The Berry phase is a geometric phase that a quantum state acquires when it undergoes adiabatic cyclic evolution along a closed loop in a parameter space [1,2]. Responsible for novel physical phenomena such as the Aharonov-Bohm effect [3–5] and the unconventional Hall effect [6,7], the Berry phase has been extensively studied to understand topological properties of quantum materials including topological insulators [8,9], spin-orbit coupled systems [10–12], and transition-metal dichalcogenides [13,14]. Graphene is a prototypical quantum material with a well-known Berry phase [6,7,15–19]. Due to the characteristic honeycomb lattice of graphene, quasiparticles in graphene close to the band touching point between the conduction and valence bands around the Brillouin zone corner, the K point, are approximately described by spinor eigenstates that give pseudospin nature to the quasiparticles. The pseudospin is locked to the momentum in graphene, directly revealing the Berry phase of quasiparticles in graphene. In other words, when the pseudospin travels along a closed loop in the momentum space, n-layer graphene (n=1 for monolayer, n=2 for bilayer) exhibits a pseudospin winding number of n, corresponding to nπ Berry phase [20–22]. The Berry phase in graphene leads to an anisotropic distribution of angle-resolved photoemission spectroscopy (ARPES) intensity, which rotates by a characteristic angle when light polarization used to excite quasiparticles is rotated by π/2 [16,18,19,23]. The π/n rotation of the ARPES intensity distribution for n-layer graphene originates from the nπ Berry phase [18].

Despite extensive experimental and theoretical studies on graphene, previous studies on the Berry phase have mainly focused on monolayer and bilayer graphene, while trilayer graphene has been rarely studied to date [24–30]. Depending on the stacking order, trilayer graphene has rhombohedral or Bernal structures, so-called ABC- or ABA-stacked graphene, whose crystalline structures are shown in Figs. 1(a)–(d). A and B sublattices of each graphene layer are colored by dark and bright balls, where the bottom, middle, and top graphene layers are denoted by 1, 2, and 3, respectively. Each trilayer graphene is theoretically predicted to exhibit a Berry phase of 3π and 2π + π [21], different from the uniquely defined Berry phases of monolayer and bilayer graphene.

Figure 1. Top views of (a) ABC- and (b) ABA-stacked trilayer graphene, where green, red, and blue balls represent carbon atoms in the bottom, middle, and top layers, denoted by 1, 2, and 3, respectively. A and B denote carbon sublattices in each graphene layer that are colored by dark and light balls. Side views of (c) ABC- and (d) ABA-stacked trilayer graphene. Dashed arrows indicate tight-binding hopping parameters of trilayer graphene, denoted by γ₀, γ ₁, γ ₃, and γ ₄. Calculated electron band structure of (e) ABC- and (f) ABA-stacked trilayer graphene along the Γ-K-M direction using the tight-binding model.

Here we report the simulation of ARPES intensity for trilayer graphene with different stacking orders within the tight-binding formalism and explore the characteristics of the Berry phase via band-selective analysis. When light polarization is rotated by π/2, the ARPES intensity distribution of all the bands close to the band touching point of ABC-stacked trilayer graphene rotates by π/3. On the other hand, two different rotational angles of about π/2 and π are observed for bands of ABA-stacked trilayer graphene. These results indicate Berry phases of 3π and 2π + π for ABC- and ABA-stacked graphene, respectively, consistent with the prediction based on a previous work on monolayer and bilayer graphene [18] and the Berry phase of trilayer graphene [21]. Surprisingly, however, in ABA-stacked trilayer graphene, the two different rotational angles of the monolayer- and bilayer-like bands converge to about π/2 upon approaching the band touching point, suggesting a Berry phase of 2π, instead of the 2π + π Berry phase. The change in the rotational angle as a function of binding energy is attributed to the hybridized wave function of the two different bands and the resultant convergence of the Berry phase. These findings will provide important insight for manipulating not only the Berry phase in condensed matter, but also exotic phenomena arising from the Berry phase.

