Resonant Tunneling of Polarized Eletrons Through
Nonmagneti III-V Semiondutor Multiple Barriers
C. MoysesAraujo, A. Ferreirada Silva,
Instituto deFsiaUniversidadeFederaldaBahia,
UFBa,Salvador,BA,Brazil
and E. A. de Andrada eSilva
InstitutoNaional dePesquisas Espaiais-INPE,
12201-900,S.J.dosCampos,SP,Brazil
Reeivedon23April,2001
Thequantumtransportofspin-polarizedeletronsarossnonmagnetiIII-Vsemiondutormultiple
barriersisonsideredtheoretially. WehavealulatedthespindependenttransmissionoeÆient,
forondutingeletronstransversinglattie-mathedIn0:53Ga0:47As/GaAs0:5Sb0:5/In0:53Ga0:47As/
InP/In
0:53 Ga
0:47
Asnanostrutureswithdierentnumbersofasymmetridoublebarriers,asa
fun-tionofeletronenergyandangleofinidene. Spin-orbitsplitresonanes,duetotheRashbaterm,
areobserved. TheenvelopefuntionapproximationandtheKanekpmodelforthebulkareused.
For anunpolarized inident beamof eletrons,we alsoobtain the spinpolarization of the
trans-mittedbeam. Theformationofspindependentminibandsofenergywithnonzerotransmissionis
observed.
Thepossibilityofeletron-spinpolarizationby
res-onanttunneling,duetotheRashbaspin-orbitoupling
in semiondutor heterostrutures, has been reently
proposed [1, 2℄. Suh spin-dependent eet is of
in-terest in the development of the soalled spintronis
[3, 4, 5℄. The material or struture paramater
op-timization of suh eet is however still missing. In
this ontribuition,wedisusstheresultsforboth
spin-dependenttransmissionoeÆientandpolarizationfor
eletronstraversingmultiplebarrierswithvarying
num-ber of asymmetri double barrier unit ells.
Spin-dependentminibandsofenergieswithnozero
transmis-sion and an inreasing maximum polarization of the
transmitted beam, for inreasing number of ells, are
obtained.
ItiswellknownthatthesoalledRashbaspin-orbit
termintheeetiveHamiltonianforeletronsonned
inasymmetriquantumwellsdependsonlyontheangle
between thegrowth diretion (bz) and the eletron ;
s
wave-vetork. Itanbewrittenas[6℄
H
SO =
d
dz
(z;E)ksin= d
dz
(z;E)k
k
: (1)
The oupling parameter as given bythe eigth band
Kanemodelreads[7℄
(z;E)= P
2
2
1
E E
v (z)
1
E E
v
(z)+(z)
(2)
where E
v
is the edge of the valene band, (z) is
thespinorbitsplittingin themaximumofthevalene
band and P is the interband momentum matrix
ele-ment. Simplespindependentboundaryonditionsfor
the envelope funtion an be derived in the presene
ofthisterm[7℄andtheproblemofthespin-dependent
quantum transportanbestudied withstandardwave
mehanisproedure (notethat for zerobias H
so =0
in eah layer of the struture and the solution there
remainsaplanewave).
Wehavethenonsideredaninomingeletronwith
energyE,wave-vetork
k
paralleltotheplanesandspin
+or (upor downwith respettoy),b andsolvedfor
thespindependenttransmissionoeÆientt
, for
dif-ferentnumberofasymmetrialdoublebarrierunitells.
Usingstandardtransfermatrixmethod thesolutionis
straightforward.Forthetwoellsase,forexample,we
mustsolvethefollowingequation
t
0
=M 2
e ikz2p
0
0 e
ikz2p
1
r
with M =B (1) e ik z w 1 0 0 e ik z w 1 B (2) e ik z w 2 0 0 e ik z w 2 (4) d and w j =d j +L j
; p=w
1 +w
2
(5)
where theB (j)
;j =1;2arethe spin-dependent
trans-fer matries orresponding to the two dierent
barri-ers,L
1
isthedistanefrombarrier1tobarrier2,while
L
2
is from barrier 2 to barrier 1, d
j
is the j th
-barrier
widthandhk
z =
p
2m
0
(E)Eosistheeletron's
mo-mentum alongthegrowthdiretion. Thetransfer
ma-trixanbeobtained diretlyfrom thespin-dependent
boundaryonditions[7℄and anbewritten as[1℄
B (j) = m 0 m j 2k z j sinh( j d j ) e ik z d j 0 0 e ik z d j P Q Q P (6) with P = 2k z j m 0 m j 1 tanh( j d j ) + i " 2k k h 2 2 ( 0 j ) 2 + k 2 z m 2 0 2 j m 2 j !# (7) and Q = 4k z k k h 2 m 0 ( 0 j )+ i " 2k k h 2 2 ( 0 j ) 2 k 2 z m 2 0 + 2 j m 2 j !# ; (8)
where, hk
k =
p
2m
0
(E)Esin is the onserved
momentum parallel to the interfaes and
j = q 2m j (E)(E j E)=h 2 +k 2 k
is the deay oeÆient of
theevanesentwaveinsidethej th
-barrier.Thewelland
barrier material parameter,fm
0 ;
0
gand fm
j ;
j g
re-spetively,areenergydependent,inaordtotheKane
model.
