Semiconductor Electronics Set-5

Q1. An n-type silicon sample at $300\ \text{K}$ has donor concentration $N_D=10^{16}\ \text{cm}^{-3}$ and intrinsic carrier concentration $n_i=1.5\times10^{10}\ \text{cm}^{-3}$. Using the relation $E_F-E_i = kT\ln\!\left(\dfrac{N_D}{n_i}\right)$ and $kT=0.0259\ \text{eV}$, calculate the shift $E_F-E_i$.

Q2. A silicon PN junction at $300\ \text{K}$ has $N_A=10^{17}\ \text{cm}^{-3}$ and $N_D=10^{16}\ \text{cm}^{-3}$. Given $n_i=1.5\times10^{10}\ \text{cm}^{-3}$, $\epsilon_s=11.7\epsilon_0$ with $\epsilon_0=8.85\times10^{-14}\ \text{F/cm}$, $q=1.6\times10^{-19}\ \text{C}$ and $kT/q=0.0259\ \text{V}$, compute the depletion width $W$ at zero bias using
$V_{bi}=\dfrac{kT}{q}\ln\!\left(\dfrac{N_A N_D}{n_i^2}\right)$ and
$W=\sqrt{\dfrac{2\epsilon_s V_{bi}}{q}\left(\dfrac{1}{N_A}+\dfrac{1}{N_D}\right)}$. Give $W$ in micrometres.

Q3. A full-wave bridge rectifier has transformer secondary $V_{\text{rms}}=12\ \text{V}$ at $50\ \text{Hz}$. The load is $R_L=1\ \text{k}\Omega$ and the smoothing capacitor is $C=100\ \mu\text{F}$. Each conducting diode drops $0.7\ \text{V}$. Using $V_{\text{peak}}=V_{\text{rms}}\sqrt{2}$, $I_{load}\approx\dfrac{V_{\text{peak}}-2V_D}{R_L}$ and the small-ripple approximation $\Delta V \approx \dfrac{I_{load}}{2fC}$, estimate the peak-to-peak ripple voltage $\Delta V$.

Q4. A step PN junction at fixed $V_{bi}$ is initially symmetric with $N_A=N_D=N$, so its depletion width is $W_{\rm old}=\sqrt{\dfrac{2\epsilon V_{bi}}{q}\left(\dfrac{1}{N}+\dfrac{1}{N}\right)}$ and junction capacitance per unit area $C_{j,\rm old}=\dfrac{\epsilon}{W_{\rm old}}$. If the acceptor concentration is increased to $N_A=4N$ while $N_D=N$ remains unchanged, what is the ratio $\dfrac{C_{j,\rm new}}{C_{j,\rm old}}$? (Use algebraic manipulation of $W$; numerical value rounded to two decimal places.)

Q5. A silicon diode carries forward current $I_F=10\ \text{mA}$ and has minority-carrier lifetime $\tau=1\ \mu\text{s}$. The diffusion capacitance can be approximated by $C_d\approx \tau\dfrac{I_F}{V_T}$ with $V_T=25.9\ \text{mV}$. The zero-bias depletion capacitance is $C_j=2\ \text{pF}$. The diode is in series with source resistance $R_s=10\ \Omega$. Using $f_c\approx\dfrac{1}{2\pi R_s C_{\rm total}}$ with $C_{\rm total}=C_d+C_j$, which statement about the small-signal $-3\ \text{dB}$ cutoff $f_c$ is correct?

Q6. A silicon diode at $T=300 \,\text{K}$ has a reverse saturation current $I_s=10^{-12}\ \text{A}$. If a forward bias $V=0.65\ \text{V}$ is applied, estimate the diode current using the diode equation $I=I_s\left(e^{V/(kT/q)}-1\right)$. Use $kT/q\approx 26\ \text{mV}$.

Q7. A varactor diode has zero-bias junction capacitance $C_0=50\ \text{pF}$ and the capacitance varies with reverse bias $V_R$ as $C(V_R)=C_0\left(1+\dfrac{V_R}{\Phi}\right)^{-m}$ with $\Phi=0.7\ \text{V}$ and $m=\tfrac{1}{2}$. This diode forms an LC tank with $L=10\ \mu\text{H}$. If the reverse bias is changed from $0\ \text{V}$ to $9\ \text{V}$, by what factor does the resonant frequency $f=\dfrac{1}{2\pi\sqrt{LC}}$ change?

