Spatially resolved quantum plasmon modes in metallic nano-films from first principles

3.1 MB·English

by  Kirsten Andersen

Spatially resolved quantum plasmon modes in metallic nano-films from first principles
Kirsten Andersen,1 Karsten W. Jacobsen,1 and Kristian S. Thygesen2,∗
1Center for Atomic-scale Materials Design (CAMD), Department of Physics,
Technical University of Denmark,DK - 2800 Kgs. Lyngby, Denmark.
2Center for Atomic-scale Materials Design (CAMD) and Center for Nanostructured Graphene (CNG),
Department of Physics, Technical University of Denmark,DK - 2800 Kgs. Lyngby, Denmark.
(Dated: December 11, 2013)
Electronenergylossspectroscopy(EELS)canbeusedtoprobeplasmonexcitationsinnanostruc-
3 tured materials with atomic-scale spatial resolution. For structures smaller than a few nanometers
1
quantum effects are expected to be important, limiting the validity of widely used semi-classical
0
response models. Here we present a method to identify and compute spatially resolved plasmon
2
modesfrom first principles based on a spectral analysis of thedynamical dielectric function. Asan

n examplewecalculatetheplasmonmodesof0.5-4nmthickNafilmsandfindthattheycanbeclassi-
a fiedas(conventional)surfacemodes,sub-surfacemodes,andadiscretesetofbulkmodesresembling
J standing waves across the film. We find clear effects of both quantum confinement and non-local

2 response. The quantum plasmon modes provide an intuitive picture of collective excitations of
confined electron systems and offer a clear interpretation of spatially resolved EELS spectra.

]
l
l PACSnumbers: 73.21.-b,78.20.-e,73.22.Lp,71.45.Gm
a
h
s- I. INTRODUCTION ne2/mǫ0 where n is the average electron density of
e pthe material2. For metal surfaces the Drude model pre-
m
Thedielectricpropertiesofamaterialaretoalargeex- dicts the existence of surface plasmons with frequency
t. tentgovernedbythecollectiveeigenmodesofitselectron ωs =ωp/√2. While the Drude model provides a reason-
a system known as plasmons1. Advances in spectroscopy abledescriptionofplasmonsinsimplemetalstructuresin
m and materials preparation have recently made it possi- the mesoscopic size regime it fails to account for the dis-
- ble to study and control light-matter interactions at the persion(wave vectordependence) of the plasmonenergy
d
nanometer length scale where particularly the surface in extended systems and predicts a divergent field en-
n
o plasmons play a key role2,3. While the ultimate goal of hancement at sharp edges. These unphysical results are
c nanoplasmonics as a platform for ultrafast and compact to some extent remedied by the semi-classical hydrody-
[ informationprocessingremainsachallengeforthefuture, namic models which can account for spatial non-locality

the unique plasmonic properties of metallic nanostruc- of ǫ and smooth charge density profiles at the metal-
v1 tures have already been utilized in a number of applica- vacuum interface2,11,12. Still, for materials other than
4 tionsincludingmolecularsensors4,5,photo-catalysis6and the simple metals and for truly nanometer sized struc-
8 thin-film solar cells7. tures, predictive modeling of plasmonic properties must
1 Electron energy loss spectroscopy (EELS) has been be based on a full quantum mechanical description13,14.
0 widely used to probe plasmon excitations in bulk ma- The calculation of plasmon energies and EELS spec-
1. terials and their surfaces. More recently, the use of tra of periodic solids from first principles is a well es-
0 highly confined electron beams available in transmis- tablished discipline of computational condensed matter
3 sion electron microscopes has made it possible to mea- physics15–18. However, the application of these powerful
1 sure the loss spectrum of low-dimensional structures on methods to systematically explore the real space struc-
v: a sub-nanometer length scale and with < 0.1 eV energy ture of plasmon excitations at the nanoscale has, to our
i resolution8. Because the loss spectrum is dominated by knowledge, not been achieved previously.
X
the plasmons this technique provides a means for visual-
ar ising the real-space structure plasmon excitations9. spaIntiathlliys preaspoelrvewdepplraessmenotnamgeondeersalfrmomethfiordstt-oprcianlcciupllaetse.
