The selection rule of graphene in a composite magnetic field

The generalized tight-binding model with exact diagonalization method is developed to calculate the optical properties of monolayer graphene in the presence of composite magnetic fields. The ratio of the uniform magnetic field and the modulated one accounts for a strong influence on the structure, number, intensity and frequency of absorption peaks, and thus the extra selection rules that are subsequently induced can be explained. When the modulated field increases, each symmetric peak, under a uniform magnetic field, splits into a pair of asymmetric peaks with lower intensities. The threshold absorption frequency exhibits an obvious evolution in terms of a redshift. These absorption peaks obey the same selection rule that is followed by Landau level transitions. Moreover, at a sufficiently strong modulation strength, the extra peaks in the absorption spectrum might arise from different selection rules.


Introduction
Monolayer graphene (MG), constructed from a single layer of carbon atoms densely packed in a hexagonal lattice, was successfully produced by mechanical exfoliation [1,2] and chemical vapor deposition [3][4][5][6][7]. This particular material constitutes an excellent system for studying two-dimensional (2D) physical properties, e.g., quantum Hall effects [8][9][10][11]. In the low-energy region |E c,v | ≤ 1 eV, MG possesses isotropic linear bands crossing at the K (K ) point and is regarded as a 2D zero-gap semiconductor, where c (v) indicates the conduction (valence) bands [12]. The linear bands are symmetric about the Fermi level (E F = 0) and become nonlinear and anisotropic at higher energy |E c,v | > 1 [12]. Most importantly, the quasiparticles related to the linear bands can be described by the Dirac-like Hamiltonian [13] that is associated with relativistic particles and dominates the low-energy physical properties [11,14,15]. This special electronic structure has been verified by experimental measurements [2,16].
MG has become a potential material candidate for nano-devices due to its exotic electronic properties. A good understanding of the behavior of MG under external fields [17][18][19][20][21][22][23][24][25][26][27] is useful for improving the characteristics of graphene-based nano-devices. In the presence of a uniform perpendicular magnetic field, the linear bands change into dispersionless Landau levels (LLs) which obey the specific relationship E c,v ∝ √ n c,v B 0 , where n c (n v ) is the quantum number of the conduction (valence) states and B 0 is the magnetic field strength. The related anomalous quantum Hall effects [11,28,29] and particular optical excitations [17] have been verified experimentally [10,11]. The anisotropic behavior of dispersive quasi-Landau levels (QLLs) and the special selection rules presented in the related optical absorption spectra are shown for a modulated magnetic field. Furthermore, Haldane predicted the existence of quantum Hall effects in MG when it is subjected to the modulated magnetic field even without any net magnetic flux through the entire space [9]. For two cases of composite fields, a uniform magnetic field combined with a modulated magnetic field and a uniform magnetic field combined with a modulated electric potential, the LL properties are drastically changed by the modulated field [30][31][32][33][34]. The broken symmetry, displacement of the localization center, and alteration of the amplitude of the LL wave functions can, in some cases, be observed [35,36].
MG is predicted to exhibit feature-rich optical absorption spectra. The spectral intensity is proportional to the frequency, but no prominent peak exists at low frequency [37]. However, a uniform perpendicular magnetic field can lead to a number of symmetric absorption peaks originating from LLs. Each peak obeys the specific selection rule ∆n = |n c − n v | = 1, which has been confirmed by magneto-transmission measurements [16,17,38]. The selection rule is ascribed to the spatial symmetric configuration of the magneto-electronic wave function. Under a modulated magnetic field, the optical spectra exhibit many asymmetric principal peaks and subpeaks. The former satisfy a selection rule similar to those pertinent to LLs, while the latter do not [39,40]. For a better understanding, it is necessary to investigate the primary features of optical excitations in composite magnetic fields with various field strength ratios between the uniform and the modulated fields.
