Intrinsic and Extrinsic Semiconductors

1. Definition and Properties of Semiconductors

Definition and Properties of Semiconductors

Fundamental Definition

Semiconductors are materials with an electrical conductivity intermediate between conductors (e.g., metals) and insulators (e.g., ceramics). Their conductivity arises from the movement of charge carriers—electrons and holes—which can be modulated by external factors such as temperature, doping, or electric fields. The defining property of semiconductors is their bandgap (Eg), the energy difference between the valence band (VB) and conduction band (CB).

$$ E_g = E_{\text{CB}} - E_{\text{VB}} $$

Key Properties

Band Structure and Charge Transport

The band structure of a semiconductor determines its electronic properties. In an intrinsic (pure) semiconductor, the Fermi level (EF) lies near the middle of the bandgap. The intrinsic carrier concentration (ni) is given by:

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

where Nc and Nv are the effective densities of states in the conduction and valence bands, respectively, k is Boltzmann’s constant, and T is temperature.

Practical Applications

Semiconductors form the backbone of modern electronics, including:

Historical Context

The study of semiconductors accelerated in the mid-20th century with the invention of the transistor at Bell Labs (1947). Early theoretical work by Alan Wilson (1931) laid the foundation for band theory, while the development of silicon purification techniques enabled the semiconductor industry’s growth.

Band Theory: Valence and Conduction Bands

Energy Band Formation in Solids

In isolated atoms, electrons occupy discrete energy levels dictated by quantum mechanics. When atoms form a crystalline solid, their outer electrons interact due to the proximity of neighboring atoms. This interaction causes the discrete energy levels to split into closely spaced states, forming energy bands. The Pauli exclusion principle prevents electrons from occupying identical quantum states, leading to band separation.

Valence and Conduction Bands

The highest energy band containing electrons at absolute zero temperature is the valence band. Electrons in this band are bound to atoms and do not contribute to conduction. Above the valence band lies the conduction band, which is either empty or partially filled. The energy gap between these bands, known as the bandgap (Eg), determines the material's electrical properties.

$$ E_g = E_c - E_v $$

where Ec is the conduction band minimum and Ev is the valence band maximum.

Bandgap Classification

Fermi Level and Carrier Statistics

The Fermi level (EF) represents the energy at which the probability of electron occupation is 50%. In intrinsic semiconductors at thermal equilibrium, EF lies near the middle of the bandgap. The probability of an electron occupying a state at energy E is given by the Fermi-Dirac distribution:

$$ f(E) = \frac{1}{1 + e^{(E - E_F)/k_BT}} $$

where kB is the Boltzmann constant and T is temperature.

Effective Mass and Density of States

Electrons and holes in a crystal behave as if they have an effective mass (me*, mh*), differing from the free electron mass due to periodic lattice potentials. The density of states in the conduction and valence bands is:

$$ g_c(E) = \frac{(2m_e^*)^{3/2}}{2\pi^2\hbar^3} \sqrt{E - E_c} $$ $$ g_v(E) = \frac{(2m_h^*)^{3/2}}{2\pi^2\hbar^3} \sqrt{E_v - E} $$

Practical Implications

Band theory underpins semiconductor device operation. For example, in solar cells, photon absorption excites electrons from the valence to the conduction band, generating electron-hole pairs. In transistors, applied voltages modulate the Fermi level, controlling carrier density in the conduction band.

Conduction Band (Ec) Valence Band (Ev) Bandgap (Eg)

Energy Gap and Its Significance

Definition and Physical Interpretation

The energy gap (Eg) in semiconductors is the minimum energy required to excite an electron from the valence band to the conduction band. In intrinsic semiconductors, this gap is a fundamental property of the material, determined by its crystal structure and bonding. The energy gap can be expressed as:

$$ E_g = E_c - E_v $$

where Ec is the conduction band minimum and Ev is the valence band maximum. The magnitude of Eg dictates whether a material behaves as an insulator (Eg > 3 eV), semiconductor (0.1 eV < Eg < 3 eV), or conductor (Eg ≈ 0).

Temperature Dependence of the Energy Gap

The energy gap is not constant but varies with temperature due to lattice vibrations (phonons) and electron-phonon interactions. For most semiconductors, Eg decreases with increasing temperature, empirically modeled by Varshni's equation:

$$ E_g(T) = E_g(0) - \frac{\alpha T^2}{T + \beta} $$

where Eg(0) is the gap at 0 K, and α, β are material-specific constants. For silicon, α ≈ 4.73 × 10−4 eV/K and β ≈ 636 K.

