Received 9 January 2008Accepted for publication 13 February 2008Published 21 April 2008Online atstacks.iop.org/STAM/9/014106
AbstractThis paper provides a brief overview of the young, but rapidly growing field of spintronics.Its primary objective is to explain how as electrons tunnel through simple insulators such asMgO, wavefunctions of certain symmetries are preferentially transmitted. This symmetryfiltering property can be converted into a spin-filtering property if the insulator is joinedepitaxially to a ferromagnetic electrode with the same two-dimensional symmetry parallel tothe interface. A second requirement of the ferromagnetic electrodes is that a wavefunctionwith the preferred symmetry exists in one of the two spin channels but not in the other. Theserequirements are satisfied for electrons traveling perpendicular to the interface forFe–MgO–Fe tunnel barriers. This leads to a large change in the resistance when the magneticmoment of one of the electrodes is rotated relative to those of the other electrode. This largetunneling magnetoresistance effect is being used as the read sensor in hard drives and mayform the basis for a new type of magnetic memory.
(Some figures in this article are in colour only in the electronic version.)
1. Spintronics
Tunneling magnetoresistance is one of several relatively newphenomena that have contributed to the growth of the youngfield of ‘spintronics’. Spintronics can be defined as the artand science of utilizing the spin of the electron (as well as itscharge) to accomplish some purpose. The birth of spintronicscan be dated from 1988 when groups led by Albert Fert andPeter Grünberg independently discovered the phenomenon ofGiant Magnetoresistance (GMR) [1, 2]. Their contributionwas recognized this year (2007) by the award of the NobelPrize in physics.
This recognition is well deserved because their workcaused a paradigm shift in the way we think about magnetic
∗ Invited paper.
materials. Before their discoveries, magnetic materials wereof interest because of the magnetic fields that they couldcreate, the forces they could exert on nearby objects andtheir ability to retain information, e.g. in the direction oftheir magnetization. The discovery of GMR reminded us thatferromagnetic materials can be considered as two differentmaterials simultaneously occupying the same space; theproperties of one of these materials being determined by theelectronic structure of the majority spin electrons and thoseof the other by the electronic structure of the minority spinelectrons. Their work showed us how to make these twodifferent personalities of a ferromagnet evident through theway they affect the transport of electrons.
The majority-spin and minority-spin electrons are some-times referred to as ‘up’ or ‘down’ spin electrons, respectively.
‘Up’ and ‘down’ here have nothing to do with the directionof the center of the earth or the heavens. The terms areused because electrons act as if they are charged spinningobjects. In contrast to macroscopic spinning objects, the spinof an electron is quantized. It can have only two values. Forconvenience these are called ‘up’ and ‘down’. A chargedspinning object will create a magnetic field. The electron isnot an exception; its magnetic (dipole) moment is one Bohrmagneton (9.274× 10−24 Amp m2). For ‘up’ spin electrons ina ferromagnet, this moment is parallel to the local direction ofmagnetization. For ‘down’ spin electrons it is anti-parallel.
1.1. Spintronic phenomena
There are now several types of spintronic phenomena. Themost important are Giant magnetoresistance which appearsin two basic versions (current in the plane and currentperpendicular to the plane), tunneling magnetoresistance andspin-torque.
Giant Magnetoresistance (GMR) is a change in theresistance of a magnetically inhomogeneous material whenan applied magnetic field brings the magnetic moments ofthe material into alignment. The magnetically inhomogeneoussystem usually consists of a magnetic multilayer in which thelayers are typically a few nanometers in thickness. It is usefuland customary to divide GMR into two major types, Currentin the Plane (CIP) GMR and Current Perpendicular to thePlane (CPP) GMR. The physics of these two types of GMRare quite different. The type of GMR first observed by theteams led by Fert and Grünberg is CIP GMR.
In CIP GMR , the current is parallel to the planes of thelayered magnetic film. Observation of CIP GMR requires atleast two ferromagnetic layers separated by a thin metallicspacer layer. To observe GMR one contrives circumstancessuch that the magnetic moments in the ferromagnetic layersin the absence of an applied magnetic field are not aligned.On application of a sufficiently strong magnetic field, themoments are pulled into alignment and GMR manifests itselfas a change in the in-plane electrical resistance. CIP GMRdepends on the fact that electrical conduction is a non-localeffect. Electrons may be accelerated by the electric field in oneferromagnetic layer and travel through the spacer layer wherethey contribute to the conductance in the second spacer layer.The alignment of the magnetic moments matters becausewhen the moments are aligned, up-spin electrons in oneferromagnetic layer will be up spin-electrons in the secondlayer whereas if the moments are anti-aligned locally up spin-electrons in one layer will be locally down-spin when theydrift into the second ferromagnetic layer.
One important contributor to both CIP and CPP GMRsis the matching of the electronic structures of the spin-channels. For example, the first observation of GMR wasin Fe–Cr multilayers. It turns out that Fe and Cr have verysimilar minority spin bands whereas their majority spin bandsare quite different. An important type of spin-valve usedin magnetic field sensors is based on layers of Cu and Co(or materials similar to Co in the majority channel). Theelectronic structure of Cu is rather similar to the electronic
structure of majority spin Co. There can be other importantcontributors to CIP GMR including a ‘wave guide’ effect thatcauses channeling of electrons within the Cu layer [3].
CPP GMR is conceptually simpler than CIP GMR. Itcan be understood even within a local picture of conduction.However, it is probably more difficult to treat quantitatively. InCPP GMR the electrons travel perpendicular to the magneticlayers. It is relatively easy to see in a qualitative picture howif the majority or minority bands of the two ferromagneticlayers match the spacer layer, electrons in one of the spinchannels can easily pass through the layers when the momentsare aligned. Furthermore, some magnetic alloys, e.g. Ni-richalloys containing Co and Fe conduct up spin electrons withvery little scattering while down spin electrons can hardlytravel more than an interatomic distance before they scatter.
CPP GMR is difficult to measure because the resistanceof a thin film (thickness measured in tens of nanometersversus cross section in square micrometers) is usuallytiny across its thickness. The thickness must remain smallcompared to the distance over which the electrons remembertheir spin which for magnetic materials is on the order ofone to twenty nanometers. For this reason, the first CPP-GMR measurements were performed using superconductingleads [4]. The first major application of CPP-GMR willprobably be extremely small magnetic field sensors (∼10 nm),so small that their lateral dimensions are comparable to thefilm thickness.
Tunneling Magnetoresistance (TMR)is geometricallysimilar to CPP-GMR. The difference is that the non-magnetic metallic spacer layer is replaced by an insulator orsemiconductor. It would appear that band-matching betweenone of the spin-channels and the spacer layer (the origin ofCPP GMR) can no longer occur because there are no bandsat the Fermi energy in an insulator. For this reason (as willbe described below) the theory of TMR was based on theFermi energy density of states of the ferromagnetic electrodes.Somewhat surprisingly, as will also be described below, itturns out that a new and different kind of band matchingcan occur and this can be used to achieve a very large ratioof tunneling conductance between parallel and anti-parallelalignment of the spins.
