Effect of 3He impurities on the mass decoupling of 4He films
Kenji Ishibashi, Jo Hiraide, Junko Taniguchi, Tomoki Minoguchi, Masaru Suzuki
aa r X i v : . [ c ond - m a t . o t h e r] D ec Effects of He impurities on the mass decoupling of He films
Kenji Ishibashi a , Jo Hiraide a , Junko Taniguchi a , Tomoki Minoguchi b , and Masaru Suzuki a ∗ a Department of Engineering Science, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan. b Institute of Physics, University of Tokyo, Meguro-ku, Tokyo 153-8902, Japan (Dated: January 1, 2020)We carried out quartz crystal microbalance experiments of a 5 MHz AT-cut crystal for superfluid He films on Grafoil (exfoliated graphite) with a small amount of He up to 0.40 atoms/nm . Wefound that the mass decoupling from oscillating substrate is considerable sensitive even in a smallamount of He doping. In a He film of 29.3 atoms/nm , we observed a small drop in resonancefrequency at T of ∼ He atoms on the He solid atomic layer. For a large amplitude, the He solid layer shows a reentrant mass decouplingat T R close to T . This decoupling can be explained by the suppression of the superfluid counterflowdue to the adsorption of He atoms on edge dislocations. As the He areal density increases, T R shifts to the lower temperature, and vanishes around a He film of 39.0 atoms/nm . I. INTRODUCTION
It is well known that the surface of graphite is atom-ically flat and helium film on graphite grows up layer-by-layer to more than five-atom thick film in layers.[1, 6]Because of both the quantum nature of helium and theideal two-dimensional system, helium film on graphitehas been attracting the attention of many researchers.The adsorbed structure,[1–3] the magnetism[4, 5] and thesuperfluidity[6, 7] are extensively studied experimentallyand theoretically.Lately the nanofriction of films, or the mass decouplingof films from oscillation, has been widely discussed.[8]Several films on metal substrates show a partial massdecoupling.[9]. In addition, it is reported that the filmtakes place the pinning-depinning transition against thedriving force of oscillating substrates.[10]In response to the study on nanofriction, we startedto study the mass decoupling of helium films on graphiteusing the quartz crystal microbalance (QCM) technique.Up to the present, we have reported the following obser-vations above two-atom thick films: [11–14](a) When the oscillation amplitude is large enough, thesolid layer of He films undergoes partial mass decouplingbelow a certain temperature T S .(b) This decoupling brings a low-friction metastable statewhen the overlayer is normal fluid. The solid layer afterthe reduction in amplitude remains in the low-frictionstate with a finite life time.(c) When the overlayer is superfluid, the mass decouplingsuddenly vanishes at T D below T S .(d) For He films, the mass decoupling shows a similarbehavior up to five-atom-thick films without an abruptsuppression due to superfluid.The inhomogeneity of films plays an important role inthis decoupling. We proposed the following scenario:[13]The motion of the edge dislocation in the solid layer isresponsible for mass transport. The mass decoupling oc-curs when the edge dislocation overcomes the potential ∗ [email protected] barriers of the substrate (Peierls potential). This ex-plains the external force threshold for mass decouplingand the low-friction state being metastable. In addition,the sudden vanishment below T D can be explained bythe cancellation of mass transport due to the superfluidcounterflow of the overlayer.The mass decoupling of helium films has shown var-ious interesting behaviors. In the present experiments,we confine ourselves to He impurity effects of He filmson Grafoil (exfoliated graphite) when the overlayer is su-perfluid. In this Paper we report a systematic study onthe mass decoupling using a MHz range AT-cut crystal.After a brief explanation on the experimental setup inII, we show In III.1 the He areal density dependence fora four-atom thick film for various oscillation amplitudes.By adding a small amount of He, it was found that themass decoupling appears again at a certain temperature T R below T D for a large amplitude. In III.2, we show the He areal density dependence for a fixed amount of He. T R decreases with increasing the He areal density, anddisappears above a certain He areal density. In additionto these observations, we discuss a possible mechanismof the reentrant mass decoupling at T R . II. EXPERIMENTAL SETUP
