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On the influence of the surface potential on the one-electron properties and spin waves in transition metal thin films. The multi-band approach
Physica B 159 (198’)) 304-310 North-Holland. Amsterdam
ON THE INFLUENCE OF THE SURFACE POTENTIAL ON THE ONE-ELECTRON PROPERTIES AND SPIN WAVES IN TRANSITION METAL THIN FILMS. THE MULTI-BAND APPROACH R. SWIRKOWICZ’ Institute
of Physics,
Received
Technical
17 February
University.
OO-662 Warsaw,
Poland
1989
Dynamic susccptibihty and spin waves in thin ferromagnetic films of transition metals are investigated numerically within the framework of the multiband model for various values of the surface one-electron potential. The tight-binding approach is used. Influence of the potential shifting on the one-electron properties and spin waves is profoundly discussed. Results, consistent with those obtained in the one-band model, show that acoustic surface modes appear for cases of increasing of the surface potential, which lead to surface magnetization enhancement.
of Ni thin films. Shifting of the core energy in the surface layer is also taken into account in model calculations based on the tight-binding approach, whereas changes of the other parameters in the surface layer are neglected [14-161. The results obtained for the ground-state properties are consistent with ab initio results for various transition metals (Fe, Co, Ni). Changes of the core potential are also important in cases of sandwiches, thin films deposited on some type of substrates and films with coated surfaces [17]. Therefore. it seems to us that, first of all, one should investigate the influence of the core shifting in the surface layer on the one-electron properties and on a degree of localization of spin waves. The detailed investigation of this problem will be presented in this paper. We will show, that analogous to the one-band approach, an appropriate increase of the core potential in the surface layer leads to an enhancement of the surface magnetization and favors an appearance of acoustic surface spin waves.
1. Introduction The dynamical susceptibility and elementary excitations in transition metal films have been profoundly discussed recently within the framework of the one-band Hubbard model [l-10]. Numerical calculations performed for thin ferromagnetic films have shown that the possibility of an appearance of surface spin waves and the degree of their localization strongly depend on values of surface parameters and surface magnetization. In the multiband approach, on the other hand, investigations have been performed only for a specially chosen set of surface parameters [ll, 121. In the present paper the influence of the surface parameters on the one-electron properties and spin waves is investigated within the framework of the multiband model for transition metal thin films. According to ab initio calculations [13], changes of the core potential in the surface layer essentially influence the properties
2. Method
of calculation
Calculations Hamiltonian:
of the susceptibility
’ Supported
University
by the
0921-4526/89/$03.50 0 (North-Holland Physics
of Lodi
of thin ferromagnetic
under
grant
CPBP
Ol.OX.Bl.1.
Elsevier Science Publishers Publishing Division)
B.V.
films are based
on the following
multiband
R. Swirkowicz
I Influence
of the surface potential on transition metal films
30s
c,~,,,,, denotes here an annihilation operator of an electron with spin (T of the orbital m at the lattice site hopping integral. The one-electron potential in the layer v, uj. HZi.(j, j’) re p resents the two-center E,, is assumed to be common to all orbitals. The elements of the matrix U describe the effective intra-atomic interactions between electrons in the layer u. According to the theory presented in paper [la], a reduced transverse susceptibility x( q, OJ) calculated within the framework of RPA with assumptions analogous to those used by Lowde and Windsor for the bulk case [19], takes the following form:
x(4, w) = (Z-
x0(4,4J-‘x”GzI w>.
The mixed representation is used in the above expression and the susceptibility depend on a two-dimensional wave vector q parallel to the surface. Elements susceptibility x” are equal to:
(2) matrices x and x0 of the free-electron
energy of an where N is the number of electrons in the film plane, ETkc denotes the Hartree-Fock is the Fermi-Dirac occupation function and electron with spin u, wave vector k and band index r, fikc coefficients of the one-electron wave functions [la]. T :y” are expansion The band structure of the thin film is determined with the assumption that the core potential EY changes its value in the surface layer only: E
= Y
E, I0
for Y in the surface for other cases .
