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Electric transport perpendicular to the planes

H. C. Herper1, 2

1

Theoretische Tieftemperaturphysik, Universität Duisburg–Essen, Duisburg Campus,

47048 Duisburg, Germany

2

Center for Computational Materials Science, TU Vienna, 1060 Vienna, Austria

Received 8 March 2006, revised 15 May 2006, accepted 11 July 2006

Published online 23 August 2006

PACS 71.15.–m, 75.47.De, 75.70.Cn

Since the discovery of the giant magnetoresistance (GMR) in magnetic multilayers, several theoretical de-

scriptions have been used to determine the resistivity of such layered structures. The resistance for the

current in direction of the planes of layers can easily be measured, and has been intensively studied theo-

retically. However, the investigation of the GMR for the current perpendicular to the planes (CPP) is

slightly more difficult. Here, a microscopic formalism for the study of the CPP GMR is reported by mak-

ing use of the Kubo–Greenwood equation. Within this method perturbations of the interfaces like inter-

diffusion, alloy formation, or impurities can easily be included, which is of importance, because the dis-

cussion of the GMR is always related to the structure of the interfaces. The presentation of the Kubo–

Greenwood formalism for CPP transport is complemented by a brief discussion of some exemplary results.

phys. stat. sol. (b) 243, No. 11, 2632–2642 (2006) / DOI 10.1002/pssb.200642107

phys. stat. sol. (b) 243, No. 11, 2632–2642 (2006) / DOI 10.1002/pssb.200642107

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Review Article

Electric transport perpendicular to the planes

H. C. Herper*, 1, 2

1 Theoretische Tieftemperaturphysik, Universität Duisburg–Essen, Duisburg Campus,

47048 Duisburg, Germany 2 Center for Computational Materials Science, TU Vienna, 1060 Vienna, Austria

Received 8 March 2006, revised 15 May 2006, accepted 11 July 2006

Published online 23 August 2006

PACS 71.15.–m, 75.47.De, 75.70.Cn

Since the discovery of the giant magnetoresistance (GMR) in magnetic multilayers, several theoretical de-

scriptions have been used to determine the resistivity of such layered structures. The resistance for the

current in direction of the planes of layers can easily be measured, and has been intensively studied theo-

retically. However, the investigation of the GMR for the current perpendicular to the planes (CPP) is

slightly more difficult. Here, a microscopic formalism for the study of the CPP GMR is reported by mak-

ing use of the Kubo–Greenwood equation. Within this method perturbations of the interfaces like inter-

diffusion, alloy formation, or impurities can easily be included, which is of importance, because the dis-

cussion of the GMR is always related to the structure of the interfaces. The presentation of the Kubo–

Greenwood formalism for CPP transport is complemented by a brief discussion of some exemplary results.

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Contents

1 Introduction 2 CPP transport from the experimental point of view 3 Electric transport within the Kubo–Greenwood formalism

3.1 Kubo–Greenwood equation

3.2 Application to layered structures 4 Some examples 5 Summary References

1 Introduction

The behavior of two ferromagnetic (FM) layers separated by a nonmagnetic (NM) material strongly

depends on the magnetic configuration of the two magnets. Usually, the resistance of such a sandwiched

system decreases if a magnetic field is applied and the magnetic moments of the two magnets are aligned

* e-mail: [email protected], Phone: + 492033793564, Fax: + 492033793665

phys. stat. sol. (b) 243, No. 11 (2006) 2633

www.pss-b.com © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Review

Article

in parallel. This effect is known as giant magnetoresistance (GMR) and has intensively been studied

during the last fifteen years [1–6]. The GMR is quite different from the natural magnetoresistance ef-

fect, i.e., the anisotropic magnetoresistance (AMR). The AMR is a small, relativistic effect, which can be

observed in every ferromagnetic metal. It originates from the orbital momentum of the conduction elec-

trons and the Lorentz force, that results in a resistance, which depends on the relative orientation of mag-

netization and current [7, 8]. Usually, the resistance measured parallel to the magnetization direction is

larger as compared to the resistance perpendicular to the magnetization direction of the film. The AMR

effect has been used in read heads of hard disk drives [9]. Today the AMR has been mostly replaced by

the GMR where the resistance differences depends on the relative orientation of the magnetic moments

in the two FM layers. Accordingly, the GMR is no elemental effect. It occurs only in layered systems in

which an external magnetic field can change the magnetic configuration of the FM layers, or, in other

words, if the coupling between the layers is not too strong or even ferromagnetic.

