Hexagon Lattices and Nanotubes
W.J. Zakrzewski
Department of Mathematical Sciences,University of Durham,
Durham DH1 3LE, UK
Work done in collaboration with:
B. Hartmann - Durham and Bremen
1
Introduction
• Nanotubes studied since their discovery in 1991.
• Structure: Carbon cylinders with a hexagonal grid - thus
fullerene related.
• Their mechanical, thermal, optical and electrical properties
depend on the diameter, chirality and length of the tube.
• A lattice distortion affects the energy band gap.
• Two ways to achieve this distortion:
– through an external force like e.g. bending, stretching or
twisting
– through an internal excitation, which interacts with the
lattice.
• Known that localised states (called solitons in what follows)
can result from the interaction of an excitation such as an
amide I vibration in biopolymers or an electron (in the case of
the Fröhlich Hamiltonian) with a lattice.
• This goes back to Davydov (1976) - to explain the dispersion
free energy transport in biopolymers.
• Recent study of Fröhlich Hamitonian on a two-dimensional,
discrete, quadratic lattice established the existence of localised
states whose properties depend on the electron-phonon coupling constant.
• In the continuum limit these discrete equations reduce to a
modified non-linear Schrödinger (MNLS) equation - which also
possesses solitons (with an additional term resulting from the
discreteness of the lattice).
• Here we report our study of a hexagonal, periodic lattice with
a large extension in x and a small extension in y directions (an
oversimplified model of nanotubes).
• Our case corresponds to the (5, 5) armchair tube.
2
The Hamiltonian and equations of motion
2.1
Hamiltonian
• The Hamiltonian H of is a sum of four sums which result from
the special features of the hexagonal grid.
• ψi,j denotes the electron field on the i-th, j-th lattice side
• ui,j and vi,j are the displacements of the i-th, j-th lattice point
from equilibrium in the x direction and y directions
H =
−
+
+
−
+
+
−
N2
N
−1 41 −3 ·
2X
X
∗
∗
(E + W )ψi,j ψi,j
− jxψi,j
(ψi+1,j+1 + ψi−1,j + ψi+1,j−1)
j−1
i−1
=0
2 =0 µ4
∗
jxψi,j ψi+1,j+1
+
µ
|ψi,j |2 c3x (ui+1,j+1
N2 N1
−2 ·
2
4X
X
∗
ψi−1,j
+
∗
ψi+1,j−1
¶
+ ui+1,j−1 − 2ui−1,j ) +
cx
√
(v
3 i+1,j+1
− vi+1,j−1)
¶¸
∗
∗
(ψi+1,j + ψi−1,j+1 + ψi−1,j−1)
(E + W )ψi,j ψi,j
− jxψi,j
j
i−2
=0
2 =1 4 µ
∗
jxψi,j ψi+1,j
+
∗
ψi−1,j+1
µ
|ψi,j |2 c3x (−ui−1,j−1
N2 N1
−1 ·
2
4X
X
+
∗
ψi−1,j−1
¶
− ui−1,j+1 + 2ui+1,j ) +
cx
√
(v
3 i−1,j+1
− vi−1,j−1)
∗
∗
(E + W )ψi,j ψi,j
− jxψi,j
(ψi+1,j+1 + ψi−1,j + ψi+1,j−1)
j
i−3
=0
2 =1 4 µ
∗
jxψi,j ψi+1,j+1
+
∗
ψi−1,j
+
∗
ψi+1,j−1
¶
¶¸
+
+
−
µ
|ψi,j |2 c3x (ui+1,j+1
N2
N
−1 41 ·
2X
X
− vi+1,j−1)
¶¸
∗
∗
(E + W )ψi,j ψi,j
− jxψi,j
(ψi+1,j + ψi−1,j+1 + ψi−1,j−1)
j−1
2 =0
i
4µ=1
∗
jxψi,j ψi+1,j
+ |ψi,j |2
+ ui+1,j−1 − 2ui−1,j ) +
cx
√
(v
3 i+1,j+1
µ
+
∗
ψi−1,j+1
cx
3 (−ui−1,j−1
+
∗
ψi−1,j−1
¶
− ui−1,j+1 + 2ui+1,j ) + √cx3 (vi−1,j+1 − vi−1,j−1)
¶¸
The phonon energy W :
W =
+
+
!
