advanced topics in semiconductor physicscmyles/phys5335/lectures/optical properties 2 - yu... · 1...

Physics Dept, HKU (Nov 2009)1

Advanced Topics in Semiconductor Physics

Peter Y. YUDept. of Physics, Univ. of California &Lawrence Berkeley National Laboratory,Berkeley, CA. 94720 USA


COURSE OUTLINE

Lecture 1: Electronic structures of Semiconductors

Lecture 2: Optical Properties of Semiconductors

Lecture 3: Defects and their effect on Semiconductor Devices


OPTICAL PROPERTIES OF SEMICONDUCTORS

OUTLINE– Optical Constants– Interband transitions & Critical Points– Exciton Effects– Quantum Confinement Effects on Optical

Properties– Polaritons


Optical Constants (in cgs units)

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 orP(ω)= χ(ω)E(ω) with χ=linear electric susceptibilityD =E+4πP= E(1+4πχ)=εE

ε(ω)= dielectric function


Optical Constants

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 niby: α=4πni/λo (λo=wavelength of light in air)The dielectric function can also be determined by reflection via Fresnel equation :

( )2

11++−+

=ir

irn inn

innR ω


Optical Constants of Si and GaAs[From Philipp & Ehrenreich 1967]

RR x 10

Si

0

20

40

60

80

R(%)

(a)

0

0-20

20

40

60

5 1510 20 250

0.5

1.0

1.5

2.0

Energy (eV)

-Im -1

-Im -1

i , ri

r

(b)

GaAs(a)

R x 10

R

0

20

40

60

R(%)

25

15

5

0

-5

0 5 10 15 20 25

0.4

0.8(b)

Energy (eV)

-Im -1

-Im -1r, i

i

rεr,εi

εr,εi

Imε-1

Imε-1

εr

εiImε-1


Imaginary Part of Dielectric Function of GaN (laser for blue-ray DVD)

Dielectric Function measured directly by method known as Ellipsometry


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 transition the interaction Hamiltonian Her between EM wave and a charge q is given by: -(qr)•EElectron 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)The extra term induced by E is therefore: Her=(e/mc) A•p



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:



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)αIwhere I=(n2/8π)|E(ω)|2

From this result one can obtain εi(ω) :


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 interband energy difference Ecv=Ec(k)-Ev(k) by defining the Joint Density of States (JDOS) Dj(Ecv) as:

( ) ( )( )∑ −−k

vc kEkE ωδ h



• 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 singularitiesThe type of singularities possible is strongly dependent on dimensionality



This band gap is a MoCP in 3D

This band gap is a M1CP in 3D but almost a Mo CP in 2D

Band Structure of Ge showing interbandtransitions labelled as Eo, E1 etc


Critical Points and the Absorption Spectrum of Ge

Agreement between Theory and Experiment is much better now

Lowest direct gap (Eo) of Ge


Absorption at Fundamental Band Gap

The lowest energy absorption occurs at the fundamental band gap which is a Mo type (or minimum) of critical point


Absorption at Fundamental Band Gap

Why at low T the absorption spectra of GaAs show peaks?


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.


Two pictures of the Excitation of Excitons

Exciton Wave functions and EnergyFrom Effective Mass Approximation:


Excitonic Absorption

Absorption of the Bound States:

Absorption of the Continuum States:


Excitonic Absorption in Cu2O

Cu2O has inversion symmetry. The conduction and valence bands have same parity so electric dipole transitions to sstates are forbidden. This series is due to transitions to the np levels


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)

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


The QW Laser

A Laser utilizing Confinement of Carrierswith the additional benefit of Photon Confinement(An idea worth a Nobel Prize in Physics in 2000)


Laser Performance with reduction in dimensionality

Adapted from Asada et al.(1986).

Quantum Dot Laser was announced by Fujitsu in 2008


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 polariton dispersion (upper branch and lower branch)

Lower BranchUpper Branch


Exciton-Polariton in CdS

( )22

2)(4ωω

πεε−

+=XX

Xb m

eN

Wavenumber (cm )-120500 20600 20700 208000

1

2

3

4

ωX = ωx(0)+[hk2/(2mx)]

22

2

2

222

2

2

22

)0()0(

)/(41

2)0(

)/(41

ωωω

επ

ωω

επωε

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+≈

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=

XXX

xbX

XX

xbX

b

mk

meN

mk

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 polariton branches

Expriment

Theory

A Exciton

B Exciton

-Log

10(T

rans

mitt

ed In

tens

ity)


Cavity-Polariton (a 2D Polariton)

Energy (meV)100 120 140 160 180

58.20

58.38

58.90

58.49

60.0060.13

60.32

61.2561.98

0.85 0.86 0.87 0.88sinθ

120

140

160

MicrocavitySample formed by air on top and AlAs at bottom

Experimental Geometry

Experimental Reflectivity Spectra with polariton dispersion in inset

Light

[Dimitri Dini, Rüdeger Köhler, Alessandro Tredicucci, Giorgio Biasiol, and Lucia Sorba. Phys. Rev. Lett. 90, 116401 (2003)]


CONCLUSIONS

Semiconductors have many applications depending on their opticalproperties, 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 gapQuantum confinement in QW will also enhance the emission probability between e and h leading to better lasersThe most fundamental approach to understand the optical properties of semiconductors is to consider polaritons.

advanced topics in semiconductor physicscmyles/phys5335/lectures/optical properties 2 - yu... · 1...

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