burakh 040816

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

[email protected]

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

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

First-Principles Modeling• Density functional theory (DFT) • Band structure: LDA/GGA functional

• Quantum ESPRESSO package

• Phonons: Density functional perturbation theory (DFPT)

• LO mode frequencies + dielectric constant: Born effective charges

SrTiO3 - bands (LDA)

SrTiO3 - phonons (LDA)

-4

-2

0

2

4

6

8

Ener

gy (e

V)

K M X K R

1

mailto:[email protected]

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

Polaron Concept

• One of the simplest, and most widely studied model for interacting electrons and (optical) phonons:

Part I

Single polaron

I. INTRODUCTION. THE ”STANDARD” THEORIES

A. The polaron concept

A charge placed in a polarizable medium is screened. Dielectric theory describes the

phenomenon by the induction of a polarization around the charge carrier. The idea of the

autolocalization of an electron due to the induced lattice polarization was first proposed by

L. D. Landau [1]. In the further development of this concept, the induced polarization can

follow the charge carrier when it is moving through the medium. The carrier together with

the induced polarization is considered as one entity (see Fig. 1). It was called a polaron by

S. I. Pekar [2, 3]. The physical properties of a polaron differ from those of a band-carrier. A

polaron is characterized by its binding (or self-) energy E0, an effective mass m∗ and by its

characteristic response to external electric and magnetic fields (e. g. dc mobility and optical

absorption coefficient).

FIG. 1: Artist view of a polaron. A conduction electron in an ionic crystal or a polar semiconductor

repels the negative ions and attracts the positive ions. A self-induced potential arises, which acts

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

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.





4

Interaction potential:

Coupling constant: ↵ =e2

~ ✏0

✓mb

2~!LO

◆1/2 ✓1

✏1� 1

✏

◆

Vq =~!LO

q

✓↵

✏0 Vcell

◆1/2 ✓~

2mb !LO

◆1/2

Weak coupling and large polaronsMacroscopic 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

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

Example Coupling Constants

Table 1: Frohlich coupling constants

Material α Material αCdTe 0.31 KI 2.5CdS 0.52 RbCl 3.81ZnSe 0.43 RbI 3.16AgBr 1.6 CsI 3.67AgCl 1.8 TlBr 2.55CdF2 3.2 GaAS 0.068InSb 0.02 GaP 0.201KCl 3.5 InAs 0.052KBr 3.05 SrTiO3 4.5

46

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

!

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





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

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


e.g.: n-doped SrTiO3 thin-films~!LO

max

' 0.1 eV

9

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

The spectral weight is quantified according to,

n3D vð Þ~2me

e2p

ðv

0s1 v0ð Þ dv0 ð2Þ

where me is the free electron mass. The results for GdTiO3 films A1and A2 and the delta-doped superlattice B2 are shown in Fig. 3(c).The difference is attributed to the delta-doping and is plotted usingthe right axis in units of a two-dimensional density per delta dopedplane according to n2D 5 dn3D/Nd. The resulting n2D is substantiallyless (,1/10) than the nominal doping density ,7 3 1014 associatedwith each SrO plane. By comparison, the modest suppression of thespectral weight of metallic two6 and three18-dimensional electrongases in SrTiO3 is attributed to an approximately three-fold enhance-ment over the bare band mass in equation (2) to ,2me

19.

DiscussionGdTiO3 is a Mott insulator. Beyond a critical doping, x , 0.211,20, thesystem Gd12xSrxTiO3 undergoes a MIT to a metal with carrier den-sity significantly larger than the doping concentration. We find thatdelta doping far exceeding the critical concentration for GdTiO3 doesnot render it metallic. Rather, as evidenced by optical and electronictransport measurements discussed in the following, the systemremains an insulator; small polaron formation occurs when substi-tuting an SrO plane for a GdO plane confining the resulting holes tothe adjacent (TiO2) planes21–23.

Our optical measurements do not detect the characteristic Druderesponse from mobile charge carriers. Rather, we observe the broadcarrier-induced absorption band characteristic of small polarons.

This absorption primarily transfers a small polaron’s electronic car-rier from the site at which it is self-trapped to a neighboring site. Nearits peak the absorption band’s profile is a modified Gaussian centeredat the difference between the final site’s electronic energy and that ofthe self-trapped carrier, 2Eb

15,24,25

s1 vð Þ<np2p1=2e2a2t2

v!h2Dexp {

2Eb{!hvð Þ2

D2

" #, ð3Þ

where Eb is the polaron binding energy.The absorption strength is determined by the polaron density, np,

the electronic transfer energy t and the inter-site separation, a. At lowtemperatures kT=!hvo the width is determined by atoms’ zero-pointvibrations, D~ 4Eb!hvoð Þ1=2, where !hvo is the energy of a represent-ative phonon mode. The sum rule for this absorption process, Eq. (2),yields , (npe2/me)(t/Eb).

Equation (3) is fit to the optical conductivity of the delta-dopedlayer in Fig. 3(b). We assume a hole density ,7 3 1014 cm2, thenominal 2D doping. The fit at 10 K (296 K) gives Eb 5 440(370) meV, t 5 110 (100) meV, D 5 440 (402) meV, assuming!hv0~110 meV. The deduced transfer energy t is comparable to thatobtained from first-principles calculations12. The summing of thesmall-polaron conductivity described by Eq. (2) is shown inFig. 3(c). Its magnitude implies that t is significantly smaller thanEb, the condition for small-polaron formation.

Recent work by Zhang et al.9 is particularly relevant here. Theyshow that the thinnest SrTiO3 quantum wells suffer interface distor-tions that will reduce t. Small transfer energy t favors localization byeither electron correlation or small polaron formation.

