# Roles of Bond Alternation in Magnetic Phase Diagram of RMnO3

\abstIn order to investigate the nature of the antiferromagnetic structures in perovskite MnO, we study a Heisenberg - model with bond alternation using analytical and numerical approaches. The magnetic phase diagram which includes incommensurate spiral states and commensurate collinear states is reproduced. We discuss that the magnetic structure with spin configuration (E-type structure) and the ferroelectricity emerge cooperatively to stabilize this phase. Magnetoelastic couplings are crucial to understand the magnetic and electric phase diagram of MnO. \kword Heisenberg model, bond alternation, E-type structure, spin spiral state, A-type structure, MnO, multiferroics

Strongly correlated electron systems show various phase transitions involving complex order parameters, in general. Typically, in perovskite manganites MnO, phases with spin, charge, orbital and lattice degrees of freedom emerges and competes with each other, through control of carrier concentrations and ionic radius of ions.[1] In these Mn ions, 3d electrons exhibit varieties of phases in nearly identical systems with a slight change in a few set of parameters.

One of the recent topics in these MnO is the presence of multiferroic properties. For compounds with La, Nd, Sm, …, a spin A-type antiferromagnetic (A) phase is observed. In this phase, spins order ferromagnetically in the plane and antiferromagnetically along axis, i.e., the magnetic ordering vector is in the orthorhombic lattice with GdFeO-type distortions. However, when one substitutes ions with smaller ones Tb, Dy, …, an incommensurate spin spiral (ICS) phase emerges accompanied by ferroelectricity.[2, 3] In this phase, magnetic propagation vectors are along the axis, where . The origin of the ferroelectricity is the inverse Dzyaloshinsky-Moriya (DM) mechanism for the ICS structures.[4, 5, 6] Here, a local electric polarization is generated through the antisymmetric magnetoelectric (ME) coupling , where is the spin direction at -th site and denotes the unit vector connecting and sites. One of the experimental evidences is that the reorientations of from to by chemical substitutions or application of magnetic fields are always accompanied by to cycloidal-plane flops.[7, 8]

Among various attempts to understand phase diagrams of MnO,[3, 9] finite temperature phase diagrams including - and -cycloid phases have only been reproduced successfully by a model described as follows, so far: The model is a classical Heisenberg model defined on an three-dimensional orthorhombic lattice with nearest neighbor (n.n.) ferromagnetic exchanges , and next nearest neighbor (n.n.n.) antiferromagnetic exchanges , as well as single ion anisotropies and DM interactions.[10, 11] Models based on the - Heisenberg model also explain spectra for magnons and electromagnons in these compounds.[12, 13, 14] Construction of a realistic and accessible microscopic model is quite appreciable from the viewpoint of comprehension of mechanisms as well as predictions and materials designs for novel phenomena.

For smaller ions at =Ho, Tm, Lu, …, where is expected to be larger, spins exhibit an E-type antiferromagnetic (E) phase with spin collinear structure along the -axis with the magnetic propagation vector . This phase also exhibits ferroelectricity and thus is multiferroic. It is considered that the ferroelectricity is driven by the E phase orderings through the symmetric ME coupling of the magnetoelastic origin,[15, 16] as illustrated in Fig. 1.

The Heisenberg model with - interactions alone, however, does not reproduce the E phase, as recently emphasized by Kaplan.[17] A possible way to stabilize the E phase is to introduce a uniaxial anisotropy which enhances collinear behaviors. The authors have studied the Heisenberg model with anisotropies and DM interactions, which successfully reproduces the A-ICS transition, in the large region.[11] Within the realistic range of parameters, however, we fail to observe the E phase. Another candidate is the biquadratic interaction which also enhances collinearity.[17] At this point, however, it is not clear weather such an interaction dominantly acts to stabilize the E phase in MnO.

In this paper, we introduce an alternative model to investigate the spin structures of MnO. Namely, we study a classical Heisenberg model with bond alternations as a result of the magnetoelastic couplings mentioned above. We clarify that the model can reproduce the A-ICS-E phase transitions in MnO, and discuss the nature of the ME phase diagram. Since MnO is a rare system which exhibits both symmetric and antisymmetric ME couplings E and ICS phases, respectively, it is also quite interesting to study the phase transition across these phases in order to make further comprehension for the ME effects in these compounds.

