ultrafast low-power spin-injection devices based on modified ferromagnetic-semiconductor junctions

Ultrafast low-power spin-injection devices based onmodified ferromagnetic-semiconductor junctions

A.M. Bratkovsky and V.V. Osipov

Abstract: The authors describe recent theoretical and experimental advances in achieving largeaccumulated spin polarisation in semiconductors and suggest new classes of low-power ultrafastdevices. Tunnelling of electrons between nonmagnetic semiconductors (S) and ferromagnets (FM)through a Schottky barrier modified by a d-doped layer at the interface is described. It is shownthat, in such reverse (forward) biased FM-S junctions, electrons with a certain spin projection canbe efficiently injected in (extracted from) S, while electrons with the opposite spin can efficientlyaccumulate in S near the interface. This occurs due to spin filtering of electrons in the tunnellingprocess, and the authors found conditions for most efficient accumulation of spin polarisation.Extraction of spin can proceed in degenerate semiconductors at low temperatures. Novel spin-valveultrafast devices with small dissipated power are described: a magnetic sensor, a spin transistor, anamplifier, a frequency multiplier, a square-law detector and a source of polarised radiation.

1 Introduction

Spin injection and manipulation in semiconductors holdspromise for the next generation of high-speed low-powerelectronic devices [1–13]. Amongst important spintroniceffects already used in practice, we can indicate a giantmagnetoresistance in magnetic multilayers and tunnelferromagnet–insulator–ferromagnet (FM–I–FM) structures[14–18]. Injection of spin-polarised electrons into semicon-ductors is of particular interest because of relatively largespin relaxation time [1, 2]. An efficient spin injection inheterostructures with ferromagnetic semiconductor (FMS)as a spin source has been reported in [9–21]. However, themagnetisation in FMS usually vanishes or is too small atroom temperature. Relatively high spin injection fromferromagnets (FM) into nonmagnetic semiconductors (S)has been recently demonstrated at low temperatures [22–24],and attempts to achieve an efficient room-temperature spininjection have faced substantial difficulties [25–30]. Theore-tical studies of the spin injection from ferromagnetic metals,started in [31–34] have been the subject of extensive researchin [8–13, 35–45].

The principal difficulty of the spin injection is that thematerials in the FM-S junction usually have very differentelectron affinity and, therefore, a high potential Schottkybarrier forms at the interface [46–48], Fig. 1 (curve 1). ForGaAs and Si the barrier height DC0.5–0.8 eV withpractically all metals, including Fe, Ni and Co, [22, 46–48]and a large barrier width l\100 nm for doping concentra-tion Ndt1017 cm�3. The spin injection corresponds to areverse current in the Schottky contact, which is saturated

and usually negligible due to such large l and D [46–48].Hence, a thin heavily doped n+�S layer between FMmetaland S is used to increase the reverse current [46–48]determining the spin injection, [9, 11, 22, 39]. This layersharply reduces the thickness of the barrier and increases itstunnelling transparency [9, 46–48]. Thus, a substantial spininjection has been observed in FM-S junctions with a thinn+-layer [22].

A usually overlooked formal paradox of the spininjection is that a current through Schottky junctions inprior theories depends solely on parameters of a semi-conductor [46–48] and cannot formally be spin-polarised.Some authors even emphasised that in Schottky junctions‘spin-dependent effects do not occur’ [37]. In earlier works[31–43] with spin transport through FM-S junctions, thespin-selective properties and nonlinear I–V characteristicshave not been actually calculated. They were described byvarious, often contradictory, boundary conditions at theFM-S interface. For example, Aronov and Pikus assumedthat a spin polarisation of current (we shall call it spininjection efficiency, the term used in the current literature,and denote it G ¼ ðJ" � J#Þ=J instead of PJ which we haveused in prior works) at the FM-S interface is a constantequal to that in the ferromagnet FM, G ¼ PFM �ðs" � s#Þ=s, where s"ð#Þ are the conductivities of up-(down-)spin electrons in the ferromagnet, s ¼ s" þ s#, andthen studied nonlinear spin accumulation in S consideringspin diffusion and drift in electric field [31, 32]. The authorsof [33–39] assumed a continuity of both the currents and theelectrochemical potentials for both spins and found that aspin polarisation of injected electrons depends on a ratio ofconductivities of an FM and S (the so-called ‘conductivitymismatch’ problem). At the same time, it has been assertedin [40–43] that the spin injection becomes appreciable whenthe electrochemical potentials have a substantial disconti-nuity (produced by, for example, a tunnel barrier [41]). Theeffect, however, was described by the unknown spin-selective interface conductances Gis, which cannot be foundwithin those theories.

We have developed a microscopic theory of the spintransport through ferromagnet-semiconductor junctions,

The authors are with Hewlett-Packard Laboratories, 1501 Page Mill Road, 1L,Palo Alto, CA 94304, USA

V.V. Osipov is also with New Physics Devices, NASA Ames Research Center,Moffett Field, CA 94305, USA

E-mail: [email protected]

r IEE, 2005

IEE Proceedings online no. 20050017

doi:10.1049/ip-cds:20050017

Paper first received 18th January and in revised form 26th May 2005

IEE Proc.-Circuits Devices Syst., Vol. 152, No. 4, August 2005 323

which include an ultrathin heavily doped semiconductorlayer (d-doped layer) between FM and S [8, 9, 11]. We havestudied nonlinear effects of spin accumulation in S nearreverse-biased modified FM-S junctions with the d-dopedlayer [9] and spin extraction from S near the modifiedforward-biased FM-S junctions [11]. We found conditionsfor the most efficient spin injection, which are opposite tothe results of previous phenomenological theories. We showthat (i) the current of the FM-S junction does depend onspin parameters of the ferromagnetic metal, but not itsconductivity, and so, contrary to the results [33–39, 41–43],the ‘conductivity mismatch’ problem does not arise for theSchottky FM-S junctions. We find also that (ii) spininjection efficiency (polarisation of current) G of the FM-Sjunction strongly depends on the current, contrary to theassumptions in [31–39, 41–43], and (iii) the highest spinpolarisation of both the injected electrons Pn and spininjection efficiency G can be realised at room temperaturesand relatively small currents in high-resistance semiconduc-tors, contrary to the claims made in [38], which are of mostinterest for spin injection devices [4–7, 9]. We show that (iv)tunnelling resistance of the FM-S junction has to berelatively small, which is opposite to the condition obtainedin the linear approximation in [41], and that (v) the spin-selective interface conductances Gis are not constants, aswas assumed in [40–43], but vary with a current J in astrongly nonlinear fashion. We have suggested a new classof spin devices on the basis of the present theory.

