Nonanalyticity of magnetization curve for Dzialoshinsky–Moriya antiferromagnet: LaMnO[sub 3] as a model magnet

Nonanalyticity of magnetization curve for Dzialoshinsky–Moriya antiferromagnet: LaMnO 3 as a model magnet F. Bolzoni and R. Cabassi Citation: Journal of Applied Physics 103, 063905 (2008); doi: 10.1063/1.2840125 View online: http://dx.doi.org/10.1063/1.2840125 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/103/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Strain driven anisotropic magnetoresistance in antiferromagnetic La0.4Sr0.6MnO3 Appl. Phys. Lett. 105, 052401 (2014); 10.1063/1.4892420 Magnetization reversal mechanism in La 0.67 Sr 0.33 MnO 3 thin films on NdGaO 3 substrates J. Appl. Phys. 107, 013904 (2010); 10.1063/1.3273409 Magnetic ordering anisotropy in epitaxial orthorhombic multiferroic YMnO 3 films J. Appl. Phys. 104, 103912 (2008); 10.1063/1.3021112 Anomalous magnetic ordering in b -axis-oriented orthorhombic Ho Mn O 3 thin films Appl. Phys. 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Bolzonia� and R. Cabassi Istituto IMEM-CNR, Parco Area delle Scienze 37/A, Parma, Italy �Received 12 July 2007; accepted 9 December 2007; published online 18 March 2008� An extensive study of the energy expression for an antiferromagnetic system with antisymmetric Dzialoshinsky–Moriya �DM� interaction and uniaxial magnetocrystalline anisotropy is presented and the main features are shown as a function of magnetic field H applied along the different crystallographic directions. The spin-flop transition is analyzed, showing that for the DM coupling constant D�Dcrit the transition changes from first to second order. Analytical results are followed by numerical computer simulations, both for single-crystal and for polycrystal with isotropic easy axis distribution. The first and second derivatives of the average magnetization with respect to H are analyzed as well, for their insight potentiality: the first derivative is found to be useful for a direct measurement of the spin-flop transition field in polycrystalline samples, while the shape of the second derivative allows us to distinguish the case D�0 from the case D=0. The obtained results are then applied to the manganite perovskite LaMnO3, attaining satisfactory interpretation of experimental data and suggestions for further investigation techniques. © 2008 American Institute of Physics. �DOI: 10.1063/1.2840125� I. INTRODUCTION Renewed interest in the interpretation of weak ferromag- netism observed in antiferromagnetic crystals by means of Dzialoshinsky–Moriya �DM� interaction1–3 has recently de- veloped, as, for example, in the case of the weak ferro- magnetism in cuprates such as La2CuO4 �Ref. 4� or La2−x−yEuySrxCuO4, 5 of spin chain compounds as �Cu,Zn�2V2O7, 6 or of the antiferromagnetic molecule-based magnet Mn�N�CN2��2 �Ref. 7� and the orthorhombic crystal YVO3. 8 Several authors9–11 suggested the DM interaction as a possible mechanism of spin-order-driven ferroelectricity in multiferroic perovskites. A prototypical example of a compound with DM inter- action is the manganese perovskite LaMnO3, 12,13 as shown by studies14–19 on antiferromagnetic single crystals and poly- crystals. The compounds of the LaMnO3 family are very impor- tant from a technological and applicative point of view. Man- ganese oxide compounds with perovskite structure show co- lossal magnetoresistance, which is associated to orbital ordering and spin correlation. The orbital ordering gives rise to anisotropic electron-transfer interaction, yielding a com- plex spin-orbital coupled state. LaMnO3 is a typical example, in which the alternate ordering of �3x2−r2� and �3y2−r2� orbitals causes the ferromagnetic spin coupling in parallel planes. The resulting magnetic structure, driven by the or- bital ordering, is called A-type, i.e., the interaction is antifer- romagnetic between adjacent planes and ferromagnetic within the plane. The interaction complexity is the origin of the antisymmetric DM energy term. There are many theoretical studies on DM interaction, but few numerical simulations of