To simulate the ARPES intensity, the Hamiltonian was considered using the tight-binding model for the pz orbital of each carbon with hopping parameters γ₀, γ₁, γ₃, and γ₄, as depicted in Figs. 1(c) and 1(d). These hopping parameters correspond to intralayer nearest-neighbor (A_i↔B_i for i={1,2,3}) hopping, interlayer nearest-neighbor interlayer (B_i↔A_i+1 for i={1,2}), and interlayer hopping between different sublattices (A_i↔B_i+1 for i={1,2}) and between the same sublattices (A_i↔A_i+1 and B_i↔B_i+1 for i={1,2}), respectively. The values of each parameter used in the simulations are γ₀ = −3.1 eV, γ₁ = 0.38 eV, and γ₄ = −0.141 eV [35]. γ₃ was chosen to be zero to minimize the trigonal wrapping effect. The basis set composed of Block sums of localized orbitals on each sublattice was introduced as follows: ϕ₁ = (A₁ − A₃) /$\sqrt{2}$, ϕ₂ = (B₁ − B₃) /$\sqrt{2}$, ϕ₃ = (A₁ + A₃) /$\sqrt{2}$, ϕ₄ = B₂, ϕ₅ = A₂, and ϕ₆ = (B₁ + B₃) /$\sqrt{2}$. The potential difference between two outer layers was set to be Δ₁ = 0.05 eV, and that of the two outer layers with respect to the middle layer was Δ₂ = −0.023 eV, while the potential difference between B₁ and A₃ sites was δ = −0.0105 eV [35]. Within this setup, the tight-binding (TB) Hamiltonians for ABC- and ABA-stacked trilayer graphene are as follows [21,32–36].

Here, $u\left(k\right)={\displaystyle \sum _{i=1}^3\text{exp}\left(ik\cdot b_i\right)}=e^{\frac{i{k}_{y}4\pi }{3\sqrt{3}}}\left(1+2\text{cos}\left(\frac{k_{x}2\pi }{3}\right)e^{-\frac{i{k}_{y}2\pi }{\sqrt{3}}}\right)$, constructed with the three vectors in the real space connecting nearest neighbor carbon atoms b₁ = b (0, 1) , $b_2=b\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$, and $b_3=b\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$, where k is the vector representation in the momentum space and b = 1.42 Å is the in-plane inter-carbon distance. The interaction Hamiltonian H_int (k, Q) coupling to the electromagnetic waves with a wave vector Q is defined as $-\frac{e}{c}\widehat{A}\cdot \widehat{v}$, where $\widehat{A}$ is the external vector potential and $\widehat{v}=\left[\widehat{r},H\right]/i\hslash$ with $\widehat{r}=i\hslash \left({\nabla }_k,\partial k_z\right)$, $\hslash$ is Planck’s constant, e is the charge of an electron, and c is the speed of light [18]. Based on this setup, the photoelectron intensity has been calculated by the absolute square of the transition matrix element $M_{sk}=〈f_{k+Q}\left|H_{\text{int}}\left(k,Q\right)\right|{\psi }_{sk}〉$, where $\left|{\psi }_{sk}\right.〉$ is the eigenstate with s = ±1 the band index and $\left|f_{k+Q}\right.〉$ is the plane-wave final state projected onto the p_z orbitals of graphene [18].

Figures 1(e) and 1(f) show calculated electron band structures close to the band touching point around the K point for ABC- and ABAstacked trilayer graphene using the full TB Hamiltonians discussed above. The ABC-stacked graphene consists of three parabolic bands, whose outermost conduction and valence bands touch slightly off the K point, while the two inner bands cross at about ±0.4 eV from the Fermi energy, E_F. The basis of ABA-stacked trilayer graphene, which has mirror reflection symmetry along the out-of-plane direction, can be recombined and block-diagonalized to [L1+L3, L2] (L=A, B) with the mirror eigenvalue = +1 and [L1-L3] with the eigenvalue = –1, resulting in a combination of bilayer-like parabolic bands (M_z = +1) and monolayer-like linear bands (M_z = –1). The outermost parabolic band and the linear band merge together as they approach toward E_F.

