theoretical investigations of quantum transport ... - World Scientific

Surface Review and Letters, Vol. 7, Nos. 5 & 6 (2000) 637–642 c World Scientific Publishing Company

THEORETICAL INVESTIGATIONS OF QUANTUM TRANSPORT THROUGH CARBON NANOTUBE DEVICES C. ROLAND and M. BUONGIORNO NARDELLI Department of Physics, North Carolina State University, Raleigh, NC 27695, USA H. GUO, H. MEHREZ and J. TAYLOR Department of Physics and Center for the Physics of Materials, McGill University, Montreal, PQ Canada H3A 2T8 J. WANG and Y. WEI Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China Received 7 July 2000 By combining a nonequilibrium Green’s function analysis with a standard tight-binding model, we have investigated quantum transport through carbon nanotube devices. For finite-sized nanotubes, transport is dominated by resonant tunneling, with the conductance being strongly dependent on the length of the nanotubes. Turning to nanotube devices, we have investigated spin-coherent transport in ferromagnetic–nanotube–ferromagnetic devices and nanotube-superconducting devices. The former shows a significant spin valve effect, while the latter is dominated by resonant Andreev reflections. In addition, we discuss AC transport through carbon nanotubes and the role of photon-assisted tunneling.

Among the many remarkable properties of singlewall carbon nanotubes (SWNT’s), it is the mechanical and electronic properties that stand out.1 The excellent resistance of carbon nanotubes to bending should lead to future applications of the tubes as a high strength, lightweight material.2 Turning to their electronic properties, SWNT’s may be either metallic or semiconducting, depending upon their helicity, and therefore have the potential of forming the basis of a future nanotube-based molecular electronics.3 To explore this exciting possibility, the electronic properties of nanotubes have been the subject of numerous experimental and theoretical investigations. Progress has been rapid, and several prototypical nanotube-based devices have already been produced with the aid of nanomanipulators.4 Recently the field of nanotube research has entered a new phase with the fabrication of hybrid device structures in which the nanotubes are contacted

electronically with other materials. This is a crucial step, as a carbon-nanotube-based electronics is only possible when the tubes are efficiently coupled to external leads. Examples of such devices are nanotube-based magnetic tunnel junctions,5 nanotube heterojunctions6 and superconducting junctions.7 These systems are interesting not only because of their potential for technological applications, but also as a fundamental testing ground for fundamental physics at the nanometer length scale. In this paper, we shall review our theoretical analysis of such hybrid device structures.8–11 Specifically, we shall treat device systems with two ferromagnetic (FM) leads (i.e. an FM/SWNT/FM system), and normal metal (N) nanotube-superconducting (S) devices (i.e. N– SWNT–S systems). In addition, we shall discuss aspects of the dynamic or AC transport through carbon nanotubes, along with the role of photonassisted tunneling.

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Our analysis is based on a proper combination of the Keldysh nonequilibrium Green’s function12 with a standard tight-binding model for the SWNT such that the coupling of the nanotubes to the leads is included via the appropriate self-energies within the retarded Green’s function Gr , i.e. Gr (E) = Pr Pr Pr 1 Pr , where = L + R represents the E−H − tube

self-energy of the left (L) and right (R) leads, respectively. The nanotube Hamiltionian Htube is modeled using a nearest neighbor π orbital tight-binding model with a bond potential Vppπ = −2.75 eV, which is known to be a reasonable, qualitative description of the electronic properties of SWNT’s.13 Before presenting our data, we briefly review quantum transport through infinite SWNT’s. Metallic nanotubes are characterized by the crossing of two bands at the Fermi level, so that the theoretical DC conductance G ≈ 2 × 2e2 /h = 12.9 kΩ. Since there are no other bands in the energy range E = ±0.8 for (10,10) tubes, the DC conductance is a constant over this energy range. At larger energies, the electrons are able to probe different energy bands giving rise to an increase in G that is proportional to the number of additional bands available for transport. The DC conductance therefore consists of a series of “down-and-up” steps in which the positions of the steps correlate with the band edges which are marked by peaks in the density of states (DOS). We now turn to transport through finite-sized nanotubes coupled to two metallic leads, as shown in Fig. 1. In this case, the behavior is strikingly different from that of the infinite SWNT. Here, the SWNT tunnel junction shows a resonance behavior, with G sharply peaked at energies where the nanotube has a transmissive level.8,11 Physically, this resonance behavior is attributed to scattering at the interface between the leads and the nanotube. It turns out that both the positions and the height of the resonance levels are sensitive to the length of the nanotubes. Specifically, for (n, n) armchair tubes, if the length L = 3N + 1, with N an integer, the tubes have large conductances with a resonance peak centered about E = 0. Other tube lengths show much smaller conductances due to a gap between the scattering states. As the length of the SWNT increases, the number of resonance peaks increases. At the same time, there is a decrease in the peak widths. Eventually, when the tube is long enough the peaks merge to form a continuous curve and the “down-and-up” structure

