shot noise in mesoscopic conductors - Low Temperature Laboratory

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8 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 To explain the origin of shot noise we consider "rst a "ctitious experiment in which only one particle is incident upon a barrier. At the barrier the particle is either transmitted with probability ¹ or re#ected with probability R"1!¹. We now introduce the occupation numbers for this experiment. The incident state is characterized by an occupation number n , the transmitted state is characterized by an occupation number n and the re#ected state by an occupation number n .If we repeat this experiment many times, we can investigate the average of these occupation numbers and their #uctuations away from the average behavior. In our experiments the incident beam is occupied with probability 1 and thus n "1. However, the transmitted state is occupied only with probability ¹ and is empty with probability R. Thus n "¹ and n "R. The#uctuations away from the average occupation number, can be obtained very easily in the following way: The mean-squared #uctuations in the incident beam vanish, (n !n )"0. To "nd the mean-squared #uctuations in the transmitted and re#ected state, we consider the average of the product of the occupation numbers of the transmitted and re#ected beam n n . Since in each event the particle is either transmitted or re#ected, the product n n vanishes for each experiment, and hence the average vanishes also, n n "0. Using this we "nd easily that the mean squares of the transmitted and re#ected beam and their correlation is given by (n )"(n )"!n n "¹R , (2) where we have used the abbreviations n "n !n and n "n !n . Such #uctuations are called partition noise since the scatterer divides the incident carrier stream into two streams. The partition noise vanishes both in the limit when the transmission probability is 1 and in the limit when the transmission probability vanishes, ¹"0. In these limiting cases no partitioning takes place. The partition noise is maximal if the transmission probability is ¹" . Let us next consider a slightly more sophisticated, but still "ctitious, experiment. We assume that the incident beam is now occupied only with probability f. Eventually f will just be taken to be the Fermi distribution function. The initial state is empty with probability 1!f. Apparently, in this experiment the average incident occupation number is n "f, and since the particle is transmitted only with probability f ¹ and re#ected with probability fR, we have n "f ¹ and n "fR. Since we are still only considering a single particle, we have as before that in each event the product n n vanishes. Thus we can repeat the above calculation to "nd (n )"¹f (1!¹f ) , (3) (n )"Rf (1!Rf ) , (4) n n "!¹Rf . (5) If we are in the zero-temperature limit, f"1, we recover the results discussed above. Note that now even in the limit ¹"1 the #uctuations in the transmitted state do not vanish, but #uctuate like the incident state. For the transmitted stream, the factor (1!¹f ) can be replaced by 1, if either the transmission probability is small or if the occupation probability of the incident carrier stream is small. We can relate the above results to the #uctuations of the current in a conductor. To do this we have to put aside the fact that in a conductor we deal not as above just with events of a single
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 9 carrier but with a state which may involve many (indistinguishable) carriers. We imagine a perfect conductor which guides the incident carriers to the barrier, and imagine that we have two additional conductors which guide the transmitted and re#ected carriers away from the conductor, such that we can discuss, as before, incident, transmitted and re#ected currents separately. Furthermore, we want to assume that we have to consider only carriers moving in one direction with a velocity v(E) which is uniquely determined by the energy E of the carrier. Consider next the average incident current. In a narrow energy interval dE, the incident current is dI (E)"ev(E)d(E), where d(E) is the density of carriers per unit length in this energy range. The density in an energy interval dE is determined by the density of states (per unit length) (E)"d/dE times the occupation factor n (E) of the state at energy E. We thus have d(E)"n (E)(E)dE. The density of states in our perfect conductors is (E)"1/(2v(E)). Thus the incident current in a narrow energy interval is simply dI (E)" e 2 n (E)dE . (6) This result shows that there is a direct link between currents and the occupation numbers. The total incident current is I "(e/2)n (E)dE and on the average is given by I "(e/2) f (E)dE. Similar considerations give for the average transmitted current I "(e/2) f (E)¹ dE and for the re#ected current I "(e/2) f (E)R dE. Current #uctuations are dynamic phenomena. The importance of the above consideration is that it can now easily be applied to investigate timedependent current #uctuations. For occupation numbers which vary slowly in time, Eq. (6) still holds. The current #uctuations in a narrow energy interval are at long times determined by dI (E, t)"(e/2)n (E, t)dE where n (E, t) is the time-dependent occupation number of states with energy E. A detailed derivation of the connection between currents and occupation numbers is the subject of an entire section of this Review. We are interested in the low-frequency current noise and thus we can Fourier transform this equation. In the low-frequency limit we obtain I()"(e/2)dEn(E, E#). As a consequence the #uctuations in current and the #uctuations in occupation number are directly related. In the zero frequency limit the current noise power is S "edES (E). In each small energy interval particles arrive at a rate dE/(2) and contribute, with a mean square #uctuation, as given by one of Eqs. (3)}(5), to the noise power. We have S (E)"(1/)nn. Thus the #uctuation spectra of the incident, transmitted, and re#ected currents are S "2 e dEf(1!f ) , (7) 2 S "2 e dE ¹f (1!¹f ) , (8) 2 S "2 e dERf(1!Rf ) . (9) 2
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Edge channels in the quantum Hall e
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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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with D ,¹ (2!¹ ). Performing t
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For completeness, we mention that i
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We note here, however, that there i
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The noise power (201) is strongly v
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Fig. 32. Experimental results of Ku
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where Ya.M. Blanter, M. Bu( ttiker
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where the quantity is expressed th
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6.3. Interaction ewects Ya.M. Blant
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one "rst goes from the non-interact
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non-thermalized electrons, the high
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Fig. 34. Geometry of the chaotic ca
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which describes strictly ballistic
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approximately e/C. Between the peak
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Boltzmann}Langevin approach (see Se
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thermal to shot noise. It is, howev
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Note added in proof Ya.M. Blanter,
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and Lee et al. [340] obtain in this
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shot noise in mesoscopic conductors - Low Temperature Laboratory

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