PHYSICAL REVIEW B 100, 085203 (2019)Editors’ Suggestion

Generalized Fourier’s law for nondiffusive thermal transport: Theory and experiment

Chengyun Hua ,1,* Lucas Lindsay,2 Xiangwen Chen,3 and Austin J. Minnich4,†

1Environmental and Transportation Science Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

3Jet Propulsion Laboratory, Pasadena, California 911094Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125,USA

(Received 26 February 2019; revised manuscript received 24 June 2019; published 21 August 2019)

Phonon heat conduction over length scales comparable to their mean free paths is a topic of considerableinterest for basic science and thermal management technologies. However, debate exists over the appropriateconstitutive law that defines thermal conductivity in the nondiffusive regime. Here, we derive a generalizedFourier’s law that links the heat flux and temperature fields, valid from ballistic to diffusive regimes and forgeneral geometries, using the Peierls-Boltzmann transport equation within the relaxation time approximation.This generalized Fourier’s law predicts that thermal conductivity not only becomes nonlocal at length scalessmaller than phonon mean free paths but also requires the inclusion of an inhomogeneous nonlocal sourceterm that has been previously neglected. We provide evidence for the validity of this generalized Fourier’slaw through direct comparison with time-domain thermoreflectance measurements in the nondiffusive regimewithout adjustable parameters. Furthermore, we show that interpreting experimental data without the generalizedFourier’s law can lead to inaccurate measurement of thermal transport properties.

DOI: 10.1103/PhysRevB.100.085203

I. INTRODUCTION

Fourier’s law is a macroscopic constitutive law of heatconduction, and it provides the definition of thermal con-ductivity and forms the basis of many methods to determineits value. However, Fourier’s law fails when a temperaturegradient exists over a length scale comparable to or smallerthan the mean free paths (MFPs) of heat carriers. In thisregime, the heat flux and temperature fields may differ fromthe predictions of heat diffusion theory based on Fourier’s law.These discrepancies have been observed at a localized hot spotcreated by a doped resistor thermometer in a suspended siliconmembrane [1] and, more recently, in optical pump-probeexperiments, including soft x-ray diffraction from nanolinearrays [2,3], transient grating [4,5], and time-domain [6–9]and frequency-domain [10] thermoreflectance methods. Inparticular, due to the absence of scattering the transport prop-erties become nonlocal [11–13], in contrast to Fourier’s law,in which the heat flux at a certain location is determined by thetemperature gradient only at that location. A constitutive lawin the nondiffusive regime is desirable to provide a rigorousdefinition of thermal conductivity and a theoretical basis forexperimental methods to characterize the associated nonlocaleffects.

Lattice thermal transport in crystals is generally describedby the Peierls-Boltzmann equation (PBE), first derived byPeierls [14], from which the thermal conductivity is given

*Author to whom correspondence should be addressed:huac@ornl.gov

†Author to whom correspondence should be addressed: amin-nich@caltech.edu

in terms of the microscopic properties of phonons [14,15].However, solving the PBE for a general space-time-dependentproblem remains a challenging task due to the high dimen-sionality of the integro-differential equation. Thus, most priorworks have determined solutions of the PBE in certain limit-ing cases.

Guyer and Krumhansl [11] first performed a linear re-sponse analysis of the PBE, deriving a space-time-dependentthermal conductivity by assuming the normal scattering rateswere much larger than umklapp scattering rates, and theyapplied their solution to develop a phenomenological cou-pling between phonons and elastic dilatational fields causedby lattice anharmonicity. Hardy and coworkers reported arigorous quantum-mechanical formulation of the theory oflattice thermal conductivity using a perturbation method thatincluded both anharmonic forces and lattice imperfections[16–18]. This quantum treatment of lattice dynamics wasthen verified both theoretically and experimentally, demon-strating the presence of Poiseuille flow and the second soundin a phonon gas at low temperatures when umklapp pro-cesses can be neglected [19–23]. The variational principlewas also used to solve the PBE with umklapp scatteringincorporated [24,25]. Levinson developed a nonlocal dif-fusion theory of thermal conductivity from a solution ofthe PBE with three-phonon scattering in the low-frequencylimit [12].

Advances in computing power have enabled the numericalsolution of the PBE with inputs from density functionaltheory, fully ab initio. For instance, bulk lattice thermalconductivities are now routinely computed from first princi-ples using an iterative solution of the PBE [26–30] or fromvariational approaches [31]. Chaput [32] presented a directsolution to the transient linearized PBE with an imposed

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HUA, LINDSAY, CHEN, AND MINNICH PHYSICAL REVIEW B 100, 085203 (2019)

constant temperature gradient. Cepellotti and Marzari [33]introduced the concept of a “relaxon,” an eigenstate of thesymmetrized scattering operator of the PBE first used byGuyer and Krumhansl [11] and Hardy [21]. They applied thistreatment to solve steady-state problems in two-dimensionalsystems with a constant temperature gradient imposed in onedirection [34].

Solving the PBE with the full collision operator, even inits linearized form, is difficult for complicated geometries.Therefore, various theoretical frameworks based on a simpli-fied PBE have been developed to describe nonlocal thermaltransport for general problems. Nondiffusive responses ob-served in experiments [4,6,10,35–37] have been explained us-ing the phonon-hydrodynamic heat equation [38], a truncatedLevy formalism [39], a two-channel model in which low-and high-frequency phonons are described by the PBE andheat equation [40], and a McKelvey-Shockley flux method[41]. Methods based on solving the PBE under the relaxationtime approximation (RTA), where each phonon mode relaxestowards thermal equilibrium at a characteristic relaxationrate, have been developed to investigate nonlocal transportin an infinite domain [13,42–44], a finite one-dimensionalslab [45,46], and experimental configurations such as tran-sient grating [44,47] and thermoreflectance experiments [39,48–50]. An efficient Monte Carlo scheme has been used tosolve the PBE under the RTA in complicated geometriesinvolving multiple boundaries [51–53].

