The equivalent electrical circuit of the Ebers-Moll type is introduced for
magnetic bipolar transistors. In addition to conventional diodes and current
sources, the new circuit comprises two novel elements due to spin-charge
coupling. A classification scheme of the operating modes of magnetic bipolar
transistors in the low bias regime is presented.Comment: 4 pages, 2 figure

Fabian, J.

Zutic, I.

English

arXiv

The equivalent electrical circuit of the Ebers-Moll-type is introduced for magnetic bipolar transistors. In addition to conventional diodes and current sources, the new circuit comprises two novel elements due to spin-charge coupling. A classification scheme of the operating modes of magnetic bipolar transistors in the low bias regime is presented. (C) 2005 American Institute Of Physics

Fabian, Jaroslav

Zutic, Igor

University of Regensburg Publication Server

The Ebers–Moll model for magnetic bipolar transistorsJaroslav Fabiana!Institute of Physics, Karl-Franzens University, Universitätsplatz 5, 8010 Graz, AustriaIgor Žutiæb!Center for Computational Materials Science, Naval Research Laboratory, Washington, D.C. 20375and Condensed Matter Theory Center, Department of Physics, University of Maryland at College Park,College Park, Maryland 20742-4111sReceived 20 September 2004; accepted 15 February 2005; published online 23 March 2005dThe equivalent electrical circuit of the Ebers–Moll-type is introduced for magnetic bipolartransistors. In addition to conventional diodes and current sources, the new circuit comprises twonovel elements due to spin-charge coupling. A classification scheme of the operating modes ofmagnetic bipolar transistors in the low bias regime is presented. © 2005 American Institute ofPhysics. fDOI: 10.1063/1.1886251gSemiconductor spintronics1 offers novel functionalitiesby combining electronics for signal processing and magne-tism for both nonvolatility and additional electronic control.At the present stage, with the fundamentals of spininjection,2–5 spin relaxation,6–8 as well as semiconductormagnetism9,10 established, there is a need for new ideas dem-onstrating practical use of the fundamental spin physics. Itwas shown in Refs. 11–13 that magnetic bipolar transistorssMBTd, which can employ both ferromagnetic and paramag-netic semiconductors swith large g-factord,1 can significantlyextend functionalities of conventional bipolar junction tran-sistors sBJTd sRef. 14d by exploiting spin-charge coupling ofthe Silsbee–Johnson-type.15,16 Current amplification inMBT’s, for example, can be modulated by magnetic fieldduring the device operation, giving rise to the phenomena ofsgiantd magnetoamplification.13In this paper we generalize the widely used Ebers–Mollequivalent circuit of BJT sRef. 17d sreprinted in Ref. 18d toMBT. Two novel electronic elements are added to the origi-nal circuit—spin diodes and spin current sources—to de-scribe spin-charge coupling. Our goal is to provide a simplecomputational scheme for MBT’s as well as to show an ex-ample how a novel spintronics device can be described bysand integrated withd a more conventional electronic cir-cuitry.MBT’s comprise two magnetic p–n junctions19,20 in se-ries. In the scheme of Fig. 1 we show an npn MBT with amagnetic base. Magnetic here means that there is an equilib-rium spin splitting 2qzb of the conduction band svalenceband would also workd, giving rise to an equilibrium spinpolarization P0b=tanhsqzb /kBTd in the base sT is temperatureand q is the proton charged. Hole spins are assumed unpolar-ized. Either a ferromagnetic semiconductor swith conductionor impurity band free carriersd or a diluted magnetic semi-conductor with a large g-factor in a magnetic field will work.Both the magnitude and the sign of P0b can be controlled byan external magnetic field sthis is what we call magneticcontrold. In