What topics are covered in UGC NET Electronic Science Unit 1?

Unit 1 covers semiconductor fundamentals (energy bands, effective mass, density of states, Fermi level), PN junction and diode equation, breakdown mechanisms, Zener and tunnel diodes, metal–semiconductor (Ohmic and Schottky) contacts, JFET and MOSFET characteristics and equivalent circuits, low-dimensional devices (quantum wells, wires, dots), HEMT, solar cells (I–V, fill factor, efficiency), LED, LCD, flexible displays, and emerging materials such as graphene, CNT, ZnO and SiC.

What is the diode equation asked in UGC NET?

The Shockley diode equation is I = I0 [exp(V/ηVT) − 1], where I0 is the reverse saturation current, η is the ideality factor (1 for Ge, ≈2 for Si at low currents) and VT = kT/q ≈ 26 mV at 300 K.

What is the difference between Zener and avalanche breakdown?

Zener breakdown is quantum-mechanical tunneling in heavily doped, narrow junctions (typically below about 5–6 V) with a negative temperature coefficient, while avalanche breakdown is impact-ionization carrier multiplication in lightly doped, wider junctions (above about 6 V) with a positive temperature coefficient.

UGC NET Electronic Science Unit 1 Notes: Semiconductors, PN Junction, FET, MOSFET, HEMT & Solar Cells

§ 1.1Introduction to Semiconductors

A semiconductor is a crystalline solid whose electrical conductivity lies between that of a conductor (σ ≈ 10⁶–10⁸ S/m) and an insulator (σ ≈ 10⁻¹⁰–10⁻²⁰ S/m), typically 10⁻⁶ to 10⁴ S/m. Its defining features are a moderate band gap (≈ 0.1–3.5 eV), conductivity that increases with temperature (negative temperature coefficient of resistance), and conduction by two carrier types: electrons and holes.

Intrinsic semiconductors

A chemically pure semiconductor (Si, Ge, GaAs) is intrinsic. At T = 0 K the valence band is full and the conduction band empty — it behaves as an insulator. At T > 0 K thermal energy breaks covalent bonds, generating electron–hole pairs, so that

Intrinsic condition $$ n = p = n_i $$

The intrinsic carrier concentration follows

Intrinsic carrier concentration $$ n_i = \sqrt{N_C N_V}\, e^{-E_g/2kT} \;\propto\; T^{3/2} e^{-E_g/2kT} $$

where $N_C$ and $N_V$ are the effective densities of states of the conduction and valence bands. At 300 K: $n_i \approx 1.5\times10^{10}\,\text{cm}^{-3}$ for Si, $2.5\times10^{13}\,\text{cm}^{-3}$ for Ge, and $\approx 2\times10^{6}\,\text{cm}^{-3}$ for GaAs (wider gap → fewer intrinsic carriers).

Extrinsic semiconductors and doping

n-type: pentavalent donors (P, As, Sb) contribute electrons; donor level $E_D$ lies ≈ 0.01–0.05 eV below $E_C$. Majority carriers: electrons; minority: holes. $n \approx N_D$.
p-type: trivalent acceptors (B, Al, Ga, In) create holes; acceptor level $E_A$ lies just above $E_V$. Majority carriers: holes. $p \approx N_A$.

Mass-action law (thermal equilibrium) $$ np = n_i^2 \quad\text{(independent of doping)} $$

Charge neutrality $$ p + N_D^+ = n + N_A^- $$

For an n-type sample with $N_D \gg n_i$: $n \approx N_D$, $p = n_i^2/N_D$. For compensated material, $n \approx N_D - N_A$ (if $N_D > N_A$).

Conductivity and mobility

Conductivity $$ \sigma = q\,(n\mu_n + p\mu_p), \qquad \rho = \frac{1}{\sigma} $$

Mobility $\mu = v_d/E$ (cm²/V·s) is limited by lattice (phonon) scattering ($\mu_L \propto T^{-3/2}$, dominant at high T) and ionized impurity scattering ($\mu_I \propto T^{3/2}/N_I$, dominant at low T), combined by Matthiessen's rule $1/\mu = 1/\mu_L + 1/\mu_I$. Typical 300 K values: Si — $\mu_n = 1350$, $\mu_p = 480$; Ge — $\mu_n = 3900$, $\mu_p = 1900$; GaAs — $\mu_n = 8500$, $\mu_p = 400$ cm²/V·s.

Einstein relation (links diffusion & drift) $$ \frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q} = V_T \approx 25.9\ \text{mV at } 300\ \text{K} $$

Drift and diffusion currents

Total current densities $$ J_n = qn\mu_n E + qD_n\frac{dn}{dx}, \qquad J_p = qp\mu_p E - qD_p\frac{dp}{dx} $$

The sign difference (− for hole diffusion) is a recurring trap in objective questions: both carriers diffuse from high to low concentration, but their charges differ.

Hall effect

A current $I_x$ in a magnetic field $B_z$ develops a transverse Hall voltage $V_H$. It identifies the carrier type (sign of $V_H$) and the carrier concentration:

Hall coefficient $$ R_H = \frac{1}{qp} \ (\text{p-type}), \quad R_H = -\frac{1}{qn}\ (\text{n-type}); \qquad V_H = \frac{B I R_H}{t} $$

UGC NET focus Frequently asked: numerical use of $np = n_i^2$; computing ρ or σ from doping; Einstein relation value at 300 K; direction/sign conventions in the Hall effect; why GaAs has high electron mobility (low effective mass, Γ-valley conduction).

§ 1.2Energy Bands in Solids

When N isolated atoms condense into a crystal, each discrete atomic level splits into N closely spaced levels forming quasi-continuous bands separated by forbidden gaps. Two complementary viewpoints explain this:

Tight-binding picture: overlap of atomic orbitals broadens levels into bands; greater overlap (smaller lattice spacing) → wider bands.
Nearly-free-electron / Kronig–Penney model: electrons in a periodic potential have Bloch-wave solutions $\psi_k(x) = u_k(x)e^{ikx}$; Bragg reflection at the Brillouin-zone boundaries $k = \pm n\pi/a$ opens energy gaps.

Key classification (at 0 K) Metal: partially filled band or overlapping bands. Insulator: full valence band, $E_g \gtrsim 3\text{–}4$ eV. Semiconductor: full valence band but small gap (Si 1.12 eV, Ge 0.66 eV, GaAs 1.42 eV at 300 K) allowing thermal excitation.

Direct vs indirect band gap

Direct vs indirect gap — a guaranteed exam comparison
Property	Direct gap (GaAs, InP, GaN, ZnO)	Indirect gap (Si, Ge, SiC, GaP)
CB minimum & VB maximum	Same k (Γ point)	Different k
Band-to-band transition	Photon only (momentum conserved)	Needs phonon assistance
Radiative efficiency	High → LEDs, lasers	Poor light emitters
Minority-carrier lifetime	Short (ns)	Long (µs–ms)

Band gap shrinks with temperature (Varshni relation $E_g(T) = E_g(0) - \alpha T^2/(T+\beta)$); for Si, $E_g$ falls from 1.17 eV (0 K) to 1.12 eV (300 K). The optical absorption edge gives $\lambda_c\,(\mu m) = 1.24/E_g\,(\text{eV})$.

