Semiconductor Diodes – Lecture 1 Comprehensive Notes

Semiconductor Material

• “Semi” implies intermediate behavior; semiconductors conduct better than insulators but worse than good conductors.
• Conductors: generous charge flow for modest voltage; Insulators: extremely low conductivity even under substantial voltage.
• Frequently-used semiconductor elements: Germanium (Ge), Silicon (Si), Gallium-Arsenide (GaAs).
• Typical resistivity comparison problem illustrated: Si samples with differing length/area and a copper sample show $R = \rho \frac{l}{A}$ leading to kΩ–MΩ values for Si vs µΩ for Cu.

Covalent Bonding & Intrinsic Materials

• Atoms consist of electrons, protons, neutrons; in solids, nuclei form a lattice with electrons in quantized shells (Bohr model).
• Valence electrons participate in bonding; tetravalent (4) → Si, Ge; trivalent (3) → Ga; pentavalent (5) → As.
• Covalent bonding: neighboring atoms share valence electrons, strengthening lattice (e.g., Si crystal, GaAs compound) and leaving no free carriers at 0 K.
• Breaking a bond via thermal or photon energy generates a free electron and a corresponding hole.
• Intrinsic carrier concentration $n_i$: at room T, per $1\,\text{cm}^3$
  • GaAs: $1.7\times10^6$
  • Si: $1.5\times10^{10}$
  • Ge: $2.5\times10^{13}$
• Electron mobility $\mu_n$ (cm²/V·s): GaAs 8500 > Ge 3900 > Si 1500 → determines conductivity.
• Temperature response: Semiconductors have negative temperature coefficient—conductivity rises as T rises (more broken bonds), opposite to metals.

Energy Bands & Levels

• Isolated atoms ➜ discrete energy levels ruled by Pauli exclusion (2 e⁻ per level with opposite spins).
• Bringing atoms together splits each level; in a crystal billions of splits form quasi-continuous energy bands.
• Key bands:
  • Valence band – highest completely filled band at 0 K.
  • Conduction band – next higher band; partly filled or empty.
• Forbidden gap $E<em>g$ separates bands. Sizes: insulator $E</em>g \gg 3\,\text{eV}$, semiconductor $\approx0.5\text{–}1.5\,\text{eV}$, conductor $E_g\approx0$ (bands overlap).
• Conduction requires available empty states near electron energy; hence valence electrons are immobile unless promoted across $E_g$.
• Relation to opto-electronics: Larger $E_g$ ⇒ photons of higher energy (shorter wavelength) emitted in LED action.
• Energy–charge relation: $W = QV$ (e.g., moving $6\,\text{C}$ through $3\,\text{V}$ needs $18\,\text{J}$).

Doping & Extrinsic Materials

• Goal: raise carrier concentration for practical current levels.
• Doping = adding ppm–ppb of impurities.
• n-type: add group V donors (As, Sb, P) ⟹ extra loosely-bound electron near conduction band. Ratio example: 1 donor per $10^7$ atoms increases carriers $10^5:1$.
• p-type: add group III acceptors (B, Ga, In) ⟹ create a hole (missing electron) in valence band.
• Carriers:
  • Majority = dopant-generated (electrons in n, holes in p).
  • Minority = thermally generated opposite-sign carriers.
• Materials remain globally neutral; uncovered ionized dopants create local electric fields but total charge balances.
• Electron vs hole flow analogy: holes propagate like gaps in a traffic jam, effectively moving opposite to electron drift.

Semiconductor Diode Biasing

• Joining p-type & n-type forms a p-n junction. Initial recombination near interface leaves fixed ionized donors/acceptors → depletion region devoid of free carriers.
• Three bias states:

1. No bias $V_D = 0$: diffusion of majority carriers balanced by drift of minority carriers; external current zero.
2. Reverse bias $V<em>D < 0$ (p at lower potential): depletion width widens, majority current nearly zero. A small minority-carrier current persists – the reverse saturation current $I</em>S$ (µA or less).
3. Forward bias V_D > 0 (≈0.6–0.7 V for Si): depletion narrows, majority carriers injected; exponential rise in current.

