NanoVision 2020

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PHYS 2235 Physics of NanoMaterials S. J. Xu D
epartment of Physics
(Lecture 8)
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Nanotubes and Nanowires

• Contents of Lecture 8
• Quantum Size Effect of Nanotubes and Nanowires
• Imaging Electron Wave Functions of Carbon
  Nanotubes
  
• Single-Electron Transistors Based on
  Nanomaterials
  
• References

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Quantum Size Effect in Nanotubes and Nanowires
Nanostructured Materials
Quantum Size Effect
Example 1 Carbon Nanotubes
In 1991, Iijima discovered carbon nanotube.
Nature 354, 96 (1991). Shortly later, theoret
ical studies predict that carbon nanotubes can be
metallic or semiconducting depending on their
diameter and the helicity of the arrangement of
graphitic rings in their walls. Scanning
tunneling microscopy (STM) offers the potential
to probe this prediction, as it can resolve
simultaneously both atomic structure and local
electronic density of states.
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The diameter and helicity of a defect-free SWNT
are uniquely characterized by the vector C na1
ma2 (n, m) that connects crystallographically
equivalent sites on a two-dimensional graphene
sheet, where a1 and a2 are the graphene lattice
vectors and n and m are integers. Two limiting
cases (n, 0) zigzag and (n, n) armchair tubes.
Fig.1. Schematic of SWNT structure and several
STM images.
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Calculations predict that armchair (n m)
tubes have bands crossing the Fermi level and are
therefore metallic. For all other tubes (chiral a
nd zigzag) there exist two possibilities. When n
- m 3l (where l is an integer), tubes are also
expected to be metallic. In the case
, tubes are predicted to be semiconducting with
an energy gap of order of 0.5 eV. This gap
should be only dependent on the diameter, that
is,
Fig. 2a. I-V curves b. dI/dV spectra c. the gap
vs. tube diameter. The solid line is fitting
curve.
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where ?0 is the C-C tight-binding overlap energy,
ac-c 0.142 nm the nearest neighbour C-C
distance, and d the diameter.
Figure 2a shows a selection of IV curves
obtained by STS on different tubes. Most curves
show a low conductance at low bias, followed by
several kinks at larger bias voltages. From all
the chiral tubes that we have investigated, we
can clearly distinguish two categories the one
has a well defined gap value around 0.50.6 eV,
the other has significantly larger gap values of
1.72.0 eV (see Table 1 and Fig. 2b). The gap
values of the first category coincide very well
with the expected gap values for
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semiconducting tubes. As illustrated in Fig. 2c,
which displays gap versus tube diameter, the
measurement agrees well with theoretical gap
values obtained for an overlap energy ?0 2.7
0.1 eV, which is close to the value ?0 2.5 eV
suggested for a single graphene sheet.
The very large gaps that we observe for the
second category of tubes, 1.72.0 eV, are in good
agreement with the values 1.61.9 eV that we
obtain from one-dimensional dispersion relations
for a number of metallic tubes (n - m 3l ) with
diameter of 1.4 nm.
In addition, sharp van Hove singularities in
the DOS are predicted at the onsets of the
subsequent energy bands, reflecting the
one-dimensional character of carbon nanotubes. Th
e derivative spectra indeed show a number of peak
structures (Fig. 3).
Fig.3. dI/dV spectra of SWNT.
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Example 2 Silicon Nanowires
Silicon nanowires are another outstanding example
of one dimensional
Fig.4. STM image and structure schematic of
single Si nanowire.
nanomaterials. Figure 4 shows STM image of
individual Si nanowire. Si is a classic
semiconducting material which fundamentally
supports modern IT industry. Si nanowires are
particularly expected to play an important role
in nanotechnology rapidly emerging. They also
provide us good chance to test quantum size
effect in nanomaterials.
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Si Nanowires
Fig. 5A. I-V curves of individual Si nanowires
B. dI/dV spectra C. Deduced band gap versus
diameter and some theoretical values.
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Imaging Electron Wave Functions of Carbon
Nanotubes
Electrons in carbon nanotube can only
propagate along the tube axis due to the quantum
confinement in the radial and circumference
directions. Carbon nanotubes therefore are
interesting systems for studying the quantum
behavior of electrons in one dimension. Limiting
the length of a carbon nanotube leads to a
particle-in-a-box quantization of the energy
levels.
Fig.6A. Example of a carbon nanotube shortened.
B. Atomically resolved image of an armchair
carbon nanotube. The arrow denotes the tube axis.
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As shown in previous sections, the armchair
nanotubes should behave like metal wires.
Figure 7A depicts the measured tunneling
current-voltage curves at 4 different positions
on the armchair nanotube shown in Fig. 6B. Notice
that there are step-like structures appearing in
the I-V characteristics. These current step
structures correspond to the quantized energy
levels of electrons. The energy positions of
these quantum levels are given by
Fig.7A. Measured I-V characteristics at different
positions on a shortened armchair nanotube shown
in Fig.6B B. Corresponding dI/dV curves C.
dI/dV as a function of position along the tube
axis.
where EF is the Fermi level, e the electron
charge, and a 1.
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The experimentally observed width of the
current plateaus between steps ranges 0.05 to
0.09 V. The plateau width is determined by the
total energy required to add an electron to the
tube. This addition energy consists of a
combination of finite-size level splitting and
the Coulomb charging energy that is due to the
small capacitance of the tube. A simple estimate
