Coexistence of spiral magnetic state and weak ferromagnetism in a multiferroic, cross-controlled by external magnetic and electric fields

Multiferroic (MF) crystals are of interest due to the coexistence of magnetism and ferroelectricity [1, 2], different phase states and phase transitions, including the known first-order transition of an antiferromagnet–insulator into a ferromagnetic 'metal glass' [3, 4]. The transition into a 'magnetic glass' depends strongly on external effects such as mechanical tension and magnetic fields [3, 5]. In several perovskite-type multiferroics, such as BiFeO₃ and BiMnO₃, electrical polarization (EP) occurs due to a symmetry change inherent in these crystals which takes place well above the Neel point T_N. However, EP in new (Eu,Y)MnO₃ multiferroics is induced by a helical spin ordering which takes place only below T_N. A rather large external magnetic field changes the spin configuration in such MFs, which radically affects their ferroelectric properties [6–8].

The perovskite type Eu_1−xY_xMnO₃ crystals have an orthorhombic structure and exhibit different magnetic and dielectric properties. They become nearly collinear antiferromagnets (AFM) and paraelectrics at T < T_N ≈ 50 K, ferroelectrics (FE) at T < T_FE ≈ 30 K (the polarization P is along the x axis), and weak ferromagnets (WFM) (m is along the c (z) axis) with a simultaneous appearance of a spiral structure (along the y axis) at T < T_WF ≈ 20 K [9]. An increase of yttrium concentration x leads to the evolution of a low-temperature magnetic state from a WFM to a spiral one and is accompanied by a sharp change of the volume ferroelectric polarization [10, 11].

Previously it was phenomenologically shown [12] that in an applied magnetic field a decrease of spontaneous EP with a simultaneous increase of the magnetic moment of the WFM was obligatory in Eu_1−xY_xMnO₃ perovskites, due to the existence of the Dzyaloshinskii–Moriya (DM) effect [13, 14].

For the lanthanide Eu_0.75Y_0.25MnO₃ it was experimentally shown [15] that the magnetoelectric effect occurs under an applied magnetic field of a specific (threshold) value. In this case, the spiral magnetic structure (SMS) and WFM are formed simultaneously as a result of first-order phase transition at a critical temperature below the point T_FE of the ferroelectric transition. The temperature range of this transition is broadened with increasing magnetic field. Within this temperature range the magnetic moment rapidly rises and the correction to EP is negative, also rapidly increasing in absolute value with increasing magnetic field above its threshold [15]. Such a behavior is explained in [12] on the basis of the phenomenological expansion of free energy in small corrections to the spontaneous polarization p (the p_x component) and in a small wavevector q (the q_y component) of the spiral structure (cycloid).

The aim of this study is the phenomenological explanation of the thermodynamic state of perovskites as multiferroics (in applied magnetic and electric fields) with due regard for the temperature dependence of sublattice magnetizations of the antiferromagnet. The role of spin–orbital interaction in the occurrence of WFM and its influence on the spontaneous polarization is clarified. The magnetoelectric effect in the absence of spontaneous polarization below T_N is also discussed. The influence of an applied electric field in different directions on the described effects is considered.

Let us give an elementary discussion of the reasons for the above phenomena. According to the Moriya microscopic approach [13], the energy of spin interaction of i and i + 1 ions, bilinear in the spin–orbital interaction and exchange interaction, in the second-order of perturbation theory is written as

$$\begin{eqnarray} {E}_{1,2}&=&\mathrm{i}\zeta {U}^{-1}\left \{ \mathop{\sum }\limits _{m}\frac{(n\vert {\vec{l}}_{i}\vert m){b}_{n{n}^{\prime}}{b}_{m n}}{{\varepsilon }_{n}-{\varepsilon }_{m}}\right .\nonumber\\ &&-\left .~\mathop{\sum }\limits _{{m}^{\prime}}\frac{({n}^{\prime}\vert {\vec{l}}_{i+1}\vert {m}^{\prime}){b}_{n{n}^{\prime}}{b}_{{m}^{\prime}n}}{{\varepsilon }_{{n}^{\prime}}-{\varepsilon }_{{m}^{\prime}}}\right \} \left [{\vec{S}}_{i}{\vec{S}}_{i+1}\right ]. \end{eqnarray} \tag{ 1 }$$

Here we consider the localized electron orbital states of magnetic ions and spin–orbital interaction as a disturbance, and the orbital states are supposed to be nondegenerate in the crystal field. The superexchange integral b_nn'b_m'n/U is a matrix in the quantum number space of i and i + 1 ions (n and n' are the basic orbital states of i and i + 1 ions, m and m' are the excited states of these ions), b_nm is the transfer integral, and U is the energy required for the electron transfer from an ion to its nearest neighbor to create a polar state. The expression in the curly brackets in equation (1) before $[{\vec{S}}_{i}{\vec{S}}_{i+1}]$ is the DM vector D [15], which depends on the quantum states of the ${\vec{l}}_{i}$ and ${\vec{l}}_{i+1}$ orbital moments.

