Phonon-assisted tunneling regimes in diatomic molecules

0.35 MB·English

by  E. Vernek

Phonon-assisted tunneling regimes in diatomic molecules
E. Vernek,1,2 E. V. Anda,2 S. E. Ulloa,1 and N. Sandler1
1Department of Physics and Astronomy, and Nanoscale and Quantum Phenomena Institute,
Ohio University, Athens, Ohio 45701-2979
2Departamento de F´ısica, Pontif´ıcia Universidade Cat´olica, Rio de Janeiro-RJ, Brazil
7
(Dated: February 6, 2008)
0
0 Electronic transport in diatomic molecules (two-level systems) connected to metallic contacts is
2 analyzedinthepresenceofcompetingelectron-electronandelectron-phononinteractions. Weshow
n that phonon emission and absorption processes are strongly modified when a Coulomb energy U is
a included, as the phonons open channels that can result in destructive or constructive interference
J effects. Resonance conditions for these processes produce dramatic effects both in the density of
5 statesatthemolecularsites,aswellasintheconductancethroughthesystem. Wefindinparticular
1 an enhanced Rabi-assisted tunneling dueto phonons, as the resonance conditions are met, which is
made more evident for increasing temperatures. These effects are controllable by voltage gating of

] themolecular sites, and should beaccessible in current experiments.
l
l
a PACSnumbers: 73.23.–b,73.63.Kv,71.38.–k,73.63.–b
h
-
s
e I. INTRODUCTION tions in the transport properties of a single-level quan-
m tum dotconnectedto twometalic contactsin the Kondo
regime.17 They found an unexpected cancelation of the
. Electronic transport properties of nanometer scale
t
a structures such as natural and artificial molecules (e.g. conductanceresultingfrominterferencebetweenthepure
m electronic and phonon-assisted tunneling channels.
quantum dot structures) attached to metallic reservoirs
- have received a great deal of attention recently. Three Ourgroupreportedontheeffectofcompetinginterac-
d
main reasons drive the active research in the area: i) tions onthe conductance ofa diatomic molecule coupled
n
these systems have a fundamental role in the devel- to metallic reservoirs in the Coulomb blockade regime
o
c opment of the field of “molecular electronics” where and for temperatures above the Kondo scale.18 That
[ molecules serve as electronic components;1 ii) they are workfocusedonaregimewheretheenergydifferencebe-

relatively “simple” structures in which to address differ- tween the two local electronic levels is much larger than
1
ent conceptual issues regardingthe nature of charge and the broadening of the levels due to the coupling to the
v
6 spin transport;2 and iii) they represent the prototypical leads. In that case, indirect coupling between the local
5 gateable structures at the nanoscale.3 levelsis negligible andinterferenceofthe differenttrans-
3 Oneofthemostdefiningcharacteristicsofthesemolec- port channels from one lead to another is not impor-
1 ular systems is the strong spatial confinement that re- tant. Utilizing the equations of motion for the various
0 sultsinenhancedelectron-electroninteraction(EEI),and Green’s functions involved, we found a splitting in the
7
profoundly modifies their transport properties. Well electronic density of states (DOS) of the molecule un-
0
studied examples of the effect of EEI in confined sys- der suitable resonance conditions. This effect was iden-
/
at tems are Coulomb blockade and Kondo effect in quan- tified as Rabi-assisted tunneling, because of the formal
m tum dots.4,5,6,7,8,9 Molecular systems may also exhibit similarity with the well-known Rabi splitting phenom-
strong electron-phonon interactions (EPI), which play ena in atomic systems,19 although in this case it is due
-
d a significant role in their physical properties. Several to the interplay of EEI and EPI. The physical origin of
n works have addressed the effects of EPI in quantum- theRabi-assistedtunnelingcanbetracedbacktotheap-
o dot molecules for different geometrical arrangements, in pearance of new channels for electron tunneling between
c
a perturbative regime.10,11,12 It is known for example the metallic reservoirs. These channels, due to EEI,pro-
:
v that phonons cause a broadening of Coulomb blockade vide new resonance conditions for phonon emission and
Xi peaks and the appearance of satellite peaks in the non- absorption. An immediate consequence of this result is
lineartransportregime. Phononsalsoproviderelaxation the possibility of important changes in the conductance
r
a mechanisms(throughinelastic scatteringprocesses)that as function of the gate potential that controls electron
affect the electronic conductance, specially in molecular occupation in the molecule, and which would be evident
systems,13,14,15aswellaspolaronicshiftsoftheelectronic in experiments. This surprising result warrants a closer
levels.16 examination of the approximations used in the solution
While the effect of these different interactions on elec- of the model and a thorough analysis of the various res-
tronic transport properties has been studied extensively onance and interference conditions stemming from the
in regimes where one of them is dominant, not much is EPI in the Hamiltonian.