To investigate the Berry phase of the trilayer graphene, the ARPES intensity distribution was simulated within the TB formalism [18]. Figures 2(a) and 2(b) show the results simulated at 1.0 eV below E_F for ABC- and ABA-stacked graphene, respectively. The white-dashed hexagon is the first Brillouin zone of graphene. For both constant energy contours, several crescent-like spectral features are observed around each K point. To examine this spectral feature in more detail, Figs. 2(c) and 2(d) show polarization-dependent ARPES intensity distribution simulated at –1.0, 0.0, and 1.0 eV with respect to E_F around the K point indicated by the white squares in Figs. 2(a) and 2(b). The left and right panels are results obtained using light polarization parallel to the k_x (X-polarization or X-pol.) and k_y (Y-polarization or Ypol.) axes of graphene, respectively. At E_F, where the conduction and valence bands touch each other, both X- and Y-polarization data exhibit a point-like shape, whereas upon moving away from E_F, concentric crescent-like shapes appear, consistent with the characteristic conical electron band structure of trilayer graphene, as shown in Figs. 1(e) and 1(f). In addition, the intensity distribution of the conduction band for X-polarization is almost the same as that of the valence band for Y-polarization. This originates from the chiral nature of quasiparticles in graphene [18]. Apart from these common features, the overall intensity distribution of ABC-stacked trilayer graphene is different from that of ABA-stacked trilayer graphene.

Figure 2. Simulated constant energy contours of (a) ABC- and (b) ABA-stacked trilayer graphene at E – E_F = –1.0 eV for the first Brillouin zone, denoted by the white-dashed hexagon. The simulation has been done for X-polarized light. Simulated constant energy contours of (c) ABC- and (d) ABA-stacked trilayer graphene for X-(left) and Y-(right) polarized light for the area denoted by the white rectangle in panels (a) and (b). The inner (left panels), middle (middle panels), and outer (right panels) bands of (e) ABC- and (f) ABA-stacked trilayer graphene simulated with X- and Y-polarized lights at E – E_F= –1.0 eV.

To better understand the anisotropic intensity distribution of the three bands of each graphene, Figs. 2(e) and 2(f) show the bandselective intensity distribution for the inner, middle, and outer bands at E – E_F = –1.0 eV for X- and Y-polarization, respectively. Each band can be displayed separately by setting the broadening effect used in the simulation to zero. The inner band of the ABC-stacked trilayer graphene for X-polarization has maximum intensity along the K-Γ direction, whereas it lies along the K-M direction for Y-polarization. The intensity distribution of the middle and outer bands shows similar polarization dependence to that of the inner band, i.e., maximum intensity along the K-M (Γ-K) direction for X-polarization changes to the K-Γ (K-M) direction for the middle (outer) band. On the other hand, ABA-stacked trilayer graphene exhibits slightly different polarization dependence from ABC-stacked graphene. Although the inner and outer bands show similar behavior to that of ABC-stacked graphene, the intensity maxima of the inner (outer) band for X-polarization (Y-polarization) deviate slightly from the K-M (K-Γ) direction. More interestingly, for the middle band, the intensity maximum along the Γ-K direction changes its direction to the K-M direction upon changing light polarization; that is neither observed for the ABC-stacked trilayer graphene nor for the other two bands of ABA-stacked trilayer graphene.

The change in the ARPES intensity distribution can be quantitatively analyzed by taking the intensity profile for each band around the K point as shown in Fig. 3. The intensity profile is taken at E – E_F = –1.0 eV, where θ is the angle with respect to the +k_x axis as shown in the inset. The ARPES intensity of the inner, middle, and outer bands is plotted with black-solid, red-dashed, and yellow-solid curves, respectively. In ABC-stacked trilayer graphene, the intensity maxima of all the bands rotate by about π/3 upon rotating light polarization by π/2. On the other hand, in ABA-stacked trilayer graphene, the intensity maximum of the middle band rotates by about π, whereas the inner and outer bands rotate by about π/2.

