- Email: [email protected]
Perpendicular transport through rough interfaces in the metallic regime
~
Pergamon
Solid-State ElectronicsVol. 37, Nos 4-6, pp. 1239-1242, 1994 0038-1101(93)E0054-5
Copyright ~ 1994ElsevierScienceLtd Printed in Great Britain.All rights reserved 0038-1101/94 $6.00+0.00
PERPENDICULAR TRANSPORT THROUGH ROUGH INTERFACES IN THE METALLIC REGIME ARNE BRATAAS'I"and GERRITE. W. BAUER Faculty of Applied Physics and DIMES, Delft University of Technology, Lorentzweg I, 2628 CJ Delft, The Netherlands Almtraet--Perpendicular transport through an interface in the metallic regime is considered. The semiclassical theory presented is based on the Landauer-Bfittiker formalism taking into account an effective mass mismatch at the interface and a non-zero average of random scattering potentials. The transmission probability for a given mode is found in terms of the effective mass and the conduction band bottom to the left and to the right of the interface, the Fermi energy, the self-energyof the electron, and the transverse wave vector of the electron. The diffuse and specular contributions to the interface roughness scattering are shown to be equally important in the weak scattering limit. Predictions for the transport properties of interfaces with a low concentration of strongly scattering defects should be accessible to verification by experiments. The theory is applied to the spin-valve effect in magnetic multilayers.
The importance of interface scattering in many areas of metal and semiconductor physics is reflected by numerous papers since the seminal work by Fuchs[1]. The overwhelming majority of the work is concerned with transport parallel to an impenetrable rough interface, e.g. in the two-dimensional electron gas[2]. Here, a theory of interface scattering based on the Landauer-Biittiker formalism[3] is presented which is concerned with transport normal to the interfaces in the metallic regime. The relation between diffuse and specular scattering at an interface is usually described by introducing a factor p which is determined empirically or derived from a microscopic model of the interface roughness[I,4]. We will show that diffuse scattering is uniquely connected to the so-called vertex correction. Our approach gives a simple relation between the specular and diffuse part of the transmitted wave and a simple formula for the specularity parameter. The conductance for a system of multiple interfaces can be found by a semiclassical concatenation of single interface transmission matrices. The theory is applied to find the magnetoconductance in magnetic multilayers, where spin-dependent interface scattering is generally believed to be responsible for the "giant" magnetoconductance or spin-valve effect[5-12]. The magnetoconductance is well described by a simple formula in terms of the mean free number of transversed interfaces for the majority and minority spin electrons. The mathematical details are presented in Ref. [13]. Here, these results are generalised to include the effect of effective mass mismatch tPermanent address: University of Trondheim, The Norwegian Institute of Technology, Faculty of Physics and Mathematics, Division of Physics, N-7034 Trondheim, Norway. SSE 37,4,~-SS
at the interface and non-vanishing average of the random scattering potential. Let us first consider scattering at a single interface. The interface roughness is modeled by short-range scatterers that are randomly distributed over the interface with density nta. The incoming and outgoing electron states are taken to be Bloch waves and are approximated by plane waves. Heterostructures are described by means of spin-dependent potential steps and effective masses. The wave function at an energy E is determined by the Schr6dinger equation: I
fiV
q
1 m---z~V + Uc(X) + v(x, y, z)j
xJ/(x,y,z)=E~b(x,y,z).
(!)
The conduction band profile, Uc(x), and the effective mass of the electron m*(x), are simply step functions at x = 0. The interface roughness gives rise to the potential V(x, y, z). By integrating out the transverse coordinates (y and z), a one-dimensional equation is obtained[14]. We assume that the particles are incident from the left with transverse momentum k~. To the left of the interface the longitudinal part of the wave function consists of the incoming and reflected waves and to the right side of the interface there are only transmitted right-going waves. For propagating modes, the longitudinal wave function is:
~k#.k i
qn ( x )
1239
=
+
~ / h Ik,RI "~"'~' ~
Xl/~ m~ I r'"'~',o-~k~ ~ x I> 0
x
~0 (2)
1240
ARNE BRATAASand GERRITE. W. BAUER 1.0
where k~ and k± are the transverse and longitudinal components of the wave vector and
k k = [2rn'~/hZ(Ev - Ut) - , ,k~~ ] ,l , 2 . k ~ = [2m ~ /h :( E F - UR ) -- k ~]L~2. Note that we have normalised the longitudinal parts of the wave function, ck, (x), so that they carry unit flux. The transmission (reflection) coefficient from state k I to state kll is tL,:.L~(rLk,.k.~).Evanescent modes correspond to complex longitudinal wave vectors, which should be treated as in Ref. [15]. For low temperatures, the conductance can now be found from the Landauer-Biittiker formalism[3]: 2e 2 prop.
