PHT.301 Physics of Semiconductor Devices
29.04.2022

Problem 1

Describe a green laser diode. What kind of semiconductor is needed to make it green? Make a sketch of the device indicating the contacts.

Solution

<hr />

<div style="display:block; width:100%;">
	<div style="width:40%; float: left; display: inline-block;">
		<p>A semiconductor material with a direct bandgap is needed, that an electron-hole pair, which is generated in the pn-junction, can emit light at its recombination. There are differences between spontaneous emission and stimulated emission. Stimulated emission is responsible for the coherent light of lasers. For stimulated emission, we need an injected current density which is above a certain threshold. </br>
		\begin{equation*}
			\tau (\omega)_{stimulated} = \tau (\omega)_{spontaneous} * \eta(\omega)_{ph}
		\end{equation*}
		where $\tau (\omega)_{spontaneous}$ is the rate of spontaneous emission and $\eta(\omega)_{ph}$ is the number of photons already in the state. This means, that the probability for stimulated emission to the state $\omega$ grows with the number of photons already present in the state $\omega$.
		</p>
	</div>
	
	<div style="width:60%; float: left; display: inline-block;">
		</br>
		<img src="1_laser_diode_injected_current_density.png" height="300" alt="picture laser diode injected current density">
		<p></p>
	</div>
</div>
<div style="display:block; width:100%;">
	<div style="width:40%; float: left; display: inline-block;">
		<p></br>
			The dimensions and the direction (Miller-indices) of the crystal must be choosen correctly, so that the desired wavelength and its propagation modes are not suppressed. The faces of the crystal are cleaved to make mirrors. The mirrors are used to trap the photons and let them propagate at one direction with the desired wavelength.
		</p>
	</div>
	
	<div style="width:60%; float: left; display: inline-block;">
		<img src="1_laser_diode_built_up.png" height="250" alt="picture laser diode functional structure">
	</div>
</div>
<div style="display:block; width:100%;">
	<div style="width:30%; float: left; display: inline-block;">
		<p>
			<u>Dimensioning a green laser diode:</u></br>
			\begin{equation*}
				\begin{aligned}
					& \lambda = 550\mathrm{nm}\\
					& h = 6.6260755 × 10^{-34}~\mathrm{Js}\\
					& c = 2.99792458 × 10^8~\mathrm{m/s}\\
					& e = 1.60217733 × 10^{-19} \mathrm{As}
				\end{aligned}
			\end{equation*}
		</p>
	</div>
	
	<div style="width:30%; float: left; display: inline-block;">
		<p></br>
			\begin{equation*}
				\begin{aligned}
				& E=h * f \;\; \mathrm{and} \;\; \lambda=\frac{c}{f} \rightarrow E=h * \frac{c}{\lambda}\\
				& E_G = \frac{E}{e} = \frac{h * c}{e * \lambda} = 2,254~\mathrm{eV}
				\end{aligned}
			\end{equation*}
			</br>
		</p>
	</div>
	<div style="width:40%; float: left; display: inline-block;">
	<p><br></br><br></br></p><br></br><br></br></div>
</div>
<p> For laser diodes with green light, one can use GaP or GaAsP as direct bandgap semiconductor material.</p>

Problem 2

(a) An n-channel MOSFET is in weak inversion. This means that the electron concentration near the semiconductor-oxide interface is smaller than the acceptor doping concentration. Draw the charge density along a line normal to the gate, oxide, and body of the MOSFET. Indicate the position of the acceptor concentration on the charge density axis.

(b) Draw the corresponding electric field that goes with the charge density drawn in (a).

(c) Consider the drain current. What is the dominant current transport mechanism when the gate voltage is above the threshold voltage and what is the dominant current transport mechanism when the gate voltage is below the threshold voltage?

Solution

<hr />
<p><b>(a) & (b) </b></br></p>
<div style="display:block; width:100%;">
	<div style="width:40%; float: left; display: inline-block;">
		<p>
			<img src="2_a_nmos_weak_inversion_cross_section.png" height="300" alt="picture nmos weak inversion cross section">
		</p>
	</div>
	
	<div style="width:40%; float: left; display: inline-block;">
		<p>
			<img src="2_a_nmos_weak_inversion_band_diagram.png" height="300" alt="picture nmos weak inversion band diagram">
		</p>
	</div>
	<div style="width: 20%; height:350; float: left; display: inline-block;">
	</div>
</div>
<div style="display:block; width:100%;">
	<div style="width:40%; float: left; display: inline-block;">
		<p>
			<img src="2_a_nmos_weak_inversion_charge_densities.png" height="300" alt="picture nmos weak inversion charge densities">
		</p>
	</div>
	
