Semiconductor Electronics: Materials, Devices, and Simple Circuits - Class 12

A cylindrical semiconductor rod containing a p-n junction is rigidly clamped between two immovable walls along its length (the $x$-axis). The lateral surface of the rod is completely free to expand. The rod has a coefficient of linear thermal expansion $\alpha$, Poisson's ratio $\nu$, and Young's modulus $Y$. The temperature of the rod is uniformly increased by $\Delta T$.

According to Deformation Potential Theory, the bandgap energy $E_g$ of the semiconductor changes with the volumetric strain $\frac{\Delta V}{V}$ as $\Delta E_g = \Xi \frac{\Delta V}{V}$, where $\Xi$ is the deformation potential constant (in Joules). Assume the built-in potential $V_{bi}$ of the junction changes solely due to this mechanically-induced bandgap shift, given by $\Delta V_{bi} = \frac{\Delta E_g}{e}$ (where $e$ is the elementary charge in Coulombs).

The depletion width of the junction is given by $W = \sqrt{\frac{2\varepsilon V_{bi}}{e N_{eff}}}$. If the initial built-in potential is $V_{bi0}$ and the initial depletion width is $W_0$, which of the following expressions correctly represents the fractional change in the square of the depletion width, $\frac{\Delta (W^2)}{W_0^2}$?

A full-wave rectifier circuit operates with an input AC frequency $f = 50 \text{ Hz}$ and produces a peak output voltage $V_m = 20 \text{ V}$. The output is filtered by a capacitor $C = \frac{250}{\sqrt{3}} \mu\text{F}$ and connected to a load which is a Positive Temperature Coefficient (PTC) thermistor. The thermistor's resistance varies with its temperature $T$ as $R(T) = R_0 [1 + \alpha (T - T_a)]$, where $R_0 = 100 \, \Omega$ is the resistance at the ambient temperature $T_a$, and $\alpha = 0.05 \text{ K}^{-1}$. The thermistor dissipates heat to the surroundings at a rate $P_{loss} = K(T - T_a)$, where $K = 0.1 \text{ W/K}$. Assuming the DC output voltage across the load is approximately equal to $V_m$ for power calculations, what is the steady-state ripple factor $\gamma$ of the output voltage?

A full-wave rectifier operates from a $50 \text{ Hz}$ AC mains supply and delivers a constant direct current $I_0 = 28 \text{ mA}$ to a load. The filtering circuit uses a parallel-plate capacitor with a non-linear ferroelectric dielectric. Due to the non-linear energy density locality in the dielectric, the total charge $q$ stored on the capacitor depends on the voltage $V$ across it as $q(V) = C_0 V + k V^2$, where $C_0 = 50 \text{ \mu F}$ and $k = 5 \text{ \mu F/V}$. The peak voltage reached by the capacitor during each charging cycle is $V_0 = 10 \text{ V}$. Assuming the discharge time is approximately equal to the time interval between consecutive voltage peaks, what is the peak-to-peak ripple voltage $\Delta V$ across the load?

A full-wave rectifier is connected to an AC source of frequency $f = 50\text{ Hz}$ and peak voltage $V_m = 150\pi\text{ V}$. The output of the rectifier is fed into an L-C filter consisting of a series inductor $L = \frac{1.25}{\pi}\text{ H}$ and a parallel capacitor $C = \frac{100}{\pi}\text{ }\mu\text{F}$ connected across a load resistor $R = 1000\text{ }\Omega$. Assuming $R \gg X_C$ at the fundamental ripple frequency, calculate the amplitude of the fundamental wattless (reactive) current drawn from the rectifier by the filter circuit.

A small-signal half-wave rectifier circuit uses a p-n junction diode biased at a constant forward DC voltage $V_0$. The circuit employs an inductor filter and a negligible load resistance such that the ripple factor $\gamma$ of the output is directly proportional to the diode's dynamic forward resistance $r_d$. The reverse saturation current of the diode is given by $I_s = C T^3 \exp(-E_g/kT)$, where $C$ is a constant, $k$ is the Boltzmann constant, and $E_g$ is the bandgap energy. Due to thermal expansion and electron-phonon interactions, the bandgap decreases with absolute temperature $T$ as $E_g(T) = E_{g0} - \alpha T$, where $E_{g0}$ and $\alpha$ are positive constants. Assuming the diode operates in the region where the forward current is $I \approx I_s \exp(eV_0/kT)$ (with $e$ being the elementary charge), what is the fractional change in the ripple factor, $\frac{\Delta \gamma}{\gamma}$, for a small increase in temperature $\Delta T$?

A full-wave rectifier circuit provides a peak output voltage of $V_0 = 100 \text{ V}$. It is connected to a filter capacitor $C = 1000 \text{ } \mu\text{F}$ in parallel with a DC motor. The motor has an armature resistance $R = 10 \text{ } \Omega$ and negligible inductance. At $t = 0$, the rectifier diodes become reverse-biased exactly as the capacitor voltage reaches its peak $V_0$. At this instant, the motor is running with a back-EMF $E = 50 \text{ V}$, which is increasing at a constant rate of $500 \text{ V/s}$ due to the motor's acceleration. Assuming ideal components, what is the instantaneous rate of change of the current flowing through the motor, $\frac{dI}{dt}$, at $t = 0^+$?