layer,
(4)
This assumption is consistent with results of ab initio calculations which show that the surface perturbations are screened very well [13]. The surface relaxation is neglected and the hopping integrals on and near the surface are assumed to be equal to those in the bulk. The strength of the effective Coulomb interaction is also assumed to be unchanged in the surface layer. It seems reasonable to limit the number of surface parameters and to consider only one of them, namely E,. Calculations performed within the framework of the one-band Hubbard model clearly show [lo] that changes of the Coulomb integral in the surface layer influence the one-electron states and the spin waves in a similar way as E,. Changes of the hopping integral in the surface layer influence also the spin waves in the same way, provided they lead to analogous modifications of the surface magnetization. We perform numerical calculations bearing Ni in mind. A thin film with fee structure consisting of 7 atomic layers is considered. The surface of the film is assumed to be perpendicular to the [0 0 l] direction. The tight-binding approach with hopping integrals estimated by Fletcher [20] is used. Analogous to other model calculations concerning the systems with surfaces (see, e.g., ref. [14]) the effect of the s-p bands is not included. The Fermi energy is calculated according to the condition that the number of electrons per atom in the middle layer is equal to 9.45. For the strength of the effective Coulomb interaction U, a value of 0.68 eV is taken, which leads to the exchange-splitting parameter A
306
R. .?wirkowicz
I Injuence
of the surface potential
on trunsition
metal films
being equal to 0.37 eV. This value of J is substantially lower than that calculated with use of the local spin-density functional method (see, e.g. ref. [13]), but on the other hand, it is rather close to values used in the spin-wave calculations for bulk nickel [21-231. Our evaluations show that an increase of d leads to an increase of spin-wave energies and therefore to an increase of spin-wave stiffness parameter D. The relation between A and D was discussed by Edwards and Muniz for bulk nickel [23]. Band structure, reduced spin susceptibility x( q, co) and spin-wave eigenfunctions are calculated here for three values of the surface parameter E,, namely for E, = 0.0. 0.15 and 0.25 eV. The obtained results are discussed in the next section.
3. Results Analogous to the one-band Hubbard model [lo], the shifting of the core energy E, in the surface strongly influences both one-electron states and spin waves. First. we will discuss the one-electron properties. The surface magnetization m, obtained for E, = O.OeV is equal to 0.12, whereas the bulk one, m,,, calculated for the central layer is equal to 0.55. Therefore, the substantial lowering of the surface magnetization is observed in this case. An increase of the surface potential leads to the increase of the surface magnetization, and m, equal to 0.38 and 0.71 is obtained for E, equal to 0.15 and 0.25 eV, respectively. The influence of the core shifting on the surface magnetization is qualitatively consistent but rather stronger than in the case considered by Hasegawa [14]. It is a result of different values of model parameters taken in these two cases. Especially, in our case the d-bandwidth is considerably smaller than in Hasegawa’s case and therefore it is more close to the results known from ARP experiments [24]. On the other hand, however, this strong reaction of the surface magnetization on the core shifting allows us to discuss spin waves for various values of the surface magnetization. In all cases considered (E, = 0.0, 0.15, 0.25 eV) the one-electron surface states can be observed, but their positions according to the Fermi level depend on values of the parameter E,. A decrease of E, leads to shifting of the localized electron states (corresponding to majority spins) into higher energies and, simultaneously, to shifting of the states with minority
spins to lower energies, so that the exchange splitting A, for the surface states decreases rather fast. We will discuss this process with help of an example of the strongly localized surface electronic states, which appear at the M point in the two-dimensional Brillouin zone for states with symmetry xy. These states lay near the Fermi level and they are the most striking surface states of Ni thin films. For E, = 0.25 eV the strongly localized M states with majority spins arc occupied and lay 0.05 eV below the Fermi level. Localized states with minority spins are empty and lay 0.43eV above the Fermi level. The exchange splitting A,, equal to 0.48 eV, is higher than the bulk one (A,, = 0.37 eV). There are no holes in the band with majority spins in this case. The surface magnetization is strongly enhanced because the number of electrons with minority spins is reduced. For E, = 0.15 eV the surface electron states at point M are empty for both spin directions and A, is equal to 0.26 eV. A small number of holes with majority spins appears in the surface layer, mainly for states with tZy symmetry. The surface magnetization is reduced as compared to the bulk one. For the case of E, = 0.0 eV, the considered surface states are also empty and lay O.OSeV and 0.16eV above the Fermi level for majority and minority spins, respectively. The exchange splitting A, equal to 0.08 eV is essentially lower than for bulk states. The holes with majority spins appear in the surface and subsurface layers. Simultaneously the number of electrons with minority spins is enhanced in the surface layer. It leads to further reduction of the surface magnetization. Changes of the parameter E, have practically no influence on positions of bulklike states of M
R. Swirkowicz
I Influence of the surface potential on transition metal films
