Advanced Topics in
Semiconductor Physics
Peter Y. YU
Dept. of Physics, Univ. of California &
Lawrence Berkeley National
Laboratory,
Berkeley, CA. 94720
USA
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Physics Dept, HKU (Nov 2009)
COURSE OUTLINE
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Lecture 1: Electronic structures of
Semiconductors
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Lecture 2: Optical Properties of Semiconductors
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Lecture 3: Defects and their effect on
Semiconductor Devices
Physics Dept, HKU (Nov 2009)
OPTICAL PROPERTIES OF
SEMICONDUCTORS
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OUTLINE
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Optical Constants
Interband transitions & Critical Points
Exciton Effects
Quantum Confinement Effects on Optical
Properties
Polaritons
Physics Dept, HKU (Nov 2009)
Optical Constants (in cgs units)
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For Maxwell’s Equations in a Macroscopic
Medium we add this constitutive equation:
P(r’,t’)=∫χ(r’,r,t’,t)E(r,t)drdt or
P(ω)= χ(ω)E(ω) with χ=linear electric
susceptibility
D =E+4πP= E(1+4πχ)=εE
ε(ω)= dielectric function
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Physics Dept, HKU (Nov 2009)
Optical Constants
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Experimentally we measure n=refractive index which is
related to ε by ε=(n)2.
To account for absorption we define n as a complex
function: n=nr+ini
The absorption coefficient α is defined by:
I(x)=Ioexp(-αx) and is related to the absorption index ni
by: α=4πni/λo (λo=wavelength of light in air)
The dielectric function can also be determined by
reflection via Fresnel equation
:
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Rn (ω ) =
nr + ini − 1
nr + ini + 1
Physics Dept, HKU (Nov 2009)
Optical Constants of Si and GaAs
[From Philipp & Ehrenreich 1967]
60
80
(a)
Si
(a)
GaAs
40
60
R(%)
R x 10
R
R(%) 40
R
R x 10
20
20
0
0
25
60
εi
40
εr,ε
i , ri
(b)
2.0
(b)
εr,εi
-Im -1 -1
Imε
1.5
i
20
1.0
0.8
-Im -1
15
i
Imε-1
r, i
Imε-1
-Im -1
-Im
0.4
5
-1
0
r
-5
εr
6
0
0.5
r
-20
0
5
0
5
10
15
20
25
Energy (eV)
10
15
Energy (eV)
20
25
0
Physics Dept, HKU (Nov 2009)
Imaginary Part of Dielectric Function of
GaN (laser for blue-ray DVD)
Dielectric Function
measured directly
by method known as
Ellipsometry
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Physics Dept, HKU (Nov 2009)
Microscopic Theory of Optical
Properties
We will use a semi-classical approach in which EM wave is
treated classically while electron is treated QM.
z If we assume an electric-dipole transition the interaction
Hamiltonian Her between EM wave and a charge q is given by: (qr)•E
z Electron in crystals are waves (Bloch states with well-defined
wave vectors k) so it will be more convenient to express Her in
terms of p (after using the Coulomb Gauge: E=-(1/c)∂A/∂t and
B=∇xA where A= vector potential and the scalar potential φ=0):
H=(1/2m)[p+(eA/c)]2+V(r)~(1/2m)p2+(e/mc)(A•p)+V(r)
z The extra term induced by E is therefore: H =(e/mc) A•p
z
er
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Physics Dept, HKU (Nov 2009)
Microscopic Theory of Optical
Properties
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Using the Fermi Golden Rule the transition rate (per unit
volume of crystal) from valence band to conduction band is
given by:
R=(2π/h)Σ|<c|Her|v>|2δ(Ec-Ev-hѡ)
Where the matrix element can be shown to be
approximated (for small k) by:
and |Pcv|2=|<c|p|v>|2.
The final result is:
Physics Dept, HKU (Nov 2009)
Microscopic Theory of Optical
Properties
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The absorption coefficient can be related to the power loss per
unit volume of crystal: Power loss=Rhω= -(dI/dt)
= -(dI/dx)(dxdt)=(c/n)αI
where I=(n2/8π)|E(ω)|2
From this result one can obtain εi(ω) :
Physics Dept, HKU (Nov 2009)
Dielectric Function and Critical Points
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The frequency dependence (or dispersion) of εi(ω)
results mainly from the summation over both initial
and final states satisfying energy and momentum
conservation:
∑ δ (E (k ) − E (k ) − hω )
c
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v
k
This summation over k can be converted into
integration over the interband energy difference
Ecv=Ec(k)-Ev(k) by defining the Joint Density of
States (JDOS) Dj(Ecv) as:
Physics Dept, HKU (Nov 2009)
Dielectric Function and Critical Points
• Dj(Ecv) contains van Hove singularities
whenever ∇k(Ecv)=0. The features, such as
peaks and shoulders, in εi(ω) and εi(ω) are
caused by these singularities
z The type of singularities possible is strongly
dependent on dimensionality
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Physics Dept, HKU (Nov 2009)
Dielectric Function and Critical Points
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Physics Dept, HKU (Nov 2009)
Dielectric Function and Critical Points
This band gap is a Mo
CP in 3D
This band gap is a M1
CP in 3D but almost a
Mo CP in 2D
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Band Structure
of Ge showing
interband
transitions
labelled as Eo,
E1 etc
Physics Dept, HKU (Nov 2009)
Critical Points and the Absorption
Spectrum of Ge
Agreement between
Theory and
Experiment is much
better now
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Lowest direct gap
(Eo) of Ge
Physics Dept, HKU (Nov 2009)
Absorption at Fundamental Band Gap
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The lowest energy absorption occurs at the
fundamental band gap which is a Mo type (or
minimum) of critical point
Physics Dept, HKU (Nov 2009)
Absorption at Fundamental Band Gap
Why at low T the absorption spectra of
GaAs show peaks?
