When the solid is perfectly ordered over hundreds of thousands of atoms, this transport obeys established physics.
Unlike bulk materials, nanoscale devices have thermal properties which are complicated by boundary effects due to small size.
is comparable to or smaller than the mean free path (which is of the order 1 μm for carbon nanostructures[5]), the continuous energy model used for bulk materials no longer applies and nonlocal and nonequilibrium aspects to heat transfer also need to be considered.
Chen et al. [7] has shown that the one-dimensional cross-over for 20 nm Si nanowire occurs around 8K, while the phenomenon was observed for temperature values greater than 20K.
Therefore, the reason of such behaviour is not in the confinement experienced by phonons so that three-dimensional structures display two-dimensional or one-dimensional behavior.
Assuming that Boltzmann transport equation is valid, thermal conductivity can be written as: where C is the heat capacity, vg is the group velocity and
Note that this assumption breaks down when the dimensions of the system are comparable to or smaller than the wavelength of the phonons responsible for thermal transport.
It was shown that all phonon modes contributing to thermal transport are excited well below the Si Debye temperature (645 K).
The best fit was found when contribution of high-frequency TA phonons is accounted as 70% of the product at room temperature.
Thermal conductivity is then expressed as: where S is the cross sectional area of the wire, az is the lattice constant.
It was shown [9] that, using this formula and atomistically computed phonon dispersions (with interatomic potentials developed in [10]), it is possible to predictively calculate lattice thermal conductivity curves for nanowires, in good agreement with experiments.
The low-temperature specific heat and thermal conductivity show direct evidence of 1-D quantization of the phonon band structure.
In general, the thermal conductivity is a tensor quality, but for this discussion, it is only important to consider the diagonal elements: where C is the specific heat, and vz and
[citation needed] Therefore, in ordinary materials, the low-temperature thermal conductivity has the same temperature dependence as the specific heat.
For instance, in graphite, the thermal conductivity parallel to the basal planes is only weakly dependent on the interlayer phonons.
[16] In 1999, Keith Schwab, Erik Henriksen, John Worlock, and Michael Roukes carried out a series of experimental measurements that enabled first observation of the thermal conductance quantum.
In 2008, a colorized electron micrograph of one of the Caltech devices was acquired for the permanent collection of the Museum of Modern Art in New York.
At high temperatures, three-phonon Umklapp scattering begins to limit the phonon relaxation time.
The value of k(T) at the peak (37,000 W/(m·K)) is comparable to the highest thermal conductivity ever measured (41,000 W/(m·K) for an isotopically pure diamond sample at 104 K).
In graphite, the interlayer interactions quench the thermal conductivity by nearly 1 order of magnitude [citation needed].
In a (10,10) tube, for instance, the six phonon bands (three acoustic and three optical) of the graphene sheet become 66 separate 1-D subbands.
A direct result of this folding is that the nanotube density of states has a number of sharp peaks due to 1-D van Hove singularities, which are absent in graphene and graphite.
Although these devices are traditionally made from bulk crystalline material (silicon), they often contain thin films of oxides, polysilicon, metal, as well as superlattices such as thin-film stacks of GaAs/AlGaAs for lasers.
[25][26] For example, a Si/Ge thin-film superlattice has a greater decrease in thermal conductivity than an AlAs/GaAs film stack [27] due to increased lattice mismatch.
Likewise, disordered or amorphous films will experience a severe reduction of thermal conductivity, since the small grain size results in numerous grain-boundary scattering effects.
Often, a resistive heater and thermistor is fabricated on the sample film using a highly conductive metal, such as aluminium.
The most straightforward approach would be to apply a steady-state current and measure the change in temperature of adjacent thermistors.