Conductivity mismatch and voltage dependence of magnetoresistance in asemiconductor spin injection deviceArunanshu M. Roy, Dmitri E. Nikonov, and Krishna C. Saraswat Citation: J. Appl. Phys. 107, 064504 (2010); doi: 10.1063/1.3319570 View online: http://dx.doi.org/10.1063/1.3319570 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v107/i6 Published by the American Institute of Physics. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Conductivity mismatch and voltage dependence of magnetoresistancein a semiconductor spin injection device

Arunanshu M. Roy,1,a� Dmitri E. Nikonov,2 and Krishna C. Saraswat11Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA2Components Research, Intel Corporation, Santa Clara, California 95052, USA

�Received 14 October 2009; accepted 18 January 2010; published online 25 March 2010�

Magnetoresistance �MR� in a semiconductor spin injection and detection device is simulated bycombining the formalisms for tunneling probabilities and spin polarized carrier diffusion. Therebydependences of resistance and spin selectivity at the ferromagnet-semiconductor interface onvoltage as well as on material parameters are determined. This leads to predicting the voltagedependence of MR of the overall ferromagnet-semiconductor device. It is found to be qualitativelysimilar to that of a magnetic tunnel junction. Similarly the dependence of the MR on the tunnelingbarrier height and thickness, and doping density are studied. Optimal material parameters fordetection of spin polarized current are thus determined, which are helpful for designing experimentson spin injection into semiconductors. © 2010 American Institute of Physics.�doi:10.1063/1.3319570�

I. INTRODUCTION

Injection of spin polarized current into semiconductorshas garnered great interest recently with the development ofspintronics,1 and proposals of semiconductor spintronic de-vices. For a critical review see Refs. 2–4. Such devices areconsidered as one of emerging logic technologies which havethe potential to work alongside electronic transistors and ful-fill application specific functions.5 Since the pioneering pro-posal of the spin transistor by Datta and Das,6 several experi-ments have confirmed the possibility of spin injection intosemiconductors like GaAs and Si. Experiments on GaAshave been based on optical and electrical detection7–10 whilethose on Si have utilized all-electrical effects such as nonlo-cal measurements,11 use of hot electron spin injectors,12 oroptical detection13 via spin-polarized fluorescence. Magne-toresistance �MR� devices based on spin injection and detec-tion through semiconductors suffer from the problem of con-ductivity mismatch14 between the ferromagnetic spin injectorand the semiconductor layer. It has been shown that by usinga suitable spin-dependent interface resistance the problem ofconductivity mismatch can be overcome.15 Spin injectioninto semiconductors using tunnel barriers and Schottky bar-riers at a ferromagnet-semiconductor interface has beendemonstrated.16–19 Dependence of spin injection and MR onthe height of the Schottky barrier and doping of the semicon-ductor has been discussed in Ref. 20. High MR of a devicewith a semiconductor channel would allow making a spinfield effect transistor �spinFET�.21 Quantum transport simu-lation of a spinFET via the Keldysh nonequilibrium Green’sfunction method has been performed.22

This work studies in more detail the MR properties of asemiconductor layer sandwiched between ferromagnetic lay-ers using spin-dependent tunneling resistance to overcomethe conductivity mismatch problem. In particular, we de-velop a numerical model to account for aspects important in

practical spintronic devices: the voltage dependent interfaceresistance and MR and the voltage and charge distribution inthe depletion region of the semiconductor. An analysis ofrelevant semiconductor and oxide parameters is performed todesign devices with an optimized MR ratio.

II. METHODOLOGY

A typical semiconductor spin injection and detection de-vice consisting of a semiconductor layer sandwiched be-tween two ferromagnetic layers is shown in Fig. 1.

The two ferromagnets can be magnetized parallel or an-tiparallel �or, in general, at an angle to each other�. MR isdefined as the following ratio of resistances, RP and RAP, inthe above two cases, respectively,

MR =RAP − RP

RP. �1�

MR depends on the conduction of the carriers with spin “up”�along the magnetization of the source� and spin “down”�opposite to it� as well as spin relaxation, which causes spinflip, i.e., transitions between the spin orientations. Tunnelbarriers serve as spin-dependent resistance elements andthereby increase MR. The total resistance of the interface iscombined from the tunneling and the Schottky barrierswhich appear at the junctions of metals and semiconductors.

a�Electronic mail: amroy@stanford.edu.

FIG. 1. Schematic diagram of semiconductor spin injection and detectiondevice. When a voltage is applied to the device, one junction is forwardbiased and the other is reverse biased.

JOURNAL OF APPLIED PHYSICS 107, 064504 �2010�

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Here we model spin-polarized carrier transport in thesemiconductor based on the one-dimensional �1D� spin dif-fusion approach proposed by Fert and co-workers.23,24 Thenovelty element that we introduce here is a more rigoroustreatment of tunneling through the barriers. The spin-dependent interface resistance is one of the most crucial pa-rameters in spin diffusion simulations as it controls nonequi-librium spin polarization �expressed by the splitting ofelectrochemical potential at an interface� and hence MR. Inthis approach we are able to obtain the voltage dependenceof the tunneling resistances and �for the first time� of theoverall MR, while in Ref. 24 the resistance and spin selec-tivity of the barrier were set constant and independent ofvoltage. Nontrivial features of the voltage dependence alsoarise from the fact that the resistances of the reverse andforward biased Schottky barriers are different. Also we areable to study the dependence of MR on the parameters of thetunneling barriers �as well as the semiconductor itself�. Sec-tions II A and II B briefly describe the mathematical equa-tions used to model spin diffusion and interface I-V charac-teristics.

A. Spin diffusion model

Using the approach of Valet and Fert,23 a macroscopicmodel for spin diffusion in a multilayer stack of materials ischaracterized by spin diffusion equations. A short review ofthe mathematical model used is provided here.

The spin diffusion is described in terms of the differencein electrochemical potentials for electrons with spin +1 /2and spin �1/2 by Eq. �2�. We will use subscripts that refer toelectrons with spins +1 /2 and �1/2. Thus, �+ and �− des-ignate the electrochemical potentials and �+ and �− desig-nate the conductivity for the carriers with spins up and down,respectively, and ��= ��+−�−� /2 is used to describe spindiffusion. Current continuity condition is expressed by Eq.�3�

�2��

�z2 =��

lsf2 , �2�

�2��+�+ + �−�−��z2 = 0. �3�

The conductivity itself is expressed in terms of a spin-dependent parameter � and a resistivity parameter �� usingEq. �4�

�↑�↓� =1

�↑�↓�= 2���1 − �+ ��� . �4�

The resistivity �↑ and �↓ refer to the resistivity for themajority and minority electrons, respectively, in the material.This notation ↑�↓ � is also used to label other variables de-scribing majority�minority� electrons. The parameter lsf iscalled the spin-flip length; it is characteristic of the materialand is related to the spin relaxation time �sf and the densityof carriers as follows �for nondegenerate semiconductors�.24

kb is the Boltzmann constant and T is the absolute tempera-ture

lsf =� kBT�sf

2ne2��. �5�

We apply the transfer matrix formalism here which per-mits writing down the solution easily even for a stack ofmultiple layers, without being encumbered by complicatedalgebraic expressions. A general solution to this set of equa-tions can be written as follows where J is the current densityand z is the distance in the direction perpendicular to thematerial interfaces.

�↑ = �1 − �2�q��Jz + K1 + �1 + ���K2 exp� z

lsf�

+ K3 exp�− z

lsf� �6�

�↓ = �1 − �2�q��Jz + K1 − �1 − ���K2 exp� z

lsf�

+ K3 exp�− z

lsf� �7�

J↑ = �1 − ��J

2+

1

2q��lsf�K2 exp� z

lsf� − K3 exp�− z

lsf�

�8�

J↓ = �1 + ��J

2−

1

2q��lsf�K2 exp� z

lsf� − K3 exp�− z

lsf�

�9�

Note that K1, K2, and K3 are constants for every layerand we need to choose appropriate values for these constantthat satisfy boundary conditions. For nonmagnetic materiallayers, the same equations apply but �=0.

