Unit 1: Semiconductors

Insulators, Semiconductors, and Metals

The electrical properties of solid materials are best understood using the Energy Band Theory. In an isolated atom, electrons occupy discrete energy levels. When atoms are brought together to form a solid crystal, these discrete levels broaden into continuous bands of allowed energy, separated by forbidden energy gaps.

• Valence Band (VB): The highest energy band that is completely filled with electrons at 0 Kelvin. Electrons in this band are involved in covalent bonding and are not free to move.
• Conduction Band (CB): The lowest energy band that is empty or partially filled at 0 Kelvin. Electrons in this band are free to move and contribute to electrical conduction.
• Forbidden Energy Gap (): The energy difference between the top of the valence band () and the bottom of the conduction band (). No allowed electron energy states can exist within this gap. .

Materials are classified based on the size of their energy gap:

1. Insulators

• Energy Gap: Very large ( eV, typically 5-10 eV).
• Band Structure: At room temperature, the valence band is completely full, and the conduction band is completely empty.
• Conduction: A very large amount of thermal or electrical energy is required to excite an electron from the valence band to the conduction band. Therefore, under normal conditions, they do not conduct electricity.
• Examples: Diamond ( eV), Silicon Dioxide (SiO₂, eV).

2. Semiconductors

• Energy Gap: Small, but finite ( to 3 eV).
• Band Structure: At 0 K, they behave like insulators. However, at room temperature, thermal energy is sufficient to excite a small number of electrons from the valence band to the conduction band.
• Conduction: When an electron moves to the CB, it becomes a free carrier. The vacancy it leaves behind in the VB is called a hole, which acts as a positive charge carrier. Conduction occurs due to the movement of both electrons and holes.
• Examples: Silicon (Si, eV), Germanium (Ge, eV), Gallium Arsenide (GaAs, eV).

3. Metals (Conductors)

• Energy Gap: Zero. The valence band and conduction band overlap ().
• Band Structure: The conduction band is partially filled with a very large number of free electrons even at 0 K.
• Conduction: A small applied electric field can easily impart energy to the free electrons, causing them to move and create a current.
• Examples: Copper (Cu), Aluminum (Al), Gold (Au).

Electrons and Holes in an Intrinsic Semiconductor

An intrinsic semiconductor is a semiconductor in its purest form, without any added impurities. Silicon (Si) and Germanium (Ge) are the most common examples.

• Crystal Structure: Si and Ge are Group IV elements, each having 4 valence electrons. They form a covalent bond with four neighboring atoms in a crystal lattice structure (diamond lattice).
• At 0 Kelvin: All valence electrons are tightly bound in covalent bonds. The valence band is full, the conduction band is empty. The semiconductor behaves as a perfect insulator.
• At T > 0 Kelvin: Thermal energy causes some covalent bonds to break.
  • An electron is liberated from the bond and becomes a free electron, moving into the conduction band.
  • The vacancy left behind in the covalent bond is called a hole.
• Electron-Hole Pair (EHP) Generation: This process of creating a free electron and a hole is called EHP generation. The energy required for this is at least the bandgap energy, .

Carrier Concentration in an Intrinsic Semiconductor

• In an intrinsic semiconductor, for every electron generated, a hole is also generated. Therefore, the concentration of free electrons () is equal to the concentration of holes ().

• This concentration is called the intrinsic carrier concentration, denoted by .

  TEXT

      n = p = n_i
      

• The intrinsic carrier concentration is highly dependent on temperature () and the bandgap energy ():

  TEXT

      n_i^2 = A_0 T^3 e^{-E_g / kT}
      

  
  or
  
  TEXT

      n_i = A T^{3/2} e^{-E_g / 2kT}
      

  
  Where:
  • A_0, A: Material-specific constants.
  • T: Absolute temperature in Kelvin.
  • E_g: Bandgap energy in eV.
  • k: Boltzmann constant ( eV/K).