Fig. 1. Conductances for (5,5) SWNT’s of different lengths as a function of energy E (eV), with EF = 0 and ΓL,R = 0.5: (a) for length N = 50; (b) for N = 200; (c) for N = 2000; and (d) for N = 2000 with ΓL,R = 2.0. Note the emergence of resonant tunneling peaks that increase in number and narrow in width as N increases.

characteristic of G of an infinite nanotube is recovered. Experimentally, there is considerable evidence for this picture as an energy gap along with conductance oscillations have actually been detected with scanning probes for finite tubes as long as at least 500 nm.14 Spin-coherent transport in a carbon nanotube magnetic tunnel junction was recently investigated experimentally with two Co leads attached to a nanotube. The data showed that the SWNT’s have very long spin-scattering lengths of at least 130 nm,5 making them ideal candidates for molecular scale magnetoelectronic devices in which both the charge and spin degrees of freedom are utilized. Essentially, the spin valve effect reported is due to the misalignment of the magnetic moments of the two Co leads, which, because of its small size, gives rise to a hysteretic magnetoresistance, even when the same material is used. To investigate spin-coherent transport through SWNT’s theoretically, an FM/SWNT/FM system was studied8 which assumed that the magnetization of the left lead points in the z direction, while the magnetization of the right lead is oriented at an angle θ away from the z direction in the x–z plane. Our main results are shown in Fig. 2, for a (5,5) armchair tube. We observe a clear spin valve effect, such

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Fig. 2. Conductance of the (5,5) SWNT device for different coupling parameters and tube lengths, as a function of the angle θ, showing the spin valve effect. We fixed Γ↑ /Γ↓ = 2.0: (a) for N = 5; (b) for N = 6; (c) for N = 7.

that the resistance R(θ) varies smoothly with θ. In agreement with the experimental results, the device has a minimum resistance when θ = 0, i.e. when the magnetic moments of the leads are parallel; and a maximum resistance when θ = π, i.e. when they are antiparallel. Physically, this variation of the resistance is a reflection of the differences in the majority and minority carrier concentration of the FM lead material. There are several aspects of the spin valve effect that are, however, unique to SWNT systems. Specifically, as shown in Fig. 2, R(θ) is an order of magnitude lower when the nanotube length is commensurate with L = 3 × N + 1, and higher when that is not the case. This variation may be understood by noting that nanotube devices for which L = 3 × N + 1 are on-resonance, while devices with an incommensurate length are off-resonance. Essentially, it turns out that the latter are more sensiPr tive to the different values of (here these values are parametrized by the corresponding line width Pr function Γασ = −2Im( ασ ), with α = L, R and σ =↑, ↓), such that larger values of Γασ give rise to a smaller resistance since the SWNT is now better contacted to the leads. The on-resonance data show just the opposite effect, because changing Γσ shifts the resonance point slightly, thereby reducing the resonant transmission and increasing the resistance. The

calculated off-resonance resistance is in the range of 40–280 kΩ, in excellent agreement with the experimental results. We now turn to a theoretical analysis of transport through nanotube-superconducting devices.9 Experimentally, such devices consist of SWNT’s bridging two superconducting electrodes. By tuning the transparency of the device, clear signals of Andreev reflections were detected via changes in the subgap resistance at a temperature of T = 4.2 K, while other transport anomalies were observed at lower temperatures.7 Although the experimental device consisted of two SWNT–S junctions, the data indicate that each of the junctions acts essentially independently. Hence, we have chosen to focus on the somewhat simpler problem of an N–SWNT–S system. Theoretical analysis shows that there are two contributions to the current flowing through the device, i.e. I = IA + I1 . Here IA represents the Andreev current or subgap contribution, while I1 represents the current contributions when the energy is greater than the gap energy. Three processes contribute to the latter: (i) the familiar contribution from the tunneling current; (ii) the branch crossing process of Blonder–Tinkham-Klapwijk theory; and (iii) the formation (or annihilation) of Cooper pairs inside the S lead by incoming electrons (or holes). 15


Fig. 3. I–V curves for the N–SWNT–S device at T = 4.2 K. The SWNT is a (5,5) tube with length N = 19. The solid curve is for the low transparency device (offresonance transmission) with parameters ΓL = ΓR = 0.8 and gate voltage Vg = 0. The dashed curve is for high transparency devices (on-resonance transmission) with Vg = 0.6 meV.

Fig. 4. Differential resistance dV /dI for a high transparency device at different temperatures. A peak emerges from the overall dip as the temperature is lowered. The parameters are Vg = 0.6 meV, ΓL = 0.003 and ΓR = 0.007.

