Damping of mechanical vibrations by free electrons in metallic nanoresonators

# Damping of mechanical vibrations by free electrons in metallic nanoresonators

Ze’ev Lindenfeld and Ron Lifshitz Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
November 2, 2012
###### Abstract

We investigate the effect of free electrons on the quality factor () of a metallic nanomechanical resonator in the form of a thin elastic beam. The flexural and longitudinal modes of the beam are modeled using thin beam elasticity theory, and simple perturbation theory is used to calculate the rate at which an externally excited vibration mode decays due to its interaction with free electrons. We find that electron-phonon interaction significantly affects the of longitudinal modes, and may also be of significance to the damping of flexural modes in otherwise high- beams. The finite geometry of the beam is manifested in two important ways. Its finite length breaks translation invariance along the beam and introduces an imperfect momentum conservation law in place of the exact law. Its finite width imposes a quantization of the electronic states that introduces a temperature scale for which there exists a crossover from a high-temperature macroscopic regime, where electron-phonon damping behaves as if the electrons were in the bulk, to a low-temperature mesoscopic regime, where damping is dominated by just a few dissipation channels and exhibits sharp non-monotonic changes as parameters are varied. This suggests a novel scheme for probing the electronic spectrum of a nanoscale device by measuring the of its mechanical vibrations.

###### pacs:
62.25.-g, 63.20.kd, 85.85.+j, 63.22.-m

## I Introduction

The design and fabrication of high- mechanical resonators is an ongoing effort that has intensified with the advent of microelectromechanical systems (MEMS) and even more with the recent progression toward nanoelectromechanical systems (NEMS).Roukes (2001); Cleland (2003); Ekinci and Roukes (2005) One requires low-loss mechanical resonators for a host of nanotechnological applications, such as low phase-noise oscillators;Kenig et al. (2012) highly sensitive mass,Ekinci et al. (2004); *Yang06; *Hanay12; Ilic et al. (2004); Jensen et al. (2008); Lassagne et al. (2008) spin,Rugar et al. (2004) and charge detectors;Cleland and Roukes (1998) and ultra-sensitive thermometersRoukes (1999) and displacement sensors;Cleland et al. (2002); *knobel; Ekinci et al. (2002); *truitt07 as well as for basic research in the mesoscopic physics of phonons,Schwab et al. (2000) and the general study of the behavior of mechanical degrees of freedom at the interface between the classical and the quantum worlds.Schwab and Roukes (2005); LaHaye et al. (2004); *naik06; O’Connell et al. (2010); Katz et al. (2007); *katz08 It is therefore of great importance to understand the dominant damping mechanisms in small mechanical resonators.

A variety of different mechanisms—such as internal friction due to bulk or surface defects,Mihailovich and MacDonald (1995); Olkhovets et al. (2000); *carr2; *evoy2; Liu et al. (2005); Mohanty et al. (2002); Zolfagharkhani et al. (2005); Seoánez et al. (2008); Remus et al. (2009); Chu et al. (2007); Unterreithmeier et al. (2010) phonon-mediated damping,Lifshitz and Roukes (2000); *lifshitzPhonon; Houston et al. (2002); De and Aluru (2006); Kiselev and Iafrate (2008) and clamping losses Cross and Lifshitz (2001); Photiadis and Judge (2004); Patton and Geller (2003); *geller1; *geller2; Schmid and Hierold (2008); Wilson-Rae (2008); *cole—may contribute to the dissipation of energy in mechanical resonators, and thus impose limits on their quality factors. The dissipated energy is transferred from a particular mode of the resonator, which is driven externally, to energy reservoirs formed by all other degrees of freedom of the system. Here, we focus our attention on electron-phonon damping, arising from energy transfer between the driven mode of the resonator and free electrons. This dissipation mechanism is avoided altogether by fabricating resonators from dielectric materials, but for different practical reasons one often prefers to fabricate MEMS and NEMS resonators from metals, such as platinum,Husain et al. (2003) gold,Buks and Roukes (2002); Venkatesan et al. (2010) and aluminum.Davis and Boisen (2005); Li et al. (2008); *hoehne; Teufel et al. (2008) Free electrons are also present in metallic carbon-nanotube resonatorsPeng et al. (2006); Eriksson et al. (2008) and in resonating nanoparticles.Hu et al. (2003); Pelton et al. (2009); Zijlstra et al. (2008) All these different resonators exhibit a wide range of quality factors, from as low as about and up to around , yet one still lacks a full understanding of their damping mechanisms.

