Ultrafast processes in Solids

1. Ultrafast processes in Solids Two general categories of processes: • 1) Electronic processes • How (fast) do “hot” electrons relax? • How (fast) do electrons and holes recombine? • How quickly is coherence lost? • 2) Structural deformations/changes • Phonons • Melting • Electron-phonon interactions

2. Solids Loosely classify solids into 5 categories: Most interesting • Crystalline insulators/semiconductors • Metals • Molecular materials • Doped glasses and insulators • Glasses (amorphous) Ultrafast processes Least interesting • For electronic processes, concentrate on semiconductors and metals • Interact reasonably strongly with light • Relevant time scales (fs-ps) • For structural changes, crystalline materials are most interesting • Phonons only occur in crystalline material • Diffraction (used to study melting) only occurs in crystalline material

3. Bands free atom solid Energy Interatomic separation The discrete atomic states in isolated atoms transform into bands in a crystalline solid as the interatomic distance decreases Optical transitions typically correspond to an electron moving from one band to another (up for absorption, down for emission). When are such transitions allowed? p anti-bonding conduction band p s anti-bonding Eg p bonding valence band s s bonding Crystal Molecule Atom

4. Band structure Taking the full crystal structure into account yields a complicated band structure • E(k) • depends on magnitude and direction of k • examples for two semiconductors shown at right • GaAs – direct gap = interacts strongly with light • Si – indirect gap = only absorbs light • Semiconductor/insulator: valence band full, conduction band empty (Fermi level in the gap) • Metal: band half full (Fermi level in a band) • For direct gap materials, often only bands near gap need to be considered • and approximated by parabolas • i.e., a free particle with effective mass • note that effective mass of electrons in valence band is negative (faster it moves, the slower it goes)

5. Optical transitions Optical excitation adds the energy & momentum of a photon to that of an electron • Must be a state (in another band) at resulting energy and momentum • Photon momentum usually negligible • Vertical approximation • equivalent to dipole approximation • Ignoring the Coulomb interaction: • Absorption spectrum due to density of states • dipole moment approximately independent of k • Yields square root dependence on E • (in 3 dimensions) • Emission: opposite, subtract photon energy and momentum • Thus the poor emission of indirect gap materials

6. Hot electrons A short pulse tuned well above gap will create a non-thermal distribution of electrons • Relaxation occurs in following steps: • Electrons relax to bottom of the conduction band, typically be emitting phonons (particular optical phonons) • Electrons then thermalize amongst themselves • Thermal distribution, but with a temperature much larger than that of the crystal lattice • Electrons thermalize with lattice • Emission of acoustic phonons • Fermi-Dirac distribution • Electrons recombine with holes • Holes underwent similar relaxation to top of valence band Lin, Schoenlein, Fujimoto and Ippen, JQE 24, 267 (1988)

7. Excitons • What about the Coulomb interaction between the electron and hole? • Results in a bound state known as an “exciton” • Relative electron-hole coordinate is described by hydrogenicwavefunction • Binding energy • ~ 4 meV (~ 44 K) in bulk GaAs • ~ 10 meV in a GaAs quantum well • ~ 25 meV in wide-gap materials (GaN) • ~ 104 reduction from hydrogen due to masses and dielectric constant • Modifies the low temperature absorption spectrum • Why do we care about excitons? • Large oscillator strength • ~ probability electron and hole are on same lattice site • Appreciable coherence time Optical Absorption with Coulomb Exciton and realistic broadening

8. Light Single particle  Pair • Can we draw an exciton “band” on the band diagram? • No – it is a single particle picture, and the exciton is a pair • But we can on a pair diagram

9. 1.4 1.4 HH exciton 1.2 1.2 1.0 1.0 0.8 0.8 Absorbance 0.6 0.6 LH exciton 0.4 0.4 0.2 0.2 0.0 0.0 1.535 1.540 1.545 1.550 Photon Energy (eV) Quantum Wells • Bulk GaAs: • Two valence bands • Heavy Hole • Light Hole • Degenerate at k = 0 • Quantum confinement breaks degeneracy due to mass difference • Excitons form between holes in each valence band • Dominant features in optics near band edge

10. Exciton formation • Free carriers can bind to form an excitons • (T.C. Damen et al, Phys. Rev. B 42, 7434 (1990))

11. Exciton formation II However, it later became a topic of interest and controversy Does a bound exciton actually have to form for luminescence to occur at exciton energy? For a different interpretation see Chatterjee, et al., Phys. Rev. Lett. 92, 067402 (2004)

12. Exciton Ionization • It is also possible to watch excitons ionize • Use pump pulse with bandwidth less than binding energy • At room temperature, excitons will be thermally ionized • Use broadband probe to monitor spectrum

13. Exciton Dephasing I

14. Exciton Dephasing II

15. Exciton Dephasing: Negative Delay signal Explanations ultimately included: 1) Local Fields 2) Excitation induced dephasing 3) Biexcitons 4) Excitation induced shift Distinguishing these was problematic Solution: 2D spectroscopy (!?!)

16. Excitonic Coherence • Excitons also have fairly long coherence time (10’s of ps) • For free carriers it is a few fs • Monitor using photon-echo/transient-four-wave-mixing • Decoherence due to • carrier scattering • phonon scattering • disorder effects • However, the coherent response is exquisitely sensitive to many-body interactions

17. Exciton Wavepackets It is also possible to form wavepackets of excitons and magnetoexcitons

18. 2D spectrum of exciton resonances Co-circular polarized excitation pulses • Presence of cross-peaks • A cross peak is the strongest feature • “Dispsersive” line shapes • Lineshapes reveal microscopic interaction mechanisms [X. Li, T. Zhang, C.N. Borca, and S.T.C., Phys. Rev. Lett. 96, 057406 (2006)] • Vertical stripes from continuum [C.N. Borca, T. Zhang, X. Li, and S.T.C., Chem. Phys. Lett. 416, 311 (2005)] Surprises:

19. Experiment – Theory Comparison Only the full theory reproduces the experiment T. Zhang, I. Kuznetsova, T. Meier, X. Li, R.P. Mirin, P. Thomas and S.T.C., Proc. Nat. Acad. Sci. 104, 14227-14232 (2007)

20. Metals • Electron dynamics in metals can be studied • difficult because band structure generally doesn’t allow vertical transitions • overcome using pump-probe photoelectron spectroscopy

21. Graphene • Graphene is a layer of carbon atoms in hexagonal structure • Graphite is many layers of graphene • Unusual band structure: linear  electrons have zero effective mass • Cooling of hot electons can be studied with ultrafast differential transmission

22. ARPES and High-Tc superconductors • Angle resolved photo-emission spectroscopy (ARPES) • Usually done at synchrotron • Can be done with harmonic of ultrafast laser • Basis for time-resolved ARPES J.D. Koralek, et al., Rev. Sci. Instr. 78, 053905 (2007)

23. Time-resolved ARPES

24. Phonons We discussed the observation of phonons in solids using impulsive stimulated Raman scattering

25. Melting • It is also possible to observe melting induced by a laser pulse • Change in reflectivity • Initial change in reflectivity due to e-h plasma • Transfer of energy to crystal causes non-thermal melting • Interpretation complex due to thin film effects • Diffraction • x-ray • electron

26. Melting observed with x-ray diffraction Crystal germanium Initial non-thermal melting Acoustic effects: shifting of diffraction peaks

27. Electron Diffraction Generate electron pulses as for molecule studies Polycrystalline aluminum Sufficient resolution to see pair correlations