Nanomechanical damping via electron-assisted relaxation of two-level systems
Olivier Maillet, Dylan Cattiaux, Xin Zhou, Rasul R. Gazizulin, Olivier Bourgeois, Andrew D. Fefferman, Eddy Collin
NNanomechanical damping via electron-assisted relaxation of two-level systems
Olivier Maillet,
1, 2, ∗ Dylan Cattiaux, Xin Zhou,
1, 3
Rasul R. Gazizulin, Olivier Bourgeois, Andrew D. Fefferman, and Eddy Collin Univ. Grenoble Alpes, CNRS, Institut N´eel, 38000 Grenoble, France Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland CNRS, Univ. Lille, Centrale Lille, Univ. Polytechnique Hauts-de-France,IEMN UMR8520, Av. Henri Poincar´e, Villeneuve d’Ascq 59650, France
We report on measurements of dissipation and frequency noise at millikelvin temperatures ofnanomechanical devices covered with aluminum. A clear excess damping is observed after switchingthe metallic layer from superconducting to the normal state with a magnetic field. Beyond thestandard model of internal tunneling systems coupled to the phonon bath, here we consider therelaxation to the conduction electrons together with the nature of the mechanical dispersion lawsfor stressed/unstressed devices. With these key ingredients, a model describing the relaxation oftwo-level systems inside the structure due to interactions with electrons and phonons with wellseparated timescales captures the data. In addition, we measure an excess 1 /f -type frequency noisein the normal state, which further emphasizes the impact of conduction electrons. Nano-electro-mechanical systems (NEMS) [1] are nowcommon tools used for ultra-sensitive detection [2] whilebeing ubiquitous model systems for the study of quan-tum foundations involving mechanical degrees of freedom[3, 4]. Both endeavours require resonators with high qual-ity factors Q [5], so as to resolve either small frequencychanges due to e.g. masses added [2], or to preserve quan-tum coherence over long enough times [6]. Yet, mecha-nisms limiting the intrinsic Q factor of nanomechanicalsystems mostly remain a puzzle despite intensive efforts[7–9], especially at low temperature where quantum ef-fects are expected to manifest themselves. Commonlyproposed mechanisms include clamping losses [10], andhigher order phonon processes [11], e.g. thermoelasticdamping [12] and Akhiezer damping [13]. While clamp-ing losses are vanishingly small for thin beam structures[10], phonon-phonon interactions are switched off at lowtemperatures. In most cases, the surviving mechanism isthought to be the coupling between the mechanical strainarising from the resonator’s motion and low-energy ex-citations in the constitutive material [7, 9, 14, 15]. Thelatter are either defects or (groups of) atoms that tunnelquantum mechanically between two nearly equivalent po-sitions in the atomic lattice, hence forming two-level sys-tems (TLS). These TLS cause damping of the mechanicalmotion through their interaction with the induced strainfield and their own energy relaxation. The initial mi-croscopic description of such a mechanism, the so-calledStandard Tunneling Model (STM), was introduced in theearly 70s [16, 17] to explain low-temperature properties ofamorphous materials and is still widely used nowadays,its importance being renewed by e.g. superconductingcircuits [18–21] or nanomechanics studies [9, 14, 15, 22].However, in the latter case, the model was unsuccess-ful in describing nano-systems which integrate resistivemetallic elements [7, 14, 23].In this Letter, we report the measurements from 10 Kdown to 30 mK of the damping rate and frequency shift of P o t en t i a l ene r g y Configurational coordinate ∆ V ∆ � d FIG. 1. Top: scanning electron micrograph of the device (sidegate electrode not used). Bottom: schematic side view ofNEMS (left) and TLS-strain coupling (right) under a macro-scopic applied force. Right panel: schematic potential en-ergy of a single TLS at equilibrium (dashed) and under strain(full line). The two minima (separated by an energy gap ∆)are coupled through a tunneling element ∆ ∝ e − d √ m a V / (cid:126) ,where V is the barrier height, d the interwell distance and m a the tunneling entity effective mass. a high Q , high stress silicon nitride NEMS beam, coveredwith a thin aluminum layer (Fig. 1). Using different mag-netic fields we can tune the metallic layer state from su-perconducting to normal below 1 K, revealing the unam-biguous contribution to nanomechanical damping of thenormal state electrons reported for low-stress nanocan-tilevers [23]. To explain this contribution, we quantita-tively include a mechanism of TLS relaxation due to theconduction electron bath [24] in parallel with phonon- a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p N o r m a li z ed i n pha s e r e s pon s e Drive frequency (MHz)
Normal Superconducting (a) (b) R e l a t i v e f r equen cy s h i ft ( x - ) Temperature (K)
