High-frequency spin valve effect in ferromagnet-semiconductor-ferromagnet structure based on precession of injected spins

0.14 MB·English

by  A. M. Bratkovsky

High-frequency spin valve effect in ferromagnet-semiconductor-ferromagnet structure
based on precession of injected spins
A.M. Bratkovsky and V.V. Osipov
Hewlett-Packard Laboratories, 1501 Page Mill Road, 1L, Palo Alto, CA 94304
(November19, 2003)
4
0
0 Newmechanismofmagnetoresistance, basedontunneling-emission ofspinpolarized electronsfrom
2 ferromagnets (FM) into semiconductors (S) and precession of electron spin in the semiconductor
layer under external magnetic field, is described. The FM-S-FM structure is considered, which
n
includes very thin heavily doped (δ-doped) layers at FM-S interfaces. At certain parameters the
a
J structure is highly sensitive at room-temperature to variations of the field with frequencies up to
100 GHz. Thecurrentoscillates with thefield,anditsrelativeamplitudeisdeterminedonlybythe
2
spin polarizations of FM-S junctions at relatively large bias voltage.
1

72.25.Hg, 72.25.Mk
]
i
c
s
-
l
r Manipulation of an electron spin may lead to break- agram of a thin FM-S-FM structure looks as a rectan-
t
m throughsinsolidstateultrafastscalabledevices[1]. Spin- gular potential barrier of a height ∆ and a thickness w.
tronic effects, like giant and tunnel magnetoresistance Hence, the current through the FM-S-FM structure is
.
t (TMR) are already widely used in read-out devices and negligible when w >30 nm. To increase a spin injection
a
m non-volatile memory cells [1,2]. Theory of TMR junc- current,a thin hea∼vilyn+ semiconductorlayerbetween
−
tions has been considered in Refs. [3,4]. A large bal- theferromagnetandsemiconductorshouldbeused[8,11].
-
d listic magnetoresistance of Ni and Co nanocontacts was This layer sharply decreases the thickness of the Schot-
n reportedinRefs.[5]. The injectionofspin-polarizedcar- tky barriers and increases their tunneling transparency
o
riers into semiconductors provides a potentially power- [14]. Recently an efficient injection was demonstrated in
c
[ ful mechanism for field sensing and other applications, FM-S junctions with a thin n+ layer [8].
which is due to relatively large spin-coherence lifetime We consider a heterostructur−e, Fig. 1, where left (L)
3
of electrons in semiconductors [1,6]. Different spintronic and right (R) δ doped layers satisfy the following op-
v devices,includingmagneticsensors,areconsideredinde- timal conditions−[11]: the thickness lL(R) < 2 nm,
3
7 tail in [1]. The efficient spin injection into nonmagnetic the donor concentration N+ > 1020cm−3, N∼+(lL)2
4 semiconductorshasbeenrecentlydemonstratedfromfer- 2εε (∆ ∆ +rT)/q2,andNd +∼(lR)2 2εε (∆d ∆ )/q≃2,
9 romagnets [7,8] and magnetic semiconductors [9]. Con- whe0re ∆− =0E F, F is thedFermi≃level i0n th−e eq0uilib-
0 c
0 −
ditions for efficient spin injection have been discussed in rium (in the left FM), E the bottom of semiconductor
3 c
Refs.[10,11]. Spindiffusionanddriftinelectricfieldhave conduction band, r 2 3, and T the temperature (we
0
≃ −
/ been studied in Refs. [12,13]. use the units of kB = 1). The value of ∆0 and the rele-
at Inthispaperwestudyanewmechanismofmagnetore- vant profile of Ec (x) can be set by choosing Nd+, lL(R),
m sistance,operationalupto100GHzfrequencies. Wecon- andadonorconcentration,N ,inthe n semiconductor.
