Journal of Low Temperature Physics manuscript No.(will be inserted by the editor)

The Thermal Conductance of Sapphire Ball Based DetectorClamps

H.D. Pinckney · G. Yacteen · A. Serafin ·S.A. Hertel · For the Ricochet Collaboration

the date of receipt and acceptance should be inserted later

Abstract In order to provide secure clamping with a low thermal conductance, some lowtemperature detectors are held with point contact sapphire ball clamps. While this method isincreasingly common, the thermal conductance across this interface has not been well stud-ied. We present a direct measurement of the thermal conductance of such clamps between200 and 600 mK, with a clamping force of approximately 0.4 ±0.2 N/clamp. The thermalconductance of a single sapphire-on-copper clamp was found to be 660+360

−210 (T/K)3.08 [nW/K].For a sapphire-on-silicon clamp the conductance was found to be 380+190

−120 (T/K)2.84 [nW/K].The conductance measured is consistent with thermal boundary resistance.

Keywords Thermal conductance, Sapphire ball clamping, detector holder

1 Introduction

The challenge of rigidly holding low-temperature sensors while thermally isolating themfrom the environment typically employs insulating materials of small dimension. One com-mon technique employs sapphire balls [1]. This method takes advantage of the low surfacearea of the point contact between the sphere and the (typically planar) sensor surface tolower the conductance. While this clamping method is increasingly common, the exact ther-mal conductance has not been well documented in the literature. In this article we presenta study of the thermal conductance of such clamps. While this study employs clamps repli-cating a Ricochet collaboration holding style, the result is generally relevant to any sapphireball clamp.

2 Experimental Setup and Procedure

The experimental setup consists of a copper bar suspended by sapphire ball based clamps,see Fig. 1. The clamps are 3.18 mm diameter sapphire balls [2] held in place by springsconstructed of 4 mm × 11 mm × 0.25 mm phosphor-bronze [3]. The clamps are secured to

Department of Physics, University of Massachusetts Amherst,Amherst, MA 01003, USAE-mail: hpinckney@umass.eduar

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an additional set of copper bars that act as an extension of the mixing chamber (MC) plate ofa Cryoconcept Hexadry UQT-B 400 dilution refrigerator. Additionally, we mounted a Cry-oconcept provided heater and RuOx thermometer on the bar to facilitate the measurement.All data was recorded at 1 second intervals via the MMR3 bridge system and logged in theCryoconcept software.

In the following subsections we describe our determination of the clamping force, andthen discuss our “steady-state” and “dynamic” data taking modes.

Ball 3.18mm ⌀

Test BarMC Bar

Clamp (4×11×0.25mm)

RuOx

MC

Bar

1

MC

Bar

2

Test BarHeater

M2 Screw

MC Plate

Fig. 1: Experimental setup. Left: Photograph of a single ball clamp during installation. Right:Photograph of the entire assembly. The central copper test bar is held by many clamps (halfare on the back side, not visible here). A single sapphire ball acts as a spacer at the bottomside, and it was observed to not make strong contact. A heater and thermometer are fixed tothe bar at top and middle respectively. In this specific test, 92 sapphire ball based clamps areinstalled, and a polished Si surface has been installed between clamps and test bar.

2.1 Clamping Force

The force of each clamp was estimated by the force of kinetic friction on the bar, and cross-checked with Euler-Bernoulli beam theory.

To perform the kinetic friction measurement, the bar was held in place with 12 clamps,and using a scale we measured that the bar required 440 ± 30 g of force to be moved.To convert this frictional force to a normal force, we require the coefficient of kinetic fric-tion of sapphire on copper. If we assume the plowing effects demonstrated in [4] are linear

The Thermal Conductance of Sapphire Ball Based Detector Clamps 3

in clamping force, an assumption to which we assign 50% uncertainty, the coefficient offriction is µk = 0.67 ± 0.34. This results in a force of approximately 0.4 ± 0.2 N/clamp.

