Experimental Evidence of Non-Diffusive Thermal
Transport in Si and GaAs
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Citation
Johnson, Jeremy A., Alexei A. Maznev, Jeffrey K. Eliason, Austin
Minnich, Kimberlee Collins, Gang Chen, John Cuffe, Timothy
Kehoe, Clivia M. Sotomayor Torres, and Keith A. Nelson.
“Experimental Evidence of Non-Diffusive Thermal Transport in Si
and GaAs.” MRS Proceedings 1347 (January 23, 2011).© 2011
Materials Research Society.
As Published
http://dx.doi.org/10.1557/opl.2011.1333
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Cambridge University Press/Materials Research Society
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Final published version
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Thu May 26 09:05:39 EDT 2016
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http://hdl.handle.net/1721.1/82598
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Mater. Res. Soc. Symp. Proc. Vol. 1347 © 2011 Materials Research Society
DOI: 10.1557/opl.2011.1333
Experimental Evidence of Non-Diffusive Thermal Transport in Si and GaAs
Jeremy A. Johnson1, Alexei A. Maznev1, Jeffrey K. Eliason1, Austin Minnich2, Kimberlee
Collins2, Gang Chen2, John Cuffe3,4, Timothy Kehoe3, Clivia M. Sotomayor Torres3,5,6,
Keith A. Nelson1
1
Dept. of Chemistry, MIT, 77 Massachusetts Ave, Cambridge, MA 02139, U.S.A.
2
Dept. of Mechanical Engineering, MIT, 77 Massachusetts Ave, Cambridge, MA 02139, U.S.A.
3
Catalan Institute of Nanotechnology, Campus de Bellaterra, Edifici CM7, ES 08192, Barcelona,
Spain.
4
Dept. of Physics, Tyndall National Institute, University College Cork, College Road, Ireland.
5
Catalan Institute for Research and Advanced Studies ICREA, 08010, Barcelona, Spain
6
Dept. of Physics, Universitat Autonoma de Barcelona, 08193 Bellaterra (Barcelona), Spain.
ABSTRACT
The length-scales at which thermal transport crosses from the diffusive to ballistic regime are
of much interest particularly in the design and improvement of nano-structured materials. In this
work, we demonstrate that the departure from diffusive transport has been observed in Si and
GaAs using an optical transient thermal grating technique where an arbitrary, experimentally set
length scale can be imposed on a material. In a transient thermal grating experiment, crossed
laser pulses interfere creating a well-defined periodic absorption and temperature profile. A
probe beam is diffracted from this transient grating and length-scale dependent thermal transport
properties can be determined from the signal decay. As the length scale is decreased to lengths
shorter than the mean free paths of heat carrying phonons, quasi-ballistic heat transport effects
become apparent allowing us to map out length scales and mean free paths relevant to nondiffusive thermal transport in Si and GaAs.
INTRODUCTION
In dielectrics and typical semiconductors relevant for thermoelectric or nano-electronic
applications, lattice excitations (phonons) are responsible for most of the thermal transport [1].
In a simplified view, thermal conductivity is directly related to the frequency dependent phonon
mean free path (MFP) and group velocity of the entire thermal distribution of phonons [2]. A
textbook estimate of the average MFP in semiconductors based on simple kinetic theory [3]
yields 1-100 nm; thus thermal transport over longer length scales is thought to be well described
by the classical thermal diffusion model and ballistic effects may only be observed over
exceedingly short length scales.
In reality, the MFP of phonons contributing significantly to thermal conductivity varies
strongly with phonon frequency and extends over several orders of magnitude. For any length
scale there will be low frequency phonons that propagate ballistically, so the important question
is the contribution of low frequency phonons to heat transport. New theoretical studies based on
molecular dynamics simulations and ab initio calculations have recently emerged for silicon
indicating that phonons with MFP exceeding 1 µm contribute 40-50% of the thermal
conductivity at room temperature [4,5]. Although quantitative discrepancies between the
different models still persist, the role of lower frequency phonons in thermal transport is evident.
On the experimental side, deviations from diffusive transport at room temperature have been
observed in sapphire and GaAs at length scales ~100 nm [6,7]. Measuring non-diffusive thermal
transport at small distances in a configuration that can be rigorously compared to theoretical
models has been a challenge for experimentalists. In order to be persuasive, an experiment
should, preferably, (i) avoid interfaces, (ii) ensure one dimensional thermal transport, (iii) clearly
define the distance of the heat transfer and provide a way to vary this distance in a controllable
manner.
A method satisfying the above requirements has in fact been well known under the name
laser-induced transient thermal gratings [8,9]. In this method, two short laser pulses are crossed
in a sample resulting in an interference pattern with period L defined by the angle between the
beams. Absorption of laser light leads to a spatially periodic temperature profile, and the decay
of this temperature grating via thermal diffusion is monitored via diffraction of a probe laser
beam.
In this paper, we present transient thermal grating measurements of in-plane heat transport in
a free standing silicon (Si) membrane and bulk gallium arsenide (GaAs). By varying the grating
period we are able to directly measure the effect of the heat transfer distance on the thermal
transport. We will then use our data to test recent models for spectrally dependent thermal
conductivity [4].