It is well-known from at least as early as the 1950s that electron-phonon scattering is a dominant source of attenuation of longitudinal sound waves in bulk metals at low temperatures,Bömmel (1954); Kittel (1955); Pippard (1955); Blount (1959); Ziman (1960); Kokkedee (1962) and indications exist that it may play a significant role in the damping of longitudinal vibrations in freely suspended bi-pyramid gold nanoparticles.Pelton et al. (2009) We note that the effect of electron-phonon scattering on electronic transport through suspended nanomechanical beams,Weig et al. (2004) carbon nanotubes,LeRoy et al. (2004); Leturcq et al. (2009); *mariani; Steele et al. (2009); *huttel09; *Laird12 fullerenes,Park et al. (2000) atomic wires,Paulsson et al. (2005); Viljas et al. (2005); de la Vega et al. (2006) and molecular junctionsPecchia et al. (2004); Kushmerick et al. (2004); Wang et al. (2004); Galperin et al. (2007); Tal et al. (2008) is well documented and intensively studied. There is also evidence for the effect of electron-phonon scattering on heat transport in nanostructures.Fon et al. (2002); Barman and Srivastava (2006) Motivated by all of these considerations, it is our aim here to estimate the contribution of electron-phonon interaction to the damping of vibrational modes in small metallic resonators, while focusing on the effects of their finite dimensions.

We describe the interaction between electrons and phonons by means of a simple screened electrostatic potential. We assume that initially both electrons and phonons are at thermal equilibrium at the same temperature, except for a single mode, which is externally excited by the addition of just a single phonon to its thermal population. This allows us to assume that the electrons remain almost thermally distributed at all times, even though they do not actually relax back to equilibrium. The decay rate of the excited mode is calculated pertubatively, using Fermi’s Golden Rule, as the difference between the rates at which phonons enter and leave the excited mode through their interaction with free electrons. This requires us to assume that the electron and phonon energies are precisely known, or in other words that the a priori lifetimes of both the electrons and the phonons are much longer than all other relevant time scales.

For the phonons this means that all other damping mechanisms must be much weaker than electron-phonon damping—although we show later that additional damping mechanisms do not significantly alter the results of our calculations. For the electrons, on the other hand, this implies that we are working in the high-frequency, or unrelaxed adiabatic limit, with , where is the vibration frequency and is the mean lifetime of the electron due to its scattering with other electrons, thermal phonons, defects, etc. In certain situations, as discussed in detail in section 5.12 of the book by Ziman,Ziman (1960) it is sufficient to satisfy the spatial version of this requirement, namely that —where is the electron mean free path and is the wavenumber of the excited mode—which is easier to satisfy because the ratio of the phonon group velocity to the electron Fermi velocity is typically very small. Intuitively speaking, it is as if the moving electron explores the elastic wave much faster than it would have, if it were standing in place and waiting for the wave to go by. In bulk metals it is difficult to satisfy the adiabatic condition, and one is inevitably required to address the relaxation of the electronic distribution by employing the Boltzmann equation or other approaches.Pippard (1955); Blount (1959); Khan and Allen (1987) However, in clean nanometer scale devices oscillating at very-high frequencies, and operating at sufficiently low temperatures, there is a greater chance of reaching the adiabatic limit. We therefore assume that this is the case, and alert the reader to the fact that our results may be less applicable at high temperatures.