Q8. In a voltage-divider biased common-emitter circuit $V_{CC}=12\ \text{V}$, $R_1=100\ \text{k}\Omega$ (to $V_{CC}$), $R_2=20\ \text{k}\Omega$ (to ground), $R_C=2\ \text{k}\Omega$, $R_E=1\ \text{k}\Omega$. Take $V_{BE}=0.7\ \text{V}$ and $V_{CE(sat)}=0.2\ \text{V}$. As the transistor current gain $\beta$ varies from small values to very large values, will the transistor ever enter saturation?

Q9. Assertion (A): A Zener diode with breakdown voltage close to $5.6\ \text{V}$ typically exhibits an almost zero temperature coefficient of its breakdown voltage.
Reason (R): This happens because at around $5.6\ \text{V}$ the diode is heavily doped so that pure Zener (tunnelling) breakdown — which has negligible temperature dependence — is the only mechanism responsible for breakdown.

Q10. A Darlington pair of two identical NPN transistors drives a load $R_L=200\ \Omega$ from $V_{CC}=12\ \text{V}$. Each transistor has small‑signal gain $\beta=100$ in active region, but in saturation the second transistor's forced gain reduces to $\beta_{sat}=10$ (first transistor acts as driver). The Darlington's total saturated collector‑emitter voltage is $V_{CE(sat,\,total)}\approx 0.9\ \text{V}$. Estimate the minimum input base current $I_{b(in)}$ required to ensure saturation of the load. Use $I_C\approx\dfrac{V_{CC}-V_{CE(sat,\,total)}}{R_L}$ and assume $I_{b(in)}$ is the base current of the first transistor.

Q11. An n-type silicon sample at 300 K has donor concentration $N_D=10^{16}\,\text{cm}^{-3}$ and electron mobility $\mu_n=1350\,\text{cm}^2\text{V}^{-1}\text{s}^{-1}$. A uniform electric field $E=1\,\text{V/cm}$ is applied. Assuming majority-electron drift dominates, the magnitude of electron drift current density $J_n$ is approximately:

Q12. A one-sided abrupt p–n junction at 300 K has $N_A=10^{17}\,\text{cm}^{-3}$ (p-side) and $N_D=10^{15}\,\text{cm}^{-3}$ (n-side). Under reverse bias $V_R=5\,\text{V}$ the total depletion width is given by $W=\sqrt{\dfrac{2\epsilon_s}{q}\left(\dfrac{1}{N_A}+\dfrac{1}{N_D}\right)(V_{bi}+V_R)}$. Take $\epsilon_s=11.7\epsilon_0$ with $\epsilon_0=8.85\times10^{-14}\,\text{F/cm}$, $q=1.6\times10^{-19}\,\text{C}$ and $V_{bi}=0.7\,\text{V}$. The depletion width $W$ (in $\mu$m) is approximately:

Q13. A 12 V (rms) transformer secondary feeds a full-wave bridge rectifier supplying a DC load current $I_L=100\,\text{mA}$. Each diode drop is $0.7\,\text{V}$. To ensure that the instantaneous output voltage never falls below $10\,\text{V}$, the minimum filter capacitance required (use mains $f=50\,\text{Hz}$ and $\Delta V\approx I_L/(2fC)$) is closest to:

Q14. A silicon p–n diode measured at low forward currents shows an apparent ideality factor near $2$.

Statement 1: "This observation necessarily indicates the diode has a large series resistance limiting current."

Statement 2: "Recombination of carriers in the depletion region contributes a current component that varies approximately as $\exp\!\left(\dfrac{qV}{2kT}\right)$, producing an apparent ideality factor of $2$."

Which one of the following is correct?

Q15. In a semiconductor at $T=300\,\text{K}$ the electron concentration varies as $n(x)=n_0 e^{\alpha x}$ with $\alpha=10^{4}\,\text{m}^{-1}$. Using the Einstein relation $D_n=\mu_n kT/q$ and the drift–diffusion expression $J_n=q n\mu_n E+qD_n\,\dfrac{dn}{dx}$, the electric field $E$ (magnitude and sign) required to make the net electron current density zero is approximately: