Phys. Status Solidi A 208, No. 12, 2719–2723 (2011) / DOI 10.1002/pssa.201127331 p s sa

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applications and materials science

Mechanical loss of calcium fluoride atcryogenic temperatures

Christian Schwarz1, Daniel Heinert1, Paul Seidel1, Andreas Tunnermann2, Giles D. Hammond3,1,and Ronny Nawrodt*,1,3

1 Friedrich-Schiller-Universitat Jena, Institut fur Festkorperphysik, Helmholtzweg 5, 07743 Jena, Germany2 Friedrich-Schiller-Universitat Jena, Institut fur Angewandte Physik, Albert-Einstein-Straße 15, 07745 Jena, Germany3 SUPA, University of Glasgow, G12 8QQ Glasgow, UK

Received 6 January 2011, revised 6 June 2011, accepted 1 July 2011

Published online 29 July 2011

Keywords calcium fluoride, cryogenic temperatures, mechanical loss, ring down

* Corresponding author: e-mail ronny.nawrodt@uni-jena.de, Phone: þ49 3641 947480, Fax: þ49 3641 947412

Low mechanical loss materials are in great demand for micro-

mechanical applications as well as high precision instrumenta-

tion. We describe low mechanical loss measurements of a

calcium fluoride (CaF2) bulk sample. A new mechanical

relaxation process was observed for frequencies in the range

from 31 to 64 kHz and temperatures of 10 to 50 K. The

activation energy of this process was found to be 10 meV. This

relaxation process increases the mechanical loss significantly

and thus sets limits to the application of this material at these

temperatures.

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Calcium fluoride (CaF2) is a veryattractive material for optical and opto-mechanical appli-cations due to its excellent thermal, mechanical and opticalproperties and additionally due to its availability in largesample size. It has attracted significant interest in opticalapplications such as UV lithography in the semiconductorindustry due to its low optical absorption in the UV spectrum[1]. Its low mechanical loss throughout a wide range oftemperatures makes it interesting for high-precision exper-iments such as gravitational wave detectors [2, 3], laserstabilisation cavities [4] and quantum non-demolitionmeasurements utilising whispering gallery mode resonators[5]. For these applications the Brownian thermal noise of theoptical components is directly related to the mechanical lossof the materials if the mechanical loss is spatially homo-geneously distributed [6]. Combining low optical absorptionand low thermal noise makes CaF2 a promising material formodern technologies. A previous mechanical loss measure-ment on CaF2 at low temperatures was carried out by [7]. Dueto a lack of resolution the measurement was not sensitiveenough to investigate the here presented loss. Suchmeasurements are essential in order to fully realise thesuitability of this material for future applications where lowthermal noise is an essential requirement. We present themeasurement of the mechanical loss over a wide temperaturerange from 5 to 300 K and with frequencies from 31 to

64 kHz. At around 25 K a mechanical loss peak was observedwhich is most likely related to a thermally activated processin CaF2.

2 Sample preparation and experimental setupThe sample used in our investigations was a CaF2 singlecrystal grown by Schott Lithotec [8]. The sample had acylindrical geometry with a diameter of 75 mm and a lengthof 75 mm. The symmetry axis of the cylinder was parallel tothe [100] axis of the single crystal. All surfaces, including thefront and back as well as the barrel and bevel area, werepolished to an optical quality by Hellma Optics [9].

The mechanical loss of the sample was obtained byexciting internal mechanical resonant modes of the crystaland recording the subsequent free ringdown. An electrostaticexcitation was used to drive the sample close to or at theresonant frequency. The vibration of the sample wasinterferometrically read-out along the symmetry axis of thecylinder. The front surface of the sample formed one end-mirror of a Michelson-like interferometer. The crystal wassuspended by means of a 50mm diameter tungsten wirearound its central plane to reduce seismic disturbances.Regarding the internal resonances of the tungsten wire thecopper made suspension structure was specially built(see Fig. 1) to provide the possibility to use the shortestpossible wire lengths resulting in a high fundamental

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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Figure 1 (online colour at: www.pss-a.com) Left: photograph of the experimental setup inside the probe chamber. Right: schematic of thesuspension, 1 sample support with heat links to the temperature controlled base plate (see No. 7), 2 tungsten suspension wires (depending onthe adjustment L1 and L2 are not necessarily equal), 3 break-off point from where on the suspension wire has no contact to the sample,4 electrostatic driving plate, 5 CaF2 bulk sample, 6 substrate catcher, 7 temperature controlled probe chamber base plate.

resonant frequency of the suspension wire. For a typicalsuspension used for this investigation the fundamentalsuspension violin mode is in the order of 10 kHz. As a resultof the high fundamental resonant frequency of the suspen-sion wire the higher order modes of the suspension wire arewell separated. This allows to reduce the unwanted effect ofresonant coupling between the substrate modes and one ofthe higher order suspension modes.

