Electronic properties of double-layer carbon nanotubes

0.31 MB·English

by  M. Pudlak

Electronic properties of double-layer carbon nanotubes
M. Pudlak1, and R. Pincak1,2,
∗ †
1Institute of Experimental Physics, Slovak Academy of Sciences,
Watsonova 47,043 53 Kosice, Slovak Republic
9
0 2Joint Institute for Nuclear Research, BLTP,
0
2 141980 Dubna, Moscow region, Russia

n
a (Dated: January 20, 2009)
J

0 Abstract
2

The electronic spectra for double-wall zigzag and armchair nanotubes are found. The influence
]
i
c
of nanotubecurvatures on the electronic spectrais also calculated. Ourfindingthat the outer shell
s
-
l
r is hole doped by the inner shell is in the difference between Fermi levels of individual shells which
t
m
originate fromthedifferenthybridization ofπ orbital. Theshiftand rotation of theinnernanotube
.
t
a
with respect to the outer nanotube are investigated. We found stable semimetal characteristics of
m
- the armchair DWNTs in regard of the shift and rotation of the inner nanotube. We predict the
d
n
shiftof k towards the bigger wave vectors with decreasing of the radius of the armchair nanotube.
o F
c
[
PACS numbers: 73.63.-b,73.63.Fg, 73.22.-f
3
v
6
4
3
4
.
2
1
7
0
:
v
i
X
r
a
∗Electronic address: [email protected]
†Electronic address: [email protected]
1
I. INTRODUCTION
Carbon nanotubes are very interesting because of their unique mechanical and electronic
properties. A single-wall carbon nanotube can be described as a graphene sheet rolled into
a cylindrical shape so that the structure is one-dimensional with axial symmetry and in gen-
eral exhibiting a spiral conformation called chirality. The primary symmetry classification
of carbon nanotubes is either achiral (symmorphic) or chiral (non-symmorphic). Achiral
carbon nanotubes are defined by a carbon nanotube whose mirror images have an identical
structure to the original one. There are only two cases of achiral nanotubes, armchair and
zigzag nanotubes. The names of armchair and zigzag nanotubes arise from the shape of the
cross-section ring at the edge of the nanotubes. Chiral nanotubes exhibit spiral symmetry
whose mirror image cannot be superposed onto the original one. There is a variety of ge-
ometries in carbon nanotubes which can change the diameter, chirality and cap structures.
The electronic structure of carbonnanotubes is derived by a simple tight-binding calculation
for the π-electrons of carbon atoms. Of special interest is the prediction that the calculated
electronic structure of a carbon nanotube can be either metallic or semiconducting, depend-
ing on its diameter and chirality. The energy gap for a semiconductor nanotube, which is
inversely proportional to its diameters, can be directly observed by scanning tunneling mi-
croscopy measurements. The electronic structure of a single-wall nanotube can be obtained
simply from that of two-dimensional graphite. By using periodic boundary conditions in the
circumferential direction denoted by the chiral vector C , the wave vector associated with
h
the C direction becomes quantized, while the wave vector associated with the direction of
h
the translational vector T along the nanotube axis remains continuous for a nanotube of
infinite length. Thus, the energy bands consist of a set of one-dimensional energy dispersion
relations which are cross sections of those for two-dimensional graphite. To obtain explicit
expressions for the dispersion relations, the simplest cases to consider are the nanotubes
having the highest symmetry, e.g. highly symmetric achiral nanotubes. The synthesis of
DWNTs has been reported recently [1, 2]. Their electronic structure was investigated by the
localdensity approximation[3, 4, 5, 6, 7]andthetight-binding model[8, 9, 10, 11]. Asimilar
method can be used to investigate the electronic spectra of the fullerene molecules [12, 13].
In this paper we are interested in the zigzag and armchair double-wall nanotubes(DWNTs)
with a small radius. In these DWNTs the difference of Fermi levels of individual nanotubes
2
has to be taken into account. We focus on (9,0) (18,0) zigzag tubules and (5,5) (10,10)
− −
armchair tubules. They are the best matched, double layer tubules.
