Advanced Topics in Semiconductor Physics

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Nov 1, 2013 (3 years and 8 months ago)

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Physics Dept, HKU (Nov 2009)
1
Advanced Topics in
Semiconductor Physics
Peter Y. YU
Dept. of Physics, Univ. of California &
Lawrence Berkeley National
Laboratory,
Berkeley, CA. 94720
USA
Physics Dept, HKU (Nov 2009)
2
COURSE OUTLINE

Lecture 1: Electronic structures of
Semiconductors

Lecture 2: Optical Properties of Semiconductors

Lecture 3: Defects and their effect on
Semiconductor Devices
Physics Dept, HKU (Nov 2009)
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OPTICAL PROPERTIES OF
SEMICONDUCTORS

OUTLINE

Optical Constants

Interbandtransitions & Critical Points

Exciton Effects

Quantum Confinement Effects on Optical
Properties

Polaritons
Physics Dept, HKU (Nov 2009)
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Optical Constants
(in cgsunits)

For Maxwell’s Equations in a Macroscopic
Medium we add this constitutive equation:

P(r’,t’)=∫χ(r’,r,t’,t)E(r,t)drdtor

P(
ω
)= χ(ω)E(
ω
)with χ=linear electric
susceptibility

D=E+4πP= E(1+4πχ)=εE
ε(ω)= dielectric function
Physics Dept, HKU (Nov 2009)
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Optical Constants

Experimentally we measure n=refractive index which is
related to εby ε=(n)2.

To account for absorption we define n as a complex
function: n=nr+in
i

The absorption coefficient αis 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 Fresnelequation :
()
2
1
1
++
−+
=
ir
ir
n
inn
inn
R
ω
Physics Dept, HKU (Nov 2009)
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Optical Constants of Si and GaAs
[From Philipp& Ehrenreich1967]
R
R x 10
Si
0
20
40
60
80
R(%)
(a)
0
0
-20
20
40
60
515102025
0
0.5
1.0
1.5
2.0
Energy (eV)
-Im
-1
-Im
-1
i
,
r
i
r
(b)
GaAs
(a)
R x 10
R
0
20
40
60
R(%)
25
15
5
0
-5
0510152025
0.4
0.8
(b)
Energy (eV)
-Im
-1
-Im
-1
r,
i
i
r
εr,εi
εr,εi
Imε-1
Imε-1
εr
εi
Imε-1
Physics Dept, HKU (Nov 2009)
7
Imaginary Part of Dielectric Function of
GaN(laser for blue-ray DVD)
Dielectric Function
measured directly
by method known as
Ellipsometry
Physics Dept, HKU (Nov 2009)
8
Microscopic Theory of Optical
Properties

We will use a semi-classical approach in which EM wave is
treated classically while electron is treated QM.
If we assume an electric-dipole transitionthe interaction
Hamiltonian Her
between EM wave and a charge q is given by: -
(qr)•E
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=∇xAwhereA= vector potentialandthescalar potentialφ=0):
H=(1/2m)[p+(eA/c)]2+V(r)~(1/2m)p2+(e/mc)(A•p)+V(r)
The extra term induced by Eis therefore:
Her=(e/mc) A•p
Physics Dept, HKU (Nov 2009)
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Microscopic Theory of Optical
Properties

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|H
er|v>|2δ(Ec-E
v-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)
10
Microscopic Theory of Optical
Properties

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)
11
Dielectric Function and Critical Points

The frequency dependence (or dispersion) of εi(ω)
results mainly from the summation over both initial
and final states satisfying energy and momentum
conservation:

This summation over k can be converted into
integration over the interbandenergy difference
Ecv=Ec(k)-Ev(k) by defining the Joint Density of
States(JDOS) Dj(E
cv) as:
(
)
(
)
(
)

−−
k
vc
kEkE
ωδ
h
Physics Dept, HKU (Nov 2009)
12
Dielectric Function and Critical Points
•D
j(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

