shot noise in mesoscopic conductors - Low Temperature Laboratory

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18 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 Table 2 Output probabilities for one particle incident in each input arm (from Ref. [8]) Probability Classical Bosons Fermions P(2, 0) R¹ R¹(1#J) R¹(1!J) P(1, 1) R#¹ R#¹!2R¹J R#¹#2R¹J P(0, 2) R¹ R¹(1#J) R¹(1!J) particles went which way at the scatterer and there is thus no interference. The outcome is classical: P(1,1)"R for an initial state with a spin up in arm 1 and a spin down in arm 2. For the same state we have P(1,1)"¹. Now let us assume that there is no way of detecting the spin state of the outgoing particles. For a given initial state we have P(1, 1)" P(1,1) where and are spin variables. If we consider all possible incident states with equal probability, we "nd P(1, 1)"¹#R#¹RJ i.e. a result with an interference contribution which is only half as large as given in Eq. (26). For further discussion, see Appendix B. The scattering experiments considered above assume that we can produce one or two particle states either in a single mode or by exciting many modes. Below we will show that thermal sources, the electron reservoirs which are of the main interest here, cannot be described in this way: In the discussion given here the ground state is the vacuum (a state without carriers) whereas in an electrical conductor the ground state is a many-electron state. 2.3. The scattering approach The idea of the scattering approach (also referred to as Landauer approach) is to relate transport properties of the system (in particular, current #uctuations) to its scattering properties, which are assumed to be known from a quantum-mechanical calculation. In its traditional form the method applies to non-interacting systems in the stationary regime. The system may be either at equilibrium or in a non-equilibrium state; this information is introduced through the distribution functions of the contacts of the sample. To be clear, we consider "rst a two-probe geometry and particles obeying Fermi statistics (having in mind electrons in mesoscopic systems). Eventually, the generalization to many probes and Bose statistics is given; extensions to interacting problems are discussed at the end of this Section. In the derivation we essentially follow Ref. [9]. 2.3.1. Two-terminal case; current operator We consider a mesoscopic sample connected to two reservoirs (terminals, probes), to be referred to as `lefta (L) and `righta (R). It is assumed that the reservoirs are so large that they can be To avoid a possible misunderstanding, we stress that the long-range Coulomb interaction needs to be taken into account when one tries to apply the scattering approach for the description of systems in time-dependent external "elds, or "nite-frequency #uctuation spectra in stationary "elds. On the other hand, for the description of zero-frequency #uctuation spectra in stationary xelds, a consistent theory can be given without including Coulomb e!ects, even though the #uctuations themselves are, of course, time-dependent and random.
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 19 characterized by a temperature ¹ and a chemical potential ; the distribution functions of electrons in the reservoirs, de"ned via these parameters, are then Fermi distribution functions f (E)"[exp[(E! )/k ¹ ]#1], "L, R (see Fig. 2). We must note at this stage, that, although there are no inelastic processes in the sample, a strict equilibrium state in the reservoirs can be established only via inelastic processes. However, we consider the reservoirs (the leads) to be wide compared to the typical cross-section of the mesoscopic conductor. Consequently, as far as the reservoirs are concerned, the mesoscopic conductor represents only a small perturbation, and describing their local properties in terms of an equilibrium state is thus justi"ed. We emphasize here, that even though the dynamics of the scattering problem is described in terms of a Hamiltonian, the problem which we consider is irreversible. Irreversibility is introduced in the discussion, since the processes of a carrier leaving the mesoscopic conductor and entering the mesoscopic conductor are unrelated, uncorrelated events. The reservoirs act as sources of carriers determined by the Fermi distribution but also act as perfect sinks of carriers irrespective of the energy of the carrier that is leaving the conductor. Far from the sample, we can, without loss of generality, assume that transverse (across the leads) and longitudinal (along the leads) motion of electrons are separable. In the longitudinal (from left to right) direction the system is open, and is characterized by the continuous wave vector k l.Itis advantageous to separate incoming (to the sample) and outgoing states, and to introduce the longitudinal energy E l"k l /2m as a quantum number. Transverse motion is quantized and described by the discrete index n (corresponding to transverse energies E _ , which can be di!erent for the left and right leads). These states are in the following referred to as transverse (quantum) channels. We write thus E"E #E l. Since E l needs to be positive, for a given total energy E only a "nite number of channels exists. The number of incoming channels is denoted N (E) in the left and right lead, respectively. We now introduce creation and annihilation operators of electrons in the scattering states. In principle, we could have used the operators which refer to particles in the states described by the quantum numbers n, k l. However, the scattering matrix which we introduce below, relates current amplitudes and not wave function amplitudes. Thus, we introduce operators a( (E) and a( (E) which create and annihilate electrons with total energy E in the transverse channel n in the left lead, which are incident upon the sample. In the same way, the creation bK (E) and annihilation bK (E) Fig. 2. Example of a two-terminal scattering problem for the case of one transverse channel.
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Physically, this corresponds to ele
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Our starting point is Eq. (51), whi
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introducing the transmission coe$ci
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investigated experimentally. A self
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and the non-equilibrium resistance
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the sample. These electrons are des
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with D ,¹ (2!¹ ). Performing t
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For completeness, we mention that i
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We note here, however, that there i
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The noise power (201) is strongly v
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Fig. 32. Experimental results of Ku
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where Ya.M. Blanter, M. Bu( ttiker
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where the quantity is expressed th
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6.3. Interaction ewects Ya.M. Blant
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one "rst goes from the non-interact
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non-thermalized electrons, the high
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Fig. 34. Geometry of the chaotic ca
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which describes strictly ballistic
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approximately e/C. Between the peak
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Boltzmann}Langevin approach (see Se
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thermal to shot noise. It is, howev
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Note added in proof Ya.M. Blanter,
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and Lee et al. [340] obtain in this
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