Respective Roles of Electron-Phonon and Electron-Electron Interactions in the Transport and Quasiparticle Properties of SrVO3

Respective Roles of Electron-Phonon and Electron-Electron Interactions

in the Transport and Quasiparticle Properties of

SrVO

David J. Abramovitch,

1,2

Jernej Mravlje ,

3,4

Jin-Jian Zhou ,

Antoine Georges ,

6,2,7,8

and Marco Bernardi

Department of Applied Physics and Materials Science, and Department of Physics,

California Institute of Technology

, Pasadena, California 91125, USA

Center for Computational Quantum Physics,

Flatiron Institute

, 162 5th Avenue, New York, New York 10010, USA

ž

ef Stefan Institute

, Jamova 39, 1000 Ljubljana, Slovenia

Faculty of Mathematics and Physics,

University of Ljubljana

, Jadranska 19, 1000 Ljubljana, Slovenia

School of Physics,

Beijing Institute of Technology

, Beijing 100081, China

Coll`

ege de France

, Paris, France

Centre de Physique Th ́

eorique, Ecole Polytechnique, CNRS,

Institut Polytechnique de Paris

, 91128 Palaiseau Cedex, France

DQMP,

Universit ́

edeGen`

eve

, 24 quai Ernest Ansermet, CH-1211 Gen`

eve, Switzerland

(Received 15 April 2024; accepted 23 September 2024; published 29 October 2024)

The spectral and transport properties of strongly correlated metals, such as SrVO

(SVO), are widely

attributed to electron-electron (

) interactions, with lattice vibrations (phonons) playing a secondary role.

Here, using first-principles electron-phonon (

-ph) and dynamical mean field theory calculations, we show

that

-ph interactions play an essential role in SVO: they govern the electron scattering and resistivity in a

wide temperature range down to 30 K, and induce an experimentally observed kink in the spectral function.

In contrast, the

interactions control quasiparticle renormalization and low temperature transport, and

enhance the

-ph coupling. We clarify the origin of the near

temperature dependence of the resistivity by

analyzing the

and

-ph limited transport regimes. Our work disentangles the electronic and lattice

degrees of freedom in a prototypical correlated metal, revealing the dominant role of

-ph interactions

in SVO.

DOI:

10.1103/PhysRevLett.133.186501

Introduction

—

Strontium vanadate, SrVO

(SVO), is a

perovskite oxide widely studied as a prototypical correlated

metal

–

. Experiments have measured transport and

spectral functions in detail in SVO, owing to advances in

growth of clean samples

[4,5]

and characterization by

angle-resolved photoemission spectroscopy

–

. There

are clear spectroscopic signatures of strong electron inter-

actions in SVO, including kinks in the quasiparticle

dispersion

[8,9]

and mass enhancement with quasiparticle

weight

≈

[6]

. In addition, transport measurements

have found a near

-dependent resistivity in broad

temperature ranges below 300 K

[10

–

14]

These findings are often attributed to strong electron-

electron (

) interactions. As a result, SVO serves as a test

bed for theoretical methods treating strongly correlated

materials, including first-principles variants of dynamical

mean field theory (DMFT) such as density functional

theory

DFT

Þþ

DMFT

[15]

DMFT

[16

–

19]

, and

linear response DMFT

[20]

, and the dynamical cluster

[21]

and dynamical vertex approximations

[22]

However, one can question whether the transport pro-

perties and spectral features observed in SVO are the

result of purely electronic interactions. In particular,

electron-phonon (

-ph) interactions may also play a role

in SVO, as they do in other correlated metals where

experiments

[23

–

25]

and theory

[26

–

28]

have highlighted

the importance of

-ph coupling for spectral kinks

[23,29]

and electronic transport

[30]

. A quantitative study combin-

ing

and

-ph interactions in SVO is needed to clarify the

microscopic origin of its electronic behavior.

In this Letter, we show calculations of spectral and

transport properties in SVO combining first-principles

-ph

interactions with DFT

DMFT

interactions

[30]

.We

find that

-ph interactions govern the resistivity and its

temperature dependence above

∼

K, and account for the

experimentally observed kinks and for most of the line-

width broadening of the spectral functions. In contrast, the

interactions control the resistivity below 20 K, and are

responsible for most of the quasiparticle mass renormal-

ization. We also find that the

interactions lead to an

enhancement of the effective

-ph coupling. Our results

provide a blueprint for quantifying electronic and lattice

contributions to the properties of correlated metals.

