Differences

This shows you the differences between two versions of the page.

--- courses:ast201:8 [2023/12/09 01:23] – [4.1 Intrinsic semiconductors] asad
+++ courses:ast201:8 [2023/12/09 22:39] (current) – [4.2 Photoabsorbers or photoconductors] asad
@@ Line 33: / Line 33: @@
 The semiconductor $c$ has electrons in the conduction band without any corresponding holes in the valence band. These are extrinsic semiconductors widely used to make electronic devices. Another class of extrinsics exhibits holes in the valence band without any corresponding electrons in the conduction band.
-==== - Temperature ====
+At room temperature, all elements of the column IVA are non-conductors, but they become capable of ionization (promotion of electrons to the conduction band) as temperature increases.
-$$ P(T,E) = \frac{1}{1+e^{(E-E_F) / (kT)}} $$
+How does an electron get enough energy to emancipate itself? One way is to collide with each other and escape from the valence band together. But the more important way is to absorb or emit a **phonon**, another name for the particle associated with the quantized lattice vibration energies. A lattice can vibrate by oscillating bond length and angle. An electron can absorb these vibrational kinetic energy and jump to the conduction band.
+At a particular temperature $T$, the number of electrons in at energy $E$ is
 $$ n_e (T,E) = P(T,E) S(E) $$
-where $S$ is the number density of available states at an energy $E$.
+where $P(T,E)$ is the probability of an electron having an energy $E$ at temperature $T$, and $S(E)$ is the number density of the available states at energy $E$. In case of a semiconductor, the valence band states are completely filled, so the electrons are in a **degenerate** condiciton. In such a condiction, the aforementioned probability is given by the famous **Fermi-Dirac distribution**:
+$$ P(T,E) = \frac{1}{1+e^{(E-E_F) / (kT)}} $$
+where $E_F$ is the **Fermi energy**, the energy at which the probability of finding an electron in a permitted state is 1/2. Fermi-Dirac reduces to Maxwell-Boltzmann at high temperatures ($T\gg 0$). At zero temperature, however,
+$$ P(E) =
+\begin{cases}
+, & \text{if } E \lt E_F \\
+, & \text{if } E \gt E_F
+\end{cases}$$
+which is shown in Panel $b$ of the diagram below.
 {{:courses:ast201:fermi.webp?nolink&750|}}
+The Panel $a$ shows the bands of Si at absolute zero where $E_F$ (dashed line) is exactly in the middle of the band gap. The solid black line of Panel $b$ makes it clear that at $T=0$ K, the probability of finding electrons at $E\lt E_F$ is always 1, meaning all the permitted states are filled. At higher temperatures, the probability distribution follows the green or blue line.
+Panel $c$ shows $S(E)$ for the electrons in the valence and conduction bands of Si. This function is almost quadratic near the edges of the bands, but vanishes in the gap. Finally, Panel $d$ shows the number density of electrons $n_e$ and also the number density of holes
+$$ n_h = [1-P(E)] S(E) $$
+at a non-zero temperature and we see that $n_e$ peaks near the edge of the valence band, but $n_h$ peaks inside the valence band just as $n_e$ peaks inside the conduction band. The total number of charge carriers can be calculated as
+$$ N_n = \int_{E_F}^\infty n_e dE $$
+$$ N_p = \int_{-infty}^{E_F} n_h dE. $$
+In intrinsic semiconductors the positive and negative charge carriers are equal in number in equilibrium; $N_p=N_n$. The number depends on temperature as
+$$ N_p = N_n = AT^3 e^{-E_G/kT}. $$
 ===== - Semiconductors =====
@@ Line 78: / Line 109: @@
 As seen above, the diamond allotrope of carbon is an insulator because its band gap is huge (5.48 eV). The allotropes graphite and carbon nanostructures are conductors.
-{{:courses:ast201:semiconductors.webp?nolink&750|}}
-==== - Intrinsic photoabsorbers ====
+==== - Photoabsorbers ====
+So far we have talked about the emancipation of electrons via lattice vibration and collision, but an a photon with a wavelength larger than the **curoff wavelength** can also emancipate an electron. The cutoff wavelength
+$$ \lambda_c = \frac{hc}{E_G} = \frac{1.24 \ \mu\text{m}}{E_G \text{ eV}}. $$
+For Si, the value of 1.1 $\mu$m. In astronomy we use semiconductors as photoabsorbers. A simple example is shown below.
+{{:courses:ast201:photoabsorber.jpg?nolink&400|}}
+The photon stream promotes electrons to the conduction band leaving behind an equal number of holes in the valence band. This is a basic **detector** or **sensor** or **receiver** that converts energy into matter, photons into electrons. The greater the stream the higher the conductivity of the detector. If the voltage across the semiconductor is constant, the electrical current $i$ through the resistor $R_L$ would depend on the number of photons absorbed per second.
+So the voltage measured at $V_o$ will be directly related to the **intensity** of light.
 ==== - Extrinsic semiconductors ====