--- title: "Carrier Statistics: Intrinsic and Extrinsic Semiconductors" module: Semiconductors moduleNumber: 7 lessonNumber: 2 order: 702 summary: > The number of mobile electrons and holes in a semiconductor follows from the density of states near each band edge and the Fermi-Dirac tail that reaches into it. This lesson derives the effective densities of states, the intrinsic concentration and its exponential gap dependence, the law of mass action, the temperature march of the Fermi level, and the freeze-out, saturation, and intrinsic regimes of a doped crystal. topics: [Semiconductors] sources: - book: Ashcroft & Mermin ref: "Ch. 28 — Homogeneous Semiconductors" - book: Kittel ref: "Ch. 8 — Semiconductor Crystals" - book: Hook & Hall ref: "Ch. 5 — Semiconductors" draft: false --- The [band picture](/condensed-matter/semiconductors/semiconductor-bands-and-junctions) sorts solids by gap width but leaves the carrier count qualitative. Making it quantitative is the task of this lesson: given the gap $E_g$, the band-edge effective masses, the doping, and the temperature, how many electrons sit in the conduction band and how many holes in the valence band? The answer rests on two inputs — the density of states near each band edge and the Fermi-Dirac occupation that reaches into it — and produces the exponential temperature laws that govern every semiconductor device. ## Carriers from the density of states Near the bottom of the conduction band the energy is parabolic in the crystal momentum, $E(\vec k) = E_c + \hbar^2 k^2 / 2 m_e^\ast$, with $m_e^\ast$ the conduction-band [effective mass](/condensed-matter/band-theory/fermi-surfaces-and-semiclassical-dynamics). A parabolic band has the same density of states as a free particle of that mass, shifted to start at $E_c$: $$ g_c(E) = \frac{1}{2\pi^2}\left(\frac{2 m_e^\ast}{\hbar^2}\right)^{3/2}\sqrt{E - E_c}, \qquad E \ge E_c. $$ Symmetrically, holes near the top of the valence band have $g_v(E) = \frac{1}{2\pi^2}(2 m_h^\ast/\hbar^2)^{3/2}\sqrt{E_v - E}$ for $E \le E_v$, with $m_h^\ast$ the hole effective mass. The electron concentration is the density of states weighted by the probability that a state is occupied: $$ n = \int_{E_c}^{\infty} g_c(E)\, f(E)\,\d E, \qquad f(E) = \frac{1}{e^{(E - E_F)/k_B T} + 1}. $$ In an undoped or lightly doped crystal the Fermi level sits deep in the gap, many $k_B T$ below $E_c$, so throughout the conduction band $E - E_F \gg k_B T$ and the Fermi-Dirac factor collapses to its **Boltzmann tail**, $f(E) \approx e^{-(E - E_F)/k_B T}$. This is the **nondegenerate** limit, valid whenever the carriers are dilute. The integral is then a standard Gamma integral: $$ n = e^{-(E_c - E_F)/k_B T}\,\frac{1}{2\pi^2}\left(\frac{2 m_e^\ast}{\hbar^2}\right)^{3/2}\int_{E_c}^{\infty}\sqrt{E - E_c}\;e^{-(E - E_c)/k_B T}\,\d E. $$ With $x = (E - E_c)/k_B T$ the integral is $(k_B T)^{3/2}\int_0^\infty x^{1/2}e^{-x}\,\d x = (k_B T)^{3/2}\,\Gamma(\tfrac32) = (k_B T)^{3/2}\tfrac{\sqrt\pi}{2}$, and the concentration takes the compact form $$ n = N_c\, e^{-(E_c - E_F)/k_B T}, \qquad N_c = 2\left(\frac{m_e^\ast k_B T}{2\pi\hbar^2}\right)^{3/2}. $$ The prefactor $N_c$ is the **effective density of states** of the conduction band: the whole band acts as though $N_c$ states were collapsed onto the edge $E_c$, a distance $E_c - E_F$ above the Fermi level. The identical argument for holes gives $$ p = N_v\, e^{-(E_F - E_v)/k_B T}, \qquad N_v = 2\left(\frac{m_h^\ast k_B T}{2\pi\hbar^2}\right)^{3/2}. $$ > **Definition (Effective density of states).** The **effective density of > states** $N_c$ (and $N_v$) is the number of conduction-band (valence-band) > states per unit volume that, if placed exactly at the band edge, would give the > actual carrier concentration through a single Boltzmann factor. It scales as > $T^{3/2}$ and as the $3/2$ power of the band-edge effective mass. $$ % caption: The electron concentration n is the overlap of the rising band-edge % density of states g_c(E) with the falling Fermi-Dirac tail f(E); the shaded % product peaks a few k_BT above E_c and gives n = N_c exp[-(E_c - E_F)/k_BT]. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % vertical energy axis \draw[->, black] (0,-0.3) -- (0,5.0) node[left] {$E$}; \draw[->, black] (0,0) -- (5.6,0) node[below] {density}; % conduction band edge \draw[black, dashed] (0,3.2) -- (5.4,3.2); \node[anchor=west, font=\scriptsize] at (4.7,3.35) {$E_c$}; % Fermi level \draw[black, dashed] (0,1.6) -- (5.4,1.6); \node[anchor=west, font=\scriptsize] at (4.7,1.45) {$E_F$}; % density of states sqrt above E_c (drawn horizontally: value vs E) \draw[acc, thick, domain=3.2:4.9, samples=40, variable=\y] plot ({1.6*sqrt(\y-3.2)},{\y}); \node[acc, anchor=west, font=\scriptsize] at (1.9,4.5) {$g_c(E)$}; % Fermi tail (exponential decreasing upward) \draw[black, thick, domain=1.6:4.9, samples=60, variable=\y] plot ({2.6*exp(-(\y-1.6)*1.3)},{\y}); \node[black, anchor=west, font=\scriptsize] at (1.3,2.05) {$f(E)$}; % product (occupied electrons) shaded lobe above E_c \draw[acc, thick, fill=acc!14, domain=3.2:4.6, samples=40, variable=\y] plot ({1.6*sqrt(\y-3.2)*exp(-(\y-1.6)*1.3)*1.9},{\y}) -- (0,4.6) -- (0,3.2) -- cycle; \node[acc, anchor=west, font=\scriptsize] at (0.55,3.75) {$n$}; \end{tikzpicture} $$ ## The intrinsic concentration and the law of mass action The product $np$ eliminates the Fermi level entirely: $$ np = N_c N_v\, e^{-(E_c - E_v)/k_B T} = N_c N_v\, e^{-E_g/k_B T}, $$ with $E_g = E_c - E_v$ the band gap. The right side depends only on the material and temperature, not on doping — this is the **law of mass action**. Defining the **intrinsic carrier concentration** $n_i$ by $n_i^2 \equiv np$, $$ n_i = \sqrt{N_c N_v}\; e^{-E_g/2k_B T} = 2\left(\frac{k_B T}{2\pi\hbar^2}\right)^{3/2}(m_e^\ast m_h^\ast)^{3/4}\,e^{-E_g/2k_B T}. $$ > **Theorem (Law of mass action).** In any nondegenerate semiconductor at thermal > equilibrium, the product of the electron and hole concentrations is fixed by the > temperature and gap alone: $np = n_i^2(T)$, independent of the doping. Raising > one carrier population by doping suppresses the other in exact reciprocal > proportion. The exponent $e^{-E_g/2k_B T}$ is the dominant factor. Because $E_g \gg k_B T$ at room temperature (silicon: $E_g/k_B T \approx 43$), $n_i$ is astronomically smaller than the atomic density $\sim 5\times10^{22}\ \text{cm}^{-3}$, and it is exquisitely sensitive to both gap and temperature. > **Worked example.** Silicon at $T = 300\ \text{K}$ has $E_g = 1.12\ \text{eV}$, > $N_c = 2.8\times10^{19}\ \text{cm}^{-3}$, and $N_v = 1.0\times10^{19}\ > \text{cm}^{-3}$. With $k_B T = 0.0259\ \text{eV}$, > > $$ > n_i = \sqrt{(2.8\times10^{19})(1.0\times10^{19})}\;e^{-1.12/(2\times0.0259)}\ \text{cm}^{-3}. > $$ > > The prefactor is $1.7\times10^{19}\ \text{cm}^{-3}$ and the exponent is > $e^{-21.6} = 4.1\times10^{-10}$, giving $n_i \approx 7\times10^{9}\ > \text{cm}^{-3}$ (the accepted value is close to $1\times10^{10}\ \text{cm}^{-3}$; > the difference is the temperature dependence of $E_g$ and band nonparabolicity). > Germanium's narrower $0.67\ \text{eV}$ gap raises $n_i$ to $\sim 2\times10^{13}\ > \text{cm}^{-3}$, and gallium arsenide's wider $1.42\ \text{eV}$ gap lowers it to > $\sim 2\times10^{6}\ \text{cm}^{-3}$ — seven orders of magnitude across three > common semiconductors. The mass-action balance is the reason a small dose of one carrier type strongly suppresses the other. In n-type silicon with $n = 10^{16}\ \text{cm}^{-3}$, the hole concentration is driven down to $p = n_i^2/n \approx 10^{4}\ \text{cm}^{-3}$, twelve orders of magnitude below the electrons. The doped carrier is the **majority** carrier; its mass-action partner is the **minority** carrier, and the minority population — small as it is — controls junction currents in the lessons that follow. $$ % caption: The law of mass action as a hyperbola np = n_i^2. Undoped material % sits on the diagonal n = p = n_i; n-type doping slides the state down the % hyperbola (n up, p down), p-type doping slides it the other way, always on the % same curve. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (5.4,0) node[below] {$n$}; \draw[->, black] (0,0) -- (0,4.6) node[left] {$p$}; % hyperbola np = const \draw[acc, very thick, domain=0.62:4.6, samples=80, variable=\x] plot ({\x},{2.6/\x}); \node[acc, anchor=west, font=\scriptsize] at (3.4,1.15) {$np = n_i^2$}; % diagonal n=p \draw[black, dashed] (0,0) -- (3.0,3.0); % intrinsic point \fill[black] (1.612,1.612) circle (2pt); \node[anchor=south west, font=\scriptsize] at (1.7,1.66) {intrinsic}; % n-type point \fill[acc] (3.8,0.684) circle (2pt); \node[acc, anchor=west, font=\scriptsize] at (3.9,0.7) {n-type}; % p-type point \fill[acc] (0.75,3.47) circle (2pt); \node[acc, anchor=west, font=\scriptsize] at (0.85,3.6) {p-type}; \end{tikzpicture} $$ ## The Fermi level in intrinsic material Setting $n = p$ in the two Boltzmann forms fixes the intrinsic Fermi level $E_i$. Equating $N_c e^{-(E_c - E_i)/k_B T} = N_v e^{-(E_i - E_v)/k_B T}$ and solving, $$ E_i = \frac{E_c + E_v}{2} + \frac{1}{2}k_B T \ln\frac{N_v}{N_c} = \frac{E_c + E_v}{2} + \frac{3}{4}k_B T \ln\frac{m_h^\ast}{m_e^\ast}. $$ At $T = 0$ the Fermi level sits exactly at midgap. When the hole mass exceeds the electron mass (as in silicon, $m_h^\ast > m_e^\ast$) the second term is positive and $E_i$ drifts upward toward the conduction band as $T$ rises; the shift is only a few $k_B T \ln(m_h^\ast/m_e^\ast)$, tens of millivolts at most, so $E_i$ stays close to midgap over the whole ordinary temperature range. ## Doping and charge neutrality A donor contributes a level $E_d$ a small energy $\varepsilon_d = E_c - E_d$ below the conduction band; an acceptor a level $E_a$ an energy $\varepsilon_a = E_a - E_v$ above the valence band. Both are shallow (silicon: $\varepsilon_d \approx 0.045\ \text{eV}$ for phosphorus, $\varepsilon_a \approx 0.045\ \text{eV}$ for boron), of order $k_B T$ at room temperature, so most dopants are ionized. A donor that has given up its electron is a fixed positive charge $N_d^+$; an acceptor that has captured one is a fixed negative charge $N_a^-$. Global **charge neutrality** ties the mobile and fixed charges together: $$ n + N_a^- = p + N_d^+. $$ The ionized-donor fraction follows from the occupation of the donor level, which carries a factor $\tfrac12$ from spin degeneracy of the bound state: $$ N_d^+ = \frac{N_d}{1 + 2\,e^{(E_F - E_d)/k_B T}}, \qquad N_a^- = \frac{N_a}{1 + 4\,e^{(E_a - E_F)/k_B T}}. $$ Consider an n-type crystal with donor density $N_d$ and no acceptors. Neutrality reads $n = p + N_d^+$. Together with $np = n_i^2$ this is a closed system for $n$ and $p$ at each temperature. Three regimes emerge as temperature rises, and they are the defining behavior of a doped semiconductor. $$ % caption: The three doping regimes on a log-n versus 1/T plot for an n-type % crystal. At low T carriers freeze onto donors (steep slope); an intermediate % saturation plateau has all donors ionized so n = N_d; at high T intrinsic % generation overwhelms the doping and n follows the steep n_i slope. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.0,0) node[below] {$\frac{1}{T}$}; \draw[->, black] (0,0) -- (0,4.4) node[left] {$\ln n$}; % intrinsic branch (steep, left = high T = small 1/T) \draw[acc, very thick] (0.4,4.0) -- (2.2,2.4); \node[acc, anchor=south west, font=\scriptsize] at (0.4,3.7) {intrinsic}; % saturation plateau \draw[acc, very thick] (2.2,2.4) -- (4.8,2.4); \node[acc, anchor=south, font=\scriptsize] at (3.5,2.5) {saturation}; \draw[black, dashed] (0,2.4) -- (2.2,2.4); \node[anchor=east, font=\scriptsize] at (-0.05,2.4) {$N_d$}; % freeze-out branch (steep down to the right) \draw[acc, very thick] (4.8,2.4) -- (6.6,0.7); \node[acc, anchor=north east, font=\scriptsize] at (6.7,1.0) {freeze-out}; % markers for regime boundaries \draw[black, dashed] (2.2,0) -- (2.2,2.4); \draw[black, dashed] (4.8,0) -- (4.8,2.4); \end{tikzpicture} $$ **Freeze-out (low $T$).