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Received 3 Dec 2013 | Accepted 26 Aug 2014 | Published 1 Oct 2014

The applicability of Fouriers law to heat transfer problems relies on the assumption that heat carriers have mean free paths smaller than important length scales of the temperature prole. This assumption is not generally valid in nanoscale thermal transport problems where spacing between boundaries is small (o1 mm), and temperature gradients vary rapidly in space. Here we study the limits to Fourier theory for analysing three-dimensional heat transfer problems in systems with an interface. We characterize the relationship between the failure of Fourier theory, phonon mean free paths, important length scales of the temperature prole and interfacial-phonon scattering by time-domain thermoreectance experiments on Si, Si0.99Ge0.01, boron-doped Si and MgO crystals. The failure of Fourier theory causes anisotropic thermal transport. In situations where Fourier theory fails, a simple radiative boundary condition on the heat diffusion equation cannot adequately describe interfacial thermal transport.

DOI: 10.1038/ncomms6075

Anisotropic failure of Fourier theory in time-domain thermoreectance experiments

R.B. Wilson1 & David G. Cahill1

1 Department of Materials Science and Engineering, Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801, USA. Correspondence and requests for materials should be addressed to R.B.W. (email: mailto:[email protected]

Web End [email protected] ).

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When heat carriers have mean free paths (MFP) shorter than important length scales of the temperature prole, heat transport is diffusive. When heat carriers

have MFPs comparable to, or larger than, important length scales of the temperature prole, heat transport is ballistic. In nonmetals, the dominant heat carriers are phonons that have a broad spectrum of MFPs and frequencies1,2. As a result, no single length scale describes the transition from diffusive to ballistic thermal transport. Instead, as length scales in a thermal transport problem decrease, Fourier theory has diminishing predictive power37.

The breakdown of Fourier theory at small length scales has implications for nanoelectronics8 and nanostructured thermoelectric devices9 and has received a great deal of recent attention17,1014. For example, deviations between Fourier theory predictions and experimental data have been reported to depend on (1) the width of the nickel heater lines in nanofabricated Ni/sapphire samples7, (2) the diameter of the heating pump laser beam, 2w0, in time-domain thermoreectance (TDTR) measurements of bulk Si between 30 and 100 K (ref. 4),(3) the spatial frequency of heating in thermal grating measurements of 400 nm thick Si membranes at 300 K (ref. 10) and (4) the frequency of periodic heating, f, in TDTR measurements of semiconductor alloys and some types of amorphous silicon between 80 and 420 K (refs 6,15). In TDTR, and the related technique of frequency-domain thermoreectance (FDTR), f sets the time interval over which heat travels and, therefore, controls an important length scale of the temperature prole. When heat ow is diffusive, this length scale is known as the thermal penetration depth, dp (L/pCf)1/2, where L is the

thermal conductivity and C is the volumetric heat capacity.

These experiments have greatly advanced understanding of thermal transport on nanometer to micron length scales. However, unresolved issues preclude a complete picture of when and how Fourier theory will fail in a nanoscale thermal transport problem. For example, why do deviations between TDTR data and Fourier theory predictions sometimes depend on f but not w0 (ref. 6), and other times on w0 but not on f (ref. 4)?

All prior TDTR and FDTR studies have interpreted data that deviates from Fourier theory predictions as phenomena intrinsic to the phonon dynamics of the material being studied and have neglected the role of the interface. TDTR and FDTR experiments require samples to be coated with a thin metal lm to serve as an optical transducer, introducing a metal/sample interface to the heat transfer problem. The interface causes a thermal resistance due to the reection/transmission of phonons, that is, interfacialphonon scattering16. This thermal resistance is typically described with a radiative boundary condition on the heat diffusion equation, known as the interfacial thermal conductance17. This description implicitly assumes the temperature is well dened at each side of the interface and that all phonons in the solid are in equilibrium with each other at all distances from the interface. Whether this assumption is justied in nanoscale thermal transport problems has long been controversial18, but it is particularly suspect in experiments where a signicant fraction of heat-carrying phonons are ballistic. Despite this, experiments that cannot be explained using a radiative boundary condition for heat transfer at the interface are rare5.

Here, we report the results of three sets of experiments that examine how the failure of Fourier theory in TDTR experiments relates to the MFP distributions in solids and the transport properties of the metal/sample interface. In our rst set of experiments, we perform TDTR measurements of Si at room temperature with offset pump and probe beams, allowing for the independent determination of the through-plane and in-plane thermal conductivities19. The beam-offset measurements

demonstrate that varying w0 in a TDTR experiment more strongly affects the applicability of Fourier theory in the in-plane direction. In our second set of experiments, we perform TDTR measurements of Si, Si0.99Ge0.01 and Si heavily doped with boron

(Si:B) as a function of w0, f and temperature. The Ge in Si0.99Ge0.01 preferentially scatters high-frequency phonons due to mass disorder20, while the boron in Si:B preferentially scatters low-frequency phonons due to hole/phonon scattering21. As a result, the MFP distributions are systematically different in Si, Si0.99Ge0.01 and Si:B, allowing us to relate differences in the phonon dynamics to differences in how Fourier theory fails. We posit that the thermal diffusivity of high-wavevector phonons determines whether a failure of Fourier theory is observable in TDTR experiments as a function of f or w0. In materials whose high-wavevector phonons have a low thermal diffusivity, Fourier theory fails more readily as a function of f than w0. In materials whose high-wavevector phonons have high thermal diffusivity, Fourier theory fails as a function of w0. In our third set of experiments, we show treating interface transport with a radiative boundary condition on the heat diffusion equation can be an inadequate description when length scales of the temperature prole are comparable to phonon MFPs.

A secondary goal of our work is to describe a ballistic/diffusive thermal model capable of explaining (1) why the failure of Fourier theory is anisotropic and (2) how interfacial-phonon scattering impacts the accuracy of Fourier theory predictions. Fourier theory fails in TDTR experiments because Fouriers law is unable to predict the heat current, J, due to long MFP phonons near temperature-prole minima and maxima (extrema). Fouriers law fails near temperature-prole extrema because it uses a rst-order Taylor-series approximation for the temperature prole that is particularly inaccurate in proximity to locations where rT changes sign.

ResultsAnisotropic failure of Fourier theory in Si. For TDTR measurements of Si at room temperature with w0o5 mm, we observe

discrepancies between Fourier theory predictions and experimental data22, see Fig. 1. From our TDTR data, we derive an apparent thermal conductivity, LA, and an apparent interface conductance, GA, for each measurement by tting the data with an isotropic diffusive model19,23 that treats the L of Si, and interfacial thermal conductance between Al and Si, G, as tting parameters. GA is the value for G that produces the best t to the shape of decay of Tin with pump-probe delay time, tD. LA is the value for L that produces the best t to the out-of-phase temperature response, Tout.

Beam-offset measurements, see Fig. 1b, suggest that Fourier theory over-predicts the thermal response of Si for a measurement with w0E1.05 mm and f 9.8 MHz because Fourier theory

over-predicts the in-plane component of the heat current, Jr.

In other words, the solution of the heat diffusion equation with an isotropic thermal conductivity of LA 105 W m 1 K 1

over-predicts the amplitude of the thermal response away from the centre of the pump beam. If we instead t the data with an anisotropic model19 that treats the through-plane and in-plane thermal conductivities, Lz and Lr, as independent tting parameters, we nd LzE140 W m 1 K 1 and

LrE80 W m 1 K 1. The heat current has two directional components, through-plane and in-plane; therefore, we can only dene apparent thermal conductivities for these directions. Adding additional tting parameters cannot improve the t because the t is already excellent.

