Phonon properties of multiferroic BiFeO3

Materials Science & Engineering B 251 (2019) 114446

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Phonon properties of multiferroic BiFeO3

T

Xian-Sheng Cao School of Mathematics and Physics, Changzhou University, 1 Gehu, Changzhou 213164, China

ARTICLE INFO

ABSTRACT

Keywords: Phonon properties Specific heat Multiferroic properties

The phonon properties of BiFeO3 have been studied by using the specific heat. It is found that the specific heat of BiFeO3 exhibits two peaks. The first peak appears at the magnetic phase transition temperature TN = 650 K. This anomaly in specific heat can be interpreted as antiferromagnetic transition. The values of the specific heat (at the magnetic phase transition temperature TN) increase with the increasing of the J1, |J2|, I and g. Another abnormal peak of specific heat appears at nearly 550 K, which is associated with the onset of structural changes. Note that the second peak shifts to lower temperature as the J1, |J2|, I and g decrease. It is considered that the main reason of leading to the structure distortion increases is due to the increase of the J1, |J2|, I and g. Those calculation conclusions are in accordance with the experimental results of other researchers.

1. Introduction

given by:

BiFeO3 (BFO) has attracted the attention of many experimental researchers in recent years due to having a high Curie temperature (TC = 1100 K) of ferroelectricity and a high Neel temperature (TN = 640 K) of G type antiferromagnetism (AFM) [1–8]. Especially, Cao et al. [9,10] have used the microscopic theoretical model to investigate the sound velocity in BFO, whereas they have not studied the effect of multiferroic on specific heat. However, specific heat measurements play an important role in the investigation of phase transitions in magnetic materials [11]. For all of the experimental investigations [12–20], only the specific heat of BFO has been studied. On the other hand, theoretical research approach of First-principle has been applied to study the properties of specific heat of BFO [21,22]. To the best of our knowledge, little theoretical work has been done on the effect of multiferroic properties on the phonon properties of BFO. For all that, the studies of phonons have played a crucial role in the research of ferroelectrics. It is well known that the phonons are also to be influenced by spin correlation and then offer a complementary tool [23]. At the same time, the thermal properties of BFO, including the heat capacity and its variation in a broad temperature range, have not been studied systematically by the approach of Green’s function so far. In this paper, we report the Green’s function method calculations of phonons and the specific heat of BiFeO3.

HM =

2. Model The model of multiferroic BFO system can be described as [9,10,24,25]:

Hmu =

HM

+

HE

+

H ME

+

HPH

(1)

HM is the Hamiltonian of the ferromagnetic subsystem (FMS), which is

J1 2

J2 2

si ·sj ij

Di (siz )2

si·sj [ij]

siz

Bz

i

(2)

i

where, the first and second summations run for the nearest-neighbor (nn) and next nearest-neighbor (nnn) sites of spins denoted by < ij > and [ij] with the exchange coupling interactions J1(> 0) and J2(< 0), respectively. si is the magnetic spin of the FMS at the site i. Bz and Di (< 0) are the external applied magnetic field and the single-site anisotropy parameter, respectively. HE is the Hamiltonian of the ferroelectric subsystem (FES), it reads:

1 2

Skx

HE = k

Ikl Skz Slz

Skz

Ez

kl

(3)

k

Here are the pseudo-spin-1/2 operators of FES at the site k. Ez denotes the external applied electric field. Ω and Ikl (> 0) are the tunneling frequency and the exchange coupling constant between the nn pseudo-spin, respectively. HME is the Hamiltonian of the coupling between FMS and FES in BFO:

Skx andSkz

H ME =

Skz Slz si sj

g ij

(4)

kl

where g is the magnetoelectric coupling parameter. The Hamiltonian of phonon is 1 HPH = F M (q) Aq s zq 2 qq RM (q, q1 ) Aq A q1 sqz1 q

F E (q ) A q S z q q +

1 + 4!

1 2!

qq1 q2

q

(Bq B

q

+

V (4) Aq1 Aq2 A

https://doi.org/10.1016/j.mseb.2019.114446 Received 13 October 2017; Received in revised form 30 July 2019; Accepted 1 November 2019 0921-5107/ © 2019 Elsevier B.V. All rights reserved.