Spin-Torque is the newest major spintronic phenom-enon. It was predicted independently by John Sloncewski andLuc Berger in 1996 [5, 6]. In a sense, it is the converse ofGMR and TMR in which the transport of electrons betweentwo ferromagnetic layers is affected by the relative alignmentof their magnetic moments. It is probably not surprising thatcurrents flowing between ferromagnetic layers can change therelative alignment of their magnetic moments. Spin-polarizedcurrents have been observed to cause precession of magneticmoments and switching from anti-parallel to parallel relativealignment and vice versa.
1.2. Spintronic applications
1.2.1 Read sensors for hard drives.The CIP-GMR spinvalve was the first major application of spintronics. Thesedevices have been used since the mid-1990s to detect the
transitions between the magnetic domains which encode theinformation stored on hard drives. The size of a read sensorin a hard drive must be comparable to the bit size. One of themajor advantages of the GMR spin-valve is that it allowedthe sensor to decrease in size keeping pace with (and enabling)the rapidly decreasing size of the magnetic bit as storagedensities have increased. In a CIP spin-valve, the momentsof one of the ferromagnetic layers (called the ‘pinned’ or‘reference’ layer) are held in a fixed direction by means ofan adjacent anti-ferromagnetic layer whose local moments aretightly coupled to those of the ferromagnetic layer. The otherferromagnetic layer (called the ‘free’ layer) is designed so thatit will respond to an applied field. Various schemes are usedto ensure that in the absence of an applied field, the momentsof the ‘free’ layer are oriented perpendicular to the momentsof the pinned layer. This is done to ensure a linear responseof the resistance to the applied field. CIP-GMR spin-valvesrapidly became the dominant technology for hard drive readsensors in the late 1990s.
TMR read sensors have recently displaced the CIP-GMR spin valve as the dominant technology for hard drivesbecause they are easier to fabricate at smaller sizes thanCIP-GMR devices and because they can provide a muchlarger signal. Because of the design of the read/write headit is easier to make a small read sensor if the current flowsperpendicular to the layers. The recent prediction, discoveryand development of symmetry filter based TMR sensors(described below) has made it possible to obtain a large signalfrom a very small TMR device with a resistance that is not sohigh that shot noise becomes a problem.
As bit sizes continue to decrease it is expected that eventhe new class of TMR sensors will encounter the shot-noiseproblem. Shot noise arises because the electronic charge isdiscrete. As tunnel devices are made smaller and smaller,and the total current becomes smaller, there will be randomfluctuations in the current as the electrons tunnel through thebarrier at random times. When shot noise becomes too large,it is expected that CPP-GMR based spin-valves may becomethe preferred technology. It should be possible to makeCPP-GMR devices that show an extremely large changein resistance between parallel and anti-paralell alignment ofthe magnetic layers using materials called half-metals. Half-metals are metals for one spin-channel, but insulators (orsemiconductors) for the other spin-channel.
1.2.2 Magnetic random access memory.TMR is also usedto read the information stored in a new type of non-volatilemagnetic memory called Magnetic Random Access Memory(MRAM). In principle MRAM has many advantages overcurrent alternatives. It can be relatively fast (especially ifit takes advantage of the large signal offered by symmetrybased TMR), it is non-volatile which gives it an advantageover conventional dynamic random access memory whichrequires refreshing and it offers an unlimited number of readand write cycles. Compared to flash memory, MRAM hasthe advantages of unlimited read and write cycles and muchshorter write times. The advantage of flash is that of a more
mature technology which can be made quite dense and ispresently much less expensive per bit.
In MRAM the information is stored in the relativeorientation of two magnetic layers. Parallel orientation mightrepresent a zero, anti-parallel might represent a one. TMR isused to sense the difference in resistance of the two states. Onedifficulty with scaling MRAM to smaller bit sizes and higherdensities is its mechanism for writing which uses magneticfields supplied by current carrying lines to switch the momentsof one of the layers, i.e. the free layer relative to a pinned layer.A possible solution to this problem that is being aggressivelyexplored is to utilize spin-torque for switching the momentsof the free layer.
1.2.3 Other applications. GMR and TMR can be usedwherever there is a need for an electrical response to amagnetic field. They are particularly appropriate when thesensors need to be small or where an array of sensors isneeded [7].
It has been suggested [8, 9] that MRAM-type cells withvery large ratios of parallel to anti-parallel conductance mightenable a new type of computer architecture in which logiccircuits are reprogrammed dynamically. Such a device wouldbe similar to a field programmable gate array that could bereprogrammed on a nanosecond timescale.
One major goal of current research in spintronics is toelectrically generate and manipulate spin-polarized currents insemiconductors. The symmetry filter effect can, in principle,be used to inject strongly spin-polarized currents intosemiconductors such as GaAs. These should be particularlylong lived if they are injected into the conduction bandbecause of the weak spin–orbit coupling near the bottom ofthe conduction band of a typical semiconductor such as GaAs.Transistors based on the precession of spins injected intosemiconductors have been proposed, but so far have not beenrealized [10]. Another possible direction for future spintronicresearch is the use of low Z and high conductivity materialssuch as carbon nanotubes and graphene for conductingspin-polarized currents. The transmission of spin-polarizedcurrents over micron length scales has been demonstrated incarbon nanotubes [11].
2. Tunneling Magnetoresistance
Tunneling magnetoresistance was first reported by Julliere in1975 [12]. Julliere made a Co–Ge–Fe sandwich and measuredthe change in electrical resistance on switching the relativealignment of the Co and Fe magnetic moments from parallelto anti-parallel. He reported a 14% increase in resistanceat a temperature of 4.2 K. This short paper is also famousfor the introduction of the Julliere model for TMR whichcontinues to be the most often used theory for analyzingthe results of TMR experiments. Julliere’s work may havebeen inspired in part by the work of Tedrow and Meservey[13, 14] who had earlier measured the spin-dependence oftunneling currents through an amorphous aluminum oxidetunnel barrier separating various ferromagnetic electrodesfrom superconducting aluminum. After the discovery of GMR
in 1988, tunneling magnetoresistance received much moreattention. In 1995 Miyazakiet al [15] and Mooderaet al [16]independently reported TMR in excess of 10% at roomtemperature. This was sufficient to make TMR interesting forapplications.
During the 1970s there was considerable interest inquantum mechanical tunneling between metallic electrodesin which at least one of the leads was a superconductor. Itwas learned during this period that aluminum oxide makesan excellent tunnel barrier. The probable reason for this isthat aluminum has a remarkable affinity for oxygen. Theaggressive oxidation process leads to amorphous but highlycoherent oxides that give complete coverage without pinholeseven when the layers are very thin. These thin aluminumoxide layers are very important for tunneling. The manyorders of magnitude difference in the conductivity of themetal and the oxide mean that even a small number ofpinholes will dominate the measured conductance. On theother hand the oxide cannot be very thick. The resistanceof a tunneling device increases very rapidly with thickness,typically doubling (or more) with every atomic layer. Becauseof the practical advantages of AlOx, TMR came to be almostsynonymous with amorphous aluminum oxide barriers.