Optical and Electrical Implications

The energy gap directly influences a semiconductor's optical absorption and electrical conductivity:

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

where Nc and Nv are the effective densities of states in the conduction and valence bands, respectively.

Bandgap Engineering in Device Design

Modern semiconductor devices often use engineered materials (e.g., heterostructures, quantum wells) to tailor Eg for specific applications:

Measurement Techniques

Experimental methods to determine Eg include:

Practical Relevance

The energy gap is critical in selecting materials for:

2. Pure Semiconductor Materials

2.1 Pure Semiconductor Materials

Pure semiconductors, also known as intrinsic semiconductors, consist of a single element or compound with a highly ordered crystalline lattice structure. Silicon (Si) and germanium (Ge) are the most widely studied intrinsic semiconductors due to their tetrahedral covalent bonding and bandgap characteristics. At absolute zero (0 K), all valence electrons are bound in covalent bonds, leaving no free charge carriers. However, as temperature increases, thermal excitation promotes electrons from the valence band to the conduction band, generating electron-hole pairs.

Crystal Structure and Bonding

In intrinsic semiconductors, each atom forms four covalent bonds with neighboring atoms in a diamond cubic (Si, Ge) or zinc blende (GaAs) lattice. The sp³ hybridization results in a tetrahedral arrangement with a bond angle of 109.5°. The periodic potential of the lattice creates allowed energy bands separated by a forbidden gap (Eg). For silicon, Eg ≈ 1.12 eV at 300 K, while germanium has Eg ≈ 0.67 eV.

Carrier Concentration in Intrinsic Semiconductors

The intrinsic carrier concentration (ni) depends on temperature and bandgap energy. The equilibrium electron (n) and hole (p) concentrations are equal (n = p = ni). The mass-action law holds:

$$ n_i^2 = np $$

The intrinsic carrier density is derived from the density of states and Fermi-Dirac statistics:

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

where Nc and Nv are the effective density of states in the conduction and valence bands, respectively, k is the Boltzmann constant, and T is the temperature in Kelvin.

Temperature Dependence

The conductivity (σ) of an intrinsic semiconductor is given by:

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

where q is the electron charge, and μn and μp are the electron and hole mobilities, respectively. Since n and p increase exponentially with temperature, while mobility decreases due to lattice scattering, the net conductivity exhibits an Arrhenius-like behavior:

$$ \sigma \propto e^{-\frac{E_g}{2kT}} $$

Practical Considerations

Ultra-high-purity silicon (>99.9999999%) is essential for intrinsic behavior, as even trace impurities can dominate conduction. Float-zone (FZ) and Czochralski (CZ) growth methods produce single-crystal ingots with minimal defects. Intrinsic semiconductors are rarely used directly in devices but serve as the baseline for doped (extrinsic) materials in diodes, transistors, and integrated circuits.

The intrinsic Fermi level (Ei) lies near the middle of the bandgap:

$$ E_i = \frac{E_c + E_v}{2} + \frac{3kT}{4} \ln\left(\frac{m_p^*}{m_n^*}\right) $$

where Ec and Ev are the conduction and valence band edges, and mp* and mn* are the effective masses of holes and electrons, respectively.

Silicon Crystal Lattice and Band Diagram A combined diagram showing the tetrahedral atomic arrangement of silicon (left) and the corresponding energy band diagram (right), with labeled valence/conduction bands, Fermi level, and bandgap. Si sp³ sp³ sp³ Silicon Crystal Lattice Conduction Band (Ec) Valence Band (Ev) Eg Ei Energy Band Diagram e⁻ h⁺
Diagram Description: The section describes crystal lattice structures and bandgap diagrams, which are inherently spatial and require visual representation to clarify the arrangement of atoms and energy levels.

2.2 Electron-Hole Pair Generation

In an intrinsic semiconductor at absolute zero temperature, all valence electrons are bound in covalent bonds, leaving no free charge carriers. However, as temperature increases, thermal energy excites some electrons from the valence band to the conduction band, creating electron-hole pairs. This process is fundamental to semiconductor operation and is governed by quantum mechanical principles.