  Key Takeaway: increases exponentially with temperature. For Silicon at room temperature (300 K), cm⁻³.

Donor and Acceptor Impurities (Doping)

The conductivity of intrinsic semiconductors is too low for most practical applications. Their conductivity can be dramatically and controllably increased by adding a small, measured amount of impurities, a process called doping. The resulting material is called an extrinsic semiconductor.

1. N-type Semiconductor (Donor Impurities)

• Dopant: A pentavalent impurity (Group V element) like Phosphorus (P), Arsenic (As), or Antimony (Sb) is added to an intrinsic semiconductor (like Si).
• Mechanism:
  • The pentavalent atom replaces a Si atom in the lattice.
  • Four of its five valence electrons form covalent bonds with the neighboring Si atoms.
  • The fifth electron is loosely bound to the parent atom. A very small amount of energy (e.g., ~0.045 eV for P in Si) is required to free this electron, making it a charge carrier in the conduction band.
  • Since this impurity atom donates a free electron, it is called a donor impurity.
  • The donor atom becomes a fixed positive ion () in the lattice.
• Energy Levels: Donor impurities introduce discrete energy levels () just below the conduction band (). This small energy gap () is easily overcome by thermal energy at room temperature.
• Majority/Minority Carriers:
  • The concentration of free electrons () becomes much greater than the concentration of holes ().
  • Electrons are the majority carriers.
  • Holes are the minority carriers.
  • The semiconductor is called n-type because the primary charge carriers are negative (electrons).

2. P-type Semiconductor (Acceptor Impurities)

• Dopant: A trivalent impurity (Group III element) like Boron (B), Gallium (Ga), or Indium (In) is added.
• Mechanism:
  • The trivalent atom replaces a Si atom.
  • Its three valence electrons form covalent bonds with three neighboring Si atoms.
  • This leaves one bond incomplete, creating a vacancy or a hole.
  • This hole can easily accept an electron from a neighboring covalent bond with very little energy, causing the hole to move through the crystal.
  • Since this impurity atom accepts an electron, it is called an acceptor impurity.
  • When the acceptor atom accepts an electron, it becomes a fixed negative ion () in the lattice.
• Energy Levels: Acceptor impurities introduce discrete energy levels () just above the valence band (). An electron from the valence band can easily be excited to this level, creating a hole in the VB.
• Majority/Minority Carriers:
  • The concentration of holes () becomes much greater than the concentration of electrons ().
  • Holes are the majority carriers.
  • Electrons are the minority carriers.
  • The semiconductor is called p-type because the primary charge carriers are positive (holes).

Charge Densities in a Semiconductor

1. The Law of Mass Action

In any semiconductor (intrinsic or extrinsic) under thermal equilibrium, the product of the electron concentration () and the hole concentration () is a constant for a given temperature. This constant is equal to the square of the intrinsic carrier concentration ().

n * p = n_i^2

• Significance: This law is fundamental. If doping increases the concentration of one type of carrier (e.g., electrons in n-type), the concentration of the other type of carrier (holes) must decrease to keep the product constant. This is due to an increased rate of recombination.

2. The Charge Neutrality Equation

A semiconductor crystal, as a whole, is electrically neutral. The total positive charge density must equal the total negative charge density.

• Positive Charges:
  • Holes ()
  • Ionized donor atoms ()
• Negative Charges:
  • Electrons ()
  • Ionized acceptor atoms ()

Assuming all impurity atoms are ionized (which is a very good approximation at room temperature), the charge neutrality equation is:

p + N_D = n + N_A

Where:

• N_D is the concentration of donor impurities.
• N_A is the concentration of acceptor impurities.

Carrier Concentrations in Extrinsic Semiconductors

By combining the Mass Action Law and the Charge Neutrality Equation, we can find the carrier concentrations.

For an N-type semiconductor (doped with donors, , ):

• Neutrality equation:
• Since provides a large number of electrons, . So we can approximate in the neutrality equation.
• Majority carrier concentration:

• Using Mass Action Law to find minority carriers:

  TEXT

      n_n \approx N_D
      p_n = n_i^2 / N_D
      

  
  (The subscript 'n' denotes n-type material).