Table 1 | Sample parameters and transport properties. From left to right: sample name, total film thickness d, number of delta-doped layersNd, the number of SrO layers tSTO, room temperature resistivity r (300 K), Arrhenius activation energy EA

a for conductivity, small polarontransport activation energy Ea, measured prefactor, and calculated small polaron prefactor using parameters consistent with the opticalconductivity. Samples A1–A3 are nominally undoped GdTiO3 films. The activation energies are from fits over the range ,200–400 K.Data for sample A3 and A1 have been published previously33,11. No transport data is available for sample A2 although it was used toobtain optical conductivity for the nominally undoped films. B1 and B2 are GdTiO3 films delta doped with single layers of SrO. B1 has asingle layer while B2 has 4 such layers separated from each other by ,4 nm of GdTiO3

(1)

Sample

(2)

d(nm)(3)

Nd

(4)

tSTO

(5)

r (300 K)

(V cm)

(6)

EAa (meV)

r0 expEA

a

kBT

$ %(7)

Ea(fit) (meV)

s~C

kBTexp {

Ea

kBT

$ % (8)

C (fit)(S/cm)J

(9)

C 5 ne2a2v0/(2p)(S/cm)J

A1 10 0 7.2 129 6 5 160 2.6 10219

A2 19.5 0A3 11 0 5.2 144 6 5 162 4.8 10219

B1 8 1 1 1.24 106 6 10 130 6 10 5.0 10219 8.45 10219

B2 21.5 4 1 0.35 93 6 5 111 6 5 1.0 10218 1.25 10218

Figure 3 | (a) Volume averaged 3D optical conductivity at 10 K (solid) and 296 K (dashed) for the indicated samples. Symbols indicate the dcconductivity 10 K (blue circle) and 296 K (red square). (b) Two-dimensional optical conductivity of the delta-doped layer after normalization togetherwith the small polaron model, displaced upward for clarity. (see text) (c) Thin lines: sum rule obtained by integrating the (3D) optical conductivityaccording to (1). Fat line: delta doping contribution to sum rule. The right axis expresses the difference in units of 2D density per delta-doping plane. Theerror bars were estimated from the uncertainty in the reflectance measurement.

www.nature.com/scientificreports

SCIENTIFIC REPORTS | 3 : 3284 | DOI: 10.1038/srep03284 3

e.g.: SrTiO3 /GdTiO3 interface: p-doping on the GdTiO3 side

Oulette et al. Sci. Rep. 3, 3284 (2013)10

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 TiO

6

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

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-consistentlyAnisimov 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

Electronic Structure of YTiO3HSE and DFT+U calculations

7

� HSE and DFT+U (U=3.70 eV calculated) in remarkable agreement. � Band gap:

𝐸𝑔 ≈ 2.07 𝑒𝑉

𝐸𝑔 ≈ 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:

𝐸𝑔 ≈ 2.07 𝑒𝑉

𝐸𝑔 ≈ 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 gapReal gap = ? probably in between..

13

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





8




Small Polaron Delocilzed Hole Small polaron Delocalized hole

Himmetoglu et al. Phys. Rev. B 90, 161102R (2014)

More stable!

14

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

Optical absorption• Site-to-site hopping (mentioned earlier) • Transition between small polaron and delocalized hole: RAPID COMMUNICATIONS

INTERBAND AND POLARONIC EXCITATIONS IN YTiO . . . PHYSICAL REVIEW B 90, 161102(R) (2014)

FIG. 3. (Color online) Configuration coordinate diagram for theformation of small polarons. The insets show the structure and chargedensity isosurfaces for the small polaron and the delocalized hole.EST represents the self-trapping energy of the small polaron, ES

the lattice energy cost, and ET the vertical transition energy. Thered arrow represents possible lower-energy excitations. The reportedenergies result from HSE calculations. The charge density isosurfacescorrespond to the delocalized and self-trapped hole wave functionsat 10% of their maximum value. Arrows in the left inset indicate thelattice distortions around the self-trapped hole.

energy differences that are within 0.02 eV. In the following wereport the results using the (1/4,1/4,1/4) point.

For a missing electron (one excess hole) in the 160-atom supercell, we find two possible configurations: onecorresponding to a delocalized hole, and another in which thehole is self-trapped. In the case of a delocalized hole, all the Tiatoms in the supercell contribute equally. The correspondingcharge distribution is extended throughout the crystal (see theinset of Fig. 3). In contrast, the self-trapped hole is localizedon a single Ti atom, changing its oxidation state from Ti+3 toTi+4, and is accompanied by a sizable local lattice distortion:The Ti-O bonds contract by 6% of the equilibrium bond lengthin HSE, or 4% in DFT+U (see the inset of Fig. 3).

We find that the self-trapped hole is significantly morestable than the delocalized hole. The self-trapping energyof the small polaron (EST), defined as the energy differencebetween the delocalized and self-trapped hole [31], is foundto be 0.52 eV in HSE calculations. The localization of thehole leads to an electronic energy gain, but costs lattice energy(corresponding to ES in the configuration coordinate diagramof Fig. 3). This cost is computed by evaluating the energydifference between the atomic configuration corresponding tothe optimized atomic coordinates of the small polaron andthe configuration corresponding to the delocalized hole in acharge-neutral configuration (i.e., in the absence of any hole),and is found to be 0.57 eV. The calculated values of theself-trapping (EST) and strain (ES) energies for the DFT+Uand HSE calculations agree to within 0.1 eV (see Sec. II of theSupplemental Material [22]).

When small hole polarons are present, electronic transitionscan occur from the occupied LHB to the empty polaronic state.The vertical excitation energy ET is the sum of the latticeenergy cost ES and the self-trapping energy EST, followingthe Frank-Condon principle. The calculated vertical transitionenergy is 1.09 eV, well below the band-gap energy, andrepresents a peak in the absorption spectrum. Furthermore,

at finite temperature, vibrational broadening will shift theabsorption onset to lower energies (depicted by the red arrowin Fig. 3). We thus propose that small polarons are responsiblefor the low-energy onset in the optical conductivity spectra ofYTiO3 [8–10].

To obtain a band gap as low as the onset of opticalconductivity, one has to use a value of U substantially smallerthan our calculated value. We have found that U = 1.5 eVyields a gap of 0.63 eV. A calculation of hole polarons withsuch a small value of U yields a self-trapping energy of20 meV; therefore optical phonons would easily destabilizethe small polarons at finite temperature (see Sec. II of theSupplemental Material [22]). Given that the presence of smallpolarons is experimentally well established based on hoppingconductivity [16], these results additionally cast doubt on theinterpretation of the 0.6 eV onset as a fundamental gap.

One might think that self-trapped electrons (small electronpolarons) could also be important in YTiO3. However, theformation of a small polaron due to an extra electron on aTi site in YTiO3 is unfavorable since the oxidation state Ti+2

is unstable against electron delocalization in the conductionband due to electron-electron repulsion. We were indeedunable to find localization for an extra electron in the YTiO3supercell.

