The surface spin waves in the multiband model

Physica 141B (1986) 199-202 North-Holland, Amsterdam

THE SURFACE SPIN WAVES IN THE MULTIBAND MODEL* R.

SWIRK~WICZand K. SWIDERCZAK

Institute of Physics,

Technical

University,

00-662 Warsaw,

Poland

Received 17 June 1985 Revised manuscript received 25 November 1985

The transverse dynamic susceptibility in thin films of ferromagnetic transition metals is calculated within the framework of the approach used by Lowde and Windsor for bulk crystals. Numerical calculations are carried out using the band structure determined in a self-consistent way according to the LCAO method. The changes of the effective Coulomb integral and of the atomic potential in the surface layers are taken into account. For wave vectors small as compared to the reciprocal lattice vector the well-defined surface spin waves with energies lower than energies of bulk modes are obtained. For greater values of the wave vector the spin wave modes are rather strongly damped.

1.

Introduction

The surface spin waves in ferromagnetic thin films of transition metals have been investigated up to now only within the framework of the one-band model [l-6]. According to numerical calculations performed in that model, it has been shown that the surface spin.waves with energies lower than energies of bulk modes exist in the case of the surface which is magnetically weaker as compared to the material inside the film [5]. The dispersion relation of the surface modes has also been found [6]. On the other hand, recently, the general theory of the dynamic susceptibility of thin films have been worked out in the multiband itinerant electron model [7]. Thus, the possibility of investigation of the surface spin waves with use of more exact multiband approach has appeared. In the present paper the susceptibility of a thin film is calculated numerically in the multiband model and the surface spin waves are profoundly discussed. To this purpose the band structure of a thin film is calculated in a self-consistent way within the framework of the LCAO method. The Lowde-Windsor approach known from the bulk problem [8] and adapted to thin film geometry [7] * Supported by the Institute for Low Temperature and Struc-

is consequently

used in our considerations.

2. The thin film susceptibility According to the theory presented in ref. 7, the transverse dynamic susceptibility of ferromagnetic thin films calculated within the framework of the multiband model with assumptions analogical to those used by Lowde and Windsor in the case of the bulk nickel [8], takes the form:

c

X,,(4,~) = Iez)12 [Z-

x0(4,WI;,'

The mixed Bloch-Wannier representation is used in the above expression; V, /J denote the indices of atomic layers and q is the wave vector parallel to the surfaces. The elements of matrix U describe the effective interactions between electrons. Z is the unit matrix and F(q) denotes an atomic form factor. The matrix x0 represents the free-electron susceptibility of the considered system. It is equal to [7]: J&(q,

ture Research of the Polish Academy of Sciences.

0378-4363 /86/ $03.50 @ Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)

w) = $

c TT’~ mm’

(zy

)*zy”T

(eq. contd.)

200

R. l?wirkowicz

’

f

fiw

-f

and K. Swiderczak

T’k+qf

;k;T
(2)

where N is the number of atoms in the film plane, E zkmdenotes the Hartree-Fock energy of an electron with spin a, wave vector k (parallel to the surface) and band index r. fTkgis the Fermi-Dirac occupation function and Tf,y’=”are expansion coefficients of the wave functions @Tk( r) (expressed by means of the localized orbitals [9]). The Hartree-Fock energies Erko and coefficients Tty” can be calculated according to the following equation (71:

I Surface spin waves

The numerical calculations will be performed with the assumption that the Coulomb integral changes its value only in the surface layers. This does not limit our considerations, but it allows us to reduce the number of elements of the matrix AU which are different from zero, i.e. the number of parameters which must be fitted. When symmetric boundary conditions are assumed and Im x1 appears to be small enough, eq. (5) can be reduced and spin wave energies can be calculated from the following equations: I-

AU

Re[x:l(q, 0) + x:,(4, w)l=0,

for symmetric modes, I-

AU

Re[x:l(q, 0) - x:,(4,

for antisymmetric

x

T;;‘” = E,,Tt;”

,

(3)

where n,, denotes the number of electrons with spin cr in the vth layer and Hr,Jk) is the Fourier transform of

e

(4) Here j is the radius vector in the film plane and V(r) is a one-particle potential.

