Strain dependencies of energetic, structural, and polarization properties in tetragonal (PbTiO3)1/(SrTiO3)1 and (BaTiO3)1/(SrTiO3)1 superlattices: a comparative study with bulks
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Strain dependencies of energetic, structural, and polarizationproperties in tetragonal (PbTiO ) /(SrTiO ) and(BaTiO ) /(SrTiO ) superlattices: a comparative study withbulks Yanpeng Yao and Huaxiang Fu
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA (Dated: August 14, 2018)
Abstract
First-principles density functional calculations are performed to investigate the interplay betweeninplane strains and interface effects in 1 × /SrTiO and BaTiO /SrTiO superlattices oftetragonal symmetry. One particular emphasis of this study is to conduct side-by-side comparisonson various ferroelectric properties in short-period superlattices and in constituent bulk materials,which turns out to be rather useful in terms of obtaining valuable insight into the different physicswhen ferroelectric bulks form superlattices. The various properties that are studied in this workinclude the equilibrium structure, strain dependence of mixing energy, microscopic ferroelectricoff-center displacements, macroscopic polarization, piezoelectric coefficients, effective charges, andthe recently formulated ~k ⊥ -dependent polarization dispersion structure. The details of our findingsare rather lengthy, and are summarized in Sec. IV. PACS numbers: 77.84.-s, 77.80.bn, 77.80.-e . INTRODUCTION Ferroelectric (FE) materials have found spread applications in microelectronics such assensors, actuators, transducers, etc.[1] In recent years, ferroelectric superlattices have at-tracted attention for their promising potential in modifying and tuning the structural andpolarization properties of FE materials. For instance, when forming superlattices withBaTiO , incipient SrTiO was found to exhibit strong ferroelectricity.[2] Meanwhile, FE su-perlattices grown with desired constituents and/or periodicity provide an important fieldto probe and understand the fundamental physics of ferroelectric materials and relatedproperties.[3] Among various FE superlattices, those with ultrashort period are of particularinterest, since the strong interface effect may lead to some properties in the superlattices thatare drastically different from those in bulk constituents. In short-period FE superlattices,one component significantly influences another, making the material properties interestingand less predictable.In the study of FE physics, another subfield of importance is to understand the straindependence of FE properties. Inplane strain, caused by either lattice mismatch or externalstress, alters the interatomic interaction in an anisotropic manner, which often gives rise tonew physics and/or phenomena. For example, inplane strains have been shown to change thecritical temperature of BaTiO by as large as 500 C.[4] Furthermore, different FE materialswere found to possess very different polarization responses to inplane strain.[5, 6] While po-larizations in BaTiO and PbTiO were found sensitive to lattice mismatch,[5] Pb(ZrTi)O nevertheless displays a surprisingly weak polarization dependence on the inplane strain.[6]More recently, it was theoretically demonstrated that when FEs are under large strains, the χ polarization is to saturate, and this polarization saturation was shown to be a generalphenomenon applicable for different materials.[7] This finding also leads to a nature expla-nation on why polarization in some FEs (not the others) displays a weak strain dependence,since the polarization in these FEs is approaching the saturation and thus is less affectedby the inplane strain.[7] While the strain influences on bulk FEs are amply studied, thestrain-induced effects in FE superlattices are relatively less understood, however.In this paper we intend to address a topic which concerns both of the above two sub-fields (namely, FE short-period superlattices as well as strain effects), by studying theproperty changes caused by epitaxial inplane strains in ferroelectric PbTiO /SrTiO and2aTiO /SrTiO superlattices with short periodic length. The topic is of interest for thefollowing reason. In short-period superlattices, the interface plays a far more important rolethan in long-period superlattices and in bulk, and consequently, will significantly alter thestructural and polarization responses to external inplane strains. The interplay—caused bystrongly interacting interface and inplane strains—makes the strain effects in short super-lattices to differ from their bulk constituents and to be potentially much more complex. Theabove viewpoint emphasizes the differences between FE superlattices and FE bulks. On theother hand, FE superlattices must bear some resemblance to the FE bulks, since superlat-tices are made of individual bulk constituents. The strain responses of the superlattices thusmust, to a varied degree, reflect and resemble the properties of bulk constituents. Based onthese considerations, one key purpose of this work is to conduct a side-by-side comparativestudy of strain-induced effects in short-period FE superlattices and in FE bulks.The advantage of a comparative study is rather obvious: by comparing bulks (with nointerface) and short-period superlattices with strong interface, one is able to obtain a directinsight and understanding on the interplay of interface effect and strain effect, and on howthe existence of one component in superlattices affects the other component under differentinplane strains. This being said, the comparative study nevertheless is not as straightforwardas it seems to be, for the following reasons. (i) In certain short-period superlattices, therotation instability of oxygen octahedral may exist.[8, 9] On the other hand, bulk PbTiO has only one stable phase of tetragonal symmetry without oxygen rotation as shown bythe lack of soft modes at the zone boundary [10], and consideration of other phases insuperlattices makes it difficult to conduct a side-by-side comparison on strain effects betweensuperlattices and bulks. (ii) Properties such as ferroelectric off-center displacements, effectivecharges, and polarization structure depend on structural symmetry. By allowing rotationinstability, most of properties in bulks and in superlattices can not be directly compared,and the advantage of comparative study will be largely lost. To enable a comparative study,we thus deliberately confine ourself to FE superlattices and bulks of tetragonal symmetrywithout oxygen rotation. For readers who are interested in superlattices with structuralphases other than tetragonal symmetry, results can be found in previous reports.