G=--ff ~ ItL.L;,I2,
(3)
k,L::
where the summation is over propagating modes. In matrix notation the transmission coefficients are determined by the equation[14] (1 + F)t = A where Aki!,kl ' =
6k,,,ki ,
~ / ~ ±
eL.L: =E,
and:
r~ * y = ~ e'x" h2 1~± ~ ~
1
e - a,i~,, ,v/-A'
(4)
with rh* = ~ and ~'l = (rnRk± , L + rn~k~)/2rn*. This result generalises eqn (4) in Ref. [13] for the case of different effective masses to the left and to the right of the interface. Current conservation and the conti-
k Rt ±k
±
Pt
0.5
\
\ \ \ \ \ X \ \
0
7~
2 Fig. 1. Fraction of electrons transmitted specularly as a function of incoming angle to the normal of the interface. The specularity factor is shown for r/,R = 0.05 and r/tR = 0.5 (dotted line). terms in the expansion and as a tool to understand the nature of the approximations ([13] and references therein). The diagrams contributing to the transmission probability can be classified as crossing or non-crossing. The crossing diagrams describe phase coherence and are, e.g., responsible for weak (Anderson) localisation of the wave function[! 6]. Neglecting crossing diagrams, i.e. phase coherent scattering between different defects, it is possible to find the transmission probability in terms of the irreducible self-energy Z[13]:
k~
(ItL"Li'Iz)=fL'~X I~z + i -_~T. -"-~ ~ + i - h TZth* 2 h2 Im(Z) ILIII ~
~c:[
k'~
. 1 m/~ ,Cl-
"
E',
r~-~*I::
(6)
k"
nuity of the wave function relate the transmission probabilities and the transmission coefficients as:
kr,k
_
/Ik'lLI Re(tL, k )
- X] [k~RI
:
(5)
where the maximum transverse wave-vectors corresponding to propagating modes are k ~ m~ and k [.,,~x. The transmission probabilities of present interest are given by the ensemble average of all impurity configurations, which cannot be treated exactly. We have chosen a perturbation approach. The relevant perturbation series are obtained by inverting the matrix relation for t, expanding (1 + F)-~ as a power series in F and by taking the configurational average. Green functions can be introduced to identify the
Q u a n t u m interference between different scattering centers is neglected. The approximation can therefore be labeled as semiclassical and breaks down when the scattering becomes too strong. Electrons are scattered specularly at the interface if the transverse component of the wave vector is conserved, which is the first term on the right hand side of eqn (6). The second term clearly represents the diffuse scattering contribution, which vanishes if the vertex correction is not taken into account. In the following we set A U c = 0 and Am* = 0 , which considerably simplifies the analytical treatment. It is interesting to make contact with the traditional treatment of interface roughness in terms of the specularity factor p. The factor Pt defined as the fraction of electrons transmitted specularly is found to be: Pt (0)2 =
cos(0) rt~R+ cos(0)'
(7)
1241
Rough interfaces in the metallic regime where r / m = - m * I m ( ~ ) / h 2 k F is a scattering parameter and 0 is the angle of incidence of an electron wave vector measured from the surface normal. This expression is exact for semiclassical transport and the specularity factor is independent of the real part of the self-energy. Diffuse scattering increases for larger incoming angles as shown in Fig. I. Electrons with wave vector perpendicular to the interface are transmitted mainly specularly, and electrons with wave vector parallel to the interface plane are completely diffused. A similar relation can be derived for reflected electrons. To lowest order in the self-energy m*~-/h2kF the conductance is:
1
0.1
~
0.Ol
0.o01
0.0001
\
n nlanntl
t
n LJnaH]
10
n ~H,i
100
n I ltlllll
1000
10,000
N/~ 2e2 Ak~ [l - 4r/IR + 2r/m], G = h--T 4n
(8)
