spin polarized tunneling in ferromagnetic...

Journal of Magnetism and Magnetic Materials 200 (1999) 248}273

Spin polarized tunneling in ferromagnetic junctions

Jagadeesh S. Moodera!,*, George Mathon"

!Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, 170 Albany st., Cambridge MA 02139, USA"Department of Mathematics, City University, London EC1V OHB, UK

Received 10 March 1999; received in revised form 1 July 1999

Abstract

Spin polarized tunneling studies by Tedrow and Meservey in the early 1970s that showed the spin conservation inelectron tunneling gave rise to the possibility of spin sensitive tunneling between two ferromagnetic (FM) "lms. Julliereput forward a quantitative model (1975) showing that tunneling in FM/I/FM junctions should lead to a largemagnetoresistance (JMR). This conjecture was realized with repeatable results only in 1995, and since then JMR values'30% have been achieved at room temperature. This recent success has led to several fundamental questions regardingthe phenomenon of spin tunneling, besides showing tremendous potential for applications as nonvolatile magneticmemory elements, read head and picotesla "eld sensors. We brie#y review the experimental results and the currenttheoretical understanding of FM}I}FM tunneling: its dependence on bias, temperature and barrier characteristics. Thein#uence of inelastic tunneling processes, metal at the interface and material properties on the JMR is discussed. Earlytheories are reviewed and their relationship to the linear response theory is presented. The future direction, both from thepoint of fundamental physics as well as applications, is also covered. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Tunneling; Magnetoresistance; Spin polarization; Magnetic tunnel junctions

1. Introduction

Tunnel experiments have been of great import-ance in the understanding of the physics of super-conductors and semiconductors as well as leadingto important applications [1,2]. With the greatadvance in tunneling between ferromagnets in1995, we can expect important advances in theunderstanding of spin transport and correspondingadvances in technology [3].

*Corresponding author. Tel.: #1-617-253-5423; fax: #1-617-253-5405.

E-mail addresses: [email protected] (J.S. Moodera),[email protected] (G. Mathon)

2. Experimental part

2.1. Spin-polarized tunneling

Spin-polarized tunneling (SPT), discovered in1970 by Tedrow and Meservey [4,5], laid the foun-dation to a new "eld of research. Meservey andTedrow [6] measured the conduction electron spinpolarization (P) of magnetic metals and com-pounds using the Zeeman-split quasi-particle den-sity of states in a superconductor as the spindetector. Tunneling from a ferromagnetic "lm, withits unequal spin distribution at the Fermi level (E

F),

into such a spin split superconducting Al "lm re-#ects the spin polarization of the tunneling elec-trons coming from the ferromagnet.

0304-8853/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 5 1 5 - 6

Table 1Measured spin polarization values for various magnetic materials

Material Ni Co Fe Ni80

Fe20

Co50

Fe50

Co84

Fe16

Polarization(from Eq. (6))

23% 35% 40% 32% * *

New values 33%! 45%! 44% 48%" 51%" 49%"

!Films grown in MBE system (J.S. Moodera et al., unpublished)."R.J.M. van de Veerdonk, J.S. Moodera, W.J.M. de Jonge, 1997 ICMFS Conf. digests 15th, pp. 74}75.

Fig. 1. Relative conductance (*G/G) versus DC bias forFe}Ge}Co junctions at 4.2 K. *G is the di!erence between thetwo conductance values corresponding to parallel and anti-parallel magnetizations of the two ferromagentic "lms (after Ref.[10]).

Table 1 has a listing of the spin polarization forvarious ferromagnetic materials measured by theabove method. Recently measured values of P arehigher due to improved junction preparation con-ditions including MBE grown samples (Moodera,unpublished). Highly polarized tunneling electronscan also be obtained through a phenomenon calledthe spin-"lter e!ect using magnetic semiconductorssuch as EuS, EuO and EuSe as tunnel barriers[7}9].

2.2. Experiments by Julliere and other work

SPT experiments by Meservey and Tedrowshowed that the conduction electrons in ferromag-netic metals are spin polarized and that the spin isconserved in the tunneling process. Julliere's model[10], for FM}I}FM tunneling, predicts that thetunnel junction magnetoresistance (JMR) is (seetheory section)

JMR"

*R

R"

RA!R

PR

A

"

2P1P

21#P

1P

2

(1)

where P1

and P2

are the spin polarization of thetwo ferromagnetic electrodes, based on SPT resultsand the analysis of Meservey and Tedrow, andR

Aand R

Prepresent the junction resistances when

the two FMs have their magnetizations (M) anti-parallel and parallel, respectively. Note that in theliterature &tunnel magnetoresistance' (TMR) is of-ten expressed using the de"nition: TMR"(*R

P/

RP)"2P

1P2/(1!P

1P2). In this work we quote all

the results in terms of JMR.Julliere [10] made the "rst reported mag-

netoresistance measurement on a FM}I}FMtrilayer junction (using oxidized amorphous Ge

(a-Ge) as the insulator) and interpreted it by statingthat the tunneling current should depend on therelative orientation of the magnetizations of theelectrodes. At that time, spin-dependent tunnelinghad already been invoked in explaining mag-netoresistance observations on granular systems[11]. For Fe}a-Ge}Co tunnel junctions, Julliereobserved a change in the conductance (*G) of 14%in zero bias at 4.2 K, which reduced to negligiblevalues when the DC bias reached 6 meV (Fig. 1),Eq. (1), whereas the expected value is 26% based onPC0"34% and P

F%"44%. The large decrease in

*G with bias was attributed to spin scattering atFM}I interfaces. Later experiments with a-Ge tun-nel barriers, failed to exhibit any spin polarizationof the tunnel current [12,13] so that Julliere's ob-servation of a 14% JMR has yet to be reproduced.

J.S. Moodera, G. Mathon / Journal of Magnetism and Magnetic Materials 200 (1999) 248}273 249

Fig. 2. Schematic of junction geomety for four-point measure-ment.

Nevertheless, the elegant model by Julliere, aswe will see later, turns out to be quite good inpredicting the magnitude of JMR seen in cleanjunctions.

Several other groups have attempted tunnelingbetween ferromagnetic "lms prior to 1995. The "rstexperiment, which showed a JMR of &3% inNi/NiO/Co at 4.2 K, supported by the magnetichysteresis loop of the corresponding FM elec-trodes, was by Maekawa and GaK fvert [14]. Thee!ect decreased rapidly as temperature increased,and a much smaller value was observed at 77 K.Several others have reproduced such e!ects, mainlywith NiO, CoO, GdO

x, and Al

2O

3barriers, and

only small changes were seen* no more than 7%at 4.2 K [15}20]. In recent times, Miyazaki andTezuka [21] have improved the room-temperatureresults to about 2% (see section on geometricale!ects).

2.3. Problems in FM/I/FM tunneling

Tunneling between two ferromagnetic "lms al-though apparently simple, was not successful inproducing high values of JMR at room temper-ature for 20 yrs. Several factors contributed to thefailed attempts by many groups until 1995, andsuccess still can be elusive. The major problems arerelated to: (i) surface roughness of the FM elec-trodes, (ii) tunnel barrier, (iii) interface quality, (iv)FM electrodes and their domain walls are dis-cussed below.

Surface roughness of the "rst FM electrode canlead to dipolar or orange peel coupling between thebottom and top FM electrodes [22] not allowingindependent switching of the magnetization of the"lms. Secondly, growing a thin insulator "lm astunnel barrier over a rough surface is very di$cultdue to nonuniform coverage. The tunnel barrier,such as Al

2O

3, must be thin ((20 As ) and yet pro-

vide a uniform coverage, and hence the necessity forthe bottom electrode to be nearly atomicallysmooth. Magnetic oxides as tunnel barriers lead tozero JMR at room temperature, although smallJMR e!ects were observed at LHe temperature inthe past. Spin scattering by magnetic oxides at theinterface has a similar destructive e!ect on JMR

and even a nonmagnetic metal layer at the interfaceleads to degraded polarization. Presence of do-main walls can lead to coupling of the magneti-zations and reduce the magnetic response of thejunction.

2.4. Successful FM}Al2O3}FM junctions

Large magnetoresistance in FM}I}FM tunneljunctions was "rst observed in 1995 by carefullyaddressing most of the problems mentioned in theprevious section [3]. It was shown that a largeJMR of over 10% could be obtained consistentlyand reproducibly.

The most important part of a junction prepara-tion is the formation of the tunnel barrier. Mag-netic tunnel junctions (MTJ) were prepared inthe authors' laboratory in a high vacuum((10~7 Torr) evaporator system, although in somecases an MBE system was utilized. The resultsobtained were nearly the same for the samplesprepared in the two systems. The preparation pro-cedure will be described below. Most other groupsprepare junctions by sputtering but basically ad-opted the method given below for the barrierformation.

MTJ were prepared in situ by thermal evapor-ation. Cryogenic deposition through shadowmasks is utilized to create a cross-geometry junc-tion structure of area 4}6]10~4 cm2 (Fig. 2). Ingeneral, the "rst FM "lm (an 80 As thick and0.2 mm wide long strip on a Si seed layer) is depos-ited onto a LN

2cooled glass substrate. To create

250 J.S. Moodera, G. Mathon / Journal of Magnetism and Magnetic Materials 200 (1999) 248}273

Fig. 3. Resistance versus applied magnetic "eld fora Co/Al

2O

3/Ni

80Fe

20junction at room temperature and 77 K,

showing JMR values of 20.2 and 27.1%, respectively. The bar-rier is formed by oxidation of a 8 As Al layer (after Ref. [23]).

the tunnel barrier, 8}16 As of Al is deposited over it,which is subsequently oxidized at room temper-ature using an oxygen plasma. Cross strips of thetop FM "lm, 100}200 As thick and 0.2}0.3 mm widewere then deposited as the second electrode. FM"lms were grown in an applied "eld of about100 Oe.

