Himmetoglu presents research on modeling polarons and their effects in complex oxide materials like YTiO3 using first-principles calculations. Small polarons form when holes become self-trapped due to strong electron-phonon coupling, leading to localized distortions around the hole. Calculations find two stable configurations for a single hole - a small polaron state and a delocalized hole state. The transition between these states is attributed to the 0.6eV onset seen in optical absorption measurements, rather than representing the fundamental band gap. Accounting for small polaron formation reconciles theoretical predictions of band gaps around 2eV with experimental observations.

1. Ab-initio Studies of Polarons,
Optical Phonons, and Transport:
Case of Complex Oxides
Burak Himmetoglu
ETS & Center for Scientific Computing
University of California Santa Barbara
bhimmetoglu@ucsb.edu
Interband and polaronic excitations
in YTiO3 from first principles
Burak Himmetoglu, Anderson Janotti, Lars Bjaalie
and Chris G. Van de Walle
Materials Department
University of California Santa Barbara
1
d and polaronic excitations
iO3 from first principles
metoglu, Anderson Janotti, Lars Bjaalie
and Chris G. Van de Walle
Materials Department
versity of California Santa Barbara
1
First-Princi
• Density functional theory (DFT)
• Band structure: LDA/GGA func
• Quantum ESPRESSO package
• Phonons: Density functional
perturbation theory (DFPT)
• LO mode frequencies + dielec
constant: Born effective charge
1

2. Electron-Phonon interactions
• Temperature dependence of resistivity in metals and
mobility in semiconductors
• Superconductivity
• Temperature dependence of band energies, band gaps
• Kohn anomalies, structural transitions
• Polaron formation
Numerous phenomena, including but not limited to:
2

3. Polaron Concept
• One of the simplest, and most widely studied model for
interacting electrons and (optical) phonons:
phenomenon by the induction of a polarization around the charge
autolocalization of an electron due to the induced lattice polarizat
L. D. Landau [1]. In the further development of this concept, the
follow the charge carrier when it is moving through the medium. T
the induced polarization is considered as one entity (see Fig. 1). I
S. I. Pekar [2, 3]. The physical properties of a polaron differ from t
polaron is characterized by its binding (or self-) energy E0, an effe
characteristic response to external electric and magnetic fields (e. g
absorption coefficient).
FIG. 1: Artist view of a polaron. A conduction electron in an ionic cryst
repels the negative ions and attracts the positive ions. A self-induced
back on the electron and modifies its physical properties. (From [4].)
9
• A conduction electron (or hole) together with its
self-induced polarization in a polar
semiconductor or ionic crystal:
Devreese, Polarons,
Encyclopedia of Applied Physics, Vol. 14, pp. 383-409 (1996)
• Depending on the radius of the ionic distortion, we can talk
about (loosely) two types of polarons i) small ii) large
• It is a quasi-particle
3

4. Weak coupling and large polarons
• Radius of the polaronic state exceeds the lattice constant
• Propagate as free electrons with enhanced effective mass
e.g.: One longitudinal optical (LO) phonon interacting with a band
electron in a crystal:
Scattering by Phonons
• A specific type of phonon scattering
dominates at RT
• Longitudinal Optical (LO) phonons:
• Transverse vs longitudinal
• TO modes charged planes slide
• LO modes charge accumulation!
Strong Coulomb scattering!!
Dominant for materials with
polar LO modes.
by Phonons
4

5. Interaction potential:
Coupling constant: ↵ =
e2
~ ✏0
✓
mb
2~ !LO
◆1/2 ✓
1
✏1
1
✏
◆
Vq =
~!LO
q
✓
↵
✏0 Vcell
◆1/2 ✓
~
2 mb !LO
◆1/2
Weak coupling and large polarons
Macroscopic polarization field due to one LO phonon interacting with
electron density:
phonon wavevector
phonon frequency
electron band mass
electronic
dielectric
constant
static
dielectric
constant
Fröhlich, Proc. Roy. Soc. A 160, 230 (1937)
“Fröhlich Coupling”
5

6. Weak coupling and large polarons
• We have an interacting electron-phonon system (non-diagonal
Hamiltonian)
• Perform a transformation to obtain the polaron state
(diagonalize the Hamiltonian) at linear oder in coupling
constant:
Polaron mass:
m⇤
' mb/(1 ↵/6)
Weak coupling:
0 < ↵ < 6
• When coupling constant is large, nonlinear effects are
important —> small polarons
6

7. Example Coupling Constants
Table 1: Fr¨ohlich coupling constants
Material α Material α
CdTe 0.31 KI 2.5
CdS 0.52 RbCl 3.81
ZnSe 0.43 RbI 3.16
AgBr 1.6 CsI 3.67
AgCl 1.8 TlBr 2.55
CdF2 3.2 GaAS 0.068
InSb 0.02 GaP 0.201
KCl 3.5 InAs 0.052
KBr 3.05 SrTiO3 4.5
Devreese, Polarons,
Encyclopedia of Applied Physics, Vol. 14, pp. 383-409 (1996)
↵ =
e2
~ ✏0
✓
mb
2~ !LO
◆1/2 ✓
1
✏1
1
✏
◆
• 3 polar LO modes in STO
• Bands are highly non-parabolic, i.e.
mb energy dependent.
Word of caution:
7
!

