You are an expert at summarizing long articles. Proceed to summarize the following text: 
compared to heat transfer by phonons in crystalline materials , heat transfer in amorphous materials is complicated by the existence of three regimes of vibrational modes.@xcite low - frequency propagons are delocalized and have a well - defined wave vector and group velocity,@xcite similar to phonons in crystals , while high - frequency locons are localized and contribute negligibly to thermal conduction.@xcite diffusons have intermediate frequencies and are delocalized , but do not have well - defined wave vectors or group velocities . the contribution of diffusons to thermal conduction can be notable , however , as they occupy the majority of the vibrational spectrum.@xcite    from kinetic theory,@xcite the contribution of an individual phonon or propagon mode to thermal conductivity is proportional to its mean free path ( mfp ) . because diffusons do not have a well - defined group velocity , it is not clear if they have a mfp or how it can be defined . their contribution to thermal conductivity can be predicted using their diffusivity , which is well - defined and can be calculated from allen - feldman theory.@xcite nevertheless , it would be insightful to identify a frequency - dependent length scale for propagons and diffusons describing the decay of the heat flux at each vibrational frequency . such a definition would lift the ( fundamentally ) arbitrary distinction between propagons and diffusons and enable a unified description of heat transfer at all vibrational frequencies .    in this paper , we apply the spectrally - decomposed mfp method@xcite to probe the non - equilibrium mfps of vibrational heat carriers in amorphous silicon ( a - si ) . this method is based on calculating the spectrally - decomposed heat current ( sdhc)@xcite in systems of different lengths using non - equilibrium molecular dynamics ( nemd ) simulations . the mfps are determined from the variation of the sdhc as a function of system length at each vibrational frequency.@xcite    we previously used the spectrally - decomposed mfp method to calculate the non - equilibrium mfps in low - dimensional systems such as carbon nanotubes@xcite and anharmonic chains.@xcite we demonstrated that the non - equilibrium mfps transparently describe the ballistic - to - diffusive transition in the length - dependence of thermal conductivity . compared to previous calculations for a - si , the spectrally - decomposed mfp method has several advantages . unlike in modal life - time calculations,@xcite we do not need to estimate the group velocities of individual modes to calculate their mfps . we also do not need to distinguish between propagons and diffusons@xcite nor resort to the harmonic approximation.@xcite in contrast to recent calculations studying the spectral conductivity of a - si in fixed - size systems , @xcite we focus on the mfps of heat carriers and the system - size dependence of thermal conductivity . the rest of the paper is organized as follows . the calculation methods are presented in sec . [ sec : setup ] and the numerical results are discussed in sec . [ sec : results ] . we also introduce a simple method for the quantum correction of thermal conductivity from classical md simulations , based on weighting the sdhc by the quantum occupation function . because this quantum correction method operates at the frequency level , it is more reasonable than quantum - correction methods that operate at the system level ( see ref . and references therein ) and allows us to compare our predictions to experimental measurements . nm . the spectral heat flux @xmath1 is calculated at the cross - section in the middle of the structure ( dashed line ) . the length @xmath2 between the langevin heat baths is varied to extract the vibrational mode mfps based on the decrease of @xmath1 as a function of @xmath2.,width=325 ]    all simulations are carried out using the lammps package@xcite with a time step of 2.5 fs . the si - si interactions are modeled by the stillinger - weber potential@xcite with the parameters of ref . . the nemd simulation geometry is shown in fig . [ fig : geom ] . the atomic coordinates for a - si are generated by following the melt - quench procedure of ref . and the final density is 2,291 kg / m@xmath3 . after the equilibration of the quenched system , atoms located within a distance @xmath4 nm from the left and right edges of the structure are coupled to langevin heat baths at temperatures @xmath5 and @xmath6 with bath relaxation times of 1 ps . to prevent sublimation , atoms at the far left and right edges are fixed to their equilibrium positions . periodic boundary conditions are applied at the boundaries transverse to the current flow . the width of the system cross - section is @xmath7 nm . system lengths @xmath2 ( i.e. , the region between the baths ) between 1 and 10 nm at intervals of 1 nm are considered . the sdhc is calculated through the plane of decomposition located halfway between the hot and cold baths ( dashed line in fig . [ fig : geom]).