Floquet states of \alpha-T_{3} lattice: Role of variable Berry phase

Floquet states of - lattice: Role of variable Berry phase

Bashab Dey and Tarun Kanti Ghosh
Department of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016, India

We consider - model illuminated by intense circularly polarized radiation in terahertz regime. We present quasienergy band structure, time-averaged energy spectrum and time-averaged density of states of - model by solving the Floquet Hamiltonian. We obtain exact analytical expressions of the quasienergies at the Dirac points for all values of and field strength. We find that the quasienergy band gaps at the Dirac point decrease with increase of . Approximate forms of quasienergy and band gaps at single and multi-photon resonant points are derived using rotating wave approximation. The expressions reveal a stark dependence of quasienergy on the Berry phase of the charge carrier. The quasi energy flat band remains unchanged in presence of radiation for dice lattice. However, it acquires a dispersion in and around the Dirac and even-photon resonant points when . The valley degeneracy and electron-hole symmetry in the quasienergy spectrum are also broken for . Unlike graphene, the mean energy follows closely the linear dispersion of Dirac cones till near the two-photon resonant points. There are additional peaks in the time-averaged density of states at the Dirac point for .

I Introduction

In recent years, dynamical effect of an intense AC field on transport and optical properties in quantum 2D materials having Dirac-like spectrum has drawn much interest Hanggi (); Eckardt (); Efetov (); Efetov1 (); Lopez (); Oka (); Oka1 (); Zhao (); Kibis (); Wu (); Schilman (); Gupta (). It is seen that intense time-periodic field substantially changes the energy band structure by photon-dressing and consequently the density of states (topological properties of materials). Inducing gap in Dirac materials is an important issue for electronic devices. A stationary energy gap appears at the Dirac points under a circularly polarized radiation Oka (); Zhao (); Kibis (). Also, the gaps appear in the quasienergy spectrum Wu () due to single-photon and multi-photon resonances, which decreases with increase of momentum. Oka and Aoki showed that photovoltaic Hall effect can be induced in graphene under intense ac field Oka (), even in absence of uniform magnetic field. The energy gap at the Dirac point closes as soon as the spin-orbit interaction in graphene monolayer is taken into account Schilman (). The optical conductivity of graphene monolayer under intense field has been reported to show multi-step-like behavior due to sideband modulated optical transitions Wu (). A photo induced topological phase transition in silicene has been proposed by Ezawa Ezawa (). The photoinduced zero-momentum pseudospin polarization, quasienergy band structure and time-averaged DoS of the charge carriers in monolayer silicene have also been studied Schilman-silicene ().

There exists an analogous lattice of graphene, known as - lattice, in which quasiparticles are described by the Dirac-Weyl equation. The - lattice is a honeycomb lattice with two sites (A,B) and an additional site (C) at the center of each hexagon. The C sites are bonded to the alternate corners of the hexagon, say B sites, as shown in Fig. 1. The hopping parameter between A and B sites is and that between C and B sites is . The sites in such a lattice can be subdivided into two categories on the basis of number of nearest neighborshub sites with coordination number 6 and rim sites with coordination number 3. The rim sites form hexagonal lattice with no bonds among them. The hub sites form a triangular lattice. Each hub site is connected to 6 rim sites out of which 3 are equivalent. The hopping parameter alternates between and among the 6 hub-rim bonds from a single hub site. The results in the honeycomb lattice resembling monolayer graphene, which corresponds to Dirac-Weyl system with pseudospin-1/2. On the other hand, leads to the well-studied or dice lattice with pseudospin-1 Sutherland (); Vidal (); Korshunov (); Rizzi (); Urban (); JDMalcolm (). Tuning of from 0 to 1 gradually allows us to study the continuous changes in the electronic properties of massless fermions.

Figure 1: (a) Sketch of the - lattice. (b) Band structure of - lattice using tight-binding model.

It is shown that cold fermionic atoms can be trapped in optical dice lattice Wolf (). The dice lattice can naturally be built by growing trilayer structure of cubic lattices in (111) direction Ran (). Recently, a list of physical quantities like orbital susceptibility Raoux (), optical conductivity Illes (); Cserti (), magnetotransport properties Tutul (); Malcolm (); Duan (), Klein tunneling Klein () and wave-packet dynamics Tutul1 () in - lattice have been studied extensively. The Berry phase has become indispensable ingredient in modern condensed matter physics due to its strong influence on magnetic, transport and optical properties. For example, the variation of the orbital susceptibility with is a direct consequence of the variable Berry phase of the -T lattice Raoux (). It has been pointed out that the quantization of the Hall plataues and behavior of the SdH oscillation change with the Berry phase of the -T lattice Tutul (); Malcolm (); Duan ().

