Complex impedance of a spin injecting junctionEmmanuel I. Rashba Citation: Applied Physics Letters 80, 2329 (2002); doi: 10.1063/1.1465527 View online: http://dx.doi.org/10.1063/1.1465527 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/80/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Efficient spin extraction from nonmagnetic semiconductors near forward-biased ferromagnetic-semiconductormodified junctions at low spin polarization of current J. Appl. Phys. 96, 4525 (2004); 10.1063/1.1788839 Spin injection across ( 110 ) interfaces: Fe ∕ GaAs ( 110 ) spin-light-emitting diodes Appl. Phys. Lett. 85, 1544 (2004); 10.1063/1.1786366 Response to “Comment on ‘Efficient electrical spin injection from a magnetic metal/tunnel barrier contact into asemiconductor’” [Appl. Phys. Lett. 81, 2130 (2002)] Appl. Phys. Lett. 81, 2131 (2002); 10.1063/1.1507361 Spin injection in ferromagnet-semiconductor heterostructures at room temperature (invited) J. Appl. Phys. 91, 7256 (2002); 10.1063/1.1446125 Efficient electrical spin injection from a magnetic metal/tunnel barrier contact into a semiconductor Appl. Phys. Lett. 80, 1240 (2002); 10.1063/1.1449530

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Complex impedance of a spin injecting junctionEmmanuel I. Rashbaa)

Department of Physics, The State University of New York at Buffalo, Buffalo, New York 14260

~Received 21 December 2001; accepted for publication 31 January 2002!

Theory of the ac spin injection from a ferromagnetic electrode into a normal conductor through atunnel or Schottky contact is developed. Diffusion and relaxation of nonequilibrium spins results ina frequency-dependent complex impedanceZ(v) controlled by the spin relaxation rates and theresistances involved. Explicit expression forZ(v) is presented. Experimental investigation of thefrequency dependence ofZ should allow measuring spin relaxation times in both conductors, theireffective resistances, and also the parameters of the contact controlling the spin injection. ©2002American Institute of Physics.@DOI: 10.1063/1.1465527#

Spin injection from magnetic contacts into semiconduc-tor microstructures is one of the basic problems of theemerging field of spintronics.1 Effective spin injection hasbeen first achieved from semimagnetic semiconductors butonly at low temperatures and in an external magnetic field.Reliability of the first reports2,3 on the observation of spininjection from ferromagnetic metals into semiconductors atthe level of about 1% has been questioned. The problemstems from the conductivity mismatch between a metal and asemiconductor, and the spin injection coefficientg at thelevel of only aboutg;sN /sF is expected from a ‘‘perfect’’ohmic contact,4 sF and sN being conductivities of a ferro-magnetic aligner and a normal conductor, respectively. If thelatter is a semiconductor and the former is a metal, thensN!sF and, therefore,g!1. However, it has been proposedthat employing a spin selective contact with the resistancer c*r F ,r N , wherer F andr N are effective resistances of bothconductors, can fix the problem.5 With spin selective con-tacts of enlarged resistance, values ofg;10% have beenrecently reported by different experimental groups.6–9

Therefore, a reliable and nondestructive control of theresistancesr c ,r F , and r N is an important problem. Physi-cally r c is the resistance of a tunnel or a Schottky barrier thatis strongly technology dependent. Effective resistances ofboth conductors depend not only on their bulk conductivitiesbut also on the spin diffusion lengths in these materials,LF

andLN , as r F5LF /sF and r N5LN /sN . Measuring the dcresistance of a junction provides only a single quantity, itstotal resistanceR. In what follows, I find an explicit expres-sion for the frequency dependence of the complex imped-anceZ(v) of an F –N junction in the diffusive regime andshow that it can provide a lot of information on the param-eters involved.

Let us consider a junction consisting of a ferromagneticconductor (F) at x,0, a spin selective contact atx50, anda normal conductor (N) at x.0. The theory is based on thesystem of standard equations5,10–12 relating the currents ofup- and down-spin electronsj ↑,↓(x,t) in theF region to theirelectrochemical potentialsz↑,↓(x,t):

j ↑,↓~x,t !5s↑,↓¹z↑,↓~x,t !, ~1!

these potentials being connected to the nonequilibrium con-centrationsn↑,↓(x,t) of both spins as

z↑,↓~x,t !5~eD↑,↓ /s↑,↓!n↑,↓~x,t !2wF~x,t !. ~2!

