FULLERÉNEK ÉS

SZÉN NANOCSÖVEK

előadás fizikus és vegyész hallgatóknak

(2008 tavaszi félév – május 07.)

Kürti Jenő

ELTE Biológiai Fizika Tanszék

e-mail: kurti@virag.elte.hu www: virag.elte.hu/kurti

FIRST PRINCIPLES CALCULATIONSFIRST PRINCIPLES CALCULATIONS

DFT: LDA

plane wave basis set, cutoff: 400 eV

G. Kresse et al

Wien

Budapest

Lancaster

arrangement: tetragonal (hexagonal for test)

distance between tubes: l = 0.6 nm (1.3 nm for test)

hexa

tetra

r1

r2r3

1

23

bond lengths

bond angles

(4,2)

d

c

56 atoms

building block

tube axis

ideal hexagonal lattice

tube axis

d increases

c decreases

tube axis

extra bond misalignment

1

GEOMETRY OPTIMIZATION

diameter

1/d vs 1/d0

DFT optimized diameter

1/d0 (nm-1)

1/d

(nm

-1)

.

ZZ

AC

CH

r0 = 0.1413 nm

(d-d0)/d0 vs 1/d0

relative change

1/d0 (nm-1)

(d-d

0)/d

0 (

%)

.

ZZ

AC

CH

r0 = 0.1413 nm

(9,0) : 1.06 ± 0.01 %

(d-d0)/d0 vs 1/d0

relative change

1/d0 (nm-1)

(d-d

0)/d

0 (

%)

.

ZZ

AC

CH

r0 = 0.1413 nm

(9,0) : 1.06 ± 0.01 %

length of the unit cell

unit cell lengths vs 1/d0

relative change

1/d0 (nm-1)

(c-c

0)/c

0 (

%)

.

ZZ

AC

CH

r0 = 0.1413 nm

ZZ triads

(9,0) : -0.05 ± 0.01 %

bond lengths

(r1-r0)/r0 vs 1/drelative change

1/d (nm-1)

(r1-

r 0)/r

0 (

%)

.

ZZ

AC

CH

r0 = 0.1413 nm

ZZ triads

(9,0) : -0.32 ± 0.004 %

(r2-r0)/r0 vs 1/drelative change

1/d (nm-1)

(r2-

r 0)/r

0 (

%)

.

ZZ

AC

CH

r0 = 0.1413 nm

ZZ triads

bond angles

bond angle 1 vs 1/d0

DFT optimized

1/d0 (nm-1)

1 (

deg)

.

ZZ

AC

CH

r0 = 0.1413 nm

pyramidalization or rehybridization

sp2 sp3

S.Niyogi et al., Acc. Chem. Res. 35, 1105 (2002)

pyramidalization angle P vs 1/d0

DFT optimized

1/d0 (nm-1)

P (

deg)

.

ZZ

AC

CH

r0 = 0.1413 nm

C60: 11.6°

BAND STRUCTURE

tight binding(nearest neighbour)

DFT (VASP)

(6,5) - DFT

(20,0)

zigzag

chiral

(19,0)(17,0)(16,0)

(14,0)(13,0)

(11,0)

(10,0)

(8,0)

(7,0)(5,0)(4,0)

(6,4)

(6,2)

(5,3)

(6,1)

(4,3)

(5,1)(4,2)

(3,2)

ZF-TB

DFT

1/d

1/d

(5,0)

ZF-TB: Eg = 2.3 eV

DFT: Eg = 0 !

(20,0)

zigzag

chiral

(19,0)(17,0)(16,0)

(14,0)(13,0)

(11,0)

(10,0)

(8,0)

(7,0)(5,0)(4,0)

(6,4)

(6,2)

(5,3)

(6,1)

(4,3)

(5,1)(4,2)

(3,2)

ZF-TB

DFT

1/d

1/d

ZF-TB METALLIC

non-armchair: zigzag, chiral

kF

kF kF - kF (d)

= f(1/d2)

K tube axis

opening of a secondary gap

secondary gap in (7,1)

0.14 eV

ZF-TB METALLIC

armchair

kF kF - kF (d)

= f(1/d2)

K

kF

tube axis

NO secondary gap

(6,6)

(4,4)

kF (d)=2/3

kF

kFF

F

AC

(3,3)

(4,4)

(5,5)

(6,6)

(7,7)(8,8)

(9,9)(10,10)

(11,11)

