Valley Plasmonics in the Dichalcogenides

3 MB·

by  R. E. Groenewald

Valley Plasmonics in the Dichalcogenides
R. E. Groenewald,1,∗ M. R¨osner,2,3,∗ G. Sch¨onhoff,2,3 S. Haas,1 and T. O. Wehling2,3
1Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA
2Institut fu¨r Theoretische Physik, Universit¨at Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany
3Bremen Center for Computational Materials Science,
Universita¨t Bremen, Am Fallturm 1a, 28359 Bremen, Germany
(Dated: January 11, 2016)
Therichphenomenologyofplasmonicexcitationsinthedichalcogenidesisanalyzedasafunction
of doping. The many-body polarization, the dielectric response function and electron energy loss
spectra are calculated using an ab initio based model involving material-realistic Coulomb inter-
6
actions, band structure and spin-orbit coupling. Focusing on the representative case of MoS , a
1 2
plethora of plasmon bands are observed, originating from scattering processes within and between
0
the conduction or valence band valleys. We discuss the resulting square-root and linear collective
2
modes,arisingfromlong-rangeversusshort-rangescreeningoftheCoulombpotential. Weshowthat

n themulti-orbitalnatureofthebandsandspin-orbitcouplingstronglyaffectsinter-valley scattering
a processes by gapping certain two-particle modes at large momentum transfer.
J

7
Introduction: Collective excitations are of great inter- teristicscanbeexperimentallysampledbymeansoffield

] est in low-dimensional materials which are characterized effect electron or hole doping [25], the resulting impact
l
l by reduced dielectric screening of Coulomb interactions. to the plasmonic dispersions is not known.
a
h As a prominent example, plasmon modes in layered sys-
- tems might form the basis to build optical devices, wave
s
e guidesorsocalledplasmoniccircuits[1–4]. Intwodimen-
m sions(2d)theplasmonicdispersionexhibitsacharacteris-
√
. ticlow-energyacousticmodeω(q)∝ q originatingfrom
t
a low-momentumelectronscattering[5,6],whichhasbeen
m
observed experimentally [7, 8] and studied extensively
- from a theoretical point of view [3, 9–12] in graphene.
d
Furthermore, it has been predicted that additional lin-
n
Figure 1: (Color online) Sketch of the Fermi surfaces in hole
o earplasmonswithω(q)∝q ariseduetohigh-momentum
(left) and electron doped (right) monolayer MoS . The dif-
c scattering processes between degenerated valleys such as 2
ferent orbital characters are indicated by red (d ) and blue
[ K and K(cid:48) in graphene [13]. Coupling of the electrons z2
1 with such intrinsic gapless bosonic modes may lead to (inddxyicaatnedddbxy2−dyi2ff)erfielnletdmsaurrkfaecres.s. Pointsofhighsymmetryare
v instabilities, such as charge density wave and supercon-
7 ducting phases [14–17], similar to the effect of phonons.
Toclosethisgap,wepresentinthisletteranextensive
0
7 An analogous but even richer phenomenology can be study of the plasmon dispersion at arbitrary momenta
1 expectedinthestructurallyrelatedmonolayertransition along paths throughout the whole Brillouin zone for
0 metal dichalcogenides (TMDCs) MX2, where M stands different doping levels. Specifically we are interested in
1. foratransitionmetalandXforachalcogenatom. These inter-valley plasmons which have not been studied in
0 materials host rich plasmonic physics including an inter- TMDCs so far. In order to highlight the multi orbital
6 play of plasmons with charge density waves [18–20] and character of the Fermi surface (see Fig. 1) and the
1
first plasmon based applications have already been pro- presence of spin-orbit coupling we consider hole and
:
v posed [21–23]. electron doped cases. In the hole doped example we
i
X Here we focus on the representative example of doped show how spin-orbit coupling affects the inter-valley
MoS whose low-energy band structure can be described plasmons while the electron doped case is used to study
r 2
a by three effective tight-binding bands. These originate the influence of the multi pocket structure of the Fermi
from the Mo d orbitals, giving rise to prominent valleys surface. Thereby we gain a comprehensive and realistic
at wave vectors K and Σ in the lowest conduction band picture of the most important contributions to the low
as well as at K in the highest valence band, leading to energy plasmon modes in monolayer TMDCs.