1

1 RE (q, qq1 2 1 2 0 A q A q )+ 3 ! q q2 A q1+ q

q1 ) Aq A qq1

z q1 Sq1

V (3) Aq A

q

q

q1 A q1 q

(5)

Materials Science & Engineering B 251 (2019) 114446

X.-S. Cao

Here, the first two terms describe the coupling between the magnetic order parameter and the phonons, the following two terms are responsible for the coupling between the ferroelectric pseudo-spins and phonons. The fifth term is the harmonic phonon Hamiltonian, where 0 is the phonon frequency of the lattice mode, Aq = 1/2 0 (bq + bq+ ¯ ), Bq = i 0 /2 (bq+ bq¯) . The last two terms are the Hamiltonian of the anharmonic phonon [26]. 3. Green’s functions The Green’s function of FMS is [27]

GkRM

(6)

sk+ (t ), sk (t )

(t , t ) =

Following the method of Callen [28], the magnetization can be solved

sz =

+ 1 + s ) 2s + 1 ( ( + 1)2s + 1

(

1

+ 1) 2s + 1

s )(

(7)

2s + 1

1

Here = N k M (k ) , M is the energy of FMS [9,10]. e 1 The pseudo-spins-1/2 of FES can be transformed to + + x ci+¯ ci ¯ )/2 , here cn+ (cn ) is the ferP = (ci ci ¯ + ci ¯ ci )/2 , P z = (ci+ci mion-type creation (annihilation) operator. Then the Green’s function of FES is. (8)

cn (t ); cn+ (t )

G RE (t , t ) =

Fig. 1. Plot of specific heat vs. T for different values of the nn exchange coupling constants J1 = 85 K, 90 K.

According to the method of Teng et al. [29], the averages of the xand z-component of the pseudo-spin are

P x = ( /2

E)

P z = ( /2

E)

tanh(

tanh(

(9a)

E /2kB T )

(9b)

E /2kB T )

+ , IP = I1 l = I/2+ I1 Here , and 2g [ij] ( + siz s jz ) . In order to get the value of A and A2 , we should calculate the phonon retarded Green functions Dq, q (t t ) [27]: Dq, q (t

2 E = si+s j

2

t)=

Aq (t ); Aq+ (t )

Plz

2

=

i (t

t ) < [Aq (t ); Aq+ (t )] > (10)

The phonon Green’s function can be derived by using Dyson equation [9,10,27]

Dqq ( ) =

1 2 q

2 0

(11)

( )

( )= + Here phonon self-energy, [V 3

(V 4

<

A2

>

2F M (k ) S z + 2F E (k ) P z

A = q1

RM (k )

Sqz1

q1

RE (k )

RM

q

Pqz1 q

Sz

RE

Pz

Fig. 2. Plot of specific heat vs. T for different values of the nnn exchange coupling constants J2 = −25 K, −20 K.

)/2] is the

Figs. 1–4 (solid line). Those parameters can produce TC = 1100 K and TN = 650 K. It is found that there are two peaks in solid lines of Figs. 1–4 (Fig. 5 displays the schematic of calculation). One peak appears at the magnetic phase transition temperature TN = 650 K, this anomaly in the specific heat can be interpreted as the antiferromagnetic phase transition [14]. In other words, the transition from antiferromagnetic order to paramagnetic disorder occurs when the temperature is higher than the magnetic phase transition temperature TN, which leads to an increase in magnetic entropy and thermal properties. However, Morozovska argued that the antiferrodistortive – antiferromagnetic effect leads to the smearing of the jump of the specific heat near the temperature of antiferromagnetic phase transition [31]. The achieved conclusion is in accordance with the experimental results [6,7]. Another anomaly peak (kink) of specific heat appears at nearly 550 K. Similar kind of behaviour has been observed in ref.15 and 20. They suggested that this kink (533 K) is associated with the onset of structural changes [15,20]. Note that the second peaks shift to lower temperature as the J1, |J2|, I and g decrease (see Figs. 1–4, from 550 K shift to 450 K, 400 K, 510 K and 455 K, respectively). This is due to the increases of the J1, |J2|, I and g,

V (3) (2n¯ q + 1)/ N +

q

V (4)

(2n¯ q + 1)/ N (12)

where n¯ q = 1/[exp( ¯ q / T )

1].