Electronic structure and electron-phonon coupling

—

calculate the electronic structure, phonon dispersions, and

-ph coupling using DFT and density functional perturba-

tion theory (DFPT) with the

QUANTUM ESPRESSO

package

[31

–

34]

. We use the experimental lattice parameter of

3.842 Å

[2,13]

and project the electronic structure onto the

Contact author: bmarco@caltech.edu

PHYSICAL REVIEW LETTERS

133,

186501 (2024)

0031-9007

133(18)

186501(7)

186501-1

orbitals of vanadium

[35]

. We use

PERTURBO

compute the

-ph interactions,

-ph self-energy, spectral

functions, and transport

[36]

. The

self-energy is

obtained with DFT

DMFT using the

TRIQS

code with

a continuous-time quantum Monte Carlo solver

[37

–

42]

and Pad ́

e analytical continuation

[38]

. We use Hubbard-

Kanamori interactions with

eV and

675

eV to obtain band renormalization and quasiparticle

weights in agreement with experiments

–

. Additional

computational details are provided in the Supplemental

Material (SM)

[43]

As shown in Fig.

1(a)

, our DFT calculations predict a

bandwidth of 2.5 eV for the t

electronic bands, which is

renormalized by a factor

≈

by DMFT, in agreement

with experiments

[6]

and previous DMFT results

[15,17]

.In

the temperature range we study (

∼

115

–

390

K), the imagi-

nary part of the

self-energy, Im

−

, follows a Fermi

liquid behavior. Figure

1(b)

shows that Im

−

within

100 meVof the Fermi energy can be fit closely by a Fermi

liquid parameterization

[44]

,Im

−

Þ¼

−

½ð

ℏ

with

≈

−

[45]

. Therefore, based on the

Kramers-Kronig relations, Re

−

and the quasipar-

ticle dispersion near the Fermi energy depend weakly on

temperature.

Figure

1(c)

shows the phonon dispersions in SVO

computed with DFPT and color-coded according to the

-ph coupling strength

Þj

, for each phonon mode

and

momentum

, averaged over the Fermi surface (see SM

[43]

). The

-ph coupling is stronger for the six highest-

energy modes, which involve distortions of the VO

octahedra, such as Jahn-Teller modes.

Spectral properties

—

We investigate the contributions of

-ph and

interactions in SVO by computing the

corresponding self-energies

[30]

. The real and imaginary

parts of the

and

-ph self-energies at 115 K are shown

in Figs.

2(a)

and

2(b)

, respectively. The

interactions

dominate quasiparticle renormalization, as seen from the

greater derivative of Re

−

compared to Re

−

within

150 meV of the Fermi energy. Accordingly, extracting

quasiparticle weights

¼f

−

∂

∂

−

with a fit near the Fermi surface, gives a weak contribution

to renormalization for

-ph interactions (

−

) and

a dominant contribution for

interactions, with

−

and

both

The imaginary part of the self-energy shows an opposite

behavior: Im

−

is much greater than Im

−

, and thus

the

-ph interactions account for the majority of electron

scattering and spectral width at low energy. The dominant

role of

interactions on quasiparticle renormalization in

SVO, despite their small effect on low energy scattering,

can be rationalized using the Kramers-Kronig relations

[44,46]

: due to the larger energy scales involved, the

interactions dominate the imaginary part of the self-energy

at higher energies (see SM

[43]

), leading to a greater

magnitude (and energy derivative) of Re

−

compared

to Re

−

at low energy.

We compute the spectral function

Þ¼

−

from the Green

’

s function

FIG. 1. (a) DFT electronic band structure (black) and the

spectral function computed with DMFT at 290 K (blue), showing

renormalization by a factor

≈

. (b) Imaginary part of the

electron self-energy due to

interactions, computed with

DMFT. The lines show a fit to the Fermi liquid form,

−

Þ¼

−

½ð

ℏ

with

≈

−

the

-ph coupling strength

Þj

averaged on the Fermi surface.