The generalized tight-binding model with exact diagonalization method is developed to resolve the optical absorption spectra. By rearranging the tight-binding functions, the giant Hermitian matrix corresponding to the experimental fields can be transformed into a band-like matrix, so that the numerical computation time is greatly reduced [41][42][43][44]. Moreover, the πelectronic structure of MG can be solved in the wide energy range of 5 eV, a solution proven valid regardless of whether a magnetic, electric or composite field is applied. In this work, we focus on the optical absorption spectra of MG under a composite magnetic field. By means of controlling the ratio between a uniform magnetic field and a modulated one, the magnetooptical properties can be thoroughly explored and then presented in detail. For an increased modulated field, each symmetric peak, under a uniform magnetic field, splits into a pair of asymmetric peaks. Redshift is an obvious evolution exhibited by the threshold absorption frequency. Perceivably, each peak obeys the same selection rule as in the situation where only the uniform magnetic field exists until the modulated field is increased beyond a certain point. As a result, important differences of the absorption peaks are observed with regard to their structure, number, intensity, frequency and selection rule when the applied magnetic field is varied among the uniform, composite and purely modulated ones.

Band-like Hamiltonian matrix in external fields
The low-frequency optical properties of MG are determined by the π-electronic structure resulting from the 2p z orbitals of the carbon atoms. The generalized tight-binding model with the exact diagonalization method is developed to characterize the electronic properties, and then the gradient approximation is applied to obtain the optical-absorption spectra. In the presence of magnetic fields, the vector potential induces Peierls phases [39,41,45,46], which are then accumulated in the Bloch wave functions. Such a phase owns the extra period R or changes the unit cell. The enlarged rectangular unit cell, marked by the green rectangle in Fig. 1, is chosen as the primitive unit cell. In this work, the major discussions focus on R along the armchair direction. MG is subjected to three kinds of magnetic fields: a uniform perpendicular magnetic field, a periodically modulated magnetic field, and a composite magnetic field comprised of both the uniform and modulated parts. In the case of the uniform magnetic field B 0 = B 0 z, the period is given by 42Å is the C-C bond length, and φ 0 (= hc/e = 4.1356 × 10 −15 [T/m 2 ]) is the unit flux quantum. In the case of the modulated magnetic field, B M = B M sin(2πx/l M ) z is exerted along the armchair direction, where B M is the field strength and l M is the period length with the modulation period R = R M = l M /3b . In the case of the composite field, the period R = R C is the least common multiple of R 0 and R M . An enlarged rectangular unit cell induced by an external field encompasses 2R a atoms and 2R b atoms. The Hamiltonian matrix is a 4R × 4R Hermitian matrix spanned by 4R TB functions. Based on the arrangement of odd and even atoms in the primitive cell, the Bloch wave function |Ψ k can be expressed as: where |a mk (|b mk ) is the TB function corresponding to the 2p z orbital of the mth a (b) atom.   where are the Peierls phases in the off-diagonal elements associated with the uniform and modulated magnetic fields, respectively.
When electrons are excited from occupied valence to unoccupied conduction bands by an electro-magnetic field, only inter-π-band excitations exist in the zero temperature case. Based on Fermi's golden rule, the optical absorption function results in the following form where f (E(k, n)) is the Fermi-Dirac distribution function, Γ (= 2 × 10 −4 γ 0 ) is the broadening parameter, P is the momentum, and m e is the bare electron mass. Meanwhile, n and n are the quantum numbers with respect to the initial and final states. The electric polarization E along y is taken into account for discussions. Within the gradient approximation [47][48][49][50][51], the velocity Equation (4) implies that the main features of the wave functions are key factors in determining the selection rules and the absorption intensity of the optical excitations. Similar gradient approximations have been successfully applied to investigate optical spectra of carbon-related systems, e.g., graphite [47], graphite intercalation compounds [51], carbon nanotubes [52], fewlayer graphenes [53], and graphene nanoribbons [54].