2 Spin accumulation and extraction

The modified FM-S junction with transparent Schottkybarrier is produced by d-doping the interface by sequentialdonor and acceptor doping. The Schottky barrier is madevery thin by using large donor doping Nþd in a thin layer ofthickness l. For reasons to become clear shortly, we wouldlike to have a narrow spike followed by the narrowpotential well with the width w and the depthBrT, where Tis the temperature in units of kB¼ 1 and rB2�3, producedby an acceptor doping Nþa of the layer w (Fig. 1). When

donor and acceptor concentrations, Nþd and Nþa , and thecorresponding thicknesses of the doping profile, l and w,satisfy the conditions:

Nþd l2q2’2ee0 D� D0 � rTð Þ; Nþa w2q2’2ee0rT ð1Þ

and ltl0, where l0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_2= 2m� D� D0ð Þ½ �

q(l0t2nm), the

remaining low (and wide) barrier will have the heightD0 ¼ ðEc0 � F Þ40, where Ec0 the bottom of conductionband in S in equilibrium, q the elementary charge, e (e0) thedielectric permittivity of S (vacuum). A value of D0 can beset by choosing a donor concentration in S,

Nd ¼ Nc exp F S � Ec0� �

=T� �

¼ Nc exp �D0=Tð Þ ¼ n ð2Þ

where FS is the Fermi level in the semiconductor bulk,

Nc ¼ 2Mcð2pm�T Þ3=2h�3 the effective density of states andMc the number of effective minima of the semiconductorconduction band; n and m� the concentration and effectivemass of electrons in S [46–48]. Owing to small barrierthickness l, the electrons can rather easily tunnel throughthe d-spike, but only those with an energy EZEc canovercome the wide barrier D0 due to thermionic emission,where Ec ¼ Ec0 þ qV . We assume here the standardconvention that the bias voltage Vo0 and current Jo0 inthe reverse-biased FM-S junction and V40 (J40) in theforward-biased FM-S junction [46–48]. At positive biasvoltage V40, we assume that the bottom of conductionband shifts upwards to Ec¼Ec0+qV with respect to theFermi level of the metal, Fig. 1. The presence of the mini-well allows the thickness of the d-spike barrier to be keptequal to ltl0 and its transparency to be kept high atvoltages qVtrT (see the following).

We assume elastic coherent tunnelling, so that the energy

E, spin s and ~kk (the component of the wave vector ~kparallel to the interface) are conserved. The exact currentdensity of electrons with spin s¼m,k through the FM-Sjunction containing the d-doped layer (at the point x¼ l,Fig. 1) can be written as [18, 49]

Js0 ¼qh

ZdE ½f E � F S

s0

� �� f E � F FM

s0

� ��Z

d2kjj2pð Þ2

Ts

ð3Þwhere Tks is the transmission probability, f(E�F) the Fermifunction, F S

s0 F FMs0

� �are the spin quasi-Fermi levels in the

semiconductor (ferromagnet) near the FM-S interface, andthe integration includes a summation with respect to a bandindex. Note that here we study a strong spin accumulationin the semiconductor. Hence, we use nonequilibrium Fermilevels, F FM

s0 and F Ss0, describing distributions of electrons

with spin s¼m,k in the FM and the S, respectively, whichis especially important for the semiconductor. Thisapproach is valid when the spin relaxation time ts is muchlarger than the relaxation time of electron energy, te, whichis met in practically all semiconductors in a wide range oftemperatures, including room temperature. In particular,the electron density with spin s in the S at the FM-Sjunction is given by

ns0 ¼ 1=2ð ÞNc exp F Ss0 � Ec

� �=T

� �ð4Þ

where Fs0 is a quasi-Fermi level at a point x¼ l, Fig. 1. Wecan see from (3) that the current Js0 ¼ 0 if we takeF FMs0 ¼ F S

s0, i.e. if we were to use the assumption of [33–39].In reality, due to very high electron density in FM metal incomparison with electron density in S, F FM

s0 differs negligiblyfrom the equilibrium Fermi level F for currents under

∆

∆0

l0F

w qVEc(x )

x

2

1

3

4

ferromagnet semiconductor

Fig. 1 Energy diagrams of ferromagnet-semiconductor hetero-structure with d-doped layerF is the Fermi level, D the height and l the thickness of an interfacepotential barrier, D0 the height of the thermionic barrier inn-semiconductor. The standard Schottky barrier (curve 1), Ec(x) thebottom of conduction band in n-semiconductor in equilibrium(curve 2), under small (curve 3), and large (curve 4) bias voltage.The spin polarised density of states in Ni is shown at xo0

324 IEE Proc.-Circuits Devices Syst., Vol. 152, No. 4, August 2005

consideration, therefore we can assume that F FMs0 ¼ F , as in

[18, 49] (see the following discussion).The current (3) should generally be evaluated numerically

for a complex band structure Eks [50, 51]. The analyticalexpressions for TsðE; kkÞ can be obtained in an effectivemass approximation, _ks ¼ msvs, where vs ¼ jrEksj=_ isthe band velocity in the metal. This applies to ‘fast’ free-liked-electrons in elemental ferromagnets [18, 52]. The presentSchottky barrier has a ‘pedestal’ with a height(Ec�F)¼D0+qV, which is opaque at energies EoEc. ForE4Ec we approximate the d-barrier by a triangular shapeand can use an analytical expression for TsðE; kkÞ [8] andfind the spin current at the bias 0o�qVtrT, including atroom temperature,

Js0 ¼ j0ds2ns0 Vð Þ

n� exp

�qVT

� �� ð5Þ

j0 ¼ a0nqvT exp �Zk0lð Þ ð6Þwith the most important spin factor

ds ¼vT vs0

v2t0 þ v2s0ð7Þ

where a0 ¼ 1:2ðk0lÞ1=3; k0� 1=l0 ¼ ð2m�=_2Þ1=2ðD� D0

�qV Þ1=2; vt0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðD� D0 � qV Þ=m�

pis the characteristic

‘tunnel’ velocity, vs ¼ vsðEcÞ the velocity of polarised

electrons in FM with energy E¼Ec, vT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3T=m�

pthe

thermal velocity. At larger reverse bias the mini-well on theright from the spike in Fig. 1 disappears and the currentpractically saturates. Note that, in the present case, thecarriers are subject to Boltzmann distribution f ðE � F Þ �exp F�E

T in (3) for energies of interest, E�F\rT, and only

small parallel momenta kktffiffiffiffiffiffiffiffiffim�Tp

=_ contribute to theintegral (see [8, 10]). Quite obviously, the tunnellingelectrons incident almost normally at the interface con-tribute most of the current, so that the peripheral areas ofspin-up and spin-down Fermi surfaces do not matter much(more careful sampling can be done in special cases like, forexample, resonant tunnelling levels in the barrier [18], ornumerically when more quantitative results for d-states withcomplex Fermi surfaces are desired [50, 51]).

We can see from (5) that the total current J ¼ J"0 þ J#0and its spin components Js0 depend on a conductivity of asemiconductor but not a ferromagnet, as in usual Schottkyjunction theories [46–48]. On the other hand, Js0 isproportional to the spin factor ds and the coefficientj0ds / v2T / T , but not the usual Richardson’s factor T2

[46–48]. Expression (5) for current in FM-S structure isvalid for any sign of the bias voltage V. Note that at V40(forward bias) it determines the spin current from S intoFM. Hence, it describes spin extraction from S [11].

Consider spin injection that occurs at reverse bias, Vo0.For �qV’rT’ð2� 3ÞT the value of expð�qV =T Þ �2ns0=n � 1 and, according to (5), the spin polarisation ofthe current, PF, and the spin current at FM-S junction areequal, respectively,

PF ¼J"0 � J#0J"0 þ J#0

¼ d" � d#d" þ d#

¼v"0 � v#0� �

v2t0 � v"0v#0� �

v"0 þ v#0� �

v2t0 þ v"0v#0� � ð8Þ

J"0 ¼ ð1þ PF ÞJ=2 ð9Þwhere vs0 ¼ vsðEcÞ with Ec¼Ec0+qV. Hence, PF dependson bias voltage V and differs from that in usual tunnellingMIM structures [18], because in the present structure PF

refers to the electron states in FM above the Fermi level atenergy E¼Ec4F. This corresponds to high-energy equili-brium electrons, which may be highly polarised (see thefollowing text). Following the pioneering work by Aronovand Pikus [31, 32], we customarily assume a boundarycondition J"0 ¼ 1þ PFMð ÞJ=2, where

PFM ¼J" � J#

J¼ s" � s#

sð10Þ

is the spin polarisation of current in FM. As there is a spinaccumulation in S near the FM-S boundary, the densityof electrons with spin s in the semiconductor isns0 ¼ n=2þ dns0, where dns0 is a nonlinear function ofthe current J, and dns0 / J at small current [31, 32] (seealso in the following text). Hence, the larger J the higher thedns0 and the smaller the current Js0 (see (5)). In other words,there is a situation where a kind of a negative feedback isrealised, which decreases the spin injection efficiency(polarisation of current) G and makes it a nonlinearfunction of J, as we show as follows. We show that thespin injection efficiency, G0, and the polarisation,Pn0 ¼ ½n"ð0Þ � n#0�=n in the semiconductor near FM-Sjunctions, essentially differ and both are small at small biasvoltage V (and current J) but increase with the current up toPF. Moreover, PF can essentially differ from PFM, and mayideally approach 100%.

The current in a spin channel s is given by the standarddrift-diffusion approximation [31, 32, 43]

Js ¼ qmnsE þ qDrns ð11Þwhere E the electric field, D and m are the diffusion constantand mobility of the electrons, D and m do not depend on theelectron spin s in the nondegenerate semiconductors. Fromcurrent continuity and electroneutrality conditions

J xð Þ ¼Xs

Js ¼ constant;

n xð Þ ¼Xs

ns ¼ constantð12Þ

we find

E xð Þ ¼ J=qmn ¼ constant; dn# xð Þ ¼ �dn" xð Þ ð13ÞAs the injection of spin-polarised electrons from FM into Scorresponds to a reverse current in the Schottky FM-Sjunction, we have Jo0 and Eo0, Fig. 1. The spatialdistribution of density of electrons with spin s in thesemiconductor is determined by the continuity equation[31, 32, 38]

rJs ¼qdns

tsð14Þ

where in the present one-dimensional case r ¼ d=dx. Withthe use of (11) and (13), and we obtain the equation fordnm(x)¼�dnk(x) [31, 32, 43]. Its solution, satisfying aboundary condition dnm-0 at x-N, is

dn" xð Þ ¼ Cn2exp � x

L

�� Pn0 exp �

xL

�ð15Þ

Linject ðextractÞ ¼1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2

E þ 4L2s

qþ �ð ÞLE

�

¼ Ls

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiJ2

J2S

þ 4

s� J

JS

!ð16Þ

where plus (minus) sign refers to forward (reverse) bias onthe junction, Ls ¼

ffiffiffiffiffiffiffiffiDtsp

is the usual spin-diffusion length,LE ¼ mjEjts ¼ LsjJ j=Js the spin-drift length. Here we have


introduced the characteristic current density

JS � qDn=Ls ð17Þand the plus and minus signs in the expression for the spinpenetration depth L (16) refer to the spin injection at areverse bias voltage, Jo0, and spin extraction at a forwardbias voltage, J40, respectively. Note that Linject4Lextract,and the spin penetration depth for injection increases withcurrent, at large currents, jJ j � JS , Linject ¼ LsjJ j=Js � Ls,whereas Lextract ¼ LsJs=J � Ls.

The degree of spin polarisation of nonequilibriumelectrons, i.e. a spin accumulation in the semiconductornear the interface is simply given by the parameter C in (15):

C ¼ n" 0ð Þ � n# 0ð Þn

¼ Pn 0ð Þ � Pn0 ð18Þ

By substituting (15) into (11) and (5), we find

J"0 ¼J2

1þ Pn0LLE

� �¼ J

2

1þ PFð Þ g� Pn0ð Þg� Pn0PF

ð19Þ

where g¼ exp(�qV/T)�1. From (19), we obtain a quadraticequation for Pn(0) with a physical solution that can bewritten fairly accurately as

Pn0 ¼PF gLE

gLþ LEð20Þ

By substituting (20) into (5), we find for the total currentJ¼ Jm0+Jk0:

J ¼ �Jmg ¼ �Jm e�qV =T � 1 �

ð21Þ

Jm ¼ a0nqvT 1� P 2F

� �d"0 þ d#0� �

e�Zk0l ð22Þfor the bias range 7qV7trT. The sign of the Boltzmannexponent is unusual because we consider the tunnellingthermoemission current in a modified barrier. Obviously,we have J40 (o0) when V40 (o0) for forward (reverse)bias.

We notice that, at a reverse bias voltage �qVCrT, theshallow potential mini-well vanishes and Ec(x) takes theshape shown in Fig. 1. (curve 3). For �qV4rT, a widepotential barrier at x4l (in S behind the spike) remains flat(characteristic length scale\100nm at Ndt1017cm�3), as inusual Schottky contacts [46–48]. Hence, the current becomesweakly dependent on V, because the barrier is opaque forelectrons with energies EoEc�rT (Fig. 1, curve 4). Thus(21) is valid only at �qVtrT and the reverse current at�qV\rT practically saturates at the value

Jsat ¼ qna0vT d"0 þ d#0� �

1� P 2F

� �exp r � Zk0lð Þ ð23Þ

With the use of (21) and (16), we obtain from (20) the spinpolarisation of electrons near the FM-S interface,

Pn0 ¼ �PF2J

2Jm þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ2 þ 4J 2

S � Jq ð24Þ

The spin injection efficiency at FM-S interface is, using (5),(19), (16) and (24),

G0 ¼J"0 � J#0J"0 þ J#0

¼ Pn 0ð Þ LLE

¼ �PF

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4J2

S þ J 2

q� J

2Jm þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ2 þ 4J 2

S

q� J

ð25Þ

We can see that G0 strongly differs from Pn0 at smallcurrents. As expected, Pn � PF jJ j=Jm ! 0 vanishes with thecurrent (Fig. 2), and the prefactor differs from thoseobtained in [31, 32, 38, 40, 42, 43].

