University of Notre DameCenter for Nano Science and Technology Optical Properties at the Nanoscale...

University of Notre Dame Center for Nano Science and Technology

Optical Properties at the Nanoscale

James L. MerzDepartment of Electrical Engineering

Department of PhysicsUniversity of Notre Dame

EE 698D – Advanced Semiconductor PhysicsNotre Dame

23 November 2004


References

► Primary reference for this talk:

Quantum Semiconductor Structures, Fundamentals and Applications,

C. Weisbuch and B. Vinter, Academic Press, Inc., San Diego, 1991.

(referred to throughout the talk as W & V.)

► The Quantum Dot, R. Turton, Oxford Univ. Press, NY, 1995.

► Quantum Dot Heterostructures, D. Bimberg, M. Grundmann, and

N.N. Ledentsov, John Wiley and Sons, Chichester, England, 1999

► Electronic and Optoelectronic Properties of Semiconductor Structures,

J. Singh, Cambridge Univ. Press, Cambridge, 2003.


References (continued)

► Many of the slides were taken from a Plenary talk by Maurice Skolnick at the International Conference on the Physics of Semiconductors (ICPS),Flagstaff, AZ (July 2004). Used with his permission.

► "Near-field Magneto-photoluminescence Spectroscopy of Composition Fluctuations in InGaAsN", A.M. Mintairov, J.L. Merz, et al, Phys. Rev. Letters 87, 277401 (31 December 2001); “Exciton Localization in InGaAsN and GaAsSbN Observed by Near-field Magnetoluminescence”, James L. Merz, A.M. Mintairov et al, Proceedings of the Spring meeting of the European Marterials Research Society, Symposium M, to be published in IEE Proceedings Optoelectronics. (referred to in the talk as M & M)


How do we make Quantum Wells (QWs)?

► Typically by MBE or MOCVD.

► Grow thin films (a few monolayers to tens of monolayers) of a narrow-bandgap semiconductor bounded by a wider-bandgap semiconductor.

► The process can be repeated several times or many times, to make multiple (non-interacting) QWs, or a superlattice of interacting QWs.


Fundamentals of Quantum-Confined StructuresQuantum Wells (1-D structures)

W & V, pg. 3, Fig.1


“Classical” Quantum Mechanical Problem:Particle in a Box

► These structures are best grown by Molecular Beam Epitaxy (MBE) or Metal-Organic

Chemical Vapor Deposition (MOCVD) ► Let the potential barriers be infinitely high

► Solve the Schrödinger Equation for this one-dimensional case

► Solutions are sinusoidal functions: Ψ(z) = sin(kz) or cos(kz), where z is growth direction.

► Solutions must vanish at the well/barrier interfaces (i.e., Ψ(z) = 0 at z = 0 and z = b,

where b is the thickness of the film)


Eigenfunctions and Eigenvalues

W & V, pg. 12, Fig. 5


Finite barrier (Vo) allows exponential penetration into the barrier

Note that this problem is mathematically equivalent to the dielectric waveguide. Energy eigenstates then become

guided optical modes.

W & V, pg. 13, Fig.6


Coupled Quantum Wells

► As two wells approach each other, their wave functions overlap

► This leads to a pair of eigenstates which split into two levels

► For many coupled wells, get multiple states → energy band

W&V, pg. 29, Fig.14


Superlattice Energy Bands

►At a given well or barrier width a, the higher energy states become broader bands. This results from the fact that the wave function overlap increases for the higher-lying energy eigenstates.

►Thus, for a = 50 Å, E1 is a discrete state, while E2, E3,E4, etc. form successively broader bands.

W & V, pg. 38, Fig.18c


AlxGa1-xAs:an ideal material to

form Quantum Wells and

Superlattices

► For 0 < x < 0.4 AlxGa1-xAs has a direct bandgap that is larger than GaAs.

► GaAs and AlGaAs are very nearly lattice matched.

