The Fermi level is one of the most important concepts in semiconductor physics because it determines the carrier concentration and hence the electrical conductivity of a semiconductor.
In semiconductors, the position of the Fermi level is not fixed — it varies depending on temperature and the type of doping applied. In an intrinsic semiconductor, the Fermi level lies approximately at the middle of the energy band gap.
However, when the semiconductor is doped to form n-type or p-type material, the Fermi level shifts toward the conduction band or valence band respectively. Understanding how the Fermi level shifts with these two factors is essential for explaining:
- Behaviour of n-type and p-type semiconductors
- Temperature regions of operation
- Performance of semiconductor devices
Variation of Fermi Level in Intrinsic Semiconductor:
Effect of Temperature on Fermi Level in Intrinsic Semiconductor
In intrinsic semiconductors, the number of electrons in the conduction band is equal to the number of holes in the valence band. Due to this balance, the Fermi level lies approximately at the center of the energy band gap at absolute zero temperature.
However, the position of the Fermi level is not completely fixed. With an increase in temperature, slight shifts may occur due to changes in carrier concentration and effective mass of electrons and holes.
The position of the intrinsic Fermi level is given by:
$$E_F=\frac{E_V+E_C}{2}+\frac{3kT}{4}log_e\left({\frac{m_h^*}{m_e^*}}\right)$$
where:
- me* = effective mass of electrons
- mh* = effective mass of holes
- k = Boltzmann constant
- T = absolute temperature
Interpretation of the Formula
- If T = 0k or mh* = me*, the second term becomes zero
→ Fermi level stays exactly at mid-gap. - If T > 0K and the hole’s effective mass is larger:
→ Fermi level shifts slightly toward the conduction band. - If T > 0 K and the electron effective mass is larger:
→ Fermi level shifts slightly toward the valence band
Variation of Fermi Level in Extrinsic Semiconductor:
Dependence of Fermi Level on Temperature
In an extrinsic semiconductor, the position of the Fermi level changes with temperature because the dominant type of charge carriers changes as temperature increases.
In n-type semiconductors
In the n-type semiconductor, EF is given by
$$E_F = \frac{E_D+E_C}{2}+\frac{kT}{2}log_e\left( \frac{N_D}{N_C} \right)$$
It is clear that at T 0K, $$E_F = \frac{E_D+E_C}{2}$$
Hence, at T = 0K, the Fermi level lies between the donor level and the bottom of the conduction band.
At very low temperatures, known as the depletion temperature region, most donor atoms are not ionized. Very few free electrons are present in the conduction band. In this region, the Fermi level lies close to the donor energy level, because electrons are still bound to donor atoms.
As the temperature increases to the extrinsic region, donor levels gradually get depleted, and almost all donor atoms become completely ionized. The electron concentration becomes approximately equal to the donor concentration (n≈ND). In this region, the Fermi level moves downward. At the depletion temperature (Td), the Fermi level coincides with the donor level.
When the temperature increases further and reaches the intrinsic region, thermal generation of electron–hole pairs becomes dominant over impurity carriers, and the Fermi level shifts downward approximately linearly.
At intrinsic temperature (Ti), the intrinsic carrier concentration becomes significant. The semiconductor loses extrinsic character and behaves like an intrinsic semiconductor.
As a result, the Fermi level moves away from the conduction band and shifts toward the middle of the forbidden energy gap, coinciding with the intrinsic Fermi level.
The variation of the Fermi level in n-type semiconductors with temperature is illustrated in the following figure:
In p type semiconductor
The behavior of a p-type semiconductor is similar, but the direction of movement of the Fermi level is opposite.
In the p-type semiconductor, EF is given by
$$E_F = \frac{E_V+E_A}{2}+\frac{kT}{2}log_e\left( \frac{N_V}{N_A} \right)$$
It is clear that at T 0K, $$E_F = \frac{E_V+E_A}{2}$$
Hence, at absolute zero temperature (T = 0K), acceptor atoms are not ionized, and no holes are present in the valence band. The Fermi level lies between the acceptor level and the valence band.
As the temperature increases slightly (freeze-out region), some acceptor atoms ionize and create a small number of holes in the valence band. Since ionization is incomplete, the Fermi level remains close to the acceptor level.
As temperature increases (extrinsic region), the acceptor level gradually fills, and the Fermi level moves upward. At saturation temperature (Ts), almost all acceptor atoms are fully ionized, and the hole concentration becomes approximately equal to the acceptor concentration (p≈NA); the Fermi level coincides with the acceptor level.