In the perovskite-type structure the D vector depends on the displacement $\vec{u}$ of oxygen atom, for example, along the x axis, and on the distance ${\vec{r}}_{i,i+1}$ between the adjacent magnetic ions, for example, along the y axis. The D vector may be written as $\vec{D}={\lambda }^{\prime}[\vec{u}{\vec{r}}_{i,i+1}]$, or phenomenologically $\vec{D}=\lambda [\vec{p}\vec{\nabla }]$, where the electric polarization p is proportional to the displacement u (p ∝ u). Here and below $\vec{u}$ and u are identical $(\mathbf{u}\equiv \vec{u})$. The dimensionalities of the pseudo-scalar constants λ' and λ are different, λ = (e₁[e₂ e₃])ζ, (e_i are the unit orts, ζ is a scalar). In a nonuniform spin structure D is constant due to invariability of $\vec{u}$ and ${\vec{r}}_{i,i+1}$ values.

In the crystal unit cell the lengths and directions of interatomic bonds can vary with temperature T and applied magnetic field H. Accordingly, the quantum states and the transfer integrals in equation (1) are varied with an external action, which leads to rotations of the spins ${\vec{S}}_{i}$ and ${\vec{S}}_{i+1}$.

There are several positions of oxygen, europium, and yttrium ions in the Eu_1−xY_xMnO₃ crystal, and the geometry of their bonds with Mn atoms is different. The change of these bonds in the (yz) plane and along the x axis may be different with temperature variation. As was suggested in [16, 17], bending of the rare earth–oxygen planes probably occurs with a temperature change in the presence of nonmagnetic Y ions (differing in size from Eu ions) located between the MnO₆ octahedra (figure 1). In this case, the O atoms are 'ejected' from the (yz) plane at the expense of a corresponding reorientation of the atomic orbitals and magnetic moments [4]. This changes the energy of the exchange interaction between the magnetic ions.

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As shown in [18], the ferroelectric moment in perovskite-type manganites (in particular, in LuMnO₃) is not associated with atomic displacements in the (yz) planes, but mainly depends on the lanthanide–oxygen bond lengths along the x axis. Because of atomic displacements the magnetic properties and polar state of the crystal proved to be mutually dependent. The more strongly these planes are bent, the more this ejection, i.e., the greater the displacement u of the O atoms and the related electric polarization p occurring due to the DM effect (p_x component). In this case, the wavenumber q_y of the SMS increases as a result of atomic displacements and orbital transformation.

Let us consider the AFM spin ordering of Mn ions in 1 and 2 sublattices along the z axis (figure 2). Small canting of spins S in the (xz) and (yz) planes in adjacent sites can be in the same direction along the z axis. The triangle formed by the moments of two sublattices M₁,M₂ and the AFM moment $\mathbf{L}={\vec{M}}_{1}-{\vec{M}}_{2}$ rotates around the z axis in the (xy) plane at large angles. The values of M₁ and M₂ are the thermodynamic averages of individual moments $\vec{\mu }~(\mu =g\sqrt{S(S+1)})$ over the 1 and 2 sublattices. As the moment L rotates, the spin cycloid along the y axis retains an equal direction of spin canting in the chains along the z axis (as shown in figure 2). In this case, the magnetization $\mathbf{m}={\vec{M}}_{1}+{\vec{M}}_{2}$ (WFM) appears along the z axis, which increases with applying magnetic field along the same axis. Thus, the uniform canting of magnetic moments M₁ and M₂ from the (xy) plane to the z axis leads to the uniform displacement $\vec{u}$ of oxygen atoms along the x axis. Due to the spin–orbital interaction, this spin canting results in the deformations and deviations of electronic orbitals which move oxygen atoms. The greater the magnetization m, the greater the displacements $\vec{u}$. As noted above, the cycloid of magnetic moments is characterized by the wavevector q along the y axis.

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Such a synchronous atomic rearrangement is, actually, the magnetoelectric effect, since the displacements u (associated with m) modify the existing spontaneous polarization P by a small value p. As shown below, in this case, P and p have opposite signs (p_x ∝ ζq_yM²). A decrease of the total polarization due to spin–orbital interaction, resulting in a correction p, is associated with the modulation of the spin structure q. An increase of the wavenumber q increases the polarization p, and, hence, the displacement u, i.e., the canting of magnetic moments. This means that an increase of the wavenumber q_y increases the spontaneous magnetization m, i.e., the weak magnetism is enhanced.

When the applied magnetic field H increases, the augmentation of magnetization $\mathbf{m}={\vec{M}}_{1}+{\vec{M}}_{2}$ is caused by the magnetic susceptibility χ_m, which increases by a value δχ_m due to the spin modulation. As shown below, this augmentation is associated with the wavevector q. If the magnetoelectric effect was not caused by the SMS, the spontaneous magnetization of a weak ferromagnetic m along the z axis would be caused by the Dzyaloshinskii field H_D:

$$\begin{eqnarray} &&m={\chi }_{m}{H}_{\mathrm{D}},\tqs {H}_{\mathrm{D}}=\left \vert \eta \right \vert L. \end{eqnarray} \tag{ 2 }$$

Here η is a small value of relativistic nature, caused by the spin–orbital interaction¹, the vector η (η ∝ u ∝ p) is parallel to the x axis, and the AFM vector $\mathbf{L}={\vec{M}}_{1}-{\vec{M}}_{2}$ is along the y axis (figure 2).

