PHYSICAL REVIEW B 88, 014116 (2013)

Ferroelectric ground state and polarization-switching path of orthorhombic YMnO3 with coexistingE-type and cycloidal spin phases

Jung-Hoon Lee,1 Seungwoo Song,1 and Hyun Myung Jang1,2,*

1Department of Materials Science and Engineering, and Division of Advanced Materials Science, Pohang University of Scienceand Technology, Pohang 790-784, Republic of Korea

2Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea(Received 21 November 2012; revised manuscript received 21 May 2013; published 31 July 2013)

Orthorhombic YMO3 (o-YMO) is currently being intensively investigated since a single-crystalline thin filmof o-YMO recently showed a pronounced degree of magnetoelectric coupling. According to first-principlescalculations, the ferroelectric ground state is represented by two degenerate E-type collinear spin configurationswith the computed polarization P of ∼1.4 μC/cm2. The bc-cycloidal spin phase is identified as the spinconfiguration that corresponds to the transition state in the +P ↔ −P polarization switching. The remarkablecoexistence of the E-type and cycloidal spin phases below ∼40 K is attributed to a small value of the Kohn-Shamenergy difference between these two phases, 2.2 meV per formula unit. A modulated spin structure, which ischaracterized by the tilting of the Mn-spin vectors to the a-axis direction of Pbnm setting, is proposed to accountfor the observed strong magnetic-field-dependent polarization in o-YMO.

DOI: 10.1103/PhysRevB.88.014116 PACS number(s): 77.80.−e, 71.15.Mb, 75.50.Ee, 75.80.+q

I. INTRODUCTION

Multiferroic materials have received a great deal of attentionowing to their potential for novel device applications.1–9

Among numerous multiferroics currently under investiga-tion, rare-earth orthorhombic manganites, such as TbMnO3,2

TbMn2O5,3 DyMnO3,5 HoMnO3,6 and YMnO3,7 have beenmost extensively studied owing to their tendency towards astrong magnetoelectric (ME) coupling which stems from spin-ordering-induced polarizations.9,10 There appear two distincttypes of the spin-ordering-induced improper ferroelectricity inrare-earth manganites: (i) the spin-current-induced improperpolarization of the antisymmetric (Si × S j )-type in a non-collinear antiferromagnetic (AFM) magnet, which can beunderstood in terms of the spin-orbit-coupling-driven reverseDzyaloshinskii-Moriya (DM) interaction,11,12 and (ii) thepolarization of the symmetric (Si•S j )-type which is inducedby the exchange-striction interaction between neighboringspins.10,13,14 Concerning the exchange-striction mechanism,both collinear and noncollinear spin structures can inducethe improper polarization by Si•S j . The most prominentexample of the exchange-striction-induced ferroelectricity in acollinear spin system is the polarization caused by the collinearE-type spin structure found in o-HoMnO3.6,13,15,16 In thiscase, the symmetry of the spin zigzag-chain E-phase in theorthorhombic perovskite allows for the occurrence of a finitepolarization, which is independent of the spin-orbit coupling.13

Orthorhombic YMO3 (yttrium orthomanganite; o-YMO)is also known to possess the E-type spin structure.17 Thecoexisting spin phases and the associated ferroelectricity originare currently being intensively investigated7,17–19 as a single-crystalline thin film of o-YMO recently showed a pronounceddegree of ME coupling.18 The ground state of o-YMO iscurrently understood in terms of the coexisting E-type andcycloidal spin phases.18,19 However, we still have severaluncertain points and unexplored issues that have to be clarified.These include the following five important points: (i) Thespin configuration that corresponds to the doubly degenerateferroelectric ground states; (ii) the relative contribution of

the (Si × S j )-type polarization arising from the noncollinearcycloidal spin structure (either bc-cycloidal or ab-cycloidalphase) with respect to the (Si•S j )-type polarization causedby the collinear E-type spin structure; (iii) the path of thepolarization switching from one of the doubly degenerateferroelectric ground states to the other state—in other words,what is the spin structure that corresponds to the transition statein the +P ↔ −P polarization switching; (iv) understandingthe uncommon coexistence of the collinear E-type spin phaseand the noncollinear cycloidal spin phase below 35 K;18

(v) a single-crystalline thin film of o-YMO showed a strongcoupling between the magnetic order and the ferroelectricpolarization as manifested by a pronounced dependence ofthe a-component polarization (Pa) on the applied magneticfield at low temperatures.18 Then, what is the modulated spinstructure responsible for the observed strong ME coupling?

