Semiconductors

Schrödinger equation

iℏ∂ψ∂t=−ℏ22m∇2ψ+Vψiℏ∂t∂ψ​=−2mℏ2​∇2ψ+Vψ

• ψ(x,t) = wavefunction; |ψ|² = probability density of finding a particle at (x,t).

• Stable solutions only exist for fixed energy eigenstates, ψₙ(x,t) → these are atomic orbitals (s, p, d, f).

Molecular orbitals

• When multiple nuclei are present, atomic orbitals combine into molecular orbitals → chemical bonds.

• Example: H₂ — two 1s orbitals combine into a bonding (lower energy, in-phase) and antibonding (higher energy, out-of-phase) orbital.

Filling of orbitals (left-step periodic table)

• Electrons have spin up/down → max 2 electrons per orbital (Pauli exclusion).

• Atoms are neutral (electrons = protons).

• Orbitals fill in increasing energy order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s...

Quantum Basis of Band Structure

• In a solid, many atoms → energy levels of eigenstates blur into continuous energy bands.

• Metals: a band is partially filled at all temperatures → very high conductivity (e.g., Na has a partially filled 3s band).

• Semiconductors: small band gap between a full valence band and empty conduction band.

  • At normal temps, some valence electrons are thermally excited into the conduction band → become free (current-carrying) electrons.

  • This leaves holes in the valence band, which behave like mobile positive charges (a neighbouring electron hops into the hole).

• Insulators: same picture as semiconductors but with a much larger band gap (negligible thermal excitation).

Energy Bands

• Above 0 K, some electrons are randomly knocked loose from Si lattice bonds → electron–hole pairs.

• Applying an electric field lets both electrons and holes contribute to current flow.

Doping — adding small amounts of impurity atoms changes conductivity dramatically:

Type	Impurity example	Effect
n-type	Antimony (Sb) — 5 valence electrons	Donates an extra free electron
p-type	Boron (B) — 3 valence electrons	Creates an extra hole (acceptor)

Semiconductors and Doping

• Joining p-type and n-type material creates a p-n junction = basic diode.

• Current flows easily in one direction (forward biased) and is blocked in the other (reverse biased).

Depletion region

• At the junction, free electrons from n-side diffuse over and recombine with holes → forms a region of immobile charged ions with no free carriers.

• Unbiased junction: depletion region ≈ 1 µm.

Forward bias (+ to p-side, – to n-side)

• Holes pushed toward junction from p-side, electrons pushed toward junction from n-side.

• Depletion region shrinks; electrons and holes recombine at junction → large continuous current.

Reverse bias (– to p-side, + to n-side)

• Carriers pulled away from junction → depletion region widens.

• Current stops except for a tiny thermally-generated reverse current (~10⁻⁸ A) until breakdown voltage (typically –20 V to –700 V).

Diode I–V characteristic

• Forward turn-on ≈ 0.5 V, significant conduction above ≈ 0.7 V (LEDs: 1.2–3.6 V).

• Reverse current: tiny and roughly voltage-independent until breakdown.

• Diode = two-terminal, nonlinear, passive device.

The P-N Junction (Diode)

Half-wave rectifier

• Single diode; only lets through the half-cycle when source exceeds diode's ~0.6 V forward drop.

• Output (V_load) is a rectified copy of the AC input's positive half-cycles only.

Full-wave bridge rectifier

• 4 diodes; current always passes through 2 diodes in series.

• V_load(max) ≈ V_source(max) – 1.2 V (two diode drops).

• V_load ≈ 0 whenever –1.2 V ≤ V_source ≤ 1.2 V (dead zone from diode drops).

Smoothing with a capacitor

• A capacitor in parallel with the load smooths the ripple: charges near the peak, discharges (supplies load current) between peaks.

• Ripple current relation: i=C dvdt ≈ Vload/Rload​​

• Ripple voltage:

  Vripple = δv ≈ (Vload/R_loadC)δt

  where δt = ripple period = 1/f (half-wave) or 1/(2f) (full-wave).

Diode Circuits — Rectifiers

• 3 doped regions: emitter, base (very thin, ~10 µm), collector (largest, dissipates most heat).

• NPN: n-type emitter / p-type base / n-type collector.

• PNP: p-type emitter / n-type base / p-type collector.

• Small base current controls a much larger collector current: I_C=βI_B (β typically 50–200) → current amplification.

• Base–emitter junction: forward biased (V_BE ≈ 0.6 V typical).

• Base–collector junction: reverse biased; large current still flows because base is thin and field is high.

