Flexural phonons in supported graphene: from pinning to localization
We identify graphene layer on a disordered substrate as a possible system where Anderson localization of phonons can be observed. Generally, observation of localization for scattering waves is not simple, because the Rayleigh scattering is inversely proportional to a high power of wavelength. The situation is radically different for the out of plane vibrations, so-called flexural phonons, scattered by pinning centers induced by a substrate. In this case, the scattering time for vanishing wave vector tends to a finite limit. One may, therefore, expect that physics of the flexural phonons exhibits features characteristic for electron localization in two dimensions, albeit without complications caused by the electron-electron interactions. We confirm this idea by calculating statistical properties of the Anderson localization of flexural phonons for a model of elastic sheet in the presence of the pinning centers. Finally, we discuss possible manifestations of the flexural phonons, including the localized ones, in the electronic thermal conductance.
Most of the research in graphene emphasizes the relativistic character of its electron spectrum. However, graphene is also interesting due to its out-of-plane (flexural) vibrational phonon modes. Flexural phonons are a unique addition that Van der Waals heterostructures have brought into microscopic physics.Geim and Grigorieva (2013) Usually, flexural phonons are considered in the context of the suspended graphene. However, Van der Waals nature of the interaction of the graphene sheet with a supporting substrate, provides an interesting possibility to observe the flexural phonons even in supported graphene. Here we argue that graphene layer placed on the top of the corrugated SiO substrate gives an opportunity to observe Anderson localization for the flexural phonons (the FPs). To get an idea, let us recall the known facts about a long-wave acoustic wave scattering on a cylinder. The output depends drastically on the boundary conditions for the velocity potential on the surface of the cylinder.Morse and Feshbach (1986) If the velocity component normal to the surface of the cylinder vanishes, i.e., , the scattering cross-section is proportional to , where is the radius of the cylinder and is the wave vector. This is the conventional Rayghley scattering result Rayleigh (1896) for two dimensional geometry. However, when pressure is constant, the boundary condition reads , and this influences forcefully the scattering. Unlike the Rayghley scattering, the zero angular harmonic is involved, and as a result the cross section diverges at small as . (The same takes place for an electro-magnetic wave scattering on a metallic cylinder.)
In graphene, the substrate cannot scatter effectively the usual acoustic waves, longitudinal and transverse, because graphene itself is one of the most rigid substances. The other thing are the out of plane phonons. From the analysis of intrinsic and extrinsic corrugation of monolayer graphene deposited on SiO substrate, it has been concluded that in this system the layer is suspended between hills of the substrate. Ishigami et al. (2007); Stolyarova et al. (2007); Geringer et al. (2009); Deshpande et al. (2009) We have checked that scattering of the FPs from areas attached to the substrate is similar to the scattering from a rigid obstacle.Norris and Vemula (1995) The zero harmonic is also involved, and the scattering cross-section diverges as . As a result, the elastic scattering rate for low-energy FPs exceeds their energy. One may, therefore, expect localization for low-energy FP modes. This is in striking contrast with localization of acoustic modes which is known to happen only at high enough frequency.John et al. (1983); Kirkpatrick (1985); Akkermans and Maynard (1985); Sepehrinia et al. (2008); Monthus and Garel (2010)
Let us comment upon the graphene layer deposited on the top of the corrugated substrate. Naively, the membrane-like layer either follows the substrate or hovers over the surface at some distance. Measurements of Ref. Conley et al., 2011 with the use of cantilevers indicate, however, towards a possibility of the detaching a graphene sheet from a substrate to relieve its strain by slipping. (This is manifest by straightening of the cantilever.) In the case of the SiO substrate, both experiment and theory agree that for typical magnitude for corrugations, the graphene layer is partially detached from the substrate. Moreover, the theoretical considerationsSabio et al. (2008); Kusminskiy et al. (2011) justify the use of a contact force that is finite when graphene is conforming to the substrate and zero otherwise. Factors that may be particular for the SiO substrate are charge-donating impurities below the graphene layer,Zhang et al. (2009); Deshpande et al. (2009, 2011) and water molecules which may lie between graphene and the substrate.Schedin et al. (2007); Moser et al. (2008) The basic experimental facts which lead to the conclusion that graphene layer deposited on SiO is partly freely suspended are as follows.Geringer et al. (2009) The long-range corrugation of the substrate with the correlation length of about 25nm is also visible on the graphene sheet, but with a smaller amplitude than on the substrate. Mesoscopic corrugations with smaller length of about 15nm not induced by the substrate were also identified. These short range corrugations are similar in height and wavelength to the ones observed on suspended graphene.Meyer et al. (2007); Laitinen et al. (2014)
In this work we study the statistical properties of the out of plane excitations in a model of a pinned-suspended flexible sheet, see Fig. 1. Whether pinning centers are located in the vicinity of the maximal heights of the substrate where the interaction with the layer is the strongest, or there are charges on the substrate which interact strongly with their images, will be not important for our purposes. For simplicity, it will be assumed that centers of pinning are located randomly. First of all, we are interested in the scattering rate , where is the concentration of the pinning centers. As we have already discussed, the scattering cross-section on a rigid obstacle is . Taking into consideration that the spectrum of the flexural phonons is quadratic, i.e., velocity is linear in , one obtains a scattering time that is finite in the low-energy limit. Thus, in contrast to acoustic phonons, for flexural phonons one may expect localization of the low-energy modes with .
