Mesoscopic systems in the quantum realm: fundamental science and applications

A schematic picture of electronic spectra is shown in figure 1, going from an isolated molecule, to a cluster/small nanoparticle and to a large nanoparticle. The localized electronic states become progressively delocalized upon the increasing overlap of wave functions as the system size increases towards the bulk limit. Some physical properties, notably electrical conductivity, appear only as a consequence of spatial extendedness on the part of the electron wave function, thereby manifesting its quantum mechanical nature. Quantum mechanics provides a mathematical/physical description at the atomic and subatomic scale where classical mechanics fails completely to do so. It provides a unified view of the performance of atomic-scale objects, including electrons, photons and other elementary particles and excitations.

Zoom In Zoom Out Reset image size

When a system size is at the atomic scale, it is only quantum mechanics that can account for the observed physical properties. In the jargon of the field, 'quantum confinement' means that the de Broglie wavelength of the particles is comparable to the size of the system that contains them. Small size implies strong quantum confinement effects. Bulk extended matter, when sufficiently curtailed in one of its dimensions, for example in the z-direction, will behave as a quasi-two-dimensional system in the complementary x–y-plane. The price to be one paid for this is to lose the long-range extendedness of the wave function in the z-direction; the energy of motion along that axis becomes quantized as the wave function is confined.

Repeating the restriction in a second direction, say the y-direction, will make the system quasi-one-dimensional in the remaining x-direction; now the wave function is deliberately forced to occupy a small region within the y–z-plane. One last restriction along the x-axis of motion would produce a quasi-zero-dimensional system, where all extendedness of the wave function has been lost, in every dimension. The entire electronic system is localized in a relatively tiny volume of x, y and z. As a result of the complete confinement, all the physical properties of the system will be drastically affected (see the schematic figure 2 below). We have made, then, an 'artificial atom' or a quantum dot.

Zoom In Zoom Out Reset image size

From this brief discussion we learn that the electronic, optical and mechanical (elastic) properties of materials are radically changed by both size and shape. Well-established technical achievements, including zero-dimensional quantum dots, have been attained through ingenious size manipulation—and for that the quantum-confinement effect is crucial.

It would be instructive to follow how the theory of quantum confinement tracks the behaviour of an exciton (a jointly bound electron and hole pair) as it crosses over to an atomic-like orbital as its host space is progressively diminished. A rather good approximation of an exciton's behaviour is the 3D model of a particle in a shrinking box. A systematic solution to this problem provides the mathematical connection between the evolution of energy states and the dimensionality of the space within which the wave function exists. It is obvious in any case that decreasing the volume, or the dimensionality, of the available space increases the energy of the states.

The following equations show the non-interacting wave function and energy of electrons:

Using the above wave functions and energies one can calculate carrier density and the density of the states (the availability of quantum states are those solutions allowed within the system). Given in figure 2 are the densities of electronic states in all space dimensions from three to zero. Those are: in 3D, (for each sub-band) in 2D, (for each sub-band) in 1D and ρ(E) as a set of discrete δ-function spikes in 0D.

Inter-electronic correlations will modify the above ρ(E) spectra. This qualitatively different part of the quantum picture is crucial to understanding the dynamics of electrons acting under an externally applied field; a problem that governs both the characterization of nanodevices and their eventual practical uses. Here we remark that if an electrically active system's length is reduced to the nanoscale, there will be considerable changes in its properties. At the bulk phase, the device interfaces are expected to control some of the macroscopically observed properties (for instance, they affect access resistance). But at the nanoscale, a system's interfaces with the 'rest of the universe' have more spectacular effects.

What is the role of a surface? A surface is said to be the first frontier or line of defence for any interaction with the outside world. In general a system always minimizes its free energy. Unless it is truly isolated thermodynamically, it must do so in the presence of its mechanical and electromagnetic coupling to the world outside its boundaries. So the question is: to what extent do the internals of the system dominate its free energy, and to what extent do all of its various interactions at the surface contribute?

The smaller the size of the system, the larger the ratio of its surface area S to volume V is. A high ratio implies a strong thermodynamic 'driving force' that speeds up many of the processes that minimize thermodynamic free energy. Chemically, the smaller the size of a material sample, the faster its reactions at a relatively large S. A porous material's chemical reactivity (e.g. a catalytic exhaust converter) is much greater because of its large surface area. For the same reason as a high S:V ratio, nano- or mesomaterials have a higher chemical reactivity compared to the bulk. For biological systems, surface-to-volume ratios are more significant still.