Preparing 72 junctions in each run allowed us togrow simultaneously control junctions and junc-tions with some preparation parameters varied.Tunnel junction resistance (R

J) ranged from

(100 ) to tens of k) depending on the Al "lmthickness, duration of glow discharge and the typeof FM used. Deposition of the seed layer, "rst FM"lm and the Al barrier "lm at cryogenic temper-ature increases the junction yield and stabilitywhereas room temperature deposition also worked,although the yield was slightly lower. Atomic forcemicroscopic observation showed smoother "lms inthe former case.

RJas a function of applied magnetic "eld H for

a Co/Al2O

3/Ni

80F20

junction with 12 As of Al2O

3is shown in Fig. 3, which also displays minimaltemperature dependence of R

J[23]. One can dis-

tinctly see the two stable and well-de"ned resist-ance states as the applied "eld is varied. The JMRseen in this case (de"ned with respect to the peakresistance) are 20.2 and 27.1% at 295 and 77 K,respectively, changing to 27.3% upon cooling to4.2 K. With a CoFe electrode, at 77 K, a JMR of

32% has been observed [24]. The peak value ofR

Jcan be maintained with H turned to zero, lead-

ing to two stable states of resistance at H"0 evenin the absence of a bias, thus giving a nonvolatile,two level memory state.

The RJ

versus H can be understood based onJulliere's model. At high H, with the two FM "lmshaving their M (indicated by the arrows) parallel toH, the tunneling probability is highest and tunnel-ing current is maximum, thereby yielding a low R

J.

Upon reversing H, MN*F%

(with the lower HC) aligns

itself in the new H direction whereas MC0

(withhigher H

C) remains magnetized in the original H di-

rection, creating anti-parallel M. In the antiparallelcon"guration, the tunneling probability and thecurrent are low, resulting in higher R

J. Taking

PC0"35% and P

N*80F%20"45%, Julliere's model

gives a JMR of 27.2%, in very good agreement withthe measured values at low temperatures. Note thatthe JMR values apparently in excess of the Julliere'smodel predictions that have been reported [25]could be due to the fact that they referred to earlier,unoptimized values of P [6]. Julliere model still setsan upper limit for JMR when adopting the updatedpolarization values.

The rotation of the magnetization of one FMwith respect to the other also supports theFM}I}FM tunneling model [26]. At a "eldvalue higher than the H

Cof one electrode, when the

sample is rotated in a magnetic "eld, the M ofthe softer "lm follows the "eld, changing the rela-tive orientation of M in the two FM "lms* switching from parallel to anti-parallel orienta-tion gradually. The periodic variation of R

Jas

a function of angle between the magnetization vec-tors h, showed the expected (cos h/2)2 dependence[27].

JMR e!ects have been investigated with anumber of FM electrodes including Co, CoCr,CoFe, Fe

0.7Pt

0.3and Ni

80Fe

20, and tunnel bar-

riers of Al2O

3, AlN and MgO. With CoFe layers

at both the interfaces, Sousa et al. [28] haverecently reported 27%, one of the highest JMRat room temperature [28,29]. Based on the po-larization of 55% for CoFe (the highest P valueamong transition metal or alloy ferromagnets), onecan expect a JMR is 46% for a good CoFe/Al

2O

3/

CoFe junction.


Fig. 4. Junction magnetoresistance plotted as a function of thethickness of the Al metal overlayer used to form the Al

2O

3barrier in (a) Co/Al

2O

3/Ni

80Fe

20and (b) Co/Al

2O

3/Co

50Fe

50tunnel junctions (after Ref. [34]).

2.5. Barrier properties

The possibility of seeing a high JMR, its stabil-ity, and the bias dependence and the junctionresistance critically depend on the quality ofthe tunnel barrier. The current technique offorming with Al

2O

3barriers, by oxidizing a

thin Al layer deposited on the ferromagnet can alsobe used for other barrier materials such as Mg andTa [3,21,24,26,30}33]. For FM}I}FM tunnelingthe most successful barrier materials have beenuntil now Al

2O

3, AlN and MgO, whereas other

barriers that have been tried are in general non-stoichiometric and/or magnetic [16,20,32,33].These latter barriers can lead to spin memory lossor spin scattering (see section on barrier dopinge!ects).

The optimum Al "lm thickness needed for bar-rier formation was extensively studied [34]. In gen-eral, for a uniform coverage the Al "lm thicknessranged in 7}18 As , depending also on the type ofFM electrodes (Veerdonk et al., to be published)there is a small range of Al thickness that yield thebest JMR for a given oxidation condition. Withthinner Al, the uncovered FM surface is oxidizedduring barrier formation; and with thicker Al, ex-cess Al metal will be left behind unoxidized. Thisnonmagnetic metal at the interface reduces the po-larization [35] and hence the JMR. This is explicit-ly seen in Fig. 4, for two types of junctions. Oneinteresting observation is that even with 4 As Alcoverage a JMR of 10% was observed. Uniformityof the Al

2O

3oxidation and its stoichiometry have

been characterized using RBS [29] and XPS[36,37].

It is customary to evaluate the barrier para-meters using either Simmons' or Brinkman'stunneling theory [38,39]. The e!ective barrierthickness d obtained for good junctions agrees withthe estimated Al

2O

3thickness from a known Al

layer thickness, showing the uniform coverage ofthe barrier Al "lm, and an average barrier heightu6 above 2.5 eV is found [23]. When dealing withFM electrodes such as Ni

80Fe

20, Ni or any of the

half-metallic FM, the I}< characteristics can besigni"cantly in#uenced by band features. Fittingthe I}< data in such junctions can at best givenominal barrier parameters.

Additionally, incomplete coverage of the bottomelectrode with good insulator will provide parallelconductance channel or even a composite barrierthrough the magnetic oxide in the barrier. In thesetwo situations the "tted u6 and d values can bemisleading. Usually good insulators such as Al

2O

3and MgO lead to u6 in the eV range, whereas themagnetic oxides have u6 of fractions of eV. Local#uctuations in the Al

2O

3thickness have been

studied on a nanoscopic scale by mapping the tun-nel current using AFM/STM, and a current histo-gram has been related to the spatial #uctuations ofthe barrier [40].

2.6. Geometrical enhancement of JMR

With junctions having measured resistance (RJ)

greater than several hundred ohms, a JMR as muchas 20.2% in ambient conditions, and up to 32% at77 K was observed. In these types of junctions, thehigh resistance of the FM thin "lm electrodes(&20 ) over the junction area of 4}6]10~4 cm2)does not in#uence R

J, in the cross geometry


Fig. 5. Apparent and corrected magnetoresistance of low-resist-ance junctions (after Refs. [24,42]).

electrode junction (see Fig. 2). However, when theactual junction resistance (R

T) becomes compara-

ble to the resistance of the lead (RL) over the junc-

tion area, then measured RJwill not represent R

T.

Current #ow in such cases becomes extremelynonuniform over the junction area, giving rise tospurious measured resistances [41]. In extremecases, a negative, 4-terminal, DC resistance can beobserved. This was qualitatively attributed to themeasuring geometry artifact. For example, shownin Fig. 5 are the R

Jversus H for junctions with

resistances in three di!erent ranges at room tem-peratures. As one moves from high R

Jtowards

lower values, the apparent JMR percentage in-creases very signi"cantly. In fact, when R

Jwas

slightly negative, it even changed sign with H, act-ing like a "eld switch. The nonvolatile memorye!ect of JMR was reported in all junctions, irre-spective of the magnitude and sign of R

J[24].

Finite element calculations performed assumingOhm's law, showed the limitation when R

Tbe-

comes comparable to RL

[42]. This leads to sig-ni"cant voltage drops in the electrodes and thecurrent #ow becomes largest in the corner wheretwo current leads meet and the apparent junc-tion resistance R

Jbecomes smaller than the

actual junction resistance RT. These e!ects lead

to an in"nite apparent JMR ratio at the pointwhere R

Japproaches zero. For example, shown in

Fig. 5, a corrected value of 15% was found for thejunction with a negative resistance, instead of theapparent value of over 1000%. Similarly, for thedata presented in Ref. [21] for a 1 mm2 area tunneljunction with a 7 m) resistance a corrected mag-netoresistance ratio of 1.2% is obtained, much re-duced from their reported 18% [21]. Later, onmicron sized junctions by the same group [43]observed that the JMR is signi"cantly smaller thantheir earlier reports on similar but macroscopicjunctions.

One necessary condition for application of MTJsis the possibility to form small junctions. Thesehave been fabricated down to micron dimensionsusing optical lithography and to submicron dimen-sions using electron beam lithography [44], withvarious aspect ratios revealing shape anisotropy asa new way to control the magnetoresistive response(Fig. 6) [31]. An upper limit of about 20 k) lm2 tothe resistance-area product of the MRAM cells isset by operational requirements on noise and ac-cess time [45] resulting in an e!ort to developbarrier growth techniques for low resistance junc-tions by various oxidation techniques [31,46,47].

Signi"cant JMR values seen in low RJjunctions

(micron sized) can perhaps be due to a continuousbut rough Al overlayer "lm where (i) the thickerparts of the Al "lm have been enough oxidized toplay only a negligible role in the conduction and (ii)oxidation of the FM below the thinner Al regionshas not yet set in, due to the di!erence of heat offormation between Al

2O

3and transition metal ox-

ides. This can explain the barrier thickness valuessmaller than the deposited Al thickness, reportedby several groups [31,47,48]. Low values of u6 re-ported for Al

2O

3means, part of the conduction can

be assumed to occur through defect states createdby impurities or nonstoichiometry. This is likely to


Fig. 6. Magnetoresistance curves at room temperature for a series of junctions with an identical area (256 lm2) but having di!erentaspect ratio (after Ref. [31]).

