electronic excitations of double-walled armchair carbon nanotubes geometric configurations magneto...
Post on 15-Jan-2016
226 views
TRANSCRIPT
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 !