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Quantum Shot Noise in Mesoscopic Systems: A de Broglie-Bohm Wave-Particle Description

X.Oriols Departament d’Enginyeria Electrònica, ETSE, Universitat Autònoma de Barcelona, 08193, BCN, Spain E.mail: [email protected]

The experimental current measured in quantum-based devices fluctuates around average values due to the quantum mechanical wave-particle duality [1]. The electron probability of being reflected or transmitted is determined by the wave nature of electrons. The fact that the electron can be either transmitted or reflected (but not both!) shows its particle-like behavior. Both characteristics of the electron introduce randomness in the transport process that leads to fluctuations in the current measured experimentally. An approach which permits the discussion of both (wave and particle) aspects is mandatory to correctly model current fluctuations, i.e. quantum shot noise.

In general, most of the approaches that predict noise values for quantum-based systems follow the path proposed by Buttiker [1-3]. He generalized the Landauer (first-quantization) scattering approach using the second quantization formalism. The basic elements of this approach are creation and annihilation operators. Due to the commuting algebra of these operators, the simultaneous presence of one electron at different contacts is not possible [3]. Therefore, this approach contains the essential wave and particle electron descriptions. The Buttiker formalism has been successfully applied to predict many experimental noise situations [1-3] and it has the additional advantage that it directly includes the fermionic/bosonic symmetries when describing many particle systems [1-3].

In this conference, we present an alternative quantum noise approach based on the de Broglie-Bohm (dBB) interpretation of the quantum theory [4-6]. Following the dBB formalism, each electron can be described by a well- defined quantum trajectory guided by a wave function. Electrons associated to the same wave function have different initials conditions and follow different path. By construction, all standard (Copenhagen) probabilistic results obtained from the wave function are exactly reproduced by an appropriate ensemble of Bohm trajectories [4- 8]. Since the dBB formalism directly deals with the wave and particle nature of electrons, it provides also an excellent framework to study quantum noise. In fact, most of the numerical simulation techniques used in the classical Monte Carlo approach, that are related with the microscopic description of electron trajectories, can be directly adapted to our quantum formalism [7-10]. Accordingly, we named it a Quantum Monte Carlo approach. Our formalism can be extended to any complex quantum system as far as we can explicitly obtain the wave function solution of the corresponding Schrödinger equation. Therefore, the Hamiltonian has to include all electron interactions which are meaningful for the particular system that we are studying (such as the tunnelling barriers [7- 8], mutual coulomb interaction [10], etc.). In addition, our approach can also be generalized to many-particle systems by simply providing antysimmetric/symmetric wave functions to guide electrons [11].

As an example of the applicability of our approach, we show the role of the electron-electron Coulomb interaction in the transport correlations properties of a typical (2nm/6nm/2nm) AlGaAs resonant tunnelling device (RTD). We solve self-consistently the Poisson and the Schrödinger (with time dependent potentials) equations [10]. The role of the Coulomb interaction as a source of electron correlation is shown in the noise properties (the Fano factor, F) of the RTD depicted in figs. 1 and 2. We have used room temperature, 300 K, for all numerical simulations. In order to be able to distinguish the effects originated by the electron-electron interaction, first in fig 1, we consider frozen potentials (i.e without using self-consistence). For this simple system, with independent electrons, the excellent agreements between our numerical values and the theoretical predictions, refs [1] and [12], provides a satisfactory test of our approach implementation.

The results depicted in figure 2 are obtained for Coulomb interacting electrons. The Poisson and the many- particle Schrödinger equations for the same RTD considered in figure 1 are solved self-consistently. The instantaneous charge is computed from the Bohm trajectories. The effect of the Coulomb correlation between electrons is clearly manifested in the Fano factor depicted in figure 2b. Just after the resonant voltage (when the resonant energy, ER, is below the conduction band at the emitter contact) the presence of one electron inside the quantum well raises ER (see inset in fig. 2b). Thus, the transmission for the next electron is highly enhanced at this voltage. Roughly speaking, the Coulomb interaction affects the electron dynamics by trying to regroup the electrons and providing a Fano factor higher than one. The noise shows a process of bunching [10-13] of electrons at this particular (with negative conductance) voltage. The results are in good agreement with experimental data [13-14].

