Zero temperature conductance of parallel T-shape double quantum dots

Abstract

We analyze the zero temperature conductance of a parallel T-shaped double quantum dot system. We present an analytical expression for the conductance of the system in terms of the total number of electrons in both quantum dots. Our results confirm that the system's conductance is strongly influenced by the dot which is not directly connected to the leads. We discuss our results in connection with similar results reported in the literature. Comment: 5 pages, revtex

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arXiv:0708.1842v1 [cond-mat.mes-hall] 14 Aug 2007
Zero temperature conductance of parallel T-shape double quantum dots
M. Crisan and I. Grosu
Department of Physics, “Babe¸ s- Bolyai” University, 40084 Cluj-Napoca, Romania
I. T ¸ifrea
Department of Physics, California State University, Fullerton, CA 92834, USA
We analyze the zero temperature conductance of a parallel T-shaped double quantum dot system.
We present an analytical expression for the conductance of the system in terms of the total number
of electrons in both quantum dots. Our results confirm that the system’s conductance is strongly
influenced by the dot which is not directly connected to the leads.
connection with similar results reported in the literature.
We discuss our results in
PACS numbers: 73.63.Kv; 73.23.-b; 72.15.Qm
I. INTRODUCTION
Recent advances in the fabrication and precise con-
trol of nanoscale electronic systems lead to an increased
interest in the study of many body effects in quantum
dot structures. The Anderson single impurity model [1]
was extensively explored to successfully understand elec-
tronic correlations in small single, or double quantum dot
structures. In general, single or double quantum dot con-
figurations provide the ideal systems to study many body
effects. For example, single dot configurations allow the
realization of the Kondo regime of the Anderson impu-
rity [2]. On the other hand, double quantum dot (DQD)
configurations provide the ideal candidate for the study
of the many body effects associated to both Kondo effect
and RKKY interaction [3]. The connection of several
quantum dots (QD) gives rise to remarkable phenomena
due to the interplay of electron correlations and interfer-
ence effects which depend on how the dots are arranged.
One possible configuration is the double quantum–dot
(DQD) system, where the dots are connected to the same
leads and between them. Recently, Dias da Silva et al. [4]
studied a DQD with one dot in the Kondo regime and the
other close to the resonance with the connecting leads.
One of the most interesting results reported in Ref. [4]
is the finite temperature analysis of the parallel T-shape
double dot configuration with one dot disconnected from
the leads (Fig. 1) using the idea of interference between
resonances. In such a configuration, the active dot A is
directly connected to the left and right leads and to a side
dot S. They showed that when the side dot S is coupled
to the leads only through the active dot A, the Kondo
resonance from the side dot S develops a sizable splitting
even if there is no magnetic field in the system. This
band filtering produced by the connected dot preserves
the Kondo singlet and at finite temperature the magnetic
moment is completely screened.
The calculation of system’s conductance is of major
interest both for single and double quantum dots con-
figurations. For double quantum dot configurations the
problem was considered extensively using different meth-
ods, however, the results presented by different authors
A
S
LR
VL
VR
t
FIG. 1: Schematic representation of the parallel T–shaped
double quantum dot system. The active dot A is connected
to the left and right leads and to the side dot S. The presence
of the side dot S, which is only connected to the active dot
A, influences the general conductance of the system.
are in agreement only partially [5, 6, 7, 8]. The system’s
electronic conductance is realized through the active dot
A, however, the presence of the side dot S will influence
the total conductance of the parallel T–shaped quantum
dot system [5].The suppression of the system’s con-
ductance at low temperatures can be understood if two
possible conduction paths are considered, a direct path
through the active dot A (L → A → R) and an indirect
path through the side dot S (L → A → S → A → R).
New features have been pointed out by Takazawa et al.
