Shytov, A.V.; Levitov, L.S.; Beenakker, C.W.J.
Citation
Shytov, A. V., Levitov, L. S., & Beenakker, C. W. J. (2002). Electromechanical noise in a
diffusive conductor. Physical Review Letters, 88(22), 228303.
doi:10.1103/PhysRevLett.88.228303
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Electromechanical Noise in a Diffusive Conductor
A. V. Shytov,1 L. S. Levitov,2 and C. W. J. Beenakker3
1Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030 2Department of Physics, Center for Materials Science & Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
3Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
(Received 29 October 2001; published 17 May 2002)
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 j) is found to be reduced below the Fermi momentum by a factor of order l兾L for shear fluctuations and共j兾L兲2for 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.
DOI: 10.1103/PhysRevLett.88.228303 PACS numbers: 85.85. +j, 73.23. – b, 73.50.Td, 77.65. – j
Impressive advances in the fabrication of miniature me-chanical oscillators provide new opportunities for research in mesoscopic physics [1,2]. The coupling of electrical and mechanical degrees of freedom is of particular inter-est. We mention the observation of thermal vibration [3] and acoustoelectric effects [4] in carbon nanotubes, the coupling of the center-of-mass motion of C60 molecules
and single-electron hopping [5], and also theoretical work [6] on the coupling between a tunneling electrical current and a localized phonon mode.
This Letter was motivated by a question posed to us by M. Roukes: Electrons in a metal collide with impuri-ties and thereby exert a fluctuating force on the lattice. In equilibrium this electromechanical force cannot be distin-guished from other sources of thermal noise. Might it be measurable out of equilibrium by driving a current through a nanoscale oscillator? To address this question one has to consider a delicate balance of forces.
We will provide both a general theory and a specific ap-plication to the electromechanical excitation of a bending mode in the geometry of Fig. 1: a thin elastic beam con-necting two massive Ohmic contacts. The beam could be a conductor or an insulator covered with a metal (e.g., a metallized suspended silicon beam [7]). We calculate the excess noise in the bending mode that arises in the presence of a dc voltage V and conclude that it should be observable in the background of the thermal noise.
Let us first discuss the order of magnitude. The noise at low temperatures is due to the N eV兾EF“noisy” electrons
within a range eV of the Fermi energy EF (with N the
total electron number in the metal). Each electron transfers to the lattice a typical momentum Dp ⯝ pFin a scattering
time t. The mean-squared momentum transfer in a time t for uncorrelated increments Dp would be
共N eV 兾EF兲 共Dp兲2共t兾t兲 ⯝ N mⴱeV共t兾t兲 ⬅ Pmaxt ,
(1) with mⴱ the electron effective mass.
We find that Pmax overestimates the fluctuations in the
transverse force. The actual noise power is of order P ⯝ 共l兾L兲2P
max, with L the length of the beam and l the
mean-free path in the metal. The reduction appears because sub-sequent momentum transfers are strongly correlated, since an electron being scattered back and forth alternatingly transfers positive and negative momentum to the lattice. The factor 共l兾L兲2 reduces the noise substantially, but we
estimate that it should be observable in an oscillator with a 10216 N兾pHz sensitivity [2,7].
We combine two independently developed theoretical frameworks: the dynamic theory of elasticity [8,9] and the kinetic theory of fluctuations [10]. We start from the Boltzmann-Langevin equation of Kogan and Shulman [10]. This is a kinetic equation with a fluctuating source dJ共r, p, t兲 that describes fluctuations in the time of the distribution function n共r, p, t兲,
共≠t 1 v ? =r 1 eE ? =p 1 S兲n 苷 dJ . (2) Here p 苷 mⴱv is the quasimomentum and E共r, t兲 is the electric field. The collision integral S for elastic scattering on impurities (with rate W) is given by
S n共p兲 苷 具W共 ˆp ? ˆp0兲 关n共p兲 2 n共p0兲兴典pˆ0. (3)
The angular brackets indicate an average over the direction ˆ
p0 of the momentum p0, with jp0j苷 jpj.
L
conductor
oscillator
u(x)
FIG. 1. Sketch of an elastic beam clamped at both ends to a contact and covered by a metal layer. A current flowing through the metal excites a bending mode u共x兲 of the beam.
The noise source dJ has zero time average and variance dJ共r, p, t兲dJ共r0, p0, t0兲 苷 d共r 2 r0兲d共t 2 t0兲d共´ 2 ´0兲n21
3关4pd共ˆp 2 ˆp0兲 具W共 ˆp ? ˆp00兲 共¯n 1 ¯n00 2 2 ¯n ¯n00兲典pˆ00 2 W共 ˆp ? ˆp0兲 共¯n 1 ¯n0 2 2 ¯n ¯n0兲兴 . (4)
Here n共´兲 is the density of states (at energy ´ 苷 p2兾2mⴱ) and ¯n is the time-averaged distribution. We have set Planck’s constant h ⬅ 1, so that n is dimensionless, and have abbreviated ¯n0 苷 ¯n共r, p0, t兲, ¯n00苷 ¯n共r, p00, t兲.