and is independent of the real part of the self-energy. The first term is the Sharvin conductance which is proportional to the sample cross section A. The second term reduces the conductance due to specular scattering. The third, diffuse term increases the conductance by opening additional channels for electron transport. The self-energy is:
.m*nm 2 -i~ T-B = niR ~Y -- Z "-h-T"~- y kF[l
l],
(9)
in the Born approximation, where the average scattering strength : = ~ 7 J n m A and the mean square value of the scattering strength y : = Z,y~/nmA are introduced. Here a ultraviolet divergence in the summation over intermediate evanescent states has been cut-off at a wave-vector ~tkF to account for the finite range of the potential. ~t >/l, but not much larger than unity for magnetic multilayers, since the range can not be shorter than a d orbital radius. The Born approximation is valid in the weak scattering limit, ]m*~,a/h2ke[ ,~1 and m*TkF/nh2~ 1, which means that the probability of strong scattering by a single impurity is small. The scattering parameter is in this approximation ~/m = nla 72/~. The specularity factor is not affected by f in the weak scattering limit, a non-zero average of scatterers has also no effect on the conductance. We will now study the situation where all scatterers have equal magnitude of strength, such that 17~]= 7. The self-energy is calculated in the single-site approximation, i.e. using the exact cross-section for isolated defects but neglecting crossed diagrams which stand for quantum interference between different defects: m* rim: -i--~--~y2kF(I -- i x / ~ -- !)
Bs = l
+~-~(l f m *7kF
_
_
i ~ ) )
2
(10)
I
This result reduces to the Born result in the weak scattering limit (m*ykF/h:Tt ~. 1). For strong scatter-
Fig. 2. Conductance for a metallic multilayer. The dotted 5ne is the conductance where only specular scattering is considered. The solid line shows the conductance where diffuse scattering is included. ing (m*TkF/h27t ~ 1), but to lowest order in n,R, we obtain the interesting result that:
G =-EL ~
~].
Here fm*fk~/h2rl~(m*~kF/h2n) 2 in the present regime has been used. The conductance is reduced by a factor proportional to the number of defects N m = A n m. Each scatterer effectively blocks one channel and the conductance becomes independent on the scattering strength. This blocking is somewhat reduced by a factor l/ct 2 via a "leak" of evanescent states. An experiment is proposed to test this expression: insert a layer (or a multilayer, see below) with strong short-range scatterers between two perfect leads. By measuring the conductance and the number of impurities the theory can be checked and the "leaking factor" ~t can be determined, which provides information about the scattering potential. Results for a single interface can be generalised to a multilayer situation where interface scattering and bulk impurity scattering is taken into account. Semiclassical concatenation of transmission probabilities[14] is consistent with the neglect of crossing diagrams in the single interface scattering. The transmission properties do not change with the distance between the interfaces in this approximation. By allowing the interfaces to be shifted on top of each other, one can convince oneself that the relation between the transmission probabilities and the transmission coefficients, eqn (5), still holds for the N-interfaces configuration. Concatenation of the diagonal transmission coefficients is straightforward and the conductance for transport through N interfaces is: GO - l - 2 ( N ~ m ) + 2 ( N ~ m ) 2 1 n I +
.
(12)
This equation has been verified by comparison with numerical results for the concatenated total transmission probabilities eqn (6). The conductance
1242
ARm~ BRATAASand GERRITE. W. BAUER 1.0
<1
0.5
-
AN=I. 8 K1
#
AN =. 1.3 R
f
AN t?
--
=0.6 I
I
5
10
N/~ Fig. 3. Magnetoconductance for an antiferromagnetically coupled magnetic multilayer. The curves illustrate the effect of spin-dependent mean free number of traversed interfaces.