4. Specific heat The specific heat can be obtained as [30]

C=

d H dE = dT dT

(13)

5. Results and discussions The temperature variation of the specific heat of BiFeO3 exhibits two peaks when J1 = 90 K, J2 = −25 K, I = 700 K, g = 25 K, E −1 1 D = −5 K, = 50 K, , FM = 3 cm−1, 0 = 340 cm , F = 5 cm RE = −25 cm−1, RM = −5 cm−1, V(3) = 0.5 cm−1 and V(4) = −5 cm−1, the spin s = 2 and the pseudo-spin S = 0.5 in 2

Materials Science & Engineering B 251 (2019) 114446

X.-S. Cao

Fig. 3. Plot of specific heat vs. T for different values of I = 700 K, 650 K.

Fig. 4. Plot of specific heat vs. T for different values of g = 20 K, 25 K.

which leads to the increase of the structure distortion. Those results are in accordance with the experimental results [11,12,20,31–36]. It is found that the temperature of the magnetic phase transition increases when J1 increases (Fig. 1). It means that ferromagnetism is enhanced, which leads to the result that TN shifts to higher temperature when J1 increases. It is well known that the specific heat is mainly affected by the lattice distortion. This result is in well agreement with the experimental specific heat data [12,20]. It is also in agreement with the theoretical results of Apostolova et al. [32]. They found that the magnetic phase transition temperature TN increases with the increasing of J1 [32]. Similar kind of behaviour has been observed in other reported data [11,31]. The result of our calculation is in agreement with the sound velocity depending on the J1 [9]. It is found that the temperature of the magnetic phase transition TN decreases as |J2| increases, whereas the specific heat increases as |J2| increases. As mentioned above, J2 (< 0) and J1 (> 0) describe antiferromagnetism and ferromagnetism, respectively. The increase of |J2| means that the antiferromagnetism is enhanced and ferromagnetism is weakened. Then, it leads to phenomenon that the Neel temperature TN shifts to higher temperature. Those results are in well agreement with the experimental results [33].

Fig. 5. Schematic of the specific heat calculation. The value of < A > can be obtained by solving Eq. (12). 3

Materials Science & Engineering B 251 (2019) 114446

X.-S. Cao

Fig. 3 presents the plot of specific heat vs. T for different values of I. It is found that the specific heat increases as I increases from 650 to 700. It is also found that TN increases when I increases. It influences the magnitude of specific heat and increases Neel temperature TN in greater extent near Neel temperature. Those results are consistent with the report of Liu et al.[34]. As we all know, the high valence Ti4+ is responsible for the charge compensation and the oxygen reducing, which leads to the enhancement of the ferroelectric properties. Similar behavior has also been observed in La doped BFO [35]. They argued that the structural modification from rhombohedral to orthorhombic is caused by La3+ doping destructs the spin cycloid of BFO, and thus lead to the shift of TN to high temperature. It is in agreement with the sound velocity depending on the I [9]. Fig. 4 shows the plot of specific heat vs. T for different values of magnetoelectric coupling constant g. It is found that the specific heat increases when g increases from 20 to 25. It is also found that TN increases with g increases. It is the increasement of g that leads to an enhanced polarization and magnetization. As we all know, the phase transition temperatures of the magnetic subsystems increasement are caused by the growth of coupling strength g. Those results are also in good qualitative agreement with the experimental result of Gu et al [36]. They argued that the variation in the structure and valence state change with Co doping, which leads to an enhancement in the ferroelectricity and ferromagnetism. The increasement of magnetization and polarization would results in the specific heat increasing, which based on Eq. (13). Other similar experimental results [37–39] and theoretical data [9,24,25,32] were also reported.

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6. Conclusions Through a Green’s function method on the basis of the anharmonic phonon-phonon interaction, we have studied the phonon properties of BiFeO3 by using the specific heat. It is found that the specific heat of BiFeO3 exhibits two peaks. The first peak appears at the magnetic phase transition temperature TN = 650 K. This anomaly in specific heat can be interpreted as antiferromagnetic transition. The specific heat values of TN increase as J1, |J2|, I and g increase. Another abnormal peak appears at nearly 550 K, which is associated with the onset of structural changes. Note, the second peak shifts to lower temperature as the J1, |J2|, I and g decrease. In other words, the increase of the J1, |J2|, I and g leads to the structure distortion increasing. Those calculation conclusions are in accordance with the experimental results of other researchers. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The financial support of this research project by National Natural Science Foundation of China (No. 11864001), the Education Department of Jiangsu Province through its Philosophy and Social Science Fund (No. 2018SJA1401), and the Open Research Fund of

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