#### Transcript Last Time - West Virginia University

Group: Linear Chain of F F: 1s22s22px22py22pz1 F F F (a) F F F (b) F F (c) F F (d) EF E(k) EF EF 0 k p/a 0 k p/a 0 EF k p/a 0 k Which of the following is the correct band structure for a linear chain of F atoms (atomic #=9)? p/a Linear F Chain • There are 4 n=2 orbitals in the unit cell (a single F atom with 1 2s + 3 2p orbitals) • There is a lower 1s band (not shown) EF E(k) EF EF 0 k p/a 0 k p/a 0 EF k p/a 0 k p/a Band Structure: Linear Chain of F Bonding 2s s Antibonding 2s s* A sigma bond shares only one electron Antibonding 2pz s* Bonding 2pz s Which one of these has lower energy? Linear F Chain • There are 4 n=2 orbitals in the unit cell (a single F atom with 1 2s + 3 2p orbitals) • The fact that the wavefunction corresponding to a p-orbital changes sign at the nucleus causes the 2p s band to run downhill (opposite of the 2s s band). Band Structure: Linear Chain of F Bonding 2s s Antibonding 2s s* A sigma bond shares only one electron Antibonding 2pz s* Bonding 2pz s Which of these two p orbitals is doubly degenerate (which will we have two of)? A pi bond shares two electrons Bonding 2px/2py p Antibonding 2px/2py p* Group: Linear Chain of F F: 1s22s22px22py22pz1 F F F (a) F F F (b) F F (c) F F (d) EF E(k) EF EF 0 k p/a 0 k p/a 0 EF k p/a 0 k How do we determine between the final remaining two? p/a Linear F Chain • Do the sigma and pi p orbitals have the same band width? What affects bandwidth? • For the same lattice parameter, the reduced spatial overlap of the p interaction causes the p bands to be narrower than the s bands. Band Structure: Linear Chain of F Antibonding 2pz s* Doubly degenerate EF Antibonding 2px/2py p* Doubly degenerate Antibonding 2s s* Bonding 2px/2py p Bonding 2pz s Bonding 2s s 0 k p/a Band Structure: Linear Chain of F A more accurate treatment of the band structure would show an avoided crossing between the 2pz s and 2s s * interactions at k=p/a. There would be mixing between these two bands (creating sphybrid like states). Antibonding 2pz s* EF Doubly degenerate Bonding 2px/2py p Bonding 2s s 0 k p/a Band Overlap Flat band diagrams • Many materials are metals due to band overlap • Often the higher energy bands become so wide that they overlap with the lower bands Band Hybridization In some cases the opposite occurs – Due to the overlap, electrons from different shells form hybrid bands, which can be separated in energy – Depending on the magnitude of the gap, solids can be insulators (Diamond); semiconductors (Si, Ge, Sn) Learning Objectives for Today After today’s class you should be able to: • Calculate effective mass and cyclotron frequency • Understand the difference between an electron and an electron quasiparticle • Distinguish between two types of excitons Eph>Eg Conduction band Energy gap The ground state is sometimes called the vacuum state Electron energy Flat band structure in semiconductors - Forbidden band Valence Band = Dirac Sea of Electrons If the photon energy is higher than the energy gap the electron can be excited Quasiparticles 2 2 k E 2m* • Quasiparticles occur when a solid behaves as if it contained free particles. • Example: as an electron travels through a semiconductor, its scatters with electrons and nuclei; however it ~behaves like an electron with a different mass traveling unperturbed through free space. This "electron" with a m* is called an "electron quasiparticle". (m* =Effective mass) • Aggregate motion of electrons in the valence band of a semiconductor is the same as if the semiconductor contained positively charged quasiparticles called holes or positrons. • • • • - Conduction band Energy gap E Electron energy Electron energy band structure in semiconductor + Forbidden band Valence band Quasiparticles: excitations of the ground state (vacuum state), behave as if particles in free space Excited electron leaves in the valence band positive hole/positron. Electron and positron are quasiparticles and charge carriers (equal) Positively charged hole interacts with negatively charged electron by Coulomb interaction. Excitons (quasiparticle) are bound electron-hole states A free electron and a free hole (empty electronic state in the valence band) exert Coulomb force on each other: hydrogen-like bound states possible: excitonic states n=3 n=2 n=1 e Coulomb force E Eb is the exciton binding energy = h energy released upon exciton formation, or Eb k energy required for exciton breakup Note: exciton can move through crystal, i.e. not bound to specific atom! Two Types of Excitons Wannier – Matt excitons (free exciton): mainly exist in semiconductors, have a large radius, are delocalized states that can move freely throughout the crystal, the binding energy ~ 0.01 eV Frenkel excitons (tight bound excitons): found in insulator and molecular crystals, bound to specific atoms or molecules and have to move by hopping from one atom to another, the binding energy ~ 0.1 -1 eV. The maximum energy of a thermally excited phonon ~ kBT = 0.025 eV (RT) Wannier – Matt A phonon is another excitons: quasiparticle stable at cryogenic temperature Frenkel excitons: stable at room temperature Excitons in most of semiconductors are not observable at room temperature, because of the low binding energy Quasiparticle in a box Light Hole and Heavy Hole bands (holes with different m*) On the other hand, excitonic emission is very important for opto-electronic applications, as it is narrow and highly energetic Confused electron Let’s focus on Effective Mass Confused hole Comparing free electrons and the electron quasiparticle Free electron 2 2 k E 2me Electron quasiparticle 2 2 k E * 2me While electrons scatter, we treat electron quasiparticles like free electrons with m*. The renormalized mass effectively takes into account all interactions (with crystal, electrons, phonons) Thus, electron quasiparticles don’t scatter. EFFECTIVE MASS Real metals: electrons still behave like free particles, but with “renormalized” effective mass m* In potassium (a metal), assuming m* =1.25m gets the correct (measured) electronic heat capacity 2 2 k E * 2me Physical intuition: m* > m, due to “cloud” of phonons and other excited electrons that slow it down (add mass) Fermi Surface Interactions with the periodic crystal, electron-electron interactions and electron-phonon interactions renormalize the elementary excitation to an “electron-like quasiparticle” of mass m* Physical Meaning of the Band Effective Mass 2 2 k E * 2m The effective mass is inversely proportional to the curvature of the energy band. Near the bottom of a nearly-free electron band m* is approximately constant, but it increases dramatically near the inflection point and even becomes negative(!) near the zone edge. What does that mean for electron near BZE? Because the energy bands are different along different directions, the effective mass depends on which direction in kspace we are “looking” Effective Mass 2 Heavy and1light holes 1 dhave E different 2 masses bands so different 2effective m* from (also different electrons) dk Group: Find the effective mass tensor for electrons in a simple cubic tight-binding band at the center , face center X, and at the corner R of the Brillouin zone. 1 1 2E 2 m * ki k j i, j x, y, z Besides Getting the Energy Bands, How Could You Measure the Effective Mass? Deflection of Electrons in a Uniform Magnetic Field (1) The force F acting on an electron in a uniform magnetic field is given by F ev B Since the magnetic force F is at a right angle to the velocity direction, the electron moves round a circular path. Deflection of Electrons in a Uniform Magnetic Field (2) F Bev ma The centripetal acceleration of the electrons is Bev a m v 2 Bev Hence a r m mv r eB which gives Cyclotron frequency Effective Mass Measure effective mass: cyclotron resonance v. crystallographic direction -Measure the absorption of radio frequency energy v. magnetic field strength. eB c m* Put sample in a microwave resonance cavity at <40 K and adjust the rf frequency until it matches the cyclotron frequency. Will see a resonant peak in the energy absorption. Effective Masses in Semiconductors v hh = 2 2 k 2mhh m* determined by cyclotron resonance k lh = 2m (rf) at low carrier k concentration soh = 2m 2 2 v lh 2 2 v soh m* Eg for direct-gap crystals For InSb, InAs, InP mc 0.015 0.026 0.073 , , m Eg 0.23 0.43 1.42 0.065, 0.060, 0.051 Prep for Homework (due Thursday after break) Hint: • How do constants affect the effective mass? • Is there a way to get around dealing with M-1? • Any way to simplify matrix given H direction? • The formula for an ellipse (the orbit) may help. Why an ellipse? Extra Slides • Depending on time, I might throw the next three slides into another lecture as it shows why a full band doesn’t contribute to the electrical conduction. • You are also welcome to look through the argument yourself. Dynamics of Bloch Electrons (electrons in a periodic potential) Metals have partially-filled upper bands, while semiconductors and insulators have filled upper bands. What is the qualitative difference between filled and partially-filled bands? J ne v (k ) For a collection of Current density for j nev k electrons: single electron: But the symmetry of the energy bands requires: E (k ) E (k ) Thus we conclude: So for a filled band, which J ne v (k ) 0 has an equal number of 1stBZ electrons with k positive and negative, v (k ) v(k ) Filled energy bands carry no current! We will see that this is true even when an electric field is applied. Note: the electrons in filled bands are not stationary…there are just the same number moving in each direction, so the net current is zero. Electron in an Electric Field An external electric field causes a change in the k vectors of all electrons: dk dk eE F eE dt dt If the electrons are in a partially filled band, this will break the symmetry of electron states in the 1st BZ and produce a net current. But if they are in a filled band, even though all electrons change k vectors, the symmetry remains, so J = 0. E pa v p a kx When an electron reaches the 1st BZ edge (at k = p/a) it immediately reappears at the opposite edge (k = p/a) and continues to increase its k value. kx As an electron’s k value increases, its velocity increases, then decreases to zero and then becomes negative when it re-emerges at k = -p/a!! Thus, an AC current is predicted to result from a DC field! (Bloch oscillations) Do We Observe This? Not until fairly recently, due to the effect of collisions on electrons in a periodic but vibrating lattice. In both experiments the periodic potential was fabricated in an artificial way to minimize the effect of collisions and make it possible to observe the Bloch oscillations of electrons (or atoms!). Discuss presentation grading • 10 minutes, no more than 5 PPT slides • Board work acceptable but visuals encouraged Graded upon: • Explains the interest of the topic to the average taxpayer including possible applications • Explains at least 2 important findings in the field with appropriate references • Relates topic to some portion of the class • All displayed visuals are at least briefly mentioned • No more than 20 words per slide • The main point per slide is stated orally and maybe briefly in words (recommend as title) Now we can derive a bandstructure (E-k diagram), what information can we get? Effective mass, group velocity. k describes electron’s response to the external force (only) Absorption/emission of photon - Vertical transition Absorption/emission of phonon - horizontal transition The onset of indirect transition should include both Eg-Ephonon, Eg+Ephonon, In any transition, K must be conserved as well as E. (a) A direct gap semiconductor; on the left is the E-K diagram, and on the right the conventional energy band diagram. (b) An indirect gap material (so called because conduction band minimum and the valence band maximum do not occur at the same value of K). Eg & m* versus T Strong dependence of Eg on T Weak dependence of m* on T Indirect Bandgap Direct Bandgap Spin-orbit split band energy ~ 10 – 1000 meV, increases with decreasing Eg.