[8, 9, 11,12] Experimentally, superlattices of tetragonal symmetry without oxygen rotation can berealized in several possible ways: (1) One can grow short-period superlattice films betweenmetallic electrodes possessing no oxygen rotation. The lack of oxygen rotation in electrodes3ill inhibit the rotation instability in the superlattices. (2) One may use compressive inplanestrains to suppress the rotation instability. It was shown in SrTiO that under sufficientlylarge inplane strain, the structure with rotation instability becomes less stable and eventuallydisappears.[7] (3) One may engineer superlattices by choosing atoms of different sizes toweaken the rotation instability. For instance, the rotation instability in BaTiO /SrTiO isconsiderably weaker than in PbTiO /SrTiO .In this study, we apply first-principles density functional theory (DFT) calculations to1 × /SrTiO (PT/ST) and BaTiO /SrTiO (BT/ST) superlattices, as well as to theindividual bulk constituents PbTiO (PT), SrTiO (ST), and BaTiO (BT). As shown below,various properties are to be investigated, which include microscopic ferroelectric off-centerdisplacements, macroscopic polarization, piezoelectric coefficients, effective charges Z ∗ , andthe recently formulated ~k ⊥ -dependent polarization dispersion structure [13]. A number ofinteresting differences between strain effects in superlattices and those effects in bulks havebeen found, the details of which are summarized in Sec.IV. II. THEORETICAL METHOD
We use first-principles density functional theory within the local densityapproximation[14] (LDA) to determine the structure response to external strains insuperlattices and in bulks. For the 1 × a , a , and a , with | a | = | a | = a and | a | = c . The optimized cell structure and atomic positions are obtainedby minimizing the total energy. More specifically, for each inplane lattice constant a , theout-of-plane c length and atomic positions are optimized. Biaxial in-plane strain is definedas η = η = ( a − a ) /a , where a is the equilibrium in-plane lattice constant. In our study,we consider compressive inplane strains.Calculations are performed using the mixed-basis pseudopotential method.[15] The norm-conserving pseudopotentials are generated according to the Troullier-Martins procedure.[16]Atomic configurations for generating pseudopotentials, pseudo/all-electron matching radii,and accuracy checking were given elsewhere[17]. The wave functions of single-particle Kohn-Sham states in solids are expanded in terms of a basis set which consists of the linearcombination of numerical atomic orbitals and plane waves. An energy cutoff of 100 Ryd is4sed throughout all calculations for both bulk materials and superlattices, which we foundsufficient for convergence.In ferroelectric crystals with tetragonal symmetry, the polarization is nonzero along theout-of-plane c axis. Polarizations are calculated using the geometric phase of the valencemanifold of electron wave functions according to the modern theory of polarization[18, 19],which we have implemented in our mixed-basis computational scheme. III. RESULTSA. Structure and polarization under zero strain
To better understand the strain effects in PT/ST and BT/ST superlattices, we first studythe equilibrium structure of the superlattices under zero external strain. By minimizing thetotal energy of these superlattices, we obtain that the unstrained superlattices have an opti-mized inplane lattice constant a =3.87˚A and tetragonality c/a =2.0 for PT/ST, and a =3.91˚Aand c/a =2.0 for BT/ST. Comparing with our theoretical inplane lattice constants of un-strained pure bulk materials—which are a =3.88˚A for PT, a =3.95˚A for BT, and a =3.86˚Afor ST, we thus see that, when transitioning from bulk to superlattices, the PT (or BT)layers are compressively strained while the ST layers are stretched.For unstrained PT/ST and BT/ST superlattices, our calculations using the modern the-ory of polarization reveal that both superlattices have zero polarization, showing that theproperties in the short-period superlattices differ significantly from bulk PT and BT con-stituents. The calculated vanishing polarization in the BT/ST superlattice does not contra-dict the experimental results[20] where the BT/ST superlattice with one unit cell of BT wasfound to be ferroelectric, since the experimental sample was grown on SrTiO substrate withan inplane lattice constant of a =3.86˚A. In our case, the BT/ST superlattice is free standingwithout substrate ( a =3.91˚A). The null polarization in equilibrium PT/ST and BT/ST su-perlattices is interesting and meanwhile puzzling, if one recognizes that bulk PT has a verylarge polarization of ∼ µC/cm when strained to a =3.87˚A (the inplane lattice constant ofthe PT/ST superlattice), and bulk BT also has a large polarization of ∼ µC/cm whenstrained to a =3.91˚A (the inplane lattice constant of the BT/ST superlattice). One maywonder why the strained PT or BT layers inside the superlattices do not polarize the ST5ayers and lead to some polarization.To understand why the unstrained superlattices have null polarization, we examine theoptimized atomic positions at zero strain, which is schematically shown in Fig.1(a). The z -axis is along the superlattice stacking direction, i.e., the direction of the tetragonal c -axis.Here we are interested in the relative atomic displacements, rather than the absolute shiftsof each atom. By placing Pb at the origin, we define a high-symmetry location along the z -direction for each atom; more specifically, the high-symmetry locations for Sr and O ′ atoms are at z = c , while those of Ti and O ′ atoms are respectively located at z = c and z = c along the z -axis. In Fig.1(b) we show the z -axis atomic displacements of theLDA-optimized structure with respect to the high-symmetry locations for different atoms.We see that the O and O ′ atoms undergo a notable displacement that is 0 .