The mechanical loss f at a given frequency f and atemperature T can be calculated:

� 20

fðf ; TÞ ¼ 1

pf t; (1)

Figure 2 (online colour at: www.pss-a.com) Calculated modeshapes for the given resonant frequencies using the finite elementanalysis software ANSYS [10]. The colours indicate the amount ofmotion parallel to the cylinder axis. Red and blue represent themaxima of motion in opposite directions, while green coloured areasremain in rest.

where t is the time it takes for the resonant mode’samplitude to decay to 1/e of its initial value.

The whole apparatus was mounted on a cold plate of acryostat in order to measure the loss at temperatures withinthe range 5–300 K. The probe chamber was evacuated tobetter than 10�3 Pa to minimise any additional damping dueto the presence of residual gas. The temperature was sensedwith an identical witness sample in the probe chamber withan accuracy of about 0.3 K. Each resonant mode (see Fig. 2)was measured several times and the lowest mechanical lossobtained for all runs at a given temperature was considered tobe the most reliable value. This is due to the fact that anydeviation from the optimum working point for the measure-ment will always result in an increased loss. A typicalmeasuring time of about 20 min for one mode is needed foreach temperature step. The temperature is then varied and anew measurement started. Further details of the setup and themeasuring procedure can be found elsewhere [7, 11].

3 Results and discussion Figure 3 shows themeasured mechanical loss of the CaF2 sample between 5and 300 K for the two resonant modes (a) and (b) in Fig. 2(31.35 and 42.05 kHz). As the temperature is reduced from300 to 50 K the loss of the observed modes decreases by

11 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

roughly one order of magnitude. Both modes show peaks inthe mechanical loss at around 87, 120 and 255 K (indicated asI, II and III).

As previously discussed the sample was suspended bymeans of a tungsten wire. The free length of the wire betweenthe break off points on the samples barrel and the upper end,where both wires were rigidly clamped (see Fig. 1) was about32 mm. According to this length we estimated a frequencydifference between the violin modes of the suspension wireof 10.2 kHz. As the temperature of the setup is changed theresonant frequency of the wire and the samples changes due

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Figure 3 (online colour at: www.pss-a.com) Overview plot of themechanical loss of the CaF2 substrate obtained for two selectedmodesbetween5and300 Kincomparisonwith thehighestpredictedthermo-elastic loss for all modes under investigation.

Figure 4 (online colour at: www.pss-a.com) Plot of the temper-ature dependent resonant frequency of the sample (black squares).Thehorizontal lines(redandblue)represent thecalculatedfrequencybehaviour of the suspension wires. At 87 and 120 K the samplesfrequency matches one suspension wire violin mode which leads totwo resonant loss peaks (see Fig. 3).

to the temperature dependence of its Young’s modulus (seeFig. 4). Within the temperature range of interest (5–300 K)the resonant frequency of the tungsten wire changes less than0.1%. The sample’s frequency shift is in the order of 2%.Having a large separation of the violin modes of thesuspension wire leads in maximum to one unwantedresonance between the wire and the substrate within thetemperature range of interest. This explains peak number Iand III in Fig. 3 as a 2nd and 3rd harmonic resonance of thefundamental frequency of the suspension wires. Peaknumber II occurs directly near another resonance with thesuspension wire. We believe that this second resonance iscaused by a slightly asymmetry of the suspension wirelengths L1 and L2. The frequency separation can be explained

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with a length difference of about 0.1 mm which is beyond theexperimenters control for such a suspension. For the 31 kHzmode this second peak is expected at temperatures above300 K which were not accessible with our setup.

An experimental verification that the loss peaks I, II andIII are caused by the suspension was done by re-suspendingthe sample with slightly different suspension wire lengthsand remeasuring it. All three peaks changed their position.

Peak number IV occurred in all investigated modes atlow temperatures around 25 K independent on the suspen-sion lengths. It is thus expected that this peak is caused by aninternal process which should be described later.

