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Anomalous sound velocity in multiferroic BiFeO3
Solid State Communications 245 (2016) 55–59
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
Anomalous sound velocity in multiferroic BiFeO3 Xian-Sheng Cao a,b,n, Gao-Feng Ji a, Xing-Fang Jiang a,b a b
School of Mathematics and Physics, Changzhou University, 1 Gehu Road, Changzhou 213164, China Institute of Modern Optical Technologies, Soochow University, 1 Shizi Street, Suzhou 215006, China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 March 2016 Accepted 20 July 2016 Available online 21 July 2016
The sound velocity in multiferroic BiFeO3 (BFO) is studied with using Green's function technology on the basis of the magnetoelectric coupling, the spin-phonon interaction and the anharmonic phonon–phonon interaction. The Heisenberg-like model is employed to describe the magnetic subsystem, and the transverse Ising model is used to explain the ferroelectric subsystem. The reduced velocity is obtained in the limit of zero wave vectors. It is shown that the reduced velocity of sound in BiFeO3 exhibits a kink at the magnetic phase transition temperature TN. This anomaly in reduced velocity can be explained as an influence of vanishing magnetic ordering above TN and the ferroelectric subsystem can not be influenced by the magnetic subsystem above TN due to TN { TC in the BFO. It is shown that the influence of the RM is only below TN in the phase where ferroelectric and magnetic properties exist together, whereas the RE influences the properties of the reduced velocity in the whole temperature region (T oTC). Above TC only the anharmonic phonon–phonon interactions remain. It can also be seen that the reduced velocities decrease with increasing temperature. The achieved conclusion is in accordance with the experimental results. & 2016 Elsevier Ltd. All rights reserved.
Keywords: A. Multiferroic C. Anharmonic phonon–phonon interaction D. Sound velocity
1. Introduction BiFeO3 (BFO) has attracted much attention in recent years because of its rare or even unique properties of having room-temperature ferroelectric and magnetic order. It has a high Curie temperature of ferroelectricity (TC ≈ 1100 K) and a high Neel temperature (TN≈ 640 K) of G type antiferromagnetism (AFM). [1–8] At room temperature, BFO appears a rhombohedrically distorted (space group R3c) perovskite structure [1]. The AFM symmetry is a cycloidal helix of spins with a long period (62 nm), and incommensurate with the lattice periodicity [2,3]. BFO shows a transition from rhombohedral to orthorhombic symmetry, around 1100 K [4] (or perhaps monoclinic [5]) and it becomes cubic around 1200 K. Theoretical researches can be implemented by either macroscopic [6–8] or microscopic [9–12] methods. Especially, Wesselinowa et al. [13,14] have used the method of Tserkovnikov [15] to investigate the phonon spectra of BFO nanoparticles. Very recently, there have been many sound velocity measurements of the multiferroic BFO at low temperatures with and n
Corresponding author at: School of Mathematics and Physics, Changzhou University, 1 Gehu Road, Changzhou 213164, China. E-mail address: [email protected] (X.-S. Cao).
without an external magnetic field [16,17]. The sound velocity has benefit from conduction electrons, lattice, and spin waves according to these experiments. The thermodynamic Green's function is a powerful technique and it gets the related macroscopic quantities at finite temperatures. On the other hand, phonon scattering is a useful tool for studying various vibration and spin excitations, lattice distortions, phase transitions, etc. In addition, little theoretical work has been done on the sound velocity in multiferroic BFO. In the present study, we report the microscopic theory of the sound velocity in BFO. For this purpose, the Heisenberg-like model is employed to describe the magnetic subsystem, and the transverse Ising model is used to explain the ferroelectric subsystem. The magnetoelectric coupling, the spin-phonon interaction and the anharmonic phonon–phonon interaction are also included in Section 2. The sound velocity is calculated in Section 3. The results and discussions are given in Section 4.