experiments are available. In the following we try to address this by analyzing the en- ergy of a two-sublattice system with DM interaction and magnetocrystalline anisotropy in order to give the resulting magnetization behavior both in the case of single crystal and polycrystalline aggregate. II. THE MODEL The energy of a Heisenberg ferromagnet with DM inter- action and magnetocrystalline anisotropy is commonly writ- ten in terms of the spin Hamiltonian, E = − K� i=1 N �Si z�2 − � i=1 N Si · h + 2� i�j JijSi · S j + � i�j Dij · Si � S j , �1� where the single terms represent the anisotropy, Zeeman, ex- change, and DM energy, respectively, and N is the number of magnetic ions per volume unit, K is the single ion anisotropy constant, S are the local spin vectors, and the algebraic signs of Jij depend on the ferromagnetic versus antiferromagnetic character of magnetic interactions. If we restrict the analysis to a two-sublattice model with mean molecular field at tem- perature T=0 K, the energy expression becomes E = K�sin2 �A + sin2 �B� − �MA + MB� · H + J�MA · MB� + D · MA � MB, �2� where MA and MB are the magnetization vectors of the two sublattices A and B, respectively. Figure 1 shows the vector and angle definition, where the Z-axis has been chosen as the easy magnetization direction. We also fix the DM vector D along the Y-axis as appropriate to LaMnO3 �see below�, and we assume that the two sublattices are fully magnetized, so that �MA�= �MB�=M =Ms /2, with Ms the saturation magneti- zation of the material. The reduced field in Eq. �1� is ha�Electronic mail: [email protected]. JOURNAL OF APPLIED PHYSICS 103, 063905 �2008� 0021-8979/2008/103�6�/063905/9/$23.00 © 2008 American Institute of Physics103, 063905-1 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.127.200.132 On: Wed, 10 Dec 2014 16:37:41 http://dx.doi.org/10.1063/1.2840125 http://dx.doi.org/10.1063/1.2840125 http://dx.doi.org/10.1063/1.2840125 =g�BH /kB, where g is the electron gyromagnetic factor, kB is the Boltzmann constant, and �B is the Bohr magnetron, so the relationships among the parameters in Eqs. �1� and �2� are easily obtained: M = g 2 �BNS , K = KN 2 S2kB, J = JAF 4z g2 kB N�B 2 , D = D2z g2 kB N�B 2 , �3� where z is the number of antiferromagnetically coupled next- neighbors of each site. The study of the system can be carried out by means of the analysis of first and second derivatives of Eq. �2� with respect to the sublattice angles �A, �A, �B, and �B for fixed angles �H, �H defining the orientation of H. In the following, this problem will be addressed by both analytical treatment and numerical computer simulations, depending on the com- plexity of the particular cases. The behavior of a generic compound with respect to its D value is particularly ana- lyzed, and in view of application to the case of LaMnO3 compound �developed in the last part of the paper, Sec. IV B 3�, the parameters for numerical simulations are chosen of the same order of magnitude as those reported in literature14,18,20,21 for LaMnO3. III. ANALYTICAL RESULTS A. D=0 If the DM vector D is equal to zero, the symmetry prop- erties of the system allow exact solutions for Eq. �2� to be obtained. Separate treatment for the case of H along the Z-axis and H perpendicular to the Z-axis will be necessary. 1. H � Z-axis The system is symmetric for rotations around the Z-axis, so that we can fix �A=�H=0 and �B=� in Eq. �2� and the equilibrium condition can be derived, obtaining the expres- sions of �E /��A and �E /��B for �H=0 and equating them to zero, which leads to the following equation: sin �A�1 + 2KMH cos �A = sin �B�1 + 2KMHcos �B . �4� Equation �4� has two trivial solutions, the first one with �A =�B=0 and corresponding energy E1�H�=JM2−2MH, and the second one with �A=0, �B=�, and corresponding energy E2=−JM 2. In addition to the trivial solutions, Eq. �4� also has a field-dependent solution �A=�B=��H�. The function ��H� can be obtained considering that the equilibrium con- dition becomes �E/���H� = MH sin ��H� + 2�− JM2 + K�sin ��H�cos ��H� = 0, �5� which yields the following expression