6% of latticelength c . Meanwhile, the Ti atom (which shares the same plane as O ) is also displaced offthe center of high symmetry, but the amount of displacement of Ti differs from that of O .As a result of the different displacements from Ti and O , our calculations thus reveal thatthere is a local dipole moment for the Ti-O plane. However, due to the fact that O and O ′ atoms move along the opposite directions with equal amount of displacements (so do Ti andTi ′ ), the local dipole moments of the Ti-O plane and Ti ′ -O ′ plane thus cancel, leading to avanishing total polarization. Since both O and O ′ shift towards Sr, the inversion symmetryby the SrO-plane is thus maintained (to the accuracy of numerical certainty). This explainswhy the superlattice has no polarization at zero strain.The opposite displacements of O and O ′ can not be naively attributed to the size dif-ference between Sr and Pb, since their numerical atomic sizes do not differ significantly.By analyzing the electron density for the optimized structure and for the high-symmetricstructure, we found that O and O ′ move towards Sr because of the strong covalent bondingbetween O and Sr. This makes sense since the covalent nature of the Sr-O bond is strongerthan that of Pb-O or Ba-O. It also implies that if one replaces Sr by other A-site atomswith less covalent nature, there may be a possibility to produce polarization. Our studythus shows that atoms in zero-strain superlattices indeed are displaced off the center, butthe displacement pattern maintains the plane-inversion symmetry.6 . Dependence of mixing energy on inplane strain Thermodynamically, when constituents A and B form the A/B superlattice, the mixingenergy is defined as ∆ E ( a ) = E A / B ( a ) − E A ( a ) − E B ( a ) , (1)where E A / B is the total energy of a 10-atom cell for the 1 × E A (or E B )is the total energy of one unit cell of bulk material A (B). All energies are calculated at thesame inplane a lattice constant, with the out-of-plane c lattice constant and atomic positionsoptimized for each structure involved in Eq.(1). Mixing energy tells us the relative stabilityof the superlattice with respect to the individual constituents. A positive mixing energymeans that the superlattice is thermodynamically less stable and will segregate into pureconstituents. Of course, this segregation may take a long time due to the possible existenceof energy barrier. While polarization properties of FE superlattices have been studied, littleis known thus far about the mixing energy, for instance, (i) What is the value of mixingenergy for prototypical superlattice such as PT/ST or BT/ST? (ii) How does the change inthe inplane strain affect the mixing energy?Figure 2 shows the mixing energies ∆ E , as well as total energies E A / B and E A + E B , forthe PT/ST and BT/ST systems at different inplane lattice constants. From Fig.2, we findthat, (i) For PT/ST, the mixing energy is always positive in the considered strain region,revealing that the superlattice is thermodynamically less stable as compared to segregatedbulk constituents and an extra energy is needed to build the PT/ST superlattice. (ii)However, the BT/ST superlattice turns out to have a negative mixing energy at smallstrains, thus being thermodynamically stable energywise. (iii) When the inplane strain issmall, the mixing energy ∆ E of PT/ST increases in a linear fashion with the deceasing a lattice constant. Interestingly, this increase does not last forever; instead ∆ E reaches itspeak value at a certain inplane lattice and then starts to decline when a is further reduced.In fact, we have calculated ∆ E down to a =3.60˚A (not shown in the figure), which confirmsthe continuous decline of ∆ E . The non-monotonous strain dependence of the mixing energyis found true also for BT/ST. (iv) For the strain region considered, the mixing energy rangesfrom 20 to 100 meV per 10-atom cell for PT/ST, while for BT/ST it is within ±
20 meV.In other words, the mixing energy is large for PT/ST, while being rather close to zero forBT/ST. As a result of the small mixing energy, BT and ST are more likely to form FE7lloys, which is indeed true in experiments.
C. Ferroelectric polarization
Previous studies on the ferroelectric polarization in superlattice largely focused on thecase that the inplane lattice constant of the superlattice is fixed to be that of substrateSrTiO . Here, our emphasis is slightly different; we examine the polarization in superlatticeunder varied inplane lattice constants, and study how the superlattice responds differently(or similarly) to the inplane strain as compared to the bulk constituents. In Fig.3(a) weshow the total (electronic+ionic) polarizations in PT/ST and BT/ST superlattices, in com-parison with the values in bulk PT, BT, and ST. On the one hand, the result in Fig.3(a) israther trivial—it shows that for a given inplane a lattice constant, polarization in BT/STsuperlattice is larger than in ST, but smaller than in BT. As a result, polarization in BT/STsuperlattice never exceeds that in bulk BT of the same a . Similar conclusion is also truefor PT/ST. On the other hand, a careful examination of the calculation results in Fig.3(a)reveals some interesting observations: (1) At a =3.86˚A (which is the inplane lattice con-stant of typical substrate SrTiO ), BT/ST has a sizable polarization while PT/ST does not,although, at this a lattice constant, bulk PT has a much larger polarization than bulk BT.This suggests that, the concept that a stronger FE component (such as PT as comparedto BT) in superlattice will polarize better the non-FE component (such as ST) does notalways work. (2) Below and above a =3.82˚A, the polarization curve of PT/ST (similarly ofBT/ST) undergoes an evident change in the slope. More specifically, the polarization