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Multilayer Heisenberg models: linear spin wave analysis
Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
Multilayer Heisenberg models: linear spin wave analysis Abdelilah Benyoussef *, Abdelrhani Boubekri, Hamid Ez-Zahraouy Laboratoire de Magne& tisme et de Physique des Hautes Energies, De& partement de Physique, B.P. 1014, Faculte& des Sciences, Rabat, Morocco Received 10 February 1998; received in revised form 26 May 1998
Abstract The effect of the interlayer and intralayer anisotropies on the energy gap and sublattice magnetization is investigated using the linear spin wave theory for multilayer Heisenberg models with odd number of coupled layers. In the isotropic case, such systems exhibit a long-range order and no energy gap, while, in the anisotropic case, they are still ordered and the energy gap opens for certain anisotropies. 1998 Elsevier Science B.V. All rights reserved. PACS: 75.10.Jm; 75.30.Ds Keywords: Anisotropy; Heisenberg model
1. Introduction In recent years, intensive studies of coupled layers, for the elucidation of the high-¹ superconductivity mechanism, have been performed [1—11]. A new kind of samples with antiferromagnetic interaction between spins within the layer and ferromagnetic interaction between layers (AF—F), GdBa Cu O (x"0.5), has \V been studied experimentally [12]. With the increase of oxygen content from x"0.5 to 1, the interaction between Gd moments changes from ferromagnetic type to antiferromagnetic one between nearest layers (AF). The spin pseudogap observed in underdoped YBa Cu O is one of the fascinating characteristics of \V the high-¹ cuprates. Neutron scattering experiments showed the decrease of low-energy magnetic excitation with decreasing temperature and found the precursor of a finite spin gap [1]. Therefore, it is speculated that the number of CuO layers between the insulating layers is essential for the formation of this gap although a successful theory has not been presented [3—8]. Furthermore, the anisotropic properties are very important since the exchange interaction in real materials is anisotropic to some extent. For instance, it is pointed that a small anisotropy exists in La Sr CuO [13—15] and YBa Cu O [16]. \V V From the theoretical point of view, it has been suggested that systems with two coupled AF isotropic layers are gapped spin liquids above a critical interlayer coupling value [9,10], while the AF—F simple cubic lattice
* Corresponding author. Tel. #212 7 77 89 73; e-mail: [email protected]. 0304-8853/98/$ — see front matter 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 2 4 2 - X
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is ordered and the sublattice magnetization of AF—F system is bigger than that of AF system [17]. The linear spin wave theory has been applied to various low dimensional systems [11,17—21]. As far as we know, however, this theory has not been applied to a multilayer Heisenberg model with odd number of antiferromagnetic anisotropic layers, with ferromagnetic interlayer coupling (AF—F), in spite of the importance of this model. Our aim in this paper is to investigate the ground state of multilayer Heisenberg models with odd number of coupled quantum spin-1/2 antiferromagnetic anisotropic layers with ferromagnetic and antiferromagnetic interlayer exchange interactions, using the linear spin wave theory [18] and to compare these models in two regions of anisotropy (a and a *1; a and a )1). , , , , 2. Models and method 2.1. Antiferromagnetic—ferromagnetic model The M-layer Heisenberg model (AF—F) is denoted as follows: H"JVW (SVSV#SWSW#a SXSX)!JVW (SVSV#SWSW#a SXSX), (1) G H G H , G H , G H G H , G H , 6G H7, 6G H7, where the sums run over the nearest neighbours 1i, j2 along the layers and 1i, j2 perpendicular to the , , layers. JVW and JVW are the positive antiferromagnetic intralayer and ferromagnetic interlayer exchange , , interactions, respectively. a "JX /JVW and a "JX /JVW are the intralayer and interlayer anisotropy para, , , , , , meters, respectively. 2.1.1. Region a and a *1 , , The antiferromagnetic spin wave theory introduced by Anderson and Kubo in Ref. [18] is applied to the Hamiltonian (1). Thus, we divide a lattice with NM sites (N is the number of the sites in a layer, and M the number of layers) into two sublattices (A) and (B) such that each site of (A) is adjacent to sites of (B) in the same layer and to sites of (A) in the nearest neighbours layers. Then S and S denote, the spin operators J K of the sublattices (A) and (B), respectively. The Holstein—Primakoff transformation is applied to these operators. This transformation is defined by SX"S!a>a , J J J
