Carrier Concentration in Semiconductor Physics: (n-type & p-type)-full explanation

 Introduction

In semiconductor physics, carrier concentration is one of the most important concepts. It refers to the number of charge carriers electrons and holes present in a semiconductor material per unit volume. These carriers are responsible for electrical conduction.

Semiconductors, such as silicon and germanium, have electrical properties that lie between conductors and insulators. Their conductivity depends strongly on the number of available charge carriers, which in turn depends on factors like temperature, impurity doping, and energy band structure.

Carrier concentration determines how well a semiconductor can conduct electricity and plays a key role in designing electronic devices such as diodes, transistors, and integrated circuits.

Types of Charge Carriers

In semiconductors, there are two types of charge carriers:

 Electrons

• Negatively charged particles
•  Located in the conduction band
•  Responsible for current flow in n-type semiconductors

Holes

•  Positively charged vacancies created when electrons leave the valence band
•  Located in the valence band
•  Responsible for conduction in p-type semiconductors

 Definition of Carrier Concentration

Carrier concentration is defined as:

• Electron concentration (n): Number of electrons per unit volume
• Hole concentration (p): Number of holes per unit volume

Units:

Carrier concentration = number of carriers per m³ or  cm³

Intrinsic Semiconductors

Definition

An intrinsic semiconductor is a pure semiconductor without any impurities.

Examples:

•  Silicon (Si)
•  Germanium (Ge)

 Carrier Generation

At absolute zero temperature:

•  No free carriers
• Acts as an insulator

At higher temperatures:

• Thermal energy breaks covalent bonds
•  Generates electron-hole pairs

 Intrinsic Carrier Concentration

In intrinsic semiconductors

n = p = n_i

Where:

 n_i = intrinsic carrier concentration

Expression for Intrinsic Carrier Concentration

The intrinsic carrier concentration is given by:

n_i = √N_c N_v e^-E_g /2kT

Where:

•  N_c = effective density of states in conduction band
• N_v  = effective density of states in valence band
• E_g  = energy band gap
•  k  = Boltzmann constant
•  T = temperature (Kelvin)

 Key Points

• n_i  increases with temperature
• Materials with smaller band gap have higher n_i
•  Example:

Germanium has higher carrier concentration than silicon

Extrinsic Semiconductors

Extrinsic semiconductors are formed by adding impurities (doping) to intrinsic semiconductors.

 Types

• n-type semiconductor
• p-type semiconductor

 Carrier Concentration in n-type Semiconductor

Doping

• Doped with pentavalent atoms (e.g., phosphorus, arsenic)
• Adds extra electrons

Majority and Minority Carriers

• Majority carriers: electrons
• Minority carriers: holes

 Electron Concentration

n ≈ N_d

Where:

•  N_d  = donor concentration

 Hole Concentration

Using mass action law:

p = n_i²/n

 Key Observations

•  Electron concentration is high
•  Hole concentration is very small
• Conductivity increases significantly

Carrier Concentration in p-type Semiconductor

Doping

• Doped with trivalent atoms (e.g., boron, aluminium)
•  Creates holes

 Majority and Minority Carriers

• Majority carriers: holes
• Minority carriers: electrons

Hole Concentration

P ≈ N_a

Where:

•  N_a = acceptor concentration

 Electron Concentration

n = n_i²/p

 Key Observations

• Hole concentration is high
•  Electron concentration is low

Mass Action Law

One of the most important relations in semiconductor physics:

  n . p = n_i²

Meaning

•  The product of electron and hole concentrations is constant at a given temperature
• Applies to both intrinsic and extrinsic semiconductors

Importance

• Helps calculate minority carrier concentration
• Essential in device analysis

 Temperature Dependence of Carrier Concentration

Carrier concentration strongly depends on temperature.

Low Temperature Region

•  Very few carriers
•  Semiconductor behaves like an insulator

 Moderate Temperature (Extrinsic Region)

• Carrier concentration depends on doping
• Nearly constant

 High Temperature (Intrinsic Region)

Intrinsic carriers dominate

 n ≈p  ≈ n_i

Conclusion

•  Increasing temperature increases carrier concentration
•  Due to increased thermal generation of electron-hole pairs

 Effective Density of States

 Conduction Band

N_c = 2 (2 π m_e^* kT/h²)^3/2

Valence Band

N_v = 2 (2 π m_h^* kT/h²)^3/2

Where:

•  m_e^* ,  m_h^*  = effective masses of electrons and holes
•  h  = Planck’s constant

Significance

•  Determines how many states are available for carriers
•  Influences intrinsic carrier concentration

 Fermi Level and Carrier Concentration

Definition

Fermi level is the energy level at which the probability of finding an electron is 50%.

Relation with Carrier Concentration

For electrons:

n = N_c e^-(E_c ^{- E}_F^)/kT

For holes:

p = N_v e^-(E_F^{– E}_V^)/kT

Where:

• E_F = Fermi energy
• E_c , E_v  = conduction and valence band energies

 Effect of Doping

•  In n-type: Fermi level moves closer to conduction band
•  In p-type: Fermi level moves closer to valence band

 Carrier Concentration and Conductivity

Electrical conductivity is given by:

  σ = q n μ_n + p μ_p

Where:

•  q  = charge of electron
• μ_n, μ_p = mobility of electrons and holes

Key Insight

• Higher carrier concentration → higher conductivity
•  Both carrier concentration and mobility affect current flow

Carrier Generation and Recombination

 Generation

• Creation of electron-hole pairs
•  Due to thermal energy or light

Recombination

• Electron recombines with a hole
•  Releases energy

Equilibrium Condition

Generation rate = Recombination rate

 Intrinsic Carrier Concentration Values

Typical values at room temperature:

 Silicon:

  n_i ≈ 1.5× 10¹⁰ cm^-3

Germanium:

  n_i ≈ 2.5 ×10¹³  cm^-3

Observation

•  Germanium has higher carrier concentration due to smaller band gap

 Importance of Carrier Concentration

Carrier concentration is crucial because:

 Device Performance

•  Determines current flow in diodes and transistors

Semiconductor Design

• Helps in selecting doping levels

Temperature Sensors

•  Used in thermistors

Solar Cells

• Affects efficiency of photovoltaic devices

 Practical Applications

Diodes

•  p-n junction behavior depends on carrier concentration

 Transistors

• Amplification depends on carrier movement

Integrated Circuits

•  Controlled doping creates complex circuits

 Sensors

•  Light and temperature sensors rely on carrier changes

Summary

Carrier concentration is a fundamental concept in semiconductor physics that defines the number of electrons and holes available for conduction.

Key points:

•  Intrinsic semiconductors: n = p = n_i
• Extrinsic semiconductors: majority and minority carriers
• Mass action law:  n p = n_i²
•  Depends strongly on temperature and doping
• Determines conductivity and device behavior

Understanding carrier concentration allows engineers and physicists to design efficient electronic devices and optimize semiconductor materials for various applications.