DOI: 10.1002/ijch.201600122

Ab initio Modelling of Plasmons in Metal-semiconductorBilayer Transition-metal Dichalcogenide HeterostructuresHuseyin Sener Sen,[a] Lede Xian,[a] Felipe H. da Jornada,[b, c] Steven G. Louie,[b, c] and Angel Rubio*[a, d]

1. Introduction

Plasmons are collective oscillations of electrons in metalsor semiconductors. Through plasmons, it is possible toconfine and control the electromagnetic energy ona smaller scale than light in vacuum and achieve highnear-field intensities.[1] Such control over the electromag-netic fields is crucial for modern information, communi-cation, and imaging technologies.[2] Because of their elec-tronic origin, plasmon excitations depend intrinsically onthe electronic properties of their host material, which canalso depend dramatically on the dimensionality.

After the isolation of graphene in 2004,[3] the interestin the field of two-dimensional crystals has risen enor-mously. Graphene has an unconventional gapless elec-tronic structure, which hosts quasiparticle states in linear-ly dispersing cones with a vanishing electronic density ofstates at the Fermi level, which can, in turn, be under-stood in terms of a massless Dirac pseudospin Hamiltoni-an.[4] This gapless electronic structure, while fascinating,[5]

poses a challenge for the application of graphene in elec-tronic devices such as transistors. However, when gra-phene is electronically doped, an optical gap and a finitedensity of states at the Fermi energy are introduced,which leads to the emergence of a carrier plasmonbranch. These plasmons show strong spatial confinementof more than 40 times[6] the wavelength of light, but theachievable carrier concentration and the energy of theplasmon restricts the applications down to the mid-infra-red region.

To overcome the necessity of doping and energy re-striction for applications, other families of two-dimension-

al crystals have been explored. A particularly interestingclass is that of monolayer two-dimensional transition-metal dichalcogenides (TMDs).[7] The chemical formulafor TMDs is MX2, where M is the transition metal, and Xis one of the following chalcogens: S, Se, or Te. A singlesheet of MX2 is composed of an array of transition-metalatoms, each bonded to two chalcogens, in a hexagonalstructure, as displayed in Figure 1. TMDs display fascinat-ing properties, such as valley-selective optical circular di-chroism,[8] coupled valley and spin degrees of freedom,

Abstract : Two-dimensional transition-metal dichalcogenides(TMDs) have attracted enormous interest, due to the rich-ness of their optical and electronic properties. Here, we con-sider two prototypical two-dimensional TMD metal-semicon-ductor bilayer heterostructures, VSe2-MoSe2 and VSe2-WSe2,and investigate the effect of the semiconducting layer onthe plasmons supported by the metallic layer using firstprinciples time-dependent density functional theory

(TDDFT) calculations. We focus on the flat region of theplasmon dispersion, where momentum transfer is largerthan 0.05 b@1 and the interband transitions gain importance.With the addition of the semiconducting layer, we show thatthe electronic band structure undergoes significant changesclose to the Fermi level, and hybridization occurs, whichleads to strengthening of the interband transitions and a sig-nificant redshift in the plasmon energy.

Keywords: ab initio calculations · collective excitations · density functional calculations · dielectric and response functions · electron energyloss spectroscopy

[a] H. S. Sen, L. Xian, A. RubioNano-Bio Spectroscopy group and ETSF Scientific DevelopmentCentreDepartamento Fisica de MaterialesUniversidad del Pais VascoCentro de Fisica de Materiales CSIC-UPV/EHU-MPC and DIPCAv. Tolosa 72E-20018 San Sebastian (Spain)e-mail: angel.rubio@mpsd.mpg.de

[b] F. H. da Jornada, S. G. LouieDepartment of PhysicsUniversity of California at BerkeleyBerkeley California 94720 (USA)

[c] F. H. da Jornada, S. G. LouieMaterials Sciences DivisionLawrence Berkeley National LaboratoryBerkeley California 94720 (USA)

[d] A. RubioMax Planck Institute for the Structure and Dynamics of MatterLuruper Chaussee 14922761 Hamburg (Germany)

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and large exciton binding energies.[9] Moreover, depend-ing on the transition metal in the structure, these materi-als can be either metallic or semiconducting.[7,10] The elec-tronic properties of TMDs also depend dramatically onthe number layers forming the material. For instance,even though the layers are bound by weak van der Waalsforces, MoS2 has an indirect band gap in its bulk form,whereas a single layer of MoS2 has a direct gap.[11]

