Chapter 1, Problem 1.3P

Calculate the intrinsic carrier concentration in silicon and germanium at (a) $T=100\text{K}$ , (b) $T=300\text{K}$ , and (c) $T=500\text{K}$ .

To determine

The intrinsic carrier concentration in silicon and germanium for the given temperature.

Answer to Problem 1.3P

The intrinsic carrier concentration for silicon at T = 100 K is

  $n_i=8.79\times {10}^{-10}{\text{cm}}^{\text{-3}}$

The intrinsic carrier concentration for germanium at T = 100 K is

  $n_i=35.9{\text{cm}}^{\text{-3}}$

Explanation of Solution

Given Information:

The given temperature is $T=100\text{K}$

Calculation:

The intrinsic carrier concentration for semiconductor material is:

  $n_i=B{T}^{\frac{3}{2}}{e}^{\left(\frac{-E_g}{2KT}\right)}\mathrm{....}\left(1\right)$

B is a constant related to the specific semiconductor material.

T = temperature (K)

  $E_g$ = energy band gap (eV)

K = Boltzmann’s constant $\left(8.6\times {10}^{-5}\raisebox{1ex}{$\text{eV}$}\!\left/ \!\raisebox{-1ex}{$\text{K}$}\right.\right)$

The value of B for silicon is $B=5.23\times {10}^{15}{\text{cm}}^{-3}{\text{K}}^{-\frac{3}{2}}$ .The value of B for germanium is $B=1.66\times {10}^{15}{\text{cm}}^{-3}{\text{K}}^{-\frac{3}{2}}$ .Also, the value of $E_g$ for silicon is 1.1 eV and for germanium is 0.66 eV.

The intrinsic carrier concentration for silicon is determined as follows:

  $\begin{array}{l}{n}_i=\left(5.23\times {10}^{15}\right){\left(100\right)}^{\frac{3}{2}}{e}^{\left(\frac{-1.1}{2\times 86\times {10}^{-6}\times 100}\right)}\\ =\left(5.23\times {10}^{18}\right)e^{\left(-63.95\right)}\\ =8.79\times {10}^{-10}{\text{cm}}^{\text{-3}}\end{array}$

The intrinsic carrier concentration for germanium is determined as follows:

  $\begin{array}{l}{n}_i=\left(1.66\times {10}^{15}\right){\left(100\right)}^{\frac{3}{2}}{e}^{\left(\frac{-.66}{2\times 86\times {10}^{-6}\times 100}\right)}\\ =\left(1.66\times {10}^{18}\right)e^{\left(-38.37\right)}\\ =35.9{\text{cm}}^{\text{-3}}\end{array}$

To determine

The intrinsic carrier concentration in silicon and germanium for the given temperature.

Answer to Problem 1.3P

The intrinsic carrier concentration for silicon is at T = 300 K

  $n_i=1.5\times {10}^{10}{\text{cm}}^{\text{-3}}$

The intrinsic carrier concentration for germanium is at T = 300 K

  $n_i=2.40\times {10}^{13}{\text{cm}}^{\text{-3}}$

Explanation of Solution

Given Information:

The given temperature is $T=300\text{K}$

Calculation:

The intrinsic carrier concentration for semiconductor material is:

  $n_i=B{T}^{\frac{3}{2}}{e}^{\left(\frac{-E_g}{2KT}\right)}\mathrm{....}\left(1\right)$

B is a constant related to the specific semiconductor material.

T = temperature (K)

  $E_g$ = energy band gap (ev)

K = Boltzmann’s constant $\left(8.6\times {10}^{-5}\raisebox{1ex}{$\text{eV}$}\!\left/ \!\raisebox{-1ex}{$\text{K}$}\right.\right)$

The value of B for silicon is $B=5.23\times {10}^{15}{\text{cm}}^{-3}{\text{K}}^{-\frac{3}{2}}$ .The value of B for germanium is $B=1.66\times {10}^{15}{\text{cm}}^{-3}{\text{K}}^{-\frac{3}{2}}$ . Also, the value of $E_g$ for silicon is 1.1 eV and for germanium is 0.66 eV.

The intrinsic carrier concentration for silicon is determined as follows:

  $\begin{array}{l}{n}_i=\left(5.23\times {10}^{15}\right){\left(300\right)}^{\frac{3}{2}}{e}^{\left(\frac{-1.1}{2\times 86\times {10}^{-6}\times 300}\right)}\\ =\left(2.718\times {10}^{19}\right)e^{\left(-21.32\right)}\\ =1.5\times {10}^{10}{\text{cm}}^{\text{-3}}\end{array}$

The intrinsic carrier concentration for germanium is determined as follows:

  $\begin{array}{l}{n}_i=\left(1.66\times {10}^{15}\right){\left(300\right)}^{\frac{3}{2}}{e}^{\left(\frac{-.66}{2\times 86\times {10}^{-6}\times 300}\right)}\\ =\left(8.626\times {10}^{18}\right)e^{\left(-12.79\right)}\\ =2.40\times {10}^{13}{\text{cm}}^{\text{-3}}\end{array}$

To determine

The intrinsic carrier concentration in silicon and germanium.

Answer to Problem 1.3P

The intrinsic carrier concentration for silicon is at T = 500 K

  $n_i=1.63\times {10}^{14}{\text{cm}}^{\text{-3}}$

The intrinsic carrier concentration for germanium is at T = 500 K

  $n_i=8.62\times {10}^{15}{\text{cm}}^{\text{-3}}$

Explanation of Solution

Given Information:

The given temperature is $T=500\text{K}$

Calculation:

The intrinsic carrier concentration for semiconductor material is:

  $n_i=B{T}^{\frac{3}{2}}{e}^{\left(\frac{-E_g}{2KT}\right)}\mathrm{....}\left(1\right)$

B is a constant related to the specific semiconductor material.

T = temperature (K)

  $E_g$ = energy band gap (eV)

K = Boltzmann’s constant $\left(8.6\times {10}^{-5}\raisebox{1ex}{$\text{eV}$}\!\left/ \!\raisebox{-1ex}{$\text{K}$}\right.\right)$

The value of B for silicon is $B=5.23\times {10}^{15}{\text{cm}}^{-3}{\text{K}}^{-\frac{3}{2}}$ .The value of B for germanium is $B=1.66\times {10}^{15}{\text{cm}}^{-3}{\text{K}}^{-\frac{3}{2}}$ .Also, the value of $E_g$ for silicon is 1.1 eV and for germanium is 0.66 eV.

The intrinsic carrier concentration for silicon is determined as follows:

  $\begin{array}{l}{n}_i=\left(5.23\times {10}^{15}\right){\left(500\right)}^{\frac{3}{2}}{e}^{\left(\frac{-1.1}{2\times 86\times {10}^{-6}\times 500}\right)}\\ =\left(5.847\times {10}^{19}\right)e^{\left(-12.79\right)}\\ =1.63\times {10}^{14}{\text{cm}}^{\text{-3}}\end{array}$

The intrinsic carrier concentration for germanium is determined as follows:

  $\begin{array}{l}{n}_i=\left(1.66\times {10}^{15}\right){\left(500\right)}^{\frac{3}{2}}{e}^{\left(\frac{-.66}{2\times 86\times {10}^{-6}\times 500}\right)}\\ =\left(1.856\times {10}^{19}\right)e^{\left(-7.674\right)}\\ =8.62\times {10}^{15}{\text{cm}}^{\text{-3}}\end{array}$