Ab-intio study of ultrafast charge dynamics in graphene

Abstract

Monolayer graphene provides an ideal material to explore one of the fundamental light-field driven interference effects: Landau-Zener-St\"uckelberg interference. However, direct observation of the resulting interference patterns in momentum space has not proven possible, with Landau-Zener-St\"uckelberg interference observed only indirectly through optically induced residual currents. Here we show that the transient electron momentum density (EMD), an object that can easily be obtained in experiment, provides an excellent description of momentum resolved charge excitation. We employ state-of-the-art time-dependent density function theory calculations, demonstrating by direct comparison of EMD with conduction band occupancy, obtained from projecting the time propagated wavefunction onto the ground state, that the two quantities are in excellent agreement. For even the most intense laser pulses we find that the electron dynamics to be almost completely dominated by the $\pi$-band, with transitions to other bands strongly suppressed. Simple model based tight-binding approaches can thus be expected to provide an excellent description for the laser induced electron dynamics in graphene.

Full text

Ab-intio study of ultrafast charge dynamics in graphene
Q. Z. Li1, P. Elliott1, J. K. Dewhurst2, S. Sharma1and S. Shallcross1
1 Max-Born-Institute for Non-linear Optics and and Short Pulse Spectroscopy,
Max-Born Strasse 2A, 12489 Berlin, Germany and
2 Max-Planck-Institut fur Mikrostrukturphysik Weinberg 2, D-06120 Halle, Germany.
Monolayer graphene provides an ideal material to explore one of the fundamental light-field driven
interference effects: Landau-Zener-St¨uckelberg interference. However, direct observation of the re-
sulting interference patterns in momentum space has not proven possible, with Landau-Zener-St¨uck-
elberg interference observed only indirectly through optically induced residual currents. Here we
show that the transient electron momentum density (EMD), an object that can easily be obtained in
experiment, provides an excellent description of momentum resolved charge excitation. We employ
state-of-the-art time-dependent density function theory calculations, demonstrating by direct com-
parison of EMD with conduction band occupancy, obtained from projecting the time propagated
wavefunction onto the ground state, that the two quantities are in excellent agreement. For even the
most intense laser pulses we find that the electron dynamics to be almost completely dominated by
the π-band, with transitions to other bands strongly suppressed. Simple model based tight-binding
approaches can thus be expected to provide an excellent description for the laser induced electron
dynamics in graphene.
Intense laser light offers the possibility to control elec-
trons in matter on femtosecond time scales. Triumphs
of this burgeoning field include tuning the optically in-
duced current in graphene via the carrier envelope phase
of light1–3, attosecond control over magnetic order in thin
films of magnetic overlayers4,5, and controlled valley ex-
citation in the semi-conducting few layer dichalcogenides
by circularly polarized light6,7 to name only a few ex-
amples. The two band Dirac cone found in graphene
provides an ideal in materials platform for studying one
of the canonical light-field driven interference effects:
Landau-Zener-St¨uckelberg (LZS) interference8,9, which
before its observation in graphene3had only been ob-
served in designed two state quantum systems10–14. This
effect occurs when an oscillating electromagnetic field
drives intraband oscillation through the Bloch acceler-
ation theorem k→k+A(t) and in the region of an
avoided crossing interband transitions occur even when
the band gap exceeds the dominant pulse frequency, so-
called Landau-Zener transitions. Upon repeated pass-
ing of the avoided crossing multiple pathways exist to
the conduction band with consequent constructive and
destructive interference of electron states. This offers
rich possibilities for controlling electron dynamics by in-
tense laser light, demonstrated by the recent observa-
tion of control over optical currents underpinned by LZS
interference3, a result anticipated theoretically in Ref. 15.
