Imaging attosecond electron dynamics in graphite (and graphene)( g p )

with inelastic x-ray scatteringPeter AbbamonteU i it f Illi i U b IL USAUniversity of Illinois, Urbana, IL, USA

James Reed, University of IllinoisBruno Uchoa, University of IllinoisDi C Ad d Ph t S

P. A., et. al., Phys. Rev. Lett. 92, 237401 (2004)P. A., et. al., Proc. Natl. Acad. Sci. 105, 12159 (2008)

Diego Casa, Advanced Photon SourceYu Gan, University of IllinoisThomas Gog, Advanced Photon SourceEduardo Fradkin, University of Illinois

Funding: Office of Basic Energy Sciences, U.S. Department of Energy #s DE-FG02-06ER46285 & DE-AC02-98-CH10886

People

Experiments, algorithms:

James Reed Young Il Joe Yu Gan Bruno Uchoa ICMTJames Reed Young Il Joe Yu Gan

Optics, front end, support:Bruno Uchoa, ICMT

Diego Casa, APS

Acknowledgements: Wei Ku, Brookhaven National Laboratory; Ken Finkelstein, CHESS / Cornell; Sol Gruner, Cornell / CHESS; Tim Graber, Advanced Photon Source; Chen Lin Yeh,

Thomas Gog, APS Rick Krakora, APS

/ Cornell; Sol Gruner, Cornell / CHESS; Tim Graber, Advanced Photon Source; Chen Lin Yeh, Tamkang University; Abhay Shukla, Université Marie et Pierre Curie; Jean-Pascal Rueff, Soleil; Yong Cai, Brookhaven National Laboratory; Serban Smadici, University of Illinois; Marcus Collins, University of Washington; Gerard Wong, University of Illinois; Robert Coridan, University of Illinois

Imaging (17th century)

Optical images may be “magnified”:

h h'

One can make a “microscope” (Johann Lippershey, 1595 AD):

Disadvantages: Not very small (limited to /2)Disadvantages: Not very small (limited to /2) Not very fast (limited by human eye ~ 30 Hz)

To do better one must develop other approaches

Å-scale imaging: x-ray “diffraction”

D(1)

n(x)

Scattered intensity ~ |D(1)(r)|2 ~ |n(q)|2

Classical oscillator model:

• “X-ray diffraction measures the FT of the electron density.”• Not true: XRD measures the electron density squared

y | ( )| | (q)|

Not true: XRD measures the electron density squared.• Must overcome the inverse scattering problem or the “phase problem.”• Soluble, though one must determine some constraints.

Periodic case: Crystallography

• Density is the sum of discrete Fourier components:

• Scattering localized to points in momentum space – “Bragg reflections”.

• Integrated intensity proportional to the Fourier coefficient, usually called a “structure factor”:

(fn = Z nominally)

Periodic case: Crystallography

KcsA potassium channelD. A. Doyle, et. al.,

Science 280, 69 (1998)

3.5 Å resolution

Phase constraint: Hg atoms in known locations of the sequence (MAD)q ( )

Aperiodic case: “speckle” imaging

Pb nanocrystalM. A. Pfeifer, et. al., Nature

442, 63 (2006)

Phase constraint:“Oversampling”. Density known to be zero outsideknown to be zero outside

the specimen.t?

Scattering generalized – IXS (Raman effect)

Photons are particles, and the EM field is an operator:

Couple into Hamiltonian via canonical transformation:

Take first Born amplitude and compute a transition rate:

Dynamic structure factor:

Can get S(k,w) from energy-resolved, inelastic x-ray scattering.

How does this give us dynamics?

The dynamic structure factor is the Fourier transform of the Van Hove space-time correlation function for the density:

(x,t)

“Diffraction”, which is energy-integrated scattering, actually measures an equal-time correlation function:

( )

These quantities contain dynamical information, but are “indirect” in that they are not causal.

(0,0)

Causality

The direct quantity is the retarded, density propagator:

Causality

Quantum mechanical version of the Fluctuation-Dissipation theorem.

“Phase problem” and the arrow of time

Cannot invert with only Im[(k,)]

To get to the dynamics one must solve the phase problem.

Im[]

Re[]

• (x t) = 0 for t < 0• (x,t) = 0 for t < 0

• Raw spectra do not really describe dynamics – no causal information

• Must assign an arrow of time to the problem. Permits retrieval of (x,t) –view dynamics explicitlyview dynamics explicitly.

• Rise of entropy arrow of time

Inelastic x-ray scattering: practical

Advanced Photon Source

Sector 9 CMC / XOR

Inelastic x-ray scattering: practical

backscattering Ge(733) analyzer

APS Undulator Apre-monochromator,

Si(111)

specimen

Si(111)secondary

monochromator, Si(333)detector

= i – fq = ki - kf

Strong correlations in graphite / graphene?

