NNSE 618 Lecture #15

1

Lecture contents

• Burstein shift

• Excitons

• Interband transitions in quantum wells

• Quantum confined Stark effect

NNSE 618 Lecture #15

2

Absorption edges in semiconductors

• Offset corresponds to bandgap

• Abs. coefficient is orders of

magnitude higher for direct

transitions

• Abs. coefficient roughly follows

density of states

NNSE 618 Lecture #15

3

Burstein-Moss shift

• Shift of absorption edge in degenerate semiconductors

• Usually in direct n-type semiconductors with low effective mass

• Due to occupation of band energy states up to: , the edge shifts:

From Seeger, 1973

Absorption edge shift in doped n-InSb

Burstein edge in degenerate n-type semiconductor

eBn

e

Tkm

k4

2 *

22

**

22 11

2he

gmm

kE

NNSE 618 Lecture #15

4

Wannier Excitons

r

erVi

2

)( Similar to hydrogen-like impurity: electron and hole

bound by screened coulomb interaction

Solution for discrete energy levels:

With reduced effective mass (electron and

hole orbiting around their center of mass):

Envelope function of the ground state

(hydrogen-like):

Bohr radius:

Free exciton can move in the crystal

as a quasiparticle with a mass

BB

a

r

a

rF exp1

)(213

0

20

2 m

emaB

22222

4 111

2 nmRy

n

eEex

Exciton Ry*

Ry* = 6 meV, aB = 100 A

**

111

he mm

For excitons in GaAs

(me*=0.07m: and = 12.6 ):

**he mmM

NNSE 618 Lecture #15

5

Exciton absorption

Exciton absorption edge in GaAs

• Exciton absorption red-shifts the absorption edge by the exciton binding energy

• Exciton edge absorption is higher than for band absorption (Sommerfeld Enhancement)

• Exciton peaks at room temperature are difficult to resolve in most materials (notable exception -

quantum wells)

• Excitons in bound states are fragile e.g., broken by colliding with phonons (e.g., in a few hundred

femtoseconds).

• By the uncertainty principle, they must then have broad linewidth

From Harris, 2004

NNSE 618 Lecture #15

6

Sommerfeld Enhancement

Lots of bound states near

the onset of continuum sum

together to give Sommerfeld

enhancement.

• Even without excitonic peaks, bandedge

absorption is enhanced due to Coulomb

interaction between electrons and holes

• The reason is an increased density of states of

excitons over the band edge

• This results in increasing of the absorption

coefficient at the band edge:

• Above the bandedge exciton contribution is due to

“mobile” excitons with nonzero wavevector k:

with

Absorption edge in direct band semiconductors

exex

Ry

n

dE

dnDOS

3

2

21

212

g

exfreegex

E

RyE

x

xeE

x

freegex

sinh

21

21

g

ex

E

Ryx

NNSE 618 Lecture #15

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Exciton absorption in “forbidden” direct band edges

Exciton absorption in Cu2O at 4 K

• “Forbidden” direct band-to-band

transition Cu2O due to even parity of

electron and hole wavefunctions

(momentum operator has odd parity)

• Higher order transition (quadruple

instead of dipole) and dipole transition

for non-zero k in confined states are

allowed

From Seeger , 1973

NNSE 618 Lecture #15

8 Absorption in Si

Low absorption (indirect)

High absorption (direct)

NNSE 618 Lecture #15

9

Interband transitions in quantum wells

From Singh, 2003

• Calculated absorption spectrum of 100A

GaAs/Al0.3Ga0.7As without exciton effects

• Strong exciton effects are present

Absorption spectra of GaAs/Al0.3Ga0.7As and

In0.53Ga0.47As/n0.52Ga0.48As QWs

Alloy broadening

Heavy-hole exciton binding

energy as a function of well size

NNSE 618 Lecture #15

10

Franz-Keldysh effect in GaAs

Modulation of interband transitions in bulk semiconductors:

Franz-Keldysh effect

From Seeger, 1973

• Concept of Franz-Keldysh effect:

solution for electron and hole

envelope wavefunctions with

constant field are Airy functions.

• Wavefunctions now "tunnel" into

the bandgap region allowing

overlap of electron and hole

wavefunctions even for photon

energies less than the bandgap

energy, hence allowing optical

absorption below the bandgap

energy.

Franz-Keldysh effect

NNSE 618 Lecture #15

11

Absorption spectrum due to Franz-Keldysh effect

Franz-Keldysh effect

Franz -Keldysh effect is a central-force problem

with perturbation:

Airy function Ai(Z)

• Z>0: electron-hole energy < electric field potential

• Z<0: electron-hole energy > electric field potential, i.e.

above bandgap oscillation wavefunction

Absorption spectrum reduce to the familiar

square root energy when field 0

NNSE 618 Lecture #15

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• With applied field, electron and hole wavefunctions are

distorted (second order perturbation)

• The intersubband separation decreases with electric

field (dominant term)

• Binding energy of excitons decreases with field;

carriers are separated by the field (few meV effect)

Modulation of interband transitions in quantum wells

QW

no electric field

QW

in electric field

Calculated variation of ground state intersubband

transition in W= 100A GaAs/Al0.3Ga0.7As QW

2

422

2

)2(1

*1

15

24

1

WemE

NNSE 618 Lecture #15

13

Electric field modulation of transmission spectra

of 100 A GaAs/AlGaAs QW at two polarizations

Modulation of intersubband transitions in quantum wells

• Absorption edge red-shifts with

electric field

• Exciton absorption strength

reduces with field because the

electron and hole wavefunctions

are separated by electric field

• Polarization rules apply due to

symmetry of electron-radiation

matrix elements

From Miller, 1986

fiferi pAmc

eH