2. Model The Hamiltonian for the multiferroic BiFeO3 (BFO) system can be presented as:
Hmu = HM + HE + HME + HPH + HPH
http://dx.doi.org/10.1016/j.ssc.2016.07.022 0038-1098/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: X.-S. Cao, et al., Solid State Commun (2016), http://dx.doi.org/10.1016/j.ssc.2016.07.022i
(1)
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X.-S. Cao et al. / Solid State Communications 245 (2016) 55–59
The Hamiltonian of the magnetic subsystem ( HM ) is given by:
J J HM = − 1 ∑ Si⋅Sj − 2 ∑ Si⋅Sj − 2 ⎡⎣ ij⎤⎦ 2 ij
∑ Di ( Siz )2 − Bz ∑ Siz i
∑ ( ω) = V ( 3)
k
(3) Skz
The Eq. (3) is the transverse Ising model (TIM), and are the spin-1/2 operators of the pseudo-spins at the site k, Ez is the external electric field along the z direction, Ikl > 0 denotes the exchange coupling constant between the nearest-neighbor pseudospin, and Ω is the tunneling frequency. The Eq. (4) is the Hamiltonian of the coupling between the magnetic and the electric subsystems ( HME ) in BFO:
= − g∑ ∑ ij
Pkz Plz Si Sj
(4)
kl
where g is the magnetoelectric coupling constant. The Hamilton of the spin-phonon interactions ( H SP − PH ) can be read
H SP − PH = −
∑ F M ( q) A q S−zq − 1 ∑ RM ( q, q1) A q A−q1Sqz1− q 2 qq1
q
−
∑ F E ( q) A q P−zq − 1 ∑ RE ( q, q1) A q A−q1Pqz1− q 2 qq1
q
(5)
+
)
(
G RE ( t , t′) =
)
= − iΘ ( t − t′)
1 ω2 − ω02 − ∑ ( ω)
(10)
)
bnσ ( t ); bn+σ ( t′)
(11)
(12)
here
ωE2 = δ2 + Ω2
(13a)
δ = IP + Ez + F M A +
1 M 2 R A 2
(13b)
(13c)
I1 = I /2 + 2g ∑ ⎡⎣ ij⎤⎦
(
Si+S −j + Siz S zj
)
(13d)
The related Green's function of FM subsystem is [21]
GkRM ( t , t′) =
Sk+ ( t ), Sk− ( t′)
(14)
( Φ + 1 + S ) Φ2S + 1 − ( Φ − S )( Φ + 1)2S + 1 ( Φ + 1)2S + 1 − Φ2S + 1
(15)
here
Φ=
1 1 ∑ N k e βωM ( k) − 1
( J10 ′ − J1k ′ + J20
(16a) − J2k + 2DΘmz ) + Bz + F M ( 0) A
1 M R ( 0) A2 2
J10 ′ = z1J1 ′
(16b) (16c)
J1k ′ = J1 ′∑ eik ⋅ ( i − j) (7)
The equations of motion for the phonon Green's function can be derived in accordance with the total Hamiltonian
Dqq ′ ( ω) =
(
P z = ( δ/2ωE ) tanh ( ωE /2kB T )
+
The phonon retarded Green function is defined as following: [19]
< ⎡⎣ A q ( t ); Aq+′ ( t′) ⎤⎦ >
q1
⎤ 1 ∑ V ( 4) 2N¯ q + 1 ⎥⎥ N q ⎦
Following the method of Teng et al. [20], the average of the zcomponent of the pseudo-spin is
(6)
3. Green's functions
A q ( t ); Aq+′ ( t′)
q1
S z − ∑ R E ( k ) Pqz1− q +
)
ωM = Skz
Dq, q ′ ( t − t′) =
∑ RM ( k )
where N¯ q = 1/⎡⎣ exp ( ω¯ q /T ) − 1⎤⎦. The pseudo-spins-1/2 of FE subsystem can be transformed to P x = bi+σ biσ¯ + bi+σ¯ biσ /2, P z = bi+σ biσ − bi+σ¯ biσ¯ /2. Here bn+σ is creation operator and bnσ is annihilation operator. They are the fermiontype operators. We use them to construct the related Green's function of FE subsystem.