3. Spin waves The energies of spin waves characteristic for thin films correspond to poles of the susceptibility given by (l), The matrix U in eq. (1) can be expressed in the form: U = ZJ, + AU, where U,, is a diagonal matrix whose elements are the same in all atomic layers and AU is a diagonal matrix describing the changes of Coulomb interactions near the film surfaces. If the matrix x1 = (I x”Uo)-l~o is introduced to the formula (1) the spin wave energies can be calculated according to the equation Redet(Z-X’AU)=O.

(5)

= 0,

modes.

Here, AU describes the change of the Coulomb integral in the surface layer and n denotes the number of atomic layers in the thin film. The first step in our numerical calculations is to determine the energies ETku and coefficients T ty”. These quantities are calculated self-consistently within the framework of the LCAO method. The computations are performed for thin nickel films analogically as in ref. 10 with the difference, however, that the Brillouin zone sums are now calculated with use of the triangle method [ll]. 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. The value 0.55 eV is assumed for the Coulomb integral U,, so that the spontaneous magnetization in the middle layer equal to 0.55 is obtained. The computations of the electronic structure are performed with such changes of the Coulomb parameter and the atomic potential in the surface layers that the condition of the local neutrality of the film is fulfilled. The local density of states, the local magnetization and the local number of electrons per atom are obtained. The local densities of states at the surface and in the middle layers calculated with AU = 0.05 eV and with the atomic potential shifted by -0.05 eV are presented in fig. 1. It can be seen that the density of states at the Fermi

+V(r)lv~‘m’).

Hzi.(j-j')=(vjml -g

w)l

(6)

R. swirkowicz and K. Swidercrak

spm4

Fig. 1. The local density of states per atom per spin as a function of energy for: Y = 1 (surface layer), Y = 4 (middle layer). AU = 0.05 eV and the atomic potential is shifted by -0.05 eV.

level in the surface layer is lower than in the middle layer. The analogical results were obtained within the framework of the tight-binding method with use of the one effective band [5] and with five d-bands [9, lo]. Therefore, such a result seems to be a general feature of simplified approaches to calculation of the band structure of thin films. The numbers of electrons obtained in the considered case for the surface and middle layers are n, = 9.46 and n,, = 9.45, respectively. The obtained surface and bulk magnetizations are m, =OSl and m,, = 0.55, respectively. The surface magnetization is therefore slightly smaller than the bulk magnetization. The recent selfconsistent ab-initio calculations give a slight enhancement of the surface magnetization [12]. It should be pointed out, however, that the behaviour of the magnetization in the surface layer is a very delicate problem and it would be rather

I Surface spin waves

201

difficult to discuss this problem within the framework of the simple model presented here. On the other hand, however, one can expect that the basic conclusions concerning the spin waves, especially the long-wavelength spin waves, should not depend essentially on the fine details of the band structure. The band structure determined above is used for the dynamic susceptibility calculations. First, the matrix x0 is computed using the Singhal’s programs [13] modified and adapted to thin films. It is more convenient to carry out those calculations in the impulse representation instead of the mixed Bloche-Wannier representation. The transformation discussed in ref. 5 is used for the direction perpendicular to the film. The computations show that, analogically as in the Hubbard model [5], the off-diagonal elements of the matrix x0 expressed in the impulse representation are very small as compared to the diagonal ones. Thus, in calculation of x1 only diagonal elements of the x0 are taken into account. The spin wave energies are calculated now for q = (~/4a) (l,O). In this case eqs. (6) can be curves corresponding to used. The Re[Xil(q, w) + xi,,(q, w)] are depicted in fig. 2. One can see that if AU = 0.05 eV and the atomic potential is shifted by -0.05 eV, the well-defined surface spin wave splitted of a bottom of the bulk spin-wave band appears. The energy of this mode is equal to 17.5 meV and is lower than the value obtained in the Hubbard model (see ref. 6). It should be pointed out that for discussing the possibility of an appearance of surface spin waves one should compare the value of the product of the effective Coulomb integral and the density of states at the Fermi level for the surface layer to the similar product for the middle layer. In the case considered, despite AU > 0, this product is lower in the surface layer than in the middle layer, as a result of a decrease of the local density of states near the Fermi level at the surface. Therefore, in view of the Stoner criterion of ferromagnetism, the surface can be treated as magnetically weaker as compared with the material inside the film and just for this case the surface spin wave with energy lower than energies of bulk modes appears. The obtained result is consistent

