electronic excitations of double-walled armchair carbon nanotubes geometric configurations magneto...

Electronic excitations of double-walled armchair carbon nanotubes

• Geometric configurations• Magneto band structures• Magneto electronic excitations• Conclusions

何彥宏‚ 林明發 教授 ( 指導教授 )

成功大學 物理系

Rx=m a1+n a2 Ry=p a1+q a2 bnmn

nmnmbx

)]2/(3[tan angle chiral

2/)(32

radius

1

22

R

Geometric configurations --- Single-wall carbon nanotube

armchair(m,m)

a1

a2

intertube distance: 3.39 Å, closed to interlayer spacing of graphite.

Geometric configurations--- Double-walled carbon nanotubes

Double-walled armchair carbon nanotubes

(5,5)-(10,10)

• 3 kinds of symmetric structures, due to translation and rotation symmetry

• 12 atoms in a primitive unit cell: (4 from inner tube) (8 from outer tube)

the tight-binding model

','

''

''

8884

4844

12555

4111

12122211

exp ,

exp,

exp , :elementmatrix

1212 :nHamiltonia

}{

}{

:ioneigenfunct

'

'

'

'

5

5

1

1

outoutininRRda

outinrrRR

outoutoutoutoutoutrrRR

ininininininrrRR

R

Rik

R

Rik

RikRikWeRRhH

RikRikRRhH

RikRikRRhH

HH

HHH

RRXec

RRXec

ccc

outin

outin

outout

inin

out

in

Intratube & intertube interactions

Vppπ=-2.66 eV (γ0)

Vppσ=6.38 eV

sinsinsincoscoscoscoscos,

:intertube

sin2

sincos2

sin2

cos,

:intratube

22'

22222'

ppppppoutinrr

ppppppininrr

VVVRRh

VVVRRh

Band structures

without intertube interaction:

• symmetric about EF , and EF=0• linear bands intersecting at EF=0 , so metallic• parabolic band with double degenercy

Band structures

with intertube interaction:

• breaks symmetry of band structures• changes energy dispersion• localization of wavefunction: △: inner tube ○: outer tube

Density of states

• linear bands →pleataues

• parabolic bands →square-root divergences

• several low-energy

divergences in S5 system

, 22,

1

2Jz

z

kJE

dk

L

D

Magnetoelectronic properties

quantumflux

2

2 2

2

2

0

0

2222

e

hc

J

rdAc

eJrdkJrdk

Ac

eikik

Vc

Aek

mHV

m

kH

B

J → J+ψ/ψ0

shift angular momentum

Band structures linear band → parabolic band,

form energy spacing.

• induce energy gap

• break state degenercy.

(0.04 ψ0~ 114 Tesla)

Density of states

• linear band

to parabolic band

→ pleataue to divergence

• break degenercy

→ more divergences

ψ-dependent energy gap

• magnetic flux induces energy gap

• intertube interactions & spin-B interactions reduces energy gap

e -

e -

φc (J,kz+q;σ,ψ)

φv (J,kz;σ,ψ)

3. Magneto electronic excitations

• energy transfer

• momentum

transfer: Δkz=q

12212

2112

222

22

112

11

0

''

, ',

1

22'

'

'

2''

2

0

)()(4),(),(

)()(4),(

)()(4),(

4.2

),(),(),;,(

)(),;,(

),;,(2

)),(()),((

,,)2(

2),,(

),,(),(),,(

rrqrKqrIeLqVLqV

qrKqrIeLqV

qrKqrIeLqV

kJEqkLJELqkJ

iLqkJ

LqkJ

kJEfqkLJEf

kJeeqkLJdk

Lq

LqLqVLq

LL

LL

LL

yh

yh

yhh

J hh

BZst

yhh

yhh

yh

yh

iy

hiLiqyy

hi

y

i

Response function

response functioninner: χ1

outer: χ2

Band structures

Response functions

Response functions

2),,(),,(),,(

)Im(

)Im(),,(

inner tubeat localized A

2

22

2

11

22221112122

22121111111

122

111

2222111222

2212111111

111

LqPLqPLqP

V

VLqP

VVVVVV

VVVVVV

VV

VV

VVVVVV

VVVVVV

VVV

BA

eff

effA

effexeffexexeff

effexeffexexeff

exex

exex

effexeffexexeff

effexeffexexeff

inexeff

2

22

2

11

22221112222

22121111121

121

222

)Im(

)Im(),,(

outer tubeat localized B

eff

effB

effexeffexexeff

effexeffexexeff

exex

exex

V

VLqP

VVVVVV

VVVVVV

VV

VV

Intertube Coulomb interactions: Random-Phase Approximation (RPA)

Loss function

• Intertube interactions enrich electron-hole excitations, thus reduce plasmon intensity

• Plasmons appears at certain q region

Loss function

• Plasmon frequencies almost unchanged

by the magnetic flux

• Plasmon intensity reduced by the magnetic flux

q-dependent plasmon frequencies

• more plasmon modes

• acoustic plasmons to optical plasmons

4. Conclusion

• The intertube interactions alter the low energy bands, enrich the low-frequency single-particle excitations.

• The main features of the low-frequency plasmons are dominated by the momentum transfer q, the intertube interactions and the symmetric geometry.

• Double-walled geometry could be determined by the electron-energy loss spectroscopy (EELS).

Thanks for your attention !