Multiferroic response to magnetic field in orthorhombic manganites
M. H. Qin,1Y. M. Tao,2S. Dong,3H. B. Zhao,1X. S. Gao,1and J.-M. Liu1,2,4,a兲
1School of Physics, South China Normal University, Guangzhou 510006, People’s Republic of China
2Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, People’s Republic of China
3Department of Physics, Southeast University, Nanjing 211189, People’s Republic of China
4International Center for Materials Physics, Chinese Academy of Science, Shenyang 110016, People’s Republic of China
共Received 5 January 2011; accepted 18 February 2011; published online 10 March 2011兲
The magnetoelectric coupling in Eu0.55Y0.45MnO3 is studied based on a microscopic spin model which includes the superexchange interaction, the single-ion anisotropy, the Dzyaloshinskii–Moriya interaction, and the cubic anisotropy. Our Monte Carlo simulation reproduces the experimentally observed multiferroic response to magnetic field B. It is demonstrated that the magnetic field can control the multiferroic behaviors by modulating the spin arrangements, leading to various flops of electric polarization. In addition, an interesting state in which both the electric polarizations along the a-axis and c-axis are activated under high B is predicted and discussed. © 2011 American Institute of Physics.关doi:10.1063/1.3565241兴
Multiferroics are attracting continuous attentions due to the interesting physics and potential applications.1In the past few years, multiferroicity has been found in a number of systems, such as spiral magnets, orthorhombically distorted perovskite manganites RMnO3 关R= Tb, Dy, Eu1−xYx, etc., crystal structure on the ab-plane is shown in Fig. 1共a兲兴,2 Ni3V2O8,3 MnWO4,4 and a conical magnet CoCr2O4.5 The ferroelectricity in these materials is induced by spiral spin order through the inverse Dzyaloshinskii–Moriya 共DM兲 mechanism 共alternatively the spin current model兲.6 In the spin current scenario, adjacent two spins共Si,Sj兲can generate a local polarization Pij⬀−eij⫻共Si⫻Sj兲witheijthe unit vec- tor connecting the two neighboring sites. Thus, polarization Pin the ab-plane cycloidal spin共ab-CS兲phase with propa- gation vector along the b-axis is induced along the a-axis while in the bc-plane cycloidal spin 共bc-CS兲 phase it is in- duced along the c-axis, as illustrated in Figs.1共b兲 and1共c兲.
RMnO3 offers the capability for magnetic control of ferro- electricity via the strong magnetoelectric共ME兲coupling. For TbMnO3 and DyMnO3, application of a magnetic field B along theb-axis共fieldBb兲flopsP from thec-axis共polariza- tion Pc兲to the a-axis 共polarization Pa兲.7 Several theoretical works in order to understand the origin for such multiferroic response and the ME coupling inRMnO3are available.8,9
Most recently, a microscopic spin model 共Mochizuki–
Furukawa model兲which includes the superexchange interac- tion, the single-ion anisotropy共SIA兲, the DM interaction, and the cubic anisotropy, was proposed and reproduced the phase diagrams of RMnO3 in the plane of temperature 共T兲 versus R-site ionic radius.9It was demonstrated that the orthorhom- bic lattice distortion mainly controlled by the R-ionic radius tunes the SIA and the DM interaction energies and in turn determines the competition between the ab-CS phase and bc-CS phase. This leads to the flop of P from thea-axis to thec-axis with reducedR-ionic radius. Subsequently, several other phenomena have been well explained based on the same or similar models.10–14 For example, the phase dia-
grams of TbMnO3and DyMnO3under magnetic fieldBhave been reproduced.11
On the other hand, several multiferroic states and strong ME effects were revealed inRMnO3共R= Eu1−xYx兲by tuning theR site ionic size.15–17 In Eu0.55Y0.45MnO3, polarizationP flops from Pa toPcwhenBis applied along thea-axis共Ba兲 while it flops fromPctoPawithBalong thec-axis共Bc兲 关see Figs.1共d兲–1共f兲兴.16These ME phenomena may help us to un- derstand the origin of the multiferroic response toB, due to the fact that this system is free from the influence of mag- netic moments ofRions. Furthermore, a multiferroic state in which P is induced via the spin exchange striction mecha- nism was observed in EuMnO3 under a field up to 30 Tesla 共T兲.10It is thus reserved to question some unrevealed phases in Eu0.55Y0.45MnO3 in the highBrange.
a兲Electronic mail: [email protected].
FIG. 1. 共Color online兲 共a兲Crystal structure共ab-plane兲ofRMnO3. The in- ducedPand spin-helicity vectorh=⌺ijSi⫻Sjin theab-CS共b兲andbc-CS 共c兲states. Experimentally obtainedB-Tphase diagrams for共d兲B储a,共e兲B储b, and共f兲B储care reproduced from Ref.16.
