Conductance Calculations for Real Systems on the Nanoscale

Transcription

1 Conductance Calculations for Real Systems on the Nanoscale Frank Grossmann,* Rafael Gutie rrez, and R diger Schmidt [a] Electron transport across molecular junctions is a rapidly growing topic at the borderline between physics and chemistry. We review calculations which were done in the Landauer transport formalism for monovalent systems, ranging from clusters to fullerenes. A realistic description of molecular conductance can be achieved by a density functional based approach to the calculation of the electronic transport properties. KEYWORDS: density functional theory Landauer theory molecular conductance Molecules as Building Blocks for Electronic Circuits Triggered by recent advances in chemical synthesis, scanning probe microscopy, and break junction techniques, the seminal idea of using molecular scale conductors as active components of electronic devices [1] has received a new impetus. [2] For example, rectification as well as negative differential resistance could already be demonstrated on the nanoscale. A tutorial overview of first experimental approaches to build elements for a future molecular computer using a bottom up approach is given in ref. [3], while more recent progress in the same direction is highlighted in ref. [4]. This rapid development of the experimental field represents a challenge to theoreticians. The understanding of the basic physical mechanisms which determine the electronic transport on the nanoscale is essential for achieving conductance control and thus for opening the possibility of device applications. Especially, a realistic description (beyond model Hamiltonian approaches) of quantum transport demands the combination of reliable electronic structure methods with a suitable transport formalism. In this concept we will first give an overview of theoretical approaches to this problem. In particular, we will review some numerical investigations of electronic transport in monovalent systems on the nanoscale. Hereby we focus on sodium clusters and an all carbon setup of nanotubes connected to a C 60 buckyball. Our computational approach is based on the combination of a tight-binding method parameterized by density functional theory (TB-DFT) with the Landauer formalism. Landauer Approach and Electronic Structure Methods Physical systems on the mesoscopic or nanoscale have linear dimensions L which are of the same order or even smaller than the electronic phase coherence length. This fact precludes a semiclassical description, where it is assumed that the elastic mean free path is of the same order or smaller than the phase coherence length. [5] The electronic transport on mesoscopic length scales will then show a strong dependence on specific sample properties like disorder distribution and boundary geometries. Thus, a description of electronic transport in terms of local (eventually disorder-averaged) intensive quantities such as the Kubo conductivity s is not possible and the basic physical quantity to be considered is rather the conductance G ˆ sl d 2, where d is the spatial dimension. A formulation that includes interference effects due to phase coherence as well as geometric effects was initially developed by Landauer [6] for a two-terminal geometry as displayed in Figure 1 and further extended by B ttiker [7] to the multiterminal case. The essential idea of the Landauer formulation is to relate the Figure 1. Schematic representation of a two-terminal device. The scattering region (enclosed in the dashed-line frame) with transmission probability T(E) is connected to semi-infinite left (L) and right (R) leads which end into electronic reservoirs (not shown) at chemical potentials m L, m R. By applying a small potential difference electronic transport will occur. The scattering region or molecule (M) may include in general parts of the leads (shaded areas). This is necessary for the description of real systems, where the surface topology may be modified as a result of relaxation or reconstruction processes. This may introduce additional scattering due to, for example, surface states. [a] PD Dr. F. Grossmann, Dr. R. Gutie rrez, Prof. Dr. R. Schmidt Institute for Theoretical Physics Dresden University of Technology, Dresden (Germany) Fax: ( 49) frank@physik.tu-dresden.de 650 ¹ 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim /02/03/08 $ /0 CHEMPHYSCHEM 2002, 3, 650 ± 654