FIG. 2. (a) Real and (b) imaginary parts of the self-energy at

115

K and

¼ð

=a;

;

, showing contributions

from

and

-ph interactions. (c) Spectral functions including

both

and

-ph interactions, (d)

interactions only, and (e)

ph interactions only. Quasiparticleweights

are indicated for each

spectral function, and the dashed lines in (c) guide the eye to the

quasiparticle dispersion near the kink. All spectral functions are

shown along the

direction near the Fermi surface at 115 K.

PHYSICAL REVIEW LETTERS

133,

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Þ¼½

−

−

−

at energy

for electron band

and momentum

. Here,

is the DFT band energy,

is the Fermi energy, and

is the electron self-energy. Following our previous work

[30]

, in separate calculations we compute this Green

’

function using the self-energy from DMFT

interactions,

the lowest-order self-energy from

-ph interactions, and

their sum

[47]

, obtaining corresponding spectral functions

capturing different combinations of interactions [Figs.

2(c)

–

2(e)

]. The spectral functions from

-ph, and those from

-ph

plus

interactions, show a kink around 60 meV from the

Fermi energy that has been observed in experiments

[8,9]

There is a corresponding sharp change in the derivative of

−

at this energy [Fig.

2(a)

], whereas this feature is

absent in Re

−

. This result shows that the 60 meV

kink observed experimentally in SVO is caused by

-ph

interactions.

Transport

—

Numerous experiments have measured a

near

temperature dependence of the resistivity in

SVO below 300 K

[2,4,11,13,48

–

52]

. Because of the

strong electronic correlations in SVO, several studies have

attributed this resistivity to

interactions in the Fermi

liquid regime

[11,48,52]

, where

behavior is expected.

An exception is recent work by Mirjolet

et al.

, who argued

that the temperature dependence is better explained by

-ph

limited resistivity with strong coupling to a dominant

phonon mode

[12]

. Recently, the growth of ultraclean

samples has enabled detailed measurements of the resis-

tivity with reduced defect scattering

[4,13]

. In these

samples, Ahn

et al.

[13]

and Brahlek

et al.

[14]

find a

near-

resistivity below 25 K and between about 100

–

300 K, together with a stronger than

temperature

dependence at intermediate temperatures.

To understand the microscopic origin of this behavior,

we compute the resistivity arising from

-ph and

interactions using the Green-Kubo formula

[30,53]

−

αβ

Þ¼

ℏ

−

;

where

αβ

is the resistivity tensor,

and

are Cartesian

directions,

is the energy derivative of the Fermi

occupation factor,

is the band velocity, and

the spectral function. The resistivity for different combi-

nations of interactions is shown in Fig.

3(a)

and compared

with experimental data

[11,13,14]

Surprisingly, we find that the resistivity is governed by

the

-ph interactions in SVO. The

-ph limited resistivity is

an order of magnitude greater than the

limited resis-

tivity, with the latter accounting for only

∼

10%

of the

experimental value. This result is in contrast with the

conventional wisdom that transport properties in SVO are

governed by purely electronic interactions. In addition, the

contributions are opposite to another prototypical strongly

correlated metal, Sr

RuO

, where the

-ph interactions

account for only

∼

10%

of the resistivity

[30]

. In SVO, the

-ph contribution is similar in magnitude to the exper-

imental value, and the total resistivity including both

interactions is in good agreement with experiments.

Interestingly, Im

−

and the

-ph limited resistivity are

similar in SVO and Sr

RuO

, a result consistent with their

similar low-energy electronic structure governed by

orbitals. Below, we show that taking into account the

electron correlation induced enhancement of the

-ph

interactions increases the resistivity and brings the results

in even better agreement with experiments.

The temperature dependence of the

-ph limited resis-

tivity is analyzed in more detail in Fig.

3(b)

, where we show

our results on a log-log plot and compare them with

experiments

[11,13]

. In that plot, the resistivity is computed

with the full (iterative) solution of the Boltzmann transport

FIG. 3. (a) Resistivity as a function of temperature calculated using the Green-Kubo formalism with

interactions,

-ph interactions,

and their combination. (b) Temperature dependence of the

-ph limited resistivity calculated using the full (iterative) solution of the

BTE. (c) Low-temperature

-ph and

limited transport. Experimental data (from which the

residual resistivity was subtracted)

are from Refs.