Fig. 7. Dynamic conductance at two temperatures as a functionof DC bias for parallel and antiparallel orientation of magneti-zation for the same junction as in Fig. 3 (after Ref. [23]).

reduce the JMR and its decay with bias voltage isstronger.

2.7. Bias voltage dependence of the JMR

Tunnel junctions are nonlinear elements: at lowbias (;barrier height), they are ohmic, whereas athigher bias with normal metals the dynamic con-ductance (G) versus DC bias has nearly a parabolicdependence [2]. However, with an FM electrode,G versus <

DCcurves can deviate noticeably from

a &standard' junction such as Al/Al2O

3/Au [2]. For

a Co/Al2O

3/Ni

80Fe

20junction, the dynamic con-

ductance variation with<DC

is shown in Fig. 7. Theslight asymmetry seen is a common feature fordissimilar FM electrodes. The sharpening of thecurves near zero bias as ¹ is decreased [2] is similarto the zero bias anomaly observed with transitionmetal electrodes or with impurities in the barrier[49]. The presence of metal particles, magnons,magnetic impurities, localization e!ects, multi-steptunneling, and states in the barrier or at theinterface can adversely a!ect the spin polarizations

of the tunneling electrons by causing spin #ip scat-tering. The dip in the conductance at <"0 ob-served at 4.2 K can be expected to be due to somecombination of these e!ects.


Fig. 8. IET spectra at three temperatures for the same junctionas in Fig. 3, measured at H"0. Similar specra are seen forjunctions where on electrode is a FM and the other electrode isAl (after Ref. [23]).

Fig. 9. JMR versus DC bias at three temperatures for the samejunction as in Fig. 3. Data shown are (a) the actual percentagesand (b) normalized at zero bias. The inset shows the JMR in thelow bias region displaying near constancy of JMR. The dashedline in (b) is the theoretically expected variation fora Fe}Al

2O

3}Fe junction with a 3 eV barrier height [56].

In the study of dynamic resistance by Lu et al.,a cusp-like feature at zero bias ((100 mV) at lowertemperatures was seen [25]. This feature was ob-served to be more pronounced in the antiparallelcon"guration of M than the parallel case. Theyattribute this to cause most of the increase in JMRat zero bias, whereas the decrease with increasing<

DCas due to magnon excitations at the FM}I

interface. Inelastic tunneling (IET) spectra mea-sured in zero H at various temperatures (see Fig. 8)showed a peak (dip) at about $100 mV and anadditional sharp feature at &17 meV at lower tem-perature (which was present even in junctionswhere only one electrode was magnetic) [23]. Formagnetic junctions, these peaks in IET spectra havebeen attributed to magnons generated in the mag-netic barrier [50] or in the FM electrodes [23].

One surprising feature exhibited by FM-junc-tions has been the DC bias dependence of JMR.Irrespective of the junction quality, JMR showsa signi"cant decrease with increasing <

DCat all

temperatures [10,25,51}54]. The bias dependenceof JMR at 295, 77 and 1 K for a junction in Fig. 9

showed a monotonic decrease in JMR as D<DC

Dincreases. The normalized data (lower plot in Fig. 9)show its temperature independence. The magnitudeof the decrease depended not only on the quality ofthe interfaces and barrier type, but also on the FMelectrode. For instance, junctions with con-taminated interfaces or having lower barrier height(such as with MgO) showed larger dependence onthe bias. Doped junctions (see the section on barrierdoping e!ects) or junctions where two-state tunnel-ing is favored by the presence of defect states in thebarrier [54] also showed a strong decrease of JMRwith D<

DCD whereas the best undoped junctions

JMR decreased only to about half their value at0.5 V. Moreover, it was observed that the junctionswith Ni or Ni

80Fe

20electrodes showed a stronger

decrease in JMR than junctions with Co or CoFeelectrodes (Moodera, unpublished data).

The DC bias dependence of JMR is not wellunderstood (see the theory section). This has been


Fig. 10. JMR curve of NiFe (17.1 nm)/Co(3.3 nm) Al}AlO

x(1.3 nm)/Co(3.3 nm)/NiFe (17.1 nm)/Fe}Mn(45 nm)/Ni}Fe

(8.6 nm)/Ni}Fe(8.6 nm) junction after annealing at 3003C for 1 h(after Ref. [58].

Fig. 11. In#uence of annealing treatment on (a) the resistance,(b) magnetoresistance, (c) e!ective barrier thickness and (d)height of low-resistance tunnel junctions (after Refs. [28,29]).

attributed to several factors: increase in the conduc-tance with bias, excitation of magnons, or energydependence of spin polarization due to band struc-ture e!ects [23,26]. Recent calculations show thata signi"cant part of the JMR decrease can be at-tributed to magnon excitation [55,56], as also seenfrom the IET spectra [23]. In a later paper, Brat-kovsky provided a model that improved the "t ofthe JMR versus <

DCdata of Nickel et al. (unpub-

lished) by including phonon contributions in addi-tion to elastic tunneling and magnon processes[57].

2.8. Exchange-biased junctions and annealing ewects

The coercive "elds of the FMs can easily bein#uenced by the "lm growth conditions and/or byan exchange bias layer. A good antiparallel align-ment of M in magnetic junctions can be obtainedby exchange biasing one of the FM "lms, as in theGMR spin valve structure. Sato and Kobayashi[58] reported one of the "rst such cases wherea FeMn layer was used to exchange the bias topFM "lm in Ni

80Fe

20/Co/Al

2O

3/Co/Ni

80Fe

20/FeMn

junctions they obtained a JMR of 19% after an-nealing the junctions at 3003C for 1 h (Fig. 10).Several other groups have reported good spin valvestructure in magnetic junctions using materialssuch as TbCo, MnRh, NiO etc. or with an antifer-romagnetic structure using Ru layer inside the Co[53,59,60]. Exchange-bias "elds from tens to

hundreds of Oe have been achieved without a lossof biasing "eld up to 2003C in some cases.

Sato and Kobayashi also studied the e!ect ofannealing on the junctions. Contrary to the expec-tations, the junctions not only survived ¹'2003C,but in fact improved their JMR values. Sousa et al.[28] have investigated the e!ect of annealing on R

J,

JMR and the barrier parameters, as shown inFig. 11. In general, with short annealing up toabout 2303C JMR improved signi"cantly, from17.4 to 27% (according to Eq. (1)), whereas thechange in R

Jdepended on its magnitude. The opti-

mum annealing temperature to achieve the max-imum JMR was inferred to be around 2303C,beyond which the junctions began to deteriorate.The observation that the value of JMR decreasesonly slightly even at an operating temperature of


Fig. 12. Temperature dependence of the normalized *G for twoferromagnetic junctions. The solid lines are the "ts to the theorybased on thermal spin-wave excitations (after Ref. [62]).

2003C is highly encouraging from the applicationpoint of view.

The improvement in the junction propertiesupon annealing has been attributed to barrier homo-genization (as seen in RBS analysis of the oxygendistribution [28,29]) and better magnetic proper-ties of the FM "lm near the interface. The can alsobe other factors, such as reduction of the defectdensity of the Al

2O

3barrier and sharpening of the

FM}I interfaces. Some of these conclusions aresupported by the slight increase of the barrierheight as well as the large increase in JMR. Atmuch higher temperatures, di!usion of the metalatoms into the barrier can occur, leading to thedegradation of the junction properties.

There has been a tendency by various investiga-tors to correlate the observed JMR with the cal-culated value of barrier height /. However, u6 isonly an e!ective value, which is in#uenced by vari-ous conduction channels in an imperfect insulatingbarrier [61], as discussed in the section on barrierdoping e!ects. The theoretical prediction of thebarrier dependence of polarization and JMR inmagnetic junctions is based on the assumption of&clean' barriers [27] (see theory section). In a realmagnetic junction d, u6 and P (and hence JMR) canbe altered due to barrier nonstoichiometry, impu-rities in the barrier, imperfect FM}I interfaces anddegraded FM surfaces. During annealing some ofthese detrimental e!ects can be reduced therebyimproving the polarization of tunneling electronsand increasing the apparent barrier height. Thuscomparing JMR with apparent barrier height andto theory can be misleading.

2.9. Temperature dependence

The observed JMR in MTJ has, at low temper-ature, reached nearly the optimum values expectedfrom Julliere's model. However, even with the bestjunctions there is a signi"cant decrease in JMR atroom temperature as compared to values at 4.2 or77 K. The ¹ dependence of R

Jis, however, not only

found for MTJs but also for standard junctionswith nonmagnetic electrodes. This fact suggestsa nonmagnetic origin of the R

Jversus ¹ behavior.

To explain this, the Julliere model has been modi-"ed by assuming that in addition to the conduc-

tance due to direct elastic tunneling, a secondconductance G

SIis present, which is taken as

unpolarized and hence independent of the relativeorientation of M [62]. The total conductance wastaken as a sum:

G(h)"GTM1#P

1P2

cos(h)N#GSI, (2)

where h is the angle between the directions of M inthe two electrodes, e.g., h"0 or 1803 for parallel oranti-parallel magnetizations, respectively, and G

Tis

the pre-factor for direct elastic tunneling witha temperature dependence of a few percent between4.2 K and room temperature as per the theory [63],arising from the broadening of the Fermi distribu-tions in the electrodes. G

SIwas assumed to be ¹ de-

pendent in a manner determined by the responsiblephysical mechanism.