known when both interactions become comparable. Re- In the present paper, we address these issues and in-
cently,twoworkshaveaddressedthisissue. Al-Hassanieh clude a complete analysis of the effects of the indirect
et al. have studied the effect of electron-phonon interac- couplingbetweenthetwoelectroniclevelsinducedbythe
2
with
U
ttt ttt Hel = (cid:20)ǫic†iσciσ + 2niσniσ¯(cid:21), (3)
ttt ttt σ,iX=α,β
hh 00
---444 ---333 ---222 ---111 ttt ttt 111 222 333 444 Hph = b†b+ 1 ~ω0, (4)
(cid:18) 2(cid:19)
and the electron-phonon interaction is given by
FIG. 1: Schematic representation of themodel system. Only
tǫh0β)eevlieactprhoonn-pohnosnoofnfrienqteureancctyionω0coannndecwtisththceoluopcalilnsgitceosn(sǫt0αan<t Hel−ph =λXσ (cid:0)b†c†ασcβσ+h.c(cid:1) , (5)
λ.
where b† (b) creates (annihilates) a phonon with energy
~ω , ǫ =ǫ0 eVi, and i=α,β. We have assumed that
metallic leads. We now explore the regime where the lo- 0 i i − g
U = U = U, and included independent gate voltages
α β
cal electronic level spacing tends to zero, which makes Vi that control the electron occupation of the molecular
g
the phase effects much more important in the coherent
sitesbyshiftingthelocalizedenergieswithrespecttothe
transport processes. We also study in detail all resonant
Fermi energy of the left and right electrodes. Also,
regimesandconductingchannelsprovidedbytheEPI.To
solvethe equationsofmotion, we use aHubbard-like de- H = t c† c +h.c. (6)
mol−leads iγ iσ γkσ
coupling procedureforallthe Green’sfunctions involved iX=α,β
in the calculation.20 This approach has proven reliable γ=R,L
in the description of properties of interacting systems at and
temperatures above their characteristic Kondo tempera-
ture. Thesecalculationsallowustoconfirmtheexistence Hleads = ǫγkc†γkσcγkσ. (7)
of the Rabi-splitting in the DOS and the appearance of Xk,σ
γ=R,L
resonances and anti-resonances not obtained previously.
We show that there are structures in the DOS depend- Hmol−leads connects the diatomic molecule to the leads;
ing upon resonanceand anti-resonanceconditions whose for simplicity, we consider tiγ = t′, for all γ = R,L and
activation is level-occupation–dependent, reflecting the i = α,β. Away from the Kondo regime, the main effect
many-body nature of the problem. of the leads is to broaden the energy levels of the dots
We further explore how the DOS structures and res- through the tunnel couplings t′. To describe the trans-
onance conditions affect the conductance through the port of electrons through the molecule, we calculate the
diatomic molecule structure. We find that different localretardedGreen’sfunctionsdefinedinthe usualway
Coulomb blockade peaks in the absence of EPI, appear as,22,23
split whenever phonons are present and this effect is en-
hanced at higher temperatures. A;B iθ(t) [A(t),B(0)] eiǫtdt, (8)
∓
hh ii ≡ Z h i
whereAandB areoperatorsand denotesthe ther-
II. MODEL AND THEORETICAL METHOD h···i
modynamic average (for T = 0) or the ground state ex-
6
pectation value (for T = 0). In the frequency domain,
Ourmodelconsistsofadiatomicmoleculewithasingle the equation of motion for Green’s functions is written
level per site (or equivalently a quantum dot pair) with as
local energies ǫ0 and ǫ0 (we assume the non-degenerate
α β
case with ǫ0 >ǫ0), as schematically shown in Fig. 1. ω A;B = [A,B]± + [A,H]−;B , (9)
β α hh ii h i hh ii
Each single-level site (or quantum dot) is connected
wherethesubindexinthebracketsdenotescommutation