type, strongly localized in the second and third layers. States with majority spins are occupied and states with minority spins are empty for all considered values of E,. The surface density of states is strongly influenced by changes of potential E, (fig. la). One can see that the increase of E, leads to the increase of the surface density of states at the Fermi level. For the cases of E, = 0.0 and 0.15 eV, the surface density of states at the Fermi level is lower than the corresponding density of states in the central layer, whereas for E, = 0.25 eV it is higher. Calculations show also that the surface density of states with symmetry e, at the Fermi level is enhanced more than for states with symmetry t,,. The density of states in the central layer is also influenced by the potential E, (fig. lb). It is a result of the fact that a very thin film is considered. Results obtained here numerically for the multiband model are consistent with qualitative considerations presented by Mathon for the case of Ni surfaces [S]. We also want to emphasize that for E, = 0.25 eV, results obtained here are consistent with results for Ni thin films known from ab initio approaches (e.g. ref. [13]). Namely, the surface density of states at the Fermi level is higher than the bulk one. The magnetization in the surface layer is enhanced and equal to 0.71, whereas in the “bulk” the magnetization is practically the same and equal to 0.58, 0.57, 0.55 in layers s - 1, s - 2 and c (central), respectively. However, due to the fact that hybridization sp-d is neglected in our calculations, a slight breaking of the charge neutrality on the surface appears. Positions of the most important surface electron states are consistent with ab initio theories. First of all, there is a strongly localized state of M3 type occupied by majority spins which lays 0.05 eV below the Fermi level. The f state is also occupied by majority spins, but it is not so strongly localized as in the ab initio approach. Well localized surface states appear also near the x points in the two-dimensional Brillouin zone. Now, we can discuss the influence of the surface potential E, on the dynamic susceptibility and spin waves. Spin-wave eigenfunctions are determined by diagonalization of matrix Re x in
307
(a)
4
2
0
6
ES = 0.25
eV i
Fig. 1. Density of states for: (a) the surface layers and (b) central layers, calculated for various values of surface potential E,.
308
R. Swirkowicz E,=0.15 r-.-
1
layer
Fig. 2. Dependences
I Influence Es=025 r
of the surface potential
eV
index
on layer index of spin-wave
eigenfunctions for various values of E, and q = ~/4a (1.0). Energies of the modes increase from the top to bottom of the figure.
the region of very low energies. The strong dependence of the spin-wave eigenfunctions on the parameter E, can be observed (fig. 2). For E,= 0.0eV modes with lowest energies are mainly localized near the central plane and they are typical bulk modes. However, two modes with highest energies have a tendency to localization in the surface layer, so optical surface modes appear in this case. For E,= 0.15 eV all modes are of bulk character and for E, = 0.25 eV spin waves with lowest energies are localized in the surface layers and acoustic surface modes appear. The change of degree of localization of spin waves with increase of surface potential is well illustrated in fig. 2. Surface parameter E, also strongly influences Im x,,( q. co). In fig. 3 is presented as a function of energy for Im X”” various atomic layers and for E,= 0.0 and 0.15 eV. respectively. In the region presented in the figure two peaks can be observed in Im x,, which correspond to spin-wave modes with lowest energies. For E, = 0.0eV the height of the spin-wave peak corresponding to the mode with the lowest energy is the greatest in the center of
on trunsition
metal films
the film and it decreases strongly from the central to the surface layers. Namely, the height of the peak is equal to 70.6 in the central layer and to 58.3, 26.4, 4.23 in the s-2, s-l and s (surface) layers, respectively. According to fig. 2 one can see that in the case of E,= 0.0eV the amplitude of the first mode is highest in the center of the film and it is considerably lower in the surface layer. Therefore. this mode is mainly localized in the central layer. We can see also that there is a close relation between the height of the spinwave peak in Im x,, and the amplitude of the mode. This relation has already been found in the one-band model [9, lo]. An increase of the surface potential E, leads to the increase in magnitude of the first spin-wave peak in the surface layer. For E, = 0.15 eV the peak corresponding to the mode with the lowest energy is considerably enhanced in the surface layer, but it is lower than in the central plane. The mode considered is still of bulk character, however its amplitude in the surface layer is greater than in the previous case and is comparable to the amplitude in the central layer. Unfortunately, such a relation cannot be illustrated for E,= 0.25 eV, when the acoustic mode appears and one could expect that the height of the first spin-wave peak should be the greatest in the surface layer. It appears, namely, that the damping of the spinwave mode, closely related to the width of the spin-wave line in Im x, depends essentially on the surface potential E,.The width of the line increases with decreasing E,. For E, = 0.25 eV the damping of the modes is much smaller than in previous cases (E,= 0.0,0.15 eV) and the widths of the lines corresponding to two modes with lowest energies approach zero within the accuracy of our calculations. The considered modes are determined very well. An analogous dependence of the damping of the modes on the surface parameters has also been found in the one-band approach [9, lo]. Energies of the lowest mode calculated for E,= 0.0and 0.15 eV, i.e., the first bulk mode are about 20-22.5 meV for q = (rr/4a)(l, 0), where u is a lattice constant. The corresponding stiffness parameter D is therefore about 400-450 meVA*. The obtained value is close to the one calculated
Swirkowicz
I Influence of the surface potential on transition metal films
ES=O.O
Es--O.15
eV
q=JT/4a(l,O)
Fig.