That Fourier theory fails anisotropically helps resolve how prior TDTR studies of Fourier theory failure relate to each other. Prior studies that observed a dependence of LA on f studied low

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Beam offset

0 1

Beam offset (w0)

Pump Probe

1 m

z(m)

Intensity

Al / Si sample

1 In-phase

Out-of-phase

10 m

0.5

0 1 2

r (w0)

T (K)

0.1

T out (K)

0.05

0.1

Bulk

Isotropic

0.02

Anisotropic

0.01 0.1 0.2 0.5 1 2 5

0.01

2 3

Time delay (ns)

Figure 1 | Anisotropic failure of Fourier theory. (a) Amplitude of measured in-phase and out-of-phase temperature response of 80 nm Al/Si at room temperature as a function of pump/probe delay time with w0 1.05 and 10.3 mm (circles), and the prediction of an isotropic diffusive model with bulk

thermal properties for Si and an Al/Si interface conductance of GA 250 MWm 2 K 1 (lines). Bulk thermal properties for Si cannot explain the

magnitude of Tout at w0 1.05 mm; a thermal conductivity of LA 105 Wm 1 K 1 is required to explain the data with an isotropic diffusive model (b)

Amplitude of the measured out-of-phase temperature response at 100 ps delay time and w0 1.05 mm, as function of offset distance between pump and

probe beam centres. An anisotropic diffusive thermal model with Lr 80 and Lz 140 Wm 1 K 1 (red curve) is in good agreement with the data, while

an isotropic value of LA 105 Wm 1 K 1 (blue curve) over-predicts the amplitude of temperature oscillations away from the centre of the pump beam.

L materials, such as semiconductor alloys and a-Si (refs 6,15). These TDTR measurements observed a failure of Fourier theory in the through-plane direction because dpoow0, meaning the in-plane heat current was negligible. On the other hand, the observation of Minnich et al.4 that LA of Si depends on w0 below 100 K was related to a failure of Fourier theory in the in-plane direction. That Fourier theory fails anisotropically has direct implications for the interpretations of prior experiments that assumed Fourier theory failed isotropically in three-dimensional heat-transfer problems1,4,7,24, see Supplementary Note 1 and Supplementary Figs 1 and 2.

Failure of fourier theory in Si, Si:B and Si0.99Ge0.01. Compar

isons between measurements of Si, Si:B and Si0.99Ge0.01 allows us

to study how Fourier theory failure in TDTR experiments relate to differences in MFP distributions because hole/phonon and point-defect/phonon scattering produce controlled differences in the MFP distributions of these materials21. We quantify the impact of hole/phonon and point-defect/phonon scattering on the L of Si with thermal conductivity relaxation time approximation (RTA) models for Si, Si:B and Si0.99Ge0.01 (see

Supplementary Methods).25 By RTA, we mean that we assume all scattering can be combined into a single frequency-dependent relaxation-time using Matthiessens rule, ignoring that normal phonon/phonon scattering is not resistive in isolation from other scattering mechanisms26. Our RTA models compare favourably to rst-principles calculations20,27,28, see Fig. 2 and Supplementary Fig. 3.

The RTA model results for Si, Si:B and Si0.99Ge0.01 are shown

in Fig. 2 three ways: the thermal conductivity accumulation function, the heat capacity spectral distribution and the thermal conductivity spectral distribution. The thermal conductivity accumulation function is the cumulative contribution to the heat current from phonons with a MFP less than L (ref. 2),

a L

1 LB

where j sums over all three polarization branches, LB is the bulk thermal conductivity, c is the MFP and v is the group velocity. The heat-capacity spectral distribution is

c q

dq kBD q

dq; 2

where kB is Boltzmanns constant, D(q) is the density of states and n(q) is the occupation number. The thermal conductivity spectral distribution, l(q), quanties the heat carried by phonons with wavevector q,

l q

dq X

dn q

dT v q

dq 13 cj q

vj q

2tjq; 3

where t(q) is the relaxation time.

In our discussion below, we refer to differences in l(q) for high- and low-wavevector phonons in different materials. We dene high-/low-wavevector phonons as those with a q larger/ smaller than q0E0.4qmax in Si and Si:B and q0E0.25qmax in

Si0.99Ge0.01. This denition of q0 is chosen so that c(q0) at room temperature is less than dp(20 MHz)/5 so that all high-wavevector phonons have MFPs much shorter than the minimum length scales of the temperature prole. This denition for high-/lowwavevector phonons leaves the majority of the total heat capacity (490%) with high-wavevector diffusive phonons (Fig. 2b), meaning the mean occupation of the high-wavevector phonons denes the temperature prole of the solid. We choose to frame our discussion in terms of high-/low-wavevector phonons instead of short/long MFPs because several of our conclusions are based on comparing results for materials with controlled differences in l(qoq0) and l(q4q0).

At room temperature, Fourier theory accurately predicts the thermal response of Si and Si:B up to f 17.6 MHz, but

inaccurately predicts the thermal response of Si0.99Ge0.01, see

Fig. 3a. (Fig. 3a also includes measurements of LA(f) for Si0.2Ge0.8 and Ge that are relevant to results presented in the next section but are not discussed here.) Phonons with c4dp (17.6 MHz)

carry 50 W m 1 K 1 of heat in Si and 30 W m 1 K 1 in

d 13 cj

; 1

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150

200

Si 1st P

[afii9825](W m1 K1 )

100

Si:B

100

Si:B

Si0.99Ge0.01

0 0.1 1 10 Mean free path (m)

100

A(W m1 K1 )

Si0.99Ge0.01

0.5

c(q)(mJ m)

Si0.2Ge0.8

0.25

10 20

0 0 0.25 0.5q (2[afii9843]/a0)

q (2[afii9843]/a0)

0.75 1

200

100

[afii9838](q)(nW K1 )

Si:B

Si0.99Ge0.01

Si:B

0 0 0.25 0.5 0.75 1

20 1 2 5

Spot-size (m)

10 20

Figure 2 | Thermal property spectra. Relaxation time approximation model predictions for the thermal properties of Si, Si0.99Ge0.01 and Si:B at room

temperature (Supplementary Equation 2). (a) Thermal conductivity accumulation functions, a. First-principles results from ref. 27 and ref. 28 for

Si are included for comparison (green dashed and solid lines). (b) Spectral distribution of the heat capacity. (c) Spectral distribution of the thermal conductivity.

1 2 5Frequency (MHz)

Figure 3 | Apparent thermal conductivities of Si, Si:B and Si0.99Ge0.01.

(a) Deviations between LA(f) for Si, Si:B, Si0.99Ge0.01, Si0.2Ge0.8 and Ge (markers) at w0 10 mm and bulk thermal conductivity values (dashed

lines). Solid lines are the predictions of a ballistic/diffusive model for thermal transport. (b) Deviations between LA(w0) for Si, Si:B, and

Si0.99Ge0.01 at f 1.1 MHz (makers) and their bulk thermal conductivity values (dashed lines). Solid lines are the predictions of a ballistic/diffusive model for thermal transport.