d
−
- sideraheterostructurecomprisingan typesemiconduc- The energy diagram of such a FM n+ n n+ FM
d − − − − −
tor (n S) layer sandwiched between two ferromagnetic structure is shown in Fig. 1. Importantly, there is a
n −
(FM)layerswithultrathinheavilyn+ doped(δ doped) shallow potential well of depth rT next to the left
o − − ≈
c semiconducting layers at the FM-S interfaces. Magne- δ spike. Presence of this mini-well allows to retain the
: toresistance of the heterostructure is determined by the th−ickness of the left δ barrier equal to lL < l and
iv following processes: (i) injection of spin polarized elec- its tunneling transparen−cy high for the bias v∼olta0ge up
X
trons from the left ferromagnet through the δ doped to qV rT. The δ spike is transparent for tunnel-
L
− ≃ −
ar layer into the n−S layer; (ii) spin ballistic transport of ing when lL(R) < l0 = ¯h2/[2m∗(∆ ∆0)], where m∗
spin polarized electrons through that layer; (iii) preces- ∼ −
is the effective mass ofqelectrons in the semiconductor.
sion of the electron spin in an external magnetic field
However, when w l , only electrons with energies
0
during a transit through the n S layer; (iv) variation of ≫
− E Ec = F + ∆0 can overcome the barrier ∆0 due
conductivity of the system due to the spin precession. ≥
to thermionic emission [11]. We assume w λ, λ be-
There are known obstacles for an efficient spin injec- ≫
ing the electron mean path in a semiconductor, so one
tioninFM-Sstructures. ASchottkybarrierwithaheight
can consider the FM-S junctions independently. We as-
∆ > 0.5 eV usually forms in a semiconductor near a
sume the elastic coherent tunneling, so that the energy
met∼al-semiconductor interface [14]. The energy band di- E and the wave vector ~k in the plane of the interface
k
1
H spin σ, n , are given by
σ
(a) M1 M2 ∆0 Nc Fσ Ec
n=N exp =N , n = exp − ,
c d σ
− T 2 T
(cid:18) (cid:19) (cid:18) (cid:19)
n-S M (2)
M 2
1 ∆ θ where Nc = 2Mc(2πm∗T)3/2h−3 is the effective density
0 of states in the semiconductor conduction band and M
c
E the number of the band minima [14]. The left (right)
c
∆ junctionsareatx=0(w),sothatinEq.(1)FL =F (0)
F 0 F σ σ
and FR = F (w). The analytical expressions for TL(R)
lL w lR σ σ kσ
can be obtained in an effective mass approximation,
¯hk = m v , where v is a velocity of electrons with
σ σ σ σ
spin σ. The present interface barriers are opaque at en-
ergiesE <E . For energies E >E we can approximate
c c
(b) the δ doped barrier by a trian∼gular shape and find [11]
H −
16αL(R)vL(R)vL(R)
TL(R) = σx x exp ηκL(R)lL(R) , (3)
kσ (vL(R))2+(vL(R))2 −
σ σ θ σx tx (cid:16) (cid:17)
H
v where E = h¯2k2/2m , vL(R) = h¯κL(R)/m the “tun-
F k k ∗ tx ∗
Ec qqVVL neling” velocity, vxL(R) = 2(E−Ec−Ek)/m∗ and
R vL(R) are the x components of electron velocities in
σx p
−
F-qV a direction of current in the semiconductor and fer-
romagnets, respectively, κL(R) = (2m /¯h2)1/2(∆ +
∗
FIG.1. EnergydiagramofatheFM-S-FMheterostructure F E + E )3/2(∆ ∆ qV )−1, αL(R) =
with δ−doped layers in equilibrium (a) and at a bias voltage − k − 0 ± L(R)
V (b),with VL (VR) the fraction of thetotal drop across the 3−1/3πΓ−2 32 κL(R)lL(R) 1/3 ≃ 1.2 κL(R)lL(R) 1/3,
left (right) δ-layer. F marks the Fermi level, ∆ the height, η = 4/3 (for a rectangular barrier α = 1 and η = 2),
lL(R) the thickness of the left (right) δ−doped layer, ∆ the whereV (cid:0)is(cid:1)t(cid:2)hevoltaged(cid:3)ropacrossthe(cid:2)left(right)(cid:3)bar-
0 L(R)
height of the barrier in the n−type semiconductor (n-S), Ec rier, Fig. 1. The preexponential factor in Eq. (3) takes
thebottomofconductionbandinthen-S,wthewidthofthe into account a mismatch between the effective masses,
n-S part. The magnetic moments on the FM electrodes M~1 m andm ,andthevelocities,v andv ,ofelectronsat
andM~2 areatsomeangleθ0 with respecttoeachother. The thσeFM-Si∗nterfaces(cf. Ref.[4])σ.xWeconxsiderqV <∆ ,
b 0
spins, injected from the left, drift in the semiconductor layer T < ∆ ∆ and E E > F +T, when Eqs. (∼1),(3)
and rotate by the angle θH in the external magnetic field H. 0 ≪ ≥ c
yield the following result for the tunneling-emission cur-
Inset: schematicofthedevice,withanoxidelayerseparating
rent density
theferromagneticfilmsfromthebottomsemiconductorlayer.
