Moiré phonons in the twisted bilayer graphene

Moiré phonons in the twisted bilayer graphene

Mikito Koshino Department of Physics, Osaka University, Toyonaka 560-0043, Japan    Young-Woo Son Korea Institute for Advanced Study, Seoul 02455, Korea

We study the in-plane acoustic phonons in twisted bilayer graphenes using the effective continuum approach. We calculate the phonon modes by solving the continuum equation of motion for infinitesimal vibration around the static relaxed state with triangular domain structure. We find that the moiré interlayer potential only affects the in-plane asymmetric modes, where the original linear dispersion is broken down into mini phonon bands separated by gaps, while the in-plane symmetric modes with their linear dispersion are hardly affected. The phonon wave functions of asymmetric modes are regarded as collective vibrations of the domain wall network, and the low-energy phonon band structure can be qualitatively described by an effective moiré-scale lattice model.


I Introduction

The twisted bilayer graphene (TBG), or a pair of graphene layers rotationally stacked on top of each otherBerger et al. (2006); Hass et al. (2007, 2008); Li et al. (2009); Miller et al. (2010); Luican et al. (2011), exhibits a variety of physical properties depending on the twist angle . In a small , in particular, a long-period moiré pattern due to a slight lattice mismatch folds the Dirac cone of graphene into a superlattice Brillouin zone Lopes dos Santos et al. (2007); Mele (2010); Trambly de Laissardière et al. (2010); Shallcross et al. (2010); Morell et al. (2010); Bistritzer and MacDonald (2011); Kindermann and First (2011); Xian et al. (2011); Lopes dos Santos et al. (2012); Moon and Koshino (2012); de Laissardiere et al. (2012); Moon and Koshino (2013); Weckbecker et al. (2016), and nearly-flat bands with extremely narrow band width emerge at some particular ’s called the magic angles. The discovery of the superconductivity and strongly correlated insulating state in the magic-angle TBGs Cao et al. (2018a, b); Yankowitz et al. (2019) attract enormous attention of this system.

While the early theoretical studies on TBG simply assumed a stack of rigid graphene layers without deformation, the actual TBG spontaneously relaxes to the energetically favorable lattice structure. There the in-plane distortion maximizes the area of AB stacking (graphite’s Bernal stacking) to form a triangular domain pattern Popov et al. (2011); Brown et al. (2012); Lin et al. (2013); Alden et al. (2013); van Wijk et al. (2015); Dai et al. (2016); Jain et al. (2016); Nam and Koshino (2017); Carr et al. (2018); Lin et al. (2018); Yoo et al. (2019); Walet and Guinea (2019), and at the same time the out-of-plane distortion leads to a corrugation of graphene layers Uchida et al. (2014); van Wijk et al. (2015); Lin et al. (2018). The electronic band structure is also significantly affected by the lattice relaxation Nam and Koshino (2017); Lin et al. (2018); Koshino et al. (2018); Yoo et al. (2019); Walet and Guinea (2019); Lucignano et al. (2019). The similar structural relaxation was also observed in moiré superlattice of graphene and hexagonal boron-nitride (h-BN)Woods et al. (2014), and the effects on the band structure was investigated San-Jose et al. (2014a, b); Jung et al. (2015).

The lattice deformation of the moiré superlattice is expected to significantly influence the phonon properties. In the intrinsic graphene, the low-energy phonon spectrum is composed of transverse acoustic (TA) modes and longitudinal acoustic (LA) modes which have linear dispersions, and the out-of-plane flexural phonons (ZA) with a quadratic dispersion Grüneis et al. (2002); Suzuura and Ando (2002); Yan et al. (2008). In the AB-stacked bilayer graphene, the phonon modes of the top layer and the bottom layer are coupled to form layer-symmetric and asymmetric modes Yan et al. (2008); Nika and Balandin (2017). The phonon spectrum of TBG was calculated by fully taking account of all the atoms in the moiré unit cells Jiang et al. (2012); Cocemasov et al. (2013); Choi and Choi (2018), where the phonon density of states is found to be close to that of the regular AB-stacked bilayer graphene insensitively to the twist angle. On the other hand, it is naturally expected that the moiré interlayer potential would cause the superlattice zone-folding and the miniband generation in the phonon spectrum. While so far such the zone-folding effect was argued for graphene h-BN superlattice Felix and Pereira (2018), the low frequency phonon mini-bands in the TBGs are not thoroughly understood although they may have important roles for low energy transport and collective phenomena Choi and Choi (2018); Wu et al. (2018, 2019).