Sz =
here ω0 is the harmonic phonon frequency of the lattice mode. Aq = 1/2ω0 aq + aq+¯ , Bq = i ω0/2 aq+ − aq¯ . The coefficients V(3) and V(4) are Fourier transforms of the anharmonic phonons of the third- and fourth-order atomic force constants [18].
(
⎡ ⎢ω − ⎢ 0 ⎣
The expression of the magnetization can be solved according to Callen method [22]
)
1 ∑ V ( 4)A q1A q2 A−q − q2 A−q1+ q 4! qq1q2
(9)
⎤ ⎡ 1 A = ⎢ 2F M ( k ) S z + 2F E ( k ) P z − ∑ V ( 3) 2N¯ q + 1 ⎥ ⎥ ⎢ N q ⎦ ⎣
l
1 ∑ Bq B−q + ω02 A q A−q + 1 ∑ V ( 3)A q A−q1A q1− q 2! q 3! qq1
(
)
IP = I1∑ Plz
here the first two terms describe the coupling between the magnetic order parameter and the phonons, while the last two terms represent the coupling between the ferroelectric pseudospins and phonons. The Hamiltonian of phonon is
HPH =
(
here
(
1 ∑ Ikl Pkz Plz − Ez ∑ Pkz 2 kl k Skx
HME
1 V ( 4) A2 − RM S z − RE P z 2
A +
(2)
i
The Eq. (2) is the Heisenberg-like Hamiltonian, Si is the Heisenberg magnetic spin at the site i, ⎡⎣ ij ⎤⎦ and ij denote summation over the nearest neighbor (nn) and the next-nearest neighbor (nnn) spins, respectively. J1 ( >0) and J2 ( <0) are the exchange coupling constants between the nn and nnn magnetic spins, respectively. Di (Do0) is the single-site anisotropy parameter. Bz is the external applied longitudinal magnetic field (along z direction). The Hamiltonian of the electrical subsystem ( HE ) reads:
HE = − Ω∑ Pkx −
where
⎡⎣ ij⎤⎦
J20 = z2 J2 J2k = J2 ∑ eik ⋅ ( i − j)
(8)
ij
Please cite this article as: X.-S. Cao, et al., Solid State Commun (2016), http://dx.doi.org/10.1016/j.ssc.2016.07.022i
(16d) (16e)
(16f)
X.-S. Cao et al. / Solid State Communications 245 (2016) 55–59
57
J1 ′ = J1 + 2g ∑ Pkz Plz ⎡⎣ kl⎤⎦
Θmz = 1 −
(16g)
1 ⎡ z z ⎣ S ( S + 1) − Sk Sk ⎤⎦ 2S
(16h)
z1 and z2 are the numbers of nn and nnn spins, respectively.
4. Sound velocity The renormalized phonon propagator in Eq. (8) has poles at frequencies ω given by
ω2 − ω02 −
∑ ( ω) = 0
(17)
ω = vq and ω0 = v0 q ( v and v0 are the normalized and bare longitudinal sound velocities) in the long wavelength limit q → 0. Then Eq. (17) reduces to [23,24]
⎛ ω ⎞2 Re ∑ ( ω, q = 0) =0 ⎟ −1− ⎜ ⎝ ω0 ⎠ ω02
Fig. 1. Plot of reduced velocity ( v˜ ) vs. temperature (T) for different values of the coupling between the ferroelectric pseudo-spin and phonon RE ¼ 25, 20.
(18)
The reduced velocity
v˜ =
v = v0
1 − H2
where H =
1 ω0
1⎡ M R 2⎣
(19) S z + RE P z − V ( 4) A2 ⎤⎦ − V ( 3) A .
