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New collective modes in multiferroic TbMnO3: The electromagnons
Solid State Communications 149 (2009) 1557–1560
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
New collective modes in multiferroic TbMnO3 : The electromagnons Shruti Shukla ∗ , Debanand Sa Department of Physics, Banaras Hindu University, Varanasi -221005, India
article
info
Article history: Received 27 February 2009 Received in revised form 22 May 2009 Accepted 29 May 2009 by Y.E. Lozovik Available online 2 June 2009
abstract Using a phenomenological Landau theory as well as equations of motion, we look for new excitations in the multiferroic TbMnO3 . Coupling these excitations to an electromagnetic field, we derived the magneto–polariton dispersion relation. Further, the frequency as well as temperature dependent dielectric susceptibility, and hence the refractive index, are obtained in this formulation. The imaginary part of the latter is computed and compared with the existing experimental data. © 2009 Elsevier Ltd. All rights reserved.
PACS: 78.20.Ls 75.80.+q 75.30.Fv 77.80.-e Keywords: A. Multiferroics D. Magneto-optical studies D. Spin density wave
Multiferroic materials are those where more than one ferroic order such as magnetism and ferroelectricity coexist together and are coupled to each other. From the application point of view, coupling between ferroelectricity and magnetism would be useful in multistate memory devices with mutual electric and magnetic control or as magnetically switchable optical devices among others. The coexistence of ferroelectric and magnetic order presents a fundamental challenge since the origins of both the orders are very different: magnetism is related to the ordering of spins of electrons in incomplete ionic shells, whereas ferroelectricity results from the relative shifts of negative and positive ions. Much of the earlier work on multiferroics was directed towards bringing them together in one material [1], which was proved to be difficult because of their mutually exclusive nature [2]. The long sought control of electric properties by magnetic fields was achieved in a class of multiferroics [3–6] called frustrated magnets, namely RMnO3 , GdMnO3 , RMn2 O5 , CoCr2 O4 , MnWO4 , etc. These materials become unique not due to the strength of their coupling (electric polarization is two to three orders of magnitude smaller than in the typical ferroelectrics) but the high sensitivity of their dielectric properties to an applied magnetic field [7–13]. This is in the heart of complex spin structure in these materials which makes the ferroelectricity of magnetic origin. For example, the multiferroic TbMnO3 , which has orthorhombically distorted perovskite
∗
Corresponding author. E-mail address: [email protected] (S. Shukla).
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.05.044
(space group Pbnm) structure at high temperature, is identical to the colossal magnetoresistive manganites in terms of the electronic configuration of the Mn3+ ion. In LaMnO3 , staggered orbital order becomes responsible for the layered antiferromagnetic (AF) order, whereas the spin structure in TbMnO3 is of sinusoidal AF type (of Mn3+ moments) below a temperature TN ≈ 41 K E (0, qs , 1). The wave vector qs is incommenwith a wave vector q surate (∼0.295) at TN , decreases with decreasing T and takes a constant value (0.28) below Tl ≈ 28 K. Neutron scattering measurements [14] revealed that the second transition corresponds to a noncollinear spiral magnetic structure where a simultaneous static polarization appears along the z-axis. This is confirmed by X-ray diffraction measurements [15], where the modulated magnetic phase is accompanied by a magnetoelastically induced lattice modulation. The existence of new elementary excitations in such magneto–electric (ME) coupled systems called electromagnons was already predicted long ago [16], and the first possible observations were made [17] in GdMnO3 and TbMnO3 . The dielectric data obtained for TbMnO3 without a magnetic field and with the ac component E k x exhibit a broad peak with characteristic frequency ω = (20 ± 3) cm−1 . The dielectric contribution of this mode increases with decreasing temperature, and below Tl its width decreases. Similar effects are observed for GdMnO3 also. The application of a static magnetic field along the z-axis suppresses the imaginary part of the dielectric function by more than a factor of 2, and also the real part. This provides a clear indication that the observed mode is related to the modulated magnetic structure.