For a P-type semiconductor (doped with acceptors, , ):

• Neutrality equation:
• Since provides a large number of holes, . So we can approximate .
• Majority carrier concentration:

• Using Mass Action Law to find minority carriers:

  TEXT

      p_p \approx N_A
      n_p = n_i^2 / N_A
      

  
  (The subscript 'p' denotes p-type material).

Fermi Level in a Semiconductor

The Fermi-Dirac distribution function, , gives the probability that an available energy state at energy will be occupied by an electron at a given temperature .

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

1. Fermi Level in an Intrinsic Semiconductor ()

• In an intrinsic semiconductor, . This means the probability of finding an electron in the CB is equal to the probability of finding a hole in the VB.

• Therefore, the Fermi level lies very close to the middle of the bandgap.

  TEXT

      E_{Fi} \approx (E_C + E_V) / 2
      

• A more precise formula accounts for the effective masses of electrons () and holes ():

  TEXT

      E_{Fi} = (E_C + E_V) / 2 + (3/4)kT * ln(m_p^* / m_n^*)
      

  
  Since and are often of similar magnitude, the second term is very small, and is considered to be at mid-gap for practical purposes.

2. Fermi Level in a Semiconductor with Impurities

Doping shifts the Fermi level.

a. N-type Semiconductor:

• Doping with donors increases the electron concentration () in the conduction band.
• According to the probability function, for the occupation probability to be high near the conduction band edge, the Fermi level must move closer to the conduction band.

• The position of the Fermi level () relative to the conduction band () is given by:

  TEXT

      E_C - E_F = kT * ln(N_C / N_D)
      

  
  or
  
  TEXT

      n = N_C * e^{-(E_C - E_F) / kT}
      

  
  Where N_C is the effective density of states in the conduction band.
  Conclusion: In an n-type semiconductor, is above and closer to .

b. P-type Semiconductor:

• Doping with acceptors increases the hole concentration () in the valence band. This means the probability of finding an electron in the valence band must decrease, creating more holes.
• For the probability of electron occupation to be low near the valence band edge, the Fermi level must move closer to the valence band.

• The position of the Fermi level () relative to the valence band () is given by:

  TEXT

      E_F - E_V = kT * ln(N_V / N_A)
      

  
  or
  
  TEXT

      p = N_V * e^{-(E_F - E_V) / kT}
      

  
  Where N_V is the effective density of states in the valence band.
  Conclusion: In a p-type semiconductor, is below and closer to .

Mobility and Conductivity

Carrier transport in semiconductors occurs through two main mechanisms: Drift and Diffusion.

1. Drift

Drift is the movement of charge carriers under the influence of an applied electric field ().

• In the absence of an electric field, carriers move randomly, and there is no net current.
• When an electric field is applied, it exerts a force () on the charge carriers, causing them to accelerate.
• This acceleration is interrupted by collisions with lattice atoms and impurities.

• The net effect is that carriers achieve an average velocity, called the drift velocity (), which is proportional to the electric field.

  TEXT

      v_d = \mu \mathcal{E}
      

• The constant of proportionality, , is called Mobility.

Mobility ()

• Definition: Mobility is a measure of how easily a charge carrier can move through a crystal lattice under the influence of an electric field.
• Units: cm²/V·s
• Formula:
• Electron and Hole Mobility: In general, electrons are lighter and move more easily than holes. Therefore, electron mobility () is typically greater than hole mobility ().
  • For Si: cm²/V·s, cm²/V·s.
• Factors Affecting Mobility:
  • Temperature: As temperature increases, lattice vibrations (phonons) increase, leading to more frequent collisions. This decreases mobility.
  • Doping Concentration: As impurity concentration increases, there are more ionized impurity centers for carriers to scatter off of. This decreases mobility.