An NPN transistor is operating in the active region within a common-emitter amplifier circuit. The base current is maintained strictly constant at $I_B = 10 \ \mu\text{A}$. The collector is connected to a DC supply $V_{CC} = 30 \ \text{V}$ through a temperature-dependent load resistor $R_C$. At a reference temperature $T_0$, the resistance is $R_C(T_0) = 5 \ \text{k}\Omega$. The resistance increases with temperature according to $R_C(T) = R_C(T_0)[1 + \alpha(T - T_0)]$, where the temperature coefficient is $\alpha = 0.02 \ ^\circ\text{C}^{-1}$.

Due to base-width modulation (the Early effect), the macroscopic current gain $\beta$ is not constant but depends on the collector-emitter voltage $V_{CE}$ as $\beta(V_{CE}) = \beta_0 \left(1 + \frac{V_{CE}}{V_A}\right)$, where the zero-bias gain is $\beta_0 = 100$ and the Early voltage is $V_A = 150 \ \text{V}$.

If the operating temperature is raised to $T = T_0 + 50 \ ^\circ\text{C}$, what is the effective macroscopic current gain $\beta_{eff} = \frac{I_C}{I_B}$ of the transistor in the steady state?

A common-emitter NPN transistor amplifier has a low-frequency small-signal current gain $\beta_0$ and a base-emitter dynamic resistance $r_\pi$. A parasitic parallel-plate capacitor is formed directly across the base-emitter terminals. The capacitor has circular plates of radius $R$ and separation $d$. The space between the plates is filled with a non-uniform dielectric whose permittivity varies radially as $\epsilon(r) = \epsilon_0(1 + \alpha r)$, where $r$ is the radial distance from the central axis. An AC input voltage $V_{be}(t) = V_0 \sin(\omega t)$ is applied across the base-emitter terminals. Assuming the intrinsic capacitances of the transistor are negligible, which of the following expressions correctly gives the maximum induced magnetic field $B_0$ at the edge of the capacitor ($r=R$) and the effective high-frequency current gain magnitude $|\beta_{eff}| = |I_c|/|I_{in}|$ of the combined structure? (Here $I_{in}$ is the total current supplied by the source and $I_c$ is the collector current).

A common-emitter amplifier uses an NPN transistor. A potentiometer of length $L = 100 \text{ cm}$ is connected across a $V_0 = 2.0 \text{ V}$ DC supply. The wiper, positioned at a distance $x$ from the grounded end, provides the DC bias voltage $V_{BE}$ to the transistor. The collector is connected to a $V_{CC} = 12 \text{ V}$ supply via a load resistor $R_C = 500 \ \Omega$.

The collector current $I_C$ (in mA) varies with $V_{BE}$ (in V) according to the empirical relation: $I_C(V_{BE}) = 4.0 + 60(V_{BE} - 0.7) - 2000(V_{BE} - 0.7)^3$ for the range $0.6 \le V_{BE} \le 0.8 \text{ V}$.

A small time-varying signal $v_i(t)$ is superimposed on the DC bias $V_{BE}$. Assuming the base current drawn by the transistor is negligible compared to the current in the potentiometer, determine the wiper position $x$ that biases the transistor for maximum small-signal voltage gain, and the magnitude of this maximum gain $|A_v|$.

An NPN transistor operates in the active region with a high-frequency AC base current $I_B(t) = I_0 \sin(\omega t)$. The time-varying voltage across the collector-base junction is given by $V_{CB}(t) = V_0 \sin(\omega t)$. The collector-base depletion region acts as an ideal parallel-plate capacitor with cross-sectional area $A$, width $d$, and uniform permittivity $\epsilon$. The transistor has a constant DC current gain $\beta$. According to the generalized Ampere-Maxwell law, the total collector current must account for both conduction and displacement currents. Assuming the emitter-base junction capacitance is negligible, which of the following expressions gives the peak value of the total emitter current $I_{E,max}$?

A silicon NPN transistor is used in a common-emitter amplifier circuit with a DC collector supply voltage $V_{CC}$ and a load resistor $R_C$. The base current $I_B$ is maintained constant. The total collector current is $I_C = \beta I_B + I_{CEO}$, where $\beta$ is the current gain (assumed temperature-independent) and $I_{CEO}$ is the reverse leakage current. Let $V_{CE}$ be the operating collector-emitter voltage. Based on the kinetic theory of charge carriers, $I_{CEO}$ depends on the junction temperature $T$ as $I_{CEO}(T) = C T^3 \exp\left(-\frac{E_g}{k_B T}\right)$, where $C$ is a constant, $E_g$ is the bandgap energy, and $k_B$ is the Boltzmann constant. The transistor dissipates heat to the ambient environment with a constant thermal resistance $\theta$ (in $\text{K/W}$). Assuming $E_g \gg 3 k_B T$, which of the following represents the condition for the onset of thermal runaway?