Allinformationabouttheplasmonsofagivenmaterial
We apply the method to Na films of a few nanome-
is contained in its frequency-dependent dielectric func-
ter thickness where effects of quantum confinement and
tion, ǫ, which relates the total potential in the material
nonlocal response are expected to be important. The
to the externally applied potential to linear order,
plasmon modes we find can be categorized as surface
modes located mainly outside the metal surface, sub-
φext(r,ω)=Z ǫ(r,r′,ω)φtot(r′,ω)dr′ (1) surface modes located just below the surface, and bulk
modes. Very intuitively, the bulk plasmon modes re-
(We have specialized to the case of longitudinal fields semble standing waves with nodes at the film surfaces.
which can be represented by scalar potentials). In the However, only plasmons with oscillation periods larger
widely used Drude model, one neglects the spatial vari- than10˚AarefoundasaresultofLandaudampingwhich
ation of ǫ and describes the frequency dependence by suppresses the strength of plasmon modes with smaller
a single parameter, the bulk plasmon frequency, ω = oscillationperiods. Finally, we calculatethe spatiallyre-
p
2
FIG. 1: (Color online) Left: Spatial profile of the plasmon modes in the direction normal to the film for in-plane momentum
transferq = 0.1˚A−1 . Thered (blue)curvesshowthepotential(density)associated with theplasmon excitation. Mode type,
k
energy, and strength (α) is shown to the left. Middle: Contour plot of the energy loss of an in-plane electron beam as it is
scannedacrossthefilm. Onlythelefthalfofthefilmisshownandtheposition oftheNaatomsisindicatedbywhitecirclesat
thebottom. Thethreebrightfeatures in thespectrumoriginate from energyloss duetoexcitation of threetypesof plasmons,
namely the surface plasmon mode S1, the subsurface mode S2, and the lowest bulk plasmon modes B1, respectively. Right:
Calculated electron energy loss spectra along thedashed lines indicated in the middlepanel.
solved EELS spectrum of the metal films and show that withself-sustainedcharge-densityoscillationsdampedby
allitsfeaturescanbetracedtoexcitationofspecificplas- electron-hole pair formations at the rate Γ .
n
mon modes. It is instructive to consider the dielectric function in
its spectral representation
II. METHOD ǫ(r,r′,ω)= ǫ (ω)φ (r,ω)ρ (r′,ω), (4)
n n n
X
n
According to Eq. (1), a self-sustained charge density
where the left and right eigenfunctions satisfy the usual
oscillation, ρ(r,ω), can exist in a material if the related
orthonormality relation
potential, satisfying the Poisson equation 2φ = 4πρ
∇ −
(atomic units are used throughout), obeys the equation φ (ω)ρ (ω) =δ . (5)
n m nm
h | i
ǫ(r,r′,ω)φ(r′,ω)dr′ =0 (2) In the appendix we show that the left and right eigen-
Z
functions are related through Poisson’s equation
In general, this equation cannot be exactly satisfied be-
2φ (r,ω)= 4πρ (r,ω), (6)
cause the dielectric function will have a finite imaginary ∇ n − n
part originating from single-particle transitions which
and thus correspond to the potential and charge den-
will lead to damping of the charge oscillation26. We
sity of the dielectric eigenmode, respectively. Physically,
therefore require only that the real part of ǫ vanishes
the condition (5) expresses the fact that the different
and use the following defining equation for the potential
dielectric eigenmodes for a given frequency ω, are elec-
associated with a plasmon mode,
trodynamically decoupled. The inverse dielectric func-
tion,ǫ−1(r,r,ω),isobtainedbyreplacingtheeigenvalues
ǫ(r,r′,ω )φ (r′,ω )dr′ =iΓ φ (r,ω ) (3)
Z n n n n n n ǫn(ω) in Eq. (4) by 1/ǫn(ω).
When the imaginarypartof the eigenvalue ǫ (ω) does
n
where Γn is a real number. Mathematically, the plas- notvarytoomucharoundtheplasmonfrequencyωn,the
mon modes are thus the eigenfunctions corresponding condition (3) is equivalent to the condition that
to purely imaginary eigenvalues of the dielectric func-
tion. Physically, they represent the potential associated Imǫ (ω )−1 is a local maximum. (7)
n n
3
This is the case for most of the plasmon modes of the (> 90%). For this reason only the real part is shown in
simplemetalfilmsstudiedinthiswork. However,forthe Figs. 1 and 3.
