3 Electronic Structure Calculations for Nanomolecular Systems

The electronic structure constitutes the fundamentals on which a reliable quantitative knowledge of the electrical properties of materials should be based. Here, we first present an overview of the methods employed to elucidate the ground-state electronic properties, with an emphasis on the results of Density Functional Theory (DFT) calculations on selected cases of (bio)molecular nanostructures that are currently exploited as potential candidates for devices. In particular, we show applications to carbon nanotubes and assemblies of DNA-based homoguanine stacks. Then, to move ahead from the electronic properties to the computation of measurable features in the operation of nanodevices (e.g., transport characteristics, optical yield), we proceed along two different lines to address two non-negligible issues: the role of excitations and the role of contacts. On one hand, for an accurate simulation of charge transport, as well as of optoelectronic features, the ground state is not sufficient and one needs to take into account the excited states of the system: to this aim, we introduce Time-Dependent DFT (TDDFT), we describe the TDDFT frameworks and their relation to the optical properties of materials. We present the application of TDDFT to compute the optical absorption spectra of fluorescent proteins and of DNA bases. On the other hand, the details of the conductor-leads interfaces are of crucial importance to determine the current under applied voltage, and one should compute the transport properties for a device geometry that mimics the experimental setup: to this aim, we introduce a novel development based on Wannier functions. The method, which is a framework for both an in-depth analysis of the electronic states and the plug-in of tight-binding parameters into the Green’s function, is described with the aid of examples on nanostructures potentially relevant for device applications. 3.1 Electronic Structure of Nanomolecular Systems The methods commonly adopted for large scale calculations of the electronic properties of molecular nanostructures were mostly adapted from the theories available for materials at the macroand mesoscopic scales. Such computational schemes are based on the solution of the many-body Schrödinger R.D. Felice et al.: Electronic Structure Calculations for Nanomolecular Systems, Lect. Notes Phys. 680, 77–116 (2005) www.springerlink.com c © Springer-Verlag Berlin Heidelberg 2005 78 R.D. Felice et al. equation. While the problem is completely defined in terms of the total number of particles N and the external potential v(r), its solution depends on 3N coordinates. This makes the direct search for either exact or approximate solutions to the many-body problem a task of rapidly increasing complexity. The non-relativistic time-independent Schrödinger equation for the system is ĤΦ(r1, ..., rN ) = EΦ(r1, ..., rN ) (3.1) where the Hamiltonian operator is (atomic units = e = me = 4π 0 = 1 are used throughout, unless explicitly stated) Ĥ = T̂0 + V̂e−i + Ûe−e + Ûi−i . (3.2) T̂0 is the kinetic energy of the electrons, V̂e−i is the potential energy of the electrons in the field of the α nuclei of charge Zα, Ûe−e (Ûi−i) is the electronelectron (ion-ion) electrostatic energy. Equation (3.1) is an eigenvalue equation for the N-electron many-body wave-function Φ, where Ĥ is hermitian. In this time-independent formulation, the Schrödinger equation (3.1) for the Hamiltonian Ĥ is equivalent to the variational principle δE[Φ] = 0. The well-known Hartree and Hartree-Fock methods correspond to searching for solutions of δE[Φ] = 0 in the subspace of the products of single-particle orbitals and in the subspace of antisymmetrized products of single-particle orbitals (Slater determinants). The Hartree-Fock theory (HF) [1] is obtained by considering the wavefunction to be a single Slater determinant: thus, the N -body problem is reduced to N one-body problems with a self-consistent requirement due to the dependence of the HF effective potential on the wave-functions. By the variational theorem, the HF total energy is a variational upper bound of the ground-state energy for its particular symmetry. The HF eigenvalues are estimates of the true excitation energies: the assumption made in this statement is that the one-electron wave function of any electron in any energy level is the same in the N and (N − 1) systems (Koopman’s theorem), e.g., the relaxation of the system upon a change in the number of particles is neglected. A better procedure to estimate the excitation energies is to perform selfconsistent calculations for the N and (N − 1) states and subtract the total energies (this is called is the “self-consistent method” for excitation energies, which has also been used in other theoretical frameworks, as DFT). Note that for extended systems this scheme gives the same result as the Koopman’s theorem, and more refined methods should be implemented to address the problem of excitation energies (quasi-particles, QPs) in solids as well as in nanostructures. The HF theory is far from being exact, because the wavefunction of the system cannot be written as a single determinant: MBPT is then required to obtain reliable energies and electronic spectra [1, 2]. HF is often used in conjunction with perturbation theory in the Møller-Plesset (MP) formalism to solve nanomolecular problems. Šponer and coworkers extensively applied these combined methodologies (2nd order MP, also referred 3 Electronic Structure Calculations for Nanomolecular Systems 79 to as MP2) to aggregates of DNA bases [3, 4]: a discussion of the theoretical framework and of its efficiency to deal with biomolecular systems where non-chemical bonding plays a strong role, can be found in their works and in the references therein. This kind of studies (HF+MP), based on the explicit knowledge of the many-body wave-function in (3.1), are still too computationally cumbersome for reasonable applications to molecular nanostructures. The alternative DFT formulation [5–11], based on the particle density rather than on the many-body wave-function, is instead feasible and has been rather successful [12]. Nanomolecular systems are interesting on one hand to study genuine fundamental quantum mechanical effects: Kondo and Coulomb blockade behaviors have been observed in aromatic molecules containing a redox center, as well as in carbon nanotubes [13–16]. On the other hand, they have a huge potential for applications in electronics and optoelectronics, in view of device miniaturization and intelligent fabrication [17, 18] based on recognition and self-assembly (supramolecular chemistry [19, 20]). In both respects, research efforts aim at unraveling their transport and optical excitation properties, which for a theoretical approach require the full accurate account of manybody electronic correlations to compute the electronic structure. However, it is possible to gain a deep knowledge of the system already at the groundstate level, and then use it as the starting point for many-body perturbation theory (MBPT). Thus, in the following we first exemplify the computation of ground-state electronic properties by Density Functional Theory (DFT) (Sect. 3.2). In the second part we describe selected approaches for the theoretical investigation of excited-state optical (Sect. 3.3) and transport (Sect. 3.4) features. 