202

R. hvirkowicz

-301 0

I 50

and K. hviderczak

I

I 100

150 ho

[meV]

Fig. 2. Plot of the Re[,y:,(q, o) + ,y:,(q, o)] versus energy for q = (rr/4a) (l,O). Straight horizontal line corresponds to l/AU. AU = 0.05 eV and the atomic potential is shifted by -0.05 eV.

with conclusions obtained on the basis of calculations performed within the framework of both the Heisenberg model [15] as well as the Hubbard model [3-51.

I Surface spin waves

pointed out, however, that the widths of the spin-wave peaks, calculated in the one-band model are practically equal to zero in a wide energy and wave vector region. It means that the modes obtained for this region are non-damped and their lifetime is practically infinite. On the other hand, according to calculations carried out within the framework of the multiband model, it appears that even for small q and low energy the imaginary part of det(Z - ,y’AU) is different from zero although rather small. It leads, therefore, to the finite value of the lifetime of low-energy magnons. The computations performed for greater q show that in the case of the multiband model the damping increases very quickly. E.g. for q = (?7/4a) (3,O) 1 ‘t IS . so strong that no spin waves can be obtained. The value of wave vector at which the spin waves characteristic for thin films completely disappear, depends essentially on the details of the band structure. The simplifications introduced in calculations of the band structure as well as of the dynamic susceptibility of thin film seem to be very important for this problem. The values of surface parameters can also show some influence on the value of this vector.

4. Conclusions References

The surface spin waves in ferromagnetic thin films are investigated within the framework of the multiband model with use of the Lowde-Windsor approach. The changes of the effective Coulomb integral and the atomic potential in the surface layers are taken into account. The values of the surface parameters are chosen to fulfil the condition of the local neutrality of the film. The approach presented in this paper is quite new in the theory of spin waves in thin films. It allows to take into account the degeneracy of the d-band and some other features of the electronic band structure characteristic for thin films. The obtained results show clearly that for q small as compared to the reciprocal lattice vector the surface spin waves with energies lower than energies of bulk modes appear in the case of the surface magnetically weaker (in view of the Stoner criterion) than the material inside the film. It is consistent with the results obtained within the framework of the Hubbard model. It should be

[l] A. Griffin and G. Gumbs, Phys. Rev. Lett. 37 (1976) 371. [2] G. Gumbs and A. Griffin, Surf. Sci. 91 (1980) 669. [3] J. Mathon, Phys. Rev. B24 (1981) 6588. [4] R. Swirkowicz and A. Sukiennicki, Acta Phys. Pal. A66 (1984) 643. [5] R. Swirkowicz, K. Swiderczak and A. Sukiennicki, Acta Phys. Pal. A67 (1985) 913. [6] R. Swirkowicz, Physica 128B (1985) 297. [7] R. Swirkowicz, Phys. Stat. Sol. (b) 129 (1985) 641. [8] R.D. Lowde and C.G. Windsor, Adv. Phys. 19 (1970) 813. (91 K. Terakura and I. Terakura, J. Phys. Sot. Jap. 39 (1975) 356. [lo] K. Swiderczak and A. Sukiennicki, Acta Phys. Pol. A62 (1982) 189. [ll] 0. Jepsen, J. Madsen and O.K. Andersen, Phys. Rev. B18 (1978) 605. [12] H. Krakauer, A.J. Freeman and E. Wimmer, Phys. Rev. B28 (1983) 610. [13] S.P. Singhal, Phys. Rev. B12 (1975) 564. [14] H. Puszkarski, in: Progress in Surface Science, Vol. 9, Part 1, S.G. Davison, ed. (Pergamon, New York, 1979) p. 191.