Semiconductor Physics Essentials

Cheatsheet Content

1. Energy Bands in Solids Energy Band Formation: When $N$ atoms form a solid, their discrete energy levels split into $N$ closely spaced levels, forming continuous energy bands due to interatomic interaction and Pauli's exclusion principle. Valence Band: The highest occupied allowed energy band, formed by valence electrons. Can be completely or partially filled. Conduction Band: The lowest unfilled permitted energy band. Electrons in this band move freely and are called conduction electrons. Forbidden Energy Gap ($E_g$): The energy band between the valence band and conduction band, representing non-permitted energy levels. It is the energy required for an electron to move from the valence band to the conduction band. 2. Classification of Solids (Band Theory) Insulators: Full valence band. Empty conduction band. Large energy gap ($E_g \approx 5 \text{ to } 10 \text{ eV}$). Very high resistivity ($\approx 10^{10} \text{ }\Omega \cdot \text{cm}$). Semiconductors: Electrical properties between insulators and conductors. Almost empty conduction band and filled valence band with narrow energy gap ($E_g \approx 1 \text{ eV}$). At $0 \text{ K}$, valence band is full, conduction band is empty. At room temperature, some electrons gain enough energy to cross $E_g$, creating electron-hole pairs (EHPs). Resistivity varies ($10^{-12} \text{ to } 10^9 \text{ }\Omega \cdot \text{cm}$). Examples: Ge, Si. Conductors: Conduction band and valence band are overlapped and partially filled. No forbidden gap. Low resistivity ($\approx 10^{-6} \text{ }\Omega \cdot \text{cm}$). Examples: Copper, silver, gold. 3. Direct and Indirect Band Gap Semiconductors Band Gap: Minimum energy difference between the top of the valence band and the bottom of the conduction band. Direct Band Gap: Top of valence band and bottom of conduction band occur at the same value of electron momentum. Efficient electron-hole pair generation and recombination (e.g., LEDs, semiconductor lasers). Indirect Band Gap: Maximum energy of valence band occurs at a different momentum value from the minimum in the conduction band. Requires interaction with a phonon for momentum change during electron-hole pair processes, making them less efficient. Example: Silicon. conduction band valence band band gap Energy Momentum band gap Momentum Energy 4. Intrinsic Semiconductors Definition: A perfect semiconductor crystal with no impurities or lattice defects. At $0 \text{ K}$, no charge carriers (valence band full, conduction band empty). At higher temperatures, electron-hole pairs (EHPs) are generated by thermal excitation across the band gap ($E_g$). Carrier Concentration: In intrinsic semiconductors, electron concentration ($n$) equals hole concentration ($p$), both equal to the intrinsic carrier concentration ($n_i$). $$n = p = n_i$$ 5. Extrinsic Semiconductors (Doping) Doping: Adding controlled impurities to increase conductivity. n-type semiconductor: Formed by adding pentavalent impurities (donor impurities like As, P) to a pure semiconductor. Donor atoms contribute an extra electron, creating an energy level ($E_D$) just below the conduction band. Electrons are majority carriers, holes are minority carriers. p-type semiconductor: Formed by adding trivalent impurities (acceptor impurities like B, Al, Ga, In) to a pure semiconductor. Acceptor atoms create a "hole" (vacancy) by accepting an electron, forming an energy level ($E_A$) just above the valence band. Holes are majority carriers, electrons are minority carriers. 6. Fermi Level and Fermi-Dirac Function Fermi-Dirac Distribution Function ($f(E)$): Gives the probability that an available energy state at energy $E$ will be occupied by an electron at absolute temperature $T$. $$f(E) = \frac{1}{1 + e^{(E - E_F) / kT}}$$ where $k$ is Boltzmann's constant, $E_F$ is the Fermi level. Fermi Level ($E_F$): The energy level at which the probability of occupancy by an electron is $1/2$. $$f(E_F) = \frac{1}{1 + e^{(E_F - E_F) / kT}} = \frac{1}{1 + e^0} = \frac{1}{2}$$ At $0 \text{ K}$, all states below $E_F$ are filled, and all states above $E_F$ are empty. $E_F$ is the highest occupied energy state. Fermi Level in Intrinsic Semiconductors The concentration of electrons in the conduction band ($n$): $$n = N_c \exp\left[-\frac{E_c - E_F}{kT}\right]$$ where $N_c = 2(2\pi m_n kT/h^2)^{3/2}$ is the effective density of states in the conduction band. The concentration of holes in the valence band ($p$): $$p = N_v \exp\left[-\frac{E_F - E_v}{kT}\right]$$ where $N_v = 2(2\pi m_p kT/h^2)^{3/2}$ is the effective density of states in the valence band. For intrinsic