addition to the equilibrium spin, there can be anonequilibrium sexcessd spin injected by external meanssproviding spin controld into the emitter and collector. Thecorresponding spin polarizations are dPe and dPc. If the biason the base-emitter sbed and base-collector sbcd junction isVbe and Vbc respectively, then the excess, dn=n−n0, wheren0 is the equilibrium number, electron densities in the base,close to the be and bc junctions, are13dnbe = n0bszbdfeqVbe/kBTs1 + dPeP0bd − 1g , s1ddnbc = n0bszbdfeqVbc/kBTs1 + dPcP0bd − 1g . s2dThe influence of the equilibrium spin is felt both by theequilibrium number of electrons in the base, n0bszbd=n0bs0dcoshsqzb /kBTd, as well as by the spin-charge cou-pling factor 1+dPP0b. The nonequilibrium spin plays a roleonly in the latter. The unpolarized hole excess densities inthe emitter and collector, close to the depletion layer with thebase, are given by the standard formulasdpe = p0eseqVbe/kBT − 1d , s3dadPresent address: Institute of Theoretical Physics, University ofRegensburg, 93040 Regensburg, Germany; electronic mail:jaroslav.fabian@physik.uni-regensburg.debdElectronic mail: zutic@dave.nrl.navy.milFIG. 1. Scheme of a npn magnetic bipolar transistor in the forward activemode. The base has an equilibrium electron spin polarization P0b, illustratedby the spin-split conduction band. Spin up sdownd electrons are pictured asdark slightd filled circles. Holes are unpolarized. The emitter has a source ofspin polarization, here shown as a circularly polarized light, giving rise to anonequilibrium spin polarization dPe. The direction of the currents isindicated.APPLIED PHYSICS LETTERS 86, 133506 s2005d0003-6951/2005/86~13!/133506/3/$22.50 © 2005 American Institute of Physics86, 133506-1Downloaded 16 Jul 2007 to 132.199.145.54. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspdpc = p0cseqVbc/kBT − 1d , s4dwhere p0e and p0c is the equilibrium number of holes in theemitter and collector.Electrical currents in MBT’s can be expressed, as inBJT’s through the excess densities dnbe and dnbc of the mi-nority carriers13je = jgbn Fdnben0b−1coshswb/Lnbddnbcn0bG + jgep dpebp0e , s5djc = jgbn F− dnbcn0b+1coshswb/Lnbddnben0bG − jgcp dpcbp0c . s6dThe base current is jb= je− jc and the electron generation cur-rent in the base isjgbn =qDnbLnbn0b cothS wbLnbD . s7dHere Dnb stands for the electron diffusion coefficient in thebase whose effective width is wb and Lnb is the electrondiffusion length in the base ssee Fig. 1d. The hole generationcurrents in the emitter, jgep , and collector, jgcp , are given simi-larly to Eq. s7d with n replaced by p and e replaced by eithere or c.The Ebers–Moll model17 is an equivalent circuit to aBJT. We will now introduce this standard model and gener-alize it to the case of MBT’s. Denote by jse and jsc the emit-ter and collector saturation currents snote that s stands forsaturation, not spind:jse = jgbn + jgep , s8djsc = jgbn + jgcp . s9dThe emitter saturation current is the emitter current thatflows if Vbe,0, Vbc=0, and only equilibrium spin presentfje=−jse, see Eq. s5dg. Similarly for the collector saturationcurrent. Denote next the forward and reverse currents stermi-nology from the forward active moded asj f = jseseqVbe/kBT − 1d , s10djr = jscseqVbc/kBT − 1d . s11dFinally, we introduce the spin-charge forward and reversecurrentsjmf = jgbn dPeP0beqVbe/kBT, s12djmr = jgbn dPcP0beqVbc/kBT. s13dHere subscript m stands for magnetic to stress that the cur-rent, that is due to spin-charge coupling across the depletionregions, appears only in magnetic transistors. These currentsflow due to the presence of nonequilibrium spin polarizationand are finite even at zero bias sspin-voltaic effect19d.The generalized Ebers–Moll model directly derives fromEqs. s5d and s6d, and readsje = j f − arjr + jmf − at jmr, s14djc = a f