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Physiea B 194--196 (1994) 999-1000 North-HoUand

Conductance

fluctuations in quantum

dots

K.B. Efetov*, V.N. Prigodin t, and S. Iida I M ax-Planck-Institut fiir FestkSrperforschung, Heisenbergstr. 1, D-7000 Stuttgart 80, Germany A complete analytical theory of conductance fluctuations in quantum dots is developed. For the first time the conductance distribution function is calculated exactly. The calculations are performed using the supersymmetry method.

The most striking feature of experiments on mesoscopic structures is an irregular dependence of the conductance on the varied parameters such as, for example, a gate voltage or magnetic field [1]. The conductance fluctuations are typically very large and therefore, in order to describe an experiment in an adequate way one needs to know not only traditional averages but the whole distribution function. This is a new and not trivial theoretical problem which requires developement of new nonperturbative methods. Below we calculate the conductance distribution function for a small system (quantum dot) weakly connected with external leads. The coupling to the leads results in a level broadening 3'In modern devices the inequality 7 <:< Ec, where E¢ ,-, D I L ~ is the Thouless energy, can be easily achieved. In this region the perturbation theory in terms of diffusion modes (Cooperons and Diffusons) [2] is no longer applicable and one encounters new very interesting phenomena. In the limit under consideration 3' << E¢ all calculations can be done using the zero dimensional supersymmetric a-model [3]. Although the a-model was derived for a system with impurities there is a very strong evidence that problems of quantum chaos are described by the zero dimensional a-model (which is equivalent to the random matrix theory) as well. Therefore our theory can be used for clean quantum dots in which electrons are scattered mainly by boundaries. We start from a model of a dot with two point

contacts. Due to this connection energy levels of the dot are no longer discrete. Inelastic scattering and other contacts with a bulk also result in broadening the levels. The Hamiltonian of the model can be written in the form H

=

i(xA +2--~'-~v i [at6(r - rl) Hc+~--~r

+

where He stands for the Hamiltonian of the closed dot. In Eq. (1) v is the mean density of states, A = (vv) -1 is the mean level spacing between the quantized levels and v is the volume of the dot. The Hamiltonian Hc contains both a regular H0 and irregular H1 part. In order to obtain physical quantities one should average over H1. The last term in Eq. (1) describes the point contacts, the dimensionless parameters cq and c~2 being dependent on the barriers between the dot and the leads. The second term in Eq. (1) differs from the third one by the absence of a spatial dependence and stands for inelastic scattering or for an additional electronic exchange with a reservoir. The fact that these two terms are purely imaginary results in broadening the energy levels. Explicit calculations can be performed for an arbitrary relation between a l , ~2 and a. However, for the sake of simplicity we consider the limit c~ >> cq, (is. In this limit the level width 7 is given by

3' = o,A/

*permanent a~ldress: Landau Institute, Moscow, Russia lpermanent address: Ioffe Institute, St. Petersburg, Russia t permanent address: Department of Physics, Faculty of Science, Ehime University, Matsuyama 790, Japan

0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E1080-6

(i)

(2)

1000 and is the same for all levels. Due to fluctuations of energy levels and wavefunctions there can be strong fluctuations of conductance. We write the conductance G using a Landauer type formula

a = 2e2

r2)G

h -'----~ (~rv) ~

(r2,n)

(3)

trl,

where G~ ,A are retarded (advanced) Green functions. The distribution function P(g) of the conductanced is defined as

P(g) -: (6(g - Gh/20))

(4)

where the angular brackets stand for averaging over irregularities of the system. In the limit of small 7 the supersymmetry method [3] is the only possibility to evaluate the average of Eq. (4). After some, by now standard, transformations Eq. (4) can be reduced to the form

P(g)

:

(5(g + a 1 a 2~¢33£ "~12'r1)qCa3£ 'f~21"r 2]}/q "\

(5)

In Eq. (5) the symbol (...) stands for a functional integral with the weight e x p ( - F ) where F is the free energy functional describing the supersymmetric a-model and Q is the supermatrix entering the a-model. The limit a <<: Ec/A considered in the present work corresponds to the zero dimensional o'model and F takes the form

F : -a/4STr(AQ)

,1+ ~ - ~ + ~

(1 "31-2g/~g) 1/2,

•~ :

e-~xsinha

]

(7)

g :

~1C~2/~

(8)

Equations (7) and (8) describe completely the statistics of the conductance fluctuations in the quantum dot. Any anverages can be calculated using the distribution function P(g). Surprisingly, the function P(g) decreases monotonously and g -- 0 is the most probable value of the conductance. From asymptotics of P(g) one can derive that the phase differences ~o~ = ~ ( r l ) ~o~(r2) of wave functions are not correlated for different /3 [4]. Moduli [ ~ [ of the wave functions at different points are not correlated and obey Gaussian statistics. One can derive also the distribution function of conductances at maxima which enables us to describe the conductance fluctuations in the Coulomb blocade regime. In a particular case when oo = o~2 we reproduce in this regime results of Ref. [5].

REFERENCES

[1] M.A. Kastner, Rev. Mod. Phys. 64 (1992) 849.

[2]

B.L. Altshuler, Pis'ma Zh. Eksp. Teor. Fiz. 41 (1985) 530 [JETP Lett. 41 (1985) 648].

[31

K.B. Efetov, Adv. Phys. 32 (1983) 53.

[4]

V.N. Prigodin, K.B. Efetov, and S. Iida, unpublished.

[5]

R.A. Jalabert, A.D. Stone, Y. Alhassid, Phys. Rev. Lett. 68 (1992) 3468.

(6)

Then one should calculate in Eq. (5) a definite integral instead of the functional one. Calculation of this integral in the unitary case using the parametrization of Ref. [2] is rather simple and we obtain 1 d [2 -~x P(g) = 2c~A dA [ ~ e cosh a +

+ (1-

where