Comprehensive Study Notes on Semiconductor Physics, Carrier Concentrations, and the Hall Effect

Fundamental Properties of Metals and Semiconductors

• Charge carrier densities differ by several orders of magnitude between metals and semiconductors:
  • Metals possess a very large density of free electrons on the order of $n \sim 10^{28}\,\text{m}^{-3}$, conferring high electrical conductivity and thermal conductivity.
  • Semiconductors have considerably lower electron densities, typically spanning $n \sim 10^6\,\text{m}^{-3}\text{ to }10^{13}\,\text{m}^{-3}$, which results in significantly lower electrical conductivities.
• Conduction mechanism distinctions:
  • In metals, only free electrons contribute to both electrical and thermal conduction.
  • In semiconductors, transport properties are governed by both electrons in the conduction band and holes in the valence band.
• Electrical parameters of representative elements and compounds:
  • Copper ($\text{Cu}$): Conductivity $\sigma = 5.98 \times 10^7\,(\Omega\text{m})^{-1}$, Resistivity $\rho = 1.67 \times 10^{-8}\,\Omega\text{m}$, Electron density $n = 8.47 \times 10^{28}\,\text{m}^{-3}$
  • Silver ($\text{Ag}$): Conductivity $\sigma = 6.30 \times 10^7\,(\Omega\text{m})^{-1}$, Resistivity $\rho = 1.58 \times 10^{-8}\,\Omega\text{m}$, Electron density $n = 5.86 \times 10^{28}\,\text{m}^{-3}$
  • Gold ($\text{Au}$): Conductivity $\sigma = 4.25 \times 10^7\,(\Omega\text{m})^{-1}$, Resistivity $\rho = 2.35 \times 10^{-8}\,\Omega\text{m}$, Electron density $n = 5.90 \times 10^{28}\,\text{m}^{-3}$
  • Zinc ($\text{Zn}$): Conductivity $\sigma = 1.68 \times 10^7\,(\Omega\text{m})^{-1}$, Resistivity $\rho = 5.95 \times 10^{-8}\,\Omega\text{m}$, Electron density $n = 13.2 \times 10^{28}\,\text{m}^{-3}$
  • Aluminum ($\text{Al}$): Conductivity $\sigma = 3.50 \times 10^7\,(\Omega\text{m})^{-1}$, Resistivity $\rho = 2.86 \times 10^{-8}\,\Omega\text{m}$, Electron density $n = 18.1 \times 10^{28}\,\text{m}^{-3}$
  • Germanium ($\text{Ge}$): Conductivity $\sigma = 2.17\,(\Omega\text{m})^{-1}$, Resistivity $\rho = 0.47\,\Omega\text{m}$, Carrier density $n = 2.2 \times 10^{13}\,\text{m}^{-3}$
  • Silicon ($\text{Si}$): Conductivity $\sigma = 1.56 \times 10^{-3}\,(\Omega\text{m})^{-1}$, Resistivity $\rho = 640\,\Omega\text{m}$, Carrier density $n = 8.72 \times 10^9\,\text{m}^{-3}$
  • Gallium Arsenide ($\text{GaAs}$): Conductivity $\sigma = 5 \times 10^{-8}\text{ to }10^3\,(\Omega\text{m})^{-1}$, Carrier density $n = 2.03 \times 10^6\,\text{m}^{-3}$

Electron Concentration in the Conduction Band

• Band occupancy conditions:
  • At absolute zero ($0\,\text{K}$), all energy states in the valence band (VB) are completely filled and all states in the conduction band (CB) are completely empty.
  • At room temperature, thermal energy provides sufficient excitation to promote a small concentration of electrons from the valence band into the conduction band across the forbidden energy gap $E_g$.