Euler-Bernoulli beam theory describes the force of a bent beam as

Fclamp =Ewt3z

4L3 , (1)

where E is the Young’s Modulus of the spring material, w is the width of the spring, t is thethickness of the spring, z is the spring deflection distance, and L is the length of the spring.Using measurements of our springs and an assumed Young’s Modulus of 100 ± 10 GPa [3]we get a force of approximately 2 ± 1.2 N/clamp. This prediction is in order-of-magnitudeagreement with the value estimated from the friction measurement.

2.2 Conductance Measurement Methods

Thermal conductance was measured in two complementary measurement methods we’llrefer to as “steady-state” and “dynamic”.

2.2.1 Steady-State Mode

In the steady-state measurement mode, we used the heater to supply a power to the clampedbar and measured the temperature difference between the bar and the MC plate (i.e., acrossthe clamps). This assumes the clamps are the dominant thermal resistance between the barand MC thermometers.

We assume a thermal conductance of the form [5]:

G = AT n =dPdT

, (2)

where G is the thermal conductance, T is the temperature of the material, P is the totalpower on the bar, and A, n are constants. Assuming this form we integrate to find a rela-tionship between the temperature difference measured, the power applied, and the thermalconductance:

P = Pheater +Pexternal =A

n+1

(T n+1

bar −T n+1MC

)(3)

where Tbar is the temperature of the bar and TMC is the temperature of the MC. We haveseparated the total power on the bar into the power applied with the heater (Pheater) and theincidental power from other external heat loads (Pexternal). This external heat load is likelydominated by the wiring to the bar’s heater and thermometry.

The data-taking procedure was to supply 10 different heater powers at a constant TMC,measure Tbar at each heater power, and fit for A, n, and Pexternal . We performed this measure-ment for a setup with both 48 and 92 clamps, and with TMC fixed at both 9 mK and 150 mK.We also tested for a dependence on substrate material by inserting pieces of polished siliconwafer between the copper bar and sapphire balls. The Si was thermally anchored to the Cubar using GE 7031 Varnish, and 92 clamps were in place for this Si interface measurement.An example of a steady-state data trace is shown in Fig. 2a

4 H.D. Pinckney et al.

0 50 100 150 200 250Time [s]

228.6

228.8

229.0

229.2

229.4

229.6

229.8

230.0

Tem

pera

ture

[mK]

(a) Steady-state mode

0 100 200 300 400 500Time [s]

310

320

330

340

350

360

Tem

pera

ture

[mK]

(b) Dynamic mode

Fig. 2: Examples of data collected in steady-state mode (a), and dynamic mode (b). Thedominant uncertainty in temperature is a 10 % calibration offset, not illustrated here. In (b),red regions indicate power on, and the green region indicates power off. (Color figure online)

2.2.2 Dynamic Mode

The dynamic measurement method observes the time-dependent temperature response aftera sudden change in applied power. The temporal response depends on a combination ofthermal conductance and heat capacity. The temporal response can be used as a measure ofthermal conductance in this case because the heat capacity of the bar can be well-estimated.

The measurement begins with the bar at near-constant temperature under constant ap-plied power (as in the steady-state measurement). The applied heater power is then abruptlyturned off, and the bar’s temperature is logged as a function of time for a few minutes.An example data trace is shown in Fig. 2b. Such data can be described by the followingdifferential equation:

CdTdt

= BTdTdt

=A4

(T 4

bar −T 4MC

)−Pexternal , (4)

where C = BT is the heat capacity of the copper test mass at low temperature [6], B is aconstant, and we have set n = 3 from Eq. 3.

3 Analysis and Results

Steady-state fitting was performed using the python iminuit package. Dynamic fittingwas done using the scipy.optimize.curve fit package.

Fig. 3 shows the fit results for the TMC = 9 and 150 mK steady-state runs with 92 clampsdirectly on the copper bar. The fits qualitatively agree with the data. Fixing the exponent ton= 3, as would be motivated by bulk phonon transport or thermal boundary resistance, doesnot dramatically change the fit results for sapphire-on-copper data, though it does qualita-tively worsen the fit at lower temperatures for the sapphire-on-silicon data. Dependence onTMC is apparent, as expected at the lower bar temperatures.