Conventionally, values for P are determinedfrom a tunneling measurement at low temperature(¹(1 K). For alloys, the observation that P scaledapproximately with the magnetic moment of thealloy leads to the assumption that P varied with¹ as does the magnetization [64]. M(¹) is de-scribed well by thermal excitation of spin waves for¹ far below ¹

C, leading to ¹3@2 dependence of

M (see the theory section). The change *G in theconductance for parallel and anti-parallel orienta-tions, where *G is proportional to P

1P2

and it isassumed that P(¹)"P

0(1!a¹3@2) directly re#ect

the ¹ dependence of P1

and P2. In Fig. 12, for


a Co/Al2O

3/Ni

80Fe

20junction, *G decreased by

nearly 30% as ¹ increased from 77 K to about400 K, indicating a substantial reduction of P. Forthe Co}Co junction, *G showed a much weakerdecay as compared to junctions having Ni

80Fe

20as

one electrode. Thus, the spin-wave-related reduc-tion of P was larger for Ni

80Fe

20are consistent

with the ¹C

for Co and Ni80

Fe20

.The material-dependent constant a is generally

larger for the surface due to surface exchange sof-tening. It has also been observed that both P

0and

a are very sensitive to surface contamination. High-er contamination at the interface can lead to highera, resulting in a more rapid decrease of P withincreasing ¹. This can partially explain many of theprevious results on MTJ. Valuable insight intothese phenomena is expected to be obtained from¹-dependent measurements in MTJ, comp-lementing other methods for determining surfacemagnetic properties, while at the same time provid-ing input for theoretical work aimed at relating thetunneling spin polarization to intrinsic propertiesof the ferromagnetic materials.

The spin-independent conductance GSI

as a func-tion of ¹ showed a G

SI(¹)J¹c power-law depend-

ence with a c of 1.35$0.15. The analysis showedthat G

SIrose much faster with ¹ than the (spin-

polarized) direct tunneling does. This explains theunusually strong reduction of the overall junctionresistance, while causing the JMR to go down with¹ even faster than due to spin-wave excitationsalone. This has probably played a role in many ofthe earlier results, where a sizable JMR was ob-served only at 4.2 K, and negligible e!ect was re-ported at RT.

Spin-independent contributions can come fromimperfections in the Al

2O

3barrier to provide

a noticeable hopping conductance through theassociated localized states due to amorphouscharacter of the Al

2O

3insulator. Theoretical work

[65] has shown that hopping through chains ofN localized states should have a power law depend-ence on ¹, the exponent being c(N)"N!2/(N#1). The temperature dependence originatesfrom phonon emission or absorption at thetransition from the "rst to the next localized chain.For N"2, c"1.33, close to the experimentalvalue.

2.10. Barrier doping ewects

Magnetic junctions allow us to study electronspin scattering in a systematic and controlled man-ner, i.e., by introducing a well-de"ned amount ofa known foriegn element into the barrier. Note thatscattering at the Fermi energy is of importancehere, as tunneling electrons generally originatefrom states in a narrow energy interval around theFermi level. For that purpose, Co/Al

2O

3/Ni

80Fe

20junctions prepared with submonolayer amounts ofdopants incorporated into the middle of the insu-lating oxide were studied [66].

When a spin-#ip event occurs in the barrier,a spin-up electron tunneling from FM1 has to entera spin-down empty state in FM2. In other words,for electrons that change their spin during tunnel-ing, it is as if the magnetization of electrode 2 hasbeen reversed, i.e. they exhibit an inverse JMR.Denoting their fraction by f, the conductance forparallel M becomes (1!f )G

P#f G

APand similarly

for the antiparallel case, leading to:

JMR"

(1!2f )JMR0

1!f JMR0(3)

where JMR0 is the magnetoresistance in the ab-sence of spin-#ip scattering ( f"0). In "rst approxi-mation, the JMR is thus expected to decreaselinearly with the fraction f.

Dopants such as Ni, Co, Pd, Au, and Cu investi-gated at the submonolayer level showed a signi"-cant reduction of JMR with increasing dopantcontent. Fig. 13 shows the near linear decrease ofJMR with the increase of dopant thickness forvarious dopants, Co showing the weakest sup-pression compared to even Cu or Au. The lineardependence as expected by Eq. (3), assumed thatthe dopant covered junction area increased linearlywith thickness t and also that the fraction f oftunneling electrons that experience spin #ip scaleslinearly with t. The weak in#uence of Co was as-cribed to the dominant presence of Co3` with nomagnetic moment, whereas Ni and Cu ions were inan oxidation state with a magnetic moment. Theseresults agreed with other studies of these ions inAl

2O

3matrix. Thus incorporation of sub-

monolayer level of dopants in the barrier in MTJs


Fig. 13. Normalized JMR versus thickness t of the layer ofimpurities present in the tunnel barrier. Data, measured at 77 K,are shown for Co ("lled circles), Pd (open squares), Cu (opencircles), and Ni ("lled squares), together with a linear "t (solidlines) (after Ref. [66]).

Fig. 14. Spin transport through a normal metal. Dependence ofJMR on the bias voltage for increasing thickness of the Auinterface layer in Co/Au/Al

2O

3/NiFe tunnel junction at 77 K:

(a) Measurements for tNM

)0.3 nm; (b) tNM

*0.4 nm (Mooderaet al., to be published).

lead to severe reduction of JMR as a result of spinscattering. In other words, anything short ofa single step tunneling appears to reduce JMR.

2.11. Junctions with nonmagnetic interface layersand inverse JMR ewects

The basic phenomena of spin transport througha normal metal layer has not been well explored.One can utilize the MTJ for such studies. Insertingan ultra-thin layer of a nonmagnetic (NM) metallayer at the FM}I interface in an MTJ drasticallydecreases the JMR, irrespective of the metal(NM"Ag, Al, Au, Cr, Cu, Pd and Pt) used (Mood-era et al., unpublished). In all cases, the JMR reach-ing negligible values with just a few monolayers ofNM layer at the interface. Similar e!ects with Cu atthe interface were observed by Sun and Freitas[67]. These results are consistent with our earlierdirect SPT measurement of the polarizationthrough Au layers using a superconducting Al "lm[35]. However, for example in Co/Au/Al

2O

3/

Ni80

Fe20

junctions, for Au (in some cases also forCu) "lm thickness in the range of 5 to 8 As , a nega-tive JMR e!ect and an unexpected bias voltagedependence was observed, as shown in Fig. 14 [68].Theoretical calculations by Vedyayev et al. [69]and Zhang and Levy [70] had predicted oscilla-tions of JMR in FM}NM}I}NM}FM systems as

a function of the normal metal (NM) thickness, theinterface layer behaving like a quantum well lead-ing to the formation of quantum well states (QWS)when a resonance condition was ful"lled. Besides,according to calculations by Zhang and Levy, theJMR suppression length in NM layer could be asmuch as even 100 As when it was #at, whereas forrougher FM/NM interface, the coherence was bro-ken thereby reducing the JMR faster with the NMthickness. However, in interpreting the experi-mental results one has to pay attention to thepossibility of interfacial mixing of the atoms (espe-cially in sputtered samples, for example Co/Cu)which would yield spurious decay length. Numer-ical calculations by our group for the presence ofQWS, based on a model "rst proposed by Slon-czewski [27], qualitatively explained the experi-mental features, including its' bias dependence[68]. Such studies may allow one to engineera special electrode with strong spin "ltering, forexample by choosing FM/NM/FM trilayer elec-trode with a suitable NM layer thickness.


Fig. 15. Magnetoresistance of a NiMnSb/Al2O

3/Ni

80Fe

20/

CoO junction at 77 K data for di!erent "eld directions: (a) alongthe [1 0 0] easy axis of the NiMnSb (b) along the [0 1 0] easyaxis and (c) 203 from the [1 0 0] axis. The Ni

80Fe

20magneti-

zation is strongly pinned in the [1 0 0]NiMnSb direction bythe CoO antiferromagnet.

Inverse JMR e!ects were observed by Sharmaet al. [71] in tunnel junctions with Ni

80Fe

20elec-

trodes and Ta2O

5or Ta

2O

5/Al

2O

3composite bar-

riers. They reported strong bias dependence ofJMR as well as polarity dependent. JMR evenchanged sign with the DC bias, whereas with onlyAl

2O

3barriers the JMR was positive at all bias

voltages. These features have been attributed to theband structure e!ects in Ta

2O

5and Ni

80Fe

20, re-

sulting in the negative polarization at theTa

2O

5/electrode interface. To explain their data

they assume di!erent band features and EF

forNi

80Fe

20at the interface of Al

2O

3or Ta

2O

5. They

also point out that the magnitude of the inverseJMR to be dependent on the barrier oxidationtime, which seem to indicate some role played bythe barrier defects in these observations, possiblyspin #ip scattering. Very recently, De Teresa et al.[72] observed as much as 50% inverse JMR inCo/SrTiO

3/La

0.7Sr

0.3MnO

3junctions with

a strong bias dependence. The inverse JMR at "niteDC bias in this case was attributed to the negativespin polarization of the d-band in Co. They explaintheir results satisfactorily with the calculated d-band features of Co density of states [73,74]. It isexpected that such interesting observations re#ect-ing the band features of especially ferromagneticmaterials such as NiMnSb, can be relevant.

2.12. Other magnetoresistive systems and geometries

The class of Heusler alloy compounds such asNiMnSb and PtMnSb, known as half-metallic fer-romagnets (HMF), due to the predicted 100% po-larized conduction electrons [75], are interestingboth from fundamental physics as well as applica-tion potential. Junctions of the kind NiMnSb/Al

2O

3/Ni

80Fe

20/CoO have been partially success-

ful in showing the JMR e!ect. In the case of epi-taxial "lms prepared in an MBE system, the JMRvalues (20%) were higher than in the polycrystal-line "lms, and magnetocrystalline anisotropy wasobserved in R

J(H) plots as shown in Fig. 15 [76].

However, for a fully polarized NiMnSb, the JMRshould be 62% for the above junction. Furtherwork is needed which clearly shows the surfacee!ects in these compounds.

