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Theory of the resistivity anisotropy in ferromagnetic metallic films
Physica 82B (1976) 339-342 0 North-Holland Publishing Company
THEORY OF THE RESISTIVITY ANISOTROPY IN FERROMAGNETIC METALLIC FILMS VU DINH KY Polytechnical Natuurkundig
University of Hanoi, Hanoi Laboratorium, Universiteit van Amsterdam,
Amsterdam,
The Netherlands
Received 27 May 1975 Revised 30 September 1975
An explicit expression for the Ap effect of a thin ferromagnetic film is given, using results obtained earlier by the author and the Fuchs-Chambers method. The theoretical results are compared with the experimental results. The qualitative agreement is satisfactory.
1. Introduction
boundary conditions are treated by the FuchsChambers method and the film surface is characterized by a Fuchs parameter p. In this report the results of the solution of the kinetic equation in ferromagnetic metals obtained in a previous paper [3] and the Fuchs-Chambers method are used to suggest a phenomenological approach, in which both the size-effect and the spinorbit coupling are considered. The numerical results obtained from the general formula are used to explain some experimental data.
It is well known that the resistivlty of the ferromagnetic metals depends on the orientation of the magnetization. This anisotropy of the resistivity which can be characterized by the difference Ap = p ,, - pI between the resistivities when the crystals are magnetized to saturation in the directions parallel and perpendicular to the current, is due to the spin-orbit interaction in ferromagnetic metals [l-3]. In ferromagnetic metallic films the Ap-effect was investigated intensively during the last years (see, for example ref. 4 and references given there and refs. 5-7). This effect was not only used to investigate the magnetic characteristics of the films [4], but possibly could also be applied in memory techniques [8]. However, the theoretical considerations needed to explain the properties of the Ap anisotropy in ferromagnetic films were lacking, until now. The main difficulty is the solution of the kinetic equation in which the spin-orbit interaction and the boundary conditions must be considered together. The spinorbit coupling for the current carriers in ferromagnetic metals was investigated in ref. 3 to obtain the expression of the Ap anisotropy. On the other hand, th: size-effect due to the boundary conditions in nonferromagnetic films was investigated in numerous papers (see, for example ref. 9 and the references given there). Usually, for nonmagnetic films the
2. The solution
of the kinetic equation
Let us consider an unidomaln ferromagnetic and isotropic crystal with planar geometry and the current in the direction of the x-axis in this plane. The z-axis is perpendicular to the sample plane. According to the results of ref. 3 for bulk metals, the solution of the kinetic equation for the cases that M II J and M 1 J can be written as
afoII f/l(k)=-$*(E.k)z
7 91 ( k),
(1)
where f:y’(/c)is the nonequilibrium part of the distribution function for the current carriers, and it 339
= ~~(1 + +M23;),
(24
340
Vu dinh Ky/Resistivity
71(k) = To( 1 + -)4@ {j),
anisotropy in ferromagnetic
x
1_
(1 - p)exp(-z + bVT”(k)4 1 - p exp(-d/T”(k)u,)
d/2
(2b)
r. is the isotropic part of the relaxation time; yll, r1 are small coefficients proportional to the second order in the intensity of the spin-orbit coupling (y”M2, +M2 < 1). M is the magnetization of the sample. The expressions for ro, y” and 9 can be written explicitly [3] , but they are not needed in the following calculation. { is the unit vector in the direction of the wave vector k. The formulae (2) show that, if using the relaxation time approximation, one can interpret the anisotropy of 7(k) as being a consequence of the spin-orbit coupling. Now we can use eqs. (1) and (2) to obtain the solution of the kinetic equation for the ferromagnetic films. Let us consider only the interval of thicknesses where the size-quantization effect can be neglected and the electron spectrum in the film is not different from the spectrum in the bulk material. Then only the surface conditions must be considered in the calculation. As in ref. 5, it is assumed that the electron scattering at the surfaces of the film gives no contribution to the anisotropy of the relaxation time. Then using the conventional Fuchs-Chambers method [lo] the solution of the kinetic equation for the ferromagnetic films can be obtained. For example in the case that M IIJ and u, > 0 we have
1’
(3)
where d is the thickness of the film and p is the Fuchs parameter of the film surfaces. For the other cases we have analogous expressions. Because in films the magnetization and, consequently, the intensity of the spin-orbit coupling depend on the film thickness, the magnetization must be taken as M(d), but not M(m) in the bulk samples.
