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A model description of the first-order phase transition in MnFeP1−xAsx
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 290–291 (2005) 658–660 www.elsevier.com/locate/jmmm
A model description of the first-order phase transition in MnFeP1xAsx O. Tegusa,c,, G.X. Linb, W. Dagulaa,c, B. Fuquana,c, L. Zhanga, E. Bru¨cka, F.R. de Boera, K.H.J. Buschowa a
Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65, Amsterdam 1018 XE, The Netherlands b Department of Physics, Xiamen University, 361005, P.R. China c Physics Department, Inner Mongolia Normal University, Huhhot 010022, P.R. China Available online 13 December 2004
Abstract We present a description of the critical behavior at the first-order phase transition in MnFeP1xAsx system in terms of the Bean–Rodbell model. Within the molecular-field approximation, the Gibbs free energy of the system is expressed in terms of the exchange interaction, the elastic energy, the entropy term, the pressure term and the Zeeman energy. A magnetic-state equation has been obtained by minimizing the Gibbs free energy with respect to the volume and the magnetization. The characteristic parameters for the phase transition observed in this system have been obtained by fitting our experimental data. The results show that the magnetoelastic coupling plays a very important role in the mechanism of the phase transition. r 2004 Elsevier B.V. All rights reserved. PACS: 71.20.Be; 75.30.Kz; 75.30.Sg Keywords: Transition metal-compounds; Phase transition-first-order; Magnetocaloric effect
Recently, a large magnetocaloric effect (MCE) has been observed in materials, which have a first-order magnetic phase transition [1–4]. The giant MCE [1] observed in the pseudo-binary Gd5(Ge1xSix)4 is related to a first-order magneto-structural transition, which can be induced by temperature and/or by magnetic field. The MCE observed in MnFeP1xAsx originates from a fieldinduced first-order magnetic phase transition from the paramagnetic (PM) state to the ferromagnetic (FM) state [2]. The MCE observed in MnAs results from a first-order magnetic phase transition from the FM state Corresponding author. Tel.: +31 20 5256310; +31 20 5255788. E-mail address: [email protected] (O. Tegus).
fax:
to the PM state. The later transition is also accompanied by a structural transformation from the NiAs-type to the MnP-type of structure [4]. A deeper understanding of the first-order phase transitions in these compounds and their influence on the MCE is important for developing new materials for magnetic refrigeration. Ranke et al. have studied the first-order transition in Gd5(Ge1xSix)4 [5] and in MnAs1xSbx [6], based on the Bean–Rodbell model [7,8]. In this work, we report on a description of the critical behavior at the first-order transition observed in MnFeP1xAsx, also based on the Bean–Rodbell model. Bean and Rodbell [7] proposed a model that describes the first-order transition in MnAs. The central assumption in their model is that the exchange interaction is
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.325
ARTICLE IN PRESS O. Tegus et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 658–660
strongly dependent on the interatomic distance. This dependence is described as V V0 TC ¼ T0 1 þ b ; (1) V0
1.0
where TC is the Curie temperature, T0 would be the Curie temperature if the lattice is not compressible, b the slope of the dependence of TC on volume, V the volume and V0 is the equilibrium volume in the absence of magnetic interactions. Taking this deformation into account, the Gibbs free energy (GFE) per unit volume, within the molecular field approximation and for an arbitrary spin J, can be written as [8] 3 J 1 V V0 2 G¼ NkB T C s2 þ 2 J þ1 2K V0 V V0 gmB JNsB; TðS l þ S m Þ þ p ð2Þ V0
0.6
where a is the thermal-expansion coefficient. The magnetic entropy can be expressed as a series of even powers of the magnetization [7]. If we consider that the localized magnetism in MnFeP1xAsx system comes from the magnetic Mn and Fe ions, which have S-state character, then the average J is estimated to be 2 per formula unit because of the experimental value of the saturation magnetization of about 4 mB/f.u. at 5 K and an assumed g ¼ 2: Inserting Eqs. (1) and (3) into Eq. (2), for the case J ¼ 2 and p ¼ 0; we obtain the magnetic part of the reduced GFE: DG ¼ ðT T 0 Þs2 þ 0:22ðT ZT 0 Þs4 6
þ 0:26 Ts 2:96 Bs: 2
ð4Þ
Here, Z ¼ 30NkB T 0 b =13 is an important parameter, involving the parameters K and b that are related to the volume change. As will be discussed below, Z controls the order of the magnetic phase transition. Minimizing the GFE given by Eq. (4) with respect to s, we obtain a magnetic-state equation, T0 3 T0 s5 þ 0:56 1 Z s þ 1:3 1 s T T B 1:7 ¼ 0: ð5Þ T As an example for the analysis of the temperature dependence of the magnetization in the MnFeP1xAsx
MnFeP0.45As0.55 0.8
J=2 B =1 T T0 = 293 K
(c) (d)
(b)
σ
(a)
0.4
0.2
η=0 η=1 η = 1.75 η=2 Experimental data
(a) (b) (c) (d)
0.0 240
260
280 T (K)
300
320
Fig. 1. Temperature dependence of the relative magnetization of MnFeP0.45As0.55 measured in a field of 1 T, and fits based on the Bean–Rodbell model with different parameters.
