PHY 752 Fall 2015 Lecture 26 1 PHY 752 Solid State Physics 111150 AM MWF Olin 103 Plan for Lecture 27 Optical properties of semiconductors and insulators Chap 7 amp 12 in GGGPP Excitons ID: 539384
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PHY 752 Fall 2015 -- Lecture 26
1
PHY 752 Solid State Physics
11-11:50 AM MWF Olin 103
Plan for Lecture 27:
Optical properties of semiconductors and insulators (Chap. 7 & 12 in GGGPP)
ExcitonsSlide2
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Interband
transitionsSlide4
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In general the matrix element
M
cv
(
k
) is a smooth function of
k
and the joint density of states often determines the frequency dependence of the optical properties:Slide5
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Real spectra and more complete analysis
From Michael
Rohlfing
and Steven Louie, PRB
62
4927 (2000)Slide6
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PHY 752 Fall 2015 -- Lecture 26
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Real spectra and more complete analysis
From Michael
Rohlfing
and Steven Louie, PRB
62
4927 (2000)Slide7
10/30/2015
PHY 752 Fall 2015 -- Lecture 26
7
Real spectra and more complete analysis
From Michael
Rohlfing
and Steven Louie, PRB
62
4927 (2000)Slide8
10/30/2015
PHY 752 Fall 2015 -- Lecture 26
8
Real spectra and more complete analysis
From Michael
Rohlfing
and Steven Louie, PRB
62
4927 (2000)Slide9
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PHY 752 Fall 2015 -- Lecture 26
9
Simple treatment of
exciton
effects in a two-band modelSlide10
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Electronic Hamiltonian
Ground state
wavefunction
Excited state from two-band model summing over
wavevectors
k’
Solving
Schroedinger
equation in this basis:Slide11
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Exciton
equations -- continued
where:
After several steps:
Ignoring
U
2
for the moment --Slide12
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Equation for
k
ex
=0:
Define an envelope functionSlide13
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Introduced electron-hole screening
Hydrogen-like
eigenstates
:Slide14
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Some details –
Considered a closed shell system with
N
electrons
We can write the effective Hamiltonian:
withSlide15
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Define:
Eigenstates
:Slide17
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More detailed treatment of
U
2
(
J
) term:
Effective dipole moment: Slide18
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Resulting equation for envelope function:
Relationships of envelope function:Slide19
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Hydrogen-like
eigenstates
:
Summary of
Wannier
exciton
analysis
Wannier
analysis is reliable for loosely bound
excitons
found in semiconductors; for
excitons
in insulators (such as
LiF
)
Frenkel
exciton
analysis applies.Slide20
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Optical absorption due to
excitons
(Chap. 12)
Transition probability from ground state
withSlide21
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For spherically symmetric
excitons
(“first class” transitions)
For
p
-like
excitons
(“second class” transitions)Slide22
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Example of Cu
2
O: