Effects of site asymmetry and valley mixing on Hofstadter-type spectra of bilayer graphene in a square-scatter array potential
Under a magnetic field perpendicular to an monolayer graphene, the existence of a two-dimensional periodic scatter array can not only mix Landau levels of the same valley for displaying split electron-hole Hofstadter-type energy spectra, but also couple two sets of Landau subbands from different valleys in a bilayer graphene. Such a valley mixing effect with a strong scattering strength has been found observable and studied thoroughly in this paper by using a Bloch-wave expansion approach and a projected effective Hamiltonian including interlayer effective mass, interlayer coupling and asymmetrical on-site energies due to a vertically-applied electric field. For bilayer graphene, we find two important characteristics, i.e., mixing and interference of intervalley scatterings in the presence of a scatter array, as well as a perpendicular-field induced site-energy asymmetry which deforms severely or even destroy completely the Hofstadter-type band structures due to the dependence of Bloch-wave expansion coefficients on the applied electric field.
Shortly after its discovery and fabrication in 2004, graphene has captured tremendous attention and generated an enormous wave of research activities due to its unique Dirac-cone-type electronic band-structures and properties. Novoselov et al. (2005); Geim and Novoselov (2007); Neto et al. (2009) This wave also includes a huge amount of research works concerned with magnetic-field behavior, electronic properties, Landau levels (LLs) Goerbig (2011); Zhang et al. (2006) and quantum Hall effect Zhang et al. (2005); Novoselov et al. (2007); Kane and Mele (2005). Nearly at the same time, bilayer graphene (BLG), which consists of two closely-located graphene sheets, was also fabricated and tested experimentally. Novoselov et al. (2004); Oostinga et al. (2008); Ohta et al. (2006) The BLG electronic properties are found significantly different depending on details of an A-B stacking process, or Bernal-stacked form, with relatively shifted carbon-atom positions in two layers. Yan et al. (2011a) Bilayer graphene revealed some highly unusual properties, e.g., unconventional quantum Hall effect Novoselov et al. (2006) and cyclotron resonance Henriksen et al. (2008).
A comprehensive theoretical study of the LL degeneracy and quantum Hall effect for BLG in Bernal stacking was reported in Ref. [McCann and Fal’ko, 2006]. Based on an effective two-dimensional Hamiltonian, it was concluded that the low-energy spectrum of BLG can be characterized as parabolic dispersion of chiral quasi-particles with a Barry phase . Meanwhile, its magnetic-filed dependent energy spectrum is found consisting of a set of nearly equidistant four-fold degenerateLLs. In this paper, we will employ such an effective-Hamiltonian approach to establish theoretical formalism for modulated LLs in the presence of a square-scatter array potential in Sec. II.
One of the most unusual and fascinating phenomena related to the electronic spectrum under a perpendicular quantizing magnetic field is the so-called Hofstadter butterfly Hofstadter (1976); Azbel (1964), theoretically predicted in 1976. Here, a recursive fractal electron spectrum was obtained as a function of prime ratio of the magnetic flux passing through a lattice unit cell to a fundamental flux quanta, and these degenerate electronic subbands split and clustered themselves into different patterns corresponding to the value of a given magnetic-flux ratio. By performing first-principles calculations for hexagonal two-dimensional graphene-type lattice, tight-binding approximation resulted in a Hofstadter-butterfly-like clustering pattern, except for an asymmetry with respect to zero wave vector Gumbs and Fekete (1997). Such types of Hofstadter band-structure were also predicted to exist in carbon nanotubes as pseudo-fractal magneto-electronic spectrum Nemec and Cuniberti (2006) and also in bilayer graphene Nemec and Cuniberti (2007).
