6.1 Intrinsic and extrinsic semiconductors

Semiconductor materials

Semiconductors sit between conductors and insulators in terms of electrical conductivity. What makes them special is that you can tune their conductivity over many orders of magnitude by adding tiny amounts of impurities. This controllability is what makes nearly all modern electronics possible.

Elemental semiconductors

The most common elemental semiconductors come from group IV of the periodic table: silicon (Si) and germanium (Ge). Both form a diamond cubic crystal structure where each atom bonds covalently with four neighbors.

• Silicon dominates the semiconductor industry because it's abundant (second most common element in Earth's crust), relatively cheap to purify, and has a stable native oxide ($SiO_2$) that's extremely useful for device fabrication.
• Germanium was used in early transistors but has a smaller band gap (0.67 eV vs. 1.1 eV for Si), which makes it more sensitive to temperature and limits its use in many applications.

Compound semiconductors

Compound semiconductors are formed by combining elements from different groups:

• III-V compounds (e.g., GaAs, InP, GaN) combine group III and group V elements. These tend to have direct band gaps, making them excellent for optoelectronic applications like LEDs and laser diodes.
• II-VI compounds (e.g., CdTe, ZnSe) combine group II and group VI elements. These are used in specialized applications such as thin-film solar cells and infrared detectors.

Compound semiconductors offer a wider range of band gaps and optical properties than elemental semiconductors, but they're generally more expensive and harder to process.

Intrinsic semiconductors

An intrinsic semiconductor is a pure semiconductor with no intentional impurities. Its electrical behavior comes entirely from the material's own crystal and electronic band structure.

Properties of intrinsic semiconductors

• The number of free electrons exactly equals the number of holes: $n = p = n_i$.
• Electrical conductivity is low compared to metals because the carrier concentration is small at room temperature.
• Carrier concentration depends strongly on temperature: heating the material generates more electron-hole pairs.

Energy band structure

Semiconductors have two key energy bands separated by a band gap ($E_g$):

• The valence band is the highest range of energies where electrons are normally found at absolute zero. These electrons participate in covalent bonding.
• The conduction band is the next higher range of allowed energies. Electrons here are free to move and carry current.
• The band gap is the energy difference between the top of the valence band and the bottom of the conduction band. For silicon, $E_g = 1.1 \text{ eV}$; for germanium, $E_g = 0.67 \text{ eV}$.

The size of the band gap determines which wavelengths of light a semiconductor can absorb and how its conductivity responds to temperature.

Charge carrier generation

When an electron absorbs enough energy (thermal, optical, or electrical) to cross the band gap, it jumps from the valence band into the conduction band. This leaves behind a hole in the valence band, which behaves as a positive charge carrier.

Each excitation event creates one electron-hole pair. Both the electron in the conduction band and the hole in the valence band contribute to electrical conduction.

Intrinsic carrier concentration

The intrinsic carrier concentration, $n_i$, is the number of free electrons (or equivalently, holes) per unit volume in a pure semiconductor at thermal equilibrium.

• For silicon at room temperature (300 K): $n_i \approx 1.5 \times 10^{10} \text{ cm}^{-3}$
• This sounds large, but silicon has about $5 \times 10^{22}$ atoms per $\text{cm}^3$, so only roughly one in every $10^{12}$ atoms contributes a free carrier at room temperature.

Temperature dependence of carrier concentration

The intrinsic carrier concentration rises exponentially with temperature:

$n_i = A T^{3/2} \exp\left(\frac{-E_g}{2k_BT}\right)$

where $A$ is a material-dependent constant, $T$ is absolute temperature, $E_g$ is the band gap, and $k_B$ is Boltzmann's constant.

The exponential term dominates: as $T$ increases, more thermal energy is available to excite electrons across the gap, so $n_i$ grows rapidly. This is why semiconductor devices can fail or behave unpredictably at high temperatures.

Fermi level in intrinsic semiconductors

The Fermi level ($E_F$) is the energy at which the probability of an electron occupying a state is exactly 0.5, according to the Fermi-Dirac distribution.