FIG. 2. a) NEMS in-phase response to a small magnetomotiveexcitation at T = 275 mK in superconducting (red dots) andnormal (blue dots) states of the metallic layer, normalizedto their peak height for better comparison. Solid lines areLorentzian fits. The resonance frequency difference is dueto the magnetomotive contribution, and the magnetomotivedamping contribution accounts for 5 % of the line width inthe normal state. b) Relative frequency shift δω /ω as afunction of temperature. The line is a logarithmic fit, whilethe dashed curve is a T empirical law (see text). assisted relaxation, for a given fraction of the TLS dis-tribution. The reasoning is formally equivalent to theone proposed in [8] where the authors introduce ad hoc aretarded (imaginary in frequency domain) Young’s mod-ulus, whose microscopic origin is addressed in our work.The data are fit in all regimes, with a minimal set of freeparameters. For comparison, the model also successfullyreproduces the nanocantilever data of Ref. [23]. In addi-tion, we measure the frequency noise as a function of tem-perature in both normal and superconducting state. Themagnitude is found to differ substantially between thetwo states, pointing again towards an electron-assistedmechanism. The results together with the model pro-vide an answer to the issue of nano-electro-mechanicaldamping in (hybrid) metallic systems at low tempera-tures, consistent with all the related results reported inthe literature so far.The main sample is a 15 µ m long silicon nitride beamcovered with 30 nm aluminum, having transverse dimen-sions e × w = 130 nm ×
300 nm (see Fig. 1), andmounted on a cold finger thermally anchored to the mix-ing chamber stage of a dilution refrigerator in cryogenicvacuum. The motion of the fundamental flexural modeof the beam at a frequency ω = 2 π × . δω /ω = C ln( T /T ) [with respect to an arbitrary reference ω , see Fig. 2b)] below 1 K, which we interpretas evidence for TLS-driven behavior [17], in particularsince C = 2 . × − agrees with commonly reportedvalues for similar structures [7, 9, 14]. This behaviourarises from the resonant interaction of applied phononsthrough magnetomotive driving and TLSs [17]. The fre-quency shift above 1 K follows a T law which we at-tribute to thermal expansion mismatch between the twolayers [26]. Meanwhile, the measured damping rate rep-resented in Fig. 3 is divided into two regimes in tempera-ture: above 1-2 K it essentially reaches a nearly constantplateau around 1 . −
800 mK, while it does not contributefurther to the resonance frequency shift within our ex-perimental accuracy (see Fig. 2b). Those observationscomplement previous results [23] obtained with a low-stress goalpost-shaped silicon nanoresonator (see Fig. 3inset), with very similar features. This suggests that themechanism at stake in the normal state is independentof geometry or mechanical properties. Between roughly150 mK and 1 K the damping in the normal state showsa sublinear power law-like behavior and reaches a satura-tion threshold for lower temperatures, consistently withprevious measurements in similar conditions [7, 14, 27]where T α , α = 0 . − . ω , and the structure undergoes an axial oscil-lating strain. The strain field changes the local potentialenergy landscape, leading to a modulation of the TLSenergy splitting ε = (cid:112) ∆ + ∆ , where ∆ is the TLSasymmetry and ∆ the coupling energy between the twopotential wells. Subsequently, the TLS returns to equi-librium by exchanging energy with the phonon bath (thethermal strain field) over a characteristic time τ . Thiscauses a lagging stress response, leading to mechanicalenergy dissipation. The mechanical damping rate result-ing from the strain modulation of a TLS ensemble writes[30]:Γ[ τ ] = Cω (cid:90) (cid:90) d ε d∆ × P ( ε, ∆) (cid:18) ∆ ε (cid:19) sech (cid:18) ε k B T (cid:19) ωτ ( ε, ∆)1 + ω τ ( ε, ∆) , (1)Here we introduce the commonly assumed [17] TLS dis- D a m p i ng R a t e ( H z ) Temperature (K) D a m p i ng R a t e ( H z ) Temperature (K)
SuperconductingNormalDoubly-Clamped String Goalpost Cantilever
FIG. 3. NEMS damping rate as a function of temperaturein normal (blue) and superconducting (red) states, with themagnetomotive contribution subtracted. Inset: data fromRef. [23]. Solid curves (main and inset) are fits using Eq. (1)with relaxation rates due to phonons [Eq. (2)] and electrons[Eq. (3)] to electrons. Dashed red and black lines correspondto low and high temperature asymptotic behaviors, respec-tively. The blue dotted line indicates a possible saturationaround 120 mK. tribution P ( ε, ∆) = P ε/ ( ε − ∆ ) where P is theTLS density of states and C = P γ π e / σ l , with σ = 1 . ± . C is theoretically identical to the logarithmic slope ofthe frequency shift [31], but its expression differs fromthe one commonly derived [31] due to the influence ofhigh tensile stress on the excited flexural mode [30].For high enough temperatures, the damping reaches aplateau πωC/
2, regardless of microscopic TLS scatter-ing mechanisms. From the measured plateau we extract C = 4 . × − , in qualitative agreement but differing bya factor of 2 from the value given by the frequency shiftmeasurement. Similar discrepancies have been reported[7, 9, 14, 23] and may be due to our simplified model thatneglects e.g. the bilayer structure or inhomogeneities inthe TLS distribution.Below 600 mK, the measured linear dependence is con-sistent with previous reports for similar beams [9, 32], butis in contradiction with the usual T dependence. How-ever, at subkelvin temperatures, the dominant phononwavelength λ = hc/ . k B T ( ∼
