quantum size effect of two couple quantum dots
Post on 04-Jun-2018
212 Views
Preview:
TRANSCRIPT
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
1/10
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
ghnzaki@yahoo.com
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
2/10
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
3/10
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)
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
4/10
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-
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
5/10
Electronic Journal of Theoretical Physics 5, No. 19 (2008) 3342 37
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
[1] T.Y.Marzin, et-al, Phys. Rev. Lett. 73 (1994) 716.
[2] S. Raymond, et-al., Phys. Rev. B54 (1996) 154.
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
6/10
38 Electronic Journal of Theoretical Physics 5, No. 19 (2008) 3342
[3] M. Grundmann, et-al., Appl.Phys. Lett. 68 (1996) 979.
[4] H. Jiang et-al., Phys. Rev.B56 (1997) 4696.
[5] M. Rontani, et-al. Appl. Phys. Lett. 72 (1998) 957.
[6] M. A. Kastner, Rev. Mod. Phys. 64 (1992) 849.
[7] K. Nakazato, et-al., Electron. Lett. 29 (1992) 384.
[8] S. Tiwari, et-al., Appl. Phys. Lett., 69 (1996) 1232.
[9] P. Mazumder, et-al, Proc. IEEE 86 (1998) 664.
[10] Gergley Zarond at al., arxiv: cond-mat / 0607255V2 (18 Oct2006).
[11] Xiufeng Cao and Hang Zheng, arxiv: cond-mat / 0701581V1 (24 Jan2007).
[12] J. Gorman et al., Phys. Rev. Lett., 95 (2005) 090502.
[13] S. Sasaki et al., Phys. Rev. Lett., 93, 017205 (2004).
[14] P. Jarillo-Herreror et al., Nature 434, 484 (2005).
[15] A. Kogan et al., Phys. Rev. B, 67 (2003) 113309.
[16] L. Yang, et-al., J. Appl. Phys. 68 (1990) 2997.
[17] J. Sune, et-al. Microelectron. Eng. 36 (1997) 125.
[18] A. A. Awadalla, A.M.Hegazy, Adel H. Philips and R. Kamel, Egypt. J. Phys. 31(2000) 289.
[19] J. S. Sun, et-al., Proc. IEEE 86(1998)641.[20] U. Meirav, et-al., Phys. Rev. Lett. 65 (1990)771.
[21] Y. Imry, Introduction to mesoscopic physics (Oxford University, New York, 1997).
[22] H. Kroemer, Quantum Mechanics, (Prentice Hall, Englewood Cliffs, New Jersey,07632 (1994)).
[23] C.W.J. Beenakker, in : Mesoscopic physics, eds. E. Ackermanns, G. Montambaisand J. L. Pickard (North-Holland, Amsterdam, 1994).
[24] R. Ugajin, Appl. Phys. Lett. 68 (1996) 2657.
[25] D. K. Ferry and S. M. Goodnick in Transport in Nanostructures CambridgeUniversity first edition 1997.
[26] Arafa H. Aly, Adel H. Phillips and R. Kamel, Egypt J. Physics, 30 (1999) 32.
[27] W.M. Van Hufflen, T. M.Klapwijk, D.R.Heslinga, Phys.Rev. B, 47 (1993) 5170.
[28] Aziz N. Mina, Adel H. Phillips, F. Shahin and N.S. Adel-Gwad, Physica C 341-348(2000).
[29] J. M. Kinaret, Physica B, 189 (1993) 142.
[30] H. Yamamoto, et-al, Appl. Phys. A50 (1990) 577.
[31] H.Yamamoto, et-al, Jpn. J. Appl. Phys. 34 (1995) 4529.[32] Y. C. Kang, et-al. Jpn. J. Appl. Phys. 34(1995)4417.
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
7/10
Electronic Journal of Theoretical Physics 5, No. 19 (2008) 3342 39
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.
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
8/10
40 Electronic Journal of Theoretical Physics 5, No. 19 (2008) 3342
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.
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
9/10
Electronic Journal of Theoretical Physics 5, No. 19 (2008) 3342 41
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.
-
8/13/2019 Quantum Size Effect of Two Couple Quantum Dots
10/10
top related