The ubiquity of the avoided crossing band structure
in 2d materials, found not only in the Dirac cone of
graphene but also in the the semi-conducting mono-
layer dichalcogesides16, phosphorene17 , silicene18 , and
stanene19, points towards the importance of LSZ inter-
ferometry in controlling electron dynamics in 2d mate-
rials. However, while interference physics can be eas-
ily probed theoretically through the conduction band
population20,21 , the experimental situation is more dif-
ficult, with to date only indirect observations of LSZ
physics in materials reported. In this paper we show
that the transient electron momentum density (EMD)
difference, defined as
∆ρ(p, tf) = ρ(p, tf)−ρ(p, t = 0) (1)
with pmomentum and ρ(p, t) the electron momentum
density22 before (t= 0) and after (tf) the pump laser
pulse, offers a tool for directly probing LZS interfer-
ence effects. As transient EMD is a practical probe
experimentally23,24, this suggests a way in which the LZS
physics may be directly observed in 2d materials, open-
ing the way to correlate indirect LZS physics such as in-
duced currents with the fundamental momentum space
interference patterns.
In contrast to previous works that have employed sim-
ple single particle tight-binding Hamiltonians to study
the LZS effect3,20,21,25–29, we will here deploy the time de-
pendent version of density functional theory (TD-DFT).
To establish the accuracy of the EMD as a record of LZS
interference we compare it with the excited electron dis-
tribution, Nex, defined within TD-DFT as30 :
Nex(k, t) =
occ
X
i
unocc
X
j
|hψik(t)|ψjk(t= 0)i|2(2)
where ψjk(t) is the time-dependent Kohn-Sham orbital
at time t, and ψik(t= 0) is the ground state orbital. In
all cases we find that the pattern of excitation in momen-
tum space generated by transient EMD and Nex is nearly
identical in the first BZ.
Finally, we consider the role of the non-π-band states
in the electron dynamics in graphene. Remarkably, de-
spite electron excitation through the whole energy range
of the π-band (up to 10 eV above the Fermi energy, an
energy range encompassing the σ∗bands as well as sev-
eral high lcharacter bands), it turns out that there oc-
cur almost no transitions to states outside the π-band
arXiv:2012.00435v1 [cond-mat.mes-hall] 1 Dec 2020

2
(a) (b)
-0.4
-0.2
0
0.2
0.4
E-field (V/Å)
Ex
-0.1
0
0.1
J (a.u.)
Jx
φ=−π/2
(c)
(d) (e)
-0.6
-0.4
-0.2
0
0.2
E-field (V/Å)
-0.2
-0.1
0
0.1
J (a.u.)
φ=0
(f)
(g) (h)
-0.4
-0.2
0
0.2
0.4
E-field (V/Å)
-0.1
0
0.1
J (a.u.)
φ=π/2
(i)
(j) (k)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
E-field (V/Å)
Ey
Ex
-0.1
0
0.1
J (a.u.)
Jy
Jx
φy=0
φx=-π/2
(l)
(m) (n)
-0.6
-0.4
-0.2
0
0.2
E-field (V/Å)
-0.2
-0.1
0
0.1
J (a.u.)
φy=0
φx=0
(o)
(p) (q)
0 2 4 68
time (fs)
-0.6
-0.3
0
0.3
0.6
E-field (V/Å)
-0.2
-0.1
0
0.1
0.2
J (a.u.)
φy=0
φx=π/2
(r)
Figure 1: Conduction band occupation as a function of
k-vector as determined directly by projection of the time-
dependent state onto the ground-state Kohn-Sham states
(first column), see Eq. (2), and, second column, the transient
electron momentum density (td-EMD) difference, see Eq. (1).
Evidently, both quantities in a consistent way capture the mo-
mentum space intensity fringes generated by Landau-Zener-
St¨uckelberg interference in the first BZ. This depends sensi-
tively on the momentum space path induced by the laser pulse
as evidenced by the dependence on carrier envelope phase φ
(CEP) and polarization. The CEP values are given in the
third column which displays the electric field of the pump
laser pulse and induced current density. All pulses have the
same full width half maximum (FWHM) of 1.935 fs, a central
frequency of 1.4 eV, and peak intensity of 5.43×1012 W/cm2.
In the first and second columns, the white hexagons represent
the boundary of the 1st BZ.
manifold. We attribute this to the near vanishing of the
corresponding dipole matrix elements. Our calculations
thus suggest that even for very significant laser excitation
tight-binding based models will provide a good descrip-
tion of the electron dynamics.
