Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance

Abstract

Non-metallic crystalline materials conduct heat by the transport of quantized atomic lattice vibrations called phonons. Thermal conductivity depends on how far phonons travel between scattering events--their mean free paths. Due to the breadth of the phonon mean free path spectrum, nanostructuring materials can reduce thermal conductivity from bulk by scattering long mean free path phonons, whereas short mean free path phonons are unaffected. Here we use a breakdown in diffusive phonon transport generated by high-frequency surface temperature modulation to identify the mean free path-dependent contributions of phonons to thermal conductivity in crystalline and amorphous silicon. Our measurements probe a broad range of mean free paths in crystalline silicon spanning 0.3-8.0 μm at a temperature of 311 K and show that 40±5% of its thermal conductivity comes from phonons with mean free path >1 μm. In a 500 nm thick amorphous silicon film, despite atomic disorder, we identify propagating phonon-like modes that contribute >35±7% to thermal conductivity at a temperature of 306 K.

ARTICLE

Received 10 Aug 2012 | Accepted 20 Feb 2013 | Published 27 Mar 2013

Non-metallic crystalline materials conduct heat by the transport of quantized atomic lattice vibrations called phonons. Thermal conductivity depends on how far phonons travel between scattering eventstheir mean free paths. Due to the breadth of the phonon mean free path spectrum, nanostructuring materials can reduce thermal conductivity from bulk by scattering long mean free path phonons, whereas short mean free path phonons are unaffected. Here we use a breakdown in diffusive phonon transport generated by high-frequency surface temperature modulation to identify the mean free path-dependent contributions of phonons to thermal conductivity in crystalline and amorphous silicon. Our measurements probe a broad range of mean free paths in crystalline silicon spanning 0.38.0 mm at a temperature of 311 K and show that 405% of its thermal conductivity comes from phonons with mean free path 41 mm. In a 500 nm thick amorphous silicon lm, despite atomic disorder, we identify propagating phonon-like modes that contribute 4357% to thermal conductivity at a temperature of 306 K.

DOI: 10.1038/ncomms2630

Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreectance

Keith T. Regner1, Daniel P. Sellan2, Zonghui Su1, Cristina H. Amon1,2, Alan J.H. McGaughey1,3

& Jonathan A. Malen1,3

1 Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA. 2 Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada. 3 Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA. Correspondence and requests for materials should be addressed to J.A.M. (email: mailto:[email protected]

Web End [email protected] ).

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Thermal conductivity (k), which relates the heat ux ( q !)

and temperature gradient (rT) in a material through the

Fourier law, q

! krT, results from the cumulative

contributions of phonons with a broad range of mean free paths (MFPs). The spectral MFP distribution is critical in nanostructured materials and devices, where size effects selectively scatter phonons or create non-Fourier conduction based on individual phonon MFPs. Such effects have an impact on heat dissipation in nanoelectronics and photonics, as well as the design of nanostructured thermoelectric materials with reduced thermal conductivity19. Because of its ubiquity in electronics, crystalline silicon (c-Si) has emerged as the prototypical material of study, yet controversy persists on what phonon MFPs dominate thermal transport, even in the bulk material. Kinetic theory denes the thermal conductivity as k CvsLG=3, where C is the volumetric

heat capacity, vs is the speed of sound and LG is the average (or grey) MFP. For c-Si, kinetic theory yields LG 41 nm at a

temperature (T) of 300 K10. This grey approximation severely underestimates the MFPs of the phonons that contribute signicantly to thermal conductivity because (i) dispersion makes vs an overestimate of the average group velocity of acoustic phonons and (ii) optical phonons contribute to C but negligibly to bulk k (ref. 11). Thermal conductivity measurements of thin silicon lms indicate that an effective MFP of 300 nm at T 300 K is more appropriate12.

