Abstract
In this letter, we present a step towards understanding the bilayer graphene (2LG) interlayer (IL)related phonon combination modes and overtones as well as their phonon selfenergy renormalizations by using both gatemodulated and laserenergy dependent inelastic scattering spectroscopy. We show that although the IL interactions are weak, their respective phonon renormalization response is significant. Particularly special, the IL interactions are mediated by Van der Waals forces and are fundamental for understanding lowenergy phenomena such as transport and infrared optics. Our approach opens up a new route to understanding fundamental properties of IL interactions which can be extended to any graphenelike material, such as MoS_{2}, WSe_{2}, oxides and hydroxides. Furthermore, we report a previously elusive crossing between ILrelated phonon combination modes in 2LG, which might have important technological applications.
Introduction
In spite of its outstanding properties, for many practical purposes, that, for example, require a band gap, single layer graphene (1LG) cannot be readily applied without the use of complex engineering procedures^{1,2,3,4,5}. One possible and interesting solution that responds constructively to several drawbacks of 1LG is to look at its multilayer graphene (MLG) counterparts^{1,2,3,4,5}. MLG systems involve weak interlayer (IL) interactions mediated by VanderWaals (VdW) forces. These IL interactions are sensitive to the number of layers and stacking order and are important for technological applications of these systems because they are important for the lowenergy electronic and vibrational properties and, therefore, will be important for phenomena such as transport, infrared optics and telecommunication bands in the infrared (IR) range^{6,7,8}. In bilayer graphene (2LG), although the effects of the IL interactions on the electronic properties are well understood^{9,10}, the present understanding of ILrelated vibrational properties, electronelectron (ee), phononphonon (phph), and electronphonon (eph) interactions is still under development^{11,12,13,14,15}.
As regards the IL vibrational properties in 2LG, C. H. Lui et al.^{8} studied the outofplane optical (ZO′) phonon mode (with frequency ω_{ZO′} = 90 cm^{−1} predicted at the Γpoint, as shown in Fig. 1b). The ZO′ mode is also known as the IL breathing modeLBM, and its combination mode LOZO′ with the longitudinal optical (LO) phonon (with frequency ω_{LO} = 1575 cm^{−1} predicted at the Γpoint, as shown in Fig. 1b) occurs in the range for ω_{LOZO′} from 1600 to 1800 cm^{−1}, for MLG with up to 6 layers, thereby explaining their frequency dependence on both the number of layers and their stacking orders^{8}. The other related LOZA combination mode (ZA is the IL outofplane acoustic mode whose frequency ω_{ZA} is zero at the Γpoint, as shown in Fig. 1b, and the 2ZO overtone (ZO is the outofplane tangential optical mode with frequency ω_{ZO} = 885 cm^{−1} predicted at the Γpoint, as shown in Fig. 1b) demand a more detailed analysis, which is still elusive. All these features involve q ≠ 0 (throughout the text, q is the phonon wavevector) intravalley (AV) processes, therefore occurring around the Γpoint in the Brillouin zone. However, only the 2ZO overtone presents two possible forward (q ≈ 0) and backward (q ≈ 2k) scattering mechanisms^{11}. Note that both the ZA and ZO modes are not Raman active at the Γpoint, where q = 0. In spite of recent advances in the study of these interlayer modes, their phonon selfenergies and eph interactions for these ILdependent modes have hardly been discussed. It is worth saying that, these modes ranging from 1600 to 1800 cm^{−1} are spectroscopic signatures for MLG and by understanding them in detail, we can understand the VdWrelated phonondependent phenomena associated with these systems. Such knowledge will have considerable impact on developing this research field, by opening a route to understanding IL interactions in similar, but more complex 2Dlayered materials, such as MoS_{2}, WSe_{2}, oxides and hydroxides.
In the present letter we use gatemodulated and laserenergy (E_{L})dependent resonant Raman spectroscopy (RRS) together to address two fundamental issues regarding ILinteractions in 2LG systems: (1) we discuss in detail the ILdependent phonon selfenergies and the eph interactions of the combination modes related to the ZA, ZO and ZO′ phonons and (2) we show that the elusive overtone phonon mode 2ZO q ≈ 2k is indeed Raman active and its phonon dispersion crosses the LOZO′ combination mode at two different energies, one at about 2.58 eV and another at about 2.78 eV.