5. Results and discussion In this section we present the numerical results based on our theoretical calculations where the following model parameters are used for the BFO: TC ¼1100 K, TN ¼ 650 K, D ¼ 5K, J1 ¼85 K, J2 ¼ 25K, I ¼650 K, Ω ¼50 K, and g¼ 20 K, ω0 = 340cm−1, FE ¼5 cm 1, FM ¼3 cm 1, RE ¼ 25 cm 1, RM ¼ 5 cm 1, V(3) ¼0.5 cm 1, V(4) ¼ 5 cm 1, the spin S¼ 2 and the pseudo-spin P ¼0.5. The temperature variation of the reduced velocity of sound of BiFeO3 exhibits a kink at the magnetic phase transition temperature TN ¼650 K. This anomaly in reduced velocity can be explained as an influence of vanishing magnetic ordering above TN in the BFO. It is well known that the ferroelectric subsystem can not be influenced by the magnetic subsystem above TN due to TN ooTC in BFO. The magnetic and ferroelectric phases coexist only below TN. The achieved conclusion is in accordance with the experimental results [16,17]. Figs. 1 and 2 present the plot of reduced velocity v˜ versus T for different values of the different anharmonic spin-phonon interaction RE, RM. It is found that the RM affects the coexisting phase of ferroelectric and magnetic properties only below TN, whereas the RE influences the properties of the reduced velocity in the whole temperature region To TC. Above TC only the anharmonic phonon– phonon interactions are remained (Figs. 3 and 4). It can be seen that the reduced velocities decrease with increasing temperature. This is in agreement with the experiment [25]. It is found that the E(1) mode softens strongly from 5 K to room temperature with using the infrared reflectivity measurements in this experiment. This behavior also agrees with the high temperature infrared [26] and the Raman measurements [27]. It can be seen that the reduced velocity depends strongly on g in the temperature between TN and TC (Fig. 5). Trimper et al. [28] argued that a larger g leads to an enhanced polarization with a more pronounced peak around the Neel temperature TN. This result is also in good qualitative agreement with the experimental result of Vijayasundaram et al. [29]. The ferroelectric and magnetic
Fig. 2. Plot of reduced velocity ( v˜ ) vs. temperature (T) for different values of the coupling between the magnetic order parameter and the phonon RM ¼ 5, 2.
Fig. 3. Plot of reduced velocity v˜ vs. T for different values of anharmonic phonons of the third -order atomic force constant V(3) ¼0.5, 0.8.
quantities are appreciably enhanced and the structural distortion is found in the rhombohedral structure (R3c) in the Co and Gd codoped BFO [29]. It is argued that the distorted crystal structure is involved in the contribution of enhanced magnetization and polarization [29], which lead to the reduced velocity increases with g increases in the temperature between TN and TC. As mentioned
Please cite this article as: X.-S. Cao, et al., Solid State Commun (2016), http://dx.doi.org/10.1016/j.ssc.2016.07.022i
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X.-S. Cao et al. / Solid State Communications 245 (2016) 55–59
only the anharmonic phonon–phonon interactions remain. It can be also seen that the reduced velocities decrease with increasing temperature. The conclusion is in accordance with the experimental results.
Acknowledgments The financial support of this research project by Open Research Fund of Institute of Modern Optical Technologies of Soochow University (KJS1405) is gratefully acknowledged.
References
Fig. 4. Plot of reduced velocity v˜ vs. T for different values of anharmonic phonons of the fourth-order atomic force constant V(4) ¼ 5, 8.
Fig. 5. Plot of reduced velocity ( v˜ ) vs. temperature (T ) for different values of the magnetoelectric coupling constant g ¼20, 10.
earlier, it is the reduced velocity which increases with the increasing of magnetization and polarization based on Eq. (19). Similar experimental results were also reported in other co-doped BFO [30–32].
