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Resonance Raman spectroscopy in twisted bilayer graphene
Solid State Communications 175-176 (2013) 13–17
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Resonance Raman spectroscopy in twisted bilayer graphene A. Righi a, P. Venezuela b, H. Chacham a, S.D. Costa a, C. Fantini a, R.S. Ruoff c, L. Colombo d, W.S. Bacsa e, M.A. Pimenta a,n a
Departamento de Física, Universidade Federal de Minas Gerais, 30123-970 Belo Horizonte, Brazil Instituto de Física, Universidade Federal Fluminense, Niteroi, Brazil c Department of Mechanical Engineering, Texas Materials Institute, The University of Texas at Austin, 1 University Station C2200, Austin, TX 78712-0292, United States d Texas Instruments Incorporated 13121 TI Blvd, MS-365 Dallas, TX 75243, United States e CEMES/CNRS, University of Toulouse, 29 rue Jeanne Marvig, 31055 Toulouse, France b
art ic l e i nf o
a b s t r a c t
Article history: Received 10 May 2013 Accepted 28 May 2013 by A.K. Sood Available online 5 June 2013
In this work we study the Raman spectra of twisted bilayer graphene samples, with different twisting angles, by changing the incident laser energy between 2.54 and 4.14 eV. The spectra exhibit a number of extra peaks, classified in different families, each one associated with bilayer graphenes with different twisting rotational angles. We theoretically analyze the laser energy dependence of these extra peaks considering a set of discrete wavevectors within the interior of the Brillouin zone of graphene, which activate special double-resonance Raman processes. Our result show a nice qualitative agreement between the experimental and simulated spectra, demonstrating that these extra peaks are indeed ascribed to an umklapp double-resonance process in graphene systems. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Twisted bilayer graphene D. Umklapp double-resonance E. Resonance Raman scattering
1. Introduction Graphene materials have attracted a lot of attention because of their potential use in technological applications, specially in electronics [1]. An increasing number of studies have dedicated to bilayer graphenes where the two layers are rotated by an arbitrary twisting angle, and it has been demonstrated that the optical and electronic properties of twisted bilayer graphene (TBG) are strongly dependent on the twisting angle [2,3]. Recent studies have shown that Raman spectroscopy is an useful tool to investigate TBG, both from the resonance behavior of the Raman G-band in graphene [4,5] and also from the appearance of new Raman peaks [6–8], which have been ascribed to the umklapp double resonance Raman process [7]. In this work, we will present a resonance Raman study of the extra new peaks, using many different laser lines, and the results are compared with the theoretical simulation of the double resonance process considering only the wavevectors of the Moiré superlattice of TBG. Raman spectroscopy has proven to be an extremely useful technique to characterize graphene, since it furnishes a rich variety of information, such as disorder, defects atomic edge structure, stacking order, number of layers, strain and electric charges [9–17]. In resonant Raman investigations, where the energy of the exciting laser can be tuned, we can also obtain
n
Corresponding author. Tel.: +55 31 34096622; fax: +55 31 34995600. E-mail address: mpimenta@fisica.ufmg.br (M.A. Pimenta).
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.05.015
further information about the electronic structure and the dispersion of phonons near the Dirac point [9,18]. It has been shown recently that Raman spectroscopy can also characterize twisted bilayer graphene samples. Hanever et al. [4] and Kim et al. [5] have reported that the G-band can be significantly enhanced for TBG with specific twisting angle, and the enhancement occurs when the incident laser energy is in resonance with van Hove singularities in the joint density of states (JDOS) that appear in the crossing region of the two Dirac cones of the top and bottom layers. In measurements on monolayer graphene folded back upon itself, Gupta et al. [6] reported the presence of a new and nondispersive Raman peak around 1380 cm−1 and interpreted the result considering that the static interlayer perturbation activates finite-wavevector double-resonant Raman scattering. A similar result was reported by Carozo et al. [8]. In a previous work [7], we have observed the appearance of the new extra peaks in TBG grown by CVD in a copper foil. The extra peaks were grouped in families of peaks, each family being associated with bilayer regions with different twisting rotational angles. We ascribed these new peaks to phonon modes within the interior of the graphene Brillouin zone that become Raman active through an umklapp double resonance (U-DR) Raman process (intra and inter-valley) associated with reciprocal lattice vectors of the Moiré pattern supercell. In this work, we present a resonance Raman study of these extra Raman peaks in TBG, obtained using many different laser
14
A. Righi et al. / Solid State Communications 175-176 (2013) 13–17
lines in the energy range 2.54–4.14 eV. We will show that these peaks are non-dispersive, and that their intensities increase and decrease when the laser energy is changed. The experimental results are compared with theoretical simulations of the double resonance Raman spectra involving only wavevectors of the Moire superlattice, supporting thus the interpretation that these extra peaks are ascribed to an umklapp double-resonance process in graphene systems.