100 nm at 1 K) becomesbigger than the transverse dimensions of the resonator.As a result TLS relax to equilibrium by exchanging en-ergy only with longitudinal ones [30], which are not con-fined and thus lie at lower energies. They realize a quasione-dimensional phonon bath with constant density ofstates l/πc l and linear dispersion relation, as proposede.g. in Refs. [9, 32] and more extensively studied in Refs.[15, 28, 29]. At first non-vanishing order in the thermalstrain field perturbation, Fermi’s Golden rule yields the relaxation rate of a single TLS due to its interaction withthe phonon bath [15, 30]: τ − ph = γ (cid:126) ρewc l ∆ ε coth (cid:18) ε k B T (cid:19) , (2)where c l ∼ ρ = 2 . × kg/m arethe longitudinal sound speed and mass density for SiN,respectively. Combining this relaxation rate with thedamping expression (1), we capture both the dampingdata in the low temperature range when the field is lowenough to maintain superconductivity in the metalliclayer, and the high temperature limit. To consistentlyfit both the low (Γ[ τ ph ] ∝ γ CT , dashed red line in Fig.3) and high (Γ[ τ ph ] ∝ C ) temperature regimes we use γ = 9 . P = 2 . × J − .m − . The fittedinteraction energy is rather high (one would expect itmore around 1 eV), but may likely reflect a non-uniformdistribution of the TLS inside the beam [22], which isout of the scope of this study. Note that our expres-sion does not fit the data in the superconducting statebetween 600 mK and 1 K, which we attribute to the sub-stantial density of quasiparticle excitations in this range,that should contribute to an excess damping through thesame mechanism as electrons in the normal state [33]: infact, above 800 mK the data in both states are identi-cal within experimental accuracy, as observed previously[23]. This shall be addressed elsewhere.Concerning the results in the normal state, it is natu-ral to consider the conduction electrons as an additionalrelaxation channel for TLS in parallel with the phononbath: when a TLS entity tunnels, the Coulomb poten-tial that scatters conduction electrons is modified, whichtranslates as an effective electron-TLS coupling. The cor-responding relaxation rate is again obtained by Fermi’sGolden Rule [30]: τ − el = 4 πK (cid:126) ∆ ε coth (cid:18) ε k B T (cid:19) , (3)where K = ( n V Ω) is a normalized electron-TLS cou-pling strength, n = 1 . × J − . m − being the elec-tronic density of states at the Fermi level for aluminum, V an averaged coupling constant, and Ω the effective in-teraction volume, which should vanish beyond the metal-lic screening length ( (cid:46) -4 -3 -2 -1 F r equen cy no i s e PS D ( H z ) Frequency (Hz) (a) C oun t A ll an de v i a t i on ( H z ) Temperature (K)
Normal
Superconducting (b)
FIG. 4. a) Frequency noise spectrum at T = 100 mK in the superconducting (red) and normal (blue) state of the metalliclayer. Solid lines are 1 /f µ type functions, with µ = 0 . ± . (cid:112) (cid:104) δf (cid:105) of frequency noise ( δf is thejump between two successive frequency measurements), extracted from the standard deviation of Gaussian fits to the jumpshistograms. Dashed lines (constant for the superconducting state, ∝ / √ T for the normal state) are guides to the eye. Inset:comparison of frequency jumps histograms in the superconducting and normal state at T = 100 mK. Solid lines are Gaussianfits with mean (cid:104) δf (cid:105) = 0 and standard deviation thus directly reflecting the Allan deviation. [9]. Therefore, electrically ”neutral” TLS interact withphonons, but not with electrons, which leads to separateaverages over charged and neutral TLS.For ”neutral” TLS and TLS within the SiN layer,which relax only due to interactions with phonons, thecrossover temperature T ∗ ph between the plateau and thepower law regime is defined by the condition ωτ ph ∼ T ∗ el is shifted down by electron-assisted relaxation be-cause for these TLS the condition ωτ ∼
1, where now τ − = τ − ph + τ − el , is modified. Thus, for T ∗ el < T < T ∗ ph ,an intermediate regime emerges. At ultra-low temper-atures, far beyond reach of conventional refrigerationmeans, the ∝ T relaxation regime should be recoveredwhen all TLS in the structure satisfy ωτ (cid:29)
1. We fit thedata in the normal state using a balanced expression ofthe damping rate Γ N = (1 − x )Γ[ τ ph ]+ x Γ[( τ − ph + τ − el ) − ]using for the two contributions the generic form of Eq.(1), by assuming a fraction x = 0 .
17 of TLS interactingwith electrons with a coupling constant K = 0 .
07 wellbelow 1, which allows us to neglect Kondo-type strongcoupling corrections [34, 35], yielding an electron-TLScoupling energy V in the 0.1 eV range. For comparison,we have also fit the data obtained on another sample,namely the goalpost-shaped silicon nanocantilever, cov-ered with aluminum similar to that of the high-stress SiNsample, measured in Ref. [23]. The dissipation in thesuperconducting state features a T / power law at lowtemperatures which might owe to a non-linear dispersionrelation of flexural mechanical modes in the device [36].The data are reproduced using a semi-phenomenologicalexpression for the TLS-phonon relaxation rate [30]. Thenormal state data were fit with parameters x = 0 .
12 and K = 0 .