Indeed, semiconducting TMDs are studied quite exten-sively, both theoretically and experimentally.[10a,11,12]

There are also very novel, interesting phenomena thattake place on metallic TMDs. For instance, it was report-ed that metallic TMDs, such as NbSe2 and TaS2, havea phase diagram where the charge density wave (CDW)and superconductivity coexist at low temperatures. TheCDW phase has a tendency to open a gap in the material,whereas the superconducting phase needs a conductor.Hence, these two many-body states are in direct competi-tion, leading to a complex phase diagram.[13] Additionally,in the metallic form, TMDs have much higher carrierconcentration than doped graphene, resulting in plasmonenergies in the near-infrared region.[14] Moreover, it wasobserved experimentally that 2H-TaS2, 2H-TaSe2, and2H-NbSe2 exhibit negative plasmon dispersions, whichare qualitatively different from the plasmon dispersionfound in a homogenous electron gas (HEG).[15] Therefore,it is crucial to include the microscopic details of theatomic arrangement when computing the dynamical re-sponse of these layered metals, which can be achieved byfirst-principles methods. Indeed, previous ab initio calcu-lations on the plasmon dispersion of metallic TMDs –both on monolayer and bilayer forms – report interestingfeatures of both high-energy[16] and low-energy plas-mons.[14c,d,f,15,17] High-energy plasmons originate from theinterband transitions from p and p+s bands at zero mo-mentum (q=0), and low-energy plasmons occur due tothe intraband transitions in the metallic d band. Thesecarrier plasmons are quite interesting, as they display

a strong dispersion for wave vectors up to ~0.05 c@1, andbecome flat for larger wave vectors.

In this study, we theoretically examine the possibility oftuning these low-energy carrier plasmons by engineeringmetal-semiconductor heterostructure bilayer TMDs,namely, VSe2-MoSe2 and VSe2-WSe2, as shown inFigure 1. We perform ab initio calculations of the lossfunction, by which we can pinpoint the plasmon excita-tions, as in Eq. (1), for different values of wave vectors q,ranging from 0.07 to 0.35 c@1, within time-dependentdensity functional theory (TDDFT). We have studied therole of the semiconducting layer on the plasmon disper-sion, and observe that the semiconducting layer is respon-sible for a significant redshift in the plasmon energy ofthe metallic monolayer TMD. This finding can be appliedto tune the plasmon dispersion in metallic monolayerTMDs.

2. Theory

To identify the plasmon excitations, we look at the peaksin the loss function L(q, w) which is the negative of theimaginary part of the inverse macroscopic dielectric func-tion (L q;wð Þ ¼ @Im e@1

M q;wð ÞE C). Here, q is the momen-

tum transfer and w is the frequency. The macroscopic die-lectric function is a complex function which can be writ-ten in terms of real and imaginary parts as eM(q, w)=e1(q, w)+ ie2(q, w). Then, the loss function becomes

Lðq;wÞ ¼ e2ðq;wÞe2

1ðq;wÞ þ e23ðq;wÞ

: ð1Þ

The peaks of the loss function are identified as plasmonexcitations, and are obtained when e1(q, w) is zero ande2(q, w) is small. The macroscopic dielectric function canbe calculated from the microscopic dielectric function asfollows:

eM ¼1

e@1G¼0;G0¼0ðq;wÞ

; ð2Þ

where G and G’ are reciprocal lattice vectors, and eG,G’ isthe dielectric matrix in reciprocal space. While the dielec-tric matrix is diagonal for a homogenous electron gas, theoff-diagonals can be quite important in real systems,giving rise to what is known as local field effects. The mi-croscopic dielectric function can be calculated withTDDFT with the use of a Dyson-like equation for thelinear-response polarizability c starting from independentparticle polarizability c0 as follows:

c ¼ c0 þ c0ðuþ fXCÞc; ð3Þ

where u is the Coulomb interaction, and fXC is the ex-change correlation kernel. Here, we used the random

Figure 1. Top and side view of 2H-bilayer TMD heterostructuresystem. The introduction of an incident electromagnetic field(black arrow), perpendicularly to the plane of the system, whichcreates a plasmon (red wave) in the metallic layer is shown in theside view.

