Semiconductor physics and band theory notes — Unit 4

In an isolated atom, electrons occupy discrete energy levels. However, in a solid crystal lattice, millions of atoms are packed closely together. Their outermost electron orbitals overlap and interact.

Due to Pauli's Exclusion Principle, no two electrons can have the same energy state. Therefore, the single discrete energy level splits into millions of closely spaced energy levels, forming a continuous Energy Band.

This mathematical model explains the origin of energy bands. It treats an electron moving in a 1D periodic crystal lattice as moving through a periodic array of rectangular potential wells (representing positive ion cores).

• By solving the Schrodinger equation for this periodic potential (using Bloch's Theorem), the model produces a complex determinant equation.
• The mathematical solutions reveal that there are ranges of energy $E$ where solutions exist (Allowed Bands).
• There are also ranges of energy where no valid solutions exist (Forbidden Gaps or Band Gaps).

Based on the energy band diagram (specifically the gap between the Valence Band and the Conduction Band), solids are classified into three types:

Intrinsic semiconductors (like pure Silicon and Germanium) are perfectly pure crystals without any chemical impurities.

At absolute zero ($0$ K), they act as perfect insulators. All electrons are locked in covalent bonds (Valence band is full, Conduction band is empty).

At room temperature, thermal energy breaks some covalent bonds. Electrons jump to the CB, leaving behind positively charged vacancies called Holes in the VB.

In intrinsic semiconductors:
n = p = n_i
Where n = electron concentration, p = hole concentration

Electrons are fermions (spin-1/2 particles obeying Pauli's principle). Their energy distribution at thermal equilibrium is governed by the Fermi-Dirac probability function.

f(E) = 1 / [1 + exp((E - E_F)/kT)]

Where:
E_F = Fermi Energy
k = Boltzmann's constant
T = Absolute temperature

$f(E)$ gives the probability that an available energy state $E$ is occupied by an electron.

The Fermi Level is a reference energy level. Let's analyze $f(E)$ at $T > 0$ K for a state exactly at the Fermi energy ($E = E_F$):

f(E_F) = 1 / [1 + exp(0)]
f(E_F) = 1 / (1 + 1) = 1/2

Thus, the Fermi level is the energy level where the probability of finding an electron is exactly 50% (0.5).

To find the total number of electrons in the Conduction Band ($n$), we integrate the product of the Density of States $N(E)$ and the Fermi Probability $f(E)$ over the CB.

After rigorous integration, the electron concentration in the CB is:

n = N_c · exp[-(E_C - E_F) / kT]

Where N_c is the effective density of states in the CB.

The probability of finding a hole is simply the probability of NOT finding an electron, which is $1 - f(E)$.

Integrating over the Valence Band yields the hole concentration ($p$):

p = N_v · exp[-(E_F - E_V) / kT]

Where N_v is the effective density of states in the VB.

⇒Mass Action Law

For any semiconductor in thermal equilibrium, the product of $n$ and $p$ is constant.

n · p = n_i²

To increase conductivity, intentional impurities are added to intrinsic semiconductors in a process called Doping. The resulting material is an Extrinsic Semiconductor.

• N-Type: Doped with Pentavalent impurities (P, As, Sb). They have 5 valence electrons. 4 form bonds, 1 is loosely bound and "donated" to the CB.
• P-Type: Doped with Trivalent impurities (B, Al, Ga). They have 3 valence electrons, creating a vacancy (hole) that can "accept" an electron.

In N-type, electrons are the majority charge carriers ($n \gg p$). The donor impurity atoms introduce a new energy level ($E_D$) just slightly below the Conduction Band ($E_C$).

Because $E_D$ is so close to $E_C$ ($\sim 0.01$ eV), thermal energy easily excites these donated electrons into the CB.

⇒Fermi Level Shift

Because the probability of finding electrons near the CB is high, the Fermi level $E_F$ shifts upwards, lying close to the Conduction Band.

In P-type, holes are the majority charge carriers ($p \gg n$). The acceptor impurity atoms introduce a new energy level ($E_A$) just slightly above the Valence Band ($E_V$).

Electrons from the VB easily jump into $E_A$, leaving behind millions of holes in the VB.

⇒Fermi Level Shift

Because the probability of finding holes near the VB is high (meaning finding electrons is low), the Fermi level $E_F$ shifts downwards, lying close to the Valence Band.

