In semiconductor physics, merely knowing the energy band structure is not sufficient to understand electrical behavior. We must also know how electrons are distributed among the available energy states. The concept that provides this information is the Fermi level.
The position of the Fermi level plays a decisive role in determining:
- Carrier concentration
- Type of semiconductor (intrinsic, n-type, or p-type)
- Electrical conductivity
In this chapter, we discuss the position and significance of the Fermi level in intrinsic and extrinsic semiconductors.
Meaning of Fermi Level
The Fermi level (EF) is defined as:
The energy level at which the probability of finding an electron is 50% at a given temperature.
Important points:
- Fermi level is a reference energy, not necessarily an actual energy state
- At absolute zero (0 K), it represents the highest occupied energy level
- At temperatures above 0 K, electrons are distributed according to Fermi–Dirac statistics
Physical Significance of Fermi Level in Semiconductors:
- Probability of Occupation: The Fermi level is defined using the Fermi-Dirac distribution function, representing the energy state where the probability of occupation is 50% (f(E) = 1/2).
- Indicator of Carrier Type (Doping):
- Intrinsic: lies near the middle of the bandgap, which indicates equal concentration of electrons and holes.
- N-type: shifts upward, closer to the conduction band, which indicates higher electron density.
- P-type: shifts downward, closer to the valence band, which indicates higher hole density.
- Temperature Dependence: The Fermi level represents the maximum energy that an electron can possess (the “sea of electrons” level). As the temperature increases, its position shifts, which reflects changes in the electron population.
- Charge Transport (Equilibrium): In a semiconductor device, the Fermi level must be constant throughout the material at thermodynamic equilibrium. If there is any difference in Fermi levels between materials, it indicates a potential difference or nonequilibrium state between the materials.
Fermi Level in Intrinsic Semiconductor
Characteristics of Intrinsic Semiconductor
An intrinsic semiconductor is:
- Pure (no intentional impurities)
- Conduction occurs due to thermally generated electron–hole pairs
- Electron concentration equals hole concentration
We can find the Fermi energy value for an intrinsic semiconductor as:
Since n = p = ni
Hence, using the electron and hole concentrations in an intrinsic semiconductor:
$$N_Cexp\left[ -\left( \frac{E_C-E_F}{kT} \right) \right] = N_Vexp\left[ -\left( \frac{E_F-E_V}{kT} \right) \right]$$
By solving the above equation, we get
$$E_F=\frac{E_C+E_V}{2}+\frac{kT}{2}log_e\left(\frac{N_V}{N_C} \right)$$
But, $$N_C=2\left( \frac{2\pi m_e^*kT}{h^2} \right)^{3/2}$$
and $$N_V=2\left( \frac{2\pi m_h^*kT}{h^2} \right)^{3/2}$$
Hence, $$E_F=\frac{E_C+E_V}{2}+\frac{3kT}{4}log_e\left(\frac{m_h^*}{m_e^*} \right)$$
Position of Fermi Level
In an intrinsic semiconductor:
- The valence band is completely filled at 0 K
- The conduction band is empty at 0 K
- A small forbidden energy gap exists between the valence band and conduction band.
The Fermi level lies approximately at the middle of the forbidden energy gap.
Where $$E_F\approx \frac{E_C+E_V}{2}$$
EC = bottom of conduction band
EV = top of valence band
Physical Significance
- Mid-gap position indicates an equal probability of electrons and holes
- Explains why electron and hole concentrations are equal
- Conductivity is low at room temperature but increases with temperature
Energy Band Diagram (Intrinsic Semiconductor)
Fermi Level in Extrinsic Semiconductor
Doping introduces impurity energy levels, which shift the Fermi level from its intrinsic position.
Extrinsic semiconductors are of two types:
- n-type
- p-type
Fermi Level in n-Type Semiconductor
Characteristics of n-Type Semiconductor
- Doped with pentavalent (donor) impurities
- Extra electrons are available for conduction
- Electron concentration is much greater than hole concentration
n >> p
Position of Fermi Level
In an n-type semiconductor:
- Donor energy level lies just below the conduction band
- Fermi level shifts upward, closer to the conduction band.
- At a certain temperature T, the Fermi energy is given by
- $$E_F=\frac{E_D+E_C}{2}+\frac{kT}{2}log_e\left({\frac{N_D}{N_C}}\right)$$
When donor concentration increases, the Fermi level moves closer to the conduction band.
Physical Explanation
- High electron concentration increases the probability of occupancy near the conduction band
- Hence, the Fermi level shifts upward
- This shift explains the high conductivity of n-type semiconductors
Energy Band Diagram (n-Type Semiconductor)
Fermi Level in p-Type Semiconductor
Characteristics of p-Type Semiconductor
- Doped with trivalent (acceptor) impurities
- A large number of holes are available.
p >> n
Position of Fermi Level
In a p-type semiconductor:
- The acceptor energy level lies just above the valence band
- Fermi level shifts downward, closer to the valence band.
- At a certain temperature T, the Fermi energy is given by
$$E_F=\frac{E_A+E_V}{2}+\frac{kT}{2}log_e\left({\frac{N_V}{N_A}}\right)$$
Higher acceptor concentration moves the Fermi level even closer to the valence band.
Physical Explanation
- Increased hole concentration increases the probability of empty states near the valence band
- Fermi level shifts downward
- Explains the dominance of hole conduction
Energy Band Diagram (p-Type Semiconductor)
Comparison of Fermi Level Positions
| Semiconductor Type | Position of Fermi Level |
|---|---|
|
Intrinsic |
Midway between VB and CB |
|
n-Type |
Near conduction band |
|
p-Type |
Near valence band |
Important Examination Questions
Short Answer Questions
- Define Fermi level.
- Where does the Fermi level lie in an intrinsic semiconductor?
- How does doping affect the Fermi level?
Long Answer Questions
- Explain the position of the Fermi level in intrinsic, n-type, and p-type semiconductors with neat energy band diagrams.
- Discuss the effect of temperature on the Fermi level in extrinsic semiconductors.
⭐ Very frequently asked in university examinations
Conceptual Questions
- Why does the Fermi level shift after doping?
- Why is the Fermi level nearer to the conduction band in an n-type semiconductor?