obtained by neglecting the diffuse scattering in eqn (6) gives a similar result, but the scattering parameter is increased by a factor 2. The diffuse contribution due to the vertex correction is therefore important. The conductance calculated with and without the diffuse part is shown in Fig• 2. A bulk system is modeled by N interfaces with an interface scattering parameter r/2~• Letting N - - * ~ and r /2D m ~ 0 but keeping Nq ~ = Lqm where L is the length of the bulk material and qm is the scattering parameter for the bulk system. The conductance for a multilayer is: GO = 1 - - 2
~-~
+2
~-~
In 1 + - ~ -
, (13)
where 57 is the mean free number of traversed interfaces given by 57 _~[2)hR+2Lqm] -~. This relation agrees with eqn ( I I ) in Ref. [9] for AUc = 0. In the large N limit a Drude-like (Ohm's law) expression is obtained for the conductivity of a thick multilayer: tr~ = Nlira ~ N L G ( ~ / A = "2e2k~ ~ - ~ N L-,
(143
which agrees with the results of Zhang and Levy[8]. The Drude result is approached rather slowly. For a magnetic multilayer it is now straightforward to find the conductance by including spin-dependent interface scattering and bulk scattering. The difference in mean free number of traversed interfaces between both spin channels is A57 and the spin-averaged result is 57. The relative magnetoconductance of an antiferromagnetically coupled multilayer AG/t7 is shown in Fig. 3, where A G = G F - G Av and (~ = (G v + G^V)/2. The relative magnetoconductance depends on the ratios AN/N and N / N . The spin-valve effect increases with the number of bilayers and saturates at the Drude limit for N >> 57 as given by Zhang and Levy[8]. In this limit of a magnetic superlattice the relative magnetoconductance is ( A G / d ) D~d~ = (A57/257) 2.
The effect of the potential steps and different effective masses in the materials on the conductance can be found by concatenation o f e q n (6). This should be done numerically since the expression for the N-layer conductance is very complicated• In summary, we have derived semiclassical expressions for perpendicular transport through disordered interfaces which are exact for the present model. The effect of a different effective mass to the right and to the left of the interface is included in the formalism. In the weak scattering limit the effect of a non-zero average of the random potential on the conductance is found to be of a higher order in the concentration of scatterers n~R and can be neglected in most cases• An experiment to check the theory is proposed which might lead to a deeper understanding of the scattering process and the microscopic structure of disordered interfaces. A simple, semiclassical formula for the giant magnetoconductance of antiferromagnetically coupled magnetic multilayers in terms of the mean free number of traversed interfaces for the majority and minority spins is derived• Acknowledgement--This work is part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie (FOM)", which is financially supported by the "Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)". REFERENCES
1. K. Fuchs, Proc. Phil. Cam. Soc. 34, 100 (19383. 2. T. Ando, A. B. Fowler and F. Stern, Rev. mod. Phys. 54, 437 (1982). 3. R, Landauer, Z. Phys. B 68, 217 (1987). 4. S. B. Softer, J. appl. Phys. 38, 1710 (19673. 5. M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen van Dau and F. Petroff, Phys. Rev. Lett. 21, 2472 (1988); G. Binasch, P. Grfinberg, F. Saurenbach and W. Zinn, Phys. Rev. B 39, 4828 (1989). For a recent review see R. Coehoorn, Europhys. News 24, 43 (19933. 6. W. P. Pratt, S. F. Lee, R. Lolee, P. A. Schroeder and J. Bass, Phys. Rev. Lett. 66, 3060 (19913. 7. M. A. M. Gijs, S. L. J. Lenczowski and J. B. Giesbers, Phys. Rev. Lett. 70, 3343 (1993). 8. S. Zhang and P. M. Levy, J, appl. Phys. 69, 4786 (1991); S. Zhang and P. M. Levy, Phys. Rer. 45, 8689 (1992); H. E. Camblong, S. Zhang and P. M. Levy, Phys. Rev. B 46, 4735 (1993). 9. G. E. W. Bauer, Phys. Rev. Lett. 69, 1676 (1992). The Erratum 70, 1733 (1993) is superseded by the present results. 10. R. Q. Hood and L. M. Falicov, Phys. Ret,. B 46, 8287 (1992). 11. T. Valet and A. Fert, Phys. Rev. B 48, 7099 (19933. 12. Y. Asano, A. Oguri and S. Maekawa, Phys. Rev. B 48, 6192 (19933. 13. A. Brataas and G. E. W. Bauer, preprint. 14. M. Cahay, M. McLennan and S. Datta, Phys. Rev. B 37, 10125 (1988). 15. P. H. Bagwell, Phys. Ret,. B 41, 10354 (1990). 16. P. A. Lee and T. V. Ramakrishnan, Rev. rood. Phys. 57, 2 (1985). 17. J. Inoue, A. Oguri and S. Maekawa, J. Phys. Soc. Japan 60, 376 (1990).