for the angle of the field-dependent solution, cos ��H� = MH 2�JM2 − K� , �6� with the corresponding field-dependent energy and magnetization E3�H� = �MH�2 − JM2 + K + 2K − JM2, m�H� = 2M cos ��H� = M2H JM2 − K . �7� In order to track the possible appearance of a spin-flop transition, we start with H=0. In this situation, the minimum energy solution is the trivial one with angles �A=0 and �B =�, and corresponding energy E=E1=−JM 2. When H is switched on, both E1�H� and E3�H� change as functions of H, but while E1�H� always decreases with increasing H, E3�H� decreases with increasing H only if K�JM2. Therefore, if K�JM2 we can only have a spin-flop be- tween the two trivial solutions, i.e., a direct jump from zero magnetization to saturation, when E1�H�=E2, corresponding to a spin-flop field Hsf =JM. On the other hand, if K�JM2 one can algebraically verify that it is always E3�H��E1�H�, and we could have a spin-flop when E3�H�=E2, corresponding to a spin-flop field FIG. 1. Vector and angle definitions. 063905-2 F. Bolzoni and R. Cabassi J. Appl. Phys. 103, 063905 �2008� [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.127.200.132 On: Wed, 10 Dec 2014 16:37:41 Hsf = 2 K�JM2 − K� M . �8� However, we must remember that Eq. �6� is physically mean- ingful only when cos ��1, and applying this condition to Eq. �8�, we are lead to the more strict constraint K �JM2 /2 for the occurrence of this spin-flop, leaving the spin-flop to saturation at Hsf=JM for K�JM 2 /2. When this constrain is satisfied, the jump is from zero magnetization to a magnetization given by Eqs. �7� and �8�, Msf = 2M KJM2 − K . �9� The saturation field is obtained by the condition cos ��Hsz� =1 applied to Eq. �6�, which yields Hsz = 2�JM2 − K� M , �10� while the susceptibility in the Z-direction for H�Hsf is �see Eq. �7� and Fig. 2� // = M 2/�JM2 − K� = �2M − Msf�/�Hsz − Hsf� . �11� In literature other expressions for Hsf can be found. Here we report for convenience some of them:22 Hsf = �2K/� � − ���1/2 �12� or Hsf = � �2Hm + HK�HK1 − �/ � � 1/2 �2HmHK�1/2, �13� where HK=K /Ms, Hm is the molecular field, and � and � are the easy axis and perpendicular susceptibility, respectively. 2. H� Z-axis We will now calculate the saturation field Hsx when H is perpendicular to the Z-axis, e.g., along the X-axis. The equi- librium condition can be derived, obtaining the expressions of �E /��A and �E /��B from Eq. �2� for �H=� /2 and �A =�B=0, and equating them to zero, which leads to the fol- lowing equation: cos �A�1 − 2KMH sin �A = − cos �B�1 − 2KMH sin �B . �14� Equation �14� has three solutions: �i� �A=�B=� /2, �ii� �A =�B=a sin�MH /2K�, and �iii� �B=�−�A=��H�. Since cal- culation of second derivatives gives �2E /��A��B=JM2�0 for both solutions �i� and �ii�, the only solution correspond- ing to an energy minimum is solution �iii�. Substitution of the condition �B=�−�A=��H� into the equation �E /��A=0 yields then �E/���H� = − cos ��H��MH − �2JM2 + K�sin ��H�� = 0, �15� which gives the expression for the dependence of the angle ��H� with respect to the applied field H, sin ��H� = MH 2�JM2 + K� . �16� Since at saturation one has ��H�=� /2, the saturation field H=Hsx is given by Hsx = 2�JM2 + K� M = HEx + HA, �17� with HEx=2JM. If we remember that M is the magnetization of a single sublattice, we see from Eq. �16� that the magne- tization curve follows the law m�H� = 2M sin ��H� = M2H JM2 + K , �18� so that the susceptibility for applied field H in whatever di- rection lying in the X-Y plane turns out to be � = M2 JM2 + K = 2M/�HEx + HA� . �19� B. DÅ0 While a full description of the whole magnetization curve for D�0 can be obtained by means of numerical methods only, an analytical treatment is still possible for the remanent magnetization with H=0 and for the saturation point with H applied along the X-direction or the Y-direction. 