J. Phys. Chem. B 1999, 103, 8671-8674
8671
Interference-Modulated Conductance in a Three-Terminal Nanotube System Gabin Treboux,*,† Paul Lapstun,‡ Zhanghua Wu,‡ and Kia Silverbrook‡ National Institute for AdVanced Interdisciplinary Research, 1-1-4 Higashi, Tsukuba-shi, Ibaraki 305, Japan, and SilVerbrook Research, P.O. Box 207, Balmain 2041, NSW, Australia ReceiVed: April 14, 1999; In Final Form: July 14, 1999
We study ballistic conductance in a three-terminal carbon nanotube system and show that conductance between two of the terminals can be controlled via the third terminal. The conductance is modulated by tuning the energy spectrum of the third terminal. Such a system may provide the core of a nanotransistor. We show more generally that the quasi one-dimensional nature of nanotubes can persist in branched topologies and that insights gained in analyzing one-dimensional systems can therefore be applied in analyzing networks of nanotubes.
Carbon nanotubes have generated a great deal of interest since their discovery in 1991 by Iijima.1 A wide variety of nanotube configurations have subsequently been experimentally observed and theoretically and computationally analyzed. The strong interest in nanotubes stems from their extraordinary mechanical and electronic properties. Ballistic transport has been experimentally demonstrated in carbon nanotubes,2 providing the basis for using nanotubes as efficient metallic wires. Rectification has also been experimentally observed in nanotubes,3 providing the first step toward realizing a nanoelectronic diode. Here, we study ballistic conductance in a three-terminal nanotube system and show that conductance between two of the terminals can be controlled via the third terminal. Such a system may provide the core of a nanotransistor. We decompose the system as finite or semi-infinite terminals joined via a central spacer. To predict the conductance of such systems, the algorithm recently proposed by Chico et al.4 has been implemented. Let us examine the configuration composed of two semi-infinite terminals, designated as input i and output o, and a finite control c terminal, all joined via a central spacer S (see Figure 1). The calculation of the corresponding single particle Green’s function g, defined by
[
g) Hcs E - Hc 0 E - Ho1,1 - Ho1,2To Hos 0 Hsc Hso E - Hs His 0 0
E-
]
-1
0 0 Hsi i H-1,-1
-
i H-1,-2 T hi
(1)
is performed using Green’s function matching method (GFM).5 Here, Hik,k represents the Hamiltonian matrix corresponding to a one-dimensional unit cell of the medium i. The index k refers to the position of this unit cell relative to the interface. In the implementation used for this article, the Hamiltonian is restricted to a tight-binding Hamiltonian with one π-electron per atom. * Correponding author:
[email protected]. † National Institute for Advanced Interdisciplinary Research. ‡ Silverbrook Research.
Figure 1. Schematic view of a three-terminal system composed of an input (i), output (o), and control (c) terminal connected via a spacer S.
Therefore, the matrix Hik,k contains pi × pi elements corresponding to the interaction between the pi π-electrons of the i of one-dimensional unit cell. The Hamiltonian matrix Hk,k-1 dimension pi × pi represents the coupling between adjacent unit cells of medium i. Similarly, the Hamiltonian matrix Hs of dimension ps × ps represents the interaction between the ps atoms of the spacer S, and the Hamiltonian matrix His of dimension pi × ps represents the coupling between the first unit cell of medium i and the spacer S. The Green’s function corresponding to a configuration where the control terminal is semi-infinite is obtained by replacing the Hamiltonian Hc by Hc1,1 - Hc1,2Tc in eq 1. The body of the calculation involves obtaining the transfer matrix of each medium according to
GMn+1,m ) TMGMn,m
(n g m)
(2)
GMn-1,m ) T h MGMn,m
(n e m)
(3)
GMn,m+1 ) GMn,mSM
(m g n)
(4)
hM GMn,m-1 ) GMn,mS
(m e n)
(5)
Here, the Green’s function matrix GMn,m of dimension p × p corresponds to the interaction between the one-dimensional unit cells n and m of the medium M (M ) i, o, c), while the matrices h M, SM, S h M of dimensions p × p are the transfer matrices TM, T of the medium M. These transfer matrices are obtained using the algorithm derived by Lopez Sancho et al.6
10.1021/jp9912105 CCC: $18.00 © 1999 American Chemical Society Published on Web 09/23/1999
8672 J. Phys. Chem. B, Vol. 103, No. 41, 1999
Treboux et al.
Figure 2. Three 1D monatomic wires joined via a single-atom central spacer. The input and output wires are semi-infinite, and the control wire is finite.
