GATE E&C Engineering Electronic Devices Syllabus

Write the total current density in a semiconductor including both drift and diffusion.

$$J = J_n + J_p$$ $$J_n = q n \mu_n E + q D_n \frac{dn}{dx}, \quad J_p = q p \mu_p E - q D_p \frac{dp}{dx}$$

State the Einstein relation linking diffusion coefficient and mobility.

$$\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q} = V_T$$ where $V_T \approx 25.9\,\text{mV}$ at $300\,K$.

Define carrier mobility $\mu$ and give its units.

Mobility is the drift velocity per unit electric field: $\mu = \frac{v_d}{E}$, with units $\text{cm}^2/(V\cdot s)$. It measures how easily carriers move under a field.

Compare electron and hole mobility in silicon and explain.

In Si, $\mu_n \approx 1350\,\text{cm}^2/Vs$ and $\mu_p \approx 480\,\text{cm}^2/Vs$. Electrons are more mobile than holes because of their smaller effective mass, so $\mu_n > \mu_p$.

How do mobility and conductivity depend on temperature and doping at high doping?

At high doping/low temperature, ionized-impurity scattering dominates ($\mu \propto T^{3/2}$, rising with T); at high temperature, lattice (phonon) scattering dominates ($\mu \propto T^{-3/2}$, falling with T). Mobility decreases with increasing doping.

Write the conductivity and resistivity of a semiconductor in terms of carrier concentrations and mobilities.

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

For an n-type extrinsic semiconductor, simplify the conductivity expression.

Since $n \approx N_D \gg p$: $$\sigma \approx q N_D \mu_n$$

What is carrier generation in a semiconductor and what supplies the energy?

Generation is the creation of electron-hole pairs by breaking covalent bonds. Energy can come from thermal vibration (thermal generation) or photon absorption (optical generation) with $h\nu \geq E_g$.

What is carrier recombination and name its main mechanisms?

Recombination is an electron falling from the conduction band into a hole, annihilating a pair. Mechanisms: (1) band-to-band (radiative), (2) trap-assisted Shockley-Read-Hall (non-radiative), and (3) Auger recombination.

Define minority-carrier lifetime $\tau$ and relate it to the recombination rate.

$\tau$ is the average time an excess minority carrier survives before recombining. For low-level injection: $$R = \frac{\Delta p}{\tau_p} = \frac{\Delta n}{\tau_n}$$

How does excess carrier concentration decay with time after the source is removed (low-level injection)?

$$\Delta p(t) = \Delta p(0)\,e^{-t/\tau_p}$$ Exponential decay governed by the minority-carrier lifetime.

State the diffusion length $L$ and its relation to lifetime and diffusion coefficient.

$$L_p = \sqrt{D_p \tau_p}, \qquad L_n = \sqrt{D_n \tau_n}$$ It is the average distance a minority carrier diffuses before recombining.

Write Poisson's equation as used in semiconductor device analysis (1-D).

$$\frac{d^2\phi}{dx^2} = -\frac{dE}{dx} = -\frac{\rho}{\varepsilon}$$ where $\rho = q(p - n + N_D^{+} - N_A^{-})$ is the charge density and $\varepsilon$ the permittivity.

Write the steady-state continuity equation for holes in one dimension.

$$\frac{\partial p}{\partial t} = -\frac{1}{q}\frac{\partial J_p}{\partial x} + (G_p - R_p) = 0 \text{ (steady state)}$$ Expanded: $D_p\frac{d^2 p}{dx^2} - \mu_p\frac{d(pE)}{dx} + G_p - R_p = 0$.

Write the minority-carrier diffusion equation (steady state, no field, no generation) for excess holes in n-region.

$$\frac{d^2 \Delta p}{dx^2} = \frac{\Delta p}{L_p^{2}}, \qquad L_p = \sqrt{D_p \tau_p}$$

What is the built-in potential of a P-N junction and its formula?

It is the equilibrium potential barrier across the depletion region: $$V_{bi} = V_T \ln\!\left(\frac{N_A N_D}{n_i^{2}}\right)$$

Write the depletion (space-charge) region width of a P-N junction.

$$W = \sqrt{\frac{2\varepsilon}{q}\left(\frac{1}{N_A}+\frac{1}{N_D}\right)(V_{bi} - V)}$$ where $V$ is the applied forward voltage.

State the ideal diode (Shockley) equation.

$$I = I_0\left(e^{V/\eta V_T} - 1\right)$$ where $I_0$ is the reverse saturation current, $\eta$ the ideality factor ($1$ to $2$), and $V_T = kT/q$.

How does the depletion-region width and junction capacitance change under reverse bias?

Reverse bias widens the depletion region ($W \propto \sqrt{V_{bi}+V_R}$) and decreases the junction (transition) capacitance $C_j = \varepsilon A / W \propto 1/\sqrt{V_{bi}+V_R}$.

Distinguish transition (junction) capacitance and diffusion capacitance of a diode.

Transition capacitance $C_T = \varepsilon A/W$ dominates under reverse bias (from depletion charge). Diffusion capacitance $C_D = \tau I/V_T$ dominates under forward bias (from stored minority charge).

