Thermal resistance at a solid/superfluid helium interface

Transcription

1 SUPPEMENTARY INFORMATION DOI: /NMAT4574 Thermal resistance at a solid/superfluid helium interface Aymeric Ramiere 1, Sebastian Volz and Jay Amrit 1,* 1 aboratoire d Informatique pour la Mécanique et les Sciences de l Ingénieur, IMSI-CNRS UPR 51, Université Paris-Sud, Rue John von Neumann, Orsay, France. aboratoire Énergétique Moléculaire et Macroscopique Combustion, EMC-CNRS UPR 88, École Centrale Paris, Grande voie des Vignes, 995 Chatenay-Malabry, France. *Correspondence to: jay.amrit@limsi.fr Change in bulk acoustic properties of superfluid 4 He with pressure Both the bulk density and the sound velocity c of superfluid 4 He increase strongly with pressure and their respective dependencies, after fitting the data of Brooks and Donnelly 1, are found to be: 5 ( P) P P in g/cm, where the pressure P is in bars; and c ( P) P P in m/s. These expressions are used in the forthcoming calculations. Note that the temperature dependences of and c are negligible below K. The dominant thermal wavelengths of phonons in superfluid 4 He are pressure and temperature dependent since ( hc ( P)/.8k T). As illustrated in Figure S1, identical wavelengths can be achieved at different pressures and temperatures in our experiment. B NATURE MATERIAS Macmillan Publishers imited. All rights reserved.

2 T (K) 1, SVP 0, (nm) Figure S1: Thermal wavelengths of phonons in superfluid 4 He. Identical values of are accessible at different temperatures and pressures, as illustrated here for four different pressures: SVP (red curve), 5 bars (green curve), 10 bars (violet curve) and 4 bars (blue curve). Silicon crystal surface (111) nm 0.00 nm Figure S: Typical image of one of the various measurements conducted by AFM over an area of 100 nm x 100 nm on our Si sample surface. The image size is 51 x 51 pixels. The minimum lateral resolution is nm. 016 Macmillan Publishers imited. All rights reserved.

3 Experimental Owing to the extremely high thermal conductivity of superfluid helium (several orders of magnitude higher than metal conductors like pure copper), thermal gradients in the superfluid do not exceed 10µK/cm for heat fluxes of ~0.1W/cm. The temperature is therefore uniform throughout the experimental cell to within much less than 0.1 mk in our experiment. While the temperature of the superfluid T in the cell is regulated to a reference value, data points are acquired at different 4 He pressures varying from the saturated vapour pressure (SVP) to ~5 bars. Figure S shows typical data recording at 0. bars, 15 bars and 4 bars, conducted at T ( ) K on different dates. Under the same heater power and PID parameters for the temperature controller, the regulation of T is highly reproducible to within less than 1%. The temperature evolutions of T 1, T and T are also clearly highly reproducible at a given pressure as the heater power is varied. Thermometers reach stationary states within a few µs after the heater power injected into the crystal is changed. Figure S: Data recordings of temperatures under identical heat fluxes (indicated by numbers in mw) at three different pressures. Each set of data was recorded on different dates for T = 0.6 K. 016 Macmillan Publishers imited. All rights reserved.

4 At zero applied power the shifts in the temperature indicated by the thermometers mounted on the crystal ( T 1, T and T ), with respect to T, are due to residual heat leaks from the current leads. This has been extensively discussed in ref. []. We emphasize the point that all corrections mentioned in ref. [] were taken into account here. In this experiment, the RuO thermometer which indicates the liquid 4 He temperature T, is subjected to strong pressure changes of the order of ~5 bars. Figure S shows no effect of these pressure changes on T. Also, the stability of the manometer readings during acquisition, which lasted about ~0 minutes for each data point shown in Figure S, confirms no detectable pressure losses due to possible leaks along the 4 He filling line. Pressure dependence of the Khalatnikov acoustic mismatch model for liquid He/solid interfaces The Kapitza resistance in the Khalatnikov acoustic mismatch 4 model is given by:, 4 RK 0 (0 Sict Si) /( kbz( P) FT ) (1) where refers to the reduced Planck constant, kb to the Boltzmann constant, Si. g/cm is the Si density and 5 ct, Si cm/s corresponds to the transverse speed of sound in the (111) direction in Si. The factor F ~ 1.6 is due to the presence of Rayleigh waves at the interface. Clearly, R K 0 contains a pressure dependency since the bulk acoustic impedance Z c varies by ~80% as the pressure changes from the saturated vapour pressure to ~5 bars. Consequently, R K 0 decreases with pressure approximately ten times faster than our experimental values of R K which vary by ~1% at the lowest temperature. The pressure dependency of R K 0 is taken into account in our analysis when determining the experimental root-mean-square surface roughness. Superfluid density variations with pressure at the silicon surface In this section we investigate the role of the superfluid 4 He density as the helium pressure is varied from SVP to ~5 bars. The bulk nature of superfluid helium maybe altered within a very fine layer of thickness when in contact with a solid surface. This mechanism depends on the strength of the van Macmillan Publishers imited. All rights reserved.