We describe the vibrational modes of the nanomechanical resonator using continuum elasticity theory, which is often employed for treating nanomechanical systems, even in the case of carbon nanotubes,Kahn et al. (2001); Suzuura and Ando (2002); De Martino et al. (2009), and also in the quantum regime.Blencowe (1999); Santamore and Cross (2001); *santamore02; Lindenfeld et al. (2011) The small size of such nanostructures may raise the question of the validity of a continuum elastic approach. However, explicit comparisons with atomistic calculations and experimental results have shown that continuum elasticity is valid down to dimensions of a few nanometers,Broughton et al. (1997); Saviot et al. (2004); Combe et al. (2007); Ramirez et al. (2008) and may indeed be used even for carbon nanotubes, as long as one uses appropriate effective parameters.Yoon et al. (2005); Chico et al. (2006)

Finally, we investigate beams with typical dimensions that are much larger than the bulk Fermi wavelength. In such systems it is usually assumed that the effect of the boundaries on the free electrons can be ignored and that the electrons are essentially unconfined. However, as we demonstrate here, the transverse dimensions of the beam set a temperature scale—between a few kelvins to more than a hundred kelvins for the beam geometries considered here—below which the confinement of the electrons can no longer be ignored, and electron-phonon damping is expected to behave qualitatively different.

The paper is organized as follows. In section II we describe the quantized vibrational modes of a nanomechanical beam. The interaction between these modes and free electrons is addressed in section III, and the resulting expressions for the dimensionless damping are derived in section IV. The reader who is not interested in the technical details of the calculations is invited to skip directly to section V, where we discuss and explain the results of the calculation and their physical implications. We conclude with a summary in section VI.

## Ii Quantized vibrational modes of a nanomechanical beam

We consider a mechanical resonator in the form of a thin elastic beam with a rectangular cross section, although we expect the essence of our calculation to be independent of the specific geometry of the resonator. In what follows we describe the classical longitudinal and flexural normal modes of vibration, obtained within standard theories of thin elastic beams,Graff (1975); Landau and Lifshitz (1986) and quantize these modes as a collection of simple harmonic oscillators. We do not consider the twist modes of the thin beam because the displacement fields of these modes have zero divergence and therefore do not couple significantly to free electrons, as explained in the next section.

We take the length of the beam to lie along the axis from to , and its transverse dimensions along the axis from to , and along the axis from to . In equilibrium, the beam is unstrained, unstressed, and at temperature everywhere. Departure of the beam from equilibrium is described by a displacement field (). The displacement field , as well as the strain and stress tensors and , are all functions of position and time. Yet, we assume that the temperature of the beam remains uniform and constant during the vibration, thus ignoring thermoelastic effects.Lifshitz and Roukes (2000) We take the surfaces of the beam to be stress free, which implies that all but the component of the stress tensor vanish on the surface. Because the beam is thin, this approximately holds in its interior as well. Hooke’s law for the thin beam then takes a rather simple form:

 uxx = 1Eσxx, (1a) uyy = uzz=−σEσxx, (1b) uxy = uyz=uzx=0, (1c)

where is Young’s modulus, and is Poisson’s ratio.

### ii.1 Quantized longitudinal modes

To describe longitudinal vibrations in the thin beam we make an additional simplifying assumption of neglecting any contribution to the dynamics that arises from having a nonzero Poisson ratio. We therefore take , take to be independent of and , and ignore any deviation of the cross section of the beam from its rectangular shape. Under these assumptions, longitudinal vibrations are governed by a standard dispersionless wave equation. Thus, a mode of wavenumber vibrates at a frequency , where is the speed of sound of longitudinal waves in the bulk, and is the mass density of the beam. Taking in Eq. (1b) would lead to the so-called Love equationGraff (1975) and to dispersive longitudinal modes, which we do not consider here.

