Chapter 7. Quantum Mechanics Part 4 - Band Gap Theory
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4. Energy Bands and Brillouin Zone
5. E-k Diagram
6. Meaning of k
7. Introduction to Effective Mass Concept
9. Applications
1. Band Gap Theory
⑴ Overview
① Definition: Theory analyzing potential energy for confined atoms, not free particles
② Explains semiconductors
⑵ Fermi-Dirac Statistics
① Fermi-Dirac Distribution
○ Definition: Probability of energy level E being occupied by particles at arbitrary temperature T
○ Electrons in atomic lattice structure follow Fermi-Dirac distribution
② Fermi Level
○ Definition: Highest energy level electrons can occupy at 0 K
○ Fermi level defined in this way leads to electrons having a 50% probability of occupancy at any temperature
○ Electrons exceeding certain energy beyond Fermi level become free electrons, ignoring atomic nucleus potential
③ Fermi Gas
○ Fermi Gas (Free Electron Gas): Ensemble of non-interacting fermions
○ According to Pauli Exclusion Principle, average energy of Fermi gas at absolute zero is greater than that of ground-state single particle
○ Degeneracy pressure: At absolute zero, Fermi gas pressure is nonzero due to Pauli Exclusion Principle
2. Bloch’s Theorem
⑴ Definition: Potential energy U(x) is periodic and U(x+a) = U(x)
⑵ Formulation
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 Formulation
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, existence of solutions → existence of solutions → non-existence of solutions → ···
④ Conclusion: Energy levels manifest 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 [Footnote:1]
Figure. 3. Formation of Energy Bands Due to Orbital Overlap [Footnote:2]
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 states exist
② Forbidden Band Gap: Continuous range without states (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 k = 0, ± π / (a + b): Slope of E-k diagram is 0 : Observed not only in Kronig-Penney model but also in reality
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 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 (Note) Real E-k diagram is complex, but generally parabolic at ends of bands
⑺ (Note) Free particles also have a parabolic E-k diagram (since E ∝ p2, p ∝ k)
6. Meaning of k
⑴ For free particles, k = wave vector, hk = momentum
⑵ For periodic potential, k = wave vector, 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
⑵ Near the bottom of all bands in E-k diagram, the curvature is downward : meff > 0 (electrons)
⑶ Near the top of all bands in E-k diagram, the curvature is upward : meff < 0 (holes)
⑷ End parts of bands generally form parabolas → Double differentiation results in a constant → 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: Metallic bonding → Single band
Input: 2019.09.08 21:15