These expressions should be compared with the resultsfor the case of a degenerate semiconductor [13], for thepolarisation

Pn0 ¼ �PF6J

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ2 þ 4J 2

S

q� J

�þ 10Jm

ð26Þ

and the spin injection efficiency

G0 ¼ �PF

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4J2

S þ J 2

q� J

2Jm þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ2 þ 4J 2

S

q� J

ð27Þ

In spite of the very different statistics of carriers in adegenerate and nondegenerate semiconductor, the accumu-lated polarisation as a function of current behaves similarlyin both cases. An important difference comes from anobvious fact that the efficient spin accumulation indegenerate semiconductors may proceed at and belowroom temperature, whereas in the present design an efficientspin accumulation in FM-S junctions with nondegenerate Scan be achieved at around room temperature only.

In the reverse-biased FM-S junctions the current Jo0and, according to (24) and (25), sign(dnm0)¼ sign(PF). Insome realistic situations, like elemental Ni, Fig. 1, thepolarisation at energies EEF+D0 would be negative, PFo0and, therefore, electrons with spin s¼k will be accumu-lated near the interface. For large currents jJ j � JS the spinpenetration depth L (16) increases with current J and thespin polarisation (of electron density) approaches themaximum value PF. Unlike the spin accumulation Pn0,the spin injection efficiency (polarisation of current) G0 does

not vanish at small currents, but approaches the value G00 ¼

PF JS= JS þ Jmð Þ � PF in the present system with transpar-ent tunnel d-barrier. There is an important difference withthe magnetic tunnel junctions, where the tunnel barrier isrelatively opaque and the injection efficiency (polarisationof current) is high, G � PF [18]. However, the polarisationof carriers Pn0 (measured in, for example, spin-LED devices

−1.0 −0.5 0 0.5 1.0

J /Jm

−1.0

0

1.0

Js /Jm

PJ / PF

Pn / PF

L / Ls

extractioninjection

Fig. 2 Spin accumulation Pn¼ (nm�nk)/n, spin polarisation ofcurrent PJ¼ (Jm�Jk)/J and relative spin penetration depth L/Ls

(broken line) in the semiconductor as functions of relative currentdensity J/Js for spin injection (Jo0) and spin extraction (J40)regimesPF is spin polarisation in the ferromagnet, the ratio Js/Jm¼ 0.2, Ls isthe usual spin diffusion depth. The spin penetration depth consider-ably exceeds Ls for the injection and smaller than Ls for the extraction


[22–24]) would be minute (see the following). Both Pn0 andG0 approach the maximum PF only when jJ j � JS , Fig. 3.The condition jJ j � JS is fulfilled at qVCrT\2T, whenJm\JS.

Thus, we have shown that (i) an efficient spin injection inthe reverse-biased FM-S junctions at room temperatureoccurs in ferromagnetic-semiconductor junctions when anultrathin heavily n+-doped semiconductor layer (d-dopedlayer) satisfying conditions (1) is formed between theferromagnet and nondegenerate n-type semiconductor; (ii)the reverse current of such modified Schottky junctions,which determines the spin injection from ferromagnets intosemiconductors, is due to tunnelling and thermionicemission of spin polarised electrons; (iii) spin injectiondepends on parameters of both a semiconductor and aferromagnet, in particular, on velocity of electrons with spins and energy ECEc and a conductivity of a semiconductor(but not a ferromagnet); (iv) spin injection efficiency(polarisation of current), G0, and polarisation of carrierdensity, Pn0, in the semiconductor are two differentquantities and both are small at low current, they increasewith the total current and reach the maximal possible value7PF7E1 only at relatively large current Jm, when the spinpenetration depth LE is much larger than the spin diffusionlength Ls, and (v) the smaller the semiconductor conductiv-ity the lower the threshold current Jm for achieving anefficient spin injection. In the following, we also show thatthe most efficient spin injection can occur when (vi) theconduction band bottom of the semiconductor,Ec ¼ Ec0’Ec0 þ rT , is close to a peak in the density ofminority electron states of elemental ferromagnets Ni, Coand Fe above the Fermi level.

Another situation is realised in the forward-biased FM-Sjunctions when J40. Indeed according to (24) and (25)at J40, the electron density distribution is such thatsign(dnm0)¼�sign(PF). If a system like elemental Ni isconsidered (Fig. 1), then PF (F+D0)o0 and dnm040, i.e. theelectrons with spin s¼m would be accumulated innonmagnetic semiconductor (NS), whereas electrons withspin s¼k would be extracted from NS (the oppositesituation would take place for PF (F+D0)40). We can seefrom (24) that 7Pn07 can reach a maximum PF only whenJ � JS . According to (21), the condition J � JS can onlybe fulfilled when Jm � JS . In this case, (24) reduces to

Pn0 ¼ �PF J=Jm ¼ �PF 1� e�qV =T �

ð28Þ

Hence, absolute magnitude of a spin polarisation ap-proaches its maximal value 7Pn07CPF at qV\2T linearlywith current (Fig. 2). The maximum is reached when Japproaches the value Jm, which depends weakly on bias V(see the following). In this case, dnm(k)(0)E8PFn/2 atPF40 (dm4dk), so that the electrons with spin s¼m areextracted, n"ð0Þ � ð1� PF Þn=2, from a semiconductor,while the electrons with spin s¼k are accumulated in asemiconductor, n"ð0Þ � ð1þ PF Þn=2, near the FM-S inter-face. The penetration length of the accumulated spin (16) atJ � JS is

L ¼ L2s=LE ¼ LsJS=J � Ls at J � JS ð29Þ

i.e. it decreases as Lp1/J (Fig. 2). We see from (25) that, atJ � JS ,

G0 ¼ PF J2S =JmJ ! 0 ð30Þ

Hence, the behaviour of the spin injection efficiency atforward bias (extraction) is very different from a spininjection regime, which occurs at a reverse bias voltage: herethe spin injection efficiency G0 remains � PF and vanishesat large currents as G0pJs/J. Hence, we come to anunexpected conclusion that the spin polarisation ofelectrons, accumulated in a nonmagnetic semiconductornear the forward-biased FM-S junction can be relativelylarge for the parameters of the structure when the spininjection efficiency is actually very small [11]. Similar, albeitmuch weaker phenomena are possible in systems with wideopaque Schottky barriers [53, 54] and have been probablyobserved [55]. Spin extraction may be observed at lowtemperature in FMS-S contacts as well [56]. Proximity effectleading to polarisation accumulation in FM-S contacts [57]may be related to the same mechanism.