j f − jr + at jmf − jmr. s15dHere a f has the meaning of the transport factor in the for-ward active mode, while ar is the transport factor in thereverse active mode in the absence of spin-charge coupling,as can be seen directly from Eqs. s14d and s15d. The transportfactor at is at=1/coshswb /Lnbd. The conventional Ebers–Moll model is recovered by putting jmf = jmr=0. As in theconventional model, the following equality holds:a f jse = arjsc. s16dThis can be verified by requiring thatjesVbe = 0,Vbc = Vd = − jcsVbe = V,Vbc = 0d , s17dfor dPe=dPc=0. In our ideal case it is straightforward toshow thata f jse = arjsc = at jgbn . s18dThe equivalent circuit to Eqs. s14d and s15d is shown inFig. 2. The current flow is the same as in Fig. 1. Let usdiscuss the emitter circuit. It consists of four elements: sid aconventional diode with the directional current j f that de-pends on Vbe, siid a conventional current source giving cur-rent arjr that depends on Vbc and on the transport factor armeasuring the amount of current injected into the emitterfrom the collector, siiid a spin diode with the forward currentjmf, and finally, sivd a spin current source at jmr. The first twoelements appear already in BJT’s. The spin diode siiid, whichappears due to spin-charge coupling, works similar to adiode in the sense that its current is rectified with jmf,expsqVbe/kBTd. The crucial difference from conventionaldiodes is that the direction of the current flow can bechanged by changing the sign of dPeP0b, see Eq. s12d. Thesymbol for the spin diode reflects this fact. The filled triangleshows the direction when dPeP0b is positive. The new func-tionality of MBT’s then lies in the ability to switch or modifythe spin diode during its operation. There is, in addition, thespin current source sivd that is due to the electron currentfrom spin-charge coupling. The current, injected into thebase from the collector, diffuses towards the emitter throughthe base sthis is why the transport factor at appearsd. Theelement is a current source because it does not depend on thevoltage drop shere Vbed across it. It is, however, a controlledcurrent source, similar to siid, because it can be controlled byVbc. Because it arises from spin-charge coupling, the magni-FIG. 2. The Ebers–Moll equivalent circuit of a MBT. The voltage sourcesare arranged for the forward active mode. The left srightd circuit is theemitter scollectord. The emitter circuit has a diode for the forward current,and a current source which depends on the bias in the collector circuit. Inaddition, there are two new elements. A spin diode whose direction can beflipped: its filled triangle points to the forward direction when dPeP0b.0,otherwise the current direction changes. A new spin current source sdashedarrowd, pointing in the direction of the current if dPeP0b.0. The directionof the current can also be flipped. Similar notation applies for the rightscollectord circuit.133506-2 J. Fabian and I. Zutic Appl. Phys. Lett. 86, 133506 ~2005!Downloaded 16 Jul 2007 to 132.199.145.54. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsptude and direction of sivd can be controlled by spin and mag-netic field, adding to the functionality of siiid. Similar de-scription applies to the collector circuit.For completeness we summarize in Table I the operatingmodes sfor a textbook discussion see, for example, Ref. 21dof both BJT’s and MBT’s, described by the Ebers–Mollmodel. Conventional transistors have four modes, with am-plification only in the forward and reverse active modes sdueto design only the forward active mode has significant cur-rent gaind. The saturation and cutoff modes are used in logiccircuits for ON and OFF states, respectively. MBT’s havea much richer structure. In the active modes bothmagnetoamplification11,13,22 due to the dependence of thesaturation