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Magnetic structures in metallic thin films K. Okazaki *, Y. Teraoka Department of Materials Science, CIAS, Osaka Prefecture University, Sakai 599-8531, Japan

Abstract The electronic and magnetic states of metallic thin films are theoretically studied on the basis of the uniform jellium model within a framework of the local spin density approximation. For a given average electron density, the magnetic structure is found to change successively as a function of the film thickness D. For r =6 a.u., the s paramagnetic, the ferromagnetic, and the antiferromagnetic states are found. In Cs and Rb, r is about 5.5, and so s we expect very thin films of Cs and Rb to be magnetic. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Alkali metals; Density functional calculations; Jellium model; Magnetic films; Magnetic phenomena; Surface electronic phenomena

1. Introduction A large number of experimental and theoretical studies on surface magnetism of transition metals have been carried out [1]. One of the most interesting findings is that the magnetic moments of surface atoms are largely enhanced, compared with the bulk value. Such an enhancement has been observed in a few experiments; one example is an increase of T in Cr fine particles [2] and in CrAl N alloys [3], and another one is an observation of ferromagnetic ordering on polycrystalline Cr films well above T [4,5]. It is theoretically concluded N that the origin is in d–d bond-breaking effect due to a reduction of the coordination number of the surface atom [6,7]. Originally, a metal atom with an open shell has a magnetic moment, irrespective of whether or not there are d electrons. Therefore, * Corresponding author. Fax: +81-722-54-9723. E-mail address: [email protected] ( K. Okazaki)

the above-mentioned phenomenon suggests that a reduction of the dimension of a system consisting of metal atoms leads to an appearance of magnetic moments. Alkali metals have no magnetic moments in the bulk or on the surface, but a single alkali metal atom has a magnetic moment. This suggests that magnetic moments may appear/disappear, and the magnetic structure can change as a function of the dimension and the shape of the system. In order to study the surface electronic structures of metals, Lang and Kohn applied the local density approximation (LDA) to the exchange– correlation energy of electrons in the uniform jellium model ( UJM ) [8–10]. The calculated work function [9] and surface energy [10] show a good agreement with experimental values for alkali metals. Therefore, we expect that the treatment is suitable for the surface electronic structure calculation of alkali metals. In addition, Schulte applied the same method to metallic thin films and found that the work function and the Fermi energy strongly depend on the film thickness [11].

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In the present work, we study the electronic and the magnetic states in metallic thin films with the use of the spin-density-functional theory [12]. The local spin density approximation (LSDA) of the Janak–Moruzzi–Williams type [13] is applied to the exchange–correlation energy of electrons in the UJM. The states are discussed as a function of the film thickness, D, and the average electron density, r . Hereafter, we use r (=(3/4pr )1/3) 0 s 0 instead of r . The theoretical treatment is discussed 0 in Section 2, the calculated results in Section 3, and Section 4 is devoted to the summary. Atomic units are used in expressions, with e (the magnitude of the charge on an electron), m (the mass of an electron), and all set equal to unity.

2. Theory The electronic and magnetic properties of an electron gas in the external field v(r) caused by a positive charge background can be obtained by the spin-density-functional theory [12]. The electron density with s spin, r (r), in a system s consisting of interacting electrons can be determined by the following equation −1 V2Ys (r)+vs (r)Ys (r)=Es Ys (r) i eff i i i 2 and setting

(1)

r (r)=∑ |Ys (r)|2, (2) s i i where the sum is to be extended over the states with a lower energy than the Fermi energy E . s F takes + or −. The effective potential vs (r) in Eq. eff (1) consists of the electrostatic potential v (r) and es the exchange–correlation potential vs (r). xc The positive charge density r (r) is assumed as B follows: |x|≤1 D 2 (3) |x|≥1 D, 2 where D is the film thickness (see Fig. 1). The film surfaces are denoted by the planes x=±D/2. The electronic states along the parallel directions, y and z, to the surfaces are treated as free electron

G

r r (r)= 0 B 0

673

Fig. 1. Positive background charge density in the uniform jellium model.

states. Thus, the solution of Eq. (1) can be written in the following way: Ys (r)=Ys (r) i n,ky,kz =ws (x) exp[−i(k y+k z)] (4) n y z where n=1, 2, 3,…k and k are the wave numbers y z along the y- and z-axes, respectively. In terms of ws (x), Eq. (1) can be rewritten as n 1 d2 ws (x)+vs (x)ws (x)=es ws (x), (5) − eff n n n 2 dx2 n where the eigenvalue es denotes the energy levels n and is connected with the eigenvalue Es of Eq. (1) i by the relation Es =es +1 (k2 +k2 ). (6) i n 2 y z The electron density with s spin has the form 1 r (x)= s 2p

∑ (E −es )|ws (x)|2 (7) F n n sn≤EF The electron and the spin densities are given by r (x)+r (x) (=r(x)) and r (x)−r (x), respec+ − + − tively. The Fermi energy, E , is so determined that F the film is electrically neutral. Since the positive charge and the electron densities depend only on x, the electrostatic potential is given by e

v (x)=−4p es

P

x

(x−s)[r(s)−r (s)]ds. B

(8)

−2 The vacuum level is zero of the energy. For the exchange–correlation part of the effective potential, the LSDA is applied [13]. The exchange–

