Universal excess noise in resonant tunneling via strongly localized ...

PHYSICAL REVIEW B

VOLUME 53, NUMBER 23

15 JUNE 1996-I

Universal excess noise in resonant tunneling via strongly localized states Yuli V. Nazarov and J. J. R. Struben Faculteit der Technische Natuurkunde, Technische Universiteit Delft, 2628 CJ Delft, The Netherlands ~Received 26 October 1995; revised manuscript received 14 March 1996! We show that in disordered resonant tunneling conductors the excess current noise is suppressed by a factor of 3/4 in comparison with its Poisson value provided the electrons tunnel via strongly localized states. Thus we reveal a class of systems exhibiting universality of its shot-noise properties. We discuss recent experiments. @S0163-1829~96!04324-X#

At sufficiently small currents, low-frequency power of current noise in any conductor obeys the Johnson-Nyquist relation,1 S54TR, R being zero-current resistance. At further increase of average current, some conductors show more current noise. Frequently a linear dependence holds for big currents I, S52e a I,

~1!

frequently, at least in highly disordered systems. The total current and current noise are contributed by many states and therefore an effective averaging over disorder takes place. We establish that, as a result of such averaging, the excess noise exhibits universal suppression with a5 43 , not depending on concrete properties of the system. Therefore we have found yet another noise universality class. To describe the system, we apply the model first introduced in Ref. 14 and improved in Ref. 15 to account for Coulomb correlations. The model has been successfully tested in recent experiments.16 In this model, the resonant center can be in two states: one with no electrons and another one with a single electron. Double occupancy of the resonant center is excluded from the consideration since, due to Coulomb repulsion, this would cost too much energy. The state with one electron is spin degenerate; this degeneracy may be lifted in magnetic field as shown in Fig. 2. Provided the temperature or voltage applied is not too low, max$eV,T%@G, the transport via each resonant center can be described in a master equation formalism. In general, there are eight rates involved:

as illustrated in Fig. 1. This is called excess noise. The most-known example of excess noise is the shot noise,2 when the electrons traverse the conductor one by one without correlations between traversing events. The current obeys Poisson statistics, with a51. We shall stress that the same value of a occurs in a variety of conductors ranging from vacuum tubes to ultrasmall tunnel junctions. These systems thus form a universality class with respect to their noise properties, noise universality class. Recently the problem of excess noise has received much attention in studies of mesoscopic systems. Generally, the excess noise is suppressed in coherent metallic conductors.3 The scattering matrix approach to the excess noise4 can be elaborated to obtain the complete noise statistics,5 to treat normal-metal–superconducting6 and fractional quantum hall9 systems. Noise in single-electron and/or resonanttunneling devices has been studied in detail.7,8 In the course of these studies, two new noise universality classes have been discovered. It has been shown10 that a5 31 for an arbitrary coherent diffusive conductor. Although this does not hold for a conductor with extended defects,11 the value of a does not depend on material properties of the conductor such as diffusivity, geometrical size and shape, etc. Very recently a strikingly simple result12 has been obtained for the contrast case of fast energy relaxation in a diffusive conductor. In this case a5)/4. Suppression of excess noise in resonant tunneling has been studied both experimentally13 and theoretically.8 The systems under consideration were semiconductor heterstructures where the resonant states were delocalized in a wide well between two artificial tunnel barriers with no resonant centers inside the barriers. In contrast to this, we consider a situation where the resonant states are randomly distributed within a single tunnel barrier having the localization radius much smaller than the barrier width. We stress that such an arrangement of resonant states has higher ‘‘entropy’’ than the one previously considered and thus would occur more

corresponding to electron hopping to and from the center. Here b561 labels the spin of the state with one electron, d51,2 labels the leads having chemical potential md , m12m25eV, the bias applied. The Fermi function f accounts

0163-1829/96/53~23!/15466~3!/$10.00

15 466

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G 0,b ; d 5G d f ~ e 2 m d 2 b B m B /2! ,

~2!

G b ,0; d 5G d @ 12 f ~ e 2 m d 2 b B m B /2!# ,

~3!