Ã
N
2 N
1
X
X
duij 2
dvij 2
1
( dt ) + ( dt )
2M
j=1 i=1
N2
N
−1 41 −3 µ
2X
X
1
M
kx[(uij − ui−1,j )2 + (vij − vi−1,j )2
2
j−1
i−1
=0
2 =0 4
(uij − ui+1,j+1)2 + (vij − vi+1,j+1)2 + (uij − ui+1,j−1)2
¶
(vij − vi+1,j−1)2] + 12 M
N 2 N1
−2 µ
2
4X
X
i−2
j
=0
2 =1 4
2
kx[(uij − ui+1,j )2 + (vij − vi+1,j )2
2
2
2
+ (uij − ui−1,j+1) + (vij − vi−1,j+1) + (uij − ui−1,j−1) + (vij − vi−1,j−1) ]
+ 12 M
N2 N1
−1 µ
2
4X
X
j
i−3
=1
4 =0
2
kx[(uij − ui−1,j )2 + (vij − vi−1,j )2 + (uij − ui+1,j+1)2
2
2
2
+(vij − vi+1,j+1) + (uij − ui+1,j−1) + (vij − vi+1,j−1) ]
+ 12 M
+ (uij −
N2
N
−1 41 µ
2X
X
j−1
i
=1
2 =0 4
ui−1,j+1)2 +(vij
¶
kx[(uij − ui+1,j )2 + (vij − vi+1,j )2
2
2
2
¶
− vi−1,j+1) +(uij − ui−1,j−1) + (vij − vi−1,j−1) ]
¶
• jx - the electron field self-interaction coupling
• cx coupling of the electron field to the displacement fields u
and v
• kx - the self-coupling of the displacement fields.
2.2
Equations of motion
• Easy to derive the equations of motion
• E.g. the (discrete Schrödinger) equation for the ψi,j field is:
ih̄
∂ψi,j
∂t
= (E + W )ψi,j − 2jx (ψi+1,j+1 + ψi−1,j + ψi+1,j−1)
·
+ψi,j c3x (ui+1,j+1
+ ui+1,j−1 − 2ui−1,j ) +
cx
√
(v
3 i+1,j+1
− vi+1,j−1)
• the equations for the displacement fields ui,j and vi,j are:
d2 ui,j
dt2
= kx (3ui,j − ui+1,j+1 − ui−1,j − ui+1,j−1)
+
d2 vi,j
dt2
cx
3M
µ
2
2
2
2|ψi−1,j | − |ψi+1,j+1| − |ψi+1,j−1|
¶
= kx (3vi,j − vi+1,j+1 − vi−1,j − vi+1,j−1)
−
√cx
3M
µ
2
|ψi+1,j+1| − |ψi+1,j−1|
2
¶
.
Rescale
τ=
jx t
h̄
, U = 3Cxu , V = 3Cxv , E0 =
Cx =
cx
9jx
, Kx =
kx h̄2
jx2
, g=
2Cx2
Es
E
jx
, W0 =
, Es =
M jx
9h̄2
W
jx
¸
and find:
i
∂ψi,j
∂τ
= (E0 + W0)ψi,j − 2 (ψi+1,j+1 + ψi−1,j + ψi+1,j−1)
·
¸
√
+ ψi,j (Ui+1,j+1 + Ui+1,j−1 − 2Ui−1,j ) + 3(Vi+1,j+1 − Vi+1,j−1)
d2 Ui,j
dτ 2
= Kx (3Ui,j − Ui+1,j+1 − Ui−1,j − Ui+1,j−1)
+
g
2
µ
2
2
2
2|ψi−1,j | − |ψi+1,j+1| − |ψi+1,j−1|
¶
and
d2 Vi,j
dτ 2
= Kx (3Vi,j − Vi+1,j+1 − Vi−1,j − Vi+1,j−1)
√
−
3
3g
2
µ
2
2
|ψi+1,j+1| − |ψi+1,j−1|
¶
Stationary limit
In the stationary limit, we have:
λψi,j + 2 (3ψi,j − ψi+1,j+1 − ψi−1,j − ψi+1,j−1)
+ψi,j [Ui+1,j+1 + Ui+1,j−1 − 2Ui−1,j +
√
3(Vi+1,j+1 − Vi+1,j−1)] = 0
where λ = E0 + W0 − 6 and
3Ui,j − Ui+1,j+1 − Ui−1,j − Ui+1,j−1 =
2
2
−|ψi+1,j+1| − |ψi+1,j−1|
− g̃2
¶
µ
2|ψi−1,j |2
√
3Vi,j − Vi+1,j+1 − Vi−1,j − Vi+1,j−1 =
with g̃ =
3.1
3g̃
2
µ
2
2
|ψi+1,j+1| − |ψi+1,j−1|
¶
g
Kx .