To clarify the spectral contributions of phonons to thermal conductivity as a function of MFP, Dames and Chen13 established the thermal conductivity accumulation function,

kaccumL* Xpolarization Z

L* 0

13 CL vL L dL 1

which identies the thermal conductivity due to phonons having MFPs less than L*. To evaluate equation (1) analytically, models for phonon scattering must be chosen to form expressions for C(L) and v(L)14,15. There is no current consensus on how to select these models, which may be different for different materials. Furthermore, direct calculations of kaccum at the

phonon mode level by molecular dynamic simulations or lattice dynamics calculations1,2,10 yield different behaviours for kaccum

than the analytical approaches.

Discrepancies among predictions and a lack of information from conventional thermal conductivity measurements have motivated innovative experiments to probe phonon MFP spectra1,1620. Koh and Cahill16 found that the thermal conductivities of InGaP, InGaAs and SiGe alloys measured by time domain thermoreectance (TDTR) depend on the surface heat ux modulation frequency (f). Periodic heat ux modulation induces an exponentially decaying temperature prole with a thermal penetration depth, Lp

k=Cpf p

, that identies the depth normal to the surface at which the temperature amplitude is e 1 of its surface amplitude. Koh and Cahill reasoned that phonons with MFPs larger than Lp travel ballistically and do not contribute to the measured thermal conductivity of the material (that is, TDTR measures kaccum where L* Lp). More recent

inquiries to the MFP spectrum have set L* as the heat source dimension through the use of nano-patterned heaters on sapphire17, the TDTR spot-size on c-Si at cryogenic temperatures1 and the grating period in transient grating experiments1820. Spot-size limitations due to the diffraction limit of light and an upper limit to the modulation frequency achievable with pulsed lasers, however, restrict TDTRs ability to probe a broad range of the MFP spectrum. Furthermore, the nano-patterning approach requires an optically transparent sample, whereas the transient grating observations of non-diffusive transport are as yet limited to thin membranes19,20.

Hence, despite its intrigue, the MFP spectrum of bulk c-Si remains elusive at device operating temperatures.

In this work, we apply broadband frequency domain thermoreectance (BB-FDTR), a new technique, to measure phonon MFP spectra through variation of Lp. The use of continuous wave (CW) lasers in BB-FDTR eliminates the multiple time scales associated with a pulsed-laser technique like TDTR21,22 and, as we show in this work, enables ne resolution of an expanded range of Lp. A schematic of BB-FDTR is shown in Fig. 1a. The 488 nm CW pump laser is intensity modulated by an electro-optic modulator and heats the sample surface at a frequency f1. The 532 nm CW probe laser senses the samples periodic temperature response based on the thermoreectance of its surface. The phase-lag of temperature (probe) to heat ux (pump) is monitored by a lock-in amplier. To vary Lp and probe a wide range of MFPs, the pump beam is modulated from 200 kHz to 200 MHz. Increasing the frequency beyond 20 MHz causes signal-to-noise issues because the amplitude of the temperature response decreases, whereas coherent and ambient noise increase. To resolve this issue, we use a novel heterodyne approach (see Methods).

ResultsHeating frequency-dependent thermal conductivity. Intrinsic c-Si (o1 1012 cm 3 impurity atoms), doped c-Si

(3 1019 cm 3 boron atoms), amorphous SiO2 (1 mm lm

thermally grown on c-Si), single crystal platinum (Pt) and amorphous silicon (a-Si) (500 nm, 1 mm and 2 mm lms sputtered on c-Si) samples were sputtered with a Cr adhesion layer and a Au transducer (see Methods). Au is our ideal transducer because it has high absorption at 488 nm (pump) and a high coefcient of thermoreectance at 532 nm (probe)23. Phase-lag data from BBFDTR is t to an analytical solution of the diffusive heat conduction equation21. The unknown tting parameters are the sample thermal conductivity and the Au-Cr/sample interface thermal conductance (G).