metallic films
d3kf;‘?*(k,
uz, z)
(4)
one obtains the resistivity anisotropy Ap = Au/o~(~), where Au(d) = u$, - u& and uo(d) is the isotropic part of the conductivity in the film with the thickness d. Because y”JM2 Q 1 one can use the approximative expansion in the integral of (4), for example
ew@l(l -___ 1 - exp(a/(l
+ 6))
exp a l+ S)) x 1 - exp a (
a6
1 - expa 1 ’
6 < 1.
(5)
Substituting finally obtain
(2) and (3) into (4) and using (5) we
(64 (6b)
(74 0)
A&
A)=
1 -&(l
-P)+(l
-PY
(
+1,,5x4
i pn-’ n=l
1
+ E,(nA)(&n4h3 - &n6A5) , I)
(84 A2(P, X)= 1 ,1-&l
-p)+(l
-?9ZnzI
3. The resistivity anisotropy Substituting the expression of the solution of the kinetic equation into the formula of the conductivity
* ux,
5
3__-n____+-
8h
8
n2h 16
n3h2 16
?+
Vu dinh KyfResistivity
+ E, (nh)
anisotropy in ferromagnetic
metallic films
341
&z2h- i’sn4j\3)
113
El(x)
=
7G dt. X
In these formulae the parameter h is defined usually by the relation X = d/Z where 1 is the mean free path of the electrons in the bulk material. Ap(m) and Ap(d) are the resistivity anisotropy in bulk material and film, respectively. Ap/p = (p” - $)/PO, p0 is the isotropic part of the resistivity.
4. Numerical results and discussions The numerical calculation of A@, A) and &I, h) are performed with a computer and some of these results are given in table I. Some graphs of A@, A) are shown, together with the graph of p(d, p = 0)/p(m), in fig. 1. One can see that for all values of Fuchs parameter p in the interval 0
0
.t
Fig. 1. The graphs of A@, A) and the graph of p(p = 0, d)/p(m).
100 A it was observed that at room temperature the resistivity anisotropy Ap does not differ from the bulk value, but the resistivity changes considerably [S--7]. Moreover, if the experiment is performed at liquid nitrogen temperature with the same films, the increase
results of A@, h) and B@, h) p=o Ati,
0.01 0.02 0.04 0.06 0.10 0.20 0.40 0.60 1.00 2.00 10.00
I *0*
6.52 4.33 2.99 2.46 1.98 1.54 1.28 1.19 1.11 1.05 1.01
p = 0.05 A)
HP, A)
Ati,
0.246 0.283 0.329 0.363 0.413 0.498 0.602 0.670 0.757 0.862 0.974
6.21 4.16 2.89 2.39 1.93 1.52 1.27 1.18 1.10 1.05 1.01
A)
p = 0.20 B@, h) 0.252 0.291 0.339 0.374 0.426 0.513 0.618 0.685 0.770 0.870 0.975
A@, 5.35 3.65 2.60 2.18 1.79 1.44 1.23 1.16 1.09 1.04 1.01
a)
p = 0.60 B@, V
A@,
0.212 0.315 0.371 0.410 0.468 0.560 0.666 0.730 0.807 0.892 0.979
3.30 2.41 1.85 1.62 1.42 1.23 1.12 1.08 1.05 1.02 1.00
h)
HP, A) 0.344 0.406 0.484 0.536 0.609 0.711 0.806 0.854 0.903 0.948 0.990
342
Vu dinh KylResistivity anisotropy in ferromagnetic
metallic films
Therefore, for films with such small thicknesses one may not assume that B is constant and, consequently, it is incorrect to use the relation Ap/p = BM2 to define M as was done in ref. 10. Therefore, one must be careful with conclusions about M in thin films when these are derived from indirect measurements by this method, although these conclusions may be correct for thick films (d 9 I).