We used the QCM technique with an AT-cut crystalto measure the mass decoupling. In the QCM technique,the coupled mass to the oscillating substrate is obtainedfrom the change in the resonance frequency ∆ f as∆ ff = − mM (1)where m is the coupled mass of film, M is the oscil-lating mass of the crystal, and f is the resonance fre-quency. When the film is decoupled from the oscillation,the coupled mass decreases and the resonance frequencyincreases.In the present experiments, the resonator is a 5.0 MHzAT-cut crystal. The crystal was commercially available,and no special treatment was applied to the Ag elec-trode. At first, Grafoil was baked in a vacuum at 900 ◦ Cfor 3 h, and a 300-nm-thick film of Ag was depositedonto it. The crystal and Ag-plated Grafoil were pressedtogether and were heated in a vacuum at 350 ◦ C for 2 h.Then, Grafoil was bonded on both sides of the Ag elec-trode. After bonding, the excess amount of Grafoil wasremoved to increase the Q value of the crystal. To keepgood thermal contact, the crystal was fixed to the metalholder with electrically conductive adhesive. After theseprocesses, the Q value remained better than 10 , andthe areal density of Grafoil was 7.30 g/m . After beingheated in 2 × − Pa at 130 ◦ C for 5 h, the crystal wasmounted in the sample cell. In the present experiments,the mass loading of He is 3.8 Hz · atoms − · nm .The resonance frequency was measured using a trans-mission circuit. In the circuit, the crystal was placed inseries with a coaxial line connecting a 50 Ω cw signal gen-erator and a RF lock-in amplifier. The frequency of thesignal generator was then controlled in order to keep theinphase output zero, and was locked to the resonance fre-quency. The quadrature output at this frequency is theresonance amplitude.In the present experiments, the He areal density is atmost up to 0.4 atoms/nm , which corresponds to 5% ofthe areal density of He one-atomic layer.
III. RESULTS AND DISCUSSIONIII.1. He areal density dependence
We carried out temperature sweep experiments of afour-atom thick He film for various oscillation ampli-tudes by changing the He areal density (Run A).Figure 1 shows the variation in resonance frequencyfor He of 29.3 atoms/nm with several He areal den-sities. The overlayer of these films undergoes superfluidat low temperatures. All data were taken during coolingwith the oscillation amplitude being fixed at 0.018 nm.In this amplitude, the superfluid onset of a pure He filmwas clearly observed at T C of 0.80 K, although it is hardlyseen in the scale of Fig. 1. As the He areal density in-creases, T C decreases gradually. For a He film with Heof 0.30 atoms/nm , T C is shifted down to 0.75 K. Byadding He, it was found that a small additional drop inresonance frequency appears at T below T C . As the Heareal density increases, this drop becomes clear. How-ever, T does not depend strongly on the He areal den-sity above 0.1 atoms/nm .In the inset, we compare the variation in resonance fre-quency and Q -value between the pure He film and the He film with He of 0.20 atoms/nm . For the pure Hefilm, the superfluid onset is observed at T C of 0.80 K, ac-companied with a small increase in ∆(1 /Q ). When Heis added by 0.20 atoms/nm , T C is slightly shifted downto 0.76 K. The resonance frequency is deviated down-wards at T of 0.41 K from the extrapolated curve fromhigh temperatures. The difference from this extrapolatedcurve increases gradually down to the lowest attainabletemperature ∼ ∼ ∆ F r e q . ( H z ) Temp. (K) T C He (atoms/nm )0.050.100.150.200.250.30T ∆ F r e q . ( H z ) ∆ ( / Q ) ( pp m ) He (atoms/nm ) C T FIG. 1. Variations in the resonance frequency of a He film of29.3 atoms/nm at an amplitude of 0.018 nm for various Heareal densities. The data are shifted vertically. Inset: Com-parison in the resonance frequency and Q -value between pure He film and He film with He of 0.20 atoms/nm . (Run A) hand, the anomaly in ∆(1 /Q ) was not observed at T within the present accuracy.It is natural that the drop below T is connected tothe addition of He. The mass loading of He is es-timated to be 2.9 Hz · atoms − · nm from that of He.The drop of ∼ ∼ for He. This value is about thedouble of He dopant. Thus, it is concluded that thedrop below T is caused not only by the sticking of Heatoms on the He solid layer, but also by preventing Heatoms from decoupling. Furthermore, it was found that T does not depend strongly on the He areal densityabove 0.1 atoms/nm , which means that a number of ad-sorption sites for He atoms is on the order of 0.1 nm − .The possible candidate of the adsorption site on the He solid layer is the edge dislocation core. Because ofthe adsorption potential of graphite, the first solid atomiclayer is about 20% denser than the second one,[1] Dueto the density difference between the solid layers, it isnaturally assumed that the top solid atomic layer consistsof commensurate domains separated by domain walls tothe first solid atomic layer. Since domain walls have thesame motif as edge dislocations,[13] we here call themedge dislocations. The local areal density of the top solidatomic layer becomes small at the edge dislocation. Fromthe difference in the zero-point energy, it is thought that He atom is adsorbed on the edge dislocation core from