We study a classical Heisenberg model with spin exchange bonds depicted in Fig. 2 as
{align}
H &= - ∑_¡i,j¿ J_ij S_i ⋅S_j
-J_2 ∑_¡i,j¿’ S_i ⋅S_j

& -J_c ∑_¡i,j¿” S_i ⋅S_j.
Here, the first summation is taken over n.n. bonds on the plane, and
takes the alternating values along -axis in the form

(1) |

as depicted in Fig. 2. Bond alternation parameter is restricted within . The second summation is for n.n.n. bonds along -axis. We take (ferromagnetic) and (antiferromagnetic). The third term of the Hamiltonian is the inter-plane antiferromagnetic couplings along -axis. At , merely create staggered stacking of -plane spin structure along -axis, and thus irrelevant for the phase diagrams with respect to and . For example, it is determined automatically that a ferromagnetic alignment of spins in the plane stack antiferromagnetically form the A-phase. Therefore, we may focus on spin structures within the plane.

In the absence of the bond alternation , the model at gives an ordinary spiral state with , where describes the angle of the spin within the spiral plane. Here, describes the position of the spin along the propagation vector . The rotation angle is determined by , where

(2) |

and the component of the propagation vector is given by .

At , staggered modulation of is introduced. Then we introduce a variational spin state with uniform and staggered component for the rotation angle

(3) |

Total exchange energy per site scaled by is described by ,
which is calculated as
{align}
ε(θ,δ)
&= -(1+Δ)cos(θ-δ)
- (1-Δ) cos(θ+δ)

& + γcos2θ- γ_c,
where .
Minimization of with respect to and
through leads to
{align}
tanδ&= Δtanθ,

cosθ&=1-Δ24γ2- Δ21-Δ2.
From Eq. (3)
we have in the limit
, which implies an A-phase, irrespective of . Similarly, at
we have which makes a collinear E phase.
Critical value of for the A-ICS boundary
is given from Eq. (3) at as

(4) |

whereas at we have the ICS-E boundary

(5) |

Since , E phase is stabilized only in the presence of the bond alternation . In Fig. 3 we show the phase diagram.

In order to justify the above analytical discussion, we perform the Monte-Carlo calculations at low temperatures. In the above, we have assumed the spin configuration of Eq. (3). On the other hand, the Monte-Carlo calculation does not restrict the spin configuration, and thus provides unbiased results. The calculation indeed confirms that only the three magnetic phases, i.e., A, ICS, and E phases are possible within our model so that the assumption of Eq. (3) is justified. As its consequence, the analytical results are precisely reproduced by the calculation.

In our numerical calculations, we also add the , which makes magnetization along the axis hard. Because of this term, the spins in the ICS phase rotate in the plane. This term makes the calculations stable by suppressing thermal fluctuations of the spiral plane without affecting the A-ICS-E transitions at . We take =1 and =0.2 in the calculation. We analyze this model using the Monte-Carlo technique for systems with 48486 sites with periodic boundaries.

The calculation successfully reproduces the analytically predicted phase diagram of Fig. 3. The circle, triangle and square symbols in Fig. 3 denote the points at which the A, E, and ICS spin structures are respectively obtained in the Monte-Carlo calculation at =0.2.

These magnetic structures are assigned from the peak position of the spin correlation functions at . We also measure the spin-helicity correlation function to identity the phases. Here, the local spin-helicity vector is defined as where and point the n.n. sites along [110] (pseudo-cubic ) and [-110] (pseudo-cubic ) directions, respectively. In spiral structures, the rotating spins give rise to a ferro-arrangement of the spin helicities, which results in a peak of the spin-helicity correlation at =0, while in collinear phases, the spin-helicity correlation has no peak structure.

Let us now show some details of the calculation. In Fig. 4, we show the temperature dependence of specific heat , as well as the spin-helicity vector where (=, and ) denotes the component of the spin-helicity vector, at (a) (, )=(0.4, 0.2), (b) (, )=(0.4, 0.6), and (c) (, )=(0.4, 1.2) where the ground states are predicted to be (a) A-type, (b) ICS, and (c) E-type, respectively —– see also the phase diagram in Fig. 3.