risesmore slowly when a < a , as compared to the case when a > a . We recognize that a actually coincides with the critical inplane lattice constant where bulk ST starts to becomeferroelectric. The critical property of bulk ST (namely, becoming FE at a ) is thus alsoreflected in superlattices . As the ST component turns ferroelectric when a < a , PT/ST orBT/ST becomes less incipient, thus showing a smaller strain-induced polarization enhance-ment. (3) When a > a , polarization in PT/ST is smaller than in BT/ST. However, when a < a , a crossover occurs, and polarization in PT/ST becomes larger than in BT/ST.When bulk constituents A and B form an A/B superlattice, one can use the polarizations8f bulk FEs and define an average of polarization¯ P ( a ) = P A ( a )Ω A ( a ) + P B ( a )Ω B ( a )Ω A ( a ) + Ω B ( a ) (2)for a given inplane a lattice constant, where P A and Ω A are the polarization and cell volumeof the A constituent, respectively. This definition is based on the fact that polarizationitself (namely, dipole moment per unit volume) is not an additive thermodynamic quantityand must be weighted by volume. We then compare the polarization P A / B of the super-lattice (calculated using optimized structure and modern theory of polarization) with ¯ P byexamining ∆ P = P A / B − ¯ P . ¯ P can be viewed as the anticipated polarization when onecombines bulk A and B constituents together into a heterostructure, each with the sameinplane lattice constant a , but without interaction between them. The ∆ P quantity thusreflects mainly the interfacial effect on the polarization, caused by various interactions suchas the polarizing (or depolarizing) field and size effect.The ¯ P and ∆ P quantities are given in Fig.3(b) for PT/ST and in Fig.3(c) for BT/ST.For PT/ST, we see in Fig.3(b) that (1) when a decreases, P of the superlattice increasesfaster than ¯ P ; (2) ∆ P is negative in the strain range considered, namely the polarizationof superlattice does not exceed ¯ P ; (3) At small compressive strains, P A/B and ¯ P differsignificantly. But at large compressive strains, they become close. For BT/ST in Fig.3(c),∆ P is negative at small strains. However, when a < . P becomes positive, revealingthat P of short-period superlattice can in fact exceed the average ¯ P of bulk constituents. Ifwe examine the magnitude of ∆ P (a quantity that indicates the gain of polarization whentwo materials form superlattice), we see that for PT/ST, the gain runs from − µC/cm to zero, while for BT/ST, the gain stays within ± µC/cm .Generally one tends to think that polarization in superlattice is to be enhanced withrespect to the average of single materials. This need be taken with caution. As shown inFig.3(b) for PT/ST, the polarization of the superlattice is considerably smaller than theaverage ¯ P of the corresponding single materials at the same inplane a ; while for BT/ST, thegain of polarization varies from negative to positive as strain increases. As an outcome of thecompeting effect between polarizing field and different covalent strengths of different A-siteatoms, the gain of total polarization in the superlattice is thus a collective result influencedby the properties of single materials, their interaction, and external strain.Technologically, one possible advantage of forming FE superlattices is to tune material9roperties. Here we examine how piezoelectric coefficients may be tuned in PT/ST ascompared to bulk PT. We are interested in the proper piezoelectric coefficient e = − e Ω dχdη c ,where χ is related to the c -axis polarization by P = e Ω χc . e reflects what magnitude ofpolarization enhancement can be achieved by applying the inplane strain η . In Fig.3(d) weshow the χ polarizations in PT/ST and in bulk PT. Fitting the χ values over the consideredstrain range yields piezoelectric coefficient e , and the resulting e values are given nearthe fitting lines in Fig.3(d). One sees that, the piezoelectric e = 19 . C/m coefficient inPT/ST is much larger than the value e = 10 . C/m in bulk PT. D. Microscopic insight: atomic displacements
To understand microscopically how different layers in superlattices interact and how theycollectively respond to inplane strains, we report and analyze in this section the atomicdisplacements occurring in different layers of the superlattices.Bulk ferroelectric perovskite ABO of tetragonal symmetry consists of two layers—theAO layer and the BO layer—alternating along the c -axis. The total electric polarization ofthe solid could be viewed as the local dipole contributions from the two individual layers,as demonstrated by the Wannier functions and local polarizations in Ref.21. The relativedisplacements of the A and B atoms with respect to the oxygen centers of the same layerare thus important quantities that reveal the origin and amplitude of the polarization inthe material. In 1 × , Ti-O , Sr-O ′ and Ti ′ -O ′ layers (see Fig.1a). For the convenience ofdiscussion, we define the z -direction relative displacement of the cation with respect tothat of oxygen in each layer as ∆ z ( P bO ) = z ( P b ) − z ( O ), ∆ z ( T iO ) = z ( T i ) − z ( O ),∆ z ( SrO ′ ) = z ( Sr ) − z ( O ′ ), and ∆ z ( T i ′ O ′ ) = z ( T i ′ ) − z ( O ′ ), where z ( A ) is the z -axisposition of atom A. ∆ z ’s in the theoretically optimized structures of PT/ST and BT/STare shown in Fig.4, where the corresponding displacements in pure bulk materials are alsogiven for comparison.Let us look at PT/ST first. It is known in bulk PbTiO that Pb has a considerable off-center displacement. As a result, PbTiO is a rather strong A-site FE. In comparison, bulkSrTiO has less ferroelectricity from the A-site. This is indeed confirmed by our calculationresults of ∆ z (AO) for bulk PT and ST (see the dotted lines in Fig.4a). However, in