[11,13,14]

PHYSICAL REVIEW LETTERS

133,

186501 (2024)

186501-3

equation (BTE)

[36]

to include backscattering and improve

the treatment of acoustic phonons. The computed

-ph

limited resistivity follows a

temperature dependence

between 100

–

200 K, in excellent agreement with the

−

dependence found in experiments in that temperature range

[11,13,14]

, and falls to

at 300 K. We identify the origin

of this nearly

temperature trend of the

-ph limited

resistivity by analyzing the contribution of different phonon

modes. Our calculations show that the increasing contri-

bution of strongly coupled optical phonons at higher

temperatures is responsible for the

dependence of the

resistivity between 100

–

200 K (see SM

[43]

Next, we focus on transport at low temperature, where

the

-ph contribution is expected to be weaker. While

DMFT calculations become difficult at low temperatures,

we obtain the

limited resistivity by extrapolating our

higher-temperature DMFT calculations with a

fit. This

approach is justified because the

scattering is in the

Fermi liquid regime below at least 400 K. Figure

3(c)

shows the computed

-ph and DMFT

limited resistiv-

ities below 100 K. We find a clear crossover between 20

–

30 K from

-ph to

dominated transport. The

-ph

limited resistivity becomes much smaller than the

limited resistivity below 20 K, showing that

scattering

governs transport at low temperature. This result indicates

that

interactions are the origin of the

resistivity

observed experimentally below 25 K, with the

-ph

contribution causing deviations from a

behavior above

25 K. Note that our DMFT resistivity underestimates the

experimental value below 25 K by a factor of 2

–

[14]

We attribute this discrepancy to limitations of single-

site DMFT

[54,55]

, which employs a local, and thus

-independent,

self-energy. Including nonlocal correla-

tions is expected to improve the description of

-dependent

scattering, which controls transport at low temperature.

Correlation-corrected electron-phonon interactions

—

Strong electronic interactions are known to significantly

modify

-ph interactions

[56,57]

. In correlated metals,

-ph

coupling is often enhanced. For example, calculations using

hybrid functionals and the

method found correlation-

enhanced

-ph coupling in unconventional superconduc-

tors, attributing the enhancement to decreased electronic

screening

[26]

. Similarly, in multiband

-electron systems

such as FeSe, treating correlations with DMFT enhances

-ph coupling, in this case by increasing the orbital

polarization response to phonon perturbations

[27]

To study the role of correlations in SVO, we compute

the

-ph interactions using Hubbard-corrected DFPT

(DFPT

)

[33,34]

, which captures the strong local

interactions between

orbitals in a static approximation

and accounts for the resulting change in orbital polarization

response. We use a Hubbard-

parameter of 3 eV, which

provides orbital polarization responses to phonon pertur-

bations similar to our DMFT settings (

eV and

) and to a calculation using a Hubbard-

parameter computed from linear-response theory

[58]

(

eV) in combination with Hund

’

s coupling

(see SM

[43]

Adding the Hubbard correction has a small effect on the

phonon dispersions, but it enhances the

-ph interactions,

as shown in Fig.

4(a)

. The enhancement is mode-dependent

and is generally higher for strongly coupled optical

phonons involving VO

distortions. The enhancement is

also higher for phonons with momenta away from the

point, suggesting a more important role of correlations for

distortions breaking lattice-translation symmetry. The spec-

tral and transport properties computed with enhanced

-ph

coupling from DFPT

give results qualitatively similar

to those from DFPT

[43]

, but with stronger

-ph effects.

Notably, the

-ph limited resistivity increases by

∼

35%

room temperature [Fig.

4(b)

], bringing the resistivity

computed with both

-ph and

interactions into very

good agreement with experiments. For example, the com-

puted resistivity at 290 K is

μΩ

cm, versus an exper-

imental value of

–

μΩ

[11,13,14]

Discussion

—

The origin of the temperature dependence

of the resistivity merits further discussion. While the

trend for

interactions is expected based on Fermi liquid

theory

[53]

, the origin of the near-

behavior of the

-ph

limited resistivity is less clear. At very low temperatures,

the

-ph limited resistivity in metals is expected to exhibit a

temperature dependence

[59]

when scattering is domi-

nated by acoustic phonons with momentum

∝

.In

our calculations, we find a

-ph limited resistivity below

∼

K, but the overall resistivity becomes

limited in

this temperature range, explaining the experimental

resistivity below 25 K.

