quantum size eﬀect of two couple quantum dots

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EJTP 5, No. 19 (2008) 3342 Electronic Journal of Theoretical Physics

Quantum Size Effect of Two Couple Quantum Dots

Gihan H. Zaki(1), Adel H. Phillips(2)and Ayman S. Atallah(3)

(1)Faculty of Science, Cairo University, Giza, Egypt(2)Faculty of Engineering, Ain-Shams University, Cairo, Egypt(3)Faculty of Science, Beni- Suef University, Beni-Suef, Egypt

Received 18 February 2008, Accepted 16 August 2008, Published 10 October 2008

Abstract: The quantum transport characteristics are studied for double quantum dots

encountered by quantum point contacts. An expression for the conductance is derived using

Landauer - Buttiker formula. A numerical calculation shows the following features: (i) Two

resonance peaks appear for the dependence of normalized conductance, G, on the bias voltage,

V0, for a certain value of the inter barrier thickness between the dots. As this barrier thickness

increases the separation between the peaks decreases. (ii) For the dependence of, G, on, Vo,

the peak heights decrease as the outer barrier thickness increases. (iii) The conductance, G,

decreases as the temperature increases and the calculated activation energy of the electronincreases as the dimension, b, increases. Our results were found concordant with those in the

literature.c Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Quantum Dots; Landauer - Buttiker Formula

PACS (2008): 73.21.La; 68.65.Hb; 61.46.Df

1. Introduction

Interest in low dimensional quantum confined structures has been fueled by the richness

of fundamental phenomena therein and the potential device applications [1-4]. In partic-

ular, ideal quantum dots can provide three-dimensional carrier confinement and resulting

discrete states for electrons and holes [5]. Interesting electronic properties related to the

transport of carriers through the bound states and the trapping of quasi-particles can

be used to realize a new class of devices such as the single electron transistor, multilevel

logic element, memory element, etc. [6-8]. Because of its high switching speed, low power

consumption, and reduced complexity to implement a given function, resonant tunneling

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Electronic Journal of Theoretical Physics 5, No. 19 (2008) 3342 35

j(x) =Ajexp(ikj.x) + Bjexp(ikj.x) (4)

Where:

kj =[2m(E Vj e2N2/4C)]1/2

(5)

is Plancks constant divided by 2. The coefficients Aj and Bj are determined by

matching the wave functions j and their first derivatives at the subsequent interface.

Now, using the transfer matrix, we get the coefficients Aj and Bj as:

AlBl

=

j=1

Rj

ArBr

(6)

Where the notations (l, r) denote left and, right regions. In Eq. (6), the coefficientsRj in the j

th region is given by:

Rj = 1

2kj

(kj+ kj+1)exp[i(kj+1 kj)xj]

(kj kj+1)exp[i(kj+1+ kj)xj]

(kj kj+1)exp[i(kj+1+ kj)xj ]

(kj+ kj+1) exp[i(kj+1 kj)xj ]

(7)

According to the present model of the device, the corresponding wave vectors in the

regions where the barriers exist are given by:

=

2m

Vo+ Vb+ e

2

N

2

4C1/2

(8)

Where, Vb is the barrier height. And also, the wave vectors in the quantum dots are

expressed as:

= [2mE]1/2

(9)

Now, the tunneling probability, (E), is given by solving Eq. (6) [22] and we get:

(E) = 1

1 + A2

B2

(10)

Where:

A=

Vo+ Vb+

e2N2

4C

sinh (b)

EVo+ Vb+

e2N2

4C

E

1/2 (11)and

B=D1D2sinh( (2b c))

sinh (b) (12)

in which the expression for D1 and D2 are:

D1 = 2 cosh (b) . cos(ka)

2E Vo Vb e2N2

4C

sinh(b)sin(ka)EVo+ Vb+

e2N2

4C E

1/2 (13)


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36 Electronic Journal of Theoretical Physics 5, No. 19 (2008) 3342

and

D2= 2 cosh (c)cos(ka)

2E Vo Vb

e2N2

4C

sinh (c)sin(ka)

EVo+ Vb+

e2N2

4C E1/2

(14)

Now, by substituting Eq. (10) for the tunneling probability, (E), into Eq. (1),

taking into consideration eqs. (11-14), and performing the integration numerically( using

Mathemtica-4), one can calculate the conductance.

3. Numerical Calculation and Discussion

In order to show that the present mesoscopic junction operates as a resonant tunneling

device, we perform a numerical calculation of the tunneling probability (E) (Eq.10).

