Charge transport through single molecules, quantum dots, and quantum wires
S. Andergassen, V. Meden, H. Schoeller, J. Splettstoesser, M. R. Wegewijs
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y TOPICAL REVIEW
Charge transport through single molecules,quantum dots, and quantum wires
S. Andergassen , V. Meden , H. Schoeller , J. Splettstoesser and M.R. Wegewijs , Institut f¨ur Theoretische Physik A, RWTH Aachen, 52056 Aachen, Germany, andJARA-Fundamentals of Future Information Technology Institut f¨ur Festk¨orperforschung - Theorie 3, Forschungszentrum J¨ulich, 52425J¨ulich, GermanyE-mail: [email protected]
Abstract.
We review recent progresses in the theoretical description of correlationand quantum fluctuation phenomena in charge transport through single molecules,quantum dots, and quantum wires. A variety of physical phenomena is addressed,relating to co-tunneling, pair-tunneling, adiabatic quantum pumping, charge and spinfluctuations, and inhomogeneous Luttinger liquids. We review theoretical many-bodymethods to treat correlation effects, quantum fluctuations, nonequilibrium physics,and the time evolution into the stationary state of complex nanoelectronic systems. harge transport through single molecules, quantum dots, and quantum wires
1. Introduction
In this review we consider different aspects of charge transport through nanoscaledevices attached to electronic reservoirs, focusing on theoretical approaches dealingwith interaction and quantum fluctuation effects. Transport experiments have shownconvincingly that many of these systems - from carbon nanotubes down to singlenanometer sized molecules - behave as quantum dots . A quantum dot is a confinedsystem of electrons, which is so small that the discreteness of energy spectrum with alevel spacing ∆ ǫ becomes important; it is therefore often referred to as an artificial atom.Due to the smallness of the quantum dot Coulomb interaction between the electrons alsoplays an important role: the charging energy U which can be of same order as the levelspacing. If the quantum dot is attached to reservoirs by tunneling contacts, electronscan leave and enter the dot and the single-particle levels are therefore broadened dueto the finite lifetime of the electrons. For sufficiently weak tunnel coupling Γ and smalltemperature T , the level spacing ∆ ǫ and the charging energy U can be resolved atstandard cryogenic temperatures. This opens up the possibility of performing transportspectroscopy by measuring the differential conductance as function of gate voltage V g and bias voltage V . The transport characteristics allow to count the discrete chargestates of such systems, and even more complex degrees of freedom of the quantum dotare revealed. As an example, the spin states can be identified with the help of theZeeman effect, and also vibrational motion of the quantum dot system can be detected.The transport behavior, however, depends strongly on the ratios of all the energyscales, and on the way transport is induced, e.g., by static or time-dependent appliedvoltages, or by measuring linear or non-linear response in the transport voltage. Inparticular, for molecular scale quantum dots the excitation spectra may be quite complexand various splittings appear. This complicates matters further but also leads to avariety of transport effects. Besides stationary properties of quantum dots, the timeevolution out of an initially prepared nonequilibrium state is another important issueof recent interest, providing the new energy scale ~ /t . The time evolution itself can beused as a tool to reveal various many-body aspects. It is also of practical importancedue to the progress in the field of solid-state quantum information processing, initiatedby the suggestion to use spin states of quantum dots as basic elements of qubits [1].In addition to the “bare” energy scales mentioned so far, due to correlation effects new effective energy scales can be generated, which describe the physics of quantumfluctuations induced by the reservoir-dot coupling. These new energy scales are complexfunctions of the above “bare” energies and we denote them in the following by T c ,where c stands for “cutoff”. Examples are the Kondo temperature describing the strongcoupling scale for spin fluctuations (usually denoted by T K ) and renormalised tunnelingcouplings generated by charge fluctuations. These scales will dominate the measuredtransport in the strong coupling regime where T c is the largest energy scale. However,also in the weak coupling regime quantum fluctuations set on and are of particularinterest close to resonances, where they can be analysed by a perturbative treatment harge transport through single molecules, quantum dots, and quantum wires renormalisation group (RG) ideas are required.RG transport theories reveal that interactions are of key importance for the generationof new effective energy scales by quantum fluctuations. Without interactions, quantumfluctuations are well understood, they just lead to an unrenormalised broadening of theenergy conservation by the tunneling coupling. Although new effects from correlationsare most pronounced in the regime of strong correlations Γ ≪ U , interesting quantumfluctuation effects already start to become visible in the regime of moderate correlationsΓ ∼ U , where perturbative and mean-field theories are often not yet applicable. Anotherinteresting regime is the case of a continuous level spectrum, i.e., where the quantum dotis replaced by a one-dimensional quantum wire. In this case, quantum fluctuations inthe wire itself lead to renormalisation effects and Luttinger liquid physics, with typicalpower-law suppression of the bulk spectral density. Similar to the case of quantum dots,the simultaneous presence of correlations and transport through inhomogeneities leadsto very rich phenomena. A prominent example is the suppression of the spectral densityat wire boundaries, which leads to vanishing conductance at zero temperature and biasvoltage if a quantum wire is weakly coupled to two leads. This has to be contrasted tothe Kondo effect in quantum dots, where conductance becomes maximal when a singlespin is placed in an “insulating” quantum dot.In this review we will illustrate these key issues and concentrate on correlationeffects in quantum dots and quantum wires. We will describe transport theoriesspecifically designed to treat the case of moderate to strong Coulomb interactionsand quantum fluctuation effects. With these methods it is possible to account forthe complex interplay of various combinations of quantum mechanical effects (levelquantization, interference, spin and angular momentum), complex excitations (e.g., dueto spin-orbit interaction and/or mechanical vibrations), Luttinger liquid physics, non-equilibrium (due to static as well as time-dependent applied voltages), and the timeevolution into the stationary state. An important aspect is that these methods applyto general models incorporating experimentally relevant microscopic details, and offerprospects for numerical implementation. Despite these common themes, the differentsections of this review are meant to be almost self-contained and each section closeswith a short outlook. We consistently use natural units ~ = k B = e = 1. The paper isorganized as follows.In section 2, we will introduce a general approach to transport through quantumdots of arbitrary complexity, aiming at the description of three terminal single-moleculejunctions. The tunneling coupling is treated perturbatively up to second order inthe coupling Γ, accounting for charge fluctuation effects, while not capturing spinfluctuations which start in higher order. Besides well-known level renormalisationand cotunneling effects, electron pair tunneling is shown to give rise to resonances,even for the most basic quantum dot model. It is shown how the complexity of, e.g.,vibrational and magnetic excitations of molecular quantum dots needs to be accountedfor in transport spectroscopy calculations. harge transport through single molecules, quantum dots, and quantum wires V ≫ T , together with the time evolution into the stationary state. Thebasic physics of weak spin and strong charge fluctuations is illustrated on recentresults for two elementary models: the anisotropic Kondo model at finite magneticfield and the interacting resonant level model. A real-time renormalisation group(RTRG) method is introduced, where, complementary to the fRG scheme presentedin section 4, an expansion in a renormalised tunneling coupling is performed whereasthe Coulomb interaction is treated non-perturbatively. This allows the description ofstrong correlation effects Γ ≪ U for moderate tunneling.Finally, in section 6 we describe transport properties of quantum wires. The statusof Luttinger liquid physics is reviewed, in particular in connection with the presence ofinhomogeneities, backscattering, interference effects, and nonequilibrium. As in section4, the fRG scheme is shown to be a unique method, capable of describing the wholecrossover from few- to multi-level dots up to the limit of continuous spectra in quantumwires. Its ability to simultaneously incorporate microscopic details is of particularimportance for the description of experiments.
2. Transport through single molecule quantum dots
Molecular quantum dots present the ultimate miniaturization of quantum dot devices,which have been realized in semi-conductor heterostructures and wires, metallicnanoparticles, and carbon-nanotube molecular wires. Various methods have beendeveloped to contact nanometer size single molecules. Transport measurements in harge transport through single molecules, quantum dots, and quantum wires | s i of the quantum dot in this case, H D = X s E s | s ih s | . (1)Often, only a few of these states are actually accessible in an experiment. Subsequently,we incorporate the non-zero transparency of the junction in the coupling H T = X s,s ′ X αkσ T ss ′ αkσ | s ih s ′ | c αkσ + h . c . (2)through the amplitudes T ss ′ αkσ for an electron with spin σ to tunnel on / off the dot from/ to one of the electronic reservoirs labeled by α = L , R. Generally, in Equation (2) allstates s and s ′ are coupled which differ by the addition of one electron. The reservoir α by itself is described by H α = X kσ ( ǫ kσ + µ α ) n αkσ , (3)where n αkσ denotes the electron number operator and k labels the orbital states. Eachelectrode is restricted to contain effectively non-interacting fermion quasi-particles withdensity of states ρ α . We make the further statistical assumption that the electro-chemical potential µ α and temperature T are well defined for each reservoir α separately.The central quantity is the reduced density operator p of the dot, obtained by partialaveraging of the full density operator over the reservoirs. From p any non-equilibriumexpectation value of a local observable can be calculated, and also (see below) thetransport current. Starting from the above general model, one can derive the kineticequation from which the time-evolution of the dot density operator can be calculated: ddt p ( t ) = − i L D p ( t ) − i Z tt dt ′ Σ( t, t ′ ) p ( t ′ ) . (4)Here the Liouvillian superoperator L D acts on a density operator by commuting it with H D , i.e., L D p ( t ) = [ H D , p ( t )]. This linear transformation generates the “free” time-evolution of the dot density matrix, i.e., in the case where it is not coupled to theelectrodes. The challenge lies in the calculation of the transport kernel Σ( t, t ′ ) whichincorporates all effects due to the coupling to the electrodes which is adiabaticallyswitched on at time t ( t < t ). The superoperator Σ( t, t ′ ) is a non-trivial functionalof three “objects”: (i) the dot Liouvillian L D (energy differences of initial, final andvirtual intermediate states of processes), (ii) the dot part of the tunnel operator (tunnelvertices) and (iii) the product of the Fermi distribution functions f (( ǫ kσ − µ α ) /T ) anddensity of states ρ α of all electrodes α . Importantly, compact diagrammatic rules for the harge transport through single molecules, quantum dots, and quantum wires U T R T L µ µ g V L R
Figure 1. (Color online) Sketch of a single-level quantum dot attached to tworeservoirs with µ L / R = ± V / calculation of this kernel in terms of these three objects are known exactly [5, 6]. In thissection and section 3 we review the basic physics which emerges from the calculation ofthe kernel by perturbation theory in the tunnel operator H T ∼ Γ to leading and nextto leading order. This is applicable to the “high-temperature” regime, where higherorder effects are present but still weak. In section 4 and 5 the low-temperature physicsdue to strong quantum fluctuations is illustrated. In particular, in section 5 the kernelis calculated non-perturbatively in a renormalisation-group framework. The technicaldetails of these two approaches can be found in [6] and [5] respectively.We now first review all possible basic types of transport resonances which can arisedue to tunneling processes of the first and second order in Γ. For this purpose weconsider a single energy level with Coulomb interaction, the so-called Anderson model,where we emphasize the need to consider a finite charging energy U [7] such that doubleoccupancy is possible (not only as a virtual intermediate state). This will set the stagefor the subsequent discussion of examples of transport involving more complex spectrain section 2.2 and the time-dependent transport considered in section 3. Most types of transport resonances observed in quantum dot experiments can beillustrated by the Anderson model sketched in figure 1 where a single orbital levelis coupled to electrodes α = L , R with tunnel amplitudes T α . The strength of thetunnel coupling is characterized by the spectral function, Γ α = 2 πρ α | T α | , whose energydependence is neglected. Denoting by N the electron number on the dot, the eigenstates | s i of the dot (when decoupled from the electrodes) are easily classified. For the emptydot ( N = 0) there is one state s = 0 with energy E = 0. For the singly occupied dot( N = 1) there are two states s = σ = ↑ , ↓ with energy E σ = ε σ . The N = 1 statesare split by a magnetic field such that s = ↓ is the ground state and s = ↑ the excitedstate. Finally, for the doubly occupied dot ( N = 2) the energy E d = P σ ε σ + U of thestate s = d includes the charging energy. One can show that due to conservation of theelectron number and the spin-projection along the magnetic field only the occupationprobabilities enter into the description of the non-equilibrium state. We collect theseinto a vector p = ( p , p ↑ , p ↓ , p d ) T . Its time-evolution is fully described by a kinetic harge transport through single molecules, quantum dots, and quantum wires Σ ( t, t ′ ), ddt p ( t ) = − i Z t −∞ dt ′ Σ ( t, t ′ ) p ( t ′ ) , (5)where we have set t = −∞ (c.f. (4)). For time-independent parameters consideredhere, we have Σ( t, t ′ ) = Σ( t − t ′ ) and the occupancies will reach a constant value in thestationary long-time limit: p ( t ) → p = constant for t − t → ∞ . With the subsidiarycondition of probability conservation P s p s = 1, this state is determined uniquely by0 = − i Σ p . (6)As expected, here the time integral over the kernel is needed, i.e., the zero-frequencyLaplace transform lim z → i0 R ∞ dte i zt Σ ( t,