HerewF(x,t) is the electric potential andD↑,↓ are diffusioncoefficients that are related to the temperature dependentdensities-of-statesr↑,↓5]n↑,↓ /](ez↑,↓) by Einstein equa-tions e2D↑,↓5s↑,↓ /r↑,↓ . If the Poisson equation is substi-tuted by the quasineutrality condition

n↑~x,t !1n↓~x,t !50, ~3!

then the continuity and charge conservation equations take aform

¹ j ↑~x,t !5en↑~x,t !/tsF , J~ t !5 j ↑~x,t !1 j ↓~x,t !, ~4!

whereJ(t) is the total current andtsF is the spin relaxation

time.In symmetric variables

zF~x,t !5z↑~x,t !2z↓~x,t !,

ZF~x,t !5 12@z↑~x,t !1z↓~x,t !#,

j F~x,t !5 j ↑~x,t !2 j ↓~x,t !, ~5!

the basic equations are

¹2zF~x,t !5zF~x,t !/LF21] tz~x,t !/DF ,

~6!¹ZF~x,t !52~Ds/2sF!¹zF~x,t !1J~ t !/sF ,

j F~x,t !52~s↑s↓ /sF!¹zF~x,t !1~Ds/sF!J~ t !, ~7!

whereDF5(s↓D↑1s↑D↓)/sF is the ‘‘bispin’’ diffusion co-efficient, sF5s↑1s↓ , Ds5s↑2s↓ , and LF

25DFtsF .

Similar equations operate in theN region.Neglecting the spin relaxation in the contact, the bound-

ary conditions at the contact are

j F~0,t !5 j N~0,t ![ j ~0,t !,

j ↑,↓~0,t !5S↑,↓@z↑,↓N ~0,t !2z↑,↓

F ~0,t !#, ~8!

whereS↑,↓ are spin selective conductivities of the contact,z↑,↓

F (0,t) and z↑,↓N (0,t) are the values ofz↑,↓

F(N)(x,t) at bothsides of the contact, and similarly forj ↑,↓(0,t). The first ofEqs.~8! ensures the continuity of the spin-polarized current,

a!Also at: Department of Physics, MIT, Cambridge, Massachusetts 02139;electronic mail: erashba@mailaps.org

APPLIED PHYSICS LETTERS VOLUME 80, NUMBER 13 1 APRIL 2002

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and the second relates the partial spin polarized currents tothe discontinuities in the corresponding electrochemical po-tentials.

If the time dependence of the ac signal is chosen asexp(2ivt), then in the diffusion equation, Eq.~6!, for theFandN regions the spin diffusion lengths should be changedto

1

LF2~v!

51

LF2 ~12 ivts

F!,1

LN2 ~v!

51

LN2 ~12 ivts

N!, ~9!

i.e., they become complex and frequency dependent. Never-theless, the procedure of solving the equations forzF(x,t)and zN(x,t) with boundary conditions of Eq.~8! is essen-tially the same as in the dc regime. BecausezF(x→2`,t)→0 and zN(x→`,t)→0, the difference ZN(xN ,t)2ZF(xF ,t) tends to the potential dropw(xF ,t)2w(xN ,t)between these points for largexF and xN . However, it isimportant to emphasize that the conditions under which Eqs.~6! and ~7! are valid in the dc and the ac regimes are verydifferent. In the dc regime they can be derived without usingthe quasineutrality condition of Eq.~3!, while in the ac re-gime Eq.~3! should be explicitly employed.

An explicit equation for the dc resistanceR of a F –Njunction that is most convenient for our purposes is as fol-lows:

R5Req1Rn–eq, Req5S215~S↑1S↓!21,

Rn–eq51

r FN$r N@r c~DS/S!21r F~Ds/sF!2#

1r cr F@~DS/S!2~Ds/sF!#2%, ~10!

r FN5r F1r N1r c .

It consists of two parts. The equilibrium resistanceReq is thelimit of R when LN ,LF→0, i.e., when spin nonequilibriumcan be neglected both in theF and N regions. It dependsonly on the contact resistance.Rn–eq is the nonequilibriumresistance controlled by spin diffusion and relaxation. Theparameters of the spin selective contact entering the nonequi-librium resistanceRn–eq are

DS5S↑2S↓ , r c5S/4S↑S↓ . ~11!

