3.2 Hall effects in inversion symmetry broken honeycomb lattices
3.2.2 Hall conductivity
We now wish to compute the quantum Hall conductivity due to each valley. The most expedite way to do this is to use Eq. (B.36) with Eq. (3.29), noting that, for a two-level massive Dirac Hamiltonian, we haveΘ =π/2andΦ = 2πτ. This yields
στH= e2 h
Φ 2πsin2Θ
2 =τ3e2 2h,
which shows that the Chern number of a two-level massive Dirac Hamiltonian has Chern number equal to1/2.
There is a subtlety regarding the integration that leads up to Eq. (3.29), which is the re- quirement to take it over the entire real plane. Likewise, Eq. (B.35) requires the integral to be taken over the entire BZ. However, this curvature is only defined in a neighbourhood of the
±K valleys, therefore, it is expected that the integral of this function would require a cut-off in crystal momentum, to account for the fact that it is obtained under a low-energy, continuum limit approximation. Thus, we are faced with the problem of determining an appropriate momentum cut-off and, if possible, of making sure that the final result does not depend on it, so that it can be taken to infinity and, thus, recover the result of Eq. (3.29).
An energy scale argument
To solve this, it is useful to go to the definition of the Berry curvature in terms of the 1st order adiabatic evolution of quantum states of Sec. B.2. There, it is found that the Berry curvature of a Bloch band can be physically interpreted as the effect of the residual overlap of the band’s states with empty states in other bands, due to the transitions induced by the time-variation of the momentum. According to the form of the time-evolved state given by Eq. (B.24), this overlap is inversely proportional to the energy difference between states. Indeed, it is found that the Berry curvature can be written as
Ωnµν =i X
m6=n
hn|∂µHˆ|mihm|∂νHˆ|ni − hn|∂νHˆ|mihm|∂µHˆ|ni
(εn−εm)2 ,
from Eq. (B.14), showing that it is inversely quadratic dependent on the energy difference.
This aspect of the Berry curvature leads us to state that the contributions to the Berry cur- vature of the Brillouin Zone are concentrated at the valence-conduction band edges - the±K points, in our case - and their immediate vicinity. This means that we can expect Eq. (3.31) to contain all the essential features of the Berry curvature, and that contributions to higher order ofkare negligible. Thus, although this result was obtained from an apparently stringent limit, it holds, to a very good approximation, within a broad range of values ofk.
In turn, this fact can be translated into the magnitude of the cut-off of the aforementioned integral, in the sense that it can be much larger than one would expect if one were only to consider that the result is derived from a low-energy approximation. Namely, the only condition is that the cut-off keeps the contour of the surface integration from going into the other vertices of the BZ, where it would be necessary to account for contributions from other valleys. These are separated by magnitudes of4π/√
3from equivalent valleys and of4π/3from the inequivalent valley. Therefore, the contour of the integral of the Berry curvature should go no farther than half the magnitudes to the nearest valleys.
Now consider the plots of the Berry curvature of Eq. (3.31) at the+Kvalley, for variousMX2,
shown in Figure 3.1. It is clear that the curvature is very near zero at even half the magnitude separating nearest valleys. Thus, to an excellent approximation, we can make the cut-off go to infinity, since contributions beyond the cut-off are negligible.
● MoS2 ■ WS2 ◆ MoSe2 ▲ WSe2
●
●
●
■
■
■
■
◆
◆
◆
◆
◆
▲
▲
▲
▲
▲
▲
▲
-2π
3 -π
3
0 π
3
2π 3
0 2 4 6
ka Ω12
a2
Figure 3.1: Plot of the Berry curvature of Eq. (3.31) in units of the lattice constant a. Values used for the parameterst≡1tvcand∆are those of Table 3.1. (Note: The curves for MoS2and MoSe2coincide).
A topological argument
An alternative argument to circumvent this problem, one which does not invoke the limited width of the curvature, is to notice that the parametersΘandΦ, the only variables that contain information specific to the model, do not depend on the half-gap∆. Thus, it should be possible to manipulate the system into closing the gap and reduce the Berry curvature to a sharp peak at theK-point, in which case we would be able to choose an arbitrarily small region of integration that would contain all the curvature of that valley. In turn, this amounts to making the cut-off tend to infinity and the result is, thus, recovered.
The independence of this result on system-specific parameters reflects the topological origin of these quantities. Indeed, in the previous section we managed to compute the Chern number
for any two-level system while containing all the model-specific details to a single numberΘ, which, taking after the particular case of the massive Dirac Hamiltonian, we can expect to not depend on any system-specific parameters. Indeed, this means that all systems described by a given family of Hamiltonians, related among each other by variation of parameters, are classified by the same number, or set of numbers, of topological origin. Due to this invariance under continuous deformations (i.e. variation of parameters), these numbers are known as topological invariants.
Considering the foregoing arguments, not only is the use of Eq. (3.29) validated, it is also possible to integrate (3.31) over all momentum space. Introducing the dimensionless vector y=atk/∆and integrating over the angular part yields
σvµν =εµντ3
e2 2h
ˆ +∞
0
d|y| 1
(1 +|y|2)3/2, (3.32)
that is solved simply by making the variable substitution x2= 1
1 +|y|2 ,0≤ |y|<+∞ ⇔ |y|=
√1−x2
x , 0< x≤1 ⇒ d|y|=− dx x2√
1−x2, for which the integral of Eq. (3.32) becomes
ˆ 1 0
dx x
√
1−x2 = 1,
so that the contribution to the Hall conductivity of the valence band from a Dirac cone indexed byτ is given by
σvµν =εµνστH=εµντ3e2
2h. (3.33)
Hence, we have recovered the result computed at the very beginning of this section. We can conclude that the Hall conductivity of each Dirac cone is quantized in units of e2/(2h), with sign determined by the valley index. Before proceeding into further discussion of this effect, note that integration of the Berry curvature over the entire BZ yields a null Chern number, since inequivalent valleys contribute an equal absolute amount of opposite sign.
Taking after the symmetry considerations of Sec. B.1.2, this result is no surprise. Indeed, crystal momentum transforms as k → −k under parity and under time-reversal, satisfying Eq. (B.16), and the system described by the Hamiltonian of Eq. (2.14) breaks only parity, pre- serving time-inversion symmetry. Thus, according to Eq. (B.20), the Chern number vanishes for every band of this system and the material is trivial in the topological sense. The fact that the valleys have a non-vanishing Chern number in no way contradicts this, since the Chern number is a global property, accumulated from the Berry curvature over the BZ, meaning that it can be locally non-vanishing, as long as the non-vanishing contributions compensate reciprocally.
Moreover, the existence of an even number of valleys is a fundamental consequence of
defining a gauge theory on a lattice, a result which is encoded in the general chiral fermion doubling theorem [112]. Thus, for any crystalline material with a non-identically vanishing Berry curvature, the sum of the Chern number contributions of the valleys will either vanish or yield an integer multiple of the quantum of conductivitye2/h.
The valley Hall effect
According to the findings of Sec. B.2, namely, Eqs. (B.29) and (B.32), the Berry curvature of the total parameter-space represents the response of the expectation value of the velocity of particles in Bloch states to the variation of an external parameter. In the particular case that the external parameters couple minimally to momentum, like an electric field through a time-varying vector-potential as in the Hamiltonian of Eq. (B.30), the parameter space reduces to the Brillouin Zone. Thus, the response of Bloch electrons in a material to a linearly weak, spatially uniform electric field depends only on the Berry curvature of the BZ.
Having this in mind, Eq. (3.31) shows a remarkable feature: under the application of an in-plane electric-field, electrons will not only move transversally to its direction, they will also move in opposite directions according to which inequivalent valley they are in, giving rise to a net valley Hall current. This is the valley Hall effect [3].
Although the quantized Hall response vanishes for this class of materials, the existence of a Berry curvature and, thus, the topological features of the valleys, are experimentally observ- able and have physical consequences. It was first proposed for inversion-symmetry broken graphene [93], where the existence of a Berry curvature would allow charge carriers from each valley to be selectively excited by circularly polarized light. This can be intuitively understood in the sense that the the valleys are connected by time reversal symmetry. Thus, only by inter- acting with the system with some time-reversal-symmetry breaking perturbation can we obtain differentiated responses from the inequivalent valleys. Exciting charge carriers from a specific valley thus requires breaking time reversal symmetry with circularly polarized light.
The fact that TMDCs, beyond being intrinsically inversion-symmetry broken, have a sizeable spin-splitting, allows excitation of charge carriers polarized in the valley as well as in the spin degrees of freedom. This has been proposed in [3] and experimentally observed in [97].
Part II
Electron-Electron Interactions
Chapter 4
Perturbation Theory and
self-energy correction to the electronic structure
This chapter explores corrections to the electronic structure due to the electron-electron inter- active problem in TMDCs based on the Hartree-Fock approximation. We will be taking a dia- grammatic approach based out of the formulation to be used throughout the remainder of this work, which is the path integral formulation of quantum field theory for quantum statistical me- chanics, presented in Appendix C.