In this work, we study the in-plane acoustic phonons in TBG and investigate the effects of the moiré superlattice structure on the phonon spectrum. The calculation is based on the continuum approach using the elastic theory and the registry-dependent interlayer potential, which was used to obtain the relaxed lattice structure with a triangular domain pattern Nam and Koshino (2017). We derive the phonon modes by solving the continuum equation of motion for infinitesimal vibration around the static relaxed state. We find that the moiré interlayer potential couples only to the in-plane asymmetric modes (i.e., the top and bottom layers slide in the opposite directions parallel to the graphene layers), while the in-plane symmetric modes (slide in the same direction) are hardly affected. In the in-plane asymmetric modes, the original linear dispersion is broken into mini phonon bands separated by gaps, where the phonon wave functions are regarded as collective vibrations of the nano-scale triangular lattice of AB- and BA-stacking domains. The formation of mini bands and gaps is more pronounced in lower twist angles. We find that the phonon band structure in the low twist angles becomes nearly invariant when renormalized by the energy scale inversely proportional to the moiré superlattice period. The universal behavior of the moiré phonon bands can be understood by an effective moiré-scale lattice model.

Ii Methods

Figure 1: Non-relaxed atomic structure of TBG with . AA spots are located at the crossing points of the grid lines.

We consider a TBG lattice with a relatively small twist angle lower than a few degree, as shown in Fig. 1. We define the primitive lattice vectors of layer 1 as , , and those of layer 2 as , where nm is graphene’s lattice constant and is the two-dimensional rotation matrix. The reciprocal lattice vectors of layer 1 are given by and , and those of layer 2 by (). When the twist angle is small, the slight mismatch of the lattice periods of two layers gives rise to a long-period moiré beating pattern. The primitive lattice vector of the moiré superlattice is given by Nam and Koshino (2017)


The lattice constant becomes


The corresponding moiré reciprocal lattice vectors satisfying are written as


The local structure of TBG approximates the non-rotated bilayer graphene with the in-plane translation. It is characterized by the interlayer sliding vector, or the shift of an layer 2’s atom at , measured from its counterpart on layer 1. Without the lattice relaxation, the interlayer sliding vector is given by


Now we introduce the in-plane lattice vibration specified by the time-dependent displacement vector, for layer . The interlayer sliding vector under the deformation is


Here we concentrate on the in-plane displacement, while the qualitative argument of out-of plane phonon modes will be presented in Sec. IV. The inter-layer binding energy of TBG is written as


where the interlayer binding energy per area of non-rotated bilayer graphene with the sliding vector Nam and Koshino (2017). It can be approximately written as


where . The function takes the maximum value at AA stacking () and the minimum value at AB and BA stacking. The difference between the binding energies of AA and AB/BA structure is per area, and this amounts to per atom where is the area of graphene’s unit cell. In the following calculation, we use (eV/atom) as a typical value Lebedeva et al. (2011); Popov et al. (2011). By using Eqs. (5) and (7), we have


where and we used the relation .

The elastic energy of strained TBG is expressed by Suzuura and Ando (2002); San-Jose et al. (2014b)


where eV/ and eV/ are graphene’s factors Zakharchenko et al. (2009); Jung et al. (2015), and is strain tensor. Lastly, the kinetic energy due to the motion of the lattice is expressed as


where kg/m is the area density of single-layer graphene, and represents the time derivative of .

The Lagrangian of the system is given by as a functional of . We define and rewrite as a functional of . The Euler-Lagrange equations for read


where is -component of . The equation of motion for is given by replacing with and removing all the terms including in Eqs. (11) and (12). Here the potential terms with vanishes because is not dependent on and then the force term is zero. Therefore, the mode is just equivalent to the original acoustic phonon of graphene with the linear band.

For , we consider a small vibration around the static equilibrium state, or


Here is the static solution of Eqs. (11) and (12), i.e., the optimized relaxed state to minimize , and is the perturbational excitation around . We define the Fourier components as


where are moiré reciprocal vectors, is the phonon frequency.

