Download - : A ferrimagnetic pyroelectric
PHYSICAL REVIEW B 90, 045129 (2014)
CaBaCo4O7: A ferrimagnetic pyroelectric
R. D. Johnson,1,* K. Cao,2 F. Giustino,2 and P. G. Radaelli11Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
2Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom(Received 3 March 2014; revised manuscript received 4 July 2014; published 23 July 2014)
Magnetoelectric coupling in pyroelectric CaBaCo4O7 is investigated using ab intio calculations and Landautheory. The former shows that exchange striction is strong enough to produce a giant change in electricpolarization upon ferrimagnetic ordering, comparable to the experimentally determined value of ∼17 mC/m2.Furthermore, Landau theory demonstrates that magnetoelastic coupling in CaBaCo4O7 is responsible for thestrong magnetoelectric coupling appearing close to the magnetic phase transition.
DOI: 10.1103/PhysRevB.90.045129 PACS number(s): 75.85.+t, 75.50.Gg, 77.70.+a, 75.10.−b
I. INTRODUCTION
Solid-state materials that adopt a polar crystal structurehave sustained interest in condensed matter physics andhave become key components in technology. For example,the change in the intrinsic bulk electric polarization ofnoncentrosymmetric pyroelectric compounds, which occursupon varying the temperature of the material, forms the basisof infrared-sensing devices. Also, research into ferroelectricmaterials, in which inversion symmetry is broken at a phasetransition giving rise to switchable ferroelectric states, has leadto the development of electronic devices such as ferroelectricrandom access memory (FeRAM). Multiferroic materialsform a subset of ferroelectrics, in which spontaneous electricpolarization is coupled to long-range magnetic order. Researchin this field has recently undergone a renaissance of interest,following the discovery of magnetic-field-switchable electricpolarization in the now-canonical systems TbMnO3 [1] andTbMn2O5 [2]—opening new routes toward the developmentof novel multifunctional devices.
To integrate multiferroic materials into technology itis necessary to identify systems that exhibit a very largemagnetically induced ferroelectric polarization close to roomtemperature. However, the largest magnetically induced ferro-electric polarization measured to date, (2870 μCm−2 observedin CaMn7O12 below 90 K [3–5]), is two orders of magnitudesmaller than that of a good ferroelectric. Recently, a muchlarger spin-assisted change in polarization of ∼17 000 μCm−2
was measured in CaBaCo4O7 below 64 K [6,7]—a very sig-nificant observation that, if confirmed, could pave the way fora new generation of magnetic ferroelectrics. In this paper, weperform first-principles calculations and a phenomenologicalanalysis to study the magnetoelectric coupling in CaBaCo4O7.We demonstrate that all single-crystal experimental data areconsistent with CaBaCo4O7 being pyroelectric rather thanferroelectric in both paramagnetic and magnetically orderedstates. The distinction here is critical if these materials are tobe considered for device construction. Both pyroelectricityand ferroelectricity have the same prerequisite symmetry(i.e., the host crystal structure must adopt one of the 10nonpolar pyroelectric point groups: 1, 2, m, mm2, 3, 3m, 4,4mm, 6, 6mm); however, a ferroelectric polarization may be
switched by an external electric field (for example, switchingopposite displacements of a perovskite B-site transition metalion), whereas a pyroelectric polarization cannot be switched(for example, rigid, coaligned dipole moments of tetrahedraltransition metal-oxygen coordinations). Energetically, ferro-electric materials have a small energy barrier between states ofopposite polarization of an order comparable to applied electricfields, whereas pyroelectric materials have an essentiallyinfinite energy barrier between polar states.
In CaBaCo4O7 the large pyroelectric currents observednear the magnetic phase transition result from an exchange-striction-driven change of �P in the paramagnetic pyroelectricpolarization Ppyr. However, �P is always coaligned in a fixrelation with Ppyr, either parallel or antiparallel depending onthe sign of the magnetostrictive constant, and neither Ppyr nor�P are switchable.