Spin-dependent tunneling in oxide ferromagnetssuch as Fe

3O

4, CrO

2and La

xSr

1~xMnO

3is in-

creasingly being studied, as charge carriers in theseoxides are believed to be fully spin polarized. Thetransport in such systems is dealt with in anotherarticle in this issue. To avoid the di$culty of ob-taining good tunnel barriers and their sensitivity tostructural and chemical defects, several other typesof systems have been investigated. Magnetoresis-tive cermets, consisting of FM metal grains disper-sed in an insulating matrix have actually been the"rst system where observation of MR has beeninterpreted in terms of spin-dependent tunneling[11]. Such systems can tolerate &metallic shorts' toa certain degree since these are very unlikely toform continuous paths. Recently, a MR of 8% atroom temperature was observed in thin "lms of Coclusters dispersed in a Al

2O

3matrix [77]. The

magneto-transport properties of such systemsresult from an interplay between spin dependenttunneling and charging e!ects, but these systemsrespond only at rather high "elds.

Other systems such as discontinuous metal-insu-lator multilayers, in order to reach lower saturation


"elds of magnetic tunnel junctions, showed an MRof only 2.2%, due to e!ects related to single elec-trons charging (Coulomb blockade) and witha strong increase of resistance at low temperatureand bias voltage dependence [78,79]. In that casethe electrodes act as soft layers and the clusters ashard layers. Evidence of Coulomb blockade, alongwith a 13% low-"eld MR, was observed whena single layer of clusters was embedded in theAl

2O

3barrier, forming a double junction structure

[80]. Magnetoresistance as high as 28% in theCoulomb blockade regime has also been measuredat very low temperature (20 mK) and voltage(0.12 mV) in planar arrays of islands connected bysmall (10~2 lm2) Ni/NiO/Co tunnel junctions[81].

Double junctions formed by a ferromagneticelectrode, a layer of ferromagnetic clusters embed-ded in Al

2O

3and a nanometer-scale ferromagnetic

point contact should also provide a powerful toolfor the detailed investigation of the interplay be-tween Coulomb blockade and TMR [82]. Junc-tions with an area below 10~2 lm2 can also beformed by electro-deposition through the pores of"ltration membranes. Ni/NiO/Co junctions grownthis way exhibited large steps of the magnetoresis-tance with ratios ranging from 4 to 33% andtwo-level #uctuations of resistance (telegraph noise)attributed to spin-dependent trapping and untrap-ping of electrons in the oxide barrier [83]. Improve-ment in barrier growth techniques now allows oneto study double junctions FM

1/I

1/M/I

2/FM

2,

where M is a continuous metallic layer [84}86].Whether some of the bias e!ects arise from reson-ant/coherent tunneling contributions can be de-duced by further study.

2.13. Application potential

In the short time ((3 yr) since the "rst results oflarge JMR were reported, there has been greatinterest in the potential application. Several mainareas of possible application for JMR devices are:nonvolatile magnetic random access memory(MRAM) elements, read head sensors, large arraysof sensors for imaging and ultra low-"eld sensors[87,88]. Some of the advantages of JMR elementsare, large signal as well as sensitivity, nonvolatile

memory, better radiation hardness and inherentsmall size. However, unreasonably high impedanceof tunnel junctions when dealing with micrometerand submicron size elements and its extreme de-pendence on the barrier thickness are some of thelimitations. Other issues are dielectric breakdown,noise, long-term stability and switching times. Inthese latter studies, MTJs were elegantly used toobtain fundamental information about the domainswitching in magnetic "lms down to the pico-sec-ond range [89]. The bias dependence of dielectricbreakdown and junction lifetime studies by Oeptset al. [90], and electrical noise measurement byNowak et al. [91], have shown encouraging results.Studies of the memory stability due to magneticswitching in junctions by Gider et al. [92], showedthe memory state was stable to at least 107 cycles.

3. Theoretical part

Tunneling from normal metals, including fer-romagnets, to superconductors is a well-developed"eld and there is an excellent recent review byMeservey and Tedrow [6] covering this subject.Tunneling between two ferromagnetic electrodesand its dependence on the relative orientation ofthe magnetizations of the left and right electrodes,i.e. tunneling junction magnetoresistance (JMR),was "rst observed by Julliere [10] and Maekawaand GaK fvert [14]. Julliere was also "rst to givea quantitative estimate [10] of the JMR ratio interms of the classical theory of tunneling which hadbeen formalized earlier by Bardeen [93]. The classi-cal theory of tunneling assumes that the left andright electrodes are two completely independentsystems and the insulating barrier is a perturbationcausing quantum transitions (tunneling) betweenthem. More recently, a di!erent approach in whichthe electrodes and the barrier are treated as a singlequantum mechanical system was developed bySlonczewski [27]. His method allows one to studytunneling through low and relatively permeablebarriers when the left and right electrodes cannotbe regarded as totally independent.

All the early theories of JMR were based ona number of simplifying assumptions. In particular,they usually employ simple parabolic bands [27]


and/or momentum and energy independent tun-neling matrix elements (e.g. Ref. [10]). Withthe recent explosion of interest in JMR stimulatedby new experiments and potential applications,discussed elsewhere in this review, a numberof more realistic calculations of JMR has appeared.In this theoretical section, we "rst review theearly theories of JMR and show that they canbe deduced from a very general and quite rigorousapproach based on the Kubo/Landauer formulafor the conductance. We then attempt to explainthe observed features of the JMR using essentiallythe simple classical theory of tunneling indicatingwhere it needs to be modi"ed or extended inthe light of recent experimental and theoreticaldevelopments.

3.1. Early theories of JMR

The conductance G(H4) of a tunnel junction with

two ferromagnetic electrodes whose magnetic mo-ments are aligned parallel in an applied saturating"eld H

4is much higher than its conductance G(0) in

zero "eld when the moments are antiparallel[3,21,25]. The e!ect is called tunneling junctionmagnetoresistance (JMR) and the relative changein the resistance of the junction, i.e. the so called&pesimistic' magnetoresistance ratio is de"ned by

JMR"

G(0)~1!G(H4)~1

G(0)~1. (3@)

It is also common to de"ne an &optimistic' TMRratio by TMR"(G(0)~1!G(H

4)~1)/G(H

4)~1. The

traditional explanation of the JMR e!ect is basedon the assumption that electrons tunneling froma ferromagnet are spin-polarized and their polar-ization P is given in terms of the spin-dependentdensity of states Dp of the ferromagnet by P"

[Dt(EF)!Ds(E

F)]/[Dt(E

F)#Ds(E

F)]. It is also as-

sumed that spin is conserved in tunneling, i.e. thetunneling current #ows in the up- and down-spinchannels as if in two wires connected in parallel.Since the classical theory of tunneling [6] statesthat the junction conductance is proportional tothe product of the densities of states of the left andright electrodes, it is easy to show that the JMRratio (3@) can be written in terms of the spin polar-

izations PL, P

Rof the left and right electrodes

RTMR

"

2PLP

R1#P

LP

R

. (4)

This is the well-known Julliere's formula [10]which is remarkably successful in predicting theTMR ratio from the observed values [6] of the spinpolarization of electrons tunneling from Fe, Ni andCo into a superconductor. However, the real test ofthe Julliere's formula is whether the observed spinpolarization P of tunneling electrons can be cor-rectly deduced from the densities of states Dp(E) forFe, Co, and Ni. One would expect that the tunnel-ing current from Fe, Co, and Ni should be domin-ated by down-spin (minority) electrons since theirdensity of states at E

Fis high, i.e. the predicted P is

negative. In fact, the polarization P observed intunneling from Fe, Ni, and Co into a superconduc-tor [6] has just the opposite sign for all threemetals, i.e. P'0. Another problem with the simpleclassical theory of tunneling [6] is that it cannotpredict the dependences of the JMR on the thick-ness and height of the insulating barrier. It alsocannot describe more complex situations such asthe e!ect on JMR of impurities in the barrier andthe e!ect of a nonmagnetic interlayer inserted be-tween one of the ferromagnetic electrodes and thebarrier.

It is worth mentioning that some of the earlytheories of tunneling between normal metals im-plied [38,94] that the tunneling current should beindependent of the densities of states of the elec-trodes (the argument was that the densities of statesof the initial and "nal states are cancelled out by thenormalization factor in the transition probability[94]). If these theories were true, there could be noJMR e!ect. However, the experiment and theoriesbased on the rigorous Kubo formula (see the nextsection) demonstrate that the spin-dependent elec-tronic structure of the electrodes manifests itself inthe tunneling current and, hence, a nonzero JMRarises. The reason why the early theories are notreliable is that they are based on oversimpli"edmodels. They assumed [37,94] that the energy oftunneling electrons could be separated into twoadditive components parallel and perpendicular tothe barrier and WKB approximation was made


(the band structure varies slowly compared withthe electron wavelength). Neither of these assump-tions is valid for a real metal}insulator}metal tun-neling junction.

Another problem with the classical theory oftunneling is that it does not treat the junction asa single quantum-mechanical system. In fact, it isassumed in the classical theory of tunneling (see,e.g. Ref. [93]) that electron waves originating in oneof the electrodes become evanescent in the barrierbut never reach the other electrode. When the bar-rier is relatively permeable (low/thin), the evan-escent wave functions of electrons from the left andright electrodes overlap in the barrier region and,therefore, need to be matched across the wholejunction. The need for matching the wave functionsacross the junction was "rst recognized by Slon-czewski [27]. He described the ferromagnet by twosimple parabolic bands (one for up-spin, the otherfor down-spin) shifted rigidly with respect to oneanother by an amount D (the exchange splitting).He then solved the SchroK dinger equation for thewave functions of up- and down-spin electrons tun-neling across a rectangular barrier and determinedthe current from the current operator. It is assumedin such a calculation that the electron momentumparallel to the junction k

,is conserved in tunneling.