** Thermal energy is too small to ionize all donors, so electrons condense back onto their donor atoms. With $n \ll N_d$ and $p$ negligible, neutrality gives $n \approx N_d^+$, and solving the occupation relation in the dilute limit yields $$ n \approx \sqrt{\tfrac12 N_c N_d}\; e^{-\varepsilon_d/2k_B T}. $$ The concentration falls off with a slope set by half the donor binding energy — the shallow analogue of the intrinsic slope, with $\varepsilon_d$ in place of $E_g$. **Saturation / extrinsic ($T$ intermediate).** Once $k_B T \gtrsim \varepsilon_d$ every donor is ionized, $N_d^+ = N_d$, while intrinsic generation is still negligible ($n_i \ll N_d$). Neutrality collapses to $n \approx N_d$: the electron concentration is pinned at the donor density and is essentially flat over a wide temperature window. This is the useful regime for devices, where the carrier count is set by doping and is nearly temperature-independent. **Intrinsic (high $T$).** When $T$ climbs high enough that $n_i > N_d$, thermal generation across the full gap dominates the dopants; $n \approx p \approx n_i$ and the doping becomes irrelevant. The concentration resumes the steep intrinsic slope $e^{-E_g/2k_B T}$. The onset temperature rises with doping and with gap, which is why wide-gap semiconductors like silicon carbide and gallium nitride keep their extrinsic behavior — and their device function — to much higher temperatures than germanium. ## The temperature march of the Fermi level Because $n = N_c e^{-(E_c - E_F)/k_B T}$, the Fermi level position is a direct readout of the carrier count: $E_c - E_F = k_B T \ln(N_c/n)$. Following $n$ through the three regimes traces $E_F$ across the gap. - **Near $T = 0$:** in n-type material $E_F$ lies **between the donor level and the conduction band**, because the highest occupied states are the donor electrons. - **Saturation:** with $n = N_d$ fixed, $E_c - E_F = k_B T \ln(N_c/N_d)$ grows linearly in $T$ — the Fermi level **descends from near $E_c$ toward midgap** as temperature rises. - **Intrinsic:** $E_F \to E_i$ near the middle of the gap, the same limit a p-type crystal approaches from below. $$ % caption: Fermi-level position versus temperature. In n-type material E_F starts % just below the conduction band, descends through the saturation regime, and % merges into the near-midgap intrinsic line E_i at high T; the p-type curve is % the mirror image rising from just above the valence band. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.0,0) node[below] {$T$}; \draw[->, black] (0,0) -- (0,4.4) node[left] {$E_F$}; % band edges \draw[black, thick] (0,3.8) -- (6.6,3.8); \node[anchor=west, font=\scriptsize] at (6.65,3.8) {$E_c$}; \draw[black, thick] (0,0.6) -- (6.6,0.6); \node[anchor=west, font=\scriptsize] at (6.65,0.6) {$E_v$}; % midgap intrinsic line \draw[black, dashed] (0,2.2) -- (6.6,2.2); \node[anchor=west, font=\scriptsize] at (6.65,2.2) {$E_i$}; % n-type: starts near E_c, descends to E_i \draw[acc, very thick, domain=0.2:6.2, samples=60, variable=\x] plot ({\x},{2.2 + 1.35*exp(-\x*0.55)}); \node[acc, anchor=south, font=\scriptsize] at (1.4,3.55) {n-type}; % p-type: mirror, rises from near E_v to E_i \draw[acc, very thick, dashed, domain=0.2:6.2, samples=60, variable=\x] plot ({\x},{2.2 - 1.35*exp(-\x*0.55)}); \node[acc, anchor=north, font=\scriptsize] at (1.4,0.85) {p-type}; \end{tikzpicture} $$ The heavily doped limit deserves a caution. When $N_d$ approaches $N_c$ the Fermi level rises into or above the conduction band, the Boltzmann approximation fails, and the semiconductor becomes **degenerate** — the electron gas behaves like that of a [metal](/condensed-matter/free-electron-fermi-gas/sommerfeld-model-and-heat-capacity), with the full Fermi-Dirac statistics and a Fermi surface. Degenerate doping is exactly what tunnel diodes and the ohmic contacts of every chip rely on. For the moderate doping of ordinary devices, however, the nondegenerate formulas above hold, and they set the stage for [carrier transport](/condensed-matter/semiconductors/carrier-transport-and-recombination): knowing how many carriers there are, the next question is how fast they move and how long they live.