Answer:
Part (i) at –70° C, intrinsic carrier concentration of silicon is 2.865 x 10⁵ carriers/cm³ and fraction of the atoms ionized is 5.37 x 10⁻¹⁸
Part (ii) at 0° C, intrinsic carrier concentration of silicon is 1.533 x 10⁹ carriers/cm³ and fraction of the atoms ionized is 3.067 x 10⁻¹⁴
Part (iii) at 20° C, intrinsic carrier concentration of silicon is 8.652 x 10⁹ carriers/cm³ and fraction of the atoms ionized is 1.731 x 10⁻¹³
Part (iv) at 100° C, intrinsic carrier concentration of silicon is 1.444 x 10¹² carriers/cm³ and fraction of the atoms ionized is 2.889 x 10⁻¹¹
Part (iv) at 125° C, intrinsic carrier concentration of silicon is 4.754 x 10¹² carriers/cm³ and fraction of the atoms ionized is 9.508 x 10⁻¹¹
Explanation:
![ni^2 = BT^3(e^{\frac{-E_g}{KT}})\\\\ni = \sqrt{ BT^3(e^{\frac{-E_g}{KT}})}](https://tex.z-dn.net/?f=ni%5E2%20%3D%20BT%5E3%28e%5E%7B%5Cfrac%7B-E_g%7D%7BKT%7D%7D%29%5C%5C%5C%5Cni%20%3D%20%5Csqrt%7B%20BT%5E3%28e%5E%7B%5Cfrac%7B-E_g%7D%7BKT%7D%7D%29%7D)
where;
B = 5.4 x 10⁻³¹
Eg = 1.12 ev
K = 8.62 x 10⁻⁵ eV/K
T = (273 + ⁰C) K
Number of atoms in silicon crystal = 5 x 10²² atoms/cm³
Part (i) For –70° C, T = (273 -70 ⁰C)K = 203 K
![ni = \sqrt{ 5.4*10^{31}*203^3(e^ \ {\frac{-1.12}{8.62*10^{-5}*203}})}} \ =2.685*10^5 \ carriers/cm^3](https://tex.z-dn.net/?f=ni%20%3D%20%5Csqrt%7B%205.4%2A10%5E%7B31%7D%2A203%5E3%28e%5E%20%5C%20%7B%5Cfrac%7B-1.12%7D%7B8.62%2A10%5E%7B-5%7D%2A203%7D%7D%29%7D%7D%20%5C%20%3D2.685%2A10%5E5%20%5C%20carriers%2Fcm%5E3)
![Fraction \ of \ atoms \ ionized = \frac{2.685*10^5}{5 *10^{22}} = 5.370 *10^{-18}](https://tex.z-dn.net/?f=Fraction%20%5C%20%20of%20%5C%20atoms%20%5C%20ionized%20%3D%20%5Cfrac%7B2.685%2A10%5E5%7D%7B5%20%2A10%5E%7B22%7D%7D%20%3D%205.370%20%2A10%5E%7B-18%7D)
Part (ii) For 0° C, T = (273 +0 ⁰C)K = 273 K
![ni = \sqrt{ 5.4*10^{31}*273^3(e^ \ {\frac{-1.12}{8.62*10^{-5}*273}})}} \ =1.533*10^9 \ carriers/cm^3](https://tex.z-dn.net/?f=ni%20%3D%20%5Csqrt%7B%205.4%2A10%5E%7B31%7D%2A273%5E3%28e%5E%20%5C%20%7B%5Cfrac%7B-1.12%7D%7B8.62%2A10%5E%7B-5%7D%2A273%7D%7D%29%7D%7D%20%5C%20%3D1.533%2A10%5E9%20%5C%20carriers%2Fcm%5E3)