2.1. Two current model
Descriptions of spintronic effects are typically based on the‘two current model’ of electron transport in solids [17].Within this model, the up-spin and down-spin electronsconduct in parallel. Consider the tunneling conductancebetween two ferromagnetic electrodes. Within the two-current model, the conductance for the cases of paralleland anti-parallel alignment of the magnetic moments ofthe electrodes are written asGP = G↑↑ + G↓↓ and GAP =
G↑↓ + G↓↑, respectively. Thus for parallel alignment of theelectrodes the up-spin electrons on the left side are still up-spin on the right after tunneling through the barrier. Similarly,the down-spin electrons on the left remain down-spin on theright. For anti-parallel alignment, however, electrons that arelocally up spin (meaning their moment is parallel to that of thelocal magnetization) find themselves in a region of oppositemagnetization which means that they are locally down-spin.Similarly, down spins on the left become up spins on the right.
The two-current model neglects the spin–orbit interac-tion, a relativistic effect that couples the electron’s spin withits orbital motion around the nucleus and through the lattice.It also neglects the possibility that there may be magnetic mo-ments (especially near the interfaces) that are not perfectlyaligned with those of their layer. Imperfect alignment canoccur because of thermal effects or because of weak exchangeinteractions. Exchange interactions are the name we giveto the quantum mechanical effect that causes the magneticmoments in ferromagnets to align in a common direction.
2.2. Julliere model
In his short, but famous 1975 paper, Julliere explained hisresults using a simple and easily applied theory that will bedescribed here. Julliere proposed that the tunneling current
should be proportional to the density of electronic states on thetransmitting side of the barrier and to the density of electronicstates on the receiving side of the barrier. This proposalmay seem reasonable since transition probabilities are oftenproportional to the densities of initial and final states. Thereare also matrix elements involved in transition probabilities,but those are difficult to calculate for a barrier like AlOx sothe effect of the matrix element was incorporated into thedensity of states through the notion that the density of statesbeing described was ‘the density of states of the tunnelingelectrons’. Presumably, some electrons can tunnel more easilythan others.
If we ignore the ambiguity in what is meant by ‘density ofstates’ in the Julliere approach, it is straight-forward to derivea simple formula for TMR. The equations,GP = G↑↑ + G↓↓
and GAP = G↑↓ + G↓↑ mathematically express the parallelconductance implied by the two current model for the case ofparallel and anti-parallel alignment of the moments of the twoelectrodes, respectively. The hypothesis that the conductanceis proportional to the density of states of the left and rightelectrodes implies,
G↑↑∝ N↑
L N↑
R, G↓↓∝ N↓
L N↓
R, G↑↓∝ N↑
L N↓
R,
G↓↑∝ N↓
L N↑
R.
Defining the tunneling magnetoresistance as the ratio of thechange in conductance to the minimum conductance, we have,
TMR =GP − GAP
GAP=
N↑
L N↑
R + N↓
L N↓
R − N↑
L N↓
R − N↓
L N↑
R
N↑
L N↓
R + N↓
L N↑
R
.
Defining the polarization of the left and right electrodes by
PL,R =N↑
L,R − N↓
L,R
N↑
L,R + N↓
L,R
=1NL,R
NL,R,
we obtain
TMR =1NL1NR
(1/2)(NL NR −1NL1NR)=
2PL PR
(1− PL PR).
Thus the TMR is expressed in terms of the ‘spin-polarization’ of the left and right electrodes. Although thisformula is much used to rationalize TMR experiments andoften seems to be useful for this purpose, its meaning isnot so clear as our simple derivation appears to imply. Thepolarization,PL,R, cannot be interpreted as the polarizationof the density of states at the Fermi energy as implied in thederivation. The reason for this is that the spin polarization ofthe tunneling current can be measured when electrons tunnelbetween a ferromagnetic electrode and a superconductingelectrode. For Co and Ni which are known to have a Fermienergy density of states that is overwhelmingly minority, it isthe majority electrons that are observed to carry the tunnelingcurrent in these experiments. Another indication that the TMRdepends not just on the barrier but also on the electrodes isthat the sign of the TMR has been observed to change whenthe barrier is changed using the same electrode materials [18].
Figure 1. Conceptual model for deriving the Landauer relationship between transmission probability and conductance.
The Julliere formula is most appropriate when comparingTMR for systems with different electrode materials butidentical barriers. When almost all barriers were AlOx, thiswas quite useful. The free electron—simple barrier model forquantum mechanical tunneling (described in the next section)can be solved analytically. The TMR is not given by theJulliere model if the polarizations are defined in terms ofthe density of states at the Fermi energy. However, the TMRcan be expressed in the Julliere form if the polarizations aredefined in terms of the probabilities for up- and down-spinelectrons in one of the electrodes to be ‘transmitted’ into thebarrier as an evanescent wave [19]. In general, however, itmust be admitted that there is not yet a good theory of TMRfor systems with amorphous barriers.
2.3. Theory of tunneling in epitaxial systems
Let us now consider tunneling for epitaxial systems.Fortunately, relatively straight-forward theoretical approachesexist for such systems. A simple and effective approachfor understanding and calculating ballistic transport wasdeveloped by Landauer [20]. In this approach one imagines(figure 1) two reservoirs of electrons, one on the left atchemicalµ1 and another on the right at chemical potentialµ2. At T = 0, electrons with energies less thanµ1 in the leftreservoir contribute to an electron current that flows from leftto right while electrons in the right reservoir with energiesless thanµ2 contribute to an electron current in the oppositedirection. The difference between these currents will be thenet current flowing through the sample for applied bias,V =
(µ1 −µ2)/e.
2.3.1 Landauer expression for tunneling conductance.Thecurrent density for right going electrons in the left lead canbe written as,J+
=eV
∑k v
+z (k) f (µ1). Here, the volumeV
is the cross-sectional area A times some lengthl along thelead. Note that we are including all of the electrons from someminimum energy up to the local chemical potential. Notealso that we are only counting the electrons whose velocityin the z-direction is positive. In equilibrium, (µ1 = µ2) therewill be an equal number of electrons at each energy goingin the opposite direction so the net current density wouldbe zero. The sum over wavevectors consists of a sum overthe components parallel to the interface and the sum overcomponents perpendicular to the interface. We can use thefact that the electron velocity is the derivative of the bandenergy with respect to wavevector,v+
z (k)= (1/h̄)(∂Ek/∂kz),and the identity that relates a sum overkz to the corresponding
0
0 0.5 1.5
ikzikz e e
0
1.5
–0.5
–1.0
–2.0 –1.5 –1.0 1.0 2.0 2.5 3.0–0.5–1.5
0
0.5
1.0
1.5
1.5
Ene
rgy,
wav
e fu
nctio
n
z
Tunneling through simple barrier
Energy
VB
e r ikzteze
Figure 2. Energy landscape and wavefunction for the simplebarrier model for tunneling.
integral,∑
kz→
l2π
∫dkz, to convert the sum overkz into an
integral,
J+=
e
A
∑k‖
1
2π
∫dkz
1
h̄
∂E(k)∂kz
f (µ1)∑
k′
T++(k, k′).
