Optical & Transport Properties of Carbon Nanotubes II

Transcription

1 Optical & Transport Properties of Carbon Nanotubes II Duncan J. Mowbray Nano-Bio Spectroscopy Group European Theoretical Spectroscopy Facility (ETSF) Donostia International Physics Center (DIPC) Universidad del País Vasco (UPV/EHU) Nanostructural Properties March 30, 2011

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3 Electron micrographs of (a) five-wall CNT with outer radius R 5 = 33.5 Å, (b) double-wall CNT with outer radius R 2 = 27.5 Å, (c) seven-wall CNT with outer radius R 7 = 32.5 Å and inner radius R 1 = 11 Å. Scanning tunneling microscopy image of a SWNT on the surface of a rope, showing a nanotube s chirality. S. Iijima, Nature 354, 56 (1991). M. S. Dresselhaus, et al., Carbon Nanotubes: Synthesis, Structure, Properties, and Applications (2001).

4 Single Walled Nanotube (SWNT) Structures (a) Pristine SWNT (b) Substitutionally B Doped SWNT (c) K Intercalated SWNT Bundle (d) Peapods C Endohedral Doping

5 2 SWNT DFT-PBE Bandstructure & DOS (6,4) Single-walled Carbon Nanotube (6,5) Single-walled Carbon Nanotube (a) Band Structure (b) Density of States (a) Band Structure (b) Density of States ε ε F [ev] 0 ε ε F [ev] Γ Χ -2 Γ Χ

6 SWNT Van Hove Singularities (6,4) Single-walled Carbon Nanotube (6,5) Single-walled Carbon Nanotube 2 (a) Band Structure (b) Density of States 2 (a) Band Structure (b) Density of States 1 1 ε ε F [ev] 0 ε ε F [ev] Γ Χ -2 Γ Χ

7 SWNT Excitations (6,4) Single-walled Carbon Nanotube (6,5) Single-walled Carbon Nanotube 2 (a) Band Structure (b) Density of States 2 (a) Band Structure (b) Density of States 1 1 ε ε F [ev] 0 ε ε F [ev] Γ Χ -2 Γ Χ

8 1 active site ((6,6) CNT doped by 3d transition metal atoms) 2 target molecule (CO, NH 3, H 2 S) 3 background (atmospheric air) 4 sensing property (resistance) J. M. García-Lastra, et al., Phys. Rev. B 81, (2010); D. J. Mowbray, et al., Phys. Status Solidi B 1-5 (2010).

9 J. M. García-Lastra, et al., Phys. Rev. B 81, (2010); D. J. Mowbray, et al., Phys. Status Solidi B 1-5 (2010).

10 J. M. García-Lastra, et al., Phys. Rev. B 81, (2010); D. J. Mowbray, et al., Phys. Status Solidi B 1-5 (2010).

11 J. M. García-Lastra, et al., Phys. Rev. B 81, (2010); D. J. Mowbray, et al., Phys. Status Solidi B 1-5 (2010).

12 J. M. García-Lastra, et al., Phys. Rev. B 81, (2010); D. J. Mowbray, et al., Phys. Status Solidi B 1-5 (2010).

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16 E loss = F ind dr = Q γ dt Φ ind t r=r0

17 E loss = F ind dr = Q γ Ẽloss(q) = Q [ π I dωω 0 dt Φ ind t r=r0 ] dte iωt Φind (q, ω)

18 E loss = F ind dr = Q γ Êloss(q) = Q [ π I dωω 0 [ ] Ẽloss(q, ω) I ε 1 m (q, ω) dt Φ ind t r=r0 ] dte iωt Φind (q, ω)

19 SWNT EELS T. Pichler et al., Phys. Rev. Lett. 80, 4729 (1998).

20 SWNT EELS T. Pichler et al., Phys. Rev. Lett. 80, 4729 (1998); D. J. Mowbray et al., Phys. Lett. A 329, 94 (2004). D. J. Mowbray Optical & Transport Properties of Carbon Nanotubes

21 Experimental EELS SWNT Experiment q (Å -1 ) ω (ev) FIG. 1: Experimental single-walled carbon nanotube (SWNT) electron energy loss spectroscopy (EELS) ( ) as a function of energy ω in ev and longitudinal momentum transfer q in Å 1 from Refs. 1 and 2 with d 20 Å. Fits to the plasmon peak positions ( ) are also provided ( ).