αqM T5/2(8m )1/2vL(R) exp( ηκL(R)lL(R))
jL(R) = c ∗ 0σ(σ′) − 0
are conserved, so the current density of electrons with σ π3/2¯h3 (vL(R))2+(vL(R))2
spinσ throughtheleftandrightjunctions,includingthe σ(σ′) t0
h i
δ doped layers, can be written as [15,4,11] FσL(R)−Ec FL(R)−Ec
− e T e T , (4)
q d2k × − !
JL(R) = dE[f(E FL(R)) f(E F )] k TL(R),
σ h − σ − − L(R) (2π)2 kσwhere κL(R) 1/lL(R) = (2m /¯h2)1/2(∆ ∆
Z Z 0 ≡ 0 ∗ − 0 ±
(1) qV )1/2, vL(R) = 2(∆ ∆ qV )/m , and
L(R) t0 − 0± L(R) ∗
where Tkσ is the transmission probability, f(E) = vσL((σR′)) =vσ(σ′)(∆0±qVL(pR)). ItfollowsfromEqs.(4)and
[exp(E F)/T +1]−1 the Fermi function, F = F and (2)that the spin currentsofelectronswith the quantiza-
− L tion axis M~ in FM with σ = ( ), Fig. 1, and M~
F =F qV istheleftandrightFermilevel,Fig. 1,qthe 1 1 2
R k ↑ ↓ k
− in FM with σ′ = through the junctions of unit area
elementarycharge,andintegrationincludesasummation 2
±
are equal to
with respect to a band index. We take into account the
sFpeirnmaicfcuunmctuiolantsiownitihntthheenseomneicqounildiburcituomrdqeusacsriibFeedrmbyiltehve- JσL =J0LdLσ eqVTL −2nσ(0)/n , (5)
seelsmFicσo.nTduhcetocronidsintioonnd∆eg0en=erEatce−, sFo >tha0tmtheaentsotthalateltehce- JσR′ =J0RdRσ′(cid:16)2nσ′(w)/n−e−q(cid:17)VTR , (6)
tron density, n = N , and a density of electrons with JL(R) = αL(R(cid:16))nqv exp( ηκL(R)lL(cid:17)(R)). (7)
d 0 − 0 T − 0
2
1/3 J (1+P )[γ 2δn (0)/n]
Here we have introduced αL(R) = 1.6 κL(R)lL(R) , JL = L L− ↑ , (14)
0 0 ↑ 2 γ 2P δn (0)/n
L L ↑
the thermal velocity v 3T/m(cid:16), and the(cid:17)spin −
factors dLσ = vTvσL TvtL0≡2+pvσL 2 ∗−1 and dRσ′ = J↑R = J2 (1+γPRR+c2oPsθR)c[oγsRθ+δn2↑δ(nw↑)(/wn)/n]. (15)
vTvσR′ vtR0 2+(vRσ′)2h−(cid:0)1. (cid:1) (cid:0) (cid:1) i Here we have introduced the spin polarization PL(R) =
Nowh(cid:0)we c(cid:1)an find theidependence of current on a mag- dL(R) dL(R) dL(R)+dL(R) −1 for the left (right)
netic configuration in FM electrodes and an external ↑ − ↓ ↑ ↓
(cid:16)contact, which(cid:17)is(cid:16)equal to (cid:17)
magneticfield. The spatialdistributionofspin-polarized
electronsisdeterminedbythekineticequationdJ /dx=
σ vL(R) vL(R) (vL(R))2 vL(R)vL(R)
tqiδmneσ/oτfs,thweheerleecδtnroσn=s innσt−hen/n2,sτesmisicsopnind-uccothoerr,eanncdeltifhee- PL(R) = (cid:16)v↑L(R)+−v↓L(R)(cid:17)h(vtL0(R))2+−v↑L(R)v↓L(R)i. (16)
− ↑ ↓ t0 ↑ ↓
current in spin channel σ is given by
(cid:16) (cid:17)h i
Importantly, this P is determined by the electron
J =qµn E+qDdn /dx, (8) L(R)
σ σ σ states in FM above the Fermi level, at E = E > F,
c