Although the PBE under the RTA has been widely adaptedto solve for the temperature field in complicated geometries,RTA descriptions of nondiffusive thermal transport have notbeen explicitly tested experimentally. Moreover, a formalismthat links heat flux and temperature for arbitrary geometriesin the nondiffusive regime is lacking. Therefore, a formaldefinition of nonlocal thermal conductivity is still underdebate.

Here, we derive a generalized Fourier’s law to describenondiffusive thermal transport for general geometries usingthe linearized PBE within the RTA. The generalized Fourier’slaw requires the inclusion of an inhomogeneous nonlocalterm arising from the source or the boundary conditionsof the particular problem. By including the inhomogeneouscontribution to the heat flux, the space- and time-dependentthermal conductivity is independent of the specific geometryor heat source. Evidence of the correctness of the generalizedFourier’s law is provided by favorable comparisons with a se-ries of time-domain thermoreflectance (TDTR) measurementsin the nondiffusive regime. We also show that neglectingthe inhomogeneous contribution to the heat flux leads toinaccurate measurement of thermal transport properties in thenondiffusive regime. Our work provides a formal definitionof nonlocal thermal conductivity in the nondiffusive regimeand a theoretical basis to study nondiffusive thermal transportusing existing experimental methods.

II. THEORY

A. Governing equation

We begin by briefly reviewing the derivation of the trans-port solution to the PBE. The mode-dependent PBE under the

relaxation time approximation for transport is given by

∂gμ(x, t )

∂t+ vμ · ∇gμ(x, t )

= −gμ(x, t ) − g0(T (x, t ))τμ

+ Q̇μ(x, t ), (1)

where gμ(x, t ) = h̄ωμ[ fμ(x, t ) − f0(T0)] is the deviationalenergy distribution function at position x and time t forphonon states μ [μ ≡ (k, s), where k is the wave vector ands is the phonon branch index]. f0 is the equilibrium Bose-Einstein distribution, and g0(T (x, t )) = h̄ωμ[ f0(T (x, t )) −f0(T0)] ≈ Cμ�T (x, t ), where T (x, t ) is the local tempera-ture, T0 is the global equilibrium temperature, �T (x, t ) =T (x, t ) − T0 is the local temperature deviation from theequilibrium value, and Cμ = h̄ωμ∂ f0/∂T |T0 is the mode-dependent specific heat. Here, we assume that �T (x, t ) issmall such that g0(T (x, t )) is approximated to be the first termof its Taylor expansion around T0. Finally, Q̇μ(x, t ) is the heatgeneration rate per mode, vμ = (vμx, vμy, vμz ) is the phonongroup velocity vector, and τμ is the phonon relaxation time.

To close the problem, energy conservation is used to relategμ(x, t ) to �T (x, t ) as

∂E (x, t )

∂t+ ∇ · q(x, t ) = Q̇(x, t ), (2)

where E (x, t ) = V −1 ∑μ gμ(x, t ) is the volumetric energy,

q(x, t ) = V −1 ∑μ gμ(x, t )vμ is the vector heat flux, and

Q̇(x, t ) = V −1 ∑μ Q̇μ(x, t ) is the volumetric heat generation

rate. Here, the sum over μ denotes a sum over all phononmodes in the Brillouin zone, and V is the volume of thecrystal. The solution of Eq. (1) yields a distribution function,gμ(x, t ), from which temperature and heat flux fields can beobtained using Eq. (2). Like the classical diffusion case, theexact expression of the temperature field varies from problemto problem. However, in a diffusion problem, the constitutivelaw that links the temperature and heat flux fields is governedby one expression, Fourier’s law. Here, we seek to identify asimilar relation that directly links temperature and heat fluxfields for nondiffusive transport, regardless of the specificproblem.

To obtain this relation, we begin by rearranging Eq. (1) andperforming a Fourier transform in time t on Eq. (1), whichgives

�μx∂ g̃μ

∂x+ �μy

∂ g̃μ

∂y+ �μz

∂ g̃μ

∂z+ (1 + iητμ)g̃μ

= Cμ�T̃ + Q̃μτμ, (3)

where η is the Fourier temporal frequency and �μx, �μy,and �μz are the directional mean free paths along the x, y,and z directions, respectively. Equation (3) can be solved bydefining a new set of independent variables ξ , ρ, and ζ suchthat

ξ = x, (4a)

ρ = �μy

�μ

x − �μx

�μ

y, (4b)

ζ = �μz

�μ

x − �μx

�μ

z, (4c)

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where �μ =√

�2μx + �2

μy + �2μz. The Jacobian of this trans-

formation is �2μx/�

2μ, a nonzero value if vμx �= 0. The physi-

cal picture of the proposed coordinate transformation is to finda set of characteristic curves that align with vector (vμx, 0, 0).If vμx is zero, there would be no x-direction mode-specificheat flux for that phonon mode. Then the correspondingcharacteristic curves do not exist, and it is unnecessary toapply the current analysis to that phonon mode.