The surface-to-volume ratio for a 3D cube can be readily obtained. The total surface area of a cube is S=6L², whereas the volume is V=L³. Therefore, S/V=6/L. As L→ larger, S/V→ smaller. For other structures such as a sphere or ellipsoid, the S/V ratio can be similarly calculated to scale as 1/L. A close-packed cubic structure of 1 ^{cm 3} contains a number of atoms of the order of Avogadro's number ∼10²³. The number of its surface atoms will be ∼6×10¹⁵, so the ratio of surface atoms to atoms within the bulk is ∼10⁻⁷; on that measure the number of surface atoms—those that mediate most of the system's interactions with the surroundings—would appear to be insignificant. On the other hand, if we take a one-nanometre cube (10 atoms in a row), the surface-to-volume atoms' ratio is ∼6×10²/10³=0.6. The fraction of boundary atoms is in every way significant. Even this most simple size argument demonstrates the potential importance of surface-mediated effects over bulk effects as the system size is reduced. Thus, the smaller the system, the more its surfaces must dominate its actual properties.

Researchers in the US recently made a surprisingly novel chemical structure that has the largest internal surface area ever observed in an ordered material. Omar Yaghi and co-workers at Michigan and Arizona State Universities fabricated a new porous metal–organic framework with an estimated surface area of 4500 ^{m 2  g −1}, which is nearly five times larger than the previous known record. The structure can bind large quantities of gas and could, therefore, find a variety of applications, including gas storage and catalysis [12].

With a decrease in size, the surface area and surface energy increase, and thereby the melting point of a sufficiently small sample decreases. For example, 3-nm-wide CdSe nanocrystals melt at 700 K, compared to a CdSe bulk that melts at 1678 K. This is another confirmation of the dominance of surface/boundary effects at the nonoscale. It is a very similar story for nanoelectronic devices: a conducting structure's coupling to the exterior circuit is via its boundary surfaces. The latter's nature can substantially modify, and indeed often dominate, observed transport behaviour.

As we indicated earlier atoms in nanostructures have a higher average energy than atoms in larger structures, because most of them are surface atoms and the uneven bonding generates new tensional forces not otherwise experienced at equilibrium in the deep bulk. Consequently, the chemical activity of a material can be exponentially improved as the material is reduced in size at the nanoscale. The properties of nanosystems are significantly affected by minor changes in size, shape or surface states of their particular structures.

In summary, at the nanoscale, properties become strongly size-dependent. Here are some examples of various properties related to some phenomena sensitive to size: (i) chemical properties—reactivity, catalysis; (ii) mechanical properties—adhesion, capillary forces; (iii) thermal properties—melting temperature; (iv) electrical properties—tunnelling current, dipole layers; (v) optical properties—absorption and scattering of light; (vi) magnetic properties—super-paramagnetic effect.

Obviously these new qualitative and quantitative properties herald entirely new applications which, being out of the reach of our earlier science of bulk materials, often lack any obvious technological precedents. They represent a truly unexplored domain.

Already, for a long time, we have been studying the physics, chemistry and biology of atoms, molecules, clusters and other collective entities; what is so special about so-called nano- or mesoscience? Viewed from the right perspective one would find that the objects mentioned above are never encountered in an isolated state—in vacuo, so to speak. Rather, we find them to be always coupled to some active environment.

This embedding in the environment (also known as the 'bath') introduces the idea of dissipation (friction) whereby the system's energy is transferred irreversibly to the bath. As first analysed by Einstein in his third famous paper from 1905, it is the evident stability of an embedded system, together with the ubiquitous presence of energy dissipation, which implies that the system is subject to fluctuating microscopic forces. The counterbalancing of the twin effects of fluctuation and dissipation induces the system to relax (settle down) to thermal equilibrium, at a characteristic energy (temperature) set by the bath.