Thermal Generation of Electron-Hole Pairs

The probability of an electron gaining sufficient energy to cross the bandgap \(E_g\) follows Fermi-Dirac statistics. At equilibrium, the intrinsic carrier concentration \(n_i\) is given by:

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

where \(N_c\) and \(N_v\) are the effective density of states in the conduction and valence bands, respectively, \(k\) is the Boltzmann constant, and \(T\) is the temperature in Kelvin. The product of electron and hole concentrations remains constant under thermal equilibrium:

$$ n_0 p_0 = n_i^2 $$

Direct vs. Indirect Bandgap Transitions

In direct bandgap semiconductors (e.g., GaAs), electrons transition directly from the valence band maximum to the conduction band minimum without a change in crystal momentum. In indirect bandgap materials (e.g., Si, Ge), phonon assistance is required to conserve momentum, making generation less efficient.

Recombination Mechanisms

Electron-hole pairs recombine through:

Impact of Doping on Carrier Generation

In extrinsic semiconductors, doping introduces additional carriers:

The modified carrier concentrations are:

$$ n \approx N_d \quad \text{(n-type)}, \quad p \approx N_a \quad \text{(p-type)} $$

where \(N_d\) and \(N_a\) are donor and acceptor concentrations, respectively.

Non-Equilibrium Carrier Generation

External energy sources (e.g., light, electric fields) can create excess carriers (\(\Delta n\), \(\Delta p\)). The continuity equation describes their dynamics:

$$ \frac{\partial \Delta n}{\partial t} = G - R + \frac{1}{q} \nabla \cdot \mathbf{J}_n $$

where \(G\) is the generation rate, \(R\) is the recombination rate, and \(\mathbf{J}_n\) is the electron current density.

In optoelectronic devices (e.g., photodiodes, solar cells), photon absorption generates electron-hole pairs with a quantum efficiency \(\eta\) dependent on the material's absorption coefficient \(\alpha\):

$$ \eta = 1 - e^{-\alpha d} $$

where \(d\) is the device thickness.

Conduction Band Valence Band e⁻ h⁺ E_g
Semiconductor Band Diagram with Electron-Hole Pair Generation Energy band diagram showing conduction and valence bands, electron-hole pair generation, and recombination mechanisms. Eg Conduction Band Valence Band e− h+ Generation Radiative Auger SRH
Diagram Description: The diagram would physically show the band structure of a semiconductor with labeled conduction and valence bands, illustrating electron-hole pair generation and recombination mechanisms.

2.3 Charge Carriers in Intrinsic Semiconductors

In intrinsic semiconductors, charge carriers arise solely due to thermal excitation of electrons from the valence band to the conduction band. The equilibrium concentrations of electrons (n) and holes (p) are equal, denoted as ni, the intrinsic carrier concentration. This process is governed by the semiconductor's bandgap energy (Eg) and temperature (T).

Thermal Generation of Electron-Hole Pairs

At absolute zero, all electrons reside in the valence band, leaving the conduction band empty. As temperature increases, some electrons gain sufficient energy to overcome the bandgap, creating electron-hole pairs. The probability of an electron occupying a state at energy E is given by the Fermi-Dirac distribution:

$$ f(E) = \frac{1}{1 + e^{(E - E_F)/kT}} $$

where EF is the Fermi level, k is Boltzmann's constant, and T is temperature. In intrinsic semiconductors, EF lies near the middle of the bandgap.

Intrinsic Carrier Concentration

The intrinsic carrier concentration ni is derived from the product of electron and hole concentrations, which depend on the effective densities of states in the conduction (NC) and valence (NV) bands:

$$ n_i^2 = np = N_C N_V e^{-E_g / kT} $$

Solving for ni yields:

$$ n_i = \sqrt{N_C N_V} e^{-E_g / 2kT} $$

For silicon at 300 K, ni ≈ 1.5 × 1010 cm−3, while germanium has a higher ni (≈ 2.4 × 1013 cm−3) due to its narrower bandgap.

Mobility and Conductivity

The conductivity (σ) of an intrinsic semiconductor depends on both carrier concentration and mobility (μ):

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

where μn and μp are electron and hole mobilities, respectively. Mobility decreases with temperature due to lattice scattering, while ni increases exponentially, leading to a net increase in conductivity at higher temperatures.

Practical Implications

Intrinsic behavior is critical in high-purity materials used in detectors and sensors. For instance, silicon radiation detectors operate by measuring electron-hole pairs generated by incident photons. However, intrinsic semiconductors are rarely used in devices due to their low conductivity—doping (extrinsic behavior) is typically employed to enhance performance.

Conduction Band Valence Band e⁻ h⁺ Eg

2.4 Temperature Dependence of Conductivity

Fundamental Temperature Effects in Semiconductors

The conductivity σ of a semiconductor depends on temperature through two primary mechanisms: the temperature dependence of carrier concentration (n or p) and the temperature dependence of carrier mobility (μn or μp). The overall conductivity can be expressed as:

$$ σ = q(nμ_n + pμ_p) $$

where q is the electronic charge. Both intrinsic and extrinsic semiconductors exhibit distinct temperature-dependent behaviors due to variations in these parameters.

Intrinsic Semiconductor Behavior

In intrinsic semiconductors, the carrier concentration follows an exponential relationship with temperature:

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

where Nc and Nv are the effective density of states in the conduction and valence bands respectively, Eg is the bandgap, and k is Boltzmann's constant. The temperature dependence of mobility in intrinsic semiconductors is primarily governed by lattice scattering:

$$ μ_L \propto T^{-3/2} $$

The net result is that intrinsic conductivity increases exponentially with temperature, as the exponential increase in carrier concentration dominates over the power-law decrease in mobility.

Extrinsic Semiconductor Behavior

Extrinsic semiconductors exhibit more complex temperature dependence, typically showing three distinct regions:

Mobility Considerations

At higher temperatures, additional scattering mechanisms become important:

$$ \frac{1}{μ} = \frac{1}{μ_L} + \frac{1}{μ_I} $$

where μL is lattice scattering mobility and μI is impurity scattering mobility. Impurity scattering becomes less significant at higher temperatures (μI ∝ T3/2), while lattice scattering dominates.

Practical Implications

This temperature dependence has critical implications for semiconductor device operation:

Mathematical Modeling

The complete temperature-dependent conductivity model combines all these effects. For an n-type semiconductor:

$$ σ(T) = q\left(N_d μ_n(T) + n_i(T)μ_n(T)\right) $$

where Nd is the donor concentration. The mobility term includes both lattice and impurity scattering components:

$$ μ_n(T) = μ_{min} + \frac{μ_0 T^{-3/2}}{1 + \left(\frac{T}{T_0}\right)^{1/2}} $$

This comprehensive model accurately predicts conductivity across the full temperature range for practical semiconductor devices.

Temperature Dependence of Semiconductor Conductivity A semi-log plot showing conductivity vs temperature for intrinsic and extrinsic semiconductors, with labeled regions (freeze-out, saturation, intrinsic) and mobility components. Temperature (T) Conductivity (σ) T₁ T₂ T₃ 10⁻⁴ 10⁻² 10⁰ 10² Extrinsic Intrinsic Freeze-out Saturation Intrinsic μ_L μ_I n_i(T) ∝ exp(-E_g/2kT)
Diagram Description: The section describes complex temperature-dependent behaviors with multiple regions and competing effects that would be clearer with a visual representation.

3. Doping: Introduction to Donor and Acceptor Impurities

3.1 Doping: Introduction to Donor and Acceptor Impurities

The electrical properties of semiconductors are fundamentally altered through doping, the deliberate introduction of impurities into an intrinsic semiconductor lattice. This process modifies charge carrier concentrations by introducing either excess electrons (n-type doping) or holes (p-type doping). The choice of dopant determines whether the semiconductor becomes electron-rich or hole-rich.

Donor Impurities (n-Type Doping)

Donor impurities are atoms with more valence electrons than the host semiconductor material. When incorporated into a silicon (Si) or germanium (Ge) lattice, these impurities donate excess electrons to the conduction band. Common donor dopants for silicon include:

The ionization energy required to release the fifth electron into the conduction band is typically small (~0.05 eV for Si), making these impurities nearly fully ionized at room temperature. The electron concentration in an n-type semiconductor is given by:

$$ n_n \approx N_D $$

where \( n_n \) is the electron concentration and \( N_D \) is the donor impurity density.

Acceptor Impurities (p-Type Doping)

Acceptor impurities have fewer valence electrons than the host semiconductor. When introduced into the lattice, they create holes by accepting electrons from the valence band. Common acceptor dopants include:

The ionization energy for hole creation is similarly small (~0.05 eV for Si), ensuring high hole concentrations at room temperature. The hole concentration in a p-type semiconductor is:

$$ p_p \approx N_A $$

where \( p_p \) is the hole concentration and \( N_A \) is the acceptor impurity density.