2 Conductance Calculations for Nanoscale Systems conductance to an elastic scattering problem and, ultimately, to transmission probabilities. The simplest way to derive this relationship is to consider a region connected to two ballistic leads, which are connected to electronic reservoirs at the chemical potentials m L and m R, see Figure 1. It is assumed that electrons entering the reservoirs do completely lose their phase coherence. As stated in ref. [8], assuming semi-infinite leads is enough to warrant that no electron exiting the scattering region will reenter it with the same phase, so that an explicit modeling of the reservoirs is not necessary. At equilibrium m L ˆ m R, but if a voltage ev ˆ m L m R is applied a nonequilibrium situation is induced and a current will flow. The scattering region is characterized by the energy-dependent transmission coefficient T(E). In the zero-temperature, linear response (ev!0) regime, the proportionality relation of Equation (1) is found to e2 XNF G ˆ ph T(E F), T ˆ j t mn j 2 (1) m;nˆ1 hold, [5] where E F is the Fermi energy of the whole system in equilibrium and the transmission amplitudes t mn describe the scattering of one electron from channel n on the left lead to channel m in the right lead. They can be extracted from the scattering matrix, for example. The sums run over all N F open channels at the Fermi level (whose number is assumed to be equal on both sides). Channels (transverse modes) appear due to the finite cross section of the leads which induces quantization of the electronic states perpendicular to the direction of current transport. For the special case of ideal transmission, namely S n j t mn j 2 ˆ 1 8m, the conductance is simply proportional to N F. Thus, G/G 0 with G 0 ˆ e 2 (ph) 1 shows steps as a function of N F. This is the well-known fact of conductance quantization, [5] shown experimentally by van Wees et al. and Wharam et al. [9] The finite contact resistance R Tˆ (G 0 N F ) kwn 1 F is a purely geometric effect related to the mismatch between the large number of channels (propagating modes) in the reservoirs and the small number of channels in the leads. Finite temperatures and finite voltages can be easily dealt with by a generalization of Equation (1), using Fermi distribution functions and an integration of the resulting expression over energy. [5] Furthermore, it is worth noting that a different expression for the conductance, including a denominator R ˆ 1 T in Equation (1), was discussed in the literature. As it became clear later on, this alternative expression is related to a four-terminal measurement, where the conductance is measured between two points located just before and behind the scattering region, thus characterizing the sample itself (the contact resistance R T is not included so that for T ˆ 1 the conductance would diverge). In contrast, the physical situation described by Equation (1) corresponds to the case where the current is measured between points deep inside the reservoirs (see also ref. [5] for a discussion of this point). For a general scattering region where inelastic effects are included, Meir and Wingreen [10] used nonequilibrium Green functions to derive an expression for the current which reduces to Equation (1) in the elastic case. An advantage of their derivation is that an explicit connection to the Green function G M of the scattering region is established. The result for the transmission probability is then given by Equation (2), [10] where G L,G R (defined below) describe the coupling of the scattering T(E F ) ˆ Tr[G M (E F ih)g L (E F )G M (E F ih)g R (E F )] (2) region to the leads and the trace is to be taken over states in the scattering region. The Green function G M is in general defined as the inverse operator (E ih H M ) 1 (h!0 ) for some suitable molecular Hamiltonian H M. Similar expressions have been derived by Fisher and Lee, and Todorov and Sutton. [11] In a seminal paper Szafer and Stone have derived the Landauer result, Equation (1) from Kubo's linear response theory. [12] Some subtle points concerning the relationship between Kubobased and Landauer-based conductances for treating quantum transport in mesoscopic systems have been recently discussed by Nikolic. [8] Furthermore, it is worthwhile to mention the close analogy between electron transport (in mesoscopic physics) and electron transfer between donor and acceptor states usually studied in the realm of theoretical chemistry. The relation between these topics has recently been reviewed and clarified by Nitzan. [13] Finally, there is also a close formal analogy between the Landauer formula of Equation (1) and chemical reaction rate expressions (see refs. in [13]). As mentioned above, only components of the Green function on the Hilbert subspace associated with the scattering region, which we will denote as the molecule from now on as a reminder that transport through molecular scale systems is the main issue to be adressed here, are needed. The relevant submatrix is denoted by G M in the following. Notice, however, that the molecule may also include some atoms belonging to the