Superconducting phases of monolayer transition-metal dichalcogenides
Energy Bands
- Δ—band splitting
- λ—spin splitting
- τ—valley index
- σ—spin index
MoS2
- at=3.15Å eV
- Δ=1.66eV
- 2λ=0.15eV
- μ=−0.83eV
Enτσ(k)=12(λτσ+n√(2at)2|k|2+(Δ−λτσ)2)
Induced Superconductivity
Intervalley pairing
- aντσ—orbital operators
- bα—quasiparticle operators
- BCS pairs in opposite valleys
- Reduces to standard BCS Hamiltonian where α=τ=σ plays the role of the spin index
- Not a singlet ground state: mixture of singlet and triplet states
HV=−∑∑′k∑∑ν,τΔνaν−τ↓†(−k)aντ↑†(k)+h.c.
H−μN=∑∑′k∑∑αλαkb†kαbkα+∑∑′k(ξk↓+λ−k).
Optical Transitions
Pτσ(k)=m0ħ⟨u+|∇kHτσ0(k)|u−⟩
Pτσ±(k)=Pτσx±iPτσy
- Right circular polarization strongly couples to τ=+ valley transitions
- Left circular polarization strongly couples to τ=− valley transitions
D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108, 196802 (2012).
SC Optical Excitations
P(k)=m0ħ⟨Ωf|∇kHτσ(k)|Ω⟩
P±(k)=Px±iPy
|Ω⟩=∏kbk↑b−k↓|0⟩
|Ωf⟩={c+α†(k)b−α(−k)|Ω⟩k>kμc+α†(k)b†−α(−k)|Ω⟩k<kμ
SC Optical Excitations
Compare to normal transitions
- Upper band excitations are now paired with lower band quasiparticle excitations
- Valley-polarization coupling is retained even in the superconducting case
- Contrast is reduced in an region around the chemical potential