From Eqs. (11) and (12), the equation for the static solution is given by Nam and Koshino (2017)


The equation of motion for the dynamical perturbation part is written as


We first derive the static solution by solving a set of the self-consistent equations, Eqs. (14), (16) and (18), by numerical iterations Nam and Koshino (2017). Using the obtained , we solve the eigenfunction Eq. (19), to obtain eigenvalues and the phonon modes as a function of in the moiré Brillouin zone. Throughout the calculation, we set the -space cut-off for the Fourier components of and , which is sufficient large for convergence.

Figure 2: Band dispersions of in-plane asymmetric modes () in various twist angle ’s. The horizontal axis is the scaled by the superlattice Brillouin zone size , and the vertical axis is scaled by (see the text). The panel (a) is the phonon dispersion when the moiré interlayer coupling is absent.
Figure 3: Three-dimensional band structure of in-plane asymmetric modes () in .
Figure 4: Phonon wave functions of the lowest five modes at in TBG of . The axis is taken as the horizontal axis, and the color code represents the local binding energy at a certain . The corresponding band structure is shown in Fig. 2(d).
Figure 5: Schematics of the domain structure in TBG. The shaded region indicate is the domain walls separating AB and BA regions.
Figure 6: (a) Phonon band structures of the effective lattice model (black solid) and the original model for from Fig. 2(e) (blue dashed). (b) Wave functions of the first and the second modes at , which are indicated by dots in (a).

Iii Phonon modes

Figure 2 shows the band dispersions of in-plane asymmetric modes () calculated for various twist angle ’s. Here the horizontal axis is the scaled by the superlattice Brillouin zone size () and the vertical axis is scaled by


Fig. 2(a) is the phonon dispersion when the moiré interlayer coupling is absent. This is equivalent to the folded band structure of intrinsic TA and LA phonons of graphene, which are , , respectively, where and .Suzuura and Ando (2002) The same dispersions are also indicated by red dashed lines in Figs. 2(b)-(d). Note that the band structures of in-plane symmetric modes () remain intact and ungapped. So, they are not drawn in this figure and hereafter also. Figure 3 is a three-dimensional plot of the phonon band in case of TBG with of which phonon dispersion along the high symmetric lines is in Fig. 2(d). It shows that the spectral gaps exist between the second and third modes and between the fifth and sixth ones over the whole moiré BZ.

For , we clearly see that the original linear bands of acoustic phonons of graphene are reconstructed into superlattice mini-bands separated by the spectral gaps. The mini-band formation is more pronounced in lower twist angles. In Fig. 4, we plot the phonon wave functions of the lowest five modes at () in . Here the axis is taken as the horizontal axis, and the color code represents the local binding energy at a certain . The first mode (a) and second mode (b) are regarded as the longitudinal and transverse acoustic modes of the superlattice, respectively, where the AA spots are ‘sites’ arranged in a triangular lattice, and the AB-BA domain walls are bonds connecting the sites. Here we notice that the longitudinal mode of the superlattice continuously connects to the transverse mode of intrinsic graphene (and vice versa) in switching off the interlayer coupling. This is because the shift of the moire pattern is actually perpendicular to the atomic movement .

Importantly, we notice that the low-energy band structure (scaled by ) is almost identical in and [Fig. 2(d) and (e)], indicating that it is converging to the universal band structure in the low twist angle limit. This is somewhat surprising because the relative energy scale of the interlayer interacting potential (constant) to the reference scale is increasing when decreasing , so that one may naively expect the phonon band gap should increase relative to as well.

The universal feature of the scaled band structure can be understood by considering the domain wall structure in the relaxed TBG, which is schematically illustrated as Fig. 5. Here triangular AB and BA regions are separated by the domain walls (gray regions) with AA stacked spots at vertices. It was shown that the characteristic width of the domain walls is almost independent of the twist angle Alden et al. (2013), and it is given by Nam and Koshino (2017)


which is about 5 nm. The domain pattern becomes clear in low twist angles such that , or lower than about 2 degree. We first qualitatively understand the scale of by the following order estimation. The area of the domain wall per the moiré unit cell is given by with the numerical factor neglected. The total interlayer binding energy (relative to the AB / BA stacking) per a moiré unit cell is then . The elastic energy is also concentrated to the domain wall regions and it is given by . Here we took as a representative scale for elastic constant, since and are in the same order of magnitude. In the domain wall, the strain tensor is of the order of since the atomic shift changes by about inside the domain wall. The relaxed state is given by the condition , and this gives , which has the same order of magnitude as Eq. (21).