It is worth mentioning that two different effective resistancesof the contact,Req and r c , appear in the equilibrium andnonequilibrium parts of the total resistanceR, respectively.

It is an important property ofRn–eq that it is alwayspositive,Rn–eq.0, as it has been already concluded in Ref.5. Therefore, there is a considerable difference between thespin injection and bipolar injection through ap–n junction.Bipolar injection always contributes to the conductivity, anda large resistance of the deplicion region plays no essentialrole.13 For aF –N junction, nonequilibrium resistanceRn–eq

resulting from spin injection is added toReq. Large positivemagnetoresistance of a spin-injecting device has been re-cently observed in weak magnetic fields and attributed to themagnetic alignment of semimagnetic electrodes enhancingspin injection.14

In the ac regime, the impedanceZ(v) can be foundfrom Eq. ~10! by the transformation LF ,LN

→LF(v),LN(v) of Eq. ~9!:

Zn–eq~v!

51

r FN~v!$r N~v!@r c~DS/S!21r F~v!~Ds/sF!2#

1r cr F~v!@~DS/S!2~Ds/sF!#2%, ~12!

where

r F~v!5LF~v!/sF , r N~v!5LN~v!/sN . ~13!

Equation~12! is the basic result of the letter.The imaginary part ofZ(v) results in a reactive conduc-

tivity that can be compared to the diffusion capacitance of ap–n junction.13 The frequency dependence of Re$Z% andIm$Z% provides a useful tool for measuring different param-eters of aF –N junction. Becauser c does not depend onv,frequency dependence comes exclusively throughr F(v) andr N(v). Some general regularities follow directly from Eqs.~9! and~12!. Spin relaxation timests

F andtsN in a ferromag-

netic aligner and a semiconductor microstructure, respec-tively, may differ by several orders of magnitude. Usuallyts

F!tsN . Therefore, two different scales, (ts

F)21 and (tsN)21,

should emerge in the frequency dependence ofZ(v). Thisproperty opens an opportunity for measuring these param-eters separately, and through them to evaluater F andr N . Athigh frequencies,v@(ts

F)21,(tsN)21, the diffusion contribu-

tion to Z vanishes andZ→Req.It is instructive to consider the low- and high-frequency

regimes in more detail. Expansion of ImZ(v) in v startswith a linear inv term, and the positive sign of Im$Z% sug-gests that Im$Z 21% can be considered as the conductivityvCdiff of a capacitor connected in parallel to the resistorR:

Cdiff5H tsNr NS r c

DS

S1r F

Ds

sFD 2

1tsFr FF r c

DS

S2~r c1r N!

Ds

sFG2J /2R2r FN

2 . ~14!

It is seen from Eq.~14! thatCdiff.0 for all parameter values.Capacitance arising from the diffusion of nonequilibriumcarriers is typical of injection devices and is well known forp–n junctions.13 However, the dependence of the low-frequency diffusion capacitance on the relaxation timet israther different for these systems. The square root depen-dence,Cdiff}ts

1/2, is typical of p–n junctions. Diffusion ca-pacitanceCdiff of a F –N junction follows this law only whenr c!r N ,r F , i.e., when spin injection into a semiconductor isstrongly suppressed. In the opposite limitr c@r N ,r F that isof major interest for spin injecting devices, it follows fromEq. ~14! that Cdiff}ts

3/2. Depending on the relative magni-tudes ofr F andr N , different combinations ofts

F andtsN can

appear inCdiff . Therefore, largetsN typical of semiconductor

microstructures should enlargeCdiff considerably.15

In the high frequency regime, whenvtsF , vts

N@1, theresistancesr F(v) and r N(v) are small and one can expandZdiff(v) in r F(v)/r c , r N(v)/r c!1. Because in this limitLF(v)'(11 i )LF /A2vts

F and similarly forLN(v), the realand imaginary parts ofZdiff(v) are nearly equal and

2330 Appl. Phys. Lett., Vol. 80, No. 13, 1 April 2002 Emmanuel I. Rashba

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Rn–eq~v!51

A2vF r N

AtNS DS

S D 2

1r F

AtFS DS

S2

Ds

sFD 2G ,

~15!

Cdiff~v!5Rn–eq~v!/vReq2 . ~16!

It is seen from Eqs.~15! and ~16! that bothRn–eq(v) andCdiff(v) remain positive even in the high frequency limit.