11, comparable with the ones used for thehigh stress SiN sample. This is consistent with a mecha-nism independent from the mechanical properties of theresonator, as the interaction lengthscale is much smallerthan any mechanical dimension.The model captures the data in the whole tempera-ture range, as seen in Fig. 3. In particular, it reproducesthe sublinear power law-like behaviour in the 0.1 K-1 Krange, consistently reported in previous works [7, 14, 23]in similar experimental conditions. Although our modelcaptures to a large extent the saturation observed below150 mK and reported in nearly every nanomechanicaldamping study for the lowest operation temperatures, itis likely that a hot electron effect also causes overheatingof the structure [37]. This saturation can be estimatedthrough the frequency shift measured in the normal state:it provides a thermometer by using the deviation from thelogarithmic shift at the lowest temperatures, indicatinga saturation near 120 mK here [see Fig. 2b), blue dots],consistent, within experimental accuracy, with the ob-served damping saturation slightly above the theoreticalfit below 100 mK [see Fig. 3, blue dotted line].The proposed modeling is fairly generic, and makesminimal assumptions on the microscopic nature and lo-cation of TLS. An educated guess based on our resultswould locate those TLS which interact with electrons atthe interfaces, between the SiN and Al layer, and be-tween the Al layer and its native oxide at the NEMSsurface: indeed, tunneling atoms are less likely to existwithin the metallic layer due to long-range order, leavingkinks on dislocations, which are only weakly interactingwith electrons, as most probable candidates for TLS inpolycrystalline aluminum, as proposed in Ref. [9, 38].As an opening for further investigations, we have mea-sured the resonant frequency noise of our device usingthe dynamical bifurcation properties of the NEMS in theDuffing regime [39, 40]. The observed spectrum is thatof a 1 /f -type noise typical of a collection of switchingtwo-level systems [41]. Notably, a visible increase of itsmagnitude is observed below 1 K when the metallic layeris switched to the normal state, as seen in Fig. 4. Sinceat these temperatures the switching, which is due to tun-neling, is mainly induced by TLS-electrons interactions,it is reasonable to expect that electron-TLS interactionscause the excess frequency noise: the tunneling events oc-curring during TLS relaxation cause local rearrangementof atoms, and may thus lead to stress (i.e., frequency)fluctuations.In conclusion, our results support the idea thatelectron-driven TLS relaxation in metallic nanomechan-ical structures is the dominant mechanism of damp-ing, through timescale decoupling between phonon- andelectron-induced TLS relaxation. This may bring an an-swer to several issues raised over the last two decadesby nanomechanical damping measurements at low tem-peratures. In addition, we expect that measurements offrequency noise may shed further light on microscopicmechanisms at work, possibly highlighting interactionsbetween TLS [42], through e.g. a careful extraction ofthe exponent of frequency noise [43].We acknowledge the use of the N´eel facility Nanofab for the device fabrication. We acknowledge support fromthe ERC CoG grant ULT-NEMS No. 647917, StG grantUNIGLASS No. 714692. The research leading to theseresults has received funding from the European Union’sHorizon 2020 Research and Innovation Programme, un-der grant agreement No. 824109, the European Mi-crokelvin Platform (EMP). ∗ olivier.maillet@aalto.fi[1] H. G. Craighead, “Nanoelectromechanical systems,” Sci-ence , vol. 290, no. 5496, pp. 1532–1535, 2000.[2] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali,and A. Bachtold, “A nanomechanical mass sensor withyoctogram resolution,”
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Phys. Rev. Lett. , vol. 101,p. 197203, Nov 2008.
Relaxational TLS-strain interaction
The following derivation is adapted from several reference texts [31], which we briefly recall while adding featuressuch as the reduced dimensionality (phonon-wise) and the string nature of the nanomechanical beam. A single TLS(see Fig. 1 of the main text) is modelled as a double-well potential, each one locally in its ground state (the intra-wellenergy spacing is on the order of the Debye frequency (cid:29) k B T / (cid:126) ). The double well is characterized by its asymmetry∆, barrier height V and distance d in space, which is typically a few ˚A. Following the WKB approximation we modelthe overlap between the two wells’ wave functions | Ψ L (cid:105) , | Ψ R (cid:105) to vanish exponentially as (cid:104) Ψ L | Ψ R (cid:105) = e − λ G where λ G = d √ m a V / (cid:126) is the Gamov factor. The coupling matrix element between the two wells write ∆ = (cid:126) Ω e − λ G ,7where Ω is a reference energy scale. The Hamiltonian of a single TLS in the position basis {| Ψ L (cid:105) , | Ψ R (cid:105)} thus writes:ˆ H ( p )0 = 12 (cid:18) ∆ ∆ ∆ − ∆ (cid:19) = ∆ σ x + ∆2 ˆ σ z , (S04)where we introduce the usual Pauli matrices σ x,z . Eigenenergies ε ± = ± (cid:112) ∆ + ∆ / ε = ε + − ε − = (cid:112) ∆ + ∆ . In typicalamorphous solids at low temperatures compared to the glass transition temperature, it is commonly assumed [31]that the distribution of TLS follows P (∆ , ∆ ) = P / ∆ . This rewrites for a different set of variables using thejacobian transform P ( ε, ∆ ) = P / (∆ (cid:112) − ∆ /ε ). Let us now assume the NEMS is driven: this corresponds to astrain perturbation which, at the level of the TLS, influences its energy landscape through local atomic displacements.The applied strain field E in the MHz range corresponds to a wavelength much larger than the typical TLS interwellspacing d and therefore can be treated as a linear perturbation to the TLS Hamiltonian, which changes the asymmetryof the TLS but negligibly affects its tunneling amplitude. In the position basis, we thus write it ˆ H ( p ) int,ph = − γ E ˆ σ z ,where γ = ∂ ∆ /∂ E is the deformation potential energy, i.e. the TLS-strain coupling constant. Rewritten in the TLSenergy basis {| − ε/ (cid:105) , | ε/ (cid:105)} ≡ {| g (cid:105) , | e (cid:105)} the interaction Hamiltonian is:ˆ H ( ε ) int,ph = − γ E ε (∆ˆ σ z − ∆ ˆ σ x ) . (S05)We then introduce the occupation probabilities of the TLS p g,e and its polarization p = p e − p g . At thermal equilibrium p = (cid:104) ˆ σ z (cid:105) T = − tanh( ε/ k B T ). Applying a classical strain field E = E cos( ωt ) modulates the energy splitting andthus leads to a change in polarization δp = p − p away from equilibrium. For small perturbations, one can write theevolution of p using the relaxation time approximation:˙ p = − p − p st τ , (S06)where p st is the instantaneous equilibrium value corresponding to a given strain magnitude E , and τ the relaxationtime of the TLS. Developing at first order in the strain field, one can link p st to the thermal equilibrium polarization p : p st ≈ p + ∂p ∂ ∆ · ∂ ∆ ∂ E E = p + γ E k B T ∆ ε sech (cid:18) ε k B T (cid:19) . (S07)Eq. (S06) can now be rewritten for δp : τ ˙ δp = − δp − γ E k B T ∆ ε sech (cid:18) ε k B T (cid:19) . (S08)Assuming a sinusoidal excitation at frequency ω , i.e. E = E e iωt , one can define a TLS susceptibility in polarization χ ( ω ) = δp/ E : χ ( ω ) = γ E k B T ∆ ε sech (cid:18) ε k B T (cid:19) − iωτ . (S09)From this we can write the power dissipated by the applied strain through a single TLS relaxation: using the interactionHamiltonian (S05), the applied strain changes the internal energy by an amount δU ( t ) = γ ∆ δp ( t ) /ε . The imaginarypart in frequency domain of the energy change, proportional to that of the TLS susceptibility (S09), corresponds tothe dissipated energy, from fluctuation-dissipation theorem. Integrating over an oscillation cycle, we thus obtain thepower dissipated by a continuously strain-driven TLS: P ε = ( γ E ) k B T (cid:18) ∆ ε (cid:19) sech (cid:18) ε k B T (cid:19) ω τ ω τ . (S010)We then introduce the reduced variables u = ε/k B T, v = ∆ /ε . The power dissipated by a collection of TLS per unitvolume thus writes P V = P ω ( γ E ) (cid:90) ∞ d u (cid:90) d v v − v sech (cid:16) u (cid:17) ωτ ( u, v, T )1 + ω τ ( u, v, T ) (cid:124) (cid:123)(cid:122) (cid:125) I ( T ) . (S011)8Note that the integral I ( T ) diverges in the general case. However, the explicit dependence on u, v of the TLSrelaxation time usually regularizes the integrand. To obtain the total power dissipated we sum over the whole NEMSspace. However, care must be taken because the strain is not uniform for our structure, a point which is not usuallyconsidered in the standard approach for TLS-driven dissipation but was addressed by Ref. [8]. Note that this isdifferent from considering the contribution of string elongation to energy loss, which can be neglected [8]. For a stringundergoing flexure [11], the macroscopic imposed strain field oscillates with an amplitude E ( x t , z ) = ∂ Ψ( z ) ∂z x x t , (S012)where x is the NEMS oscillation amplitude at mid abscissa, x t the local coordinate spanning the beam thickness from − e/ e/ z ) is the excited mode shape. For a high-stress doubly clamped beam, the fundamental flexuralmode shape writes: Ψ( z ) = cos( πz/l ). The total power is thus P = w × (cid:82) d z d x t P V ( x t , z ). We equate it with theexpression obtained by macroscopic arguments from Newton’s second law P = mω x Γ, where m = ρewl/ ρ being its mass density. We finally obtain the generic expression of the TLS contributionto the NEMS damping rate:Γ[ τ ] = Cω (cid:90) (cid:90) d ε d∆ P ( ε, ∆) (cid:18) ∆ ε (cid:19) sech (cid:18) ε k B T (cid:19) ωτ ( ε, ∆)1 + ω τ ( ε, ∆) , (S013)where C = P γ π e / σ l , σ = 1 . ± C can be seen as a dimensionlessTLS-strain coupling strength. In particular, at high enough temperature, the damping rate (S013) reaches a plateau,regardless of the microscopic TLS relaxation mechanism:Γ ≈ ωτ (cid:28) πωC . (S014)Interestingly, the C value obtained from the plateau in damping has been experimentally found to vary only little(10 − − − ) for a wide range of macroscopic amorphous solids, which has led to a debate on a possible universalityof glasses at