Dynamical conductivity of gated AA-stacking multilayer graphene with spin–orbital coupling

Abstract

An efficient method, without the numerical diagonalization of a huge Hamiltonian matrix and calculation of a tedious Green’s function, is proposed to calculate the exact energy spectrum and dynamical conductivity in gated AA-stacking N-layer graphene (AANLG) with intrinsic spin–orbital coupling (SOC). The 2N × 2N tight-binding Hamiltonian matrix, velocity operator and Green’s function representation of AANLG are simultaneously reduced to N 2 × 2 diagonal block matrices through a proper transformation matrix. Gated AANLG with intrinsic SOC is reduced to N graphene-like layers. The energy spectrum of a graphene-like layer is E = ε_⊥ ± ε_∥, and ε_⊥ depends on the interlayer interaction, gated voltage and layer number. , where E_MG is the energy spectrum of a graphene monolayer and Δ is the magnitude of the intrinsic SOC. More importantly, by inserting the diagonal block velocity operator and Green’s function representation in the Kubo formula, the exact dynamical conductivity of AANLG is shown to be , the sum of the dynamical conductivity of N graphene-like layers. The analytical form of σ_j is presented and the dependence of σ_j on ε_⊥, Δ, and chemical potential is clearly demonstrated. Moreover, the effect of the Rashba SOC on the electronic properties of AANLG is explored with the exact energy spectrum presented.

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1 Introduction

Graphene and its members of the graphene family, including AA-, AB- and ABC-stacking graphene, have long attracted great attention due to their striking physical properties. Graphene, a pure two-dimensional (2D) system, is an atomic sheet peeled off from graphite.^1,2 Carbon atoms are brought together and packed into a hexagonal lattice to form a graphene sheet. Such a geometrical structure consequently brings about a pair of low-lying linear energy bands. Electrons on the graphene sheet behave like relativistic massless particles. This linear dispersion induces a variety of unique electronic properties, such as electron–hole symmetry, Klein tunneling, high mobility at room temperature, non-zero conductivity, and anomalous quantum Hall effect.^3–17 Owing to its manifestation of fascinating effects, graphene is a promising material expected to play a vital role in technological applications, e.g., display screens, electric circuits, solar cells, analog electronics and photonics/optoelectronics.^18–25

Multilayer graphene is a pile of several graphene layers held together by van der Waals forces. The low-energy physical properties depend strongly both on the stacking order and on the number of layers.^26–31 The most studied multilayer graphene is AB-stacking bilayer graphene^32–40 and ABC-stacking trilayer graphene.^41–44 AB-stacking bilayer graphene shows four parabolic bands around the Dirac points. The contact between the valence and conduction bands makes AB-stacking bilayer graphene a zero gap semimetal. A band gap is opened by the application of a vertical electric field.⁴⁰ The low energy dispersions of ABC-stacked trilayer graphene are described by two remarkably flat bands. The two-fold degeneracy in the band structure can be readily lifted by a perpendicular electrical field. Due to progress in the fabrication and manipulation of graphene layers, AA-stacking graphite and AA-stacking multilayer graphene have been produced.^45,46 Following this, theoretical and experimental studies were conducted in order to explore the electronic properties of AA-stacking bilayer and multilayer graphene, e.g. infrared spectra, Raman spectra, Landau-level energies, absorption spectra, magneto absorption spectra, static polarization, and dynamical conductivity.^45–53

The aforementioned study shows that with different stacking sequences, the energy dispersions around the Dirac point of AA-stacking bilayer graphene present two pairs of linear bands, which are significantly distinct from those of AB-stacking bilayer graphene exhibiting four parabolic bands. The application of a vertical electric field induces a band gap in AB-stacking bilayer graphene. The external electric field causes a different effect on the energy dispersions of AA-stacking bilayer graphene. It does not destroy the characteristics of the linear bands and chiefly shifts the linear bands upward or downward. The effects of the stacking sequence on the electronic properties are also revealed in the low energy dispersions of AAA-, ABA-, and ABC-stacking trilayer graphene. AAA-stacking trilayer graphene possesses three pairs of linear bands. The energy spectrum of trilayer graphene with ABA-stacking consist of the superposition of one pair of linear bands and four parabolic bands, similar to the spectrum of one monolayer graphene and one modified bilayer graphene. ABC-stacking trilayer graphene exhibits approximately cubic bands. The gated field produces an electrically tunable band gap in ABC-stacking trilayer graphene, while the external electric field can not give rise to a band gap in AAA- and ABA-stacking trilayer graphene.

