We
recently measured the thermal contact
resistance and thermal conductivity of graphene encased within layers
of
silicon dioxide -- a configuration of great importance for possible
applications in post-silicon microelectronics.
(Collaboration with Prof. C. N. Lau, UCR Physics)
The coefficient of thermal expansion of graphene is negative. As part of a collaboration with Prof. C. N. Lau of UCR Physics, we
measured this coefficient for the first time and showed how it can
result in ripples upon thermal cycling (the latter also with Prof. S. B. Cronin of USC EE).
We
have also been developing a detailed model of
thermal rectification in nanostructures via ballistic phonons (Work
in progress.)
Measurements
of individual nanowires and nanotubes
Microfabrication approach
Various electrical and thermal measurements require electrical contact
to an individual nanowire or nanotube. One
technique for making contacts is to use electron-beam lithography to
pattern metallic contacts that are aligned to a nanowire. In
the example shown below, many silicon nanowires are first dispersed
randomly over a substrate that has been prepatterned with a fiducial
grid. Then a target nanowire is identified and
located with respect to the grid. Finally a custom script
("MATKIC")
is used to automatically generate a pattern for
the metallic leads, which is fed into the e-beam lithography tool and
used to define the contacts:
+
To minimize heat losses from the periphery of the nanowire/nanotube, it
is generally necessary to isolate the nanostructure from the
substrate.
This can be done by etching away a portion of the substrate,
leaving the nanostructure suspended:
Nano-positioner approach
(with Shuo Chen and Tom Harris)
An alternative technique for contacting a single nanowire or nanotube
uses a nano-positioner inside a scanning electron microscope (SEM) or
transmission electron microscope (TEM). An existing
TEM-nanopositioner at Boston College was modified with a special
home-built hot-wire
thermal probe. The probe of the nanopositioner is used to
pick out a target carbon nanotube, and then touch it to the
thermal probe. This technique is in some ways simpler than
the microfabrication approach described above, and may also
offer higher throughput.
After making the contacts, this configuration can then be used to
measure various thermoelectric properties, including electrical
conductivity, thermal conductivity, and the Seebeck coefficient:
It is also possible to perform electron-beam induced deposition (EBID)
inside the TEM chamber to trace out almost arbitrary shapes, such as
the letters "TEM" (deposition by Shuo Chen):
Modeling classical and quantum size effects
Classical size effects
In insulators and most semiconductors most of the heat is carried by
phonons, the quanta of sound waves. Thermal conduction can be
though of as the diffusion of phonons down a temperature gradient.
The thermal conductivity is less than infinity because
the phonons collide with defects, impurities, and other
phonons, thus limiting the phonon mean free path. In a
nanowire, the phonons also experience many collisions with the walls of
the nanowire, further reducing the thermal conductivity. This
is known as the classical size effect:
Similar scattering effects can lead to reduced thermal conductivity in
superlattices. For this type of analysis that treats phonons
as incoherent classical particles, the transition from bulk
conductivity to reduced conductivity is determined by
comparing the nanowire diameter to the bulk mean free path (perhaps
modified by Ziman's specularity parameter, p). Calculations
yield a regime map revealing the limiting behaviors of bulk,
nanowire, superlattice, and superlattice nanowire:
To estimate the onset of these transitions it is very important to have
a good
understanding of the mean free path of phonons in bulk.
Usually a single value is cited for the mean free path of a
bulk phonon in a given material, for example 200 nm for phonons in
bulk silicon at 300 K. However, more detailed calculations
reveal that there is a broad distribution
of mean free paths. For example, in bulk silicon at 300 K the
important part of the "long tail" of the thermal-conductivity-weighted
phonon distribution can extend out well beyond 10 microns:
This
means that classical phonon size effects are likely to be important for
larger nanostructures than might normally be realized.
By taking this full phonon distribution into account, we obtain a model
which compares favorably with available experimental data from the
literature:
Notice that the thermal conductivity of silicon
nanowires can be reduced by a factor of 1000 compared to bulk silicon,
and that the model (lines) is in good agreement with the available
experiments (points) at both low and high temperatures.
Quantum size effects
For very small structures and/or very low temperatures, a second type
of confinement phenomenon becomes important: quantum size effects.