APPLIED PHYSICS LETTERS98, 102510共2011兲
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In this work, we study the Mochizuki–Furukawa model with Mn spin S= 2 on a cubic lattice.9The Hamiltonian can be written as H=Hex+HSIA+HDM+Hcub+HZeeman. The first term Hex=⌺具i,j典Jij⫻共Si·Sj兲 denotes the spin exchange inter- actions, where Jab= −0.8 and Jb= 0.8 are the coupling con- stants in the Mn–Mn bonds on the ab plane 关Fig. 1共a兲兴, Jc
= 1.25 is the antiferromagnetic 共AFM兲 exchange in the Mn–Mn bonds along thec-axis. Here the energy unit is mil- lielectron volt. The second term is the SIA, which consists of two parts as HSIA=D·⌺iSii+E·⌺i共−1兲ix+iy·共Si
i −Sii兲 withD
= 0.25, E= 0.30. Here, i, i, i are the tilted local axes at- tached to theith MnO6octahedron, as clearly given in Ref.9.
For their direction vectors, we use the experimental data of EuMnO3.18 The third termHDM represents the DM interac- tions expressed by HDM=⌺具i,j典di,j⫻共Si⫻Sj兲. Here the DM vectors di,j are determined by five DM parameters, 共␣ab,ab,␥ab兲=共0.10, 0.10, 0.14兲 and 共␣c,c兲=共0.30, 0.30兲.
The fourth termHcub=A·⌺i共Sxi4+Syi4+Szi4兲/S共S+ 1兲represents the cubic anisotropy with coupling constant A= 0.0162. The last term HZeeman= −BBg⌺iSi stands for the Zeeman cou- pling. Here g= 2 is the Lande factor, and B is the Bohr magneton.
Our Monte Carlo simulation is performed on a 36⫻36
⫻6 cubic lattice with periodic boundary conditions using the standard Metropolis algorithm and temperature exchange method.19–21 The selected parameters reproduce well the magnetic states of Eu0.55Y0.45MnO3 in the absence of B.
With decreasing T, the system successively exhibits the paramagnetic共PM兲phase, the sinusoidal collinear antiferro- magnetic 共sc-AFM兲 order with Mn spins along the b-axis, the bc-CS phase, and the ab-CS phase. The specific heat C共T兲=共具H2典−具H典2兲/NkBT2 and spin-helicity vector h␥共T兲
=具兩⌺iSi⫻Si+b兩典/NS2 共␥=a,b,c兲 are calculated to determine the transition points and spin structures, hereNis the number of Mn ions, kB is the Boltzmann constant, and the brackets denote thermal and configuration averaging. The spin and spin-helicity correlation functions in the momentum space,
⌽␥共k,T兲=⌺ij具S␥i·S␥j典exp关ik·共ri−rj兲兴/N2 and ⌿␥共k,T兲
=⌺ij具h␥i
b·h␥bj典exp关ik·共ri−rj兲兴/N2for␥=a,b,care also calcu- lated in order to characterize the spin structures.
The calculated phase diagram in the B-T plane with B储a-axis is shown in Fig. 2共a兲, which reproduces the ob- servedPflop from thea-axis to thec-axis, in associated with the flop of the spiral-spin plane from the ab-plane to the bc-plane. In the low field range 共Ba= 3.0 T兲, the simulated C共T兲 curve shows three specific-heat peaks, indicating the successive three phase transitions with decreasing T, as shown in Fig. 2共b兲. The first one is the transition from the PM phase to the sc-AFM phase. When T falls down to the second transition point, spin-helicity vector ha共T兲 increases whilehb共T兲andhc共T兲remain small, fingering a transition to the bc-CS order. At the third transition, hc共T兲 steeply in- creases, accompanied with the sudden drop of ha共T兲, as a sign of spin spiral flop from thebc-plane to theab-plane. In addition, the third transition point shifts toward the low-T side asBaincreases, indicating that the spiral-plane gradually flops from the bc-plane to the ab-plane at low T. As Ba increases up to 5 T and above, the system exhibits only two transitions at low T. For instance, at Ba= 6 T, C共T兲 shows two peaks and ha共T兲 is small over the whole T-range, as shown in Fig. 2共c兲. This simply indicates that the bc-CS
order component if any is completely suppressed and the ab-CS order occupies the whole T-range below the second transition point, in agreement with experiments. It is well known that for an isotropic AFM or spiral spin system, field B tends to align the spins in perpendicular toB. The flop of spiral spin order into the bc-plane from the ab plane under highBa becomes physically reasonable. Surely, such spiral- plane flop must be accompanied with the reorientation of P.
