Transcript No Slide Title

Patterns of field electron emission
from carbon nanotubes
J. Pengx , C. J. Edgcombe*, V. Heine*
xSun
Yat-Sen University, Guangzhou, P.R. China
*Dept of Physics, University of Cambridge, UK
Introduction
Experiment:
Field emission microscopy of carbon nanotubes (CNTs)
+1000 V
vacuum
few nm
0V
CNT
~10 mm
V
W
E
s
barrier
The pattern observed on screen
sometimes1,2,3 consists of sets of
dark-centred rings that together
form a pentagon or hexagon, with
fringes between adjacent rings.
Fig. 1 Hata et al (Ref. 3)
Explanations have been proposed for the fringes2,3,4. The fact
that the centres of the rings are dark, when simple
interference theory suggests they should be bright, does not
seem to have been explained hitherto.
Pentagons in the cap
Formation of a closed cap on a carbon nanotube
requires the presence of at least 6 pentagons of
carbon atoms. The direction of the surface normal
changes more rapidly near these pentagons than
over the hexagons forming the rest of the CNT.
Hence the local field and the induced charge density
are greatest, and the potential barrier is thinnest, near
the pentagons. Where the barrier is thinnest, the
wave function is attenuated least. So the current
density emitted from pentagons is likely to be much
greater than from the rest of the CNT structure.
Fig. 2 Hata et al
(Ref 3)
We interpret the individual rings seen in Fig. 1 as coming from different
pentagons in the cap, distributed approximately as in Fig. 2. In a real CNT,
the positions of pentagons may be distributed without exact symmetry.
Why does each ring have a dark centre ?
The pairs of dark bands seen between adjacent rings
suggest interference between waves associated with the
pentagons. Thus the emission from adjacent pentagons seems
coherent. Then we can expect wavefronts from all parts of a
pentagon to travel the same distance to any point on its axis, so we
might expect wavefronts to reinforce at the axis and produce the
strongest current density on the axis.
But the image shows that the centres of all rings are darker than
the surroundings – that is, the current density on each ring axis is
smaller than the density off-axis. How does this happen ?
Band structure for pentagons
Each pentagon has bonds to surrounding atoms, but to simplify
the description, the effect of neighbouring structure is ignored
here; only the wave function of a pentagon is considered here.
Symmetry and theory for a simple ring suggest that the
wavefunctions of p orbitals on a pentagon have a regular phase
variation around the axis that is non-zero above the ground state.
For states varying as exp (imf), the relation between orbital
energy and m is as shown in Fig. 3.
In the absence of applied field,
each pentagon has
5
pelectrons which fill the m = 0
states and some of the m = ±1
states of the possible orbitals.
States described by exp ±imf
Occupation
can be combined to give real
functions cos mf and sin mf.
Fig. 3 Band structure for pentagon
Emission from the states m = ±1
and occupation of levels
comes from a superposition of
these functions and so the
density is independent of f.
Electrons can tunnel from states m = 0 and m = ±1, but because the m
= 0 state is lower in energy, the potential barrier seen by an electron
with m = 0 is both wider and higher than that seen by an electron with
m = ±1. The density of m = 0 electrons is thus much more heavily
attenuated by tunnelling than the m = ±1 density. The stream emerging
from the barrier can thus be expected to contain mainly electrons with
m = ±1.
1.5
-cos(2pi*m/5)
1
0.5
0
-0.5
-1
-1.5
-3
-2
-1
0
m
1
2
3
Calculated electron density at the CNT apex
In general it’s difficult to calculate a steady-state distribution of charge
outside the CNT with DFT codes, since they show correctly that, under
realistic emission conditions, the charge all falls to the anode.
The code ONETEP allows charge to be confined by use of Wannier
functions. This can represent the relatively high charge density in the barrier
region while ignoring the smaller density outside it.
Fig. 4 Surfaces of constant induced charge density (increasing l - r) for
(18,0) nanotube with 6 pentagons at outer circumference, in applied field of
28 V/mm, calculated by DFT code ONETEP (Ref.5).
Analysis using the complex wave function shows that the phase
does not change much in the barrier region. Hence the relative
phase of emission from different parts of the pentagon persists
during an electron-wave’s travel through the potential barrier at the
tube tip.
V
W
E
s
barrier
Beyond the barrier the phase changes with k0 = (2emV(s))1/2/ h .
From potential barrier to anode
Electron-wave propagation from the outside of the potential barrier to
the anode can be described (1) approximately by optical theory; (2)
approximately by ballistic theory; or (3) by solving Schrödinger’s
equation.
Optical theory is approximate since it assumes constant k0 = w / c.
Realistically, the electrons start with low speed and have k0 ~
V(s)1/2, so their phase evolution differs from the optical model.
Numerical solution of Schrödinger’s equation for realistic geometries
is inaccurate, because the anode-cathode distance is very large
compared with the required step length.
However, Schrödinger’s equation can be solved analytically for some
simple geometries.
Propagation from a ring in planar geometry
We can find an analytic solution for a simplified geometry consisting
of a circular emitting ring of radius a , in a planar cathode surface,
subject to a uniform accelerating field F. The solution can be found
for a general f-periodicity of the form eimf .
r
2a
O
axis
s
F
Fig. 5 Emitting ring in planar geometry.
The source is assumed to be y = d(r - a)
at s = 0.
Assume
y = R(r) Z(z) exp imf ,
with z = k2b2 - (s - s0) / b , b = (h2/ 2 m e F)1/3 ~ 3 Å , s0 = W / e F, W =
work function and F = electric field.
The general solution of S’s eqn can be written in the form
y = eimf  c(k) (Bi(z) + i Ai(z)) Jm(k r) k dk
Here k is the phase constant for transverse (not axial) variation. The Airy
functions Ai & Bi give well-defined solutions through V(s) - E = 0.
Now y has the form of a Hankel transform, so can be inverted to define
c(k), if the source distribution is known. With the assumed source
condition y = d(r - a) at s = 0 (z = k2b2 + s0 / b),
imf
y  ae