Acknowledgements
Fig. 2. The graphs of B(p, A).
of Ap becomes significant [S] . This fact can be explained by the decrease of h due to the increase of the electron mean free path at lower temperatures, so that A@, h) becomes larger (see fig. 1). Some graphs of B(p, X) are shown in fig. 2. We see that B(p, A) decreases with decreasing h for all values of p in the interval 0 < p < 1. For films with thicknesses such that one can neglect the dependence of the magnetization on the thickness, B(p, A) shows the same behaviour as Ap/p. A decrease of (Ap/p) similar to B(p, A) in fig. 2 was observed [5] at room temperature as well as at liquid nitrogen temperature. B(p, A) depends strongly on X for small values of X.
The author wishes to express his thanks to Dr. R. Gersdorf for his program for the computer calculations. The very kind hospitality of the Natuurkundig Laboratorium of the Universiteit van Amsterdam to the author is gratefully acknowledged.
References [l] [2] [3] [4]
L. Berger, Physica 30 (1961) 1141. J. Kondo, Progr. Theor. Phys. 27 (1962) 772. Vu dinh Ky, Phys. stat. sol. 15 (1966) 739. P. Suda, Phys. stat. sol. 2 (1962) 1227. [ 5] Vu dinh Ky, Phys. stat sol. 26 (1968) 565. [6] E. Mitchell et al, .I appl. Phys. 35 (1964) 2604. [ 71 T. T. Chen and V. A. Marsocci, Physica 59 (1972) 498. [ 81 B. Carpuat et al., Rev. phys. Appl. 4 (1969) 533. [9] R. F. Greene, Solid State Surface Science, M. Green,. ed. vol. 1 (Marcel Dekker, New York, 1969). [lo] W. HellenthaJl, Z. Phys. 151 (1958) 421.
THEORY OF THE RESISTIVITY ANISOTROPY IN FERROMAGNETIC METALLIC FILMS VU DINH KY Polytechnical Natuurkundig
University of Hanoi, Hanoi Laboratorium, Universiteit van Amsterdam,
Amsterdam,
The Netherlands
Received 27 May 1975 Revised 30 September 1975
An explicit expression for the Ap effect of a thin ferromagnetic film is given, using results obtained earlier by the author and the Fuchs-Chambers method. The theoretical results are compared with the experimental results. The qualitative agreement is satisfactory.
1. Introduction
boundary conditions are treated by the FuchsChambers method and the film surface is characterized by a Fuchs parameter p. In this report the results of the solution of the kinetic equation in ferromagnetic metals obtained in a previous paper [3] and the Fuchs-Chambers method are used to suggest a phenomenological approach, in which both the size-effect and the spinorbit coupling are considered. The numerical results obtained from the general formula are used to explain some experimental data.
It is well known that the resistivlty of the ferromagnetic metals depends on the orientation of the magnetization. This anisotropy of the resistivity which can be characterized by the difference Ap = p ,, - pI between the resistivities when the crystals are magnetized to saturation in the directions parallel and perpendicular to the current, is due to the spin-orbit interaction in ferromagnetic metals [l-3]. In ferromagnetic metallic films the Ap-effect was investigated intensively during the last years (see, for example ref. 4 and references given there and refs. 5-7). This effect was not only used to investigate the magnetic characteristics of the films [4], but possibly could also be applied in memory techniques [8]. However, the theoretical considerations needed to explain the properties of the Ap anisotropy in ferromagnetic films were lacking, until now. The main difficulty is the solution of the kinetic equation in which the spin-orbit interaction and the boundary conditions must be considered together. The spinorbit coupling for the current carriers in ferromagnetic metals was investigated in ref. 3 to obtain the expression of the Ap anisotropy. On the other hand, th: size-effect due to the boundary conditions in nonferromagnetic films was investigated in numerous papers (see, for example ref. 9 and the references given there). Usually, for nonmagnetic films the
2. The solution
of the kinetic equation
Let us consider an unidomaln ferromagnetic and isotropic crystal with planar geometry and the current in the direction of the x-axis in this plane. The z-axis is perpendicular to the sample plane. According to the results of ref. 3 for bulk metals, the solution of the kinetic equation for the cases that M II J and M 1 J can be written as
afoII f/l(k)=-$*(E.k)z
7 91 ( k),
(1)
where f:y’(/c)is the nonequilibrium part of the distribution function for the current carriers, and it 339
= ~~(1 + +M23;),
(24
340
Vu dinh Ky/Resistivity
71(k) = To( 1 + -)4@ {j),
anisotropy in ferromagnetic
x
1_
(1 - p)exp(-z + bVT”(k)4 1 - p exp(-d/T”(k)u,)
d/2
(2b)
r. is the isotropic part of the relaxation time; yll, r1 are small coefficients proportional to the second order in the intensity of the spin-orbit coupling (y”M2, +M2 < 1). M is the magnetization of the sample. The expressions for ro, y” and 9 can be written explicitly [3] , but they are not needed in the following calculation. { is the unit vector in the direction of the wave vector k. The formulae (2) show that, if using the relaxation time approximation, one can interpret the anisotropy of 7(k) as being a consequence of the spin-orbit coupling. Now we can use eqs. (1) and (2) to obtain the solution of the kinetic equation for the ferromagnetic films. Let us consider only the interval of thicknesses where the size-quantization effect can be neglected and the electron spectrum in the film is not different from the spectrum in the bulk material. Then only the surface conditions must be considered in the calculation. As in ref. 5, it is assumed that the electron scattering at the surfaces of the film gives no contribution to the anisotropy of the relaxation time. Then using the conventional Fuchs-Chambers method [lo] the solution of the kinetic equation for the ferromagnetic films can be obtained. For example in the case that M IIJ and u, > 0 we have
1’
(3)
where d is the thickness of the film and p is the Fuchs parameter of the film surfaces. For the other cases we have analogous expressions. Because in films the magnetization and, consequently, the intensity of the spin-orbit coupling depend on the film thickness, the magnetization must be taken as M(d), but not M(m) in the bulk samples.
metallic films
d3kf;‘?*(k,
uz, z)
(4)
one obtains the resistivity anisotropy Ap = Au/o~(~), where Au(d) = u$, - u& and uo(d) is the isotropic part of the conductivity in the film with the thickness d. Because y”JM2 Q 1 one can use the approximative expansion in the integral of (4), for example
ew@l(l -___ 1 - exp(a/(l
+ 6))
exp a l+ S)) x 1 - exp a (
a6
1 - expa 1 ’
6 < 1.
(5)
Substituting finally obtain
(2) and (3) into (4) and using (5) we
(64 (6b)
(74 0)
A&
A)=
1 -&(l
-P)+(l
-PY
(
+1,,5x4
i pn-’ n=l
1
+ E,(nA)(&n4h3 - &n6A5) , I)
(84 A2(P, X)= 1 ,1-&l
-p)+(l
-?9ZnzI
3. The resistivity anisotropy Substituting the expression of the solution of the kinetic equation into the formula of the conductivity
* ux,
5
3__-n____+-
8h
8
n2h 16
n3h2 16
?+
Vu dinh KyfResistivity
+ E, (nh)
anisotropy in ferromagnetic
metallic films
341
&z2h- i’sn4j\3)
113
El(x)
=
7G dt. X
In these formulae the parameter h is defined usually by the relation X = d/Z where 1 is the mean free path of the electrons in the bulk material. Ap(m) and Ap(d) are the resistivity anisotropy in bulk material and film, respectively. Ap/p = (p” - $)/PO, p0 is the isotropic part of the resistivity.