independently to two external metallic leads. We in-
( )andanti-commutation(+)relations. Theexpression
cludelocal(on-site)electron-electroninteractionsandan −
for the local electronic Green’s function is:
electron-phonon inter-site interaction. As described the
model can represent various experimental geometries; in Gσ(ω) = c ;c† , i=α,β. (10)
ii hh iσ iσii
particular, this model was studied in the limit of non-
interacting electrons, to explore interference effects.21 The assumption of identical leads allows the introduc-
The total Hamiltonian is then written as tionofsymmetricandantisymmetricfermionicoperators
c , c
Skσ Akσ
H = H +H +H , (1)
T mol leads mol−leads 1
c = (c +c ) (11a)
Skσ Rkσ Lkσ
where √2
1
c = (c c ). (11b)
Hmol = Hel+Hph+Hel−ph, (2) Akσ √2 Rkσ − Lkσ
3
After this transformation, H and H read as functions that involve operators for both levels α and
mol mol−leads
β (“mixed Green’s functions”), we close the equations
H˜ = ǫ c† c . (12) by considering the following approximations:
leads k γkσ γkσ
Xk,σ
γ=S,A
b†c† c c ;c† b†c† c c ;c† (15a)
and hh ασ¯ βσ¯ ασ ασii ≈ h ασ¯ βσ¯ihh ασ ασii
bc† c c ;c† bc† c c ;c† (15b)
H˜ = t˜′c† c +h.c., (13) hh βσ¯ ασ¯ ασ ασii ≈ h βσ¯ ασ¯ihh ασ ασii
mol−leads iσ Skσ b†bc ;c† b†b c ;c† . (15c)
Xk,σ hh ασ ασii ≈ h ihh ασ ασii
i=α,β
where we have assumed ǫ = ǫ = ǫ , and t˜′ = √2t′. The first two equations are the standard Hubbard de-
Rk Lk k
coupling procedure. At this level of the approxima-
In the symmetric-antisymmetric basis, the coupling be-
tion, correlations between electron operators with oppo-
tween the molecule and the leads occurs only through
site spins are neglected. The third equation decouples
the symmetric channel. The otherterms inthe Hamilto-
the phonon mode from the electron operators, as sug-
nian are not affected by this transformation. We obtain
gested by Zubarev.23 Finally, by keeping up to second
the relevant Green’s functions by using the equation of
ordertermsinλintheexpressionoftheself-energies,we
motion technique. As an illustrative example, let us de-
obtain:
scribe the steps taken in the calculation of the Green’s
function for level α :
−1
(ǫ ǫ )(ǫ ǫ U)
(ω−ǫα)hhcασ;c†ασii = 1+Uhhnασcασ;c†ασii Gii(ǫ)=(cid:26)[ǫ −ǫi i U(−1 i−niσ¯ )] −Σiσ(ǫ)(cid:27) , (16)
+λ b†c ;c† − − −h i
hh βσ ασii
+Xk t˜′hhcSkσ;c†ασii. (14) withΣiσ(ǫ)=Σi(σel)(ǫ)+Σi(σph)(ǫ). HereΣi(σel)(ǫ)is(tphhe)elec-
tronself-energyinleveliduetotheleads,andΣ (ǫ)is
iσ
Toobtainaclosedsetofequations,weapplyaHubbard- the phonon-related self-energy. They have the following
like decoupling procedure. In particular, for Green’s expressions:
[ǫ ǫ U(1 n )]t˜′2g˜(ǫ)
Σ(el)(ǫ) = t˜′2g˜(ǫ) 1+ − j − −h jσ¯i , (17)
iσ (cid:26) (ω ǫ )(ǫ ǫ U) [ǫ ǫ U(1 n )]t˜′2g˜(ǫ)(cid:27)
j j j jσ¯
− − − − − − −h i
λ2 b†b + n [ǫ ǫ +~ω U(1 n )]
Σ(ph)(ǫ) = h i h βσi − β 0− −h βσ¯i , (18)
ασ (ǫ ǫ +~ω )(ǫ (cid:2)ǫ U +~ω (cid:3)) [ǫ ǫ +~ω U(1 n )]t˜′2g˜(ǫ+~ω )
β 0 β 0 β 0 βσ¯ 0
− − − − − − −h i
b†b +1 n [ǫ ǫ ~ω U(1 n )]
Σ(ph)(ǫ) = λ2 h i −h ασi − α− 0− −h ασ¯i . (19)
βσ (ǫ ǫ ~ω )((cid:2)ǫ ǫ U ~ω )(cid:3) [ǫ ǫ ~ω U(1 n )]t˜′2g˜(ǫ ~ω )
α 0 α 0 α 0 ασ¯ 0
− − − − − − − − − −h i −
Inalltheseexpressionsg˜(ǫ)referstotheGreen’sfunction in the DOS peaks, to shift their positions by an amount
oftheleadsatenergyǫ(bothleadshavethesameGreen’s ReΣ(el), and to give rise to transport interference effects
function in equilibrium). In Eq. (17), the expression for due to the topology of the system.
