Charge dynamics effects in conductance through a large semi-open quantum dot

0.97 MB·English

by  Piotr Stefanski

Charge dynamics effects in conductance through a large semi-open quantum dot
Piotr Stefan´ski,1,∗ Arturo Tagliacozzo,2 and Bogdan R. Bu lka1
1Institute of Molecular Physics, Polish Academy of Sciences,
ul. M. Smoluchowskiego 17, 60-179 Poznan´, Poland
2Coherentia-INFM, Unit´a di Napoli and Dipartimento di Scienze Fisiche,
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Universit´a di Napoli ”Federico II”, Monte S. Angelo-via Cintia, I-80126 Napoli, Italy
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(Dated: February 2, 2008)
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2 Fano lineshapes in resonant transmission in a quantum dot imply interference between localized
n andextendedstates. Theinfluenceof thechargeaccumulated at thelocalized levels, which screens
a the external gate voltage acting on the conduction channel is investigated. The modified Fano q
J parameter and the resonant conduction is derived starting from a microscopic Hamiltonian. The
7 latest experiments on ”charge sensing” and “Coulomb modified Fano sensing “ compare well with
1 theresults of thepresent model.

] PACSnumbers: 73.23.Hk,73.63.Kv
l
l
a
h I. INTRODUCTION tothecontacts,t (r=L,R),theCBstructureiseventu-
r
-
allywashedout. Weassumethatasingleedgechannelof
s
e Conductance peaks and Coulomb Blockade (CB) val- energy ǫ0 is centered at µ, providing a broad single par-
m leys mark the charge transport across a small quantum ticle resonance peak in the transmission. The electron
. dot(QD),weaklycoupledtothecontacts,whenthegate flow acrossthe dotis essentiallyuncorrelated. Neverthe-
t
a voltage V is varied. However, in a large QD the multi- less, the dynamics of the charge in the core levels does
g
m level structure cannot be ignored. The dot may display influence the resonant tunnelling, as we show in Section
- a “core” level structure including levels ǫ which are lo- II of the present work.
d γ
calized at different places of the dot area. Hence, they The ǫγ levels hybridize with the contacts only indi-
n
o may not hybridize among themselves, nor directly with rectly, by weakly coupling to the ǫ0 resonance, with a
c the contact leads, but they may be weakly coupled to hoppingmatrixelementtγ. Wealsoincludesomeon-site
[ one or more conduction channels. The latter could be CoulombrepulsionU at the core sites, which produces a

1 of “edge” type and extend from the left (L) to the right lower Hubbard level ǫγ (with single occupancy) and an
v (R)contact. Theexistenceofsuchanedgeandcorelevel upper Hubbard level (with double occupancy) ǫγ +U.
5 structure in large QDs has been proven experimentally Byincludingthechargepiledupintheǫγ stateswithin
8 by Single Electron Capacitance Spectroscopy2. In linear the Hartree-Fock (HF) approximation, we find that the
3 transport measurements, the coupling of localized and main features of the Fano landscape are not lost. There
1 delocalizedstates,called”bouncingstates”in3,hasbeen aretwomainalterations,however: a)thechargescreens
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invoked to justify the asymmetry of the CB peaks. the gate voltage Vg by acting on the ǫ0 level with a ca-
5
On the other hand, Fano resonances are ubiquitous in pacitive interaction and it shifts the resonances; b) the
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/ the conductance of nanodevices. They arise when local- presence of the charge modifies the Fano q factor and
at izedlevelsareembeddedinacontinuumofsingleparticle induces a marked asymmetry in the peaks.
m extended states which contribute to transport4. Linear InSectionIIIweelaborateonmodelingtheexperimen-
conductanceonasemi-openQDdisplaysmanytransmis- tal setup that has been investigatedrecently by Johnson
-
d sion peaks with the characteristic Fano shape, eachtime et al.7. The conductance measured through a quantum
n a core level crosses the chemical potential µ5. pointcontact(QPC)”senses”theadditionofchargetoa
o In a previous paper6 we have shown that a Fano-like QD, which is capacitively coupled, parallel to the QPC.
c
pattern also appears when the continuum of conduction Weshowthatourmodelapplieswithsmallmodifications.
:
v states at µ originates from electron correlations, as is Inthiscasetheedgechannelisnotinternaltothedotbut
Xi the case of a broad Kondo resonance1, due to a strongly itdescribes the QPCconductance,while the ǫγ levelslo-
hybridized level ǫ , lying deeply below µ. A bunch of calizedwithin the dotarefedwith chargebyanexternal
r 0
a localized core levels ǫ , weakly coupled to the ǫ level, reservoir. We neglect U which implies only minor mod-
γ 0
imprints the broad Kondo peak with Fano lineshapes. ification when the static screening approximation holds,
The Kondo conductance tends to freeze the charge sit- butweinclude asmallhybridizationtγ ofthe corestates
ting on the ǫ level, in competition with the Fano mech- to the current carrying QPC state. This allows us to
0
anism which causes particle number oscillations. Kondo monitor the full range from bare capacitive coupling or
correlationsstick the chargeoccupancy of the ǫ level to “charge sensing “ (CS), to Fano resonant tunneling.
0
the value one, and the Fano factor q close to zero. It Anincreasinggatevoltageappliedtothedotshiftsthe
follows that the transmission shows a dip in form of an discrete structure of energy levels across µ and the core
antiresonance. levels are gradually filled by electrons. The charge ac-
In opening the QD further, by increasing the coupling cumulated in the core levels capacitatively modifies the
2
gate voltage acting on the conducting channel. In the II. MODEL HAMILTONIAN OF THE
“chargesensing “ case, sudden jumps appear in the con- QUANTUM DOT AND CALCULATIONS
ductance, monitoring the discreteness of the charge ac-
cumulation process. Additionally, when t 6= 0, a set of
γ
Fano resonances emerges in the conductance across the
dot as a result of quantum interference. We show that
the shapeofFanoresonancesis markedlyaffectedbythe The Hamiltonian of the system under consideration is
charge trapping within the core levels as found in the composed of parts describing the edge state of the dot,
experiment7. H , the core QD levels, H and the electrodes H :
edge core el
H =H +H +H ,
edge core el
Hedge = ǫ0−Vg +(U′/2) nγσ′a†σaσ+ tα(Vg)[c†kασaσ+h.c],
σ γσ′ kσα
X X X
H = (ǫ −V )d† d +  Un n + t [d† a +h.c.],
core γ g γσ γσ γ↑ γ↓ γ γσ σ
γσ γ γσ
X X X
H = ǫ c† c . (1)
el kr krσ krσ
kσ,r=L,R
X
The edge QD state of energy ǫ is hybridized with elec- (n = d† d is the occupation number on the core
0 γσ γσ γσ
trodes. The matrix element t , (r =L, R) has a given state |γσi ).
r
dependence on the gate voltage V , very much like what
g
happens in a QPC, which opens up when the gate volt- The retarded Green function for the edge state is ob-
ageincreases8. The capacitivecoupling is describedby a tained by the Equation of Motion Method (EOM) with
cross-interactionU′ betweentheedgeandthecorestates anadditionalHFdecouplingofthecapacitiveinteraction:
−1
t2
G (ω,V ;hn i)= h0|[ω−H (V ;hn i)]−1|0i− γ . (2)
0,σ g γσ¯ edge g γσ¯
ω−ǫ +V
( γ,σ g)
γ
X
The first term on the r.h.s. is an average on the ground r.h.s. in Eq. 4. Charge accumulation in the core states
state of the system |0i (zero temperature is assumed). themselvesshifts thepositionoftheselevelswithrespect
It includes hybridization with the contact leads. The to the chemical potential even further, but we neglect
occupancyofthecorelevelsisselfconsistentlydetermined this capacitative modification of their energy location in
from hn i = (−1/π) ℑmG (ω)dω, where the Green the calculation, since it is inessential, as core levels do
σ γ,σ
function of a γ-levelwith spin σ is givenin the Hubbard not participate directly in the transport across the dot.
approximation9: R The linear conductance (i.e. inzero limit ofthe drain-
source voltage) in units of the quantum conductance
(ω−ǫ˜γ,σ)(ω−˜ǫγ,σ−U) −1 2e2/~, is10:
G (ω)= +iΓ . (3)
γ,σ γ,σ
ω−ǫ˜ −U(1−hn i)
(cid:20) γ,σ γσ¯ (cid:21) ∞ ∂f(ǫ)
G = Γ (ǫ) − ρ (ω,V ;hn i)dǫ, (5)
σ 0,σ g γσ¯
Thus, G has finite width ∂ǫ
γ,σ σ Z−∞ (cid:18) (cid:19)
X
Γγ,σ =−ℑm tγ h0|[ω−Hedge(Vg;hnγσ¯i)]−1|0itγ (4) where f(ǫ) is the Fermi distribution function, Γσ(ǫ) =
Γ (ǫ)Γ (ǫ)/[Γ (ǫ) + Γ (ǫ)] is the hybridization
γ Lσ Rσ Lσ Rσ
X of the edge state with the contact leads (Γ = Γ
L R
duetoindirectinteractionwithelectrodes. Thebareǫ − is further assumed) and the spectral density ρ =
γ 0,σ
V isalsoadditionallyshifted→ǫ˜ bytherealpartofthe −(1/π)ℑmG .
g γ 0,σ
3
Itis instructiveto introduceFanoparameterq defined
0,15
as11,12: 2,0
ℜeG (ω =0,V ;hn i) 0,10 1,5
0,σ g γσ¯
qσ(Vg)=− , (6) 1,0
ℑmG (ω =0,V ;hn i) n
0,σ g γσ¯ 0,05 0,5
with gate voltage screened by the trapped charge: eV] 0,0
VgSC =Vg −(U′/2) hnγσ(Vg)i, (7) SC [Vg 0,00 -V0g,1 [e0V,0] 0,1 0,2 0,3 0,4
-0,05
Xγσ t=0.15 U
Letusconsideronesinglecorelevelforthetimebeing. -0,10 t=0.3 U
TheV ,VSC arethebareandscreenedvoltageactingon t=0.6 U
g g -0,15
the edge level, but they move the core level as well. In -0,15 -0,10 -0,05 0,00 0,05 0,10 0,15
crossingµ,thecorelevelisgraduallyfilledwithelectrons. Vg [eV]
The occupancy of the γ level is shown in the inset of
FIG. 1: Gate voltage screened by the filling of the γ level
Fig.(1) for various t ’s. In turn, while the occupancy
γ
when the latter is moved below µ = 0 ( α = 1). The inset
of the core level changes, a step appears in the screened
voltage, resulting from the capacitive coupling between showstheselfconsistentlycalculatedtotaloccupancynγ. The
straight line is the bare Vg and is drawn for comparison.
the core and the edge state.
VSC is plotted in Fig. (1) vs the bare V . Two such
g g
kinks appear,correspondingto the single anddouble oc-
2,0
cupancy of the γ level. Their energy separation is of the
order of the Coulomb repulsion U (taken as U = 0.1eV
in the calculation). The dependence of the hybridiza-
1,5
tion parameter on the voltage is implemented by mul-
tiplying the matrix element tmr ax = 2U by the function h]
θth(Vegs)h=arp{nexesps[(osf−thVeg)k/iδn]k+,1a}n−d1,sw(=her-0e.δ2(i=n c0a.5lc)uclaotnitornosl)s 2 G [e/1,0 1,5
1,0
ihtysbproidsiitziaotnio.nTthγebkeintwkeiesnsmǫ0oaonthdedǫγwbitehcaiunsceretahseinegleocftrtohne 0,5 -Fano q000,,,055 ==01.8
hasanincreasingpreferencetotunnelresonantlythrough -1,0 =0.4
ǫ0 insteadofdwellingontheǫγ level6 whenǫ0 crossesthe -1,5-0,1 0,0Vg [eV]0,1 0,2 =0
chemical potential. Here α = U′/U is the charge sensi- 0,0
-0,1 0,0 0,1 0,2 0,3
tivity parameter. It can increase to unity when the level Vg [eV]
broadening in QD becomes larger than the level spacing
andthecross-interactionU′ approacheson-siteCoulomb FIG.2: ConductancevsVg for tγ =0.15U at variouscharge
repulsion U. sensitivities α of the filling of the γ-level. In the inset the
InFig. (2)weplotthecorrespondingconductance,for Fano q,Eq. (6), is shown.