9 0 36 0 0.7010 0.7084 -0.043 330.3 317.8 318.2

8 2 56 10.9 0.7139 0.7209 -0.021 324.6 316.5 314.9

7 4 124 21.1 0.7512 0.7580 -0.051 308.7 307.1 303.7

10 0 40 0 0.7789 0.7852 0.008 298.0 294.5 290.7

6 6 24 30 0.8095 0.8158 -0.058 286.8 284.3 283.6

11 0 44 0 0.8568 0.8629 -0.051 271.2 268.2 264.6

12 0 48 0 0.9347 0.9401 -0.013 248.9 242.8 242.2

7 7 28 30 0.9444 0.9499 -0.053 246.3 246.2 243.4

13 0 52 0 1.0126 1.0172 0.006 230.0 228.9 225.5

8 8 32 30 1.0793 1.0843 -0.046 215.8 216.5 213.1

14 0 56 0 1.0905 1.0951 -0.026 213.7 212.4 209.7

15 0 60 0 1.1684 1.1725 -0.007 199.6 196.4 195.6

9 9 36 30 1.2142 1.2186 -0.037 192.0 192.7 189.4

16 0 64 0 1.2463 1.2501 0.002 187.2 186.4 183.7

17 0 68 0 1.3241 1.3281 -0.023 176.2 175.6 173.0

10 10 40 30 1.3491 1.3531 -0.031 172.9 173.6 170.7

18 0 72 0 1.4020 1.4054 -0.005 166.5 164.4 163.5

19 0 76 0 1.4799 1.4831 0.001 157.8 157.3 155.1

11 11 44 30 1.4840 1.4874 -0.024 157.3 158.0 155.2

20 0 80 0 1.5578 1.5613 -0.019 149.9 149.3 147.1

4 0 16 0 0.3116 0.3341 -1.293 700.4 651.3 642.3

3 2 76 23.4 0.3395 0.3531 0.277 662.7 651.1 648.6

4 1 28 10.9 0.3569 0.3732 -0.343 627.0 589.5 584.1

5 0 20 0 0.3895 0.4043 -0.274 578.8 544.9 536.1

3 3 12 30 0.4047 0.4176 -0.096 560.3 541.1 551.4

4 2 56 19.1 0.4122 0.4252 -0.196 550.4 539.5 536.3

5 1 124 8.9 0.4337 0.4462 -0.172 524.4 487.1 493.2

6 0 24 0 0.4673 0.4793 -0.188 488.2 463.6 458.5

4 3 148 25.3 0.4738 0.4833 0.101 484.1 479.4 476.0

5 2 52 16.1 0.4864 0.4970 -0.063 470.8 451.5 448.8

6 1 172 7.6 0.5108 0.5216 -0.192 448.6 437.8 432.8

4 4 16 30 0.5396 0.5489 -0.028 426.3 424.3 419.2

5 3 196 21.8 0.5452 0.5546 -0.084 421.9 417.4 413.5

7 0 28 0 0.5452 0.5546 -0.043 421.9 411.0 405.3

6 2 104 13.9 0.5617 0.5704 -0.008 410.3 404.5 400.2

7 1 76 6.6 0.5881 0.5971 -0.090 391.9 374.6 373.1

6 3 84 19.1 0.6182 0.6263 -0.038 373.6 364.5 363.1

8 0 32 0 0.6231 0.6319 -0.119 370.3 363.6 358.5

5 5 20 30 0.6746 0.6821 -0.058 343.1 337.3 338.9

6 4 152 23.4 0.6790 0.6864 -0.043 340.9 336.1 333.4

n m N Θ0 d0 dDFT c/c0 234/dDFT DFT *DFT n m N Θ0 d0 dDFT c/c0 234/dDFT DFT *

DFT

REZGÉSI TULAJDONSÁGOK

D band

Radial Breathing Mode

(5,3)

quadratic fit force constant RBM-frequency

DFT

RBM vs 1/d0

1/d0 (nm-1)

(c

m-1)

linear fit for large diameters

Alarge_d= 233.1

ZZ

AC

CH

RBM vs 1/ddeviation from linear fit

1/d (nm-1)

(

cm-1)

ZZ

AC

CH

7,0

5,3

d=0.5546 nm

ZZ

AC

CH

AZZ= 232

AAC= 236

COUPLING of

TOTALLY SYMMETRIC MODES

(RBM + G (HFM))

radial tangential

1 for achiral

2 for chiral

ZZ

AC

CH

Raman

Stokes: 2 = 1 –

(Anti-Stokes: 2 = 1 +

1

1

2 = 1 –

ba

2 = 1 +

1

1

hin hin hout houthout hin

(incoming) resonance RamanC. V. Raman

(a) RBM spectra of HiPCO produced carbon nanotubes at different excitation energies.

The spectra are vertically offset for clarity. From top to bottom the laser energy increases between 1.51 and 1.75 eV. Each peak arises from a different (n,m) nanotube.

(b) Resonance profiles for the peaks marked in a by vertical lines. The dots are experimental data; the lines are fits.

Two-dimensional plot of the radial-breathing-mode range vs. laser excitation energy. Note the various laola-like resonance enhancements, from which we can determine both the optical transition energies and the approximate diameter of the nanotubes. The spectra were each calibrated against the Raman spectrum of CCl4.

2D RBM

Contour plot of 2D RBM

Contour plot of 2D RBM

A. Jorio et al., (in press)