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Adv. Nat. Sci.: Nanosci. Nanotechnol. 1 (2010) 043001 (11pp) doi:10.1088/2043-6262/1/4/043001

Mesoscopic systems in the quantumrealm: fundamental scienceand applicationsMukunda P Das

Department of Theoretical Physics, Research School of Physics and Engineering,The Australian National University, Canberra ACT 0200, Australia

E-mail: mukunda.das@anu.edu.au

Received 20 September 2010Accepted for publication 22 November 2010Published 16 December 2010Online at stacks.iop.org/ANSN/1/043001

AbstractIn mesoscopic systems three significant issues are of pivotal importance: (i) due to thesmallness of their size, quantum effects are crucial, (ii) for the same reason, thesurface-to-volume ratio is large and this characteristic feature induces certain unique andfascinating effects, and (iii) the system remains in active contact with its environment, whichinduces a variety of novel properties. In view of the enormity of this subject, we shall limit ourdiscussions to certain essential and fundamental science, with a number of non-trivialexamples. We shall highlight current activity on various issues mainly related to electrontransport in meso/nano systems. Models relevant to the latter have some real application tomolecular electronics.

Keywords: meso- and nanoscopic systems, inelastic, elastic and phase-breaking lengths,non-equilibrium statistical mechanics, dissipation in quantum systems, conductancequantization, molecular electronics

Classification numbers: 2.00, 2.02, 3.00, 4.00

1. Retrospection

Michael Faraday, pre-eminent in the history of physicsthrough his groundwork on the basis of electromagnetism,was also one of the greatest experimental physicists andchemists of all time. Less widely known is the fact thatFaraday carried out, in 1857, the very first experimentson colloidal gold suspensions [1]. He discovered andcharacterized small metallic-particle suspensions called ‘goldsol’, whose optical properties differ markedly from those ofbulk metals.

Faraday noted, without offering an explanation for thephenomenon, that variations in the size of the colloidalparticles gave rise to a corresponding variety of colours. Thisbehaviour is clearly quite distinct from what is seen in normalmetallic states (with the notable exception of very thin goldfilms, whose thickness is indeed comparable to the size ofFaraday’s colloidal particles). It is perhaps fair to see, inthese observations by Faraday, the birth of nanoscience andnanotechnology: a realm of condensed matter where bulk

properties cease to be the norm and requires us to reconsider,with due care, all of our familiar ways of thinking.

In recent times the notion of ‘nanotechnology’ hasbeen attributed to the celebrated physicist Richard Feynman.In his 1959 APS talk, ‘There’s Plenty of Room at theBottom’ [2], he speculated about all the possible ways inwhich miniaturization, information technology and physicscould be used to explore—and exploit—physics at the limitsof the microscopic world. His idea was bravely picked upand expanded by K Eric Drexler, whose 1986 book [3],Engines of Creation: The Coming Age of Nanotechnology,finally brought the field to wide popular attention. The term‘nanotechnology’ itself had been coined even earlier, in1974, by Norio Taniguchi from the Tokyo Science Universityto describe semiconductor processes, such as thin filmdeposition and ion beam milling, with control of size downto the order of nanometres.

Experimental nanotechnology, however, could not trulycome into its own before the advent of sufficiently sensitiveand delicate diagnostic instruments. An enormous step came

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Adv. Nat. Sci.: Nanosci. Nanotechnol. 1 (2010) 043001 M P Das

in 1981 when the IBM scientists Gerd Binnig, HeinrichRohrer and their collaborators in Zurich built the first scanningtunnelling microscope (STM) [4]. This permitted, for the firsttime, single atoms in situ to be seen by scanning a tiny atomicprobe over the surface of a silicon crystal. Previously, onehad to rely on brute-force imaging using high-energy probebeams, such as x-rays or electrons, at hundreds of keV (someof us might recall Erwin Müller’s famous image from the1950 s of atoms on a tungsten needle tip, produced usingfield-ion microscopy—but obtained at fields of the order of100 kVcm).

In 1990, IBM scientist Don Eigler and his co-workers,this time at Almaden, discovered how to use an STM tomanipulate individual xenon atoms on a nickel surface. In aniconic experiment, and with an inspired eye for marketing,they moved 35 atoms to spell out IBM [5]. Further techniqueshave since been developed to capture images at the atomicscale; these include the atomic force microscope (AFM),magnetic resonance imaging (MRI) and even a kind ofmodified optical microscope.

Other significant advances were made in 1985, whenchemists from Rice University in the USA and SussexUniversity in the UK discovered how to create a soccer-ballshaped molecule of 60 carbon atoms, which is known asbuckminsterfullerene (also known as C60 or Bucky balls) [6].In 1991 [7], tiny, super-strong rolls of carbon atoms knownas carbon nanotubes (also called buckytubes) were createdby Sumio Iijima at NEC in Japan. These tubes are sixtimes lighter, yet a hundred times stronger, than steel. Bothmaterials have important applications as nanoscale buildingblocks. Nanotubes have been made into fibres, long threads(of up to 15 cm) and fabrics, and used to create toughplastic composites, computer chips, toxic-gas detectors andnumerous other novel materials and devices. Another amazingmaterial in this class, invented by Andre Geim’s group inManchester, is graphene (the parent structure of normalgraphite), made from a sheet of interconnected carbon atomswith a honeycomb structure. It is a single layered materialwith very peculiar electronic properties due to its zero energygaps at special points in the Brillouin zone. It is now a hottopic of research both in theory and experiment because of itsvery special and novel electronic properties.

