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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