Si0.99Ge0.01 (Fig. 2a), making it difcult to explain the different

behaviours for LA(f) as only a length-scale effect. The most signicant difference between Si and Si0.99Ge0.01 is l(q) for q4q0

(Fig. 2c). Therefore, we suggest a second important factor in observing a failure of Fourier theory in the through-plane direction, aside from low-wavevector phonons with MFPs longer than experimental length scales, is for high-wavevector phonons to have a small thermal conductivity, L0 R

qmax

q0 l q

dq, and

therefore small thermal diffusivity.

Prior experimental studies that observed a dependence of LA on f emphasized the importance of temperature prole length scales being comparable to MFPs1,6,24. As we demonstrate in more detail in subsequent sections, another factor causing LA to

deviate from LB in the through-plane direction is interfacialphonon scattering. The interfacial-phonon scattering of long MFP phonons reduces the effective thermal conductivity, Kz(z) |Jz(z)/rzT(z)|, of the sample near the interface,

analogous to how boundary scattering reduces L of nanostructured materials2. (Here, we use a different symbol for thermal conductivity, K instead of L, to distinguish this effective property from the apparent values dened earlier.) The dependence of LA on f is evidence that the interfacial thermal resistance is not isolated to the interface at z 0 as is assumed in

standard analysis of TDTR and FDTR experiments.

An equivalent way to consider the effect of the interface is as a boundary condition on the heat current. The spectral distribution of the heat current across the interface is proportional to v(q)c(q) (ref. 17), while the spectral distribution of the heat current in the solid is l(q). This creates a spatial mismatch in the spectral distribution of the heat current, resulting in a nonequilibrium between high- and low-wavevector phonons near z 0. We label

this effect as an interfacial nonequilibrium thermal resistance, GNE 1. GNE 1 quanties the resistance between the high-wavevector phonons that carry the heat across the interface and the low-wavector phonons that carry the heat in the solid.

In this context, the thermal diffusivity of high-wavevector phonons affects LA(f) two ways. A small value for L0

qmax

q0 l q

dq corresponds to a larger GNE 1 (ref. 3). In addition, a

small thermal diffusivity increases the measurement sensitivity to Kz near z 0, where the high- and low-wavevector phonons are

not in equilibrium. The dependence of LA on f occurs because the sensitivity to GNE 1, which is localized near z 0, increases with

increasing f and shorter dp. The predictions of a ballistic/diffusive model (described in Methods) that accounts for both shortened length scales and interfacial-phonon scattering is in reasonable agreement with our data (solid lines in Fig. 3a). In our ballistic/

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms6075 ARTICLE

2,000

diffusive model, the heat current from low-wavevector phonons is calculated with a nonlocal expression that reduces to Fouriers law if MFPs are shorter than the temperature prole length scales. The model makes no assumptions regarding how low-wavector phonon MFPs compare with temperature prole length scales. Supplementary Figure 4 shows how interfacial/phonon scattering and f control the Kz(z) predicted by our ballistic/diffusive model.

At room temperature, Fourier theory does not accurately predict the thermal response of Si and Si:B for spot sizes below 5 mm, see Fig. 3b. Beam-offset measurements of Si:B and Si at f 9.8 MHz yield LrE7410 W m 1 K 1 and

LrE8410 W m 1 K 1 at w0 1 mm, respectively. Alterna

tively, we can resolve no dependence of LA on w0 in Si0.99Ge0.01, despite E25 W m 1 K 1 of heat being carried by phonons with c4w0 (Fig. 2a). For comparison, Si and Si:B have E50 and25 W m 1 K 1 of heat carried by phonons with c4w0.

The lack of a spot-size dependence for LA of Si0.99Ge0.01

suggests a second condition for an observable failure of Fourier theory in the in-plane direction, aside from low-wavevector phonons having c4w0, is for high-wavevector phonons to have a large L0 and therefore high thermal diffusivity. TDTR measurements are more sensitive to Kr(rEw0) |Jr(rEw0)/rrT(rEw0)|

in high thermal diffusivity materials where the following condition is met: the magnitude of Jr(rEw0) is comparable to the average magnitude of Jz across 0ozodp. Making w0 small enough to meet this condition in low thermal diffusivity materials is challenging given diffraction constraints on w0, but we expect that LA for Si0.99Ge0.01 would be reduced at smaller spot size. The

predictions of our ballistic/diffusive model are in reasonable agreement with our data (solid lines Fig. 3b).

The temperature dependence of l(q) in Si, Si:B and Si0.99Ge0.01

are systematically different. Therefore, varying the ambient temperature offers another avenue for relating Fourier theory failure in TDTR experiments to l(q). In Si, both l(q4q0) and

c(qoq0) increase rapidly with decreasing temperature. In Si0.99Ge0.01, mass disorder prevents a large increase in l(q4q0) with decreasing temperature. In Si:B, hole-phonon scattering prevents a large increase in c(qoq0) with decreasing temperature.

Figure 4 shows measurements of LA(w0) for Si, Si:B and Si0.99Ge0.01 between 40 and 300 K. In Si, the spot-size dependence of LA is large, but the frequency dependence between 1 and10 MHz is insignicant. We credit the increase in the spot-size dependence of LA as temperature decreases to large increases in both the thermal diffusivity of high-wavevector phonons and the MFPs of low-wavevector phonons. The longer MFPs of lowwavevector phonons increases the percentage of ballistic phonons. The increase in thermal diffusivity increases measurement sensitivity to Jr and decreases measurement sensitivity to Kz near the interface. In Si:B, LA is weakly dependent on w0 and has no observable f dependence because hole-phonon scattering prevents a rapid increase in c(qoq0). The similarity in the temperature dependence of LA for Si versus Si:B supports the hypothesis that when w0 is small, Fourier theory is over predicting the heat current from long MFP phonons, as Minnich et al.4 posited.

In Si0.99Ge0.01, LA below 100 K does not approach the pre

diction of the RTA model for LB at any w0 or f, indicating that Fourier theory is grossly inaccurate for this heat transfer problem because the majority of heat-carrying phonons have MFPs longer than temperature prole length scales. We attribute the increase in spot-size dependence of LA for Si0.99Ge0.01 with decreasing temperature to a rapid increase of c(qoq0), along with the more slowly increasing thermal diffusivity of low-wavevector phonons.

Beam-offset measurements conrm the failure of Fourier theory is anisotropic at low temperatures. For example, tting beam-offset data for Si at 77 K, 9.8 MHz and w0 4.7 mm yields

Lz E900 and Lr E300 W m 1 K 1. Similarly, tting beam-offset data for Si0.99Ge0.01 at 120 K, 1.1 MHz and w0 5 mm yields

LzE50 and LrE30 W m 1 K 1.

Our results for LA of Si agree with ref. 4 if compared as a function of the root mean square of the pump and probe spot sizes, but not if compared as a function of only pump spot size. (Our pump and probe spot sizes are equal, but ref. 4 used different pump and probe spot sizes.) In a linear heat transfer problem, an exchange symmetry exists between thermometers and heaters. Therefore, the TDTR measured thermal response must by symmetric to the exchange of the pump and probe beams. The exchange symmetry between thermometers and heaters exists because a linear problem can be solved with a Greens function solution, g(r,t;r0t0), where r demarks position, and Greens functions possess the following symmetry: g(r,t;r0t0) g(r0, t0;r, t) (ref. 29). That the length scale that

governs LA includes the probe spot size is an important result, as any theoretical effort to derive MFP distributions from LA(w0)

ignoring probe spot size is incomplete.