II. CRYSTAL AND MAGNETIC STRUCTURES ANDELECTRICAL PROPERTIES
The crystal structure of CaBaCo4O7, shown in Fig. 1, wasfound to adopt the polar space group Pbn21 at all temperaturesbelow 400 K [8]. The structure comprises interleaved kagomeand triangular layers of CoO4 tetrahedra, which are buckledwith respect to a high symmetry, high temperature trigonalpolar phase (space group P 31c) common to other membersof the RBaCo4O7 series [9,10] (R = rare earth, calcium, oryttrium), but yet to be observed in CaBaCo4O7. In both spacegroup symmetries, CaBaCo4O7 is likely to be a nonswitchablepyroelectric material, since atoms in inversion-related struc-tures are separated by large distances. Furthermore, it followsthat a high-temperature phase transition to a centrosymmetricgroup is extremely unlikely to occur below the melting point.
The geometric frustration intrinsic to both kagome andtriangular lattices, well known to give rise to exotic magneticground states [11,12], is lifted as a result of the CoO4
buckling. This structural distortion is reported to be largestin CaBaCo4O7 [8], removing the frustration, and promotingferrimagnetic order developing at Tc = 64 K. The magneticstructure [8,13] is shown in Fig. 2. There are four symmetryinequivalent cobalt sites in the unit cell, labeled Co1, Co2, Co3,and Co4, and colored green, blue, red, and pink, respectively, inaccordance with the color scheme in Ref. [13]. The magneticmoments of the four sites lie within the ab plane with Co1and Co4 moments approximately antiparallel to those of Co2
1098-0121/2014/90(4)/045129(7) 045129-1 ©2014 American Physical Society
R. D. JOHNSON, K. CAO, F. GIUSTINO, AND P. G. RADAELLI PHYSICAL REVIEW B 90, 045129 (2014)
FIG. 1. (Color online) Left: The crystal structure of CaBaCo4O7.Calcium, barium, cobalt, and oxygen atoms are shown as black, green,blue, and red spheres, respectively. The CoO4 tetrahedra are shadedblue. Right: The triangular and kagome CoO4 layers in the ab plane.
and Co3. CaBaCo4O7 is a mixed valance system, with greatercharge, and hence larger magnetic moments, located on theCo1 and Co4 sites; a primary component to the ferrimagnetism.
Measurements on a powder sample of CaBaCo4O7 [6]showed that a pyrocurrent signal, corresponding to a changein electrical polarization, coincided with an anomaly in thedielectric constant at the ferrimagnetic ordering transitionTc. The pyroelectric signal switches sign with the externalelectric field, and the results were therefore interpreted as
FIG. 2. (Color online) The ground state magnetic structure ofCaBaCo4O7. Cobalt ions are shown as spheres colored yellow orblack if charge rich or charge poor, respectively. Magnetic momentsare colored in accordance with the scheme adopted in Ref. [13].The 12 unique nearest-neighbor exchange paths are shown by blackarrows.
evidence for ferroelectricity and multiferroicity [6,7]. We note,however, that the magnetically induced change in polarization(�P ∼ 80 μCm−2 was found to be extremely small withrespect to the “pyroelectric polarization” Ppyr, with whateverdefinition might be adopted for it (see below); these results,therefore, have to be interpreted with caution, as they couldeasily arise from an artefact. Similar measurements on single-crystal samples showed much larger pyroelectric currentsdeveloping at Tc, consistent with a giant change in polarizationof �P ∼ 17 000 μCm−2 [7]. However, no switching behaviorwas reported for the single-crystal sample.