3.2 Selected Applications of Ground-State Electronic Structure Calculations by DFT Within DFT, the ground-state energy of an interacting system of electrons in an external potential can be written as a functional of the ground-state electronic density [5–8]. When comparing to standard quantum chemistry methods, this approach is particularly appealing, because it does not rely on the complete knowledge of the N -electron wave-function, but only on the electronic density. However, although the theory is exact, the energy functional contains an unknown quantity called the exchange-correlation energy, Exc[n], that must be approximated in practical implementations. Although failures of DFT are known, its use continues to increase due to the better scaling with the number of atoms and the fact that failures are connected with a particular choice of the local-functional, with the possibility of improving the accuracy as more and more sophisticated exchange-correlation functionals are generated. 80 R.D. Felice et al. Since excellent presentations of DFT are available in the existing literature, in both the static and dynamical formulations, we completely skip here the formal treatment. The computer packages [21–26] employed for the different examples are explicitly cited below. We present few illustrative examples based on the experience of the authors. (i) In carbon nanotubes (CNTs), where all the bonds imply reactive chemistry (e.g., the formation of orbitals filled with electrons shared by different atoms), the structure and relative energetics between different conformations are well reproduced at the DFT level. (ii) In DNA base assemblies instead, where interactions that do not imply making or breaking of bonds (e.g., H-bonding, Van der Waals) play a role, correlations beyond DFT are important already at the ground-state level to predict the structure, whereas their effect on the electronic spectrum has not yet been assessed. However, we show here what can be learnt at the ground-state level if one gives up predicting the structure and energetics. 3.2.1 Carbon Nanotubes Nanotubes are of both fundamental and technological importance: being quasi-one-dimensional (1D) structures, they possess a number of exceptional properties. While the peculiar electronic structure – metallic versus semiconducting behavior – of CNTs depends sensitively on the diameter and the chirality [27, 28], boron nitride (BN) tubes display a more uniform behavior with a wide band-gap (larger than 4 eV), almost independent of diameter and chirality [29,30]. The potential applications of nanotubes in nano-technology are numerous [31–33], ranging from (opto)electronic to mechanical devices. Given their importance and their broad range of electronic characteristics, CNTs have been an ideal playground for DFT simulations to investigate a number of features. Scanning tunnelling microscopy (STM) and spectroscopy (STS) have been among the most powerful experimental techniques to study and manipulate nanotubes. STM is a local probe that allows to extract information about the spatial localization. First-principle DFT simulations enable the computation of STM images of nanostructures, starting from the knowledge of their ground-state electronic structure. In the following, we focus on the DFT-based simulation of STM images of selected CNTs, according to a basic formalism presented elsewhere [34]. We focus on the role of the local environment in the electronic properties of CNTs. We do not discuss here the effect of tube-substrate coupling [35]. As a first issue, we show how tube-tube interactions modify the electronic structure. In Fig. 3.1 we present the calculated STM/STS image obtained for an external voltage of +0.5 eV for a bundle made of three (8,8) nanotubes, and compare the electronic structure to that of an isolated (8,8) tube. The STM image Several other DFT software packages, besides those cited here, are distributed. Most of them are publicly available on the world wide web for academic institutions. 3 Electronic Structure Calculations for Nanomolecular Systems 81 is quite different from the typical patterns of isolated tubes [35]. The intertube coupling clearly modifies the spectral features (right panel of Fig. 3.1). There are two main effects. (i) The presence of tube-tube interaction opens a “pseudogap” close to the Fermi level (see the well at E = 0), as already predicted for randomly oriented nanotube ropes [36] (pseudogap of ∼ 0.1 eV). The bundle remains metallic. We remark that such predictions found experimental demonstration in low-temperature STM measurements [37], for metallic tubes and ropes similar to that shown in Fig. 3.1. (ii) The interaction between different tubes in a bundle makes the electron-hole asymmetry in the DOS more accentuated. The spike structure of the van Hove singularities is smoothed out. The fact that the position in energy of the peaks is not strongly modified explains the success of using isolated single-wall nanotube (SWNT) spectra to describe the experimental data, even though most experiments are performed on bundles. However, Fig. 3.1 also shows that the shape of the spectra (relative intensities) is significantly different for bundles with respect to isolated tubes. Therefore, care must be adopted in quantitative comparisons. Fig. 3.1. STM image (left) and DOS (right) for a small carbon nanotube-rope formed by three (8,8) SWNTs (with 1.09 nm diameter) packed in a triangular lattice, with an inter-tube distance of 0.345 nm. The DOS plot clearly shows the opening of a “pseudogap” of about ∼ 0.1 eV around the Fermi level. The DOS of the nanotube rope is compared in the same plot to the DOS of an isolated (8,8) SWNT [Adapted from [38] with permission; Copyright 1999 by Springer-Verlag] As a second issue, we address the role of topological defects [39], focusing on Stone-Wales (SW) defects [40, 41]. STM images for an applied bias potential of ±1.5 eV are presented in Fig. 3.2 for a (10,10) CNT (a) and for a zig-zag (6,0) BNT (b). Such images illustrate that the SW deformation creates a very localized modification of the STM pattern as compared to a 82 R.D. Felice et al.