Calculation of Electrical Conductivity and Giant Magnetoresistance within the Free Electron Model
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z direction within each layer. It can be shown that variations of the real part of the selfenergy contribute very little to the GMR. Therefore we assume that the real part of E(z) is a constant throughout space. The imaginary part of E(z) is determined by the mean free path within each layer, /CF (1)
Im.= 2ff'
where kF is the Fermi momentum, and the subscript I denotes the layer I. The quantum solution to such a multilayer cannot be obtained analytically. To obtain a numerical solution we embed a finite number of multilayers (about 1000A thick) into an infinite square well, and calculate the Green function using[3] G(k 1j; z, z') = -L(Z)W
(2)
where k1l is the parallel component of the wave vector, and ,bL and differential equation, 2
h
[E +
ObR
are solutions to the
82
2-(•z
k2)
-
-
1E(z)]O(z)
(3)
= 0,
and satisfy the boundary conditions on the left and right sides of the system, respectively. For a multilayer system of total thickness d we used the boundary conditions, 4OL(O) = 0 and tkR(d) = 0. W is the Wronskian of bL and ObR. The conductivity for current-in-plane (CIP) can be calculated from the Kubo formula which gives,
S e2h
dz
d2kj1k2ImG(kjj; z, z')ImG(kll; z', z).
dz'
(4)
We first use this to calculate the conductivity of a simple multilayer system and compare with the semi-classical results and those obtained using the theory of ZLF. The comparison is shown in Fig. 1, as a function of the thickness of one period. In these calculations it is assumed that the scattering rates for the two layers correspond to bulk mean free paths of 36.0555 and 360.555 atomic units (1 a.u.=0.529A) and that the thickness of the dirty layers is twice that of the clean layers. No additional scattering at the interfaces is included. In all of the multilayer calculations we used a sufficient number of periods of the multilayer to avoid the physical quantum size effects for the exact results and the large unphysical size effects that occur for the semi-classical and ZLF theories. There are two limits that all theories approach correctly: The thin limit in which the layer thicknesses are small compared to the mean free path, and the thick limit in which the layer thicknesses are much larger than the mean free path. In the thin limit the conductivity is determined by the average of the scattering rate, which gives,
1__
t
d 11"
=
E(5)
In the thick limit, the mean free paths are averaged,
Ic = I
.(6)
There is a surprisingly good agreement between the semi-classical theory and the FERPS model. On the other hand, the ZLF theory seems to approach the thin limit too fast. 324
55
-
. I . I. "I
1"
1
' I I . .. . .
Thick limit
50 45 40 15
ar (10 /sec)
Camblong-Levy/ Fuchs-Sondheimer
35
Zhang-Levy-Fert
30 Exact
25 20 15.........
.
Thin limit ,,..i
10
, .
100 1000 Thickness (a.u.)
10000
Figure 1: Conductivity as a function of the total thickness of a period of a multilayer system. The period contains two layers, with thicknesses 2/3 and 1/3 of the period, and mean free paths 36.0555a.u. and
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