PHY410: Low Dimensional Semiconductors

M S Skolnick, 2nd Semester 2010/11Syllabus

1 Summary of key properties of semiconductors and motivation for low1. Summary of key properties of semiconductors and motivation for low dimensional structures

2. Alloy semiconductors, lattice matched and mismatched structures

3. Growth techniques for quantum wells

4. Effect of 2, 1 and zero dimensional quantisation on properties of electrons and holes. Wavefunctions, density of statesy

5. Optical properties of wells, wires and dots. Absorption and emission spectra. What they tell us and what controls their properties

6 Coupled quantum wells and superlattices6. Coupled quantum wells and superlattices

7. Resonant tunnelling and the quantum cascade laser

8. Quantum confined Stark effect

9. Transport properties. Modulation doping

10. Growth techniques for quantum dots

111. Modern day physics and applications of quantum dots

Suitable textbooks:Suitable textbooks:

1 J Hook and H Hall chapter 14 and pages 192-1961. J Hook and H Hall chapter 14 and pages 192-196

2. The Physics of Low Dimensional Semiconductors (J H Davies, Cambridge)

3. Physics of Semiconductors and their Heterostructures (J Singh, Wiley)

4. Electronic and Optical Properties of Semiconductor Structures (J Singh C b id )Cambridge)

5. Quantum Wells, Wires and Dots, (P Harrison, Wiley)

6 Low Dimensional Semiconductors (M J Kelly Oxford)6. Low Dimensional Semiconductors, (M J Kelly, Oxford)

7. Quantum Semiconductor Structures (C Weisbuch and B Vinter Academic Press)

2

The influence of semiconductor devices is all-pervasive in life in the 21st century The range of applications includes to name just

1 Integrated circuits in computers

the 21st century. The range of applications includes, to name just a few :

1. Integrated circuits in computers

2. High frequency components in mobile and satellite communications

3. Light emitting diodes for displays, car indicators and break-lights and3. Light emitting diodes for displays, car indicators and break lights and lighting

4. Lasers in cds, dvds, blue-ray, laser pointers

5. Lasers for fibre optic communications

6. Any others??

Many of these vital parts of everyday life rely on semiconductor structures of reduced dimensionality.

The understanding of the basic physics underlying such structures, how they are made and their key properties forms the basis of this course.

3We first begin with a summary of the basic properties of semiconductors required to underpin much of the course

5nm10nm5nm

25nm

45nm transistors Intel web site

Quantum dot

Multi-45nm transistors. Intel web sitequantum well

Bookham tunable t l i tiV ti l it d d itti l telecommunications laser

Vertical cavity and edge emitting lasers

(Ledentsov 2002)

4Internet, optical data storage, lighting, displays, wireless and satellite communications

Key Properties of SemiconductorsKey Properties of Semiconductors

1. They have a band gap separating valence and conduction bands. These lie in the range 0.2 to 3.5eV approximately. Typical examples include ……

2. Direct and indirect band gaps

3. Direct gap semiconductors are important for optical applications, and some electronic applications

4. Silicon and germanium have indirect gaps. Silicon dominates S co a d ge a u a e d ect gaps S co do ateselectronics applications

5. Their conductivity can be controlled by doping

6. Examples of dopants

7. Their band gaps can be controlled by composition and by control of layer thicknesslayer thickness

8. We will be mainly concerned in this course with III-V semiconductors – see periodic table

5

Periodic TableGroup III

Group IVGroup VGroup III Group V

6

Can vary band gap by controlling composition

The following graph plots band gap versus lattice constant for a variety of III-V semiconductors

7

Multi Quantum Well Structure

GaAsGaAs

8

By controlling compositions of different layers can create potential wells for electrons and holes. Reduced dimensionality results in size quantisation.

This is the basis of low dimensional structures: control of size to give new physical properties

Total difference in band gaps distributed Quantum well g pin ratio of ~60:40 between conduction and valence bands

For Al composition of 0.33

Ec ~ 240meV

E 160 VEv ~ 160meV

9barrier well barrier

Transmission electron microscope cross-sections

InGaAs-InP multi quantum well structure

GaAs-AlGaAs multi quantum well structure

10Need layer thicknesses to be order of de Broglie wavelength of electrons (<~30nm) to realise quantum effects

Lattice matched and lattice mismatched structuresLattice matched and lattice mismatched structures

Lattice matching: aim to have lattice constant of material A same as that for material B

11

Can vary band gap by controlling composition Lattice Matched Growth: Growth of material layers one

The following graph plots band gap versus lattice constant for a variety of III-V semiconductors

Growth of material layers one on top of the other with same lattice constant, and with same lattice constant as substratesemiconductors

1. GaAs, AlAs and AlGaAs

lattice constant as substrate (see slide 7) e.g.

