Quantum Transport Modeling of Current Noise in Quantum Devicesdownloads.hindawi.com/archive/2001/085054.pdf · 2019. 8. 1. · tunneling andthe noise suppression is morethan halfofthe

VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 369-373Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Quantum Transport Modeling of CurrentNoise in Quantum DevicesTANROKU MIYOSHI, TETSUO MIYAMOTO

and MATSUTO OGAWA*

Department of Electrical and Electronics Engineering, Kobe University, 1 Rokkodai,Nada, Kobe, 657-8501, Japan

We have studied the dependence of noise characteristics on the dimension of electronconfinement of quantum devices at low temperature. By using the nonequilibriumGreen’s function method, we have found that in a double barrier resonant tunnelingdiode the shot noise is suppressed only around the bias voltage of the resonanttunneling and the noise suppression is more than half of the full shot noise in case ofsymmetric structures with thin barriers. On the other hand, in the Coulomb staircasecharacteristics of a quantum dot with equal barriers, the shot noise is suppressed onan average about half of the full shot noise while further drops are observed at thecurrent-step voltages.

Keywords: Suppression of shot noise; Pauli principle; Interaction of electrons; NonequilibriumGreen’s function; Quantum dot; Resonant tunneling diode

INTRODUCTION

Shot-noise has attracted a lot of attention inmesoscopic tunneling devices because it givesinformation on the temporal correlation of theelectrons, which is not contained in the time-averaged transport characteristics such as con-ductance [1-3]. When the tunneling events arecompletely random and described by Poissonstatistics, the shot noise has its maximum value2eI, where I is the time-averaged current and e isthe electron charge. However, when the tunnelingevent of one electron is correlated to those of other

electrons, the shot noise will be suppressed below2eL There exist two types of correlations ofelectrons in semiconductor quantum devices. Oneis the Pauli exclusion principle, which forbidsmultiple occupancy of the same single particlestate. Coulomb interactions are considered an-other source of correlations among electrons.

In this paper, we have studied the current noisecharacteristics of quantum devices at low tem-perature by employing the tight-binding Green’sfunction method based on Keldysh’s perturbationtheory [4, 5]. In particular, the dependence of shotnoise on the dimension of electron confinement

*Corresponding author. Tel./Fax: + 81-78-803-6071, e-mail: [email protected]

369

370 T. MIYOSHI et al.

will be discussed by comparing the noise suppres-sion of the two tunneling devices: a quantum dotand a double barrier resonant tunneling diode(DBRTD). We discussed previously the currentnoise in semiconductor quantum dots in [6], wherethe interaction between electrons was assumed toextend over all sites in a dot and it was representedby the retarded self-energy within the Hartree-Fock approximation. We found that the shotnoise at peaks of Coulomb oscillations reducedsharply close to zero when the two barriers wereequal, and the Coulomb interaction between theelectrons hardly affected the noise in Coulomboscillations. Furthermore, we demonstrated thatthe shot noise in Coulomb staircase was alsosuppressed less than half of the full shot noise, andthe noise suppression for the unequal barriers wasalways smaller than that for equal barriers on anaverage.On the other hand, a theory of noise character-

istics of a DBRTD system was presented in [1]based on the nonequilibrium Green’s functionapproach, where unfortunately the interaction ofelectrons was not considered in the calculation, butthe computed noise characteristics were qualita-tively reasonable. They concluded that the shotnoise in a DBRTS was not full, but it wassuppressed to a certain degree that depended onthe symmetry parameter of the structure. In ex-tremely asymmetric structures, the current ismainly controlled by one barrier and thus theshot noise should be full. On the contrary, itbecomes only half of the full shot noise in asymmetric structure. In our paper, to improve thetheory [1] the interaction of electrons will beconsidered within the Hartee approximation byusing the quantum mechanical calculation of thecharge in an iterative self-consistent solution of thePoisson’s equation. It will be shown quantitativelythat in a symmetric structure the noise suppressionof a DBRTD will not always be limited half of thefull shot noise, but will be more than half aroundthe current peak voltage against the prediction ofthe theory [1].

