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interband transitions in semiconductors

M. Fox, Optical Properties of Solids, Oxford Master Series in Condensed Matter Physics

interband transitions in quantum wells

Atomic wavefunction of carriers in the conduction and valence band haveparity differing by 1, hence only transitions with ∆n = 0 are dipole-allowed

for a rectangular potential with infinite wallsonly transitions with ∆n = 0 are possible.

This selection rule is weakened for „real“ quantumwells with finite barrier heights but still the transitionswith ∆n = 0 dominate the spectra

J. H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press (1998)

quantum well photoluminescence

exciton binding energy and Bohr radius


2D: quantum well excitons


exciton correction to the absorption continuum: Sommerfeld factor


laser applications of semiconductor heterostructures

Z. I. Alferov, Nobel Lecture (2000)

quantum well applications: quantum cascade laser (QCL)

Unlike typical interband semiconductor lasers that emit electromagnetic radiation through the recombination of electron–hole pairs across the material band gap, QCLs are unipolar and laser emission is achieved through the use of intersubband transitions in a repeated stack of semiconductor multiple quantum well heterostructures

QCL emission wavelength 2.75 – 250 µm (~ 500 – 5 meV)

3

2

1

3221 Γ>>Γ=LOω

Quantum Cascade Laser invented by Bell Labs physicists; Cover illustration for Science,

April 22,1994.

http://en.wikipedia.org/wiki/Quantum_cascade_laser

interband transitions


QCL: principle and experimental realization

E32 = 291 meV = 2347 cm-1 ~ 4.26 µm

interband transitions:double heterojunction laser

http://britneyspears.ac/physics/fplasers/fplasers.htm

laser applications of semiconductor heterostructures: quantum well LED and laser


quantum well LED and laser



quantum well laser


Confinement in heterostructures

system dimension:

0D

yx

z

Ly

Lx

Lz

3D

Lx,Ly,Lz>>λF

dz

2D dy1D dx

dz ≈ λF dy,dz ≈ λF dx,dy,dz ≈ λF

density of states:

E

D(E) 3D

E

D(E)

const.

E1 Ei

2D

E

D(E) 1D

E

D(E) 0D~ E

1

EijkE111

~δ(E-Eijk)

~ E

E11 Eij

Quantum dots

Stranski-Krastanow QDs

1.28 1.29 1.30 1.31 1.32 1.330

500

1000

1500

Energy (eV) P

L (c

ount

s in

150

s)

monolayer fluctuations QDs

0D dx

dx,dy,dz ≈ λF

E

D(E) 0D

EijkE111

~δ(E-Eijk)

Self-assembly of quantum dotsInAs

< 1.5 ML • Filmmolecular beam epitaxy film growth (InAs on GaAs)mismatch between lattice parameter ⇒ stressed film

GaAs

~ 1.5 ML • Quantum DotsStranski-Krastanov growth of InAs dots is a result from equilibrium between mechanical stressand surface energy~ 2 ML

dots: "rings":

0 50100 150

200nm

0 50100 150

200nm

0 50100 150

200nm

0 50100 150

200nm

height ~ 2 - 6 nmdiameter ~ 20 – 50 nm10% size variation

0 50100 150

200nm

> 2.5 ML

dislocations

• DislocationsFurther growth relaxes excess energy through creation ofdislocations

0D excitons: quantum dots

0 50100 150

200nm

0 50100 150

200nm

0 50100 150

200nm

• InAs quantum dots~ 6 nm high~ 20nm diameter, 10% size variation

0 50100 150

200nm

0 50100 150

200nm

• Quantum rings (Partially Covered InAs Islands) ~ 1 to 2 nm high~ 50nm diameter, 30% size variation

• Vertical coherent growth: double layer of dots

Electron and hole confinement in quantum dots

r (nm)-100 -50 0 50 100

~ 300 meV

~ 150 meV

1445 m e Vs

pd

spd

CB

VB

p

p

d

d

s

s

• Confinement energies:electron ~ 50 meV, holes ~ 25 meV

GaAs

InAsGaAs

Energy Ec

Evx, y

z

Energy

• Capping with GaAs: electronic barrier material

• Quasi-parabolic confinement

Localized states in a self-assembled quantum dot

axial confinement:rectangular quantum well

lateral confinement:parabolic quantum well

GaAs InAs GaAs GaAs InAs GaAs

InAsCBE

z1E

=enE +InAs

CBE +z1E

GaAsCBE

xynE

0xyn ω)1n(E h+=

1st energy levelof a quantum well

energy spectrum of a 2D harmonic oscillatorwith degeneracy m=2(n+1) (2 because of spin)

n=2

-1 +10m= -2 +2

d

pn=1

n=0 s

Localized states in a self-assembled quantum dot

-40 -20 20 40

0.2

0.4

0.6

0.8

1.0

-40 -20 20 40

0.2

0.4

0.6

0.8

1.0

solutions of a 2D harmonic problem:(e.g. Cohen-Tannoudji, Quantum mechanics)

χ |χ|2

n=0m=0

-40 -20 20 40

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-40 -20 20 40

-0.6

-0.4

-0.2

0.2

0.4

0.6

n=1m=+/- 1

-40 -20 20 40

0.2

0.4

0.6

0.8

1.0

-40 -20 20 40

-1.0

-0.8

-0.6

-0.4

-0.2

0.2

0.4

n=2m=0

-40 -20 20 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-40 -20 20 40

0.1

0.2

0.3

0.4

0.5

n=2m=+/-2

Shell structure of quantum dots in spectroscopy

shell structure of artificial atoms:

p

p

d

d

s

s

Abs

orpt

ion

0

1e-4

2e-4

3e-4

Energy (eV)0.9 1.0 1.1 1.2 1.3 1.4

Abs

optio

n

0

1e-4

2e-4

3e-4

Rings(4.2K)

Dots(4.2K)

Absorption measured on ~ 107 quantum dots

s-s

p-p d-d

R.J. Warburton et al. PRL 79, 5282 (1997)

Energy (eV)1.30 1.35 1.40

PL in

tens

ity

0

1

2s-s

p-p

d-d

Emissionmeasured on ~ 107 quantum rings

p

p

d

d

s

s

inhomogeneous broadening ~ 30 meV

Energy scales

Exciton binding energy meV20E h,es,s ≈

Quantization energies meV25meV50

h

e

<ω<ω

h

h

Inhomogeneous broadening meV30≈Ensemblespectroscopy

Excitons in bulk semiconductors


compare with in quantum dots

quantum confinement enhances Coulomb correlations (e.g. exciton binding energy)

meV22Eehss =

0

2000

4000

6000confocal

PL

(cou

nts

in 3

0 s)

1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.350

500

1000

1500

Energy (eV)

300 nm aperture

PL

(cou

nts

in 1

50 s

)

Ensemble and single dot photoluminescence

~ 100 -1000 dots

5 – 10 dots

Al

Quantum dot biexciton cascade: source of entangled photons

Electrical source of entangled photons

Toshiba Research Europe Ltd., Cambridge Research Laboratory

Salter et al, Nature 465, 594–597 (03 June 2010)

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