1. GaAs, AlAs and AlGaAs alloys

2. InGaAs alloy on InP with specific composition ofspecific composition of InGaAs. 53% In composition

I G A ll I P ithInGaAs alloy on InP with specific composition of InGaAs

GaAs, AlAs 0.3% difference in lattice constant

12

lattice constant

InAs, GaAs 7% difference+GaN ~3.5eV

Lattice matched growth: schematic

Layers A, B have same lattice constantlattice constant

Lattice matched growth

13

But lattice mismatched growth is also possibleBut lattice mismatched growth is also possible

At least one of layers is then strained – termed strained layer growth

Restriction on thickness that can be grown (‘critical thickness’)Restriction on thickness that can be grown ( critical thickness )

In-plane lattice constant is same as that of substratep

• Material is compressed in planeplane

• Extended vertically

14Davies p97

Constraint:

St i b ild ith thi k• Strain energy builds up with thickness

• Until strain energy becomes greater than energy to form dislocations in gylattice

• Dislocations highly undesirable

• But mismatched growth gives high quality up to critical thickness

Very great flexibility in band gaps and layer properties if structures designed correctly

Combination of alloy composition, suitable lattice matching, controlled degree of mismatch and use g, gof quantum wells gives control of emission wavelength used in modern day light emitters

15

Crystal Growth Techniques for Low Dimensional Structures

Need control on 1-30nm scale (why?)

1. Molecular Beam Epitaxy (MBE)

2. Metal organic chemical vapour deposition (MOCVD) or Metal Organic Vapour Phase Epitaxy (MOVPE)

16

MBE SchematicsMBE Schematics f

17

Molecular beam epitaxy growthMolecular beam epitaxy growth• Molecular beams

• Evaporation sources Knudsen cells• Evaporation sources Knudsen cells

• Layer by layer growth (~1m/hour)Layer by layer growth ( 1m/hour)

• Nanometre scale thickness control

• Surface analysis techniques (RHEED reflection high y q ( genergy electron diffraction)

• Shutters on sources

• Sample rotation

• Ultra high vacuum ~10-10 Torr

18• Heated substrate ~600oC

Molecular Beam Epitaxy V90 Production Geometry ReactorGeometry Reactor

19http://www.shef.ac.uk/eee/nc35t/MBEepi.pdfSee also

Metal organic chemical vapour deposition (MOCVD)Metal organic chemical vapour deposition (MOCVD)

• Layer by layer growth

• Flowing gases over heated substrateo g gases o e ea ed subs a e

• Typical reaction

3CHG AA HGCH

• Near atmospheric pressure

4333 3CHGaAsAsHGaCH

• Reaction takes place over substrate

• Somewhat less complex equipment than MBE

• Gases switched to change composition by mass flow controllers

G th t ( 1 5 /h )• Growth rate (~1-5m/hour)

• Nanometre scale thickness control

20

G (CH ) Al(CH ) Z (C H )

H2H2H2

Sili

for acceptors

v v v

G (CH ) Al(CH ) Z (C H )

H2H2H2

Sili

for acceptors

v v v

MOCVD schematicsGa(CH3)3 Al(CH3)3 Zn(C2H5)2

Waste gases

Automated

Silica Reactor

H2

x x

x x x

vv

• • • • •

Ga(CH3)3 Al(CH3)3 Zn(C2H5)2

Waste gases

Automated

Silica Reactor

H2

x x

x x x

vv

• • • • •

MOCVD schematics

AsH3/H2 H2Se/H2

substrate (~600oC)

valves

R.F. induction heating

for donors

v

x x

v

• • • • •

AsH3/H2 H2Se/H2

substrate (~600oC)

valves

R.F. induction heating

for donors

v

x x

v

• • • • •

H2 H2

for donors

H2 H2

for donors

21

Metal-organic chemical vapour deposition reactorvapour deposition reactor chamber

Switching

22manifold

See also http://www.shef.ac.uk/eee/nc35t/MOVPEepi.pdf

Q t ll i d d tQuantum wells, wires and dots

Quantum well – quantisation in one dimension

Quantum wire – quantisation in two dimensions

Quantum dot – quantisation in all three dimensions

Quantum wells require layer by layer growth (MBE or MOCVD growth as described above, with switching between layers)

Wires and dots need additional techniques

Will describe growth techniques for dots later

23

Quantum WellsQuantum Wells

• Infinite well• Infinite well

• Finite wellFinite well

• Density of statesDensity of states

Applications: LEDs, lasers, modulatorspp , ,

24

Infinite well: Wavefunctions zero at boundaries of well

Hook and Hall, p400

25

Infinite potential well (Schiff p38)