Scientists actively working on the nanoscale havealready created a multitude of nanoscale components anddevices. The never-ending list includes molecular transistors,quantum dots, quantum wires, nanodiodes, nanosensors andbiomolecular devices.

Many more applications can be found in theubiquitous Wikipedia entry for Nano-technology (http://en.wikipedia.org/wiki/Nanotechnology).

2. What is meso-/nanoscience?

As a definition, ‘Nanoscience’ is the study of materials whosephysical size is on the nanometre scale in the range of(1–1000 nm). While ‘nano’ means precisely small, ‘meso’ isa broader term, being intermediate between the microscopic(molecular) and macroscopic (bulk) scales. In practice,the ‘mesoscopic’ regime partly overlaps the description of‘nanoscopic’.

In mesoscopic physics the concept of ‘quantumcoherence’ is widely used. To practitioners in this field,a mesoscopic system inevitably means one that sustainsphase-coherent transport. Within a single-particle picture ofthe situation, a one-electron wave function remains coherentacross the entire system of interest. It is only in the presenceof elastic (energy-conserving) scattering that coherence of thewave function’s phase can be retained.

• Diffusive transportWhen host to a large concentration of impurities, ameso-conductor can show diffusive current conduction.In such cases it has sometimes been argued that astatic configurational average over the location of theimpurities will destroy its coherence. Indeed it does;but purely within that level of description, not as amatter of real physics. The mathematical analogy forthis viewpoint is drawn from wave scattering in chaoticcavities [8].

• Quantization phenomenaSome interesting physics has been discussed, atlength, regarding (i) interference, (ii) quantum-sizeand (iii) charging effects. The complex pattern ofdiffraction, arising from scattering off a random impuritydistribution, induces interference effects that modulatethe flow of electrons as a function of, for example, appliedvoltage. This appears in the conductance fluctuations ofmeso-systems: a universal feature in that the size of thefluctuations is sample-independent. Quantum-size effectsare a manifestation of quantum-well confinement towell-defined energy levels (sub-bands); the conductanceplateaux in a one-dimensional wire are examples ofthis effect. If the confinement is so that strong it bindsan electron within a localized state, as envisaged in aquantum dot, the attempt to add a further electron byelectrical or optical means can give rise to chargingeffects (Coulomb blockade). An analogous mechanism,involving electron spin, is also important for the Kondoeffect in a quantum dot.

• Physical decoherenceA carrier’s quantum state will decohere intrinsically ifand only if it undergoes inelastic scattering, openingup a channel by which the particle loses part of itsenergy. For example, it may admit electron–electron orelectron–phonon interactions and these are essentiallymany-particle scattering events. To describe this classof decoherence effects more quantitatively, Nico vanKampen [9] introduced the idea of an inelastic scatteringlength scale into the mesoscopic picture.

Inelastic scattering is a process arising out of energynonconserving collisions that involve, not the scattering ofelectrons off static and passive objects, but rather collisionsbetween it and other active, dynamical players in the transportprocess. Let us recall the various length scales that regulatethe fate of a charge carrier:

(a) the elastic scattering length le, occurs due to staticimpurity scattering, where energy is conserved by theelastic collision;

(b) the Fermi wavelength λF = 2π/kF, kF being the Fermiwave-vector relating to electron density;

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Adv. Nat. Sci.: Nanosci. Nanotechnol. 1 (2010) 043001 M P Das

(c) the inelastic scattering length, lin arising fromelectron-electron and electron-phonon scattering,which dissipates the carrier’s energy into the thermalenvironment;

(d) the phase-breaking length lφ which plays an essentialrole in the description of coherent effects in complexsystems [10]. The classic demonstration of lφ is theobservation of the Aharonov–Bohm effect (interferenceof electron beams) in metals in the presence of a uniformmagnetic field.

Additionally, Thouless length is defined as LT =√(1/3)linle. This is the characteristic path length through

which an electron wave propagates diffusively (analogous to aclassical random walk) before losing its phase coherence. Alltypical mesoscopic conductors at low temperatures satisfy thecondition le < LT < lin [11]. More about coherence later.

In terms of these lengths and L (being the system length),one can define these three limits:

Ballistic limit λF < L � le, lin,Diffusive limit λF � le � L < LT,Macroscopic limit λF � le � LT � L .

With this information, and its defining length scales, wehighlight the fact that an evolving mesosystem (interactingwith its surroundings and/or by inelastic scattering insidethe system itself) cannot avoid losing whatever intrinsiccoherence it may start with. That is how a normalconductive material exhibits resistive dissipation. In astatistical mechanical sense there are some generic anduniversal features observed for the properties of mesosystems.These are: conductance quantization, quantization of charges,universal conductance fluctuations, etc. We return to thesequestions below.

On the other hand, nanosystems are often non-generic,with the rich variety of their chemical characteristics playinga crucial role. Their properties are highly sample-specificand their interaction with the environment varies in anon-universal way. Typically, their experimentally observedproperties inhabit the grey area between two distinct aspects:physical generality and material specificity.