In quantum dynamics there is yet another feature beyond those two dynamical drivers: the system can display coherent effects, i.e. long-range interference phenomena over space and time. Coherence, by its nature, introduces a very high level of orderly correlations. The rate of spatial decay in a particle's quantum-state correlations is characterized by the coherence length. As mentioned earlier, in mesosystems, the coherence length is much larger than the inter-particle separation but still smaller than the system size. However, once a system's inevitable coupling to the environment is taken into account, the correlations are degraded; an effect known, logically enough, as 'decoherence' or 'dephasing'. It is precisely this interaction of the system with the enveloping bath that makes nanoscience non-trivial from a fundamental point of view. In other words, it is the essential quantum nature of nanoscopic matter intimately interacting with its much bigger, macroscopic surroundings that defines its 'nanoscopic' aspect in the first place. We make this point explicit in the following sections.

2.3.1. Closed systems: equilibrium states

2.3.2. Open system: interacting

Now consider a nano-object embedded in its interacting environment. We are still in an equilibrium scenario; the big difference is that the nano-object, in its interactions with the environment, undergoes changes in its electronic distribution, vibrational properties, excitation energies, etc. However, we should keep in mind that the system, by its very openness to the bath, loses its total 'ownership' of these attributes; one needs an integral approach in which the bath is an explicit player.

Generally, interfacial properties are drastically affected when a nanostructure is embedded in an interacting environment. Here we present a simple example of an acetone molecule [^CH ₃]₂ ^CO (figure 3) both in gas and in a solvent. Results are obtained from a time-dependent density-functional-theory/effective-fragment-potential approach (quantum molecular dynamical calculations) [15], showing the change of some parameters as the environment changes, table 1 below).

Zoom In Zoom Out Reset image size

Table 1. The average CO distance (R ^_CO ) as well as the average values of the vertical excitation (ω₁), the highest occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO) energies, which are obtained from the equilibrated MD trajectories. Energies are given in eV and distances in Å.

R ^_CO HOMOLUMO

Acetone in gas	1.222	4.382	−6.744	−0.443
Acetone in solvent	1.233	4.593	−7.124	−0.634
Change Δ	0.011	0.211	−0.380	−0.191

The relative change in parameters with the environment, such as a shift in the HOMO energy level, can be substantial. More importantly, non-trivial effects will be discussed in the next section for nanosystems contacted with large metallic leads for the transport of electrical current.

This is a very elementary theory of quantum transport. One assumes that single-particle states are identical to the actual (collective) excitations of a many-body system. The best example is the single-particle orbitals found in the Kohn–Sham density functional theory. In the special case of non-interacting (or even Hartree-like mean-field interacting) quasi-particles, the conductivity is expressed as a Fermi Golden Rule formula that involves matrix elements of the current operator, which is naïvely interpreted to be W_nm ∼[ψ _n ^* ∇ ψ _m −∇ ψ _n ^* ψ _m ]. Here ψ's are Kohn–Sham effectively single-particle wave functions. Technically, this picture corresponds to an approximation that incorporates only the first bubble diagram of conductivity [27, 28], which lands one in the several serious difficulties of consistency that we discuss below.

We remark that in strongly metallic systems the independent-particle picture can be placed on a firm theoretical footing (the Fermi-liquid, or quasi-particle, theory of Landau). This succeeds in incorporating real scattering effects subject to the assumption that the basic transport states are totally delocalized and have intrinsic non-zero current associated with them. In a driven system the observed current is the outcome of the slight induced imbalance in the occupancy of counter-propagating states. But each such state is itself distributed throughout the system; that is the essence of metallic charge transport.

In contrast to metallic transport, charge transfer by quantum tunnelling takes place by the hopping of single charges from one isolated conducting island on the left of a device to another island on its right. These islands, though in proximity to each other, are separated by a finite-width potential energy barrier. States are localized to one or other of the islands and therefore can sustain different amounts of local occupancy. Note that this is a huge qualitative difference from metallic states, which may be filled or empty but always delocalized across the entire system and carrying an intrinsic current.

In deriving the Bardeen formula, one encounters the conceptual problem that the eigenstates of the left island and the right island do not together form a complete orthogonal set of total Hamiltonian for tunnelling. Despite this basic formal weakness, Bardeen's formula is at least a physically consistent model with a successful repertoire of useful applications [29]. It is entirely different, however, from metallic transport, which depends on the existence of completely delocalized particle states.

The Landauer formula [30] is a popular formula that deals with coherent transport in a wide span of mesoscopic systems. In the Landauer picture the mesoscopic system is attached to two reservoirs. These left and right reservoirs are assumed to be in a thermodynamic equilibrium state with local but dissimilar chemical potentials from/to which the electrons, in single-particle states, are injected/collected at both ends.