Multiplying J+ by the area to getI +, we obtain,
I +=
e
h
∑k‖
∫dkz
∂E(k)∂kz
f (µ1)∑
k′
T++(k, k′).
We also included the probability that the electron will betransmitted through the sample region to statesk′ in lead2. We can now convert the integral overkz into an integralover energy, obtaining an expression forI +, the current ofelectrons, in the left lead being transmitted to the right,
I +=
e
h
µ1∫dE
∑k‖,k′
‖
T++(k, k′).
A similar procedure yields the corresponding expressionfor I −, the current of electrons in the right lead beingtransmitted to the left,
I −=
e
h
µ2∫dE
∑k‖,k′
‖
T−−(k, k′).
The difference of these two currents gives the net currentthrough the sample and for small differences between the
Figure 3. Energy bands in MgO along the (100)-direction. The red,green and blue bands were calculated for MgO. The black bandrepresents the simple barrier model with free electron mass.
Figure 4. Plot of (kz1z)2 versus energy showing how one of thevalence bands (red) continues as an evanescent state into the gapand then becomes the conduction band. The black line gives(kz1z)2 for the simple barrier model. For the bands shown herek‖
was set to zero. One Hartree is 27.2 eV.
chemical potentials can be written as,
I = I +− I −
=e2
h
∑k‖,k′
‖
T++(k‖, k′
‖)µ1 −µ2
e.
The relationshipT++= T−− from time reversal symmetry
was also used. Finally, dividing the net current by the voltagedifference across the sample,V = (µ1 −µ2)/e, we obtain theLandauer expression for the conductance,
G =I
V=
e2
h
∑k‖,k′
‖
[T++(k‖, k′
‖)]EF.
For the systems that we will consider, two-dimensionalperiodicity is retained across the sample. This allows us towrite,
G =e2
h
∑k‖
[T(k‖)]EF,
Mg-sMg-s
O-pz
MgO
z
Mg-sMg-s
Mg-sMg-s
O-pzO-pz
MgO
z
Figure 5. Simple model for the11 valence and conduction bandsof MgO. The simplified Hamiltonian includes the magnesiums-state represented by the blue circles and the oxygen pz stateindicated by the red (ψ < 0) and green (ψ > 0) ellipses.
Figure 6. Energy as a function of wavevector for11 bands inMgO. The red lines are the results of first-principles calculationsusing DFT. The blue dotted lines result from the simple two orbitalapproximation described above.
expressing the conductance as a sum over the transversenormal modes of the leads, sometimes called conductionchannels. In this case, the conduction channels are thecomponents of the wavevectors parallel to the interfaces, i.e.perpendicular to the current direction.
2.4. Simple barrier model for tunneling
Now that we know how to calculate the conductance from thetransmission probability we can consider some examples. Letus begin by considering the simple barrier model for tunneling(figure 2). Because this model is so easily solved, it usuallyforms the basis for our understanding of tunneling and TMR.
Figure 7. Square of wavevector as a function of energy. The redcurve with crosses is the first-principles result, the blue line is thetwo-orbital approximation.
In this review, I will contrast the simple barrier model with amore realistic model that results when one includes the atomsthat make up the electrodes and barrier. Understanding thebarrier model, however, makes it easier to understand someof the results of the more realistic models.
In the barrier model, one envisions free electrons withenergyE propagating in some direction determined by thewavevectork, incident on a barrier of heightV (greaterthan E) that extends fromz = 0 to z = t . The barrier isassumed to have infinite extent inx and y. In this case,of electrons incident fromz = −∞, the x-component of thewavevector is assumed to be positive. This problem can besolved analytically in closed form by considering the form ofthe wavefunction in the three different regions and requiringcontinuity of the wavefunction and its derivative at bothinterfaces. The results are summarized in appendix A, wherethe transmission probability is given.
Qualitatively, the transmission probability is highest forelectrons traveling perpendicular to the barrier and fallsoff rapidly as the component ofk parallel to the interfaceincreases. One can understand this result in two ways.Classically, one can imagine a beam of particles that decaysat a certain rate as it propagates through a barrier region.Because the particles traveling at an angle different fromperpendicular must go a greater distance, the decay ontraversing the barrier will be proportionately greater. Ratherthan distance,t , the beam must travel a distancet (1 +k2‖/k2
z)1/2 where kz is the component of the wavevector
perpendicular to the barrier andk‖ is the component parallelto the barrier. Therefore, the transmission probability for anelectron with wavevectork = (k‖, kz) will be proportional toT0 exp[−2kzt (1 +k2
‖/k2
z)1/2] compared toT0 exp(−2kzt) for
electrons traveling toward the barrier.The quantum mechanical picture is slightly different, but
it gives the same answer. In this picture, one has a waveof energy E. In the barrier, the wavefunction for wavestraveling perpendicular to the layer will be proportional toexp(−κz) whereκ2
= (2m/h̄2)(V − E). For electrons witha wavevector component parallel to the layers the decay
Figure 8. Two-dimensional Bloch states symmetries compatiblewith a square lattice in thex–y plane.
will be faster, since the wavefunction will be proportionalto exp(−z)exp(ikxx + ikyy), whereκ2 will be given byκ2
=
(2m/h̄2)(V − E)+ k2‖. The quantum mechanical picture is
that the oscillations of the wavefunction in the plane of thebarrier due to thek‖ component of the wavevector representan energy (̄h2k2
‖/2m) associated with lateral motion of the
electron. This energy is not available for getting through thebarrier. Thus the expression forκ2 can be written asκ2
=
(2m/h̄2)(V − E′), where E′= E − h̄2k2
‖/2m. We shall see
that oscillations of the wavefunction in the plane of the barriertend to increase the rate of decay of the wavefunction in thebarrier [21].
2.5. Evanescent states
Let us next consider a more realistic picture of the barrier.In reality, the barrier will be an insulating material made upof atoms. We usually consider an insulator to be a materialwith no electronic states within an interval in energy (theenergy gap) around the Fermi energy. Figure3 shows the bandstructure of MgO for the (100)-direction. The meaning of theenergy bands is that they give the energy of the propagatingBloch states as a function of the quasi-momentum,k wherek, describes how the wavefunction in one cell is relatedto the wavefunction in a neighboring cell. If the system isperiodic, one can show that the wavefunction (solution to theSchrödinger equation) at pointr in one cell is proportionalto the wavefunction at the corresponding pointr + a in anadjacent cell,ψk(r + a)= cψk(r) wherea is a lattice vector.A simple argument supporting this claim is given in appendixB. The constant of proportionality,c can be any complexnumber. However, if the wavefunction is to be normalizedover a system of infinite extent,c∗c must be unity, otherwisethe wavefunction will diverge either forr + na or r − na asn → ∞. States of this type, which can be normalized overa solid of infinite extent, are the propagating Bloch states.The Bloch condition for propagating states can be writtenasψk(r + a)= exp(ik · a) ψk(r), where the vectork is calledthe wavevector. The Bloch condition, simply describes themost general way that wavefunctions on neighboring sites can
Figure 9. Energy bands for Fe calculated from a first-principles electronic structure code.
