shot noise in mesoscopic conductors - Low Temperature Laboratory

More documents

Recommendations

Info

80 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 In what follows, the #uctuation dissipation theorem for the non-interacting system also ensures this theorem for the interacting system. Current conservation is restored only if interactions are taken into account. Below we assume that charging e!ects are the only manifestation of interactions [157,131]. We "rst present the general result which is, within the limitations stated above, valid for arbitrary frequencies. We then consider in detail the low-frequency expansion of this result which can be expressed in physically appealing quantities: an electrochemical capacitance and a charge relaxation resistance. 3.4.1. General result Our starting point are the particle current operators in the left and right lead (43), IK (t) and IK (t). Their #uctuation spectra are determined by Eq. (135). Now we must take into account that the total currents are in fact not just the particle currents but contain an additional contribution generated by the #uctuating electrostatic potential on the capacitor plates. We introduce the operators of the potential on the left u( (t) and right u( (t) plate. The #uctuation of the total current through the lead can be written in operator form as follows: IK (t)"IK (t)# dt (t!t)u( (t), "L, R . (137) Here we introduced IK (t)"IK (t)!I and u( (t)"u( (t)!u . Furthermore, is the response function which determines the current generated at contact in response to an oscillating potential on the capacitor plate. For the simple case considered here, it can be shown [157] that this response function is directly related to the ac conductance, Eq. (136), for non-interacting electrons, which gives the current through the lead in response to a voltage applied to the same lead, ()"!g (). The minus sign is explained by noting that the current #uctuation is the response to !u rather than u . The operators IK , and not IK , determine the experimentally measured quantities. The total current in this system is the displacement current. In the capacitance model the charge of the capacitor QK is given by QK "C(u( !u( ). Note that this is just the Poisson equation expressed with the help of a geometrical capacitance. To the extent that the potential on the capacitor plate can be described by a uniform potential this equation is valid for all frequencies. Then the #uctuations of the current through the left and right leads are IK (t)" R R QK (t)"C Rt Rt [u( (t)!u( (t)] , (138) and IK (t)"!IK (t). Thus, the conservation of the total current is assured. In contrast to the non-interacting problem, the currents to the left and right are now completely correlated. Eqs. (137) and (138) can be used to eliminate the voltage #uctuations. The result is conveniently expressed in the frequency representation, iC IK "!IK " g g !iC(g #g ) [g IK !g IK ].
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 81 The noise power S,S becomes C S()" g g !iC(g #g ) [g S #g S] . (139) Eq. (139) expresses the frequency-dependent noise spectrum of the interacting system in terms of the conductances g (), g () and the noise spectra S, S of the problem without interactions. Together with Eq. (135), this is the result of the "rst step of our calculation. Note that the noise of the capacitor depends only implicitly, through the scattering matrices and the Fermi functions, on the stationary (dc) voltage di!erence across the capacitor. Eq. (139) is thus valid independently on whether the potentials in the two reservoirs are the same or not. So far, our consideration is valid in the entire frequency range up to the frequencies at which the concept of a single potential no longer holds: ;e/d, where and d are the static susceptibility and the distance between the capacitor plates, respectively. 3.4.2. Low-frequency expansion Now we turn to the low-frequency expansion of Eq. (139), leaving only the leading term. The expansion of g can be easily obtained [157], g ()"!ie A #O() , A "! 1 2idE Rf RE Trs (E) Rs (E) , (140) RE where is the density of states per unit area and A is the area of the cross-section of the plate . The fact that the density of states can be expressed in terms of the scattering matrix is well known [160,161]. Expanding Eq. (135), we write S "ek ¹ dE !Rf RE Trs (E)Rs (E) RE #O() . Now we introduce the electrochemical capacitance, C ,C#(e A )#(e A ) , (141) and the charge relaxation resistances, R " 4eA dE !Rf RE Trs (E) Rs (E) RE . (142) These two quantities, which determine the RC-time of the mesoscopic structure, now completely specify the low-frequency noise of the capacitor. A little algebra gives [157,131] S"4k ¹C (R #R )#O() . (143) For zero temperature, the expansion of Eq. (135) starts with a term proportional to rather than . As a result, 4k ¹ is replaced by 2 in the "nal expression (143) for noise S.
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 13 and 14:
Page 15 and 16:
Page 17 and 18:
and the initial state of our two-pa
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
2.3.3. Multi-terminal case We consi
Page 25 and 26:
Note that in the absence of time-de
Page 27 and 28:
Page 29 and 30: equilibrium and shot noise. For low
Page 31 and 32: Ya.M. Blanter, M. Bu( ttiker / Phys
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: investigated experimentally. A self
Page 85 and 86: and the non-equilibrium resistance
Page 87 and 88: the sample. These electrons are des
Page 93 and 94: with D ,¹ (2!¹ ). Performing t
Page 97 and 98: For completeness, we mention that i
Page 107 and 108: We note here, however, that there i
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 123 and 124: one "rst goes from the non-interact
Page 125 and 126: non-thermalized electrons, the high
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?