This is associated with the presence in the free energy of a constant correction in the form

$$\begin{eqnarray} &&\vec{\eta }\left [\vec{m}\vec{L}\right ]={\eta }_{x}{m}_{z}{L}_{y}, \end{eqnarray} \tag{ 3 }$$

which is a scalar because L changes sign under inversion of all coordinates, including the coordinates of neighboring spins [19]. When applying an external magnetic field H along the z axis, the effective field is H_eff = H_D + H. Weak ferromagnetism occurs in the field H_eff, and the corresponding magnetic moment is m = χ_mH_eff, if L remains constant in the absence of spin modulation.

The properties of such systems near the transition to the paramagnetic phase (in an external magnetic field) can be described in terms of the Landau theory [19]. Thermodynamic averages m,L and M are found by minimizing the free energy and, in general, should be considered along with the macroscopic parameters q and p. However, in perovskite Eu_1−xY_xMnO₃, the transition temperature in the cycloid MF state with q ≠ 0 and p ≠ 0 is significantly less than the temperature of transition to the AFM state. Then, describing MF, we assume that the M values in all the phenomenological expressions are averaged parameters almost independent of temperature.

The WFM moment m remains constant when a spiral twist of M_1,2 and L moments appears in the plane (xy). In the presence of the SMS, the L_y component is a sign-alternating value, and the above correction of equation (3) is absent, since the averaging over the coordinate y at constant $\vec{\eta }$ and $\vec{m}$ values gives $\langle \vec{\eta }[\vec{m}\vec{L}]\rangle =0$, where $\vec{L}\equiv \mathbf{L}={\vec{M}}_{1}-{\vec{M}}_{2}$ is a function periodically varying with y. Thus the existence of the cycloid of the magnetic moments is not contrary to the presence of WFM in the effective magnetic field.

As shown in [15], both these states—WFM and cycloid with WFM moment—can transform into each other by a first-order phase transition due to the peculiarities of the crystal structure, in particular, the vitreous. We do not consider this phase transformation in this paper.

The correction δχ_m related to the wavevector q is nonzero only with due regard for quadratic terms $\langle (\vec{\nabla }[\vec{m}\vec{L}])^{2}\rangle \not = 0$ in the free energy. Taking into account that a rise of q_y ∝ p_x increases m, a corresponding correction in energy should be negative and proportional to the value $-{q}_{y}^{2}{M}^{2}{m}^{2}$. With the contribution $-\vec{m}\vec{H}$ in the applied magnetic field H, it takes the form $-\beta {q}_{y}^{2}{M}^{2}{H}^{2}$, where the parameter β is positive and the positive value δχ_m is $\beta {q}_{y}^{2}{M}^{2}$ [20–22]² (see comment in footnote 2).

Thus, the space modulation of the sublattice magnetic moments in AFM increases the magnetic susceptibility proportional to the square of the wavenumber q_y of the cycloid arising in a MF along with an electric moment.

Let us write the free energy F by considering a temperature change of the sublattice moment M ≈ L/2 in the absence of an external magnetic field

$$\begin{eqnarray} F&=&{t}_{1}{P}_{\mathrm{s}}^{2}+\frac{1}{2}{b}_{1}{P}_{\mathrm{s}}^{4}+2{b}_{1}{P}_{\mathrm{s}}{p}_{x}^{3}+\frac{1}{2{\chi }_{\mathrm{e}0}}{p}_{x}^{2}\nonumber\\ &&+~\frac{1}{2}{b}_{1}{p}_{x}^{4}+{t}_{2}{q}_{y}^{2}{M}^{2}+\frac{1}{2}b{q}_{y}^{4}{M}^{2}+\zeta {q}_{y}{M}^{2}{p}_{x}. \end{eqnarray} \tag{ 4 }$$

Equation (4) is written on the assumption of the spontaneous polarization P_s existing below T_N. Here P_x = P_s + p_x is the total perovskite polarization, ${P}_{\mathrm{s}}=\sqrt{-{t}_{1}/{b}_{1}}$ is the spontaneous polarization, p_x is the correction to P_s,χ_e0 is the electric susceptibility, ζ is a scalar, |λ| = |ζ|,t₁ = a₁(T − T₁), t₂ = a(T − T₂),T₁ ≡ T_FE, T₂ is the temperature of transition to a modulated state, a,a₁,b,b₁ are the positive phenomenological constants. From (4) we approximately obtain

$$\begin{eqnarray} &&{p}_{x}\approx -{\chi }_{\mathrm{e}0}\zeta {q}_{y}{M}^{2}, \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&F={F}_{0}+{\tilde {t}}_{2}{q}_{y}^{2}{M}^{2}-{b}_{2}{M}^{2}{q}_{y}^{3}+\frac{1}{2}B{q}_{y}^{4}{M}^{2}, \end{eqnarray} \tag{ 6 }$$