high energy bulk modes we found that the variation in
Imǫ (ω)canshiftthelocalmaximumofImǫ (ω)−1 away
n n
from the point where Reǫ (ω) = 0. (The effect can be III. RESULTS
n
even stronger of more complex materials with interband
transitions.) In such cases the condition (7) rather than In Fig. 1(a) we show the plasmon modes φ (z) (red)
n
(3) should be used to define the plasmon energy. Im- andcorrespondingchargedensitiesρ (z)(blue)obtained
n
portantly, however, the eigen functions φn(r,ω) do not fora10atomthickNafilmterminatedby(100)surfaces.
change significantly when ω is varied between the fre- Fig. 2 shows the real part of the eigenvalues of ǫ for the
quencies given by the two criteria and thus the spatial 10 layer film. The red dots indicate the plasmon fre-
form of the plasmon remains well defined. quencies,ω ,wheretherealpartofaneigenvaluecrosses
n
In the case of metallic films where the in-plane vari- the real axis from below. Note that the point where an
ation of ǫ and φn can be assumed to have the forms eigenvalue crosses the real axis from above corresponds
exp(iqk · (rk − rk′)) and exp(iqk · rk), respectively, Eq. roughly to the energy of the individual single-particle
(3) can be written transitions contributing to the plasmon state. Thus the
distance between the two crossing points, α˜, represents
ǫ(q ,z,z′,ω )φ (z′,ω )dz′ =iΓ φ (z,ω ) (8) the Coulombic restoring force of the plasma oscillation.
Z k n n n n n n The eigenfunctions of ǫ corresponding to the indicated
eigenvaluesarethe plasmonmodesshowninFig. 1. The
We stressthatdespite the atomicvariationinthe poten-
energies and strengths, α , of the plasmon modes are
tial we have found that local field effects are negligible n
listedto the left. The strengthis determinedby fitting a
within the plane justifying this assumption.
single-pole model
Allthecalculationswereperformedwiththeelectronic
structure code GPAW20,21 using the Atomic Simulation α
n
Environment22. TheNafilmsweremodeledinasupercell ǫn,1p(ω)=1− ω ω +iγ (9)
n,0 n
with periodic boundary conditions imposed in all direc- −
tions. The minimal Na unit cell was used in the plane of to the value and slope of the relevant eigenvalue branch
thefilmand30˚Aofvacuumwasincludedinthedirection of ǫ at the point ω =ω , see Fig. 2.
n
perpendiculartothefilmtoseparatetheperiodicimages. Returning to Fig. 1, we see that the lowest lying plas-
Thesingle-particlewavefunctionsandenergieswerecom- mon modes (S ) are the symmetric and anti-symmetric
1
putedonarealspacegridwithagridspacing0.18˚Aand surface plasmons also predicted by the classical Drude
the Brillouin zone was sampled by 64x64 k-points. The model. The two sets of modes denoted S and S are
2 3
LDA functional was used for exchange and correlation. also localized at the surface, although they penetrate
The non-interacting density response function, χ0, was more into the bulk, and we therefore refer to them as
calculatedfromtheDFTsingle-particlestatesusinga50 subsurface modes. The subsurface character of the S
3
eV energy cutoff for the plane wave basis and including modeisperhapsmoreevidentforthe 20layerfilmwhere
statesupto15eVabovetheFermilevel. Thefrequency- it is more clearly separated from the bulk modes, see
dependenceofthedielectricfunctionwassampledonthe Fig. 4. Experimental loss spectra of simple metal sur-
realfrequencyaxisusingagridfrom0to20eVwitha0.1 faces have shown a small peak between the surface and
eV spacing. We checked that our results were converged bulk peaks which was assigned to a subsurface plasmon
withrespecttoalltheparametersofthecalculation. The (inthatworkdenotedmultipolesurfacemode)10. Sucha
microscopicdielectricfunctionwascalculatedfromχ0us- peakatintermediateenergieswasalsoobservedinapre-
ing the randomphase approximation(RPA). We note in vious RPA calculation for a jellium surface23. However,
passing that this DFT-RPA level of theory yields highly the complete analysis of the plasmons modes presented
accuratebulkandsurfaceplasmonenergies(withinafew here shows that more than one subsurface mode exists.