After changing the coordinates from (x, y, z) to the newcoordinate system (ξ, ρ, ζ ), (vμξ = vμx, 0, 0) is the set ofelements for the velocity vector vμ in the new coordinates,and Eq. (3) becomes a first-order partial differential equationwith only one partial derivative,

�μξ

∂ g̃μ

∂ξ+ αμg̃μ = Cμ�T̃ + Q̃μτμ, (5)

where αμ = 1 + iητμ. Assuming that ξ ∈ [L1, L2], Eq. (5) hasthe following solution:

g̃+μ(ξ, ρ, ζ , η) = P+

μ e−αμ

ξ−L1�μξ +

∫ ξ

L1

Cμ�T̃ + Q̃μτμ

�μξ

× e−αμ

ξ−ξ ′�μξ dξ ′, vμξ > 0, (6a)

g̃−μ(ξ, ρ, ζ , η) = P−

μ eαμ

L2−ξ

�μξ −∫ L2

ξ

Cμ�T̃ + Q̃μτμ

�μξ

× e−αμ

ξ−ξ ′�μξ dξ ′, vμξ < 0. (6b)

P+μ and P−

μ are functions of ρ, ζ , and η and are determinedby the boundary conditions at ξ = L1 and ξ = L2, respec-tively. Using the symmetry of vμξ about the center of theBrillouin zone, i.e., vμξ = −v−μξ , Eqs. (6a) and (6b) can becombined into the following form:

g̃μ(ξ, ρ, ζ , η) = Pμe−αμ

ξ

�μξ

+∫

�

Cμ�T̃ + Q̃μτμ

|�μξ | e−αμ| ξ ′−ξ

�μξ|dξ ′, (7)

where

Pμ =⎧⎨⎩

P+μ e

αμL1

�μξ if vμξ > 0,

P−μ e

αμL2

�μξ if vμξ < 0,(8)

and

� ∈{

[L1, ξ ) if vμξ > 0,

(ξ, L2] if vμξ < 0.(9)

The mode-specific heat flux along the ξ direction is given byvμξ g̃μ, expressed as

q̃μξ = Pμvμξ e−αμ

ξ

�μξ +∫

�

Q̃μ(ξ ′, ρ, ζ , η)e−αμ| ξ−ξ ′

�μξ|dξ ′

+∫

�

Cμvμξ

|�μξ | �T̃ (ξ ′, ρ, ζ , η)e−αμ| ξ−ξ ′

�μξ|dξ ′. (10)

Applying integration by parts to the third term in Eq. (10),we can write the heat flux per mode as

q̃μξ = −∫

�

κμξ (ξ − ξ ′)∂T̃

∂ξ ′ dξ ′ + Bμ(ξ, ρ, ζ , η), (11)

where

Bμ(ξ, ρ, ζ , η)

= Pμvμξ e−αμ

ξ

�μξ + Cμ|vμξ |αμ

e−αμ

ξ

�μξ

[�T̃ e

αμξ ′

�μξ

]�

+ sgn(vμξ )∫

�

Q̃μ(ξ ′, ρ, ζ , η)e−αμ| ξ−ξ ′

�μξ|dξ ′ (12)

is solely determined by the boundary condition and the vol-umetric heat generation rate. κμξ (ξ ) is the modal thermalconductivity along the ξ direction, given by

κμξ (ξ ) = Cμvμξ�μξ

e−αμ| ξ

�μξ|

αμ|�μξ | . (13)

Equation (11) is the primary result of this work. This equa-tion links the temperature gradient to the mode-specific heatflux for a general geometry. Since this constitutive equation ofheat conduction is valid from ballistic to diffusive regimes, wedenote it as a generalized Fourier’s law. It describes that for aspecific phonon mode μ, heat flows only in the ξ direction inthe new coordinate system (ξ , ρ, ζ ) since the velocities in theρ and ζ directions are zero.

There are two parts in Eq. (11). The first part representsa convolution between the temperature gradient along the ξ

direction and a space- and time-dependent thermal conduc-tivity κμξ (ξ ). As reported previously, this convolution indi-cates the nonlocality of the thermal conductivity [13,43,46].However, a second term exists that is determined by the in-homogeneous term originating from the boundary conditionsand source terms. Similar to the first term, the contributionfrom the external heat generation to the heat flux, given

by∫�

Q̃μ(ξ ′, ρ, ζ , η)e−αμ| ξ−ξ ′

�μξ|dξ , is nonlocal, meaning the

contribution at a given point is determined by convolving theheat source function with an exponential decay function witha decay length of �μx.

While the nonlocality of thermal conductivity wasidentified by earlier works on phonon transport [6,11–13,20,39,43,54], the contribution from the inhomoge-neous term has been neglected. Recently, Allen and Pere-beinos [43] considered the effects of external heating andderived a thermal susceptibility based on the PBE that linksexternal heat generation to temperature response and a ther-mal conductivity that links temperature response to heat flux.However, their derived thermal susceptibilities and thermalconductivities depend on the specific choice of the externalheat generation. In this work, we demonstrate that there existsa general relation between heat flux and temperature distri-bution without specifying the geometry of the problem. Thespace- and time-dependent thermal conductivity in the firstterm of Eq. (11) is independent of boundary conditions andexternal heat generation. The dependence of heat flux on thespecific problem is accounted for by the inhomogeneous term.

Although the derived generalized Fourier’s law does notresolve the inherent challenges in solving the PBE, the

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simplicity of this formula allows us to explore the physicalorigins of nonlocal thermal transport. To obtain the total heatflux in the original coordinate system, e.g., qx, qy, and qz inCartesian coordinates, all the functions involved in Eq. (11)must be mapped from the coordinate system (ξ , ρ, ζ ) to (x, y,z). The mapping between the coordinate systems is problemspecific and typically requires numerical rather than analyticaltreatment. Analytical mappings exist for three special casescorresponding to three typical experimental configurationsused to study nondiffusive thermal transport. We now considerthese cases.