5 der Waals forces which creates a pressure gradient 5 in the x-direction perpendicular to the surface according to the equation : / mv x P x where and m are the density and the atomic mass of helium. The van der Waals potential energy of interaction between a 4 He atom and the surface has the form: V A / x, where A is a constant. The density variation at a point x within the layer of thickness can be Taylor expanded as ( x) ( x) / PP / x density profile can be cast as: where. Hence the He, S x d0 ( x) () 1 T d0 x x He, S refers to the density of solid 4 He, d 0 corresponds to the distance from the Si crystal surface over which the helium density is, P He, S 1 is the compressibility of liquid T helium and depends on physical constants. Based on the work by Dzyaloshinskii et al. 6, Pikhista and Salistra 7 conducted a more detailed calculation taking into account electromagnetic properties of ( 0 both the solid and helium. They showed that /16 )( 1) where 0 is the dielectric constant of helium and is the resonant frequency in the solid material. Substituting this expression for in equation () and taking as determined by Chase et al. 8 and as measured in silicon by Dresselhaus et al. 9, we plot the density profile at different pressures in Figure S4, with the boundary condition imposing a density of liquid helium lower than that of solid 0 J helium He, S g.cm -. Pressure dependencies of and T were taken respectively from Brooks and Donnelly 1, and Grilly 10. Firstly, Figure S4 shows that the helium density varies nonlinearly as the solid surface is approached. Bulk density values are retrieved at distances as small as ~0.8 nm from the Si surface. Secondly, at distances smaller than d nm from the Si surface, the density reaches a constant value ( ), that is, it becomes independent of the pressure in x He, S liquid helium. Noting that d nm is of the same order of magnitude as the hard core separation between two helium atoms (0.64 nm), the constant density limit therefore appears to be due to the size of the helium atoms. Thirdly, the value of d 0 corresponds only to a fraction of the thickness of a Macmillan Publishers imited. All rights reserved.

6 helium monolayer, which is 0. 6nm. This means that the van der Waals forces are not strong enough to bind 4 He atoms to the Si surface so as to form a complete solid 4 He layer. We also remark that in our experiment all measurements are done under heat currents ranging from a few µw to tens of mw, directed from the Si to liquid helium. A heat flux would induce a pressure gradient in the opposite direction to that due to the van der Waals forces. Johnson and ittle 11 determined the expression for the van der Waals interaction energy between a 4 He atom and a Si surface to be W 0.068x where the distance from the Si surface x is in angstroms and W in ev. For x d nm, we find W 0. 08eV and introducing our heat flux values Q, we confirm that Q dt W. These arguments strongly disprove that helium within a thickness of d nm from the Si surface (into the liquid) can be assimilated to a partial solid helium layer in our experiment. A clear proof also lies in our observation of the variation of 1 K. RK with pressure at temperatures lower than As is evident from Figure S4, d 0 is much less than the rms surface roughness which comes into play in the Adamenko and Fuks model. The dominant wavelengths of thermal phonons in liquid helium, which vary from ~1.5 nm to ~8 nm, largely exceed d 0 by more than an order of magnitude. Thus the effectiveness of the phonon-surface roughness interaction mechanism, as discussed in the main section, remains unaltered. Finally, we examine the possible influence of the density variation of liquid 4 He close to the Si wall and demonstrate that it has no impact on the surface roughness-phonon interaction analysis, conducted in the previous sections. As the Si wall is approached, liquid He becomes acoustically better matched to the Si surface since the density (x), given by equation (), increases as shown in Fig. S4. Consequently, the 4 He sound velocity c ( ( x)) increases linearly with the liquid density (x). Accordingly, the acoustic impedance close to the wall is expected to deviate from the bulk acoustic impedance Z c as the wall is approached. The average liquid 4 He acoustic impedance close to the Si wall is assumed to be defined by Z ( x) c ( ), where the averaging must be performed over a distance equivalent to at least a wall Macmillan Publishers imited. All rights reserved.