currents on the equilibrium spin polarization andgiant magnetoamplification13 due to spin-charge coupling ap-pear. In contrast to conventional transistors, MBT’s providecurrent gain even in the saturation mode, due to spin-chargecoupling. Furthermore, the transistor can act as a spin switch,switching the current direction by flipping the spin.23 In thecutoff mode MBT’s are OFF and spin effects are inhibitedfsee Eqs. s12d and s13dg. Finally, a qualitatively new mode,spin-voltaic, appears, due to spin-charge coupling. In thismode, with no applied biases, the currents that flow are dueonly to the presence of nonequilibrium spin swhich providespin emfd and MBT’s act as spin switches.In summary, we have generalized the Ebers–Moll modelto include spin-charge coupling and cover magnetic bipolartransistors. We have classified different operating modes ofthe transistors. In most modes MBT’s offer new functional-ities such as spin switches or magnetoamplifiers, which mayhave potential for signal processing, logic circuits and non-volatile memories.This work was supported by the U.S. ONR, NSF, andDARPA. I. Ž. acknowledges financial support from the Na-tional Research Council.1I. Žutiæ, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 s2004d.2R. Fiederling, M. Kleim, G. Reuscher, W. Ossau, G. Schmidt, A. Waag,and L. W. Molenkamp, Nature sLondond 402, 787 s1999d.3D. K. Young, E. Johnston-Halperin, D. D. Awschalom, Y. Ohno, and H.Ohno, Appl. Phys. Lett. 80, 1598 s2002d.4B. T. Jonker, Y. D. Park, B. R. Bennett, H. D. Cheong, G. Kioseoglou, andA. Petrou, Phys. Rev. B 62, 8180 s2000d.5X. Jiang, R. Wang, S. van Dijken, R. Shelby, R. Macfarlane, G. S. So-lomon, J. Harris, and S. S. P. Parkin, Phys. Rev. Lett. 90, 256603 s2003d.6Optical Orientation, edited by F. Meier and B. P. Zakharchenya sNorth–Holland, New York, 1984d.7J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80, 4313 s1998d.8R. I. Dzhioev, V. L. Korenev, B. P. Zakharchenya, D. Gammon, A. S.Bracker, J. G. Tischler, and D. S. Katzer, Phys. Rev. B 66, 153409 s2002d.9T. Dietl, Semicond. Sci. Technol. 17, 377 s2002d.10H. Ohno, Science 281, 951 s1998d.11J. Fabian, I. Žutiæ, and S. Das Sarma, cond-mat/0211639 s2002d.12J. Fabian, I. Žutiæ, and S. Das Sarma, Appl. Phys. Lett. 84, 85 s2004d.13J. Fabian and I. Žutiæ, Phys. Rev. B 69, 115314 s2004d.14W. Shockley, M. Sparks, and G. K. Teal, Phys. Rev. 83, 151 s1951d.15R. H. Silsbee, Bull. Magn. Reson. 2, 284 s1980d.16M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790 s1985d.17J. J. Ebers and J. L. Moll, Proc. IRE 42, 1761 s1954d.18J. J. Ebers and J. L. Moll, in Semiconductor Devices: Pioneering Papers,edited by S. M. Sze sWorld Scientific, Amsterdam, 1991d, pp. 276–287.19I. Žutiæ, J. Fabian, and S. Das Sarma, Phys. Rev. Lett. 88, 066603 s2002d.20J. Fabian, I. Žutiæ, and S. Das Sarma, Phys. Rev. B 66, 165301 s2002d.21S. Dimitrijev, Understanding Semiconductor Devices sOxford UniversityPress, New York, 2000d.22N. Lebedeva and P. Kuivalainen, J. Appl. Phys. 93, 9845 s2003d.23J. Fabian and I. Žutiæ, Acta Phys. Pol. A 106, 109 s2004d.TABLE I. Operational modes of BJT’s and MBT’s. Forward sFd and reversesRd bias means positive and negative voltage, respectively. Symbols MA andGMA stand for magnetoamplification and giant magnetoamplification, whileON and OFF are modes of small and large resistance, respectively; SPSWstands for spin switch.Mode Vbe Vbc BJT MBTForward active F R amplification MA, GMAReverse active R F amplification MA, GMASaturation F F ON ON, GMA, SPSWCutoff R R OFF OFFSpin-voltaic 0 0 OFF SPSW133506-3 J. Fabian and I. Zutic Appl. Phys. Lett. 86, 133506 ~2005!Downloaded 16 Jul 2007 to 132.199.145.54. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

The Ebers-Moll model for magnetic bipolar transistors

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