Superconducting phases of monolayer transition-metal dichalcogenides

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$$\newcommand{\vect}[1]{\mathbf{#1}} \newcommand{\vK}{\mathbf{k}} \newcommand{\exOfK}{\left( \vK \right)} \newcommand{\exOfMK}{\left( - \vK \right)} \newcommand{\hc}{\text{h.c.}} \newcommand{\exInteration}[2]{ {c^-_{#1}}^† \left( \vK' \right) {c^-_{#2}}^† \left( - \vK' \right) c^-_{#2} \left( - \vK \right) c^-_{#1} \left( \vK \right) } $$

Outline

Dichalcogenides overview
Induced superconducting phase
Optical transitions
Superconducting optical excitations

Effective Hamiltonian

$ \mathrm{MoS_2} $, $ \mathrm{WS_2} $, $ \mathrm{MoSe_2} $, $ \mathrm{WSe_2} $
Similar to monolayer graphene: two inequivalent valleys: $ \vect{K} $, $ \vect{K}' $
Strong spin-orbit coupling and inversion symmetry breaking
Leads to opposite valley Berry curvature
Tight binding model: $ d_{z^2}, d_{xy}, d_{x^2 - y^2} $

$$ H_0^{τ σ} \exOfK = a t \left(τ k_x σ_x + k_y σ_y \right) ⊗ I_2 + \frac{Δ}{2} σ_z ⊗ I_2 - λ τ \left(σ_z - 1 \right) ⊗ S_z $$

$$ H_0^{τ σ} \exOfK = \left[ \begin{matrix} \dfrac{Δ}{2} & a t \left( τ k_x - i k_y \right) \\ a t \left( τ k_x + i k_y \right) & λ τ σ - \dfrac{Δ}{2} \end{matrix} \right] $$

D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108, 196802 (2012).

Energy Bands

$ Δ $—band splitting
$ λ $—spin splitting
$ τ $—valley index
$ σ $—spin index

$ \mathrm{MoS_2} $

$ a t = 3.15 \: \text{Å eV} $
$ Δ = 1.66 \: \text{eV} $
$ 2 λ = 0.15 \: \text{eV} $
$ μ = -0.83 \: \text{eV} $

$$ E_{τ σ}^n \exOfK = \frac{1}{2} \left( λ τ σ + n \sqrt{ (2 a t)^2 \left\lvert \vect{k} \right\rvert^2 + \left( Δ - λ τ σ \right)^2 } \right) $$

Induced Superconductivity

Intervalley pairing

BCS pairs for induced superconducting states.

$ a^ν_{τ σ} $—orbital operators
$ b_α $—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

$$ \begin{equation} H_V = - \sideset{}{'}∑_{\vK} \sideset{}{}∑_{ν, τ} Δ_ν {a^ν_{-τ ↓}}^† \exOfMK {a^ν_{τ ↑}}^† \exOfK + \hc \end{equation} $$

$$ \begin{equation} H - μ N = \sideset{}{'}∑_{\vK} \sideset{}{}∑_α λ_{\vK}^α b_{\vK α}^† b_{\vK α} + \sideset{}{'}∑_{\vK} \left(ξ_{\vK ↓} + λ_{\vK}^- \right) . \end{equation} $$

Optical Transitions

$ \vect{P}^{τ σ} \exOfK = \frac{m_0}{ħ} \left\langle u_+ \right\rvert ∇_{\vK} H_0^{τ σ} \exOfK \left\lvert u_- \right\rangle $

$ P_±^{τ σ} \exOfK = P_x^{τ σ} ± i P_y^{τ σ} $

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

Induced superconducting optical transition rates for $ τ = + $.

$ \vect{P} \exOfK = \frac{m_0}{ħ} \left\langle Ω_f \right\rvert ∇_{\vK} H^{τ σ} \exOfK \left\lvert Ω \right\rangle $

$ P_± \exOfK = P_x ± i P_y $

$ \left\lvert Ω \right\rangle = ∏_{\vK} b_{\vK ↑} b_{-\vK ↓} \left\lvert 0 \right\rangle $

$ \left\lvert Ω_f \right\rangle = \begin{cases} {c^+_α}^† \exOfK b_{-α} \exOfMK \left\lvert Ω \right\rangle & k > k_μ \\ {c^+_α}^† \exOfK b_{-α}^† \exOfMK \left\lvert Ω \right\rangle & k < k_μ \end{cases} $

SC Optical Excitations

Compare to normal transitions

Strong induced superconducting optical transition rates. Dashed lines are $ H_0 $ transitions.

Weak induced superconducting optical transition rates. Dashed lines are $ H_0 $ 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

Outline

Effective Hamiltonian

Energy Bands

\( \mathrm{MoS_2} \)

Induced Superconductivity

Intervalley pairing

Optical Transitions

SC Optical Excitations

SC Optical Excitations

Compare to normal transitions