B. Diffusive limit

We first examine the diffusive limit. In this limit, the spatialand temporal dependence of thermal conductivity disappears,and it asymptotically approaches its bulk value. To demon-strate this limit, we first identify two key nondimensionalparameters in Eq. (11): the Knudsen number, Knμ ≡ �μξξ

−1,which compares the phonon MFP with a characteristic length,in this case ξ , and a transient number, �μ ≡ ητμ, whichcompares the phonon relaxation times with a characteristictime, in this case η−1. In the diffusive limit, both � and Knare much less than unity. Then, Eq. (7) is simplified to Cμ�T̃ ,and Eq. (11) becomes

q̃μξ = −∫

�

Cμvμξ�μξ

∂T̃

∂ξ ′ δ(ξ − ξ ′)dξ ′ = −κμξ

∂T̃

∂ξ(14)

since in this limit we can perform the following simplifica-tions:

lim�→0

αμ ≈ 1, (15)

lim�, Kn→0

e−αμ

ξ

|�μξ | ≈ 0, (16)

lim�, Kn→0

e−αμ| ξ−ξ ′

�μξ|

2|�μx| ≈ δ(ξ − ξ ′). (17)

The equation of energy conservation becomes

−∑

μ

κμξ

∂2T̃

∂ξ 2+ iη

∑μ

Cμ�T̃ =∑

μ

Q̃μ. (18)

Since ∂∂ξ

= ∂∂x + ∂

∂y�μy

�μx+ ∂

∂z�μz

�μx, Eq. (18) can be mapped

back to Cartesian coordinates as

−κx∂2T̃

∂x2− κy

∂2T̃

∂y2− κz

∂2T̃

∂z2+ iη�T̃

∑μ

Cμ =∑

μ

Q̃μ,

(19)

where κi = ∑μ Cμvμi�μi is the thermal conductivity along

axis i = x, y, or z. Here, we assume that the off-diagonalelements of the thermal conductivity tensor are zero, i.e.,κi j = ∑

μ Cμvμi�μ j = 0 when i �= j. Equation (19) is theclassical heat diffusion equation.

C. Transient grating experiment

We now check another special case of Eq. (11) by applyingit to the geometry of a one-dimensional transient grating ex-periment [4,55]. Since it is a one-dimensional (1D) problem,

ξ in Eq. (11) is equivalent to x. In this experiment, the heatgeneration rate has a spatial profile of eiβx in an infinitedomain, where β ≡ 2π/L and L is the grating period. Theboundary term vanishes, i.e., ξ ∈ (−∞,∞), and both thedistribution function and temperature field exhibit the samespatial dependence. Then, the total heat flux is expressed as

q̃x(x, η) = iβT̃ (η)eiβx∑μx>0

κμx

α2μ + �2

μxβ2

+∑μx>0

Qμ

δ

eiβxαμ�μx

α2μ + �2

μxβ2, (20)

where the total volumetric energy deposited on a sample isgiven by

∑μ Qμ and the duration of the energy deposition is

δ. A derivation of Eq. (20) is given in Appendix A.The timescale of a typical transient grating (TG) experi-

ment is on the order of a few hundred nanoseconds while re-laxation times of phonons are typically less than a nanosecondfor many semiconductors at room temperature. Therefore, weassume that � 1, and Eq. (20) is simplified to

q̃x(x, η) = iβT̃ (η)eiβx∑μx>0

κμx

1 + �2μxβ

2

+∑μx>0

Qμ

δ

eiβx�μx

1 + �2μxβ

2, (21)

which is consistent with our earlier work [40,55]. The firstpart of Eq. (21) represents the conventional understandingof nonlocal thermal transport, a Fourier-type relation with areduced thermal conductivity given by

κx =∑μx>0

κμx

1 + �2μxβ

2, (22)

while the second part of the equation represents the contribu-tion from the heat source to the total heat flux, which increasesas the Knudsen number �μx/L increases. In a TG experiment,the presence of a single spatial frequency simplifies the con-volutions in Eq. (11) into products, and the only time depen-dence of the heat flux comes from the temperature. Therefore,the decay rate of the measured transient temperature profile isdirectly proportional to the reduced thermal conductivity. Ingeneral, the spatial dependence of the temperature field is lesscomplicated in a TG experiment than in other experiments,making the separation of the intrinsic thermal conductivitycontribution from the inhomogeneous contribution easier.

D. Generalized Fourier’s law with infinite transverse geometries

The third special case considered here is when the yand z directions extend to infinity. The analytical mappingof Eq. (11) to Cartesian coordinates can be completed viaFourier transform in y and z. After Fourier transform, Eq. (11)becomes

q̃x(x, fy, fz, η) = −∫ L2

L1

κx(x − x′, fy, fz, η)∂T

∂x′ dx′

+∑

μ

B̃μ(x, fy, fz, η), (23)

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where thermal conductivity κx is given by

κx(x, fy, fz, η)

=∑

μx>0,μy,μz

κμxe− 1+i�μ+i fy�μy+i fz�μz

|�μx | x

(1 + i�μ + i fy�μy + i fz�μz )|�μx| .

(24)

fy and fz are the Fourier variables in the y and z directions,respectively, and B̃μ(x, fy, fz, η) is the Fourier transform ofBμ(x, ρ, ζ , η) with respect to fy and fz. The exact expressionof B̃μ and a derivation of Eq. (23) are given in Appendix B.In this case, both the temperature field and the inhomogeneousterm have spatial and temporal dependence. Their dependenceon the boundary conditions and heat source should be ac-counted for when extracting the intrinsic thermal conductivityfrom the observables such as total heat flux or an averagetemperature.

