Superconducting van der Waals materials

Gapped trivial superconductors :
- They are inherently a conventional superconductors.
- Examples - NbSe2,
Nodal superconductivity and spin-triplet superconductivity
- They are not SC by themselves.
- Example - Twisted graphene multilayers
- Very low DOS near fermi level.
- Not enough electrons.
- Can be unconventional SC
- M.F can even enhance SC (spin-triplet SC) : Reenterant SC
Gapped topological superconductors
- Example - CrBr3/NbSe2

Electronic interactions are responsible for symmetry breaking. 1. Broken time-reversal symmetry : Classical magnets ($M \rightarrow -M$) 2. Broken crystal symmetry : CDW ($r \rightarrow r + R$) 3. Broken gauge symmetry : Superconductors ($<c_\uparrow c_\downarrow> \rightarrow e^{i\phi}<c_\uparrow c_\downarrow>$)

$\(H = \sum_{i,j} t_{ij}c_i^\dagger c_j + \sum_{ijkl} V_{ijkl} c_i^\dagger c_j c_K^\dagger c_l$\) What are these interactions coming from? - Electronic (repulsive) interactions - Mediated by other quasiparticles(phonons, magnons, plasmons) - The net effective interaction can be attractive or repulsive. SC requires attractive interactions

With a mean field description : Approximate quadratic Hamiltonian and Effective single particle description. Weakly correlated matter. $\(H \approx \sum_{i,j}\bar{t}_{ij}c_i^\dagger c_j + \sum_{ij} \Delta_{ij} c_i c_j$\)

Origin of attractive interactions¶

Conventional superconductors¶

Phonons

Unconventional superconductors¶

Antiferromagnetic magnons
Ferromagnetic magnons
Plasmons
Valence fluctuations
Charge fluctuations

A simple interacting Hamiltonian¶

\[H = \sum_{ij} t_{ij}[c_{i\uparrow}^\dagger c_{j\uparrow} + c_{i\downarrow}^\dagger c_{j\downarrow}] + \sum_i U c_{i\uparrow}^\dagger c_{i\uparrow} c_{i\downarrow}^\dagger c_{i\downarrow}$$ $U < 0 \rightarrow 0$ : Superconductivity Mean field approximation $$U c_{i\uparrow}^\dagger c_{i\uparrow} c_{i\downarrow}^\dagger c_{i\downarrow} \approx U <c_{i\uparrow}^\dagger c_{i\downarrow}^\dagger> c_{i\uparrow} c_{i\downarrow} + h.c\]

$\(U c_{i\uparrow}^\dagger c_{i\uparrow} c_{i\downarrow}^\dagger c_{i\downarrow} \approx \Delta c_{i\uparrow} c_{i\downarrow} + h.c$\) $\Delta \sim <c_{i\uparrow}^\dagger c_{i\downarrow}^\dagger>$ is the superconducting order. This is a s-wave superconducting order which is uniform in momentum space when we do Fourier transform.

If we have a Hamiltonian $H \sim \Delta c_{i\uparrow} c_{i\downarrow} + h.c$, then we understand that this term destroys (creates if $c_{i\uparrow}^\dagger c_{i\downarrow}^\dagger$) two electrons. This means that this state cannot have constant number of electrons. $\(H|GS> = E_{GS} |GS>$\) The ground state($|GS>$) cannot have a well-defined number of electrons. $\(|GS> \sim |2e> + |4e> + |6e> + \cdots$\) $\(|GS> \sim |1e> + |3e> + |5e> + \cdots$\)

How does superconducting order ($\Delta = <c_{i\uparrow}^\dagger c_{i\downarrow}^\dagger>$ ) transform under a gauge transformation?¶

The phases do not cancel out, instead it adds up($\Delta \rightarrow e^{-2i\phi}\Delta$). Superconducting order is an order parameter which changes under gauge transformation. Hence superconductivity breaks Gauge Symmetry. Under Nambu representation a superconducting gap opens up for a non-zero $\Delta$.

Impact of Superconductivity in the electronic structure¶

The following images show the electronic structure for triangular lattice for a varying chemical potential($\mu$).

For $\Delta = 0$

For $\Delta = 0.5$ triangular-lattice-delta_0.5.png

A non-zero uniform (in momentum space) superconducting order parameter will always open a gap with same magnitude, irrespective of what the chemical potential is or how does the dispersion looks like.

Gapped and gapless superconductivity¶

The electronic structure is modified differently depending on the type of superconductivity.