Fermi surfaces as depicted in Fig. 1. Furthermore, there
is substantial spin-orbit coupling (SOC) in these materi- Method: The collective plasmon modes are described
als [24], with a primary effect on the low-energy physics by the polarization and dielectric functions, which we
by introducing a splitting of the Σ valleys in the low- evaluate in several steps, starting with a G W calcula-
0 0
est conduction band and of the K valleys in the highest tion to determine the electronic band structure for the
valenceband. AlthoughalloftheseFermisurfacecharac- undoped system. We then obtain an effective 3-band
2
Figure 2: (Color online) Real and imaginary parts of the polarization functions (d /d channel) and EELS spectra for hole
xy xy
dopedMoS without(toprow)andwith(bottomrow)spinorbitcoupling. Theinsetsin(a)and(d)illustratetheFermisurface
2
pockets around K and K(cid:48).
model by projecting to a Wannier basis spanned by the where q and k are wave vectors from the first Bril-
Mo d , d and d orbitals, which has been found louin zone, λ band indices, fσ(k) Fermi functions for
to accxu2−rayt2elyxdyescribezt2he highest valence band and the the energies Eiσ (k) and iδ a sλmiall broadening parame-
λi
two lowest conduction bands with tight-binding hopping ter. The overlap matrix elements are given by Mλ1λ2 =
αβ
matrix elements tαβ, where α and β are the orbital in- c¯λ1(k)cλ1(k)c¯λ2(k+q)cλ2(k+q),wherecλi(k)istheex-
dices. The same projection is used to obtain the static α β β α α
pansion coefficient of the eigenfunction corresponding to
part of the Coulomb interaction in the Wannier basis,
Eσ (k)intheorbitalbasis. Here,wealreadyreducedthe
which is screened by all bands including those, which λi
polarization tensor of 4th order to a matrix to describe
are not included in the minimal 3-band-model [38].
density-density correlations only. Hence, we neglect or-
This procedure leads to an effective material-specific
bitalexchange(Fock-like)matrixelementsaswellasele-
model with screened Coulomb U and hopping t
αβγδ αβ ments with three or even four different orbital contribu-
matrix elements in the orbital basis, describing the un-
tions. Adetailedanalysisofthefullbackgroundscreened
dopedsysteminitsgroundstate. Wefindthatthistreat-
Coulomb tensor U shows, that these elements are in
αβγδ
ment is essential to derive material realistic plasmonic
generaloneorderofmagnitudesmallerorevenvanishdue
dispersions upon doping. In contrast to simplified k·p
to symmetries, which convinces us to stay with density-
models [26, 27], which utilize bare Coulomb matrix ele-
density like elements.
ments at this stage, our interaction matrix elements are
Using the full density-density polarization Π(q,ω) =
strongly reduced due to screening effects from the elec-
Π↑(q,ω) + Π↓(q,ω) the dielectric function is obtained
tronicbandswhichareneglectedinthek·pmodels. Asa
via the following matrix equation
resultofthe2dlayergeometry,thesedielectricproperties
cannot be modeled by a simple dielectric constant but ε(q,ω)=1−U(q)Π(q,ω), (2)
havetobedescribedasaq-dependentdielectricfunction
[28, 29]. where the background screened Coulomb interaction en-
ters via U(q). By including an effective spin-orbit cou-
In order to obtain the dynamic response in the doped
pling[30]thespindegeneracyisremovedbuttimerever-
system, we determine the dynamic susceptibility within
sal symmetry is preserved. Then, the spin resolved band
the 3-orbital basis by evaluating the polarization in the
structure still obeys E↑(k) = E↓(−k) and the total po-
random phase approximation (RPA), which is given for λ λ
larization including the spin summation can be written
a single spin channel σ by
as Π(q,ω)=Π↑(q,ω)+Π↑(−q,ω).
Πσ (q,ω)= (cid:88) Mαλβ1λ2(cid:2)fλσ2(k+q)−fλσ1(k)(cid:3) , (1) maTthriexdVie(leqc,tωri)c=funεc−ti1o(nq,dωes)cUr(ibqe)satnhdesicmrepelniceidtlyCoduelfionmebs
αβ ω+iδ+Eσ (k+q)−Eσ (k)
λ1λ2k λ2 λ1 the plasmonic dispersions by εm(q,ω) = 0, where εm is
3
the macroscopic part of the dielectric function [31]. The same as the polarization function itself. Thus for mo-
most promising experimental method to map these plas- menta away from Γ it is sufficient to study the polariza-
mon modes is electron energy loss spectroscopy (EELS), tion function to understand how the resulting plasmon
measuring the imaginary part of the inverse dielectric dispersion will behave.
function Of special interest are damping effects, which are
(cid:18) (cid:19) known to attenuate plasmon modes which merge with
1
EELS(q,ω)=−Im , (3) the particle-hole continuum. Here the square-root mode
ε (q,ω)
m around Γ behaves in a distinctly different manner com-
which is sensitive to both collective and single-particle pared to the linear modes originating at K. At suffi-
excitations(visibleasmaximaintheEELSspectra)[32]. cientlysmallmomentumtransfersq <qc thesquare-root
The combination of our material realistic description modes are more separated from the nearby particle-hole
of the undoped system and the very accurate band continua [Fig. 2 (c) and (f)], and therefore better pro-
structure for the RPA evaluation yields indeed quite tectedfromdecompositionviahybridizationandLandau
accurate plasmon dispersions compared to full ab initio damping [expressed as non-vanishing imaginary parts of
results, as we show for NbS in the supplement. the polarization as shown in Fig. 2 (b) and (e)] com-
2
pared to the linear modes originating at finite momenta.
