Chapter 4. Topological analysis of the effective theory
4.2 Chern number and Z 2 number of topological insulator
The Berry phase or the Berry curvature discussed in previous section enable us to study the topology of the energy band structures. By Eq. (4.5), we can define the Berry curvature of the n-th band, Ω𝑛𝑛(𝐤𝐤), with the varying parameter being the crystal momentum, 𝐤𝐤. Since the crystal momentum is defined on the first Brillouin zone with the closure property, 𝐤𝐤 + 𝐆𝐆 = 𝐤𝐤, the domain of Ω𝑛𝑛(𝐤𝐤) is a close surface (manifold).
According to the Chern theorem, the integral of the Berry curvature over a closed manifold is quantized in unit of 2𝜋𝜋, which defines the Chern number of the nth band
1
2 . . ( )
n B Z n
C = π
∫
Ω k kd ∈.The total Chern number defined by summing over all occupied bands, 𝐶𝐶 =
∑𝑛𝑛∈valance𝐶𝐶𝑛𝑛, is an invariance provided there is a finite gap separating the valance and conduction bands within the whole Brillouin zone. Thus, 𝐶𝐶 is a topological order that characterize the topology of the bands. Historically, Thouless et al utilize Kubo formula to calculate 𝜎𝜎𝑥𝑥𝑦𝑦 of a quantum Hall system and recover 𝜎𝜎𝑥𝑥𝑦𝑦 = 𝐶𝐶 𝑒𝑒2⁄ which explains ℎ the robustness of the quantization of 𝜎𝜎𝑥𝑥𝑦𝑦, and provides a topological understanding of quantum Hall effect (QHE). Following this, the field of topological insulator (or topological phase) arises. General speaking, a topological insulator is an insulator with bulk gap, however, the conducting edge states arise at the boundaries if it is connected to other type of insulator. The Chern number categories the classes of topological phase and the topological phase transition is accompanied by a process of gap closing during the varying of parameters in the Hamiltonian. In this sense, the topological phases are robust under perturbation to the Hamiltonian.
Topological quantities are practically significant in characterizing the electrical transport properties in quantum Hall effects of 2DEG. It has been proposed earlier that the quantum Hall (QH) effect is associated with a topological invariant integer known as Chern number. The value of Chern number 𝑛𝑛, which gives the quantized Hall conductivity for each band𝐶𝐶𝑒𝑒2ℏ, is given by the integral of Bloch wave functions over the magnetic Brillouin zone. Since the Hall conductivity does not obey the time-reversal symmetry (TRS), recently people have proposed that the SOI in a single plane of graphene in absence of magnetic field results in TRS quantum spin Hall (QSH) state which has a bulk energy gap and a pair of gapless spin filtered edge states on the boundary. The QSH effect is analogous to QH effect but it doesn’t break TRS. The QH states are specified by Chern numbers, while the QSH states are characterized by 𝑍𝑍2 numbers. As TRS rising, for instance, in a 2D model [8] (π-electron tight-binding model with mirror-symmetry about the plane), the perpendicular spin component 𝑆𝑆𝑧𝑧
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TRS requires the Chern number 𝐶𝐶 = 𝐶𝐶↑ + 𝐶𝐶↓ is equal to zero, distinguish from ordinary insulator and offers a new class of topological invariant which is classified as 𝑍𝑍2 topological order. The 𝑍𝑍2 invariant is shown as
𝜈𝜈 = 𝐶𝐶𝜎𝜎 mod 2.
The value 𝜈𝜈 = 0 is trivial insulator and 𝜈𝜈 = 1 is topological insulator (QSH insulator).
At the end of this section, we want to discuss the intuitive 2-band model that manifests the topology of the system. The Hamiltonian of the 2-band model is, generally, expressed in the combination of Pauli matrices:
ˆ( ) ( ) ˆ z x y
where + and – stand for conduction and valance bands, respectively, and θ, φ are the polar and azimuth angles of the unit vector, 𝒏𝒏� = ‖𝐡𝐡‖−1𝐡𝐡(𝐤𝐤) = (sin𝜃𝜃cos𝜑𝜑, sin𝜃𝜃sin𝜑𝜑, cos𝜃𝜃), defined over the Bloch sphere. Assume there is an overall gap separating the conduction and valance bands, thus it’s straightforward that
2 2
Therefore the Berry curvature of the valance band would be
( )
and the Chern number is
( )
Since 𝐧𝐧� is a unit vector which defines the mapping from Brillouin zone to the normal vector at Bloch sphere, 𝐧𝐧� ⊥ ∂𝜇𝜇 𝐧𝐧�; the integrand of Eq. (4.6) immediately interpreted as the solid angle spanned by ∂1𝐧𝐧�𝑑𝑑𝑘𝑘𝑥𝑥 and ∂2𝐧𝐧�𝑑𝑑𝑘𝑘𝑦𝑦. Integrate over the whole Brillouin zone and the Chern number, here, defines the number of times 𝐧𝐧� wrapping around the Bloch sphere as Brillouin zone mapping to 𝐧𝐧�.32
Chapter 4. Topological analysis of the effective theory
4.3 Topological analysis of the effective theory
In this section, the topological quantities such as Berry curvature and Chern (or Z2) number of the muffin-tin potential lattice with SOI and general magnetic field, B, is discussed. In the former thesis of our group [15], we have demonstrated the successful of effective theory in calculating the Berry curvature and Chern number. Therefore, we will continually employ the effective Hamiltonian in Eq. (3.10) and analytically discuss the band topology.
Particularly, the effective Hamiltonian we considering here is, by Eq. (3.10),
SO
and immediately, we have
{ }
which is useful in calculating
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( ) ( )
From Eq. (4.10) and the definition of Berry curvature in Eq. (4.4), we have
( ) ( )
and the Berry curvature in Eq. (4.11) is further reduced( ) ( )
The Chern number is straightforwardly,
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Chapter 4. Topological analysis of the effective theory
valley-independent, therefore, it concludes the Chern number, 𝐶𝐶𝑛𝑛 , is also 𝜏𝜏 -independent. Finally, the Chern number of the band label by 𝜂𝜂 and 𝑖𝑖 iswhich is decomposed into two uncoupled 2 × 2 matrices. Therefore the general eigen states of Eq. (4.16) would be
Following similar derivation from Eq. (4.10) to Eq. (4.11), it is easy to show the Chern numbers here are zero for all the four bands.
In conclusion, we have derived the Chern numbers that manifest the band topology of the MTP system. The Chern numbers are characterized by the parameters 𝛾𝛾⊥𝜀𝜀⊥ which are introduced in Eq. (4.7). If only in-plane magnetic field is presence (i.e. 𝜀𝜀⊥ = 0), the Chern numbers are zero, independent of the magnitude and direction of 𝐁𝐁∥. However, for the presence of both 𝐁𝐁∥ and 𝐁𝐁⊥, the Chern numbers become 12𝑖𝑖𝛾𝛾⊥, depending on the direction of 𝐁𝐁⊥ (still independent of the magnitude of magnetic field). Fig 4.1 (a), (b), (c) and (d) label the Chern numbers for the lower bands under the different conditions of magnetic field.
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Fig 4.1 The Chern numbers of the lowest 4 bands near the K point. (a) 𝐁𝐁∥ ≠ 0, 𝐁𝐁⊥ = 0 (b) 𝐁𝐁∥≠ 0, 𝐁𝐁⊥ = |𝐁𝐁⊥|𝑧𝑧̂ (c) 𝐁𝐁∥ ≠ 0, 𝐁𝐁⊥ = −|𝐁𝐁⊥|𝑧𝑧̂ (d) 𝐁𝐁∥ = 0, 𝐁𝐁⊥ = −|𝐁𝐁⊥|𝑧𝑧̂. The Chern numbers of the band are indicated by the numbers with corresponding color. The SOI are present for all of them. The parameters we employ here are 𝑚𝑚∗ = 0.023𝑚𝑚𝑒𝑒; the MTP strength 𝑈𝑈0 = 165meV with diameter, 𝑑𝑑 = 0.663𝑎𝑎, where 𝑎𝑎 = 40 nm is the lattice constant. The SOI coupling constant is 𝜆𝜆 = 120Å2.
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