A theory of spin-polarized transport in inhomogeneous magnetic semiconductors
is developed and applied to magnetic/nonmagnetic p-n junctions. Several
phenomena with possible spintronic applications are predicted, including
spinvoltaic effect, spin valve effect, and giant magnetoresistance. It is
demonstrated that only nonequilibrium spin can be injected across the
space-charge region of a p-n junction, so that there is no spin injection (or
extraction) at low bias.Comment: Minor Revisions. To appear in Phys. Rev. Let

B. T. Jonker

I. Žutić

Igor Žutić

J.-E. Wegrowe

Jaroslav Fabian

M. E. Flatte

M. Johnson

P. R. Hammar

R. Fiederling

S. Das Sarma

S. Datta

S. Tiwari

T. Dietl

Y. Ohno

English

arXiv

A theory of spin-polarized transport in inhomogeneous magnetic semiconductors is developed and applied to magnetic/nonmagnetic p-n junctions. Several phenomena with possible spintronic applications are predicted, including spin-voltaic effect, spin valve effect, exponential and giant magnetoresistance. It is demonstrated that only nonequilibrium spin can be injected across the space-charge region of a p-n junction, so that there is no spin injection (or extraction) at low bias

Zutic, I.

Fabian, J.

Das Sarma, S.

MPG.PuRe

Spin-polarized transport in inhomogeneous magnetic semiconductors: Theory of magnetic/nonmagnetic p-n junctions