What is a Zener diode and in which region is it operated?

A Zener diode is a heavily doped P-N junction designed to operate in reverse breakdown, where it maintains a nearly constant voltage. It is used as a voltage regulator/reference.

Compare Zener breakdown and Avalanche breakdown.

Zener breakdown: heavy doping, thin depletion region, high field tunneling, $V_Z \lesssim 5\,V$, negative temperature coefficient. Avalanche breakdown: lighter doping, wide depletion, impact ionization, $V_Z \gtrsim 6\,V$, positive temperature coefficient.

In a BJT, name the three regions and their relative doping levels.

Emitter (heavily doped), Base (lightly doped and very thin), Collector (moderately doped, largest area). The asymmetry maximizes emitter injection efficiency and collector dissipation.

Define the BJT current-gain parameters $\alpha$ and $\beta$ and relate them.

$\alpha = I_C/I_E$ (common-base gain, $<1$); $\beta = I_C/I_B$ (common-emitter gain). $$\beta = \frac{\alpha}{1-\alpha}, \qquad \alpha = \frac{\beta}{\beta+1}$$

Write the fundamental BJT terminal current relationship.

$$I_E = I_C + I_B$$ and in active region $I_C = \alpha I_E = \beta I_B$.

State the bias conditions of the BJT junctions in the four operating regions.

Active: EB forward, CB reverse. Saturation: both forward. Cutoff: both reverse. Reverse-active: EB reverse, CB forward.

Write the Ebers-Moll active-region collector current expression for a BJT.

$$I_C \approx I_S\,e^{V_{BE}/V_T}$$ where $I_S$ is the saturation current; $I_C$ is nearly independent of $V_{CB}$ in the active region (ignoring Early effect).

What is the Early effect in a BJT?

Base-width modulation: increasing reverse collector-base voltage widens the CB depletion region, narrows the effective base, and slightly increases $I_C$. Output curves extrapolate back to $-V_A$ (the Early voltage).

Describe the structure of a MOS capacitor.

A MOS capacitor is a metal (gate) / oxide (insulator, e.g. $\text{SiO}_2$) / semiconductor (e.g. p-Si) sandwich, with an ohmic back contact. The oxide capacitance per area is $C_{ox} = \varepsilon_{ox}/t_{ox}$.

Name the three operating regions of a MOS capacitor (p-type substrate) as gate voltage varies.

Accumulation (negative $V_G$: holes accumulate at surface), Depletion (small positive $V_G$: surface depleted), and Inversion (large positive $V_G > V_T$: electron inversion layer forms).

Define the threshold voltage condition (strong inversion) for a MOS capacitor.

Strong inversion occurs when the surface potential equals twice the bulk potential: $$\phi_s = 2\phi_F, \qquad \phi_F = V_T \ln\!\left(\frac{N_A}{n_i}\right)$$

Write a MOSFET drain current expression in the triode (linear) region.

$$I_D = \mu_n C_{ox}\frac{W}{L}\left[(V_{GS}-V_T)V_{DS} - \frac{V_{DS}^{2}}{2}\right], \quad V_{DS} < V_{GS}-V_T$$

Write the MOSFET drain current in saturation and the saturation condition.

$$I_D = \frac{1}{2}\mu_n C_{ox}\frac{W}{L}(V_{GS}-V_T)^{2}, \qquad V_{DS} \geq V_{GS}-V_T$$

What is channel-length modulation in a MOSFET and how is it modeled?

In saturation, increasing $V_{DS}$ shortens the effective channel, raising $I_D$. Modeled as: $$I_D = \frac{1}{2}\mu_n C_{ox}\frac{W}{L}(V_{GS}-V_T)^{2}(1+\lambda V_{DS})$$

Differentiate enhancement-mode and depletion-mode MOSFETs.

Enhancement mode: no channel at $V_{GS}=0$, a channel must be induced ($V_T>0$ for NMOS). Depletion mode: channel exists at $V_{GS}=0$, gate voltage depletes it ($V_T<0$ for NMOS).

Compare BJT and MOSFET as devices (control, impedance, type).

BJT: current-controlled ($I_C=\beta I_B$), bipolar (both carriers), low input impedance. MOSFET: voltage-controlled ($V_{GS}$), unipolar (one carrier), very high input impedance, easier to scale/integrate.

What is an LED and what physical process produces its light?

A Light Emitting Diode is a forward-biased P-N junction made from a direct band-gap semiconductor. Injected carriers undergo radiative recombination, emitting photons of energy $h\nu \approx E_g$.

Give the relation between LED emission wavelength and band gap.

$$\lambda = \frac{hc}{E_g} \approx \frac{1240}{E_g(\text{eV})}\,\text{nm}$$ A larger band gap gives a shorter (bluer) wavelength.

Why is silicon not used to make LEDs, and which materials are used instead?

Silicon is an indirect band-gap semiconductor, so recombination is non-radiative (phonon-assisted) and inefficient. LEDs use direct band-gap compounds such as GaAs, GaAsP, GaN and InGaN.