Increasing the layer number, the appearance of interlayer interactions, and the application of an external field will lead to more difficulties in exploring the electronic properties of multilayer graphene. For instance, a 2N × 2N tight-binding (TB) Hamiltonian matrix is constructed and used to describe an N-layer graphene with the nearest neighbor interactions taken into account. The exact diagonalization of a 2N × 2N Hamiltonian matrix will be cumbersome as the layer number N increases. A high-rank Hamiltonian matrix gives rise to more tasks in calculation of the Green’s function, which is generally adopted to study the minimal and dynamical conductivities. Moreover, the mirror-symmetry-breaking, caused by a vertical electric field applied to a multilayer graphene, also increases the difficulty of the diagonalization of the Hamiltonian matrix. Most studies focus on the exploration of the physical properties with the layer number N < 3, e.g., AB-stacking bilayer graphene, and ABC-stacking trilayer graphene. Recently, investigations of the dynamical conductivity of AA-stacking graphene and the static polarization of AAA-stacking graphene have been reported.^52,53 A model that can deal with physical properties of multilayer graphene in different stacking orders or various layer numbers N under an external field is envisioned and desired. We previously presented analytical modes to exactly describe the minimal conductivity of AB-stacking multilayer graphene⁵⁴ and exact Landau levels of AA-stacking multilayer graphene.⁵⁵

In this work, an analytical model is proposed in order to derive the dynamical conductivity and energy spectrum in gated AANLG with intrinsic SOC. The 2N × 2N Hamiltonian matrix of AANLG is decomposed into N 2 × 2 diagonal block matrices. AANLG is decoupled into N graphene-like layers. Thus, a closed form of the energy spectrum is obtained. Application of the current analytical model to the study of the dynamical conductivity of AANLG is conducted. It is shown that the dynamical conductivity of AANLG is equal to the sum of the dynamical conductivities of N graphene-like layers with/without intrinsic SOC. Above all, the presented model can efficiently and exactly output the energy spectrum and dynamical conductivity in gated AANLG with intrinsic SOC and avoid the diagonalization of a huge Hamiltonian matrix and the calculation of the associated Green’s function.

2 Gate-tuned energy spectrum of AANLG with spin–orbital coupling

Graphene is a two dimensional atomic sheet made up of carbon atoms, which are precisely packed in a planar hexagonal lattice, viewed as a bipartite lattice composed of two interpenetrating triangular sublattices. The carbon–carbon bond length is b = 1.42 and the lattice vector is equal to . A primitive cell contains two atoms denoted as A and B. With SOC taken into consideration, the Hamiltonian H_MG of monolayer graphene is^56,57

H_MG = h₀ + h_ISO + h_R, (1)

where the first term, , is the Hamiltonian operator of monolayer graphene without SOC. c_i⁺(c_j) is the creation (annihilation) operator and creates (annihilates) an electron at the site i(j). α₀ is the intralayer nearest-neighbor hopping between atoms A and B on the same graphene layer, as illustrated in Fig. 1. The second term h_ISO is the intrinsic spin–orbit interaction. The third term is the Rashba SOC, which is induced by the external perpendicular electric field or the interaction with the substrate. The Hamiltonian operator of the Rashba SOC is , where λ_R is the magnitude of the Rashba SOC. S is the Pauli vector, the subscripts μ and ν represent the spin index, and d_i,j is the unit vector pointing from atom site i to its nearest neighbor j.

Fig. 1 The geometric structure of AA-stacking multilayer graphene and the intralayer and interlayer interactions.

Without the Rashba SOC (λ_R = 0), the TB Hamiltonian matrix, spanned by the periodic Bloch functions ∣A〉 and ∣B〉, is⁵²

where . b_j represents the three nearest neighbors on the same graphene plane and k is the in-plane wave vector. Δ is the strength of ISOC and s_z = ±1 represents the up or down spin. τ_z = ±1 at the Dirac points K and K′. The energy dispersions are .

Furthermore, the TB Hamiltonian matrix of the Hamiltonian H_MG = h₀ + h_ISO + h_R, acting on the periodic Bloch functions ∣A↑〉, ∣B↑〉, ∣A↓〉, ∣B↓〉, is⁵⁷

where the Rashba SO interaction between ∣A↑〉 and ∣B↓〉 (∣B↑〉 and ∣A↓〉) is neglected because it is much weaker than λ_R.⁵⁷ The analytical energy dispersions are

2.1 Energy spectra of AANLG with intrinsic SOC

By stacking N layers of graphene directly on each other with an interlayer distance between graphene layers c = 3.35 Å,⁵⁸ an AANLG is formed, as shown in Fig. 1. In the stacking direction, N atoms of A(B) form a linear chain. The primitive unit cell contains 2N atoms, denoted as A₁, A₂, …, A_N, B₁, B₂, …, B_N. The first Brillouin zone is the same as that of a single graphene layer. The Hamiltonian of AANLG is given by

H_AANLA = H₀ + H_ISO + H_R, (5)

where H₀ is the Hamiltonian operator of AANLG without SOC, and H_ISO and H_R are caused by the ISOC and Rashba SOC interactions. Following the discussion in the subsection above, we first neglect the Rashba effect, i.e., H_R = 0. In the presence of an electric field, the Hamiltonian representation of AANLG with intrinsic SOC, spanned by periodic Bloch functions ∣A₁↑〉, ∣A₂↑〉, …, ∣A_N↑〉, ∣B₁↑〉, ∣B₂↑〉, …, ∣B_N↑〉, is a 2N × 2N matrix, readingwhere H_AA, H_AB, H_BA, and H_BB are N × N matrices. The Hamiltonian operators H_AA and H_AB describe an N-site linear chain with the intrinsic SOC subjected to a parallel electric field.

Recent research,^59,60 using ab initio calculations and the TB method, shows that in Bernal stacking bilayer graphene and ABC stacking trilayer graphene, many intralayer and interlayer SOC parameters are included to fully describe the spin–orbital interactions in the TB model. The numerical fitting of the TB parameters to the results of the ab initio calculation shows that the magnitudes of the intrinsic (Δ) and Rashba spin–orbital interactions (λ_R) are layer-position-dependent. What’s more, the strength of the interlayer intrinsic SOC is much weaker than that of the intralayer intrinsic SOC. Accordingly, we neglect the interlayer intrinsic spin–orbital interactions in our case and then take the intralayer intrinsic and Rashba spin–orbital interactions into account. Thus, the two 4 × 4 H_AA and H_BB Hamiltonian matrices of AA-stacking quad-layer graphene, for instance, are expressed as the following,

where Δ_j (j = 1, 2, 3 and 4) is the intralayer intrinsic spin–orbital interaction of the j-layer graphene. V = ∣e∣Fc is the effective electric potential difference between the adjacent layers caused by the external electric field. Then, V is denoted as the gate voltage. In multilayer graphene, the potential drop between two adjacent layers might be affected by the screening processes.⁶¹ For simplicity, we also assume that the potential drop is the same on each graphene layer. The interlayer hopping parameter, α₁, couples the two A (or B) atoms from two adjacent layers [Fig. 1]. α₃, the interlayer interaction between atoms A and B from the two adjacent layers, results in a weak electron–hole asymmetry in AANLG.^55,58 The values of the hopping integrals are α₁ = 0.361 eV and α₃ = −0.032 eV.⁵⁸ Only the main interlayer interaction α₁ is taken into consideration because α₃ ≪ α₁. The matrix element H_AB, resulting from the intralayer interaction, reads