3. The
spin-wave
0.65
(eV) peaks
in Im x for two modes
by Edwards and Muniz for bulk Ni [23] as well as the value measured in the neutron-scattering experiments [25]. It shows therefore, that the spinwave energy does not critically depend on the fine details of the band structure. Calculations show that the energy of the first bulk mode strongly depends on the bulk exchange splitting A,, but it practically does not depend on the A, and the surface parameter E,. The energy of the acoustic surface mode for E, = 0.25 eV is a little lower and it is about 17.5-20.0meV.
4. Discussion Results obtained in the paper for the multiband model are fully consistent with those obtained within the framework of the one-band Hubbard approach [lo]. Numerical calculations performed for various values of the surface one-
eV
q=Tr/40(1,0)
0.05
energy
309
with
lowest
energies
o.io
energy
(eV)
calculated
for E, = 0.0 and 0.15 eV.
electron potential E, confirm also conclusions drawn by Mathon [5] on the basis of qualitative considerations performed in the one-band model that the ratio of ms/mb is essentially related to the conditions of appearance of surface spin waves. In the case of the significant reduction of the surface magnetization (E, = 0.0 eV) we obtain optical surface modes, whereas in the case of enhancement of the surface magnetization (E, = 0.25 eV) the acoustic surface modes appear. Therefore, in thin Ni films with free surfaces, acoustic surface modes should be observed. The situation can be different in cases of thin films deposited on some substrates and for some type of sandwiches. The influence of the substrate or overlayer will, first of all, lead to a potential shifting and to modifications of the magnetization near the interfaces which, by turns, will lead to the change of conditions of appearance of surface modes. Strong depen-
310
R. Swirkowicz
I InJluence
of the surface potential on transition metal films
dence of the character of the spin waves observed in Ni films deposited on various substrates on the type of substrate was found by Whitting
P61.
References [l] G. Gumbs and A. Griffin, Surf. Sci. 91 (1980) 669. [2] J. Mathon, Phys. Rev. B 24 (1981) 6588. [3J 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] J. Mathon and S.B. Ahmad, Phys. Rev. B 37 (1988) 660. [7] J. Mathon, Physica B 149 (1988) 31. [8] J. Mathon, Rep. Prog. Phys. 51 (1988) 1. [9] R. Swirkowicz, Physica B 154 (1989) 373. [lOI R. Swirkowicz and A. Sukiennicki. J. de Phys.. to be published.
[ 111 R. Swirkowicz and K. Swiderczak, Physica B 141 (1986) 199. [12] R. Swirkowicz and A. Sukiennicki, Physica B 149 (1988) 37. 1131 E. Wimmer. A.J. Freeman and H. Krakauer, Phys. Rev. B 30 (1984) 3113. [14] H. Hasegawa, J. Phys. F 17 (1987) 679. [15) H. Hasegawa. J. Phys. F 17 (1987) 165. [16] H. Hasegawa, J. Phys. F 16 (1986) 1555. [17] H. yasegawa, Surf. Sci. 182 (1987) 591. [18] R. Swirkowicz, Phys. Stat. Sol. (b) 129 (1985) 641. [19] R.D. Lowde and C.G. Windsor, Adv. Phys. 19 (1970) 813. [20] G.C. Fletcher, Proc. Phys. Sot. A 65 (1952) 192. [21] J.F. Cooke, J.A. Blackman and T. Morgan, Phys. Rev. Lett. 54 (1985) 718. [22] J. Callaway, A.K. Chatterjee, S.P. Singhal and A. Ziegler, Phys. Rev. B 28 (1983) 3818. [23] D.M. Edwards and R.B. Muniz. J. Phys. F 15 (1985) 2339. [24] M.A. Thompson and J.L. Erskine, Phys. Rev. B 31 ( 1985) 6832. [25] P.W. Mitchell and D. MckPaul, Phys. Rev. B 32 (1985) 3272. [26] J.S.S. Whiting, IEEE Trans. Magn. Mag-18 (1982) 709.