In a recent experiment, Hofstadter’s butterfly and fractal quantum Hall effect have been extended to Moire superlattices, which are formed as BLG or flakes are coupled to a rotationally aligned hexagonal boron nitride layer Dean et al. (2013); Ponomarenko et al. (2013); Woods et al. (2014) inside a van der Waals heterostructure sample. Hunt et al. (2013) The main idea involved in such experiments is that an elementary lattice-unit cell through which the magnetic flux was measured Hofstadter (1976) will be replaced by a much bigger supercell of the Moire lattice, Schmidt et al. (2014); Yang et al. (2016); Wang et al. (2015) so that the butterfly is expected to be seen at a much lower magnetic field. Additionally, the theory for such butterfly structures in twisted BLG was proposed in Ref. [Bistritzer and MacDonald, 2011], in which long-period spatial patterns can be created precisely at small twist angles. Later, the coexistence of both fractional-quantum-Hall and integral-quantum-Hall states associated with fractal Hofstadter spectrum was confirmed experimentally within such twisted-bilayer structures. Wang et al. (2012) Moreover, specific subband gaps of a Hofstadter’s butterfly were also found for interacting Dirac fermions in graphene. Apalkov and Chakraborty (2014)
On the other hand, in the absence of a magnetic field, a periodic electrostatic field gives rise to new zero-energy states with minigaps and chirality Brey and Fertig (2009), and their composite wave functions can still satisfy the required Bloch periodic condition. Apart from this, new massless Dirac fermions with strong anisotropic properties Park et al. (2008a) are realized in graphene subjected to a slowly-varying periodic potential. Park et al. (2008b) In contrast, a spatially-uniform interaction of Dirac electron with an off-resonant optical field can lead to the formation of either gapped Kibis (2010); Iurov et al. (2011, 2013) or anisotropic dressed Kibis et al. (2017) states depending on polarizations of an imposed irradiation.
Very interestingly, two unique features associated with BLG system have been found. The first property is the intervalley mixing and the quantum interference effect coming from two valleys in the presence of a two-dimensional scattering-lattice potential, while the second property results from a site-energy asymmetry induced by a perpendicular electric field. Here, the latter factor is able to destroy the Hofstadter-type fractal band structures established by an in-plane scattering-lattice potential and an out-of-plane quantizing magnetic field, resulting in strongly deformed self-repeated patterns. Such a phenomena is attributed to the dependence of Bloch-wave expansion coefficients on an applied electric field, leading to an electro-modulation of the Hofstadter-type subband splittings.
The rest of the paper is organized as follows. In Sec. II we present theoretical formalism and acquire a set of characteristic equations, describing electron energy spectrum and corresponding eigenstates for BLG in the presence of both a perpendicular quantizing magnetic field and a two-dimensional periodic electrostatic modulation potential. These results expand the previously studies for a two-dimensional electron gas Kühn et al. (1993) and for a monolayer graphene Gumbs et al. (2014a, b). In Sec. III, we display and discuss our numerical results demonstrating fractal Hofstadter band-structures in different ranges of magnetic field of interest and with various modulation strengths in a close up view for separate LLs and self-repeated superstructures as well. Finally, a brief summary with remarks is given in Sec. IV.
Ii Model and Theory
By considering and valleys, where , and Å, and including sublattices and as well as bilayer structure, the four-component wave functions for each valley can be formally written as McCann and Fal’ko (2006)
where and label the bonds in the bottom layer and and label the bonds in the top layer. For each valley, the graphite tight-binding Hamiltonian matrix within the -plane for Bernal-stacking Yan et al. (2011b) bilayer takes the form
where represents the () or () valley, cm/s is the intralayer (monolayer) Fermi velocity, characterizes the effective mass of electrons in the parabolic band, measures the strength of the interlayer coupling, represents the bias-induced on-site energies of bilayer, with electric field and bilayer separation , and corresponds to a symmetrical bilayer. In addition, we have introduced canonical momentum operators and , where the Landau gauge is chosen for a uniform magnetic field along the vertical direction. The potential of a two-dimensional (2D) scatter array in Eq. (2) is assumed as
where is an integer, stands for the scattering-potential strength, and are the two array periods in the and directions, respectively.
