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The Curie temperature of a spin-one Heisenberg ferromagnet with biquadratic exchange and uniaxial anisotropy
Volume 70A, number 5, 6
PHYSICS LETTERS
2 April 1979
THE CURIE TEMPERATURE OF A SPIN-ONE HEISENBERG FERROMAGNET WITH BIQUADRATIC EXCHANGE AND UNIAXIAL ANISOTROPY Takashi IWASHITA and Norikiyo URYU Department ofApplied Science, Faculty of Engineering, Kyushu University, Fukuoka 812, Japan Received 12 December 1978
With the pair model approximation, the Curie temperature of a spin-one Heisenberg ferromagnet with biquadratic exchange interaction J’ and uniaxial anisotropy term D is calculated for both positive as well as negative values ofJ’ and D.
The biquadratic exchange interaction has turned out to have significant effects on the Curie temperature and other magnetic properties of the Heisenberg ferromagnet. previous studies by [1], the change ofIn thethe Curie temperature has Brown been calculated only for the case of negative value of the biquadratic exchange interaction. It may be worthwhile to extend the calculation for both cases of positive as well as negative values of the biquadratic exchange interactionJ’ and to investigate the variation of the Curie temperature with J’. In the present paper, the results of such a calculation including also the effects of uniaxial single-ionic anisotropy term D will be reported for some crystal lattices having different coordination number, Let us consider a ferromagnetic spin system with a Heisenberg exchange interaction and a biquadratic one between nearest-neighbour spin pairs and anuniaxial anisotropy term. Then the spin hamiltonian is given as follows: 2 H = J (1/> (S1 S1) + (S. S)
~I
.
~
/
+D~S~+ES7~)
.
(1)
1
The hamiltonian (1) may be approximated by making use of the pair model (Oguchi approximation [21) as
464
S.)
= 1
J
+
J’(S. S.)2 I
+
D(S2 LZ
+
S2) JZ
l)J(S~)(S~~ +S 2 + 1~) S~) (2) + 1)J’(SZ )2(S LZ JZ where z is the number of nearest neighbour spins and (SZ) denotes the thermal average of the spin operator ~ From the hamiltonian (2), we have nine eigenvalues and with the use of these eigenvalues the thermal average (SZ) can be calculated. Then from the consistency condition that assures the non-vanishing solution of (S 2~)the Curie temperature Tc can be calculated. The J’-dependence of Tc is obtained by numerical calculations. The results are shown in fig. 1 for a sc lattice (z = 6) and in fig. 2 for a fcc lattice (z = 12), respectively, by curve A. In this case the value of D was taken to be zero. As can,be seen, the value of Tc increases with increasing J (J = J -* --J) in the range of J’ = (—J)—J. The behaviour of T~in the negative range ofbyJ’Brown (J’ = 0—f) qualitatively the results [1]. agrees The change of Tc forwith negative values of J’ is larger than that for positive value of J’. The results of the calculation of the D.dependence of Tc are shown in fig. 1 for sc and in fig. 2 for fcc lattices, respectively, by curve B. In this case the value of f’ was taken to be zero. It is noted that the Curie temperatures for both lattices decrease with increasing D in the range of the present investigation (D = (—J) — J). The spin system with large negative D is characterized by the Ising type and the one with large positive D is + (z
—
—
Volume 70A, number 5, 6
PHYSICS LETTERS
2 April 1979
42 Zr6 105
Fig. 1. The variation of kBTc/IJI with J’ and D for a simple cubic lattice, The J’-dependence is represented by curve A and the D-dependence is represented by curve B.
characterized by the XY type. Therefore, the behavior of Tc forD <0 may be compared with the results for the Ising model and shows reasonable agreement [3]. Now, let us denote the Curie temperature in the case of vanishing / ani D as T~.The variation of the ratio Tc/T~in the range of J’ = (—.1)—f and D = 0 for various values of z is shown in fig. 3. It can be seen from the figure that at a certain fixed value of J’ in the
78/
—J
Fig. 3. The J’-dependence of Tc/T°~ with D = 0 for various values of z. T°cdenotes the Curie temperature of the spin system with J only. The broken line represents the case z = 4 with no long.range order.
range of J’ = 0—f the value of ~Tc/T~ = ITc T~/ T~° decreases with increasing z. In this range of J’, we may describe thez-dependence of 1~Tc/T~approximately as ~Tc/Tc°~ (z + 1)—i. On the other hand, the z-dependence of ~Tc/T~ is complicated in the range of f’ = 0—(J). This fact may suggest an unusual situation of the spin system with biquadratic exchange of
_
—0.5J
0 D.J
0.5J
J
Fig. 2. The variations of kBTc/IJI with J’ and D for a facecentered cubic lattice. The J’-dependence is represented by curve A and the D-dependence is represented by curve B.
