Superconducting Phases of Monolayer TMDs

Session R15: 2D Materials: Superconductivity and Correlations I

APS March Meeting 2016

Baltimore, MD, US

Transition metal dichalcogenides

TMD overview and tight-biding, low-energy, two-valley model
Intrinsic and induced superconductivity
Intervalley and intravalley paring channels
Optical valley selection rules and Berry curvature

TMD monolayers

Direct band gap semiconductors
Break inversion symmetry ⇒ gapped out spectrum
Two inequivalent valleys ⇒ new degree of freedom
Strong spin-orbit coupling ⇒ large valance band spin-splitting
Opposite valley Berry curvature
Optical valley probe and valley Hall effect
Two state tight binding model: $ d_{z^2} $, and $ d_{xy} \pm i d_{x^2 - y^2} $

$$ H_0^{\tau s} \left( \mathbf{k} \right) = a t \left(\tau k_x s_x + k_y s_y \right) \otimes I_2 $$$$ + \frac{\Delta}{2} s_z \otimes I_2 - \lambda \tau \left(\sigma_z - 1 \right) \otimes S_z $$

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

Energy bands

$ \Delta $—band splitting
$ \lambda $—spin splitting
$ \tau $—valley index
$ s $—spin index

$ a t = 3.15 $ Å eV
$ \Delta = 1.66 $ eV
$ 2 \lambda = 0.15 $ eV
$ \mu = -0.83 $ eV

Superconducting sources

Substrate induced

$$ V = - \sum_{\mathbf{k} \nu \tau} \Delta_{\nu} {d^{\nu}_{\tau}}^{\dagger} \left( \mathbf{k} \right) d^{\nu}_{-\tau} \left( \mathbf{k} \right) $$

Intrinsic (density-density interaction)

$$ V = \frac{1}{2} \sum_{\mathbf{R} \mathbf{R'}} v_{\mathbf{R} \mathbf{R'}} :n_{\mathbf{R}} n_{\mathbf{R'}}: $$

Assume even pairing interaction

Superconducting channels

Project into upper-valance bands ($ \tau = s $)
Mean field theory BCS-type solutions: gap function $ \Delta_{\mathbf{k}} $
$ m = 0 $ intervalley channel always dominates
Both classes give constant gap function for dominant paring

Induced

$$ \Delta_{\mathbf{k}} = \frac{1}{2} \left( \Delta_+ + \Delta_- \right) - \frac{1}{2} \left( \Delta_+ - \Delta_- \right) \cos{\theta_{\mathbf{k}}} $$

Intrinsic

$$ \Delta_{\mathbf{k}} = g_{\mathbf{k}} v_0 \sum_{\mathbf{k}'} g^*_{\mathbf{k}'} \langle c_{-\mathbf{k}' \alpha'} c_{\mathbf{k}' \alpha} \rangle $$

Optical coupling

Harmonic perturbation ⇒ correlated phase's optical transition rates
$ \mathbf{A} / |\mathbf{A}| = \mathbf{\epsilon} e^{i \omega t} + \mathbf{\epsilon}^* e^{-i \omega t} $, $ \mathbf{k} \rightarrow \mathbf{k} + e \mathbf{A} $
Valley selection remains coupled to circularly polarized light

Berry curvature

BCS ground state has zero net curvature
Optical pair-breaking ⇒ selectively excite carriers in single valley
Berry curvature retains relative sign relations
Anomalous Hall velocity $ \mathbf{v} = - \dot{\mathbf{k}} \times \mathbf{\Omega} \left( \mathbf{k} \right) $