UGC NET focus Expect questions on: identifying direct/indirect materials; cutoff wavelength $ \lambda = 1.24/E_g $; why Si cannot make efficient LEDs; band-gap values of Si, Ge, GaAs, GaN (3.4 eV), SiC (~3.2 eV for 4H), ZnO (3.37 eV).

§ 1.3Concept of Effective Mass

Inside a crystal an electron responds to an external force as if its mass were modified by the periodic lattice potential. The effective mass packages all internal lattice forces into a single parameter so that Newtonian equations still apply:

Effective mass — definition $$ m^{*} = \frac{\hbar^{2}}{\dfrac{d^{2}E}{dk^{2}}} $$

$m^*$ is inversely proportional to band curvature: sharply curved bands → light carriers (fast devices); flat bands → heavy carriers.
Near the conduction-band minimum $d^2E/dk^2 > 0$: electrons have positive $m^*$.
Near the valence-band maximum $d^2E/dk^2 < 0$: electrons have negative $m^*$; these states are described instead as holes with positive mass $m_h^* = -m_e^*$ and charge +q.

Group velocity of a Bloch electron: $v_g = \frac{1}{\hbar}\frac{dE}{dk}$; acceleration under force F: $a = F/m^*$.

Representative effective masses (units of m₀)
Material	mₙ*	mₚ*	Consequence
Si	1.08 (DOS)	0.56	moderate mobility
Ge	0.55	0.37	higher µ than Si
GaAs	0.067	0.48	very high µₙ → HEMT, MESFET

UGC NET focus Standard numericals give an E–k relation, e.g. $E = Ak^2$, and ask for $m^* = \hbar^2/2A$. Also remember: hole = empty state at top of valence band behaving as +q, positive-mass particle; light effective mass of GaAs explains its use in high-frequency devices.

§ 1.4Density of States

The density of states g(E) counts the number of available quantum states per unit volume per unit energy. Solving the particle-in-a-box problem in k-space, allowed states fill a sphere of radius k with spacing $2\pi/L$ per dimension, two spins per state:

Derivation — 3D density of states

Number of states with wavevector ≤ k (volume V = L³):

$$ N(k) = 2 \times \frac{\tfrac{4}{3}\pi k^{3}}{(2\pi/L)^{3}} = \frac{V k^{3}}{3\pi^{2}} $$

Using the parabolic relation $E = \hbar^2k^2/2m^*$ → $k = \sqrt{2m^*E}/\hbar$, substitute and differentiate w.r.t. E per unit volume:

$$ g_{3D}(E) = \frac{1}{2\pi^{2}}\left(\frac{2m^{*}}{\hbar^{2}}\right)^{3/2}\sqrt{E - E_C} $$

Dimensionality fingerprint (memorize) $$ g_{3D}(E) \propto \sqrt{E}, \qquad g_{2D}(E) = \frac{m^{*}}{\pi\hbar^{2}}\ (\text{step}), \qquad g_{1D}(E) \propto \frac{1}{\sqrt{E}}, \qquad g_{0D}(E) \propto \delta(E - E_n) $$

This E-dependence is the single most examined fact: bulk → √E; quantum well → staircase (constant per subband); quantum wire → inverse-square-root spikes (van Hove singularities); quantum dot → discrete delta functions (atom-like levels).

Effective density of states

Integrating g(E)f(E) over the conduction band with Boltzmann statistics defines:

Effective DOS $$ N_C = 2\left(\frac{2\pi m_n^{*} kT}{h^{2}}\right)^{3/2}, \qquad N_V = 2\left(\frac{2\pi m_p^{*} kT}{h^{2}}\right)^{3/2} \;\propto\; T^{3/2} $$

For Si at 300 K: $N_C \approx 2.8\times10^{19}$, $N_V \approx 1.04\times10^{19}\ \text{cm}^{-3}$.

§ 1.5Fermi Level and Carrier Statistics

Electrons obey Fermi–Dirac statistics. The probability that a state at energy E is occupied:

Fermi–Dirac distribution $$ f(E) = \frac{1}{1 + e^{(E - E_F)/kT}} $$

At $E = E_F$: $f = \tfrac12$ at any temperature — the standard definition of the Fermi level.
At T = 0 K: step function — all states below $E_F$ filled, above empty.
For $E - E_F \gg 3kT$: $f(E) \approx e^{-(E-E_F)/kT}$ (Maxwell–Boltzmann approximation, valid for non-degenerate semiconductors).
Symmetry: $f(E_F + \Delta) = 1 - f(E_F - \Delta)$.

Equilibrium carrier concentrations

Carrier concentrations $$ n = N_C\, e^{-(E_C - E_F)/kT}, \qquad p = N_V\, e^{-(E_F - E_V)/kT} $$

Position of the Fermi level

Intrinsic Fermi level $$ E_{Fi} = \frac{E_C + E_V}{2} + \frac{kT}{2}\ln\!\frac{N_V}{N_C} = \frac{E_C+E_V}{2} + \frac{3kT}{4}\ln\!\frac{m_p^{*}}{m_n^{*}} $$

$E_{Fi}$ sits essentially at mid-gap, displaced slightly toward the band with the lighter effective mass.

Doped material $$ E_F - E_{Fi} = kT\ln\frac{n}{n_i} = kT\ln\frac{N_D}{n_i}\ (\text{n-type}); \qquad E_{Fi} - E_F = kT\ln\frac{N_A}{n_i}\ (\text{p-type}) $$

n-type: $E_F$ moves toward $E_C$; heavier doping → closer to (or into) the conduction band (degenerate, n⁺).
p-type: $E_F$ moves toward $E_V$.
Temperature ↑: extrinsic material trends toward intrinsic behaviour; $E_F \to E_{Fi}$.
In equilibrium the Fermi level is flat (constant) throughout any connected system — the key to drawing band diagrams of junctions.
Under bias/illumination, separate quasi-Fermi levels $E_{Fn}, E_{Fp}$ describe each population; their split equals the applied junction voltage: $E_{Fn} - E_{Fp} = qV$, and $pn = n_i^2 e^{qV/kT}$.

UGC NET focus (1) Probability questions: occupancy at $E_F \pm kT$ → $f = 1/(1+e^{\pm1}) = 0.269/0.731$. (2) Locating $E_F$ for given $N_D$. (3) Statement "Fermi level is constant at equilibrium across a junction" — used to derive built-in potential. (4) Degenerate semiconductor: $E_F$ inside a band (basis of the tunnel diode).

§ 1.6PN Junction

A PN junction is formed within a single crystal where doping changes from acceptor to donor type. On contact, the large carrier-concentration gradients drive diffusion: holes p→n, electrons n→p. The uncovered ionized dopants (NA⁻ on the p-side, ND⁺ on the n-side) form the space-charge / depletion region, whose built-in field opposes further diffusion. Equilibrium = drift current exactly balancing diffusion current; the Fermi level becomes flat.

Built-in potential $$ V_{bi} = \frac{kT}{q}\,\ln\!\frac{N_A N_D}{n_i^{2}} $$

Typical: ≈ 0.6–0.8 V for Si, ≈ 0.3 V for Ge, ≈ 1.2 V for GaAs. Note $V_{bi}$ cannot be measured directly with a voltmeter (contact potentials cancel it).