In the limit of an infinitely long thin beam it is convenient to describe the longitudinal modes as travelling waves leading to a phonon field operator of the form

 ux=∑q√ℏ2Mωqeiqx(bq+b†−q), (2)

where and are bosonic creation and annihilation operators, is the mass of the beam, is its volume, and we use a large volume normalization of the phononic wave functions as and tend to infinity along with . In this limit, the sum is to be interpreted as an integral .

For beams of finite length we consider either doubly-clamped boundary conditions, with at both ends of the beam, which is a common experimental geometry, or stress-free boundary conditions, with at both ends, which may be suitable for describing freely suspended resonators in solution.Hu et al. (2003); Zijlstra et al. (2008); Pelton et al. (2009) The phonon field operator is then given by

 ux=∞∑n=1√ℏMωqn(Ansinqnx+Bncosqnx)(bqn+b†qn), (3)

where , and for doubly-clamped modes

 [An,Bn]={[0,1],odd n,,even n, (4)

while for stress-free modes

 [An,Bn]={[1,0],odd n,,even n. (5)

### ii.2 Quantized flexural modes

We describe flexural vibrations in the thin beam by the transverse motion of the neutral axis of the beam in the direction. We make the usual Euler-Bernoulli assumption that the transverse dimensions of the beam, and , are sufficiently small compared with the length of the beam and the radius of curvature of the bending that any plane cross section, initially perpendicular to the axis of the beam, remains perpendicular to the neutral axis during bending. We further assume that the rectangular shape of the cross section remains unaltered during the bending motion. Such an approximation is justified for small deflections since the error it introduces is only on the order of the transverse beam dimension divided by the radius of curvature of the bending. Since we assume a non-deformable cross-section, there is in fact a neutral surface running through the length of the beam, at , which suffers no extension or contraction during its bending. One can then showLandau and Lifshitz (1986) that the longitudinal strain component , a distance away from the neutral surface, is equal to . By replacing the curvature of the beam with we express the non-zero components of the strain field in the beam as

 uxx = −(z−b2)∂2w∂x2, (6a) uyy = uzz=σ(z−b2)∂2w∂x2. (6b)

Using this strain field it is possible to write down the Lagrangian density of the beam and derive its equation of motion. This equation contains two kinetic terms, one that is associated with the transverse motion of the beam and one that is associated with the rotation of the cross-section. The latter is smaller by a factor of , and is therefore usually neglected, leading to the Euler-Bernoulli equation of motion

 ρA∂2w∂t2+∂2∂x2(EI∂2w∂x2)=0, (7)

where is the area of the cross-section, and is the moment of inertia of the cross section. The resulting flexural modes possess a quadratic dispersion, , where .

In the limit of an infinitely long beam, as in Eq. (2), it is again convenient to describe the transverse modes as travelling waves with a quantized phonon field operator of the form

 w=∑q√ℏ2Mωqeiqx(bq+b†−q). (8)

For beams of finite length we consider doubly clamped boundary conditions, taking at both ends. The phonon field operator is then given by

 w=∞∑n=1√ℏ2Mωqn(Ansinqnx+Bncosqnx+Cnsinhqnx+Dncoshqnx)(bqn+b†qn), (9)

where the wavenumbers are solutions of the transcendental equation , with tending to odd-multiples of as increases. The numerical coefficients are determined through the boundary conditions and the normalization of the phonon wave functions, where by symmetry for odd , and for even .

## Iii Electron-phonon interaction

We assume a standard screened static interaction potential Fetter and Walecka (1971)

 v(r−r′)=−4πZe2q2TFδ(r−r′), (10)

between the negative charge of the electron density and the positive charge of the disturbance in the density of the ionic background, induced by the vibration. Here is the Thomas-Fermi wavenumber, is the magnitude of the electron charge, is the electron spin (not to be confused with Poisson’s ratio), is the number of valence electrons per atom in the material, is the atomic density, and is the local volume change induced by the vibration. The electron-phonon interaction Hamiltonian, derived from this potential, is

 He−ph=−g∫V∑σψ†σ(r)ψσ(r)δv(r)d3r, (11)

where

 g=4πZn0e2q2TF. (12)

For small deformations, to first order in the displacement field, the local change in volume . For twist modes giving , which is why we have ignored them here; for longitudinal modes , where is given by either Eq. (2) or Eq. (3); and for flexural modes we find from Eqs. (6) that

 ∇⋅u=(2σ−1)(z−b2)d2wdx2, (13)

where is given by either Eq. (8) or Eq. (9).

