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  {\Large{\bf {Practice Problems- Band Theory of solids and Semiconductors}}}\\
  
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%\rightline{Due date: 06-9-16, before 6 PM.}
\begin{enumerate}
\item Refer to the class notes on band structure of solid. In the case of periodically placed delta function potential, show that the wave function for a particle in the
periodic delta function potential can be written in the form

\[\psi(x)=C[sin(kx)+e^{-iKa}sin(k(a-x))],\qquad 0\leq x \leq a.\]

\item Find the energy at the top of the first allowed band, for the case
$\beta = 5$, correct to three significant digits. For the sake of argument, assume $\alpha/a = 1$.

\item Find the Fermi energy of copper metal. The electron density in copper is $8.5 \times 10^{28} electrons/m^3$.\\

[Ans. $7.04$ eV]

\item An electron beam strikes a crystal of cadmium sulfide (CdS). Electrons scattered by 
the crystal move at a velocity of $4.4 \times 10^5$
 m/s. Calculate the energy of the incident beam. Express your result in eV. CdS is a semiconductor with a band gap, $E_g$, of $2.45$ eV.\\ (Hint: $E_{incident\, e}=E_{emitted\, \gamma} + E_{scattered\, e}$ ).   
 
\item  The number of electron-hole pairs in intrinsic germanium (Ge) is given by: 
\\
\[n_i= 9.7 \times 10^{15}\times T^{3/2} \times e^{\frac{-E_g}{2 K T}}\, cm^{-3}\] 
  $(E_g   = 0.72 eV)$\\
 
(a) What is the density of pairs at T = 20 deg-C? \\

(b) Will undoped Ge be a good conductor at 200 deg-C? If so, why?

\item If no electron-hole pairs were produced in germanium (Ge) until the temperature
reached the value corresponding to the energy gap, at what temperature would Ge
become conductive? ($E_{th} = 3/2 kT$)

\item Given $\mu_p=470 cm^2/V.s$ for Si, what is Ihe hole drift velocity at ${\bf \cal{E}}=10^3 V/cm$. What is $\tau_{mp}$ and what is the average distance Iraveled between collisions, i.e.,
the mean free path ? (Ans. mean free path =22nm).

\item What is the hole diffusion constant in a piece of silicon doped with $3 \times 10^{15} cm^{–3}$
of donors and $7 \times 10^{15} cm^{–3}$ of acceptors at $300$ K? at $400$ K? [Ans. $7.6\, cm^2/V.s$]

\end{enumerate}

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