where ${\tilde {t}}_{2}\approx {t}_{2}-\frac{1}{2}{\chi }_{\mathrm{e}0}{\zeta }^{2}{M}^{2},{b}_{2}=2{b}_{1}{P}_{\mathrm{s}}(\zeta {\chi }_{\mathrm{e}0})^{3}{M}^{4}$, B = [b + b₁(χ_e0ζM²)⁴], and the parameter F₀ does not depend on the wavenumber q_y or the sublattice magnetization M. Note that this approach makes sense for negative ${\tilde {t}}_{2}$ values, allowing a phase transition from the AFM state to the SMS state with a finite wavenumber q_y. It is possible only for rather large M values: M² > 2t₂/(χ_e0ζ²).

Hence we find the value of the wavenumbers ${q}_{y}^{+}$ of the appearing helical structure. Below the transition point to the SMS state the temperature behavior of ${q}_{y}^{+}$ is determined by ${\tilde {t}}_{2}$:

$$\begin{eqnarray} &&{q}_{y}^{+}=\frac{3{b}_{2}+\sqrt{9{b}_{2}^{2}-1 6 B{\tilde {t}}_{2}}}{4 B}. \end{eqnarray} \tag{ 7 }$$

The sign of p_x depends on the sign of the wavenumber q_y of the spiral. Sign (+) for q_y in equation (7) corresponds to the sign (+) before the square root in this expression and to the minimum free energy (6), if the DM parameter ζ is positive. Sign (−) corresponds to the sign (−) before the square root in the expression for ${q}_{y}^{-}$ (see equation (9)) and to the free energy minimum (6), if the parameter ζ is negative. Values ${q}_{y}^{\pm }$ describe the branches F₊ and F₋ as the results of those solutions for q_y (see equation (8) and figure 3).

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According to equation (5), under coordinate inversion the sign of vector p is reversed, i.e., the signs of vectors P_s and p remain opposite.

In the applied magnetic field the coefficient ${\tilde {t}}_{2}$ in equations (4) and (7) effectively changes from ${\tilde {t}}_{2}$ to ${\tilde {t}}_{2 H}={\tilde {t}}_{2}-\beta \hspace{0.167em} {H}^{2}$, and the first-order phase transition takes place at the reduced temperature ${\tilde {t}}_{2 H c}=\frac{{b}_{2}^{2}}{2 B}$, at which the number q_y undergoes a jump from zero to ${q}_{c}^{+}={b}_{2}/B$. In this point the polarization also changes by a jump: ${P}_{x c}={P}_{\mathrm{s}}-\Lambda {\chi }_{\mathrm{e}0}{q}_{c}^{+}{M}^{2}$.

The system is able to overheat due to hysteresis effects in which this modulated (MF with WFM) and FE phases coexist. Here 'overheating' means a non-equilibrium coexistence of the SMS with WFM phase above the point ${\tilde {t}}_{2 H c}$ of phase transition in a magnetic field.

The value of thermal hysteresis (ΔT)_hys can be estimated from the dependence F(T), on the basis of equation (6). The branches ΔF₊ ≡ F₊ − F₀ and ΔF₋ ≡ F₋ − F₀ are shown in figure 3 (the signs ± correspond to signs for ${q}_{y}^{\pm }$). Here $(\Delta T)_{\mathrm{hys}}\sim {b}_{2}^{2}/a B$, and the region of MF overheating is

$$\begin{eqnarray} &&\Delta {T}_{+}\approx {b}_{2}^{2}/1 6 a B,\tqs \nonumber\\ &&\Delta {F}_{-}({\tilde {t}}_{2 H c})={M}^{2}{\beta }^{4}/3 2{B}^{3},\tqs \nonumber\\ &&\Delta {F}_{+}(0)=-2 7{M}^{2}{\beta }^{4}/6 4{B}^{3},\nonumber\\ &&\Delta {F}_{+}(9{b}_{2}^{2}/1 6 B)=\Delta {F}_{-}(9{b}_{2}^{2}/1 6 B)=2 7{M}^{2}{\beta }^{4}/5 1 2{B}^{3}.\nonumber\\ \end{eqnarray} \tag{ 8 }$$

The overheating of the SMS existing simultaneously with WFM phase in a magnetic field was observed in Eu_0.75Y_0.25MnO₃ [15].

The coexistence regions and changes in the relative volume fraction of the MF phase (below T_FE) and the WFM phase in different T ranges are shown in the phase diagram of figure 3 in relation to the history of the action of magnetic and electric fields. It was established [15] that the WFM phase without the SMS coexisted with the FE phase also below the temperature T_f of spin freezing (vitrification), which corresponds to the first-order phase transition into a so-called 'magnetoelectric glass' [15]. In the absence of magnetic field the WFM phase was not observed below T_f, and only the FE phase was presented.