The principal result of Slonczewski's calculationis that the polarization P of tunneling electronsnow depends on the height of the barrier<

"through an imaginary wave vector ii in the

barrier de"ned by +i"[2m(<"!E

F)]1@2

P"

kt!ks

kt#ksi2!ktks

i2#ktks. (5)

Here, kt, ks are the Fermi wave vectors in the up-and down-spin bands. Using the result ktJDt(E

F), ksJDs(E

F), which holds for parabolic

bands, it is easy to see that the "rst factor(kt!ks)/(kt#ks) is the polarization obtained inthe classical theory of tunneling but the secondfactor A"(i2!ktks)/(i2#ktks) is new. Sincei ranges from 0 (low barrier) to R (high barrier),we have !1(A(1. It follows that, for a highbarrier, the polarization P given by Eq. (5) reducesto Julliere's result but P can even change sign whenthe barrier is low. However, the change of sign may

be just an artefact of the simple parabolic bandmodel. It will be shown later that a realistic bandstructure is essential for the correct sign of theelectron polarization.

While Slonczewski's model treats the ferromag-net/barrier interface more realistically than theclassical theory of tunneling, the method of match-ing the wave functions is not easily generalizablebeyond a simple parabolic band. It is for thesereason that virtually all recent theories of JMRare based on the very versatile and quite rigorousKubo/Landauer formula for the conductance.

3.2. Linear-response theory of the tunnelingmagnetoresistance

It was shown by Landauer [95] that the conduc-tance Gp in a spin channel p of any sample (metallicor insulating) sandwiched between two electrodescan be written in terms of its total transmissioncoe$cient. This very general result is exact withinthe linear-response theory, i.e. in the low-bias re-gime and is known to be equivalent [96] to theKubo formula [97].

The simplest case to which the Kubo/Landauerformula can be applied is that of coherent tunnelingwhen the electron wave vector parallel to the bar-rier k

,is conserved. Experimentally, coherent tun-

neling would occur for a barrier grown epitaxiallyon a ferromagnetic electrode or for tunnelingacross a vacuum gap between two ferromagneticelectrodes. To apply the Kubo/Landauer formulato a tunneling junction, it is convenient to usea tight-binding parametrization of an ab initioband structure of the junction. Since the current isconserved, the transmission coe$cient can be cal-culated between any two neighboring atomicplanes in the junction. We shall, therefore, choosean imaginary cleavage plane separating the junc-tion into left and right halves and refer to the twoneighboring planes in question as the left (L) andright (R) planes. In the case of tunneling acrossa vacuum gap, the left and right planes are simplythe surface planes of the left and right electrodes,and the separation of the two electrodes by a vac-uum gap is real. When the tunneling is through aninsulator, one can choose as the left and rightplanes of any two neighboring atomic planes in the


insulating barrier. In either case, the total conduc-tance Gp in a spin channel p can be expressed [98]in terms of the one-electron Green's functionsgpL(E

F, k,), gp

R(E

F, k,) in the left and right planes

Gp"4e2

h+k,

Tr([Tp Im gpR(E

F, k,)]

) [T-

p Im gpL(E

F, k,)]), (6)

where bold upright letters indicate matrices in theorbital space. The summation in Eq. (6) is over thetwo-dimensional Brillouin zone and the trace isover the orbital indices corresponding to s, p, d or-bitals which are required in a tight-binding par-ametrization of the electrodes and the barrier. Thematrix Tp is given by

Tp"t(k,)[I!gp

R(E

F, k,)ts(k

,)gp

L(E

F, k,)t(k

,)]~1,

(7)

where I is a unit matrix in the orbital space andt(k

,) is the matrix of tight-binding hopping inte-

grals connecting atomic orbitals in the left andright planes between which the tunneling current iscalculated. We "rst discuss the results obtainedfrom the Kubo/Landauer formula for coherent tun-neling.

3.2.1. JMR due to tunneling through a potentialbarrier

It is useful to discuss "rst JMR for a single-orbital tight-binding model since they are phys-ically transparent. The model assumes that thejunction consists of two ferromagnetic electrodesseparated by N atomic planes of an insulator withan on-site potential <

*/4chosen so that the Fermi

level EF

lies outside its band of allowed energies.Such a model has been used to determine the de-pendences of the JMR on the height and width ofthe insulating barrier [98] and also to investigatethe e!ect on JMR of disorder at the ferromag-net/insulator interface [99] and of impurities in theinsulator [100]. In the case of coherent tunnelingthrough a high barrier, the Kubo/Landauer for-mula takes the following simple form:

Gp+A4e2

h Be~2i0aN

]+k,

Im gpL

Im gpR

D1!(gpL#gp

R)e~i0a#gp

LgpRe~2i0aD2

, (8)

where gpL(E

F, k,) and gp

R(E

F, k,) are the surface

Green's functions of the isolated left and right elec-trodes, i

0is the value of the imaginary wave vector

i in the barrier averaged over the two-dimensionalBrillouin zone, and a is the lattice constant. Eq. (8)holds in the limit i

0a<1, i.e. when electrons in the

barrier are strongly attenuated over distances ofseveral lattice constants.

The tunneling conductance given by the Kubo/Landuer formula (8) has a very simple physicalinterpretation. Since !(1/p) Im gp

L(E

F, k,) and

!(1/p) Im gpR(E

F, k,) are the one-dimensional sur-

face densities of states (DOS) in a channel (p, k,) for

the left and right electrodes, the current in everychannel (p, k

,) is proportional to the product of the

one-dimensional surface DOSs of the two elec-trodes (as in the classical theory of tunneling) butthe product is scaled by the denominator in Eq. (8)which describes the mutual interaction of the twoelectrodes due to an overlap of the electron wavefunctions.

It can be seen from Eq. (8) that the JMR ratio inthe high-barrier limit i

0a<1 becomes indepen-

dent of the barrier height and width. Neither is truewhen the barrier is low. The dependence of theoptimistic ratio R

TMRon <

*/4/= (= is the band-

width) determined numerically [98] from Eq. (6) isshown in Fig. 16 for three thickness of the insulat-ing barrier N"1, 3, and 5 atomic planes. Thevalues of the ferromagnet parameters were chosento mimic a junction with Co electrodes which willbe discussed later using a fully realistic band struc-ture of Co. It should be noted that the bandwidth= in this single-band model is taken to be that ofthe d-band, i.e. 3}5 eV. The height of the barriergiving the best JMR ratio is estimated (see theexperimental part) to be about 3 eV.

It can be seen from Fig. 16 that the RTMR

increaseswith increasing barrier height and reaches a satura-tion value for barrier heights <

*/4of the order of the

bandwidth (saturation is reaches most rapidly forthe narrow barrier N"1). This is in qualitativeagreement with recent experimental results of Sousaet al. [28,29] which indicate that R

TMRincreases with

increasing barrier height. Quantitative agreement isnot to be expected for a single-orbital model andthat is also the reason why the calculated values ofthe TMR ratio are too high.


Fig. 16. Dependence of the tunneling magnetoresistance on theheight <

*/4of an insulating barrier between the ferromagnetic

electrodes for a barrier whose thickness is one (squares), three(triangles) and "ve (circles) atomic planes. Single-orbital tight-binding model (= is the bandwidth).

Fig. 17. Dependence of the tunneling magnetoresistance on thenumber of atomic planes in an insulating barrier for threeheights of the barrier <

*/4measured in units of the bandwidth

=: <*/4

/="2.0 (squares); <*/4

/="1.0 (circles); <*/4

/="

0.58 (triangles). Single-orbital tight-binding model.

The dependence of the RTMR

on the barrier widthN (measured in atomic planes) is shown in Fig. 17for three heights of the tunneling barrier:<

*/4/="

1.0, 2.0, and also for a very low barrier<

*/4/="0.58 (E

Fjust outside the insulator band).

As expected from Eq. (8), the dependence ofR

TMRon N is weak for a high potential barrier

(<*/4

/="2.0) but the TMR ratio decreases rapidlywith N when the insulating barrier is very low(<

*/4/="0.58). As discussed in the experimental

part of this review, the observed dependence of theJMR on the barrier width is very weak, whichwould seem to indicate that most JMR experimentsare performed in the high-barrier regime.

3.2.2. Inyuence of the electrode band structure onJMR

Although the classical theory of tunneling [6]and linear-response theories based on a one-bandmodel provide useful insight they can predict nei-ther the magnitude of JMR nor the correct sign ofthe spin polarization. Key to this problem lies inthe multi-orbital band structure of the ferromag-netic electrodes. It was pointed out by Stearns[101] a long time ago that it is s}p rather than

d-electrons that tunnel from transition metal fer-romagnets. One should not, therefore, look at thetotal DOS which is dominated by d-electrons butonly at those portions of the Fermi surface that areof s}p character. She then argued that such por-tions of the Fermi surface lead to the correct sign ofthe tunneling current. Numerical evaluation of theKubo formula using fully realistic band structure ofthe ferromagnetic electrodes allows us to test thisidea. There are three such calculations dealing withtunneling between Co electrodes across a vacuumgap [98], tunneling between Fe electrodes througha simple step barrier [102], and tunneling from Feand Co to an s-band through a barrier modeled bytwo s-bands separated by a gap [103].

