shot noise in mesoscopic conductors - Low Temperature Laboratory

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140 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 the asymmetry of the spin-up and spin-down propagation, and the asymmetry of the barrier, respectively. For a given spin projection, the current is proportional to R R /(R #R ), while the shot noise is proportional to R R (R#R )/(R #R ). Here R and R must be taken for each spin projection separately; the total current (shot noise) is then expressed as the sum of the currents (shot noises) of spin-up and spin-down electrons. Evaluating in this way the Fano factor, we obtain F "(1#)/(1#) , (259) tt which is precisely the double-barrier Fano factor (78), since the Fano factors for spin-up and spin-down electrons are the same. Similarly, for the anti-parallel (!N!) orientation we write R "R , R "R , t s R "R , and R "R . The Fano factor is t s 1# F " ts (1#) ## 1 1 # (#)1# # , (260) and depends now on . In the strongly asymmetric junction, ,
approximately e/C. Between the peaks (in the valleys), as we implicitly assumed, the zerotemperature current is due to quantum tunneling, and is exponentially small. We concluded therefore, that the Fano factor is 1 } the shot noise is Poissonian. In reality, however, there are cotunneling processes } virtual transitions via the high-lying state in the dot } which are not exponentially suppressed and thus give the main contribution to the current. Cotunneling is a genuinely quantum phenomenon, and cannot be obtained by means of a classical approach. It is clear that the cotunneling processes may modify the Fano factor in the valleys; moreover, it is a good opportunity to study quantum e!ects in the shot noise. This problem, among many others, remains unaddressed. 7.2. Anderson and Kondo impurities Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 141 7.2.1. Anderson impurity model In the context of mesoscopic physics, the Anderson impurity model describes a resonant level with a Hubbard repulsion. It is sometimes taken as a model of a quantum dot. Commonly the entire system is described by the tight-binding model with non-interacting reservoirs and an interaction ;n( s n( t on the site i"0 (resonant impurity), with n( s and n( t being the operators of the number of electrons on this site with spins down and up, respectively. Tunneling into the dot is described, as in the non-interacting case, by partial tunneling widths and , which may be assumed to be energy independent. The on-site repulsion is important (in the linear regime) when ;
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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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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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Page 93 and 94: with D ,¹ (2!¹ ). Performing t
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