- Email: [email protected]
Spin wave spectra in metamagnet multilayers
Physica B 324 (2002) 217–222
Spin wave spectra in metamagnet multilayers J.M. de Souzaa,b, E.L. Albuquerquea,*, D.H.A.L. Anselmoc, G.A. Fariasb a
Departamento de F!ısica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal RN, Brazil b ! 60451-970 Fortaleza-CE, Brazil Departamento de F!ısica, Universidade Federal do Ceara, c ! Departamento de F!ısica, Universidade do Estado do Rio Grande do Norte, 59600-900 Mossoro-RN, Brazil Received 1 April 2002; accepted 19 June 2002
Abstract A microscopic theory is employed to investigate the spin wave spectra in binary multilayer structures, where one of their components is a metamagnetic material, the other one being a simple cubic ferromagnetic crystal. The metamagnet consists of ferromagnetically ordered layers, with the intralayer ferromagnetic exchange interactions being much stronger than the weak antiferromagnetic interaction between adjacent layers. The calculations are carried out for the exchange-dominated regime within the framework of the Heisenberg model and taking into account the random phase approximation. For numerical calculations, we consider as the metamagnet material the well-known FeCl2 and FeBr2 ; Fe being the ferromagnet spacer. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.30.Ds; 75.30.Gw; 75.30.Kz; 75.70.i Keywords: Magnetic films and multilayers; Spin dynamics
As a result of recent advances in fabrication techniques, magnetic multilayer structures of impressive quality are now synthesized from films composed of a wide variety of crystals. They form an intriguing new class of materials, in that their macroscopic properties are subject to design or control by varying the thickness or composition of the constituent films; in fact, some of these properties may be unique to the multilayer structure (for a review see Ref. [1]). Theoretical problems involving spin waves excitations in magnetic multilayers have also been extensively studied by considering the nature of *Corresponding author. Tel.: +55-84-2153993; fax: +55-842153791. E-mail address: [email protected] (E.L. Albuquerque).
the solutions for the appropriate wave field in each film. They are then linked together through appropriate boundary conditions, and the assistance of Bloch’s theorem. The surfaces and interfaces in these layered structures play an important role in the properties of the entire system, and indeed, most of the interesting properties of these excitations are due to the surface and interface effects. Many of these works have been concerned with the spin-wave excitations at the low-temperature regime, where at least one of the components is a ferromagnetic or an antiferromagnetic material. Furthermore, depending on the relative importance of the magnetic dipole–dipole and exchange interactions, different models for the magnetic behavior can be employed. For instance, for sufficiently small values of the
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 3 1 5 - 7
218
J.M. de Souza et al. / Physica B 324 (2002) 217–222
excitation wave vector, dipolar effects are dominant and magnetostatic modes should propagate in such structures [2,3]. On the other hand, at large excitation wave vectors typically greater than 108 m1 in a ferromagnet, exchange interaction, which is the restoring force for spin waves, will be dominant [4,5]. On the experimental side, the use of Brillouin scattering spectroscopy has proved to be an important tool to probe experimentally some theoretical predictions of these excitations [6,7]. In this paper we intend to extend previous studies on this subject by considering the propagation of spin waves in metamagnet multilayer structures. The metamagnetic material consists of ferromagnetically ordered layers, with the intralayer ferromagnetic exchange interactions being much stronger than the weak antiferromagnetic interaction between adjacent layers. We also consider the presence of a weak external magnetic ~ 0 applied perpendicular to the layers. In the field H regime of low temperatures and for small values of ~ 0 ; the adjacent layers the external magnetic field H of the metamagnet material order antiparallel to one another, giving the antiferromagnetic (AFM) ~ 0 large enough to phase. On the other hand, for H overcome the interlayer antiferromagnetic coupling, the overall ordering is ferromagnetic (FM phase). In between the metamagnet material, we consider a simple cubic ferromagnetic crystal (see Fig. 1). As examples of metamagnet material we consider FeCl2 and FeBr2 ; in which the magnetic anisotropy intralayers are sufficiently large compared with the interlayer weak antiferromagnetic coupling. Their bulk samples have been studied experimentally by techniques like neutron scattering [8] and light scattering [9], mainly in the AFM phase. Theoretical studies were also done, leading to satisfactory agreement with experiment [10]. They typically have a ferromagnetic resonance below 1 cm1 in which all sublattices precess in phase and, in addition, one or more higher-frequency exchange resonances, in which some of the sublattices are not in phase. By comparison with metallic ferromagnets, these additional resonance frequencies have been far less studied, and we intend here to put some light on these features.
Fig. 1. Schematic representation of the metamagnet superlattice considered in this paper.