S\"(2Sa> J J
(2a) a>a 1! J J, 2S
SX "!S#b>b , K K K
b>b S\"(2S 1! K Kb , K 2S K
(2b) (2c) (2d)
where a , a>, b and b> are the creation and annihilation operators of spin deviations for sublattices (A) and J J K K (B), respectively, satisfying the Bose commutation relations, namely [a , a>]"d , J JY J JY [b , b> ]"d , K KY K KY [a , b>]"[a>, b ]"[a , b ]"[a>, b>]"0. J K J K J K J K
(3a) (3b) (3c)
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323
We substitute Eqs. (2a), (2b), (2c) and (2d) into Eq. (1) and keep the part of only lowest order terms denoted by H*152. Then H*152"JVW +(a b #a> b> )#a (a> a #b> b ), , G N H N G N H N , G N G N H N H N 6G H7 N + #a> a )!a (a> a #a> a ), !JVWS +(a> a L N L N> L N> L N , L N L N L N> L N> , L , N +\ !JVWS +(b> b #b> b )!a (b> b #b> b ), , K N K N> K N> K N , K N K N K N> K N> K , N +\ !2MNJVWSa !N(M!1)JVWSa . , , , ,
(4)
We apply the Fourier transformation
2 a" e\GIJa , I J MN I
(5a)
2 b " eGIKb , (5b) I K MN I where k is a vector in a reciprocal lattice of a sublattice. Using Eqs. (5a) and (5b), the commutation relations become [a , a>]"[b , b>]"d , I IY I IY I IY
(6a)
[a , b ]"[a>, b ]"[a , b>]"[a>, b>]"0. I IY I IY I IY I IY
(6b)
Then Eq. (4) is given by H*152"4JVWS , I
M!1 b a # (a !c,) (a>a #b>b )#c,(a b #a>b>) , I I I I I I I I I I M 2 ,
M!1 b !MN JVWS2 a # a , , , M 2 ,
(7)
where b"JVW/JVW, c,"[cos(k )#cos(k )] and c,"cos(k ). V W I X , , I This Hamiltonian can be diagonalized by the Bogoliubov transformation: º"exp[! h (a>b>!a b )], I I I I I I a "ºa º>"a cosh(h )#b> sinh(h ), I I I I I I b>"ºb>º>"a sinh(h )#b> cosh(h ). I I I I I I
(8a) (8b) (8c)
As Eqs. (8a), (8b) and (8c) is a canonical transformation, the commutation relations (6) are preserved, namely [a , a>]"[b , b>]"d , I IY I IY I IY
(9a)
[a , b ]"[a>, b ]"[a , b>]"[a>, b>]"0. I IY I IY I IY I IY
(9b)
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Then the Hamiltonian (7) becomes
M!1 b (a !c,) #¼*152(k) (a>a #b>b )#¼*152(k) H*152" !4JVWS a # , , I I I I I M 2 , I M!1 b !MN JVWS2 a # a , , , M 2 ,
(10)
where the dispersion relation is given by
¼*152(k)"4JVWS ,
M!1 b a # (a !c,) !c, . I , I M 2 ,
(11)
Therefore, a , a>, b and b> are the creation and annihilation operators of the elementary excitation, and the I I I I ground state is their vacuum state. Then the spin deviation averaged in the ground state has the form
1 1 p p dk dk 1 V W *S"1a>a 2"! # J J 2 2 (2p) M \p \p IX\p>Hp + H + The ground state energy per spin is given by
M!1 4JVWS a # b(a !c,) , , , I 2M ¼*152(k)
E*152 M!1 "!2JVW a # ba S (S#1) , , , MN 2M
1 p p dk dk 1 M!1 V W ¼*152(k)#4SJVW bc, # , 2M I 2 (2p) M \p \p IX\p>Hp + H + The energy gap has the form E "¼*152(k "p, k "n, k "0)"4JVWS V W X ,
.
.
(12)
(13)
M!1 a # b(a !1) !1 . , , 2M
(14)
2.1.2. Region a and a )1 , , In this region, spins in the two sublattices are oriented antiparallel in the easy plane in the classical limit. If we choose the new z-axis for each spin in the direction of its classical orientation and the new y-axis along the direction of the original z-axis, we have
1 1 H"!JVW SXSX# (1#a ) (S>S>#S\S\)# (1!a ) (S>S\#S\S>) G H 4 G H G H , , G H , G H 4 6G H7, 1 1 !JVW SXSX# (1!a ) (S>S>#S\S\)# (1#a ) (S>S\#S\S>) . (15) , G H 4 , G H G H , G H G H 4 6G H7, Using the linear spin wave theory defined above, the dispersion relation, the spin deviation averaged in the ground state and the ground state energy per spin are given, respectively, by ¼*152(k)"2JVWS ,
M!1 b (2!(1!a )c,)# (2!(1#a )c,) , I , I M 2
M!1 b ! (1#a )c,# (1!a )c, , , I , I M 2
(16)
A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
1 1 p p dk dk V W *S"! # 2 2 (2p) \p \p
1 ; M
IX\p>Hp + H +
M!1 1 p p dk dk E*152 "!2JVW 1# V W b S (S#1)# , MN 2M 2 (2p) \p \p
;
1 M
M!1 2JVWS (2!(1!a )c,)# b(2!(1#a )c,) , , I , I 2M ¼*152(k)
325
,
(17)
M!1 ¼*152(k)#2JVWS (1!a )c,# (1#a )bc, , , I , I 2M X I \p>Hp + H +
.
(18)
2.2. Antiferromagnetic model The M-layer Heisenberg model (AF) is denoted as follows: H"JVW (SVSV#SWSW#a SXSX)#JVW (SVSV#SWSW#a SXSX), G H G H , G H , G H G H , G H , 6G H7, 6G H7,
(19)
where the sums run over the nearest neighbours 1i, j2 along the layers and 1i, j2 perpendicular to the , , layers. JVW and JVW the positive antiferromagnetic exchange intralayer and interlayer constants, respectively. , , a "JX /JVW and a "JX /JVW are the intralayer and interlayer anisotropy parameters, respectively. , , , , , , 2.2.1. Region a and a *1 , , We divide a lattice with NM sites into two sublattices (A) and (B) such that each site of (A) is adjacent only to sites of (B). In this region, the dispersion relation, the spin deviation averaged in the ground state, the ground state energy per spin and the energy gap are given, respectively, by ¼*152(k)"4JVWS ,
1 1 p p dk dk 1 V W *S"1a>a 2"! # J J 2 2 (2p) M \p \p X I \p>Hp + H +
M!1 b M!1 b ! c,# , a # a c, , , I I M 2 M 2
(20)
M!1 4JVWS a # ba , , , 2M ¼*152(k)
,
(21)
M!1 1 p p dk dk 1 E*152 V W "!2JVW a # ba S (S#1)# ¼*152(k) , , , , (2p) M MN 2M 2 \p \p IX\p>Hp + H + E "¼*152(k "p, k "p, k "p)"4JVWS V W X ,
M!1 b M!1 b a # a ! 1# . , M 2 , M 2
(22)
(23)
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2.2.2. Region a and a )1 , , Using the same procedure as in Section 2.1.2, the dispersion relation, the spin deviation averaged in the ground state and the ground state energy per spin are given, respectively, by
¼*152(k)"2JVWS ,
M!1 b (2!(1!a )c,)# (2!(1!a )c,) , I , I M 2
M!1 b ! (1#a )c,# (1#a )c, , , I , I M 2
(24)
1 1 p p dk dk V W *S"! # 2 2 (2p) \p \p
;
1 M
X I \p>Hp + H +
M!1 b(2!(1!a )c,) 2JVWS (2!(1!a )c,)# , I , I , 2M ¼*152(k)
E*152 M!1 1 p p dk dk "!2JVW(1# V W b) S (S#1)# , MN 2M 2 (2p) \p \p
;
1 M
,
(25)
M!1 ¼*152(k)#2JVWS (1!a )c,# (1!a )bc, , , I , I 2M IX\p>Hp + H +
.
(26)
3. Results and discussion The spin wave theory results for multilayer Heisenberg models with odd number of coupled quantum spin-1/2 antiferromagnetic anisotropic layers are given for the isotropic and anisotropic cases. The sublattice magnetization, which is displayed in Fig. 1, shows that the multilayer Heisenberg model with odd number of layers with ferromagnetic interlayer coupling (AF—F) is ordered for any value of b"JVW/JVW at fixed values of the interlayer and intralayer anisotropies, a "a "a"1. Thus, for infinite M, , , , , a simple cubic lattice presents a long-range order. The result is in excellent agreement with experimental results obtained by Chattopadhyay et al. [12]. Furthermore, there exists a critical value b below which the sublattice magnetization increases with increasing number of coupled layers. Fig. 2 shows that the sublattice magnetization of AF—F isotropic system is bigger than that of the AF isotropic one. The increase of the strength of the interlayer coupling (relative to the intralayer coupling) favours the magnetic order for the AF—F case. In the AF case, the magnetic order decreases for large enough interlayer coupling. This phenomenon is due to the enhancement of quantum fluctuation by the interlayer coupling similar to the bilayer model [9,10]. The effect of the anisotropy on the sublattice magnetization is given in Figs. 3 and 4. In fact, the sublattice magnetizations of AF—F and AF systems, which are presented in Fig. 3 at a fixed value of b"1, show that these systems are ordered for any value of the interlayer and intralayer anisotropies (a "a "a). Indeed, , , when increasing a, the sublattice magnetizations of AF—F and AF systems decrease, for a)1, and increase, for a*1. From Fig. 4, it is clear that the sublattice magnetization increases with increasing value of b"JVW/JVW at a fixed value of the anisotropy (a "a "a). , , , ,
A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
327
Fig. 1. The dependence of the sublattice magnetization on b"JVW/JVW at fixed values of the interlayer and intralayer anisotropies , , (a "a "a"1) for AF—F spin-1/2 system. The number accompanying each curve denotes the number of coupled layers. , ,
Fig. 2. The dependence of the sublattice magnetization on b"JVW/JVW at fixed values of the interlayer and intralayer anisotropies , , (a "a "a"1), S"1/2 and M"3. , ,
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Fig. 3. The dependence of the sublattice magnetization on the anisotropy (a "a "a) at fixed values of b"1, S"1/2 and M"3. , ,
Fig. 4. The dependence of the sublattice magnetization on the anisotropy (a "a "a) at a fixed value of M"3 for AF—F spin-1/2 , , anisotropic system. The number accompanying each curve denotes the value of b"JVW/JVW. , ,
A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
329
Fig. 5. The dependence of the ground state energy on b"JVW/JVW at fixed values of the interlayer and intralayer anisotropies , , (a "a "a"1) and S"1/2. The number accompanying each curve denotes the number of coupled layers. , ,
Fig. 6. The dependence of the energy gap E on the anisotropy (a "a "a) at fixed values of b"1, S"1/2 and M"3. , ,
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A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
Fig. 7. The dependence of the energy gap E on the anisotropy (a "a "a) at a fixed value of M"3 for AF—F spin-1/2 anisotropic , , system. The number accompanying each curve denotes the value of b"JVW/JVW. , ,
Fig. 8. The dependence of the energy gap E on b"JVW/JVW at fixed values of the intralayer and interlayer anisotropies (a "1 and , , , a "1.3), S"1/2 and M"3. ,
A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
331
The ground state energies of AF—F and AF systems, given in Fig. 5, decrease with increasing number of coupled layers or increasing value of b. For b"0, we recover the usual spin-wave energy per spin for the isotropic layer (E"!0.658) which is in qualitative agreement with Monte Carlo calculations (E "!0.669) [22]. In the isotropic case, Fig. 5 shows that the ground state energy of AF system is lower than that of the AF—F one, which means that the system is more stable in the AF state. Figs. 6 and 7 show that the AF—F and AF systems are gapless in the isotropic case (a "a "a"1). The , , effect of the a and a anisotropies on the energy gap is displayed in Figs. 6—8. Indeed, the energy gap of , , AF—F and AF systems opens for the values of interlayer and intralayer anisotropies a "a "a'1 at , , a fixed value of b"1 (Fig. 6). The energy gap of AF—F and AF systems vanishes, for a)1, while, for a'1, it increases with increasing a. The energy gap of AF system is bigger than that of the AF—F one. Fig. 7 shows that the energy gap increases with the increase of b. However, for a "1 and a '1, the energy gap of AF—F , , and AF systems presented in Fig. 8 increases with the increase of b. In addition, when b"0, we obtain the known results (gapless) of the antiferromagnetic isotropic layer [11,23]. In conclusion, using the linear spin wave theory, we have studied the sublattice magnetization and energy gap of the ground state of multilayer Heisenberg models with odd number of weakly or strongly coupled layers. The energy gap of AF—F and AF systems vanishes, in the case a "a "a)1, and opens, in the case , , a "a "a'1. In all cases, our systems exhibit a long-range order within the present approximation. The , , known results of both single layer and simple cubic systems have been obtained.
References [1] J.M. Tranquada, P.M. Gehring, G. Shirane, S. Shamoto, M. Sato, Phys. Rev. B 46 (1992) 5561. [2] K. Yamada, Y. Endoh, C.H. Lee, S. Wakimoto, M. Arai, K. Ubukata, M. Fujita, S. Hosoya, S.M. Bennington, J. Phys. Soc. Japan 64 (1995) 2742. [3] B.L. Altshuler, L.B. Ioffe, Solid State Commun. 82 (1992) 253. [4] M. Ubbens, P.A. Lee, Phys. Rev. B 50 (1994) 438. [5] A.J. Millis, H. Honien, Phys. Rev. Lett. 70 (1993) 2811; A.J. Millis, H. Honien, Phys. Rev. B 50 (1994) 16606. [6] A. Sokol, D. Pines, Phys. Rev. Lett. 71 (1993) 2813. [7] A.W. Sandvik, A.V. Chubukov, S. Sachdev, Phys. Rev. B 51 (1995) 16483. [8] A.V. Chubukov, D.K. Morr, Phys. Rev. B 52 (1995) 3521. [9] T. Miyazaki, I. Nakamura, D. Yoshioka, Phys. Rev. B 53 (1996) 12206. [10] K. Kong Ng, F. Zhang, Michael Ma, Phys. Rev. B 53 (1996) 12196. [11] A. Benyoussef, A. Boubekri, H. Ez-Zahraouy, Phys. Scripta 58 (1998) 189. [12] T. Chattopadhyay, H. Malleta, W. Wirges, K. Fischer, P.J. Brown, Phys. Rev. B 