6. Conclusion By using a Green's function technique on the basis of the magnetoelectric coupling, the spin-phonon interaction and the anharmonic phonon–phonon interaction, we have calculated the reduced velocity as a function of temperature. The Heisenberg-like model is employed to describe the magnetic subsystem, and the transverse Ising model is used to explain the ferroelectric subsystem. It is shown that the reduced velocity of sound of BiFeO3 exhibits a kink at the magnetic phase transition temperature TN. This anomaly reduced velocity can be explained as an influence of vanishing magnetic ordering above TN and the magnetic subsystem have no influence on the ferroelectric above TN due to TN { TC in the BFO. It is shown that the influence of the RM is only below TN in the phase where ferroelectric and magnetic properties exist together, whereas the RE influences the properties of the reduced velocity in the whole temperature region ToTC. Above TC
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Please cite this article as: X.-S. Cao, et al., Solid State Commun (2016), http://dx.doi.org/10.1016/j.ssc.2016.07.022i
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
Anomalous sound velocity in multiferroic BiFeO3 Xian-Sheng Cao a,b,n, Gao-Feng Ji a, Xing-Fang Jiang a,b a b
School of Mathematics and Physics, Changzhou University, 1 Gehu Road, Changzhou 213164, China Institute of Modern Optical Technologies, Soochow University, 1 Shizi Street, Suzhou 215006, China
art ic l e i nf o
a b s t r a c t
Article history: Received 16 March 2016 Accepted 20 July 2016 Available online 21 July 2016
The sound velocity in multiferroic BiFeO3 (BFO) is studied with using Green's function technology on the basis of the magnetoelectric coupling, the spin-phonon interaction and the anharmonic phonon–phonon interaction. The Heisenberg-like model is employed to describe the magnetic subsystem, and the transverse Ising model is used to explain the ferroelectric subsystem. The reduced velocity is obtained in the limit of zero wave vectors. It is shown that the reduced velocity of sound in BiFeO3 exhibits a kink at the magnetic phase transition temperature TN. This anomaly in reduced velocity can be explained as an influence of vanishing magnetic ordering above TN and the ferroelectric subsystem can not be influenced by the magnetic subsystem above TN due to TN { TC in the BFO. It is shown that the influence of the RM is only below TN in the phase where ferroelectric and magnetic properties exist together, whereas the RE influences the properties of the reduced velocity in the whole temperature region (T oTC). Above TC only the anharmonic phonon–phonon interactions remain. It can also be seen that the reduced velocities decrease with increasing temperature. The achieved conclusion is in accordance with the experimental results. & 2016 Elsevier Ltd. All rights reserved.
Keywords: A. Multiferroic C. Anharmonic phonon–phonon interaction D. Sound velocity
1. Introduction BiFeO3 (BFO) has attracted much attention in recent years because of its rare or even unique properties of having room-temperature ferroelectric and magnetic order. It has a high Curie temperature of ferroelectricity (TC ≈ 1100 K) and a high Neel temperature (TN≈ 640 K) of G type antiferromagnetism (AFM). [1–8] At room temperature, BFO appears a rhombohedrically distorted (space group R3c) perovskite structure [1]. The AFM symmetry is a cycloidal helix of spins with a long period (62 nm), and incommensurate with the lattice periodicity [2,3]. BFO shows a transition from rhombohedral to orthorhombic symmetry, around 1100 K [4] (or perhaps monoclinic [5]) and it becomes cubic around 1200 K. Theoretical researches can be implemented by either macroscopic [6–8] or microscopic [9–12] methods. Especially, Wesselinowa et al. [13,14] have used the method of Tserkovnikov [15] to investigate the phonon spectra of BFO nanoparticles. Very recently, there have been many sound velocity measurements of the multiferroic BFO at low temperatures with and n
Corresponding author at: School of Mathematics and Physics, Changzhou University, 1 Gehu Road, Changzhou 213164, China. E-mail address: [email protected] (X.-S. Cao).
without an external magnetic field [16,17]. The sound velocity has benefit from conduction electrons, lattice, and spin waves according to these experiments. The thermodynamic Green's function is a powerful technique and it gets the related macroscopic quantities at finite temperatures. On the other hand, phonon scattering is a useful tool for studying various vibration and spin excitations, lattice distortions, phase transitions, etc. In addition, little theoretical work has been done on the sound velocity in multiferroic BFO. In the present study, we report the microscopic theory of the sound velocity in BFO. For this purpose, the Heisenberg-like model is employed to describe the magnetic subsystem, and the transverse Ising model is used to explain the ferroelectric subsystem. The magnetoelectric coupling, the spin-phonon interaction and the anharmonic phonon–phonon interaction are also included in Section 2. The sound velocity is calculated in Section 3. The results and discussions are given in Section 4.
