Practice MCQ

Unit 1 - Notes

CSE212 8 min read

Unit 1: Semiconductors

1. Insulators, Semiconductors, and Metals

The electrical properties of materials are primarily determined by their band structure. According to the Energy Band Theory, electrons in solid materials occupy distinct energy bands, specifically the Valence Band (VB) and the Conduction Band (CB). The gap between these bands is known as the Forbidden Energy Gap ( $E_g$ ).

Metals (Conductors)

Band Structure: The valence band and conduction band overlap, or the conduction band is partially filled.
Energy Gap: $E_g = 0 \text{ eV}$ .
Conductivity: Highly conductive at room temperature. Even a tiny applied electric field gives electrons enough energy to move freely and constitute a current.
Temperature Coefficient: Positive temperature coefficient of resistance (resistance increases as temperature increases due to increased lattice scattering).

Insulators

Band Structure: The valence band is completely full, and the conduction band is completely empty at absolute zero ( $0 \text{ K}$ ).
Energy Gap: Large forbidden energy gap ( $E_g > 5 \text{ eV}$ ). E.g., Diamond has an $E_g$ of ~6 eV.
Conductivity: Extremely poor conductivity. Normal thermal energy at room temperature is insufficient to jump the wide bandgap.
Temperature Coefficient: Negative temperature coefficient (though practically, they break down before conducting significantly).

Semiconductors

Band Structure: Similar to insulators at $0 \text{ K}$ (full VB, empty CB), but with a much narrower bandgap.
Energy Gap: Typically around $1 \text{ eV}$ . Examples: Silicon (Si) $\approx 1.12 \text{ eV}$ , Germanium (Ge) $\approx 0.67 \text{ eV}$ at room temperature.
Conductivity: At absolute zero, they behave as perfect insulators. At room temperature, thermal energy is enough to excite some electrons across the bandgap into the conduction band, allowing moderate conductivity.
Temperature Coefficient: Negative temperature coefficient of resistance (conductivity increases exponentially with temperature).

2. Electrons and Holes in an Intrinsic Semiconductor

An intrinsic semiconductor is a pure semiconductor crystal (like extremely pure Si or Ge) with no impurities added.

Covalent Bonding: In pure Silicon, each atom has 4 valence electrons and shares them with 4 neighboring atoms to form strong covalent bonds. At $0 \text{ K}$ , all bonds are intact.
Thermal Generation: As temperature rises above $0 \text{ K}$ , thermal energy breaks some covalent bonds.
Electron-Hole Pairs (EHPs):
- When an electron breaks free, it jumps to the Conduction Band and becomes a free electron.
- The vacancy left behind in the Valence Band is called a hole. It acts as a positively charged particle because neighboring electrons can move into this vacancy, causing the hole to "move" in the opposite direction.
Carrier Concentration: In an intrinsic semiconductor, the number of thermally generated electrons () exactly equals the number of holes ().
- $n = p = n_i$
- Where $n_i$ is the intrinsic carrier concentration.

3. Donor and Acceptor Impurities

To increase the conductivity of intrinsic semiconductors, specific impurities are deliberately added in a process called doping, creating extrinsic semiconductors.

Donor Impurities (N-Type Semiconductors)

Dopants: Group V elements (Pentavalent), such as Phosphorus (P), Arsenic (As), and Antimony (Sb).
Mechanism: These atoms have 5 valence electrons. When a pentavalent atom replaces a Silicon atom, 4 electrons form covalent bonds with adjacent Si atoms. The 5th electron is loosely bound and easily "donated" to the conduction band at room temperature.
Energy Level: The donor energy level ( $E_D$ ) lies just below the Conduction Band edge ( $E_C$ ).
Charge Carriers: Electrons become the majority carriers, while holes (thermally generated) are the minority carriers.

Acceptor Impurities (P-Type Semiconductors)

Dopants: Group III elements (Trivalent), such as Boron (B), Gallium (Ga), and Aluminum (Al).
Mechanism: These atoms have 3 valence electrons. When a trivalent atom replaces a Silicon atom, it leaves one bond incomplete (a hole). It can "accept" an electron from a neighboring Si atom to complete the bond, thereby allowing the hole to move freely through the lattice.
Energy Level: The acceptor energy level ( $E_A$ ) lies just above the Valence Band edge ( $E_V$ ).
Charge Carriers: Holes become the majority carriers, while electrons are the minority carriers.

4. Charge Densities in a Semiconductor

In any semiconductor at thermal equilibrium, the Mass Action Law holds true:
$n \cdot p = n_i^2$
(where $n$ is electron concentration, $p$ is hole concentration, and $n_i$ is intrinsic concentration).

Law of Electrical Neutrality

A semiconductor crystal is electrically neutral overall. The total positive charge density must equal the total negative charge density:
$p + N_D^+ = n + N_A^-$

Where:

$p$ = hole concentration
$n$ = electron concentration
$N_D^+$ = ionized donor concentration (approximately total $N_D$ )
$N_A^-$ = ionized acceptor concentration (approximately total $N_A$ )

Calculating Charge Densities

For N-Type Semiconductors:
Assume $N_A = 0$ and $N_D \gg n_i$ .

Majority carrier (electrons): $n \approx N_D$
Minority carrier (holes): $p = \frac{n_i^2}{N_D}$

For P-Type Semiconductors:
Assume $N_D = 0$ and $N_A \gg n_i$ .

Majority carrier (holes): $p \approx N_A$
Minority carrier (electrons): $n = \frac{n_i^2}{N_A}$

5. Fermi Level in a Semiconductor Having Impurities

The Fermi Level ( $E_F$ ) represents the energy level where the probability of finding an electron is exactly 50%. The probability distribution is governed by the Fermi-Dirac distribution function.

Intrinsic Semiconductor

In a pure semiconductor, $E_F$ lies exactly in the middle of the forbidden energy gap.
$E_F = \frac{E_C + E_V}{2}$

N-Type Semiconductor

Adding donor impurities introduces energy states just below $E_C$ . This drastically increases the probability of finding electrons near the conduction band.

Position: The Fermi level shifts upwards, closer to the Conduction Band ( $E_C$ ).
Equation: $E_F = E_C - kT \ln\left(\frac{N_C}{N_D}\right)$
(Where $N_C$ is the effective density of states in the conduction band, $k$ is Boltzmann's constant, $T$ is temperature).

P-Type Semiconductor

Adding acceptor impurities introduces energy states just above $E_V$ , increasing the probability of empty states (holes) near the valence band.

Position: The Fermi level shifts downwards, closer to the Valence Band ( $E_V$ ).
Equation: $E_F = E_V + kT \ln\left(\frac{N_V}{N_A}\right)$
(Where $N_V$ is the effective density of states in the valence band).

Note: At very high temperatures, thermal generation overwhelms the dopant concentration, and extrinsic semiconductors revert to behaving like intrinsic ones; $E_F$ shifts back to the center of the bandgap.

6. Mobility and Conductivity

When an electric field ( $E$ ) is applied to a semiconductor, the charge carriers do not accelerate indefinitely due to collisions with the crystal lattice (scattering). Instead, they reach a steady average velocity called the drift velocity ( $v_d$ ).

Mobility ( $\mu$ )

Mobility is defined as the drift velocity acquired by a charge carrier per unit electric field strength.
$\mu = \frac{v_d}{E}$

Units: $cm^2 / V\cdot s$ or $m^2 / V\cdot s$ .
Electrons vs. Holes: Electrons are in the conduction band and move more freely than holes, which move by valence electrons jumping between bonds. Therefore, electron mobility is always greater than hole mobility ( $\mu_n > \mu_p$ ).

Drift Current Density ( $J$ )

The drift current density is directly proportional to the applied electric field.
$J = \rho v_d = n q v_d = n q \mu E = \sigma E$
Where:

$\rho$ = volume charge density
$q$ = elementary charge ( $1.6 \times 10^{-19} \text{ C}$ )
$\sigma$ = conductivity

7. Conductivity of a Semiconductor

The total electrical conductivity ( $\sigma$ ) of a semiconductor is the sum of the conductivities contributed by both electrons and holes.

General Conductivity Equation:
$\sigma = q (n \mu_n + p \mu_p)$
Where:

$n$ and $p$ are electron and hole concentrations.
$\mu_n$ and $\mu_p$ are electron and hole mobilities.

Conductivity in Intrinsic Semiconductors:
Since $n = p = n_i$ :
$\sigma_i = n_i q (\mu_n + \mu_p)$

Conductivity in Extrinsic Semiconductors:

N-Type: Electrons vastly outnumber holes ( $n \approx N_D$ , $p \approx 0$ ).
$\sigma \approx q N_D \mu_n$
P-Type: Holes vastly outnumber electrons ( $p \approx N_A$ , $n \approx 0$ ).
$\sigma \approx q N_A \mu_p$

8. Diffusion and Life Time

Diffusion

While drift current is caused by an electric field, diffusion current is caused by a concentration gradient. Charge carriers naturally move from a region of higher concentration to a region of lower concentration, even without an external electric field.

Diffusion Current Densities:

Electron diffusion current density: $J_{n(diff)} = q D_n \frac{dn}{dx}$
Hole diffusion current density: $J_{p(diff)} = -q D_p \frac{dp}{dx}$
(The negative sign for holes is because they diffuse down the gradient ( $-dp/dx$ ), and carrying positive charge, the current is in the direction of diffusion).

Einstein Relationship:
At thermal equilibrium, carrier mobility and diffusion constants are related:
$\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = V_T$
Where $V_T$ is the thermal voltage ( $V_T = \frac{kT}{q} \approx 26 \text{ mV}$ at room temperature).

Life Time ( $\tau$ )

When excess electron-hole pairs are created (e.g., by light or forward biasing a junction), they do not exist indefinitely. Electrons will eventually fall back into empty bonds, annihilating both the electron and the hole. This process is called recombination.

Carrier Lifetime ( $\tau_n$ for electrons, $\tau_p$ for holes): The mean time an excess charge carrier exists before recombining.
Recombination Rate ( $R$ ): Proportional to the excess carrier concentration ( $\Delta n$ or $\Delta p$ ) divided by the lifetime. $R = \frac{\Delta p}{\tau_p}$ .
Diffusion Length ( $L$ ): The average distance a carrier diffuses before recombining.
$L_n = \sqrt{D_n \tau_n}$ and $L_p = \sqrt{D_p \tau_p}$

Understanding diffusion and lifetime is critical for analyzing the dynamic behavior of PN junctions, Bipolar Junction Transistors (BJTs), and optoelectronic devices.

Unit 2