Derivation — depletion width (abrupt junction, depletion approximation)

Solve Poisson's equation $ \dfrac{d^2V}{dx^2} = -\dfrac{\rho}{\varepsilon_s} $ with $\rho = +qN_D$ (0<x<x_n) and $\rho = -qN_A$ (−x_p<x<0). The field is triangular, peaking at the metallurgical junction:

$$ E_{max} = \frac{qN_D x_n}{\varepsilon_s} = \frac{qN_A x_p}{\varepsilon_s} $$

Charge neutrality gives $N_A x_p = N_D x_n$ — the depletion region extends deeper into the lightly doped side. Integrating the field over the region:

$$ W = x_n + x_p = \sqrt{\frac{2\varepsilon_s}{q}\left(\frac{1}{N_A}+\frac{1}{N_D}\right)(V_{bi} - V)} $$

with V > 0 for forward bias (W shrinks) and V < 0 for reverse (W grows as $\sqrt{V_{bi}+|V_R|}$).

Junction (depletion) capacitance $$ C_j = \frac{\varepsilon_s A}{W} = \frac{C_{j0}}{\left(1 - V/V_{bi}\right)^{m}}, \quad m = \tfrac12\ (\text{abrupt}),\ \tfrac13\ (\text{linearly graded}) $$

$C_j \propto V_R^{-1/2}$ for an abrupt junction — the operating principle of the varactor diode (voltage-tuned capacitance; used in VCOs and AFC). A plot of $1/C_j^2$ vs $V_R$ is a straight line whose intercept gives $V_{bi}$ and slope gives doping.

Diffusion (storage) capacitance

Under forward bias, injected minority-carrier charge $Q = I\tau$ must change with voltage:

Diffusion capacitance $$ C_D = \frac{dQ}{dV} = \frac{\tau I}{\eta V_T} \gg C_j \ \ (\text{forward bias}) $$

$C_D$ dominates in forward bias and sets switching speed (reverse-recovery time); $C_j$ dominates in reverse bias.

UGC NET focus Memorize: $W \propto \sqrt{V_{bi}+V_R}$; depletion penetrates the lightly doped side; $E_{max} = 2(V_{bi}-V)/W$ for the triangular profile; varactor relation $C \propto V^{-1/2}$; one-sided junction (p⁺n) formulas with only $N_D$.

§ 1.7Diode Equation and Equivalent Circuit

Forward bias lowers the junction barrier to $q(V_{bi}-V)$, exponentially increasing minority-carrier injection (law of the junction: $p_n(0) = p_{n0}e^{V/V_T}$). Solving the diffusion equation for the injected profiles and summing electron and hole components yields the Shockley ideal diode equation:

Diode equation $$ I = I_0\left(e^{\,V/\eta V_T} - 1\right), \qquad V_T = \frac{kT}{q} \approx 26\ \text{mV at } 300\ \text{K} $$

Reverse saturation current $$ I_0 = qA\left(\frac{D_p\, p_{n0}}{L_p} + \frac{D_n\, n_{p0}}{L_n}\right) = qA\,n_i^{2}\left(\frac{D_p}{L_p N_D} + \frac{D_n}{L_n N_A}\right), \qquad L = \sqrt{D\tau} $$

Ideality factor η: 1 for ideal diffusion current (Ge), ≈ 2 when recombination in the depletion region dominates (Si at low current).
Temperature dependence: $I_0$ approximately doubles every 10 °C (∝ $n_i^2$); forward voltage at constant current falls ≈ −2 to −2.5 mV/°C for Si.
Cut-in voltage: ~0.3 V (Ge), ~0.7 V (Si), ~1.2 V (GaAs).
Reverse current in real Si diodes is dominated by depletion-region generation current ∝ $n_i W$, which grows with reverse bias — so it does not perfectly saturate.

Small-signal (AC) equivalent circuit

Dynamic resistance $$ r_d = \frac{dV}{dI} = \frac{\eta V_T}{I} \;\approx\; \frac{26\ \text{mV}}{I\,(\text{mA})}\ \Omega \ \ (\eta = 1) $$

The complete small-signal model: $r_d$ in parallel with ($C_j + C_D$), in series with bulk/contact resistance $r_s$. Large-signal piecewise model: ideal diode + cut-in source $V_\gamma$ + slope resistance $R_f$; reverse: open circuit (or $R_r$ very large).

UGC NET focus Classic numericals: $r_d = 26/I_{mA}$; ratio of currents for ΔV (e.g., +60 mV → ×10 at η=1); I₀ doubling per 10 °C; identifying η from the slope of ln I vs V.

§ 1.8Breakdown in Diodes & the Zener Diode

Beyond a critical reverse voltage $V_{BR}$ the reverse current rises abruptly. Breakdown is not destructive if power dissipation is externally limited. Two mechanisms:

Avalanche vs Zener breakdown — the most repeated Unit-1 table
Feature	Avalanche breakdown	Zener breakdown
Mechanism	Impact ionization → carrier multiplication $M = 1/[1-(V/V_{BR})^n]$	Quantum-mechanical band-to-band tunneling
Doping / depletion width	Light doping, wide W	Very heavy doping, W < ~10 nm
Typical voltage	> ≈ 6 V	< ≈ 5 V (4–6 V: mixed)
Field required	~3×10⁵ V/cm (Si)	~10⁶ V/cm
Temperature coefficient	Positive (phonon scattering impedes ionization)	Negative (gap shrinks, tunneling easier)
I–V knee	Sharp	Soft

Around $V_Z \approx 5\text{–}6$ V the two coefficients cancel — the basis of temperature-compensated reference diodes. (A third, destructive mechanism — thermal runaway — occurs in rectifiers without current limiting.)

Zener diode as voltage regulator

Operated in reverse breakdown, the Zener holds $V_Z$ nearly constant. For the standard shunt regulator (source $V_S$, series $R_S$, load $R_L$):

Shunt regulator design $$ I_S = \frac{V_S - V_Z}{R_S} = I_Z + I_L; \qquad \text{regulation holds while } I_{Z(min)} \le I_Z \le I_{Z(max)} = \frac{P_{Z,max}}{V_Z} $$

Line regulation: ΔV_out/ΔV_in ≈ $r_z/(R_S + r_z)$, where $r_z$ is the small Zener dynamic resistance (a few Ω).
Load regulation: output change as $I_L$ varies; worst case when $I_L$ max and $V_S$ min — check $I_Z \ge I_{Z(min)}$.

UGC NET focus Numericals on choosing $R_S$ and verifying Zener power; conceptual: sign of temperature coefficients; "both mechanisms present near 5–6 V"; Zener used in clippers and as reference, not for rectification.

§ 1.9Tunnel (Esaki) Diode

A tunnel diode is a PN junction with degenerate doping on both sides (≈10¹⁹–10²⁰ cm⁻³): $E_F$ lies inside the conduction band (n-side) and inside the valence band (p-side). The depletion region is so thin (~5–10 nm) that electrons tunnel directly through the gap.