The calculation of tunneling between Co elec-trodes across a vacuum gap [98] is based on nu-merical evaluation of the Kubo/Landauer formula(6) using tight-binding bands of Co "tted to an abinitio band structure [104]. Within the tight-bind-ing scheme, one can model a vacuum gap by turn-ing o! gradually the tight-binding hopping matrixt between the Co electrodes [94]. It is useful tostudy tunneling across a vacuum gap since it hasbeen demonstrated [98] that such a model leads to


Fig. 18. Dependence of the spin polarization of electrons tunnel-ing across a vacuum gap between two cobalt electrodes on thereciprocal of electron hopping t~1 between the electrodes (widthof the gap).

the same JMR ratio as tunneling through a highinsulating barrier. The conductances Gt, Gs of themajority- and minority-spin electrons in the sat-urating "eld (magnetizations parallel) and the con-ductance Gt,s in zero "eld (magnetizationsantiparallel) were determined in Ref. [98] as func-tions of electron hopping across the vacuum gap.We have used these data to plot in Fig. 18 thedependence of the spin polarization of tunnelingelectrons P"(Gt!Gs)/(Gt#Gs) on the width ofthe vacuum gap. The width of the gap is character-ized by the reciprocal of electron hopping t betweens orbitals (see Eq. (7)) measured in units of bulkhopping in Co. (Note that t is dimensionless andt;1 corresponds to the tunneling limit, whereast+1 is the metallic limit.) For a small vacuum gapof the order of the lattice constant (1/t+1), theconductance is dominated by d-electrons and P hasthe &wrong' sign P(0 consistent with the totalDOS argument of the classical theory of tunneling[6]. However, there is a rapid crossover to P'0 asthe width of the gap increases. It can be seen fromFig. 3 that the calculated P for Co not only has thecorrect sign in the tunneling regime 1/t<1 but itsmagnitude 30}40% is in excellent agreement withthe observed P+35% [6]. The correspondingvalue of the calculated optimistic TMR ratio is+65%. The crossover from negative to positiveP occurs because the overlap of d-orbitals de-creases with increasing gap much faster than that ofs-orbitals and it is, therefore, s-electrons that deter-mine the conductance in the tunneling regime. The

same pattern emerges from the tight-binding calcu-lations [103] of the conductance of Co and Fe dueto tunneling to an s-band provided only sp-bond-ing at the metal/insulator interface is included. Thecalculated values of P+35% and 45% for Co andFe are again in excellent agreement with the ob-served results [6]. One may, therefore, concludethat sd-bonding between the ferromagnet andAl

2O

3barrier must be weak. The calculation of

tunneling between Fe electrodes through a simplestep barrier [102], based on layer Korringa}Kohn}Rostoker method, also gives the correct signof P but the calculated values of P are much higherthan observed.

3.2.3. Ewect of disorder on JMR * towardsa generalized Jullie% re's formula

All the theories we have discussed assume coher-ent tunneling, i.e. conservation of the momentumk,

parallel to the junction. This is almost certainlynot satis"ed for Al

2O

3barriers which are amorph-

ous. The theories of noncoherent tunneling fall intotwo categories. Either a simpli"ed treatment ofdisorder is combined with a realistic band structureor simple (one-band) model is used but disorder istreated realistically. The simplest treatment of dis-order in the barrier is provided by the classicaltheory of tunneling [6] (Julliere's formula), whichassumes that tunneling from any occupied state ofthe left electrode to any unoccupied state in theright electrode is equally probable. This is a reason-able model of disorder but it leads to an incorrectsign of the tunneling current. It also fails to describeJMR in junctions with a nonmagnetic metallic in-terlayer. To "nd out how the Julliere's formula canbe corrected, it is necessary to examine the approxi-mations that are made to derive it from the exactKubo/Landauer formula. There are three approxi-mations involved:

(i) It is assumed that in the tunneling regimet+0 (electron hopping between the electrodes isweak), the Kubo formula can be linearized, i.e.Tp"t(I!gp

Rtsgp

Lt)~1Pt;

(ii) only tunneling between the same orbitals isconsidered and assumed to be equally probable, i.e.the hopping matrix t is replaced by t

0I, where I is

a unit matrix in the orbital space and t0

is a singletunneling matrix element independent of k

,;


(iii) complete loss of coherence across the barrier(vacuum gap) is imposed, i.e. it is assumed thata state k

,tunnels with an equal probablity to any

other state k@,.

With these approximations, the Kubo formulatakes the form

Gp"4e2

hN,

Dt0D2C+k

,

Tr Im gpR(E

F, k,)D

]C+k@,

Tr Im gpL(E

F, k@,)D, (9)

where N,

is the number of atoms in the plane of thejunction. Since the expressions in the brackets are(up to a factor 1/p) the total densities of states of theright and left electrodes, Eq. (9) reduces to the usualexpression for the conductance obtained in theclassical theory of tunneling [6].

It will be seen in the next section that the lineriz-ation (approximation (i)) of the Kubo formula is thereason for the failure of the Julliere's formula toexclude from the tunneling current the spuriouscontribution of quantum well states that are for-med in junctions with a nonmagnetic metallic inter-layer.

The approximation (ii) is responsible for the in-correct sign of the spin polarization P predicted byEq. (9) for Fe, Co and Ni. This is because theformula (9), based on the total DOS, allocates equalweights to tunneling via d-states and s}p states. Inreality, tunneling via s}p states dominates.

The approximation (iii) is useful since it providesthe simplest way of dealing with loss of coherencein tunneling (nonconservation of k

,).

Based on this analysis, it is clear how the Jul-liere's formula should be corrected to eliminate theaforementioned problems. First of all, bound(quantum well) states, which do not contribute totransport of charge, must be omitted from the sumover k

,and k@

,in Eq. (9).

The second approximation cannot be made sinceit leads to an incorrect sign of the tunneling current.It is, therefore, necessary to keep the dependence ofthe hopping matrix on the orbital indices.

Finally, complete loss of coherence implies thathopping between the electrodes is a constantmatrix (independent of the wave vector), which can

be approximated by t(0), where t(0) is the value ofthe diagonal hopping matrix element for k

,"0.

This is reasonable since the perpendicular tunnel-ing with k

,"0 is expected to dominate.

The generalization of the Julliere's formula (9)incorporating all the corrections discussed abovetakes the form

Gp"4e2

hN,

TrC+@k,

t(0) Im gpR(E

F, k,)D

]C+@k@,

ts(0) Im gpL(E

F, k@,)D, (10)

where the prime indicates that all the quantum wellstates are excluded and the trace is again over allthe orbital indices.

The structure of Eq. (10) is very similar to that ofthe Julliere's formula. However, there are two im-portant di!erences. All the nonpropagating (quan-tum well) states are removed from the DOSs of theleft and right electrodes and the DOS of each elec-trode is multiplied by the hopping (tunneling)matrix t. The latter means that the trace over theorbital indices can no longer be factorized as in theoriginal Julliere's formula (9). The generalized Jul-liere's formula (10) provides the simplest physicallyplausible description of tunneling in the presence ofdisorder [105].

Rigorous studies of the e!ect of disorder on tun-neling, based on a single-orbital tight-bindingmodel and the Kubo formula [99,100], show that,in addition to a mixing of k

,channels, disorder also

leads to resonant tunneling via localized electronicstates which are formed in the barrier in the pres-ence of impurities or defects [55,56,106]. Resonanttunneling results [100] in quasi-one-dimensionalhigh-conductance channels which dominate tun-neling when disorder is high and the barrier isthick. It follows that the tunneling current is deter-mined not only by the intrinsic properties of theferromagnet, such as DOS for a given spin, but alsoby the type and degree of disorder in the barrier. Inspite of that, the numerical results of Ref. [100]indicate that the TMR ratio is approximately givenby the Julliere's formula. This suggests that, even inthe presence of resonant tunneling, a generalizedJulliere's formula (10) should be a reasonable ap-proximation.


Fig. 19. TMR (%) as a function of Cu interlayer thickness. Thecontinuous line represents the TMR evaluated from the fullKubo formula for coherent tunneling. The broken line repres-ents the TMR evaluated from the generalized Julliere's formula(8) for incoherent tunneling.

3.2.4. JMR of a junction with a nonmagneticinterlayer

A tunnel junction with a thin nonmagnetic me-tallic interlayer, such as Cu, Ag, or Au insertedbetween one of the ferromagnetic electrodes andthe insulating barrier is an interesting system fortesting the theory. The conventional Julliere's for-mula (9) predicts zero JMR since the density ofstates of the Cu layer adjacent to the barrier is spinindependent. This contradicts the experiment (seethe experimental section). A nonzero JMR waspredicted by Vedyaev et al. [66] and Zhang andLevy [67] for a simple parabolic band model.Mathon and Umerski [105] used the Kubo for-mula and the generalized Julliere's formula (10) tocalculate the dependence of the TMR ratio on thethickness of the Cu interlayer in a Co junction withvacuum gap. Their results are reproduced inFig. 19. It can be seen that TMR oscillates due toquantum interference of electron waves on the Culayer, which leads to a negative TMR for a verythin Cu layer. The physical explanation of a non-zero magnetoresistance is that the nonmagneticlayer acts as a spin "lter. This is best explainedfor a Cu interlayer in a junction with Co elec-trodes. Since the Fermi surfaces of Cu and of the

majority-spin electrons in Co are very similar (theCo majority d-band lies below E

F), majority-spin

electrons cross easily the Co/Cu interface and par-ticipate in tunneling as if there were no interveningCu layer. On the other hand, there is a poor matchbetween the Cu bands and the minority-spin bandsin Co, which results in formation of down-spinquantum well states in the Cu overlayer [107}109].Since the quantum well states do not contribute totransport of charge and they occur only in thedown-spin channel, their loss from transport givesrise to a spin asymmetry (nonzero polarization P)of the tunneling current and, hence, nonzero TMR.Interfacial roughness and/or scattering from impu-rities may allow quantum well states to evolve intopropagating states, which destroys JMR for thickernonmagnetic interlayers.