Obviously one must note here that physics, chemistryand materials science, with their respective realms ofunderstanding, each have their own vital perspectiveand method of pinpointing the essentials of a givenmeso/nanoscopic device problem. They must work togetherto obtain meaningful results when phenomena can occur atall length scales from short to long, over energy scales frommeV to tens of eV, and involving properties ranging from thegeneric to quite specific. Many specialists are already workingin the respective areas (and perforce learning to speak oneanother’s language). Below we examine just a few of theingredients that are needed for a true nanoscience.

2.1. Role of quantum mechanics

A schematic picture of electronic spectra is shown infigure 1, going from an isolated molecule, to a cluster/smallnanoparticle and to a large nanoparticle. The localizedelectronic states become progressively delocalized upon theincreasing overlap of wave functions as the system size

Figure 1. Energy levels with increase of size. In large nanoparticlesthe energy levels become dense to form quasi-continuum bands.

increases towards the bulk limit. Some physical properties,notably electrical conductivity, appear only as a consequenceof spatial extendedness on the part of the electronwave function, thereby manifesting its quantum mechanicalnature. Quantum mechanics provides a mathematical/physicaldescription at the atomic and subatomic scale whereclassical mechanics fails completely to do so. It providesa unified view of the performance of atomic-scale objects,including electrons, photons and other elementary particlesand excitations.

When a system size is at the atomic scale, it is onlyquantum mechanics that can account for the observed physicalproperties. In the jargon of the field, ‘quantum confinement’means that the de Broglie wavelength of the particles iscomparable to the size of the system that contains them.Small size implies strong quantum confinement effects. Bulkextended matter, when sufficiently curtailed in one of itsdimensions, for example in the z-direction, will behaveas a quasi-two-dimensional system in the complementaryx–y-plane. The price to be one paid for this is to losethe long-range extendedness of the wave function in thez-direction; the energy of motion along that axis becomesquantized as the wave function is confined.

Repeating the restriction in a second direction, saythe y-direction, will make the system quasi-one-dimensionalin the remaining x-direction; now the wave function isdeliberately forced to occupy a small region within they–z-plane. One last restriction along the x-axis of motionwould produce a quasi-zero-dimensional system, where allextendedness of the wave function has been lost, in everydimension. The entire electronic system is localized in arelatively tiny volume of x, y and z. As a result of the completeconfinement, all the physical properties of the system will bedrastically affected (see the schematic figure 2 below). Wehave made, then, an ‘artificial atom’ or a quantum dot.

From this brief discussion we learn that the electronic,optical and mechanical (elastic) properties of materials areradically changed by both size and shape. Well-establishedtechnical achievements, including zero-dimensional quan-tum dots, have been attained through ingenious sizemanipulation—and for that the quantum-confinement effectis crucial.

It would be instructive to follow how the theory ofquantum confinement tracks the behaviour of an exciton(a jointly bound electron and hole pair) as it crosses overto an atomic-like orbital as its host space is progressivelydiminished. A rather good approximation of an exciton’s

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Adv. Nat. Sci.: Nanosci. Nanotechnol. 1 (2010) 043001 M P Das

Figure 2. Matter in spatial dimensions (from 3 to 0) and theircorresponding density of states. Zero dimension is called a quantumdot, one dimension a quantum wire and two dimensions is known asa 2D electron gas.

behaviour is the 3D model of a particle in a shrinking box.A systematic solution to this problem provides themathematical connection between the evolution of energystates and the dimensionality of the space within which thewave function exists. It is obvious in any case that decreasingthe volume, or the dimensionality, of the available spaceincreases the energy of the states.

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

ψnx ,ny ,nz =

√8

L x L y L zsin

(nxπx

L x

)sin

(nyπy

L y

)sin

(nzπ z

L z

),

Enx ,ny ,nz =h2π2

2m

[(nx

L x

)2

+

(ny

L y

)2

+

(nz

L z

)2].

Using the above wave functions and energies one cancalculate carrier density and the density of the states (theavailability of quantum states are those solutions allowedwithin the system). Given in figure 2 are the densities ofelectronic states in all space dimensions from three to zero.Those are: ρ(E)∼

√E in 3D, ρ(E)∼ constant (for each

sub-band) in 2D, ρ(E)∼ 1/√

E (for each sub-band) in 1Dand 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 quantumpicture is crucial to understanding the dynamics of electronsacting under an externally applied field; a problem thatgoverns both the characterization of nanodevices and theireventual practical uses. Here we remark that if an electricallyactive system’s length is reduced to the nanoscale, there willbe considerable changes in its properties. At the bulk phase,the device interfaces are expected to control some of themacroscopically observed properties (for instance, they affectaccess resistance). But at the nanoscale, a system’s interfaceswith the ‘rest of the universe’ have more spectacular effects.

2.2. Surface-to-volume ratio

What is the role of a surface? A surface is said to be the firstfrontier or line of defence for any interaction with the outsideworld. In general a system always minimizes its free energy.Unless it is truly isolated thermodynamically, it must do so inthe presence of its mechanical and electromagnetic coupling

to the world outside its boundaries. So the question is: to whatextent do the internals of the system dominate its free energy,and to what extent do all of its various interactions at thesurface contribute?