NON-EQUILIBRIUM GREEN’SFUNCTION APPROACH

Figure shows the schematic structure of anA1GaAs/GaAs/A1GaAs double barrier resonanttunneling diode used in the tight-binding analysis,which consists of two semi-infinite electrodes withtwo nondoped spacers, nondoped spacers, non-doped two barriers and a well. In the single-bandtight-binding model, the steady state current den-sity Jt and the electron density nt at a layer arerepresented by using the Fourier transformedcorrelation function G <

2e/dE < (k E)}Jl -2Re{tl,+lG+l,! II,2i J" dEn AA -{G1, (kli,E)} (2)

where kll is a transverse wave vector, and tt,denotes the hopping energy between the /th andmth layers. A and A are the cross section of thedevice and the layer spacing, respectively.The power spectrum of noise current obtained

by the Fourier transform of the current correlation

Left Device RightReservoir Reservoir

k D

(o) a(t)

b(N) /

(N+I)

FIGURE Schematic structure of a AIGaAs/GaAs/A1GaAsdouble barrier resonant tunneling diode (DBRTD) used in thetight-binding Green’s function analysis.

QUANTUM TRANSPORT MODELING 371

function is given at low frequency as

4e2 t,l+l ./_o dESt(O) A 2r

+G < (E,)G>(E,1+1,l+1 kll t,t klll,l+1 kll 1,1+1 kll

GI_ (E, > (E, )] (3)

In the numerical calculation, the quantum me-chanical electron density given by Eq. (2) is usedin the iterative self-consistent solution of Poisson’sequation to consider the interaction of electronswithin the Hartree approximation.On the other hand, the analysis of the current

fluctuation of a quantum dot is much the same asmentioned above, but there are two major dif-ferent points. One is the perfect quantization ofelectron energy in a quantum dot, which removesthe calculation of the sum in terms of kll in Eqs.(1)-(3). Second, since the interaction of electronsinside the dot plays a crucial role in the transport,the intradot interaction is well incorporated interms of self-energy within the Hartree-Fockapproximation. In addition, to model the quantumtransport in the barrier more precisely than thatimplemented in [5], we proposed to consider thefact that the transfer coefficient through the barrierdepends on the energy and further, an electroninside the barrier can make excursions into thelead [6].

NOISE CHARACTERISTICSOF QUANTUM DEVICES

2.5

1.0

0.5

00

5.0

4.0

3.0

2.0

1.0

0

(a)

0.1 0.2 0.3 0.4 0.5

Bias Voltage [V]

(b)

0.1 0.2 0.3 0.4 0.5

Bias Voltage [V]

(c)

0.1 0.2 0.3 0.4 0.5

Bias Voltage [V]

Double Barrier Resonant Tunneling Diode

Figures 2(a)-(c) show the calculated I-V char-acteristics, the shot noise power, and the noisepower ratio, respectively, for a A10.4Ga0.6As/GaAs/A10.4Ga0.6As DBRTD with symmetricstructure. Here, the noise power ratio is defined

FIGURE 2 I-V characteristics of a double barrier resonanttunneling diode with symmetric structure (a), the shot noisepower (b), and the noise power ratio defined by the noise powerdivided by full shot noise (c).

by the noise power divided by full shot noise. Thethickness of barrier and that of well are 8 layersand 20 layers, respectively. In the calculation, the

372 T. MIYOSHI et al.

5.0

3.0

2.0

1.0

0

8.0

"r" 6.0

."-. 4.0

1.1

m 0.9

0.8O

" 0.7’< 0.6C) 0.5Z

0.4

0.3

0 0.1 0.2 0.3Bias Voltage [V]

0.4 0.5

BW=6,6BW=9,9BW=12,12BW=15,15

(b)

I lI 1

0 0.1 0.2 0.3 0.4 0.5Bias Voltage [V]

(c)

-BW=6,6BW=9,9BW=12,12BW=15,15

0 0.1 0.2 0.3 0.4 0.5Bias Voltage [V]

FIGURE 3 Barrier thickness dependence of noise character-istics for symmetric double barrier resonant tunneling diodes.

doping density in the electrodes is assumed 1.01018[cm-3], and the temperature is 4.2K. Thesuppression of shot noise is observed only aroundthe bias voltage of the resonant tunneling. Thenoise suppression is found to be maximum at a

