Dielectric, piezoelectric, and elastic properties of BaTiO 3 /SrTiO 3 ferroelectric superlattices from first principles
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l Dielectric, piezoelectric, and elastic properties of BaTiO /SrTiO ferroelectricsuperlattices from first principles Alexander I. Lebedev ∗ Physics Department, Moscow State University,119991 Moscow, Russia (Dated: October 29, 2018)The effect of epitaxial strain on the phonon spectra, crystal structure, spontaneous polarization,dielectric, piezoelectric, and elastic properties of (001)-oriented ferroelectric (BaTiO ) m /(SrTiO ) n superlattices ( m = n = 1–4) was studied using the first-principles density-functional theory. Theground state of free-standing superlattices is the monoclinic Cm polar phase. Under the in-planebiaxial compressive strain, it transforms to tetragonal P mm polar phase, and under the in-planebiaxial tensile strain, it transforms to orthorhombic Amm . Sr . TiO solidsolution modeled using two special quasirandom structures SQS-4 with the mixing enthalpy of thesuperlattices reveals a tendency of the BaTiO –SrTiO system to short-range ordering and showsthat these superlattices are thermodynamically quite stable. PACS numbers: 64.60.-i, 68.65.Cd, 77.84.Dy, 81.05.Zx
I. INTRODUCTION
The success in creating of ferroelectric superlatticeswith a layer thickness controlled with an accuracy ofone monolayer offers new opportunities for design of newferroelectric multifunctional materials with high spon-taneous polarization, Curie temperature, dielectric con-stant, and large dielectric and optical nonlinearities. Be-cause of many problems encountered in the growth andexperimental studies of ferroelectric superlattices, first-principles calculations of their physical properties can beused to reveal new promising fields of investigations andapplications of these materials.Earlier studies of thin epitaxial films of ferroelectricswith the perovskite structure have shown that their prop-erties differ strongly from those of bulk crystals. Itwas established that substrate-induced strain (epitaxialstrain) has a strong influence on the properties of films.Due to strong coupling between strain and polarization,this strain changes significantly the phase transition tem-perature and can induce unusual polar states in thinfilms.
To date, the most experimentally studied ferroelec-tric superlattice is the BaTiO /SrTiO (BTO/STO)one. Studies of these superlattices from first princi-ples have established main factors responsiblefor the formation of their polar structure. The specificfeature of the superlattice is that the strains induced init by the lattice mismatch between BaTiO and SrTiO and by the substrate result in concurrency of equilibriumpolar structures in neighboring layers, so that the po-lar structure of the superlattice can be tetragonal, mon-oclinic, or orthorhombic, depending on the mechanicalboundary conditions at the interface with the substrate.Although some properties of BTO/STO superlatticeshave been already studied, a number of problems re- main unresolved. For instance, first-principles studyof dielectric properties of these superlattices havefound only P mm and Cm polar phases, whereas the Amm , SrTiO , and for PbTiO /PbZrO super-lattices, was not observed. The piezoelectric propertieswere calculated only for PbTiO /PbZrO superlattice; for BTO/STO superlattices these data are absent. Fi-nally, the elastic properties of ferroelectric superlatticesand their behavior at the boundaries between differentpolar phases have not been studied at all.In this work, first-principles density-functional cal-culations of the phonon spectra, crystal structure,spontaneous polarization, dielectric, piezoelectric, andelastic properties for polar phases of (001)-oriented(BTO) m /(STO) n superlattices (SL m/n ) with m = n =1–4 are performed. The influence of compressive andtensile epitaxial strain on the structure and properties ofpolar phases is studied in details for (BTO) /(STO) su-perlattice. The stability ranges of tetragonal, monoclinic,and orthorhombic phases are determined. The criticalbehavior of static dielectric, piezoelectric, and elastic ten-sors at the boundaries between different polar phases arestudied. The ferroelastic type of the phase transitionsbetween the polar phases is