Phase-lag versus modulation frequency data are shown in Fig. 1b for intrinsic c-Si (T 311 K) and SiO2 (T 307 K). Also

plotted are the best-t lines when the entire frequency range is t to a constant value of thermal conductivity. For SiO2, this t yields a thermal conductivity of 1.40.2 W m 1 K 1, which agrees with bulk values24 and has a low mean square error (MSE 0.04). In intrinsic c-Si, however, the constant

thermal conductivity t underpredicts the bulk value (143 W m 1 K 1 (ref. 25)) as 996 W m 1 K 1 with a large

MSE of 2.4. Consequently, we divide the frequency range into overlapping windows of 13 points, and t the thermal conductivity in each section (see Supplementary Methods and Supplementary Note 1). As shown in Fig. 1b, a frequency-dependent thermal conductivity is obtained. Window-tting the phase data for SiO2, however, yields a frequency-independent thermal conductivity. In both cases, G is assumed to be frequency independent (see Methods).

Modelling the diffusive to ballistic transition. To interpret the frequency dependence of the thermal conductivity measurements, the Boltzmann transport equation (BTE), which simultaneously considers diffusive and ballistic phonon transport, was solved using the lattice Boltzmann method (LBM) for an FDTR-like system (see Methods)11,26. A modulated heat ux is applied to the surface of a large silicon medium, and the time evolution of the temperature within the entire domain is calculated. All phonons are assumed to have the same group velocity and MFP (that is, the grey approximation). From this data, we plot the steady-state temperature oscillation amplitude (DT) as a function of sample depth for diffusive and ballistic transport regimes in Fig. 2a, DT at

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a

b

CW

Intensity

20

Time

Pump electro-optic modulator, f1

Heterodyne electro-optic modulator, f2

Phase response,

[p10] pump [p10] probe

40

60

c-Si SiO2

Constant k fit

CW

Intensity

Thermal conductivity,

k(W m1 K1 )

102

Probe, 532 nm

Pump, 488 nm

Time

101

f1

Intensity

Sample

100

Time

107 108 Heating frequency, f1 (Hz)

106

c

f1

1 0

Intensity

Low f1

Normalized temperature

High f1

Time

Penetration depth,

Photodiode

Band pass filter

f1f2, f1 + f2

Lp = k

C[afii9843]f1

Low-pass filter

To lock-in, f1f2

Intensity

Diffusive transport, (Lp>MFP)

Quasi-ballistic transport, (Lp<MFP)

Time

Figure 1 | BB-FDTR experiments reveal a heating frequency-dependent thermal conductivity. (a) Schematic of BB-FDTR technique used to measure phonon MFP spectra. (b) Phase-lag versus modulation frequency data for intrinsic c-Si and SiO2 shown with a constant thermal conductivity t over the entire experimental frequency range. Fitting SiO2 yields a thermal conductivity value of 1.40.2 Wm 1 K 1, and window-tting produces a heating frequency-independent thermal conductivity. Fitting c-Si over the entire frequency range yields a tted thermal conductivity below bulk(996 Wm 1 K 1), and window-tting shows a modulation frequency-dependent thermal conductivity. (c) Illustration of diffusive and ballistic transport.

At low heating frequencies, when the thermal penetration depth is greater than the phonon MFPs, there is diffusive thermal transport, and a bulk value of thermal conductivity is measured. At high heating frequencies, the thermal penetration depth decreases below the MFPs of some phonons, which travel ballistically through the thermally affected zone (white arrows). These ballistic phonons do not contribute to the measured value of thermal conductivity.

the surface (DTS) as a function of Lp in Fig. 2b and the perceived thermal conductivity based on DTS in Fig. 2c. The temperature amplitudes predicted by the Fourier law are plotted for comparison. At low heating frequencies, where Lp4MFP, the

BTE solution matches the Fourier prediction. As frequency increases and Lp decreases, however, the BTE-predicted DT becomes larger than that predicted by the Fourier law, indicating an onset of ballistic phonon transport. In this case, phonons can travel ballistically through the thermally affected zone without scattering, as depicted in Fig. 1c. In the context of the BB-FDTR experiments, the BTE results should be interpreted as an observation of the transition of one phonon mode from diffusive to ballistic transport as the heating frequency is increased. Consistent with the BB-FDTR experiments, the BTE predicts a reduced thermal conductivity compared with bulk as the heating frequency is increased.