In the base-emitter P-N junction of an N-P-N transistor at temperature $T$, the depletion region is formed by uniform doping. This results in a position-dependent electric field that peaks at the metallurgical junction and drops linearly to zero at the edges of the depletion region. Under zero bias, the maximum electric field is $E_0$ and the total depletion width is $W_0$. When a forward bias $V_{BE}$ is applied, the maximum electric field reduces to $E_1$ and the depletion width becomes $W_1$.

The emitter current $I_E$ is governed by the probability of electrons overcoming the potential barrier, yielding $I_E \approx I_0 \exp(e V_{BE} / k_B T)$ for $V_{BE} \gg k_B T/e$, where $I_0$ is the reverse saturation current, $e$ is the elementary charge, and $k_B$ is the Boltzmann constant.

Using the work-energy theorem to evaluate the potential barrier from the electric field profile, which of the following expressions correctly represents the emitter current $I_E$?

A logic circuit consists of a two-input NAND gate with infinite input impedance. The gate registers a logic HIGH (1) if the input voltage $V \ge 2.0 \text{ V}$ relative to ground, and a logic LOW (0) if $V < 2.0 \text{ V}$. Input A is connected to the cathode of photoelectric Cell A, and Input B is connected to the cathode of photoelectric Cell B. The anodes of both cells are grounded. Cell A has a work function $\Phi_A = 2.5 \text{ eV}$ and is illuminated by light of variable frequency $\nu_A$. Cell B has a work function $\Phi_B = 2.2 \text{ eV}$ and is illuminated by light of fixed wavelength $\lambda_B = 310 \text{ nm}$ with a continuously increasing intensity $I_B$. Initially, $\nu_A$ is zero. What is the minimum frequency $\nu_A$ required to switch the NAND gate output from HIGH (1) to LOW (0)? (Take $hc = 1240 \text{ eV}\cdot\text{nm}$ and Planck's constant $h = 4.14 \times 10^{-15} \text{ eV}\cdot\text{s}$)

A composite rod is formed by joining two rods of equal length $L = 1 \text{ m}$ end-to-end. Rod 1 has Young's modulus $Y_1 = Y$ and coefficient of linear expansion $\alpha_1 = 2\alpha$. Rod 2 has $Y_2 = 2Y$ and $\alpha_2 = \alpha$. The composite rod has a uniform cross-sectional area $A = 10^{-4} \text{ m}^2$ and is rigidly clamped between two immovable walls at $T_0 = 300 \text{ K}$. Given $Y = 10^{11} \text{ N/m}^2$ and $\alpha = 10^{-5} \text{ K}^{-1}$. A thermistor is attached to the rod, whose resistance varies as $R(T) = C T^{-3/2} \exp(E_g / 2kT)$, where $E_g$ is the band gap energy and $k$ is the Boltzmann constant. At $T_0 = 300 \text{ K}$, its resistance is $R_0$. A safety valve is controlled by a logic circuit with an XOR gate: $Z = X \oplus Y$. The valve opens if $Z = 1$. Input $X$ becomes HIGH (1) when the compressive thermal force in the rod $F \ge 60 \text{ kN}$. Input $Y$ becomes HIGH (1) when $R(T) \le R_0 2^{-3/2} e^{-2}$. As the system is slowly heated from $300 \text{ K}$, the safety valve NEVER opens. What is the value of $E_g / k$?

A series RLC circuit consists of a resistor $R = 100 \ \Omega$, an inductor $L = 1 \text{ H}$, and a capacitor $C = 100 \ \mu\text{F}$, driven by an AC voltage source $V(t) = 10 \sin(\omega t) \text{ V}$. Two ideal peak detectors are connected across the inductor and the capacitor to measure their respective voltage amplitudes, $V_{L,peak}$ and $V_{C,peak}$. The outputs of these detectors are fed into a digital logic circuit. Input A of a 2-input OR gate receives logic '1' if $V_{L,peak} \ge 11 \text{ V}$, and logic '0' otherwise. Input B of the same OR gate receives logic '1' if $V_{C,peak} \ge 11 \text{ V}$, and logic '0' otherwise. Let $Y$ be the Boolean output of the OR gate. Determine the sequence of outputs $(Y_1, Y_2, Y_3)$ at three driving frequencies: $\omega_1 = 50\sqrt{2} \text{ rad/s}$, $\omega_2 = 100 \text{ rad/s}$, and $\omega_3 = 100\sqrt{2} \text{ rad/s}$.

A deep-space probe of initial rest mass $M$ is equipped with two sensor panels: one perfectly absorbing and one perfectly reflecting. A logic circuit on the probe has two inputs, $A$ and $B$. Input $A$ registers '1' if the probe's internal energy increases by exactly $U_0$. Input $B$ registers '1' if the probe's final kinetic energy is exactly $K_0$. A single laser pulse of unknown energy $E$ is fired at the probe, illuminating both panels simultaneously. The reflected portion of the pulse travels back along the incident line. For this specific pulse, the logic gate $Y = A \cdot B$ outputs '1'. Assuming the probe is initially at rest in free space, what is the initial energy $E$ of the laser pulse?

(Note: $c$ is the speed of light in vacuum. Do not assume $K_0$ or $U_0$ are small compared to $Mc^2$).