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K. Okazaki, Y. Teraoka / Surface Science 433–435 (1999) 672–675

correlation potential is given by ∂e (x) ∂e (x) v± (x)=e (x)+r(x) xc ± xc [1Af(x)], xc xc ∂r(x) ∂f(x) (9) where e is the exchange–correlation energy per xc electron in a homogeneous electron gas. In our numerical calculation, we adopt the Janak– Moruzzi–Williams-type exchange–correlation energy for e . f(x) is the local spin polarization xc given by {r (x)−r (x)}/{r (x)+r (x)}. The + − + − electron density, spin density, and effective potential are obtained by solving self-consistently Eqs. (5), (7)–(9). A few magnetic states, including a paramagnetic state appear, and so we should compare the total energies with one another in order to determine the ground state. The total energy per surface area, E , is given by tot 1 ∑ [E2 −(es )2]− vs (x)r (x)dx E =∑ tot eff s F n s 4p e sn≤EF 1 + v (x)[r(x)−r (x)]dx+ e (x)r(x)dx, B xc 2 es

G

P

P

Fig. 2. Spin density of the F state (the solid curve) and the AF state (the broken curve) at D=15 for r =6. s

H

P

(10) where the first term is the kinetic energy, the second term is the electrostatic energy, and the third term is the exchange–correlation energy.

3. Results First, we discuss the magnetic state for r =6 as s a function of the film thickness D (5≤D≤70). There are three types of state: the paramagnetic (P), the ferromagnetic (F ), and the antiferromagnetic (AF ) states. In the P state, there is no spin polarization at any value for x. Such a solution is found for all Ds, but magnetic (F and AF ) states are found only in some regions of D; F in D= 5~8, 12~18, and 23~27 and AF in D=12~15. The spin density profiles in F and AF along the x-axis are shown in Fig. 2, where D=15. In the y–z plane, there is uniform spin polarization depending on x in F and AF. Of course, the

Fig. 3. Energy levels as a function of D for r =6. The broken s and solid curves indicate the P and the F states, respectively. The dotted curve represents E ( F ). F

electron density depends only on x, too. Such magnetic states are no longer found at D≥28. The energy levels es and the Fermi energy, E , n F in P and F are shown as a function of D for r =6 in Fig. 3. In order to keep the charge neutrals ity, the number of states with a lower energy than E is increased with an increase of D. This makes F cusps in the es versus D and E versus D curves. n F The phenomenon was found in the P state by Schulte [11]. Now, we suppose that the system is in the P state. The lowest unoccupied energy level intersects with E with an increase of D. At the F intersecting point, the energy level splits off into two levels with + and – spins. The level with + spin becomes occupied and that with – spin

K. Okazaki, Y. Teraoka / Surface Science 433–435 (1999) 672–675

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4. Summary

Fig. 4. Total energy as a function of r . The solid and broken s curves indicate the P and F states, respectively.

remains unoccupied. Such a situation is found around D=12 and 23. With a further increase in D, the splitting is enhanced, decreases and then disappears. At 5≤D≤8, there is only one energy level below E , and so only electrons with + F spin exist. The total energy calculation shows that the lowest energy state is F at 5≤D≤8, P at 9≤D≤11, F at 12≤D≤17, P at 18≤D≤22, F at 23≤D≤26, and P at D≥27. The F state is most stable due to the exchange–correlation energy gain. The AF state is unstable due to the kinetic energy loss. In addition, no other types of AF state are found for a small r . s For r =6, the F state is most stable in D= s 5~8. Now, we investigate how the stability of the F state depends on r for D=7 and 8. The total s energies of the F and the P states as a function of r are shown in Fig. 4. The F state is more stable s at r ≥5.4 for D=7, and at r ≥5.7 for D=8. s s Therefore, we expect that alkali metals with r =5~6, such as a Cs and Rb, are F in very thin s films. With a decrease in D, the F state becomes more stable at a smaller r . s

The electronic and magnetic states in metallic thin films have been investigated using the LSDA applied to the UJM. It has been found there are three types of magnetic state for r =6; the paras magnetic, the ferromagnetic, and the antiferromagnetic states. As a function of D, the magnetic structure is varied successively. The total energies have been calculated and compared with one another in order to determine the ground state. The ferromagnetic state is the most stable in very thin films. This suggests that alkali metals with r =5~6, such as Cs and Rb, are ferromagnetic in s very thin films. In addition, the antiferromagnetic state is found at D=12~15. However, the total energy is higher than those of the paramagnetic and the ferromagnetic states due to the kinetic energy loss. Other types of the antiferromagnetic state are not found for a small r . s

References [1] A.J. Freeman, R. Wu, Prog. Theor. Phys. 106 (1991) 397. [2] S. Matsuo, I. Nishida, J. Phys. Soc. Jpn. 49 (1980) 1005. [3] S. Arajs, N.L. Reeves, E.E. Anderson, J. Appl. Phys. 42 (1971) 1691. [4] P.E. Ferguson, J. Appl. Phys. 49 (1978) 2203. [5] L.E. Klebanoff, S.W. Robey, G. Liu, D.A. Shirley, Phys. Rev. B 30 (1984) 1048. [6 ] Y. Teraoka, J. Kanamori, Inst. Phys. Conf. Ser. 39 (1978) 588. [7] G. Allan, Phys. Rev. B 19 (1979) 4774. [8] N.D. Lang, F. Seitz, D. Turnbull, H. Ehrenreich ( Eds.), Solid State Physics Vol. 28 , Academic Press, New York, 1973, p. 255. [9] N.D. Lang, W. Kohn, Phys. Rev. B 1 (1970) 4555. [10] N.D. Lang, W. Kohn, Phys. Rev. B 3 (1971) 4555. [11] F.K. Schulte, Surf. Sci. 55 (1976) 427. [12] U. von Barth, L. Hedin, J. Phys. C 5 (1972) 1629. [13] J.F. Janak, V.L. Moruzzi, A.R. Williams, Phys. Rev. B 12 (1975) 1257.