- Email: [email protected]
On the possibility of appearance of surface spin waves in transition metal thin films
Physica B 154 (1989) 373-378 North-Holland, Amsterdam
ON THE POSSIBILITY OF APPEARANCE OF SURFACE SPIN WAVES IN TRANSITION METAL THIN FILMS* R. SWIRKOWICZ Institute of Physics, Technical University, 00-662 Warsaw, Poland Received 1 March 1988 Revised manuscript received 22 November 1988
The transverse dynamic susceptibility of thin ferromagnetic film and spin-wave eigenfunctions are calculated within the framework of the Hubbard model. For various values of surface parameters the localization of the spin-wave modes, their energies and damping of the modes are discussed. The calculations show that a decrease of the effective Coulomb integral in the surface layer without any change of other surface parameters causes the substantial lowering of the surface magnetization and lead to appearance of optical surface modes. These modes, however, are strongly damped. An increase of the effective integral in the surface layer leads to an increase of the surface magnetization and favors appearance of the acoustic surface modes. The changes of the core energies in the surface layer also favor appearance of acoustic surface modes provided the surface magnetization is close or larger than the bulk one.
I. Introduction
Recently, the dynamic susceptibility and surface spin waves in thin films of transition metals have been investigated within the framework of the itinerant electron model [1-7]. The one-band Hubbard model has mainly been used [1-5], although some results have been also obtained in the multiband model [6, 7]. With use of various approaches the dynamic susceptibility have been calculated and the possibility of the appearance of surface spin-wave modes has been discussed. The complete dispersion relation of the surface modes has been also found [4]. Recently, numerical calculations of the dynamic susceptibility in the mixed representation have also been presented [8]. The wave functions of spin-wave modes characteristic for thin films have been found numerically and a localization of the modes has been discussed. These calculations have been performed for a specially chosen set of surface parameters only. It seems to be inter* Supported by the University of /:,6di under grant CPBP 01.08.Bl.1.
esting to investigate the influence of changes of surface parameters on the possibility of the appearance of the surface spin-wave modes as well as their energies. In the present paper the numerical calculations of the dynamic susceptibility and the wave functions of spin-wave modes are performed for various values of surface parameters. The influence of the changes of the effective Coulomb integral in the surface layer on the surface spin waves is profoundly and systematically investigated. The calculations performed within the framework of the one-band Hubbard model allow us to state, that an increase of the value of the Coulomb integral in the surface leads to an increase of the surface magnetization and therefore it favors the appearance of the acoustic surface spin waves. The optical modes are obtained in the case, when the Coulomb integral diminishes in the surface and other surface parameters remain unchanged. The surface magnetization in this case is much lower than the bulk one. The influence of the changes of the surface Coulomb integral on the spin-wave energies and damping of the modes is also investigated.
0921-4526/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
374
R, Swirkowicz / Surface spin waves in metal films
2. The dynamic thin film susceptibility in the Hubbard model
The transverse dynamic susceptibility of thin magnetic film calculated in the Hubbard model within the framework of RPA takes the following form [6]:
the energy of an electron with spin o-, twodimensional wave vector k and band index ~-. f,k~ denotes the Fermi-Dirac occupation function and T k~.~are the coefficients of expansion of the one-electron wave functions [6].
3. Results X ( q , w) : [ I - X ° ( q , w ) U ]
1X°(q, to).
(1)
All quantities in the above equation are matrices, which dimensions are determined by the number of atomic layers in the film. The susceptibility matrices X and X ° are expressed in the mixed representation and depend on a twodimensional wave vector q parallel to the surface. I denotes the unit matrix and the element U,v of matrix U describes the effective intraatomic Coulomb interaction between electrons in the atomic layer v (v = 1,2 . . . . . n). According to ab initio calculations performed in ref. [9], the surface perturbations are screened very well, so that the effective modifications of a charge density and a one-particle potential take place in the surface layer only. Therefore, the following assumption seems to be justified: {Uo+AU
U
=
Uo
for v = l , n , for v ~ l , n .
