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Elements of Solid State Physics

J.P. Srivastava

Chapter 9

Semiconductors - all with Video Answers

Educators

Chapter Questions

05:25

Problem 1

Assuming the resistivity of an intrinsic Ge crystal as $47 \mathrm{ohm} \mathrm{cm}$ and electron and hole mobilities as $0.39$ and $0.19 \mathrm{~m}^{2} \mathrm{~V}^{-1} \mathrm{~s}^{-1}$ respectively, calculate its intrinsic carrier density at room temperature.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
13:03

Problem 2

A $p$ -type semiconductor has an acceptor density of $10^{18} \mathrm{~cm}^{-3}$. If $E_{\mathrm{D}}-E_{\mathrm{v}}=0.2 \mathrm{eV}$ and $m_{\mathrm{e}}^{*}=m_{\mathrm{h}}^{*}=m$ (the free electron mass),
(a) show that the intrinsic conduction in the crystal is negligible at room temperature;
(b) estimate the conductivity of the crystal at room temperature, assuming the hole mobility to be equal to $0.01 \mathrm{~m}^{2} \mathrm{~V}^{-1} \mathrm{~s}^{-1}$ at room temperature.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
15:10

Problem 3

Plot the intrinsic carrier concentration as a function of $1 / T$ on a semilog graph paper over $10-500 \mathrm{~K}$ for:
(a) GaAs $\left(E_{\mathrm{g}}=1.5 \mathrm{eV}, m_{\mathrm{c}}^{*}=0.1 \mathrm{~m}, m_{\mathrm{h}}^{*}=0.4 \mathrm{~m}\right)$
(b) $\operatorname{lnSb}\left(E_{\mathrm{g}}=0.22 \mathrm{eV}, m_{\mathrm{e}}^{*}=0.013 m, m_{\mathrm{h}}^{*}=0.18 \mathrm{~m}\right)$
For which electron and hole concentrations is the conductivity of a semiconductor minimum? To which value of the net impurity content $\left|N_{\mathrm{D}}-N_{\mathrm{A}}\right|$ does this correspond?

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
09:03

Problem 4

In an InSb crystal, $E_{\mathrm{g}}=0.23 \mathrm{eV}$ : the dielectric constant $\left(\epsilon_{\mathrm{s}}\right)=18$ and $m_{\mathrm{c}}^{*}=0.015 \mathrm{~m} .$ Calculate (i) $E_{\mathrm{d}}$ and (ii) the radius of the ground state orbit. What should be the minimum donor concentration so as to produce an appreciable overlap between the orbits of adjacent impurity atoms?

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
13:17

Problem 5

The effective mass of electrons at the lower conduction band edge of a semiconductor is three times higher than that of holes at the upper valence band edge. How far is the Fermi level located from the middle of the forbidden energy gap, assuming that the semiconductor is intrinsic? Explain why $E_{\mathrm{g}}$ should be greater than $8 k_{\mathrm{B}} T$ for your calculation.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
01:41

Problem 6

Calculate $N(\mathrm{c})$ and $N(\mathrm{v})$ in silicon at room temperature. Ir" the conduction band of silicon there are six minima with $m^{*}=0.97 m, m_{1}^{*}=0.19 m$. The valence band can be approximated by two independent bands coincident at the band edge with $m_{1}^{*}=0.5 m$ and $m_{2}^{*}=0.16 m$.

Suzanne W.
Suzanne W.
Numerade Educator
01:41

Problem 7

In a two-dimensional intrinsic semiconductor the density of states is defined as
$$
\begin{gathered}
D_{\mathrm{c}}(\varepsilon)=\frac{4 \pi m_{\mathrm{c}}^{*}}{h^{2}}, \quad \varepsilon>E_{\mathrm{c}} \\
D(\varepsilon)=0, \quad E_{\mathrm{v}}<\varepsilon<E_{\mathrm{c}} \\
D_{\mathrm{v}}(\varepsilon)=\frac{4 \pi m_{\mathrm{b}}^{*}}{h^{2}}, \quad \varepsilon<E_{\mathrm{v}}
\end{gathered}
$$
Use Fermi-Dirac statistics without making any approximation to show that
$$
\varepsilon_{\mathrm{F}}=E_{\mathrm{c}}-\frac{1}{2} E_{\mathrm{g}}+\frac{k_{\mathrm{B}} T}{2} \ln \frac{8}{3}-k_{\mathrm{B}} T \ln \cos \frac{\phi}{3}
$$
with $\phi=\tan ^{-1}\left[\left(\frac{32}{27}\right) \exp \left(-E_{\mathrm{g}} / k_{\mathrm{B}} T\right)-1\right]^{1 / 2}$