Pergamon
Solid-State ElectronicsVol. 37, Nos 4-6, pp. 1239-1242, 1994 0038-1101(93)E0054-5
Copyright ~ 1994ElsevierScienceLtd Printed in Great Britain.All rights reserved 0038-1101/94 $6.00+0.00
PERPENDICULAR TRANSPORT THROUGH ROUGH INTERFACES IN THE METALLIC REGIME ARNE BRATAAS'I"and GERRITE. W. BAUER Faculty of Applied Physics and DIMES, Delft University of Technology, Lorentzweg I, 2628 CJ Delft, The Netherlands Almtraet--Perpendicular transport through an interface in the metallic regime is considered. The semiclassical theory presented is based on the Landauer-Bfittiker formalism taking into account an effective mass mismatch at the interface and a non-zero average of random scattering potentials. The transmission probability for a given mode is found in terms of the effective mass and the conduction band bottom to the left and to the right of the interface, the Fermi energy, the self-energyof the electron, and the transverse wave vector of the electron. The diffuse and specular contributions to the interface roughness scattering are shown to be equally important in the weak scattering limit. Predictions for the transport properties of interfaces with a low concentration of strongly scattering defects should be accessible to verification by experiments. The theory is applied to the spin-valve effect in magnetic multilayers.
The importance of interface scattering in many areas of metal and semiconductor physics is reflected by numerous papers since the seminal work by Fuchs[1]. The overwhelming majority of the work is concerned with transport parallel to an impenetrable rough interface, e.g. in the two-dimensional electron gas[2]. Here, a theory of interface scattering based on the Landauer-Biittiker formalism[3] is presented which is concerned with transport normal to the interfaces in the metallic regime. The relation between diffuse and specular scattering at an interface is usually described by introducing a factor p which is determined empirically or derived from a microscopic model of the interface roughness[I,4]. We will show that diffuse scattering is uniquely connected to the so-called vertex correction. Our approach gives a simple relation between the specular and diffuse part of the transmitted wave and a simple formula for the specularity parameter. The conductance for a system of multiple interfaces can be found by a semiclassical concatenation of single interface transmission matrices. The theory is applied to find the magnetoconductance in magnetic multilayers, where spin-dependent interface scattering is generally believed to be responsible for the "giant" magnetoconductance or spin-valve effect[5-12]. The magnetoconductance is well described by a simple formula in terms of the mean free number of transversed interfaces for the majority and minority spin electrons. The mathematical details are presented in Ref. [13]. Here, these results are generalised to include the effect of effective mass mismatch tPermanent address: University of Trondheim, The Norwegian Institute of Technology, Faculty of Physics and Mathematics, Division of Physics, N-7034 Trondheim, Norway. SSE 37,4,~-SS
at the interface and non-vanishing average of the random scattering potential. Let us first consider scattering at a single interface. The interface roughness is modeled by short-range scatterers that are randomly distributed over the interface with density nta. The incoming and outgoing electron states are taken to be Bloch waves and are approximated by plane waves. Heterostructures are described by means of spin-dependent potential steps and effective masses. The wave function at an energy E is determined by the Schr6dinger equation: I
fiV
q
1 m---z~V + Uc(X) + v(x, y, z)j
xJ/(x,y,z)=E~b(x,y,z).
(!)
The conduction band profile, Uc(x), and the effective mass of the electron m*(x), are simply step functions at x = 0. The interface roughness gives rise to the potential V(x, y, z). By integrating out the transverse coordinates (y and z), a one-dimensional equation is obtained[14]. We assume that the particles are incident from the left with transverse momentum k~. To the left of the interface the longitudinal part of the wave function consists of the incoming and reflected waves and to the right side of the interface there are only transmitted right-going waves. For propagating modes, the longitudinal wave function is:
~k#.k i
qn ( x )
1239
=
+
~ / h Ik,RI "~"'~' ~
Xl/~ m~ I r'"'~',o-~k~ ~ x I> 0
x
~0 (2)
1240
ARNE BRATAASand GERRITE. W. BAUER 1.0
where k~ and k± are the transverse and longitudinal components of the wave vector and
k k = [2rn'~/hZ(Ev - Ut) - , ,k~~ ] ,l , 2 . k ~ = [2m ~ /h :( E F - UR ) -- k ~]L~2. Note that we have normalised the longitudinal parts of the wave function, ck, (x), so that they carry unit flux. The transmission (reflection) coefficient from state k I to state kll is tL,:.L~(rLk,.k.~).Evanescent modes correspond to complex longitudinal wave vectors, which should be treated as in Ref. [15]. For low temperatures, the conductance can now be found from the Landauer-Biittiker formalism[3]: 2e 2 prop.
