Large Chern Number Quantum Anomalous Hall Effect In Thin-film Topological Crystalline Insulators
Abstract
Quantum anomalous Hall (QAH) insulators are two-dimensional (2D) insulating states exhibiting properties similar to those of quantum Hall states but without external magnetic field. They have quantized Hall conductance , where integer is called the Chern number, and represents the number of gapless edge modes. Recent experiments demonstrated that chromium doped thin-film (Bi,Sb)Te is a QAH insulator with Chern number . Here we theoretically predict that thin-film topological crystalline insulators (TCI) can host various QAH phases, when doped by ferromagnetically ordered dopants. Any Chern number between can, in principle, be reached as a result of the interplay between (a) the induced Zeeman field, depending on the magnetic doping concentration, (b) the structural distortion, either intrinsic or induced by a piezoelectric material through proximity effect and (c) the thickness of the thin film. The tunable Chern numbers found in TCI possess significant potential for ultra-low power information processing applications.
A quantum anomalous Hall state is a 2D topological insulating state that has quantized Hall conductance in the form of where is an integer, and possesses gapless edge modes along any 1D edge. These properties are shared by the well-known quantum Hall statesKlitzing et al. (1980). Nevertheless, there is no external magnetic field in a QAH state, which makes it ‘anomalous’. Hence, the nontrivial topology in QAH does not come from the topology of the Landau levels, but rises from the band structure of electrons coherently coupled to certain magnetic orders, e.g., spin orders and orbital current orders. The first theoretical model that shows this phase is given in Ref.[Haldane, 1988], which is followed by other models and experimental proposals in various systemsOnoda and Nagaosa (2003); Qi et al. (2008); Yu et al. (2010); Nomura and Nagaosa (2011); Jiang et al. (2012); Wang et al. (2013a). Very recently, experimentalists have adapted one of the proposals and realized a QAH state with in chromium doped thin-film (Bi,Sb)Te, which is a 3D topological insulator (TI)Chang et al. (2013).
We first recapitulate the basic idea underlying the realization of QAH insulators with in a thin-film 3D topological insulatorQi et al. (2008); Yu et al. (2010); Nomura and Nagaosa (2011); Jiang et al. (2012). Each surface of a 3D TI is a gapless 2D Dirac spin-split semi-metalHasan and Kane (2010); Qi and Zhang (2011), as opposed to spin-degenerate Dirac semi-metals such as grapehene. The surface is spin-split except at the Dirac point where double-degeneracy is protected by time-reversal symmetry, and spectral flow into the bulk conduction and valence bands occurs away from the Dirac point. Upon the application of a Zeeman field along the perpendicular direction, induced by ferromagnetic dopants, a gap is opened at the Dirac point, giving rise to a massive Dirac cone. Such a massive Dirac cone has been well known to contribute Hall conductance of Qi et al. (2008); Bernevig and Hughes (2013), or, a Chern number of . Moreover, since a thin film has two surfaces (top and bottom), the total Chern number is . An identical effect would take place in bulk samples - thin films are being used here only because they allow tuning of the Fermi level in the gap by gating. Here we use a symmetry-based analysis to show that the topological crystalline insulatorsFu (2011); Hsieh et al. (2012); Xu et al. (2012); Dziawa et al. (2012); Tanaka et al. (2012); Fang et al. (2012); Wang et al. (2013b); Liu et al. (2013a); Okada et al. (2013) [such as (Pb,Sn)(Te,Se)] are much richer compounds to explore QAH physics. As thin films of (Pb,Sn)(Te,Se) have already been grownElleman and Wilman (1948); Bylander (1966); Taskin et al. (2013) and various magnetic dopants have been successfully dopedMathur et al. (1970); Inoue et al. (1976); Nielsen et al. (2012), we believe our proposal is experimentally realizable. The existence of such a widely tunable topological phase transition in the TCI class of materials may form the basis for new types of information processing devices which consume much less power compared to current technology.
