# 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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**ABSTRACT:**In a recent paper [Physica E 39 (2007) 214, arXiv:0708.1842v1] Crisan, Grosu, and Tifrea revisited the problem of the conductance through a double-quantum-dot molecule connected to electrodes in a T-shape configuration. The authors obtained an expression for the conductance that disagrees with previous results in the literature. We point out an error in their derivation of the conductance formula and show that it gives unphysical results even for non-interacting quantum dots.Physica E Low-dimensional Systems and Nanostructures 06/2008; 40:2844-2845. · 1.86 Impact Factor

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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].

Page 5

5

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