Topological defects in graphene nanostructures

Transcription

1 J. Smotlacha International Conference and Exhibition on Mesoscopic & Condensed Matter Physics June 22-24, 2015, Boston, USA Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 1 / 20

2 Carbon Nanostructures: graphene fullerene nanocone wormhole nanotubes, nanotoroids, etc. Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 2 / 20

3 Basic structure: Hexagonal plane lattice it is composed of 2 inequivalent sublattices, A and B Topological defects: n-sided polygons n 5 (positive curvature) n 7 (negative curvature) Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 3 / 20

4 Electronic properties Important characteristics: Local density of states number of states per the unit interval of energy and per the unit area of surface at each energy level that is available to be occupied by electrons Calculation of LDoS for low energies: periodical structures: from the low energy electronic spectrum using Schrödinger equation describing the electron motion [1,2] aperiodical structures: from the continuum limit of the gauge field theory using Dirac-like equation describing the motion of massless fermion [3] graphene wormhole: using Dirac-like equation describing the motion of massive fermion [4] Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 4 / 20

5 Periodical structures: plain graphene, nanotubes (continuous spectrum), fullerene, nanoribbons (discrete spectrum) the electron which is bounded on the molecular surface satisfies the Schrödinger equation [1]: Hψ = Eψ, ψ = C A ψ A + C B ψ B where A, B represent the particular sublattices solution: Bloch function ψ k ( r) = e i k r u k ( r), where u k ( r) has the lattice periodicity [5] tight-binding approximation: ψ A(B) = A(B) exp[i k r A(B) ]X( r r A(B) ), where X( r) is the atomic orbital function (this solution satisfies the Bloch theorem) assumption: X( r r A )X( r r B )d r = 0 Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 5 / 20

6 case of graphene we denote H ab = ψa Hψ b d r, S = ψaψ A d r = ψbψ B d r, a, b A, B, then ( HAA H AB H BA H BB ) ( CA C B ) ( CA = ES C B the rotational symmetry gives H AA = H BB, then, putting H ab = H ab /S, we get from the secular equation E = H AA ± H AB we consider H AA to be the Fermi level, then, after substitution the corresponding expansion into H AB, we get E( k) = ±γ cos kya kya kxa cos cos, where γ 0 = X ( r ρ )HX( r )d r, ρ joining the given site A with the nearest site B Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 6 / 20 )

7 the LDoS we get as LDoS(E, k) = δ(e E(k)), D(E) where π D(E) = lim 2Im η 0 π k dk E E(k) iη Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 7 / 20

8 case of nanoribbon [6] matrix elements: H ij = ψa i Hψ Aj d r, S = unit cell: ψa i ψ Ai d r = ψa j ψ Aj d r, i, j {1,..., n max}, then H A1 A 1 H A1 A H A1 Anmax H A2 A 1 H A2 A H A2 Anmax C A1 C A = ES C A1 C A , H Anmax A 1 H Anmax A H Anmax Anmax C Anmax C Anmax where H A1 A 1 =... = H Anmax A nmax Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 8 / 20

9 electronic spectrum: LDoS: zigzag edge armchair edge zigzag armchair Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 9 / 20

10 Aperiodical structures: close vicinity of the defects, nanocone, wormhole due to the aperiodicity, the eigenfunctions in the Schrödinger equation can t be labelled by the wave-vector k in the corresponding Hamiltonian, the aperiodicity is present by the potential U( r), using the k p perturbation theory and the continuum gauge field theory, the original Schrödinger equation for the bounded electron is transformed to the Dirac-like equation for the massless fermion [7] iσ α e µ α[ µ + Ω µ ia µ ia W µ ia µ]ψ = Eψ, ψ = ( ψa ψ B ) e µ α - zweibein, ω µ - spin connection, Ω µ = 1 8 ωαβ µ [σ α, σ β ] - spin connection in the spinor representation a µ, a W µ - ensuring the circular periodicity A µ - magnetic field Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 10 / 20

11 solution: using the substitution ( ) ψa ψ = ψ B = 1 ( 4 gϕϕ uj (ξ)e iϕj v j (ξ)e iϕ(j+1) ), j = 0, ±1,..., so ξ u j j u j = Ev, gξξ gϕϕ ξv j j v j = Eu, gξξ gϕϕ where j = j + 1/2 aϕ a W ϕ A ϕ, g ξξ, g ϕϕ the metric coefficients LDoS(E, ξ 0 ) = u 2 (E, ξ 0 ) + v 2 (E, ξ 0 ) Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 11 / 20

12 close vicinity of the defects [7, 10] (hyperboloidal geometry): heptagonal defects (1-fold hyperboloid) magnetic field: B = 0 B = 0.5Φ 0 pentagonal defects (2-fold hyperboloid) magnetic field: B = 0 B = 0.5Φ 0 Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 12 / 20