Chai Santi
Chai Santi
Numerade Educator
11:37

Problem 8

A semiconductor cuboid crystal ( $1 \mathrm{~cm} \times 5 \mathrm{~mm} \times 1 \mathrm{~mm}$ ) has a resistivity of $12.5 \mathrm{ohm} \mathrm{cm}$. A Hall voltage of $5 \mathrm{mV}$ across the $5 \mathrm{~mm}$ width is measured when the applied magnetic field is 2000 gauss and a current of $1 \mathrm{~mA}$ flows along the length of the crystal. Calculate the carrier concentration and Hall mobility in the crystal.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
04:16

Problem 9

Hall measurements are made on a $p$ -type semiconductor bar $500 \mu \mathrm{m}$ wide and $20 \mu \mathrm{m}$ thick. The Hall contacts $\mathrm{A}$ and $\mathrm{B}$ are displaced $2 \mu \mathrm{m}$ with respect to each other in the direction of current flow of $3 \mathrm{~mA}$. The voltage between $\mathrm{A}$ and $\mathrm{B}$ with a magnetic field of $10 \mathrm{kG}$ pointing out of the plane of the sample is $3.2 \mathrm{mV}$. When the magnetic field direction is reversed the voltage changes to $-2.8 \mathrm{mV}$. What is the hole concentration and the mobility?

Keshav Singh
Keshav Singh
Numerade Educator
10:57

Problem 10

In an acceptorless $n$ -type semiconductor with $10^{13}$ donors $\mathrm{cm}^{-3}$, the donor ionization energy is $1 \mathrm{meV}$. Taking the effective mass as $0.01 \mathrm{~m}$ and assuming that $E_{\mathrm{g}} \gg>k_{\mathrm{B}} T$, calculate (a) the density of conduction electrons at $4 \mathrm{~K}$ and (b) the Hall coefficient.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
06:50

Problem 11

Show that the Hall coefficient of a $p$ -type semiconductor is zero when the excess density of acceptors over donors is $$
N_{\mathrm{A}}-N_{\mathrm{D}}=n_{i}\left(\frac{b^{2}-1}{b}\right)
$$
where $n_{\mathrm{i}}$ is the intrinsic density and $b=\mu_{n} / \mu_{p}$

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
02:22

Problem 12

Can the longitudinal magnetoresistance of $n$ -type germanium be zero for any orientation of the magnetic field? If yes, find the direction.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
11:34

Problem 13

Prove that the minimum conductivity of an extrinsic semiconductor is given by
$$
\sigma=2 n_{i} e\left(\mu_{n} \mu_{p}\right)^{1 / 2}
$$
Show that the conductivity minimum occurs when
$$
N_{\mathrm{A}}-N_{\mathrm{D}}=n_{i}\left[\left(\frac{\mu_{n}}{\mu_{p}}\right)^{1 / 2}-\left(\frac{\mu_{p}}{\mu_{n}}\right)^{1 / 2}\right]
$$

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
16:34

Problem 14

Prove that the barrier step height of an unbiased $p-n$ junction is given by
$$
e V_{\mathrm{B}}=k_{\mathrm{B}} T \ln \left(\frac{n_{n}}{n_{p}}\right)
$$
where all of the symbols have their usual meaning.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
10:32

Problem 15

Calculate the built-in voltage $V_{\mathrm{B}}$ for a $p-n$ junction formed by diffusing boron $\left(n_{p}=\right.$ $\left.10^{15} \mathrm{~cm}^{-3}\right)$ into one end of an $n$ -type silicon chip $\left(n_{n}=3.87 \times 10^{16} T^{3 / 2} \exp \left(-E_{8} / 2 k_{\mathrm{B}} T\right) \mathrm{cm}^{-3}\right.$
$E_{\mathrm{g}}=1.1 \mathrm{eV}$ ) at room temperature and $127^{\circ} \mathrm{C}$.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
02:23

Problem 16

Explain why Esaki diodes do not show high electrical conductivity in spite of having very large carrier concentrations.

Shazia Naz
Shazia Naz
Numerade Educator