Charge Neutrality and Doping Concentration

In a doped semiconductor, charge neutrality must be maintained. For an n-type material, the positive charge from ionized donors balances the negative charge from conduction electrons:

$$ n_n = N_D^+ $$

Similarly, for p-type material, the negative charge from ionized acceptors balances the positive holes:

$$ p_p = N_A^- $$

In compensated semiconductors where both donor and acceptor impurities exist, the net carrier concentration depends on the difference between donor and acceptor densities:

$$ n = N_D - N_A \quad (\text{if } N_D > N_A) $$ $$ p = N_A - N_D \quad (\text{if } N_A > N_D) $$

Practical Considerations in Doping

Doping techniques include:

Modern semiconductor devices rely on controlled doping profiles to achieve desired electrical characteristics, such as in transistors, diodes, and solar cells. The ability to tailor carrier concentrations through doping enables the precise engineering of device performance.

Acceptor (B) Donor (P) 3 Valence e⁻ 5 Valence e⁻ Si Lattice
Donor and Acceptor Impurities in Silicon Lattice Atomic schematic of n-type (donor) and p-type (acceptor) doping in a silicon lattice, showing valence electrons and charge states. N-Type (Donor) Si Si Si Si P Conduction e⁻ Ionized Donor (P⁺) P-Type (Acceptor) Si Si Si Si B Hole Ionized Acceptor (B⁻) Legend Silicon Atom (4 valence e⁻) Phosphorus Donor (5 valence e⁻) Boron Acceptor (3 valence e⁻) Valence Electron Conduction Electron Hole
Diagram Description: The diagram would physically show the atomic structure of donor and acceptor impurities within a silicon lattice, highlighting their valence electrons and charge states.

3.2 N-Type Semiconductors: Properties and Behavior

Doping Mechanism and Charge Carriers

N-type semiconductors are formed by doping an intrinsic semiconductor (typically silicon or germanium) with pentavalent impurities such as phosphorus, arsenic, or antimony. These donor atoms introduce additional electrons into the conduction band, significantly increasing the material's conductivity. The doping process can be quantified by the donor concentration \( N_d \), which determines the majority carrier density.

$$ n_n \approx N_d $$

where \( n_n \) is the electron concentration in the n-type material. The minority carrier (hole) concentration \( p_n \) is given by the mass-action law:

$$ p_n = \frac{n_i^2}{N_d} $$

where \( n_i \) is the intrinsic carrier concentration.

Energy Band Structure

The introduction of donor impurities creates discrete energy levels just below the conduction band, known as donor levels (\( E_d \)). At room temperature, thermal excitation causes these donor electrons to transition into the conduction band, leaving behind positively ionized donor atoms. The Fermi level \( E_F \) in an n-type semiconductor shifts closer to the conduction band edge \( E_c \):

$$ E_F = E_c - kT \ln\left(\frac{N_c}{N_d}\right) $$

where \( N_c \) is the effective density of states in the conduction band, \( k \) is Boltzmann's constant, and \( T \) is temperature.

Conductivity and Mobility

The conductivity \( \sigma \) of an n-type semiconductor is dominated by electron mobility \( \mu_n \):

$$ \sigma = q n_n \mu_n $$

where \( q \) is the electron charge. Mobility is influenced by lattice scattering (dominant at high temperatures) and impurity scattering (dominant at low temperatures). The temperature dependence of mobility follows:

$$ \mu_n \propto T^{-3/2} \quad \text{(lattice scattering)} $$ $$ \mu_n \propto T^{3/2} \quad \text{(impurity scattering)} $$

Practical Applications

N-type semiconductors are foundational in modern electronics, including:

Temperature Effects

At low temperatures, carrier freeze-out occurs as electrons remain bound to donor atoms. As temperature increases, ionization raises conductivity until intrinsic carriers dominate at very high temperatures, reversing the n-type behavior. The critical temperature \( T_{\text{transition}} \) where intrinsic carriers overtake extrinsic ones is given by:

$$ n_i(T) = N_d $$

This transition is avoided in most devices by operating below 150–200°C for silicon.

N-Type Semiconductor Energy Band Diagram Energy band diagram of an N-type semiconductor showing conduction band, valence band, donor levels (Ed), Fermi level (EF), and ionized donor atoms. Energy Ec Ev Ed EF + + + N-Type Semiconductor Energy Band Diagram Conduction Band (Ec) Valence Band (Ev) Donor Level (Ed) Fermi Level (EF)
Diagram Description: The energy band structure and doping mechanism are highly visual concepts that require spatial representation of donor levels, conduction/valence bands, and Fermi level shifts.