It follows from equations (6) and (7) that, if the point ${\tilde {t}}_{2 H c}$ is close to aT_f in the presence of the magnetic field, then the spontaneous EP on cooling in the region ${\tilde {t}}_{2 H c}\lt {\tilde {t}}_{2 H}$ can increase by the simple law $\sqrt{{T}_{\mathrm{FE}}-T}$, which seems to be observed in the experiment for Eu_0.75Y_0.25MnO₃ [15].

Above, it was implicitly assumed that the parameter ζ is positive. If it is negative ζ < 0, the coefficient b₂ in equation (6) changes sign and, instead of equation (7), the solution is the wavenumber

$$\begin{eqnarray} &&{q}_{y}^{-}=\frac{3{b}_{2}-\sqrt{9{b}_{2}^{2}-1 6 B{\tilde {t}}_{2}}}{4 B}\lt 0. \end{eqnarray} \tag{ 9 }$$

Thus, according to (9), the sign of the spiral is actually reversed. However, it follows from equation (5) that the sign of the polarization p is unchanged for ζ < 0. The jump ${q}_{c}^{-}={b}_{2}/B$ becomes negative in the transition point to the spiral-cycloid magnetic state, and the polarization p_x undergoes a similar negative jump of $-\zeta {\chi }_{\mathrm{e}0}{q}_{c}^{-}{M}^{2}$.

Using equations (6) and (7) we define the correction p_xH to EP and the correction δm = (δχ_m)H to the magnetic moment at the expense of the applied magnetic field

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle m={\chi }_{m}H=\left ({\chi }_{m 0}+\delta {\chi }_{m}\right )H:\\ \displaystyle {p}_{H}=-\zeta {\chi }_{\mathrm{e}0}{M}^{2}{q}_{y H}^{+},\\ \displaystyle \delta m=\beta (H-{H}_{\mathrm{th}}){M}^{2}\left ({q}_{y H}^{+}\right )^{2}. \end{array} \end{eqnarray} \tag{ 10 }$$

Here H_th is the threshold field associated with the magnetic anisotropy, and q_yH is the wavenumber taking account of an external magnetic field.

Taking into account the data [12, 15], the 'magnetic' correction to the spontaneous EP in the intermediate temperature range between the temperature of 'vitrification' T_f and the point of ferroelectric transition T_FE is shown in figure 4. In this region the first-order phase transition from an AFM ferroelectric into a spiral MF takes place in the magnetic field at a temperature ${\tilde {T}}_{2 H c}={T}_{2}+{\tilde {t}}_{2 H c}/a$.

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Our reasoning for a positive coefficient β was given above. The magnetic field is assumed to be sufficiently large to overcome the threshold H_th value associated, for example, with the 'freezing' of the system or/and with the magnetic anisotropy. We suppose that q_y = 0 and the field η is very weak (if at all), since a finite q_y was not observed in the experiment in the absence of an external magnetic field. Here η is the magnetic field occurring due to spin–orbital interaction (see equation (2)).

Equation (10) shows that at low temperature the polarization p_H should increase rather rapidly with the magnetic moment M, proportionally to ${M}^{2}\sqrt{-{\tilde {t}}_{2 H}}$, if the parameter b₂ ∝ ζ³ is small and B ≈ b. In this case, the wavenumber ${q}_{y}^{+}\propto \sqrt{-{\tilde {t}}_{2 H}}$ rises more slowly. These nuances of behavior of Eu_1−xY_xMnO₃ type perovskites require experimental verification.

Now we write by analogy with equation (4) the expression for the thermodynamic potential in the absence of spontaneous polarization P_s below T_N and in the presence of an external magnetic field H

$$\begin{eqnarray} \Phi &=&A{L}^{2}+\frac{1}{2}C{L}^{4}+B{m}^{2}-D{q}_{y}^{2}{L}^{2}{m}^{2}+\frac{1}{2{\chi }_{\mathrm{e}0}}{p}_{x}^{2}\nonumber\\ &&+~\frac{1}{2}{b}_{1}{p}_{x}^{4}-d{m}^{2}{p}_{x}^{2}+{t}_{2}{q}_{y}^{2}{M}^{2}\nonumber\\ &&+~\frac{1}{2}b{q}_{y}^{4}{M}^{2}+\zeta {q}_{y}{M}^{2}{p}_{x}-H m-\frac{{H}^{2}}{8 \pi }. \end{eqnarray} \tag{ 11 }$$

Here A,B,C and D are the phenomenological parameters. A changes sign at the point of phase transition to the AFM state, and thus the difference of sublattice magnetizations L depends on the temperature. B,C and D are positive, the sum of magnetizations m and field H are directed along the z axis. The vector L cyclically changes its direction in the (xy) plane (see figure 2), so that L is its root-mean-square value.

Assuming that the p_x,q_y,ζ, and H values are small, we obtain after minimization of Φ for m that

$$\begin{eqnarray} &&m\approx \frac{H}{2 B}\left (1+\frac{D}{B}{q}_{y}^{2}{L}^{2}+\frac{d}{B}{p}_{x}^{2}\right ). \end{eqnarray} \tag{ 12 }$$

Whereas the minimization of Φ for p_x gives a value

$$\begin{eqnarray} &&{p}_{x}\approx -{\chi }_{\mathrm{e}0}\zeta {q}_{y}{L}^{2}\left (1+\frac{{\chi }_{\mathrm{e}0}d{H}^{2}}{{B}^{2}}\right ). \end{eqnarray} \tag{ 13 }$$

Then, according to (11)–(13), the thermodynamic potential approximately takes the form

$$\begin{eqnarray} \Phi &\approx &A{L}^{2}+\frac{1}{2}C{L}^{4}-\frac{D{H}^{2}{q}_{y}^{2}{L}^{2}}{2{B}^{2}}-\frac{1}{2}{\chi }_{\mathrm{e}0}{\zeta }^{2}{q}_{y}^{2}{L}^{4}\nonumber\\ &&+~{t}_{2}{q}_{y}^{2}{L}^{2}+\frac{1}{2}b{q}_{y}^{4}{L}^{2}-\frac{{H}^{2}}{8 \pi }-\frac{{H}^{2}}{4 B}. \end{eqnarray} \tag{ 14 }$$

It follows from (14) that the temperature at which the spin modulation occurs due to the spin–orbital interaction ζ, can be found from the equation

$$\begin{eqnarray} &&{t}_{2}-\frac{D{H}^{2}}{2{B}^{2}}-\frac{1}{2}{\chi }_{\mathrm{e}0}{\zeta }^{2}{L}^{2}=0, \end{eqnarray} \tag{ 15 }$$

and the square of the wavenumber of occurred spin modulation is

$$\begin{eqnarray} &&{q}_{y}^{2}\approx {b}^{-1}\left (-{t}_{2}+\frac{D{H}^{2}}{2{B}^{2}}+\frac{1}{2}{\chi }_{\mathrm{e}0}{\zeta }^{2}{L}^{2}\right ). \end{eqnarray} \tag{ 16 }$$

We obtain from (14) and (16)

$$\begin{eqnarray} \Phi &\approx &A{L}^{2}-\frac{1}{2 b}\left ({t}_{2}-\frac{D{H}^{2}}{2{B}^{2}}\right )^{2}{L}^{2}+\frac{1}{2}C{L}^{4}\nonumber\\ &&+~\frac{{\chi }_{\mathrm{e}0}{\zeta }^{2}}{2 b}\left ({t}_{2}-\frac{D{H}^{2}}{2{B}^{2}}\right ){L}^{4}-\frac{{H}^{2}}{8 \pi }-\frac{{H}^{2}}{4 B}. \end{eqnarray} \tag{ 17 }$$

From (17) the square of the parameter of transition L² from the paramagnetic to the AFM state with modulated spin structure is defined. Thus, according to (15)–(17), the function L(T), which describes the phase transition with temperature to the AFM state in the presence of a magnetic field, experiences a disturbance due to the influence of the magnetic field and spin modulations. In particular, T_N shifts to higher temperatures in an external magnetic field according to the equation for T_N:

$$\begin{eqnarray} &&A({T}_{\mathrm{N}})-\frac{1}{2 b}\left ({t}_{2}-\frac{D{H}^{2}}{2{B}^{2}}\right )^{2}=0. \end{eqnarray} \tag{ 18 }$$

According to (15) and (18) T_N coincides with the temperature of occurrence of modulated spin structure at A = 0 and ${t}_{2}-\frac{D{H}^{2}}{2{B}^{2}}=0$, that is at

$$\begin{eqnarray} &&{T}_{\mathrm{N}}={T}_{2}+\frac{D{H}^{2}}{2 a{B}^{2}},\tqs \text{where }{t}_{2}=a\left (T-{T}_{2}\right ). \end{eqnarray} \tag{ 19 }$$

Equation (16) shows that the wavenumber q_y has two equivalent values, differing in sign. Accordingly, the electric polarization p_x also has two values opposite in sign. Therefore, in the absence of an external electric field, such a crystal, on average, has no polarization in the presence of the DM effect. It is possible that a similar situation arises in the langasite crystals in the absence of spontaneous polarization above T_N [23–25]. Application of an electric field permits one to choose the specific q_y and p_x values for such a multiferroic.

Now suppose that an electric field E is applied along the x axis in the presence of spontaneous polarization P_s which is parallel to the x axis (at T < T_FE < T_N). We consider two cases: when the field E coincides with the direction of vector p (that is E↑↑p and E_x < 0), and when it is opposite to p (E↑↓p and E_x > 0). Then equation (4) for the free energy can be written as

$$\begin{eqnarray} F&=&{F}_{0}+{\tilde {t}}_{2 H}{M}^{2}{q}_{y}^{2}-{b}_{2}{M}^{2}{q}_{y}^{3}\nonumber\\ &&+~\frac{1}{2}B{M}^{2}{q}_{y}^{4}+{\chi }_{\mathrm{ e}0}{M}^{2}\zeta {q}_{y}{E}_{x}. \end{eqnarray} \tag{ 20 }$$

In deriving (20) we used equation (5) and the phenomenological parameters from equation (6). Here p_x ≈ χ_e0E_x − χ_e0ζM²q_y,ζ > 0, and q_y > 0, and the sign (+) is omitted. The first-order transition from AFM into the modulated spin structure takes place, if at the transition point F(q_y > 0) = F(q_y = 0) and ∂F(q_y ≠ 0)/∂q_y = 0. In this case, to find the wavenumber q_y and the transition temperature point ${\tilde {t}}_{2 H}$, we obtain two equations. The first one is

$$\begin{eqnarray} &&{q}_{y}=\frac{{\tilde {t}}_{2 H}+\sqrt{{\tilde {t}}_{2 H}^{2}+3{\chi }_{\mathrm{e}0}{b}_{2}\zeta {E}_{x}}}{{b}_{2}}. \end{eqnarray} \tag{ 21 }$$

The second equation, in general, is

$$\begin{eqnarray} {q}_{y}&=&\frac{{b}_{2}}{2 B}+\sqrt[3]{-\frac{Q}{2}+\sqrt{\frac{{Q}^{2}}{4}+\frac{{U}^{3}}{2 7}}}\nonumber\\ &&+~\sqrt[3]{-\frac{Q}{2}-\sqrt{\frac{{Q}^{2}}{4}+\frac{{U}^{3}}{2 7}}}, \end{eqnarray} \tag{ 22 }$$

where $Q=\frac{{b}_{2}^{3}}{{B}^{3}}(\frac{t}{2}-\frac{1}{4}+\frac{\Sigma }{2})$, $U=(\frac{{b}_{2}}{B})^{2}(t-\frac{3}{4})$, and ${\tilde {t}}_{2 H}=\frac{{b}_{2}^{2}}{B}t$, $\Sigma =\frac{{B}^{2}}{{b}_{2}^{3}}\left ({\chi }_{\mathrm{e}0}\zeta E\right )$, where t and Σ are dimensionless variables. At E = 0, it follows from (22) that $t=\frac{1}{2}$ and ${q}_{y}=\frac{{b}_{2}}{B}$, with $Q=0,U=-\frac{1}{4}(\frac{{b}_{2}}{B})^{2}$, and the sum of the roots of the third degree in (22) is the sum of complex-conjugate numbers $\frac{{b}_{2}}{2\sqrt{3}B}\exp \left (\mathrm{i}\pi /6\right )$ and $\frac{{b}_{2}}{2\sqrt{3}B}\exp (-\mathrm{i}\pi /6)$, which is equal to $\frac{{b}_{2}}{2 B}$.

From (21) and (22) at small E values we can obtain that the point of first-order phase transition into a spiral MF is described by the values

$$\begin{eqnarray} &&{q}_{y}\approx \frac{{b}_{2}}{B}+\frac{B}{{b}_{2}^{2}}\left ({\chi }_{\mathrm{e}0}\zeta {E}_{x}\right )\tqs \text{and}\tqs \nonumber\\ &&{\tilde {t}}_{2 H}\approx \frac{{b}_{2}^{2}}{2 B}-\frac{B}{{b}_{2}}\left ({\chi }_{\mathrm{e}0}\zeta {E}_{x}\right ). \end{eqnarray} \tag{ 23 }$$

We recall that here ${\tilde {t}}_{2 H c}$ is the reduced critical temperature of the transition, which is equal to ${\tilde {t}}_{2 H c}=\frac{{b}_{2}^{2}}{2 B}$ at E_x = 0.

It is evident from (23) that at the transition point, the wavenumber q_y increases when the applied electric field E_x is positive (that is, E↑↓p) and decreases for negative E_x (E↑↑p). In such a case, at positive E_x, the polarization p_x ≈ χ_e0E_x − χ_e0ζM²q_y decreases in its modulus due to the increase of the term χ_e0E_x, if the field correction to the wavenumber $\frac{B}{{b}_{2}^{2}}\left ({\chi }_{\mathrm{e}0}\zeta {E}_{x}\right )$ is small and $\frac{\zeta B{M}^{2}}{{b}_{2}^{2}}\lt 1$, i.e. the parameter of spin–orbital interaction ζ is small. The essential decrease in the polarization p_x was observed in the Eu_0.75Y_0.25MnO₃ crystal [15] on the application of a positive E_a (i.e., E_a = E_x > 0).

According to (23), the temperature of the transition to the spiral MF decreases for a positive E_x, and increases for a negative E_x. As follows from equations (20) and (21), the transition point ${\tilde {t}}_{2 H}$ becomes zero at

$$\begin{eqnarray} &&\left ({\chi }_{\mathrm{e}0}\zeta \right ){E}^{\ast }=\frac{1 6}{2 7}\frac{{b}_{2}^{3}}{{B}^{2}}. \end{eqnarray} \tag{ 24 }$$

If the electric field exceeds the threshold E* value the transition temperature ${\tilde {t}}_{2 H}$ formally becomes negative:

$$\begin{eqnarray} &&{\tilde {t}}_{2 H}\approx -\frac{3 B}{4{b}_{2}}{\chi }_{\mathrm{e}0}\zeta E+\frac{4{b}_{2}^{2}}{9 B}. \end{eqnarray} \tag{ 25 }$$

This means that the correction p is formed at a lower temperature if E↑↓p. In this case the electric field suppresses the polarization p at large positive values of E_x, since E↑↓p. Equations (24) and (25) for the transition point to a spiral MF in the electric field are valid for certain intermediate values of the applied field.