► Thus, AlGaAs is an excellent potential barrier for GaAs quantum wells.

Energy GapEg (eV)

H.C. Casey and M.B. Panish, J. Appl. Phys. 40, 4910 (1969).


Optical absorption of multiple uncoupled quantum wells – comparison with bulk

W & V, pg. 64, Fig.30e

GaAs: Eg = 1.43 eVAl.3Ga.7As: Eg = 1.79 eV

50 periods of 100 Å GaAs = .5 mComparison with 1 m bulk GaAs


What are the differences??

► For quantum wells, the absorption edge is at higher energy than the bulk absorption edge, due to increased energy of the

confined state in the GaAs quantum well.

► Bulk GaAs shows √E energy dependence due to direct gap of GaAs (3-D bulk density of states).

► GaAs quantum well absorption shows “stair case” dependence on energy (2-D density of states).

► Sharp peaks are due to exciton absorption.

► Some corrections must be made for e-h correlation effects.


What is the Density of States?

► The density of states is the number of states per unit volume per unitenergy interval that are available for occupation by electrons (or holes).

► Optical absorption must be proportional to the density of states, because a photon cannot be absorbed if there is no final state available for the electronic transition.

► For 3-D parabolic bands (bulk), ICBST N(E)= (1/22)(2m*/ħ2)3/2√E,where m* is the effective mass of the electron.

► For a 2-D quantum well, ICBST N(E)= m*/ħ2, independent of E. For each quantum state in the quantum well, there will be a step in the density of states.


Absorption of Bulk Semiconductors:Direct and Indirect Bandgap

► Direct Bandgap: h – Eg)1/2

► Indirect Bandgap: Must have momentum conservation

Absorb a phonon: ah – Eg + ħ)2

Emit a phonon: eh – Eg – ħ)2


2-D and 3-D Density of States

W & V, pg. 21, Fig.10


Excitons

W & V, pg.26, Fig.13

► Exciton: photon produces electron-hole pair.

► Electron-hole pair is bound by Coulomb attraction, losing energy and creating sharp energy states (analogous to H2 atom)

► The 2-D Rydberg is 4x greater than the 3-D Rydberg.

► Sommerfeld factor is due to electron-hole correlation in unbound states.


Excitons and Shallow Impurities

► Hydrogen atom: e2/err term gives series of energy states:

En = ERydb/n2,

where ERydb = e4mo/2ħ2 = 13.6 eV,

and the Bohr radius aB = ħ2/moe2 = 0.529 Å.

► For a donor, must correct for the electron mass and the dielectric constant:

En = (me*/er2)ERydb → 10-20 meV

► For an exciton, must use reduced mass: 1/ = 1/me* + 1/mh*

En = (mr*/er2)ERydb → 5-20 meV


The 2-D Rydberg

► Can solve the problem for an infinite potential model

ERydb2-D = ERydb

3-D * 1/(n – ½)2 = ERydb3-D * 4/(2n-1)2

Ground state: n = 1 → ERydb2-D = 4 * ERydb

3-D


Optical absorption of multiple uncoupled quantum wells – comparison with bulk

W & V, pg. 64, Fig.30e

GaAs: Eg = 1.43 eVAl.3Ga.7As: Eg = 1.79 eV

50 periods of 100 Å GaAs = .5 mComparison with 1 m bulk GaAs


Dependence of exciton binding energy (EBX)on quantum well thickness

W & V, pg. 25, Fig.12

----- Light-hole exciton_____ Heavy-hole exciton

► For an infinite well, EBX increases as the well gets thinner, just as does the single-electron state.

► For a finite well, EBX reaches a max. and then decreases because e and h wave functions spread out into the barrier.

► As d becomes very large, the light hole binding energy > heavy hole because EBX ~ 1/mass.