II. (9,0) (18,0) ZIGZAG TUBULES
−
Firstly, we describe the model for the zigzag nanotubes. The π electronic structures are
calculated from the tight-binding Hamiltonian
H = ǫ ϕout ϕout + γ ϕin ϕin +h.c + ǫ ϕin ϕin + γ ϕin ϕin +h.c
| i ih i | ij | i ih j | | i ih i | ij | i ih j |
i i,j i i,j
X X (cid:0) (cid:1) X X (cid:0) (cid:1)
e e
+ W ϕin ϕout +h.c , (1)
ln | l ih n |
l,n
X (cid:0) (cid:1)
ǫ and ǫ are Fermi energies of the outer and inner nanotubes; ϕout , ϕin are π orbitals on
| i i | i i
site i at the outer and inner tubes; γ , γ are the intratube hopping integrals; W are the
ij ij ij
e
intertube hoping integrals which depends on the distance d and angle θ between the π
ij ij i
e
and π orbitals (see [14, 15, 16] for details).
j
γ
W = 0 cos(θ )e(ξ dij)/δ, (2)
ij ij −
8
where θ is an angle between the ith atom of the inner shell and the jth atom of the outer
ij
shell, d is the interatom distance and ξ is a intertube distance. The characteristic length
ij
δ = 0.45˚A.
To describe the parameter which characterized the zig-zag tubules, we start from the
graphene layer [17] where we can define the vectors connecting the nearest neighbor carbon
atoms for zigzag nanotubes in the form:
1
τ = a(0; ),
−→1
√3
1 1
τ = a( ; ),
−→2
2 −2√3
1 1
τ = a( ; ). (3)
−→3
−2 −2√3
The distance between atoms in the unit cell is d = τ = a . Following the scheme in Figs
|−→i| √3
1,2 [18] we want to find solution to the double-layer graphene tubules in the form:
ψ( r ) = ψ ( r )+ψ ( r ), (4)
−→ out −→ in −→
3
FIG. 1: The outer shell part of the unit cell in the case of zigzag nanotubes.
where
ψ ( r ) = C ψ +C ψ +C ψ +C ψ
out −→ A1 A1 A2 A2 B1 B1 B2 B2
+C ψ +C ψ +C ψ +C ψ , (5)
A‘1 A‘1 A‘2 A‘2 B1‘ B1‘ B2‘ B2‘
and
ψ ( r ) = C ψ +C ψ +C ψ +C ψ . (6)
in −→ A A B B A‘ A‘ B‘ B‘
We want to find solution to the above equation in the form of the Bloch function
1
ψ (−→k , r ) = ei→−k(−r→n+→−dα) ϕ( r r −→d ) , (7)
α →− −→ −→n α
√M | − − i
n
X
where α denotes A or B atoms. Here −→d is the coordinate of the α atom in the unit cell
α
and r is a position of a unit cell, M is a number of the unit cell; ϕ(~r) is a π orbital which
−→n
| i
is generally different for the outer and inner shell. We denote
ǫ = ϕout(r A ) H ϕout(r A ) = ϕout(r B ) H ϕout(r B ) , (8)
i i i i
h − | | − i h − | | − i
ǫ = ϕin(r A ) H ϕin(r A ) = ϕin(r B ) H ϕin(r B ) . (9)
i i i i
h − | | − i h − | | − i
e 4
FIG. 2: The inner shell part of the unit cell in the case of zigzag nanotubes.
