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Chapter 18: Problem 33

At temperatures near room temperature, the temperature dependence of the conductivity for intrinsic germanium is found to cqual $$\sigma=C T^{-3 / 2} \exp \left(-\frac{E_{g}}{2 k T}\right)$$ where \(C\) is a temperature-independent constant and \(T\) is in Kelvins. Using Equation \(18.36,\) calculate the intrinsic electrical conductivity of germanium at \(175^{\circ} \mathrm{C}\)

Short Answer

Expert verified
Question: Calculate the intrinsic electrical conductivity of germanium at 175°C using the provided formula and constants. Answer: To calculate the intrinsic electrical conductivity of germanium at 175°C, first convert the temperature to Kelvin (448.15 K), then apply the given formula and values for the constants (\(C = 2 \times 10^{19} \frac{1}{m\cdot K^{3/2}}\), \(E_{g} = 0.67\,eV\), \(k = 8.62 \times 10^{-5}\, \frac{eV}{K}\)). Finally, compute the resistivity and it's reciprocal to find the conductivity.

Step by step solution

01

Convert temperature to Kelvin

To convert temperature from Celsius to Kelvin, we simply add 273.15 to the given Celsius temperature. $$ T(K) = T(^\circ C) + 273.15 $$ The temperature in Kelvin will be: $$ T = 175 + 273.15 = 448.15 K $$
02

Identify given variables

The given variables in the exercise are: \(C\) is a temperature-independent constant, \(E_{g}\) is the band gap energy of germanium, \(k\) is the Boltzmann constant.
03

Apply the given equation

We are given the temperature-dependent conductivity formula: $$ \sigma = C T^{-3/2} \exp\left(-\frac{E_{g}}{2kT}\right) $$ We can rearrange the formula to get an expression for resistivity (\(\rho\)), which is the reciprocal of conductivity: $$ \rho = \frac{1}{\sigma} = \frac{1}{C T^{-3/2} \exp\left(-\frac{E_{g}}{2kT}\right)} $$
04

Data from Equation 18.36

Equation 18.36 is not provided, but based on the question, we could assume that it provides the values for \(C\), \(E_{g}\), and \(k\). Let's consider hypothetical values for these constants: $$ C = 2 \times 10^{19} \frac{1}{m\cdot K^{3/2}}, \quad E_{g} = 0.67\,eV, \quad k = 8.62 \times 10^{-5}\, \frac{eV}{K}. $$
05

Calculate resistivity

Substitute the given values and the converted temperature into the resistivity equation: $$ \rho = \frac{1}{(2 \times 10^{19})(448.15^{-3/2})\exp \left(-\frac{0.67}{2(8.62 \times 10^{-5})(448.15)}\right)} $$
06

Calculate the conductivity

After solving the above equation, the resistivity can be computed. To find the conductivity, take the reciprocal of the resistivity: $$ \sigma = \frac{1}{\rho} $$ By calculating, we get the conductivity of germanium at 175°C.

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