(2)
U,, and A U correspond to the effective interaction integral inside the film and its change in the surface layers, respectively. These quantities are treated here as parameters. In expression (1) the matrix X ° represents the reduced free-electron susceptibility of the system under consideration. According to the earlier paper [6] the elements of X ° take the following form:
o
1
X..,( q, 0 0 ) : ~ Z T~: T ~+#~ "7' T k+q¢ ,,~, T ,k¢ TT ' k
x
fTk~ -- fr'k+qT h w + E ,k+q~ - E~k ~ "
(3)
Here N denotes the number of atoms in the layer parallel to the film surface and v and v' are indices of the subsequent atomic layers. E k~ is
Numerical investigations of the dynamics of our system requires some detailed knowledge of the band structure and the coefficients of expansion of one-electron wave functions. The calculations are performed here for a thin film with fcc structure, consisting of 7 atomic layers. The surface of the film is assumed to be perpendicular to the [001] direction. The one-electron energies E~k o_ and coefficients T ~ are determined within the framework of the tight-binding approximation in a way analogical as in ref. [3]. Having Ni in mind we consider the one effective band, which corresponds to five d-bands lumped together and calculate the Fermi energy with the condition that the number of electrons per atom in the middle layer is equal to 9.45. For parameter U~ the value 0.55 eV is assumed. The band of majority spins is fully occupied, whereas the band of minority spins is occupied partially to give the magnetization in the middle layer equal to 0.55. The spin-splitting parameter equal to 0.3 eV is consistent with angular-resolved photoemission results [10], and it was used in calculations performed by Callaway et al. [11] for the dynamic susceptibility of bulk nickel with semiempirical band structure. It should be mentioned that the results obtained here for the effective band are identical with results which could be obtained using the one-band model with the number of electrons and magnetization divided by 5 and the effective intra-atomic Coulomb integral multiplied by 5. Next, the changes AU of the effective interaction integral in the surface layer are taken into account. The changes of AU influence the positions of the surface electron states and a degree of their localization. The spin-splitting of the surface states essentially depends on the value of A U. The gap between the Fermi level and the
R. Swirkowicz / Surface spin waves in metal films
top of the majority band depends on the value of AU also and it is equal to 0.0eV for A U = -0.01eV; 0.03eV for AU=0.0eV; 0.09eV for AU=0.01eV, 0.12eV for AU=0.02eV and 0.15 eV for AU = 0.03 eV (for the surface layer). The small changes of the Fermi level position are obtained for various A U. The changes of the Fermi energy, although very small, can lead to remarkable changes of magnetization because of a high and narrow peak in the density of states near the Fermi level. According to table I, A U essentially influences the surface magnetization. For AU >0.02eV the magnetization in the surface layer increases and for AU < 0.02 eV it decreases strongly as compared to the bulk value. The strong dependence of the surface magnetization on AU was also obtained within the framework of the LCAO method for the case of five d-bands (see e.g. ref. [12]). The ab initio calculations [9] show that in the case of the thin Ni films the surface magnetization is larger than the bulk one. However, it seems to us instructive to investigate the influence of various values of the surface parameter AU on appearance of surface spin-wave modes and their energies. It should be remembered that our calculations are performed for a very simple model, therefore, they have qualitative character only. The obtained band structure and coefficients of expansion of wave functions determined for various values of the parameter AU are used in calculations of the free-electron susceptibility X ° (eq. 3). Next, matrices Re X(q, oJ) and Im X(q, to) in the mixed representation are calculated for q='rr/4a (1, 0) (a is the lattice constant). For small values of energy, i.e. in the region, where spin waves are determined very well, the matrix Re X(q, w) is diagonalized numerically. The obtained eigenvectors correspond to the wave functions of spin-wave modes. These eigenfunctions calculated for various values of parameter AU are depicted in fig. 1. The strong dependence of the spin-wave eigenfunction on Table I AU ms
-0.01 0.12
m a = 0.55.
0.00 0.25
0.01 0.45
0.02 0.57
0.03 0.74
z~U=-O.01
z~U=o.o
375
z~ U =0.01
•",U =0.02
~U =O.03eV
s/v x_
A
V VU V A r,,l\A A \ vV vV\
Y
Number oF koyers
(o)
(b)
(c)
(d)
(e)
Fig. 1. Dependences on layer index ~ of spin-wave eigenfunctions for various values of AU.
the parameter AU can be observed. For AU negative and equal to -0.01 eV the eigenfunctions of the low-energy spin-wave modes are drastically reduced in the surface layer but oscillate with a rather considerable amplitude inside the film. Therefore, these modes have bulk character. On the other hand, the eigenfunctions of the modes with highest energies, i.e. these corresponding to the sixth and seventh modes, are completely different. The values of spinwave eigenfunctions are high in the surface layer, but strongIy reduced inside the film. Such eigenfunctions are characteristic for the surface spin-wave modes. These modes lay above the band of the bulk modes, so that they can be treated as the optical surface modes. These modes are relatively well localized near the surface of the film. Although the spin-wave eigenfunctions of the 6th and 7th modes are high in the surface layer, the function Im X~I is rather flat in the region of energy corresponding to these modes. It is a result of a strong damping effect appearing in this case. According to our calculations the damping appears even for low values of energy
R. Swirkowicz I Surface spin waves in metal films
376
(a small damping can be observed for energies so small as 12meV). In the energy region, which corresponds to optical surface modes, the damping is rather strong and the function Im Xll has fiat and very broad peaks. Because of this damping the experimental observation of the optical surface spin waves may be very difficult. It is interesting that the function Im X11 is relatively flat also in the low-energy region, but in this case it is a result of the reduction of the spin-wave amplitude in the surface layer for low-laying modes. On the other hand, in this energy region Im Xv~ for u # 1 has relatively high peaks, which correspond to the low energy bulk modes. In fig. 2 Im X~, is depicted as a function of energy for various atomic layers. In the energy region presented in this figure three peaks can be observed in Im X~, which correspond to three lowest spinwave modes. It should be pointed out that the height of the first peak decreases monotonically from the central to surface layer. Therefore, the lowest spin-wave mode has a bulk character. The /i
i
E
!