Now we define the intratube hopping integrals
ϕout(r A ) H ϕout(r B ) = γ ,
1 1 0
h − | | − i
ϕout(r A ) H ϕout(r B ) = γ β = ϕout(r A ) H ϕout(r B‘) , (10)
h − 1 | | − 2 i 0 h − 1 | | − 2 i
and
ϕin(r A) H ϕin(r B) = γ ,
0
h − | | − i
ϕin(r A) H ϕin(r B′) = γ β, (11)
0
h − | | − i
where γ0 is the hoping integral in the graphene and β(β) ies part which depends on the
surface curvature and will be computed latter. So in a tight-binding approximation we get
e
the systems of equations as showing in Appendix A.
Firstly, we solve the equations in Appendix A assuming that W is the perturbation.
ij
So we can decouple these 12 equations. We get 8 equations for the outer shell and 4
for the inner shell. If we express the state of the outer shell (Eq.5) in the form ψ =
out
(C ,C ,C ,C ,C ,C ,C ,C ), we get the solutions to the outer shell in the form
A1 B1 A2 B2 A‘1 B1‘ A‘2 B2‘
mπ √3ka mπ 1
E (k) = ǫ γ (1+4βcos cos +4β2cos2 )2,
1,2 0
± N 2 N
5
1
ψ = 1; e iϕ1;1; e iϕ1,1; e iϕ1;1; e iϕ1 (12)
1,2 − − − −
√8 ± ± ± ±
(cid:0) (cid:1)
mπ √3ka mπ 1
E (k) = ǫ γ (1 4βcos cos +4β2cos2 )2,
3,4 0
± − N 2 N
1
ψ = 1; e iϕ2; 1; e iϕ2,1; e iϕ2; 1; e iϕ2 (13)
3,4 − − − −
√8 ± − ∓ ± − ∓
(cid:0) (cid:1)
mπ √3ka mπ 1
E (k) = ǫ γ (1+4βsin cos +4β2sin2 )2,
5,6 0
± N 2 N
1
ψ = 1; e iϕ3; i; ie iϕ3, 1; e iϕ3;i; ie iϕ3 (14)
5,6 − − − −
√8 ± − ∓ − ∓ ±
(cid:0) (cid:1)
mπ √3ka mπ 1
E (k) = ǫ γ (1 4βsin cos +4β2sin2 )2,
7,8 0
± − N 2 N
1
ψ = 1; e iϕ4;i; ie iϕ4, 1; e iϕ4; i; ie iϕ4 (15)
7,8 − − − −
√8 ± ± − ∓ − ∓
(cid:0) (cid:1)
where, for instance,
ei√ka3 +2βcos mπe−i2k√a3
eiϕ1 = N . (16)
(1+4βcos mπ cos √3ka +4β2cos2 mπ)12
N 2 N
Similar results for the electronic spectra in the case of inner nanotubes were found in the
form (ψ = (C ,C ,C ,C ))
in A B A‘ B‘
mπ √3ka mπ 1
E (k) = ǫ˜ γ (1+4βcos cos +4β2cos2 )2,
9,10 0
± N 2 N
1e e
ψ = 1; e iϕ5;1; e iϕ5 (17)
9,10 − −
√4 ± ±
(cid:0) (cid:1)
mπ √3ka mπ 1
E (k) = ǫ˜ γ (1 4βcos cos +4β2cos2 )2,
11,12 0
± − N 2 N
1
ψ = e1; e iϕ6; 1; e iϕ6 e. (18)
11,12 − −
√4 ± − ∓
(cid:0) (cid:1)
Since the radii of the outer and inner nanotubes are different β = β. Here k = k and
y
6
π < k < π is the first Brillouin zone. As we have a curved surface, the local normals
−√3a √3a e
on the neighboring sites are no longer perfectly aligned and this misorientation also changes
the transfer integral. The change can be calculated using the curvature tensor b [19]. The
αβ
result is
δt 1
a = b bγτβτα, (19)
t −2 γβ α a a
6
where the only nonzero term is b bx = 1/R2. So we have
xx x
δt
1
= 0, (20)
t
δt 1 1
2 = b bx(τx)2 = (τx)2, (21)
t −2 xx x 2 −2R2 2
δt 1 1
3 = b bx(τx)2 = (τx)2. (22)