Dr—J
D:0
DZJ
Fig. 4. The D-dependence of Tc/Tg with J’ 0 for various values of z, The broken line represents the case z = 2 with no long-range order.
465
Volume 70A, number 5, 6
PHYSICS LETTERS
positive f’ and is consistent with our previous results obtained by means of the spin.wave theory [4]. The canted spin structure which has been predicted in our previous investigation as the stable state in a certain range of f’ may be related to this abnormal behavior of T~. The variation of the ratio Tc/Tc° in the range of D = (—J) —J and J’ = 0 for various values of z are shown in fig. 4. As can be seen from the figure, the effect of D on Tc is larger than that of J’. Also in this 0 may be described case, the z-dependence of~T/T 0 c c 1 approximately as ~Tc/]~ cr(z 1) Therefore, the z-dependence of ~Tc/T~~’versus D has turned out to be larger than that Of~Tc/T~versusf’. —
466
.
2 April 1979
In the present report, we have described how the biquadratic exchange f’ and the single-ionic anisotropy D affect the Curie temperature T~of the Heisenberg ferromagnet. Along the same line, the effects of J’ and D on the temperature dependences of the magnetization and the magnetic susceptibility are calculated and the results will be reported elsewhere. References [1] [2] [3] [4]
H.A. Brown, Phys. Rev. B4 (1971) 115. T. Oguchi, Prog. Theor. Phys. 13 (1955) 148. R. Peierls, Proc. Camb. Phil. Soc. 32(1936)477. T. Iwashita and N. Uryft, J. Phys. Soc. Japan 40 (1976) 1288; Phys. Rev. B14 (1976) 3090.
PHYSICS LETTERS
2 April 1979
THE CURIE TEMPERATURE OF A SPIN-ONE HEISENBERG FERROMAGNET WITH BIQUADRATIC EXCHANGE AND UNIAXIAL ANISOTROPY Takashi IWASHITA and Norikiyo URYU Department ofApplied Science, Faculty of Engineering, Kyushu University, Fukuoka 812, Japan Received 12 December 1978
With the pair model approximation, the Curie temperature of a spin-one Heisenberg ferromagnet with biquadratic exchange interaction J’ and uniaxial anisotropy term D is calculated for both positive as well as negative values ofJ’ and D.
The biquadratic exchange interaction has turned out to have significant effects on the Curie temperature and other magnetic properties of the Heisenberg ferromagnet. previous studies by [1], the change ofIn thethe Curie temperature has Brown been calculated only for the case of negative value of the biquadratic exchange interaction. It may be worthwhile to extend the calculation for both cases of positive as well as negative values of the biquadratic exchange interactionJ’ and to investigate the variation of the Curie temperature with J’. In the present paper, the results of such a calculation including also the effects of uniaxial single-ionic anisotropy term D will be reported for some crystal lattices having different coordination number, Let us consider a ferromagnetic spin system with a Heisenberg exchange interaction and a biquadratic one between nearest-neighbour spin pairs and anuniaxial anisotropy term. Then the spin hamiltonian is given as follows: 2 H = J (1/> (S1 S1) + (S. S)
~I
.
~
/
+D~S~+ES7~)
.
(1)
1
The hamiltonian (1) may be approximated by making use of the pair model (Oguchi approximation [21) as
464
S.)