[6] in connection with the inter–dot coupling strength t
and the values of the energies of the active dot EAand
the energy of the side dot ES . For the case EA= ES
the occurrence of a non trivial suppression of the con-
ductance was associated with a Fano–like effect between
two distinct channels, i.e., the direct Kondo resonance of
the active dot A and the indirect resonance via both the
active and side dots. An interesting feature of the system
was discussed by Cornaglia and Grempel [7] and by Zitko
and Bonca [8] using the new idea introduced in Ref. [9]
known as the two-stage Kondo effect, a behavior obtained
for a dot in a strong magnetic field. The T–shape DQD
close to half–filling has a similar behavior for small inter–
dot coupling t, the possibility of a two-stage Kondo effect
leading to a nonmonotonic behavior of the conductance
as function of the gate voltage and magnetic field. At
large inter–dot coupling t the magnetic moments of the

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two quantum dots form a “local” molecular spin–singlet
and the conductance varies monotonically at low temper-
ature. One of the main results from [7] is the calculation
of the conductance G(T) in terms of the spectral den-
sity of the active dot A interacting with the side dot S.
The zero temperature conductivity depends on the total
number of electrons in the two dots, i.e., the active dot
A and the side dot S.
Here, we present a T = 0 K calculation for the dc
conductance of the T-shaped quantum dot system. Our
analysis will start from a general Hamiltonian describing
the possible interactions in the double quantum dot sys-
tem, i.e., interactions inside each component dot, inter–
dot interactions, and interactions with the reservoirs.
Previous results obtained by Cornaglia and Gempel [7]
give the system’s conductance in terms of the total elec-
tron density in the system. However, as we will prove
later in the paper, there are additional contributions re-
lated to the inter–dot electron–electron interaction which
were not included in Ref. [7]. We will also consider the
system’s conductance in the presence of a magnetic field
whose role is to remove the spin degeneracy for the pos-
sible bound states in the active and side dots. All our
calculations are performed in the T = 0 K limit, so finite
temperature effects will be neglected. The relevance of
temperature effects due to the different Kondo regimes
can be evaluated by calculating the self energies of the
electrons using the equation of motion method with an
appropriate decoupling, however, the finite temperature
conductance of the system will be the subject of another
investigation [12].
II. THE MODEL
The general hamiltonian of the T–shape double quan-
tum dot configuration is
H = HD+ HE+ HDE. (1)
Here, HDdescribes both the active A and side S dots
HD =
?
i=A,S
[ǫi(ni↑+ ni↓)+ Uini↑ni↓]
+t
?
σ
(d†
AσdSσ+ d†
SσdAσ) , (2)
where
Coulomb interaction and t describes the coupling be-
tween the active and side dots. The operators d†
diσ(i = A,S) are the standard electron creation and an-
nihilation operators. The electrons in the left (L) and
right (R) electrodes are described by
Ui
representsthe on–siteelectron–electron
iσand
HE=
?
k,σ,j
Ejc†
kσjckσj, (3)
where the index j = L,R; c†
lates) an electron with momentum k and spin σ in the j
kσj(ckσj) creates (annihi-
electrode of the configuration. The coupling between the
T–shape DQD and the leads is described by the Hamil-
tonian HD−Ewhich has the form:
HD−E=
?
k,σ,j
Vkj(d†
Aσckσj+ c†
kσjdAσ) . (4)
All the properties of T-shape DQD configuration can be
obtained from the Green function of the d–electrons. The
d–electron’s Green function can be obtained by differ-
ent methods including the equation of motion method
(EOM) or the perturbation theory. In the following we
will explore the EOM to extract the electronic Green
function and thereafter the configuration’s total conduc-
tance.
The properties of the T-shape DQD can be expressed
in terms of a 2 × 2 Green-function matrix according to
the Dyson equation
G−1
σ(ω) = G−1
0(ω) − Σσ(ω) ,(5)
where G0is the noninteracting Green function
G−1
0(ω) =
?ω − EA+ i∆t
tω − ES
?