For T > Ts, the Fermi level shifts upward approximately linearly. At intrinsic temperature (Ti), the semiconductor behaves intrinsically, and the Fermi level approaches the intrinsic value.
The variation of the Fermi level in p-type semiconductors with temperature is illustrated in the following figure:
Dependence of Fermi Level on Impurity Concentration
When impurity atoms are added (doping), the carrier concentration changes significantly, and consequently, the position of the Fermi level also shifts.
Thus, the position of the Fermi level is not fixed — it depends strongly on:
Type of impurity (donor or acceptor)
Magnitude of impurity concentration
Temperature
In this section, we discuss how the Fermi level varies with impurity concentration in both n-type and p-type semiconductors.
In n-type semiconductors
Consider an n-type semiconductor doped with a donor concentration ND. In the extrinsic region (moderate temperature and non-degenerate doping), the electron concentration is approximately:
$$n\approx N_D$$
Substituting it in the electron concentration equation:
$$N_D= N_C exp\left[ -\left( \frac{E_C-E_F}{kT} \right) \right]$$
Taking natural logarithm:
$$E_C-E_F =kTlog_e\left( \frac{N_C}{N_D} \right)$$
Thus:
$$E_F =E_C-kTlog_e\left( \frac{N_C}{N_D} \right)$$
As donor concentration ND increases, ln(NC/ND) decreases
Hence, EC – EF decreases; therefore, the Fermi level moves closer to the conduction band
Hence, in an n-type semiconductor, the addition of donor impurities creates discrete donor levels below the conduction band. As impurity concentration increases, donor levels broaden and may form an impurity band. With heavy doping, this band may overlap with the conduction band, and the Fermi level shifts closer to or into the conduction band.
In p type semiconductor
For a p-type semiconductor with acceptor concentration NA:
$$p\approx N_A$$
Using the hole concentration equation:
$$N_A= N_V exp\left[ -\left( \frac{E_F-E_V}{kT} \right) \right]$$
Taking natural logarithm:
$$E_F -E_V=kTlog_e\left( \frac{N_V}{N_A} \right)$$
As acceptor concentration NA increases, ln(NV/NA) decreases
- Hence, EF – EV decreases
- Therefore, the Fermi level moves closer to the valence band
In a p-type semiconductor, acceptor levels broaden with increasing impurity concentration and may overlap with the valence band. The Fermi level shifts closer to the valence band and may enter it at very high doping levels.
The variation of the Fermi level in p-type semiconductors with impurity concentration is illustrated in the following figure:
Combined Effect: Impurity Concentration and Temperature
| Factor | Effect on Fermi Level |
|---|---|
|
Increase in donor concentration |
Moves upward (toward CB) |
|
Increase in acceptor concentration |
Moves downward (toward VB) |
|
Increase in temperature (high T) |
Moves toward mid-gap |
|
Low temperature |
Moves toward impurity level |
Physical Significance:
- Explains why conductivity changes with temperature
- Helps in selecting doping levels for devices
- Determines the operating region of a semiconductor
- Essential for p–n junction and transistor analysis
Important Examination Questions
Short Answer Questions
- How does impurity concentration affect the Fermi level?
- What happens to the Fermi level at high temperatures?
Long Answer / Derivation Questions
- Derive expressions showing the dependence of the Fermi level on impurity concentration in n-type and p-type semiconductors.
- Explain the variation of the Fermi level with temperature in extrinsic semiconductors.
Very frequently asked in university examinations
Conceptual Questions
- Why does the Fermi level move toward the mid-gap at high temperature?
- Why is the Fermi level nearly constant in the extrinsic region?
Reference: A Textbook of Engineering Physics (S. Chand Publications) and AI
FAQs
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1. What is the Fermi level in a semiconductor?
The Fermi level is the energy level at which the probability of finding an electron is 50% at absolute zero temperature.
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2. Where is the Fermi level located in an intrinsic semiconductor?
In an intrinsic semiconductor, the Fermi level lies approximately at the center of the energy band gap.
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3. Where is the Fermi level in an n-type semiconductor?
In an n-type semiconductor, the Fermi level shifts closer to the conduction band.
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4. What happens to the Fermi level with heavy doping in n-type materials?
With heavy doping, the Fermi level moves even closer to the conduction band and may enter it in extreme cases.
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5. Where is the Fermi level in a p-type semiconductor?
In a p-type semiconductor, the Fermi level shifts closer to the valence band.
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6. How does doping affect the Fermi level?
Doping shifts the Fermi level toward the conduction band (n-type) or valence band (p-type).