Once the Green’s function for the matched system is obtained, we define the scattering matrix between input and output terminals through
[ ] [ ] φo,out φo,in ) [S] φi,out φi,in
(6)
where φin and φout denote the incoming and outgoing wave function of the corresponding infinite media i or o and where the scattering matrix S is calculated from
[S] )
[ ][
][ ] [
Tom 0 goo - go goi Som 0 n g gii - gi 0 Sh n × hi 0 T io i Gom,m-1 0 Gin,n-1 0
]
(7)
Here gM is the Green’s function of the infinite medium M projected on the p atoms forming the input (unit cell labeled -1 in Figure 1) or output (unit cell labeled 1 in Figure 1) p interfaces using the projector I ) ∑k)1 |k〉〈k| where |k〉 represents the atomic orbital of an atom k of the input or output interface. goo, gii, and goi are obtained from eq 1. The conductance of the system input-spacer-output is obtained from the generalization of the Landauer conductance formula corresponding to the two-probe experiment.7
Γ(E) )
2e2
() Vo
∑ h jk V
|〈φoj|Soi(E)|φik〉|2
(8)
i
The asymptotic solution of eq 8 is obtained for n and m far from the interface by setting incident amplitudes as the eingenvectors of the Hamiltonian of part i and using the conservation of the current in the ballistic regime. Here, we focus on the energy band(s) of the system closest to the Fermi level and calculate the energy-dependent conductance within this energy window. The first system we consider consists of three one-dimensional (1D) monatomic wires joined via a singleatom central spacer (Figure 2). The input and output terminals are semi-infinite, while the control terminal is finite. This gives rise to the possibility of an interference effect between the control terminal and the output terminal. The conductance of the system is shown in Figure 3 for specific lengths of the control terminal. When the length of the finite control terminal is zero, the system degenerates to a 1D wire, and we obtain a constant conductance corresponding to a single ballistic channel. When the length of the finite control terminal is nonzero, the conductance between the input and output terminals is zero wherever the energy coincides with an energy level on the control terminal.
Figure 3. Conductance between the semi-infinite input and output terminals of a three-terminal monatomic wire system. When the length of the finite control terminal is zero, we recover a single ballistic channel (dots). As the length of the finite control terminal is increased, the conductance between the input and output terminals is zero wherever the energy coincides with an energy level on the control terminal. Conductance curves are shown for control terminal lengths of one atom (solid circles) and two atoms (open circles). The tight-binding parameter β is set to -1 eV.
For example, when the control terminal contains a single atom and therefore has a single energy level at E ) 0, we obtain a corresponding zero conductance at E ) 0. The conductance then increases as the energy increases. When the control terminal contains two atoms and therefore has two energy levels at E ) (β, we obtain corresponding zero conductance at E ) (β. This phenomenon of interference is general in branched topologies. Sautet et al.8 have found similar interference produced by a benzene embedded in an infinite polyacetylene chain, and Weisshaar et al.9 have analyzed the phenomenon in terms of quantum waveguide structures. The effect of the control terminal length on the conductance between the input and output terminals raises the possibility of implementing a nanotransitor where a variable-length control terminal acts as the gate. To avoid Peierls distortion, the experimental realization of a three-terminal system using 1D terminals could utilize atomic metallic wires deposited on a semiconductor surface. Such 1D wires have been fabricated in recent experiments.10 The length of the 1D control terminal can be varied by moving a single atom. Various techniques have been proposed for implementing such a switching atom,10 with switching frequencies in the terahertz range predicted using first principles molecular dynamics.11 Another way to avoid Peierls distortion is to realize the threeterminal system using single-wall carbon nanotubes (SWNTs). The geometrical structure of a SWNT is uniquely determined by the chiral vector C ) na1 + ma2, where a1 and a2 are graphene sheet lattice translation vectors. The (n,n) “armchair” nanotube and the (n,0) “zigzag” nanotube both have a nonchiral configuration. The (n,n) armchair nanotubes are metallic, while the (n,m) tubes are semimetallic if n - m is a nonzero multiple of 3, and semiconducting otherwise.12 Branched nanotube systems have been found experimentally,13 and their stability has been explored theoretically.14,15 We examine the case of a metallic zigzag nanotube (Figure 4). All metallic nanotubes have two conducting channels in the energy window considered here. In the zigzag nanotube, these
Three-Terminal Nanotube System
Figure 4. Three-terminal system consisting of (12,0) zigzag nanotube terminals joined via a central spacer with 3-fold symmetry. Exactly two atoms lie on the 3-fold axis of the spacer. The input and output terminals are semi-infinite, and the control terminal is finite.