various α. The filling of the ǫ level has a marked effect
γ
on the shape of the Fano resonances because it modifies
the Fano q factor (see inset). The non zero value of q H ≡H + ǫBb† b + t [b† d +h.c]
dot core k,σ k kσ kσ kγσ B kσ γσ
and its sign are responsible for the asymmetry of the and H ≡H +H . We have added an additional
QPC edge el
Fano dips within the broad conductance peak centered reservoir B, in conPtact with the doPt, with levels ǫB cor-
k
at ǫ ∼0. respondingtosingleparticleoperatorsb . Afeatureless
0 kσ
and broad density of states for the reservoir is assumed.
ThedotissuppliedwithelectronsfromthebathBdueto
III. CHARGE AND FANO “SENSING” IN THE the hoppingparametert . The levelǫ correspondsnow
B 0
QPC CONDUCTANCE to the QPC, having a gate voltage dependent hybridiza-
tion with the leads. In Figs. (3, 4) we report the results
Our model can also be applied straightforwardly to for a multilevel dot with γ= 13 discrete levels equally
describe recent transport measurements through a QPC spaced by ∆ = 0.2 eV and symmetrically located with
“sensing” the charge piled up in a QD placed at its side respect to µ = 0 for V = 0. In this hypothesis we can
g
andcapacitivelycoupledtoit7. Weshowthatitispossi- assume that ∆ is the addition energy of the dot, inde-
ble to continuously move from a “capacitive sensing” to pendent of the electron number N. Thus, we drop the
a “Fano sensing” by increasing the hybridization of the interaction term ∝ U in H and the relevant energy
core
single particle levels ǫ of the dot with the conduction scale for charge sensing is now ∆ itself. A static screen-
γ
channel ǫ of the QPC. ing in the multi-electron system is expected to be quite
0
To match with the experimental setup we use the effective, if the dot is large. The screened gate voltage
Hamiltonian of Eq. (1) by reinterpreting it as follows: acting on the QPC, sensing the charge on the dot, takes
4
broadresonancecenteredatǫ = V = 0. Its widthand
0 g
2,0
characteristicasymmetryisgovernedbythe gatevoltage
dependent coupling to the L,R contacts. By increasing
α,themaximumoftheresonanceismovedtohigherV ’s,
1,5 =0 g
=0.5 because of the voltage screening. The conductance peak
=1 acquires a sawtooth pattern, which grows sharper when
h]
2 G [e/1,0 αtiminecraeanseews. eTlehcetrpoantteenrnterfoslltohwesdtohte.juTmhpessohfaVpgesco,feathche
overall conductance curve is similar to the one seen in
0,5 the experiment (see Fig. (2a) of7).
In Fig.(4), the conductance is reported for finite hop-
ping between QD andQPC (t =0.15 ∆), atα=0.0,0.5,
0,0 γ
-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 1.0. Thecaseα=0isthepureFanocase. Themaximum
Vg [eV] of the conductance is still centered at Vg ≈0 because tγ
has been chosen to be rather small. The zero of the real
FIG. 3: Conductance through the QPC with ”charge sens-
part of the QPC Green’s function G defines the loca-
ing” only (tγ =0) for various α parameter and tB/∆=0.15. tion of the maximum of the resonan0c,σe, and the zero of
The curve for α=0 without charge sensing is also shown for
the Fano q as well, according to Eq.(6). The symmetric
comparison.