Now let us consider an oscillation of the moiré lattice around the relaxed state. We consider a simple excitation such as the Fig. 4(a) or (b), and assume the in-plane displacement of the AA positions (‘lattice points’) at is . During the oscillation, the binding energy and elastic energy are shown to be approximately proportional to the total length of the domain walls. In Figs. 4(a) and (b), we actually see that the domain walls (‘bonds’) are elongated (or shortened), and here the graphene’s atomic lattice is not stretched in the same direction, but the area of the same local atomic structure is just increased, so that the total energy change is proportional to the domain wall length. The change of the total domain-wall length from the relaxed state is of the order of per a moiré unit cell, noting that the linear term in can be negative and positive place by place and vanishes in total. Therefore, the changes in the binding energy and the electric energy are given by and , which are of the same order because as argued above. The kinetic energy is estimated by considering the atomic motion. Here the change of the moiré lattice is actually caused by a change of the interlayer atomic shift around the domain wall. Here we have a relation , because a change of the atomic lattice of the order of is magnified to a change of moiré lattice by . Considering the atoms are oscillating only around the domain wall regions, the kinetic energy per a moiré unit cell is given by where is the oscillation frequency. By assuming the kinetic energy and the potential energy and are of the same order of magnitude, we end up with


which correctly gives the order of .

Based on the above argument, we can construct a simple effective model to describe the lowest two modes. We consider a triangular lattice composed of masses (corresponding to the AA spots) and bonds (domain walls) connecting the neighboring masses. We assume that the bonds are always straight and the energy of each bond is proportional to its length. We consider a vibration around the perfect triangular lattice with the lattice constant , assuming that the total area of the system is fixed. If the in-plane displacement of the mass is given by , the length change of the bonds summed over the whole system is given by


where , the vector runs over the lattice points, and we defined . The total energy of the bonds is given by , where is a numerical factor of the order of 1 to match the energy scale with the original model. By the Fourier transformation , it is written as


where is the dynamical matrix defined by


where . The kinetic energy is given by , where is the effective mass. Finally the equation of motion is given by . It is straight forward to see that the equation can be transformed to a dimensionless form by scaling the frequency by . Figure 6(a) plots the phonon band structure of this model, where the blue dashing line represents the original band structure of . Here we choose for the best fitting in the long wave-length region near . We see that the effective model qualitatively reproduce the whole dispersion relation of the original model for the first and second low energy modes, and we can also show that the phonon wave functions are also correctly reproduced in the whole Brillouin zone. Figure 6(b) illustrates the wave functions of the first and the second modes at , which agree with Figs. 4(a) and (b), respectively.

The vibration of the moiré lattice is different from that in an ordinary spring-mass model, in that the potential energy of a bond (a domain wall) is proportional to its length, but not to the squared length. This is directly related to the important property that the longitudinal mode [Fig. 4(a)] has the lower frequency than the transverse mode [Fig. 4(b)] unlike in the usual elastic system. In the ordinary spring-mass model, the strain energy of the longitudinal mode as in Fig. 4(a) is mostly contributed from the bonds parallel to the wave vector (horizontal bonds in this figure), where the length change gives the tensile energy cost. In the present system, however, the horizontal bonds do not change the total energy because elongated bonds and shortened bonds just cancel in the total length, while the diagonal bonds contributes to a relatively small increase in the total length. In the effective lattice model given by Eq. (25), it is straight forward to check that the phonon frequency of the transverse mode is times as large as that of the longitudinal mode in the long wave limit. In the original model, the ratio of the second band frequency to the first is about 1.8 near point, indicating that the effective model is valid.

The linear dependence of energy for elongation of bonds implies that there is a constant force for all stretching modes, while such modes are prohibited in the present calculation because the moiré unit cell size is fixed. In a real TBG, we actually has an instability to expand the moiré unit cell, and it is nothing but decreasing the twist angle toward the non-rotated AB-stacked bilayer. Here the total energy decreases in elongation of the unit cell, because the area of the AB region increases so that the area of the domain wall to the total area is decreasing. In experiments, a TBG with a finite twist angle can also exist as a meta-stable phase, where certain external constraints caused by local pinning mechanisms should prohibit the global rotation. The calculation fixing moiré unit cell size should be justified in such a situation.