established. In addition, animportant question about the thermodynamic stabilityof BTO/STO superlattices is considered.The remainder of this paper is organized as follows. InSec. II, we give the details of our calculations. Next, wepresent the results for the ground state (Sec. III A) andthe polarization (Sec. III B) of (BTO) n /(STO) n superlat-tices. Dielectric, piezoelectric, and elastic properties of(BTO) /(STO) superlattice are described in Secs. III C,III D and III E, respectively. The thermodynamic sta-bility of the superlattices is analyzed in Sec. III F. Theobtained results are discussed in Sec. IV. TABLE I. Parameters used for construction of pseudopotentials. Non-relativistic generation scheme was used for Sr, Ti, andO atoms, and scalar-relativistic generation scheme was used for the Ba atom. All parameters are in Hartree atomic units exceptfor the energy V loc , which is in Ry.Atom Configuration r s r p r d q s q p q d r min r max V loc Sr 4 s p d s s p d s s p d s s p d II. CALCULATION DETAILS
The calculations were performed within the first-principles density-functional theory (DFT) with pseu-dopotentials and a plane-wave basis set as implementedin
ABINIT software. The local density approximation(LDA) for the exchange-correlation functional wasused. Optimized separable nonlocal pseudopotentials were constructed using the OPIUM software; to improvethe transferability of pseudopotentials, the local poten-tial correction was added according to Ref. 53. Param-eters used for construction of pseudopotentials are givenin Table I; the results of testing of these pseudopotentialsand other details of calculations can be found in Ref. 48.The plane-wave cut-off energy was 30 Ha (816 eV). Theintegration over the Brillouin zone was performed witha 8 × × · − Ha/Bohr (0.25 meV/˚A).The lattice parameters calculated using the pseudopo-tentials were a = 7 . and a = 7 . c = 7 . . Slight underestima-tion of the lattice parameters (in our case by 0.4–0.7%compared to the experimental data) is a known problemof LDA calculations.The calculations were performed on two structures:supercells of 1 × × n perovskite unit cells for(001)-oriented (BTO) n /(STO) n superlattices ( n = 1–4) and two special quasirandom structures (SQS) forBa . Sr . TiO solid solution; the construction of SQSsis described in Sec. III F.Phonon spectra, dielectric, piezoelectric, and elasticproperties of the superlattices were calculated within theDFT perturbation theory. Phonon contribution to the di-electric constant was calculated from phonon frequenciesand oscillator strengths. The Berry phase method was used to calculate the spontaneous polarization P s .As the layers of the superlattices are epitaxially grownon (001)-oriented substrate with a cubic structure, thecalculations were performed for pseudotetragonal unitcells in which two in-plane translation vectors have thesame length a and all three translation vectors are per-pendicular to each other. This means that for monoclinicand orthorhombic phases (with Cm and Amm ◦ (which were typically less than0.07 ◦ ) were neglected. As was checked, this does notinfluence much the results. III. RESULTSA. Ground state of epitaxially strainedsuperlattices
The lattice mismatch between BaTiO and SrTiO creates tensile biaxial strain in SrTiO layers and com-pressive biaxial strain in BaTiO layers of free-standingBTO/STO superlattice. If the layers were isolated, thesestrains would result in appearance of the in-plane sponta-neous polarization in SrTiO ( Amm lay-ers ( Cm or P mm space groups). As the polar statewith strong local variations of polarization is energet-ically unfavorable, the structure of the polar groundstate of the superlattice requires special consideration.Earlier studies of BTO/STO and PbTiO /PbZrO superlattices have demonstrated that both the magni-tude and orientation of polarization depend also on thesubstrate-induced (epitaxial) strain in superlattices.The ground state of BTO/STO superlattice wassearched as follows. For a set of in-plane lattice param-eters a , which were varied from 7.35 to 7.50 Bohr, wefirst calculated the equilibrium structure of the paraelec-tric phase with P /mmm space group by minimizing theHellmann-Feynman forces. The phonon frequencies atthe Γ point were then calculated for these structures. Itis known that the ground