2. Experimental details Graphene films used in this work were grown by chemical vapor deposition (CVD) on a copper-foil enclosure at high temperature (1035 1C). The sample showed a high density of twisted bilayer graphene (TBG) regions, each one with a different twisting angle between the top and bottom layers [7,19]. The Raman measurements were performed in the as-grown sample, using a triple monochromator spectrometer (JY T64000) and a DILOR XY UV spectrometer and UV microscope (objective 40 UV grade A) with a back illuminated and cooled CCD detector. We have used the 3.2 eV (325 nm) and 2.81 eV (441.8 nm) lines of a He–Cd laser, the UV lines at 3.40 eV (364.0 nm), 3.53 eV (351.1 nm), 3.69 eV (335.8 nm) and 4.13 eV (300.3 nm) of a 20 W Argon laser, the 2.54 eV (488 nm), 2.60 eV (476.5 nm) and 2.67 eV (465 nm) lines of an Ar–Kr laser and the 3.28 eV (378 nm) and 3.04 eV (408 nm) lines of a diode solid-state laser.
3. Results and discussion 3.1. New peaks activated by the Moiré patterns Fig. 1 shows the Raman spectra obtained in four different regions of the twisted bilayer graphene sample, with different twist angle, using the 3.82 eV laser line [9,18]. The most important feature in this figure is the so-called G band, around 1580 cm−1, which is the only first-order Raman mode in graphene, and is
Fig. 1. Raman spectra obtained in four regions of the twisted bilayer sample, each one with a different twisting angle. The spectra were recorded with the 3.82 eV laser line.
related to the vibration of the sublattice A against the sublattice B. Notice that the intensity of the G-band is strongly dependent on the twist angle, and this phenomenon is ascribed to resonance of the incident laser with van Hove singularities in their electronic density of states, whose separation depend on the twist angle [4,5]. In addition to the G band, the spectra also exhibit a number of sharp extra peaks in the spectral regions 1250–1450 cm−1 and 1590–1620 cm−1. The D-band recorded with the 3.81 eV laser line should appear as a broad band around 1420 cm−1 [7], and its absence in the spectra can be ascribed to the high crystalline quality of the sample in these regions. The other important feature in Fig. 1 is the so-called 2D band, in the range 2800–2900 cm−1, that involve two phonons within the interior of the Brillouin that satisfy the double and triple resonance Raman processes [20].
3.2. Resonance Raman behavior of the Moiré peaks An important characteristic of the D and 2D bands in graphene systems is the fact that they are dispersive, that is, their positions change when the laser energy is varied. In order to check the laser energy dependence of the extra peaks, we measured the same region of the TBG using different laser lines in the range 2.54– 4.14 eV. Fig. 2 shows the Raman spectra in the 1200–1700 cm−1 range, obtained with many different laser lines, where we can observe the peak around 1380 cm−1 associated with the TBG. Notice in Fig. 2 that the position of the extra peak does not depend significantly on the laser energy ðElaser Þ, whereas its intensity is strongly dependent on Elaser , the maximum intensity occurring around 3.04 eV. In order to understand the resonance behavior shown in Fig. 2, we need to consider the double resonance process involving the Bravais lattice superstructures of exact Moire patterns of two rotated graphene sheets. These Moiré patterns give rise to the folding of the graphene Brillouin zone (BZ), and, therefore, a number of wavevectors within the BZ of graphene are folded into the center of the new BZ of the Moiré pattern superstructure. Fig. 3 shows the direct lattice and the reciprocal space of a Moiré pattern generated by two layers of graphene rotated by a given angle with respect to each other. Each point within the interior of the Brillouin zone (BZ) of graphene corresponds to an allowed wavevector of the reciprocal lattice of the superstructure. The
Fig. 2. Raman spectra in the 1200–1700 cm−1 range, obtained with different laser lines, showing the resonance behavior of the sharp extra peak around 1380 cm−1. The peaks marked with asterisks are broadened since the laser lines of the diode laser were very broad.
A. Righi et al. / Solid State Communications 175-176 (2013) 13–17
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Fig. 3. (a) Moiré pattern in two twisted graphene layers and (b) wavevectors of the Moiré pattern superlattice and two possible DR Raman process (intervalley and intravalley).
Fig. 4. Simulated umklapp double-resonance Raman spectra, IðωÞ, for two possible twisting angles between graphene layers.