The position of the Fermi level is highly sensitive to temperature.

• At 0 K: $E_F$ in N-type lies exactly halfway between $E_D$ and $E_C$. In P-type, exactly between $E_A$ and $E_V$.
• As T increases: More intrinsic electron-hole pairs are generated across the full band gap. The extrinsic majority carriers get swamped by the intrinsic carriers.
• At High T: The semiconductor behaves like an intrinsic semiconductor, and $E_F$ shifts back to the middle of the band gap.

Discovered by Edwin Hall (1879). When a current-carrying semiconductor is placed in a transverse magnetic field, an electric field (Hall Voltage) is induced perpendicular to both the current and the magnetic field.

⇒Mechanism

The magnetic field exerts a Lorentz force ($F = qvB$) on the moving charge carriers, pushing them to one face of the slab. This charge separation creates an opposing transverse electric field ($E_H$), until equilibrium is reached ($qE_H = qvB$).

The Hall Coefficient ($R_H$) is defined as $E_H / (J \cdot B)$.

R_H = 1 / (n·q)

⇒Crucial Applications of Hall Effect:

• Determine Semiconductor Type: The sign of $R_H$ is negative for N-type (electrons) and positive for P-type (holes).
• Calculate Carrier Concentration: Since $R_H = 1/(nq)$, finding $R_H$ gives $n$ directly.
• Calculate Mobility ($\mu$): Mobility $\mu = \sigma \cdot R_H$, where $\sigma$ is conductivity.
• Measure Magnetic Field: Hall effect sensors are widely used to measure $B$.

When a p-type and n-type semiconductor are metallurgically joined, a p-n junction is formed. It is the basis of all modern electronics (diodes, transistors, ICs).

⇒Formation of the Depletion Region

Due to the concentration gradient, holes from the p-side diffuse to the n-side, and electrons diffuse from n to p. When they cross, they recombine and annihilate each other.

They leave behind immobile ionized atoms: positive ions on the n-side and negative ions on the p-side. This creates a region depleted of mobile charge carriers, known as the Depletion Region.

The uncovered positive ions on the n-side and negative ions on the p-side create a built-in electric field pointing from n to p.

This field exerts a force opposing any further diffusion of majority carriers. The voltage equivalent of this field is called the Barrier Potential ($V_0$).

• For Silicon, $V_0 \approx 0.7$ Volts at room temp.
• For Germanium, $V_0 \approx 0.3$ Volts.

Without external voltage, the diffusion current is perfectly balanced by the drift current (driven by the built-in field), so net current is zero.

Forward bias means connecting the Positive terminal of the battery to the p-type, and Negative to the n-type.

• The applied voltage opposes the built-in barrier potential.
• The effective barrier height is reduced to $(V_0 - V)$.
• The width of the depletion region decreases.
• Majority carriers have enough energy to cross the junction, resulting in a large forward current (measured in milliAmperes).

Reverse bias means connecting the Negative terminal to the p-type, and Positive to the n-type.

• The applied voltage acts in the same direction as the built-in barrier.
• The effective barrier height increases to $(V_0 + V)$.
• The width of the depletion region increases.
• Majority carriers are pulled away from the junction. Current is almost zero.
• A very small reverse leakage current (microAmperes) flows due to the drift of thermally generated minority carriers.

The V-I characteristic of a p-n junction is non-linear and governed by Shockley's Diode Equation:

I = I₀ [exp(qV / ηkT) - 1]

⇒Optoelectronics

• LED (Light Emitting Diode): A heavily doped forward-biased p-n junction. When electrons recombine with holes, they drop from the CB to the VB, releasing energy as photons. Band gap must be $>1.8$ eV (visible light).
• Solar Cell: An unbiased p-n junction with a large surface area. Incoming photons ($h\nu > E_g$) generate electron-hole pairs, which are separated by the built-in depletion field, generating a voltage.

⇒Unit 4 Mastery

• Can you explain the origin of energy bands using the Kronig-Penney model qualitatively?
• Write the Fermi-Dirac function and define the Fermi level.
• How does the Fermi level shift with temperature in N-type and P-type semiconductors?
• State the Hall Effect and list its four main applications.
• Explain the formation of the depletion region and barrier potential.
• Draw the energy band diagram of a p-n junction under zero, forward, and reverse bias.