Diode I–V Characteristics & Equations

• Shockley equation (forward & reverse, ideal $n=1\text{–}2$):
  $I<em>D = I</em>S\left(e^{\frac{V<em>D}{nV</em>T}} - 1\right)$
  where $V_T = \frac{kT}{q} \approx 25\,\text{mV}$ at 300 K.
• Reverse saturation current doubles roughly every $10\,^{\circ}\text{C}$ rise for Si.
• Exercises include solving for $I<em>D$ at 0.6 V, 20 °C vs 100 °C with increased $I</em>S$, and negative-bias region verification.

Zener Region (Breakdown)

• Under sufficient reverse bias, current sharply increases (negative voltage region):
  • Avalanche breakdown: minority carriers accelerated, generate more carriers via impact ionization ($W<em>k = \tfrac12 mv^2$). • Zener breakdown: at high doping (thin depletion, strong electric field), quantum tunneling across $E</em>g$ at low $|V_Z| \lesssim 5\,\text{V}$.
• Resultant nearly-constant voltage $V_Z$; used for regulation.
• Peak Inverse Voltage (PIV/PRV): max reverse voltage before breakdown; series diodes raise PIV, parallel diodes raise current capability.
• Temperature effects: sign of $dV<em>Z/dT$ depends on breakdown mechanism (negative for avalanche >6 V, positive for Zener

Resistance Levels of Diodes

1. DC (static) resistance: $R<em>D = \frac{V</em>D}{I<em>D}$ at a fixed operating point. Higher current ⇒ lower $R</em>D$.
2. AC (dynamic) resistance (small-signal, about Q-point): $r<em>d = \frac{\Delta V</em>D}{\Delta I<em>D} \approx \frac{nV</em>T}{I<em>D}$ (practical rule $26\,\text{mV}/I</em>D$ at 300 K for Si and $n\approx1$).
3. Average AC resistance (large-signal swing): straight-line between max/min excursion points: $r<em>{av}=\frac{\Delta V</em>D}{\Delta I<em>D}|</em>{pt\,to\,pt}$. Lower swing current ⇒ larger resistance.

Diode Equivalent Circuits

• Purpose: simplify analysis in chosen region.

1. Piecewise-linear: forward knee voltage $V<em>K\approx0.7\,\text{V}$ in series with small $r</em>f$ (slope resistance) for conduction; open circuit in reverse.
2. Simplified: ideal diode + fixed $V_K$ (no slope resistance).
3. Ideal: perfect short in forward, open in reverse.

Transition & Diffusion Capacitances

• Reverse bias: depletion-region acts as parallel-plate capacitor $C<em>T$ (“transition”); varies inversely with $\sqrt{V</em>R+1}$.
• Forward bias: stored minority charge yields “diffusion” capacitance $C<em>D$ (dominant at high current), proportional to $I</em>D$.
• Important in RF design and fast switching.

Reverse Recovery Time

• When switching from forward to reverse bias, stored charge must be removed.
• $t<em>{rr} = t</em>s + t<em>t$ • $t</em>s$ – storage time (diode still conducts opposite direction due to stored charge).
  • $t_t$ – transition time (carriers removed, current decays to zero).
• High-speed diodes feature $t<em>{rr}$ from few ns to $\mu s$; exercise given with $t</em>t = 2 t<em>s$ and $t</em>{rr}=9\,\text{ns}$ (so $t<em>s=3\,\text{ns}, t</em>t=6\,\text{ns}$) to sketch current reversal waveform.