The effect of B on the multiferroicity revealed above also applies to the case with B储c-axis 共Bc兲. The calculated Bc-T phase diagram is displayed in Fig.3共a兲. The magnetic field applied along the c-axis suppresses the bc-CS order while it enhances theab-CS order, resulting in the flop ofP from thec-axis to thea-axis. The ab-CS order overwhelms the bc-CS order atBc⬃3.0 T, coinciding with experiments.
For details, the calculated C共T兲andh␥共T兲for Bc= 6.0 T are shown in Fig. 3共b兲, indicating that the ab-plane spiral spin order is completely suppressed.
Subsequently, we look at the case of B储b-axis 共Bb兲. A prominent feature is that the magnetic phases in Eu0.55Y0.45MnO3 show little dependence on Bb up to Bb
= 7.0 T, which is also reproduced in our simulation. At low field, the three magnetic transitions remain essentially un- changed and in fact no changes in the transition points 共not shown here兲. One notes that for RMnO3, theac-CS order is unfavorable due to the fact that it cannot be stabilized by the DM interaction and the SIA. A lowBbcannot flip the spins into theac-plane from the initialab-plane andbc-plane, sug- gesting the robustness of the ab-CS or bc-CS orders. For high field case, as an example, we present the simulatedC共T兲 andh␥共T兲atBb= 9 T in Fig.3共c兲. The first and second tran- sitions remain roughly unchanged while the third transition shifts toward the low-T side. In addition, below the second transition point, both ha共T兲 andhc共T兲 have large values, in- dicating the coexistence of the bc-CS order and the ab-CS
FIG. 2. 共Color online兲 共a兲 Calculated Ba-T phase diagram of Eu0.55Y0.45MnO3. Here, the high-temperature PM phase is denoted by PM, accompanied by the paraelectric phase PE, CS order denotes cycloidal spin order, sc-AFM stands for sinusoidal collinear antiferromagnetic order. Spe- cific heatC共T兲and spin-helicity vectorh␥共T兲 共␥=a,b,c兲as a function ofT under variousBa:共b兲Ba= 3.0 T, and共c兲Ba= 6.0 T.
102510-2 Qinet al. Appl. Phys. Lett.98, 102510共2011兲
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order. In this case, the spins have both a-axis components andc-axis components. At the same time, the DM interaction with vectors on the in-plane Mn–O–Mn bonds stabilizes the ab-CS order while the DM interaction with vectors on the out-of-plane bonds stabilizes the bc-CS order. This leads to the simultaneous appearance of these two types of spiral spin orders. Around the third transition point, ha共T兲 suddenly drops to nearly zero, indicating the disappearance of the bc-CS order. AsBbincreases up to 12 T, the system exhibits only two transitions, as shown in Fig. 3共d兲. The former is a transition from the PM phase into the sc-AFM phase, and the latter is a transition into a magnetic phase in which the ab-CS order and the bc-CS order coexist. According to the spin-current model, both Pa andPcwill be observed in the state with the coexistingab-CS andbc-CS orders.
The Mochizuki–Furukawa model, proposed in the clas- sical Heisenberg spin framework, shows surprisingly good consistency with experiments. In particular, our simulations reveal the coexistence of the ab-CS andbc-CS orders under high magnetic field along the b-axis, implying the coexist- ence of the a-axis and c-axis polarization components. In fact, the corresponding magnetic structures are also con- firmed in our calculated spin-helicity correlations ⌿␥. Fig- ures 4共a兲 and 4共b兲 show the simulated ⌿␥共␥=a,c兲 under Bb= 15 T atT= 5 K. Both⌿aand⌿chave their peak loca- tions at k=共0 , 0 , 0兲, indicating the coexistence of the ab-CS and bc-CS orders. The simulated ⌽␥ also characterize this spin structures. However, the predicted phase was not ob- served in earlier experiments in which the high field phase diagrams of Eu1−xYxMnO3共x= 0 and 0.4兲underBbwere stud- ied in pulsed magnetic fields. This inconsistence between the theory and experiment may be due to the fact that the actual
system is hard to be relaxed toward the equilibrium state at lowTbecause of the high potential barrier between the equi- librium state and the quasi-static state under high B. Of course, this issue remains to be checked further.
This work was supported by the Natural Science Foundation of China 共Grant Nos. 50832002, 51072061, 51031004, and 11004027兲, the National Key Projects for Basic Research of China 共Grant No. 2011CB922101兲, and the China Postdoctoral Science Foundation 共Grant No.
20100480768兲.
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C共T兲 and spin-helicity vector h␥共T兲 as a function of T under: 共b兲 Bc
= 1.3 T,共c兲Bb= 9.0 T, and共d兲Bb= 12 T.
FIG. 4. 共Color online兲Calculated spin-helicity correlation functions共a兲⌿a
and共b兲⌿cunderBb= 15 T atT= 5 K.
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