We employ a transfer-matrix model to describe the spin waves in the multilayers structure, similar to the one used with success in dealing with the theory of superlattice plasmon–polaritons [11]. Experimental systems are likely to be more complicated than the model described here, with less simple crystal structures and possible different ordering at interface layers. However, these facts would influence only the detailed form of the transfer-matrix, and neither the general method nor the qualitative form of the dispersion relations should be affected. As it is depicted in Fig. 1, we consider a magnetic multilayer structure in which nA layers of a ferromagnet crystal A alternate with nB layers of a metamagnet material B: The size of the multilayer unit cell is L ¼ ðnA þ nB Þa; where a is the lattice parameter of both materials. A static ~ 0 is assumed to be in the applied magnetic field H z-direction. The two materials are characterized by single-ion uniaxial anisotropy parameters DA and
J.M. de Souza et al. / Physica B 324 (2002) 217–222
DB : The exchange constant across each interface A B is equal to I: The Heisenberg Hamiltonian for the bulk ferromagnet specimen is X X ~j gmB H0 ~i S H ¼ ð1=2Þ Jij S Sz i
X
i;j
i
DA ðSiz Þ2 :
ð1Þ
i
Here, Jij is the exchange term, Si denotes the spin operator at magnetic site i; g is the usual Lande! factor and mB is the Bohr magneton. The dispersion equation for the bulk spin wave is found within the random-phase-approximation (RPA) from the equation of motion for the operator Siþ ¼ Six þ iSiy : We can prove that this equation is X ð_o DF ÞSiþ ¼ SA JA ðSiþ Sjþ Þ; ð2Þ
and the anisotropy Hamiltonian is expressed by " # X X z 2 z 2 ðSi Þ þ ðSj Þ : ð6Þ HA ¼ DB i
j
The effect of the Ising anisotropy can be included if the parameter s is non-zero. Here i and i0 denote sites on one sublattice (i.e. one type of layer in the metamagnet crystal), while j and j 0 denote sites on the other sublattice (the set of adjacent layers). Also Jij is the interlayer antiferromagnetic exchange interaction and Jii0 ; Jjj 0 are the intralayer ferromagnetic exchange terms. Using again the Heisenberg equation of motion for the spin operator we get, after a bit of algebra, the following expression for the spin-wave dispersion relation for the metamagnetic material: (a) ferromagnetic (FM) phase: cosðkB aÞ ¼ ½1 þ 3ðJ1 þ J2 Þ=J3 ðJ1 =J3 Þg ðk~jj Þ M
j
where we have made a Fourier transform to frequency o: Also, DF ¼ gmB H0 þ 2DA SA Z; with Z ¼ ½1 ð2SA Þ1 ; where SA is the RPA approximation for Siz : Considering plane-wave solution for the operators, and either by diagonalizing Hamiltonian (1) or by solving (2) directly, we can find that the spinwave dispersion relation in an infinite ferromagnet is given by cosðkA aÞ ¼ 3 g ðk~jj Þ þ ðDF _oÞ=2JA SA : ð3Þ F
Here, gF ðk~jj Þ ¼ cosðkx aÞ þ cosðky aÞ; with k~jj ¼ ðkx ; ky ; 0Þ: On the other hand, the spin Hamiltonian for uniaxial metamagnets can be written as [12,13]: 1X ~i S ~j þ sS z Sz Þ H¼ Jij ðS i j 2 i;j 1X ~i S ~i0 þ sS z S z0 Þ Jii0 ðS i i 2 i;i0 1X ~j S ~j0 þ sSz S z0 Þ Jjj 0 ðS j j 2 j;j 0 þ HZ þ HA ;
ð4Þ
where the Zeeman Hamiltonian is given by " # X X z z Si þ Sj ; HZ ¼ gmB H0 i
219
j
ð5Þ
þ ðDþ M _oÞ=2J3 SB ; where gM ðk~jj Þ ¼ cosðkx aÞ þ 2 cosðkx a=2Þ cosðky a pffiffiffi þ 2ðJ2 =J1 Þ½cosðky a 3Þ pffiffiffi þ 2 cosðkx a3=2Þcosðky a 3=2Þ
ð7Þ pffiffiffi 3=2Þ
ð8Þ
and D7 M ¼ gmB H0 7ð2SB 1ÞDB : Here, J1 and J2 are the dominant nearest-neighbor exchange and the weaker next-nearest-neighbor exchange terms, respectively. Also, J3 is the weak antiferromagnetic interlayer exchange term. (b) antiferromagnetic (AFM) phase (two branches): cosðkB aÞ ¼ 7ð1=2SB J3 Þf½l þ ðDþ M _oÞ 1=2 ; ½l ðD M _oÞg
ð9Þ
where l ¼ 2J1 SB fð1 þ sÞ½3ð1 þ J2 =J1 Þ þ ðJ3 =J1 Þ gM ðk~jj Þg: We now turn to the magnetic multilayer structure depicted in Fig. 1. At non-zero temperature, the equilibrium configuration must exhibit the analog of surface reconstruction. This implies that the mean spin SJ (J ¼ A or B), in both materials is a function of its distance from the nearest A B interface. However, although this effect is important, we can overcome it by restricting our attention to the low-temperature
220
J.M. de Souza et al. / Physica B 324 (2002) 217–222
regime, that is T5Tc ; at which the spins are fully ordered. The spin-wave dispersion equation can then be found by solving the RPA equations of motion for the spin operators. A spin that is not in an interface layer, like those labeled a; b; g; and d; in Fig. 1, has the same nearest-neighbor environment and therefore the same equation of motion as a spin in the corresponding bulk medium. Thus, the spin wave amplitudes should be given, within each bulk material, by a linear combination of the positive- and negative-going solutions, i.e.: SJþ ¼ An expðikJ zÞ þ Bn expðikJ zÞ;