semiconductors, $n=p=n_i$. Equating $n$ and $p$ and solving for $E_F$: $$E_F = \frac{E_c + E_v}{2} + \frac{kT}{2} \ln\left(\frac{N_v}{N_c}\right)$$ If $m_n = m_p$, then $N_c = N_v$, and the Fermi level lies exactly in the middle of the band gap: $$E_F = \frac{E_c + E_v}{2}$$ Fermi Level in Extrinsic Semiconductors n-type material: With donor concentration $N_D$, assuming $p=0$ and $N_A=0$, $n=N_D$. $$N_D = N_c \exp\left[-\frac{E_c - E_F}{kT}\right]$$ Solving for $E_F$: $$E_F = E_c - kT \ln\left(\frac{N_c}{N_D}\right)$$ The Fermi level lies slightly below the conduction band. As $N_D$ increases, $E_F$ moves up. As $T$ increases, $E_F$ moves down towards the intrinsic level. p-type material: With acceptor concentration $N_A$, assuming $n=0$ and $N_D=0$, $p=N_A$. $$N_A = N_v \exp\left[-\frac{E_F - E_v}{kT}\right]$$ Solving for $E_F$: $$E_F = E_v + kT \ln\left(\frac{N_v}{N_A}\right)$$ The Fermi level lies slightly above the valence band. As $N_A$ increases, $E_F$ moves down. As $T$ increases, $E_F$ moves up towards the intrinsic level. 7. Law of Mass Action for Semiconductors For any semiconductor (intrinsic or extrinsic) at a given temperature, the product of electron and hole concentrations is constant and equal to the square of the intrinsic carrier concentration ($n_i^2$). $$np = n_i^2$$ This implies that if $n$ increases (e.g., due to doping), $p$ decreases to maintain the constant product, and vice versa. Derivation: $$n = N_c \exp\left[-\frac{E_c - E_F}{kT}\right]$$ $$p = N_v \exp\left[-\frac{E_F - E_v}{kT}\right]$$ $$np = N_c N_v \exp\left[-\frac{E_c - E_F}{kT}\right] \exp\left[-\frac{E_F - E_v}{kT}\right]$$ $$np = N_c N_v \exp\left[-\frac{E_c - E_v}{kT}\right] = N_c N_v \exp\left[-\frac{E_g}{kT}\right]$$ Since $n_i^2 = N_c N_v \exp\left[-\frac{E_g}{kT}\right]$, we have $np = n_i^2$. 8. Drift Velocity, Mobility, and Conductivity Drift Velocity ($v_d$): The average velocity acquired by charge carriers in the presence of an applied electric field ($E$). $$v_d \propto E \quad \Rightarrow \quad v_d = \mu E$$ Mobility ($\mu$): The constant of proportionality between drift velocity and electric field. It is the drift velocity per unit electric field. Units: $\text{m}^2 / (\text{V} \cdot \text{s})$. Current Density due to electrons ($J_n$): $$J_n = -nq v_{dn}$$ Since $v_{dn} = -\mu_n E$ (electron moves against field): $$J_n = -nq(-\mu_n E) = nq\mu_n E$$ Current Density due to holes ($J_p$): $$J_p = pq v_{dp}$$ Since $v_{dp} = \mu_p E$ (hole moves in field direction): $$J_p = pq\mu_p E$$ Electrical Conductivity ($\sigma$): For electrons: $\sigma_n = nq\mu_n$ For holes: $\sigma_p = pq\mu_p$ Total conductivity: $\sigma = \sigma_n + \sigma_p = q(n\mu_n + p\mu_p)$ For Intrinsic Semiconductors: $n=p=n_i$ $$\sigma = qn_i(\mu_n + \mu_p)$$ 9. p-n Junction Formation: Formed by joining p-type and n-type semiconductors. Depletion Region (Space Charge Region): Region near the junction devoid of mobile charge carriers, formed by diffusion of holes from p-side to n-side and electrons from n-side to p-side, followed by recombination. Leaves behind uncompensated acceptor and donor ions. Built-in Potential Barrier ($V_0$): An electric field is created across the depletion region, opposing further diffusion. This potential difference is necessary to maintain equilibrium. 10. Hall Effect and its Applications Definition: When a current-carrying conductor is placed in a magnetic field, a potential difference (Hall voltage) is developed across the material perpendicular to both the current and the magnetic field. Lorentz Force: Total force on a charge $q$ moving with velocity $\vec{v}$ in an electric field $\vec{E}$ and magnetic field $\vec{B}$: $$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$ Hall Electric Field ($E_y$): In steady state, for current in x-direction and magnetic field in z-direction, an electric field $E_y$ is established in the y-direction to balance the magnetic force: $$E_y = v_x B_z$$ Hall Voltage ($V_{AB}$): $V_{AB} = E_y w$, where $w$ is the width of the bar. Hall Coefficient ($R_H$): For holes (p-type): $$R_H = \frac{1}{q p_0}$$ For electrons (n-type): $$R_H = -\frac{1}{q n_0}$$ The sign of $R_H$ indicates the type of semiconductor (positive for p-type, negative for n-type). Applications: Measurement of Carrier Concentration ($p_0$ or $n_0$): By measuring Hall voltage, current density, and magnetic field: $$p_0 = \frac{J_x B_z}{q E_y} = \frac{I_x B_z}{q t V_{AB}}$$ where $t$ is the thickness. Measurement of Mobility ($\mu_p$ or $\mu_n$): Conductivity $\sigma = q p_0 \mu_p$. Using $p_0 = 1/(q R_H)$: $$\mu_p = \frac{\sigma}{q p_0} = \frac{1}{\rho q p_0} = \frac{R_H}{\rho}$$ where $\rho$ is resistivity.