Schrodinger equation in one dimension

d

22

V=0 for a < x < a

EVdxd

m

22

V

V=0 for –a < x < a

Ed

22-a a

Solution of form

Edxm

22

xBxA cossin

xBxAExBxAmdx

dm

cossincossin22

222

2

22

mdxm 22

2122 2

mEE

262,

2

m

E

V t ± i 0 t ±V = ∞ at x = ±a requires = 0 at x = ±a

aBaA cossin0 (1)

To satisfy (1) A = 0, cosa = 0

aBaA cossin0 (2)

(2) B = 0, sina = 0

For (1) a = n/2 n oddFor (2) a = n/2 n even

Thus (x) = B cos nx/2a n odd( ) A i /2(x) = A sin nx/2a n even

222222222 2

222

2

222

2

222

2842 mdn

man

an

mE

where d = 2a is the width

27

Alternative simple derivation for infinite well

Electron wavelength = 2d/n (quantisation of electron wavelength)

Where d is idth of ellWhere d is width of well

Wavevectordπn

2d2ππ

λ2πk

Energy

d2dλ

2

22222 nπkE 22md2m

E

For 10nm well in GaAs (m* ~ 0.07me), E1 = 45meV

28

For finite well, wavefunctions leak into barrier

Effect on energy?

Quantum well

barrier barrier

Kelly p252

29

y p

Alternative simple derivation for infinite well

Electron wavelength = 2d/n (quantisation of electron wavelength)

Where d is idth of ellWhere d is width of well

Wavevectordπn

2d2ππ

λ2πk

Energy

d2dλ

2

22222 nπkE 22md2m

E

For 10nm well in GaAs (m* ~ 0.07me), E1 = 45meV

For finite well, E1 = 29meV

30

Wavefunctions are quantised along z but there is dispersion inWavefunctions are quantised along z, but there is dispersion in the plane

222 22

2 yxe

ntot kkm

EE

Quantised energy along z, the growth direction (x, y are in plane)

Consequences for density of statesdensity of states, optical absorption

One dimensional quantisation

31

quantisation

p135, Davies

Density of States (DOS) in Systems of ReducedDensity of States (DOS) in Systems of Reduced Dimensionality

Important property which underlies new functions

Fi t ll DOS i 3DFirst recall DOS in 3D

Free particle wavefunction satisfying periodic boundary conditions:Free particle wavefunction satisfying periodic boundary conditions:

Ner rik

k 2

)( .

LN

Lk

.......2,0

32

In 3D, one allowed value of wavevector in volume element of (2/L)3, ( )

Thus up to energy E, wavevector, number of states

NkVkL

F

3

2

33

33422

(1)

N is number of particles, V volume

B t k 22But

Substitute in (1) for k, mkE

2

23

222

3

mEVN

Density of states, number of states per unit energy range:

3

2123

222

2)( EmV

dEdNED

33

222dE

Density of States in Two Dimensional System

In 2D, one allowed value of k in area element (2/L)2

2222 Nk

L

212 nk

L

2

ANn

Constant independent of i 2D

dm

nE

energy in 2D

The states which contribute are the in-plane states which are 2

)(

mdEdnED

34not quantised

(see slide 31)

3D

2D

Hook and Hall p403p

35

Transition Energies and Spectra

These are what are actually measured – not energy levels densityThese are what are actually measured not energy levels, density of states directly, but they give clear experimental evidence for above concepts

36

Transition Energies

Both electron and hole states are quantised

Quantum well

37barrier well barrier

Predicted absorption spectra (a and c) and densities of states (b)p p ( ) ( )

Peaks in a and c are due to excitonic effects

Exciton: bound electronExciton: bound electron-hole pair, bound by mutual Coulomb interaction

As for donors, binding energy is given by

2224 2 neE 2

0

/6.13

2

eVE

neE

x

x

But is now the electron-hole reduced mass

Exciton binding energies are

38

Exciton binding energies are enhanced in GaAs quantum wells relative to bulk (from 4 to ~10meV)

Discussion of variation of Ex with d

Excitons areExcitons are stable at room temperature in quantum wells, inquantum wells, in marked contrast to bulk semiconductors

Mill t l A l Ph

39

Miller et al Appl Phys Lett 41, 679, 1982

GaAs-AlGaAs QWs Dingle Phys Rev Lett 33, 827, 1974

Key features?