@xcite the sdhc @xmath8 between particles @xmath9 and @xmath10 located on opposite sides of this plane is given by the pair - wise sdhc equation @xcite @xmath11 where @xmath12 is the simulation time , @xmath13 is the angular frequency , and the interatomic force constant @xmath14 is defined as @xmath15 the velocities @xmath16 and @xmath17 are the discrete fourier transforms of the atomic velocities @xmath18 and @xmath19 ( the exact definitions are in ref . ) , where @xmath20 and @xmath21 are the displacements of atoms @xmath9 and @xmath10 from their equilibrium positions in directions @xmath22,@xmath23 . in eq .  , @xmath24 is the interatomic potential energy function . the spectral flux through the plane of decomposition is obtained from eq .   by summing over all pairs of atoms ( one of the left side , denoted by @xmath25 , and one on the right side , denoted by @xmath26 ) within the potential cut - off distance of each other and dividing by the interface area @xmath27 : @xmath28 while eq . is the first - order approximation to the inter - particle sdhc,@xcite we have confirmed that the contribution of higher - order terms is negligible for a - si by comparing the integral of eq . , which we denote as @xmath29 , to the total flux determined from the work done by the heat baths . the two results agree within 4% . we attribute this good agreement to the stiffness of the interatomic bonds in a - si , which ensures that the first - order term in the current ( proportional to the harmonic force constants @xmath30 ) dominates the higher - order terms that are related to anharmonic force constants . this restriction to the first - order current term at the plane of decomposition does not , however , mean that anharmonic scattering in the bulk is neglected , because all anharmonic effects are included in the nemd simulations.@xcite    frequency - dependent mfps @xmath31 are calculated by determining @xmath32 for different system lengths @xmath2 and fitting the length - dependent @xmath32 to the equation @xcite @xmath33 } , \label{eq : mfp}\ ] ] where @xmath34 is the spectral flux when the baths are in contact ( @xmath35 ) . both @xmath34 and @xmath31 are determined from the fitting procedure . while @xmath34 depends on the details of the heat baths , the mfps extracted from the length - dependence are not expected to depend on the bath details.@xcite the frequency - dependent mfps determined from eq . are mode - averaged and projected along the direction of heat transfer.@xcite the mfps are independent of the system length and therefore correspond to the bulk values.@xcite we note that the mfps determined from eq . correspond to the decay length of the heat flux . this definition does not necessarily coincide with the conventional definition of the mfp as the decay length of a wave packet.@xcite this distinction is important for diffusons , which do not have a well - defined wave - vector so that the traditional definition can not be applied . once the spectral mfps are determined , the thermal conductivity @xmath36 for length @xmath2 can be determined from@xcite @xmath37}. \label{eq : kappa_est}\ ] ] the length - dependence of the thermal conductivity can be intuitively understood by writing eq . in the equivalent form @xmath38 where the `` effective '' mfp @xmath39 has been introduced , which is similar to the well - known matthiessen rule.@xcite the effective mfp accounts for boundary scattering through the additional @xmath40 term and is limited to below this value . finally , eq .   allows for a simple quantum correction to the thermal conductivity prediction , because the contributions of different frequencies can be weighted by the vibrational mode energy and occupation , as in the landauer - bttiker formalism @xcite . we define the quantum corrected thermal conductivity as @xmath41 } \frac{\hbar \omega}{k_\mathrm{b } } \frac{\partial f_{\mathrm { be}}(\omega , t)}{\partial t } , \label{eq : kappa_quantum}\ ] ] where @xmath42^{-1}$ ] is the bose - einstein distribution function , @xmath43 is the boltzmann constant , and @xmath44 is the planck constant divided by @xmath45 . by defining the dimensionless , length - dependent bath - to - bath transmission function as @xmath46},\ ] ] eq . can be written in the familiar landauer - bttiker form as @xcite @xmath47 the proposed quantum correction accounts for the quantum specific heat of the modes at each frequency , but does not account for quantum effects in the dynamics . the method is thus similar to the one recently introduced by lv and henry,@xcite who weight the modal contributions to the equilibrium green - kubo thermal conductivity by the quantum population function . all our nemd simulations are performed at a mean temperature of @xmath48 k with temperature bias @xmath49 k. choosing a relatively large temperature bias allows for very good signal - to - noise ratio in the spectral heat flux , suppressing the statistical noise . we checked that halving the bias to @xmath50 k does not change the spectral mfps . in addition , we checked that the heat flux is not sensitive to the exact arrangement of atoms arising from the melt - and - quench procedure , which we attribute to the large cross - section of the system giving rise to spatial averaging in the spectral currents . therefore , we performed a single melt - quench for each system length . for different system lengths @xmath2 . increasing the system length reduces the heat flux , especially at high frequencies , where the mfps are shorter . at frequencies below 2 thz , the heat current is independent of system length , suggesting ballistic transport.,width=325 ]    