38 (1988) 838. [13] K. Hanzawa, J. Phys. Soc. Japan 63 (1994) 264. [14] C.J. Peters, R.J. Birgeneau, M.A. Kastner, H. Yoshizawa, Y. Endoh, J. Tranquada, G. Shirane, Y. Hidaka, M. Oda, M. Suzuki, T. Murakami, Phys. Rev. B 37 (1988) 9761. [15] B. Keimer, N. Belk, R.J. Bergeneau, A. Cassanho, C.Y. Chen, M. Greven, M. Kastner, A. Aharony, Y. Endoh, R.W. Erwin, G. Shirane, Phys. Rev. B 46 (1992) 14034. [16] J.M. Tranquada, G. Shirane, B. Keimer, S. Shamoto, M. Sato, Phys. Rev. B 40 (1989) 4503. [17] A. Du, G.Z. Wei, Phys. Stat. Sol. (b) 177 (1993) 243. [18] P.W. Anderson, Phys. Rev. 86 (1952) 694; R. Kubo, Phys. Rev. 87 (1952) 568. [19] T. Matsuda, K. Hida, J. Phys. Soc. Japan 59 (1990) 2223; K. Hida, J. Phys. Soc. Japan 59 (1990) 2230. [20] A. Benyoussef, A. Boubekri, H. Ez-Zahraouy, Phys. Lett. A 226 (1997) 117. [21] A. Benyoussef, A. Boubekri, H. Ez-Zahraouy, Phys. Lett. A 238 (1998) 398. [22] J. Carlson, Phys. Rev. B 40 (1989) 846; N. Trivedi, D.M. Ceperley, Phys. Rev. B 41 (1990) 4552. [23] A. Parola, S. Sorella, Q.F. Zhong, Phys. Rev. Lett. 71 (1993) 4393.
Multilayer Heisenberg models: linear spin wave analysis Abdelilah Benyoussef *, Abdelrhani Boubekri, Hamid Ez-Zahraouy Laboratoire de Magne& tisme et de Physique des Hautes Energies, De& partement de Physique, B.P. 1014, Faculte& des Sciences, Rabat, Morocco Received 10 February 1998; received in revised form 26 May 1998
Abstract The effect of the interlayer and intralayer anisotropies on the energy gap and sublattice magnetization is investigated using the linear spin wave theory for multilayer Heisenberg models with odd number of coupled layers. In the isotropic case, such systems exhibit a long-range order and no energy gap, while, in the anisotropic case, they are still ordered and the energy gap opens for certain anisotropies. 1998 Elsevier Science B.V. All rights reserved. PACS: 75.10.Jm; 75.30.Ds Keywords: Anisotropy; Heisenberg model
1. Introduction In recent years, intensive studies of coupled layers, for the elucidation of the high-¹ superconductivity mechanism, have been performed [1—11]. A new kind of samples with antiferromagnetic interaction between spins within the layer and ferromagnetic interaction between layers (AF—F), GdBa Cu O (x"0.5), has \V been studied experimentally [12]. With the increase of oxygen content from x"0.5 to 1, the interaction between Gd moments changes from ferromagnetic type to antiferromagnetic one between nearest layers (AF). The spin pseudogap observed in underdoped YBa Cu O is one of the fascinating characteristics of \V the high-¹ cuprates. Neutron scattering experiments showed the decrease of low-energy magnetic excitation with decreasing temperature and found the precursor of a finite spin gap [1]. Therefore, it is speculated that the number of CuO layers between the insulating layers is essential for the formation of this gap although a successful theory has not been presented [3—8]. Furthermore, the anisotropic properties are very important since the exchange interaction in real materials is anisotropic to some extent. For instance, it is pointed that a small anisotropy exists in La Sr CuO [13—15] and YBa Cu O [16]. \V V From the theoretical point of view, it has been suggested that systems with two coupled AF isotropic layers are gapped spin liquids above a critical interlayer coupling value [9,10], while the AF—F simple cubic lattice
* Corresponding author. Tel. #212 7 77 89 73; e-mail: [email protected]. 0304-8853/98/$ — see front matter 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 2 4 2 - X
322
A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