Transition energies

xhelg EnEnEE )()(

Ex is exciton binding energy (~10meV)

40

Absorption spectrum in two dimensional system (quantum well)

1. Resembles step function density of states

2. As well gets wider, absorption features get closer together in energy

3. Absorption strength from each sub-band the same

4. Exciton features appear due to Coulomb interaction between photo-excited electron and hole

5 Exciton effects are enhanced in 2D relative to 3D since electron and hole5. Exciton effects are enhanced in 2D relative to 3D, since electron and hole are forced closer together by confinement

6. Exciton binding energy enhanced in quantum wells relative bulk. Hence excitons are stable at room temperature

7. Strongly allowed transitions are between electrons and holes with same sub-band index (since states have same parity and have large overlap. ( p y g p(n=0 selection rule)

41

Allowed transitions in quantum wellsAllowed transitions in quantum wells

Hook and Hall p404

n=0 selection rule

Hook and Hall p404

Good approximation for symmetric wells

T iti t b t t t f th42

Transitions strong between states of the same parity and with large wavefunction overlap

Density of States in 3D, 2D, 1D, 0D y , , ,Evolution of DOS as dimensionality is reduced

Bulk (3D) Well (2D)DOS

f(E)

DOSf(E)

Bulk (3D) Well (2D)DOS

f(E)

DOSf(E)

DOSf(E)

DOSf(E)

DOSf(E)

DOSf(E)

DOSf(E) DOS – density of

states

ribut

ion f(E)

n(E)

( )

kTn(E)rib

utio

n f(E)

n(E)

( )

kTn(E)

f(E)

n(E)

f(E)

n(E)

( )

kTn(E)

( )( )

kTn(E)

states

f(E) Fermi-Dirac distribution function

Wire (1D)er d

istr

Dot (0D)

≈ kT( )

Wire (1D)er d

istr

Dot (0D)

≈ kT( )

≈ kT( )

≈ kT( )

n(E) injected carrier distribution as a function of energy

Car

ri

DOSf(E)DOSf(E)C

arri

DOSf(E)DOSf(E)DOSf(E) DOSf(E)DOSf(E) DOSf(E)

function of energy

n(E) n(E)n(E) n(E)n(E)n(E) n(E)n(E)

43EnergyEnergy

• Modification of density of states as dimensionality is reduced is highly beneficial since e.g. in laser devices as carriers injected, they are increasingly confined to be at the same energy. This leads to higher gain, lower threshold and more favourable laser properties.

• Furthermore quantisation leads to additional degree of control over wavelengthwavelength

• What else gives control?• What else gives control?

44

Applications of Quantum WellsApplications of Quantum Wells

Main applications of quantum wells are as light emitting elements in light emitting diodes or lasersg gTelecommunications wavelengths: InGaAs/AlGaInAs (1.3, 1.55m)

Near infrared: (In)GaAs/AlGaAs (0.8-1.0 m)

Red GaInP/AlGaInP (630-680nm)

Green InGaN/GaN (540-560nm)

Blue InGaN/GaN (380-440nm)

Design considerations include those of lattice matching, or controlled mismatch that we discussed earlier

Other applications:

Modulators detectors

45

Modulators, detectors

Coupled Quantum Wells and Superlatticesp p

For real quantum wells with finite barriers wavefunctions leak out ofFor real quantum wells with finite barriers, wavefunctions leak out of well, as we have seen earlier. Tunnelling of wavefunctions into barriers. (see also section on tunnelling)

If wells grown one on top of the other, with thin barriers then wavefunctions from adjacent wells overlap and coupled quantum wells (2 wells) and superlattices (many wells) can be formedwells (2 wells) and superlattices (many wells) can be formed

46

Weisbuch and Vinter p25

dzψVψV 2112

• V12 is the coupling potential – determines how energy of electron in one well is modified by presence of other

• Without coupling the energies in the two wells are theWithout coupling, the energies in the two wells are the same

• Degeneracy is lifted by coupling, by amount 2V12

• Resultant wavefunctions have symmetric, antisymmetric symmetry

• Coupling arises due to leakage of wavefunctions into

47

Coupling arises due to leakage of wavefunctions into barrier and interaction

Coupling of many wells results in formation of superlatticeCoupling of many wells results in formation of superlattice

Similar to tight binding model for formation of energy bands in solids

Here coupling of say N wells results in band with (2)N statesHere coupling of say N wells, results in band with (2)N states

As barrier width LB decreases, band width increases due to greater interaction between wells

What gives rise to dependence on well width

48Weisbuch and Vinter p27

New three dimensional band structure created by coupling wells into superlatticesuperlattice

New bands termed minibands, separated by mini-gaps – see section on quantum cascade lasers

miniband

minigap

49Davies p182

Resultant change in density of states

Davies p183

Application of superlattices in quantum cascade lasers. Efficient current transport and precise electron injection in p p jcomplex multilayer structures

