Optical properties of deformed few-layer graphenes with AB stacking

043509-3 Lee et al. J. Appl. Phys. 108, 043509 2010FIG. 2. Color online The low-energy bands of few-layer graphenes nearthe K point along k̂. y Different stains are shown for a N=1, b N=2, cN=3, and d N=4.FIG. 3. Color online The same plot as Fig. 2, but shown along k̂. x Energybands crossing the Fermi level are also plotted by the dashed curves forcomparison.term. f is the Fermi–Dirac distribution function. =0.4 meV is the broadening parameter due to variousde-excitations. The subbands away from the Fermienergy are assigned to the index J=1,2,...,N,correspondingly. The velocity matrix element, h k x ,k y ,JÊ· P /m e h k x ,k y ,J, is evaluated within thegradient approximation. 29,30 It can be approximated as h k x ,k y ,JÊ · P /m e h k x ,k y ,J= a ll,lh a l h l Ê · P /m e l l,lah l a H h l,ll .k xH l,l /k x includes the intralayer and the interlayer atomicinteractions, as seen in Eq. 3. The former dominate theabsorption spectra, since they are much stronger.III. LOW-ENERGY ELECTRONIC AND OPTICALPROPERTIESFirst, a simple review is given for the low-energy-electronic structure of monolayer graphene, and is followedby a detailed discussion about N-layer AB-stackedgraphenes. Single-layer graphene possesses two isotropic linearbands black lines in Figs. 2a and 3a intersecting atthe Fermi level; therefore, it is a zero-gap system. The Fermimomentum k F is located exactly at the K point black arrow.Deformation changes the K point and the Fermi momentum.The coordinate of the former moves from2/3b,2/3 3b to 2/3b1+,/ 3b1−/65−1−/6/3 3b1+ 2 red and blue arrows. The energydispersions near the K point remain linear along ̂k ygreen and red lines in Fig. 2a, whereas transform intoparabolic ones along ̂k x with an energy gap green and redcurves in Fig. 3a. Notice that there is still a linear intersectionalong ̂k x at kF green and red dashed lines in Fig.3a. The isotropy is also destroyed because of the slightlydifferent slopes of the linear bands along ̂k x and ̂. ky As for theFermi momentum, it varies from 2/3b,2/3 3b to2/3b1+,2 cos −1 0 /2 0 / 3b1−/6. kF alonĝk y blueshifts under uniaxial tension.The interlayer atomic interactions have significant influenceon the band structures. For instance, the low-energybands of multilayer graphenes becomes anisotropic. The ABbilayergraphene exhibits two pairs of parabolic energybands. Without deformation, one pair has overlapping nearE F =0, and the other pair is located at higher energy 1 ;not shown. A semimetallic overlap appears when depictedthrough the K point along ̂k y black curves in Fig. 2b;while along ̂, k x another overlap intersecting at EF is locatednear the K point black dashed curves in Fig. 3b. The localmaxima in the former are actually the local minima along ̂. k xThis means that they are saddle points in the energy-wavevectorspace, marked by triangle heads in Fig. 2b. Thedeformation causes the band structures to be more anisotropic.Under the uniaxial tension compression, the overlappingalong ̂k y grows disappears red green curves inFig. 2b. The local extreme values are still saddle pointstriangle heads in Fig. 2b. These saddle points contributeDownloaded 07 Sep 2010 to 140.116.46.83. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

043509-4 Lee et al. J. Appl. Phys. 108, 043509 2010FIG. 4. Color online The JDOS for few-layer graphenes with a N=1, bN=2, c N=3, and d N=4.to the high density of states, resulting in the special structuresin the optical spectra. As to the overlapping along ̂, k x itis enhanced and still intersects at the Fermi level whenuniaxial compression is applied green dashed curves in Fig.3b, but tension causes it to disappear. Meanwhile, the augmentationof deformation increases the energy spacing betweenthe conduction and valence bands at the saddle points.In addition to the saddle points, the Fermi momenta play animportant role in the optical excitations as well. The k F ’s aredetermined by the intersections of energy bands and E F . Theincreasing tension enlarges the spacing between the conductionand valence bands at k F ’s red curves in Fig. 2b.While the increase in compression doesn’t open up a bandgap, it only changes the slope of the linear bands intersectingat E F green dashed curves in Fig. 3b. The Fermi momentaare expected to decide the initial threshold frequency of opticalspectra, because the vertical excitations of the least energytake place at these states.The 3- and 4-layer AB-stacked graphenes both exhibittwo pairs of conduction and valence bands near the Fermilevel. The former has one pair of linear bands and one pair oftwisted parabolic bands, where more band-edge states existblack curves in Fig. 2c. The latter has two pairs of parabolicbands, one of which shows twisting as well blackcurves in Fig. 2d. The interlayer atomic interactions leadto the different Fermi-momentum states and band-edgestates. Some of the band-edge states are saddle points andmake important contributions to the special structures of theabsorption spectra. Others relate to the local maxima orminima, causing insignificant structures. The effects ofuniaxial stress in the case of N=3 and 4 are similar to thosein bilayer graphene, such as, the increasing energy spacingbetween the conduction and valence bands at the Fermi momentaand saddle points red and green curves