Zutic, Igor

Fabian, Jaroslav

Sarma, S. Das

University of Regensburg Publication Server

VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002066603-1Spin-Polarized Transport in Inhomogeneous Magnetic Semiconductors:Theory of Magnetic/Nonmagnetic p-n JunctionsIgor Žutic´,1 Jaroslav Fabian,1,2,* and S. Das Sarma11Department of Physics, University of Maryland at College Park, College Park, Maryland 20742-41112Max-Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany(Received 5 June 2001; published 29 January 2002)A theory of spin-polarized transport in inhomogeneous magnetic semiconductors is developed andapplied to magnetic/nonmagnetic p-n junctions. Several phenomena with possible spintronic applicationsare predicted, including spin-voltaic effect, spin valve effect, exponential and giant magnetoresistance.It is demonstrated that only nonequilibrium spin can be injected across the space-charge region of a p-njunction, so that there is no spin injection (or extraction) at low bias.DOI: 10.1103/PhysRevLett.88.066603 PACS numbers: 72.25.Dc, 72.25.MkSemiconductor spintronics is an emerging field promis-ing device applications with more functionality and bet-ter integrability with traditional semiconductor technologythan metal-based spintronics [1]. Spin injection into asemiconductor has been demonstrated [2,3], but an im-portant question, what this injected spin is useful for,remains largely unanswered (see, however, [4,5]). ThisLetter formulates the problem of spin and charge bipolar(electron and hole) transport in inhomogeneously dopedmagnetic semiconductors (such as CdMnSe or GaMnAs,whose carrier g factors are large due to magnetic doping).In particular, we show that a magnetic/nonmagnetic p-njunction has large magnetoresistance and displays spin-voltaic and spin valve effects: a current, whose directionchanges with the direction of magnetic field, can flow with-out any applied bias. These effects should be useful forsensing magnetic fields, and for probing spin polarizationand spin relaxation.To introduce the equations for spin and charge bipo-lar transport in inhomogeneously doped magnetic semi-conductors, consider a semiconductor doped with Naracceptors and Ndr donors, and with magnetic impuri-ties, whose density varies in space and whose presenceleads to large g factors for electrons and holes, gnr andgp r. In a homogeneous magnetic field B the carrier en-ergies are Zeeman split: spin up (l  1 or ") and spindown (l  21 or #) electrons have their energy shiftedby 2lqznr  2lgnrmBB2, where mB is Bohr mag-neton and q is the proton charge; the energy of holeschanges by lqzpr  lgprmBB2. Carrier charge cur-rent densities, resulting from the electric field E  2=f,nonuniform magnetic “potentials” z , electron (hole) den-sities c  n (c  p), areJcl  qmclclE 6 qDcl=cl 2 qlmclcl=zc , (1)where the upper (lower) sign is for electrons (holes), mand D stand for mobility and diffusivity, and the “mag-netic” drift q=z forces carriers with opposite spins to goin opposite directions. Equation (1) follows from the re-quirement that carrier densities in an inhomogeneous envi-ronment have the quasiequilibrium form [6]. From Eq. (1)0031-90070288(6)066603(4)$20.00the carrier charge and spin current densities J  J" 1 J#and Js  J" 2 J# areJc  scE 2 ssc=zc 6 qDc=n 6 qDsc=sc , (2)Jsc  sscE 2 sc=zc 6 qDsc=c 6 qDc=sc . (3)Here c  c" 1 c# and sc  c" 2 c#, and we introducedcarrier charge and spin conductivities sc  qmcc 1mscsc and ssc  qmscc 1 mcsc, where mc  mc" 1mc#2 and msc  mc" 2 mc#2, and similarly fordiffusivities. Equation (2), which is a generalization of theJohnson-Silsbee magnetotransport equations [7], reflectsthe spin-charge coupling in bipolar transport in inhomo-geneous magnetic semiconductors: a spatial variationin spin density, as well as in q=z , can cause chargecurrents, while spin currents can flow as a result of aspatial variation of carrier densities and f.Generation and recombination of electrons and holes areassumed to be mostly due to band-to-band processes. Fur-thermore, electrons (holes) with a given spin are assumedto recombine with holes (electrons) of either spin. The sta-tionary continuity equation for carriers is= ?Jclq 6wclclc¯ 2 cl0c¯0 6cl 2 c2l 2 ls˜c2T1c.