But growth is very challenging, since barrier widths typically of thickness 1nm to get large band widths and

50

typically of thickness 1nm to get large band widths and efficient vertical transport

R t T lliResonant Tunnelling

As opposed to classical particles, electron wavefunctions can penetrate through potential barriers due to the phenomenon of quantum mechanical tunnelling

Can construct such tunnelling structures by controlled layer by layer growth

A li ti id i f d itt d hi h f ill tApplications as mid infrared emitters and as high frequency oscillators

51

Schiff, p104

Electron waves incident on barrier

Shows finite probability for transmission through barrier

Earlier example for coupled QWs and superlattices

Zero for classical particle for E<V0

The single barrier tunnelling structure

p

52

The single barrier tunnelling structure

Perfect transmission when barrier contains integral number of half wavelengths

Double barrier resonant tunnelling structure

V = 0 V > 0

1. Low voltage emitter energy less than E, I=0

2. Voltage (V1) such that emitter

V1 V2

aligned with E1. Peak in current

3. V>V1. I 0

534. V=V2. Second current peak

when emitter aligned with E2

Further pictorial figure to assist in the understanding of previous page

Davies p174

54

Negative differential resistance regions give rise to application asNegative differential resistance regions give rise to application as high frequency oscillator up to 500GHz.

V1 V2

Major application as quantum cascade lasers

55

The Quantum Cascade Laser (QCL)• A device based on 2D subbands (and quantum wells and barriers)• A device based on 2D subbands (and quantum wells and barriers)

• As opposed to all other semiconductor lasers which rely on conduction to valence recombination, in a QCL transitions are only between confined conduction band statesstates

• Wavelength is determined by width of layers, by design

• Composed of injector and active regions which are then repeated

• Structures have up to 500 layers, individual layers ~1nm thick). Each electron gives rise to up to 25 photons by emitting a photon in each period of the device

• Formation of superlattice minibands and electron tunnelling underlies operation• Formation of superlattice minibands and electron tunnelling underlies operation

Diagram shows 2 periods of >25 period structure

Population inversion occurs between levels

Sirtori et al Pure Appl. Opt. 7 (1998) 373–381

56

occurs between levels 3 and 2.Frequently now

superlattice injector or active regions

ActiveRegion

ActiveRegion

InjectorRegion

InjectorRegion

ActiveRegion

ActiveRegion

InjectorRegion

InjectorRegion

MinibandMiniband

electronsE3

Mi iLaser

electronsE3

Mi iLaserE32

E21

E2

Miniband

MinigapLaser TransitionE32

E21

E2

Miniband

MinigapLaser Transition

21

photon

E1

Minigap

21

photon

E1

Minigap

EnergyEnergy

electronsDistance electronsDistance

Transitions between conduction

57

Transitions between conduction sub-bands: in mid infrared spectral region 4-20m

Sirtori et al Pure Appl. Opt. 7 (1998) 373–381

Quantum cascade laser with superlattice injector region and ‘vertical’ lasing transition

58

Layer structure of QCL grown in Sheffieldgrown in Sheffield

Layer thicknesses are of order ~1nm to achieve effective tunnelling

Resonant tunnelling is fundamental toResonant tunnelling is fundamental to device operation

Lasers operate in mid infra-red and terahert range 4 to 50 m rangeterahertz range, 4 to 50m range

Applications gas sensing, environmental monitoring

50nm

59

The Quantum Confined Stark Effect

Quantum well in an electric field

60

Applied electric field gives rise to perturbation in quantum well Hamiltonian of

eEzV

V is potential, E electric field and z is position

Leads to tilting of energy bands with two main resultsLeads to tilting of energy bands with two main results (see next page)

61

TransitionTransition energy reduced by applied field

D i 259Davies p259

Selection rules changed

Now inversion70kV/cm?

Now inversion asymmetric

Parity no longer d t

62

good quantum number

Singh p429

Further application of quantum wells: exploitation of Quantum Confined Stark Effect as modulator in telecommunications systemsStark Effect as modulator in telecommunications systems

By applying modulation to Stark effect modulator

absorption

effect modulator impose signal on light transmitted into fibre

1.3 or 1.55m

63

Inter-sub-band infra-red absorption

n=1 selection ruleAlGaAs GaAs

n=1 selection rule

O ti l b ti b t d ti b b dOptical absorption between conduction sub-bands

Spacing between first two sub-bands is ~150meV for a 4nm well (corresponds to wavelength of ~8m)( g )

Permits absorption in mid IR region in wide band gap material

New property not possessed by host semiconductor

Further application of quantum wells: as mid infrared detectors

64In emission related structures: quantum cascade lasers

Note that selection rules are different for intra-band (conduction band to valence band) and inter-sub-band transitions )

n=0 and n=1 in the two cases

Hint: the total electron wavefunction in a quantum well is:

ruze knrik .

n(z) is the slowly varying envelope function

uk(r) is the Bloch function which labels the band (s-uk(r) is the Bloch function which labels the band (slike conduction band, p-like valence band)

Electric dipole transition: operator e r , odd parityElectric dipole transition: operator e r , odd parity

For transition within conduction band, Bloch function does not change, and transition between envelope functions, which must thus have opposite parity. Thus p , pp p yn=1.

