Supporting Information
1. The scanning electron microscope (SEM) operations on carbon
nanotubes (CNTs)
The SEM examination of CNTs #1-7 was carried out using the SEM mode of
Carl Zeiss Nvision 40 and its in-lens detector exclusively for secondary electrons
(SEs). The accelerating voltage of the electron beam (e-beam) was 1 kV. The pressure
in the SEM sample chamber was around 1×10-6 mbar. The working distance was
adjusted to make sure that the leakage current, i", of any oxide area without any CNTs
was around zero. For each SE measurement, the following process was used to assure
that any influence of imaging history was avoided. Initially, an unscanned fresh
segment of a measured CNT was located quickly with a fast scanning and a low
magnification. Then, we increased quickly the magnification to get a desirable value
of the size, L, of the scanned area, and the e-beam was blanked instantly. Only after
the reading of the current meter returned to zero, we unblanked the e-beam and started
to measure the leakage current, i', with a moderate scanning speed. After this, the
same process was repeated on an unscanned fresh oxide surface area near the CNT
and without any CNTs. i", L and i' are defined by Eq. (1) in the main text.
2. The SEM imaging mechanism of CNTs
In SEM, the e-beam scans any sample surface point by point, and the signal
from an SE detector is used to form an image. SEM is a powerful instrument to image
individual CNTs.1-6 Figure S1 shows that the full width at half maximum of the
intensity profile of the SEM image of CNT #1 before breaking was about 86 nm,
much larger than its real diameter of 1.48±0.09 nm measured by atomic force
microscope (AFM). In order to analyze this, Fig. S2 illustrates the interaction between
1
the e-beam and the samples, where the amorphous carbon (aC) covering the oxide
surface is generated by the decomposition of residual hydrocarbon gases in vacuum
induced by the illumination of the e-beam and its thickness is around 1 nm measured
by AFM. The aC and the CNT are both very thin and light, and so the high-energy ebeam penetrates through them without much backscattering. However, a portion of
the incident electrons will be backscattered by the oxide, as shown by the pink arrows
in Figs. S2(d-f). Both the e-beam and the beam of the backscattered electrons (BSEs)
can excite the bombarded samples to emit SEs.7,8 Generally, these two beams have
larger diameters than CNTs, as illustrated in Fig. S2. Only a fraction of the electrons
of the two beams reach the CNT. Thus, when the e-beam hits a point in Fig. S2(a) or
S2(b) or S2(c) and dwells on it, the current balance gives:9
I b  I SE  I BSE  i
(1)
I SE   SiO2 I b   aC ( I b  I BSE )   CNT ( I b CNT  I BSECNT )
(2)
I BSE  SiO2 I b
(3)
where Ib, ISE and IBSE denote the currents of the e-beam, the SEs and the BSE beam,
respectively. i denotes the leakage current flowing through the sample to ground. δ
and η denote the SE and the BSE yields, respectively. Ib-CNT and IBSE-CNT denote the
currents of the fractions of the e-beam and the BSE beam hitting the CNT,
respectively.
2
FIG. S1. Intensity profile and its fitting of SEM image (inset, cut from an intact image) of
CNT #1 before it broke. The profile is the average of six profiles taken at six different
positions in the SEM image. The transversal black line in the inset shows a profiling position.
The SEM image in the inset was collected under the condition that the accelerating voltage of
the e-beam was 1 kV and no charge existed in the oxide surface.
FIG. S2. Schematic images for the interaction between the e-beam and the samples at three
different moments, where the distance between the CNT axis and the centre of the e-beam
spot changes from zero in (a) to d0 in (b) and then 2d0 in (c). d0 is the size of each scanned
point on the oxide surface and the points are represented by the blue squares. The yellow
arrows in (a) denote the scanning sequence of the e-beam, which runs from left to right and
from top to bottom. The semitransparent green circles denote the amorphous carbon. The
large black rectangle denotes the CNT. The red circle denotes the e-beam spot. The pink
circle denotes the BSE beam spot. (a-c) Top views. (d-f) Cross sections corresponding to (a-c),
respectively. The cross-cutting position is shown by the dashed white line in (a). The CNT is
electrically connected with the Si part.
The magnitudes of Ib-CNT and IBSE-CNT are determined by the distance between
the CNT axis and the e-beam spot and the density profiles of the e-beam and the BSE
3
beam, which can be described by the Gaussian distribution.10,11 So, the densities, Jb
and JBSE, of the two beams are given by:
x2  y2
J b  J b 0 exp( 
)
wb2
J BSE  J BSE0 exp( 
(4)
x2  y 2
)
2
wBSE
(5)
where Jb0 and JBSE0 are both constants, x and y are the coordinates along the x and the
y directions in Fig. S2(a), and wb and wBSE are the Gaussian radii of the e-beam and
the BSE beam, respectively. The e-beam and the BSE beam spots have the same
centre,7 as shown in Fig. S2. The centre is used as the origin of the x, y coordinates.
wBSE is generally much larger than wb.7,11 Moreover, there are equations between the
currents and the densities as
Ib  
 

J b dxdy  wb2 J b 0
 
I BSE  
 

 
(6)
2
J BSEdxdy  wBSE
J BSE0
(7).
Because the axis of the CNT is along the y direction, Ib-CNT and IBSE-CNT are
given as
I b CNT  
 x  r

 x  r
I BSECNT  
 x  r

(8)
J b dxdy
 x  r
(9)
J BSE dxdy
where r is the radius of the CNT and x is the distance between the CNT axis and the
centre of the e-beam spot. When wb is much larger than r, Eqs. (8, 9) evolve to
I b  CNT 
I b dCNT
x2
exp(  2 )
wb
 wb
I BSE  CNT 
(10)
I BSE dCNT
x2
exp(  2 )
wBSE
 wBSE
(11)
4
where dCNT is the CNT diameter and equal to 2r. Thus, Eq. (2) can be rewritten as
I SE   SiO2 I b   aC ( I b  I BSE )   CNT ( I b CNT  I BSE CNT )
 I 0   CNT ( I b CNT  I BSECNT )
I BSE d CNT
x2
x2
 I 0   CNT [
exp(  2 ) 
exp(  2 )]
wb
wBSE
 wb
 wBSE
I b d CNT
 I 0   CNT [
 I0  Ib
I b d CNT
 wb
exp( 
 SiO2 I b d CNT
x2
x2
)