In addition to the experiments described above, we collected FDTR data for Si at room temperature between 1.1 and 17.6 MHz with w0 2.55 and 4.7 mm. Our FDTR data agree with the

predictions of a thermal model using the LAand GA derived from TDTR data collected at the same w0 (see Supplementary Fig. 5).

1,000

Al/Si

25 m

10 m

5 m

500

200

A(W m1 K1 )

100

Al/Si0.99Ge0.01

Al/Si:B

1.1 MHz

9.8 MHz

10 30 50 100 200 300

T (K)

Figure 4 | Low-temperature apparent thermal conductivities of Si, Si:B and Si0.99Ge0.01. Apparent isotropic thermal conductivity of Si (circles),

Si:B (squares) and Si0.99Ge0.01 (down triangles) for w0 5, 10 and 25 mm

and f 9.8 MHz, as a function of temperature. For Si0.99Ge

0.01, LA(T) at

1.1 MHz is also shown (up triangles). Solid lines are literature values for the thermal conductivity of bulk Si and a 3-mm thick thin lm of Si:B with a1 1019 cm 3 doping level21,31. Dashed lines are the prediction of our RTA

model for Si0.99Ge0.01, for which there are no prior low-temperature

thermal conductivity measurements. The spot size used to measure LA is

indicated by colour, red for 25 mm, black for 10 mm and blue for 5mm.

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500

Si0.99Ge0.01

G A (MW m2 K1 )

200

100

50 1 2 5 10 20

Frequency (MHz)

Si0.2Ge0.8

Figure 5 | Effect of mass disorder on apparent interface conductance. Apparent interface conductance of Si (black squares), Si0.99Ge0.01 (red

circles), Si0.2Ge0.8 (green circles) and Ge (orange squares) along with the predictions of a ballistic/diffusive model (solid lines). The value of GA for Si shown here differs from the value reported in Fig. 1 because the data in Fig. 1 is for a sample that did not have the native oxide removed before Al deposition. The low thermal diffusivity of Si0.99Ge0.01 and Si0.2Ge0.8 results

in a large interfacial nonequilibrium thermal resistance that effects the value of GA at low f, resulting in a lower value of GA than is observed for Si or Ge.

Our FDTR results therefore strongly disagree with BB-FDTR measurements of a Au/Cr/Si sample in the same frequency range1. We attribute the discrepancy to ref. 1 neglecting weak Au electronphonon coupling in their data analysis, see Supplementary Note 2 and Supplementary Fig. 6.

Role of the interface. Standard analysis of TDTR and FDTR data treats interfacial transport with a radiative boundary condition on the heat diffusion equation, Jz(z 0) G0DT(z 0), where G0 is

the interface conductance of diffusive phonons17,

1 4

j q vj q

: 4

Here, Rj is the probability that a phonon with polarization j impinging on the interface from the sample side reects and is believed to be determined by interfacial bonding strength, interface disorder and morphology and the vibrational properties of both the sample and metal transducer16,30. Using equation (4) assumes that the thermal resistance from interfacial/ phonon scattering is localized to z 0, not spread across a nite

length scale determined by phonon MFPs. This assumption is overly simplistic; interfacial/phonon scattering should affect the heat current over a length scale comparable to the MFPs of the heat-carrying phonons.

Measurements of GA for Si, Si0.99Ge0.01, Si0.2Ge0.8 and Ge

crystals coated with Al after native oxide removal demonstrate that equation (4) is an inadequate description of interface transport for Al/Si0.99Ge0.01 and Al/Si0.2Ge0.8. The measurement

results, shown in Fig. 5, have two features inconsistent with the description offered by equation (4): the dependence of GA on f and the signicant differences between GA of Si versus Si0.99Ge0.01 and of Si0.2Ge0.8 versus Ge.

The dependence of GA on f for Si0.99Ge0.01 and Si0.2Ge0.8 is

consistent with a reduced effective thermal conductivity of the sample, Kz |Jz/rzT|, near the interface due to an interfacial

nonequilibrium thermal resistance, GNE 1. For each TDTR measurement, we have two pieces of data that determine GA and LA: the shape of decay of Tin(tD) and the magnitude of the out-of-phase thermal response, Tout (Fig. 1). The shape of the decay of Tin(tD) depends on G and Kzzod. Here, d is

the distance heat travels in 3.5 ns, the maximum delay time. Tout,

which determines LA, depends on the average value of Kz over the distance heat travels in 1/f, Kzzodp. As f increases, Tout

becomes more sensitive to Kzzod because dp decreases, and

therefore LA decreases (Fig. 3). In the thermal model we use to derive LA and GA (ref. 23), we assume solid L is homogenous, that is, that KzzodpKzzodLA. If this is false, the only

way the diffusive thermal model can still t the frequency-independent shape of Tin(tD), which it must to t the data, is for

GA to depend on f. Therefore, the dependence of GA on frequency suggests an inhomogenous Kz(z). GA and LA that increase and decrease, respectively, with increasing frequency suggests Kz(zod)oLA and G04GA, consistent with the ballistic/

diffusive model predictions (Supplementary Fig. 4).

The large differences in the magnitude of GA for Si, Si0.99Ge0.01, Si0.2Ge0.8 and Ge are consistent with our hypothesis that a GNE 1 is related to observations of Fourier theory failure in the through-plane direction3. In most TDTR experiments, dp is much larger than the length scale of the nonequilibrium near the interface3, and therefore GNE 1 causes a reduced value of GA in addition to any reduced value of LA. Our ballistic/diffusive model predicts GNE 1 is largest, and therefore GA will be smallest, for materials whose high-wavevector phonons have small L0. This is consistent with our results: GA for Si0.2Ge0.8 is smaller than GA for either Ge or Si, GA for Si0.99Ge0.01 is smaller than GA for Si.

To summarize, we explain the difference in magnitude and convergence/divergence of GA(f) for Si versus Si0.99Ge0.01 and Ge versus Si0.2Ge0.8 with a GNE 1 whose length scale and magnitude are largely independent of f. What changes with f is measurement sensitivity. The magnitude of GNE 1 is included in the value of GA of the alloys at low f (Fig. 5), but moves to LA at high f (Fig. 3) because dp becomes comparable to the length scale of nonequilibrium.

Thus far, we have posited that interfacial-phonon scattering is related to observations of Fourier theory failure in the through-plane direction based on results for LA and GA of SiGe alloys.

Now, we present another set of experiments that supports this hypothesis: measurements of LA for Si and MgO at low temperature with different metal transducers and interface conditions.

In Fig. 6, we show the impact of the interface on LA of Si at 300 K and w0 10.3 mm. For w0 10.3 mm, JrooJz, and therefore

LAELz. TDTR measurements of Al/11 nm SiO2/Si and Ta/2 nm SiO2/Si show LA values that deviate outside our error bars from the bulk thermal conductivity value for Si of 142 W m 1 K 1 (refs 22,31).

In Fig. 7, we compare the effect of the interface and spot size for Si and MgO at 48, 83 and 144 K. The magnitude of transducer dependence of LA for both Si and MgO at these temperatures is on the order of 100 W m 1 K 1. In MgO, this is an B30%

change, comparable to the effect of changing from a 25 to 5 mm spot size. The large transducer dependence in MgO is evidence that signicant heat is carried by non-diffusive phonons, and that Ta serves as a better thermal sink for these phonons than Al does (see Supplementary Note 3).