III. METHODS
Our first-principles calculations were based on density-functional theory implemented in the Vienna ab initio sim-ulations package (VASP) [14,15]. We used the spin-polarizedgeneralized gradient approximation with onsite Coulomb in-teractions, U , included for cobalt 3d orbitals (GGA + U)[16].By fixing the Hund coupling constant J = 1 eV and testingseveral U values we found that the experimental ground-state electronic structure becomes metallic if U < 3 eV. Wetherefore only present the results for U = 4 eV, as CaBaCo4O7
is known to be an insulator. We also performed calculationswith U = 6 eV, which produced very similar results. Theprojector augmented-wave (PAW) [17] method with a 500-eVplane-wave cutoff was used throughout, and a 6 × 4 × 4k-point mesh converges the calculation very well. Calculationswere performed on a single unit cell of the experimentalatomic structure unless otherwise stated. In case of structuralrelaxation, structural parameters were left to vary until changesin total energy in the self-consistent calculations were less than10−7 eV and the remnant forces were less than 1 meV/A.The electric polarization was calculated using the Berry phasemethod [18,19].
The Berry phase method determines the polarization towithin a factor (f qeR/�), where f is a band-filling factor, qe isthe electron charge, � is the unit cell volume, and R is a latticevector. According to the modern theories of polarization, only achange in electric polarization is well defined. In a ferroelectricmaterial, one can calculate electric polarization by referringto the change in polarization with respect to a “reference”centrosymmetric structure for which one can define the Berryphase polarization, P = 0. However, in a pyroelectric materialit is often impossible to define a physically meaningfulreference structure for the calculation of absolute electricpolarization. In Sec. IV we calculate the change of polarizationfrom the paramagnetic phase of pyroelectric CaBaCo4O7 tothe experimentally determined ferrimagnetic phase, withouthaving to define a centrosymmetric reference structure. Here,the magnetically induced change in polarization is small withrespect to (qeR/�), and hence well defined.
IV. MODELING MAGNETISM AND PYROELECTRICITY
To gain a microscopic insight into the coupling betweenmagnetism and electric polarization in CaBaCo4O7 we mustfirst develop a model for the paramagnetic phase. We turnto the symmetric Heisenberg model to describe the magneticinteractions in CaBaCo4O7. The expression for the magnetic
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Hamiltonian is simplified to
Hm =∑ij
Jij Si · Sj , (1)
where Jij are exchange integrals between cobalt spins Si andSj . For this model to be valid, the magnetic moment magnitudeof any given cobalt spin must be independent of the spinconfiguration. To check this, we performed calculations on60 randomly generated collinear (RGC) spin configurations.Small variations in the calculated magnetic moments of lessthan ±0.05 μB were found for each Co atom, in supportof using a Heisenberg spin model to describe the magneticinteractions in CaBaCo4O7. In addition, we performed non-collinear spin calculations, which yielded results comparableto the collinear case.
In the empirical atomic structure there are 12 nonequivalentnearest-neighbor (NN) Co-Co bonds and 25 next-nearest-neighbor (NNN) bonds. For simplicity, we consider only theNN magnetic exchange interactions, labeled J1 − J12 in Fig. 2,which are expected to be dominant over the NNN interactions.We test this NN approximation by fitting all 12 NN exchangeinteractions to the ab initio-calculated energies of 30 RGC spinconfigurations. The fitted values for the NN exchange integralswere then substituted into Eq. (1), and the energy, EHeisenberg,of another set of 30 RGC spin configurations was calculated.The energies of the second set of 30 spin configurations werethen independently determined through ab initio calculations,Eab initio. Figure 3(a) shows an excellent agreement betweenEHeisenberg and Eab initio, both plotted relative to the energyof the PM state calculated below. The largest discrepancieswere found to be <1 meV per spin, indicating that the NNmodel provides an accurate approximation of the magneticinteractions in CaBaCo4O7.