Figure 3. Angle-dependent intensity profiles of (a) ABC- and (b) ABA-stacked trilayer graphene extracted from Figs. 2(e) and 2(f). Black-solid, red-dashed, and yellow-solid curves are for inner, middle, and outer bands, respectively. The rotational angle θ is defined as the angle with respect to the +k_x axis, as shown in the inset. (c) Pseudospin textures of ABC-(upper panel) and ABA-(lower panel) stacked trilayer graphene.

For ABC- and ABA-stacked trilayer graphene, the spinor eigenstates are analytically obtained when the momentum with respect to the K point is assumed to be very small, allowing us to determine the pseudospin texture of each graphene, as shown in Fig. 3(c) [21,22]. The pseudospin of ABC-stacked graphene, denoted by blue arrows in the upper panel, shows a winding number of 3 when traveling along a closed loop in the momentum space. This indicates a Berry phase of 3π, which is also applied to the other two bands of ABC-stacked graphene [21,24–26,33–37]. The pseudospin of ABA-stacked trilayer graphene exhibits different winding numbers of 1 and 2 for the linear (middle) and parabolic (outer and inner) bands as denoted by green and red arrows, which gives π and 2π Berry phase, respectively [21,25,26,33].

The rotation of the ARPES intensity distribution in graphene upon rotating light polarization is explained by the Berry phase effect in monolayer and bilayer graphene [18]. The spectral intensity of n-layer graphene (n=1 for monolayer, n=2 for bilayer) rotates by π/n due to the nπ Berry phase when the light polarization is rotated by π/2 [18]. Analogously, the rotational angle of the ARPES intensity distribution is predicted to be π/3 for ABC-stacked trilayer graphene, whereas it is π/2 and π for the parabolic and linear bands for ABA-stacked trilayer graphene, due to their Berry phases of 3π and 2π + π, respectively. This prediction is manifested roughly in the simulations for the ARPES intensity distribution for trilayer graphene shown in Figs. 2 and 3.

The rotational angle of the outer and inner bands of ABA-stacked graphene, however, deviates slightly from π/2. This discrepancy is likely due to the asymmetric line-shape of the intensity profile, which may distort angular information. Interestingly, the asymmetry of the line-shape is more pronounced in the middle band, plotted by the reddashed curve in the lower panel of Fig. 3(b). For Y-polarization, although the spectral intensity exhibits its maximum at 0 or 2π, a clear shoulder of the intensity appears at about π/2 or 3π/2. The unexpected spectral intensity suggests the possibility of an additional effect that the isolated monolayer and bilayer graphene bands and their Berry phases cannot describe.

To find the origin of the asymmetric line-shape, the ARPES intensity distribution was simulated for ABA-stacked trilayer graphene as a function of energy relative to E_F, as shown in Fig. 4. The positions of the intensity maxima of the inner and outer bands do not notably change, as denoted by white arrows in Figs. 4(a) and 4(b). On the other hand, the positions of the intensity maxima of the middle band for Ypolarization dramatically change as denoted by white arrows in Fig. 4(c), while they remain the same for X-polarization. The rotational angle of the intensity maxima as a function of E – E_F is summarized in Fig. 4(d) for the inner, outer, and middle bands with black, yellow, and red symbols, respectively. The rotational angle of the inner band (black symbol) above E – E_F = –0.7 eV cannot be extracted as the valence band maximum of the inner band is about –0.6 eV, as shown in Fig. 1(f).

Figure 4. Constant energy contours of the (a) inner, (b) outer, and (c) middle bands of ABA-stacked trilayer graphene at several different energies with respect to E_F. White arrows denote the positions of maximum spectral intensity. (d) Rotational angle of the maximum intensity upon rotating light polarization by π/2 as a function of E – E_F in the upper panel and band-selective Berry phase as a function of E – E_F converted from the rotational angle in the lower panel. Black, red, and yellow colors denote the inner, middle, and outer bands, respectively.

