*2.3.2 Thermal capacitance of CNT absorber*

To determine the heat capacity of a carbon nanotube, we first determine the internal vibrational energy of the CNT, and then take the derivative with respect to temperature. The internal energy is found by calculating the energy of each vibrational mode, multiplying by the probability that the mode is populated using Bose-Einstein statistics, and then summing over all the allowed modes.

Since the number of allowed modes is dependent on the diameter and wrapping angle of the CNTs, we must take a statistical sample. After multiplying the heat capacity of the individual CNTs by the number of CNTs in the film to provide a reasonable value for the heat capacity of the film, and inserting numerical values for physical constants, we arrive at the following average numerical value for the thermal capacity of a CNT:

$$\mathbf{C}\_{\text{vt}} \approx \mathbf{1.4} \times \mathbf{10}^{-18} Ld \tag{5}$$

where *L* is the CNT length in microns and *d* is the CNT diameter in nanometers.

### *2.3.3 CNT thermal diffusion coefficient*

In addition to the thermal capacitance, we need to determine the thermal diffusion coefficient and then the thermal resistance of a single CNT. Experiments on isolated CNTs have involved fitting the coefficient of thermal diffusion to data, obtaining the following expression [16]:

$$\kappa(L, T) = \left[ 3.7 \times 10^{-7} T + 9.7 \times 10^{-10} T^2 + \frac{9.3}{T^2} \left( 1 + \frac{0.5}{L} \right) \right]^{-1} \tag{6}$$

From *κ* we can obtain the thermal resistance *RT* of a single CNT using the following definition:

$$R\_T = \frac{4L}{\kappa \pi d^2} \tag{7}$$

Using typical values for a CNT gives the following numerical value for its thermal resistance:

$$R\_T \approx \dots \approx 10^8 \text{(L/}d^2\text{)}\tag{8}$$

where *RT* is in units of K/W. Thus, a CNT that is 1 μm long and 1 nm in diameter will have a thermal resistance of ~5 × 108 K/W.

#### *2.3.4 Net IR radiation power absorbed*

Now that the thermal diffusion and capacitance are known, to begin solving the above heat flow equation in order to determine the bolometer temperature we first need to determine the net IR power absorbed by the bolometer, *Hnet*. This parameter can be established using the Stefan-Boltzmann law for blackbody radiation, which relates the net power absorbed to the temperatures of the subject and of the bolometer using the following expression:

$$H\_{net} = \sigma A \varepsilon \left( T\_{obj}^4 - T\_b^4 \right) \tag{9}$$

**77**

**Figure 6.**

*absorber [14].*

*Nanostructure Technology for EO/IR Detector Applications*

*2.3.5 Calculating the bolometer temperature distribution*

of position and time throughout the bolometer absorber.

readout integrated circuit (ROIC) can be clearly seen in **Figure 7**.

36.5°C [16]. Cooling the bolometer down to 30°C below room temperature enables significantly more power to be absorbed, which can lead to a much stronger signal.

Using the above expressions for *Cvt*, *κ*, and *Hnet*, we can convert the heat flow equation above into a thermal network like that illustrated in **Figure 6** [14]. Each resistor in the figure represents the thermal resistance of the CNTs in series with the thermal resistance between adjacent CNTs. The capacitors represent the thermal capacity of the individual CNTs, while the current sources represent the net IR

This thermal network contains thousands of nodes, where there is a specific equation relating the thermal resistance, capacitance, and net power for each node. This system of equations can be solved to determine the temperature as a function

The output data from these calculations are shown plotted in **Figure 7** for different CNT network types [16]. Here, we assume the pixel tightly packed with CNTs and *Hnet* of 1 nW. Two different thermal resistance values were modeled for the nanotubes:

As expected, a higher thermal resistance results in a greater temperature difference from the ambient. We note that in general the thermal resistance also rises with increasing temperature, resulting in further heating of hot spots. The color-mapped temperature gradients of the contact legs that connect the film to the

Networks of SWCNTs are considered as potential replacements for VOx and amorphous silicon in uncooled microbolometer-based infrared focal plane arrays

*Schematic of superimposed thermal network for calculating temperature map of CNT-based bolometer* 

K/W, shown in **Figure 7(a)** and **(b)**, respectively.

*DOI: http://dx.doi.org/10.5772/intechopen.85741*

radiation absorbed by each CNT.

K/W and 1 × 109

**2.4 CNT-based microbolometer pixels**

5 × 108

where *σ*, *A*, *ε*, *Tobj* and *Tb* are the Stefan-Boltzmann constant, cross-sectional area, emissivity, object of interest temperature, and bolometer absorber temperature, respectively. **Figure 5** shows the net IR power absorbed by the absorber material as a function of bolometer temperature for objects radiating at 20.0°C and

**Figure 5.** *Net power received by bolometer as a function of bolometer temperature [16].*

*Nanorods and Nanocomposites*

κ(*L*,*T*) = [

ing definition:

thermal resistance:

3.7 × 10−7

*RT* = \_\_\_\_ <sup>4</sup>*<sup>L</sup>*

*RT* ≈ 5 × 108(*L*/*d*<sup>2</sup>

will have a thermal resistance of ~5 × 108

*2.3.4 Net IR radiation power absorbed*

eter using the following expression:

*Hnet* = *A*(*Tobj*

*Net power received by bolometer as a function of bolometer temperature [16].*

*T* + 9.7 × 10−10*T*<sup>2</sup> + \_\_\_

From *κ* we can obtain the thermal resistance *RT* of a single CNT using the follow-

where *RT* is in units of K/W. Thus, a CNT that is 1 μm long and 1 nm in diameter

K/W.

Now that the thermal diffusion and capacitance are known, to begin solving the above heat flow equation in order to determine the bolometer temperature we first need to determine the net IR power absorbed by the bolometer, *Hnet*. This parameter can be established using the Stefan-Boltzmann law for blackbody radiation, which relates the net power absorbed to the temperatures of the subject and of the bolom-

where *σ*, *A*, *ε*, *Tobj* and *Tb* are the Stefan-Boltzmann constant, cross-sectional area, emissivity, object of interest temperature, and bolometer absorber temperature, respectively. **Figure 5** shows the net IR power absorbed by the absorber material as a function of bolometer temperature for objects radiating at 20.0°C and

<sup>4</sup> − *Tb* 4

Using typical values for a CNT gives the following numerical value for its

9.3 *T*<sup>2</sup>(<sup>1</sup> <sup>+</sup> \_\_\_ 0.5 *<sup>L</sup>* )] −1

*d*<sup>2</sup> (7)

) (8)

) (9)

(6)

**76**

**Figure 5.**

36.5°C [16]. Cooling the bolometer down to 30°C below room temperature enables significantly more power to be absorbed, which can lead to a much stronger signal.