I Results
i.1 Unperturbed Hamiltonian on the -surface
Consider the symmetries of such a thin film. (Pb,Sn)(Te,Se) crystalizes into a face-centered-cubic lattice with point group . Below a critical temperature, depending on composition, the cubic symmetry spontaneously breaks into either rhombohedral or orthorhombic symmetries, resulting in a small lattice distortion. Here we assume that the lattice has cubic symmetry and treat the small distortion as perturbative strain. The thin-film sample is terminated on the -plane, where reduces to 2D point group . The bulk system also has time-reversal symmetry and inversion symmetry, which relates the top and the bottom surfaces in the absence of asymmetric surface terminations. The in-plane translational symmetry allows the definition of the surface Brillouin zone (SBZ), which is centered at and bounded by along the -direction and along -direction [Figure 1(a)]. Four Dirac points close to the Fermi energy have been observed in experimentsXu et al. (2012); Tanaka et al. (2012); Dziawa et al. (2012). Two Dirac points, denoted by , are located along , close to and symmetric about ; two others, denoted by , are located along , close to and symmetric about . The band dispersion around any of the four Dirac points is linear in all directions to first order, resulting in four copies of a spin-split Dirac semi-metal, related to each other by 90-degree rotations [Figure 1(b)]. Recently, scanning tunneling spectroscopy (STM) measurements suggestOkada et al. (2013) that in the rhombohedral phase, two of the four Dirac points are gapped [Figure 1(c)].
We assume that the Fermi level is exactly at the Dirac point energy. While this is not true in bulk samples due to intrinsic impurity doping, in thin-film samples the Fermi level can be tuned anywhere in the bulk gap. Since the change in the Chern number only depends on the electronic states near the gap-closing points, i.e., the four Dirac points, we start by deriving the effective theories for each Dirac cone and then consider their coupling to gap-opening perturbations. The minimal model for each Dirac cone , where , is a two-band model, due to the double-degeneracy at . The form of is determined by how the doublet at transforms under the little group at , i.e., a subgroup of the full symmetry group which leaves invariant. For example, consider : the little group is generated by the mirror reflection about the -plane, denoted by and a combined operation of a 180-degree rotation about the -direction followed by time-reversal, denoted by . This little group has only one 2D irreducible representation (see Sec.III.1): and , where means complex conjugation, and are Pauli matrices. It restricts to the form
(1) |
up to the first order of . decomposes into two components and , where is the unit vector along the -direction. The parameters can be fixed by matching the dispersion of equation (1), to the measured Fermi velocities along - and -directions [eVÅ]. The Dirac cones centered at can be related to the cone centered at by symmetry. This automatically gives the effective theories of the other Dirac cones: , and (see Sec.III.2 for a formal proof).
i.2 The effect of induced Zeeman field
We assume a Zeeman field in the sample along -direction, induced by ferromagnetically ordered dopants. In order to couple this field to the electrons in the models, we add an additional term to and note the following facts: (i) magnetization along -direction changes sign under both and and (ii) it is invariant under 90-degree rotations about -direction. Using these facts, we have:
where is the field strength of the Zeeman field, which is proportional to the Curie temperature, , of the ferromagnetism. The sign of depends on the direction of the magnetization. The Hamiltonian for each cone with the induced Zeeman field is , which has a gap of size at each Dirac point [see Figure 2(b)].
i.3 The effect of intrinsic and applied strain
Now we consider the effect of intrinsic and external strains. Depending on Sn and Se concentration, the cubic lattice can have spontaneous distortions into either rhombohedral or the rhombohedral symmetries. One may also cap the top surface of the film with a piezoelectric material such as BaTiO, to control the strain on the top surface. A general strain tensor is given by a symmetric matrix where , written in the frame spanned by . In order to represent couplings to the strain tensor in the models, we need to determine the transform of each component under the symmetry group and time-reversal (Table I). Using these relations, we obtain the following strain induced terms for the four Dirac cones, to the zeroth order of :
where are electro-phonon couplings.
Consider the full Hamiltonian for each Dirac cone under both Zeeman field and strain, . In , only terms proportional to open gaps in the spectrum while others move the position of the Dirac point . The gap at each , i.e., the coefficient before the term in the Hamiltonians, denoted below by , is:
where we have defined . Each gapped Dirac cone contributes
(2) |
to the Hall conductance (see Sec.III.3 for formal proof).
i.4 The effect of finite thickness
We have so far assumed that the top and the bottom surfaces are isolated from each other, and hence the total Hall conductance is
(3) |
where superscript denotes the top/bottom surface. When the thickness is comparable to the decay length of the surface states, the hybridization gap between the two surfaces, denoted by , becomes significant, and the total Hall conductance is generically not given by equation (3). Diagonalizing each Hamiltonian with hybridization (see Sec.III.4 for the explicit forms of the band dispersion)
(4) |
we have two scenarios. (i) If (where denotes the gap at top/bottom surface), as increases, the gap at closes at and reverses [see Figure 3], and at , the total contribution to vanishes; (ii) if , there is no quantum phase transition as increases, and the total contribution to Hall conductance stays at zero. The complete expression for the Hall conductance is therefore
(5) |
where is the Heaviside step function.