As a final point, we address the differences between ourresults and previous calculations for the electronic structure ofYTiO3. Theoretical studies based on dynamical mean fieldtheory (DMFT) [14,15], using a Hubbard U parameter of5 eV, yielded a band gap of approximately 1 eV, i.e., 1 eVsmaller than our HSE and DFT+U results [32]. These studieswere based on an effective Hamiltonian that contains only asubset of the bands that comprise the LHB and UHB. WhileRefs. [14,15] considered full structural relaxations (at the levelof LDA), they assumed that in YTiO3 the splittings of thed orbitals are approximately determined by the octahedralcrystal field, which would result in doubly degenerate higher-energy eg and triply degenerate lower-energy t2g states. Thesingle electron per Ti site would then justify the use of aHamiltonian based on t2g-derived bands only.

However, in YTiO3, the crystal field does not have a simpleoctahedral form. Even in the case of a cubic crystal, withuntilted TiO6 octahedra, the Y atoms modify the crystal fieldand split the t2g states. The octahedral rotations break theremaining degeneracies, yielding nondegenerate d orbitals.Our full band-structure calculations confirm this argument,and reveal that the LHB contains a large contribution from oneof the eg states.

Figure 4 shows that the LHB is not purely composed of dzx ,dzy , and dxy (the t2g states), but has an important contributionfrom other orbitals as well [33]. If the Ti d states contributingto the LHB had a purely t2g character, the resulting chargedensity would exclusively exhibit lobes that point in betweenthe Ti-O bonds. Instead, the charge density in the inset ofFig. 4 is distinctly nonzero along the Ti-O bonds, providingclear evidence of contributions from dz2 states. In fact, thealignment of d orbitals on different Ti sites forming the LHBis in overall agreement with the orbital polarization observed inRefs. [14,15], which is a consequence of octahedral rotations.The only difference in the orbital polarization picture is thefull inclusion of eg orbitals.

161102-3

• 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

Optical absorption

symmetry was broken by slightly displacing the O atomsaround a given Ti atom, followed by the relaxation of theatomic positions in the supercell. A 400 eV cutoff for theplane-wave expansion was used, and the integrations overthe Brillouin zone were performed with a (1/4, 1/4, 1/4) spe-cial k-point.

The mid- to near-IR conductivity, Re(r(x)), at 10 K isshown in Fig. 1. In all samples, a 67 meV phonon mode asso-ciated with Ti-O stretching is observed as a sharp peak. Thebroad feature close to 0.1 eV is not understood at this time,but we note that the substrate dielectric constant is changingvery rapidly in this frequency region and may introduce arti-facts. The temperature dependence of the conductivity wasnot investigated, but we expect no change in the features upto 300 K, as found in the previous work on undoped anddoped GdTiO3.6,7 With increasing doping (x¼ 0.13), a broadfeature, labeled as A, appears at energies around 1 eV, simi-lar to what has been observed in bulk titanates.26 This featureis interpreted in the following as being due to a small po-laron. The difference in small hole polaron concentrationbetween the x¼ 0 and x¼ 0.04 samples is not large enoughto be able to detect a difference, given the magnitude of theerror bar.

To explain the behavior of feature A, we consider possi-ble excitation mechanisms. One of these is the excitation ofthe small polaron out of its self-trapping potential well via ahopping process, with the structure subsequently relaxing bycreating a lattice distortion at a neighboring site [Fig. 2(a)].In this process, the optical absorption is expected to peaknear twice the polaron self-trapping energy, EST [defined inFig. 2(b)].12 Considering the probability of hopping to an ad-jacent site within perturbative solutions to the Holsteinmodel, the optical absorption peak is expected to be fourtimes as large as the DC conductivity activation energy.27–29

Therefore, we also expect the self-trapping energy to betwice as large as the activation energy.

We compare the incoherent absorption below the Mott-Hubbard gap to an approximate expression for the opticalconductivity of a small polaron10–12

r xð Þ ¼ npp1=2e2

mxt

Dexp $ 2EST $ !hxð Þ2

D2

" #

; (1)

where np is the polaron density and t ¼ !h2

2ma2 is the electronicbandwidth parameter with a being the lattice constant and mthe effective mass. At low temperatures, kT % !hx0, thewidth is determined by the zero-point phonon motion,D ¼ ð4EST!hx0Þ1=2, where x0 is the frequency of the relevantphonon mode. The factor t/D represents a transition probabil-ity and should be replaced by unity in the adiabatic limit,t>D.12

To separate the polaron contribution from the Mott-Hubbard feature at higher energy, the optical conductivity ofthe undoped GdTiO3 film was subtracted from that of theGd0.87Sr0.13TiO3 film at 10 K and the difference was fitted toEq. (1). The fit is shown as the dashed line in Fig. 1. Theresulting self-trapping energy is EST¼ 0.58 eV, about twice aslarge as the 0.24 eV activation energy that has been reportedfor bulk samples.2 The broadening factor, D¼ 0.51 eV, corre-sponds to a phonon energy !hx0 ¼ 110 meV, which is largerthan the highest-frequency optical phonon mode (67 meV)reported for GdTiO3.30 That there is some difference isexpected, since Eq. (1) is an approximate expression involvingmultiple fitting parameters. The presence of SrGd substitu-tional impurities will also lead to an increased mode energy,as will be discussed further below. Taking the polaron density

FIG. 1. Real part of the optical conductivity of Gd1$xSrxTiO3 thin films at10 K. The “Difference-13%” curve (red) represents the difference betweenthe spectra of Gd0.87Sr0.13TiO3 and of GdTiO3, and the “Difference-4%”curve (purple) the difference between Gd0.96Sr0.04TiO3 and GdTiO3. Thedashed line is the fit of the “Difference-13%” curve to the small polaronmodel (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 parabolicfits. The dashed horizontal lines correspond to the vibronic ground state inthe starting configurations. Eh

T and ET are the transition energies of the twoprocesses, EST is the small-polaron self-trapping energy and Em is the po-laron migration barrier energy. Note that the atomic displacements are dif-ferent for (a) and (b), giving different generalized coordinates Qa and Qb. In(c) and (d), the calculated broadening of each transition is plotted.

232103-2 Bjaalie et al. Appl. Phys. Lett. 106, 232103 (2015)

Reuse 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 Apr2016 18:42:25

• Consider both type of processes in

GdTiO3:

• GdTiO3 very similar to YTiO3 (structure,

gap)

as np¼ x¼ 0.13, we obtain a bandwidth t¼ 0.18 eV and massm¼ 1.4 me, where me is the free-electron mass.