In this work, we study quasienergy band structure, time-averaged energy spectrum and time-averaged density of states of - lattice irradiated by circularly polarized light. We provide exact and approximate analytical expressions of the quasienergies at the Dirac points as well as at discrete points for all values of , respectively. We establish a direct connection between the photoinduced band gaps and the variable Berry phase. The quasienergy gap at the Dirac point decreases with . The behavior of the time-averaged quasienergy and time-averaged density states for are completely different from that of graphene case.

This paper is oraganized as follows. In section II, we present preliminary information of the - lattice. In section III, we solve Floquet eigensystem for - lattice driven by circularly polarized light. In particular, we present numerical and analytical results of quasienergy bands and the corresponding band gaps. In section IV, the results of time-averaged energy spectrum and time-averaged density of states are presented. In section V, we discuss main results of our study.

Ii Basic information of model

The tight-binding Bloch Hamiltonian of the system considering only nearest neighbor (NN) hopping is given by


where is the NN hopping amplitude, is parameterized by the angle as and . Here, ’s are the position vectors of the three nearest neighbors with respect to the rim site. Diagonalising the Hamiltonian gives three energy bands () independent of : and . Here correspond to the conduction, flat and valence bands, respectively. A unique feature of its band structure is that a flat band is sandwiched between two dispersive bands which have electron-hole symmetry. The nondispersive band also appears in the Lieb Dagotto (); Shen (); Apaja (); Goldman () as well as Kagome models Green (). The flat band remains dispersion-less for all values of and . On the other hand, the dispersion of the conduction and valence bands is identical to that of graphene. The full band structure is shown in Fig. 1b.

The low-energy Hamiltonian around the two inequivalent Dirac points and can be written as


where , with refers to the K and valleys, respectively and the components of the spin matrix are defined as


In the vicinity of the two Dirac points, are linear in i.e. , implying massless excitations around the Dirac points, as in the case of graphene.

In contrast to the band structure, the normalized eigen vectors depend on and are given by

where . Moreover, the elements of the spinors from top to bottom represent the probability amplitude of staying in sublattices A (rim), B (hub) and C (rim), respectively. The flat band wavefunction exhibits that the probability amplitude of an electronic wave function centered over the hub sites is always zero. Hence, electrons in the flat band remain localized around the rim sites.

For , Eq. (2) reduces to the pseudospin-1 Dirac-Weyl Hamiltonian where are the standard spin-1 matrices.

Berry phase: The topological Berry phase for is simply which is independent of the valleys. For , the -dependent Berry phase in the conduction and valence bands is given by


and for the flat band is given by


Note that the Berry phase is different in the and valleys except for . The Berry phase is smoothly decreasing with increase of and becomes zero at . Later, we will show how the Berry phase appears in quasienergy gaps.

Iii Floquet eigensystem for - model

We consider a circularly polarized electromagnetic radiation propagating perpendicular to the - lattice placed in the - plane. The corresponding vector potential is given by where with is the amplitude of the electric field and is the frequency of the radiation. Also, denotes counter-clockwise and clockwise rotations of the circularly polarized light, respectively. The frequency of the driving is small compared to the bandwidth of the system. The vector potential satisfies the time periodicity: with the time-period . The minimal coupling between the charge carrier and the electric field is obtained through the Peierls substitution: with being the electronic charge. The Hamiltonian for the coupling between the charge carriers and the electromagnetic field can be written as


where the matrices are and the dimensionless parameter characterizes the strength of the coupling between electromagnetic radiation and charge carrier with in the THz frequency regime. The dimensionless parameter is less than 1 for the typical intensity of lasers available in the THz frequency regime. In the semiclassical picture, is the energy gained by the charge carrier while travelling a distance with the speed during one period of the radiation. On the other hand, is the minimal photon energy which is being absorbed by the charge carrier.