Within the NN Heisenberg model, the paramagnetic (PM)phase can be described as a disordered phase with a vanishingNN pair correlation function, � = 〈Si · Sj 〉 = 0. Typically,this is achieved by employing the special quasirandomstructure (SQS) method [20], which was initially developedto treat disordered alloy systems and then generalized tostudy paramagnetic phases. Alternatively, one may use a con-ventional magnetic sampling method (MSM) with randomly
-40 -20 0 20 40ΔE
ab-initio [meV / spin]
-40
-20
0
20
40
ΔEH
eise
nber
g [m
eV /
spin
]
0 10 20 30 40 50 60MSM spin configuration
-60
-40
-20
0
20
40
60
ΔE [
meV
/ sp
in]
(a) (b)
FIG. 3. (Color online) (a) Ab initio-calculated energy of 30 RGCspin configurations compared to those calculated using Eq. (1) withpreviously fitted J ’s. The solid red line is a guide to the eye.(b) Ab initio-calculated energy of 60 RGC spin configurations (blackpoints) with the corresponding accumulated average (red solid line)compared to the average energy of the four spin configurations (greendashed line) used to approximate the PM phase.
generated magnetic configurations [21]. The SQS methodinvolves generating a quasirandom magnetic structure in alarge supercell that satisfies � = 0 for spin pairs within des-ignated coordination shells. The requirement for calculationsover large supercells makes this technique well suited to thetreatment of simple magnetic systems, e.g., CrN [21], but notto that of complex systems such as CaBaCo4O7 where thecomputational expense is prohibitively large. Furthermore,while the MSM method can accurately determine the totalenergy of a paramagnetic phase, it is again too computationallyexpensive to be used in calculations of the electric polarizationof complex systems, especially when it is vital to includeatomic relaxation, as described below. Instead, a PM phasemay be modeled by averaging several nonrandom spin configu-rations in such a way that individual NN exchange interactionscancel, i.e., � = ∑
k Si(k) · Sj (k) = 0, where k denotes thespin configuration—a method reliably used to calculate spin-phonon coupling in the PM phase of ZnCr2O4 [22].
In CaBaCo4O7 the PM phase was modeled with fourcollinear spin configurations that we label FM, AFM1, AFM2,AFM3, respectively. FM is a ferromagnetic structure, whilethe other three are antiferromagnetic structures of the type(− − + +), (− + − +), (− + + −) for (Co1,Co2,Co3,Co4),respectively. The values of exchange interaction energies (Hm)for the four spin configurations are given in Table I, whereeach component exactly cancels within the Heisenberg spinmodel approximation, described above. We follow a similarstrategy employed in Ref. [21], based upon the MSM method,to justify the use of this scheme in the case of CaBaCo4O7.In accordance with the MSM method, we generated a largenumber of RGC spin configurations with an average energyrepresentative of that of the disordered state. The individualenergies of the 60 spin configurations relative to the averageenergy of the four spin configuration PM model are shownin Fig. 3(b). The cumulative average is also plotted, whichshows rapid convergence to an energy <2 meV/spin below thatof the PM model. Compared to the fluctuation of individualenergies (as large as ±35 meV/spin), this result demonstratesthat the average energy of the four spin configuration model
TABLE I. Atom pair, bond length (l), and exchange energies of asingle unit cell for each spin configuration, given for all NN exchangeinteractions. Note that in the calculations, the magnetic moments andexchange parameters are normalized to 1.
J Co pair l (A) FM AFM1 AFM2 AFM3
J1 Co1-Co2 3.099 4 4 −4 −4J2 Co1-Co3 3.165 4 −4 4 −4J3 Co1-Co4 3.107 4 −4 −4 4J4 Co1-Co3 3.187 4 −4 4 −4J5 Co1-Co2 3.207 4 4 −4 −4J6 Co1-Co4 3.226 4 −4 −4 4J7 Co2-Co3 3.299 4 −4 −4 4J8 Co2-Co4 3.231 4 −4 4 −4J9 Co2-Co3 3.007 4 −4 −4 4J10 Co2-Co4 3.162 4 −4 4 −4J11 Co3-Co4 3.262 4 4 −4 −4J12 Co3-Co4 3.012 4 4 −4 −4
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R. D. JOHNSON, K. CAO, F. GIUSTINO, AND P. G. RADAELLI PHYSICAL REVIEW B 90, 045129 (2014)
TABLE II. The calculated electric polarization in units mC/m2
of the four spin configurations that average to give the net PMpolarization and that calculated for the experimentally determinedferrimagnetic structure (EFM). For each row, the polarization in theFM state is taken as a reference. �P = P (EFM) − P(PM).