PT/ST10uperlattice, the ∆ z displacements are remarkably close for the Pb-O layer and for the Sr-O ′ layer (see two solid lines in Fig.4a). Also note that ∆ z ( SrO ′ ) in PT/ST superlattice ismuch larger than the counterpart in bulk SrTiO , for a fixed inplane lattice constant. Theseresults are interesting and tell us that, by forming a superlattice, the SrTiO componentbecomes a much stronger A-site FE, as compared to bulk ST. Regarding ∆ z ( P bO ) [orsimilarly ∆ z ( SrO ′ )], we further recognize that this quantity should be identical to zero ifthe 1 × c -axis. On the other hand, once the inversion symmetry is broken by the appearance offerroelectricity, ∆ z ( P bO ) becomes nonzero. This is indeed verified by our numerical resultsin Fig.4(a), where ∆ z ( P bO ) is zero for a > . a < . z ( P bO ) serves as a microscopic order parameter for ferroelectricity in the 1 × z ( T iO ) and ∆ z ( T i ′ O ′ ), withrespect to oxygen in PT/ST. Unlike the A cations where ∆ z ( P bO ) and ∆ z ( SrO ′ ) arealmost identical in different layers, the Ti-O relative displacements in two TiO layers areevidently different in Fig.4b. At zero strain ( a =3.87˚A), since the O atom moves up and theO ′ atom moves down due to the fact that the Sr-O bond has a stronger covalent nature thanthe Pb-O bond as described in a previous section (see Fig.1a), ∆ z ( T iO ) and ∆ z ( T i ′ O ′ )appear to be equal but with opposite sign. With increasing strain, the O atom starts to movedownwards as ferroelectricity is developed, which causes ∆ z ( T iO ) to change from negativeto positive in Fig.4a. Interestingly, even for very large inplane strains (e.g., at a =3.75˚A), thedifference between ∆ z ( T iO ) and ∆ z ( T i ′ O ′ ) still exists, showing that the stronger covalentnature of Sr-O bond continues to manifest itself in the microscopic picture.From Fig.4(a) and (b), one thus sees that, even at large compressive inplane strains,the relative atomic displacements in PT/ST are considerably smaller than the counterpartsin bulk PT, and meanwhile much larger than in bulk ST. This demonstrates the stronginfluence between two constituents when they form superlattice. On the other hand, withinthe PT/ST superlattice, atomic displacements are rather uniform in different layers, exceptfor the slight difference in Ti-O displacements caused by the different covalency in A sites.We next examine the situation in BT/ST as shown in Fig.4(c) and (d). At small strainin Fig.4(c), the relative displacements of Ba-O and Sr-O ′ are small and similar. As strain11ncreases, the difference increases. This is in difference from what we have previously seenin PT/ST where ∆ z ( P bO ) and ∆ z ( SrO ′ ) are close over a wide range of considered strains.The difference could be attributed to the large size of Ba atom which, as the inplane a constant decreases, will push away more strongly the oxygen atom on the BaO plane, leadingto a larger difference in ∆ z ( BaO ) than in ∆ z ( SrO ′ ). Regarding the Ti-O displacement inBT/ST, we see in Fig.4(d) that, as compressive strain increases, ∆ z ( T iO ) and ∆ z ( T i ′ O ′ )become gradually close to each other. At a =3.75˚A, ∆ z ( T iO ) exceeds ∆ z ( T i ′ O ′ ), showinga crossover that does not occur in PT/ST. E. Effective charges
In this part of the section we study how the effective charges of atoms are modified whenforming superlattices. For each inplane lattice constant, we compute the effective Z ∗ byfinite difference Z ∗ = Ω e ∆ P ∆ r z , where ∆ r z is chosen to be 0.002 c . All effective charges aregiven in unit of one electron charge.For equilibrium structures of zero strain, the calculated effective charges in superlatticesand in bulk materials (each at its own equilibrium) are given in Table I. The most notableresults in Table I are: (i) Z ∗ of Ti atom in bulk PT is merely 5.76. However, its valuedrastically increases to 7.31 in the PT/ST superlattice. Therefore, Z ∗ s in bulk materialscan not and should not been used in superlattices. In bulk PT, the Ti atom is stronglybonded to only one of the nearby O atoms due to the strong tetragonality. In PT/ST, theTi atom is bonded to both O and O ′ , leading to a large Z ∗ . (ii) while Z ∗ s of the A, Ti,or O sites change significantly from bulk to superlattice, Z ∗ s of the O site are similar inbulk and in superlattice. (iii) In PT/ST at equilibrium, Z ∗ s of two non-equivalent Ti atoms,namely Z ∗ ( T i ) and Z ∗ ( T i ′ ), are identical, so are Z ∗ ( O ) and Z ∗ ( O ′ ). This is causedby the planar inversion symmetry in equilibrium structure. Similar conclusion is true forBT/ST except for some numerical uncertainty. (iv) In PT/ST, Z ∗ s of two non-equivalentO atoms—i.e., Z ∗ ( O ) and Z ∗ ( O ′ )—are very different. The O ′ atom on the SrO layer hasa much larger Z ∗ than the O atom on the PbO layer.Under the application of strain, effective charge for each atom is given in Fig.5(a) forPT/ST. First, we see that, as the inplane lattice constant decreases by 0 . a =3.88˚A to ∼ a =3.74˚A, demon-12trating a wide range of tunability. Similar scale of tunability also occurs to the O and O ′ atoms. In contrast, the Z ∗ charges of Pb, Sr, O and O ′ atoms subject to relatively smallchanges (around 0.3). Second, under compressive inplane strains, the Z ∗ charges of the O and O ′ atoms are no longer identical, unlike the zero-strain case. At zero strain, the O andO ′ atoms are symmetric due to the planar inversion symmetry. Under compressive strains,the symmetry between O and O ′ is broken as ferroelectricity develops. Meanwhile, theTi-O bond is considerably weakened as a result of the increasing tetragonality, which is re-sponsible for the sharp decline of Z ∗ ( T i ). Third, we recognize that the absolute magnitudeof the Z ∗ charge decreases for most of atoms such as Pb, Ti, O and O , once the impressivestrain is applied. One exception is Sr. As a decreases, Z ∗ of Sr increases instead, probablydue to the strain strengthened Sr-O ′ bond. Many of these conclusions are also true for theeffective charges in BT/ST which are shown in Fig.5(b), except for two evident differences:(1) As a decreases, Z ∗ of Ba increases, unlike Pb; (2) Z ∗ ( O ) and Z ∗ ( O ′ ) in BT/ST arevery similar over the considered strain range. F. Polarization structure