The atomic structure of a small hole polaron, as obtainedfrom our first-principles calculations, is shown in Fig. 3. Weconsider two types of excitation processes: (a) the hoppingprocess described above and (b) the excitation of a smallhole polaron to a delocalized-hole configuration. The calcu-lated configuration coordinate diagrams associated withthese two mechanisms are shown in Fig. 2. The generalizedcoordinate Q describes the displacement of the atoms withrespect to the initial state (Q¼ 0) weighted by the mass ofeach atomic species

Q2 ¼X

a

maðRa # Ri;aÞ: (2)

Here, ma are the atomic masses of the atoms, labeled withindex a. The atomic positions of the intermediate configura-tions were obtained by interpolation between the initial andfinal configurations. A parabola was fitted to the data points.

Based on the configuration coordinate diagrams, thebroadening of the transition energies due to lattice vibrationswas calculated using the formalism developed by Huang andRhys.31,32 Each configuration corresponds to a harmonic os-cillator, with quantized levels corresponding to differentvibronic states. The vibrational problem is therefore approxi-mated by a single effective phonon frequency. The calcu-lated vibrational modes are 75 meV for the harmonicoscillators 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 polaronself-trapping energy, consistent with the Holstein model.The calculated peak energy for the excitation of an electronfrom the lower Hubbard band to the polaron state in the gapis 1.12 eV (ET), and for the excitation of the polaron out ofits self-trapping potential well via a hopping process it is1.09 eV (Eh

T). Both of these values are in good agreementwith the observed 1 eV peak (Fig. 1), and once again

consistent with the Holstein model, being about twice aslarge as the polaron self-trapping energy.

The calculated absorption curves for the two mecha-nisms are shown in Figs. 2(c) and 2(d). Since the experi-ments were performed at 10 K, we only take transitions fromthe first vibronic state of the initial configuration intoaccount. The onsets are at a slightly higher energy than inexperiment. We attribute this to the presence of the SrGd sub-stitutional impurities in the samples, which is likely to leadto a broadening of the absorption peak that is not included inthe calculations. This is consistent with the phonon modefound by fitting the experimental data to Eq. (1) being larger(110 meV) than the calculated modes [75 meV for transition(a) and 65 meV for transition (b)].

In summary, we have identified small hole polarons inGd1#xSrxTiO3 thin films. The polaron self-trapping energy isfound to be 0.6 eV, both from optical conductivity measure-ments as well as from first-principles calculations.Calculations for excitation of a small hole polaron to adelocalized-hole configuration and for excitation of the smallpolaron out of its self-trapping potential well via a hoppingprocess both yield an optical excitation peak at 1.1 eV, ingood agreement with the experimental peak at 1 eV. We con-clude that this feature in the optical conductivity spectra iscaused by small hole polarons and not by excitations acrossthe 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 supportedthrough the Center for Energy Efficient Materials, an EnergyFrontier 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 theCenter for Scientific Computing at the CNSI and MRL (anNSF MRSEC, DMR-1121053) (NSF CNS-0960316), and bythe Extreme Science and Engineering DiscoveryEnvironment (XSEDE), supported by NSF (ACI-1053575).

1M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039(1998).

2H. D. Zhou and J. B. Goodenough, J. Phys.: Condens. Matter 17, 7395(2005).

3Y. Tokura, Y. Taguchi, Y. Okada, Y. Fujishima, T. Arima, K. Kumagai,and Y. Iye, Phys. Rev. Lett. 70, 2126 (1993).

4T. Katsufuji, Y. Taguchi, and Y. Tokura, Phys. Rev. B 56, 10145 (1997).5D. A. Crandles, T. Timusk, J. D. Garrett, and J. E. Greedan, Physica C201, 407 (1992).

6D. G. Ouellette, P. Moetakef, T. A. Cain, J. Y. Zhang, S. Stemmer, D.Emin, and S. J. Allen, Sci. Rep 3, 3284 (2013).

7P. Moetakef, D. G. Ouellette, J. Y. Zhang, T. A. Cain, S. J. Allen, and S.Stemmer, J. Cryst. Growth 355, 166 (2012).

8L. Bjaalie, A. Verma, B. Himmetoglu, A. Janotti, S. Raghavan, V.Protasenko, E. H. Steenbergen, D. Jena, S. Stemmer, and C. G. Van deWalle, “Determination of the Mott-Hubbard gap in GdTiO3” (submitted).

9B. Himmetoglu, A. Janotti, L. Bjaalie, and C. G. Van de Walle, Phys. Rev.B 90, 161102(R) (2014).

10M. I. Klinger, Phys. Lett. 7, 102 (1963).11H. G. Reik and D. Heese, Phys. Chem. Solids 28, 581 (1967).12D. Emin, Phys. Rev. B 48, 13691 (1993).13A. Janotti, C. Franchini, J. B. Varley, G. Kresse, and C. G. Van de Walle,

Phys. Status Solidi RRL 7, 199–203 (2013).14P. 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 bondlength, as indicated by the dashed arrows.

232103-3 Bjaalie et al. Appl. Phys. Lett. 106, 232103 (2015)

Reuse 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 Apr2016 18:42:25

Small polaron

Bjaalie et al. Appl. Phys. Lett. 106, 232103 (2015)16

Carrier transport: SrTiO3 & KTaO3Background

Son et al. Nat. Mater. 9, 482 (2010)

• Device applications: transport • Mobility: how fast charge carriers move !• Epitaxial SrTiO3: high mobility at low T • Room T mobility very low: problem for devices • Fundamental understanding of transport

needed. • Designing high mobility complex oxides

Cain et al. Appl. Phys. Lett. 102, 182101 (2013)

Background

Son et al. Nat. Mater. 9, 482 (2010)

• Device applications: transport • Mobility: how fast charge carriers move !• Epitaxial SrTiO3: high mobility at low T • Room T mobility very low: problem for devices • Fundamental understanding of transport

needed. • 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

Carrier transport• Large polarons: can treat e-p interaction at linear order • Compute carrier lifetimes:

Case of SrTiO3


• 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

⌧�1nk =

2⇡

~X

q⌫,m

|gq⌫ |2n(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 scattering rate

gq⌫ !

se2~!⌫

2✏0 Vcell q2

r1

✏1� 1

✏

⌧�1nk

• Electron-phonon scattering: • Coupling constant for LO modes:

• Scattering rate for 3 conduction bands in SrTiO3:

n = 1020 cm�3 T = 300K

• First-principles density functional theory (DFPT) • Model from large polaron theory

• electron-phonon (ep) coupling constant: gq⌫

18

Carrier transport• Boltzmann Transport + Relaxation time approximation:

�↵� = e2X

n

Zd3k ⌧nk

✓�@fnk@✏nk

◆vnk,↵ vnk,�

vnk,↵ =1

~@✏nk@k↵

Band velocities:derivative of FD distribution

electron lifetime

Evaluation of Transport Integrals

1. Madsen et al. Comput. Phys. Commun. 175, 67 (2006) 2. Pizzi et al. Comput. Phys. Commun. 185, 422 (2014)

• Transport integrals require very fine sampling around Fermi surface

!• Very large grids: band-structure

interpolation techniques:

�↵� = e2X

n

Zd3k ⌧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

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

AO

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

SrTiO3 & KTaO3

5

0

0.1

0.2

0.3

0.4

0.5

0.6

1e+18 1e+19 1e+20 1e+21

n F (e

V-1)

n (cm-3)

STOKTO-collin

KTO-ncollin

FIG. 4. Densities of states at the Fermi level. For KTO, bothcollinear and noncollinear calculations are displayed.

at EF compared to the collinear case in Fig. 3(a). This isdue to the fact that the Fermi energies for the consideredelectron densities are much lower than the split-off bandenergies, leading to a reduction of its density of states.The overall reduction of scattering rates when spin-orbitcoupling is strong has in fact been proposed before16 andKTO serves as an example for this behavior.

V. TRANSPORT INTEGRALS

For calculation of transport coefficients, we computethe following two integrals:

I(1)αβ =1

Vcell

!

n,k

τnk

"

−∂fnk∂ϵnk

#

vnk,α vnk,β

I(2)αβ =1

Vcell

!

n,k

τnk

"

−∂fnk∂ϵnk

#

(ϵnk − EF ) vnk,α vnk,β .

(5)

In terms of these integrals, the conductivity (σ) and See-beck (S) tensors are given by

σαβ = 2 e2 I(1)αβ , Sαβ = −1

eTI(1)−1αν I(2)νβ , (6)

where EF is the Fermi energy, α,β, ν are cartesian in-dices (repeated indices are summed over), and the factor2 accounts for the spin (in case of the noncollinear calcu-lation, the factor 2 is not included and the band indices nalso contain the total angular momentum quantum num-ber j = l ± 1/2). The band velocities vnk are definedas

vnk,α ≡1

h

∂ϵnk∂kα

. (7)

In this study, we compute band velocities in a very densegrid (60×60×60) using finite differences. For the cases of

KTO and STO, the conduction band structure is rathersimple (i.e. no band crossings), therefore this approachleads to accurate band velocities. In more complex bandstructures, it is necessary to use interpolation techniquesfor accurate calculations.49,50 Since KTO and STO arecubic, the conductivity and the Seebeck tensors are di-agonal with equal entries. Therefore, we will refer to thescalar quantities of conductivity (σ) and Seebeck coeffi-cient (S) from now on. The mobility is defined throughthe ratio of the conductivity to the electron density as

µ =σ

ne. (8)

Using the scattering rates computed from Eqn.(3), basedon the electron-phonon couplings of Eqn.(1), we calcu-late the transport integrals in Eqn.(5) for KTO and STO.For KTO, we perform the integration both for collinearand noncollinear band structures while for STO we onlyuse the collinear band structure. We have explicitlychecked that the noncollinear calculations in STO leadto only negligible changes in the transport integrals. Inthe 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(c

m2 /V

s)

n (cm-3)

STO

KTO-collin

KTO-ncollin

FIG. 5. Calculated mobilities of STO and KTO at 300K. ForKTO, both collinear and noncollinear calculations are dis-played.

A. Electron mobilities

Fig. 5 shows the calculated mobilities as a function ofelectron concentration at 300 K. KTO with spin-orbit in-teractions taken into account (i.e. noncollinear) displaysthe highest mobility across the range of electron densi-ties we have considered, whereas the mobilities in STOare significantly lower 51. As discussed previously, inclu-sion of spin-orbit splitting reduces the effective scatteringrate, leading to an increase in the mobilities, which is thereason for higher values in the noncollinear calculation.In the case of the collinear calculations, where the bandsare not split due to spin-orbit coupling, the mobilities

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 theΓ point, in units of cm−1.

LO TOLDA 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

Energ

y (e

V)

Γ M X Γ R

-0.1

0

0.1

0.2

0.3

0.4

0.5

Energ

y (e

V)

Γ M X Γ R

Δso

(b)

-0.1

0

0.1

0.2

0.3

0.4

0.5

En

erg

y (e

V)

Γ M X Γ R

(c)

FIG. 1. Conduction band structure for KTO, (a) in collinearcalculation, and (b) in noncollinear calculation. Band struc-ture of STO is shown in (c) for comparison.

pling depends on the dielectric constants and the LOphonon frequency. In STO and KTO, there are multi-ple LO modes that contribute to the macroscopic electricfield. In this case, the electron-phonon coupling matrix

element squared is given by17,45,46

|gq ν |2 =

1

q2

!

e2 hωLν

2ϵ0 Vcell ϵ∞

"

#Nj=1

$

1−ω2

Tj

ω2

Lν

%

#

j =ν

$

1−ω2

Lj

ω2

Lν

% , (1)

where N is the number of LO/TO modes, ωLν are LOmode frequencies, ωTν are TO mode frequencies, Vcell isthe unit cell volume, and ϵ∞ is the electronic dielectricconstant. This equation is obtained using the electron-phonon Hamiltonian in Ref. 45 and the Lyddane-Sachs-Teller (LST) relation ϵ(ω) = ϵ∞

#Nj=1 (ω

2 − ω2L,j)/(ω

2 −

ω2T,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 inTable II. Here, we define

Cν =$ a02π

%2q2 |gqν|

2, (2)

where a0 is the cubic lattice parameter. Note that there

TABLE II. Calculated electron-phonon couplings in eV2

based on Eqns.(1) and (2).

