1. Summary of last lecture
Main properties of semiconductors can be described by values of bandgap and effective mass
Heterojunctions can be formed by epitaxial growth of different semiconductors
They can result in spatial confinement strong enough to split band into subbands k G E3 E2 E1 z
k||

2. This lecture: Look at more complicated heterostructures
Quantum wells (or films)
Quantum wires
Quantum boxes (or dots)
Reminder of last week’s Homework:
Revise Density of States for 3D system (see level 2 semiconductors 2SP, level 3 Physics of Stars 3PS etc)
Find out Density of states for 2D, 1D and 0D system.
Bring drawings and formulas to lecture

3. Species of Heterostructures
GaAs
InP
Type I
Type II
e- and h+ confined in same material
AlxGa1-xAs
e- and h+ confined in different materials
InAlAs
InGaAs
InAs
Type III
Type I
Semiconductor basics and bonding.pdf
e- and h+ spontaneously created at interface
GaSb
InP
Alignment can be described by valence band offset….

4. Which species depends on bandgaps and sign and size of valence band offset

5. Material Parameters for AlxGa1-xAs
Lattice constant a = x (Angstroms)
direct if x<0.45
Bandgap EG = x (eV)
Valence band discontinuity: ΔEv = x (eV)
Conduction band discontinuity: ΔEc = 0.79x (eV)
Effective electron mass me*/m0 = x
Effective hole masses mhh*/m0 = x mlh*/m0 = x
Direct Indirect
Semiconductor basics and bonding.pdf
0.45 1 x
See for other parameters and other semiconductors

6. Semiconductor quantum wells
QWs and QDs.pdf

7. Infinite quantum wells
Motion in x-y plane is free
Ψ(z)
Ψ2(z) E E
n=3 E3
n=2 E2
Semiconductor basics and bonding.pdf
n=1 E1
z/L
k||
Q: Estimate lowest energy transition for GaAs/AlGaAs well, 10nm wide, Eg = 1.424eV, m*c = 0.067

8. Finite barrier QW Solution to Schrodinger in 1D
In well: sum of forward and backward travelling waves
In barrier: sum of increasing and decaying evanescent waves Vb E
-d/ d/2

9. Parity Symmetry of well means solutions have definite parity
Even parity
Odd parity

10. Boundary conditions Continuity of ψ
Continuity of dψ/dz (particle current)
Divide:

11. Graphical solution Equation of a circle
Solve these equations graphically (or numerically)

12. Example GaAs/Al0.3Ga0.7As Show that the lowest conduction state is
d = 10 nm
Conduction band: Vb = 0.3 eV, mw*= 0.067me
Show that the lowest conduction state is
E1 ~ 31.5 meV
Find a better approximation using Newton-Raphson method

13. Further refinements to infinite barrier model
m* discontinuities:
if masses in well and barriers differ this must be taken into account when matching wavefunction at boundaries
Homework: find new solution if m*w ≠m*b, and the new B.C. is continuity of (1/m*) dψ/dz
non-parabolicity:
we neglected terms in the energy higher than k2 but they are important at high energy. Including these is like having an effective mass that increases with energy. This usually requires computer iteration of the wavefunction solutions
multiple bands:
in the valence band there are typically two hole bands (light and heavy) that must be considered. They anti-cross if they overlap.
strained wells:
strain will shift bands (e.g. split the light and heavy hole bands)

14. Optical transitions in Quantum Well
Normal incidence means light polarised in x-y plane
Parity selection rule for symmetric wells: Dn = 0
Q: Estimate (infinite approx) lowest energy transition for GaAs/AlGaAs well, 10nm wide, Eg = 1.424eV, m*c = 0.067, m*v = 0.4
QWs and QDs.pdf

15. Experimental absorption in GaAs wells
QWs and QDs.pdf
Excitonic effects enhanced in quantum wells
Pure 2-D: RX2D = 4 × RX3D
Typical GaAs quantum well: RX ~ 10 meV ~ 2.5 × RX (bulk GaAs)
Splitting of heavy and light hole transitions (different mass) h+ e-