According to Runge-Gross theorem31, which extends
the Hohenberg-Kohn theorem into the time domain, with
common initial states there will be a one to one corre-
spondence between the time-dependent external poten-
(a) (b)
-0.2
-0.1
0
0.1
0.2
E-field (V/Å)
Ex
-0.03
0
0.03
Jx
ω=1.3eV
I=1012W/cm2
J (a.u.)
(c)
(d) (e)
-0.5
0
0.5
E-field (V/Å)
-0.2
-0.1
0
0.1
0.2
J (a.u.)
ω=1.3eV
I=1013W/cm2
(f)
(g) (h)
-1.5
0
1.5
E-field (V/Å)
-3
-1.5
0
1.5
3
J (a.u.)
ω=1.3eV
I=1014W/cm2
(i)
(j) (k)
-0.2
-0.1
0
0.1
0.2
E-field (V/Å)
-0.03
0
0.03
J (a.u.)
ω=2.2eV
I=1012W/cm2
(l)
(m) (n)
-0.5
0
0.5
E-field (V/Å)
-0.4
-0.2
0
0.2
0.4
J (a.u.)
ω=2.2eV
I=1013W/cm2
(o)
(p) (q)
0510 15 20
time (fs)
-3
-2
-1
0
1
2
3
E-field (V/Å)
-0.4
-0.2
0
0.2
0.4
J (a.u.)
ω=2.2eV
I=1014W/cm2
(r)
Figure 2: Dependence of Landau-Zener-St¨uckelberg interference
fringes on pulse intensity and frequency. Each row represents a
different pulse with the frequency and intensity shown in the inset
of the third column panels (in all cases the full width half max-
imum (FWHMs) was 2.758 fs, and carrier envelope phase (CEP)
was π
2). Shown in the first column is the conduction band occupa-
tion as a function of k-vector determined by directly projecting the
time propagated wavefunction onto the ground-state Kohn-Sham
orbitals (first column), see Eq. (2). The second columns displays
the transient electron momentum density (td-EMD) difference, see
Eq. (1) for definition. Evidently, both quantities capture in a con-
sistent way the momentum space intensity fringes resulting from
Landau-Zener-St¨uckelberg interference. The third column shows
the residual current density induced by the pulse, along with the
pump pulse electric field, E(t). In the first and second columns,
the white hexagons indicate the boundaries of the 1st BZ.
tials and densities32,33. Based on this theorem, a system
of non-interacting particles can be chosen such that the
density of this non-interacting system is equal to that
of the interacting system for all times, with the wave
function of this non-interacting system represented by
a Slater determinant of single-particle orbitals. These
time-dependent Kohn-Sham (KS) orbitals are governed
by the Schr¨odinger equation (for the spin degenerate
case):

3
i∂tψj(r, t) = "1
2−i∇+1
cAext(t)2
+vs(r, t)#ψj(r, t).
(3)
In the above equation Aext(t) is the vector potential
representing the applied laser field, the effective poten-
tial vs(r, t) is given by vs(r, t) = vext(r, t) + vH(r, t) +
vxc(r, t), where vext (r, t) is the external potential, vH(r, t)
the Hartree potential, and vxc (r, t) is the exchange-
correlation (xc) potential. For the latter we have used the
adiabatic local density approximation. From the Fourier
transform of the Kohn-Sham states, ψik(r), the elec-
tron momentum density can be constructed as ρ(p) =
Pik|ψik(p)|2. This EMD constructed from KS states
has been found to provide excellent agreement with that
obtained from Compton scattering22.
All the calculations are performed by employing the
state-of-the art all-electron full potential linearized aug-
mented plane wave (LAPW) method34, as implemented
in the ELK code35 . We have used 30×30 k-point set; for
further details of the implementation of TD-DFT within
the LAPW basis we referee the reader to Refs. 36 and
37.
LZS interference probed by 2D td-EMD: the patterns
of excited charge in momentum space that most directly
characterise Landau-Zener-St¨uckelberg interference are
generally presented by plotting the conduction band oc-
cupation over the first Brillouin zone. However this in-
formation, while easy to obtain theoretically, is difficult
to obtain experimentally. We thus look at an alternative
quantity, the change in electron momentum density due
to the laser pulse.
