Quartz tuning fork as a probe of surface waves
Igor Todoshchenko, Alexander Savin, Miika Haataja, Jukka-Pekka Kaikkonen, Pertti Hakonen
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Quartz tuning fork as a probe of surface waves.
Quartz tuning fork as a probe of surface waves.
I. Todoshchenko, A. Savin, M. Haataja, J.-P. Kaikkonen, and P. J. Hakonen Low Temperature Laboratory, Deptartment of Applied Physics, Aalto University,Finland a) (Dated: 13 October 2018) Quartz tuning forks are high-quality mechanical oscillators widely used in low temperature physics as vis-cometers, thermometers and pressure sensors. We demonstrate that a fork placed in liquid helium near thesurface of solid helium is very sensitive to the oscillations of the solid-liquid interface. We developed a double-resonance read-out technique which allowed us to detect oscillations of the surface with an accuracy of 1 ˚A in10 sec. Using this technique we have investigated crystallization waves in He down to 10 mK. In contrastto previous studies of crystallization waves, our measurement scheme has very low dissipation, on the orderof 20 pW, which allows us to carry out experiments even at sub-mK temperatures. We propose to use thisscheme in the search for crystallization waves in He, which exist only at temperatures well below 0.5 mK.Keywords: Solid helium, Interface in quantum systems, Displacement detection, Tuning fork, Effective massInterfaces in quantum fluids and solids display a varietyof physical phenomena. In addition to usual capillary-gravity waves they are mobile enough to support alsophase waves, like crystallization waves or massless phasewaves on the A-B interface of He. Besides the factthat these phase waves are very interesting and unusualon their own, they present a unique tool to investigatesurface bound states. Surface quasiparticles contributedirectly to the surface tension as well as to the dissipationof the surface waves, so that they can be detected bymeasuring the dispersion relation of the waves.Usually, the contribution of surface excitations to thedissipation of the surface waves is masked by the inter-action with particles in bulk phases. However, at lowtemperatures the contribution from the bulk particlesdecreases as T or even faster. Due to their 2D nature,surface excitations experience slower decay upon cooling,and at a low enough temperature they provide the largestcontribution to the dissipation of the waves. Helium isthe only substance which remains liquid at ultralow tem-peratures and thus allows to investigate surface wavesand various surface bound states.Manninen et al. have performed pioneering experi-ments on capillary-gravity waves on the free surface of liq-uid He well below the superfluid transition temperatureof 1 mK. Their measurements suggest that at tempera-tures below 0.2 mK the damping of the capillary-gravitywaves is no longer determined by the bulk quasiparticles,the density of which vanishes exponentially with decreas-ing temperature.
Interestingly, different modes of thesurface waves had different temperature dependence ofdissipation, which may indicate wave-number dependentinteraction with the surface states.Crystallization waves in He were predicted by An-dreev and Parshin in 1978 and discovered two years laterby Keshishev et al. at temperatures below 0.5 K. Bymeasuring the frequency of crystallization waves at crys-tal surfaces of different orientations, Rolley et al. have a) Electronic mail: igor.todoshchenko@aalto.fi observed, as predicted by Landau, a singularity in thesurface tension at the orientation of the basal facet. In He, an interfacial wave between solid and super-fluid phases is an even more intriguing object to study be-cause it has to accommodate to the magnetically orderedphase of the solid and the p-wave-paired order parame-ter of He superfluid. Due to the magnetically orderedsolid, the crystallization wave is accompanied by spin su-percurrents which contribute to the inertia of this wavealong with the regular mass flow, and in high enoughmagnetic fields the whole inertial mass of the wave is ofa magnetic origin. However, due to the entropy related tomagnetic degrees of freedom, crystallization waves in Heare strongly damped and can be observed only at tem-peratures well below 0.5 mK.
