Fig. 6.1 Semiconductor quantum wells

GaAs AlGaAs GaAs quantum wells crystal growth GaAs AlGaAs substrate direction

z substrate d b d C.B. e- C.B.

E AlGaAs GaAs EGaAs AlGaAs g Eg g Eg V.B. V.B. h+ Single quantum well MQW or superlattice growth • Molecular beam epitaxy (MBE) methods • Metal-organic chemical vapour deposition (MOCVD) Infinite quantum well Fig. 6.2

) 10 2 n = 3 d * • kn =nπ/d m

/8 2 2 2 2 2 5 • En = (= π /2m*d ) n

h n = 2 ( 1/2 E • = (2/d) sin (k z +n /2) n = 1 ψn n π 0 z −d/2 0 d/2

• symmetry about z = 0 ⇒ wave functions have definite parity

• ψn has (n–1) nodes

• En depends on m*, hence heavy and light holes split Figs 6.3–4 Finite quantum well

E y = 0.85 (13.2−x2)½ / x V 0 E 2 n = 2 8 y = tan(x) E1 n = 1 0 z 4 – d 0 d y 2 2 0 x • Wave functions tunnel into the 0 24 barrier Example : GaAs/AlGaAs • wave function still identified by V = 0.3 eV, d = 10 nm parity and number of nodes 0 mw*= 0.067me, mb*= 0.092me • Confinement energy reduced E = 31.5 meV compared to infinite well 1 c.f. infinite well: E1 = 57 meV • graphical solution to find En Figs 6.5–6 Optical transitions quantum well conduction band

=ω n = 1 n = 2

z valence band

• Light polarized in x,y plane for normal incidence • Parity selection rule: ∆n = even number • Infinite well selection rule: ∆n = 0 Figs 6.7–8 2-D absorption

conduction band E 2-D 3-D n = 3 Eg =ω n = 2 n = 1 0 Absorption k 0510 0 xy (=ω−E ) in units of (h 2/8d 2µ) valence band g

• Absorption ∝ density of states 2 • Density of states constant in 2-D: g2D(E) = m / π=

• Thresholds whenever =ω exceeds (Eg + Een+ Ehn)

• Band edge shifts to (Eg + Ee1+ Ehh1) Figs 6.9–10 GaAs quantum wells

GaAs/AlAs MQW, d = 7.6 nm GaAs/AlGaAs MQW d = 10 nm lh n = 1 n = 2 hh 10 bulk

1.0 ) 4 hh hh hh lh n = 3 −1 n = 2 lh m lh 0.5 5 5 2 (10

T = 6 K α n =1

Absorption (au) 300 K 0.0 0 0 1.6 1.8 2.0 2.2 1.4 1.5 1.6 Photon energy (eV) Photon Energy (eV) • Excitonic effects enhanced in quantum wells: strong at room temp 2D 3D • Pure 2-D: RX = 4 × RX

• Typical GaAs quantum well: RX ~ 10 meV ~ 2.5 × RX (bulk GaAs) • Splitting of heavy and light hole transitions The quantum confined Stark effect

GaAs MQW, d = 9.0 nm, 300 K =ω pi n MQW lh1→e1 hh1→e1 hh2→e2 εz (a) 0 V V0 1.5×106 V/m photocurrent

• Red shift of excitons • Excitons stable to high (b) −10 V hh1→e2 7 fields (c.f. Fig 4.5) 1.1×10 V/m Photocurrent (arb. units) • Parity selection rule hh2→e1 broken 1.4 1.5 1.6 1.7 • used to make modulators Energy (eV) Figs 6.11–12 Figure 6.13 Emission spectrum

Zn0.8Cd0.2Se/ZnSe quantum well E = 2.55eV (10K) d = 2.5 nm g E = 2.45eV (300K) 10 K g 300 K

• Emission energy shifted from

PL intensity Eg to (Eg + Ee1 + Ehh1) 0 • Tune λ by changing d 2.4 2.5 2.6 2.7 • Brighter than bulk due to Energy (eV) improved electron-hole overlap • Used in laser diodes and LEDs Figure 6.14 Intersubband transitions

n-type quantum well • Need z polarized light hν n = 2 • Parity selection rule: electrons ∆n = odd number n = 1

• Transition energy ~ 0.1 eV (~ 10 µm, infrared) • Absorption used for infrared detectors • Emission used for infrared lasers (Quantum cascade lasers) Quantum dots Figs 6.15–16 CdS quantum dots Discrete atomic-like density of states versus dot size d at 4.2K 0-D 0-D 0-D 2 1.2 nm 1.5 nm 3-D 1 2.3 nm 2-D 33 nm Density of states

0 510 Optical density (E−E ) in units of (h 2/8d 2m*) 0 g 2.5 3.0 3.5 4.0 Energy (eV) Examples: 1. Semiconductor doped glass (Colour glass filters & stained glass) 2. Self-organized III-V quantum dots (eg InAs/GaAs)