The force density f共r, t兲 exerted by the electrons on the lattice is the divergence of a symmetric tensor P that can be decomposed into an isotropic pressure P0 and a
traceless shear tensor S:
fa 苷 2=bPab, Pab 苷 P0dab 1 Sab. (5)
In the approximation of a deformation-independent effec-tive mass, one has [9,11]
Pab 苷 mⴱ
Z
dp yaybn . (6)
The time-averaged force (5) vanishes, since it contains a derivative of the spatially uniform time-averaged distri-bution ¯n. The electrical current drag on impurities (the so-called “wind force”) is cancelled by the electric field force exerted on the ions [12]. Since f is a total derivative the net fluctuating force vanishes as well at low frequencies (ignoring boundary contributions). Although the center of mass does not move, there are fluctuating compres-sion modes (driven by P0) as well as torsion and bending
modes (driven by S). The driving force F共t兲 for each of these modes is obtained by weighing f共r, t兲 with a sensi-tivity function g共r兲 proportional to the displacement field of the mode, F 苷 Z dr f ? g苷Z dr µ P0= ? g 1 Sab ≠gb ≠ra ∂ . (7) The two contributions P0and S can be separated by
ex-panding n共r, p, t兲 in spherical harmonics n共q兲共r, ´, t兲 with respect to the direction ˆp of the momentum. It is conve-nient to write the spherical harmonics in Cartesian (rather than spherical) coordinates,
n苷 ` X q苷0 ˆ pa1· · · ˆpaqn 共q兲 a1···aq 苷 n共0兲 1 ˆp an共1兲a 1 µ ˆ papˆb 2 1 3 dab ∂ nab共2兲 1 . . . . (8)
Here ˆpq is the traceless part of the symmetric tensor ˆ
pa1· · · ˆpaq. These tensors form an orthonormal set [13],
具 ˆpn
ˆ
pm典pˆ 苷 dnm m!
共2m 1 1兲!!D共m兲. (9) The tensor D共m兲projects onto the traceless symmetric part of a tensor of rank m. We will need D共1兲ab 苷 dab and
D共2兲aba0b0 苷 1 2 daa0dbb0 1 1 2 dab0dba02 1 3 dabda0b0. (10) In view of the orthogonality of different spherical har-monics, one has
P0苷 1 3 Z d´ 2´nn共0兲, Sab 苷 2 15 Z d´ 2´nnab共2兲 . (11) The two harmonics n共0兲and n共2兲have to be found from the kinetic equation (2). We first consider the harmonic n共2兲 that determines the shear tensor S, and then discuss the harmonic n共0兲 and resulting pressure P0.
To obtain an equation for n共2兲we multiply both sides of Eq. (2) by ˆppˆ and perform an angular average. Employ-ing the diffusion approximation on length and time scales larger than l and t, we neglect the derivatives with respect to t and r in Eq. (2). Also, in the linear response approxi-mation, we neglect the derivative with respect to p, since it gives a term bilinear in E and j. What remains is a local relation between n共2兲and the second harmonic J共2兲 of the fluctuating source,
n共2兲 苷 t2dJ共2兲. (12)
The momentum transport time t2is defined by
1 t2 苷 3 4 Z 1 21 dj W共j兲 共1 2 j2兲 . (13) For anisotropic scattering the time t2 is larger
than the charge transport time t, defined by 1t 苷
1 2
R1
21dj W共j兲 共1 2 j兲. (For isotropic scattering t2苷
t 苷 1兾W.) The correlator of dJ共2兲 follows in the same way from Eq. (4). In the diffusion approximation we replace ¯n, ¯n0, and ¯n00by ¯n共0兲. Using Eq. (9) we arrive at dJab共2兲共r, ´, t兲dJ共2兲a0b0共r0, ´0, t0兲 苷
15 nt2
D共2兲aba0b0d共t 2 t0兲d共r 2 r0兲d共´ 2 ´0兲¯n共0兲共r, ´兲 关1 2 ¯n共0兲共r,´兲兴 . (14)
Since ¯n共0兲共r, ´兲 differs from 0 or 1 only in a narrow range
near the Fermi level, we ignore the energy dependence of n and t2and evaluate them at ´苷 EF.
