- Email: [email protected]
Curie temperature of body-centered-tetragonal Ni
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) e301–e303 www.elsevier.com/locate/jmmm
Curie temperature of body-centered-tetragonal Ni Ying Zhua, Ping Yua, Xiaofeng Jina,, Ding-sheng Wangb a
Surface Physics Laboratory and Department of Physics, Fudan University, Shanghai 200433, China b Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China Available online 3 November 2006
Abstract Body-centered-cubic (bcc) Ni, which does not exist in nature, has recently been achieved as a thin film on GaAs (0 0 1) via molecular beam epitaxy. Its Curie temperature ðT c Þ is determined to be 456 K, much lower than that of face-centered-cubic (fcc) Ni (627 K). In order to understand how T c depends on the lattice structure, we have conducted a systematic study on the Curie temperature as a function of the tetragonal distortion ðc=aÞ, i.e., the gradual transformation from bcc-Ni ðc=a ¼ 1Þ to fcc-Ni ðc=a ¼ 1:414Þ, using the selfconsistent all electron linearized augmented plane-wave (LAPW) method in the local spin-density approximation (LSDA) together with the Monte Carlo (MC) simulations. The result shows that T c does increase monotonously about 180 K as the c=a ratio changes from bcc to fcc-Ni, which clearly explains and confirms the experimental observations. r 2006 Elsevier B.V. All rights reserved. PACS: 75.50.y; 31.15.Ar; 87.53.Wz; 77.80.Bh Keywords: Ab initio; LAPW; Monte Carlo; Bct Ni; Increase monotonously
Much effort has been devoted to the study of Ni, one of the top three important ferromagnetic elements in the periodic table, and prominent successes have been made in recent years. Body-centered-cubic (bcc) Ni, which does not exist in nature, has recently been achieved as a thin film on GaAs (0 0 1) via molecular beam epitaxy [1]. Its Curie temperature ðT c Þ is determined to be 456 K, much lower than that of face-centered-cubic (fcc) Ni (627 K). To develop a deep understanding of how T c depends on the lattice structure, it is of much attraction to study the transformation of Ni from fcc to bcc, that is bodycentered-tetragonal (bct) Ni. We have conducted a systematic study on the Curie temperature as a function of the tetragonal distortion ðc=aÞ, i.e., the gradual transformation from bcc-Ni ðc=a ¼ 1Þ to fcc-Ni ðc=a ¼ 1:414Þ. Here we present a theoretical study of the Curie temperature for bct Ni by using an ab initio-Monte Carlo simulation approach [2]. First-principles total energy Corresponding author. Tel.: +86 21 65642960; fax: +86 21 65104949.
E-mail address: [email protected] (X. Jin). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.240
calculations on bct Ni with a set of frozen collinear spin configurations are performed by means of the selfconsistent all electron linearized augmented plane-wave (LAPW) method in the local spin-density approximation (LSDA). The Curie temperatures can be determined by the Monte Carlo (MC) simulations with the exchange integrals extracted from the ab initio total energies. Finally, the calculated T c as a function of c=a ratio is obtained and comparisons with the experiments are made. All electron linearized augmented plane-wave (LAPW) method in the local spin-density approximation (LSDA) is employed in the present total energy calculations for the collinear spin configurations [3]. The fcc Ni canpbe ffiffiffi regarded as a special bct with c ¼ 3:52 A˚ and a ¼ c= 2. Here c and a are the out-plane and in-plane lattice parameters of the bct structure. By fixing the volume per Nipatom, the lattice distortion c=a ratio ffiffiffi changes from 1 to 2, over bcc and fcc lattice structures. For every bct Ni in the calculations two kinds of magnetic supercells are chosen: along h1 0 0i and h1 1 1i directions while the magnetic supercells contain six atomic layers. The total energy is calculated for one ferromagnetic (FM) and
ARTICLE IN PRESS Y. Zhu et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e301–e303
e302
<111>
6
3 2
5 4
a3
a3
a2 2
a2 1
6 4 5
1
3
0.6
a1
FM <100> AF1 AF2 <111> AF3
1 2 3 4 5 6
Exchange parameter J (m Ry)
<100>
a1
Bct Ni J1 J2 J3
0.4 0.2 0.0 -0.2 -0.4 -0.6
fcc
bcc
Fig. 1. Two kinds of magnetic supercells in the calculations: along h1 0 0i and h1 1 1i directions while the magnetic supercells contain six atomic layers.
1.0
1.1
1.2 c/a
1.3
1.4
1.5
Fig. 2. Exchange parameters as functions of the c=a ratio.
i
iaj
For simplicity where ~ ri is the unit vector of the ith spin, J r denotes the exchange integral of the rth nearest neighbor and rðijÞ identifies the distance between atom i and j to the rth nonequivalent pair. This model works well in the cubic cases that have been discussed in Zhou et al. [5]. In Fig. 2, the exchange parameters J are shown as the functions of the c=a ratio. For bct structure, we consider here the first three nearest neighbor interaction (J 1 , J 2 , J 3 Þ. These parameters are deduced from the following equation by counting the number of antiparallel pairs in one unit cell
350
Curie Temperature (K)
three antiferromagnetic (AF) spin configurations as shown in Fig. 1. Since the collinear spin configurations with antiparallel spins cannot be solved self-consistently, we use the FSM technique in an implementation similar to the one of Uhl et al. [4]. In this implementation, an auxiliary local magnetic field is introduced which is determined by the requirement of a fixed local magnetic moment. To gain the same accuracy, the number of k points in the irreducible Brillouin zone for every bct Ni is set to 50–60, keeping the same density. The convergence measured by rms difference between input and output is better than 0:02 me=au3 for the charge density and the spin density. Thus the total energy is stabilized to within 0.02 mRy/ atom. First all the total energies of listed collinear spin configurations are calculated and the total-energy differences between the collinear spin configuration and the ferromagnetic state with the same local magnetic moments are used to derive the exchange interaction constants fitting to the classical isotropic Heisenberg model for every bct Ni. In the present structure, the total energy measured with respect to the FM state can be expressed as X X ET ¼ EM J rðijÞ~ rj . (1) ri ~
fcc
300
250
200
150 bcc 100
1.0
1.1
1.2 c/a
1.3
1.4
1.5
Fig. 3. Calculated Curie temperature as functions of the c=a ratio.
in the AF states: 0
10
J1
1
0
E AF 1 E FM
1
0
Ds1
1
32
0
16
B @ 64 48
0 32
CB C B C B sC 16 A@ J 2 A ¼ @ E AF 2 E FM A ¼ @ D2 A. Ds3 J3 E AF 3 E FM 16 (2)
In order to accurately determine the transition temperature T c , the fourth-order cumulant U L ¼ 1 hM 4 iL =3hM 2 i2L is calculated theoretically. For ToT C , U L tends to 23, and for T4T C , U L decreases toward zero [6]. This behavior of the cumulant makes it very useful to estimate T c . Using the exchange parameters shown in Fig. 2 classical MC simulations are carried out in various system sizes (spin on each cube edge n ¼ 16, 24, 32) with periodic boundary conditions and T c is estimated
ARTICLE IN PRESS Y. Zhu et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e301–e303
from the common intersection point of the U L curves. The Curie temperature of bct Ni is shown as the functions of the c=a ratio in Fig. 3. The result shows that T c does increase monotonously about 180 K as the c=a ratio changes from bcc to fcc-Ni, which confirms the experimental observations, although the absolute T c values calculated here are too low in comparison with the experimental ones. This will be discussed in more detail somewhere else [7].
e303
References [1] [2] [3] [4] [5] [6]
C.S. Tian, X.F. Jin, et al., Phys. Rev. Lett. 94 (2005) 137210. Xing Gao, et al., J. Magn. Magn. Mater. 251 (2002) 29. Y.S. Shi, et al., J. Magn. Magn. Mater. 277 (2004) 71. M. Uhl, et al., Phys. Rev. B. 55 (1997) 2995. Y.M. Zhou, D.S. Wang, et al., Phys. Rev. B. 59 (1999) 8387. K. Binder, D.W. Heermann, Monte Carlo Simulation in Statistical Physics, Springer, Berlin, 1998. [7] P. Yu, X.F. Jin, et al., to be published.
Journal of Magnetism and Magnetic Materials 310 (2007) e301–e303 www.elsevier.com/locate/jmmm
Curie temperature of body-centered-tetragonal Ni Ying Zhua, Ping Yua, Xiaofeng Jina,, Ding-sheng Wangb a
Surface Physics Laboratory and Department of Physics, Fudan University, Shanghai 200433, China b Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China Available online 3 November 2006
Abstract Body-centered-cubic (bcc) Ni, which does not exist in nature, has recently been achieved as a thin film on GaAs (0 0 1) via molecular beam epitaxy. Its Curie temperature ðT c Þ is determined to be 456 K, much lower than that of face-centered-cubic (fcc) Ni (627 K). In order to understand how T c depends on the lattice structure, we have conducted a systematic study on the Curie temperature as a function of the tetragonal distortion ðc=aÞ, i.e., the gradual transformation from bcc-Ni ðc=a ¼ 1Þ to fcc-Ni ðc=a ¼ 1:414Þ, using the selfconsistent all electron linearized augmented plane-wave (LAPW) method in the local spin-density approximation (LSDA) together with the Monte Carlo (MC) simulations. The result shows that T c does increase monotonously about 180 K as the c=a ratio changes from bcc to fcc-Ni, which clearly explains and confirms the experimental observations. r 2006 Elsevier B.V. All rights reserved. PACS: 75.50.y; 31.15.Ar; 87.53.Wz; 77.80.Bh Keywords: Ab initio; LAPW; Monte Carlo; Bct Ni; Increase monotonously
Much effort has been devoted to the study of Ni, one of the top three important ferromagnetic elements in the periodic table, and prominent successes have been made in recent years. Body-centered-cubic (bcc) Ni, which does not exist in nature, has recently been achieved as a thin film on GaAs (0 0 1) via molecular beam epitaxy [1]. Its Curie temperature ðT c Þ is determined to be 456 K, much lower than that of face-centered-cubic (fcc) Ni (627 K). To develop a deep understanding of how T c depends on the lattice structure, it is of much attraction to study the transformation of Ni from fcc to bcc, that is bodycentered-tetragonal (bct) Ni. We have conducted a systematic study on the Curie temperature as a function of the tetragonal distortion ðc=aÞ, i.e., the gradual transformation from bcc-Ni ðc=a ¼ 1Þ to fcc-Ni ðc=a ¼ 1:414Þ. Here we present a theoretical study of the Curie temperature for bct Ni by using an ab initio-Monte Carlo simulation approach [2]. First-principles total energy Corresponding author. Tel.: +86 21 65642960; fax: +86 21 65104949.
E-mail address: [email protected] (X. Jin). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.240
calculations on bct Ni with a set of frozen collinear spin configurations are performed by means of the selfconsistent all electron linearized augmented plane-wave (LAPW) method in the local spin-density approximation (LSDA). The Curie temperatures can be determined by the Monte Carlo (MC) simulations with the exchange integrals extracted from the ab initio total energies. Finally, the calculated T c as a function of c=a ratio is obtained and comparisons with the experiments are made. All electron linearized augmented plane-wave (LAPW) method in the local spin-density approximation (LSDA) is employed in the present total energy calculations for the collinear spin configurations [3]. The fcc Ni canpbe ffiffiffi regarded as a special bct with c ¼ 3:52 A˚ and a ¼ c= 2. Here c and a are the out-plane and in-plane lattice parameters of the bct structure. By fixing the volume per Nipatom, the lattice distortion c=a ratio ffiffiffi changes from 1 to 2, over bcc and fcc lattice structures. For every bct Ni in the calculations two kinds of magnetic supercells are chosen: along h1 0 0i and h1 1 1i directions while the magnetic supercells contain six atomic layers. The total energy is calculated for one ferromagnetic (FM) and
ARTICLE IN PRESS Y. Zhu et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e301–e303
e302
<111>
6
3 2
5 4
a3
a3
a2 2
a2 1
6 4 5
1
3
0.6
a1
FM <100> AF1 AF2 <111> AF3
1 2 3 4 5 6
Exchange parameter J (m Ry)
<100>
a1
Bct Ni J1 J2 J3
0.4 0.2 0.0 -0.2 -0.4 -0.6
fcc
bcc
Fig. 1. Two kinds of magnetic supercells in the calculations: along h1 0 0i and h1 1 1i directions while the magnetic supercells contain six atomic layers.