8. Strong coupling: small polarons
• A macroscopic description of polarization field is invalid
• Typically happens for e-/h+ states with small band-widths, i.e.
localized states.
• Strong crystal distortion around the localized e-/h+ state
8
Rationalizing 0.6 eV onset: small
polarons
Experimental evidence for unintentional p-type doping in RTiO3.
Zhou et.al JCPM 17, 7395 (2007)
Supercell calculations with one hole (i.e. one electron missing)
Two possible solutions are found:
Small Polaron Delocilzed Hole
8
Rationalizing 0.6 eV onset: small
polarons
Experimental evidence for unintentional p-type doping in RTiO3.
Zhou et.al JCPM 17, 7395 (2007)
Supercell calculations with one hole (i.e. one electron missing)
Two possible solutions are found:
Small Polaron Delocilzed Hole
e.g.: h+ polaron in rare-earth titanates (RTiO3):
vs.
Localized h+ accompanied
by strong lattice distortion
within a unit cell
Delocalized h+ band,
small lattice distortion,
over many unit cells length
9
Optical absorption
Holes are self-trapped as
polarons.
Crystal strain energy (ES)
Vertical transition from polaron
to delocalized hole (ET)
Broad absorption below the
fundamental gap.
Configuration coordinate diagram
Atomic deformations around
small polaron
Atomic deformations
around the polaron
8

9. Effects on carrier transport
• Large-polarons: More phonons —> More scattering:
n =
1
e~!LO/kT 1
(kT << ~!LO)
µLO ⇠ 1/n ⇠ e~!LO/kT
Low T: LO phonons
not occupied
High T: LO phonon scattering
effective
Case of SrTiO3
Verma et al. Phys. Rev. Lett. 112, 216601 (2014)
• Temperature dependence signals
various scattering mechanisms
• They are effective at different
regimes of T
• RT mobility: scattering by crystal
vibrations (phonons)
Ziman, “Electrons and phonons: the theory of
transport phenomena in solids”
(Oxford University Press, 2001)
phonon
✏k + ~ ! = ✏k+q
Verma et al. Phys. Rev. Lett. 112, 216601 (2014)
e.g.: n-doped SrTiO3 thin-films
~!LOmax
' 0.1 eV
9

10. Effects on carrier transport
• Small-polarons: More phonons —> More inter-site hopping:
µ ⇠ e ~!LO/kT (kT << ~!LO) (thermally activated)
• Optical conductivity (absorption exp.):
(!) / exp
✓
(2Eb ~!)2
2
◆
polaron
binding energy
photon
energy
Width: Atom site’s
zero point energy: ' (4Eb ~!LO)
1/2
Discussion
GdTiO3 is a Mott insulator. Beyond a critical doping, x ,
system Gd12xSrxTiO3 undergoes a MIT to a metal with
sity significantly larger than the doping concentration.
delta doping far exceeding the critical concentration for G
not render it metallic. Rather, as evidenced by optical an
transport measurements discussed in the following,
remains an insulator; small polaron formation occurs w
tuting an SrO plane for a GdO plane confining the resul
the adjacent (TiO2) planes21–23
.
Our optical measurements do not detect the characte
response from mobile charge carriers. Rather, we obser
carrier-induced absorption band characteristic of sm
Figure 3 | (a) Volume averaged 3D optical conductivity at
conductivity 10 K (blue circle) and 296 K (red square). (b) T
e.g.: SrTiO3 /GdTiO3 interface:
p-doping on the GdTiO3 side
Oulette et al. Sci. Rep. 3, 3284 (2013)
10

11. Small Polarons in RTiO3
Electronic and structural properties
Ti
Y O
Structure:
Orthorhombic perovskite structure with octahedral
distortions.
Electronic properties:
Ferromagnetic ordering of electrons on Ti+3 .
Mott insulator: strongly correlated.
Value of the band gap not clear:
Optical absorption features down to ~ 0.6 eV.
Gössling et.al PRB 78, 075122(2008)
Combined photo-emission data and TiO6
cluster calculations: band gap of 1.77 eV.
Bocquet et.al PRB 53, 1161(1996)
3
• e.g.: YTiO3
• Orthorhombic perovskite with octahedral rotations
Structure:
Electronic Structure:
• Ferromagnetic ordering of electrons on Ti+3
• Mott insulator, strongly correlated
• Value of band gap ?
• Optical absorption features down to ~0.6 eV
• Combined photoemission + TiO6 cluster calculations:
band gap ~ 1.77 eV
Gössling et al. Phys. Rev. B 78, 075122 (2008)
Bocquet et al. Phys. Rev. B 53, 1161 (1996)
11

12. DFT based calculations
• Standard approximate DFT functionals (e.g. LDA, GGA)
• Poor description of e-e interactions
• Suffer from self-interaction
• Tend to over-delocalize electron density
• Hubbard Model based corrective functionals (DFT+U)
• Effective on-site repulsion parameter U on localized orbitals (e.g. d & f)
• U parameter computed self-consistently
Anisimov et al. Phys. Rev. B 44, 943 (1991) and Phys. Rev. B 52, R5467 (1995)
Cococcioni et al. Phys. Rev. B 71, 035105 (2005)
Himmetoglu et al. Int. J. Quantum. Chem. 114, 14 (2013)
• Hybrid functionals (e.g. HSE)
• Partial cancellation of self-interaction
Heyd et al. J. Chem. Phys. 118, 8207 (2003) and J. Chem. Phys. 121, 1187 (2004)
12