Experimental data for Eu_0.75Y_0.25MnO₃ [15] show that small changes of E_x and δm are opposite in sign, and δm changes linearly with E_x. Taking into account of the fact that the magnetic spiral is destroyed when E_x increases, the correction to the magnetic susceptibility δχ_m in the presence of the electric field can contain a negative term −ζ'q_yE_x, where the positive parameter ζ' is related to the spin–orbital interaction. In this case the magnetic moment δm ∝− ζ'q_yE_xH, which corresponds to the experiment [15].

Since by equation (23), the transition temperature increases for negative values of E_x, i.e. E↑↑p, then the system 'unfreezes' and is in the temperature range where the phase transition occurs over the wavenumber q_y (i.e., over the correction to the polarization p_x). In this case, the magnetization m increases sharply due to the DM effect (proportional to ${q}_{y}^{2}$) [12]. At the same time the number q_y increases dramatically, and the modulus of the negative correction p_x increases when the temperature drops below the transition point (see figure 4).

Summarizing the results, we can conclude that the orthorhombically distorted perovskites, which are ferroelectrics below T_N, can claim to be effective multiferroics with mutually perpendicular vectors of the electric polarization and the spiral of magnetic moments (both) perpendicular to the main c axis. In the perovskite-type crystals in the absence of a center of symmetry, the DM effect works in the plane perpendicular to the c axis. Because of this, an additional polarization and spiral modulation of the magnetic moments with mutually perpendicular directions appear in the ab(xy) plane. This additional EP is proportional to the square of the magnetization (as an inherent parameter of transition to the AFM state at T_N).

In the applied magnetic field the appearance of electric polarization p at T < T_2Nc (due to DM) must be accompanied by the appearance of a WFM moment m along the c axis. In this case, the cycloid of magnetic moments is preserved. In an applied field H, the enhancement of the wavenumber q_y in WFM with decreasing temperature leads to an increase in the spontaneous magnetization m. At the same time, a positive correction to magnetic susceptibility appears which is quadratic in the wavevector q_y and sublattice magnetization M.

On this basis, the thermodynamics of the first-order phase transition of an AFM ferroelectric into a spin-cycloid MF with WFM and jumps of electrical polarization and magnetization is described. Analysis of the thermodynamic potential in an external magnetic field indicates the possibility of the appearance of the SMS below T_N in the absence of spontaneous EP. The function of the difference in the sublattice magnetizations L(T) is found, which describes the phase transition from the paramagnetic to the AFM state. The function L(T), dependent on the magnetic field and spin modulation, shows that the point T_N shifts to higher temperatures in an external magnetic field.

Of particular interest is the dependence of the phenomena on the an applied electric field E. In the absence of an electric field, the crystal with the spin-modulated structure, on average, does not have EP. However, an applied electric field significantly affects the polarization and magnetic structure. For example, if the electric field is applied along the x axis, the polarization p_x appears and the wavevector q_y changes in such a multiferroic.

We have shown that in the field E the correction p to the spontaneous polarization (which appears due to the DM effect) increases if p coincides with the direction of E. Simultaneously decreases the wavenumber q of the magnetic spiral. In the field E, the temperature of the p onset increases. Otherwise (i.e., when E↑↓p) with increasing E_x, the modulus |p_x| of the negative correction p_x is reduced, thus increasing the overall polarization P_x. Then the temperature of appearance of the correction p_x decreases. Simultaneously, when E_x increases, a WFM moment m (in the presence of a constant magnetic field) gradually increases if p and E are parallel and decreases if p and E are antiparallel.

The application of a sign-alternating electric field causes a linear-in-field change of the magnetic moment, opposite in sign to the change of the electric field, it is also possible due to the DM effect. This effect has been observed experimentally in the crystal Eu_0.75Y_0.25MnO₃ [15].

Hysteresis phenomena greatly complicate the described effects. In this paper we define the temperature ranges of overheating and overcooling of MF under the application of magnetic and electric fields. In this case we take into account the proximity of the transition points in the ferroelectric, multiferroic and weak ferromagnetic states, as well as to the magnetoelectric glassy state. It is shown that overheating of the MF occurs in a much narrower temperature range than its overcooling. This corresponds to the experimental data [15].

We thank Dr K O Funtov for his help in the design of the figures. This study is supported by the Russian Foundation for Basic Research project Nos 11-02-12089 and 11-02-00636a and by the Program of the Russian Academy of Sciences 'Strongly correlated electronic systems'.