20i
t
i
! 1S ¸
I
10
i
; i ! IW
0.02
f2
l
!
1:,'.,-; 2! J';-"-
0.05
~
,..,'X !-~/
0.10
v
!
~ o (eV)
Fig. 2. The spin-wave peaks in Im X(q, tg) for the three lowest modes in the subsequent atomic layers for A U = - 0 . 0 1 eV and AD = 0.0eV.
peak, which corresponds to the second, antisymmetric mode, is not seen in the imaginary part of the susceptibility of the central layer. The height of this peak is greatest in the second layer, where the spin-wave eigenfunction has a maximum value (fig. la). For the third mode the oscillations of the magnitude of spin-wave peaks, which correspond to oscillations of the eigenfunction of the mode, can be observed. Unfortunately, such an analysis is much more difficult for higher values of w. Because of damping, the function Im Xv, is very flat and broad. In any case however, the results presented above allow us to state that there is a close relation between the magnitudes of the spin-wave peaks in the imaginary part of the susceptibility and the eigenfunctions of the corresponding mode. The analogical relation was obtained in the multiband approach [81. In fig. lb the eigenfunctions of the spin waves calculated for A U = 0 . 0 e V are depicted. The modes obtained in this case have a bulk character only. In comparison with the previous case (AU = - 0.01 eV, fig. la) one can observe now an increase of the amplitudes of spin waves in the surface layer for the low-energy modes. On the other hand, the amplitudes of the highest modes are strongly decreased near the surface, so that the modes are mainly localized near the center of the film, i.e. they are bulk modes. Further increase of the parameter A U leads to an increase of the amplitudes of low-energy spinwave modes in the surface layer. For A U = 0.01 eV the lowest mode is a bulk one, but its amplitude in the surface layer is only slightly smaller than the amplitude in the central plane (fig. lc). On the other hand, for AU = 0.02 eV (fig. ld), two relatively well localized acoustic surface modes appear, i.e. symmetric and antisymmetric ones. The higher modes are the bulk ones. Their amplitudes in the surface layer decrease, when the energy of the mode increases. Therefore, the high-energy modes have a tendency to localization near the centre of the film. For AU = 0.03 eV (fig. le) the two lowest modes are the surface ones too. In all cases considered here, the energy of the lowest, symmetric mode is about 27.5 to 32 meV,
R. Swirkowicz / Surface spin waves in metal films
so that it depends very weakly on the value of AU. On the other hand, the energies of higher modes slightly increase with increasing values of A U. Therefore, the width of the band corresponding to bulk modes slightly increases when AU increases. The changes of AU have essential influence on I m X v , . For A U < 0 damping is strong and it appears even at low values of energy (about 12 meV). In this case the lowest spin-wave modes have a finite life-time. For A U = 0 . 0 e V the damping can be also observed but for higher energy values (about 37 meV) the corresponding peaks are narrower than in the previous case. Further increase of the p a r a m e t e r AU leads also to increase of the value of energy, for which the damping can be observed. It is e.g. 110 meV for A U = 0 . 0 1 e V , 130meV for A U = 0 . 0 2 e V and 160meV for AU = 0.03 eV. Thus in the case of greater values of A U the spin-wave modes are well defined even for the relatively high energies. To investigate the influence of changes of other surface parameters on surface waves, the susceptibility calculations are p e r f o r m e d for the case when AU = - 0.01 eV and the core energy is changed in the surface layer. The value of the core shift AD is chosen in such a way that the mean numbers of electrons per atom in the subsequent layers are close to values obtained for AU = 0.02 eV without any change of the core energy (in the last case the numbers of electrons in the central and surface layers are close to each other, what is consistent with results of ab initio calculations [9], so that this case can be treated as the nearest to the real situation). Calculations p e r f o r m e d for AU = - 0 . 0 1 eV and AD = 0.15 eV show that as far as spin waves are concerned the obtained results are consistent with these for AU=0.02eV and A D = 0 . 0 e V . Two lowest modes are localized near the surface (acoustic surface modes) and the higher modes are the bulk ones (fig. 3). In these two cases ( A U = 0.02 eV, AD = 0.0 eV and AU = - 0.01 eV, AD = 0.15 eV) the spin-wave energies of the corresponding modes are close to each other. Moreover, for A U = - 0 . 0 1 e V and AD = 0 . 1 5 e V the damping is observable only for energies greater than 115eV, so that the spin waves are well
377
2 o~
h7
tO I
Number oF Layers Fig. 3. Dependence on layer index u of eigenfunctions of spin-wave modes for AU = - 0 . 0 1 eV and AD = 0.15 eV.
