shot noise in mesoscopic conductors - Low Temperature Laboratory

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128 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 A similar problem, a ballistic degenerate conductor in the presence of a nearby gate, was posed by Naveh et al. [269]. We conclude this subsection by recalling that the e!ect of interactions on noise arises not only in long ballistic structures but already in samples which are e!ectively zero dimensional, like resonant tunneling diodes [78] or in quantum point contacts [98]. The added complication in extended structures arises from the long-range nature of the Coulomb interaction. 6.6. Boltzmann}Langevin method for shot noise suppression in chaotic cavities with diwusive boundary scattering Now we turn to the classical derivation of the -shot noise suppression in chaotic cavities. In standard cavities, which are regular objects, the chaotic dynamics arises due to the complicated shape of a surface. Thus, scattering at the surface is deterministic and in an individual ensemble member scattering along the surface of the cavity is noiseless. Thus, it is not obvious how to apply the Boltzmann}Langevin equation. However, recently a model of a random billiard } a circular billiard with di!usive boundary scattering } was proposed [270,271] to emulate the behavior of chaotic cavities. It turned out that the model can be relatively easily dealt with, and Refs. [270,271] used it to study spectral and eigenfunction properties of closed systems. Ref. [99] suggests that the same model may be used to study the transport properties of the open chaotic cavities and presents the theory of shot noise based on the Boltzmann}Langevin approach. We consider a circular cavity of radius R connected to the two reservoirs via ideal leads; the angular positions of the leads are ! /2(( # /2 (left) and ! /2(( /2 (right), see Fig. 34; is the polar angle. The contacts are assumed to be narrow, ;1, though the numbers of the transverse channels, N "p R /, are still assumed to be large compared to 1. Inside the cavity, motion is ballistic, and the average distribution function fM (r, n) obeys the equation nfM (r, n)"0 . (242) At the surface (denoted by ) we can choose a di!usive boundary condition: the distribution function of the particles backscattered from the surface is constant (independent of n) and "xed by the condition of current conservation, fM (r, n)" (Nn) fM (r, n)dn, Nn(0 , (243) Nn where r3, N is the outward normal to the surface, and dn"1. Furthermore, we assume that the electrons coming from the leads are described by the equilibrium distribution functions, and are Earlier, a similar model was numerically implemented to study spectral statistics in closed square billiards [272,273]. This is the simplest possible boundary condition of this kind. For a review, see Ref. [274]. We expect similar results for any other di!usive boundary condition.
Fig. 34. Geometry of the chaotic cavity with di!usive boundary scattering. emitted uniformly into all directions. Explicitly, denoting the cross-sections of the left and the right leads by and , we have fM (r, n)"f , r3 , Nn(0 . (244) Now we can "nd the average distribution function. Since motion away from the boundary is ballistic, the value of the distribution function, Eq. (242), at a point away from the boundary, is determined by the distribution function at the surface associated with the trajectory that reaches this point after a scattering event at the surface. With the boundary conditions (243) and (244), we can then derive an integral equation for fM (), fM () " 1 4 fM () sin ! d (245) 2 subject to the additional conditions fM () "f . This exact equation may be considerably simpli"ed in the limit of narrow leads, ;1. Integrals of the type F()d can now be replaced by F(0). This gives for the distribution function fM ()" f # f # # g(0)!g( ) ( ! ) 4 ( # ) ( f !f ) with the notation # g()!g(! ) 4 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 129 ( # ) ( f !f ) , (246) g()" cos l 1 " (3!6#2), 0442 . l 12 The "rst part of the distribution function (246) does not depend on energy and corresponds to the random matrix theory (RMT) results for the transport properties. The second two terms on
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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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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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Physically, this corresponds to ele
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Our starting point is Eq. (51), whi
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