Polarization structure [13] reveals how the geometrical phase φ ( ~k ⊥ ) of individual ~k ⊥ string contributes to the electronic polarization ~P el , as described by the modern the-ory of polarization[18, 19] in the equation ~P el = e (2 π ) R d~k ⊥ φ ( ~k ⊥ ), where φ ( ~k ⊥ ) = i P Mn =1 R G k dk k h u n~k | ∂∂k k | u n~k i . Like band structure, the φ ( ~k ⊥ ) ∼ ~k ⊥ polarization structurecontains various important microscopic insight into the polarization properties. Further-more, it was shown that the polarization structure is determined by, and thus can reveal,the fundamental interaction among Wannier functions.[13] While the φ ( ~k ⊥ ) ∼ ~k ⊥ dispersionof bulk ferroelectric has been studied previously,[13] the polarization structure of FE super-lattices remains interesting and unknown. For instance, when bulk BT and ST form BT/STsuperlattice, the total polarization is known (Fig.3a) to decline as compared to bulk BT.However, it is not clear at which ~k ⊥ points the φ ( ~k ⊥ ) phases suffer more; will the ~k ⊥ pointsnear the zone center or near the zone boundary suffer most? Also, how is the polarizationdispersion in superlattice to be affected by the inplane strain?The polarization structures of the BT/ST superlattice at two different inplane latticeconstants, a =3.86˚A and a =3.82˚A, are shown in Fig.6, where the counterpart polarization13tructures of bulk BT and ST at the same a length are also made available for comparison.Our calculation results in Fig.6a show that, at a =3.86˚A, the band width of the polarizationstructure in BT/ST is far smaller than that in BT. When transitioning from BT to BT/ST,the reduction of the φ ( ~k ⊥ ) phase occurs mainly near the X and X points. In other words,the φ ( ~k ⊥ ) phases near the zone boundary are most affected when forming FE superlattices.Based on the consideration that (1) bulk BT and bulk ST have very different polarizationdispersion at a fixed a lattice constant, and (2) the φ ( ~k ⊥ ) phase is inversely proportional tothe c -lattice length,[13] one valid approach to compare, at a given ~k ⊥ point, the φ ( ~k ⊥ ) phasesin superlattice with those in bulk constituents is to define an average phase as ¯ φ ( ~k ⊥ , a ) = c A ( a )+ c B ( a ) [ φ A ( ~k ⊥ , a ) c A ( a ) + φ B ( ~k ⊥ , a ) c B ( a )], where c i ( a ) is the c -lattice length of bulk i atthe inplane lattice constant a , and c A/B is the c -lattice length of the superlattice. Allquantities in the above equation are calculated at the same a lattice constant. ¯ φ is alsodepicted in Fig.6. At a =3.86˚A, we find that φ ( ~k ⊥ ) in BT/ST can be described rather wellby ¯ φ . However, this is not the case for a =3.82˚A. In Fig.6b, one sees: (i) Though bulkST still has zero polarization with φ ( ~k ⊥ )=0 for all ~k ⊥ strings, the φ ( ~k ⊥ ) dispersion in theBT/ST superlattice is nevertheless notably similar to that of bulk BT. (ii) φ ( ~k ⊥ ) in BT/STis considerably larger than the average ¯ φ phase, demonstrating that the strong interactionbetween the BT layer and the ST layer makes the BT/ST superlattice no longer resemblingthe average of two bulk materials. The strong interaction takes place at a =3.82˚A but notat a =3.86˚A, since SrTiO at a is near the critical point of becoming ferroelectric, and canthus be easily polarized by the electric field arising from the polarization in the BaTiO layer. IV. SUMMARY
Density-functional calculations were performed to study a variety of properties in 1 × /SrTiO and BaTiO /SrTiO superlattices of tetragonal symmetry under compres-sive inplane strains. An emphasis is placed on the side-by-side comparison of these propertiesin superlattices and in bulks, which is particularly useful in terms of obtaining insight intothe rather complicated interplay between inplane strains and interface effects. The inves-tigated properties include equilibrium structure, strain dependence of mixing energy, ferro-electric polarization, microscopic atomic displacements, effective charges, and dispersion of14olarization structure. Our main findings are summarized in the following.(i) In zero-strain superlattices without oxygen rotation, while atoms are indeed displacedoff the centers, the displacements nevertheless maintain a plane inversion symmetry. As aresult, the superlattices show no polarization. The planar inversion symmetry (and thusthe vanishing polarization) in zero-strain superlattices originates from the strong covalentbonding between Sr and O. (ii) The mixing energy is found small and on the order of 20meV for the BT/ST superlattice. For PT/ST, this mixing energy is relatively large andranges from 20 to 100 meV in the considered strain region. The small mixing energy inBT/ST is consistent with the fact that BT and ST are more likely to form ferroelectricalloys. (iii) Under small inplane strains, the mixing energy is revealed to increase linearlywith the decreasing inplane lattice constant. However, at a certain (large) inplane strain,the mixing energy starts to decline with the decreasing inplane a lattice constant (Fig.2).As a result, the mixing energy ∆ E exhibits a non-monotonous behavior.On ferroelectric polarization under strains, our calculations show: (iv) At the inplanelattice constant of SrTiO substrate ( a =3.86˚A), BT/ST has a sizeable polarization whilePT/ST does not, although at this a lattice constant bulk PT has a much larger polarizationthan bulk BT. This indicates that a stronger FE constituent (such as PT with respect toBT) does not always polarize better the non-FE component in short-period superlattices.