![Fraction \ of \ atoms \ ionized = \frac{1.533*10^9}{5 *10^{22}} = 3.067 *10^{-14}](https://tex.z-dn.net/?f=Fraction%20%5C%20%20of%20%5C%20atoms%20%5C%20ionized%20%3D%20%5Cfrac%7B1.533%2A10%5E9%7D%7B5%20%2A10%5E%7B22%7D%7D%20%3D%203.067%20%2A10%5E%7B-14%7D)
Part (iii) For 20° C, T = (273 + 20 ⁰C)K = 293 K
![ni = \sqrt{ 5.4*10^{31}*293^3(e^ \ {\frac{-1.12}{8.62*10^{-5}*293}})}} \ =8.652*10^9 \ carriers/cm^3](https://tex.z-dn.net/?f=ni%20%3D%20%5Csqrt%7B%205.4%2A10%5E%7B31%7D%2A293%5E3%28e%5E%20%5C%20%7B%5Cfrac%7B-1.12%7D%7B8.62%2A10%5E%7B-5%7D%2A293%7D%7D%29%7D%7D%20%5C%20%3D8.652%2A10%5E9%20%5C%20carriers%2Fcm%5E3)
![Fraction \ of \ atoms \ ionized = \frac{8.652*10^9}{5 *10^{22}} = 1.731 *10^{-13}](https://tex.z-dn.net/?f=Fraction%20%5C%20%20of%20%5C%20atoms%20%5C%20ionized%20%3D%20%5Cfrac%7B8.652%2A10%5E9%7D%7B5%20%2A10%5E%7B22%7D%7D%20%3D%201.731%20%2A10%5E%7B-13%7D)
Part (iv) For 100° C, T = (273 + 100 ⁰C)K = 373 K
![ni = \sqrt{ 5.4*10^{31}*373^3(e^ \ {\frac{-1.12}{8.62*10^{-5}*373}})}} \ =1.444*10^{12} \ carriers/cm^3](https://tex.z-dn.net/?f=ni%20%3D%20%5Csqrt%7B%205.4%2A10%5E%7B31%7D%2A373%5E3%28e%5E%20%5C%20%7B%5Cfrac%7B-1.12%7D%7B8.62%2A10%5E%7B-5%7D%2A373%7D%7D%29%7D%7D%20%5C%20%3D1.444%2A10%5E%7B12%7D%20%5C%20carriers%2Fcm%5E3)
![Fraction \ of \ atoms \ ionized = \frac{1.444*10^{12}}{5 *10^{22}} = 2.889 *10^{-11}](https://tex.z-dn.net/?f=Fraction%20%5C%20%20of%20%5C%20atoms%20%5C%20ionized%20%3D%20%5Cfrac%7B1.444%2A10%5E%7B12%7D%7D%7B5%20%2A10%5E%7B22%7D%7D%20%3D%202.889%20%2A10%5E%7B-11%7D)
Part (v) For 125° C, T = (273 + 125 ⁰C)K = 398 K
![ni = \sqrt{ 5.4*10^{31}*398^3(e^ \ {\frac{-1.12}{8.62*10^{-5}*398}})}} \ =4.754*10^{12} \ carriers/cm^3](https://tex.z-dn.net/?f=ni%20%3D%20%5Csqrt%7B%205.4%2A10%5E%7B31%7D%2A398%5E3%28e%5E%20%5C%20%7B%5Cfrac%7B-1.12%7D%7B8.62%2A10%5E%7B-5%7D%2A398%7D%7D%29%7D%7D%20%5C%20%3D4.754%2A10%5E%7B12%7D%20%5C%20carriers%2Fcm%5E3)
![Fraction \ of \ atoms \ ionized = \frac{4.754*10^{12}}{5 *10^{22}} = 9.508 *10^{-11}](https://tex.z-dn.net/?f=Fraction%20%5C%20%20of%20%5C%20atoms%20%5C%20ionized%20%3D%20%5Cfrac%7B4.754%2A10%5E%7B12%7D%7D%7B5%20%2A10%5E%7B22%7D%7D%20%3D%209.508%20%2A10%5E%7B-11%7D)