low temperature [24, 44]. Note that since ω ∝ √ σ and C ∝ /σ , in the high-stress limit the dampingwill essentially follow Γ ∝ / √ σ , that is, an axial stress load reduces the dissipation induced by TLS.Let us also mention that our derived value of C differs from the usual expression C = P γ /ρc , where c = (cid:112) E/ρ is the typical sound speed in the material, E being the material Young’s modulus. However, we stress that thediscrepancy is due to the string nature of the high-stress mechanical resonator combined with its high aspect ratio.The link between our value and the commonly derived one C is: C = (cid:16) πel (cid:17) E σ C . (S015)This result is quite interesting: it directly links the damping rate to the aspect ratio of the string, quadratically, ascompared with a bulk macroscopic body.In the low temperature limit, TLS typically relax to equilibrium on a timescale longer than the driving period, i.e. ωτ (cid:29)
1. In that limit, the damping rate is an explicit function of τ and therefore depends on the microscopic TLSrelaxation mechanisms: Γ = C (cid:90) ∞ d u (cid:90) d v v − v sech (cid:16) u (cid:17) τ − ph ( u, v, T ) , (S016)We first consider relaxation only to a phonon bath. The generic relaxation rate can be obtained using perturbationtheory in the interaction Hamiltonian (S05), where now E is the strain operator relative to all phonon modes ofwavevector k and polarization s . We make a few assumptions: first, the beam confines transverse modes, suchthat their lowest energies are (cid:126) c/w, (cid:126) c/e (cid:29) k B T below 1 K (this is equivalent to saying that the dominant thermalwavelength is larger than these modes’ wavelengths). As a result, these modes are not thermally excited and thuswill not contribute to the TLS relaxation. Second, since shear produces a much smaller deformation locally than alongitudinal strain, we assume that torsional modes, that emerge from shear, couple much less to TLS than longitudinalmodes do. This leaves only longitudinal modes and the two flexural families. Let us address first the longitudinalmodes. In a second quantized form [45]: (cid:12)(cid:12)(cid:12) ˆ E (cid:12)(cid:12)(cid:12) = (cid:88) k (cid:115) (cid:126) ρV ω k k (cid:16) ˆ a † k + ˆ a − k (cid:17) . (S017)9The matrix element Q emk (∆ , ε ) for such an interaction, considering a TLS emission event into the phonon mode, is | Q emk (∆ , ε ) | = (cid:12)(cid:12)(cid:12)(cid:68) g, n k + 1 (cid:12)(cid:12)(cid:12) ˆ H ( ε ) int,ph (cid:12)(cid:12)(cid:12) e, n k (cid:69)(cid:12)(cid:12)(cid:12) = γ ∆ kε (cid:115) (cid:126) ( n k + 1)2 ρV ω k , (S018)while the matrix element for the absorption process writes | Q absk | = | Q emk | (cid:112) n k / ( n k + 1). Using thermal average overthe squared matrix element and through Fermi’s Golden Rule, one then obtains the relaxation rate of a single TLSto the phonon bath: τ − ph = 2 π (cid:126) (cid:88) k (cid:104)(cid:68) | Q emk (∆ , ε ) | (cid:69) T δ ( ε − (cid:126) ω k ) + (cid:68)(cid:12)(cid:12) Q absk (∆ , ε ) (cid:12)(cid:12) (cid:69) T δ ( ε + (cid:126) ω k ) (cid:105) . (S019)Going to the continuum limit, and using the Debye approximation for the phonon modes ω k = c l k , ( c l = (cid:112) E/ρ isthe longitudinal sound speed) the relaxation rate reads: τ − ph = γ (cid:126) ρewc l ∆ ε coth (cid:18) ε k B T (cid:19) . (S020)Note that this result was derived independently in Ref. [15]. It may be instructive to write τ − ph in reduced variables: τ − ph = γ (cid:126) ρewc l (1 − v ) uk B T coth (cid:16) u (cid:17) , (S021)where we see that the u dependence outside the coth term must be the same as the one in T because of the transfor-mation u = ε/k B T . This temperature dependence directly reflects the mechanical damping temperature dependenceat very low temperature. We use this property to disregard the contribution of flexural modes: for these, the decom-position of the strain field writes:ˆ E f = (cid:88) n (cid:115) (cid:126) ρew (cid:82) Ψ n ( z )d zω n (cid:0) ˆ a † n + ˆ a n (cid:1) y ∂ Ψ n ( z ) ∂z . (S022)In the string limit, the mode shapes write Ψ n = cos[( n +1) πz/l ] and frequencies are linked to wave numbers k n = nπ/l through a linear dispersion relation ω n = c f k n , with c f = (cid:112) σ /ρ the sound speed for the two flexural families (in thestring limit they have the same sound speed and mode profiles). Thus (cid:82) Ψ n = l/ x t , z ) in the plan parallel to flexural motion leadsto a spatially dependent relaxation rate to the flexural modes: τ − ph,f ( ε, ∆ , x t , z ) = 2 πγ ∆ x t ρewlc f ε (cid:88) n k n cos ( k n z ) coth (cid:18) (cid:126) c f k n k B T (cid:19) δ ( ε − (cid:126) c f k n ) . (S023)The spatial average of this rate, which has to be done in order to obtain the damping rate in the low temperaturelimit [Eq. (S016)], removes the cos term. Besides, since the fundamental flexural mode is ∼ . (cid:80) → (cid:82) d kl/π , one obtains: (cid:104) τ − ph,f ( ε, ∆ ) (cid:105) ∝ ∆ ε coth (cid:18) ε k B T (cid:19) , (S024)which, written in reduced variables, yields a T dependence of the damping, essentially negligible at low temperaturecompared to the ∝ T damping due to TLS relaxation to longitudinal modes. Phonon-driven TLS relaxation in the goalpost cantilever case