0) = Σ . This kernel has been calculated inclosed form up to order Γ for an arbitrary dot model Hamiltonian in [6], allowingarbitrarily complex transport spectra to be calculated. From a similar equation for themeasurable tunnel current from electrode αI α = − i Z t −∞ dt ′ Tr D Σ α ( t, t ′ ) p ( t ′ ) → − i Tr D Σ α p , (7)the stationary current can be found. This requires an additional zero-frequency kernel Σ α which takes account of both the amount and the direction of the electrons transferredfrom electrode α . Here Tr D denotes the trace over the dot degrees of freedom.In figure 2 we show exemplary results for the Anderson model of the calculationsoutlined above. A complete overview of non-linear transport resonances is obtainedwhen plotting dI/dV (quantifying changes in the current as new transport processesbecome energetically allowed) as function of the static applied bias V (which drives thecurrent) and gate voltage V g (which uniformly shifts all energy levels ǫ σ without changingthe quantum states) ‡ . In the lower half of the panel we plot the differential conductance dI/dV , whereas in the upper half of the panel the occupation p ↑ of the spin-excited stateof the quantum dot is shown. This figure illustrates the insight provided by transporttheory, linking the measured tunnel current to what goes on in the quantum dot. Toemphasize this correspondence of the two panels, the bias axes in the lower panel ismirrored with respect to the upper.To the far left and right of the plot the dot is in the N = 0 and N = 2 staterespectively around zero bias. Moving towards the center from there, dI/dV showsa peak along the first encountered yellow skewed lines (marked “SET”) where thecondition µ α = ε ↓ and µ α = ε ↑ + U are met, respectively. Beyond this line a spin ↓ electron can now enter or leave the dot in a sequential or single-electron tunnel (SET)process which is of first order in Γ. In this way the N = 1 ground state s = ↓ can bereached starting from N = 0 and N = 2 respectively. Once beyond the next skew yellowline (marked “SET*”), defined by µ α = ε ↑ and µ α = ε ↓ + U , respectively, an electronwith spin ↑ can tunnel onto the dot. This first order process makes the N = 1 excitedstate s = ↑ accessible, as the upper panel clearly shows. ‡ Here one assumes to a good first approximation that the bias and gate electric fields are spatiallyuniform. Corrections to this picture have been calculated in [8] for molecular junctions. harge transport through single molecules, quantum dots, and quantum wires Figure 2. (Color online) Occupation of excited spin-state, p ↑ (upper panel) anddifferential conductance (lower panel) for the single-level Anderson model, plotted asa function of bias voltage V and gate voltage βV g , where β is the gate coupling factor.The spin degeneracy is lifted by an applied magnetic field: ε ↑ − ε ↓ = 50 T where T is the electron temperature. The dot is symmetrically coupled to the left and rightelectrodes: Γ L = Γ R = 10 − T = 5 × − U . The horizontal green line indicates theinelastic cotunneling threshold, which equals the Zeeman energy. The skewed greenlines indicate where the sequential relaxation of the spin-excited state becomes possible(COSET). Moving further towards the center, there is a large triangular region of the dI/dV map where only the N = 1 ground state is accessible: sequential tunneling processes outof this state are suppressed to first order since U ≫ T , which is referred to as Coulombblockade. However, a second order process, referred to as inelastic cotunneling, allowsthe excited state to become occupied [9, 10] when the bias exceeds the excitation energyfor a spin flip, µ L − µ R = V > ε ↑ − ε ↓ . This defines a gate-voltage independentline (horizontal), reflecting the fact that the charge state is left unaltered by the spin-fluctuation process which flips the spin. Below this onset voltage only elastic cotunnelingis possible, giving a smaller conductance and current. In the small triangular regionmarked “ICOT” in the upper panel relaxation of the excited state is possible only byinelastic cotunneling. Above this, in the region marked “COSET”, the spin excited state,populated by slow inelastic excitation, can relax by a much faster sequential relaxationprocess. It therefore is depleted again and the dI/dV shows a small peak as signatureof this cotunneling-assisted sequential tunneling (COSET) [11].The above processes are well-known and have been experimentally observed invarious types of quantum dots, see, e.g., [9, 12, 13] and the reviews [2–4, 14]. However, harge transport through single molecules, quantum dots, and quantum wires electron pair tunneling : it showsup along a skewed line in the dI/dV map marked “PT”, the change in the contrastindicating a step in the conductance as function of bias V . The resonance condition forthis is 2 µ α = P σ ε σ + U , indicating that the energy is paid (gained) by two electronscoming (going) from reservoir α . This electron pair can be added to (extracted from)the dot in one coherent process, thereby changing the charge from N = 0 to N = 2 (orvice versa). The electro-chemical potential µ α = ( P σ ε σ + U ) at which this happensequals the average to the above mentioned SET resonances, i.e., due to quantum chargefluctuations the Coulomb energy paid (gained) per electron is reduced by .Finally, we note that the resonance positions of the sequential tunneling peaks areslightly shifted with respect to the above mentioned values µ α = ǫ ↓ , ǫ ↑ + U due to thesecond order energy level renormalisation which is consistently accounted for. Althoughthis may seem unimportant here, in section 3 it will be shown that in time-dependenttransport this level renormalisation effect may actually generate and dominate the fullcurrent. Molecular quantum dots are clearly more complex than the above considered model,for instance by their ability to mechanically vibrate. One important general aspect ofthis complexity is that the only conserved quantity in a tunnel process is the electronnumber § . In contrast, the quantum numbers of, e.g., various vibrational modes ofmolecular quantum dot may change when an electron tunnels onto it. This impliesthat the transport theory has to account explicitly for the density matrix off-diagonalelements between different excited states with the same charge number N . As is wellknown, this is important already in first order in Γ when there are degeneracies of theenergy levels E s in the same charge ( N ) state on the scale set by the tunneling [15–17].In the case where there are no such degeneracies only the probabilities matter to order Γ.However, it was shown only recently, that even in this simplest case this is no longer truein second (or higher) order in Γ and Equation (6) can no longer be used. Instead, onecan derive from (5) an effective stationary state master equation for the probabilities ofthe same form as (6), but with an effective kernel which contains important correctionsfrom the off-diagonal elements [6, 18]. An example where these corrections are found tobe very important is the Franck-Condon effect. This effect arises in its most elementaryform in the Anderson-Holstein model, where the dot is described by an Anderson level(discussed in section 2.1) plus a single vibrational mode with frequency ω : H D = X σ ε σ n σ + U n ↑ n ↓ + λ √ Q X σ n σ + ω P + Q ) . (8) § Conserved here refers to the system of reservoirs and dot, including their interaction harge transport through single molecules, quantum dots, and quantum wires Figure 3. (Color online) Differential conductance map vs. applied bias V and gatevoltage V g ( β = gate coupling factor) for a molecular level coupled to a vibrationalmode with frequency ω and strong local interaction U ≫ T, V, λ [6]. With increasingelectron vibration coupling λ , the inelastic cotunneling steps (red arrows in the λ = 2figure indicate 1 and 2 phonon absorption) become relatively important with respectto the suppressed SET processes. The white lines / regions correspond to negative dI/dV , which cannot be plotted due to the logarithmic color scale. Strikingly, thecurrent thus displays a peak in the Coulomb blockade regime, which is very uncommon.In all plots we have chosen ω = 40 T = 10 Γ and conduction bandwidth D = 250 ω .The conductance is scaled to Γ M , the maximal sequential tunneling rate, i.e., Γ timesthe maximal Franck-Condon overlap factor (which is less than 1). Here Q = ( b † + b ) / √ P = i( b † − b ) / √ ∝ λ the vibrational mode is displaced when charging the molecule. For λ > N . As a result the tunnel rates calculated tolowest order in Γ are suppressed and the sequential tunneling current is blocked, evenfor gate voltages close to the resonance [19–21], an effect also called “Franck-Condonblockade” [22]. This implies that second order processes become of crucial importanceeven at resonance [23]. The general approach developed in [6] consistently accountsfor all second order effects, i.e., not just cotunneling rates as in [23]. In figure 3we show the differential conductance calculated to second order where the vibrationalmode is shifted by several multiples of its zero-point amplitude. Whereas for λ = 1 thestandard Coulomb blockade picture is still identifiable, for λ = 4 it is drastically altered:inelastic cotunneling (threshold at µ L − µ R = ω, ω, . . . ) and COSET [24] resonancesproliferate in the transport spectrum. Experimentally, vibration assisted transport inthree terminal molecular devices has been reported [25–27]. Significant Franck-Condonblockade has been observed in suspended carbon nanotubes and compared with theory harge transport through single molecules, quantum dots, and quantum wires S > / N is H ( N )D = − D N S z + E N (cid:0) S x − S y (cid:1) , (9)where the uni-axial ( D N ) and transverse ( E N ) anisotropy parameters describe thesplitting and mixing of spin states, respectively. Recently, three-terminal measurementson magnetic molecules have been reported, including comparison with theory [45–48],see for a review [49]. Calculations of the sequential [50–55] and cotunneling [56–60]regimes for models including the anisotropy have identified distinctive transportsignatures and possibly useful effects. An important goal is to gain control overthe spin of a single molecule by electric means [61, 62]. Recently, the magneticanisotropy splittings were measured in a three terminal junction, similar to two terminalSTM measurements [63, 64]. Importantly, the magnetic field evolution of inelasticcotunneling [48] was measured in different charge states. It was found that the magnetic D N parameter depends strongly on the charge state N , which can be electricallycontrolled with the gate voltage.Beyond the issue of control over single-molecule magnetism, is the understandingof coupled molecular magnets or magnetic centers. Apart from spin-blockade effects,expected for large spin molecular ground states [65], the effect of spin-orbit interactionon the exchange coupling between two magnetic centers inside a magnetic doublequantum dot is of interest. For instance, the spin-orbit induced Dzyaloshinskii-Moriya orantisymmetric exchange interaction was shown to give rise to a characteristic dependenceof the transport on an externally applied magnetic field and the polarizations of magneticelectrodes [66].Higher order tunneling effects for magnetically anisotropic molecular quantumdots described by (9) have attracted attention [67–69] after the prediction of a novel harge transport through single molecules, quantum dots, and quantum wires D N ) (section 4), but italso generates intrinsic spin-tunneling (due to E N ). This interplay leads to interestinganisotropic effective exchange interaction with electrons transported through a molecularmagnet, as will be discussed in section 5.Besides their own interesting transport signatures, spin and vibrations can alsointeract or influence each other in transport. For instance, a mechanism of vibration-induced spin-blockade of transport was predicted for a mixed-valence dimer moleculartransistor (“double dot”) [71, 72].Finally, we mention that level spectrum complexity may also deceive one whenanalyzing experimental data. A particularly clear example is offered by recent transportdata on carbon-nanotube “peapods”, i.e. fullerene molecules in a host nanotube [73].Here weakly gate-dependent dI/dV peaks were observed which are strongly reminiscentof inelastic cotunneling (second order in Γ, see section 2.1). However, detailed transportcalculations have conclusively shown that the complex measured spectrum can beassigned to sequential tunneling processes only, i.e., of first order in Γ. This deceptivecase occurs because the two coupled quantum dots, the host nanotube and the fullerene“impurities” have a very different gate voltage dependence. Fortunately, the significanthybridization of the two dots gives a characteristic interference effect in the first ordercurrent, distinguishing it clearly form second order effects. In view of applications itis interesting that the analysis in [73] indicates that the charge state of the fullereneimpurities is electrically tunable, independently of that of the host carbon nanotube. In summary, we have illustrated that general methods for spectroscopy calculationsfor molecular quantum dot models are essential for the experimental characterizationand ultimately the design of nanoscale quantum dot devices. The complex spectradue to various strong many-body interactions play a crucial but complicating role,at the same time however, enabling interesting new possibilities. An exciting futuredirection important for further progress is to efficiently account for renormalisationeffects in complex models, in particular as outlined in section 5. Also, the approachoutlined here can be extended to other transport quantities, such as spin and heat [74].The molecule and junction models used here can be parametrized based on atomisticelectronic structure calculations, as for instance, in [17, 75, 76]. Pursuing this further isanother important step in understanding molecular quantum dots.