Under the assumption that current continuity is main-tained and there is no spin flipping at an interface, the dis-continuity in the electrochemical potential of majority spinelectrons ��↑=�↑�z0

+�−�↑�z0−� and minority spin electrons

��↓=�↓�z0+�−�↓�z0

−� at interface z=z0 is modeled using aninterface resistance area product parameter rb

� and spin selec-tivity parameter �.

��↑�z0� = 2qrb��1 − ��J↑ �10�

��↓�z0� = 2qrb��1 + ��J↓ �11�

The above equations are further expressed in terms ofmaterial parameters relating the coefficients of the solutionK1, K2, and K3 in adjacent layers. This constitutes the trans-fer matrix formalism. The details of it are outlined in Appen-dix. This makes it suitable for studying multilayer deviceswith interfaces and also semiconductors graded dopingwhere the spin diffusion properties vary with semiconductordoping density.

B. Model for the tunneling I-V characteristics

As shown in Fig. 2 a simple model for a ferromagnetband structure consists of majority and minority bands whichare split due to exchange interaction. In this figure, � is the

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exchange interaction energy splitting, Ef is the Fermi energyof the ferromagnetic metal, m is the metal work function, s

is the semiconductor electron affinity, and ox is the electronaffinity of the tunnel barrier.

The current for each band is calculated by treating themajority and minority bands independently. Using the Tsu–Esaki model,25 the current through the interface for one bandis given by Eq. �12�.

Jtot =2�meffq

h3 Emin

Emax

�� z�Ntot� z�d z, �12�

where meff is a density of states effective mass, h is Planck’sconstant, � is the electron tunneling probability, �z is theelectron energy perpendicular to the interface, and Ntot is asupply function which depends on the Fermi energy in metaland semiconductor. Emin is the minimum energy where elec-tron states are present on both sides of the barrier. Emax inthis case can be taken to infinity but practically the Fermidistribution causes the current contribution of electron statesto fall very rapidly for energies above the Fermi energy level.Under the assumptions of parabolic bands, Fermi–Dirac sta-tistics for electrons in the conduction bands, and transmis-sion probability independent of the parallel wave vector atthe interface, the supply function Ntot is given by

Ntot = �kbT�ln� 1 + exp� fm − z

kbT�

1 + exp� fs − z

kbT� � , �13�

where fm is the Fermi energy in the metal and fs is theFermi energy in the semiconductor. To calculate the tunnel-ing probability for a given applied voltage, we determine thedepletion region in the semiconductor by solving the 1DPoisson equation and using equilibrium charge concentra-tions consistent with the band bending. The value of thetunneling transmission probability is then obtained using thetransfer matrix formalism for tunneling problems. Figure 3shows the calculated majority and minority currents for spininjection into n-type Si with an Al2O3 tunnel barrier. Theinterface resistance depends upon the tunneling barrier thick-ness and the semiconductor Schottky barrier. The spin de-pendence of the tunneling currents is mainly due to the lack

of minority states near the Fermi energy in the ferromagnet.

III. INTERFACE RESISTANCE PARAMETERS NEARZERO VOLTAGE

At low voltages, both the forward and the reverse biasedjunctions will see a very small potential drop and will havethe same value of rb

� and � defining the interface spin-dependent resistance at the two interfaces. Following themethodology described in Sec. II we determine the spin-dependent interface resistance parameters near V=0 usingthe Tsu–Esaki model for structures with different dopingdensities, oxide thicknesses, oxide material, and metal workfunctions. We present here some results showing the varia-tion in the spin-dependent interface resistance parametersand discuss how they will relate to the MR of the entiredevice. For these simulations the ferromagnetic metal hasparameters �=2.2 eV and Ef =2.2 eV. The ferromagnet isthus almost half metallic. The minority spin electron statesonly start at the Fermi energy level. The majority spin elec-tron states present below the Fermi energy thus contribute toan additional current which gives rise to spin injection. Thesimulation parameters used for different insulators �dielectricconstant �, bandgap Eg, barrier height to conduction bandelectrons in n-Si �ox−s�, and tunneling effective mass me�are reported in Table I. Important simulation parameters usedfor Si are relative permittivity �s=11.9, electron affinity s

=4.05 eV. Other simulation parameters specific to simula-tions are reported with the corresponding figures. All simu-lations are for temperature T=300 K.