defined in a wide region of energy like in the case AU = 0.02 eV, AD = 0.0 eV.
4. Discussion Analysis of the results obtained in this p a p e r for the one-band H u b b a r d model leads us to the conclusion that there is a relation between the value of magnetization in the surface layer and the possibility of appearance of spin waves localized near the surface. Namely, the acoustic surface modes are obtained in the cases (AU = 0.02 eV, AD = 0.0 eV; AU = 0.03 eV, AD = 0.0eV; AU=-0.01eV, AD=0.15eV), for which the value of the surface magnetization is close to or greater than the value of the bulk magnetization. Analogical calculations performed within the f r a m e w o r k of the multiband model [8] show that there is a weakly localized
378
R. Swirkowicz / Surface spin waves in metal films
acoustic spin-wave mode in the case for which the surface magnetization is only slightly smaller than the bulk one. On the other hand, according to the calculations presented in this p a p e r tlqere are no acoustic surface modes in the cases of significant reduction of surface magnetization. For AU = 0.01 eV and AD = 0.0 eV (m s = 0.45) the lowest m o d e is a bulk one but the amplitude of the m o d e in the surface layer is only slightly lower than in the central layer. For A U = - 0 . 0 1 eV and AD = 0 . 0 e V (m s = 0 . 1 2 ) the appearance of optical surface spin waves is in favor. It is however rather difficult to determine quantitatively the proper values of the surface parameters which are necessary for the appearance of surface spin-wave modes. Considerations p e r f o r m e d by Mathon within the framework of the one-band H u b b a r d model [5] show that the optical surface modes appear when the surface magnetization is lower than the bulk one, namely in the case m s < 0.88m B. In the present paper the optical modes are obtained for AU = - 0 . 0 1 eV and AD = 0.0eV when the surface magnetization is much lower than the bulk one, but these modes are d a m p e d strongly. It seems that as far as the possibility of appearance of the optical surface modes is concerned there is qualitative agreement between the results obtained here and in ref. [5]; however, Mathon does not take into account any damping effect. On the other hand, Mathon does not expect appearance of any acoustic surface spin waves. It should be pointed out, however, that his calculations are p e r f o r m e d for a simple cubic structure only. It may well be that considering other types of lattice structures, for example fcc structures, it would be possible to obtain the acoustic modes also in his approach.
On the basis of the results obtained here within the framework of the band model it is still difficult to formulate any quantitative criterion of the possibility of the appearance of surface spinwave modes. The results of the present p a p e r show that the increase of the effective Coulomb integral in the surface layer leads to increase of the surface magnetization and favors the appearance of the surface acoustic modes. More detailed investigations however are necessary for formulation of the exact criterion.
Acknowledgement The author is indebted to Prof. A. Sukiennicki for valuable discussions and for critical reading of the manuscript.
References [1] G. Gumbs and A. Griffin, Surf. Sci. 91 (1980) 669. [2] J. Mathon, Phys. Rev. B 24 (1981) 6588. [3] R. Swirkowicz, K. Swiderczak and A. Sukiennicki, Acta. Phys. Pol. A 67 (1985) 913. [4] R. Swirkowicz, Physica B 128 (1985) 297. [5] J. Mathon, Phys. Rev. B 34 (1986) 1775. [6] R. Swirkowicz, Phys. Stat. Sol. b 129 (1985) 641. [7] R. Swirkowicz and K. Swiderczak, Physica B 141 (1986) 199. [8] R. Swirkowicz and A. Sukiennicki, Physica B 149 (1988) 37. [9] E. Wimmer, A.J. Freeman and H. Krakauer, Phys. Rev. B 30 (1984) 3113. [10] M.A. Thompson and J.L. Erskine, Phys. Rev. B 31 (1985) 6832. [11] J. Callaway, A.K. Chatterjee, S.P. Singhal and A. Zieg,ler, Phys. Rev. B 28 (1983) 3818. [12] K. Swiderczak and A. Sukiennicki, Acta Phys. Pol. A 62 (1982) 189.
