excellent agreement without any fitting parameters. We do observe a slight discrepancy in the
d = 27 nm case, likely connected to the different
type of fringe pattern that was used to measure l
in this device compared with the other two devices (13). The local approximation predicts
plasmon velocities falling below nF for the 5.5-nm
device, in contrast to the full theory, which is
forbidden from this region (for reasons explained
Figure 4A depicts the three layers of our non-local theory, based on dominant effects known
from electron liquid theory (1, 2). Including all
nonlocal corrections, the conductivity takes the
following convenient form (for frequency and
wave vector below Fermi values, as in this
sðw; qÞ ¼ e2nF
where f(z) is a dimensionless function that describes the nonlocal response
f ðzÞ ¼ 2
ð1 ; z2Þ;1=2 ; 1 þ d
where the dimensionless d will be used to introduce one of the corrections (see below). Using
this functional form, we can gradually introduce
the different layers of nonlocal response, which
are plotted in Fig. 3B. The local approximation
consists of ignoring the q-dependence [which
amounts to setting f (z) = 1], yielding a Drude
response s º i/w from Eq. 2.
The first layer of nonlocal response is to consider the response of noninteracting electrons
(14, 15) [via random phase approximation (RPA)],
which is the case of Eq. 3 with d = 0. In the RPA,
conductivity s increases with q, due to the change
in Fermi surface deformations. This is closely
related to Landau-Bohm-Gross dispersion (21)
in classical plasma physics: Some of the electrons, those travelling with a velocity that nearly
matches np, can interact longer with each passing wavefront and thereby provide enhanced
response (Fig. 4A). Classically, this nonlocal
dispersion would come along with Landau
damping due to fast thermal electrons that fully
match the plasma velocity and dissipate energy;
this does not occur in a quantum degenerate
system due to the narrowly distributed electron
velocity (the Fermi velocity), which instead yields
a divergent intraband contribution to the conductivity as q → w/nF (i.e., z → 1), and no Landau
damping before this point. This divergence results in the nonlocal plasmon velocity never
falling below the Fermi velocity (as can be seen
in Fig. 3), in contrast to the prediction of a local
The second and third layers of our nonlocal
theory involve microscopic electron-electron
interactions (many-body effects). We have cal-
culated these many-body corrections fully con-
sistently, including the realistic screening by
capacitance C(w,q) (figs. S1 and S2) (13). The
major many-body effect is renormalization of
band structure, which in graphene amounts
to an increase in Fermi velocity (Fig. 4B). Al-
though the value vF = 1.0 × 106 m/s is nominally
assumed in graphene plasmon studies, the Fermi
velocity actually varies logarithmically with car-
rier density, from its bare value of 0.85 × 106 m/s
up to as much as 3.0 × 106 m/s for very low car-
rier densities (22–25). Because our experiments
enter a regime of relatively low electron den-
sities, it is crucial to include this ns-dependent
velocity renormalization. The secondary many-
body effect has to do with electron liquid cor-
relations, which produce a Pauli-Coulomb hole
(1, 2) around each reference electron (Fig. 4C).
We capture this by including a local field
factor G(w,q), via s;1 ¼ s;1 RPA þ q2G=ðiwCÞ, that
forces consistency between the dynamic re-
sponse and the isothermal compressibility (2).
In our experimental regime, this ultimately
introduces a factor d = 1 − (k0/k) into Eq. 3
[see the supplementary materials for further
details (13)], where k0 is the RPA compressibility
and k is the proper isothermal compressibility
Figure 4B shows how we can isolate the graph-
ene conductivity function s(w,q) to directly ob-
serve the nonlocality. This is possible due to Eq. 1,
which implies that a determination of the plas-
mon wave vector, qp = qp(w), produces a measure-
ment of the dynamical conductivity at that wave
vector: ℑsðqp; wÞ ¼ ðw=q2 p ÞCðqp; wÞ. We are able
to exactly calculate C(w,q) from Maxwell equa-
tions (13), and hence this approach of introduc-
ing variable d (causing variable C and variable
qp) allows us to map out ℑsðqpÞ as shown in Fig.
4. Each device (differing in d) thus provides a
distinct probe of the functional dependence of the
conductivity on wave vector q under otherwise-identical parameters (w,ns). It can be seen in Fig.
3B that our data are only matched by theory after
taking into account all three layers of quantum
corrections and that the measured conductivity
of graphene shows considerable departures from
the local theory (a horizontal line).
The recipe set forth in this work can be transferred to probe other electron systems with
exotic physical properties. Not only does this
technique reveal the collective excitation (
plasmon), but we have also shown how one may
isolate the electronic response from its environment, quantitatively mapping out the underlying
response function (nonlocal conductivity) as a
function of wavelength. This kind of spatial spectroscopy forms a valuable counterpart to the
traditional temporal (frequency) spectroscopy,
and the marriage of these two approaches into
a precision spatiotemporal spectroscopy—a full
determination of s(w,q)—would provide an unprecedented window into electron physics. This
may allow a greatly enriched understanding of
electron correlation physics such as those underlying fractional quantum Hall effects [e.g., magneto-rotons (28, 29)] and the binding mechanism in
superconductors (30), as well as probing the nonlocality of Fermi-surface deformations in unusual
band structures [e.g., Weyl fermions (31)].
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F.H.L.K., M.P., and R.H. acknowledge support by the European
Union Seventh Framework Programme under grant agreement no.
696656 Graphene Flagship. M.P. acknowledges support by
Fondazione Istituto Italiano di Tecnologia. F.H.L.K. acknowledges
financial support from the European Union Seventh Framework
Programme under the ERC starting grant (307806, CarbonLight)
and project GRASP (FP7-ICT-2013-613024-GRASP). F.H.L.K.
acknowledges support from the Spanish Ministry of Economy and
Competitiveness, through the “Severo Ochoa” Programme for
Centres of Excellence in R&D (SEV-2015-0522), support by
Fundacio Cellex Barcelona, CERCA Programme/Generalitat de
Catalunya, the Mineco grants Ramón y Cajal (RYC-2012-12281),
Plan Nacional (FIS2013-47161-P and FIS2014-59639-JIN), and
support from the Government of Catalonia through the SGR grant
(2014-SGR-1535). R.H. acknowledges support from the Spanish
Ministry of Economy and Competitiveness (national project MAT-
2015-65525-R). P.A-G. acknowledges financial support from the
national project FIS2014-60195-JIN and the ERC starting grant
715496, 2DNANOPTICA. K. W. and T. T. acknowledge support from
the Elemental Strategy Initiative conducted by the MEXT, Japan,
and JSPS KAKENHI grant numbers JP26248061, JP15K21722, and
JP25106006. Y.G., C. T., and J.H. acknowledge support from the
U.S. Office of Naval Research N00014-13-1-0662. C. T. was supported
under contract FA9550-11-C-0028 and awarded by the Department
of Defense, Air Force Office of Scientific Research, National
Defense Science and Engineering Graduate (NDSEG) Fellowship,
32 CFR 168a. This research used resources of the Center for
Functional Nanomaterials, which is a U.S. Department of Energy
Office of Science Facility at Brookhaven National Laboratory under
contract no. DE-SC0012704. B.V.D. acknowledges support from
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