Quality Factor Thickness (nm) Quality Factor Thickness (nm) Quality Factor 10

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1 Oxygen-Terminated Fluorine-Terminated Length = 24 mm Length = 2 mm Length = 1 mm Length = 12 mm Length = 8 mm 8 mm Width, K 12 mm Width, K 1 mm Width, K 8 mm Width, K 12 mm Width, K 1 mm Width, K Supplementary Figure 1: Thickness-dependence of Q-factor, separated by length, width, surface termination, and temperature. 2

2 a Quality factor x K (el SCD) K (el SCD) Oxygen-terminated b Quality factor 12 9 Oxygen-terminated x K (o SCD) K (o SCD) c Quality factor 12 9 x K (o SCD) K (o SCD) Fluorine-terminated Cantilever Length (µm) Cantilever Length (µm) Cantilever Length (µm) Supplementary Figure 2: Length-dependence of mechanical dissipation. a. Quality factor vs. length for oxygen-terminated diamond cantilevers fabricated from electronic-grade (el-scd) substrate. b. Quality factor vs. length for oxygen-terminated diamond cantilevers fabricated from optical-grade (o-scd) substrate. c. Quality factor vs. length for fluorine-terminated diamond cantilevers fabricated from opticalgrade substrate.

3 a Quality factor x K (el SCD) Oxygen-terminated K (el SCD) b Quality factor x K (o SCD) K (o SCD) Oxygen-terminated c Quality factor x K (o SCD) K (o SCD) Fluorine-terminated Cantilever Width (µm) Cantilever Width (µm) Cantilever Width (µm) Supplementary Figure : Width-dependence of mechanical dissipation. a. Quality factor vs. width for oxygen-terminated diamond cantilevers fabricated from electronic-grade substrate. b. Quality factor vs. width for oxygen-terminated diamond cantilevers fabricated from optical-grade substrate. c. Quality factor vs. width for fluorine-terminated diamond cantilevers fabricated from optical-grade substrate. 4

4 a b Quartz Sandwich O term. DOI O term. Quartz Sandwich, F term. 4 2 Temperature (K) Quartz Sandwich O term. 2 Temperature (K) Supplementary Figure 4: Temperature dependence of mechanical dissipation between and K. a. Quality factor vs. temperature for diamond cantilevers fabricated from optical-grade substrate. b. Quality factor vs. temperature for diamond cantilevers fabricated from electronic-grade substrate.

5 1 f/f(4k) el SCD (28 nm) o SCD( nm) UNCD (27 nm) SC Si (1 nm) Temperature (K) Supplementary Figure : Cantilever frequency change of the studied materials vs. temperature. This figure shows the frequency variation with temperature for the cantilevers shown in Figure 4 in the main manuscript. Among the cantilevers measured, diamond resonators exhibited the lowest frequency variation in the temperature range of the experiments, consistent with diamond having the highest Debye temperature. In excellent agreement with literature results [1], we observed a monotonous frequency variation of.()% over the K- K temperature range for cantilevers thicker than about 2 nm. For thinner cantilevers, such as the optical-grade device shown here, frequency variation up to.11% is seen. Debye temperature of diamond was obtained via least-squares fitting of the data for several thicker cantilevers using Equation (2) given in a paper by Orson L. Anderson [2], and was found to range beween 9 K- K, with an average of 17 K. These values are consistent with known values for diamond []. The large uncertainty is partly a result of the limited temperature range, compared to diamond Debye temperature, surveyed in the experiment.

6 8 a 1 b power law TLS.8 Q ( ) 4 1/Q ( ) T res (K) 1 1/T res (K -1 ) Supplementary Figure : Fit of low-temperature data to two dissipation models. a. Q vs. T plot. b. 1/Q vs. 1/T plot. 7

7 Q factor ( ) a b c Q factor ( ) 1 T refrigerator = 9 mk Refrigerator temperature (K) Laser power incident at cantilever (W) 4 2 T refrigerator = 8 mk Resonator temperature (K) T refrigerator = 8 mk Laser power incident at cantilever (W) Supplementary Figure 7:.1- K measurements of the Q factor for the resonator shown in Figure in the main manuscript. a. Quality factor vs. temperature plot obtained by sweeping refrigerator temperature between.8- K. b. Quality factor vs. laser power plot for two refrigerator temperatures. c. Resonator temperature vs. laser power at refrigerator base temperature inferred from b as described in the text. 8

8 Displacement spectral density (m 2 /Hz) a T bath = 29 K Frequency (khz) Displacement spectral density (m 2 /Hz) b -1 T bath = K T noise = 9 K Frequency (khz) Supplementary Figure 8: Thermal noise measurement of a representative nanoresonator. Displacement power spectral density is shown as a function of frequency for the electronic-grade cantilever shown in Figure 1 and Table 1 in the main manuscript. a. At 294 K bath temperature. b. At K bath temperature. 9