= 1
J
+
J’(S. S.)2 I
+
D(S2 LZ
+
S2) JZ
l)J(S~)(S~~ +S 2 + 1~) S~) (2) + 1)J’(SZ )2(S LZ JZ where z is the number of nearest neighbour spins and (SZ) denotes the thermal average of the spin operator ~ From the hamiltonian (2), we have nine eigenvalues and with the use of these eigenvalues the thermal average (SZ) can be calculated. Then from the consistency condition that assures the non-vanishing solution of (S 2~)the Curie temperature Tc can be calculated. The J’-dependence of Tc is obtained by numerical calculations. The results are shown in fig. 1 for a sc lattice (z = 6) and in fig. 2 for a fcc lattice (z = 12), respectively, by curve A. In this case the value of D was taken to be zero. As can,be seen, the value of Tc increases with increasing J (J = J -* --J) in the range of J’ = (—J)—J. The behaviour of T~in the negative range ofbyJ’Brown (J’ = 0—f) qualitatively the results [1]. agrees The change of Tc forwith negative values of J’ is larger than that for positive value of J’. The results of the calculation of the D.dependence of Tc are shown in fig. 1 for sc and in fig. 2 for fcc lattices, respectively, by curve B. In this case the value of f’ was taken to be zero. It is noted that the Curie temperatures for both lattices decrease with increasing D in the range of the present investigation (D = (—J) — J). The spin system with large negative D is characterized by the Ising type and the one with large positive D is + (z
—
—
Volume 70A, number 5, 6
PHYSICS LETTERS
2 April 1979
42 Zr6 105
Fig. 1. The variation of kBTc/IJI with J’ and D for a simple cubic lattice, The J’-dependence is represented by curve A and the D-dependence is represented by curve B.
characterized by the XY type. Therefore, the behavior of Tc forD <0 may be compared with the results for the Ising model and shows reasonable agreement [3]. Now, let us denote the Curie temperature in the case of vanishing / ani D as T~.The variation of the ratio Tc/T~in the range of J’ = (—.1)—f and D = 0 for various values of z is shown in fig. 3. It can be seen from the figure that at a certain fixed value of J’ in the
78/
—J
Fig. 3. The J’-dependence of Tc/T°~ with D = 0 for various values of z. T°cdenotes the Curie temperature of the spin system with J only. The broken line represents the case z = 4 with no long.range order.
range of J’ = 0—f the value of ~Tc/T~ = ITc T~/ T~° decreases with increasing z. In this range of J’, we may describe thez-dependence of 1~Tc/T~approximately as ~Tc/Tc°~ (z + 1)—i. On the other hand, the z-dependence of ~Tc/T~ is complicated in the range of f’ = 0—(J). This fact may suggest an unusual situation of the spin system with biquadratic exchange of
_
—0.5J
0 D.J
0.5J
J
Fig. 2. The variations of kBTc/IJI with J’ and D for a facecentered cubic lattice. The J’-dependence is represented by curve A and the D-dependence is represented by curve B.
Dr—J
D:0
DZJ
Fig. 4. The D-dependence of Tc/Tg with J’ 0 for various values of z, The broken line represents the case z = 2 with no long-range order.
465
Volume 70A, number 5, 6
PHYSICS LETTERS
positive f’ and is consistent with our previous results obtained by means of the spin.wave theory [4]. The canted spin structure which has been predicted in our previous investigation as the stable state in a certain range of f’ may be related to this abnormal behavior of T~. The variation of the ratio Tc/Tc° in the range of D = (—J) —J and J’ = 0 for various values of z are shown in fig. 4. As can be seen from the figure, the effect of D on Tc is larger than that of J’. Also in this 0 may be described case, the z-dependence of~T/T 0 c c 1 approximately as ~Tc/]~ cr(z 1) Therefore, the z-dependence of ~Tc/T~~’versus D has turned out to be larger than that Of~Tc/T~versusf’. —
466
.
2 April 1979
In the present report, we have described how the biquadratic exchange f’ and the single-ionic anisotropy D affect the Curie temperature T~of the Heisenberg ferromagnet. Along the same line, the effects of J’ and D on the temperature dependences of the magnetization and the magnetic susceptibility are calculated and the results will be reported elsewhere. References [1] [2] [3] [4]
H.A. Brown, Phys. Rev. B4 (1971) 115. T. Oguchi, Prog. Theor. Phys. 13 (1955) 148. R. Peierls, Proc. Camb. Phil. Soc. 32(1936)477. T. Iwashita and N. Uryft, J. Phys. Soc. Japan 40 (1976) 1288; Phys. Rev. B14 (1976) 3090.