1558
S. Shukla, D. Sa / Solid State Communications 149 (2009) 1557–1560
In this study, we considered a Landau free energy (from symmetry considerations) to address both the magnetic phases in multiferroic TbMnO3 . Using equations of motion, in addition to Maxwell’s equation, we coupled the electromagnetic radiation to the excitations in these system and obtained the magneto–polariton dispersion. In order to compare our finding with the available experiments, we derived an expression for the frequency and temperature dependent dielectric constant. The imaginary part of the dielectric function is computed and is compared with the observed spectrum. In almost all the multiferroics listed above, complex magnetic structures and phase diagrams are observed which show strong interplay between magnetic and dielectric phenomena [18]. Thus, the key question would be how the magnetic ordering induces ferroelectricity. This question has been addressed by Katsura et al. [10], which is based on the idea of the spin current induced between noncollinear spins yielding an electric dipole moment. This can be regarded as the effect of either Dzyaloshinskii–Moriya (DM) interaction [19,20] or spin–lattice interaction [13]. In the present work, we consider the phenomenological treatment of this mechanism. The coupling between ferroelectric polarization and magnetization is governed by the symmetries of these order parameters. The minimal coupling which could yield ferroelectricity is nonlinear [12]. Thus, we start with a free energy which has magnetic and ferroelectric components, and with the coupling between them given as F =V
−1
Z
dE r (Fs + Fl + Fs−l )
(1)
1
Fs =
2
E2 + aM 1
1 2
4
1 2
2
bPz2 − Ez Pz
(3)
and Fs−l = ηPz (Mz ∂y My − My ∂y Mz ).
(4)
In constructing the free energy, the details of the unit cell for TbMnO3 have not been taken into account. Here, the parameters a = a0 (T − T0 ), a0 , u, b, α are positive and γ is negative. The parameters γ and α are responsible for the modulation of the magnetic order parameter. The anisotropy parameter w is considered to be positive. The parameter η represents the coupling between the ferroelectricity and the magnetization. Considering the equilibrium AF states in TbMnO3 to be My0 = M1 cos ky and Mz0 = M2 sin ky, the above free energy is minimized with respect to the order parameters M1 , M2 , Pz and the modulation vector k. Thus, one finds the sinusoidal collinear (I) and the noncollinear spiral (II) magnetic phases, which are given as follows: M2 = 0, Pz0 = 0, M12 = −4L1 /3u, L1 = a − ac , ac = γ 2 /4α, k2 = −γ /2α.
2 M1 = (L2 − 3L1 + 1 )/2u, M22 = (L1 − 3L2 + 2 )/2u, II : L2 = a − ac + w, 1,2 = 2k2 η2 (3L1,2 − 5L2,1 )/ub, Pz0 = ηkM1 M2 /b.
(5)
(6)
In the minimization, we have assumed that the magnetoelectric coupling is weak; that is,
η2 k2 ub
(8)
+ η(2pz ∂y My0 + My0 ∂y pz ) = 0. 1
2
λ¨pz + bpz − Ez pz + η(mz ∂y My0 − My0 ∂y mz ) = 0
1
E4 wMz2 + uM
+ γ {(∂y My )2 + (∂y Mz )2 } + α{(∂y2 My )2 + (∂y2 Mz )2 }, (2)
I:
δF =0 δ pz (7) δF ¨ z + δ2 m ˙z + =0 µm δ mz where µ and λ are the density parameters which characterize the kinetic energy of the system. In the limit of no damping (δ1 = 0 = δ2 ), these equations can explicitly be written as λ¨pz + δ1 p˙ z +
2 ¨ z + (a + w + uMy0 µm − γ ∂y2 + α∂y4 )mz
where
Fl =
η2 k2 w
6 uba such that Tl < TN . Thus, the above minimization scheme 0 yields a phase diagram where, on lowering T , one obtains the sinusoidal AF phase below TN , and on further lowering (below Tl ) it is the noncollinear spiral magnetic phase which prevails. The ferroelectricity of magnetic origin appears below Tl (see the last expression in Eq. (6)). Since the system under consideration is ME coupled, the excitations in such a system are investigated using equations of motion method. In order to do this, we consider the fluctuations E −M E 0 ) and the electric polarization E = M in the AF vector (m E E (Ep = P − P0 ) with respect to their equilibrium values. Further, these excitations can be coupled to the z-component of the electromagnetic field (Ez ), propagating along the y-direction. This is taken care of by Maxwell’s equations since its electric field component Ez excites polarization along the z-direction. Thus, the Maxwell’s equations, along with the equations of motion in the linear approximation, give rise to a coupled set of equations for pz and mz which are written as
1. The transition temperature ac 3w and Tl ≈ TN − 2a + a0 0
TN and Tl are obtained as TN = T0 +
In order to get the modes of excitations, these equations are solved P in Fourier space, and are taken to be mz (y, t ) = l ml exp[i{(lk+q) y − ωt }] and Ez (pz )(y, t ) = E0 (p0 ) exp[i(qy − ωt )]. Here, q is the wave vector associated with the electromagnetic field propagation and is assumed to be much smaller than that of the wave vector of magnetic order parameter which has sinusoidal variation (q k). In such an approximation, the higher harmonics in l except the first one can be neglected. Moreover, using Maxwell’s equation, E0 can be eliminated from the above equation by expressing in terms of p0 ; that is, 4π p0 = E0 (n2 − 1), n(= qc /ω) being the refractive index of the material. Thus, the above equations can be reduced to a coupled equation in terms of p0 , m1 and m−1 , written as B iD−k iDk
−iDk −R
−iD−k
L−
−R
!