FIG. 2. �Color online� The exchange splitting between the electrons in themajority and minority bands of the ferromagnet gives rise to a spin-dependent contact resistance.

−1 −0.5 0 0.5 110

4

106

108

1010

Voltage (V)

Cur

rent

dens

ity(A

/m2 )

Majority band currentMinority band current

FIG. 3. �Color online� Simulated currents for majority and minority bands.The structure used is metal �m=4 eV, Ef =2.2 eV, �=2.2 eV� Al2O3

�8 Å� Si �n-type doping 1016 cm−3�.

TABLE I. Simulation parameters.

Insulator �Eg

�eV�ox−s

�eV� me

Al2O3 10 8.8 2.6 0.2HfO2 20 5.8 1.5 0.2SiO2 3.9 9 3.1 0.5

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A. Dependence on doping density

As seen in Fig. 4, changing the doping density does notaffect the resistance area product parameter rb

� significantlybut the spin selectivity parameter � increases at higher valuesof doping density. The variation in rb

� for different oxides isdue to different barrier heights of the oxides to conductionband electrons. Spin injection at the ferromagnet contact isdue to the additional majority spin electron density of statespresent below the Fermi energy level. The faster is the decayof transmission probability as we go to lower energies, thelower is the spin selectivity of the barrier. Thus a tunnelbarrier with a higher tunneling effective mass will lead tolower spin selectivity since the transmission probability at-tenuates faster inside it and the relative contribution of theadditional majority spin states is lower. Thus we can see thatSiO2 has a lower spin selectivity than Al2O3 and HfO2. Bar-rier height also plays a role in determining spin selectivityhowever the effect is less significant than that of tunnelingeffective mass as we can see that Al2O3 and HfO2 have simi-lar spin selectivity. At higher doping densities, the Schottkybarrier is reduced thereby increasing the transmission prob-ability at energies near the additional majority spin electronstates. This leads to improved spin selectivity at higher dop-ing densities.

B. Dependence on oxide thickness

Increasing the oxide thickness leads to an exponentialincrease in the parameter rb

� as shown in Fig. 5�a� due to theincrease in tunneling resistance. Thus rb

� increases rapidlyand for large oxide thicknesses will lie outside the favorableregion for spin injection and detection. Increasing the oxidethickness also leads to a fall in the spin selectivity of thebarrier as shown in Fig. 5�b�. As the oxide thickness in-creases, the transmission probability falls faster as we go tolower electron energies. As explained in Sec. III B, this leadsto a lower relative contribution of the additional majorityspin electron states and results in a lower spin selectivity.This indicates that thin oxides are required for good MRcharacteristics. It is important to note that this predictiondoes not hold for crystalline MgO where the spin selectivityincreases with oxide thickness. This is because the simpletunneling model used here is insufficient to describe the tun-neling through a crystalline MgO barrier. It is important totake into account the detailed band structure of the metal andMgO while calculating tunneling currents in this case be-cause the electrons will couple to different states in the insu-lator depending on the symmetry of the majority and minor-ity electron bands. Practically at very low thickness too thespin selectivity of the oxide barrier is reduced due to defectsin the thin oxide film and spin flipping at the interface can nolonger be neglected.

FIG. 4. �Color online� Dependence of spin-dependent interface resistancearea product parameters rb

� and � on the Si n-type doping density for differ-ent oxides of thickness 1 nm. Ferromagnet work function m=4.0 eV.