Experimental phonon MFP spectra. By tting our experimental data (which includes ballistic effects) with a purely diffusive

model, we nd an effective thermal conductivity. Our interpretation is that this effective thermal conductivity is kaccum from

equation (1), where only diffusive phonons that have MFPoLp contribute (that is, we are physically imposing L* Lp). This

interpretation is the same as that of Koh and Cahill16, and consistent with Minnich et al.1 and Johnson et al.19,20 who instead used the laser spot diameter and transient grating period as the cutoff dimensions. To generate a phonon MFP spectrum, the measured thermal conductivity at the median frequency of the ith tting window (f1,i) is plotted as a function of the corresponding penetration depth, Lp;i

ki=Cpf1;i

p(see Supplementary Methods).

Phonon MFP spectra for SiO2, intrinsic c-Si, doped c-Si, a-Si (500 nm lm) and single crystal Pt near room temperature are compared in Fig. 3. The thermal conductivities are normalized by their bulk values24,25,2729. Shaded regions indicate uncertainty due to uncertainty in the thickness and thermal conductivity of the Au-Cr transducer, the laser spot-size, G, and the measured phase response (see Methods). In SiO2, Lp,i from 60900 nm yield a constant value of thermal conductivity, which suggests that any

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3

1.2

&

BTE

Fourier, T(z) = Ts,Fourier exp(z/Lp)

Temperature

oscillation amplitude, T (K)

2

1

Ballistic transport

Diffusive transport

1.0

0.8

0

0

5

10

Normalized sample depth, z/MFP

k/k bulk

0.6

oscillation amplitude, T s(K)

1

3

Ballistic Transitional

Diffusive

Ts,BTE

Ts,Fourier = q Lp /(2kbulk)

Surface temperature

2

0.4

0.2

SiO2, kbulk = 1.4 W m1 K1

a-Si (500 nm), kmax = 1.7 W m1 K1

Doped c-Si, kbulk = 120 W m1 K1

c-Si, kbulk = 143 W m1 K1

Pt, kbulk = 72 W m1 K1

1 2 3

0 0 5

4

Normalized penetration depth, Lp/MFP

0

1.2

101 100 101

kFourier /kbulk

kBTE /kbulk = ( Ts,Fourier /Ts,BTE)2

Lp, MFP (m)

1.0

Figure 3 | Material-dependent phonon MFP spectra. MFP spectra for intrinsic c-Si, doped c-Si, a-Si (500 nm lm), amorphous SiO2 and Pt near room temperature. The thermal conductivity of SiO2 is independent of Lp, suggesting that the MFPs of energy carriers are shorter than 60 nm. The

MFP spectrum of Pt is independent of Lp because of short electron MFPs (B10 nm) and strong electronphonon coupling in the metal. The MFP spectra of intrinsic c-Si and doped c-Si increase with Lp and show that micron-long MFPs contribute signicantly to bulk thermal conductivity at

T 300 K. The MFP spectrum of the 500 nm a-Si lm shows that

propagating phonons with MFPs 4100 nm contribute 357% to its thermal conductivity (for a-Si, we have used our maximum value for normalization).

0.8

k/k bulk

0.6

0.4

0.2

1 2 3 4

0 0 5

Normalized penetration depth, Lp/MFP

Figure 2 | BTE predictions for the BB-FDTR experiment. (a) Spatial variation of the temperature oscillation amplitude for diffusive transport (corresponding to the lowest heating frequency in b and c) and ballistic transport (corresponding to the highest heating frequency in b and c) from the Fourier law (dashed line) and the LBM solution to the BTE for a grey material (solid line), all for a periodic surface heat ux. (b) Amplitude of the surface temperature oscillation and (c) perceived thermal conductivity plotted versus normalized penetration depth. When Lp4MFP, the Fourier and BTE predictions match, and BB-FDTR experiments measure a bulk thermal conductivity. When LpoMFP, the Fourier law underpredicts the surface temperature oscillation amplitude, which is perceived as a reduced thermal conductivity, here and in BB-FDTR experiments.

phonon-like modes have a MFP spectrum that lies below 60 nm. The thermal conductivity of amorphous materials also has a contribution from non-propagating modes that have been called diffusons30. Because these modes do not propagate, there will be no observable transition between diffusive and ballistic transport as the BB-FDTR heating frequency is increased. This frequency-independent data is consistent with measurements of SiO2 thin lms where there was no observed thermal conductivity reduction from bulk due to boundary scattering24,31. In Pt, electrons with MFPs B10 nm (ref. 32) are the dominant heat carriers, and strong electronphonon coupling ensures that they are in thermal equilibrium with the lattice33. Thus, the thermal conductivity of Pt shows no Lp dependence.