The smaller the size of the system, the larger the ratioof its surface area S to volume V is. A high ratio implies astrong thermodynamic ‘driving force’ that speeds up manyof the processes that minimize thermodynamic free energy.Chemically, the smaller the size of a material sample, thefaster its reactions at a relatively large S. A porous material’schemical reactivity (e.g. a catalytic exhaust converter) is muchgreater because of its large surface area. For the same reasonas a high S:V ratio, nano- or mesomaterials have a higherchemical reactivity compared to the bulk. For biologicalsystems, surface-to-volume ratios are more significant still.

The surface-to-volume ratio for a 3D cube can be readilyobtained. The total surface area of a cube is S = 6L2,whereas the volume is V = L3. Therefore, S/V = 6/L . AsL → larger, S/V → smaller. For other structures such as asphere or ellipsoid, the S/V ratio can be similarly calculatedto scale as 1/L . A close-packed cubic structure of 1 cm3

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—thosethat mediate most of the system’s interactions with thesurroundings—would appear to be insignificant. On the otherhand, if we take a one-nanometre cube (10 atoms in a row), thesurface-to-volume atoms’ ratio is ∼6 × 102/103

= 0.6. Thefraction of boundary atoms is in every way significant. Eventhis most simple size argument demonstrates the potentialimportance of surface-mediated effects over bulk effects asthe system size is reduced. Thus, the smaller the system, themore its surfaces must dominate its actual properties.

Researchers in the US recently made a surprisinglynovel chemical structure that has the largest internal surfacearea ever observed in an ordered material. Omar Yaghiand co-workers at Michigan and Arizona State Universitiesfabricated a new porous metal–organic framework with anestimated surface area of 4500 m2 g−1, which is nearly fivetimes larger than the previous known record. The structure canbind large quantities of gas and could, therefore, find a varietyof applications, including gas storage and catalysis [12].

With a decrease in size, the surface area and surfaceenergy increase, and thereby the melting point of a sufficientlysmall sample decreases. For example, 3-nm-wide CdSenanocrystals melt at 700 K, compared to a CdSe bulkthat melts at 1678 K. This is another confirmation of thedominance of surface/boundary effects at the nonoscale. It isa very similar story for nanoelectronic devices: a conductingstructure’s coupling to the exterior circuit is via its boundarysurfaces. The latter’s nature can substantially modify, andindeed often dominate, observed transport behaviour.

As we indicated earlier atoms in nanostructures have ahigher average energy than atoms in larger structures, becausemost of them are surface atoms and the uneven bondinggenerates new tensional forces not otherwise experienced atequilibrium in the deep bulk. Consequently, the chemicalactivity of a material can be exponentially improved as thematerial is reduced in size at the nanoscale. The properties

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Adv. Nat. Sci.: Nanosci. Nanotechnol. 1 (2010) 043001 M P Das

of nanosystems are significantly affected by minor changes insize, shape or surface states of their particular structures.

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

Obviously these new qualitative and quantitativeproperties herald entirely new applications which, being outof the reach of our earlier science of bulk materials, oftenlack any obvious technological precedents. They represent atruly unexplored domain.

2.3. Equilibrium versus dynamical properties: effect of theinterface for nano/meso systems

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

This embedding in the environment (also known as the‘bath’) introduces the idea of dissipation (friction) wherebythe system’s energy is transferred irreversibly to the bath. Asfirst analysed by Einstein in his third famous paper from 1905,it is the evident stability of an embedded system, togetherwith the ubiquitous presence of energy dissipation, whichimplies that the system is subject to fluctuating microscopicforces. The counterbalancing of the twin effects of fluctuationand dissipation induces the system to relax (settle down) tothermal equilibrium, at a characteristic energy (temperature)set by the bath.

In quantum dynamics there is yet another feature beyondthose two dynamical drivers: the system can display coherenteffects, i.e. long-range interference phenomena over space andtime. Coherence, by its nature, introduces a very high level oforderly correlations. The rate of spatial decay in a particle’squantum-state correlations is characterized by the coherencelength. As mentioned earlier, in mesosystems, the coherencelength is much larger than the inter-particle separation butstill smaller than the system size. However, once a system’sinevitable coupling to the environment is taken into account,the correlations are degraded; an effect known, logicallyenough, as ‘decoherence’ or ‘dephasing’. It is preciselythis interaction of the system with the enveloping bath thatmakes nanoscience non-trivial from a fundamental point ofview. In other words, it is the essential quantum nature ofnanoscopic matter intimately interacting with its much bigger,macroscopic surroundings that defines its ‘nanoscopic’ aspectin the first place. We make this point explicit in the followingsections.

2.3.1. Closed systems: equilibrium states. Let us first try toconceive of the nano-object as being strictly closed, in contrast

with genuine matter, which always dwells in (and so mustinteract with) a large open environment. In the former scenariothe system can be said, almost trivially, to be in its ownthermodynamic equilibrium. We have both phenomenologicaland microscopic methods to study that idealized state ina self-contained and satisfactory way. Generally, for amolecular system, the energy levels are discrete: separatedby energy gaps (see figure 1). For a nanosystem, the energylevels are still in principle discrete, but those levels are nownumerous and spaced closely together in quasi-continuousenergy bands. Gaps remain between such bands, which mayeither decrease or increase; in any case we can calculate allthis using standard electronic structure techniques. In the bulklimit the energy bands become continuous spectra, while finitegaps separate the distinct bands. Bands may be (a) completelyfilled by all available electrons (an insulating band), or (b)partly occupied (metallic), or indeed (c) completely empty(insulating—by default). With reference to figure 1, in smallsystems most of the occupied states are bound (fully spatiallyconfined) states. The wave functions are well localized. Inthe bulk system—by very sharp contrast—the wave functionsare spatially extended throughout the system. If the numberof electrons contributed by the constituent atoms is less thanthe number of possible states in the band, then the highestoccupied electron state lies well inside the permitted energyspread of the band. This corresponds to cases where oneobtains an electrically conductive metallic system.

