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

2nx kx  Lx 2n k y 2 2 y  2D k space: Ly Area per k point: L L 2nz 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

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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*

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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

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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

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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

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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 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 and therefore can be excited by either a photon or an electron

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v2 2 m e e  2 2 2 r r n h 2 rn  2 2  n a0 nh 4 m e v  e 2mer 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 

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 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

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h

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h

Chem 253, UC, Berkeley

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Chem 253, UC, Berkeley

h

h h

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Implication in solar cell

Chem 253, UC, Berkeley Thin film PV Si PV

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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

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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:

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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 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(kxt) 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 : f(-x)=f(x) odd parity: f(-x)=-f(x)

Initial/final wavefunctions must have different parities for allowed electric dipole transition!

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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!

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Selection Rules

Symmetric function (gerade): (x)  (x) Asymmetric function (ungerade): (x)  (x)

The operator of the : -(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

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