Modeling of optical properties of 2D crystals: Silicene, germanene and stanene

Transcription

1 Modeling of optical properties of 2D crystals: Silicene, germanene and stanene Friedhelm Bechstedt 1 collaboration: L. Matthes 1 and O. Pulci 2 1 Friedrich-Schiller-Universität Jena, Germany 2 Università di Roma Tor Vergata, Rome, Italy

2 Motivation: 2D materials beyond graphene? Graphene Other 2D materials? Silicene (Graphene-like Si) K. Takeda, K. Shiraishi, PRB 50, (1994) G. G. Guzmán-Verri, et al., PRB 76, (2007) Theory: Silicene is energetically stable and has similar electronic properties as graphene S. Cahangirov et al., PRL 102, (2009) Is silicene the next graphene? L.C. Lew Yan Voon et al., MRS Bulletin 39, 366 (2014) but problem: no graphite for Si, Ge and Sn

3 Silicon-based 2D honeycomb crystals Top view + functionalization silicene +H Side view silicane =0.44 Å Experiment: P. Vogt et al., PRL 108, (2012) Not planar!!! larger atomic radii =0.70 Å Theory: L.C.L. Yan Voon, APL 97, (2010) realization (Ge): E. Bianco et al., ACS Nano 7, 4414 (20013)

4 2D honeycomb crystals: Prototype: Silicene Atomic structure Brillouin zone sp 2 /sp 3 -bonding not flat (buckling) Dirac cones at K, K (because of symmetry)

5 Band structure of σ* bands silicene π* bands Dirac cones π bands σ bands

6 Outline 1. Modeling / general results 2. Infrared optical absorbance 3. Boundary conditions / global properties 4. Summary

7 (Ab initio) Modeling of atomically thin crystals? artificial superlattice (hexagonal crystal with a and L 20 Å) density functional theory electronic bands and wave functions dielectric function / (ω) (independent-particle / QP approximation) optical properties of isolated sheet (= ^ 2D optical conductivity) Compensation of QP blue shift and excitonic red shift L. Yang et al., PRL 103, (2009) / / σ2d( ω ) = εω i 0 L ( ω) 1

8 Quasiparticle band structures graphene silicene germanene stanene Dirac cones at K, K despite sp 3 and buckling opening of small gaps due to SOC L. Matthes,, F.B., J. Phys. CM 25, (2013)

9 Freestanding 2D group-iv allotropes stable in DFT (QP) computations property graphene silicene germanene tinene lattice constant (Å) buckling (Å) v F (10 6 m/s) 0.83 (1.01) 0.53 (0.65) 0.52 (0.62) 0.48 (0.55) SOC-induced gap (mev) 0.0 (0.0) 1.6 (1.9) 24 (33) 73 (101) L. Matthes, F.B., J. Phys. CM 25, (2013) germanene band structure with Dirac cones (modified conical linear bands) massive Dirac particle)

10 Optical conductivity in units of ac conductivity graphene stanene silicene germanene L. Matthes et al., New J. Phys. 16, (2014)

11 Infrared absorbance A(ω) of graphene Fine Structure Constant Defines Visual Transparency of Graphene R.R. Nair, 1 P. Blake, 1 A.N Grigorenko, 1 K.S. Novoselov, 1 T.J. Booth, 1 T. Stauber, 2 N.M.R. Peres, 2 A.K. Geim 1 * Science 320, 1308 (2008) AA = ππππ 2.3% (Sommerfeld fine structure constant α) for massless Dirac fermions and transversal gauge Same optical absorbance for other 2D group-iv honeycomb crystals despite sp 3 hybridization and buckling?

12 Joint band structure and density of states graphene silicene germanene For vanishing photon energies still linear dispersion and Dirac cones near K Van Hove singularities M 0 or M 1 near Γ and M.

13 Band structure and optical matrix elements normalized to v F graphene silicene germanene Interband energies and optical transition matrix elements near K (K ) are rather independent of group-iv element if normalization with v F F. Bechstedt et al., APL 100, (2012) L. Matthes et al., PRB 87, (2013)

14 AA ωω = αα Independent-particle approximation (vertical interband transitions) ħ 1 mm 2 ωω 2 cckk pp AA kk cc,vv jj=xx,yy jj vvkk 2 δδ(εε cc kk εε vv kk ħωω) RRRRσσ 2DD (ωω) germanene silicene graphene A(0) = πα A(ω) A(ω) independent of buckling, hybridization, element, and gauge quadratic increase with ω deviations in the range of interband transitions F. Bechstedt et al., APL 100, (2012)

15 Problem: Modeling of boundaries for atomically thin sheets 1 only σ 2D (ω) boundary condition with surface charge density ρ = j q /ω T. Staufer, PRB 78, (2008) L. Matthes, New J. Phys. 16, (2014) σ 2D 2 3 σ 2D (ω) transfer matrix method Tianvong Zhan, J. Phys. CM 25, (2013) σ 2D (ω) and σ 2D (ω) transfer matrix method Xin-Hua Deng, EPL 109, (2015) L. Matthes, PhD thesis (2015) analogous d L d = + L L 1 d 1 L d 1 = + L L sheet SL environment sheet SL environment R, T, A independent for normal incidence but different for grazing incidence and p-polarization

16 Freestanding sheets at normal incidence Reflectance Transmittance RR = TT = σσ 1 + σσ σσ 2 1-πα significant effects due one atomically thin layer (especially: in resonances) Absorbance AA = 2RRRR σσ 1 + σσ 2 πα = 1 RR TT with σ = σ 2D (ω)/2σ 0

17 Optical properties graphene stanene R T 1 A Re σ 2D (ω)/σ 0 silicene germanene L. Matthes et al., New J. Phys. 16, (2014) huge effect in resonances graphene resonances near π and π + σ plasmons first resonance graphene & silicene M 1 saddle point (M point) first resonance germanene & stanene M 0 minimum (Γ point)

18 Experiment Kin Fai Mak et al., PRL 106, (2011) 4.62 ev saddle point exciton Fano lineshape (excitons of M 1 with continuum of interband transitions near K)

19 Modeling Summary / Conclusions Isolated sheet from superlattice (σ 2D (ω), σ 2D (ω)) IR properties A(ω) = πα: Vertical π π * transitions near K with vanishing interband energy (spin-orbit?!) Global properties problem: boundary conditions frequency variation: van Hove singularities huge effects on R, T, A near resonances