tenths on an electron volt) of the simple metals18. The bulk plasmons (B) occur at higher energies and
To obtain the plasmon modes, the dielectric function resemble standing waves across the film. The fact that
is diagonalized in a plane-wave basis on each point of a only a discrete set of bulk modes are observed is clearly
uniformfrequencygrid. Tosimulateanisolatedfilm,the aresultofconfinementofthe electrongaswhichrequires
obtained eigenmode potentials were corrected to remove the density to vanish at the boundaries of the film. The
theeffectofinteractionswithfilmsintheothersupercells, reason only a finite number of modes are found is that
i.e. the periodic boundary conditions from the DFT cal- the damping of the modes due to single-particle transi-
culation(this is mostimportantfor small q vectors). In tions increases for smaller oscillation periods, i.e. larger
k
generalthe eigenmodeswerefoundtobe complexvalued wave number. Consequently, the strength of the bulk
reflecting the spatial variation in the phase of the plas- plasmons decreases with increasing wave number until
mon potential. This phase variation is, however, rather the point where the realpart of ǫ does not cross the real
weak for the present system, and by a suitable choice axis. This is evident in Fig. 2 where a series of local
of the overall phase the modes can be made almost real minimaintheeigenvaluespectrumcanbe seenathigher
4
3 layers
1 Mode | ħωn(eV)| αn(eV)
S2a 5.2 <0.1
0 α˜1 S2s 5.2 1.1
S1a 4.7 1.3
S1s 3.0 1.3
−1
Reǫ FIG.3: (Coloronline)Spatialprofileoftheplasmonmodesof
Reǫ1p(ω) a3 layerNa filmsin thedirection normal to thefilm for mo-
−2 mentum transfer q = 0.1 ˚A−1. The red (blue) curves show
0 2 4 6 8 10 12 k
ħω(eV) the potential (density) associated with the plasmon excita-
tion,andthemodetype,energyandstrength(α)isshownto
FIG. 2: (Color online) Real part of the eigenvalues of the theleft.
microscopic dielectric function, ǫ(ω), for a 10 layer Na film.
Thefrequencieswhereaneigenvaluecrossestherealaxisfrom
below (red dots) define the plasmon frequencies, and the
corresponding eigenfunctions represent the plasmon modes electron density of Na. For small qk the agreement with
shown by the red curves in Fig. 1a. The blue curve shows the classical result is striking for the symmetric mode.
the real part of the single-pole model used to extract the On the other hand, the quantum results for the anti-
strength of the plasmon. The distance between the zero- symmetric mode are significantly red shifted compared
points, α˜ = pα2−4γ2, increases with increasing strength to the Drude result with deviations up to 1 eV for the
of the mode.
thinnest film. For large q the quantum plasmons show
k
a q2 dispersion whereas the classical result approaches
k
the asymptotic valueω /√2. This failureofthe classical
p
frequencies. Forallthefilms,thehighestlyingbulkmode model is due to the neglect of the spatial non-locality of
has a wave number around 0.5 ˚A−1. This is very close ǫ , i.e. the (r r′)-dependence.
to the thresholdq where Re[ǫ(q,ω)] becomes positive for −
We notethatwhenexchange-correlationeffects arein-
all ω in bulk Na.
cluded through the adiabatic local density approxima-
As already mentioned, the main mechanism of energy
tion (ALDA) kernel, a larger deviation from the classi-
lossoffastelectronspropagatingthroughamaterialisvia
cal model is observed. In particular the anti-symmetric
excitationof plasmons. In general,the energy dissipated
modeisshiftedevenfurtherdownforsmallq . Asimilar
k
to the electron systemdue to an applied potential of the
behavior, i.e. downshift of plasmon energies, is observed
form φ (r,t)=φ (r)exp(iωt), is
ext ext forsimple metalbulk systemswherethe ALDAalsopre-
dictsanegativedispersionforsmallq2. Thespatialform
P(ω)= φ (r)χ (r,r′,ω)φ (r′)drdr′. (10) of the plasmon modes obtained with the ALDA is, how-
Z Z ext 2 ext
ever, identical to those obtained at the RPA level.
Here χ is the imaginary part of the density response In Fig. 6 we show the same data as shown in Fig. 5
2
but with the plasmon energy plotted relative to the di-
function, χ. In the case of a fast electron, the external
potentialis simply thatof a pointchargemovingatcon- mensionlessparameterdqk,wheredisthefilmthickness.