We take two approaches for describing the electron field , where from here onwards we suppress the spin index. As discussed in section I, treating the electrons as bulk-like is a common approximation for structures of the size considered in this work. Thus we can use a simple free electron field of the form

 ψ(r)=∑k1√Veik⋅rck, (14)

where is a fermionic annihilation operator. Within this approximation it is permissable to treat the transverse dimensions of the beam as being infinite from the point of view of the electrons. Accordingly, the sum is replaced by a 3-dimensional integral . This leads to a simplification of the calculations and the resulting expressions below.

Alternatively, we do not neglect the finite transverse dimensions (but the length of the beam is still considered to be infinite as far as the electrons are considered), and view the electrons as being geometrically confined. We then obtain the “particle in a box” field operators

 ψ(x,y,z)=∑k2√Veikxxsinkyysinkzz ck, (15)

with , , and In both cases the energy of the free electrons is given by

 εk=ℏ22m(k2x+k2y+k2z)=ℏ2k22m, (16)

but when the lateral confinement of the electrons is not neglected it takes the explicit form of parabolic bands

 εk=εminny,nz+ℏ2k2x2m,with\ εminny,nz=ℏ2π22m(n2ya2+n2zb2). (17)

### iii.1 Interaction of free electrons with longitudinal phonons

To obtain the interaction Hamiltonian (11) for longitudinal phonons with unconfined free electrons in an infinite beam we use the derivative of Eq. (2) for the quantized elastic displacement field, and the field operator (14) for unconfined electrons. This yields

 He−ph = −igV∑k,k′,q√ℏ2Mωqq∞∫−∞dyei(ky−k′y)y∞∫−∞dzei(kz−k′z)z∞∫−∞dxei(kx−k′x+q)xc†k′ck(bq+b†−q) (18) = −ig∑k,q√ℏ2Mωqqc†k+qck(bq+b†−q),

where , and where we have performed the sum over to eliminate the three delta functions that appear as exponential integrals in the first line of Eq. (18). We note that the integrations in the and directions involve only the unconfined electron wavenumbers. Thus, the width and thickness of the beam can effectively be taken to be infinite, which leads to momentum conservation in the and directions.

When the confinement of the electrons is not neglected, the electronic field (14) is replaced with the one given by Eq. (15). When longitudinal phonons in an infinite beam are considered, the replacement of the continuous electron spectrum (16) with the set of parabolic bands given by Eq. (17) yields the same formal expression for the interaction Hamiltonian as above in Eq. (18). The only difference is that the summations over and become discrete.

To obtain the interaction Hamiltonian of longitudinal modes with unconfined electrons in a beam of finite length we use the derivative of the elastic displacement field given by Eq. (3). This then yields

 He−ph = (19) = −g2∑k,k′x,qn√ℏqnMcl[(An−iBn)sinc(qn+Δkx)l2+(An+iBn)sinc(qn−Δkx)l2]c†k+Δkck(bqn+b†qn),

where , with , and where the functions replace the momentum conservation delta function that exists for the infinite beam in Eq. (18).

We note that by taking the limit of in the first line of Eq. (19) we obtain the interaction Hamiltonian for an infinitely long beam as written in the basis of standing waves instead of the basis of traveling waves, used in Eq. (18). Using this Hamiltonian it is possible to calculate the decay rate of a standing wave mode in an infinitely long beam, which is identical to the one obtained from the Hamiltonian in Eq. (18) for a traveling wave. Alternatively, by converting the creation and annihilation operators of the standing waves in Eq. (19) into creation and annihilation operators of traveling waves, it is possible to show that the Hamiltonian in Eq. (19) is identical to that of Eq. (18) in the limit of an infinite beam and continuous phonon wavenumbers.