3. Adiabatic transport through a quantum dot due to time-dependent fields
Charge transport through quantum dots is not necessarily due to an applied static biasvoltage: in the absence of a bias, a current can be obtained by the time-dependentmodulation of fields, applied externally to a mesoscopic device. This is of interest for harge transport through single molecules, quantum dots, and quantum wires adiabatic .The system then remains in a quasi-equilibrium state, meaning that the system is notbrought to an excited state by the time-dependent modulation k .A prominent example for transport in the absence of a bias voltage is adiabaticpumping : the cyclic variation of at least two of the system parameters, see figure 4,leads to a dc charge transfer through the quantum device. In this case the pumpedcharge does surprisingly not depend on the detailed time evolution of the pumpingcycle; this can be shown by formulating it in terms of geometric phases [83–85].Theoretically, adiabatic pumping has been extensively studied in the non-interactingregime [86–92]. In this case a scattering matrix approach is convenient [86, 93] and hasbeen broadly used. While indeed Coulomb interactions are of minor importance in somesetups [77, 78, 94], they play an important role in many quantum-dot systems and leadto a variety of interesting results. Several approaches have been used to address theproblem of adiabatic pumping through interacting quantum dots in different regimes,including weak Coulomb interaction [95, 96], pumping in Luttinger liquids [97], anda quite general approach using nonequilibrium Green’s functions [98–100]. The latterworks are particularly appropriate for the study of weak interaction or for the calculationof spin and charge pumping in the Kondo regime [99] and the role of elastic and inelasticscattering processes has been studied in detail [100]. The treatment of the tunnelcoupling is non-perturbative in these works and they are in this sense complementaryto the results presented in the following.It turns out that the interplay of Coulomb interaction and coherent tunnelingbetween a quantum dot and leads, leading to an effective renormalisation of the energylevel, can be at the origin of the pumping mechanism. The result is that this energyrenormalisation, which is a pure Coulomb interaction effect vanishing when Coulombinteractions are negligible, is accessible via the pumped charge.Here we present results in the regime in which the coupling between the dot andthe leads is moderate; this means that the energy scale Γ approaches temperature T but is still smaller than T so that charge fluctuations start to become important. Incontrast, the energy scale T K of the Kondo temperature setting the scale for the onsetof spin fluctuations, is taken to be much smaller than Γ and T . To approach the effectdescribed above we use a perturbative expansion in the tunnel coupling between the k The slow variation of the externally applied fields becomes experimentally particularly importantfor delicate systems as molecules, see section 2, where problems related to heating, surface excitations(plasmons) and laser-induced chemical reactions (photo-bleaching) should be avoided. harge transport through single molecules, quantum dots, and quantum wires ε (t) + U Γ R (t) Γ (a) L (t)L R ε (t) X(t)Y(t) (b)
Figure 4. (Color online) (a) Sketch of a single-level quantum dot with time-dependentparameters attached to two reservoirs. (b) Two of the system parameters are timedependent as indicated in (a) and perform a closed cycle in parameter space. dot and the leads up to second order in Γ, which is appropriate in this regime. Theaim is the understanding of the influence of finite and arbitrary Coulomb interaction onthe pumping characteristics and the identification of the physical origin of the variouscontributions to the pumped charge. In order to achieve this, a diagrammatic real-time technique [101–104], which has been developed to describe non-equilibrium DCtransport through an interacting quantum dot, see section (2), was extended for thetreatment of adiabatically time-dependent fields [105].
Even though the approach is not restricted to specific systems, here the focus is put on asingle-level quantum dot as shown in figure 4. The quantum dot has a time-dependentlevel ε ( t ) = ¯ ε + δε ( t ) and is coupled to leads α = L , R with time-dependent tunnelamplitudes T α ( t ). The strength of the tunnel coupling is characterized by the time-dependent quantity, Γ α ( t ) = 2 πρ α T ∗ α ( t ) T α ( t ) = ¯Γ α + δ Γ α ( t ), with the lead’s constantdensity of states, ρ α . As described in section 2, the eigenstates of the decoupledsystem are given by s = { , ↑ , ↓ , d } . Tracing out the lead degrees of freedom thesystem is described by its reduced density matrix, where here only the occupationprobabilities p = ( p , p ↑ , p ↓ , p d ) T are of interest. Their time-evolution in the long-timelimit, t → −∞ , is fully described by the generalized Master equation, Equation (5).The goal being the description of slowly varying system parameters, it is useful toperform an adiabatic expansion [105]: the frequency of the variation therefore has tobe much smaller than the dwell time of electrons in the system, Ω ≪ Γ. The effect ofthe system parameters varying slowly in time is that the density matrix of the systemat time t is not only described by the instantaneous values of the parameters, but itlags a bit behind the parameter change. Therefore an expansion around this time t suggests itself. To obtain such an expansion as a first step a Taylor expansion of p ( t ′ )is performed around t up to linear order in the integrand on the right hand side ofEquation (5). This is related to the fact that memory effects of the kernel have to betaken into account. Furthermore, an adiabatic expansion of the kernel Σ ( t, t ′ ) itself is harge transport through single molecules, quantum dots, and quantum wires -20 -10 0 10 ε/Γ Q Γ L , ε [ η / Γ ] U=0.1 Γ U=4 Γ U=8 Γ -40 -30 -20 -10 0 10 ε/Γ -0.06-0.040.040.06 Q Γ L Γ R [ η / Γ ] U=0.1 Γ U=4 Γ U=20 Γ U=30 Γ Figure 5. (Color online) (a) Pumped charge due to a parameter variation of the dotlevel and of one of the barriers as a function of the average dot level. The dominantcontribution to the pumped charge is due to first order tunneling processes. (b)Pumped charge due to the pure variation of the barrier strengths as a function ofthe average dot level. The dominant contribution to the pumped charge is due to second order tunneling processes only. In both plots the temperature equals T = 2Γ. performed. The zeroth-order term, Σ ( i ) t ( t − t ′ ), is indicated with the superscript ( i ) for instantaneous . The subscript t to emphasize that the system parameters X ( τ ) → X ( t )are frozen at time t , i.e., the functional dependence on X ( τ ) is replaced by X ( τ ) → X ( t ).The first-order term is obtained by linearizing the time dependence of all parameters X ( τ ) with respect to the final time t , i.e., X ( τ ) → X ( t ) + ( τ − t ) ddτ X ( τ ) | τ = t , andretaining only linear terms in time derivatives. This linear correction to the kernel isindicated by the superscript ( a ) for adiabatic . Finally, it is necessary to perform anadiabatic expansion for the occupation probabilities in the dot, p ( t ) → p ( i ) t + p ( a ) t (10) Σ ( t, t ′ ) → Σ ( i ) t ( t − t ′ ) + Σ ( a ) t ( t − t ′ ) (11)The instantaneous probabilities p ( i ) t are the solution of the time-independent problemwith all parameter values fixed at time t . They are obtained by solving the stationaryMaster equation, Equation (6), with all parameters replaced by time-dependentparameters evaluated at time t . Once the instantaneous probabilities p ( i ) t are known,the adiabatic corrections p ( a ) t are found from the Master equation in first order in Ωobtained from the expansion described above. Similarly an equation for the current canbe developed, see equation (7), and adiabatically expanded, where a current kernel Σ L takes account for the amount and the direction of electron transfer.On top of the adiabatic expansion a perturbative expansion in the tunnel couplingstrength Γ is performed up to second order, allowing the actual evaluation of thekernel using diagrammatic rules as elaborated in Refs. [101–105], see section 2. Thisperturbative expansion is valid in the regime of moderate couplings with respect totemperature, i.e. Γ . T . harge transport through single molecules, quantum dots, and quantum wires We are interested in the time-resolved pumping current as well as the charge pumpedthrough the dot per pumping cycle. The pumped charge, Q = R T I L ( t ) dt , in the bilinearresponse regime, i.e. for modulations with small amplitudes, is proportional to the areaspanned in parameter space, see figure 4. As a first step the time-resolved pumpingcurrent is calculated in lowest order in the tunnel coupling. It is straightforward torelate the pumped current to the dynamics of the average instantaneous charge of thedot h n i ( i ) t and one finds I (0)L ( t ) = − Γ L ( t )Γ( t ) ddt h n i ( i, t , (12)where h n i ( i, t denotes the instantaneous occupation in lowest order in Γ. This suggeststhe following interpretation. As the dot occupation in lowest order, h n i ( i, t , is changed intime by varying the pumping parameters the charge moves in and out of the quantum dotgenerating a current from/into the leads. Note that one of the pumping parameters mustbe the level position since h n i ( i, t is given by Boltzmann distributions and is thereforeindependent of the tunnel-coupling strengths. This means that ddt h n i ( i, t = 0 if the levelposition is constant. The charge moving in and out of the quantum dot is divided tothe left and to the right, depending on the time-dependent relative tunnel couplingsΓ α ( t ) / Γ( t ).The next step is to calculate the next-to-leading order correction to the lowestorder result shown in Equation (12). Also the next-to-leading order can be written ina compact way I (1)L ( t ) = − (cid:26) ddt (cid:16) h n i ( i, broad , L) t (cid:17) + Γ L ( t )Γ( t ) ddt h n i ( i, ren) t (cid:27) . (13)The first term of Equation (13) contains the contribution due to the correction ofthe average dot occupation induced by lifetime-broadening. It contains a total timederivative, and as parameters are periodically changing in time, it will not lead to a netpumped charge after the full pumping cycle. The second term has the same structureas the zeroth-order contribution, Equation (12). It can be understood as the correctionterm induced by the combined influence of charge fluctuations and correlation effectsgiving rise to a renormalisation of the level position, ε ( t ) → ε ( t ) + σ ( ε ( t ) , Γ( t ) , U ).The level renormalisation, σ ( ε, Γ , U ), is positive when the level is in the vicinity ofthe Fermi energy of the leads and negative for ε + U being close to the Fermi energy.The sign of the renormalisation thus indicates if transitions take place between emptyand singly occupied dot or between single and doubly occupied dot. The energy levelrenormalisation may be time dependent via time-dependent tunnel couplings or a time-dependent level. The result is that a finite charge can be pumped by means of levelrenormalisation. This is a pure Coulomb interaction effect, it vanishes for U →
0. Wenote that correction terms from the renormalisation of the tunneling couplings do notcontribute in first order in Γ. For the DC current, different contributions from higherorder processes are present at the same time, which makes it challenging to identify them harge transport through single molecules, quantum dots, and quantum wires dominant contributionto the pumped charge is due to time-dependent level renormalisation. These resultssuggest that adiabatic pumping can be used to directly access the level renormalisationin quantum dots.The results for the pumped charge per pumping cycle are shown in figure 5, where η is the enclosed surface in parameter space. Modulating the level position and one of thetunnel couplings the pumped charge has a maximum contribution at the resonances,see figure 5(a). Importantly the contribution from sequential tunneling is dominanthere. Figure 5(b) shows the pumped charge obtained by modulating the two tunnelcouplings. As described above, the pumped charge is due to level renormalisation andtherefore vanishes for vanishing Coulomb interaction. The sign change between the twocontributions at the resonances reflects the opposite sign of the level renormalisation forthe two resonances.The method described here is generally formulated and it is therefore extendableto a variety of systems in which Coulomb interaction is important and to which time-dependent fields are applied. Charge and spin pumping through an interacting quantumdot has been studied in the presence of ferromagnetic leads [106]. Pumping throughone or two metallic interacting islands with a continuous density of states has beenexamined in [107]. In [108] the validity of the frequency regime has been extended tofaster modulations and in [109] the influence of quantum interference was studied onthe pumped charge through one or two quantum dots embedded in an Aharonov-Bohmgeometry. Finally this approach has been used to study the capacitive and the relaxationproperties of a driven quantum dot figuring as a mesoscopic capacitor in [110]. In this section we have presented results on adiabatic pumping, where interesting effectsdue to quantum charge fluctuations and finite Coulomb interaction are revealed. Theunderlying method shown here uses an adiabatic expansion of the Master equation basedon a real-time diagrammatic technique; this method is applicable whenever a mesoscopicsystem is exposed to a certain number of adiabatically time-dependent fields. Theintriguing results found for systems with a relatively simple spectrum together with the general formulation of the adiabatic expansion of the Master equation, motivate furtherinvestigations. A challenging question is, e.g., to study the impact of time-dependentfields on molecular devices with a complex energy spectrum, as introduced in section 2.Another interesting development concerns the combination of the formalism of adiabaticquantum pumping with renormalisation group methods, as described in sections 4 and harge transport through single molecules, quantum dots, and quantum wires