7 phonon wavelength from the Si wall. From the expression ( hc ( P))/(.8k T), we deduce that the dominant wavelength has the smallest value of 1. 5 nm under SVP at T. 1K. We note d, min B that is larger than the thickness of approximately monolayers of 4 He, that is, 0. 7 nm. d, min This value covers the entire range within which (x) varies before reaching the bulk value of the solid. Furthermore, Z wall calculated over d, min corresponds to a maximum value of Z wall, max and the ratio Z / ) 1.. This ratio decreases and tends to unity at lower temperatures and/or ( wall, max Z higher pressures. The maximum correction due to this effect on the rms surface roughness determined from our measurements using 1 R 1 R K / wall Z bulk, is proportional to Z / Thus, in the temperature range 0.8- K and for all pressures up to ~5 bars, the density variation of liquid 4 He close to the Si wall has a negligible effect on heat transmission at the interface. 0,19 iquid helium density (g.cm - ) 0,18 0,17 0,16 0,15 He "solid" d o = 0.4 nm svp 0,14 0 0,5 1 1,5 Distance from silicon surface (nm) Figure S4: Calculated liquid helium density profiles as a function of the distance from the silicon surface, for He pressures ranging from SVP to 5 bars Macmillan Publishers imited. All rights reserved.

8 Alternate analysis of interface resistance demonstrating resonant scattering and surface characteristics Following a remark made by one of the referees, in this alternate approach to analyse to our interface resistance data, we assume the correlation length to be one third the dominant phonon wavelength at all temperatures and pressures. This assures the applicability of the AF theory. Using equation (5) (see Methods) we determined the ( / ) values to fit the experimental data. The roughness heights were then calculated using relationship 0.. The results are in very fine agreement with our analysis (see main text). The roughness heights, plotted as a function of the thermal wavelengths in Fig. S5 nicely reproduces Fig. in the main text. The values fall within the range (0.5-4) nm as found in our analysis presented in the main text. Our AFM analysis confirmed their presence on our sample surface. Clearly, the criterion that the roughness heights are one third of the dominant phonon wavelengths for resonant scattering to occur is corroborated. The values fitting our experimental data are shown in Fig. S6. These values fall within the range 1. Figure S6 also indicates that as the wavelengths increase, with temperature and/or pressure, the values decrease. The values vary at most by a factor of Macmillan Publishers imited. All rights reserved.

9 Surface roughness height (nm) 4,5,5 1,5 1 0, K 0.618K 0.78K 0.991K 1.485K 1.819K Thermal wavelength (nm) Figure S5: Roughness heights selected by thermal wavelengths in resonant scattering process at different pressures and temperatures. The dashed line clearly highlights the selection criterion ( / ) K, K l 0.618K 0.417K = l l 1.485K 0.78K 1, Thermal wavelength (nm) Figure S6: Evolution of ( / ) with the thermal wavelength. The curves to the data are added to show tendencies. For each temperature, as the wavelength increases with pressure the Macmillan Publishers imited. All rights reserved.

10 values decrease, and the relationship between and shifts from to. The sensitivity of the scattering mechanism on surface roughness is clearly apparent. Analysis of Kapitza resistance at a solid 4 He/silicon crystal interface using the acoustic mismatch model The Khalatnikov acoustic mismatch (AM) model requires two physical conditions to be met, namely (i) the conservation of phonon frequency across the interface and (ii) the continuity of the parallel components of the phonon wave vectors in the plane of the interface. These conditions define 1 a critical cone of angle c, j sin ( che, j / csi, j ) in the less dense medium (solid 4 He in our case), where c is the speed of sound of phonons and j (,T ) refers to the longitudinal and transverse j phonon branches respectively. The number of phonons of branch j incident on the interface and susceptible of being transmitted into Si must lie within the critical cone. This number is given by d j N( j, T) where N ( j, T) g( j ) nbe is the number of phonons having the frequency j, 4 g 1 / c j 1 corresponds to the usual D density of states and e 1 ( j ) j j n BE is the Bose-Einstein phonon occupation number with T 1. k B d j 4 refers to the fraction of phonons falling within the solid angle d j formed by the critical cone. The heat flux carried by a given branch j of phonons going across a hexagonal close packed 4 He crystal, grown on a (111) surface of a Si crystal can be written as: where d 4 j HeSi, j N( j, T ) jc j j d j () Q j is the transmission coefficient from 4 He to Si. When j is independent of frequency, equation () simplifies to: 4 c, j at HeSi, j j c j 0 Q sin d, where 4 a ( / 60 ). For the sake of j j k B simplicity, c, c, T c and the thermal boundary resistance at the solid 4 He/Si interface Macmillan Publishers imited. All rights reserved.