For valence to conduction band transition (interband transition), transition is

65between Bloch functions which conserves parity. Envelope functions must thus have same parity and thus n=0.

Valence to conduction band, interband absorption – electric dipole transition between Bloch functions, multiplied by overlap integral between

l f tienvelope functions

Inter-sub-band absorption – electric dipole transition between envelope p p pfunctions

Breakdown of selection rules in applied electric field – since parity no longer a good quantum number

66

M d l ti D iModulation Doping

C ti f t ll f l t b d i l• Creation of quantum wells for electrons by doping alone

• Charge transfer across heterostructure interface

• Suppression of electron scattering• Suppression of electron scattering

• High frequency devices with important applications in mobile and satellite communications

67

Heterojunction between materials of two different compositions

But this is not anBut this is not an equilibrium situation

Electrons from donor Two layers not in contactdopants will transfer to narrower gap material

y

Conduction bands aligned relative to vacuum level

68

relative to vacuum level

Electrons flow from wider to narrower gap material until equilibrium is established i e until chemicaluntil equilibrium is established, i.e. until chemical potentials on two sides are equalised

For Ec = 0, Ev = 0 (conduction and valence band discontinuities zero between AlGaAs and GaAs) i.e GaAs homojunction (n, p regions both GaAs), we have

F

Band bending is due to

69

Band bending is due to space charge

AlGaAs GaAs

In case of modulation doped heterojunction, electrons transfer from wider gap (doped)

t i l t t i lmaterial to narrower gap material

Leave positive space charge behind in AlGaAs. Negative space charge in GaAs.g p g

From Poisson’s equation, this spatially varying charge distribution gives rise to bending of the bands

Open symbols, ionised bending of the bands.impurities

2DEG – two2DEG two dimensional electron gas

02

2

dx

Vd

Weisbuch and Vinter p39

Quantum well for electrons formed at interface: one

70dimensional confinement

Understanding band bendingUnderstanding band bending• Positive space charge, d2V/dx2 is negative

• d2V/dx2 is positive on electron band diagramAlGaAs GaAs

• d2V/dx2 is positive on electron band diagram (since electron charge negative)

• i.e. upward curvature of bands (positive 2nd

derivative) in AlGaAs region

• Electrons transfer to narrower gap region2 2• Negative space charge, d2V/dx2 is negative (on

electron band diagram)

• Slope decreases with x on GaAs sidep

• Quantum well for electrons formed by triangular potential of electrostatic origin

• Transfer of charge occurs until F on two sides equalised

• Triangular potential result of band bending and

71

• Triangular potential result of band bending and band discontinuity between two materials of differing band gap

• Importantly have achieved separation of electrons from dopant atoms hence modulation dopingdopant atoms – hence modulation doping

• Thus can suppress ionised impurity scattering of electrons, which in heavily doped semiconductors is the dominant scattering mechanism, particularly at low temperature

• Can minimise scattering by use of ‘spacer’ layers to further separate electrons from donorsseparate electrons from donors

• See next page for figures

72

Mobility versus temperature in bulk (3D) G A

Suppression of ionised impurity i i d l i d d(3D) GaAs scattering in modulation doped

structure

Pfeiffer, Appl Phys Ionised impurity scattering dominates in heavily doped

73

Lett 55, 1888, 1989dominates in heavily doped material

For mobility of 12,000,000 cm2/Vsecy

Electron mean free path is >50m

‘Ballistic’ transport of electrons

Such structures used to observe the quantum and fractional quantum Hall effects, and as the basis for room temperature high frequency (2-10GHz) transistorsfrequency (2 10GHz) transistors

Technically high channel transconductance (high n and high ) is the reason for excellent high frequency performance and important

t d li ti i bil i ti f lpresent day applications in mobile communications for example

74

Fermi energy of two dimensional electron gas

Typical electron density in channel may be 1016 m-2

D it f t t i t di i i 2* /)( EDDensity of states in two dimensions is

Thus for two dimensional electron gas of density n

2/)( mED

Thus for two dimensional electron gas of density ns,

VE 342

16 2

meVnm

E sF 34*

for n = 1016 m-2

75

S i d t Q t D tSemiconductor Quantum Dots

Fully confined systems in all three dimensions

Atom like systems in the solid stateAtom-like systems in the solid state

Will consider self assembled quantum dots

Applications in fundamental physics (access to single quantum states), and in telecommunications wavelength lasers to name just two