which may be substantially more polarized compared to
whereD andµarediffusionconstantandmobilityofthe
the states at the Fermi level [11]. In general, the param-
electrons, E the electric field [12,14]. From conditions of
eters c , V are determined by Eqs. (10), (5), (6),
continuity of the total current, J =J +J =const and 1(2) L(R)
↑ ↓
and (12)-(15).
n = n +n = const, it follows that E(x) = J/qµn =
↑ ↓
The current through the structure is ohmic at small
const and δn = δn . Note that J < 0, thus E < 0.
↑ ↓
− bias, J V at qV < T, and saturates at larger bias
With the use of the kinetic equation and (8), we obtain
voltages.∝Indeed,|fro|m Eqs. (14),(15) at qV >2T we
the equation for δn (x) [12,13]. Its general solution is R(L)
↑ find JL = J(1+P ) and ∼
↑ 2 L
δn (x)=(n/2)(c e−x/L1 +c e−(w−x)/L2), (9)
↑ 1 2
J (1+P )(1+2δn (w)/n)
JR = θ ↑ . (17)
where L1(2) = (1/2) L2E +4L2s+(−)LE , Ls = ↑ 2 1+2Pθδn↑(w)/n
√Dτs and LE = µ|E|τs(cid:16)pare the spin diffusion(cid:17)and drift where Pθ ≡ PRcosθ. We can obtain the current at
lengths [12,13]. Substituting Eq. (9) into (8), we obtain x=0 (w) from Eq. (10), equate it to JL (JR) assuming
↑ ↑
negligible spin relaxation at the interface, and then find
J↑(x)=(J/2) 1+b1c1e−x/L1 +b2c2e−(w−x)/L2 (10) the unknown c1(2) with the use of (9). At LE Ls,
≫
we have b = 1, b = L2/L2 and c = P , c =
h i 1 2 − s E 1 L 2
where b = L /L ( L /L ). We consider the case P (1 P2)/(1 P P ). Thus, according to Eqs. (9),
1(2) 1 E − 2 E − θ − L − L θ
whenw L andthetransittimet w2/(D+µE w) the spin densities at the two interfaces are
1 tr
≪ ≃ | |
of the electrons through the n semiconductor layer is
shorter than τs. In this case−spin ballistic transport 2δn↑(0)/n=PL−e−w/L2Pθ(1−PL2)/(1−PLPθ), (18)
takes place, i.e. the spin of the electrons injected from 2δn (w)/n=(P P )/(1 P P ). (19)
↑ L θ L θ
− −
the FM layer is conserved in the semiconductor layer,
1
σ′ = σ. Probabilities of the electron spin σ = to have The spin-polarized density profile nσ(x) is shown in
the projections along ±M~2 are cos2(θ/2) and s↑in2(θ/2), Fig. 2 (bottom panel) for w ≪ Ls. One can realize
respectively, where θ is the angle between vectors σ = from Eqs. (9),(18) and (19) that large accumulation of
and M~ . Therefore, the spin current through the righ↑t the majority injected spin occurs when the moments on
2
junction can be written, using Eq. (6), as the magnetic electrodesare antiparallel,M~1 M~2, and
k−
relatively small accumulation occurs in the case of the
J↑R(↓) =J0R 2n↑(↓)(w)/n−exp(−qVR/T) parallel configuration, M~1 kM~2, Fig. 2.