Figure 10. 11 energy bands for Fe. Green dashed lines represent abroad s-band and a narrow d3z2−r 2 band. Hybridization between thebands results in upper and lower11 bands (solid blue lines)separated by a gap. Bands calculated from first-principles arerepresented by the red pluses for comparison with the bandscalculated from the simple model used here.
differ by a phase. It also allows us to make a connection withthe free electron model for which the wavefunction can bewritten as,ψk(r)= V1/2 exp(ik · r), where the pre-factor ofthe square root of volume is needed so that the wavefunctionwill be normalized to unity over the volumeV .
In addition to the solutions to the Schrödinger equationthat differ by a phase from one lattice constant to the next therewill be solutions that differ by a factor with absolute valuedifferent from unity. Such wavefunctions are not allowedsolutions if the system is infinite, because they will diverge inone direction or the other and therefore cannot be normalized.However, they are allowed if the system is finite. In fact, theyare usually necessary to ensure that the wavefunction andits derivative are continuous at the boundaries between thebarrier and the electrodes. These states are called evanescentstates because they vanish away from the boundaries. Both thepropagating states and the evanescent states come in pairs. Forevery propagating Bloch state describing an electron moving
Figure 11. Solution of secular equation for cos(kza/π) (blue solidlines) Fe11. First principles results are red crosses and pluses. Therate of decay of the wavefunction in the11 gap is shown as thegreen dashed line. Near the center of the gap, the wavefunction willdecay by a factor of approximately 0.6 for every atomic layer.
in the +z-direction, there will be a corresponding state withthe same energy describing an electron moving in the−z-direction. Similarly every evanescent state that decays in the+z-direction will have a partner decaying in the−z-direction.
To relate, the simple barrier model to the more realisticband model for MgO, one traditionally assumes that thetunneling current occurs through virtual transitions fromthe Fermi energy of the electrodes to the conduction bandof the MgO. The dispersion relation for this model wouldbe given by E = Ec + (h̄2k2)/(2me). Or since k2
= k2z +
k2‖
and k2z = (2me/(h̄2))(E − Ec)− k2
‖. The corresponding
wavefunction isψk(r)= V−1/2 exp(ik · r) which we writeasψk(r)= V−1/2 exp(ik‖ · ρ + ikzz). This wavefunction anddispersion relation would be valid forE > Ec. For E < Ec,relevant to tunneling,k2
Figure 12. Absolute square of the wavefunction for incident electrons of different symmetries as a function of position. The two top panelsdescribe tunneling for parallel alignment of the magnetic moments of the electrodes. The upper left panel describesk‖ = 0 contributions toG↑↑. The upper right panel describesk‖ = 0 contributions toG↓↓. Similarly, the bottom two panels describe contributions toG↑↓ andG↓↑.
valence bands as well as a conduction band. Electrons can alsotunnel through the barrier via the valence bands; this wouldbe described as ‘hole’ tunneling because it would describea process in which a valence electron makes an upwardtransition from the top of the valence band to the Fermi energyof the right electrode simultaneously with the transition ofan electron at the Fermi energy of the left electrode makinga downward transition that fills the hole. The situation iseven more interesting, because as we shall show, the redvalence band (labeled11) is really the same band as the redconduction band (also labeled11).
The first-principles based layer KKR approach was usedto calculatekz as a function ofE as shown in figure4. Justas for the simple barrier model, energies exist for which nostates with realkz(E, k‖) exist. However, for these energies,states exist which are imaginary or complex. In fact all of thepropagating Bloch states continue as evanescent states whenthey are no longer valid propagating states. In particular, forMgO, the valence band with11 symmetry continues throughthe gap region as an evanescent state for whichkz is imaginaryand re-emerges at the top of the gap as the conduction band.An excellent approximation for this11 state in the vicinity ofthe gap is
1
k2z
=−h̄2
2m∗v(E − Ev)
+h̄2
2m∗c(E − EC)
,
whereEv andm∗v represent the energy and effective mass at
the top of the valence band whileEc and m∗c represent the
analogous quantities at the bottom of the conduction band. ForMgO,m∗
v andm∗c seem to be the same so that the interpolation
formula for the evanescent band in the gap can besimplified to
h̄2k2z
2m∗=(E − Ev)(E − Ec)
(Ec − Ev).
The minimum value ofk2z is [k2
z]min = −m∗(Ec−Ev)
2h̄2 and occursat mid-gap, E = (Ec + Ev)/2. For comparison, the simplebarrier model would givek2
z = −me(Ec−Ev)
h̄2 at mid-gap.The continuation of the bands into the gap region as
evanescent states can be understood on very elementary termsfor MgO. A minimal model for the electronic structure in thevicinity of the gap would include the Oxygen p-states and theMg-s state. The tight-binding Hamiltonian matrix forkx = 0andky = 0, can be written as
H(kz)=(Es w[exp(ikza/2)− exp(−ikza/2)]
w[exp(−ikza/2)− exp(ikza/2)] Ep
).
The diagonal elements of the matrix represent the atomicenergies of the Mg-s and oxygen-pz states (possibly modifiedby terms independent ofkz). The off diagonal matrix elementsrepresent interactions between the Mg-s and nearest neighbor
Figure 13. Calculated transmission probability for majority electrons for parallel alignment of the Fe electrodes for three thicknesses of theMgO barrier. The thicknesses are (top to bottom) four, eight and 12 atomic MgO layers.
oxygen pz states. The form can be understood from thefact that each Mg-s will interact with nearest-neighborO-atoms located at±a/2 in thez-direction. The sign of thetwo exponentials can be understood from the fact that theO-pz state has odd-parity while the Mg-s has even parity.This causes the two overlaps with nearest neighbors to haveopposite signs. The secular equation for the tight-bindingSchrödinger equation,
Det
∣∣∣∣ E − Es wi sin(kza/2)−wi sin(kza/2) E − Ep
∣∣∣∣ = 0,
is (E − Es)(E − Ep)−w2 sin2(kza/2)= 0. This can be
solved forE,
E =Es + Ep
2±
√(Es − Ep
2
)2
+w2 sin2(kza/2),
as shown in figure6. It can also be solved forkz
sin2(kza/2)=(E − Es)(E − Ep)
w2,
as shown in figure7.