In intrinsic c-Si, we probe Lp from 0.38.0 mm and nd that phonons with MFPs longer than 1 mm contribute 405% to the bulk thermal conductivity. We also note that 956% of the bulk thermal conductivity is obtained at the lowest heating frequency (200 kHz, LpE8 mm). This result underscores the importance of using low heating frequencies or steady-state measurements when attempting to measure bulk thermal conductivities. Relative to direct thermal conductivity measurements of c-Si thin lms12 and nanowires8, kaccum at the lm thickness or wire diameter is lower.

In these nanostructures, phonons with MFPs greater than the limiting dimension are not excluded (as they are in BB-FDTR) and contribute to thermal conductivity with a MFP similar to the limiting dimension. Compared with intrinsic c-Si, the MFP spectrum of doped c-Si has a reduced slope, indicating that dopants broaden the MFP spectrum by adding an additional phonon scattering mechanism. Phonons scattered by dopants are forced to contribute to the thermal conductivity at a shorter MFP then they would in the intrinsic crystal.

The nature of thermal transport in amorphous solids is a longstanding question in solid-state physics34. Often, as in our results for SiO2, the thermal conductivity can be described in terms of diffusons (that is, non-propagating modes). In contrast, our MFP spectrum for the 500 nm a-Si lm shows that 357% of its bulk

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1.0

0.8

G(MW m2 K 1 )

250

200

150

100 200 300 400

k/k bulk

Temperature (K)

417 K, kbulk = 100 W m1 K1

311 K, kbulk = 143 W m1 K1

155 K, kbulk = 350 W m1 K1

81 K, kbulk = 1,200 W m1 K1

Esfarjani and Chen2 PRB, 2011

Minnich et al.1, PRL, 2011

0.6

0.4

0.2

300 K

150 K

100 K 60 K

0 102

101 100 101 102

Lp, MFP (m)

Figure 4 | Temperature-dependent phonon MFP spectra for c-Si. MFP spectra for intrinsic c-Si at T 81, 155, 311 and 417 K compared with MFP

predictions from rst principles calculations. Inset shows the temperature dependence of interface thermal conductance between the Au-Cr transducer and the intrinsic c-Si sample. The spectrum of phonon MFPs relevant to thermal conductivity shifts to lower values with increasing temperature due to increased phononphonon scattering. At T 300 K, phonons with MFP41 mm contribute 405% to bulk thermal conductivity. Comparison with

predictions from Minnich et al.1 and Esfarjani and Chen2 are favourable at all temperatures.

thermal conductivity comes from propagating (that is, phonon-like) modes with MFPs longer than 100 nm. Propagating phonon-like modes in a-Si were rst inferred by a disagreement between thermal conductivity measurements and modelling predictions that was resolved by the consideration of low-frequency Debye modes35. More recent simulations and experiments have also pointed to the existence of long MFP phonon-like modes in a-Si34,3638. Using atomistic calculations, He et al.36 dene modes with MFPs o10 nm as diffusons, which contribute approximately half to the bulk thermal conductivity. Phonon-like modes with MFPs 410 nm account for the remaining half. Our range of

Lp,i (401,000 nm) in a-Si accesses these phonon-like modes. The spectrum is at below 100 nm as all of the propagating modes ballistically transmit through the domain, whereas the non-propagating modes remain diffusive. A steep transition occurs from 100500 nm before the MFP spectrum asymptotes to a constant value as Lp,i exceeds the lm thickness (500 nm). Similar features exist in thicker lms, though the transition is delayed to larger Lp, indicating that kaccum is inuenced by the a-Si lm boundaries (see Supplementary Discussion).