Importance of temperature-prole extrema. Here, we use our ballistic/diffusive model, described in equations (5)(10) of Methods, to gain insight into the anisotropic failure of Fourier theory in TDTR measurements. The predictions of the ballistic/ diffusive model support our conclusions that (1) the Fourier theory failure in TDTR experiments at small spot sizes is due to an inability of Fouriers law to accurately predict Jr and (2) Kz is lowered near the interface because of interfacial-phonon scattering.

A simple approach for determining whether Fourier theory is accurate for a given heat transfer problem is to compare the

Z dq 1 Rj q

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predictions of ballistic and diffusive expressions for the heat current. In Fig. 8, we compare the predictions of diffusive and ballistic expressions for the heat current (Fouriers law and equation (9) in Methods) for the measurement of Al/Si shown in Fig. 1. For these calculations, we approximate the temperature prole by solving the heat diffusion equation using an Al/Si interface conductance of G0 250 MW m 2 K 1 and thermal

conductivity of Si of LB 142 W m 1 K 1. This represents

the expected temperature prole if Fourier theory did not fail at small w0.

Fouriers law fails most signicantly near temperature-prole extrema for three types of long MFP subpopulations of the Si phonons (Fig. 8ad): (1) phonons travelling in opposite directions in the in-plane direction across r 0, (2) phonons travelling in

opposite directions in the through-plane direction that reect at the interface at z 0 and (3) phonons travelling in opposite

directions in the through-plane direction that transmit across the interface at z 0. The rst-order Taylor series approximation of

the temperature-prole Fouriers law uses is grossly inadequate for cases (1) and (2) due to the sign change of dT/dr at r 0 and

dT/dz at z 0. (The plane of z 0 is a virtual extrema in the

through-plane direction because reected phonons see a mirror image of the temperature prole.) As a result, Fouriers law over predicts the heat current from phonons described by cases

(1) and (2). Fouriers law is also inaccurate for phonons travelling in the through-plane direction that transmit at the interface, case(3). However, the under-prediction of the heat current is less than in cases (2) and (3), because this group of phonons still see a signicant temperature difference, DT. While Fig. 8bd only illustrates the failure of Fouriers law for phonons with a MFP of 2 mm at two specic positions, the illustrations in Fig. 8bd are representative of why Fourier theory fails near temperature-prole extrema.

The illustrations in Fig. 8 qualitatively explain why the failure of Fourier theory is anisotropic in TDTR measurements of Si even when w0 and dp are comparable. The majority of phonons with MFPs longer than the in-plane length scale traverse the hot region without scattering (Fig. 8b), meaning these phonons wont carry signicant heat. In contrast, a signicant percentage of long MFP phonons, 1 R, travelling in the through-plane direction

are emitted from or absorbed in the hot metal lm, (Fig. 8d), and will therefore carry signicant heat.

Because Fouriers law over-predicts Jr near r 0 (Fig. 8b), and

over-predicts Jz near z 0 for phonons that reect from the

interface (Fig. 8c), the failure of Fourier theory is anisotropic, spatially inhomogeneous, and depends on the probability of phonons reecting from the interface. We show this in Fig. 8eh by comparing the average heat current magnitude from phonons with a 1 mm MFP predicted by Fouriers law to the predictions of our ballistic/diffusive model in the adiabatic (R 1) and radiative

(R 0) limits. In Supplementary Note 4, we discuss how altering

w0 and f has the dual effect of changing length scales and measurement sensitivity to nonequilibrium near extrema.

DiscussionOur ballistic/diffusive model predicts that the spectral distribution of heat carried by low-wavevector ballistic phonons, |JNL|, is

not equal to l(qoq0) within a length scale of the interface comparable to c because of interfacial-phonon scattering. For sufciently short through-plane length scales of the temperature prole, TDTR is sensitive to this region where the spectral distribution of JNL is not equal to l(qoq0), explaining the dependence of LA on f in alloys.

Prior TDTR and FDTR studies have explained deviations between LA and LB by assuming phonons with MFPs longer than some experimental length scale are ballistic and do not contribute to LA (refs 1,6,11). This is equivalent to the explanation offered by our ballistic/diffusive model in the limit that TDTR is

Al/Si

A B (W m1 K1 )

Ta/2 nm SiO2/Si

Ta/Si

R=0

Al/11 nm SiO2/Si

R=1

60 1 2

Frequency (MHz)

5 10

Figure 6 | Effect of interface on Si apparent thermal conductivity. Deviations of LA from LB 142 Wm 1 K 1 for Al/Si and Al/ 11 nm

SiO2/Si (open markers), Ta/Si and Ta/2 nm SiO2/Si (lled markers) at w0 10.3 mm as a function of f. The solid and dashed lines are the

predictions of a ballistic/diffusive model that assumes all low-wavevector Si phonons that impinge on the interface reect (R 1) or transmit (R 0).

5 10

Spot size (m)

48 K

83 K

144 K

1,000

[Ref 4]

500

Ta/Si

A(W m1 K1 )

Al/Si

Ta/2nm SiO2/Si

200

Ta/MgO

Al/MgO

100

5 10 20

Figure 7 | Effect of interface on low-temperature Si and MgO apparent thermal conductivity. Effect of spot size and transducer/interface on LA for Si and MgO at 48, 83 and 144 K. For comparison, we include the results of ref. 4 plotted as a function of the root mean square of the pump and probe spot sizes used in their experiment. In MgO, the interface has an effect on the value of LA comparable to that of w0.

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Temperature-profile of 80 nm Al / Si

z(m)

JF T

1.5

T(K)

z(m)

JF / Jmax

2 0 1

3 3 0r (m)

0 1r (m)

r (m)

JNL T, R = 1

J T 2 T

1.0

0.5

T(K)

z(m)

2 0 1 2

3 0r (m)

r (m)

0 1r (m)

JNL / JF

JNL T, R = 0

0.5

R = 1

T(K)

z(m)

3 0z (m)

2 0 1r (m)

0 1r (m)

R = 0

T(K)

NL / J F

0.5

r (1, 0.15)

z (0, 0.5)

J 0.2

0.1

3 0z (m)

0.05

0.02

0.2

0.1

Mean free path (m)

0.5

1 2

Figure 8 | Ballistic versus diffusive predictions for the heat current. (a) Amplitude of temperature oscillations predicted by the heat diffusion equation for 80 nm Al/Si sample undergoing sinusoidal surface heating at f 10 MHz and w0 1 mm. (bd) Qualitative comparison between the rst-order

Taylor series approximation of the temperature prole used in Fouriers law (red lines), 2crT, and the temperature prole actually traversed by 2 mm

MFP phonons (black lines) with the trajectories illustrated by the arrows in a. The rst-order Taylor series that Fouriers law uses (equation (8)) is a poor approximation for the temperature prole that is traversed by phonons that are (b) travelling in the in-plane direction across r 0 and (c) travelling in the

through-plane direction after reecting from the interface at z 0. (e) Average magnitude of Jr and Jz predicted by Fouriers law for the temperature prole

in a. The colour axis is normalized to the maximum value predicted by Fouriers law, which occurs in the z direction at r z 0 and in the r direction at

z 0 and rEw0. TDTR measurements are most sensitive to transport properties in the regions where the heat current is largest. (f,g) Ratio of the average

magnitude of Jz and Jr predicted by equation (9), to the predictions of Fouriers law for 1 mm MFP phonons. The heat current proles shown in f assume all 1 mm MFP phonons reect at the interface, R 1 (adiabatic limit). The heat current proles shown in g assume 1 mm MFP phonons transmit at the

interface, R 0 (radiation limit). (h) Magnitude of the heat current predicted by equation (9) relative to Fouriers law at two representative points. The red

curves are for Jr at r 1 mm and z 150 nm. The blue curves are for the Jz at r 0 and z 500 nm. These two points are selected because they are

representative of how Fouriers law over-predicts the heat current in the regions the measurement is most sensitive to the effective thermal properties. Solid lines were calculated assuming R 1 (adiabatic limit), while the dashed lines were calculated assuming R 0 (radiation limit).