G=--ff ~ ItL.L;,I2,
(3)
k,L::
where the summation is over propagating modes. In matrix notation the transmission coefficients are determined by the equation[14] (1 + F)t = A where Aki!,kl ' =
6k,,,ki ,
~ / ~ ±
eL.L: =E,
and:
r~ * y = ~ e'x" h2 1~± ~ ~
1
e - a,i~,, ,v/-A'
(4)
with rh* = ~ and ~'l = (rnRk± , L + rn~k~)/2rn*. This result generalises eqn (4) in Ref. [13] for the case of different effective masses to the left and to the right of the interface. Current conservation and the conti-
k Rt ±k
±
Pt
0.5
\
\ \ \ \ \ X \ \
0
7~
2 Fig. 1. Fraction of electrons transmitted specularly as a function of incoming angle to the normal of the interface. The specularity factor is shown for r/,R = 0.05 and r/tR = 0.5 (dotted line). terms in the expansion and as a tool to understand the nature of the approximations ([13] and references therein). The diagrams contributing to the transmission probability can be classified as crossing or non-crossing. The crossing diagrams describe phase coherence and are, e.g., responsible for weak (Anderson) localisation of the wave function[! 6]. Neglecting crossing diagrams, i.e. phase coherent scattering between different defects, it is possible to find the transmission probability in terms of the irreducible self-energy Z[13]:
k~
(ItL"Li'Iz)=fL'~X I~z + i -_~T. -"-~ ~ + i - h TZth* 2 h2 Im(Z) ILIII ~
~c:[
k'~
. 1 m/~ ,Cl-
"
E',
r~-~*I::
(6)
k"
nuity of the wave function relate the transmission probabilities and the transmission coefficients as:
kr,k
_
/Ik'lLI Re(tL, k )
- X] [k~RI
:
(5)
where the maximum transverse wave-vectors corresponding to propagating modes are k ~ m~ and k [.,,~x. The transmission probabilities of present interest are given by the ensemble average of all impurity configurations, which cannot be treated exactly. We have chosen a perturbation approach. The relevant perturbation series are obtained by inverting the matrix relation for t, expanding (1 + F)-~ as a power series in F and by taking the configurational average. Green functions can be introduced to identify the
Q u a n t u m interference between different scattering centers is neglected. The approximation can therefore be labeled as semiclassical and breaks down when the scattering becomes too strong. Electrons are scattered specularly at the interface if the transverse component of the wave vector is conserved, which is the first term on the right hand side of eqn (6). The second term clearly represents the diffuse scattering contribution, which vanishes if the vertex correction is not taken into account. In the following we set A U c = 0 and Am* = 0 , which considerably simplifies the analytical treatment. It is interesting to make contact with the traditional treatment of interface roughness in terms of the specularity factor p. The factor Pt defined as the fraction of electrons transmitted specularly is found to be: Pt (0)2 =
cos(0) rt~R+ cos(0)'
(7)
1241
Rough interfaces in the metallic regime where r / m = - m * I m ( ~ ) / h 2 k F is a scattering parameter and 0 is the angle of incidence of an electron wave vector measured from the surface normal. This expression is exact for semiclassical transport and the specularity factor is independent of the real part of the self-energy. Diffuse scattering increases for larger incoming angles as shown in Fig. I. Electrons with wave vector perpendicular to the interface are transmitted mainly specularly, and electrons with wave vector parallel to the interface plane are completely diffused. A similar relation can be derived for reflected electrons. To lowest order in the self-energy m*~-/h2kF the conductance is:
1
0.1
~
0.Ol
0.o01
0.0001
\
n nlanntl
t
n LJnaH]
10
n ~H,i
100
n I ltlllll
1000
10,000
N/~ 2e2 Ak~ [l - 4r/IR + 2r/m], G = h--T 4n
(8)