Optical Absorption of Coupled Wells

W & V, pg.73,Fig.35Dingle et al, Phys.Rev.Lett. 34, 1327 (1975)Dingle et al, Phys.Rev.Lett. 33, 827 (1974)

► (a) Single well: one state, split into heavy & light hole.

► (b) Double well: two states (bonding and antibonding), each of which is split into heavy & light hole.

► (c) Triple well: three states, each split into heavy & light hole.


Effect of Quantum Well Thickness

W & V, pg.76, Fig.38

► In layer-to-layer growth mode, one expects thickness variations of ~0.5 monolayer from average monolayer thickness.

► Intralayer thickness fluctuations cause variations of confining energies.

► For thin wells (51Å), these variations cause a larger relative effect, hence lines broaden.

► Note increase in photon energy of absorption edge as well gets thinner, as expected.


Franz-Keldysh Effect for Bulk Material

W & V, pg.89, Fig.46

► Bulk material Applied field E ≠ 0 Franz-Keldysh Effect: bands are tilted.

► Absorption below Eg because of exponential wave-function tails.

► Oscillations above Eg due to wave-function interference.


Quantum-Confined Stark Effect (QCSE)

W & V, pg.89, Fig.46

► Quantum Well, E = 0 Usual case seen before.

► Quantum Well, E ≠ 0 Bands tilt and energy of quantum states decreases → “red” shift of luminescence energy.

Wave function overlap decreases → reduction of luminescence intensity.


Optical Absorption due to QCSE

W & V, pg.90, Fig.47

► Values of applied electric field: (i) E = 0 (ii) E = 60 kV/cm (iii) E = 110 kV/cm (iv) E = 150 kV/cm (v) E = 200 kV/cm

► The predicted red shift and intensity decrease are both observed with increasing electric field.


Wannier-Stark Localization (WSL)

► QCSE is observed for a single QW.

► WSL is observed for a QW superlattice.

► At E = 0 the discrete QW states form bands in a superlattice, and the electron can be anywhere in the superlattice.

► As E increases, the superlattice bands tilt, and the energy eigenstates no longer overlap. Bands get narrower and then coalesce into sharp states.

► At high electric fields, the electron is completely localized.

Mendez et al, Phys.Rev.Lett. 60, 2426 (1988)


Formation of “Stark Ladders” in WSL

At low E-field, e is delocalized.With increasing E, up to 5 statesare seen, which reduce to oneat high field.

Plot of photocurrent peak energiesvs. E. Stark ladders from outlying states are clearly seen, which quicklydisappear at high field.

QCSE red shift

Mendez et al, Phys.Rev.Lett. 60, 2426 (1988)


n-i-p-i Structures

W & V, pg.52, Fig.27

► These are n-i-p-i homojunctions.

► Electrons from n region fall into holes in p regions, leaving ionized impurities.

► Resulting space charge distribution modulates the bands, forming energy eigenstates.

► The result is a homojunction superlattice.


n-i-p-i Light Modulation Phenomena

In the dark, the n-i-p-i structure has reduced the effective bandgap of thestructure, as shown by the energy arrow in (a). If above-bandgap monochromatic light is incident on the structure, electrons and holes are produced which reduce the space charge, flattening the bands and increasing the radiative recombinationenergy, as shown by the energy arrow in (b). Thus, changes in the intensity of monochromatic light shining on the sample changes the wavelength of the emission.

Turton, pg.134, Fig.8.8


Semiconductor Lasers

► electrons and holes are injected into GaAs active region by p-n junction.

► Carriers are confined to active region by potential barriers.

► Photons are confined to active region by refractive index difference.