FM AFM1 AFM2 AFM3 PM EFM �P
Empirical crystal structure0 −0.16 0.36 1.06 0.31 0.74 0.43
Atomic position relaxation0 −14.0 −10.4 1.1 −5.8 −0.6 5.2
Full atomic position and lattice relaxation0 −17.71 −21.7 −3.1 −11.2 −5.3 5.9
is a good approximation of the PM phase. We also tried othercombinations with the same cancelation relation as satisfiedby the FM, AFM1, AFM2, and AFM3 spin configurations, andall the calculated average energies were found to be consistent.
It is well established that in typical exchange-striction-typemultiferroics [23,24], the local magnetically induced electricpolarization is proportional to the NN Si · Sj spin interaction.Therefore, in our � = 0 approximation of the PM phase anylocal electric polarization induced through magnetostrictionwill exactly cancel, leaving only that intrinsic to the crystalstructure. This PM phase can therefore be used as an accuratereference for spin polarized calculations of the magneticallyinduced change in electric polarization, �P .
In the following, we calculate and compare the electricpolarization of the PM and EFM structures. We note thatthe effect of spin-orbit coupling was found to be negligibleand so has been omitted from the final results. Initially, weperformed calculations on a fixed, experimentally determinedatomic geometry. The results, given in the first part ofTable II, show only small variations for all spin configurations,with �P approximately 1 mC/m2—one order of magnitudesmaller than that measured by experiment. To again checkthe reliability of our PM phase approximation we returned tothe MSM tests performed above. The cumulative average ofthe electric polarization calculated in the fixed experimentalatomic geometry for all 60 RGC spin configurations showedsimilar behavior to the energies given in Fig. 3, rapidlyconverging to a value consistent with that found for our PMmodel.
In other ab initio studies of multiferroic and magneto-electric materials [25–27], it has been shown that an ionicrelaxation contribution is necessary to account for the fullmagnetoelectric polarization. Therefore, further contributionsto the electric polarization were taken into account by fixingthe experimental lattice parameters and relaxing the ionicpositions. The results are shown in the second part of Table II,where it can be seen that both the relative variations foundacross the model magnetic structures, and the final �P , aresignificantly enhanced. Finally, we allow for magnetic straincoupling by performing full relaxations (both atomic positionsand lattice parameters set free to vary) under each magneticconfiguration. The results, shown in the third part of Table II,give rise to a value of �P comparable with both previouscalculations and experiment. We note that the sign of the
polarization in the EFM phase changes sign upon relaxation,however the relative change, �P , remains positive.
The correct approximation of the PM phase is essential inevaluating �P . To double check our model, we also generateda model PM state with a set of noncollinear spin configurationsthat satisfy the same cancelation relation. In this case, fullyrelaxing the atomic geometry gives �P = 4.6 mC/m2 withU = 4 eV, which is in good agreement with that presentedin Table II. All calculations show that exchange-strictioneffects alone can give rise to a giant enhancement of theelectric polarization of the order 1–6 mC/m2, comparable tothe experimental observations. There remains a discrepancybetween the actual magnitude of calculated and experimentallydetermined change in electric polarization. However, thiseffect is typical in such calculations due to the limitationof current DFT in the treatment of the strong correlationeffects of, in this case, cobalt 3d electrons. Furthermore,the approximation of the paramagnetic phase required forcalculations might reduce �P since the real PM phase iscompletely disordered, while our model is constructed fromordered states.
V. H − T PHASE DIAGRAM
At variance with ferroelectric systems, CaBaCo4O7 ispyroelectric, with no known (or predicted) centrosymmetriccrystal structure present in the phase diagram. Furthermore,one cannot generate a physically viable, model centrosym-metric phase that maintains the chemical connectivity of thepolar crystal structure. As such, it is impossible to directlycalculate the absolute value of the electric polarization in suchcompounds. However we chose to define it, the pyroelectricpolarization cannot be switched with an electric field, althoughone can reverse it by simply rotating the crystal upside down.