and is independent of the real part of the self-energy. The first term is the Sharvin conductance which is proportional to the sample cross section A. The second term reduces the conductance due to specular scattering. The third, diffuse term increases the conductance by opening additional channels for electron transport. The self-energy is:
.m*nm 2 -i~ T-B = niR ~Y -- Z "-h-T"~- y kF[l
l],
(9)
in the Born approximation, where the average scattering strength : = ~ 7 J n m A and the mean square value of the scattering strength y : = Z,y~/nmA are introduced. Here a ultraviolet divergence in the summation over intermediate evanescent states has been cut-off at a wave-vector ~tkF to account for the finite range of the potential. ~t >/l, but not much larger than unity for magnetic multilayers, since the range can not be shorter than a d orbital radius. The Born approximation is valid in the weak scattering limit, ]m*~,a/h2ke[ ,~1 and m*TkF/nh2~ 1, which means that the probability of strong scattering by a single impurity is small. The scattering parameter is in this approximation ~/m = nla 72/~. The specularity factor is not affected by f in the weak scattering limit, a non-zero average of scatterers has also no effect on the conductance. We will now study the situation where all scatterers have equal magnitude of strength, such that 17~]= 7. The self-energy is calculated in the single-site approximation, i.e. using the exact cross-section for isolated defects but neglecting crossed diagrams which stand for quantum interference between different defects: m* rim: -i--~--~y2kF(I -- i x / ~ -- !)
Bs = l
+~-~(l f m *7kF
_
_
i ~ ) )
2
(10)
I
This result reduces to the Born result in the weak scattering limit (m*ykF/h:Tt ~. 1). For strong scatter-
Fig. 2. Conductance for a metallic multilayer. The dotted 5ne is the conductance where only specular scattering is considered. The solid line shows the conductance where diffuse scattering is included. ing (m*TkF/h27t ~ 1), but to lowest order in n,R, we obtain the interesting result that:
G =-EL ~
~].
Here fm*fk~/h2rl~(m*~kF/h2n) 2 in the present regime has been used. The conductance is reduced by a factor proportional to the number of defects N m = A n m. Each scatterer effectively blocks one channel and the conductance becomes independent on the scattering strength. This blocking is somewhat reduced by a factor l/ct 2 via a "leak" of evanescent states. An experiment is proposed to test this expression: insert a layer (or a multilayer, see below) with strong short-range scatterers between two perfect leads. By measuring the conductance and the number of impurities the theory can be checked and the "leaking factor" ~t can be determined, which provides information about the scattering potential. Results for a single interface can be generalised to a multilayer situation where interface scattering and bulk impurity scattering is taken into account. Semiclassical concatenation of transmission probabilities[14] is consistent with the neglect of crossing diagrams in the single interface scattering. The transmission properties do not change with the distance between the interfaces in this approximation. By allowing the interfaces to be shifted on top of each other, one can convince oneself that the relation between the transmission probabilities and the transmission coefficients, eqn (5), still holds for the N-interfaces configuration. Concatenation of the diagonal transmission coefficients is straightforward and the conductance for transport through N interfaces is: GO - l - 2 ( N ~ m ) + 2 ( N ~ m ) 2 1 n I +
.
(12)
This equation has been verified by comparison with numerical results for the concatenated total transmission probabilities eqn (6). The conductance
1242
ARm~ BRATAASand GERRITE. W. BAUER 1.0
<1
0.5
-
AN=I. 8 K1
#
AN =. 1.3 R
f
AN t?
--
=0.6 I
I
5
10
N/~ Fig. 3. Magnetoconductance for an antiferromagnetically coupled magnetic multilayer. The curves illustrate the effect of spin-dependent mean free number of traversed interfaces.
