Symmetry is essential to classify different electronic states of matter. The Landau theory of the phase transition^[1] is the key concept to differentiate the states of matter, this theory is based on the local order parameter. For example, ferro/paramagnetism and superconductivity/superfluidity states can be understood by the aforementioned principle. But after 1980, when the quantum Hall effect (QHE)^[2,3] was discovered, the Landau theory failed to explain this new state of matter because the quantum Hall (QH) states, in principle, are distinct states but break no symmetry and hence cannot be explained under the shadow of the Landau theory of phase transition. So a new modified band theory was introduced known as topological band theory,^[4] which takes the concept of topological invariants (Chern number) instead of the local order parameter. In principle, the QHE is the first example of a topological insulator (displaying insulating behavior in the bulk but has robust conducting channels at the edges).^[5–7] The Chern-number can be expressed as an integral of a well-defined quantity known as the Berry curvature (analogous to magnetic field in electrodynamics) over the first Brillouin zone (BZ).^[8]

In 1982, Using Kubo formalism, Thouless and his collaborators^[8] made an efficacious attempt to link the topology (Chern-number) to the band structure of the two-dimensional band insulator having non-interacting electrons exposed to a strong magnetic field. The quantization of the Hall conductance in terms of Chern-number confirms the topological nature of the QH state; the edge channels cannot be removed unless the bulk band gap closes. Due to this major contribution, Thouless was awarded a Nobel prize in 2016. The next important development in the field was made by Haldane in 1988,^[9] who came up with a theoretical model, where he demonstrated that one can think of topological protection without an applied magnetic field. It is in principle related to the composition of the band structure of materials. Initially, the citations of the Haldane model were very low because of its impractical realization but it increased rapidly after 2005, when Kane and his collaborator Male^[10,11] presented a physical realization, for the first time, of the Haldane model by demonstrating that one can visualize the role of intrinsic spin–orbit (ISO) interaction as a magnetic field in the momentum space (momentum-space Berry curvatures), which has the opposite sign for opposite spins. This model introduced a new topological state requiring no external magnetic field and having spin current on the boundaries; this state of matter is now identified as quantum spin Hall (QSH) state^{[1,5,10–15]} in the literature. Haldane was also awarded a noble prize shared with Thouless and Kosterlitz in 2016 for his contribution.

In this new electronic phase, called the quantum spin Hall effect (QSHE), the edge states are uniquely designed in the sense that they are insensitive to the non-magnetic perturbations because of their helical nature (momentum and spin are locked with each other).^[16] This phase preserves the time reversal (TR) symmetry and hence is known as a topological insulator. The topological invariant used to classify the QSH phases is known as Z ₂ topological invariant^[17] or spin Chern parity which can possess only two values ν = 0,1, where 0/1 for an even/odd number of Kramer pairs at the boundaries.^[18,19]

Impurities, which preserve the TR symmetry (non-magnetic impurities), do not posses the ability to back scatter the electrons, as for the reversal of momentum the flipping of spin is needed, which does not lie under the capacity of non-magnetic impurities.^[20] The QSHE is observed only in those materials which have strong essence of intrinsic spin–orbit (ISO) coupling, but unfortunately the ISO coupling in graphene is so weak ( –10⁻³ meV^[4,21,22]) that the quantum spin Hall (QSH) state is difficult to be observed experimentally in graphene and its layers, however the QSH state in graphene provides a good theoretical framework to observe this topological state in other two-dimensional (2D) materials having a strong essence of the spin–orbit coupling. In 2005, Molenkamp and his team were successful in the experimental confirmation of QSHE in HgTe quantum wells sandwiched between HgCdTe layers.^[23,24] Another 2D material is known as silicene, which has strong SOC than graphene (and its layers) also hosts the QSH state.^[25,26] In contrast to QSHE, the quantum-valley Hall effect (QVHE)^[27–29] originates as a result of the broken space inversion-symmetry. In this quantum state of matter, Dirac-fermions that corresponds to different valleys, move to opposite transverse edges in the presence of an in-plane electric field. Recently, a great deal of attention has been paid to the topological insulators in the study of condensed matter physics,^[30–41] because some new properties are believed to be extracted from it, which could be useful for technological usages, such as electronics, spintronics, valleytronics, and quantum computation.

