Topological and trivial gaps in graphene-based van der Waals heterostructures under strain and disorder

László Oroszlány

Department of Physics of Complex Systems, Eötvös Loránd University

and MTA-BME Lendület Topology and Correlation Research Group

Duncan and Kekulé

Charlie, Gene vs. August

Strained relationship

Kane, Mele, Phys. Rev. Lett. 95, 226801 (2005)

Act I

heterostructures, strain, gaps

Graphene is ...

... a good template!

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

BiTeX: gigant spin-orbit coupling

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

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

Bulk

exfoliatin of a single trilayer

BiTeX/graphene heterostructures

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

\(K \leftarrow \Gamma \rightarrow M \)

Pressure→I, inplane strain →TI !?!

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

BiTeBr

BiTeCl

BiTel

Effective model

t_2

t_3

t_1

im_z\sigma_z

Tajkov el al. Phys. Rev. B 101, 235146 (2020)

take away message:

Kekulé gap is killed by strain
KM gap is insensitive to strain

\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}

low energy Dirac picture

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

Act II

A colorfull mess

Adatoms

adatoms distort the lattice, alter hopping around a hexagon

adatoms introduce Kane-Mele type spin-orbit interaction on a plaquette

Promising candidates: Tl and dimers

Phys. Rev. X 1, 021001 (2011)

Phys. Rev. Lett. 109, 266801 (2012)

E[eV]

Hidden order of adatoms via RKKY

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

\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 SOI