Table Of Content

Quantum Transport through Nanostructures in the Singular-Coupling Limit Maximilian G. Schultz and Felix von Oppen Institut fu¨r Theoretische Physik, Freie Universität Berlin, Fachbereich Physik, 14195 Berlin, Germany (Dated: January 15, 2009) Geometric symmetries cause orbital degeneracies in a molecule’s spectrum. In a single-molecule junction, these degeneracies are lifted by various symmetry-breaking effects. We study quantum transport through such nanostructures with an almost degenerate spectrum. We show that the master equation for the reduced density matrix must be derived within the singular-coupling limit 9 asopposed totheconventionalweak-couplinglimit. Thisresultsinstrongsignaturesofthedensity 0 matrix’s off-diagonal elements in thetransport characteristics. 0 2 PACSnumbers: 73.23.Hk,81.07.Nb,05.60.Gg n a J Introduction.—The Anderson model of localized, inter- and its competition with the transport dynamics due to 5 acting electronic states coupled to two Fermi-gas elec- electrontunneling. Ourapproachshowsthatthiscompe- 1 trodes is the archetypical model for studying electronic titioncausesrichandinterestingphysicsinthetransport transport through quantum nanostructures. The model characteristics. ] l accurately describes a wide range of experimentally ac- The singular-coupling limit also remedies the shortcom- l a cessibletransportregimesofquantumdotsincluding the ings of the Bloch–Redfield equation, which is a master h Coulomb blockade and the Kondo regime. It has been equation that explicitly keeps the coherences between - s successfully extended to describe single-molecule junc- non-degenerate states in the density matrix [5] but is e tionsbycouplingthe electronstothevibrationaldegrees known to produce negative probabilities [6]. The master m of freedom ofthe molecule. The study of molecular elec- equation in the singular-coupling limit is included in the . t tronicsusingthismodelhasreceivedenormousattention general theoretical framework of master equations and a in recent years [1]. rate equations. It closes the gap between the descrip- m An important issue that in contrast to the vibrational tion by rate equations and degenerate master equations - d structure has been studied much less arises from the or- enablingustostudyhigh-temperaturesequentialtunnel- n bital degeneracies due to the geometric symmetries of ing for all possible energy regimes. o molecules. Thetransportpropertiesduetoorbitaldegen- We illustrate the physics of an orbital near-degeneracy c eraciesdifferfromthoseduetospindegeneracy,whichare withinaminimalmodel,atwo-level,interactingbutspin- [ being studied extensively in the quantum-dot literature. less Anderson impurity coupled to two electrodes 2 Sincethere,theunderlyingSU(2)symmetryisrespected v Ω both by the electronsin the reservoirsandby the tunnel H =eV (n +n )+ (n −n )+Un n 1 g ↑ ↓ 2 ↑ ↓ ↑ ↓ 9 couplings to the localized system, the low-temperature (1) 4 phenomenology is characterizedby Kondo physics [2]. + εkc†kαckα+ tασc†kαdσ+h.c. 1 Thesymmetry-inducedorbitaldegeneraciesofamolecule Xkα kXασ . 2 areingeneralliftedbybindingittometallicelectrodesor The on-site electronic orbitals are labeled by a pseudo- 1 the interaction with an underlying substrate. Asymme- spinσ =↑,↓andareseparatedinenergybyΩ. Thissplit- 8 tries in the coupling of the orbitals to the leads cause a ting is assumed to be due to symmetry-breaking mech- 0 : tunneling-inducedsplittingofthedegeneratelevelsasthe anisms other than electronic tunneling. Double occupa- v result of perturbation theory. The problem with orbital tion of the system is suppressed by Coulomb repulsion i X degeneracies therefore naturally extends to studying the of strength U, and both orbitals are coupled to Fermi- r role of near-degeneracies in quantum transport theory. gas electrodes α = L,R, held in thermal equilibrium at a Theimportanceofcoherentsuperpositionsofdegenerate temperature k T, via lead- and orbital-dependent am- B localized levels for quantum transport has already been plitudes t . ασ mentionedintheliterature[3,11]. Inthispaper,wegen- In Fig. 1 (a) we show the stationary current through eralize the discussion from the non-generic case of exact the model system for eV = 0. It is evaluated both for g degeneracyandshowhowthebreakingofsymmetriesand vanishing Ω using a master equation that includes the the lifting of molecular degeneracies can be consistently off-diagonal elements of the reduced density matrix ρ, accountedforinamasterequationformalismbyemploy- and for finite Ω, which is small compared with k T, us- B ing the “singular-coupling limit” [4] in the derivation of ing a rate equation for the diagonal elements of ρ. The the kinetic equation for sequential tunneling. Contrary qualitative difference between the two results is