The fact that the magnetic configuration in such materials can easily be switched by a magnetic field

makes them interesting for technological applications. Today, the GMR is used in read heads of hard

disk drives and for magnetic data storing – MRAM technology [10, 11]. It should be mentioned that for

technical purposes sometimes systems with semi-conducting spacers are preferred, which means making

use of the TMR effect [6].

As mentioned above, the GMR occurs if a magnetic field forces the magnetic moments of neighbored

FM layers in parallel, which changes the scattering of the conduction electrons [8]. The actual properties

of electric transport depend on the composition of the multilayer, but they are also influenced by the

preparation technique and temperature [12], because these parameters affect the structure of the

interface, e.g., the occurrence of interdiffusion, alloy formation, and roughness. Although the GMR is

determined by the relative orientation of the FM layers its size is also a function of the thickness of the

NM layer. Closely related to these topics is the question of the interlayer exchange coupling in such

systems, because it has been shown that AF coupling supports the occurence of spin-dependent transport

and gives possibilities for practical applications [13]. Besides, the size of the GMR in a layered structure

also depends on the experimental setup. The common way to measure a resistance in a multilayer is

given by the current in plane (CIP) geometry, which is used in present GMR devices. However, future

devices may demand a higher recording density or suitability for high frequency application [14]. Re-

lated to this is the idea of GMR devices with current perpendicular to the plane (CPP) geometry. The

CPP GMR effect may be realized in devices for high density recording. However, there are still some

technical problems concerning the appearence of superparamagnetism on the nanometer scale [15].

Today, several preparation techniques exist which make use of the CPP GMR and related phenomena

in magnetic multilayers, see Ref. [16, 17]. Nonetheless, there is still no real consense what the main

cause of the GMR is: The connection between the electronic band-structure and the magnetic moments

of the layers or the spin-dependence of the single-site scattering potentials [18]. Further questions con-

cern the constitution of the particular system: How does interdiffusion influence the size of the GMR and

how are GMR and interlayer exchange coupling related? The calculational approaches used to investi-

gate the resistances range from the semi-classical Boltzmann theory [7] to quantum-mechanical descrip-

tions, like the Kubo–Greenwood equation [18–20]. Furthermore, the Landauer-Büttiker method has

been established for ballistic CPP transport and the TMR effect in tunnel-junctions [21, 22]. The Lan-

dauer-Büttiker method describes transport properties on a mesoscopic scale, which means that the results

depend on the size of the sample. Usually, the resistance is calculated far away from the FM/NM inter-

faces. Therefore, a description of effects near the interfaces is somewhat difficult to include. Recently,

some progress concerning the handling of disorder within the Landauer approach has been achieved [23,

24], where disorder is described by using large lateral super cells and the coherent potential approxima-

tion (CPA). However, interdiffusion and disorder can be described quite easily within the Kubo–

Greenwood approach. Alloy formation, impurities and other effects can be included in the calculation,

which is important for the comparison with measurements and interpretation of experimental findings. A

detailed description of the Kubo–Greenwood equation and its application to layered structures is pre-

sented in Section 3. Section 2 gives an overview of relevant experimental results and techniques. Finally,

2634 H. C. Herper: Electric transport perpendicular to the planes

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

some results concerning CPP transport in Fe/T/Fe with (T = Si, Cr, V) sandwiches is discussed in Sec-

tion 4 in order to demonstrate the advantages of this approach.

2 CPP transport from the experimental point of view

In this section some fundamental experimental results concerning the GMR effect are presented. The

GMR effect can be measured between a FM and any disordered magnetic state of the multilayer. Usu-

ally, the GMR ratio is defined as the change of the resistance by a magnetic field H (FM solution) rela-

tive to the resistance of the zero field state, R0,

0 H

exp

0

R RR

R

-

= . (1)

Today, some authors use the saturation state instead of the zero field state, because in this case the con-

tribution from spin disorder is minimized [8]. In theoretical calculations it is convenient to assume that

the zero field state corresponds to a perfect AF arrangement of the magnetic moments of the leads, where

effects of spin disorder are neglected. The GMR is then defined by the difference between the parallel

(FM) and anti-parallel (AF) solution, whereby R0 is replaced by RFM:

AF FM

AF1

R RR R

R

-= , £ . (2)

Here, the difference of the resistances is divided by the value for the anti-ferromagnetic configuration AF

R , which gives a bounded solution provided that AF FMR R> (so-called pessimistic version of the

GMR). It is also common to use an unbounded solution dividing by FMR instead of by AF

R .

