Eletron-Phonon Interation in Eletroni Tunneling:
From Sequential Rate Equations to a Coherent Desription
L.E.F. Foa Torres a
, H.M. Pastawski a
, and S. S.Makler b;
a
FaMAF,UniversidadNaional deCordoba, CiudadUniversitaria,5000Cordoba,Argentina
b
InstitutodeFsia,UniversidadedoEstadodoRiode Janeiro, Riode Janeiro, Brazil
Institutode Fsia, UniversidadeFederalFluminense,CampusdaPraia Vermelha,24210-340Niteroi,Brazil
Reeivedon23April,2001
We disuss the main theoretial approahes for the phonon-assisted tunneling in double barrier
resonanttunnelingdeviesandintrodueaquantumoherenttreatmentbasedonthemappingof
the many-bodyproblemintoahigherdimensional one-body system. Conditionsfor amaximized
phonon-emissionareestablished.
Muh progress in semionduting and moleular
eletroni devies[1,2℄is inspiredby Landauer'sview
[3, 4℄ of ondutane as transmittane. However, the
eletron-eletron (e-e) and the eletron-phonon(e-ph)
interations add substantial omplexity to the
ele-troni problem, limitingits appliation. The rst has
reeivedmuh attentionin dierent ontexts. In
on-trast, after the observation of opti phonon-assisted
tunneling,[5℄interestone-ph interationremained
fo-used in double barrier Resonant Tunneling Devies
(RTD). However, the reent observation of related
eletro-mehanial eets in moleular eletronis [6℄
requiresareonsiderationoftheproblem.
In a AlGaAs-GaAs RTD, besidesthe usual elasti
resonane peak, asatellite peak rises in the valley of
theurrent-voltage(I-V ) urve. This ours whenthe
groundstatein thewellisonelongitudinalopti (LO)
phonon energy below the Fermi level of the emitter.
Thus, an eletron with kinetienergy ""
F
and
po-tentialenergyeVintheemitterdeaysintoaneletron
with energy "+eV ~!
0
in the olletor plus a
LO-phonon. Atthis point, anumberofquestions emerge:
Is it possible to extend the Landauer's piture to
in-ludeinelasti sattering? Whatis theroleofe-ph
in-teration intheeletroni dephasing? Inthisworkwe
disuss the main theoretial approahes used to treat
thee-phinterationinRTDsandtrytoshedlightover
these questions. Besides, the onept of resonane in
e-phFokspaeisintrodued.
The rst solution of transport in a RTD
inlud-ing strongly inelasti e-ph sattering [7℄ onsidered a
singleeletron state in the well interating with opti
phonons. Thesattering problemwassolved,in aone
eletronapproximation, byomputingthe many-body
suhasenergyindependentouplingstotheeletrodes
[7℄ (broad band approximation). Inthis oherent
pi-ture,atight-bindingmodel[8℄yieldssimilarresults.
Aoneptuallydierentapproah,seeRefs. [9℄and
[10℄,onsideredthee-phinterationasasoureof
deo-hereneandthermalizationfortheeletronsby
adopt-ing a omplex self-energy orretion to the eletroni
states. Thus, in this desription, the phonon system
ats in a way analogous to the \voltage probes" in
theButtiker'sformulationofLandauer'spiture. Only
eletrons that do not interat with phonons maintain
oherene with its soure. This line, whih nds full
formalsupport within theKeldyshformalism[11℄, has
beenfurtherdeveloped[12℄toinludestronglyinelasti
proessesandoriginatedomputationalodes[13℄that
simulate mesosopidevies.
Most frequently rate equations [14, 12℄ are used.
The alulationof the ratetransition probabilities
re-liesontheappliationoftheFermiGoldenRule(FGR)
at two stages: a) Todesribe tunneling into the well.
Quantum oherent eets are ignored sine it is
as-sumed that the phase of the eletroni wave funtion
is randomizedby somemehanism. Then, within this
sequentialtunneling piture, theeletron tunnels into
the welland, after losingmemory ofits phase,it
tun-nels out of the well. b) Toprodue phonon emission.
It requires aweake-ph oupling witha densephonon
spetrumjustifying theFGRand the eletroni
deo-herene.
An alternative approah was introdued in Refs.
[15℄ and [16℄. There, the many-body problem of one
eletroninteratingwithphononswasexatlymapped
Tox ideas,letusonsiderasimpleHamiltonian:
H =
X
j fE
j
+
j
j V
j;j+1 (
+
j
j+1 +
+
j+1
j )g+
+~!
0 b
+
b V
g X
j2well
+
j
j (b
+
+b); (1)
Thersttermrepresentsanearest-neighbortight
bind-ing Hamiltonianfor theeletrons,where +
j and
j are
eletronoperatorsatsitej ona1-dhainthatinludes
a number of sites in the barriers and the well. The
hopping parameters are V
j;j+1
= V. The site
ener-gies E
j
model the potential prole. The seond and
third terms represent the phonon and the e-ph
on-tributions. b +
and b are the phonon operators and
V
g
is the e-ph oupling that is limited to the w
ellre-gion. Then, if weonsider the Fok spae expanded
byjj;ni= +
j (b
+
) n
= p
n!j 0i; themany-bodyproblem
mapstothe2-dimensionalone-bodyproblemshownin
Fig. 1 a). The vertialdimensionis thenumber nof
phonons [15, 16℄. Themodel an befurther simplied
byadeimationproedureifoneonsidersonlythe
ele-tronigroundstateinthewell. Then,onegetsamodel
fortheRTDasaentralsiteweaklyoupledtotheleads
that interatwith the phonons(see Fig 1(b)). Then,
E
0
isthewell'sgroundstatewhihisshiftedbythe
ele-tri eld and V
0;1 = V
R and V
1;0 =V
L (V
L(R) V)
whihxthetunnelingratesthroughthebarriers.