Even in the absence of the scatter potential (i.e., ), the eigen-energies and eigen-states correspond to the Hamiltonian in Eq. (2) can only be calculated numerically. For low-energy states of electrons (with kinetic energy less than ), however, the Hamiltonian in Eq. (2) can be projected onto a one. For such a situation, the wave functions in Eq. (1) for each valley also reduce to a two-component form
and the projected effective Hamiltonian matrix becomes
where in the last term is the identity matrix, the first, second and the rest two terms represents the intralayer, interlayer and bias effects, respectively.
By taking in Eq. (5) as a start, in the strong-field limit, i.e., with a cyclotron frequency , we can formally set in Eq. (5). Based on this simplification, we obtain the analytical form of the eigen-energy levels for each valley ()
where , and correspond to electron and hole energy levels at each valley, respectively, each energy level is spin degenerate, and the lowest two energy levels are four-fold degenerate with respect to both spins and electron-hole pseudospins. If , we get from Eq. (6), which becomes eight-fold degenerate now. The corresponding eigen-states to these electron (hole) energy levels () are calculated as
where is the sample length in the direction, () for (), is the harmonic-oscillator wave functions with a guiding center , the magnetic length, and two coefficients
Assuming , we have and for , which becomes independent of and . On the other hand, for and we obtain
After the scatter array has been included in the strong-field limit, the wave function of the system can be expanded as
where , corresponds to electron and hole states, , and for the first magnetic Brillouin zone, is the number of unit cells spanned by and in the direction, () is the sample length in the direction, is the reciprocal lattice vector in the direction, and is a new quantum number for labeling split subbands from a -degenerated LL in the absence of scatters. Importantly, the above constructed wave function satisfies the usual Bloch condition, i.e.,
where and .
Now, by taking into account of the term in Eq. (5), a tedious but straightforward calculation leads to an explicit expression for the matrix elements of the potential , yielding
where correspond to electron and hole levels, respectively. From Eq. (13) we find two valleys for bilayer graphene can be coupled to each other, which is different from the monolayer graphene Gumbs et al. (2014b). Here, the terms with come from the intravalley contribution, whereas the terms with stand for the intervalley coupling which presents an interference effect. Moreover, we have defined in Eq. (13) two intervalley () coupling factors
where the binomial expansion coefficient for is
Finally, we have introduced in Eq. (14) the following three self-defined functions
with and being the integers prime to each other, is the magnetic flux per unit cell, is the flux quanta, , , is the associated Laguerre polynomial, , , ,
and characterizing the intervalley coupling for and the interference effect as well. Here, the range of extends to all magnetic Brillouin zones in this direction for Umklapp scatterings.
The energy dispersion of the th magnetic band around each valley for this modulated system is a solution of the eigenvector problem with elements of the coefficient matrix given by
where for (i.e., degenerate electron-hole levels) and for , is a composite index, and is an orthonormal eigenvector. Furthermore, the eigenvalues of the system are determined by roots of the characteristic equation .
Iii Numerical Results and Discussions
iii.1 Two-Dimensional Electron Gas and Monolayer Graphene
As a starting point, we first briefly discuss the effect of a two-dimensional (2D) periodically-modulated scattering-lattice potential in Eq. (3) on a 2D electron gas (EG) under a perpendicular quantizing magnetic field . In the absence of this scattering-lattice potential, 2DEG will be quantized into a series of discrete LLs: with , as the cyclotron frequency, and as the effective mass of electrons. These uncoupled LLs are highly degenerate with respect to their guiding centers (or with different cyclotron orbits), where is the magnetic length. In the presence of the scattering-lattice potential, however, these degenerate LLs are strongly coupled to each other and expand into a set of split Landau bands, as shown in Fig. 1. Furthermore, a close-up view in Fig. 1 reveals that a self-similar pattern occurs within the fourth () Landau band at low , just as predicted early by Hofstadter in his seminal work Hofstadter (1976).