2.5

1.5

.0

17.5

0.4 0.6Bias Voltage [Itl/e]

(a)

0.8

2.5

.5

(b)

0.2 0.4 0.6 0.8Bias Voltage [Itl/e]

1.1

0.9

0.8

0.7

0.6

0.50.40.3

(c)

0 0.2 0.4 0.6 0.8Bias Voltage [It}/e]

FIGURE 4 Coulomb staircase characteristics of a quantumdot with equal barriers (a), the shot noise power (b), and thenoise power ratio (c).

bias voltage close to the current peak voltage.Furthermore, in the study of the barrier thicknessdependence of noise characteristics in symmetric

QUANTUM TRANSPORT MODELING 373

structure, we have found that the noise suppres-sion is enhanced as the thickness of barrier de-creases as shown in Figure 3, where the barrierthickness changes from 6 layers to 15 layers. Moreinterestingly, the noise suppression of a DBRTD isnot always limited to half of the full shot noise, butreaches more than half against the prediction ofthe simple theory [1].

Quantum Dot

In Figure 4, the Coulomb staircase characteristics,the shot noise power, and the noise power ratio fora dot with equal barriers are shown as functions ofbias voltage, where used as the unit of energydenotes the hopping energy. The averaged I-Vcharacteristics have step like increases of currentwith voltage. At each subsequent step in current,the number of current-carrying electrons increasesby one. The tunneling current is always flowingabove the bias voltage where the current starts toflow after overcoming the initial Coulomb block-ade. In this bias region, since the tunneling of oneelectron is correlated to those of other electronsdue to the Pauli exclusion principle, the shot noiseis suppressed on an average about half of the fullshot noise while further drops in noise ratio areobserved at the current step voltages as shown inFigure 4(c).

CONCLUSION

We have studied the current noise characteristicsof quantum devices at low temperature, particu-larly, the dependence of noise suppression on the

dimension of electron confinement by compar-ing the two tunneling devices: a quantum dotand a double barrier resonant tunneling diode(DBRTD). By using the nonequilibrium Green’sfunction method, we have found that the suppres-sion of shot noise is observed in DBRTD onlyaround the bias voltage of the resonant tunneling.The noise suppression is found to be maximum ata bias voltage a little lower than the current peakvoltage. Furthermore, we have found that thenoise is suppressed more than half of the full shotnoise in case of the symmetric DBRTD with thinbarriers. On the other hand, in the Coulombstaircase characteristics of a quantum dot withequal barriers, the shot noise is always suppressedon an average about half of the full shot noisewhile further drops in noise ratio are observed atthe current step voltages.

References

[1] Chen, L. Y. and Ting, C. S. (1991). Theoretical investiga-tion of noise characteristics of double-barrier resonant-tunneling systems, Phys. Rev., B 43(5), 4534-4537.

[2] Birk, H., de Jong, M. J. M. and Schonenberger, C. (1995).Shot noise suppression in the single-electron tunnelingregime, Phys. Rev. Lett., 75(8), 1610-1613.

[3] Hershfield, S., Davies, J. H., Hyldgaard, P., Stanton, C. J.and Wilkins, J. W. (1993). Zero frequency current noise forthe double-tunnel-junction Coulomb blockade, Phys. Rev.,B 47, 1967-1979.

[4] Lake, R., Klimeck, G., Brown, R. C. and Jovanovic, D.(1997). Single and multiband modeling of quantum electrontransport through layered semiconductor devices, J. Appl.Phys., 81(12), 7845-7869.

[5] Henrickson, L. E., Glick, A. J., Bryant, G. W. and Barbe,D. F. (1994). Nonequilibrium-Green’s function theory oftransport in interacting quantum dots, Phys. Rev., B 50(7),4482-4496.

[6] Wang, Z., Iwanaga, M. and Miyoshi, T. (1998). Currentnoise in semiconductor quantum dots, Jpn. J. Appl. Phys.,37(11 ), 5894- 5901.

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Quantum Transport Modeling of Current Noise in Quantum Devicesdownloads.hindawi.com/archive/2001/085054.pdf · 2019. 8. 1. · tunneling andthe noise suppression is morethan halfofthe