L+
p0 m −1 m1
!
0 0 0
!
=
(9)
where B = λω2 − b + 4π /(n2 − 1), L± = µω2 − a − w − γ (q ± k)2 − α(q ± k)4 − uM12 /2, Dk = ηM1 (k − q/2) and R = uM12 /4. As already mentioned above, q k, so we take q = 0 (optic modes only), which simplifies the parameters as D−k = −Dk , L+ = L− and m−1 = −m1 . Thus, the above equation simplifies to a 2 × 2 matrix equation in terms of p0 and m (m = m1 = −m−1 ) whose solution is obtained by solving the determinant. These yields the energies/frequencies of new q = 0 excitations called electromagnons, which are given as
ω
2 1,2
=
1 2
ω +ω ∓ 2 p
2 0
q
(ω − ω ) + 8η 2 p
2 2 0
2 k2 M 2 1
(µλ)−1
,
(10)
where ω0 and ωp are respectively the magnetic and dipole frequencies, defined as ω02 = µ1 [w + 2(a − ac )/3] = 3µ0 (T − Tl ) + 2a
η2 k2
4 ub w and ωp2 = b/λ. It should be noted here that the magnetic µ frequency ω0 depends on temperature and it takes the lowest value at T = Tl .
S. Shukla, D. Sa / Solid State Communications 149 (2009) 1557–1560
Fig. 1. Variation of the imaginary part of the dielectric function with frequency. The calculated value of 2 (ω, T ) is fitted with the dielectric data of Pimenov et al. at 20 K (filled circles).
1559
Fig. 2. Variation of the imaginary part of the dielectric function with temperature at ω = 6 cm−1 and ω = 4 cm−1 .
The frequency and temperature dependent dielectric function
((ω, T )) (which is the square of the refractive index n) in such a formulation is calculated to be
(ω, T ) =
(ω2 − Ω12 )(ω2 − Ω22 ) , (ω2 − ω12 )(ω2 − ω22 )
(11)
where the expression for Ω12,2 is the same as that of ω12,2 with ωp replaced by Ωp = (b+λ4π) . This result is contrasted with the polariton problem in conventional ferroelectrics where the longitudinal and the transverse optic mode frequencies appear as zeros and poles respectively in the frequency dependent dielectric function. One of the transverse optic mode softens toward the phase transition and is condensed according to the Lyddane–Sachs–Teller (LST) relation [21]. This conventional viewpoint is not relevant in the case of multiferroics since the lattice displacement is not essential to the electronic polarization. Therefore, it is important to identify the relevant collective modes [22] that are responsible for ferroelectricity of magnetic origin and to study the dynamical magneto–electric effect. The dielectric function (ω, T ) can in general be a complex quantity. This is realized from Eq. (10) by making ω → ω + iδ (δ being the imaginary part). Thus, writing (ω, T ) = 1 (ω, T ) + i2 (ω, T ) (1 and 2 are respectively the real and imaginary part of the dielectric function), one can compute 2 (ω). The computed value of 2 (ω, T ) is fitted with the dielectric data of Pimenov et al. [17] at 20 K (see Fig. 1), which yields the frequencies as ω1 = 23.2 cm−1 , Ω1 = 26.3 cm−1 , ω2 = 78.5 cm−1 and Ω2 = 231.2 cm−1 . From this, the magnetic and lattice frequencies are estimated to be ω0 (T = 20 K) = 67.5 cm−1 , ωp = 75.7 cm−1 and Ωp = 230.6 cm−1 . In the process of fitting, the dimensionless η2 k2
ME coupling came out to be small, that is, ub = 0.031, whereas the width parameter δ turned out to be considerably large (δ = 11.0 cm−1 ), which is a case of overdamping. However, on lowering T , the Tb spins order at T = 7 K, and it is unclear at this moment whether the narrowing of excitations results from such an ordering. Using the above parameters, the variation of the imaginary part of the dielectric function with temperature at ω = 6 cm−1 and ω = 4 cm−1 is shown in Fig. 2. It is clear from the figure that 2 (ω, T ) shows a peak at Tl , and its width decreases once the frequency is lowered. From the above discussions, it is obvious that the magnetic frequency (ω0 ) is smaller than the lattice frequencies (ωp , Ωp ). Of both the electromagnon modes ω2 and ω1 , the latter takes a minimum value (∼(η2 k2 /ub)2 ) at T = Tl . This is the signature of a soft mode (see Fig. 3) which is associated with the second
Fig. 3. Variation of electromagnon frequencies ω1 and ω2 with temperature, showing the soft mode.