InFig. 1weshowtheresultsofthespindependent
transmission probability T
= t t
as a funtion of
eletronenergyintheaseofamultipledouble-barrier
struturewiththreeunitells,orrespondingtoa
stru-turewithsixbarriers.Thebandparametersusedinthe
alulationare listedin Table1. Oneansee that
in-steadofthebroadresonanesfoundforonedouble
bar-rier[1,2℄,oneanseeinthismultiplebarriersystemthe
nonzerotransmission. Itisinterestingtonotethat the
miniband width is muh largerthanthe spin-splitting
obtained with just one unit ell. For this ase with
L
1 =L
2
,asweadd moreand moreellsthestruture
losesgraduallyitsinversionasymmetryandtheoposite
spinminibandstendtooverlapompletely,
reestablish-ingthespindegenerayofthesymmetristruture.
Figure 1. Transmission oeÆient for eletrons arriving
with an angle = =4 with respet to the growth
dire-tion, and rossing three (3X) asymmetri double-barrier
unit ells of lattie-mathed In
0:53 Ga 0:47 As/InP/In 0:53 Ga 0:47 As/GaAs 0:5 Sb 0:5 /In 0:53 Ga 0:47
As. Wehaveused
L
1 =L
2
=20nm andd
1 =d
2
=3nm Theband
parame-ters usedarelisted inTable 1andthebandosets forthe
ondutionbandwere0.18eVand0.36eVforInGaAs/InP
andInGaAs/GaAsSbrespetively,whihwerereently
mea-sured[8℄.
E
g
(eV) (eV) m e (m e ) In 0:53 Ga 0:47
As 0.75 0.36 0.041
InP 1.42 0.11 0.079
GaAs
0:5 Sb
0:5
0.81 0.75 0.040
Table1. Bandparametersfor thebulkmaterialsused
in thealulation[9℄.
If one onsiders an unpolarized beam of inoming
onduting eletronsandalulates thepolarizationof
thetransmittedbeamdenedby
P(E;)= T
+
(E;) T (E;)
T
+
(E;)+T (E;)
; (9)
onends,forsmallnumberofells,asshoowninFig. 2,
Wehaveplottedin Fig. 2theobtainedpolarizationas
a funtion of eletron'senergy for transmissionaross
1,2 and 3 asymmetri double barrier unit ells. The
inreasedpolarizationwithmoreellsours,however,
atenergieswithaorrespondingdereasedtransmission
probability,betweentheinnitesuperlattieminibans.
Figure2.Polarizationofthetransmittedbeamaross
stru-tureswithone(1X),two(2X)andthree(3X)repetitionsof
the asymmetridouble-barrier unitell, with the
parame-tersasinFig.1.
In onlusion,wehave studied the spin-dependent
quantumoherenttransportofspinpolarizedeletrons
along nonmagneti III-V semiondutor multibarrier
nanostrutures. We have observed the formation of
spin dependent energy minibands with nonzero
trans-missionandaninreasingmaximumpolarizationofthe
transmittedbeamwiththenumberofells,betweenthe
minibands, whih asmentioned beome spindegenare
in the superllatie limit. Nevertheles, the interesting
situation would be that of minibands strongly
depen-general L
1 6= L
2
ase, orresponding to superllaties
withoutinversionsymmetry. Suhstudy,aswellasthe
analysis of the polarization of the reeted beam, is
workin progressandwillbepublishedelsewhere.
Aknowledgments
This work was partially supported by CNPq,
CAPES, and FAPESP. We aknowledge helpfull
dis-ussionswithProf. G.C.LaRooa.
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