STO KTOC1 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. Oneof the LO modes in Table I is not polar– the mode withfrequency 253 cm−1 appears both in the LO and TOsectors, and since it is nonpolar, it does not split up. Theconstants Cν are enumerated in order of increasing LOmode frequency. The electron-phonon couplings reportedhere for STO are lower than those reported in Ref 16.This is due to the fact that the electron-phonon couplingconstant used in Ref 16 was only valid for the case ofone LO mode, overestimating their strength when threemodes were present.The coupling constants listed in Table II are also in

agreement with previous results in literature availablefor STO.17 For comparison, we computed values of Cν

resulting from the parameters reported in Ref 17. Us-ing the dimensionless constants αν , experimental phononfrequencies, and the band mass of 0.81me, from Ref 17,one obtains: C1 = 1.26×10−5 eV 2, C2 = 1.25×10−3 eV 2

and C3 = 9.50 × 10−3 eV 2. The first two coupling con-stants are in good agreement with our results for STOin Table II. Our constant C3 is slightly lower than thatof Ref 17. The effective mass enters into the definitionof the dimensionless constants αν defined in Ref 17, butcancels out when Cν are calculated, so its value does not

3



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

Energ

y (e

V)

Γ M X Γ R

-0.1

0

0.1

0.2

0.3

0.4

0.5

Energ

y (e

V)

Γ M X Γ R

Δso

(b)

-0.1

0

0.1

0.2

0.3

0.4

0.5

En

erg

y (e

V)

Γ M X Γ R

(c)




|gq ν |2 =

1

q2

!

e2 hωLν

2ϵ0 Vcell ϵ∞

"

#Nj=1

$

1−ω2

Tj

ω2

Lν

%

#

j =ν

$

1−ω2

Lj

ω2

Lν

% , (1)


#Nj=1 (ω

2 − ω2L,j)/(ω

2 −



Cν =$ a02π

%2q2 |gqν|

2, (2)




STO KTOC1 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





3



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

Energ

y (e

V)

Γ M X Γ R

-0.1

0

0.1

0.2

0.3

0.4

0.5

Energ

y (e

V)

Γ M X Γ R

Δso

(b)

-0.1

0

0.1

0.2

0.3

0.4

0.5

En

erg

y (e

V)

Γ M X Γ R

(c)




|gq ν |2 =

1

q2

!

e2 hωLν

2ϵ0 Vcell ϵ∞

"

#Nj=1

$

1−ω2

Tj

ω2

Lν

%

#

j =ν

$

1−ω2

Lj

ω2

Lν

% , (1)


#Nj=1 (ω

2 − ω2L,j)/(ω

2 −



Cν =$ a02π

%2q2 |gqν|

2, (2)




STO KTOC1 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





KTO-collin KTO-ncollin STO-ncollin21

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

SrTiO3 & KTaO3

• Band velocities (near EF) and/or band masses:

e.g.: Parabolic bands: ✏nk =~2k22mb

vnk,↵ =~k↵mb

Smaller mass, larger mobility (but d0 perovskites have highly non-parabolic bands)

6

are higher in KTO than in STO because the higher bandvelocities. Since Ta 5d orbitals lead to wider conductionbands compared to those of Ti 3d, such an increase isexpected. These findings are in agreement with the ex-perimental observation of higher room-temperature mo-bilities in KTO compared to STO.2,9

We note that our calculations overestimate the ob-served room temperature mobilities of both STO, whichis less than 10 cm2/Vs ,4,14,15 and KTO, which is around30 cm2/Vs,2 for a range of electron concentrations. Thisis due to the fact that our calculations contain only LOphonon scattering mechanism, and ignores others such asimpurities,12 TO-phonon11 and electron-electron scatter-ing.13

Further insight into the behavior of mobility shown inFig.(5) can be gained by calculating the average valuesof the Fermi velocities, which we define as

v2F ≡

!

n,k δ(EF − ϵnk) v2nk!

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 KTOcompared to STO. While KTO in the noncollinear cal-culation has a higher band-width (as can seen in Figs. 1and 4), the averaged values of the Fermi velocities areslightly 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 andlow velocity regions, averaging out to a similar numericalvalue for both collinear and non-collinear calculations.Instead, lower scattering rates in the noncollinear calcu-lations lead to the enhancement of mobilities as seen inFig. 5.

0.2

0.3

0.4

1e+18 1e+19 1e+20 1e+21

a 0 v

F (e

V)

n (cm-3)

STOKTO-collin

KTO-ncollin

FIG. 6. Averaged Fermi velocities (rescaled with the latticeparameter a0). For KTO, both collinear and noncollinearcalculations are displayed.

An additional feature which deserves attention is theincrease in mobility with increasing electron density nas seen in Fig. 5. This may seem counterintuitive, sincethe effective scattering rates at EF also increases with n(see Fig. 3). However, band velocities vnk increase as a

function of electron energy in the conduction band. Theincrease in v2nk dominates over the increase in τ−1

nk leadingto an overall increase of the mobility as a function of n.

B. Seebeck coefficients

Fig. 7 shows the calculated Seebeck coefficients as afunction of electron concentration at 300 K. The non-collinear calculation for KTO displays the lowest Seebeck(absolute value) coefficients across the range of electrondensities we have considered. STO on the other handdisplays the highest Seebeck coefficients, with collinearKTO displaying slightly lower values. This behavior canbe understood through the dependence of the Seebeckcoefficient on EF . The Seebeck coefficient is inverselyproportional to EF

22,23,44 (for a parabolic band approxi-mation in a metal, it is inversely proportional to EF ). Fora given electron concentration n, EF is lowest for STO,since its conduction band width is smaller than KTO,which in turn leads to a higher Seebeck coefficient |S|.The difference between collinear and noncollinear calcu-lations in KTO mainly results from the effective numberof conduction bands. For the noncollinear calculation,the split-up band is higher in energy, leading to an ef-fective 2-conduction-band system (see Fig. 1(b)). Fora given n, EF increases when the number of conductionbands decrease; when there are less bands which are closein energy, electrons occupy higher lying bands. Thus, |S|in the noncollinear case is lower than that of the collinearone, due to the reduction of the number of conductionbands.23

0

200

400

600

1e+18 1e+19 1e+20 1e+21

-S (µ

V/K)

n (cm-3)

STOKTO-collin

KTO-ncollin

FIG. 7. Calculated Seebeck coefficients of STO and KTO at300K. For KTO, both collinear and noncollinear calculationsare displayed.

Compared to experimental results, our calculationsslightly underestimate the Seebeck coefficients. For STO,experiments22 yield 147 < |S| < 380 µV/K in the range1021 > n > 8.8 × 1019 cm3, while our calculations yield80 < |S| < 220 µV/K in the range 1021 > n > 1020 cm3.Similarly for KTO, experiments9 yield 200 < |S| < 460

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

2DEGs & screening LO modes out

with

KLO ¼!h

2b!hxLO

e

mP

me

mP

! "2

f bð Þ; (3)

where !h is the reduced Planck’s constant, e is the electroncharge, kB is the Boltzmann constant, !hxLO is the energy ofthe LO phonon, b is the electron-phonon coupling constant,me is the electron effective mass (taken to be 1.8 times thefree electron mass29), mP ¼ með1þ b=6Þ is the polaronmass, and f ðbÞ is a function with a value of approximatelyone.28 KLO is approximately 0.12 eV cm2/V s, see Ref. 28.

