Semiconductors and Atomic Models

Introduction to Semiconductors

• Controlling electron movement in solids forms the basis of semiconductor device engineering
• Modern electronic systems, like computers and communication networks, rely on semiconductors

Atomic Models

• Democritus, around 460-370 BC, had early atomic ideas
• Dalton's model appeared in 1805
• Rutherford's model was introduced in 1911
• JJ Thomson's plum pudding model came out in 1904
• Bohr's model dates to 1913
• The quantum theory of atoms emerged in the late 1920s, with key figures like Heisenberg and Schrödinger
• Chadwick experimentally demonstrated the existence of neutrons in 1932

Bohr Model for Hydrogen Atom

• The force (F) equation is F = (Q1 * Q2) / (4 * pi * epsilon_0 * r^2) = -q^2 / (4 * pi * epsilon_0 * r^2)
• Potential energy F = -VEp = - dEp/dr
• Change in potential energy calculates as dEp = dEp(r) = -Fdr = (q^2 dr) / (4 * pi * epsilon_0 * r^2)
• At infinite distance Ep(r = ∞) is equal to Evac
• The potential energy equals Ep = Evac - q^2 / (4 * pi * epsilon_0 * r)
• Derived from the radius equation, rn = (4 * pi * epsilon_0 * n^2 * h^2) / (m * q^2)
• The Bohr radius is at the nth state

Quantized Energy States

• Energy is quantized, having discrete values linked to the quantum number n.
• n = 1 refers to the smallest radius and energy in the Bohr model.
• n = 2 represents the next larger values, and so on.
• The first four Bohr energies for hydrogen atom are: E1 = Evac - 13.6 eV, E2 = Evac - 3.40 eV, E3 = Evac - 1.51 eV, E4 = Evac - 0.850 eV.

Covalent and Ionic Bonding in Semiconductors

• Silicon bonds covalently
• Compound semiconductors, like gallium arsenide (GaAs), consist of regular arrangements of different elements
• In GaAs, gallium from column III has three outer electrons, while arsenic from column V has five, forming bonds with eight electrons

Energy Bands

• Vacuum energy is Evac
• Electron affinity is χ
• Ionization energy is γ
• Energy gap is Eg
• At absolute zero (0 Kelvins), every electron is in the lowest possible energy state
• In a perfect semiconductor, the valence band is completely occupied; the conduction band is empty
• Examples of electron affinity and band gap values: Si (4.05 eV, 1.12 eV), GaAs (4.07 eV, 1.43 eV), Ge (4.0 eV, 0.67 eV)
• Empty states in the valence band are called "holes" that can move around
• An insulator, semiconductor and metal have different energy band diagrams

Homogeneous Semiconductors

• Homogeneous semiconductors consist of one uniform material, such as pure silicon or silicon with uniformly distributed impurities
• Quasi-free electron concept is used to explore some of the electronic properties of homogeneous semiconductors
• Classical mechanics isn't applicable when understanding the behavior of electrons in crystals
• Velocity of the electron in the crystal is v = (1 / h) * (dE / dK)
• Formula m* = h^2 * [d^2E / dK^2] ^-1

Conduction and Valence Band Structure

• E-K diagrams exist for common semiconductors like GaAs, Silicon, and Germanium
• For an element such as GaAs: the slope of the E-K curve must be zero at the Brillouin zone edge, unless multiple bands coincide there.
• The bottom of the conduction band is designated Ec
• The top of the valence band is designated Ev

Intrinsic Semiconductors

• Extrinsic semiconductors incorporate impurity atoms via doping
• In intrinsic semiconductors, conduction band electron concentration equals valence band holes concentration
• A semiconductor is n type if no > po, meaning current is carried by negatively charged electrons
• A semiconductor is p type if po > no, meaning current is carried by positively charged holes

Donors and Acceptors

• Donors are impurities that, when added to a semiconductor, increase the number of free electrons
• Acceptors are impurities that, when added to a semiconductor, create holes

The Concept of Holes

• Hole charge is described by the following equations:
• J = -qn
• J = -(q/volume)*Σvi
• With unopposed electrons, the equations become:
• J = -(q/volume)*Σvui
• J = -(q/volume) * Σ[vi - vhi] = -(q/volume) * [Σvi - Σvhi]
• J = +(q/volume)*Σvhi
• J = +qp