measuring Kz(zoc), that is, the through-plane temperature prole length scale is less than c, and the heat carried by diffusive phonons is much greater than the heat carried by ballistic phonons, |L0rT|c|JNL|, where L0 is the thermal

conductivity of the high-wavevector diffusive phonons.Whether the condition that |L0rT|c|JNL| is met depends on

the phonon dynamics of the material being studied. The amount of heat carried by ballistic phonons in the through-plane direction near the interface, |JNL (0ozoc)|, depends on the spectral distribution of c(q)v(q) for low-wavevector ballistic-phonons (see equations (9) and (10) in Methods), while L0 depends on l(q) for high-wavevector diffusive phonons. For materials like Si0.2Ge0.8, where only phonons with qo0.1qmax have MFPS longer than the length scales of the temperature prole, the assumption that |L0rT|c|JNL| is reasonable since

phonons with qo0.1qmax have little heat capacity (Fig. 2b). In

materials such as Si, Si0.99Ge0.01 and MgO, where the majority of

heat is carried by phonons with qo0.1qmax, signicant deviation from Fourier theory implies phonons with a non-negligible c(q)v(q) are ballistic. Therefore, JNL may or may not be negligible depending on L0 and R.

Our results are relevant for efforts to use TDTR and FDTR as a MFP spectroscopy technique1,4,24. Observations of Fourier theory failure in the through-plane and in-plane direction are both sensitive to the MFP distribution of solids (Fig. 8h). In materials where a measurement of Lr is possible with laser spot sizes on the order of microns, for example, materials with a high thermal diffusivity, it will be simpler to extract information concerning MFP distributions from measurements of Lr(w0) rather than measurements of Lz because Jr is less sensitive to interface effects than Jz. That Jr is less sensitive to R than Jz can be seen in Fig. 8fh. For example, the difference between the adiabatic and

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For TDTR measurements with laser beams that have 1/e2 radii less than 5 mm, in situ measurements of the beam intensity proles on the sample were necessary to prevent systematic errors in w0 that can cause result in spot size and frequency-dependent LA values19.

When performing FDTR measurements, our experimental setup remains nearly the same as for TDTR measurements; however, the Ti:sapphire oscillator used to generate the train of pump and probe pulses is dropped out of mode lock, thereby replacing the train of pulses with a CW beam.

Si and MgO samples were prepared with different surface treatments and different metal lm coatings. The geometry of the samples that were characterized are Al/2 nm SiO2/Si, Al/11 nm SiO2/ Si, Ta/2 nm SiO2/Si, Ta/Si, Al/3 nm SiO2/

Si0.99Ge0.01, Ta/MgO and Al/MgO. The Si samples with 23 nm of SiO2 were untreated commercial wafers with a native oxide. Thicker layers of thermal oxide were grown by heating Si wafers in an oven to 950 C for 5 min at a ramp rate of 5 C per min. The thickness of SiO2 was measured using variable angle spectroscopic ellipsometry. The Si substrate for the Ta/Si sample was HF treated to remove the native oxide and then loaded into a high vacuum sputtering chamber. The thin metal lms were deposited on the Si, Si:B, Si0.99Ge0.01 and MgO substrates

under high vacuum. MgO samples were annealed at E800 C under high vacuum for 1 h before Al and Ta deposition. Si samples that were coated with Al lms were heated to E600 C and then allowed to cool to room temperature before deposition to provide cleaner interfaces. The Ta was deposited on the MgO substrate at E700 C to the form the a (bcc) phase34. On Si, the Ta lm was deposited slightly below 600 C to prevent the formation of an extensive Ta-Si silicide layer at the interface. Picosecond acoustics16,35 and Rutherford backscattering spectroscopy conrmed that TaSi2 or Ta2O5 layers of appreciable thickness were not formed. For control experiments, Al and Ta lms were also deposited on Si wafers that had 105 nm layers of thermal oxide on the surface, and Al lms were deposited on single crystal Al, Cu, and Ni substrates, see Supplementary Fig. 7. The boron doping concentration in the Si:B sample was1.6 1019 cm 3, determined by secondary ion mass spectroscopy. We determined

the relative sensitivity factor for boron in our secondary ion mass spectroscopy measurement by measuring a reference sample of Si with a known concentration prole of ion implanted boron.

Control TDTR measurements of Al/Ni and Al/SiO2 samples were performed as a function of w0 and f, see Supplementary Fig. 7. We observed no dependence of LA on w0 or f outside of our estimated experimental uncertainty of 8%. A radial thermal conductivity for Ni within 5% of the through-plane value was derived from beam-offset measurements at w0 1.05 mm and f 1.1 and 10 MHz, see

Supplementary Fig. 8. We also performed control measurements to conrm the temperature reading of our cryostat for low-temperature experiments, see Supplementary Fig. 9.

In addition to the samples described above, a separate set of Al-coated Si, Si0.99Ge0.01, Si0.2Ge0.8 and Ge samples were prepared specically for the apparent interface conductance measurements shown in Fig. 5. These samples were rst HF treated before being loaded in a high vacuum sputtering chamber to remove as many extrinsic differences between the interfaces as possible, such as different thicknesses of native oxide. In addition to the HF dip, the Si and Si0.99Ge0.01

samples were heated to E700 C and the Si0.2Ge0.8 and Ge samples were heated to E450 C in high vacuum for 30 minutes and then allowed to cool to room temperature before Al deposition.

Ballistic/diffusive model for thermal transport. The thermal response measured in TDTR experiments in heat transfer problems where Fourier theory is invalid can be understood with a ballistic/diffusive model. By ballistic phonons, we mean phonons with MFPs that are comparable to, or larger than, the important length scales of the temperature prole and therefore require a nonlocal expression for the heat current. By diffusive phonons, we mean phonons whose heat current is well described by Fourier theory because their MFPs are shorter than the important length scales of the temperature prole. The new ballistic/diffusive framework we describe here is based on the derivation presented in Aschroft and Mermin for L of a Drude metal36, and builds on our prior work in ref. 3. Our ballistic/diffusive model draws on concepts outlined in two-uid models for phonon transport described by Armstrong37 and more recently by Maznev et al.12

High-wavevector phonons (q4q0), which contain the vast majority of the solids heat capacity (Fig. 2b), form a thermal reservoir12,37. In our model, the phonon wavevector that divides the high- and low-wavevector phonons, q0, must meet the requirement that c(q0)oow0 and c(q0)oodp so that all high-wavevector phonons are diffusive and the mean occupation of all high-wavevector phonons at all positions is well described with a single temperature prole, T(r, z).

The dominant inelastic scattering mechanism for low-wavevector phonons is three-phonon processes that involve two high-wavevector phonons12,38. Therefore, we follow Maznev et al. and neglect coupling between low-wavevector phonons and assume that low-wavevector phonons are only coupled to high-wavevector phonons. Then, the thermal reservoir transports heat both diffusively and by radiating and absorbing the lower frequency phonons with qoq0 and the total heat current at position r due to all phonons is

Jr L0rT r

radiative limits for Jz in Fig. 8h is much larger than for Jr. That interfacial-phonon scattering impacts Jr less than Jz is analogous to boundary scattering having less impact on the in-plane L of a thin lm than on the through-plane L2.