The layer-position-dependent intralayer intrinsic spin–orbital interaction and Rashba SOC destroy the inversion symmetry of AANGL. The breaking of the inversion symmetry complicates the analysis and discussion. For simplicity, the intralayer intrinsic spin–orbital interaction is assumed to be independent of the vertical positions; that is, Δ_j = Δ and λ_R,j = λ_R. Notably, the H_AA(H_BB) is the sum of two matrices H_V and H_ISO and reads

where is an N × N identity matrix. H_V describes an N-site linear chain without SOC subjected to a parallel electric field. H_AA(H_BB) commutes with H_V; that is, H_AA(H_BB) and H_V share the same eigenfunctions. The eigenenergy ε_j and associated eigenfunction ∣S_j〉 of H_V are easily obtained through the diagonalization of the eigenvalue equation⁵⁵

H_V∣S_j〉 = ε_j∣S_j〉, (8)

where j = 1, 2, …, N. The transpose of ∣S_j〉 is ∣S_j〉^T = ∣s_j,1, s_j,2, s_j,3, …, s_j,N〉 and the component s_j,l, is the site amplitude of atom A or B located at the lth layer.

With column vectors ∣S_j〉, the eigenfunctions, an N × N unitary transformation matrix Û_V = (∣S₁〉, ∣S₂〉, …, ∣S_N〉) is then constructed and used to diagonalize H_V, i.e., , where is a unit matrix. The eigenenergies of H_AA and H_BB are ε_j + Δ and ε_j − Δ after the diagonalization of eqn (7a) and (7b), respectively.

To acquire the energy spectrum of AANLG, a 2N × 2N unitary transformation matrix

is built to transform the Hamiltonian matrix (eqn (6a)) into a simple form. After the unitary transformation, a reduced matrix [script letter H]_red = U^†HU has the formwhere 1 is an N × N unit matrix. Then, the reduced Hamiltonian matrix can be rearranged into block diagonal form, [script letter H]_red = [script letter H]₁ ⊕ [script letter H]₂ ⊕ … ⊕ [script letter H]_N, where each 2 × 2 block diagonal matrix [script letter H]_j is expressed as followsThat is to say, AANLG can be decomposed into N subsystems, [script letter H]_j. The exact energy spectrum of each subsystem is
where ε_j(= ε_⊥) depends on the magnitude of the interlayer interaction, gated voltage and layer number. is the energy spectrum of monolayer graphene with SOC.

Around the Dirac point K, the diagonal block for k = K + q is

and is the Fermi velocity. The low-lying energy dispersions associated with [script letter H]_j are , where .

2.2 Energy spectrum of AANLG with intrinsic and Rashba spin–orbital interactions

If we take both the intrinsic and Rashba spin–orbital interactions into consideration, the TB Hamiltonian matrix of AANLG subject to a perpendicular electric field, acting on the periodic Bloch functions ∣A₁↑〉, ∣B₁↑〉, ∣A₁↓〉, ∣B₁↓〉, …, ∣A_j↑〉, ∣B_j↑〉, ∣A_j↓〉, ∣B_j↓〉, …, ∣A_N↑〉, ∣B_N↑〉, ∣A_N↓〉, ∣B_N↓〉, is a 4N × 4N Hermitian matrix and expressed as followswhere H_j and H_T are 4 × 4 blocks. The off-diagonal block originates in the main interlayer interaction α₁. The diagonal block is the Hamiltonian matrix of the j-layer graphene in the presence of the gated potential V_J, which has the formIt is easy to diagonalize the block through a 4 × 4 unitary transformation matrix [Doublestruck U], which transforms H_MG into a diagonal matrix, i.e., [Doublestruck U]⁺H_MG[Doublestruck U] = diag(Λ₊₊, Λ₊₋, Λ₋₋, Λ₋₊) (eqn (4)). The eigenvalues of H_j are Λ_J,_±± = V_J + Λ_±±. Then, a 4N × 4N unitary transformation matrix U = diag([Doublestruck U], [Doublestruck U], …, [Doublestruck U]) is constructed and used to transform H_AANLG into a diagonal block form. After the operation, we obtain U⁺H_AANLGU = H₊₊ ⊕ H₊₋ ⊕ H₋₋ ⊕ H₋₊, where [script letter H]_ηξ (η = ±, ξ = ±) is an N × N matrix.