the spectral heat flux @xmath1 for selected system lengths @xmath2 as a function of frequency @xmath51 is plotted in fig . [ fig : flux ] . the spectral distribution of the heat flux for @xmath52 nm was recently analyzed in detail by zhou and hu,@xcite so we focus here on its length - dependence . as expected , increasing the system length reduces the heat current throughout the whole frequency range because of increased phonon - phonon scattering . the reduction is strongest at high frequencies , where the mfps are shorter compared to low frequencies . at frequencies less than 2 thz , the spectral current is nearly independent of system length . such nearly ballistic conduction suggest that the low frequency mfps are notably longer than the system sizes considered here . equation suggests that the inverse of the spectral flux will be linearly proportional to the system length , with the slope given by @xmath53^{-1}$ ] . to determine the mfps , we calculated the spectral flux for system sizes @xmath54 nm , plotted @xmath55 versus @xmath2 and fitted a linear function at each frequency using least squares fitting.@xcite a linear function accurately reproduces the length - dependence of @xmath55 ( not shown ) , as previously also observed for other systems.@xcite        because of the high computational cost associated with calculating the spectral heat fluxes for large systems , we limited our study to systems at most 10 nm long . this restriction precludes extracting mfps longer than 10 nm accurately , limiting the current analysis to frequencies greater than 2 thz . the mfps @xmath31 extracted from the linear fitting procedure are shown in a log - log plot in fig . [ fig : mfps ] . at frequencies below 5 thz , the mfps obey a power - law scaling @xmath56 . this scaling agrees with modal life - time calculations on a - si.@xcite at frequencies greater than 5 thz , the power - law scaling breaks down and the mfps increase with increasing frequency , giving rise to a local maximum around 8 thz . a similar maximum for a - si has been observed in effective mfps @xcite and in lifetimes.@xcite at higher frequencies , the mfps decrease again and fall below 1 nm , which is on the order of the silicon - silicon bond length . at such high frequencies , the uncertainty is large because of the sensitivity of the spectral flux to the system size . larkin and mcgaughey reported a propagon - diffuson transition frequency of 1.8 thz,@xcite such that the frequency range considered in fig . [ fig : mfps ] mostly corresponds to diffuson - like modes . in the analysis below , we assume that the scaling @xmath56 ( solid line in fig . [ fig : mfps ] ) remains valid at frequencies below 2 thz . with such scaling , the mfps exceed 100 nm below 530 ghz and 1 @xmath57 m below 170 ghz . while the @xmath58 scaling may break down in real situations because of defects , boundary scattering , or even the onset of a rayleigh - like @xmath59 scaling at very low frequencies , @xcite we assume it to hold for simplicity . we now investigate the length - dependence of the thermal conductivity using eq .  . the calculated thermal conductivity ( continuous line ) is compared with that determined directly from nemd simulations ( data points ) for lengths up to @xmath60 nm in fig . [ fig : kappa ] . in the evaluation of the integral in eq . , the mfps have been assumed to scale as @xmath61 at frequencies below 2 thz . these calculations were carried out without the quantum correction as the nemd simulations are classical . equation combined with the mfp data of fig . [ fig : mfps ] reproduces the length - dependence of thermal conductivity up to lengths @xmath62 nm to within 2% . this close agreement ( i ) supports the assumption of @xmath61 scaling at low frequencies , ( ii ) shows that the mfp data in fig . [ fig : mfps ] , which were determined from simulations of systems shorter than @xmath63 nm , can be reliably used to estimate the relative contributions of different vibrational frequencies to thermal transport in much larger systems , and ( iii ) provides support for the accuracy of eq . in describing the length - dependence .    because the debye temperature of a - si is @xmath64 k @xcite , which is well above room temperature , we need to apply the quantum correction to compare the predicted temperature - dependence of thermal conductivity to experimental data . to do so , we use eq .   and evaluate the integral as a function of temperature using the mfps from fig . [ fig : mfps ] , again assuming the scaling @xmath61 at frequencies below 2 thz . a full quantum - corrected analysis would require determining the mfps at each temperature . for simplicity and based on the recent results of lv and henry @xcite , we assume that the mfps calculated at a temperature of 300 k remain valid at other temperatures . below frequencies of 2 thz and to be independent of temperature . also plotted is the thermal conductivity of a 520 nm thick hydrogenated a - si thin film measured by cahill _ @xcite.,width=325 ]    the quantum - corrected thermal conductivity [ eq . ] is plotted as a function of temperature for system lengths of 50 , 250 , and 1000 nm in fig . [ fig : kappa_qm ] . as noted above , assuming finite @xmath2 in eq . limits the mfps to @xmath40 . experimental data from cahill _ et al . _ for a 520 nm thick film of hydrogenated a - si with one atomic percent hydrogen content are also plotted.