In Fig. 1 we consider a few cycle pulse of full width
half maximum 1.935 fs, central frequency 1.4 eV and a
peak pulse intensity of 5.43 ×1012 W/cm2. We vary the
pulse carrier envelope phase (the angular difference be-
tween the E-field and pulse envelope maxima) for various
pulse polarizations, y-polarized (shown in Fig. 1 (a-i)),
circularly polarized (Fig. 1 (j-l) and (p-q)), and linearly
polarized (Fig. 1(m-o)). The magnitude of the electric
field is of the order of 5 V/nm, which places these pulses
in the strong non-perturbative regime for graphene. The
A-field can drive several passes of the gap that occurs on
any constant momentum slice that does not pass through
the Dirac point. These repeated crossings of the gap
mimina generate multiple electron pathways from valence
to conduction band and concomitant interference effects.
In the first and second columns of Fig. 1 are shown the
number of excited electrons Nex (Eq. (2)) and the EMD
(Eq. (1)) respectively. Evidently, these two objects con-
vey consistent information concerning the excited charge
distribution.
A striking asymmetry of charge excitation in the 1BZ
as a function of the carrier envelope phase (CEP) can
be observed, compare panels (a), (d), and (g) of Fig. 1.
Carrier envelope phases (CEP) of φ= +π/2 and −π/2
0
0.03
0.06
0.09
0.12
Partial DOS
px, py (t=0)
pz (t=0)
px, py (after pulse)
pz (after pulse)
(a)
-10 -5 0 5 10 15
Energy (eV)
0
0.03
0.06
0.09
0.12
Partial DOS
(b)
Figure 3: Time dependent partial density of states (PDOS) pro-
jected onto the l= 1 spherical harmonics. Here the PDOS (in
states/atom/eV) is shown both at t= 0 before the pulse, and at
the end of the silulation after the pulse has been applied. The
pump pulse for panels (a) and (b) is polarized in the x-direction,
with intensities 1012 W/cm2and 1014 W/cm2respectively. As
can be seen, even for almost complete excitation of the π-band in
which charge is excited from the π-band minima up to the π∗-band
maxima, there is no excitation into states of pxor pycharacter.
displays pronounced excitation for kx<0 and kx>0 re-
spectively, while for φ= 0 the excitation is symmetric in
kx(we here measure momentum from the high symmetry
K point). This asymmetry is a consequence of the fact
that for non-zero CEP the A(t) field executes motion in
momentum space breaking the kxmirror symmetry3. In-
terestingly, it can also be seen that the momentum space
excitations weakly break symmetry in kyas well.
In Fig. 2 are displayed the Nex and EMD for a pulses
of central frequency 1.3 eV and intensities 1012 W/cm2,
1013 W/cm2, 1014 W/cm2(Fig. 2 (a-i)) and a frequency
2.2 eV with the same intensities (Fig. 2 (j-r)). Once again
the very good agreement between the Nex and EMD can
be noted. Increasing the intensity results in the LSZ in-
terference fringes (the lines of high probability density in
momentum space) extending increasingly far from the BZ
boundary, a fact that follows from the increase in the vec-
tor potential maximum which, by the Bloch acceleration
theorem, results in electron trajectories extending fur-
ther into momentum space. Taken together, these results
demonstrate that EMD generates a map of momentum
space excitations carrying nearly identical information to
the conduction band occupancy in the 1st BZ, and thus
represents an ideal experimental tool for studying the
fundamental interference patterns of the LZS effect.
The residual current: In the past asymmetric LZS in-
terference has been indirectly accessed by means the net

4
Figure 4: Band structure of graphene showing the πand σband
character. Negative and positive numbers indicate dominance by
π- and σ-character respectively.