Diode Specification Parameters

• Data sheet (example: BAY73)
  • Working Inverse Voltage $WIV=100\,\text{V}$; Breakdown voltage $BV\ge125\,\text{V}$ @ $100\,\mu A$.
  • Forward voltage $V<em>F$ ranges 0.60–1.00 V across various currents (1 mA to 200 mA). • Reverse current $I</em>R$ in nA–µA range depending on $V<em>R$ & temperature. • Capacitance $C</em>J\approx8\,\text{pF}$ at $f=1\,\text{MHz}, V<em>R=0$. • Reverse recovery $t</em>{rr}\approx3\,\text{ps}$ @ $I_F=10\,\text{mA}$ – extremely fast.
  • Absolute maximum ratings: junction 175 °C, power 500 mW (derate 3.33 mW/°C over 25 °C), surge current 1–4 A.
• Package outline DO-35 glass; diode polarity: anode = p, cathode = n (band end, also K).

Zener Diodes – Special Considerations

• Symbol: diode with bent bar; usually reverse-biased (anode negative) with current $I<em>Z>0, V</em>Z\approx\text{constant}$ as long as $P<em>D=I</em>Z V_Z$ < rated.
• Temperature coefficient calculation: $TC<em>{V</em>Z}(\%/^{\circ}C)=\frac{\Delta V<em>Z}{V</em>Z}\frac{100}{\Delta T}$.
• Example (Table 1.4): 10 V Zener, $TC=+0.072\%/^{\circ}C$, find $V<em>Z$ at 100 °C: $\Delta V</em>Z = 0.00072\times10\times75 = 0.54\,\text{V}$ ⇒ $V_Z=10.54\,\text{V}$.
• Reverse problem: for $V_Z=10.75\,\text{V}$ determine required temperature by rearranging same formula.

Worked-Example Highlights

• Resistivity problems (Si & Cu) reinforce $R=\rho\frac{l}{A}$ scaling.
• DC/AC resistance example: At $I<em>D=2\,\text{mA}, V</em>D=0.7\,\text{V}$ ⇒ $R<em>D=350\,\Omega$ but dynamic $r</em>d=27.5\,\Omega$; disparity widens at high current (25 mA ⇒ $R<em>D=31.6\,\Omega, r</em>d=2\,\Omega$).
• Shockley exercise: Room-temperature $I<em>D$ computation given $V</em>D, I<em>S, n, V</em>T$. Elevated temperature case shows dramatic rise due to exponential + larger $I_S$.
• Reverse-bias example with large negative voltage returns $I<em>D\approx- I</em>S$ (matches expectation of saturation).

Formulas & Physical Constants Summary

• Resistivity relation: $R = \rho\frac{l}{A}$.
• Thermal voltage: $V_T = \frac{kT}{q} = 0.02585\,\text{V}$ at 300 K; rule of thumb 26 mV.
• Shockley: $I<em>D = I</em>S\left(e^{V<em>D/(nV</em>T)} - 1\right)$.
• Dynamic resistance: $r<em>d = \frac{nV</em>T}{I<em>D} \;(\approx26\,\text{mV}/I</em>D)$.
• Reverse recovery: $t<em>{rr}=t</em>s+t_t$.
• Zener temperature coefficient: $TC<em>{V</em>Z}=\frac{\Delta V<em>Z}{V</em>Z\Delta T}\times100\%$.
• Kinetic energy: $W_k=\frac12 m v^2$.
• Energy–charge: $W=Q V$; $1\,\text{eV}=1.6\times10^{-19}\,\text{J}$.
• Boltzmann constant $k=1.38\times10^{-23}\,\text{J·K}^{-1}$; electronic charge $q=1.6\times10^{-19}\,\text{C}$.

Practical/Philosophical Connections

• Diode “decision-making” ability underpins rectification, logic, RF detection, protection, LED emission.
• Transistors expand on diode concept: three-terminal control of larger currents.
• Ethical/environmental implication: semiconductor manufacturing relies on ultra-high-purity processes and can generate hazardous chemicals; responsible handling & recycling are essential.
• Real-world relevance: temperature sensitivity necessitates heat-sinking, derating, and compensation networks in analog & digital designs (e.g., band-gap references).