Our study provides insight into a long standing issue regarding the denition of temperature and phonon mediated heat transfer across interfaces18. The concept of an interfacial thermal conductance implies an abrupt temperature drop at crystal boundaries, which is consistent with molecular dynamics simulations that dene the temperature of individual atoms with their kinetic energy. However, whether an equilibrium concept such as temperature can be rigorously dened on length scales shorter than the MFPs of heat-carrying phonons has long been controversial18. Our ballistic/diffusive framework offers insight into this dilemma. High-wavevector phonons are in thermal equilibrium with each other on short length scales, and it is primarily the occupation of high-wavevector phonons that determine the kinetic energy of atoms (Fig. 2b). Molecular dynamics simulations are measuring Geff |Jz/DT|z

0, which is

different from GA when GA includes a signicant interfacial nonequilibrium thermal resistance, GNE 1, as is the case for SiGe alloys.

In conclusion, we performed TDTR experiments on Si, Si0.99Ge0.01, Si:B and MgO as a function of w0, f and temperature.

These experiments provide three insights into how Fourier theory fails in TDTR experiments. First, in measurements with small w0,

the primary effect of the small spot size is to cause a failure of Fourier theory in the in-plane direction. Second, the thermal diffusivity of high-wavevector phonons determines whether measurements can resolve a failure of Fourier theory in the in-plane or through-plane directions for fo20 MHz and w041 mm

at room temperature. Third, an interfacial thermal conductance is not adequate in a ballistic/diffusive heat transfer problem. To accurately describe the effect of the interface on ballistic/diffusive heat transfer problems, it is necessary to consider how the reection and transmission of phonons prevents thermal equilibrium between heat-carrying phonons near the boundary (Fig. 8c,d).

While this paper was under review, a paper was published by Ding et al.32 theoretically examining three-dimensional heat-transfer in an Al/Si sample on timescales between 1 ps and 1 ns using the Boltzmann transport equation. Their model indicates that only Jr, and not Jz, is affected by a reduced spot size, consistent with our beam-offset measurements (Fig. 1); however, their models prediction that interfacial scattering does not affect Jz appears inconsistent with our experimental observations that the interface plays an important role in through-plane transport (Figs 57).

Methods

Experiment. In TDTR, the thermal response of a sample to a train of pump pulses periodically modulated at frequency f is characterized by measuring temperature-induced changes in the intensity of a reected probe beam23. The experimental data consists of the in-phase and out-of-phase voltages recorded by a Si photodiode connected to an RF lock-in that picks out the signal components at the pump modulation frequency f. In our implementation of TDTR, the pump and probe spot sizes are equal. Temperature oscillations that are in-phase with the pump beam, Tin, are to a good approximation the time-domain response to pulsed heating. Temperature oscillations that are out-of-phase with the pump beam, Tout,

are, to a good approximation, the out-of-phase thermal response of the sample at frequency f. Since most of the sensitivity of a TDTR measurement to the thermal properties of a sample comes from Tout, 1/f is the most important timescale in our experiment. Example TDTR data is shown in Fig. 1a. The parameters LA and GA are convenient proxies for our data that represent how much our measurements deviate from the predictions of Fourier theory (Fig. 1a). We do not view LA and GA as effective thermal properties when they depend on f or w0. In other words, the solution to the heat diffusion equation with L LA and G GA is only an accurate

description of the surface temperature, not the entire temperature prole. Further details of our technique can be found in refs 19, 23 and 33.

XqoqJNL q; r ; 5

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where j sums over polarizations, L0 is the total thermal conductivity due to phonons with q4q0 and JNL(c(q)) is the nonlocal heat current due to phonons with a MFP c being radiated and absorbed from the thermal reservoir. The nonlocal heat current across a plane, q, in a homogenous solid due to a lowwavevector phonon excitation is equal to the product of the difference in the number density of thermally excited phonons travelling in the positive and negative directions, Dn(q)D(q)/2, the excitations group velocity, v(q) and the difference in energy between excitations travelling in the positive and negative directions, o(q), that is, JNL oDnD(q)v/2, where n is the BoseEinstein

occupation number and D is the volumetric density of states. Here, the difference in thermally occupied states depends on the differences in temperature of the solid where and when the low-wavevector phonons travelling in the positive and negative directions last scattered. Because the most important timescale in our experiment, 1/f, is much longer than phonon lifetimes, we can neglect any temporal lag when relating Dn to T. Then, for a one-dimensional problem3,

JNLq

2 C q

: 6 Here, DTq is the average temperature difference between phonons with MFP

c(q) travelling in the positive and negative direction.A physically intuitive approximation for DT; at position r is36

Tr; T r

T r

; 7 because, on average, phonons have travelled a distance c since last scattering. A more general expression for DT that also accounts for the probability of scattering at an interface is derived in Supplementary Methods. When the temperature prole near r can be accurately described by a rst-order Taylor series36, that is,

Tr; 2rT; 8 the nonlocal expression in equation (6) for the heat current reduces to the local expression known as Fouriers law. In other words, Fouriers law will accurately predict the heat current due to phonon with wavevector q provided equation (8) is a reasonable approximation.

When the heat transfer problem in a TDTR or FDTR experiment is purely diffusive, the distance heat can diffuse in one period of oscillation determines the through-plane length scale of the temperature prole and is known as the thermal penetration depth, dp

L=pCf p

. In a ballistic/diffusive transport problem, this length scale of the temperature prole is decreased because ballistic phonons carry less heat than Fouriers law predicts. Therefore, we can bound the through-plane temperature prole length scale as less than dp but larger than d0p

L0=pCf p

where L0 is the thermal conductivity of high-wavevector phonons.

When the heat transfer problem is not one-dimensional, we must integrate over all directions and equation (6) becomes

JNL r; ; R; i

T r; f0; y0; ; R

vi;j: 9

where j labels phonon polarization, y1, y2, f1 and f2 are determined by direction, i,R is the probability of reection for a Si phonon impinging on the metal/Si interface and vi,j is the component of the group velocity of a phonon with polarization j in direction i. In the radial direction, y1 0, y2 p, f1 p/2,

p/2 and vi,j vj cos f sin y. In the through-plane direction, y1 0, y2 p/2,

0, f2 2p and vi,j vj cos y.

For the group of phonons that satisfy the limit ccdp, equation (9) predicts that the heat current in through-plane direction is14

JBz

1 R

df0

dy0 sin y0D

vLCL vTCT

4 TM; 10

where CL and CT are the collective volumetric heat capacities of longitudinal and transverse phonons that satisfy ccdp, vL and vT are the average longitudinal and transverse group velocities of ballistic phonons and TM is the temperature of the metal transducer. In the r direction, the heat carried by phonons with ccw0 is zero.

Ballistic and diffusive phonons in our model are affected by the presence of a boundary in different ways. For the diffusive channel, we assume the boundary condition Jz(z 0) G0DTz0, is an adequate description of the effect of the

interface on diffusive thermal transport. Here, G0 is the diffusive thermal conductance. For the ballistic channel, the heat current across the interface is given by equation (9) with the exclusion of phonons travelling toward the metal surface, and can be viewed in terms of a ballistic conductance, Jz z0

GBDT, where

GB (1 R)C(qoq0)v/4 is the conductance of ballistic phonons. In addition, the

presence of a boundary alters the heat carried by long MFP phonons in the substrate within a distance c of the interface, because the reection and transmission of phonons alters DT (Fig. 8). In the main text, we labelled this latter effect an interfacial nonequilibrium thermal resistance because the interface is preventing thermal equilibrium between phonons near z 0.