state of any crystal is character-ized by positive values of all optical phonon frequenciesat all points of the Brillouin zone. So, if the structureunder consideration exhibited unstable phonons (withimaginary phonon frequencies), the atomic positions init were slightly distorted according to the eigenvector ofthe most unstable mode, and a new search for the equi-librium structure was initiated. The phonon frequenciescalculation and the search for equilibrium structure wererepeated until the structure with all positive phonon fre-quencies was found.It should be noted that the only unstable mode in theparaelectric P /mmm phase of (BTO) /(STO) super- FIG. 1. Phonon spectra for (a) the P mm phase of(BTO) /(STO) superlattice grown on SrTiO substrate( a = 7 . Cm phase of the same, free-standing superlattice ( a = 7 . lattice is the ferroelectric one at the Γ point. The well-known structural instability of SrTiO associated withthe R phonon mode at the boundary of the Brillouinzone disappears in the superlattice: the frequency of thecorresponding phonon at the M point of the folded Bril-louin zone (to which the R point transforms when dou-bling the c lattice parameter) is 55 cm − for 1/1 free-standing superlattice and 61 cm − for 1/1 superlatticegrown on SrTiO substrate (see Fig. 1).The phonon spectra calculations show that in 1/1 su-perlattice grown on SrTiO substrate (compressive epi-taxial strain, the in-plane lattice parameter a is equalto that of cubic strontium titanate) the tetragonal po-lar phase with P mm space group is the ground state(Fig. 1). For free-standing superlattice, the P mm struc-ture is unstable and transforms to monoclinic Cm polarone. Under tensile epitaxial strain ( a = 7 .
46 Bohr), theorthorhombic
Amm a (for example, by growing the superlattice on differentsubstrates) can be used to control the polar state of thesuperlattice.In order to determine accurately the location of theboundaries between P mm and Cm phases and between Cm and Amm a , and for each of these structures the phonon fre-quencies at the Γ point were computed. The dependenceof four lowest phonon frequencies on the a parameter is FIG. 2. (Color online) Frequencies of four lowest phononmodes at the Γ point for polar phases of (BTO) /(STO) su-perlattice as a function of the in-plane lattice parameter a .The labels near the points indicate the symmetry of modes.Vertical lines indicate the phase boundaries. plotted in Fig. 2. It is seen that the frequency of a doublydegenerate E mode decreases critically when approach-ing the boundary between P mm and Cm phases fromthe tetragonal phase. After transition to the monoclinicphase two non-degenerate A ′ and A ′′ soft modes appearin the phonon spectrum; the first of these modes becomessoft again when approaching the boundary between Cm and Amm
Amm B symmetry.Extrapolation of the squared frequencies of soft ferro-electric modes ( E mode in the P mm phase, A ′ modein the Cm phase and B mode in the Amm a to zero gives the in-plane lattice pa-rameters corresponding to the boundaries between dif-ferent polar phases. The P mm – Cm boundary is at a = 7 . Cm – Amm a = 7 . a = 7 . − /(STO) and (BTO) /(STO) superlatticesshows that the same P mm , Cm , and Amm , free-standing superlattices, and superlatticesgrown on the substrate with a = 7 .
46 Bohr.
FIG. 3. The in-plane ( P ⊥ ) and out-of-plane ( P z ) componentsof polarization for (BTO) /(STO) superlattice as a functionof the in-plane lattice parameter a . Vertical lines indicatethe phase boundaries. B. Spontaneous polarization
The calculated spontaneous polarization for free-standing and substrate-supported superlattices, tetrag-onal BaTiO , and disordered Ba . Sr . TiO solid solu-tion modeled using SQS-4 structures (see Sec. III F) aregiven in Table II.As was established in Sec. III A, the tetragonal P mm phase with the polarization vector normal to the layersis the ground state for BTO/STO superlattices grown onSrTiO substrate. The calculations shows that in thesesuperlattices the spontaneous polarization P s increasesmonotonically from 0.277 C/m to 0.307 C/m as thelayer thickness is increased from n = 1 to 4 unit cells(see Table II). The obtained P s values agree well with thevalue of 0.28 C/m estimated from the data of Ref. 33for the superlattice with equal thickness of BaTiO andSrTiO layers; the P s value of 0.259 C/m for tetrag-onal BaTiO agrees well with the value of 0.250 C/m reported in Ref. 33. As follows from Table II, for allsuperlattices the P s values are larger than those forBa . Sr . TiO solid solution; for superlattices grown onSrTiO substrates they are even larger than P s of tetrag-onal BaTiO . These results agree with experiment andresults of previous calculations. For free-standing superlattices, the monoclinic Cm phase is the ground state; the components of the polar-ization vector in this phase are also given in Table II. It isseen that the polarization is rotated continuously in the(¯110) plane and the magnitude of polarization increaseswith increasing n .For biaxially stretched superlattices, the Amm P /mmm structure of the paraelectric phase.The in-plane and out-of-plane components of the po-larization for 1/1 superlattice are plotted as a function FIG. 4. Eigenvalues of the static dielectric constant tensor ε ij for (BTO) /(STO) superlattice as a function of the in-plane lattice parameter a . Vertical lines indicate the phaseboundaries. of the in-plane lattice parameter a in Fig. 3. Extrap-olation of the P ⊥ and P z dependence on a to zerogives the positions of the P mm – Cm and Cm – Amm a = 7 . a = 7 . C. Dielectric properties
The eigenvalues of the static dielectric constant tensor ε ij ( i, j = 1, 2, 3) for (BTO) /(STO) superlattice as afunction of the in-plane lattice parameter a are shown inFig. 4. In the tetragonal phase, the eigenvectors of the ε ij tensor coincide with crystallographic axes and ε = ε .So, the dielectric properties of this phase are describedby two nonzero independent parameters, ε and ε .In the monoclinic phase, the polarization vector rotatesmonotonically in the (¯110) plane; all three eigenvectorsof the ε ij tensor are different and do not coincide withcrystallographic axes of the reference tetragonal struc-ture. As all nine components of the ε ij tensor in thiscoordinate system are nonzero for the Cm phase, themost compact way to describe the properties of this ten-sor is to present its eigenvalues. In the Cm phase, thedirection of the eigenvector corresponding to the smallesteigenvalue is close, but do not coincide with the directionof the polarization.In the orthorhombic phase, the eigenvectors of the ε ij tensor are oriented along the [110], [1¯10] and [001] direc-tions of the reference tetragonal structure. The directionof the eigenvector corresponding to the smallest eigen-value coincides with the polarization vector and the di-rection of eigenvector corresponding to the largest eigen-value is [001].As follows from Fig. 4, at least one of the eigenvalues ofthe ε ij tensor diverges critically at the P mm – Cm and TABLE II. Spontaneous polarization (in C/m ) for BTO/STO superlattices with different thickness of layers, two SQS-4structures used for modeling of disordered Ba . Sr . TiO solid solution, and tetragonal barium titanate. The in-plane latticeparameters a (in Bohr) for superlattices are also presented.Structure SL 1/1 SL 2/2 SL 3/3 SL 4/4 SQS-4a SQS-4b BaTiO P s orientation [ xxz ] [001] a [ xxz ] [001] a [ xxz ] [001] a [ xxz ] [001] a [111] [001] [001] P z P x = P y a a Superlattices grown on SrTiO substrate. TABLE III. Largest piezoelectric moduli for monoclinic phaseof free-standing (BTO) /(STO) superlattice, for tetragonalphase of the same superlattice grown on SrTiO substrate,and for tetragonal barium titanate.Structure SL 1/1 BaTiO P s orientation [ xxz ] [001] a [001] e , C/m ( d , pC/N) 31.9 (460) 7.1 (49) 6.3 (42) e , C/m ( d , pC/N) − −
19) 3.2 (31) − − b Superlattice grown on SrTiO substrate. Cm – Amm a is changed. When approaching the P mm – Cm boundary from the tetragonal phase, the ε = ε com-ponents of this tensor diverge as the polarization vector P s k [001] becomes less stable against its rotation in the(¯110) plane. When approaching the Cm – Amm ε value divergesas the polarization vector P s k [110] becomes less stableagainst its rotation in the same plane. D. Piezoelectric properties