the self-energy Σ(el)(ǫ) of an electron in level i, involves Equations (18) and (19) are the phonon self-energies
iσ
energiesandoccupationnumbersofelectronsintheother calculated up to second order in the electron-phonon in-
level j. These occupation numbers njσ have to be cal- teraction strength λ. They involve phonon occupation
h i
culated self-consistently. Since we are interested in the numbers b†b , which for simplicity can be calculated
h i
non-magnetic regime, the occupation numbers are taken fromthe free-bosonHamiltonian. Although this approx-
to be spin-independent ( njσ¯ = njσ ) and they will be imation is not essential, it is convenient to simplify the
h i h i
denotedjustby nj . Noticethattheelectronself-energy numerical calculation, and it does not introduce any ar-
h i
includes the indirect coupling between the two levels due tifact in the results, as it is a frequency independent c-
to the presence of the leads through the second term in- number.
sidethebracketsinthedenominatorofEq.(17). Therel-
From Eq. (16), we obtain the total DOS by using
evance of this term reduces as the inter-level separation ρ (ǫ) = ( 1/π)[ImGσ (ǫ) + ImGσ (ǫ)]. In the limit
in the dots increases. As a consequence, the net effect σ − αα ββ
wheretheelectron-phononcouplingisnegligible(λ 0),
of the electron self-energy is to produce a finite width →
the DOS presents four Coulomb blockade peaks coming
4
the molecule. As the four terms of Eq. (22) are complex
quantities with different phases, their contributions to
the current can interfere constructively or destructively
as we discuss below. The off-diagonal Green’s functions
(G and G ) in the conductance are also calculated
αβ βα
by equation of motion techniques at the same level of
approximation described above. Finally, notice that we
have neglected inelastic processes in the present calcula-
tion because we assume that the system is essentially at
thermodynamical equilibrium, at the small bias voltages
of interest.
FIG. 2: Schematic representation of the various resonances III. NUMERICAL RESULTS
andtheactivelevels. Thefigurescorrespondtotheresonance
conditions (a) 20a, (b) and (c) 20b and (d) 20c.
Equations(16)-(19)aresolvednumericallyandusedto
calculatethe DOSandconductancein(21), asdescribed
above. For convenience, the Fermi energy is set to zero
from the original atomic levels located at ǫ , ǫ + U,
α α (ǫ =0). In what follows, all the physical quantities are
ǫ and ǫ + U. As the electron-phonon interaction is F
β β giveninunitsoftheinter-levelseparation∆ǫ. Thelevels
turned on, these peaks show Rabi splittings, at char-
ǫ0 andǫ0 are set at 1.0and 2.0,respectively (above ǫ ).
acteristic energies, corresponding to phonon-absorption α β F
TheeffectofanincreaseinthegatevoltageV islowering
and/or phonon-emission processes at resonance condi- g
these levels (initially towards the Fermi energy). In the
tions. These resonance conditions occur when the
nextfewfigures,weassumeequalgatingofbothsites,so
phonon energy ~ω matches the energy difference be-
0 that Vα =Vβ =V .
tween the two energy levels. These resonance conditions g g g
Letus considerfirstthe results for the interactingsys-
occur when
tem in the absence of electron-phonon ineraction (U =
6
~ω +U =∆ǫ (20a) 0;λ = 0). Figure (3) shows the results for the DOS
0
(top) and the conductance (bottom) for the particular
~ω =∆ǫ (20b)
0 values U = 0.4 and T = 0.0025 (energies in units of
~ω U =∆ǫ, (20c)
0− ∆ǫ = ǫβ ǫα). As expected, the conductance exhibits
−
peaks located at gate voltages near ǫ , ǫ +U, ǫ and
where ∆ǫ = ǫ ǫ . Figure 2 shows a schematic repre- α α β
sentation of thβe−effαective levels active in each resonance. ǫβ+U;thesearethewell-knownCoulombblockadepeaks
that appear whenevereachlevel is in resonance with the
Notice that the resonance conditions do not explicitly
Fermi energy. Notice slight shifts from the anticipated
depend on the electron occupation number (unlike the
energy values, caused by the non-vanishing real part of
simpler approximation results in [18]). However, when
the electron self-energy discussed above.