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phase approximation (RPA), in which the exchange andcorrelation effects are neglected, and a bare Coulomb in-teraction is assumed in the screening Dyson-like equationfor the response of the system; hence, fXC =0. c0 can becalculated in Fourier space with the help of the Kohn@Sham wavefunctions @KS

n;q and eigenvalues eKSn;q obtained in

the ground-state calculation as:

c0GG0 ¼

1V

Xknn0

@KSn0 ;k@qr e@iðqrþGÞ:r44 44@KS

n;k

D E@KS

n;k eiðqrþGÞ:r44 44@KSn0 ;k@qr

D EG0ðn; n0; k; qr;wÞ;

ð4Þ

in which

G0ðn; n0;k; qr;wÞ ¼

fn0k@qrð2@ fnkÞ

1

(hwþ eKSn0k@qr

@ eKSnk

0 /þ ih

@ 1

(hwþ eKSnk @ eKS

n0k@qr

0 /: ih

24 35;ð5Þ

where fnk is the occupation number of the correspondingKohn@Sham orbital and h is the broadening, whichshould be infinitesimal. Therefore, starting from ground-state Kohn@Sham wavefunctions and eigenvalues, the lossfunction and the collective electronic excitations can becalculated.

3. Computational Details

To compute the plasmon dispersion, we first obtained theground-state charge density of the heterostructures withindensity functional theory (DFT),[18] as implemented in theABINIT electronic structure code.[19] The calculationswere performed using a plane-wave basis in a supercellarrangement, where we kept a large distance of 40 c be-tween repeated heterostructures in neighboring supercellsto avoid spurious interaction. We used norm-conservingTroullier@Martins pseudopotentials, including semicorestates, and performed our calculations within the localdensity approximation (LDA).[20] The systems were re-laxed to their ground state, with a k-point mesh of 24 X24X 1, and a cutoff energy of 75 Ha, until two consecutivesteps had an energy difference of less than 10@7 Ha. Even-tually, the lattice constants, a, and the separation betweenthe layers, d, of VSe2-MoSe2 and VSe2-WSe2 bilayer het-erostructures were found to be a=3.31 c, d=2.94 c, anda=3.29 c, d=3.11 c, respectively. Next, we computedthe loss function of these heterostructures withinTDDFT,[21] making use of the computational codeYambo.[22] The calculations were performed withinRPA,[23] where the exchange correlation kernel was zero(fxc =0 in Eq. (3) above). For the evaluation of the macro-scopic dielectric function, we included 120 bands in ourcalculations, with a k-point mesh of 32X 32X 1. The dielec-tric matrix was expanded, and subsequently inverted in

a plane-wave basis set of 500 G-vectors with all local-field effects (LFE) included. To exclude artificial long-range Coulomb interaction between the periodic imagesin the direction perpendicular to the layers, we truncatedthe Coulomb potential.[24]

4. Results and Discussion

4.1 Band Structure

Before studying the plasmon excitations, we first examinehow the semiconducting layer affects the electronic bandstructure. In Figure 2a, we show the electronic band struc-tures of the VSe2 single layer with red dashes. For com-parison, we also show the electronic band structure of theVSe2-MoSe2 bilayer heterostructure with black lines. Sim-ilarly, in Figure 2b, the blue dashed lines represent theband structure of the MoSe2 single layer. We notice thatthe electronic bands of the monolayer structures can betraced in the bands of the bilayer heterostructure withslight shifts and insignificant changes. However, when weanalyzed deeply, we observed that there are significanthybridizations in the bands just below the Fermi level.Therefore, although the general shapes of the electronicbands are not changed much with respect to their single-layer forms, the combination of the two band structures isextremely important for the plasmon excitations. Specifi-cally, we observe that most of the changes in plasmonicproperties occurring due to the addition of the semicon-ducting layer, for the wave vectors studied, can be under-stood in terms of just two bands: the metallic half-occu-pied band which crosses the Fermi energy, and the firstfully occupied band that is just below in energy, andwhich we refer to as “the first valence band” of themetal-semiconducting heterostructure. These two bandsare highlighted in Figure 2c, where the inset figures fur-ther show how the partial charge density due to each par-ticular Kohn@Sham state is distributed in real space foreach high-symmetry point. From the figure, it becomesclear that the metallic band (band 1 in Figure 2c) origi-nates from the metallic layer, while the first valance band(band 2) is a hybrid band that has varying contributionsfrom each layer at different high-symmetry points. At theG point, 42 % of the wavefunction from band 2 originatesfrom the semiconducting layer; this contribution decreas-es to 26% at the M point. As we move in the Brillouinzone away from the M point along the T line, the wave-function becomes more localized on the semiconductinglayer until it becomes 95% concentrated at the semicon-ducting layer at point K. It is not surprising to find contri-butions to the hybridization of band 2 from the metalliclayer around the G and M points, since there are alreadytraces of the electronic bands of this layer in the proximi-ty, as in Figure 2a. However, the hybridization around theK valley (blue area in Figure 2c) is not an obvious phe-nomenon at first, and it plays a very important role for

