Communication in Physics 16, 193 (2006)

CURRENT AND SHOT NOISE IN 1D-ARRAYS OF QUANTUM DOT Nguyen Hoai Nam1 , Pham Tuan Anh1 , Nguyen Viet Hung2 1 Faculty

of Physics, Hanoi University of Education, 136 Xuanthuy, Caugiay, Hanoi 2 Institute of Physics and Electronics, 10 Daotan, Badinh, Hanoi

Abstract. We study current and shot noise in 1D-arrays of N-quantum dots coupled with leads. The negative differential conductance and multi-coulomb gaps could be observed with specific parameters. Shot noise is always sub-poissonian and enhanced with decreasing differential conductance. Fano factor is greater than the value of 1/(N+1), and reaches to this value only in the case of identical array

I. INTRODUCTION During last decade, correlated single-electron tunneling [1] has remained an attractive topic. The shot noise (SN) in single-electron tunneling is due to the randomness of tunneling events and reveals information on transport properties which are not available through the conductance measurement alone [2, 3]. SN is a consequence of charge quantization and becomes relatively important in nano-devices. In the case of uncorrelated current the SN has the full or Poissonian value 2eI, where e is the elementary charge and I is the average current. Because of correlations in the motion of charge carriers, deviations either suppression or enhancement from this value. The measure of the deviations is the Fano factor F , which is defined as the ratio of the actual noise spectral density to the full SN-value. There are mainly two kinds of correlation: the Pauli exclusion principle always causes a suppression of SN, while the Coulomb interaction may suppress or enhance the noise depending on the conduction regime. The non-Poissonian noise has been most extensively studied in double barrier resonant tunneling diodes, where the SN is partially suppressed in the positive differential conductance (PDC) regime and becomes super-Poissonian in the negative differential conductance (NDC) regime [4]. The super-Poissonian noise accompanied by an NDC has been also predicted and observed in quantum dot (QD) devices [5]. However for several structures of metallic QD such as the single-electron transistor (SET) and single electron pump (SEP), the different conductance either positive for SET or negative for SEP but noise is shown to be sub-Poissonian [7, 8, 9]. We extend to study structures of N-QDs in series coupled with leads for several cases. By using master equation approach and/or Monte-Carlo (MC) method we can observe negative differential conductance and multi-coulomb gaps with specific parameters. Shot noise is always sub-poissonian and enhanced with decreasing differential conductance.

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Nguyen Hoai Nam, Pham Tuan Anh, Nguyen Viet Hung

R 2 ,C 2

R 1 ,C 1

R 3 ,C 3

R N ,C N

R N+1 ,C N+1 Vr

Vl C g1

C g2

Vg1

C gN

Vg2

VgN

Figure 1: Schematic diagram of a one-dimensional array of N quantum dots. QD i is coupled to QD i+1 by a tunnel barrier with capacitance Ci+1 and tunneling resistance Ri+1 , and to a gate electrode by an insulating barrier with capacitance C gi . The first (last) QD couples to the left (right) electrode by a tunnel barrier C1 , R1 (CN +1 , RN +1 ). Fano factor is greater than the value of 1/(N+1), and reaches to this value only in the case of identical array. II. GENERAL CONSIDERATION AND CALCULATING METHOD The system under consideration is shown schematically in Fig. 1. Within the orthodox model, the state of the system is described by the numbers ni of excess electrons on the ith QD, which we combine in a vector: ~n = (n1 , n2 , ..., nN ) [10]. The free energy is defined as: F (~n) =

N N +1 1X 1X Cgi (φi − Vgi )2 + Ci (φi − φi−1 )2 2 2 i=1

i=1

− V l Ql − V r Qr −

N X

Vgi Qgi .