Results
Interlayerrelated E_{L}dependent analysis
Figure 1a shows the phonon combination modes and overtones observed in 2LG in the spectral range 1600 to 1800 cm^{−1}. The insets give the phonon vibration symmetries together with their respective E_{L}dependent frequency dispersions for the LOZA (P1) and the two LOZO′ (P2) peaks (upper box in Fig. 1a), and the two 2ZO (P3) peaks (lower box in Fig. 1a). As regards the dispersion relations and phonon peak assignments, our findings for the LOZA and LOZO′ combination modes agree well with those reported in Ref. 8. The LOZA mode (P1 peak in Fig. 1a) comes from a q ≈ 2k intravalley phonon scattering process (q ≈ 2k AV) showing a frequency dispersion ∂ω_{LOZA}/∂E_{L} = 26.1 cm^{−1}/eV. By looking at the feature P2 in Fig. 1a, we observe that the LOZO′ mode (q ≈ 2k AV process) splits into two peaks, LOZO′(+) and LOZO′(−), whose frequency dispersions are ∂ω_{LOZO′(+)}/∂E_{L} = 55.1 cm^{−1}/eV and ∂ω_{LOZO′(−)}/∂E_{L} = 34.2 cm^{−1}/eV, respectively. As schematized in Fig. 2c, the two P2 peaks do not arise from the phonon dispersion but rather, they come from different resonant regimes of the LOZO′ combination mode with the two electronic valence bands (π_{1} and π_{2}) and the two electronic conduction bands ( and ) of 2LG. In other words, the two peaks, LOZO′(+) and LOZO′(−) observed in the Raman spectra, come from the same phonon combination mode LOZO′ but probed at two different points of its phonon dispersion^{8}. Indeed, the LOZO′(+) comes from a resonance process involving the bands, while the the LOZO′(−) comes from a resonance process involving the bands (a process similar to the wellestablished doubleresonance process explaining the G′(2D)band in 2LG where the iTO phonon exhibits different resonances with the bilayer electronic dispersion^{16}). These resonance conditions (see Fig. 2c) require the phonon momentum q for the LOZO′(+) mode to be larger than that for the LOZO′(−) mode (q_{LOZO′(+)} > q_{LOZO′(−)}). As a consequence the phonon energies are such that .
Next, we discuss the two 2ZO features. Sato et al.^{11} predicted through tightbinding calculations (see Fig. 1d) that, in 2LG systems, the 2ZO overtone should be observed for forward (q ≈ 0) and backward (q ≈ 2k) AV scattering, where the q ≈ 2k mode presents a negative frequency dispersion (for clarity, in Ref. 11, the abbreviations for the phonon modes are as follows: oTO stands for ZO, M stands for 2ZO, while oTA stands for ZA and ZO stands for ZO′). These authors also predicted that the 2ZO (q ≈ 2k) would cross the LOZO′ dispersion at E_{L} = 2.6 eV (Fig. 1d). Although some information for the 2ZO (q ≈ 0) mode was reported by C. H. Lui et al.^{8}, the existence of the 2ZO (q ≈ 2k) mode remained elusive. Here, as shown in Fig. 1c, we report the 2ZO (q ≈ 2k) mode which was found to show a negative frequency dispersion ∂ω_{2ZO(q≈2k)}/∂E_{L} = −48.1 cm^{−1}/eV. Surprisingly, as indicated in Fig. 1c, the 2ZO (q ≈ 2k) mode crosses the LOZO′(+) at 2.58 eV (predicted in Ref. 11 and is also in good agreement with our observations). However, the 2ZO (q ≈ 2k) mode has another cross point with the LOZO′(−) mode at 2.78 eV, according to the estimate based on the phonon dispersion observed in the present work. The second crossing at 2.78 eV was not predicted in Ref. 11 and is a consequence of the different dispersions observed for the LOZO′ phonon when this phonon is in resonance with the or with the bands (only the resonance process with the bands was considered by Sato et al.^{11}). Indeed, these phonon mode crossings could affect both the dynamics of photoexcited carriers and the thermal properties of many systems, since they rely on relaxation processes mediated by highenergy optical and acoustic phonons^{17,18}. Recently, it has been shown that the control over the interactions between individual phonon modes and combinations of phonon modes plays an important role in the thermal conductivity of PbTe materials^{18}. This is an interesting concept that could be applied to graphenelike materials. It is also important to say that, from Fig. 1c, at higher E_{L} (4.15 eV according to our observations) the LOZA combination mode might cross the LOZO′ mode too.