ON THE INFLUENCE OF THE SURFACE POTENTIAL ON THE ONE-ELECTRON PROPERTIES AND SPIN WAVES IN TRANSITION METAL THIN FILMS. THE MULTI-BAND APPROACH R. SWIRKOWICZ’ Institute
of Physics,
Received
Technical
17 February
University.
OO-662 Warsaw,
Poland
1989
Dynamic susccptibihty and spin waves in thin ferromagnetic films of transition metals are investigated numerically within the framework of the multiband model for various values of the surface one-electron potential. The tight-binding approach is used. Influence of the potential shifting on the one-electron properties and spin waves is profoundly discussed. Results, consistent with those obtained in the one-band model, show that acoustic surface modes appear for cases of increasing of the surface potential, which lead to surface magnetization enhancement.
of Ni thin films. Shifting of the core energy in the surface layer is also taken into account in model calculations based on the tight-binding approach, whereas changes of the other parameters in the surface layer are neglected [14-161. The results obtained for the ground-state properties are consistent with ab initio results for various transition metals (Fe, Co, Ni). Changes of the core potential are also important in cases of sandwiches, thin films deposited on some type of substrates and films with coated surfaces [17]. Therefore. it seems to us that, first of all, one should investigate the influence of the core shifting in the surface layer on the one-electron properties and on a degree of localization of spin waves. The detailed investigation of this problem will be presented in this paper. We will show, that analogous to the one-band approach, an appropriate increase of the core potential in the surface layer leads to an enhancement of the surface magnetization and favors an appearance of acoustic surface spin waves.
1. Introduction The dynamical susceptibility and elementary excitations in transition metal films have been profoundly discussed recently within the framework of the one-band Hubbard model [l-10]. Numerical calculations performed for thin ferromagnetic films have shown that the possibility of an appearance of surface spin waves and the degree of their localization strongly depend on values of surface parameters and surface magnetization. In the multiband approach, on the other hand, investigations have been performed only for a specially chosen set of surface parameters [ll, 121. In the present paper the influence of the surface parameters on the one-electron properties and spin waves is investigated within the framework of the multiband model for transition metal thin films. According to ab initio calculations [13], changes of the core potential in the surface layer essentially influence the properties
2. Method
of calculation
Calculations Hamiltonian:
of the susceptibility
’ Supported
University
by the
0921-4526/89/$03.50 0 (North-Holland Physics
of Lodi
of thin ferromagnetic
under
grant
CPBP
Ol.OX.Bl.1.
Elsevier Science Publishers Publishing Division)
B.V.
films are based
on the following
multiband
R. Swirkowicz
I Influence
of the surface potential on transition metal films
30s
c,~,,,,, denotes here an annihilation operator of an electron with spin (T of the orbital m at the lattice site hopping integral. The one-electron potential in the layer v, uj. HZi.(j, j’) re p resents the two-center E,, is assumed to be common to all orbitals. The elements of the matrix U describe the effective intra-atomic interactions between electrons in the layer u. According to the theory presented in paper [la], a reduced transverse susceptibility x( q, OJ) calculated within the framework of RPA with assumptions analogous to those used by Lowde and Windsor for the bulk case [19], takes the following form:
x(4, w) = (Z-
x0(4,4J-‘x”GzI w>.
The mixed representation is used in the above expression and the susceptibility depend on a two-dimensional wave vector q parallel to the surface. Elements susceptibility x” are equal to:
(2) matrices x and x0 of the free-electron
energy of an where N is the number of electrons in the film plane, ETkc denotes the Hartree-Fock is the Fermi-Dirac occupation function and electron with spin u, wave vector k and band index r, fikc coefficients of the one-electron wave functions [la]. T :y” are expansion The band structure of the thin film is determined with the assumption that the core potential EY changes its value in the surface layer only: E
= Y
E, I0
for Y in the surface for other cases .