Accordingly, the main purpose of this paper is to clarify theabove-mentioned uncertainties by exploiting first-principlesdensity functional theory (DFT) calculations. To this end, wehave first identified the spin structure corresponding to theferroelectric ground state and elucidated that the bc-cycloidalphase is the unique spin structure which corresponds to thetransition state in the +P ↔ −P polarization switching.

II. COMPUTATIONAL METHODS

We performed DFT calculations on the basis of thegeneralized gradient approximation implemented with theprojector augmented wave pseudopotential20 using the Viennaab initio simulation package.21 We adopted (i) a 6 × 2 × 4Monkhorst-Pack k-point mesh centered at �,22 (ii) a 500 eVplane-wave cutoff energy, and (iii) the tetrahedron method withthe Blochl corrections for the Brillouin zone integrations.23

The Hubbard Ueff of 4.0 eV for Mn was chosen on the basisof empirical corrections. We explicitly treated 11 valenceelectrons for Y (4s24p64d15s2), seven for Mn (3d54s2),and six for O (2s22p4). The structural optimizations wereperformed for the 40-atoms cell, which corresponds to a

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JUNG-HOON LEE, SEUNGWOO SONG, AND HYUN MYUNG JANG PHYSICAL REVIEW B 88, 014116 (2013)

(b) bc-cycloidal

θ

θ

(c) ab-cycloidal

(a) E1 E)d()(MFAepyt- 2-type AFM ( )

θ = 0o θ = 180o

θ = 90o

YMn

O

+

FIG. 1. (Color online) Schematic magnetic unit-cell structures of o-YMnO3 for prototypical spin configurations. (a) E1-type spinconfiguration that corresponds to the symmetric (Si•S j )-type polarization with −P 0

a of 1.4 μC/cm2. (b) bc-cycloidal spin structure thatcorresponds to the antisymmetric (Si × S j )-type polarization with a nonzero Pc of ∼10 nC/cm2, but Pa = 0. (c) ab-cycloidal spin structurewith a nonzero Pa of ∼10 nC/cm2, but Pc = 0. (d) E2-type spin configuration with +P 0

a of 1.4 μC/cm2.

(a × 2b × c) supercell. This supercell closely mimics theground-state incommensurate wave-vector of k = (0, 0.458,0).15 All the structural relaxation was performed with aGaussian broadening of 0.05 eV.24 The ferroelectric polar-ization was calculated using the Berry-phase method.25

III. RESULTS AND DISCUSSION

A. Doubly degenerate E-type spin configurations

Four prototypical magnetic unit-cell structures of o-YMOare depicted in Fig. 1. According to DFT calculations, theferroelectric ground state is represented by two degeneratecollinear spin configurations. They are shown in Figs. 1(a) and1(d) and are denoted by E1-type AFM (−P 0

a ) and E2-typeAFM (+P 0

a ) structures, respectively. Here, +P 0a or −P 0

a

appearing inside parentheses denotes the equilibrium polar-ization induced by the E-type collinear spin structure, wherethe subscript a signifies that this (Si•S j )-type polarization isparallel to the a axis of Pbnm setting. Strictly speaking, thepolar crystal symmetry that can induce ±P 0

a by the E-type spinordering is P 21nm.17 We will show that these two degenerateE-type spin configurations E1 and E2 correspond to the twominima in the ferroelectric double-well potential.

As described previously, the ground state of o-YMO iscurrently understood in terms of the coexisting E-type and twocycloidal spin configurations.18,19 Thus, we also depict the twopossible cycloidal spin states in Fig. 1. In the bc-cycloidal spinstructure [Fig. 1(b)], all of the Mn-spin vectors lie on the b-cplane (with no a-axis component) in Pbnm setting. On the otherhand, all the Mn-spin vectors lie on the a-b plane in the ab-cycloidal spin configuration [Fig. 1(c)]. In Fig. 1, the relativeorientation or the canting angle of the progressively rotatingMn spins with respect to the fixed Mn spins is denoted by θ .16

Thus, the canting angle for the E-type spin configuration is

either 0◦ (E1 configuration) or 180◦ (E2 configuration). Onthe contrary, the canting angle for the two cycloidal spin stateswhich originate from the spin-orbit-coupling-driven reverseDM interaction is 90◦. It should be emphasized here that thepolar direction of the improper polarization arising from theab-cycloidal spin configuration is parallel to the a axis ofPbnm setting. This contrasts with the improper polarizationcaused by the bc-cycloidal spin configuration in which thepolar direction is parallel to the c axis.