Interlayerrelated phonon selfenergy renormalizations
Having understood the origin of the IL combination of modes and overtones, we next study their phonon selfenergy renormalizations, which have not yet been explored. The IL interactions in 2LG rely on the interlayer hopping among equivalent and inequivalent carbon atoms and, therefore, will be directly related to the phonon selfenergy and to the eph coupling regarding the ZO, ZO′ and ZA modes^{3,19}. It is, however, important to note that the LO mode is dependent on the intralayer hopping between two inequivalent carbon atoms, and the LO mode remains essentially unchanged when changing the IL interactions^{8,9,10,11,12,13,14,15}. From now on, this manuscript is focused on ILrelated phonon selfenergy renormalizations. As depicted in the inset of Fig. 2c, the LOZO′ combination mode relates to a q ≈ 2k AV process. Note that, Figs. 2a and 2b show, respectively, the E_{F} dependence of ω_{LOZO′(+)} and ω_{LOZO′(−)} when V_{g} is varied. Both, ω_{LOZO′(+)} and ω_{LOZO′(−)} soften with increasing E_{F} which is controlled by increasing V_{g}. Correspondingly, as shown in the insets of Figs. 2a and 2b, the phonon line widths γ_{LOZO′(+)} and γ_{LOZO′(−)} broaden with increasing E_{F}. Analogously, Fig. 3a shows that ω_{LOZA} (γ_{LOZA}) softens (broadens) with increasing E_{F}, while for the 2ZO overtone, a negligible dependence on V_{g} is observed for both ω_{2ZO} and γ_{2ZO} (Fig. 3b).
As a reminder, this behavior is opposite to what happens to the q = 0 phonons at the Γpoint^{20}, as is the case of the Gband feature, where ω_{q} hardens and γ_{q} narrows when E_{F} increases as a consequence of electronhole (eh) pair creation (annihilation) due to phonon absorption (emission). For q = 0 phonons at the Γpoint, the eh pair creation (annihilation) is halted by the Pauli principle, when . As explained by Araujo et al. for monolayer graphene^{19}, for both intravalley AV and intervalley EV q≠ 0 phonons, instead of the Pauli principle, the density of electronic and vibrational states together with energy and momentum conservation requirements will be responsible for halting eh pair production. Indeed, for E_{F} = 0 the density of states is almost null and the electronic structure slope (∂E(k)/∂k) is larger than the vibrational structure slope (∂ω(q)/∂q) so that no phonon can couple to any electrons^{19}. In this case no phonon selfenergy renormalization due to eh pair creation (annihilation) is observed. However, when E_{F} increases, the density of states increases and the difference between the slopes of the electronic and vibrational dispersions decreases^{19}. In this situation, phonon selfenergy corrections are observed since the density of states is nonnull and a phonon with momentum q can connect two electronic states k and k′ while fulfilling the energy and momentum requirements. In order to quantify the ZO′ phonon selfenergy corrections, we measured the V_{g} dependence of ω_{2LO} and γ_{2LO} for the 2LO overtone (see Fig. 2d), which is known as the 2D′ band around 3244 cm^{−1}. In analogy to the LOZO′ combination mode about the Γpoint, the 2LO overtone is a q≈ 2k AV double resonant process and is a fruitful choice for unraveling the twophonon selfenergy contributions that are merged in the LOZO′ combination mode.