ð10Þ
where J is A or B: The wave vector kA is given by Eq. (3), while kB is given by Eq. (7) for the ferromagnetic phase, and by Eq. (9) for the antiferromagnetic phase. Then, following the procedure described in Ref. [14], the equations at the interface layers can be cast in matrix forms. Making use of the transfer-matrix treatment and Bloch’s theorem, we find the required spin wave dispersion relation, i.e: cosðQLÞ ¼ ð1=2ÞTr T;
layers. On the other hand, for FeCl2 ; the stacking of the layers is staggered, leading to three nearest neighbors in each of the adjacent layers. Fig. 2(a) shows the spin-wave dispersion relation for bulk spin waves in a multilayer structure whose constituents are Fe (ferromagnet) and FeBr2 (metamagnet), in the FM phase. The metamagnet material has spin SB ¼ 1 and critical temperature Tc ¼ 14:2 K: The approximate values of the exchange and anisotropic parameters are known from Raman scattering experiments [8,9], i.e., J1 ¼ 5:07 cm1 ; J2 ¼ 1:2 cm1 ; J3 ¼ 1:45 cm1 ; DB ¼ 7:34 cm1 ; and s ¼ 0:28: The spin-wave analysis is more straightforward in this case which applies for the external magnetic field H0 greater than a bulk critical field, given
ð11Þ
where Q is the Bloch wave vector, and the transfer matrix T is defined by T ¼ NA1 MA NB1 MB : Here the forms of the matrices MJ and NJ (J ¼ A or B) can be found elsewhere [14], if one neglects the non-uniaxial anisotropy terms. Thus, once T is evaluated, the required spin wave spectra can be determined. We now discuss our analytical results in more details for the specific cases of the metamagnets FeBr2 and FeCl2 ; with Fe being the ferromagnet spacer, in both the FM and AFM phases. These metamagnets have competing ferromagnetic and antiferromagnetic exchange terms, and large single-ion uniaxial anisotropy which prevents the appearance of the spin-flop phase intermediate between the AFM and FM phases. Besides, they have different crystallographic arrangements of the magnetic ions, leading to differences in the spin-wave spectra. Although they have the same trigonal arrangement of the magnetic ion Fe2þ within each ferromagnetically ordered layer, the two materials differ in the stacking arrangement of the layers. In FeBr2 ; the Fe2þ ions in one layer are directly above and below those in the adjacent
Fig. 2. (a) Dispersion relation of bulk (shaded areas) spin waves in the Fe=FeBr2 superlattice for the FM phase as a function of the in-plane dimensionless wavevector kx a; for the applied magnetic field gmB H0 ¼ 4:0 cm1 : (b) Same as in (a), but now for O against the dimensionless Bloch wavevector QL:
J.M. de Souza et al. / Physica B 324 (2002) 217–222
approximately by gmB Hc ¼ 2ð1 þ sÞSB ; because the direction of net spin alignment is the same in both sublattices. We take the applied magnetic field gmB H0 ¼ 4:0 cm1 and the interlayer exchange term between the two materials as I=JA ¼ 1:2: We plot the reduced frequency O ¼ o=JA SA (we consider _ ¼ 1) against kx a (here we take for simplicity kjj ¼ kx ), which is a good quantum number not only for a single film but also for a layered structure [15]. This spectrum is much more sophisticated than the one presented in pure metamagnets, with and without non-uniaxial anisotropy field [12,13]. Besides, as a characteristic of the zone folding behavior found in superlattice structures, the spectrum presents several broad bulk bands, which are here shown shaded, limited by the curves QL ¼ 0 and p; with many resonance frequencies. This is a consequence of the oscillatory profile of the appropriate wavefunction in both the ferromagnet and metamagnet material. In between these curves there are gap regions, where the surface modes can propagate. To some extent, it resembles the electronic band structure in solids, where the electronic bands develop from single atomic levels as a result of inter-atomic interactions. Fig. 2(b) shows the O versus QL spectrum for the same materials. Now one can see clearly the zone folding effect and its influence in the so-called stop (in the vicinity of the Brillouin zone’s center) and pass (closed to the Brillouin zone’s edge) frequency bands. In both Fig. 2(a) and (b), the gap between the allowed bulk bands (or modes) is also influenced by the anisotropic field HA : On the other hand, for the O kx a spectra, the widths of the pass bands are strongly determined by the value of J3 ; the antiferromagnetic exchange term. On a single metamagnetic crystal [12,13], this effect also occurs, but now we have a cooperative (or effective) multilayer exchange interaction, which in turn yields broader pass bands and narrower stop bands, especially in the FM phase. For completeness, we have also shown in Fig. 3(a) the allowed spin-wave bulk bands for an extended Brillouin zone, considering the case where the metamagnet material is FeCl2 in its AFM phase. We justify this choice because at the low-temperature limit this is the metamagnet’s