13. Electronic Structure of YTiO3
HSE and DFT+U calculations
7
HSE and DFT+U (U=3.70 eV calculated) in remarkable agreement.
Band gap:
𝐸 𝑔 ≈ 2.07 𝑒𝑉
𝐸 𝑔 ≈ 2.20 𝑒𝑉
HSE
DFT+U
LHB UHB
O p
HSE band structure
HSE projected DOS
HSE and DFT+U calculations
7
HSE and DFT+U (U=3.70 eV calculated) in remarkable agreement.
Band gap:
𝐸 𝑔 ≈ 2.07 𝑒𝑉
𝐸 𝑔 ≈ 2.20 𝑒𝑉
HSE
DFT+U
LHB UHB
O p
HSE band structure
HSE projected DOS
• HSE & DFT+Usc are almost identical.
d1 occupied
Ti sites
empty d states
Eg ' 2.20 eV
Eg ' 2.07 eV
Usc = 3.70 eV
(DFT+Usc)
(HSE)
Significantly larger than 0.6 eV gap
Real gap = ? probably in between..
13

14. 0.6 eV onset: Small Polarons
• Experimental evidence for unintentional p-type doping in RTiO3
Zhou et al. J. Phys. Condens. Matter.: 17, 7395 (2007)
• DFT modeling of small-polarons:
• Supercell calculations with one h+ (i.e. one e- missing)
• Optimize electronic and structural d.o.f
• Two possible solutions
8
Rationalizing 0.6 eV onset: small
polarons
Experimental evidence for unintentional p-type doping in RTiO3.
Zhou et.al JCPM 17, 7395 (2007)
Supercell calculations with one hole (i.e. one electron missing)
Two possible solutions are found:
Small Polaron Delocilzed Hole
8
Rationalizing 0.6 eV onset: small
polarons
Experimental evidence for unintentional p-type doping in RTiO3.
Zhou et.al JCPM 17, 7395 (2007)
Supercell calculations with one hole (i.e. one electron missing)
Two possible solutions are found:
Small Polaron Delocilzed HoleSmall polaron Delocalized hole
Himmetoglu et al. Phys. Rev. B 90, 161102R (2014)
More
stable!
14
At
sm

15. Optical absorption
• Site-to-site hopping (mentioned earlier)
• Transition between small polaron and delocalized hole:RAPID COMMUNICA
ERBAND AND POLARONIC EXCITATIONS IN YTiO . . . PHYSICAL REVIEW B 90, 161102(R) (2014)
FIG. 3. (Color online) Configuration coordinate diagram for the
mation of small polarons. The insets show the structure and charge
ity isosurfaces for the small polaron and the delocalized hole.
represents the self-trapping energy of the small polaron, ES
lattice energy cost, and ET the vertical transition energy. The
arrow represents possible lower-energy excitations. The reported
gies result from HSE calculations. The charge density isosurfaces
espond to the delocalized and self-trapped hole wave functions
at finite temperature, vibrational broadening will shift the
absorption onset to lower energies (depicted by the red arrow
in Fig. 3). We thus propose that small polarons are responsible
for the low-energy onset in the optical conductivity spectra of
YTiO3 [8–10].
To obtain a band gap as low as the onset of optical
conductivity, one has to use a value of U substantially smaller
than our calculated value. We have found that U = 1.5 eV
yields a gap of 0.63 eV. A calculation of hole polarons with
such a small value of U yields a self-trapping energy of
20 meV; therefore optical phonons would easily destabilize
the small polarons at finite temperature (see Sec. II of the
Supplemental Material [22]). Given that the presence of small
polarons is experimentally well established based on hopping
conductivity [16], these results additionally cast doubt on the
interpretation of the 0.6 eV onset as a fundamental gap.
One might think that self-trapped electrons (small electron
polarons) could also be important in YTiO3. However, the
formation of a small polaron due to an extra electron on a
Ti site in YTiO is unfavorable since the oxidation state Ti+2
• Holes are self-trapped as small polarons
• Crystal strain energy (ES)
• Vertical transition: polaron to delocalized hole (ET)
• Broad absorption below the gap
Polaron binding energy
(self-trapping)
Himmetoglu et al. Phys. Rev. B 90, 161102R (2014)
15