1.0 0.8
η=1
η = 0.5
η = 1.5
η = 1.75
MnFeP0.45As0.55
η=0
0.6 σ
where N is the number of magnetic atoms per unit volume, kB the Boltzmann constant, s ¼ M=gmB JN the relative magnetization, K the compressibility, Sl the lattice entropy, Sm the magnetic entropy, p the pressure, g the Lande´ factor, B the applied field. Inserting Eq. (1) into Eq. (2) and minimizing the GFE with respect to V, we obtain V V0 3 J NkB bKT 0 s2 pK þ aT; ¼ (3) 2 J þ1 V0
659
0.4 0.2 0.0
J=2 B=0T T0 = 293 K
200
220
T1
240 260 T (K)
280
T2
300
Fig. 2. Calculated curves of the relative magnetization of MnFeP0.45As0.55in zero field for different Z values.
system, we have selected MnFeP0.45As0.55 for a detailed analysis. Fig. 1 shows the relative magnetization of MnFeP0.45As0.55 as a function of temperature for different values of the parameter Z. The parameters that yield the best match with the experimental M(T) curve measured in a field of 1 T are Z ¼ 1:75 and T 0 ¼ 293 K: On the basis of these fitting parameters, we have calculated the M(T) curves in zero field. This is plotted in Fig. 2. It can be seen that Z ¼ 1 separates the firstorder and the second-order transition. The curves with Zo1 correspond to a continuous change in magnetization. In this case, the temperature T1 is the Curie temperature, also being the PM Curie temperature. When Z41; then the transition is of first order. A discontinuous change occurs in the magnetization (indicated by dashed vertical lines). The temperature
ARTICLE IN PRESS O. Tegus et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 658–660
660
Table 1 Composition dependence of the parameters Z and T0 and the Curie temperature [9] for MnFeP1xAsx
2 MnFeP0.45As0.55
1
∆G
302 K 301 300
0
299
x
0.35
0.45
0.55
0.65
TC (K) T0 (K) Z
213 200 1.87
240 229 1.80
300 293 1.75
332 322 1.46
298
-1
297
J=2
296
T0 = 293 K
295
= 1.75
-2 -0.8
-0.6
-0.4
-0.2
0.0 σ
0.2
0.4
0.6
0.8
Fig. 3. Reduced GFE isotherms vs. the relative magnetization in the vicinity of the Curie temperature for MnFeP0.45As0.55.
T2 is the limit of the Curie temperature. Up to this temperature, with increasing temperature, the system is found in the FM state. Using the parameters obtained from the fits in Fig. 3, we plot the magnetic part of the reduced GFE isotherms as a function of magnetization in the vicinity of the Curie temperature, which is about 300 K [9] for MnFeP0.45As0.55. We may illustrate some of the features of the first-order transition from the evolution of the GFE. Just above TC, the reduced GFE has two shallow minima besides the absolute minimum at s ¼ 0; indicating the existence of a metamagnetic state. These free-energy minima, which are separated by an energy barrier, determine the metastable state of the magnetization and are strongly dependent on temperature. The minima disappear when the temperature increases. When the temperature decreases from high temperature where the system is in the stable PM phase down to T1, the PM phase becomes metastable. So T1 is the limiting temperature of the PM state when temperature decreases. The FM state and the PM state coexist in the temperature interval T 1 oToT 2 : In this temperature interval, a temperature TC exists where the stability of the two phases is equal, and which represents the Curie temperature of the first-order transition. From Fig. 3, we estimate this TC to be around 300 K, which is in good agreement with the experimental value of 300 K. The discontinuity in the relative magnetization that occurs at TC at the first-order transition is found to be 0.55. We have carried out the same fittings for other MnFeP1xAsx compounds and obtained the composition dependence of the parameters Z and T0 as listed in Table 1. All values of Z are larger than 1, indicating that these compounds exhibit a first-order phase transition. This is consistent with the experimental observations, and in agreement with the reported values Z ¼ 1:62 and T 0 ¼ 250 K for MnFeP0.5As0.5 [10].