In the present work, our aim is to study the trilayer graphene (ABC stacking) in the presence of intrinsic spin–orbit interaction (SOI) to probe QSHE while using a two-band Hamiltonian approach. Activation of spin index in the trilayer graphene introduces a new energy term in the Hamiltonian that corresponds to intrinsic SOI, this extra term gaps the spectrum in the bulk at the Dirac points with conducting spin-polarized edges. Unfortunately, this extra energy contribution is very small because of weak ISO interaction in the trilayer graphene^[42] but recent development in this field shows that it can be strengthened by numerous means, such as curvature,^[43] impurities^[44] or Coulomb interaction,^[45] this makes it possible to observe the QSH state. In addition, the QSHE in trilayer can be used as an approach to understand the coupling among three layers with each one having a topological nature. An attempt has been made to also discuss the topological configuration of the band structure of the trilayer graphene through the plotting of a three-dimensional (3D) unit vector on the unit sphere. Our consideration also revolves around the study of trilayer graphene in the presence of both ISO coupling and an applied electric field. Besides this, we have also incorporated a discussion regarding what happens when one considers a coupling between a metal and topological insulator. To do this, we present a theoretical model where only one of the outer layers has finite SOC while it is zero in the other two layers. This sort of tenability of the trilayer graphene is unrealizable because till now there has been no known way to tune the SO coupling in each sheet independently. However, this model provides a simple theoretical framework to understand SO proximity effects in other systems.^[46] The stability of the QSH state against various perturbations such as Rashba SOI and exchange magnetization is also part of the discussion in this paper.

This paper is systematized as follows. In Section 2, we investigate the QSH effect in ABC trilayer layer graphene using the two-bands low energy approach and calculate the spin Chern-number for each valley and spin. Section 3 comprises discussions about the topological structure of the QSH state in the trilayer graphene. In Section 4, we highlight the main features of the trilayer graphene in the presence of both ISO coupling and perpendicular electric field. In Section 5, we examine the stability of the QSH state in the trilayer graphene against various perturbation terms such as Rashba spin–orbit (RSO) coupling and Zeeman term M. In Section 6, we consider a theoretical model in which only one of the outer layers of the trilayer graphene has nonzero ISO coupling while the other two layers have zero ISO coupling. In the last section, we summarize our results and outcomes.

The band structure of ABC trilayer graphene is composed of six energy bands having cubic dispersion, two of them are gapless at the valley points and (at zero energy) and are known as low energy bands, these bands originate as a result of coupling between A₁–B₃ sites, while the remaining four bands do not touch the Fermi surface and are known as high energy bands.^[47] Since the strength of the ISO interaction in the trilayer graphene is weak, it cannot change the topology of the high energy bands. In other words, the contribution from high energy bands in the first Chern-number is very small so it is a convention to involve only the low energy states for the spin Chern-number calculations. So in the light of the above discussion we use effective Hamiltonian^[48] in our calculations. The two-level effective Hamiltonian of the ABC trilayer graphene in the basis , can be modeled^[42,48] as, 1 where ( ) corresponds to the electronic part (ISO interaction part), which is given by 2 3 where is the Fermi velocity, a = 1.42 Å is the lattice constant of monolayer graphene, describes the nearest neighbor intra-layer (strong inter-layer) coupling between A _j–B _j for j = 1,2,3 (B ₁–A ₂ and B ₂– , , i = 0,1,2,3 represents the Pauli matrices in the space (A ₁,B ₃), η is the valley index which is η = +1 (η = −1) for valley ( ), s _z is the real spin index which is ( ) for spin ( ), is the strength of the ISO interaction, and is the momentum and . In this model the spin is a good quantum number. Now to obtain the band dispersion relation, we diagonalize the above Hamiltonian, 4 where is the band index and We see that the system turns out to be gaped when the SO interaction is introduced. In addition, the energy does not depend on the spin index which ensures the TR symmetry in the system. In order to further investigate whether either this gaped system is a trivial or non-trivial band insulator, we calculate the first Chern number (winding number) for the said system. The Hamiltonian (1) can be modeled in terms of 3D unit vector as, 5 where the three-component unit vector is defined as 6 Here and . The Chern-number (winding number) in terms of this unit vector for the Hamiltonian (1) is given by^[49–52] 7 where . Physically, this number corresponds to how many times the 3D unit vector covers the entire unit sphere upon covering the whole BZ. This is related to the topology of the band structure and may have different values for different band structures. The quantity can be calculated as 8 with where, we have used, Now substituting Eq. (8) into Eq. (7), 9 The total Chern-number can be calculated as 10 This ensures that the QSH state is a TR symmetrical electronic phase of the matter. Additionally, it ensures the fact that there exists zero charge current flowing on the boundaries of the said phase. Another interesting result can be seen by summing the contribution from both valleys for a given spin 11 These results imply that the Chern-number for a given spin (up or down) has a quantized value, which is a justification of the statement that each filled band of a system having non-interacting electrons must have a quantized Chern-number.^[53] The spin Chern-number plays the role of topological invariant for the systems having spin rotational symmetry about the z axis and is defined in the following fashion,^[10,19] 12 where the summation runs over the occupied bands. Thus the spin Chern-number for our trilayer graphene system is . Like the charge Hall conductivity, the spin Hall-conductivity can be written in terms of the spin Chern-number as, 13 We see that the first Chern-number of the trilayer graphene is three times the single layer graphene, therefore accordingly the trilayer system has the edge channels, three times more than a single layer because the spin Chern-number corresponds to the number of Kramer pairs on a given edge of the sample.^[18]

