--- title: The p-n Junction in Depth module: Semiconductors moduleNumber: 7 lessonNumber: 4 order: 704 summary: > Joining p-type and n-type silicon aligns their Fermi levels and leaves a depletion region of fixed charge with a built-in potential. This lesson derives the space-charge field and potential from Poisson's equation in the depletion approximation, the built-in voltage from Fermi-level alignment, the Shockley diode equation from minority-carrier diffusion, junction and diffusion capacitance, and the avalanche and Zener breakdown mechanisms. topics: [Semiconductors] sources: - book: Ashcroft & Mermin ref: "Ch. 29 — Inhomogeneous Semiconductors" - book: Kittel ref: "Ch. 8 — Semiconductor Crystals" - book: Hook & Hall ref: "Ch. 5 — Semiconductors" --- The p-n junction is a single crystal doped p-type on one side and n-type on the other. The [survey lesson](/condensed-matter/semiconductors/semiconductor-bands-and-junctions) sketched its equilibrium and its rectifying diode curve; this lesson derives them. The [transport equations](/condensed-matter/semiconductors/carrier-transport-and-recombination) supply the tools — Poisson's equation for the field, minority-carrier diffusion for the current — and the result is the Shockley diode equation, the foundation of every semiconductor device. ## Equilibrium and the built-in potential When the two regions are joined, electrons pour from the electron-rich n side into the p side and holes the other way. Each departing electron leaves behind a fixed, ionized donor $N_d^+$; each departing hole leaves a fixed, ionized acceptor $N_a^-$. A **space-charge** double layer builds up — negative on the p side, positive on the n side — and its field opposes further diffusion. Equilibrium is reached when the **Fermi level is flat** across the whole crystal, the thermodynamic condition for no net particle flow. Flattening the Fermi level forces the bands to bend. Far into the n side the conduction band lies close below $E_F$; far into the p side it lies well above. The total bend is the **built-in potential** $V_{bi}$. Its value follows from the [carrier statistics](/condensed-matter/semiconductors/intrinsic-and-extrinsic-semiconductors): in the neutral n region $n = N_d = n_i\,e^{(E_F - E_i)/k_B T}$, and in the neutral p region $p = N_a = n_i\,e^{(E_i - E_F)/k_B T}$. The difference in the intrinsic level between the two sides is $e V_{bi}$, giving $$ V_{bi} = \frac{k_B T}{e}\ln\frac{N_a N_d}{n_i^2}. $$ > **Worked example.** A silicon junction with $N_a = N_d = 10^{16}\ \text{cm}^{-3}$ > and $n_i = 10^{10}\ \text{cm}^{-3}$ at $300\ \text{K}$ has > > $$ > V_{bi} = (0.0259\ \text{V})\ln\frac{10^{16}\cdot 10^{16}}{(10^{10})^2} = (0.0259\ \text{V})\ln(10^{12}) = 0.72\ \text{V}. > $$ > > Doping each side more heavily raises $V_{bi}$ only logarithmically; the built-in > potential of a silicon junction is always a few tenths of a volt, always below > the gap $E_g/e = 1.12\ \text{V}$. ## The depletion approximation To find the field and potential, model the space-charge region with the **depletion approximation**: the transition region $-x_p < x < x_n$ is swept entirely clear of mobile carriers, leaving only the fixed dopant charge, while the neutral regions beyond are perfectly screened. The charge density is then a step, $$ \rho(x) = \begin{cases} -e N_a, & -x_p < x < 0, \\[2pt] +e N_d, & 0 < x < x_n, \end{cases} $$ and zero elsewhere. Overall neutrality of the depletion region requires equal total charge on the two sides, $$ N_a\,x_p = N_d\,x_n, $$ so the depletion layer extends