The important point to notice is that the valence band andconduction bands are really the same band, connected by anevanescent state. The empirical result that
h̄2k2z
2m∗=(E − Ev) (E − Ec)
(Ec − Ev)
is reproduced by the two-orbital approximation if
Ec = Es, Ev = Ep and m∗=
2h̄2(Ec − Ev)
a2w2.
2.6. MgO as a symmetry filter
Returning to figure4, one can see that the evanescent statesat the Fermi energy will decay at very different rates. The11
states will decay most slowly, the15 states will decay muchfaster and the12′ states will decay faster still. The12 state(not shown) decays even faster. The symbols,11, 15, 12′ ,
Figure 14. Minority conductance for parallel alignment of the magnetic moments in the Fe electrodes. The left-hand panels show thetransmission probability plotted as the vertical axis as a function ofkx andky. The panels on the right show the same data in a contour plot.Brighter areas have higher conductance.
12, etc. label wavefunction symmetries compatible with thesquare symmetry of the two-dimensional lattice of bcc Fe orrock salt structure MgO when viewed along the 001-direction.These symmetries are depicted in figure8.
11 symmetry is that of a circle. Atomic orbitalscompatible with this symmetry include s, pz and d3z2−r 2. 15
symmetry alternates in sign in either thex- or y-directions. Forthis reason15 states are always doubly degenerate. Atomicorbitals compatible with15 symmetry are px and py and dxz
and dyz. States with12 and12′ symmetry have more in-plane sign changes than11 or 15. 12 and12′ symmetriesare not compatible with atomic s- or p-orbitals. Note thatthere is a qualitative correlation between the rate of decayof the evanescent states in MgO and the number of in-planeoscillations of the wavefunction.
Figure 4 shows that MgO can act as a symmetry filter.Incident wavefunctions of different symmetries will decay at
different rates within an MgO barrier if it is epitaxial on anelectrode and the two-dimensional symmetry is maintained atthe interface. In particular, incident states with11 symmetryshould be transmitted with much higher probability than othersymmetries. One way to convert this symmetry filter intoa spin filter, is to find materials compatible with MgO thathave propagating Bloch states with11 symmetry at the Fermienergy for one spin channel but not for the other.
2.7.11 Bloch states in Fe
Figure9 shows the majority and minority bands in the (001)-direction for Fe. The Fermi energy at approximately 5 eVintersects the11,15 and12′ bands for the majority channel,and the12, 15 and12′ bands for the minority spin channel.The up- and down-spin bands are very similar for the two spinchannels. The primary difference is a downward shift of the
Figure 15. Calculated conductance for the anti-parallel alignment of the moments of the Fe electrodes.
d-bands in the majority relative to the minority. Thus Fe isa possible system to use with MgO to take advantage of itssymmetry filtering effect to make a spin-filter.
The reason that the11 band is present at the Fermi energyin the majority, but not in the minority is the11 gap thatseparates a lower11 band from a higher11 band for bothspin-channels. Before discussing the spin-filtering effect ofFe–MgO tunnel junctions, it may be helpful to consider theorigin of this 11 gap. As mentioned previously,11 statesare compatible with atomic orbitals with angular symmetries,s, pz and d3z2−r 2. In most of the ferromagnetic transitionmetals, the p-states are significantly higher in energy thanthe s and d-states. Therefore, a minimal model for the11
states will consist of the s and d3z2−r 2 orbitals. For simplicity,we retain only nearest neighbor interactions, so that a tight-binding Hamiltonian to describe the system can be written for
electrons propagating along (001) as,
H(kz)=
(Es −ws cos(kza/2) −wsdcos(kza/2)−wsdcos(kza/2) Ed −wd cos(kza/2)
).
Here Es and Ed are the positions of the centers of thes- and d-bands. The parameterws describes the interactionbetween s-orbitals on nearest neighbor sites in the bcc latticeand determines the width of the s-band. The parameter,wd,plays a similar role for the d3z2−r 2 sub-band. The parameter,wsd describes the interaction between s- and d3z2−r 2 orbitalson neighboring sites. The energy bands can be obtained bysetting the determinant of(E1− H(kz)) to zero and solvingthe simple quadratic equation forE as a function ofkz.
Figure 16. Absolute square of the wavefunction fork‖ = 0 Bloch states as a function of atomic layer number for Co–MgO–Co magnetictunnel junctions. The left panel is for parallel alignment of the Co moments. The right panel is for anti-parallel alignment.
If wsd were zero, there would be separate s and d-bands asshown by the dotted (green) lines in figure10. The presenceof the interactionwsd prevents the crossing of the bandsand opens a hybridization gap as shown by the solid lines.The first-principles bands are shown as the red pluses forcomparison.
The secular equation can also be solved forkz as afunction of E. It is interesting to do this because we want toknow what will happen to11 electrons incident at an energywhere there are no propagating11 states. In figure11, thesolution of the same secular equation as in figure10is plotted,but in this instance we plot cos(kza/π) versusE rather thanEversuskza/π . Within the11 gap, cos(kza/π) andkz becomecomplex. In this region, the absolute value of the ratio of
Bloch states on neighboring sites,∣∣∣ψ(r+a)ψ(r)
∣∣∣, is plotted. The
probability density will decay as the square of this ratio asan electron with11 symmetry and an energy within the11
gap traverses a layer.
2.8. Tunneling conductance for Fe–MgO–Fe magnetictunnel junctions
Appendix A describes in detail how the transmissionprobability can be calculated in the simple barrier model. It ispossible to extend this approach [19, 21–24] to more realisticsystems in which the incident, reflected and transmittedelectrons are represented by propagating Bloch waves. Theresult of one such calculation is shown in figure12 forthe particular case of Bloch waves of different transversesymmetries incident from the left in an Fe electrode fork‖ = 0.
For majority to majority tunneling, both electrodes havepropagating Bloch states with similar symmetries,11, 15
and12′ . Each of these Bloch states will be transmitted witha probability that depends on the interfacial transmissionamplitudes but primarily on the rate of decay of thewavefunction in the barrier. This rate of decay can be seen
to vary dramatically with the transverse symmetry of theincident Bloch state with12′ decaying much faster than15,which in turn decays much faster than11. These decay rates,which can be accurately estimated from the slopes of thelines within the barrier are the same as those predicted fromthe imaginary wavevectors that can be read off of figures4and7. The primary difference between the two upper panelsdescribingG↑↑ andG↓↓ is the absence of11 symmetry in theminority–minority channel. For this reason, the conductancefor parallel alignment is dominated by the majority spin-channel conductance.
For anti-parallel alignment (bottom two panels) we mustconsider contributions to the conductance fromG↑↓ (left) andG↓↑ (right). ForG↓↑, there is no11 state so the conductancewill be small. ForG↑↓, however, there is a11 state in theleft electrode at the Fermi energy which decays relativelyslowly in the barrier. However, it does not contribute to theconductance because it cannot propagate in the minority spinchannel. It can be seen that the wavefunction continues todecay exponentially in the minority Fe electrode on the right-hand side of the barrier.