t −2 xx x 3 −2R2 3
With using the unit vectors we have (τx)2=(τx)2=a2. We found the radius of the inner
2 3 4
nanotube from the expression 2πR = Na. The nonzero terms are δt2 = δt3 = 1(π)2. The
t t 2 N
same holds for the outer nanotube. The parameters β, β can be expressed in the form
δt 1 π
β = 1 2 = 1 ( e)2, (23)
− t − 2 9
e
and
δt 1 π
β = 1 2 = 1 ( )2. (24)
− t − 2 18
Now we calculate the values ǫ and ǫ˜ which are different because the inner and outer shell
radii are different. Due to the curvature the coordinates of τ in space are
−→i
τ = d(0;1;0),
−→1
√3 1 √3
τ = d( cosθ; ; sinθ),
−→2
2 −2 − 2
√3 1 √3
τ = d( cosθ; ; sinθ), (25)
−→3
− 2 −2 − 2
where sinθ = a/4R; R is the radius of the nanotube. Now one can construct three hybrids
along the three directions of the bonds. These directions are
e = (0;1;0),
−→1
√3 1 √3
e = ( cosθ; ; sinθ),
−→2
2 −2 − 2
√3 1 √3
e = ( cosθ; ; sinθ). (26)
−→3
− 2 −2 − 2
The requirement of the orthonormality of the hybrid wave functions determines uniquely
the fourth hybrid, denoted by π , which corresponds to the p orbital in graphite. The
z
| i
hybridization of the σ bonds therefore changes from the uncurved expression to
σ = s s + 1 s2 p ,
| 1i 1| i − 1| yi
q
7
√3 1 √3
σ = s s + 1 s2 cosθ p p sinθ p ,
| 2i 2| i − 2 2 | xi− 2| yi− 2 | zi
!
q
√3 1 √3
σ = s s + 1 s2 cosθ p p sinθ p ,
| 3i 3| i − 3 − 2 | xi− 2| yi− 2 | zi
!
q
π = D s +D p +D p +D p . (27)
1 2 x 3 y 4 z
| i | i | i | i | i
The mixing parameters s ,D can be determined by the orthonormality conditions σ σ =
i j i j
h | i
δ , π σ = 0, π π = 1. We get
ij i
h | i h | i
1 1
σ = s + 1 p ,
1 y
| i √3cos2θ | i − 3cos2θ | i
r
3cos2θ 1 2 1 √3 1 √3
σ = − s + cosθ p p sinθ p ,
2 x y z
| i s3(cos2θ+1) | i r3 cosθ 2 | i− 2| i− 2 | i!
3cos2θ 1 2 1 √3 1 √3
σ = − s + cosθ p p sinθ p ,
3 x y z
| i s3(cos2θ +1) | i r3 cosθ − 2 | i− 2| i− 2 | i!
3cos2θ 1 tanθ √cos2θ
π = tanθ − s + p + p . (28)
y z
| i 3cos2θ | i √3cos2θ | i cosθ | i
r
Now we can find the expression for the π orbital to the lowest order in a/R
a a
π s + p + p , (29)
y z
| i ≈ 2√6R| i 4√3R | i | i
and so we get
a2 a2
ε = π H π s H s + p H p + p H p . (30)
h | | i ≈ 24R2h | | i 48R2h y| | yi h z| | zi
Due to a/2R = π/N,(N = 9) we have
1 π2 1 π2
˜ǫ = s H s + p H p + p H p , (31)
6N2h | | i 12N2h y| | yi h z| | zi
and
1 π2 1 π2
ǫ = s H s + p H p + p H p . (32)
24N2h | | i 48N2h y| | yi h z| | zi
In the case m = 3 we find
√3ka 1
E (k) = ǫ γ (1 2βcos +β2)2, (33)
3,4 0
± − 2
√3ka 1
E (k) = ǫ˜ γ (1 2βcos +β2)2, (34)
11,12 0
± − 2
e e
8
where k = 0 is a Fermi point for both the inner and outer nanotubes in the case β = β = 1.