double resonance Raman process involving one phonon and one wavevector of the Moiré superlattice is also represented in Fig. 3. Normally, a defect is needed for momentum conservation in the double resonance Raman process that gives rise to the D and D′ bands in graphene materials [18]. However, in the presence of Moiré patterns, momentum conservation is provided by one wavevector of the Moiré superlattice, and we can assign this mechanism to an umklapp double resonance (UDR) Raman process. Given the reciprocal lattice vectors GB of a Moire pattern, we can simulate the umklapp double-resonance Raman spectrum IðωÞ. Specifically, the double-resonance Raman spectrum IðωÞ is calculated following the methodology of Ref. [20]. However, here instead of summing up in the whole BZ we consider only the lattice vectors GB of a Moire pattern. In this way our calculation model the twisted bilayer graphene as a monolayer of graphene interacting with a network of defects corresponding to the lattice vectors GB . The phonon dispersion is obtained from ab-initio
density functional theory calculations with many-body corrections as in Refs. [21,20]. We only consider the LA, TO and the LO branches. For the inter-valley double-resonant processes, we consider the LA and the TO branches, which are allowed by symmetry. For such processes, involving a photon energy Elaser, maximum scattering intensity occurs for phonon wavevectors near a specific one, qinter , which has the minimum possible modulus. This is schematically shown in Fig. 3. For the LO branch, we will consider intra-valley double-resonant processes. In this case, maximum scattering intensity occurs at the phonon wavevectors with modulus qintra, as also shown in Fig. 3. Fig. 4 shows the simulated umklapp double-resonance Raman spectra, IðωÞ, for two different twisting angles (7.341 and 13.17 1), and for different incident laser energies. Notice that the intensity of the peak around 1380 cm−1 is small for small values of Elaser . However, it increases with increasing Elaser , reaches a maximum and then decreases for larger values of Elaser . In agreement with the experimental results, the Raman peaks activated by the
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A. Righi et al. / Solid State Communications 175-176 (2013) 13–17
umklapp double resonance process are non-dispersive, contrarily to the case of the D band, and only their intensities depend resonantly on the incident laser energy 3.3. Raman excitation profile of the Moiré peaks
IM/IG
Fig. 5 shows the Raman excitation profile (REP) of the intensity of extra peak around 1380 cm−1, normalized by the intensity of the G-band. The black squares correspond to the experimental results presented in Fig. 2 and the open and black circles correspond to the data of the simulated spectra for the TBG superlattice shown in Fig. 4. Notice that there is a nice qualitative agreement between the experimental and simulated results, proving that these extra peaks are indeed ascribed to an umklapp double-resonance process. The full width at half maximum (FWHM) of the experimental REP is about 0.7 eV, which is slightly larger than the REP of the simulated spectra. Let us now discuss the physical origin of the wide REP observed for the intensity of the extra peak, shown in Fig. 5, and reproduced in our quantum mechanical calculations. As we shall see, there is a remarkably simple physical interpretation that explains (i) the origin of the laser energy resonance, (ii) the energy position of the resonance peak, and (iii) the width of the Raman excitation profile (REP), which is very large in comparison with the small widths of the Raman peaks. As discussed in a previous work [7], our theoretical interpretation of the new Raman peaks in Fig. 1 considers a periodical perturbation ΔV to graphene originated from the twisted-doublelayer Moiré pattern. The perturbation allows for elastic scattering from electron states ψðkÞ to ψðk′Þ, described by matrix elements M kk′ ¼ 〈ψðk′ÞjΔVjψðkÞ〉. This leads to a double resonance (DR) Raman scattering process that is similar, in its physical origin, to that of the D band, where the electron is (i) optically excited, (ii) inelastically scattered by a phonon of wavevector q, (iii) backscattered by the elastic scattering process with wavevector −q, and (iv) radiatively de-excited. In the cases of the D band, the local perturbation ΔV, which is due to defects, leads to non-null values of M kk′ in a continuum range of q ¼ k−k′. This leads to the dispersive behavior of the D band observed experimentally [9,18]. In contrast, the periodic Moiré perturbation ΔV leads to allowed scattering (non-null M kk′ ) only for a discrete set of k′−k ¼ G, where G is the set of the reciprocal lattice vectors of
