Electron-lattice coupling contributions to polarization switching incharge-order-induced ferroelectrics

Yubo Qi, and Karin M. RabeDepartment of Physics & Astronomy, Rutgers University,

Piscataway, New Jersey 08854, United States

In this study, we carry out density functional theory calculations to elucidate the polarizationswitching mechanism in charge-order-induced ferroelectrics. Based on the investigations about(SrVO3)1(LaVO3)1 superlattice, we demonstrate that the charge ordering state couples strongly tolattice modes, and charge-transfer induced polarization switching has to be associated with changesof lattice distortions. Based on the type of lattice mode strongly coupled to charge ordering states,we classify the charge ordering materials in two type, namely polyhedral breathing and off-centeringdisplacement types. We further demonstrate that only in off-centering displacement type chargerordering material, the polarization is switchable under an external field. The implications of thistheory to experimental observations are also discussed and we successfully explain the differentelectrical behaviors in LuFe2O4 and Fe3O4. This study aims to provide guidance for searching anddesigning charge ordering ferroelectrics with switchable polarization.

In strongly correlated materials, the charges in tran-sition metal atoms can disproportionate and form anordered superlattice. This charge ordering (CO) effectbreaks the symmetry and may induce ferroelectricity [1–4]; this behavior has been experimentally observed inFe3O4 [5, 6]. Searching for such a kind of novel ma-terials attracts intensive research interest. The pro-posed systems include PrxCa1−xMnO3 [2, 7–14], rareearth (R) nickelates (RNiO3) [15, 16], rare earth man-ganites RMn2O5 [17–23], and so on. Specifically, the-oretical design based on first-principles calculations hasworked as a powerful tool for discovering CO-driven fer-roelectrics [24, 25]. The basic design principles are thateach unit call should possess two multiple-valence atoms,and their ordering arrangement can the break symmetryforbidding polarization. Our previous first-principles cal-culations demonstrate that the 1:1 superlattice of SrVO3

and LaVO3 [denoted by (SrVO3)1(LaVO3)1] works as avery simple example satisfying these criteria [24]. Atlow temperature, the (SrVO3)1(LaVO3)1 superlattice isa Mott insulator, in which vanadium atoms dispropor-tionate into V3+ and V4+ and form a layered stack. CO-driven ferroelectrics have promising functional applica-tions, since the polarization switching, which is associ-ated with the inter-ionic electronic transfer, is expectedto be much faster than the lattice relaxation in conven-tional ferroelectrics [5, 26, 27]. However, the CO stateis coupled to lattice distortion modes. Such a couplingcan reduce the polarization switching speed back to thelattice relaxation time scale, and can even prohibit theswitching in some cases. In this work, we carry out first-principles calculations on the (SrVO3)1(LaVO3)1 super-lattice to investigate the coupling between lattice modesand CO states. We demonstrate that the polarizationswichability of a CO material is determined by the char-acter (whether it is infrared active) of the primary lat-tice mode coupling to the CO states. Next, we apply themodel to two other representative CO materials, Fe3O4

and LuFe2O4. We demonstrate that this model can suc-

cessfully explain the electrical behaviors in CO materials,and subsequently provide guidance for designing ferro-electric CO materials with switchable polarization.

In the (SrVO3)1(LaVO3)1 superlattice, the V3+ andV4+ ions are layered stacked, leading to a non-zero po-larization along the z direction (Supplementary Materi-als (SM) section III. A [28]). Since the ionic radius ofV3+ is larger than that of V4+, the oxygen polyhedronassociated an V3+ ion elongates and the one associatedan V4+ ion contracts, leading to a polyhedral breathing(PB) lattice distortion [Fig. 1 (a)], whose amplitude canbe measured by the ratio of the heights of the two poly-hedra as

R =L1

L2. (1)

In addition, the V atoms are displaced off-center from thesurrounding octahedron [Fig. 1 (b)], which can be quan-tified by the distance between the center of a polyhedronand position of the V atom associated with it (denotedby QOD). Since the V3+ ion is larger, it correspondsto a smaller Goldschmidt tolerance factor and a smallerQOD. In the relaxed structure, the favored direction forthe OD points from the LaO layer to the SrO layer, re-sulting in antiparallel displacements on V1 and V2 (SMsection II). This means that one of the OD will be alongthe polarization direction and one will be opposite.

