Electromechanical Noise in a Diffusive Conductor

Abstract

Electrons moving in a conductor can transfer momentum to the lattice via collisions with impurities and boundaries, giving rise to a fluctuating mechanical stress tensor. The root-mean-squared momentum transfer per scattering event in a disordered metal (of dimension L greater than the mean-free path l and screening length xi) is found to be reduced below the Fermi momentum by a factor of order l/L for shear fluctuations and (xi/L)(2) for pressure fluctuations. The excitation of an elastic bending mode by the shear fluctuations is estimated to fall within current experimental sensitivity for a nanomechanical oscillator.

Carbon nanotubes as nanoelectromechanical systems

“…5 From a fundamental point of view, NEM physics is an unexplored field in which new phenomena are likely to be found. Examples include tunneling through moving barriers, 6 additional sources of noise, 7 and shuttling mechanism for transport. [8][9][10] Studies with NEMS have mostly been performed in devices made with silicon technology.…”

Momentum noise in a quantum point contact

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Ballistic elections flowing thiough a constuction can transfei momentum to the lattice and excite a Vibration of a free-standing conductor We show (both numencally and analytically) that the electromechanical noise power P does not vanish on the plateaus of quantized conductance -in contiast to the current noise The dependence of P on the constnction width can be oscillatoiy or stepwise, depending on the geometry The stepwise mcrease amounts to an approximate quantization of momentum noise DOI 10 1103/PhysRevB 66 2413XX PACS number(s) 85 85 +j, 73 23 Ad, 73 50 Td, 73 63 Rt Not long after the discovety of conductance quantization in a balhstic constnction 1 it was predicted that the quantization is noiseless ~ The time dependent current fluctuations should vanish at low tempeiatuies on the plateaus of quantized conductance and they should peak m the tiansition from one plateau to the next The conclusive expenmental venfication of this piediction followed many years latei, 3 delayed by the difficulty of ehmmating extraneous sources of noise The notion of noiseless quantum balhstic tiansport is now well estabhshed 4 The ongm of noiseless transpoit lies in the fact that the eigenvalues T n of the tiansmission matnx pioduct tt^ take only the values 0 01 l on a conductance plateau The cunent noise power at zeio tempeiatuie Ρ/ α Σ,,Γ,,(1 -Τ η ) then vanishes 5 In othei woids, cuirent fluctuations lequire paitially filled scatteimg channels, which are incompatible with a quantized conductanceIn this papei we pomt out that the notion of noiseless quantum balhstic tianspoit does not apply if one consideis momentum transfer mstead of Charge tiansfei Momentum noise cieated by an electucal cunent (socalled electiomechanical noise) has been studied m the tunneling legime 6 and m a diffusive conductor, 7 but not yet in connection with balhstic tiansport Our analysis is based on a recent scatteimg matnx repiesentation of the momentum noise powei P, accoiding to which P depends not only on the ttansmission eigenvalues but also on the eigenvectoi s 8 This makes it possible for the elections to generate noise even m the absence of partially filled scatteimg channelsThe geometiy is shown schematically m Fig l We considei a two-dimensional election gas channel m the x-y plane The width of the channel m the y duection is W and the length in the χ direction is L The channel contams a nanow constnction of length SL-^L and width ciH 7 ·^ W located at a distance L' fiom the left end (We choose x = 0 at the middle of the constnction, so that the channel extends fiom -L'<x<L -L' ) A voltage ydnves a cunent thiough the constnction, excitmg a vibiation of the channel We seek the low-frequency noise powei. , ~ t

(D of the fluctuatmg foice 8F(t) = F(t)-F that dnves the vibiation

The noise power is piopottional to the vanance of the momentum ΔΡ(ί) transfened by the electrons to the channel in a long time t We assume that the election gas is deposited on top of a doubly clamped beam extended along the χ axis and free to vibiate m the y duection The solution of the wave equation is u(r,t)=yu(x)cos ωί, with ω the mode fiequency and u(x) the mode piofile Both u and duldx vanish at the ends of the beam and u(x) is noimahzed such that it equals to l at the point XQ at which the amphtude is measuied 7 We choose x 0 -0 so that F corresponds to a pomt foice at the location of the constnction

