shot noise in mesoscopic conductors - Low Temperature Laboratory

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104 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 where and are transition rates through the left barrier, from the reservoir to the well and from the well to the reservoir, respectively. We have " A dE dE (E !E ) f (E #E #e;!e
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 105 Up to now, we have only discussed average quantities. The idea of the Langevin approach is that the (current) #uctuations can be calculated from the same rate equations (189), if random currents (Langevin sources) are added to their right-hand sides. We write QQ "I ! (Q !Q )# (t) , QQ "I # (Q !Q )# (t) , where the Langevin sources (t) have the following properties. They are zero on average, (t)"0, "L, R. Furthermore, they are correlated only for the same barrier, and the correlation function describes Poissonian shot noise at each barrier, (192) (t) (t)"eI(t!t) , (193) where I is the average current. To "nd the noise power, we do not need to specify higher cumulants of the Langevin sources. As we mentioned above, the de"nition (193) is intuitive rather than the result of a formal derivation. However, the results we obtain in this way coincide with those found by ensemble averaging the quantum-mechanical results. Eqs. (192) are linear and can be easily solved. The average current is I"( I # I )/ , (194) where again " # .The"nite frequency shot noise power is found to be c#c 2 S()"2eI # c #() c c c ! , (195) with the de"nition c,c #c . For zero frequency, Eq. (195) gives the result (78), F"(#)/. For high frequencies < we have S()"2eI(c#c)/c. This expression can be obtained from Eq. (191) based on the assumption that the particle currents QQ and QQ #uctuate independently and according to the Poisson shot noise. The crossover frequency between these regimes is &/: In accordance with general expectations, the frequency dependence of the shot noise is governed by time-scales inherent to the conductor. We note "nally that for a symmetric double barrier, c "c and " , the noise power (195) is frequency independent and equal to S()"eI. Note also that Eq. (195) does not contain the high-frequency Nyquist noise (129), which is proportional to : zero-point noise is quantum-mechanical and cannot be reproduced by a classical discussion. The classical derivation of Eq. (195), based on the master equation approach, was given independently by Chen and Ting [134] (c "c , zero frequency limit) and by Davies et al. [54] (general case), and later by Chen [221] (c "c , arbitrary frequency) and MuK ller et al. [222] ("0). Sun and Milburn [223,224] and Milburn [225] developed a quantum master equation approach and also derived the same result (195). The Langevin approach was applied to doublebarrier structures in Ref. [226]; here we have given an extended version of the derivation. We also remark that the frequency dependence of the noise of the double-barrier structure (195) was obtained by Runge [57] (c "c ) and by Lund B+ and Galperin [58] (general case) using the non-equilibrium Green's functions method.
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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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and Lee et al. [340] obtain in this
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