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Supplementary Material of Thermal and electronictransport characteristics of highly stretchablegraphene kirigami†

Bohayra Mortazavi,a‡ Aurélien Lherbier,b‡ Zheyong Fan,c,d‡ Ari Harju,d Timon Rabczuk,e

and Jean-Christophe Charlierb

1 Electronic diffusivityIn order to display the complex dynamics of charge carrierstaking place in the kirigami structures one discusses herethe directional, energy and time dependent diffusivities. Thediffusivities along the x direction (Dx(E, t)) and along the ydirection (Dy(E, t)) are presented in Fig.1 for five selectedenergies E =-0.4, -0.2, 0.0, +0.2, and +0.4 eV with respect tothe Fermi energy (EF ). The top panels of Fig.1 correspond toθc=0◦, the middle panels to θc=90◦, and the bottom panels toθc=180◦. First, one notices that Dy(E, t) is systematically lowerthan Dx(E, t) by one or two orders of magnitude, indicating aclear anisotropy in the two transport directions. Second, themagnitude of the diffusivity is dependent on the energy whichcan be understood from the complex DOS containing numerousvan Hove singularities and possible band gaps as observed inmain text in Fig.??. Some diffusivity curves exhibit significantundulations which can introduced by the numerical derivatives

a Institute of Structural Mechanics, Bauhaus-Universität Weimar, Marienstr. 15, D-99423 Weimar, Germany; E-mail: bohayra.mortazavi@gmail.comb Université catholique de Louvain, Institute of Condensed Matter andNanosciences, Chemin des étoiles 8, 1348 Louvain-la-Neuve, Belgium; E-mail:aurelien.lherbier@uclouvain.bec School of Mathematics and Physics, Bohai University, Jinzhou, Chinad COMP Centre of Excellence, Department of Applied Physics, Aalto University, Helsinki,Finland; E-mail: zheyong.fan@aalto.fie College of Civil Engineering, Department of Geotechnical Engineering, Tongji Univer-sity, Shanghai, China† Electronic Supplementary Information (ESI) available: [See SupMat.tex Sup-Mat.pdf]. See DOI: 10.1039/b000000x/‡ These authors contributed equally to this work

of wave packet quadratic spreading ( ∂∆X2(E,t)∂ t , ∂∆Y 2(E,t)

∂ t ), inparticular for long cut lengths lc =80 and 160 nm. In somecases, it even induces a negative value of the diffusivity becausein the localization regime the saturating quadratic spreading isno more increasing monotonically and may slightly oscillate,i.e. has a small negative slope. This is a possible artifact of thecalculation. One also observes that the long time ballistic regimediscussed in the main text in section ?? is easily observed forlc =10 nm (red curves) but is not clearly obtained for the othersystems. For (θc, lc)=(0◦,20 nm), the diffusivities are found tobe anomalously low compared to other systems. However, thiscan be rationalized by the fact that a rather large band gap withlocalized states were observed in the vicinity of the Fermi levelfor this (θc, lc)=(0◦,20 nm) graphene kirigami. Some diffusivitycurves display a diffusive regime i.e. a constant value of D(t) (seefor instance Dx(E, t) and Dy(E, t) of the (θc, lc)=(180◦,160 nm)for E=-0.4, +0.2 and +0.4 eV), which results from the scatteringat the cut edges. In other cases, a localized regime is observedi.e. a zero value of D(t) corresponding to constructive quantuminterferences in scattering loops inducing localization of thewave packet. However, as mentioned in main text in section ??,at longer propagation time the diffusivity should finally recovera ballistic behavior because of the periodicity. Overall, Fig.1demonstrates the complex transient regimes that can be obtainedin these graphene kirigami structures.

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Electronic Supplementary Material (ESI) for Nanoscale.This journal is © The Royal Society of Chemistry 2017

Fig. 1 Energy and time dependent diffusivity (Dx(E, t) and Dy(E, t)) fordifferent kirigami structures with cut angles θc=0, 90, and 180◦, cutlengths lc=10, 20, 40, 80, and 160 nm.

2 Heat Flux profiles

To better explain the reason why graphene kirigami structurespresent highly anisotropic thermal conductivity, i.e. a muchhigher thermal conductivity along the transverse direction thanthe longitudinal one, we present in Fig.2 samples of heat fluxprofiles obtained using the finite element method. For the longi-tudinal heat transfer, the cuts directly obstacle the heat transferand as a result some parts of the material do not involve in theheat transfer. On the other side, for the heat transfer along thetransverse direction, the cuts are parallel to the heat flow and inthis case the material are very uniformly involved in the heat fluxtransfer.

Fig. 2 Sample of heat flux profiles computed using the finite elementmethod for the thermal transport along the transverse and longitudinaldirections.

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