An LED and a photodiode are fabricated from the same intrinsic semiconductor material. The energy bandgap $E_g$ (in $\text{J}$) of the material depends on the absolute temperature $T$ (in $\text{K}$) as $E_g(T) = E_0 - \alpha T$, where $E_0$ (in $\text{J}$) and $\alpha$ (in $\text{J/K}$) are positive constants. The threshold voltage for the LED is given by $V_{th} = E_g/e$ (where $e$ is the elementary charge in $\text{C}$). The dark current $I_d$ (in $\text{A}$) of the reverse-biased photodiode is proportional to $\exp\left(-\frac{E_g}{2k_BT}\right)$, where $k_B$ is the Boltzmann constant (in $\text{J/K}$). Assuming the proportionality constant for $I_d$ is independent of temperature, which of the following correctly represents the pair $\left( \Delta V_{th}, \; \frac{\Delta I_d}{I_d} \right)$ when the temperature is increased by a small amount $\Delta T$?

A p-n junction diode is forward-biased at a steady DC operating point with a current $I_{DC} = 50 \text{ mA}$ and a voltage $V_{DC} = 0.7 \text{ V}$. The diode's current-voltage relationship is governed by the Shockley equation $I = I_0 (e^{qV/nkT} - 1)$, where the ideality factor is $n = 2$ and the thermal voltage is $\frac{kT}{q} = 25 \text{ mV}$. A small-signal AC photocurrent $i(t) = I_m \sin(\omega t)$ with peak amplitude $I_m = 2 \text{ mA}$ is superimposed on the DC current due to a modulated light source. Assuming $I_{DC} \gg I_0$, what is the peak amplitude of the AC voltage signal across the diode?

An LED is used as the monochromatic light source in a Young's Double Slit Experiment (YDSE) with a slit separation $d = 0.10 \text{ mm}$ and a screen placed at a distance $D = 1.0 \text{ m}$. Initially, the LED is forward biased with a current $I_1 = 10 \text{ mA}$. At this current, its junction temperature is $T_1 = 300 \text{ K}$ and it emits light corresponding exactly to its bandgap energy $E_g = 2.000 \text{ eV}$. When the forward current is increased to $I_2 = 50 \text{ mA}$, the junction temperature increases due to Joule heating. The thermal characteristics of the LED setup are such that the junction temperature rises at a constant rate of $1.0 \text{ K}$ per $\text{mA}$ of forward current increase above $10 \text{ mA}$. The bandgap energy of the LED material decreases with temperature at a rate $\frac{dE_g}{dT} = -4.0 \times 10^{-4} \text{ eV/K}$. Assuming the emission peak always corresponds to the bandgap energy, what is the magnitude of the shift in the position of the $10^{\text{th}}$ order bright fringe on the screen? (Take $hc = 1240 \text{ eV}\cdot\text{nm}$)

A photo-diode is modeled as an ideal AC current source $i_{ph}(t) = I_0 \sin(\omega t)$ in parallel with its voltage-dependent junction capacitance $C_j$. The internal shunt resistance of the diode is assumed to be infinite. The junction capacitance varies with the applied DC reverse bias voltage $V_R$ (in Volts) according to the relation $C_j = \frac{C_0}{\sqrt{1 + V_R/V_{bi}}}$, where $C_0$ (in Farads) is the zero-bias capacitance and $V_{bi}$ (in Volts) is the built-in potential. The photo-diode is connected to an external series combination of an inductor $L$ (in Henrys) and a resistor $R$ (in Ohms). An ideal biasing network maintains the DC reverse bias $V_R$ across the diode without affecting the AC impedance of the circuit. The intensity of the incident light is modulated at an angular frequency $\omega$ (in rad/s). At a specific reverse bias voltage $V_R = V_{res}$, the power factor of the equivalent AC circuit, as seen by the internal photo-current source, becomes unity. Which of the following expressions correctly gives $V_{res}$?

A nuclear battery contains $N_0$ radioactive nuclei at $t=0$, which decay with a decay constant $\lambda$ (in $\text{s}^{-1}$). Each decay event emits exactly one photon of wavelength $\lambda_p$ (in $\text{m}$). All emitted photons are incident on a semiconductor photovoltaic cell with a band-gap energy $E_g$ (in $\text{J}$), where $E_g < \frac{hc}{\lambda_p}$ ($h$ is Planck's constant, $c$ is the speed of light). Due to thermalization, the excess energy of each photon is lost as heat, and the maximum electrical energy generated per absorbed photon is $E_g$. The instantaneous electrical power output of the cell is given by $P(t) = \eta(t) \Phi(t) E_g$, where $\Phi(t)$ is the instantaneous incident photon flux (photons per second). The dimensionless efficiency factor $\eta(t)$ is directly proportional to the incident photon flux $\Phi(t)$. At $t=0$, this factor is $\eta_0$. What is the total electrical energy generated by the cell from $t=0$ to $t \to \infty$?

A monochromatic plane wave of wavelength $\lambda$ in vacuum is incident normally on an opaque screen with a circular aperture of diameter $D$. The screen is coated directly onto the horizontal flat top surface of a transparent encapsulant block of refractive index $n_1 = 2$. Inside the block, the light strikes a planar solar cell interface having a refractive index $n_2 = \sqrt{2}$. The normal to the solar cell interface makes an angle $\alpha = 15^\circ$ with the vertical axis of the aperture. To ensure that the entire central maximum (Airy disk) of the diffracted light enters the solar cell without any total internal reflection at the encapsulant-cell interface, what is the minimum required diameter $D$ of the aperture?