3.3 P-Type Semiconductors: Properties and Behavior

Formation and Doping Mechanism

P-type semiconductors are created by doping an intrinsic semiconductor (typically silicon or germanium) with acceptor impurities, such as boron (B), aluminum (Al), or gallium (Ga). These impurities introduce holes in the valence band, acting as majority charge carriers. The doping process can be described by the following reaction in silicon:

$$ \text{Si} + \text{B} \rightarrow \text{Si}^- + \text{h}^+ $$

Here, boron (a Group III element) replaces a silicon atom, creating an electron-deficient site (hole). The ionization energy of boron in silicon is approximately 0.045 eV, making it easy for holes to form at room temperature.

Charge Carrier Concentration

In p-type semiconductors, the majority carriers are holes (p), while minority carriers are electrons (n). The equilibrium hole concentration (p0) is given by:

$$ p_0 = N_a $$

where Na is the acceptor doping concentration. The minority electron concentration (n0) is derived from the mass-action law:

$$ n_0 = \frac{n_i^2}{p_0} $$

where ni is the intrinsic carrier concentration.

Conductivity and Mobility

The conductivity (σ) of a p-type semiconductor is dominated by hole mobility (μp):

$$ \sigma = q p_0 \mu_p $$

where q is the electron charge. Hole mobility is lower than electron mobility due to the valence band's heavy effective mass. For silicon at 300 K, typical values are:

Fermi Level Position

The Fermi level (EF) in p-type semiconductors shifts toward the valence band edge (EV). For non-degenerate doping, its position is:

$$ E_F = E_V + kT \ln\left(\frac{N_a}{N_V}\right) $$

where NV is the valence band effective density of states (~1.04×10¹⁹ cm⁻³ for Si at 300 K). Under heavy doping (>10¹⁹ cm⁻³), the Fermi level may enter the valence band, leading to degenerate behavior.

Temperature Dependence

P-type semiconductors exhibit three distinct regimes:

Applications in Devices

P-type semiconductors are critical in:

For example, in a silicon PN diode, the p-side's hole concentration determines the forward-bias current via the Shockley diode equation:

$$ I = I_0 \left( e^{\frac{qV}{kT}} - 1 \right) $$

where I0 depends on p0 and the minority carrier diffusion length.

Energy Band Diagram of P-Type Semiconductor Illustration of the energy band structure of a p-type semiconductor, showing the conduction band, valence band, Fermi level, acceptor energy level, and holes. Energy (E) E_C (Conduction Band) E_F (Fermi Level) E_A (Acceptor Level) E_V (Valence Band) h+ (holes) Energy Gap (E_g) Acceptor Ionization
Diagram Description: The diagram would show the energy band structure of a p-type semiconductor, illustrating the Fermi level shift toward the valence band and the acceptor impurity states.

3.4 Majority and Minority Charge Carriers

In doped semiconductors, the equilibrium concentrations of electrons and holes are not equal. The dominant charge carriers are termed majority carriers, while the less abundant ones are called minority carriers. Their relative densities govern conductivity, recombination dynamics, and device behavior.

Carrier Concentrations in Extrinsic Semiconductors

For an n-type semiconductor doped with donor concentration ND, the majority carrier (electron) concentration at thermal equilibrium is approximately:

$$ n_n \approx N_D $$

where nn denotes electron density in the n-type material. The minority carrier (hole) concentration is derived from the mass-action law:

$$ p_n = \frac{n_i^2}{N_D} $$

Similarly, for a p-type semiconductor with acceptor doping NA:

$$ p_p \approx N_A $$ $$ n_p = \frac{n_i^2}{N_A} $$

Here, ni is the intrinsic carrier concentration, typically ~1.5×1010 cm-3 in silicon at 300K.

Non-Equilibrium Carrier Injection

Under external bias or optical excitation, minority carrier concentrations can exceed equilibrium values. The excess carrier density Δn (or Δp) follows the continuity equation:

$$ \frac{\partial \Delta n}{\partial t} = D_n \nabla^2 \Delta n - \frac{\Delta n}{\tau_n} + G_L $$

where Dn is the diffusion coefficient, τn the minority carrier lifetime, and GL the generation rate due to light.

