρ
−
1
RTA
= 2
e
2
X
n
Z
d
k
v
2
n
k
τ
n
k
−
∂f
0
n
k
(
T
)
∂ε
n
k
,
(2)
where
ε
n
k
is the band energy,
v
n
k
is the band velocity,
τ
n
k
= (2ImΣ
e
−
ph
n
k
(
ω
=
ε
n
k
))
−
1
is the electron relaxation time
due to
e
-ph interactions, and
f
0
n
k
(
T
) is the Fermi-Dirac electron occupation at temperature
T
.
If we assume the band and momentum dependence of
e
-ph scattering is not important for understanding the
temperature dependence, it is informative to simplify the RTA with an effective relaxation time
τ
and electron
velocity
v
, defined as:
τ
=
⟨
τ
n
k
⟩
=
P
n
R
d
k
τ
n
k
v
2
n
k
−
∂f
0
n
k
(
T
)
∂ε
n
k
P
n
R
d
k
v
2
n
k
−
∂f
0
n
k
(
T
)
∂ε
n
k
(3)
and
v
2
=
⟨
v
2
n
k
⟩
=
P
n
R
d
k
v
2
n
k
−
∂f
0
n
k
(
T
)
∂ε
n
k
P
n
R
d
k
−
∂f
0
n
k
(
T
)
∂ε
n
k
(4)
and replace the band and momentum indices with energy to rewrite the RTA resistivity as
ρ
−
1
RTA
=
e
2
τv
2
Z
dεN
(
ε
)
−
∂f
0
(
ε,T
)
∂ε
≈
e
2
τv
2
N
(
ε
F
)
,
(5)
where
N
(
ε
F
) is the electron density of states at the Fermi energy. Because the band velocity
v
2
is weakly temperature
dependent, the temperature dependence of
ρ
RTA
is set by that of
τ
−
1
.
In Fig. S3(a) below, we calculate the effective scattering rate
τ
−
1
using only phonons with energy up to
ω
ph
. (This is
done by evaluating the
e
-ph self-energy, and consequently
τ
n
k
in Eq. (3) above, using only a subset of phonons.) This
“cumulative” scattering rate shows that transport is dominated by low-energy phonons with energies between 10
−
30
meV at 50
−
100 K, and by optical phonons with energy greater than 30 meV at temperatures above
∼
100 K. Higher-
energy phonons give a greater relative contribution to
e
-ph scattering as temperature increases between 50
−
200 K,
while above 200 K the relative phonon contributions remain nearly unchanged. As shown in Fig. S3(b), this leads
to a scaling of the effective scattering rate
τ
−
1
somewhat faster than
T
2
in the 50
−
200 K temperature range where
higher-energy phonons become progressively occupied, explaining the
T
2
trend of the
e
-ph limited resistivity in
that temperature range. Conversely, between 200
−
300 K the relative contributions of different phonons are nearly
temperature independent, and the scattering rate becomes closer to linear in
T
.
This explanation is confirmed by computing the RTA resistivity including only phonons with energy up to
ω
ph
[see
Fig. S3(c)]. In this case, the resistivity exhibits the
T
2
behavior up to
∼
200 K only when higher-energy phonons are
included.
5
FIG. S3. (a) Cumulative effective scattering rate
τ
−
1
(
ω
ph
) computed using phonons with energy up to
ω
ph
in the weighted
average in Eq. (3). The results are expressed as a fraction of the total
τ
−
1
computed using all phonons, and are shown
for temperatures between 50
−
300 K. (b) Total
τ
−
1
as a function of temperature, showing a stronger (near
T
2
) temperature
dependence between 50
−
200 K than in the 200
−
300 K temperature range. (c) RTA resistivity as a function of temperature
computed using only phonons up to a certain energy, which is indicated for each curve.
6
V. DFPT+
U
CALCULATIONS
We calculate Hubbard-corrected phonon dispersions and
e
-ph coupling with DFPT+
U
using the
Quantum
Espresso
package [1, 17] with a Hubbard-
U
parameter of 3 eV for the vanadium
d
electrons. This value is lower than
then Hubbard-
U
parameter computed
ab initio
by linear response [18] (
∼
5.1 eV) and is chosen for several reasons.
First, a lower value of
U
is needed to compensate for the lack of a Hund’s
J
parameter in the DFPT+
U
calculation
(see Fig. S4a), as DFPT+
U
+
J
is currently not available. Second, due to the technical differences between DFT+
U
and DMFT, including different Hubbard orbitals and the static approximation inherent to DFT+
U
, a different value
of
U
may be needed to best align with DMFT predictions.