One typical (basic) experimental result for the GMR is shown in Fig. 1. The size of the GMR oscil-

lates with the thickness of the NM layer. Very distinct oscillations have been observed for Co/Cu multi-

layers by Mosca et al. (Fig. 1). It has been argued that these oscillations are related to the interlayer ex-

change coupling of the two leads [5]. Furthermore, the GMR of a multilayered structure depends on the

number of bilayers, that is the repetition of the two components. In case of Fe/Cr, it has been observed

that the size of the CPP GMR strongly increases with growing number of repetitions N, whereas the CIP

GMR is not much affected by N [26].

0 10 20 30 40

Cu layer thickness, t(Å)

0

20

40

60

80

100

GM

R (

%)

T = 4.2 KT = 300 K

30*[Co(15Å)/Cu(tÅ)]

AF

AF

AF

FM FM

Fig. 1 Measured GMR of Co/Cu multilayers with varying thickness of the Cu layers, t. The oscillations

can be related to the interlayer exchange coupling of the Co layers. Data are taken from [25].

phys. stat. sol. (b) 243, No. 11 (2006) 2635


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3 Electric transport within the Kubo–Greenwood formalism

3.1 Kubo–Greenwood equation

In order to determine the resistance of a layered system described in the previous section, we make use

of Kubo’s method [19]. The Kubo formula describes the response of a quantum mechanical system to an

external field, for instance the change of the electric current due to an electric field. Assuming a small

perturbation gives a linear relation between the field and the current. This means the Hamiltonian of the

original system is replaced by

0

ˆ ˆ ˆ= + ,¢H H H (3)

where 0

ˆH is the Hamiltonian of the unperturbated system and ˆ

¢H describes the perturbation. The pertur-

bation is caused by the periodic electric field

( )

0 ei i tω δ- +

= ,E E (4)

leading to

ˆ = -¢H pE (5)

with the electric dipole-moment p. The present choice of E ensures that the perturbation vanishes for

t = - • and 0δ Æ . In linear response theory the electric current at point ¢r is related to the electric field

at r by the two-point conductivity tensor ( ) ,σ , ¢r r

3( ) d ( ) ( )r= , .¢ ¢ ¢Új r r r E rs (6)

In the case of static, homogeneous electric fields (i.e., in the zero frequency limit) the Kubo–Greenwood

approach yields for the conductivity [19, 27, 28]

0

1( ) lim ( , )

ω

ω

ωÆ

, = - , ,¢ ¢r r r rs P (7)

with ( )ω, ,¢r rP being the current–current correlation function

0

ˆ ˆ( ) d e [ ( ) ( )]it t

ωτ

ω τ τ⟨ ⟩

•

, , = , + , , .¢ ¢Úr r j r j rP (8)

Here, ⟨ ⟩ denotes the expectation value over all states of the system at zero temperature and ˆ( )t,j r is the

quantum mechanical current operator

0 0ˆ ˆ †eˆ ˆ ˆ( ) e ( ) e ( ) ( ) ( )

iH t iH tt

miψ ψ

-

, = , = .r

j r j r j r r r— (9)

Finally, Eq. (8) leads to the Kubo–Greenwood equation for the electric conductivity

F F

( ) ( )mn nm m n

mn

j jV

µ ν

µν

πσ δ ε ε δ ε ε

= - - ,Â (10)

with µ and ν denoting the Cartesian indices x y z, , , and Fε being the Fermi energy [19, 27]. V is the

product of the number of atoms and the atomic volume. The elements of the current operator are given

by

mnj m j nµ µ

⟨ ⟩= , (11)



where m⟩ is an eigenstate of the particular configuration and jµ is the µ th component of the current

operator. The expression in Eq. (9) only holds for NM metals. This problem can be overcome by intro-

ducing spin-dependent current densities, see for example Refs. [20, 29]. Charge rearrangement is auto-

matically taken into account by the self-consistent calculation of the electronic structure (this, however,

is different from the charge accumulation effects arising from nonlinear, non-equilibrium current effects,

which are not described by the linear response theory [30]).