is ordered and the sublattice magnetization of AF—F system is bigger than that of AF system [17]. The linear spin wave theory has been applied to various low dimensional systems [11,17—21]. As far as we know, however, this theory has not been applied to a multilayer Heisenberg model with odd number of antiferromagnetic anisotropic layers, with ferromagnetic interlayer coupling (AF—F), in spite of the importance of this model. Our aim in this paper is to investigate the ground state of multilayer Heisenberg models with odd number of coupled quantum spin-1/2 antiferromagnetic anisotropic layers with ferromagnetic and antiferromagnetic interlayer exchange interactions, using the linear spin wave theory [18] and to compare these models in two regions of anisotropy (a and a *1; a and a )1). , , , , 2. Models and method 2.1. Antiferromagnetic—ferromagnetic model The M-layer Heisenberg model (AF—F) is denoted as follows: H"JVW (SVSV#SWSW#a SXSX)!JVW (SVSV#SWSW#a SXSX), (1) G H G H , G H , G H G H , G H , 6G H7, 6G H7, where the sums run over the nearest neighbours 1i, j2 along the layers and 1i, j2 perpendicular to the , , layers. JVW and JVW are the positive antiferromagnetic intralayer and ferromagnetic interlayer exchange , , interactions, respectively. a "JX /JVW and a "JX /JVW are the intralayer and interlayer anisotropy para, , , , , , meters, respectively. 2.1.1. Region a and a *1 , , The antiferromagnetic spin wave theory introduced by Anderson and Kubo in Ref. [18] is applied to the Hamiltonian (1). Thus, we divide a lattice with NM sites (N is the number of the sites in a layer, and M the number of layers) into two sublattices (A) and (B) such that each site of (A) is adjacent to sites of (B) in the same layer and to sites of (A) in the nearest neighbours layers. Then S and S denote, the spin operators J K of the sublattices (A) and (B), respectively. The Holstein—Primakoff transformation is applied to these operators. This transformation is defined by SX"S!a>a , J J J
S\"(2Sa> J J
(2a) a>a 1! J J, 2S
SX "!S#b>b , K K K
b>b S\"(2S 1! K Kb , K 2S K
(2b) (2c) (2d)
where a , a>, b and b> are the creation and annihilation operators of spin deviations for sublattices (A) and J J K K (B), respectively, satisfying the Bose commutation relations, namely [a , a>]"d , J JY J JY [b , b> ]"d , K KY K KY [a , b>]"[a>, b ]"[a , b ]"[a>, b>]"0. J K J K J K J K
(3a) (3b) (3c)
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323
We substitute Eqs. (2a), (2b), (2c) and (2d) into Eq. (1) and keep the part of only lowest order terms denoted by H*152. Then H*152"JVW +(a b #a> b> )#a (a> a #b> b ), , G N H N G N H N , G N G N H N H N 6G H7 N + #a> a )!a (a> a #a> a ), !JVWS +(a> a L N L N> L N> L N , L N L N L N> L N> , L , N +\ !JVWS +(b> b #b> b )!a (b> b #b> b ), , K N K N> K N> K N , K N K N K N> K N> K , N +\ !2MNJVWSa !N(M!1)JVWSa . , , , ,
(4)
We apply the Fourier transformation
2 a" e\GIJa , I J MN I
(5a)
2 b " eGIKb , (5b) I K MN I where k is a vector in a reciprocal lattice of a sublattice. Using Eqs. (5a) and (5b), the commutation relations become [a , a>]"[b , b>]"d , I IY I IY I IY
(6a)
[a , b ]"[a>, b ]"[a , b>]"[a>, b>]"0. I IY I IY I IY I IY
(6b)
Then Eq. (4) is given by H*152"4JVWS , I
M!1 b a # (a !c,) (a>a #b>b )#c,(a b #a>b>) , I I I I I I I I I I M 2 ,
M!1 b !MN JVWS2 a # a , , , M 2 ,
(7)
where b"JVW/JVW, c,"[cos(k )#cos(k )] and c,"cos(k ). V W I X , , I This Hamiltonian can be diagonalized by the Bogoliubov transformation: º"exp[! h (a>b>!a b )], I I I I I I a "ºa º>"a cosh(h )#b> sinh(h ), I I I I I I b>"ºb>º>"a sinh(h )#b> cosh(h ). I I I I I I
(8a) (8b) (8c)
As Eqs. (8a), (8b) and (8c) is a canonical transformation, the commutation relations (6) are preserved, namely [a , a>]"[b , b>]"d , I IY I IY I IY
(9a)
[a , b ]"[a>, b ]"[a , b>]"[a>, b>]"0. I IY I IY I IY I IY
(9b)
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Then the Hamiltonian (7) becomes
M!1 b (a !c,) #¼*152(k) (a>a #b>b )#¼*152(k) H*152" !4JVWS a # , , I I I I I M 2 , I M!1 b !MN JVWS2 a # a , , , M 2 ,
(10)
where the dispersion relation is given by
¼*152(k)"4JVWS ,
M!1 b a # (a !c,) !c, . I , I M 2 ,
(11)
Therefore, a , a>, b and b> are the creation and annihilation operators of the elementary excitation, and the I I I I ground state is their vacuum state. Then the spin deviation averaged in the ground state has the form
1 1 p p dk dk 1 V W *S"1a>a 2"! # J J 2 2 (2p) M \p \p IX\p>Hp + H + The ground state energy per spin is given by
M!1 4JVWS a # b(a !c,) , , , I 2M ¼*152(k)
E*152 M!1 "!2JVW a # ba S (S#1) , , , MN 2M
1 p p dk dk 1 M!1 V W ¼*152(k)#4SJVW bc, # , 2M I 2 (2p) M \p \p IX\p>Hp + H + The energy gap has the form E "¼*152(k "p, k "n, k "0)"4JVWS V W X ,
.