As discussed below, we use supercell finite-difference calculations to estimate the effect of the Hund’s parameter
J
and find that the electronic response to a selected lattice perturbation at the
ab initio
value
U
= 5
.
1 eV together with
an optimal nonzero value of
J
is similar to the response at
U
= 3
−
4 eV and
J
= 0 eV. Furthermore, we a perform
(charge self-consistent) DMFT finite displacement calculation with the values of
U
= 4
.
5 eV and
J
= 0
.
675 eV in the
wannier orbital basis as used in our other DMFT calculations. This produces and orbital occupation response very
similar to
U
= 3 eV and
J
= 0 eV. Therefore, we elect to use
U
= 3 eV in our DFPT+
U
calculations. We note that
this choice is somewhat empirical by necessity, because neither DFPT+
U
+
J
or DMFT
e
-ph coupling calculations
currently exist.
A. Effect of the Hubbard
U
Parameter
To better understand the effect of the Hubbard-
U
parameter, we carry out finite displacement calculations for the
q
= M phonon mode with
∼
65 meV energy (the second highest energy mode at M). This mode shows strong
e
-ph
coupling with significant correlation enhancement. Physically, it involves VO
6
octahedra stretching and contracting
along the in-plane
x
and
y
axes in alternating unit cells. We prepare a 2
×
2
×
2 supercell with atoms displaced according
to this phonon mode by 0
.
005
a
, where
a
is the cubic lattice parameter of a single unit cell; this displacement is
comparable to the displacements associated with a single phonon. We perform Hubbard-corrected DFT in
Quantum
Espresso
[1] and calculate the
t
2
g
Wannier orbital occupation using using TRIQS/DFTTools [8] for values of
U
from
0 to 5 eV and for two values of
J
, respectively
J
= 0 and
J
= 0
.
15
U
. We also perform (charge self-consistent) DMFT
calculations on the same supercell with TRIQS [19][20] with
U
= 4
.
5 eV,
J
= 0
.
675 eV as in the main text.
The occupation of
t
2g
d
-orbitals obtained from these finite-displacement calculations is shown as a function of
U
in Fig. S4a, for both
J
= 0 and
J
= 0
.
15
U
. Changes in
d
-orbital occupation characterize the orbital polarization
response to the phonon mode. For
J
= 0, we find a smoothly increasing
d
-orbital occupation as a function of
U
up
to
U
= 4 eV with a larger change at higher values. The inclusion of a finite Hund’s coupling parameter
J
(dashed
FIG. S4. (a) Occupation of the three vanadium
t
2g
wannier orbitals as a function of Hubbard parameter
U
, for
J
= 0 (solid
curves) and
J
= 0
.
15
U
(dashed curves). The presence of the Hund’s parameter
J
decreases the orbital polarization response
to the phonon. (b) Same as (a), but with the DFT+
U
+
J
calculations replaced by horizontal lines indicated the orbital
occupation found with (charge self-consistent) DMFT calculations performed with TRIQS
7
curves in Fig. S4a below) decreases the orbital polarization. In particular, the orbital polarization for
U
= 5 eV and
J
= 0
.
75 eV (a value of
U
close to the 5.1 eV value computed
ab initio
) is similar to the polarization at
U
≈
3
−
4 eV
with
J
= 0. For this reason, we choose
U
= 3 eV for our DFPT+
U
calculations. We have also verified that in the
absence of atomic displacements there is no orbital polarization for any value of
U
studied.
In Fig. S4b, we show the
t
2g
occupation for the DFT+
U
(
J
= 0) calculations along with the orbital occupation
from DMFT. The DMFT orbital occupation is very similar to that found with DFT+
U
with
U
= 3 eV.
Finally, the band structure of the t
2g
orbitals in supercells with and without finite phonon displacements shows a
similar trend. Similar to the case of orbital polarization, the band shift caused by the phonon displacements increases
as a function of
U
, and is lower when a finite
J
is included. Again, the shifts with
U
= 5 eV and
J
= 0
.
75 eV are
closest to those with
U
= 3
−
4 eV and
J
= 0.
VI. SELF-ENERGY AND SPECTRAL FUNCTIONS WITH DFPT+
U
E-PH COUPLING
FIG. S5. Self-energies and spectral functions as in Fig. 2 of the main text but calculated with
e
-ph coupling from DFPT+
U
.
(a) Real and (b) imaginary parts of the self-energy, showing contributions from
e
-
e
and
e
-ph interactions. (c) Spectral function
including both
e
-
e
and
e
-ph interactions, (d)
e
-
e
interactions only, and (e)
e
-ph interactions only. Quasiparticle weights
Z
are
indicated for each spectral function. The spectral function are computed along Γ
−
X
at 115 K.
8
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