Practical Implications

n-type p-type Majority (e⁻) Minority (h⁺) Minority (e⁻) Majority (h⁺)

Temperature Dependence

At high temperatures, intrinsic carrier concentration ni grows exponentially:

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

where Nc and Nv are effective density of states in the conduction and valence bands, respectively. This causes minority carrier concentrations to rise dramatically, eventually making extrinsic semiconductors behave intrinsically.

4. Conductivity Differences

4.1 Conductivity Differences

Fundamental Conductivity Mechanisms

The conductivity of a semiconductor arises from the movement of charge carriers—electrons in the conduction band and holes in the valence band. In intrinsic semiconductors, the carrier concentration is solely determined by thermal excitation across the bandgap, while extrinsic semiconductors exhibit enhanced conductivity due to deliberate doping with impurities.

The total conductivity σ is given by:

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

where q is the electronic charge, n and p are the electron and hole concentrations, and μn and μp are their respective mobilities.

Intrinsic Semiconductor Conductivity

In intrinsic semiconductors (e.g., pure silicon or germanium), the electron and hole concentrations are equal (n = p = ni), where ni is the intrinsic carrier density. The conductivity is thus:

$$ \sigma_i = q n_i (\mu_n + \mu_p) $$

The intrinsic carrier density ni depends exponentially on temperature and bandgap energy Eg:

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

where Nc and Nv are the effective density of states in the conduction and valence bands, respectively.

Extrinsic Semiconductor Conductivity

Extrinsic semiconductors are doped with donor (n-type) or acceptor (p-type) impurities, significantly altering their conductivity. For n-type semiconductors, the electron concentration dominates (n ≈ Nd), where Nd is the donor concentration. The conductivity becomes:

$$ \sigma_n \approx q N_d \mu_n $$

Similarly, for p-type semiconductors (p ≈ Na), where Na is the acceptor concentration:

$$ \sigma_p \approx q N_a \mu_p $$

Temperature Dependence

Conductivity in extrinsic semiconductors exhibits distinct temperature-dependent regimes:

Mobility Effects

Carrier mobility is influenced by scattering mechanisms:

The net mobility can be approximated using Matthiessen's rule:

$$ \frac{1}{\mu} = \frac{1}{\mu_{\text{lattice}}} + \frac{1}{\mu_{\text{impurity}}} $$

Practical Implications

Doping allows precise control over semiconductor conductivity, enabling the fabrication of devices with tailored electrical properties. For instance, heavily doped regions (Nd > 1018 cm-3) exhibit near-metallic conductivity, essential for ohmic contacts in integrated circuits.

Temperature Dependence of Extrinsic Semiconductor Conductivity A graph showing the conductivity of an extrinsic semiconductor as a function of temperature, with labeled regions: ionization, saturation, and intrinsic. Conductivity (σ) Temperature (T) 10³ 10⁴ 10⁵ 10⁶ 10⁷ T₁ T₂ T₃ Ionization Saturation Intrinsic N_d
Diagram Description: A diagram would visually show the temperature dependence of conductivity in extrinsic semiconductors across the three distinct regions (ionization, saturation, intrinsic).

4.2 Charge Carrier Concentrations

Intrinsic Carrier Concentration

In an intrinsic semiconductor, the charge carrier concentrations are governed by thermal excitation across the bandgap. The intrinsic carrier concentration ni is derived from the density of states in the conduction and valence bands and the Fermi-Dirac distribution. For a semiconductor with parabolic bands, the intrinsic carrier concentration is given by:

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

where Nc and Nv are the effective density of states in the conduction and valence bands respectively, Eg is the bandgap energy, k is Boltzmann's constant, and T is the absolute temperature. The temperature dependence of ni is crucial for understanding semiconductor behavior in different operating conditions.

Extrinsic Carrier Concentration

In extrinsic semiconductors, doping introduces additional charge carriers. For an n-type semiconductor doped with donor concentration Nd, the electron concentration n at room temperature (where nearly all donors are ionized) is:

$$ n \approx N_d $$

Similarly, for a p-type semiconductor with acceptor concentration Na, the hole concentration p is:

$$ p \approx N_a $$

The minority carrier concentrations are determined through the mass-action law:

$$ np = n_i^2 $$

Complete Ionization vs. Freeze-out

At different temperature regimes, the behavior of extrinsic semiconductors changes dramatically:

Non-degenerate vs. Degenerate Conditions

The standard equations assume non-degenerate conditions where the Fermi level is at least 3kT away from either band edge. In heavily doped (degenerate) semiconductors:

Practical Implications

Understanding charge carrier concentrations is critical for:

$$ \mu_n = \frac{q\tau_n}{m_n^*} $$

where μn is electron mobility, τn is the mean free time between collisions, and mn* is the effective mass of electrons. Similar relationships exist for holes.

