Transcript Lecture 5
The Ig Nobel Prizes are “Booby Prizes”! Electronic Band Structures BW, Ch. 2; YC, Ch 2; & S, Ch. 2 ~ Follow BW & YC + input from S & outside sources where appropriate Band Structure E(k) • We are interested in the behavior of electronic energy levels in crystalline materials (any, not just semiconductors!) as a function of wavevector k or momentum p = ħk • Group Theory: The math of symmetry that is useful to simplify calculations of E(k). • A Mathematical Subject! Detailed math coverage & discussion will be kept to a minimum. Results & PHYSICS will be emphasized over math! • Many calculational methods exist: – They are highly sophisticated & computational! Basic knowledge assumed for this discussion: 1. You know that electron energies are quantized. 2. You’ve seen the simple Schrödinger Equation (from quantum mechanics) Note: If you are weak on this or need a review, please get & read an undergraduate quantum mechanics book! 3. You are familiar with the crystal structures of some simple crystals, such as those we recently discussed. Note: If you are weak on this or need a review, please get & read an undergraduate solid state book! 4. In a solid, the electronic energy levels form into regions of allowed energy (bands) & forbidden energy (gaps). See any undergraduate solid state physics book! • Overview: A qualitative/semi-quantitative discussion now. Later, a detailed & quantitative discussion. Electronic Energy Bands E(k) These describe the behavior of the electronic energy levels in crystalline materials as function of wavevector k or momentum p = ħk. • Recall that for FREE electrons, the energies have the simple form: E(k) = (p)2/2mo = (ħk)2/2mo (mo= free electron mass) • However, in crystalline solids: E(k) forms into regions of allowed energies (bands) & regions of forbidden energies (gaps). • As we’ll see soon, Often in solids, a good Approximation is: E(k) = (ħk)2/2m* m* is not mo! m* = m*(k) “effective mass” However, the true E(k) are complicated! • Accurate calculations of E(k) require sophisticated quantum mechanics, AND they also require sophisticated computational methods to obtain them numerically. Qualitative Picture of Bands in “r-Space” (as functions of position in the solid) One way to distinguish between solid types is by how the electrons fill the bands & by the band gaps. Electronic Energy Bands Qualitative Picture Allowed band Forbidden gap Allowed band Forbidden gap Allowed band Number of Atoms Electrons occupying quantized energy states around a nucleus must obey the Pauli Exclusion Principle. This prevents more than 2 electrons (of opposite spin) from occupying the same level. Semiconductors, Insulators, Conductors Full Band Empty Band All energy levels are occupied All energy levels are empty (contain electrons) It can be shown that Neither full nor empty bands participate in electrical conduction. • Consider the case of an Intrinsic Semiconductor (a material with no impurities, so there are no acceptors or donors). That is, there is no doping. • For such a material, the highest energy band that is completely filled with electrons (at T = 0 K) is called the highest Valence Band. • The next highest band is completely empty of electrons (at T = 0K) & is called the lowest Conduction Band. • The Energy Difference between the bottom of the Conduction Band & the top of the Valence Band is called the Fundamental Band Gap. (Sometimes also called “Band Gap” or “Bandgap”, or “Gap” ) • So, for such an Intrinsic Semiconductor, for energies near the Band Gap, A Qualitative Picture of the Bands in “rSpace” (as functions of position ) looks like the figure below. • Consider the Conduction Process for this material. Suppose that somehow, some electrons are added to the conduction band & some holes are added to the valence band. Electron Conduction • In an applied electric field, the electrons in the conduction band will move almost like free particles. Hole Conduction • Holes are treated similarly to positively charged particles in the valence band. In an applied field, they behave almost like free particles. Intrinsic Semiconductors • Consider a nominally pure semiconductor at T = 0 K. • There are no electrons in the conduction band • At T > 0 K a small fraction of electrons is thermally excited into the conduction band, “leaving” the same number of holes in the valence band. Calculated Si Bandstructure in k Space GOALS After this chapter, you should: Eg 1. Understand the underlying Physics behind the existence of bands & gaps. 2. Understand how to interpret this figure. 