From diffusive to ballistic Stefan–Boltzmann heat transport in thin non-crystalline films

Today, different theoretical models exist to describe heat transport in ultra-thin films with a thickness approaching the phonon mean free path length. Due to the influence of parasitic effects, the experimental assessment of heat transport in these ultra-thin films with the required sensitivity is extremely challenging. In this work, the heat transport through thin non-crystalline metal-oxide films is studied using scanning near-field thermal microscopy, which allows to minimize parasitic thermal effects and therefore provides uttermost sensitivity even for the study of thermal transport in ultra-thin films. For the first time, we provide experimental evidence of enhanced out-of-plane heat dissipation in these ultra-thin metal-oxide films by ballistic thermal phonon transport according to the Stefan–Boltzmannlike heat transport model.


Introduction
In electrical insulators, phonons are the predominant species for heat transport. For bulk materials the heat uxq is classically described by the Fourier law, 1 where in general T(r) is a given temperature distribution and l ! is the tensor of thermal conductivity: 2q There are steady state as well as transient techniques within the space and time/frequency domain to analyze thermal transport. [3][4][5][6][7] If the absorbed power in the sample is known, the thermal conductivity can be directly determined in steady state approaches. In the dynamic case, the density r as well as the specic heat c of the material must be considered. Heat ow is described by the general equation of heat conduction: 8 here, q E indicates the heat generation per unit time and unit volume. The heat can be generated for example by a laser, by heated resistive metal strips directly deposited onto the sample, etc. Since only the time derivative of T(r) appears in this equation, the knowledge of the absolute temperature and the absorbed power is not required. To get access to the threedimensional material property l ! , directional heat transport investigations have been carried out using various experimental techniques. 7 Furthermore, the heat ux strongly depends on the dimensions of the materials and is of growing general interest in the case of a shrinking device size and in thin lm technology. Hence, lms on substrates with either low or high thermal conductivity have been investigated in order to study in-plane or out-of-plane (also called cross-plane) heat ux (Fig. 1). 7 While there is extensive experimental work on the in-plane heat transport in thin lms down to lm thicknesses of around 20 nm, it must be pointed out that out-of-plane transport is experimentally far less explored and studies are typically limited to lm thicknesses larger than 100 nm. [9][10][11][12][13][14] Within these works, a decrease of the thermal conductivity with decreasing lm thickness d lm from 2 mm (ref. 15) to 300 nm (ref. 16) has been found for dielectrics and semiconductors. On the contrary, we have found the thermal conductivity of an organic Alq 3 thin lm with d lm down to 20 nm to be constant. 17 For ultra-thin lms with a lm thickness below the phonon mean-free path L (i.e. d lm < L), also referred to as Casimir limit, 18 some further ambiguities exist. In some reports, these ultra-thin lms are considered to be thermal insulators, where phonons are claimed to be trapped between the layer boundaries. Trapped phonons are not able to transport thermal energy. 10 In addition the Fourier law is no longer valid in the case of heat source dimensions to values smaller than L, which can be hundreds of nanometers in crystalline materials at room temperature. 19 A signicant decrease in energy transport occurs even into bulk material at nanoscale heated regions compared with Fourier-law predictions due to quasi-ballistic thermal transport effects (quasi-ballistic: with and without scattered phonons). On the contrary could be demonstrated, that surface phonon polaritons excited by a laser beam dramatically enhance energy transfer between two surfaces at small gaps (30 nm) by measuring radiation heat transfer between a microsphere and a at surface. 20 An extremely enhanced heat ux by blackbody radiative transfer of energy from the heater can only be considered under extreme near-eld conditions. Although experimental advances have enabled elucidation of heat transfer in gaps as small as 20 nm, quantitative analysis of enhanced radiative heat transfer in the extreme near-eld (gaps less than 10 nm) have been carried out just very recently. 21 By a similar token, theoretical work based on the Boltzmann transport equation (BTE) for lms with d lm < L (below the Casimir limit) considered heat transport to be ballistic and resulted in a description similar to that known from radiative heat transport according to the Stefan-Boltzmann law. [22][23][24] As of yet, the experimental verication of this theory is missing.