### iii.2 Interaction of free electrons with flexural phonons

For flexural vibrations interacting with laterally confined electrons in an infinite beam, we substitute the divergence (13) of the quantized elastic displacement field (8) together with the field operator (15) for laterally confined electrons into the general expression (11) of the electron-phonon interaction Hamiltonian. We then obtain

 He−ph = 4g(2σ−1)V∑k,k′,q√ℏq22MGa∫0dysinkyysink′yyb∫0dz(z−b2)sinkzzsink′zz∞∫−∞dxei(kx−k′x+q)xc†k′ck(bq+b†−q) (20) =8g(1−2σ)bπ2∑k,k′z,q√ℏq22MGnzn′z(n2z−n′2z)2c†k+Δkck(bq+b†−q),

where now , with , and where the last sum is restricted to pairs of integers and that are of different parity. Note that owing to the transverse nature of the vibration the electron momentum changes both in the and in the directions.

The interaction Hamiltonian between the flexural modes and unconfined free electrons in an infinitely long beam is

 He−ph = g(1−2σ)V∑k,k′,q√ℏq22MG∞∫−∞dxei(kx−k′x+q)x∞∫−∞dyei(ky−k′y)yb∫0dz(z−b2)ei(kz−k′z)zc†k′ck(bq+b†−q) (21) = ig(1−2σ)b∑k,k′z,q√ℏq22MG2sinΔkzb2−ΔkzbcosΔkzb2Δk2ze−iΔkzb2c†k+Δkck(bq+b†−q),

where as above , with . The linear dependence of on the distance from the neutral axis of the beam, which is a result of the Euler-Bernoulli thin-beam approximation, precludes the replacement of the finite limits of integration in the direction with integration over an infinite range, even though we consider unconfined electronic states, since it leads to an unphysical divergence of the decay rate. A more realistic model for the flexural modes may saturate this effect and lead to finite results even in the limit of large .

The interaction Hamiltonian for flexural vibrations in a finite beam is obtained by replacing the quantized displacement field of the infinite beam (8) with the one given by Eq. (9), yielding

 (22)

where here , and

 Ln(κ)=1ll2∫−l2eiκx(Ansinqnx+Bncosqnx−Cnsinhqnx−Dncoshqnx)dx (23)

replaces the momentum conservation delta function that exists in the infinite beam (20).

## Iv Inverse quality factor

The damping of mechanical vibrations is estimated by assuming that both electrons and phonons are in thermal equilibrium except for a single vibration mode, which is externally excited by adding just a single phonon to its thermal population. The rate at which the excited mode decays is evaluated as the difference between the rates at which phonons leave it and enter it due to their scattering with electrons. These rates are calculated with Fermi’s golden-rule using the interaction Hamiltonians derived above. The decay rate of an externally excited vibration mode of wavenumber is thus given by

 Γq=2πℏ∑k,k′,σ|⟨f−|He−ph|i⟩|2δ(εk−εk′+ℏωq)−2πℏ∑k,k′,σ|⟨f+|He−ph||i⟩|2δ(εk−εk′−ℏωq), (24)

where is the state with one less phonon and is the state with one more phonon, namely

 |i⟩ = |nk,nk′,Nq+1⟩, (25a) |f−⟩ = |nk−1,nk′+1,Nq⟩, (25b) |f+⟩ = |nk−1,nk′+1,Nq+2⟩, (25c)

where, without any loss of generality, for infinitely long beams we consider modes with a positive wavenumber . The dimensionless damping, or inverse quality factor, of a mode with wavenumber is then defined as . In Eqs. (25) stands for the Fermi-Dirac distribution and for the Bose-Einstein distribution. We note that we do not take into account the change in the electronic chemical potential as the temperature is increased above zero, but rather assume that it remains equal to the Fermi energy of the beam. The latter is calculated for each specific thickness and width of the beam when the confinement of the electrons is taken into account.