As an example, take an isolated fullerene molecule,which is a system of 60 bonded carbon atoms on thesurface of a sphere. We have ab-initio methods (like thedensity functional theory (DFT)) to calculate the electronic,geometrical and vibrational properties of this large assembly.When compared with experimentally measured quantities,these theoretically obtained properties often provide excellentagreement (see for example [13]). Next consider a fullerenemolecule contacted with a pair of gold electrodes subject toa low bias. The fullerene molecule in this new environmentwill be conducting with a considerable amount of electriccurrent, which arises from the non-equilibrium dynamicalcondition [14]. A conventional static DFT is unsuitable toprovide the correct answer to this problem.

Many such instances can be found in the literature.The moral is, that as far as purely isolated systems areconcerned, we have a reasonably good understanding of theirelectronic properties. But while this is certainly a necessaryprecondition for building up a sound body of knowledgeof meso/nanoscience, it is far from adequate. The key isstill in unavoidable dynamic interaction processes with theenvironment; it can be argued that our knowledge of these keyeffects is largely incomplete.

2.3.2. Open system: interacting. Now consider anano-object embedded in its interacting environment.We are still in an equilibrium scenario; the big difference isthat the nano-object, in its interactions with the environment,undergoes changes in its electronic distribution, vibrationalproperties, excitation energies, etc. However, we should keepin mind that the system, by its very openness to the bath,loses its total ‘ownership’ of these attributes; one needs anintegral approach in which the bath is an explicit player.

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Adv. Nat. Sci.: Nanosci. Nanotechnol. 1 (2010) 043001 M P Das

Figure 3. Acetone molecule [CH3]2CO (O in red, H in white and Cin green).

Table 1. The average CO distance (RCO) as well as the averagevalues of the vertical excitation (ω1), the highest occupiedmolecular orbital (HOMO) and the lowest-unoccupied molecularorbital (LUMO) energies, which are obtained from the equilibratedMD trajectories. Energies are given in eV and distances in Å.

RCO ω1(n → π∗) HOMO LUMO

Acetone in gas 1.222 4.382 −6.744 −0.443Acetone in solvent 1.233 4.593 −7.124 −0.634Change 1 0.011 0.211 −0.380 −0.191

Generally, interfacial properties are drastically affectedwhen a nanostructure is embedded in an interactingenvironment. Here we present a simple example of anacetone molecule [CH3]2CO (figure 3) both in gas and in asolvent. Results are obtained from a time-dependent density-functional-theory/effective-fragment-potential approach(quantum molecular dynamical calculations) [15], showingthe change of some parameters as the environment changes,table 1 below).

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 thenext section for nanosystems contacted with large metallicleads for the transport of electrical current.

3. Special topic on electron transport in mesoscopicsystems

In the remaining part of the paper we choose electron transportas a special topic, because it is relatively more difficultboth for calculations and in understanding compared toequilibrium or ground state properties. Furthermore, this topichas been very actively researched during the past two decadesboth on the basic and applied aspects of nano/mesosystems.There have been a number of specialized monographs, editedvolumes and review articles. See for example, some recentones [16–26].

We shall discuss here two main types of model underlyingthe current understanding of electron transport; quantumtunnelling and metallic conduction. While the physicalrelevance of both tunnelling and metallic transport is notin question, the more subtle aspects of these two kinds

of description and their mutual, logical compatibility needfurther clarification. The most important question is: whatare the physical conditions for a valid application of eachapproach?

The response to such a question rests on the basicprinciple that all quantum transport effects are necessarilymany-body in nature. Even classically, they are not reducibleto a description of transport by strictly independent, individualcarriers because correlations among them are inherent intransport physics. The correlations may be direct or mediatedby the bath (phonons, photons, etc) and unavoidable in everycase. The quantitative expression of this principle is thecelebrated fluctuation dissipation theorem (FDT).

3.1. Quantum transport—independent particle viewpoint

This is a very elementary theory of quantum transport.One assumes that single-particle states are identical tothe actual (collective) excitations of a many-body system.The best example is the single-particle orbitals found inthe Kohn–Sham density functional theory. In the specialcase of non-interacting (or even Hartree-like mean-fieldinteracting) quasi-particles, the conductivity is expressed asa Fermi Golden Rule formula that involves matrix elementsof the current operator, which is naïvely interpreted tobe Wnm ∼ [ψ∗

n ∇ψm − ∇ψ∗nψm]. Here ψ’s are Kohn–Sham

effectively single-particle wave functions. Technically, thispicture corresponds to an approximation that incorporatesonly the first bubble diagram of conductivity [27, 28], whichlands one in the several serious difficulties of consistency thatwe discuss below.