the Moiré pattern. This process can therefore be described as an umklapp DR process for the Moiré lattice. From momentum conservation we have q ¼ G, and therefore the only relevant phonon scattering events in such umklapp-Moiré double resonance process are those for phonons with wavevectors G. As a consequence, instead of a wide and dispersive Raman resonance as that of the D band, the umklapp Moiré resonances will be characterized by several sharp, non-dispersive resonances found in the same energy region as that of the D and D′ bands. Let us now assume, for simplicity, that the periodic perturbative potential of the Moire lattice can be approximated by a periodic array of localized perturbations, each perturbation being akin to the defect perturbations that leads to the D band. Within this approximation, the umklapp Moiré resonances become, essentially, a “filtered” D band. That is, each resonance is placed at the energy of a phonon with wavevectors G, with its intensity given by the intensity of the D band at that same energy. Therefore, within this approximation, our analysis of the laser energy dependence of the intensity of a given Moiré resonance peak at phonon frequency ω, I M ðω; Elaser Þ, can be mapped into the laser energy dependence of the intensity I D ðω; Elaser Þ of the D band at the same frequency ω. Both experiments [9,18] and calculations [20] show that the peak position ω0 of the D band, as a function of the laser energy Elaser (that is, the value of ω for the maximum of I D ðω; Elaser Þ at a given Elaser) can be reasonably approximated by a linear relation ω0 ¼ a þ bElaser . From Ref. [18], we can estimate the values a ¼ 1237 cm−1 and b ¼ 45 cm−1 =eV. The same experiments and calculations indicate that the width of the D band is approximately constant with Elaser. Such phenomenology for the D band, together with the proposed ID −IM mapping, lead us to propose the following ansatz for the Moiré peak intensity I M ðω0 ; Elaser Þ ¼ f ðω0 −a−bElaser ÞgðElaser Þ
ð1Þ
where ω0 is frequency of the Moiré peak, Elaser is the laser energy, f (x) is a function with a maximum at x ¼0 with value f ð0Þ ¼ 1, and gðElaser Þ is the Moiré peak intensity for a given laser energy. Now, considering that g is a slowly varying function (as compared to f), the above ansatz leads to two important results. The first result is that, due to the specific form of the argument of f, the value Epeak of the laser energy at the peak in I M ðω0 ; Elaser Þ is related to ω0 through ω0 ¼ a þ bEpeak . This theoretical result is consistent with our experimental results: using the experimental value of Epeak in Fig. 5, Epeak ¼ 3.09 eV, and the a and b values mentioned in the preceding paragraph, we obtain ω0 ¼ 1376 cm−1 , which near the observed phonon frequency for this peak, ω0 ¼ 1380 cm−1 . The second result obtained from the ansatz is that (again from the argument of f in the ansatz) the peak width ΔE in I M ðω0 ; Elaser Þ for a given ω0 is related with the width Δω of the D band through Δω ¼ bΔE. Again, this theoretical result is consistent with our experiments: using the experimental value of ΔE in Fig. 5, ΔE ¼0.7 eV, and the b value mentioned in the preceding paragraph, we obtain Δω ¼ 31 cm−1 , which is nice agreement with the D bandwidth.
4. Conclusion
2.6
2.8
3.0
3.2 3.4 Laser energy (eV)
3.6
3.8
Fig. 5. Laser energy dependence of the relative intensity of the extra M band with respect to the G-band intensity. The black squares correspond to the experimental results presented in Fig. 2 and the open and black circles correspond to the data of the simulated spectra shown in Fig. 4.
In summary, we have studied the Raman spectra of twisted bilayer graphene samples, with different twisting angles, by changing the incident laser energy between 2.54 and 4.14 eV. The spectra exhibit a number of extra and sharp Raman peaks, which are grouped in families of peaks, each one corresponding to a different twisting angle. From the resonance Raman measurements, we were able to determine the Raman excitation profile of these extra peaks.
A. Righi et al. / Solid State Communications 175-176 (2013) 13–17
We theoretically analyze the laser energy dependence of these extra peaks considering a set of discrete wavevectors within the interior of the Brillouin zone of graphene, which activate special selective double-resonance Raman modes. We show that there is a nice qualitative agreement of the experimental and simulated spectra, showing that these extra peaks are indeed ascribed to an umklapp double-resonance process in graphene systems. Acknowledgment This work was supported by the Instituto Nacional de Ciencia e Tecnologia (INCT) em Nanomateriais de Carbono - MCT, CNPq, CAPES and FAPEMIG (Brazil). References [1] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [2] A. Luican, G. Li, A. Reina, J. Kong, R.R. Nair, K.S. Novoselov, A.K. Geim, E. Y. Andrei, Phys. Rev. Lett. 106 (2011) 126802. [3] A.H. MacDonald, R. Bistritzer, Nature 474 (2011) 453. [4] R.W. Havener, H. Zhuang, L. Brown, R.G. Hennig, J. Park, ACS Nano 12 (2012) 3162–3167. [5] K. Kim, S. Coh, L.Z. Tan, W. Reagan, J.M. Yuk, E. Chaterjee, M.F. Crommie, M.L. Cohen, S.G. Louie, A. Zettl. Phys Rev. Lett. 108 (2012) 246103.