(1)

i=1

We denote by φi the electrochemical potential of QD i (φ0 ≡ Vl and φN +1 ≡ Vr ) and QL = CL (VL − V1 ) + enL , QR = CR (VR − V2 ) + enR , Qgi = Cgi (Vgi − Vi ); nL (nR ) by the number of electrons that have entered the structure from the left (right). The electrochemical potential of QDs is defined by solving the following equation: ~=Q ˆφ ~ d, M

(2)

~ ≡ (φ1 , φ2 , ..., φN ), Q ~ d ≡ (Q ˜ 1, Q ˜ 2 , ..., Q ˜ N ), and Q ˜ 1 = en1 + Cg1 Vg1 + C1 Vl , Q ˜i = where φ ˜ eni + Cgi Vgi (i=2,...N-1), QN = enN + CgN VgN + CN +1 Vr ; ni is the number of excess ˆ is defined as follows: electrons on QD i; matrix M   ˜ C1 −C2 0 ... 0  −C2 C˜2 −C3 ... 0    ˆ ˜3 ... 0  , (3) M = 0 −C C 3    ... ... ... ... ...  0 0 0 ... C˜N

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CURRENT AND SHOT NOISE

in which, C˜i = Ci + Cgi + Ci+1 . From the free energy (1) we can define the rate of an electron tunneling: 1

Γ=

e2 Rt

∆F exp [∆F/kB T ] − 1

(4)

where ∆F is the change in free energy defined from eq.(1), Rt is tunneling resistance of the junction in which electron tunnels through, and the rate is defined at temperature T . By using eqs. (1-4), in principle, we can solve the master equation (ME) or perform Monte-Carlo (MC) simulations to calculate the current and further the noise. In practice, in this case, for the structures of multi-QDs, the MC-simulation is much more efficient. First, we need to defined the net-current in dependence on electron tunneling events: I (t) =

N +1 X

gi Ii (t)

(5)

i=1

Ii (t) is the tunneling current measured at ith-junction, the factors gi are defined (assume Cgi is very small in comparison with Ci ) by: ³ ´ g1 = −CN +1 Zˆ −1 N +1,2 ¸ ·³ ´ ³ ´ −1 −1 ˆ ˆ gj = C 1 Z − Z j = 2, ..., N (6) 1,j+1 1,j ³ ´ gN +1 = C1 Zˆ −1 1,N +1

where, 

   Zˆ =    

1 1 1 1 C1 −C2 0 0 0 C2 −C3 0 0 0 C3 −C4 ... ... ... ... 0 0 0 0

... 1 ... 0 ... 0 ... 0 ... ... ... −CN +1



   .   

(7)

Then, the power spectral density of a real random process I(t) is defined as the Fourier transform of the correlation function: S (ω) = 2

Z∞

−∞

£ ¤ dte−iωt hI (t) I (0)i − I 2

(8)

In eq.(8), we have used the notation h...i to denote ensemble averages and I ≡< I(t) >. For focusing on the shot noise at zero-frequency, we calculate the fano-factor as [6]: DP E P 2 δq δq k j − h k δqk i k,j S (0) P = . (9) 2eI e h k δqk i

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Nguyen Hoai Nam, Pham Tuan Anh, Nguyen Viet Hung

(a)

(b)

Figure 2: Numerical results of single electron transistor (SET): the current, calculated from eq. (5) (dashed line, see the left axis) and the Fano factor, Fn = S(0)/2eI, calculated from eq. (9) (solid line, see the right axis), are plotted against the bias voltage V. Fig. 2(a) is for the symmetric device and fig. 2(b) is the asymmetric one. We collect a few thousand time evolutions and such a collection is used to evaluate the averages in eq.(9) and, therefore, to calculate the Fano factor. In calculations as well as in MC-simulations, the elementary charge e, the capacitance C (≡ C1 ) and the tunneling resistance R (≡ R1 ) are chosen as the basic units. The voltage, the current, and the frequency are then measured in the units of e/C, e/CR, and 1/CR, respectively. The process of calculating current was mentioned in Ref [11, 12, 13] and references therein. III. NUMERICAL RESULTS AND CONCLUSION Figure 2 shows numerical results for single electron transistor (SET). The current, calculated from Eq. 5 (dashed line, see the left axis) and the Fano factor, F n = S(0)/2eI, calculated from Eq. 9 (solid line, see the right axis), are plotted against the bias voltage V. The I-V characteristics have linear forms in the case of symmetric coupling to leads whereas noise reduces gradually with increasing bias (Figure 2(a)). Staircase is clearly observed in current in the case of asymmetry (figure 2b), noise oscillates with period of staircase (e/C). These features are determined by the struggle between two mechanism: charge accumulation (CA) and charge quantization (CQ). In the symmetrical device, CA is weak so that current has a linear form and noise is enhanced due to CQ [12]. Therefore fano factor gradually reduces whereas increasing bias. Nevertheless, in the asymmetrical device, CA is strong and plays the major roll that causes staircase form in I-V curve and enhancement/oscillation of noise at low differential conductance. The results always show 1/2 ≤ Fn ≤ 1 and the lower limit value is reached in the case of symmetry. Moreover, NDC is not appearance for both symmetric and asymmetric case. This result is coincident with other studies [7, 8]. NDC appears for QDs in series which contain not less than two dots (N ≥ 2) by adjusting structure parameters. The numerical results for single electron pump (SEP) are