1. Case H=0 For symmetry reasons, the two vectors MA and MB will lay in the X-Z plane, i.e., �A=�B=0, and the equilibrium condition �E /��A=�E /��B=0 has two solutions: �i� �B =� /2+�A and �ii� �B=�−�A. Since solution �i� gives �2E /��A��B=−MD2�0, we identify �ii� as the minimum en- ergy solution, and we can express the energy as E = 2K sin2 �B − JM 2 cos�2�B� − DM2 sin�2�B� . �20� The lowest energy solution of Eq. �20� corresponds to FIG. 2. Magnetization field dependence for H applied along the Z-direction and in the X-Y plane. The parameters used are MA=MB=304 emu /cm3, K=8.7�106 erg /cm3, J=1266, and D=0. The characteristic expressions are reported for convenience. 063905-3 F. Bolzoni and R. Cabassi J. Appl. Phys. 103, 063905 �2008� [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.127.200.132 On: Wed, 10 Dec 2014 16:37:41 tan �B = − �JM2 + K� + �JM2 + K�2 + D2M4 DM2 . �21� Therefore the remanent magnetization Mr is along the X-direction and its value is Mr = 2M tan �B, �22� with tan �B given by Eq. �21�. The modulus of the DM vec- tor D when the Mr value is known is given by D = �JM2 + K� M2�1 − �Mr/2M�2� Mr M . �23� In the special case K=0, one has tan �B = − J D + J2 D2 + 1 or tan�2�B� = D J . �24� 2. Saturation for H � X-axis We will now calculate the saturation field Hs with H applied along the X-direction. In this case �B=�−�A for symmetry, and the first derivative of the total energy in Eq. �2� gives the equilibrium condition dE/d�B = �2JM2 + 4K�sin�2�B� − 2DM2 cos�2�B� − 2HM cos �B. �25� Since near saturation �B→� /2, we can develop the expres- sion in power series of =� /2−�B and obtain at first order Hs = 2�JM2 + K� M + DM , �26� which clearly shows that Hs → →0 �, i.e., the magnetization is asymptotic to saturation. 3. Saturation for H � Y-axis We now analyze the case with magnetic field H applied along the Y-direction. For H close to the saturation point �A=�B=� and �B=�−�A, so that the first derivative of the total energy in Eq. �2� gives the equilibrium condition: �E/��B = 4K sin �B cos �B + 2JM2 sin�2�B� + 2DM2 cos�2�B�cos � − 2HM cos �B sin � = 0, �E/�� = − DM2 sin�2�B�sin � − 2HM sin �B cos � = 0. �27� Since �B→� /2 and �→� /2 when saturation is approached, we can develop the expression in power series of �=� /2 −�, =� /2−�B and obtain at first order 4K + 4JM2 − 2DM2� − 2HM = 0, − DM − H� = 0, �28� and it is easy to see that for field H�DM, then �� can be neglected and we obtain Hs = 2�JM2 + K� M , �29� which is exactly the expression found for the case D=0. 4. Saturation for H � Z-axis For the case with H applied in the Z-direction, the satu- ration field will be calculated numerically in Sec. IV. C. The fixed point An interesting feature that can be shown for H on the X-Z plane is that the magnetization curve for the D=0 case shares a fixed point with all the D�0 curves. In order to prove it for the case of H along the Z-axis, we compare the derivative of the total energy for D=0 and for D�0 when H�Hsf, remembering that �A=�B=0 and �A=−�B=−� as shown above, and we obtain dED/d� = 4DM2�cos2 � − sin2 �� = 0, �30� from which it follows that �=� /4. The magnetic field cor- responding to the angle �=� /4 is obtained, solving the de- rivative of the total energy with respect to H, �dE/d���=�/4 = �4K sin � cos � − 4JM2 sin � cos � + 2HM sin ���=�/4 = 0, �31� which gives HZ � = 2�JM2 − K�/M = �HEx − HA�/ 2, MZ � = 2M . �32� A similar calculation can be done for H applied along the X-direction, in this case �A=�B=0, �B=�−�A. The obtained values are HX � = 2�K + JM2�/M = �HEx + HA�/ 2, MX � = 2M . �33� The points PZ�HZ � ,MZ �� and PX�HX � ,MX �� are independent of the value of D, in other words the families of curves obtained by varying the modulus of the vector D have a common fixed point. Computer simulations �see inset of Fig. 5� show that this property holds also for whatever H-direction laying in the X-Z plane. IV. COMPUTER SIMULATIONS A. Single crystal In order to find the minimum of the energy surface for a generic parameter set we resort to numerical computer meth- ods, viz., to the Broyden–Fletcher–Goldfarb–Shannon �BFGS� code,23 and study the above equations as functions of the various involved parameters. We start analyzing the magnetization curves for the case D=0, applying the field in the X-Y plane and in the Z-direction. The results are shown in Fig. 2. The magnetization curve in the X-Y plane follows a straight line up to saturation, while for applied field H along the Z-direction, a spin-flop transition is observed at a field Hsf, and for higher fields a linear behavior is recovered. 