current induced by the laser pulse3. For non-zero carrier
envelope phase, i.e. a lightwave in which the E-field max-
imum is displaced from the pulse envelope maximum, the
asymmetric momentum space trajectories generate exci-
tations that result in a net current. To date this rep-
resents the only observation in experiment of LSZ in a
material3. Underpinning this residual current is an early
time coherent steady current, that ultimately at longer
time scales generates heating, and the diffusive residual
current measured in experiment. This coherent current
(current per unit cell) induced by the laser pulse is dis-
played in the third column of Figs. 1 and 2. All pump
pulses with a non-zero CEP can be seen to induce such
a steady state current, with the change of sign of the
current with CEP reflecting the asymmetric momentum
space excitations seen in the first two columns of Figs. 1
and 2. As the momentum space trajectories travel further
from the Dirac point (e.g by increasing the intensity of
the pulse), the current initially increases, correlating with
an increased asymmetry in the excited charge (see Fig. 2
d-f), before widespread excitation throughout the BZ re-
sults in overall cancellation of current carrying states and
a very small residual current for the most intense 1 eV
pulse (see Fig. 2 g-i). A similar picture can be seen for
the 2 eV pulse, however here the decreased asymmetry in
the excited charge results in a reduced current. We thus
see that the magnitude of the induced current correlates
well with the asymmetry of excited charge in momentum
space, although as stressed by Motlagh et al.38 the opti-
cally induced current is in general not governed solely by
the momentum space distribution of excited charge.
Dominance of the π-manifold in electron dynamics:
the results for the momentum resolved conduction band
occupation shown in the previous sections, correspond
very closely to results obtained on the basis of model π-
band only tight-binding Hamiltonians. This raises the
question of whether this is due simply to the relatively
low energies of the excited charge (in Fig 1 and Fig. 2 the
excited charge resides predominantly at the K point and
the K-M-K line) or whether, for a more general reason,
the π-band will always dominate ultrafast laser induced
electron dynamics in graphene.
To explore this in Fig. 3 we display the partial den-
sity of states calculated before and after the laser pulse.
As can be seen, see Fig. 3a, for the pulse of intensity
1012 W/cm2, the partial DOS after the pulse shows con-
duction band occupation only up to 2.5 eV. At these en-
ergies, see Fig. 4, only the π-band is available for excited
charge. Remarkably, when we consider a very strong
pulse of intensity 1014 W/cm2the excited electrons are
again only of pzcharacter, Fig. 3b, despite the fact that
the laser pulse is sufficiently strong to excite charge from
the minima of the π-manifold up to the maxima of the
π∗-manifold. As may be noted from the band structure,
Fig. 4, within this energy range exist many other bands
that would, in principle, be expected to be involved in
the electron dynamics at such high energies.
Examination of the relevant dipole matrix elements re-
veals that transitions from πto σ∗and π∗to σare neg-
ligible for laser pulses with in-plane polarization. Thus
even in the highly non-perturbative regime transitions
from the ground state to the σ∗manifold will be strongly
suppressed. It might be argued that the partial DOS,
a projection within (touching) muffin tins, does not ac-
count for excitation to delocalized bands of high lchar-
acter. Comparison of the interstitial density of states
before and after the pulse shows that there is indeed an
increase in interstitial charge at around 9 eV, possibly
indicating transitions from the π∗manifold to delocal-
ized bands (note the intersections between π∗and high l
character bands on the M-Γ line), however this is a rather
small effect. It would thus appear that the model π-band
only tight-binding Hamiltonians provide an excellent de-
scription of the electron dynamics even for very intense
laser pulses.
To summarize we have investigated ab-initio the laser
induced electron dynamics in monolayer graphene. This
system provides a canonical example of a material for
which Landau-Zener-St¨uckelberg interferometry can be
explored, and we have shown that direct visualisation of
the interference fringes in momentum space is possible
via the transient electron momentum density (EMD), es-
tablishing transient EMD as an excellent experimental
tool for exploring LZS interference in 2d materials.
Examination of the excited state partial density of
states reveals that the π-band manifold decisively domi-
nates ultrafast laser induced dynamics in graphene, justi-
fying the deployment of the popular H¨uckel tight-binding
model. Whether this remains true for the complex few
layer graphene systems, for which such an approach is
the only one that can reasonably be envisioned, remains
an open question.
I. ACKNOWLEDGEMENTS
QZL would like to thank DFG for funding through
TRR227 (project A04). SS would like to thank DFG for
funding through SH498/4-1 and PE acknowledges fund-

5
ing from DFG Eigene Stelle project 2059421. The au-
thors acknowledge the North-German Supercomputing
Alliance (HLRN) for providing HPC resources that have
contributed to the research results reported in this paper.
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