Uniform magnetic field combined with a modulated magnetic field
Low-energy band structures of MG exhibit rich features in the presence of magnetic fields. The uniform perpendicular magnetic field causes the states to congregate and induces dispersionless Landau levels (LLs), indicated by the red curves in Fig. 2 for B 0 = 4 T. The unoccupied LLs and occupied LLs are symmetric about the Fermi level (E F = 0). Each LL is characterized by the quantum number n c,v , which corresponds to the number of zeros in the eigenvectors of harmonic oscillators [42,55]. Each LL is fourfold degenerate for each (k x , k y ) state without considering the spin degeneracy. Its energy can be approximated by a simple square-root [8,56], which is valid only for |E c,v n | ≤ 1 eV [8]. The main characteristics of the LLs are affected by the modulated magnetic field, as shown in Fig. 2  curvature with a larger density of states (DOS). On the contrary, fewer states congregate at k α be , a situation leading to the larger band curvature and a smaller DOS.
The band structure may be considerably modified by increasing the strength of the modulated magnetic field. As B M increases to the magnitude of B 0 , more complex energy spectra are introduced, as shown in Fig. 2 Fig. 2). Therefore, each type of band-edge state corresponds to two different energies. The band-edge states belonging to the α (β ) type own the two energies k α− be and k α+ be (k β − be and k β + be ), as indicated by the orange and green triangles (blue and red circles) in the figure. Furthermore, the band structure in the composite fields displays band-edge state energies similar to those found under the influence of only a modulated magnetic field. In the presence of a pure modulated magnetic field, energy bands are partial flat bands at E F = 0 and parabolic bands elsewhere, as shown in Fig. 3(b). Each parabolic subband has one specific band-edge state k sp be , respectively. The latter k e+ be and k e− be for the modulation strength B M own the same energy as k α+ be and k α− be , respectively. The quantum number n c,v corresponding to the partial flat bands at E F = 0 is defined as zero, and n c,v ≥ 1 are associated with the n-th (n = 1, 2, 3 . . .) conduction and valence parabolic subbands. The definition of n c,v is discussed in the following paragraph and graphically illustrated in Fig. 5.
The above-mentioned features of band structures influenced by the modulated magnetic fields would be reflected in the density of states (DOS). In the presence of a uniform magnetic field, DOS exhibits many delta-function-like peaks resulting from LLs, as shown by the red curve for B 0 = 4 T in Fig. 4(a). Such peaks suggest that LLs possess 0D energy levels. With an applied modulated magnetic field, each peak, except for the n c,v = 0 LL at E F = 0, splits into a pair of square-root divergent peaks ω α and ω β , as shown by the black curve in Fig. 4(a) for R M = 500 and B M = 0.5 T. The square-root divergence implies that the 0D energy level becomes the 1D energy band. Each pair of peaks ω α and ω β corresponds to the k α be and k β be states, respectively, and the peak frequencies are identical to those of LLs at B 0 + B M and B 0 − B M . However, the two peaks ω α and ω β possess different peak heights owing to the distinct band curvatures at the two kinds of band edge states. The frequency and intensity are strongly dependent on the modulation strength. The peak heights decline and the spacing between ω α and ω β rises for a large modulation strength B M = 4 T, as shown in Fig. 4(b). This reflects the fact that the greater curvatures and wider amplitudes of the oscillation subbands are a result of the increasing B M . As the modulated field strength is further raised to B M = 32 T (blue dashed curves in Fig.  4(c)), the DOS displays some features similar to those of the DOS in the modulated magnetic field where R M = 500 and B M = 32 T (black solid curves). The modulated magnetic field alone leads to a symmetric delta-function-like peak at ω = 0 (inset in Fig. 4(c)), as well as asymmetric square-root divergent peaks. The former comes from the partial flat bands at E F = 0. The latter can be further divided into weak subpeaks (ω n s ) and strong principal peaks (ω n p ) that are, respectively, dominated by the extra band-edge states (k e+ be and k e− be ) and specific bandedge states (k sp be ). In the cases of a composite magnetic field, the frequencies and intensities of the subpeaks are almost the same as those in the modulated magnetic field alone. However, the principal peaks possess pair structures of ω n P+ and ω n