W & V, pg.166, Fig.88


Confinement of Optical Guided Wave

W& V, pg.168, Fig.89

► Double Heterostructure – Optical mode well confined. Too many electron states available. ~ 1

► Single Quantum Well Optical mode poorly confined. Electrons well confined to single state. n d2

► Separate Confinement Heterostructure Optical mode moderately confined to the quantum well. Electrons well confined. ~ n d


Confinement Factor ()

d/2 ∞

= ∫│E (z)│2dz / ∫│E (z)│2dz

-d/2 -∞


Threshold Current Density (Ith)

W & V, pg.171, Fig.92

►Want to minimize Ith. ►GSCH is Graded-index Separate Confinement Heterostructure (also called GRINSCH)


Vertical Cavity Surface Emitting Laser (VCSEL)

W & V, pg.183, Fig.102a

► Active layer: GaInAs/GaAs QWs

► Mirrors: GaAs/AlGaAs multiple layers with thickness ~ optical wavelengths → Distributed Bragg Reflectors (DBRs)

► Advantages: • Low Ith due to QW active layer (Ith≤400 A/cm2) • Very low I (<70 A) due to small cross-sectional area (8 m) • Low output diffraction → can couple efficiently into optical fibers • Can make large arrays

Huffaker & Deppe, Appl. Phys. Letters 70, 1781 (1997).


VCSEL Arrays

J. Jewell et al, Appl. Phys. Letters 55, 2724 (1989)


Thus far:

► We solved the Schrödinger Equation for the one-dimensional “particle in a box” ► These results were appropriate for a two-dimensional semiconductor structure

i.e., a quantum well

► Now we are interested in confinement in two and three directions, leading to structures that are one-dimensional and zero-dimensional, respectively

i.e., quantum wires and quantum dots or boxes. ► We also said that we could derive the so-called “density of states”, and that this

concept is very important to understand electrical and optical phenomena of these

quantum-confined structures.

► Now the concept of density of states becomes increasingly important, and for quantum wires and dots the experimental techniques of single-electron

transport (Snider) and near-field optics (Merz) become increasingly important.


2-D and 3-D Density of States

W & V, pg. 21, Fig.10


Quantum Wires (QWires)


Early Prediction about Quantum Wires

► Hiroyuki Sakaki1 predicted, more than two decades ago,that ideal 1-D electrons moving at the Fermi level in quantum wires would require very large momentum changes (k = 2kF, where kF is the Fermi wave-vector) to undergo any scattering,

► The result would be that electron scattering would be strongly forbidden.

► This is a consequence of the fact that in one dimension, electrons can scatter only in one of two directions: forward and 180o backwards.

► With this large reduction in scattering, electrons would achieve excellenttransport properties (e.g., very high mobility).

1 H. Sakaki, J. Vac. Sci. Technol. 19(2), 148 (1981)


Quantum Dots (QDs)


Early Prediction about Quantum Dots

► A year later, Arakawa and Sakaki1 predicted significant increases in thegain, and decreases in the threshold current, of semiconductor lasersutilizing quantum dots or boxes in the active layer of the laser.

► Highly efficient, low power lasers could be the consequence of these predictions.

► These predictions set off an intense effort worldwide to fabricate such structures.

1 Arakawa and Sakaki, Appl. Phys. Letters 40, 939 (1982)


Low-dimensional Semiconductor LaserPerformance Calculations

Asada, Miyamoto, & Suematsu,IEEE J. Quantum Electronics QE-22, 1915 (1986)


• SAQDs important for both physics and applications

• Strong confinement and high radiative efficiency

• Quasi-0D systems in the solid state. ‘Atom-like’

• Embedded in semiconductor matrix. Wide variety of semiconductor devices, processing technology

Why are Quantum Dots Important

from Skolnick


Modification of Density of States by Reduction of Dimensionality

(a) (b)

(c) (d)

0

0

Energy

Den

sity

of

Sta

tes

3D 2D

1D0D

dot

wire

wellbulk

from Skolnick


General approaches to QD synthesis

Colloidal growth of CdSe dots

Artificial patterning

Self-assembled quantum dots (SAQDs) by MBE

Bimberg et al, pg.5, Fig.1.3


Colloidal CdSe QDs

Colloids of CdSe QDs are fluorescent at various frequencies within the visible range. The highly tunable nature of the QD size yields a broad range of colors in the visible spectrum.E. Karreich, Nature 413, 450 (2001)