Fermi-Dirac Statistics

• The probability of an electron occupying a state at energy E is given by the Fermi-Dirac probability function: f(E) = 1 / [1 + e^(E-Ef)/kT]

Degenerate Semiconductors

• Degenerate semiconductors have a high doping level and behave more like metals
• Non-degenerate semiconductors have a moderate doping level
• Non-degenerate semiconductors follow the equation: n0p0 = ni^2
• Degenerate semiconductors follow the the equation: n0p0 = ni^2 * e^(ΔEf/kT)
• If material is sufficiently degenerate, the Fermi level lies within the conduction band (n type) or within the valence band (p type)

Temperature Dependence of Carrier Concentration

• For Nondegenerate Semiconductors assume that all donors and acceptors are ionized, following equations:
• n0 = ND - NA, where ND > NA (n type)
• p0 = NA - ND, where NA > ND (p type)
• The minority carrier concentration was then determined from the relation -- n0p0 = ni^2

Temperature Dependence of Eg

• Intrinsic carrier concentration n₁ equation: n₁ = √(NcNv) * e^(-Eg/2kT)
• Nc = 2.54 x 10^19 * (mdse/m0)^(3/2) * (T/300)^(3/2) cm^-3
• Nv = 2.54 x 10^19 * (mdsh/m0)^(3/2) * (T/300)^(3/2) cm^-3
• Temperature dependence of Eg for Si: Eg(T) = 1.170 – (4.73 x 10^-4 * T^2) / (T + 636) eV

Impurity Band Mobility

• Majority carriers in n-type semiconductors can drift in the impurity band with reduced mobility

Temperature Dependence of Mobility

• At low temperatures, impurity scattering dominates.
• At high temperatures, lattice vibrations dominate.

Currents

• Types of currents in a semiconductor are drift current and diffusion current
• Drift current results when electrons and holes are in an electric field
• Diffusion current arises when there is a variation in the concentration of carriers with position
• Diffusion describes mobile particles moving from high to low concentration regions due to random motion

Drift Current

• The motion of an electron in a crystal changes direction randomly whenever it makes a collision
• Total drift current is the sum of electron and hole drift currents: I(drift) = In(drift) + Ip(drift)
• Drift velocity is influenced by impurity concentrations, temperature and minority/majority carrier types

Electron and Hole Mobility

• Hole Mobility equation is μp = vdp / E
• Electron Mobility equation is μη = vdn / E
• Total conductivity equation is σ = qημη + qpup
• Electron and hole mobilities are dependent on: Impurity concentrations of donors and acceptors, temperature, and minority / majority carrier types
• Empirical formula for mobility: μ = μ0 + μ1 / [1 + (N / Nref)^α]

Majority/Minority Carriers

• At low impurity concentrations, majority and minority carrier electron mobilities approach the same values: around 1330 cm²/V • s.
• A similar result holds for holes: around 495 cm²/V · s.
• Electron and hole mobilities (both majority and minority carrier) reduce monotonically with increasing impurity concentration.
• For a given doping level, minority carrier mobilities for electrons and holes are greater than corresponding majority carrier mobilities.
• These fractional differences increase with increased doping.

Carrier Scattering and Recombination

• In generation, a valence electron acquires extra energy and moves into the conduction band
• Generation is creating electron-hole pairs (exciting electron from valence band to conduction band)
• Recombination (process where electron from conduction band is moved to the valence band), annihilates an electron-hole pair
• With two-step generation (electron and hole is generated), there is a Two-step process involving trap state or acceptor states

Optical Processes

• Optical processes in semiconductors involve absorption and emission, affecting current due to excess carrier concentrations.

Minority Carrier Lifetime

• Measures the average time, τn, an electron spends in the conduction band of a p-type semiconductor

Carrier Scattering

• Mobility is dependent on the drift velocity in the formulas:
• μp = vdp / E
• μη = vdn / E
• The carrier drift velocity is influenced by scattering events, change in direction and/or energy of a carrier by collisions with a particle

Scattering Mechanisms

• Carriers can be scattered by interactions with "particles" such as ionized impurity atoms and by phonons
• Vibrating atoms create acoustic waves in the crystals (phonons)
• The drift velocity and thus the mobility are dependent on the mean free time between collisions
• Mobility is related to effective mass and scattering time
• By single scattering mechanism like ionized impurity scattering and phonon scattering, mobility (u) is proportional to the mean free time (T).
• High T means High u
• Equation for electron mobility (scattering rates): un=qTn/mce*