ON THE POSSIBILITY OF APPEARANCE OF SURFACE SPIN WAVES IN TRANSITION METAL THIN FILMS* R. SWIRKOWICZ Institute of Physics, Technical University, 00-662 Warsaw, Poland Received 1 March 1988 Revised manuscript received 22 November 1988
The transverse dynamic susceptibility of thin ferromagnetic film and spin-wave eigenfunctions are calculated within the framework of the Hubbard model. For various values of surface parameters the localization of the spin-wave modes, their energies and damping of the modes are discussed. The calculations show that a decrease of the effective Coulomb integral in the surface layer without any change of other surface parameters causes the substantial lowering of the surface magnetization and lead to appearance of optical surface modes. These modes, however, are strongly damped. An increase of the effective integral in the surface layer leads to an increase of the surface magnetization and favors appearance of the acoustic surface modes. The changes of the core energies in the surface layer also favor appearance of acoustic surface modes provided the surface magnetization is close or larger than the bulk one.
I. Introduction
Recently, the dynamic susceptibility and surface spin waves in thin films of transition metals have been investigated within the framework of the itinerant electron model [1-7]. The one-band Hubbard model has mainly been used [1-5], although some results have been also obtained in the multiband model [6, 7]. With use of various approaches the dynamic susceptibility have been calculated and the possibility of the appearance of surface spin-wave modes has been discussed. The complete dispersion relation of the surface modes has been also found [4]. Recently, numerical calculations of the dynamic susceptibility in the mixed representation have also been presented [8]. The wave functions of spin-wave modes characteristic for thin films have been found numerically and a localization of the modes has been discussed. These calculations have been performed for a specially chosen set of surface parameters only. It seems to be inter* Supported by the University of /:,6di under grant CPBP 01.08.Bl.1.
esting to investigate the influence of changes of surface parameters on the possibility of the appearance of the surface spin-wave modes as well as their energies. In the present paper the numerical calculations of the dynamic susceptibility and the wave functions of spin-wave modes are performed for various values of surface parameters. The influence of the changes of the effective Coulomb integral in the surface layer on the surface spin waves is profoundly and systematically investigated. The calculations performed within the framework of the one-band Hubbard model allow us to state, that an increase of the value of the Coulomb integral in the surface leads to an increase of the surface magnetization and therefore it favors the appearance of the acoustic surface spin waves. The optical modes are obtained in the case, when the Coulomb integral diminishes in the surface and other surface parameters remain unchanged. The surface magnetization in this case is much lower than the bulk one. The influence of the changes of the surface Coulomb integral on the spin-wave energies and damping of the modes is also investigated.
0921-4526/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
374
R, Swirkowicz / Surface spin waves in metal films
2. The dynamic thin film susceptibility in the Hubbard model
The transverse dynamic susceptibility of thin magnetic film calculated in the Hubbard model within the framework of RPA takes the following form [6]:
the energy of an electron with spin o-, twodimensional wave vector k and band index ~-. f,k~ denotes the Fermi-Dirac occupation function and T k~.~are the coefficients of expansion of the one-electron wave functions [6].
3. Results X ( q , w) : [ I - X ° ( q , w ) U ]
1X°(q, to).
(1)
All quantities in the above equation are matrices, which dimensions are determined by the number of atomic layers in the film. The susceptibility matrices X and X ° are expressed in the mixed representation and depend on a twodimensional wave vector q parallel to the surface. I denotes the unit matrix and the element U,v of matrix U describes the effective intraatomic Coulomb interaction between electrons in the atomic layer v (v = 1,2 . . . . . n). According to ab initio calculations performed in ref. [9], the surface perturbations are screened very well, so that the effective modifications of a charge density and a one-particle potential take place in the surface layer only. Therefore, the following assumption seems to be justified: {Uo+AU
U
=
Uo
for v = l , n , for v ~ l , n .