Hole doped MoS : We fix the chemical potential such Incontrast,thelinearplasmonmodesaremuchcloserto
2
that there are holes in the valence band in the K and theirneighboringcontinua[Fig. 2(c)],whichleadstoat-
K(cid:48) valleys only. The resulting Fermi surfaces consists of tenuation effects, reflected in reduced oscillator strength
circle-like areas around the K points (see Fig. 1), which and broadening of the peaks. There is a significant dif-
have mainly d /d character and depend on spin- ference in the oscillator strengths of these modes, which
xy x2−y2
orbit coupling. Hence, we expect low energy plasmon can be several orders of magnitude apart as can be seen
modes for q≈Γ (intra-valley) and q≈K (inter-valley), in Fig. 2 (c) and (f). Hence, in order to clearly detect
which are possibly influenced by SOC. theselinearplasmonmodesinexperiments,itmayprove
InFig. 2(a)weshowanintensityplotoftherealpart practical to use a logarithmic scale to shield the domi-
of the polarization function for scattering within d or- nant square-root mode around q = Γ, as shown in Fig.
xy
bitals along the complete path Γ → K without SOC 2 (c) and (f).
[39]. Next to some band-like structures (red) we clearly When we account for spin-orbit coupling the relative
see the particle-hole continuum (blue). In comparison depth of the K and K(cid:48) pockets shifts. In this case mo-
to the corresponding EELS data in Fig. 2 (c), we see mentum transfer of q = K no longer connects points
that for higher momentum transfers (away from Γ) the on the Fermi surface belonging to different hole pockets,
EELSmaximacloselyfollowtheband-likecharacteristics which results in two clearly visible characteristics in the
of the polarization function. For small momenta around polarization of Fig. 2 (d): (1) At q = K the scatter-
Γ we find a clearly separated band in the EELS spectra, ing process is possible only for a finite energy difference,
which can not be seen in the real part of the polariza- which opens a finite energy gap of ≈ 250meV. (2) The
√
tion. Thisseparatedbandarisesfromthewellknown q- FermisurfacesatK andK(cid:48) arenowofdifferentsizesbut
dispersiveintra-valley plasmonmodein2d[26],whilewe can still be connected with slightly smaller and larger
find a linear-dispersive mode around K stemming from q, resulting in gap-less linear modes originating slightly
an inter-valley plasmon [13]. These activation laws are shifted from K as seen in Fig. 2 (d).
consistent with the generalized expression for the plas- Weconcludethattheplasmonicfeaturesinholedoped
mondispersionrelationdefinedbythedielectricfunction MoS2 are qualitatively similar to graphene as long as
via [15], SOC is not taken into account and the K valley is occu-
pied solely. Upon inclusion of SOC the linear plasmon
(cid:115)
[N U(q)]2 mode around K is shifted leading to a gapped excitation
ω(q)=(cid:126)v q 1+ 0 , (4)
F (1/4)+N U(q) spectra at this point.
0
Electron doped MoS : The lowest conduction band is
2
where v is the Fermi velocity, N the density of states characterized by two prominent minima around K and
F 0
at the Fermi level and U(q) the macroscopic background Σ. Without SOC these minima are separated by only
screenedCoulombinteractionoftheundopedsystem. In 90meV. Hence, in contrast to the hole doped case, small
the long-wavelength limit (q → 0), the Coulomb poten- variations in the electron doping can change the Fermi
tialremainsunscreened,i.e. inleadingorderU ∝1/q,re- surface drastically. In order to study these changes,
sulting in a square-root renormalization of the otherwise we will neglect the SOC for the beginning and choose
linear dispersion. However, in the opposite short-range two doping levels, resulting in Fermi surfaces compara-
limit, i.e. at the zone boundary, the screened Coulomb ble to the hole doped case (i.e. K valley occupation
potential approaches a constant, and therefore the re- only) and a surface with additional pockets at Σ, la-
sulting dispersion of the dielectric function is linear in q, beled by low- and high-doping respectively (see Supple-
4
Figure 3: (Color online) (a) The polarization function for d /d scattering at low electron doping concentration (only K
z2 z2
valleys are partially occupied) without SOC. (b) The polarization function for d /d scattering at elevated electron doping
xy xy
concentration(K andΣvalleysarepartiallyoccupied)withoutSOC.In(c)thesamesituationasin(b)isshown,butwiththe
effect of spin-orbit interaction. The insets show illustrations of the Fermi surfaces.