3.2 Application to layered structures

The general expressions discussed in Section 3.1 can be simplified assuming a layered structure which

provides two-dimensional translational symmetry. Suppose the layers grow along the z-axis and the

fields in the (x– )y -plane are homogeneous, then Eq. (6) reduces to

( ) d ( ) ( )z z z z zσ= , .¢ ¢ ¢Új E (12)

Usually transport measurements are carried out for current in the (x–y)-plane (CIP) or perpendicular to

the plane (CPP) geometry. In the case of CIP geometry, the calculation of the in-plane component of the

conductivity matrix σ is straightforward, because the electric field in z-direction is constant. This leads

to

( ) d ( )z E z z zσ = , .¢ ¢Új (13)

Until now the current is a microscopic quantity. The measured current corresponds to the expectation

value of the current divided by the system size, which is here the length of the system in growth direc-

tion, L,

( ) d d ( )CIP

Ej z z z z z E

Lσ σ⟨ ⟩ = , = .¢ ¢Ú (14)

The situation becomes more complex if we apply the electric field perpendicular to the planes. In this

case the electric field in z-direction ( )E z is no longer constant. However, it will be shown that the trans-

port in CPP geometry can be handled similarly to the CIP case assuming steady state conditions [8]. In

order to discuss perpendicular transport, it is more convenient to use the inverse of Eq. (12),

( ) d ( ) ( )E z z z z j zρ^

= , .¢ ¢ ¢Ú (15)

In the steady state the electronic density is time-independent, 0tρ∂ /∂ = . Therefore, the continuity equa-

tion

0j

t t z

ρ ρ∂ ∂ ∂+ — = + =

∂ ∂ ∂j (16)

can only be fulfilled if the current is constant. In this case, Eq. (15) can be rewritten as

( ) d ( )E z j z z zρ^

= , .¢ ¢Ú (17)

The microscopic quantities in Eq. (17) can now be used to express the measured electric field, which is

the average of ( )E z divided by the system size L

( )CPP

d d , ,j

E z z z z jL

ρ ρ⟨ ⟩^

= =¢ ¢Ú (18)

where CPP

ρ corresponds to the measured resistivity.

phys. stat. sol. (b) 243, No. 11 (2006) 2637


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The last step is to determine the resistivity or sheet resistance of a layered system, which will be dis-

cussed in the following section. Since the Kubo–Greenwood formula [Eq. (10)] provided the conductiv-

ity instead of the resistivity, we must solve the following integral equation in order to obtain the resistiv-

ity CPP

ρ [Eq. (18)] of the system

( )d , ( , ) ( ) ,z z z z z z zσ ρ δ^ ^

= -¢¢ ¢¢ ¢¢ ¢ ¢Ú (19)

which means inverting the conductivity matrix

1( ) ( ( ))z z z zρ σ-

^ ^, = , .¢ ¢ (20)

3.3 Sheet resistance

In our calculations we use the so called sheet resistance which is product of resistivity and system length

CPP

d d ( )r L z z z zρ ρ^

= = , ,¢ ¢Ú (21)

where z and z ¢ are continuous variables. The purpose of the following transformation is to map the con-

ductivity tensor ( )z zσ , ¢ on a discrete expression for layered systems zz

pqσ

¢ , with p and q denoting planes

of atoms,

( ) ( )zz

pqz z nσ σ→

¢

, ,¢ (22)

where n is the total number of layers taken into account. Due to the symmetry of the problem, the map-

ping concerns only the z-components, the index z is suppressed in the following discussion. The mapping

should conserve the formal structure of the problem, which is equivalent to

d ( ) ( ) ( ) ( ) ( )pr rq pq

z z z z z z z n nσ ρ δ σ ρ δ→, , = - = .¢¢ ¢¢ ¢¢ ¢ ¢ ÂÚ (23)