(2)
U,, and A U correspond to the effective interaction integral inside the film and its change in the surface layers, respectively. These quantities are treated here as parameters. In expression (1) the matrix X ° represents the reduced free-electron susceptibility of the system under consideration. According to the earlier paper [6] the elements of X ° take the following form:
o
1
X..,( q, 0 0 ) : ~ Z T~: T ~+#~ "7' T k+q¢ ,,~, T ,k¢ TT ' k
x
fTk~ -- fr'k+qT h w + E ,k+q~ - E~k ~ "
(3)
Here N denotes the number of atoms in the layer parallel to the film surface and v and v' are indices of the subsequent atomic layers. E k~ is
Numerical investigations of the dynamics of our system requires some detailed knowledge of the band structure and the coefficients of expansion of one-electron wave functions. The calculations are performed here for a thin film with fcc structure, consisting of 7 atomic layers. The surface of the film is assumed to be perpendicular to the [001] direction. The one-electron energies E~k o_ and coefficients T ~ are determined within the framework of the tight-binding approximation in a way analogical as in ref. [3]. Having Ni in mind we consider the one effective band, which corresponds to five d-bands lumped together and calculate the Fermi energy with the condition that the number of electrons per atom in the middle layer is equal to 9.45. For parameter U~ the value 0.55 eV is assumed. The band of majority spins is fully occupied, whereas the band of minority spins is occupied partially to give the magnetization in the middle layer equal to 0.55. The spin-splitting parameter equal to 0.3 eV is consistent with angular-resolved photoemission results [10], and it was used in calculations performed by Callaway et al. [11] for the dynamic susceptibility of bulk nickel with semiempirical band structure. It should be mentioned that the results obtained here for the effective band are identical with results which could be obtained using the one-band model with the number of electrons and magnetization divided by 5 and the effective intra-atomic Coulomb integral multiplied by 5. Next, the changes AU of the effective interaction integral in the surface layer are taken into account. The changes of AU influence the positions of the surface electron states and a degree of their localization. The spin-splitting of the surface states essentially depends on the value of A U. The gap between the Fermi level and the
R. Swirkowicz / Surface spin waves in metal films
top of the majority band depends on the value of AU also and it is equal to 0.0eV for A U = -0.01eV; 0.03eV for AU=0.0eV; 0.09eV for AU=0.01eV, 0.12eV for AU=0.02eV and 0.15 eV for AU = 0.03 eV (for the surface layer). The small changes of the Fermi level position are obtained for various A U. The changes of the Fermi energy, although very small, can lead to remarkable changes of magnetization because of a high and narrow peak in the density of states near the Fermi level. According to table I, A U essentially influences the surface magnetization. For AU >0.02eV the magnetization in the surface layer increases and for AU < 0.02 eV it decreases strongly as compared to the bulk value. The strong dependence of the surface magnetization on AU was also obtained within the framework of the LCAO method for the case of five d-bands (see e.g. ref. [12]). The ab initio calculations [9] show that in the case of the thin Ni films the surface magnetization is larger than the bulk one. However, it seems to us instructive to investigate the influence of various values of the surface parameter AU on appearance of surface spin-wave modes and their energies. It should be remembered that our calculations are performed for a very simple model, therefore, they have qualitative character only. The obtained band structure and coefficients of expansion of wave functions determined for various values of the parameter AU are used in calculations of the free-electron susceptibility X ° (eq. 3). Next, matrices Re X(q, oJ) and Im X(q, to) in the mixed representation are calculated for q='rr/4a (1, 0) (a is the lattice constant). For small values of energy, i.e. in the region, where spin waves are determined very well, the matrix Re X(q, w) is diagonalized numerically. The obtained eigenvectors correspond to the wave functions of spin-wave modes. These eigenfunctions calculated for various values of parameter AU are depicted in fig. 1. The strong dependence of the spin-wave eigenfunction on Table I AU ms
-0.01 0.12
m a = 0.55.
0.00 0.25
0.01 0.45
0.02 0.57
0.03 0.74
z~U=-O.01
z~U=o.o
375
z~ U =0.01
•",U =0.02
~U =O.03eV
s/v x_
A
V VU V A r,,l\A A \ vV vV\
Y
Number oF koyers
(o)
(b)
(c)
(d)
(e)
Fig. 1. Dependences on layer index ~ of spin-wave eigenfunctions for various values of AU.
the parameter AU can be observed. For AU negative and equal to -0.01 eV the eigenfunctions of the low-energy spin-wave modes are drastically reduced in the surface layer but oscillate with a rather considerable amplitude inside the film. Therefore, these modes have bulk character. On the other hand, the eigenfunctions of the modes with highest energies, i.e. these corresponding to the sixth and seventh modes, are completely different. The values of spinwave eigenfunctions are high in the surface layer, but strongIy reduced inside the film. Such eigenfunctions are characteristic for the surface spin-wave modes. These modes lay above the band of the bulk modes, so that they can be treated as the optical surface modes. These modes are relatively well localized near the surface of the film. Although the spin-wave eigenfunctions of the 6th and 7th modes are high in the surface layer, the function Im X~I is rather flat in the region of energy corresponding to these modes. It is a result of a strong damping effect appearing in this case. According to our calculations the damping appears even for low values of energy
R. Swirkowicz I Surface spin waves in metal films
376
(a small damping can be observed for energies so small as 12meV). In the energy region, which corresponds to optical surface modes, the damping is rather strong and the function Im Xll has fiat and very broad peaks. Because of this damping the experimental observation of the optical surface spin waves may be very difficult. It is interesting that the function Im X11 is relatively flat also in the low-energy region, but in this case it is a result of the reduction of the spin-wave amplitude in the surface layer for low-laying modes. On the other hand, in this energy region Im Xv~ for u # 1 has relatively high peaks, which correspond to the low energy bulk modes. In fig. 2 Im X~, is depicted as a function of energy for various atomic layers. In the energy region presented in this figure three peaks can be observed in Im X~, which correspond to three lowest spinwave modes. It should be pointed out that the height of the first peak decreases monotonically from the central to surface layer. Therefore, the lowest spin-wave mode has a bulk character. The /i
i
E
!