The topological configuration of the QSH state can be best visualized by looking at the plot of 3D unit vector as a function of momentum over a 3D unit sphere. Observing the structure of vector on a unit sphere, one can extract some interesting physics in the form of topological objects, which we call triple-vortex merons. In topological terminology, the valley index η defines the vorticity (superfluidity analogy), η = 1 for right vortex and η = −1 for left vortex. The unit vector depends on the index therefore we have four flavors which are In all flavors, at asymptotically large momentum, the z component of the unit vector vanishes and thus it becomes confined to the x–y plane, i.e., winds around the equator of the unit sphere and displays vortex-like winding on the circle [see Fig. 1]. While in the vortex core, where the momentum is zero, the z component of the becomes unity and we obtain, . This shows that, in the vortex core the unit vector points to the north pole of the unit sphere for while it points to the south pole in the flavors . This clarifies that (i) for the flavors , the mapping starts to tilt gradually from the north pole towards the equator of the sphere as we increase the momentum, and is completely shifted to the x–y plane when the momentum becomes asymptotically large. In this way the unit vector covers the upper hemisphere (three times) upon scanning the whole BZ. We call such a configuration triple-vortex merons with associated topological charge [see Figs. 1(a) and 1(c)]. (ii) Similarly, for the flavors , the vector starts to flip gradually from the south pole towards the equator of the 3D unit sphere, when we increase the momentum it is confined to the equator at asymptotically large momentum. This implies that the unit vector scans the lower hemisphere upon covering the entire BZ. Thus the mapping presents anti-triple-vortex meron-like configuration with associated topological charge [see Figs. 1(b) and 1(d)]. In principle, a meron is a half of a skyrmion having unit topological charge.^[54] One more thing that is important to be noted is that unlike superfluids, where the vortex core has a singularity, these vortices do not have a singularity at the origin (vortex core).^[50]

Fig. 1.

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Fig. 1. (color online) Four flavors of triple-vortex merons. (a) Flavor with valley and : the unit vector covers the upper hemisphere and displays a triple vortex meron-like configuration with associated topological charge . (b) Flavor with valley and : covers the lower hemisphere and displays a meron-like structure with . (c) Flavor with valley and : the mapping covers the upper hemisphere and displays a triple vortex anti-meron-like configuration with associated topological charge . (d) Flavor with valley and : covers the lower hemisphere and displays an anti-meron-like structure with .

When the trilayer graphene system is exposed to a perpendicular electric field, inversion symmetry breaks in the system and the model Hamiltonian takes the form^[46] 14 By diagonalizing the above Hamiltonian we gain 15 where is the band index with value +1 or −1 represents the conduction band and valence band, respectively, and . We see that the energy of spin ( ) electron in valley is not equal to that of spin electron in valley , which is a justification of the statement that the external electric field breaks the inversion symmetry in the trilayer graphene system. It is also quite evident that when , the system remains gaped. But when the strength of the applied electric field is increased the gap between spin-up bands gets decreases (Fig. 2(b)) and completely vanishes when the strength of the electric field becomes equal to the ISO coupling strength [see Fig. 2(c)). Thus for the case , the trilayer system exhibits a metallic state. When we further increase the applied electric field, the gap between spin-up bands reopens and hence the trilayer graphene goes through a phase transition from QSH state to QVH state [Fig. 2(d)].

Fig. 2.

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	Fig. 2. (color online) Band structure of the trilayer graphene for different values of electric field and ISO coupling. Different values of and are mentioned inside the plots.

The Hamiltonian (14) can be parameterized in terms of a three-component unit vector as 16 where the three-component unit vector is defined as 17 where and . Repeating the same process as in Section 2, we obtain 18 Hence we find, 19 We observe a nice interplay between ISO coupling and external electric field. When , we gain & , and the system exhibits the QSH state. While for the case , we obtain & , and the system exhibits the QVH state. We recover the results of Section 2, when the electric field is switched-off, the corresponding spin/valley Hall conductivity is given by, 20 21 Now, we discuss a more realistic case, when the Fermi level exists in one of the bands. When the Fermi level exists in the conduction band we obtain with 22 The spin and valley Hall conductivities become, 23 24

We obtain the same results when the Fermi level is in the valence band because of the electron–hole symmetry in our system. We plot Eqs. (23) and (24) in Fig. 3, the QSH/QVH conductivity versus the applied electric field for a fixed-value of ISO coupling. By varying the electric field, we may tune the SHE/VHE. We observe from Fig. 3 that the trilayer graphene also goes through a phase transition in the case when the Fermi level is in the conduction or the valence band.

