○ A semiconductor has electrical conductivity between that of a conductor and an insulator, and its conductivity can be controlled depending on temperature, purity, etc.

⑵ Fermi–Dirac Statistics

① Fermi–Dirac distribution

○ Definition: The probability that an energy level E is occupied by a particle at an arbitrary temperature T.

○ Valence electrons in a crystal lattice follow the Fermi–Dirac distribution.

② Fermi level

○ Definition: The highest energy level that electrons in a solid can have at 0 K.

○ The Fermi level defined this way corresponds to an energy where, at an arbitrary temperature, the probability of electron occupation is one-half (50%).

○ A certain fraction of electrons with energies higher than the Fermi level ignore the atomic nucleus’s potential and become free electrons.

○ Conductor: The Fermi level lies at the boundary between the conduction band and the valence band.

○ Insulator: The Fermi level exists in the middle between the conduction band and the valence band.

○ Semiconductor: The Fermi level exists in the middle between the conduction band and the valence band.

③ Fermi gas

○ Fermi gas (free-electron gas): An ensemble of non-interacting fermions.

○ Due to the Pauli exclusion principle, at absolute zero temperature the average energy of a Fermi gas is greater than that of a single particle in the ground state.

○ Degeneracy pressure: Due to the Pauli exclusion principle, the pressure of a Fermi gas at absolute zero is nonzero.

⑶ Connection to Circuit Theory

① The size of the energy gap determines electrical conductivity and resistivity

○ Conductor: The valence band and the conduction band overlap.

○ At room temperature, many electrons can move freely between atoms.

○ Example 1: Silver has resistivity ρ = 1.59 × 10^-6 Ω·cm

○ Example 2: Copper has resistivity ρ = 1.67 × 10^-6 Ω·cm

○ Semiconductor: The band gap is small.

○ Example 1: Germanium has resistivity ρ = 50 Ω·cm

○ Example 2: Silicon has resistivity ρ = 250,000 Ω·cm; the energy gap is 1.12 eV.

○ Insulator: The band gap is large.

○ Example 1: Diamond has resistivity ρ = 10¹² Ω·cm

○ Example 2: Mica has resistivity ρ = 9 × 10¹² Ω·cm

② Resistance change with temperature

○ Conductor: Temperature increase → increased atomic vibration → resistance increases.

○ Semiconductor: Temperature increase → increased free electrons and holes → resistance decreases.

○ Insulator: Temperature increase → electrons detach from atoms → resistance decreases.

○ In semiconductors and insulators, the resistance-decreasing effect is larger than the resistance-increasing effect caused by increased atomic vibration.

○ Semiconductor-based devices (e.g., computer components) can experience temperature increases that induce overcurrent, which is why cooling systems are necessary.

2. Bloch’s Theorem

⑴ Definition: Potential energy U(x) is periodic and U(x+a) = U(x)

⑵ Formula

3. Kronig-Penney Model

⑴ Definition

Figure 1. Kronig-Penney Model

⑵ Assumptions

① Assumes crystal is infinite

② Assumes crystal structure has step-like potential

⑶ Formula

4. Energy Bands and Brillouin Zone

⑴ Interpretation 1: Mathematical Interpretation

① Potential energy f(ξ) oscillates

② cos k(a+b) takes values between -1 and 1

③ As ξ increases, non-existence of solutions → existence of solutions → non-existence of solutions → ···

④ Conclusion: Energy levels appear as bands.

⑵ Interpretation 2: Pauli Exclusion Principle

① Pauli Exclusion Principle: No more than one electron with the same quantum numbers can exist in a single orbital

② Many atoms together cause energy levels to overlap and shift slightly, forming energy bands

Figure 2. Splitting of Energy Levels Due to Orbital Overlap

Figure 3. Formation of Energy Bands Due to Orbital Overlap

⑶ Interpretation 3: Coulomb’s Law

① Electrons repel each other, causing slight splitting of energy levels

② Many atoms together cause energy levels to shift slightly, forming energy bands

⑷ Classification of Energy Bands

① Energy Band: Continuous range where solutions exist.

② Forbidden Band Gap: Continuous range without solutions (except for the initial range)

③ Band Gap: Energy required to transition electrons from valence band to conduction band

④ Energy bands further categorized into valence band and conduction band

⑤ (Note) Valence and Conduction Bands of a Single Atom

○ Magnesium atom’s valence band includes 1s, 2s, 2p, 3s orbitals

○ Magnesium atom’s conduction band includes 3p, 4s, 3d orbitals beyond 3s

5. E-k Diagram

Figure 4. E-k Diagram

⑴ k ∈ [-π / (a + b), π / (a + b)]

⑵ At k = 0, ± π / (a + b), the slope of the E-k diagram is zero: this is observed not only in the Kronig–Penney model but also in real materials.

⑶ E-k solutions outside the range selected by periodicity overlap with solutions within the selected range

⑷ Only two k values possible per allowed energy (due to bilateral symmetry at the origin)

⑸ When expanding the E-k diagram, it resembles a parabola and converges as energy increases

Figure 5. Expanded E-k Diagram

⑹ The actual E-k diagram is very complex, but near the band edges it generally takes a parabolic shape at both the top and the bottom of the band.

⑺ Free particles also have a parabolic E-k diagram (since E ∝ p², p ∝ k)

6. Meaning of k

⑴ For free particles, k = wavenumber, hk = momentum p

⑵ For periodic potential, k = wavenumber, hk = crystal momentum

⑶ Represents a constant related to interactions and momentum within the crystal, not actual electron momentum

7. Introduction to Effective Mass Concept

⑴ Effective Mass

⑵ Near the bottom of all bands in E-k diagram, the curvature is downward : m_eff 0 (electrons)

⑶ Near the top of all bands in E-k diagram, the curvature is upward : meff < 0 (holes)

⑷ Near the band edges, the dispersion is generally parabolic → taking the second derivative gives a constant → therefore the effective mass is constant.

Figure 6. Approximation of Band End as a Parabola

8. Carriers and Current

⑴ N atoms per band, 2 electrons per atom, at room temperature

Figure 7. N atoms per band, 2 electrons per atom, at room temperature

⑵ Band 4: All empty

⑶ Band 3: Mostly empty

⑷ Band 2: Mostly full

⑸ Band 1: All full

⑹ Band 4 has no electrons, net current = 0

⑺ Band 1 is fully occupied by electrons, net current = 0

⑻ Bands 2 and 3 must break symmetry for net current to exist

9. Applications

⑴ Why graphite conducts electricity: Electrons can move easily from HOMO to LUMO, similar to Na

⑵ Na has higher electrical conductivity than Mg: in metallic bonding, the orbitals form a single band.

Input: 2019.09.08 21:15

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Chapter 7. Quantum Mechanics Part 4 - Band Gap Theory

1. Band Gap Theory