exp(

)]
2
wb2
wBSE
 wBSE
 CNT d CNT
  d
x2
x2
exp(  2 )  I b CNT SiO2 CNT exp(  2 )
wb
wBSE
 wb
 wBSE
(12)
where I0 denotes the contribution from SiO2 and aC. The second and the third terms
are contributed by the SEs from the CNT excited by the e-beam and the BSE beam,
respectively. In SEM instrument, only a portion of the SEs can reach the SE detector.
So, the current ISE-d of the SEs reaching the detector is
I SE  d  C 0 I 0  C1 I b
 CNT d CNT
  d
x2
x2
exp(  2 )  C 2 I b CNT SiO2 CNT exp(  2 )
wb
wBSE
 wb
 wBSE
(13)
where C0-2 denote the ratios of the SEs reaching the detector. Because the detector
converts this portion of SEs to an imaging signal with the intensity proportional to the
amount of the SEs in this portion, the total signal intensity S of the point hit by the ebeam and with the distance x from the CNT axis is
S  C 3 I SE  d
 CNT d CNT
 CNT  SiO2 d CNT
x2
x2
 C 3 C 0 I 0  C 3C1 I b
exp(  2 )  C 3 C 2 I b
exp(  2 )
wb
wBSE
 wb
 wBSE
(14)
where C3 denotes the conversion ratio. For simplicity, Eq. (14) can be rewritten as
B1
B2
x2
x2
S  S0 
exp(  2 ) 
exp(  2 )
wb
wBSE
wb
wBSE
(15).
5
The first, the second and the third terms are the contributions from the SEs of
SiO2 and aC, the ones of the CNT excited by the e-beam and the ones of the CNT
excited by the BSE beam, respectively. Because the e-beam scans any sample surface
point by point, we get
x  nd 0
(16)
where n is the amount of the scanned points along the x distance and d0 is the size of
each point.
Equations (15, 16) fit well with the intensity profile in Fig. S1, giving
wb=4.2±1.5 nm and wBSE=53.9±0.7 nm. In order to confirm the validity of the above
analysis, the value of wb was measured independently under the same SEM conditions
by a sharp edge method.12 Figures S3(a-c) show a membrane windown grid. This grid
is actually a Si chip with a window at its centre. A Si3N4 membrane with low stress,
the root mean squared roughness of 0.2 nm and the thickness of 50 nm is suspended
across the window. Although there were pores produced by the manufacturer in the
membrane, we used the focused ion beam (FIB) of Carl Zeiss Nvision 40 to drill new
square holes in the membrane. Because the angle between the e-beam and the FIB
was known, we tilted the grid with the membrane to acquire that the edges of the
square holes were exactly parallel with the incident direction of the e-beam. After this,
firstly we measured the leakage current from an area of the membrane without any
holes to ground and found it to be 0±49 fA, exactly at the zero point. This indicated
that the membrane was completely opaque to electrons and no charge existed on it.
Then we increased the magnification quickly and collected the SEM image of the
edge of a square hole, as shown in the inset of Fig. S3(d). From the intensity profile of
a line across the edge image, we measured the value of w, the distance between the
two points corresponding to 75% and 25% of the highest intensity, as shown in Fig.
6
S3(d). Because wb = 1.045w,12 the value of wb was obtained as 6.7±0.7 nm, close to
the value of 4.2±1.5 nm from the fitting within the error bars. This consistency was
repeated on the two sections of CNT #1 after breaking and a second CNT (CNT #2),
as shown in Table I. These indicate that the above theoretical analysis is consistent
with observations. We conclude that the increase in diameter observed in SEM images
of CNTs compared to their true diameters is caused by the Gaussian density profiles
of e-beams and BSE beams. It is noted that the above theoretical analysis and
conclusion are also applicable to CNTs unconnected with electron reservoir, because
the difference between the cases of the unconnected and the connected CNTs is only
that the values of the leakage currents are different due to the different flowing paths
of the leakage currents. This applicability is proved by the data of the unconnected
sections of CNTs #1 and 2 in Table I.
FIG. S3. Measurement of wb of the e-beam by the sharp edge method. (a) Top-view SEM
image of a membrane window grid purchased from the company of SPI Supplies, where there
is a Si3N4 membrane suspended acorss the window in the centre. (b) Top-view close-up of the
Si3N4 membrane, where the round pores were produced by the company and the square hole
was made by the focused ion beam (FIB) of Carl Zeiss Nvision 40. (c) Cross-sectional
7
schematic (not to scale) of the grid, where the square hole is not shown. (d) Intensity profile
of the blue line across the edge of the square hole in the SEM image in the inset. w is the
distance between the two points on the blue line, which are corresponding to 75% and 25% of
the highest intensity, respectively.
Table I. Data of the analyses on the SEM imaging mechanism of the CNTs.
CNT#
dCNT (nm)
1
1
1
2
2
1.48±0.09
1.48±0.09
1.48±0.09
2.5±0.3
2.5±0.3
wb (nm) from the
fitting
4.2±1.5
2.6±0.7
2.5±1.2
12.3±2.4
5.8±3.7
wb (nm) from the
edge measurement
6.7±0.7
4.7±0.6
4.6±0.7
11.0±1.5
9.0±0.6
Connected
with Si
Yes
Yes
No
Yes
No
Notes
Before breaking
After breaking
After breaking
After breaking
After breaking
3. Theoretical analysis on the SE yields of CNTs
When the e-beam hits a point and dwells on it, Eqs. (1-3) give:
I b   SiO2 I b  SiO2 I b   aC ( I b  I BSE )   CNT ( I b CNT  I BSECNT )  i
 I 0   CNT ( I bCNT  I BSECNT )  i
(17)
I 0   SiO2 I b  SiO2 I b   aC ( I b  I BSE )
(18).
So, during the dwelling of the e-beam on this point, the charge flowing
through the sample to ground is
q  it d
(19)
where td is the dwell time of the e-beam on the point and equal to 100×2p-1 ns. p is the
ranking number of the nominal scanning speed in SEM and can be selected among the
integers of 1~15. Thus, the total charge flowing through the sample to ground during
scanning a whole intact SEM image is
Q   (it d )  td  i
x, y
(20).
x, y
An intact SEM image consists of M (x-directional) × N (y-directional) scanned
points. Because the axis of a CNT is along the y direction, as shown in Figs. S2(a-c), a
8
whole intact SEM image can be divided into N identical lines parallel to the x
direction and perpendicular to the y direction. Thus, Eq. (20) can be rewritten as
Q  Nt d  i
(21).
x
If td is too long in experiments, obvious drift of sample, overcharge and
deposition of too much aC will happen in taking SEM images. A short td is used,
which does not leave enough time to measure i, however, the average leakage current
i' from scanning a whole intact SEM image can be measured. We can get
Q  i T  i   M  Nt d
(22)
where T is the period of scanning a whole intact SEM image. If no CNT exists in an
intact SEM image, Eq. (17) becomes
I b  I 0  i
(23).
where i" is the leakage current flowing through the sample to ground when no CNT
exists in the SEM image. Based on Eqs. (3, 10, 11, 16, 17, 21, 22, 23), we get
 CNT 
M (i   i )
 ( I bCNT  I BSECNT )
x

