Magnetization of Coupled Ultrathin Ferromagnetic Films

Transcription

1 Commun. Theor. Phys. (Beijing, China) 38 (2002) pp c International Academic Publishers Vol. 38, No. 1, July 15, 2002 Magnetization of Coupled Ultrathin Ferromagnetic Films WANG Huai-Yu, 1 ZHOU Yun-Song, 2 and WANG Chong-Yu 1 1 Department of Physics, Tsinghua University, Beijing , China 2 Department of Physics, Capital Normal University, Beijing , China (Received June 8, 2001; Revised December 13, 2001) Abstract The magnetization of coupled ferromagnetic films is calculated by Green s function method. The coupling can either be ferromagnetic or antiferromagnetic. For the latter case, a concept of pseudo-spin is suggested to make calculation possible. A pseudo-spin is actually an anti-spin with its properties being analogue to other known antiparticles such as a hole. The decreasing of Curie point as the coupling strength decays is computed. It is noted that with the same strength, antiferromagnetic coupling has higher Curie point than ferromagnetic coupling. PACS numbers: Ak, q, m Key words: magnetization, pseudo-spin, coupled ferromagnetic films 1 Introduction Recently, the coupling between ferromagnetic ultrathin films across a nonmagnetic metal spacer has attracted considerable attention. In experiments, the coupling can be oscillating, that is alternatively ferromagnetic or antiferromagnetic as a function of spacer thickness. [1 4] To explain the phenomenon, some works established models to derive the oscillatory interaction between the ferromagnetic films. [5,6] Experimentally, the coupling has an effect on Curie point of the ferromagnetic films, [7,8] especially, the oscillating coupling leads directly the oscillation of the Curie point. [8] However, it has not been investigated theoretically yet that how the coupling affects the magnetization curves and the Curie point. This paper is devoted to the research of the spontaneous magnetization and the effect on the Curie point of the films by either ferromagnetic or antiferromagnetic coupling in such systems with quantum statistical method. To this end, we develop a method to calculate the magnetization for both ferromagnetic and antiferromagnetic coupling cases. For the latter case, we have to resort to a concept of the pseudo-spin. It is shown that to study a magnetic system antiferromagnetic exchange is involved, the concept of the pseudo-spin is useful. We will use the method of Green s function. [9] The definition of the retarded Green s function is [10] G ij (t t ) = A i ; B j = iθ(t t ) A i B j B j A i. (1) In this paper, the operators are constructed by spins (or pseudo-spins which will be introduced in Sec. 2), A i = S + i, B j = (S z j ) n S j. (2) The Green s function is Fourier time transformed and put into the equation of motion, ω A i ; B j = 1 2π [A i, B j ] + [A i, H]; B j. (3) Magnetization can be calculated using the following formula [11] i B j A i = [A i, B j ] 2πN k dω e βω 1 [ g(k, ω + i0 + ) g(k, ω i0 + )], (4) or the corresponding expression for A i B j. Here we use the bold lowercase letter k as the wave vectors. The integration of wave vector k can be in one, two or three dimensions depending on the system studied. Equations (1) (4) are our theoretical basis in this paper. In the following discussions, the Heisenberg exchange Hamiltonian is used. Only the nearest neighboring interaction is considered. All the parameters are in unit of J, the nearest Heisenberg exchange integral. β = 1/T, the temperature T is in unit of J/k B and k B is the Boltzman constant. 2 Pseudo-Spin Let us first recall the concept of spin. The spin operators are satisfied with commutators [S +, S ] = 2S z, [S ±, S z ] = S ±. (5) From the commutators and the relationship S 2 = 1 2 (S+ S + S S + ) + S z S z = S(S + 1), (6) we get and S + S = S(S + 1) (S z ) 2 + S z, (7) S S + = S(S + 1) (S z ) 2 S z. (8) The project supported by National Natural Science Foundation of China