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the plasmon dispersion. Since band 2 has an energy dis-persion almost parallel to that of band 1, and since thereis a sizable hybridization, there is a significant spatialoverlap between the states in these two bands, so we cananticipate an important contribution to the polarizabilitycoming from this transition. Due to the low energy of thistransition, and a van Hove singularity of the joint densityof states for transitions between band 2 to band 1, weexpect this transition to explain most changes in the plas-mon dispersion as we go from the monolayer structure tothe heterostructure.

4.2 Plasmons

If we take the intraband contribution out of the sum inEq. (4), the remaining part is the interband contribution,

so we can rewrite the independent particle polarizabilityas: c0 ¼ cintra þ cinter. Since the metallic band is very simi-lar, both in the VSe2 single layer and the VSe2-MoSe2 bi-layer heterostructure, the main difference in the c0 ofthese systems is due to cintra, especially transitions fromtransition band 2 to transition band 1. In Figure 3, we

compare the loss functions of the VSe2-MoSe2 bilayer het-erostructure and the VSe2 single layer for q=0.07 c@1

along the G–M direction, with only intraband transitionsincluded in the calculation. Since there is no interbandtransition contribution to this calculation, and the shapesof the metallic bands of the two systems are very similar,we get almost identical loss functions. As we mentionedearlier, after the addition of the semiconducting layer,band 2 hybridizes, which facilitates the overlap betweenband 2 and band 1. Also, band 1 and band 2 have almostthe same shape at every point around the K valley, result-ing in the enhancement of the excitation of the electronsfrom band 2 to band 1 for a small energy window around0.62 eV for the VSe2-MoSe2 and 0.65 eV for the VSe2-WSe2 bilayer heterostructures, which are the averageenergy differences between band 2 and band 1 aroundthe K valley, respectively. Therefore, the value of the frac-tion:

ðfn0k@qr@ fnkÞ

(hwþ eKSn0k@qr

@ eKSnk

0 /þ ih

ð6Þ

in Eq. (4), together with the spatial overlap between band2 and band 1, increases significantly. In fact, this featureappears as a shoulder in the loss functions of both theVSe2-MoSe2 and the VSe2-WSe2 bilayer heterostructures,as shown in Figure 4a and Figure 4b with green arrows,respectively. Notably, there is no shoulder in the lossfunction of the metallic single layer. We assign thisshoulder to the interband transitions from band 2 to band1, because when this transition is excluded from the po-

Figure 2. Electronic band structures of the: a) VSe2-MoSe2 bilayerheterostructure and VSe2 single layer; and the b) VSe2-MoSe2 bilay-er heterostructure and MoSe2 single layer. c) The dispersion of theconduction (band 1) and the first valance bands (band 2) of theVSe2-MoSe2 bilayer heterostructure, with partial charge densitiesfor high-symmetry k points as an inset. The top layer in the insetrepresents the semiconducting layer, while the bottom layer repre-sents the metallic layer.

Figure 3. Calculated loss function of the VSe2-MoSe2 bilayer heter-ostructure and the VSe2 single layer, excluding the interband transi-tions for q = 0.07 a@1 along the G–M direction.