The key issue that remains to be addressed is whether �P
is switchable, since a positive answer would indicate a veryunusual kind of spin-assisted ferroelectricity. In multiferroics,the magnetically induced polarization changes sign uponreversing the polarity of the magnetic structure, which musttherefore be acentric. For CaBaCo4O7, reproducing this mech-anism is highly problematic: although the crystal is highlypolar, the cobalt sublattice is almost centrosymmetric and thecobalt magnetic structure is also quasicentrosymmetric, theinverted EFM structure being essentially identical to the EFMstructure. For this very reason, ab intio calculations alone areof limited use in assessing whether �P can switch, sincethere is no obvious way to construct a magnetic domainthat would support a reversed �P . We therefore adopt adifferent approach based upon a phenomenological Landautheory and demonstrate that �P does not switch. Furthermore,this minimal model is found to be sufficient to describethe experimental behavior of CaBaCo4O7 when tuned bytemperature and magnetic field close to Tc. We will employa scalar notation throughout. Extension to tensor notation isstraightforward but does not add to the essential physics of theproblem.
We start by defining a simple Landau free energy for theparamagnetic phase that accounts for its electrical propertiesin the vicinity of the equilibrium pyroelectric structure. We
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CaBaCo4O7: A FERRIMAGNETIC PYROELECTRIC PHYSICAL REVIEW B 90, 045129 (2014)
can write the free energy as
Fp = −α
2P 2 + β
4P 4 − PE, (2)
where α and β are positive constants, which entirely deter-mine the polarizability χ0 and hyperpolarizability h0 of theparamagnetic phase—both measurable quantities:
χ0 = ∂P
∂E
∣∣∣∣E=0
= 1
2α(3)
h0 = 1
2
∂2P
∂E2
∣∣∣∣E=0
= ∓√
9
2χ5
0 β.
Since we are only interested in the behavior close to themagnetic transition, we ignore the temperature dependenciesof α and β, which would give rise to conventional pyroelectriccurrents upon heating. From Eq. (2) we can also extract theequilibrium value of P :
P0 = ±√
α
β. (4)
We stress that P0 is not a measurable polarization and doesnot correspond to a density of dipole moments, Ppyr, estimatedusing any choice of the unit cell. In fact, this would entailextending Eq. (2) much beyond the range in which the quarticapproximation is valid. However, changes in P in the vicinityof the equilibrium position are well defined and would result ina measurable current in the standard experimental setup. Belowwe present an estimation of P0 from ab initio calculations.
We first consider how the Landau free energy in Eq. (2) canbe modified in the magnetically ordered phase. The point groupsymmetry of CaBaCo4O7 changes from mm2.1′ to m′m2′ atthe magnetic ordering transition. Only time-reversal symmetryis broken, with all spatial symmetry operations preserved.The lowest order magnetoelectric coupling invariant in theLandau expansion of the free energy is γPM2, where γ isa coupling constant, P = P0 + �P , and �P is the changein electric polarization upon magnetic ordering as before. M
is the magnetic order parameter, which, although primarilyantiferromagnetic, is coupled with the magnetic field throughthe ferrimagnetic component. γPM2 is time reversal even andparity odd, and as such is allowed prima facie given the polar,paramagnetic parent phase point group. However, the Landaufree energy must also be invariant by any continuous or discretefree-space operator applied to the crystal as a whole. If oneapplies an inversion operator (not part of the crystal symmetry)the magnetic structure remains almost invariant, as explainedabove, while P changes sign. Since this is an approximaterelation, we can only conclude that the γPM2 term mustbe very small. An alternative, and perhaps more intuitiveinterpretation is to consider that CaBaCo4O7 undergoes ahypothetical nonpolar to polar structural phase transition atvery high temperatures. The term γPM2 is not invariant in anyof the hypothetical parent phases and is therefore rigorouslyexcluded.