We remark that in strongly metallic systems theindependent-particle picture can be placed on a firmtheoretical footing (the Fermi-liquid, or quasi-particle, theoryof Landau). This succeeds in incorporating real scatteringeffects subject to the assumption that the basic transportstates are totally delocalized and have intrinsic non-zerocurrent associated with them. In a driven system the observedcurrent is the outcome of the slight induced imbalance in theoccupancy of counter-propagating states. But each such stateis itself distributed throughout the system; that is the essenceof metallic charge transport.

3.2. Bardeen formula

In contrast to metallic transport, charge transfer by quantumtunnelling takes place by the hopping of single charges fromone isolated conducting island on the left of a device toanother island on its right. These islands, though in proximityto each other, are separated by a finite-width potential energybarrier. States are localized to one or other of the islands andtherefore can sustain different amounts of local occupancy.Note that this is a huge qualitative difference from metallicstates, which may be filled or empty but always delocalizedacross the entire system and carrying an intrinsic current.

In deriving the Bardeen formula, one encounters theconceptual problem that the eigenstates of the left island andthe right island do not together form a complete orthogonal setof total Hamiltonian for tunnelling. Despite this basic formalweakness, Bardeen’s formula is at least a physically consistentmodel with a successful repertoire of useful applications [29].

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Adv. Nat. Sci.: Nanosci. Nanotechnol. 1 (2010) 043001 M P Das

It is entirely different, however, from metallic transport, whichdepends on the existence of completely delocalized particlestates.

3.3. Landauer formula

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

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

In the absence of scattering, the Landauer formula forstates in a single sub-band gives a remarkable result forconductance: G = 2e2/h, where e and h are the quantum ofelectronic charge and Planck’s constant, respectively. With Nconducting sub-bands the right hand side of the formula ismultiplied by N since each band is an independently openchannel conducting in parallel with the rest. This phenomenonis known as quantized ballistic conductance.

If there is scattering in a ballistic system, it takes placeat and within the interfaces with the reservoir leads: byassumption the inner system itself is free of impurities.The single-particle nature of the Landauer formula dealsonly with elastic scattering. Using a one-body potentialscattering theory, the conductance formula is modified toread G = (2e2/h)T , where T is the transmission function orfactor [30]. The function T is the probability that a singleelectron will traverse the ballistic wire. As a probability,its eigenvalues are bounded between 0 to 1. Where thetransmission factor is zero, the system does not conduct, whilefor unity transmission factors the system is ideally ballistic.

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

(i) How can a system with ballistic (i.e. strictly elastic)transport have finite conductance without any dissipationof energy?

(ii) The notion of reservoirs attached to the system isredundant to some extent; it is incorporated within theboundary conditions for any electrical circuit.

(iii) The two quite distinct modalities of conduction,tunnelling (T → 0) and metallic charge flow (T → 1), asoutlined above, are applied interchangeably in Landauerpicture. No physical distinction is made between them.This begs the question as to when and how a Bardeen-likeinsulating barrier (as subsumed in the zero-transmissionlimit of the Landauer model) is able to morph into anideal ballistic conductor whose states extend right acrossthe structure—as for a metal.

The situation is made clear in the following figure 4.Conductance is shown as a function of barrier thickness

Figure 4. Conductance (normalized to e2/h as barrier (rectangular)thickness (in Å) varies see [29]).

in both the Landauer and Bardeen models [29]. For largebarrier thicknesses both the models coincide and give thesame result. However, for small barrier thicknesses theBardeen conductance diverges (as logically it should) whilethe Landauer result goes to unity as the ‘barrier’ thicknessvanishes. The tunnelling formula is inappropriate for avery small separation and, for the Landauer picture, it isunphysical. If transport becomes ballistic (ideally metallic),the underlying states are extended and the assumptionof different local occupancies of the reservoirs has littlemeaning.

Since for normal resistive conduction (finite G) theintroduction of incoherence is an essential objective,Büttiker [30] introduced into the ballistic system, in a fullyphenomenological way, additional probes with adjustablechemical potentials: places for the carriers to instantaneouslyrelax and lose their coherent phase memory.

The Landauer formula includes transmission via aone-body potential scattering method, identical to an elasticscattering process [30]. We have seen that a natural andunavoidable physical process in mesoscopics is scattering bymany-body interactions. This renders inoperative any conceptstrictly reliant on single-particle description alone. Therefore,the simple inclusion of passive reservoirs and leads, andeven of additional phenomenological voltage probes, cannotsave an exclusively single-particle treatment of physics at thenano/meso level [27, 28, 31]. Rather more is needed.

3.4. Anderson localization and mesoscopics

In the late 1970s P W Anderson and co-workers (popularlyknown as The Gang of Four) established that in anon-interacting, elastically disordered system there is aninsulator–metal transition only in 3D, while all otherlow-dimensional systems must be insulators. In the languageof ‘one-body scattering theory’ many have concluded thatthe quantum-coherent discrete conductance for a 1D system,as predicted (among others) by Landauer’s transport theory,and amply experimentally confirmed, is a contradiction toThe Gang of Four’s insulator result. See for example,Anderson in ‘50 years of Anderson localisation’ published in

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IJMPB 24 (2010) 1501. In this area ‘weak localization andmesoscopics’ have some commonality in popular literature(See for example, M. Büttiker and M. Moskalets in the abovereference p. 1555).

