Wikipedia:Tutorial (historical)/Wikipedia links/sandbox

Testing out linking to other wikipedia articles. I'm curious to see what happens with various capitalization schemes within links. What happens if I link to lol? Cool!

what happens if i link to WP:SW

Now let's try a link to a specific section: Section headings. Just have to get the name of the article right!

<Page Created for 598EP> wikipedia

As the devices continue to shrink further into the sub-100nm regime following the trend predicted by Moore’s law, the topic of thermal properties and transport in such nanoscale devices becomes increasingly important. Display of great potential by nanostructures for thermoelectric applications also motivates the studies of thermal transport behaviors in such devices. These fields, however, generate two contradictory demands: high thermal conductivity to deal with heating issues in sub-100nm devices and low thermal conductivity for thermoelectric applications. These issues can be addressed with phonon engineering, once nanoscale thermal behaviors have been studied and understood.

Unlike bulk materials, nanoscale devices have thermal properties which are complicated by boundary effects due to small size. It has been shown that phonon-boundary scattering effects dominate the thermal conduction processes, resulting in reduced thermal conductivity.

It is likely in nanostructures that the phonon mean free path values (Λ) may be comparable or larger than the dimensions. In such cases, the continuous energy model used for bulk materials no longer applies and nonlocal and nonequilibrium aspects to heat transfer also need to be considered. In general, the bulk model and data can be used for devices with feature sizes L > Λ. When L is comparable to or smaller than the mean free path, size effects must be considered. More severe changes in thermal behavior are observed when the feature size L shrinks further down to the wavelength of phonons.

The interatomic potentials given in cannot correctly model the effects of transverse acoustic (TA) phonons at high frequency. Such TA phonons can contribute to heat conduction process at symmetry directions. Better accuracy can be achieved by using improved interatomic potentials.

Phonon Relaxation Times

Phonons can scatter through several mechanisms as they travel through the material. These scattering mechanisms are: phonon-phonon scattering, phonon-impurity scattering, phonon-electron scattering, and phonon-boundary scattering.

These processes can all be taken into account using Matthiessen’s rule. Then the combined relaxation time $\tau _{C}$ can be written as:

The parameters $\tau _{U}$, $\tau _{M}$, $\tau _{B}$, $\tau _{ph-e}$ are due to Umklapp phonon-phonon scattering, mass-difference impurity scattering, boundary scattering and phonon-electron scattering respectively.

For phonon-phonon scattering, effects by normal processes are ignored in favor of umklapp processes. Since normal processes vary linearly with ω and umklapp processes vary with $\omega ^{2}$, Umklapp scattering dominates at high frequency. At low frequencies, boundary scattering dominates; hence, it is acceptable to account only for umklapp processes. $\tau _{U}$ is given by:

where $\gamma$ is Gruneisen anharmonicity parameter, μ is shear modulus, Vo is volume per atom and $\omega _{D}$ is Debye frequency.

Mass-difference impurity scattering is given by:

where Γ is measure of impurity scattering strength. Note that $v_{g}$ is dependent of the dispersion curves.

Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation time is given by:

where D is the dimension of the system and p represents the surface roughness parameter. The value p=1 means a smooth perfect surface that the scattering is purely specular and the relaxation time goes to ∞; hence, boundary scattering does not affect thermal transport. The value p=0 represents a very rough surface that the scattering is then purely diffusive which gives:

This equation is also known as Casimir limit. Phonon-electron scattering can also contribute when the material is lightly doped. The corresponding relaxation time is given as:

The parameter $n_{e}$ is conduction electrons concentration, ε1 is deformation potential, ρ is mass density and m* is effective electron mass. It is usually assumed that contribution to thermal conductivity by phonon-electron scattering is negligible.