16. Optical absorption
mmetry was broken by slightly displacing the O atoms
ound a given Ti atom, followed by the relaxation of the
omic positions in the supercell. A 400 eV cutoff for the
ane-wave expansion was used, and the integrations over
e Brillouin zone were performed with a (1/4, 1/4, 1/4) spe-
al k-point.
G. 1. Real part of the optical conductivity of Gd1ÀxSrxTiO3 thin films at
K. The “Difference-13%” curve (red) represents the difference between
e spectra of Gd0.87Sr0.13TiO3 and of GdTiO3, and the “Difference-4%”
rve (purple) the difference between Gd0.96Sr0.04TiO3 and GdTiO3. The
shed line is the fit of the “Difference-13%” curve to the small polaron
odel (Eq. (1)).
FIG. 2. Calculated one-dimensional configuration-coordinate diagrams for
(a) the excitation of a small hole polaron to a nearest-neighbor site and (b)
the excitation of a small hole polaron to a delocalized-hole configuration.
Symbols correspond to calculated values and the solid lines are parabolic
2103-2 Bjaalie et al. Appl. Phys. Lett. 106, 232103 (2015)
• Consider both type of processes in
GdTiO3:
• GdTiO3 very similar to YTiO3 (structure,
gap)
Rhys. Each configuration corresponds to a harmonic os-
cillator, with quantized levels corresponding to different
vibronic states. The vibrational problem is therefore approxi-
mated by a single effective phonon frequency. The calcu-
lated vibrational modes are 75 meV for the harmonic
oscillators in transition (a), and 65 meV for those of transi-
tion (b), which are both close to the highest-frequency opti-
cal phonon mode in bulk GdTiO3 (67 meV).30
As seen in the configuration coordinate diagram in Fig.
2(b), we find a polaron self-trapping energy of 0.55 eV (EST),
and in Fig. 2(a), we find the energy barrier for polaron migra-
tion to be 0.29 eV (Em). This is close to half of the polaron
self-trapping energy, consistent with the Holstein model.
The calculated peak energy for the excitation of an electron
from the lower Hubbard band to the polaron state in the gap
is 1.12 eV (ET), and for the excitation of the polaron out of
its self-trapping potential well via a hopping process it is
1.09 eV (Eh
T). Both of these values are in good agreement
with the observed 1 eV peak (Fig. 1), and once again
clude that this feature in the optical conductivity spectra is
caused by small hole polarons and not by excitations across
the Mott-Hubbard gap.
This work was supported by the NSF MRSEC program
(No. DMR-1121053). P.M. was supported by NSF Grant No.
DMR-1006640. Experimental work by T.A.C. was supported
through the Center for Energy Efficient Materials, an Energy
Frontier Research Center funded by the DOE (Award No.
DE-SC0001009). B.H. was supported by ONR (N00014-12-
1-0976). Computational resources were provided by the
Center for Scientific Computing at the CNSI and MRL (an
NSF MRSEC, DMR-1121053) (NSF CNS-0960316), and by
the Extreme Science and Engineering Discovery
Environment (XSEDE), supported by NSF (ACI-1053575).
1
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(1998).
2
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(2005).
3
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and Y. Iye, Phys. Rev. Lett. 70, 2126 (1993).
4
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5
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201, 407 (1992).
6
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Emin, and S. J. Allen, Sci. Rep 3, 3284 (2013).
7
P. Moetakef, D. G. Ouellette, J. Y. Zhang, T. A. Cain, S. J. Allen, and S.
Stemmer, J. Cryst. Growth 355, 166 (2012).
8
L. Bjaalie, A. Verma, B. Himmetoglu, A. Janotti, S. Raghavan, V.
Protasenko, E. H. Steenbergen, D. Jena, S. Stemmer, and C. G. Van de
Walle, “Determination of the Mott-Hubbard gap in GdTiO3” (submitted).
9
B. Himmetoglu, A. Janotti, L. Bjaalie, and C. G. Van de Walle, Phys. Rev.
B 90, 161102(R) (2014).
10
M. I. Klinger, Phys. Lett. 7, 102 (1963).
11
H. G. Reik and D. Heese, Phys. Chem. Solids 28, 581 (1967).
12
D. Emin, Phys. Rev. B 48, 13691 (1993).
13
A. Janotti, C. Franchini, J. B. Varley, G. Kresse, and C. G. Van de Walle,
Phys. Status Solidi RRL 7, 199–203 (2013).
14
P. Moetakef, J. Y. Zhang, S. Raghavan, A. P. Kajdos, and S. Stemmer,
J. Vac. Sci. Technol., A 31, 041503 (2013).
FIG. 3. Atomic configuration and charge-density isosurface (10% of maxi-
mum value) for a small hole polaron in GdTiO3. The Ti-O bonds surround-
ing the Ti atom where the polaron resides shrink relative to the bulk bond
length, as indicated by the dashed arrows.
use of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 128.111.243.137 On: Mon, 04 Apr
Small polaron
Bjaalie et al. Appl. Phys. Lett. 106, 232103 (2015)
16

17. Carrier transport: SrTiO3 & KTaO3
und
Son et al. Nat. Mater. 9, 482 (2010)vices
ain et al. Appl. Phys. Lett. 102, 182101 (2013)
Son et al. Nat. Mater. 9, 482 (2010)
Mobility: how fast charge carriers move
pitaxial SrTiO3: high mobility at low T
Room T mobility very low: problem for devices
undamental understanding of transport
eeded.
Designing high mobility complex oxides
Cain et al. Appl. Phys. Lett. 102, 182101 (2013)
Son et al. Nat. Mater. 9, 482 (2010) Cain et al. Appl. Phys. Lett. 102, 182101 (2013)
• Discussed in context of device applications
• Rapid fall of mobility at high T in epitaxial STO detrimental
• Scattering mechanisms
• Factors that determine transport properties
• Designing higher mobility oxides
DFT based simulations:
17