22 Experimental EELS Setup q (Å -1 ) 0.0 SWNT Experiment Dispersive Peak ω~ 5 12 ev Non Dispersive Peak ω ~ 5.25 ev ω (ev) FIG. 1: Experimental single-walled carbon nanotube (SWNT) electron energy loss spectroscopy (EELS) ( ) as a function of energy ω in ev and longitudinal momentum transfer q in Å 1 from Refs. 1 and 2 with d 20 Å. Fits to the plasmon peak positions ( ) are also provided ( ). 1. What is the origin of the non-dispersive peak in the loss function? 2. Does curvature play a role in the EELS dispersion, and if so, at what radii? 3. Can EELS experiments yield detailed information on the chirality, diameter, and electronic structure of small SWNTs?

23 GPAW Grid-based Projector-Augmented Wave-function (GPAW) Real-space Density Functional Theory (DFT) Code wiki.fysik.dtu.dk/gpaw

24 TABLE I: Calculated (n, m) SWNT diameters in Å from DFT-PBE d PBE and the formula d a π n2 + nm + m 2, where a 2.46 Å. SWNT Type d PBE (Å) d (Å) Atoms/Cell (10,10) armchair (17,0) zigzag (6,6) armchair (10,0) zigzag (3,3) armchair (5,0) zigzag

25 Fermi Surface z (a) Graphene Conduction Band Band Structure Armchair Direction 2.0 Zigzag Direction x 0.5 (e) (3,3) SWNT ky (Å 1) (a) (5,0) SWNT 0 ε εf (ev) 4 5 Γ Κ (b) (10,0) SWNT 0 (f) (6,6) SWNT Non Dispersive Band in M M Direction (Zigzag SWNTs) -4 4 ε εf (ev) 6 Μ kx (Å 1) (b) Graphene Valence Band (c) (17,0) SWNT 0 1 (g) (10,10) SWNT Μ ky (Å 1) (d) (,0) Graphene 0 (h) (, ) Graphene k (Å ) k (Å ) 0.8 FIG. 4: Band energies ε in ev relative to the Fermi level εf versus momentum k in Å 1 along the (a d) armchair direction for the (a) (5,0), (b) (10,0), and (c) (17,0) semiconducting zigzag SWNTs and (d) graphene; and (e h) zigzag direction for the (e) (3,3), (f) (6,6), and (g) (10,10) metallic armchair SWNTs and (h) graphene. Armchair Direction y 3 Γ Κ z x Zigzag Direction D. J. Mowbray ε ε F (ev) -4 ε εf (ev) ε ε F (ev) ε εf (ev) y kx (Å 1) FIG. 5: Fermi surfaces ε in ev relative to the Fermi level εf for (a) the conduction band of graphene, and (b) the valence band of graphene, calculated over the Brillouin zone, with reciprocal lattice vectors k x and ky in Å 1 to the zigzag and armchair directions in graphene, respectively. Optical & Transport Properties of Carbon Nanotubes

26 TDDFT-RPA Time Dependent Density Functional Theory (TDDFT) Random Phase Approximation (RPA) wiki.fysik.dtu.dk/gpaw /tutorials/dielectric_response/dielectric_response.html

