115 年 國立中央大學電機工程學系碩士班乙：固態組《半導體元件》

For germanium (Ge), the effective density of states at 300 K ($k_B T \approx 0.025$ eV) is $N_c \approx 1.3 \times 10^{19}$ cm$^{-3}$, $N_v \approx 6.0 \times 10^{18}$ cm$^{-3}$. The bandgap energies are $E_g(300 \text{ K}) = 0.66$ eV, $E_g(77 \text{ K}) = 0.75$ eV. At 77 K, the measured electron concentration is $n = 5 \times 10^{16}$ cm$^{-3}$. Assume negligible acceptor concentration and a donor energy level is $E_c - E_D = 0.012$ eV. Which answer of $N_D$ at 77 K ($k_B T \approx 0.006$ eV) is more reasonable?
(#hint: $n = N_c(T) \exp\left(-\frac{E_c - E_F}{k_B T}\right)$, $N_c(T) = N_c(300) \times 0.13$, $N_D \approx n \times \left(1 + 2 \exp\left(\frac{E_F - E_D}{k_B T}\right)\right)$. You may determine $E_c - E_F$ from $n/N_c$, then compute $E_F - E_D$, $\exp(2) \approx 7.39$)
(A) $1.1 \times 10^{14}$ cm$^{-3}$
(B) $5.9 \times 10^{14}$ cm$^{-3}$
(C) $3.1 \times 10^{15}$ cm$^{-3}$
(D) $1.5 \times 10^{16}$ cm$^{-3}$
(E) $7.2 \times 10^{16}$ cm$^{-3}$

An n-type silicon sample has $N_D = 1 \times 10^{13}$ cm$^{-3}$. At 400 K ($k_B T \approx 0.035$ eV), the intrinsic carrier concentration is $n_i = 1 \times 10^{13}$ cm$^{-3}$. Assuming complete ionization, calculate the electron concentration $n_0$, hole concentration $p_0$, and Fermi-level shift $E_F - E_i$?
(#hint: $\ln(1.0 \times 10^{13}) \approx 29.93$, $\ln(1.62 \times 10^{13}) \approx 30.42$; $\ln(2.0 \times 10^{13}) \approx 30.63$, $\ln(6.17 \times 10^{12}) \approx 29.45$, $\sqrt{5} \approx 2.23$)
(A) $n_0 = 1.0 \times 10^{13}$, $p_0 = 1.0 \times 10^{13}$, $E_F - E_i = 0$;
(B) $n_0 = 1.62 \times 10^{13}$, $p_0 = 6.17 \times 10^{12}$, $E_F - E_i = 0.017$ eV;
(C) $n_0 = 2.0 \times 10^{13}$, $p_0 = 5.0 \times 10^{12}$, $E_F - E_i = 0.035$ eV;
(D) $n_0 = 1.0 \times 10^{13}$, $p_0 = 1.0 \times 10^{12}$, $E_F - E_i = 0.05$ eV;
(E) $n_0 = 6.17 \times 10^{12}$, $p_0 = 1.62 \times 10^{13}$, $E_F - E_i = -0.017$ eV;

A silicon sample at 300 K contains acceptors and donors with concentrations: $N_A = 1.2 \times 10^{16}$ cm$^{-3}$, $N_D = 1.0 \times 10^{16}$ cm$^{-3}$. The intrinsic carrier concentration is $n_i = 1.0 \times 10^{10}$ cm$^{-3}$. Assume complete ionization and carrier mobilities $\mu_p = 420$ cm$^2$/V.s, $\mu_n = 1300$ cm$^2$/V.s. Calculate the electrical resistivity $\rho$ of the sample? (#hint: $\sigma = q(n_0\mu_n + p_0\mu_p)$)
(A) $0.1 \, \Omega \cdot \text{cm}$
(B) $1.6 \, \Omega \cdot \text{cm}$
(C) $3.2 \, \Omega \cdot \text{cm}$
(D) $7.4 \, \Omega \cdot \text{cm}$
(E) $15.8 \, \Omega \cdot \text{cm}$

An Au-doped silicon with concentration $N_t = 8 \times 10^{15}$ cm$^{-3}$. Au introduces deep-level recombination centers. The hole/electron capture coefficients are: $r_p = 1.2 \times 10^{-7}$ cm$^3$/s and $r_n = 5.0 \times 10^{-8}$ cm$^3$/s, assuming low-level injection: Calculate the minority carrier lifetime in (i) n-type Si and (ii) p-type Si, respectively?
(A) $\tau_p = 1.0 \times 10^{-10}$ s, $\tau_n = 2.2 \times 10^{-9}$ s;
(B) $\tau_p = 8.3 \times 10^{-10}$ s, $\tau_n = 2.5 \times 10^{-9}$ s;
(C) $\tau_p = 1.0 \times 10^{-9}$ s, $\tau_n = 2.5 \times 10^{-9}$ s;
(D) $\tau_p = 2.5 \times 10^{-9}$ s, $\tau_n = 8.3 \times 10^{-10}$ s;
(E) $\tau_p = 4.6 \times 10^{-9}$ s, $\tau_n = 1.0 \times 10^{-10}$ s;

Consider an ideal n-channel MOSFET operating in saturation with channel length of $L = 1.0 \, \mu\text{m}$, electron mobility of $\mu_n = 600$ cm$^2$/V.s, oxide capacitance of $C_{ox} = 7.5 \times 10^{-8}$ F/cm$^2$, threshold voltage of $V_T = 0.7$ V. As gate-source voltage is given by $V_{GS} = 4.5$ V, what channel width is required to achieve a saturation current $I_D = 3$ mA? (Assume long-channel behavior and neglect channel-length modulation). (#$(V_{GS}-V_T)^2 = 14.44$)
(A) $0.7 \, \mu\text{m}$
(B) $1.2 \, \mu\text{m}$
(C) $4.6 \, \mu\text{m}$
(D) $9.2 \, \mu\text{m}$
(E) $18.0 \, \mu\text{m}$

Consider a silicon pn junction at $T = 300$ K with doping concentrations of $N_A = 10^{16}$ cm$^{-3}$ and $N_D = 10^{15}$ cm$^{-3}$. When a reverse-biased voltage $V_R = 5$ V is applied, calculate the Space Charge Width in the pn junction. $W = ?$
(#hint: $k T = 0.025$ eV, $\ln((10^{11})/2.25) = 24.52$, $\varepsilon_s = 11.7 \varepsilon_0$, $\varepsilon_0 = 8.85 \times 10^{-14}$ F / cm, $W = \sqrt{\frac{2\varepsilon_s}{q(N_A+N_D)}(V_{bi} + V_R)}$)
(A) $1.83 \, \mu\text{m}$
(B) $2.83 \, \mu\text{m}$
(C) $3.83 \, \mu\text{m}$
(D) $4.83 \, \mu\text{m}$
(E) $5.83 \, \mu\text{m}$

Assume that the intercept and the slope of the curve in Fig. 1 are $V_{bi}=0.725\ \text{V}$ and $6.15\times10^{15}\ (\text{F}/\text{cm}^2)^{-2}\ (\text{V})^{-1}$, respectively, for a silicon $p^+n$ junction at $T=300\ \text{K}$. Determine the impurity doping concentration $N_a$ in the $p^+n$ junction. $N_a=$? (Hint: $\exp(0.725/0.0259)=1.44\times10^{12}$)

🖼️【此處有附圖，請對照原卷】

Figure 1. $[1/C]^2$ versus $V_R$ of a uniformly doped pn junction.

(A) $1.64\times10^{17}\ \text{cm}^{-3}$
(B) $2.64\times10^{17}\ \text{cm}^{-3}$
(C) $3.64\times10^{17}\ \text{cm}^{-3}$
(D) $4.64\times10^{17}\ \text{cm}^{-3}$
(E) $5.64\times10^{17}\ \text{cm}^{-3}$

What are the $N_a$ and $N_d$? (11%)（複選題，全對才給分）

Design a PN junction diode to produce particular electron and hole current densities at a given forward-bias voltage. Consider a silicon pn junction diode at $T=300\ \text{K}$. Design the diode such that $J_n=20\ \text{A}/\text{cm}^2$ and $J_p=5\ \text{A}/\text{cm}^2$ at $V_a=0.65\ \text{V}$. $N_a=?$ $N_d=?$

(Hint: $J_n\propto\frac{n_i^2}{N_a}\left(\exp\left(\frac{V_a}{V_T}\right)-1\right)$, $\exp(0.65/0.0259)=7.93\times10^{10}$)

(A) $N_a=1.01\times10^{15}\ \text{cm}^{-3}$
(B) $N_a=2.01\times10^{15}\ \text{cm}^{-3}$
(C) $N_a=3.01\times10^{15}\ \text{cm}^{-3}$
(D) $N_d=1.55\times10^{15}\ \text{cm}^{-3}$
(E) $N_d=2.55\times10^{15}\ \text{cm}^{-3}$

Calculate the ideal reverse-saturation current densities of a Schottky barrier diode. Consider a tungsten barrier on silicon with a measured barrier height of $\phi_{Bn}=0.67\ \text{eV}$. The effective Richardson constant is $A^*=114\ \text{A}/(\text{K}^2\cdot\text{cm}^2)$. Let $T=300\ \text{K}$.

(Hint: $\exp(0.67/0.0259)=1.72\times10^{11}$)

(A) $1.98\times10^{-5}\ \text{A}/\text{cm}^2$
(B) $2.98\times10^{-5}\ \text{A}/\text{cm}^2$
(C) $3.98\times10^{-5}\ \text{A}/\text{cm}^2$
(D) $4.98\times10^{-5}\ \text{A}/\text{cm}^2$
(E) $5.98\times10^{-5}\ \text{A}/\text{cm}^2$

For ideal C–V characteristics of the MOS capacitor（複選題，全對才給分）:

What is the capacitance in the high-frequency inversion mode?