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[6] A.K. Gupta, Y.J. Tang, V.H. Crespi, et al., Source: Phys. Rev. B 82 (2010) 241406. [7] A. Righi, S.D. Costa, H. Chacham, C. Fantini, P. Venezuela, C. Magnuson, L. Colombo, W.S. Bacsa, R.S. Ruoff, M.A. Pimenta, Phys. Rev. B 84 (2011) 241409. [8] V. Carozo, et al., Nano Lett. 11 (2011) 4527–4534. [9] M.A. Pimenta, G. Dresselhaus, M.S. Dresselhaus, L.G. Cancado, A. Jorio and, R. Saito, Phys. Chem. Chem. Phys. 9 (2007) 1276. [10] L.G. Cancado, M.A. Pimenta, B.R.A. Neves, M.S.S. Dantas, A. Jorio, Phys. Rev. Lett. 93 (2004) 247401. [11] L.G. Canado, K. Takai, T. Enoki, M. Endo, Y.A. Kim, H. Mizusaki, N.L. Speziali, A. Jorio, M.A. Pimenta, Carbon 46 (2008) 272. [12] A.C. Ferrari, J.C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K.S. Novoselov, S. Roth, A.K. Geim, Phys. Rev. Lett. 97 (2006) 187401. [13] T.M.G. Mohiuddin, A. Lombardo, R.R. Nair, A. Bonetti, G. Savini, R. Jalil, N. Bonini, D.M. Basko, C. Galiotis, N. Marzari, K.S. Novoselov, A.K. Geim, A. C. Ferrari, Phys. Rev. B 79 (2009) 205433. [14] M. Huang, H. Yan, C. Chen, D. Song, T.F. Heinz, J. Hone, Proc. Natl. Acad. Sci. 106 (2009) 7304. [15] S. Pisana, M. Lazzeri, C. Casiraghi, K.S. Novoselov, A.K. Geim, A.C. Ferrari, F. Mauri, Nature Mater. 6 (2007) 198. [16] J. Yan, Y. Zhang, P. Kim, A. Pinczuk, Phys. Rev. Lett. 98 (2007) 166802. [17] A. Das, S. Pisana, B. Chakraborty, S. Piscanec, S.K. Saha, U.V. Waghmare, K.S. Novoselov, H.R. Krishnamurthy, A.K. Geim, A.C. Ferrari, A.K. Sood, Nature Nanotech. 3 (2008) 210. [18] L.M. Malard, M.A. Pimenta, G. Dresselhaus, M.S. Dresselhaus, Phys. Rep. 473 (2009) 51. [19] X. Li, C.W. Magnuson, A. Venugopal, R.M. Tromp, J.B. Hannon, E.M. Vogel, L. Colombo, R.S. Rouff, J. Am. Chem. Soc. 133 (2010) 2816. [20] P. Venezuela, M. Lazzeri, F. Mauri, Phys. Rev. B 84 (2011) 035433. [21] M. Lazzeri, C. Attaccalite, L. Wirtz, F. Mauri, Phys. Rev. B (2008) 081406.
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Resonance Raman spectroscopy in twisted bilayer graphene A. Righi a, P. Venezuela b, H. Chacham a, S.D. Costa a, C. Fantini a, R.S. Ruoff c, L. Colombo d, W.S. Bacsa e, M.A. Pimenta a,n a
Departamento de Física, Universidade Federal de Minas Gerais, 30123-970 Belo Horizonte, Brazil Instituto de Física, Universidade Federal Fluminense, Niteroi, Brazil c Department of Mechanical Engineering, Texas Materials Institute, The University of Texas at Austin, 1 University Station C2200, Austin, TX 78712-0292, United States d Texas Instruments Incorporated 13121 TI Blvd, MS-365 Dallas, TX 75243, United States e CEMES/CNRS, University of Toulouse, 29 rue Jeanne Marvig, 31055 Toulouse, France b
art ic l e i nf o
a b s t r a c t
Article history: Received 10 May 2013 Accepted 28 May 2013 by A.K. Sood Available online 5 June 2013
In this work we study the Raman spectra of twisted bilayer graphene samples, with different twisting angles, by changing the incident laser energy between 2.54 and 4.14 eV. The spectra exhibit a number of extra peaks, classified in different families, each one associated with bilayer graphenes with different twisting rotational angles. We theoretically analyze the laser energy dependence of these extra peaks considering a set of discrete wavevectors within the interior of the Brillouin zone of graphene, which activate special double-resonance Raman processes. Our result show a nice qualitative agreement between the experimental and simulated spectra, demonstrating that these extra peaks are indeed ascribed to an umklapp double-resonance process in graphene systems. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Twisted bilayer graphene D. Umklapp double-resonance E. Resonance Raman scattering
1. Introduction Graphene materials have attracted a lot of attention because of their potential use in technological applications, specially in electronics [1]. An increasing number of studies have dedicated to bilayer graphenes where the two layers are rotated by an arbitrary twisting angle, and it has been demonstrated that the optical and electronic properties of twisted bilayer graphene (TBG) are strongly dependent on the twisting angle [2,3]. Recent studies have shown that Raman spectroscopy is an useful tool to investigate TBG, both from the resonance behavior of the Raman G-band in graphene [4,5] and also from the appearance of new Raman peaks [6–8], which have been ascribed to the umklapp double resonance Raman process [7]. In this work, we will present a resonance Raman study of the extra new peaks, using many different laser lines, and the results are compared with the theoretical simulation of the double resonance process considering only the wavevectors of the Moiré superlattice of TBG. Raman spectroscopy has proven to be an extremely useful technique to characterize graphene, since it furnishes a rich variety of information, such as disorder, defects atomic edge structure, stacking order, number of layers, strain and electric charges [9–17]. In resonant Raman investigations, where the energy of the exciting laser can be tuned, we can also obtain
n
Corresponding author. Tel.: +55 31 34096622; fax: +55 31 34995600. E-mail address: mpimenta@fisica.ufmg.br (M.A. Pimenta).