221
Fig. 3. (a) The bulk modes for spin waves in the Fe=FeCl2 superlattice, in the AFM phase. Here, the applied magnetic field is gmB H0 ¼ 1:6 cm1 ; and we show an extended Brillouin zone. (b) Same as in (a), but now the spectrum is shown as O versus QL:
stable phase when H0 oHc : We take the applied magnetic field gmB H0 ¼ 1:6 cm1 : The other physical parameters, in accordance with data from neutron and Raman scattering [10] are: J1 ¼ 5:5 cm1 ; J2 ¼ 1:2 cm1 ; J3 ¼ 0:28 cm1 ; DB ¼ 9:6 cm1 ; and s ¼ 0:2: In this case their bands are narrower than those in the correspondent FM phase (not shown here). The appropriated wavefunction is more localized and decays exponentially. As a consequence, the spin-wave mode is almost confined in the layers, with little overlap, and the bulk band width shrinks. This is mainly influenced by the fact that for FeCl2 the bulk layers are staggered from one to another, providing a weaker ðJ3 ¼ 0:28 cm1 Þ antiferromagnetic
222
J.M. de Souza et al. / Physica B 324 (2002) 217–222
exchange constant. This fact can be more appreciated in Fig. 3(b), depicting the reduced frequency O against the dimensionless wave vector QL; for the propagating modes in the Fe=FeCl2 multilayer. It is quite clear that the spectrum now present narrow pass and stop bands, as well as a higher number of allowed modes, compared to the FM phase. The most appropriate experimental technique for studying the multilayers spin waves is the inelastic light scattering spectroscopy of Raman and Brillouin type, using a backscattering geometry to scatter light from the free surface of the superlattice (for details see Ref. [16]). Techniques involving magnetic resonance (for example, ferromagnetic resonance, standing spin-wave resonance, etc.) can also be used, and indeed they have previously been successfully applied to surface and bulk spin waves in various magnetic microstructures [1]. Other experimental techniques, like the grating coupling and Attenuated Total Reflection (ATR), were proposed by a number of authors (for a review see Ref. [17]). However there are some difficulties in probing surface modes on ferromagnets, due to the fact that their resonance frequencies are quite low (less than 1 cm1 in YIG, for instance). Therefore the samples should be very large and, in addition, the frequencies are below the range of conventional far-infrared spectrometers. Nevertheless, we hope that our theoretical predictions can be tested through experimental measurements.
Acknowledgements We would like to thank the Brazilian Agencies CNPq, CT-Petro and CAPES-Procad for partial financial support. References [1] See the papers published in the special issue of J. Magn. Magn. Mater. 200 (1999). [2] R.E. Camley, M.G. Cottam, Phys. Rev. B 35 (1987) 189. [3] R.E. Camley, M.G. Cottam, D.R. Tilley, Solid State Commun. 81 (1992) 571. [4] E.L. Albuquerque, P. Fulco, E.F. Sarmento, D.R. Tilley, Solid State Commun. 58 (1986) 41. [5] J. Barn!as, J. Phys. C 21 (1988) 1021,8097. [6] I.K. Schuller, M. Grimsditch, J. Appl. Phys. 55 (1985) 2594. [7] C.A. Ramos, D. Lederman, A.R. King, V. Jaccarino, Phys. Rev. Lett. 65 (1990) 2913. [8] W.B. Yelon, C. Vettier, J. Phys. C 8 (1975) 2760. [9] G.C. Psaltakis, G. Mischler, D.J. Lockwood, M.G. Cottam, A. Zwick, S. Legrand, J. Phys. C 17 (1984) 1735. [10] G.C. Psaltakis, M.G. Cottam, J. Phys. C 15 (1982) 4847. [11] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 233 (1993) 67. [12] J.H. Baskey, M.G. Cottam, Phys. Rev. B 42 (1990) 4304. [13] D.H.A.L. Anselmo, E.L. Albuquerque, M.G. Cottam, J. Appl. Phys. 83 (1998) 6955. [14] D.H.A.L. Anselmo, E.L. Albuquerque, Phys. Stat. Sol. B 198 (1996) 827. [15] J. Barn!as, in: M.G. Cottam (Ed.), Linear and Nonlinear Spin Waves in Magnetic Films and Superlattices, World Scientific, Singapore, 1994. [16] M.G. Cottam, D.J. Lockwood, Light Scattering in Magnetic Solids, Wiley, New York, 1986. [17] K. Abraha, D.R. Tilley, Surf. Sci. Rep. 24 (1996) 125.