This sub-mK tempera-ture requirement sets strong limitations for the excitationof the waves because the electrical capacitors utilized togenerate the waves may produce significant heat load tothe helium sample.In this Letter we suggest a new, extremely sensitiveand low dissipation scheme for detection of surface waves.We have tested this scheme by successfully observingcrystallization waves in He down to oscillation ampli-tudes of few nanometers. The obtained results show thatthe developed technique can be used to investigate sur-face waves in the sub-millikelvin range.Excitation of surface waves at low temperatures is avery difficult experimental task. Generally, one needsto oscillate pressure at / near the surface. Moreover, theoscillating pressure should be applied locally because aspatially uniform pressure variation will oscillate the sur-face as a whole ( q = 0), instead of producing waves onit. However, the speed of sound c = 366 m / s in liq-uid helium at the melting pressure is much faster thanthe speed of surface waves which means that the neededpressure variation cannot be applied via liquid by any me-chanical transducer. Instead, electrical capacitors partlyimmersed in helium can be used to create local pressurevariation. Due to its electrical polarizability, helium ac-quires additional energy in the electric field. The electri-cal energy is equivalent to additional pressureuartz tuning fork as a probe of surface waves. 2 δp = ε ( ε − E / . (1)According to the Clausius-Mossotti relation, the per-mittivity ε relates to the polarizability α by: ε = (1 + π αv ) / (1 − π αv ) ≈ π αv where v is the molar volume.Among all atoms, helium atom has the smallest polar-izability, α He = 0 . / mol, which is three timessmaller than that of hydrogen. In addition, liquid he-lium is quite a rarefied medium with a molar volume ofabout 30 cm / mol at p = 25 bar, almost twice that ofwater. These result in the very small value of ( ε −
1) forcondensed helium phases (0.054 for liquid He and 0.052for liquid He at zero bar) and, correspondingly, in verylow electrostatic pressure δp from Eq. (1). To exemplify,an electrical field of 2 MV / m is needed to lift the liquid-vapour helium interface on a capacitor by 1 mm. To de-crease the voltage needed for such a high electric field onecan decrease the spacing d in the capacitor. However, inthis way the volume of the capacitor also decreases, whichleads to a reduction of the relative share of the electricenergy. Practical values of the spacing are thus set bythe above limitations to the range 20 ...100 µ m, and thecorresponding voltages are a few hundred of volts.In their experiments on crystallization waves Rolley etal. utilized an interdigital capacitor on borosilicate glasswith a periodicity of 80 µ m. They measured the dielec-tric losses in the substrate of the capacitor to be about30 µ W at typical drives (170 V peak-to-peak @ 1 kHz),and such a high heat leak limited the temperature oftheir experiment to 40 mK. To reduce dielectric losses,we made our capacitor using a double winding of 60 µ mthin superconducting wire on a copper holder. Eddy cur-rent heating due to the displacement currents are mini-mized in this design because currents in neighboring wiresflow in opposite directions. The amount of dielectric wasminimized by choosing wires with a thin (5 µ m) insula-tion layer and using thin cigarette paper and very dilutedGE-varnish. The height and the width of the capacitorwere also kept small with H = 1 . W = 4 mm,respectively, thus reducing the volume of dielectric ma-terial and the induced losses in proportion.The solid-liquid interface of helium is even more diffi-cult to drive because solid helium wets solid surfaces verypoorly. The capillary forces preventing the solid from fill-ing the narrow gap of large field at the capacitor can beestimated as follows. The equilibrium balance of liquidand solid phases near the capacitor is set by the equal-ity of chemical potentials, ( p l − p ) /ρ l = ( p s − p ) /ρ s where p is the equilibrium pressure at the flat referenceinterface far from the capacitor. Gravity and electricfield contribute to the pressure in the liquid as p l − p = − ρ l gh + ε ( ε l − E /
2, where h is the vertical position ofthe surface with respect to the reference level. The pres-sure within the solid having a curvature R is larger thanin the liquid by the Laplace pressure p s = p l + α/R where α is the surface tension and R ∼ d reflects the size of thecapacitor spacing. Hence, one obtains the following equa- FIG. 1. a) Calculated areas of high ( >
10 MV / m) field in thevicinity of two neighboring wires of the capacitor at differentvoltages. b) Schematic illustration of the experimental celland the read-out scheme using two lock-in amplifiers. Thefork is placed in the liquid just above the surface of solidand driven at its resonance frequency f = 32125 Hz. Thefork current is measured by the first lock-in amplifier. HighAC voltage applied to the capacitor at frequency F/ F which leadsto a periodical detuning of the fork. The detuning resultsin the oscillation of the quadrature current through the forkwhich is measured by another lock-in amplifier at frequency F = 2 ...