The Raman scattering process involving overtone or combinations of phonon modes will conserve both the frequency and momentum, so that in our case (for the Stokes process) ω_{s} = ω_{L} − (ω_{LO} + ω_{j}) and k_{s} = k_{L} − (q_{LO} + q_{j}), where ω_{s}(k_{s}) is the frequency (momentum) of the Stokes scattered light, ω_{L}(k_{L}) is the frequency (momentum) of the incident light, ω_{LO}(q_{LO}) is the frequency (momentum) of the LO phonon mode and ω_{j}(q_{j}) is the frequency (momentum) of the ZO, ZO′ or ZA phonon modes. Since the electron is vertically excited from the valence to the conduction band by the absorption of a photon, we have k_{s} = k_{L} and, therefore, the phonon momenta will be such that q_{LO} ≈ q_{j}. This scattering process is understood as follows: the electron is first scattered by one of the phonons, let us say the LO phonon, and then the electron is scattered again by a second phonon; LO, if an overtone is observed or ZO, ZO′ and ZA if a combination mode is observed. On top of this, the phonon selfenergy renormalization relies on each specific phonon mode, being described by^{21}: where k and k′ are, respectively, wavevectors for the initial and final electronic states; q ≡ k − k′ is the phonon wavevector; is the eh pair energy; ω_{q} is the phonon frequency; γ_{q} is the phonon linewidth; f_{h}(f_{e}) is the Fermi distribution function for holes (electrons) and V_{kk′} gives the elph coupling matrix element. The renormalization will happen independently for each phonon, so that, as a first approximation for a combination mode or overtone, the renormalization will be Π(ω_{LO+j}, E_{F}) = Π(ω_{LO}, E_{F})+Π(ω_{j}, E_{F}), where j is a LO mode if an overtone is considered or j is a ZO, ZO′ and ZA if a combination mode is considered. Moreover, the IL interactions governing the ZO, ZO′ and ZA modes will not change the intralayer dependent LO mode^{8,9,10,11}. Therefore, what we are observing in the gatemodulated Raman experiment for overtone and phonon combination modes is, indeed, the summation of the individual phonon selfenergy corrections of each phonon participating of the scattering process^{21}. The phonon selfenergy renormalization strengths will be quantified by the difference between the frequencies at V_{g} = 0 and ω_{q} for V_{g} ≠ 0, i.e., (the renormalization strength could also be obtained from the phonon linewidth, which will be reported elsewhere), which is the real part of equation 1. As explained above, at V_{g} = 0 describes the system with no renormalizations that are associated with electronhole (eh) pair formation.
Discussion
By inspecting Figs. 2a and 2b, we find that Δω_{LOZO′(+)} = 9 cm^{−1} and Δω_{LOZO′(−)} = 7 cm^{−1}, respectively. On the other hand, by looking at Fig. 2d we see that Δω_{2LO} = 5 cm^{−1}, which means that the LO frequency renormalization for this AV process is Δω_{LO} ≈ 2.5 cm^{−1}. The selfenergy corrections regarding the LO mode will be the same for the LO contribution for both the LOZO′(+) and LOZO′(−) features. Therefore the phonon selfenergy correction Δω_{ZO′(+)} for the ZO′(+) mode will be given by Δω_{LOZO′(+)} − Δω_{LO} = 6.5 cm^{−1} while the phonon selfenergy correction Δω_{ZO′(−)} for the ZO′(−) mode will be given by Δω_{LOZO′(−)} − Δω_{LO} = 4.5 cm^{−1}. From the analysis above and remembering that, the larger the selfenergy corrections Δω_{q}, the stronger are the eph couplings, we deduce that the IL eph coupling mediating the renormalizations for the ZO′ mode is stronger than the renormalization for the LO mode. It is noteworthy that the selfenergy renormalizations for LOZO′(+) and LOZO′(−) are different even though they involve the same LO phonon. We understand these differences as follows: the phonon selfenergy corrections for q ≠ 0 phonons rely on the density of electron and phonon states^{19}. The density of phonons states will be the same because the same phonon is involved. However, as shown in Fig. 2e, for energies smaller than 2 eV, the density of electronic states for is always smaller than that for . This means that the phonon selfenergy corrections are weaker for the ZO′(−) in comparison to that for the ZO′(+) not due to a different eph coupling symmetry, but because the density of electronic states for is smaller in comparison to that for , as seen in Fig. 2e. By following the same strategy, we could also estimate the phonon selfenergy corrections for the ZA mode, whose LOZA combination mode frequency (linewidth) also hardens (broadens) as expected for q ≠ 0 AV processes. As shown in Fig. 3a, Δω_{LOZA} = 8 cm^{−1}. Therefore, the ZA mode selfenergy corrections Δω_{ZA} will be given by Δω_{LOZA} − Δω_{LO} = 5.5 cm^{−1}. It is interesting to note that, the renormalization for the ZA mode is similar to that ruling the ZO′ mode. Moreover, we note that this is the first time the phonon selfenergy renormalizations of an acoustic mode are reported.