ment). Since the K valley is described by d orbitals each of these subsets are mutually connected by 2π/3
z2
andtheΣvalleypredominatelybyd andd states, rotations and remain equivalent after inclusion of SOC,
xy x2−y2
we focus on corresponding diagonal orbital channels in while the degeneracy of Σ and Σ(cid:48) is lifted by SOC. As a
Π in the following. Off-diagonal elements between d consequence, the phase space for Σ↔Σ(cid:48) is lost and the
αβ z2
and d /d orbitals are negligible here (off-diagonal gap-less excitations at q ≈ Σ and q ≈ K must vanish,
xy x2−y2
termsbetweend /d statesaresimilar). Thecorre- but Σ ↔ Σ scattering processes are still possible. Con-
xy x2−y2
spondingpolarizationfunctionsareshownalongthepath sequently, we see in the corresponding polarization for
Γ−Σ−K−M −Γ through the whole Brillouin zone in the d channel with SOC in Fig. 3 (c) gap-less modes
xy
Fig. 3. only at Γ and M. Since the Fermi surface around K
Analogous to the hole doped case, we observe around is not changed drastically upon SOC, the corresponding
q = Γ the expected resonances arising from intra-valley polarizationforthedz2 channelisverysimilartotheone
scattering. Thisisnaturallypresentinbothhighandlow obtained without SOC (see Supplement).
electron doping cases. By inspection of Fig. 1 we can Conclusions: Wefoundthatthelowenergydynamical
understandthestructureofthepolarizationforlargerq. screening in MoS is controlled by both inter- and intra-
2
The momenta q = Σ,K and M connect different Σ valley scattering processes. These give rise to plasmons
valleys. Therefore,inthehighelectrondopingcase(with withasquarerootdispersionatsmallqandlineardisper-
Σ partially occupied) we expect additional inter-valley sion for higher momentum transfers which connect sepa-
plasmon branches close to these momenta. ratevalleysontheFermisurface. Ingeneral, inter-valley
At q ≈ K we observe plasmon bands in both, high plasmon modes are observable, although their oscillator
and low doping cases since this momentum transfer al- strengthsarestronglyreducedincomparisontozonecen-
lowsinter-valleyscatteringbetweenK andΣpockets. As ter modes. Due to the multi-orbital character of the
wearecalculatingorbitalresolvedpolarizationfunctions, wave functions and spin-orbit coupling, which leads to
the observed low energy excitation in Fig. 3 (a) is due spin-valley coupling in monolayer TMDCs, not all inter-
to K ↔ K(cid:48) and thus d scattering, whereas in Fig. 3 valley scatteringprocessesareallowed. Asaconsequence
z2
(b) it is due to Σ ↔ Σ and correspondingly d /d of spin-valley coupling some inter-valley plasmon modes
xy x2−y2
scattering (K ↔ K(cid:48) scattering is obviously still present, areshiftedandgappedout,whilethe2π/3rotationsym-
but can only be seen in the d polarization as shown in metryprotectscertainlowenergymodesatM. Wespec-
z2
the Supplement). ulate this selective gapping out of collective modes could
Finally, momentum transfers q = M and Σ can con- haveconsequencesfortherealizationofmany-bodyinsta-
nect different Σ valleys, and therefore we find a gap-less bilities towards superconducting or charge density wave
linearinter-valley plasmonmodeoriginatingatthispoint phases in monolayer TMDCs.
only in the high doping case. In the low doping case, we Acknowledgments: We are grateful for useful discus-
observe a gapped (≈ 0.1eV) excitation at q = M, origi- sions with A.V. Balatsky, A. Bill, F. Guinea as well as
nating from a K ↔Σ excitations. B.Normand. S.H.wouldliketothethanktheHumboldt
While the SOC has a negligible effect on the d val- Foundationforsupport. Thisworkwassupportedbythe
z2
ley at K it splits the d /d valleys at Σ resulting European Graphene Flagship and by the Department of
xy x2−y2
in minima at comparable energies. The corresponding EnergyunderGrantNo. DE-FG02-05ER46240. Thenu-
Fermi surface for a single spin component is indicated merical computations were carried out on the University
in the inset of Fig. 3 (c). The six Σ points decompose of Southern California high performance supercomputer
into two distinct sets, Σ and Σ(cid:48). Fermi pockets within cluster and the Norddeutscher Verbund zur F¨orderung
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