The mismatch in chemical potentials is identified by the externally applied voltage difference; a crucial assumption which means that the left and right reservoirs behave as in the Bardeen model: they have different local occupancies. Further, the electron reservoirs are adiabatically (seamlessly) connected to three-dimensional macroscopic leads, where the electrons are completely free to propagate.

In the absence of scattering, the Landauer formula for states in a single sub-band gives a remarkable result for conductance: G=2e ² /h, where e and h are the quantum of electronic charge and Planck's constant, respectively. With N conducting sub-bands the right hand side of the formula is multiplied by N since each band is an independently open channel conducting in parallel with the rest. This phenomenon is known as quantized ballistic conductance.

If there is scattering in a ballistic system, it takes place at and within the interfaces with the reservoir leads: by assumption the inner system itself is free of impurities. The single-particle nature of the Landauer formula deals only with elastic scattering. Using a one-body potential scattering theory, the conductance formula is modified to read G=(2e ² /h)T, where T is the transmission function or factor [30]. The function T is the probability that a single electron will traverse the ballistic wire. As a probability, its eigenvalues are bounded between 0 to 1. Where the transmission factor is zero, the system does not conduct, while for unity transmission factors the system is ideally ballistic.

This simple picture suffers from a set of conceptual deficiencies. Obvious ones are:

• How can a system with ballistic (i.e. strictly elastic) transport have finite conductance without any dissipation of energy?
• The notion of reservoirs attached to the system is redundant to some extent; it is incorporated within the boundary conditions for any electrical circuit.
• The two quite distinct modalities of conduction, tunnelling and metallic charge flow , as outlined above, are applied interchangeably in Landauer picture. No physical distinction is made between them. This begs the question as to when and how a Bardeen-like insulating barrier (as subsumed in the zero-transmission limit of the Landauer model) is able to morph into an ideal ballistic conductor whose states extend right across the structure—as for a metal.

The situation is made clear in the following figure 4. Conductance is shown as a function of barrier thickness in both the Landauer and Bardeen models [29]. For large barrier thicknesses both the models coincide and give the same result. However, for small barrier thicknesses the Bardeen conductance diverges (as logically it should) while the Landauer result goes to unity as the 'barrier' thickness vanishes. The tunnelling formula is inappropriate for a very small separation and, for the Landauer picture, it is unphysical. If transport becomes ballistic (ideally metallic), the underlying states are extended and the assumption of different local occupancies of the reservoirs has little meaning.

Zoom In Zoom Out Reset image size

Since for normal resistive conduction (finite G) the introduction of incoherence is an essential objective, Büttiker [30] introduced into the ballistic system, in a fully phenomenological way, additional probes with adjustable chemical potentials: places for the carriers to instantaneously relax and lose their coherent phase memory.

The Landauer formula includes transmission via a one-body potential scattering method, identical to an elastic scattering process [30]. We have seen that a natural and unavoidable physical process in mesoscopics is scattering by many-body interactions. This renders inoperative any concept strictly reliant on single-particle description alone. Therefore, the simple inclusion of passive reservoirs and leads, and even of additional phenomenological voltage probes, cannot save an exclusively single-particle treatment of physics at the nano/meso level [27, 28, 31]. Rather more is needed.

In the late 1970s P W Anderson and co-workers (popularly known as The Gang of Four) established that in a non-interacting, elastically disordered system there is an insulator–metal transition only in 3D, while all other low-dimensional systems must be insulators. In the language of 'one-body scattering theory' many have concluded that the quantum-coherent discrete conductance for a 1D system, as predicted (among others) by Landauer's transport theory, and amply experimentally confirmed, is a contradiction to The Gang of Four's insulator result. See for example, Anderson in '50 years of Anderson localisation' published in IJMPB 24 (2010) 1501. In this area 'weak localization and mesoscopics' have some commonality in popular literature (See for example, M. Büttiker and M. Moskalets in the above reference p. 1555).

Kubo's linear response theory represents the first full quantum-mechanical formalism in modern kinetics [32]. It connects the irreversible processes prescribed for the non-equilibrium state to the thermal fluctuations observed in equilibrium. Kubo's formula is more popularly known, in fact, as the fluctuation dissipation theorem (FDT).