Although we are not aware of a fully comprehensive theory that could explain from first principles the T / dependence of the damping rate measured in Ref. [23], this dependence can be obtained by making some assumptions10of a functional form of the phonon-driven TLS relaxation rate. Following Eq. (S013), we write in the low temperaturelimit, where ωτ ph (cid:29)
1: Γ = C (cid:48) (cid:90) ∞ d u (cid:90) d v v − v sech (cid:16) u (cid:17) τ − ph ( u, v, T ) , (S025)where C (cid:48) = (cid:82)(cid:82)(cid:82) V P γ E / C derived earlier, due todifferent strain fields associated with goalpost cantilever modes. For the same reason, we cannot explicit the phonondriven relaxation rate for lack of a consistent hypothesis about the phonon modes involved in the TLS relaxationprocess. Nevertheless, from Eq. (S025) we notice that once τ ph is written in reduced variables ( u, v ), its dependence in T is directly reflected by that of the NEMS damping rate. Therefore, based on the damping rate measurements of thegoalpost cantilever [23], we infer a dependence τ − ph ( u, v, T ) ∝ T / . This implies, from the variable transformation u = ε/k B T (see ending discussion of the previous paragraph), that a factor u / appears, since any temperature dependenceoutside the coth term must be canceled when written in explicit variables. In addition, phonon emission and absorptionrates, which are summed to obtain the total relaxation rate, must obey detailed balance, i.e. τ − ph,em /τ − ph,abs = e ε/k B T .This leads to writing the total relaxation rate τ − ph ( u, v, T ) = f ( v ) u / coth (cid:0) u (cid:1) T / . Finally, the action of theinteraction Hamiltonian (S05) on the TLS states Hilbert space is independent from the phonon bath characteristicsand thus is unaffected by the thermal average. The matrix element Q em,abs derived in the previous section for thestring case is thus still ∝ ∆ /ε = √ − v . To sum up, one can write the phonon-driven TLS relaxation rate in asemi-phenomenological way: τ − ph = α (1 − v ) u / coth (cid:16) u (cid:17) T / , (S026)where α is a parameter that contains the TLS-strain coupling energy and ultimately depends on the NEMS geometryand the thermal strain field spatial dependence. Combining the obtained phonon-driven TLS relaxation rate (S026)with Eq. (S025) and performing the integration, one can explicit further the NEMS damping rate:Γ = (1 − √ / √ πζ (cid:18) (cid:19) αC (cid:48) T / ≈ . αC (cid:48) T / , (S027)from which α and thus τ ph may be numerically obtained, using the value of C (cid:48) = 1 . × deduced from the dampingplateau at high temperatures and leaving α as a fit parameter to match the low temperature T / asymptotic law ofthe damping data in the superconducting state (here we obtain α ≈ K − / .s − ). These values can then be usedwhen fitting the data in the normal state to our model including electron-driven TLS relaxation (see main text andsection below). TLS-electron interaction
An incoming conduction electron with wave vector k may be scattered due to the atomic potential to which theTLS contributes. In addition, it may also trigger TLS tunneling (electron-assisted tunneling) by modifying the TLSdouble-well potential barrier. The Hamiltonian of the TLS-electron interaction writes in the electron Fock state { (cid:78) k,η | n k,η (cid:105)} ⊗ TLS position state basis {| Ψ L (cid:105) , | Ψ R (cid:105)} [34]:ˆ H ( p ) int,el = (cid:88) k,q,η (cid:0) V zk,q ˆ c † q,η ˆ c k,η ˆ σ z + V xk,q ˆ c † q,η ˆ c k,η ˆ σ x (cid:1) , (S028)where ˆ σ x,z are the usual Pauli matrices, ˆ c ( † ) k,η is the fermionic annihilation (creation) operator for an electron of wavevector k and spin η , and V x,zk,q are longitudinal (z) and transverse (x) interaction constants. One can show that theelectron-assisted tunneling interaction parameter satisfy V xk,q ∼ (cid:104) Ψ L | Ψ R (cid:105) V zk,q (cid:28) V zk,q and can therefore be neglected.Let us consider a possible microscopic mechanism of TLS relaxation to the electron bath: for an excited TLS ofenergy splitting ε to relax to its ground state, tunneling must occur, which leads to a re-arranging the local atomicconfiguration and thus to a modification of the local Coulomb potential. An incoming electron with wavevector k will be sensitive to such a potential landscape change within a radius given by the typical screening length λ of themedium. This leads to an excitation transfer between the TLS and the electron. The Hamiltonian (S028) effectivelycaptures this scenario: rewritten in the TLS energy basis,ˆ H ( ε ) int,el = − (cid:88) k,q,η V zk,q ε ˆ c † q,η ˆ c k,η (∆ˆ σ z − ∆ ˆ σ x ) , (S029)11which makes TLS transitions appear when electrons are (inelastically) scattered. The transition amplitude associ-ated to e.g. an absorption event and involving an incoming electron of wave vector k writes, assuming state | q, η (cid:105) isintially unoccupied: M k,q,η ( ε, ∆ ) ≡ (cid:10) n q,η + 1 , n k,η − , e (cid:12)(cid:12) ˆ H ( ε ) int,el (cid:12)(cid:12) n q,η , n k,η , g (cid:11) = V zk,q ∆ ε (cid:113) n k,η ( n q,η + 1) (S030)Using Fermi’s Golden Rule, taking into account both absorption and emission, we thus obtain the relaxation rate ofa TLS with energy splitting ε to the electron ensemble: τ − el ( ε, ∆ ) = 2 π (cid:126) (cid:88) k,q,η (cid:10) | M k,q,η ( ε, ∆ ) | (cid:11) T (cid:2) δ ( E k − E q + ε ) + δ ( E k − E q − ε ) (cid:3) , (S031)where (cid:104) ... (cid:105) T denotes thermal average over the electron reservoir at thermal equilibrium. Applied to the matrix element(S030) squared, it makes the product f ( E k )[1 − f ( E q )] appear, where f ( E ) = 1 / [exp([ E − µ ] /k B T ) + 1] is the Fermi-Dirac distribution of the electron bath at equilibrium with temperature T and chemical potential µ . Going to thecontinuum limit and introducing the (single-spin) electronic density of states (DoS) n , one obtains: τ − el ( ε, ∆ ) = 4 π ∆ Ω (cid:126) ε (cid:90) ∞ n ( E ) f ( E ) (cid:0) n ( E + ε )[1 − f ( E + ε )] V E,E + ε + n ( E − ε )[1 − f ( E − ε )] V E,E − ε (cid:1) d E, (S032)where Ω ∼ λ is the effective interaction volume, λ (cid:46) f ( E )[1 − f ( E ± ε )]),only quasiparticle excitations within a bandwidth k B T (cid:28) µ around the Fermi level effectively interact with the TLS,which is also a low-energy excitation such that ε (cid:28) µ . Therefore we can make the approximations n ( E ) ≈ n ( n ≈ . × J − · m − is the single-spin DoS at Fermi level), V E,E (cid:48) ≈ V . We are left with integrals (cid:82) d Ef ( E )[1 − f ( E ± ε )]which are analytical. In the limit k B T (cid:28) µ , we finally obtain: τ − el ( ε, ∆ ) = 4 π ∆ K (cid:126) ε coth (cid:18) ε k B T (cid:19) . (S033) Additional data
Here we present additional measurements of a SiN/Aluminum nanomechanical string [46] having the same mechan-ical properties as the beam presented in the main text aside from its length (50 µ m). This nanostring, resonating at3 .
79 MHz, is embedded in a 6 GHz superconducting niobium microwave LC resonator enabling its dispersive readout.This allows us to make measurements free of magnetomotive loading, but prevents operation above 1 K and quenchingof the aluminum layer covering the beam to the normal state, since a strong magnetic field cannot be used in theseconditions. Yet, we use the frequency shift and damping data to test the STM prediction for pure phonon-assistedrelaxation of TLS.The resonance frequency shift and damping rate measurements are summed up in Fig. S1. The data are presentedwith the back-action contributions (optomechanical spring and damping) [47] carefully removed. The frequency shift[Fig. S1a)] shows a logarithmic dependence in temperature, pointing again towards a TLS mechanism. The extracted C = 5 . × − is low compared to values reported for bulk materials but is reasonably close to values measured inthe NEMS literature [9]. The damping rate data are captured with our model using Eq. (S013) and the relaxationrate (S020). We fit the data to our model [solid dark red curve in Fig. S1b)] with the same TLS-strain couplingconstant γ = 9 . P = 7 . × J − · m − , whichhas the same order of magnitude as the one used for the main sample. The parameters choice is somewhat loose,in the sense that we cannot measure and fit the damping up to a plateau due to an operation limited to very-lowtemperature. Note that the value of C extracted by extrapolating our data to a high temperature plateau differs fromthe one measured by frequency shift by only a factor 1.6 (close to that obtained for the main sample), again reflectingpossible refinements due to inhomogeneous TLS distribution.For completeness, we have represented a curve based on a hypothetic electron-assisted damping if one were able toquench the Al layer, using identical interaction strength K and fraction x parameters as those of the main sample.While an excess damping is clearly present below 300 mK, it is not significantly contributing at higher temperatures,being contained in the experimental accuracy. Therefore, a contribution arising from quasiparticles (which will besmaller since their density is smaller than that of normal state electrons) is essentially negligible here, except around1K, precisely where the purely phononic model falls a little below the measured damping.12
10 100 1000-2.5x10 -5 -2.0x10 -5 -1.5x10 -5 -1.0x10 -5 -5.0x10 -6 R e l a t i v e F r equen cy S h i ft ( H z ) T (mK) D a m p i ng r a t e ( H z ) T (mK) normal state electroniccontribution
FIG. S1. a) Resonance frequency shift of a 50 µ m long SiN nanostring as a function of temperature. The solid line is alogarithmic fit C ln( T /T ) b) Damping rate of the NEMS resonator as a function of temperature. The dark red solid line isa fit using Eq. (1) of the main text, with the phonon-driven relaxation rate of TLS derived in this section. The red dashedline is the low temperature asymptotic law, linear in temperature. The blue dashed line represent the extra damping due toconduction electrons if the Aluminum layer were quenched into its normal state. The black dashed line is an empirical ∝ T .65