S. Bellucci, A. Sindona, D. Mencarelli, L. Pierantoni Electrical conductivity of graphene: a timedependent density functional theory study

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1 S. Bellucci, A. Sindona, D. Mencarelli, L. Pierantoni Electrical conductivity of graphene: a timedependent density functional theory study INFN Laboratori Nazionali Frascati (LNF), Italy Univ. Calabria, Cosenza, Italy Università Politecnica delle Marche, Ancona, Italy bellucci@lnf.infn.it FANEM 2015 Minsk

2 OUTLINE Graphene and Graphene NanoRibbons: an Electromagnetic Characterization through the eyes of Linear Response Density Functional Theory THz plasmonics in graphene by Density functional theory (DFT) and Kubo derived formulation

3 Graphene and Graphene NanoRibbons An Electromagnetic Characterization through the eyes of Linear Response Density Functional Theory

4 a = nm The Geometry An ideal Graphene Sheet. a = nm a =0.09nm and two simple Armchair and Zig- Zag Graphene Ribbons (Edges passivated by Hydrogen atoms)

5 Imagine we replicate the structures and obtain the following 3D Crystals Armchair GNR Zig-Zag GNR Graphene

6 Electron Density New electron density Then. we perform a 3D Density Functional Calculation Plane Wave (PW) DFT Bloch s Theorem FD Statistics Initial Guess Effective Kohn-Sham (KS) Potential Hartree b = band index k = wave vector in the 1 st BZ G = reciprocal lattice vectors Converged? No: Start Back Yes Atomic Nuclei Ground State Properties:Total Energy, Forces, Stresses Exchange and Correlation KS Equations

7 Graphene: Full electronic structure (DFT) Dirac s Electrons Brillouin ZONE (k Space) Exp: A. Bostwick et al, Nature 3, 36 (2007)

8 Graphene.zooming at the Dirac points (on the THz Scale)

9 Graphene.zooming at the Dirac points (on the THz Scale) The valence and conduction bands are asymmetrical above 120 THz Conical Approx. (ok below 100 THz)

10 Time Dependent DFT (Linear Response) Macroscopic Average Inverse Permittivity Tensor Resistivity Tensor

11 Observables Plasmon Resonances Absorption Spectrum Loss Function (Plasmon Spectrum) Surface Resistance Surface Reactance

12 0T Permittivity Room T Permittivity

13 Conical Approximation & Optical Limit Kubo-Drude Formula Permittivity (LR-DFT vs KD)

14 Conductivity-Resistivity (LR-DFT vs KD)

15 Conductivity vs Frequency Conductivity vs Time Resisitvity vs Frequency

16 Permittivity response of graphene vs an armchair nanoribbon Only LR-DFT can be used (no conical approx.)

17 We have presented advanced tools to study the linear electromagnetic response of graphene and graphene-like materials on the THz scale. Starting from an atomistic point of view, we have defined an ab initio approach in which the ground state properties of the material, i.e., energies, occupations, and one-electron wave-functions are computed by plane-wave DFT. These information are plugged in the relations of linear response theory to predict the EM response of the system, in the optical limit. Although several permittivity simulations have been performed, following similar guidelines, on pristine graphene on the ev scale, here we have defined a procedure to properly sample the electronic structure on the THz scale. At the same time, we have tested the reliability of the widely-used KD approach, operating in the same frequency range. Upon comparison of DFT-results with those obtained by the KD formulation, some significant differences have been pointed out. Nevertheless the KD formula seems to reasonably capture the main quantum features of graphene for EM applications. However, the proposed ab initio tool can be feasibly adapted to describe graphenelike systems with a more complex electronic structure than graphene, such as graphene multilayers, nano-ribbons, or nanotubes. More importantly, it has the potential to properly account for the role of metal contacts and substrate contacts. This is the object of current and future work.

