3.3 Fermi level and doping

Fermi Level in Semiconductors

The Fermi level represents the energy at which the probability of an electron occupying a quantum state is exactly 0.5. Its position relative to the conduction and valence band edges directly determines the electron and hole concentrations in a semiconductor, making it the single most important quantity for understanding doping, carrier statistics, and device behavior.

Fermi-Dirac Distribution

The Fermi-Dirac distribution gives the probability that an electron occupies a state at energy $E$ for a given temperature $T$:

$f(E) = \frac{1}{1 + e^{(E - E_F)/kT}}$

where $E_F$ is the Fermi energy, $k$ is Boltzmann's constant ($8.617 \times 10^{-5}$ eV/K), and $T$ is the absolute temperature.

A few things to notice about this function:

• At $E = E_F$, the exponential term equals 1, so $f(E_F) = 0.5$. That's the defining property of the Fermi level.
• At $T = 0$ K, the distribution becomes a sharp step function: every state below $E_F$ is filled ($f = 1$), and every state above is empty ($f = 0$).
• As $T$ increases, the step softens. States a few $kT$ above $E_F$ gain partial occupancy, and states a few $kT$ below lose some. The "smearing" region has a width of roughly $4kT$ to $6kT$.

At room temperature (300 K), $kT \approx 0.026$ eV. Since typical semiconductor band gaps are 0.5–3 eV, the Fermi level usually sits far enough from both band edges that the Boltzmann approximation (replacing the Fermi-Dirac function with a simple exponential) works well for non-degenerate semiconductors.

Electron and Hole Concentrations

When $E_F$ is more than a few $kT$ away from the band edges, the carrier concentrations simplify to:

• Electron concentration: $n = N_c \cdot e^{-(E_c - E_F)/kT}$
• Hole concentration: $p = N_v \cdot e^{-(E_F - E_v)/kT}$

Here $N_c$ and $N_v$ are the effective densities of states in the conduction and valence bands, respectively. They depend on the effective masses and temperature (both scale as $T^{3/2}$). For silicon at 300 K, $N_c \approx 2.8 \times 10^{19}$ cm$^{-3}$ and $N_v \approx 1.04 \times 10^{19}$ cm$^{-3}$.

A critical relationship follows from multiplying $n$ and $p$:

$np = N_c N_v \cdot e^{-E_g/kT} = n_i^2$

This is the mass-action law. It holds in equilibrium regardless of doping. For silicon at 300 K, $n_i \approx 1.5 \times 10^{10}$ cm$^{-3}$. If you know one carrier concentration, you can always find the other from $n_i^2/n$ or $n_i^2/p$.

Temperature Dependence of Fermi Level

The Fermi level shifts with temperature, and the behavior differs between intrinsic and extrinsic materials:

• Intrinsic semiconductors: The Fermi level sits near mid-gap. More precisely, $E_F = \frac{E_c + E_v}{2} + \frac{kT}{2}\ln\frac{N_v}{N_c}$. The second term is small, so $E_F$ is approximately at mid-gap but shifts slightly toward the band with the smaller effective density of states as $T$ rises.
• Extrinsic semiconductors: At low temperatures, carriers freeze out onto dopant sites and $E_F$ sits near the dopant level. In the extrinsic range (including room temperature for typical doping), $E_F$ is close to the majority-carrier band edge. At very high temperatures, intrinsic carriers overwhelm the dopants and $E_F$ migrates back toward mid-gap.

The key takeaway: as temperature increases, every extrinsic semiconductor eventually behaves like an intrinsic one.

Intrinsic vs. Extrinsic Semiconductors

Intrinsic Semiconductor Properties

An intrinsic semiconductor is an undoped, pure crystal (e.g., pure Si or Ge). Electron-hole pairs are generated only by thermal excitation across the band gap, so:

$n = p = n_i = \sqrt{N_c N_v} \cdot e^{-E_g / 2kT}$

Because $n_i$ depends exponentially on $-E_g/2kT$, materials with larger band gaps have far fewer intrinsic carriers. At 300 K, silicon ($E_g = 1.12$ eV) has $n_i \approx 1.5 \times 10^{10}$ cm$^{-3}$, while germanium ($E_g = 0.66$ eV) has $n_i \approx 2.4 \times 10^{13}$ cm$^{-3}$.

Extrinsic Semiconductor Doping

Extrinsic semiconductors are created by intentionally adding impurity atoms (dopants) to the crystal. Doping shifts the Fermi level away from mid-gap and dramatically increases one type of carrier concentration while suppressing the other (via the mass-action law).

n-type vs. p-type Doping

n-type doping uses donor atoms from Group V (e.g., phosphorus, arsenic in silicon). Each donor has one extra valence electron compared to Si. That electron is loosely bound, with an ionization energy of only ~45 meV for P in Si. At room temperature, virtually all donors are ionized, contributing their electrons to the conduction band.