063905-4 F. Bolzoni and R. Cabassi J. Appl. Phys. 103, 063905 �2008� [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.127.200.132 On: Wed, 10 Dec 2014 16:37:41 The value of Hsf depends on the parameters as asserted by Eq. �8�. It is interesting to observe �Fig. 3� the variation of the spin-flop transition at Hsf due to the tilting of the mag- netic field H with respect to the Z-axis in the X-Z plane. We note that the transition occurs only in a limited range of values for the tilting angle �H of the field direction, in agree- ment with Rohrer,24 while the corresponding spin-flop field Hsf increases a little, i.e., Hsf=56 Oe or 0.24% for the whole spin-flop transition range of the example. According to the literature,24 the spin-flop occurs for tilting angle values such that tan��H� � K JM2 − K , �34� which for small angles can be approximated by �H � 56.2K/�JM2� = 56.2�HA/HEx� �35� degrees. For the parameters used in the example of Fig. 3, the maximum tilting angle to see the transition turns out to be �H�4.5°, in agreement with Eq. �34�, which gives �H =4.58°. The amplitude of the magnetization discontinuity at Hsf decreases with increasing �H, until the curve becomes continuous, starting from the angle given by Eq. �34�. After the spin-flop transition the magnetization increases linearly, with saturation field depending on the H-direction and fall- ing into the range �Hsz ,Hsx�. It is also interesting to study the variation of the spin- flop transition field Hsf for different values of the DM vector modulus D= �D�. The examples in Fig. 4 show that the tran- sition field Hsf decreases with increasing D, while the mag- netization discontinuity at the transition decreases in ampli- tude and the magnetization curve acquires an upward curvature for H�Hsf. The transition can be both of first or second order, in agreement with the analysis already devel- oped in Ref. 25, as shown in the inset of Fig. 4. An experi- mental example of second-order spin-flop transition is given by the case of SmFeO3. 26 In our notation the critical value Dcrit, above which the transition becomes of second order, transforms as Dcrit = 18 M3 K3 J . �36� It is evident that Hsf decreases by decreasing the anisotropy constant K, and for K=0 the transition occurs at Hsf=0. The figure shows also the fixed point PZ�HZ � ,MZ ��, with coordi- nates H� and M� given by Eqs. �32�, which is common to all the magnetization curves obtained by variation of D and leaving unchanged the other parameters. Figure 5 shows examples of magnetization curves for the case in which the magnetic field is applied along the X-direction. The remanence and the curvature increase with increasing D, the value of the saturation field is clearly dis- cernible for the case D=0 only, while for D�0 an asymptotic behavior is present in agreement with Eq. �26�. The variation of the fixed point H� for different H-directions in the X-Z plane is displayed inside the inset. FIG. 3. Magnetization field dependence for H applied along different direc- tions �H with respect to the Z-axis. The parameters used are MA=MB =304 emu /cm3, K=8.7�106 erg /cm3, J=1266, and D=0. The inset shows the zoom around spin-flop transitions at Hsf. FIG. 4. Magnetization field dependence for H applied along the Z-axis direction, for different modulus values of the DM vector D applied along the Y-direction. The parameters used are MA=MB=304 emu /cm3, K=8.7 �106 erg /cm3, and J=1266. The inset shows two curves with K=1.7 �106 erg /cm3 and different D values, exhibiting the crossing from first- to second-order transition. FIG. 5. Magnetization field dependence for H applied along the X-axis direction, for different modulus values of the DM vector D applied along the Y-direction. The parameters used are MA=MB=304 emu /cm3, K=8.7 �106 erg /cm3, and J=1266. In the inset the value of the magic field H� function of the H angle with respect to the Z-direction is given. 