The total Hamiltonian of a charge carrier near the Dirac point in presence of the electromagnetic radiation is which is periodic in time. By Floquet theory, the solution of time-dependent Schrodinger equation


is given by


Here are the time-periodic pseudo-spinors and are the corresponding quasienergies. There are three independent quasienergy branches along with the three corresponding eigenstates indexed by . Substituting Eq. (9) into Eq. (8), the time-periodic spinor becomes the eigenstate of the Floquet Hamiltonian with the eigenvalue :


Multiplying a phase with being an integer to Eq. (9) and substituting it back to Eq. (10), we obtain


This is also an eigenvalue equation as Eq. (10) but with a shifted quasienergy . Equations (10) and (11) yield the same Floquet mode, with quasienergies differing by an integer multiple of photon energy . Hence, the index corresponds to a whole class of solutions indexed by having a discrete spectrum of quasienergies . Thus, a given Floquet state has multiple quasienergy values repeating in the intervals of . For - model, we have three independent values of quasienergy for a given momentum, which can be attributed to the three independent eigenvalue equations . Due to the infinite spectrum without physical distinguishability, the quasienergies can also confined to a reduced Brillouin zone in energy space with .

In order to calculate the quasienergies and the corresponding states of the Floquet Hamiltonian, we consider the Fourier expansion of


which follows from the temporal periodicity of the Floquet mode. Using Eq. (12), the time-dependent differential Eq. (10) reduces to the time-independent eigensystem problem as


where the diagonal Floquet Hamiltonian in the Floquet basis is


and the off-diagonal interaction Hamiltonian


couples various Fourier modes. Thus, by Floquet matrix theory, we numerically compute the Floquet quasienergies in units of and the corresponding Floquet states of the Floquet Hamiltonian . The following parameters have been used in the numerical calculation: THz, kV/cm, m/s and . Also, are considered for all the plots unless otherwise stated.

iii.1 Exact analytical expressions of quasienergies and band gap at the Dirac points

First, we present exact analytical results of quasienergies and band gaps at the Dirac points. At the Dirac points (), the time-dependent Hamiltonian (in units of ) can be written as


where .

The corresponding Floquet Hamiltonian can be written explicitly as


Let us define an unitary operator given by


where is the identity matrix. By performing the unitary transformation , an effective time-independent Floquet Hamiltonian is obtained


The quasienergy spectra are given by


where Arg and with Arg gives the argument of the complex number . The parameter can be expressed in terms of the Berry phase given by Eqs. (5) and (6). Thus, the quasienergy is directly related to the Berry phase acquires during a cyclic motion of the charge carriers in presence of a circularly polarized radiation.

The three Floquet quasienergy branches may be labelled as , where represent three branches and the Floquet index. The corresponding quasienergy . The quasienergies of the three branches in the first energy Brillouin zone are given by and . The variation of these quasienergies with for fixed value of is shown in Fig. 3.

Figure 2: Plots of quasienergies at the Dirac point vs for various combinations of and : (a) , (b) , (c) , (d) . The green, red and blue dotted curves correspond to the three independent quasienergy branches , respectively.

The quasienergy variations for the combinations of and for which [Fig. 8(a) and (b)] or [Fig. 8(c) and (d)] are identical to each other apart from the interchange of branches. Also, the quasienergy structure for the cases of and are inverted copies of each other for . This implies that spectrum undergoes a flipping on- (i) switching between valleys K and for a given sense of circular polarization and (ii) changing sense of rotation of the polarization for a given valley. The flipping of quasienergies is trivially symmetric for graphene (=0) and dice lattice (=1) on switching of valleys or polarization.

For , the quasienergies within the first energy BZ are obtained as


The same results are obtained by Oka and Aoki Oka () for irradiated graphene. On the other hand, the quasienergies for the dice lattice () are and


Equations (23) and (24) can be combined to write a general form for quasienergy at the Dirac point as


where is the pseudospin of the underlying lattice. The energy gap at the Dirac point for the pseudospin is . The energy gap for graphene is and that for dice lattice is . It can be easily checked from Fig. 3(a) as well as from Eq. (25) that .

Figure 3: Plots of variation of exact quasienergy at the Dirac point with (a) for fixed and (b) for different values of . Exact numerical results are denoted by the triangles. In Fig. (b), the solid green, blue and red lines are for , the dotted lines for and the purple-orange pair is for .

In Fig. (3), we show the variation of the three quasienergy branches with and . In Fig. 3(a), we see electron-hole symmetry is broken for . The quasienergy gap at the Dirac point for graphene is higher than that of dice lattice. Thus, the flat band has a shielding effect on the dipole coupling between the elctron-photon levels.

iii.2 Floquet quasienergy branches and band gaps for large values of momentum

Figure 4: Floquet quasienergy bands for different values of : (a) , (b) , (c) .