18. Carrier transport
• Large polarons: can treat e-p interaction at linear order
• Compute carrier lifetimes:
Verma et al. Phys. Rev. Lett. 112, 216601 (2014)
various scattering mechanisms
• They are effective at different
regimes of T
• RT mobility: scattering by crystal
vibrations (phonons)
Ziman, “Electrons and phonons: the theory of
transport phenomena in solids”
(Oxford University Press, 2001)
phonon
✏k + ~ ! = ✏k+q
⌧ 1
nk =
2⇡
~
X
q⌫,m
|gq⌫|2
n
(nq⌫ + fm,k+q) (✏m,k+q ✏nk ~!q⌫)
+(1 + nq⌫ fm,k+q) (✏m,k+q ✏nk + ~!q⌫)
o
Fermi’s Golden Rule:
Full sc
• Electron-phonon scattering
• First-principles density functional theory (DFPT)
• Model from large polaron theory
• electron-phonon (ep) coupling constant: gq⌫
18

19. Carrier transport
• Boltzmann Transport + Relaxation time approximation:
↵ = e2
X
n
Z
d3
k ⌧nk
✓
@fnk
@✏nk
◆
vnk,↵ vnk,
vnk,↵ =
1
~
@✏nk
@k↵
Band velocities:derivative of
FD distribution
electron
lifetime
valuation of Transport Integrals
sen et al. Comput. Phys. Commun. 175, 67 (2006)
i et al. Comput. Phys. Commun. 185, 422 (2014)
port integrals require very fine
ling around Fermi surface
arge grids: band-structure
olation techniques:
↵ = e2
X
n
Z
d3
k ⌧nk
@f
(0)
nk
@✏nk
!
vnk,↵ vnk,
• All quantities can be calculated
from band structure
• Fine sampling of the Brillouin Zone
needed for accurate integration
19

20. SrTiO3 & KTaO3
• Both KTO and STO are cubic perovskites (no rotations)
at Room T.
• d0 conduction band: Do not expect very strong
correlation effects.
Background
Moetakef et al. Appl. Phys. Lett. 99, 232116 (2011)
• Complex oxides: ABO3
• Magnetism, ferroelectricity, superconductivity
rich physics!
!
• High quality epitaxial growth
• 2deg at interface between two complex
oxides
• e.g. SrTiO3/LaAlO3
• e.g. SrTiO3/GdTiO3
!
• Device applications:
• field effect transistors
• high power electronics
A
O
B
t2g
eg
• Room T mobility of KTO is higher than STO:
µRT ' 10 cm2
/Vs
µRT ' 30 cm2
/VsKTO:
STO:
Wemple, Phys. Rev. 137, A1575 (1965)
cubic crystal
field
d0
triply degenerate
conduction band*
20

21. SrTiO3 & KTaO3
mpared to the collinear case in Fig. 3(a). This is
fact that the Fermi energies for the considered
ensities are much lower than the split-off band
leading to a reduction of its density of states.
ll reduction of scattering rates when spin-orbit
s strong has in fact been proposed before16
and
es as an example for this behavior.
V. TRANSPORT INTEGRALS
culation of transport coefficients, we compute
ing two integrals:
1
ell
n,k
τnk −
∂fnk
∂ϵnk
vnk,α vnk,β
1
ell
n,k
τnk −
∂fnk
∂ϵnk
(ϵnk − EF ) vnk,α vnk,β.
(5)
of these integrals, the conductivity (σ) and See-
tensors are given by
β = 2 e2
I
(1)
αβ , Sαβ = −
1
eT
I(1)−1
αν I
(2)
νβ , (6)
is the Fermi energy, α, β, ν are cartesian in-
eated indices are summed over), and the factor
s for the spin (in case of the noncollinear calcu-
use the collinear band structure. We have explicitly
checked that the noncollinear calculations in STO lead
to only negligible changes in the transport integrals. In
the following two subsections, we report the the room-
temperature mobility and Seebeck coefficients.
80
100
120
140
160
180
200
220
1e+18 1e+19 1e+20 1e+21
µ(cm
2
/Vs)
n (cm
-3
)
STO
KTO-collin
KTO-ncollin
FIG. 5. Calculated mobilities of STO and KTO at 300K. For
KTO, both collinear and noncollinear calculations are dis-
played.
A. Electron mobilities
Fig. 5 shows the calculated mobilities as a function of
Himmetoglu et al. J. Phys. Condens. Matter.:
28, 065502 (2016)
• KTO has higher mobility
• Spin-orbit interaction relevant for KTO
• Calculations involving spin-orbit lead to
higher mobility
• Predicted mobilities higher than
experiment
spin-orbit
no spin-orbit
T = 300 K
3
TABLE I. Calculated and experimental longitudinal optical
(LO) and transverse optical (TO) phonon frequencies at the
Γ point, in units of cm−1
.
LO TO
LDA Exp. LDA Exp.
176 188a
, 184-185b,c
105 88a
, 81-85b
, 88c
253 290a
, 279b
, 255c
186 199a
, 198-199b,c
403 423a
, 421-422b,c
253 290a
, 279b
, 255c
820 833a
, 826-838b,c
561 549a
, 546-556b
, 574c
a
Ref. 41, b
Ref. 42, c
Ref. 43.
(a)
-0.1
0
0.1
0.2
0.3
0.4
0.5
Energy(eV)
Γ M X Γ R
0.4
0.5
(b)
element squared is given by17,45,46
|gq ν |
2
=
1
q2
e2
¯h ωLν
2ϵ0 Vcell ϵ∞
N
j=1 1 −
ω2
T j
ω2
Lν
j̸=ν 1 −
ω2
Lj
ω2
Lν
, (1)
where N is the number of LO/TO modes, ωLν are LO
mode frequencies, ωT ν are TO mode frequencies, Vcell is
the unit cell volume, and ϵ∞ is the electronic dielectric
constant. This equation is obtained using the electron-
phonon Hamiltonian in Ref. 45 and the Lyddane-Sachs-
Teller (LST) relation ϵ(ω) = ϵ∞
N
j=1 (ω2
− ω2
L,j)/(ω2
−
ω2
T,j). In this expression, the dispersion of the optical
phonon frequencies are ignored and in our calculations,
we use their value at the zone center. Alternative formu-
lations of the LO phonon coupling based on Born effec-
tive charges, which take into account their full dispersion,
have also been discussed in the literature.47,48
The cal-
culated electron-phonon coupling elements are listed in
Table II. Here, we define
Cν =
a0
2π
2
q2
|gqν|2
, (2)
where a0 is the cubic lattice parameter. Note that there
TABLE II. Calculated electron-phonon couplings in eV 2
based on Eqns.(1) and (2).
3
TABLE I. Calculated and experimental longitudinal optical
(LO) and transverse optical (TO) phonon frequencies at the
Γ point, in units of cm−1
.
LO TO
LDA Exp. LDA Exp.
176 188a
, 184-185b,c
105 88a
, 81-85b
, 88c
253 290a
, 279b
, 255c
186 199a
, 198-199b,c
403 423a
, 421-422b,c
253 290a
, 279b
, 255c
820 833a
, 826-838b,c
561 549a
, 546-556b
, 574c
a
Ref. 41, b
Ref. 42, c
Ref. 43.
(a)
-0.1
0
0.1
0.2
0.3
0.4
0.5
Energy(eV)
Γ M X Γ R
-0.1
0
0.1
0.2
0.3
0.4
0.5
Energy(eV)
Γ M X Γ R
Δso
(b)
0.4
0.5
(c)
element squared is given by17,45,46
|gq ν |
2
=
1
q2
e2
¯h ωLν
2ϵ0 Vcell ϵ∞
N
j=1 1 −
ω2
T j
ω2
Lν
j̸=ν 1 −
ω2
Lj
ω2
Lν
, (1)
where N is the number of LO/TO modes, ωLν are LO
mode frequencies, ωT ν are TO mode frequencies, Vcell is
the unit cell volume, and ϵ∞ is the electronic dielectric
constant. This equation is obtained using the electron-
phonon Hamiltonian in Ref. 45 and the Lyddane-Sachs-
Teller (LST) relation ϵ(ω) = ϵ∞
N
j=1 (ω2
− ω2
L,j)/(ω2
−
ω2
T,j). In this expression, the dispersion of the optical
phonon frequencies are ignored and in our calculations,
we use their value at the zone center. Alternative formu-
lations of the LO phonon coupling based on Born effec-
tive charges, which take into account their full dispersion,
have also been discussed in the literature.47,48
The cal-
culated electron-phonon coupling elements are listed in
Table II. Here, we define
Cν =
a0
2π
2
q2
|gqν|2
, (2)
where a0 is the cubic lattice parameter. Note that there
TABLE II. Calculated electron-phonon couplings in eV 2
based on Eqns.(1) and (2).
STO KTO
C1 1.43 × 10−5
1.82 × 10−5
C2 1.33 × 10−3
1.46 × 10−3
C3 6.77 × 10−3
7.50 × 10−3
are only three coupling constants listed in Table II. One
of the LO modes in Table I is not polar– the mode with
frequency 253 cm−1
appears both in the LO and TO
sectors, and since it is nonpolar, it does not split up. The
constants Cν are enumerated in order of increasing LO
TABLE I. Calculated and experimental longitudinal optical
(LO) and transverse optical (TO) phonon frequencies at the
Γ point, in units of cm−1
.
LO TO
LDA Exp. LDA Exp.
176 188a
, 184-185b,c
105 88a
, 81-85b
, 88c
253 290a
, 279b
, 255c
186 199a
, 198-199b,c
403 423a
, 421-422b,c
253 290a
, 279b
, 255c
820 833a
, 826-838b,c
561 549a
, 546-556b
, 574c
a
Ref. 41, b
Ref. 42, c
Ref. 43.
(a)
-0.1
0
0.1
0.2
0.3
0.4
0.5
Energy(eV)
Γ M X Γ R
-0.1
0
0.1
0.2
0.3
0.4
0.5
Energy(eV)
Γ M X Γ R
Δso
(b)
-0.1
0
0.1
0.2
0.3
0.4
0.5
Energy(eV)
Γ M X Γ R
(c)
FIG. 1. Conduction band structure for KTO, (a) in collinear
calculation, and (b) in noncollinear calculation. Band struc-
ture of STO is shown in (c) for comparison.
element squared is give
|gq ν |
2
=
1
q2
e2
¯h
2ϵ0 V
where N is the numbe
mode frequencies, ωT ν
the unit cell volume, a
constant. This equatio
phonon Hamiltonian in
Teller (LST) relation ϵ(
ω2
T,j). In this expressi
phonon frequencies are
we use their value at th
lations of the LO phon
tive charges, which take
have also been discuss
culated electron-phono
Table II. Here, we defin
Cν =
where a0 is the cubic la
TABLE II. Calculated
based on Eqns.(1) and (2
S
C1 1.43
C2 1.33
C3 6.77
are only three coupling
of the LO modes in Ta
frequency 253 cm−1
a
sectors, and since it is n
constants Cν are enum
mode frequency. The el
here for STO are lowe
This is due to the fact t
constant used in Ref 1
one LO mode, overesti
modes were present.
The coupling consta
agreement with previo
for STO.17
For compa
resulting from the par
ing the dimensionless c
frequencies, and the ba
KTO-collin KTO-ncollin STO-ncollin21