La

Sr

V4+

V3+

O

(a) (b)

V1

V2 L2

L1

QOD(V2)

QOD(V1)

FIG. 1. Schematic plots of the (a) PB and (b) OD latticemodes.

In this study, we perform density functional theory

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30

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(DFT) calculation on the (SrVO3)1(LaVO3)1 superlat-tice (see SM section I for computational details), andacquire that the mode magnitudes of the optimized(SrVO3)1(LaVO3)1 superlattice structure are R = R0 =0.91, QOD(V3+) = 0.04 A and QOD(V4+) = 0.09 A.This suggests that it is favorable to put the V4+ ionon the site with the smaller octahedron and the largerQOD. Moreover, we would like to emphasize that it isthe arrangement of V4+ and V4+ (the CO state) that pri-marily determines the polarization (V1,V2 = V3+,V4+

for the up-polarized state or V1,V2 = V4+,V3+ for thedown-polarized state), which tells us how electric fieldcan change the energy difference between the up anddown polarized state. The OD can influence the stabilityof a particular CO state (as we will discuss later), butsuch displacements themselves have much less contribu-tions to the polarization, compared with the CO effect.

To investigate the coupling between the lattice modesand CO-state stability, we generate the electronic energylandscapes (see SM section I) at different R values inthe plane of QOD(V1) and QOD(V2) [29, 30]. In Fig. 2(a), we begin with the energy landscape for R = 1, inwhich case the PB character is completely eliminated andthe stability of each CO state is determined by the off-centering displacements. In the upper part, we have

QOD(V1) < QOD(V2)⇒ V1 = V3+ and V2 = V4+, (2)

which corresponds to an up-polarized CO state. Sim-ilarly, the lower part corresponds to a down-polarizedCO state. Because of the symmetry imposed by R = 1,the energies of the two local minima corresponding toup- and down-polarized states are identical, and thedifference ∆E = 0. As R decreases to 0.994 [Fig. 2(b)], the oxygen polyhedron associated with V1 con-tracts, favoring a V4+ state (at the original point withQOD(V1) = QOD(V2) = 0, where the OD character iscompletely eliminated, V4+ is on V1 site, correspond-ing to down polarization). As reflected in the energylandscape, the dashed line, which represents the bound-ary of the up- and down-polarized regions, moves towardthe left corner, indicating that QOD(V2) need to exceedQOD(V1) by a certain amount, in order to stabilize theup-polarized state. The region corresponding to the up-polarized CO state shrinks and the energy difference be-tween the two local minima ∆E = Eup−Edown increasesto 24 meV/f.u.. As R decreases to 0.987 [Fig. 2 (c)], ∆Eincreases to 49 meV/f.u., and the local minimum cor-responding to the up-polarized CO state becomes quiteshallow. If R decreases further to 0.981 (not shown inFig. 2), the up-polarized CO state completely loses itsstability; No matter what QOD(V1) and QOD(V2) thestructure has, the up-polarized CO state can not be alocal minimum. These results indicate that the stabilityof a CO state is a competition between the PB and ODmodes, and the PB lattice mode has a more profoundinfluence. .

We should note that this critical R value is still faraway from the value for the relaxed structure R0 = 0.913.

Since there is no local minimum corresponding to the up-polarized CO state for R = R0, we can not calculate ∆Edirectly. To get an estimation, we plot R vs. ∆E withavailable data in Fig. 2 (d), extrapolate the line, andinfer that ∆E = 340 meV/f.u. at R = 0.913. To inducean electronic transfer between V1 and V2, such a energydifference has to been compensated by the an electricalenthalpy term as

∆E = ∆P ·Eel, (3)

where ∆P is the difference between the polarization ofthe two CO states, and Eel is the applied electric field.For ∆E = 340 meV/f.u., we have

Eel = ∆E/∆P = 16 MV/cm. (4)

large compared to the breakdown field of most per-ovskites [31, 32].