9 Length Width Thickness f K f 4K f mk Q K Q 4K Q mk ( µm) ( µm) (nm) (Hz) (Hz) (Hz) Supplementary Table 1: Mechanical frequencies f and quality factors Q of additional electronic grade resonators measured at T = K, 4 K and refrigerator base temperature (mk). The table collects data from four additional electronic grade resonators that were cooled to the refrigerator base temperature (8 mk). Due to laser heating by the nw interferometer laser the resonator temperature for these measurements was significantly higher than the bath temperature, as is explained in Supplementary Note 1 below. Although we do not precisely know the resonator temperature for the millikelvin measurements in the above table, we estimate that T 4 mk based on the calibration of the device shown in Fig. 2(b) of the main manuscript. The data show that some, but not all devices display an increase in mechanical Q below 1 K. The variability could be easily attributed to the fact that single crystal diamond often has growth sectors of different quality even in the same crystal [4], or to variability in processing. On the other hand, not much change is seen in Q down 4 mk for the extensively studied device from Fig. 2(b). As a side note, we have never observed significant increase in Q in the.1 1 K regime for single crystal silicon devices.

10 Material Density ρ [kg/m ] Young s modulus E [GPa] ρe [kg/(m2 s)] Single-crystal diamond, Polycrystalline diamond ca., ca Single-crystal silicon 2, Supplementary Table 2: Densities and Young s moduli of resonator materials. 11

11 Supplementary Note 1: Dissipation and fitting of the data at < 1 K We have fitted the low temperature ( mk 1 K) Q vs. T data to two simple functional forms, a power law and a model for a thermally-activated two-level system. Power law. Power-law fitting used the following equation, Q 1 = Q 1 [1 + (T/T ) ɛ ], (1) where Q, T and ɛ are free parameters. For the data shown in Fig. in the main manuscript the fit yields Q =.8, T =. K, and ɛ = 1.. Note that this ɛ is considerably larger than the ɛ =. predicted by general standard tunneling model (STM) theory of surface friction [], indicating that this model does not fully describe the dissipation in diamond nanoresonators at low temperatures. Thermally-activated ensemble of two-level systems. Alternatively, we have fit the entire dataset ( mk 2 K) to model involving an ensemble of two-level systems (TLS), such as bulk or surface defects. In this model we assume that the harmonic oscillator interacts with an ensemble of like TLS that have one single energy separation E. Energy transfer occurs due to re-population between the two TLS states as the energy splitting is slightly modulated by the resonator oscillation. Energy transfer is at a maximum if the relaxation rate between the TLS states matches the resonator frequency. Moreover, dissipation is proportional to the occupation probability of the excited TLS state. As temperature is lowered below T = E/k B, the excited TLS state becomes depopulated and dissipation is suppressed. Dissipation in this model will have the following functional form, e β E Q 1 = Q 1 + Q 1 1 e β E, (2) + e +β E where Q is the quality factor for T and Q 1 1 = Q 1 (2Q ) 1, with Q being the quality factor for T. β = (k B T ) 1 is thermal energy. A fit of this equation to our data is shown as a blue curve in Supplementary Figure, and yields Q =.9, Q = 1.2, and T =. K. The corresponding energy splitting is about E/h = 1 GHz. 12

12 Supplementary Note 2: Scaling of resonator parameters and thermal force noise with geometry and material Resonance frequency f c and spring constant k c can be calculated for a singly-clamped rectangular cantilever beam as (Ref. []) f c =.12 t E l 2 ρ, () k c = wt E, (4) 4l where t is the thickness, w is the width, l is the length of the cantilever, ρ is the density of the material, and E Young s modulus. Literature values for density and Young s moduli of single-crystal diamond, polycrystalline diamond and single-crystal silicon are collected in Supplementary Table 2. Using Eqs. () and (4), thermal force noise per unit bandwidth can be expressed as a function of t, w and l, F min = ( ) 4kB T k 1/2 c = 2πf c Q (.17kB T wt 2 ) 1/2 ( Eρ Eρt Ql Q ) 1/2 ( wt l ) 1/2 ( ) wt 1/2 α 1/2, () l where α = Eρt/Q is the geometry-independent dissipation factor that appears with Fig. 4 in the main manuscript. The last expression assumes that Q scales linearly with thickness, and can be used to predict the scaling of force sensitivity with device dimensions at a given temperature. α is related to the mechanical dissipation or loss parameter γ [] as γ α(wt 2 /l). 1