The quasienergy band structures within the first two energy Brillouin zones are displayed in Fig. 3 for three different values of . The dotted lines indicate the spectrum for zero intensity of radiation, which are identical for all values of . In the - model, the quasienergies pertaining to different satisfy for and for .

The known results of graphene are reproduced by setting . The quasienergy band structures for and 1 are displayed in Fig. 3. For better visualization, the quasienergy band for is shown separately in Fig. 4. With increase in , the flat band acquires small spikes at and at with The spikes get completely wiped out at . The height of the spikes decreases with increases of the momentum. The band structure gets inverted about the axis on changing the rotation of electric field vector of the circularly polarized light. The band remains flat for all values of when applied radiation is linearly polarized. It is to be noted that there is no splitting in the flat band since it does not have any partner band.

Figure 5: Quasienergy band for at different values of : (i) (green), (ii) (red), (iii) (purple) and (blue).

The gaps between the bands () open up at and at with . Note that the origin of the gap at the Dirac points () is different from that of at finite . The exact analytical expression of the gap at the Dirac point is described in the previous subsection. Here we show the comparison between exact analytical and numerical results.

Figure 6: Plots of variation of gaps with for (a) , (b) and (c) .

The gap at arises due to the AC Stark splitting occurrs due to the multiphoton resonances Gupta (); Home (); Grifoni (); Faisal (); Hsu (); Acosta (). There is a set of Bloch states lying on a circle in the vicinity of the Dirac point -space with radius such that energy difference between the bands is multiples of photon energy: . On illumination, new electron-photon states with energy (conduction band state with photons) and (valence band state with photons) are formed. When i.e , the the degenerate levels split due to the coupling between the electron and the radiation field and the gap opens up at . All the gaps tend to diminish at higher values of momentum.

For example, the gap at corresponds to one-photon resonance decreases monotonously with increase of and becomes vanishingly small for . The flat band acts as a shield for the odd-photon resonances so that the coupling between electron-photon states by the dipole interaction is reduced.

The variation of the gaps with and are shown in Fig. 6 and Fig. 7, respectively. All the gaps increase monotonically with . Moreover, (Solid blue) and (dashed purple) get reduced at higher . In contrast, (dotted red) is found to increase with . The gap (dashed-dotted green) increases very slowly with .

Figure 7: Plots of variation of gaps with for .

Iv Time-averaged energies and density of states

In this section, we discuss about time-averaged quantities such as mean energy and the corresponding density of states. The mean energy is a single-valued quantity, which is independent of the choice of the quasienergy of the Floquet state. The mean energy helps to understand whether the Floquet states are occupied or unoccupied Gupta (); Faisal (); Hsu (). For example, the Floquet states having lower mean quasienergy will be accommodated first. Also, the mean quasienergy can be used to characterize whether the state is electron-like or hole-like Wu ().

Time-averaged/mean energies: The expectation value of the Hamiltonian in a Floquet state is a periodic function of time. This helps us to formulate the energy averaged over a full cycle of a periodic driving and is given by


Incorporating the Fourier series (Eq. (12) into the above equation, we get


Hence, the averaged energy can be viewed as the weighted average of energies posessed by the Fourier harmonics of the Floquet modes.

Figure 8: Time-averaged energy of the three Floquet branches for different values of : (a) , (b) , (c) .

The time-averaged energy corresponding to the Floquet states of the three branches for different values of are shown in Fig. 5. This may be viewed as a modified band structure in presence of radiation. The blue, green and red bands represent the quasielectron, flat band and quasihole states respectively. As , we obtain the field-free Dirac cones shown by the dotted lines. For , the mean energy goes to zero near one-photon and two-photon resonant points due to crossover between quasielectron and quasihole states Wu (). For (graphene), the mean energy goes to zero near - and - resonant points due to crossover between quasielectron and quasihole states Wu (). The vanishing of mean energies at these resonant points is also observed for . But, dice lattice has a non-zero mean energy near - resonance. This can be attributed to the fact that the gap at - resonance becomes vanishingly small and mimics the radiation-free case for . A finite gap exists at the Dirac point for all values of . The three-fold degeneracy at the Dirac point is lifted by the radiation. Careful examination reveals that the symmetric nature of bands (electron-hole symmetry) is slightly disrupted near the Dirac point for . The symmetry is restored at . A distortion occurs in the mean energy spectrum of the flat band at , similar to that obtained in the quasienergy spectrum. The distortions flatten out at .