2. Model The Hamiltonian for the multiferroic BiFeO3 (BFO) system can be presented as:
Hmu = HM + HE + HME + HPH + HPH
http://dx.doi.org/10.1016/j.ssc.2016.07.022 0038-1098/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: X.-S. Cao, et al., Solid State Commun (2016), http://dx.doi.org/10.1016/j.ssc.2016.07.022i
(1)
56
X.-S. Cao et al. / Solid State Communications 245 (2016) 55–59
The Hamiltonian of the magnetic subsystem ( HM ) is given by:
J J HM = − 1 ∑ Si⋅Sj − 2 ∑ Si⋅Sj − 2 ⎡⎣ ij⎤⎦ 2 ij
∑ Di ( Siz )2 − Bz ∑ Siz i
∑ ( ω) = V ( 3)
k
(3) Skz
The Eq. (3) is the transverse Ising model (TIM), and are the spin-1/2 operators of the pseudo-spins at the site k, Ez is the external electric field along the z direction, Ikl > 0 denotes the exchange coupling constant between the nearest-neighbor pseudospin, and Ω is the tunneling frequency. The Eq. (4) is the Hamiltonian of the coupling between the magnetic and the electric subsystems ( HME ) in BFO:
= − g∑ ∑ ij
Pkz Plz Si Sj
(4)
kl
where g is the magnetoelectric coupling constant. The Hamilton of the spin-phonon interactions ( H SP − PH ) can be read
H SP − PH = −
∑ F M ( q) A q S−zq − 1 ∑ RM ( q, q1) A q A−q1Sqz1− q 2 qq1
q
−
∑ F E ( q) A q P−zq − 1 ∑ RE ( q, q1) A q A−q1Pqz1− q 2 qq1
q
(5)
+
)
(
G RE ( t , t′) =
)
= − iΘ ( t − t′)
1 ω2 − ω02 − ∑ ( ω)
(10)
)
bnσ ( t ); bn+σ ( t′)
(11)
(12)
here
ωE2 = δ2 + Ω2
(13a)
δ = IP + Ez + F M A +
1 M 2 R A 2
(13b)
(13c)
I1 = I /2 + 2g ∑ ⎡⎣ ij⎤⎦
(
Si+S −j + Siz S zj
)
(13d)
The related Green's function of FM subsystem is [21]
GkRM ( t , t′) =
Sk+ ( t ), Sk− ( t′)
(14)
( Φ + 1 + S ) Φ2S + 1 − ( Φ − S )( Φ + 1)2S + 1 ( Φ + 1)2S + 1 − Φ2S + 1
(15)
here
Φ=
1 1 ∑ N k e βωM ( k) − 1
( J10 ′ − J1k ′ + J20
(16a) − J2k + 2DΘmz ) + Bz + F M ( 0) A
1 M R ( 0) A2 2
J10 ′ = z1J1 ′
(16b) (16c)
J1k ′ = J1 ′∑ eik ⋅ ( i − j) (7)
The equations of motion for the phonon Green's function can be derived in accordance with the total Hamiltonian
Dqq ′ ( ω) =
(
P z = ( δ/2ωE ) tanh ( ωE /2kB T )
+
The phonon retarded Green function is defined as following: [19]
< ⎡⎣ A q ( t ); Aq+′ ( t′) ⎤⎦ >
q1
⎤ 1 ∑ V ( 4) 2N¯ q + 1 ⎥⎥ N q ⎦
Following the method of Teng et al. [20], the average of the zcomponent of the pseudo-spin is
(6)
3. Green's functions
A q ( t ); Aq+′ ( t′)
q1
S z − ∑ R E ( k ) Pqz1− q +
)
ωM = Skz
Dq, q ′ ( t − t′) =
∑ RM ( k )
where N¯ q = 1/⎡⎣ exp ( ω¯ q /T ) − 1⎤⎦. The pseudo-spins-1/2 of FE subsystem can be transformed to P x = bi+σ biσ¯ + bi+σ¯ biσ /2, P z = bi+σ biσ − bi+σ¯ biσ¯ /2. Here bn+σ is creation operator and bnσ is annihilation operator. They are the fermiontype operators. We use them to construct the related Green's function of FE subsystem.