The arguments of the preceding paragraphs applyrigorously only for k‖ = 0, i.e. for electrons travelingperpendicular to the interfaces. Figure13shows the calculatedtransmission probability as function ofk‖ for the majorityspin channel for parallel alignment of the electrodes. It canbe seen that the transmission probability decreases as thethickness of the MgO barrier increases. It is also evident thatthe transmission becomes more strongly concentrated neark‖ = 0.
The transmission probability for minority electrons forparallel alignment of the moments in the electrodes is shownin figure 14. The rapid decay of thek‖ = 0 wavefunctions inthe MgO allows other contributions to become important forour idealized case with perfect two-dimensional periodicity.The minority channel conductance is strongly influenced bysurface resonance states. These cause the strong peaks seen
Figure 17. Absolute square of the wavefunction fork‖ = 0 Bloch states as a function of atomic layer number for FeCo–MgO–FeComagnetic tunnel junctions. The left panel is for parallel alignment of the FeCo moments. The right panel is for anti-parallel alignment.
the kx-ky plane. Similar peaks can be seen in the calculatedtransmission spectrum for the anti-parallel conductance(figure 15). Because of the assumed symmetry, the majorityand minority channel conductances are the same for anti-parallel alignment.
2.9. Tunneling conductance for bcc Co–MgO bcc Co andCoFe–MgO–CoFe magnetic tunnel junctions
The conductance has also been calculated [25] for bcc Coelectrodes (figure16) and for CoFe electrodes (figure17)with MgO tunnel barriers similarly to those for Fe–MgOsystems. The ordered B2 phase was assumed for CoFe. Forboth of these systems, there is only a single majority bandin the (001)-direction at the Fermi energy. This band has11
symmetry. The propagating minority Bloch states atk‖ = 0have15 and12′ symmetries. As a consequence, thek‖ = 0decay rates for the wavefunctions in the barrier can bedescribed quite simply.
For parallel alignment of the electrodes, the majoritywavefunction with11 symmetry decays much more slowlywithin the MgO barrier than the minority wavefunctionswhich have either15 or12′ symmetry. Thus the conductancewill be dominated by the majority spin channel. For anti-parallel aligment of the electrodes, on the other hand, majorityelectrons in the left electrode will enter the MgO similarlyto the case of parallel alignment, but cannot propagate inthe right electrode because there are no11 minority statesavailable. Similarly, the minority electrons in the left electrodewith 15 or 12′ symmetries will become locally majoritywhich means that they cannot propagate because only11
majority states are available.The description of the wavefunction decay in the
barrier is essentially identical for FeCo as for bcc Co. Forboth of these cases, systems with perfect two-dimensionalsymmetry at zero temperature in the absence of spin–orbitcoupling would have zero contribution to the conductance
for anti-parallel alignment of the electrodes. For othervalues of k‖, of course there will be contributions to theconductance. Nevertheless, an integral overk‖ indicates amuch larger conductance for parallel alignment than for anti-parallel. For a system with a barrier consisting of eightMgO atomic layers, the ratio of parallel to anti-parallelconductance for a perfect system in the absence of spin–orbitcoupling was predicted [25] to be approximately 50 forFe–MgO–Fe, approximately 150 for bcc Co–MgO–bcc Co,and approximately 350 for CoFe–MgO–CoFe. It should beemphasized, however, that real systems are never perfectlyordered and that spin–orbit coupling is always present.
2.10. Experimental confirmation
Because the mechanism for tunneling magnetoresistancedescribed here is quite different from that previously observedusing amorphous oxide barriers, obtaining the requiredstructures with local two-dimensional periodicity requiredsignificant effort. It was necessary, for example, to preventoxidation of the Fe layer at the Fe–MgO interface [26], thereason for this is that O atoms in the interfacial Fe layersignificantly degrade the overlap between the11 states at theFermi energy in Fe and in the MgO. In the perfect structure,the Fe electrode terminates in a square lattice of Fe atoms,immediately above each of these Fe atoms will be an O atomin the first atomic layer of MgO. The Mg atoms will sitin the centers of the squares formed by the O atoms. Thisconfiguration allows strong overlap between the Fe d3z2−r 2
and the O pz states. If, however, O atoms enter the interfacialFe layer, this strong overlap is significantly degraded andthe 11 electrons are much less likely to enter the MgO.This is illustrated in figure18, which shows a DFT supercellcalculation calculation for Fe–MgO [27].
In 2003, TMR values of 100% at low temperature and67% at room temperature were achieved [28] using molecularbeam epitaxy (MBE). In late 2004 observations of TMR in
Figure 18. Absolute square of a11 wavefunction near the Fermienergy in an Fe–MgO supercell calculation. The penetration anddecay of the wavefunction for the perfect Fe–MgO interface isshown by the solid black line. The gray dashed line shows that thewavefunction is effectively prevented from entering the MgO ifthere is oxygen in the interfacial Fe layer on the left-hand side of theMgO barrier. The supercell contained seven atomic layers of Fe and11 of MgO.
the range of 200% at room temperature were announced[29, 30] using both MBE grown Fe–MgO–Fe [29] andsputtered CoFe–MgO–CoFe [30]. The current record (in late2007) seems to be 1010% at low temperature and 500% atroom temperature [31]. In addition, the predicted high TMRof bcc Co–MgO–bcc Co has been confirmed [32].
The prediction and discovery of symmetry filter basedTMR has had a significant impact on current and potentialspintronic devices. Three important aspects of these havehelped to drive their rapid commercialization: (i) the highTMR is very important for most spintronic devices; (ii) therelatively low resistance resulting from the slow decay ofthe11 wavefunction in the MgO is also important in manytypes of devices such as read sensors for hard drives andspin–torque switched magnetic random access memory; and(iii) the discovery [30] that these systems could be grownby sputtering allowed them to be readily deposited usingdeposition tools widely available in industrial laboratories.Growth by sputter deposition requires that the system beannealed after deposition to achieve a locally crystallinebarrier and electrodes.
Presently most (if not all) new hard drives appear toutilize this technology in the magnetic field sensor used forreading the information stored on disk. These materials setsare also at present the leading candidates for a new type ofmagnetic random access memory that is written using thespin–torque effect.
3. Summary
In summary, a brief overview of the young, but rapidlygrowing field of spintronics was presented. The manner in
which electrons tunnel through simple insulators was alsodescribed and the physical phenomena that cause electronswhose wavefunctions are most symmetric to be preferentiallytransmitted were explained. It was also explained how thissymmetry filtering effect can be made into a spin filteringeffect by combining a symmetry filtering barrier material withelectrodes that have wavefunctions with the preferentiallytransmitted symmetry in one spin channel, but not in the other.The resultant large spin-filtering effect has been applied inread sensors for hard drives and may soon be applied to makea new type of non-volatile solid state magnetic memory.