leads, see Figure 1. This will be the case when investigating real systems, where the surface atomic structure of the leads is explicitly taken into account (clean surfaces are usually energetically unstable, so that upon structural relaxation the surface topology may be modified and this will introduced additional scattering). From the full Green function G of the open (infinite) system consisting of the leads plus the molecule it is possible to extract G M using projector operator techniques. [14] In order to do so, one partitions the whole system into three components as shown in Figure 1, where a left electrode, the molecule, and a right electrode are depicted. The full associated Hamiltonian matrix (in a suitable basis representation) can then be formally written as Equation. The matrices V L M (V R M ) couple atoms belonging to the left (right) leads to the molecule and it has been assumed that no direct lead ± lead coupling exists. Notice that H L(R) are infinite dimensional submatrices. Introducing an operator 2 P M (P M ˆ 1), which projects onto the molecular subspace of dimension M, it can be shown that G M ˆ P M GP M satisfies the M-dimensional matrix of Equation (4), where S M is the (zs M H M S L (z) S R (z))g M (z) ˆ 1, z ˆ E ih, h!0 (4) CHEMPHYSCHEM 2002, 3, 650 ±

3 F. Grossmann et al. overlap matrix for the general case of a nonorthogonal basis set. The energy-dependent self-energies S L (S R ) include the coupling to the leads as well as information on the electronic structure of the leads. They are given by Equation (5). S L(R) (z) ˆ (zs y L R M V y L R M g L(R)(z)(zS L(R) M V L(R) M ) (5) The matrices S L(R) are overlap matrix elements between molecule atoms and lead atoms and g L(R) (z) are lead Green functions. Since the coupling matrices are in general shortranged they will eliminate all contributions coming from atoms other than those nearest to the molecule. Hence, only surface Green functions will be usually needed. Finally, the quantities G L(R) are given by i[s L(R) (E ih) S L(R) (E ih)]. We would like to stress that Equations (1) and (2) are only valid in the case that inelastic processes in the scattering region can be completely neglected. Otherwise no simple relationship between conductance and transmission can be obtained. A typical example where electron ± electron interactions are decisive are quantum dots. There, the scattering region (the dot) is weakly coupled to the leads so that the coupling-induced level broadening will be much smaller than the charging energy. Hence, electron interaction effects leading to, for example, Coulomb blockade phenomena should be included in the description of quantum transport. [15] At this point we are led to the issue of characterizing the electronic structure of the molecule as well as of the leads. If we only focus on the essential physics, some kind of model Hamiltonians can be used. [16] However, if real situations are adressed where the knowledge of the detailed electronic structure is important, the use of more realistic computational schemes is unavoidable. From the point of view of electronic structure calculations, three classes of approaches may be roughly distinguished: 1) Semiempirical or empirical tight-binding (TB) schemes, such as (extended) H ckel Hamiltonians, where the matrix elements are fitted to experiments or to first-principle calculations. 2) First-principles or abinitio approaches like Hartree ± Fock and density functional theory (DFT). 3) Schemes which combine some elements of points (1) and (2) in first-principles parametrized tight-binding Hamiltonians as [17, 18] it is the case for TB-DFT methods. All of these methods have been implemented for quantum transport calculations. H ckel Hamiltonian approaches were [19, 20] combined with a scattering formalism by Datta et al. to study linear transport in xylyl-dithiol molecules connected to gold electrodes, by Magoga and Joachim to investigate transport through molecular wires, [21] and by Emberly and Kirczenow to study 1,4-benzene dithiolate between gold electrodes. [22] p- Orbital based tight-binding Hamiltonians have also been applied to study transport in C 60 and in carbon nanotube devices. [23] Concerning abinitio approaches, a DFT-based scheme was developed by Lang to study quantum transport through atomic wires and organic molecules. [24] A drawback of this approach is the use of jellium electrodes, because it neglects the structural details of the lead ± molecule interface which are extremely important for conductance control. Yaliraki et al. [25] applied Hartree ± Fock theory to compare the transport properties of xylyl-dithiol and benzene 1,4-dithiol between gold leads. Recently, new DFT-based approaches have been proposed to investigate charge transfer effects in