d(cid:2)+(−)cos2(θ/2)+d−(+)sin2(θ(cid:3)/2) . (11) At qV >T the current saturates at the value
×
Itfollowsf(cid:2)romEqs.(5)and(11)thatthe tot(cid:3)alcurrent J =J0 1−PR2cos2θ (1−PLPRcosθ)−1, (20)
J = J↑L + J↓L = J↑R + J↓R through the left and right where J =(cid:0)JR(d +d ),(cid:1)as follows from Eqs. (18) and
interfaces is equal, respectively, 0 0 + −
(13). Fortheoppositebias,qV < T,thetotalcurrentJ
−
J =JL(d +d )[γ 2P δn (0)/n], (12) isgivenbyEq.(20)withthe replacementPL ⇀↽PR.The
0 ↑ ↓ L− L ↑ current J is minimal for antiparallel (AP) moments M~
J =JR(d +d )[γ +2P cosθδn (w)/n], (13) 1
0 − + R R ↑ and M~ in the electrodes when θ =π and near maximal
2
where γL =eqVL/T 1 and γR =1 e−qVR/T, and forparallel(P)magneticmomentsM~1andM~2. Theratio
− −
3
1.0
where K is the magneto-sensitivity coefficient. For ex-
H
ample, K 2 10−3J P P A/Oe for m /m = 14
H 0 L R 0 ∗
J/J00.5 P AP (GaAs) and≃g =×2, ttr ∼ 10−11s, and the angle θH = π
at H 1 kOe. Thus, J 1 mA at J = 25 mA,
H 0
≃ ≃
P P 0.2, and H 100 Oe. The maximum op-
H/∆H L R ≃ ≃
0.0 erating speed of the field sensor is very high, since re-
0 2 4 6
distribution of nonequilibrium injected electrons in the
1.0
Antiparallel M and M semiconductor layer occurs over the transit time ttr =
0.8 σ 1 2 w/µE = Jswτs/(JLs), ttr < 10−11s for w < 200
nm,|τ | 3 10−10s, and J/J∼ > 10 (D 25∼cm2/s
n 0.6 σ at T s ∼300×K [14]). Thus, tshe∼operatin≈g frequency
/nσ 0.4 σ Parallel M and M f = 1≃/ttr > 100 GHz (ω = 2π/ttr 1 THz) would
σ 1 2 be achievab∼le at room temperature. ≃
0.2
0.0 Antiparallel M1 and M2 funWcteioenmspohfatshizeebtihaastVthe paarnadm∆ete.rsTκhL0e(Re)ffi, cPiLen(Rt)spairne
0.0 0.2 0.4 0.6 0.8 1.0 L(R) 0
x/w injectioncanbeachievedwhenthebottomofconduction
band in a semiconductor E near both FM-S junctions
FIG. 2. (Color online) Oscillatory dependence of the cur- c
is close to a peak in a density of spin polarized states,
rent J through the structure on the magnetic field H (top
e.g. of minority electrons in the elemental ferromagnet
panel) for parallel (P) and antiparallel (AP) moments M
1
like Fe, Co, Ni (cf. [11]). For instance, in Ni and Fe the
and M2 on the electrodes, Fig. 1, and PL = PR = 0.5. Spa-
tialdistributionofthespinpolarized electronsn↑(↓)/ninthe peak is at F +∆↓, ∆↓ ≃0.1 eV [16].
