Energy Band 7 In free electron model, electrons occupy positive energy levels from E=0 to higher values of energy. They are valence electron so called “Valence band”. The free electrons and other electrons can be thought as belonging to the whole crystal → band rather than the discrete energy level There are three ways to look at bands. 1. Bands resulting from a periodic potential. 2. Bands resulting from interacting atoms. <tight binding interaction> 3. Bands resulting from periodic perturbation of free electrons.<weak binding approx.> (a) bound electrons (b) free electrons (c) electrons in solid

One-Dimensional Periodic Potential

One-Dimensional Periodic Potential In a periodic lattice with V(x+a) = V(x), it is expected that the wavefunction solution will also show this periodicity. Since yk has the form of exp(ikx), we expect, Apply this requirement to values of x=-b and x=(a-b),

Kronig-Penney approximation: One-Dimensional Periodic Potential Kronig-Penney approximation:

Kronig-Penney approximation One-Dimensional Periodic Potential Kronig-Penney approximation

Kronig-Penney approximation One-Dimensional Periodic Potential Kronig-Penney approximation

Kronig-Penney approximation One-Dimensional Periodic Potential Kronig-Penney approximation 2. Width of energy band increases with energy 3. P is a measure of tightness of binding. Width of bands decreases as P↑ P=0 : free electron → no band (continuum) P → ∞ : discrete levels → no bands higher bands are wider energy bands for more tightly bound (ionic atom) electrons are narrower allowed energy bands end at

The Tight-binding Approximation As atoms come together Isolated atom a Discrete level Electrons : identified with a particular atom → effectively belong to the whole crystal (solid) No two electrons can have same energy → N atoms : energy level breaks up into slightly different levels

The Tight-binding Approximation N atoms Less tight → wider More tight → narrower Extend of split depends on the binding tightness N atoms → n different levels Belong to the whole crystal

The Tight-binding Approximation Na : 3s and 3p energy overlapping gives continuous band E 0.3 Rydberg 2.8 Rydberg 5.1 Rydberg (-63.4 ev) 81 Rydberg (-1041 ev) E a r Equilibrium position

Effect of Periodic Potential on Free Electrons Free electrons superimposed by a small periodic potential on the free electrons (introduce a small periodic potential as a perturbation in the Schroedinger equation) + small periodic potential → particular value of k can not propagate through the crystal because of the Bragg reflection Waves with is represented by Standing Wave → Open up the energy gaps Start from C.B. Purturbation → band (7.14) (7.15)

Effect of Periodic Potential on Free Electrons Periodic nature of the crystalline structure Translate to “Extended” representation “Reduced” representation : The first Brillouin Zone

Effect of Periodic Potential on Free Electrons Free electron like behavior Point : extrema in energy band

3-D Brillouin Zones Simple cubic bcc fcc

Density of States in a Band For free electrons, define the density of states N(E) such that N(E)dE as the number of orbital states lying between E and E+dE, as Nearly free electron model: → nearly parabolic upward at the bottom of the band approximately parabolic downward at the top. near the bottom of the band Near the top of the band : effective mass (b : bottom, t : top) : maximum energy of the band : corresponding value of

Density of States in a Band Consider the example of a typical band describing an s-state in a one-dimensional simple cubic crystal as calculate by the tight-binding model

Density of States in a Band (Fig. 7.6) Near the bottom of the band Near the top of the band In the middle equal energy surface is not spherical → distortion

Equal energy surfaces 2-dimensional crystal FCC

Density of States in a Band Metal Semiconductor or Insulator or Partially occupied Bands overlap The upper most band is filled

Summary of different band representation spherical

Electron Velocity E vs k curves correspond to the dispersion relationship for electron waves, the group velocity can be calculated, (7.21) + - (7.22) (7.23) (7.24) Velocity is zero at band extrema In free electron system, goes to infinite or increase as In thermal equilibrium, there are equal number of electron occupied states with positive velocity as there are with negative velocity; In completely filled band, equal number of → no net charge transport under an electric field → Insulator

Effective mass Effect of external force on electrons in solids ↑ what is the proportionality constant? If for 1-D The effective mass of an electron is the reciprocal of the curvature of the E vs. k plot. For 3-D,

Effective mass E ② ③ ① Greater the curvature, smaller the effective mass E E + - -

Effective mass For a free electron, (electrons at the extrema have effective mass like the inertia mass) at the bottom of a band at the top of a band A negative mass implies that the induced acceleration is on the opposite direction to the force that caused it The existence of an negative effective mass is the result of Bragg reflection effects coming from the crystal potential in which an electron acted on by force in one direction is actually accelerated in the opposite direction because it undergoes reflection at the zone face.

Holes Holes: missing electrons in a nearly filled band, have a positive effective mass and a positive charge. In a semiconductor, the band structure looks like In the presence of an electric field, electrons in the bottom of the conduction band and holes at the top of the valence band move in the opposite directions in real space (same sign mass but different sign charge), whereas electrons and holes both at the top of the valence band move in the same direction (different sign mass cancels different sign charge) E C.B V.B Holes (+m*, +q) “Bubble in water”

Holes Apply an electric field E, then the force is Initially located at in conduction band (CB) moves toward the negative direction, where the velocity ( dE/dk) is negative. Initially located at in the valence band (VB) moves toward the direction, where the velocity is positive. The holes at the top of the valence band have their k changed toward positive values of the hole E vs. k diagram, and hence they acquire velocity toward +x direction in the presence of the an electric field in the +x direction. -