FIG. 5. �Color online� Dependence of interface resistance area product pa-rameters rb

� and � on the oxide thickness for different oxides on Si withn-type doping of 1017 cm−3. Ferromagnet work function m=4.0 eV.

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C. Dependence on metal work function

Increasing the metal work function increases the tunnelbarrier height as well as the Schottky barrier height in thesemiconductor leading to an increase in the rb

� parameter asshown in Fig. 6�a�. However, since the tunnel barriers usedin these devices are very thin, the contribution of theSchottky barrier to the increasing rb

� parameter is most sig-nificant. As seen in Fig. 6�a�, since Si has a work function of4.05 eV there is an exponential increase in rb

� for metal work

functions above that due to the increasing Schottky barrier.As seen in Fig. 6�c� the spin selectivity � also falls when themetal work function is greater than the semiconductor workfunction due to the increased Schottky barrier resistance. Theincreased Schottky barrier strongly attenuates the transmis-sion probability for the additional majority spin electronstates. Fermi pinning of the metal work function has beenneglected here. It has been observed that by incorporating athin insulator layer between the metal and the semiconductor,it is possible to alleviate the problem of Fermi pinning at ametal semiconductor interface.26,27

IV. VOLTAGE DEPENDENCE OF MR

Since previously published simulations23,24 used a con-stant spin-dependent interface resistance, they failed to cap-ture the effect of the voltage dependence of a tunnel barrierinterface resistance and also the fact that I-V characteristicsof a tunnel barrier are not necessarily symmetric for positiveand negative voltages. In these works the spin-dependent in-terface resistance is modeled in terms of constant parametersrb

� �interface resistance area product� and � �spin selectivity�.Thus the interface resistance area product is given by 2rb

��1

FIG. 6. �Color online� Dependence of spin-dependent interface resistancearea product parameters rb

� and � on the metal work function for differentoxides of thickness 1 nm and Si n-type doping 1017 cm−3.

FIG. 7. �Color online� Voltage dependence of resistance area product fordevice in Fig. 1. The parallel configuration has a lower resistance than theanti-parallel configuration.

−1.5 −1 −0.5 0 0.5 1 1.50

10

20

30

40

50

60

Voltage applied (V)

Mag

neto

resi

stan

cera

tio(%

)

FIG. 8. �Color online� Voltage dependence of MR for device in Fig. 1. Theinterface resistance and spin selectivity is calculated using the current volt-age characteristics shown in Fig. 3. Length of the semiconductor channel is200 nm.

FIG. 9. �Color online� Voltage dependence of interface resistance area prod-uct for simulations in Fig. 3. Reverse biased junctions show a substantialincrease in interface resistance area product with applied voltage.

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−�� for the majority electrons and 2rb��1+�� for the minority

electrons. In the present work, the computed I-V character-istics for majority and minority bands are used self-consistently with the majority and minority electrochemicalpotentials in the spin diffusion simulations to compute rb

� and� �a voltage and spin-dependent interface resistance for eachinterface�. As shown in Figs. 7 and 8, the resistance areaproduct of these devices increases with applied voltage andthe MR of these devices is maximal when the potential dropacross the structure nears zero volt.

This is similar to the experimentally observed character-istics of magnetic tunnel junctions.28 Figures 9 and 10 showthat both interface resistance area product and the spin selec-tivity of the forward and reverse biased contacts are verydifferent and they vary substantially with voltage applied tothe device. This is because the Schottky barrier resistancesare intrinsically voltage dependent.

To see how the properties of the spin injection and de-tection device affect the voltage dependence of MR we studydevices with different semiconductor doping densities, oxidethickness, and metal work function. In Fig. 11 we see that aswe increase the semiconductor doping density the maximumMR seen near zero volts first increases and then decreasesagain. In Fig. 12 we see that the oxide thickness also affectsthe maximum MR and there will exist an optimum value ofoxide thickness where the MR is maximal. Figure 13 shows

that a lower value of metal work function will invariablylead to better MR characteristics and the MR value can satu-rate near a high value even when applied voltage is large.