in Figs. 2c,2d, 3c, and 3d. These changes can be seen in opticalexcitations.The main characteristics of absorption spectra A aregoverned by the JDOS and the velocity matrix element. TheJDOS represents the total number of optical excitation channels.In the absence of stress, the JDOS of monolayergraphene increases linearly from zero with the growing Fig. 4a. The uniaxial tension compression increases decreasesthe JDOS. It is because the uniaxial tension compressionresults in the smaller larger slope of linear energybands, inducing more less electronic states and generatingmore less transition channels. As for the bilayer graphene,the parabolic energy bands exhibit saddle points triangleheads in Fig. 2b near the K point, which lead to many newexcitation channels and a prominent peak in the low-energyJDOS Fig. 4b. Under uniaxial tension, the JDOS is vanishingnear 0, and it keeps almost zero within a frequencyrange red curve in Fig. 4b. Such range is definedas the transition gap tg , the vertical excitation energy betweenthe Fermi-momentum states and the occupied or theunoccupied states, depending on which one is the smallest.That is to say, it is due to the vertical excitation channelsbetween the conduction and valence bands at the Fermi momentumred curves in Fig. 2b. On the contrary, the JDOSat =0 and 0 black and green curves in Fig. 4b exhibitsalmost zero transition gaps because the energy bandsintersecting near E F provide the low-energy excitation channelsblack curve in Fig. 2b and green dashed curves in Fig.3b.The interlayer atomic interactions in trilayer and tetralayergraphenes induce richer electronic properties, creatingmore special structures in the JDOS. The former has aprominent peak and a transition gap black curve in Fig.4c, owing to the saddle points and the Fermi momenta ofthe twisted parabolic bands, respectively black curve in Fig.2c. Both the peak frequency and transition gap are enlargedby tension and compression. Furthermore, a slightshoulder structure at 0.02 eV owes to the contributionfrom the Fermi momenta of the linear bands black curve inFig. 2c. For the tetralayer graphene, more than one peakare induced black curve in Fig. 4c. These pronouncedpeaks are related to the saddle points, and other minor peaksand shoulder structures come from the local minima ormaxima. In the presence of stress, more minor peaks andshoulder structures exist in the JDOS and the uniaxial compressioneven causes a prominent structure with twin peaksgreen curve in Fig. 4d.The optical spectra have the similar special structures asthe JDOS does. However, the velocity matrix element alsoinfluences A drastically, such as the spectrum intensity.For monolayer graphene, the linear -dependence of Arelates to that of the JDOS. Compression leads to strongerspectrum intensity than tension green and red lines in insetof Fig. 5a, as the opposite is true for the JDOS green andred lines in inset of Fig. 4a. This result clearly illustratesthat the velocity matrix element is of great importance concerningA.Downloaded 07 Sep 2010 to 140.116.46.83. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

043509-5 Lee et al. J. Appl. Phys. 108, 043509 2010FIG. 6. The strain-dependence of a the frequencies of the most prominentpeaks and b the transition gaps.FIG. 5. Color online The absorption spectra with different strains for aN=1, b N=2, c N=3, and d N=4.As for multilayer graphenes, A is affected significantlyby the interlayer atomic interactions. Spectrum intensitiesare determined by the competition between the JDOSand the velocity matrix element. In the absence of deformation,the bilayer graphene possesses a pronounced peakblack curve in Fig. 5b due to the vertical excitations fromthe saddle points of the parabolic bands black curve in Fig.2b. In addition, such a symmetric peak is divergent in thelogarithmic form at the zero broadening width. The parabolicbands intersecting at the Fermi level offer efficient verticalexcitation channels near 0 and, as a result, no transitiongap. The trilayer and the tetralayer graphenes exhibit differentabsorption frequencies and intensities black curves inFigs. 5c and 5d. Their transition gaps differ from eachother as well. In the tetralayer graphene, the most pronouncedpeak at 0.011 eV black curve in Fig. 5dassociates with the second highest JDOS peak black curvein Fig. 4d. The highest JDOS peak correlates to the shoulderstructure at 0.016 eV, mainly owing to the weakervelocity matrix element.Deformation leads to great variations in absorption spectraof multilayer graphenes. No simple relation between theuniaxial stress and spectrum intensities, as seen in the monolayergraphene, could be obtained. Deformation enhances thefrequencies and the intensities of pronounced peaks. Moreshoulder structures and minor peaks are generated because ofnew vertical excitations from the Fermi momenta and thenew local minima or maxima, respectively. Compressioneven creates a prominent structure with twin peaks in tetralayergraphene green curve in Fig. 5d. The main reason isthat the excitation energies from the two saddle points areclose but yet distinguishable in A. Besides, tension inducesa transition gap in bilayer graphene tg 0.008 eV;red curve in Fig. 5b for the vertical transitions from theFermi momenta with nonzero excitation energy. However,A remains no transition gap under compression greencurve in Fig. 5b. For trilayer and tetralayer graphenes, tgincreases in the presence of stress red and green curves inFigs. 