We proceed to demonstrate that a change in bulk electricpolarization can occur via a magnetoelastic contribution tothe free energy described by the higher order term − c
2M2P 2,which can result in a fractional change �P of the electricpolarization.
The lowest-order, stable Landau expansion of the freeenergy may be written as
F (M,P,B)=F0 − BM + a
2(T − T ∗)M2 + b
4M4 − c
2M2P 2
+ d
6M6 − α
2P 2 + β
4P 4 − PE, (5)
where a, b, d, and T ∗ are constants of the purely magnetic partof the free energy, c is the magnetoelastic coupling constantthat may be positive or negative, and all other constants aredefined such that positive values stabilize the free energy.
In zero applied electric field, two equilibrium conditionsfollow:
∂F
∂M= −B + a(T − T ∗)M + bM3 − cP 2M + dM5 = 0,
(6)and
∂F
∂P= −αP + βP 3 − cM2P = 0. (7)
Substiting Eq. (4) into Eq. (7) and rearranging the termsgives the solution
P 2 = P02 + c
βM2, (8)
where M is found by solving Eq. (6), as described below. Equa-tion (8) contains the essence of the physics of CaBaCo4O7: atthe magnetic ordering temperature, M becomes nonzero, andan additional contribution to the polarization �P develops dueto magnetostriction. The sign of �P in relation to P0 is fixedonce and for all by the sign of the coupling constant c and cannever be switched.
In principle, it is possible to determine the values of α
and β, and therefore derive P0 from ab initio calculations ofthe paramagnetic phase in applied electric field. However, theaforementioned difficulty of performing accurate calculationsin the paramagnetic phase makes this approach impractical.We therefore chose a different method, exploiting the factthat P0 enters as a parameter in the free-energy expansionas a function of �P around the magnetic ground state. Asbefore, all calculations here are performed with a Coulombinteraction energy of U = 4 eV. We first estimate the ab initioatomic structure of the paramagnetic phase (corresponding to�P = 0) by averaging the relaxed atomic structures in thefour spin configurations (FM, AFM1, AFM2, AFM3) withfixed experimental lattice parameters. The ground-state atomicstructure (corresponding to �P = �Pmax) was previouslydetermined by relaxing the atomic positions with the ground-state magnetic structure (EFM). By interpolating betweenthese two extremes we can estimate atomic positions, andtherefore calculate ground-state energies, as a function of�P . In these calculations, it was assumed that M does notvary, a reasonable approximation for values of �P closeto the ground state. These energies were then fitted to theexpression E = E0 + a′(P0 + �P )2 + b′(P0 + �P )4, whichclosely follows the form of our Landau theory.
The results of the calculations are plotted as closedcircles in Fig. 4 as a function of P/�Pmax, together with thefitted expression, plotted as a line. From the fit, we extracta value P0/�Pmax = 26.51. Taking the reported value of
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R. D. JOHNSON, K. CAO, F. GIUSTINO, AND P. G. RADAELLI PHYSICAL REVIEW B 90, 045129 (2014)
0 5 10 15 20 25 30 35P (ΔP normalized to 1)
-320
-310
-300
Ene
rgy
[eV
]
26.5 27 27.5 28 28.5P (ΔP normalized to 1)
Ene
rgy
[eV]
M = 1M = 0
ΔP
FIG. 4. (Color online) Fit of the polarization free energy to P ,determined at points close to the energetic minimum by ab initiocalculations. The energy for M = 0 and M = 1 is shown by red andblack lines, respectively.
�Pmax = 17 mC/m2, this gives P0 � 450 mC/m2. We notethat the calculated EvP curve is symmetrical about the origin(not shown in Fig. 4). Hence, there exists a single energyminima for each ±P0 domain, respectively. The energy neededto reverse �P , i.e., switch between energetic minima, whilepreserving the magnetization is about 411 meV, consistentwith a value of 464 meV from direct ab initio calculations.This result clearly indicates that �P is not switchable for agiven P0 domain.