In an intrinsic semiconductor, the Fermi level sits approximately at the midpoint of the band gap. More precisely, it shifts slightly from the exact center due to the difference in effective masses of electrons and holes, but for most purposes you can treat it as centered.

The position of $E_F$ relative to the band edges controls the equilibrium carrier concentrations.

Extrinsic semiconductors

An extrinsic semiconductor is created by intentionally adding impurity atoms (dopants) to an intrinsic semiconductor. Doping lets you control the type and concentration of charge carriers with high precision.

N-type vs P-type doping

• N-type doping: You add atoms with one extra valence electron compared to the host (e.g., phosphorus or arsenic in silicon, which have 5 valence electrons vs. silicon's 4). The extra electron is loosely bound and easily freed into the conduction band. Electrons become the majority carriers; holes are the minority carriers.
• P-type doping: You add atoms with one fewer valence electron (e.g., boron or gallium in silicon, which have 3 valence electrons). This creates an empty state that readily accepts an electron from the valence band, generating a hole. Holes become the majority carriers; electrons are the minority carriers.

Donor and acceptor impurities

• Donor impurities (group V in silicon) "donate" an extra electron to the conduction band. They create an energy level just below the conduction band edge.
• Acceptor impurities (group III in silicon) "accept" an electron from the valence band, creating a hole. They create an energy level just above the valence band edge.

Energy levels of dopants

Dopant energy levels sit very close to the relevant band edge:

• Phosphorus in silicon: donor level about 45 meV below the conduction band
• Boron in silicon: acceptor level about 45 meV above the valence band

Since thermal energy at room temperature is $k_BT \approx 26 \text{ meV}$, these shallow levels are easily ionized. At 300 K, nearly all dopant atoms are ionized and contributing carriers.

Carrier concentration in extrinsic semiconductors

At room temperature with full ionization, the majority carrier concentration approximately equals the dopant concentration:

• N-type: $n \approx N_D$ (where $N_D$ is the donor concentration)
• P-type: $p \approx N_A$ (where $N_A$ is the acceptor concentration)

The minority carrier concentration is found using the law of mass action:

$n_0 \cdot p_0 = n_i^2$

For example, if $N_D = 10^{16} \text{ cm}^{-3}$ in silicon, then $n \approx 10^{16} \text{ cm}^{-3}$ and $p = n_i^2 / n \approx (1.5 \times 10^{10})^2 / 10^{16} = 2.25 \times 10^{4} \text{ cm}^{-3}$. The minority carrier concentration is twelve orders of magnitude smaller than the majority carrier concentration.

Majority vs minority carriers

• Majority carriers: the more abundant type (electrons in n-type, holes in p-type). They dominate electrical conduction.
• Minority carriers: the less abundant type (holes in n-type, electrons in p-type). Despite their low concentration, minority carriers play a critical role in devices like p-n junctions, BJTs, and solar cells.

Fermi level shifts in extrinsic semiconductors

Doping moves the Fermi level away from the middle of the band gap:

• N-type doping shifts $E_F$ toward the conduction band.
• P-type doping shifts $E_F$ toward the valence band.

Higher doping concentrations push $E_F$ closer to the respective band edge. The exact position depends on doping concentration and temperature.

Temperature dependence of extrinsic carrier concentration

The carrier concentration in an extrinsic semiconductor goes through three regimes as temperature increases:

1. Freeze-out region (low T): Not all dopants are ionized. The carrier concentration is below $N_D$ (or $N_A$) and increases with temperature as more dopants ionize.
2. Extrinsic (saturation) region (moderate T): All dopants are ionized. The carrier concentration plateaus at approximately $N_D$ (or $N_A$) and stays roughly constant.
3. Intrinsic region (high T): Thermal generation of electron-hole pairs across the band gap overwhelms the dopant contribution. The semiconductor behaves like an intrinsic material.

Most devices are designed to operate in the extrinsic saturation region.