Figure 5. Conductance between the semi-infinite input and output terminals of a three-terminal (12,0) zigzag nanotube system with a finite control terminal. We confine our calculations to the energy window within which the corresponding infinite (12,0) nanotube has two ballistic channels (horizontal line). As the length of the finite control terminal is increased, the conductance between the input and output terminals is at a minimum value wherever the energy coincides with an energy level of the appropriate symmetry on the control terminal. Conductance curves are shown for control terminal lengths of three 1D (12,0) unit cells (solid circles), six unit cells (dashes), and nine unit cells (crosses). The tight-binding parameter β is set to -2.66 eV.
channels belong to the same E-irreducible representation of their corresponding point group symmetry. We can therefore expect an interference effect between the control terminal and the output terminal similar to the one seen in the 1D system. We treat the input and output terminals as semi-infinite and vary the length of the control terminal. As the length of the control terminal is increased, we gain additional energy levels on the control terminal, some of which have a symmetry that correlates with the bands of interest on the input and output terminals. The conductance between the input and output terminals is therefore expected to be zero wherever the energy coincides with such energy levels on the control terminal. In Figure 5 we show the conductance between the semi-infinite input and output terminals for various lengths of the finite control terminal. Because of molecular orbital distortions induced by the central spacer, the conductance diminishes significantly but does not vanish entirely where the control terminal energy levels interfere with the conductance path.
J. Phys. Chem. B, Vol. 103, No. 41, 1999 8673
Figure 6. Three-terminal systems composed of three identical semiinfinite monatomic or (12,0) nanotube terminals: (a, b) with an atomcentered spacer; (c, d) with a six-member-ring-centered spacer. The 3-fold axes of the nanotube systems are marked. Part b is the infinite counterpart of the system in Figure 4.
Figure 7. Conductance between the semi-infinite input and output terminals of a three-terminal monatomic wire system with a semiinfinite control terminal. The single-channel conductance is split equally between the output terminal and the control terminal. Note that the reflectance induced by the spacer is not zero. In the atom-centered system this accounts fully for the conductance (solid circles). In the six-member-ring-centered system the conductance vanishes at E ) 0 because of interference (open squares). The tight-binding parameter β is set to -1 eV.
In the design of a working device, modifying the length of the finite control terminal is impractical. A more realistic approach, where the energy levels of the terminal are shifted by distorting the nanotube mechanically using a microscopic tip, can be considered, however. Such an approach allows the conductance of the three-terminal device to be varied in a controlled way, providing a possible basis for a nanotransistor. Mechanically controlled conduction through C60 has already been demonstrated,16 and a corresponding amplifier has been proposed.17 Finite length SWNTs have been experimentally obtained in aqueous solution.18 Attempts to connect them to form nanodevices have been reported, and recent experiments have shown that, as in the case of C60, SWNTs can be addressed with the use of a scanning tunneling microscope.19,20 The quasi-1D nature of nanotubes is well-known. The analysis of the nanotube-based three-terminal system shows that this 1D behavior can persist in a branched topology, indicating that
8674 J. Phys. Chem. B, Vol. 103, No. 41, 1999
Treboux et al. We have reported an interference effect in 1D three-terminal systems whereby ballistic transport from the input terminal to the output terminal can be controlled by the positions of the energy levels of the control terminal. We have shown that this effect persists in a three-terminal system implemented using zigzag nanotubes. We have proposed controlling the energy levels of the control terminal by varying its length or by subjecting it to mechanical stress. The possibility of using the control terminal directly as an electronic gate is of obvious interest. However, this would require calculating electrostatic potentials through the terminals, which in turn requires selfconsistent solution of the charge density and Poisson equations. State-of-the-art ballistic conductance models need to be modified before such calculations can be predictive. References and Notes
Figure 8. Conductance between the semi-infinite input and output terminals of a three-terminal (12,0) nanotube system with a semi-infinite control terminal. We confine our calculations to the energy window within which the corresponding infinite (12,0) nanotube has two ballistic channels. The double-channel conductance is split equally between the output terminal and the control terminal. Note that the reflectance induced by the spacer is not zero. Note also that the conductance does not converge to zero at the edge of the energy band because of the existence of additional energy bands further from the Fermi level. In the atom-centered system this accounts fully for the conductance (solid circles). In the six-member-ring-centered system the conductance vanishes at E ) 0 because of interference (open squares). The tightbinding parameter β is set to -2.66 eV.
insights gained in analyzing one-dimensional systems can be applied in analyzing networks of nanotubes. As a further example, we now consider three-terminal systems where all the terminals are now infinite but where we vary the topology of the central spacer. In Figure 6 we consider atom-centered and six-member-ringcentered spacers for both monatomic wire systems and (12,0) nanotube systems. The atom-centered nanotube systems considered here is the infinite counterpart of the system in Figure 4. For each type of spacer there is a strong similarity between the behavior of the monatomic wire system and the behavior of the nanotube system (Figures 7 and 8, respectively), showing that the 1D behavior of the nanotube system persists.
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