V. EFFECT OF MATERIAL PARAMETERS ONMR

Using the MR at zero volts as a metric, we study theeffect of parameters such as oxide thickness, semiconductordoping, and ferromagnet band structure. As we have seen inthe Sec. IV the device properties need to be optimized to getthe best MR characteristics. The spin selectivity of tunnelbarrier is affected most strongly by the ferromagnet bandstructure. As we approach a half metallic ferromagnet withthe minority conduction band minimum at or above theFermi level, the spin selectivity improves greatly since theFermi distribution ensures that minority currents are low.The semiconductor doping determines the required interfacecontact resistance area product which itself is determined bythe metal work function, the oxide thickness and the semi-conductor doping. We study the effect of these parameters onMR. In Fig. 14 we see that the RA product for the entiredevice is a strong function of oxide thickness and also de-pends on the doping density of the semiconductor. In Figs.15 and 16 the variation in MR at zero volts with oxide thick-ness, semiconductor doping, and metal work function is

FIG. 10. �Color online� Voltage dependence of spin selectivity of the inter-faces shows that the forward and reverse biased junction have very differentspin selectivity which varies significantly with the applied voltage.

FIG. 11. �Color online� Voltage dependence of MR for simulations of dif-ferent values of n-type doping. Oxide thickness �Al2O3�=1 nm and metalwork function m=4 eV.

FIG. 12. �Color online� Voltage dependence of MR for simulations of dif-ferent values of oxide �Al2O3� thickness. n-type doping=1017 cm−3 andmetal work function m=4 eV.

FIG. 13. �Color online� Voltage dependence of MR for simulations of dif-ferent values of metal work function. n-type doping=1017 cm−3 and oxidethickness �Al2O3�=1 nm.

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shown. Different tunnel barrier materials Al2O3, SiO2, andHfO2 have been simulated. The semiconductor is n-type Siand the ferromagnet is nearly half metallic with the bottomof the minority conduction band coinciding with the Fermienergy. A lower metal work function increases the MR sig-nificantly and for a given work function the plots show the

optimal values of doping and oxide thickness. Among theoxides simulated here, HfO2 appears to show the best MRcharacteristics while SiO2 shows poor MR characteristicsdue to the high barrier height it has for electrons.

VI. CONCLUSION

We demonstrate that the voltage dependence of interfaceresistance captured by this numerical model, significantly af-fect the resulting MR. The voltage dependence of MR ismoderate and comparable to that in magnetic tunnel junc-tions. Our models and implementation are both general andefficient. They can be used to study the effect of materialparameters as well as doping effects, bringing us closer todesigning a working spinFET.

ACKNOWLEDGMENTS

The authors are grateful to Intel Corp. for providing sup-port for this work and wish to acknowledge Professor ShanWang, Dr. Tejas Krishnamohan, J. Jason Lin, and DonkounLee for helpful discussions.

FIG. 14. �Color online� Parallel resistance area product �� m2� variationusing Al2O3. Ferromagnet work function=4 eV. Increasing the oxide thick-ness increases the resistance area product.

FIG. 15. �Color online� MR variation using �a� Al2O3, �b� SiO2, and �c�HfO2. Ferromagnet work function=4 eV. Increasing the resistance areaproduct reduces the MR. Higher doping values need lower oxide thickness.

FIG. 16. �Color online� MR variation using �a� Al2O3, �b� SiO2, and �c�HfO2. Ferromagnet work function=3.8 eV. A low work function giveshigher MR due to a lower resistance area product and higher spin selectivityat the interfaces.