• Mathematical formulation of electron concentration:
  • The concentration of electrons $n$ in the conduction band is obtained by integrating the product of the density of states $g(E)\,dE$ and the Fermi-Dirac distribution function $f(E)$:     $n = \int g(E) f(E)\,dE$
  • The density of states in the conduction band, where energy states commence at the band edge $E_c$, is modified by replacing $E$ with $(E - E_c)$:     $g(E)\,dE = \frac{4\pi}{h^3} (2m_e^*)^{\frac{3}{2}} (E - E_c)^{\frac{1}{2}}\,dE$     where $m_e^*$ denotes the effective mass of an electron and $h$ is Planck's constant.
  • The Fermi-Dirac distribution function defines the probability of occupancy:     $f(E) = \frac{1}{1 + e^{\frac{E - E_F}{kT}}}$
  • At room temperature, the condition $(E - E_F) \gg kT$ holds for all states in the conduction band, allowing the exponential term to dominate the denominator:     $f(E) \approx e^{-\frac{E - E_F}{kT}}$
• Integration over the conduction band:
  • Energy levels in the conduction band extend from the bottom edge $E_c$ upwards to a maximum level defined by the work function $\beta$ ($E = E_c + \beta$). Because the number of available states up to $\beta$ is extremely large, the upper limit is approximated as $\infty$:     $n = \int_{E_c}^{\infty} \frac{4\pi}{h^3} (2m_e^*)^{\frac{3}{2}} (E - E_c)^{\frac{1}{2}} e^{-\frac{E - E_F}{kT}}\,dE$
  • Applying a change of variable: let $x = E - E_c$, yielding $dE = dx$. The term $E - E_F$ becomes $(E_c - E_F) + x$, transforming the integral limits from $0$ to $\infty$:     $n = \frac{4\pi}{h^3} (2m_e^*)^{\frac{3}{2}} e^{-\frac{E_c - E_F}{kT}} \int_0^{\infty} x^{\frac{1}{2}} e^{-\frac{x}{kT}}\,dx$
  • Using standard Gamma function evaluations, the integral is solved as:     $\int_0^{\infty} x^{\frac{1}{2}} e^{-\frac{x}{kT}}\,dx = (kT)^{\frac{3}{2}} \frac{\sqrt{\pi}}{2}$
  • Incorporating this result into the concentration expression:     $n = 2 \left( \frac{2\pi m_e^* kT}{h^2} \right)^{\frac{3}{2}} e^{-\frac{E_c - E_F}{kT}}$
  • Defining $N_c$ as the effective density of states in the conduction band:     $N_c = 2 \left( \frac{2\pi m_e^* kT}{h^2} \right)^{\frac{3}{2}}$$n = N_c e^{-\frac{E_c - E_F}{kT}}$
• Concentration of holes in the valence band:
  • An analogous derivation evaluates states in the valence band ending at the top edge $E_v$:     $p = N_v e^{-\frac{E_F - E_v}{kT}}$     where $N_v$ represents the effective density of states in the valence band, defined with hole effective mass $m_h^*$:     $N_v = 2 \left( \frac{2\pi m_h^* kT}{h^2} \right)^{\frac{3}{2}}$

Intrinsic Carrier Concentration and Fermi Level

• Carrier concentration balance:
  • In an intrinsic semiconductor, every electron excited to the conduction band leaves behind an identical hole in the valence band, establishing $n = p = n_i$.
  • By the law of mass action:     $n \times p = n_i^2$$n_i^2 = N_c N_v e^{-\frac{E_c - E_F}{kT}} e^{-\frac{E_F - E_v}{kT}} = N_c N_v e^{-\frac{E_c - E_v}{kT}}$
  • Substituting the forbidden band gap energy $E_g = E_c - E_v$:     $n_i^2 = N_c N_v e^{-\frac{E_g}{kT}}$
  • Under the assumption that $m_e^* = m_h^* = m$ (where $m$ is the free electron rest mass):     $N_c N_v = 4 \left( \frac{2\pi m kT}{h^2} \right)^3 = 4 \left( \frac{2\pi m k}{h^2} \right)^3 T^3$$n_i^2 = 2.322 \times 10^{43} \times T^3 e^{-\frac{E_g}{kT}}$
  • Expressed in terms of a constant $A$:     $n_i^2 = A T^3 e^{-\frac{E_g}{kT}}$     where $A = 2.322 \times 10^{43}\text{ m}^{-6}\text{K}^{-3}$.
  • Taking the square root gives the intrinsic carrier concentration $n_i$:     $n_i = A_0 T^{\frac{3}{2}} e^{-\frac{E_g}{2kT}}$     where $A_0 = \sqrt{A} = 4.819 \times 10^{21}\text{ m}^{-3}\text{K}^{-\frac{3}{2}}$.
• Position of the Fermi level in an intrinsic semiconductor:
  • Setting electron and hole concentrations equal ($n = p$):     $N_c e^{-\frac{E_c - E_F}{kT}} = N_v e^{-\frac{E_F - E_v}{kT}}$$\frac{e^{-\frac{E_c - E_F}{kT}}}{e^{-\frac{E_F - E_v}{kT}}} = \frac{N_v}{N_c}$$e^{\frac{2E_F - (E_c + E_v)}{kT}} = \frac{N_v}{N_c}$
  • Taking the natural logarithm on both sides:     $\frac{2E_F - (E_c + E_v)}{kT} = \ln\left( \frac{N_v}{N_c} \right)$$E_F = \frac{E_c + E_v}{2} + \frac{kT}{2} \ln\left( \frac{N_v}{N_c} \right)$
  • Substituting the ratio of effective densities of states:     $\frac{N_v}{N_c} = \left( \frac{m_h^*}{m_e^*} \right)^{\frac{3}{2}}$$E_F = \frac{E_c + E_v}{2} + \frac{3}{4} kT \ln\left( \frac{m_h^*}{m_e^*} \right)$
  • When effective masses are equivalent ($m_e^* = m_h^*$):     $\ln(1) = 0$$E_F = \frac{E_c + E_v}{2}$
  • The Fermi energy level in an intrinsic semiconductor lies precisely in the center between the top of the valence band and the bottom of the conduction band.