We take AA-stacking trilayer graphene as a study model. The Hamiltonian matrix H_AATLG and unitary transform matrix U are

After the operation U⁺H_AATLGU, H_AATLG is arranged into the block diagonal form H_AATLG = H₊₊ ⊕ H₊₋ ⊕ H₋₋ ⊕ H₋₊, where each H_ηξ is a 3 × 3 matrix and . The latter term H_V, eqn (7a), describes an N(= 3)-site linear chain without SOC subjected to a parallel electric field. The energy spectrum related to H_ηξ are E = Λ_ηξ + ε_j = ε_∥ + ε_⊥, where Λ_ηξ = ε_∥ and .⁵⁵

3 Electronic properties and discussion

The energy dispersions of AANLG with SOC in the presence of a gated potential are easily obtained through the calculation of the energy spectrum of each subsystem by using the analytical formula E_j = ε_⊥ + ε_∥. For example, the energy spectrum of AA-stacking bilayer graphene is described as , where . Since the energy dispersions are symmetrical about E = 0, only the energy spectrum for E > 0 is shown in Fig. 2. In the absence of the gated potential and SOC, the energy dispersions around the Dirac point K illustrate one pair of linear bands crossing at E = α₁ (dashed curves in the inset). The gated potential V = 0.4α₁ shifts the linear bands upward (red solid curves in the inset). In AB-stacking bilayer graphene, the intrinsic SOC parameter is Δ ∼ 10⁻⁵ eV and hence Δ/α₁ ∼ 10⁻⁴.⁵⁹ For convenience of the numerical analysis, we use Δ/α₁ = 0.1 in this work. The analytical model and numerical results are relevant and applicable to the exploration of physical properties in multilayer graphene-like systems. The inclusion of the intrinsic SOC Δ = 0.1α₁ changes the linear bands (red solid curves) into the parabolic bands (blue solid curves), which are described by . The maximum (minimum) of the parabolic band, located at the Dirac point K, is . The green curves are the energy spectrum of gated AA-stacking bilayer graphene with a Rashba SOC interaction λ_R = 0.05α₁. The Rashba SOC interaction λ_R destroys the degeneracy of the linear bands and produces four parabolic bands (green curves). The middle two parabolic bands touch each other at .

Fig. 2 Calculated energy dispersions around the Dirac point K of AA-stacking bilayer graphene for different gated potential V, intrinsic SOC Δ, and Rashba SOC λ_R. The dashed curves in the inset: (V, Δ, λ_R) = (0, 0, 0); red solid curves: (0.4, 0, 0)α₁; blue solid curves: (0.4, 0.1, 0)α₁; green solid curves: (0.4, 0.0, 0.05)α₁.

The energy dispersions of AA-stacking trilayer graphene were also evaluated with the formula E = Λ_±± + ε_j, where . Without SOC, the energy spectrum E₂ = Λ_±± + ε₂ is independent of the magnitude of the gated potential, as shown by the black dashed and red solid curves in the inset of Fig. 3. Two linear bands cross over at E = 0. The intrinsic SOC Δ changes the linear into the parabolic bands, as illustrated by dashed blue curves, which is simulated by . The maximum (minimum) of the parabolic band is determined by the strength of Δ = 0.1α₁. There are four parabolic bands after the inclusion of the Rashba SOC (green curves). The middle two parabolic bands do not touch at E = 0 due to the intrinsic SOC.

Fig. 3 Calculated energy dispersions of AA-stacking trilayer graphene for different V, Δ, and λ_R. The dashed curves in the inset: (V, Δ, λ_R) = (0, 0, 0); red solid curves: (0.4, 0, 0)α₁; blue solid curves: (0.4, 0.1, 0)α₁; green solid curves: (0.4, 0.1, 0.05)α₁.