Sz =
here ω0 is the harmonic phonon frequency of the lattice mode. Aq = 1/2ω0 aq + aq+¯ , Bq = i ω0/2 aq+ − aq¯ . The coefficients V(3) and V(4) are Fourier transforms of the anharmonic phonons of the third- and fourth-order atomic force constants [18].
(
⎡ ⎢ω − ⎢ 0 ⎣
The expression of the magnetization can be solved according to Callen method [22]
)
1 ∑ V ( 4)A q1A q2 A−q − q2 A−q1+ q 4! qq1q2
(9)
⎤ ⎡ 1 A = ⎢ 2F M ( k ) S z + 2F E ( k ) P z − ∑ V ( 3) 2N¯ q + 1 ⎥ ⎥ ⎢ N q ⎦ ⎣
l
1 ∑ Bq B−q + ω02 A q A−q + 1 ∑ V ( 3)A q A−q1A q1− q 2! q 3! qq1
(
)
IP = I1∑ Plz
here the first two terms describe the coupling between the magnetic order parameter and the phonons, while the last two terms represent the coupling between the ferroelectric pseudospins and phonons. The Hamiltonian of phonon is
HPH =
(
here
(
1 ∑ Ikl Pkz Plz − Ez ∑ Pkz 2 kl k Skx
HME
1 V ( 4) A2 − RM S z − RE P z 2
A +
(2)
i
The Eq. (2) is the Heisenberg-like Hamiltonian, Si is the Heisenberg magnetic spin at the site i, ⎡⎣ ij ⎤⎦ and ij denote summation over the nearest neighbor (nn) and the next-nearest neighbor (nnn) spins, respectively. J1 ( >0) and J2 ( <0) are the exchange coupling constants between the nn and nnn magnetic spins, respectively. Di (Do0) is the single-site anisotropy parameter. Bz is the external applied longitudinal magnetic field (along z direction). The Hamiltonian of the electrical subsystem ( HE ) reads:
HE = − Ω∑ Pkx −
where
⎡⎣ ij⎤⎦
J20 = z2 J2 J2k = J2 ∑ eik ⋅ ( i − j)
(8)
ij
Please cite this article as: X.-S. Cao, et al., Solid State Commun (2016), http://dx.doi.org/10.1016/j.ssc.2016.07.022i
(16d) (16e)
(16f)
X.-S. Cao et al. / Solid State Communications 245 (2016) 55–59
57
J1 ′ = J1 + 2g ∑ Pkz Plz ⎡⎣ kl⎤⎦
Θmz = 1 −
(16g)
1 ⎡ z z ⎣ S ( S + 1) − Sk Sk ⎤⎦ 2S
(16h)
z1 and z2 are the numbers of nn and nnn spins, respectively.
4. Sound velocity The renormalized phonon propagator in Eq. (8) has poles at frequencies ω given by
ω2 − ω02 −
∑ ( ω) = 0
(17)
ω = vq and ω0 = v0 q ( v and v0 are the normalized and bare longitudinal sound velocities) in the long wavelength limit q → 0. Then Eq. (17) reduces to [23,24]
⎛ ω ⎞2 Re ∑ ( ω, q = 0) =0 ⎟ −1− ⎜ ⎝ ω0 ⎠ ω02
Fig. 1. Plot of reduced velocity ( v˜ ) vs. temperature (T) for different values of the coupling between the ferroelectric pseudo-spin and phonon RE ¼ 25, 20.
(18)
The reduced velocity
v˜ =
v = v0
1 − H2
where H =
1 ω0
1⎡ M R 2⎣
(19) S z + RE P z − V ( 4) A2 ⎤⎦ − V ( 3) A .
