11.5 Semiconductors and Doping

Electronic Structure of Semiconductors

Semiconductors sit between conductors and insulators in terms of electrical behavior. Their conductivity can be precisely tuned by introducing impurities, a process called doping. This tunability is what makes semiconductors the foundation of nearly all modern electronics, from transistors to solar cells.

Band Structure and Energy Levels

In a crystalline semiconductor, atomic energy levels broaden into continuous bands. Two bands matter most:

• The valence band is the highest energy band that's fully occupied by electrons at absolute zero.
• The conduction band is the next band up, completely empty at absolute zero.
• The band gap (or forbidden energy gap) is the energy range between these two bands where no electron states exist.

The Fermi level in an intrinsic (pure) semiconductor sits approximately at the midpoint of the band gap. This level represents the energy at which the probability of finding an electron is 50%.

Band gap energies for common semiconductors typically fall between 0.1 eV and 4 eV. Silicon, the most widely used semiconductor, has a band gap of about 1.1 eV at room temperature. Germanium's is smaller (~0.67 eV), which is why Ge devices are more sensitive to temperature changes.

Intrinsic semiconductors are pure crystalline materials, most commonly from Group IV (Si, Ge), though compound semiconductors from Groups III-V (like GaAs) and II-VI (like CdTe) are also widely used.

Thermal Excitation and Conductivity

At absolute zero, a pure semiconductor acts like an insulator because no electrons have enough energy to cross the band gap. As temperature rises, thermal energy excites some electrons from the valence band into the conduction band. Each excited electron leaves behind a hole in the valence band, so carriers are always generated in electron-hole pairs.

Semiconductor conductivity falls in the range of roughly $10^{-8}$ to $10^{3} \; (\Omega \cdot m)^{-1}$, between metals and insulators.

The intrinsic carrier concentration is given by:

$n_i = \sqrt{N_c N_v} \, e^{-E_g / 2kT}$

• $n_i$: intrinsic carrier concentration (carriers per unit volume)
• $N_c$, $N_v$: effective density of states in the conduction and valence bands
• $E_g$: band gap energy
• $k$: Boltzmann constant ($8.617 \times 10^{-5}$ eV/K)
• $T$: absolute temperature (K)

The exponential dependence on $-E_g / 2kT$ is the key takeaway: even small changes in temperature or band gap produce large changes in carrier concentration.

Doping and Semiconductor Properties

Doping Process and Impurities

Doping means intentionally introducing a small number of impurity atoms into the semiconductor crystal lattice to change its electrical properties. The impurity atoms replace host atoms at lattice sites but have a different number of valence electrons.

There are two types of dopants:

• Donor impurities (for n-type doping): These atoms have one more valence electron than the host. In silicon (Group IV), common donors are phosphorus and arsenic (Group V). The extra electron is loosely bound and easily promoted to the conduction band.
• Acceptor impurities (for p-type doping): These atoms have one fewer valence electron than the host. In silicon, common acceptors are boron and gallium (Group III). The missing bond creates a hole that can accept an electron from the valence band.

Dopant concentrations are small, typically on the order of one impurity atom per $10^6$ to $10^9$ host atoms. Despite these tiny concentrations, doping dramatically changes conductivity because it introduces energy levels inside the band gap:

• Donor levels sit just below the conduction band (within ~0.05 eV for common Si dopants).
• Acceptor levels sit just above the valence band.

Because these levels are so close to a band edge, thermal energy at room temperature is enough to ionize nearly all dopant atoms.

Effects on Electrical Properties

Doping increases conductivity by orders of magnitude because it adds majority charge carriers without requiring thermal excitation across the full band gap. It also shifts the Fermi level:

• In n-type material, the Fermi level moves up toward the conduction band.
• In p-type material, the Fermi level moves down toward the valence band.

The conductivity of a doped semiconductor is:

$\sigma = q(n\mu_n + p\mu_p)$

• $\sigma$: conductivity
• $q$: elementary charge ($1.6 \times 10^{-19}$ C)
• $n$, $p$: electron and hole concentrations
• $\mu_n$, $\mu_p$: electron and hole mobilities

In a doped material, one carrier type dominates. For n-type, $n \gg p$, so $\sigma \approx qn\mu_n$. For p-type, $p \gg n$, so $\sigma \approx qp\mu_p$. This is what gives engineers precise control over device behavior.