III. GENERALIZED FOURIER’S LAW IN TDTR

We now discuss the application of the generalized Fourier’slaw to a TDTR experiment, an optical pump-probe methodthat is able to create local heating with variable length scalesand probe the resulting thermal transport. In a TDTR exper-iment, the sample typically consists of two layers, a metaltransducer film with a finite thickness d and a semi-infinitesubstrate. The in-plane directions are regarded as infinite,and thus, Fourier transforms in the y and z directions arejustified. The cross-plane heat flux is then described byEq. (23) with x ∈ [0, d] in the transducer film and x ∈ [d,∞)in the substrate. Prescribed boundary conditions are appliedto determine the unknown boundary coefficients Pμ that arecarried over from Eq. (8). At x = 0, a specular boundary isassumed, given by

q+1μx(x = 0, fy, fz, η) = q−

1μx(x = 0, fy, fz, η), (25)

where the subscript 1 indicates the transducer layer. Thesuperscript + and − correspond to the mode-specific heat fluxcontaining P+

μ and P−μ , respectively. At x = ∞, the heat flux

disappears. Mathematically, this condition is given as

q+2μx(x = ∞, fy, fz, η) = 0, (26)

where the subscript 2 indicates the substrate layer. At theinterface, x = d , we use the elastic transmission interfacecondition given by

q+2μx = T12(μ)q+

1μx + [1 − T21(μ)]q−2μx, (27)

q−1μx = T21(μ)q−

2μx + [1 − T12(μ)]q+1μx. (28)

The spectral transmissivity from layer 1 to 2, T12(μ), isrelated to transmissivity from layer 2 to 1, T21(μ), throughthe principle of detailed balance, given by

T12(μ)C1μv1μ = T21(μ)C2μv2μ. (29)

Plugging the heat flux expression into the equation of energyconservation, the temperature field can be obtained. The de-tails of the interface treatment and procedures to obtain thetemperature response in a TDTR experiment can be found in

Ref. [9]. The surface temperature response (x = 0) in a TDTRexperiment has the following mathematical form:

Z (t, T12(μ)) =∫ ∞

−∞

∫ ∞

−∞

∑μ

Hμ0( fy, fz, T12(μ), t )

× Qμ( fy, fz )dfydfz. (30)∑

μ Hμ0(T12(μ), t ) is the surface temperature response to aDirac delta function δ(y, z).

We consider a sample consisting of an aluminum film ona silicon substrate. Phonon dispersions for Al and Si andlifetimes for Si were obtained from first principles usingdensity functional theory (DFT) [56]. We assumed a constantMFP for all modes in Al; the value �Al = 60 nm is chosento yield a lattice thermal conductivity κ ≈ 123 W m−1 K−1.We have explicitly shown that the phonon MFPs of Al havelittle effect on the final TDTR signals in Ref. [9], where moredetails on the justification of this approach can be found.

In the same work [9], we have treated the spectral transmis-sivity profile of phonons across the Al/Si interface T12(μ) asunknown fitting parameters and extracted an optimal profileby minimizing the objective function |Z (T12(μ), t ) − Z0(t )|,where the surface temperature Z0(t ) is the measurable quan-tity in time under a uniform film heating ( fy = fz = 0). Inthat work, we assumed the generalized Fourier’s law was thecorrect description of nondiffusive transport. However, evenif the constitutive relation is wrong, Z (T12(μ), t ) can still befit to Z0(t ) by changing the transmissivity profile. To test ifthe relation itself is correct or not, we can take advantageof the decoupling of the in-plane and cross-plane directionsvia the interface and incorporate the y and z dependency intothe problem.

To include the in-plane spatial dependency, we compare thecalculated TDTR responses using the generalized Fourier’slaw with pump-size-dependent TDTR measurements on thesame Al/Si sample measured in Ref. [9], where the 1/e2

diameter D of a Gaussian pump beam was varied between5 and 60 μm at different temperatures. As the pump diameterdecreases and becomes comparable to the thermal penetra-tion depth along the cross-plane direction [the x axis of theschematic in Fig. 1(a)], in-plane thermal transport is no longernegligible and requires a three-dimensional transport descrip-tion. P+

μ ( fy, fz, η) is determined from the interface conditionsgiven by Eqs. (27) and (28), and for a given fy, fz, and η it isdetermined by the spectral transmissivity of phonons as givenin Ref. [9].

IV. RESULTS

We now provide experimental evidence of the generalizedFourier’s law by comparing the predicted and measured sur-face thermal responses to incident heat generation in TDTRexperiments. In the diffusive regime, energy conservation,Fourier’s law, and the boundary conditions can fully de-scribe a transport problem. In the nondiffusive regime, thereplacement for Fourier’s law is the generalized Fourier’slaw described in this work. If the generalized Fourier’s lawis the correct description of nondiffusive transport, it willaccurately predict the surface temperature response for thevarious parameters in TDTR experiments such as heating

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FIG. 1. Experimental (a) amplitude and (b) phase versus time (symbols) from TDTR for an Al/Si sample with a pump beam size of15 μm at T = 300 K for modulation frequencies of 0.97 and 4.77 MHz. Predictions from the generalized Fourier’s law (shaded regions) andoriginal Fourier’s law (curves) are also shown. The shaded regions around the PBE simulations correspond to the likelihood of the measuredtransmissivity possessing a particular value (darker area corresponds to more probable). Using the measured transmissivity profile from uniformheating [9], the prediction from the generalized Fourier’s law agrees well with the TDTR measurements, while the Fourier results overestimatethe phase and underestimate the amplitude at later times. The inset in (a) shows the schematic of the sample subject to a Gaussian beamheating. The inset in (b) shows the measured transmissivity of longitudinal phonons TSi→Al(ω) that is obtained with 1D transport in Ref. [9].

geometry, modulation frequency, and ambient temperature ofthe sample.

We compared the measured signals directly to predictionsfrom the nonlocal transport governed by the generalizedFourier’s law and the strictly diffusive transport governed byFourier’s law. To ensure a consistent comparison between theconstitutive relations, the thermal conductivity of silicon isobtained using the same DFT calculations, and the interfaceconductance G is given by [57]

1

G= 4∑

μ CSiμ vSi

μ TSi→Al− 2∑

μ CAlμ vAl

μ

− 2∑μ CSi

μ vSiμ

, (31)

where TSi→Al is the mode-dependent phonon transmissivityfrom Si to Al. This expression was first derived by Chen andZeng, who considered the nonequilibrium nature of phonontransport at the interface within the phonon transmissivity[57].

Figures 1(a) and 1(b) show the measured TDTR amplitudeand phase versus delay time for a pump size of 15 μm atroom temperature along with predictions from the generalizedFourier’s law and original Fourier’s law. As in Ref. [9], theintensity of the shaded regions corresponds to the likelihoodof the measured transmissivity profile plotted in the inset ofFig. 1(b). A higher likelihood of a transmissivity profile isindicated by a higher intensity of the shaded area, and thus,the PBE simulation using a transmissivity profile with higherlikelihood better fits the experimentally measured TDTRsignals. Excellent agreement between predictions from thegeneralized Fourier’s law and experiments is observed. Onthe other hand, Fourier’s law fails to accurately account forthe experimental data, overestimating the phase and underes-timating the amplitude after 2 ns in delay time. Note that thedifferent transmissivity profiles in the inset of Fig. 1(b) give a

value of interface conductance G = 223 ± 10 MW m−2 K−1

using Eq. (31), and this deviation in G leads to uncertaintiesin the TDTR signals that are within the linewidth of the plottedcurves.

Using the framework of the generalized Fourier’s law, weare able to identify the source of the discrepancy betweenFourier’s law and the generalized Fourier’s law. For thecase of Al/Si, the discrepancy originates from the interfaceconditions. The low-frequency phonons (less than 6 MHz)have transmissivity close to unity, while the high-frequencyphonons are mostly reflectd by the interface. Examining theform of Eq. (23), low-frequency phonons in the first term(nonlocal thermal conductivity) are suppressed due to nondif-fusive transport; however, they carry most of the heat acrossthe interface, which increases their contribution to the secondterm in Eq. (23). These two factors cancel each other. Thus,the TDTR experiments routinely give a value close to the bulkSi thermal conductivity. This cancellation is not guaranteed tooccur for all systems. We first highlighted this finding qualita-tively in Ref. [9]. However, the complicated numerics requiredto solve for the temperature field in that work masked the un-derlying physical effects that occur in nondiffusive transport.The framework of the generalized Fourier’s law reveals howthe boundary condition affects the heat flux without solvingfor the temperature field and provides a straightforward wayto examine if a Fourier-type relation with a reduced thermalconductivity is valid in a particular experiment setting.

In Figs. 2(a) and 2(b), comparisons of phases at differentpump diameters between the generalized Fourier’s law, orig-inal Fourier’s law, and experimental data are given at 300 K.In Figs. 2(c) and 2(d), we compare the measured phase versusmodulation frequency at a fixed pump diameter to predictionsfrom the generalized Fourier’s law and original Fourier’s law

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GENERALIZED FOURIER’S LAW FOR NONDIFFUSIVE … PHYSICAL REVIEW B 100, 085203 (2019)

FIG. 2. Measured (symbols) and predicted (shaded regions) phases versus pump size at room temperature and a fixed delay time of (a) 1.5and (b) 6 ns for modulation frequencies of 0.97 and 4.77 MHz. At temperatures of 100, 150, 200, and 250 K, measured and predicted phases areplotted versus modulation frequency at a fixed delay time of (c) 1.5 and (d) 6 ns for a pump diameter of 15 μm. The lines show the predictionby Fourier’s law with temperature-dependent thermal conductivities from DFT and the interface conductance given by Eq. (31).

at 100, 150, 200, and 250 K. The data are given for two differ-ent delay times, 1.5 and 6 ns. The seeming discontinuities andabrupt changes in curvature in Figs. 2(a) and 2(b) are artifactsof interpolations between a finite number of calculated pointsin the curve. Figure 2 shows that predictions from the originalFourier’s law do not reproduce the experimental results. Thedeviation of Fourier’s law predictions from the experimen-tal results becomes larger when the temperature decreases,modulation frequency increases, or pump diameter decreases,all indicating that nondiffusive effects increase with thesechanges. On the other hand, predictions from the generalizedFourier’s law agree well with the experimental measurementsfor the various temperatures, modulation frequencies, andpump sizes, providing evidence for its validity to describe thenonlocal thermal transport in different regimes.

V. DISCUSSION

The above comparisons between the simulations and ex-periments with different heating geometries and at differenttemperatures provide evidence that the generalized Fourier’slaw is an appropriate replacement for Fourier’s law in thenondiffusive regime. We now use this formalism to examineTDTR measurements on boron arsenide (BAs).

Boron arsenide has recently attracted attention because ofits ultrahigh thermal conductivity determined from measure-ments including the TDTR technique and reported by severalgroups [58–60]. Moreover, pump-size-dependent measure-ments have also been conducted in an attempt to accessinformation of the phonon MFPs in BAs [59]. The thermalconductivity measurements are based on interpreting the rawTDTR data as fit to a diffusion model based on Fourier’s lawwith thermal conductivity of BAs and interface conductancebetween Al and BAs as two fitting parameters.

