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||

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

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….

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

Material Parameters for AlxGa1-xAs Lattice constant a = 5.6533+0.0067x (Angstroms) direct if x<0.45 Bandgap EG = 1.424+1.247x (eV) Valence band discontinuity: ΔEv = - 0.46x (eV) Conduction band discontinuity: ΔEc = 0.79x (eV) Effective electron mass me*/m0 = 0.0637+0.083x Effective hole masses mhh*/m0 = 0.50+0.29x mlh*/m0 = 0.087+0.063x Direct Indirect Semiconductor basics and bonding.pdf 0.45 1 x See http://www.ioffe.rssi.ru/SVA/NSM/Semicond/ for other parameters and other semiconductors

Semiconductor quantum wells QWs and QDs.pdf

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

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/2 0 d/2

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

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

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

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

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)

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

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-

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)

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)

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

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

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

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

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

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)

Quantum wires: cleaved edge overgrowth Introduction_to_Semiconductor_Nanostructures

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

QDs: alternatives to etching Selective Epitaxy Precipitate in Glass Matrix Ref: Dr. Marius Grundmann (http://sol.physik.tu-berlin.de) Aerosol Synthesis Colloid Nanocrystals

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

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

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

Colloidal (chemically synthesised) QDs CdSe dots Introduction_to_Semiconductor_Nanostructures

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