Fig. 3.

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	Fig. 3. (color online) Quantum spin/valley Hall conductivities as a function of applied electric field for a fixed value of the ISO coupling (4 meV).

Considering a theoretical model in which only one of the outer layers of the trilayer graphene has a finite intrinsic SO coupling while it is zero in the other two layers. This sort of tenability is un-realizable because the intrinsic SOC is difficult to control from outside like the RSO coupling. However, this model provides a simple theoretical framework to understand what happens when one considers a coupling between a metal and a topological insulator. For the case in which only one of the layers is affected by the intrinsic SO coupling, the Hamiltonian (1) takes the form, 33 By diagonalizing the above equation, we obtain 34 where with +1 (−1) represents the conduction (valence) band. From Eq. (34), we see that the energy for spin ( ) electron with valley does not equal that of a spin electron with valley , i.e., and , this justifies the existence of the inversion asymmetry in the trilayer graphene having intrinsic SO coupling in only one layer. We also observe that and , this ensures the TR symmetry in the system. Now combining both of these conditions we concluded that, 35 This implies that the energy for each valley depends on the electron spin and is a consequence of the absence of the inversion symmetry in the system. Basically, we have four branches of the spectrum for each valley: two for valence band and two for conduction band . From the dispersion relation it looks clear that the two branches are gapless and two have a gap at the Dirac point (p = 0), i.e., 36 All these bands are plotted in Fig. 6. Now the Hamiltonian (33) can be parameterized in terms of a three-component unit vector as 37 where the three-component unit vector is defined as 38 where and . The form of the unit vector is similar to that of the trilayer graphene with SO coupling in both layers and hence this system will have the same Chern-number as that of the trilayer graphene with SO coupling in both layers, i.e., . But this result goes against our dispersion relation (34) which shows that our system is not a true insulator at all but an indirect zero gap semiconductor [see Fig. 6]. From Fig. 6, we see that when both spins and (or) both valleys are taken into account then the spectrum becomes gapless. However, it is clear that for each spin and valley we have an energy gap equal to at p = 0: Thus, we conclude that the trilayer graphene with only one layer having the essence of the ISO coupling is not a quantum spin Hall system because of having the gapless bulk spectrum.

Fig. 6.

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	Fig. 6. (color online) Low energy spectrum of the trilayer graphene with only one layer has the essence of ISO interaction. Panel (a) (panel (b)) corresponds to the bands near valley ( ). The introduction of intrinsic SO coupling in only layer opens an energy gap having an amount of with different signs for spin-up and spin-down as well as for valley and .

In conclusion, we studied the two bands low energy description of the ABC trilayer graphene system in the presence of the intrinsic SO coupling to probe a topological insulator. Our results show that the trilayer graphene goes through a phase transition from zero gap semiconductor to a QSH phase when the intrinsic SO coupling is introduced. We analytically calculated that the net spin Chern number for each spin in the trilayer graphene system is three times that of the single layer graphene. This confirms that the trilayer graphene has triple edge states as compared to a single layer. The spin Chern number is odd, which confirms that our proposed system is a strong topological insulator. The band topology of the trilayer graphene was also discussed by plotting as a function of momentum over a unit sphere, which shows that displays a triple-vortex meron-like configuration for each spin and valley. Besides, we found that the trilayer graphene system goes through a phase transition from QSH state to QVH state, in the presence of the electric field. In addition, we introduced various perturbations such as Rashba SO coupling and Zeeman field, and investigated the stability of the QSH phase; in the presence of these perturbations, the following results were obtained: (i) non-trivial topological phase when , ; (ii) the QVH phase when ; (iii) the metallic state when . Besides this, we investigated that the trigonal warping does not affect the spin Chern number.

Moreover, we considered a situation in the trilayer graphene in which only one of the out layers has the essence of the intrinsic SO coupling. Although the nonzero Chern number for each valley signals that the system is a topological insulator, but in fact, it is not a topological insulator because the spectrum of the system is not gapped as a whole.

The author Majeed Ur Rehman acknowledges the support from the Chinese Academy of Sciences (CAS) and TWAS for his Ph. D. studies at the University of Science and Technology, China in the category of 2016 CAS-TWAS President’s Fellowship Awardee (Grant No. 2016-156).