4. Numerical results and discussions The numerical calculation of A@, A) and &I, h) are performed with a computer and some of these results are given in table I. Some graphs of A@, A) are shown, together with the graph of p(d, p = 0)/p(m), in fig. 1. One can see that for all values of Fuchs parameter p in the interval 0
0
.t
Fig. 1. The graphs of A@, A) and the graph of p(p = 0, d)/p(m).
100 A it was observed that at room temperature the resistivity anisotropy Ap does not differ from the bulk value, but the resistivity changes considerably [S--7]. Moreover, if the experiment is performed at liquid nitrogen temperature with the same films, the increase
results of A@, h) and B@, h) p=o Ati,
0.01 0.02 0.04 0.06 0.10 0.20 0.40 0.60 1.00 2.00 10.00
I *0*
6.52 4.33 2.99 2.46 1.98 1.54 1.28 1.19 1.11 1.05 1.01
p = 0.05 A)
HP, A)
Ati,
0.246 0.283 0.329 0.363 0.413 0.498 0.602 0.670 0.757 0.862 0.974
6.21 4.16 2.89 2.39 1.93 1.52 1.27 1.18 1.10 1.05 1.01
A)
p = 0.20 B@, h) 0.252 0.291 0.339 0.374 0.426 0.513 0.618 0.685 0.770 0.870 0.975
A@, 5.35 3.65 2.60 2.18 1.79 1.44 1.23 1.16 1.09 1.04 1.01
a)
p = 0.60 B@, V
A@,
0.212 0.315 0.371 0.410 0.468 0.560 0.666 0.730 0.807 0.892 0.979
3.30 2.41 1.85 1.62 1.42 1.23 1.12 1.08 1.05 1.02 1.00
h)
HP, A) 0.344 0.406 0.484 0.536 0.609 0.711 0.806 0.854 0.903 0.948 0.990
342
Vu dinh KylResistivity anisotropy in ferromagnetic
metallic films
Therefore, for films with such small thicknesses one may not assume that B is constant and, consequently, it is incorrect to use the relation Ap/p = BM2 to define M as was done in ref. 10. Therefore, one must be careful with conclusions about M in thin films when these are derived from indirect measurements by this method, although these conclusions may be correct for thick films (d 9 I).
Acknowledgements
Fig. 2. The graphs of B(p, A).
of Ap becomes significant [S] . This fact can be explained by the decrease of h due to the increase of the electron mean free path at lower temperatures, so that A@, h) becomes larger (see fig. 1). Some graphs of B(p, X) are shown in fig. 2. We see that B(p, A) decreases with decreasing h for all values of p in the interval 0 < p < 1. For films with thicknesses such that one can neglect the dependence of the magnetization on the thickness, B(p, A) shows the same behaviour as Ap/p. A decrease of (Ap/p) similar to B(p, A) in fig. 2 was observed [5] at room temperature as well as at liquid nitrogen temperature. B(p, A) depends strongly on X for small values of X.
The author wishes to express his thanks to Dr. R. Gersdorf for his program for the computer calculations. The very kind hospitality of the Natuurkundig Laboratorium of the Universiteit van Amsterdam to the author is gratefully acknowledged.
References [l] [2] [3] [4]
L. Berger, Physica 30 (1961) 1141. J. Kondo, Progr. Theor. Phys. 27 (1962) 772. Vu dinh Ky, Phys. stat. sol. 15 (1966) 739. P. Suda, Phys. stat. sol. 2 (1962) 1227. [ 5] Vu dinh Ky, Phys. stat sol. 26 (1968) 565. [6] E. Mitchell et al, .I appl. Phys. 35 (1964) 2604. [ 71 T. T. Chen and V. A. Marsocci, Physica 59 (1972) 498. [ 81 B. Carpuat et al., Rev. phys. Appl. 4 (1969) 533. [9] R. F. Greene, Solid State Surface Science, M. Green,. ed. vol. 1 (Marcel Dekker, New York, 1969). [lo] W. HellenthaJl, Z. Phys. 151 (1958) 421.