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
Most interesting
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
The discrete atomic states in isolated atoms transform into bands in a crystalline solid as the interatomic distance decreases
Interatomic separation
Energy
solid
free
atom
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
s anti-bonding p Eg
valence band
p bonding 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. and realistic broadening
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 hydrogenic wavefunction
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. Single particle  Pair Light
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
Light

9. 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
Absorbance
Photon Energy (eV)
HH exciton
LH exciton
1.535
1.540
1.545
1.550
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4

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, (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
Surprises:
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, (2006)]
Vertical stripes from continuum
[C.N. Borca, T. Zhang, X. Li, and S.T.C., Chem. Phys. Lett. 416, 311 (2005)]

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, (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
FIG. 1 (color). Sample structure and energy dispersion curves of doped and undoped graphene layers. The sample has a buffer layer (green) on the SiC substrate followed by 1 heavily doped layer (red) and approximately 20 undoped layers (blue) on top. The Fermi level is labeled with a dashed line (brown) lying 348 meV (from the later data) above the Dirac point of the doped graphene layer and passing through the Dirac point of the undoped graphene layers. The blue solid line shows the transitions induced by the 800-nm optical pump pulse; the three dashed lines correspond to probe transitions at different energies with respect to the Fermi level
FIG. 2 (color). DT spectrum normalized by transmission (T). (a) DT spectrum on epitaxial graphene at 10 K, with 500-W 800-nm pump (photon fluence of 1: photons=cm2 per pulse), at probe delays of 10, 5, 2, 1, 0.5 ps, and background (50 ps before the pump arrives). The arrows at 1.78 and 2:35 m indicate the DT zero crossings. (b) DT time scan of the two probe wavelengths marked in (a) at the red (1:85 m) and blue side (1:75 m) of the 1:78 m DT zero crossing. (c) Time scan of the two probe wavelengths marked in (a) at the red (2:40 m) and blue side (2:25 m) of the 2:35 m DT zero crossing. In all figures, the dashed line (brown) marks where the DT signal is zero. The DT tails in (b) and (c) are simply fitted by a sigmoidal curve.

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, (2007)

23. Time-resolved ARPES
Fig. 1. Typical ARPES dispersions before and after pumping for nodal (f = 45°) and gapped (f = 31°) regions of k-space. The incident pump fluence was 5 mJ/cm2. (A) Equilibrium (t = −1 ps) and (B) transient (t =
0.6 ps) energy-momentum maps for the nodal state. Data are shown with identical color scales. The inset shows the location of the cut. The arrow marks the position of the dispersion kink. (C) Subtraction between (A) and (B). Blue indicates intensity gain and red intensity loss. (F to J) Same as (A to E) but for a gapped (off-nodal)momentum cut. Spectra have been corrected for detector nonlinearity. The diagonal line in the lower right portion of (F) and (G) is the edge of the detector.
Fig. 2. Evolution of the superconducting gap after pump excitation. Symmetrized EDCs at kF for f =32°at low(A) and higher (B) fluence. The gap is obtained by fitting to a phenomenological model (12), but can be approximated by halving the distance between positive and negative peaks. Bold curves correspond to t = 0. (C andD) Analogous EDCs for a cut at f =27

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
The x-ray pulses are generated by focusing the output of titanium sapphire laser (repetition rate 10 Hz) on a thin moving wire of titanium metal to produce a microplasma. The energy, duration and wavelength of the laser pulses are 100 mJ, 120 fs, and 800 nm, respectively.

27. Electron Diffraction Generate electron pulses as for molecule studies
Polycrystalline aluminum
Sufficient resolution to see pair correlations
Fig. 2. Snapshots of the time-dependent pair correlation function. (A) Comparison of the structure factor K(s) at pump-probe delays before and after the phase transition. These curves were obtained from diffraction patterns containing the cumulative diffracted intensity of 1200 electron pulses. (B) Radial density functions, H(r), computed from the measured structure factors shown. The T 6 and 50 ps curves have been scaled by a factor of 4 to better show the structure. The correspondence between the peaks in H(r) at T –1 ps and the interatomic spacings present in the fcc Al lattice (inset) are shown for the first four peaks. Atoms of a given color are the same distance from the central black atom. The peak in H(r) that each interatomic separation produces is labeled with a circle of the same color. Only short-range correlations in atomic position are present at T 6 ps, and the highly modulated structure of H(r) that is due to the fcc lattice is replaced by the simple coordination shell structure of liquids. Subtle changes occur between 6 and 50 ps; namely, a shift in the correlation peaks to larger distance and a signi.cant reduction in the magnitude of the second peak at r 5 Å.