Notice that the point here is not in flexural phonons as such, but it is their softness that allows to realize scattering by effectively rigid obstacles which leads to a non-vanishing scattering rate in the limit of small energy . Let us touch upon this point in more detail. Usually by a rigid obstacle one understands an inclusion with the Young’s modulus much higher than that in surrounding area. For graphene, which itself is very rigid, this is not an issue. However, the pinning potential introduces a barrier of a finite height for the flexural modes:
Here is displacement in the out of plane direction; the term describing the barrier is on the right-hand side of the above equation. As a result, a flexural phonon with an energy smaller than cannot enter the area of pinning. We have checked that such an inclusion is equivalent to a rigid obstacle. For not small , the cross-section .See Supporting Materials () Interestingly enough, for , . The limiting cross-section is , i.e., twice larger than the width of the obstacle.Morse and Feshbach (1986) (This is, of course, valid only when the energy of the phonon is much less than the pinning potential, i.e., .)
To understand the general properties of flexural phonons in the presence of random scatterers, we study a model of random pinning centers. Relying on the existing experimental data,He et al. (2013) we assume that the pinning potential is about few meV. Hence, at the pinning centers the graphene sheet can be considered as completely attached. We solve for eigenmodes of discretized LHS of Eq. (1) with condition at randomly chosen pinned sites,See Supporting Materials () so that one pinned site represents an attached area of the size . We estimate the size of an attached area to be . Typical distance between the pinning centers is around ; we will assume that . The representative fraction of the pinned sites is . Correspondingly, we studied ”samples” with of the pinned sites to determine statistical properties of the eigenmodes and eigenvalues. In what follows, we measure the energy eigenvalues in units of which approximately equals for the parameters mentioned above. We found out that typical energy scale for strong localization for concentration of pinned sites in the range is a fraction of 1K. Eigenmodes at two representative energies are shown on the Fig. 2 for a sample (only fragments are shown).
Iii Phonon ”Conductance”
As is well known, localization is a quantum critical phenomenon. The peculiarity of is that the critical point is at , where is electrical conductance per square measured in units . At a finite , statistical properties are determined by the localization length , which is analogue of a correlation length at a quantum phase transition. A sample of size is in the regime of criticality which may take place in a very broad range of the sample sizes, because for small the localization length is exponentially large.Abrahams et al. (1979) A consequence of strong fluctuations of wave function amplitude in the critical region is the multifractality (that is when an eigenstate is extended but the occupied volume is noticeably smaller than the volume of the sample).
Following questions will be addressed now for disordered flexural phonons: Is there a transition to delocalized states at a certain energy (i.e., the ”metal-insulator” transition with the mobility edge), or there is a crossover from strong to weak localization? If to compare with the electrons propagating in a disordered lattice (Anderson model), will the observed behavior statistically the same or different?