M (i   i )
I b d CNT
I BSE d CNT
x2
x2
x [  w exp(  w 2 )   w exp(  w 2 )]
b
BSE
b
BSE

i   i 

Ib

i   i 

Ib
M 

1
x2
x2
d CNT  [ exp(  2 )  SiO2 exp(  2 )]
wb
wBSE
wb
wBSE
x
M 
M
2
d CNT
1
 [w
n
M
2
b
exp( 
(24).
n d

n d
)  SiO2 exp(  2 )]
2
wBSE
wb
wBSE
2
2
0
2
9
2
0
This equation can be simplified further. When wb >> d0, 
Md 0
Md 0
…
are
2wb
2wb
quasi-continuous. Thus, we can get
M
2
M
n 2 d 02
n 2 d 02
1
1
2 [
[
exp(

)]

exp(

)]dn
M w
2
2
 M2 wb
w
w
b
b
b
n
2
Md 0
2 wb
Md 0

2 wb



d0
[
d n
1
exp( V 2 )]dV , V  0
d0
wb
erf (
Md 0
)
2 wb
(25).
where erf is the Gauss error function. When M/2 >> wb/d0, we further get

d0
erf (
Md 0


)
erf () 
2wb
d0
d0
(26).
This mathematical process can be also used for
M
2

M
n
2
[
1
w BSE
exp( 
n 2 d 02
2
w BSE
)] . Thus,
we can simplify Eq. (24) as
 CNT 
Md 0
i   i 
i   i 
L
, L  Md 0



Ib
d CNT (1   SiO2 )
Ib
d CNT (1   SiO2 )
(27),
where L is the x-directional size of the scanned area on the sample. This equation’s
condition is L/2 >> wb >> d0. We’ve used the experimental data to check Eqs. (24, 27)
and found that they are exactly equal to each other in the ranges of our experimental
data. In fact, it is found by our data that even when wb ≈ d0, Eqs. (24, 27) are also
close to each other.
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