Osc. amp. (nm)0.560.250.180.100.0560.018 T ∆ F r e q . ( H z ) T D T R T C T S ∆ F r e q . ( H z ) Temp. (K) S T D T C FIG. 2. Variations in the resonance frequency of a He filmof 29.3 atoms/nm with He of 0.20 atoms/nm for variousoscillation amplitudes. The data are shifted vertically. Inset:Variations in the resonance frequency of a pure He film of29.3 atoms/nm . (Run A) the liquid overlayer. Here, it should be noted that thethickness of the liquid overlayer is at most one atomiclayer and that He atoms may not be bounded on the freesurface in contrast to bulk He.[18] In fact, it is revealedthat He atoms are trapped on the dislocation core withthe adsorption potential of 0.7 K in the case of He- Hesolids.[17]Figure 2 shows that the amplitude dependence for a He film with He of 0.20 atoms/nm . All data weretaken during cooling. As shown in Fig. 1, for the am-plitude of 0.018 nm, the superfluid onset and the dropin frequency are observed at T C of 0.76 K and T of0.41 K, respectively. As the amplitude increases, the in-crease in resonance frequency due to the superfluid on-set is smeared out. In contrast, for the amplitudes of0.18, 0.25 and 0.56 nm, the resonance frequency increasesclearly at T S , and this increase is terminated abruptly at T D .As shown in the inset, these behaviors are also observedfor a pure He film, which is attributed to the decouplingand sticking of the He solid layer. By adding a smallamount of He, a new phenomenon appears. Below T D ,the resonance frequency rises up at a certain tempera-ture T R , which means that the He solid layer undergoesdecoupling again. It was found that the reentrant massdecoupling temperature T R is close to T , i.e., the tem-perature where He atoms are trapped at the adsorption T S T D T R T C He (atoms/nm )0.20.6 T e m p . ( K ) slip I slip IIstick IIstick I 0 He (atoms/nm )0.050.150.100.200.250.300.40T S T D T R ∆ F r e q . ( H z ) Temp. (K)
FIG. 3. Variations in the resonance frequency of a He filmof 29.3 atoms/nm at the amplitude of 0.25 nm for various He areal densities. The data are shifted vertically. Inset:Phase diagram of the decoupling and sticking behaviors. T S , T D and T R are from Fig. 3, while T C from Fig. 1. (Run A) He edge dislocation counterflow 1st atomic layer 2nd atomic layer (a) slip I (b) stick II (c) slip II
FIG. 4. Cartoon for He- He mixture films, (a) slip I, (b)stick II and (c) slip II. site on the He solid layer. For the amplitude of 0.56 nm, T D and T R disappear and the decoupling of the He solidlayer remains at low temperatures.To clarify the He areal density dependence of T S , T D and T R , we carried out temperature sweep experimentswith the amplitude of 0.25 nm for several He areal den-sities. Figure 3 shows the variation in resonance fre-quency. All data were taken during warming. When Heof 0.05 atoms/nm is added, the decoupling and stick-ing behaviors are drastically changed from the pure Hefilm. T S is lowered down to 0.69 K from 0.74 K of thepure He film. In contrast, T D of 0.50 K does not changegreatly. As the temperature decreases, the resonance fre-quency increases gradually below 0.4 K, and rises up at T R of 0.33 K. With further decreasing temperature downto 0.2 K, it decreases gradually again. Above He of0.10 atoms/nm , the increase in frequency at T R becomessharp. As the He areal density increases, T R increasesgradually and T D does not change greatly.The inset shows a phase diagram of decoupling andsticking behaviors. This diagram is divided into four re-gions. At high temperature, the He solid layer sticks tothe oscillating substrate (stick I). As the temperature de-creases, this layer undergoes decoupling below T S (slip I),and sticks suddenly at T D (stick II), regardless whetheror not the film contains He. By adding He, the reen-trant mass decoupling appears below T R (slip II).We discuss a possible mechanism of the reentrant massdecoupling. It should be noted that T D and T R show upwhen the overlayer of these films becomes superfluid. Forthe pure He film, the sticking at T D can be explained bya mechanism in which the mass transport caused by themotion of edge dislocations is cancelled by the superfluidcounterflow of the overlayer.