B. Polarization switching via the bc-cycloidal spin state

The Kohn-Sham (K-S) energy of o-YMO is plotted inFig. 2(a) as a function of the canting angle θ . This double-well-type potential indicates that the doubly degenerate ferro-electric ground states are represented by the two E-type spinconfigurations, namely, E1-type AFM with θ = 0◦ and E2-typeAFM with θ = 180◦. As half of the Mn spins progressivelyrotate from θ = 0◦ to θ = 90◦ (or equivalently from θ =180◦ to θ = 90◦), the K-S energy increases with the spinrotation [Fig. 2(a)], and o-YMO undergoes a transition in thespin configuration from the collinear E-type spin state to thenoncollinear cycloidal spin state (either ab or bc).

According to the Berry-phase calculations,25 the polar-ization arising from the E-type spin configuration has thea-axis component only. The θ -dependent polarization Pa

presented in Fig. 2(b) indicates that, under an applied electricfield, the two ground-state E-type polarizations undergo aswitching between +P 0

a and −P 0a via the transition state

having the bc-cycloidal spin configuration. This cycloidaltransition state provides a low-energy pathway for the +P 0

a ↔−P 0

a polarization switching [∼2.2 meV per formula unit;Fig. 2(a)]. In Fig. 2(b), we deliberately removed “ab-cycloidalstate” at θ = 90◦ since the ab-cycloidal spin configurationalways leads to a nonzero Pa . In other words, the ab-cycloidal

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0 90 180-1.8

-1.2

-0.6

0.0

0.6

1.2

1.8

Pa (μ

C/c

m2 )

θ (degree)

bc-cycloidalE2 AFM

E1 AFM

0 90 180

-2

-1

0

Δ EK

S (m

eV/f.

u.)

E2E1

+Pa-Pa

ab- or bc- cycloidal(a)

(b)

FIG. 2. (Color online) (a) The Kohn-Sham energy and (b) thea-axis component of the ferroelectric polarization Pa computed as afunction of the canting angle θ of the Mn spin. The point at θ = 90◦

corresponds to the bc-cycloidal spin state with Pa = 0.

spin state cannot be the transition state in the +P 0a ↔ −P 0

a

polarization switching. The DFT value of Pc for the bc-cycloidal configuration (at θ = 90◦) is ∼10 nC/cm2, whilePa is strictly zero. Thus, the (Si•S j )-type polarization causedby the E-type spin structure (∼1.4 μC/cm2) overwhelminglydominates over the (Si × S j )-type polarization originatingfrom the cycloidal spin configuration (bc or ab).

C. Stabilization of orthorhombic phase by spin ordering

We have also examined the effect of the spin ordering onthe insulating property and the Kohn-Sham energy of o-YMO.According to the DFT calculations by adopting the Hubbardon-site Coulomb energy of 4.0 eV ( = Ueff), both the collinearE-type spin ordering and the noncollinear cycloidal spinordering remarkably increase the band gap Eg: Eg = ∼0.5 eVfor a disordered spin state (no spin) versus Eg = ∼1.5 eV forboth E-type and cycloidal spin states at the highest-symmetry� point. More importantly, we have found that the spinordering itself greatly stabilizes the orthorhombic systemthermodynamically. Indeed, the difference in the Kohn-Shamenergy (EKS) between the E-type spin-ordered state and thespin-disordered state of o-YMO is 40 meV per formula unit(f.u.) or per Mn ion which is ∼18 times bigger than the

EKS difference between the E-type spin-ordered state andthe cycloidal spin state [2.2 meV per f.u. ≡ �Ea; Fig. 2(a)].