Side by side with the diffusion capacitanceCdiff , therealways exists a geometrical capacitanceCgeom'«/4pX,whereX is the contact thickness and« is the dielectric per-meability. Cgeom depends on the specific geometry of a tun-nel or Schottky contact and cannot be found in a generalform. However, it does not depend on the frequencyv andthis property is critical for eliminating the bypassing effect ofthe geometric capacitanceCgeom and measuringZdiff(v). Itis the frequency dependence of the diffusion impedanceZdiff(v) with two characteristic frequencies (ts

F)21 and(ts

N)21 that should facilitate separating the diffusion andgeometric contributions to the total impedanceZ(v). Be-cause Eq.~12! provides explicit expression forZdiff(v), aquantitative comparison of experimental data with the theoryis possible in the wide region of frequencies.

In a similar way, a frequency-dependent complex spininjection coefficientg(v)5 j F(v)/J(v) can be found

g~v!5@r F~v!~Ds/sF!1r c~DS/S!#/r FN~v!. ~17!

It describes the frequency dependence of the magnitude ofthe spin-polarized currentj F(t) and the phase shift betweenj F(t) and J(t). Nonequilibrium spins related to the currentj F(t) can also be detected in optical experiments. Opticaldata should not be obscured by the bypass current flowingthrough the geometric capacityCgeom.

Because the frequency dependence of the impedanceZ(v) comes completely from the frequency dependence ofthe diffusion lengthsLF(v) andŁN(v) that are the param-eters of the bulk, a similar approach can be applied to theimpedance ofF –N–F junctions. However, the equation for

the dc resistanceR of a F –N–F junction is much morecumbersome than Eq.~10!.

In conclusion, an explicit expression for the compleximpedanceZ(v) of a spin injecting junction is presented.The diffusion contribution toZ(v) shows strong frequencydependence with two characteristic frequencies correspond-ing to the inverse spin relaxation times in the ferromagneticemitter and the normal conductor into which spins are in-jected. Electrical and~or! optical detection of the time de-pendent spin injection should allow measuring spin relax-ation times and effective resistances controlling theefficiency of spin injection.

Support from DARPA/SPINS by the Office of Naval Re-search Grant No. N000140010819 is gratefully acknow-ledged.

1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. vonMolnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science294, 1488~2001!.

2P. R. Hammar, B. R. Bennett, M. J. Yang, and M. Johnson, Phys. Rev.Lett. 83, 203 ~1999!.

3S. Gardelis, C. G. Smith, C. H. W. Barnes, E. H. Linfield, and D. A.Ritchie, Phys. Rev. B60, 7764~1999!.

4G. Schmidt, D. Ferrand, L. W. Mollenkamp, A. T. Filip, and B. J. vanWees, Phys. Rev. B62, R4790~2000!.

5E. I. Rashba, Phys. Rev. B62, R16267~2000!.6H. J. Zhu, M. Ramsteiner, H. Kostial, M. Wassermeier, H.-P. Scho¨nherr,and K. H. Ploog, Phys. Rev. Lett.87, 016601~2001!.

7P. R. Hammar and M. Johnson, Appl. Phys. Lett.79, 2591~2001!.8A. T. Hanbicki, B. T. Jonker, G. Itskos, G. Kioseoglou, and A. Petrou,cond-mat/0110059.

9V. F. Motsnyi, V. I. Safarov, J. De Boeck, J. Das, W. Van Roy, E. Goo-vaerts, and G. Borghs, cond-mat/0110240.

10P. C. van Son, H. Kempen, and P. Wyder, Phys. Rev. Lett.58, 2271~1987!.

11T. Valet and A. Fert, Phys. Rev. B48, 7099~1993!.12S. Hershfield and H. L. Zhao, Phys. Rev. B56, 3296~1997!.13W. Shockley, Bell Syst. Tech. J.28, 435 ~1949!.14G. Schmidt, G. Richter, P. Grabs, C. Gould, D. Ferrand, and L. W. Mol-

lenkamp, Phys. Rev. Lett.87, 227203~2001!. The equation for the resis-tance proposed in that letter is questionable and does not ensure thatRn–eq.0.

15J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett.80, 4313~1998!.

2331Appl. Phys. Lett., Vol. 80, No. 13, 1 April 2002 Emmanuel I. Rashba

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