QOD(V1)

QO

D(V

2)Up

R = 1

QOD(V1)

QO

D(V

2)

R = 0.994

QOD(V1)

QO

D(V

2)

R = 0.987

(a) (b)

(c)(d)

280

360

200120

200

280

120120

200

280

Down

Up

Down

Up

Down

Energy Energy

Energy

FIG. 2. Energy landscapes as functions of QOD(V1) andQOD(V2), with (a) R = 1, (b) R = 0.994, and (c) R = 0.987.The dashed line represents the boundary between the regionscorresponding to the up- and down-polarized states. (d) Rvs. ∆E with available data, based on which we estimate that∆E = 340 meV/f.u. at R = R0 = 0.913.

To investigate the polarization switchability of the(SrVO3)1(LaVO3)1 superlattice, we relaxed the up-polarized and the down-polarized structures in nonzeroelectric fields. We find that the up-polarized state is un-stable for electric field in the up direction all the way to12.6 MV/cm, the largest field for which we could obtainconverged results. This is consistent with our analysisabove, which showed that for R0 = 0.913, polarizationswitching requires an electric field of at least 16 MV/cm.For the relaxed down-polarized structure, the change inthe R value with increasing electric field was negligible.This is reasonable because the PB mode is infrared in-active and thus does not couple directly to an electricfield.

3

The analysis above suggests that the polarization inPB-type CO material will be difficult to switch with anapplied electric field, because the polarization switchinghas to be accompanied with a change of the PB latticemode, which does not couple directly to an electric field.However, the situation is different for an OD-type COmaterial, since the relevant mode is infrared active. Toillustrate this point, we consider the (SrVO3)1(LaVO3)1superlattice with R artificially fixed to 1. As a result,the PB character is completely eliminated, and in zeroapplied field, the CO pattern, and thus the polarization,couples to the OD modes only. We begin with a down-polarized structure, in which the 4+ valence state is onthe V2 site and the larger OD is along the polarizationdirection. The up-directed electric field is increased from0 to 7.2 MV/cm in steps of 1.8 MV/cm. At each step,the structure is optimized with R fixed to 1. We findthat the system remains in the down-polarized state, in-dicating that this CO state is a local minimum duringthis electric field application process. However, the up-directed electric field favors the CO state in which the4+ valence state is on the V1 site, and for large enoughfields, this CO state might be lower in electric enthalpyeven if the atomic displacements are kept fixed to thoseof the down-polarized state.

To investigate the competition between these two po-larization states with increasing field, for the down-polarized state, we plot the changes of QOD(V1 = V4+)and QOD(V2 = V3+) with respect to the electric field inFig. 3 (a). Since V1 and V2 have off-centering displace-ments parallel and antiparallel to the field, respectively,QOD(V1) increases and QOD(V2) decreases, and theirvalues cross at Eel = 6.5 MV/cm (QOD(V1) becomeslarger than QOD(V2)).

Next, we calculated the polarization of the down-polarized state of each structure [Fig. 3 (b)], as discussedabove the V cations both displace upward under the influ-ence of the field. In addition, the A-site cations displaceupward. The up-polarized state we consider has the sameatomic positions as the down-polarized state. The polar-ization difference comes from the transfer of the electronfrom the V1 to the V2 site, and is approximately inde-pendent of electric field.