layer,
(4)
This assumption is consistent with results of ab initio calculations which show that the surface perturbations are screened very well [13]. The surface relaxation is neglected and the hopping integrals on and near the surface are assumed to be equal to those in the bulk. The strength of the effective Coulomb interaction is also assumed to be unchanged in the surface layer. It seems reasonable to limit the number of surface parameters and to consider only one of them, namely E,. Calculations performed within the framework of the one-band Hubbard model clearly show [lo] that changes of the Coulomb integral in the surface layer influence the one-electron states and the spin waves in a similar way as E,. Changes of the hopping integral in the surface layer influence also the spin waves in the same way, provided they lead to analogous modifications of the surface magnetization. We perform numerical calculations bearing Ni in mind. A thin film with fee structure consisting of 7 atomic layers is considered. The surface of the film is assumed to be perpendicular to the [0 0 l] direction. The tight-binding approach with hopping integrals estimated by Fletcher [20] is used. Analogous to other model calculations concerning the systems with surfaces (see, e.g., ref. [14]) the effect of the s-p bands is not included. The Fermi energy is calculated according to the condition that the number of electrons per atom in the middle layer is equal to 9.45. For the strength of the effective Coulomb interaction U, a value of 0.68 eV is taken, which leads to the exchange-splitting parameter A
306
R. .?wirkowicz
I Injuence
of the surface potential
on trunsition
metal films
being equal to 0.37 eV. This value of J is substantially lower than that calculated with use of the local spin-density functional method (see, e.g. ref. [13]), but on the other hand, it is rather close to values used in the spin-wave calculations for bulk nickel [21-231. Our evaluations show that an increase of d leads to an increase of spin-wave energies and therefore to an increase of spin-wave stiffness parameter D. The relation between A and D was discussed by Edwards and Muniz for bulk nickel [23]. Band structure, reduced spin susceptibility x( q, co) and spin-wave eigenfunctions are calculated here for three values of the surface parameter E,, namely for E, = 0.0. 0.15 and 0.25 eV. The obtained results are discussed in the next section.
3. Results Analogous to the one-band Hubbard model [lo], the shifting of the core energy E, in the surface strongly influences both one-electron states and spin waves. First. we will discuss the one-electron properties. The surface magnetization m, obtained for E, = O.OeV is equal to 0.12, whereas the bulk one, m,,, calculated for the central layer is equal to 0.55. Therefore, the substantial lowering of the surface magnetization is observed in this case. An increase of the surface potential leads to the increase of the surface magnetization, and m, equal to 0.38 and 0.71 is obtained for E, equal to 0.15 and 0.25 eV, respectively. The influence of the core shifting on the surface magnetization is qualitatively consistent but rather stronger than in the case considered by Hasegawa [14]. It is a result of different values of model parameters taken in these two cases. Especially, in our case the d-bandwidth is considerably smaller than in Hasegawa’s case and therefore it is more close to the results known from ARP experiments [24]. On the other hand, however, this strong reaction of the surface magnetization on the core shifting allows us to discuss spin waves for various values of the surface magnetization. In all cases considered (E, = 0.0, 0.15, 0.25 eV) the one-electron surface states can be observed, but their positions according to the Fermi level depend on values of the parameter E,. A decrease of E, leads to shifting of the localized electron states (corresponding to majority spins) into higher energies and, simultaneously, to shifting of the states with minority
spins to lower energies, so that the exchange splitting A, for the surface states decreases rather fast. We will discuss this process with help of an example of the strongly localized surface electronic states, which appear at the M point in the two-dimensional Brillouin zone for states with symmetry xy. These states lay near the Fermi level and they are the most striking surface states of Ni thin films. For E, = 0.25 eV the strongly localized M states with majority spins arc occupied and lay 0.05 eV below the Fermi level. Localized states with minority spins are empty and lay 0.43eV above the Fermi level. The exchange splitting A,, equal to 0.48 eV, is higher than the bulk one (A,, = 0.37 eV). There are no holes in the band with majority spins in this case. The surface magnetization is strongly enhanced because the number of electrons with minority spins is reduced. For E, = 0.15 eV the surface electron states at point M are empty for both spin directions and A, is equal to 0.26 eV. A small number of holes with majority spins appears in the surface layer, mainly for states with tZy symmetry. The surface magnetization is reduced as compared to the bulk one. For the case of E, = 0.0 eV, the considered surface states are also empty and lay O.OSeV and 0.16eV above the Fermi level for majority and minority spins, respectively. The exchange splitting A, equal to 0.08 eV is essentially lower than for bulk states. The holes with majority spins appear in the surface and subsurface layers. Simultaneously the number of electrons with minority spins is enhanced in the surface layer. It leads to further reduction of the surface magnetization. Changes of the parameter E, have practically no influence on positions of bulklike states of M
R. Swirkowicz
I Influence of the surface potential on transition metal films
