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Prezentacja Programu Powerpoint Quasiparticles in the Solid State – standard model of quasi-Universe The Faculty of Physics, University of Wrsaw [email protected] Google: Jacek Szczytko Google: Jacek Szczytko Login: student Hasło: ******* 2016-05-29 2 The polariton laboratory attocube CFM 1.5-320K, 0.0-9.0T 700-1000nm 420nm, 532nm, 633nm… Kasia Lekenta Dr Barbara Piętka Mateusz Król Rafał Mirek The polariton laboratory Appl. Phys. Lett. 107, 201109 (2015) MBE growth: Rafał Rudniewski, Dr Wojciech Pacuski, Jean-Guy Rousset Magneto-optical properties Rafał Mirek Katarzyna Lekenta Mateusz Król Dr Barbara Piętka 2016-05-29 4 Laboratory of SQUID magnetometry Andrzej Twardowski Andrzej Majhofer Anita Gardias Jarosław Rybusiński Maciej Marchwiany (Monte Carlo) 0.0-7.0T, 1.5-800.0K, photomagnetism 2016-05-29 5 Mathematics and the Nature The conversation with the Nature must be carried out in the language of mathematics, otherwise nature does not answer our questions. prof. Michał Heller Dialog z przyrodą musi być prowadzony w języku matematyki, w przeciwnym razie przyroda nie odpowiada na nasze pytania. prof. Michał Heller Particles 2016-05-29 7 Elements 2016-05-29 8 Elementary Particles LHC CERN 2016-05-29 9 Elementary Particles 2016-05-29 10 Particles - quarks 2016-05-29 11 Elementary Particles 2016-05-29 12 Elementary Particles 2016-05-29 13 Kinetic Energy 푚 ≠ 0 푚푣2 푝Ԧ2 퐸 푝Ԧ = = 퐸 푝Ԧ = 푚2푐4 + 푝2푐2 2 2푚 2016-05-29 14 Kinetic Energy 퐸 푝Ԧ , 푝Ԧ = ℏ푘 푚 ≠ 0 푚 = 0 푚푣2 푝Ԧ2 퐸 푝Ԧ = = 퐸 푝Ԧ = 푚2푐4 + 푝2푐2 퐸 푝Ԧ = 푐 푝Ԧ 2 2푚 퐸 푘 푣 푐 푘 −푐 2016-05-29 15 Many-body interaction 2016-05-29 16 Many-body interaction atoms.. chemical bonds… symmetry, structure… defects, impurities… junctions, processing, 2D, 1D, 0D… external fields 퐸, 퐵, light ℎ휈, stress, heat… 2016-05-29 17 The electronic band structure W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics 2016-05-29 18 Basis of solid state Born-Oppenheimer approximation Max Born Jacob R. Oppenheimer (1882-1970) (1904-1967) S. Harris S. 2016-05-29 19 Periodic potential Bloch’s theorem When potential is periodic 푉 푟Ԧ = 푉 푟Ԧ + 푅 then solutions of Schrödinger equation Bravais vectors 푝2 + 푉 푟Ԧ 휑 푟Ԧ = 퐸 휑 푟Ԧ 2푚 푛,푘 푛,푘 푛,푘 are in the form of: 푖푘푟Ԧ 휑푛,푘 푟Ԧ = 푒 푢푛,푘 푟Ԧ Plane wave Envelope where Bloch function: 푢푛,푘 푟Ԧ = 푢푛,푘 푟Ԧ + 푅 = 푢푛,푘+퐺Ԧ 푟Ԧ 5/29/2016 20 Effective mass approximation 푝Ԧ2 퐸 푝Ԧ = 2푚 Crystal = periodic potential ax 2016-05-29 21 The electronic band structure 푖푘푟Ԧ 휑푛,푘 푟Ԧ = 푒 푢푛,푘 푟Ԧ Plane wave Envelope 2016-05-29 22 The electronic band structure bands ℏ2푘2 Expanding 퐸 푘 = 퐸 − near extremum, e.g. 