3.2.5. Temperature dependence of JMRAll the theories described so far are applicable

only at zero temperature. It is necessary to extendthem to "nite temperatures to explain the observeddecrease of the JMR with increasing temperature(see the experimental part of this review). There aretwo principal mechanisms that may cause a tem-perature dependence of the TMR: spin-#ip scatter-ing of tunneling electrons from magnetic impuritiesin the barrier and a reduction of the magneticmoment M(¹) in the ferromagnet due to excitationof magnons. We discuss "rst the e!ect of magneticimpurities. Since electron scattering from magneticimpurities may lead to a spin #ip, a fraction ofelectrons with a given spin orientation in the leftelectrode end up with the opposite spin orientationin the right electrode. Inoue and Maekawa [110]pointed out that such electrons give rise to aninverse (positive) magnetoresistance which has signopposite to the usual (negative) JMR due to spin-conserving tunneling. As already discussed, spin-conserving tunneling leads to a negative JMR sincethe saturation "eld conductance G(H

4)JDt

LDt

R#

DsLDs

Ris higher than the zero-"eld conductance

G(0)JDtLDs

R#Ds

LDt

R. On the other hand, since the

role of up- and down-spin "nal DOSs is reservedfor electrons undergoing spin-#ip scattering inthe barrier, G(0) and G(H

4) are interchanged. JMR

due to such electrons is, therefore, positive. Thetotal JMR, which is the sum of the nonspin-#ip


(negative) and spin-#ip (positive) contributions isreduced. Since spin-#ip scattering is an inelasticprocess, the fraction of electrons contributing to theinverse JMR increases with increasing temperature.It follows that the total JMR decreases with in-creasing temperature. However, it is di$cult toestimate reliably the temperature dependence ofJMR due to this mechanism since the energy re-quired to #ip the impurity spin is not known.

The temperature dependence of the JMR due toa reduction of the magnetic moment in the fer-romagnet was "rst discussed by Moodera. He pro-posed [23] that the spin polarization P should beproportional to the magnetization M(¹) andshowed that this assumption used in Julliere's for-mula can account well for the observed temper-ature dependence of JMR(¹). His rather empiricalapproach was justi"ed more rigorously by Mac-Donald et al. [111] and we describe here theirargument. According to the modern theory of itin-erant ferromagnets [112}118], the majority- andminority-spin bands at "nite temperatures are lo-cally split at each atomic site but the direction ofa unit vector which de"nes the local spin quanti-zation axis along which the bands are split #uctu-ates from site to site following the direction of thelocal magnetization. We recall that the direction ofthe local moment, i.e. the local spin quantizationaxis at zero temperature is the z-axis for all atomicsites. The above picture then means that, at "nitetemperatures, the direction of the local exchangesplitting of bands #uctuates from site to site but themagnitude of the splitting remains the same as atzero temperature. At low temperatures, long-wavelength magnons are the dominant temper-ature-induced #uctuations. One can picture a mag-non of an in"nite wavelength as a uniformprecession of the spin quantization axis about theglobal direction of the magnetization (the z-axis). Inother words, each thermally excited magnon tiltsthe magnetization away from the z-axis. The tiltingof the local quantization axis at ¹'0 means thatan electron with spin up (down) at zero temper-ature is now in a state which is a superposition ofup (down and down (up) spin states weighted withweights (1#M(¹)/M(0))/2 and (1!M(¹)/M(0))/2,respectively. It follows that PJM(¹), as assumedby Moodera. To explain the observed JMR(¹) one

needs further to assume that the generalized Jul-liere's formula (10) is approximately valid. One alsoneeds to take account of the fact that the surfacemagnetization decreases with ¹ more rapidly thanin the bulk. In fact, the surface M(¹) obeys theusual ¹3@2 law but with a prefactor which is a factorof two or more larger than in the bulk [119}121].

3.2.6. Bias dependence of JMRAn e!ect which is of great importance from the

point of view of applications is the bias dependenceof JMR. As discussed in the experimental part ofthis review, it is observed that JMR decreases rap-idly with increasing bias. A decrease of JMR isobtained in a simple Slonczewski-type parabolicband model. The conductance in such a modelincreases with increasing bias simply because thebias shifts the band of the electrode into whichelectrons tunnel downward, i.e. towards higherDOS [37,38]. That alone decreases the JMR ratiowith increasing bias. However, the initial decrease[55,122] of JMR is much slower than observed.The failure of the parabolic band model does notrule out entirely the mechanism which relies ona shift of bands and modi"cation of the barrierpotential due to an applied bias since band struc-ture e!ects may be important. To test this ideawould require a calculation with realistic bandsgoing beyond the linear-response formalism. Un-fortunately, such a calculation remains to be done.

An alternative mechanism is scattering frommagnons [110,123]. The e!ect of magnons on JMRis best explained [110] using again the concept ofinverse (positive) JMR. Since magnons are spin onequasiparticles, creation (annihilation) of a magnonin the collision with an electron #ips the electronspin. It is assumed here that such a collision takesplace at the ferromagnet/barrier interface and theelectron whose spin has been reversed undergoestunneling to the other electrode. The e!ect of suchspin-#ip scattering on JMR is completely analog-ous to the e!ect of spin-#ip scattering from impu-rities in the barrier discussed earlier. Since the roleof the "nal DOSs for up- and down-spin electronsis reversed for electrons undergoing scattering frommagnons, inverse (positive) JMR results. Since thephase space available for electron}magnon scatter-ing increases with increasing bias, the total JMR


(sum of the spin-conserving and spin-#ip contribu-tions) decreases with increasing bias. Reasonable"ts to the observed bias dependence of the JMRhave been obtained [123]. However, since the "tsrely on adjustable parameters, it is not clear thatmagnon scattering is the complete explanation ofthe e!ect.

3.2.7. Coulomb blockade ewectsAll the theories discussed so far apply to junc-

tions of macroscopic transverse dimensions. Whenone has a small grain of a ferromagnetic materialseparated by insulating barriers from ferromagneticleads, tunneling to the grain may be strongly in-#uenced by the charging energy in the grain. This isbelieved to occur for magnetic granular systemssuch as Co particles in Al

2O

3[124]. For a very

small grain, the electrostatic energy increases bye2/2C when an electron tunnels into the grain (e isthe electron charge and C the capacitance of thegrain). Tunneling is, therefore, blocked unless thecharging energy is overcome by bias voltage orthermal energy (Coulomb blockade). Tunnelingcurrent for small grains is clearly determined by theresistance of the junction and capacitance of thegrain. For nonmagnetic grains, discrete chargingleads to a characteristic Coulomb staircase in thecurrent}voltage characteristic. For a magneticgrain, the resistance of the junction depends onthe relative orientation of the grain and electrodemagnetizations. One, therefore, has two distinctcharging e!ects for the parallel and antiparallelcon"gurations of the magnetizations. Barnas andFert predicted [125] that this e!ect should lead tooscillations of the JMR ratio as a function of theapplied bias.

Another interesting e!ect predicted byTakahashi and Maekawa [126] is an enhancementof the JMR in the Coulomb blockade regime. Fora double junction consisting of small grains, se-quential tunneling (two independent tunneling ef-fects in a double junction) is suppressed at lowtemperatures due to Coulomb blockade. However,a coherent tunneling through both junctions viaa virtual intermediate state (co-tunneling) is ener-getically favorable and, therefore, dominates at lowtemperatures. Takahashi and Maekawa pointedout that, for sequential tunneling, the e!ective res-

istance of a double junction is proportional to thesum of the resistances of the two junctions whereas,for co-tunneling, the total resistance is proportionalto the product of the junction resistances. As aresult, the ratio of the resistances in the antifer-romagnetic and ferromagnetic con"gurations in theco-tunneling regime is the square of the corre-sponding ratio in the sequential tunneling regime.It follows that at low temperatures, when co-tun-neling dominates, the JMR ratio is enhanced but itreverts to the usual bulk JMR ratio at high temper-atures when sequential tunneling dominates (ab-sence of Coulomb blockade). An extension of theseideas to tunneling between two large grainsthrough several small grains exhibiting Coulombblockade e!ect provides an explanation of an en-hancement of the JMR observed in granular sys-tems at low temperatures [127].

3.2.8. Future directionsThe success of spin-dependent tunneling between

FM "lms has exposed several new possibilitiesof study. For example, investigation of the spintransport through a normal metal, both at lowenergy as well as in the ballistic regime canbe done [128,129]. The theoretical prediction ofresonant- or co-tunneling needs to be experi-mentally veri"ed. The in#uence of Coulomb block-ade and/or spin blockade in spin tunneling isespecially rich in physics. Studies on double junc-tions have just started and is promising. There istremendous application potential in the "eld ofhalf-metallic based tunnel devices, but clearly morework is needed to obtain well-controlled "lms andinterfaces.

The development of devices based on hot elec-tron transport across spin-valves [130] should alsobene"t from the global improvement of the know-ledge on the preparation of tunnel barriers. The"eld of hybrid FM-semiconductor-FM hetero-structures, is yet take to o!, but promises newphysics and potential for spin-polarized devices[131}133]. Of late there has been a thrust in utili-zing dilute magnetic semiconductors as ferromag-netic layers [134,135]. In tunnel junction structuressuch as (Ga,Mn)As/AlAs/(Ga,Mn)As about 28%MR was observed in a "elds of about 1 T, but thissystem is far from optimization.


Spin-polarized tunneling on an atomic scale isperhaps in too early a stage to be technologicallyviable, but may soon contribute to fundamentalstudies. Among the various approaches, using op-tically pumped GaAs tip enables spin-polarizedvacuum tunneling and thus imaging of magneticdomain structure of FM "lms [136}138]. Anotherapproach was chosen in the use of the exchange-split surface state of a FM to study surface magnet-ism by spin-polarized tunneling [139}141]. There isgreat promise in these techniques.

Acknowledgements

This review was made possible by the valuablecontributions from R. Meservey, P. Tedrow, J.Nowak, R. Jansen, C. Tanaka, R.J.M. van de Veer-donk, C.H. Shang and G.P. Berera. The help fromJ. Nassar of University of Paris-Sud at Orsay hasbeen commendable in the preparation of thismanuscript. Special gratitude from J.S.M. goes toL.R. Kinder, P.R. LeClair, T.M. Wong, B. Davisand S. Gupta, the MIT undergraduates and thehigh school students L.F. Gallagher and K.Z.Robinson for their input and enthusiasm through-out the development of this "eld in the last fewyears. The research program at MIT is supportedby ONR grant No. 0014-92-J-1847 and NSF grantsDMR 9423013 and 9730908.

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