T
2, 0
T
0, 0
2
1
0
n
..
.
..
.
a)
b)
c)
0.1
0.2
0.3
0.01
0.1
1
I
[a
rb.
uni
ts
]
V [Volts]
G
R
G
L
G
R
G
L
~
Figure1.a)EahsiteisastateintheFokspae:Thelower
roware eletronistatesindierent siteswithnophonons
inthewell,thesitesinblakareinthebarriers.Higherrows
orrespondtohighernumberofphonons. Straightlinesare
hoppings and wavy lines are e-ph ouplings. b) Pitorial
representationof the entangledproesses of the rst two
polaronistates. )Calulated I-Vurveshowing satellite
peaksofoneandtwophononproesses.
Within this equivalentproblem, the transmission
probability of eletrons between inoming and
outgo-be alulated exatly from the Shrodingerequation.
Oneanprune theFokspae andinlude onlystates
withinsomerangeofnallowingavariational,non
per-turbative,alulation. Thus, wearenotrestritedtoa
weake-phoupling.Itmustbeemphasizedthatinthis
approah, nophase randomization aused of the e-ph
interationisassumed.Insteadofalulatingtransition
rates,the omplexquantum amplitudes for eah state
intheFokspaeareobtained. Toalulatethe
trans-mittanes between dierent hannels severalmethods
anbeadopted. Onepossibilityistosolveforthewave
funtion iteratively[16℄. An alternative is to obtain
Green's funtions whose onnetion with the
satter-ingmatrix wasestablishedby Fisherand Leeand
ex-tendedformultileadtight-bindingsystemsbyD'Amato
and Pastawski [10℄. Here the powerof the Green's
funtions tehniquesanbeanalytiallyexploited and
transformedinto omputationally eÆient algorithms.
Inthisase,thehorizontaldanglinghainsanbe
elim-inated througha deimation proedure [10,17℄in
tro-duingomplexself-energiesintheorrespondingsites.
Onethetransmittanesareobtained,thequestion
of howto omputethe urrentsnaturally emerge. In
theLandauer'spiture, theview isthat of orthogonal
satteringstatesextendedalongtheondutorfromthe
emittertotheolletor. Thisorthogonalityimpliesthat
thePauliexlusionprinipledoesnotenterinthe
alu-lationoftheurrents. Inthepreseneofinelasti
sat-tering,eletronsfrom dierentinomingstatesan
o-upythesameoutgoingstate.Thus,ifoneusesasingle
eletrontransmittanestorepresentthemany-eletrons
system,thesemustbeomplementedwithsomefators
aountingforthePauliexlusion[18℄. Otherwise,there
may be an overow of the nal states. An attempt
to solvethis problem is the implementation of a
self-onsistent proedure for the non-equilibrium eletron
distributions [19℄. However,for theexperimental ase
oflowtemperaturesand~!
o >"
F
,thereisno-overow
intherightleadsineeletronswithenergiesupto"
F
annot ompete for the same nal state. Then, the
urrentsanbeomputedasinamultileadLandauer's
piture. The total urrent from left to right is asum
of theurrents througheah ofthe leadson the right
orrespondingtodierentnumberofphonons:
I
tot =
X
n I
n
; (2)
where,forhighbias(eV>"
F ),
I
n =(
2e
h )
Z
"
F
0 T
n;0
(")d": (3)
T
n;0
isthetransmissionprobabilityfrom theleft
han-nelwithnophononsto thehannelwithnphononsin
theolletor.
al-spaeandtoidentifytheontrolparametersinanRTD
(devie geometry, voltage) that optimize the oherent
proesses leading to the phonon emission. To
illus-trate this, we onsider the states orresponding to 0
and 1 phonons. We found that the peak valueof the
inelasti transmission probability at the satellite peak
is maximized when the in-sattering rate equals the
out-sattering rate at the state with 1 phonon. The
in-satteringrate ~
L
is equalto therate of inometo
the state with no phonons,
L
, redued by a fator
(V
g = ~!
o )
2
(see Fig. 1b)). For (V
g = ~!
o )
2
1and
L +
R <~!
o
, theinelasti urrentis
I
1 '
e
~ 4
e
L R
( e
L +
R )
"
2
artan
"
F
2( e
L +
R )
!#
(4)
' (
e
~ 4
e
L R =(
e
L +
R
) for"
F (
e
L +
R )
2e
h T
1;0 "
F
for"
F (
e
L +
R )
:
Then, when "
F
(
e
L +
R
) the inelasti urrent
beomes geometry independent in the wide range of
"
F
R >
e
L
. IntheoppositeaseI
1
; andhenethe
poweremittedasphonons~!
0 I
1
=e;beomesdetermined
bythetransmittaneatresonane,whihismaximized
by the generalizedsymmetry ondition e
L =
R . An
I-VurvemaximizingphononemissionisshowninFig.
1 ). This optimization an be useful for the
genera-tionoftheprimarylongitudinalopti(LO)phononsin
aSASERdevie [15,20℄.
Finally, bynotingthat even if~!
o
!0the
outgo-ing urrents in Eq. (2) annot interfere, we
apprei-atehowthee-phinterationintrodues\deoherene"
on the former singlepartile desription. Within this
formulation, deoherene arises beause the inlusion
of eah phonon mode inreases the \dimensionality "
of theHilbert spae preventingtheinterferene ofthe
outgoingeletronstates.
Aknowledgments
WeaknowledgenanialsupportfromCONICET,
SeCyT-UNC,ANPCyTandaninternationalgrantfrom
Andes-Vitae-Antorhas. HMP and LEFFT are
aÆli-atedwithCONICET.
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