2 108 WANG Huai-Yu, ZHOU Yun-Song, and WANG Chong-Yu Vol. 38 Callen [12] showed that the magnetization S z can be calculated by S z = (S Φ)(1 + Φ)2S+1 + (S Φ)Φ 2S+1 (1 + Φ) 2S+1 Φ 2S+1, (9) Φ = 1 N k 1 e βω(k) 1. (10) For any S, it is obvious from Eq. (9) that as Φ 0, S z S. This is the case of saturated spontaneous magnetization when temperature is near 0 K. When Φ, S z 0. This is the case that the spontaneous magnetization of ferromagnetic systems vanishes at the Curie point. Φ varies in the interval [0, ]. As a simplest example, let us consider a ferromagnetic system in an external field. The Hamiltonian is H = ij J ij S i S j h i S z i, (11) in the second term, Bohr magnon and g factor are merged into the factor h which is also in the unit of J. With the basic formulas (1) (4), the retarded Green s function S + ; S can be calculated under the random phase approximation (RPA). The energy spectrum is [11] ω(k) = 2 S z [J(0) J(k)] + h, (12) J(k) is the Fourier component of J ij. It is seen that the energy of a ferromagnetic system is always positive. Now we introduce the concept of pseudo-spin. if S is a spin, then the pseudo-spin is defined as S p = S. It can also be called as anti-spin as it formally appears. Hereafter, we use a superscript or a subscript p to denote a quantity that is related to the pseudo-spin. The commutators for the pseudo-spin operators should be [S p+, S p ] = 2S pz, [S p±, S pz ] = ±S p±. (13) Notice the difference between Eqs. (5) and (13). We can also define the pseudo-spin as a vector that meets commutators (13) or the relationship of a pseudo-angular momentum S p S p = is p. Here we do not intend to explore the properties of either an angular momentum or a pseudo-angular momentum itself. What we concern is merely the usefulness of the concept of the pseudo-spin. With equation (6), we have S p+ S p = S(S + 1) (S pz ) 2 S pz, (14) S p S p+ = S(S + 1) (S pz ) 2 + S pz. (15) Following Callen s routine, [12] we obtain S pz = (S Φ p)(1 + Φ p ) 2S+1 + (S Φ p )Φ 2S+1 p (1 + Φ p ) 2S+1 Φ 2S+1 p, (16) Φ p = 1 N k 1 e βωp(k) 1. (17) Note that as Φ p 1, S pz S. While at the Curie point, Φ p and S pz 0. Φ p varies within the interval [, 1]. If one makes the energy spectrum ω(k) ω(k) in Eq. (10), the relationship between S z and Φ in Eq. (9), is converted to that between S pz and Φ p in Eq. (16). This suggests that a pseudo-spin system should have a negative energy spectrum. It is indeed so as we think the simplest instance of a ferromagnetic system. It is obvious that a ferromagnetic system can be alternatively thought that it is composed of paralleled pseudospins. The Hamiltonian (11) is rewritten as H = ij J ij S p i Sp j h i S pz i. (18) By the standard method of the equation of motion for the Green s function, the resulting energy spectrum under RPA is that ω p (k) = 2 S pz [J(0) J(k)] h. (19) Here we note that the energy spectrum is always negative, contrary to the positive energy spectrum of a spin system, Eq. (12). Thus it is suggested that the concept of pseudo-spin or anti-spin is similar to that of an antiparticle such as a hole. If one electron is extracted from a fermion sea, we say that there losses an electron. On the other hand, we can also think that there is a hole created with negative energy. The case is the same for pseudospin. If a spin is extracted from ferromagnetic system, we can also think that a pseudo-spin is added to the system. According to Eq. (12), the lower energy a spin system has, the stabler it will be. While for a pseudo-spin system, from Eq. (19), the higher energy represents the stabler state as a hole behaves. This consideration is also applied to the case when there is a magnetic field H. The Zeeman energy of a pseudo-spin should still be written as H S p, because this is the state in which it has the highest energy in magnetic field. Although starting from the Hamiltonian (11), we can also get the energy spectrum Eq. (19) by constructing appropriate Green s function S ; S +. Its physical meaning is not clear. One cannot explain why there appear two spectra for one system. The energy spectrum of a system should be unique unless a novel concept is introduced. Equation (12) is a real energy spectrum of spin wave. With the concept of pseudo-spin, it is easily understood that the spectrum, Eq. (19), is related to a pseudo-spin system. Equation (19) can be termed as a pseudo-energy spectrum. The pseudo-energy spectrum is also revealed in an antiferromagnetic system. We assume that the lattice, such