Due to high sensitivity of both magnitude and orienta-tion of the polarization vector in superlattices to epitax-ial strain, we can expect them to be good piezoelectrics.It is known that anomalously high piezoelectric mod-uli found in some ferroelectrics like PbZr − x Ti x O nearthe morphotropic phase boundary are due to the easeof strain-induced rotation of polarization in the interme-diate monoclinic phase. A similar situation appearsin the monoclinic phase of BTO/STO superlattice. Toour knowledge, the piezoelectric properties of BTO/STOsuperlattices have not been studied so far neither exper-imentally, nor theoretically. The only superlattice, forwhich some piezoelectric properties were calculated, isthe PbTiO /PbZrO which was usedto simulate the properties of PbTi . Zr . O solid solu-tion.The largest piezoelectric stress moduli e iν ( i = 1,2, 3; ν = 1–6) calculated for the P mm phase of(BTO) /(STO) superlattice grown on SrTiO substrate FIG. 5. (Color online) Components of the piezoelectric ten-sor e iν in polar phases of (BTO) /(STO) superlattice as afunction of the in-plane lattice parameter a . Vertical linesindicate the phase boundaries. and for the Cm phase of the same free-standing super-lattice are given in Table III. It is seen that in tetragonalphases of the superlattice and BaTiO the e modulido not differ much. However in the monoclinic phase,which is the ground state for free-standing superlattice,the e value is five times larger. Even stronger effectcan be seen for the d piezoelectric strain coefficient( d iν = P µ =1 e iµ S µν ), which is a result of an 1.5-foldincrease in the elastic compliance modulus S in themonoclinic phase (see Sec. III E).The piezoelectric moduli e iν in polar phases of(BTO) /(STO) superlattice as a function of the in-planelattice parameter are shown in Fig. 5. According to thesymmetry, in the tetragonal phase the piezoelectric ten-sor has three independent and five nonzero components: e = e , e , and e = e . In our superlattice onlytwo of them have large values: e and e . When ap-proaching the P mm – Cm boundary from the tetrago-nal phase, the e value increases monotonically whereasthe e value diverges critically and reaches the value of80 C/m (not shown).In the orthorhombic phase (in the coordinate systemof the reference tetragonal structure) the e = e , e = e , e = e , e = e , and e = e moduliare nonzero, and the total number of independent pa-rameters is 5. Among them the e , e , e , and e moduli have the largest values (see Fig. 5). The onlymodulus that behaves critically at the boundary between Cm and Amm e one, with a maximumvalue reaching 192 C/m (not shown).The most complex behavior of piezoelectric moduli isobserved in the monoclinic phase because the change ofthe in-plane lattice parameter a results in monotonic ro-tation of the polarization vector in the (¯110) plane. Inthe coordinate system of the reference tetragonal struc-ture, all 18 components of the e iν tensor are nonzero (thetotal number of independent parameters is 10). As fol-lows from Fig. 5, in the monoclinic phase the e = e , e = e , e = e , e = e , e = e , and e = e moduli diverge critically at the boundary between Cm and P mm phases, and e , e = e , e = e , and e moduli diverge critically at the boundary between Cm and Amm ∼ e modulus (at the Cm – P mm boundary) and in e and e moduli (at the Cm – Amm
E. Elastic properties
It is known that ferroelectric phase transitions betweentwo polar phases are often the improper ferroelastic ones,which means that they are accompanied by appearanceof soft acoustic modes and spontaneous strain, but thestrain is not the primary order parameter. This occurswhen the strain tensor and the polar vector transformaccording to the same irreducible representation of thehigh-symmetry phase. As such phase transitions occurin BTO/STO superlattices when changing the in-planelattice parameter a , it was interesting to study the influ-ence of these phase transitions on the elastic propertiesof superlattices, especially taking into account that theseproperties of superlattices have not been studied so far.In the tetragonal P mm phase, the elastic compliancetensor S µν ( µ, ν = 1–6) has six independent and ninenonzero components. In the orthorhombic Amm Cm phase it has13 independent and 21 nonzero components.The components of the elastic compliance tensor S µν for polar phases of (BTO) /(STO) superlattice are plot-ted as a function of the in-plane lattice parameter a inFig. 