Next, we discuss the V_{g} dependence of the 2ZO overtone for both, q ≈ 0 and q ≈ 2k AV processes. Interestingly, the results in Figs. 3b and 3c show that, for both cases, the phonon selfenergy corrections to the phonon frequency Δω_{2ZO} and to the line width Δγ_{2ZO}, are weak and as a consequence, the ω_{2ZO} and γ_{2ZO} renormalizations show a constant behavior with increasing E_{F}. This result is understood as follows: for phonon selfenergy corrections, the phonon energies themselves will determine where in the Dirac cones the eh pair creation (annihilation) will be happening and, therefore, will determine the initial (final) density of electronic and vibrational states. On top of this, the momentum conservation requirement q = k − k′, which is mostly determined by the slopes in the electronic and vibrational dispersion relations, must be obeyed in order to observe a strong coupling. Comparing all the cases, the ZO phonon mode would create (annihilate) an eh pair at much higher energies ~ 110 meV and, therefore, at a much higher density of electronic and vibrational states compared to the energies of the ZA (~ 3.1 meV) and the ZO′ (~ 11.2 meV) modes. Because phonon renormalizations can be observed for both the ZA and ZO′ modes, the authors understand that the reason behind the weak renormalization observed for the ZO mode is due to the lack of a phonon momentum q such that q = k − k′, and this lack prevents any resonant renormalization from happening. This is confirmed by symmetry arguments since the deformation potential mediating the eph coupling related to the ZO mode, which is an antisymmetric IL vibration, is not expected to allow coupling of orthogonal electronic states since its vibration breaks the lattice symmetry, which implies V_{kk′} = 0 in equation 1. Thus, no renormalizations are expected for the 2ZO mode^{22}.
It is worth commenting that 1LG and 2LG are essentially different as regards the electronic structure but very similar as regards the vibrational structure (except for the LBM). Nevertheless, the phonon renormalization phenomena happen similarly in 1LG and 2LG for AV q ≠ 0 processes where ω_{q} (γ_{q}) softens (broadens) with increasing V_{g}. This is interesting since under the application of V_{g}, 1LG still remains a zerogap material around the Kpoint with massless carriers, contrary to the case of 2LG, in which a bandgap is opened when there is an asymmetry between the two layers. Another point is that 2LG presents a new scenario where the symmetry of the vibration is important to determine whether or not the eph coupling renormalizations will take place, which was not the case for 1LG (since the ZO mode is Ramaninactive in 1LG).
Finally, the micromechanical cleavage of bulk graphite (socalled Scotchtape method) gives 2LG graphene samples with mostly ABstacked structures and the strongest IL interactions. For this reason, the 2LG system obtained from exfoliation is the best system to provide the fundamental physics behind the IL related phonon selfenergy renormalizations. Another commonly used synthesis method is the Chemical Vapor Deposition (CVD). Through the CVD method, in general, the 2LG system is grown in two steps where in step one a graphene layer is grown and in step two, the second graphene layer is epitaxially deposited over the first. Recently, selected area electron diffraction (SAED) and Raman scattering have been used to show that the resultant 2LG systems grown by the CVD methods are ABstacked^{23,24}. Indeed, the Raman G′(2D)/G intensity ratio (≈ 1) for the CVD grown 2LG system agrees well with the Scotchtape sample and also the G′band linewidth (≈ 60 cm^{−1}) is similar. In fact, the lineshape for the G′ band can be interpreted in terms of four peaks whose frequencies (ranging from 2640 to 2715 cm^{−1}) and linewidths (≈ 30 cm^{−1}) are basically the same as the ones found for the Scotchtape sample. However, the G′ lineshape, which is determined by the Raman crosssections of each of the four peaks, is slightly different mainly because the little shoulder around 2645 cm^{−1} observed for the Scotchtape samples^{16} is not evident in the CVD grown samples. This shows that the interlayer interactions are quite similar for both production methods but not exactly the same and further studies are clearly needed to explain these small differences in detail. It is important to say that, although some similarities are expected between different methods of producing 2LG samples, these different methods will produce 2LG samples with IL interactions which are different. However, it is noteworthy that the phonon behavior in which ω_{q} (γ_{q}) softens (broadens) with increasing V_{g} is related to the fact that the phonon wave vector is different of zero (q ≠ 0). This phenomenon is explained in detail by the authors in a previous publication^{20}.