In the high-temperature limit, based on the tempera-

ture dependence of the phonon occupations, one expects a

-linear

-ph limited resistivity

[59]

. However, this requires

that all phonon modes contribute equally to

-ph scattering,

with no mode frozen out. While our computed

-ph limited

resistivity becomes nearly

-linear well above 300 K, it is

close to a

behavior between

∼

100

–

200

K, in agreement

FIG. 4. (a) Phonon dispersions and

-ph coupling as in Fig.

1(c)

but calculated with DFPT

. Note the change in

-ph coupling

scale. (b) Transport as in Fig.

3(a)

but calculated with DFPT

phonons and

-ph couplings, showing improved agreement with

experiments

[11,13]

PHYSICAL REVIEW LETTERS

133,

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186501-4

with experiments. As discussed above, this

trend is due to

the increasing contribution of higher-energy optical pho-

nons with strong

-ph coupling for increasing temperatures

[43]

. Note also that

-ph scattering above

∼

KinSVO

involves phonons with all momenta, ruling out momentum-

dependent mechanisms resulting in

behavior

[60]

Finally, we analyze two approximations made in the

-ph

transport calculations (see results in SM

[43]

). First, we

examine the use of the lowest-order

-ph self-energy

[61,62]

by computing the resistivity with a cumulant

diagram-resummation method capable of treating delocal-

ized polarons

[63]

. Including polaron effects leads to a

small increase in the resistivity, showing that lowest-order

-ph interactions are adequate to describe SVO. Second, we

examine the effect of vertex corrections to the current-

current correlation function in the Green-Kubo formalism

[44,53]

. Vertex corrections improve the description of

backscattering and the momentum dependence of

-ph

scattering, which is particularly important at low temper-

ature

[53]

. We assess their role above 100 K in the

semiclassical limit by comparing the full solution of the

BTE, which includes vertex corrections, to the relaxation

time approximation, which neglects them. We find that

vertex corrections in the BTE give only a small increase in

the resistivity and its temperature dependence. This analy-

sis shows that higher-order

-ph interactions and vertex

corrections play a minor role in SVO and do not affect our

conclusions.

Conclusion

—

In summary, we have shown that

-ph

interactions play an essential role in the transport and

spectral properties of a prototypical correlated metal, SVO.

In this material, electronic correlations control other aspects

of the low energy physics, including the quasiparticle mass

renormalization and transport at low temperature. We also

found that electronic correlations lead to an effective

enhancement of the

-ph interactions. This suggests that

SVO may serve as a test bed for investigating the interplay

between electron correlations and

-ph interactions. Our

results highlight the potential of first-principles calculations

combining

and

-ph interactions in a consistent way as

an emerging tool to study correlated materials. This work

paves the way for a quantitative description of transport and

spectral properties in broad classes of correlated quantum

materials.

Acknowledgments

—

We thank Andrew Millis, Jennifer

Coulter, and Roman Engel-Herbert for helpful discussions.

D. J. A. is supported by the National Science Foundation

Graduate Research Fellowship under Grant No. 2139433.

This work was also supported by the National Science

Foundation under Grant No. OAC-2209262, which pro-

vided for code development. D. J. A. and M. B. were

partially supported by the AFOSR and Clarkson

Aerospace under Grant No. FA95502110460. J.-J. Z.

acknowledges support from the National Key R&D

Program of China (Grant No. 2022YFA1403400) and

the National Natural Science Foundation of China

(Grant No. 12104039). J. M. is supported by the

Slovenian Research Agency (ARIS) under Grants

No. P1-0044 and No. J1-2458. This research used resour-

ces of the National Energy Research Scientific Computing

Center, a DOE Office of Science User Facility supported by

the Office of Science of the U.S. Department of Energy

under Contract No. DE-AC02-05CH11231 using NERSC

award NERSC DDR-ERCAP0026831. The Flatiron

Institute is a division of the Simons Foundation.

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