We quantify the transverse momentum noise through the correlator of the shear tensor,
Caba共2兲 0b0共r,r0兲 苷 2
Z ` 2`
dt Sab共r,0兲Sa0b0共r0, t兲 . (15)
Combining Eqs. (11), (12), and (14), we obtain the result C CC共2兲共r, r0兲 苷 8 15共m ⴱy2 F兲 2t 2nd共r 2 r0兲K共r兲D共2兲, (16) K共r兲 苷Z d´ ¯n共0兲共r, ´兲 关1 2 ¯n共0兲共r, ´兲兴 . (17) The kernel K is given by [14] K 苷 共eV兾L兲x共1 2 x兾L兲, where V is the voltage applied between the two contacts at
x 苷 0, L. The parabolic profile K共x兲 requires kBT ø eV
and the absence of inelastic scattering.
We now turn to the pressure fluctuations. Instead of Eqs. (12) and (14) we have the fluctuating drift-diffusion equation [14]
j 1 D=r 2 sE苷 etZ d´ nydJ共1兲 ⬅ dI , (18) dIa共r, t兲dIb共r0, t0兲 苷 2sdabd共r 2 r0兲d共t 2 t0兲K共r兲,
(19) which relates the fluctuations in the charge density r 苷
eRd´ nn共0兲 and the current density j 苷 13eRd´ nyn共1兲.
Once we know the charge density fluctuations we can find the fluctuating pressure from
P0 苷 共D兾m兲dr , (20)
cf. Eq. (11). The diffusion constant is D 苷 13yF2t, the
conductivity s 苷 e2nD, and the mobility m 苷 et兾mⴱ.
The correlator C共0兲of the pressure fluctuations is defined as in Eq. (15), with S replaced by P0. To close the
prob-lem we need the continuity equation, ≠r兾≠t 1 = ? j 苷 0, and the Poisson equation, k= ? E 苷 dr (with dielectric constant k). The time derivative of r in the continuity equation may be omitted in the low-frequency regime. The fluctuations in the electron density then obey
D=2dr 2共s兾k兲dr 苷 = ? dI. (21) The current fluctuations create a fluctuating charge dipole that is screened over a length j 苷 共kD兾s兲1兾2 苷 共k兾e2n兲1兾2. On length scales ¿j, one may neglect the
diffusion term in Eq. (21) and use the local relation [15] dr 苷 2共k兾s兲= ? dI. Equations (19) and (20) then yield
C共0兲共r, r0兲 苷 4sj 4 m2 ≠ ≠r ? ≠ ≠r0 d共r 2 r 0兲K共r兲 . (22) The next step is to use the results (16) and (22) to es-timate the low-frequency noise power P 苷 2R`2`dt 3
F共0兲F 共t兲 of the fluctuating force F 共t兲 that drives a par-ticular oscillator mode [16]. To that end the correlator (15) is integrated over r and r0, weighted by the sensitivity function of the mode as in Eq. (7). For a bending mode we use Eq. (16), which gives the noise power
P共2兲 苷 8 15共m ⴱy2 F兲2t2n Z dr K共r兲D共2兲aba0b0 ≠gb ≠ra ≠gb0 ≠ra0 . (23) For a compression mode in a metal of size ¿j we use
Eq. (22) and find P共0兲 苷 4sj
4
m2
Z
dr K共r兲 j== ? gj2. (24) For an order of magnitude estimate, we take K ⯝ eV, g⯝ 1, and estimate spatial derivatives by factors 1兾L, and the volume integral by a factor V . For simplicity we as-sume isotropic impurity scattering, so that t2 苷 t. Then
the noise power due to fluctuations in the shear tensor is of order P共2兲 ⯝ 共mⴱyF2兲2tneV V L22, and the noise
power due to pressure fluctuations is of order P共0兲 ⯝
sj4m22eV V L24. It is instructive to write these two
estimates in the same form, using s兾em 苷 13mⴱyF2n 苷
N兾V ⬅ ne, with nethe electron density. One finds
P共0兲 ⯝ 共j兾L兲4P
max, P共2兲 ⯝ 共l兾L兲2Pmax, (25)
with Pmax 苷 N mⴱeV兾t being the noise power for
inde-pendent momentum transfers mentioned earlier.
The experimental observation of the shear tensor fluc-tuations looks more promising than the observation of the pressure fluctuations, first, because j is typically ø共lL兲1兾2
so that P共0兲 ø P共2兲, and, second, because a typical oscil-lator operates in a bending or torsion mode rather than in a compression mode. For that reason we will now limit the more quantitative calculation to P共2兲. We consider a bend-ing mode u共x兲 cosv0tin the geometry of Fig. 1. The
sensi-tivity function g共x兲 苷 u共x兲兾u共x0兲 equals the displacement
(in the y direction) normalized by the value at a reference point x0. We choose x0苷 L兾2, so that F is equivalent
to a point force at the beam’s center. Equation (23) now takes the form
P 苷 4 5 nepF共lA兾L兲 Z L 0 dx L K共x兲 关Lg 0 共x兲兴2 , (26) with A苷 V 兾L the cross-sectional area of the metal layer.