However, due to the presence of phonons with long MFPscompared to the TDTR thermal length scale, Fourier’s lawis no longer valid at the length scales probed by TDTR. Aspredicted by DFT-based PBE calculations [61,62], more than70% of phonons in single-crystalline BAs have mean freepaths longer than 1 μm. Therefore, properly interpreting thedata requires using the generalized Fourier’s law.

Equation (11) needs close examination to understand themicroscopic information contained in the surface temperatureresponses measured in experiments. In a two-layer structurelike the one used in TDTR, the second term in Eq. (11)does not vanish and has a nonlocal nature through the sourceterm. This nonlocality implies that even though only the tran-sient temperature at the metal surface is observed, the mea-surement contains information from the entire domain. We

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HUA, LINDSAY, CHEN, AND MINNICH PHYSICAL REVIEW B 100, 085203 (2019)

5 10 20 30 40 50 60( m)

0.7

0.8

0.9

1.0

1.1

1.2

Nor

mal

ized

ther

mal

con

duct

ivity

0.51 MHz5.52 MHz10.1 MHz

Profile 1

Profile 2

0 2 4 6 8 10Phonon frequency (THz)

0.1

0.3

0.5

0.7

Tran

smis

sivi

ty

LA

TA

TA

0 2 4 6 8 10Phonon frequency (THz)

0.05

0.15

0.25

0.35

Tra

nsm

issi

vity

TA

LA

TA

5 10 20 30 40 50 60( m)

0.7

0.8

0.9

.0

0.51 MHz5.52 MHz10.1 MHz

Profile 1:

G = 115 MWm-2K-1

Profile 2:

G = 253 MWm-2K-1

(a)

(b)

(c)

Pump diameter

FIG. 3. Spectral transmissivity profiles from BAs to Al versus phonon frequency that give an interface conductance of (a) 115 and(b) 253 MW m−2 K−1 using Eq. (31). The profiles are used to generate the synthetic TDTR data. (c) Normalized thermal conductivity versuspump diameter, obtained by fitting the synthetic TDTR data at modulation frequencies of 0.51 MHz (solid curves), 5.52 MHz (dashed curves),and 10.1 MHz (dotted curves), using the transmissivity profiles in (a) (solid symbols) and (b) (open symbols) to a diffusion model based onFourier’s law. Both three-phonon and four-phonon scatterings are included in the DFT calculations of single-crystalline BAs. The calculatedbulk thermal conductivity of BAs is 1412 W m−1 K−1 [61,62].

demonstrated in Ref. [63] that the spectral distribution of thesource term alters the surface temperature response. In otherwords, even though the first term in Eq. (11) remains the same,the observable at the surface is altered by the inhomogeneoussource term originating from the interface.

To illustrate this point, we choose two transmissivity pro-files TBAs→Al, as shown in Figs. 3(a) and 3(b). The profilesshare a similar dependence on phonon frequency but withdifferent magnitudes. The nominal interface conductance iscalculated to be 115 and 253 MW m−2 K−1, respectively,using Eq. (31). Along with the ab initio properties of BAsat room temperature, we calculate the TDTR responses at theAl surface with different pump diameters at three modulationfrequencies. The calculated TDTR responses are then fit tothe typical diffusion model based on Fourier’s law to extractthe effective thermal conductivity, as was performed in Refs.[58–60].

The results are shown in Fig. 3(c). The key observation isthat the effects of modulation frequency and pump size on theeffective thermal conductivity depend on the transmissivityprofiles. A decrease in the effective thermal conductivity isobserved in both profiles as the pump diameter decreases orthe modulation frequency increases. However, the magnitudeof the reduction and the absolute value compared to thebulk value depend on the transmissivity. While the effectivethermal conductivity seems to be approaching the bulk valueat a large pump diameter and low modulation frequencyusing profile 1, the effective thermal conductivity using profile2 exceeds the bulk value under the same conditions. Thereduction of the effective thermal conductivity using profile1 as pump diameter decreases is less than 5% at a givenmodulation frequency, while the reduction using profile 2 canbe as much as 40%.

Our calculations therefore indicate that in the nondiffusiveregime, simply interpreting measurements from a method

such as TDTR using Fourier’s law with a modified thermalconductivity may yield incorrect measurements. Not only isFourier’s law unable to describe the nonlocal nature of thermalconductivity, but it also does not include the effects of inho-mogeneous terms. Therefore, when interpreting TDTR mea-surements of high-thermal-conductivity materials, Fourier’slaw is not the appropriate constitutive relation. Although thisinsight was previously discussed, under the framework of thegeneralized Fourier’s law we have a simple equation showinghow microscopic interface conditions enter the constitutiverelation and thus clearly demonstrates that extra caution isnecessary when interpreting TDTR measurements of high-thermal-conductivity materials.

VI. CONCLUSIONS

In summary, we derived a generalized Fourier’s law usingthe Peierls-Boltzmann equation under the relaxation timeapproximation. The new constitutive relation consists of twoparts, a convolution between the temperature gradient anda space- and time-dependent thermal conductivity, and aninhomogeneous term determined from boundary conditionsand heat sources. By comparing predictions from this newconstitution law to a series of time-domain thermoreflectancemeasurements in the nondiffusive regime, we provide experi-mental evidence that the generalized Fourier’s law accuratelydescribes thermal transport in a range of transport regimes. Wealso show that interpreting nonlocal thermal transport usingFourier’s law can lead to erroneous interpretation of measuredobservables. To correctly extract microscopic phonon infor-mation from the observation of nonlocal thermal transport, itis necessary to separate the inhomogeneous contribution fromthe nonlocal thermal conductivity based on the generalizedFourier’s law.