[13]In developing this scenario, we can explain the reen-trant mass decoupling. A cartoon for He- He mixturefilms is shown in Fig. 4. Since He atoms are dissolved,or are spread over the fluid overlayer at high tempera-ture, the He solid atomic layer shows the decoupling at T S and the sticking at T D as the same manner as pure He film (Figs. 4(a) and (b)). As above-mentioned, thesticking at T D means that the superfluid counterflow be-tween edge dislocations cancels the mass transport. Asthe temperature decreases, He atoms start to adsorb onthe edge dislocation at around T , and prevent the ex-change between liquid and solid He atoms (Fig. 4(c)),i.e., the superfluid counterflow is ceased. Then, the Hesolid atomic layer undergoes decoupling again at T R . III.2. He areal density dependence
In a different series of experiments from III.1., wecarried out temperature sweep experiments for a fixedamount of He by changing the He areal density(Run B). Figure 5 shows the variation in resonancefrequency for several He areal densities with He of0.20 atoms/nm . The oscillation amplitude was fixed at0.18 and 0.018 nm during the temperature sweep.In the amplitude of 0.18 nm, all data were taken dur-ing warming. For He of 28.5 atoms/nm , the resonancefrequency increases at T S of 0.64 K, which can be at-tributed to the mass decoupling of the He solid layer.For He of 29.0 atoms/nm , the mass decoupling is ter-minated abruptly at T D of 0.52 K and the reentrant massdecoupling occurs at T R of 0.39 K, as the same manneras Figs. 2 and 3. As the He areal density increases, themass decoupling connected to T S disappears, while T R ∆ F r e q . ( H z ) He (atoms/nm )28.529.039.037.0 35.032.030.029.5Osc. amp. = 0.018 nmT C T ∆ F r e q . ( H z ) He (atoms/nm )28.529.039.0 37.035.032.030.029.5Osc. amp. = 0.18 nmT R T C T S T D (a)(b) FIG. 5. Variations in the resonance frequency at the ampli-tudes of (a) 0.18 nm and (b) 0.018 nm for various He arealdensities with He of 0.20 atoms/nm . The data are shiftedvertically. (Run B) remains. As further increasing the He areal density, T R shifts to the lower temperature and vanishes at around He of 39.0 atoms/nm .In the amplitude of 0.018 nm, all data were taken dur-ing cooling. For He of 28.5 atoms/nm , it is difficultto definitely determine T , i.e., the sticking temperatureof He atoms, while it is observed at T of 0.40 K for He of 29.0 atoms/nm . As the He areal density in-creases, T shifts to the lower temperature. Above Heof 35.0 atoms/nm , it is difficult to determine T again.Here, it should be noted that T has the same He arealdensity dependence as T R , although T is observed in thelimited range. On the other hand, the superfluid onset isalso observed for these areal densities. As the He arealdensity increases, T C moves to the higher temperatureand reaches 1.23 K at He of 39.0 atoms/nm .Figure 6 shows the phase diagram of the sticking anddecoupling behaviors for He of 0.20 atoms/nm . T S , slip IIslip I stick IIstick I He ( atoms/nm ) T e m p . ( K )
30 4000.51.01.5 four-atomthick film five-atomthick filmT S T C T D T R T He : 0.20 atoms/nm FIG. 6. Phase diagram of the sticking and decoupling behav-iors for He of 0.20 atoms/nm . T S , T D and T R are taken fromthe amplitudes of 0.18 nm, T C and T from the amplitudes of0.018 nm. T D and T R are obtained from the amplitudes of 0.18 nm,while T C and T at the amplitudes of 0.018 nm. In con-trast to the phase diagram of Fig. 3, both regions ofStick I and II and of Slip I and II connect continuously.As mentioned above, T is close to T R . This supportsstrongly the scenario mentioned in III.1., i.e., the ad-sorption of He atoms on the edge dislocation causes thereentrant mass decoupling.Furthermore, the vanishment of T R at a high He arealdensity may be explained by the competition betweenthe adsorption on the edge dislocation and on the freesurface. For bulk He, it is well known that He atoms arebounded on the free surface at low temperature.[18] Thebound energy primarily comes from the difference in thezero-point energy between in bulk He liquid and on thefree surface. Thus, we can propose the following scenario:In the case of an atomic-thin overlayer, He atoms arelocated on the He solid layer because of no advantageof the zero-point energy on the free surface. As the Heareal density increases, i.e., the overlayer becomes thick, He atoms move to the free surface, and the adsorptionof He atoms no longer occurs.