(4)Here c¯ is pn if c is np, c0 is the equilibrium carrierdensity, w is the (generally spin dependent) band-to-bandrecombination rate, and T1 is the spin relaxation time.Spin relaxation equilibrates carrier spins while preserv-ing nonequilibrium carrier density, so, for a nondegen-erate semiconductor, s˜c  ac0c, where ac0  sc0c0 tanhzcVT  is the equilibrium carrier spin polarization(VT  kBTq, with kB being the Boltzmann constant andT temperature). Equations (1) [or (2) and (3)], (4), andPoisson’s equation = ? eE  r, where r  qp 2 n 1Nd 2 Na and e is the dielectric constant of the semicon-ductor, determine the distributions of charge and spin in amagnetic semiconductor under applied bias V.We have made a number of simplifying assumptions inour theory which are not strictly valid in real magnetic© 2002 The American Physical Society 066603-1VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002semiconductors. Most importantly, we treat carrier statesas spin doublets and neglect both the effect of magneticimpurities on the band structure and the response of themagnetic ions to the magnetic field of polarized carriers[8]. Our formalism can easily be tailored to include thesemore complicated effects on a case-by-case basis. Ourapproach is to consider a simple but reasonable model toillustrate new phenomena that can occur in inhomogeneousmagnetic semiconductors.We now apply the above theory to the problem of amagnetic/nonmagnetic p-n junction with nondegeneratecarrier densities and with only electrons magnetically ac-tive (zp  0). Two cases are considered (Fig. 1): (i) themagnetic n side to study spin injection, and (ii) the mag-netic p side, to study spin extraction and magnetoresis-tance phenomena. We assume perfect Ohmic contacts(both carriers and spins at equilibrium) as boundary condi-tions, except when we study the dependence of the currenton nonequilibrium spin, in which case we keep the ohmiccontacts only for carrier densities.Specifically we consider an L  12 mm long GaAsp-n junction at room temperature, doped with Na  3 31015 cm23 acceptors to the left, Nd  5 3 1015 cm23donors to the right [5], and with magnetic impurities in-ducing electronic g factor which follows the shape of Na(Nd) for the magnetic p (n) region. Holes are unpolarized,so we omit the label n when dealing with spin-relatedquantities, and reserve that symbol for the n-region-relatedvariables. Figure 1 depicts our geometry and the distribu-tion of electrons and holes for various cases consideredbelow. The materials parameters are the electron andhole diffusivities, Dn  10Dp  103.6 cm2 s21, and mo-x=0 Lxpxmagneticpmagneticmagneticppnnonmagneticnonmagneticspin−polarizednnnnonmagneticDA BCNnpFIG. 1. Band-energy schemes for magnetic/nonmagnetic p-njunctions with magnetically active electrons (arrows). Holes(circles) are unpolarized. (A) Electrons from the magnetic nregion (note the band split) can be injected into the nonmagneticp region only at large bias. (B) Similarly for spin extraction ifthe p region is magnetic. (C) If the p region is magnetic andthere is a nonequilibrium spin in the n region, a giant magne-toresistance and spin-voltaic effects arise. The former can beobserved in a scheme (D), where electron spin is injected froma magnetic N (through a heterostructure) into the nonmagneticn region (p is magnetic).066603-2bilities mn  10mp  4000 cm2 V21 s21. The intrinsic(nonmagnetic) carrier density is ni  1.8 3 106 cm23,the dielectric constant e  13.1 of the vacuum permit-tivity; the calculated built-in voltage is 1.1 V at B  0.Recombination rate w  13 3 1025 cm3 s21, and thespin relaxation time T1  0.2 ns. The minority [9] diffu-sion lengths are [5] Ln  1 mm, Lp  0.25 mm, and theelectron spin diffusion length in the np region is Lsn 1.4 mm (Lsp  0.8 mm). Figure 2 illustrates the dopingprofile and a typical profile of carrier and spin densities.We first ask the important question whether spin can beinjected and extracted into/from the nonmagnetic region.Figure 3 shows the results of our numerical calculations,where we plot spin (density, not current) polarizationa  sn. At small bias (below the built-in value) thereis no significant spin injection or extraction. As the biasincreases, the injection and extraction become large andintensify with increasing T1. The reason why there is nospin injection or extraction at small bias is that althoughthere are exponentially more, say, spin up than spin downelectrons in the magnetic side, the barrier for crossing thespace-charge region is exponentially larger for spin up thanfor spin down electrons (see Fig. 1). Those two exponen-tial effects cancel out, leaving no net spin current flowingthrough the space-charge region. At large bias (where nu-merics is indispensable) spin injection is possible, a resultof building up a nonequilibrium spin at the space-charge region.The current through a magnetic/nonmagnetic p-n junc-tion depends on magnetic field. This dependence has twosources. First, a magnetic field makes the band gap spindependent, leading to the increased density of the minoritycarriers. For example, the equilibrium density of electrons0 2 4 6 8 10 12x (µm)10−21001021041061081010101210141016carrier and spin density (cm−3 )pnsp−nonmagnetic n−magneticFIG. 