The central wavelength is an average of all the wave-
lengths weighted against the phonon energy density,
and is representative of the wavelengths of most pop-
ulated phonons (Chen, 1997). The coherence length
is inversely proportional to the effective bandwidth of
phonons. It is an indicator whether the wave nature is
ture is much larger than the coherence length, phonons
can be treated as particles.
In Figure 1, the estimated coherence length, the cen-
tral wavelength, and the mean free path of phonons
in bulk GaAs as a function of temperature are shown.
As the characteristic lengths of nanostructures, i.e., the
diameter of nanoparticles and nanowires, and the thick-
ness of thin ﬁlms, become comparable to the mean
free path or even the coherence length, the diffusion
approximation underlying the Fourier law is no longer
valid. Size and interface effects must be taken into
consideration. Depending on the relative magnitude
of the characteristic length of the nanostructures and
the characteristic lengths of the energy carriers, one
may treat the energy carriers as particles or as waves.
The latter requires consideration of the phase informa-
tion of the energy carriers. Different transport regimes
and the governing principles for photons, electrons,
and phonons are summarized in Table 1. The basic
Figure 1. Characteristic length scales of phonons in bulk gallium arsenide.
principles governing the transport of energy carriers are
very similar and analogy can often be made to under-
stand the particularities of phonon transport by exam-
ining the photon and electron transport.
Applicability of the Fourier law
In general, the Fourier theory applies only to the diffu-
sion regime as indicated in Table 1. The failure of the
Fourier law in nanostructures may occur to all variables
in the constitutive relation, namely the temperature and
its gradient, and the thermal conductivity.
First, temperature is an equilibrium concept.
Although heat transfer is intrinsically a nonequilib-
rium process, the deviation from equilibrium is usually
small and a local thermal equilibrium is assumed. The
establishment of equilibrium requires enough scatter-
ing among phonons to thermalize their energy. When
the structure is small compared to the mean free path,
the equilibrium cannot be established and temperature
cannot be deﬁned in the conventional sense. A simple
example is illustrated in Figure 2(a), which shows a
thin ﬁlm sandwiched between two thermal reservoirs.
Consider now the limit that no internal scattering exists
inside a thin ﬁlm. There will be two groups of phonons
Table 1. Transport regimes for three common energy carriers: Device characteristic length (such as thickness) is h; O
denotes the order-of-magnitude of a length scale; the listed MFP and coherence lengths are typical values but these values
are strongly material and temperature dependent
Length scale Regimes Photon Electron Phonon
Coherence length, L
photon: 1 µm–1 km
electron: 1–100 nm
phonon: ∼ 1–10 nm
h < O(L) wave regime Maxwell EM theory Quantum mechanics Quantum mechanics
h ∼ O(L) Optical coherence Electron coherence Phonon coherence
partial coherence regime theory theory theory
Mean free path,
photon: 10 nm–1 km
electron: ∼100 nm
phonon: ∼100 nm
h < O( ) ballistic regime Ray tracing Ballistic transport Ray tracing
h ∼ O( ) Radiative transfer Boltzmann transport Boltzmann transport
quasi-diffusive regime equation equation equation
h > O( ) Diffusion Ohm’s and Fourier’s law
diffusion regime approximation Fourier’s laws
Figure 2. Illustration of nonequilibrium nature of ballistic
phonon transport across a thin ﬁlm in the absence of internal scat-
tering, (a) counter-propagating phonons do not interact with each
other due to the absence of scattering and (b) equivalent local tem-
perature if those phonons interact and were thermalized locally
(temperature independent speciﬁc heat assumed).
propagating inside the ﬁlm: one is at temperatureT1 and
the other at temperature T2. The local energy density
spectrum deviates signiﬁcantly from the equilibrium
distribution. If these phonons were to interact and
become thermalized, an equivalent local temperature
can be deﬁned. The temperature distribution accord-
ing to such a deﬁnition inside the ﬁlm is shown in
Figure 2(b). At the boundaries, temperature jump hap-
pens due to the nonequilibrium nature of phonons and
the artiﬁcial deﬁnition of local temperature.
Clearly, in the above example, the concept of temper-
ature gradient is useless. The proportionality between
the local heat ﬂux and local temperature gradient as
seen in the Fourier law is indicative of diffusion pro-
cesses, while the above-example is a purely ballistic
transport process. In such a process, the origin of
phonons, which determines their spectral energy den-
sity, is most important, while in the diffusive trans-
port regime, the origin of the heat carrier is of no
signiﬁcance. For the intermediate cases between the
totally ballistic transport and totally diffusive transport,
the heat ﬂux depends on the trajectory of phonons.
Mathematically, it is a path integral, as dictated by the
Boltzmann transport equation.
The failure of the Fourier law is reﬂected in the ther-
mal conductivity of nanostructures. Strictly speaking,
unless a local temperature gradient can be established
such that the Fourier law is applicable, the concept of
thermal conductivity is meaningless. In reality, how-
ever, it is still a convenient parameter to measure and
has been used to gauge the size effects in nanostruc-
tures. The thermal conductivity is no longer an intrinsic
material property but a structure property, and it may
depend on how the heat source is applied. Generally,
because the interface imposes additional resistance to
the heat ﬂow compared to bulk materials, the effective
thermal conductivity of nanostructures is smaller than
those of their corresponding bulk materials.
Clearly, the Fourier theory is not valid if one is
interested in the local temperature and their gradients.