3.5. Linear response theory of Kubo

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

The Kubo theory is completely general and encompassesnot only bulk quantum systems but also meso/nanosystemsequally. In principle, the study of transport does not limitus to those non-equilibrium states lying sufficiently close toequilibrium. Still, computation of linear kinetic coefficients iseasiest to carry out, since the final expression involves purelyequilibrium expectation values of the relevant dynamicalvariables, a much simpler procedure than extracting anycorresponding far-from-equilibrium quantities.

In the context of mesoscopic systems, the Kuboformulation dates back to the 1980 s with Fisher and Leeand others [33], who derived the Landauer formula in thenon-interacting (independent particle) limit of the Kuboformula. Our derivation of conductance [31] from the Kuboformula also produces the Landauer formula as a naturaloutcome, but the transmission function includes an inelasticpart (as demanded by the physics of dissipation) and notpurely elastic as in independent-particle approaches. In detail,transmission is a much more complicated concept and invokesmany-body scattering, in which elastic and inelastic processeswork together.

An analogous microscopic analysis of mesoscopictransport has been made by Soree and Magnus [34],who derived conductance quantization purely using themethod of non-equilibrium statistical operators and withoutany mismatched-reservoir phenomenology. In a differentlanguage, Di Ventra and Vignale [27] show that the Landauerformula is incomplete, being devoid of essential many-bodyeffects. There has been use and misuse of the Landauerformula in many papers over time. We have criticallyexamined a few examples in a recent viewpoint paper [35].

3.6. Non-equilibrium Green function (NEGF) theoryof Keldysh

A rigorous, microscopic and completely general theory ofquantum transport can be based on the NEGF formalismdeveloped by Keldysh [36] and Kadanoff and Baym [37].It has been applied to device problems since then (forapplications of NEGF see [38]). This method allows oneto systematically solve the interactions within the electronpropagator (Green function) under fully non-equilibriumconditions. Therefore, in principle, all scattering mechanismsarising out of many-body correlations can be taken intoaccount systematically, in a well-controlled way, in theevaluation of the current.

When a system is driven out of equilibrium by anapplied bias, a standard quasi-equilibrium perturbation theory

to a finite order is not suitable to describe the transportproperties; for the system response can be strongly nonlinear.The resultant net flux of current, sustained by the externalbias, is evidence that the system is not in equilibrium. Inparticular, equilibrium-state theory is incapable of describingreal exchanges of energy between electronic and phonon-bathsubsystems, and therefore has no means of capturing thephysics of heating and dissipation. This is obvious, since inequilibrium there cannot be a steady net flux of energy out ofthe electron system into the vibrational degrees of freedom.

In its current popularly accepted form, the so-calledNEGF formalism does not drastically differ from anequilibrium theory. The principal technical differenceis that all time-dependent functions are defined fortime-arguments located on a Keldysh contour (in the complextime-plane) [36]. The advantage of the current versions of theNEGF is that one can systematically improve approximationsby taking into account various physical processes. Thedisadvantage (though this is true only for the popularlyadopted form and not for the genuine NEGF method) isthat the Ward identities are not guaranteed, and so neither ismicroscopic conservation.

We cannot review here such a broad and rather technicalfield. Useful descriptions of the NEGF theory can be foundin Haug and Jauho [39] and Langreth [40]. We shall discusssome salient issues of this theory in the forthcoming 16thVietnam School of Physics in Hanoi.

3.7. Electronic contacts

Contacts in an electrical circuit are very important because,self-evidently, the current must pass through them to drivethe system of interest. The physics of contacts becomesmore subtle when the system is in a non-equilibriumstate. The contacts can be of different types: bimetallic orSchottky (i.e. contact between a metal and a non-metal).Passage of electrons through the contact decides theconducting nature of the device. As an example, we pointto an interesting molecular nanodevice (known as theAviram–Ratner Diode [41]) which has a rectifying property.Rectification takes place because of (i) the specific nature ofthe 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 simplestelectronic meso- or nanodevice will have three parts to it: twoleads on either side to make contact with the third item: thecentral, active region. The central region may be a quantumdot, a molecule, a nanotube or a 1D wire, etc. One can createthis device, conceptually, by correspondingly partitioning thetotal Hamiltonian into three parts, with couplings betweenthe parts representing the action of the contacts. This is apopular model, due to Meir and Wingreen [42]. In figure 5, wehave shown a representative example of a molecular diode—atwo-terminal device attached to two leads on either side. Thisrectifying diode is current carrying [43].

Partitioning of the Hamiltonian in the non-equilibriumproblem has severe difficulties, which seem to have beencomprehensively ignored [44, 45]. The steady-state current inthe NEGF approach is obtained in a form that looks simpleenough; yet the NEGF and the non-equilibrium self-energy

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Figure 5. A molecular diode biased to carry a current (schematic).A dipyrimidinyl–diphenyl molecule at the centre is connected totwo leads (green slabs). Sulfur in yellow, carbon in green andnitrogen in blue (see [43]).

of the particle states still need to be computed before onecan extract a workable and self-contained calculation. Thisis undoubtedly a formidable task, if it is to be performedseriously.