Fanodipcorrespondingtoq =0iscenteredontopofthe
conductance maximum, because a γ level sits exactly at
2,0 Vg = 0. When α increases, the condition ǫ0−VgSC ∼ 0
2
0 nowmarksthelocationoftheconductancemaximum,as
1,5 Fano q---642 ==00 .5 =1 wTehlulsa,stthheecpolnadcuecwtahnecree qen∼vel0op(seeemtohveesinasgeatinoftFoigla.(r4g)e)r.
-8 =1
-1-01,5-1,0-0,50,0 0,5 1,0 1,5 voltages,together with the Fano-like structures, and the
Vg [eV]
h] conductance peak broadens as in Fig. (3). Appearance
2 G [e/1,0 of sharp peaks for large negative gate voltages is caused
by rapid decrease of q (see inset) due to the closing of
=0.5
QPC, Eq. (6). Our case with α = 1.0 compares well
0,5
withFig.(2b)of7,displayingCoulombmodifiedFanores-
=0 onances. Both ref.7 and in ref.14 propose a model con-
0,0 ductance based on the grand canonical average of dot
-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5
configurations, which can be fitted to the experimental
Vg [eV] data. Ourcalculationshowsthatthedynamicsoftheres-
onant tunnelling together with some minor assumptions
FIG.4: ConductancethroughtheQPCwith”chargesensing”
andfinitehybridization with thedot(tγ=0.15∆) forvarious providesimplerresults,inqualitativeagreementwiththe
α parameter. Inset shows corresponding Fano q, the straight experimental data.
lineisforα=0andconstantcouplingtoelectrodestr =tmr ax. To conclude, we have demonstrated that a Coulomb
modified Fano resonant tunneling arises from charge
sensing in structures like a QPC with a dot aside, or in-
now the same form as Eq. (7), but with U′ replaced side a large dot where core levels coexist with extended
by the charging energy of the dot ∆. The α parameter edge states and both can be described within the same
now describes the sensitivity of the capacitive coupling quantum mechanical model with minor modifications.
between the QD and QPC which can be controlled by
appropriate gates7. Again, the particle number on the
dot n (V ) is determined by means of the Friedel
γ,σ γ,σ g
sum rule13. It shows the usual steps as in the inset of Acknowledgments
P
Fig. (1) when the gate voltage increases and the dot is
filledbyelectrons. Forα=1theCSisequaltothecharg- This work was supported by the Polish State Com-
ingenergy,eachtime anelectronhopsontothe dot. Fig. mittee for Scientific Research (KBN) under Grant no.
(3) shows the conductance through the QPC when the PBZ/KBN/044/P02/2001 and the Centre of Excellence
dot is only capacitively coupled to it (t =0), for various for Magnetic and Molecular Materials for Future Elec-
γ
α’s. For α=0 the dot has no effect on the QPC con- tronics within the European Commission Contract No.
ductance. The transmission across the QPC displays a G5MA-CT-2002-04049.
∗ Electronic address: [email protected] 1 D. Goldhaber-Gordon et al.,Nature, 391, 156 (1998).
5
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5 C.Fu¨hner et al.,cond-mat/0307590(2003); Phys.Rev.B 12 P. Stefan´ski, Solid St. Commun. 128, 29 (2003).
66, 161305 (2002). 13 A.C. Hewson, The Kondo Problem to Heavy Fermions,
6 P. Stefan´ski et al.,Phys.Rev. Lett.93, 186805 (2004). (CambridgeUniv.Press,Cambridge1993),D.C.Langreth,
7 A.C. Johnson et al., Phys.Rev.Lett. 93, 106803 (2004). Phys. Rev. 150, 516 (1966).
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