This procedure presupposes that the electron and phonon energies that appear in the delta functions in Eq. (24) are exact, or equivalently that the a priori lifetimes of electrons and phonons are infinite. For this to be valid we require (1) that all other vibration damping mechanisms be much weaker than electron-phonon damping; and (2) that scattering rates of electrons—with other electrons, thermal phonons, defects, or surface imperfections—be much slower than the frequency of the mechanical vibration. As we demonstrate in section V.4, removal of the first requirement does not significantly affect the contribution of electron-phonon scattering to the overall damping, as long as the different damping mechanisms are assumed to be independent of each other. The second requirement implies that our calculation is valid in the adiabatic limit, , where is the mean lifetime of the electron. This requirement will be better satisfied for cleaner and smaller—hence, higher-frequency—devices at sufficiently low temperatures.

In what follows we give the derivation of the exact expressions for the inverse quality factors, using Eq. (24) along with the interaction Hamiltonians of section III. The reader who is not interested in these rather technical derivations is welcome to skip to the next section where we discuss the physical consequences of these expressions.

### iv.1 Damping of longitudinal vibrations by unconfined electrons in an infinite beam

Using the Hamiltonian of Eq. (18) and the general expression (24) for the decay rate we obtain

 Q−1q= πg2Mc2l × ∑k,σ[nk(1−nk+q)(Nq+1)δ(εk−εk+q+ℏωq) −nk(1−nk−q)(Nq+2)δ(εk−εk−q−ℏωq)], (26)

where . We convert the sum over wave vectors in Eq. (IV.1) into three integrals and change the energy conservation delta functions into momentum delta functions

 δ(εk−εk±q±ℏωq)=mℏ2qδ(kx−(mclℏ∓q2)). (27)

After performing the integration over and and summing over the spin index we obtain

 Q−1q −(Nq+2)Nq(βℏωq−ln1+eγ+1+eγ−)], (28)

where

 γ±=βℏ22m[(clmℏ±q2)2−k2F]. (29)

Both logarithmic terms in Eq. (28) are negligible as long as and are small compared to (which is the case for all reasonable values of ). For such vibrational wavenumbers and the logarithms vanish. Combining the two terms in the square brackets in Eq. (28) then yields

 Q−1q=g2m22πℏ3ρcl, (30)

which is the same as the result for the damping of bulk longitudinal acoustic waves in the adiabatic limit (Ziman,Ziman (1960) Eq. 8.10.9, and KokkedeeKokkedee (1962)). The damping in Eq. (30) is independent of the mode wavenumber, the temperature, and the geometry of the beam. In fact, depends only on the material parameters of the beam, and typically varies between and for different metals.

### iv.2 Damping of longitudinal vibrations by unconfined electrons in a finite beam

We substitute the Hamiltonian of Eq. (19) into Eq. (24) and again convert the sums over the electronic wave vectors into integrals. After summing over the spin index we obtain

 Q−1qn = g2n16π2ρc2lqn∫dkdk′xnk(1−nk+Δk)Wn(kx−k′x) (31) ×[(Nqn+1)δ(ϵk−ϵk+Δk+ℏωqn) −(Nqn+2)δ(ϵk−ϵk+Δk−ℏωqn)],

where , and

 Wn(κ)=An(sinc(κ−qn)l2+sinc(κ+qn)l2)2+Bn(sinc(κ−qn)l2−sinc(κ+qn)l2)2. (32)

We perform the integration over and and change the energy delta functions into a delta function of the variable

 δ(ϵk−ϵk+Δk+ℏωqn)=mℏ2k′0x[δ(k′x−k′0x)+δ(k′x+k′0x)], (33a) and a delta function of the variable kx δ(ϵk−ϵk+Δk−ℏωqn)=mℏ2k0x[δ(kx−k0x)+δ(kx+k0x)], (33b)

where , and . We integrate over for the first delta function and over for the second delta function and obtain

