Carrier concentration equations (n, p, ni)

1. From first principles: what are we integrating?

WHY the limits? Electrons only conduct if they sit above $E_c$ (conduction band). Holes are missing electrons below $E_v$ (valence band). So each integral covers only its own band.

2. Deriving the density of states $g_c(E)$

Near the band edge the electron behaves like a free particle with an effective mass $m_n^*$: $E - E_c = \frac{\hbar^2 k^2}{2 m_n^*}$

Why? The parabolic approximation: the bottom of a band always looks like a parabola, and curvature $\to$ effective mass.

Count $k$-states in a sphere of radius $k$ (spin factor 2, volume $V$): $N(k) = 2 \cdot \frac{V}{(2\pi)^3}\cdot \frac{4}{3}\pi k^3$

Convert to energy using $k=\sqrt{2m_n^*(E-E_c)}/\hbar$ and differentiate $g_c(E)=\frac{1}{V}\frac{dN}{dE}$:

$\boxed{g_c(E) = \frac{1}{2\pi^2}\left(\frac{2m_n^*}{\hbar^2}\right)^{3/2}\sqrt{E-E_c}}$

3. The Boltzmann approximation (WHY it simplifies everything)

For a non-degenerate semiconductor, $E_F$ sits deep in the gap, so for $E \ge E_c$ we have $E-E_F \gg kT$: $f(E)=\frac{1}{1+e^{(E-E_F)/kT}} \approx e^{-(E-E_F)/kT}$

4. The master result: $n$ and $N_c$

Plug $g_c$ and Boltzmann $f$ into the integral: $n = \int_{E_c}^{\infty}\frac{1}{2\pi^2}\left(\frac{2m_n^*}{\hbar^2}\right)^{3/2}\sqrt{E-E_c}\;e^{-(E-E_F)/kT}dE$

Substitute $x=(E-E_c)/kT$, use $\int_0^\infty \sqrt{x}\,e^{-x}dx=\tfrac{\sqrt\pi}{2}$:

$\boxed{n = N_c\, e^{-(E_c-E_F)/kT}}, \qquad N_c = 2\left(\frac{2\pi m_n^* kT}{h^2}\right)^{3/2}$

By the identical argument for holes (using $1-f\approx e^{-(E_F-E_v)/kT}$):

$\boxed{p = N_v\, e^{-(E_F-E_v)/kT}}, \qquad N_v = 2\left(\frac{2\pi m_p^* kT}{h^2}\right)^{3/2}$

5. The intrinsic concentration $n_i$ and the mass-action law

Multiply $n$ and $p$ — the $E_F$ terms cancel: $np = N_c N_v\, e^{-(E_c-E_v)/kT} = N_c N_v\, e^{-E_g/kT}$

This is independent of $E_F$ → true for any doping. In an intrinsic crystal $n=p\equiv n_i$, so:

$\boxed{n_i = \sqrt{N_c N_v}\; e^{-E_g/2kT}} \quad\Rightarrow\quad \boxed{np = n_i^2}$

Location of intrinsic level $E_i$ (set $n=p$): $E_i = \frac{E_c+E_v}{2} + \frac{kT}{2}\ln\!\frac{N_v}{N_c}$ So $E_i$ sits near midgap, nudged by the small $\ln(N_v/N_c)$ term.

Rewriting $n,p$ around $E_i$ (very handy form): $n = n_i\, e^{(E_F-E_i)/kT}, \qquad p = n_i\, e^{(E_i-E_F)/kT}$

6. Worked examples

7. Feynman + Mnemonic

8. Forecast-then-Verify checkpoints

Flashcards

Connections

• Fermi-Dirac distribution — the occupancy factor $f(E)$ we approximated.
• Density of states — origin of $g_c(E)\propto\sqrt{E-E_c}$.
• Effective mass — sets $m_n^*, m_p^*$ inside $N_c, N_v$.
• Doping and charge neutrality — combines with $np=n_i^2$ to solve for $n,p$.
• Fermi level position — computed from $n$ vs $N_c$.
• Intrinsic vs extrinsic semiconductors — regimes where these equations apply.
• Band gap Eg — controls the exponential in $n_i$.

Concept Map

Hinglish (regional understanding)