I–V characteristic

Small forward bias: filled CB states (n) align with empty VB states (p) → tunneling current rises to a peak ($I_P$ at $V_P \approx$ 50–65 mV for Ge).
Increasing bias: bands de-align; tunneling falls → current drops to a valley ($I_V$ at $V_V \approx$ 350 mV). This region exhibits negative differential resistance (NDR): $dI/dV < 0$.
Larger bias: ordinary thermionic diffusion current takes over and rises exponentially.

Figure of merit Peak-to-valley current ratio $I_P/I_V$: ≈ 10 : 1 (Ge), higher in GaAs/GaSb. Larger ratio → better oscillator/switch.

Tunneling is a majority-carrier, quantum process — essentially instantaneous → operation to tens of GHz (microwave oscillators, amplifiers, fast switches/flip-flops).
No minority-carrier storage → negligible reverse recovery.
In reverse bias the tunnel diode conducts heavily (no breakdown knee); a related device biased for this property is the backward diode (used as a zero-bias detector).
Drawbacks: low voltage swing, low power, two-terminal device (no isolation between input and output).

UGC NET focus NDR region between $V_P$ and $V_V$; degenerate doping requirement; comparison with Gunn diode (bulk NDR, transferred-electron, Unit-VII) — tunnel diode NDR is a junction tunneling effect.

§ 1.10Metal–Semiconductor Junctions: Schottky & Ohmic Contacts

Let $\phi_m$ = metal work function, $\phi_s$ = semiconductor work function, $\chi_s$ = electron affinity. For an n-type semiconductor:

$\phi_m > \phi_s$: electrons spill into the metal, leaving a depletion layer → Schottky (rectifying) contact.
$\phi_m < \phi_s$: electrons accumulate at the interface → ohmic contact. (Conditions reverse for p-type.)

Schottky barrier height (ideal, n-type) $$ \phi_B = \phi_m - \chi_s, \qquad V_{bi} = \phi_m - \phi_s $$

(In practice interface states "pin" $\phi_B$ nearly independent of the metal — Fermi-level pinning.)

Schottky diode current — thermionic emission

Thermionic-emission equation $$ I = A^{*}A\,T^{2}e^{-q\phi_B/kT}\left(e^{\,qV/kT}-1\right) = I_S\left(e^{\,qV/kT}-1\right) $$

where $A^{*}$ is the effective Richardson constant (~110 A·cm⁻²K⁻² for n-Si).

Schottky diode vs PN diode
Property	Schottky	PN junction
Carriers	Majority only (electrons)	Minority injection
Storage / reverse recovery	Negligible → ultrafast (ps–ns)	Significant ($t_{rr}$)
Cut-in voltage	0.2–0.3 V	0.7 V (Si)
Reverse leakage	Higher	Lower
Applications	SMPS rectifiers, RF detectors/mixers, clamp in Schottky-TTL, solar bypass	General rectification

Making practical ohmic contacts

Since suitable work-function metals rarely exist, real ohmic contacts use heavy n⁺/p⁺ doping under the metal: the barrier becomes so thin that carriers tunnel through (field emission), giving a linear, symmetric, low-resistance I–V. Specific contact resistance $\rho_c\,(\Omega\cdot\text{cm}^2)$ is the figure of merit. Examples: Al on p-Si; Au–Ge/Ni on n-GaAs; Ti/Al stacks on n-GaN.

UGC NET focus Conditions $\phi_m \gtrless \phi_s$ for n- and p-type (four-case table is frequently asked); Schottky = majority-carrier, no storage delay; n⁺ tunneling route to ohmic behaviour; $T^2$ prefactor identifies thermionic emission.

§ 1.11JFET — Characteristics and Equivalent Circuit

The junction field-effect transistor is a voltage-controlled, unipolar device: current of one carrier type flows through a doped channel (source → drain) whose cross-section is squeezed by the depletion regions of reverse-biased gate PN junctions. The n-channel JFET takes $V_{GS} \le 0$.

Operation

Ohmic (triode) region ($V_{DS}$ small): channel acts as a voltage-controlled resistor.
Pinch-off: as $V_{DS}$ rises, the depletion region widens near the drain; at $V_{DS} = V_{GS} - V_P$ the channel pinches off and current saturates (does not become zero — carriers are swept across the pinched region).
Pinch-off voltage $V_P$ (= $V_{GS(off)}$, negative for n-channel): the gate voltage that fully depletes the channel at $V_{DS}\to 0$. For a channel of half-width a, doping $N_D$: $ |V_P| = \dfrac{qN_D a^2}{2\varepsilon_s} - V_{bi}$ (internal pinch-off $V_{P0}=qN_Da^2/2\varepsilon_s$).

Transfer characteristic — Shockley's square law $$ I_D = I_{DSS}\left(1 - \frac{V_{GS}}{V_P}\right)^{2} \quad (V_P \le V_{GS} \le 0) $$

Transconductance $$ g_m = \frac{\partial I_D}{\partial V_{GS}} = -\frac{2I_{DSS}}{V_P}\left(1-\frac{V_{GS}}{V_P}\right) = g_{m0}\left(1-\frac{V_{GS}}{V_P}\right) = \frac{2\sqrt{I_{DSS}\,I_D}}{|V_P|} $$

Small-signal equivalent circuit

Between gate and source: open circuit (reverse-biased junction, input resistance ~10⁸–10¹⁰ Ω, plus $C_{gs}$). Between drain and source: current source $g_m v_{gs}$ in parallel with output resistance $r_d = (\partial I_D/\partial V_{DS})^{-1}$ (25 kΩ–1 MΩ); feedback capacitance $C_{gd}$ bridges input and output (Miller effect at high frequency).

Amplification factor $$ \mu = g_m\, r_d $$

JFET vs BJT — favourite comparison
	JFET	BJT
Control	Voltage (field)	Current (base)
Carriers	Unipolar (majority)	Bipolar
Input impedance	Very high	Low–moderate
Noise	Low	Higher
Thermal runaway	No (negative temp. coefficient of I_D)	Possible
g_m	Lower	Higher (= I_C/V_T)

UGC NET focus Square-law numericals (find $I_D$ or $V_{GS}$); $g_m$ at a given bias; saturation begins at $V_{DS(sat)} = V_{GS} - V_P$; JFET is depletion-mode only and gate must never be forward-biased appreciably.

§ 1.12MOSFET — Characteristics and Equivalent Circuit

The metal–oxide–semiconductor FET controls channel charge capacitively through a gate insulator (SiO₂ / high-k). It is the workhorse of VLSI (fabrication and scaling details appear again in Unit-II).

MOS capacitor regimes (p-substrate, NMOS)

Accumulation ($V_G < 0$): holes pile up at the surface.
Depletion (small $V_G > 0$): holes repelled; fixed acceptor space charge.
Inversion ($V_G > V_T$): surface electron layer forms; strong inversion when surface potential $\psi_s = 2\phi_F$, $\phi_F = (kT/q)\ln(N_A/n_i)$.

Threshold voltage $$ V_T = V_{FB} + 2\phi_F + \frac{\sqrt{2\varepsilon_s q N_A (2\phi_F)}}{C_{ox}}, \qquad C_{ox} = \frac{\varepsilon_{ox}}{t_{ox}} $$

$V_{FB} = \phi_{ms} - Q_{ox}/C_{ox}$ is the flat-band voltage. $V_T$ is raised by source–body reverse bias (body effect): $ V_T = V_{T0} + \gamma(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}) $.

I–V characteristics (NMOS, enhancement)

Triode / linear region (V_DS < V_GS − V_T) $$ I_D = \mu_n C_{ox}\frac{W}{L}\left[(V_{GS}-V_T)V_{DS} - \frac{V_{DS}^{2}}{2}\right] $$

Saturation region (V_DS ≥ V_GS − V_T) $$ I_D = \frac{1}{2}\,\mu_n C_{ox}\frac{W}{L}\,(V_{GS}-V_T)^{2}\,(1 + \lambda V_{DS}) $$

λ models channel-length modulation (Early-effect analogue, output resistance $r_o = 1/\lambda I_D$). For very small $V_{DS}$, the device is a gate-controlled resistor $R_{on} = [\mu_n C_{ox}(W/L)(V_{GS}-V_T)]^{-1}$.

Transconductance (saturation) $$ g_m = \mu_n C_{ox}\frac{W}{L}(V_{GS}-V_T) = \sqrt{2\mu_n C_{ox}\frac{W}{L} I_D} = \frac{2I_D}{V_{GS}-V_T} $$

Device types & symbols

Enhancement (normally OFF): channel induced only for $|V_{GS}| > |V_T|$; NMOS $V_T>0$, PMOS $V_T<0$.
Depletion (normally ON): built-in channel; conducts at $V_{GS}=0$, turned off by opposite-polarity gate voltage; characteristics resemble the JFET but allow both enhancement and depletion operation.

Small-signal model & figures of merit

Gate: purely capacitive input ($C_{gs} \approx \tfrac{2}{3}WLC_{ox}$ in saturation, plus overlap $C_{gd}$). Output: $g_m v_{gs}$ ∥ $r_o$. Unity-current-gain frequency:

Cut-off frequency $$ f_T = \frac{g_m}{2\pi (C_{gs}+C_{gd})} \;\approx\; \frac{\mu_n (V_{GS}-V_T)}{2\pi L^{2} \cdot \tfrac{2}{3}} \ \Rightarrow\ f_T \propto \frac{1}{L^{2}} $$

— the fundamental reason channel-length scaling boosts speed (full scaling rules: Unit-II).

UGC NET focus (1) Boundary $V_{DS(sat)} = V_{GS}-V_T$; (2) square-law numericals with W/L; (3) body-effect direction (V_T increases); (4) MOSFET input gate current ≈ 0 (insulated gate) vs JFET (junction leakage); (5) handle MOS devices with antistatic care — thin oxide breaks down near 10⁷ V/cm.

§ 1.13Low-Dimensional Devices: Quantum Wells, Wires, Dots

When a semiconductor layer's physical size approaches the de Broglie wavelength of carriers (~10 nm) or the exciton Bohr radius, motion in that direction is quantized into discrete subbands — quantum confinement.

Particle-in-a-box energy levels (infinite well, width L) $$ E_n = \frac{n^{2}\pi^{2}\hbar^{2}}{2m^{*}L^{2}} = \frac{n^{2}h^{2}}{8m^{*}L^{2}}, \quad n = 1,2,3\ldots $$

Smaller L → larger level spacing → blue-shift of emission. Effective gap of a confined structure: $E_g^{eff} = E_g^{bulk} + E_{1,e} + E_{1,h}$.

Dimensionality summary (memorize the DOS column)
Structure	Confined / free directions	g(E) shape	Devices
Bulk (3D)	0 / 3	$ \propto \sqrt{E} $	conventional
Quantum well (2D)	1 / 2	staircase, $ m^*/\pi\hbar^2 $ per subband	QW lasers, HEMT 2DEG, QWIP detectors
Quantum wire (1D)	2 / 1	$ \propto 1/\sqrt{E} $ spikes	nanowire FETs, ballistic conductors
Quantum dot (0D)	3 / 0	δ-functions ("artificial atom")	QD lasers, QLED displays, single-photon sources, qubits

Quantum well: thin low-gap layer (GaAs) between wide-gap barriers (AlGaAs) — a type-I heterostructure. QW lasers achieve lower threshold current and temperature-stable, tunable wavelength versus double-heterostructure lasers.
Resonant tunneling diode (RTD): double-barrier QW; current peaks when the emitter Fermi level aligns with a well subband → NDR at THz speeds.
Quantum wire: conductance quantized in units of $G_0 = 2e^{2}/h \approx 77.5\ \mu S$ (Landauer formula) in the ballistic limit.
Quantum dot: size-tunable colour (CdSe: red→blue as diameter shrinks ~6→2 nm); single-electron charging energy $e^2/2C$ → Coulomb blockade, single-electron transistor.

UGC NET focus DOS shape vs dimensionality (asked repeatedly); $E_1$ numericals from $h^2/8m^*L^2$; blue-shift with decreasing size; conductance quantum $2e^2/h$; exciton Bohr radius criterion for confinement.

§ 1.14High Electron Mobility Transistor (HEMT)

The HEMT (also MODFET — modulation-doped FET) exploits a heterojunction, classically AlGaAs/GaAs, or AlGaN/GaN for power. The wide-gap barrier is doped (modulation doping) while the narrow-gap channel is left undoped; the conduction-band discontinuity $\Delta E_C$ forms a roughly triangular quantum well at the interface that traps a two-dimensional electron gas (2DEG).

Central idea Electrons are spatially separated from their parent donors (often by an undoped spacer layer), so ionized-impurity scattering nearly vanishes → extremely high mobility (≥ 8500 cm²/V·s at 300 K in GaAs; >10⁶ cm²/V·s cryogenic) and high sheet density simultaneously — a combination impossible in a uniformly doped channel.

Structure (top→bottom): metal gate (Schottky) / n-AlGaAs barrier / undoped AlGaAs spacer / undoped GaAs channel with 2DEG / semi-insulating GaAs substrate.
Gate modulates the 2DEG sheet density $n_s$; drain current and $g_m$ expressions parallel the MOSFET with $C_{ox}$ replaced by the barrier capacitance $ \varepsilon_b/d $.
pHEMT (pseudomorphic, strained InGaAs channel) and mHEMT (metamorphic) extend performance; GaN HEMTs use polarization-induced 2DEG (no doping needed) with sheet densities ~10¹³ cm⁻² and very high breakdown fields → RF power amplifiers, 5G base stations, fast EV chargers, pulsed-power switches.
Figures of merit: very high $f_T$/$f_{max}$ (hundreds of GHz–THz for InP HEMTs), lowest noise figure of any transistor family → LNAs for satellite receivers (DBS), radio astronomy, radar.

UGC NET focus Why mobility is high (donor–electron spatial separation); 2DEG = quantum-well confinement at a heterointerface; AlGaAs/GaAs material pair; application keyword "low-noise microwave amplifier".

§ 1.15Solar Cells — I–V Characteristics, Fill Factor, Efficiency

A solar cell is a large-area PN junction operated in the fourth quadrant (V > 0, I < 0 by diode convention → power delivered). Photons with $h\nu \ge E_g$ generate EHPs; the built-in field separates them (photovoltaic effect), driving photocurrent $I_L$ opposite to the diode forward current.