The necessary condition for the transformation in Eq. (23) is that the Cauchy convergence criterion is

fulfilled, which corresponds to

lim ( )n

r r n n Nη+

Æ•

- < , Œ , (24)

with η being an infinitesimal small number. In this case, the integral in Eq. (21) can be replaced by the

sum over the planes of atoms

1

( ) ( )

n

pq

p q

r n nρ

, =

= .Â (25)

If Eqs. (22)–(24) are fulfilled, then the sheet resistance in CPP geometry of a layered system with a

given magnetic configuration can be derived from Eq. (25). However, it should be kept in mind that the

actual quantity of interest in electric transport is the change of the ressistance due to an applied magnetic

field, see Section 1. Therefore, it has to be taken into account that the sheet resistance also depends on

the magnetic configurations FM, AFC = of the leads. The two configuration considered in the calcula-

tions are displayed in Fig. 1. There, the leads consist of semi-infinite systems, i.e.; bulk potentials or

vacuum, which cover the spacer layers [18]. The FM configuration agrees with the experimental solution

for an applied magnetic field, and the AF case corresponds to the zero field situation, see Section 2. Due

to technical resons the k-space integration and the handling of the surface Green’s function, in particular

the Kubo–Greenwood equation, is evaluated for a small but finite imaginary part of the Fermi energy

F FE iε δ= - . The actual sheet resistance for a magnetic configuration C and a fixed system length n is

given in the limit 0 ,δ Æ

0 0

, 1

( , ) lim ( , , ) lim ( , , ) .

n

pq

p q

r C n r C n C nδ δ

δ ρ δÆ Æ

=

= = Â (26)



Accordingly, using the definition of the GMR from Eq. (2) within the Kubo–Greenwood formalism, the

CPP GMR is given by

(AF, ) (FM, )

( ) .(AF, )

r n r nR n

r n

-

= (27)

In addition to Eq. (26), a layer-resolved sheet resistance can be defined by

1

( , ) ( , ) .

n

p q

q

r C n C nρ

=

=Â (28)

This expression can be used for the dicussion of the resistance, see also Section 4. The current operators,

which enter the conductivity matrix, can be obtained from multiple scattering methods using Green’s

functions. In the Kubo formalism the electric conductivity contains different vertex corrections occur, if

we replace the impurity averages of the two particle propagators by the product of impurity averanged

one particle propagators [6]. The former type is included in the calculation by using the following in-

version process of the conductivity matrix. The inversion is made for the electric currrent at point z,

which can be expressed by the resistivity ( ),z zρ ¢ [Eq. (15)] and the electric current j at a different place

z ¢ . Assuming steady state conditions the current is independent of z ¢ [Eq. (17)]; this type of vertex is

included because of current conservation. The second type of vertex corrections is more difficult to de-

termine for layered structures and has not been included so far. There exist a few calculations by

Weinberger et al. which allow to estimate that these vertex corrections may be quite small [18, 31].

There is a way to avoid the problem of vertex corrections namely a real space approach in which disorder

is treated by averaging over random contributions. This type of approach is computationally costly, al-

though there exist some implementations within tight-binding models [32]. Two further aspects should

be noticed when calculating the CPP GMR using Eq. (27). First, charge accumulation effects may occur

at the bulk-spacer interface. In order to partially account for this effect within the Kubo–Greenwood

formalism a number of buffer layers is included in the part of the multilayer, which is calculated self-

consistently (dark regions in Fig. 2). It can be shown that, for a sufficient large number of lead layers and

a given value of δ , the sheet resistance depends linearly on n. For 0 ,δ Æ the sheet resistance is then

independent of the number of lead layers, for details see [33]. A typical example is displayed in Fig. 3a for

a 10 4 10

Fe /V /Fe (110) trilayer and and 1 5δ = . mRy.

Second, it is obvious from Eq. (26) that the sheet resistance is a function of the imaginary part of the

Fermi energy. However, the sheet resistance basically varies linearly with δ provided the number of

buffer (lead) layers is sufficiently large (Fig. 3b). Therefore, the actual sheet resistance can be obtained

from calculations for finite δ [33].