In this paper, we study out-of-plane heat transport from a nanoscale heat source through ultra-thin non-crystalline metal-oxide lms, e.g. Al 2 O 3 and TiO 2 , with a thickness down to 4 nm by means of dynamic scanning near-eld thermal microscopy (SThM). We have prepared our ultra-thin amorphous like lms by atomic layer deposition (ALD), as it is known to allow for uttermost control of layer thickness, conformity, and homogeneity conrmed by former X-ray diffraction and transmission electron microscopy studies. 25 For the rst time, we provide experimental evidence that the out-of-plane heat transport in ultra-thin lms is dominated by ballistic heat transport and can therefore be described by a Stefan-Boltzmann like model.

Experimental
For the out-of-plane heat transport analysis we used silicon as substrate material with a relatively high thermal conductivity of 150 W (m À1 K À1 ). Non-crystalline Al 2 O 3 and TiO 2 lms were prepared in a step-like matrix (Fig. 2), where all lms exhibited a direct step from the lm to the Si-substrate to directly assess the heat ux through the lm for each lm thickness as described below.
The SThM in the present work is installed in the analysis chamber of an Environmental Scanning Electron Microscope (ESEM), thus our SThM measurements are carried out under vacuum (5 Â 10 À6 mbar). 26 Effects of thermal transport due to convection or moisture related interfacial effects can therefore be excluded. The signal detection in our specic setup has been explained in detail elsewhere. 27 Briey, we applied a transient measurement technique within the frequency domain thus only the derivative of T(r, u) in the swung-in state will be discussed in the following. A constant low force between SThM probe and sample was applied during these measurements to guarantee a constant resulting thermal contact resistance and in order to prevent mechanical inuences/damages to the lms. Further details can be found in the ESI. † The resistive thermal probe of the SThM is powered by an AC current I 0 sin(ut) with angular frequency u. As demonstrated earlier, 28,29 the probe can be considered in far-eld as a point like heat source and in close proximity as a line-shaped nanoscale heat source fullling the demands for high resolution near-eld microscopy 30 by generating a heat wave of frequency 2u with cylindrical symmetry. This wave will diffuse into the substrate and is exponentially damped in the radial direction. According to Carslaw and Jaeger, the resulting amplitude of the temperature oscillationsT(r) in the sample at a distance r from the heat source depends on the amplitude of the generated heat powerP per unit length owing into the sample and the thermal conductivity l as: 31T K 0 is the zeroth-order modied Bessel function and s(u) denes the complex thermal wave number accounting for a phase difference between temperature and AC heating power of the source. The magnitude of the complex quantity 1/s(u) is the thermal penetration depth, i.e. the characteristic depth of heat diffusion during one cycle of the ac power heating the sample. According to Cahill 31 and Fiege et al. 32 in SThM, one may approximate: here, a ¼ l/rc denotes the thermal diffusivity of the material. Since heater and thermometer are combined in one element, the thermal conductivity of the sample can be determined quantitatively by keepingP constant and measuring the resulting 3u component of the voltage drop across the SThM probeÛ 3u at two different frequencies u 1 and u 2 .
The term dR dT represents the differential thermal coefficient of the probe. Unfortunately, in SThMP depends on the thermal properties (e.g. thermal conductivity) 27,33 of both the sample and the probe design. This means that factors like heat dissipation into the supply lines of the thermal probe etc. must also be considered. Furthermore, the heat transfer mechanisms between the probe and the sample must be calibrated. The tip-sample thermal contact resistance can be affected by heat conduction and thermal radiation near the contact region as discussed in detail by Wilson et al. 33 in order to perform thermal conductivity measurements in the 3u mode using a Wollaston wire. This renders the application of the conventional 3u method difficult for the quantitative determination of thermal conductivity in dependence on the lm thickness using VITA-SThM probes. Thus, in order to perform measurements of out-of-plane heat transport from a nanoscale heat source through thin lms, we determine the difference of the amplitude of the temperature oscillations of the probe on the lm and on the substrate DT 1 T lm ÀT substrate as illustrated in Fig. 2.