(6)
with ∆ = 2πN(0) < |Vkj|2> and Σσ(ω) is the self–
energy matrix as it results from the Coulomb electron–
electron interactions, U. In the most general form the
self–energy matrix can be written as
Σσ(ω) =
?Σσ
Σσ
AA(ω) Σσ
SA(ω) Σσ
AS(ω)
SS(ω)
?
, (7)
a form which accounts both for electron–electron inter-
actions in each of the two dots and for electron–electron
interactions between the two dots of the configuration.
The exact Green’s function and the self–energy of the
system satisfy the Luttinger theorem:
?0
−∞
dω Tr
?∂ Σσ(ω)
∂ω
Gσ(ω)
?
= 0 , (8)
where TrA represents the trace of the matrix. The knowl-
edge of the electronic Green’s function permits the cal-
culation of the total electron density in the system as:
ndσ= Im
?0
−∞
dω
π
TrGσ(ω) .(9)
The above expression can be simplified to
ndσ=1
πcot−1Re?detG−1
Im?detG−1
σ(0)?
σ (0)? .(10)
III. CONDUCTANCE
Confinement of electronic systems in small quantum
dot configurations may result in very interesting trans-
port properties. Here, we calculate the T-shape DQD

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system’s transport properties following the general for-
malism introduced by Meir and Wingreen [10]. Accord-
ing to Ref. [10] the current through a quantum dot sys-
tem in the presence of an external bias voltage is given
by
I =
e
h
?
σ
?
dω
?
f(ω) − f
?
ω +eV
h
??
× Im[Tr(ΓGσ(ω))] , (11)
where f(ω) represents the Fermi–Dirac distribution func-
tion, and
Γ =
?−∆ it
it0
?
. (12)
In the zero temperature limit, T = 0 K, we can evaluate
the conductance of the T-shape DQD configuration (G =
∂I(V )/∂V ) as
G = G0Im[Tr(ΓGσ(ω = 0))] , (13)
where G0 = 2πe2/h2. The calculation of the system’s
conductance as function of the total number of electrons
(n) is relatively simple, and a general formula can be
given as
g(n) =
G
G0
=∆2E2
[E2
A(0) − 2t2E2
A(0)E2
A(0)E2
S(0) + 2t4
S(0)
S(0) − t2]2+ ∆2E2
, (14)
where EA(0) = EA− ReΣA(0) and ES(0) = ES −
ReΣS(0) are the renormalized energies of the bound
states in the active, respectively side, quantum dots of
the configuration. Eq.(14) is an exact result which
shows that the dc conductance of the system depends
on two coupling parameters, t - the coupling between the
active and side dots and ∆ - the coupling between the
active dot and the leads, and the value of the system’s
self–energy. Accordingly, the behavior of the system’s
conductance depends on the selection of the constituent
dots and on the external applied bias. For example, in
Figure 2 we plotted the value of the relative conductance,
g(n) = G/G0, as function of the relative inter dot cou-
pling t/∆ for various values of the relative energy level of
the side dot and a fixed value of the relative energy level
in the active dot. Such graphic representations of the
T-shape DQD conductance as function of various inter-
action energies in the system allow the optimal selection
of the active and side dots. Our plotting assumes fixed
values for the energy levels inside the active and side
dot. This assumption may be questionable as for many-
body effects in the system the initial energy of the bound
states in the two component dots will be changed; how-
ever, in most of the real situations the corrections due to
the self-energy on the value of the two bound states, EA
and ES, are small, and in a first approximation they can
be neglected. Tsvelik and Wiegmann calculated the self-
energy due to the Coulomb interaction U [13] and proved
that the real part of the self-energy depends linearly on
frequency ω. At T = 0, in the Fermi liquid approxima-
tion, the main contribution to the self-energy comes from
the term ω = 0 and accordingly it can be neglected.
On the other hand Eq. (10) gives a direct relation be-
tween the system’s self energy and the total number of
electrons in the constituent dots. As a result, the con-
ductance of the T-shape DQD system can be expressed
also using the total number of electrons in the active and
side dots. A straightforward calculation, which implies
Eqs. (10) and (14), gives
Im[Tr(ΓGσ(0))] = sin2(πndσ) +
t2
∆ES(0)sin(2πndσ) ,
(15)
and in terms of the total number of electrons the system’s
conductivity becomes
g(n) =1
2
?