In order to obtain a first-order phase transition, the fourth power term in Eq. (4) must be negative. The distortion of the lattice and the change of TC due to this distortion compete in the fourth power term. As a result, for a large volume dependence of TC an energy barrier forms in the free-energy curves, as shown in Fig. 3. According to Eq. (3), the magnetization is one of the factors controlling the distortion. Here, we mainly consider the effect of the magnetization on the volume change. Therefore, magneto-elastic effects play an important role in the critical behavior of the first-order phase transition in MnFeP1xAsx. At a certain temperature, the applied field shifts the energy minimum of the FM state to values lower than that of the PM state above TC, resulting in the metamagnetic transition. Our conclusion is that the critical behavior of the transition can be well understood on the basis of the Bean–Rodbell model. This work has been financially supported by the Dutch Technology Foundation, and partly has been carried out within the cooperation program between The Netherlands and P.R. China.
References [1] V.K. Pecharsky, K.A. Gschneidner Jr., Phys. Rev. Lett. 78 (1997) 4494. [2] O. Tegus, E. Bru¨ck, K.H.J. Buschow, F.R. de Boer, Nature 415 (2002) 150. [3] A. Fujita, S. Fujieda, Y. Hasegawa, K. Fukamichi, Phys. Rev. B 67 (2003) 104416. [4] H. Wada, Y. Tanade, Appl. Phys. Lett. 79 (2001) 3302. [5] P.J. von Ranke, N.A. de Oliveira, S. Gama, J. Magn. Magn. Mater. 277 (2004) 78. [6] P.J. von Ranke, N.A. de Oliveira, S. Gama, Phys. Lett. A 320 (2004) 302. [7] C.P. Bean, D.S. Rodbell, Phys. Rev. 126 (1962) 104. [8] R.W. de Blois, D.S. Rodbell, Phys. Rev. 130 (1963) 1347. [9] E. Bru¨ck, L. Zhang, Dagula, K.H.J. Buschow, F.R. de Boer, Physica B 319 (2002) 174. [10] R. Zach, M. Guillot, J. Tobola, J. Appl. Phys. 83 (1998) 7237.
Journal of Magnetism and Magnetic Materials 290–291 (2005) 658–660 www.elsevier.com/locate/jmmm
A model description of the first-order phase transition in MnFeP1xAsx O. Tegusa,c,, G.X. Linb, W. Dagulaa,c, B. Fuquana,c, L. Zhanga, E. Bru¨cka, F.R. de Boera, K.H.J. Buschowa a
Van der Waals-Zeeman Instituut, Universiteit van Amsterdam, Valckenierstraat 65, Amsterdam 1018 XE, The Netherlands b Department of Physics, Xiamen University, 361005, P.R. China c Physics Department, Inner Mongolia Normal University, Huhhot 010022, P.R. China Available online 13 December 2004
Abstract We present a description of the critical behavior at the first-order phase transition in MnFeP1xAsx system in terms of the Bean–Rodbell model. Within the molecular-field approximation, the Gibbs free energy of the system is expressed in terms of the exchange interaction, the elastic energy, the entropy term, the pressure term and the Zeeman energy. A magnetic-state equation has been obtained by minimizing the Gibbs free energy with respect to the volume and the magnetization. The characteristic parameters for the phase transition observed in this system have been obtained by fitting our experimental data. The results show that the magnetoelastic coupling plays a very important role in the mechanism of the phase transition. r 2004 Elsevier B.V. All rights reserved. PACS: 71.20.Be; 75.30.Kz; 75.30.Sg Keywords: Transition metal-compounds; Phase transition-first-order; Magnetocaloric effect
Recently, a large magnetocaloric effect (MCE) has been observed in materials, which have a first-order magnetic phase transition [1–4]. The giant MCE [1] observed in the pseudo-binary Gd5(Ge1xSix)4 is related to a first-order magneto-structural transition, which can be induced by temperature and/or by magnetic field. The MCE observed in MnFeP1xAsx originates from a fieldinduced first-order magnetic phase transition from the paramagnetic (PM) state to the ferromagnetic (FM) state [2]. The MCE observed in MnAs results from a first-order magnetic phase transition from the FM state Corresponding author. Tel.: +31 20 5256310; +31 20 5255788. E-mail address: [email protected] (O. Tegus).