FIG. 1. Noise power of the disordered resonant tunneling conductor versus the current. The asymptotic slope corresponds a5 43. © 1996 The American Physical Society

53

BRIEF REPORTS

Pˆ 5Re

FIG. 2. Resonant tunneling. Electrons traverse the barrier via intermediate resonant centers, their degeneracy being lifted by external magnetic field. The active centers are concentrated near the center of the barrier.

for electron filling in the leads; e is the energy of the resonant level. Since the states are assumed to be well localized in the barrier, the transition rates exhibit exponentially strong dependence on the distance between the resonant center and the corresponding lead. If the origin of the coordinates is chosen in the middle of the barrier, we have, for a general symmetric barrier, G1,25G0 exp [ f (6x)]. The criterion of strong localization is that k[[d f (0)/dx] 21 !d, d d being the barrier width. As shown in Ref. 14, the resonant centers contributing to the transport are concentrated close to the middle of the barrier, u x u .k, and have the energies close to Fermi level. This justifies a simple assumption about statistics of the disorder: the centers are uniformly distributed within the barrier and in energy space, with the density n. Stationary noise in systems obeying a master equation can be treated rigorously and in a straightforward manner. Albeit, the resulting expressions do not look anymore compact. We refer the reader to Refs. 7 and 8 for related mathematical details. In Ref. 7 one can find, for instance, a convenient matrix formulation of the problem. Adopting this formulation to our particular problem, we can obtain a general relation for the frequency-dependent noise power generated by each resonant center, ˆ Pˆ ~ v ! M ˆ u1&, S ~ v ! 5S shot1 ^ p 0 u M S shot /e 2 5

~ p 0 G 0,b ; d 1p b G 0,b ; d ! . ( b,d

Here the following 333 matrices have been used: ˆ5 M

e 2

S

0

~4!

G 0,1;1 2G 0,1;2 G 0,21;1 2G 0,21;2 2G 1,0;1 1G 1,0;2 0 0 2G 21,0;1 1G 21,0;2 0 0

~5!

D

~6!

15 467

1 i v 1Gˆ

; Gˆ 5

S

2G 0,21 2G 0,1 G 1,0 G 21,0

G 0,1

2G 1,0 0

G 0,21 0

2G 21,0

D

. ~7!

Here G0,b5(dG0,b;d. The three-vector ^p 0u is composed of equilibrium probabilities p 0,61 to find the center in each of three possible states; all components of u1& are equal to 1. The relation ~5! gives the noise power of each resonant center in terms of the rates ~2!. The total noise power comprises incorrellated contributions of all centres. To evaluate this, we have to average the expression ~5! over all possible center positions and energies. We can, however, obtain the main result of the present work without complicated mathematical exercises. In the limit of large bias eV@T the master equation becomes very simple since the electrons hop only in one direction. The active resonant centers can be subdivided onto ‘‘degenerate,’’ for which both spin states are available for tunneling, and ‘‘nondegenerate,’’ for which only a single spin state is active. Let us consider first nondegenerate states. In order to get the noise power, we shall estimate the correlator of two electron hops to the center separated by a time t. It equals the probability to have an empty center times the rate G1 of the first hop times the probability to have an empty center at the moment t times the rate of the second hop G1 . The latter probability, 12exp@2t~G11G2!#G2/~G11G2!, is suppressed in comparison with its equilibrium value G2/~G11G2! giving rise to negative correlations of hopping events. The corresponding irreducible correlator, G 21G 22/~G11G2!2exp@2t~G1 1G2!#, determines suppression of the noise power in comparison with its shot-noise value. The deficit of the noise power is proportional to the Fourier component of the irreducible correlator. Integrating over the time, we thus obtain S52eI24e 2

G 21 G 22 ~ G 1 1G 2 !

2 3 52e

G 1 G 2 ~ G 21 1G 22 ! ~ G 1 1G 2 ! 3

,

~8!

the average current being given by I5eG 1 G 2 /~G11G2!. This coincides with the result of Ref. 8 obtained with a more complicated technique. The expressions shall be averaged with respect to the center positions. To perform such an averaging, we expand the tunneling action in x near the most relevant position x50. This yields G1,25G0 exp~6x/k!. Substituting x-dependent G’s we obtain for the average quantities S52e 2 n ~ 0 ! G 0

E

I5en ~ 0 ! G 0

`

2`

E

dx

`

2`

cosh~ 2x/ k ! , 4 cosh3 ~ x/ k !

dx . 2 cosh~ x/ k !

~9!

~10!

Here n~0! is an average concentration of impurities at x50, the integration over x can be extended to infinite limits since k!d and the integrand vanishes at distances .k. Performing these two simple integrations, we compare results and obtain that S52e 34 I.

~11!

BRIEF REPORTS

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For degenerate centers we obtain the same expression ~8! with G1 replaced by 2G1 , since there are two states available for tunneling. The integrals over x appear to be the same if x is shifted: x→x1ln 2/2k. Therefore after the averaging over center positions the noise power and the current of degenerate centers satisfy the same relation ~11!. This is why this relation holds for the total current. We shall stress here that the applicability range of the model and of the result is in fact wider than it might seem from the previous outlining. For instance, the detailed distribution of the centers over the energy is not important. It does not matter whether the barrier is wide and rectangular. The only essential feature, which is general for any kind of tunneling, is the strong dependence of tunneling rates on the distance. This makes it relevant to approximate the rates by exponentials near the center of the barrier. We stress that this does not depend on a concrete form of the wave-function tail: it could be exponential as well as a Gaussian. Therefore we believe that the universality of a is preserved for most of the resonant-tunneling conductors. The noise-current dependence in the whole range of the currents is important for comparison with future experiments. Making use of the general scheme ~5! we have computed this dependence in two limiting cases of nondegenerate ( m B@T) and degenerate ( m B!T) resonant centers. For the degenerate case, we have derived a simple analytical formula, S ~ I ! 5 23 eI coth

S D

eIR T 1 , 2T R

~12!