4 Green’s function and velocity operator

After the introduction of the Rashba effect, as illustrated in the section above, the 4N × 4N Hamiltonian matrix can be divided into four N × N diagonal blocks. This would complicate the discussion and block us from pursuing a simple analytical form of the conductivity of AANLG. We, then, switch off the Rashba effect and consider the intrinsic SOC (ISOC) alone in the following work. Now, we demonstrate that the Green’s function and velocity operator associated with AANLG can be transformed into diagonal block matrices. With the Hamiltonian matrix H, it is straightforward to calculate the Green’s function through . The larger Hamiltonian matrix gives rise to more complex tasks in the calculation of the inverse matrix of zI − H. To reduce the tasks, we use the unitary operator U, which causes [script letter H]_red = U^†HU, to transform the Green’s function. After the operation, we have . [script letter H]_red is a block diagonal matrix and so is (zI − [script letter H]_red). Now, the Green’s function [capital G, script] is also in a block diagonal form; that is, [capital G, script] = [capital G, script]₁ ⊕ [capital G, script]₂ ⊕ … ⊕ [capital G, script]_N. Moreover, each sub-Green’s function is and it is a 2 × 2 matrix,with the corresponding elements

The velocity operator, , is approximated as the derivative of the Hamiltonian with respect to the momentum ħk, based on the gradient approximation. According to eqn (6a), the velocity matrix related to AANLG is

Since H_AA and H_BB are independent of the wave vector k, and are equal to zero. As a result, the velocity matrix ishere is an N × N identical matrix and is the Fermi velocity. After the action of the transformation matrix, [scr V, script letter V] = U^†V_xU, the transferred velocity matrix is a diagonal block matrix, which is in the form of [scr V, script letter V] = [scr V, script letter V]₁ ⊕ [scr V, script letter V]₂ ⊕ … ⊕ [scr V, script letter V]_N. Each [scr V, script letter V]_j, a 2 × 2 matrix, isThe unitary transformation matrix U, diagonal block Green’s function representation [capital G, script], and velocity operator [scr V, script letter V] are now utilized to derive the analytical form of the dynamical conductivity of AANLG.

5 Dynamical conductivity of AANLG

The finite frequency conductivity is studied by using the Kubo formula. The conductivity is written in terms of the imaginary part of retarded current–current correlation function divided by the frequency Ω as , where Π_αβ(Ω) is also referred to as the polarization function. Furthermore, the polarization function can then be written in the bubble approximation aswhere V_α is the velocity operator in the direction α = x or y and G(iω_n, k) is the Green’s function. With the spectral function representationthe real part of the conductivity, at zero temperature, T = 0, is expressed aswhere f(x) = 1/[exp(x/T) + 1] is the Fermi function and μ is the chemical potential. Following the aforementioned method, the AC conductivity for AANLG can be directly calculated by putting the 2N × 2N Green’s function representation (or spectral function representation) and 2N × 2N velocity operator in eqn (23). The larger the Green’s function (or spectral function representation) is, the more calculation tasks there are.

To reduce the complexity, we first utilize the relation, TrM = Tr[U^†MU], the invariant of the trace of a matrix (or operator) under a unitary transformation. Then, with a proper unitary transformation matrix, both the Green’s function (or spectral function representation) and velocity operator V_α are reduced to diagonal block matrices. As a result, the analytical form of the real part of the conductivity of AANLG can be easily accessible. The details are as below. First, by setting M = V_αG(ω + Ω, k)V_βG(ω, k), the trace of M is TrM = Tr[U^†V_αGV_βGU]. Then, insert the identical matrix UU^† = I between the velocity operator V_β and the Green’s function G, and the result

