Graphene has attracted a great deal of attention for its unique and promising electronic properties since it was discovered experimentally in 2004.^[1,2] Graphene is an allotrope of carbon in the form of a two-dimensional, atomic-scale, and hexagonal lattice. These properties enable researchers to realize electronic devices with novel quantum properties. In recent years, great theoretical and experimental research interests have been focused on multilayer graphene systems. Different from the monolayer graphene, the electronic and transport properties of multilayer graphene can be modified and tuned by changing the number of layers of graphene and stacking order in a system. Among the family of multilayer graphene, the trilayer graphene has drawn intensive research interests due to its intriguing band dispersion. Recently, experimental developments in trilayer graphene have been reported.^[3–7] Experimental and theoretical researches have shown that trilayer graphene has a very high mobility, and exhibits quantum Hall effect. Scherer et al.^[8] have shown that quantum spin Hall state can occur in interacting trilayer honeycomb lattices. Zhang et al.^[9] observed the predicted unconventional sequence of quantum Hall effect plateaux in trilayer graphene. Anomalous and spin Hall states were studied in weakly disordered ABC-stacked trilayer graphene.^[10] Using high-magnetic fields, Kumar et al.^[11] observed compelling evidence of the integer quantum Hall effect in trilayer graphene. When interlayer displacement is zero, fractional quantum Hall states at filling factors 2/3 and −11/3 were observed in BN-encapsulated trilayer graphene.^[12] Barlas^[13] predicted the fractional parity Hall states in trilayer graphene with mirror symmetry. When the degeneracy of the bands is lifted by Coulomb interactions, topological bands in trilayer graphene can lead to anomalous quantum Hall phases.^[14]

On the other hand, achievements in laser technology and microwave techniques have made possible manipulation of quantum systems with a high-frequency electromagnetic field to explore Hall effect, which is based on the Floquet theory of periodically driven quantum systems. Particularly, electronic and transport properties of graphene are currently in the focus of attention in condensed matter physics.^[15–21] However, quantum Hall phases in trilayer graphene exposed to a circularly polarized light have not been extensively and deeply studied so far. Trilayer graphene has two natural stable allotropes: (1) ABA (Bernal stacking), where the atoms of the topmost layer lie exactly on top of those of the bottom layer; (2) ABC (rhombohedral stacking), where one sublattice of the top layer lies above the center of the hexagons in the bottom layer.^[22,23] In this paper, we use the Floquet theory to analyze the effect of circularly polarized light on topological properties of ABA trilayer graphene in a high-frequency limit.

The paper is organized as follows. In Section 2, we present a brief account of ABA-stacked trilayer graphene irradiated by a circularly polarized light by the Floquet theory. The band structure of the irradiated trilayer graphene are studied. In Section 3, the Hall conductivities in the irradiated trilayer graphene and corresponding Chern numbers are studied. Conclusions are given in Section 4.

The effective low-energy Hamiltonian of ABA trilayer graphene can be written as^[13,24]

Considering the trilayer graphene irradiated by circularly polarized light, the vector potential introduced into the Hamiltonian by substitution k → k + eA(t) is

Diagonalizing straightforwardly the Hamiltonian (8), we obtain

Based on Eqs. (13) and (14), we plot the band structures of trilayer graphene in units of ħ v = 688 meV for τκ = −1 in Fig. 1. Figure 1(a) shows that the dispersion of trilayer graphene is a combination of the linear dispersion of monolayer graphene and the quadratic relation of bilayer graphene.^[4]. The on-site sublattice potential can induce the energy gap in trilayer graphene (see Fig. 1(b) for m_s = 3 meV, m_b = 8 meV and γ₁ = 310 meV). When considering the circularly polarized light irradiating onto the ABA trilayer graphene sheet, the band structure evolution with light intensity eA/ħ of 0.186, 0.392, and 0.45 at τκ = −1 are shown in Figs. 1(c)–1(e), respectively. Based on Eqs. (8)–(12), and denote the virtual photon absorption and emission process.^[25] They lead to the term and . Thus the light leads to the closing and reopening of energy gap for the fixed sublattice potential, and process of a band gap closing and reopening happen twice. Both the processes are due to the emission and absorption of a single photon and two virtual photons, respectively. Figure 2 shows the energy gap as a function of light intensity eA/ħ. When τκ = −1, the light can lead to the closing and reopening of the energy gap, while the light only amplifies the energy gap for τκ = 1.