4. Quantum fluctuations in linear response
In this section we will discuss the physics of strong quantum fluctuations in combinationwith correlation effects in quantum dots. We will concentrate on the linear response andstatic regime, the dependence on finite bias voltage together with the time evolutionwill be discussed in section 5. One of the most prominent examples of the drasticeffects of spin fluctuations in quantum dots is the experimental observation of theKondo effect [111]. In bulk solids a small amount of magnetic impurities leads toan increased magnetic scattering of the electrons at low temperature, which resultsin an increased resistance. In quantum dots, the mesoscopic realization of a singlespin coupled to two leads displays instead a zero-bias peak in the conductance [112].The particular advantage of quantum dots is that they allow for a very flexible tuningof the parameters and can easily be extended to study the effects of more compleximpurities, where orbital Kondo, quantum critical and interference effects may arise.Moreover, additional environmental degrees of freedom in presence of ferromagneticor superconducting reservoirs coupled to the quantum dot as well as finite-size effectsaffect the transport properties. For spin fluctuations, the characteristic energy scale T c is given by the Kondo temperature T K , which, for the special case of a single-level dotwith Coulomb energy U and tunneling coupling Γ, is given by T K ∼ √ U Γ e − πU . Thisscale is exponentially small for large Coulomb energy and, therefore, the Kondo effect inquantum dots is only visible for strongly coupled leads. In contrast, the importance ofcharge fluctuations is controlled by Γ and signatures are already significant for Γ ∼ T .Already in the last section, we showed how renormalised level positions can be identifiedin the adiabatically pumped charge. Experimentally, effects from charge fluctuationshave been detected in transmission phase lapses through multi-level quantum dots inAharonov-Bohm geometries [113, 114]. New light was shed on this long-outstandingpuzzle by the insight that the combined influence of broadening and renormalisationeffects induced by charge fluctuations is responsible for this effect [115, 116].The analysis of the signatures of correlation and quantum fluctuation effectsrequires appropriate methods for their theoretical description. The importance of thedevelopment of analytical as well as numerical techniques for their treatment constitutestherefore an important issue. In recent years functional renormalisation-group (fRG)methods [117] have been established as a new computational tool in the theory ofinteracting Fermi systems. These methods are particularly powerful in low dimensions.The low-energy behavior usually described by an effective field theory can be computedfor a concrete microscopic model by solving a differential flow with the energy scaleas the flow parameter. Thereby also the nonuniversal behavior at intermediate scalesis obtained. For applications to quantum dots and wires the fRG scheme turned outto be an efficient approach for the description of the single-particle spectral properties harge transport through single molecules, quantum dots, and quantum wires The appearance of an upturn followed by a low-temperature saturation of the resistivityin metals containing diluted magnetic impurities was explained by Kondo in 1960 as anenhanced scattering due to the screening of the local impurity spins by the conductionelectrons [118]. The Fermi liquid nature of the ground state [119] and the varioustheoretical treatments, including Wilson’s numerical renormalisation group (NRG) [120],make the Kondo problem one of the best understood many-body phenomenon incondensed matter physics, as well as a paradigm for correlated electron physics [121].In the late 90’s, the prediction by Ng and Lee [122], as well as Glazman and Raikh [112]of the Kondo effect in quantum dots led to a revival of the subject, and has beenobserved in 2D GaAs/AlGaAs heterostructure quantum dots [111, 123, 124], siliconMOSFET’s, as well as in carbon nanotubes [125] and contacted single molecules[126, 127]. Confinement and electrostatic gating enabled to investigate the Kondoeffect in an artificial Anderson impurity through electrical transport measurements,characterized by a unitary conductance 2 e h at zero temperature [128]. Besides providinga paradigm for a variety of physical effects involving strong electronic correlations, theKondo effect in nanostructures allows for the realization of spintronic device constituentsas spin filters or quantum gates. The understanding and control of the transport of spin-polarized currents are of fundamental importance for the advancement of semiconductorspintronic device technology [129].The coupling of a quantum dot with spin-degenerate levels and local Coulombinteraction to metallic leads gives rise to Kondo physics [121]. A small quantum dot withlarge level spacing is described by the single-impurity Anderson model (SIAM) in theregime in which only a single spin-degenerate level is relevant. At low temperatures andfor sufficiently high barriers the local Coulomb repulsion U leads to a broad resonanceplateau in the linear conductance G of such a setup as a function of a gate voltage V g which linearly shifts the level positions [112, 122, 130–132]. It replaces the Lorentzian ofwidth Γ found for noninteracting electrons. On resonance the dot is half-filled implyinga local spin- degree of freedom responsible for the Kondo effect [121]. In the limit oflarge U ≫ Γ the charge degrees of freedom can be integrated out and the spin physics isdescribed by the Kondo model, which will be discussed in section 5. For the SIAM the harge transport through single molecules, quantum dots, and quantum wires U in theconductance is due to the pinning of the Kondo resonance in the spectral function at thechemical potential for − U . V g . U (here V g ≡ ǫ + U = 0 corresponds to the half-filleddot case) [121, 132]. Kondo physics in transport through quantum dots was confirmedexperimentally [111, 134], and theoretically using the Bethe ansatz [130, 132] and theNRG technique [120, 135]. However, both methods can hardly be used to study morecomplex setups. In particular, the extension of the NRG to more complex geometriesbeyond single-level quantum-dot systems [131, 132, 136] is restricted by the increasingcomputational effort with the number of interacting degrees of freedom. Alternativemethods which allow for a systematic investigation are therefore required. Here the fRGapproach is proposed to study low-temperature transport properties through mesoscopicsystems with local Coulomb correlations.A particular challenge in the description of quantum dots is their distinct behavioron different energy scales, and the appearance of collective phenomena at new energyscales not manifest in the underlying microscopic model. An example of this is theKondo effect where the interplay of the localized electron spin on the dot and the spinsof the lead electrons leads to an exponentially small (in U/ Γ) scale T K . This diversityof scales cannot be captured by straightforward perturbation theory. One tool to copewith such systems is the renormalisation group: by treating different energy scalessuccessively, one can often find an efficient description for each one. The fermionicfRG [137] is formulated in terms of an exact hierarchy of coupled flow equations forthe vertex functions (effective multi-particle interactions) as the energy scale is lowered.The flow starts directly from the microscopic model, thus including nonuniversal effectson higher energy scales from the outset, in contrast to effective field theories capturingonly the asymptotic behavior. As the cutoff scale is lowered, fluctuations at lower energyscales are successively included in the determination of the effective correlation functions.This allows to control infrared singularities and competing instabilities in an unbiasedway. Truncations of the flow equation hierarchy and suitable parametrizations of thefrequency and momentum dependence of the vertex functions lead to new approximationschemes, which are devised for moderate renormalised interactions. A comparison withexact results shows that the fRG is remarkably accurate even for sizeable interactions.To exemplify the effect of correlations, we first focus on a quantum dot with spin-degenerate levels. For simplicity only a single level with a local Coulomb repulsion U described by the SIAM [121] is considered, see figure 1. The hybridization of the dotwith the leads broadens the levels on the dot by Γ α = 2 πT α ρ α , where ρ α is the localdensity of states at the end of lead α = L, R assumed to be independent of frequencyhere. The energy level is determined by the gate voltage V g . Integrating out the leaddegrees of freedom, the bare Green function of the dot [121] is G ( iω ) = 1 iω − V g + i Γ2 sgn( ω ) , (14)where Γ = Γ L + Γ R . By solving the interacting many-body problem a self-energy harge transport through single molecules, quantum dots, and quantum wires G / ( e / h ) U/ Γ =0.5U/ Γ =5U/ Γ =12.5 -4 -2 0 2 4 V g /U n -4 -2 0 2 40 Figure 6. (Color online)
Upper panel: conductance as a function of gate voltage fordifferent values of U/ Γ. Lower panel: average number of electrons on the dot. contribution Σ( iω ) is obtained dressing the bare propagator through the Dyson equation.The fRG is used to provide a self-energy describing the physical properties of the T = 0linear conductance obtained by G ( V g ) = e h π Γ ρ (0) [133] in terms of the dot spectralfunction ρ ( ω ) = − π Im G ( ω + i + ). In order to implement the fRG, an infrared cutoffin the bare propagator is introduced G Λ0 ( iω ) = G ( iω )Θ( | ω | − Λ). As the cutoff scale Λis gradually lowered, more and more low-energy degrees of freedom are included, untilfinally the original model is recovered for Λ →