The type of singularities possible is strongly
dependent on dimensionality
Physics Dept, HKU (Nov 2009)
13
Dielectric Function and Critical Points
Physics Dept, HKU (Nov 2009)
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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
Band Structure
of Geshowing
interband
transitions
labelledas E
o,
E1
etc
Physics Dept, HKU (Nov 2009)
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Critical Points and the Absorption
Spectrum of Ge
Agreement between
Theory and
Experiment is much
better now
Lowest direct gap
(Eo) of Ge
Physics Dept, HKU (Nov 2009)
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Absorption at Fundamental Band Gap

The lowest energy absorption occurs at the
fundamental band gapwhich is a M
o
type (or
minimum) of critical point
Physics Dept, HKU (Nov 2009)
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Absorption at Fundamental Band Gap
Why at low T the absorption spectra of
GaAs show peaks?
Physics Dept, HKU (Nov 2009)
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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 neutralover 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.
Physics Dept, HKU (Nov 2009)
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Two pictures of the Excitation of Excitons
Exciton Wave
functions and Energy
From Effective Mass
Approximation:
Physics Dept, HKU (Nov 2009)
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ExcitonicAbsorption
Absorption of the Bound States:
Absorption of the Continuum States:
Physics Dept, HKU (Nov 2009)
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ExcitonicAbsorption in Cu
2O
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 nplevels
Physics Dept, HKU (Nov 2009)
22
Enhancement of Electron-Radiation
Interaction by Quantum Confinement

Absorption at exciton is
enhanced into by Coulomb
attraction between e and h.
Absorption will also be
enhanced if both e and h are
physically confined together
Photon
Confined electron
Confined hole
Transition Probability~
|<Φconduction|er•E|Φvalence>|2(|Ψ(0)|2)(JDOS)
Ψis 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
Physics Dept, HKU (Nov 2009)
23
The QW Laser

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)
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Laser Performance with reduction in
dimensionality
Adapted from Asada et al.
(1986).
Quantum Dot Laser
was announced by
Fujitsu in 2008
Physics Dept, HKU (Nov 2009)
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Coupled EM-Polarization Waves
(Polariton)
Photon
Exciton
0
I
I
ωT
ωL
WAVEVECTOR
Two degenerate waves: photon
and exciton
Any Interaction due to Her
will split
this degeneracy. The results are
two “mixed waves”or polariton.
There are two branches to the
polaritondispersion (upper branch
and lower branch)
Lower Branch
Upper
Branch
Physics Dept, HKU (Nov 2009)
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Exciton-Polaritonin CdS
()
22
2
)(4
ωω
π
εε

+=
XX
X
b
m
eN
Wavenumber (cm )
-1
20500
2060020700
20800
0
1
2
3
4
ωX
= ω
x(0)+[hk2/(2mx)]
2
2
2
2
2
2
2
2
2
22
)0()0(
)/(4
1
2
)0(
)/(4
1
ωωω
επ
ωω
επ
ωε

















+
+≈

















+
+=
X
XX
xbX
X
X
xbX
b
m
k
meN
m
k
meNkc
hh
Combine
with
Exciton-
Polariton
Dispersion
Experimental transmission Spectrum of CdS from
Dagenais, M. and Sharfin, W. Phys. Rev. Lett. 58, 1776-
1779 (1987). Oscillations due to interference between the
two polaritonbranches
Expriment
Theory
A
Exciton
B
Exciton
-Log
10
(Transmitted Intensity)
Physics Dept, HKU (Nov 2009)
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Cavity-Polariton(a 2D Polariton)
Energy (meV)
100120
140160
180
58.20
58.38
58.90
58.49
60.00
60.13
60.32
61.25
61.98
0.850.860.870.88
sin
θ
120
140
160
Microcavity
Sample formed
by air on top
and AlAsat
bottom
Experimental
Geometry
Experimental Reflectivity Spectra
with polaritondispersion in inset
Light
[Dimitri Dini, Rüdeger Köhler, Alessandro Tredicucci, Giorgio
Biasiol, and Lucia Sorba.
Phys. Rev. Lett. 90, 116401 (2003)]
Physics Dept, HKU (Nov 2009)
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CONCLUSIONS

Semiconductors have many applications depending on their optical
properties, such as lasers, LED, solar cells, image sensors etc.

In the near infrared, visible and uvregion the optical properties of
semiconductors are determined by interbandtransitions 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.