22. SrTiO3 & KTaO3
~!LO
SO
• Recall the energy conserving
terms:
(✏m,k+q ✏nk ± ~!q⌫)
k
k + q
• Inter-band scattering forbidden for
split-off band ( )SO >> ~!LO
• Ta is heavier than Ti, stronger spin-
orbit splitting.
• KTO is effectively 2-band, while
STO is 3-band.
• Effective number of conduction bands:
• Can strain be used to split bands or modify LO phonon frequencies?
22

23. SrTiO3 & KTaO3
• Band velocities (near EF) and/or band masses:
e.g.: Parabolic bands: ✏nk =
~2
k2
2 mb
vnk,↵ =
~k↵
mb
Smaller mass, larger mobility (but d0 perovskites have highly non-parabolic bands)
v2
F ≡
n,k δ(EF − ϵnk) v2
nk
n,k δ(EF − ϵnk)
, (9)
where the index n runs over the conduction bands.
Shown in Fig. 6 are vF , which have higher values in KTO
compared to STO. While KTO in the noncollinear cal-
culation has a higher band-width (as can seen in Figs. 1
and 4), the averaged values of the Fermi velocities are
slightly lower than that of KTO in the collinear calcu-
lation. This is most probably due to the highly non-
parabolic Fermi surface of KTO, which has high and
low velocity regions, averaging out to a similar numerical
value for both collinear and non-collinear calculations.
Instead, lower scattering rates in the noncollinear calcu-
lations lead to the enhancement of mobilities as seen in
Fig. 5.
0.2
0.3
0.4
1e+18 1e+19 1e+20 1e+21
a0vF(eV)
n (cm
-3
)
STO
KTO-collin
KTO-ncollin
mation in a metal, it is inversely proportional to EF ).
a given electron concentration n, EF is lowest for S
since its conduction band width is smaller than K
which in turn leads to a higher Seebeck coefficient
The difference between collinear and noncollinear ca
lations in KTO mainly results from the effective num
of conduction bands. For the noncollinear calcula
the split-up band is higher in energy, leading to an
fective 2-conduction-band system (see Fig. 1(b)).
a given n, EF increases when the number of conduc
bands decrease; when there are less bands which are c
in energy, electrons occupy higher lying bands. Thus
in the noncollinear case is lower than that of the colli
one, due to the reduction of the number of conduc
bands.23
0
200
400
600
1e+18 1e+19 1e+20 1e+2
-S(µV/K)
n (cm
-3
)
STO
KTO-collin
KTO-ncollin
FIG. 7. Calculated Seebeck coefficients of STO and KT
vFComputed
• 5d states of Ta have larger band-
width than 3d of Ti
• Leads to higher band velocities
• Can strain be used to modify band
masses/velocities ?
• Can we use even heavier atoms ?
23

24. 2DEGs & screening LO modes out
The solid line in Fig. 1(a) shows a fit using Eqs. (1)–(3),
which accurately describes the experimental data. The fit
parameters were l0, a, and xLO. In SrTiO3, carriers are
believed to couple strongly to two LO phonon modes18
and
the value for xLO determined here is an average, effective
value. Fits for the more highly doped samples are shown in
the supplementary material30
and yield similarly good
descriptions. Figures 1(b) and 1(c) show the results for a and
xLO as a function of N. As N increases, xLO and a both
increase. As a result, lLO increases while le-e decreases and
the total mobility becomes increasingly limited by electron-
electron scattering. This is summarized in Fig. 1(c), which
shows le-e and lLO calculated at room temperature using the
extracted values for a and xLO, the total mobility calculated
using these two terms, and the experimentally measured Hall
mobility. At N $ 6 Â 1018
cmÀ3
, the electron-phonon scatter-
ing limited regime crosses over to an electron-electron
scattering limited regime; as a result, the mobility remains
<10 cm2
VÀ1
sÀ1
for all N.
The increase of lLO with N is in agreement with first-
principles calculations,17
where it has been shown that the
electron-phonon scattering rates decrease away from the
Brillouin zone center. Thus with increasing N, as the Fermi
surface enlarges, the scattering rate becomes smaller, leading
to an increase in lLO.
The results shown in Fig. 1 confirm earlier results that
62102-2 Mikheev et al. Appl. Phys. Lett. 106, 062102 (2015)
scattering contribution in the 2DELs. The absence of LO
phonon scattering explains the slightly larger room-
temperature mobility in the 2DELs. The inset in Fig. 2 shows
a as a function of SrTiO3 thickness. For the thinnest layers,
the three-dimensional carrier density in the 2DEL is
increased (the layer width is substantially smaller than the
roughness scattering, cause a drop in the mobility below the
le-e limit.
In summary, we have shown that the room temperature
mobility in high-carrier-density 2DELs in SrTiO3 is not lim-
ited by LO phonon scattering, in sharp contrast to bulk,
FIG. 2. Temperature dependence of the inverse of the mobility l in 2DELs
at GdTiO3/SrTiO3 interfaces. The samples consisted of epitaxial bilayers
with SrTiO3 different thicknesses (GdTiO3 is the top layer) on an insulating
substrate, from Ref. 11. The inset shows the extracted parameter a from fits
to lÀ1
¼ aT2
.
FIG. 3. Room temperature mobility as a function of the extracted electron-
electron scattering parameter a for the 2DELs (diamonds) and three-
dimensional electron gases (squares). The dashed line is calculated accord-
ing to lÀ1
¼ aT2
. The mobility of the three-dimensional electron gases falls
below the dashed line due to the significant contribution of LO phonon scat-
tering, which is absent in the 2DELs. The extremely confined 2DELs (two
right-most data points) fall below the line due to contributions from interface
roughness scattering even at room temperature.
062102-3 Mikheev et al. Appl. Phys. Lett. 106, 062102 (2015)
Bulk doped STO 2DEG on STO thin-film
• Mobility fits to a quadratic T dependence instead of
exponential in 2DEG
• High density 2DEG could screen the polarization field and
remove the LO modes
Mikheev et al. Appl. Phys. Lett. 106, 062102 (2015)
24