An important contribution to the overall mechanical lossin crystalline materials is thermo-elastic damping [12].While the sample is excited to resonant modes parts of thesample become compressed and other parts becomeexpanded. This local volume change is linked by means ofthe coefficient of thermal expansion with a local temperaturechange. The local temperature changes cause a heat fluxwhich is increasing entropy and thus is a loss process. Adetailed description of a numerical procedure to calculatethermo-elastic damping in the test samples under investi-gation can be found in Ref. [13]. The method involves thenumerical solution of the heat equation with a finite elementanalysis tool to predict the temperature distribution withinthe body. The mechanical loss can then be related to theenergy loss, DE, dissipated during one cycle of oscillationusing:

fTE ¼ 1

2p

DE

Etot

¼ 1

4pfT0Etot

ZlðrTÞ2

dV ; (2)

where Etot represents the total elastic energy of the mode, Tthe internal temperature distribution with average value T0, fthe oscillation frequency and l is the thermal conductivity(which is assumed isotropic for the cubic CaF2 system). Theintegration in Eq. (2) was numerically evaluated over thewhole volume of the sample. Amongst all modes underinvestigation the 31 kHz mode had the highest thermo-elastic loss. The predicted level of thermo-elastic loss of thismode is plotted in Fig. 3 for comparison. It is clear from thisfigure that thermo-elastic damping will only be significant atvery low temperatures well below 25 K. Thus, the origin ofthe observed loss peak at 25 K must belong to anothermechanism.

The results of a detailed measurement of the frequencyand temperature dependence of the loss peak are shown inFig. 5. The temperature of the loss maximum is clearly linkedto the resonant frequency of the mode under investigation.The larger the resonant frequency the larger is thetemperature of the observed loss maximum. This indicatesa thermally excited process in the material that can bedescribed by a Debye like relaxation peak [14]:

f ¼ ~DvtR

1 þ v2t2R

; (3)

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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Figure 5 (online colour at: www.pss-a.com) Detailed investiga-tion of the mechanical loss peak between 10 and 50 K. The temper-ature of the maxima for different frequencies are a guide to the eye.

with an Arrhenius-like behaviour of the thermal relaxationtime tR [15]:

Figulosssym

� 20

tR ¼ tR;0eDE=kBT ; (4)

where ~D is the relaxation strength, DE the activation energyof the process and tR,0 is the relaxation constant. Theobserved loss has a maximum if tR¼ 1/v with the angularfrequency v of the vibration. Plotting the natural logarithmof the angular frequency against the reciprocal of the peaktemperature Tpeak, will result in a linear dependency (alsocalled an Arrhenius plot). Figure 6 shows the results of theArrhenius plot for all modes under investigation.

The parameters of the linear fit give the activation energyDE of the process and the relaxation constant tR,0. The valuesobtained from Fig. 6 are DE ¼ ð10:0 � 0:1Þ meV and

re 6 (online colour at: www.pss-a.com) Arrhenius plot of thepeak. The error of lnv is small compared to the size of thebols used for this graph.

11 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

tR;0 ¼ ð100 � 30Þ � 10�9 s, respectively. The activationenergy and the relaxation times are in the range of typicalactivation energies of processes that involve impurities incrystalline materials [14, 16–18]. The relaxation strength ofthe different loss peaks is given by its height [see Eq. (3)]. It isobvious from Fig. 5 that the relaxation strength is differentfor different modes. The reason for this behaviour is thedifferent stress and strain distribution within a given modeshape (see Fig. 2).

4 Conclusions We have presented mechanical lossmeasurements of CaF2 in a temperature range from 5 to300 K and at frequencies in the range from 31 to 63 kHz. Wehave observed a mechanical relaxation peak in CaF2 havingan activation energy of (10.0� 0.1) meV and a relaxationtime of tR;0 ¼ ð100 � 30Þ � 10�9 s. This loss mechanismwas contributed to an intrinsic effect in CaF2. The activationenergy is in the order of typical impurity driven processes.The observed loss peak might limit applications which relyon low mechanical loss materials like laser stabilisationcavities or opto-mechanical devices. The origin of the losspeak is still unknown and will be within the focus of ourfuture investigations. The correlation between the relaxationstrength and the strain distribution for different mode shapescan give information about the impurity symmetry within theCaF2 lattice and might thus be a valuable tool to identify thedefect. This correlation is within the focus of our currentresearch.

Acknowledgements This work was supported by theGerman Science Foundation under contract SFB Transregio 7.We would like to thank colleagues at the Institute for GravitationalResearch (SUPA, University of Glasgow) for their interest inthis work and the opportunity for GDH to take a sabbaticalat the Friedrich-Schiller-Universitat Jena to collaborate on thiswork.

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