1.5 Intrinsic and extrinsic semiconductors

Intrinsic semiconductors

Pure semiconductor materials, with no intentional doping or impurities, are called intrinsic semiconductors. At absolute zero, they behave as insulators because the valence band is completely filled and the conduction band is empty. As temperature rises, some electrons gain enough thermal energy to jump across the bandgap into the conduction band, leaving behind holes. Each jump creates one electron-hole pair.

Energy band structure

Intrinsic semiconductors have two key energy bands separated by a bandgap ($E_g$):

• The valence band is the highest energy band that's fully occupied at absolute zero.
• The conduction band is the lowest energy band that's empty at absolute zero.
• The bandgap is the minimum energy an electron needs to move from the valence band into the conduction band.

Silicon (Si) has a bandgap of 1.12 eV, while germanium (Ge) has a smaller bandgap of 0.67 eV. The smaller the bandgap, the easier it is for thermal energy to promote electrons into the conduction band.

Electron and hole concentrations

In an intrinsic semiconductor, every electron that reaches the conduction band leaves behind exactly one hole. That means the electron concentration equals the hole concentration:

$n_i = p_i = \sqrt{N_c N_v} \exp(-E_g / 2k_BT)$

• $N_c$ and $N_v$ are the effective density of states in the conduction and valence bands
• $k_B$ is the Boltzmann constant
• $T$ is the absolute temperature

For silicon at room temperature (300 K), the intrinsic carrier concentration is approximately $1.5 \times 10^{10} \, cm^{-3}$. That sounds like a lot, but compared to the number of atoms in silicon (~$5 \times 10^{22} \, cm^{-3}$), only a tiny fraction of electrons are free carriers.

Fermi level position

The Fermi level ($E_F$) is the energy at which the probability of finding an electron is exactly 50%. In an intrinsic semiconductor, it sits near the middle of the bandgap:

$E_F = \frac{E_c + E_v}{2} + \frac{k_BT}{2} \ln\left(\frac{N_v}{N_c}\right)$

Here $E_c$ and $E_v$ are the conduction and valence band edges. If $N_c$ and $N_v$ were exactly equal, the Fermi level would land precisely at midgap. In practice, the slight difference between $N_c$ and $N_v$ shifts it a small amount, but it remains very close to the center.

Temperature dependence

Temperature has a dramatic effect on intrinsic carrier concentration because of the exponential term in the equation. The full temperature dependence is:

$n_i(T) \propto T^{3/2} \exp(-E_g / 2k_BT)$

The $T^{3/2}$ factor comes from the temperature dependence of the effective density of states, but the exponential term dominates. Even a modest temperature increase can cause a large jump in carrier concentration. This is why semiconductor devices can behave very differently at elevated temperatures.

Carrier transport mechanisms

Charge carriers in intrinsic semiconductors move through two mechanisms:

• Drift occurs when an external electric field pushes electrons and holes in opposite directions. The drift velocity is proportional to the field strength and the carrier's mobility.
• Diffusion occurs when carriers move from regions of high concentration to low concentration due to random thermal motion, even without an applied field.

The total current density combines both contributions:

$J_{total} = J_{drift} + J_{diffusion}$

The Einstein relation connects diffusion and mobility:

$\frac{D}{\mu} = \frac{k_BT}{q}$

where $D$ is the diffusion coefficient, $\mu$ is the carrier mobility, and $q$ is the elementary charge. This relation is useful because if you know the mobility, you can immediately find the diffusion coefficient (and vice versa).

Extrinsic semiconductors

Extrinsic semiconductors are created by intentionally adding impurity atoms (dopants) into an intrinsic semiconductor. Doping gives you precise control over carrier type and concentration, which is what makes semiconductor devices possible. The two types are n-type and p-type.

n-type vs p-type doping

• n-type doping uses impurity atoms with one more valence electron than the host. For silicon (Group IV), a common n-type dopant is phosphorus (Group V). The extra electron from each phosphorus atom is loosely bound and easily enters the conduction band, so electrons become the dominant carriers.
• p-type doping uses impurity atoms with one fewer valence electron than the host. Boron (Group III) in silicon is a classic example. Each boron atom creates a "missing bond" that acts as a hole, so holes become the dominant carriers.

Donor and acceptor impurities

Donor impurities (like phosphorus, arsenic, or antimony in silicon) "donate" electrons to the conduction band. They introduce energy levels just below the conduction band edge, so very little thermal energy is needed to ionize them at room temperature.

Acceptor impurities (like boron, aluminum, or gallium in silicon) "accept" electrons from the valence band. They introduce energy levels just above the valence band edge. When a valence electron fills one of these levels, it leaves a hole behind in the valence band.