If the 2DEG is replaced by a monolayer graphene, a different set of LLs with appears in the absence of a scattering-lattice potential, where , is the Fermi velocity of graphene, and () corresponds to electrons (holes), respectively. In this case, we find that the LL sits at the zero-energy Dirac point instead of for 2DEG, and but not proportional to for 2DEG. After the scattering-lattice potential in Eq. (3) has been employed, these guiding-center degenerated energy levels also expand into a Landau band through mutual couplings, as seen in Fig. 2. However, the mirror symmetry with respect to the band center is lost in Fig. 2 for monolayer graphene, as discussed in details recently by us Gumbs et al. (2014b). Here, one crucial difference between 2DEG and monolayer graphene is the LL separation for graphene, in contrast with a uniform one, , for 2DEG. Consequently, the graphene energy-level separation will decrease with increasing , and therefore, overlaps of many Hofstadter butterflies will show up for higher values as in Fig. 2.
iii.2 Bilayer Graphene
Now, Let us turn our attention to discussions on development of Landau bands in a bilayer graphene. For bilayer graphene subjected to a scattering-lattice potential given by Eq. (3) and under a perpendicular quantizing magnetic field at the same time, our numerical solutions for the eigenvalue equation in Eq. (22) are presented in Figs. 3 - 5 with various scattering strengths . As a whole, we find that degenerate LLs with different guiding centers tend to couple to each other and lead to band-center asymmetric Landau bands within which a fractal Hostadter structure is seen for high magnetic fields . Furthermore, the developed Landau bands for two valleys () are coupled to each other in a bilayer graphene through an Umklapp scattering process across whole magnetic Brillouin zones, which is in contrast with the case for a monolayer graphene where the Landau bands are found independent of a valley.
As indicated in Section II, the valley mixing and interference effect contained in the modulation potential in Eq. (13) are described explicitly by the wave number , where , and nm. The integer power , which measures the peak sharpness of the scattering potential in Eq. (3), is selected as . In the absence of the 2D scattering-lattice potential, each LL under the magnetic flux ratio has a -fold degeneracy for magnetic subbands. We have taken , and , respectively, in Figs. 35. For all three graphs, we only show the lowest four Landau bands for both electrons and holes. All the numerical results which display self-repeated Hofstadter butterfly structures are presented as a function of . Here, all energy levels, except for and , are shifted upwards by a fixed energy offset for . Therefore, we have to made an adjustment to our plots in Figs. 35 so that the electron-hole symmetry can be restored with respect to the zero-energy point. With fixed lattice period , a magnetic-flux ratio can be uniquely related to a magnetic-field strength . The upper bound of in Figs. 35 for observing Hofstadter spectra is found within the range of .
The unperturbed LL spectrum is shown in Eq. (6). In our numerical calculations, we have set and so that the LL structure consists of a few pairs of extremely closely-located levels, corresponding to for two valley indexes. This on-site energy-level separation () depends on or . Additionally, two groups of LaLs associated with and are nearly degenerate due to their very small separations , as found from the inset of Fig. 3. Furthermore, the spin degeneracy in these LLs is kept since none of them depends on spin index. All scaled higher levels staring for have the same dependence which becomes nearly equidistant as and in contrast with the monolayer graphene. Here, the pseudospin index hints a complete electron/hole symmetry for these LLs. After the scattering-lattice potential given by Eq. (3) has been introduced to bilayer graphene, the previously uncoupled and highly-degenerate LLs expand into many magnetic bands with self-similar structures, as can be verified directly from Fig. 3. Since the higher LLs become almost equally separated in bilayer graphene, we expect similar self-repeated structures within a magnetic band for large values.
Because the mixing of LaLs depends on , we present comparisons in Figs. 4 and 5 for strong and intermediate scattering strengths . When the strong scattering strength is , the mixing of and Landau bands is severe, as seen in Fig. 4. In addition, the band mixing is found to increase with magnetic field in this case. If the scattering strength, , is weak, on the other hand, no band mixing appears, as can be verified from Fig. 4.