3 Conditions for efficient spin injection andextraction

According to (22) and (17), the condition for maximalpolarisation of electrons Pn can be written as

Jm\JS ð31Þor, equivalently, as a condition

b � a0vT d"0 þ d#0� �

1� P 2F

� �e�Zl=l0ts=Ls\1 ð32Þ

Note that, when l\l0, the spin injection efficiency at small

current is small G00 ¼ PF =ð1þ bÞ � PF , because, in this

case, the value b’ðd"0 þ d#0Þa0vT ts=Ls � 1 for realsemiconductor parameters. The condition b� 1 can besimplified and rewritten as a requirement for the spin-relaxation time

ts � DD� D0

2a0v2s0T

� �2

exp2Zll0

ð33Þ

x /Ls

0

0.2

0.4

0.6

P/P

F a

nd P

n/P

F

J / Js

0

0.2

0.4

0.6

0.8

pola

risat

ion

at x

= 0

Pn(0) /PF

P (0) /PF

J /Js = 10

Pn /PF

P /PF

J /Js = 5

J /Js = 1 and 0.5

Ls /vT �s = 0.2

0 102 4 6 8

0 1 2 3 4 5

Fig. 3 Spin polarisation of current P¼ (Jm�Jk)/J (solid line) andspin accumulation Pn¼ (nm�nk)/n (broken line) in the semicon-ductor as functions of relative current density (J/Js) (top panel) andtheir spatial distribution for different densities of total current J/Js

(bottom panel) at Ls/vTts ¼ 0.2Js¼ qnLs/ts, PF is the spin polarisation in the ferromagnet (see themain text)


It can be met only when the d-doped layer is very thin,ltl0�k�10 . With typical semiconductor parameters atTC300K (DE25cm2/s, (D�D0)C0.5 eV, vs0C108 cm/s[46–48]) the condition (33) is satisfied at ltl0 when thespin-coherence time t� 10�12 s. It is worth noting that itcan certainly be met: for instance, ts can be as large asB1ns even at TC300K (e.g. in ZnSe [58–61]).

Note that the higher the semiconductor conductivity,sS¼ qmnpn, the larger the threshold current J4Jmpn (22)for achieving the maximal spin injection. In other words,the polarisation Pn0 reaches the maximum value PF at lowercurrent in high-resistance lightly doped semiconductorscompared to heavily doped semiconductors. Hence, the‘conductivity mismatch’ [37, 41, 42] is actually irrelevant forachieving an efficient spin injection.

The necessary condition jJ j � Js can be rewritten at lowvoltages, jqV j � T , as

rc � Ls=sS ð34Þ

where rc ¼ ðdJ=dV Þ�1 is the tunnelling contact resistance.Here we have used the Einstein relation D/m¼T/q fornondegenerate semiconductors. We emphasise that (34) isopposite to the condition found by Rashba in [41] for smallcurrents. Indeed, at small currents, the spin injectionefficiency may be large, G0 ¼ PF =ð1þ bÞ’PF only whenb� 1, i.e. rc � Ls=ss (cf. [41]). This is exactly the situationwith magnetic tunnel junctions, where the current throughthe structure varies a great deal depending on the mutualorientation of moments on the ferromagnetic electrodes[18]. However, at such large tunnelling contact resistance rc

(near the opaque tunnel barrier) the saturation current Jsat

of the FM-S junction is much smaller than Js. Hence, thedegree of spin accumulation in the semiconductor is verysmall, Pn0 � 1, but this Pn0 is exactly the characteristic thatdetermines the main spin effects [1, 2, 4–9]. Note that theconditions (32) and (34) do not depend on the electronconcentration in the semiconductor, n, and are valid also forheavily doped degenerate semiconductors. We notice thatthe quasi-Fermi level F FM

s0 for electrons with spin s in FMdiffers very little from equilibrium-Fermi level F. It is easyto see that jF FM

s0 � F j � jF Ss0 � F j at current JtJsat,

because n=nFM � 1, where nFM is the electron density inFM metal. Thus, the assumption used from the precedingtext that F FM

s0 ¼ F indeed holds.

The spin factor ds / v�1s0 in the effective mass approx-

imation, because usually vs0\vt0. In a metal v�1s0 / gs0 ¼gs ðEcÞ [62], so that dspgs(Ec) where gs0¼ gs(Ec) is thedensity of states of the d-electrons with spin s and energyE¼Ec in the ferromagnet. Thus, taking ms ¼ m we find,from (8), PF � ðg"0 � g#0Þ=ðg"0 þ g#0Þ. We assume that thesame proportionality between the polarisation and thedensity of states approximately holds in the general case ofmore complex band structures. Note that the polarisation ofd-electrons in elemental ferromagnets Ni, Co and Fe isreduced by the current of unpolarised s-electrons rJs, wherero1 is a factor (roughly the ratio of the number of s-bandsto the number of d-bands crossing the Fermi level).Together with the contribution of s-electrons the totalpolarisation is, approximately,

PF ¼J"0 � J#0

J"0 þ J#0 þ Js0’ g"0 � g#0

g"0 þ g#0 þ 2rgsð35Þ

Such a relation for PF can be obtained from a usual‘golden-rule’ type approximation for tunnelling current(cf. [46–48, 63–68]. The density of states gk for minorityd-electrons in Fe, Co and Ni has a peak at E ¼ EF þD#ðD#’0:1 eVÞ which is much larger than gm for the

majority d-electrons and gs for s-electrons [69, 70] (seeFig. 1). The FM-S junction in Fig. 1 can be tailored toadjust the cutoff energy EcCEF+Dk to the peak in thedensity of states of minority electrons. Thus, if we selectD0 ¼ D# þ qV’D# þ rT , then g#0 � g#0 � gS , and, ac-cording to (35), the polarisation PF may be close to 100%(note that, in present case, the polarisation PF is negative,PFE�1). We emphasise that the spin injection in structuresconsidered in the literature [6, 7, 22–43] has been dominatedby electrons at the Fermi level and, according to calculation[69, 70], gk(F) and gm(F) are such that PFt40%. We alsonotice that the condition (32) for parameters of theFe/AlGaAs heterostructure studied in [22–24] (lC3nm,l0C1nm and D0¼ 0.46 eV) is satisfied when ts\5 10�10 sand can be fulfilled only at low temperatures. Moreover, forthe concentration n¼ 1019 cm�3 Ec lies below F, so that theelectrons with energies ECF are involved in tunnelling, butfor these states the polarisation is PFt40%. Hence, theauthors of [22–24] were indeed able to estimate the observedspin polarisation as being E32% at low temperatures.

Better control of the injection can be realised inheterostructures where a d-layer between the ferromagnetand the n-semiconductor layer is made of very thin heavilydoped n+-semiconductor with larger electron affinity thanthe n-semiconductor. For instance, FM–n+-GaAs�n-Ga1�xAlxAs, FM�n+-GexSi1�x�n-Si or FM�n+-Zn1�xCdxSe–n-ZnSe heterostructures can be used for thispurpose. The GaAs, GexSi1�x or Zn1�xCdxSe n+-layermust have width lo1nm and donor concentrationNþd 41020cm�3. In this case, the ultrathin barrier formingnear the ferromagnet-semiconductor interface is transparentfor electron tunnelling. The barrier height D0 at Gex-

Si1�x�Si, GaAs�Ga1�xAlxAs or Zn1�xCdxSe�ZnSe inter-face is controlled by the composition x and can be selectedas D0 ¼ 0:05� 0:15 eV. When the donor concentration inSi, Ga1�xAlxAs or ZnSe layer is Ndo1017 cm�3, the injectedelectrons cannot penetrate relatively low and wide barrierD0 when the width l0410nm.