18 OUTLINE Graphene and Graphene NanoRibbons: an Electromagnetic Characterization through the eyes of Linear Response Density Functional Theory THz plasmonics in graphene by Density functional theory (DFT) and Kubo derived formulation (i.e. going form theory to engineering device application)

19 Motivations: THz plasmonics Science, Pendry et al., 305 (5685): o o possibility to have subwavelength confinement of the e.m. field possibility to bridge from Microwave to Optics (Conventional, i.e. non graphene like) methods to achieve THz plasmons: spoof surface plasmons (SSP): structuring metals on a subwavelength scale (periodic corrugation, holes, etc.) control dispersion by changing the geometry of the patterns use of highly doped materials, such as semiconductors and conducting polymers the conductivity can be altered by light, heat or a reversible reactions

20 THz graphene plasmonics The first plasmon observation in graphene dates back to 2012 [J. Chen et al., Nature 487, 2012] THz and mid infrared nanoimaging in graphene T. Low and P. Avouris, ACS Nano

21 1. Common Kubo-Drude formula for graphene Graphene je j fd fd (,,, T ) c d (1 j2 ) fd 2 fd d 2 (1 j2 ) 4 / (non-dispersive case) Some assumptions/approximations: Conical dispersion of the electron energy Plane-wave like wavefunctions

22 Linear Response Density Functional Theory (LR-DFT): a few formulas

23 1. In plane-wave DFT approaches to solid crystals, the (pseudo) band-electrons are probed the wave-functions ψ nk r = Ω 1/2 c nk+g ej k+g r G Eigenenergies ε nk Eigenbasis ψ nk 2. The corresponding band levels ε nk are populated according to the Fermi-Dirac distribution f εnk. 3. The electric displacement response to the applied electric field is given by the permittivity matrix 0 ε GG q, ω ± = ε 0 δ GG ε 0 V G q χ GG q, ω ± ε 0 V G q = q + G 2 where the independent-particle polarizability is 0 GG kq kq ( f f ) ( ) ( ) n n nn nn ( ) k k q G G q, ( ) k, nn nk n k q with the correlation terms kq ρ G nn = cnk+g c nk+q+g+g G

24 The central point is that: the DFT-based approach makes use of real electron wavefunctions Using Tight Binding wavefunctions could also be a good choice, but there are strong limits when extending the analysis to: nanoribbons, substrate effects, presence of defects, ecc.

25 DFT accuracy: experimental comparison Energy loss spectrum (directly related to the polarizability) Very good accordance up to high energies Relaxation time ~2.20 fs, and electron energy loss spectrum taken from the experiment, [T. Eberlein et al., Phys. Rev. B 77, (2008)]

26 Comparison of the surface impedance of graphene, by DFT and Kubo-derived formula pristine and doped graphene at 300 K Imag (Zs) Real (Zs) Kubo formula DFT

27 Comparison of the plasmon quality factor, by DFT and Kubo-derived formula We define the plasmon quality factor as: Imag(Zs)/Real(Zs) Kubo formula DFT

28 Numerical example: plasmon excitation

29 Plasmon excitation by the near field of a wire antenna antenna Increasing losses (changing the mean life time) plasmon Graphene patch Some curves from the literature [2013 IEEE IELNANO, Balaban et al.]: dipole excitation of graphene disk (for different doping levels)

30 Plasmon excitation Arrows: plasmon excitation Plasmon distribution vs im(zs) strongly perturbed radiation S11 vs im(zs) high reflection (detuning)

31 Field radiated by the antenna (iso-surface) Low detuning High detuning through plasmon strong coupling antenna-patch Im(Zs)=1700 Ohm Im(Zs)=2000 Ohm E( z, r, ) G J j I 2 2 jk r r' z z' e 2 k 2 r r' z z' 2 J( z', r', ' ) dv '

32 Plasmon radial profile: e.m. simulation using Kubo and DFT responses Normalized electric field Fermi level ev 10 THz Kubo DFT 0 0 R/2 R Radial Position DFT plasmon with a linear increase of the charge density from the center to the edge n Normallized electric field Fermi level ev 16 THz with losses lossy neglected Increase of plasmon wavelength 0.2 R 0 0 R/2 R Radial position

33 A step forward - allowed by DFT - will be Plasmonics in 2D materials beyond graphene (in the following example: silicene) Band dispersions DOS for planar and buckled Silicene (DFT vs TB) DFT band and DOS for buckled (periodic tilt of atoms to give structural stability) Silicene and Graphene

34 Notes and conclusion 1. Up to a few THz, Kubo-Drude and DFT approaches provide similar results, but, above, differences cannot be neglected 2. Spatial dispersion (i.e. going beyond the optical limit) cannot be neglected at THz frequencies 1,2 - so the next work step: comparison between dispersive DFT and semiclassical dispersive approaches (like the Bhatnagar Gross Krook approximated model) [1] G. Lovat, G. W. Hanson t, R. Araneo and P. Burghignoli, EUCAP 2013 [2] D. C.-Serrano, J. S. G.-Diaz, and A. A.-Melcon, IEEE AWPL, Vol. 13, 2014