Quantum Size Effect of Two Couple Quantum Dots

EJTP 5, No. 19 2008) 33 42 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, V 0, 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 electron increases 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 particular, 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

34 Electronic Journal of Theoretical Physics 5, No. 19 2008) 33 42 diodes RTDs) have candidates for digital circuit application [9]. Quantum dots can be built as single-electron transistors [10], a charge Quantum bit and double quantum charge qubit [11, 12]. They can serve as artificial atoms quantum dots) and artificial molecules coupled quantum dots) [13-15]. In parallel to technological efforts aimed toward searching compact circuit architecture, great deal of attention has been dedicated to model and simulate RTDs [16-18], as a way to optimize device design and fabrication [19] and also to understand mesoscopic transport properties of these devices. In the present paper, it is desired to investigate the quantum size effect on the electron transport of mesoscopic devices whose dimensions are less than the mean free path of electrons. Such a device will be modeled as two series-coupled quantum dots based semiconductor - heterostructure separated by an inner barrier of width, c. These quantum dots are coupled weakly to two conducting leads via quantum point contact. As we shall see from the treatment of this model, that such device will operate as a resonant tunneling device. 2. The Model The resonant tunneling device could be constructed as two series coupled quantum dots, each of diameter, a, and separated by an inner barrier of width, c. Also, these quantum dots are separated from two leads by outer tunnel barriers from the corresponding sides, each of width, b. Electron transport through these quantum dots could be affected by the phenomenon of Coulomb blockade [20]. An expression for the conductance, G, of the present device could be derived using Landauer- Buttiker formula [21]: G = 4e2 h de ΓE) f ) sin φ 1) E Where Γ E) is the tunneling probability, φ is the phase angle of the tunneled electrons, h is Planck s constant and e is the electron charge. The derivative of the Fermi-Dirac distribution is given by: f [ ] E E =4k BT ) 1 cosh 2 EF ) 2) 2k B T Where E F is the Fermi Energy, k B is Boltzmann s constant and T is the absolute temperature. The tunneling probability, Γ E) of electrons through such device could be determined by using the method of a transfer matrix [22] as follows: The Schrödinger equation describing electron transport in the j th region is given by [21]: ) 2 d 2 ψ j V 2m dx + 2 j + e2 N 2 ψ j = Eψ j 3) 4C Where m* is the effective mass of the electron, V j is the potential energy of the j th region, e 2 N 2 /4C is the charging energy of each quantum dot [18,20,21], in which C is its capacitance and N is the number of electrons entering each quantum dot. The eigenfunctions ψ j x) in the j th region corresponding to the Schrödinger equation 3) is expressed as [18, 21, and 23]:

Electronic Journal of Theoretical Physics 5, No. 19 2008) 33 42 35 ψ j x) =A j expik j.x)+b j exp ik j.x) 4) Where: k j = [2m E V j e 2 N 2 /4C)] 1/2 5) is Planck s constant divided by 2π. The coefficients A j and B j are determined by matching the wave functions ψ j and their first derivatives at the subsequent interface. Now, using the transfer matrix, we get the coefficients A j and B j as: A l B l = j=1 R j A r B r 6) Where the notations l, r) denote left and, right regions. In Eq. 6), the coefficients R j in the j th region is given by: R j = 1 2k j k j + k j+1 ) exp[ik j+1 k j )x j ] k j k j+1 ) exp[ik j+1 + k j )x j ] k j k j+1 ) exp[ik j+1 + k j )x j ] 7) k j + k j+1 ) exp[ik j+1 k j )x j ] According to the present model of the device, the corresponding wave vectors in the regions where the barriers exist are given by: )] 1/2 [2m V o + V b + e2 N 2 4C κ = 8) Where, V b is the barrier height. And also, the wave vectors in the quantum dots are expressed as: κ = [2m E] 1/2 9) Now, the tunneling probability, Γ E), is given by solving Eq. 6) [22] and we get: Where: A = ΓE) = and sinh κ 2b c)) B = D 1 D 2 sinh κb) in which the expression for D 1 and D 2 are: D 1 = 2 cosh κb). cos ka) 1 1+A 2 B 2 10) ) V o + V b + e2 N 2 sinh κb) 4C ) ) E Vo + V b + e2 N 2 1/2 11) 4C E 2E V o V b e2 N 2 4C ) sinh κb) sin ka) 12) E Vo + V b + e2 N 2 4C E)) 1/2 13)

36 Electronic Journal of Theoretical Physics 5, No. 19 2008) 33 42 and D 2 = 2 cosh κc) cos ka) 2E V o V b e2 N 2 4C ) sinh κc) sin ka) E Vo + V b + e2 N 2 4C E)) 1/2 14) Now, by substituting Eq. 10) for the tunneling probability, Γ 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 Γ E) Eq.10). 1- Figure 1 shows the variation of the tunneling probability Γ E), with the bias voltage, V o, 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 1 st quantum dot to the 1 st and the 2 nd 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, Γ 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 double quantum 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, V o, 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-

Electronic Journal of Theoretical Physics 5, No. 19 2008) 33 42 37 culated at different values of the diameter, a, is shown in fig. 4. It s 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 = G o exp -E / k B. T)], the activation energy of the electron is calculated and arranged in table 1. It s 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 highspeed nanodevices. References [1] T.Y.Marzin, et-al, Phys. Rev. Lett. 73 1994) 716. [2] S. Raymond, et-al., Phys. Rev. B54 1996) 154.

38 Electronic Journal of Theoretical Physics 5, No. 19 2008) 33 42 [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 861998)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. Ackermann s, G. Montambais and 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 Cambridge University 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 341-3482000). [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. 341995)4417.

Electronic Journal of Theoretical Physics 5, No. 19 2008) 33 42 39 Fig. 1 The variation of the tunneling probability ΓE), with the bias voltage V o ev) at: a) different values of the inner barrier, c. b) different values of the outer barrier, b. c) different values of the diameter, a. Fig. 2 The variation of the normalized conductance with the bias voltage, V o ev), at different values of the inner barrier width between the two dots, c.

40 Electronic Journal of Theoretical Physics 5, No. 19 2008) 33 42 Fig. 3 The variation of the normalized conductance with the bias voltage, V o ev), at different values of the outer barrier width, b. Fig. 4 The variation of the normalized conductance with the bias voltage calculated for different values of the diameter of the quantum, a. Fig. 5 The variation of the normalized conductance with the absolute temperature K o )at different values of the outer barrier width, b.

Electronic Journal of Theoretical Physics 5, No. 19 2008) 33 42 41 Fig. 6 The variation of the logarithm of the normalized conductance, Ln G, with the reciprocal of the absolute temperature, 1/T, at different values of the outer barrier width, b.