In the figure, \(\rm \ln{(r_i)}\) is plotted as a function of \(\
In the figure, \(\rm \ln{(r_i)}\) is plotted as a function of \(\rm 1/T\), where \(\rm r_i\) is the intrinsic resistivity of silicon, \(\rm T\) is the temperature, and the plot is almost linear
The slope of the line can be used to estimate
A. <span style="font-family:arial,sans-serif">Band gap energy of silicon E<sub>g</sub></span>
B. sum of electron and hole mobility in silicon <strong>μ<sub>n</sub> + μ<sub>p</sub> </strong>
C. reciprocal of the sum of electron and hole mobility in silicon <strong>(μ<sub>n</sub> + μ<sub>p</sub>)<sup>-1</sup></strong>
D. intrinsic carrier concentration of silicon <span class="math-tex">\(\rm (n _i)\)</span>
Please scroll down to see the correct answer and solution guide.
Right Answer is: A
SOLUTION
Concept:
We know that the conductivity of Intrinsic semiconductor is given by:
σi = ni q (μn + μp)
Also, the intrinsic carrier concentration is defined as:
\({n_i} = \sqrt {{A_0}} {T^{\frac{3}{2}}}{e^{ - {E_{G0}}/2KT}}\)
The intrinsic conductivity will be:
\({\sigma _i} = \sqrt {{A_0}} {T^{3/2}}{e^{ - {E_{G0}}/2KT}}q\left[ {{\mu _n} + {\mu _p}} \right]\)
Also, we know that:
\({\mu _n} \propto {T^{ - \frac{3}{2}}}\) and \({\mu _p} \propto {T^{ - \frac{3}{2}}}\)
\({\mu _n} = {\mu _{n0}}{T^{ - \frac{3}{2}}}\) and \({\mu _p} = {\mu _{{p_0}}}{T^{ - \frac{3}{2}}}\)
The conductivity Equation (1) can now be written as:
\({\sigma _i} = \sqrt {{A_0}} {e^{ - {E_{G0}}/2KT}}q{T^{\frac{3}{2}}}\left[ {\mu_{n_0} + \mu_{p_0}} \right]{T^{ - 3/2}}\)
\({\sigma _i} = \sqrt {{A_0}} {e^{ - {E_{G0}}/2KT}}q\left[ {{\mu _{{n_0}}} + {\mu _{{p_0}}}} \right]\)
Since the Resistivity is the inverse of conductivity, we can write:
\({p_i} = \frac{1}{{{\sigma _i}}}\)
\({p_i} = \frac{1}{{\sqrt {{A_0}} {e^{ - {E_{G0}}/2KT}}q\left( {\mu_{n_0} + \mu_{p_0}} \right)}}\)
\(ln{p_i} = ln1 - ln\left[ {\sqrt {{A_0}} {e^{ - \frac{{{E_{G0}}}}{{2KT}}}}V\left( {{\mu _{{n_0}}} + {\mu _{{p_0}}}} \right)} \right]\)
\(ln{p_i} = 0 - \left[ {ln\sqrt {{A_0}} - \frac{{{E_{G0}}}}{{2KT}} + lnV\left( {{\mu _{{n_0}}} + {\mu _{{p_0}}}} \right)} \right]\)
\(ln{p_i} = \frac{{{E_{G0}}}}{{2K}}\frac{1}{T} + ln\left[ {\sqrt {{A_0}} V\left( {{\mu _{{n_0}}} + {\mu _{{p_0}}}} \right)} \right]\)
Comparing this with the equation of the line: y = m x + C, we get the slope as:
\(m = \frac{{{E_{G0}}}}{{2K}}\) = Band Gap energy of silicon.