Heavy vs light doping

• Lightly doped: dopant concentration typically below $\sim 10^{16} \text{ cm}^{-3}$. Lower conductivity, wider depletion regions in p-n junctions.
• Heavily doped (often denoted $n^+$ or $p^+$): dopant concentration above $\sim 10^{18} \text{ cm}^{-3}$. Higher conductivity, narrower depletion regions.

Heavy doping is used for ohmic contacts and low-resistance regions, while light doping is used where you need wide depletion regions (e.g., the base of a photodiode).

Degeneracy and Fermi level

When doping is very heavy, the Fermi level can move into the conduction band (n-type) or into the valence band (p-type). At that point, the semiconductor is called degenerate.

A degenerate semiconductor behaves more like a metal: it has high conductivity, and the standard Boltzmann approximation for carrier statistics no longer applies. You need the full Fermi-Dirac distribution to describe carrier occupation.

Compensation doping

If both donor and acceptor impurities are present in the same material, they partially cancel each other out. The net behavior depends on which type has the higher concentration:

• If $N_D > N_A$: the material is n-type with an effective donor concentration of $N_D - N_A$.
• If $N_A > N_D$: the material is p-type with an effective acceptor concentration of $N_A - N_D$.

Compensation doping is used to fine-tune carrier concentrations or to create semi-insulating substrates (when $N_D \approx N_A$).

Charge transport in semiconductors

Current in semiconductors arises from two distinct mechanisms: drift (driven by electric fields) and diffusion (driven by concentration gradients). Both contribute simultaneously in most device situations.

Drift current

When you apply an electric field $E$ across a semiconductor, charge carriers accelerate in response but frequently scatter off lattice vibrations and impurities. The net result is a steady average velocity called the drift velocity, proportional to the field:

$v_d = \mu E$

where $\mu$ is the carrier mobility. The total drift current density is:

$J_{\text{drift}} = q(n\mu_n + p\mu_p)E$

Here $q$ is the elementary charge, $n$ and $p$ are electron and hole concentrations, and $\mu_n$ and $\mu_p$ are their respective mobilities.

Diffusion current

If carriers are unevenly distributed in space, they naturally spread from high-concentration regions to low-concentration regions. This produces a diffusion current:

$J_{\text{diff}} = qD_n\frac{dn}{dx} - qD_p\frac{dp}{dx}$

where $D_n$ and $D_p$ are the diffusion coefficients for electrons and holes. Note the sign difference: electrons diffuse down their concentration gradient (positive contribution for positive $dn/dx$), while the hole diffusion term carries a minus sign because holes are positive carriers diffusing down their own gradient.

The diffusion coefficient and mobility are related by the Einstein relation: $D = \frac{k_BT}{q}\mu$.

Carrier mobility

Mobility ($\mu$) quantifies how easily carriers move through the lattice under an electric field. It depends on:

• Temperature: Higher temperatures increase lattice vibrations (phonon scattering), which generally reduces mobility.
• Doping concentration: More ionized impurities means more Coulomb scattering, reducing mobility at high doping levels.
• Scattering mechanisms: Lattice (phonon) scattering and ionized impurity scattering are the two dominant mechanisms.

Typical values for silicon at room temperature with low doping:

• Electron mobility: $\mu_n \approx 1400 \text{ cm}^2/\text{Vs}$
• Hole mobility: $\mu_p \approx 450 \text{ cm}^2/\text{Vs}$

Electrons are more mobile than holes in most semiconductors because of differences in the effective mass associated with the conduction and valence band curvatures.

Conductivity in intrinsic vs extrinsic semiconductors

Conductivity ($\sigma$) measures how well a material carries current:

• Intrinsic: $\sigma = qn_i(\mu_n + \mu_p)$
• N-type (extrinsic): $\sigma \approx qN_D\mu_n$
• P-type (extrinsic): $\sigma \approx qN_A\mu_p$

Even modest doping dramatically increases conductivity. For example, doping silicon to $N_D = 10^{16} \text{ cm}^{-3}$ increases the electron concentration by a factor of about $10^6$ compared to intrinsic silicon, boosting conductivity by a similar factor.

Hall effect in semiconductors

The Hall effect is a key experimental technique for characterizing semiconductors. Here's how it works:

1. Pass a current through a semiconductor sample.
2. Apply a magnetic field perpendicular to the current direction.
3. The magnetic force (Lorentz force) deflects carriers to one side, building up charge and creating a transverse electric field.
4. This transverse voltage is the Hall voltage ($V_H$).

The Hall coefficient is defined as:

$R_H = \frac{E_H}{JB}$

where $E_H$ is the Hall electric field, $J$ is the current density, and $B$ is the magnetic field.

From the Hall measurement you can determine:

• Carrier type: $R_H < 0$ for n-type, $R_H > 0$ for p-type.
• Carrier concentration: $|R_H| = 1/(nq)$ for a single carrier type.
• Mobility: Combine the Hall coefficient with a conductivity measurement: $\mu = |R_H| \sigma$.

Applications of semiconductors

The ability to control carrier type and concentration through doping is what makes the enormous variety of semiconductor devices possible. Below are the major device categories that build directly on the intrinsic/extrinsic semiconductor concepts covered above.

PN junction diodes

A p-n junction forms when p-type and n-type regions meet. At the interface, electrons and holes diffuse across and recombine, creating a depletion region with a built-in electric field.

• Forward bias (positive voltage on p-side): reduces the barrier, allowing current to flow.
• Reverse bias (positive voltage on n-side): widens the depletion region, blocking current (only a tiny leakage current flows).

This asymmetric behavior is called rectification and is the basis for diodes.

Bipolar junction transistors (BJTs)

A BJT consists of three doped regions forming two back-to-back p-n junctions (either npn or pnp). The three terminals are the emitter, base, and collector.

A small current injected into the thin, lightly doped base controls a much larger current flowing from emitter to collector. This current amplification makes BJTs useful as amplifiers and switches.

Field-effect transistors (FETs)

FETs control current through a semiconductor channel using an electric field applied via a gate electrode. The two main types are:

• JFETs: The gate is a reverse-biased p-n junction.
• MOSFETs: The gate is insulated from the channel by a thin oxide layer ($SiO_2$).

MOSFETs are the dominant transistor type in integrated circuits because of their low power consumption, high input impedance, and excellent scalability to nanometer dimensions.

Light-emitting diodes (LEDs)

LEDs are forward-biased p-n junctions made from direct band gap semiconductors. When electrons and holes recombine across the junction, they release energy as photons.

The color of the emitted light corresponds to the band gap energy: GaAs emits infrared (~1.4 eV), GaN emits blue/UV (~3.4 eV), and InGaN alloys can be tuned across the visible spectrum.

Solar cells and photovoltaics

Solar cells work as the reverse process of LEDs. Incoming photons with energy greater than the band gap generate electron-hole pairs. The built-in electric field at the p-n junction separates these carriers, driving electrons to the n-side and holes to the p-side, producing a photocurrent.

Common materials include crystalline silicon (most of the market), GaAs (high efficiency), and emerging perovskite materials.

Semiconductor lasers

Semiconductor lasers are similar to LEDs but use an optical cavity to achieve stimulated emission, producing coherent, monochromatic light. Materials like GaAs, InGaAs, and GaN are commonly used.

These lasers are compact and efficient, making them essential for fiber-optic communication, optical disc drives, barcode scanners, and laser pointers.

Integrated circuits and microelectronics

Integrated circuits (ICs) pack millions to billions of transistors, diodes, resistors, and capacitors onto a single silicon chip. Advances in lithography and fabrication have driven Moore's Law scaling, enabling:

• Microprocessors with billions of transistors
• Memory chips (DRAM, flash) with enormous storage density
• System-on-chip (SoC) designs combining processing, memory, and communication on one die

The entire IC industry rests on the ability to precisely dope different regions of a silicon wafer to create the n-type and p-type regions needed for each device.