Spin wave spectra in metamagnet multilayers J.M. de Souzaa,b, E.L. Albuquerquea,*, D.H.A.L. Anselmoc, G.A. Fariasb a
Departamento de F!ısica, Universidade Federal do Rio Grande do Norte, 59072-970 Natal RN, Brazil b ! 60451-970 Fortaleza-CE, Brazil Departamento de F!ısica, Universidade Federal do Ceara, c ! Departamento de F!ısica, Universidade do Estado do Rio Grande do Norte, 59600-900 Mossoro-RN, Brazil Received 1 April 2002; accepted 19 June 2002
Abstract A microscopic theory is employed to investigate the spin wave spectra in binary multilayer structures, where one of their components is a metamagnetic material, the other one being a simple cubic ferromagnetic crystal. The metamagnet consists of ferromagnetically ordered layers, with the intralayer ferromagnetic exchange interactions being much stronger than the weak antiferromagnetic interaction between adjacent layers. The calculations are carried out for the exchange-dominated regime within the framework of the Heisenberg model and taking into account the random phase approximation. For numerical calculations, we consider as the metamagnet material the well-known FeCl2 and FeBr2 ; Fe being the ferromagnet spacer. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.30.Ds; 75.30.Gw; 75.30.Kz; 75.70.i Keywords: Magnetic films and multilayers; Spin dynamics
As a result of recent advances in fabrication techniques, magnetic multilayer structures of impressive quality are now synthesized from films composed of a wide variety of crystals. They form an intriguing new class of materials, in that their macroscopic properties are subject to design or control by varying the thickness or composition of the constituent films; in fact, some of these properties may be unique to the multilayer structure (for a review see Ref. [1]). Theoretical problems involving spin waves excitations in magnetic multilayers have also been extensively studied by considering the nature of *Corresponding author. Tel.: +55-84-2153993; fax: +55-842153791. E-mail address: [email protected] (E.L. Albuquerque).
the solutions for the appropriate wave field in each film. They are then linked together through appropriate boundary conditions, and the assistance of Bloch’s theorem. The surfaces and interfaces in these layered structures play an important role in the properties of the entire system, and indeed, most of the interesting properties of these excitations are due to the surface and interface effects. Many of these works have been concerned with the spin-wave excitations at the low-temperature regime, where at least one of the components is a ferromagnetic or an antiferromagnetic material. Furthermore, depending on the relative importance of the magnetic dipole–dipole and exchange interactions, different models for the magnetic behavior can be employed. For instance, for sufficiently small values of the
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 3 1 5 - 7
218
J.M. de Souza et al. / Physica B 324 (2002) 217–222
excitation wave vector, dipolar effects are dominant and magnetostatic modes should propagate in such structures [2,3]. On the other hand, at large excitation wave vectors typically greater than 108 m1 in a ferromagnet, exchange interaction, which is the restoring force for spin waves, will be dominant [4,5]. On the experimental side, the use of Brillouin scattering spectroscopy has proved to be an important tool to probe experimentally some theoretical predictions of these excitations [6,7]. In this paper we intend to extend previous studies on this subject by considering the propagation of spin waves in metamagnet multilayer structures. The metamagnetic material consists of ferromagnetically ordered layers, with the intralayer ferromagnetic exchange interactions being much stronger than the weak antiferromagnetic interaction between adjacent layers. We also consider the presence of a weak external magnetic ~ 0 applied perpendicular to the layers. In the field H regime of low temperatures and for small values of ~ 0 ; the adjacent layers the external magnetic field H of the metamagnet material order antiparallel to one another, giving the antiferromagnetic (AFM) ~ 0 large enough to phase. On the other hand, for H overcome the interlayer antiferromagnetic coupling, the overall ordering is ferromagnetic (FM phase). In between the metamagnet material, we consider a simple cubic ferromagnetic crystal (see Fig. 1). As examples of metamagnet material we consider FeCl2 and FeBr2 ; in which the magnetic anisotropy intralayers are sufficiently large compared with the interlayer weak antiferromagnetic coupling. Their bulk samples have been studied experimentally by techniques like neutron scattering [8] and light scattering [9], mainly in the AFM phase. Theoretical studies were also done, leading to satisfactory agreement with experiment [10]. They typically have a ferromagnetic resonance below 1 cm1 in which all sublattices precess in phase and, in addition, one or more higher-frequency exchange resonances, in which some of the sublattices are not in phase. By comparison with metallic ferromagnets, these additional resonance frequencies have been far less studied, and we intend here to put some light on these features.
Fig. 1. Schematic representation of the metamagnet superlattice considered in this paper.