2. Calculated electron (n), hole (p), and spin (s) densityprofiles for a magnetic p-n junction with qz  0.5kBT in the(magnetic) n region, under forward applied bias of V  0.8 V.The thin dashed line indicates the doping profile Nd 2 Na (notto scale).066603-2VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 20020 2 4 6 8 10 12x (µm)−0.4−0.20.00.20.4α−0.20.00.20.40.60.8α 0.81.21.51.5, T1−>100 T1p−nonmagnetic n−magneticn−nonmagneticp−magneticFIG. 3. Calculated spin polarization profiles for different for-ward bias and spin relaxation rates. Spin injection (top) fromthe magnetic n side into the nonmagnetic p side occurs only atlarge bias (indicated by the numbers in volts). The largest in-jection in the graph is for V  1.5 V, with spin relaxation timeof 100T1. Similar behavior is observed for spin extraction fromthe nonmagnetic n region into the magnetic p region (bottom).The magnetic splitting is qz  0.5kBT .in a magnetic p region (Fig. 1C) is n0  n2i coshzVT Na [in equilibrium, n0p0  n2i coshzVT ]. Thecurrent through a p-n junction is proportional to the den-sity of the minority (not majority) carriers, and so it willgrow exponentially with B for z * VT . The exponentialmagnetoresistance for the system in Fig. 1B is shownin Fig. 4. The effect diminishes with increasing bias.The second way a current can depend on B is if anonequilibrium spin with polarization aL is introduced(optically or by electrical spin injection) into the sample(Figs. 1C and 1D). The exponential magnetoresistancebecomes giant (meaning, the resistance changes whenthe orientation of the magnetic moment in the magneticregion with respect to the orientation of the injected spinchanges) as seen in Fig. 4. The current is exponentiallysensitive to both B and aL, being large (small) if theyhave the same (opposite) sign.To explain analytically our numerical results, wegeneralize the Shockley approximation [6] for the p-njunction to include nonequilibrium spin, which is given,in the quasineutral regions, by the spin-diffusion equationDn=2s  wsp 2 s0p0 1 s 2 s˜T1 , (5)where s˜  a0n, of which n is the solution of the cor-responding carrier diffusion equation. We consider themagnetic p region, and impose a nonequilibrium spin po-larization aL at the right boundary (see Fig. 1C, wherealso the boundary points used below are indicated). Wepostulate as the second boundary condition for spin, in thespirit of the quasiequilibrium approximation [6], that spin066603-3−1 −0.5 0 0.5 1α(L)10−310−1101103I (A.cm−2 )0.811.5−5 −2.5 0 2.5 5B/B*10−310−1101103I (A.cm−2 )0.811.5FIG. 4 (color). Calculated I-B (top) and I-a (bottom) char-acteristics for the magnetic p-n junction with the magneticp region, under different bias (indicated in volts). We useB  2kBTgmB (900g Tesla at room temperature) as thescale for magnetic field (BB  qzVT ). In the top graph thered line is for aL  0, and the violet (cyan) line for a  1and spin relaxation time of 100 (104) T1 . The bottom graph isfor BB  3, and the red, violet, and cyan lines are for the spinrelaxation times of T1, 100T1, and 104T1 . The dashed lines arethe analytical calculations. In both graphs, at V  1.5 V, linesfor 100T1 and 104T1 almost coincide.current vanishes at xn, the n-side boundary of the space-charge region. The spin polarization in the n region isax  aL coshh 2 tanhhn sinhh , (6)where h  L 2 xLsn and hn  L 2 xnLsn. Atx  xn, an  axn  aL coshL 2 xnLsn. Inthe p region a0  a0 (Ohmic boundary condition forspin) and the second boundary condition, that at x  xp ,the p-side boundary of the space-charge region, whichcan be obtained from the usual quasiequilibrium conditionof constant chemical potentials (applied to nondegeneratestatistics) from xp to xn, isap  a0 1 an1 1 a0an.The nonequilibrium minority carrier density at x  xp is066603-3VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002nxp  n01 1 a0an expVVT . With these boundaryconditions the carrier and spin profiles in the magnetic pregion arenxn0 1 1sinhxLnsinhxpLneVVT 1 1 ana0 2 1 , (7)sxs0nxn01sinhxLspsinhxpLspana01 2 a20 eVVT . (8)Having the profiles, we calculate the current in themagnetic p-n junction. We distinguish equilibrium-spinelectron Jn and hole Jp currents, and nonequilibrium-spin-induced current J 0n, so that the total charge currentis J  Jn 1 Jp 1 J 0n. The individual contributions areJn  qDnLnn2iNacoshµzVT∂cothµxpLn∂eVVT 2 1 , (9)Jp  qDpLpn2iNdcothµL 2 xnLp∂eVVT 2 1 , (10)J 0n  qDnLnn2iNacothµxpLn∂sinhµzVT∂aneV VT . (11)Figure 4 shows how well this analytical model compareswith our numerical calculation. The small discrepancy iscaused by the neglect of the recombination processes inthe space-charge region. The model breaks down at largebias, where the Shockley approximation is not valid [6].We note that if the spin is injected by the bias contact atx  L, one needs to consider a model in which aL de-pends on J (since it then obviously vanishes at J  0),and solve Eqs. (9)–(11) for J. Otherwise, the above equa-tions describe either the case of an independent spin in-jection or the region of J where aL is independent of J.The large exponential magnetoresistance effect, J expjz jVT  at large jz j (or B), comes from the increase ofthe minority electron population with jz j. However, once anonequilibrium spin population (finitean) is maintained inthe space-charge region, a giant magnetoresistance (GMR)should be observed, J  expzVT  at large jz j and an !61. The GMR coefficient is then exp2zVT . In addi-tion to sensing B, these effects can be used for an all-electrical probing of the injected spin polarization aLand spin-diffusion length Lsn, as they both determine an(a possible device scheme is proposed in Fig. 1D). The keyto employing the large magnetoresistance effects in spin-tronics is the development of materials with large g factors.Even for g  100, the GMR coefficient at T  100 K andB  1 T is about exp0.65, which is close to 200%. Amagnetic impurity that would give g  500 would yield2500% (300% at room temperature). The device potentialof magnetic p-n junctions is enormous, but to take fulladvantage we need materials with large g factors at roomtemperature [10].Equations (9)–(11) reveal another interesting phenome-non, a spin-voltaic effect. The current (J 0n stays finiteeven for V  0, if an ﬁ 0. A current flows at zero bias.066603-4This is an analogue of the photovoltaic effect, where areverse current flows if carriers are photogenerated in thespace-charge region. In the spin-voltaic effect both reverseand forward currents can flow, depending on the relativeorientation of B and an; the device can function as aspin valve. The physics of the spin-voltaic effect is thatnonequilibrium spin in the space-charge region disturbsthe balance between the generation and recombination cur-rents (Fig. 1C). If z . 0, and more spin up electrons arepresent at xn (an . 0), the barrier for them to cross theregion is smaller than the barrier for the spin down elec-trons, so more electrons flow from n to p than from p to n,and positive charge current results. If an , 0, the currentis reversed.In summary, we have studied bipolar spin-polarizedtransport in magnetic/nonmagnetic p-n junctions. We pre-dict novel phenomena of exponential and giant magnetore-sistance, and of spin-voltaic effect, if a nonequilibrium spinis added into the junction. We have demonstrated that nospin injection or extraction is possible under low bias, andthat only nonequilibrium spin can be transferred throughthe space-charge region. Finally, we have introduced amodified Shockley model which explains our numericalfindings.This work was supported by DARPA and the U.S. ONR.*Present address: Institute for Theoretical Physics, Karl-Franzens University, 8010 Graz, Austria.[1] S. Das Sarma, J. Fabian, X. Hu, and I. Žutic´, IEEE Trans.Magn. 36, 2821 (2000); Superlattices Microstruct. 27, 289(2000); Solid State Commun. 119, 207 (2001).[2] R. Fiederling, M. Kleim, G. Reuscher, W. Ossau, G.Schmidt, A. Waag, and L. W. Molenkamp, Nature (Lon-don) 402, 787 (1999); Y. Ohno, D. K. Young, B. Beschoten,F. Matsukura, H. Ohno, and D. D. Awschalom, Nature(London) 402, 790 (1999); B. T. Jonker, Y. D. Park,B. R. Bennett, H. D. Cheong, G. Kioseoglou, and A.Petrou, Phys. Rev. B 62, 8180 (2000).[3] P. R. Hammar, B. R. Bennett, M. J. Yang, and M. Johnson,Phys. Rev. Lett. 83, 203 (1999).[4] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990);M. E. Flatte and G. Vignale, Appl. Phys. Lett. 78, 1273(2001).[5] I. Žutic´, J. Fabian, and S. Das Sarma, Phys. Rev. B 64,121201 (2001); Appl. Phys. Lett. 79, 1558 (2001).[6] S. Tiwari, Compound Semiconductor Device Physics(Academic Press, San Diego, 1992).[7] M. Johnson and R. H. Silsbee, Phys. Rev. B 35, 4959(1987); J.-E. Wegrowe, Phys. Rev. B 62, 1067 (2000).[8] For ferromagnetic semiconductors z and carrier spin den-sity need to be evaluated self-consistently.[9] Term minority (majority) refers to the relative populationof electrons and holes.[10] At low temperatures Cd0.95Mn0.05Se has g . 500:T. Dietl, in Handbook of Semiconductors, edited byT. S. Moss and S. Mahajan (North-Holland, New York,1994), Vol. 3, p. 1279.066603-4

Spin-Polarized Transport in Inhomogeneous Magnetic Semiconductors: Theory of Magnetic/Nonmagnetic p-n Junctions

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Spin-Polarized Transport in Inhomogeneous Magnetic Semiconductors: Theory of Magnetic/Nonmagnetic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="italic">p</mml:mi><mml:mi>−</mml:mi><mml:mi mathvariant="italic">n</mml:mi></mml:math>Junctions

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Spin-polarized transport in inhomogeneous magnetic semiconductors:
  theory of magnetic/nonmagnetic p-n junctions

Spin-polarized transport in inhomogeneous magnetic semiconductors: theory of magnetic/nonmagnetic p-n junctions

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