The electric enthalpy for these two polarization statesis shown in Fig. 3 (b). The enthalpy of the down-polarized state increases with the field while the enthalpyof the up-polarized state decreases with the field. Thisis a consequence of the field-induced changes in both thepolarization and off-centering displacements. Since theV ion with a larger OD prefers to be V4+, the down-polarized CO state becomes less energetically favorableunder an up-directed electric field. If we assume thatas the electric field is increased, a CO state that be-comes metastable will quickly decay to the lowest en-thalpy CO state, the electron-transfer-induced polariza-tion switching in this constrained (SrVO3)1(LaVO3)1 su-perlattice occurs at Eel = 2.8 MV/cm, above which theup-polarized CO state has a lower enthalpy. We thus

demonstrate that the polarization in an OD-type CO ma-terial is switchable, because the change in lattice modethat accompanies the polarization switching is the OD,which we have shown couples directly to the electric field.

-0.4

-0.3

0 1 2 3 4 5 6 7 8

-100

0

100

200

down-polarizedup-polarized

0.3

0.4(b)

(b)

Electric field (MV/cm)

0 1 2 3 4 5 6 7 80.03

-0.4

-0.3

0

100

200

down-polarizedup-polarized

0.3

0.40 1 2 3 4 5 6 7 8

0.03

0.06

0.09

0.12

V2

= V4+

V1

= V3+

-0.3

0.3

0.4

Electric field (MV/cm)

(a)

(b)

down-polarizedup-polarized

Electric field (MV/cm)

Electric field (MV/cm)

H (

meV

/f.u.

)|Q

OD|

FIG. 3. (a) The changes of QOD(V1 = V4+) and QOD(V2 =V3+) with respect to an applied electric field. (b) The changesof H in the up- and down-polarized states with respect to anapplied electric field.

Our results based on the (SrVO3)1(LaVO3)1 super-lattice structure demonstrate that PB- and CO-typesCO materials have different electrical behaviors, becausetheir characteristic modes respond differently to an elec-tric field. In the following part, we will extend this modelto other CO materials, such as LuFe2O4 and Fe3O4, bydiscussing the implications of this theory to the differencein their experimentally-observed electrical behaviors.

First, we discuss polarization switchability inLuFe2O4. In 2005, Ikeda et al. demonstrated thatLuFe2O4 has two oppositely polarized states, whichcan be obtained by cooling from a high-temperaturecentrosymmetric structure under oppositely directedelectric fields [33–35]. The emergence of polarizationwas attributed to the disproportionation and orderedarrangement of Fe atoms [33, 34, 36, 37]. These resultsmade LuFe2O4 a promising candidate for CO-inducedferroelectricity. However, electric-field-induced polariza-tion switching in LuFe2O4 has not been experimentallydemonstrated, leading to a long-lasting debate aboutwhether LuFe2O4 is indeed ferroelectric [38–40].

In the following, we perform a structural analysis thatdemonstrates that LuFe2O4 is a PB-type CO material,whose polarization is difficult to switch with an elec-tric field (SM section III. B). To quantify the polyhedralbreathing distortion, we define QPB as

QPB(Fei) =∑j

(rij − r) , (5)

4

where j runs over all the oxygen atoms bonded with theFe ion i, rij is the length between the center of the poly-hedron containing the Fe ion i and the oxygen atom j,and r is the average of all the rij in the entire unit cell. Anegative QPB corresponds to a smaller polyhedron, anda positive QPB corresponds to a larger polyhedron. InFig. 4 (a), we show QPB for (SrVO3)1(LaVO3)1 super-lattices, with the R previously defined being the ratioof the heights of polyhedra containing the two V atoms,for comparison. In Fig. 4 (b), we plot QPB and QOD

for the six Fe ions in a LuFe2O4 primitive cell [37]. Wecan see that in LuFe2O4, QPB for Fe2+ is noticeablylarger than QPB for Fe3+, with a ratio even larger thanthat of the fully relaxed (SrVO3)1(LaVO3)1 superlattice[Fig. 4 (a) R = R0]. In this system, in contrast to the(SrVO3)1(LaVO3)1 superlattice, the PB in LuFe2O4, ismainly in the in-plane directions. From this we concludethat the CO states in LuFe2O4 are strongly coupled tothe PB mode. To understand the OD values, in Fig. S2,we show the primitive unit cell of LuFe2O4 [37], witha double layer of Fe and O atoms. In the figure, thetop layer is Fe3+-rich and the bottom layer is Fe2+-rich,leading to a non-zero out-of-plane polarization. Both Felayers are displaced outward and the displacement of theFe2+-rich layer, which is in the direction opposite to thepolarization, is systematically larger.