Whenplottedinthis wayallthe classicaldispersionsfall
stant velocity. We have calculated the loss function for
on the same universal curve. In the regime dq < 2, it
high energy electron beams directed along lines parallel k
is clear that the quantum results for the anti-symmetric
to the film. The resulting EELS spectrum is seen in the
mode are not converged to the classical result even for
middle panel of Fig. 1. It is clear that the loss spectrum
the thickest slab.
iscompletelydominatedbythreetypesofexcitationscor-
responding to the surface, subsurface, and bulk plasmon In Fig. 7 we present the dispersion of all plas-
modes shownto the left. The intensity of the subsurface mon modes (conventional surface, sub-surface, and bulk
modesisratherweakinagreementwiththe lowstrength modes) for the 20 layer film. The bulk modes show a
(α) of these modes. weakq2 dispersion. Theenergyoffsetbetweenthediffer-
k
Fig. 5showsalltheenergiesofthesymmetricandanti- ent bulk modes arises from the different wave lenght of
symmetricsurfaceplasmons(subsurfaceplasmonsnotin- the plasmons in the normal direction. The energy of the
cluded)foundforfourdifferentfilmthicknessesasafunc- lowest bulk mode (B ) in the q = 0 limit is < 0.1 eV
1 k
tionofthein-planewavevector,q 24. Thefullcurvesare higher than the bulk value of ~ω . This small deviation
k p
the classical Drude results for a 2D metal film with the is due to the finite wavelengthofthe plasmononthe di-
5
Surface mode dispersion
ħωP6
20 layers
Mode | ħωn(eV)| αn(eV) 5
B10 7.5 1.7 ħωP
4 2
(cid:0)
B9 7.3 1.5 V)
ω(e3
B8 7.1 1.8 ħ QQQQuuuuaaaannnnttttuuuummmm CCCCllllaaaassssssssiiiiccccaaaallll
3 layers
2
B7 7.0 2.4 6 layers
1 10 layers
B6 6.8 2.4 20 layers
0
0.0 0.1 0.2 0.3 0.4 0.5
B5 6.6 2.7 q (A◦−1)
∥
S3a 6.4 1.1 FIG. 5: (Color online) Dispersion of surface plasmon ener-
gies for different Na film thicknesses. The symbols represent
S3s 6.4 1.0 thefirst-principlesRPAresultsfortheenergyofthesymmet-
ric(lowerbranch)andanti-symmetric(upperbranch)surface
B4 6.4 2.8 plasmons as function of the in-plane wave number qk . The
full lines are theresult of a classical Drudemodel.
B3 6.3 2.9
B2 6.2 3.0 modes (yellow color) is very similar to the bulk disper-
sion. The second sub-surface mode is rather weak and
B1 6.1 2.9 forq >0.2˚A−1 its eigenvalue, ǫ (ω), doesnotcrossthe
k n
real axis at any frequency. For both subsurface modes
S2s 5.2 0.6 the splitting between the symmetric and anti-symmetric
modes is rather small. We ascribe this to the weaker
S2a 5.2 0.5 strength of the electric field associated with the subsur-
face mode compared to the conventional surface mode:
S1s 4.2 1.7 Whilethechargedistribution,ρnassociatedwiththelat-
terhasmonopolecharacterinthedirectionperpendicular
S1a 3.9 1.6 to the film, the subsurface modes have dipole character,
see Fig. 4.
FIG.4: (Coloronline)Spatialprofileoftheplasmonmodesof IV. CONCLUSION
a20layerNafilmsinthedirectionnormaltothefilmsformo-
mentum transfer q = 0.1 ˚A−1. The red (blue) curves show
k We have introduced a method for calculating spa-
the potential (density) associated with the plasmon excita-
tion, and the mode type, energy and strength (α) is shown tially resolved plasmon modes in nanostructured mate-
to the left. In addition to the conventional surface modes rials from first principles. For the case of 2D Na films
(S1), the 20 layer slab supports two sets of subsurface plas- of thickness 0.5-4 nm we found that the modes could be
monmodes(S2,S3)withenergybetweenthesurfaceandbulk classified as either surface modes, subsurface modes or
modes bulk modes. In contrast to previous studies, the direct
computationofeigenmodesrevealedthatseveralsubsur-
face modes canexist atthe surface ofsimple metals. We
found clear effects of quantum confinement on the sur-
rection perpendicular to the film and this indicates that
face plasmon energies. In particular, the anti-symmetric
quantum size effects have a very small influence on the
surface mode of the thinner films were significantly red
bulk modes of the 20 layer film.