3 No. 1 Magnetization of Coupled Ultrathin Ferromagnetic Films 109 as a simple cubic (s.c.) and a body center cubic (b.c.c.), can be divided into two sub-lattices. The spins are aligned parallelly in each sub-lattice but anti-parallelly between the two sub-lattices. The Hamiltonian of the system is H = J i 1,i 2 S i1 S i2. (20) Here we merely consider the case the magnetic field is absent. i 1 and i 2 label the two sub-lattices. On the other hand, we can also think that the system is composed of a spin sub-lattice and a pseudo-spin sub-lattice. Thus the Hamiltonian is rewritten as H = J i 1,i 2 S i1 S p i 2. (21) Now we need to calculate the Green s functions G 1 = S 1 + ; S 1 and G 2 = S p+ 2 ; S 1 by making use of the Hamiltonian Eq. (21). The Fourier transformation of G 1,2 is denoted as g 1,2. The results are and g 1 (ω, k) = Sz ( ω0 (k) + zj S z ω 0 (k) ω ω 0 (k) ω 0(k) zj S z ), (22) ω + ω 0 (k) g 2 (ω, k) = Sz ( zj S z ω 0 (k) ω ω 0 (k) zj Sz ), (23) ω + ω 0 (k) ω 1,2 = ±ω 0 (k), ω 0 (k) = zj S z 1 (γ(k)) 2, γ(k) = 1 e ik a. (24) z a Here we note that there are two branches of energy spectra, ω 1,2 = ±ω 0 (k). We can say that ω 1 = ω 0 (k) > 0 belongs to spins, while ω 2 = ω 0 (k) < 0 belongs to pseudospins. Each spectrum has its own physical meaning. It seems that there are pseudo-spins and pseudo-energy spectra. A new concept can help one to realize things with a physical significance they should have. In fact, Pu et al. [13] first dealt with the antiferromagnetic system and obtained the energy spectra of Eqs. (24) using a skill by introducing an S which was defined as S z = S z, S + = S, S = S +. (25) The S is equivalent to S p. It meets relationship S S = is. Comparing S and S p, their three Cartesian axes constitute a left-hand system, so that either can be thought as a pseudo-spin. There are two slight differences between them. One is that when space is inverted for S, there is only one axis, z axis, reverses direction, while for S p all three axes are reversed. The second difference is that S meets usual commutators as in Eqs. (5) while S p meets Eqs. (13) which can be called pseudo-commutators. It seems better that a pseudo-angular momentum meets pseudo-commutators. We think our definition of pseudospin is more natural and has a clearer physical signification. 3 Model and Method 3.1 Ferromagnetic Coupling (FC) Between Ferromagnetic Films The studied film is an s.c. lattice and has the thickness of L monolayers (ML). Its normal is [001] direction, i.e. the perpendicular direction to the film (xy plane). The film is composed of two layers containing L 1 and L 2 monolayers (L 1 + L 2 = L), respectively. One layer, which is simply called film 1, includes the first to the L 1 -th monolayer. The other, called film 2, includes the (L 1 + 1)-th to the L-th monolayer. The nearest neighbor exchange interaction in film 1 is J 1 and that in film 2 is J 2. J 1, J 2 < 0, so that both are ferromagnetic. The coupling between the L 1 -th monolayer and the (L 1 + 1)-th monolayer is J int with J int > 0, which means ferromagnetic coupling. The Hamiltonian is L 2 H 1 = J 1 H = H 1 + H 2 + H int, (26) µ,ν=1 S iµ S jν ij L 2 [D µ (Siµ) z 2 + h µ=1 i i S z iµ ] (27) is the Hamiltonian of the film 1. Here the Greek letter subscripts denote the monolayers and English letters subscripts denote sites in one monolayer. iµ means the i-th site in the µ-monolayer. The second term in Eq. (27) represents the anisotropy which usually exists in film systems. H 2 has the same form as H 1 except that L 1 +1 µ, ν L and J 1 is replaced by J 2. H int = J int S il1 S il1+1, J int < 0. (28) i H int is the interfacial interaction between the two layers and can be called the coupling Hamiltonian. To calculate the magnetization of each monolayer Sµ, z one should calculate the Green s functions { S + lµ ; B mν },