structurefordifferentconfigurationsofthemagneticmoments In conclusion, we have showed that (i) the present
M andM inthelimitofsaturatedcurrentdensityJ,w=60 heterostructure can be used as a sensor for an ultrafast
1 2
nm, L2 =100 nm (bottom panel). nanoscale reading of an inhomogeneous magnetic field
profile, (ii) it includes two FM-S junctions and can be
used for measuring spin polarizations of these junctions,
Jmax(P) = 1+PLPR is the same as for the tunneling FM-
Jmin(AP) 1−PLPR and (iii) it is a multifunctional device where current de-
I-FM structure [3,4], hence, the structure may be also
pendsonmutualorientationofthemagnetizationsinthe
used as a memory cell.
ferromagnetic layers, an external magnetic field, and a
The present heterostructure has an additional degree
(small) bias voltage, thus it can be used as a logic ele-
of freedom, compared to tunneling FM-I-FM structures,
ment,amagneticmemorycell,oranultrafastreadhead.
which can be used for a magnetic sensing. Indeed, spins
of the injected electrons can precess in an external mag-
netic field H during the transit time t of the elec-
tr
trons through the semiconductor layer (t < τ ). In
tr s
Eqs. (12), (20) the angle between the electron spin and
the magnetization M~ in the FM layer is in general
2 2
θ = θ + θ , where θ is the angle between the mag- [1] S. A. Wolf et al., Science 294, 1488 (2001); Semicon-
0 H 0
ductorSpintronicsandQuantumComputation,editedby
netizations M and M , and θ is the spin rotation
1 2 H
D. D. Awschalom, D. Loss, and N. Samarth (Springer,
angle. The spin precesses with a frequency Ω = γH,
Berlin, 2002).
where H is the magnetic field normal to the spin direc-
[2] M. Baibich et al.,Phys. Rev.Lett. 61, 2472 (1988); J.S.
tion, and γ = qg/(m c) the gyromagnetic ratio, g the
∗ Moodera et al.,ibid. 74, 3273 (1995).
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H 0 tr 0 ∗ 0 [3] J. C. Slonczewski, Phys. Rev.B 39, 6995 (1989).
−
the mass of a free electron, γ0g = 1.76 107 Oe−1c−1 [4] A. M. Bratkovsky,Phys. Rev.B 56, 2344 (1997).
×
for g = 2 (in some magnetic semiconductors g 1). [5] N. Garcia et al., Phys. Rev. Lett. 82, 2923 (1999); N.
≫
Accordingto Eq.(20),with increasingH the currentos- Garcia et al.,ibid. 85, 3053 (2000).
cillates with an amplitude (1+P P )/(1 P P ) and [6] J. M. Kikkawa et al.,Science 277, 1284 (1997).
L R L R
−
period ∆H = (2πm )(γ gm t )−1, Fig. 2 (top panel). [7] P.R.Hammaretal.,Phys.Rev.Lett.83,203(1999);H.
∗ 0 0 tr
Study of the current oscillations at various bias voltages J. Zhu et al., ibid. 87, 016601 (2001).
[8] A.T.Hanbickiet al.,Appl.Phys.Lett.82,4092(2003);
allows to find P and P .
L R
For magnetic sensing one may choose θ =π/2 (M~ A. T. Hanbickiet al.,ibid.80, 1240 (2002).
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M~ ). Then, it follows from Eq. (20) that for θ 1
2 H ≪ al., ibid.402, 787 (1999).
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J =J [1+P P γ gHt (m /m )]=J +J , (21)
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K =dJ/dH =J P P γ gt (m /m ), (22) (1997); G. Schmidt et al., ibid. 62, R4790 (2000); E.I.
H 0 L R 0 tr 0 ∗
4
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5