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GENERALIZED FOURIER’S LAW FOR NONDIFFUSIVE … PHYSICAL REVIEW B 100, 085203 (2019)

ACKNOWLEDGMENTS

C.H. and L.L. acknowledge support from the Labora-tory Directed Research and Development Program of OakRidge National Laboratory, managed by UT-Battelle, LLC,for the US Department of Energy. A.J.M. acknowledges sup-port from the National Science Foundation under Grant No.CBET CAREER 1254213. This research used resources ofthe National Energy Research Scientific Computing Center(NERSC), a US Department of Energy Office of Science UserFacility operated under Contract No. DE-AC02-05CH11231.

APPENDIX A: DERIVATION OF EQUATION (20)

In a one-dimensional (1D) problem, Eq. (11) becomes

q̃μx = −∫

�

Cμvμx�μxe−αμ| x−x′

�μx|

αμ|�μx|∂T̃

∂x′ dx′

+∫

�

Q̃μ(x′)e−αμ| x−x′�μx

|dx′, (A1)

where

� ∈{

[−∞, ξ ) if vμξ > 0,

(ξ,∞] if vμξ < 0.

In a 1D transient grating experiment, both the temperatureprofile and mode-specific heat generation rate have a spatialdependence of eiβx, i.e., T̃ (x, η) = eiβxT̃ (η) and Q̃μ(x) =Qμδ−1eiβx. Summing Eq. (A1) over all phonon modes andusing the symmetry of vμx about the center of the Brillouinzone, i.e., vμx = −v−μx, the total heat flux is expressed as

q̃x = −iβ�T̃ (η)∑μx>0

∫ ∞

−∞Cμvμx�μx

eiβx′e−αμ| x−x′

�μx|

αμ|�μx| dx′

+∑μx>0

∫ ∞

−∞

Qμ

δeiβx′

e−αμ| x−x′�μx

|dx′. (A2)

Now we define y = x′ − x. Then the above equation becomes

q̃x = −iβ�T̃ (η)eiβx∑μx>0

∫ ∞

−∞

∑μx>0

Cμvμx�μxeiβye−αμ| y

�μx|

αμ|�μx| dy

+ eiβx∑μx>0

∫ ∞

−∞

Qμ

δeiβye−αμ| y

�μx|dy

= −iβT̃ (η)eiβx∑μx>0

κμx

α2μ + �2

μxβ2

+ eiβx∑μx>0

Qμ

δ

αμ�μx

α2μ + �2

μxβ2. (A3)

APPENDIX B: DERIVATION OF EQUATION (23)

When the y and z directions can be regarded as infinite,the analytical mapping to the Cartesian coordinates can becompleted via Fourier transform in y and z. To show it, we firstdefine g(x, y, z) = f (ξ, ρ, ζ ) with the coordinate transformgiven by Eq. (4). G and F are the functions after Fouriertransform in y and z. Using the affine theorem of the two-dimensional Fourier transform, we obtain

G(x, fy, fz )

= e−i( fy�μy�μx

+ fz�μy�μx

)xF

(x,− fy

�μ

�μx,− fz

�μ

�μx

)�2

μ

�2μx

, (B1)

where fy and fz are the Fourier variables in the y and zdirections, respectively.

Applying Eq. (B1) to Eq. (10) gives

q̃μξ = P∗μ( fy, fz, η)vμxe− 1+iγμ

�μxx

+ sgn(vμx )∫

�

Q̃μ(x, fy, fz, η)e− 1+iγμ

|�μx | |x−x′|dx′

+∫

�

Cμvμx

|�μx| �T̃ (x′, fy, fz, η)e− 1+iγμ

|�μx | |x−x′|dx′, (B2)

where γμ = ητμ + fy�μy + fz�z and P∗μ( fy, fz, η) is deter-

mined as∫∫P(ρ, ζ , η)ei fyy+i fzzdydz

= Pμ

(− fy

�μ

�μx,− fz

�μ

�μx, η

)�2

μ

�2μx

e−i( fy�μy�μx

+ fz�μz�μx

)x

= P∗μ( fy, fz, η)e−i( fy

�μy�μx

+ fz�μz�μx

)x. (B3)

Applying integration by parts to the third term in Eq. (B2)and summing over phonon modes gives

q̃x(x, fy, fz, η) = −∫ L2

L1

κx(x − x′, fy, fz, η)∂T

∂x′ dx′

+ B̃(x, fy, fz, η), (B4)

where thermal conductivity κx is given by

κx(x, fy, fz, η) =∑

μx>0,μy,μz

κμxe− 1+iγμ

|�μx | x

(1 + iγμ)|�μx| (B5)

and

B̃(x, fy, fz, η) =∑

μ

P∗μ( fy, fz, η)vμxe− 1+iγμ

�μxx +

∑μx>0,μy,μz

Cμ|vμx|1 + iγμ

[�T (L2)e− 1+iγμ

�μx(L2−x) − �T (L1)e− 1+iγμ

�μx(x−L1 )]

+∑

μx>0,μy,μz

|�μx|1 + iγμ

[Qμ(L2)e− 1+iγμ

�μx(L2−x) − Qμ(L1)e− 1+iγμ

�μx(x−L1 )]

−∑

μx>0,μy,μz

|�μx|1 + iγμ

∫ L2

L1

∂Qμ

∂x′ e− 1+iγμ

|�μx | |x′−x|dx′. (B6)

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HUA, LINDSAY, CHEN, AND MINNICH PHYSICAL REVIEW B 100, 085203 (2019)

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