푘 = 0: 푛 푛 2푚 Landolt-Boernstein 2016-05-29 23 Effective mass approximation 3 3 2 1 ℏ 푘푖푘푗 퐸 푘 = 퐸 0 + ෍ ෍ + ⋯ 푛 푛 푚∗ 2 푖=1 푗=1 푖푗 We replace MANY BODY INTERACTION by the effective mass tensor: 휕 3 휕 3 ׬ 푢푛,0 푢푙,0 푑 푟 ⋅ ׬ 푢푛,0 푢푙,0 푑 푟 2 1 훿푖푗 2ℏ 휕푥푖 휕푥푗 ∗ = + 2 ෍ 푚 푚 푚 퐸푛 0 − 퐸푙 0 푖푗 푙≠푛 2 2 2 2 ℏ 푘1 푘2 푘3 퐸푛 푘 ≈ 퐸푛 0 + ∗ + ∗ + ∗ 2 푚1 푚2 푚3 Haris ∗ 푚 = 0.01 − 1000 푚0 S. 2016-05-29 24 Effective mass approximation 푚∗ > 0 푚∗ = 0 퐸 퐸 ℏ2푘2 퐸 푘 = ℏ푐ǁ 푘 퐸 푘 = 2푚∗ 푘 푘 푚푣2 푝Ԧ2 ℏ2푘2 퐸 푝Ԧ = = = 2 2푚 2푚∗ 2016-05-29 25 Effective mass approximation 푚∗ < 0 푚∗ = 0 (i 푚∗ < 0) 퐸 퐸 푘 푘 ℏ2푘2 퐸 푘 = 2푚∗ 퐸 푘 = ℏ푐ǁ 푘 2016-05-29 26 k·p perturbation theory – effective mass The energy En(k) around extremum for the uniaxial crystal (np. GaN): 2 2 2 2 ℏ 푘1 + 푘2 푘3 퐸푛 푘 = 퐸푛 0 + ∗ + ∗ 2 푚⊥ 푚∥ For a cubic crystal: ℏ2푘2 퐸 푘 = 퐸 0 ± 푛 푛 2푚∗ 2016-05-29 27 k·p perturbation theory – effective mass Na, K, Co, Al – elektrony Zn, Cu, Au - ??? Pasmo prawie całkowicie zapełnione elektronami. 2016-05-29 28 Effective mass approximation Many body system: cb cb cb electron in cb hole in vb vb vb vb Ground state Excited state Excited state 2016-05-29 29 Effective mass approximation Many body system: cb cb cb electron in cb hole in vb vb vb vb Ground state Excited state Excited state We „created” quasi-particles, which are non-interacting (at least „not too strong”) („free electrons”, „effective mass”) – the same for phonons, polarons, plasmons, excitons, trions, bi- excitons 2016-05-29 30 Quasi-particles creator (you!) 2016-05-29 31 Quasi-particles (standard model) cb ℏ2푘2 Fermions Bosons 퐸 = ∗ 2푚∗ 푚 ≠ 0 Photon 퐸 = ℎ휈 electron in cb hole in vb vb 퐸 = 푐ǁ푘 푚∗ = 0 Phonon 퐸 = ℏ휔 Magnon 퐸 = ℏ휔 2016-05-29 32 Elementary Particles 3D 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 -1 1 1 0 훾 ½ 풆 1/2 풍풉 3/2 풉풉 1 electron light hole heavy hole photon 0 0 ℏ휔 E 1 cb phonon 0 0 1 ℏΩ Eg k magnon hh lh 2016-05-29 33 Elementary Particles 3D 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 -1 1 1 0 훾 ½ 풆 1/2 풍풉 3/2 풉풉 1 electron light hole heavy hole photon 0 0 1 ℏ휔 phonon 0 0 1 ℏΩ magnon + dimension 2016-05-29 34 Composed particles FIRST: ∗ Coulomb potential in 3D in the semiconductor of dielectric constant 휀푟, effective mass 푚 : 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 -1 1 1 0 훾 ½ 풆 1/2 풍풉 3/2 풉풉 1 electron light hole heavy hole photon 0 0 훾 1 photon 0.0-1000풎ퟎ 0.1-1000풎ퟎ -1 1 ½ 풆 3/2 풉풉 electron heavy hole 2016-05-29 35 Composed particles FIRST: ∗ Coulomb potential in 3D in the semiconductor of dielectric constant 휀푟, effective mass 푚 : 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 -1 1 1 0 훾 ½ 풆 1/2 풍풉 3/2 풉풉 1 electron light hole heavy hole photon 0 0 훾 1 photon 0.0-1000풎ퟎ 0.1-1000풎ퟎ -1 EXCITON 1 ½ 풆 3/2 풉풉 electron heavy hole 2016-05-29 36 Composed particles FIRST: ∗ Coulomb potential in 3D in the semiconductor of dielectric constant 휀푟, effective mass 푚 : 