We employ a transfer-matrix model to describe the spin waves in the multilayers structure, similar to the one used with success in dealing with the theory of superlattice plasmon–polaritons [11]. Experimental systems are likely to be more complicated than the model described here, with less simple crystal structures and possible different ordering at interface layers. However, these facts would influence only the detailed form of the transfer-matrix, and neither the general method nor the qualitative form of the dispersion relations should be affected. As it is depicted in Fig. 1, we consider a magnetic multilayer structure in which nA layers of a ferromagnet crystal A alternate with nB layers of a metamagnet material B: The size of the multilayer unit cell is L ¼ ðnA þ nB Þa; where a is the lattice parameter of both materials. A static ~ 0 is assumed to be in the applied magnetic field H z-direction. The two materials are characterized by single-ion uniaxial anisotropy parameters DA and
J.M. de Souza et al. / Physica B 324 (2002) 217–222
DB : The exchange constant across each interface A B is equal to I: The Heisenberg Hamiltonian for the bulk ferromagnet specimen is X X ~j gmB H0 ~i S H ¼ ð1=2Þ Jij S Sz i
X
i;j
i
DA ðSiz Þ2 :
ð1Þ
i
Here, Jij is the exchange term, Si denotes the spin operator at magnetic site i; g is the usual Lande! factor and mB is the Bohr magneton. The dispersion equation for the bulk spin wave is found within the random-phase-approximation (RPA) from the equation of motion for the operator Siþ ¼ Six þ iSiy : We can prove that this equation is X ð_o DF ÞSiþ ¼ SA JA ðSiþ Sjþ Þ; ð2Þ
and the anisotropy Hamiltonian is expressed by " # X X z 2 z 2 ðSi Þ þ ðSj Þ : ð6Þ HA ¼ DB i
j
The effect of the Ising anisotropy can be included if the parameter s is non-zero. Here i and i0 denote sites on one sublattice (i.e. one type of layer in the metamagnet crystal), while j and j 0 denote sites on the other sublattice (the set of adjacent layers). Also Jij is the interlayer antiferromagnetic exchange interaction and Jii0 ; Jjj 0 are the intralayer ferromagnetic exchange terms. Using again the Heisenberg equation of motion for the spin operator we get, after a bit of algebra, the following expression for the spin-wave dispersion relation for the metamagnetic material: (a) ferromagnetic (FM) phase: cosðkB aÞ ¼ ½1 þ 3ðJ1 þ J2 Þ=J3 ðJ1 =J3 Þg ðk~jj Þ M
j
where we have made a Fourier transform to frequency o: Also, DF ¼ gmB H0 þ 2DA SA Z; with Z ¼ ½1 ð2SA Þ1 ; where SA is the RPA approximation for Siz : Considering plane-wave solution for the operators, and either by diagonalizing Hamiltonian (1) or by solving (2) directly, we can find that the spinwave dispersion relation in an infinite ferromagnet is given by cosðkA aÞ ¼ 3 g ðk~jj Þ þ ðDF _oÞ=2JA SA : ð3Þ F
Here, gF ðk~jj Þ ¼ cosðkx aÞ þ cosðky aÞ; with k~jj ¼ ðkx ; ky ; 0Þ: On the other hand, the spin Hamiltonian for uniaxial metamagnets can be written as [12,13]: 1X ~i S ~j þ sS z Sz Þ H¼ Jij ðS i j 2 i;j 1X ~i S ~i0 þ sS z S z0 Þ Jii0 ðS i i 2 i;i0 1X ~j S ~j0 þ sSz S z0 Þ Jjj 0 ðS j j 2 j;j 0 þ HZ þ HA ;
ð4Þ
where the Zeeman Hamiltonian is given by " # X X z z Si þ Sj ; HZ ¼ gmB H0 i
219
j
ð5Þ
þ ðDþ M _oÞ=2J3 SB ; where gM ðk~jj Þ ¼ cosðkx aÞ þ 2 cosðkx a=2Þ cosðky a pffiffiffi þ 2ðJ2 =J1 Þ½cosðky a 3Þ pffiffiffi þ 2 cosðkx a3=2Þcosðky a 3=2Þ
ð7Þ pffiffiffi 3=2Þ
ð8Þ
and D7 M ¼ gmB H0 7ð2SB 1ÞDB : Here, J1 and J2 are the dominant nearest-neighbor exchange and the weaker next-nearest-neighbor exchange terms, respectively. Also, J3 is the weak antiferromagnetic interlayer exchange term. (b) antiferromagnetic (AFM) phase (two branches): cosðkB aÞ ¼ 7ð1=2SB J3 Þf½l þ ðDþ M _oÞ 1=2 ; ½l ðD M _oÞg
ð9Þ
where l ¼ 2J1 SB fð1 þ sÞ½3ð1 þ J2 =J1 Þ þ ðJ3 =J1 Þ gM ðk~jj Þg: We now turn to the magnetic multilayer structure depicted in Fig. 1. At non-zero temperature, the equilibrium configuration must exhibit the analog of surface reconstruction. This implies that the mean spin SJ (J ¼ A or B), in both materials is a function of its distance from the nearest A B interface. However, although this effect is important, we can overcome it by restricting our attention to the low-temperature
220
J.M. de Souza et al. / Physica B 324 (2002) 217–222
regime, that is T5Tc ; at which the spins are fully ordered. The spin-wave dispersion equation can then be found by solving the RPA equations of motion for the spin operators. A spin that is not in an interface layer, like those labeled a; b; g; and d; in Fig. 1, has the same nearest-neighbor environment and therefore the same equation of motion as a spin in the corresponding bulk medium. Thus, the spin wave amplitudes should be given, within each bulk material, by a linear combination of the positive- and negative-going solutions, i.e.: SJþ ¼ An expðikJ zÞ þ Bn expðikJ zÞ;