It may, however, be applied to certain heat conduc-
tion conﬁgurations in nanostructured materials with a
modiﬁed structural-dependent thermal conductivity, as
long as the domain of interests is much larger than
the phonon mean free path inside the nanostructure.
Examples are heat conduction along long nanowires or
thin ﬁlm planes, in which local temperature gradient
can be established in the heat ﬂux direction, and heat
conduction in macroscopic structures made of nano-
materials such as nanoparticles.
Heat transfer in nanostructures
Studies under the name ‘microscale heat transfer’
have surged in the heat transfer community over the
last decade (Tien, 1997), although the length scale at
which deviation from the classical Fourier law in solids
typically occurs at nanoscale. Some of the research is
Heat transfer in thin ﬁlms has received most
attention. Experimental and theoretical studies have
been carried out to investigate thermal conductivity of
various thin ﬁlms used in microelectronics, photonics,
and thermoelectrics (Goodson & Ju, 1999; Chen,
2000). In amorphous materials with short phonon mean
free path, the size effect is typically less important
than that of those caused by microstructural variations.
The most dramatic size effects are observed in differ-
ent crystalline superlattices with nanometer periodicity
(Yao, 1987), which show orders of magnitude reduc-
tion in thermal conductivity compared to the prediction
of the Fourier diffusion theory.
Studies of thermal transport in nanowires are few at
this stage but will become increasingly important as
new technologies such as carbon nanotubes are being
implemented. A few experimental studies have been
reported on the electron–phonon interaction in and the
thermal conductance of nanowires at low temperatures
(Potts et al., 1991; Seyler & Wybourne, 1992; Tighe
et al., 1997), including the observation of quantized
phonon conductance (Roukes, 1999). Thermal con-
ductivity of carbon nanotube bundles have also been
reported (Hone et al., 1999; Yi et al., 1999). Boltzmann
transport equation and molecular dynamics simulation
of thermal conductivity of nanowires was carried out,
but the modeling results await experimental conﬁrma-
tion (Walkauskas et al., 1999; Volz & Chen, 1999).
Heat conduction away from nanoparticles and
between interconnected nanoparticles is representative
of many applications such as nanoparticles or nano-
devices embedded in a host medium, thermal insula-
tion, and opal structures for light manipulation. The
thermal pathway from a heated nanoparticle to the
ambient depends on the coupling of the nanoparticle
phonon spectrum with the surrounding medium, and
depends on the ratio of the nanoparticle diameter to
the phonon mean free path in the surrounding medium.
The former determines the interface thermal resistance
and the later resembles the rareﬁed gas conduction
external to an object. Due to the phonon rarefaction,
the effective thermal conductivity that the nanoparticle
‘feels’ about the surrounding medium is signiﬁcantly
reduced (Chen, 1996), as shown in Figure 3. Intercon-
nected nanoparticles may form regular porous struc-
tures such as opals and random porous structures such
as nanoporous Si. There have been several studies of
heat transfer in these nanostructures (Arutyunyan et al.,
1997; Gesele et al., 1997; Chung & Kaviany, 2000).
Since energy dissipation is a fundamental process, the
particularities of heat transfer in nanostructures have
important implications to the rapidly developing nan-
otechnology. Size effects on phonons often lead to
reduced thermal pathways from the hot region to the
cold region. This can be detrimental to microelectronic
and photonic devices and the lessons learnt from the
theoretical and experimental studies could be used to
optimize the design of those devices. On the other
hand, many other applications require low thermal
conductivity materials, such as in thermoelectrics,
thermal protective coatings, and microelectromechani-
cal systems. Various nanofabrication and characteriza-
tion techniques, data storage devices, also call for the
understanding of nanoscale heat transfer phenomena
to control the fabrication, writing, and interpretation of
Taking thermoelectrics as an example, it is well
known that the Peltier effect and the Seebeck effect of
electrons and holes can be used for refrigeration and
power generation due to the energy associated with
the motion of these charged carriers (Goldsmid, 1964).
The solid-state Peltier coolers are commercially avail-
able but their applications have been limited by their
low efﬁciency. A good thermoelectric material requires
several attributes such as a large Seebeck coefﬁcient
for maximum cooling capability, a high electrical con-
ductivity to reduce the Joule heating, and a low thermal
conductivity to minimize thermal leakage from the hot
to the cold side. Satisfying those requirements have
been proven to be a difﬁcult task (DiSalvo, 1999) in
bulk materials. Nanostructures offer new pathways by
manipulating electron and phonon transport through
size effects (Dresselhaus et al., 1999; Chen, 2000).
Figure 3. Effective thermal conductivity as a function of the particle size parameter (particle radius normalized by the phonon mean free
path in the surrounding medium) for heat conduction in the vicinity of a nanoparticle embedded in a host material. As the particle size
becomes smaller than the phonon mean free path in the host material, the effective thermal conductivity that the particle feels about the
surrounding is reduced due to rareﬁed gas effect (after Chen, 1996).
Another interesting possibility is to utilize size
effects between nanoscale heterogeneous materials
for the design of low thermal conductivity protective
coatings. It has been shown that phonon reﬂection at
interfaces can drastically reduce thermal conductivity
of superlattices and we can infer from those studies
that coatings made of heterogeneous nanoparticles may
have superior thermal insulating characteristics and
thermal stability, compared to single phase nanoparti-
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