13 Supplementary Note : Estimation of cantilever temperature Cantilever temperature was measured using two different experimental schemes. For elevated temperatures (T =.4 K), refrigerator temperature was slowly swept and the Q value continously measured. At very low temperatures (T < 2 K), where significant heating of the cantilever due to interferometer laser absorption was observed, refrigerator temperature was kept fixed and the laser power incident at the resonator varied. Moreover, the resonator mode temperature was inferred by integrating the thermal noise spectral density around the resonance frequency..4 - K measurements. The standard method for measuring cantilever parameters as a function of temperature involved slow sweeping of the refrigerator temperature while continuously recording the Q value. This method assumes that resonator and refrigerator are always at thermal equilibrium. This method worked well for about T >.4 K when using slow sweep rates (dt/dt <.2 K/min) and low laser powers (P < nw incident at the resonator). mk - 2 K measurements. At even lower temperatures, absorption of interferometer laser light combined with the T decreasing thermal conductivity of diamond leads to resonator heating, and resonator temperature is typically higher than refrigerator temperature. Since the temperature dependence of the thermal conductivity of diamond is well known [7], we can infer the resonator temperature by recording a series of measurements at different laser powers and refrigerator temperatures. This approach turned out to work well in the temperature range T =.1 2 K. In particular, we found excellent agreement of both measurement techniques in the overlap region between.4-2 K. To estimate the resonator temperature from heat absorption measurements, we consider thermal conduction through a slab of uniform cross-section A, given by the width times the thickness of the cantilever, and length L, given by the distance from the interferometer laser spot to the cantilever base. According to Fourier s law of thermal conduction, P = k(t )A T L () where P is absorbed power, k(t ) is the temperature-dependent thermal conductivity, and T/ L the temperature gradient. At low temperatures (< K), the thermal conductivity of diamond scales as T, such that k(t ) αt [7]. We can integrate Eq. () up to L, P L T dl = αa T dt, (7) T b P L = αa 4 (T 4 Tb 4 ), (8) where T b is the bath temperature (at the base of the cantilever) and T the temperature at the location of the interferometer laser spot (the resonator temperature). We obtain an expression for T as a function of absorbed laser power P, T = ( 4L αa P + T 4 b ) 1/4 = ( ɛp + Tb 4 ) 1/4 (9) An approximate value for ɛ is found by comparing quality factor measurements made at T bath = 8 mk and T bath = 9 mk (Supplementary Figure 7(b)). From the T bath = 9 mk curve we find that Q 1.44 for vanishing laser power (P = 1 nw), and at this low laser power, resonator temperature T res will be very close to the bath temperature. From the T bath = 8 mk curve we see that the laser power required to obtain the same Q value is approximately P = µw; thus, at this laser power, the resonator temperature is also about 9 mk. Together these yield ɛ = (T 4 res T 4 bath )/P. 4 K 4 /W with T res = 9 mk, T bath = 8 mk, and P = µw. Note 14

14 that there is an excellent agreement between the two Q vs. T measurement methods in the.4-2 K overlap region (see Figure 2(b) in the main manuscript). Using Eq. (9), we thus find that the resonator temperature at the lowest investigated laser power (.nw) and T bath = 8 mk is about T res = 9 mk. 1

15 Supplementary Note 4: Measurements of resonator thermal noise In thermal equilibrium, a harmonic oscillator exhibits a random (Brownian) motion with rmsamplitude x rms = k B T/k c, where k B T is thermal energy and k c is the spring constant. By measuring x rms we can thus derive the noise temperature T noise = k c x 2 rms/k B of the oscillating mode, and determine whether it is in thermal equilibrium with the physical temperature of the resonator. Alternatively, measurement of x rms at known temperature can be used to infer the spring constant k c. For the experiments presented in this study we performed a thermal noise measurement at room temperature to calibrate the spring constant k c. (This measured k c can be compared to a theoretical k c calculated from resonator geometry an material parameters, see Eq. (4), with good overall agreement). We also performed a thermal noise measurement at low temperatures (typ. K) and found mode temperatures to be in thermal equilibrium down to about K (depending on the particular device), but to be enhanced at lower temperatures due to slight mechanical disturbance by the refrigerator pumps. (We note that slight mechanical excitations do not change the Q factor, as that one is set by the physical temperature of the resonator. An example of a thermal noise spectrum is given in Supplementary Figure 8. 1

16 Supplementary References [1] McSkimin, H. J., & Andreatch, P., Jr., Elastic Moduli of Diamond as a Function of Pressure and Temperature. J. Appl. Phys. 4, 2-7 (1972) [2] Anderson, O. L., Derivation of Wachtman s Equation for the Temperature Dependence of Elastic Moduli of Oxide Compounds Phys. Rev. 144, -7 (19) [] Mukherjee, B. & Boyle A. J. F., On the Debye Characteristic Temperature of Diamond. Phys. Stat. Sol. 22, K11-K14 (197) [4] Martineau, P. M. et al., High crystalline quality single crystal chemical vapour deposition diamond. J. Phys. Cond. Matt. 21, 42 (29). [] Seoanez, C., Guinea, F. & Castro, A. H., Surface dissipation in nanoelectromechanical systems: unified description with the standard tunneling model and effects of metallic electrodes. Phys. Rev. B 77, 127 (28). [] Yasumura, K. Y. et al., Quality factors in micron- and submicron-thick cantilevers. J. Microelectromech. Syst. 9, 117 (2). [7] Berman, R., Simon, F. E. & Ziman, J. M., The thermal conductivity of diamond at low temperatures. Proc. R. Soc. A. 22, (19). 17