0. Changing the cutoff scale leads to aninfinite hierarchy of flow equations for the vertex functions. In the static approximationthe flow equation for the effective level position V Λ = V g + Σ Λ reads ∂ Λ V Λ = U Λ V Λ /π (Λ + Γ2 ) + ( V Λ ) (15)with the initial condition V Λ= ∞ = V g [138]. In first approximation the two-particlevertex U Λ ≡ U Λ= ∞ = U equals the bare Coulomb interaction. At the end of the flow,the renormalised potential V = V Λ=0 determines the conductance G ( V g ) = 2 e h Γ V +Γ .Results for different values of U/ Γ are shown in figure 6, together with the occupationof the dot.For Γ ≪ U the resonance exhibits a plateau [132]. In this region the occupationis close to 1 while it sharply rises/drops to 2/0 to the left/right of the plateau. Forasymmetric barriers the resonance height is reduced to 2 e h L Γ R (Γ L +Γ R ) [121, 132]. Focusingon strong couplings U ≫ Γ, the solution of the above flow equation V ≃ V g e − Uπ Γ describes the exponential pinning of the spectral weight at the chemical potential forsmall | V g | and the sharp crossover for a V of order U . While already the first orderin the flow-equation hierarchy captures the correct physical behavior, the inclusion ofthe renormalisation of the two-particle vertex improves the quantitative accuracy ofthe results. For details on the parametrization and extensions including dynamical harge transport through single molecules, quantum dots, and quantum wires G ( V g ) at V g = 0, providing a definition ofthe Kondo scale as the magnetic field required to suppress the total conductance to onehalf of the unitary limit. For the single dot at T = 0 the conductance and transmissionphase are related by a generalized Friedel sum rule to the dot occupancy [121] by G = 2 e h sin ( π h n i ) and α = π h n i . For gate voltages within the conductance plateauthe dot filling is 1 and the phase is π . The description of transmission phases for morecomplex setups will be discussed in the next subsection. The following application to a multi-level quantum dot illustrates the strength of the fRGapproach, the flexibility and simple implementation, in the description of the intriguingphase-lapse behavior observed in experiments by the group of Heiblum at the WeizmanInstitute [113, 114]. The transmission amplitude and phase T = | T | e iα of electronspassing through a quantum dot embedded in an Aharonov-Bohm geometry showeda series of peaks in | T | as a function of a plunger gate voltage V g shifting the dot’ssingle-particle energy levels. Across these Coulomb blockade peaks α ( V g ) continuouslyincreased by π , as expected for Breit-Wigner resonances. In the valleys the transmissionphase revealed “universal” jumps by π for large dot occupation numbers. In contrast,for small dot fillings the appearance of a phase lapse depends on the dot parameters inthe “mesoscopic” regime. From the theoretical side, the observed behavior is capturedby an fRG computation of the transport through a multi-level quantum dot with localCoulomb interactions [115, 116]. An essential aspect in the description of the genericgate-voltage dependence of the transmission lies in the feasibility of a systematic analysisof the whole parameter space. The interaction is taken into account approximately, buta comparison to numerically exact NRG data for special parameters proves the fRGresults to be reliable as long as the interaction parameter and the number of almostdegenerate levels do not become too large simultaneously. The results are shown infigure 7, for spin-degenerate levels we refer to Ref. [116].Assuming level spacings of the order of the level width or smaller for large dotfillings (similar to a hydrogen atom with decreasing level spacing for increasing quantumnumber) and well separated levels for dots occupied by only a few electrons, theobtained results are consistent with the experimental findings in both regimes. Inthe “mesoscopic” regime the Coulomb blockade yields an increase in the separationof the transmission peaks. The behavior in the valleys depends on the details of thelevel-lead couplings, and can be continuous or discontinuous with a phase lapse of π (see upper panels in figure 7). For several almost degenerate levels in the “universal”regime the hybridization leads to a single broad and several narrow effective levels (Dickeeffect [143]). In presence of the Coulomb repulsion, the gate-voltage dependence of thebroad level is significantly reduced, in contrast to the narrow levels crossing the broad harge transport through single molecules, quantum dots, and quantum wires α / π -24 0 2401 | T | α / π -6 0 6 V g / Γ | T | UNIV. δ ε /Γ= δ ε /Γ= Figure 7. (Color online) Transmission amplitude | T ( V g ) | and phase α ( V g ) for U/ Γ = 1and N = 4 equidistant levels with spacing δ ǫ . Decreasing δ ǫ / Γ leads to a crossoverfrom mesoscopic to universal behavior. level close to the chemical potential. The combination of these effects induces wellseparated transmission peaks, absent without interaction. The “universal” phase lapsecan be understood as a result of the Fano effect of the effective renormalised levels,where π phase lapses appear in resonance phenomena (see lower panels in figure 7).The mechanism leading to the phase lapses reflects also in the dot level occupancies,the broad level being filled and emptied via the narrow levels [144, 145]. The detailedunderstanding of the charging of a narrow level coupled to a wide one in presence ofinteractions is of interest also in connection with experiments on charge sensing [146].From an fRG analysis emerges a single-parameter scaling behavior of the characteristicenergy scale for the charging of the narrow level with an interaction-dependent scalingfunction [147]. The described fRG approach presents a reliable and promising tool for the investigationof correlation and quantum fluctuation effects in quantum dots, the computational effortbeing comparable to a mean-field calculation. In contrast to the latter the fRG doesnot lead to unphysical artefacts. Besides the Kondo effect and transmission phases inmulti-level dots, as explained above, also the competition between the Kondo effectand interference phenomena in more complex quantum-dot systems can be described[138,148–151]. Furthermore, correlation effects for a large number of single-particle levels harge transport through single molecules, quantum dots, and quantum wires U/ Γ. A fundamental question concerns thecombination with the Keldysh formalism to describe nonequilibrium problems addressedin the next section.
5. Quantum fluctuations in nonequilibrium and time evolution
In this section we discuss quantum fluctuations in quantum dots or molecules in thepresence of a finite bias voltage together with the time evolution into the stationarystate. As in the previous section the quantum fluctuations are induced by the couplingto the reservoirs. If the maximum of temperature T and the distance to resonances δ decreases, correlations effects from the Coulomb interaction lead to an increasedrenormalisation of the couplings and the excitation energies h i of the quantum dot(like e.g. magnetic field, level spacing, single-particle energy, etc.). Renormalisationgroup (RG) methods, which expand systematically in the reservoir-system coupling (incontrast to the fRG method described in the previous section 4, where an expansion inthe Coulomb interaction is performed), show that these renormalisations are typicallylogarithmic or power laws. In nonequilibrium or for the time evolution, the voltage V and the inverse time scale 1 /t are two new energy scales, which can cut off theRG flow. New physical phenomena emerge, which we will illustrate in this section byconsidering very basic two-level quantum dots, like two spin states (Kondo model) ortwo charge states (interacting resonant level model (IRLM)). Conceptually, one has todistinguish between the two different regimes of strong and weak quantum fluctuations.In strong coupling, an expansion in the renormalised coupling is no longer possible,and, except for the case of strong charge fluctuations in the IRLM or for the caseof moderate Coulomb interactions (which can be treated with fRG methods), thenonequilibrium case remains a fundamental yet unsolved problem. Especially for thecase of strong spin fluctuations, represented by the nonequilibrium Kondo model or thenonequilibrium single-impurity Anderson model, an analytical or numerical solution isstill lacking. Even very basic questions, like e.g. the splitting of the Kondo peak inthe spectral density by a finite bias voltage are not yet clarified. In weak coupling,where a controlled expansion in the renormalised coupling is possible, many resultsare already known. The simplest ansatz is to take the bare perturbation theory, asexplained in section 2, use a Lorentzian broadening of the energy conservation (inducedby relaxation and decoherence rates), and replace the bare couplings and excitationenergies by the renormalised ones obtained from standard equilibrium poor man scaling(PMS) RG equations cut off by the maximum Λ c = max { T, h i , V, /t } of all physicalenergy scales. However, it turns out that this approach is not sufficient. In contrast harge transport through single molecules, quantum dots, and quantum wires δ i to resonances, whereRG enhanced contributions occur. Furthermore, at resonance δ i = 0, these contributionsare cut off by relaxation and decoherence rates Γ i , i.e. one should use the energy scale | δ i + i Γ i | as cutoff parameter for the couplings. An RG approach has to be developed toreveal how the various cutoff parameters influence the couplings. It turns out that therates Γ i are transport rates , which have to be determined from a kinetic (or quantumBoltzmann) equation, in contrast to rates describing the decay of a local wave functioninto a continuum. Therefore, the RG has to be combined with kinetic equations. Thetransport rates Γ i depend also on voltage and are important parameters to prevent thesystem from approaching the strong coupling regime for voltages larger than the strongcoupling scale T c . Furthermore, it turns out that terms can occur in the renormalisedperturbation theory which are not present in the bare one, like e.g. the renormalisationof the magnetic field in linear order in the coupling for the Kondo model (similar effectscan also happen in higher orders for generic models). For the time evolution, theseterms are of particular interest in the short-time limit, since in this regime they are cutoff by the inverse time scale 1 /t and lead to universal time evolution. At large times,non-Markovian parts of the dissipative kernel of the kinetic equation lead to many otherinteresting effects. Among them are unexpected oscillation frequencies involving thevoltage, unexpected decay rates, power-law behavior, and a different cutoff behavior atresonances compared to stationary quantities. In particular, for metallic reservoirs witha constant density of states, it can be shown that all local physical observables decayexponentially accompanied possibly by power-law behavior. Since all these effects occuralready for very basic two-level systems, it has to be expected in the future that manymore interesting effects will be found with respect to the physics of quantum fluctuationsin nonequilibrium.From a technical point of view the description of correlated quantum dots in thepresence of quantum fluctuations in nonequilibrium is a very challenging problem and aplayground for the development of new analytical and numerical methods. Concerningnumerical methods, numerical renormalisation group (NRG) with scattering waves[157], time-dependent NRG (TD-NRG) [68, 158–160], time-dependent density matrixrenormalisation group (TD-DMRG) [161, 161–165], quantum Monte Carlo (QMC) withcomplex chemical potentials [166], QMC in nonequilibrium [167–169], and iterativepath-integral approaches (ISPI) [170] have been developed. However, the efficiencyof these methods is often not satisfactory in the regime of either strong Coulombinteraction, large bias voltage, or long times. Exact analytic solutions exist for somespecial cases [162, 171–174] and scattering Bethe-ansatz methods have been appliedto the IRLM [175]. Other analytical methods are mainly based on RG approaches.Besides conventional PMS [176], improved frequency-dependent RG schemes [177–179],flow equations [180], and fRG methods [141, 142, 181–184] have been used. Whereasthe latter expands systematically in the Coulomb interaction, we will introduce in this harge transport through single molecules, quantum dots, and quantum wires The aim of RTRG-FS is to combine the diagrammatic expansion in Liouville space,as described in section 2, with RG to resum systematically infrared divergences. Weconsider a time translational invariant system and use as an initial condition at t = 0that the dot and the reservoirs are decoupled. The kinetic equation (4) can be writtenin Laplace space as˜ p ( z ) = iz − L effD ( z ) p ( t = 0) , (16)where ˜ p ( z ) = R ∞ dt e izt p ( t ) is the reduced density matrix of the dot in Laplace spaceand L effD ( z ) = L D + ˜Σ( z ), where ˜Σ( z ) denotes the Laplace transform of the kernelΣ( t − t ′ ) = Σ( t, t ′ ). The effective dot Liouville operator L effD ( z ) contains all reservoirdegrees of freedom and is of dissipative nature. The qualitative dynamics can beanalyzed from the analytic structure of ˜ p ( z ), which is an analytic function in the upperhalf of the complex plane with poles and branch cuts only in the lower half. Thesingle poles located at z i p = h i − i Γ i correspond to exponential decay with oscillationfrequencies h i and decay rates Γ i . From the pole at z = 0, the stationary state canbe obtained via the solution of L effD ( i + ) p = 0. Due to non-Markovian terms arisingfrom the z -dependence of the Liouvillian, the weight of these poles is changed by Z -factors and branch cuts can occur. For a constant density of states in the leads (normalmetallic case) it can be shown generically [190, 191] that the Z -factor leads to universalshort-time behavior, whereas the branch cuts give rise to an exponential decay ∼ e − iz i b t accompanied by power-law behavior. z i b denotes the position of the branching point,which is shifted from some pole position by multiples of the electrochemical potentials ofthe reservoirs leading to oscillation frequencies involving the voltage. From the point ofview of error correction schemes in quantum information processing it is quite importantto understand these corrections to Markovian behavior [192–194].It is unique to the RTRG-FS method that it provides direct access to the importantquantity L effD ( z ). By systematically integrating out the energy scales of the reservoirsstep by step, a formally exact RG equation can be derived for L effD ( z ) Λ , where all reservoirenergy scales beyond Λ are included. This RG equation is coupled to other RG equations harge transport through single molecules, quantum dots, and quantum wires µ R L µ Γ LL U µ R L µ ε R U Γ R S z / z / JJ (a) (b)