The wave functions aie lepresented by scattenng states The incident wave has the foim <^n(r) = \fik n Im" \~mexp(ik n x)<& n (y), wheie ιη λ is the effective mass, n =1,2, , N is the mode mdex, Φ,, the transveise wave function, and k n = ±(2m H /ft 2 ) 1/2 (£ F -£") I/2 the longitudmal wave vector (at Feimi eneigy E r laigei ...

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Quantum theory of electromechanical noise and momentum transfer statistics

“…In the context of Charge tiansfer statistics theie exist two approaches a fully quantum-mechanical appioach usmg Keldysh Gieen functions 110 and a semiclassical approach using the Boltzmann-Langevin equation u Heie we take the formei appioach, to anive at a quantum theoiy of momentum tiansfei statistics As a lest, we show that the second moment calculated fiom Keldysh Gieen functions comcides in the semiclassical hmit with the lesult obtamed from the Boltzmann Langevm equation by Shytov, Levitov, and one of the authois 8 A calculation of the complete cumulant geneiating function of tiansfeired momentum (01, equivalently, of oscillatoi displacement) is piesented foi the case of a single-channel conductoi with a locahzed scatteiei The geneiating function in this case can be wntten entirely m terms of the tiansmis sion piobability Γ of the scatteier In the moie geneial multichannel case one also needs a knowledge of the wave func tions This is an essential diffeience fiom the chaige tiansfei pioblem, which can be solved in teims of transmission eigenvalues foi any numbei of channels At zeio tempeiature the momentum statistics is binomial, just äs foi the chaige At finite tempeiatuie it is multmomial, even m the hmit Γ ->0, diffeient fiom the double-Poissoman distnbution of chaige…”

“…The outline of the papei is äs follows In See II we foimulate the problem in a way that is suitable foi furthei analy sis The key techmcal step in that section is a unitaiy tiansfoimation which ehmmates the dependence of the electionphonon coupling Hamiltoman on the (unknown) scattenng potential of the disordeied lattice The lesultmg coupling Hamiltoman contams the electron momentum flow and the phonon displacement In See II we use that Hamiltoman to denve a geneial foimula foi the generating function of the distnbution of momentum transfened to a phonon (äs well äs the distnbution of phonon displacements) It is the analog of the Levitov-Lesovik foimula foi the chaige-tiansfei distnbution ' Foi a locahzed scatterei we can evaluate this statistics in teims of the scattenng matiix We show how to do this in See IV, and give an apphcation to a single-channel conductoi m See V In Sees VI and VII we turn to the case that the scattenng legion extends thioughout the conductoi We follow the Keldysh appioach to denve a geneial formula for the generating function, and check its vahdity by icdeiiving the result of Ref 8 We conclude in See VIII with an ordei-of-magnitude estimate of highei-order cumulants of the momentum-ti ansfei statistics…”

“…Electncal current is the tiansfei of chaige fiom one end of the conductoi to the other The statistics of this chaige tiansfei was investigated by Levitov and Lesovik ' It is bmomial for a single-channel conductoi at zeio tempeiature and double Poissonian at finite tempeiatuie in the tunneling legime~ The second cumulant, the noise powei, has been measured m a vanety of Systems 3 Ways of measunng the thnd cumulant have been proposed, 24 but not yet camed out Electncal cunent also tiansfers momentum to the lattice The second cumulant, the electiomechamcal noise powei, determmes the mean-square displacement of an oscillatoi thiough which a current is driven It has been studied theoretically, 5 " 8 and is expected to he withm the lange of sensitivity of nanomechanical oscillatoi s 9 No theoiy exists for highei ordei cumulants of the transfened momentum (which would deteimine higher cumulants of the oscillatoi displacement) It is the puipose of the piesent papei to piovide such a theoiy…”