25. Summary & Outlook
• Polarons (small & large) are ubiquitous in many
perovskite oxide systems.
• Impact carrier transport, optical properties
• DFT based calculations give good insight
• Boltzmann transport + relaxation time
approximation yield good qualitative description
• Code available (model e-p + transport)
• Development: Full e-p + other scattering
mechanisms
https://github.com/bhimmetoglu/transport_new
25

26. LO phonon interaction
• Fröhlich interaction for multiple modes (using Born effective
charges):
ρ(−q) =
k
ˆck ˆck−q
(8) in Eq.(7), we arrive at
Hei = i
q, ν, k
gq ν (ˆa†
−q ν + ˆaq ν) ˆc†
k ˆck−q
tron-phonon coupling gq ν defined as
gq ν ≡
e2
ϵ0 V
s, α, β, λ
2 Ms ωL ν
1/2
1
q
ˆqα (ϵ−1
∞ )αβ
Z∗ β λ
s es λ(q; ν)
Implementation
mentation, the dispersion of polar LO modes will be neglected, meaning that ωL ν and es λ(
h their values at Γ. This is a good approximation for such modes.
.: Phonons and related crystal properties from DFPT, Rev. Mod. Phys. 73, 515 (2001)
• Simplifications:
Author: Burak H
Details on the transport code implementation
g rates:
τ−1
nk =
2π
qν,m
|gqν|2
(1 − ˆvnk · ˆvmk+q) (nq ν + fm,k+q) δ(ϵm,k+q − ϵnk − ωLν)
+(1 + nq ν − fm,k+q) δ(ϵm,k+q − ϵnk + ωLν)
|gq ν|
2
=
1
q2
e2
ωLν
2ϵ0 Vcell ϵ∞
N
j=1 1 −
ω2
T j
ω2
Lν
j̸=ν 1 −
ω2
Lj
ω2
Lν
he delta-function is replaced by a Gaussian smearing with adaptive smearing width:
δ(Fnk, mk+q) =
1
√
π σ
exp −
F2
nk, mk+q
σ2
σ = ∆Fnk, mk+q = a
∂Fnk, mk+q
∂k
∆k
• For a single mode, the above expression is equivalent to
Fröhlich.
26

27. TransportTransport Theory
f(r, k, t) d3
r d3
k
• Evolution of the distribution function f under
applied E-field:
• Equilibrium distribution:
f
(0)
nk =
1
e(✏nk Ef )/kT + 1
EF
SrTiO3
Fermi surface
(Fermi-Dirac)
27

28. TransportTransport Theory
↵ = e2
X
n
Z
d3
k ⌧nk
@f
(0)
nk
@✏nk
!
vnk,↵ vnk,
• Boltzmann Transport Equation
• Solution in terms of transport integrals Conductivity:
Band velocities:
Scattering rates:
vnk,↵ =
1
~
@✏nk
@k↵
⌧ 1
nk
• Scattering times/rates determined by collisions:
Electron-defect
Electron-electron
Electron-phonon
28

29. Scattering by Phonons
• A specific type of phonon scattering
dominates at RT
• Longitudinal Optical (LO) phonons:
• Transverse vs longitudinal
• TO modes charged planes slide
• LO modes charge accumulation!
Strong Coulomb scattering!!
Dominant for materials with
polar LO modes.
Phonon scattering
29

30. Electronic structure of strongly
correlated systems
ℋ = −𝑡 𝑐𝑖𝜎
†
𝑐𝑗𝜎 + ℎ. 𝑐 + 𝑈 𝑛𝑖↑ 𝑛𝑖↓
𝑖<𝑖,𝑗>,𝜎
Independent e- e- - e- interaction
Small Coulomb repulsion: U large hopping amplitude: t
4
𝑡 ≫ 𝑈
Strongly correlated systems
30

31. Strongly correlated systems
Electronic structure of strongly
correlated systems
ℋ = −𝑡 𝑐𝑖𝜎
†
𝑐𝑗𝜎 + ℎ. 𝑐 + 𝑈 𝑛𝑖↑ 𝑛𝑖↓
𝑖<𝑖,𝑗>,𝜎
Independent e- e- - e- interaction
Large Coulomb repulsion: U small hopping amplitude: t
5
𝑡 ≪ 𝑈
31