A much-advocated ad hoc simplification is to dropthe essential non-equilibrium interacting parts. As aresult the conductance formula re-emerges as a Landauerformula [42, 46]. In this formula the effective transmissionfunction Teff is given by Teff = Tr[0LGr0RGa]. Here Tr standsfor a trace, 0L,R are imaginary parts of the single-particleself-energy, coming from the left and right couplings ofleads to the central region. Gr,a are retarded and advancednormal Green functions. They are all functions of energy.This attractive formula is routinely used in a number ofcomputer codes to calculate current-voltage characteristicsand conductance as a function of various applied gatevoltages [47].

• What has been included in this formula that is new?By partitioning the device into distinct leads andchannels, one set up has the current-inhibitingeffects associated with the interfaces: details of themicroscopic effects at the interfaces are arrived atusing density functional theory. Some of the Landauermodel’s modern proponents believe that this interfaceresistance is none other than the actual quantizeddevice resistance. Attempts have now been made toinclude electron-correlations and inelastic scattering ina so-called Landauer-like formulation [27, 47]. Unlesssuch attempts are genuinely many-body in nature, thereis again no guarantee of control over charge and currentconservation.

• What is importantly missing in this formula?When implementing this formula one has to be carefulabout some essential sum rules, such as particle numberconservation, gauge invariance etc. The first question is:does the transmission factor Teff preserve unitarity in theelastic channel? To answer that one has to show

06 Teff = Tr[0LGr0RGa]6 1.

The second question is: does the formulation satisfycurrent conservation (Kirchhoff’s rule) 6α J α = 0?

Figure 6. Conductance as a function of chemical potential [31],reproduced with permission.

We need to secure identity at a microscopic level

6α J α = 1/h∫

dωTr[Σα,<(ω)G>(ω)Σα,>(ω)G<(ω)

≡ 0.

(Here Σ and G are the Keldysh self-energy and Greenfunctions, respectively.)

A theory that violated these conservation constraintswould be mathematically and physically unsustainable,whatever its potential appeal and success in reachinggood agreement with the experiments.

• Working methods for the NEGF theory to transportNon-equilibrium quantum transport formulations, asdiscussed above, have been implemented by a numberof authors (see for example [47] for references). Thestarting point of the implementations is the basis ofelectronic states of the system. To capture them, densityfunctional theory is commonly employed. First, onesolves the Poisson equation with appropriate boundaryconditions to obtain the self-consistent Hartree potential.The exchange-correlation potential is added from achosen form (local density approximation, generalizedgradient approximation, etc) and then Kohn–Sham-likeeigenvalues εi and eigenfunctions ψi are computednumerically. These computed quantities, εi and ψi , areused to obtain the self-energies and Green functions.Details of the technique commonly used in severalcomputer codes vary from one to another [47].

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

3.8. Inelasticity and its consequences

As pointed out before, for normal dissipative transport it isessential to adopt a paradigm where inelasticity has a crucialrole. Coherent transport is no longer in sole, exclusive playfor actual dissipative systems. In the last 5–6 years manypapers have appeared, treating inelasticity as an intrinsically

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Adv. Nat. Sci.: Nanosci. Nanotechnol. 1 (2010) 043001 M P Das

Figure 7. A benzene-dithiol molecule attached to two gold electrodes (inset). Left: calculated IV curve. Right: effective potential at a biasvoltage of 3 V [52], reproduced with permission.

many-body problem. At a basic level, electron–phononand electron–electron interactions are treated within modelHamiltonian approaches [48–51]. The electron–phonon andelectron-electron interactions not only directly affect thespectral function, the density of states and, consequently, thetransport current; they also determine momentum-scatteringtime and thus the conductance.

Figure 6 shows the conductance steps of a quantumwire as a function of gate voltage, including inelasticscattering, in a linear response approach [31]. The latterclearly depresses the conductance by enhancing dissipation;yet the conductance steps survive. This demonstratesthat the steps are not the unique signature of an ideallyballistic, purely elastic system. Another example is given ofEnkovaara et al [52], who have studied a molecular transistorAu/benzene-ditheolate by a non-equilibrium electrontransport method in conjunction with a time-dependent DFT.In figure 7 (left) the calculated I–V result is shown with thecurrent steps as the bias voltage increases/decreases. On theright panel the effective potential is shown at a bias voltageof 3 V. The molecular bonds are clearly seen on both sides ofthe molecule.

4. Conclusion

Mesoscopic and nanoscopic science is a very broad areacovering physics, chemistry, materials science, biology andseveral domains of engineering. In all of these directionsa remarkably intense amount of activity is taking place.In this paper we have attempted a largely pedagogicreview of mesoscopic systems in the quantum realm, withparticular emphasis on the relevant scientific fundamentals. Insections 1–3, we have covered most of the issues related toa basic understanding of mesoscience. In the last section wehave focused on current topics in quantum electron transportand discussed some technical points in detail. Since this isa brief review, no attempt has been made to cover everydevelopment in this vast area. For the specialist’s interestone has to consult the appropriate literature. The limitedreferences cited in this review simply reflect the context of my

own developed, and developing, understanding of the subject.I wish to conclude on the cautiously optimistic note that weare progressing steadily to witnessing a major development inthe application of nano-electronics in the coming years.

Acknowledgments

It is a great honour to participate in this InternationalWorkshop for the 1000th year celebration of the city of ThangLong-Hanoi. I am grateful to Professor Nguyen Van Hieufor the kind invitation and warm hospitality that I enjoyimmensely when coming to Hanoi. I wish to thank FredGreen for useful suggestions while this manuscript was inpreparation.

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