27 TDDFT-RPA Calculations of the loss function have been performed within the Casida methodology [8], so that the non-interacting densitydensity function χ 0 GG (q, ω) for momentum transfer q at energy ω is given by χ 0 GG (q, ω) = 1 Ω k n,n fnk fn k+q ω + εnk εn k+q + iγ ΩCell drψ nk (r)e i(q+g) r ψn k+q(r) ΩCell dr ψ nk (r )e i(q+g ) r ψn k+q(r ). (1) Here the sum is over reciprocal lattice vectors k and band numbers n and n, with εnk the eigenenergy of the n th band at k, fnk the occupation of the n th band at k, γ the peak broadening, G and G the reciprocal unit cell vectors, and ψnk(r) the real-space Kohn-Sham wavefunctions for the n th band with reciprocal lattice-vector k. In the optical limit q 0, G = 0, the overlap matrices become functions of the momentum operator r, so that lim q 0 χ0 GG (q, ω) 1 fnk fn k Ω ω + k n,n εnk εn k + iγ q dr ψ nk (r) 2 rψn k(r) ΩCell εn k. (2) εnk Including local field effects, we may write the macroscopic dielectric function ε 1 (q, ω) within the random phase approximation (RPA) in terms of the non-interacting density-density response function χ 0 GG (q, ω) as [ 1 ε 1 (q, ω) δgg 4π q + G 2 χ0 GG (q, ω)], (3) G=G =0 where δgg is the Kronecker delta. In this case the loss function is given by [ I{ε 1 (q, ω)} I δgg 1 4π q + G 2 χ0 GG (q, ω)] G=G =0 (4) when local field effects are included.

28 Armchair Direction Zigzag Direction (a) (5,0) SWNT (e) (3,3) SWNT π * Plasmon Exhibits Weaker Dispersion q (Å -1 ) and is Shifted Down in Energy for Smaller Diameter SWNTs q (Å -1 ) (b) (10,0) SWNT (f) (6,6) SWNT (c) (17,0) SWNT (g) (10,10) SWNT Non Dispersive Peak Found Only in Armchair Direction (Zigzag SWNTs) q (Å -1 ) q (Å -1 ) (d) (,0) Graphene (h) (, ) Graphene ω (ev) ω (ev) D. J. Mowbray FIG. 6: Loss function Optical I{ε & Transport 1 (q, ω)} as Properties a function of of energy Carbon ω in Nanotubes ev 1

29 1. The non-dispersive peak in the loss function originates from transitions to the conduction band with momentum transfer along a direction between M and M in graphene, where the π band is nearly flat. The axis of zigzag SWNTs is parallel to these directions, allowing these excitations, while the axis of armchair SWNTs is not, forbidding these excitations.

30 2. SWNTs with d 13 Å have both similar band structure and energy loss to graphene. As the SWNT diameter decreases, the π peak in the loss function exhibits weaker dispersion and is shifted to lower energies.

31 3. Our results suggest that by careful analysis of both the presence of the non-dispersive peak (to differentiate zigzag or chiral SWNTs from armchair SWNTs) and the dispersion and peak position of the π plasmon, we may obtain detailed information as to the chirality, diameter, and electronic structure of small SWNTs (d 13 Å).

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33 X-ray Adsorption Spectroscopy (XAS) wiki.fysik.dtu.dk/gpaw /tutorials/xas/xas.html

34 C B 1s core hole exciton B 1s core hole exciton C C 1s core hole exciton Intensity C1s Normalized Intensity x1000 B1s Energy (ev) C1s Energy (ev) Energy (ev) FIG. 3: Experimental x-ray adsorption spectroscopy (XAS) measurements as a function of energy in ev for B doped single-walled carbon nanotubes (SWCNTs). The C 1s onset, π, and σ peaks (left), a close-up of the π peak (right), and a 1000 magnification of the B 1s peak (inset, blue) are shown. The van Hove singularies (vhs) in the SWNT density of states (DOS) are evident in the XAS response (green arrow), clearly indicating that the samples are quite clean.

35 Experimental ω (ev) X ray Absorption Spectroscopy (XAS) XAS Experiment for Semiconducting SWNTs Loss Function Calculation for (17,0) SWNT ½ Core-Hole Calculation for (17,0) SWNT ε F XAS Response 1s hν Theoretical 286 (ev) Theoretical ω (ev) FIG. 2: X-ray adsorption spectroscopy (XAS) response as a function of energy ω in ev from experiment on semiconducting single-walled carbon nanotube (SWNT) samples of diameter d 14Å ( ) from Ref. 3, calculated from the all-electron loss function I{ε 1 (q, ω)} for a (17,0) SWNT ( ), and a 1 /2 core-hole calculation for a (17,0) SWNT ( ). The calculated onset (lower axis) has been shifted up in energy by 1.5 ev to coincide with the experimental onset (upper axis).