Chapter 3 - Semiconductors

Lattice Structure

Silicon atoms have four valence electrons and form strong covalent bonds with four neighboring atoms. At extremely low temperatures (0 K), all of these bonds are stable, and no electrons are free to move. This makes silicon an electrical insulator.

At room temperature, however, thermal energy is enough to break some of these bonds. When a bond breaks, a free electron is released, leaving behind a positively charged “hole” where the electron used to be. Both the free electron and the hole are able to move through the crystal and conduct electricity. As the temperature increases, more bonds break, creating more electron-hole pairs and increasing the silicon’s conductivity.

Because this process, called thermal generation, always creates a free electron for every hole, the number of free electrons and holes are always equal in pure silicon. This concentration is measured by the number of charge carriers per unit volume ($cm^3$).

Thermal Equilibrium

Recombination is the process whereby electrons fill holes. Recombination is proportional to the thermal generation rate, which is a function of temperature. At thermal equilibrium,

\[n=p=n_i\]

where $n$ is the concentration of free electrons, $p$ the hole concentration, and $n_i$ (the intrinsic carrier density) denotes the number of free electrons and holes in a unit of volume at a given temperature:

\[n_i = BT^\frac{3}{2}e^{-E_g/2kT}\]

with $B$ being a material dependent parameter equal to $7.3\times10^{15}cm^{-3}K^{-3/2}$ for Si; $T$ the temperature in kelvins; $E_g$ the bandgap energy ($1.12 eV$ for Si); and $k$, Boltzmann’s constant ($8.62\times10^{-5}eV/K$).

$E_g$ is the minimum energy required to break a covalent bond $\implies$ generate an electron-hole pair.

\[pn = n_i^2\]

And at room temperature, $n_i(Si)\approx 1.5\times10^{10}/cm^3$.

Doped Semiconductors

Doping is a process used to change the carrier concentration in a semiconductor substantially and in a precise manner; impure atoms are introduced in order to increase the concentration of free electrons or holes with little to no change to Si’s crystal properties.

To increase free electrons, $n$, Si is doped with an element with a valence of $5$, such as phosphorus $P$. The result is n-type Si.
To increase hole concentration, $p$, Si is doped with an element with a valence of $3$, such as boron $B$. The result is p-type Si. We observe that a piece of n or p-type Si is electrically neutral; the charge of the majority free carriers is neutralized by the bound charges associated with the impurity atoms.

N-Type Si

If the concentration of donor atoms is $N_D$ where $N_D \gg n_i$,

\[n_n \approx N_D\] \[p_n \approx \frac{n_i^2}{N_D}\]

and hence electrons are said to be the majority charge carriers and holes the minority charge carriers.

P-Type Si

If the concentration of acceptor atoms is $N_A$ where $N_A \gg n_i$,

\[n_p \approx \frac{n_i^2}{N_A}\] \[p_p \approx N_A\]

and hence electrons are said to be the minority charge carriers and holes the majority charge carriers.

Drift

When an electrical field $E$ is established in a semiconductor crystal, holes are accelerated in the direction of $E$, and free electrons are accelerated in the direction opposite to that of $E$. The holes acquire a velocity of

\[v_{p-drift} = \mu_pE\]

where $\mu_p$ is called hole mobility. For intrinsic Si, $\mu_p = 480 cm^2/V\cdot s$

The free electrons acquire a drift velocity:

\[v_{n-drift} = -\mu_nE\]

where $\mu_n$ is called electron mobility. For intrinsic Si, $\mu_n = 1350cm^2/V\cdot s$

Diffusion

In a semiconductor, carrier diffusion is the movement of charge carriers (like holes or electrons) from an area of high concentration to an area of low concentration. This movement, driven by the density gradient, creates a diffusion current. It’s similar to how ink spreads out in water.

Thermal Voltage $V_T$

The thermal voltage $V_T$ may be expressed as the Einstein relationship:

\[V_T = \frac{D_n}{\mu_n}=\frac{D_p}{\mu_p}\]

where

\[V_T = \frac{kT}{q}\]

and $D_n$, $D_p$ are the diffusivity of electrons and holes, respectively. At room temperature, $T \approx 300 K$ and $V_T = 25.9 mV$.

pn Junctions

A pn junction consists of a p-type semiconductor brought into contact with a n-type semiconductor. In actual practice, one wafer of silicon possesses regions of different dopings, resulting in one or more pn junctions.

Because of the electron-hole concentration gradient across the junction, holes and electrons diffuse across the junction, resulting in a carrier-depletion (aka space-charge) region in the center. The uncovered charges on both sides cause an electric field $E$ to be established across the region, resulting in a potential difference with the n-side at a positive voltage relative to the p-side.

The voltage drop across the region acts as a barrier that has to be overcome for holes to diffuse into the n-region, and electrons into the p-region. The larger the barrier voltage, the smaller the number of carriers that will be able to overcome the barrier, and hence the lower the magnitude of diffusion current.