4 110 WANG Huai-Yu, ZHOU Yun-Song, and WANG Chong-Yu Vol. 38 µ = 1, 2,..., L. Here B mν is an operator in the form as in Eqs. (2). Higher order Green s functions are decoupled by RPA, S z i S+ i ; B mν = S z i S+ i ; B mν, i i, (29) and [14] S z i S + i + S + i Sz i ; B mν = Γ i S z i S + i ; B mν, (30) Γ i = 3(S z i ) 2 S(S + 1) / S z i 2. (31) By commutator of S + lµ and the Hamiltonian, we get linear equations L [ω(k)δ µλ P µλ (k)]g λν (ω, k) λ=1 = δ µν 2π [S+, (S z ) n S ]. (32) Now the Green s functions have been further Fourier transformed about two-dimensional (2D) real space. With equation (4), there is a relation [9] S(S + 1)(S z µ) n (S z µ) n+1 (S z µ) n+2 = Φ ν { 2S z µ(s z µ 1) n + [(S z µ 1) n (S z µ) n ][S(S + 1) S z µ (S z µ) 2 ] }, (33) Φ ν is the following integral Φ ν = 1 4π 2 dk x dk y L α=1 U 2 να e ωα(k)/t 1. (34) The integration is carried out in 2D first Brillouin zone. ω(k) and U µν are eigenvalues and eigenvectors of matrix P µλ (k). The vectors U µν are orthogonal and normalized. Compared to Eq. (10), the integration Φ ν in Eq. (34) has been extended to the summation of the integrations for all L branch energy spectra. Here because all interactions are ferromagnetic, that is, the whole system is composed by spins, all the L branches of spectra ω µ (k) are positive. The spontaneous magnetization of the ν-th monolayer, S z ν, is calculated by Φ ν using Eq. (9). Note that S z ν is different from monolayers to monolayers. 3.2 Antiferromagnetic Coupling (AFC) Between Ferromagnetic Films The AFC means that the coupling between the L 1 -th and the (L 1 + 1)-th monolayer is negative. That makes the magnetization of film 2 antiparallel to that of film 1 under the Curie point when the external field is absent. Now the Hamiltonian still has the form as Eq. (26) and H 1 remains unchanged. But H 2 and H int are changed. Utilizing the pseudo-spin operator, H 2 is easily put down as one substitutes S in Eq. (26) by S p, and J 1 by J 2. The coupling Hamiltonian should be H int = J int S il1 S p i(l, J 1+1) int > 0. (35) i That J int > 0 represents AFC. This time we need to calculate S + lµ ; B mν, µ = 1, 2,..., L 1 and S p+ lµ ; B mν, µ = L 1 + 1, L 1 + 2,..., L. The derivation is the same as in subsection 3.1. It is worth while noting that the spontaneous magnetization S z µ in film 1 is calculated by Φ µ using Eq. (9). While for film 2, S pz µ is expressed by Φ pµ in Eq. (16). 4 Result and Discussion Because the coupling between films is antiferromagnetic, that is, the whole system is composed by spins (in film 1) and pseudo-spins (in film 2), L 1 branches of the spectra ω µ (k) are positive and other L 2 branches are negative. In this paper, we only consider spin S =1. The anisotropy strength is set as D 1 = D L = Otherwise, D µ = 0. We have investigated the case of L 1 = L 2, and J 1 = J 2 = 1. That is to say, the two sub-films are identical and there is a symmetry along the z direction, the normal of the film. Consequently, the spontaneous magnetization is of the symmetry Sµ z = SL+1 µ z. First, let us consider the case when the external field is absent. Figure 1 shows the spontaneous magnetization versus temperature for film thickness L 1 =1 to 4 with J int = 1. If J int = 1, the FC case, the spontaneous magnetizations are the same as in AFC in film 1. As long as the interfacial interaction strength for FC and AFC are the same, the spontaneous magnetizations in film 1 are exactly the same. The only difference is that in FC case, there is a symmetry that Sµ z = SL+1 µ z. While for AFC, there is an antisymmetric Sµ z = SL+1 µ z, that is, the spontaneous magnetization in film 2 is antiparallel to that in film 1. The magnetizations of all monolayers vanish at the same temperature which is the Curie point. Suppose that one of the monolayers has a nontrivial magnetization, then the adjacent monolayer will be certainly affected as though they feel an internal magnetic field, and also exhibit nontrivial magnetizations, and so on. Hence, the whole film must reach zero magnetization at the same temperature. For each film, the deeper monolayers have a higher magnetization in the intermediate temperature region. For instance, for L 1 =4, the magnetization of the fourth monolayer is the largest as it can easily be seen in Fig. 1. Figure 2 shows that the Curie point of the ferromagnetic film increases with the film thickness. This is in line with the experiments. [15,16] The Curie point also increases with the increased interfacial coupling.