σ
sin2(πndσ) +
t2
2∆ES(0)
?
σ
sin(2πndσ) .
(16)
Note that ES(0) depends also on the system’s self-energy,
and implicitly on the total number of electrons in the
system. However, for the side dot, which is not directly
connected to the leads, the changed of the bound state
energy is much smaller that for the active dot, and such
a dependence can be neglected. Eq. (16) is an exact
result which describe the dependence of the system’s dc
conductance on the total number of conduction electrons.
From the experimental point of view the total number of
electrons in the system is controlled using the applied
external bias.
In the absence of an external magnetic field, when the
two considered dots are unpolarized (ndσ=nd−σ) the nor-
?
??
?
?????
?
?????
?
?????
FIG. 2: The conductance dependence on the relative coupling
between the active and side quantum dots, t, for various val-
ues of the side dot energy level (full line - ∆/ES(0)=−0.1,
dashed line - ∆/ES(0)=−0.2, dotted line - ∆/ES(0)=−0.3)
and fixed energy value for the active dot (∆/EA(0) = −0.1).

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malized conductance g(n) = G/G0can be written as
g(n) = sin2?πn
2
?
+
t2
∆ES(0)sin(πn) ,(17)
where n = 2ndσis the total number of electrons in the ac-
tive and side dots. It is easy to see that the zero temper-
ature conductance of the T-shape DQD system vanishes
when the total number of available electrons in the active
and side dots is even. This result was already predicted
in Ref. [7]. Our calculation for the system’s conduc-
tance also accounts for the intra and inter dots electron–
electron interactions. For a finite occupancy of the T-
shape DQD system, i.e., integer number of electrons oc-
cupy the two possible energy levels, the correction term
will cancel (sin(πn) = 0 for n-integer). However, this
fact is not an indication that the electron-electron in-
teractions in the system can be neglected, the available
number of electrons on the active and side dots being
strongly dependent on these interactions and on the ap-
plied, external bias.
A different situation occurs when the degeneracy of the
active and side quantum dots is removed. For example,
when a small magnetic field is applied, the T-shape DQD
becomes polarized and the possible energy levels become
spin dependent. In this case, the system’s dc conductance
becomes
g(n,m) =
1
2[1 − cos(πn)cos(πm)]
t2
∆ES(0)sin(πn)cos(πm) ,
+
(18)
where m = nd↑− nd↓ represents the system’s magneti-
zation. For any even number of electrons in the system
(n = 0, n = 2, and n = 4), the magnetization value re-
duces to the one presented in Ref. [7] for the n = 2 case,
i.e., g(n,m) = sin2(πm/2). On the other hand, when
the number of electrons in the system is odd, a different
behavior of the magnetization should be expected, i.e.,
g(n,m) = 1 − sin2(πm/2). Once again, the interaction
term in the conductance will vanish for an integer number
of electrons in the system.
IV. DISCUSSIONS
In this work we analyzed the conductance of a T-shape
DQD. Our main result is a generalization of the unitary
rule of the single-level Anderson impurity problem for the
case of a T-shape DQD . The system’s conductance de-
pends on various interactions inside the component dots,
and on the value of the energy levels in the active and
side dots. Tuning these interactions allow a direct control
of the system conductance. We also presented a calcula-
tion of the system conductance in terms of the occupancy
of the two possible energy levels in the active and side
dots. A similar calculation was also presented in Ref. [7],
however, our result includes contributions related to the
inter-dot interaction, t. In the t = 0 limit we recover the
result presented in Ref. [7]. Additionally, we proved that
such a term can be disregarded when the energy levels
in the T-shape DQD system are occupied by an integer
number of electrons. On the other hand, even at T = 0
K or at small, but finite temperatures, when fluctuations
are important, the occupancy of the energy levels can be
non-integer, and in such situations the interaction cor-
rection to the system’s conductance becomes important.