J. B. Johnson, Phys. Rev. 29, 37 ~1927!; H. Nyquist, ibid. 32, 110 ~1928!. 2 J. R. Pierce, Bell Syst. Tech. J. 27, 15 ~1948!. 3 G. B. Lesovik, JETP Lett. 49, 592 ~1989!; M. Bu¨ttiker, Phys. Rev. Lett. 65, 2901 ~1990!. 4 See for review: T. Martin, in Coulomb and Interference Effects in Small Electronic Structures, edited by D. C. Glattli, M. Sanquer, J. Tran Thanh Van ~Editions Frontieres, Singapore, 1994!, p. 400. 5 L. S. Levitov and G. B. Lesovik ~unpublished!; JETP Lett. 58, 235 ~1993!. 6 B. A. Muzykantskii and D. E. Khmelnitskii, in Quantum Dynamics of Submicron Structures, edited by H. A. Cerdeira, B. Kramer, and G. Scho¨n ~Kluwer, Dordrecht, 1995!, p. 359. 7 A. N. Korotkov, Phys. Rev. B 49, 10 381 ~1994!, and references therein. 8 L. Y. Chen and C. S. Ting, Phys. Rev. B 43, 4534 ~1991!; 46, 4714 ~1992!. 1

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that bridges between equilibrium noise and big current limit. For the nondegenerate case, an analytical expression cannot be obtained and we computed the curve numerically. To our surprise, the computed curve deviates by less than 0.0004 from the nondegenerate case. This is why there are two curves plotted in Fig. 2: the difference is not visible. It appears that due to the play of numerical factors, magneticfield dependence of the noise is very weak even in the range eIR.T. Recent experiments17 clearly showed a substantial excess noise power suppression in a disordered mesoscopic system. The authors have reported accurate measurements in the metallic regime, which seem to be in agreement with the theory,12 with a'0.43. We would like to emphasize that the universality class of the system could be changed by tuning the gate voltage. With further depletion of electron density in the constriction region, one should expect the system to become a resonant-tunneling conductor with strongly localized states. It would show a5 43 . Experiments in this region are still to be done. Another natural suggestion is to perform noise measurements with the systems used in Ref. 16, and also in strongly disordered semiconductor heterostuctures. In conclusion, we have demonstrated that the disordered resonant-tunneling conductors form a different universality class with respect to their noise properties, with suppression coefficient a53/4. In fact, it is a very wide class of mesoscopic systems. We hope that our results will stimulate further experimental studies of their noise properties. The authors are indebted to M. Devoret, V. A. Kozub, M. M. de Jong, and G. E. W. Bauer for stimulating discussions.

C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 72, 724 ~1994!. C. W. J. Beenakker and M. Bu¨ttiker, Phys. Rev. B 46, 1889 ~1992!. 11 Yu. V. Nazarov, Phys. Rev. Lett. 73, 134 ~1994!. 12 A. Steinbach, J. Martinis, and M. Devoret, Bull. Am. Phys. Soc. 40, 400 ~1995!; ~unpublished!; V. I. Kozub and A. M. Rudin, Pis’ma Zh. Eksp. Teor. Fiz. 62, 45 ~1995! @JETP Lett. 62, 49 ~1995!#. 13 Y. P. Li, A. Zaslavsky, D. C. Tsui, M. Santos, and M. Shayegam, Phys. Rev. B 41, 8388 ~1990!. 14 A. I. Larkin and K. A. Matveev, Zh. Exp. Teor. Fiz. 93, 1030 ~1987! @Sov. Phys. JETP 66, 580 ~1987!#. 15 L. I. Glazman and K. A. Matveev, Pis’ma Zh. Eksp. Teor. Fiz. 48, 403 ~1988! @JETP Lett. 48, 445 ~1988!#. 16 H. Bahloi, K. A. Matveev, D. Ephron, and M. R. Beasley, Phys. Rev. B 49, 2989 ~1994!, Phys. Rev. B 49, 14 496 ~1994!. 17 F. Liefrink, J. I. Dijkhuis, M. J. M. de Jong, L. Molenkamp, and H. van Houten, Phys. Rev. B 49, 14 066 ~1994!. 9

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