@xcite because available thermal conductivity measurements for a - si contain significant scatter,@xcite we use the data of cahill _ et al . _ to check the trend of our predictions , but do not expect agreement . differences may also exist due to the use of the stillinger - weber potential and the classical nature of the nemd simulations . the increase of thermal conductivity with increasing temperature is well described by the quantum - corrected thermal conductivity . at temperatures higher than 300 k , the experimentally measured thermal conductivity increases slightly slower as a function of temperature than our prediction , but this disagreement may be related to our approximation that the mfps are independent of temperature . at such high temperatures , anharmonic scattering will reduce mfps and therefore decrease the thermal conductivity . without the quantum - correction , the predicted thermal conductivity would depend only very weakly on temperature ( due to the weak temperature - dependence of the mfps ) , precluding reasonable agreement with the trend of the experimental data . we caution that this quantum correction has only been examined for a - si here and that its application to other systems warrants further investigation . we investigated vibrational heat transfer in a - si by determining the sdhc and mfps from nemd simulations . the calculated mfps directly reflect the decay of the heat flux at each vibrational frequency and do not rely on the existence of a well - defined modal wave vector , thereby avoiding the separate treatment of diffusons and propagons . as shown in fig . [ fig : mfps ] , the mfps exhibit @xmath58 scaling at frequencies above 2 thz and below 5 thz . at frequencies higher than 10 thz , the mfps fall below 1 nm , corresponding to strongly localized vibrations . the length - independent mfps can be used to accurately predict the thermal conductivity in systems as long as 100 nm ( fig . [ fig : kappa ] ) . weighting the sdhc by the frequency - dependent quantum occupation function provides a simple method for a quantum - correction of thermal conductivity and is able to reproduce the experimentally measured temperature - dependence of thermal conductivity , as shown in fig . [ fig : kappa_qm ] .    in the future , it would be useful to calculate the sdhc for systems longer than those considered here , enabling direct extraction of mfps at frequencies below 2 thz . such an analysis could inform the ongoing discussion@xcite of the low - frequency scaling of mfps in a - si . it will also be important to compare the non - equilibrium mfps to those calculated from equilibrium molecular dynamics simulations . this work was initiated during the research visit of k.s . to carnegie mellon university . the computational resources were provided by the finnish it center for science and aalto science - it project . the work was partially funded by the aalto energy efficiency research programme ( aef ) and the academy of finland . 23ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ]  + 12$12  & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1080/13642819908223054 [ * * ,   ( ) ] link:\doibase 10.1103/physrevb.48.12589 [ * * ,   ( ) ] link:\doibase 10.1103/physrevb.64.094201 [ * * , ( ) ] @noop _ _  ( ,  ) link:\doibase    10.1103/physrevb.48.12581 [ * * ,   ( ) ] link:\doibase 10.1103/physrevb.91.115426 [ * * ,   ( ) ] link:\doibase 10.1103/physrevb.90.134312 [ * * ,   ( ) ] link:\doibase    10.1103/physrevb.92.245411 [ * * ,   ( ) ] @noop * * ,   ( ) link:\doibase 10.1103/physrevb.89.144303 [ * * ,   ( ) ] http://stacks.iop.org/1367-2630/18/i=1/a=013028 [ * * ,   ( ) ] link:\doibase    10.1103/physrevb.92.195205 [ * * ,   ( ) ] link:\doibase 10.1103/physrevb.79.224305 [ * * ,   ( ) ] \doibase    http://dx.doi.org/10.1006/jcph.1995.1039 [ * * ,   ( ) ] link:\doibase 10.1103/physrevb.31.5262 [ * * , ( ) ] \doibase    http://dx.doi.org/10.1016/s0022-3093(01)00342-8 [ * * ,   ( ) ] http://stacks.iop.org/0953-8984/26/i=5/a=055011 [ * * ,   ( ) ] @noop _ _  ( , ) link:\doibase 10.1103/physrevlett.81.232 [ * * ,   ( ) ] \doibase doi : 10.1006/spmi.1997.0561 [ * * ,   ( ) ] link:\doibase 10.1103/physreve.88.012128 [ * * ,   ( ) ] \doibase    http://dx.doi.org/10.1016/0038-1098(84)90589-1 [ * * ,   ( ) ] link:\doibase    10.1103/physrevb.50.6077 [ * * ,   ( ) ]
the frequency - dependent mean free paths ( mfps ) of vibrational heat carriers in amorphous silicon are predicted from the length dependence of the spectrally decomposed heat current ( sdhc ) obtained from non - equilibrium molecular dynamics simulations . 
 the results suggest a ( frequency)@xmath0 scaling of the room - temperature mfps below 5 thz . 
 the mfps exhibit a local maximum at a frequency of 8 thz and fall below 1 nm at frequencies greater than 10 thz , indicating localized vibrations . 
 the mfps extracted from sub-10 nm system - size simulations are used to predict the length - dependence of thermal conductivity up to system sizes of 100 nm and good agreement is found with separate molecular dynamics simulations . weighting the sdhc by the frequency - dependent quantum occupation function provides a simple and convenient method to account for quantum statistics and provides reasonable agreement with the experimentally - measured trend and magnitude .