3. Have a rough, general idea about how realistic bands are calculated. 4. Be able to calculate energy bands for some simple models of a solid. Note Si has an indirect band gap! Brief Quantum Mechanics (QM) & Solid State (SS) Review • QM results that I must assume that you know! The Schrödinger Equation (time independent; see next slide) describes electrons – The solutions to the Schrödinger Equation result in quantized (discrete) energy levels for electrons. • SS results that I must assume that you know! – Solutions to the Schrödinger Equation in a periodic crystal give bands (allowed energy regions) & gaps (forbidden energy regions). Quantum Mechanics (QM) • The Schrödinger Equation: (time independent) Hψ = Eψ This is a differential eigenvalue equation. H Hamiltonian operator for the system (energy operator) E Energy eigenvalue, ψ wavefunction Particles are QM waves! |ψ|2 probability density; ψ is a function of ALL coordinates of ALL particles in the problem! The Physics Behind E(k) E(k) Solutions to the Schrödinger Equation for an electron in a solid. QUESTIONS Why (qualitatively) are there bands? Why (qualitatively) are there gaps? Bands & Gaps • These can be understood from two (very different) qualitative pictures! • The two pictures are models (& Opposite Limiting Cases) of the true situation. • An electron in a perfectly periodic crystalline solid: – The potential seen by this electron is perfectly periodic The existence of this periodic potential is the cause of the bands & the gaps! Qualitative Picture #1 “A Physicist’s viewpoint”- The solid is looked at “collectively” Almost Free Electrons (done in detail in Kittel’s Ch. 7!) For free electrons: E(k) = (p)2/2mo = (ħk)2/2mo Almost Free Electrons: • Start with the free electron E(k), add small (weakly perturbing) periodic potential V. • This breaks up E(k) into bands (allowed energies) & gaps (forbidden energy regions). • Gaps: Occur at the k where the electron waves (incident on atoms & scattered from atoms) undergo constructive interference (Bragg reflections!) Qualitative Picture #1 Forms the basis for REALISTIC bandstructure computational methods! • Starting from the almost free electron viewpoint & adding a high degree of sophistication & theoretical + computational rigor: Results in a method that works VERY WELL for calculating E(k) for metals & semiconductors! • An “alphabet soup” of computational techniques: – OPW: Orthogonalized Plane Wave method – APW: Augmented Plane Wave method – ASW: Antisymmetric Spherical Wave method – Many, many others The Pseudopotential Method (the modern method of choice!) Qualitative Picture #2 “A Chemist’s viewpoint” The solid is looked at as a collection of atoms & molecules. Atomic / Molecular Electrons • Atoms (with discrete energy levels) come together to form the solid. • Interactions between the electrons on neighboring atoms cause the atomic energy levels to split, hybridize, & broaden. Quantum Chemistry! • First approximation: Small interaction V! Occurs in a periodic fashion (the interaction V is periodic). Qualitative Picture #2 Quantum Chemistry! • First approximation: Small interaction V! Occurs in a periodic fashion (interaction V is periodic). • Groups of levels come together to form bands (& gaps). • The bands E(k) retain much of the character of their “parent” atomic levels (s-like & p-like bands, etc.) Gaps: • Also occur at the k where the electron waves (incident on atoms & scattered from atoms) undergo constructive interference (Bragg reflections!) Qualitative Picture #2 • Starting from the atomic / molecular electron viewpoint & adding a high degree of sophistication & theoretical & computational rigor Results in a method that works VERY WELL for calculating E(k) (mainly the valence bands) for insulators & semiconductors! (Materials with covalent bonding!) • An “alphabet soup” of computational techniques: – LCAO: Linear Combination of Atomic Orbitals method – LCMO: Linear Combination of Molecular Orbitals method – The “Tightbinding” method & many others. The Pseudopotential Method (the modern method of choice!) Bandstructure Theories in Crystalline Solids Pseudopotential Method Tightbinding (LCAO) Method Electronic Interaction Semiconductors, Metals Insulators Almost Free Molecular Electrons Electrons Isolated Atom, Free Atomic Electrons Electrons