6. It is seen that at the boundary between P mm and Cm phases the components of S µν tensor exhibit astep-like change ( S = S , S , S , S = S , S , S = S , S = S moduli), a critical divergencefrom the monoclinic side ( S = S , S , S = S , S = S , S moduli), or critical divergences from bothsides of the boundary ( S = S modulus). In the mon-oclinic phase, the S , S , S , S , S , S , and S FIG. 6. (Color online) Components of the elastic compliancetensor S µν for polar phases of (BTO) /(STO) superlatticeas a function of the in-plane lattice parameter a . Verticallines indicate the phase boundaries. moduli become nonzero. In the orthorhombic phase, the S , S , S , and S moduli vanish again whereas theother moduli remain nonzero. At the boundary between Cm and Amm S , S , and S moduli exhibit step-like changes, the S , S , S , S , S , S , S , and S moduli exhibit weak divergence from the monoclinicside, and the S and S moduli exhibit weak divergencefrom both sides of the boundary.The critical divergence of the S modulus at both P mm – Cm and Cm – Amm
F. Thermodynamic stability
Thermodynamic stability of ferroelectric superlatticesis very important for their possible applications. Ther-modynamic stability of BTO/STO superlattice is deter-mined by the mixing enthalpy of the superlattice andits relationship with the mixing enthalpy of the disor-dered Ba . Sr . TiO solid solution. The most complexpart of the first-principles calculation of thermodynamicstability is the calculation of the mixing enthalpy for asolid solution because its simulation using supercells witha large number of randomly distributed atoms makes itextremely time-consuming.A conceptually new approach to this problem was pro-posed by Zunger et al . In this approach, a disorderedsolid solution A x B − x is modeled using a special quasir-andom structure (SQS)—a short-period superstructure,whose statistical properties (numbers of N AA , N BB , and N AB atomic pairs in few nearest shells) are as close aspossible to those of ideal disordered solid solution (at thesame time the sites of the superstructure are determin- TABLE IV. Translation vectors, superlattice axis, stacking sequence of atomic planes, and correlation functions Π ,m for SQS-4structures used for modeling of disordered Ba . Sr . TiO solid solution.Structure Translation vectors Axis and stacking sequence Π , Π , Π , Π , SQS-4a [2¯11], [1¯12], [1¯21] [1¯11]
AABB − AABB − a , a and a translation vectors.The atoms of one of two types, A or B , occupy the sites ofone sublattice of the perovskite structure lying on the planes(shown by blue), which are perpendicular to the superlat-tice axis (shown by red arrow). The stacking sequence of theplanes along the superlattice axis is AABB . istically filled with A and B atoms). This method hasbeen widely used to study the electronic structure andphysical properties of semiconductor solid solutions andthe ordering phenomena in metal alloys. To study theproperties of ferroelectric solid solutions this approachwas used quite rare. The structure of the disordered Ba . Sr . TiO solidsolution was modeled using two special quasirandomstructures SQS-4 constructed with the gensqs programfrom ATAT toolkit. One of these structures is sketchedin Fig. 7. The translation vectors, superlattice axis,and stacking sequence of the planes filled with the sameatoms, Ba or Sr (denoted by A and B ), are given in Ta-ble IV. The pair correlation functions Π ,m ( m is the shellnumber), which describe the deviation of statistical prop-erties of these SQSs from those of an ideal solid solution,are also given in this table. For x = 0 . ,m is simply(2 N AA /N m − N m is a number of neighbors inthe m th shell. As follows from this table, for the SQS-4a structure strong deviation from an ideal solid solutionappears only in the forth shell; for the SQS-4b structuredeviations appear in the second and fourth shells, but aresmaller. The mixing enthalpy ∆ H for all studied struc-tures X (superlattices with different periods and SQSstructures) was calculated using the formula∆ H = E tot ( X ) − [ E tot (BaTiO ) + E tot (SrTiO )] / E tot (per five-atomformula unit) for free-standing fully relaxed paraelectric P m m and P /mmm phases. The obtained values of TABLE V. The mixing enthalpy (in meV) for five(BTO) n /(STO) n superlattices with different periods andtwo SQS-4 structures used for modeling of disorderedBa . Sr . TiO solid solution.SL 1/1 SL 2/2 SL 3/3 SL 4/4 SL 5/5 SQS-4a