Theory of spindependent tunneling and transport in magnetic nanostructures (invited)S. Maekawa, J. Inoue, and H. Itoh Citation: Journal of Applied Physics 79, 4730 (1996); doi: 10.1063/1.361654 View online: http://dx.doi.org/10.1063/1.361654 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/79/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin-dependent transport in antiferromagnetic tunnel junctions Appl. Phys. Lett. 105, 122403 (2014); 10.1063/1.4896291 Spin-dependent transport in II-VI magnetic semiconductor resonant tunneling diode J. Appl. Phys. 110, 034303 (2011); 10.1063/1.3610442 Effect of Rashba spin-orbit coupling on the spin-dependent transport in magnetic tunnel junctions withsemiconductor interlayers J. Appl. Phys. 107, 103722 (2010); 10.1063/1.3415532 Role of interface bonding in spin-dependent tunneling (invited) J. Appl. Phys. 97, 10C910 (2005); 10.1063/1.1851415 Spindependent interface transmission and reflection in magnetic multilayers (invited) J. Appl. Phys. 79, 5805 (1996); 10.1063/1.362195

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

130.113.111.210 On: Sat, 20 Dec 2014 07:08:20

Theory of spin-dependent tunneling and transport in magneticnanostructures (invited)

S. Maekawa, J. Inoue, and H. ItohDepartment of Applied Physics, Nagoya University, Nagoya 464-01, Japan

We study the linear response theory of the electric transport in magnetic nanostructures. Theconductance is expressed by using the wave functions of electrons on the Fermi level in the meanfield theory at zero temperature. The theory is extended to the system with single resistive layer inwhich there exists electron–electron interaction and the conductance is derived in the single-siteapproximation. The theoretical results may be applied to a variety of magnetic nanostructures suchas magnetic multilayers and ferromagnetic tunneling junctions. ©1996 American Institute ofPhysics.@S0021-8979~96!50308-4#

I. INTRODUCTION

Spin-dependent transport phenomena have receivedmuch attention since the discovery of the giant magnetore-sistance~GMR! in magnetic multilayers.1 GMR was also ob-served in granular alloys.2,3 In addition, there is growing in-terest in the study of magnetoresistance devices of magneticnanostructures. These activities renewed the study of ferro-magnetic metal/insulator/ferromagnetic metal tunnelingjunctions which were the first of the observation of magne-toresistance in the magnetic nanostructures.4,5 The magne-toresistance in the tunneling junctions is called the tunnelingmagnetoresistance~TMR!. Recent experiments have shownthat TMR is as large as GMR in magnetic multilayers.6,7Wenote that TMR is also seen in magnetic granularsemiconductors.8,9

Although interesting magnetoresistance phenomena havebeen observed in magnetic nanostructures, the theoreticalstudy is far from complete. This is partly because it is noteasy to take into account the nanostructure and random scat-tering potentials on an equal footing.

In this article, we formulate the conductance in magneticnanostructures. Starting with the Kubo formula for the con-ductance, we find that the conductance depends only on thevoltage at leads attached on the sample and does not dependon the detailed electric field distribution in it in the meanfield theory at zero temperature, indicating that the theory forthe nonlocal conductivity by Kane, Serota, and Lee10 is validin the magnetic systems as well. We extend this result andexamine the transport in a system with single resistive layerin which there exists electron–electron interaction. In thesingle-site coherent potential approximation~CPA!, we ob-tain a formula for the conductance due to electron–electroninteraction.

Since the results obtained in this article are not limited ina particular model, they may be applied to a variety of mag-netic nanostructures.

II. LINEAR RESPONSE THEORY

Let us first consider a magnetic multilayer at zero tem-perature. The thickness of each layer is of the order of 10 Å.The mean free path of an electron, which is the relaxationtime of momentum3 Fermi velocity, is estimated to be ofthe order of 100 Å from the value of the resistivity. Another

important length scale is the spin-diffusion length, which isthe relaxation time of spin memory3 Fermi velocity. Al-though it is not easy to estimate it, this length is consideredto be much longer than the mean free path. The above esti-mate indicates that the spin memory of an electron is con-served during the propagation through many layers. In otherwords, since each spin component of electrons contributes tothe electric current almost independently, the so-called two-current model11 works well. In the following, we study theconductance in the two-current model.

As seen in Fig. 1, the electric current is in thez directionand the total current (I ) is given by integrating over the crosssectional area in thexy plane by

I5E dS~z!E dr 8(n

szn~r ,r 8!En~r 8!, ~1!