4 High-frequency spin-valve effect

Here we describe a new high-frequency spin valve effect thatcan be observed in a FM-S-FM device with two back-to-back modified Schottky contacts, see Fig. 4. We find thedependence of current on a magnetic configuration in FMelectrodes and an external magnetic field. The spatialdistribution of spin-polarised electrons is determined by thecontinuity equation (14) and the current in spin channel s isgiven by (11). Note that Jo0, thus Eo0 in a spin injectionregime. With the use of the kinetic equation and (11), weobtain the equation for dnm(x), (14) [17]. Its general solutionis

dn" xð Þ ¼ n2

c1e�x=L1 þ c2e� w�xð Þ=L2

�ð36Þ

where L1ð2Þ ¼ ð1=2ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2

E þ 4L2s

qþ ð�ÞLE

h iis the same as

found earlier in (16). Substuting (36) into (11), we obtain

J" xð Þ ¼ J=2ð Þ 1þ b1c1e�x=L1 þ b2c2e�ðw�xÞ=L2

h ið37Þ

where b1(2)¼L1/LE (�L2/LE).Consider the case when w� 1 and the transit time

ttrCw2/(D+m7E7w) of the electrons through the n-semi-conductor layer is shorter than ts. In this case, a spinballistic transport takes place, i.e. the spin of the electronsinjected from the FM1 layer is conserved in the semicon-ductor layer, s0 ¼ s. Probabilities of the electron spin

s¼m to have the projections along ~M2 are cos2(y/2) and


sin2(y/2), respectively, where y is the angle between vectors

s¼m and ~M2. Accounting for this, we find that theresulting current through the structure saturates at biasvoltage �qV4T at the value

J ¼ J01� P 2

R cos2 y1� PLPR cos y

ð38Þ

where J0 is the prefactor similar to (6). For the oppositebias, the total current J is given by (38) with the replacementPL2PR. The current J is minimal for antiparallel (AP)

moments ~M1 and ~M2 in the electrodes, when y¼ p, and near

maximal for parallel (P) magnetic moments ~M1 and ~M2, seeFig. 5. The ratio

Jmax Pð ÞJmin APð Þ ¼

1þ PLPR

1� PLPR

is the same as for the tunnelling FM-I-FM structure [17,18], hence the structure may be also used as a memory cell.

The present heterostructure has an additional degree offreedom, compared to tunnelling FM-I-FM structures thatcan be used for magnetic sensing. Indeed, spins of the

injected electrons can precess in an external magnetic field Hduring the transit time ttr of the electrons through thesemiconductor layer (ttrots). The angle between the

electron spin and the magnetisation ~M2 in the FM2 layerin (38) is in general y¼ y0+yH, where y0 is the anglebetween the magnetisations M1 and M2, and yH is the spinrotation angle. The spin precesses with a frequency O¼ gH,where H is the magnetic field normal to the spin directionand g¼ qg/(m*c) is the gyromagnetic ratio, g is the g-factor.Hence, yH¼ g0gHttr(m0/m*), where m0 the mass of a freeelectron, g0g¼ 1.76 107Oe�1s�1 for g¼ 2 (in somemagnetic semiconductors g� 1). According to (38), withincreasing H the current oscillates with an amplitude(1+PLPR)/(1�PLPR) and period DH¼ (2pm*)(g0gm0ttr)

�1,Fig. 6. Study of the current oscillations at various biasvoltages allows us to find PL and PR.

For magnetic sensing, we may choose

y0 ¼ p=2 ~M1? ~M2

� �, then it follows from (38) that, for

yH � 1,

J ¼ J0 1þ PLPRg0gHttr m0=m�ð Þ½ � ¼ J0 þ JH ð39Þ

KH ¼ dJ=dH ¼ J0PLPRg0gttr m0=m�ð Þ ð40Þwhere KH is the magnetosensitivity coefficient. For example,KHC2 10�3J0PLPR A/Oe for m0/m* ¼ 14 (GaAs) andg¼ 2, ttrB10�11 s, and the angle yH¼ p at HC1kOe. Thus,JHC1mA at J0¼ 25mA, PLPRC0.2 and HC100Oe. Themaximum operating speed of the field sensor is very high,because redistribution of nonequilibrium injected electrons

lRlL

∆0F F

∆θ0

M2

w

M1

M1

M2

n−S

H

Ec

F qVL

qVR

F- qV

H

θHσ σv

Ec

a

b

Fig. 4 Energy diagram of FM-S-FM heterostructure with d-dopedlayers in equilibrium (a) and at a bias voltage V (b), with VL (VR)the fraction of the total drop across the left (right) d-layerF marks the Fermi level, D the height, lL(R) the thickness of the left(right) d-doped layer, D0 the height of the barrier in the n-typesemiconductor (n–S), Ec the bottom of conduction band in the n–S, wthe width of the n–S part. The magnetic moments on the FM

electrodes ~M1 and ~M2 are at some angle y0 with respect to each other.The spins, injected from the left, drift in the semiconductor layer androtate by the angle yH in the external magnetic field H. Inset: schematicof the device, with an oxide layer separating the ferromagnetic filmsfrom the bottom semiconductor layer

0 0.2 0.4 0.6 0.8 1.0

x /w

n σ/n

0

0.2

0.4

0.6

0.8

1.0antiparallel M1 and M2

antiparallel M1 and M2

parallel M1 and M2

σ

σσ

σ

Fig. 5 Spatial distribution of spin polarised electrons nm(k)/n in thestructure for different configurations of the magnetic moments M1

and M2 in the limit of saturated current density J, w¼ 60 nm,L2¼ 100 nm

0 2 4 60

0.5

1.0

J/J

0

H /∆H

P AP

Fig. 6 Oscillatory dependence of current J through the structureon magnetic field H for parallel (P) and antiparallel (AP) momentsM1 and M2 on the electrodes, Fig. 1, and PL¼PR¼ 0.5


in the semiconductor layer occurs over the transit timettr ¼ w=mjEj ¼ Jswts=ðJLsÞ, ttrt10�11 s for wt200nm,tsB3 10�10 s and J/Js\10 (DE25cm2/s at TC300K[46–48]). Hence, the operating frequency f¼ 1/ttr\100GHz(o¼ 2p/ttrC1THz) may be achievable at room tempera-ture.

We see that (i) the present heterostructure can be used asa sensor for an ultrafast nanoscale reading of aninhomogeneous magnetic field profile, (ii) it includes twoFM-S junctions and can be used for measuring spinpolarisations of these junctions, and (iii) it is a multi-functional device where current depends on mutual orienta-tion of the magnetisations in the ferromagnetic layers, anexternal magnetic field and a (small) bias voltage, thus itcan be used as a logic element, a magnetic memory cell, oran ultrafast read head.