To figure this out, we first studied the dependence of the Inverse Participation Ratio (IPR) on the sample size for various phonon energies.See Supporting Materials () We used samples of the size up to sites. From our simulations, it is clear that low-energy modes are localized: the IPR scales with the sample size to a finite value. For higher energies, the behavior of the wave functions changes, see Fig. 2, because the localization length starts to exceed the sample size. We are particularly interested in studying the flexural phonons in this region when . Note for comparison that although the Anderson model does not constitute a truly critical system, thanks to exponentially large localization length at , the criticality takes place in a very broad range of the system sizes, . Therefore, samples with large share many common properties with systems at the critical point of the metal-insulator transition. As we shall see, similar physics holds also for our system of the FPs.
We proceed as follows: From the size dependence of the IPR at a given energy as shown on the Fig. 3, we extracted an energy-dependent fractal dimension. For disordered electronic system of a given symmetry class, the fractal dimension is determined by the conductance . For example, in the case of the Gaussian Orthogonal Ensemble (GOE), the size dependence of the IPR is described by the fractal dimensionWegner (1980) equal to ,See Supporting Materials () where dependence of on is due to the weak-localization (WL) correction. Thus, for each concentration of the pinned sites we can prescribe for different energies the corresponding value of the phonon ”conductance” , using the expression for the fractal dimension for electrons.
For disordered electrons in , the well developed theory connects the behavior of various physical quantities with the value of the conductance.Mirlin (2000) We have calculated numerically the same quantities for flexural phonons, using the values of extracted from IPR, and found a very good agreement with the theoretical predictions existing for the disordered electrons in the case of the GOE. Below we present two examples of such calculations.
Iv Statistical properties
In Fig. S2See Supporting Materials () the distribution function of the level spacing is shown for localized states, which has almost Poissonian statistics, while for metallic states it is of the Wigner-Dyson form. This is rather standard.
In it becomes especially interesting to study the variance , which is a two-level correlation function characterizing the fluctuations of the number of levels in a strip of width around the energy . The reason why it is of particular interest is that, in contrast to and , in two dimensions this quantity is directly related to the WL corrections. Kravtsov and Lerner (1995) Fig. 4 demonstrates the level number variance as a function of the ratio , where is the average level spacing. It starts with the ergodic behavior described by the Random Matrix Theory (RMT). The ergodic regime holds up to about the Thouless energy. For larger there is a noticeable deviation: the variance starts to increase rapidly. As Fig. 4 shows, it is in full accord with the expression obtained by us for . See Supporting Materials () (To the best of our knowledge, this is the first demonstration of the mesoscopic fluctuations of the number of levels in . For Anderson model, the function was studied in Ref. Braun and Montambaux, 1995.)
Let us turn from the eigenstates to the statistical properties of the eigenmodes in the discussed model. We have already exploited the fundamental property of the multifractality to establish a quantitative connection with the disordered Anderson model for electrons. Another important property is the distribution of the amplitudes of the eigenmodes, , which is called the wave function intensity distribution , where in our case is . For metallic granulas, in the ergodic regime described by the RMT, the intensity distribution is given by the Porter-Thomas distribution . Owing to the fluctuations in the diffusive motion, there appear deviations from the ergodic behavior (one may think that the deviations from RMT and multifractality of the eigenstates have the same origin, but in fact the two are not direct consequences of each other). When calibrated with respect to , the function yields a curve with a very specific non-monotonous shape. As Fig. 5 shows, an excellent agreement with the theory developed in Ref. Fyodorov and Mirlin, 1995 is found; for review, see also Ref. Mirlin, 2000.
V Discussion of the numerical study
A general question has been addressed in the numerical study discussed above: If to compare with the electrons propagating in a disordered lattice, will the statistical properties of the eigemodes of the pinned elastic layer be the same or different? The question makes sense because for phonons in a pinned-suspended sheet there is no analogue of the on-site disordered potential . Instead, there is concentration of the pinned sites. Furthermore, the FPs are described by the square of the Laplacian, rather than by the Laplacian in the case of electrons.
We have calculated numerically a number of quantities characterizing statistical properties of FPs using the values for extracted from the data for the IPR, and found a very good agreement with the theoretical predictions existing for the disordered Anderson model electrons in the case of the Orthogonal Class of Universality. Furthermore, the theoretical expressions for the number of variance and the wave function intensity , both are intimately connected with the effects of the WL originating from the Cooperons. The excellent agreement demonstrated in Figs. 4 and 5 justifies that in the discussed model the regime of WL is the same as in the Anderson model in . We believe that the reason for the observed universal behavior is that the FPs in the lattice with pinned sites are eventually described with the same Non-Linear -model as disordered electrons in the Orthogonal Class of Universality.