ð10Þ
where J is A or B: The wave vector kA is given by Eq. (3), while kB is given by Eq. (7) for the ferromagnetic phase, and by Eq. (9) for the antiferromagnetic phase. Then, following the procedure described in Ref. [14], the equations at the interface layers can be cast in matrix forms. Making use of the transfer-matrix treatment and Bloch’s theorem, we find the required spin wave dispersion relation, i.e: cosðQLÞ ¼ ð1=2ÞTr T;
layers. On the other hand, for FeCl2 ; the stacking of the layers is staggered, leading to three nearest neighbors in each of the adjacent layers. Fig. 2(a) shows the spin-wave dispersion relation for bulk spin waves in a multilayer structure whose constituents are Fe (ferromagnet) and FeBr2 (metamagnet), in the FM phase. The metamagnet material has spin SB ¼ 1 and critical temperature Tc ¼ 14:2 K: The approximate values of the exchange and anisotropic parameters are known from Raman scattering experiments [8,9], i.e., J1 ¼ 5:07 cm1 ; J2 ¼ 1:2 cm1 ; J3 ¼ 1:45 cm1 ; DB ¼ 7:34 cm1 ; and s ¼ 0:28: The spin-wave analysis is more straightforward in this case which applies for the external magnetic field H0 greater than a bulk critical field, given
ð11Þ
where Q is the Bloch wave vector, and the transfer matrix T is defined by T ¼ NA1 MA NB1 MB : Here the forms of the matrices MJ and NJ (J ¼ A or B) can be found elsewhere [14], if one neglects the non-uniaxial anisotropy terms. Thus, once T is evaluated, the required spin wave spectra can be determined. We now discuss our analytical results in more details for the specific cases of the metamagnets FeBr2 and FeCl2 ; with Fe being the ferromagnet spacer, in both the FM and AFM phases. These metamagnets have competing ferromagnetic and antiferromagnetic exchange terms, and large single-ion uniaxial anisotropy which prevents the appearance of the spin-flop phase intermediate between the AFM and FM phases. Besides, they have different crystallographic arrangements of the magnetic ions, leading to differences in the spin-wave spectra. Although they have the same trigonal arrangement of the magnetic ion Fe2þ within each ferromagnetically ordered layer, the two materials differ in the stacking arrangement of the layers. In FeBr2 ; the Fe2þ ions in one layer are directly above and below those in the adjacent
Fig. 2. (a) Dispersion relation of bulk (shaded areas) spin waves in the Fe=FeBr2 superlattice for the FM phase as a function of the in-plane dimensionless wavevector kx a; for the applied magnetic field gmB H0 ¼ 4:0 cm1 : (b) Same as in (a), but now for O against the dimensionless Bloch wavevector QL:
J.M. de Souza et al. / Physica B 324 (2002) 217–222
approximately by gmB Hc ¼ 2ð1 þ sÞSB ; because the direction of net spin alignment is the same in both sublattices. We take the applied magnetic field gmB H0 ¼ 4:0 cm1 and the interlayer exchange term between the two materials as I=JA ¼ 1:2: We plot the reduced frequency O ¼ o=JA SA (we consider _ ¼ 1) against kx a (here we take for simplicity kjj ¼ kx ), which is a good quantum number not only for a single film but also for a layered structure [15]. This spectrum is much more sophisticated than the one presented in pure metamagnets, with and without non-uniaxial anisotropy field [12,13]. Besides, as a characteristic of the zone folding behavior found in superlattice structures, the spectrum presents several broad bulk bands, which are here shown shaded, limited by the curves QL ¼ 0 and p; with many resonance frequencies. This is a consequence of the oscillatory profile of the appropriate wavefunction in both the ferromagnet and metamagnet material. In between these curves there are gap regions, where the surface modes can propagate. To some extent, it resembles the electronic band structure in solids, where the electronic bands develop from single atomic levels as a result of inter-atomic interactions. Fig. 2(b) shows the O versus QL spectrum for the same materials. Now one can see clearly the zone folding effect and its influence in the so-called stop (in the vicinity of the Brillouin zone’s center) and pass (closed to the Brillouin zone’s edge) frequency bands. In both Fig. 2(a) and (b), the gap between the allowed bulk bands (or modes) is also influenced by the anisotropic field HA : On the other hand, for the O kx a spectra, the widths of the pass bands are strongly determined by the value of J3 ; the antiferromagnetic exchange term. On a single metamagnetic crystal [12,13], this effect also occurs, but now we have a cooperative (or effective) multilayer exchange interaction, which in turn yields broader pass bands and narrower stop bands, especially in the FM phase. For completeness, we have also shown in Fig. 3(a) the allowed spin-wave bulk bands for an extended Brillouin zone, considering the case where the metamagnet material is FeCl2 in its AFM phase. We justify this choice because at the low-temperature limit this is the metamagnet’s