Drift Current Density ()

• The flow of charge due to drift constitutes the drift current.
• Current density () is the current per unit cross-sectional area.

• The total drift current density is the sum of the contributions from both electrons and holes.

  TEXT

      J_{drift} = J_n + J_p = (n q v_{dn}) + (p q v_{dp})
      

  
  Substituting :
  
  TEXT

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

  
  Note: Both electron and hole currents add up, even though they move in opposite directions, because their charges are opposite.

2. Conductivity () and Resistivity ()

• Ohm's Law in point form is .
• Comparing this with the drift current equation, we get the expression for the conductivity of a semiconductor.

Conductivity of a Semiconductor ()

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

• Units: or Siemens/cm.
• Conductivity depends on the concentration and mobility of both carrier types.

Special Cases:

• Intrinsic:
• N-type: Since ,
• P-type: Since ,

Resistivity ()

Resistivity is the reciprocal of conductivity.

\rho = 1 / \sigma

• Units:

Diffusion and Lifetime

1. Diffusion

Diffusion is the net movement of particles from a region of higher concentration to a region of lower concentration. This process does not require an electric field; it occurs due to the random thermal motion of particles.

• Concentration Gradient: Diffusion occurs if there is a non-uniform concentration of carriers, i.e., a concentration gradient ( or ).
• Diffusion Current: This net movement of charge carriers constitutes a diffusion current.

Diffusion Current Density ()

• The diffusion current density is proportional to the concentration gradient.

• For Electrons: Electrons diffuse from high concentration to low concentration. Since electrons have a negative charge, the conventional current flows in the opposite direction of electron flow.

  TEXT

      J_{n,diff} = q D_n (dn/dx)
      

• For Holes: Holes diffuse from high concentration to low concentration. Since holes have a positive charge, the conventional current flows in the same direction as hole flow.

  TEXT

      J_{p,diff} = -q D_p (dp/dx)
      

  Where:

  • D_n, D_p are the Diffusion Constants or Diffusivity for electrons and holes, respectively.
  • Units of D: cm²/s.
  • The negative sign for holes indicates that current flows in the direction of decreasing hole concentration (i.e., opposite to the positive gradient).

Total Current in a Semiconductor

The total current is the sum of both drift and diffusion components for both carrier types.

J_n = J_{n,drift} + J_{n,diff} = nq\mu_n\mathcal{E} + qD_n(dn/dx)
J_p = J_{p,drift} + J_{p,diff} = pq\mu_p\mathcal{E} - qD_p(dp/dx)
J_{total} = J_n + J_p

2. Einstein Relation

There is a fundamental relationship between the diffusion constant () and mobility () of a charge carrier, as both phenomena are rooted in the random thermal motion and scattering of particles. This is the Einstein Relation:

D / \mu = kT / q = V_T

• V_T is the Thermal Voltage.
• At room temperature (T = 300 K), mV.

This shows that carriers with higher mobility also diffuse more readily.

3. Recombination and Lifetime

• Recombination: The process where a free electron from the conduction band falls back into a hole in the valence band, annihilating the electron-hole pair. Energy (equal to the bandgap) is released, typically as heat (phonons) or light (photons).
• Generation: The reverse process, where electron-hole pairs are created.
• In thermal equilibrium, the rate of generation () equals the rate of recombination ().
• If excess carriers are introduced (e.g., by shining light on the semiconductor), the carrier concentrations rise above their equilibrium values. The recombination rate will then exceed the generation rate until equilibrium is restored.

Carrier Lifetime ()

• Definition: The excess carrier lifetime (or simply lifetime) is the average time an excess electron or hole exists before it recombines.
• It is denoted by for electrons and for holes.

• If is the excess electron concentration above the equilibrium value, the recombination rate is given by:

  TEXT

      R = \delta n / \tau_n
      

• Lifetime is a critical parameter in devices like solar cells, LEDs, and bipolar transistors, as it determines how far minority carriers can diffuse before they recombine. This distance is called the diffusion length ().