Self-Assembled Crystal Growth of QDs

in Strained Systems by MBE

InAs-GaAs 7% lattice mismatch

GaAs

InAs

Stranski-Krastanow growth

Embedded in crystal matrix – like any other semiconductor laser or light emitting diode

Note wetting layer

from Skolnick

Most important system:


Quantum Dots and the Wetting layer

QDs WL

UHV-STM cross sections

PM Koenraad, TU Eindhoven

20nm

from Skolnick


Quantum Dots and Anti-dots using Gates

W & V, pg.194, Fig.104

► Fabricate a square grid of electrodes on the surface of a sample having a 2DEG.

►Apply a positive voltage to the grid.

► With increasing positive voltage to the periodic gate, electrons are attracted to it, causing EF to rise, forming QDs.

► With increasing bias, the QDs merge into a sea of electrons, with periodic islands which electrons cannot occupy (antidots).


•Energy Level Energy Level StructureStructure

• Discrete energy levels – atom-like

• Electron energy level splitting 20-70meV, hole levels spaced by ~10meV.

• Favourable for room temperature operation

2D state

0D states

~2

0n

m

~10nm

~ HO like potential

p-shelln=1. l=±1

QW like potential

s-shelln=0, l=0

d-shelln=2. l=0,±1,±2

z

x,y

Photon emitted

from Skolnick


emission

Absorption

Optical Spectra

Studies of large numbers ~107 dots.

Linewidth ~30meV due to shape and size fluctuations

from Skolnick


1. Directly: Measure transmission. Simple conceptually, but sometimes difficult in practice due to weak absorption. One step process.

2. Photocurrent: exciton created by resonant absorption, electron and hole tunnel out of dot and give current. Two step process.

3. Photoluminescence excitation spectroscopy: absorption, relaxation and then recombination. Excited states. Coupling to environment. Three step process.

Resonantexcitation

rec~1nsRecombination

Electron tunnellingtun

Hole tunnelling

tun

PC

excitedetect

PLE

Three ways to measure Absorption Properties

from Skolnick


Asymmetric Stark shifts Asymmetric Stark shifts (QCSE)(QCSE)

-300 -200 -100 0 100 200 3001.04

1.05

1.06

1.07

1.08

T=200K

Tra

nsiti

on E

nerg

y (e

V)

Electric Field (kV/cm)

F

p - i - nn - i - p

F

• Quadratic Stark shifts, E,

asymmetric about zero field.

E = aF + bF2

• Linear term implies existence of permanent dipole moment.

PRL 84, 733, 2000from Skolnick


From the sign of the dipole deduce that In composition increases from base to apex

Interdiffusion

Increasing indium along growth direction

InIn0.50.5GaGa0.50.5AsAs

InAsInAs

15.5nm

5.5n

m

Polarisability - height

Theory, JA Barker and EP O’Reilly PR B61, 13840, 2000

QDs are intermixed

from Skolnick


Emission: Spatially resolved PLEmission: Spatially resolved PLEmission: Spatially resolved PLEmission: Spatially resolved PL

► Emission spectrum breaks up into very

sharp lines with homogeneous linewidth ~ 1µeV

► Ground (s-shell) and excited state (p

shell) emission observed

► Single dots may be optically isolated

using apertures or mesas with ~2 or 3

within a size of 500nm.

from Skolnick


How do we measure single QDs?

► Etch mesas whose size is ~ 1 m and see discrete lines from individual dots.

► Cover sample with mask and open holes of order 1 m.

► Use NEAR-FIELD OPTICS: Near-field Scanning Optical Microscopy (NSOM).


