Shot Noise in the Presene of Spin-Flip Sattering
F.G.Brito and J.C. Egues
Departamentode Fsiae Informatia,Institutode Fsiade S~aoCarlos,
UniversidadedeS~aoPaulo,13560-970S~aoCarlos,S~aoPaulo,Brazil
Reeivedon23April,2001
Shot noise is a time-dependent urrent utuation due to the disrete harater of the eletron
harge. Hereweaddressshotnoiseinaspin-resolvedtunnelingsystemundertheinueneof
spin-ip satteringwithinamasterequationapproah. Wendthattheaverageurrent hIiandFano
fator (\normalized noise") present ontrasting behavior for diering spin-iptime ratios: hIi
dereases while inreases for
"# >
#"
as omparedto the
"# =
#"
ase and vie versa for
"# <
#" .
Spin-dependenttransportintunnelingstruturesis
a fasinating subjet. So far the bulk of
experimen-tal and theoretialinvestigationshas beenon metalli
layered systems. This fous in now turning to more
versatilesemiondutor heterostrutures. More
speif-ially, spin-polarized transport { an obvious
possibil-ityinspin-dependentgeometries[1℄{hasjust reently
beenahievedinMn-basedII-VIandIII-Vdoped
semi-ondutordiodes[2, 3, 4℄. These seminal experiments
openupnewvenuesforfurtheradvanesin the
emerg-ing eld of spintronis in semiondutors{ where the
spinofthearrier,ratherthanitsharge,runstheshow
[5,6℄.
Sofarspininjetion[2,3,4℄hasbeendemonstrated
onlyat lowtemperatures andin thediusive limit[7℄.
Spin injetion at room temperatures, relevantfor real
devie appliations, remainsahallenge. However,
in-vestigationsatlowtemperaturesarefundamentally
im-portant for understanding general features/properties
of spin-polarized transport whih may turn out
rele-vant at high temperatures. Coneivably, some
detri-mentalspin-ip eetsontaminatinglowtemperature
transportmayverywellsurviveathighertemperatures.
Here we investigate spin-ip eets on transport
through a spin-resolved tunneling system [8℄, Fig. 1.
More speially, we are interestedin spin-ip eets
onshotnoise. Shot noiseisadynami urrent
utu-ation arising from the partiular (or granular) nature
of the eletron harge. Shot noiseinvestigations have
beomeamajorsubjetofresearh in thelast deade
orso [9, 10℄. Common wisdom has it that shot noise
measurementssupplementinformationontainedinthe
averageurrent. Here in partiular we ndthat
spin-ipproessesaettheaverageurrentandshotnoise
in aontrasting fashion; in alimitedparameterrange,
while shot noise is suppressed the average urrent is
enhanedandvieversa,Fig. 2.
EMITTER
COLLECTOR
CENTRAL
ISLANDS
c
1
τ
↓
τ
↑
1
c
τ
↑↓
1
τ
↓↑
1
1
e
τ
↓
τ
↑
1
e
Figure1. Shematirepresentationofaresonant-tunneling
system with a semimagnetiquantum well/dot. Spin up
eletrons fromtheemitter anhopinto theupperspin-up
levels (\island up") and subsequently eitherspin ip into
the lower spin-downlevels (\island down") or hop out to
the olletor. Spin downeletrons perform similar
transi-tions. All these hops are desribed by \transition times"
's; the orresponding transition ratesare proportionalto
1='s as indiated. Note that hopingbetween the entral
islandsinvolvesspinipproessesharaterizedby"#and
#".
Physialsystem. WeonsideraII-VIdouble-barrier
tunneling struture with aManganese-basedquantum
well/dot. Inthepreseneofanexternalmagnetield,
thes-d exhangeinterationbetweenatraversing
ele-tronandthed eletronsoftheMngivesrisetoa
spin-dependent quantum well/dot potential. This in turn
lifts thespin-up/spin-downdegenerayoftheresonant
level. Eah exhange-splitresonantlevel aepts
ele-trons with onlyawell-denedspin omponent.Fig. 1
shematially shows the system geometry. In Fig. 1,
the many possible eletron transitions onneting the
emitter, spin-split levels(\islands" up and down)and
olletoraredesribedbytransitionratesproportional
to 1=, where is, roughly speaking, the respetive
transition time. Observethat thehops between
spin-resolvedlevelsin Fig. 1involvespinip. Here weuse
distintspin-iptimesfortheup-to-downand
down-to-uptransitions betweentheislands, i.e.,
"# 6=
#" ;this
isertainlytheasewhentherelevantspin-ip
meha-nism onnetingthe islands is thes-d exhange
inter-ation[11℄. Ourdesription,however,doesnotrelyon
anyspei formofthespin-ipmehanism.
Semilassial desription. We model spin-resolved
tunnelingthroughaquantumwell/dotintermsof
las-sialhopsamongemitter,spin-splitlevelsandolletor
as shown in Fig. 1 (see also inset in Fig. 2). Our
approah is based on rate equations for the ensemble
averaged number of eletrons n
"
(t) and n
#
(t) in the
islands up and down, respetively. These equations
arediretlyderivedfromthesystempopulationmaster
equation desribingtheprobabilityp(n
" ;n
#
;t)of
nd-ingn
" andn
#
eletronsintheislandsattimet. Island
up(down)holdsatmostN
" (N
#
). Ourtransitionrates
takeinto aountthat noeletronsanhopinto afull
island(\Paulipriniple")oroutofanemptyone. This
typeoforrelationdoesaeturrentutuations
aris-ingfrom thedisretenessoftheeletronharge.
Averageurrentanditsutuations. Fromtherate
equationsmentionedaboveweaneasilydeterminethe
averageurrentthroughthesystem. Inthesteadystate
hIi=ehn
" i= " +ehn # i= #
(e: eletronharge)andthe
average oupations hn
";# i = n
";#
(t ! 1) depend on
the several hoping times 's and N
";#
. The urrent
utuationisdened intermsoftheensembleaverage
urrent-urrent orrelation funtion hÆi(t+t 0
)Æi(t 0
)i.
Experimentally, the relevant quantity is the spetral
power of the shot noise measured (usually) at zero
frequeny, i.e., S(0) = 2 R
hÆi(t+t 0
)Æi(t 0
)idt. By
introduing appropriate \hop-hop" orrelation
fun-tions, whih desribe how likely a partiular hop
fol-lows a previous one, we an relate S(0) to the
vari-ane matrix of the system [12℄. This 22 matrixis
diretlyobtainedfrom themasterequationof the
sys-tem. The resulting expressions for thezero-frequeny
shotnoiseS(0)andtheelementsofthevarianematrix
(hÆn
" Æn
#
i=hÆn
# Æn " i, h(Æn " ) 2
i, andh( Æn
# )
2
i)aretoo
lengthyto beshownhere[13℄.
Results and disussions. Fig. 2 displays urves
for both hIi=e (symbols) and the Fano fator =
S(0)=2ehIi(lines)fortheresonant-tunnelingsystemin
Fig. 1, with diering spin-ip time ratios. The
nor-malizing fator 2ehIi, usually alled `lassial orfull'
shot noise (Shottky), denotes the shot noise power
spetraldensityfor astream of unorrelated eletrons
with average urrenthIi: The Fano fator is a
onve-nientmeanstoassesshow orrelationsaetshotnoise.
In Figure 2 both average urrent and Fano fator are
morelearlyobserved for
" =
e"
in the range1 to 10
(or, equivalently, 0 to 1 in the horizontal log axis of
Fig. 1)in whih theowof eletronsfrom `island-up'
totheolletorissuppressed. Inthisase,toarriveat
theolletoraneletronhasto neessarilygothrough
`island-down'thusundergoingspinip. Inthis
partiu-larrange,shotnoise(Fanofator)andaverageurrent
exhibitontrastingbehaviorfordiering
"# =
#" ratios;
hIi dereases while inreases for
"# >
# "
as
om-paredtothe
" # =
#"
aseandvieversafor
"# <
#" .
Thistrendisreversedin the[ 1;0℄range. Heneshot
noiseandtheaverageurrentaredistintivelysensitive
tospinip.
-2
-1
0
1
2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
<I>
γ
<I>
γ
<I>
γ
1
=
τ
↑↓
τ
↓↑
=
0.1
τ
↑↓
τ
↓↑
=
10
τ
↑↓
τ
↓↑
τ
↑↓
τ
↓↑
τ
e
↑
τ
c
↑
c
τ
↓
↑
↓
e
τ
↓
)
/
(
log
10
τ
c
↑
τ
e
↑
γ
,
<
I>
/
e
Figure2.AverageurrentandFanofatorforthe
resonant-tunnelingsystem inFig. 1withdieringspin-iptime
ra-tios. We use N
" = N # =1, e" = e# = #
= 1, inall
urves(symbolsforaverageurrentsandlinesforFano
fa-tors). WeseethathIianddependonspin-ipsattering.
Allurves arepartiularly sensitive to the
"# =
#" ratio.
Inthe[0,1℄range(horizontalaxis),theaverageurrentand
Fanofatorexhibitontrastingbehavior;hIidereaseswhile
inreasesfor"#>#"asomparedtothe"#=#"ase
andvieversafor"#<#".
Sumarizing, we have disussed spin-ip eets on
spin-dependent transport in semimagneti juntions
with spin-split resonant levels. Our semilassial
de-sriptionin termsofrate andmaster equationsallows
ustoalulateboththeaverageurrentandshotnoise;
thelatterintermsofthevarianematrixofthesystem.
Interestinglyenough,wendthat the average urrent
and shot noise are sensitive to spin ip (between the
spin-splitlevels)inaontrastingway. Inalimited
pa-rameterrange,whiletheaverageurrentdereases
(in-reases)theshotnoiseinreases(dereases)asfuntion
oftheratio
"# =
#" .
This work has been supported by FAPESP. F. G.
Britoaknowledgesafellowship fromCNPq.
Present address: University of Basel, Institute of
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