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.05.015
further information about the electronic structure and the dispersion of phonons near the Dirac point [9,18]. It has been shown recently that Raman spectroscopy can also characterize twisted bilayer graphene samples. Hanever et al. [4] and Kim et al. [5] have reported that the G-band can be significantly enhanced for TBG with specific twisting angle, and the enhancement occurs when the incident laser energy is in resonance with van Hove singularities in the joint density of states (JDOS) that appear in the crossing region of the two Dirac cones of the top and bottom layers. In measurements on monolayer graphene folded back upon itself, Gupta et al. [6] reported the presence of a new and nondispersive Raman peak around 1380 cm−1 and interpreted the result considering that the static interlayer perturbation activates finite-wavevector double-resonant Raman scattering. A similar result was reported by Carozo et al. [8]. In a previous work [7], we have observed the appearance of the new extra peaks in TBG grown by CVD in a copper foil. The extra peaks were grouped in families of peaks, each family being associated with bilayer regions with different twisting rotational angles. We ascribed these new peaks to phonon modes within the interior of the graphene Brillouin zone that become Raman active through an umklapp double resonance (U-DR) Raman process (intra and inter-valley) associated with reciprocal lattice vectors of the Moiré pattern supercell. In this work, we present a resonance Raman study of these extra Raman peaks in TBG, obtained using many different laser
14
A. Righi et al. / Solid State Communications 175-176 (2013) 13–17
lines in the energy range 2.54–4.14 eV. We will show that these peaks are non-dispersive, and that their intensities increase and decrease when the laser energy is changed. The experimental results are compared with theoretical simulations of the double resonance Raman spectra involving only wavevectors of the Moire superlattice, supporting thus the interpretation that these extra peaks are ascribed to an umklapp double-resonance process in graphene systems.
2. Experimental details Graphene films used in this work were grown by chemical vapor deposition (CVD) on a copper-foil enclosure at high temperature (1035 1C). The sample showed a high density of twisted bilayer graphene (TBG) regions, each one with a different twisting angle between the top and bottom layers [7,19]. The Raman measurements were performed in the as-grown sample, using a triple monochromator spectrometer (JY T64000) and a DILOR XY UV spectrometer and UV microscope (objective 40 UV grade A) with a back illuminated and cooled CCD detector. We have used the 3.2 eV (325 nm) and 2.81 eV (441.8 nm) lines of a He–Cd laser, the UV lines at 3.40 eV (364.0 nm), 3.53 eV (351.1 nm), 3.69 eV (335.8 nm) and 4.13 eV (300.3 nm) of a 20 W Argon laser, the 2.54 eV (488 nm), 2.60 eV (476.5 nm) and 2.67 eV (465 nm) lines of an Ar–Kr laser and the 3.28 eV (378 nm) and 3.04 eV (408 nm) lines of a diode solid-state laser.
3. Results and discussion 3.1. New peaks activated by the Moiré patterns Fig. 1 shows the Raman spectra obtained in four different regions of the twisted bilayer graphene sample, with different twist angle, using the 3.82 eV laser line [9,18]. The most important feature in this figure is the so-called G band, around 1580 cm−1, which is the only first-order Raman mode in graphene, and is
Fig. 1. Raman spectra obtained in four regions of the twisted bilayer sample, each one with a different twisting angle. The spectra were recorded with the 3.82 eV laser line.
related to the vibration of the sublattice A against the sublattice B. Notice that the intensity of the G-band is strongly dependent on the twist angle, and this phenomenon is ascribed to resonance of the incident laser with van Hove singularities in their electronic density of states, whose separation depend on the twist angle [4,5]. In addition to the G band, the spectra also exhibit a number of sharp extra peaks in the spectral regions 1250–1450 cm−1 and 1590–1620 cm−1. The D-band recorded with the 3.81 eV laser line should appear as a broad band around 1420 cm−1 [7], and its absence in the spectra can be ascribed to the high crystalline quality of the sample in these regions. The other important feature in Fig. 1 is the so-called 2D band, in the range 2800–2900 cm−1, that involve two phonons within the interior of the Brillouin that satisfy the double and triple resonance Raman processes [20].
3.2. Resonance Raman behavior of the Moiré peaks An important characteristic of the D and 2D bands in graphene systems is the fact that they are dispersive, that is, their positions change when the laser energy is varied. In order to check the laser energy dependence of the extra peaks, we measured the same region of the TBG using different laser lines in the range 2.54– 4.14 eV. Fig. 2 shows the Raman spectra in the 1200–1700 cm−1 range, obtained with many different laser lines, where we can observe the peak around 1380 cm−1 associated with the TBG. Notice in Fig. 2 that the position of the extra peak does not depend significantly on the laser energy ðElaser Þ, whereas its intensity is strongly dependent on Elaser , the maximum intensity occurring around 3.04 eV. In order to understand the resonance behavior shown in Fig. 2, we need to consider the double resonance process involving the Bravais lattice superstructures of exact Moire patterns of two rotated graphene sheets. These Moiré patterns give rise to the folding of the graphene Brillouin zone (BZ), and, therefore, a number of wavevectors within the BZ of graphene are folded into the center of the new BZ of the Moiré pattern superstructure. Fig. 3 shows the direct lattice and the reciprocal space of a Moiré pattern generated by two layers of graphene rotated by a given angle with respect to each other. Each point within the interior of the Brillouin zone (BZ) of graphene corresponds to an allowed wavevector of the reciprocal lattice of the superstructure. The
Fig. 2. Raman spectra in the 1200–1700 cm−1 range, obtained with different laser lines, showing the resonance behavior of the sharp extra peak around 1380 cm−1. The peaks marked with asterisks are broadened since the laser lines of the diode laser were very broad.