Nanowires

Experimental Thermal Conductivity Measurements

The first experimental measurement of thermal conductivity (k) in silicon nanowires was published in 2003 . Assuming that Boltzmann Transport Equation is valid, thermal conductivity can be written as:

where C is the heat capacity, $v_{g}$ is the group velocity and $\tau$ is the relaxation time. 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. In our case, phonon wavelengths are generally in the 1nm range and the nanowires under consideration are within tens of nanometers range, the assumption is valid.

Figure 1 shows the measured k values for varying nanowire diameters: d = 22, 37, 56, 115nm. These k values are much lower than the reported bulk silicon thermal conductivity (148 W/mK). This can be explained by increased phonon-boundary scattering, which decreases the relaxation time τ, and this in turn results in reduction in k. Since the boundary scattering gets more severe as the dimension size goes down, the thermal conductivity decreases as the wire diameter shrinks as seen in Figure 1.

In addition, the peak of the curves shift as the wire diameter decreases, which also indicates the increasing dominance of boundary scattering for higher temperatures, which is usually dominated by umklapp scattering.

Figure 2 shows the measured k values at low temperatures. The interesting thing to notice is that the expected T3 dependence of k is only observed in the larger 56nm and 115nm wires. Thermal conductivity k shows T2 dependence in 37 nm and T dependence in 22nm wires. Li et al. explained this phenomenon as possible modification of phonon dispersion curves and group velocities due to confinement.

Some may claim that this behavior is due to the stricter confinement experienced by phonons such that these three-dimensional structures display two-dimensional or one-dimensional behaviors. Chen et al. however contradicts that this is not the case. [CHEN] It is argued that one-dimensional cross-over for 20 nm Si nanowire occurs around 8K but the phenomenon was observed for temperature values greater than 20K.

The table below shows the diameter of nanowire samples used in Chen et al.

It was observed in Figure 1 by Li et al. that the thermal conductivity of 22nm wire does not flatten out as expected but Chen et al. [CHEN] reported the expected saturation behavior for comparable nanowire samples (Figure 3). The bottom plot of Figure 3 also shows that the experimental data from [12] also displays the linear dependence of k on temperature. (Note that thermal conductance (G) and k are related by: $k=GA/s$, where parameters A and s are cross-sectional area and diameter of the wire.)

Phonon Modes Contributing to Heat Conduction

Analyzing the experimental data for silicon nanowires of different diameters from , we can study the different phonon mode contributions to heat conduction. Figure 1 shows the thermal conductivity data from for silicon nanowires of diameters 22, 37, 56 and 115 nm. These results were then used in to extract the $Cv_{g}$ product for analysis.

Figure 3 shows the extracted product $Cv_{g}$ values with respect to temperature. The saturation of the product $Cv_{g}$ occurs well below the Debye temperature of Silicon (645K), which indicates all the phonon modes contributing to thermal transport are all excited well below the Debye temperature. From the thermal conductivity equation, we can write the product $Cv_{g}$ for each isotropic phonon branch i.

where $x=h\omega /k_{B}T$ and $v_{p,i}$ is the phonon phase velocity, which is less sensitive to phonon dispersions than the group velocity $v_{g}$.

Many models of phonon thermal transport ignores the effects of transverse acoustic phonons (TA) at high frequency due to their small group velocity. (Optical phonon contributions are also ignored for the same reason.) This can be seen in Figure 4 where the dispersion curves for TA phonons generally flatten out near the Brillouin zone boundaries. Note however in Figure 4 that the upper branch of TA phonons have non-zero group velocity at the Brillouin zone boundary along the Γ-Κ direction. These in fact behave similarly to the longitudinal acoustic phonons (LA) and can contribute to the heat transport.

Then, the possible phonon modes contributing to heat conduction are both LA and TA phonons at low and high frequencies. Using the corresponding dispersion curves, the Cvg product can then be calculated and fitted to the data extracted from Li et al. as shown by the dotted line in Figure 3. The best fit was found when contribution of TA phonons is accounted as 70% of the product at room temperature. The remaining 30% is contributed by the LA and TA phonons at low-frequency.

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