• Majority carriers: electrons
• $n \approx N_D$ (donor concentration), $p = n_i^2 / N_D$

p-type doping uses acceptor atoms from Group III (e.g., boron, gallium in silicon). Each acceptor is missing one valence electron, creating a state just above the valence band (~45 meV for B in Si). Electrons from the valence band fill these states, leaving behind holes.

• Majority carriers: holes
• $p \approx N_A$ (acceptor concentration), $n = n_i^2 / N_A$

Doping Effects on Fermi Level

Donor and Acceptor Energy Levels

Dopant atoms create discrete energy levels within the band gap:

• Donor levels ($E_D$) sit just below the conduction band edge, typically 10–50 meV below $E_c$. At room temperature, $kT \approx 26$ meV, which is comparable to or larger than these ionization energies, so most donors are ionized.
• Acceptor levels ($E_A$) sit just above the valence band edge, typically 10–50 meV above $E_v$. Similarly, most acceptors are ionized at room temperature.

These small ionization energies are why dopants in Si and Ge are called shallow impurities. Deep-level impurities (like gold or iron in silicon) have levels near mid-gap and act as recombination centers rather than effective dopants.

Fermi Level Shift with Doping Concentration

You can find the Fermi level position directly from the carrier concentration equations. For an n-type semiconductor with full ionization:

$E_c - E_F = kT \ln\left(\frac{N_c}{N_D}\right)$

For p-type:

$E_F - E_v = kT \ln\left(\frac{N_v}{N_A}\right)$

As doping increases, the logarithmic term shrinks, and $E_F$ moves closer to the relevant band edge. For example, in n-type Si at 300 K:

• $N_D = 10^{15}$ cm$^{-3}$: $E_c - E_F \approx 0.27$ eV
• $N_D = 10^{17}$ cm$^{-3}$: $E_c - E_F \approx 0.15$ eV
• $N_D = 10^{19}$ cm$^{-3}$: $E_c - E_F \approx 0.03$ eV

Degenerate Semiconductors

When the doping concentration approaches or exceeds $N_c$ (or $N_v$), the Fermi level enters the band itself. At this point:

• The Boltzmann approximation breaks down, and you must use the full Fermi-Dirac integral.
• The semiconductor is called degenerate because $E_F$ lies inside the conduction band (n-type) or valence band (p-type).
• Carrier behavior becomes metallic: conductivity is high, and its temperature dependence weakens.
• Band gap narrowing occurs due to heavy doping, which slightly reduces the effective $E_g$.

Degeneracy typically sets in above roughly $10^{19}$ cm$^{-3}$ in silicon. Heavily doped regions (often labeled $n^+$ or $p^+$) in devices frequently operate in this regime.

Carrier Concentration and Fermi Level

Majority and Minority Carriers

In any extrinsic semiconductor at equilibrium, one carrier type dominates:

• n-type: electrons are majority carriers ($n \approx N_D$), holes are minority carriers ($p = n_i^2/N_D$)
• p-type: holes are majority carriers ($p \approx N_A$), electrons are minority carriers ($n = n_i^2/N_A$)

The ratio of majority to minority carriers scales as $(N_D/n_i)^2$ or $(N_A/n_i)^2$. For typical doping levels in silicon, this ratio is enormous, which is why minority carriers are so few yet so important for device operation (they control current in p-n junctions and bipolar transistors).

Carrier Concentration Calculations

Here's a systematic approach for finding carrier concentrations in a doped semiconductor at thermal equilibrium:

1. Identify the dopant type and concentration ($N_D$ for donors, $N_A$ for acceptors, or both for compensated material).

2. Check for compensation. If both donors and acceptors are present, the net doping is $|N_D - N_A|$, and the semiconductor type is determined by whichever dopant has the higher concentration.

3. Assume full ionization (valid at room temperature for shallow dopants).

4. Find the majority carrier concentration: $n \approx N_D - N_A$ (if n-type) or $p \approx N_A - N_D$ (if p-type).

5. Find the minority carrier concentration from the mass-action law: $p = n_i^2/n$ or $n = n_i^2/p$.

6. Locate the Fermi level using $E_c - E_F = kT\ln(N_c/n)$ or $E_F - E_v = kT\ln(N_v/p)$.

Fermi Level and Carrier Concentration Relationship

The exponential dependence of $n$ and $p$ on Fermi level position means that small shifts in $E_F$ produce large changes in carrier concentration. Moving $E_F$ by just $kT$ (26 meV at 300 K) changes the carrier concentration by a factor of $e \approx 2.72$. Moving it by $60$ meV changes the concentration by roughly a factor of 10.

