Simons kick-off Semiconductors PIs: Jim Speck Claude Weisbuch Svitlana Mayboroda

1. Simons kick-off Semiconductors PIs: Jim Speck Claude Weisbuch Svitlana Mayboroda Marcel Filoche

2. background papers on impact of InGaN alloy disorder New theory to calculate effects of disorder Landscape Localization theory of disordered potentials M. Filoche and S. Mayboroda. PNAS, vol. 109, no. 37 (2012) D. N. Arnold, G. David, D. Jerison, M. Filoche and S. Mayboroda. Phys. Rev. Lett., 116, 056602 (2016) PHYSICAL REVIEW B 95, 144204 (2017) PHYSICAL REVIEW B 95, 144205 (2017) PHYSICAL REVIEW B 95, 144206 (2017) Evidence of nanoscale Anderson localization induced by intrinsic compositional disorder in InGaN/GaN quantum wells by scanning tunneling luminescence spectroscopy W. Hahn, J.-M. Lentali, P. Polovodov, N. Young, S. Nakamura, J. S. Speck, C. Weisbuch, M. Filoche, Y.-R. Wu, M. Piccardo, F. Maroun, L. Martinelli, Y. Lassailly, and J. Peretti PHYSICAL REVIEW B 98, 045305 (2018)

3. What is a light emititng diode (LED) ? A Light Emitting Diode (LED) produces light of a single color by combining holes and electrons in a semiconductor. Source of Holes (p-type Layer) External Source of Electrons (Battery) Light Out Combining of Holes and Electrons (Active / Emitting Layer) Source of Electrons (n-type Layer) Substrate (Foundation)

5. Semiconductors: electrons, holes, band structures Semiconductor are materials where electrons fill completely the available energy levels in the valence band. The next energy band, the conduction band, is empty of electrons. Under normal conditions a semiconductor does not conduct electricity, or it does it “poorly”. At finite temperatures some electrons are excited from the valence band to the conduction band, leaving behind them a hole. Both the electrons in the conduction band and the holes in the valence band can be accelerated and conduct electricity. energy energy Conduction band. Conduction band. for Forbidden bandgap No available quantum states for electrons Bandgap energy Eg valence band. valence band. position position

6. Semiconductors and Light: absorption and recombination Energy relaxation Phonon emission Semiconductor can absorb a photon if its energy is greater than gap energy: it creates a free electron in the conduction band and a free hole in the valence band Emitted photon hν ≈ Eg Incident photon hν >Eg A conduction electron can recombine with a hole in the valence band by emitting a photon with energy ≈ bandgap Eg A direct macroscopic measurement of a quantum mechanical phenomenon, the bandgap - Electrons and holes emit light by recombining together - How to obtain electrons and holes in a semiconductor? - Carrier injection in a p – n junction

7. Semiconductor band structures Wavefunctions: Bloch waves energy wavector position (a) The periodicpotential. (b) The Bloch wavefunction. (c) The periodic part of the Bloch wavefunction. (d) The sinusoidalenvelope part of the Bloch wavefunction

8. Principle of operation of LEDs at strong bias energy n-doped semiconductor p-doped semiconductor Strong bias, "flat band potential" V applied ≈ V bi ≈ Eg bandgap Carriers are distributed along a carrier diffusion length thickness Carrier density is too small to have good recombination probability’ proportional to carreir densities qV ≈ EG hν ≈ EG Need to concentrate carriers => Use double heterostructures So far, only one semiconductor, with spatially different dopings  "homostructures" Now, semiconductors with different chemical compositions "Heterostructures" – "double" because sandwich position energy hν ≈ EG2 EG2 EG1 EG1 position

9. Heroes of semiconductor light emitters: the heterostructures Large bandgap material Large bandgap material Small bandgap material The Nobel Prize in Physics 2000 Zhores I. Alferov and Herbert Kroemer"for developing semiconductor heterostructures used in high-speed- and opto-electronics" x z y energy position Allowed room temperaturecw lasers withlowtheshold (mA) [before, 77K and Amperes] Made optical communications possible

10. The next (smaller) step: quantum wells stillbetterLEDs, better lasers energy Infinite well approximation LL Very thin layer Few to tens of nm L x z y position energy π22 π22 Econf,h = Econf,e = 2meL2 2mhL2 position