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Fig. 3. (a) Moiré pattern in two twisted graphene layers and (b) wavevectors of the Moiré pattern superlattice and two possible DR Raman process (intervalley and intravalley).
Fig. 4. Simulated umklapp double-resonance Raman spectra, IðωÞ, for two possible twisting angles between graphene layers.
double resonance Raman process involving one phonon and one wavevector of the Moiré superlattice is also represented in Fig. 3. Normally, a defect is needed for momentum conservation in the double resonance Raman process that gives rise to the D and D′ bands in graphene materials [18]. However, in the presence of Moiré patterns, momentum conservation is provided by one wavevector of the Moiré superlattice, and we can assign this mechanism to an umklapp double resonance (UDR) Raman process. Given the reciprocal lattice vectors GB of a Moire pattern, we can simulate the umklapp double-resonance Raman spectrum IðωÞ. Specifically, the double-resonance Raman spectrum IðωÞ is calculated following the methodology of Ref. [20]. However, here instead of summing up in the whole BZ we consider only the lattice vectors GB of a Moire pattern. In this way our calculation model the twisted bilayer graphene as a monolayer of graphene interacting with a network of defects corresponding to the lattice vectors GB . The phonon dispersion is obtained from ab-initio
density functional theory calculations with many-body corrections as in Refs. [21,20]. We only consider the LA, TO and the LO branches. For the inter-valley double-resonant processes, we consider the LA and the TO branches, which are allowed by symmetry. For such processes, involving a photon energy Elaser, maximum scattering intensity occurs for phonon wavevectors near a specific one, qinter , which has the minimum possible modulus. This is schematically shown in Fig. 3. For the LO branch, we will consider intra-valley double-resonant processes. In this case, maximum scattering intensity occurs at the phonon wavevectors with modulus qintra, as also shown in Fig. 3. Fig. 4 shows the simulated umklapp double-resonance Raman spectra, IðωÞ, for two different twisting angles (7.341 and 13.17 1), and for different incident laser energies. Notice that the intensity of the peak around 1380 cm−1 is small for small values of Elaser . However, it increases with increasing Elaser , reaches a maximum and then decreases for larger values of Elaser . In agreement with the experimental results, the Raman peaks activated by the
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umklapp double resonance process are non-dispersive, contrarily to the case of the D band, and only their intensities depend resonantly on the incident laser energy 3.3. Raman excitation profile of the Moiré peaks
IM/IG
Fig. 5 shows the Raman excitation profile (REP) of the intensity of extra peak around 1380 cm−1, normalized by the intensity of the G-band. The black squares correspond to the experimental results presented in Fig. 2 and the open and black circles correspond to the data of the simulated spectra for the TBG superlattice shown in Fig. 4. Notice that there is a nice qualitative agreement between the experimental and simulated results, proving that these extra peaks are indeed ascribed to an umklapp double-resonance process. The full width at half maximum (FWHM) of the experimental REP is about 0.7 eV, which is slightly larger than the REP of the simulated spectra. Let us now discuss the physical origin of the wide REP observed for the intensity of the extra peak, shown in Fig. 5, and reproduced in our quantum mechanical calculations. As we shall see, there is a remarkably simple physical interpretation that explains (i) the origin of the laser energy resonance, (ii) the energy position of the resonance peak, and (iii) the width of the Raman excitation profile (REP), which is very large in comparison with the small widths of the Raman peaks. As discussed in a previous work [7], our theoretical interpretation of the new Raman peaks in Fig. 1 considers a periodical perturbation ΔV to graphene originated from the twisted-doublelayer Moiré pattern. The perturbation allows for elastic scattering from electron states ψðkÞ to ψðk′Þ, described by matrix elements M kk′ ¼ 〈ψðk′ÞjΔVjψðkÞ〉. This leads to a double resonance (DR) Raman scattering process that is similar, in its physical origin, to that of the D band, where the electron is (i) optically excited, (ii) inelastically scattered by a phonon of wavevector q, (iii) backscattered by the elastic scattering process with wavevector −q, and (iv) radiatively de-excited. In the cases of the D band, the local perturbation ΔV, which is due to defects, leads to non-null values of M kk′ in a continuum range of q ¼ k−k′. This leads to the dispersive behavior of the D band observed experimentally [9,18]. In contrast, the periodic Moiré perturbation ΔV leads to allowed scattering (non-null M kk′ ) only for a discrete set of k′−k ¼ G, where G is the set of the reciprocal lattice vectors of