D. Uncommon coexistence of collinearand noncollinear spin states

We now address the remarkable coexistence of the collinearE-type spin state with the noncollinear cycloidal spin state(either ab or bc) below ∼40 K (Tc, ferroelectric transitiontemperature).18 For this, let us first consider a rapid equilibriumbetween the E-type spin state and the cycloidal spin statebelow Tc [Fig. 3(a)]. Herein, the cycloidal spin state (eitherab or bc) can be regarded as a metastable transition statewith respect to the stable E-type spin state. According to thetransition-state theory,26 the hopping frequency (ν) betweenthese two states is given by ν = kBT

he−�Ea/kBT , where h is

the Planck constant, kB is the Boltzmann constant, and �Ea

is the free-energy difference (effectively, Kohn-Sham energydifference) between the E-type ground spin state and thecycloidal spin state [Fig. 3(a)]. We then obtain the followingexpression by applying the Boltzmann’s relation:

fcy

fE

= �cye−Ecy/kBT

�Ee−EE/kBT= 2e−Ecy/kBT

e−EE/kBT, (1)

a

Koh

n-Sh

am E

nerg

ycycloidal

E-type

0 10 20 30 40 500.00

0.25

0.50

0.75

1.00

fcy

Phas

e Fr

actio

n(f)

Temperature (K)

fE

Tco

(a)

(b)

FIG. 3. (Color online) (a) A rapid equilibrium between the E-typespin state and the doubly degenerate metastable cycloidal spin statesbelow Tc. (b) Temperature-dependent relative phase fraction of thecollinear E-type spin state fE and the noncollinear cycloidal spin statefcy , showing the crossover temperature TCO where fE = fcy = 1/2is in the vicinity of 35 K. This prediction coincides well with theexperimental result.18

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JUNG-HOON LEE, SEUNGWOO SONG, AND HYUN MYUNG JANG PHYSICAL REVIEW B 88, 014116 (2013)

where fcy denotes the fraction of the noncollinear cycloidalspin phase, fE is the fraction of the collinear E-type spinphase, and Ecy can be identified as the Kohn-Sham energyof the cycloidal spin phase. In Eq. (1), �cy denotes theconfigurational degeneracy of the cycloidal spin phase and isequal to 2 since the cycloidal spin state is characterized by thedoubly degenerate ab- and bc-cycloidal spin configurations.On the other hand, �E = 1 [Fig. 3(a)]. Since �Ea is defined asEcy − EE (thus, + 2.2 meV per f.u.), Eq. (1) can be rewrittenin terms of �Ea as

fcy

fE

= 2e−�Ea/kBT . (2)

One can eliminate fcy in Eq. (2) using the requirement thatfE + fcy = 1. Thus, we obtain

fE = 1 − fcy = 1

1 + 2e−�Ea/kBT. (3)

Equation (3) predicts that fE is 1.0 at 0 K (as expected) andfE = fcy = 1/2 at 35 K, which remarkably coincides with theexperimentally reported temperature18 of the crossover (TCO)between the E-type and cycloidal spin phases [Fig. 3(b)].According to Eq. (3), fcy is predicted to be as high as0.36 even at 20 K. Inspecting the ratio given in Eq. (2), onecan readily conclude that a small value of �Ea (2.2 meV perf.u. according to the DFT calculation) is primarily responsiblefor this uncommon coexistence of the two distinct spinphases at a temperature substantially below TCO(=35K).Equation (3) also tells us that fE does not decay to 0 butis substantial (>1/3) even at the ferroelectric transition point(Tc ≈ 40 K), which predicts a substantial value of Pa evenat Tc. Contrary to this prediction, the measured Pa is nearlyzero at 40 K (Tc).18 This contradiction suggests that �Ea

does not remain constant ( + 2.2 meV per f.u.) but becomesnegative with increasing temperature. For temperatures above

θ=60oθ=0o

θ=90o

0.01 Å

O1

(002) Plane

0.02 Å

0.01 Å

0.003 Å

-Pa

Mn

θ=180o+Pa

(a) (b)

(c) (d)

a

b

O2

a

b

FIG. 4. (Color online) Off-centering ferroelectric displacements and electron-density distributions projected on (002) for several selectedcanting angles of the Mn spin.28 (a) θ = 0◦, (b) θ = 60◦, (c) θ = 90◦, and (d) θ = 180◦.