type, strongly localized in the second and third layers. States with majority spins are occupied and states with minority spins are empty for all considered values of E,. The surface density of states is strongly influenced by changes of potential E, (fig. la). One can see that the increase of E, leads to the increase of the surface density of states at the Fermi level. For the cases of E, = 0.0 and 0.15 eV, the surface density of states at the Fermi level is lower than the corresponding density of states in the central layer, whereas for E, = 0.25 eV it is higher. Calculations show also that the surface density of states with symmetry e, at the Fermi level is enhanced more than for states with symmetry t,,. The density of states in the central layer is also influenced by the potential E, (fig. lb). It is a result of the fact that a very thin film is considered. Results obtained here numerically for the multiband model are consistent with qualitative considerations presented by Mathon for the case of Ni surfaces [S]. We also want to emphasize that for E, = 0.25 eV, results obtained here are consistent with results for Ni thin films known from ab initio approaches (e.g. ref. [13]). Namely, the surface density of states at the Fermi level is higher than the bulk one. The magnetization in the surface layer is enhanced and equal to 0.71, whereas in the “bulk” the magnetization is practically the same and equal to 0.58, 0.57, 0.55 in layers s - 1, s - 2 and c (central), respectively. However, due to the fact that hybridization sp-d is neglected in our calculations, a slight breaking of the charge neutrality on the surface appears. Positions of the most important surface electron states are consistent with ab initio theories. First of all, there is a strongly localized state of M3 type occupied by majority spins which lays 0.05 eV below the Fermi level. The f state is also occupied by majority spins, but it is not so strongly localized as in the ab initio approach. Well localized surface states appear also near the x points in the two-dimensional Brillouin zone. Now, we can discuss the influence of the surface potential E, on the dynamic susceptibility and spin waves. Spin-wave eigenfunctions are determined by diagonalization of matrix Re x in
307
(a)
4
2
0
6
ES = 0.25
eV i
Fig. 1. Density of states for: (a) the surface layers and (b) central layers, calculated for various values of surface potential E,.
308
R. Swirkowicz E,=0.15 r-.-
1
layer
Fig. 2. Dependences
I Influence Es=025 r
of the surface potential
eV
index
on layer index of spin-wave
eigenfunctions for various values of E, and q = ~/4a (1.0). Energies of the modes increase from the top to bottom of the figure.
the region of very low energies. The strong dependence of the spin-wave eigenfunctions on the parameter E, can be observed (fig. 2). For E,= 0.0eV modes with lowest energies are mainly localized near the central plane and they are typical bulk modes. However, two modes with highest energies have a tendency to localization in the surface layer, so optical surface modes appear in this case. For E,= 0.15 eV all modes are of bulk character and for E, = 0.25 eV spin waves with lowest energies are localized in the surface layers and acoustic surface modes appear. The change of degree of localization of spin waves with increase of surface potential is well illustrated in fig. 2. Surface parameter E, also strongly influences Im x,,( q. co). In fig. 3 is presented as a function of energy for Im X”” various atomic layers and for E,= 0.0 and 0.15 eV. respectively. In the region presented in the figure two peaks can be observed in Im x,, which correspond to spin-wave modes with lowest energies. For E, = 0.0eV the height of the spin-wave peak corresponding to the mode with the lowest energy is the greatest in the center of
on trunsition
metal films
the film and it decreases strongly from the central to the surface layers. Namely, the height of the peak is equal to 70.6 in the central layer and to 58.3, 26.4, 4.23 in the s-2, s-l and s (surface) layers, respectively. According to fig. 2 one can see that in the case of E,= 0.0eV the amplitude of the first mode is highest in the center of the film and it is considerably lower in the surface layer. Therefore. this mode is mainly localized in the central layer. We can see also that there is a close relation between the height of the spinwave peak in Im x,, and the amplitude of the mode. This relation has already been found in the one-band model [9, lo]. An increase of the surface potential E, leads to the increase in magnitude of the first spin-wave peak in the surface layer. For E, = 0.15 eV the peak corresponding to the mode with the lowest energy is considerably enhanced in the surface layer, but it is lower than in the central plane. The mode considered is still of bulk character, however its amplitude in the surface layer is greater than in the previous case and is comparable to the amplitude in the central layer. Unfortunately, such a relation cannot be illustrated for E,= 0.25 eV, when the acoustic mode appears and one could expect that the height of the first spin-wave peak should be the greatest in the surface layer. It appears, namely, that the damping of the spinwave mode, closely related to the width of the spin-wave line in Im x, depends essentially on the surface potential E,.The width of the line increases with decreasing E,. For E, = 0.25 eV the damping of the modes is much smaller than in previous cases (E,= 0.0,0.15 eV) and the widths of the lines corresponding to two modes with lowest energies approach zero within the accuracy of our calculations. The considered modes are determined very well. An analogous dependence of the damping of the modes on the surface parameters has also been found in the one-band approach [9, lo]. Energies of the lowest mode calculated for E,= 0.0and 0.15 eV, i.e., the first bulk mode are about 20-22.5 meV for q = (rr/4a)(l, 0), where u is a lattice constant. The corresponding stiffness parameter D is therefore about 400-450 meVA*. The obtained value is close to the one calculated
Swirkowicz
I Influence of the surface potential on transition metal films
ES=O.O
Es--O.15
eV
q=JT/4a(l,O)
Fig.
3. The
spin-wave
0.65
(eV) peaks
in Im x for two modes
by Edwards and Muniz for bulk Ni [23] as well as the value measured in the neutron-scattering experiments [25]. It shows therefore, that the spinwave energy does not critically depend on the fine details of the band structure. Calculations show that the energy of the first bulk mode strongly depends on the bulk exchange splitting A,, but it practically does not depend on the A, and the surface parameter E,. The energy of the acoustic surface mode for E, = 0.25 eV is a little lower and it is about 17.5-20.0meV.
4. Discussion Results obtained in the paper for the multiband model are fully consistent with those obtained within the framework of the one-band Hubbard approach [lo]. Numerical calculations performed for various values of the surface one-
eV
q=Tr/40(1,0)
0.05
energy
309
with
lowest
energies
o.io
energy
(eV)
calculated
for E, = 0.0 and 0.15 eV.
electron potential E, confirm also conclusions drawn by Mathon [5] on the basis of qualitative considerations performed in the one-band model that the ratio of ms/mb is essentially related to the conditions of appearance of surface spin waves. In the case of the significant reduction of the surface magnetization (E, = 0.0 eV) we obtain optical surface modes, whereas in the case of enhancement of the surface magnetization (E, = 0.25 eV) the acoustic surface modes appear. Therefore, in thin Ni films with free surfaces, acoustic surface modes should be observed. The situation can be different in cases of thin films deposited on some substrates and for some type of sandwiches. The influence of the substrate or overlayer will, first of all, lead to a potential shifting and to modifications of the magnetization near the interfaces which, by turns, will lead to the change of conditions of appearance of surface modes. Strong depen-
310
R. Swirkowicz
I InJluence
of the surface potential on transition metal films
dence of the character of the spin waves observed in Ni films deposited on various substrates on the type of substrate was found by Whitting
P61.
References [l] G. Gumbs and A. Griffin, Surf. Sci. 91 (1980) 669. [2] J. Mathon, Phys. Rev. B 24 (1981) 6588. [3J 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] J. Mathon and S.B. Ahmad, Phys. Rev. B 37 (1988) 660. [7] J. Mathon, Physica B 149 (1988) 31. [8] J. Mathon, Rep. Prog. Phys. 51 (1988) 1. [9] R. Swirkowicz, Physica B 154 (1989) 373. [lOI R. Swirkowicz and A. Sukiennicki. J. de Phys.. to be published.
[ 111 R. Swirkowicz and K. Swiderczak, Physica B 141 (1986) 199. [12] R. Swirkowicz and A. Sukiennicki, Physica B 149 (1988) 37. 1131 E. Wimmer. A.J. Freeman and H. Krakauer, Phys. Rev. B 30 (1984) 3113. [14] H. Hasegawa, J. Phys. F 17 (1987) 679. [15) H. Hasegawa. J. Phys. F 17 (1987) 165. [16] H. Hasegawa, J. Phys. F 16 (1986) 1555. [17] H. yasegawa, Surf. Sci. 182 (1987) 591. [18] R. Swirkowicz, Phys. Stat. Sol. (b) 129 (1985) 641. [19] R.D. Lowde and C.G. Windsor, Adv. Phys. 19 (1970) 813. [20] G.C. Fletcher, Proc. Phys. Sot. A 65 (1952) 192. [21] J.F. Cooke, J.A. Blackman and T. Morgan, Phys. Rev. Lett. 54 (1985) 718. [22] J. Callaway, A.K. Chatterjee, S.P. Singhal and A. Ziegler, Phys. Rev. B 28 (1983) 3818. [23] D.M. Edwards and R.B. Muniz. J. Phys. F 15 (1985) 2339. [24] M.A. Thompson and J.L. Erskine, Phys. Rev. B 31 ( 1985) 6832. [25] P.W. Mitchell and D. MckPaul, Phys. Rev. B 32 (1985) 3272. [26] J.S.S. Whiting, IEEE Trans. Magn. Mag-18 (1982) 709.