Fermi Level, Ec, Ev, Eg Explained

Fermi Level, Ec, Ev, Eg Explained in Semiconductor Physics

Semiconductor physics is based on energy band theory. The most important concepts are:

• Ec → Conduction band energy
• Ev → Valence band energy
• Eg → Bandgap energy
• EF → Fermi level

These concepts explain how electrons move inside semiconductors like silicon, germanium, and gallium arsenide.

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1. Energy Band Diagram

In solids, discrete atomic energy levels combine to form continuous energy bands.

The two important bands are:

• Valence Band
• Conduction Band

The region between them is called the forbidden energy gap.

SVG Energy Band Diagram

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2. Valence Band (Ev)

The valence band contains bonded electrons. These electrons participate in atomic bonding and normally cannot move freely.

At absolute zero temperature, the valence band is completely filled.

Ev = Top of valence band

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3. Conduction Band (Ec)

The conduction band contains free electrons that can move through the material and conduct current.

If an electron gains enough energy, it jumps from the valence band into the conduction band.

E ≥ Ec

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4. Bandgap Energy (Eg)

The energy difference between conduction band and valence band is called the bandgap.

Eg = Ec - Ev

This energy must be supplied to free an electron.

Typical Bandgap Values

Material	Bandgap
Silicon	1.12 eV
Germanium	0.66 eV
Gallium Arsenide	1.43 eV

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5. Fermi Level (EF)

The Fermi level is one of the most important concepts in semiconductor physics.

The Fermi level is the energy level where the probability of electron occupancy is 50%.

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6. Fermi-Dirac Probability Function

The probability that an energy state contains an electron is:

f(E) = 1 / [1 + e^((E - EF)/kT)]

Where:

Symbol	Meaning
f(E)	Probability of occupancy
E	Energy level
EF	Fermi level
k	Boltzmann constant
T	Temperature

At:

E = EF

The probability becomes:

f(EF) = 1/2

Meaning:

States at the Fermi level have 50% probability of containing an electron.

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7. Electron and Hole Concentration

Electron Concentration

n = Nc e^(-(Ec - EF)/kT)

Hole Concentration

p = Nv e^(-(EF - Ev)/kT)

Where:

• n = electron concentration
• p = hole concentration
• Nc = effective density of states in conduction band
• Nv = effective density of states in valence band

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8. Intrinsic Semiconductor

In a pure semiconductor:

EF ≈ (Ec + Ev) / 2

The Fermi level lies approximately at the middle of the bandgap.

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9. n-Type Semiconductor

Donor impurities add extra electrons.

Therefore:

EF → Ec

The Fermi level shifts upward toward the conduction band.

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10. p-Type Semiconductor

Acceptor impurities create holes.

EF → Ev

The Fermi level shifts downward toward the valence band.

How the Fermi Level Changes with Carrier Density

The Fermi level changes with carrier density because it represents the energy level at which the probability of finding an electron is 50% under thermal equilibrium. As the concentration of electrons or holes changes, the Fermi level shifts to maintain equilibrium within the semiconductor.

Intrinsic Semiconductor

In an intrinsic (pure) semiconductor:

• Electron concentration (n) equals hole concentration (p).
• The Fermi level is located approximately at the center of the band gap.

Effect of Increasing Electron Density (n-Type Semiconductor)

When donor impurities are added to a semiconductor, the electron concentration increases. As a result, the Fermi level shifts upward toward the conduction band.

Fermi Level Equation for n-Type Semiconductor

EF = Ei + kT ln(n / ni)

Where:

• E_F = Fermi level
• E_i = Intrinsic Fermi level
• n = Electron concentration
• n_i = Intrinsic carrier concentration
• k = Boltzmann constant
• T = Absolute temperature (Kelvin)

Since the logarithmic term increases with electron concentration, a higher electron density causes the Fermi level to move closer to the conduction band.

Effect of Increasing Hole Density (p-Type Semiconductor)

When acceptor impurities are introduced, the hole concentration increases. Consequently, the Fermi level shifts downward toward the valence band.

Fermi Level Equation for p-Type Semiconductor

EF = Ei - kT ln(p / ni)

As the hole concentration increases, the logarithmic term becomes larger, causing the Fermi level to move downward toward the valence band.

Relationship Between Carrier Density and Fermi Level

Carrier Density	Fermi Level Movement	Direction
Electron concentration increases	Toward the conduction band	Upward
Electron concentration decreases	Toward the middle of the band gap	Downward
Hole concentration increases	Toward the valence band	Downward
Hole concentration decreases	Toward the middle of the band gap	Upward

Why Does the Fermi Level Shift?

The carrier concentration depends exponentially on the position of the Fermi level. The relationships are:

n ∝ exp[(EF − EC) / kT]

p ∝ exp[(EV − EF) / kT]

Raising the Fermi level significantly increases the number of electrons in the conduction band, while lowering it increases the number of holes in the valence band. This exponential relationship explains why even a small shift in the Fermi level results in a large change in carrier density.

Key Takeaways

• The Fermi level is near the center of the band gap in intrinsic semiconductors.
• Increasing electron concentration shifts the Fermi level upward.
• Increasing hole concentration shifts the Fermi level downward.
• The Fermi level controls the probability of electron occupancy at different energy states.
• Carrier density and Fermi level are related through exponential functions, making the shift highly sensitive to doping concentration.

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