Semiconductors: Diodes, LEDs & Logic Gates

Intrinsic and Extrinsic Semiconductors

Semiconductors (Si, Ge) have a small band gap (Si: 1.1 eV, Ge: 0.67 eV) between the valence and conduction bands. At 0 K, the valence band is completely filled and the conduction band is empty — the material is an insulator. At room temperature, thermal excitation promotes some electrons to the conduction band, creating electron-hole pairs.
In an intrinsic (pure) semiconductor, $n_{e}$ = $n_{h}$ = $n_{i}$ (intrinsic carrier concentration).
Conductivity: $\sigma$ = $n_{i}$e($\mu_{e}$ + $\mu_{h}$), where $\mu_{e}$ and $\mu_{h}$ are electron and hole mobilities. Electron mobility is always greater than hole mobility.

Doping creates extrinsic semiconductors. n-type: pentavalent dopant (P, As, Sb) donates extra electrons.
Majority carriers = electrons, minority = holes.
$n_{e}$ >> $n_{h}$, but $n_{e}$ * $n_{h}$ = $n_{i}^{2}$ (mass action law). p-type: trivalent dopant (B, Al, In, Ga) creates holes.
Majority = holes, minority = electrons. Both n-type and p-type are electrically neutral overall. The Fermi level shifts toward the conduction band in n-type and toward the valence band in p-type.

p-n Junction

When p-type and n-type are joined, electrons diffuse from n to p and holes from p to n. This diffusion leaves behind immobile ions, creating a depletion region with a built-in electric field (from n to p) that opposes further diffusion. At equilibrium, diffusion current = drift current. The depletion region has no free carriers and acts as an insulator. The potential barrier (contact potential) is ~0.7 V for Si and ~0.3 V for Ge.

Forward and Reverse Bias

Forward bias: p connected to positive, n to negative terminal. External voltage opposes the barrier, reducing the depletion width. Above the threshold voltage (~0.7 V Si, ~0.3 V Ge), current increases exponentially: I = $I_{0}$*(e^(eV/kT) - 1). The junction has low resistance.

Reverse bias: p connected to negative, n to positive. External voltage reinforces the barrier, widening the depletion region. Only a tiny reverse saturation current $I_{0}$ (due to minority carriers) flows. At sufficiently high reverse voltage, breakdown occurs: Zener breakdown (thin junction, strong field pulls electrons from bonds) or avalanche breakdown (thick junction, carriers gain enough energy to ionize atoms through collisions).

I-V Characteristics

The diode equation: I = $I_{0}$*(e^(eV/nkT) - 1), where n is the ideality factor (1-2). Forward: exponential rise above threshold. Reverse: small constant $I_{0}$ until breakdown voltage. The diode is essentially a one-way valve for current. Dynamic resistance $r_{d}$ = dV/dI (very low in forward, very high in reverse).

Zener Diode

Heavily doped p-n junction designed to operate in reverse breakdown. The breakdown voltage $V_{Z}$ is sharply defined and the current can vary widely while $V_{Z}$ remains nearly constant. Used as a voltage regulator: connected in reverse bias with a series resistor $R_{s}$.
Output voltage $V_{out}$ = $V_{Z}$ (constant) as long as the input voltage exceeds $V_{Z}$.
The series resistance absorbs voltage fluctuations: $I_{s}$ = ($V_{in}$ - $V_{Z}$)/$R_{s}$.
Load current $I_{L}$ = $\frac{V_{Z}}{R_{L}}$.
Zener current $I_{Z}$ = $I_{s}$ - $I_{L}$ must remain positive for regulation.

Optoelectronic Devices

LED (Light Emitting Diode): Forward-biased p-n junction where electron-hole recombination releases energy as photons (instead of heat). The photon energy ≈ band gap: E = hc/$\lambda$. GaAs (IR, 1.4 eV), GaAsP (red-yellow), GaN (blue-UV). LEDs require forward voltage of 1.5-3 V depending on color.

Photodiode: Reverse-biased p-n junction. Incident photons create electron-hole pairs in the depletion region. The reverse current increases with light intensity, proportional to incident power. Operates in the third quadrant of the I-V curve. Used in optical communication, light detection.

Solar cell: Essentially an unbiased photodiode. Photons generate electron-hole pairs; the built-in field separates them, creating a voltage (photovoltaic effect). Open circuit voltage ~0.5-0.6 V per cell for Si.
Fill factor = ($V_{mp}$ * $I_{mp}$)/($V_{oc}$ * $I_{sc}$) measures efficiency.

Transistor Basics

A transistor (BJT) has three doped regions: emitter (heavily doped), base (thin, lightly doped), and collector (moderately doped). npn: emitter injects electrons into the thin base; most pass through to the collector.
$I_{E}$ = $I_{B}$ + $I_{C}$.
Current gain: $\beta$ = $\frac{I_{C}}{I_{B}}$ (typically 20-200), $\alpha$ = $\frac{I_{C}}{I_{E}}$ (close to 1).
Relation: $\beta$ = $\alpha$/(1-$\alpha$). The transistor amplifies: small changes in $I_{B}$ produce large changes in $I_{C}$.

Logic Gates

Logic gates implement Boolean operations on binary inputs (0 = low voltage, 1 = high voltage).

OR gate: Output Y = A + B. Y = 1 if any input is 1. AND gate: Output Y = A.B. Y = 1 only if all inputs are 1. NOT gate (inverter): Output Y = A'. Y is the complement of input. NAND gate: Y = (A.B)'. NOT of AND. Universal gate — any logic function can be built from NAND gates alone. NOR gate: Y = (A+B)'. NOT of OR. Also a universal gate.

De Morgan's theorems: (A.B)' = A' + B' and (A+B)' = A'.B'. These are essential for converting between gate types.

The key problem-solving concept is understanding the p-n junction as a one-way current valve: forward bias lowers the barrier (current flows), reverse bias raises it (no current), and this principle underlies all semiconductor devices.

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