A solar cell is illuminated by a modulated light source such that it acts as an ideal AC current source $i(t) = I_0 \sin(\omega t)$ in parallel with an internal shunt capacitance $C$. The cell is connected across an external purely resistive load $R$. Given $I_0 = 20 \text{ mA}$, $\omega = 10^4 \text{ rad s}^{-1}$, $C = 2 \text{ \mu F}$, and $R = 100 \text{ \Omega}$, what is the time-averaged power supplied by the current source to the circuit?

A specialized $p^+-n$ junction solar cell operates under a reverse bias voltage $V_R = 1.20 \pm 0.06 \text{ V}$. The built-in potential of the junction is $V_0 = 0.80 \text{ V}$ (assume exact). The depletion region width $W$ is determined by the donor doping concentration $N_D$ in the lighter doped $n$-region, given by $W = \sqrt{\frac{2\epsilon (V_0 + V_R)}{e N_D}}$, where $\epsilon$ and $e$ are exact constants. The photoelectric conversion efficiency $\eta$ of the cell scales with the depletion width as $\eta = \alpha W^{4/3}$, where $\alpha$ is a constant. If the manufacturing tolerance (maximum percentage error) in the donor doping concentration $N_D$ is $6\%$, what is the maximum percentage error in the efficiency $\eta$?

A semiconductor photocathode has a threshold energy for photoemission equal to its temperature-dependent band gap $E_g(T)$. At low temperatures, the band gap follows the relation $E_g(T) = E_0 - \gamma T^2$, where $E_0$ and $\gamma$ are positive constants. The photocathode is illuminated by a light source whose spectral photon flux (number of photons per unit time per unit energy interval) is given by $\frac{d\Phi_p}{dE} = \frac{K}{E^4}$ for $E > 0$, where $K$ is a positive constant. Assuming a constant quantum efficiency for all incident photons with energy $E \ge E_g(T)$, what is the magnitude of the relative change in the saturation photocurrent, $\left| \frac{\Delta I_s}{I_s} \right|$, due to a small relative change in temperature $\frac{\Delta T}{T} = \delta$?

Consider an intrinsic semiconductor at room temperature with an intrinsic carrier concentration $n_i = 3.0 \times 10^{10} \text{ cm}^{-3}$. The recombination of electrons and holes is a bimolecular process, meaning the recombination rate per unit volume is given by $R = \beta n p$, where $n$ and $p$ are the electron and hole concentrations, respectively, and $\beta = 1.0 \times 10^{-10} \text{ cm}^3 \text{s}^{-1}$ is the recombination coefficient. In thermal equilibrium, the thermal generation rate $G_{th}$ balances the recombination rate. The semiconductor is now continuously illuminated, providing an additional, steady external generation rate of electron-hole pairs $G_{ext} = 1.6 \times 10^{11} \text{ cm}^{-3} \text{s}^{-1}$. Assuming uniform illumination and no spatial gradients, what is the steady-state excess electron concentration $\Delta n$ in the semiconductor?

Consider a non-degenerate n-type semiconductor at thermal equilibrium at an absolute temperature $T$. The donor impurity concentration varies spatially along the x-axis ($x > 0$) as $N_D(x) = N_0 e^{-\alpha x^2}$, where $N_0$ and $\alpha$ are positive constants. Assume complete ionization such that the electron concentration $n(x) \approx N_D(x)$, and $N_D(x) \gg n_i$ everywhere, where $n_i$ is the intrinsic carrier concentration. Let $k_B$ be the Boltzmann constant and $q$ be the elementary charge. Which of the following expressions correctly represents the built-in electric field $\vec{E}(x)$ and the minority hole concentration $p(x)$ as a function of position?

Consider a semiconductor at absolute temperature $T$ with a band gap $E_g$ such that $E_g \gg k_B T$, where $k_B$ is the Boltzmann constant. The density of states for the conduction and valence bands are given by $g_c(E) = A_c \sqrt{E - E_c}$ for $E \ge E_c$ and $g_v(E) = A_v \sqrt{E_v - E}$ for $E \le E_v$, respectively. The intrinsic carrier concentration, calculated assuming classical Maxwell-Boltzmann statistics for both bands, is $n_i$. The semiconductor is heavily doped n-type such that its Fermi level $E_F$ exactly coincides with the conduction band edge $E_c$. Let the actual product of electron and hole concentrations in this degenerate state be denoted by $(np)_{\text{actual}}$. Given the definite integral ratio $\frac{\int_0^\infty \frac{\sqrt{x}}{1+e^x} dx}{\int_0^\infty \sqrt{x} e^{-x} dx} = \alpha$, which of the following expressions correctly represents $(np)_{\text{actual}}$?