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TCO ( = 35 K), a negative value of �Ea is postulated as thecycloidal phase should be more stable than the E-type spinphase. If this is the case, the computed result presented inFig. 3(b) is wrong for temperatures higher than TCO . Underthis circumstance, Pa becomes zero at Tc since the bc-cycloidalspin phase (with Pa = 0) completely dominates over the E-typespin phase.

E. Off-centering displacements and electron-density contour

Having identified the ferroelectric ground state and thepolarization-switching path, we now examine the off-centeringferroelectric displacements and the associated electron-densitycontour. The three yellow arrows appeared at the bottom ofeach panel (except for θ = 90◦) denote the net direction ofthe ferroelectric polarization with the magnitude being propor-tional to the Berry-phase polarization.25 On the other hand, thered and green arrows in Fig. 4, respectively, represent the di-rections of the off-centering atomic displacements for Mn andO with respect to the centrosymmetric Pbnm structure. It canbe shown that the sum of the displacements resulting from theMn and O sublattices has a nonzero component only along thea axis. It is interesting to note that all the off-centering Mn dis-placements disappear in the cycloidal spin state where θ = 90◦.

The right-hand-side figure of each panel in Fig. 4 illustratesthe electron-density distribution by plotting the differencein the computed electron localization function [δELF(r)]between the paraelectric and ferroelectric states.27 Thus,δELF(r) at each location signifies the asymmetric electronlocalization associated with the para-to-ferroelectric phasetransition.27 It is interesting to note that the degree of theasymmetric electron condensation at the O1 sublattice isnoticeably more pronounced than that at the O2 sublatticewhich is characterized by a smaller off-centering displacement(OCD): OCD of 0.02 A at the O1 site versus OCD of 0.01 Aat the O2 site. In addition, we have separately estimated12 thepurely electronic polarization and the lattice contribution tothe total polarization caused by the E-type spin configurationand found that the lattice (ionic) polarization (∼40%) iscomparable to the electronic contribution (∼60%). This resultis very similar to that of o-HoMnO3.16

F. Polarization change by an external magnetic field

We are now in a position to examine the atomic-scalemechanism responsible for the strong ME effect of o-YMO.This issue can be recast by the following question: What isthe modulated spin structure that accounts for the observedstrong magnetic-field dependence of the polarization? Thepronounced dependence of the a-component polarization(P H

a ) on the applied magnetic field along the a axis H a can beproperly explained by the proposed magnetic-field-dependentspin structure presented in Fig. 5(a). The modulated spinstructure projected on the (002) plane is characterized bythe tilting-angle (ϕ) which is defined as the angle betweenthe Mn-spin vector and the b-axis line as depicted in theright-hand-side panel of Fig 5(a). The tilting-angle dependenceof the polarization as presented in Fig. 5(b) predicts that P H

a

decreases gradually with increasing ϕ, but decreases ratherrapidly beyond a certain critical ϕ (ϕc ≈ 20◦). According to

0 10 20 30 400.4

0.8

1.2

1.6

PH a (μμC

/cm

2 )

ϕ (degree)

P0a

(a)

(b)

a

b

φ

(002) Plane

H-field // a

FIG. 5. (Color online) (a) Proposed structure of the Mn spinsunder a bias magnetic field along the a axis. This modulated spinstructure projected on (002) is characterized by the tilting-angleϕ which is a measure of the deviation from the E-type spinconfiguration. (b) The calculated electric polarization (P H

a ) as afunction of ϕ.

a recent report on the single-crystalline o-YMO thin-film,18

the normalized polarization reduction is as high as 20% at amagnetic-field strength of 14 T. Combining this result withthe theoretical calculations [Fig. 5(b)], one can predict thatthe tilting-angle (ϕ) which corresponds to H a of 14 T isapproximately 30◦, and this angle can produce a pronouncedME effect of ∼20% in the field-dependent polarization.

IV. CONCLUSIONS

The ferroelectricity of o-YMO in its ground state isstemming from the exchange-striction interaction betweenneighboring spins having the collinear E-type structure. Thebc-cycloidal spin phase is identified as the spin configurationthat corresponds to the transition state in the +P ↔ −P polar-ization switching. A modulated Mn-spin structure is proposedto properly account for the observed strong magnetic-field-dependent polarization reduction in o-YMO.