III.3. The model calculation for He adsorption
We discuss whether the He areal density dependenceof T can be explained by a simple adsorption model. Tobuild the model, we can refer the previous experimentsfor He- He mixture thin films.[19, 20]Saunders and co-workers have carried out heat ca-pacity experiments of He above 0.4 atoms/nm in a He film of 33.5 atoms/nm on Grafoil. They have re-ported that He atoms in a thin He film behave as thetwo-dimensional (2D) Fermi gas.[19] On the other hand,Hallock and co-workers have carried out NMR experi-ments for 0.1 monolayer of He in thin He films on He ( atoms/nm ) T e m p . ( K ) N a (sites/nm ) 0.8741.2681.5662.4040.100.070.060.061.5 m m ε a (K) n ( a t o m s / n m ) Temp. (K) 1.000.07 He ( atoms/nm )1.5 m m FIG. 7. Comparison between the model calculation and T .Open and solid circles denote T in Fig. 1 and T R in Fig. 3,respectively. Again the He areal density is 29.3 atoms/nm .The lines represent the temperature at which n equals to0.05 atoms/nm with several different parameters. In calcula-tion, the parameters were chosen where the lines pass through0.43 K at 0.20 atoms/nm . Nuclepore.[20] They have reported that a part of Heatoms are immobile below a critical He areal density.As the He areal density increases, He atoms experiencea mobility edge.The present observations are quite similar to those ofHallock and co-workers’ experiments, i.e., a small amountof He atoms are localized in thin He films, and this lo-calization vanishes at a certain He areal density. Sandersand co-workers concluded that He atoms in a thin Hefilm are not adsorbed on Grafoil and are extended. Wethink, however, that there is a possibility that a smallamount of He atoms are adsorbed because the heat ca-pacity is independent of the areal density of the 2D Fermigas.From these considerations, we consider the followingmodel: He atoms in the overlayer behave as the 2DFermi gas with the hydrodynamic effective mass m ∗ . Inaddition, there exits a surface binding state with theadsorption site density N a and the binding energy ε a measured from the bottom of the 2D Fermi gas. In thismodel, the adsorption density n is obtained as n = N a e − β ( − ε a − µ ) e − β ( − ε a − µ ) + 2(2 π ) Z ∞ πk dke β ( ε − µ ) + 1 , (2)where β = 1 /k B T is the inverse temperature, ε = ~ k / m ∗ is the kinetic energy of the Fermi gas, and µ is the chemical potential which is determined from the He areal density.Here, we may adopt m ∗ /m ∼ . He film of 33.5 atoms/nm ,[19] al-though m ∗ in thinner He films is still unknown. Theinset of Fig. 7 shows a typical calculation of n as afunction of temperature for several He areal densitieswith m ∗ /m = 1 . N a = 0 .
06 sites/nm and ε a =1 .
268 K. Here, the parameters were chosen where n =0 .
05 atoms/nm at 0.43 K for He of 0.20 atoms/nm .For comparison, we plotted a curve of m ∗ /m = 10 at0.20 atoms/nm .As seen in the inset, n increases gradually from hightemperature and becomes nearly equal to N a below a cer-tain temperature. As the He areal density increases, n shifts to the higher temperature. Although it is not clearthat which value of n corresponds to T , it is assumedhere that n C = 0 .
05 atoms/nm is T . We plotted thetemperature where n = 0 .
05 atoms/nm as a function of He areal density in Fig. 7. It was found that the calcu-lated lines has a stronger He areal density dependencethan that of the observations. This behavior does not de-pend strongly on the parameters of N a , ε a and n C . Thus,we may conclude that the simple adsorption model doesnot explain the areal density dependence of T .Here, we would like to make a comment on the model.As shown in Fig. 7, when we choose m ∗ /m = 10, n varies rapidly in a small temperature range and the arealdensity dependence of T becomes weaker. i.e., when thenumber of density just above the surface binding state islarge enough, T does not depend strongly on He arealdensity. This may suggest that He atoms in a very thinoverlayer are nearly localized on Grafoil. Although itwas reported that m ∗ is enhanced with decreasing Heareal density for Nuclepore.[20], this is only speculationfor Grafoil. Furhermore, when there exists an atractiveinteraction between the adsorption sites, n varies morerapidly. These are for future study.Here, we make a comment on a thicker overlayer. T was not observed clearly above He of 33.0 atoms/nm .It is, however, natural that T is nearly equal to T R . This means that T tends to zero around He of39.0 atoms/nm . As mentioned in III.2., He atoms arebounded on the free surface of bulk He.[18] The bindingenergy ε S was obtained to be 2.22 ± ε a in the simple adsorption model is smaller than ε S if the model explains the He adsorption.