Fig. 1.

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	Fig. 1. Band structure of ABA trilayer graphene (a) with and (b) without sublattice potential in units of ħ v. Band structure evolution of ABA trilayer graphene under the irradiation of polarized light with a light intensity eA/ħ of (c) 0.186, (d) 0.392, and (e) 0.45 at τκ = −1.

Fig. 2.

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	Fig. 2. Energy gap as a function of light intensity eA/ħ. The dashed blue line and the solid red line correspond to τκ = 1 and τκ = −1, respectively.

In this section, we investigate anomalous Hall conductivities and Chern numbers in irradiated ABA trilayer graphene. According to the linear response theory, the intrinsic Hall conductance in zero temperature for valley η and circularity κ is given by the integral of the Berry curvature over occupied states^[27]

Figure 3 shows the calculated conductivity for the two valleys K and K′ as a function of Fermi energy E_F under the irradiation of left-handed circularly polarized light (κ = −1) with light intensities (a) eA/ħ = 0.1, (b) eA/ħ = 0.2, and (c) eA/ħ = 0.5. When Fermi energy locates in the energy gap, the anomalous Hall conductivity for valley K (η = 1) is quantized and unchanged with the increasing light intensity. However, the gap region becomes greater with increasing light intensity. For valley K′ (η = −1), the quantized anomalous Hall conductivities are 2e²/h, 0, −2e²/h with light intensities eA/ħ = 0.1, eA/ħ = 0.2, and eA/ħ = 0.5, respectively. Based on Eqs. (17) and (18), we can find that the changed anomalous Hall conductivities are due to the energy gap changing with the light intensity at different parameters τ and κ. This means that there are topological transitions between anomalous Hall states and Valley Hall states. We analyze Chern numbers in the following paragraphs to confirm the results.

Fig. 3.

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	Fig. 3. Anomalous Hall conductivities as a function of Fermi energy EF under the irradiation of left-handed circularly polarized light κ = −1 in units of e2/h with light intensities: (a) eA/ħ = 0.1, (b) eA/ħ = 0.2, and (c) eA/ħ = 0.5. Blue-dashed lines and red-solid lines denote Hall conductivities in K and K′, respectively.

When the fermi energy is in the band gap, the direct correspondence between the Chern number and the Hall conductance is characterized by σ_xy = Ce²/h,^[27] and the Chern number for fixed κ is calculated by

In Fig. 4, we plot C_AH and C_VH versus light intensity A for κ = −1. This is consistent with the results shown in Fig. 2. Before the first band gap closing and reopening, the Chern numbers of the two valley sectors in two blocks take the same value 1. As a whole, the system shows a quantized anomalous Hall state (C_AH = 4 and C_VH = 0). With increasing A, the band gap closing and reopening occur in monolayer-like block, and the corresponding Chern numbers of the two valley sectors take the values 1 and −1, respectively. The Chern numbers in the bilayer-like block keep unchanged. Therefore, the system shows coexistence of a quantized anomalous Hall state and a valley Hall state (C_AH = 2 and C_VH = 2). After the band gap closing and reopening in the bilayer-like block, the Chern numbers of the two valley sectors in the two blocks take the values of 1 and −1, respectively. The system shows a quantized valley Hall state (C_AH = 0 and C_VH = 4). Consequently, the sign of the Chern number is changed only in one valley but remains the same in the other valley in the two blocks. Hence, we can realize an anomalous Hall state to the valley Hall state topological phase transition.

Fig. 4.

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	Fig. 4. Chern number as a function of A for κ = −1. The solid red line and the dashed blue line denote the Chen number and the valley Chern number corresponding to anomalous Hall and valley Hall states, respectively.

To conclude, we have analyzed Hall conductivities and Chern numbers of ABA trilayer graphene under the application of a circularly polarized light. We have employed the Floquet theory to obtain the effective Hamiltonian with a circularly polarized light in high-frequency regime. We find a process of band gap closing and reopening due to irradiation of light, and it happens twice. The study on the corresponding Chern numbers confirms the light-induced topological phase transition between anomalous Hall state and valley Hall state. The interplay between effective sublattice potential and light-induced circularity-dependent effective potential is responsible for the topological phase transition.