Majority and minority carriers

Every semiconductor has both electrons and holes, but doping makes one type far more abundant:

• In n-type: electrons are the majority carriers, holes are the minority carriers ($n \gg n_i$)
• In p-type: holes are the majority carriers, electrons are the minority carriers ($p \gg n_i$)

The minority carriers are few in number, but they play a critical role in devices like diodes and transistors.

Fermi level shifts

Doping moves the Fermi level away from midgap:

• In n-type semiconductors, the Fermi level shifts toward the conduction band because the added electrons populate states near the conduction band edge.
• In p-type semiconductors, the Fermi level shifts toward the valence band because the added holes correspond to empty states near the valence band edge.

Higher doping concentrations push the Fermi level closer to the respective band edge. Temperature also affects the shift, since at very high temperatures the intrinsic carriers can overwhelm the dopant contribution.

Carrier concentration control

The majority carrier concentration is approximately equal to the dopant concentration (assuming full ionization at room temperature):

• n-type: $n \approx N_D$
• p-type: $p \approx N_A$

where $N_D$ is the donor concentration and $N_A$ is the acceptor concentration.

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

$np = n_i^2$

So for an n-type semiconductor, $p = n_i^2 / N_D$. This tells you that increasing the majority carrier concentration actually decreases the minority carrier concentration.

Compensation doping

When both donor and acceptor impurities are present in the same semiconductor, they partially cancel each other out. This is called compensation doping.

• 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 $N_A - N_D$.
• If $N_D = N_A$, the material behaves like an intrinsic semiconductor.

Compensation doping is sometimes used deliberately to fine-tune electrical properties, but it also reduces carrier mobility because the total number of ionized impurities (which scatter carriers) is $N_D + N_A$, not just the net concentration.

Electrical properties

The electrical behavior of semiconductors depends on carrier concentrations, mobilities, temperature, and scattering. Understanding these properties is essential for predicting how devices will perform.

Conductivity and resistivity

Conductivity ($\sigma$) measures how well a material carries current. In a semiconductor, both electrons and holes contribute:

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

where $\mu_n$ and $\mu_p$ are the electron and hole mobilities. Resistivity is simply the inverse:

$\rho = 1/\sigma$

Extrinsic semiconductors have much higher conductivity than intrinsic ones because doping increases the carrier concentration by many orders of magnitude. For example, doping silicon with $10^{16} \, cm^{-3}$ phosphorus atoms increases the electron concentration by a factor of about $10^6$ compared to intrinsic silicon.

Carrier mobility

Carrier mobility ($\mu$) describes how quickly carriers move through the material per unit electric field. It depends on the carrier's effective mass and how often it gets scattered.

In silicon at room temperature:

• Electron mobility: ~1,400 $cm^2/V \cdot s$
• Hole mobility: ~450 $cm^2/V \cdot s$

Electrons are roughly 3x more mobile than holes in silicon because electrons have a smaller effective mass. This difference matters for device design: n-channel MOSFETs are faster than p-channel ones for this reason.

Hall effect measurements

The Hall effect is a standard experimental technique for determining carrier type, concentration, and mobility in a semiconductor.

1. Pass a current ($J_x$) through the semiconductor sample.
2. Apply a magnetic field ($B_z$) perpendicular to the current.
3. The magnetic force deflects carriers to one side, building up a transverse voltage called the Hall voltage.
4. Measure the Hall voltage to determine the Hall coefficient:

$R_H = \frac{1}{nq}$ (n-type) or $R_H = \frac{-1}{pq}$ (p-type)

The sign of the Hall voltage tells you whether the material is n-type or p-type, and the magnitude gives you the carrier concentration. Combined with a resistivity measurement, you can also extract the mobility.

Temperature dependence

The electrical properties of semiconductors change significantly with temperature:

• Intrinsic semiconductors: Conductivity increases with temperature because carrier concentration grows exponentially.
• Extrinsic semiconductors go through three regimes as temperature rises:
  1. Freeze-out regime (low T): Not all dopants are ionized yet, so carrier concentration is below $N_D$ or $N_A$.
  2. Extrinsic regime (moderate T): Essentially all dopants are ionized. Carrier concentration is roughly constant at the doping level.
  3. Intrinsic regime (high T): Thermally generated carriers overwhelm the dopant contribution, and the material behaves more like an intrinsic semiconductor.
• Mobility generally decreases with increasing temperature because lattice vibrations (phonons) scatter carriers more frequently.

Scattering mechanisms

Scattering limits how fast carriers can move, directly reducing mobility. The main mechanisms are:

• Lattice (phonon) scattering: Carriers interact with thermal vibrations of the crystal lattice. Dominates at higher temperatures because phonon density increases with $T$.
• Ionized impurity scattering: Carriers are deflected by the Coulomb potential of ionized dopant atoms. More significant at low temperatures (where phonon scattering is weak) and at high doping concentrations.
• Neutral impurity scattering: Carriers interact with un-ionized impurity atoms. Generally a minor effect compared to the other two.