Phonon MFP spectra of intrinsic c-Si at temperatures of 81, 155, 311 and 417 K are shown in Fig. 4 normalized by bulk values25,29. The associated values of G, shown in the inset, are consistent with prior measurements of a Cr/Si interface39. At lower temperatures, longer MFP phonons contribute more signicantly to the measured thermal conductivity. At temperatures of 81 and 155 K, the heat capacity is at 20 and 45% of saturation, respectively40. High-frequency phonon modes are not populated, and phononphonon scattering is reduced relative to T 311 K. At temperatures of 311 and 417 K, the

population effects are less critical because the heat capacity is at 75 and 83% of saturation, respectively40. Instead, the shift in the MFP spectrum from T 311 K to T 417 K comes primarily

from increased phononphonon scattering (that is, increased anharmonicity).

DiscussionAlthough our data compare favourably with the rst principles predictions by Esfarjani and coworkers1,2,41 at all temperatures, one unresolved discrepancy is that a frequency-dependent thermal conductivity was not observed by TDTR experiments for c-Si at T 300 K (ref. 16). Plausible explanations include (i)

that a narrower frequency range (0.510 MHz compared with our range of 0.2200 MHz) was considered, (ii) that the transducer material may affect the perceived frequency dependence (TDTR uses Al, whereas BB-FDTR uses Au-Cr) and (iii) that the radii of the laser beams, known to inuence TDTR measurements of thermal conductivity at cryogenic temperatures1, was different(6.515 mm for TDTR relative to 3.4 mm for BB-FDTR).

Our discovery that micron-long MFPs contribute substantially

to k at room temperature and above suggests that silicon-based devices may have thermal management challenges in structures much larger than traditionally expected10,12. BB-FDTRs continuous resolution of the MFP spectrum indicates how the dissipative ability of c-Si will continuously decrease as transistor size decreases to uphold Moores law. Likewise, BB-FDTR can aid in the understanding of thermal transport in amorphous solids by identifying propagating phonon modes and generating thermal conductivity accumulation functions. Our general experimental approach can be applied to measure the phonon MFP spectra of other semiconducting and insulating solids. Such bulk phonon MFP spectra can be extended to predict the thermal conductivity of thin lms and nanostructures using the approaches of Yang and Dames9.

Methods

Experimental details. We use a novel heterodyne approach to reject ambient noise and coherent noise that are prohibitive to high-frequency measurements. A second electro-optic modulator induces an additional modulation on the probe beam at frequency f2 after it is reected from the sample surface (see Fig. 1a). This method heterodynes the signal held by the probe beam into signals at frequencies of f1 f2

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and f1 f2. An optical band pass lter attenuates the pump beam so that only the

probe beam reaches the photodiode. The photodiode output is passed through a low-pass electronic lter to reject the signal at f1 f2 and is then connected to an

SR-830 lock-in amplier. We concurrently sweep f1 and f2 such that f1 f2 is

maintained at 86 kHz, which was chosen to minimize 1/f noise without exceeding the low-pass cutoff of our lter. An electronic mixer is used to generate the reference for the lock-in amplier at 86 kHz, and the phase-lag of the sample surface temperature with respect to the heat ux at f1 are thus measured at 86 kHz.

Our heterodyne approach enables robust high-frequency measurements limited only by the electro-optic modulators bandwidth.

Phase-lag data are taken at 60 values of f1, evenly spaced on a log scale that spans 200 kHz to 200 MHz. For the frequency-dependent ts, the frequency range was broken into overlapping sections of 13 points, and G is determined by minimizing the MSE in the high-frequency range, where the phase response is most sensitive to its value (see Supplementary Methods). A 1/e2 radius laser spot of 3.40.1 mm, measured using a knife-edge proling technique, was used in this work (see Supplementary Methods). Uncertainty, displayed as shaded regions in Figs 3 and 4, was calculated using the procedure in Malen et al.42 by considering uncertainty in the Au-Cr transducer thickness (5%), the Au-Cr transducer thermal conductivity (5%), G (sample dependant), spot radius (3.40.1 mm) and the propagation of error in the measured phase (see Supplementary Note 2). For SiO2 and a-Si, a 5% change in lm thickness was also considered.

