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# 物理代写|半导体物理代写Semiconductor Physics代考|EEE6355 Donors and Acceptors

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## 物理代写|半导体物理代写Semiconductor Physics代考|Donors and Acceptors

When a semiconductor is doped with donor or acceptor impurities, impurity energy levels are introduced that usually lie within the energy gap. A donor impurity has a donor level which is defined as being neutral if filled by an electron, and positive if empty. Conversely, an acceptor level is neutral if empty and negative if filled by an electron. These energy levels are important in calculating the fraction of dopants being ionized, or electrically active, as discussed in Section 1.4.3.

To get a feeling of the magnitude of the impurity ionization energy, we use the simplest calculation based on the hydrogen-atom model. The ionization energy for the hydrogen atom in vacuum is
$$E_H=\frac{m_0 q^4}{32 \pi^2 \varepsilon_0^2 \hbar^2}=13.6 \mathrm{eV} .$$
The ionization energy for a donor $\left(E_C-E_D\right)$ in a lattice can be obtained by replacing $m_0$ by the conductivity effective mass of electrons ${ }^5$
$$m_{c e}=3\left(\frac{1}{m_1^}+\frac{1}{m_2^}+\frac{1}{m_3^*}\right)^{-1}$$
and by replacing $\varepsilon_0$ by the permittivity of the semiconductor $\varepsilon_s$ in Eq. 31:
$$E_C-E_D=\left(\frac{\varepsilon_0}{\varepsilon_s}\right)^2\left(\frac{m_{c e}}{m_0}\right) E_H .$$

## 物理代写|半导体物理代写Semiconductor Physics代考|Calculation of Fermi Level

The Fermi level for the intrinsic semiconductor (Eq. 27) lies very close to the middle of the bandgap. Figure 11a depicts this situation, showing schematically from left to right the simplified band diagram, the density of states $N(E)$, the Fermi-Dirac distribution function $F(E)$, and the carrier concentrations. The shaded areas in the conduction band and the valence band represent electrons and holes, and their numbers are the same; i.e., $n=p=n_i$ for the intrinsic case.

When impurities are introduced to the semiconductor crystals, depending on the impurity energy level and the lattice temperature, not all dopants are necessarily ionized. The ionized concentration for donors is given by ${ }^{36}$
$$N_D^{+}=\frac{N_D}{1+g_D \exp \left[\left(E_F-E_D\right) / k T\right]}$$
where $g_D$ is the ground-state degeneracy of the donor impurity level and equal to 2 because a donor level can accept one electron with either spin (or can have no electron). When acceptor impurities of concentration $N_A$ are added to a semiconductor crystal, a similar expression can be written for the ionized acceptors
$$N_A^{-}=\frac{N_A}{1+g_A \exp \left[\left(E_A-E_F\right) / k T\right]}$$
where the ground-state degeneracy factor $g_A$ is 4 for acceptor levels. The value is 4 because in most semiconductors each acceptor impurity level can accept one hole of either spin and the impurity level is doubly degenerate as a result of the two degenerate valence bands at $\boldsymbol{k}=0$.

## 物理代写|半导体物理代写Semiconductor Physics代考|捐助者和接受者

$$E_H=\frac{m_0 q^4}{32 \pi^2 \varepsilon_0^2 \hbar^2}=13.6 \mathrm{eV} .$$

$$m_{c e}=3\left(\frac{1}{m_1^}+\frac{1}{m_2^}+\frac{1}{m_3^*}\right)^{-1}$$

$$E_C-E_D=\left(\frac{\varepsilon_0}{\varepsilon_s}\right)^2\left(\frac{m_{c e}}{m_0}\right) E_H .$$

## 物理代写|半导体物理代写半导体物理学代考|计算费米能级

.计算费米能级

$$N_D^{+}=\frac{N_D}{1+g_D \exp \left[\left(E_F-E_D\right) / k T\right]}$$

$$N_A^{-}=\frac{N_A}{1+g_A \exp \left[\left(E_A-E_F\right) / k T\right]}$$

## MATLAB代写

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