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)

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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)

4

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)

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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)

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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)

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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)

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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)

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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)

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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)

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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.

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