further into the more lightly doped side. Poisson's equation $\d E/\d x = \rho/\varepsilon$ (with $\varepsilon$ the semiconductor permittivity) integrates the step charge into a **triangular field** that peaks at the metallurgical junction $x = 0$ and falls linearly to zero at each depletion edge: $$ E_{\max} = \frac{e N_d x_n}{\varepsilon} = \frac{e N_a x_p}{\varepsilon}. $$ Integrating once more, $\d\varphi/\d x = -E$, gives a potential that varies quadratically through the depletion region, its total drop equal to the area of the field triangle: $$ V_{bi} = \tfrac12 E_{\max}(x_p + x_n) = \frac{e}{2\varepsilon}\!\left(N_a x_p^2 + N_d x_n^2\right). $$ Solving with the neutrality condition for the total **depletion width** $W = x_p + x_n$, $$ W = \sqrt{\frac{2\varepsilon\,V_{bi}}{e}\left(\frac{1}{N_a} + \frac{1}{N_d}\right)}. $$ For the symmetric silicon junction above, $W \approx 0.3\ \mu\text{m}$ — far narrower than the tens of microns of minority-carrier diffusion length, confirming that recombination happens in the neutral regions, not the depletion layer. $$ % caption: The depletion region in three stacked profiles. The fixed charge is a % step (negative acceptors on the p side, positive donors on the n side); its % integral is a triangular field peaking at the junction; the field's integral is % the smooth potential step of height V_bi. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % --- charge density panel (top) --- \begin{scope}[yshift=5.4cm] \draw[->, black] (-2.8,0) -- (3.0,0) node[below, font=\scriptsize] {$x$}; \draw[->, black] (0,-1.0) -- (0,1.1) node[left, font=\scriptsize] {charge}; \draw[acc, very thick] (-2.4,0) -- (-1.6,0) -- (-1.6,-0.8) -- (0,-0.8) -- (0,0.8) -- (1.4,0.8) -- (1.4,0) -- (2.6,0); \node[font=\scriptsize, anchor=north] at (-1.9,-0.35) {p: neg}; \node[font=\scriptsize, anchor=south] at (0.7,0.35) {n: pos}; \end{scope} % --- field panel (middle) --- \begin{scope}[yshift=2.7cm] \draw[->, black] (-2.8,0) -- (3.0,0) node[below, font=\scriptsize] {$x$}; \draw[->, black] (0,-1.3) -- (0,0.4) node[left, font=\scriptsize] {$E$}; \draw[acc, very thick] (-2.4,0) -- (-1.6,0) -- (0,-1.1) -- (1.4,0) -- (2.6,0); \end{scope} % --- potential panel (bottom) --- \begin{scope} \draw[->, black] (-2.8,0) -- (3.0,0) node[below, font=\scriptsize] {$x$}; \draw[->, black] (0,-0.2) -- (0,1.5) node[left, font=\scriptsize] {potential}; \draw[acc, very thick] (-2.4,0) -- (-1.6,0) .. controls (-0.8,0.05) and (-0.4,1.05) .. (1.4,1.1) -- (2.6,1.1); \draw[black, dashed] (-1.6,1.1) -- (1.4,1.1); \node[font=\scriptsize, anchor=east] at (-1.65,0.9) {$V_{bi}$}; \end{scope} \end{tikzpicture} $$ ## Band bending under bias An external voltage $V$ drops almost entirely across the high-resistance depletion region, adding to or subtracting from the built-in step. Every equilibrium formula carries over with $V_{bi} \to V_{bi} - V$ (taking $V > 0$ for forward bias, p side positive): $$ W(V) = \sqrt{\frac{2\varepsilon(V_{bi} - V)}{e}\left(\frac{1}{N_a} + \frac{1}{N_d}\right)}. $$ **Forward bias** ($V > 0$) lowers the barrier to $V_{bi} - V$, narrows the depletion layer, and lets carriers diffuse across in large numbers. **Reverse bias** ($V < 0$) raises the barrier, widens the depletion layer, and chokes off diffusion. The junction conducts one way — the rectifying action. $$ % caption: Band bending in three bias conditions. At equilibrium the flat Fermi % level forces a barrier eV_bi. Forward bias raises the p-side bands, shrinking the % barrier and flooding the junction; reverse bias lowers them, widening the barrier % and blocking diffusion. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % --- equilibrium --- \begin{scope} \draw[acc, very thick] (0,2.0) -- (0.9,2.0) .. controls (1.4,2.0) and (1.5,0.9) .. (2.0,0.9) -- (2.9,0.9); \draw[black, dashed] (0,1.2) -- (2.9,1.2); \node[anchor=south, font=\scriptsize] at (1.45,2.3) {equilibrium}; \node[black, anchor=west, font=\scriptsize] at (2.4,1.2) {$E_F$}; \end{scope} % --- forward --- \begin{scope}[xshift=3.9cm] \draw[acc, very thick] (0,1.6) -- (0.9,1.6) .. controls (1.4,1.6) and (1.5,0.9) .. (2.0,0.9) -- (2.9,0.9); \node[anchor=south, font=\scriptsize] at (1.45,2.3) {forward}; \draw[acc, ->, thick] (0.4,1.75) -- (1.1,1.75); \end{scope} % --- reverse --- \begin{scope}[xshift=7.8cm] \draw[acc, very thick] (0,2.6) -- (0.9,2.6) .. controls (1.4,2.6) and (1.5,0.9) .. (2.0,0.9) -- (2.9,0.9); \node[anchor=south, font=\scriptsize] at (1.45,2.9) {reverse}; \end{scope} \end{tikzpicture} $$ ## The Shockley diode equation The current follows from minority-carrier injection. Lowering the barrier by $eV$ raises the minority concentration at each depletion edge by the Boltzmann factor $e^{eV/k_B T}$ — the **law of the junction**. On the n side the hole concentration at the edge becomes $p_n(x_n) = p_{n0}\,e^{eV/k_B T}$, an excess $$ \delta p_n = p_{n0}\!\left(e^{eV/k_B T} - 1\right), \qquad p_{n0} = \frac{n_i^2}{N_d}. $$ These excess holes diffuse into the neutral n region and recombine over a diffusion length $L_p$, exactly the injection profile derived from the [continuity equation](/condensed-matter/semiconductors/carrier-transport-and-recombination). The diffusion current at the edge is $J_p = e D_p\,\delta p_n / L_p$. Adding the symmetric electron injection into the p side gives the total current density, $$ J = e\!\left(\frac{D_p\,p_{n0}}{L_p} + \frac{D_n\,n_{p0}}{L_n}\right)\!\left(e^{eV/k_B T} - 1\right) = J_0\!\left(e^{eV/k_B T} - 1\right). $$ > **Theorem (Shockley diode equation).** An ideal p-n junction carries current > $I = I_0\,(e^{eV/k_B T} - 1)$, where the **saturation current** > $I_0 = eA(D_p p_{n0}/L_p + D_n n_{p0}/L_n)$ is fixed by minority-carrier > diffusion. Forward bias gives an exponentially rising current; reverse bias > ($V \ll -k_B T/e$) saturates at the small reverse current $-I_0$, set by the > minority concentrations $n_i^2/N$ and therefore by the gap through $n_i^2 \propto > e^{-E_g/k_B T}$. The saturation current is the diode's Achilles heel and its thermometer: because $I_0 \propto n_i^2 \propto e^{-E_g/k_B T}$, it roughly doubles every $10\ \text{K}$, which is why germanium diodes ($E_g = 0.67\ \text{eV}$) leak far more than silicon ($1.12\ \text{eV}$) and why high-temperature electronics turns to wide-gap materials. $$ % caption: The ideal-diode characteristic I = I_0(exp(eV/k_BT) - 1). Forward bias % rises exponentially past the turn-on near V_bi; reverse bias saturates at the % tiny current I_0, magnified in the inset scale, giving one-way conduction. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-3.4,0) -- (3.0,0) node[below] {$V$}; \draw[->, black] (0,-1.0) -- (0,3.2) node[left] {$I$}; \draw[acc, very thick, domain=0:1.55, samples=60, variable=\x] plot ({\x*1.5},{0.24*(exp(\x*2.0)-1)}); \draw[acc, very thick] (0,0) -- (-3.0,-0.24); \node[acc, anchor=south, font=\scriptsize] at (1.9,2.5) {forward}; \node[acc, anchor=north, font=\scriptsize] at (-1.8,-0.28) {reverse}; \draw[black, dashed] (-3.0,-0.24) -- (0,-0.24); \node[anchor=west, font=\scriptsize] at (0.1,-0.5) {saturation $I_0$}; \end{tikzpicture} $$ ## Junction and diffusion