Fermi Level in Extrinsic Semiconductors

• Properties of n-type semiconductors:
  • Produced by doping an intrinsic semiconductor such as Silicon ($\text{Si}$) with pentavalent impurities (e.g., Phosphorus $\text{P}$, Arsenic $\text{As}$, Antimony $\text{Sb}$).
  • After four electrons participate in covalent bonds within the crystal lattice, the fifth valence electron remains extra and loosely bound to the donor atom with an ionization energy of $\sim 0.01\,\text{eV}$.
  • Ambient room temperature thermal energy ($kT \sim 0.025\,\text{eV}$) is sufficient to excite this fifth electron into the conduction band; hence, pentavalent impurities are called donor impurities.
  • Donor energy levels ($E_d$) are positioned slightly beneath the conduction band edge ($E_c$).
  • For Silicon, the separation between the bottom of the conduction band and the donor level is $E_c - E_d \sim 0.03\,\text{eV}$.
  • At $0\,\text{K}$, the Fermi level $E_F$ lies midway between the lowest level of the conduction band and the donor energy level:     $E_F = \frac{E_c + E_d}{2}$

• Properties of p-type semiconductors:
  • Produced by doping an intrinsic semiconductor with trivalent impurities (e.g., Boron $\text{B}$, Aluminum $\text{Al}$, Gallium $\text{Ga}$).
  • Covalent bonds are incomplete due to a deficiency of one valence electron, creating a hole that freely moves across the lattice.
  • At $0\,\text{K}$, all states in the valence band are filled, while the excess holes occupy energy levels directly above the top of the valence band ($E_v$), termed acceptor levels ($E_a$).
  • In Silicon, acceptor levels reside approximately $0.01\text{ to }0.04\,\text{eV}$ above the valence band edge ($E_a - E_v \sim 0.01\text{ to }0.04\,\text{eV}$).
  • At $0\,\text{K}$, the Fermi level lies midway between the top of the valence band and the acceptor energy level:     $E_F = \frac{E_v + E_a}{2}$

Electrical Conductivity and Band Gap Determination

• General conductivity relations:
  • Electrical conductivity in a semiconductor includes contributions from both electrons ($\sigma_n$) and holes ($\sigma_p$):     $\sigma = \sigma_n + \sigma_p = n e \mu_e + p e \mu_h$     where $n$ and $p$ are the respective carrier densities, $e$ is the elementary electron charge, and $\mu_e$ and $\mu_h$ denote electron and hole mobilities.
  • In an intrinsic semiconductor ($n = p = n_i$):     $\sigma_i = n_i e (\mu_e + \mu_h)$
• Temperature dependence of mobilities:
  • Mobilities decrease with rising temperature according to lattice scattering mechanisms:     $\mu_e = \alpha T^{-\frac{3}{2}}$$\mu_h = \beta T^{-\frac{3}{2}}$$\mu_e + \mu_h = (\alpha + \beta) T^{-\frac{3}{2}}$
  • Substituting $n_i = A_0 T^{\frac{3}{2}} e^{-\frac{E_g}{2kT}}$ into the conductivity expression:     $\sigma_i = \left[ A_0 T^{\frac{3}{2}} e^{-\frac{E_g}{2kT}} \right] e \left[ (\alpha + \beta) T^{-\frac{3}{2}} \right]$$\sigma_i = A_0 (\alpha + \beta) e \cdot e^{-\frac{E_g}{2kT}}$
  • Collecting constant terms into $B = A_0 (\alpha + \beta) e$:     $\sigma_i = B e^{-\frac{E_g}{2kT}}$
• Intrinsic resistivity and band gap evaluation:
  • Resistivity $\rho_i$ is the reciprocal of conductivity:     $\rho_i = \frac{1}{\sigma_i} = \frac{1}{B} e^{\frac{E_g}{2kT}} = C e^{\frac{E_g}{2kT}}$     where $C = \frac{1}{B}$.
  • Taking the natural logarithm of both sides yields a linear equation:     $\ln(\rho_i) = \ln(C) + \frac{E_g}{2kT} = \ln(C) + \left( \frac{E_g}{2k} \right) \left( \frac{1}{T} \right)$