for the energy level splitting for a tube with
length L30 nm is given by ?Eh?F/2L0.06 eV,
where ?F 8.1105 m/s is the Fermi velocity and h
is the Plancks constant. The capacitance C of a
nanotube lying on a metallic substrate can be
approximated by the formula
where e0 is the vacuum dielectric constant, d the
distance from the tube axis to the plane of
substrate, R the tube radius. If R0.65 nm, d0.9
nm, then C 2.0 aF. The charging energy
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Notice that the height of dI/dV peak varies
periodically with the position x along with the
tube axis, as shown in Fig.7C. The period of
these oscillations is 0.4 nm, which clearly
differs from the lattice constant of 0.25 nm.
The periodic variation of dI/dV versus x
reflects the spatial distribution of electronic
wave functions in the nanotube.
The wave functions of several adjacent energy
levels can be displayed simultaneously by
plotting the differential conductance dI/dV as a
function of voltage V and position x along with
the tube axis, as shown in Fig.8A.
Fig.8A. dI/dV against bias voltage and the
position B. dI/dV profiles of the four resolved
electron energy levels C. Topographic height
profile z(x) along the tube.
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The experimental quantity dI/dV is a measure
of the squared amplitude of the quantized
electron wave function . The dashed
curves in Fig. 8B are the fitting lines with a
trial function of the form
Notice that the separation of 0.4 nm between
peaks in the experimental dI/dV curve corresponds
to a half the wavelength ?because dI/dV measures
the square of the wave function. The dashed
curves in Fig.8B are the fitting lines with a
trial function of the form.
A short metallic carbon nanotube resembles the
textbook model for a particle in a box!
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Carbon Nanotube Single Electron Transistors
Fabrication of single electron transistors being
able to work at room temperature is a DREAM of
scientists!
Room-temperature single-electron transistors
(RTSET) are realized within individual metallic
single-walled carbon nanotube molecules. The
devices feature a short (down to 20 nanometers)
nanotube section that is created by inducing
local barriers into the tube with an atomic force
microscope. Coulomb charging is observed at room
temperature, with an addition energy of 120
millielectron volts, which substantially exceeds
the thermal energy.
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Fig.9. Fabrication of a room-temperature
single-electron transistor within an individual
metallic carbon nanotube by manipulation with an
AFM. (A) Nanotube between Au electrodes on top of
a Si/SiO2 substrate with a gate-independent
resistance of 50 kilohm. After imaging by
scanning the AFM tip over the sample in tapping
mode, the tip is pressed down onto the substrate
and moved along the path indicated by the arrow,
thus dragging the nanotube into a new
configuration. Bar, 200 nm. (B) Nanotube after
creation of a buckle. The dragging action has
resulted in a tube that is bent so strongly that
it has buckled. A second dragging action is
performed as indicated by the arrow. (C)
Double-buckle nanotube device. (D) Enlarged image
of the double-buckle device. The image shows a
height increase at the buckling points, as
expected.
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Fig. 10. Differential conductance dI/dV of the
RTSET as a function of bias and gate voltage at
various temperatures. (A) At 300 K, the
differential conductance shows a thermally
smeared gap around V 0 with gate voltage Vgate
-0.7 V (lower trace). When the gate voltage is
changed to -0.8 V, the gap is closed.
(B) Conductance oscillations as a function of
gate voltage at 260 K. (Inset) dI/dV in an intens
ity plot. Blue represents low dI/dV, red correspo
nds to high dI/dV. The gap is periodically opened
and closed as a function of the gate voltage,
which results in diamond-shaped modulations.
(C) dI/dV at 30 K, showing distinct peaks as
indicated by
the lines. The peaks in the black trace at Vgate
-1 V shift up in bias voltage when Vgate is
increased to -0.95 V (red trace). (D) Gray-scale
image of dI/dV, where shifting peaks are
indicated by the dashed lines. White represents
dI/dV
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0, whereas darker shading correspond to higher
values of dI/dV. (E) Gray-scale image of d2I/dV2,
showing the presence of conductance peaks.
Fig.11. (A) Zero-bias conductance versus gate
voltage for a single conductance peak. Data are
shown for T 4, 20, 30, 40, 70, 80, and 90 K.
The solid lines are fits to the function Gmax(T
)/cosh22(arccosh )Vgate/w(T). The T
dependence of the conductance maximum Gmax and
the full width at half maximum w is shown in Fig.
12. (B) Multiple conductance peaks on a
logarithmic scale for an extended range of gate
voltages at T 4, 40, 60, 70, 100, 170, 200,
220, and 240 K.
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Fig. 12. Power-law temperature dependence of the
conductance, demonstrating correlated sequential
tunneling through the nanotube SET device. Lower
data (right-hand scale) show the peak height
Gmax(T ) for the conductance peak in Fig.11A,
following a power-law function with exponent
0.68 (?). The conductance integrated over the
gate voltage range in Fig. 3B, G(T ) (left-hand
scale), also follows a power-law function with
exponent 1.66 (?). Note the double-logarithmic
scales. The inset shows the peak width w versus
T, which displays a linear behavior.
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References

• J. W. G. Wildöer, L. C. Venema, A. G. Rinzler, R.
  E. Smalley C. Dekker, Nature 391, 59 (1998).
  
• D. D. D. Ma, C. S. Lee, F. C. K. Au, S. Y. Tong,
  S. T. Lee, Science 299, 1874 (2003).
  
• L. C. Venema, J. W. G. Wildöer, J. W. Janssen, S.
  J. Tans, H. L. J. Temminck Tuinstra, L. P.
  Kouwenhoven, C. Dekker, Science 283, 52 (1999).
  
• H. W. Ch. Postma, T. Teepen, Z. Yao, M. Grifoni,
  C. Dekker, Science 293, 76 (2001)
  

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