N-type vs P-type Semiconductors

N-type Semiconductors

N-type semiconductors are created by doping with donor impurities (phosphorus, arsenic in silicon). Each donor atom contributes one extra electron that isn't needed for covalent bonding.

• Majority carriers: electrons
• Minority carriers: holes
• The Fermi level shifts toward the conduction band.

At moderate temperatures where essentially all donors are ionized, the electron concentration is approximately:

$n \approx N_D + \frac{n_i^2}{N_D}$

• $N_D$: donor concentration
• $n_i$: intrinsic carrier concentration

When $N_D \gg n_i$ (the usual case at room temperature), this simplifies to $n \approx N_D$. The minority hole concentration is then $p = n_i^2 / N_D$, which is very small.

P-type Semiconductors

P-type semiconductors are created by doping with acceptor impurities (boron, gallium in silicon). Each acceptor atom has one fewer valence electron, creating a hole.

• Majority carriers: holes
• Minority carriers: electrons
• The Fermi level shifts toward the valence band.

The hole concentration follows an analogous expression:

$p \approx N_A + \frac{n_i^2}{N_A}$

• $N_A$: acceptor concentration

Again, when $N_A \gg n_i$, this simplifies to $p \approx N_A$.

A useful relationship that holds in both doped and intrinsic semiconductors at thermal equilibrium is the mass action law: $np = n_i^2$. This means increasing one carrier type necessarily decreases the other.

Semiconductor Junctions and Devices

When n-type and p-type materials are brought into contact, a pn junction forms. At the interface, electrons from the n-side diffuse into the p-side and recombine with holes (and vice versa), creating a depletion region with no free carriers and a built-in electric field.

The built-in potential across this junction is:

$V_{bi} = \frac{kT}{q} \ln\left(\frac{N_A N_D}{n_i^2}\right)$

This junction is the building block for:

• Diodes: Allow current to flow easily in one direction (forward bias) and block it in the other (reverse bias).
• Transistors: Use two junctions (or a gate-controlled channel) to amplify or switch signals.
• Solar cells: Absorb photons to generate electron-hole pairs, which the junction's built-in field separates to produce current.

Temperature Dependence of Carrier Concentration

Intrinsic Semiconductors

Carrier concentration in a pure semiconductor depends exponentially on temperature, following:

$n_i \propto e^{-E_g / 2kT}$

At low temperatures, very few electrons have enough thermal energy to cross the band gap, so $n_i$ is extremely small. As temperature increases, $n_i$ grows exponentially because more electron-hole pairs are thermally generated. For silicon at room temperature (300 K), $n_i \approx 1.5 \times 10^{10} \; \text{cm}^{-3}$, which is tiny compared to the atomic density of about $5 \times 10^{22} \; \text{cm}^{-3}$.

Doped Semiconductors

Doped semiconductors show three distinct temperature regimes:

1. Freeze-out region (very low $T$): Thermal energy is insufficient to ionize dopant atoms. Carrier concentration drops rapidly as temperature decreases, and the material becomes resistive.
2. Extrinsic region (moderate $T$): Nearly all dopant atoms are ionized, so carrier concentration is roughly constant and equal to the dopant concentration. This is the normal operating range for most devices.
3. Intrinsic region (high $T$): Thermally generated carriers ($n_i$) become comparable to or exceed the dopant concentration. The material starts behaving like an intrinsic semiconductor, and the distinction between n-type and p-type fades.

This temperature dependence directly affects device parameters. For example, the reverse saturation current in a diode increases with temperature (since it depends on $n_i^2$), and transistor gain can shift as carrier mobilities and concentrations change.

Design Implications

Reliable semiconductor devices must account for temperature variation, especially in demanding environments like automotive or aerospace applications where operating temperatures can range widely.

Common strategies include:

• Bandgap reference circuits that produce a voltage nearly independent of temperature
• Temperature-dependent biasing to compensate for shifts in device characteristics
• Thermal management (heat sinks, active cooling) in high-power devices to keep junction temperatures within safe limits

Choosing a semiconductor with an appropriate band gap also matters: wider band gap materials (like SiC or GaN) maintain extrinsic behavior at higher temperatures, making them suitable for high-temperature and high-power applications.