transition (T = Tl ), which is from the collinear AF phase to the noncollinear spiral magnetic structure where ferroelectricity appears. In contrast, the electromagnon mode ω2 is almost temperature independent and the energy associated with it is high compared to ω1 . Using Eq. (10), the magneto–polariton dispersion spectrum (ω2 vs. q) is analyzed. It yields a sixth-order algebraic equation whose solutions are shown in Fig. 4. The role of ME coupling is to lower the magnetic frequency (ω0 ) to ω1 , whereas it pushes up the lattice frequency (ωp ) to ω2 . From the dispersion it is obvious that there are two frequency regions near ω1 and ω2 where the electromagnons interact very strongly with the electromagnetic radiation. The nature of the excitations near ω2 is dominated by the electric polarization mode whereas near ω1 the prominent excitations are of magnetic origin. Since ω2 ω1 and the lowenergy physics in such a system is dictated by ω1 -excitations, the window (see the inset in Fig. 4) where the resonant interaction of electromagnetic radiation with the ω1 -electromagnon occurs is calculated as
η2 k2 M12 Ω1 − ω 1 ≈ ω0 λµ
"
1
ωp2 − ω02
−
1
Ωp2 − ω02
# ,
(12)
which is very small. Since ω1 takes a minimum value at T = Tl , the above interval is largest at Tl compared to any other temperature. Further, this window vanishes in the limit of either η → 0 or k → 0. Since, in the present formulation, one can excite the electromagnons by applying an ac electric field, the ac magnetization mz
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S. Shukla, D. Sa / Solid State Communications 149 (2009) 1557–1560
coupling the excitations to electromagnetic radiation. By solving the dynamical equations, the energy of excitations as well as the magneto–polariton dispersion is obtained. In order to compare our finding with the available experiments, we derive an expression for the frequency and temperature dependent dielectric constant. The imaginary part of the dielectric function is computed and compared with the observed spectrum. We believe that such excitations can be exploited for the design of magneto–optical devices. References
Fig. 4. Magneto–polariton dispersion (ω2 vs. q) using the parameters from the fit in Fig. 1. The values shown along the y-axis are the corresponding ω2 ’s. The inset corresponds to a zoomed version of the gap between ω12 and Ω12 .
can be calculated using Eq. (8) as
ω2 − Ωp2 mz = E . (4π /λ)(2ηkM1 ) 0
(13)
Considering the frequency to be ω ∼ 20 cm−1 , which corresponds to terahertz region, the highest electric field intensity is E0 ∼ 103 V/cm. Putting theses values along with the parameters estimated above, one can calculate the magnitude of the ac ×10 magnetization as mz = ( 0.5√ µλ
−2
V ) cm . In our calculation above, we
have used the parameters 4λπ = 47 470 cm−2 and w = 4.1 × µ 103 cm−2 . By choosing w = 0.001 (in proper units), the value of mz is obtained as 0.4 × 10−2 µB /Mn atom, which is comparable with the experimental value [3]. In conclusion, we summarize the main findings of the paper. We consider a Landau free energy (from symmetry considerations) to address both the magnetic phases in multiferroic TbMnO3 . We show that in ME materials, new hybrid spin–lattice excitations exist which can be excited by ac electric fields. This is done by
[1] G.A. Smolenskii, I.E. Chupis, Sov. Phys. Usp. 25 (1982) 475. [2] D.I. Khomskii, Bull. Am. Phys. Soc. C 21 (2001) 002. [3] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, Y. Tokura, Nature 426 (2003) 55. [4] N. Hur, S. Park, P.A. Sharma, J.S. Ahn, S. Guha, C.-W. Cheong, Nature 429 (2004) 392. [5] Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima, Y. Tokura, Phys. Rev. Lett. 96 (2006) 207204. [6] K. Taniguchi, N. Abe, T. Takenobu, Y. Iwasa, T. Arima, Phys. Rev. Lett. 97 (2006) 097203. [7] G. Lawes, A.B. Harris, T. Kimura, N. Rogado, R.J. Cava, A. Aharony, O. EntinWohlman, T. Yildrim, M. Kenzelmann, C. Broholm, A.P. Ramirez, Phys. Rev. Lett. 95 (2005) 087205. [8] V.G. Bar’yachtar, I.E. Chupis, Sov. Phys. Solid State 10 (1969) 2818. [9] V.G. Bar’yachtar, V.A. L’vov, D.A. Jablonskii, JETP Lett. 37 (1983) 673. [10] H. Katsura, N. Nagaosa, A.V. Balatsky, Phys. Rev. Lett. 95 (2005) 057205. [11] A.B. Harris, T. Yildrim, A. Aharony, O. Entin-Wohlman, Phys. Rev. B 73 (2006) 184433. [12] M. Mostovoy, Phys. Rev. Lett. 96 (2006) 067601. [13] I.A. Sergienko, E. Dagotto, Phys. Rev. B 73 (2006) 094434. [14] M. Kenzelmann, A.B. Harris, S. Jonas, C. Broholm, J. Schefer, S.B. Kim, C.L. Zhang, S.-W. Cheong, O.P. Vajk, J.W. Lynn, Phys. Rev. Lett. 95 (2005) 087206. [15] T. Kimura, S. Ishihara, H. Shintani, T. Arima, K.T. Takahashi, K. Ishizaka, Y. Tokura, Phys. Rev. B 68 (2003) 060403(R). [16] V.G. Bar’yachtar, I.E. Chupis, Sov. Phys. Solid State 11 (1970) 2628. [17] A. Pimenov, A.A. Mukhin, V.Yu. Ivanov, V.D. Travkin, A.M. Balbashov, A. Loidl, Nat. Phys. 2 (2006) 97. [18] S.-W. Cheong, M. Mostovoy, Nat. Mater. 6 (2007) 13. [19] I. Dzyaloshinskii, J. Phys. Chem. Solids 4 (1958) 241. [20] T. Moriya, Phys. Rev. 120 (1960) 91. [21] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Harcourt Asia Pte Ltd., Singapore, 1976. [22] H. Katsura, A.V. Balatsky, N. Nagaosa, Phys. Rev. Lett. 98 (2007) 027203.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
