: A ferrimagnetic pyroelectric
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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
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  andTbMn2O5 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 Cm2 observedin CaMn7O12 below 90 K ), is two orders of magnitudesmaller than that of a good ferroelectric. Recently, a muchlarger spin-assisted change in polarization of 17 000Cm2was 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 ofP 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 norP are switchable.
II. CRYSTAL AND MAGNETIC STRUCTURES ANDELECTRICAL PROPERTIES
The crystal structure of CaBaCo4O7, shown in Fig. 1, wasfound to adopt the polar space groupPbn21 at all temperaturesbelow 400 K . The structure comprises interleaved kagomeand triangular layers of CoO4 tetrahedra, which are buckledwith respect to a high symmetry, high temperature trigonalpolar phase (space group P31c) 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 CoO4buckling. This structural distortion is reported to be largestin CaBaCo4O7 , 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. . 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 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. .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 80Cm2 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 000Cm2 . However, no switching behaviorwas reported for the single-crystal sample.
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).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 forU = 4 eV, as CaBaCo4O7is known to be an insulator. We also performed calculationswith U = 6 eV, which produced very similar results. Theprojector augmented-wave (PAW)  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 than107 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/), wheref 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 referencecentrosymmetric 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
CaBaCo4O7: A FERRIMAGNETIC PYROELECTRIC PHYSICAL REVIEW B 90, 045129 (2014)
Hamiltonian is simplified to
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 B 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
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 0.43Atomic position relaxation0 14.0 10.4 1.1 5.8 0.6 5.2Full 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/m2one 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 , 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 resul...