P− , which respectively correspond to the two different band-edge states k With a large modulation strength B M B 0 , the wave function is characterized by a narrow distribution and an increased amplitude, as shown in Figs. 5(i)-5(l). For n c,v = 0, the symmetry of the wave functions is almost restored. The amplitude B c,v o , owing to the fact that it has the same number of zeros as A c,v o , also contributes to the wave function. This means that carrier transfers occur between the aand b-atoms. For n c,v ≥ 1, the spatial symmetry is broken and the zero mode is changed. The wave functions at k α+ be and k α− be are different combinations of two subenvelope functions, of which the location centers are at zero net field strength, as indicated by the green and orange curves, respectively. These features are also obtained in the pure modulated magnetic field case. The wave functions associated with k y = k 1 for B M = 32 T and R M = 500 in Fig. 4(b) have two nearly overlapping subenvelope functions φ 1 and φ 0 with respect to 1 and 0 zero points, as shown in Figs. 5(m)-5(p) by the green and orange dashed curves. The quantum number n c,v is defined by the larger number of zeros of the subenvelope functions. As the wave vector moves toward the extra band-edge states k e+ be and k e− be , the two subenvelope functions φ 1 and φ 0 shift to the center located at zero net field strength and overlap with each other. For example, the overlapping wave functions A c o correspond to k e+ be and k e− be superpositioned by the different combinations of φ 0 + φ 1 and −φ 0 + φ 1 , indicated by the green and red curves, respectively. The low-frequency optical absorption spectrum of the LLs presents numerous features. Many delta-function-like peaks with an identical intensity exist, as shown in Fig. 6(a) for B 0 = 4 T by the red curves. A single peak ω nn LL comes from two equivalent transition channels: from the valence LL of n v (n v ) to the conduction LL of n c (n c ). The quantum numbers related to the transition between the LLs must obey the specific selection rule ∆n = |n c(v) − n v(c) | = 1. The main reason for this is that the velocity matrix M cv has a non-zero value only when A c,v o and B c,v o possess the same number of zeros and it is a dominant factor for the optical excitations of the prominent peaks. However, the modulated magnetic field modifies the number and intensities of the absorption peaks. The delta-function-like peak splits into two kinds of square-root-divergent peaks ω nn α and ω nn β with lower intensities, as shown by the black curves. The divergences of the former and the latter occur near the right-and left-hand sides of ω nn LL . Each ω nn α (ω nn β ) originates from the transitions of k α be (n v ) → k α be (n c + 1) and k α be , its absorption frequency is same as that generated from the LLs at B 0 − B M = 3.5 T (B 0 + B M = 4.5 T). This leads to a redshift for the threshold absorption frequency. The peak intensity of ω nn α is lower than that of ω nn β since the DOS of k α be is weaker than that of k β be . These absorption peaks obey the selection rule ∆n = 1, similarly to uniform magnetic field case. This is due to the fact that the simple relation between A c,v o and B c,v o of the wave functions is almost preserved, even though the modulated field does have some effects on the wave functions.
As the modulated field strength B M is increased to B 0 = 4 T, the frequency, intensity, peak number and selection rules in the optical absorption spectrum have evident variety, as shown in Fig. 6(b). The frequencies of ω nn α 's are largely reduced, but the opposite is true for ω nn β 's. The fact that the α and β band-edge states, respectively, approach or depart from E F = 0 is the main reason. This might lead to an abnormal relationship of absorption frequencies in that ω nn α 's with the large n c,v 's occur at a lower frequency than ω nn β with the small n c,v 's. For example, ω 12 α , ω 67 α and ω 9 10 α are smaller than ω 01 β , ω 12 β and ω 23 β , respectively. The ratio between the intensities of ω nn α and ω nn β is further diminished with increasing B M . This can be clearly observed in transitions with small n c,v 's, i.e., ω 01 β is ten times stronger than ω 01 α in terms of the intensity. Moreover, more absorption peaks exist when B M is increased. In addition to the peaks ω nn α and ω nn β obeying the selection rule ∆n = 1, two extra peaks with ∆n = 2 and 3, ω 02 α and ω 03 α , are generated. These two peaks reflect the fact that the wave functions of the LLs with n c,v = 0 are drastically changed by the modulated magnetic field.