New collective modes in multiferroic TbMnO3 : The electromagnons Shruti Shukla ∗ , Debanand Sa Department of Physics, Banaras Hindu University, Varanasi -221005, India
article
info
Article history: Received 27 February 2009 Received in revised form 22 May 2009 Accepted 29 May 2009 by Y.E. Lozovik Available online 2 June 2009
abstract Using a phenomenological Landau theory as well as equations of motion, we look for new excitations in the multiferroic TbMnO3 . Coupling these excitations to an electromagnetic field, we derived the magneto–polariton dispersion relation. Further, the frequency as well as temperature dependent dielectric susceptibility, and hence the refractive index, are obtained in this formulation. The imaginary part of the latter is computed and compared with the existing experimental data. © 2009 Elsevier Ltd. All rights reserved.
PACS: 78.20.Ls 75.80.+q 75.30.Fv 77.80.-e Keywords: A. Multiferroics D. Magneto-optical studies D. Spin density wave
Multiferroic materials are those where more than one ferroic order such as magnetism and ferroelectricity coexist together and are coupled to each other. From the application point of view, coupling between ferroelectricity and magnetism would be useful in multistate memory devices with mutual electric and magnetic control or as magnetically switchable optical devices among others. The coexistence of ferroelectric and magnetic order presents a fundamental challenge since the origins of both the orders are very different: magnetism is related to the ordering of spins of electrons in incomplete ionic shells, whereas ferroelectricity results from the relative shifts of negative and positive ions. Much of the earlier work on multiferroics was directed towards bringing them together in one material [1], which was proved to be difficult because of their mutually exclusive nature [2]. The long sought control of electric properties by magnetic fields was achieved in a class of multiferroics [3–6] called frustrated magnets, namely RMnO3 , GdMnO3 , RMn2 O5 , CoCr2 O4 , MnWO4 , etc. These materials become unique not due to the strength of their coupling (electric polarization is two to three orders of magnitude smaller than in the typical ferroelectrics) but the high sensitivity of their dielectric properties to an applied magnetic field [7–13]. This is in the heart of complex spin structure in these materials which makes the ferroelectricity of magnetic origin. For example, the multiferroic TbMnO3 , which has orthorhombically distorted perovskite
∗
Corresponding author. E-mail address: [email protected] (S. Shukla).
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.05.044
(space group Pbnm) structure at high temperature, is identical to the colossal magnetoresistive manganites in terms of the electronic configuration of the Mn3+ ion. In LaMnO3 , staggered orbital order becomes responsible for the layered antiferromagnetic (AF) order, whereas the spin structure in TbMnO3 is of sinusoidal AF type (of Mn3+ moments) below a temperature TN ≈ 41 K E (0, qs , 1). The wave vector qs is incommenwith a wave vector q surate (∼0.295) at TN , decreases with decreasing T and takes a constant value (0.28) below Tl ≈ 28 K. Neutron scattering measurements [14] revealed that the second transition corresponds to a noncollinear spiral magnetic structure where a simultaneous static polarization appears along the z-axis. This is confirmed by X-ray diffraction measurements [15], where the modulated magnetic phase is accompanied by a magnetoelastically induced lattice modulation. The existence of new elementary excitations in such magneto–electric (ME) coupled systems called electromagnons was already predicted long ago [16], and the first possible observations were made [17] in GdMnO3 and TbMnO3 . The dielectric data obtained for TbMnO3 without a magnetic field and with the ac component E k x exhibit a broad peak with characteristic frequency ω = (20 ± 3) cm−1 . The dielectric contribution of this mode increases with decreasing temperature, and below Tl its width decreases. Similar effects are observed for GdMnO3 also. The application of a static magnetic field along the z-axis suppresses the imaginary part of the dielectric function by more than a factor of 2, and also the real part. This provides a clear indication that the observed mode is related to the modulated magnetic structure.