Modes of operation (NPN)

Mode	Condition	Behaviour
Active	V_C > V_B > V_E (base-emitter forward, base-collector reverse)	Current amplifier: I_C = βI_B
Saturation	V_C < V_B > V_E	Switch ON — high conduction C→E
Cut-off	V_C > V_B < V_E	Switch OFF — no conduction
Reverse-active	V_C < V_B > V_E (rarely used)	—

Determining the mode — procedure

1. Is base-emitter forward biased (V_BE > 0.6 V or I_B > 0)?

   • No → cut-off: I_C = I_B = I_E = 0.

   • Yes → active or saturation:

2. Compute: I_C,max=V_CC−V_BE/R_C, I_B=V_BB−V_BE/R_B​

3. If I_Bβ>I_C,max → saturation (I_C = I_C,max, acts as closed switch). If I_Bβ<I_C,max → active (I_C = βI_B, acts as amplifier).

Worked examples (β=100, V_CC=10 V, V_BB=5 V, R_B=22 kΩ):

• R_C = 1 kΩ → I_C,max = 9.4 mA; I_B = 200 µA; I_Bβ = 20 mA > 9.4 mA → Saturation, I_C = 9.4 mA (switch ON).

• R_C = 100 Ω → I_C,max = 94 mA; I_Bβ = 20 mA < 94 mA → Active, I_C = 20 mA (amplifier).

BJT as a current source (LED driver)

• Choose R_B to set I_B = I_C/β for desired LED current.

• Example: I_C = 20 mA, V_CC=V_BB=5 V, β=100 → I_B = 200 µA → R_B = (5–0.6)/200µA = 22 kΩ.

• To stay in active mode: R_C<V_CC−V_LED−V_BE/I_C​​ (e.g. < 190 Ω).

Emitter follower

• Output taken at emitter; V_out ≈ V_in – 0.6 V (follows input, one diode drop down).

• High input impedance, low output impedance → current/power gain but no voltage gain (unity-gain buffer).

Common emitter amplifier

• DC collector voltage: V_C=V_CC−I_CR_C​

• AC input signal capacitively coupled; C chosen so C≥1/2πf(R1∥R2)​ (passes frequencies of interest).

• AC base perturbation v_in = v_B ≈ v_E (since V_BE ≈ const), and since β is large, i_C ≈ i_E = v_E/R_E.

• Output: v_out=v_C=−i_CR_C= −v_inR_C/R_E​

• Voltage gain = –R_C/R_E (inverting amplifier).

Bipolar Junction Transistor (BJT)

• 3 terminals: gate, drain, source (+ substrate/body).

• Gate = insulating oxide layer grown on p- or n-type substrate → forms a planar capacitor.

• No current flows through the gate (insulated) — voltage-controlled device, unlike current-controlled BJT.

• n-channel MOSFET: n-type drain/source in p-type substrate. p-channel: reverse.

Gate operation (n-channel)

• V_GS = 0: no depletion, no channel.

• 0 < V_GS < V_th: depletion layer forms (holes pushed away, immobile negative ions) — still no conducting channel.

• V_GS > V_th (threshold voltage): inversion layer forms — high concentration of electrons at the oxide/semiconductor interface → conducting channel exists.

Three modes of operation (n-channel)

Mode	Condition	Behaviour
Cutoff	V_GS < V_th	No conduction — switch OFF
Linear (triode)	V_GS > V_th and V_DS < V_GS – V_th	Acts like a voltage-controlled resistor
Saturation	V_GS > V_th and V_DS > V_GS – V_th	Current flows but only weakly depends on V_DS; mainly set by V_GS. Pinch-off: inversion layer separates from drain as V_DS increases

• V_th typically 0.5 V – 5 V, device-specific.

• I_D vs V_DS curves: for each V_GS – V_th, current rises linearly then flattens off (saturates) — family of curves fan out with increasing V_GS – V_th.

MOSFET as a digital switch (LED driver example)

• V_th = 2 V. Gate driven digitally (0 V / 5 V).

  • V_GS = 0 V → cutoff (LED off).

  • V_GS = 5 V → saturation (LED on), MOSFET behaves as a near-zero-resistance switch.

• LED current set by supply and series resistor:

  R_D=V_CC−V_LED/I_D

  e.g. V_CC = 5 V, I_D = 20 mA → R_D = (5 – 0.6)/0.02 = 220 Ω.

MOSFET (Metal-Oxide-Semiconductor FET)

	BJT	MOSFET
Control terminal	Base — current controlled	Gate — voltage controlled
Input current	Significant (I_B)	~Zero (insulated gate)
Key equation	I_C = βI_B	I_D depends on V_GS – V_th
"On" state	Saturation	Linear/Saturation depending on use
Typical use	Analog amplification, switching	Switching (digital logic, power), amplification

BJT vs MOSFET