The Kubo theory is completely general and encompasses not only bulk quantum systems but also meso/nanosystems equally. In principle, the study of transport does not limit us to those non-equilibrium states lying sufficiently close to equilibrium. Still, computation of linear kinetic coefficients is easiest to carry out, since the final expression involves purely equilibrium expectation values of the relevant dynamical variables, a much simpler procedure than extracting any corresponding far-from-equilibrium quantities.

In the context of mesoscopic systems, the Kubo formulation dates back to the 1980 s with Fisher and Lee and others [33], who derived the Landauer formula in the non-interacting (independent particle) limit of the Kubo formula. Our derivation of conductance [31] from the Kubo formula also produces the Landauer formula as a natural outcome, but the transmission function includes an inelastic part (as demanded by the physics of dissipation) and not purely elastic as in independent-particle approaches. In detail, transmission is a much more complicated concept and invokes many-body scattering, in which elastic and inelastic processes work together.

An analogous microscopic analysis of mesoscopic transport has been made by Soree and Magnus [34], who derived conductance quantization purely using the method of non-equilibrium statistical operators and without any mismatched-reservoir phenomenology. In a different language, Di Ventra and Vignale [27] show that the Landauer formula is incomplete, being devoid of essential many-body effects. There has been use and misuse of the Landauer formula in many papers over time. We have critically examined a few examples in a recent viewpoint paper [35].

A rigorous, microscopic and completely general theory of quantum transport can be based on the NEGF formalism developed by Keldysh [36] and Kadanoff and Baym [37]. It has been applied to device problems since then (for applications of NEGF see [38]). This method allows one to systematically solve the interactions within the electron propagator (Green function) under fully non-equilibrium conditions. Therefore, in principle, all scattering mechanisms arising out of many-body correlations can be taken into account systematically, in a well-controlled way, in the evaluation of the current.

When a system is driven out of equilibrium by an applied bias, a standard quasi-equilibrium perturbation theory to a finite order is not suitable to describe the transport properties; for the system response can be strongly nonlinear. The resultant net flux of current, sustained by the external bias, is evidence that the system is not in equilibrium. In particular, equilibrium-state theory is incapable of describing real exchanges of energy between electronic and phonon-bath subsystems, and therefore has no means of capturing the physics of heating and dissipation. This is obvious, since in equilibrium there cannot be a steady net flux of energy out of the electron system into the vibrational degrees of freedom.

In its current popularly accepted form, the so-called NEGF formalism does not drastically differ from an equilibrium theory. The principal technical difference is that all time-dependent functions are defined for time-arguments located on a Keldysh contour (in the complex time-plane) [36]. The advantage of the current versions of the NEGF is that one can systematically improve approximations by taking into account various physical processes. The disadvantage (though this is true only for the popularly adopted form and not for the genuine NEGF method) is that the Ward identities are not guaranteed, and so neither is microscopic conservation.

We cannot review here such a broad and rather technical field. Useful descriptions of the NEGF theory can be found in Haug and Jauho [39] and Langreth [40]. We shall discuss some salient issues of this theory in the forthcoming 16th Vietnam School of Physics in Hanoi.

Contacts in an electrical circuit are very important because, self-evidently, the current must pass through them to drive the system of interest. The physics of contacts becomes more subtle when the system is in a non-equilibrium state. The contacts can be of different types: bimetallic or Schottky (i.e. contact between a metal and a non-metal). Passage of electrons through the contact decides the conducting nature of the device. As an example, we point to an interesting molecular nanodevice (known as the Aviram–Ratner Diode [41]) which has a rectifying property. Rectification takes place because of (i) the specific nature of the contacts and (ii) the degree of symmetry of the molecule, which is the active device bridging the contacts.

Now we return to the main discussion. The simplest electronic meso- or nanodevice will have three parts to it: two leads on either side to make contact with the third item: the central, active region. The central region may be a quantum dot, a molecule, a nanotube or a 1D wire, etc. One can create this device, conceptually, by correspondingly partitioning the total Hamiltonian into three parts, with couplings between the parts representing the action of the contacts. This is a popular model, due to Meir and Wingreen [42]. In figure 5, we have shown a representative example of a molecular diode—a two-terminal device attached to two leads on either side. This rectifying diode is current carrying [43].

Zoom In Zoom Out Reset image size

Partitioning of the Hamiltonian in the non-equilibrium problem has severe difficulties, which seem to have been comprehensively ignored [44, 45]. The steady-state current in the NEGF approach is obtained in a form that looks simple enough; yet the NEGF and the non-equilibrium self-energy of the particle states still need to be computed before one can extract a workable and self-contained calculation. This is undoubtedly a formidable task, if it is to be performed seriously.

A much-advocated ad hoc simplification is to drop the essential non-equilibrium interacting parts. As a result the conductance formula re-emerges as a Landauer formula [42, 46]. In this formula the effective transmission function T ^_eff is given by T ^_eff =Tr[Γ ^_L G ^r Γ ^_R G ^a ]. Here Tr stands for a trace, Γ _^L,^R are imaginary parts of the single-particle self-energy, coming from the left and right couplings of leads to the central region. G ^r,a are retarded and advanced normal Green functions. They are all functions of energy. This attractive formula is routinely used in a number of computer codes to calculate current-voltage characteristics and conductance as a function of various applied gate voltages [47].

• What has been included in this formula that is new? By partitioning the device into distinct leads and channels, one set up has the current-inhibiting effects associated with the interfaces: details of the microscopic effects at the interfaces are arrived at using density functional theory. Some of the Landauer model's modern proponents believe that this interface resistance is none other than the actual quantized device resistance. Attempts have now been made to include electron-correlations and inelastic scattering in a so-called Landauer-like formulation [27, 47]. Unless such attempts are genuinely many-body in nature, there is again no guarantee of control over charge and current conservation.
• What is importantly missing in this formula? When implementing this formula one has to be careful about some essential sum rules, such as particle number conservation, gauge invariance etc. The first question is: does the transmission factor T ^_eff preserve unitarity in the elastic channel? To answer that one has to show
  
  The second question is: does the formulation satisfy current conservation (Kirchhoff's rule) Σ _α J^α=0? We need to secure identity at a microscopic level
  
  (Here and are the Keldysh self-energy and Green functions, respectively.)A theory that violated these conservation constraints would be mathematically and physically unsustainable, whatever its potential appeal and success in reaching good agreement with the experiments.
• Working methods for the NEGF theory to transport Non-equilibrium quantum transport formulations, as discussed above, have been implemented by a number of authors (see for example [47] for references). The starting point of the implementations is the basis of electronic states of the system. To capture them, density functional theory is commonly employed. First, one solves the Poisson equation with appropriate boundary conditions to obtain the self-consistent Hartree potential. The exchange-correlation potential is added from a chosen form (local density approximation, generalized gradient approximation, etc) and then Kohn–Sham-like eigenvalues ε_i and eigenfunctions ψ_i are computed numerically. These computed quantities, ε_i and ψ_i , are used to obtain the self-energies and Green functions. Details of the technique commonly used in several computer codes vary from one to another [47].

Calculations are done for I–V characteristics, conductance as a function of source-drain voltage and/or gate voltage, self-consistent potential, charge-density distribution, etc.

As pointed out before, for normal dissipative transport it is essential to adopt a paradigm where inelasticity has a crucial role. Coherent transport is no longer in sole, exclusive play for actual dissipative systems. In the last 5–6 years many papers have appeared, treating inelasticity as an intrinsically many-body problem. At a basic level, electron–phonon and electron–electron interactions are treated within model Hamiltonian approaches [48–51]. The electron–phonon and electron-electron interactions not only directly affect the spectral function, the density of states and, consequently, the transport current; they also determine momentum-scattering time and thus the conductance.

Figure 6 shows the conductance steps of a quantum wire as a function of gate voltage, including inelastic scattering, in a linear response approach [31]. The latter clearly depresses the conductance by enhancing dissipation; yet the conductance steps survive. This demonstrates that the steps are not the unique signature of an ideally ballistic, purely elastic system. Another example is given of Enkovaara et al [52], who have studied a molecular transistor Au/benzene-ditheolate by a non-equilibrium electron transport method in conjunction with a time-dependent DFT. In figure 7 (left) the calculated I–V result is shown with the current steps as the bias voltage increases/decreases. On the right panel the effective potential is shown at a bias voltage of 3 V. The molecular bonds are clearly seen on both sides of the molecule.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size