SAMPLES AND METHODS
Sample Preparation
Freestanding Si membranes were fabricated by backside etching of Silicon On Insulator
(SOI) wafers. In this process, the underlying Si substrate and buried oxide layer is removed
through a combination of dry and wet etching techniques to leave a top layer of suspended
silicon (see figure 1(a)).
The initial SOI wafers were 625 µm thick in total, with a top Si layer thickness of 1.5 ± 0.1
µm, and a buried oxide thickness of 1 ± 0.01 µm. The thickness of the top Si layer was reduced
to approximately 400 nm through oxidation of the wafer at elevated temperatures in water vapor.
The areas of the membranes were defined by photolithography on the backside of the wafer,
involving spin coating of a photoresist, exposure, and development. A mask of Si3N4 and SiO2
was used for wet-etching the Si substrate with subsequently KOH and TMAH. After the wet
etching process, the top, bottom, and buried oxides were removed by a wet etch of hydrofluoric
acid to release the free-standing Si membrane.
The semi-insulating GaAs wafer required no sample preparation.
Transient Thermal Grating Measurements
In the transient gating experiments as depicted in figure 1(b), a short-pulsed excitation laser
beam (λe = 515 nm, 60 ps pulse duration, 1 kHz repetition-rate), derived through second
harmonic generation of an amplified Yb:KGW laser system, was split with a diffractive optic (a
binary phase mask pattern) into two beams, which passed through a two-lens telescope and were
focused to a 300 µm 1/e beam radius and crossed in the sample with external angle θe.
Interference between the two beams created a spatially periodic intensity and absorption pattern
with the interference fringe spacing
2π
λe
.
(1)
L=
=
q 2sin(θ e 2)
Optical absorption in the silicon membrane leads to excitation of hot carriers, which
€
promptly reach an equilibrium electronic temperature and transfer energy to the lattice [10].
Energy is deposited with a sinusoidal intensity profile resulting in a transient thermal “grating”
with fringe spacing L and grating wavevector magnitude q; hot carriers and heat subsequently
diffuse from grating peak to null parallel to the surface. Temperature and carrier concentration
induced changes in the transmissivity give rise to time-dependent diffraction of an incident
continuous wave probe beam (λp = 532 nm, single-longitudinal-mode, intracavity frequencydoubled Nd:YAG laser output). The probe beam was split into two parts (probe and reference)
which were recombined at the sample (focused to 150 µm beam radius) using the same
diffractive optic and two-lens telescope used for the excitation beams, ensuring that the probe
beam was incident on the spatially periodic transient thermal “grating” at the Bragg angle for
diffraction and that the diffracted signal was superposed with the reference beam for heterodyne
detection [11]. Thus equation 1 holds for excitation parameters as well as the probe and
reference wavelength λp and angle of intersection θp (i.e. scattering angle). The signal and
references beams were directed to a fast detector and the time-dependent thermal decay was
recorded on an oscilloscope.
Experiments were also performed on bulk GaAs with identical experimental parameters,
except in reflection geometry. In this case, the dynamics of the probed transient thermal grating
at the surface are determined by heat diffusing from grating peak to null and into the depth of the
material. The signal is also complicated due contributions from displacement and
thermoreflectance, but using heterodyne detection the thermal decay can be isolated and
analyzed [12].
(a)
(b)
Figure 1. (a) Schematic of the backside etching process for the fabrication of free-standing Si
membranes from SOI wafers. (b) Schematic of transient thermal grating experiment. A
diffractive optic, a binary phase-mask (PM), splits pump and probe into ±1 diffraction orders.
Pump beams are focused and crossed in the silicon membrane, generating the transient thermal
grating. Diffracted probe light is combined with an attenuated (ND) reference beam and directed
to a fast detector. The relative phase difference between probe and reference beams is controlled
by adjusting the angle of a glass slide (Phase Adjust) in the probe beam path.
RESULTS AND DISCUSSION
Data was collected at ~12 transient grating periods ranging from 2.4 to 25 µm in two separate
€
€
€
free standing silicon membranes with thicknesses of 392 and 390 nm, and ranging from 2.0 to 18
µm in GaAs. Figure 2(a) shows traces collected in the 390 nm membrane with transient grating
periods from 3.2 to 18 µm. As the grating period increases, it takes longer for heat to diffuse
from grating peak to null, and the signal decay is slower. The inset in figure 2(a) shows the full
7.5 µm transient grating period trace, where we see a negative spike at short times due to excited
carriers; at long times, the decay is governed solely by the thermal transport. Figure 2(b) shows
similar data for the GaAs sample with dashed line demonstrating fits to equation 4 below.
Solving the one-dimensional thermal diffusion equation with a spatially periodic heat source
shows the signal will decay as
T ∝ exp( − t τ ) ,
(2)
where the inverse thermal decay time constant 1/τ is
1 4π 2
= 2 α = q 2α .