(v) At a =3.82˚A, the polarization-vs- a curve undergoes an evident change in slope for bothPT/ST and BT/ST superlattices, due to the fact that the incipient ST component starts toturn ferroelectric. For both superlattices, the polarization rises more slowly when a < a ,as compared to the region when a > a . (vi) The polarization in PT/ST is smaller than inBT/ST, when a > a . But for a < a , a crossover occurs, and the polarization in PT/STbecomes stronger than in BT/ST. (vii) By defining the average polarization ¯ P using thevalues of spontaneous polarizations in bulks, we find that ∆ P = P A/B − ¯ P is negative forPT/ST in a wide range of considered inplane strains, revealing that the polarization in 1 × P of bulks. On the otherhand, for BT/ST, ∆ P is found becoming positive when a < . P ranges from -40 to0 µC/cm for PT/ST, and varies within ± µC/cm for BT/ST. (viii) The piezoelectric e coefficient of PT/ST is calculated to be 19.1 C/m , much larger than the value of 10.6 C/m of bulk PT.Regarding the atomic displacements in PT/ST superlattice, our study reveals that (ix)15n 1 × z ( P bO ) or ∆ z ( SrO ′ ) acts like a microscopicorder parameter for the appearance of ferroelectricity. Since this order parameter is thechange in atomic positions, it can thus be probed using x-ray diffraction. (x) The relativeatomic displacements ∆ z ( P bO ) and ∆ z ( SrO ′ ) in PT/ST are found to be very close, overa wide range of inplane strains. The large values in our calculated ∆ z ( SrO ′ ) indicate that,by forming superlattices with PT, the ST component in PT/ST becomes a rather strongA-site FE as compared to bulk ST. (xi) The Ti-O displacements ∆ z ( T iO ) and ∆ z ( T i ′ O ′ ) inPT/ST differ, however, owing to the fact that O and O ′ atoms have a tendency to move inopposite directions in order to form stronger covalent bonds with the Sr atom. Calculationresults further show that this tendency continues to manifest itself even at very large inplanestrains. (xii) For a given inplane lattice constant, the atomic displacements in bulk PT andbulk ST are considerably different. However, after PT and ST form a superlattice, thedisplacements in adjacent PT and ST layers are rather uniform, demonstrating the stronginfluence between two constituents. (xiii) In BT/ST superlattice, since Ba and Sr atomshave different sizes, ∆ z ( BaO ) and ∆ z ( SrO ′ ) deviate significantly from each other at highinplane strains.On effective charges, the calculation results tell us: (xiv) in PT/ST under zero strain, Z ∗ of Ti is 7.31, much larger than Z ∗ =5.76 in bulk PT. Furthermore, Z ∗ s of O and O ′ in PT/ST are found to be very different, more specifically, Z ∗ ( O )=-5.36 and Z ∗ ( O ′ )=-6.38. (xv) With application of increasing inplane strains, the magnitude of Z ∗ drasticallydecreases for Ti and O atoms, showing a wide range of tunability. Meanwhile, as a decreases, | Z ∗ | of Sr is shown to increase whereas | Z ∗ | s of Pb, Ti, O and O atoms all decrease.Finally, the investigation on the polarization dispersion structure demonstrates (xvi)when bulks BT and ST form the BT/ST superlattice, it is the φ ( ~k ⊥ ) phases near the zoneboundary that are most affected. (xvii) At a =3.86˚A, the φ ( ~k ⊥ ) phase in BT/ST is revealedto be much smaller than the φ ( ~k ⊥ ) phase in bulk BT of the same a , and is quantitativelyclose to the averaged ¯ φ phase. (xviii) However, at a =3.82˚A, the φ ( ~k ⊥ ) phase in BT/STsuperlattice is interestingly similar to that of bulk BT, despite the fact that bulk ST stillshows no polarization at this inplane lattice constant. Furthermore, our calculations showthat, when bulk ST is near the critical point of becoming ferroelectric, the strong interactionbetween the BaTiO layer and the SrTiO layer makes the φ ( ~k ⊥ ) dispersion in BT/ST nolonger resembling the average ¯ φ phase. 16 CKNOWLEDGMENTS
This work was supported by the Office of Naval Research. [1] M.E. Lines and A.M. Glass, Principles and applications of ferroelectrics and related materials(Clarendon, Oxford, 1979).[2] J. B. Neaton and K. M. Rabe, Appl. Phys. Lett. , 10(2003).[3] M. Dawber, K. M. Rabe, and J. F. Scott, Rev. Mod. Phys. , 1083 (2005).[4] K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen,X. Q. Pan, V. Gopalan, L.-Q. Chen, D. G. Schlom, and C. B. Eom, Science , 1005 (2004).[5] C. Ederer and N. A. Spaldin, Phys. Rev. Lett. , 257601 (2005).[6] H. N. Lee, S. M. Nakhmanson, M. F. Chisholm, H. M. Christen, K. M. Rabe, and D. Vander-bilt, Phys. Rev. Lett. , 217602 (2007).[7] Y. Yao and H. Fu, Phys. Rev. B , 035126 (2009).[8] E. Bousquet, M. Dawber, N. Stucki, C. Lichtensteiger, P. Hermet, S. Gariglio, J.-M. Triscone,and P. Ghosez, Nature (London) 452, 732 (2008).[9] M. Fornari and D. J. Singh, Phys. Rev. B , 092101 (2001).[10] A. Garcia and D. Vanderbilt, Phys. Rev. B , 3817 (1996).[11] K. Johnston, X. Huang, J. B. Neaton, and K. M. Rabe, Phys. Rev. B , 100103 (2005).[12] L. Kim, J. Kim, U. Waghmare, D. Jung, and J. Lee, Phys. Rev. B , 214121 (2005).[13] Y. Yao and H. Fu, Phys. Rev. B , 014103 (2009).[14] P. Hohenberg and W. Kohn, Phys. Rev. , B864 (1964); W. Kohn and L. J. Sham, Phys.Rev. , A1133 (1965).[15] H. Fu and O. Gulseren, Phys. Rev. B , 214114 (2002).[16] N. Troullier and J.L. Martins, Phys. Rev. B , 1993 (1991).[17] Details of Pb, Ti, and O pseudopotentials were described in Ref.15. Configuration 5 s p d . and matching radii r s,p,d =2.0Bohr are used for Ba; configuration 4 s p d . and matchingradii r s,p,d =1.50, 1.50, 2.0 Bohr are used for Sr.[18] R. D. King-Smith and D. Vanderbilt, Phys. Rev. B , 1651 (1993).[19] R. Resta, Rev. Mod. Phys. , 889 (1994).