Discrete equation
• In the stationary limit these equations lead to one localised
equation for ψ (ie we can eliminate U and V ).
• This is not true for a rectangular lattice.
To see this look at the case i = 1 + 4k. We have:
∆(1)Uij =
g̃
2
µ
2
2
2
2|ψi−1,j | − |ψi+1,j+1| − |ψi+1,j−1|
¶
where
∆(1)Uij = Ui+1,j+1 + Ui−1,j + Ui+1,j−1 − 3Ui,j
. Also:
√
3g̃
2
∆(1)Vij =
µ
2
2
|ψi+1,j−1| − |ψi+1,j+1|
¶
∆(1)Vij = Vi+1,j+1 + Vi−1,j + Vi+1,j−1 − 3Vi,j
.
Next note that:
∆(1)Ui+1,j+1 =
g̃
2
µ
2
2
2
|ψi,j | + |ψi,j+2| − 2|ψi+2,j+1|
¶
,
∆(1)Ui+1,j−1 =
∆(1)Ui−1,j =
g̃
2
µ
g̃
2
2
2
|ψi,j−2| + |ψi,j | − 2|ψi+2,j−1|
µ
2
2
2
¶
|ψi−2,j−1| + |ψi−2,j+1| − 2|ψi,j |
2
,
¶
for the U field and
√
∆(1)Vi+1,j+1 =
3g̃
2
√
µ
3g̃
2
∆(1)Vi+1,j−1 =
2
2
|ψij | − |ψi,j+2|
µ
2
¶
2
|ψi,j−2| − |ψij |
,
¶
for the V field.
Then define:
Za = Ui+1,j+1 + Ui+1,j−1 − 2Ui−1,j +
√
3(Vi+1,j+1 − Vi+1,j−1)
and find that
µ
∆(1)Za = g̃ 6|ψij |2 − |ψi,j+2|2 − |ψi+2,j+1|2
2
2
2
2
−|ψi,j−2| − |ψi+2,j−1| − |ψi−2,j−1| − |ψi−2,j+1|
¶
The right hand side of this is a 7-point Laplacian ∆(2)|ψij |2.
Thus we have
∆(1)Za = −g̃∆(2)|ψij |2 .
It is easy to find one possible solution of this equation, namely:
µ
2
2
2
2
Za = −g̃ |ψi+1,j+1| + |ψi+1,j−1| + |ψi−1,j | + 3|ψij |
¶
.
• On a quadratic lattice a similar equation has no simple solution.
Inserting our solution we get:
λψi,j + 2 (3ψi,j − ψi+1,j+1 − ψi−1,j − ψi+1,j−1)
− g̃ψi,j [|ψi+1,j+1|2 + |ψi+1,j−1|2 + |ψi−1,j |2 + 3|ψij |2] = 0
ie
µ
2
λψi,j − 2∆(1)ψij − g̃ψi,j ∆(1)|ψi,j | + 6|ψij |
2
¶
= 0.
This equation constitutes our discrete nonlinear Schrödinger
(DNLS) equation.
3.2
Continuum limit
• Look at the continuum limit of our equations.
• Thus introduce:
2
2
∂ψ
∂ ψ
2∂ ψ
1
ψi±1,j+1 = ψ ± δx± ∂ψ
∂x + δy ∂y + 2 (δx± ) ∂x2 ± δyδx± ∂x∂y
2
3
3
3
∂ ψ
+ 12 (δy)2 ∂∂yψ2 ± 18 (δx±)3 ∂∂xψ3 + 12 (δx±)2δy ∂x∂ 2ψ∂y ± 21 δx±(δy)2 ∂x∂y
2
3
+ 81 (δy±)3 ∂∂yψ3 ± ...
and
2
2
∂ψ
∂ ψ
2∂ ψ
1
ψi±1,j−1 = ψ ± δx± ∂ψ
∂x − δy ∂y + 2 (δx± ) ∂x2 ∓ δyδx± ∂x∂y
2
3
3
3
∂ ψ
+ 12 (δy)2 ∂∂yψ2 ± 18 (δx±)3 ∂∂xψ3 − 12 (δx±)2δy ∂x∂ 2ψ∂y ± 12 δx±(δy)2 ∂x∂y
2
3
− 81 (δy±)3 ∂∂yψ3 ± ...
where for i = 1 + 4k and i = 3 + 4k we have δx+ = 1/2,
δx− = 1, and for i = 2 + 4k and i = 4 + 4k we have δx+ = 1,
δx− = 1/2.
Moreover, δy =
√
3/2.
