interband transitions in semiconductors M. Fox, Optical Properties of Solids, Oxford Master Series in Condensed Matter Physics

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1 interband transitions in semiconductors M. Fox, Optical Properties of Solids, Oxford Master Series in Condensed Matter Physics

2 interband transitions in quantum wells Atomic wavefunction of carriers in the conduction and valence band have parity differing by 1, hence only transitions with n = 0 are dipole-allowed for a rectangular potential with infinite walls only transitions with n = 0 are possible. This selection rule is weakened for real quantum wells with finite barrier heights but still the transitions with n = 0 dominate the spectra J. H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press (1998)

3 quantum well photoluminescence

4 exciton binding energy and Bohr radius M. Fox, Optical Properties of Solids, Oxford Master Series in Condensed Matter Physics

5 2D: quantum well excitons J. H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press (1998)

6 exciton correction to the absorption continuum: Sommerfeld factor J. H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press (1998)

7 laser applications of semiconductor heterostructures Z. I. Alferov, Nobel Lecture (2000)

8 quantum well applications: quantum cascade laser (QCL) Unlike typical interband semiconductor lasers that emit electromagnetic radiation through the recombination of electron hole pairs across the material band gap, QCLs are unipolar and laser emission is achieved through the use of intersubband transitions in a repeated stack of semiconductor multiple quantum well heterostructures QCL emission wavelength µm (~ mev) Quantum Cascade Laser invented by Bell Labs physicists; Cover illustration for Science, April 22,1994. ω LO = Γ21 >> Γ 32

9 interband transitions J. H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press (1998)

10 QCL: principle and experimental realization E 32 = 291 mev = 2347 cm -1 ~ 4.26 µm

11 interband transitions: double heterojunction laser

12 laser applications of semiconductor heterostructures: quantum well LED and laser Z. I. Alferov, Nobel Lecture (2000)

13 quantum well LED and laser Z. I. Alferov, Nobel Lecture (2000) J. H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press (1998)

14 quantum well LED and laser Z. I. Alferov, Nobel Lecture (2000) J. H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press (1998)

15 quantum well laser Z. I. Alferov, Nobel Lecture (2000)

16 quantum well laser Z. I. Alferov, Nobel Lecture (2000)

17 quantum well laser Z. I. Alferov, Nobel Lecture (2000)

18 Confinement in heterostructures system dimension: z 3D 2D 1D d y d z 0D d x x L z L y L x y L x,l y,l z >>λ F d z λ F d y,d z λ F d x,d y,d z λ F density of states: D(E) 3D ~ E D(E) 2D const. D(E) 1D 1 ~ E D(E) 0D ~δ(e-e ijk ) E E 1 E i E E 11 E ij E E 111 E ijk E

19 Quantum dots 0D d x monolayer fluctuations QDs Stranski-Krastanow QDs d x,d y,d z λ F 1500 D(E) 0D E 111 ~δ(e-e ijk ) E ijk E PL (counts in 150 s) Energy (ev)

20 Self-assembly of quantum dots InAs GaAs < 1.5 ML Film molecular beam epitaxy film growth (InAs on GaAs) mismatch between lattice parameter stressed film ~ 1.5 ML Quantum Dots Stranski-Krastanov growth of InAs dots is a result from equilibrium between mechanical stress and surface energy ~ 2 ML dots: "rings": height ~ 2-6 nm diameter ~ nm 10% size variation nm nm dislocations > 2.5 ML Dislocations Further growth relaxes excess energy through creation of dislocations

21 0D excitons: quantum dots InAs quantum dots ~ 6 nm high ~ 20nm diameter, 10% size variation nm Quantum rings (Partially Covered InAs Islands) ~ 1 to 2 nm high ~ 50nm diameter, 30% size variation nm Vertical coherent growth: double layer of dots

22 Electron and hole confinement in quantum dots 4~ 150 mev15mevenergy InAs GaAs Capping with GaAs: electronic barrier material Energy GaAs Ec z Ev x, y Quasi-parabolic confinement Confinement energies: electron ~ 50 mev, holes ~ 25 mev d p s d p s ~ 300 mev CB s p d s d p VB r (nm)

23 Localized states in a self-assembled quantum dot axial confinement: rectangular quantum well lateral confinement: parabolic quantum well GaAs E CB GaAs InAs GaAs GaAs InAs GaAs InAs E CB z E 1 xy E n e n InAs CB z E 1 E = E + + E xy n = ( n + 1) hω 0 1st energy level of a quantum well energy spectrum of a 2D harmonic oscillator with degeneracy m=2(n+1) (2 because of spin) n=2 n=1 n=0 s m= d p

24 Localized states in a self-assembled quantum dot solutions of a 2D harmonic problem: (e.g. Cohen-Tannoudji, Quantum mechanics) 1.0 χ χ n=0 m= n=1 m=+/ n=2 m= n=2 m=+/

25 Shell structure of quantum dots in spectroscopy shell structure of artificial atoms: Absorption measured on ~ 10 7 quantum dots Absorption 3e-4 2e-4 1e-4 0 Rings (4.2K) d p s Absoption 3e-4 2e-4 1e-4 Dots (4.2K) s-s p-p d-d s p d R.J. Warburton et al. PRL 79, 5282 (1997) Energy (ev) Emission measured on ~ 10 7 quantum rings PL intensity 2 1 s-s p-p d-d Energy (ev) d p s s p d inhomogeneous broadening ~ 30 mev

26 Energy scales Quantization energies Inhomogeneous broadening hω hω e h < < 50meV 25meV 30meV Ensemble spectroscopy Exciton binding energy E e, h s, s 20meV

27 Excitons in bulk semiconductors M. Fox, Optical Properties of Solids, Oxford Master Series in Condensed Matter Physics compare with E eh ss = 22meV in quantum dots quantum confinement enhances Coulomb correlations (e.g. exciton binding energy)

28 Ensemble and single dot photoluminescence ~ dots PL (counts in 30 s) confocal 5 10 dots PL (counts in 150 s) nm aperture Al Energy (ev)

29 Quantum dot biexciton cascade: source of entangled photons

30 Electrical source of entangled photons Toshiba Research Europe Ltd., Cambridge Research Laboratory Salter et al, Nature 465, (03 June 2010)

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