	<div style="width:40%; float: left; display: inline-block;">
		<p>
			<img src="2_b_nmos_weak_inversion_electric_field.png" height="300" alt="picture nmos weak inversion electric field">
		</p>
	</div>
	<div style="width: 20%; height:350; float: left; display: inline-block;">
	</div>
</div>
	
<hr>
<p><b>(c)</b></br></br>
When the gate voltage is below the threshold voltage and the semiconductor surface is only weakly inverted, </br> the corresponding drain current is called the subthreshold current. In the subthreshold region, the drain current </br> is dominated by diffusion instead of drift.</br></br>

When the gate voltage is above the threshold voltage, the semiconductor surface is strongly inverted. </br> At that point, the channel resistance is low. Applying a voltage between drain and source leads to an electric field </br> and involves a drift current. In this case, the drain current is dominated by a drift current. 
</p>

Problem 3

Draw the minority carrier concentration in a pnpn thyristor biased in the forward blocking mode. Make another drawing of the minority carrier concentration in forward conducting mode. Indicate the depletion widths in the drawings.

Solution

<hr />

<p>A thyristor consists of three pn junctions in series. The anode is p+ and the n-region next to the anode has a light doping and is chosen to be wider than the diffusion length so that at low forward bias voltages the minority holes that are injected from the anode recombine in the n-region and the minority carrier concentration returns to the equilibrium concentration in the middle of the lightly doped n-region. Under these conditions, the device acts like three diodes in series and the middle diode is reverse biased. This is the forward blocking mode.</p>

<p>As the applied voltage is increased, the depletion width of the reverse biased diode becomes wider and the neutral n-region becomes narrower than the diffusion length. Not all of the minority holes injected from the anode will recombine and the minority carrier concentration becomes a straight line like it is for a bipolar transistor. In this limit the thyristor acts like pnp and npn transistors coupled together. The transistors switch each other on. This is the forward conducting mode.</p>

Problem 4

(a) A pnp bipolar transistor is used in a common base configuration where $|V_{CE}|= 20$ V. What are approximately the voltages at the emitter and the collector? Assume that the voltage at the base is zero. Indicate the polarity (+ or -) of the emitter and collector voltages.

(b) If the base doping of this transistor is increased, what will happen to the gain factor $\alpha$ and the Early effect? Explain why these changes take place.

(c) The collector is lightly doped. How is a Schottky contact avoided between the collector and the metal contact? What determines how wide the lightly doped part of the collector should be?

Solution

<hr />

<p>(a) The emitter-base junction will be forward biased with one diode drop so the voltage at the emitter will be about 0.6 V. The rest of the voltage will drop across the base-collector junction which is reverse biased so the voltage is about -19.4 V.</p>

<p>(b) If the base doping is increased, the recombination in the base will increase so the base transport factor will decrease. The current of the minority electrons injected into the emitter will increase so the emitter efficiency will decrease. Both of these effects will result in a lower α gain factor. </p>

<p>The Early effect comes from the base width modulation. For higher doping the depletion width will be narrower and there will be less base width modulation so the Early effect will be decreased by doping the base more heavily.</p>

<p>(c) Since the collector is lightly doped, a heavily doped p-region under the metal contact is needed to form a tunnel contact. The distance of the tunnel contact from the base-collector junction depends on the voltage rating of the transistor. The distance should be far enough that at the maximum voltage the depletion region in the collector does not reach the tunnel contact and cause the electric field to exceed the breakdown field. However, the tunnel contact should be placed as close as possible to avoid an extra series resistance.</p>

Quantity	Symbol	Value	Units
electron charge	e	1.60217733 × 10^-19	C
speed of light	c	2.99792458 × 10⁸	m/s
Planck's constant	h	6.6260755 × 10^-34	J s
reduced Planck's constant	$\hbar$	1.05457266 × 10^-34	J s
Boltzmann's constant	k_B	1.380658 × 10^-23	J/K
electron mass	m_e	9.1093897 × 10^-31	kg
Stefan-Boltzmann constant	σ	5.67051 × 10^-8	W m^-2K^-4
Bohr radius	a₀	0.529177249 × 10^-10	m
atomic mass constant	m_u	1.6605402 × 10^-27	kg
permeability of vacuum	μ₀	4π × 10^-7	N A^-2
permittivity of vacuum	ε₀	8.854187817 × 10^-12	F m^-1
Avogado's constant	N_A	6.0221367 × 10²³	mol^-1

PHT.301 Physics of Semiconductor Devices29.04.2022

PHT.301 Physics of Semiconductor Devices
29.04.2022