Table Of ContentSpin-dependent tunnelling through a symmetric barrier
V.I. Perel’, S.A. Tarasenko,∗ and I.N. Yassievich
A.F. Ioffe Physico-Technical Institute, 194021 St.Petersburg, Russia
S.D. Ganichev, V.V. Bel’kov, and W. Prettl
3 Fakult¨at fu¨r Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany
0
0 The problem of electron tunnelling through a symmetric semiconductor barrier based on zinc-
2 blende-structurematerialisstudied. Thek3 DresselhaustermsintheeffectiveHamiltonian ofbulk
semiconductor of thebarrier are shown to result in a dependenceof thetunnellingtransmission on
n
the spin orientation. The difference of the transmission probabilities for opposite spin orientations
a
can achieve several percents for thereasonable width of the barriers.
J
9
PACSnumbers: 72.23.-b,73.63.-b
]
l
l Lately spin polarized electron transport in semicon-
a
h ductors attracts a great attention.1 One of the major (I) m1 (II) m (III) m1
2
- problems of general interest is a possibility and meth-
s
me owdasy otof aspchinievinejescptiinonoriienntotatsieomnicinonedxupcetroimrs.entAisntahteuirna-l r, -kz V t, k
z
jection of spin polarized carriers from magnetic mate-
at. rials. Although significant progress has been made re- kz
m cently,2,3,4,5 reliable spin-injection into low-dimensional
electrons systems is still a challenge. Schmidt et al. 0 a z, [001]
-
d pointedoutthatafundamentalobstacleforelectricalin-
n jection from ferromagnetic into semiconductor was the
o
conductivitymismatchofthemetalandthesemiconduc- FIG. 1: Electron tunnelling through the potential barrier V
c
tor structure.6 However, Rashba showed that this prob- of width a.
[
lem could be resolved by using tunnelling contact at
2 the metal-semiconductor interface.7 On the other hand
v Voskoboynikov et al.8 proposed that asymmetric non- of the barrier in effective mass approximation contains
8 the spin-dependent k3 term (Dresselhaus term)13
magnetic semiconductor barrier itself could serve as a
9
spinfilter. Itwasdemonstratedthatspin-dependentelec-
0 Hˆ =γ σˆ k (k2 k2)+σˆ k (k2 k2)+σˆ k (k2 k2) ,
1 tron reflection by inequivalent interfaces resulted in the D x x y − z y y z − x z z x− y
(1)
0 dependenceofthetunnellingtransmissionprobabilityon (cid:2) (cid:3)
whereσˆ arethePaulimatrices,γ isamaterialconstant
3 the orientation of electron spin. This effect is caused by α
0 interface-induced Rashba spin-orbit coupling9 and can (seeTable1),andthecoordinateaxisx,y,z areassumed
/ to be parallel to the cubic crystallographic axis [100],
t be substantial for resonant tunnelling through asym-
a metricdouble-barrier10,11 ortriple-barrier12 heterostruc- [010],[001],respectively. Inthecaseoftunnellingalongz
m oneshouldconsiderk intheHamiltonianasanoperator
tures. However, in the case of symmetric potential bar- z
- riers, the interface spin-orbit coupling does not lead to a i∂/∂z. We assume the kinetic energy of electrons to
d −
be substantial smaller than the barrier high V, then the
n dependence of tunnelling on the spin orientation.
Hamiltonian (1) is simplified to
o
In this communication we will show that the process
c
v: othfattuannceolnlisnigdeirsabspleinspdineppeonldaerniztaittisoenlfc.anWbeedeexmpoecntsetdraatet HˆD =γ(σˆxkx−σˆyky)∂∂z22 . (2)
i
X tunnelling of electrons even through a single symmetric
barrierifonlythebarriermateriallacksacenterofinver- One can note that essentially Hˆ induces a spin-depen-
r D
a sion like zinc-blende structure semiconductors. The mi- dent correction to the effective electron mass along z in
croscopicoriginoftheeffectistheDresselhausk3terms13 the barrier.
intheeffectiveHamiltonianofthebulksemiconductorof
the barrier.
GaSb InAs GaAs InP InSb
We considerthe transmissionofelectronswiththeini-
γ, eV˚A3 187 130 24 8 220
tial wave vector k = (kk,kz) through a flat potential ·
barrier of height V grown along z [001] direction (see m∗/m0 0.041 0.023 0.067 0.081 0.013
k
Fig. 1); k is the wave vector in the plane of the barrier,
k
and k is the wave vector normal to the barrier pointing Table 1: Parametersof band structure of various A B
z 3 5
in the direction of tunnelling. The electron Hamiltonian semiconductors.14,15
2
The Hamiltonian (2) is diagonalized by spinors
S S
+ [010] - [010]
1 1
χ± = √2 e−iϕ ! , (3) k|| [110] k|| [110]
∓
which correspond to the electron states ”+” and ” ” of [100] [100]
−
the opposite spindirections. Hereϕis the polar angleof
the wave vector k in the plane xy, being
[110] [110]
k=(k cosϕ, k sinϕ, k ). (4)
k k z
FIG. 2: Spin orientation of eigen states ”+” and ”−” as a
Transmissionprobabilities for the electrons of eigen spin
functionoftheorientationofthein-planeelectronwavevector
states ”+” and ” ” (3) are different due to spin-orbit
term (2). The orie−ntations of spins s in the states ”+” kk.