AF

z

FM

ub

lk

ub

lk

spacerbuffer buffer

Fig. 2 Schematic drawing of a trilayer. The NM spacer is covered by two semi-infinitite FM leads

(bulk), where some additional lead layers are used as buffer layers in the self-consistent calculation (dark

part). The two magnetic configurations used for the calculation of the GMR are marked by the arrows.

phys. stat. sol. (b) 243, No. 11 (2006) 2639


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0 1 2 3 4δ, imaginary part of εF

12

16

20

24

r(C

,δ)

(10-1

5Ω

m2 )

FMAF

0 2 4 6 8 10n, number of Fe layers

4

8

12

16

20

24r(

C,δ

) (1

0-15

Ωm

2 ) δ = 1.5 mRy (a) (b)

Fig. 3 Variation of the sheet resistance ( , , )r C n δ with the number of Fe lead layers n (a) and with the

imaginary part of the Fermi energy ( )bδ for a 10 4 10

Fe V Fe triayer. In both cases, the results are shown for

the AF (triangles) and FM (circles) alignment of the leads.

4 Some examples

In this section some applications of the Kubo–Greenwood approach for CPP transport in layered struc-

tures (see Section 3.2) are presented. All numerical input, which is necessary for the calculation of the

sheet resistance Eq. (25), has been obtained from the fully-relativistic spin-polarized version of the

screened Korringa–Kohn–Rostoker (SKKR) method for layered systems developed by Weinberger and

Szunyogh [35, 36]. Technical details concerning the calculation of the electric and magnetic properties

can be found in [33, 37]. Basically, the GMR is calculated for different numbers of NM spacer layers. A

typical example is shown in Fig. 4a which displays the GMR of Fe/Cr /Fen

as a function of

( )Cr 4 42 .n£ £ The GMR decreases with increasing number of Cr layers from 28% for 4 monolayers to

5% for the largest Cr layer thickness. However, the decrease is accompanied by oscillations, which de-

pend on the number of Cr layers. In order to understand the oscillations we have calculated the energy

difference between the FM and AF state, (AF) (FM)E E ED = - in dependnce of n. This has been done

by applying the magnetic force theorem [35, 37], i.e. only the FM reference state is calculated self-

consistently. The results are shown in Fig. 4b. The oscillations of the interlayer coupling have been com-

pared by checking whether GMR (n) < GMR (n + 1) or vice versa holds. In the first case, the value of the

oscillation for the (n + 1)-layer system is set to 1, whereas in the second case it is chosen to be 0. A simi-

lar procedure has been used for the interlayer exchange coupling choosing 0 for AF coupling and –1 for

FM coupling, respectively. Figure 4c shows that there is indeed a connection between the oscillations of

the GMR and the short, two-monolayer period of oscillation in the exchange coupling of Fe/Cr/Fe. The

local minima of the GMR correspond to FM coupling and the maxima to AF coupling, respectively. In

addition, the phase slips occurring every 15–16 monolayers are also reflected in the GMR, but they are

slightly shifted and smeared out.

These results demonstrate that the Kubo–Greenwood formalism is a powerful tool to describe the

electric transport properties in CPP geometry. Furthermore, this method can be used to examine systems

with some configurational disorder, which may occur at the interfaces by interdiffusion. Disorder is then

conveniently taken into account by using the CPA, which was shown to be of sufficient accuracy for

layered systems by Weinberger et al. [18]. Take for example, Fe/Si, in which interdiffusion effects are

important. This system tends to build CsCl-like FeSi alloys near the interfaces [38]. In our calculations

we have considered an interdiffusion region that is restricted to the direct interface [37]

1 1 2 1 1

. . . Fe Fe Si Fe Si Si Fe Si Fe Si Fe . . .c c c c n c c c c- - - - -

/ / / / / /



0 4 8 12 16 20 24 28 32 36 40 440

5

10

15

20

25

30

GM

R (

%)

0 4 8 12 16 20 24 28 32 36 40 44-20

-10

0

10

20

IEC

(m

eV)

0 4 8 12 16 20 24 28 32 36 40 44

n, number of Cr layers

-1

0

1

osci

llatio

n

(a)

(b)

(c)GMR

IEC

The GMR obtained from Eq. (2) for different interdiffusion concentrations c is shown in Fig. 5. The

calculations have been performed for three systems with 6, 9 and 12 monolayers of Si taking into ac-

count a two-layer interdiffusion. For c = 0, the GMR is extremely large (50%), which has not been ob-

served in any experiment. However, independent from the number of spacer layers, the GMR immedi-

ately breaks down, if one allows for alloy formation at the interface (Fig. 5). If the inerdiffusion reaches