The magnitude of the IL interactions can strengthen/weaken the phonon selfenergy renormalizations since the electronphonon coupling mediating these renormalizations relies on the IL interactions magnitude. Indeed, our results are obtained from exfoliated samples which means a perfect stacking between the two layers forming the 2LG system but we still cannot assert how the results of this work will apply for small angles. However, as explained by Kim et al.^{25}, when the orientation in between the two layers in a turbostratic 2LG is higher than a certain critical angle (around 13°), the two graphene layers behave as if they are independent from each other. This means that the interlayer interaction at that point is negligible and no interlayer related modes are supposed to appear. For angles smaller than the critical angle, where there are some meaningful interactions, we expect to observe the interlayer related modes whose spectral features will depend on the stacking angle. For different coupling magnitudes between the top and the bottom layer, we should find different magnitudes for the phonon selfenergy renormalizations, which will be smaller if the interlayer interaction is likewise smaller.
Methods
The E_{L}dependent measurements were done in the back scattering configuration using E_{L} values ranging from 2.10 to 2.54 eV with a 100× objective. The laser power was kept around 1.5 mW to avoid heating effects on our graphene samples. The samples were produced by the micromechanical cleavage of graphite on a Si substrate covered with 300 nm of SiO_{2}. The gatedependent measurements were performed with the 2.33 eV laser in devices fabricated by liftoff patterning of thermally evaporated Cr/Au (5 nm/80 nm, respectively). Back gate measurements were done near 300 K with voltages ranging from −70 to 70 V (Fermi level variation E_{F} ≈ 150 meV). The E_{F} is related to V_{g} by , where γ_{0} ≈ 3 eV is the intralayer carbon atoms hopping, γ_{1} ≈ 0.4 eV is the interlayer carbon atoms hopping, a = 2.46Å is the lattice constant, C_{g} = 1.15 × 10^{−8} F/cm^{2} is the geometric capacitance of the system and V_{0} is the gate voltage corresponding to the charge neutrality point, which is monitored by the V_{g} dependence of the G band frequency.
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Acknowledgements
P.T.A. acknowledges ONRMURIN000140911063 and J.F. RodriguezNieva for helpful discussions. D.L.M. acknowledges CNPqBrazil financial support. R.S. and K.S. acknowledge MEXT grant No.20241023 and No.23710118, respectively. M.S.D. and J.K. acknowledge NSFDMR 1004147 and 0845358, respectively.
Author information
Affiliations
Department of Electrical Engineering and Computer Sciences, Massachusetts Institute of Technology, Cambridge, MA 021394307, USA
 Paulo T. Araujo
 , Daniela L. Mafra
 , Jing Kong
 & Mildred S. Dresselhaus
Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 30123970 Brazil
 Daniela L. Mafra
Department of Physics, Tohoku University, Sendai 9808578, Japan
 Kentaro Sato
 & Riichiro Saito
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 021394307, USA
 Mildred S. Dresselhaus
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Contributions
P.T.A. and D.L.M. designed the experiments, performed the measurements and analyzed the experimental data. All the authors discussed the results and wrote the manuscript.
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The authors declare no competing financial interests.
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Correspondence to Paulo T. Araujo.
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Further reading

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