Under this condition, what is the charge distribution?

(A) 🖼️【此處有附圖，請對照原卷】
(B) $\frac{1}{C'(\text{depl})}=\frac{1}{C'_{\text{ox}}}+\frac{1}{C'_{\text{sn}}}=\frac{t_{\text{ox}}+\left(\frac{\varepsilon_{\text{ox}}}{\varepsilon_{\text{si}}}\right)x_{\text{DT}}}{\varepsilon_{\text{ox}}}$
(C) 🖼️【此處有附圖，請對照原卷】
(D) 🖼️【此處有附圖，請對照原卷】
(E) $\frac{\varepsilon_{\text{ox}}}{t_{\text{ox}}}$

Calculate $C'_{\min}$ for a MOS capacitor. Consider a p-type silicon substrate at $T=300\ \text{K}$ doped to $N_a=10^{16}\ \text{cm}^{-3}$. The oxide is silicon dioxide with a thickness of $t_{\text{ox}}=18\ \text{nm}=180\ \text{Å}$, and the gate is aluminum. (Hint: $\varepsilon_{\text{ox}}=3.9$, $\ln(10^{16}/(1.5\times10^{10}))=13.4$, $W_D\propto\sqrt{\phi_F/N_a}$)

(A) $C'_{\min}=1.93\times10^{-7}\ \text{F}/\text{cm}^2$
(B) $C'_{\min}=2.93\times10^{-7}\ \text{F}/\text{cm}^2$
(C) $C'_{\min}=1.93\times10^{-8}\ \text{F}/\text{cm}^2$
(D) $C'_{\min}=2.93\times10^{-8}\ \text{F}/\text{cm}^2$
(E) $C'_{\min}=6.93\times10^{-8}\ \text{F}/\text{cm}^2$

Determine the increase in drain current due to short-channel modulation. Consider an n-channel MOSFET with a substrate doping concentration of $N_a=2\times10^{16}\ \text{cm}^{-3}$, a threshold voltage of $V_T=0.4\ \text{V}$, and a channel length of $L=1\ \mu\text{m}$. The device is biased at $V_{GS}=1\ \text{V}$ and $V_{DS}=2.5\ \text{V}$. Determine the ratio of actual drain current compared to the ideal value. $I'_d/I_d=$?

(Hint: $\ln(2\times10^{16}/(1.5\times10^{10}))=14.1$)

(A) $0.62$
(B) $0.92$
(C) $1.22$
(D) $1.52$
(E) $1.82$

Calculate the threshold voltage shift due to short-channel effects. Consider an n-channel MOSFET with $N_a=3\times10^{16}\ \text{cm}^{-3}$, $L=1.0\ \mu\text{m}$, $r_j=0.3\ \mu\text{m}$, and $t_{\text{ox}}=20\ \text{nm}=200\ \text{Å}$.

(Hint: $\ln(3\times10^{16}/(1.5\times10^{10}))=14.51$)

(A) $-0.0726\ \text{V}$
(B) $-0.0526\ \text{V}$
(C) $-0.0326\ \text{V}$
(D) $0.0326\ \text{V}$
(E) $0.0526\ \text{V}$

Design the channel width that will limit the threshold voltage shift because of narrow channel effects to a specified value. Consider a silicon n-channel MOSFET with $N_a=3\times10^{16}\ \text{cm}^{-3}$ and $t_{\text{ox}}=20\ \text{nm}=200\ \text{Å}$. Let $\xi=\pi/2$. Assume that the threshold voltage shift is to be limited to $\Delta V_T=0.2\ \text{V}$.

(A) $0.908\ \mu\text{m}$
(B) $0.108\ \mu\text{m}$
(C) $0.308\ \mu\text{m}$
(D) $0.508\ \mu\text{m}$
(E) $0.708\ \mu\text{m}$

Consider a uniformly doped silicon npn bipolar transistor with the base width modulation and following parameters: $N_B=5\times10^{16}\ \text{cm}^{-3}$, $N_C=2\times10^{15}\ \text{cm}^{-3}$, $x_{B0}=0.7\ \mu\text{m}$, and $D_B=25\ \text{cm}^2/\text{s}$. Assume that $x_{B0}\ll L_B$, $V_{BE}=0.60\ \text{V}$, and the collector–base voltage is $2\leq V_{CB}\leq10\ \text{V}$. Estimate the Early voltage ($V_A$) from the change in collector current with a change in neutral base width.

(Hint: For $V_{CB}=2\ \text{V}$, $J_C=3.195\ \text{A}/\text{cm}^2$; for $V_{CB}=10\ \text{V}$, $J_C=3.469\ \text{A}/\text{cm}^2$)

(A) $V_A=20.7\ \text{V}$
(B) $V_A=30.7\ \text{V}$
(C) $V_A=70.7\ \text{V}$
(D) $V_A=90.7\ \text{V}$
(E) $V_A=110.7\ \text{V}$