1558
S. Shukla, D. Sa / Solid State Communications 149 (2009) 1557–1560
In this study, we considered a Landau free energy (from symmetry considerations) to address both the magnetic phases in multiferroic TbMnO3 . Using equations of motion, in addition to Maxwell’s equation, we coupled the electromagnetic radiation to the excitations in these system and obtained the magneto–polariton dispersion. In order to compare our finding with the available experiments, we derived an expression for the frequency and temperature dependent dielectric constant. The imaginary part of the dielectric function is computed and is compared with the observed spectrum. In almost all the multiferroics listed above, complex magnetic structures and phase diagrams are observed which show strong interplay between magnetic and dielectric phenomena [18]. Thus, the key question would be how the magnetic ordering induces ferroelectricity. This question has been addressed by Katsura et al. [10], which is based on the idea of the spin current induced between noncollinear spins yielding an electric dipole moment. This can be regarded as the effect of either Dzyaloshinskii–Moriya (DM) interaction [19,20] or spin–lattice interaction [13]. In the present work, we consider the phenomenological treatment of this mechanism. The coupling between ferroelectric polarization and magnetization is governed by the symmetries of these order parameters. The minimal coupling which could yield ferroelectricity is nonlinear [12]. Thus, we start with a free energy which has magnetic and ferroelectric components, and with the coupling between them given as F =V
−1
Z
dE r (Fs + Fl + Fs−l )
(1)
1
Fs =
2
E2 + aM 1
1 2
4
1 2
2
bPz2 − Ez Pz
(3)
and Fs−l = ηPz (Mz ∂y My − My ∂y Mz ).
(4)
In constructing the free energy, the details of the unit cell for TbMnO3 have not been taken into account. Here, the parameters a = a0 (T − T0 ), a0 , u, b, α are positive and γ is negative. The parameters γ and α are responsible for the modulation of the magnetic order parameter. The anisotropy parameter w is considered to be positive. The parameter η represents the coupling between the ferroelectricity and the magnetization. Considering the equilibrium AF states in TbMnO3 to be My0 = M1 cos ky and Mz0 = M2 sin ky, the above free energy is minimized with respect to the order parameters M1 , M2 , Pz and the modulation vector k. Thus, one finds the sinusoidal collinear (I) and the noncollinear spiral (II) magnetic phases, which are given as follows: M2 = 0, Pz0 = 0, M12 = −4L1 /3u, L1 = a − ac , ac = γ 2 /4α, k2 = −γ /2α.
2 M1 = (L2 − 3L1 + 1 )/2u, M22 = (L1 − 3L2 + 2 )/2u, II : L2 = a − ac + w, 1,2 = 2k2 η2 (3L1,2 − 5L2,1 )/ub, Pz0 = ηkM1 M2 /b.
(5)
(6)
In the minimization, we have assumed that the magnetoelectric coupling is weak; that is,
η2 k2 ub
(8)
+ η(2pz ∂y My0 + My0 ∂y pz ) = 0. 1
2
λ¨pz + bpz − Ez pz + η(mz ∂y My0 − My0 ∂y mz ) = 0
1
E4 wMz2 + uM
+ γ {(∂y My )2 + (∂y Mz )2 } + α{(∂y2 My )2 + (∂y2 Mz )2 }, (2)
I:
δF =0 δ pz (7) δF ¨ z + δ2 m ˙z + =0 µm δ mz where µ and λ are the density parameters which characterize the kinetic energy of the system. In the limit of no damping (δ1 = 0 = δ2 ), these equations can explicitly be written as λ¨pz + δ1 p˙ z +
2 ¨ z + (a + w + uMy0 µm − γ ∂y2 + α∂y4 )mz
where
Fl =
η2 k2 w
6 uba such that Tl < TN . Thus, the above minimization scheme 0 yields a phase diagram where, on lowering T , one obtains the sinusoidal AF phase below TN , and on further lowering (below Tl ) it is the noncollinear spiral magnetic phase which prevails. The ferroelectricity of magnetic origin appears below Tl (see the last expression in Eq. (6)). Since the system under consideration is ME coupled, the excitations in such a system are investigated using equations of motion method. In order to do this, we consider the fluctuations E −M E 0 ) and the electric polarization E = M in the AF vector (m E E (Ep = P − P0 ) with respect to their equilibrium values. Further, these excitations can be coupled to the z-component of the electromagnetic field (Ez ), propagating along the y-direction. This is taken care of by Maxwell’s equations since its electric field component Ez excites polarization along the z-direction. Thus, the Maxwell’s equations, along with the equations of motion in the linear approximation, give rise to a coupled set of equations for pz and mz which are written as
1. The transition temperature ac 3w and Tl ≈ TN − 2a + a0 0
TN and Tl are obtained as TN = T0 +
In order to get the modes of excitations, these equations are solved P in Fourier space, and are taken to be mz (y, t ) = l ml exp[i{(lk+q) y − ωt }] and Ez (pz )(y, t ) = E0 (p0 ) exp[i(qy − ωt )]. Here, q is the wave vector associated with the electromagnetic field propagation and is assumed to be much smaller than that of the wave vector of magnetic order parameter which has sinusoidal variation (q k). In such an approximation, the higher harmonics in l except the first one can be neglected. Moreover, using Maxwell’s equation, E0 can be eliminated from the above equation by expressing in terms of p0 ; that is, 4π p0 = E0 (n2 − 1), n(= qc /ω) being the refractive index of the material. Thus, the above equations can be reduced to a coupled equation in terms of p0 , m1 and m−1 , written as B iD−k iDk
−iDk −R
−iD−k
L−
−R
!
L+
p0 m −1 m1
!
0 0 0
!
=
(9)
where B = λω2 − b + 4π /(n2 − 1), L± = µω2 − a − w − γ (q ± k)2 − α(q ± k)4 − uM12 /2, Dk = ηM1 (k − q/2) and R = uM12 /4. As already mentioned above, q k, so we take q = 0 (optic modes only), which simplifies the parameters as D−k = −Dk , L+ = L− and m−1 = −m1 . Thus, the above equation simplifies to a 2 × 2 matrix equation in terms of p0 and m (m = m1 = −m−1 ) whose solution is obtained by solving the determinant. These yields the energies/frequencies of new q = 0 excitations called electromagnons, which are given as
ω
2 1,2
=
1 2
ω +ω ∓ 2 p
2 0
q
(ω − ω ) + 8η 2 p
2 2 0
2 k2 M 2 1
(µλ)−1
,
(10)
where ω0 and ωp are respectively the magnetic and dipole frequencies, defined as ω02 = µ1 [w + 2(a − ac )/3] = 3µ0 (T − Tl ) + 2a
η2 k2
4 ub w and ωp2 = b/λ. It should be noted here that the magnetic µ frequency ω0 depends on temperature and it takes the lowest value at T = Tl .
S. Shukla, D. Sa / Solid State Communications 149 (2009) 1557–1560
Fig. 1. Variation of the imaginary part of the dielectric function with frequency. The calculated value of 2 (ω, T ) is fitted with the dielectric data of Pimenov et al. at 20 K (filled circles).
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Fig. 2. Variation of the imaginary part of the dielectric function with temperature at ω = 6 cm−1 and ω = 4 cm−1 .
The frequency and temperature dependent dielectric function
((ω, T )) (which is the square of the refractive index n) in such a formulation is calculated to be
(ω, T ) =
(ω2 − Ω12 )(ω2 − Ω22 ) , (ω2 − ω12 )(ω2 − ω22 )
(11)
where the expression for Ω12,2 is the same as that of ω12,2 with ωp replaced by Ωp = (b+λ4π) . This result is contrasted with the polariton problem in conventional ferroelectrics where the longitudinal and the transverse optic mode frequencies appear as zeros and poles respectively in the frequency dependent dielectric function. One of the transverse optic mode softens toward the phase transition and is condensed according to the Lyddane–Sachs–Teller (LST) relation [21]. This conventional viewpoint is not relevant in the case of multiferroics since the lattice displacement is not essential to the electronic polarization. Therefore, it is important to identify the relevant collective modes [22] that are responsible for ferroelectricity of magnetic origin and to study the dynamical magneto–electric effect. The dielectric function (ω, T ) can in general be a complex quantity. This is realized from Eq. (10) by making ω → ω + iδ (δ being the imaginary part). Thus, writing (ω, T ) = 1 (ω, T ) + i2 (ω, T ) (1 and 2 are respectively the real and imaginary part of the dielectric function), one can compute 2 (ω). The computed value of 2 (ω, T ) is fitted with the dielectric data of Pimenov et al. [17] at 20 K (see Fig. 1), which yields the frequencies as ω1 = 23.2 cm−1 , Ω1 = 26.3 cm−1 , ω2 = 78.5 cm−1 and Ω2 = 231.2 cm−1 . From this, the magnetic and lattice frequencies are estimated to be ω0 (T = 20 K) = 67.5 cm−1 , ωp = 75.7 cm−1 and Ωp = 230.6 cm−1 . In the process of fitting, the dimensionless η2 k2
ME coupling came out to be small, that is, ub = 0.031, whereas the width parameter δ turned out to be considerably large (δ = 11.0 cm−1 ), which is a case of overdamping. However, on lowering T , the Tb spins order at T = 7 K, and it is unclear at this moment whether the narrowing of excitations results from such an ordering. Using the above parameters, the variation of the imaginary part of the dielectric function with temperature at ω = 6 cm−1 and ω = 4 cm−1 is shown in Fig. 2. It is clear from the figure that 2 (ω, T ) shows a peak at Tl , and its width decreases once the frequency is lowered. From the above discussions, it is obvious that the magnetic frequency (ω0 ) is smaller than the lattice frequencies (ωp , Ωp ). Of both the electromagnon modes ω2 and ω1 , the latter takes a minimum value (∼(η2 k2 /ub)2 ) at T = Tl . This is the signature of a soft mode (see Fig. 3) which is associated with the second
Fig. 3. Variation of electromagnon frequencies ω1 and ω2 with temperature, showing the soft mode.
transition (T = Tl ), which is from the collinear AF phase to the noncollinear spiral magnetic structure where ferroelectricity appears. In contrast, the electromagnon mode ω2 is almost temperature independent and the energy associated with it is high compared to ω1 . Using Eq. (10), the magneto–polariton dispersion spectrum (ω2 vs. q) is analyzed. It yields a sixth-order algebraic equation whose solutions are shown in Fig. 4. The role of ME coupling is to lower the magnetic frequency (ω0 ) to ω1 , whereas it pushes up the lattice frequency (ωp ) to ω2 . From the dispersion it is obvious that there are two frequency regions near ω1 and ω2 where the electromagnons interact very strongly with the electromagnetic radiation. The nature of the excitations near ω2 is dominated by the electric polarization mode whereas near ω1 the prominent excitations are of magnetic origin. Since ω2 ω1 and the lowenergy physics in such a system is dictated by ω1 -excitations, the window (see the inset in Fig. 4) where the resonant interaction of electromagnetic radiation with the ω1 -electromagnon occurs is calculated as
η2 k2 M12 Ω1 − ω 1 ≈ ω0 λµ
"
1
ωp2 − ω02
−
1
Ωp2 − ω02
# ,
(12)
which is very small. Since ω1 takes a minimum value at T = Tl , the above interval is largest at Tl compared to any other temperature. Further, this window vanishes in the limit of either η → 0 or k → 0. Since, in the present formulation, one can excite the electromagnons by applying an ac electric field, the ac magnetization mz
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S. Shukla, D. Sa / Solid State Communications 149 (2009) 1557–1560
coupling the excitations to electromagnetic radiation. By solving the dynamical equations, the energy of excitations as well as the magneto–polariton dispersion is obtained. In order to compare our finding with the available experiments, we derive an expression for the frequency and temperature dependent dielectric constant. The imaginary part of the dielectric function is computed and compared with the observed spectrum. We believe that such excitations can be exploited for the design of magneto–optical devices. References
Fig. 4. Magneto–polariton dispersion (ω2 vs. q) using the parameters from the fit in Fig. 1. The values shown along the y-axis are the corresponding ω2 ’s. The inset corresponds to a zoomed version of the gap between ω12 and Ω12 .
can be calculated using Eq. (8) as
ω2 − Ωp2 mz = E . (4π /λ)(2ηkM1 ) 0
(13)
Considering the frequency to be ω ∼ 20 cm−1 , which corresponds to terahertz region, the highest electric field intensity is E0 ∼ 103 V/cm. Putting theses values along with the parameters estimated above, one can calculate the magnitude of the ac ×10 magnetization as mz = ( 0.5√ µλ
−2
V ) cm . In our calculation above, we
have used the parameters 4λπ = 47 470 cm−2 and w = 4.1 × µ 103 cm−2 . By choosing w = 0.001 (in proper units), the value of mz is obtained as 0.4 × 10−2 µB /Mn atom, which is comparable with the experimental value [3]. In conclusion, we summarize the main findings of the paper. We consider a Landau free energy (from symmetry considerations) to address both the magnetic phases in multiferroic TbMnO3 . We show that in ME materials, new hybrid spin–lattice excitations exist which can be excited by ac electric fields. This is done by
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