• Experimental determination of $E_g$:
  • Measuring $\rho_i$ across a range of temperatures $T$ and plotting $\ln(\rho_i)$ against $\frac{1}{T}$ produces a straight line.
  • The slope of this line directly yields the energy band gap $E_g$:     $\text{Slope} = \frac{E_g}{2k}$$E_g = 2k \times \text{Slope}$     where $k$ is the Boltzmann constant.

The Hall Effect: Principles and Mathematical Derivation

• Definition:
  • When a current-carrying material is placed within a transverse magnetic field applied perpendicular to the direction of electric current, an electric field is induced along a direction perpendicular to both the current and the magnetic field.

• Theory and derivation for an n-type semiconductor:
  • Consider a rectangular semiconductor specimen configured along Cartesian axes:
  • Current $I$ flows along the $+x$-axis.
  • An external magnetic field $B$ is directed along the $+z$-axis.
  • Electrons represent majority charge carriers, moving with drift velocity $V$ in the $-x$-direction.

• Force balance and equilibrium:
  • Electrons moving through magnetic field $B$ experience a downward Lorentz force $F_L$:     $F_L = -B e V$
  • By the right-hand rule, electrons drift toward the lower face (face 1), charging face 1 negatively relative to the top face (face 2).
  • This asymmetric charge accumulation creates an electrostatic potential difference $V_H$ (Hall voltage) between the faces, producing an induced Hall electric field $E_H$ directed along the $+y$-direction.
  • The induced Hall electric field exerts an opposing electrostatic force $F_H$ on electrons:     $F_H = -e E_H$
  • At dynamic equilibrium, the magnetic Lorentz force balances the electrostatic force:     $F_L = F_H$$-B e V = -e E_H$$E_H = B V$
• Derivation of Hall voltage:
  • If $d$ represents the separation distance between the top and bottom faces (along the $y$-axis), the uniform Hall electric field is related to the potential difference by:     $E_H = \frac{V_H}{d}$$V_H = E_H d = B V d$
  • Let $w$ represent the slab dimension along the $z$-axis (thickness parallel to the magnetic field). The cross-sectional area through which current flows is $A = w d$.
  • Current density $J$ is given by:     $J = \frac{I}{w d}$
  • In terms of carrier concentration $n$ or charge density $\rho$:     $J = n e V = \rho V$$\rho V = \frac{I}{w d} \implies V = \frac{I}{\rho w d}$
  • Substituting this expression for drift velocity $V$ into the Hall voltage formula:     $V_H = B \left( \frac{I}{\rho w d} \right) d$$V_H = \frac{B I}{\rho w}$
  • By measuring $V_H$ for known values of current $I$, magnetic field $B$, and dimension $w$, the charge density $\rho$ can be calculated.
• The Hall Coefficient ($R_H$):
  • The induced Hall field $E_H$ is directly proportional to both current density $J$ and magnetic field $B$:     $E_H \propto J B$$E_H = R_H J B$     where $R_H$ is defined as the Hall coefficient.
  • Equating $E_H = B V$ with $V = \frac{J}{\rho}$:     $E_H = B \left( \frac{J}{\rho} \right) = \left( \frac{1}{\rho} \right) J B$$R_H = \frac{1}{\rho} = \frac{1}{n e}$
  • Expressing Hall voltage in terms of $R_H$:     $V_H = \frac{R_H B I}{w}$
• Practical applications of the Hall effect:
  • Identifying semiconductor type: inspecting the polarity of accumulated charges across faces distinguishes n-type materials (negative bottom face) from p-type materials (positive bottom face).
  • Quantifying carrier concentrations: calculating the density of majority carriers ($n$ or $p$) from $R_H = \frac{1}{n e}$.
  • Determining carrier mobility: measuring conductivity $\sigma = n e \mu$ alongside the Hall coefficient yields carrier mobility via $\mu = \sigma R_H$.
  • Measurement of magnetic fields: using known semiconductor sensors to compute magnetic field strength $B$ from measured Hall voltages.