2nd NanoSpain Worshop March 14-17, 2005 Barcelona-Spain

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4 6 (a) Our approach (a) Our approach Esaki results [12] 5 Visual guided line 3 4 A) A) µ µ 2 3

EFE 2 ER Current ( Current 1 Emitter Colector ( Current E FC 1 L x=0 x=L FROZEN POTENTIALS SELF-CONSISTENT POTENTIALS 0 0

1.4 1.2 Our approach Our approach Buttiker results [1] Visual guided line ER FROZEN 1.2

1.0 ER FROZEN Fano Factor

Fano Factor Fano 1.0 (b) FROZEN POTENTIALS (b) SELF-CONSISTENT POTENTIALS 0.8 0.8 0.0 0.1 0.2 0.3 0.4 0.00.10.20.30.4 Applied bias (Volts) Applied bias (Volts)

Fig.1: (a) Current voltage characteristic for the standard Fig.2: (a) Current-voltage characteristic for the RTD RTD diode of the text. The inset shows the static (frozen) diode of fig. 1 when the Poisson equation is solved self- potential profile. In dashed line, current results obtained consistently with the time-dependent Schrödinger equation. from the Esaki expression [12]. (b) In circles, numerical (b) Fano factor for each bias point computed with our values of F for each bias points computed using our approach. The insets show the effect of the electron Quantum Monte Carlo approach. In dashed line, Buttiker correlation on the transmission, before and after the formalism [1]. resonant voltage.

In conclusion, in this conference we will show that quantum noise can be naturally understood within the de Broglie-Bohm interpretation of the quantum theory. In addition, our approach has the technical advantage that the N-particle wave-function does not have to be computed for all points of the N-dimensional configuration space, but only along the configuration points of Bohm trajectories. The present approach opens a new path for studying electron transport in mesoscopic systems, in general, and quantum noise, in particular.

References: [1] Y. Blanter and M.Büttiker, Phys. Rep., 336, 1, (2000) ; M.J.M. de Jong and C.W. J. Bennakker, NATO ASI Series E, 345 Kluwer Academic Press (1997) [2] M.Büttiker, Phys. Rev. Lett., 65, 2901 (1990); Phys. Rev. B. 46(19), 12485 (1992). [3] Ya. M. Blanter and M.Buttiker, Phys. Rev. B, 59(15), 10217, (1999). [4] D. Bohm, Phys. Rev. 85, 166, (1952). [5] X. Oriols et al. Phys. Rev. A, 54, 2594, (1996). [6] J.Suñé and X. Oriols, Phys. Rev. Lett.,.85, 894 (2000). [7] X. Oriols et al., Appl. Phys. Lett., 79, 1703, (2001). [8] X. Oriols et al. , Appl. Phys. Lett., 80, 4048 (2002). [9] X.Oriols, IEEE Trans. on Electron Devices, 50(9), 1830 (2003). [10] X. Oriols et al. Appl. Phys. Lett. 85(16), 3596 (2004). [11] X.Oriols, Phys. Rev. A , 71, 017801 (2005). [12] L.Esaki and L.L.Chang, Phys. Rev. Lett., 33, 495, (1974). [13] G.Iannaccone, M.Macucci and B.Pellegrini, Phys. Rev. B, 55(7), 4539 (1997); G.Iannaccone, G.Lombardi, M.Macucci and B.Pellegrini, Phys. Rev. Lett, 80(5), 1054 (1998). [14] W. Song, E.E. Méndez, V. Kuznetsov and B. Nielsen, Appl. Phys. Lett, 82(10), 1568 (2003).

2nd NanoSpain Worshop March 14-17, 2005 Barcelona-Spain