5. Results and discussion In this section we present the numerical results based on our theoretical calculations where the following model parameters are used for the BFO: TC ¼1100 K, TN ¼ 650 K, D ¼ 5K, J1 ¼85 K, J2 ¼ 25K, I ¼650 K, Ω ¼50 K, and g¼ 20 K, ω0 = 340cm−1, FE ¼5 cm 1, FM ¼3 cm 1, RE ¼ 25 cm 1, RM ¼ 5 cm 1, V(3) ¼0.5 cm 1, V(4) ¼ 5 cm 1, the spin S¼ 2 and the pseudo-spin P ¼0.5. The temperature variation of the reduced velocity of sound of BiFeO3 exhibits a kink at the magnetic phase transition temperature TN ¼650 K. This anomaly in reduced velocity can be explained as an influence of vanishing magnetic ordering above TN in the BFO. It is well known that the ferroelectric subsystem can not be influenced by the magnetic subsystem above TN due to TN ooTC in BFO. The magnetic and ferroelectric phases coexist only below TN. The achieved conclusion is in accordance with the experimental results [16,17]. Figs. 1 and 2 present the plot of reduced velocity v˜ versus T for different values of the different anharmonic spin-phonon interaction RE, RM. It is found that the RM affects the coexisting phase of ferroelectric and magnetic properties only below TN, whereas the RE influences the properties of the reduced velocity in the whole temperature region To TC. Above TC only the anharmonic phonon– phonon interactions are remained (Figs. 3 and 4). It can be seen that the reduced velocities decrease with increasing temperature. This is in agreement with the experiment [25]. It is found that the E(1) mode softens strongly from 5 K to room temperature with using the infrared reflectivity measurements in this experiment. This behavior also agrees with the high temperature infrared [26] and the Raman measurements [27]. It can be seen that the reduced velocity depends strongly on g in the temperature between TN and TC (Fig. 5). Trimper et al. [28] argued that a larger g leads to an enhanced polarization with a more pronounced peak around the Neel temperature TN. This result is also in good qualitative agreement with the experimental result of Vijayasundaram et al. [29]. The ferroelectric and magnetic
Fig. 2. Plot of reduced velocity ( v˜ ) vs. temperature (T) for different values of the coupling between the magnetic order parameter and the phonon RM ¼ 5, 2.
Fig. 3. Plot of reduced velocity v˜ vs. T for different values of anharmonic phonons of the third -order atomic force constant V(3) ¼0.5, 0.8.
quantities are appreciably enhanced and the structural distortion is found in the rhombohedral structure (R3c) in the Co and Gd codoped BFO [29]. It is argued that the distorted crystal structure is involved in the contribution of enhanced magnetization and polarization [29], which lead to the reduced velocity increases with g increases in the temperature between TN and TC. As mentioned
Please cite this article as: X.-S. Cao, et al., Solid State Commun (2016), http://dx.doi.org/10.1016/j.ssc.2016.07.022i
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X.-S. Cao et al. / Solid State Communications 245 (2016) 55–59
only the anharmonic phonon–phonon interactions remain. It can be also seen that the reduced velocities decrease with increasing temperature. The conclusion is in accordance with the experimental results.
Acknowledgments The financial support of this research project by Open Research Fund of Institute of Modern Optical Technologies of Soochow University (KJS1405) is gratefully acknowledged.
References
Fig. 4. Plot of reduced velocity v˜ vs. T for different values of anharmonic phonons of the fourth-order atomic force constant V(4) ¼ 5, 8.
Fig. 5. Plot of reduced velocity ( v˜ ) vs. temperature (T ) for different values of the magnetoelectric coupling constant g ¼20, 10.
earlier, it is the reduced velocity which increases with the increasing of magnetization and polarization based on Eq. (19). Similar experimental results were also reported in other co-doped BFO [30–32].
6. Conclusion By using a Green's function technique on the basis of the magnetoelectric coupling, the spin-phonon interaction and the anharmonic phonon–phonon interaction, we have calculated the reduced velocity as a function of temperature. The Heisenberg-like model is employed to describe the magnetic subsystem, and the transverse Ising model is used to explain the ferroelectric subsystem. It is shown that the reduced velocity of sound of BiFeO3 exhibits a kink at the magnetic phase transition temperature TN. This anomaly reduced velocity can be explained as an influence of vanishing magnetic ordering above TN and the magnetic subsystem have no influence on the ferroelectric above TN due to TN { TC in the BFO. It is shown that the influence of the RM is only below TN in the phase where ferroelectric and magnetic properties exist together, whereas the RE influences the properties of the reduced velocity in the whole temperature region ToTC. Above TC
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