1- Figure 1 shows the variation of the tunneling probability (E), with the bias voltage,Vo, in energy units (eV) which have two main resonant peaks at certain values of V o.

voltage. These main peaks are due to the sequential resonant tunneling of the electron

from the ground state of the 1st quantum dot to the 1st and the 2nd excited states of the

adjacent quantum dot. It is noticed from figure (1) that the tunneling probability has

different behaviors when the dimensions of the device [c, b, and a] are varied as follows:

i) The peak separation decreases as the inter barrier width, c, increases and at certain

value of c, we have only one peak, (see fig. 1-a).

ii) The resonant peak heights decrease as the outer tunnel barrier width, b, increases

without any shift in peak position (see fig. 1-b).iii) The peak heights decrease with an observable shift in peak positions to higher

bias voltages as the diameter of the quantum dot, a, increases (see fig. 1-c). Behavior of

the tunneling probability, (E), has been observed by other authors [16, 24].

2-a: Fig. 2 Shows the variation of the normalized conductance with the bias voltage,

Vo, measured at different values of the inner barrier width between the two quantum

dots, c. It is noticed from the figure that the peaks separation decreases as the barrier

width, c, increases. At a certain value of, c, the two peaks becomes one. This may be

attributed to the decrease in the degree of splitting of the conductance-energy state as

the inner barrier width, c, increases and at a certain value of, c, the presented doublequantum dot system becomes a system of two isolated quantum dots rather than coupled

or superlattice system [25]. The same behavior is also noticed for the dependence of the

conductance on the diameter of the dot, a, calculated at different values of the parameter,

c.

b: The dependence of the conductance, G, on the bias voltage, Vo, at different values

of the outer barrier width, b, between the reservoirs and the quantum dots is shown in

fig. 3. No shift for the peak positions occurs, but the peak heights decrease as the value

of the barrier width, b, increases. A similar behavior is also noticed for the dependence of

the conductance on the diameter of the quantum dot, a, for different values of the outer

barrier width, b.

c: The dependence of the normalized conductance, G, on the bias voltage, Vo, cal-


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culated at different values of the diameter, a, is shown in fig. 4. Its noticed that the

peak height is decreased as the diameter of the dot increases due to the coulomb blockade

effect. Also, the peak heights shift to higher bias voltage as the diameter of the quantum

dot increases. This behavior is concordant to that of the tunneling probability.3: The dependence of the conductance, G, on the temperature, T, measured at

different values of the barrier width, b, is shown in fig. 5. The conductance, G, decreases

as the temperature increases. This agrees well with those published in literatures [26,

27, 28, and 29]. Also, the variation of, Ln G, versus, 1/T, is plotted in fig. 6. By

using the Arrhenious relation [G = Go exp (-E / kB.T)], the activation energy of the

electron is calculated and arranged in table 1. Its observed that the activation energy

of the electron increases as the value of the dimension, b, increases. This increase in the

activation energy is to overcome the increase in the resistance of the presented double

quantum dot system as the value of, b, increases.Table 1:

The activation energy of

the electron (meV)

The value of b

(nm)

0.314 1.0

0.385 1.10

0.405 1.15

0.410 1.20

It is seen from the results that the transmission spectrum is Lorentzian in shape for

such present junction with multiple barrier structure. The features of confining effects at

resonance levels are seen from our results. The dependence of the resonant level width

on various parameters such as the quantum dot diameter and the two barrier width b,

and c, are shown from our results which show the coupling effect between quantum dots.

Our results are found concordant with those in the literature [30-32].

Conclusion

In this paper, we derived a formula for the conductance of two coupled quantum dots

and analyzing its characteristic on the bias voltage, the barrier widths b, and c. We

conclude from our results that this device operates as resonant tunneling device in the

mesoscopic regime. Such quantum coherent electron device is promising for future high-

speed nanodevices.

References

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Fig. 1The variation of the tunneling probability (E), with the bias voltage Vo(eV) at:a) different values of the inner barrier, c. b) different values of the outer barrier, b. c) differentvalues of the diameter, a.

Fig. 2The variation of the normalized conductance with the bias voltage, Vo (eV), at differentvalues of the inner barrier width between the two dots, c.


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Fig. 3The variation of the normalized conductance with the bias voltage, Vo (eV), at different

values of the outer barrier width, b.

Fig. 4The variation of the normalized conductance with the bias voltage calculated for differentvalues of the diameter of the quantum, a.

Fig. 5 The variation of the normalized conductance with the absolute temperature (K

o

) atdifferent values of the outer barrier width, b.


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Fig. 6The variation of the logarithm of the normalized conductance, Ln G, with the reciprocalof the absolute temperature, 1/T, at different values of the outer barrier width, b.


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quantum size eﬀect of two couple quantum dots

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