Our previous analysis has demonstrated that the polar-ization in this PB-type CO materials cannot be switchedby an external electric field. Therefore, we conclude thatthe CO-induced polarization in LuFe2O4 is unswitchableunder an applied electric field. However, each CO statecan be obtained by annealing the high-symmetry R3mstructure from high temperatures down to T < TCO witha biased field [33], consistent with experimental observa-tion.

Next, we consider magnetite (Fe3O4), which is knownto be a CO ferroelectric with switchable polarization [5].Below TV = 125 K, Fe3O4 adopts a monoclinic Cc struc-ture [41]. Previous first-principles calculations have iden-tified the Fe2+ and Fe3+ at the 16 inequivalent 6-foldsites of this structure and shown that they are non-centrosymmetrically arranged, leading to a non-zero po-larization (SM section III. C). In Fig. 4 (c), we plot theQPB and QOD of the 6-fold Fe ions, with the descendingorder of their QPB . We note that the 8 Fe2+ ions havepositive QPB and 8 Fe3+ ions have negative QPB , whichfurther supports that the PB mode has a larger influenceon the stability of CO states. The value of QPB influ-ences the bond valences of these Fe ions, which we plotin Fig. 4 (c). We find that the bond valences of Fe2+

ions are systematically smaller than those of Fe3+ ions,indicating that relative magnitudes of bond valences canreflect the oxidation states [42]. We now focus on the twoions in the middle of the range in Fig. 4 (c), which we noteare neighboring in the crystal structure. While they havequite similar QPB , they have QOD of opposite sign. So,an electric field along the z direction will tend to increaseQOD for one and decrease it for the other. The resulting

2

2.5

2

2.5

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

(a) (SrVO3)1(LaVO3)1 (b) LuFe2O4

(c) Fe3O4

R=R0 R=1

BV

QOD

QPB

QOD

QPB

FIG. 4. QOD and QPB of the ions in (a) unconstrained (R =R0) and constrained (R = 1) (SrVO3)1(LaVO3)1 superlattice,and (b) LuFe2O4, (c) Fe3O4. The bond valences in Fe3O4 arealso plotted. The blue and orange bars correspond to thelower valence (V3+ or Fe2+) and higher valence states (V4+

or Fe3+) respectively.

displacements can result in a valence switch. More specif-ically, a 0.3 A displacement along the negative z directionswitches the order of magnitudes of the bond valences ofthe two Fe ions, which, as we have noted above, is cor-related with the valence. The change of polarization dueto this valence switching is 3 µC/m2, which is the sameorder of magnitude as the experimental result (5 µC/m2

in Ref. [43]).

In summary, we demonstrate that the switching ofcharge-ordering induced polarization is in general cou-pled to the changes of lattice distortions, especiallythe polyhedral breathing and off-centering displace-ment modes. Based on our first-principles analysis of(SrVO3)1(LaVO3)1 superlattices, we demonstrate thatcharge-ordering-induced polarization is switchable onlyin a system that has at least a subset of sites for which thestrongest coupling is to the off-centering displacementsrather than to the polyhedral breathing. Our theory can

5

successfully explain the experimentally observed differ-ent switchabilities in LuFe2O4 and Fe3O4. This studyaims to provide more insights and strategies for search-ing charge-ordering materials with electric-field switch-able polarization.

ACKNOWLEDGEMENT

We thank S. Y. Park for valuable discussions. Thiswork was supported by ONR N00014-17-1-2770. First-

principles calculations were performed by using the com-putational resources provided by the Rutgers Univer-sity Parallel Computing (RUPC) clusters and the High-Performance Computing Modernization Office of the De-partment of Defense.

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