The expression for JNL(c(q)) in equation (9) converges to the Fourier theory prediction for the heat current when c(q) is much smaller than the length scales of the temperature prole. Therefore, the total heat current predicted by equation (5) will not be sensitive to the value of q0 chosen as long as equation (8) is satised for c(q4q0).

Self-consistent criteria for choosing the value of q0 for a given material can now be outlined. The requirement that equation (8) is valid for c(q0) provides a restriction on how small q0 can be. A restriction on how large q0 can be is provided by our two assumptions that low-wavevector phonons have a negligible collective heat capacity in comparison with high-wavevector phonons, C(qoq0)ooC(q4q0), and that the dominant scattering mechanism for low-wavevector phonons is absorption/emission from the thermal reservoir. The criterion that C(qoq0)ooC(q4q0) will typically provide a stricter limit on the upper bound of q0 than the absorption/emission criterion because the spectral distribution of the heat capacity only depends on the density of occupied states, while the probability of a low-wavevector phonon scattering with two other phonons depends on both the density of occupied states and the three-phonon matrix element, and the three-phonon matrix element is larger for scattering events that involve high-wavevector phonons12. We set q0 in all materials so that c(q0) dp (20 MHz)/5 for both

longitudinal and transverse phonons, see supplementary Methods. This satises the requirement that the total heat current predicted by equation (5) is insensitive to q0. In addition, it leaves more than 90% of the total heat capacity with the high-wavevector phonons in the materials we study here.

To compare with our experimental data, we use our ballistic/diffusive model to calculate effective thermal conductivities, and then use these effective thermal conductivities to generate theoretical values of GA and LA as a function of w0 and f (solid lines in Figs 3, 5 and 6). Further details of our calculation are provided in Supplementary Methods. Supplementary Fig. 10 compares the assumptions of specular versus diffuse reection of phonons from the interface. Supplementary Figs 1113 illustrate how with properly dened effective in-plane and through-plane thermal-conductivities, Kr(z) and Kz(z), the predictions of Fouriers law and equation (9) agree.

To summarize how we xed our model parameters, once q0 is xed so that all high wave vector phonons are diffusive, and the RTA models are used to x the MFPs and heat capacities of the short wavevector phonons, the only free parameters in the ballistic/diffusive model are G0 and R. We xed R 0 for all

samples and calculations with the exception of Figs 6 and 8. Figures 6 and 8 include both the adiabatic and radiative limits of our models predictions, and is representative of the effect that varying R has on the calculations. We xed G0 for

Si and Si0.99Ge0.01 based on TDTR measurements of Si. We xed G0 for Ge and Si0.2Ge0.8 based on TDTR measurements of Ge. Therefore, our model predictions for LA of Si, Ge, Si0.99Ge0.01 and Si0.2Ge0.8 involve no tting parameters. Our model

predictions for GA of Si and Ge involve one tting parameter (G0) and our model predictions for GA of Si0.99Ge0.01 and Si0.2Ge0.8 involve no tting parameters. The

LA predicted by our ballistic/diffusive model is insensitive to q0 and G0; for example, a 5% change in either results in less than a 0.2% change in LA predicted for Si0.99Ge0.01 at 10 MHz. GA predicted by our model is sensitive to G0; for example, a 5% change in the value of G0 results in a 3% change in GA predicted for Si0.99Ge0.01 at 10 MHz.

The accuracy of our ballistic/diffusive model predictions depends on four factors: (1) the accuracy of our RTA models predictions for the thermal conductivity spectral distribution l(q), (2) the accuracy of our assumptions regarding interfacial-phonon scattering, (3) the accuracy of our assumption that only diffusive phonons store a signicant amount of heat and (4) the accuracy our nonlocal expression for the heat current equation (9). Therefore, the accuracy of quantitative predictions could be improved in future work with a more sophisticated model. For example, our models treatment of interfacial transport assumes, without rigorous justication that low-wavevector phonons impinging on the interface from the nonmetal side have a constant reection probability (usually zero), and that a radiative boundary condition is a valid description of interfacial transport for high-wavevector phonons. Our models requirement that the vast majority of a solids heat capacity is due to phonons with MFPs much shorter then temperature prole length scales limits its applicability to problems where the vast majority of thermally excited phonons do not require a nonlocal expression for the heat current, precluding our ballistic/diffusive model from making predictions at low temperatures. Our nonlocal expression for the heat current does not distinguish between elastic and inelastic scattering, which should have different effects on the nonequilibrium length scale11. Finally, another weakness in our model predictions is that to predict the thermal response of our samples at 1/f timescales, we used our ballistic/diffusive model to derive effective thermal properties, Kz(z) and Kr(z), as inputs for the heat diffusion equation. A fully rigorous calculation would be self-contained by directly calculating the thermal response of the solid on 1/f timescales.

References

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2. Yang, F. & Dames, C. Mean free path spectra as a tool to understand thermal conductivity in bulk and nanostructures. Phys. Rev. B 87, 035437 (2013).

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Acknowledgements

This work was supported by AFOSR contract number AF FA9550-12-1-0073 and was carried out in part in the Frederick Seitz Materials Research Laboratory Central Research Facilities, University of Illinois. R.B.W. thanks the Department of Defense for the NDSEG fellowship that supported him during this work, Joseph P. Feser for sharing his preliminary data for MgO and his assistance with beam-offset measurements and Dongyao Li for helpful discussions regarding heater/thermometer symmetry in heat transfer problems.

Author contributions

R.B.W. designed and carried out the experiments, conducted the data analysis and wrote the manuscript with the assistance and supervision of D.G.C.

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How to cite this article: Wilson, R. B. et al. Anisotropic failure of fourier theory in time-domain thermoreectance experiments. Nat. Commun. 5:5075doi: 10.1038/ncomms6075 (2014).

NATURE COMMUNICATIONS | 5:5075 | DOI: 10.1038/ncomms6075 | http://www.nature.com/naturecommunications

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The applicability of Fourier's law to heat transfer problems relies on the assumption that heat carriers have mean free paths smaller than important length scales of the temperature profile. This assumption is not generally valid in nanoscale thermal transport problems where spacing between boundaries is small (<1 μm), and temperature gradients vary rapidly in space. Here we study the limits to Fourier theory for analysing three-dimensional heat transfer problems in systems with an interface. We characterize the relationship between the failure of Fourier theory, phonon mean free paths, important length scales of the temperature profile and interfacial-phonon scattering by time-domain thermoreflectance experiments on Si, Si_0.99 Ge_0.01 , boron-doped Si and MgO crystals. The failure of Fourier theory causes anisotropic thermal transport. In situations where Fourier theory fails, a simple radiative boundary condition on the heat diffusion equation cannot adequately describe interfacial thermal transport.

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Title

Anisotropic failure of Fourier theory in time-domain thermoreflectance experiments

Author

Wilson, R B; Cahill, David G

Pages

5075

Publication year

2014

Publication date

Oct 2014

Publisher

Nature Publishing Group

e-ISSN

20411723

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms6075

ProQuest document ID

1566627560

Anisotropic failure of Fourier theory in time-domain thermoreflectance experiments

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