13 case of wormhole [9, 10]: composed of two graphene sheets which are in the places of wormhole bridges connected with the help of very short connecting nanotube described with the help of the polar-like coordinates r ±, ϕ satisfying r +r = a 2, where the sign ± denotes the upper and lower graphene sheet, resp. and a is the wormhole ( radius ) 1 0 metric tensor: g µν = Λ 2 (r ±) 0 r± 2, where Λ(r ±) = (a/r ±) 2 θ(a r ±) + θ(r ± a) due to the relativistic efects, the corresponding Dirac-like equation involves mass terms and using the substitution method, the radial part of the system can be replaced by one differential equation of the form ( ξξ 1 2g ξξ ξ g ξξ + j gξξ 2 gϕϕ 3 ξ g ϕϕ j 2 g ξξ + (E 2 M 2 )g ξξ g ϕϕ ) u j = 0, where the coordinate ξ stands for r ± Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 13 / 20

14 LDoS: spin-orbital interaction: comes from the presence of the connecting nanotube, it causes next splitting of the energy levels; because of the presence of the zero modes, as the result, these modes should be detected for the chiral massive fermions in the real material - actually, this could serve as a proof of the successful fabrication Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 14 / 20

15 possible modifications: perturbed wormholes pillared graphene (a) 2 defects, (b) 4 defects, (c) 6 defects, (d) 8 defects, (e) 10 defects Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 15 / 20

16 case of nanocone [11]: the form of the Hamiltonian is Ĥ s = v { iσ 2 r σ 1 r 1 [ (1 η) 1 ( is ϕ η) σ3]}, s = ±1, η = N 6, the solution are the Bessel functions involvement of the spin-orbital interaction (SOI) [12]: this interaction is negligible in the plain graphene, but it seems to be significant in the case of graphene nanotubes as well as in the investigated conical structure; we perform the transformations r r I i δγ 4γR σx( r), i ϕ i ϕ I + s(1 η)ay σy Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 16 / 20

17 solution: ψ(r, ϕ) = e ijϕ f j (r) f j (r) g j (r) g j (r) 0 0 r + F i C r r 0 0 i D r r + F r r + F 1 i D 0 0 r r i C F 1 r r r sj η, where the radial part must satisfy f j (r) f j (r) g j (r) g j (r) with F = 3 + 1, C = ξx ξy, D = ξx + ξy, 1 η 2 1 η 2 ξ x, ξ y - the spin-orbital coefficients LDoS: = E f j (r) f j (r) g j (r) g j (r) Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 17 / 20

18 finite dimensional case: quantization of the continuous spectrum, the energy shift caused by the SOI can be found with the help of the Green function by the comparison of the LDoS without and with SOI: LDoS(E) = 1 π Im G(E + i0), G = G 0 + G 0 ˆV G0 + G 0 ˆV G0 ˆV G0 +..., where G denotes the case with SOI, ˆV stands for the interaction potential of SOI and G 0 which represents the case without SOI has the form G 0 (r, ϕ, r, ϕ ; E) = 1 ( ) e i(n+s/2)(ϕ ϕ ) a 11 n (r, r ) an 12 (r, r ) 1 2π an 21 (r, r ) an 22 (r, r ; ) n Z a kl n (r, r ), k, l = 1, 2 are computed in [11]: an kl (r, r dp p ) = 2 υ 2 p 2 E 2 akl n,p(r, r ), 0 a 11 n,p(r, r ) = 2E J νn (pr )J νn (pr ), a 12 n,p(r, r ) = 2 υp J νn+1(pr )J νn (pr ), a 21 n,p(r, r ) = 2 υp J νn (pr )J νn+1(pr ), a 22 n,p(r, r ) = 2E J νn+1(pr )J νn+1(pr ) Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 18 / 20

19 Bibliography 1 P. R. Wallace, Phys.Rev. 71 (1947) J. C. Slonczewski and P. R. Weiss, Phys.Rev. 109 (1958) C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95 (2005) J. Gonzalez, F. Guinea and J. Herrero, Phys. Rev. B 79, (2009). 5 J. Callaway, Quantum Theory of the Solid State (Academic, New York, 1974). 6 K. Wakabayashi, K. Sasaki, T. Nakanishi and T. Enoki, Sci. Technol. Adv. Mater. 11 (2010) E. A. Kochetov, V. A. Osipov and R. Pincak, J. Phys.: Condens. Matter 22 (2010) J. Smotlacha, R. Pincak and M. Pudlak, Eur. Phys. J. B 84 (2011) R. Pincak and J. Smotlacha, Eur. Phys. J. B 86: 480 (2013). 10 J. Smotlacha and R. Pincak, Quantum Matter 4 (2015). 11 Yu. A. Sitenko, N. D. Vlasii, Nucl. Phys. B 787 (2007) R. Pincak, J. Smotlacha and M. Pudlak, Eur. Phys. J. B, 88 1 (2015) 17. Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 19 / 20

20 Thank you for your attention Jan Smotlacha (BLTP JINR, Dubna, Russia) Topological defects in graphene nanostructures June 22-24, 2015, Boston, USA 20 / 20