Fig. 4 compares the results of all measurements using both the steady-state and dynamicmethods, and Table 1 displays the average fit values for each method. We see that the per-clamp conductance amplitude agrees well between the 48 and 92 clamp tests. We also see

The Thermal Conductance of Sapphire Ball Based Detector Clamps 5

3 × 102 4 × 102 5 × 102

Temperature [mK]

102

103Po

wer [

nW]

92 Clamps on Copper9 mK Fit150 mK Fit9 mK Data150 mK Data

Fig. 3: Example fits to the steady-state data with 92 clamps directly on copper (data sets 2and 3 in Fig. 4). The fit function is given by equation 3. (Color figure online.)

that the change from a copper to a silicon interface surface decreased the thermal conduc-tance by nearly a factor of 2. This could be due to material hardness; we expect a smallerarea of contact between the ball and the silicon. Additionally, we notice that the steady-stateand dynamic data sets agree to within a factor of approximately 3. Given the simplifiedthermal model used in the dynamic analysis, it is remarkable that the two methods agreeon the same order of magnitude. In addition to improving the thermal model, a systematiceffect keeping these from agreeing could be unaccounted for heat capacities from screws,washers, thermometry, and other hardware. An increased heat capacity would not affect thesteady-state analysis, but would increase the dynamic thermal conductance measurement.

We can use the acoustic mismatch theory presented in [7] to then make a predictionof the thermal conductance due to boundary resistance and compare with these measuredvalues. We estimate from a scratch test that the contact area between the sapphire and copperwas 3+0.5

−1.5 × 10−5 cm2, and taking the boundary resistance between sapphire and copperto be 18.5 K4/W1/cm2 from [7], we can estimate that the conductance per clamp due toboundary resistance should be 1600 +300

−800 nW/K4. Given the uncertainty in our area estimatethis agrees with our measurement, and we attribute the thermal resistance to be dominated byboundary resistance. Comparison between theory and experiment was not made for sapphireon silicon because of a large uncertainty in the area of contact.

4 Conclusions

We have measured the thermal conductance of sapphire ball clamps and have shown that theinterface appears to be dominated by a thermal boundary resistance. The thermal conduc-tance was found to be 660+360

−210 (T/K)3.08 [nW/K] for sapphire on copper, and 380+190−120 (T/K)2.84 [nW/K]

for sapphire on silicon. Additionally, in conjunction with simulations in [8], this measure-ment allows us to feel comfortable holding a detector with approximately 9 clamps withoutfear of degraded energy resolution.

6 H.D. Pinckney et al.

0 1 2 3 4 5Dataset Index

0

250

500

750

1000

1250

1500

1750

2000Am

plitu

de [n

W/K

n+

1 /Cla

mp]

Thermal Conductivity Amplitude Per Clamp

Fig. 4: Thermal conductance amplitude per clamp for the various data sets and selecteduncertainties. Black circles represent data taken with 48 clamps directly on copper. Blue tri-angles represent data taken with 92 clamps directly on copper. Green squares represent datataken with 92 clamps on silicon. Filled markers are the result of steady-state fits, hollowmarkers are the result of dynamic fits. The red band indicates the acoustic mismatch modelof thermal boundary resistance between copper and sapphire, with uncertainty dominatedby our area estimate. Datasets 0, 2, and 4 were taken with the MC at 9 mK, and datasets 1,3, and 5 were taken with the MC at 150 mK. The uncertainties plotted are the systematicuncertainty given by scaling the thermometry by 10 % in either direction. Scaling the tem-perature down increased the conductance amplitude for steady-state and dynamic fits, andscaling the temperature up decreased the conductance amplitude. (Color figure online.)

Steady-state

CuA 660 nW/Kn+1/clampn 3.08Pexternal 16 nW

SiA 380 nW/Kn+1/clampn 2.84Pexternal 13 nW

Dynamic Cu A 200 nW/K4/clampPexternal 15 nW

Table 1: Mean results of the fits. Steady-state results are averaged over the 9 mK and 150 mKbase temperatures.

5 Data Availability

The datasets analysed during the current study are available from the corresponding authoron request.

Acknowledgements This work was partially supported by DOE QuantISED award DE-SC0020181. Tem-plate v.1 by KLL - June 18, 2015.

The Thermal Conductance of Sapphire Ball Based Detector Clamps 7

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