Let us estimate energy of FPs at which a crossover from strong to WL occurs. Strong localization, , holds for momenta that for our choice of yields . The corresponding energy is about . So far, we didn’t consider the effect of strain. The strain , ignoring anisotropy, is known to add the term into the equation of motion, Eq. (1), where is velocity of the longitudinal phonons. In the isotropic approximation this yields . One has to keep in mind that scattering of a FP from a high enough barrier doesn’t depend on details, and the cross-section remains , if . Then, for the linear spectrum the condition for strong localization is , which is similar to what we have got above. Typically, is , and the effects of bending and strain are of comparable strength for the discussed scales.
Note that our estimate for the scale of strong localization is conservative. In reality the size of the attached areas can be comparable with the distances between them. Then, owing to the factor , the energy of the strongly localized FPs can be few times larger. Furthermore, effects of the weak localization noticeably expand localization of the FPs. One may easily show that, as compared to the strong localization, the weak localization increases momenta of the FPs which undergo localization by a factor . (In this estimate, it is necessary to take into consideration the factor in the scattering cross-section.) Correspondingly, weak localization boosts the energy of the localized FPs by a factor . For a standard micron size sample, . As a result, the energy of localized FPs increases up to few K.
In this paper we studied an effective model for flexural phonon localization. In our discretization of Eq. (1), one pinned site represents an attached area of the size of the order of nm. We have neglected effects of anharmonicity and ripples. According to the analysis of Ref. Ochoa et al., 2011 the momenta at which the anharmonicity becomes relevant for FPs are about 5 times smaller than the thermal momentum. Therefore, at any temperature only a small fraction of the phase space of FPs is modified by anharmonicity. Concerning the ripples, their size was evaluated Fasolino et al. (2007) to be around nm at . Correspondingly, at temperatures about K ripples are expected to be too large and too smooth to influence motion of the FPs, which we are interested in.
Localization of the flexural phonons opens a new avenue for understanding measurements that are not yet explained. In particular, the temperature behavior of overheating in graphene at low temperatures Borzenets et al. (2013); Fong et al. (2013) has not been understood. Here, we argue that, in fact, the presence of the FPs, has been already observed in these experiments on SiO-supported graphene.
Let us first mention the thermal transport measurements in Ref. Seol et al., 2010 performed in graphene supported by SiO. According to the theoretical estimates of this paper, the contribution of the flexural modes to the heat conductivity is dominant at temperatures about 100 K. This study, while supporting the picture of partially supported graphene, was limited to relatively high temperatures above .
Now let us turn to cooling of electrons in graphene which has been analyzed in terms of two mechanisms. The one is diffusion of electrons connected with the electric conductivity via the Wiedemann-Franz (WF) law, while the other is electron-lattice cooling in which a flux of heat is directed from electrons to optical and acoustic phonons.Bistritzer and MacDonald (2009); McKitterick et al. (2016) The flux has a power-law form ; there is, of course, a reversed flux from phonons to electrons . As a result, the heat exchange rate , where . To be effective, the electron-lattice cooling demands disorder-assisted processes in order to overcome the momentum conservation constraint, which is potentially very restrictive at high temperatures. A process, called ”supercollisions”, explainsSong et al. (2012); Song and Levitov (2015) the observed energy transport in graphene at high enough temperatures.Strait et al. (2011); Graham et al. (2013); Betz et al. (2013); Somphonsane et al. (2013); Laitinen et al. (2014) However, at low temperatures the screening of the deformation potential leads to an extra power of temperature. As a result, theory yields Chen and Clerk (2012) the flux with , in contradiction to Refs. Borzenets et al., 2013 and Fong et al., 2013, where was observed far away from the neutral point in the regime of temperatures much lower than the Bloch-Gruneisen temperature for acoustic phonons, , where is velocity for acoustic phonons in graphene. The assumption that screening at concentrations may be not effective at low temperatures does not seem realisticSee Supporting Materials (): is about , therefore for temperatures momentum transferred from acoustic phonons to electrons becomes smaller than , and screening of the deformation potential leads to extra powers in in the heat exchange rate.