i.5 Proposals of materials and experiments
Depending on the parameter set of , the Chern number of the system takes each integer from to . In a realistic system, however, not all parameters are easily tunable, so the range of the Chern number is generically restricted. We propose a system shown in Figure 2(a): a thin-film PbSnTe doped with Mn or Cr, grown on a substrate, e.g., NaCl or KCl, with its top surface deposited with piezoelectric crystal such as BaTiO. Below K, the (Cr,Mn) moments develop ferromagnetism, inducing a small Zeeman gap meV in the sampleMathur et al. (1970). The external strain on the top surface may be tuned by the piezoelectric. Assuming that the strain in BaTiO be completely transferred to the top surface of the film, we estimate thatBerlincourt and Jaffe (1958); Bierly et al. (1963); Littlewood et al. (2010) the meVmV and meVmV. Since the sample with such composition has zero or negligible intrinsic distortion at low temperatures, . In the thick limit (nm), meV and is negligibleLin (2013). From equation (2), the bottom surface always contributes . There are three possible scenarios for the top surface, resulting in respectively: (i) , (ii) and (iii) , where we have assumed without loss of generality. The dispersion of the four gapped cones for the three scenarios are plotted in Figure 2(b-d). The total Chern number can thus be tuned between , and , plotted against and in Figure 2(e). In a thiner film with thickness nm, the hybridization gap is meVLiu et al. (2013b), from which we take meV as a typical value and we plot the Chern number against and in Figure 2(f). From this Figure, we see that around the critical field strength Vm, the Chern number can be electrically tuned to , or . If the length and width of the sample are both nm, this means that the Chern number can be tuned by varying within 10mV. The ability to tune the topological phase transition with such a small electric field offers hope that such a logic devices based on piezoelectric deformation of a TCI could possess on/off ratios and sub-threshold slopes which far exceed current logic device technologies.
Ii Discussion
In the derivation of the main results, we have ignored physical factors of (i) the impurities and (ii) the electron-electron interaction. The mirror Chern number of a TCI is only well defined in the presence of mirror planes. In a system with a random impurity configuration, mirror symmetries are broken and the mirror Chern number is not a good quantum number, and consistently, the gapless modes at the Dirac points are gapped by impurity scattering. This mirror symmetry breaking by impurity has, however, no effect on the Chern number in a ferromagnetically doped system, as long as the intensity of the random potential is much smaller compared with the Zeeman gap. This is because the Chern number, unlike the mirror Chern number, does not presume any symmetry, as long as the surface is gapped. Weak interactions smaller than the Zeeman gap do not have any effect on the quantized Hall conductance either, because the Chern number is also a good quantum number of an interacting gapped 2D systemX.L. Qi et al. (2006); Kohmoto (1985).
It is also interesting to discuss other surface terminations besides the -surface. On the -surface of SnTe, first principles calculationLiu et al. (2013a) shows that there are two Dirac cones centered at two Dirac points that are close to and symmetric about along in the surface BZ. The two Dirac points are protected by the mirror plane and have equal energy due to the mirror plane. A Zeeman field along gaps both Dirac points and results in a QAH phase with Chern number of . A strain along -direction breaks both the and the mirror planes, opening two gaps of opposite signs at the two Dirac points. When both the strain and the Zeeman field are present, a discussion similar to the one given in Sec.I.3 shows that the Chern number can be either or . On the -surface, there are four Dirac cones centered at and three ’s. The three Dirac points at have the same energy due to the threefold rotation symmetry about the -axis, while the one at generically has a different energy. This energy difference among the Dirac points, which has been measured to be meV in Ref.[Taskin et al., 2013], makes it hard to have a fully gapped surface using an induced Zeeman field, because the Zeeman gap is generically much smaller than meV. Therefore, an insulator with quantized Hall conductance on the -surface is not possible using the current scheme.
