shot noise in mesoscopic conductors - Low Temperature Laboratory

More documents

Recommendations

Info

122 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 6.3.2. Electron}phonon interactions In contrast to electron}electron scattering, electron}phonon interactions do change the total momentum of electrons and cause a "nite resistance and noise by themselves. Noise in ballistic, one-channel quantum wires due to electron}phonon interactions within the Boltzmann}Langevin approach was studied by Gurevich and Rudin [253,254], who start directly from Eq. (212) with the electron}phonon collision integral on the right-hand side (no impurity scattering). They consider only the situation of weak interactions, when the electron distribution function is not modi"ed by inelastic scattering. They discover that, for this case, the main e!ect is the absence of noise when the Fermi energy lies below the threshold energy E "2p s, where s is the sound velocity. The appearance of this threshold is due to the fact that the maximal wave vector of acoustic phonons which interact with electrons is 2p /. ForE 'E shot noise grows. The suppression of shot noise in long wires, which is a consequence of the equilibration of the electron distribution function due to strong interactions, was beyond the scope of Refs. [253,254]. In disordered systems, one has to take into account three e!ects. First, elastic scattering modi"es the electron}phonon collision integral [255], which assumes di!erent forms depending on the spatial dimension and degree of disorder. Then, it a!ects the distribution function of electrons. Finally, interactions modify the resistance of the sample, and the expression for the shot noise does not have the simple form of Eq. (223). The distribution function of phonons, which enters the electron}phonon collision integral, must, in principle, be found from the Boltzmann equation for phonons, which couples with that for electrons. The standard approximations used to overcome these di$culties and to get reasonable analytical results are as follows. First, for low temperatures, the contribution to the resistance due to electron}phonon collisions is much smaller than that of electron-impurity scattering. Thus, one assumes that Eq. (223) still holds, and electron}phonon collisions only modify the distribution function for electrons and the form of the collision integral. Furthermore, phonons are assumed to be in equilibrium at the lattice temperature. This approach was taken by Nagaev, who calculates the e!ects of electron}phonon scattering on the shot noise of metallic di!usive wires for the case when electron}electron scattering is negligible [75], and subsequently for the case of hot electrons [248]. In addition, careful numerical studies of the role of the electron}phonon interaction in noise in metallic di!usive conductors (starting from the Boltzmann}Langevin approach) are performed by Naveh et al. [256], and Naveh [145]. The results depend on the relation between temperature, applied voltage, and electron}phonon interaction constant, and we refer the reader to Refs. [75,248,256,145] for details. The only feature we want to mention here is that for constant voltage e
one "rst goes from the non-interacting regime F" to the hot-electron regime, F"3/4. For even longer wires electron}phonon collisions play a role, and the Fano factor decreases down to zero. 6.4. Frequency dependence of shot noise Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 123 While discussing the classical approach to the shot noise suppression, we explicitly assumed that the sample is locally charge neutral: charge pile-up is not allowed in any volume of any size. As a result, we obtained white (frequency independent) shot noise. In reality, however, there is always a "nite (though small) screening radius, which in the case when the system is locally three-dimensional has the form "(4e). As Naveh et al. [140] and Turlakov [257] point out, if one of the dimensions of a disordered sample becomes comparable with , the pile-up of the charge may modify the frequency dependence of noise, though it leaves the zero-frequency noise power unchanged. If all the dimensions of the sample exceed the screening radius (which is typically the case for metallic, and often also for semiconducting) mesoscopic systems, the charge pile-up inside the sample is negligible, and noise stays frequency independent until at least the plasma frequency, which in three-dimensional structures is very high. The situation is di!erent if the sample is capacitively connected to an external gate. As we have seen in the framework of the scattering approach (Section 3), the fact that the sample is now charged, leaves the zero-frequency noise unchanged, but strongly a!ects the frequency dependence of the shot noise. An advantage of the Boltzmann}Langevin approach is that it can treat these e!ects analytically, calculating the potential distribution inside the sample and making use of it to treat the current #uctuations. The general program is as follows. Instead of the charge neutrality condition, one uses the full Poisson equation, relating potential and density #uctuations inside the sample. In their turn, density #uctuations are related to the current #uctuations via the continuity equation. Finally, one expresses the current #uctuations via those of the potential and the Langevin sources. Thus, the system of coupled partial di!erential equations with appropriate boundary conditions needs to be solved. The solution is strongly geometry dependent and has not been written down for an arbitrary geometry. Particular cases, with a simple geometry, were considered by Naveh et al. [140,142], Nagaev [141,143], and Naveh [144,145]. Without even attempting to give derivations, we describe here the main results, referring the reader to these papers for more details. A conductor in proximity to a gate can be charged vis-a` -vis the gate. We can view the conductor and the gate as the two plates of a capacitor. In the limit where the screening length is much larger than the wire radius, only a one-dimensional theory is needed. It is this atypical situation which is considered here. For a wire of cross-section A and a geometrical capacitance c per unit length its low-frequency dynamics is characterized by the electrochemical capacitance c"c#(eA) which is the parallel addition of the geometrical capacitance and the quantum capacitance (eA). Here we have assumed that the potential is uniform both along the wire and more importantly also in the transverse direction of the wire. Any charge accumulated in the wire can dynamically relax via the reservoirs connected to the wire and via the external circuit For a discussion of electrochemical capacitance and ac conductance see Refs. [157,165]; see also Section 3.
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:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
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 43 and 44:
Page 45 and 46:
Page 47 and 48:
we include weak localization e!ects
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
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 65 and 66:
Page 67 and 68:
Page 69 and 70:
Physically, this corresponds to ele
Page 71 and 72: Ya.M. Blanter, M. Bu( ttiker / Phys
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 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 121: Ya.M. Blanter, M. Bu( ttiker / Phys
Page 125 and 126: non-thermalized electrons, the high
Page 129 and 130: Fig. 34. Geometry of the chaotic ca
Page 133 and 134: which describes strictly ballistic
Page 141 and 142: approximately e/C. Between the peak
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
show all

shot noise in mesoscopic conductors - Low Temperature Laboratory

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?