Chapter 1, Problem 1.1TYU

Determine the intrinsic carrier concentration in silicon, germanium, and GaAs at (a) $T=400\text{K}$ and (b) $T=250\text{K}$ . (Ans. (a) $\text{Si:}{n}_i=4.76\times {10}^{12}{\text{cm}}^{-3}$ , $\text{Ge:}{n}_i=9.06\times {10}^{14}{\text{cm}}^{-3}$ , $\text{GaAs:}{n}_i=2.44\times {10}^9{\text{cm}}^{-3}$ ; (b) $\text{Si:}{n}_i=1.61\times {10}^8{\text{cm}}^{-3}$ , $\text{Ge:}{n}_i=1.42\times {10}^{12}{\text{cm}}^{-3}$ , $\text{GaAs:}{n}_i=6.02\times {10}^3{\text{cm}}^{-3}$ )

To determine

The intrinsic carrier concentration for the given following materials.

Answer to Problem 1.1TYU

The intrinsic carrier concentration for Silicon, Germanium and Gallium Arsenide:

  $\begin{array}{l}\text{Si:}\\ n_i=4.76\times {10}^{12}{\text{cm}}^{-3}\\ \text{Ge:}\\ n_i=9.06\times {10}^{14}{\text{cm}}^{-3}\\ \text{GaAs:}\\ n_i=2.44\times {10}^9{\text{cm}}^{-3}\end{array}$

Explanation of Solution

Given information:

The given materials are:

Silicon, Germanium and Gallium Arsenide.

Temperature, T=400K.

Calculation:

The intrinsic carrier concentration $\left(n_i\right)$ for semiconductor material is given by:

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

Where,

B is the specific material coefficient.

T is the temperature in kelvin.

  $E_g$ is the bandgap energy.

k is the Boltzman’s constant.

The value of semi-conductor material coefficient for silicon, $B=5.23\times {10}^{15}$ .

And the energy gap for silicon, $E_g=1.1\text{eV}$ .

Now, evaluating the intrinsic carrier concentration:

  $\begin{array}{l}{n}_i=5.23\times {10}^{15}\left({400}^{3/2}\right)e^{\left(\frac{-1.1}{2\left(86\times {10}^{-6}\right)\left(400\right)}\right)}\\ =5.23\times {10}^{15}\left(8000\right)e^{\left(\frac{-1.1}{2\left(86\times {10}^{-6}\right)\left(400\right)}\right)}\\ =5.23\times {10}^{15}\left(8000\right)\left(e^{-15.988}\right)\\ =4.76\times {10}^{12}{\text{cm}}^{-3}\end{array}$

Similarly, for Germanium:

The value of semi-conductor material coefficient for Germanium, $B=1.66\times {10}^{15}$ .

And the energy gap for Germanium is, $E_g=0.66\text{eV}$ .

The intrinsic carrier concentration is

  $\begin{array}{c}{n}_i=1.66\times {10}^{15}\left({400}^{3/2}\right)e^{\left(\frac{-0.66}{2\left(86\times {10}^{-6}\right)\left(400\right)}\right)}\\ =1.66\times {10}^{15}\left(8000\right)e^{\left(\frac{-0.66}{2\left(86\times {10}^{-6}\right)\left(400\right)}\right)}\\ =9.057\times {10}^{14}{\text{cm}}^{-3}\\ \approx 9.06\times {10}^{14}{\text{cm}}^{-3}\end{array}$

Similarly, for Gallium Arsenide:

The value of semi-conductor material coefficient for Gallium Arsenide, $B=2.1\times {10}^{14}$ .

And the energy gap for Gallium Arsenide, $E_g=1.4\text{eV}$ .

The intrinsic carrier concentration is

  $\begin{array}{c}{n}_i=2.1\times {10}^{14}\left({400}^{3/2}\right)e^{\left(\frac{-1.4}{2\left(86\times {10}^{-6}\right)\left(400\right)}\right)}\\ =2.1\times {10}^{15}\left(8000\right)e^{\left(\frac{-1.4}{2\left(86\times {10}^{-6}\right)\left(400\right)}\right)}\\ =2.4429\times {10}^9{\text{cm}}^{-3}\\ \approx 2.443\times {10}^9{\text{cm}}^{-3}\end{array}$

To determine

The intrinsic carrier concentration for the given following materials.

Answer to Problem 1.1TYU

The intrinsic carrier concentration for Silicon, Germanium and Gallium Arsenide:

  $\begin{array}{l}\text{Si:}\\ n_i=1.61\times {10}^8{\text{cm}}^{-3}\\ \text{Ge:}\\ n_i=1.42\times {10}^{12}{\text{cm}}^{-3}\\ \text{GaAs:}\\ n_i=6.02\times {10}^3{\text{cm}}^{-3}\end{array}$

Explanation of Solution

Given information:

The given materials are:

Silicon, Germanium and Gallium Arsenide.

Temperature, T=250K

Calculation:

The intrinsic carrier concentration $\left(n_i\right)$ for semiconductor material is given by:

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

Where,

B is the specific material coefficient.

T is the temperature in kelvin.

  $E_g$ is the bandgap energy.

k is the Boltzman’s constant.

The value of semi-conductor material coefficient for silicon, $B=5.23\times {10}^{15}$ .

And the energy gap for silicon, $E_g=1.1\text{eV}$ .

Now, evaluating the intrinsic carrier concentration:

  $\begin{array}{c}{n}_i=5.23\times {10}^{15}\left({250}^{3/2}\right)e^{\left(\frac{-1.1}{2\left(86\times {10}^{-6}\right)\left(250\right)}\right)}\\ =5.23\times {10}^{15}\left(3952.8470\right)e^{\left(\frac{-1.1}{2\left(86\times {10}^{-6}\right)\left(250\right)}\right)}\\ =5.23\times {10}^{15}\left(3952.8470\right)\left(7.7758\right)\\ =1.6075\times {10}^8{\text{cm}}^{-3}\\ =1.6\times {10}^8{\text{cm}}^{-3}\end{array}$

Similarly, for Germanium:

The value of semi-conductor material coefficient for Germanium, $B=1.66\times {10}^{15}$ .

And the energy gap for Germanium, $E_g=0.66\text{eV}$ .

Now, evaluating the intrinsic carrier concentration:

  $\begin{array}{c}{n}_i=1.66\times {10}^{15}\left({250}^{3/2}\right)e^{\left(\frac{-0.66}{2\left(86\times {10}^{-6}\right)\left(250\right)}\right)}\\ =1.66\times {10}^{15}\left(3952.8470\right)e^{\left(\frac{-0.66}{2\left(86\times {10}^{-6}\right)\left(250\right)}\right)}\\ =1.4173\times {10}^{12}{\text{cm}}^{-3}\\ \approx 1.42\times {10}^{12}{\text{cm}}^{-3}\end{array}$

Similarly, for Gallium Arsenide:

The value of semi-conductor material coefficient for Gallium Arsenide, $B=2.1\times {10}^{14}$ .

And the energy gap for Gallium Arsenide, $E_g=1.4\text{eV}$ .

Now, evaluating the intrinsic carrier concentration:

  $\begin{array}{c}{n}_i=2.1\times {10}^{14}\left({250}^{3/2}\right)e^{\left(\frac{-1.4}{2\left(86\times {10}^{-6}\right)\left(250\right)}\right)}\\ =2.1\times {10}^{15}\left(3952.8470\right)e^{\left(\frac{-1.4}{2\left(86\times {10}^{-6}\right)\left(250\right)}\right)}\\ =6.016\times {10}^3{\text{cm}}^{-3}\\ \approx 6.02\times {10}^3{\text{cm}}^{-3}\end{array}$