Illuminated I–V (superposition) $$ I = I_L - I_0\left(e^{\,qV/\eta kT} - 1\right) $$

Key parameters

Short-circuit current $I_{SC} = I_L$ (V = 0) — proportional to illumination intensity and cell area.
Open-circuit voltage (I = 0):
Open-circuit voltage $$ V_{OC} = \frac{\eta kT}{q}\ln\!\left(\frac{I_L}{I_0} + 1\right) $$
$V_{OC}$ rises logarithmically with intensity and increases with $E_g$ (smaller $I_0$); it decreases with temperature (~−2.2 mV/°C for Si) — overall efficiency drops on hot panels.
Maximum power point (MPP): the ($V_m, I_m$) maximizing P = VI; tracked electronically (MPPT) in real systems.

Fill factor $$ FF = \frac{V_m I_m}{V_{OC} I_{SC}} \quad (\text{typically } 0.7\text{–}0.85 \text{ for good Si cells}) $$

Conversion efficiency $$ \eta = \frac{P_{max}}{P_{in}} = \frac{FF \cdot V_{OC} I_{SC}}{P_{in}}, \qquad P_{in} = 100\ \text{mW/cm}^2\ \text{(AM1.5G standard)} $$

Loss mechanisms and design

Spectral losses: photons with $h\nu < E_g$ are not absorbed; excess energy $h\nu - E_g$ is lost as heat (thermalization). The trade-off yields an optimum gap ≈ 1.1–1.5 eV; single-junction detailed-balance (Shockley–Queisser) limit ≈ 33% (~1.34 eV). Tandem/multijunction cells exceed it (>47% under concentration).
Parasitic resistances: series $R_s$ (fingers, bulk — flattens the knee, lowers FF) and shunt $R_{sh}$ (edge leakage — sags the flat part). Ideal: $R_s \to 0$, $R_{sh}\to\infty$.
Optical design: anti-reflection coating (SiN, ~λ/4), surface texturing, back-surface field (p⁺) to repel minority carriers from the rear contact.
Technologies: mono/poly-crystalline Si (~20–26%), thin film (CdTe, CIGS), perovskites (lab >26%), a-Si.

UGC NET focus Definitions of $I_{SC}$, $V_{OC}$, FF, η and a numerical combining them; effect of intensity (I_SC linear, V_OC logarithmic); effect of temperature (η falls); fourth-quadrant operation; why E_g ≈ 1.4 eV is optimal.

§ 1.16LED, LCD and Flexible Display Devices

Light-Emitting Diode (LED)

A forward-biased direct-gap PN junction emitting by injection electroluminescence — radiative recombination of injected minority carriers.

Emission wavelength $$ \lambda\,(\mu m) \approx \frac{hc}{E_g} = \frac{1.24}{E_g\,(\text{eV})} $$

Materials: GaAs (~870 nm IR), GaAsP (red–yellow), GaP:N (green), InGaN/GaN (blue, ~465 nm — 2014 Nobel; enables white LEDs via yellow YAG:Ce phosphor), AlGaInP (red–amber).
Si/Ge cannot serve as LEDs — indirect gap, phonon-assisted (non-radiative-dominated) recombination.
Efficiencies: internal quantum efficiency $\eta_{int} = \tau_{nr}/(\tau_r+\tau_{nr})$-type ratio of radiative recombination; extraction efficiency limited by total internal reflection (critical angle $\theta_c = \sin^{-1}(1/n)$, n ≈ 3.5 → dome encapsulation); luminous efficacy (lm/W).
LED drive: always with series current-limiting resistor $R = (V_S - V_F)/I_F$; forward drop 1.8–3.3 V depending on colour (higher $E_g$ → higher $V_F$).
OLED: organic emissive layers between electrodes — self-emissive, high contrast, flexible-capable; lifetime/burn-in is the drawback.

Liquid Crystal Display (LCD)

Passive light modulator (does not emit; needs backlight or ambient light) exploiting the electro-optic response of nematic liquid crystals.
Twisted-nematic (TN) cell: 90°-twisted LC between crossed polarizers. OFF: the twist guides polarization → light passes (normally white). ON (field applied): molecules align with the field, twist destroyed → light blocked.
Driven by AC voltage (DC causes electrochemical degradation); extremely low power (µW/cm²) → watches, calculators, meters.
Active-matrix (AM) LCD / TFT-LCD: one thin-film transistor (a-Si, LTPS or IGZO — a ZnO-based oxide semiconductor) per pixel eliminates cross-talk and enables large high-resolution panels. IPS/VA modes improve viewing angle over TN.
Comparison with LED/OLED displays: LCD = modulator (needs backlight, limited contrast); OLED = emissive (true black, wider angle, flexible).

Flexible display devices

Built on plastic substrates (PET, PEN, polyimide) or ultrathin glass instead of rigid glass; require low-temperature processing.
Key enabling technologies: flexible OLED/AMOLED (foldable phones), electrophoretic E-paper (bistable, reflective, near-zero static power — e-readers), TFT backplanes of IGZO/LTPS or organic transistors, transparent conductors replacing brittle ITO (graphene, CNT films, Ag nanowires, PEDOT:PSS), thin-film encapsulation against O₂/H₂O.
Mechanical metric: bending radius; classes — bendable → rollable → foldable → stretchable.

UGC NET focus λ = 1.24/E_g numericals; why indirect-gap Si is unsuitable; LCD is a passive device driven by AC; TN cell with crossed polarizers; phosphor route to white light; ITO replacement materials connect to §1.17.

§ 1.17Emerging Materials: Graphene, CNT, ZnO, SiC

Graphene

Single sp²-bonded atomic layer of carbon (honeycomb lattice); isolated 2004 (Geim & Novoselov, Nobel 2010).
Zero band gap semimetal with linear E–k dispersion near the Dirac points: $E = \hbar v_F |k|$, $v_F \approx 10^6$ m/s — carriers behave as massless Dirac fermions.
Record properties: mobility up to ~2×10⁵ cm²/V·s (suspended), thermal conductivity ~5000 W/m·K, Young's modulus ~1 TPa, ~97.7% optical transparency (absorbs πα ≈ 2.3% per layer).
Device hurdle: no gap → poor ON/OFF ratio in logic FETs. Gap engineering: nanoribbons, bilayer graphene under vertical field, strain. Strengths: RF transistors, transparent electrodes, sensors (every atom is surface), spintronics; exhibits anomalous/half-integer quantum Hall effect.

Carbon Nanotubes (CNT)

Rolled graphene cylinders (Iijima, 1991); single-walled (SWCNT, 0.4–3 nm) and multi-walled (MWCNT).
Chirality vector (n, m) decides character: metallic if (n − m) mod 3 = 0 (armchair n = m always metallic); otherwise semiconducting with $E_g \approx 0.8/d\,(\text{nm})$ eV.
Quasi-1D ballistic conductors (mean free path ~µm); current density ~10⁹ A/cm² (1000× Cu); ideal subjects of the Landauer picture (§1.13).
Applications: CNT-FETs (gate-all-around, sub-10 nm channels), interconnects and vias, field-emission tips, gas/bio sensors, composite reinforcement, supercapacitor electrodes. Challenge: chirality-pure sorting and placement.

