Chem 253, UC, Berkeley Fermi Surface E 2k 2 F E(k) x 2m K KF a2 r 2 2 r 0.4a Chem 253, UC, Berkeley With periodic boundary conditions: ik L eikx Lx e y y eikz Lz 1 2nx kx Lx 2n k y 2 2 y 2D k space: Ly Area per k point: L L 2nz x y kz Lz 2 2 2 8 3 3D k space: Area per k point: Lx Ly Lz V V A region of k space of volume will contain: allowed 8 3 8 3 k values. ( ) V 1 Chem 253, UC, Berkeley For divalent elements: free –electron model a2 r 2 r 0.56a Chem 253, UC, Berkeley 2 Chem 253, UC, Berkeley For nearly free electron: 1. Interaction of electron with periodic potential opens gap at zone boundary 2. Almost always Fermi surface will intersect zone Boundaries perpendicularly. 3. The total volume enclosed by the Fermi surface depends only on total electron concentration, not on interaction Chem 253, UC, Berkeley Alkali Metal Na, Cs: spherical Fermi surface a2 r 2 2 r 0.4a Alk. Earth metal: Be, Mg:: nearly spherical Fermi surface a2 r 2 r 0.56a 2D case 3 Chem 253, UC, Berkeley Chem 253, UC, Berkeley 4 Chem 253, UC, Berkeley Chem 253, UC, Berkeley 5 Chem 253, UC, Berkeley Chem 253, UC, Berkeley 6 Chem 253, UC, Berkeley Chem 253, UC, Berkeley 7 Chem 253, UC, Berkeley Brillouin Zone of Diamond and Zincblende Structure (FCC Lattice) Sign Convention Zone Edge or surface : Latin alphabets Interior of Zone: Greek alphabets Center of Zone or origin: Chem 253, UC, Berkeley Band Structure of 3D Free Electron in FCC in reduced zone scheme Notation: <=>[100] direction X<=>BZ edge along [100] direction <=>[111] direction E(k)=( 2/2m) (k 2+ k 2 + k 2) x y z L<=>BZ edge along [111] direction 8 Chem 253, UC, Berkeley Comparison between Free Electron and Real Electron Band Structure of Si Chem 253, UC, Berkeley v e d E m* 9 Chem 253, UC, Berkeley h Chem 253, UC, Berkeley 10 Chem 253, UC, Berkeley Wave packet made of wavefunctions near a particular wavevector k 2 2 k E Ec * 2me 2k 2 E E v * Motion of carrier in field: 2mh Parabolic Group velocity: transmission velocity of a wave packet d 1 dE vg dk dk Acceleration: 2 2 d g 1 d E 1 d E dk a 2 dt dkdt dk dt Chem 253, UC, Berkeley Wave packet made of wavefunctions near a particular wavevector k 2 2 k E Ec * 2me Motion of carrier in field: 2 2 k E Ev * 2mh Group velocity: transmission velocity of a wave packet Parabolic d 1 dE vg dk dk d 1 d 2E 1 d 2E dk Acceleration: g a 2 dt dkdt dk dt Effective mass 2 p m*v k 1 1 d E dv dk 2 2 dt m* dt m * dk 11 Chem 253, UC, Berkeley Chem 253, UC, Berkeley 1 1 d 2E 2 2 m * dk d 2E Positive m*: the band has upward curvature 0 dk 2 If the energy in a band depend only weakly on k, then m* very large d 2E m*/ m 1 When very small. dk 2 Heavy carrier 12 Chem 253, UC, Berkeley 1 1 d 2E 2 2 m * dk 2k 2 2 E(k) (k 2 k 2 k 2 ) 2m 2m x y z Chem 253, UC, Berkeley 1 1 d 2E 2 2 E m * dk k 13 Chem 253, UC, Berkeley Heavy hole Light hole Chem 253, UC, Berkeley Group Material Electron me Hole mh Si (300K) 1.08 0.56 IV Ge 0.55 0.37 GaAs 0.067 0.45 III-V InSb 0.013 0.6 ZnO 0.29 1.21 II-VI ZnSe 0.17 1.44 14 Chem 253, UC, Berkeley Excitons The annihilation of a photon in exciting an electron from the valence band to the conduction band in a semiconductor can be written as an equation: e+h. Since there is a Coulomb attraction between the electron and hole, the photon energy required is lowered than the band gap by this attraction To correctly calculate the absorption coefficient we have to introduce a two-particle state consisting of an electron attracted to a hole known as an exciton Chem 253, UC, Berkeley – Excitons represent the elementary excitation of a semiconductor. In the ground state the semiconductor has only filled or empty bands. The simplest excitation is to excite one electron from a filled band to an empty band and so creating an electron and a hole – Exciton is neutral over all but carries an electric dipole moment and therefore can be excited by either a photon or an electron 15 Chem 253, UC, Berkeley v2 2 m e e 2 2 2 r r n h 2 rn 2 2 n a0 nh 4 m e v e 2mer o r1=Bohr radius =a0=0.529 A 1 e2 E m v2 ( ) 2 e r e2 2 2m e4 k 1 e2 e2 E e ( ) n 2r n2h2 n2 2 r r n e2 2r k= 13.606 eV n2h2 Chem 253, UC, Berkeley r n2a H n 2 2 0 4 mee e2 2 2m e4 E e Excitons n 2 2 2 2rn n h 2k 2 2 e 2 2 2 E(k) (kx k y kz ) U (r) 2m 2m r e4 En E 2 2 2 2 n 1 1 1 Reduced mass * * me mh Binding energy: e4 En E 2 2 2 13.6ev 2 2 n me 2 Exciton Bohr Radius: h 1 1 me rex 2 2 ( * * ) 0.529 4 e me mh 16 Chem 253, UC, Berkeley Rx E 13.6ev n 2m m r 0.529 Chem 253, UC, Berkeley Energy gap Effective mass Dielectric Semiconductor (eV) m*/m constant at 273 K Electrons Holes Ge 0.67 0.2 0.3 16 Si 1.14 0.33 0.5 12 InSb 0.16 0.013 0.6 18 InAs 0.33 0.02 0.4 14.5 InP 1.29 0.07 0.4 14 GaSb 0.67 0.047 0.5 15 GaAs 1.39 0.072 0.5 13 17 Chem 253, UC, Berkeley E 13.6ev n 2m m r 0.529 Exciton Semiconductor Eg ........ / m Rx or Eex rex * * (me / me ;mh / mh ) meV nm Si 1.11 0.33; 0.50 14.7 4.9 Ge 0.67 0.2; 0.3 4.15 17.7 GaAs 1.42 0.0616 4.2 11.3 (0.066, 0.5) CdSe 1.74 (0.13, 0.45) 15 5.2 Bi 0 0.001 small >50 ZnO 3.4 (0.27, ?) 59 3 GaN 3.4 (0.19, 0.60) 25 11 Chem 253, UC, Berkeley Exciton binding energy h h h otherwise dissociates Related to DOS 18 Chem 253, UC, Berkeley h Chem 253, UC, Berkeley 19 Chem 253, UC, Berkeley h Chem 253, UC, Berkeley 20 Chem 253, UC, Berkeley Chem 253, UC, Berkeley h h h 21 Chem 253, UC, Berkeley Implication in solar cell Chem 253, UC, Berkeley Thin film PV Si PV 22 Chem 253, UC, Berkeley Chem 253, UC, Berkeley 23 Chem 253, UC, Berkeley Chem 253, UC, Berkeley 24 Chem 253, UC, Berkeley Chem 253, UC, Berkeley h 25 Chem 253, UC, Berkeley Fermi’s Golden Rule: •Time-dependent perturbation theory: treat excitations which depend on time •Optical transition: view the solid with unperturbed Hamiltonian H0 as being perturbed by the time-dependent EM field H’(t) generated by the incident photon flux. H H 0 H '(t) Transition Rate 2 2 ml m H 'l (El Em ) is the photon energy +: emission -: absorption Atkins, Molecular Quantum Mechanics, Oxford Chem 253, UC, Berkeley 26 Chem 253, UC, Berkeley Fermi’s Golden Rule: Assume the state m and l are the valence and conduction band states, then m H 'l v H 'c H 'vc Define: Joint density of states Chem 253, UC, Berkeley S is the surface of all possible direct optical transitions with h = Ec - Ev Fermi’s Golden Rule: S Let's introduce an energy surface S in k-space such that Ec − Ev = dkn dk =dSdkn 2 vc () 3 dk (Ec (k) Ev (k) ) 8 27 Chem 253, UC, Berkeley Fermi’s Golden Rule: At critical point where 2 2 k E Ec * 2me 2 2 Large JDOS contribution. k E Ev * 2mh Chem 253, UC, Berkeley Fermi’s Golden Rule: 28 Chem 253, UC, Berkeley h h h Chem 253, UC, Berkeley Fermi’s Golden Rule: Spontaneous 2 2 emission rate () M if () Density of states M: transition matrix elements M V dv if i f Operator for the physical interaction that couples the initial and final states 29 Chem 253, UC, Berkeley Selection Rule: Electric Dipole (E1) Transition Light interaction with dipole moment (p=ex): H E p eEx Light as harmonic EM plane wave i(kxt) E(x,t) E0e In general, the wavelength of the type of electromagnetic radiation which induces, or is emitted during, transitions between different atomic energy levels is much larger than the typical size of an atom. 2 x kx x 0 ikx Electric dipole approximation e 1 Transition dipole moment H P (0) eE x dV mn m n V M x d 3r 12 1 2 M y d 3r For x, y, z polarized light 12 1 2 M z d 3r 12 1 2 Chem 253, UC, Berkeley 2 2 () M Selection Rule: Electric Dipole (E1) Transition M x d 3r 12 1 2 M y d 3r Dipole Moment 12 1 2 M z d 3r 12 1 2 Matrix element (dipole moment) is non-zero allowed electric dipole transition Parity of wavefunction: sign change under inversion about the origin even parity: f(-x)=f(x) odd parity: f(-x)=-f(x) Initial/final wavefunctions must have different parities for allowed electric dipole transition! 30 Chem 253, UC, Berkeley Electronic Transitions in H atoms Hydrogen atom: lowest state 1S, optical transition between 1S & 2S? Both states are symmetric, angular momentum l=0 No electronic transition between 1S and 2S! Chem 253, UC, Berkeley Electronic Transitions in H atoms Hydrogen atom: lowest state 1S, optical transition between 1S & 2P? 2P is asymmetric, angular momentum l=1 Electronic transition between 1S and 2P is allowed! 31 Chem 253, UC, Berkeley Selection Rules Symmetric function (gerade): (x) (x) Asymmetric function (ungerade): (x) (x) The operator of the electric field: -(x)=-x Transition between two gerade functions: P H 21(0) gugdV udV 0 Forbidden Transition between gerade and ungerade functions: P Allowed H 21(0) uugdV gdV 0 Selection rule for electronic transition: l 1 Chem 253, UC, Berkeley Electronic Transitions: Particle in a box 32.