Bifurcation of topological phases in two dimensional Ising superconductor NbSe2 Monolayer

Zoltán Tajkov

APCOM, June 21, 2024

Centre of Low Temperature Physics, Institute of Experimental Physics, Slovak Academy of Sciences

This work was supported by the Slovak Academy of Sciences project IMPULZ IM-2021-42

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F. Soto et al. / Physica C 460–462 (2007) 789–790

M. R. Sahu et al. Phys. Rev. B 94, 235451 2016

\( T_c \approx 7 K \)

\( 2\Delta \approx 2.7 \, \mathrm{meV} \)

Xi, X., et al. Nature Phys 12, 139–143 (2016).

Ising superconductor

APCOM, June 21, 2024

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APCOM, June 21, 2024

Tight binding 

description

  • \( d \) characteristics
  • Fitted to DFT up to 7 neighbors
  • 1 electron / site + spin

Superconductivity from 

Bogoliubov–de Gennes Hamiltonian

This gives us \( H_\mathrm{e}(\mathbf{k}) \rightarrow \)

\begin{equation*} \mathcal{H}_{\mathbf{k}}^{\mathrm{BdG}} = \begin{pmatrix} H_\mathrm{e}(\mathbf{k}) & \Delta_{\mathbf{k}}\\ \Delta^{\dag}_{\mathbf{k}}& -H_\mathrm{e}^{\mathrm T}(-\mathbf{k}) \end{pmatrix} \end{equation*}
\Delta_{\mathbf{k}}

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APCOM, June 21, 2024

\Delta_{\mathbf{k}}

What is

Special thank to

Marko Milivojević

General formulation

\begin{aligned} \Delta_{A_1}^s(p,q) &= \sum_{g \in {\bf C}_{3v}} A^*_1(g) \cos{({\bf k} \cdot (g^{-1} {\bf r}_{p,q}))} d_0, \\ \Delta_{A_2}^s(p,q) &= \sum_{g \in {\bf C}_{3v}} A^*_2(g) \cos{({\bf k} \cdot (g^{-1} {\bf r}_{p,q}))} d_0, \\ \Delta_{E,1/2}^s(p,q) &= \sum_{g \in {\bf C}_{3v}} (E_{11/22}(g))^* \cos{({\bf k} \cdot (g^{-1} {\bf r}_{p,q}))} d_0. \end{aligned}
\begin{array}{c|l} A_1 & \Delta^{s}_{A_1}(1,0)=\left[2\cos{(k_x a)}+4\cos{\frac{k_x a}{2}}\cos{\frac{\sqrt{3}k_y a}{2}}\right]d_0 \\ & \Delta^{t,z}_{A_1}(1,0)=\left[\left(\cos{\frac{k_xa}{2}}-\cos{\frac{\sqrt{3}k_y a}{2}}\right)\sin{\frac{k_xa}{2}}d_z\right] \\ & \Delta^{t,xy}_{A_1}(1,1)=\left[ 3\cos{\frac{k_x a}{2}} \sin{\frac{\sqrt{3} k_ya}{2}}\right]d_x -\left[\sqrt{3}\left(2\cos{\frac{k_xa}{2}}+\cos{\frac{\sqrt{3}k_y a}{2}}\right)\sin{\frac{k_xa}{2}}\right]d_y \\ \hline A_2 & \Delta^{s}_{A_2}(2,-1)=2 \left[\sin{\frac{k_xa}{2}}\sin{\frac{3\sqrt{3}k_ya}{2}}-\sin{(2k_xa)}\sin{(\sqrt{3}k_ya)}+\sin{\frac{5k_xa}{2}}\sin{\frac{\sqrt{3}k_ya}{2}}\right]d_0 \\ & \Delta^{t,z}_{A_2}(1,-1)=\left[\sin{(\sqrt{3}k_ya)}-2\sin{\frac{\sqrt{3}k_ya}{2}}\cos{\frac{3k_xa}{2}}\right]d_z \\ & \Delta^{t,xy}_{A_2}(1,0)=\left[\left(2\cos{\frac{k_xa}{2}}+\cos{\frac{\sqrt{3}k_y a}{2}}\right)\sin{\frac{k_xa}{2}}\right]d_x +\left[\sqrt{3}\cos{\frac{k_x a}{2}} \sin{\frac{\sqrt{3} k_ya}{2}}\right]d_y \\ \hline E & \Delta_{E,{1/2}}^s(1,0)=\left[\cos{(k_xa)}-\cos{\frac{k_xa}{2}}\cos{\frac{\sqrt{3}k_ya}{2}}\pm{\rm i}\sqrt{3}\sin{\frac{k_xa}{2}}\sin{\frac{\sqrt{3}k_ya}{2}}\right] d_0 \\ & \Delta^{t,z}_{E,1/2}(1,0)=\left[\sin{(k_x a)}+\sin{\frac{k_xa}{2}}\cos{\frac{\sqrt{3}k_y a}{2}}\pm{\rm i}\sqrt{3}\cos{\frac{k_xa}{2}}\sin{\frac{\sqrt{3}k_ya}{2}}\right]d_z \\ & \begin{aligned} \Delta^{t,xy}_{E,1/2}(1,2)=&\left[2\sin{(\sqrt{3}k_y a)}-\cos{\frac{3k_x a}{2}}\sin{\frac{\sqrt{3}k_ya}{2}}\pm{\rm i}\sqrt{3}\sin{\frac{3k_x a}{2}}\cos{\frac{\sqrt{3}k_y a}{2}}\right]d_x+\\ & \left[\sqrt{3}\sin{\frac{3k_xa}{2}}\cos{\frac{\sqrt{3}k_ya}{2}}\mp3{\rm i}\cos{\frac{3k_x a}{2}}\sin{\frac{\sqrt{3}k_y a}{2}}\right]d_y \end{aligned} \end{array}

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APCOM, June 21, 2024

\begin{aligned} \Delta_{A_1}^s(p,q) &= \sum_{g \in {\bf C}_{3v}} A^*_1(g) \cos{({\bf k} \cdot (g^{-1} {\bf r}_{p,q}))} d_0, \\ \Delta_{A_2}^s(p,q) &= \sum_{g \in {\bf C}_{3v}} A^*_2(g) \cos{({\bf k} \cdot (g^{-1} {\bf r}_{p,q}))} d_0, \\ \Delta_{E,1/2}^s(p,q) &= \sum_{g \in {\bf C}_{3v}} (E_{11/22}(g))^* \cos{({\bf k} \cdot (g^{-1} {\bf r}_{p,q}))} d_0. \end{aligned}

One more parameter

the chemical potential \( \mu \)

Now we have

  • The representation: \( A_1,  A_2,  E_1,  E_2 \)
  • The strength of the pairing functions, 3 for each rep.
  • The chemical potential

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APCOM, June 21, 2024

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APCOM, June 21, 2024

Is this even possible?

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APCOM, June 21, 2024

Is this even possible?