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 -1 1 1 0 훾 ½ 풆 1/2 풍풉 3/2 풉풉 1 EXCITON electron light hole heavy hole photon 2016-05-29 37 Composed particles FIRST: ∗ Coulomb potential in 3D in the semiconductor of dielectric constant 휀푟, effective mass 푚 : 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 -1 1 1 0 훾 ½ 풆 1/2 풍풉 3/2 풉풉 1 EXCITON electron light hole heavy hole photon EXCITON0.0-1000풎ퟎ -1 ½ 풆 electron 2016-05-29 38 Composed particles FIRST: ∗ Coulomb potential in 3D in the semiconductor of dielectric constant 휀푟, effective mass 푚 : 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 -1 1 1 0 훾 ½ 풆 1/2 풍풉 3/2 풉풉 1 EXCITON electron light hole heavy hole photon EXCITON 0.1-1000풎ퟎ 1 3/2 풉풉 heavy hole EXCITON0.0-1000풎ퟎ -1 ½ 풆 electron 2016-05-29 39 Composed particles FIRST: ∗ Coulomb potential in 3D in the semiconductor of dielectric constant 휀푟, effective mass 푚 : 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 -1 1 1 0 훾 ½ 풆 1/2 풍풉 3/2 풉풉 1 EXCITON electron light hole heavy hole photon EXCITON 0.1-1000풎ퟎ 1 EXCITON 3/2 풉풉 heavy hole EXCITON EXCITON0.0-1000풎ퟎ -1 ½ 풆 electron 2016-05-29 40 Composed particles FIRST: ∗ Coulomb potential in 3D in the semiconductor of dielectric constant 휀푟, effective mass 푚 : 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 -1 1 1 0 훾 ½ 풆 1/2 풍풉 3/2 풉풉 1 EXCITON electron light hole heavy hole photon Charged EXCITON EXCITON0.1-1000풎 푿+ ퟎ 1 풉풉 EXCITON 3/2 Bi EXCITON heavy hole EXCITON Charged Charged EXCITON0.0-1000풎ퟎ EXCITON 푿− -1 ½ 풆 electron etc… 2016-05-29 41 Composed particles FIRST: ∗ Coulomb potential in 3D in the semiconductor of dielectric constant 휀푟, effective mass 푚 : Charged 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 EXCITON -1 1 1 0 훾 EXCITON ½ 풆 1/2 풍풉 3/2 풉풉 1 푿+ electron light hole heavy hole photon Charged Charged EXCITON 0.1-1000풎ퟎ 푿− EXCITON 1 푿+ 3/2 풉풉 heavy hole Bi EXCITON Charged Bi EXCITON − EXCITON 푿 Bi 0.0-1000풎ퟎ -1 EXCITON ½ 풆 electron EXCITON etc… 2016-05-29 42 Composed particles FIRST: ∗ Coulomb potential in 3D in the semiconductor of dielectric constant 휀푟, effective mass 푚 : Charged 0.0-1000풎ퟎ 0.0-1풎ퟎ 0.1-1000풎ퟎ 0 EXCITON -1 1 1 0 훾 EXCITON ½ 풆 1/2 풍풉 3/2 풉풉 1 푿+ electron light hole heavy hole photon Charged Charged EXCITON 0.1-1000풎ퟎ 푿− EXCITON 1 푿+ 3/2 풉풉 heavy hole Bi EXCITON Charged Bi EXCITON − EXCITON 푿 Bi 0.0-1000풎ퟎ -1 EXCITON ½ 풆 electron EXCITON etc… 2016-05-29 43 Composed particles Quantum well J. Szczytko et al. 2016-05-29 44 Composed particles CB p quantum dot s s PL Intensity p VB 2016-05-29 45 Potencjał harmoniczny 2D CB p s s Zależność od mocy pobudzania widm PL Intensity fotoluminescencji otrzymanych w temperaturze p bliskiej temperatury ciekłego helu (ok.
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