shot noise in mesoscopic conductors - Low Temperature Laboratory

More documents

Recommendations

Info

26 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 (where as before the trace is taken over transverse channel indices, and N is the number of channels in the lead ), we "nd S " ek ¹ dE Rf ! RE [2N !Tr(s s #s s )] . (53) This is the equilibrium, or Nyquist}Johnson noise. In the approach discussed here it is a consequence of the thermal #uctuations of occupation numbers in the reservoirs. Comparing Eqs. (45) and (53), we see that S "2k ¹(G #G ) . (54) This is the manifestation of the #uctuation-dissipation theorem: equilibrium #uctuations are proportional to the corresponding generalized susceptibility, in this case to the conductance. For the time-reversal case (no magnetic "eld) the conductance matrix is symmetric, and Eq. (54) takes the form S "4k ¹G , which is familiar for the two-terminal case, S"4k ¹G, with G being the conductance. From Eq. (54) we see that the #uctuation spectrum of the mean squared current at a contact is positive (since G '0) but that the current}current correlations of the #uctuations at di!erent probes are negative (since G (0). The sign of the equilibrium current}current #uctuations is independent of statistics: Intensity}intensity #uctuations for a system of bosons in which the electron reservoirs are replaced by black-body radiators are also negative. We thus see that equilibrium noise does not provide any information of the system beyond that already known from conductance measurements. Nevertheless, the equilibrium noise is important, if only to calibrate experiments and as a simple test for theoretical discussions. Experimentally, a careful study of thermal noise in a multi-terminal structure (a quantum Hall bar with a constriction) was recently performed by Henny et al. [12]. Within the experimental accuracy, the results agree with the theoretical predictions. 2.4.2. Shot noise: zero temperature We now consider noise in a system of fermions in a transport state. In the zero temperature limit the Fermi distribution in each reservoir is a step function f (E)"( !E). Using this we can rewrite Eq. (52) as S " e 2 dE Tr[s s s s ] f (E)[1!f (E)]#f (E)[1!f (E)] . (55) We are now prepared to make two general statements. First, correlations of the current at the same lead, S , are positive. This is easy to see, since their signs are determined by positively de"ned quantities Tr[s s s s ]. The second statement is that the correlations at diwerent leads, S with For bosons at zero temperature one needs to take into account Bose condensation e!ects. These quantities are called `noise conductancesa in Ref. [9].
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 27 O, are negative. This becomes clear if we use the property s s "0 and rewrite Eq. (55) as S "! e dE Tr s s f (E) s s f (E) . The integrand is now positively de"ned. Of course, current conservation implies that if all cross-correlations S are negative for all di!erent from , the spectral function S must be positive. Actually, these statements are even more general. One can prove that cross-correlations in the system of fermions are generally negative at any temperature, see Ref. [9] for details. On the other hand, this is not correct for a system of bosons, where under certain conditions cross-correlations can be positive. 2.4.3. Two-terminal conductors Let us now consider the zero-temperature shot noise of a two-terminal conductor. Again we denote the leads as left (L) and right (R). Due to current conservation, we have S,S " S "!S "!S . Utilizing the representation of the scattering matrix (30), and taking into account that the unitarity of the matrix s implies rr#tt"1, we obtain after some algebra S" e Tr(rrtt)e< , (56) where the scattering matrix elements are evaluated at the Fermi level. This is the basis invariant relation between the scattering matrix and the shot noise at zero temperature. Like the expression of the conductance, Eq. (38), we can express this result in the basis of eigen-channels with the help of the transmission probabilities ¹ and re#ection probabilities R "1!¹ , S " e< ¹ (1!¹ ) . (57) We see that the non-equilibrium (shot) noise is not simply determined by the conductance of the sample. Instead, it is determined by a sum of products of transmission and re#ection probabilities of the eigen-channels. Only in the limit of low-transparency ¹ ;1inall eigen-channels is the shot noise given by the Poisson value, discussed by Schottky, S " e< ¹ "2eI . (58) It is clear that zero-temperature shot noise is always suppressed in comparison with the Poisson value. In particular, neither closed (¹ "0) nor open (¹ "1) channels contribute to shot noise; the maximal contribution comes from channels with ¹ "1/2. The suppression below the This statement is only valid for non-interacting systems. Interactions may cause instabilities in the system, driving the noise to super-Poissonian values. Noise in systems with multi-stable current}voltage characteristics (caused, for example, by a non-trivial structure of the energy bands, like in the Esaki diode) may also be super-Poissonian. These features are discussed in Section 5.
Page 1 and 2: Ya.M. Blanter, M. BuK ttiker / Phys
Page 3 and 4: 8. Concluding remarks, future prosp
Page 5 and 6: text; they are usually extensions o
Page 7 and 8: such reviews are not readily availa
Page 9 and 10: Ya.M. Blanter, M. Bu( ttiker / Phys
Page 11 and 12: The above considerations are, of co
Page 17 and 18: and the initial state of our two-pa
Page 23 and 24: 2.3.3. Multi-terminal case We consi
Page 25: Note that in the absence of time-de
Page 29 and 30: equilibrium and shot noise. For low
Page 39 and 40: and S" e A e< dE ¹(E )[1!¹
Page 41 and 42: arrier, situated in the region !w/2
Page 47 and 48: we include weak localization e!ects
Page 59 and 60: Edge channels in the quantum Hall e
Page 61 and 62: Fig. 19. (a) Geometry of a ring thr
Page 63 and 64: a manifestation of interference e!e
Page 69 and 70: Physically, this corresponds to ele
Page 73 and 74: Our starting point is Eq. (51), whi
Page 77 and 78:
introducing the transmission coe$ci
Page 79 and 80:
investigated experimentally. A self
Page 81 and 82:
Ya.M. Blanter, M. Bu( ttiker / Phys
Page 83 and 84:
Page 85 and 86:
and the non-equilibrium resistance
Page 87 and 88:
the sample. These electrons are des
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
with D ,¹ (2!¹ ). Performing t
Page 95 and 96:
Page 97 and 98:
For completeness, we mention that i
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
We note here, however, that there i
Page 109 and 110:
Page 111 and 112:
The noise power (201) is strongly v
Page 113 and 114:
Fig. 32. Experimental results of Ku
Page 115 and 116:
where Ya.M. Blanter, M. Bu( ttiker
Page 117 and 118:
where the quantity is expressed th
Page 119 and 120:
6.3. Interaction ewects Ya.M. Blant
Page 121 and 122:
Page 123 and 124:
one "rst goes from the non-interact
Page 125 and 126:
non-thermalized electrons, the high
Page 127 and 128:
Page 129 and 130:
Fig. 34. Geometry of the chaotic ca
Page 131 and 132:
Page 133 and 134:
which describes strictly ballistic
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
approximately e/C. Between the peak
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Boltzmann}Langevin approach (see Se
Page 151 and 152:
thermal to shot noise. It is, howev
Page 153 and 154:
Note added in proof Ya.M. Blanter,
Page 155 and 156:
and Lee et al. [340] obtain in this
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
show all

shot noise in mesoscopic conductors - Low Temperature Laboratory

Create successful ePaper yourself

Delete template?

Save as template?