At higher binding energies, e.g., E – E_F = –1.1 eV, the rotation angles of the inner, outer, and middle bands are 0.67π, 0.61π, and π, respectively. While these angles are similar to 0.5π, 0.5π, and π predicted from the eigenstates obtained by the simple approximation of |k – K| ≪ 1 [21], these results suggest that the Berry phase of each parabolic band of the ABA-stacked trilayer graphene is 1.5π and 1.65π, and that of the linear band is π, following the analogy obtained from the monolayer and bilayer [18] and from ABC-stacked trilayer graphene in this study. Upon changing E – E_F, the rotational angle of the inner band remains almost the same, suggesting that it has a robust Berry phase of 1.5π. On the other hand, the rotational angle of the outer band gradually decreases from 0.61π at E – E_F = –1.1 eV to 0.52π at E – E_F = –0.3 eV. Surprisingly, the rotation angle of the middle band abruptly changes above E – E_F = –0.8 eV until it decreases to 0.52π at E – E_F = –0.3 eV, which is very close to the angle of the outer band. The equivalent rotational angles of the outer and middle bands suggest that the Berry phases of these two bands converge to 1.9π, which is very close to the Berry phase of 2π predicted from the bilayer-like outer band of the ABA-stacked trilayer graphene. The difference of the outer and middle bands compared to the inner band is that the latter remains intact independent of the energy relative to E_F, whereas the former two merge together upon decreasing |E – E_F| as shown in Fig. 1(f). The merging of the two bands close to E_F can lead to enhanced hybridization between quasiparticles from the two bands, giving rise to a single quantum mechanical phase for the quasiparticles from two different bands. Meanwhile, due to non-zero Δ₁, Δ₂, and δ, H_ABA has non-zero off-diagonal terms when block diagonalized by basis transformation, leading to hybridization between the parabolic and linear bands. As a result, the Berry phases of the inner, middle, and outer bands deviate from those of unperturbed bands with zero Δ₁, Δ₂, and δ, and converge around the band merging point between the parabolic and linear bands, as manifested in Fig. 4(d). One can also notice that the intensity profile of the ABC trilayer graphene shown in Fig. 3(a) also shows slight asymmetry, e.g., the middle band for Y-polarization. Since the rotational angle of the intensity maxima of this system is described well by the Berry phase effect of π/3, the asymmetry can also have an additional contribution other than the merging band effect discussed above, such as trigonal wrapping of the electron band structure, which can also be applied to the ABA trilayer graphene.

These results not only show that the band-selective Berry phase can be investigated via a simulation of the ARPES intensity distribution within the TB formalism, but also suggest that the Berry phase can be controlled through the hybridization of electron bands. This efficient way of tuning the Berry phase provides a plausible methodology to manipulate intriguing physical phenomena stemming from the Berry phase such as the Aharonov-Bohm effect [3–5], the unconventional Hall effect [6,7], and the topological properties of topologically protected surface states [8,9].

The band-selective ARPES intensity distribution of trilayer graphene with different stacking structures has been simulated within the tight-binding formalism. For ABC-stacked trilayer graphene, when the light polarization is rotated by π/2, the photoelectron intensity rotates by π/3, which is consistent with the theoretical prediction of the 3π Berry phase. On the other hand, for ABA-stacked trilayer graphene, although the ARPES intensity distribution is expected to rotate by π and π/2 due to the π and 2π Berry phases, the outer and middle bands rotate by about π/2 close to E_F, suggesting a single Berry phase of about 2π. The rotational angle gradually approaches the theoretically predicted values corresponding to the π and 2π Berry phases as the electron band structures move away from E_F. This unusual spectral feature is attributed to hybridization between quasiparticles from the outer and middle bands, which is enhanced by their mergence upon approaching E_F.

We gratefully acknowledge J. Hwang and H. C. Park for helpful discussions. This work was supported by the 2-Year Research Grant from Pusan National University.

The authors declare no conflicts of interest.