Nanotubes have no gap and have a semiconductor character. If we impose a curvature
e
correction, we get a gap
π 2 γ a 2
0
E = 2(1 β) = γ = , (35)
g 0
− 2N 4 2R
(cid:16) (cid:17) (cid:16) (cid:17)
for the outer nanotube and
π 2 γ a 2
0
E = 2(1 β) = γ = , (36)
g 0
− N 4 R
(cid:16) (cid:17) (cid:16) (cid:17)
for the inner nanotube. Here R is theeradius of the inner tube and 2R is the radius of the
outer tube. So we get the same gap as was computed in [20] where the rehybridized orbital
methodwas used. Forγ 3eV weget E 0.365eV fortheinner tubeandE 0.091eV
0 g g
≈ ≈ ≈
for the outer tube. Now we want to estimate the difference between ”Fermi levels” of the
inner and the outer shell. We have [21]
s H s 12eV, (37)
h | | i ≈ −
p H p 4eV, (38)
y y
h | | i ≈ −
and the difference is
1 π 2 π 2 1 π 2 π 2
ǫ ǫ˜= s H s + p H p . (39)
y y
− 6 2N − N h | | i 12 2N − N h | | i
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
From the expression above we finally get the value for the energy gap
ǫ ǫ˜ 0.21eV. (40)
− ≈
Now we use the eigenstates ψ to find the solution when the interaction between shells is
i
imposed. We assume the symmetric geometry of zig-zag DWNT. It means that the atoms
A,A and B,B are directly one above another in the neighboring shells [10]. We take into
1 1
account only the interactions
γ
0
W = W = . (41)
A,A1 B,B1 8
We look for solution in the form
12
Ψ = ζ ψ . (42)
i i
i=1
X
We have secular equations
12
ψ H ψ ζ = E˜ζ , (43)
i j j i
h | | i
j=1
X
9
where
ψ H ψ = δ E , (44)
i j ij i
h | | i
for i,j = 1,...8 and i,j = 9,...12, and the interaction between shells is described by the
terms ψ H ψ for i = 1,...8 ; j = 9,...12 and vice versa. We have, for instance,
i j
h | | i
1 γ
ψ H ψ = 0 1+ei(ϕ5 ϕ1) , (45)
9 1 −
h | | i 4√2 8
(cid:0) (cid:1)
1 γ
ψ H ψ = 0 1 ei(ϕ5 ϕ1) . (46)
9 2 −
h | | i 4√2 8 −
We get the eigenvalues E˜ with eigenvectors whic(cid:0)h can be exp(cid:1)ressed in the form
i
12
Ψ = ζ ψ . (47)
i i,j j
j=1
X
The eigenvalues of Eq.(55) for some values of √3ka/2 near the point k = 0 are depicted on
Fig.3 where E and E are conductive and valence band. The band structure for zig-zag
c v
DWNT without intertube interactions is also shown for comparison (Fig.4). At the point
0,8
0,6
0,4
0,2
Ev
E 0,0 Ec
-0,2
-0,4
-0,6
-0,8
-0,20 -0,15 -0,10 -0,05 0,00 0,05 0,10 0,15 0,20
3ka/2
FIG. 3: Spectra of zigzag DWNT with the intertube interactions.
k=0 we get the wave function of the valence band
Ψ 0.6ψ +0.8ψ . (48)
v 3 11
≃ −
ψ (ψ ) is π state of the outer(inner) nanotube. We get a minimum gap E 90 meV
3 11 ∗ g
≃
between the valence and conductive band of the DWNTs at the wave vectors √3ka/2
≃
0.05. At these points the wave function has the form
±
Ψ 0.263iψ +0.838ψ (0.14+0.45i)ψ +(0.29 0.09i)ψ . (49)
v 3 4 11 12
≃ − − −
10