Figure 8. (color online) Two fundamental quantum dot models. (a) is the Kondomodel, a spin- coupled via exchange couplings J z, ⊥ to two reservoirs. (b) is the IRLM,a spinless 1-level quantum dot coupled via tunneling rates Γ L , R and Coulomb couplings U L , R to two reservoirs. The electrochemical potentials are given by µ L / R = ± V / for the couplings. Similar schemes can be developed for the calculation of the transportcurrent [5] and correlation functions [189]. All RG equations involve resolvents similarto the one occurring in (16) where z is replaced by Λ together with other physical energyscales. As a consequence, it can be shown that, besides temperature, each term of theRG equation has a specific cutoff scale Λ i , which is generically of the formΛ i = | E + X j n j µ α j − h i + i Γ i | ≡ | δ i + i Γ i | . (17)Here, E is the real part of the Laplace variable, n j are integer numbers, and µ α denotesthe electrochemical potential of reservoir α . It shows that the cutoff scale is given by thedistance δ i to resonances. Furthermore, it provides the generic proof that, at resonance δ i = 0, the cutoff scale is given by the corresponding rate Γ i . This issue was underdebate for some time because it was speculated that electrons tunneling in and out viathe same reservoir correspond to low-energy processes, which could possibly lead to astrong coupling fixed point even in the presence of a finite bias voltage [195]. However,it was argued that voltage-induced decay rates prevent the system from approachingthe strong coupling regime [177, 196, 197]. The microscopic inclusion of decay rates ascutoff scales into nonequilibrium RG methods was achieved within RTRG [185–187],flow equation methods [180], and RTRG-FS [5]. The two models used to illustrate the basic physics of spin and charge fluctuations aresketched in figure 8. One model is the Kondo model at finite magnetic field h alreadydiscussed in section 4, where a spin-1 / J z/ ⊥ to the spins of two reservoirs. We have assumed a symmetric coupling to the leads andnote that during the exchange it is also allowed that a particle is transferred between thereservoirs. The model results from the Coulomb blockade regime of a quantum dot withone level, where charge fluctuations are frozen out and only the spin can fluctuate. Thisleads to an effective band width of the reservoirs of the order of the charging energy U .Anisotropic exchange couplings can be realized for a molecular magnet, see section 2.The other model is the IRLM, where the quantum dot consists of a single spinless energylevel at position ǫ . The dot interacts with the reservoirs via tunneling processes, which harge transport through single molecules, quantum dots, and quantum wires α = 2 πρ α | T α | , with α = L , R. In addition,there is a Coulomb interaction u α between the first site of the reservoir leads and thequantum dot, which are characterized by the dimensionless parameter U α = ρ α u α . Inthe following we denote the bare parameters by a super-index (0) .The two models have in common that due to spin/charge conservation the effectiveLiouvillian has the same matrix structure. There are three nonzero poles of the resolvent(16) at z = − i Γ and z ± p = ± h − i Γ . For the Kondo model, Γ / corresponds to thespin relaxation/decoherence rate and h is the renormalised magnetic field. For theIRLM, Γ is the charge relaxation rate, Γ describes the broadening of the local level,and h ≡ ǫ is the renormalised level position. Denoting the two eigenstates of the dotby ± ≡↑ , ↓≡ ,
0, the matrix elements L effD ( i + ) ss,s ′ s ′ = − iss ′ W − s ′ of the Liouvillianinvolve the rates W s for the process − s → s . The stationary occupations are givenby p s = W s /W , which contain already most of the interesting nonequilibrium physics.Similar rates can be defined to calculate the current. We consider temperature T = 0from now on to reveal the physics of quantum fluctuations. We consider first the stationary case in the weak coupling regimeΛ c = max { V, h } ≫ T K , where the Kondo temperature T K ≡ T c is the strong couplingscale for spin fluctuations. The renormalised couplings from PMS cut off at thescale Λ c are denoted by J z/ ⊥ . For the isotropic case, they are explicitly given by J = 1 / (2 ln(Λ c /T K )), with T K ∼ D √ J (0) e − / (2 J (0) ) ( D ∼ U denotes the effective bandwidth of the reservoirs).In lowest order in J z/ ⊥ , one obtains a “golden rule” like expression for the rates [188] W s = π J ⊥ { hδ s − + ( V − h ) θ Γ ( V − h ) } , (18)where h, V > θ Γ ( ω ) = 1 / /π ) arctan( ω/ Γ) is a step function broadened byΓ. The latter corresponds to a Lorentzian broadening of the energy conservation lawby quantum fluctuations. The spin relaxation/decoherence rates and the renormalisedmagnetic field are given byΓ = π J ⊥ { h + ( V − h ) θ Γ ( V − h ) } , (19 a )Γ = 12 Γ + π V J z , (19 b )where Γ = Γ − Γ and h = (cid:0) − J z + J (0) z (cid:1) h (0) − hJ ⊥ ln Λ c | h + i Γ | + 12 ( V − h ) J ⊥ ln Λ c | V − h + i Γ | , (20)where unimportant terms ∼ O ( J ) without a logarithm have been left out for h .The expression for the rates W s up to O ( J ) can be interpreted as follows. Athigh energies Λ > Λ c , the voltage and magnetic field are not relevant and the exchangecouplings are renormalised according to the PMS equations. This leads to an effectiveband width Λ c for the reservoirs with renormalised exchange couplings cut off at Λ c . harge transport through single molecules, quantum dots, and quantum wires g h (0) / V Figure 9. g-factor g = 2 dh/dh (0) for the isotropic Kondo model with V = 10 − D , T K = 10 − D . The precise value of Λ c is not important since the renormalisation is logarithmic. Afterthis step one uses lowest order perturbation theory with broadened energy conservationleading to the result (18) for the rates. However, this interpretation does not work forthe rates in O ( J ) and it fails for the renormalised magnetic field h already in O ( J ) and O ( J ). We see that h contains a term linear in J z , which does not occur in perturbationtheory, and was also discussed e.g. in [198]. It shows that it is generically not possibleto calculate coefficients of certain orders in the renormalised couplings by comparingwith bare perturbation theory. In addition, we find logarithmic corrections in O ( J ).As was already discussed generically by Equation (17), they occur at resonances h = 0or V = h . As was emphasized in [176], they are of O ( J ln J ) at resonance and remain aperturbative correction for J ≪
1. They occur because the exchange couplings are notcompletely cut off by Λ c but partially by smaller cutoff scales. The calculation of theprefactor of the logarithmic terms is quite subtle and follows from the structure of theRG equations, where each term has its own cutoff scale Λ i , see Equation (17). Similarlogarithmic corrections occur in O ( J ) for the rates and for Γ / . In turns out thatthe logarithmic terms for h and Γ / contain an unexpected cutoff scale Γ = Γ − Γ .This is generic for all quantities entering the time evolution. It occurs because the polepositions of the resolvent (16) have to be calculated self-consistently.Logarithmic enhancements have been studied for the magnetic susceptibility and forthe conductance also in [178, 179, 199], using slave particle and flow equation methods.The logarithmic terms in the magnetic field have first been calculated in [188] usingRTRG-FS. They lead to a suppression of the renormalised g-factor as function of h/V at h = 0 and h = V , i.e. a nonequilibrium induced effect, see figure 9. Experimentally,it is proposed to be measured by using a three-lead setup, where the third lead is aweakly coupled probe lead [188]. A similar interesting nonequilibrium effect occursfor the logarithmic enhancement of the magnetic susceptibility χ ( h/V ) at h = 0, asfirst proposed in [178]. All spin-spin correlation functions were calculated analyticallyin [189] using RTRG-FS. Here, the frequency variable of the correlation function enters harge transport through single molecules, quantum dots, and quantum wires t ≪ Λ − , the PMS equations are cut off bythe energy scale 1 /t leading to time dependent exchange couplings J tz/ ⊥ . In O ( J ) onlythe Z -factor from the linear z -dependence of L effD ( z ) is important. In terms of the Paulimatrices σ , the local density matrix can be written as p ( t ) = + h S i ( t ) σ with h S i ( t ) = Z t h S i (0) = (cid:8) − J tz − J (0) z ) (cid:9) h S i (0) . (21)Inserting the solution of the PMS equations, one obtains universal logarithmic (powerlaw) time evolution for the isotropic (anisotropic) case. A similar result has been foundfor the longitudinal spin dynamics in [200, 201] for the special case of the ferromagneticKondo model, which was also confirmed by TD-NRG calculations [68]. In [190] it wasalso shown that the short-time behavior of the conductance can be calculated from thegolden rule expression and replacing the exchange couplings by J tz/ ⊥ .In the long-time limit t ≫ Λ − , the cutoff scale for the PMS couplings is given byΛ c . In leading (Markovian) order one obtains the usual exponential behavior from thesingle poles of the resolvent (16), where the longitudinal/transverse spin decays withΓ / and the transverse spin oscillates with h . Interesting non-Markovian correctionsoccur in O ( J ), where logarithmic contributions ∼ J ( z − z i b ) ln(Λ c / ( z − z i b )) in L effD ( z )lead to branch cuts. The branching point of the logarithm is generically given by z i b = z j p + nV , i.e. is shifted by multiples of the voltage from some pole position. Asa consequence, an exponential behavior ∼ J e − iz j p t e − inV is obtained, accompanied bypower-law behavior ∼ /t from the branch cut integral. The result explains why thevoltage occurs generically in the oscillation frequency, which is consistent with exactsolutions at special Thoulouse points of two-level systems [173]. Furthermore, it showsthat all terms decay exponentially but with unexpected decay rates and oscillationfrequencies relative to the Markovian terms. This is illustrated in figure 10 by thetime evolution of the transverse spin for the strongly anisotropic case Γ ≫ Γ , whichis typical for molecular magnets. For short times, one obtains the expected result ofoscillations with the magnetic field and decay with the spin decoherence rate Γ . Theseterms decay quickly for large decoherence rate. For longer times, a crossover is obtainedto an oscillation with the voltage V and a decay with the opposite spin relaxation rate Γ .Finally, we mention that close to resonances, i.e. for | δ | ≪ Λ c , /t , with | δ | = | V − h | , h ,logarithmic terms ∼ J δt ln | ( δ + i Γ ) t | occur for the time evolution of the transversespin. Similar to the logarithmic terms in h and Γ / , they are cut off by Γ = Γ − Γ . For the IRLM the PMS equations lead to arenormalised tunneling rate, given by the power-law Γ α = Γ (0) α ( D/ Λ) g α , with exponent g α = 2 U α − P β U β . Cutting off this scale at T c = Γ L + Γ R , one defines self-consistentlythe strong coupling scale T c for the importance of charge fluctuations. The level position ǫ ≡ ǫ (0) and the Coulomb interaction U α ≡ U (0) α remain unrenormalised [202]. In lowest harge transport through single molecules, quantum dots, and quantum wires T K t -10-50 l og ( | S x ( t ) | ) c =0.2, Γ / Γ =0.24 c =0.25, Γ / Γ =0.1 c =0.3, Γ / Γ =0.04 Figure 10. (Color online) | S x ( t ) | ≡ h| S x ( t ) |i in the anisotropic Kondo model for V = 2 h (0) = 100 T K and various values of the anisotropy c = ( J (0) z ) − ( J (0) ⊥ ) , with S x (0) = 1 / S y (0) = 0. The dips have their origin in the oscillations of S x ( t ). order in Γ α , one obtains a golden-rule like expression for the rates W s = X α W sα , W sα = Γ α θ Γ ( s ( µ α − ǫ )) , (22)where µ L / R = ± V / = Γ / = P α Γ α .Since a single tunneling process changes the charge, there is a unique cutoff parameterΛ α c = | µ α − ǫ + i Γ | for the PMS tunneling rates. This cutoff scale will be used inthe following to define Γ α . For | µ α − ǫ | ∼ T c , we are in the regime of strong chargefluctuations. We note the essential difference to the Kondo model, where severalcutoff parameters are relevant. The current in lead α can be calculated from the rateequation I α = e h π ( W + α p − W − α p ) with the occupation probabilities p = W + / Γ and p = W − / Γ . These results are quite specific to the IRLM and are due to its elementaryform. They have been conjectured in [202] and later were confirmed microscopicallyby RTRG-FS [191] and fRG [184]. It is surprising that these results are consistentwith NRG [202] and TD-DMRG [162] calculations even for values of U α ∼ O (1),and capture all features obtained by field theoretical [162, 174] and scattering Betheansatz [175] approaches. In particular, negative differential conductance is obtained forlarge voltages due to the power-law suppression of the tunneling rates.Defining Λ c = max { Λ Lc , Λ Rc } and expanding in U α , one finds logarithmicenhancements close to the resonances Λ α c = 0Γ α = Γ (0) α (cid:18) D Λ c (cid:19) g α (cid:18) U α ln Λ c | µ α − ǫ + i Γ | + O ( U ) (cid:19) . (23)It is important to notice that if the level is in resonance with one of the reservoirsit is not with the other, i.e. exactly at resonance the cutoff scale is Γ for one rateand the voltage V for the other rate. This has to be contrasted to speculations thatboth rates are cut off by the geometric average √ Γ V [203], which leads to incorrectpower-law exponent for the on-resonance current as function of the voltage. Therefore, a harge transport through single molecules, quantum dots, and quantum wires h i + P j n j µ α j , with integer numbers n j ,i.e. the excitation energies h i of the dot are shifted by arbitrary multiples of the chemicalpotentials of the reservoirs. For the IRLM it turns out that the algebraic part of thetime-evolution is ∼ (1 /t ) − g α , i.e. the exponent depends on the Coulomb interactionstrength. The status of the field of quantum fluctuations in nonequilibrium for strongly correlatedquantum dots is that powerful techniques have been developed to study the weak-coupling limit in a controlled way. Elementary models of spin and charge fluctuationsare well understood in this limit and the methods can now be applied to more complexquantum dot models, such as discussed in section 2.2. However, two fundamentalissues are still open. First, the case of strong spin fluctuations, represented in its mostelementary form by the isotropic Kondo model in the limit max { T, h, V } ∼ T K , is oneof the most fundamental unsolved problems. The case of strong charge fluctuationsat resonances has so far only been understood for the IRLM, where it seems thatthe broadening of the level together with a PMS renormalisation of the tunnelingcouplings captures the essential physics. Whether this holds also for more complicatedmodels has to be tested in the future. Secondly, most of the methods used todescribe nonequilibrium properties of quantum dots are parametrized by the many-bodyeigenstates of the isolated quantum dot. Therefore, they can not be extended easily tolarger systems like multi-level quantum dots or quantum wires. Two exceptions arethe flow-equation method [180] and nonequilibrium fRG methods. [141, 142, 181–184].In fRG, as already described in section 4, a perturbative expansion in the Coulombinteraction is used and the vertices are parametrized in terms of single-particle levels.Therefore, although fRG is restricted to the regime of moderate Coulomb interactions,the potential of the method lies particularly in the possibility to treat multi-levelquantum dots and quantum wires. Preliminary studies have tested the fRG for the IRLMand the single-impurity Anderson model (SIAM). For the IRLM, a static version wasused, where the frequency dependence of the Coulomb vertex was neglected [184, 191].An agreement was found with RTRG-FS and TD-DMRG [162]. Recently, a dynamicscheme with frequency-dependent 2-particle vertices has been developed and was usedto analyze the nonequilibrium SIAM in the regime of strong spin fluctuations [141, 142].A good agreement with TD-DMRG, ISPI and QMC results was obtained for moderateCoulomb interactions [204]. As will be described in the next section, the static versionof nonequilibrium fRG has also been applied to quantum wires and it is a challenge for harge transport through single molecules, quantum dots, and quantum wires
6. Correlation effects in quantum wires
A different type of correlation physics than discussed so far is found in (quasi) one-dimensional (1d) quantum wires at low temperatures. By this we mean electron systemsconfined in two spatial directions such that only the lowest 1d subband is occupied.Compared to the quantum dots considered above they contain many correlated degreesof freedom and the single-particle level spacing of the wire becomes the smallest energyscale (“almost” continuous spectrum; see below). The dominant quantum fluctuationsare now driven by the interaction U itself. In fact, below we will first discuss isolated,translationally invariant wires. Still, as discussed in the main part of this section,the physics becomes even more interesting if the coupling to leads (or the couplingbetween two wires) is included. Quantum wires are realized in single-walled carbonnanotubes [205, 206], specifically designed semiconductor heterostructures [207, 208],and in atom chains which form on certain surfaces [209]. A different class of quasi1d systems are highly anisotropic (chain-like) bulk materials (for a recent review seee.g. [210]). Compared to the first type of systems it is less clear how a single wire ofthis class can be incorporated as an element in an electronic transport nanodevice andthey will thus play a minor role in the present discussion.Although being far from experimentally realizable, translationally invariant modelsof 1d electrons with sizeable two-particle interaction (Coulomb repulsion) were alreadystudied in the fifties and sixties. Exactly solvable models were constructed leadingto a rather quick (compared to higher-dimensional correlated systems) gain in theunderstanding of correlation effects. Tomonaga showed that all excitations of a 1dcorrelated electron system at (asymptotically) low energies are collective in nature(“plasmons”) rather than single-particle-like as in the three dimensional counterpart(Landau quasi-particles) [211]. In particular, collective spin and charge densityexcitations travel with different speed. Quite often this difference in velocities ofcollective excitations is mistaken as being characteristic for 1d systems although it mightalso occur in higher dimensional Fermi liquids . Only when adding, that no additionalquasi-particle excitations are possible spin-charge separation becomes a unique featureof 1d correlated electrons. As a second (related) interaction effect certain correlationfunctions display power-law behavior as a function of energy ω and momentum k as firstshown for the single-particle momentum distribution function n ( k ) by Luttinger [212]and Mattis and Lieb [213]. Using the concepts of statistical physics one can view 1dchains of electrons as being (quantum) critical, although the exponents depend on thedetails of the underlying single-particle model (band structure, band filling) as well as onthe interaction strength. S´olyom [214] and Haldane [215] showed using renormalisationgroup (RG) arguments that the physics of the two exactly solvable models—being harge transport through single molecules, quantum dots, and quantum wires Tomonaga-Luttingermodel —is generic for 1d interacting electrons as long as correlations do not drive thesystem out of the metallic phase (e.g. into a Mott insulator phase). This led to the term
Luttinger liquid (LL) physics for the above correlation effects. In particular, Haldaneargued that all the exponents appearing in correlation functions of a spin-rotationalinvariant system with repulsive two-particle interaction can be expressed in terms ofa single model parameter dependent number
K <
1, with K = 1 for noninteractingelectrons. Together with the velocities of the charge- and spin-density excitations v c and v s , K completely determines the low-energy physics. For a given microscopic model thestrategy to obtain the low-energy physics is thus the following: one has to determinethe three Luttinger liquid parameters as functions of the model parameters (ways toachieve this are e.g. discussed in [216]) and plug them into the analytic expressions forcorrelation functions computed within the Tomonaga-Luttinger model.Despite the intense effort (see e.g. [205–210]) a commonly accepted experimentwhich shows LL behavior beyond any doubts is still pending. This has to be contrastedto the situation in quantum dots discussed earlier where the appearance of Kondo physicsas a consequence of local correlations has convincingly been shown (see above). Thelack of clear cut experiments is partly related to the status of the theory. LL theoryonly makes predictions for the asymptotic low-energy behavior but does not providethe scales on which this sets in. In the construction of the Tomonaga-Luttinger modelall microscopic scales as e.g. given by the band curvature [217] and the shape of thetwo-particle interaction [218] are disregarded. One can imagine that the upper energyscale T c beyond which LL physics can be found becomes so small that power lawsare masked by other scales such as the single-particle level spacing set by the lengthof the wire. A profound comparison to experiments requires a knowledge of T c andpossible lower bounds for LL physics while methods which allow to extract these scalefor microscopic models are rare. It is thus mandatory to develop new methods whichcapture LL physics and can directly be applied to microscopic models. Further down,we will return to this issue. We note in passing that the standard ab initio method—density functional theory—cannot be applied as the existing approximation schemes failwhen it comes to LL physics.When using a quantum wire as part of a low-temperature nanodevice one has tobe alert that correlation effects might alter the electronic properties. To understandthis let us first imagine an electron tunneling from the tip of a scanning tunnelingmicroscope into the bulk of a 1d quantum wire. Using Golden Rule-like arguments itbecomes clear that the differential conductance dI/dV computed from the tunnelingcurrent I is determined by the product (more precisely the convolution) of the single-particle spectral function ρ ( ω ) (the “local density of states”) of the wire and the tip.As the latter is constant at small energies, dI/dV displays the LL power law of thewire spectral function ρ ( ω ) ∼ | ω − µ | α bulk , with the wires chemical potential µ and α bulk = ( K + K − − / α end = ( K − − / harge transport through single molecules, quantum dots, and quantum wires k F -backscattering component, with k F being the Fermi momentum,strongly affects the low-energy physics. A first step to substantiate this expectation is tocompute the 2 k F -component of the static density-density-response function χ ( q ) of theTomonaga-Luttinger model [222]. For a noninteracting 1d electron system it divergeslogarithmically when q approaches 2 k F (Lindhard function). The divergence is enhancedand becomes power-law like χ ( q ) ∼ | q − k F | K − if the interaction is turned on. Thisshows that the inhomogeneity strongly couples to the system and linear response theorybreaks down. This physics was further investigated applying different RG methods todifferent models [223–227]. These studies show that even a single impurity at lowenergy scales effectively acts as if the chain is cut at the position of the impuritywith open boundary conditions at the two end points. E.g. for temperatures T → G ( T ) across the impurity is suppressed in a power-law fashion G ( T ) ∼ T α end as can again be understood using Fermis Golden Rule and the power-law scaling of the local spectral function at the end of a quantum wire (tunnelingfrom end to end of two LLs). Remarkably, the exponent is independent of the barescattering potential. Also a Breit-Wigner resonance of G as a function of the levelposition (modified by) V g showing up in the double-barrier geometry is strongly alteredby the interaction. In case of a “perfect” resonance with peak conductance e /h (perspin), that is for equal left and right barriers (which is experimentally difficult to realize),the resonance width becomes zero, while the resonance completely disappears, that is G → all V g , in all other cases.The discussed physics of inhomogeneous LLs can be understood in terms of aneffective single particle problem. Using RG techniques one can show that the interplayof a local inhomogeneity and two-particle correlations leads to an effective oscillatoryand slowly decaying ( ∼ /x , with the distance x from the inhomogeneity) single-particlescattering potential of range 1 /δ , where δ is the largest of the relevant energy scales(e.g. temperature). The wave-length of the potential is set by the (common) chemicalpotential µ of the leads. Scattering off this (so-called Wigner-von Neumann potential)leads to the discussed power laws in transport with the exponent given by the amplitudeof the potential. [224–227]In the remaining part of this section on coherent transport through quantum wireswe describe four examples of recent attempts (by three of the present authors togetherwith varying colleagues) to gain a detailed understanding of the interplay of correlations harge transport through single molecules, quantum dots, and quantum wires U ′ = 1 U ′ = 0 . U ′ = 0 . U ′ = 0 . U ′ = 0 T G / ( e / h ) − − − − − Figure 11. (Color online) Temperature dependence of the linear conductance for theextended Hubbard model with 10 sites and a single site impurity (“ δ -impurity”) ofstrength V = 10, for a Hubbard interaction U = 1 and various choices of U ′ ; thedensity is n = 1 /
2, except for the lowest curve, which has been obtained for n = 3 / U ′ = 0 .
