*2.3.1 Heat flow equation*

*Nanorods and Nanocomposites*

*I* = *q*∫<sup>0</sup>

barrier height and electric field.

application.

silicon material on silicon. Using carbon-based CNT detector elements in IR microbolometer pixels is attractive due to their high mobility, large thermal resistance, and intrinsically high TCR values [4]. The high thermal conductivity, high strength, and other optical and electrical properties of CNTs that can be varied over a wide spectral range likewise contribute towards their usefulness in this

TCR values can be calculated using the following expression:

<sup>∞</sup> *Tt*

and electric field, and these results are shown in **Figure 4** [16].

The low electrical resistance and corresponding high conductivity of CNTs are mainly attributed to electrons tunneling between adjacent nanotubes and corresponding unimpeded flow of electrical current. The tunnel barrier height between the CNTs directly impacts the amount of heating that takes place for a given electric field. Assuming that electrons can transport across it, this barrier is governed by Fowler-Nordheim-type tunneling or thermionic emission. The expected associated

where *Tt*, *v*, *DOS*, and *f* are the transmission coefficient, thermal velocity, density of states, and distribution function, respectively, for electrons in the CNT. We have performed this calculation of TCR and current as a function of barrier height

The theoretical TCR values for a CNT film are plotted in **Figure 4(a)** as a function of the electric field between the nanotubes and the barrier height. The calculations predict a comparatively very large TCR, which can be attributed to the relatively large barrier height between adjacent SWCNTs. If a relatively low barrier height on the order of 0.06 eV is assumed, then a TCR of approximately 2.5% is obtained. It is noted that a TCR of this magnitude is associated with very low output currents. **Figure 4(b)** shows the plotted bolometer current density as a function of

Research on this CNT absorber material has provided proof-of-principle demonstrations of extraordinary TCRs in devices [7]. Within the temperature region where the composite material undergoes a volume phase transition, considerable changes in resistance are observed. Negative TCRs of magnitude larger than −10%/°C were determined, and other composite structures with thinner suspended

*(a) Contour plot of TCR vs. electric field and barrier height between SWCNTs of the film (color bar scale in units of %TCR); (b) bolometer current as a function of electric field and barrier height (color bars scale in* 

(*E*)*v*(*E*)*DOS*(*E*)*f*(*E*,*T*)*dE* (3)

**74**

**Figure 4.**

*units of amperes) [16].*

To determine the temperature of the material in the presence of IR radiation, we start with the heat flow equation. This partial differential equation relates the time rate of change in temperature to the position and the rate of net heat that is absorbed by the material as a function of time and position:

$$\mathbf{C}\_{\nu} \frac{\partial T}{\partial t} = \mathbf{x} \nabla^{2} T + H\_{\text{net}} \tag{4}$$

where *Cv* is the thermal capacitance (J/K cm3 ), *κ* is the thermal diffusion coefficient (W/K cm), and *Hnet* is the net power absorbed by the material (W/cm3 ). To solve this equation for the CNT bolometer, we must first determine *Cv* and *κ* for the CNT and CNT film.