fax:
to the PM state. The later transition is also accompanied by a structural transformation from the NiAs-type to the MnP-type of structure [4]. A deeper understanding of the first-order phase transitions in these compounds and their influence on the MCE is important for developing new materials for magnetic refrigeration. Ranke et al. have studied the first-order transition in Gd5(Ge1xSix)4 [5] and in MnAs1xSbx [6], based on the Bean–Rodbell model [7,8]. In this work, we report on a description of the critical behavior at the first-order transition observed in MnFeP1xAsx, also based on the Bean–Rodbell model. Bean and Rodbell [7] proposed a model that describes the first-order transition in MnAs. The central assumption in their model is that the exchange interaction is
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.11.325
ARTICLE IN PRESS O. Tegus et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 658–660
strongly dependent on the interatomic distance. This dependence is described as V V0 TC ¼ T0 1 þ b ; (1) V0
1.0
where TC is the Curie temperature, T0 would be the Curie temperature if the lattice is not compressible, b the slope of the dependence of TC on volume, V the volume and V0 is the equilibrium volume in the absence of magnetic interactions. Taking this deformation into account, the Gibbs free energy (GFE) per unit volume, within the molecular field approximation and for an arbitrary spin J, can be written as [8] 3 J 1 V V0 2 G¼ NkB T C s2 þ 2 J þ1 2K V0 V V0 gmB JNsB; TðS l þ S m Þ þ p ð2Þ V0
0.6
where a is the thermal-expansion coefficient. The magnetic entropy can be expressed as a series of even powers of the magnetization [7]. If we consider that the localized magnetism in MnFeP1xAsx system comes from the magnetic Mn and Fe ions, which have S-state character, then the average J is estimated to be 2 per formula unit because of the experimental value of the saturation magnetization of about 4 mB/f.u. at 5 K and an assumed g ¼ 2: Inserting Eqs. (1) and (3) into Eq. (2), for the case J ¼ 2 and p ¼ 0; we obtain the magnetic part of the reduced GFE: DG ¼ ðT T 0 Þs2 þ 0:22ðT ZT 0 Þs4 6
þ 0:26 Ts 2:96 Bs: 2
ð4Þ
Here, Z ¼ 30NkB T 0 b =13 is an important parameter, involving the parameters K and b that are related to the volume change. As will be discussed below, Z controls the order of the magnetic phase transition. Minimizing the GFE given by Eq. (4) with respect to s, we obtain a magnetic-state equation, T0 3 T0 s5 þ 0:56 1 Z s þ 1:3 1 s T T B 1:7 ¼ 0: ð5Þ T As an example for the analysis of the temperature dependence of the magnetization in the MnFeP1xAsx
MnFeP0.45As0.55 0.8
J=2 B =1 T T0 = 293 K
(c) (d)
(b)
σ
(a)
0.4
0.2
η=0 η=1 η = 1.75 η=2 Experimental data
(a) (b) (c) (d)
0.0 240
260
280 T (K)
300
320
Fig. 1. Temperature dependence of the relative magnetization of MnFeP0.45As0.55 measured in a field of 1 T, and fits based on the Bean–Rodbell model with different parameters.
1.0 0.8
η=1
η = 0.5
η = 1.5
η = 1.75
MnFeP0.45As0.55
η=0
0.6 σ
where N is the number of magnetic atoms per unit volume, kB the Boltzmann constant, s ¼ M=gmB JN the relative magnetization, K the compressibility, Sl the lattice entropy, Sm the magnetic entropy, p the pressure, g the Lande´ factor, B the applied field. Inserting Eq. (1) into Eq. (2) and minimizing the GFE with respect to V, we obtain V V0 3 J NkB bKT 0 s2 pK þ aT; ¼ (3) 2 J þ1 V0
659
0.4 0.2 0.0
J=2 B=0T T0 = 293 K
200
220
T1
240 260 T (K)
280
T2
300
Fig. 2. Calculated curves of the relative magnetization of MnFeP0.45As0.55in zero field for different Z values.