CURRENT AND SHOT NOISE

(a)

(b)

(c)

(d)

(e)

5

(f )

Figure 3: Current (dashed line, the left axis) Fano factor (solid line, the right axis) of a quantum dot array are plotted against the bias voltage V. Figs. (a) and (b) is for single electron pump (SEP), figs. (c) and (d) for 3 QDs and figs. (e) and (f) for 4 QDs device. shown in figure 3. The figure 3(a) shows a clear second Coulomb gap with the Fano factor Fn < 1. By changing parameters to create an asymmetric structures, the second Coulomb gap is narrow and tend to be disappear. The value of Fano factor for the first peak of net-current seems to be greater than it is in the case of symmetry, but it is still sub-poissonian. A similar result is found for the structures containing three QDs in series: the second coulomb gap and NDC disappear whereas parameters changing from symmetric to asymmetric case (Figure 4). However the highest value of Fn decreases from ≈ 0.95 to ≈ 0.64 and it oscillates uncertainly for both cases. In the structures of N-QDs, it is existence of junctions whereas tunneling rate inverses proportional to bias. NDC and multi-coulomb gaps can be observed in the case

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Nguyen Hoai Nam, Pham Tuan Anh, Nguyen Viet Hung

of asymmetry. When current is low (after a coulomb gap), CQ is strong and causes enhancement of noise. Nevertheless, when CA is strong, noise is enhanced and differential conductance is reduced. We found 1/(N + 1) ≤ S(0)/2eI ≤ 1 when exploring devices with parameters and the lower limit value reached in the case of symmetry. We have studied the sequential electron tunneling through metallic 1D-array of quantum dots. Using the master equation and Monte-Carlo approach, the current and the shot noise have been calculated for different device configurations, focusing on the role of the device asymmetry. An asymmetry in structure devices may produce a variety in current and noise (magnitude and character). NDC in I-V current is not appearance due to the asymmetry for SET but may be shown for structures containing not less than two QDs. Noise may be enhanced with decreasing the NDC but the limit value of deviation is less than 1. ACKNOWLEDGMENT This work is supported by the research project SPHN-06-64 of Hanoi University of Education. REFERENCES 1. D.V Averin and K.K Likharev, in Mesoscopic Phenomena in Solids, edit by Altshuler (1991) 173. 2. Yu. M. Blanter and M. Buttiker, Phys. Rep., 336 (2000) 1. 3. H. Birk, M. J. M. de Jong, and C. Schonenberger, Phys. Rev. Lett.,75 (1995) 1610. 4. W. Song, E. E. Mendez, V. Kuznetsov, and B. Nielsen, Appl. Phys. Lett., 82 (2003) 1568. 5. A. Thielmann, M. H. Hettler, J. Konig, and G. Schon, Phys. Rev. B, 71 (2005) 045341. 6. M. Gattobigio, G. Iannaccone, and M. Macucci, Phys. Rev. B, 65 (2002) 115337. 7. S. Hershfield, J.H. Davies, P. Hyldgaard, C.J. Stanton, and J.W. Wilkins, Phys. Rev. B, 47 (1993) 1967. 8. A.N. Korotkov, Phys. Rev. B, 49 (1994) 10381. 9. V. Hung Nguyen, V. Lien Nguyen, and P. Dollfus, Appl. Phys. Lett., 87 (2005) 123107. 10. A. A. Bakhvalov, G. S. Kazacha, K. K. Likharev, and S. I. Serdyukova, Sov. Phys.- JETP, 68 (1989) 581. 11. V. Hung Nguyen, V. Lien Nguyen, and H. Nam Nguyen, J. Phys.:Condens. Matter, 17 (2005) 1157. 12. V. Hung Nguyen and V. Lien Nguyen, Phys. Rev. B, 73 (2006) 165327. 13. V. Lien Nguyen, T. Dat Nguyen, H. Nam Nguyen, Phys. Lett. A, 291 (2001) 150.