5 No. 1 Magnetization of Coupled Ultrathin Ferromagnetic Films 111 that the magnetization increases with the film thickness. In each case, only the magnetization of the first monolayer is depicted. Fig. 1 Spontaneous magnetization S z vs. temperature for film thickness L 1 = 1 to 4 with J int = 1. To be clear, the results of other film thicknesses are not shown. Only the magnetizations of film 1 are shown. When J int = 1, the curves are the same. For FC, there is a symmetry S z µ = S z L+1 µ, and for AFC, there is a symmetry S z µ = S z L+1 µ. For each film, deeper monolayers have a higher magnetization in the intermediate temperature region, but the magnetizations of all monolayers vanish at the same temperature which is the Curie point. Fig. 3 Magnetization curves versus external magnetic field of ferromagnetically coupled case when J int = 0.1. The solid line is for thickness L 1 = 1 and temperature T = 0.1. Other lines are for the film thicknesses L 1 = 1 to 5 and for T = 1.0. In each case, only the magnetizations of the first monolayer of the film 1 is shown. Fig. 2 Curie point vsersus film thickness L 1 for coupling strengths J int = 1.0 and 0.5. The lines are just to connect the symbols. Higher coupling strength results in higher Curie point. When an external field is applied, the magnetization curves are computed. Figure 3 shows the case of FC, J int = 0.1. As the field is increased from zero, the magnetization increases. When the temperature is low, T = 0.1, the spontaneous magnetization Sµ z is nearly 1. Only the case of thickness L 1 = 1 is shown in Fig. 3. For thicker films (not shown), the magnetization curves are closer to 1. At higher temperature, T = 1.0, the spontaneous magnetization is less than that in lower temperature. As the field is applied, the magnetization increases. Figure 3 depicts the cases of thickness L 1 = 1 to 5. Note Fig. 4 Magnetization curves versus external magnetic field of AFC case when J int = 0.1 for the film thicknesses L 1 = 1, 3, and 5. In each case only the magnetizations of the first monolayer of film 1 (dotted lines) and the last monolayer of the film 2 (dashed lines) are shown. The curves with magnetization near to 1 (or 1) are for temperature T = 0.1 and others are for T = 1.0. Figure 4 shows the results of AFC case with J int =0.1. There appears three features in Fig. 4. The first is that with the absence of the external field, the spins in the two sub-films are equal in value and antiparallel in direction. As the field is applied, the magnetization in film 1 gradually increases as in the FC case. Under a weak field, the magnetization in film 2 decreases in value. As the field goes to a strength, the field energy will be enough to

6 112 WANG Huai-Yu, ZHOU Yun-Song, and WANG Chong-Yu Vol. 38 overcome the antiferromagnetic interaction and to make spins in film 2 reverse. The spins in every monolayer reverse under the same field which is called the reverse field h r. For L 1 = 1 5, the reverse field h r is 0.087, , , , and for temperature T = 1.0, is , , , , and for temperature T = 0.1, respectively. The h r decreases with the film thickness. This is because AFC only exists across the interface of the two sub-films, while in each sub-film the interaction is FC. As the film becomes thicker, the number of ferromagnetic film increases and the spins in film 2 reverse more easily. The