capacitance A junction stores charge two ways, and each is a capacitance. In reverse and small forward bias the stored charge is the fixed dopant charge in the depletion layer, $Q = eA N_d x_n$. Differentiating with the depletion width gives the **junction (depletion) capacitance** $$ C_j = \frac{\d Q}{\d V} = \frac{\varepsilon A}{W(V)} = A\sqrt{\frac{e\varepsilon}{2(V_{bi} - V)}\frac{N_a N_d}{N_a + N_d}}, $$ the same form as a parallel-plate capacitor of gap $W$, but with $W$ tunable by the reverse bias. A junction used this way is a voltage-controlled capacitor, the **varactor** that tunes radio-frequency oscillators; a plot of $C_j^{-2}$ versus $V$ is a straight line whose slope gives the doping. Under strong forward bias a second, larger capacitance dominates: the stored excess minority charge $Q = I\tau$ changes with the current, giving the **diffusion capacitance** $C_d = \d Q/\d V = (e/k_B T)\,I\tau$, proportional to the forward current and to the lifetime. Diffusion capacitance is what slows a diode's switching from conducting to blocking — the stored charge must be removed first. ## Breakdown Beyond a critical reverse voltage the tiny saturation current turns sharply into a large reverse current — **breakdown**. Two distinct mechanisms produce it. - **Avalanche breakdown.** In the strong field of a wide depletion region a carrier gains enough kinetic energy between collisions to knock a bound electron across the gap by **impact ionization**, creating a new electron-hole pair; the offspring are accelerated in turn, and the multiplication runs away. Avalanche dominates in lightly doped junctions, has a breakdown voltage that **rises** with temperature (hotter lattice scatters carriers before they reach ionizing energy), and sets the reverse rating of power and high-voltage diodes. - **Zener breakdown.** In a heavily doped, narrow junction the depletion layer is so thin that valence electrons on the p side **tunnel** directly into empty conduction states on the n side once the bands overlap under reverse bias. Zener (tunneling) breakdown dominates below about $5\ \text{V}$ in silicon and has a breakdown voltage that **falls** with temperature. Zener diodes exploit the sharp, stable breakdown edge as a voltage reference. $$ % caption: The reverse breakdown region. Below the breakdown voltage only the % saturation current flows; at V_BR the current turns down steeply as avalanche % multiplication or Zener tunneling sets in, clamping the voltage across the diode. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-5.2,0) -- (1.2,0) node[right] {$V$}; \draw[->, black] (0,-3.0) -- (0,1.0) node[left] {$I$}; % small forward hint \draw[acc, very thick, domain=0:0.7, samples=30, variable=\x] plot ({\x*1.2},{0.5*(exp(\x*1.6)-1)}); % reverse saturation flat \draw[acc, very thick] (0,-0.12) -- (-4.0,-0.28); % breakdown knee going down \draw[acc, very thick] (-4.0,-0.28) .. controls (-4.35,-0.9) and (-4.4,-1.8) .. (-4.45,-2.8); \draw[black, dashed] (-4.0,0) -- (-4.0,-0.28); \node[anchor=south, font=\scriptsize] at (-4.0,0.05) {$V_{BR}$}; \node[acc, anchor=west, font=\scriptsize] at (-3.9,-1.8) {breakdown}; \end{tikzpicture} $$ The junction physics assembled here — the built-in field, the exponential injection current, the stored charge, and the breakdown edge — is the vocabulary of every device. Cascading two junctions makes a transistor, and running a junction as a light source or a photon collector makes the optoelectronic devices of the [final lesson](/condensed-matter/semiconductors/transistors-and-optoelectronics).