±
and” ”depend onthe in-plane wavevectorofelectrons
−
and are given by
and m (i = 1,2) are the effective masses outside and
i
inside the tunnelling barrier, respectively. Taking into
s =( cosϕ, sinϕ, 0). (5)
±
∓ ± account the boundary conditions, which require that
Fig. 2 demonstrates the orientations of spins s and s
+ −
1 ∂u
forvariousdirectionsofthein-planeelectronwavevector u and ± (10)
±
k , i.e. as a function of polar angle ϕ. If k is directed m ∂z
k k
along a cubic crystal axis ([100] or [010]) then the spins
arecontinuousattheinterfaces,asystemoflinearequa-
are parallel (or antiparallel) to k , while s are perpen-
k ± tionsfort , r , A ,andB canbe derived. Solutionof
± ± ± ±
diculartok ifthein-planewavevectorisdirectedalong
k the system allows one to calculate the coefficients of the
the axis [1¯10] or [110]. transmission t . For the real case m k γ/¯h2 1 they
± 2 k
Electrons of the eigen spin states ”+” and ” ” prop- ≪
− are derived to be
agate through the barrier with conserving of the spin
orientation. Since the wave vector in the plane of the m k
2 k
t =t exp γ aq , (11)
barrierkk isfixed,wavefunctionsofthe electronscanbe ± 0 ± ¯h2 0
(cid:18) (cid:19)
written in the form
where t is the transmission coefficient when the spin-
0
Ψ±(r)=χ±u±(z)exp(ikk ρ), (6) orbit interaction (1) is neglected,
·
where ρ= (x,y) is a coordinate in the plane of the bar- m k q
2 z 0
rrieeflre.ctTedhewfauvnecs,tisoeneFui(gz.)1i)n,ItIhaenrdegIIioIn(strIan(sinmciotmteidngwaavned) t0 =−4im1(q0−ikzm2/m1)2 exp(−aq0), (12)
has the form
a is the width of the barrier. The general problem of
u(I)(r)=[exp(ik z)+r exp( ik z)] , (7) tunnelling of an electron with arbitrary initial spinor χ
± z ± − z can be solved by expanding χ to the eigen spinors χ±.
Itisconvenienttointroduceapolarizationefficiency
P
u(II)(r)=[A exp(q z)+B exp( q z)] , that determines the difference of tunnelling transmission
± ± ± ± − ± probabilitiesforthespinstates”+”and” ”throughthe
−
barrier
u(III)(r)=t exp(ik z),
± ± z t 2 t 2
+ −
= | | −| | . (13)
respectively. Here t± and r± are the transmission and P |t+|2+|t−|2
reflectioncoefficientsforspinstatesχ ,respectively,and
±
In our case it has the form
the wave vectors under the barrier q are given by
±
m k
q =q 1 γ2m2kk −1/2 , (8) P =tanh 2γ ¯h22k aq0 . (14)
± 0 ± ¯h2 (cid:18) (cid:19)
(cid:18) (cid:19)
Atagiveninitialwavevectorofelectrons,k,thepolar-
where q is the reciprocal length of decay of the wave
0 ization efficiency drastically increases with the strength
function in the barrier for the case when the spin-orbit
of the Dresselhaus spin-orbit coupling γ (see Eq. 14),
interaction (1) is neglected
andthebarrierwidtha. Howeverwhileincreasethebar-
rierwidthaincreasesthetunnellingefficiencyoneshould
q = 2m2V k2 m2 k2 m2 1 , (9) keep in mind that the barrier transparency decreases si-
0 s ¯h2 − z m1 − k(cid:18)m1 − (cid:19) multaneously (see Eq. (12)). In Fig. 3 the efficiency P
3
m∗ = m2 used in the calculations are given in the Ta-
20% 100 ble1. Onecanseethatforareasonablesetofparameters
2
P |0 given in the figure caption it is possible to achieve spin
fficiency, 1100--21 parency, |t pptnhaeoelrllaeimnrdigozoasbntteieoctffinahusecsioebefnasotsefivbsteahorreafrlileGaprreagmrSecbaevtnaoetrlrsui.aietlTssouffsonotrnhlueselptlipiionnrngossdeblusaeecrcetrtmiieγvrmestotp∗u.rbnee--
n e 10% GaSb ns Thepolarizationstronglydependsontheelectronwave
o 10-3 a
zati InAs InSb er tr svuecgtgoerstkskapadreavlliecletfootrhsepbinarirniejerc(tsieoenEiqn.to(1q4u))a.nTtuhmisrweesulllst.
ari GaAs 10-4 rri Let’sassumetwoquantumwellsseparatedbyatunneling
ol InP Ba barrier, and a current flowing along one of the quantum
P
0% 10-5 wells. The in-plane current results in non-zero average
00 1 22 3 44 5
electronwavevectork and,duetotheconsideredeffect,
Barrier width, q a k
0 in a spin polarization of carriers.
Inconclusion,wehavedemonstratedthatthek3 Dres-
FIG. 3: Coefficient of thepolarization efficiency P as a func- selhaus terms in the effective Hamiltonian of semicon-
tion of barrier width, aq0, for various barrier materials, and ductors lacking a center of inversionyield a considerable
k =2·106cm−1. spin polarization of electrons tunnelling through barri-
k
ers. The effect could be employed for creating spin fil-
ters, eg. on the base of type-II strained heterostructures
and the barrier transparency t 2 are plotted as a func- like InAs/GaSb, InSb/GaSb and GaSb/GaAs.
0
| |
tion of the barrier width, q a, for various barrier ma- The workwassupported by the RFBR, the Presidium
0
terials. The material parameters γ and effective mass of the RAS, and grants of the DFG and INTAS.
∗ Electronic address: [email protected]ffe.rssi.ru 58, 15397 (1998).
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