20%, the GMR is already reduced to a fifth of the value for c = 0. It is known from the literature, that a

0 5 0 5Si Fe. .

alloy is quite likely to occur at the Fe/Si interfaces [38, 39]. The experimentally determined

GMR of Fe/Si systems lies below 2.2% [40, 41]. The present calculations for c = 0.5 (Fig. 5) are there-

0 0.1 0.2 0.3 0.4 0.5c, interdiffusion concentration

0

10

20

30

40

50

GM

R (

%)

n = 6 n = 9n = 12

...Fe/Fe1-cSic/Fe

cSi1-c/Si

n-2/FecSi1-c/Fe1-cSi

c/Fe...

2.19- 2.87 %

Fig. 4 (a) GMR of Fe/Crn/Fe trilayers depending

on the number of Cr layers n. (b) Interlayer exchange

coupling (IEC) for Fe/Crn/Fe obtained from the band

energy difference vs. the number of spacer layers n.

(c) Comparison of the oscillations of the GMR and

the IEC of the trilayers shown above, for details see

text. Data are taken from Ref. [34].

Fig. 5 GMR for Fe/Si /Fen

trilayers vs. the interdifu-

sion concentration c for different spacer thicknesses n.

Data are taken from Ref. [37].

phys. stat. sol. (b) 243, No. 11 (2006) 2641


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0

0.4

0.8

0

0.4

0.8

∆rp (

c) /

∆rto

t(c)

I II III IV V0

0.4

0.8

c = 0.00

c= 0.10

c= 0.20

fore in a good agreement with the experimental findings, which allows the conclusion that the small

GMR of Fe/Si multi- or trilayers is related to the formation of Fe/Si alloys at the interfaces.

The GMR obtained from Eqs. (25) and (27) only provides information for the whole trilayer. In order

to check which layers or parts contribute mostly to the GMR, it is helpful to use sheet resistance frac-

tions. These fractions can be obtained from Eq. (28) by investigating the differences of the layer-resolved

sheet resistances. Therefore, the system is split into characteristic regions s: The leads (region I and V),

the spacer (III), and the interfaces (II and IV). The sheet resistance fraction is then defined by

tot

1

( (AF, ) (FM, ) )( )

, .( )

(AF, ) (FM, )

s s

ss

n

p p

p

r n r n

r n

s n

r n

r n r n

=

-D

= <D

-

Â

Â (29)

A typical example is shown in Fig. 6 for the case of 12 6 12

Fe Si Fe/ / . It turns out that the main contribution

of the sheet resistance fraction, and therefore of the GMR, stems from the interfaces. Even for c = 0 there

still exists a reasonably large contribution (20%) from the spacer. With increasing interdiffusion, the

latter becomes smaller and vanishes for c = 0.2. The contributions from the leads are negligible.

5 Summary

In this work I have shown that the GMR values for CPP geometry can reliably be calculated using the

Kubo–Greenwood formalism instead of the Landauer–Büttiker method. Although we use equilibrium

Green’s functions and steady state conditions, we expect that aspects of charge accumulation are mini-

mized by allowing for smooth transition from the spacer layer region to the lead (bulk) region. The width

of the transition region, in which changes of magnetization are taken into account by electronic structure

calculations, may be considered as an additional parameter. A condition for the number of layers belong-

ing to this region is the dependence of the sheet resistance on the number of buffer layers. For a suffi-

ciently large transition region the sheet resistance becomes constant.

Acknowledgment This work has been partially funded by the RT-Network Computational Magnetoelectronics and the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 491 Magnetic Heterostructures: Structure and Electronic Transport. The author would like to thank P. Weinberger for his kind support.

Fig. 6 Normalized fractions of the layer-resolved sheet resistance differences

prD for characteristic regions p of

12 6 12Fe /Si /Fe . In the upper panel, the results for the system with ideal interfaces are shown. The results for interface alloying are displayed in the middle and bottom panel. Roman numbers mark particular regions of the system: I left lead, II left lead interface, III spacer, IV right interface, and V right lead.



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