11 R AM dq dt 1 1 can be written as: R at ( / c ) ( / c ) 1 cos HeSi AM He, T He, T c, after including all three branches. Using the following values for solid helium: He g/cm, che, cm/s and 4 4 che, T. 610 cm/s; and the following values for Si: Si. g/cm, 5 c Si, cm/s and 5 c, 5.10 cm/s; we obtain Si T He, zsi, ) /( zhe, zsi, ) (4z and T (4z He, T zsi, T ) /( zhe, T zsi, T ) Finally, the acoustic mismatch prediction for the solid helium/si interface is: R AM 1.78 cm K.W -1 (4) T (1 cos ) c Our measured value of the Kapitza resistance between solid 4 He and silicon is (4 8) R K, S cm K/W at T 0. 78K. This experimental value can be explained by the AM model if we extend the critical cone angle from (calculated by the sound velocities) to 17 (see Fig. b, main c c text). This value is highly plausible due to the non-ideality of the surface state at scale lengths of the order of phonon wavelengths in solid 4 He. Increasing the critical angle implies that the transmission of phonons of longer wavelengths (and therefore smaller frequencies) is facilitated across the interface by the resonant scattering mechanism as proposed by Adamenko and Fuks. Although surface non-ideality appears to be a very likely reason here, we stress that solid 4 He is a quantum solid with a very large (~5%) zero point-motion and is presenting giant plasticity 1. Adding the presence of dislocations/kinks 1,14 whose motions are affected by thermal phonons, supplementary heat exchange mechanisms may be possible. References 1. Brooks, J. S. & Donnelly, R. J. The calculated thermodynamic properties of superfluid helium- 4. J. Phys. Chem. Ref. Data 6, 51 (1977).. Wilks, J. The properties of liquid and solid helium. (Claredon Press, 1967).. Amrit, J. Impact of surface roughness temperature dependency on the thermal contact resistance between Si(111) and liquid 4 He. Phys. Rev. B 81, 0540 (010) Macmillan Publishers imited. All rights reserved.

12 4. I. M. Khalatnikov. An Introduction to the Theory of Superfludity. (Addisson-Wesley, 1989). 5. Franchetti, S. On the Problem of the Static Helium Film. I.- General Considerations and Density Distribution in the Film. Nuovo Cim. 4, (1956). 6. Dzyaloshinskii, I. E., ifshitz, E. M. & Pitaevshii,. P. General theory of van der Waals forces. Sov. Phys. Uspekhi 7, (1961). 7. Pikhitsa, P. V & Salistra, G. I. Theory of Adsorbed thin films of helium. J. Sov. Phys. 4, (198). 8. Chase, C. E., Maxwell, E. & Millett, W. E. The dielectric constant of liquid helium. Physica 7, (1961). 9. Dresselhaus, G. & Dresselhaus, M. S. Fourier Expansion for the electronic energy bands in silicon and germanium. Phys. Rev. 160, 649 (1967). 10. Grilly, E. R. Pressure-Volume-Temperature Relations in iquid and Solid 4He. J. ow Temp. Phys. 11, 5 (197). 11. Johnson, R. C. & ittle, W. A. Experiments on the Kapitza Resistance. Phys. Rev. 10, (196). 1. Haziot, A., Rojas, X., Fefferman, A. D., Beamish, J. R. & Balibar, S. Giant plasticity of a quantum crystal. Phys. Rev. ett. 110, 18 (01). 1. Balibar, S. The enigma of supersolidity. Nature 464, (010). 14. Balibar, S. Supersolid Helium: Stiffer but flowing. Nat. Phys. 5, 54 (009) Macmillan Publishers imited. All rights reserved.