shot noise in mesoscopic conductors - Low Temperature Laboratory

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118 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 Multiplying Eq. (218) by and integrating over the whole volume we obtain the #uctuation of the current through the contact n. The potential #uctuations (r) actually play no role and are eliminated due to the boundary condition that they vanish at the contacts. At zero temperature, and to linear order in the applied voltage, this is exact: Internal potential #uctuations play a role only in the non-linear voltage dependence of the shot noise and in its temperature dependence. Taking into account the form of the correlation function (219), we "nd the noise power, S "4 dr (r) (r)(r) . (223) Eq. (223), together with Eq. (221) for the distribution function fM , is the general result for the multi-terminal noise power within the classical approach. At equilibrium (r)"k ¹, and Eq. (223) reproduces the #uctuation-dissipation theorem. We next apply Eq. (223) to calculate noise suppression in metallic di!usive wires, for the case when the inelastic processes are negligible, IM "0. We consider a wire of length ¸ and width =;¸, situated along the axis x between the point x"0 (reservoir L) and x"¸ (R). The voltage < is applied to the left reservoir. There are only two characteristic potentials, "1! "1!x/¸ , (224) which obey the di!usion equation and do not depend on the transverse coordinate. The average distribution function is found as f (x)" (x) f # (x) f , and thus the quantity for zero temperature is expressed as (x)"e< (x)[1! (x)] . Subsequently, we "nd the conductance G "=/¸, and the shot noise S " 4e=< ¸ dx x ¸1!x ¸ "2eI . (225) 3 As we mentioned earlier, this expression is due to Nagaev [75]. The Fano factor is , in accordance with the results found using the scattering approach (Section 2). For purely elastic scattering the distribution function fM in an arbitrary geometry quite generally can be written as fM (r, E)" (r) f (E) . (226) This facilitates the progress for multi-probe geometries. Sukhorukov and Loss [113,114] obtain general expressions for the multi-terminal noise power and use them to study the Hanbury Brown}Twiss e!ect in metallic di!usive conductors. The quantum-mechanical theory of the same e!ect can be found in Ref. [78]. We use two-dimensional notations, d"2.
6.3. Interaction ewects Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 119 Interaction e!ects are relatively easy to deal with in the Boltzmann}Langevin approach, in contrast to the di$culties encountered by the scattering theory. 6.3.1. Electron}electron interactions An important feature of electron}electron interactions is that they do not change the total momentum of the electron system. Generally, therefore, electron}electron scattering alone cannot cause transport, and in particular it cannot cause noise. Technically, this is manifested in the fact that the form of current}current #uctuations is given by the same expression (223), as in the non-interacting case. However, electron}electron scattering processes alter the distribution function fM , and thus the value of the shot noise. The inelastic collision integral for electron}electron interactions IM [ f ] (which we denote IM [ f ]) generally has the form IM [ f (E)]" dE d K() f (E) f (E)[1!f (E!)][1!f (E#)] !f (E!) f (E#)[1!f (E)][1!f (E)] , (227) where the kernel K() for disordered systems must be found from a microscopic theory. For three-dimensional metallic di!usive systems it was obtained by Schmid [245]; for two- and one-dimensional systems the zero-temperature result may be found in Ref. [246]. For "nite temperatures, a self-consistent treatment is needed [247]. This kernel turns out to be a complicated function of disorder and temperature. In particular, in 1D for zero temperature it diverges as K()J for P0. The strength of the interaction is characterized by a time , which we call the electron}electron scattering time. Thus, one needs now to solve Eq. (221) for fM , calculate the quantity (r), and substitute it into the expression (223) to obtain the noise. Up to now, only two limiting cases have been discussed analytically. First, for weak interactions D/¸< (with ¸ being the typical size of the system) the collision integral in Eq. (221) may be treated perturbatively. Since electron}electron interactions are of minor importance, the distribution function fM in this case is still given by Eq. (226). Nagaev [248], assuming a speci"c form of K(), found that the shot noise power is enhanced due to the electron}electron interactions compared to the non-interacting result. Another regime where progress is possible is that of strong scattering, D/¸;. In this situation, electrons undergo many scattering events before leaving the system, and one can expect that the distribution function at every point is close to the equilibrium distribution. Indeed, in the leading order, the di!usion term in Eq. (221) can be neglected, and the distribution function must then be one for which the collision integral is zero. The collision integral (227) vanishes identically for the Fermi distribution, and thus the distribution function assumes the form of a local This is only correct if Umklapp processes can be neglected. The in#uence of Umklapp processes on shot noise in mesoscopic systems has not been investigated.
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Ya.M. Blanter, M. BuK ttiker / Phys
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8. Concluding remarks, future prosp
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text; they are usually extensions o
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such reviews are not readily availa
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Ya.M. Blanter, M. Bu( ttiker / Phys
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The above considerations are, of co
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and the initial state of our two-pa
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2.3.3. Multi-terminal case We consi
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Note that in the absence of time-de
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equilibrium and shot noise. For low
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and S" e A e< dE ¹(E )[1!¹
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arrier, situated in the region !w/2
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we include weak localization e!ects
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Edge channels in the quantum Hall e
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Fig. 19. (a) Geometry of a ring thr
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a manifestation of interference e!e
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