(3)
τ
L
Therefore, given a material thermal diffusivity α=k/ρc, where k is the thermal conductivity, ρ is the density, and c is the heat capacity, the inverse decay time should scale with q2. To
determine the thermal transport properties and account for the short time electronic response in
the Si membrane samples, the traces were fit to a bi-exponential decay. The transient grating
period and slower relaxation time were used to determine the thermal diffusivity for all
membrane data sets. Because the heat transport is two-dimensional in the GaAs reflection
measurements, the thermal decay was fit to
T ∝ t −1 2 exp( − t τ ) ,
(4)
-1/2
where 1/τ is defined as above and the t term accounts for signal decay due to diffusion into the
depth of the material [13]. As shown in figure 2(b), this form shows good agreement with the
thermal decay in GaAs.
Figure 2. (a) Experimentally measured thermal decay traces in a Si membrane from transient
grating periods ranging from 3.2 to 18 µm. The inset shows the complete trace for the 7.5 µm
period. (b) Thermal decay traces in GaAs at four grating periods with fits to Eq.4. The inset
shows the inverse relaxation time plotted versus q2.
In figure 3(a) we have plotted the determined thermal conductivity, scaled by the bulk Si
value, as a function of grating period for the two Si membranes. We note that if the diffusion
model is valid for the given experimental length scale, the measured thermal conductivity should
not change with the grating period, but we see clear deviations as the grating period is reduced
below ~10 µm. This deviation from the diffusion model is most clearly illustrated in the inset to
figure 3(a). As stated above, the inverse thermal decay constant will scale linearly with q2. The
solid line in the inset indicates this linear dependence, but as the wavevector is increased,
deviations are apparent; the thermal conductivity we measure is lower and lower. Similar effects
are also seen in bulk GaAs, as shown in the departure from q2 dependence in the inset to figure
2(b), and in figure 3(b), where the measured conductivity scaled to the bulk value is decreasing
with transient grating period.
Theoretical treatment of thermal transport over micron scales at room temperature is
challenging, as phonons contributing to thermal conductivity may have their MFP both longer
and shorter than the length scale, and the transport will vary from purely ballistic to purely
diffusive over the phonon spectrum. It has been shown that ballistic phonons contribute less to
the thermal transport than the diffusion model [14]. The simplest model is to simply ignore the
contribution of phonons with MFP longer than the heat transport distance [15]. In this
approximation, in transient grating measurements, the ballistic phonons simply spread their
energy out evenly across grating peaks and nulls, and therefore do not contribute to the observed
decay of the thermal grating. Therefore, the thermal transport we measure is solely due to those
phonons with MFP shorter than half the transient grating period (<L/2). In reality ballistic
phonons also contribute to the thermal grating decay and the degree of this contribution is
currently being developed in more detail [16]. To test the validity of this simple idea we
compare our measured data with the advanced calculations of the thermal conductivity in room
temperature Si of Ref [4].
Figure 3. (a) The Si membrane scaled thermal conductivity for different transient grating periods
compared with first principles calculations from Ref [4]. The inset shows the inverse relaxation
time versus wavevector squared showing departure from the predicted diffusive behavior.
(b) The GaAs scaled thermal conductivity determined at different transient grating periods.
Henry and Chen [4] reported the thermal conductivity accumulation as a function of MFP, given
by
Λc
€
k ( Λ c ) k total = ∫ 0 k ( Λ) dΛ .
(5)
k(Λ) is the differential thermal conductivity as a function of MFP (Λ), and Λc is some cutoff
MFP, which in our comparison we equate with half the transient grating period (L/2).
In-plane phonon transport in thin films has been studied extensively [17] and the theoretical
€
foundation has been laid for an effective MFP [18] reduced by surface scattering:
Λʹ′ = ΛF ( d Λ)
,
(6)
3
3 ∞ ⎛ 1 1 ⎞ − χt
F ( χ) = 1 −
+
−
e
dt
⎜
⎟
∫
3
5
8 χ 2 χ 1 ⎝ t
t ⎠
where d is the membrane thickness. By replacing Λ by Λ’ in equation 5, the scaled Si membrane
thermal conductivity is calculated using the membrane thickness and the differential thermal
conductivity from [4]. There is quite good agreement with our transient grating results as seen in
figure 4(a), lending credence to our simple assumption that ballistic phonons are excluded from
the experimentally measured thermal transport, directly revealing effects of ballistic thermal
transport in Si over micron distances. Similar calculations are underway for GaAs, where the
comparison will be more direct due to the lack of membrane size effects.
ACKNOWLEDGMENTS
This material is based upon work supported as part of the “Solid State Solar- Thermal Energy
Conversion Center (S3TEC),” an Energy Frontier Research Center funded by the U.S.
Department of Energy, Office of Science, Office of Basic Energy Sciences under Award
Number: DE-SC0001299/DE-FG02-09ER46577 (G.C., K.N.). This work was also partially
supported by the projects: NANOPOWER, contract number 256959; TAILPHOX, contract
number 233883; NANOFUNCTION, contract number 257375; ACPHIN, contract number
FIS2009-150; AGAUR, 2009-SGR-150. The Si membranes were fabricated using facilities from
the “Integrated nano and microfabrication Clean Room” ICTS, which is funded by the MICINN.
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