100 Hz. tion for the capillary-gravity-electrical equilibrium shape:(∆ ρ/ρ l )[ − ρ l gh + ε ( ε l − E /
2] = α/R .From this equation we can separate the electric pres-sure needed to compensate for capillary forces, ε ( ε l − E c / ρ l / ∆ ρ )( α/R ) and the electric pressure neededto lift the crystal surface ε ( ε l − E g / ρ l gh . Sub-stituting values for He, α = 1 . × − J / m and ρ l / ∆ ρ ≈
10 we obtain for the critical field E c ≈
10 MV / m ≫ E g . Fig. 1a displays areas near the capac-itor’s neighbouring wires where the E > E c at differentvoltages U applied across the two capacitor electrodes.One can see that at voltages U <
100 V the gap withthe high field is still too small to overcome the capillaryforces, and first at U ∼
200 V applied to the capacitorthe gap becomes of the order of d to fit the solid. Af-ter the capillary forces have been overcome, solid fills thevolume of the capacitor, and the level of solid decreasesin the rest of the cell.To measure a sub-micron change of the level of solidwe have introduced a double-resonance method with aquartz tuning fork as a sensitive element. Tuning forksare nowadays widely exploited in helium low tempera-ture experiments as thermometers, viscometers, pressuresensors, turbulence detectors etc. In our measure-ment scheme resembling the so-called near-field acousticmicroscopy an oscillating quartz tuning fork is placedin the vicinity of the surface of the solid. A change inthe distance z between the interface and the fork distortsthe velocity field in the liquid around the oscillating forkand thus changes its effective hydrodynamic mass. Thiscauses a change in the resonance frequency of the fork.As the quality factor of the fork at low temperatures maybe as high as 10 , a very tiny detuning of the resonancefrequency can be detected.Our measurement scheme is shown in Fig. 1b. Auartz tuning fork as a probe of surface waves. 3 FIG. 2. a) Calculated absolute velocities of the liquid inthe central plane around oscillating fork placed at differ-ent distances z from the solid wall. b) Calculated detuning∆ f ≡ f res ( z ) − f of the fork as a function of the distance z .Insert: sensitivity of the measurement scheme to the changeof the position of the interface. high AC voltage U = U sin πF t at a low frequency F/ ...
50 Hz is applied to the capacitor which drivesthe interface at the frequency F because the electrostaticpressure is proportional to the square of the field. Thefork is driven at frequency f = 32125 Hz close to (oscil-lating) resonance frequency f res ( z ) and the quadratureof the fork output current is demodulated using anotherlock-in detector synchronized with U at frequency F .When F coincides with the standing surface wave reso-nance frequency, a maximum appears in the quadratureoscillation amplitude which is proportional to the ampli-tude of the surface oscillation.The experimental cell is a vertical cylinder of 35 mmdiameter directly joined to a heat exchanger on a coppernuclear demagnetization stage. Silver sinter for heat ex-change is baked on copper plates in the form of stackedhorizontal layers of 2 mm thickness with a 10 mm holein the center. The semi-open helium resonator is madeusing four thin copper plates to separate the space. Oneof the plates holds a wire-wound capacitor to generatewaves. Plates are 1.5 mm high and 5 mm wide each. Allfour plates are tilted by 45 degree to compensate for thewetting angle of solid helium. The fork is placed at adistance of 0.5 mm from the center towards the middle ofthe wave generator. A schematic horizontal cross-sectionof the cell is shown at the insert of Fig. 3b. The ex-periments were performed on a Bluefors LD-400 dilutionrefrigerator.Crystals were grown using the so-called blocked capil-lary method in which the cell is pressurized at tem-perature T ≈ x = V s /V tot can be calcu-lated from x = [ ρ l ( T ) − ρ l (0)] / [ ρ s (0) − ρ l (0)]. In ourexperiments the solid fraction x = 0 .
20 was calculated toprovide solid right below the fork, which corresponds to ρ l ( T ) = 176 . / m . We chose to pressurize the cell at T = 1 . K where the pressure providing the needed den-sity was calculated to be 27.18 bar. After cooling downbelow 200 mK the fork became frozen in solid. Then wecarefully let some helium to release from the cell untilthe resonance re-appeared. Releasing helium was madevery slowly in order not to melt too much solid because it was crucial for sensitive measurements that the forkwas placed as close to the crystal surface as possible.Since the velocity field around fork prongs decays veryfast with the distance, the detuning of the fork caused bythe perturbation of the velocity field also decreases fastwith the distance from the solid. This is illustrated inFig. 2a showing the calculated distribution of the velocityof the liquid around the fork at different distances z fromthe solid, and in Fig. 2b showing the detuning ∆ f ≡ f res ( z ) − f of the fork as a function of the distance.The calculations were done using finite elementmethod. The fork employed in our experiment had2.45 mm long, 0.10 mm wide and 0.24 mm thick prongswith a gap of 0.12 mm between the prongs. The simula-tion space was a 6mm × × ∼ × tetrahedral domainelements. The domain elements were more highly con-centrated near the fork and its tip, where changes in thevelocity field are more precipitous. The fork was placedalong the central z -axis, while the bottom xy -plane wasmade into a ”solid” by applying a zero-flux von Neumannboundary condition (BC). The same BC was applied onthe fork surfaces, except for the sides with their normalparallel to the prong’s movement. These sides had themovement implemented by applying non-zero flux BCs,which increased closer to the tip according to the fork’svelocity profile. The kinetic energy of the liquid wasthen calculated and the added hydrodynamical mass wasobtained. The corresponding shift of the resonance fre-quency of the fork as a function of distance z is shownin Fig. 2b together with the fitted empirical relation sug-gested by Callaghan et al. In the insert of Fig. 2b, weplot the sensitivity dz/d ∆ f of the fork to a displacementof the crystal surface as a function of the distance z .Fig. 3a displays the measured detuning of the fork dueto a change in the He solid-liquid interface position asa function of the DC voltage applied to the capacitor.As is seen from the figure, there is a threshold voltageof about 210 V above which the crystal starts filling thecapacitor, and the level the of solid near the fork de-creases. The value of the threshold is in good agreementwith the estimations of the capillary forces in the pre-vious section and with the calculations of the electricalfield shown in Fig. 1a. At higher voltages, the detuningincreases slowly because the field decays exponentiallywith the distance from the capacitor, and, correspond-ingly, the volume of the high field region increases onlylogarithmically with the voltage. The estimated volumeof the capacitor is about 0.05 mm which corresponds toa 0.5 µ m change of the level of solid in the rest of the cell.The sensitivity dz/d ∆ f of the measurement scheme isthus 0.5 µ m /
20 mHz = 2 . × − m / mHz, from whichwe infer that the fork is about 0.2 mm above the level ofthe solid, according to the calculated values shown in theinsert of Fig. 2b.The spectrum of crystallization waves in He measuredat 10 mK at U = 280 V AC-drive is shown in Fig. 3b.The integration time of the second lock-in amplifier mea-uartz tuning fork as a probe of surface waves. 4 FIG. 3. a) Change in the detuning of the fork as a functionof the DC voltage applied to the capacitor. On the right wepresent a displacement scale estimated from the volume ofthe capacitor divided by area of the crystal surface and thecalculated detuning. b) Spectrum of crystallization waves in He. Light: in-phase response; dark: out-of-phase response.Numbers ”1”, ”2”, and ”3” mark the fundamental resonanceand its harmonics which involve both the inner and outerregions A and B, respectively. Insert: schematic view of thehorizontal cross-section of the cell. suring detuning of the fork at the frequency of the U AC was 10 s. The first two peaks at 13 and 22 Hz corre-spond to the fundamental resonance and its first har-monic across the whole crystal-liquid interface (our wavegenerator excites the interface on both sides and the in-ner and outer regions are strongly coupled on the liquidside). The third peak at ∼
48 Hz is assigned to the sec-ond harmonic, which is weakly coupled to the fork owingto its asymmetric mode shape with respect to the oscil-lation detector. One can also notice that the efficiency ofa generation of the waves decreases with increasing fre-quency F , probably due to the limited growth rate of Hecrystals.