This most commonly occurs when the phonon wavelength is
comparable to the nanostructure size. In "large" or
"medium"-sized nanowires, the phonon wavelength is small compared to
the wire diameter, and so the phonons can generally be modeled as the
classical, incoherent particles described above. However, for
sufficiently-small nanowires, the phonon wavelength is comparable to
the wire diameter:
In this case the constraints on the allowed phonon wavelengths
lead to
significant changes in the phonon dispersion relations (the
relationship between energy and momentum). This is analogous
to waveguiding of light in an optical fiber.
The altered dispersion relations also change the phonon density of
states (basically the number of unique phonon modes at a given energy,
or frequency). The example below shows the calculated density
of
states (DOS) for a nanotube. The smaller plots to the right
show
that bulk, three dimensional (3D) materials have a DOS proportional to
the square of the frequency (in this regime). On the other
hand,
the DOS for thin films (2D) has a series of steps, each linear in
frequency, and the DOS for a nanowire (1D) has a series of
singularities. Interestingly, the DOS for the nanotube shows
features from both 3D, 2D, and 1D, which can be understood in terms of
the 3 length scales of length, diameter, and wall thickness.
The principles of solid state physics relate the DOS to the specific
heat. The solid curves below are model calculations for the
specific heat of titanium dioxide bulk (black lines) and nanotubes (red
lines). The most important feature is that the bulk and
nanotube
specific heats are the same at high and moderate temperatures, while at
very low temperatures the nanotube specific heat is enhanced compared
to bulk. These predictions are largely borne out by
experimental
measurements on a pellet of TiO2 nanotubes (red points).
1 omega, 2 omega,
and 3 omega methods for thermal measurements
"Three omega" methods are an important class of techniques for
measuring the thermal properties of wires, films, liquids, and bulk
samples. The 3-omega methods were popularized by Cahill
and by Birge
and Nagel
, who observed that for a certain class of thermal experiments driven
by a current at frequency "omega", the voltage at the third harmonic
("three omega") contains useful information about the thermal
properties of the system. It turns out that the second and
first
harmonics also contain the same information, and are in some cases
easier to use than the 3rd harmonic.
As shown below, in the traditional 3 omega method, the sample has a
metal heater that also acts as a thermometer (because its electrical
resistance depends on temperature). The heater is driven by a
sinusoidal current at frequency omega. The resulting joule
heating is at the second harmonic ("2 omega") and leads to a
temperature fluctuation at 2 omega. This temperature
fluctuation
causes a ripple in the heater's electrical resistance, also at
2
omega. Finally, the product of current (at 1 omega) and
resistance (including a 2 omega ripple) leads to a voltage contribution
at 3 omega, due to the mixing of sinusoids at different frequencies.
This 3-omega voltage is then used to extract the thermal
properties of the system.
However, as shown below it turns out that the analysis is richer than
described above, particularly if the driving current is given a DC
offset. In this case the joule heating, temperature, and
resistance all include components at DC, 1 omega, and 2 omega.
As
a result, the output voltage includes 1, 2, 3 omega, and even DC
("zero-omega"), all of which contain information about the thermal
properties of the system. Even when the DC offset is omitted
from
the driving current, for example in traditional 3-omega, the 1-omega
output voltage can easily be used to study the thermal properties.
A general solution of this new 1, 2, and 3 omega problem can
be
obtained in terms of a "thermal transfer function", Z. This
transfer function is the frequency-dependent counterpart to the thermal
resistance used in standard heat transfer analysis, and is analogous to
complex impedances in electrical circuits that include resistors as
well as capacitors and/or inductors. General equations are
derived to relate the thermal transfer function (ratio of temperature
to heating) to the electrical transfer functions (ratios of voltage to
current). Experimentally, you measure the electrical transfer
function, and then back-calculate to evaluate the thermal properties.
For example, as shown below-left, the thermal transfer function
for a self-heated platinum wire has a bump (imaginary part,
red)
and monotonic decay (real part, blue). Measurements
of the
electrical transfer functions (points) show qualitatively similar
behaviors at 1 omega, 2 omega, and 3 omega. By fitting this
data
for any one of the harmonics (solid lines), the thermal conductivity
and specific heat of the platinum wire can be extracted. The
results from 1 omega and 2 omega are consistent with the results from 3
omega, and with literature values.
Because the 1, 2, and 3 omega transfer function equations are
completely general, they can also be applied to perhaps the most
traditional 3-omega application: measuring the thermal conductivity of
a substrate. As shown below, a "slope method" can be applied
independently to the electrical transfer functions at 1, 2, and 3
omega, in each case yielding a similar estimate for the thermal
conductivity of the substrate (in this case, Pyrex).