So far, the role of the FPs in the processes of cooling has not been considered seriously, despite the fact that the flux from electrons to the FPs is knownSong et al. (2012) to be (i.e., as observed). By contrast, we consider the heat exchange with the FPs as realistic explanation of the exponent at low temperatures. Moreover, we believe that the existence of localized FPs may also explain experimental results of Ref. Fong et al., 2013 for cooling rate of graphene at the lowest temperatures . In this regime, comparing contribution of the heat diffusion along the sample, , with which was assumed to be , the authors concluded that diffusion dominates. However, the Lorenz number estimated in this way, was found to be 35% above its nominal value . It should be noted that neither the Fermi liquid nor renormalization-group corrections in disordered electron systems Schwiete and Finkel’stein (2014, 2016); Schwiete and Finkelâstein (2016) can modify the WF law at low temperatures, and this result requests for explanation. As we argue below, heat exchange of electrons with the localized FPs may resolve this problem: the heat exchange of electrons with the localized phonons acquires the form imitating the heat diffusion contribution, . We may thus attribute the seemingly deviation from the WF law to the contribution of localized FPs to electron-phonon heat exchange.
Let us comment upon . First of all, we recall that interaction of an electron with the FPs is described by the two-phonon processes. Correspondingly, the heat exchange between the electrons and FPs, which is proportional to square of the two-phonon amplitude, contains four powers of the FP-momenta. However, the localized FPs are not goldstone modes anymore. For localized FPs, the momenta that enter into the matrix elements of the electron-FP interaction should be substituted by the inverse of the localization length. As a result, two powers of frequencies in the expression for the heat flux saturate at . This, however, leads to a dramatic consequences. The factor describing the dependence on the occupation numbers in the case of the two-phonon processes for temperatures larger than the energy of the phonons diverges like . The diverging integration should be cut-off at energies typical for the localized FPs. As a result one gets a contribution to the cooling rate of the order . The obtained correction to the heat flux has just the form of the WF heat diffusion. In order to obtain experimentally observed magnitude of the deviation of from , one has to suggest . This is in full correspondence with our expectations of the scale energies where localization of the FPs takes place.
To conclude, we have argued that graphene layer placed on the top of a corrugated SiO substrate provides an interesting opportunity to observe Anderson localization for the FPs. Let us note that we have already shown that FPs give a significant contribution to dephasing rate of electrons in graphene.Tikhonov et al. (2014) Interesting perspective opens for thermal transport as heat conductance of the localized FP may remind hopping conductivity of localized electrons.
The authors thank Eva Andrei, Kirill Bolotin, Igor Gornyi, Alexander Mirlin, Daniel Prober, Valentin Kachorovskii and Eli Zeldov for useful discussions. The work at the Landau Institute for Theoretical Physics (KT) was supported by the Russian Science Foundation under the grant No. 14-42-00044. The work at the Texas AM University (AF) is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0014154. A part of this research was conducted with the advanced computing resources provided by Texas A&M High Performance Research Computing.
Appendix A Numerical procedure
We solve the equation of motion for the out-of-plane displacements,
using finite difference method on a 2d square lattice. We discretize the domain of the partial differential equation and then set a large sparse matrix to be diagonalized. Using the 12-point stencil we obtain:
where is the energy of the flexural phonon (FP), and is the lattice constant introduced for discretization ( corresponds to in the main text). For the dimensionless energy used in the figure captions in the main text, one gets from the above expression , where . The pinning is realized in this model by the condition
with pinned sites chosen randomly.
We search for all eigenvalues and eigenvectors of the resulting matrix with periodic boundary conditions. In doing so, we considered samples of the sizes from 3030 up to 200200 sites, with random realizations of 5%, 10% and 20% of the pinned sites.
Appendix B Scattering of a flexural phonon by a rigid obstacle
According to Ref. Norris and Vemula, 1995, the scattering cross-section of the FP by a rigid obstacle of the radius equals where is given by the following expression:
where . The function with asymptotes and is shown in the Fig. 6. For short wave-lengths the limiting cross-section is twice larger than the width of the obstacle. Note that the factor is important for the effectiveness of the weak localization in samples of large size.