Zinc Oxide (ZnO)

II–VI wurtzite semiconductor; direct wide gap 3.37 eV; huge exciton binding energy 60 meV (> kT at 300 K) → robust room-temperature UV emission (~380 nm).
Naturally n-type (O vacancies / Zn interstitials, H); reliable p-type doping remains the central difficulty, limiting homojunction LEDs.
Piezoelectric (non-centrosymmetric) → SAW devices, piezotronic nanogenerators (nanowire arrays); transparent conducting oxide when doped (AZO = Al:ZnO, an ITO alternative); varistors (grain-boundary nonlinearity) for surge protection; UV photodetectors, gas sensors; IGZO TFT channels for display backplanes (links to §1.16).

Silicon Carbide (SiC)

IV–IV compound with ~250 polytypes; device-grade 4H-SiC: indirect gap ≈ 3.26 eV.
Wide-band-gap power-device champion: critical (breakdown) field ~3×10⁶ V/cm (≈10× Si), thermal conductivity ~4.9 W/cm·K (>3× Si), high saturation velocity (2×10⁷ cm/s), operation >300 °C; Baliga figure of merit $ \varepsilon \mu E_c^3 $ far above Si.
Devices: Schottky barrier diodes (no reverse recovery), power MOSFETs (650 V–3.3 kV+), used in EV traction inverters and on-board chargers, solar inverters, traction drives. The only compound semiconductor with a native thermal oxide (SiO₂) — enables MOS technology; channel mobility and oxide reliability are the engineering challenges.
GaN vs SiC (frequent comparison): GaN → higher-frequency/lower-voltage RF & fast chargers (lateral HEMT); SiC → higher-voltage/high-power vertical devices.

UGC NET focus Graphene: zero gap, linear dispersion, 2.3% absorption. CNT: (n−m) mod 3 metallicity rule. ZnO: 3.37 eV + 60 meV exciton, p-doping problem, piezoelectricity. SiC: 4H polytype, ~3.2–3.3 eV, 10× breakdown field, power electronics keyword.

§ 1.18Unit-1 Formula Sheet

One-stop formula table — Unit 1
Topic	Formula	Notes
Mass-action law	$ np = n_i^2 $	equilibrium only
Intrinsic concentration	$ n_i = \sqrt{N_C N_V}\,e^{-E_g/2kT} $	∝ T^{3/2}e^{−E_g/2kT}
Conductivity	$ \sigma = q(n\mu_n + p\mu_p) $	—
Einstein relation	$ D/\mu = kT/q = V_T $	≈ 26 mV @ 300 K
Hall coefficient	$ R_H = \pm 1/qn $	sign → carrier type
Effective mass	$ m^* = \hbar^2 / (d^2E/dk^2) $	inverse curvature
3D DOS	$ g(E) = \tfrac{1}{2\pi^2}(2m^*/\hbar^2)^{3/2}\sqrt{E} $	2D: step; 1D: E^{−1/2}; 0D: δ
Fermi function	$ f(E) = [1+e^{(E-E_F)/kT}]^{-1} $	f(E_F) = ½
Carrier concentrations	$ n = N_C e^{-(E_C-E_F)/kT} $	Boltzmann regime
Fermi-level shift	$ E_F - E_{Fi} = kT\ln(n/n_i) $	n- vs p-type sign
Built-in potential	$ V_{bi} = V_T \ln(N_AN_D/n_i^2) $	—
Depletion width	$ W = \sqrt{\tfrac{2\varepsilon_s}{q}\big(\tfrac1{N_A}+\tfrac1{N_D}\big)(V_{bi}-V)} $	∝ √(V_bi+V_R)
Junction capacitance	$ C_j = \varepsilon_s A/W \propto (V_{bi}-V)^{-1/2} $	varactor
Diode equation	$ I = I_0(e^{V/\eta V_T}-1) $	η = 1–2
Dynamic resistance	$ r_d = \eta V_T/I \approx 26/I_{mA}\ \Omega $	—
Diffusion length	$ L = \sqrt{D\tau} $	—
Schottky current	$ I = A^*AT^2e^{-q\phi_B/kT}(e^{qV/kT}-1) $	thermionic emission
JFET transfer	$ I_D = I_{DSS}(1 - V_{GS}/V_P)^2 $	g_m = 2√(I_DSS·I_D)/\|V_P\|
MOSFET (sat.)	$ I_D = \tfrac12 \mu C_{ox}\tfrac{W}{L}(V_{GS}-V_T)^2 $	triode: see §1.12
Threshold voltage	$ V_T = V_{FB} + 2\phi_F + \sqrt{2\varepsilon_s qN_A 2\phi_F}/C_{ox} $	body effect adds γ-term
QW energy levels	$ E_n = n^2h^2/8m^*L^2 $	blue-shift as L ↓
Conductance quantum	$ G_0 = 2e^2/h \approx 77.5\ \mu\text{S} $	ballistic 1D
Solar cell I–V	$ I = I_L - I_0(e^{qV/\eta kT}-1) $	4th quadrant
Open-circuit voltage	$ V_{OC} = \tfrac{\eta kT}{q}\ln(I_L/I_0 + 1) $	log of intensity
Fill factor / efficiency	$ FF = \tfrac{V_mI_m}{V_{OC}I_{SC}},\ \ \eta = \tfrac{FF\,V_{OC}I_{SC}}{P_{in}} $	AM1.5G: 100 mW/cm²
Photon wavelength	$ \lambda(\mu m) = 1.24/E(\text{eV}) $	LED / absorption edge
Graphene dispersion	$ E = \hbar v_F\|k\| $	v_F ≈ 10⁶ m/s
CNT gap	$ E_g \approx 0.8/d(\text{nm}) $ eV	metallic if (n−m) mod 3 = 0

Constants & material data worth memorizing
Quantity	Value
kT/q at 300 K	0.0259 V (use 26 mV)
E_g: Ge / Si / GaAs / GaN / ZnO / 4H-SiC	0.66 / 1.12 / 1.42 / 3.4 / 3.37 / 3.26 eV
n_i (300 K): Si / Ge / GaAs	1.5×10¹⁰ / 2.5×10¹³ / ~2×10⁶ cm⁻³
µ_n (300 K): Si / Ge / GaAs	1350 / 3900 / 8500 cm²·V⁻¹·s⁻¹
ε_r: Si / SiO₂ / GaAs	11.7 / 3.9 / 13.1

§ 1.19Quick Revision Notes — Unit 1 in 25 Points

Rapid-fire recap (last-day revision)