For intermediate scattering strength in Fig. 5, we find the band mixing still happen, but it occurs at a higher magnetic field. For lower values of , on the other hand, such band mixing is completely negligible, as found from Fig. 5. Therefore, in order to observe Hofstadter butterflies and band mixing effects simultaneously, a stronger scattering strength is preferred. More importantly, the large value of also brings down the required magnetic field for observation to an experimentally accessible level.
iii.3 Effect of Breaking Down of Inversion Symmetry
For a monolayer graphene, the group of wavevector associated with the or point within the crystal first Brillouin zone is found isomorphic to the point group Malard et al. (2009) . For a bilayer graphene with a Bernal stacking, on the other hand, this point group is downgraded to with a lower symmetry. Furthermore, in the presence of a vertical bias field, these two point groups Malard et al. (2009) become and , respectively, for a gated monolayer graphene and a biased bilayer graphene. The loss of an inversion symmetry for a bilayer graphene under a vertical electric field has a profound effect on the formation of fractal Landau subbands in the presence of a square-scatter array potential, as can be seen from Eqs. (8) and (13) where both LL-coupling coefficients and are dependent and the intervalley coupling also becomes possible.
Compared with a monolayer graphene, a bilayer graphene can bring into additional valley mixing and site asymmetry after a perpendicular electric field has been applied. Such intervalley interference and electro-modulation effects can be seen clearly from Eq. (13) for the matrix elements of the scattering potential, i.e., the summation over for fixed and changing coefficients and with and for . In Figs. 3 - 5, only a negligible electric field is employed (), and therefore, no visible distortions of the Hofstadter butterfly, which results from a square 2D periodically-modulated scattering-lattice potential, can be resolved. However, as is slightly increased from to in Fig. 6 for a very weak modulation with and , we find from Fig. 6 that the previously found self-similar patterns within the third Landau band under is very strongly distorted, and therefore disappears.
Moreover, as is further increased from to in Fig. 7 for but , we find from a direct comparison between Fig. 6 and Fig. 7 that the previously observed self-repeated patterns within the fourth Landau band under is destroyed completely. Meanwhile, the mixing of the third and fourth Landau bands is seen clearly even for such a small modulation amplitude in contrast with the result in Fig. 4 for .
Iv Brief Summary
In conclusion, we have developed a theoretical formalism to demonstrate the Hofstadter-type fractal band structure for bilayer graphene in the presence of a two-dimensional periodic electrostatic modulation. The current work can be viewed as a generalization of the previous reported results based on Bloch-wave expansion approach applied to both a two-dimensional electron gas Kühn et al. (1993) and a monolayer graphene Gumbs et al. (2014b). As in previous studies Kühn et al. (1993); Gumbs et al. (2014b), this work includes explicitly deriving a non-perturbative eigenvalue equation, finding numerical solutions which display self-repeated split Landau subbands as a function magnetic flux and periodic subband dispersions as a function of electron wave number in a full magnetic Brillouin zone. Both Hofstadter butterflies and band mixing effects can be displayed simultaneously for a strong scattering strength which further reduces a required magnetic field for such observations to an accessible level.
Interestingly, we find two unique features for the bilayer-graphene system in this study. The first one is related to a bias-modulated mixing of and an interference from two valleys (i.e., non-vanishing intervalley scattering with ) in the presence of a scattering-lattice potential. The second one, however, is associated with a lost inversion symmetry due to a perpendicular electric field, which tends to distort and even destroy the Hofstadter-type fractal band structures established by this scattering-lattice potential, as seen from Figs. 6 and 7. The dependence of Bloch-wave expansion coefficients on the applied electric field directly leads to an electro-deformation of the Hofstadter-type subband splittings, resulting in strongly distorted or even destroyed self-repeated patterns.
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