Time-averaged density of states: The time-averaged density of states over a driving cycle is defined as


Here, the factor appears due to the spin degeneracy. On converting the sum over k to integral, i.e and using the azimuthal symmetry of quasienergy band structure for circularly polarized light, we get the density of states per unit area as


where 10 meV m. and are the dimensionless quasienergies such that . Using the property of the Dirac-delta function, the above integral can be further simplified as


where is a set of quantum numbers and is the -th positive root of for a given .

Figure 9: Time-averaged density of states of quasielectron abnd quasihole states at different values of : (a) , (b) , (c)


Time-averaged density of states of electronlike and holelikes quasienergy bands for three different values of is shown in Fig. 6. The DoS for the flat band quasienergy is shown separately in Fig. 7.

Figure 10: Time-averaged density of states of the flat band very close to the Dirac point for .

Fig. 6(a) shows the DoS for , which is similar to the results obtained by other groups Oka (); Wu (). The peaks resemble the Van Hove singularities occuring due to the extrema in the quasienergy band structure. Apart from large peaks, there are spikes around and with vanishingly small DoS. This is because of the photoinduced gaps at the boundaries of the energy Brillouin zones. A small but finite contribution of DoS in these energy ranges comes from higher momenta, where the quasienergy gaps are effectively closed. Additional peaks are born at the Dirac point for finite as seen in Figs. 7(b) and 7(c). The separation between the peaks centred around decreases with , while that around increases with . This is related to the fact that decreases (increases) with .

V Conclusion

We have investigated the Floquet quasienergy spectrum numerically and analytically for the - lattice driven by circularly polarized radiation. Exact analytical expressions of the quasienergy at the Dirac points for all values of and field strength are provided. The band gap at the Dirac point appears due to the circularly polarized radiation for all values of . The quasienergy gap at the Dirac point decreases with the increase of . Within the rotating wave approximation, we are able to get approximate expressions of quasienergy at single-photon and multi-photon resonant points. Approximate results match very well with the numerical results based on Floquet method. The expressions reveal that the quasienergy is directly related to the Berry phase acquires during a cyclic motion driven by the rotating electric field. The quasi energy flat band remains dispersionless in presence of radiation for dice lattice. However, dispersive spikes appear in and around the Dirac and even- photon resonant points for . The mean energy is non-vanishing around single-photon resonance point for dice lattice unlike . In contrast to graphene, we find that additional peaks appear in the time-averaged density of states at the Dirac point for . The pattern of the DoS near the single-photon and two-photon resonant points varies significantly with .

Vi Acknowledgement

We would like to thank Sonu Verma for useful discussions.

Vii Appendix

vii.1 Analytical results within rotating wave approximation

In this appendix, we shall derive analytical expressions of the quasienergy branches within rotating wave approximation. The analytical expressions help us to understand the Berry phase dependency of the quasienergy bands and band gaps.

The time-periodic Hamiltonian can be transformed in the basis formed by the eigenvectors of the low-energy Hamiltonian with the help of the unitary operator given by

The transformed Hamiltonian reads as


where with being the -component of the spin-1 matrix and


where .

The Schrodinger equation is then given by


We solve the Schrodinger equation by omitting the interband term and get the following solutions:


where . Note that is also a time-periodic function. The quasienergy of is exactly the same as the zero field case. It tells us that all the quasienergy gaps appear due to the interband term .

Let the solution of Eq. (33) be of the form:

we get


Taking and as the discrete Fourier transform of the periodic functions and respectively, we have






where with ( is defined as


The exact expressions of and are obtained as


where is the -th order Bessel fuction, and . It is not possible to solve Eqs. (38), (39), and (40) in closed analytical form. However, owing to the high frequency of radiation, standard rotating wave approximation (RWA) can be used to obtain closed form expressions.

There are two frequency detuning terms namely and , due to presence of an additional dispersionless band. Near the resonance points, , the momentum values are such that the energy difference between the bands equals multiples of photon energy .

For even (excluding 0), the terms and are retained in their respective series. But, for odd integer , we see that retaining the -th term from series allots an odd integer value. So within RWA, all the terms in the series will be rapidly oscillating, allowing us to discard this series altogether. Hence, for odd , we retain only . This leads to two distinct cases for even and odd integers, each of which produces separate systems of coupled differential equations for the determination of Floquet quasienergies.

Case I: For even case, Eqs. (38), (39) and (40) become


Note that Eq. (47) is redundant for case. Equations (46) and (48