Appendix C Weak multifractality of eigenfunctions
The spatial distribution of wave functions is conveniently characterized by inverse participation ratios (IPR):
After sample average, shows the following scaling behavior with the system size :
where is some exponent that can be expressed via . Obviously, in the insulating state , while in a metal . At a critical point, is a fractional, which leads to anomalous scaling behavior in . This is a manifestation of the wave function multifractality. Next, one can introduce anomalous dimensions via
in order to describe the deviation of the scale dependencies at the critical point from the one in the metallic phase. In fact, determines the spatial correlations of the wave function. In particular, governs the spatial correlations of the intensity :
In , the IPRs scale as
Here, the deviation of from dimension is determined by a small parameter . In analogous to weak localization, the phenomena is coined ”weak multifractality”. The above result (10) was first obtained by Wegner via the renormalization group calculations Wegner (1980).
Note that, because of the absence of the genuine critical point in , becomes size-dependent, i.e., where is the mean-free path of FPs at a given energy, and . This implies that for each scale one can use the standard formula given above, but with slowly varying in the exponent. This is possible because corrections to are not large in a finite size sample, and owing to the slow dependence of on spatial scale . With this procedure, we have obtained an excellent agreement between the theory of the logarithmic corrections to the conductivity and our numerical results as it is shown in Fig. 3 of the main text.
Appendix D Energy Level Statistics
Several quantities are introduced to measure the fluctuations of energy levels , such as the distribution function of level spacing , and the two-level correlation function of the Density of States (DOS) .
Random Matrix Theory (RMT) could be used to describe ergodic systems (e.g., electrons in metallic granules). Here, we will focus on the case of Gaussian Orthogonal Ensemble (GOE). Then the distribution function of level spacing is well described by the Wigner surmise:
where and is mean level-spacing. In the localized phase the level correlations are absent, and the distribution function of level spacings is Poissonian: . The crossover from the localized to delocalized behavior in our system is illustrated on Fig. 7.
The correlation function is defined as
where is the DOS, where is the average DOS, which is related to the mean level-spacing as .
Another interesting quantity, characterizing the fluctuations of the number of levels in a strip of width around the energy , is defined as follows:
In the RMT, the level number variance increases logarithmically with . For , it varies as
A peculiarity of in was realized by Kravtsov and Lerner Kravtsov and Lerner (1995), who found that in , unlike and , the level correlation function is governed entirely by the weak localization corrections. They found that with the diffusion constant , where
Appendix E Statistical properties of the wave functions
Porter and Thomas were first who studied the distribution of eigenfunction amplitudes within the RMT framework. Their result shows simply Gaussian distribution, which leads to the following distribution of the intensities :
Here is normalized in such a way that .
The supersymmetric field theory was applied to the study of the eigenfunction statistics in a -dimensional disordered system. The eigenfunction intensity in a point is distributed as:
For , and not too large , one can calculate perturbatively the deviations from the RMT distribution . The corrected distribution function reads Fyodorov and Mirlin (1995)
with . Notice that the expression in the square brackets is non-monotonous. A peculiar behavior of this expression as a function can be seen in Fig. 5 of the main text.
Appendix F Electron-phonon interaction at low temperatures
Some recent experiments have interpreted the cooling rate in terms of the weakly screened electron-phonon (e-ph) interaction down to very low temperatures. In particular, in Ref. Fong et al., 2013 the law, which is characteristic to the dirty regime with negligible screening of the e-ph deformation potential Chen and Clerk (2012), has been observed down to K for sample D3. We believe, however, that applicability of the screenless approximation for the e-ph interaction at such low temperatures is highly questionable.
To illustrate our point, let us estimate following Ref. Chen and Clerk, 2012 temperature above which screening becomes irrelevant in the dirty regime . Comparing the expressions for the energy flux at strong and weak screening Chen and Clerk (2012), we find that crossover from the to behavior should happen at satisfying
with for the Bloch-Gruneisen temperature and for an effective dielectric constant (for graphene on SiO substrate, ). Here we expressed the result in terms of using the fact that in graphene the inverse of the screening radius is of the order of the Fermi momentum. The unscreened behavior may occur at temperatures higher than assuming that it is less than . (For samples D1 and D3 in Ref. Fong et al., 2013 one has K.)
For the aforementioned sample D3, . Thus, one may expect for the crossover temperature K. It is clear that pushing down to K region would require an unrealistic value of . We, therefore, believe that this mechanism may be discarded for explanation of the scaling of the electron-phonon heat flux observed at low temperatures.
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