At any given temperature, the total mobility is determined by all active scattering mechanisms combined, following Matthiessen's rule: $\frac{1}{\mu_{total}} = \frac{1}{\mu_{lattice}} + \frac{1}{\mu_{impurity}} + \cdots$

Optical properties

The way semiconductors interact with light is directly tied to their band structure. These optical properties are the foundation of optoelectronic devices like solar cells, LEDs, and lasers.

Absorption and emission

Absorption happens when a photon with energy $\geq E_g$ is absorbed, promoting an electron from the valence band to the conduction band and creating an electron-hole pair. The absorption coefficient ($\alpha$) describes how strongly a material absorbs at a given photon energy.

Emission is the reverse: an electron in the conduction band recombines with a hole, releasing a photon with energy approximately equal to the bandgap. Emission can be:

• Spontaneous: Random recombination events (the basis of LEDs)
• Stimulated: A photon triggers recombination, producing a second identical photon (the basis of lasers)

Direct vs indirect bandgaps

This distinction has huge practical consequences:

• Direct bandgap semiconductors (e.g., GaAs, GaN): The conduction band minimum and valence band maximum occur at the same crystal momentum (same point in k-space). An electron can recombine with a hole by simply emitting a photon. This makes absorption and emission efficient.
• Indirect bandgap semiconductors (e.g., Si, Ge): The conduction band minimum and valence band maximum are at different crystal momenta. A phonon must be involved alongside the photon to conserve momentum. This extra requirement makes radiative transitions much less likely.

This is why GaAs is used for LEDs and laser diodes, while silicon (despite being the dominant semiconductor) is a poor light emitter.

Photoconductivity

Photoconductivity is the increase in a semiconductor's conductivity when light shines on it. Absorbed photons generate extra electron-hole pairs, increasing the free carrier concentration.

The change in conductivity is:

$\Delta\sigma = q(\mu_n + \mu_p)G\tau$

where $G$ is the carrier generation rate (proportional to light intensity) and $\tau$ is the carrier lifetime (how long carriers survive before recombining). This effect is the operating principle behind photodetectors and contributes to current generation in solar cells.

Luminescence mechanisms

Luminescence is light emission from a semiconductor due to electron-hole recombination. Several pathways exist:

• Band-to-band recombination: An electron drops directly from the conduction band to the valence band, emitting a photon at the bandgap energy.
• Excitonic recombination: A bound electron-hole pair (exciton) recombines. The emitted photon has slightly less energy than the bandgap because of the Coulomb binding energy between the electron and hole.
• Impurity-related recombination: Electrons and holes recombine through energy levels introduced by impurities or crystal defects. These transitions produce photons with energies below the bandgap.

Quantum efficiency

Quantum efficiency measures how effectively a semiconductor converts photons to carriers (in detectors) or carriers to photons (in emitters).

• External quantum efficiency (EQE): Ratio of collected carriers (or emitted photons) to incident photons. Accounts for all losses including reflection.
• Internal quantum efficiency (IQE): Ratio of collected carriers (or emitted photons) to absorbed photons. Only counts photons that actually enter the material.

IQE is always higher than EQE because it excludes reflection and other optical losses. Factors that reduce quantum efficiency include surface reflection, non-radiative recombination, and carrier trapping at defects.

Applications

The ability to control carrier type, concentration, and bandgap through doping and material selection makes semiconductors the foundation of modern electronics and optoelectronics.

Semiconductor devices

The core building blocks of semiconductor technology include:

• Diodes: Two-terminal devices built from a p-n junction. They allow current in one direction and block it in the other, useful for rectification and voltage regulation.
• Transistors: Three-terminal devices that amplify or switch signals. They're the basis of all digital logic and computing.
• Optoelectronic devices: Solar cells, LEDs, and lasers convert between light and electrical energy, relying on the optical properties discussed above.

Photovoltaic cells

Solar cells convert sunlight into electricity through the photovoltaic effect:

1. Photons with energy $\geq E_g$ are absorbed, generating electron-hole pairs.
2. The built-in electric field at a p-n junction separates the electrons and holes.
3. Electrons flow toward the n-side and holes toward the p-side, creating a photocurrent.
4. Connecting an external circuit allows this current to do useful work.

Solar cell efficiency depends on the bandgap (which determines how much of the solar spectrum can be absorbed), carrier mobility, recombination rates, and surface reflection losses. Silicon cells currently achieve commercial efficiencies of around 20-24%.

Light-emitting diodes (LEDs)

LEDs work by the reverse principle of solar cells: they convert electrical energy into light. When a forward voltage is applied across a p-n junction, electrons and holes are injected into the junction region where they recombine and emit photons. The color of the emitted light is determined by the bandgap of the semiconductor material. Direct bandgap materials like GaN (blue/UV), InGaN (green), and AlGaInP (red) are used because they emit light efficiently. LEDs are now widely used in displays, lighting, and optical communication.