As seen from Fig. 3b, the resolution of thismeasurement scheme is about 1 ˚A. Equally good resolu-tion has been reached earlier in measurements on crystalgrowth with a much more complicated optical setup.In order to test the heating by the capacitor, we madea separate cooldown in which we condensed He in thesame cell. At the melting pressure and at the lowesttemperature of 0.39 mK = 0 . T c , we have measured thedissipation produced by the driving capacitor at differentfrequencies. The dissipation was found to depend on thefrequency as f indicating that the main heat source aredielectric losses in the isolating materials of the capaci-tor. Eddy current heating would be proportional to f since the electromagnetic induction E in the copper plateis proportional to the time derivative of the displace-ment current in the capacitor, E ∝ ˙ I = (2 πf ) CU , andthe dissipation is proportional to E . The heat releasedat the maximum frequency of 50 Hz and U = 280 Vamplitude warmed the helium sample from 0.39 mK to0.40 mK in about one minute. Estimating the heat ca-pacity of our 1 mole helium sample at this temperature as C ∼ R exp ( − ∆ /T ) = 5 × − J /K we find a heat leak of P ∼
20 pW. This very small value of the heat leak shouldbe compared with the heat leak from the interdigital ca-pacitor in other works on crystallization waves in He, where, extrapolated to our frequencies and voltage, itwould be 70 nW, i. e. more than three orders of magni- tude larger. The reason for such a strong reduction ofdissipation was, we believe, the use of a very tiny 6 mm wound capacitor with a minimum possible amount of di-electric material.To summarize, we have developed a new very sensitivetechnique for measuring oscillations of the solid-liquidinterface of helium (crystallization waves). This double-resonance technique has been demonstrated to detectamplitudes of surface oscillations as small as 1 ˚A. Min-imizing the dielectric losses in the capacitor resulted invery small power dissipation of 20 pW. Such an ultra-low-dissipation technique allows experiments with solid-liquid interface of helium-3 well below 0.5 mK, wherecrystallization waves have been predicted to appear. Thetechnique can also be used effectively to probe waves atthe free surface of superfluid or at the interface betweentwo different superfluid phases. ACKNOWLEDGEMENTS
This work was supported by the Academy of Finland(grant no. 284594, LTQ CoE), by the European ResearchCouncil (grant no. 670743), and by Vilho, Yrj¨o and KalleV¨ais¨al¨a Foundation of the Finnish Academy of Scienceand Letters. This research made use of the OtaNano– Low Temperature Laboratory infrastructure of AaltoUniversity, that is part of the European Microkelvin Plat-form. A. F. Andreev and A. Ya. Parshin, Sov. Phys. JETP , 763(1978). I. Todoshchenko, Phys. Rev. B , 134509 (2016). M. S. Manninen et al.
Phys. Rev. B , 224502 (2014). M. S. Manninen et al.
J. Low Temp. Phys. , 399 (2016). K. O. Keshishev, A. Ya. Parshin, and A. V. Babkin, JETPLett. , 56 (1980). L. D. Landau, Collected papers, Pergamon Press, Oxford (1971). E. Rolley, S. Balibar, and F. Graner, Phys. Rev. E , 1500(1994). E. Rolley et al.
J. Low Temp. Phys. , 851 (1995). A. F. Andreev, Czech. J. Phys. , 3043 (1996). S. Balibar, H., Alles, A. Ya. Parshin, Rev. Mod. Phys. , 317(2005). E. R. Grilly, J. Low Temp. Phys. ,615 (1971). B. M. Abraham et al. , Phys. Rev. A , 250 (1970). M. Bla˘zkov´a et al. , J. Low Temp. Phys. , 525 (2008). S. L. Ahlstrom et al. , J. Low Temp. Phys. , 725 (2014). P. G¨unther, U. Ch. Fischer, and K. Dransfeld, Appl. Phys. B ,89 (1989). R. Steinke et al. , Appl. Phys. A , 19 (1997). I. Todoshchenko et al. , Rev. Sci. Instrum. , 085106 (2014). R. H. Salmelin et al. , J. Low Temp. Phys. , 83 (1989). S. Balibar, C. Guthmann, and E. Rolley, J. Phys. I France , 1475(1993). C. A. Swenson, Phys. Rev. , 538 (1953). F. D. Callaghan, X. Yu, and C. J. Mellor, Appl. Phys. Lett. ,916 (2002). P. Nozier`es, in
Solids far from equilibrium , Cambridge UniversityPress, edited by C. Godr`eche (1989). V. Tsepelin et al. , J. Low Temp. Phys. , 489 (2002). J. P. Ruutu et al. , Phys. Rev. Lett. , 2514 (1996). J. P. Ruutu et al. , J. Low Temp. Phys.112