As the modulated field strength is further raised to B M = 32 T (red dashed curves in Fig.  7), the absorption spectrum displays certain important features in the pure modulated magnetic field where R M = 500 and B M = 32 T (black solid curves in Fig. 7). When the absorption peaks are dominated by the modulated magnetic field, they can be divided into principal peaks ω nn+1 pal peaks ω nn+1 P with right-hand divergence are much stronger than the subpeaks. They are attributed to the higher DOS consisting of the β type edge-states (specific band-edge states) with the quantum numbers n and n + 1 in the composite field B M B 0 (pure modulated field). These absorption peaks obey the selection rule ∆n = 1, similarly to the uniform magnetic field case. The principal peaks, however, possess a pair of peaks with ω nn+1 P− and ω nn+1 P+ , which respectively correspond to two different field strengths, |B 0 − B M | = 28 T and |B 0 + B M | = 36 T, and thus the difference between the two field strengths leads to distinct absorption frequencies.
For B M B 0 , one can anticipate that the frequency discrepancy between the pair ω nn+1 P− and ω nn+1 P+ becomes very small and eventually merges into a single peak, ω nn+1 P , i.e., the absorption spectrum is restored to the status of the pure modulated magnetic field case.
The absorption frequency dependence on the modulated field strength is shown in Fig. 8 for R M = 500 and B 0 = 4 T. In the range B M ≤ B 0 , the absorption peaks can be further divided into two categories based on the different field dependence. The frequencies of ω nn α grow with increasing field strength, while the frequencies of ω nn β decline. Each of the absorption peaks ω nn α and ω nn β has a linear relationship with B M . This reflects the fact that the subband amplitudes are proportional to B M within the range. It should be noted that the abnormal relationship of the absorption frequency, the intersection of ω nn α 's with large n c,v 's and ω nn β 's with small n c,v 's, occurs at a sufficiently large modulated field. However, in the higher absorption frequency region or for the field range B M > B 0 , the linear relationship is broken since the subband amplitudes are no longer linearly magnified by B M .
In addition to the modulated magnetic field, the modulated electric potential also induces rich features in the magneto-absorption spectra. In both cases, the four-fold degenerate LLs, except the n c,v = 0 LL for the modulated magnetic field case, change into periodic oscillation subbands with double-degeneracy. The energy dispersions become stronger for an increased modulated field strength. With regard to the wave functions, the symmetry breaking of the LL spatial distributions is enhanced as the modulation potential or the quantum number grows. However, the asymmetry of the LL wave functions only at n c,v = 0 is revealed when B M is comparable to B 0 . This implies that the optical absorption spectrum simultaneously exhibits the original peaks of ∆n = 1 and the extra peaks of ∆n = 1. In the case of the modulated magnetic field, the extra selection rules, i.e., ∆n = 2 and ∆n = 3, come into existence only when B M approximately equals B 0 . On the other hand, extra selection rules induced by the electric potential would be generated when the modulation strength is increased. The extra peaks of ∆n = 1 are relatively easily observed at higher frequency.

Conclusion
The low-frequency optical absorption spectra in the presence of composite magnetic fields are studied by the generalized tight-binding model and gradient approximation. By means of controlling the ratio between B M and B 0 , systematic research on the magneto-optical spectra of MG can be thoroughly carried out. Under a uniform magnetic field, many delta-function-like peaks with uniform intensities exist in the optical spectrum. Each peak is generated from the LLs and satisfies the specific selection rule ∆n = 1. At B M B 0 , each LL splits into two periodic oscillatory subbands with two kinds of band-edge states k α be and k β be , except for the LL with n c,v = 0. Nevertheless, the spatial symmetry of the wave functions remains unchanged. The simple relationship between A c,v o and B c,v o of the wave functions is almost preserved. Optical spectra exhibit many pairs of square-root-divergent peaks ω nn α and ω nn β that obey the selection rule ∆n = 1. The two peak frequencies associated with k α be and k β be are the same as those generated from the LLs at B 0 − B M and B 0 + B M , respectively. The implication is that a redshift occurs in the threshold absorption frequency. When B M is increased to the magnitude of