 Q−1qn = ng2m28πℏ4ρc2lβqn (34) × ⎡⎢ ⎢⎣(Nqn+1)2∞∫−∞dkxk′0x⎛⎜ ⎜⎝βℏωqn+ln1+eβℏ22m(k2x−k2F)1+eβℏ22m(k2x+2mclqnℏ−k2F)⎞⎟ ⎟⎠[Wn(kx−k′0x)+Wn(kx+k′0x)] − Nqn(Nqn+2)∞∫−∞dkxk′0x⎛⎜ ⎜⎝βℏωqn+ln1+eβℏ22m(k2x+2mclqnℏ−k2F)1+eβℏ22m(k2x−k2F)⎞⎟ ⎟⎠[Wn(k′0x−kx)+Wn(−k′0x−kx)]⎤⎥ ⎥⎦,

where we have dropped the prime from the integration variable.

The integrands in Eq. (34) nearly vanish for most values of unless it is relatively close to . For these values of (and as long as is small compared to ) both and are nearly equal to and the logarithmic functions in Eq. (34) can be neglected. Finally, by using the fact that we find that

 Q−1qn=ng2m28πℏ3ρcl∞∫−∞dkxk′0x[Wn(kx−k′0x)+Wn(kx+k′0x)]. (35)

We note that the damping for the infinite beam in Eq. (30) and for the finite beam in Eq. (35), due to interaction with unconfined electrons, are both independent of temperature and of the length of the beam for a given mode.

### iv.3 Damping of longitudinal vibrations by laterally confined electrons in an infinite beam

As noted in section III, the Hamiltonian for the interaction between longitudinal phonons and laterally confined electrons is formally identical to the one describing the interaction between longitudinal phonons and unconfined free electrons, with the integrals over and in Eq. (18) replaced by discrete sums over the allowed values of and . These discrete sums yield

 Q−1q= g2mℏ2EAq∑ky,kz [n(k−x,ky,kz)(1−n(k+x,ky,kz))(Nq+1) −n(k+x,ky,kz)(1−n(k−x,ky,kz))(Nq+2)], (36)

where

 k±x=mclℏ±q2. (37)

### iv.4 Damping of flexural vibrations by laterally confined electrons in an infinite beam

The Hamiltonian of Eq. (20) which describes the interaction between flexural modes in an infinite beam with confined electrons together with the general expression for the decay rate (24) yield

 Q−1q= 64πg2(2σ−1)2G2b2M∑k,k′z,σk2zk′2z(k2z−k′2z)4 ×[nk(1−nk+q+Δk)(Nq+1) ×δ(εk−εk+q+Δk+ℏωq) −nk(1−nk−q+Δk)(Nq+2) ×δ(εk−εk−q+Δk−ℏωq)], (38)

where here .

We sum over the spin index and change the delta functions in Eq. (38) into delta functions of the variable

 δ(ϵk−ϵk+Δk±q±ℏωq)=mℏ2qδ(kx−κ±), (39)

where

 κ±=mGqℏ±k2z−k′2z2q∓q2. (40)

We express the sum over as an integral, and use Eqs. (39) and (40) to obtain

 Q−1q = 768g2m(2σ−1)2π4ℏ2EAq∑ky,kz,k′zn2zn′2z(n2z−n′2z)4 (41) × [n(κ+,ky,kz)(1−n(κ++q,ky,k′z))(Nq+1) − n(κ−,ky,kz)(1−n(κ−−q,ky,k′z))(Nq+2)].

### iv.5 Damping of flexural vibrations by laterally confined electrons in a finite beam

The inverse quality factors of flexural modes in a finite beam are obtained in a similar way but without exact momentum conservation. Thus, the momentum delta functions are replaced with the functions, given by Eq. (23), and the integral over remains in the final expression

 Q−1qn = 384(2σ−1)2g2ml2π5ℏ2EA∑ky,kz,k′2z