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larization matrix, the feature disappears, as shown in Fig-ure 4c (compare black and red lines). It is known that in-terband transitions cause a redshift in plasmon dispersionin TMDs,[14d] so a further redshift of 188 meV of the plas-mon peak in the heterostructure with respect to that ofthe monolayer, as shown in Figure 4a, upon the inclusionof the semiconducting layer is not surprising. Similarly,the energy of the plasmon is blueshifted by a largeamount (75 meV) when we block transitions from band 2to band 1, while it remains redshifted with respect to themetallic single-layer case. This means that the extra polar-izability due to transitions from band 2 to band 1 is themost important mechanism behind the change in the plas-mon in the heterostructure, even though the extra transi-tions from all other bands still slightly contribute to theshift of the plasmon energy. To validate this interpreta-tion and rule out what could be an important polarizabili-ty channel, even for the monolayer structure, we per-formed another calculation, where we did not include the

transition from the first valance band to the metallic bandof the VSe2 single layer in the polarization matrix (com-pare blue and green lines). In this case, we do not seea significant shift in the plasmon energy, as in the bilayercase. Moreover, since the polarizability is zero whenthere is no spatial overlap between different transitions, itis safe to conclude that the newly introduced hybridizedpart of band 2 around the K valley is responsible for theobserved shift in the plasmon energy. The peak in theloss function is at an energy of 0.45 eV for VSe2-MoSe2

and at 0.47 eV for VSe2-WSe2 for q=0.07 c@1 along theG–M direction. These peaks are plasmon resonances thatderive from the intraband transitions within the metallicband, where the real part of the dielectric function iszero. In Figure 4d, we show the plasmon dispersions ofthe VSe2-MoSe2 bilayer heterostructure, with and withoutthe electronic transitions from band 2 to band 1, as wellas the plasmon dispersion for a single-layer VSe2. Fromthe figure, it is clear that the electronic transitions from

Figure 4. Comparison between the loss functions of: a) the VSe2-MoSe2 bilayer heterostructure and the VSe2 single layer (rescaled byV 0.36); b) the VSe2-WSe2 bilayer heterostructure and the VSe2 single layer (rescaled by V 0.43), with all transitions included; and c) theVSe2-MoSe2 bilayer heterostructure and the VSe2 single layer (rescaled by V 0.36) with full transitions, the VSe2-MoSe2 bilayer heterostructureexcluding the interband transitions from band 2 to band 1, and the VSe2 single layer (rescaled by V 0.36) excluding the interband transi-tions from the first valance band to the conduction band for q = 0.07 a@1 along the G–M direction. d) Dispersion of the plasmon energy asa function of momentum transfer along the G–M direction for the systems in c), except for the VSe2 single layer, excluding the interbandtransitions from the first valance band to the conduction band.

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band 2 to band 1 explain most differences between theplasmon dispersion from the single-layer case and theheterostructure, especially for larger wave vectors. The in-clusion of the semiconducting layer decreases the plas-mon energy significantly, by about 0.2 eV, whereas theslope of the dispersion is not affected much.

5. Conclusion

We investigated the effect of the semiconducting layer onthe plasmon in a metal-semiconductor TMD heterostruc-ture. Although the electronic band structure is very simi-lar to the sum of the band structure of two separatedlayers, with the addition of a semiconducting layer, thebands just below the Fermi level, which are close inenergy to the bands of an isolated semiconducting TMDmonolayer, get considerably hybridized in the hetero-structure. Together with the low energy required to pro-mote an electron from this band to an unoccupied band,this band ends up contributing to significant changes inthe polarizability as the metallic monolayer TMD isbrought to the heterostructure environment. This en-hancement is so important that it can be identified in theloss function as a shoulder and at high momentum trans-fers, and it is responsible for almost all the effects of thesemiconducting layer on the plasmon. Due to this transi-tion, the energy of the plasmon is decreased significantly,by roughly 75 meV for q=0.07 c@1 along the G–M direc-tion. Our work shows that metal-semiconductor hetero-structures can be interesting vehicles to tune the plasmonenergy in monolayer TMD metals.

Acknowledgements

We acknowledge financial support from the EuropeanResearch Council (ERC-2015-AdG-694097), GruposConsolidates (IT578-13), Spanish grant (FIS2013-46159-C3-1-P), AFOSR Grant No. FA2386-15-1-006, AOARD144088, and COST Action MP1306 (EUSpec). StevenG. Louie and Felipe H. da Jornada acknowledge supportby National Science Foundation EFRI Program GrantNo. EFMA-1542741.

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Received: September 21, 2016Published online: January 24, 2017

Isr. J. Chem. 2017, 57, 540 – 546 T 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.ijc.wiley-vch.de 546

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