5c and 5d.The frequency of the most prominent peak mp and thetransition gap are strongly dependent on strain. Uniaxialstress raises mp red and green curves in Figs. 5b–5dthrough the enhancement of energy spacing at the saddlepoints red and green curves in Figs. 2b–2d. The smallestvertical excitation energy will rise proportionally when thestress grows. For this reason, mp is proportional to Fig.6a. It also arises with the increase of N. Besides, the strongerdeformation enlarges the transition gaps in trilayer andtetralayer graphenes Fig. 6b. With the increasing uniaxialstress, the transition gaps become larger as a result of thebigger energy spacing between the Fermi-momentum statesand the occupied or unoccupied states red curves in Figs.2c and 2d. Similar phenomena also take place when raisingthe uniaxial tension in bilayer graphene. On the otherhand, the transition gap almost remains zero at 0. Thatthe linear bands crossing the Fermi level provides the lowenergyexcitation channels green dashed curves in Fig. 2baccounts for this result. The predicted strain-dependence ofabsorption frequency could be verified by the experimentalmeasurements on optical spectra.Under the influence of electric and magnetic fields, thefew-layer AB-stacked graphenes have been investigated.Uniaxial stress has analogous as well as diverse effects onthe electronic properties and absorption spectra. Electricfields and the stress have similar effects, such as the subbandcrossing, the changes in subband spacing, and extra band-Downloaded 07 Sep 2010 to 140.116.46.83. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

043509-6 Lee et al. J. Appl. Phys. 108, 043509 2010edge states. 21 Moreover, the low-energy energy bands becomemore anisotropic. As for absorption spectra, the mostprominent peak presents the divergent logarithmic form atthe vanishing broadening parameter. There is no special opticalselection rule. However, the magnetic fields induce theisotropic Landau levels and result in the delta-function-likespectrum peaks. Such peaks are much more eminent than thelogarithmic ones. They obey certain selection rules; furthermore,the field-dependent absorption frequencies are verycomplex. 23IV. CONCLUDING REMARKSThis work studies the electronic properties of few-layerAB-stacked graphenes under uniaxial stress by the tightbindingmodel. The gradient approximation is utilized to calculatethe absorption spectra when the graphene is deformed.The interlayer atomic interactions, the layer number, and theuniaxial stress all have great effects on the electronic structuresand optical excitations. The frequency of the mostprominent peak and transition gap strongly depend on thestrain.Monolayer graphene owns isotropic linear bands in thelow energy. The interlayer atomic interactions destroy theisotropy and induce parabolic bands in multilayer ABstackedgraphenes. The band-edge states of these parabolicbands, such as the saddle points and the local minima andmaxima, lead to special structures in absorption spectra. Inthe presence of the uniaxial stress, the parabolic bands becomemore anisotropic. Deformation enhances the overlappingbetween the valence and conduction bands, changes theenergy spacing at the band-edge states, and generates moreband-edge states. In addition, the energy spacing at thesaddle points and the Fermi momenta is predominated by thestress strength except for the latter in AB-bilayer grapheneunder compression.As for absorption spectra, the prominent peaks are inducedby the saddle points, and the shoulder structures arerelated to the local extremes or the Fermi momenta. Thetransition gap is dominated by the smallest energy spacingbetween the occupied and unoccupied states at the Fermimomenta. Under the uniaxial stress, the enlarged energyspacing at the saddle points enhances the pronounced peakfrequencies, and the one at the Fermi momenta expands thetransition gaps. In tetralayer graphene, deformation generatesmore band-edge states and thus results in the new minorpeaks and shoulder structures. Compression causes theprominent twin-peak structure. The layer number affects thefrequencies and intensities of pronounced peaks, and createsnew shoulder structures. The strain is linearly related to thefrequency of the most prominent peak. Also, it is closelyconnected with the transition gap excluding bilayer grapheneunder compression. The effects of electric fields are similarto those of the uniaxial stress, such as energy dispersions andspecial structures in A. On the contrary, magnetic fieldshave very different consequences. These predicted resultscould be demonstrated by optical measurements.ACKNOWLEDGMENTSThis work was supported by the NSC and NCTS of Taiwan,under the Grant Nos. NSC 98-2112-M-006-013-MY4,NSC 98-2811-M-006-019, and NSC 99-2811-M-110-006.1 J. S. Bunch, Y. Yaish, M. Brink, K. Bolotin, and P. L. McEuen, Nano Lett.5, 287 2005.2 C. Berger, Z. Song, T. Li, X. Li, A. Y. Ogbazghi, R. Feng, Z. Dai, A. N.Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, J. Phys. Chem.B 108, 19912 2004.3 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V.Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 6662004.4 C. P. 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