063905-5 F. Bolzoni and R. Cabassi J. Appl. Phys. 103, 063905 �2008� [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.127.200.132 On: Wed, 10 Dec 2014 16:37:41 The above simulations in the X-Z plane were carried out for K�0, i.e., the Z-axis is the easy direction for what con- cerns the anisotropy, but what happens if K�0? In this case we turn from an easy axis to an easy plane, namely, the X-Y plane. According to the K�R transformation,27 the easy plane simulation gives the same magnetization curve for the case of the easy axis, putting R=−K in the energy expres- sion. In other words, applying the magnetic field in the di- rection �H in the X-Z plane for a given parameter set in the energy expression, we obtain the same magnetization curve as applying the field in the complementary direction � /2 −�H but changing the sign of K. The K�R transformation is no longer valid for directions out of the X-Z plane. B. Polycrystal We now focus on the magnetization curves of a poly- crystalline specimen. First we consider our sample as an ag- gregate of noninteracting, single-phase grains randomly ori- ented. The resulting magnetization curve is obtained by summation of all the different curves, corresponding to every possible grain orientation with respect to the direction of the applied magnetic field H. For an isotropic distribution �M�H,�H� = � M��H,�H�d�H � d�H , �37� where sin �H, cos �H define the direction of H with respect to the grain crystallographic axes, d�H is the elementary solid angle, and M��H ,�H� is the magnetization along the direc- tion of H. 1. H=0 When H=0, an analytical result is still easily available, since each grain magnetization is along its X-axis and is given by Eq. �22�. Taking into account the symmetry prop- erties of the system, which allow limitation of the integration angles up to � /2, Eq. �37� becomes �M�H=0�� = � 0 �/2 �2M tan �B sin ��cos � sin �d�d� � 0 �/2 sin �d�d� = Mr/2. �38� We can see therefore that the remanent magnetization of an- isotropic polycrystal with DM interaction is one half of the remanence of the single crystal with H applied in the easy direction. The analysis of the polycrystal with DM interac- tion D=0 gives zero remanence for H=0. For H�0 no analytical solutions are found and com- puter simulations are needed. 2. D=0 The case D=0 corresponds to conventional antiferro- magnetism, wherein the system symmetry lowers the dimen- sionality of the integrals in Eq. �37� to the only angle �H. The magnetization and susceptibility results for an isotropic poly- crystal are shown in Fig. 6, where the following features are noteworthy: �i� for fields H�Hsf the magnetization is always not null and the susceptibility exhibits an upward curvature, reaching a sharp peak close to H�Hsf, which results from the main contribution of grains having easy magnetization axes near the H-direction; �ii� in the intermediate field range the magnetization follows a linear behavior; �iii� when ap- proaching saturation a downward curvature in the magneti- zation appears for Hsz�H�Hsx, with two more singularities in the susceptibility at H=Hsx and H=Hsz: this behavior re- sults from the summing up of the single grain curves having increasing �decreasing� saturation fields �susceptibility dis- continuity� in this region, as evident from comparison with the single curves plotted in Fig. 3. Extrapolation of the initial magnetization slope would give an overestimated value for the saturation field. We suggest to the reader that a pulsed field technique would be a useful tool28,29 for the detection of such singularities. FIG. 6. Magnetization field dependence and first derivative of magnetization with respect to field for a polycrystalline sample with D=0. The parameters used are MA=MB=304 emu /cm3, K=8.7�106 erg /cm3, and J=1266. FIG. 7. Magnetization field dependence for H applied along different direc- tions. The parameters used are: MA=MB=319 emu /cm3, K=8.7 �106 erg /cm3, J=1165, and D=110. In the inset the low field values and selected experimental points of LaMnO3 taken from Ref. 14 are given. 