Diffusion Current $I_D$

Because the concentration of holes is high in the p region and low in the n region, holes diffuse across the junction from the p side to the n side. Similarly, electrons from the n side diffuse to the p side. These two current components add together to form the diffusion current $I_D$ which flows from the p side to the n side.

Drift Current $I_S$

In a semiconductor, electron-hole pairs are continuously being generated due to thermal energy. The electrons generated in the p-type material (minority carriers) near the junction and the holes generated in the n-type material (minority carriers) near the junction are swept across the depletion region by the strong electric field.

The movement of minority carriers under the influence of the electric field is called drift. It constitutes a current $I_S$ which flows from the n side to the p side. It is also known as the reverse saturation current.

pn Junction with open circuit / Zero Bias

Under open-circuit conditions,

\[I_D = I_S = 0\]

The diffusion current, which is strong at first due to the large concentration difference, is trying to pull charge carriers across the junction. As these charges move, they build up the internal electric field. This electric field creates the drift current, which pulls charge carriers in the opposite direction.

This process continues until the repulsive force from the electric field (which drives the drift current) becomes strong enough to exactly counteract the diffusive force (which drives the diffusion current). At this point, the flow of charge carriers in one direction due to diffusion is precisely balanced by the flow of charge carriers in the opposite direction due to drift. The net current is zero, and the junction is in equilibrium.

In this equilibrium state, even though individual charge carriers are still moving back and forth across the junction, the total current flow is zero. This is a state of dynamic equilibrium, where the two opposing currents are equal in magnitude and opposite in direction.

Junction Built-In Voltage

With no external voltage applied, the barrier voltage $V_0$ across the pn Junction may be expressed as

\[V_0 = V_T \ln(\frac{N_AN_D}{n_i^2})\]

Typically, for Si at room temperature, $0.6V \leq V_0 \leq 0.9V$.

Width and Charge of the Depletion Region

If we denote the width of the depletion region in the p side as $x_p$, and in the n side as $x_n$, the magnitude of the charge on the n side is

\[|Q_+|=qAx_nN_D\]

and on the p side

\[|Q_-| = qAx_pN_A\]

where $A$ is the cross sectional area of the junction in the plane perpendicular to the frame of reference. Then it follows that

\[\frac{x_n}{x_p}=\frac{N_A}{N_D}\]

The depletion width $W$ is

\[W = x_n + x_p = \sqrt{\frac{2\epsilon_s}{q}\cdot(\frac{1}{N_A}+\frac{1}{N_D})\cdot V_0}\]

where $\epsilon_s$ is the electrical permittivity of Silicon. Typically, $0.1\mu m \leq \epsilon_s \leq 1\mu m$.

Then, in terms of $W$,

\[x_n = W\frac{N_A}{N_A+N_D}\]

and

\[x_p = W\frac{N_D}{N_A+N_D}\]

Such that the junction’s charge $Q_J$ is

\[Q_J = A\sqrt{2\epsilon_s q\cdot(\frac{N_AN_D}{N_A+N_D})\cdot V_0}\]

One-sided Junctions

Usually, one side of the junction is more doped than the other, and the depletion region exists almost entirely on the more lightly doped side.

If $N_A \gg N_D$ then
- $x_n \gg x_p$
- depletion region is almost entirely on the n-side $\implies W\approx x_n$
If $N_D \gg N_A$ then
- $x_p \gg x_n$
- depletion region is almost entirely on the p-side $\implies W\approx x_p$

Forward Bias

Applying a positive voltage $V_D$ to the p-side reduces the barrier:

\[V_0 - V_D\]

The diffusion current $I_D$ equals

\[I_D = I_S(e^\frac{qV_D}{kT}-1)\]

where $I_S$ is the reverse saturation current; however, we approximate it as:

\[I_D = I_Se^{V_D/V_T}\]

for an ideal diode when $V_D\gg V_T$.

Reverse Bias

Under reverse bias conditions, the reverse-bias voltage $V_R$ adds to the barrier voltage, increasing the effective barrier voltage to

\[(V_0 + V_R)\]

The diffusion current $I_D$ equals

\[I_D = I_S(e^\frac{qV_D}{kT}-1)\]

but we approximate it as:

\[I_D = -I_S\]

for an ideal diode; for when $V_R \gg V_T$, the exponential term $\approx 0$.

pn Junction Summary

Condition	Net Current $I_D$	Barrier Potential $V_{\text{barrier}}$	Depletion Width $W$	Notes
Open Circuit	$0$	$V_0$	Normal	Drift = Diffusion
Forward Bias	$\approx I_S e^{V_D / V_T}$	$V_0 - V_D$	Decreases	High current
Reverse Bias	$\approx -I_S$	$V_0 + V_R$	Increases	Very small current