Interpreting non-semielliptical complex bands.

Complex band structure (CBS) emerges when translational symmetry is broken and material states with complex wavevectors become admissible. The resulting complex bands continuously connect conventional bands and their shapes are directly related to measurable physical quantities. To date, interpretations of complex bands usually assume they are semielliptical because this is the shape produced by the Su-Schrieffer-Heeger (SSH) model. However, numerous studies have reported CBSs with distinctly non-semielliptical shapes, including loops (essentially deformed, asymmetric semiellipses), spikes, and vertical lines. The primary goal of this work is to explore the phenomenology of these shapes such that deeper physical insight can be obtained from a qualitative inspection of a material's CBS. By using several variations on the SSH model, we find that (i) vertical lines are unphysical numerical artifacts, (ii) spikes indicate perfectly evanescent states in the material that couple adjacent layers but do not transfer amplitude, and (iii) asymmetric loops result from hybridization. Secondarily, we also develop a strategy for eliminating any unphysical vertical lines from calculations, thereby improving computational techniques for CBS.

Complex band structure and electronic transmission eigenchannels.

It is natural to characterize materials in transport junctions by their conductance length dependence, ß. Theoretical estimations of ß are made employing two primary theories: complex band structure and density functional theory (DFT) Landauer transport. It has previously been shown that the ß value derived from total Landauer transmission can be related to the ß value from the smallest |ki| complex band; however, it is an open question whether there is a deeper relationship between the two. Here we probe the details of the relationship between transmission and complex band structure, in this case individual eigenchannel transmissions and different complex bands. We present calculations of decay constants for the two most conductive states as determined by complex band structure and standard DFT Landauer transport calculations for one semi-conductor and two molecular junctions. The molecular junctions show that both the length dependence of the total transmission and the individual transmission eigenvalues can be, almost always, found through the complex band structure. The complex band structure of the semi-conducting material, however, does not predict the length dependence of the total transmission but only of the individual channels, at some k-points, due to multiple channels contributing to transmission. We also observe instances of vertical bands, some of which are the smallest |ki| complex bands, that do not contribute to transport. By understanding the deeper relationship between complex bands and individual transmission eigenchannels, we can make a general statement about when the previously accepted wisdom linking transmission and complex band structure will fail, namely, when multiple channels contribute significantly to the transmission.

A unified perspective of complex band structure: interpretations, formulations, and applications.

Complex band structure generalizes conventional band structure by also considering wavevectors with complex components. In this way, complex band structure describes both the bulk-propagating states from conventional band structure and the evanescent states that grow or decay from one unit cell to the next. Even though these latter states are excluded by translational symmetry, they become important when translational symmetry is broken via, for example, a surface or impurity. Many studies over the last 80 years have directly or indirectly developed complex band structure for an impressive range of applications, but very few discuss its fundamentals or compare its various results. In this work we build upon these previous efforts to expose the physical foundation of complex band structure, which mathematically implies its existence. We find that a material's static and dynamic electronic structure are both completely described by complex band structure. Furthermore, we show that complex band structure reflects the minimal, intrinsic information contained in the material's Hamiltonian. These realizations then provide a context for comparing and unifying the different formulations and applications of complex band structure that have been reported over the years. Ultimately, this discussion introduces the idea of examining the amount of information contained in a material's Hamiltonian so that we can find and exploit the minimal information necessary for understanding a material's properties.

Estimating the Landauer-Büttiker transmission function from single molecule break junction experiments.

When investigating the electronic response properties of molecules, experiments often measure conductance whereas computation predicts the transmission probability. Although Landauer-Büttiker theory usually relates the two, comparison between experiment and computation remains difficult because experimental data (specifically those from break junctions) are statistical and computational results are deterministic. In this work we develop tools to quantitatively estimate-with error bars-the shape of the Landauer-Büttiker transmission function directly from experimental statistics on conductance and thermopower (if the latter is also available). We subsequently apply these tools to existing data, demonstrating a rigorous statistical comparison between experimental and computational results on molecular electron transport.

Quantitative Interpretations of Break Junction Conductance Histograms in Molecular Electron Transport.

We develop theoretical and computational tools for extracting quantitative molecular information from experimental conductance histograms for electron transport through single-molecule break junctions. These experimental setups always measure a combination of molecular conductance and direct electrode-electrode tunneling; our derivations explicitly incorporate the effects of such background tunneling. Validation of our models to simulated data shows that background tunneling is crucial for quantitative analyses (even in cases where it appears to be qualitatively negligible), and comparison to experimental data is favorable. Finally, we generalize these ideas to the case of molecules with a destructive interference feature and discuss potential signatures for interference in a conductance histogram.

Communication: Finding destructive interference features in molecular transport junctions.

Associating molecular structure with quantum interference features in electrode-molecule-electrode transport junctions has been difficult because existing guidelines for understanding interferences only apply to conjugated hydrocarbons. Herein we use linear algebra and the Landauer-Büttiker theory for electron transport to derive a general rule for predicting the existence and locations of interference features. Our analysis illustrates that interferences can be directly determined from the molecular Hamiltonian and the molecule-electrode couplings, and we demonstrate its utility with several examples.

Response to "Comment on 'Rethinking first-principles electron transport theories with projection operators: the problems caused by partitioning the basis set'" [J. Chem. Phys. 140, 177103 (2014)].

The thesis of Brandbyge's comment [J. Chem. Phys. 140, 177103 (2014)] is that our operator decoupling condition is immaterial to transport theories, and it appeals to discussions of nonorthogonal basis sets in transport calculations in its arguments. We maintain that the operator condition is to be preferred over the usual matrix conditions and subsequently detail problems in the existing approaches. From this operator perspective, we conclude that nonorthogonal projectors cannot be used and that the projectors must be selected to satisfy the operator decoupling condition. Because these conclusions pertain to operators, the choice of basis set is not germane.

Rethinking first-principles electron transport theories with projection operators: the problems caused by partitioning the basis set.

We revisit the derivation of electron transport theories with a focus on the projection operators chosen to partition the system. The prevailing choice of assigning each computational basis function to a region causes two problems. First, this choice generally results in oblique projection operators, which are non-Hermitian and violate implicit assumptions in the derivation. Second, these operators are defined with the physically insignificant basis set and, as such, preclude a well-defined basis set limit. We thus advocate for the selection of physically motivated, orthogonal projection operators (which are Hermitian) and present an operator-based derivation of electron transport theories. Unlike the conventional, matrix-based approaches, this derivation requires no knowledge of the computational basis set. In this process, we also find that common transport formalisms for nonorthogonal basis sets improperly decouple the exterior regions, leading to a short circuit through the system. We finally discuss the implications of these results for first-principles calculations of electron transport.

The role of dimensionality in the decay of surface effects.

We computationally investigate the decay of surface effects in one-, two-, and three-dimensional materials using two-band tight-binding models. These general models facilitate a direct comparison between materials of differing dimensionality, which reveals that material dimensionality (not material-specific chemistry/physics) is the primary factor controlling the decay of surface effects. Our results corroborate more sophisticated, material-specific studies, finding that surface effects decay after ∼10, ∼25, and ≳ 100 layers in three-dimensional, two-dimensional, and one-dimensional materials, respectively. Physically, higher-dimensional materials screen surface effects more efficiently, as theoretically described by integration over each layer's Brillouin zone. Finally, we discuss several implications of these results.

Signatures of cooperative effects and transport mechanisms in conductance histograms.

We present a computational investigation into the line shapes of peaks in conductance histograms, finding that they possess high information content. In particular, the histogram peak associated with conduction through a single molecule elucidates the electron transport mechanism and is generally well-described by beta distributions. A statistical analysis of the peak corresponding to conduction through two molecules reveals the presence of cooperative effects between the molecules and also provides insight into the underlying conduction channels. This work describes tools for extracting additional interpretations from experimental statistical data, helping us better understand electron transport processes.

Model for adsorption of ligands to colloidal quantum dots with concentration-dependent surface structure.

A study of the adsorption equilibrium of solution-phase CdS quantum dots (QDs) and acid-derivatized viologen ligands (N-[1-heptyl],N'-[3-carboxypropyl]-4,4'-bipyridinium dihexafluorophosphate, V(2+)) reveals that the structure of the surfaces of the QDs depends on their concentration. This adsorption equilibrium is monitored through quenching of the photoluminescence of the QDs by V(2+) upon photoinduced electron transfer. When modeled with a simple Langmuir isotherm, the equilibrium constant for QD-V(2+) adsorption, K(a), increases from 6.7 × 10(5) to 8.6 × 10(6) M(-1) upon decreasing the absolute concentration of the QDs from 1.4 × 10(-6) to 5.1 × 10(-8) M. The apparent increase in K(a) upon dilution results from an increase in the mean number of available adsorption sites per QD from 1.1 (for [QD] = 1.4 × 10(-6) M) to 37 (for [QD] = 5.1 × 10(-8) M) through desorption of native ligands from the surfaces of the QDs and through disaggregation of soluble QD clusters. A new model based on the Langmuir isotherm that treats both the number of adsorbed ligands per QD and the number of available binding sites per QD as binomially distributed quantities is described. This model yields a concentration-independent value for K(a) of 8.7 × 10(5) M(-1) for the QD-V(2+) system and provides a convenient means for quantitative analysis of QD-ligand adsorption in the presence of competing surface processes.