Out-of-plane heat dissipation in thin non-crystalline metaloxide lms Diffusive out-of-plane heat transport. As the silicon substrate has a relatively high thermal conductivity,T substrate is relatively small (eqn (3)). If the amplitude of electrical power supplied to the probe is kept constant and if only diffusive heat transport in a lm with low thermal conductivity is assumed, the temperature oscillation of the probe on top of the lmT lm is larger thanT substrate . In the limit where the thermal conductivity of a lm would trend to zero, no heat would diffuse into the lm. In this case, the generated joule heat would exclusively be transported away from the SThM probe via its supply lines.
An exemplary result of DT vs. d lm for Al 2 O 3 lms is shown in Fig. 3. The uncertainties of temperature and lm thickness measurements (for lms below 30 nm) are AE0.5 K and AEd lm 10% nm respectively.
To understand the dependence of DT vs. d lm , different regimes of thermal transport must be considered. For d lm > 100 nm, DT saturates at a level DT max , which indicates that in this regime the Al 2 O 3 layer can be considered as bulk material. Thus, heat transport is diffusive and follows the Fourier law. 2 Using the 3u-technique here (see eqn (5)), we determined the thermal conductivity to l Al 2 O 3 ¼ 0.8 W m À1 K À1 on an average lm thickness of 338 nm AE 5 nm using the line source model of the probe. This value is in agreement with previous reports for ALD grown non-crystalline Al 2 O 3 layers. 34 The advantages of this 3u-technique on thermal conductivity analyses of amorphous solids are discussed in detail elsewhere. 35 Towards thinner lms (d lm < 100 nm), DT decays and nally levels off for d lm < 30 nm.
Lee and Cahill 16 used a one-dimensional description to model the out-of-plane heat ow through a lm with thicknesses d lm far less than the thermal penetration depth and much smaller than the width of the heater/thermometer w. If  the thermal conductivity of the lm is substantially smaller than that of the substrate, this model yields a linear relation of DT and d lm as follows: A frequency independent offset of a temperature oscillation amplitude DT is added to the thermal 3u response of the substrate. Note, as DT scales with d lm /w, the measurement of ultra-thin lms with microscopic probes (d lm ( w) becomes very challenging. This might be the reason why the out-of-plane heat transport is less explored for ultra-thin lms using macroscopic probes (with a large w). 9,10 On the other hand, in the case of a SThM the ultra-small w on the order of 50 nm (ref. 36) should be perfectly suited to assess thermal transport in ultra-thin lms.
It is not directly obvious from eqn (6) thatT lm would become independent of d lm for ultra-thin lms (d lm < 30 nm in the example shown in Fig. 3). According to eqn (6), the region of constant DT could be indicative of a thickness dependent thermal conductivity: l lm f d lm . This assumption would lead to the conclusion, that ultra-thin lms become thermal insulators, as claimed by Lee and Cahill. 16 Later Lee and Cahill revised this model by introducing a thermal contact resistance (R resistive ) to be responsible for the observation ofT lm becoming constant. 16 They assumed an apparent thermal conductivity l a which depends on the lm thickness d lm , the thermal resistance R resistive and the intrinsic thermal conductivity l i : As we will discuss below this assumption does not allow us to explain our experimental results shown in Fig. 3.