11. 200 AlN 6 5 MgS 300 ZnS 4 MgSe Forbidden bandgap (eV) GaN wavelength (nm) MgTe SiC 400 3 AlP 500 ZnSe AlAs ZnTe GaP 2 CdSe CdTe GaAs InP 1000 AlSb Si 1 GaSb InN 2000 Ge InSb InAs 0.30 0.55 0.65 0.60 0.35 Crystal lattice constant (nm)

12. What is the materials composition ?

13. Reconstructed Dataset and Mass Spectrum UCSB m-GaN LED (60 million atom dataset) n-GaN InGaN Quantum Wells AlGaN EBL Doped Mg p-GaN GaN Barrier Layers UCSB work on nitride alloys – InGaN, AlGaN Random alloys – statistics follow binomial distribution Random alloys Large naturalcompositional fluctuations

14. GaN-based visible light emitters InGaN quantum wells • Wavelengths from UV to IR • Compressively strained • Increases with indium composition • Piezoelectric field effects • Eventual relaxation Wurtzite (Al,In,Ga)N InxGa1-xN - GaN Energy (eV) Compressive strain InGaN GaN Lattice Constant (Å) InGaN QW on GaN M. R. Krames et al., J. Disp. Technol. 3, 160 (2007).

15. Internal electric fields : piezo and spontaneous fields Lattice matching is a major issue The crystal structure, the spontaneous and the piezo- electric polarizations, the polarization-induced charge density of AlGaN/GaN (AlN/GaN) heterostructure From Supryadkina

16. Electric field effects in nitride heterostructures MQW EBL p-GaN n-GaN

18. Algorithm to generate indium composition map Grid generation for mapping Cubic grid:a=2.833 Å Determine Indium comp. along the well Random process to determine Indium distribution in the well σ=2.0a (~0.6 nm) Gaussian average method Mesh nodes find corresponding grid indium value by linear interpolation

19. Band parameters for calculation Band offsets between GaN/InGaN and GaN/AlGaN conduction bands are 63% of the bandgap difference.

20. New Mathematical Method for Disordered Systems Disorder implies carrier localization M. Filoche and S. Mayboroda, PNAS 109 (37), (2012) D.N. Arnold, G. David, D.Jerison, S. Mayboroda, and M. Filoche, Phys. Rev. Lett. 116, 056602(2016) What does it bring to LEDs simulations? A new tool: the localization landscape

21. Localization Landscape Randompotential V Schrödinger equation Landscape equation u(r) ≤ E-1 Valleys in u controls the localization sub-regions Theorem Localizationlandscape u M. Filoche and S. Mayboroda, PNAS 109 (37), (2012)

22. 3D representation of the original disorderedpotential V • 3D representation of the landscape u solvingEq. 2. • The valleylines of the landscape u (black lines) delimit the variouslocalizationregions. • Effective localizationpotential W ≡ 1/u. The valleylines of u are now the crestlines of W, and the localizationsubregionsoutlined in (c) are the basins of W.

23. Landscapetheory of disorderedsystemsapplied to nitridesemiconductors Calculate e-h potential maps Calculate e-h landscape maps Calculate compositional maps electrons holes Instead of solving Schrödinger equation Solve Landscape equation for the same disordered potential 1/u gives map of localized states in disordered material This is a linear problem, not any more an eigenvalue problem! Solves ≈ 100-1000 times faster M. Filoche and S. Mayboroda, PNAS, vol. 109, no. 37 (2012) D. N. Arnold et al., Phys. Rev. Lett. 116, 056602 (2016)

24. Electric fields calculation 3D continuum strain-stress model solved by the finite element method (FEM) to calculate the strain distribution over the entire LED before solving the Poisson and DD equations Spontaneous polarization Piezo electric field Input to Poisson equation with other free charges

25. Implementing the theory of localization: computing the overlap integral in the absorption coefficient Absorption As u is a good approximation of the exact wavefunction in the localizationregion, and decaysveryfastoutside, the overlapintegralisgiven by the overlap of ue and uhwithin a region We Overlap Wh M. Filoche et al., Phys. Rev. B 95, 144204 (2017). M. Piccardo et al, Phys. Rev. B 95, 144205 (2017).

26. Fitting of Experimental results The normal QW without disorder effect and only considers the thickness fluctuation cannot explain the urbach tail slope well Using the random indium fluctuation and thickness fluctuation give us a better fit to the experimental result. M. Piccardo, C. Weisbuch, Y-R. Wu et. al, SPIE Photonic West 2016. paper is submitted.