221
Fig. 3. (a) The bulk modes for spin waves in the Fe=FeCl2 superlattice, in the AFM phase. Here, the applied magnetic field is gmB H0 ¼ 1:6 cm1 ; and we show an extended Brillouin zone. (b) Same as in (a), but now the spectrum is shown as O versus QL:
stable phase when H0 oHc : We take the applied magnetic field gmB H0 ¼ 1:6 cm1 : The other physical parameters, in accordance with data from neutron and Raman scattering [10] are: J1 ¼ 5:5 cm1 ; J2 ¼ 1:2 cm1 ; J3 ¼ 0:28 cm1 ; DB ¼ 9:6 cm1 ; and s ¼ 0:2: In this case their bands are narrower than those in the correspondent FM phase (not shown here). The appropriated wavefunction is more localized and decays exponentially. As a consequence, the spin-wave mode is almost confined in the layers, with little overlap, and the bulk band width shrinks. This is mainly influenced by the fact that for FeCl2 the bulk layers are staggered from one to another, providing a weaker ðJ3 ¼ 0:28 cm1 Þ antiferromagnetic
222
J.M. de Souza et al. / Physica B 324 (2002) 217–222
exchange constant. This fact can be more appreciated in Fig. 3(b), depicting the reduced frequency O against the dimensionless wave vector QL; for the propagating modes in the Fe=FeCl2 multilayer. It is quite clear that the spectrum now present narrow pass and stop bands, as well as a higher number of allowed modes, compared to the FM phase. The most appropriate experimental technique for studying the multilayers spin waves is the inelastic light scattering spectroscopy of Raman and Brillouin type, using a backscattering geometry to scatter light from the free surface of the superlattice (for details see Ref. [16]). Techniques involving magnetic resonance (for example, ferromagnetic resonance, standing spin-wave resonance, etc.) can also be used, and indeed they have previously been successfully applied to surface and bulk spin waves in various magnetic microstructures [1]. Other experimental techniques, like the grating coupling and Attenuated Total Reflection (ATR), were proposed by a number of authors (for a review see Ref. [17]). However there are some difficulties in probing surface modes on ferromagnets, due to the fact that their resonance frequencies are quite low (less than 1 cm1 in YIG, for instance). Therefore the samples should be very large and, in addition, the frequencies are below the range of conventional far-infrared spectrometers. Nevertheless, we hope that our theoretical predictions can be tested through experimental measurements.
Acknowledgements We would like to thank the Brazilian Agencies CNPq, CT-Petro and CAPES-Procad for partial financial support. References [1] See the papers published in the special issue of J. Magn. Magn. Mater. 200 (1999). [2] R.E. Camley, M.G. Cottam, Phys. Rev. B 35 (1987) 189. [3] R.E. Camley, M.G. Cottam, D.R. Tilley, Solid State Commun. 81 (1992) 571. [4] E.L. Albuquerque, P. Fulco, E.F. Sarmento, D.R. Tilley, Solid State Commun. 58 (1986) 41. [5] J. Barn!as, J. Phys. C 21 (1988) 1021,8097. [6] I.K. Schuller, M. Grimsditch, J. Appl. Phys. 55 (1985) 2594. [7] C.A. Ramos, D. Lederman, A.R. King, V. Jaccarino, Phys. Rev. Lett. 65 (1990) 2913. [8] W.B. Yelon, C. Vettier, J. Phys. C 8 (1975) 2760. [9] G.C. Psaltakis, G. Mischler, D.J. Lockwood, M.G. Cottam, A. Zwick, S. Legrand, J. Phys. C 17 (1984) 1735. [10] G.C. Psaltakis, M.G. Cottam, J. Phys. C 15 (1982) 4847. [11] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 233 (1993) 67. [12] J.H. Baskey, M.G. Cottam, Phys. Rev. B 42 (1990) 4304. [13] D.H.A.L. Anselmo, E.L. Albuquerque, M.G. Cottam, J. Appl. Phys. 83 (1998) 6955. [14] D.H.A.L. Anselmo, E.L. Albuquerque, Phys. Stat. Sol. B 198 (1996) 827. [15] J. Barn!as, in: M.G. Cottam (Ed.), Linear and Nonlinear Spin Waves in Magnetic Films and Superlattices, World Scientific, Singapore, 1994. [16] M.G. Cottam, D.J. Lockwood, Light Scattering in Magnetic Solids, Wiley, New York, 1986. [17] K. Abraha, D.R. Tilley, Surf. Sci. Rep. 24 (1996) 125.