the Moiré pattern. This process can therefore be described as an umklapp DR process for the Moiré lattice. From momentum conservation we have q ¼ G, and therefore the only relevant phonon scattering events in such umklapp-Moiré double resonance process are those for phonons with wavevectors G. As a consequence, instead of a wide and dispersive Raman resonance as that of the D band, the umklapp Moiré resonances will be characterized by several sharp, non-dispersive resonances found in the same energy region as that of the D and D′ bands. Let us now assume, for simplicity, that the periodic perturbative potential of the Moire lattice can be approximated by a periodic array of localized perturbations, each perturbation being akin to the defect perturbations that leads to the D band. Within this approximation, the umklapp Moiré resonances become, essentially, a “filtered” D band. That is, each resonance is placed at the energy of a phonon with wavevectors G, with its intensity given by the intensity of the D band at that same energy. Therefore, within this approximation, our analysis of the laser energy dependence of the intensity of a given Moiré resonance peak at phonon frequency ω, I M ðω; Elaser Þ, can be mapped into the laser energy dependence of the intensity I D ðω; Elaser Þ of the D band at the same frequency ω. Both experiments [9,18] and calculations [20] show that the peak position ω0 of the D band, as a function of the laser energy Elaser (that is, the value of ω for the maximum of I D ðω; Elaser Þ at a given Elaser) can be reasonably approximated by a linear relation ω0 ¼ a þ bElaser . From Ref. [18], we can estimate the values a ¼ 1237 cm−1 and b ¼ 45 cm−1 =eV. The same experiments and calculations indicate that the width of the D band is approximately constant with Elaser. Such phenomenology for the D band, together with the proposed ID −IM mapping, lead us to propose the following ansatz for the Moiré peak intensity I M ðω0 ; Elaser Þ ¼ f ðω0 −a−bElaser ÞgðElaser Þ
ð1Þ
where ω0 is frequency of the Moiré peak, Elaser is the laser energy, f (x) is a function with a maximum at x ¼0 with value f ð0Þ ¼ 1, and gðElaser Þ is the Moiré peak intensity for a given laser energy. Now, considering that g is a slowly varying function (as compared to f), the above ansatz leads to two important results. The first result is that, due to the specific form of the argument of f, the value Epeak of the laser energy at the peak in I M ðω0 ; Elaser Þ is related to ω0 through ω0 ¼ a þ bEpeak . This theoretical result is consistent with our experimental results: using the experimental value of Epeak in Fig. 5, Epeak ¼ 3.09 eV, and the a and b values mentioned in the preceding paragraph, we obtain ω0 ¼ 1376 cm−1 , which near the observed phonon frequency for this peak, ω0 ¼ 1380 cm−1 . The second result obtained from the ansatz is that (again from the argument of f in the ansatz) the peak width ΔE in I M ðω0 ; Elaser Þ for a given ω0 is related with the width Δω of the D band through Δω ¼ bΔE. Again, this theoretical result is consistent with our experiments: using the experimental value of ΔE in Fig. 5, ΔE ¼0.7 eV, and the b value mentioned in the preceding paragraph, we obtain Δω ¼ 31 cm−1 , which is nice agreement with the D bandwidth.
4. Conclusion
2.6
2.8
3.0
3.2 3.4 Laser energy (eV)
3.6
3.8
Fig. 5. Laser energy dependence of the relative intensity of the extra M band with respect to the G-band intensity. The black squares correspond to the experimental results presented in Fig. 2 and the open and black circles correspond to the data of the simulated spectra shown in Fig. 4.
In summary, we have studied the Raman spectra of twisted bilayer graphene samples, with different twisting angles, by changing the incident laser energy between 2.54 and 4.14 eV. The spectra exhibit a number of extra and sharp Raman peaks, which are grouped in families of peaks, each one corresponding to a different twisting angle. From the resonance Raman measurements, we were able to determine the Raman excitation profile of these extra peaks.
A. Righi et al. / Solid State Communications 175-176 (2013) 13–17
We theoretically analyze the laser energy dependence of these extra peaks considering a set of discrete wavevectors within the interior of the Brillouin zone of graphene, which activate special selective double-resonance Raman modes. We show that there is a nice qualitative agreement of the experimental and simulated spectra, showing that these extra peaks are indeed ascribed to an umklapp double-resonance process in graphene systems. Acknowledgment This work was supported by the Instituto Nacional de Ciencia e Tecnologia (INCT) em Nanomateriais de Carbono - MCT, CNPq, CAPES and FAPEMIG (Brazil). References [1] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [2] A. Luican, G. Li, A. Reina, J. Kong, R.R. Nair, K.S. Novoselov, A.K. Geim, E. Y. Andrei, Phys. Rev. Lett. 106 (2011) 126802. [3] A.H. MacDonald, R. Bistritzer, Nature 474 (2011) 453. [4] R.W. Havener, H. Zhuang, L. Brown, R.G. Hennig, J. Park, ACS Nano 12 (2012) 3162–3167. [5] K. Kim, S. Coh, L.Z. Tan, W. Reagan, J.M. Yuk, E. Chaterjee, M.F. Crommie, M.L. Cohen, S.G. Louie, A. Zettl. Phys Rev. Lett. 108 (2012) 246103.
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