Carrier Concentration vs. Temperature in Semiconductors A semi-log plot showing the relationship between carrier concentration and inverse temperature (1/T) in semiconductors, highlighting intrinsic, extrinsic, and freeze-out regions. 1/T (K⁻¹) log(n) Freeze-out Extrinsic Intrinsic N_d N_a n_i T₁ T₂
Diagram Description: The temperature-dependent behavior of carrier concentrations and the transition between complete ionization and freeze-out regions would be clearer with a visual representation.

4.3 Applications in Electronic Devices

Diodes and Rectifiers

Intrinsic and extrinsic semiconductors form the foundation of p-n junction diodes, where a p-type semiconductor is fused with an n-type semiconductor. The resulting depletion region creates a potential barrier that permits current flow predominantly in one direction. The current-voltage (I-V) characteristic of an ideal diode is given by the Shockley diode equation:

$$ I = I_0 \left( e^{\frac{qV}{nkT}} - 1 \right) $$

where I0 is the reverse saturation current, q is the electron charge, V is the applied voltage, n is the ideality factor, k is Boltzmann's constant, and T is the temperature. Practical applications include rectifiers in power supplies, where diodes convert AC to DC by allowing current flow only during the positive half-cycle.

Bipolar Junction Transistors (BJTs)

BJTs leverage extrinsic semiconductors to amplify signals. An npn transistor consists of a p-doped base sandwiched between two n-doped regions (emitter and collector). The emitter injects electrons into the base, where a small base current controls a larger collector current. The current gain β is defined as:

$$ \beta = \frac{I_C}{I_B} $$

BJTs are integral to analog circuits, such as operational amplifiers and radio frequency (RF) stages, due to their high gain and linearity. In digital circuits, they form the basis of transistor-transistor logic (TTL).

Field-Effect Transistors (FETs)

FETs, including MOSFETs, utilize an electric field to control conductivity in a channel. An n-channel MOSFET has a p-type substrate with two n+ doped regions (source and drain). A voltage applied to the gate creates an inversion layer, enabling current flow. The drain current ID in saturation is:

$$ I_D = \frac{\mu_n C_{ox}}{2} \frac{W}{L} (V_{GS} - V_{th})^2 $$

where μn is electron mobility, Cox is oxide capacitance, W/L is the width-to-length ratio, and Vth is the threshold voltage. FETs dominate digital integrated circuits (ICs) due to their scalability and low power consumption.

Optoelectronic Devices

Extrinsic semiconductors enable optoelectronic devices like light-emitting diodes (LEDs) and photodiodes. In LEDs, electron-hole recombination in direct bandgap materials (e.g., GaAs) emits photons. The emitted wavelength λ is:

$$ \lambda = \frac{hc}{E_g} $$

where h is Planck's constant, c is the speed of light, and Eg is the bandgap energy. Photodiodes operate inversely, converting light into current via the photovoltaic effect.

Integrated Circuits (ICs)

Monolithic ICs integrate thousands to billions of transistors on a single silicon chip. Doping profiles are precisely controlled to create NMOS and PMOS transistors in complementary MOS (CMOS) technology. CMOS logic gates, such as inverters, combine n-channel and p-channel MOSFETs to achieve low static power dissipation:

$$ P_{static} = I_{leakage} \cdot V_{DD} $$

where Ileakage is the subthreshold leakage current and VDD is the supply voltage. Modern microprocessors and memory chips rely on CMOS scaling to enhance performance and reduce feature sizes below 5 nm.

Power Electronics

Wide-bandgap semiconductors (e.g., SiC, GaN) exploit extrinsic doping for high-power applications. SiC Schottky diodes exhibit lower reverse recovery losses than silicon counterparts, enabling efficient high-frequency converters. The specific on-resistance Ron,sp of a power MOSFET scales with bandgap Eg:

$$ R_{on,sp} \propto E_g^{2.5} $$

These materials are critical for electric vehicle inverters and renewable energy systems, where high breakdown voltages and thermal stability are paramount.

5. Online Resources and Tutorials

5.2 Online Resources and Tutorials

5.3 Advanced Topics for Further Study