The transition from thin lms to bulk materials (including transition from one-dimensional heat transport across the lm to three-dimensional heat transport in bulk material) has been described by the introduction of a saturating exponential function, 17 that for a given frequency u nally leads to: þT contact (8) here, DT max is the difference of temperature oscillation (T lm À T substrate ) max found in the limit of thick lms, r contact is the radius of the effective contact area of the probe, andT contact is the temperature offset caused by a constant thermal contact resistance, as r contact and the thermal contact resistance are constant for a certain range of sample thermal conductivity 33 which is typically neglected in eqn (6). Note, aside fromT contact , in the limit of ultra-thin lms (one-dimensional heat transport) eqn (8) becomes identical to eqn (6) (assuming w ¼ 2r contact and DT max ¼P l film as frequency independent). As the radius of an effective contact area of the probe is known to be on the order of 50 nm and DT max was measured to be approximately 5.5 K in the measurement shown in Fig. 3, DT(d lm ) can be derived from eqn (8). Note,P l film is assumed to be constant and we setT contact ¼ 1 K. Obviously, this calculation only ts to the measurement for lm thicknesses larger than 80 nm (a similar treatment for TiO 2 thin lms can be found in the ESI †). The deviation of calculation and measurement indicates a substantially higher heat ux for thinner lms (d lm < 80 nm) than expected by eqn (8). We want to stress that the previous assumption of a thickness dependent thermal conductivity (l lm f d lm ) as well as a decrease of the effective l lm calculated recently by a variational approach to solving the BTE 24 does not allow for a better t to the data, but this assumption would rather increase the deviation. Note, phonon trapping between the layer boundaries affecting the thermal conductivity at a lm thickness on the order of d lm ¼ 30 nm as well as a decrease of the effective l lm is not very likely in our samples, as the phonon mean free path is estimated to be approximately two orders of magnitude less for non-crystalline Al 2 O 3 (L Al 2 O 3 ¼ 0.47 nm) and L Al 2 O 3 ¼ 3.5 nm for sintered polycrystalline Al 2 O 3 . 37 Out-of-plane Stefan-Boltzmann-like heat dissipation by ballistic thermal phonon transport. An alternative explanation for the constant DT towards ultra-thin lms and for the elevated higher heat ow in the region below 80 nm could be related to a constant thermal contact resistance and a signicant contribution of Stefan-Boltzmann like heat transport of phonons (which cannot be described by the classical eqn (3)-(8) for diffusive heat transport). It has to be noted, that our measurements are carried out under vacuum conditions, thus heat transport by convection and other parasitic mechanisms related to air and moisture are negligible. Furthermore, effects of radiative blackbody heat transfer are only important in the extreme near-eld at a distance far less than 10 nm (ref. 21) and for elevated probe temperatures signicantly above those used in our experiment (<400 K).
In order to prove Stefan-Boltzmann like heat transfer of phonons the SThM probe temperatures were measured at different applied electrical powers P electr to heat the probe. The temperatures were detected under steady state conditions for the SThM probe in vacuum (T vac ) and in contact on a 10 nm thick lm (T lm ), respectively (Fig. 4a). This thickness was used to avoid a thermal ux due to blackbody radiative heat transport (for gaps less than 10 nm), and to guarantee one dimensional heat transport across the lm, as well as to keep contributions due to diffusive heat transport (in case of TiO 2 for d lm > 17 nm) negligible.
Thereby, the heat dissipated into the supply lines of the thermal probe can be taken into consideration as the same electrical power P electr is supplied to the SThM probe in vacuum and when in contact with the lm (more details are available in the ESI †). At moderate temperatures the thermal capacity of the supply line of the thermal probe C suppl is temperature independent and the radiation heat transfer between our probe and chamber walls is negligible (see ESI †). Finally we derive: here, l suppl , A suppl , and l suppl are the effective thermal conductivity, the cross-sectional area, and the length of the supply lines and T 0 is the ambient temperature in the lab (room temperature, thermal reservoir). The term on the right describes the Stefan-Boltzmann like contribution of phonons multiplied with its relative weight given by the ratio of lm thickness and phonon mean free path (i.e. e À dfilm Lfilm ). 22 A Stefan-Boltzmann contact is the effective contact area for ballistic heat transport. s is the Stefan-Boltzmann constant for phonons, which can be related to the specic heat, 31 the temperature, and the velocities of sound, 38 which depends on the pressure. 39 Thus, as described before in the "Experimental" section, the leading force of the probe was kept as low as possible and moderate SThM temperatures were used.
Stefan-Boltzmann like heat transport of phonons is becoming more dominant with decreasing lm thickness according to the calculations presented in the ESI. † To prove the validity of this Stefan-Boltzmann model for the heat transport in our ultra-thin lms we measured T lm at varied electrical heating powers P electr , applied to the thermal probe (see Fig. 4b).