27. Two classes of problems Carrier transport: find how carriers travel in an heterostructure when an electric field is applied Carrier recombination: find how carriers recombine by emitting light Solutions to thesetwoproblems are stronglymodified by disorder

28. Carrier transport : drift diffusion vs. hopping transport Electric field No electric field Collisions: phonons, impurities, defects, ..

29. Carrier recombination: delocalized vs localized states our picture so far… But electrons are poorly localised! Alloy disorder electron Defects photon Radiative hole Auger

30. The self-consistent Poisson-Schrödinger approach Schrödinger equation Band edges Quantum states Localizationlandscapes Poisson equation Electrostaticpotential Carrier densities

31. The self-consistent Poisson-Schrödinger approach Effective potential Localizationlandscape Poisson equation Electrostatic potential Electronic density

32. The new self-consistent scheme

33. The self-consistent scheme In Nitrides calculate Solve Poisson equation Solve drift-diffusion equations Solve landscape equations No C.-K. Li et al, Phys. Rev. B 95, 144206 (2017). Converged? Yes

34. We solve these equations in a network of very small nodes as the material is inhomogeneous (quantum wells+ composition fluctuations) Mesh grid size: - in x-y plane: 0.5 nm x 0.5 nm - in z-direction: gradual mesh from 0.12 nm to 20 nm. C.-K. Li et al, Phys. Rev. B 95, 144206 (2017).

35. The self-consistent scheme Parameters used in calculation C.-K. Li et al, Phys. Rev. B 95, 144206 (2017).

36. Percolation paths for current • Carriers find percolation paths through lower potential regions • Such percolation paths with high current densities lead to further reduced turn-on voltage caused by In fluctuations - induced disordered potential, compared to x-y averaged QW landscape energy. Effective quantum potentials 1/ue and corresponding z-component current JZ C.-K. Li et al, Phys. Rev. B 95, 144206 (2017).

37. Comparison of I-V & potential for different methods As the landscape model with random indium fluctuations accounts for quantum confinement and tunneling effects, it better fits the I-V curve by increasing current for a given bias. C.-K. Li et al, Phys. Rev. B 95, 144206 (2017).

38. Comparison table: computation time • ARPACK Eigen solver • PARDISO sparse inverse solver • 2 Intel Xeon E5-5650V2 8 cores 2.6 GHz CPUs with 396 GB memory. • The landscape model coupled to the Poisson-DD equations is much more computationally efficient with respect to state-of-the-art quantum solvers, while still incorporating quantum effects such as tunneling and quantum confinement. [1] D. Watson-Parris et al., PRB83, 11 (2011) [2] D. Watson-Parris, thesis, University of Manchester (2011) [3] S. Schulz et al., PRB 91, 35439 (2015) [4] M. A. D. Maur, PRL 116, 2 (2016) C.-K. Li et al, Phys. Rev. B 95, 144206 (2017).

39. Our experimental approach: Scanning Tunneling Luminescence V STM tip STM electrons QW Contact photons optical fiber Sample W. Hahn et al. , Phys.Rev B 98, 045305 (2018) spectrometer

40. Experimental approach STL V Linescans: Probing different localization regions should be visible as changes in the luminescence spectra STM electrons 1 QW 30 2.8 eV photons 20 nm 2.7 10 spectrometer 2.6 Measuring spots 5 nm apart – 1s integration time 20 30 10 nm

41. Spectral width of local emission spectrometer resolution: 35 meV Linewidth 36 meV x0.25 51 meV • Fluctuations of emission spectra at alloy disorder scale W. Hahn et al. , Phys.Rev B 98, 045305 (2018)

42. First feedbacks to theory How do we get electron localization from LL theory ? LL does not give the wavefunction, indicatesregions of localization if theyexist Several full Schrödinger computations state thatelectronlocalizationonlyoccurs if one takeintoaccountwell-width fluctuations. But what are these? Can wedefine an interface? Could localisation of electronsbe due to the Coulomb interaction between free electrons and ldisorderocalisedholes? Needed: wavefunctions in LL theory ground and excited states Needed: energy levels of ground and excited states Applying LL theory with spatially varying energy origin ? At what concentration (disorder ) do we have localisation? How to treat weak localisation?

43. Petr Polovodov Wiebke Hahn Jacques Peretti Claude Weisbuch Lucio Martinelli Fouad Maroun Abdullah Alhassan Yuh Renn Wu Nathan Young Shuji Nakamura Bastien Bonef James Speck Yves Lassailly Marco Piccardo Svitlana Mayboroda Jean-Marie Lentali Marcel Filoche