Two assumptions are critical to the MFP spectra generated using this approach: (i) the transducer/sample interface thermal conductance is modulation frequency independent and (ii) the volumetric heat capacity is modulation frequency independent. Regarding (i), the relaxation time of electrons inAu is 3 10 14 s at T 273 K (ref. 43). Because the time scale of our modulation

frequency is minimally ve orders of magnitude larger, the electrons and phonons will be in local equilibrium within the transducer over the entire modulation frequency range. Hence, the equilibrium of phonons and electrons at the interface is modulation frequency independent. Furthermore, this assumption is consistent with prior studies1,16. Regarding (ii), the long MFP phonons probed by BB-FDTR contribute substantially to thermal conductivity, but notto heat capacity (o1%)2.

Sample preparation. The transducer layer for all samples was prepared by depositing a B5 nm Cr adhesion layer and B55 nm of Au using a Perkin

Elmer 6J sputtering system (see Supplementary Note 3). Before the sputtering process, the native oxide layers on intrinsic c-Si and doped c-Si sampleswere removed with a buffered hydrouoric acid (HF) solution. The a-Si samples were sputtered on c-Si wafers in a low-pressure, argon-lled chamber. The c-Si wafers were rst successively cleaned ultrasonically in acetone, isopropyl alcohol and deionized water and then submerged in a buffered HF solution to remove the native oxide layer. During the sputtering process, both the target and the wafer were cooled with recycling water. Low angle X-ray diffraction shows no crystalline peaks and conrms that the a-Si lms are amorphous (see SupplementaryNote S4). Prolometry was used to measure thicknesses of the a-Si lms,and transducer layers were measured using X-ray reectivity (see Supplementary Note S3).

BTE solution methodology. We use the LBM to discretize and solve the one-dimensional phonon BTE under the relaxation time approximation. The LBM provides the time and spatial dependence of phonon occupation numbers. From these phonon occupation numbers, temperature can be calculated using Bose Einstein statistics. Details of our LBM methodology and numerical recipe are available in Nabovati et al (ref. 26).

To use the LBM to solve the BTE for a BB-FDTR-like system, we consider a c-Si lm under the grey approximation that is initially set to a uniform temperature of 300 K. The volumetric heat capacity is C 1.66 MJ m 3 K 1, the speed of sound is

vs 6,733 m s 1 and the bulk phonon MFP is LG 40 nm. Similar to the BB

FDTR experiment, a modulated heat ux of amplitude q0 and frequency f were applied at the free surface (z 0). The heat ux modulation amplitude q0 is chosen

such that the maximum surface temperature oscillation amplitude remains below3 K for all frequencies. The boundary opposite to the heated surface is maintained at T 300 K. We choose sample thicknesses such that the propagating thermal

wave decays to T 300 K well before it reaches the boundary opposite of the heated

surface. We nd that an 8 mm sample is sufciently large for the penetration depths considered. We allow the system to evolve in time and report steady-state temperature oscillation amplitudes in Fig. 2a and surface temperature oscillation amplitudes in Fig. 2b.

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Acknowledgements

J.A.M., K.T.R. and Z.S. are thankful for the support provided by the NSF CAREER Award (ENG-1149374). A.J.H.M. is thankful for the support provided by the AFOSR

(FA95501010098). D.P.S. and C.H.A. are thankful for the support provided by the NSERC. We thank Keivan Esfarjani for providing calculated phonon properties of Si at 300 K and Ankit Jain for performing calculations of heat capacity as a function of phonon MFP.

Author contributions

K.T.R., Z.S. and J.A.M. developed the BB-FDTR technique. Z.S. prepared and characterized the samples. K.T.R. took all BB-FDTR measurements. D.P.S. performed lattice Boltzmann method calculations. K.T.R., D.P.S. and J.A.M. wrote the paper. All authors discussed the data and edited the manuscript.

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How to cite this article: Regner, K.T. et al. Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreectance. Nat. Commun. 4:1640 doi:10.1038/ncomms2630 (2013).

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