Thus we obtain:
λψ − 32 ∆ψ − g̃ψ
µ
2
3
4 ∆|ψ|
¶
+ 6|ψ|2 = 0
ie
µ
2
λ̃ψ + ∆ψ + 4g̃ψ |ψ| +
2
1
8 ∆|ψ|
¶
=0
• We have thus derived a modified nonlinear Schrödinger (MNLS)
equation with an extra term, which can stabilise the soliton:
i ∂ψ
∂τ
µ
2
+ ∆ψ + 4g̃ψ |ψ| +
2
1
8 ∆|ψ|
¶
= 0.
• This equation has the conserved energy:
E=
¶
Z µ
~ 2 − 2g̃|ψ|4 + g̃ (∆|ψ|2)2 dxdy .
|∇ψ|
4
• If we approximate its soliton solution by a Gaussian of the
form ψ(x, y) =
√κ
π
2
exp(− κ2 (x2 + y 2)) we find that the value
of κ minimising the energy is
κ2min = 2(1 − πg̃ )
• Thus we have an estimate of the critical g̃, ie g̃cr ∼ π.
Solutions of Schrödinger equation (without the extra term)
2
i dϕ
dτ + ∆ϕ + 2g|ϕ| ϕ = 0
collapse but we have this extra term due to the lattice.
This extra term stabilises the solitons
Consider the square of the size of the soliton-like configuration
2
R =
Z
|ϕ(x, y)|2(x2 + y 2)dxdy
Differentiate and get
dR2
dτ
Z
= − (x2 + y 2)(ϕ∆ϕ∗ − ϕ∗∆ϕ)dxdy.
and so
d2 R2
dτ 2
where
= 8(E + δ)
δ=
g
12
Z µ
¶
2 2
∆|ϕ|
dxdy.
Note E < 0 but δ > 0 so shrinking and expansion - hence
stabilisation.
4
MODIFIED NLSE in D DIMENSIONS
Study
µ
2
2
¶
iϕt + ∆ϕ + 2 g|ϕ| + G∆|ϕ| ϕ = 0.
Here G = αg/12, where α, typically, takes values in the range
1 ≤ α ≤ 4.
Conservation Laws:
1. the norm functional
N=
Z
dxD |ϕ|2
2. Energy
H=
Z
D
dx
µ
~ 2 − g|ϕ|4 + G(∂|ϕ|
~ 2 )2
|∂ϕ|
¶
3. Momentum
I~ =
Z
D~
dx j,
jµ =
− 2i
Ã
∂ϕ
ϕ∗ ∂x
µ
−
∗
ϕ ∂ϕ
∂xµ
!
.
4. Angular momentum
Lµν =
Z
dxD (xµjν − xν jµ)
Can introduce also eigen-energy
Λ=
Z
µ
∗¶
∂ϕ
dxD 2i ϕ∗ ∂ϕ
∂t − ϕ ∂t
Then for solution of the form (stationary solutions)
ϕ = φe−iλt
we have
Λ = λN
For soliton to exist we need (for stability) H < 0 (not Λ < 0)
(note - they are related but not equal).
Find
D
2
3
4
5
Numerical gcr
5.85
26.4094
82.6714
254.964
Ansatz gcr
2π ≈ 6.2832
3π(5π/2)1/2 ≈ 26.4129;
8π 2 ≈ 78.957
5(π)5/2(35/18)3/2 ≈ 237.16
• There exist solitonic solutions for g > gcr .
• gcr depends on D (dimension)
• gcr grows as D increases
Further Comments:
• As g increases - solitons are narrower and more bound (energy
is more negative).
• A good approximation is given by a Gaussian ansatz.
• There exist further unstable solutions (which become stable as
g increases).
• at D = 2 there exist also states with nonzero angular momentum l. Their gcr also increases with the value of l.
• Soliton can be made to move - so can study their dynamics.
5
Numerical results
5.1
Continuous, modified non-linear Schrödinger (MNLS) equation
Consider the radially symmetric Ansatz:
ψ(r, t) = eiαtR(r),
and solve the equation with the boundary conditions
∂R(r)
∂r |r=0
= 0 , R(r = ∞) = 0 .
Fine
• Critical value of g̃ = g̃cr ≈ 2.94
• At that value of g - R(0) tends to zero; ie. soliton is spread
out.
• This agrees well with our Gaussian approx. g̃cr = π.
• α, the frequency of a rotation vanishes at threshold.
• This is similar to what has been observed for “q-balls”
5.2
5.2.1
Discrete equations
Full system of equations
• Convenient to “squeeze” the lattice.
• Have used the periodic grid with N1 = 160 and N2 = 20.