Molecular conduction through adlayers: cooperative effects can help or hamper electron transport.

We use a one-electron, tight-binding model of a molecular adlayer sandwiched between two metal electrodes to explore how cooperative effects between molecular wires influence electron transport through the adlayer. When compared to an isolated molecular wire, an adlayer exhibits cooperative effects that generally enhance conduction away from an isolated wire's resonance and diminish conductance near such a resonance. We also find that the interwire distance (related to the adlayer density) is a key quantity. Substrate-mediated coupling induces most of the cooperative effects in dense adlayers, whereas direct, interwire coupling (if present) dominates in sparser adlayers. In this manner, cooperative effects through dense adlayers cannot be removed, suggesting an optimal adlayer density for maximizing conduction.

Guidelines for choosing molecular "alligator clip" binding motifs in electron transport devices.

We employ a one-electron, tight-binding model of an electrode-molecule-electrode junction to explore the fundamental relationship between adsorption geometry and electron transport, producing exact results (within this model). By varying the chemisorption location (e.g., atop a surface atom or in a hollow site between surface atoms) and the molecule-electrode coupling, we find that the largest currents are realized when the molecule (i) is highly coordinated by the surface and (ii) has favorable overlap with electrode states near the Fermi level. We also show the importance of electrode-induced molecular level shifting for certain adsorption geometries, which can cause molecular levels far from the Fermi level to conduct better than those near the Fermi level. Since all of these factors are greatly influenced by the chemical moiety used to link the molecule to an electrode, these results present a set of guidelines to help choose "alligator clips" for molecular electronic devices.

Closed-form Green functions, surface effects, and the importance of dimensionality in tight-binding metals.

Closed-form expressions for all elements of a d-dimensional tight-binding metal's Green function matrix are presented and used to explore edge effects of a surface. We find that, when moving from the surface into the bulk, the number of layers passed before the surfaced substrate behaves like the bulk decreases with dimensionality. In particular, the surface of a one-dimensional substrate becomes indistinguishable from the bulk after O(10(1)-10(2)) layers, a two-dimensional substrate after O(10(1)) layers, and a three-dimensional substrate after O(10(0)) layers. Finally, the effects of substrate dimensionality on molecule-substrate interactions are discussed.

A fast method for solving both the time-dependent Schrödinger equation in angular coordinates and its associated "m-mixing" problem.

An efficient split-operator technique for solving the time-dependent Schrödinger equation in an angular coordinate system is presented, where a fast spherical harmonics transform accelerates the conversions between angle and angular momentum representations. Unlike previous techniques, this method features facile inclusion of azimuthal asymmetries (solving the "m-mixing" problem), adaptive time stepping, and favorable scaling, while simultaneously avoiding the need for both kinetic and potential energy matrix elements. Several examples are presented.

Carborane-based pincers: synthesis and structure of SeBSe and SBS Pd(II) complexes.

We report the synthesis of several unique, boron-rich pincer complexes derived from m-carborane. The SeBSe and SBS pincer ligands can be synthesized via two independent synthetic routes and are metalated with Pd(II). These structures represent unique coordinating motifs, each with a Pd-B(2) bond chelated by two thio- or selenoether ligands. This class of structures serves as the first example of boron-metal pincer complexes and possesses several interesting electronic properties imposed by the m-carborane cage.

Molecular transport junctions with semiconductor electrodes: analytical forms for one-dimensional self-energies.

Analytical self-energies for molecular interfaces with one-dimensional, tight-binding semiconductors are derived, along with analytical solutions to the electrode eigensystems. These models capture the fundamental differences between the transport properties of metals and semiconductors and also account for the appearance of surface states. When the models are applied to zero-temperature electrode-molecule-electrode conductance, junctions with two semiconductor electrodes exhibit a minimum bias threshold for generating current due to the absence of electrode states near the Fermi level. Molecular interactions with semiconductor electrodes additionally produce (i) non-negligible molecular-level shifting by mechanisms absent in metals and (ii) sensitivity of the transport to the semiconductor-molecule bonding configuration. Finally, the general effects of surface states on molecular transport are discussed.

Laser field alignment of organic molecules on semiconductor surfaces: toward ultrafast molecular switches.

An ultrafast, nanoscale molecular switch is proposed, based on extension of the concept of nonadiabatic alignment to surface-adsorbed molecules. The switch consists of a conjugated organic molecule adsorbed onto a semiconducting surface and placed near a scanning tunneling microscope tip. A low-frequency, polarized laser field is used to switch the system by orienting the molecule with the field polarization axis, enabling conductance through the junction. Enhancement and spatial localization of the incident field by the metallic tip allow operation at low intensities. The principles of nonadiabatic alignment lead to switch on and off time scales far below rotational time scales.