 J m (ka J m (kr 
0
Bi(z   iAi(z 
Bi(  iAi( 
K02
K02
k dk
k is the phase constant for transverse (radial) variation.
Only a small range of the radial phase constant k near zero
contributes substantially to the result. This provides some
justification for ignoring transverse variation in the conventional
WKB approximation for the current transmitted through the barrier.
The resulting amplitude of y at the anode varies with r as
 (2emV ( s ) 1 / 2 a r 

Jm


2
h
s


where m is the f-periodicity at the source, s is the anodecathode spacing and V(s) is the voltage at s. Clearly y = 0 at r = 0
for all non-zero values of m .
The radial scale of the current density pattern at the anode
increases as s and as V(s)-1/2. The result is like that for a ballistic
particle with constant transverse velocity in a uniform axial field, but
broadened.
So, when emission at the cathode is from a hollow ring,
(1) when there is no phase variation around the emitting ring
(m=0) then on axis (r = 0) the waves add in phase to give
maximum charge density;
(2) when the phase around the emitting ring varies with non-zero
m, the charge density on axis at the anode is zero;
(3) the pattern of current density distribution becomes broader as
it moves away from the cathode.
The observed pattern at the screen, with low intensity at the
centres of rings, shows that little emission comes from m = 0
orbitals. The observed distribution seems consistent with m = 1
emission (transverse variation as J1(const . r)) .
Why is there coherence between pentagons ?
In a CNT tip, the pentagons are separated by many bond lengths,
and with such a large separation, it is not obvious that their wave
functions should have a definite phase relation.
However, the
calculation of transit above shows that the principal contribution to
the observed current comes from waves with very small transverse
momentum, that is with k ~ 0. This means that electrons of the
same energy will have the same axial momentum (at equal values
of potential) and so phase relations existing on exit from the
potential barrier will be preserved on transit to the anode. This may
provide sufficient coherence for interference to be observed.
Some results of optical modelling
(a) experimental current density
distribution (Ref. 4);
(b) result of modelling using
optical rays, with single phase
at source (Ref. 4);
(c) result of modelling with
optical rays, with one cycle of
phase around each circular ring
at source.
Conclusions
1. For a simple pentagon, available states have phase factors
exp ±imf,
where f is the azimuthal angle around the axis of the pentagon.
Tunnelling and field emission can occur from states with m = 0 and ±1, but
the current from m = 0 is more strongly attenuated by the larger barrier
seen because it has lower energy. Thus the observed emission pattern is
dominated by electrons from orbitals with phase factors exp ±if (or cos f
and sin f).
2. The wave function calculated for charge accelerated from a ring in
planar geometry shows a transverse variation at the anode as
Jm(const x a r Va1/2 / s). Thus (in planar geometry) current from orbitals
with m = 0 would have greatest density at r = 0, but current from orbitals
with any other value of m has zero density on the axis.
3. The observed lower density at the centres of rings observed in Refs. 1
and 3 is consistent with emitted current coming mainly from orbitals with
phase factor exp ±if, as expected for the Fermi level of a simple
pentagonal ring.
Acknowledgments
Thanks to Prof. R. Haydock and Dr. S. Kos for valuable
discussions.
References
1. Saito Y, Hata K & Murata T, Japan J Appl Phys 39 L271 (2000)
2. Oshima C, Matsuda K, Kona T, Mogami Y, Komaki M, Murata Y, Yamashita
T, Kuzumaki T & Horiike Y, Phys. Rev. Lett. 88 038301 (2002)
3. Hata K, Takakura A., Miura K, Ohshita A and Saito Y., J Vac Sci Tech B
22, 1312 (2004)
4. Kruit P, Bezuijen M & Barth J E, J Appl. Phys. 99, 024315 (2006)
5. Skylaris C-K, Haynes P, Mostofi A & Payne M, J Chem Physics 122,
084119 (2005)