TrM = Tr[[scr V, script letter V]_α[capital G, script][scr V, script letter V]_β[capital G, script]] = Tr[scr M, script letter M]

is acquired. [capital G, script] = U^†GU is the unitary transformation of G and it is related to the spectral function representation [scr A, script letter A] in the following manner:[scr V, script letter V]_α = U^†V_αU is the unitary transformation of V_α. Thirdly, after the unitary transformation, both [scr V, script letter V] and [scr A, script letter A] are diagonal block matrices. That is to say, the operator [scr M, script letter M] = [scr V, script letter V]_α[scr A, script letter A][scr V, script letter V]_β[scr A, script letter A] is also a block diagonal matrix, given by, [scr M, script letter M] = [scr M, script letter M]₁ ⊕ [scr M, script letter M]₂ ⊕ [scr M, script letter M]₃ ⊕…. Each [scr M, script letter M]_j = [scr V, script letter V]_j,α[capital G, script]_j[scr V, script letter V]_j,β[capital G, script]_j is a two by two matrix. Finally, the relation Tr([scr M, script letter M]) = Tr([scr M, script letter M]₁) + Tr([scr M, script letter M]₂) + Tr([scr M, script letter M]₃) + … is used to obtain the AC conductivity for AANLG:It is shown that the AC conductivity of AANLG is equal to the summation of the AC conductivity of each subsystem, and the σ_j,αβ(Ω) of each graphene-like layer is

The AC conductivity of AANLG can be analytically specified. As the 2N × 2N Hamiltonian is decomposed into N 2 × 2 reduced Hamiltonian matrices, the effective Hamiltonian of each subsystem is described as eqn (13). Furthermore, the Green’s function, spectral function representation and velocity operator associated with each subsystem are 2 × 2 matrices. Thus, the analytical form of the AC conductivity of each graphene-like layer without ISOC is⁵²

σ_j,xx(Ω) = σ_intra + σ_inter, without ISOC, (26a)

where intra and inter represent the contributions resulting from the intraband and interband transitions, respectively. With the ISOC taken into consideration, the analytical form of AC conductivity of each subsystem reads

σ_j,xx(Ω) = σ(Ω, |ε_j − μ|), with ISOC, (27a)

where σ(Ω, |ε_j − μ|) is the conductivity for massive Dirac particles,^62,63 and it is expressed asThe dependence of σ_j,xx(Ω) on the chemical potential μ, ε_j (or ε_⊥), and strength of the ISOC is clearly revealed through the previously presented formula.

The numerically calculated conductivity σ_xx(Ω) and the associated conductivity of each subsystem σ_j,xx(Ω) (denoted as sub-conductivity) of AA-stacking trilayer graphene (TLG) are presented in Fig. 4. Both the intraband and interband transitions contribute to the AC conductivity. A delta peak at frequency Ω = 0, caused by the intraband transition, is not shown here; that is, only the conductivity resulting from the interband transitions is shown. The conductivities σ_1,xx(Ω), σ_2,xx(Ω) and σ_3,xx(Ω) of the subsystems are illustrated in the dashed curves. The AC conductivities of TLG are presented in the solid curve, which is , the superposition of the AC conductivities of the subsystems. According to eqn (26b) and (27b), the profile of each sub-conductivity, σ_j,xx(Ω) = σ₀Θ(Ω − 2|ε_j − μ|), is governed by the step function Θ(|ε_j − μ|), where the ε_j related to TLG are . In the absence of a gated potential (V = 0) and at μ = 0, both σ_1,xx and σ_3,xx show an absorption edge at . σ_2,xx(Ω) contributes a constant background conductivity, which is equal to σ₀ (dashed curves in Fig. 4(a)). The AC conductivity of TLG (solid curve Fig. 4(a)) at high frequency is equal to a constant value, three times σ₀. At μ = 0.1α₁, σ_1,xx, σ_2,xx and σ_3,xx in Fig. 3(b)) show step edges at the frequencies . As a result, there are three steps in the AC conductivity (solid curve in Fig. 4(b)). With the application of the gated potential V = 0.4α₁, the absorption edges of σ_1,xx and σ_3,xx occur at (Fig. 4(c)). The intrinsic SOC (Δ = 0.1α₁) enhances the strength of the sub-conductivity σ_2,xx in the region 0.2 < Ω/α₁ < 0.6. For comparison, the solid curves in Fig. 4(a)–(d) are plotted in Fig. 4(e). The characteristics of σ_xx(Ω) of TLG are dependent on μ, V, and the strength of the ISOC.