This extreme sensitivity is exactly what makes doping so powerful: adding even parts-per-million of dopant atoms shifts $E_F$ enough to change carrier concentrations by many orders of magnitude.

Fermi Level and Band Diagram

Band Structure of Semiconductors

The band diagram plots electron energy (vertical axis) versus position (horizontal axis). The key features are:

• Conduction band edge ($E_c$): the lowest energy an electron can have in the conduction band
• Valence band edge ($E_v$): the highest energy an electron can have in the valence band
• Band gap ($E_g = E_c - E_v$): the forbidden energy range with no allowed states in a perfect crystal

For silicon, $E_g = 1.12$ eV at 300 K. For GaAs, $E_g = 1.42$ eV. The band gap determines the wavelength of light a semiconductor can absorb and the intrinsic carrier concentration.

Fermi Level Position in Band Diagram

The Fermi level appears as a horizontal dashed line on the band diagram (horizontal because $E_F$ is constant throughout a semiconductor in equilibrium). Its position tells you the semiconductor type at a glance:

• Intrinsic: $E_F$ near mid-gap
• n-type: $E_F$ in the upper half of the gap, closer to $E_c$
• p-type: $E_F$ in the lower half of the gap, closer to $E_v$
• Degenerate n-type: $E_F$ above $E_c$
• Degenerate p-type: $E_F$ below $E_v$

When two differently doped regions are in contact (as in a p-n junction), the requirement that $E_F$ be constant at equilibrium forces the bands to bend, creating built-in electric fields.

Band Gap and Fermi Level

The band gap sets the "playing field" for the Fermi level. A few practical consequences:

• Smaller $E_g$ means higher $n_i$, which means the semiconductor becomes intrinsic at lower temperatures. Germanium devices struggle above ~70°C; silicon works to ~150°C; SiC ($E_g = 3.3$ eV) operates well above 300°C.
• Larger $E_g$ means the Fermi level can be moved over a wider energy range by doping before degeneracy sets in, giving more control over carrier concentrations.
• Band gap engineering through alloying (e.g., $\text{Al}_x\text{Ga}_{1-x}\text{As}$) or strain allows you to tailor both $E_g$ and the Fermi level position for specific device applications.

Fermi Level and Semiconductor Devices

p-n Junction and Fermi Level

When p-type and n-type regions are joined, their different Fermi levels must equalize at equilibrium. This drives the following sequence:

1. Electrons diffuse from the n-side (high $E_F$) to the p-side (low $E_F$), and holes diffuse the opposite way.
2. The diffusing carriers leave behind ionized dopants, creating a depletion region with a net space charge.
3. The space charge creates a built-in electric field that opposes further diffusion.
4. Equilibrium is reached when $E_F$ is flat (constant) across the entire junction. The bands bend to accommodate this.

The built-in potential is directly related to the Fermi level difference before contact:

$V_{bi} = \frac{kT}{q}\ln\left(\frac{N_A N_D}{n_i^2}\right)$

For a silicon junction with $N_A = N_D = 10^{16}$ cm$^{-3}$, $V_{bi} \approx 0.72$ V. This built-in potential governs rectification, junction capacitance, and breakdown behavior.

Fermi Level in Equilibrium and Non-Equilibrium

At equilibrium, a single Fermi level describes the entire system. There is no net current, and $np = n_i^2$ everywhere.

Under non-equilibrium conditions (applied voltage, illumination, injection), the single Fermi level splits into two quasi-Fermi levels:

• $E_{Fn}$ for electrons: $n = N_c \cdot e^{-(E_c - E_{Fn})/kT}$
• $E_{Fp}$ for holes: $p = N_v \cdot e^{-(E_{Fp} - E_v)/kT}$

The separation $E_{Fn} - E_{Fp}$ measures how far the system is from equilibrium. In a forward-biased p-n junction, this splitting equals $qV$ (the applied voltage times the electron charge). In a solar cell under illumination, the quasi-Fermi level splitting determines the open-circuit voltage.

Fermi Level Pinning in Devices

Fermi level pinning occurs when a high density of surface or interface states locks $E_F$ at a particular energy, regardless of doping or applied bias. Common causes include:

• Dangling bonds at semiconductor surfaces
• Interface states at metal-semiconductor contacts (Schottky barriers)
• Defect states at oxide-semiconductor interfaces

When pinning is strong, the Schottky barrier height becomes nearly independent of the metal work function, which limits your ability to engineer contact properties. Techniques to mitigate pinning include surface passivation (e.g., thermal oxide on Si, sulfide treatments on III-Vs), inserting thin insulating interlayers, and careful interface engineering during fabrication.