The differences between our calculation and the one pre-
sented in Ref. [7] are a consequence of how the general
current passing through the system was calculated. The
result presented by Cornaglia and Grempel [7] only ac-
counts for the electron density in the active dot, and ac-
cordingly the conductance is calculated considering the
inter-dot interaction small, the system behaving like a
single dot with an electron occupancy nd = nA+ nS.
Such a result will be valid only in the limit t2≪ ∆ES(0).
Sweeping the gate voltage into and trough an odd val-
ley in this model, corresponds to sweeping the energy of
the active dot from above ∆ to below −(U + ∆) in the
course of which the total number of electrons ndchanges
smoothly from 0 to 2. We can obtain three different
regimes for the system as function of various interaction
parameters: (i) the “empty-orbital” when nd = 0 and
G(nd) = 0; (ii) the “mixed-valence” in which the number
of electrons ndbegin to increase due to strong charge fluc-
tuations; and (iii) the “local moment” regime, in which
the number of electrons nd approaches 1 and the local
levels acts like a local spin in one of the dots, or in both.
The latter regime can give rise to Kondo correlations,
and the system has to be studied at finite temperature.
Our calculation is done in the T = 0 K limit, however
it may be extended to finite temperatures, T → 0 K. In
this limit, the imaginary part of the self-energy
ImΣ(ω) = [(ω2) + (πT)2]/TK−→ 0 , (19)
when ω=0 and the real part of the self-energy is constant.
The temperature scale, TK, is set by the Kondo effect and
for the active dot can be calculated following the method
proposed in Ref. [11].
The T-shape DQD system is a very promising quantum
dot configuration, both from the fundamental physics
and possible applications point of view. Systems involv-
ing single or multiple quantum dots may provide many
opportunities for strong interaction effects studies. Also
they may provide the optimal environment for the study
of the Kondo effect, or of the Ruderman-Kittel-Kasuya-
Yoshida interaction among local spins. Different other
finite temperatures regimes can be considered by calcu-
lating the electronic self-energy using the equation of mo-
tion along with an appropriate decoupling [12].

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[1] P.W. Anderson, Phys. Rev. 124 (1961) 41.
[2] S.M. Cronenwett, T.H. Oosterkamp , and L.P. Kouwen-
hoven, Science 281 (1998) 540.
[3] N.J. Craig, J.M. Taylor, E.A. Lester, C.M. Marcus, M.P.
Hanson, and A.C. Gossard, Science 304 (2004) 565.
[4] L.G.G.V. Dias da Silva, N.P. Sandler, K. Ingersent, and
S.E. Uolla, Phys. Rev. Lett. 97 (2006) 096603.
[5] T.–S. Kim and S. Hershfield, Phys. Rev. 63 (2001)
245326.
[6] K. Takazawa, Y. Imai, and N. Kawkami, J. Phys. Soc.
Japan. 71 (2002) 2234.
[7] P.S. Cornaglia and D.R. Grempel, Phys. Rev. B 71
(2005) 075305.
[8] R. Zitko and J. Bonca, Phys. Rev. B 74 (2006) 045312.
[9] W.G. van der Wiel, S. De Franceschi, J.M. Elzerman, S.
Tarucha, L.P. Kouwenhowen, J. Motohisa, F. Nakajima,
and T. Fukumi, Phys. Rev. Lett. 88 (2002) 126803.
[10] Y. Meir and N.S. Wingreen, Phys. Rev. Lett. 68 (1992)
2512.
[11] V. Kaschcheyevs, A. Aharony, and O. Entin–Wolman ,
Phys. Rev. 73 (2006) 125338.
[12] M. Crisan, I. Grosu, and I. T ¸ifrea (in preparation).
[13] A. M. Tsvelik and P. G. Wiegmann, Adv. Phys 32, 453
(1983).