.
(12)
(13)
M!1 a # b(a !1) !1 . , , 2M
(14)
2.1.2. Region a and a )1 , , In this region, spins in the two sublattices are oriented antiparallel in the easy plane in the classical limit. If we choose the new z-axis for each spin in the direction of its classical orientation and the new y-axis along the direction of the original z-axis, we have
1 1 H"!JVW SXSX# (1#a ) (S>S>#S\S\)# (1!a ) (S>S\#S\S>) G H 4 G H G H , , G H , G H 4 6G H7, 1 1 !JVW SXSX# (1!a ) (S>S>#S\S\)# (1#a ) (S>S\#S\S>) . (15) , G H 4 , G H G H , G H G H 4 6G H7, Using the linear spin wave theory defined above, the dispersion relation, the spin deviation averaged in the ground state and the ground state energy per spin are given, respectively, by ¼*152(k)"2JVWS ,
M!1 b (2!(1!a )c,)# (2!(1#a )c,) , I , I M 2
M!1 b ! (1#a )c,# (1!a )c, , , I , I M 2
(16)
A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
1 1 p p dk dk V W *S"! # 2 2 (2p) \p \p
1 ; M
IX\p>Hp + H +
M!1 1 p p dk dk E*152 "!2JVW 1# V W b S (S#1)# , MN 2M 2 (2p) \p \p
;
1 M
M!1 2JVWS (2!(1!a )c,)# b(2!(1#a )c,) , , I , I 2M ¼*152(k)
325
,
(17)
M!1 ¼*152(k)#2JVWS (1!a )c,# (1#a )bc, , , I , I 2M X I \p>Hp + H +
.
(18)
2.2. Antiferromagnetic model The M-layer Heisenberg model (AF) is denoted as follows: H"JVW (SVSV#SWSW#a SXSX)#JVW (SVSV#SWSW#a SXSX), G H G H , G H , G H G H , G H , 6G H7, 6G H7,
(19)
where the sums run over the nearest neighbours 1i, j2 along the layers and 1i, j2 perpendicular to the , , layers. JVW and JVW the positive antiferromagnetic exchange intralayer and interlayer constants, respectively. , , a "JX /JVW and a "JX /JVW are the intralayer and interlayer anisotropy parameters, respectively. , , , , , , 2.2.1. Region a and a *1 , , We divide a lattice with NM sites into two sublattices (A) and (B) such that each site of (A) is adjacent only to sites of (B). In this region, the dispersion relation, the spin deviation averaged in the ground state, the ground state energy per spin and the energy gap are given, respectively, by ¼*152(k)"4JVWS ,
1 1 p p dk dk 1 V W *S"1a>a 2"! # J J 2 2 (2p) M \p \p X I \p>Hp + H +
M!1 b M!1 b ! c,# , a # a c, , , I I M 2 M 2
(20)
M!1 4JVWS a # ba , , , 2M ¼*152(k)
,
(21)
M!1 1 p p dk dk 1 E*152 V W "!2JVW a # ba S (S#1)# ¼*152(k) , , , , (2p) M MN 2M 2 \p \p IX\p>Hp + H + E "¼*152(k "p, k "p, k "p)"4JVWS V W X ,
M!1 b M!1 b a # a ! 1# . , M 2 , M 2
(22)
(23)
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2.2.2. Region a and a )1 , , Using the same procedure as in Section 2.1.2, the dispersion relation, the spin deviation averaged in the ground state and the ground state energy per spin are given, respectively, by
¼*152(k)"2JVWS ,
M!1 b (2!(1!a )c,)# (2!(1!a )c,) , I , I M 2
M!1 b ! (1#a )c,# (1#a )c, , , I , I M 2
(24)
1 1 p p dk dk V W *S"! # 2 2 (2p) \p \p
;
1 M
X I \p>Hp + H +
M!1 b(2!(1!a )c,) 2JVWS (2!(1!a )c,)# , I , I , 2M ¼*152(k)
E*152 M!1 1 p p dk dk "!2JVW(1# V W b) S (S#1)# , MN 2M 2 (2p) \p \p
;
1 M
,
(25)
M!1 ¼*152(k)#2JVWS (1!a )c,# (1!a )bc, , , I , I 2M IX\p>Hp + H +
.