20i
t
i
! 1S ¸
I
10
i
; i ! IW
0.02
f2
l
!
1:,'.,-; 2! J';-"-
0.05
~
,..,'X !-~/
0.10
v
!
~ o (eV)
Fig. 2. The spin-wave peaks in Im X(q, tg) for the three lowest modes in the subsequent atomic layers for A U = - 0 . 0 1 eV and AD = 0.0eV.
peak, which corresponds to the second, antisymmetric mode, is not seen in the imaginary part of the susceptibility of the central layer. The height of this peak is greatest in the second layer, where the spin-wave eigenfunction has a maximum value (fig. la). For the third mode the oscillations of the magnitude of spin-wave peaks, which correspond to oscillations of the eigenfunction of the mode, can be observed. Unfortunately, such an analysis is much more difficult for higher values of w. Because of damping, the function Im Xv, is very flat and broad. In any case however, the results presented above allow us to state that there is a close relation between the magnitudes of the spin-wave peaks in the imaginary part of the susceptibility and the eigenfunctions of the corresponding mode. The analogical relation was obtained in the multiband approach [81. In fig. lb the eigenfunctions of the spin waves calculated for A U = 0 . 0 e V are depicted. The modes obtained in this case have a bulk character only. In comparison with the previous case (AU = - 0.01 eV, fig. la) one can observe now an increase of the amplitudes of spin waves in the surface layer for the low-energy modes. On the other hand, the amplitudes of the highest modes are strongly decreased near the surface, so that the modes are mainly localized near the center of the film, i.e. they are bulk modes. Further increase of the parameter A U leads to an increase of the amplitudes of low-energy spinwave modes in the surface layer. For A U = 0.01 eV the lowest mode is a bulk one, but its amplitude in the surface layer is only slightly smaller than the amplitude in the central plane (fig. lc). On the other hand, for AU = 0.02 eV (fig. ld), two relatively well localized acoustic surface modes appear, i.e. symmetric and antisymmetric ones. The higher modes are the bulk ones. Their amplitudes in the surface layer decrease, when the energy of the mode increases. Therefore, the high-energy modes have a tendency to localization near the centre of the film. For AU = 0.03 eV (fig. le) the two lowest modes are the surface ones too. In all cases considered here, the energy of the lowest, symmetric mode is about 27.5 to 32 meV,
R. Swirkowicz / Surface spin waves in metal films
so that it depends very weakly on the value of AU. On the other hand, the energies of higher modes slightly increase with increasing values of A U. Therefore, the width of the band corresponding to bulk modes slightly increases when AU increases. The changes of AU have essential influence on I m X v , . For A U < 0 damping is strong and it appears even at low values of energy (about 12 meV). In this case the lowest spin-wave modes have a finite life-time. For A U = 0 . 0 e V the damping can be also observed but for higher energy values (about 37 meV) the corresponding peaks are narrower than in the previous case. Further increase of the p a r a m e t e r AU leads also to increase of the value of energy, for which the damping can be observed. It is e.g. 110 meV for A U = 0 . 0 1 e V , 130meV for A U = 0 . 0 2 e V and 160meV for AU = 0.03 eV. Thus in the case of greater values of A U the spin-wave modes are well defined even for the relatively high energies. To investigate the influence of changes of other surface parameters on surface waves, the susceptibility calculations are p e r f o r m e d for the case when AU = - 0.01 eV and the core energy is changed in the surface layer. The value of the core shift AD is chosen in such a way that the mean numbers of electrons per atom in the subsequent layers are close to values obtained for AU = 0.02 eV without any change of the core energy (in the last case the numbers of electrons in the central and surface layers are close to each other, what is consistent with results of ab initio calculations [9], so that this case can be treated as the nearest to the real situation). Calculations p e r f o r m e d for AU = - 0 . 0 1 eV and AD = 0.15 eV show that as far as spin waves are concerned the obtained results are consistent with these for AU=0.02eV and A D = 0 . 0 e V . Two lowest modes are localized near the surface (acoustic surface modes) and the higher modes are the bulk ones (fig. 3). In these two cases ( A U = 0.02 eV, AD = 0.0 eV and AU = - 0.01 eV, AD = 0.15 eV) the spin-wave energies of the corresponding modes are close to each other. Moreover, for A U = - 0 . 0 1 e V and AD = 0 . 1 5 e V the damping is observable only for energies greater than 115eV, so that the spin waves are well
377
2 o~
h7
tO I
Number oF Layers Fig. 3. Dependence on layer index u of eigenfunctions of spin-wave modes for AU = - 0 . 0 1 eV and AD = 0.15 eV.