Fig. 4 AC conductivity σ_xx (solid curves) and sub-conductivities (σ₁, σ₂, σ₃) (dashed curves) of AA-stacking trilayer graphene for different V, Δ, and μ are presented. (a) (V, Δ, μ) = (0, 0, 0). (b) (V, Δ, μ) = (0, 0, 0.1)α₁. (c) (V, Δ, μ) = (0.4, 0, 0.1)α₁. (d) (V, Δ, μ) = (0.4, 0.1, 0.1)α₁. The solid curves from (a)–(d) are plotted in (e).

Variation of the layer number N has a great influence on the AC conductivity of AANLG. Fig. 5 displays σ_xx(Ω) and the associated σ_j,xx(Ω) of AA-stacking quad-layer graphene (QLG). At high frequency, the AC conductivity of QLG shows a constant value, which is equal to four times σ₀. There are two steps in the AC conductivity of QLG at V = 0 (the solid blue curve in Fig. 5(a)). The location of the absorption edge of each sub-conductivity σ_j,xx(Ω) is controlled by the step function Θ(|ε_j − μ|). The gated-potential-dependent energy dispersions ε_j related to QLG are , where and . In the absence of a gated potential (V = 0), ε₊₊ = −ε₋₊ = 2.618α₁ and ε₊₋ = −ε₋₋ = 0.382α₁. The first and second absorption edges appear at Ω/α₁ = 0.76 and Ω/α₁ = 5.2 (Fig. 5(a)). σ_1,xx(Ω) (σ_3,xx(Ω)) is identical to σ_2,xx(Ω) (σ_4,xx(Ω)). At μ = 0.3α₁, absorption edges occur at Ω/α₁ ≈ 0.2, 1.4, 4.6, and 5.8. The AC conductivity features four steps (the solid cyan curve in Fig. 5(b)). The gated potential V = 0.3α₁ modifies ε₊₊ = 2.881α₁ and ε₊₋ = 0.344α₁ and changes the locations of the step edges (Fig. 5(c)). The intrinsic SOC Δ = 0.1α₁ increases the AC conductivity around Ω/α₁ ≈ 0.2. The solid curves in Fig. 5(a)–(d) are plotted in Fig. 5(e) to illustrate that the effects caused by variation of N, μ, V, and the strength of the ISOC on the σ_xx(Ω) of AANLG are easily and clearly revealed through the analytical formula.

Fig. 5 Same plot as Fig. 4 but for AA-stacking quad-layer graphene. (a) (V, Δ, μ) = (0, 0, 0). (b) (V, Δ, μ) = (0, 0, 0.3)α₁. (c) (V, Δ, μ) = (0.3, 0, 0.3)α₁. (d) (V, Δ, μ) = (0.3, 0.1, 0.3)α₁. The solid curves from (a)–(d) are displayed in (e).

6 Conclusions

In this work, we proposed an analytical model to derive the exact energy spectrum and dynamical conductivity of AANLG in the presence of a bias voltage and spin–orbital coupling at the same time. First, a proper transformation matrix was built and used to transform the 2N × 2N tight-binding Hamiltonian matrix of AANLG into N 2 × 2 diagonal block matrices. Then, AANLG was reduced to N graphene-like layers. Thus, the exact energy spectrum of a graphene-like layer is E = ε_⊥ ± ε_∥. ε_⊥, the effective on-site energy of the graphene-like layer, is controlled by the interlayer interaction, gated potential, and layer number. ε_∥ is the energy spectrum of monolayer graphene with SOC. Furthermore, we analytically studied the dynamical conductivity of AANLG, which was shown to be the sum of the dynamical conductivity of N graphene-like layers with/without SOC. The dependence of the dynamical conductivity of each graphene-like layer on the chemical potential, ε_⊥, and the strength of the SOC is clearly demonstrated. Above all, our model could efficiently and exactly calculate the energy spectrum and dynamical conductivity for gated AANLG with SOC.

Acknowledgements

The author gratefully acknowledges the support of the Taiwan National Science Council under the Contract no. NSC 102-2112-M-165-001-MY3.

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