Iii Methods
iii.1 Derivation of using the little group at
The full symmetry group of the thin film in the absence of applied fields is . The little group at a Dirac point is the subgroup of all operations that leave invariant. The little group therefore consists of a mirror plane that passes , and their combinations. Taking as example, the little group is generated by and . In a general spin- system we have: and , where is the 180-rotation about -direction. Therefore the two generators satisfy (i) (ii) . There is only one 2D irreducible representation up to a basis rotation: and . Physically, relates the Hamiltonian to and commutes with ; or mathematically, and . The irreducible representation of the little group along with the symmetry constraints determine the form of shown in equation (1).
In general, the model is given by
(6) |
which must satisfy the symmetry constraints:
(7) | |||||
These symmetry constraints give that (1) is even under , (2) is odd under and (3) to arbitrary order. We expand them to the second order in :
(8) | |||||
(9) | |||||
(10) |
These terms make the dispersion deviate from perfectly linear and may be understood as the ‘warping’ terms; they also make corrections to the wave functions at each . It should be noted that holds up to arbitrary orders and this means there is no out-of-plain pseudo-spin component at any . While including higher order terms explains the shape-changing of the equal energy contours from perfect ellipsoids, the Lifshitz transition cannot be described in the framework of any two-band theory. To do so, the model must be extended a four-band one, in order to account for the hybridization between nearest cones, as discussed in Ref.[Fang et al., 2012].
iii.2 Relating the four Dirac cones by symmetry
In the main text, we mention that by 90-degree rotations the effective theories for the four cones can be related. This is an intuitive statement yet to be made precise. In fact, theories are always written with respect to a chosen basis, which is our case is furnished by (the periodic part of) the two Bloch states that are degenerate at the Dirac point. Due to the degeneracy, there is a gauge degree of freedom in the choice. Here the choice is made by fixing the little group representation at : and . If we denote the two basis states by and , we then fix the bases at to be , and , respectively. Mark that here is the matrix representing the 90-degree rotation in both orbital space (including spin). Defining the Bloch wave function at as , it is easy to check that , and . Here is the single particle operator acting in the Hilbert space, which is the combination of the orbital rotation plus rotation , where is a lattice point and the rotation center is also placed at a lattice point. The full single Hamiltonian, projected to the states at the vicinities of the four Dirac points, is given by
(11) |
symmetry implies , which immediately leads to , and , confirming the intuitive relations appearing in the main text.
iii.3 Calculation of the Chern number of the top/bottom surface
In the text we refer to the Chern number contributed by one massive Dirac cone, which is not mathematically well-defined. In fact, the integrated Berry’s curvature of a gapped Dirac cone is non-quantized in any finite -space, hence possesses no well-defined Chern number. The Chern number of a whole 2D surface (top surface for example) is, however, a well-defined quantity (if periodic boundary is taken for the other two directions), which may be calculated. Suppose we are interested in the Chern number, , at some Zeeman field . Then since time-reversal reverses the Chern number, we know for , the Chern number must be . Consider a 3D space spanned by and , then from Gauss’s law, the Chern number change from to equals the total monopole charge between these two planes in the 3D parameter space. The monopole, or gap closing point, is always at , around which the Hamiltonian is that of 3D Weyl fermions: , where . The charge of such a monopole is , and since there are in total four such monopoles between , we have the difference in Chern number , or . All Chern numbers obtained in the text are derived using this method.
iii.4 Diagonalizing the Hamiltonian in equation (4)
A Hamiltonian that describes isolated top and surface states around is
(12) |
and hybridization is equivalent to adding an off-diagonal block term, resulting in, to the lowest order in ,
(13) |
Diagonalizing directly, we obtain four bands:
(14) | |||||
(15) | |||||
(16) | |||||
(17) |
Straightforward algebraic work shows that the only solution for , i.e., a gap-closing point, exists at when .
Parallel discussion for proceeds and we conclude that a topological phase transition happens when
(18) |
whereas the Chern number contributed by the cone at changes from , depending on the sign of , to zero. Mark that on the right hand side of equation (18), if , the transition cannot happen at any .
Acknowledgements CF and BAB thank A. Yazdani, R. J. Cava, N. P. Ong, and A. Alexandradinata for helpful discussions. CF specially thanks J. Liu and H. Lin for providing useful information on thin-film samples. CF is supported by ONR-N00014-11-1-0635. MJG acknowledges support from the AFOSR under grant FA9550-10-1-0459 and the ONR under grant N0014-11-1-0728. BAB was supported by NSF CAREER DMR- 095242, ONR-N00014-11-1-0635, Darpa- N66001-11-1-4110, David and Lucile Packard Foundation, and MURI-130-6082.
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