We now turn to the magnetization. Substituting Eq. (8) intoEq. (6) gives
−B + a(T − Tc)M + bM3 + dM5 = 0, (9)
where Tc = T ∗ + caP0
2 and b = b − c2
β. These expressions
capture two further key aspects of the physics of CaBaCo4O7.Coupling to the pyroelectric polarization present in the param-agnetic phase will increase the magnetic ordering temperature,and furthermore, may induce a first-order (negative b) mag-netic phase transition, as opposed to a second-order transition(positive b) expected in magnetic systems. Magnetizationdata measured parallel to the b axis for B � 0 has beenreproduced from Ref. [7] (Fig. 5), which in the following wetake to represent the thermal evolution of the magnetic-orderparameter. Despite no evidence for magnetic hysteresis atthe phase transition [6], there occurs a sharp jump in themagnetization at Tc—evidence for first-order behavior. Thiswas confirmed by fitting Eq. (9) to the data, in units of emu/g,and with B = 0. The constants b and d were allowed to varyfreely in the fit, with a set to unity having factored out ascaling parameter, ξ , to be determined later. The best fit isshown in Fig. 5, with a = ξ , b = −0.112ξ , d = 0.00385ξ , andTc = 62.0 K. We note that b takes a negative value, indicatingthat the magnetic phase transition is indeed first order due tocoupling to P . This result might be verified experimentally byinvestigating magnetic hysteresis.
Having established the temperature dependence of M ,setting the scaling parameter ξ to 0.4 and the ratio of constantsc and β that couple M and P to c
β= 210, gave the best
10 20 30 40 50 60 70 80Temperature [K]
0
5
10
15
M [
emu
g-1]
FIG. 5. (Color online) The temperature dependence of the mag-netization in zero applied magnetic field. Data points (black circles)have been extracted from Ref. [7]. The fit to the data, described indetail in the text, is shown as a red line.
qualitative agreement with the magnetic field dependenceof the electric polarization (Fig. 6). This result clearlydemonstrates that the experimentally determined H -T phasediagram close to Tc can be explained by our phenomenologicalmodel based solely on magnetoelastic coupling.
Finally, we again consider the energy barrier associatedwith the hypothetical switching of �P . The difference inenergy of two ±�P ferroelectric domains can be written as
�E = − c
2M2(P0 + �P )2 + c
2M2(P0 − �P )2
= −2cP0�PM2, (10)
i.e., there is a large energy barrier to switching �P thatscales with P0, consistent with the results of the calculationsdescribed in the above.
-10 -5 0 5 10B [T]
450
455
460
465
P [
mC
m-2
]
62.5 K
65 K
68 K
75 K
60 K
FIG. 6. (Color online) The magnetic field dependence on electricpolarization at five temperatures close to the phase transition. Datapoints (circles) have been extracted from Ref. [7]. Simulations of thedata are shown as solid lines, which are colored according to therespective temperatures. Note that P0 has been added throughout.
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CaBaCo4O7: A FERRIMAGNETIC PYROELECTRIC PHYSICAL REVIEW B 90, 045129 (2014)
VI. CONCLUSIONS
To summarize, we have demonstrated through ab initiocalculations that in CaBaCo4O7, the giant change in electricpolarization observed at the phase transition from pyroelectricparamagnetic to pyroelectric ferrimagnetic can arise as a resultof exchange-striction effects alone. The change in polarizationwas found to be an enhancement, i.e., in the same direction asthe polarization of the paramagnetic phase, and not switchable.Furthermore, such large changes are only predicted when one
considers the relaxation of ionic positions. Our ab initio resultsare supported by Landau theory, which predicts the correctmagnetoelectric behavior apparent close to Tc.
ACKNOWLEDGMENT
This work was funded by EPSRC Grant No. EP/J003557/1,entitled “New Concepts in Multiferroics and Magneto-electrics.”
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