Acknowledgments
This work would not have been possible without the helpof numerous colleagues, especially Xiaoguang Zhang, JamesMacLaren, Thomas Schulthess and Mairbek Chshiev. Thiswork was supported by NSF grants DMR 0123985 andECS-0529369 and by the Information Storage IndustryConsortium.
Appendix A. Simple barrier model for tunneling
In this model, which is a prototype for studies of ballistictunneling in real materials, an incident electron (on theleft) generates both a reflected wave (also on the left) anda transmitted wave (on the right). In the barrier region,which separates the two regions with propagating states, thewavefunction is evanescent. The wavefunctions on the left andright are normalized so that they eache±ik1z
√k1
, eik2z√
k2carry unit
flux, thus
J =−i
2
(ψ∗∂ψ
∂z−∂ψ∗
∂zψ
)= 1
for each of the three wavefunctions in the leads (omitting theprefactorsr and t which are the reflection and transmissionamplitudes). With this normalization the sum of the reflectionprobability, R = r ∗r , and the transmission probability,T =
t∗t , is unity. Thus the total outgoing fluxR+ T equals theincident flux, unity.
The wavevectors that enter these expressions are thez-components of the wavevectors of the electrons in the leadsand in the barrier,
k1 =
√2m(E − V1)
h̄2 − k2‖, κ0 =
√2m(V0 − E)
h̄2 + k2‖,
k2 =
√2m(E − V2)
h̄2 − k2‖.
They depend on both the energy,E and on the componentof the wavevector parallel to the layers,k‖. By requiring thatthe total wavefunction in each region, join continuously andsmoothly to the wavefunctions in the adjacent region, oneobtains four equations (2 each forz = 0 andz = a) in fourunknowns (r, t, A andB). Note that the total wavefunction in
Figure A.1. Simple barrier model for tunneling in which incidentelectrons with energyE in uniform potentialV1 encounter a barrierof heightV0 and widtha. Transmitted electrons encounter apotentialV2 on the other side of the barrier.
region 1 on the left is the sum of the incident and reflectedwaves. The four equations in four unknowns can be solvedsimply for the four unknowns. In particular the transmissionprobability is given by,
T = |t |2 =
8κ20k1k2
(k21 +κ2
0)(k22 +κ2
0) cosh(2κ0a)+ 4κ20k1k2 − (k2
1 − κ20)(k
22 − κ2
0).
When the barrier is thick enough that e2κ0a� 1, the expres-
sion for the transmission probability simplifies considerably,
T =16κ2
0k1k2e−2κ0a
(k21 +κ2
0)(k22 +κ2
0)=
4κ0k1
(k21 +κ2
0)
4κ0k2
(k22 +κ2
0)e−2κ0a
= T1T2e−2κ0a.
Written in this way, it is clear that the transmissionprobability for the simple barrier model can be factorized intothree factors, one that depends on the left interface, one thatdepends on the right interface and one that describes the decayof the wavefunction in the barrier.T1 andT2 in this formulacan be interpreted as the probability that an electron incidenton an infinite barrier will be transmitted as an evanescentwave. Note that this factorization is sufficient to derive aJulliere-type model in which the polarizations are defined notin terms of the electrode density of states but in terms of thethese interfacial transmission probabilities. Thus the TMR forthe simple barrier model is given by,
TMR =2P1P2
1− P1P2, where P1 =
T↑
1 − T↓
1
T↑
1 + T↓
1
and
P2 =T↑
2 − T↓
2
T↑
2 + T↓
2
for each value ofk‖.
Appendix B. Propagating and evanescent states inperiodic systems
Consider the Schrödinger equation for a periodic system,
[−
h̄2
2m∇
2 + V(r)− E
]ψ(r)= 0.
Here V(r) can be the effective potential used in a first-principles DFT calculation. With complete equivalence,because the system is periodic, we can express theSchrödinger equation relative to an adjacent lattice site,
[−
h̄2
2m∇
2 + V(r + a)− E
]ψ(r + a)= 0.
Since the potential is (by assumption) periodic,V(r + a)=
V(r). Thus, [H(r)− E]ψ(r)= 0 and [H(r)− E]ψ(r + a)=
0. Sinceψ(r) and ψ(r + a) are two solutions to the samehomogeneous differential equation with the same boundaryconditions, they must be equal up to a constant factor. Thusψ(r + a)= cψ(r).
References
[1] Baibich M N, Broto J M, Fert A, Van Dau F N, Petroff F,Eitenne P, Greuzet G, Friederich A and Chazelas J 1988Phys. Rev. Lett.612472
[2] Binasch G, Grünberg P, Saurenbach F and Zinn W 1989Phys.Rev.B 394828
[3] Butler W H, Zhang X-G, Nicholson D M C, Schulthess T Cand MacLaren J M 1996Phys. Rev. Lett.763216
[4] Pratt W P, Lee S-F, Slaughter J M, Loloee R, Schroeder P Aand Bass J 1991663060
[5] Slonczewski J C 1996J. Magn. Magn. Mater.1591[6] Berger L 1996Phys. Rev.B 549353[7] Daughton J M 2000IEEE Trans. Magn.362773[8] Moodera J S and LeClair P 2003Nat. Mater.2 707[9] Ney A, Pampuch C, Koch R and Ploog K H 2003Nature425
485[10] Datta S and Das B 1990Appl. Phys. Lett.56665[11] Hueso L E, Pruneda J M, Ferrari V, Burnell G, Valdés-Herrera
J P, Simmons B D, Littlewood P B, Artacho E, Fert A andMathur N D 2007Nature445410
[12] Julliere M 1975Phys. Lett.A 54225[13] Tedrow P M and Meservey R 1973Phys. Rev.B 7 318[14] Meservey R and Tedrow P M 1994Phys. Rep.238173[15] Miyazaki T and Tezuka N 1995J. Magn. Magn. Mater.139
L231[16] Moodera J S, Kinder L R, Wong T M and Meservey R 1995
Phys. Rev. Lett.743273[17] Mott N F 1936Proc. R. Soc.153699[18] de Teresa J M, Barthélémy A, Fert A, Contour J P, Lyonnet R,
Montaigne F, Seneor P and Vaurès A 1999Phys. Rev. Lett.824288
[19] Zhang X-G and Butler W H 2003J. Phys.: Condens. Matter15R1603
[20] Landauer R 1957IBM J. Res. Develop.1 223[21] Butler W H, Zhang X-G, Schulthess T C and MacLaren J M
2001Phys. Rev.B 63092402[22] MacLaren J M, Zhang X-G, Butler W H and Wang X-D 1999
Phys. Rev.B 595470[23] Butler W H, Zhang X-G, Schulthess T C and MacLaren J M
2001Phys. Rev.B 63054416[24] Velev J and Butler W H 2004J. Phys.: Condens. Matter16
R637[25] Zhang X-G and Butler W H 2004Phys. Rev.B 70172407[26] Meyerheim H L, Popescu R, Kirshcner J, Jedrecy N,
Sauvage-Simkin M, Heinrich B and Pinchaux R 2001Phys.Rev. Lett.87076102
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