76

Density of States in 3D, 2D, 1D, 0D y , , ,Evolution of DOS as dimensionality is reduced

Bulk (3D) Well (2D)DOS

f(E)

DOSf(E)

Bulk (3D) Well (2D)DOS

f(E)

DOSf(E)

DOSf(E)

DOSf(E)

DOSf(E)

DOSf(E)

DOSf(E) DOS – density of

states

ribut

ion f(E)

n(E)

( )

kTn(E)rib

utio

n f(E)

n(E)

( )

kTn(E)

f(E)

n(E)

f(E)

n(E)

( )

kTn(E)

( )( )

kTn(E)

states

f(E) Fermi-Dirac distribution function

Wire (1D)er d

istr

Dot (0D)

≈ kT( )

Wire (1D)er d

istr

Dot (0D)

≈ kT( )

≈ kT( )

≈ kT( )

n(E) injected carrier distribution as a function of energy

Car

ri

DOSf(E)DOSf(E)C

arri

DOSf(E)DOSf(E)DOSf(E) DOSf(E)DOSf(E) DOSf(E)

function of energy

n(E) n(E)n(E) n(E)n(E)n(E) n(E)n(E)

77EnergyEnergy

Q t d t thQuantum dot growth

Characteristic of lattice mis-matched systems

Initially layer by layer

Islands form to relieve strain (lowers total energy of system)

So-called Stranski Krastanov techniques

Self assembly process

78

Self-Assembled Crystal Growth in Strained Systems (schematic) MBE Growth

Stranski-Krastanow growth

InAs-GaAs 7% lattice i t hmismatch

N t ttiInAs

Note wetting layer

GaAs

Embedded in crystal matrix like any other79

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

InAs Quantum DotsInAs Quantum Dots

1E10

cm-2

)

Critical Thickness

200nm200nm

1 65ML 1 68ML1E9

D D

ensi

ty (c

1m

1.65ML 1.68ML

1.5 1.6 1.7 1.8 1.9

QD

I A C (ML)InAs Coverage (ML)

M Hopkinson Sheffield

D t f b lf bl l ti200nm 200nm

1 75ML 1 9ML

Dots form by self assembly – nucleation at sites on surface

There is a size and composition di t ib ti f d t ( lt f lf

80

1.75ML 1.9MLdistribution of dots (result of self assembly process)

10nm5nm10nm5nm

(a)Higher growth rateHigher growth rate, 0.3 monolayers/sec, density ~5x1010cm-2

10 nmdensity 5x10 cm

(b) Low growth rate 0.01 l / )

10

monolayers/sec), density ~1x109cm-2

8110 nm

M Hopkinson, AG Cullis, Sheffield

Quantum dot energy levels and transitions

x,y Very strong confinement in z-

p-shell

d-shell

y gdirection

Atom-like energy levels from in plane x y confinement

s-shell

in-plane x, y confinement

s, p, d shells like atom (but degeneracies not exactly the

Photon emitted

same), since ??Photoluminescence:

Electrons and holes excited atemitted Electrons and holes excited at high energy

Relax in energy and fill dot states

In accord with requirements of Pauli principle

82

Dot (e.g. InAs) Surrounding matrix material

Surrounding matrix material matrix material

(e.g. GaAs)matrix material

Photon emission – (photo)-luminescencePhoton emission – (photo)-luminescence

Photon absorption – optical absorption

83

Optical Spectra

10

Sample M1638T = 300 K

EL2Photoluminescence

E

EL2

(a.u

.)

nt (p

A)5

E3

E2

E1

EL1

Inte

nsity

ocur

ren

1E

0 EL

Phot

o

0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.300

Energy (eV)Absorption

Excitation area ~100mLarge numbers ~107 dots. Transitions observed between n=1, 2 and 3

Energy (eV)

84Linewidth ~30meV due to shape and size fluctuations

electron and hole levels

n=0 selection rule as for quantum wells

Spatially resolved emissionSpatially resolved emissionSpatially resolved emissionSpatially resolved emission

T = 10 K

• Broadening due to space and size fluctuations

200nm Single Dotb. u

nits

)

g p

removed by isolating individual dots

200nm

500

g

Spatially Resolved PL

>1 QDnsity

(arb • Single dots optically isolated using apertures

of ~500nm size.

b k h

200

500nm 1 QD

Far Field PL~107 QDsP

L In

ten •Emission spectrum breaks up into very sharp

lines

Li idth ( 1 V) li it d b di ti

1250 1300 1350 1400

200µm

Energy (meV)

•Linewidth (~ 1µeV) limited by radiative lifetime (Heisenberg uncertainty principle?)

