Field-induced magnetization of a free-electron gas in thin films

Transcription

1 Materials Science-Poland, Vol. 5, No., 007 ield-induced magnetization of a free-electron gas in thin films T. BALCERZAK 1* 1 Department of Solid State Physics, University of Łódź, Pomorska 149/153, Łódź, Poland A free-electron model in thin film emeded in an external magnetic field is considered. Based on the paramagnetic susceptiility, a formula for magnetization of the electron gas in the uniform magnetic field is derived. Selected results are presented for the films with the thickness of several atomic planes, and with the electron density corresponding to copper. Key words: thin film; free-electron gas; paramagnetic susceptiility; magnetization 1. Introduction Studies of comined electronic and magnetic properties of thin films are important from the point of view of modern technology [1]. As far as the theoretical description is concerned, the model of free electrons placed in a quantum well turned out to e very useful for the discussion of various physical properties of thin films [1, ]. In the present paper, this model will e further analyzed in order to study the magnetization of the electronic gas in thin films when the magnetization is induced y the external field. Thin film is understood here as a set of n monoatomic layers perpendicular to the z-axis, with each layer having its own thickness d. As a result, the total thickness of the film is given y L z = nd, and the total volume is V = L z S, where S is the film surface area ( S >> L z ). Apart from the thickness, the important parameter is a dimensionless electron density ρ, defined y the formula: ρ = (N e /V)d 3, where N e is the total numer of electrons in the sample, and the volume is expressed in d 3 units. The considerations of single-particle electronic states in such a model system lead to the discretization of the ermi surface, as discussed in detail in [3]. In particular, the formulas for the ermi wave vector k in thin film and the parameter τ have een * Corresponding author, t_alcerzak@uni.lodz.pl

2 448 T. BALCERZAK derived, where τ is the highest numer of the standing-wave mode in the ground state. In order to avoid a redundant repetition of the formalism, for the method of determination of k and τ adopted in this paper the reader is referred to Ref. [3].. Theory The magnetization of the electron gas (i.e., the magnetic moment per unit volume) induced y the external field at a point ( z, r ) of the thin film is given y the expression: σ 1 π V p q L z ( z, r) = χ pq, H pq, exp i pz exp[ iqr] (1) where r denotes the position of a given point in the film plane, whereas z denotes the coordinate perpendicular to the film (0 z L z ). H p, q is the ourier component of the external field and χ p, q stands for the paramagnetic susceptiility. In Eq. (1), a quasicontinuous summation over q is performed in the film plane, and the discrete summation over p (p = 0, ±1, ±,...) concerns the perpendicular modes. The paramagnetic electronic susceptiility in a thin film is given y the formula [4]: τ m 1 χ = g μ f + ( q ) + f ( q ) pq, e B τ, τ p τ, τ p τ = 1 () where f τ,τ' (q) is a generalized Lindhard function: S aττ, fττ, ( q) = ττ, ( q) θ ( 1 ττ, ( q) ) 4π q (3) where and a ττ, π ( ) ττ, = τ τ Lz ( q) k 1 τ = q a ττ, 1+ q (4) (5) k π τ = (6) τ k Lz

3 ield-induced magnetization of a free-electron gas in thin films 449 It has een shown in Ref. [4] that Eq. (3) presents a generalization of the ordinary Lindhard function as eing discussed in [5] for a two-dimensional system. Assuming in Eq. (1) that the external field is spatially uniform, we have: H = Hδ δ (7) pq, p,0 q,0 With the help of Eqs. (7) and () (6) we can then otain from Eq. (1) the magnetization induced y this field in the form: 1 1 m τ σ( zr, ) = χ0,0h = g e μb H σ V 4π L z (8) or such a case it is seen from Eq. (8) that the magnetization of electron gas in the film is also uniform (we denote it y σ). As a system with the reference magnetization, we shall assume the ulk material in which the magnetization of the electron gas induced y the same external field is given y the formula: σ 1 1 m 0 = PH g e BkH V χ = 4π μ (9) In Eq. (9) χ P is the Pauli paramagnetic susceptiility [6] and k is the ermi wave vector of the ulk material. By dividing Eqs. (8) and (9) y sides, we finally otain the relative field-induced magnetization in the form: σ τ 1 π σ = 0 ndk (10) The aove magnetization depends on the film thickness L z and the electron density ρ (via dependence of τ and k upon ρ). In the limiting case (when n ) we have πτ /nd k (see [3]) and thus σ σ Numerical results and discussion The numerical results are otained ased on Eq. (10). The value of d = Å is assumed to correspond to the interplanar distance of the (100) planes in copper. In ig. 1, the relative magnetization σ/σ 0 is shown for several film thicknesses n, when the dimensionless density ρ is freely changed from 0 to 1. The value of ρ = 0.5 corresponds exactly to copper. It can e seen that for the monolayer (with n = 1), magnetization is a monotonously decreasing function of ρ (whereas σ > σ 0 ). It is connected with the fact that for n = 1 we have the only value τ = 1 in this range of ρ. However, for n = and n = 3 the curves are no longer monotonic. or instance, for n =, a jump

4 450 T. BALCERZAK of magnetization can e oserved at ρ = 0.589, which is connected with the change from τ = 1 to τ =, whereas ρ increases. In the same way, two jumps are apparent for n = ; the first at ρ = is connected with the change from τ = 1 to τ =, and the second at ρ = is the change from τ = to τ = 3. In particular, it can e seen in ig. 1 that for ρ = 0.5 the magnetization of the trilayer system (with n = 3) is etween those of n = 1 and n =. ig. 1. Dependence of the field-induced relative magnetization σ/σ 0 of the electron gas upon the dimensionless density ρ. The three curves laelled y n = 1 (solid), n = (dashed) and n = 3 (dotted) correspond to various film thicknesses ig.. Dependence of the field-induced relative magnetization σ/σ 0 of the electron gas upon the film thickness n, when ρ = 0.5. The interplanar distance is d = 1.805Å (the same as in ig. 1), which corresponds to the (100) film of copper igure corresponds to the case when ρ = 0.5 is constant ut the thickness n changes. One can see that the magnetization is an oscillating, saw-toothed function of the film thickness. The amplitude of the oscillations decreases as n increases. A detailed analysis shows that this ehaviour is the same as the oscillations of the density of states at the ermi surface, as found for this model in [3]. It is worth noting that the oscillating ehaviour of the density of states has een found experimentally ([] and references therein). Thus, we see that the ehaviour of the field-induced magnetization in thin films is directly correlated with the density of states at the ermi level. Such a correlation could e expected to some extent ecause an analogous relationship is well estalished in ulk materials. However, in the ulk material there are no size-dependent oscillations. The final conclusion which can e drawn from the paper

5 ield-induced magnetization of a free-electron gas in thin films 451 is of a general character: despite of the relatively simple model considered here, the predicted results seem to e interesting and non-trivial. Acknowledgements The author is indeted to Mrs. Teresa Rychtelska for her valuale help in the preparation of the figures. References [1] DAVIES J.H., The Physics of Low-Dimensional Systems, Camridge Univ. Press, Camridge, 000. [] HIMPSEL.J., J. Phys.: Cond. Matter, 11 (1999), [3] BALCERZAK T., Thin Solid ilms, 500 (006), 341. [4] BALCERZAK T., Thin Solid ilms, 515 (006), 814. [5] BEAL-MONOD M.T., Phys. Rev. B, 36 (1987), [6] YOSIDA K., Theory of Magnetism, Springer, Berlin, Received 7 May 006 Revised 1 Septemer 006