PHT.301 Physics of Semiconductor Devices
02.05.2025

Problem 1
A PIN diode has a p-layer, an intrinsic layer, and an n-layer.

(a) Draw the charge density of a PIN diode at zero bias. Be sure your drawing satisfies charge neutrality.

(b) Draw the electric field of the PIN diode at zero bias.

(d) How does $V_{bi}$ depend on the thickness of the intrinsic layer?

Solution

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(a) At zero bias, the electrons diffuse from the high concentration on the n-side towards the p-side, and the holes diffuse from a high concentration on the p-side towards the n-side. The charge imbalance causes an electric field pointing from n to p that pushes the charge carriers back and establishes a drift current that cancels the diffusion current. In the depletion region, there are negative acceptors on the p-side, no charge in the intrinsic section, and positive donors on the n-side. To ensure charge neutrality, the area under the charge density curve on the p-side should be the same as the area under the charge density curve on the n-side.
 
 <img src="02_05_25_1a.png" width="400">

(b) The electric field is the integral of the charge density. The electric field must go to zero at the edges of the depletion region.

(c) The electrostatic potential is minus the integral of the electric field, and the bands have the same shape as minus the electrostatic potential. The slope of the bands is proportional to the electric field. 
 
 <img src="02_05_25_1c.png" width="400">

(d) The built-in voltage, $V_{bi}$ is the integral of the electric field in a pn-junction at zero bias. It is equal to the band offset, which is illustrated in the figure for part (c). This band offset depends on the doping of the n-side and the p-side but does not depend on the thickness of the intrinsic section. A wider intrinsic section would result in a lower maximum electric field.

Problem 2

(a) Draw an n-channel MESFET.

(b) Draw the band diagram (conduction band, Fermi energy, valence band) along a line that goes from the gate to the channel for zero applied gate voltage. Indicate the Schottky barrier in your drawing.

(d) Why is the drain current almost constant in saturation?

Problem 3
The capacitance is measured between the gate and the body contact of a MOSFET as a function of the gate-body voltage.

(a) Why doesn't a constant current flow between the gate and the body?

(b) Why does the capacitance change as the gate-body voltage changes?

(d) How does the capacitance depend on the doping of the body?

Solution

Problem 4
A npn bipolar transistor is made starting with epitaxially grown layers. The lowest layer is the subcollector, then the collector, then the base, and finally, the emitter is the top layer.

(a) Make a drawing of the transistor including the metal contacts. Be sure there are no parasitic Schottky diodes in your drawing.

(b) Why shouldn't you start with the emitter on the bottom?

(d) What limits the maximum allowed collector-emitter voltage?

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 Devices02.05.2025

PHT.301 Physics of Semiconductor Devices
02.05.2025