•Ground (s shell) and excited state (p shell)Energy (meV)

400nm aperture in metal film

•Ground (s-shell) and excited state (p-shell) emission observed

•Single dot applications up to ~50K

85400nm400nm

in metal film •Single dot applications up to ~50K

Filling of levels under optical illuminationFilling of levels under optical illumination

• s-shells, 2 electrons in conduction band, 2 holes in valence band x,y x,y x,y

• Exciton is electron-hole pair

• Thus if one electron hole pair created by external radiation can accommodate both in respective s

p-shell

d-shell

p-shell

d-shell

p-shell

d-shell

radiation, can accommodate both in respective s-shells

• Can add second e-h pair but with opposite spins.

s-shell

Ph

s-shell

Ph

s-shell

Ph• s-shell is then full

• Next carriers must then go in p-shell

Photon emittedPhoton emittedPhoton emitted

• Next slide experimental evidence

86

Evidence for level filling in Quantum Dots

• Excitation power density (Pex) controls average excitoncontrols average exciton occupancy (NX)

N 1 i l li• Nx 1 – single line• NX ~ 1-2 – two groups of lines :

s-shell (~1345meV) p-shell (~1380meV)

E id fEvidence for

• Degeneracy of QD levels

• Forbidden transitions• Forbidden transitions

87Physical Review B63, 161305R, 2001

Applications of Quantum DotsApplications of Quantum Dots

Will pick out twoWill pick out two

Quantum dot lasersQuantum dot lasers

Single photon sources

(also much fundamental physics of type carried out in(also much fundamental physics of type carried out in Sheffield)

88

Quantum Dots for Telecommunications Lasers

Ed ittiEdge emittingVertical cavity

• Very low thresholds

after Ledentsov et al, IEEE J. Select. T i Q El 6 439 2000

y

• Very small temperature dependence of threshold current

89

Topics Quant. Electron. 6, 439 2000 • 1.3m lasing on GaAs substrates

Semiconductor Laser Performance Versus Year

QD laser Sh ffi ldSheffield

90after Ledentsov et al, IEEE J. Select. Topics Quant. Electron. 6, 439 2000

1 3m quantum dot lasers for telecoms applications1.3m quantum dot lasers for telecoms applications (InGaAs-based quantum dots)

QD laser T=300KGround state lasing

(arb

)Above threshold

L In

tens

ity (

1100 1200 1300 1400

EL

W l th ( )

Below threshold

Wavelength (nm)

Liu et al IEEE Phot. Tech. Lett. 17, 1139, 2005

91

300

Comparison of QD and quantum well lasers:

250

300ity

(Acm

-2)

QD laser: temperature dependence of threshold current

quantum well lasers: temperature dependence of threshold current (DJ Mowbray lecture notes)

150

200

urre

nt D

ens current

50

100

Thre

shol

d C

u

50 100 150 200 250 300 350 4000T

Temperature (K)

Jth nearly independent of temperature up to 300K – characteristic of 0D density of states • Very low thresholds

• Very small temperature dependence of threshold current

92

of threshold current

• 1.3m lasing on GaAs substrates

Single photon sources for quantum cryptography applications

Single photon source provides secure basis for quantum communication

Resistant to eaves-dropping

Single quantum dots well-suited to fabrication of single photon sources

Acknowledgement DJ Mowbray

Shifts in emission energy due to Coulomb interactions each time an exciton is added to a dot

93

If QD excited by pulsed laser for each laser pulseIf QD excited by pulsed laser, for each laser pulse, only one photon will be detected at the single exciton energy.

Hence single photon source

Second order correlation functionfunction

Two detectors

First detector detects firstFirst detector detects first photon

Time delay till second detector detects a photon is recorded

Very small peak at timeVery small peak at time ‘t=0’ shows very good single photon source

94Time at which photon is detected on first detector triggers counting electronics

Summary of applications of low dimensional structures

Quantum Wells (extended from slide 45) ( )

Telecommunications wavelengths: InGaAs/AlGaInAs (1.3, 1.55m)

Near infrared: (In)GaAs/AlGaAs (0.8-1.0 m)

Red GaInP/AlGaInP (630-680nm)

Green InGaN/GaN (540-560nm)

Blue InGaN/GaN (380-440nm)

Mid infra-red quantum cascade lasers (3-50m)

Modulators, detectors

High electron mobility transistors (GHz range)

Quantum DotsTelecommunications wavelength sources

95Quantum optics, cryptography applications