the system is found in one of these resonances the split-
Figure 4 shows the DOS as a function of gate voltage
ting δ ReΣ(ph) for level α (and δ ReΣ(ph) for level
α ∝ α β ∝ β Vg and energy ω in the presence of electron-phonon in-
β), depends on the gate voltage through the occupation teraction. The different plots refer to different values of
numbers. thephononenergy~ω . Inallcases,theelectron-electron
0
Using the expressionsabove for the Green’s functions, interaction is set at U = 0.4, the electron-phonon cou-
we proceed to calculate the effect of electron-phonon in- pling strength λ = 0.2, and T = 0.0025. The values for
teractions on the DOS and the conductance of the sys- the phonon energies, ~ω = 0.6,1.0,1.4, have been cho-
0
tem. Transport properties are calculated in linear re- sen to fit the resonance conditions given by Eqs. (20a),
sponse in the fully interacting regime (U = 0). The (20b) and (20c), respectively. From the figure, it can
6
conductance has the following expressionin the Keldysh be observed that in the first resonance condition for the
formalism:22 smaller ~ω (= 0.6, top panel) the peaks correspond-
0
ing to the levels ǫ +U and ǫ are split, mixed by the
G=4π2t4ρL(ω)ρR(ω)|G¯11(ω)|2 ω=ǫF. (21) phononchannels. Tαhesplittingβforthepeakscorrespond-
(cid:12)
where (cid:12) ing to the level ǫα +U is ≈ 2δα (≈ 0 at a gate voltage
V 1.4), while the splitting for the level ǫ is 2δ
g β β
≈ ≈
G¯11(ω)=[g˜(ω)t′]2[Gαα+Gββ +Gαβ +Gβα] (22) (≈0.05 at Vg ≈2.0, for example). Notice that an analy-
sis of Eqs.(17)-(19), revealsthat when levelǫ coincides
α
is the Green’s function that describes the dynamics of with ǫ the splitting 2δ becomes small. This can be
F α
≈
an electron with energy ω from the site 1 to 1 (rep- understood in terms of the occupation number for the
−
resenting the leads, see Fig. 1) and g˜(ω) is the Green’s level β which in these conditions remains close to zero
function of the leads, calculated as disconnected from (this is seen by analyzing the value of the ReΣ , which
α
5
Fig. 5), indicating that the interference effects involve
two pairs of levels (as in Fig. 2b and 2c).
For the last resonance condition, ~ω =1.4, the levels
0
involved are ǫ and ǫ +U. These levels are quite far
α β
apart, separated by ∆ǫ+U, which results in the split-
tings in the DOS and the conductance to be barely ob-
servable, except for the sharp dip/splitting in the first
and last Coulomb blockade peaks (lower panel, Fig. 5).
We notice that the Rabi splitings are generally less im-
portantforthelasttworesonanceconditions. Thereason
for this behavior is that larger ~ω means that the orig-
0
inal levels participating in the phonon processes are far
apart, decreasing the spliting energies δ and δ . This
α β
is clearly reflected in the respective conductance curves,
where the peak splittings are very small.
We havealsoanalyzedthe dependence ofthese results
on temperature. The top panel in Fig. 5 shows the con-
ductance as a function of gate voltage for three different
temperatures, T = 0.0025, 0.025, and 0.1. A relatively
1
weak dependence on temperature is observed, consistent
0.9
with the fact that the resonance conditions are not ex-
h) 0.8 plicitly dependent of the electronoccupations. However,
2/ 0.7
2e theself-consistencyofthecalculationinducesanindirect
e ( 0.6 occupation dependence, resulting in splittings that be-
anc 0.5 come more pronounced for increasing temperature (i.e.,
ct 0.4
u larger phonon numbers).
nd 0.3
o Inordertofullyprobealltheresonanceconditions,we
C 0.2
fix ~ω andU andcontrolthe energyspacing ∆ǫ by gat-
0.1 0
ingonlyoneofthelocalizedlevels. Figure6showsacolor
0
0.5 1 1.5 2 2.5 3 map of the DOS as function of frequency and energy of
V /∆ε the variable levelǫβ. In this plot ǫβ is shifted downfrom
g
above to below ǫ as V increases; ǫ is kept fixed at a
F g α
point just below the Fermi energy, ǫ = 0.001, so that
α
−
∆ also changeswith V . Since the energyspacing is not
ǫ g
constant,wechooseinstead∆=D/8astheenergyunit,
FIG. 3: (Color online) Density of states map as function of
whereD isthebandwidthintheleads. Letusanalyzein
gatevoltageandenergy(toppanel)andconductanceasfunc-
detail the various peaks in this figure. Following a ver-
tion of gate voltage (bottom) for a molecule with electron-
tical line, such as ǫ = 2.25∆, one finds DOS peaks at
electron interaction, but no electron-phonon interaction. In β
ǫ , ǫ +~ω and ǫ +U ( 0, 1, and 1.4), respectively.
~uωni0ts=o0f.6∆,ǫt˜′==ǫ√β2−/5ǫ,αǫ,0αth=e1p,aarnadmTete=rs0a.0r0e2λ5.=No0t,icUee=ne0r.g4y, Tαherαe is one0 moreαpeak at≈ǫα+U +~ω0 (≈ 2.4), which
scale in top panel is shifted by ǫα. is out of range of the vertical axis in this panel. Note
that peaks at both ǫ 0 and ǫ +U appear since the
α α
≈
site α is in the mixed valence regime. Now following a
approacheszero). Asimilaranalysisshowsthatwhenthe horizontalline, at ω = ∆, we find peaks corresponding
level ǫβ crosses ǫF the splitting 2δβ is also small, since tothelevelsǫβ−~ω0,ǫβ−,ǫβ+U−~ω0andǫβ+U,respec-
ReΣ 0, a consequence of the levels ǫ and ǫ + U tively. One notices anticrossings of levels whenever one
β α α
≈
being far below ǫ (almost full levels). The splitting of ofthelevelsisaphononreplica,andcrossingswhenboth
F
the levels in the DOS is clearly, and most importantly, levels are“purely”electronic. The anticrossingsreflect a
manifested in the conductance as a function of the gate direct interaction between the electronic levels mediated
voltage V , as shown in Fig. 5. by phonons. The crossings appear because the levels in-
g
In the case of the second resonance condition, ~ω = teract only indirectly through the leads, an interaction
0
1.0 (middle panel in Fig. 4), there are two pairs of levels that is energy dependent.
involved: ǫ ,ǫ andǫ +U,ǫ +U. Theeffectoftheres- ThebottomofFig.6showstheconductanceofthesys-
α β α β
onanceconditionsontheconductanceresultstobemuch tem. Note the appearance of antiresonances in the con-
less important than in the previous case, as the splitting ductance whenever a crossing or anticrossing coincides
in the DOS at the Fermi level is much smaller here. No- with the Fermi level. The antiresonances localized near
tice,however,thatinthiscaseallfourconductancepeaks 1 and 0.4 are due to the anticrossingsin the density of
−
exhibit a certain degree of splitting (see middle panel in states discussed above. They result from the absence of
6
level ǫ , as it is always kept at the Fermi level. The full
α
uninhibited α-level conductance is evident for ǫ > 1.5,
β
where the β-level is empty. Notice yet, that the total
conductance exhibits dips and even complete vanishing
due to the complex pattern of interference arising from
competing channels of electrons traversing the molecule
through both arms of the structure whenever ǫ is near
β
theFermilevel. Inparticular,thevanishingconductance
at ǫ 0.5 appears from the strong level anticrossing
β
≃ −
lying rightatthe Fermi level(see upper panelin Fig.6).
In Fig. 7 we plot the conductance of the system as
function of ǫ , where the level α is shifted to a posi-
β
tion below the Fermi energy (ǫ = 0.1∆), in such a
α
−
way that it essentially does not contribute to the con-
ductance. As we apply a gate voltage to change ǫ , we
β
observe a conductance curve consisting of three peaks.