system, we have selected MnFeP0.45As0.55 for a detailed analysis. Fig. 1 shows the relative magnetization of MnFeP0.45As0.55 as a function of temperature for different values of the parameter Z. The parameters that yield the best match with the experimental M(T) curve measured in a field of 1 T are Z ¼ 1:75 and T 0 ¼ 293 K: On the basis of these fitting parameters, we have calculated the M(T) curves in zero field. This is plotted in Fig. 2. It can be seen that Z ¼ 1 separates the firstorder and the second-order transition. The curves with Zo1 correspond to a continuous change in magnetization. In this case, the temperature T1 is the Curie temperature, also being the PM Curie temperature. When Z41; then the transition is of first order. A discontinuous change occurs in the magnetization (indicated by dashed vertical lines). The temperature
ARTICLE IN PRESS O. Tegus et al. / Journal of Magnetism and Magnetic Materials 290–291 (2005) 658–660
660
Table 1 Composition dependence of the parameters Z and T0 and the Curie temperature [9] for MnFeP1xAsx
2 MnFeP0.45As0.55
1
∆G
302 K 301 300
0
299
x
0.35
0.45
0.55
0.65
TC (K) T0 (K) Z
213 200 1.87
240 229 1.80
300 293 1.75
332 322 1.46
298
-1
297
J=2
296
T0 = 293 K
295
= 1.75
-2 -0.8
-0.6
-0.4
-0.2
0.0 σ
0.2
0.4
0.6
0.8
Fig. 3. Reduced GFE isotherms vs. the relative magnetization in the vicinity of the Curie temperature for MnFeP0.45As0.55.
T2 is the limit of the Curie temperature. Up to this temperature, with increasing temperature, the system is found in the FM state. Using the parameters obtained from the fits in Fig. 3, we plot the magnetic part of the reduced GFE isotherms as a function of magnetization in the vicinity of the Curie temperature, which is about 300 K [9] for MnFeP0.45As0.55. We may illustrate some of the features of the first-order transition from the evolution of the GFE. Just above TC, the reduced GFE has two shallow minima besides the absolute minimum at s ¼ 0; indicating the existence of a metamagnetic state. These free-energy minima, which are separated by an energy barrier, determine the metastable state of the magnetization and are strongly dependent on temperature. The minima disappear when the temperature increases. When the temperature decreases from high temperature where the system is in the stable PM phase down to T1, the PM phase becomes metastable. So T1 is the limiting temperature of the PM state when temperature decreases. The FM state and the PM state coexist in the temperature interval T 1 oToT 2 : In this temperature interval, a temperature TC exists where the stability of the two phases is equal, and which represents the Curie temperature of the first-order transition. From Fig. 3, we estimate this TC to be around 300 K, which is in good agreement with the experimental value of 300 K. The discontinuity in the relative magnetization that occurs at TC at the first-order transition is found to be 0.55. We have carried out the same fittings for other MnFeP1xAsx compounds and obtained the composition dependence of the parameters Z and T0 as listed in Table 1. All values of Z are larger than 1, indicating that these compounds exhibit a first-order phase transition. This is consistent with the experimental observations, and in agreement with the reported values Z ¼ 1:62 and T 0 ¼ 250 K for MnFeP0.5As0.5 [10].
In order to obtain a first-order phase transition, the fourth power term in Eq. (4) must be negative. The distortion of the lattice and the change of TC due to this distortion compete in the fourth power term. As a result, for a large volume dependence of TC an energy barrier forms in the free-energy curves, as shown in Fig. 3. According to Eq. (3), the magnetization is one of the factors controlling the distortion. Here, we mainly consider the effect of the magnetization on the volume change. Therefore, magneto-elastic effects play an important role in the critical behavior of the first-order phase transition in MnFeP1xAsx. At a certain temperature, the applied field shifts the energy minimum of the FM state to values lower than that of the PM state above TC, resulting in the metamagnetic transition. Our conclusion is that the critical behavior of the transition can be well understood on the basis of the Bean–Rodbell model. This work has been financially supported by the Dutch Technology Foundation, and partly has been carried out within the cooperation program between The Netherlands and P.R. China.
References [1] V.K. Pecharsky, K.A. Gschneidner Jr., Phys. Rev. Lett. 78 (1997) 4494. [2] O. Tegus, E. Bru¨ck, K.H.J. Buschow, F.R. de Boer, Nature 415 (2002) 150. [3] A. Fujita, S. Fujieda, Y. Hasegawa, K. Fukamichi, Phys. Rev. B 67 (2003) 104416. [4] H. Wada, Y. Tanade, Appl. Phys. Lett. 79 (2001) 3302. [5] P.J. von Ranke, N.A. de Oliveira, S. Gama, J. Magn. Magn. Mater. 277 (2004) 78. [6] P.J. von Ranke, N.A. de Oliveira, S. Gama, Phys. Lett. A 320 (2004) 302. [7] C.P. Bean, D.S. Rodbell, Phys. Rev. 126 (1962) 104. [8] R.W. de Blois, D.S. Rodbell, Phys. Rev. 130 (1963) 1347. [9] E. Bru¨ck, L. Zhang, Dagula, K.H.J. Buschow, F.R. de Boer, Physica B 319 (2002) 174. [10] R. Zach, M. Guillot, J. Tobola, J. Appl. Phys. 83 (1998) 7237.