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APPENDIX: TRANSFER MATRIX FORMULATION

We define the following variables for use in the transfermatrix formalism:

�avg�z� =�+�z� + �−�z�

2, �A1�

���z� =�+�z� − �−�z�

2, �A2�

��J�z� = q��lsf�J+�z� − J−�z�� , �A3�

�J�z� = q��lsf�J+�z� + J−�z�� , �A4�

�̄�z� = �avg�z����z���J�z��J�z�

� . �A5�

The constants �� and lsf depend on the material of thelayer. Note that ��J and �J are obtained from currents andhave the same dimensions as the electrochemical potential.The new variables as defined above may be written in termsof the constants K1, K2, and K3 for each layer as follows:

�avg�z� = K1 + K2� exp� z

lsf� + K3� exp�− z

lsf� + �1 − �2�

�� z

lsf��J, �A6�

���z� = K2 exp� z

lsf� + K3 exp�− z

lsf� , �A7�

��J�z� = K2 exp� z

lsf� − K3 exp�− z

lsf� − ��J, �A8�

K̄ = K1

K2

K3

q��lsfJ� . �A9�

We can now see how these variables change from thestart of one layer to the end of the same layer. An extravariable has been defined even though there are only threeunknown constants so that the transfer matrix like formalismcan be maintained.

�̄�z� = TK̄→�̄�z�K̄ , �A10�

TK̄→�̄�z� = 1 �ez/lsf �e−z/lsf�1 − �2�z

lsf

0 ez/lsf e−z/lsf 0

0 ez/lsf − e−z/lsf − �

0 0 0 1� . �A11�

Also at the beginning of the layer �z=0�

�̄�0� = TK̄→�̄�0�K̄ , �A12�

K̄ = TK̄→�̄

−1 �0��̄�0� , �A13�

�̄�z� = TK̄→�̄�z�TK̄→�̄

−1 �0��̄�0� , �A14�

Tlay = 1 � cosh

z

lsf� sinh

z

lsf�2 sinh

z

lsf

− � + �1 − �2�z

lsf

0 coshz

lsfsinh

z

lsf� sinh

z

lsf

0 sinhz

lsfcosh

z

lsf�cosh

z

lsf

− �

0 0 0 1

� . �A15�

Now that we have a relation between our defined variables atthe beginning and end of the layer, we focus on the boundaryconditions at the interfaces. Under the assumption that cur-rent continuity is maintained and there is no spin flipping atan interface, we have the following boundary conditions atan interface written in a matrix form that will help in imple-mentation:

�J�z� = �↑�z��↓�z�J↑�z�J↓�z�

� , �A16�

�J�z0+� =

1 0 2qrb��1 − �� 0

0 1 0 2qrb��1 + ��

0 0 1 0

0 0 0 1��J�z0

−� .

�A17�

These boundary variables are related to our previously de-fined variables as follows:

�̄�z� = 1

2

1

20 0

1

2

− 1

20 0

0 0 q���z�lsf� �z� − q���z�lsf

� �z�0 0 q���z�lsf

� �z� q���z�lsf� �z�

��J�z� .

�A18�

Using these relations, we get the transfer matrix at theboundary

�̄�z0+� = Tint�̄�z0

−� , �A19�

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Tint = 1 0

− �rb�

���z0−�lsf�z0

−�rb

�

���z0−�lsf�z0

−�

0 1rb

�

���z0−�lsf�z0

−�− �rb

�

���z0−�lsf�z0

−�

0 0���z0

+�lsf� �z0

+����z0

−�lsf� �z0

−�0

0 0 0���z0

+�lsf� �z0

+����z0

−�lsf� �z0

−�

� . �A20�

We can now use Eqs. �A11� and �A12� to find the change inthe system variables from the first interface to the last inter-face. If we have semi-infinite ferromagnetic materials at thefirst and last interfaces then just before the first interface

�avg

��

��J

�J

� = �K2

K2

− �q��lsfJ + K2

q��lsfJ� . �A21�

Beyond the last interface

�avg

��

��J

�J

� = K1 + �K3

K3

− �q��lsfJ − K3

q��lsfJ� . �A22�

Since we have three unknowns here and we have threeindependent equations that we will get relating the systemvariables before the first interface to the system variablesbeyond the last interface, we can solve for all three and thusknow the polarization of the current in any layer.

1I. Žutić, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 �2004�.2G. I. Bourianoff, P. A. Gargini, and D. E. Nikonov, Solid-State Electron.51, 1426 �2007�.