• Have chosen the boundary conditions identifying the fields at
(i = 0, j) with those at (i = imax, j).
• Hence such nanotube is a (5, 5) armchair tube which is metallic.
• Used units with the carbon bond length in the nanotube being
1 (corresponding to 0.1415 nm) - thus the physical lengths of
our tube is 19.16 nm, while the diameter is 0.6756 nm.
• Starting configuration: an exponential-like excitation ψi,j extended typically over the lattice points i = 78 − 83 and j =
3 − 7 with the lattice being at its equilibrium everywhere, i.e.
ui,j = 0 and vi,j = 0 for all i, j.
• Studied the existence of solitons and their dependence on the
value of the coupling constant cx.
• Set jx = kx = 1, M = 20 and E = 0.142312.
• Note that the choice of M = 20 is a reflection of the physical
fact that the mass of carbon ≈ 20 · 10−24g.
• To absorb the energy have introduced damping terms ν
and ν
dvi,j
dt ,
dui,j
dt
respectively, into the equations for U and V . Its
value was ν = 0.25 − 0.75.
• Results
– Found that solitons exist for cx > 20.
– For larger values of cx, the soliton forms very quickly, decreasing cx increases the time needed for a soliton to form
(due to the less strong coupling between the dynamics of
the lattice itself and the excitation).
– For cx = 19 soliton has not formed until t ≈ 8000.
– In all cases - very little displacement of the lattice from the
equilibrium.
– At the location of the soliton the lattice becomes squeezed
(i.e. the lattice sites move towards the sites at which the
soliton is located).
– Perturbations: Have found that perturbations cane change
the size of solitons. Suggests that the system has many
solutions close in energy to each other.
5.2.2
Modified, discrete non-linear Schrödinger (DNLS) equation
• Studied also the DNLS - including some absorption (more complicated).
• Similar initial configuration - ψi,j being exponential and nonzero
over i = 78 − 83 and j = 3 − 7.
• Determined the value of g̃ for which a soliton exists.
• See our plots; as g̃ less than g̃cr ≈ 2.295. the soliton disappears.
∗
• At this critical coupling is (ψi,j ψi,j
)max ≈ 0.227.
• This agrees with our numerical study of the continuous MNLS
equation (g̃cr ≈ 2.94) and the analytic study which gave g̃cr =
π.
• Confirmed also the independence of our results of the form
of the initial settings (used eg a starting configuration with
two exponential-like excitations being located at i = 78 − 83,
j = 3 − 7 and i = 138 − 143 and j = 13 − 17),
• Have found that for values g̃ > 3 all results agree (always one
soliton) but the time needed to reach it depends on the initial
configuration.
• Have tested also the dependence of these results on the size of
the lattice - finding no dependence (except when solitons are
very broad).
• Have also looked at solitons for smaller values of g̃ and found
several regimes
– region of one soliton - narrow
– regime of one soliton - broad
– regime of soliton ring
– regime of no solitons
5.2.3
Comparison of results
• Compare two systems and find that g̃ is given in terms of the
coupling constants
g̃ =
2 c2x
9 M j x kx
which, for our choice of coupling constants, gives:
g̃ =
c2x
90
.
• So a critical value of cx ≈ 20 implies g̃cr ≈ 4.4.
• Discrepancy? There might exist additional terms A for which
∆(1)A = 0 and/or ∆(2)A = 0.
• such terms would change the comparison of the solutions.
• Difficult problem - under study.
• Have also looked at other nanotubes (different chiralities) no
substantial difference - work in progress.
6
Conclusions
• Have studied a model of nanotubes - based on a 2-dimensional
hexagonal lattice periodic in x and y directions with a large
extention in one (the x) direction.
• Have found that nn the stationary limit, the full system of
equations in which the electron excitation is coupled to the
displacement fields of the lattice can be replaced by a modified
discrete non-linear Schrödinger (DNLS) equation.
• Such a replacement is not possible for square lattices etc.
• Concentrated on determining the value of the critical phononelectron coupling constant above which soliton solution exist.
• For DNLS have found a unique solution (except close to the
critical value of g̃)
• We do understand this critical value quite well.
• For the full system of equations we have found a number of
solutions for each choice of the coupling constants.
• The critical value of the electron-phonon coupling is of the
same order of magnitude as in the case of the DNLS.
• Have speculated on the origin of a small discrepancy between
both critical values.
• Have also started looking at a genuine 3 dim nanotube (with
3 deformation fields) - work in progress.
• Have also looked at a field on a circle (intermediate case); again
we see solitons
• ....????