(26)
3. Results and discussion The spin wave theory results for multilayer Heisenberg models with odd number of coupled quantum spin-1/2 antiferromagnetic anisotropic layers are given for the isotropic and anisotropic cases. The sublattice magnetization, which is displayed in Fig. 1, shows that the multilayer Heisenberg model with odd number of layers with ferromagnetic interlayer coupling (AF—F) is ordered for any value of b"JVW/JVW at fixed values of the interlayer and intralayer anisotropies, a "a "a"1. Thus, for infinite M, , , , , a simple cubic lattice presents a long-range order. The result is in excellent agreement with experimental results obtained by Chattopadhyay et al. [12]. Furthermore, there exists a critical value b below which the sublattice magnetization increases with increasing number of coupled layers. Fig. 2 shows that the sublattice magnetization of AF—F isotropic system is bigger than that of the AF isotropic one. The increase of the strength of the interlayer coupling (relative to the intralayer coupling) favours the magnetic order for the AF—F case. In the AF case, the magnetic order decreases for large enough interlayer coupling. This phenomenon is due to the enhancement of quantum fluctuation by the interlayer coupling similar to the bilayer model [9,10]. The effect of the anisotropy on the sublattice magnetization is given in Figs. 3 and 4. In fact, the sublattice magnetizations of AF—F and AF systems, which are presented in Fig. 3 at a fixed value of b"1, show that these systems are ordered for any value of the interlayer and intralayer anisotropies (a "a "a). Indeed, , , when increasing a, the sublattice magnetizations of AF—F and AF systems decrease, for a)1, and increase, for a*1. From Fig. 4, it is clear that the sublattice magnetization increases with increasing value of b"JVW/JVW at a fixed value of the anisotropy (a "a "a). , , , ,
A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
327
Fig. 1. The dependence of the sublattice magnetization on b"JVW/JVW at fixed values of the interlayer and intralayer anisotropies , , (a "a "a"1) for AF—F spin-1/2 system. The number accompanying each curve denotes the number of coupled layers. , ,
Fig. 2. The dependence of the sublattice magnetization on b"JVW/JVW at fixed values of the interlayer and intralayer anisotropies , , (a "a "a"1), S"1/2 and M"3. , ,
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Fig. 3. The dependence of the sublattice magnetization on the anisotropy (a "a "a) at fixed values of b"1, S"1/2 and M"3. , ,
Fig. 4. The dependence of the sublattice magnetization on the anisotropy (a "a "a) at a fixed value of M"3 for AF—F spin-1/2 , , anisotropic system. The number accompanying each curve denotes the value of b"JVW/JVW. , ,
A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
329
Fig. 5. The dependence of the ground state energy on b"JVW/JVW at fixed values of the interlayer and intralayer anisotropies , , (a "a "a"1) and S"1/2. The number accompanying each curve denotes the number of coupled layers. , ,
Fig. 6. The dependence of the energy gap E on the anisotropy (a "a "a) at fixed values of b"1, S"1/2 and M"3. , ,
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Fig. 7. The dependence of the energy gap E on the anisotropy (a "a "a) at a fixed value of M"3 for AF—F spin-1/2 anisotropic , , system. The number accompanying each curve denotes the value of b"JVW/JVW. , ,
Fig. 8. The dependence of the energy gap E on b"JVW/JVW at fixed values of the intralayer and interlayer anisotropies (a "1 and , , , a "1.3), S"1/2 and M"3. ,
A. Benyoussef et al. / Journal of Magnetism and Magnetic Materials 190 (1998) 321—331
331
The ground state energies of AF—F and AF systems, given in Fig. 5, decrease with increasing number of coupled layers or increasing value of b. For b"0, we recover the usual spin-wave energy per spin for the isotropic layer (E"!0.658) which is in qualitative agreement with Monte Carlo calculations (E "!0.669) [22]. In the isotropic case, Fig. 5 shows that the ground state energy of AF system is lower than that of the AF—F one, which means that the system is more stable in the AF state. Figs. 6 and 7 show that the AF—F and AF systems are gapless in the isotropic case (a "a "a"1). The , , effect of the a and a anisotropies on the energy gap is displayed in Figs. 6—8. Indeed, the energy gap of , , AF—F and AF systems opens for the values of interlayer and intralayer anisotropies a "a "a'1 at , , a fixed value of b"1 (Fig. 6). The energy gap of AF—F and AF systems vanishes, for a)1, while, for a'1, it increases with increasing a. The energy gap of AF system is bigger than that of the AF—F one. Fig. 7 shows that the energy gap increases with the increase of b. However, for a "1 and a '1, the energy gap of AF—F , , and AF systems presented in Fig. 8 increases with the increase of b. In addition, when b"0, we obtain the known results (gapless) of the antiferromagnetic isotropic layer [11,23]. In conclusion, using the linear spin wave theory, we have studied the sublattice magnetization and energy gap of the ground state of multilayer Heisenberg models with odd number of weakly or strongly coupled layers. The energy gap of AF—F and AF systems vanishes, in the case a "a "a)1, and opens, in the case , , a "a "a'1. In all cases, our systems exhibit a long-range order within the present approximation. The , , known results of both single layer and simple cubic systems have been obtained.
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