65 (leading to a very small backscattering interaction); the dashed line isa power law fit for the latter parameter set. and local inhomogeneities in systems with different geometries. We believe that ourinsights should be kept in mind when designing transport setups to search for LL physics.They might also be of relevance when quantum wires (in the above defined sense) areused as elements in future nanoelectronic devices. In this case correlation effects mighteither be used to enhance the functionality or must be tuned away if they corrupt thelatter.In all the examples we use the functional renormalisation group (fRG) method [137](for a brief review see also [228]) to (approximately) treat the Coulomb interaction. Itwas already introduced in section 4. Compared to other methods it has the advantagesthat it (i) can directly be applied to microscopic models (continuum or lattice; flexiblemodeling), (ii) captures all energy scales (not restricted to the asymptotic low-energyregime), and (iii) can be applied to systems with many correlated degrees of freedom.In the present context (iii) means chains of realistic length (in the micrometer range)coupled to effectively noninteracting (Fermi liquid) leads. The aspects (i) and (ii)together assure that T c and additional energy scales affecting LL physics can beinvestigated. As mentioned in section 4, the approximations inherent to the fRGapproach are justified up to moderate renormalised two-particle interactions. In theimplementation (static self-energy) used for quantum wires (i) the LL exponents comeout correctly (only) to leading order in the two-particle interaction and (ii) inelasticprocesses generated by the two-particle interaction are neglected. We will return to thelatter in the outlook. harge transport through single molecules, quantum dots, and quantum wires In 1d systems at low energies and incommensurate fillings only forward (momentumtransfer q ≈
0) and backward (momentum transfer q ≈ k F ) two-particle scattering isactive. All other processes are suppressed due to momentum conservation and phasespace restrictions (Fermi surface consists of only two points). The relative importance ofthe two processes is determined by the (real-space) range of the two-particle interaction,which in turn is set by the screening properties of the “environment”, e.g. the substrateon which the carbon nanotube is placed. If screening is strong, the two-particleinteraction becomes short ranged in real space and backscattering is sizable and viceversa. Backscattering processes involving a spin up and a spin down electron are notpart of the Tomonaga-Luttinger model. For a translationally invariant LL it was shownusing an RG analysis [214] that these processes do not modify the asymptotic low-energy physics (they are “RG irrelevant”), but that they become irrelevant only on ascale which is exponentially small in the bare backscattering strength (they flow to zeroonly logarithmically). From this one can expect that the LL power laws only occur onexponentially small scales T c also for inhomogeneous wires. For the Hubbard model withopen boundaries it was shown that T c ∼ exp ( − πv F /U ), where U is the local Coulombrepulsion and v F the Fermi velocity [229]. In figure 11 this behavior is exemplified forthe T -dependence of the linear conductance G of a wire with a single impurity describedby the extended Hubbard model [155]. This lattice model consists of a standard tight-binding chain with nearest-neighbor hopping t , a local two-particle interaction U aswell as a nearest-neighbor one U ′ . Throughout the rest of this section we use t as theunit of energy (that is we set t = 1). For a fixed band filling the relative strengthof the forward and backward scattering can be modified by varying the ratio U ′ /U .The considered size of N = 10 lattice sites corresponds to wires in the micrometerrange, which is the typical size of quantum wires available for transport experiments.For U ′ = 0 (strong backscattering) due to logarithmic corrections the conductance increases as a function of decreasing T down to the lowest temperatures in the plot, T c being smaller than the latter. For increasing nearest-neighbor interactions U ′ the(relative) importance of backscattering decreases and a suppression of G ( T ) at low T becomes visible, but in all the data obtained at quarter-filling n = 1 / α end . By contrast, the suppression is much stronger and follows the expectedpower law more closely if parameters are chosen such that two-particle backscatteringbecomes negligible at low T , as can be seen from the conductance curve for n = 3 / U ′ = 0 .
65 in figure 11. The value of K for these parameters almost coincides with theone for another parameter set in the plot, n = 1 / U ′ = 0 .
75, but the behavior of G ( T ) is completely different. To avoid interference effects discussed next, in the presentsetup the two leads are coupled “adiabatically” to the wire, such that in the absenceof the impurity the conductance would be 2 e /h for all temperatures smaller than theband width [227]. harge transport through single molecules, quantum dots, and quantum wires T ∼ πv F /N finite size effects set in (the level structurebecomes apparent), as can be seen at the low T end of some of the curves in the figure. In most transport experiments the wires are not end-contacted and the leads do not terminate at the contacts. This has to be contrasted to the modeling in which almostexclusively end-to-end contacted wires are considered. It is thus crucial to understandhow overhanging parts of the wire and the leads alter the transport characteristics.Before tackling this problem we have to discuss transport through an interacting wirewith two contacts—above we only mentioned the single contact case with power-law scaling G ( T ) ∼ T α end . One might be tempted to argue, that two contacts canbe understood as two resistors in series, which then have to be added leading to G ( T ) ∼ T α end also for a (clean) wire with two contacts to semi-infinite leads. Althoughthe result is correct, as can be shown using (coherent) scattering theory [227, 230], thissimple argument ignores, that adding of the resistance (of resistors in series) only holdsif the transport is incoherent , which is not the case at low temperatures as consideredhere.We here exemplify the role of overhanging parts by considering overhanging leads.In figure 12 G ( T ) is shown for a spinless tight-binding wire with nearest-neighborinteraction U ′ coupled to two semi-infinite leads via tunneling hoppings T L / R . Forspinless models the low-energy LL physics can be described by K and the chargevelocity v c . Several correlation functions are again characterized by power-law scaling.Compared to models with spin the analytic dependence of exponents on K is modified(see e.g. [216]). Three setups with different overhanging parts are considered (see the leftinset). As becomes clear each non-zero number of overhanging lattice sites N L / Rlead infers anew energy scale πv F /N L / Rlead and the simple power law (with exponent α end ; dashed line)is divided in subsections separated by extensive crossover regimes. If clearly developedthe power laws in the subsections all have exponent α end (see the inset of figure 12).Overhanging parts of the wire lead to the same effect [231]. There is no reason to believethat adding the spin degree of freedom will change this. The same holds for the twoeffects discussed next and we thus stick to the spinless lattice model for the rest of thissection.One can conclude that in a generic experimental setup with overhanging parts thepower law is much more difficult to observe than suggested by most theoretical studiesconsidering end-to-end coupling. We now exemplify the very rich interference effects which occur when consideringcoherent linear transport through interacting wires with several inhomogeneities. A harge transport through single molecules, quantum dots, and quantum wires -4 -3 -2 -1 T G / ( e / h ) -4 -2 T -0.4-0.200.2 e xpon e n t +120 1101086 Figure 12. (Color online) Main plot: Linear conductance G of an interacting wirewith U ′ = 0 . N = 10 + 1 as function of the temperature T . The coupling tothe leads is located at the end of the wire and has a small amplitude T R = T L = 0 . N L / Rlead = 0, that is end-to-end contacted wire. Dashed-dotted line: N Llead = 0 , N
Rlead = 110. Dotted line: N Llead = 20 , N
Rlead = 1086. Thevertical dashed lines terminating at the different curves indicate the correspondingcrossover scales. At the lowest scale T ∼ − the power law is cut off by the levelspacing ∼ /N of the disconnected wire. Left inset: Setups studied (excluding theend-to-end contacted one). Right inset: Effective exponents obtained from taking thelogarithmic derivative of G ( T ). The solid horizontal line indicates α end . simple setup of this class is a wire with two end-to-end tunneling contacts to semi-infinite leads and an additional impurity in its bulk. As mentioned in the last subsectionthe concept of “adding resistances” cannot be applied in the present context. Usingaveraging over the (two) scattering phases one can show that for sufficiently large T (ofthe order 10 − in figure 13) the total linear conductance is given by G ∼ √ G G G with the conductances G i of the individual barriers. For the present setup we have G / ( T ) ∼ T α end and G ( T ) ∼ T α end , leading to G ( T ) ∼ T α end ¶ . As is shown infigure 13 the dashed-dotted curve obtained by phase averaging nicely follows the baredata (solid and dashed line) down to T p ≈ O (10 − ) and a power law with the expectedexponent develops for T p / T ≪ D . For T < T p details of the relative energy-levelstructure of the two “dots” defined by the three barriers matter and might even lead toresonances (see the solid line). The “dots” energy-level structure is set by the positionof the bulk impurity and the solid and dashed line in figure 13 display data obtainedfor different positions. The example clearly shows that interference due to multipleinhomogeneities can set a lower bound to power-law scaling of the conductance. As our final example we study transport at finite bias voltages . In particular weconsider the steady state current through a clean wire end-to-end coupled to two ¶ Accidentally the exponent 2 α end would also follow from the unjustified adding of resistances. harge transport through single molecules, quantum dots, and quantum wires -3 -2 -1 T -5 -4 -3 G / ( e / h ) Figure 13. (Color online) Linear conductance G of an interacting tight-binding wirewith tunnel couplings T L = T R = 0 .
1, an additional hopping impurity (single bondwith reduced hopping) of strength t ′ = 0 .
1, and nearest-neighbor interaction U ′ = 1 oflength N = 10 as function of the temperature T . The solid and dashed curve showdata obtained for different positions of the bulk impurity. Dashed-dotted line: resultobtained by phase averaging. leads (reservoirs) which are hold on different electrochemical potentials µ L / R = ± V / superposition of two decaying oscillations with wave-lengths set by µ L and µ R . The respectiveamplitudes are proportional to T / R / ( T + T ) = Γ L / R / (Γ L + Γ R ), with Γ α ∼ T α defined as in the preceding sections. Accordingly, the non-equilibrium local spectralfunction close to either of the two contacts displays power-law suppressions at µ L and µ R , ρ ( ω ) ∼ | ω − µ L / R | α L / R , with exponents proportional to the respective couplings α L / R = Γ L / R α end / (Γ L + Γ R ). In contrast to the linear response regime the exponentsnow dependent on the strength of the inhomogeneities. This behavior is shown infigure 14. The spectral weight remains finite even at µ L / R as the power-laws are cutoff by the single-particle level spacing ∼ /N of the disconnected wire. A transportgeometry in which the two power laws show up was proposed in [183]. This short review on correlation effects in 1d quantum wires gives a flavor of the richand partly surprising physics resulting from the interplay of the two-particle interactionand local (single-particle) inhomogeneities. An important ingredient missing in thecurrent description using the (approximate) fRG for microscopic models are inelasticscattering processes resulting from the (Coulomb) interaction and leading to decoherenceand dephasing at intermediate to large energies. We note that they are also only partlykept in the standard modeling which is based on the Tomonaga-Luttinger model. Onecan expect that inelastic processes set an upper energy scale T c for power-law scalingsmaller than the maximal scale given by the band width. They will also play a prominent harge transport through single molecules, quantum dots, and quantum wires -0.1 µ R µ L ω ~ ρ ( ω ) α R =4 α end /5 α L = α end /5 Figure 14. (Color online) Solid line: Local spectral function (close to the left contact)in the steady-state of an interacting tight-binding wire of length N = 2 · with tunnelcouplings T L = 0 . T R = 0 .
15, and nearest-neighbor interaction U ′ = 0 . ω . The voltage is V = 0 . .role when further considering non-equilibrium transport. First attempts to includesuch inelastic two-particle scattering for quantum dots in and out of equilibrium werementioned in section 4 and are discussed in [139–141]. List of symbols
In this section we provide a list of the most relevant symbols used:∆ ǫ level spacingΓ tunnel coupling V g gate voltage V bias voltage dI/dV differential conductance t time T K Kondo temperature U charging energy ~ = k B = e = 1 units T temperature | s i exact many-body level spectrum and states H D dot Hamiltonian H T tunneling Hamiltonian T ss ′ αkσ tunnel amplitudes σ spin α = L , R reservoirs H α reservoir Hamiltonian n αkσ electron number k label of the orbital state harge transport through single molecules, quantum dots, and quantum wires ρ α density of states µ α electro-chemical potential p ( t ) dot density operator L D dot Liouville operatorΣ( t, t ′ ) transport kernel I α tunnel current ǫ σ level energy N charge number ω frequency D N magnetic parameter λ electron vibration couplingΩ dwell time of electrons in the system Q pumped charge G (0) (bare) Green function of the dotΣ( iω ) self-energy G conductance δ i distance to resonances z i poles of the reduced dot density matrix ˜ p in Laplace space h magnetic field J z/ ⊥ exchange couplings W s rates for the process − s → sχ susceptibilityΛ flow parameterΛ c cutoff K Luttinger liquid parameter v c / s velocities of the charge/spin-density excitations k F , v F Fermi momentum, Fermi velocity U ′ nearest-neighbor interaction α bulk power-law exponent of the wire spectral function α end power-law exponent of an electron tunneling into the end of a LL Acknowledgments
We acknowledge all our collaborators in the various works presented in this review.This work is supported by the DFG-FG 723, FG 912, SPP-1243, the NanoSci-ERA, theMinistry of Innovation NRW, the Helmholtz Foundation and the FZ-J¨ulich (IFMIT).
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