063905-6 F. Bolzoni and R. Cabassi J. Appl. Phys. 103, 063905 �2008� [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.127.200.132 On: Wed, 10 Dec 2014 16:37:41 3. DÅ0: LaMnO3 We have finally carried out numerical computer simula- tions for the LaMnO3 compound �S=2, antiferromagnetic coordination number z=2�, in order to reproduce the pub- lished experimental data and, hopefully, to make predictions regarding the appearance of singularities in both single- crystal and polycrystalline samples. In LaMnO3 the D vec- tor couples Mn magnetic ions lying on the adjacent antiferromagnetic-interacting sublattice planes of the A-type magnetic structure, and its orientation may be determined applying symmetry considerations3 to the crystallographic structure described in Fig. 6 of Ref. 15. The DM interaction arises because the tilting of the oxygen octahedra removes the center of inversion between the Mn ion pairs, and the presence of a mirror plane between adjacent Mn ions re- quires that D lie parallel to the sublattice planes. In order to apply the results of the two-sublattice model, which in prin- ciple would not strictly require the presence of sublattice magnetic planes, we need to assign the XYZ-axes to the LaMnO3 crystallographic axes. For this purpose, start con- sidering that the ab crystallographic planes, as shown in Ref. 14, contain both D and the magnetocrystalline anisotropy TABLE I. Summary of relationships for spin-flop transition, saturation point, and fixed point in a uniaxial single crystal with an easy magnetization axis along the Z-axis.a D=0 H� Z-axis H� Z-axis K�JM2 /2 K�JM2 /2 Hs 2�JM 2+K� M =HEx+HA 2�JM2−K� M =HEx−HA JM M 2 �JM2+K� = 2MHEx+HA �2M−Msf� �Hs−Hsf� = M 2 JM2−K Hsf 2 KJ− � KM �2= 4KJ−HA2 JM Msf HsfM 2 �JM2−K� =2M K JM2−K 2M Spin-flop condition �H�56.2� KJM2 � tan��H�� K JM2−K D=Dy �0 H� X-axis H� Y-axis H� Z-axis Hs � 2�JM 2+K� M Spin-flop condition D� 18 M3 K3 J H� 2 JM2+KM = HEx+HA 2 2 JM2−KM = HEx−HA 2 M� 2M 2M D �JM 2+K� M2�1−�Mr/2M�2� Mr M aHs: saturation field; Hsf: spin-flop field; Msf: spin-flop magnetization; H � ,M�: fixed point coordinates; HEx =2JM; HA=2K /M; MR: remanent magnetization; and saturation magnetization Ms=2M. FIG. 8. Magnetization field dependence and first derivative of magnetization with respect to field for a LaMnO3 polycrystalline sample. The parameters used are MA=MB=319 emu /cm3, K=8.7�106 erg /cm3, D=110, and J =1165. FIG. 9. Second derivative d2M /dH2 vs applied field H of polycrystalline isotropic samples. The parameters used are MA=MB=319 emu /cm3, K =8.7�106 erg /cm3, and J=1165. Panel �a� D=0, panel �b� D=110 corre- sponding to LaMnO3 polycrystal. 063905-7 F. Bolzoni and R. Cabassi J. Appl. Phys. 103, 063905 �2008� [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.127.200.132 On: Wed, 10 Dec 2014 16:37:41 vector K; therefore, the ab planes correspond to X-Z planes of Fig. 1. In absence of D, the spins would be constrained along the easy direction, i.e., the b-axis, and MA and MB would be along the Z-axis; therefore, a and b correspond to Y and Z, respectively. The introduction of DM interaction pushes the spins out from the b-axis on the bc plane, result- ing in a magnetic remanence along the c-axis, i.e., MA and MB out from the Z-axis on the X-Z plane with remanence along the X-axis. The XYZ coordinate system translates into cab according to the notation of Ref. 14, and to bac accord- ing to the notation of Refs. 15 and 21. Neutron-scatter- ing measurements18 give K=1.92�0.10 K; JFM = −9.6�0.48 K and JAF=6.7�0.24 K for the ferromagnetic intraplane and antiferromagnetic interplane exchange cou- pling constant, respectively, while remanent magnetization perpendicular to the sublattice planes14 Mr=0.18�B / ion and Eq. �23� as expressed in the kB=1 unit system �see Table II give D=1.2 K, which is of the same order of magnitude of experimental estimates for perovskite manganites from other authors.9,14,20,21 These data together with the relationship19,30 g /g�D /2J also allow the estimation of g�2.1. These val- ues can be converted