3D. E. Nikonov and G. I. Bourianoff, J. Supercond. Novel Magn. 21, 479�2008�.

4J. A. Hutchby, R. Cavin, V. Zhirnov, J. E. Brewer, and G. Bourianoff,Computer 41, 28 �2008�.

5Semiconductor Industry Association, International Technology Roadmapfor Semiconductors, http://public.itrs.net/, 2007.

6S. Datta and B. Das, Appl. Phys. Lett. 56, 665 �1990�.7R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A. Waag,and L. W. Molenkamp, Nature �London� 402, 787 �1999�.

8X. Lou, C. Adelmann, S. A. Crooker, E. S. Garlid, J. Zhang, K. S. M.Reddy, S. D. Flexner, C. J. Palmstrøm, and P. A. Crowell, Nat. Phys. 3,197 �2007�.

9H. J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier, H.-P. Schönherr,and K. H. Ploog, Phys. Rev. Lett. 87, 016601 �2001�.

10Y. Ohno, D. K. Young, B. Beschoten, F. Matsukura, H. Ohno, and D. D.Awschalom, Nature �London� 402, 790 �1999�.

11O. M. J. vant Erve A. T. Hanbicki, M. Holub, C. H. Li, A. Awo-Affouda,P. E. Thompson, and B. T. Jonker, Appl. Phys. Lett., 91, 212109 �2007�.

12I. Appelbaum, B. Huang, and D. J. Monsma, Nature �London� 447, 295�2007�.

13B. T. Jonker, G. Kioseoglou, A. T. Hanbicki, C. H. Li, and P. E. Thomp-son, Nat. Phys. 3, 542 �2007�.

14G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. vanWees, Phys. Rev. B 62, R4790 �2000�.

15E. I. Rashba, Phys. Rev. B, 62, R16267 �2000�.16A. T. Hanbicki, B. T. Jonker, G. Itskos, G. Kioseoglou, and A. Petrou,

Appl. Phys. Lett. 80, 1240 �2002�.17V. F. Motsnyi, J. De Boeck, J. Das, W. Van Roy, G. Borghs, E. Goovaerts,

and V. I. Safarov, Appl. Phys. Lett. 81, 265 �2002�.18X. Jiang, R. Wang, R. M. Shelby, R. M. Macfarlane, S. R. Bank, J. S.

Harris, and S. S. P. Parkin, Phys. Rev. Lett. 94, 056601 �2005�.19S. A. Crooker, M. Furis, X. Lou, C. Adelmann, D. L. Smith, C. J. Palm-

strøm, and P. A. Crowell, Science 309, 2191 �2005�.20B.-C. Min, K. Motohashi, C. Lodder, and R. Jansen, Nature Mater. 5, 817

�2006�.21S. Sugahara and M. Tanaka, Appl. Phys. Lett. 84, 2307 �2004�.22T. Low, M. S. Lundstrom, and D. E. Nikonov, J. Appl. Phys. 104, 094511

�2008�.23T. Valet and A. Fert, Phys. Rev. B 48, 7099 �1993�.24A. Fert and H. Jaffrès, Phys. Rev. B 64, 184420 �2001�.25R. Tsu and L. Esaki, Appl. Phys. Lett. 22, 562 �1973�.26M. Kobayashi, A. Kinoshita, K. Saraswat, H.-S. P. Wong, and Y. Nishi, J.

Appl. Phys. 105, 023702 �2009�.27D. Connelly, C. Faulkner, P. A. Clifton, and D. E. Grupp, Appl. Phys. Lett.

88, 012105 �2006�.28M. Hosomi, H. Yamagishi, T. Yamamoto, K. Bessho, Y. Higo, K. Yamane,

H. Yamada, M. Shoji, H. Hachino, C. Fukumoto, H. Nagao, and H. Kanoetal., Tech. Dig. - Int. Electron Devices Meet. 2005, 459.

064504-9 Roy, Nikonov, and Saraswat J. Appl. Phys. 107, 064504 �2010�

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