Al-C 60 systems [26] and conductance oscillations in atomic wires. [27] A scheme combining DFT with nonequilibrium Green functions have been used to study aluminum ± carbon nanotube hybrids [28] and organic molecules. [29] Finally, concerning the third class mentioned above, a computational scheme has been developed in ref. [31], which combines a DFT-parametrized TB approach [17, 18] with the Landauer formalism to study the electronic transport properties of sodium atomic chains, [30] small sodium clusters, [31] carbon-based molecular junctions, [33] as well as to simulate scanning ± tunneling spectroscopy experiments on organic molecules. [34] The TB- DFT scheme relies on a representation of the electronic eigenstates of the system within a nonorthogonal localized basis set, usually taken as a valence basis. The many-body Hamiltonian is then approximately represented by a two-center tight-binding Hamiltonian. The matrix elements, however, are calculated numerically, avoiding the introduction of empirical parameters as in conventional TB approaches. This computational method has the advantage of being much faster than conventional abinitio approaches while achieving a comparable degree of accuracy (see refs. [17, 18] for a more detailed account of the method). In the next Section we will discuss some of the applications of this combined scheme. Sodium Clusters In this Section we review numerical results for the conductance of different isomers of sodium clusters from Na 4 to Na 9. [31] In order to be specific, we considered the conductance values along the principal axes of the tensor of inertia for each cluster, contacted by two semi-infinite tight binding sodium chains. The minimal resistance R min ˆ (G max ) 1 obtained in this way is plotted in Figure 2 for the different isomers as a function of the cluster size N. Two general features are observed in R min (N): 1) For a given cluster size N, R min depends sensitively on the cluster geometry (different isomers). 2) There is a general odd ± even oscillation as a function of the electron number, equivalently cluster size, only violated for the second Na 5 isomer. The strong isomer dependence of R min, point 1 above, is a direct consequence of the different electronic structure as function of the cluster geometry and it has also been predicted for organic molecules. [22, 25] Qualitatively the odd ± even oscillations, point 2, can be understood by considering the transmission function T(E) and the total DOS as a function of the incoming electron energy for clusters containing an even and an odd number of atoms. The essential point is the relative position of highest occupied and lowest unoccupied molecular orbitals (HOMO, LUMO) of the cluster with respect to the Fermi level of the composed sodium system: For an odd number of electrons (in sodium the total number of valence electrons and the 652 CHEMPHYSCHEM 2002, 3, 650 ± 654

4 Conductance Calculations for Nanoscale Systems Figure 2. Minimal resistance of sodium clusters as a function of the cluster size for the ground state and the first two isomers. number of atoms in the cluster coincide) the HOMO is halfoccupied and the Fermi energy lies quite close to it. For even electron numbers, however, the HOMO is doubly occupied and E F lies in the HOMO ± LUMO gap. After attaching the clusters to the leads, if the cluster ± lead coupling is not so strong as to induce a large renormalization (shifting and broadening) of the cluster eigenstates, the transmission spectrum will correlate with the DOS of the isolated cluster. Thus, a large transmission (low resistance) is expected for an odd number of atoms and, correspondingly, a low transmission (high resistance) for an even number of atoms. Finally, it is worth mentioning that similar oscillations in the conductance have also been found in simple models of molecular wires, [16] in sodium chains, [27, 30] as well as in atomic carbon chains of up to nine atoms. [24] We want to emphasize that in contrast to our work, in the sodium work of Sim et al. [27] the conductance has been calculated by using the Friedel sum rule [32] for symmetric molecular conductors. A Carbon-Based Molecular Junction As another case of a monovalent cluster, we studied a single C 60 molecule bridging two single-wall metallic (5,5) carbon nanotubes (CNT), which were taken symmetric with respect to the plane through the center of mass of C 60 and perpendicular to the CNT cylinder axes (see upper panel of Figure 3). The central aim of ref. [33] was to exploit the sensitivity of electron transport to the topology of the molecule ± electrode interface in the proposed system. In this pure carbon system, charge transfer effects will be negligible. The Fermi level of the whole system will therefore lie within the