16. Quantum confined Stark Effect
Absorption spectrum can be controlled with electric field
Red shift of excitons
Excitons stable to high fields
Parity selection rule broken
used to make modulators
QWs and QDs.pdf
Energy (eV)

17. Luminescence Emission
In QW, emission energy is shifted from Eg to (Eg + Ee1 + Ehh1)
Tune λ by changing d
Brighter than bulk due to improved electron-hole overlap
Used in laser diodes and LEDs
Homework: using peak energy, well thickness and quoted Eg, estimate electron effective mass in well
QWs and QDs.pdf
Eg = 2.55eV (10K)
Eg = 2.45eV (300K)

18. Intersubband transitions
Need z polarized light
Parity selection rule: Δn = odd number
Q: Show that transition energy ~ 0.1 eV (~ 10 μm, infrared) for GaAs well with typical width
Absorption used for infrared detectors
Emission used for infrared lasers (Quantum cascade lasers)
n-type quantum well
QWs and QDs.pdf

19. Double barrier resonant tunnelling diode
A device with unusual resistance , whose differential is negative
Useful for oscillators etc

23. From wells to wires and dots
By engineering
Lithography + etching
Cleaved-edge overgrowth
Confinement induced by
electrostatics (gate)
STM tip, ..
strain
By self-assembly
Colloidal quantum dots
Epitaxial quantum dots
Introduction_to_Semiconductor_Nanostructures

24. E-beam Lithography and Dry Etching
4) Lift-off
1) E-beam patterning of PMMA
5) Plasma etching
2) Development of PMMA
6) Removal of mask
3) Deposition of mask

25. Quantum wires: Electrical confinement
1DEG = one-dimensional electron gas

26. E-beam Lithography and Dry Etching
Ref: K. Tai, T. R Hayes, S.L. McCall, T.W. Tsang, Appl. Phys. Lett. 53, 302 (1988)
- Single nanometer scale resolution: NO
- Patterning of large areas: NO
- Structure uniformity: Limited
- High crystalline quality: NO
- Compatible with established technologies: Limited
Ref: Maile et al., Appl. Phys. Lett. 54, 1552 (1989)

27. Quantum wires: cleaved edge overgrowth
Introduction_to_Semiconductor_Nanostructures

28. V-groove epitaxial Wires
1) Photolithography of mask
2) Anisotropic etching of groove
Ref: Dr. Marius Grundmann Int. J. of Nonlin. Opt. Phys. and Mat. 4, 99 (1995)
3) Growth of QWR material
4) Capping of wire

29. QDs: alternatives to etching
Selective Epitaxy
Precipitate in Glass Matrix
Ref: Dr. Marius Grundmann (
Aerosol Synthesis
Colloid Nanocrystals

30. Quantum Dots and the Wetting layer
QDs WL
Introduction_to_Semiconductor_Nanostructures
UHV-STM cross sections
PM Koenraad, TU Eindhoven

31. Quantum Confinement in 3D (or 0D of freedom)
For 3D infinite potential boxes
Complications: as for QWs, plus
Confinement is not rectangular, smooth profile (x,y,z not separable
Spherical confinement, harmonic oscillator potential
Only a single electron
Capacitance of small sphere is very small
C=4pi epsilon0 r (indep of rel epsilon) Q=CV

32. Consequences of QD discrete DoS
emission
absorption
linewidth indep of T
Introduction_to_Semiconductor_Nanostructures
CdS QDs
versus size at 4.2K

33. Colloidal (chemically synthesised) QDs
CdSe dots
Introduction_to_Semiconductor_Nanostructures

34. Homework What is capacitance of CdS sphere of radius 1nm?
What is the energy of the first excited state?
What is the charging energy for a single electron?
[Dielectric constant = 8.9, refractive index =2.5, Energy gap =2.42 eV, effective mass 0.21 me]
QWs and QDs.pdf