5 Spin-injection devices

The high-frequency spin-valve effect, described in thepreceding text, can be used for designing a new class ofultrafast spin injection devices like an amplifier, a frequencymultiplier and a square-law detector [9]. Their operation isbased on injection of spin-polarised electrons from oneferromagnet to another through a semiconductor layer andspin precession of the electrons in the semiconductor layerin a magnetic field induced by a (base) current in anadjacent nanowire. The base current can control the emittercurrent between the magnetic layers with frequencies up toseveral 100GHz. Here we shall describe a spintronicmechanism of ultrafast amplification and frequency con-version, which can be realised in heterostructures compris-ing a metallic ferromagnetic nanowire surrounded by asemiconductor (S) and ferromagnetic (FM) thin shells,Fig. 7a. Practical devices may have various layouts, withtwo examples shown in Figs. 7b and 7c.

Let us consider the principle of operation of thespintronic devices shown in Fig. 7a. When the thickness wof the n-type semiconductor layer is not very small(w\30nm), tunnelling through this layer would benegligible. The base voltage Vb is applied between the endsof the nanowire. The base current Jb, flowing through thenanowire, induces a cylindrically symmetric magneticHb¼ Jb/2pr in the S layer, where r is the distance fromthe centre of nanowire. When the emitter voltage Ve isapplied between FM layers, the spin-polarised electrons areinjected from the first layer (nanowire FM1) through thesemiconductor layer into the second (exterior) ferromag-netic shell, FM2. The FM1-S and FM2-S junctions arecharacterised by the spin injection efficiencies P1 and P2,respectively. We assume that the transit time ttr of theelectrons through the S layer is less than the spin relaxationtime ts (i.e. we consider the case of a spin ballistic transport).The exact calculation gives a current Je, (38), through thestructure as a function of the angle y between the

magnetisation vectors ~M1 and ~M2 in the ferromagneticlayers. At small angles y or P1¼PL or P2¼PR, (38) isreduced to

Je ¼ J0e 1þ PLPR cos yð Þ ð41Þwhere y¼ y0+yH, y0 is the angle between ~M1 and ~M2, andyH is the angle that the spin precesses with the frequencyO¼ gH>, where H> is the magnetic field componentnormal to the spin and g is the gyromagnetic ratio. We cansee from Fig. 7a that H>¼Hb¼ Jb/(2pr). Thus, the angleof the spin rotation is equal to yH ¼ gHbttr ¼ gttrJb=2prs,where rs is the characteristic radius of the S layer. Then,according to (41),

Je ¼ Je0 1þ P1P2 cosðy0 þ kjJbÞ� �

ð42Þwhere kj¼ gttr/2prs¼ g/ors and o¼ 2p/ttr is the frequencyof a variation of the base current, Jb¼ Js cos(ot).

Equation (42) shows that, when the magnetisation M1 isperpendicular to M2, y0¼ p/2, and yH � p,

Je ¼ Je0 1þ kjP1P2Jb� �

; G ¼ dJe=dJb ¼ Je0kjP1P2 ð43ÞHence, the amplification of the base current occurs with thegain G, which can be relatively high even for o\100GHz.Indeed, g ¼ q=ðm�cÞ � 2:2ðm0=m�Þ105 m=ðA � sÞ, wherem0 is the free electron mass, m* the effective mass ofelectrons in the semiconductor, and c the velocity of light.

− −M1

M2

n−S

JeJb

vv

w

a

a

H

H

FM1

FM2

×

a

M1

M2

FM1 FM2n−SNW

Je

Jb

+−Ve

l

l

substrate

b

M1

H

M2

FM1 FM2

ll

Jb

Jb

Jeve−

NW

NW

n−Sn-S

δδ

�H

�0

c

Fig. 7 Schematic of the spin injection-precession devices havingcylindrical (a) semicylindrical (b) and planar shapes (c)FM1 and FM2 are the ferromagnetic layers; n-S the n-typesemiconductors layer; w the thickness of the n–S layer; d the d-dopedlayers; NW the highly conductive nanowires; I the insulating layers.

The directions of the magnetisations ~M1 and ~M2 in the FM1 and FM2

layers, as well as the electron spin s, the magnetic field Hb and theangle of spin rotation y in S are also shown


Thus, the factor kjC103A�1, when rSC30nm, m0/m* ¼ 14(GaAs) and o¼ 100GHz, so that G41 at Je40.1mA/(P1P2).

When M1 is collinear with M2 (y0¼ 0, p) and yH � p,then, according to (42), the emitter current is

Je ¼ Je0 1 P1P2ð Þ � 1

2Je0P1P2k2j J2

b ð44Þ

therefore the time-dependent component of the emittercurrent dJeðtÞ / J2

b ðtÞ, and the device operates as asquare-law detector. When Jb(t)¼ Jb0 cos(o0t), the emittercurrent has a component dJeðtÞ / cosð2o0tÞ, and thedevice works as a frequency multiplier. When JbðtÞ ¼Jh cosðohtÞ þ Js cosðostÞ, the emitter current has thecomponents proportional to cos(oh7os)t, i.e. the devicecan operate as a high-frequency heterodyne detector withthe conversion coefficient K¼ Je0JhP1P2kj

2/4. Forkj¼ 103A�1, we obtain K41 when Je0Jh44 (mA)2/(P1P2).

6 Source of polarised radiation

The spin extraction effect can be used for making anefficient source of (modulated) polarised radiation. Con-sider a structure containing an FM-S junction with d-dopedlayer and a double p–n0–n heterostructure, where n0-region ismade from narrower gap semiconductor, Fig. 8. We showthat the following effects can be realised in the structurewhen both FM-S junction and the heterostructure arebiased in forward direction and electrons are injected fromthe n-semiconductor region into FM and p-region. Owingto a spin selection property of the FM-S junction [7], spin-polarised electrons appear in the n-region with a spatialextent LtLs near the FM-S interface, where Ls is the spindiffusion length in NS. When thickness of the n-region wis smaller than L, the spin-polarised electrons from then-region and holes from the p-region are injected andaccumulated in a thin narrow-gap n0-region (quantum well)where they recombine and emit polarised photons.

The conditions for maximal polarisation are obtained asfollows. When the thickness of n-region is woL, we canassume that dnm(x)Cdnm0 and PnCPn0. In this case,integrating (14) over the volume of the n-semiconductorregion (with area S and thickness w), we obtain

I"FS þ I"pn � Ic" ¼ qdn"0wS=ts ¼ Pn0ISw=2Ls ð45Þwhere IS¼ JSS, ImFS¼ Jm0S and Impn are the electroncurrents with spin s¼m flowing into the n-region from FMand the p-region, respectively, Imc is the spin current out ofthe n-region in a contact, Fig. 8a. The current Impn isdetermined by injection of electrons with s¼m from then-region, into the p-region equal to I"pn ¼ Ipnn"0=n ¼Ipnð1þ Pn0Þ=2, where Ipn is the total current in the p–njunction. The current of metal contact Ic is not spinpolarised, hence Icm¼ (Ipn+IFS)/2, where IFS is the totalcurrent in the FM-S junction. Hence we can rewrite (45)with the use of (6) as

Pn0 ¼ �gPF þ Pn0 gþ Pn0PFð Þ ISw=Ls � Ipn� �

I�1FS ð46ÞThe current in the FM-S junction IFS approaches amaximal value Im¼ JmS at rather small bias, becausegC1 at qVFS42T. When Ipn � IFS’Im and Im � ISw=Ls,we obtain, from (46), Pn0C�PF.

The way to maximise PF is evident from (8) showingthat it depends on vs0 ¼ vsðD0 þ qVFSÞ and

vt0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 D� D0 � qVFSð Þ=m�

p, i.e. it corresponds to an

electron energy E¼Ec0+qVFS4F, Fig. 8. Hence, PF

depends on bias voltage VFS of the FM-S junction andwe may be able to maximise a spin extraction and

accumulation by adjusting VFS. The maximal 7Pn07 canbe achieved for the process of electron tunnelling throughthe d-doped layer, when the bottom of the conduction bandin a semiconductor Ec¼F+D0+qVFS is close to a peak inthe density of states of minority carriers in the elementalferromagnet, Fig. 8c (curve g). Indeed, the density of statesg# / v�1

#0, for minority d-electrons in Fe, Co and Ni at

E¼EF+Dk(DkC0.1 eV), is much larger than that of themajority d-electrons, gm and s-electrons, gs [68, 69]. Hence,owing to g# � g" � gs, the polarisation 7Pn07CPF may belarge, ideally close to 100%.

Polarisation of injection luminescence is determined byrecombination of injected spin-polarised electrons and holesinside the n0-region of the double p–n0–n heterostructure ofthickness d � Ls, Fig. 8. The injection current of electronswith s¼m into the n0-region is

I"pn ¼ Ipn 1þ Pn0ð Þ=2 ¼ qdn0"d=t0s þ qn0"d=tn ð47Þ

where n0"ð#Þ ¼ n0=2þ ð�Þ dn0" is the density of electrons

with spin s ¼"; #, n0 ¼ n0" þ n0#, t0s ðtnÞ is the time of spin

relaxation (recombination) of electrons in the n0-region. We

∆ ∆0

F F

M

x = 0

Ec 0 (x)

Ec 0

EV0

EV0 (x )

+Vpn+VFM

awl

IpnIcIFS

GaAlAs GaAlAs

p−S

n'−S

(G

aAs)

n+−S

g(E )

qVqVpn

∆0+qVmin.

maj.

Ee(x )

Ee0 − qVpn

EV (x )

� = 12

� = − 32

a

b

c

�

Fig. 8 Schematic of structure (a) and band diagram of polarisedphoton source containing FM-S junction with d-doped layer and adouble n+–n0–p heterostructure without (b) and under the biasvoltage V (c)Minority spin electrons are extracted from n+�S semiconductor layerand the remaining (majority) electrons are recombined in n0�Squantum well. F is the Fermi level, D the height and l the thickness ofthe d-doped layer, D0 the height of a barrier in the n-typesemiconductor, Ec(x) the bottom of conduction band in thesemiconductor. The spin density of states is shown at xo0, with ahigh peak in minority states at E¼F+D0, typical of elemental Ni, asan example


notice that in III–V compound semiconductors, includingGaAs and GaAlAs, there are light holes with spin s¼71/2and heavy holes with spin s¼73/2. Recombination oflight and heavy holes with electrons having spin s¼ 1/2results in radiation of photons with an opposite polarisation(l¼�1). To exclude this undesirable effect, the thickness dand the hole potential well in the narrow-gap n0-region,Fig. 8, should be such that the light holes, unlike the heavyholes, cannot be localised inside the well. In this case, wecan neglect the light holes, assuming that the concentrationof heavy holes in the well is much larger than that of thelight holes, then the rate of polarised radiation recombina-tion is Rs ¼ qn0sd=tR and the polarisation of radiation isp ¼ ðR" � R#Þ=ðR" þ R#Þ ¼ 2dn0"=n0. As Ipn ¼ qn0d=tn,

we find, from (47), 2dn0"=n0 ¼ Pn0t0sðt0s þ tnÞ�1, so that

p ¼ Pn0t0sðt0s þ tnÞ�1. Thus, the radiation polarisation pcan approach maximum p’jPF j at large current ICIm,when tot0s. The latter condition can be met at highconcentration n0, when the time of radiation recombinationtR’tnot0s. For example, in GaAs, tRC3 10�10 s atn\5 1017 cm�3 [71, 72] and t0s can be larger than tR

[58–61]. We emphasise that spin injection efficiency near aforward-biased FM-S junction is very small. Indeed,according to our discussion in Section 2 (see [10]) G0 ¼ðI"0 � I#0Þ=ðI"0 þ I#0Þ ¼ PF LIs=LsIm � PF at Im � Is asLoLs. Thus, polarisation of the recombination radiationwould be high, p¼ 7PF7, while the spin injection efficiencyin the FM-S junction would be small.

Practical structures may have various layouts, with oneexample shown in Fig. 9. It is clear that the distribution ofdn"ð r!Þ in such a two-dimensional structure is characterisedby the length LtLs in the direction x, where the electricalfield E can be strong, and by the diffusion length Ls in the(y, z) plane, where the field is weak. Hence, the spin densitynear FM and p–n junctions will be close to dnm0, when thesize of the p-region is doLs. Thus this derivation and theresults for the one-dimensional structure, Fig. 8, are alsovalid for the more complex geometry shown in Fig. 9. Thepredicted effect should also exist for a reverse-biased FM-Sjunction where the radiation polarisation p can approach+PF.

7 Conclusion

We have described a method of facilitating an efficient spininjection/accumulation in semiconductors from standard

ferromagnetic metals at room temperature. The main idea isto engineer the band structure near the ferromagnet-semiconductor interface by fabricating a d-doped layerthere, thus making the Schottky barrier very thin andtransparent for tunnelling. Long spin lifetime in asemiconductor allows us then to suggest a few interestingnew devices like field detectors, spin transistors, square-lawdetectors and sources of the polarised light described in thepresent text. This development opens up new opportunitiesin potentially very important novel spin-injection-basedtechnology.

8 References

1 Wolf, S.A., Awschalom, D.D., Buhrman, R.A., Daughton, J.M., vonMolnar, S., Roukes, M.L., Chtchelkanova, A.Y., and Treger, D.M.:‘Spintronics: A spin-based electronics vision for the future’, Science,2001, 294, p. 1488

2 Awschalom, D.D., Loss, D., and Samarth, N. (Eds.).: ‘Semiconductorspintronics and quantum computation’ (Springer, Berlin, 2002)

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ultrafast low-power spin-injection devices based on modified ferromagnetic-semiconductor junctions

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