In all cases, we can see a single phase transition from paramagnetic (PM) phase at high temperatures to each ordered phase at low temperatures, at which exhibits a sharp peak. We also confirm that the choice of gives a sufficiently low-temperature state.

In the present calculation, the spins in the ICS phase rotate in the plane since we incorporate the hard-axis type spin anisotropy along the axis in the Hamiltonian. Under this circumstance, has a large value, while and are almost zero. We can indeed see that starts increasing at the transition to the ICS phase in Fig. 4(b). On the other hand, as shown in Figs. 4(a) and 4(c), component of the helicity is as small as other components in the collinear spin phases.

In Fig. 5, we show the calculated dependence of the momentum as well as the spin-helicity for (a) =0.6, (b) =0.4, and (c) =0.2. The data for the spin-helicity are obtained by extrapolating the calculated temperature profile of to . Those data as functions of consistently exhibit A-ICS-E phase transitions, at least qualitatively.

Furthermore, let us compare these data with the variational spin structure given in Eq. (3). The spin structure gives and , where and are determined by Eqs. (3) and (3). As plotted in Fig. 5, the Monte-Carlo data for and agree well with the analytical results given by solid and dashed lines, respectively.

So far we have confirmed that the spin structure given in Eq. (3) as well as the A-ICS-E phase transition derived from that is quite valid. Then we proceed to consider the case that the bond alternation is driven by lattice distortions to minimize the total energy. We assume that a lattice distortion creates the bond alternation with being the coupling constant. Total energy is given by a sum of the spin exchange energy and an elastic energy , or equivalently

(6) |

Here, is the dimensionless spin-lattice coupling constant. Hereafter, we refer to the model as Peierls - model.

The spin exchange energy is derived by applying Eqs. (3) and (3) to Eq. (3). In the A phase at , the minimized energy is derived from as

(7) |

while in the E phase at , we have so that

(8) |

In the ICS phase at , we have

(9) |

Bond alternation is determined by minimizing with respect to within the range . The result in the region can be summarized as follows. In Fig. 6, we schematically depict as a function of for various . We observe a transition from ICS phase to E phase at a critical coupling given by

(10) |

Here, have degenerate energy minimums at (ICS region) and at (E region). As is changed across , we have a jump in the value of for the energy minimum and thus the transition is first ordered. Furthermore, there exists a characteristic coupling

(11) |

where E phase has a local minimum in its energy with respect to at . This implies an existence of a metastable E state at . Similarly, we also have

(12) |

where an ICS state at is metastable at . Therefore, in the case , we may have coexistence of the ICS and the E phases at .

Similarly, in the case , 2nd order transition between the A and the ICS phases is observed at , while the metastable E state exist at . The A states is always metastable at in this case. The forms for and coincide with those for the previous case given in Eqs. (10) and (11), respectively. The phase diagram is summarized in Fig. 6.

Let us now compare these results with the magneto-electric properties in MnO. In these compounds, commensurate E phase is always accompanied with ferroelectricity, which is consistent with our result that E phase is stabilized only at . Although it has previously been discussed that ferroelectricity is triggered by magnetic E phase through lattice distortions, our result suggests that the magnetic E phase and ferroelectricity cooperatively emerge to stabilize themselves. Therefore, Peierls-type spin-lattice couplings are essentially important to understand the electric as well as the magnetic phase diagram of MnO. Recent report shows that, in MnO there exists a region of possible coexistence for ICS and E phases.[18] The Peierls model discussed here indeed shows such a coexistence at around the 1st-order transition points between ICS and E phases. Experimental data in further details should give crucial tests for our present results. Theoretical approaches based on more realistic models including anisotropies, DM and biquadratic interactions should also be important to understand the whole phase diagrams of MnO in detail.

The author would like to thank S. Miyahara for stimulating discussions, as well as S. Ishiwata, Y. Taguchi and Y. Tokura for suggestions from experimental points of view. This work was partially supported by Grant-in-Aid for Scientific Research as well as High Tech Research Center Project from the MEXT, Japan.

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