IV. SUMMARY
We report quartz crystal microbalance experiments us-ing a 5 MHz AT-cut crystal for He- He mixture films on Grafoil. In the present experiments, the He areal den-sity is at most up to 0.4 atoms/nm . In a four-atom thick He film of 29.3 atoms/nm , we observed following be-haviors: (a) For a small amplitude of 0.018 nm, a smalldrop in resonance frequency occurs at T . (b) For a largeamplitude of 0.25 nm, the mass decoupling at T S andsticking at T D was observed as the same manner as pure He films. In addition to T S and T D , a reentrant massdecoupling occurs at T R close to T . Here, it was foundthat both of T and T R do not depend strongly on the He areal density above 0.1 atoms/nm , and are ∼ He films, we haveproposed the following scenario: the mass decoupling be-low T S results from the motion of edge dislocations be-tween the first and second solid layers. The mass stickingat T D is caused by the cancellation of mass transport dueto the superfluid counterflow of the overlayer.[13] As anextension of this scenario, the observed behaviors can beexplained as follows. He atoms which are mobile at hightemperature are localized on the edge dislocation at T .These He atoms prevent the exchange between liquidand solid He atoms, and the reentrant mass decouplingoccurs by the cease of the superfluid counterflow.From experiments changing the He areal density for He of 0.2 atoms/nm , it was found that T R decreaseswith increasing He areal density and vanishes above Heof 29.0 atoms/nm . This behavior can be interpreted bythe competition of the adsorption between on the edgedislocation and on the free surface.The above explanation about the reentrant mass de-coupling below T R naturally leads to the model that Heatoms adsorb on the He solid layer. However, this modelcannot explain a weak He areal density dependence of T R using the known hydrodynamic effective mass of Hein the overlayer. This is for future study.
ACKNOWLEDGMENTS
One of the authors (TM) wishes to express his thanksfor the financial support of Yamaguchi Educational andScholarship Foundation. [1] S. Greywall and P. A. Busch, Phys. Rev. Lett. , 3535(1991); D. S. Greywall, Phys. Rev. B , 309 (1993).[2] M. Pierce and E. Manousakis, Phys. Rev. B , 3802(1999).[3] P. Corboz, M. Boninsegni, L. Pollet and M. Troyer, Phys.Rev. B , 245414 (2008).[4] M. Neumann, J. Nye’ki, B. Cowan and J. Saunders, Sci-ence , 1356 (2007).[5] H. Fukuyama, J. Phys. Soc. Jpn. , 111013 (2008). [6] A. Crowell and J. D. Reppy, Phys. Rev. B , 2701(1996).[7] J. Nyeki, A. Phillis, A. Ho, D. Lee, P. Coleman, J. Parpia,B. Cowan and J. Saunders, Nat. Phys. , 455 (2017).[8] J. Krim, Adv. Phys. , 155 (2012).[9] A. Dayo, W. Alnasrallah, and J. Krim, Phys. Rev. Lett. , 1690 (1998) ; M. Highland and J. Krim, Phys. Rev.Lett. , 226107 (2006).[10] L. Bruschi, A. Carlin, and G. Mistura, Phys. Rev. Lett. , 046105 (2002) ; A. Carlin, L. Bruschi, M. Ferrari,and G. Mistura, Phys. Rev. B , 045420 (2003).[11] N. Hosomi, A. Tanabe, M. Suzuki, and M. Hieda, Phys.Rev. B , 064513 (2007).[12] N. Hosomi, M. Suzuki, Phys. Rev. B , 024501 (2008).[13] N. Hosomi, J. Taniguchi, M. Suzuki, and T. Minoguchi,Phys. Rev. B , 172503 (2009).[14] N. Hosomi, M. Suzuki, J. Low Temp. Phys. , 773(2007).[15] G. A. Cs´athy and M. H. W. Chan, Phys. Rev. Lett. ,045301 (2001).[16] N. Hosomi, J. Taniguchi, M. Suzuki, and T. Minoguchi,J. Phys: Conference Series , 180103(R) (2014).[18] D. O. Edwards and W. F. Saam, Progress in Low Tem-perature Physics
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