As can be concluded from the graph, which was reproducibly obtained on different scan-areas for an Al 2 O 3 lm with d lm ¼ 10 nm, there is an excellent t of (T vac À T lm ) $ T lm 4 . A temperature offset of the SThM probe in vacuum is related to the laser heating of the deection method for obtaining the topography. In this case the scattering of the measured data is attributed to different scan-areas of our SThM probe on the lm. This agreement, for the rst time provides clear experimental evidence that the out-of-plane heat transport mechanism from a nanoscale heat source in ultra-thin lms is dominated by ballistic heat transport of phonons and can be described by a Stefan-Boltzmann like model. In the discussion of DT vs. d lm (Fig. 3), the regime of ultra-thin lms is therefore dominated by Stefan-Boltzmann like heat transport. It should be noted that the contribution of ballistic heat transport is dominant even when the probability of Stefan-Boltzmann like transport is relatively small. TheT contact found in this case is associated with ballistic heat transport through A Stefan-Boltzmann contact . With increasing lm thickness, the contribution of ballistic transport decreases exponentially. This will nally lead to a signicant increase of DT >T contact with increasing lm thickness (>30 nm) as observed in our previous measurements (Fig. 3).
From diffusive to ballistic Stefan-Boltzmann-like out-ofplane heat transport. As the relative contribution of Stefan-Boltzmann heat transport depends exponentially on d lm , the phonon mean free path L lm can be estimated from the transition of the dominating ballistic regime, which is indicated by a constant DT ¼T contact , to the regime where heat transport through the thin lm will become increasingly diffusive (Fig. S3 †). The lm thickness for which the contribution of diffusive and ballistic thermal transport is equal can be determined to be d lm ¼ 26 nm in the case of Al 2 O 3 and d lm ¼ 17 nm for TiO 2 .
In this context, the contribution of various phonon modes must be taken in to account in the Stefan-Boltzmann like heat transfer, as already discussed especially for amorphous materials. 40 Note, in our case is d lm [ L lm . Consequently we considered the non-crystalline lms with d lm > 4 nm as bulk materials. Hence, in comparison to acoustic phonons the contribution of optical phonons to the thermal transport is negligible at room temperature. 41 Finally, it should be noted that optical phonons, which are usually neglected in thermal conduction for their relatively small group velocities compared with acoustic phonons, can signicantly enhance heat transport at room temperature in extremely thin lms (d lm < L lm ). 42 At low temperatures and in isotropic condensed matter, s was already calculated by Swartz and Pohl 38 in studies of thermal boundary resistances and is given by: here, k B is Boltzmann's constant, ħ is the reduced Planck constant, and c i is the velocity of sound in longitudinal and transverse direction. While transverse acoustic phonons are the primary carriers of energy at low temperatures (T < 200 K), longitudinal acoustic phonons carry more energy at higher temperatures. 43 Finally, L lm of longitudinal acoustic phonons can be estimated from the transition where diffusive and ballistic thermal transport is equal by use of eqn (9) and (10)

Conclusions
In summary, we performed out-of-plane heat transport measurements of a nanoscale heat source through thin noncrystalline Al 2 O 3 and TiO 2 lms for thicknesses from 450 nm down to 4 nm by use of scanning neareld thermal microscopy (SThM). No change of the thermal conductivity and a pure diffusive heat transport were found for Al 2 O 3 lms with a thickness larger than 80 nm. At decreasing lm thicknesses a signicant enhancement of heat transport was detected, which could not be explained by diffusive thermal transport alone. As we have clearly evidenced for the rst time, in this regime heat dissipation has a signicant contribution of ballistic Stefan-Boltzmann like heat transport of phonons. From the transition of dominating ballistic out-of-plane heat transport to increasing probability of diffusive mechanism the mean-free path of longitudinal acoustic phonons have been estimated to be 0.63 nm and 0.43 nm for Al 2 O 3 and TiO 2 respectively.
These results will signicantly inuence thin lm technology and the thermal management of modern devices, as heat dissipation is at this time the key limiting factor for power electronics, including high-power RF devices as well as in highpower laser diodes and high-brightness light-emitting diodes (LEDs). In contradiction to former estimations, that ultra-thin lms become thermal insulators, an enhanced heat dissipation by ballistic phonon transport occurs with decreasing lm thicknesses even far above the Casimir limit.