The solid line in Fig. 1(a) shows a fit using Eqs. (1)–(3),which accurately describes the experimental data. The fitparameters were l0, a, and xLO. In SrTiO3, carriers arebelieved to couple strongly to two LO phonon modes18 andthe value for xLO determined here is an average, effectivevalue. Fits for the more highly doped samples are shown inthe supplementary material30 and yield similarly gooddescriptions. Figures 1(b) and 1(c) show the results for a andxLO as a function of N. As N increases, xLO and a bothincrease. As a result, lLO increases while le-e decreases andthe total mobility becomes increasingly limited by electron-electron scattering. This is summarized in Fig. 1(c), whichshows le-e and lLO calculated at room temperature using theextracted values for a and xLO, the total mobility calculatedusing these two terms, and the experimentally measured Hallmobility. At N% 6& 1018 cm'3, the electron-phonon scatter-ing limited regime crosses over to an electron-electronscattering limited regime; as a result, the mobility remains<10 cm2 V'1 s'1 for all N.

The increase of lLO with N is in agreement with first-principles calculations,17 where it has been shown that theelectron-phonon scattering rates decrease away from theBrillouin zone center. Thus with increasing N, as the Fermisurface enlarges, the scattering rate becomes smaller, leadingto an increase in lLO.

The results shown in Fig. 1 confirm earlier results thatelectron-electron scattering must be accounted for in describ-ing the temperature-dependence of the mobility of SrTiO3,18

and furthermore show that it becomes the limiting mecha-nism at high doping. It is well known that for simple metalswith a single conduction band, electron-electron scatteringleads to resistance only for Umklapp processes at linearorder transport theory.27,31 This is due to the fact that thetotal crystal momentum is conserved in an electron-electronscattering event for normal (non-Umklapp) processes. InSrTiO3, the increased electron-electron scattering rate athigh doping [Fig. 1(b)] can also be attributed to Umklappprocesses. With increasing N the Fermi surface expands,reaching closer to the Brillouin zone boundaries, enablingUmklapp scattering. Furthermore, when there are multipleconduction bands, as is the case for SrTiO3,32,33 electron-electron scattering can proceed through both intraband andinterband transitions. For an interband scattering event, thetotal current can change via transfer of electrons betweenbands with different effective masses, leading to resistancefrom electron-electron scattering even for normal processes,in addition to Umklapp ones.34 This leads to an increasedelectron-electron scattering rate compared to simple metals.In addition, short-range Coulomb interactions are strong forthe conduction bands derived from the localized 3d orbitalsof Ti, which also leads to an enhancement of electron-electron scattering.2 These interactions increase with N (asthe probability of two electrons occupying the same siteincreases).

Figure 2 shows the Hall mobility as a function of tem-perature for the 2DELs in structures with different SrTiO3

thicknesses. As shown by the dashed lines, the mobility canbe completely described over the entire temperature rangeup to room temperature as l'1 ¼ l'1

0 þ l'1e-e. Thus, unlike

for bulk doped samples, there is no detectable LO phonon

FIG. 1. (a) Temperature dependence of the Hall mobility (l) of SrTiO3

doped with La to a carrier density of 8& 1017 cm'3 (from Ref. 21). The solidline is a fit to Eq. (1), with l'1

e-e¼ aT2 and lLO as given by Eqs. (2) and (3),while the dashed lines show the temperature dependence of the terms in Eq.(1) obtained from the fit. (b) and (c) Extracted parameters a and xLO fromfits to the samples with different doping concentrations (N). The individualfits are shown in the supplementary material.30 (d) Comparison of the meas-ured room temperature mobility (solid circles), le-e (diamonds), and lLO

(squares) calculated using the extracted fit parameters, and ðl'1e-e þ l'1

LOÞ'1

(hexagons). The close agreement between the measured l andðl'1

e-e þ l'1LOÞ'1 shows that the room temperature mobility is controlled by

electron-electron and LO phonon scattering terms, with a crossover in thedominant contribution near N% 6& 1018 cm'3.

062102-2 Mikheev et al. Appl. Phys. Lett. 106, 062102 (2015)

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scattering contribution in the 2DELs. The absence of LOphonon scattering explains the slightly larger room-temperature mobility in the 2DELs. The inset in Fig. 2 showsa as a function of SrTiO3 thickness. For the thinnest layers,the three-dimensional carrier density in the 2DEL isincreased (the layer width is substantially smaller than thespatial extent of the 2DEL) and the strength of electron-electron scattering (a) is increased.

The results indicate a complete screening of the LO pho-non mode in the 2DELs. In addition to the screening by thelarge density of carriers in the 2DEL, electronic screeningalso depends on the density of states (DOS). Confinement ina 2DEL increases the DOS at the Fermi level.35 An indicatorof a substantial increase of the DOS in GdTiO3/SrTiO3

2DELs is the observed itinerant ferromagnetism,36 as it isrequired to satisfy the Stoner criterion to cause spin polariza-tion in the conduction bands.37 Once the conduction elec-trons screen the ionic potentials effectively, there is no longrange Coulomb interaction that splits the LO and TO modefrequencies, and therefore the polar optical mode scatteringdisappears, explaining the disappearance of the LO phononscattering in the 2DEL.

Figure 3 summarizes the findings by showing the roomtemperature values of l as a function of a for the 2DELs andthe bulk doped SrTiO3 samples. The dotted line indicates theelectron-electron scattering limited mobility, calculated asl!1

e-e¼ aT2. In bulk SrTiO3, LO phonon scattering reducesthe mobility below the le-e limit. To increase the room tem-perature mobility of SrTiO3, LO phonon scattering should besuppressed. This is achieved for the 2DELs with thickerSrTiO3, which fall on the theoretical l!1

e-e line. In this limit,to improve the mobility, a should be reduced. If this provespossible, significant mobility gains (by a factor of 2 or more)could be achieved with relatively small reductions in a.For the extremely confined 2DELs (few nm SrTiO3 layers,right-most data points), other factors, including interface

roughness scattering, cause a drop in the mobility below thele-e limit.

In summary, we have shown that the room temperaturemobility in high-carrier-density 2DELs in SrTiO3 is not lim-ited by LO phonon scattering, in sharp contrast to bulk,doped SrTiO3. To develop approaches to improve the mobil-ity in the 2DELs, a quantitative, theoretical understanding ofcontributions to a in SrTiO3 2DELs must be developed. Theresults emphasize that the transport physics of 2DELs incomplex oxides is significantly different from that of con-ventional semiconductors, where electron-electron scatteringonly plays a minor role and electron-phonon scattering is thedominant mechanism limiting the room temperaturemobility.