Semiconductor resistance falls with temperature (more EHPs); metal resistance rises.
$np = n_i^2$ holds at equilibrium regardless of doping; doping moves $E_F$, not the product.
Fermi level: f(E_F) = ½ always; flat across any system in equilibrium.
n-type: E_F near E_C; p-type: near E_V; heating drives E_F → mid-gap (intrinsic).
$m^* = \hbar^2/(d^2E/dk^2)$: sharp band curvature = light, fast carriers (GaAs: 0.067 m₀).
DOS fingerprints: 3D √E · 2D staircase · 1D 1/√E · 0D δ-spikes.
Hall effect gives carrier type and concentration; mobility needs σ as well.
Depletion region extends mostly into the lightly doped side; $N_Ax_p = N_Dx_n$.
$W \propto \sqrt{V_{bi}+V_R}$; $C_j \propto W^{-1}$ → varactor: capacitance tuned by reverse bias.
Diode: I₀ doubles per 10 °C; V_F drifts −2 mV/°C (Si); r_d = 26 mV / I.
Zener (<5 V): tunneling, heavy doping, negative TC. Avalanche (>6 V): impact ionization, positive TC. ~5–6 V: both → temperature-stable references.
Tunnel diode: degenerate doping, NDR between peak and valley, GHz-class majority-carrier device; conducts in reverse.
Schottky: majority-carrier rectifier — 0.2–0.3 V drop, no reverse-recovery storage → SMPS & RF.
Practical ohmic contact = metal on n⁺/p⁺ layer (tunneling), not a work-function accident.
JFET: depletion-mode only; $I_D = I_{DSS}(1-V_{GS}/V_P)^2$; saturation starts at $V_{DS} = V_{GS}-V_P$.
MOSFET square law $ \tfrac12\mu C_{ox}\tfrac{W}{L}(V_{GS}-V_T)^2 $; body bias raises V_T; gate draws ≈ zero DC current.
$f_T \propto 1/L^2$ — why scaling makes MOSFETs faster.
Quantum confinement when size ≲ de Broglie λ / exciton Bohr radius; $E_n = n^2h^2/8m^*L^2$; smaller dot → bluer emission.
HEMT: modulation doping separates electrons from donors → 2DEG with very high mobility; the low-noise microwave transistor; GaN HEMT = polarization 2DEG for power/RF.
Solar cell: 4th-quadrant PN junction; $I_{SC} \propto$ intensity (linear), $V_{OC} \propto \ln$(intensity); efficiency drops with temperature.
FF = V_mI_m / V_OCI_SC (0.7–0.85); Shockley–Queisser single-junction limit ≈ 33% near E_g ≈ 1.34 eV.
LED needs a direct gap; λ(µm) = 1.24/E_g(eV); blue InGaN + yellow phosphor = white.
LCD: passive, AC-driven, TN cell between crossed polarizers; TFT (a-Si/LTPS/IGZO) active matrix for large panels.
Flexible displays: polyimide substrates, OLED/E-paper frontplanes, ITO replaced by graphene/CNT/Ag-nanowires, thin-film encapsulation.
Materials one-liners — Graphene: zero-gap Dirac semimetal, 2.3% absorption. CNT: metallic iff (n−m) mod 3 = 0. ZnO: 3.37 eV, 60 meV exciton, p-doping problem. SiC: 4H, ~3.26 eV, 10× Si breakdown field → EV power electronics.

↑ Back to top of Unit 1

Structure	Confined / free directions	g(E) shape	Devices
Bulk (3D)	0 / 3	\( \propto \sqrt{E} \)	conventional
Quantum well (2D)	1 / 2	staircase, \( m^*/\pi\hbar^2 \) per subband	QW lasers, HEMT 2DEG, QWIP detectors
Quantum wire (1D)	2 / 1	\( \propto 1/\sqrt{E} \) spikes	nanowire FETs, ballistic conductors
Quantum dot (0D)	3 / 0	δ-functions ("artificial atom")	QD lasers, QLED displays, single-photon sources, qubits

Topic	Formula	Notes
Mass-action law	\( np = n_i^2 \)	equilibrium only
Intrinsic concentration	\( n_i = \sqrt{N_C N_V}\,e^{-E_g/2kT} \)	∝ T^{3/2}e^{−E_g/2kT}
Conductivity	\( \sigma = q(n\mu_n + p\mu_p) \)	—
Einstein relation	\( D/\mu = kT/q = V_T \)	≈ 26 mV @ 300 K
Hall coefficient	\( R_H = \pm 1/qn \)	sign → carrier type
Effective mass	\( m^* = \hbar^2 / (d^2E/dk^2) \)	inverse curvature
3D DOS	\( g(E) = \tfrac{1}{2\pi^2}(2m^*/\hbar^2)^{3/2}\sqrt{E} \)	2D: step; 1D: E^{−1/2}; 0D: δ
Fermi function	\( f(E) = [1+e^{(E-E_F)/kT}]^{-1} \)	f(E_F) = ½
Carrier concentrations	\( n = N_C e^{-(E_C-E_F)/kT} \)	Boltzmann regime
Fermi-level shift	\( E_F - E_{Fi} = kT\ln(n/n_i) \)	n- vs p-type sign
Built-in potential	\( V_{bi} = V_T \ln(N_AN_D/n_i^2) \)	—
Depletion width	\( W = \sqrt{\tfrac{2\varepsilon_s}{q}\big(\tfrac1{N_A}+\tfrac1{N_D}\big)(V_{bi}-V)} \)	∝ √(V_bi+V_R)
Junction capacitance	\( C_j = \varepsilon_s A/W \propto (V_{bi}-V)^{-1/2} \)	varactor
Diode equation	\( I = I_0(e^{V/\eta V_T}-1) \)	η = 1–2
Dynamic resistance	\( r_d = \eta V_T/I \approx 26/I_{mA}\ \Omega \)	—
Diffusion length	\( L = \sqrt{D\tau} \)	—
Schottky current	\( I = A^*AT^2e^{-q\phi_B/kT}(e^{qV/kT}-1) \)	thermionic emission
JFET transfer	\( I_D = I_{DSS}(1 - V_{GS}/V_P)^2 \)	g_m = 2√(I_DSS·I_D)/\|V_P\|
MOSFET (sat.)	\( I_D = \tfrac12 \mu C_{ox}\tfrac{W}{L}(V_{GS}-V_T)^2 \)	triode: see §1.12
Threshold voltage	\( V_T = V_{FB} + 2\phi_F + \sqrt{2\varepsilon_s qN_A 2\phi_F}/C_{ox} \)	body effect adds γ-term
QW energy levels	\( E_n = n^2h^2/8m^*L^2 \)	blue-shift as L ↓
Conductance quantum	\( G_0 = 2e^2/h \approx 77.5\ \mu\text{S} \)	ballistic 1D
Solar cell I–V	\( I = I_L - I_0(e^{qV/\eta kT}-1) \)	4th quadrant
Open-circuit voltage	\( V_{OC} = \tfrac{\eta kT}{q}\ln(I_L/I_0 + 1) \)	log of intensity
Fill factor / efficiency	\( FF = \tfrac{V_mI_m}{V_{OC}I_{SC}},\ \ \eta = \tfrac{FF\,V_{OC}I_{SC}}{P_{in}} \)	AM1.5G: 100 mW/cm²
Photon wavelength	\( \lambda(\mu m) = 1.24/E(\text{eV}) \)	LED / absorption edge
Graphene dispersion	\( E = \hbar v_F\|k\| \)	v_F ≈ 10⁶ m/s
CNT gap	\( E_g \approx 0.8/d(\text{nm}) \) eV	metallic if (n−m) mod 3 = 0