# How to derive the effective Hamiltonian of two-dimensional TMDCs monolayers?

TMDs are transition metal dichalcogenides and have the chemical formula MX$$_2$$ where M is the transition metal and X is the chalcogen. An example of a TMD is MoSe$$_2$$.

I would like to demonstrate that the Hamiltonian of TMDs (transition metal dichalcogenides) is given by the following formula:

$$\begin{equation} \hat{H}_0=at(\tau k_x\hat{\sigma}_x+k_y\hat{\sigma}_y)+\frac{\Delta}{2}\hat{\sigma}_z-\lambda\tau\frac{\hat{\sigma}_z-1}{2}\hat{s}_z \tag{1} \end{equation}$$

where $$\tau=\pm1$$ is the valley index, $$a$$ is the lattice constant, $$\hat{\sigma}$$ are Pauli matrices, $$t$$ is the hopping integral, and $$\Delta$$ is the energy gap.

From what I've seen in articles and presentations, the first Hamiltonian term is similar to graphene, the second term is related to the breaking of symmetry that opens a band gap and the last term is the spin orbit coupling.

The basis wave functions chosen for the conduction and valence bands at K are related by time-reversal symmetry, consisting of hybridized d-orbitals from the transition metal with magnetic quantum numbers of m=0 and m=2$$\tau$$, respectively: $$\begin{equation} |\phi_c\rangle=|d_{z^2}\rangle\tag{1} \end{equation}$$ $$\begin{equation} |\phi_{v}^{\tau}\rangle=\frac{1}{\sqrt{2}}(|d_{x^2-y^2}\rangle+i\tau|d_{xy}\rangle) \end{equation}$$

Questions:

• How do you get the Hamiltonian formula for transition metal dichalcogenides? How to derivate the expression above (equation (1))?
• What does the parameter $$t$$ mean? I don't understand what they mean by "hopping integral".
• Why did you choose this basis wave functions?
• What is $$\lambda$$ in the mathematical formula?

Reference: Modtland, Brian, "Exploring Valleytronics in 2D Transition Metal Dichalcogenides" (MIT) (page 48 - equation (2.1)) (click here to see the document)

• @Anyon Thanks for the warning. I just added the document where I found the Hamiltonian in the references. Jul 27 '20 at 22:15
• @CarmenGonzález There's a few problems here: What is "k.p" in the title? That's a bit confusing! Also it doesn't seem you're looking for a "proof" but a "derivation". Also the thesis that you linked to says "Using appropriate basis wavefunctions for the conduction and valence bands based on the crystal symmetry, a two-band Hamiltonian can be written using the Pauli matrices :" before presenting this equation. Why not give us that information too, so that we have some more context without having to download a big PDF? Also, did you look at Ref. 34? Might be good to link that here too. Jul 28 '20 at 20:51
• For the term "hopping" and the meaning of $t$, this article should be enough I think: en.wikipedia.org/wiki/Hubbard_model, so perhaps this question can be made easier the "volunteers" here to answer, by reducing it to just "how is Eq. 1 derived", after giving more of the context from the thesis and the Ref.  in the thesis. Jul 28 '20 at 20:53
• Are you sure that this is a 2-band model? If the Hamiltonian is using standard notation (in which $\sigma$, $\tau$, and $s$ are all "Pauli matrices") then this looks more like an 8-band model... Jul 29 '20 at 15:39
• @NikeDattani I edited the answer to be more complete, with some information I have read on the subject. Thank you for your help. Even reading Wikipedia, I still didn't quite understand what they mean by "hopping integral". Jul 30 '20 at 22:15

How do you get the Hamiltonian formula for transition metal dichalcogenides? How to derivate the expression above (equation (1))?

• You can use the $$k \cdot p$$ method to derive this effective Hamiltonian. Please take a look at this paper for the details.

• $$k \cdot p$$ theory for two-dimensional transition metal dichalcogenide semiconductors.

What does the parameter 𝑡 mean? I don't understand what they mean by "hopping integral".

• You can just view it as a parameter of this tight-binding Hamiltonian (just a convention), which needs to be fitted by first-principles calculations.

Why did you choose this basis wave functions?

• It is decided by the group of the wave vector at the band edges ($$K$$) is $$C_{3h}$$.

What is $$𝜆$$ in the mathematical formula?

• $$2\lambda$$ is the spin splitting at the valence band top caused by the spin-orbit coupling, which can be decided with first-principles calculations.

• Useful references:

• Giving you a +1 here as always. I wanted to mention this because the other answer has more upvotes even though 50% of that answer is "do what Jack said", and probably what happened is that people upvoted the other answer from the "review queue" for the user's "first post", which means they were simply asked to vote on the user's first post, and so the voters did not see that there was also this answer. Dec 1 '20 at 6:18
• Yes, I already upvoted @Jack's answer too. I just wanted to explain what the hopping integral is, but I didn't have enough reputation to add a comment (until recently)... Dec 1 '20 at 17:18
• As @Jack suggested, try the $$\vec{k} \cdot \vec{p}$$ method.
• The hopping integral $$t_{ij}$$ is "borrowed" from the Hubbard model. It is the kinetic term in the Hamiltonian that explains electrons being able to "hop" from one atomic site to another. In real space, it is an integral of the orbital overlap between the two neighboring atomic sites located at $$\vec{x}$$ and $$\vec{x}'$$, respectively. $$t_{ij} = \int \phi_i (\vec{x}) \phi_j (\vec{x}') \mathrm{d}^3 \vec{x}$$