The wave equation for transverse waves is biharmonic,
d4u兾dx4 苷 k4u. The solution for doubly clamped bound-ary conditions is [17]
u共x兲 苷 共sinkL 2 sinhkL兲 共coskx 2 coshkx兲
2 共coskL 2 coshkL兲 共sinkx 2 sinhkx兲 , (27) with the resonance condition coskL coshkL苷 1. We use the lowest resonance at kL 苷 4.73. Substituting K 苷 共eV兾L兲x共1 2 x兾L兲 and integrating, we obtain the excess noise P 苷 45nepF共lA兾L兲 3 0.83eV. If we insert values
typical for a metal, ne 苷 1029 m23, pF 苷 10224Ns, l 苷
100 nm, and choose typical dimensions A兾L 苷 10 nm, then the force spectral density at V 苷 1 mV is P 苷 10232N2兾Hz, well above the thermal noise power at low
temperatures (of order 10234 N2兾Hz at T 苷 1 K [2]).
It is instructive to apply the result (26) to a system in thermal equilibrium, when K共x兲 苷 kBT for all x. In this
case an independent estimate of the noise P0is provided by
the fluctuation-dissipation theorem: P0 苷 4kBTMv0兾Q0,
with M the active mass of the oscillator and 1兾Q0the
Equation (26) gives 1 Q0 苷 1 5 共l兾L兲 2N mⴱ Mv0t Z L 0 dx L 关Lg 0共x兲兴2. (28)
This electromechanical quality factor might be measurable in a superconducting metal, as an increase in the overall quality factor when T drops below the critical tempera-ture. One can also calculate Q0 directly as an
“absorp-tion of ultrasound” by conduc“absorp-tion electrons [9], providing a consistency check on our analysis. However, the non-equilibrium noise (26) with an x-dependent kernel K共x兲 cannot be obtained from acoustic dissipation. The elec-tromechanical part (28) of the overall quality factor can be significant. We estimate Q0 ⯝ 5 3 104 for the above
metal parameters with L苷 1 mm, v0兾2p 苷 100 MHz,
and M兾N 苷 1000mproton. This is comparable to the
re-ported values Q ⯝ 103 105[2].
Before concluding we mention an altogether different mechanism for electromechanical noise, which is the cou-pling of a fluctuating surface charge dq共t兲 on the metal to the electromagnetic environment. In the presence of an electric field E0between the metal surface and the substrate
(e.g., due to a mismatch in work functions), the charge fluc-tuations will give rise to a fluctuating transverse force with noise power
Penv ⯝ E2
0dq2⯝ E02C2dV2⯝ E02C2Rmax共kBT , eV兲 .
Here C is the capacitance to the ground and R 苷 L兾As is the resistance of the metal [18]. The ratio Penv兾P共2兲 ⯝
共E0CL2兾elN 兲2 is quite small for typical parameter
val-ues. The reason is that the environmental charge fluctua-tions are a surface effect, while the whole bulk of the metal contributes to P共2兲. Although the noise per electron is small in l兾L, the total noise power P共2兲 is big due to the large number N eV兾EF of contributing electrons.
In summary, we have addressed the fundamental ques-tion of the excitaques-tion of an elastic mode in a disordered metal out of equilibrium, as a result of the fluctuating mo-mentum that an electrical current transfers to the lattice. The effect is small but measurable. The characteristic linear dependence of the electromechanical noise on the applied voltage should distinguish it from other sources of noise. We believe that a measurement is not only feasible but worth performing. Indeed, the nonequilibrium elec-tric current noise has proven to be a remarkably powerful tool in the study of transport properties [19], precisely be-cause it contains information that is not constrained by the fluctuation-dissipation theorem. The noise considered here could play a similar role for mechanical properties.
We mention one such application. Just as electrical shot noise measures the effective charge of the carriers, me-chanical noise could be used to measure their momentum. This should be most intriguing in 1D electron systems, such as quantum wires and nanotubes, where strong elec-tron interaction invalidates the Fermi liquid description. Electromechanical noise could thus be employed to mea-sure the Luttinger liquid equivalent of the Fermi
momen-tum. We are not aware of any other technique that would allow us to do such a measurement.
This work was motivated by discussions with M. Roukes during the Nanoscience program at the Institute for Theo-retical Physics in Santa Barbara. We thank M. Kinder-mann, Yu. V. Nazarov, and B. Spivak for discussions, and acknowledge support by the National Science Foundation under Grants No. PHY99-07949 and No. DMR98-08941 (MRSEC program), and by the Netherlands Science Foun-dation NWO/FOM.
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