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Physics Dept, HKU (Nov 2009)
Correction to the One-Electron Picture
• When a photon excites an electron and hole pair
there is a Coulomb attraction between the e and h
(Final State Interaction) resulting in the formation of
a two-particle bound state known as an exciton
• Exciton is neutral over all but carries an electric
dipole moment. Exciton has been compared to a
hydrogen atom or positronium. Actually exciton is
more than just an “atom”. Since the electron and
hole in the exciton are Bloch waves the exciton is a
polarization wave.
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Physics Dept, HKU (Nov 2009)
Two pictures of the Excitation of Excitons
Exciton Wave
functions and Energy
From Effective Mass
Approximation:
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Physics Dept, HKU (Nov 2009)
Excitonic Absorption
Absorption of the Bound States:
Absorption of the Continuum States:
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Physics Dept, HKU (Nov 2009)
Excitonic Absorption in Cu2O
Cu2O has inversion
symmetry. The conduction
and valence bands have
same parity so electric
dipole transitions to s
states are forbidden. This
series is due to transitions
to the np levels
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Physics Dept, HKU (Nov 2009)
Enhancement of Electron-Radiation
Interaction by Quantum Confinement
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Absorption at exciton is
enhanced into by Coulomb
attraction between e and h.
Photon
Absorption will also be
enhanced if both e and h are
physically confined together
Confined electron
Confined hole
Transition Probability~
|<Φconduction|er•E|Φvalence>|2(|Ψ(0)|2)(JDOS)
Ψ is the envelope function and describes the overlap of
the Electron and Hole wave functions. Confinement
leads to increase in overlap of e and h wave functions
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Physics Dept, HKU (Nov 2009)
The QW Laser
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A Laser utilizing
Confinement of
Carriers
with the additional
benefit of Photon
Confinement
(An idea worth a
Nobel Prize in
Physics in 2000)
Physics Dept, HKU (Nov 2009)
Laser Performance with reduction in
dimensionality
Quantum Dot Laser
was announced by
Fujitsu in 2008
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Adapted from Asada et al.
(1986).
Physics Dept, HKU (Nov 2009)
Coupled EM-Polarization Waves
(Polariton)
Upper
Branch
I
Lower Branch
Photon
ωL
I
Two degenerate waves: photon
and exciton
Any Interaction due to Her will split
this degeneracy. The results are
two “mixed waves” or polariton.
Exciton
ωT
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There are two branches to the
polariton dispersion (upper branch
and lower branch)
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WAVEVECTOR
Physics Dept, HKU (Nov 2009)
Exciton-Polariton in CdS
4πN X (e) 2
ε = εb +
m X ω X2 − ω 2
(
c 2k 2
-Log10(Transmitted Intensity)
ε bω
2
4πN X e 2 /(ε b mx )
⎡
⎛ hk 2
⎢ω X (0) + ⎜⎜
⎝ 2m X
⎣⎢
2
⎞⎤
⎟⎥ − ω 2
⎟
⎠⎦⎥
≈ 1+
ωX = ωx(0)+[hk2/(2mx)]
4πN X e 2 /(ε b mx )
⎡ 2
⎛ hk 2 ⎞ ⎤
⎟⎥ − ω 2
⎢ω X (0) + ω X (0)⎜⎜
⎟
⎝ m X ⎠⎦⎥
⎣⎢
ExcitonPolariton
Dispersion
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3
2
A
Exciton
B
Exciton
Experimental transmission Spectrum of CdS from
Dagenais, M. and Sharfin, W. Phys. Rev. Lett. 58, 17761779 (1987). Oscillations due to interference between the
two polariton branches
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Expriment
Theory
0
20500
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= 1+
)
Combine
with
20600
Wavenumber
20700
(cm-1 )
20800
Physics Dept, HKU (Nov 2009)
Cavity-Polariton (a 2D Polariton)
160
140
120
Light
0.85 0.86 0.87 0.88
sinθ
61.98
61.25
60.32
60.13
60.00
58.49
58.90
58.38
Microcavity
Sample formed
by air on top
and AlAs at
bottom
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58.20
Experimental
Geometry
[Dimitri Dini, Rüdeger Köhler, Alessandro Tredicucci, Giorgio
Biasiol, and Lucia Sorba. Phys. Rev. Lett. 90, 116401 (2003)]
100
120
140
160
180
Energy (meV)
Experimental Reflectivity Spectra
with polariton dispersion in inset
Physics Dept, HKU (Nov 2009)
CONCLUSIONS
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Semiconductors have many applications depending on their optical
properties, such as lasers, LED, solar cells, image sensors etc.
In the near infrared, visible and uv region the optical properties of
semiconductors are determined by interband transitions between
their valence band and conduction band.
Coulomb attraction between e and h enhanced the absorption near
the fundamental band gap
Quantum confinement in QW will also enhance the emission
probability between e and h leading to better lasers
The most fundamental approach to understand the optical properties
of semiconductors is to consider polaritons.
Physics Dept, HKU (Nov 2009)