HOMO ± LUMO gap of the isolated C 60. Therefore, the electronic transport will be mainly mediated by the overlap of the tails of the molecular resonances within the HOMO ± LUMO gap of C 60. The key problem we addressed was how severely orientational effects do influence the electronic transport. To this end several possible orientations of the C 60 (depicted by the polygon(s) facing the tube symmetry axis in the lower panel of Figure 3) have been considered for a fixed distance between the molecule and the tubes. For the sake of comparison, structurally Figure 3. Geometric configuration of the carbon molecular junction. A C 60 molecule bridges two (5,5) CNTs. The lower panel represents schematically the different orientations of C 60 with respect to the surface cross-sections of the nanotubes (for example, the upper panel geometry corresponds to orientation (1)). The nanotube symmetry axis is depicted by a cross inside a circle. unrelaxed and relaxed molecular junctions were considered. The basic results are displayed in Figure 4 for both relaxed and unrelaxed structures. Surprisingly, at fixed distance, just an atomic scale rotation of the highly symmetric C 60 molecule T(E) (2) (1) (2) (1) E E F / ev Figure 4. Transmission (dimensionless) results for both unrelaxed (upper panel) and relaxed (lower panel) configurations. The tube ± tube distance d is fixed at 0.93nm. Numbers indicate different molecular orientations as depicted in the lower panel of Figure 3. induces a large variation of the transmission at the Fermi energy by several orders of magnitude. This can be seen in Figure 4 b for three different orientations with maximum, minimum, and one intermediate value of T(E F ). As can be seen in Figure 4 a, neglecting relaxation decisively influences the transmission properties of the molecular junction. This shows up as a much different and less smooth behavior of the transmission. The qualitative difference is related to the presence of dangling bond (a) (b) CHEMPHYSCHEM 2002, 3, 650 ±

5 F. Grossmann et al. states on the CNT surfaces. Such states usually lie within a gap (a similar situation as that found in, for example, semiconductor surfaces), in this case the HOMO ± LUMO gap of the isolated molecule. They lead to the oscillatory behavior in the transmission for unrelaxed junctions. Upon relaxation these states are partly saturated or they rehybridize, moving away from the middle of the gap. However, some of them may still lie just above the HOMO or below the LUMO of C 60, giving some contribution to the transmission within the gap. It is interesting to note that such a dramatic influence of the relaxation on the conductance has not been reported before for other pure carbon systems. The results for the relaxed structures reveal that, at the Fermi energy, the pentagon configuration has a transmission lower by about three orders of magnitude than configuration (2). This fact could possibly be exploited in an electronic switching device on the nanoscale, as manipulation of fullerenes by using STM or atomic force microscope tips is becoming a standard technique in the field. [35] Future Perspectives We have presented a brief overview of quantum transport calculations in realistic monovalent systems on the nanoscale. The sensitivity of the conductance on the electronic structure of the scattering region (odd ± even oscillations in the resistance of sodium clusters and chains) and on the details of the leadscattering region interface (dangling bond states and structural relaxation effects in CNT ± C 60 hybrids) has been demonstrated for selected monovalent systems. These effects will evidently be also of relevance for heteroatomic systems. In this case, however, a new fact must be considered: charge transfer effects between leads and molecule. This can strongly change the occupation of the molecular states and thus shift the position of the Fermi level, [20] which may now lie at a molecular resonance. Moreover, in the calculations of current ± voltage characteristics beyond the linear regime the self-consistent calculation of the voltage drop across the sample becomes crucial, see for example ref. [28] for the case of carbon nanotube junctions. One further topic which is of great importance is the inclusion of correlation and inelastic effects, mainly electron ± electron [36] and electron ± phonon interactions, [13, 37] in realistic quantum transport calculations, namely beyond the model Hamiltonian level. They are for instance important in the Coulombblockade regime or for long molecular wires, respectively. This research was supported by the Deutsche Forschungsgemeinschaft through the Forschergruppe Nanostrukturierte Funktionselemente in makroskopischen Systemen. 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