E~r 8!52¹8V~r 8!, ~2!

with E andV being the electric field and electro-static po-tential, respectively. Here, the total current is independent ofthe positionz because of the current conservation. The non-local conductivity10 is given by the commutator of the cur-rent operator,

smn~r ,r 8!5 limv→0

1

v E0

`

dt exp ~ ivt !^@ j m~r ,t !, j n~r 8,t !#&,

~3!

j ~r !5e\

2im$c†~r !¹c~r !2@¹c†~r !#c~r !%, ~4!

where^•••& denotes the expectation value in the ground stateande andm are the electric charge and the effective mass ofan electron, respectively. The field operatorc~r ! is expressed

FIG. 1. Sample and leads for measuring conductance.

4730 J. Appl. Phys. 79 (8), 15 April 1996 0021-8979/96/79(8)/4730/3/$10.00 © 1996 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

130.113.111.210 On: Sat, 20 Dec 2014 07:08:20

in the mean field theory by using the wave-functionfns~r !and annihilation operatorcns in the eigen-staten with spinsas

c~r !5(n

(s

fns~r !cns . ~5!

Inserting Eq.~5! into Eq. ~4! and rewriting Eq.~3!, we have

smn~r ,r 8!5p\e2(m,n

(s

Jnmsm ~r !Jmns

n ~r 8!

3d~eF2ens!d~eF2ems!, ~6!

Jnmsm ~r !5

\

2im Ffns* ~r !]

]xmfms~r !

2S ]

]xmfns* ~r ! Dfms~r !G , ~7!

whereeF andens are the Fermi energy and eigen-value in thestaten with spin s, respectively. We note that the wave-function satisfies the Schro¨dinger equation,

S 2\2

2m¹21Us~r ! Dfns~r !5ensfns~r !, ~8!

whereUs~r ! is the spin-dependent potential. As seen in Eq.~6!, the conductivity is given by electrons on the Fermi levelat zero temperature.

In one-dimension, examining the derivative,dJnms/dz,we find that the nonlocal conductivity is constant;s(z,z8)5s0. Then, we obtain

I5s0V, ~9!

whereV is the voltage between leads.In three-dimension, examining the derivative,

(m]Jnmsm ~r !/]xm , we find

(m

]

]xmsmn~r ,r 8!5(

n

]

]xn8smn~r ,r 8!50. ~10!

Therefore, integrating Eq.~1! by parts, changing the integralover the volume to that of the boundary of the sample andnoting that the current flows out of the leads but not of in-sulating boundary, we obtain

I5VE dxdyE dx8dy8szz~r ,r 8!, ~11!

where the total current depends on neitherz nor z8. As seenin Eq. ~11!, the total current is independent of the detaileddistribution of the electric fieldE in the sample but dependsjust on the voltage between leads as also shown in one-dimension.

The conductance in Eq.~11! is given by the wave-functions of electrons on the Fermi level, which depend onthe nanostructure as well as magnetic structure in the sample.Once the geometry of the sample and the magnetic structureare given, the conductance is calculated. We note that thesample may include an insulating layer. Thus, Eq.~11! isapplied to tunneling junctions as well.

III. EFFECTS OF ELECTRON–ELECTRONINTERACTION

In Sec. II, the conductance has been derived in the meanfield theory. In this section, effects of the electron–electroninteraction on the conductance is examined. For this purpose,we set up a model: we consider the tight binding model in asimple cubic lattice which contains a single resistive layer atz50 as shown in Fig. 2. Thez axis is chosen to be parallel tothe ~001! axis of simple cubic structure. In the layer, thereexists the electron–electron interaction on each site given bythe Hamiltonian,

H85U(lnl↑nl↓ , ~12!

whereU.0 andnls is the number operator with spins atsite l in the layer.

We calculate the self-energy with spins, Ss~eF1i0!, inthe single site coherent potential~or Hubbard! approximation~CPA!. In the approximation,Ss~eF1i0! is finite in the layer,independent of the momentum in thexy plane and diagonalin the spin space. Thus, the effective Hamiltonian may bewritten as

H52t(~ i , j !

(s

cis† cjs1(

l(s

Sscls† cls , ~13!

where t is the hopping parameter between nearest neighborsites. Using the effective Hamiltonian@Eq. ~13!#, the trans-mission and reflection coefficients of electrons are calculatedin the usual way12 as

Tcohs 5

4t2 sin2 uk~2t sin uk1 iSs!~2t sin uk2 iSs* !

, ~14!

Rcohs 5

SsSs*

~2t sin uk1 iSs!~2t sin uk2 iSs* !, ~15!

wherek5(kx ,ky) is the wave number in thexy plane andukis defined as22t cosuk5eF12t~coskx1cosky!. Here,Tcoh

s

andRcohs are the coherent part of transmission and reflection

coefficients, respectively. Since there exists the imaginarypart in the self-energy in the resistive layer, electrons arecaught in the layer and then diffuse from it. Such diffusivemotion of electrons is incoherent. When the incoherent trans-mission and reflection coefficients are written asTin

s andRins ,

respectively, the current conservation is written as

Tcohs 1Rcoh

s 1Tins 1Rin

s 51. ~16!

