Fermi Level Calculator

What Is the Fermi Level?

The Fermi level (Ef) is the energy level where the probability of finding an electron is 50%.

• At absolute zero (0 K), all energy levels below the Fermi level are filled.
• All levels above it are empty.
• At higher temperatures, electrons spread out, but the Fermi level still acts as a reference point.

In simple terms, the Fermi level tells us how full the available electron states are inside a material.

Why the Fermi Level Matters

The Fermi level helps explain:

• Why metals conduct electricity well
• Why semiconductors can be tuned with doping
• How electronic devices like diodes and transistors work
• How temperature changes electrical behavior

Without the Fermi level, modern electronics would be guesswork.

What This Fermi Level Calculator Does

Your calculator is designed to work in two material regimes:

1. Metals (Free Electron Gas Model)
2. Semiconductors (Band Theory Model)

The user selects the material type, enters physical parameters, and the calculator returns the Fermi level and related quantities.

This makes it useful for:

• Physics students
• Electrical and materials engineers
• Researchers doing quick estimates
• Anyone learning solid-state concepts

Mode 1: Metal (Free Electron Gas)

What Assumptions Are Used?

In metal mode, the calculator assumes:

• Electrons behave like a free electron gas
• Electron density is very high
• Temperature effects are ignored (T = 0 K)

This model works well for common metals like copper and aluminum.

Required Input

Electron density (n)

• Unit: electrons per cm³
• Example: Copper ≈ 8.47 × 10²² cm⁻³

This value controls everything in the calculation.

What the Calculator Computes

1. Fermi Energy (Ef)
   • Calculated using the quantum free-electron formula
   • Output in electron volts (eV)
2. Fermi Temperature (Tf)
   • Shows the temperature equivalent of the Fermi energy
   • Usually much higher than room temperature
3. Fermi Velocity (vf)
   • The maximum electron speed at the Fermi level

Why These Results Matter

• A high Fermi energy explains why metals conduct so well
• Fermi temperature shows why room temperature barely affects metals
• Fermi velocity is useful in transport and scattering analysis

Mode 2: Semiconductor (Band Theory)

How This Mode Is Different

Semiconductors behave very differently from metals. Here:

• Energy bands matter
• Temperature plays a key role
• Doping shifts the Fermi level

This mode uses band theory and Boltzmann statistics, which are valid for non-degenerate semiconductors.

Inputs Used in Semiconductor Mode

1. Temperature (K)
   Controls carrier distribution
2. Band Gap (Eg in eV)
   Energy difference between conduction and valence bands
3. Effective Mass of Electrons (m*/m₀)
   Adjusts how electrons respond inside the crystal
4. Effective Mass of Holes (m*/m₀)
   Same idea, but for holes
5. Donor Concentration (Nd)
   Creates n-type material
6. Acceptor Concentration (Na)
   Creates p-type material

All concentrations are in cm⁻³, which matches standard semiconductor practice.

What the Calculator Computes

1. Effective Density of States
   • Conduction band density (Nc)
   • Valence band density (Nv)
2. Intrinsic Carrier Concentration (ni)
   • Number of electrons and holes in pure material
3. Intrinsic Fermi Level (Ei)
   • Fermi level in undoped material
4. Final Fermi Level (Ef)
   • Shifted based on doping type and amount
5. Material Type
   • Intrinsic
   • n-type extrinsic
   • p-type extrinsic

How Doping Affects the Fermi Level

• n-type material
  Fermi level moves upward, closer to the conduction band
• p-type material
  Fermi level moves downward, closer to the valence band

The calculator automatically detects this using the net doping (Nd − Na).

Key Physics Models Used

The calculator relies on well-known approximations:

• Fermi–Dirac statistics (general foundation)
• Boltzmann approximation (non-degenerate semiconductors)
• Parabolic energy bands
• Free electron gas model for metals

These models balance accuracy with speed and are widely used in education and industry.

Why This Calculator Is Practical

This tool is especially useful because:

• It avoids long hand calculations
• It uses realistic physical constants
• It separates metal and semiconductor physics clearly
• It provides extra context, not just one number

You do not just get Ef. You get insight.

Limitations to Keep in Mind

Like all models, this calculator has boundaries:

• Not valid for heavily degenerate semiconductors
• Does not include band bending or interfaces
• Assumes equilibrium conditions
• Uses simplified band structures

For advanced device simulations, more complex tools are needed. For learning and estimation, this calculator is ideal.