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1、 1.Introduction4.ConclusionsFirst-principles calculations and model analysis of plasmon excitations in graphenePengfei Li1,2,Xinguo Ren1,2,Lixin He1,2(1.Key Laboratory of Quantum Information,Hefei,Anhui,2300262.Synergetic Innovation Center of Quantum Information and Quantum Physics,Hefei,Anhui,23002
2、6,Email:)Bibliography In recent years the plasmon excitations in graphene and graphene-related materials have attracted considerable attention both theoretically and experimentally,due to their importance for basic physics and technological applications.Derived from the unique electronic band struct
3、ure,the plasmon excitation spectra of graphene span a wide energy range,falling into three distinct regimes.At low energies(0-2 eV)and under finite electron doping,graphene can sustain the so-called Dirac plasmons,originating from the intraband transitions of Dirac fermions in the vicinity of K poin
4、ts of the Brillouin zone(BZ).At higher energies(4-15eV),there exist intrinsic plasmons,arising from the collective excitations of electrons from to bands.The plasmons are very dispersive,starting at 4 eV.for small momentum transfer qs up to 8 eV for large qs.At even higher energies,bands start to co
5、ntribute,the mixture of the and transitions leads to yet another set of distinct plasmon peaks,usually denoted as+plasmons.As such,the rich plasmon physics in graphene renders the system distinct from normal metals,conventional 2-dimensional electron gas,and dope semiconduct-ors.The low-energy Dirac
6、 plasmon excitations are only present under finite electron or hole dopings,which can be readily achieved by chemical means or by electric gating.This type of plasmons attract enormous interest for potential technological applications due to the possi-bility of achieving both strong field localizati
7、on and low energy loss simultaneously.Furthermore,this type of plasmon is highly flexible in the sense that its frequency can be tuned from Terahertz to middle infrared by varying the doping level,and can be“engineered”by encapsulating graphene in between other two-dimensional(2D)layered materials.B
8、ecause of this unique property,graphene has been considered as a promising material for fabricating nano-plasmonic devices.The and+plasmons at higher energies are also of significant scientific interest and have been under intensive theoretical and experimental.investigations.It should be noted that
9、 these latter types of plasmons are also present in the parent material of graphene graphite.However,it turned out that the dispersion behavior of plasmonsin graphene is very different from their counterpart in graphite,especially for small qs.Consequently,the interest here is often on the dispersio
10、n behavior of the plasmon peaks as a function of the momentum transfer q,as well as the influence of the substrates and/or theinterlayer interactions inside graphene-based heterostructures or multi-layer graphene.1.Z.Yuan and S.Gao,Comp.Phys.Commun.180,466(2009)2.M.Chen,G.-C.Guo,and L.He,J.Phys.:Con
11、dens.Matter 22,445501(2010)3.P.Li,X.Liu,M.Chen,P.Lin,X.Ren,L.Lin,C.Yang,and L.He,Comput.Mater.Sci.112,503(2016)4.The ABACUS software webpage:http:/In this work,we developed a first-principles TDDFT-RPA module1 within the ABACUS software package234.Regarding the controversial nature of the dispersion
12、 behavior of plasmons,our first-principles results and model analysis indicate that a theoretically more sound dispersion relation should be at small qs.The essential physics behind this is that the plasmons in graphene arise from collective interband excitations in a region of the BZ that has a fin
13、ite energy gap,and hence the asymptotic behavior at q 0 is qualititively different from that of Dirac plasmons,which stem from collective excitations in a BZ section that has no gap.Finally,we demonstrated that,to extract accurate lifetime from the computed spectra,care must be taken to extrapolate
14、the results to the limit of 0,where is the technical broadening parameter used in dynamical response function calculations.2.Dispersion behavior 2.1 Overall featuresFig 1,The loss spectra for graphene(a),hBN b),andgraphene/hBN bilayer(c)systems for different momentumtransfer q along the M direction.
15、The Fermi level is shifted up by 0.05 Ry in graphene and graphene/hBN calculations to mimic the effect of finite dopings.The three different(Dirac,and+)plasmon modes in graphene are labeled in panel(a).The obtained loss functions of doped graphene andgraphene/hBN clearly show three distinct plasmonm
16、odes,corresponding respectively to the Dirac plasmons,plasmons,and+plasmons from low to high excitation energies.The Dirac plasmon peaks are verysharp and pronounced,while plasmon peaks are muchbroader and carry more spectral weights.The peaks of+plasmons are even broader,and multiple sub-peakstruct
17、ures within this regime are clearly visible.As men-tioned in the Introduction,the low-energy Dirac plasmonexcitations are only present for extrinsic graphene withfinite doping,but the and+are intrinsic and canbe activated at both finite and zero dopings.2.2 Dirac plasmonFig.2(a):The loss spectra cor
18、responding to Dirac plasmons along the K direction in the BZ for doped graphene(upper panel)and graphene/hBN(lower panel).(b):The Dispersion curves of the Dirac plasmons of along both K and M directions.Results for the acoustic mode along the K direction are also shown.2.3 plasmonFig.3 The Dispersio
19、n curves of the plasmons of graphene(squares)and graphene/BN(circles)as a function of q along both the M(solid symbols)and K(open symbols)directions.For graphene/BN,the dispersion of the majority peak(cf.Fig.1)is plotted.Fig.4 The model dispersion curve of plasmons with Eg,M=4 eV and =90 eV2A,in com
20、parison with the dispersion curves obtained from TDDFT-RPA calculations in this work,and theexperimental results of Liou et al and Lu et al.3.LifetimeFig.5 The extrapolated(0)FWHM of Dirac(a)and (b)plasmons along the M direction.For Dirac plasmons,results are shown for both freestanding graphene andgraphene/hBN.