20] D. A. Tenne, A. Bruchhausen, N. D. Lanzillotti-Kimura, A. Fainstein, R. S. Katiyar, A.Cantarero, A. Soukiassian, V. Vaithyanathan, J. H. Haeni, W. Tian, D. G. Schlom, K. J.Choi, D. M. Kim, C. B. Eom, H. P. Sun, X. Q. Pan, Y. L. Li, L. Q. Chen, Q. X. Jia, S. M.Nakhmanson, K. M. Rabe, and X. X. Xi, Science , 1614 (2006).[21] X. Wu, M. Stengel, K. M. Rabe, and D. Vanderbilt, , 087601 (2008). ABLE I: Effective charges Z ∗ of atoms in PT/ST superlattice (the 2nd column), in BT/STsuperlattice (the 3rd column), and in bulk PT, BT, and ST (the 4th-6th columns). In 1 × ′ ) 7.45 (Ti) 7.44 (Ti ′ ) 5.76 7.02 7.32O site -5.36 (O ) -6.38 (O ′ ) -5.65 (O ) -6.10 (O ′ ) -4.90 -5.61 -5.77O site -2.28 (O ) -2.28 (O ′ ) -2.22 (O ) -2.21 (O ′ ) -2.28 -2.13 -2.06FIG. 1: (a) Schematic illustration of atomic positions and the direction of atomic displacements(by arrows) in the PbTiO /SrTiO superlattice at equilibrium. Individual atoms are labeled, forthe sake of convenience of discussion. (b) Atomic displacements in the LDA-optimized structurewith respect to the positions of high-symmetry. The displacements are in units of lattice constant c . .70 3.75 3.80 3.85-0.9-0.6-0.3-0.4-0.20.0 0.000.050.10-0.020.000.02 (b) a ( ¯ ) E + . ( e V ) BT/ST
BT+ST
PT/ST PT+ST (a) E E ( e V ) E + . ( e V ) E E ( e V ) FIG. 2: (a) The total energy E A/B of the superlattice (solid squares), E A + E B of the constituentbulks (solid dots), and the mixing energy ∆ E (empty triangles) as a function of the inplane latticeconstant, for the PT/ST system. E A/B and E A + E B are plotted using the left vertical axis, and∆ E is plotted using the right vertical axis. Symbols are the calculation results; lines are guides foreyes. (b) The same as (a), but for the BT/ST system. .70 3.75 3.80 3.85050100 3.70 3.75 3.80 3.850306090-0.06 -0.04 -0.02 0.000.00.30.60.9 3.70 3.75 3.80 3.850306090 -30-150-10-505PT/STBT/ST BTST P ( C / c m ) a ( ¯ ) (a) PT (b) P P(PT/ST) P ( C / c m ) a ( ¯ ) PT/STPT ST (d) P o l a r i z a t i on (c) P P(BT/ST) P ( C / c m ) a ( ¯ ) P P ( C / c m ) P P ( C / c m ) FIG. 3: (a) Total polarizations as a function of the inplane lattice constant, for PT/ST and BT/STsuperlattices, as well as for bulk PT, BT, and ST. (b) Comparison between total polarization P (LDA-calculated) and average polarization ¯ P (defined in Eq.2 using the polarizations of bulkmaterials), for the PT/ST system under different inplane a lattice constants. The difference ∆ P = P − ¯ P is also shown. P and ¯ P are plotted using the left vertical axis, and ∆ P is given usingthe right vertical axis. (c) The same as (b), but for the BT/ST system. In (a)-(c), symbols arethe calculation results, and lines are guides for eyes. (d) The χ polarizations as a function of theinplane η strain for superlattice PT/ST, bulk PT, and bulk ST. The numbers given near thefitting straight lines are the piezoelectric e coefficient. .00.30.6 3.7 3.8 3.90.00.30.6 3.7 3.8 3.9 SrO I PbO (c) ST z ( A O ) ( ¯ ) PT (a) PT/ST BT/ST
SrO I BaO ST BT Ti I O I TiO (b) a ( ¯ ) PTST z ( T i O ) ( ¯ ) Ti I O I TiO (d) BTST a ( ¯ ) FIG. 4: Left panel: relative displacements for the PT/ST system; right panel: relative displace-ments for the BT/ST system. (a) Relative displacements ∆ z ( P bO ) and ∆ z ( SrO ′ ) in the PT/STsuperlattice under different inplane lattice constants (the calculation results are depicted as thesymbols on the solid lines). The counterpart displacements in bulk PT and in bulk ST underdifferent lattice constants are also shown for comparison (by the symbols on the dotted lines). (b)Relative displacements ∆ z ( T iO ) and ∆ z ( T i ′ O ′ ) in PT/ST under different inplane lattice con-stants (see the symbols on the solid lines). The counterpart displacements in bulk PT and in bulkST are also shown (see the symbols on the dotted lines). (c) Similar as (a), but for the BT/STsuperlattice. (d) Similar as (b), but for the BT/ST superlattice. .73.03.3-6-5-4-34567 3.7 3.8 3.9-2.2-2.0-1.8 3.7 3.8 3.9 (b)(a) O O I BT/ST Z e ff * PT/ST
Ti O O I Ti I BaPb O O I SrSr
TiTi I a ( ¯ ) a ( ¯ ) O I O FIG. 5: Effective charges Z ∗ of the non-equivalent atoms in PT/ST superlattice (the left pannel)and in BT/ST superlattice (the right panel), as a function of inplane lattice constant. IG. 6: (a) Polarization structures φ ( ~k ⊥ ) of the BT/ST superlattice, bulk BT, and bulk ST, all atthe same inplane lattice constant a =3.86˚A. The average ¯ φ phase (empty circles) is also shown forcomparison. The ~k ⊥ plane of the Brillouin zone is shown as an inset. (b) Similar as (a), but forthe inplane lattice constant a =3.82˚A.=3.82˚A.