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32 {\LARGE {\bf Materials Physics I}\\}
37 {\Large\bf Tutorial 5}
40 \section{Charge carrier density of intrinsic semiconductors}
43 \item Recall the free electron in a box.
44 Write down an expression for the density of states $D(E)$
45 of the free electron gas.
46 {\bf Hint:} The density of states is a function of internal energy $E$
47 such that $D(E)dE$ is the number of states
48 (allowed $k$-values) with energies
49 between $E$ and $E+dE$.
50 For large values of $L$ (side length of the box)
51 the states are quasi-continuous and
52 sums in $k$-space can be replaced by integrals.
53 First calculate the amount of states $dZ'$
54 in-between $k$ and $k+dk$.
55 Therefor calculate the volume of the sperical shell
56 and the volume of a single allowed $k$-point.
57 Neglect terms of the order $(dk^2)$.
58 After that express $dk$ and $k$ by $dE$ and $E$
59 and insert these expressions into $dZ'$.
60 By definition $D(E)=dZ/dE$,
61 where $dZ$ is $dZ'$ devided by the box volume
62 (states per crystal volume).
63 Adjust the expression taking into account
64 the spin of an electron.
65 \item The conduction and valence band in a semiconductor can be approximated
66 by the same parabolic functions of $k$ close to the bandedges.
67 The mass of the electron is replaced by an effective mass
68 of the electron in the conduction band or the hole in the valence band.
69 Show the relation of the effective mass and the curvature of the band.
70 {\bf Hint:} The curvature of a function $f(x)$ is given by the second
71 derivative of this function with respect to $x$.
72 \item Sketch the density of states, the Fermi function and the resulting
73 density of charge carriers (electrons: $m_n$, holes: $m_p$)
74 for $m_n=m_p$ and for $m_n\ne m_p$ for non-zero temperatures.
75 {\bf Hint:} The density of states is given by
76 $D_c(E)=\frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}
77 (E-E_c)^{1/2}$ for electrons in the conduction band and
78 $D_v(E)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
79 (E_v-E)^{1/2}$ for holes in the valence band.
80 $E_c$ is the lowest energy level of the conduction and
81 $E_v$ the highest energy level of the valence band.
82 Thus the bandgap energy is given by $E_g=E_c-E_v$.
83 The density of charge carriers is the product of $D(E)$ and
84 the Fermi function $f(E)$.
85 The Fermi energy $E_F$ adjusts itself in such a way that
86 the amount of electrons and holes equals.
87 Keep in mind that the distribution valid for the holes is
91 \section{'Density of state mass' of holes in silicon}
93 The valence band of silicon is composed by three subbands.
94 Two of them contact at the $\Gamma$-point ($k=0$),
95 the one for heavy holes ($m_{ph}$) and the one for light holes ($m_{pl}$).
96 An additional split-off hole band ($m_{pso}$) is located
97 shortly below the first two bands (see Figure).
100 \item Assume parabolic bands near $k=0$.
101 Write down the total density of states
102 near the maximum of the valence band.
103 Only consider heavy and light holes.
104 \item Write the above result in terms of a density of states expression
105 of a parabolic band with a single uniform effective mass $m_p$.
106 Determine this 'density of state mass' $m_p$.
107 Calculate $m_p$ using the values $m_{ph}=0.49 \, m_e$ and
108 $m_{pl}=0.16 \, m_e$ in which $m_e$ is the electron rest mass.
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