Near-Field Spectroscopy of Quantum Structures


Low-temperature near-field optical scanningmicroscope


Temperature dependence of near-field spectra of InGaAsN

1.06 1.08 1.10 1.12 1.14 1.16

C2

C1A

A

C5

C4

C3

C2C1

65K

60K

55K

50K

45K

40K

35K

30K

20K

10K

=514.5 nmP=20 W

Energy, eV

Nea

r-fie

ld P

L in

tens

ity

0 50 100 150 200 250 300

1.06

1.08

1.10

1.12

Ene

rgy,

eV

A

Temperature

C1

C2

In content = 8%N content = 3%

Band A Weak localizationLocalization energy = ~ 40 meV

C Lines Strong localization(Quantum Dots)

Localization energy ~ 50-60 meV

M & M


1.06 1.07 1.08 1.09 1.10

5

10

1.080 1.085

a

C6l C6

h

C7l C7

h

C6 C7

0T

4T

6T

8T

10T

T=5K

Energy, eV

Ne

ar-

fie

ld P

L i

nte

ns

ity

0 2 4 6 8 10 12

1.0825

1.0830

1.0835

1.0840

C7l

C7hb

En

erg

y,

eV

Magnetic field, T

1.06 1.08 1.10

C7C6

Energy, eV

PL

in

ten

sit

y

5

c

r, n

m

15

Energy, eV

,

eV

/T2

Magnetic Field EffectsZeeman Splitting and Diamagnetic Shift

M & M


NSOM Images of Individual QDs

1120 1140 1160 1180

0

10

20

30

40

50

Density ~100 m -3

(x=0.2, y=1.4)

(x=0.8, y=0.4)

(x=1.8, y=1.8)

ed

cba

e

c

d

b

a

Wavelength, nm

NP

L in

ten

sity

, cp

s

0 1 20

1

2

x, m

y, m

M & M


+eVreset

READ / RESET

p-substrate

i-GaAs Buffer

AlGaAs Barrier F

+Vapp

-eVstorep-substrate

i-GaAs Buffer

AlGaAs Barrier F

STORAGE

-Vapp

p+

Contact

Metal

AlGaAsBarrier

QDs

Digital Information Storage J. J. Finley et al, APL, 73, (1998) M. Kroutvar et al, APL 83 (2003)

from Skolnick


Applications (much physics as well!)

1. QD lasers

2. Mid infra-red detectors

3. Memory devices

4. Quantum information

5. Single photon sources

Ensembles

Single dots


QD lasers first discussed Arakawa and Sakaki, APL 40, 939 (1982)First report Kirstaedter, Bimberg et al, El Lett 30, 1416, 1994

Low threshold, temperature insensitive threshold current etc

Breakthrough application, 1.3m lasers for local area networks

Distinct potential advantages over installed InP-based lasers (GaAs-based technology, large area substrates, good thermal conductivity, Bragg mirrors, low temperature dependence of threshold current)

Major contributors: Berlin, Ioffe, New Mexico etc

Quantum Dot Lasers

from Skolnick


Semiconductor Laser Performance Versus Year

Ledentsov et al, IEEE J. Select.

Topics Quant. Electron. 6, 439 2000

17A/cm2, cw, 300K, 1.31m, Liu, Sellers, Mowbray et alfrom Skolnick


Emission spectra from 1.3m quantum dot lasers

Record threshold currents, continuous wave 17A/cm2 at 300K at true 1.3m.

Very encouraging temperature performance from Skolnick


1-D Structures -- Conclusions 1. Self assembled quantum dots have led to a huge

variety of new physics and applications

2. Key points, high radiative efficiency 0D states in a solid state matrix

3. Good approximation to atoms in the solid state, but there are exceptions

4. Challenges include higher uniformity, predetermined positions, larger binding energy materials, longer coherence times, incorporation in very high finesse, small volume cavities for e.g. cavity QED

from Skolnick


Octopus Optics Lab

Goodbye – It has been fun!

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