second feature is that a thicker film has a larger spontaneous magnetization in each monolayer, which is consistent with Fig. 1. The third feature is that after reverse, the magnetizations in the two sub-films become the same and increase toward saturation with the further increase of the field. Before h r, the magnetization in film 1 will slightly decrease to receive the other branch. The latter two features are more remarkable at lower temperature T = 0.1 than at T = 1.0. This is because at low temperature, the magnetization is nearly saturated for all film thicknesses. In conclusion, we have presented a theory about the pseudo-spin. In the treating systems in which there is the antiferromagnetic coupling, the concept of the pseudo-spin is necessary and effective. In dealing with pure ferromagnetic systems, one can think that the pseudo-spin and spin are equivalent to each other. Magnetization of the two coupled ferromagnetic films with and without external field is studied. The coupling between the two ferromagnetic films can be either ferromagnetic or antiferromagnetic. With the same coupling strength, both the FC and AFC have the same Curie point. The dependence of the Curie point on film thickness and coupling strength is calculated. Under an external field, the magnetization curves are calculated. In the AFC case, the magnetization in film 2 reverses when the field is strong enough. And the regulation of the reverse is analyzed. When a ferromagnetic film is coupled with an antiferromagnetic film, there appears some usual phenomena such as exchange bias and minus coercive force. With the help of the theory presented in this paper, we are able to deal with these problems. These works are in undertaking. References [1] S.S.P. Parkin, N. More, and K.P. Roche, Phys. Rev. Lett. 64 (1990) 2304; S.S.P. Parkin, R. Bhadra, and K.P. Roche, Phys. Rev. Lett. 66 (1990) 2152; D.H. Mosca, et al., J. Magn. Magn. Mater. 94 (1991) L1. [2] J. Unguris, R.J. Celotta, and D.T. Piece, Phys. Rev. Lett. 67 (1991) 140; J. Unguris, R.J. Celotta, and D.T. Piece, Phys. Rev. Lett. 79 (1997) [3] S.T. Purcell, W. Folkerts, M.T. Johnson, N.W.E. McGee, K. Jager, J. aan de Stegge, W.B. Zeper, W. Hoving, and P. Grunberg, Phys. Rev. Lett. 67 (1991) 903. [4] M.T. Johnson, S.T. Purcell, N.W.E. McGee, R. Coehoorn, J. aan de Stegge, and W. Hoving, Phys. Rev. Lett. 68 (1992) [5] D.M. Edwards, J. Mathon, R.B. Muniz, and M.S. Phan, Phys. Rev. Lett. 67 (1991) 493. [6] P. Bruno and C. Chappert, Phys. Rev. Lett. 67 (1991) [7] U. Bovensiepen, F. Wilhelm, P. Srivastava, O. Poulopoulos, M. Farle, A. Ney, and K. Baberschke, Phys. Rev. Lett. 81 (1998) [8] A. Ney, F. Wilhelm, M. Farle, P. Poulopoulos, P. Srivastava, and K. Baberschke, Phys. Rev. B59 (1999) R3938. [9] S.V. Tyablikov, Methods in the Quantum Theory of Magnetism, Plenum, New York (1967). [10] N.N. Bogolyubov and S.V. Tyablikov, Sov. Phys. Dokl. 4 (1959) 589. [11] Huai-Yu WANG, Green s Function Method in Physics, Science, Education and Culture Press, Hong Kong (1998). [12] H.B. Callen, Phys. Rev. 130 (1963) 890. [13] Qingqi Zheng and Fucho Pu, Acta Phys. Sin. 20 (1964) 624, and the references therein. [14] M.E. Lines, Phys. Rev. 156 (1967) 534. [15] F. Huang, M.T. Kief, G.J. Mankey, and R.F. Willis, Phys. Rev. B49 (1994) [16] Renjun Zhang and Roy F. Willis, Phys. Rev. Lett. 86 (2001) 2665, and the references therein.