1.0
1.1
1.2 c/a
1.3
1.4
1.5
Fig. 2. Exchange parameters as functions of the c=a ratio.
i
iaj
For simplicity where ~ ri is the unit vector of the ith spin, J r denotes the exchange integral of the rth nearest neighbor and rðijÞ identifies the distance between atom i and j to the rth nonequivalent pair. This model works well in the cubic cases that have been discussed in Zhou et al. [5]. In Fig. 2, the exchange parameters J are shown as the functions of the c=a ratio. For bct structure, we consider here the first three nearest neighbor interaction (J 1 , J 2 , J 3 Þ. These parameters are deduced from the following equation by counting the number of antiparallel pairs in one unit cell
350
Curie Temperature (K)
three antiferromagnetic (AF) spin configurations as shown in Fig. 1. Since the collinear spin configurations with antiparallel spins cannot be solved self-consistently, we use the FSM technique in an implementation similar to the one of Uhl et al. [4]. In this implementation, an auxiliary local magnetic field is introduced which is determined by the requirement of a fixed local magnetic moment. To gain the same accuracy, the number of k points in the irreducible Brillouin zone for every bct Ni is set to 50–60, keeping the same density. The convergence measured by rms difference between input and output is better than 0:02 me=au3 for the charge density and the spin density. Thus the total energy is stabilized to within 0.02 mRy/ atom. First all the total energies of listed collinear spin configurations are calculated and the total-energy differences between the collinear spin configuration and the ferromagnetic state with the same local magnetic moments are used to derive the exchange interaction constants fitting to the classical isotropic Heisenberg model for every bct Ni. In the present structure, the total energy measured with respect to the FM state can be expressed as X X ET ¼ EM J rðijÞ~ rj . (1) ri ~
fcc
300
250
200
150 bcc 100
1.0
1.1
1.2 c/a
1.3
1.4
1.5
Fig. 3. Calculated Curie temperature as functions of the c=a ratio.
in the AF states: 0
10
J1
1
0
E AF 1 E FM
1
0
Ds1
1
32
0
16
B @ 64 48
0 32
CB C B C B sC 16 A@ J 2 A ¼ @ E AF 2 E FM A ¼ @ D2 A. Ds3 J3 E AF 3 E FM 16 (2)
In order to accurately determine the transition temperature T c , the fourth-order cumulant U L ¼ 1 hM 4 iL =3hM 2 i2L is calculated theoretically. For ToT C , U L tends to 23, and for T4T C , U L decreases toward zero [6]. This behavior of the cumulant makes it very useful to estimate T c . Using the exchange parameters shown in Fig. 2 classical MC simulations are carried out in various system sizes (spin on each cube edge n ¼ 16, 24, 32) with periodic boundary conditions and T c is estimated
ARTICLE IN PRESS Y. Zhu et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e301–e303
from the common intersection point of the U L curves. The Curie temperature of bct Ni is shown as the functions of the c=a ratio in Fig. 3. The result shows that T c does increase monotonously about 180 K as the c=a ratio changes from bcc to fcc-Ni, which confirms the experimental observations, although the absolute T c values calculated here are too low in comparison with the experimental ones. This will be discussed in more detail somewhere else [7].
e303
References [1] [2] [3] [4] [5] [6]
C.S. Tian, X.F. Jin, et al., Phys. Rev. Lett. 94 (2005) 137210. Xing Gao, et al., J. Magn. Magn. Mater. 251 (2002) 29. Y.S. Shi, et al., J. Magn. Magn. Mater. 277 (2004) 71. M. Uhl, et al., Phys. Rev. B. 55 (1997) 2995. Y.M. Zhou, D.S. Wang, et al., Phys. Rev. B. 59 (1999) 8387. K. Binder, D.W. Heermann, Monte Carlo Simulation in Statistical Physics, Springer, Berlin, 1998. [7] P. Yu, X.F. Jin, et al., to be published.