Castro- Neto, et. al., arXiv:0709.1163 (2008)

Ek = vF |k|

E. H. Hwang, et. al. Phys. Rev. Lett. 99,

226801 (2007)

Strong coulomb interactions in graphene? Velocity divergence:

~ 2

No anomaly seen in theNo dispersion seen in the electronic compressibility:J. Martin, et. al., Nature Physics 4

anomaly seen with ARPES: A. Bostwick, et. al., Nature Physics 3, 36

What is the fine structure constant of graphene (or graphite for that matter)?

Nature Physics 4, 144 (2008)(2006)

Graphite data – Im[k] compton scattering

= 0

= 15o= 15

= 30o

Graphite data – Im[k]

= 0

k,

)]

|q| = 0.5 Å-1 = 0

= 15o2 /k2I

m[

(k

| | Å 1

= 15

–4e

2 |q| = 2.0 Å-1

= 30o• Two peaks from van Hove singularities in (1) the kz=0 plane and (2) near the M face Call them “plasmons” Many screening mechanisms besides Dirac

Energy (eV)

M face. Call them plasmons . Many screening mechanisms besides Dirac Fermions.

• Plasmon from Fermi surface pockets not visible. This is all “background” screening (fine structure constant)screening (fine structure constant).

• t = 0.4 eV no significant distinction between graphite and graphene as far as background screening is concerned.

Problems

Problem #1:

Im[(k,)] must be defined on infinite interval for continuous time interval

Solution:

Extrapolate.

(eV)

Side effects:

• (x,t) defined on continuous (infinitely narrow) time intervals.

• Time “resolution” tN = /max [P. Abbamonte, PNAS, 105, 12159 (2008)]

• max plays role of pulse width.

Problems

Problem #2:

Discrete points violate causality

I [ (k )] t b d fi d ti i t l P i di it i tibl ithIm[(k,)] must be defined on continuous interval. Periodicity incompatible with causality.

S l tiSolution:

Analytic continuation (interpolate)

Side effects:

• (x,t) defined forever. Vanishes for t < 0.

• R t ith i d T 2 / 13 8 f t d• Repeats with period T = 2/ = 13.8 femtoseconds

• plays role of rep rate

Frame-by-frame: complex interference among neighbors

tN = 20.7 as xN = 0.533 Å

• In 20 as light travels 6 nm in vacuumg

• Causality Analytic properties Rise of entropy Arrow of time

Fine structure constant of graphite (and graphene)

Static impurity:

Static dielectric constant:0,

0)]

Re[(

0 q ~ 0.16 A-1

~ 22

D. E. Sheehy, et. al., Phys. Rev. Lett. 99, 226803 (2007), = 1E H Hwang et al Phys Rev Lett 99 226801 (2007) = 1 75

22

E. H. Hwang, et. al., Phys. Rev. Lett. 99, 226801 (2007), 1.75I. F. Herbut, et. al., Phys. Rev. Lett. 100, 046403 (2008), = 6

Y. Barlas, et. al., Phys. Rev. Lett. 98, 236601 (2007), = 1R. Asgari, et. al., Phys. Rev. B 77, 125432 (2008), = 1

~ 0.1

Weaknesses: What this method does not do

Completely passive, linear imaging technique• No nonlinear response (SHG, etc.)• No high-field physics• No coherent control.• Electron density only (No spin excitations, Wannier excitons, etc.)y y ( )• No “slow” excitations (t ~ 1 ps or greater)• In simplistic approach gives spatially averaged response

P. Abbamonte, 0904:0795v2 (Possible extension with standing wave bba o te, 090 0 95 ( oss b e e te s o t sta d g a etechniques)

Strengths:Strengths:• Really really really fast (zeptosecond resolution possible)• True imaging, with Å – scale spatial resolution• Sources, superposition, relation to optical constants, etc.

Nyquist’s (critical sampling) theorem

f(t)|f()|2

t

max max

N = 2 max

Nyquist frequency

too small t 20 7 too small aliasing

tN = 20.7 as

xN = 0.533 Å

Not everyone believes me.

Summary

• Graphene exhibits no vF divergence because of internal screening.screening.

• Improvements in x-ray sources allow explicit reconstruction of density propagator: “direct” imaging of electron d idynamics.

• Å spatial, attosecond temporal resolution (zeptophysics?).

plasmon

• Density only; no nonlinearity, no coherent-control, no high field-physics.

• Interest in attoscience from CM community strengthens plasmon• Interest in attoscience from CM community strengthens case for all approaches, including laser-based.