defined in a wide region of energy like in the case AU = 0.02 eV, AD = 0.0 eV.
4. Discussion Analysis of the results obtained in this p a p e r for the one-band H u b b a r d model leads us to the conclusion that there is a relation between the value of magnetization in the surface layer and the possibility of appearance of spin waves localized near the surface. Namely, the acoustic surface modes are obtained in the cases (AU = 0.02 eV, AD = 0.0 eV; AU = 0.03 eV, AD = 0.0eV; AU=-0.01eV, AD=0.15eV), for which the value of the surface magnetization is close to or greater than the value of the bulk magnetization. Analogical calculations performed within the f r a m e w o r k of the multiband model [8] show that there is a weakly localized
378
R. Swirkowicz / Surface spin waves in metal films
acoustic spin-wave mode in the case for which the surface magnetization is only slightly smaller than the bulk one. On the other hand, according to the calculations presented in this p a p e r tlqere are no acoustic surface modes in the cases of significant reduction of surface magnetization. For AU = 0.01 eV and AD = 0.0 eV (m s = 0.45) the lowest m o d e is a bulk one but the amplitude of the m o d e in the surface layer is only slightly lower than in the central layer. For A U = - 0 . 0 1 eV and AD = 0 . 0 e V (m s = 0 . 1 2 ) the appearance of optical surface spin waves is in favor. It is however rather difficult to determine quantitatively the proper values of the surface parameters which are necessary for the appearance of surface spin-wave modes. Considerations p e r f o r m e d by Mathon within the framework of the one-band H u b b a r d model [5] show that the optical surface modes appear when the surface magnetization is lower than the bulk one, namely in the case m s < 0.88m B. In the present paper the optical modes are obtained for AU = - 0 . 0 1 eV and AD = 0.0eV when the surface magnetization is much lower than the bulk one, but these modes are d a m p e d strongly. It seems that as far as the possibility of appearance of the optical surface modes is concerned there is qualitative agreement between the results obtained here and in ref. [5]; however, Mathon does not take into account any damping effect. On the other hand, Mathon does not expect appearance of any acoustic surface spin waves. It should be pointed out, however, that his calculations are p e r f o r m e d for a simple cubic structure only. It may well be that considering other types of lattice structures, for example fcc structures, it would be possible to obtain the acoustic modes also in his approach.
On the basis of the results obtained here within the framework of the band model it is still difficult to formulate any quantitative criterion of the possibility of the appearance of surface spinwave modes. The results of the present p a p e r show that the increase of the effective Coulomb integral in the surface layer leads to increase of the surface magnetization and favors the appearance of the surface acoustic modes. More detailed investigations however are necessary for formulation of the exact criterion.
Acknowledgement The author is indebted to Prof. A. Sukiennicki for valuable discussions and for critical reading of the manuscript.
References [1] G. Gumbs and A. Griffin, Surf. Sci. 91 (1980) 669. [2] J. Mathon, Phys. Rev. B 24 (1981) 6588. [3] R. Swirkowicz, K. Swiderczak and A. Sukiennicki, Acta. Phys. Pol. A 67 (1985) 913. [4] R. Swirkowicz, Physica B 128 (1985) 297. [5] J. Mathon, Phys. Rev. B 34 (1986) 1775. [6] R. Swirkowicz, Phys. Stat. Sol. b 129 (1985) 641. [7] R. Swirkowicz and K. Swiderczak, Physica B 141 (1986) 199. [8] R. Swirkowicz and A. Sukiennicki, Physica B 149 (1988) 37. [9] E. Wimmer, A.J. Freeman and H. Krakauer, Phys. Rev. B 30 (1984) 3113. [10] M.A. Thompson and J.L. Erskine, Phys. Rev. B 31 (1985) 6832. [11] J. Callaway, A.K. Chatterjee, S.P. Singhal and A. Zieg,ler, Phys. Rev. B 28 (1983) 3818. [12] K. Swiderczak and A. Sukiennicki, Acta Phys. Pol. A 62 (1982) 189.