This work was supported by the Extreme ElectronConcentration Devices (EXEDE) MURI program of theOffice of Naval Research (ONR) through Grant No.N00014-12-1-0976. Film growth experiments (A.P.K.) weresupported by the UCSB MRL, which is supported by theMRSEC Program of the U.S. National Science Foundationunder Award No. DMR-1121053, and which also supportedthe central facilities used in this work. A.P.K. alsoacknowledges support from the U.S. National ScienceFoundation through a Graduate Research Fellowship (GrantNo. DGE-1144085).

1S. Stemmer and S. J. Allen, Annu. Rev. Mater. Res. 44, 151 (2014).2P. Moetakef, C. A. Jackson, J. Hwang, L. Balents, S. J. Allen, and S.Stemmer, Phys. Rev. B 86, 201102(R) (2012).

3C. A. Jackson, J. Y. Zhang, C. R. Freeze, and S. Stemmer, Nat. Commun.5, 4258 (2014).

4S. Okamoto and A. J. Millis, Nature 428, 630 (2004).5R. Pentcheva and W. E. Pickett, Phys. Rev. Lett. 99, 016802 (2007).6R. Chen, S. Lee, and L. Balents, Phys. Rev. B 87, 161119(R) (2013).

FIG. 2. Temperature dependence of the inverse of the mobility l in 2DELsat GdTiO3/SrTiO3 interfaces. The samples consisted of epitaxial bilayerswith SrTiO3 different thicknesses (GdTiO3 is the top layer) on an insulatingsubstrate, from Ref. 11. The inset shows the extracted parameter a from fitsto l!1 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 l!1¼ aT2. The mobility of the three-dimensional electron gases fallsbelow the dashed line due to the significant contribution of LO phonon scat-tering, which is absent in the 2DELs. The extremely confined 2DELs (tworight-most data points) fall below the line due to contributions from interfaceroughness scattering even at room temperature.

062102-3 Mikheev et al. Appl. Phys. Lett. 106, 062102 (2015)

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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

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

https://github.com/bhimmetoglu/transport_new

LO phonon interaction• Fröhlich interaction for multiple modes (using Born effective

charges):

2

Now, we express the electron density using the electron creation and annihilation operators (assuming free electron-likewavefunctions), which give

ρ(−q) =!

k

c†k ck−q (8)

Inserting Eq.(8) in Eq.(7), we arrive at

Hei = i!

q, ν,k

gq ν (a†−q ν + aq ν) c

†k ck−q (9)

with the electron-phonon coupling gq ν defined as

gq ν ≡e2

ϵ0 V

!

s,α, β,λ

"

!

2Ms ωL ν

#1/2 1

qqα (ϵ−1

∞ )αβ Z∗ β λs esλ(q; ν) (10)

Implementation

For implementation, the dispersion of polar LO modes will be neglected, meaning that ωL ν and esλ(q; ν) will bereplaced with their values at Γ. This is a good approximation for such modes.

1 Baroni et al.: Phonons and related crystal properties from DFPT, Rev. Mod. Phys. 73, 515 (2001)

• Simplifications:

1

Author: Burak Himmetoglu

Details on the transport code implementation

Scattering rates:

τ−1nk =

2π

!

!

qν,m

|gqν |2 (1− vnk · vmk+q)

"

(nq ν + fm,k+q) δ(ϵm,k+q − ϵnk − !ωLν)

+(1 + nq ν − fm,k+q) δ(ϵm,k+q − ϵnk + !ωLν)

#

(1)

where

|gq ν |2 =1

q2

$

e2 !ωLν

2ϵ0 Vcell ϵ∞

%

&Nj=1

'

1− ω2

Tj

ω2

Lν

(

&

j =ν

'

1− ω2

Lj

ω2

Lν

( (2)

In Eq.(1) the delta-function is replaced by a Gaussian smearing with adaptive smearing width:

δ(Fnk,mk+q) =1√π σ

exp

)

−F 2nk,mk+q

σ2

*

σ = ∆Fnk,mk+q = a

+

+

+

+

∂Fnk,mk+q

∂k

+

+

+

+

∆k

Fnk,mk+q = ϵm,k+q − ϵnk ± !ωLν (3)

where a is a dimensionless parameter that determines the width of the smearing. The natural choice is a = 1.0,however larger values of a produces smoother densities of states.

In order to compute the electron-phonon scattering strength at the Fermi level, we define

Dn τ−1n (EF ) ≡

!

k

δ(EF − ϵnk) τ−1nk . (4)

The above integral can be carried out in a reduced grid where the k-points are restricted to the region where theenergy eigenvalues satisfy

|ϵnk − EF | < c× kT (5)

where T is the temperature and c is a dimensionless parameter(we refer to it as ”cut”). This results is a band-dependent grid size. This choice is related to the fact that the transport integrals contain the derivative of theFermi-Dirac distribution, which is peaked around EF . For example, in case of a 16× 16× 16 grid, a choice of c = 10.0reduces the integration grid from a the full 4096 points, to 438 for band 1, 57 for band 2 and 33 for band 3.

• For a single mode, the above expression is equivalent to Fröhlich.

26

TransportTransport Theory

f(r,k, t) d3r d3k

• 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

TransportTransport Theory

�↵� = e2X

n

Zd3k ⌧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↵

⌧�1nk

• Scattering times/rates determined by collisions: Electron-defect Electron-electron Electron-phonon

28





Phonon scattering

29

Electronic structure of strongly correlated systems

ℋ = −𝑡 𝑐 𝑖𝜎† 𝑐 𝑗𝜎 + ℎ. 𝑐 + 𝑈 𝑛 𝑖↑𝑛 𝑖↓𝑖<𝑖,𝑗>,𝜎

Independent e- e- - e- interaction

Small Coulomb repulsion: U large hopping amplitude: t

4

𝑡 ≫ 𝑈

Strongly correlated systems

30

Strongly correlated systemsElectronic structure of strongly

correlated systems ℋ = −𝑡 𝑐 𝑖𝜎† 𝑐 𝑗𝜎 + ℎ. 𝑐 + 𝑈 𝑛 𝑖↑𝑛 𝑖↓

𝑖<𝑖,𝑗>,𝜎

Independent e- e- - e- interaction

Large Coulomb repulsion: U small hopping amplitude: t

5

𝑡 ≪ 𝑈

31