into the two-sublattice model by means of Eqs. �3�, and the declared experimental uncertainty bars allow enough freedom for a choice of numerical simulation parameters which will yield a good best-fit of magnetization curves of Ref. 14, namely, M =319 emu /cm3, K=8.7 �106 erg /cm3, J=1165, and D=110. In Fig. 7 we simulate the single-crystal magnetization curves for different field directions, from which the effect of DM contribution to remanence in the X-direction is clear. The inset of Fig. 7 shows the expanded scale near the origin together with selected experimental points from Fig. 1 of Ref. 14. The simulations are very good, with the exception of the Z-direction for fields below the spin-flop transition; this small discrepancy could be ascribed to the approximation given in our model �the � is neglected�, or, more reasonably, to sample misalignment and/or trace of ferromagnetic clus- ters due to oxygen fluctuation at the sample boundary, the latter hypothesis being corroborated by the zero field remanence. Computer simulations for polycrystals are shown in Fig. 8, where one can see that the main differences with respect to the case D=0 �see Fig. 6� are a smoothing of the suscepti- bility singularities in the saturation region and the appear- ance of a small remanence of �14 emu /cm3 at H=0, in agreement with Eq. �38�, but very difficult to appreciate experimentally. An analysis of the effect of the DM interaction on the shape of the second derivative d2M /dH2 is shown in Fig. 9. Panel �a� shows the case D=0, with a first negative jump corresponding to Hsz=HEx−HA and a second positive jump corresponding Hsx=HEx+HA. Panel �b� shows the case of D=110, with a negative peak corresponding to the dM /dH inflection point �see Fig. 8�, originating from the contribution of grains having a Y-axis parallel to the H-direction �see Eq. �29� and Fig. 7�. The different shape of the two peaks allows us, in principle, to discriminate between the cases D=0 and D�0, provided that high enough intensities of the magnetic field are accessible by the experimental setup. Although the experimental peaks could be rounded by a local demagnetiz- ing effect, grain boundaries, and domain walls, nevertheless the difference between the two peaks could remain evident. TABLE II. Summary of relationships for spin-flop transition, saturation point, and fixed point in a uniaxial single crystal with easy magnetization axis along the Z-axis expressed in terms of the reduced magnetic field h in the kB=1 unit system. a D=0 h� Z-axis h� Z-axis K�Jz K�Jz hs 2S�2zJ+K� 2S�2zJ−K� 2zJS hsf 2S K�2zJ−K� 2zJS msf K2zJ−K 1 Spin-flop condition �h�56.2� K 2zJ � tan��h�� K 2zJ−K D=Dy �0 h� X-axis h� Y-axis h� Z-axis hs � 2S�2zJ+K� Spin-flop condition D� 18z K 3 2zJ h� 2S�2zJ+K� 2S�2zJ−K� m� 1 2 1 2 D 2zJ+K z mR 1−mR 2 ahs: saturation field; hsf: spin-flop field; msf=Msf /Ms: reduced spin-flop magnetization; h�, m�=M� /Ms: reduced fixed point coordinates; mR=MR /Ms: reduced remanent magnetization; and z: coordination number. 063905-8 F. Bolzoni and R. Cabassi J. Appl. Phys. 103, 063905 �2008� [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 129.127.200.132 On: Wed, 10 Dec 2014 16:37:41 V. CONCLUSIONS We have studied the magnetization curves of an antifer- romagnet with an antisymmetric Dzialoshinsky–Moriya in- teraction and single-ion uniaxial magnetocrystalline aniso- tropy by means of analytic treatment and numerical computer simulations, considering both single-crystal and polycrystalline specimens. The most useful among the de- rived analytical relationships for the single-crystal case are collected for convenience in Tables I and II, expressed in the forms they assume for the two-sublattice model and for the spin Hamiltonian, respectively. The results for the polycrys- tal case predict noteworthy singularities in the susceptibility and inspection of the second derivative d2M /dH2, providing a method to experimentally discriminate the occurrence of measurable Dzialoshinsky–Moriya interaction in a specimen under examination. 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