SQS-4b2.9 8.9 11.4 12.6 13.4 16.8 11.0 ∆ H for these structures are given in Table V.An unexpected result of our calculation is the fact that∆ H values for two shortest-period superlattices (1/1 and2/2) appeared smaller than ∆ H values for both realiza-tions of disordered solid solution. This means that atendency to short-range ordering of components existsin the BaTiO –SrTiO system. Low values of ∆ H forthese superlattices ( < A cations to order in a lay-ered manner in perovskites, in contrast to the B cations,which prefer a rock-salt ordering. One can add thata similar effect was observed in our studies of (001)-oriented (PbTiO ) n /(SrTiO ) n superlattices, where neg-ative values of ∆ H for n = 1–3 and positive values forlarger n were observed.Our values of the mixing enthalpy for BTO/STO su-perlattices are much smaller than ∆ H value obtained inRef. 65 (42 meV per formula unit). Analysis of the cal-culation technique used in Ref. 65 shows that atomic po-sitions in the superstructures were not relaxed and thesuperstructures were assumed to be cubic when calculat-ing the energies of different atomic configurations. So,the calculated mixing enthalpy in this paper includes alarge energy of excess strain. IV. DISCUSSION
Our results on the influence of epitaxial strain on theground state of BTO/STO superlattice agree only par-tially with the results obtained in Refs. 24, 34, 36, and37. The results coincide in that: 1) the ground state forfree-standing (BTO) /(STO) superlattice is the mono-clinic Cm phase, 2) under the compressive strain, thesuperlattice undergoes the phase transition from Cm to P mm phase, and 3) the dielectric constant diverges atthe P mm – Cm phase boundary. At the same time, incontrast to the results of Refs. 36 and 37, we succeededto observe the phase transition to the Amm and SrTiO thin films, the results of recent atom-istic calculations of the strain–temperature phase dia-gram for (BTO) /(STO) superlattice, and with resultsobtained for another superlattice, PbTiO /PbZrO . Inall these systems the same phase sequence, P mm – Cm – Amm
2, was observed as the in-plane lattice parameterwas increased.The increase in the polarization P s in (BTO) n /(STO) n superlattices grown on SrTiO substrates with increasingthe layer thickness n (Table II) agrees with the resultsof Ref. 42 in which the explanation of this phenomenonwas proposed. In free-standing superlattices, the P z com-ponent of polarization also increased with increasing n ,but the P x and P y components decreased with increas-ing n , in contrast to the results observed for 3/3, 4/4and 5/5 superlattices with fixed in-plane lattice parame-ter equal to 1.01 times the lattice parameter of SrTiO . We attribute these changes to the decrease of the in-planelattice parameter a for free-standing superlattices withincreasing n (see Table II).Unfortunately, only a few data points presented inRef. 37 for the dielectric constant at the P mm – Cm boundary for (BTO) /(STO) superlattice did not en-abled us to make detailed comparison between our re-sults. However, the comparison of our dielectric datawith those calculated for (PbTiO ) /(PbZrO ) super-lattice shows that in the monoclinic phase all eigen-values of the ε ij tensor for BTO/STO superlattice arehigher, and so this superlattice may be more promisingfor different applications.To obtain large piezoelectric moduli necessary fortechnological applications, the epitaxial strain in theBTO/STO superlattice should be tuned to a value atwhich e iν and S µν properties of the superlattice diverge.As was shown in Secs. III D and III E, this appears atthe phase boundaries. The calculations shows that in thevicinity of the P mm – Cm boundary the d piezoelectriccoefficient reaches a maximum value of 2300 pC/N in themonoclinic phase and the d coefficient reaches a valueof 6200 pC/N in the tetragonal phase. In the vicinityof the Cm – Amm d piezoelectric coeffi-cient reaches a value of 920 pC/N in the monoclinic phaseand the d coefficient reaches a value of 10500 pC/Nin the orthorhombic phase. For comparison, the maxi-mum piezoelectric coefficient obtained experimentally onsingle crystals of the Pb(Zn / Nb / )O –PbTiO systemwas 2500 pC/N. Anomalous increase in the calculated d and d coefficients up to ∼ FIG. 8. Frequencies of acoustic modes in the vicinity of theΓ point in polar phases of (BTO) /(STO) superlattice as afunction of the in-plane lattice parameter a . Vertical linesindicate the phase boundaries. for hydrostatically stressed PbTiO in the vicinity of the P mm – Cm phase boundary. Consider now the elastic properties of BTO/STO su-perlattice and the results indicating the appearance ofimproper ferroelastic phase transitions. According toRef. 