As a final remark, we notice that the third lowest band in Fig. 2 becomes flatter in lowering the twist angle, resulting in a significant enhancement of phonon density of states as decreases. The corresponding wave function in Fig. 4(c) can be viewed as a ‘domain-breathing’ mode where triangular AB and BA domains expand and shrink in opposite phases. One may also view this mode as in-plane optical mode of moiré lattice. Here we see that the position of the corner points of triangles are fixed during the oscillation. This means that the oscillation of a triangle does not affect that of neighboring triangles, so the system can be approximately viewed as independent oscillators. We presume that it is the reason for the flat band nature, i.e. the constant frequency independent of the wave number. The fourth and the fifth modes [Fig. 4(d) and (e)] exhibit complicated wave patterns including the bending of the domain walls and also the distortion of AA spots.

Iv Discussions

While the current model only considered the in-plane phonons, the actual TBG also has the flexural phonon modes vibrating in the out-of-plane direction. Here we consider the effect of moiré interlayer coupling on the flexural phonons in TBG by the following simple qualitative argument. We consider three-dimenisional displacement from the non-distorted configuration in which two flat graphenes are stacked with interlayer spacing (the distance for AB-stacked bilayer). It is known that the relaxed TBG is corrugated in such a way that the spacing is minimum at AB/BA stacking regions and maximum at AA stacking regions Uchida et al. (2014); van Wijk et al. (2015); Lin et al. (2018). We can define the optimized interlayer spacing as a function of the two-dimensional sliding vector [Eq. (5)]. The interlayer binding potential for the vertical displacement can then be written as a quadratic form , where . The total binding energy is a sum of and the in-plane part, Eq. (7). The potential curvature is shown to be nearly independent of the stacking structure, Lee et al. (2008) so we assume is constant. The effect of the moiré superlattice comes only from the potential term through . The kinetic part and electric part are unchanged from the intrinsic graphene’s and do not depend on . In the equation of motion, we have the force term , and the optimized corrugated structure is given by . When we consider an excitation from the optimized state , the force term becomes , and does not depend on anymore, i.e., the superlattice effect vanishes in the equation of motion for .

From these arguments, we presume that flexural phonon does not exhibit well-pronounced miniband structure unlike the in-plane asymmetric modes, and the corresponding phonon density of states remains mostly unchanged from the intrinsic graphene. Actually the mini gap formation was not observed in a phonon calculation fully including all the atomic sites in TBG with the registry-dependent interlayer interaction Choi and Choi (2018). It implies that the flexural phonons remain almost intact, considering that the density of states in the low-energy region of graphene is dominated by the flexural phonons Lindsay et al. (2010). Rigorously speaking, the corrugated structure should cause some finite coupling between the in-plane modes and out-of-plane modes, and we leave the detailed study of full three-dimensional phonon problem for future works.

V Conclusion

We studied the in-plane acoustic phonons of low-angle TBGs using the continuum approach. We found that the moiré superlattice effect only affects the in-plane asymmetric modes where the linear dispersion of acoustic phonon is reconstructed into minibands separated by the gaps, while the in-plane symmetric modes remains intact. The phonon wave functions for in-plane asymmetric modes can be regarded as vibrations of the triangular superlattice, and they are qualitatively understood by the effective lattice model of the moiré scale.

Our calculation shows that 50% of the linear acoustic modes of the TBG are completely reconstructed by the moiré superlattice coupling and their phonon velocities are significantly lowered. Moreover, it is also shown that as the twist angle decreases, the characteristic flat phonon mode with diverging density of states is generated. We expect that these effects should affect the thermal conductivity directly and also phonon-related thermal phenomena. The electron-phonon coupling between the flat-band electrons and the moiré phonons, and its effect on the superconducting and correlating states are also important problems. We leave these issues left for future study.

Vi Acknowledgments

M. K. thanks the fruitful discussions with Pablo Jarillo-Herrero, Philip Kim and Debanjan Chowdhury. M. K. acknowledges the financial support of JSPS KAKENHI Grant Number JP17K05496. Y.-W.S. was supported by National Research Foundation of Korea (Grant No. 2017R1A5A1014862, SRC program: vdWMRC center).

Note added: After completion of the present study, we have come to notice that a recent preprint which reports the full phonon calculations in TBG and the coupling to the electronic states Angeli et al. (2019)


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