Two of them, located near ǫ = 0 and ǫ = 1.4∆ are
β β
−
the usual Coulombblockade peaks, which are essentially
not affected by the EPI. The effect of phonons, how-
ever, can be seen most clearly in the appearance of an
additionalpeaknear 0.5∆,duetophonon-assistedpro-
−
cesses. Thispeakarisesfromtheanticrossingofthelevels
ǫ and ǫ +U ~ω , opening a new channel for electron
α β 0
−
transport.
IV. CONCLUSION
In summary, we have studied in detail a diatomic
model when both EEI and EPI are taken into account.
We have presented an accurate description of the Rabi-
assistedtunnelingphenomenastudyingtheGreen’sfunc-
tionsofthesystemintheCoulombblockaderegimeabove
the Kondo temperature. We have obtained well defined
resonance conditions for the system. A detailed analy-
sis of the various resonance conditions and interacting
regimes shows that the most dramatic changes occur for
weakEEI(U,~ω <∆ǫ),sincetheEPIisenhancedwhen
0
the states involved in the phonon emission and absorp-
tionarecloseinenergy. Theseresultsemphasizetherich
physics involved in phonon-assisted phenomena, which
we believe should be acessible in experiments. The fact
FIG. 4: (Color online) Map of the DOS as function of gate that the phonon-induced resonance conditions are sub-
voltage and energy in the EEI and EPI competing regime. tly temperature dependent makes the effect strong and
From top (in units of ∆ǫ), with ~ω0 = 0.6, ~ω0 = 1 and sensitive to be probed in experiments.
~ω0=1.4. Inallpanels,U =0.4, t˜′=√2/5,λ=0.2,ǫ0α =1,
and T = 0.0025. The energy axes are shifted by ǫα, so that
level α appears always at zero.
Acknowledgments
statesattheFermilevel. Ontheotherhandthedipsnear WeacknowledgesupportfromFAPERJ,CNPq(CIAM
1.4and0areduetolevelcrossings. Noticethatthereis project) and CAPES in Brazil, as well as from the NSF-
−
a non-zerocontributionto the conductance coming from IMC grant 0336431in the US.
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8
1
T=0.1000
0.9
T=0.0250
0.8 T=0.0025
h) 0.7
22e/ 0.6 -hω=0.6
e ( 0
nc 0.5
a
uct 0.4
d
n
Co 0.3
0.2
0.1
0
000...555 000...777555 111 111...222555 111...555 111...777555 222 222...222555 222...555 222...777555 333
VVV///∆∆∆εεε
ggg
1
0.9
0.8
-hω=1.0
h) 0.7 0
22e/ 0.6
e (
nc 0.5
a
uct 0.4
d
n
Co 0.3
0.2
0.1
0
0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
V/∆ε
g
1
0.9
0.8
-hω=1.4
h) 0.7 0
22e/ 0.6
e (
nc 0.5
a
uct 0.4
d
n
Co 0.3
0.2
0.1
0
0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
V/∆ε
g
FIG. 5: (Color online) Conductance as function of gate volt-
age for system as in Fig. 4. From top to bottom, ~ω0 =0.6,
~ω0 = 1 and ~ω0 = 1.4; parameters as above. Top panel
also shows T dependence of conductance for ~ω0 = 0.6; no-
tice splittings in middle two peaks increase with increasing
temperature. Inthetoppanelthecurvesareoffsetforclarity.
9
1
T=0.0025
0.8
h)
2nce (2e/ 0.6
ucta 0.4
d
n
o
C
0.2
0
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
εβ/∆
FIG. 6: (Color online) Top: Map of the density of states as
function of energy (vertical axis) and level position ǫβ (hor-
izontal axis). Bottom: Conductance as function of energy
position ǫβ. Energies in units of ∆ = D/8. Other parame-
ters are ~ω0 = 1 ∆, and U = 1.4∆, t˜′ = √2/5, λ = 0.2,
·
and T = 0.0025∆. Level α is kept fixed at ǫα = 0.001∆,
−
essentially at the Fermi level and always contributing to the
conductance and interference. Notice ǫβ scale runs opposite
each other in both panels.
10
1
0.8
h)
2/
e
e (2 0.6
c
n
a
ct 0.4
u
d
n
o
C
0.2
0
-2 -1.5 -1 -0.5 0 0.5 1
εβ/∆
FIG.7: Conductanceasfunctionofpositionoflevelǫβ forthe
same case as in Fig. 6, but with the ǫα level well below the
Fermienergy,ǫα = 0.1/Delta,sothatitdoesnotcontribute
−
to theconductance.