A semiconductor rod of length $L$ is placed between two heat reservoirs maintained at temperatures $T_1$ (at $x = 0$) and $T_2$ (at $x = L$). The thermal conductivity of the rod is uniform and constant, and steady-state heat flow is established. The electron concentration in the rod is maintained as $n(x) = n_0 e^{-\alpha x}$, where $n_0$ and $\alpha$ are positive constants. The generalized one-dimensional electron charge current density is given by $J = e n \mu E + e \frac{d}{dx}(D n)$, where $e$ is the elementary charge, $\mu$ is the constant electron mobility, $E$ is the electric field, and $D$ is the diffusion coefficient. The diffusion coefficient satisfies the local Einstein relation $D(x) = \frac{k_B T(x)}{e} \mu$, where $k_B$ is the Boltzmann constant. Assuming the net electron current density is zero in the steady state, what is the built-in potential difference $\Delta V = V(L) - V(0)$ across the rod?

A dilute colloidal suspension of spherical semiconductor particles, each of radius $r$ and positive charge $q$, is confined in a fluid medium between $x = 0$ and $x = L$. A uniform electric field $E$ is applied in the $+x$ direction. The fluid is subjected to a steady temperature gradient such that the local absolute temperature is $T(x) = T_0(1 + \alpha x)$, where $T_0$ and $\alpha$ are positive constants. The dynamic viscosity of the fluid varies with temperature as $\eta(T) = \eta_0 \frac{T_0}{T}$.

The particle flux is governed by the Nernst-Planck equation $J = n(x)v_d(x) - D(x)\frac{dn}{dx}$, where $n(x)$ is the local particle concentration, $v_d(x)$ is the local drift velocity determined by Stokes' drag, and $D(x)$ is the local diffusion coefficient given by the Stokes-Einstein relation $D(x) = \frac{k_B T(x)}{6\pi\eta(x) r}$.

Assuming steady-state conditions with zero net flux ($J = 0$), which of the following expressions correctly represents the concentration profile $n(x)$?

In a specific semiconductor photocathode, the photoelectric threshold energy is exactly equal to its temperature-dependent band gap energy $E_g(T)$. The band gap varies with absolute temperature $T$ according to the Varshni empirical relation: $E_g(T) = E_0 - \frac{\alpha T^2}{T+\beta}$ where $E_0$ is the band gap at $0\text{ K}$, and $\alpha, \beta$ are positive material constants. If $\lambda_0$ is the threshold wavelength for photoelectric emission at temperature $T$, which of the following expressions correctly represents the temperature sensitivity of the threshold wavelength, $\frac{d\lambda_0}{dT}$? (Here $h$ is Planck's constant and $c$ is the speed of light in vacuum).

The energy band gap $E_g$ of a semiconductor decreases with absolute temperature $T$ due to non-linear thermal expansion of the lattice. This dependence is modeled as $E_g(T) = E_0(1 - \beta T^2)$, where $E_0$ is the band gap at $0\text{ K}$ and $\beta$ is a positive constant with units of $\text{K}^{-2}$. A spherical sample of this semiconductor is heated. At a specific temperature $T_0$, the peak wavelength of its thermal blackbody emission spectrum exactly matches the threshold wavelength required to excite an electron across its band gap. Assuming Wien's displacement constant is $b$, Planck's constant is $h$, and the speed of light in vacuum is $c$, which of the following expressions correctly gives the temperature $T_0$?

A semiconductor device has a temperature-dependent resistance given by $R_S(T) = R_0 \exp\left(\frac{E_g}{2k_B T}\right)$, where $E_g$ is the energy band gap, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature. The device is connected in series with a fixed load resistor $R_L$ and an ideal DC voltage source $V_0$. The device dissipates heat to the environment at a rate $P_{diss} = K(T - T_a)$, where $K$ is the thermal conductance and $T_a$ is the ambient temperature. The system is currently operating in thermal equilibrium at a temperature $T$ such that $R_S > R_L$. Thermal runaway occurs if a small positive temperature fluctuation results in the rate of electrical heat generation increasing faster than the rate of heat dissipation. What is the critical minimum value of the thermal conductance $K$ required to prevent thermal runaway at this operating point?

In a particular p-n junction diode, Zener (quantum tunneling) and Avalanche (impact ionization) breakdown mechanisms compete. The diode enters breakdown when the reverse bias voltage reaches the lower of the two threshold voltages, $V_Z$ or $V_A$.

The Zener threshold voltage $V_Z$ is directly proportional to the semiconductor's energy band gap $E_g(T)$. The Avalanche threshold voltage $V_A$ increases linearly with temperature due to enhanced phonon scattering, given by $V_A(T) = V_0 [1 + \beta (T - T_0)]$.

At $T_0 = 300\text{ K}$, both mechanisms have an identical threshold voltage of $V_0 = 6.00\text{ V}$. The energy band gap at $300\text{ K}$ is $E_g(300\text{ K}) = 1.20\text{ eV}$, and it varies with temperature as $E_g(T) = E_g(T_0) - \alpha (T - T_0)$, where $\alpha = 4.0 \times 10^{-4}\text{ eV/K}$. The temperature coefficient for the Avalanche mechanism is $\beta = 2.0 \times 10^{-3}\text{ K}^{-1}$.

What is the actual breakdown voltage of the diode at $T = 400\text{ K}$ and $T = 200\text{ K}$, respectively?

Consider a two-dimensional (2D) intrinsic semiconductor at thermal equilibrium. The effective masses of electrons in the conduction band and holes in the valence band are isotropic and equal ($m_e^* = m_h^* = m^*$). The charge carriers can be modeled as a classical 2D ideal gas. The density of states per unit area for both bands is constant and given by $g(E) = \frac{m^*}{\pi \hbar^2}$, where $\hbar$ is the reduced Planck constant. If the average speed of the charge carriers is measured to be $v_0$ and the bandgap energy is $E_g$, which of the following correctly represents the intrinsic areal carrier concentration $n_i$?

A solid semiconductor sphere of radius $R$ and relative permittivity $\epsilon_r$ is in thermal equilibrium at temperature $T$. The sphere contains a fixed positive background charge distribution with volume charge density $\rho_f(r)$ and mobile electrons of charge $-e$. Due to internal redistribution, the mobile electron number density is observed to be $n(r) = n_0 e^{-\alpha r^2}$, where $n_0$ and $\alpha$ are positive constants, and $r$ is the radial distance from the center ($r < R$). Assuming the Einstein relation $D = \mu k_B T / e$ holds (where $D$ is the diffusion coefficient, $\mu$ is the mobility, and $k_B$ is the Boltzmann constant), determine the fixed positive background charge density $\rho_f(r)$. Let $\epsilon_0$ be the permittivity of free space.

An n-type semiconductor occupies the region $x \ge 0$. In thermal equilibrium at temperature $T$, the electron concentration profile is maintained as $n(x) = n_0 (1 + \alpha x)^2$, where $n_0$ and $\alpha$ are positive constants. The hole concentration is negligible. Assuming the permittivity of the semiconductor is $\epsilon$ and using the Einstein relation $\frac{D_n}{\mu_n} = \frac{kT}{q}$ (where $k$ is the Boltzmann constant, $q$ is the elementary charge, $D_n$ is the diffusion coefficient, and $\mu_n$ is the mobility), determine the net space charge density $\rho(x)$ required in this region to maintain this profile.

A thin n-type semiconductor ring of radius $R$, radial width $w$, and axial thickness $t$ ($w, t \ll R$) lies in the $xy$-plane, centered at the origin. A magnetic dipole of moment $\vec{m} = m \hat{i}$ is fixed at the origin. A total steady current $I$ is driven around the ring in the $+\hat{\phi}$ direction. The charge carriers have mobility $\mu$ and charge magnitude $e$.

Due to the spatially varying magnetic field, a Hall electric field $E_z$ develops in the axial direction. To satisfy steady-state boundary conditions ($\nabla \times \vec{E} = 0$), the azimuthal current density $J_\phi$ must vary across the thickness of the ring.

Assuming the variation in drift velocity is small compared to its average value, find the magnitude of the difference in azimuthal current density, $\Delta J = |J_\phi(z=t/2) - J_\phi(z=-t/2)|$, at the angular position $\phi = \pi/2$ (on the $y$-axis).

A semiconductor junction is excited by a time-varying optical pulse. The instantaneous drift current responds directly to the pulse as $I_{\text{drift}}(t) = I_0 e^{-t/\tau_1}$ for $t \ge 0$. The diffusion current $I_{\text{diff}}(t)$ is driven by the excess carriers generated by the same pulse and satisfies the continuity equation $\frac{d I_{\text{diff}}}{dt} = k I_{\text{drift}}(t) - \frac{I_{\text{diff}}(t)}{\tau_2}$, where $\tau_2$ is the effective carrier lifetime and $k$ is a coupling constant. Given $I_0 = 2 \text{ mA}$, $\tau_1 = 1 \text{ \mu s}$, $\tau_2 = 0.5 \text{ \mu s}$, $k = 4 \text{ \mu s}^{-1}$, and $I_{\text{diff}}(0) = 0$. At what time $t$ does the total current $I(t) = I_{\text{drift}}(t) + I_{\text{diff}}(t)$ reach its maximum value?

Consider a 1D space charge region (depletion region) in a semiconductor device spanning $0 \le x \le W(t)$. The electric potential at $x=0$ is maintained at $0$, and at the moving boundary $x=W(t)$ it is maintained at a constant value $V_0 > 0$. The electric field at the boundary $x=W(t)$ is always zero. Due to a transient recombination process, the uniform positive space charge density in this region decays exponentially with time as $\rho(t) = \rho_0 e^{-\lambda t}$, where $\rho_0$ and $\lambda$ are positive constants. Assuming the permittivity of the medium is $\epsilon$, determine the magnitude of the displacement current density $J_d$ at $x=0$ as a function of time $t$.

A polar molecule with a constant dipole moment $\vec{p} = p \cos\theta \hat{i} + p \sin\theta \hat{j}$ is initially located at $x = 0$ in a depletion region. The electric field in this region is non-uniform and given by $\vec{E}(x) = E_0 \left(1 - \frac{x}{2W}\right) \hat{i}$ for $0 \le x \le W$. Initially, the dipole is aligned with the electric field ($\theta = 0$). An external agent moves the molecule quasi-statically from $x = 0$ to $x = W$. During this translation, the agent simultaneously rotates the dipole such that its angle $\theta$ with the $+x$-axis increases linearly with $x$, reaching $\theta = \pi/2$ at $x = W$. What is the work done by the electric torque on the dipole during this process?