obtained by neglecting the diffuse scattering in eqn (6) gives a similar result, but the scattering parameter is increased by a factor 2. The diffuse contribution due to the vertex correction is therefore important. The conductance calculated with and without the diffuse part is shown in Fig• 2. A bulk system is modeled by N interfaces with an interface scattering parameter r/2~• Letting N - - * ~ and r /2D m ~ 0 but keeping Nq ~ = Lqm where L is the length of the bulk material and qm is the scattering parameter for the bulk system. The conductance for a multilayer is: GO = 1 - - 2
~-~
+2
~-~
In 1 + - ~ -
, (13)
where 57 is the mean free number of traversed interfaces given by 57 _~[2)hR+2Lqm] -~. This relation agrees with eqn ( I I ) in Ref. [9] for AUc = 0. In the large N limit a Drude-like (Ohm's law) expression is obtained for the conductivity of a thick multilayer: tr~ = Nlira ~ N L G ( ~ / A = "2e2k~ ~ - ~ N L-,
(143
which agrees with the results of Zhang and Levy[8]. The Drude result is approached rather slowly. For a magnetic multilayer it is now straightforward to find the conductance by including spin-dependent interface scattering and bulk scattering. The difference in mean free number of traversed interfaces between both spin channels is A57 and the spin-averaged result is 57. The relative magnetoconductance of an antiferromagnetically coupled multilayer AG/t7 is shown in Fig. 3, where A G = G F - G Av and (~ = (G v + G^V)/2. The relative magnetoconductance depends on the ratios AN/N and N / N . The spin-valve effect increases with the number of bilayers and saturates at the Drude limit for N >> 57 as given by Zhang and Levy[8]. In this limit of a magnetic superlattice the relative magnetoconductance is ( A G / d ) D~d~ = (A57/257) 2.
The effect of the potential steps and different effective masses in the materials on the conductance can be found by concatenation o f e q n (6). This should be done numerically since the expression for the N-layer conductance is very complicated• In summary, we have derived semiclassical expressions for perpendicular transport through disordered interfaces which are exact for the present model. The effect of a different effective mass to the right and to the left of the interface is included in the formalism. In the weak scattering limit the effect of a non-zero average of the random potential on the conductance is found to be of a higher order in the concentration of scatterers n~R and can be neglected in most cases• An experiment to check the theory is proposed which might lead to a deeper understanding of the scattering process and the microscopic structure of disordered interfaces. A simple, semiclassical formula for the giant magnetoconductance of antiferromagnetically coupled magnetic multilayers in terms of the mean free number of traversed interfaces for the majority and minority spins is derived• Acknowledgement--This work is part of the research programme of the "Stichting voor Fundamenteel Onderzoek der Materie (FOM)", which is financially supported by the "Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)". REFERENCES
1. K. Fuchs, Proc. Phil. Cam. Soc. 34, 100 (19383. 2. T. Ando, A. B. Fowler and F. Stern, Rev. mod. Phys. 54, 437 (1982). 3. R, Landauer, Z. Phys. B 68, 217 (1987). 4. S. B. Softer, J. appl. Phys. 38, 1710 (19673. 5. M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen van Dau and F. Petroff, Phys. Rev. Lett. 21, 2472 (1988); G. Binasch, P. Grfinberg, F. Saurenbach and W. Zinn, Phys. Rev. B 39, 4828 (1989). For a recent review see R. Coehoorn, Europhys. News 24, 43 (19933. 6. W. P. Pratt, S. F. Lee, R. Lolee, P. A. Schroeder and J. Bass, Phys. Rev. Lett. 66, 3060 (19913. 7. M. A. M. Gijs, S. L. J. Lenczowski and J. B. Giesbers, Phys. Rev. Lett. 70, 3343 (1993). 8. S. Zhang and P. M. Levy, J, appl. Phys. 69, 4786 (1991); S. Zhang and P. M. Levy, Phys. Rer. 45, 8689 (1992); H. E. Camblong, S. Zhang and P. M. Levy, Phys. Rev. B 46, 4735 (1993). 9. G. E. W. Bauer, Phys. Rev. Lett. 69, 1676 (1992). The Erratum 70, 1733 (1993) is superseded by the present results. 10. R. Q. Hood and L. M. Falicov, Phys. Ret,. B 46, 8287 (1992). 11. T. Valet and A. Fert, Phys. Rev. B 48, 7099 (19933. 12. Y. Asano, A. Oguri and S. Maekawa, Phys. Rev. B 48, 6192 (19933. 13. A. Brataas and G. E. W. Bauer, preprint. 14. M. Cahay, M. McLennan and S. Datta, Phys. Rev. B 37, 10125 (1988). 15. P. H. Bagwell, Phys. Ret,. B 41, 10354 (1990). 16. P. A. Lee and T. V. Ramakrishnan, Rev. rood. Phys. 57, 2 (1985). 17. J. Inoue, A. Oguri and S. Maekawa, J. Phys. Soc. Japan 60, 376 (1990).