The salient results of our calculations are shown in Fig. 3, which presents the I–V characteristics of high and low transparency devices. These curves are in good semiquantitative agreement with the experimental results. Clearly, a higher slope is observed for the I–V curves within the subgap range for the onresonance devices (dashed curve), which is reflected in terms of a dip in the calculated differential resistance. This resistance dip has a value close to h/(2 × 4e2 ) = 3.2 KΩ, which is precisely the expected value of the Andreev reflection process in an SWNT–S junction with two transmitting modes. Low transparency or off-resonance devices (solid line) show the opposite trend. In this case the differential resistance displays a large peak, which again is in good agreement with the experimental results. The data presented have so far been for temperatures of T = 4.2 K, so that features reflecting smaller energy scales are completely washed out. However, at a lower temperature of T = 2 K, the experimental

data show that a narrow peak emerges in the dV /dI curves for zero bias, which is superimposed on the Andreev resistance dip. Such anomalous behavior has previously been ascribed to the strong electron–electron interactions characteristic of Luttinger liquids.16 Surprisingly, our analysis shows that these features emerge naturally at lower temperatures, even within the context of a single-electron theory as presented here. This is shown in Fig. 4, which shows the emergence of a narrow peak in the dip as the temperature is lowered. Theoretically, this result may be understood in terms of the formation of two nondegenerate poles in the expression for the Andreev transmission when ΓR > ΓL . So far, we have analyzed quantum transport in hybrid devices based on the DC conductance. What about the AC response of carbon nanotubes? AC conductance of material systems is complicated by the presence of time-dependent fields that can take the system out of equilibrium. Under AC conditions,

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electrodynamics shows that displacement currents are induced.17 These need to be accounted for, in order to conserve the total current in the system, and in order to maintain gauge invariance. Another important feature associated with the AC response of a conductor is that of photon-assisted tunneling: in the presence of a time-varying potential, electrons can absorb photons and thereby inelastically tunnel through other levels. Photon-assisted tunneling has been studied both experimentally abd theoretically in semiconductor quantum dots, superconducting tunnel junctions and superlattice systems.18 Figure 5 shows typical AC conductances for a (10,10) armchair nanotube for different frequencies ~ω.10 Since the conductance is now complex, we plot the real (dissipative) and imaginary (nondissipative) parts separately. We focus on ~ω ≤ 4 eV, which is well below the plasma frequency of the nanotubes. In the limit of ~ω → 0, the expected DC results are smoothly recovered. Indeed, there is little AC effect for small frequencies. Larger differences are observed

Fig. 5. Real and imaginary parts of the conductance g12 (~ω) for a (10,10) metallic tube at different frequencies in units of 2e2 /h. Solid line, ~ω = 0 eV; dashed line, 0.1 eV; dotted line, 1.0 eV; a dot–dashed line, 2.0 eV. The DOS (arb. units) is shown underneath and marks the band edges.

when the frequency is increased: for 0.1 ≤ ~ω ≤ 1 eV, there is a general reduction in the value of the AC conductance, along with a loss of the step structure. For larger values of ~ω, however, the AC conductance increases again, so that at 2 eV the conductance has not only completely recovered but actually exceeds its initial value. These results may be understood as follows. The imposition of a time-varying field in the lead induces a displacement current, even inside pristine nanotubes. This displacement current acts to oppose the change of the normal conduction current, thereby leading to an overall decrease in the AC conductance. However, if the frequency is large enough, then photon-assisted transport can place the electrons into higher subbands, thereby increasing the conductance. The result is that the conductances, after a small initial decrease, are greatly enhanced. Note that the emergence of an imaginary component signals that the nanotubes acquire capacitive and/or inductive behavior, depending upon the frequency. These results have important implications for nanotube-based devices, for which one can expect photon-assisted resonance transmission to occur. Note that the AC frequencies examined are well in the IR-to-optical range, and hence it is very useful to explore the possibilities of nanotube-based optoelectronic devices when combined with semiconductor technology. In summary, we have theoretically examined quantum transport through a variety of nanotubebased device structures with a tight-binding based nonequilibrium Green’s function formalism.8–11 Transport through finite-sized nanotubes coupled to metallic leads was shown to be dominated by a resonant transmission which is strongly lengthdependent. Spin-coherent tunneling gives rise to a clear spin valve effect in nanotube FM devices that is in good agreement with the experimental results, while nanotube-superconducting devices are dominated by resonant Andreev reflections. A study of the dynamic conductances of nanotubes shows that because of a photon-assisted tunneling, significant increases in the conductances are to be expected for nanotubes irradiated with photons in the IR-tooptical range. While all of our results are particular for carbon nanotube systems, we expect that the general features of transport uncovered are valid for other materials at the nanometer length scale.


Acknowledgments We gratefully acknowledge financial support from the NSERC of Canada and the FCAR of Quebec (H. G.); RGC grant (HKU 7115/98P) for the Hong Kong SAR (J. W.); ONR N00014-98-1-0597 and NASA NAG8-1479 (C. R.). We also thank the North Carolina Supercomputing Center (NCSC) for extensive computer support.

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