A spherical p-n junction has a depletion region extending from an inner radius $r = a$ to an outer radius $r = b$. The metallurgical junction is located at $r = R = \frac{a+b}{2}$. The effective volume charge density $\rho(r)$ within the depletion region is modeled as $\rho(r) = \frac{A \epsilon}{r^2}$ for $a \le r < R$, and $\rho(r) = -\frac{A \epsilon}{r^2}$ for $R < r \le b$, where $A$ is a positive constant with units of Volts, and $\epsilon$ is the permittivity of the semiconductor. Assuming the electric field is zero for $r < a$ and $r > b$, what is the magnitude of the built-in potential difference $\Delta V$ across the depletion region?

A Zener diode is fabricated from a semiconductor crystal rod of length $L$. The rod is rigidly fixed between two immovable, perfectly insulating walls at an initial temperature $T_0 = 300 \text{ K}$. The diode is connected in series with a resistor $R = 100 \ \Omega$ and a DC voltage source $V_S = 12.0 \text{ V}$. At $T_0$, the diode operates in the breakdown region with a Zener voltage $V_{Z0} = 6.0 \text{ V}$.

The intrinsic temperature coefficient of the Zener voltage (at zero stress) is $C_T = \frac{\partial V_Z}{\partial T} = -4.0 \times 10^{-3} \text{ V/K}$. The crystal has a Young's modulus $Y = 1.5 \times 10^{11} \text{ N/m}^2$ and a coefficient of linear thermal expansion $\alpha = 2.0 \times 10^{-6} \text{ K}^{-1}$. The Zener voltage also depends on the longitudinal compressive stress $\sigma_c$ developed in the rod due to bandgap deformation, with a coefficient $K = \frac{\partial V_Z}{\partial \sigma_c} = +2.0 \times 10^{-8} \text{ V/Pa}$.

If the temperature of the system is uniformly raised to $T = 350 \text{ K}$, calculate the new steady-state Zener current $I_Z$. Assume the walls remain perfectly rigid and the diode remains in the breakdown region.

A circuit consists of a cell (electromotive force $E = 10 \text{ V}$, internal resistance $r = 1 \text{ }\Omega$), a switch $S$, an inductor of inductance $L = 10 \text{ H}$, and a resistor of resistance $R = 4 \text{ }\Omega$ all connected in series. A Zener diode with a reverse breakdown voltage $V_Z = 20 \text{ V}$ is connected in parallel across the switch $S$. The cathode of the Zener diode is connected to the node between the cell's positive terminal and the switch. The switch $S$ is initially closed for a long time, allowing the current to reach a steady state. At $t = 0$, the switch $S$ is suddenly opened. Assuming the Zener diode enters avalanche breakdown instantly, calculate the time taken for the current in the circuit to drop to zero.

A circuit consists of a variable DC voltage source $V_0$, a resistor $R = 40 \ \Omega$, and a special avalanche diode connected in series. The diode is reverse-biased. The diode exhibits a 'snap-back' breakdown characteristic analogous to the transition from static to kinetic friction. It conducts zero current until the voltage across it reaches a static breakdown threshold $V_{BR} = 120 \text{ V}$. Immediately upon reaching $V_{BR}$, the diode breaks down and its voltage-current relationship discontinuously shifts to a dynamic state given by $V_D = V_H + I r_d$, where $V_H = 70 \text{ V}$ is the holding voltage, $r_d = 10 \ \Omega$ is the dynamic resistance, and $I$ is the current. If the source voltage $V_0$ is slowly increased from zero, what is the power dissipated in the diode immediately after the breakdown occurs?

A biasing circuit for a transistor uses a voltage divider consisting of a fixed ideal resistor $R_1 = 2.0 \text{ k}\Omega$ and a temperature-sensitive resistor $R_2$ connected in series across a constant DC supply $V_{CC} = 10.0 \text{ V}$. The output voltage $V_{out}$ is taken across $R_2$. To provide thermal compensation, $R_2$ (initial resistance $R_0 = 2.0 \text{ k}\Omega$ at $T_0$) is a thin-film resistor perfectly bonded to the top surface of a bimetallic strip. The strip consists of two metals of equal thickness with linear expansion coefficients $\alpha_1 = 3.0 \times 10^{-5} \text{ K}^{-1}$ (top metal) and $\alpha_2 = 1.0 \times 10^{-5} \text{ K}^{-1}$ (bottom metal). When the temperature increases by $\Delta T$, the strip bends, inducing a longitudinal tensile strain $\epsilon = \frac{3}{4}(\alpha_1 - \alpha_2)\Delta T$ on its top surface. The resistor $R_2$ has an intrinsic temperature coefficient of resistance $\alpha_\rho = 1.0 \times 10^{-4} \text{ K}^{-1}$ and a strain gauge factor $G = \frac{\Delta R_2 / R_0}{\epsilon} = 4.0$. Assuming the total fractional change in resistance is the sum of the intrinsic thermal and mechanical strain contributions, what is the rate of change of the output voltage with respect to temperature, $\frac{dV_{out}}{dT}$, at $T = T_0$?