ACKNOWLEDGMENTS

This work was supported by the Basic ScienceResearch Program (Grant No. 2012R1A1A2041628) through

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JUNG-HOON LEE, SEUNGWOO SONG, AND HYUN MYUNG JANG PHYSICAL REVIEW B 88, 014116 (2013)

the National Research Foundation of Korea funded by theMinistry of Education, Science and Technology. Computa-

tional resources provided by KISTI Supercomputing Centre(Project No. KSC-2012-C2-22) are gratefully acknowledged.

*Corresponding author: hmjang@postech.ac.kr1J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu,D. Viehland, V. Vaithyanathan, D. G. Schlom, U. V. Waghmare,N. A. Spaldin, K. M. Rabe, M. Wuttig, and R. Ramesh, Science299, 1719 (2003).

2T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, andY. Tokura, Nature 426, 55 (2003).

3N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha, and S. W. Cheong,Nature 429, 392 (2004).

4T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, andM. Fiebig, Nature 430, 541 (2004).

5F. Kagawa, M. Mochizuki, Y. Onose, H. Murakawa, Y. Kaneko,N. Furukawa, and Y. Tokura, Phys. Rev. Lett. 102, 057604 (2009).

6B. Lorenz, Y. Q. Wang, and C. W. Chu, Phys. Rev. B 76, 104405(2007).

7S. Ishiwata, Y. Kaneko, Y. Tokunaga, Y. Taguchi, T. H. Arima, andY. Tokura, Phys. Rev. B 81, 100411(R) (2010).

8W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature 442, 759(2006).

9S. W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 (2007).10P. Toledano, Phys. Rev. B 79, 094416 (2009); P. Toledano,

W. Schranz, and G. Krexner, ibid. 79, 144103 (2009).11H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95,

057205 (2005).12A. Malashevich and D. Vanderbilt, Phys. Rev. Lett. 101, 037210

(2008).13I. A. Sergienko, C. Sen, and E. Dagotto, Phys. Rev. Lett. 97, 227204

(2006).14J. H. Lee, Y. K. Jeong, J. H. Park, M. A. Oak, H. M. Jang, J. Y. Son,

and J. F. Scott, Phys. Rev. Lett. 108, 219702 (2012).

15M. Mochizuki, N. Furukawa, and N. Nagaosa, Phys. Rev. Lett. 105,037205 (2010).

16S. Picozzi, K. Yamauchi, B. Sanyal, I. A. Sergienko, and E. Dagotto,Phys. Rev. Lett. 99, 227201 (2007).

17D. Okuyama, S. Ishiwata, Y. Takahashi, K. Yamauchi, S. Picozzi,K. Sugimoto, H. Sakai, M. Takata, R. Shimano, Y. Taguchi,T. Arima, and Y. Tokura, Phys. Rev. B 84, 054440 (2011).

18M. Nakamura. Y. Tokunaga, M. Kawasaki, and Y. Tokura, Appl.Phys. Lett. 98, 082902 (2011).

19H. Wadati, J. Okamoto, M. Garganourakis, V. Scagnoli, U. Staub,Y. Yamasaki, H. Nakao, Y. Murakami, M. Mochizuki,M. Nakamura, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 108,047203 (2012).

20P. E. Blochl, Phys. Rev. B 50, 17953 (1994); G. Kresse andD. Joubert, ibid. 59, 1758 (1999).

21G. Kresse and J. Hafner, Phys. Rev. B 47, R558 (1993); G. Kresseand J. Furthmuller, ibid. 54, 11169 (1996).

22H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).23P. E. Blochl, O. Jepsen, and O. K. Andersen, Phys. Rev. B 49, 16223

(1994).24C. Elsasser, M. Fahnle, C. T. Chan and K. M. Ho, Phys. Rev. B 49,

13975 (1994).25R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993);

D. Vanderbilt and R. D. King-Smith, ibid. 48, 4442 (1993).26See, for example, H. Eyring, S. H. Lin, and S. M. Lin, Basic

Chemical Kinetics (John Wiley & Sons, New York, 1980),Chap. 4.

27M. A. Oak, J. H. Lee, H. M. Jang, J. S. Goh, H. J. Choi, and J. F.Scott, Phys. Rev. Lett. 106, 047601 (2011).

28K. Momma and F. Izumi, J. Appl. Crystallogr. 41, 653 (2008).

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