59, in crystals with P mm space group the phasetransition 4 mm → m should be of the ferroelastic type.The spontaneous strain at this phase transition is char-acterized by one or both nonzero u and u componentsof the strain tensor, and a soft transverse acoustic (TA)mode with a wave vector parallel to the polar axis shouldappear in the phonon spectrum in the vicinity of theΓ point. Direct calculations of frequencies of acousticmodes in the tetragonal phase at the point with a re-duced wave vector q = (0, 0, 0.05) confirmed this (seeFig. 8): when approaching the P mm – Cm boundary,the frequency of a doubly degenerate TA phonon withthe Λ symmetry decreased by 5 times. The softeningof this mode is directly related to the divergence of the S = S components of the elastic compliance tensor.At the boundary between Cm and Amm mm → m is accompa-nied by spontaneous strain with one nonzero of three u , u , u components. In our coordinate system with un-usual for orthorhombic phase orientation of the polar axis(along the [110] direction) there should be a softening ofTA phonon with a wave vector oriented along this axis.This was confirmed by direct calculations of frequenciesof acoustic modes at the point in the Brillouin zone with areduced wave vector q = (0.035, 0.035, 0) (see Fig. 8). Incontrast to the tetragonal phase in which the soft acous-tic mode is doubly degenerate, in the orthorhombic phasethe only phonon mode that softens at the phase bound-ary is the TA mode polarized along the [001] axis. In ouropinion, this difference is the reason why the anomaliesin the elastic properties at the ferroelastic Cm – Amm P mm – Cm phase transition. Unusual orienta-tion of the polar axis in our coordinate system resultsin coupling of some components of the elastic compli-ance tensor: for example, the S and S moduli, whichdiverge in the orthorhombic phase, satisfy the relation S − S ≈ const in this phase.Negative value of the acoustic phonon frequency in the Cm phase, which is seen in a narrow wave vector regionin Fig. 1, is an artifact of calculations. Computationof the phonon dispersion curves in the vicinity of the Γpoint revealed three acoustic branches ω ( q ), whose fre-quencies increased monotonically with increasing q , butgave negative values of ω (0) (about 8 cm − ) in the limit q → ω (0) restored its zero value, butthe derivative dω/dq near q = 0 for the softest modebecame negative. So, there is no contradiction betweenthe phonon spectra and the positive definiteness of theelastic moduli matrix C ij calculated in Sec. III E. V. CONCLUSIONS
In this work, the properties of (001)-oriented(BaTiO ) m /(SrTiO ) n superlattices with m = n =1–4 were calculated using the first-principles density-functional theory. For free-standing superlattices, the ground state is the monoclinic Cm polar phase. Underthe in-plane compressive epitaxial strain, it transformsto tetragonal polar P mm phase, and under in-planetensile strain, it transforms to orthorhombic Amm ε ij ), the piezoelectric tensor ( e iν ), and the elastic com-pliance tensor ( S µν ) were calculated as a function of thein-plane lattice parameter for 1/1 superlattice. The crit-ical behavior of some components of ε ij , e iν , and S µν tensors at the boundaries between different polar phaseswas observed. It was shown that the phase transitionsbetween different polar phases are of the improper fer-roelastic type. The possibility of obtaining ultrahighpiezoelectric moduli using fine tuning of the epitaxialstrain in superlattices was demonstrated. The compar-ison of the mixing enthalpy calculated for superlatticesand disordered Ba . Sr . TiO solid solution modeled us-ing two special quasirandom structures SQS-4 revealed atendency of the BaTiO –SrTiO system to short-rangeordering of cations and showed that short-period super-lattices are thermodynamically quite stable. ACKNOWLEDGMENTS
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