FIG. 2. Model of single resistive layer. Shaded layer denotes resistive layer.

4731J. Appl. Phys., Vol. 79, No. 8, 15 April 1996 Maekawa, Inoue, and Itoh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

130.113.111.210 On: Sat, 20 Dec 2014 07:08:20

Following Stone and Lee,13 we assume that electrons caughtin the layer diffuse incoherently into the both sides equallyand have

Tins 5Rin

s 5 12~12Tcoh

s 2Rcohs !. ~17!

Therefore, using the Landauer formula, we obtain the con-ductance as

szz5e2

h (k

(s

~Tcohs 1Tin

s !, ~18!

Tins 5

i t sin uk~Ss2Ss* !

~2t sin uk1 iSs!~2t sin uk2 iSs* !. ~19!

We can show that Eq.~18! is also derived from the Kuboformula @Eq. ~1!# in the single site CPA~Ref. 12! withoutintroducing the above assumption for the incoherent motionsof electrons. In this case, the self-energy and vertex correc-tion must be calculated consistently to satisfy the currentconservation. When the single site CPA is used in the systemwith random scattering potentials, the elastic scattering isreplaced by the inelastic one and the phase coherence ofelectrons is destroyed. However, it has been shown12 that theconductance@Eq. ~18!# agrees with that given by the numeri-cal simulation with any strength of random scattering poten-tials. Therefore, the CPA may be used for the calculation ofthe conductance.

IV. DISCUSSION

We have studied the conductance in magnetic nanostruc-tures. When the electron–electron interaction is neglected,the conductance depends only on the voltage between leads.Therefore, it is calculated once the nanostructure and mag-netic structure are given in the sample. Although it is noteasy to obtain the analytical solution of the conductance, thenumerical simulation has been done extensively in a varietyof the nanostructures.14–16

In the tunneling junctions, the insulating barrier is usu-ally made by oxides so that the energy gap is caused by theelectron correlation. We have introduced the electron–electron interaction in the single resistive layer and obtainedthe conductance due to the interaction. The application ofEq. ~18! to the realistic junctions will be presented sepa-rately.

Finally, the theory obtained in this article is limited inthe two-current model. Although the model works well atlow temperatures, the spin-flip scattering will not be ne-glected as temperature increases. Effects of the scatteringwill also be studied separately.

ACKNOWLEDGMENTS

This work has been supported by Priority-Areas Grantsfrom the Ministry of Education, Science and Culture of Ja-pan, and the New Energy and Industrial Technology Devel-opment Organization~NEDO!.

1M. N. Baibich, J. M. Broto, A. Fert, Nguyen Van Dau, F. Petroff, P.Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett.61,2472 ~1988!.

2A. E. Berkowitz, J. R. Mitchell, M. J. Carey, A. P. Young, S. Zhang, F. E.Spada, F. P. Parker, A. Hutten, and G. Thomas, Phys. Rev. Lett.68, 3745~1992!.

3J. Q. Xiao, J. S. Jiang, and C. L. Chien, Phys. Rev. Lett.69, 3220~1992!.4M. Julliere, Phys. Lett. A54, 225 ~1975!.5S. Maekawa and U. Gefvert, IEEE Trans. Magn.MAG-18, 707 ~1982!.6T. Miyazaki and N. Tezuka, J. Magn. Magn. Mater.139, L231 ~1995!.7J. S. Mooderaet al., Phys. Rev. Lett.74, 3273~1995!.8H. Fujimori, S. Mitani, and S. Ohnume, Mater. Sci. Eng. B31, 219~1995!; S. Mitani et al. ~to be published!.

9J. Inoue and S. Maekawa~to be published!.10C. L. Kane, R. A. Serota, and P. A. Lee, Phys. Rev. B37, 6701~1988!.11P. M. Levy, Solid State Phys.47, 367 ~1994!.12H. Itoh, J. Inoue, and S. Maekawa~to be published!.13A. D. Stone and P. A. Lee, Phys. Rev. Lett.54, 1196~1985!.14A. Oguri, Y. Asano, and S. Maekawa, J. Phys. Soc. Jpn.61, 2652~1992!.15Y. Asano, A. Oguri, and S. Maekawa, Phys. Rev. B48, 6192~1993!.16Y. Asano, A. Oguri, J. Inoue, and S. Maekawa, Phys. Rev. B49, 12831

~1994!.

4732 J. Appl. Phys., Vol. 79, No. 8, 15 April 1996 Maekawa, Inoue, and Itoh [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

130.113.111.210 On: Sat, 20 Dec 2014 07:08:20