shiftedcomparedtotheclassicalDruderesult. Finally,it
The first sub-surface modes (green color) follows a q2
k was demonstrated how the different features in the cal-
dispersion until around 0.2 ˚A−1 where it enters the bulk culated spatially resolved EELS spectrum of the metal
mode energy range. From this point the dispersion of films could be unambiguously ascribed to the excitation
the sub-surface mode is reduced and follows that of the of specific plasmon modes. Apart from providing an in-
bulk modes. The dispersion of the second sub-surface tuitive and visual picture of the collective excitations of
6
Conversion(CNEEC) at Stanford University, an Energy
Surface mode dispersion
ħωP6 Frontier Research Center funded by the US Department
of Energy, Office of Science, Office of Basic Energy Sci-
5 ences under Grant no. de-sc0001060, is also acknowl-
ħωP edged.
4 2
(cid:0)
V)
ω(e3 Appendix A: Spectral representation of ǫ
ħ QQQQuuuuaaaannnnttttuuuummmm CCCCllllaaaassssssssiiiiccccaaaallll
2 3 layers Let φn(ω) denote the eigenfunctions of the dielectric
6 layers functio|n, i
1 10 layers
20 layers ǫˆ(ω)φn(ω) =ǫn(ω)φn(ω) (A1)
| i | i
0
0 2 4 6 8 10 12 (wehavesuppressedtherdependencefornotationalsim-
dq
∥
plicity). Since ǫˆ(ω) is a non-hermitian operator, the
eigenvalues are complex and the eigenfunctions are non-
FIG. 6: (Color online) Like Fig. 5 but plotted as function of orthogonal. Inthiscasethespectralrepresentationtakes
the dimensionless parameter dq where d is the thickness of
k the form
theNa film.
ǫˆ(ω)= ǫ (ω)φ (ω) ρ (ω) (A2)
n n n
| ih |
X
n
Dispersion of modes, 20 layers
9
The set ρ (ω) is the dual basis of φ (ω) and satisfies
n n
| i | i
8
φ (ω)ρ (ω) =δ . (A3)
7 h n | n i nm
ħωP6 From the spectral representation it follows directly that
V)5
ω(eħ4 ħω2P ǫˆ(ω)†|ρn(ω)i=ǫn(ω)∗|ρn(ω)i (A4)
(cid:0)
3 In the following we show that the dielectric eigenfunc-
Classical model
tion φ (ω) and its dual function ρ (ω) constitute a
2 surface 1. order | n i | n i
surface 2. order potential-density pair, i.e.
1 surface 3. order
bulk 2φ (r,ω)= 4πρ (r,ω) (A5)
n n
0 ∇ −
0.0 0.1 0.2 0.3 0.4 0.5
q(A◦−1)
∥ Within the RPA, the dielectric function is related to
the non-interacting polarizationfunction, χ (r,r′,ω), by
0
FIG. 7: (Color online) Dispersion of all the plasmon modes
found for the20 atomic layer Na film. ǫˆ(ω)=ˆ1 vˆχˆ (ω) (A6)
0
−
where vˆ = 1/r r′ is the Coulomb interaction, ˆ1 =
δ(r r′). Expl|oit−ingt|hat, under time reversalsymmetry
a nanostructure, the spatially resolved plasmon modes
−
χ (r,r′)=χ (r′,r), we have
should be useful as a basis for the constructionofsimple 0 0
models for the full non-local dielectric function.
ǫˆ(ω)† =ˆ1 χˆ (ω)∗vˆ (A7)
0
−
V. ACKNOWLEDGEMENTS We now make the ansatz ρn(ω)=vˆ−1φn(ω)∗ and evalu-
ate
K. S. T. acknowledges support from the Danish Re-
ǫ(ω)†vˆ−1φ (ω)∗ = v−1φ (ω)∗ χˆ (ω)∗φ (ω)∗(A8)
search Council’s Sapere Aude Program. The Center n n − 0 n
for Nanostructured Graphene CNG is sponsored by the = vˆ−1[ǫˆ(ω)φn(ω)]∗ (A9)
Danish National Research Foundation. Support from = ǫ (ω)∗vˆ−1φ∗ (A10)
n n
The Catalysis for Sustainable Energy (CASE) initiative
and the Center of Nanostructuring for Efficient Energy which concludes the proof.
7
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