Competition of trivial and topological phases in graphene based hybrid systems

László Oroszlány

Department of Physics of Complex Systems, Eötvös Loránd University

and MTA-BME Lendület Topology and Correlation Research Group

Duncan and Kekulé

Duncan, Charlie, Gene and Kekulé

Strained relationship

A colourful mess!

Graphene is ...

... a good template!

A. K. Geim, I. V. Grigorieva Nature 499, 419 (2013)

BiTeX: gigant SOC

Ishizaka et al. Nature Materials 10, 521 (2011)

BiTeX monolayer

Fülöp et al. 2D Mater. 5, 031013 (2018)

BiTeX/graphene heterostructures

BiTeI sandwich: Kou et al. ACS Nano, 8 10448 (2014)

trivial

topological

Z. Tajkov el al. Nanoscale 11, 12704 (2019)

Mechanical distortions

in graphene

Weiss et al. Advanced Materials, 24 5782(2012)

Pressure→I, inplane strain →TI !?!

Z. Tajkov el al. Appl. Sci. 9, 4330 (2019)

BiTeBr

BiTeCl

Effective model and fit

t_1+i\vec{m_1}\cdot\vec{\sigma}

t_2+i\vec{m_2}\cdot\vec{\sigma}

t_3+i\vec{m_3}\cdot\vec{\sigma}

im_z\sigma_z

=\mathrm{e}^{-\beta\left(l/l_0-1\right)}

TB model predicts phase transition well!

topological

trivial

\begin{array}{c} \hat{H}=v\left( \hat{\boldsymbol{p}}\cdot\boldsymbol{\sigma}\otimes\tau_{z} -\boldsymbol{A}\cdot\boldsymbol{\sigma}\otimes\tau_{0} \right)\\ +\Delta_{\text{Kek}}\sigma_0\otimes\tau_x\\ +\Delta_{\text{KM}}\sigma_z\otimes\tau_0 \end{array}

\boldsymbol{A}=\underbrace{\frac{\beta}{2a_{0}}\varepsilon\left(1+\rho\right)}\left(\begin{array}{c} \cos2\vartheta\\ -\sin2\vartheta \end{array}\right)

|\Xi|=\sqrt{\Delta^2_{\text{Kek}}-\Delta^2_{\text{KM}}}

\Xi/v

Strain can close the gap at \(k=0\) if

Effective model

Gamayun et al. New J. Phys. 20, 023016 (2018)

Andrade el al. Phys. Rev. B 99, 035411 (2019)

Strain kills Kekulé but KM gap is resilient!

Hidden order of adatoms via RKKY

V.V.Cheianov et al. Solid State Commun. 149, 1499 (2009)

"Ferro"

"Para"

\hat{H}=v\hat{\boldsymbol{p}}\cdot\boldsymbol{\sigma}\otimes\tau_{z}

V^{\mathrm{RKKY}}_{ij}=\int_{-\infty}^{\infty}\text{d}\omega\text{Tr}\left[\hat{V}_{i}\hat{G}(\omega)\hat{V}_{j}\hat{G}(-\omega)\right]

\hat{V}_{i}=\delta(\boldsymbol{r}-\boldsymbol{r}_i)\left( \Delta_{\text{Kek}}\sigma_{0}\otimes\boldsymbol{u}_{i}\cdot\boldsymbol{\tau}+ \Delta_{\text{KM}}\sigma_{z}\otimes\tau_{0} \right )

V^{\mathrm{RKKY}}_{ij}=-\frac{\Delta_{\text{Kek}}^{2}\boldsymbol{u}_{i}\cdot\boldsymbol{u}_{j}+\Delta_{\text{KM}}^{2}}{8\pi r_{ij}^{3}v}

\boldsymbol{u}_{i}=\left( \begin{array}{c} \cos\left(\frac{2\pi}{3}\nu_{i}\right) \\ \sin\left(\frac{2\pi}{3}\nu_{i}\right) \end{array}\right),\,\nu_i\in[-1,0,1]

Large adatom \( \Rightarrow \) strong SOC