striking. to the weak-coupling limit, the singular-coupling limit While the rate equation yields a simple steplike increase canproperlydescribethecoherentdynamicsinthenear- of the current, the master equation produces additional degenerateorbitalsubspaceofthereduceddensitymatrix structure: a suppression of the current and pronounced 2 2 been summed over; the density of states ν0 in the electrodes is assumed to be uniform. We also set δΓ := 1 (a) (b) Γ −Γ . Theparameter∆:=Γ Γ −γ2isproportionalto ↑ ↓ ↑ ↓ I 0 det(Σ). A vanishing ∆ allows for a complete decoupling −1 of one electronic level from both electrodes by a unitary transformation of the Hamiltonian [7, 8]. We therefore −2 −10 −5 0 5 10 focus on the generic situation ∆6=0 in the following. eVsd The stationary coherences of the non-degenerate system (c) (d) are described by the equal-time correlator A (ω)dω. R ↑↓ In the weak-coupling limit, when Γ → 0, these are ex- R pected to vanish for every finite Ω. To confirm this explicitly, we rescale t 7→ λt and shift the energy ασ ασ ξ := ω − Ω. As λ tends to zero, the rescaled spectral 2 function Aλ (ξ) convergesto zero almost everywhereex- ↑↓ cept at the four roots of its denominator, FIG. 1: (Color online) Numerical evaluation of the station- arycurrentEq.(9)throughaninteractingtwo-levelAnderson λ2 Ω Ω 2 Tmhoedetlu,ntLn↑el=amtRp↓li=tud1,estLt↓ασ=atnRd↑ =the1.s5p,lUitt=ing2.Ω35,arkeBmTe=as0u.r0e2d. ξ1,2 = 4 δΓ−2λ2 −ıΓ±2s(cid:18)λ2(cid:19) −γ2  (3) in units of 2~πν0. Fig. (a) illustrates the difference between   the description by the degenerate master equation, Eqs. (7, and ξ = ξ∗ . By use of the residue theorem, we find 3,4 1,2 8),(blue)andtheΩ=0rateequation,whichisobtainedfrom that also the contribution of these poles to the integral themasterequationbysettingρ↑↓=0(red). Bothmodelsare converges to zero such that Aλ (ξ)dξ → 0 as λ → 0. evaluatedateVg=0inordertohighlight thequantitativeas R ↑↓ When, on the contrary, both Γ and Ω are of the same wellasqualitativediscrepancy. InFigs.(c)and(d),weshowa R fullscanofthe(eVg,eVsd)-planeforbothequations,illustrat- order,wehavetorescalenotonlytασ butalsoΩ7→λ2Ω, ingthefundamentaldifferenceofthedynamicsduetothetwo suchthatΓασ/Ωandγα/Ωremainfixedinthelimitpro- equationsintheparameterspace. Whiletherate-equationre- cess. This simultaneous scaling is known as “singular- sultisthefamiliarsequenceofstepsinthestationarycurrent, couplinglimit” incontrastto the weak-couplinglimit, in themaster-equationresultshowsastrongcurrentsuppression whichΩ is keptconstant[4]. Repeating the abovecalcu- for voltages below the double-charging threshold. Using the lation, the roots of the spectral function’s denominator masterequationinthesingular-couplinglimit asitisderived are now proportional to λ2, and the integral inthetext,weinterpolatethetwodifferentcurvesofFig.(a) by letting Ω→∞ in Fig. (b). 2γδΓ2 Aλ (ξ)dξ =2π (4) R ↑↓ Γ Γ2+4(Ω2−γ2) Z negative differential conductance. The phenomenologi- is independent of the scaling(cid:0)parameter and fi(cid:1)nite. Al- cal discrepancy between the two curves of Fig. 1 (a) il- though the system is non-degenerate, some coherences lustrates that there can be interesting and non-intuitive between the electronic levels are retained. As we let physics in the cross-over regime where Ω is of the order Ω → ∞, these converge to zero. In this limit, our ap- of the tunneling-induced broadening Γ of the electronic proach reproduces the weak-coupling result. orbitals. Master Equation.—We now derive the master equation The Singular-Coupling Limit.—The emergence of addi- for the near-degenerate, interacting two-level Anderson tional physics in the cross-over regime Ω ∼ Γ can al- model in the singular-coupling limit. The underlying readybeunderstoodwiththehelpofthenon-interacting, challenge of the regime Ω ∼ Γ is that the tunneling dy- U =0, versionof the two-levelAnderson model, Eq.(1). namics takes place on the very same time scale as the Duetotheabsenceofinteractions,theretardedpropaga- coherent on-site dynamics due to the almost degenerate torofthemodelcanbecomputedexactly,andastraight- electronic levels. We consider the Hamiltonian, rescaled forwardcalculationyields for the off-diagonalelement of according to the previous observation, Hλ = λ2HS + the spectral function HE+λHS–E. The term λ2HS describes the system, that isthenear-degenerateelectronicorbitals. WeseteV =0 g γ ω− Ω ω+ Ω − ∆ forconvenience—itcaneasilyberestoredlateron. HE is A (ω)= 2 2 4 . (2) ↑↓ ω− Ω(cid:0)+(cid:0) ıΓ↑ (cid:1)ω(cid:0) + Ω +(cid:1)ıΓ↓ (cid:1)+ γ2 2 thebath,inourcasetheFermi-gaselectrodes,andλHS–E 2 2 2 2 4 is the system–bath interaction, the tunnel Hamiltonian. (cid:12)(cid:16) (cid:17)(cid:16) (cid:17) (cid:12) We rewrite the von Neumann equation as an integral (cid:12) (cid:12) The self ene(cid:12)rgy Σ = ıπ~ν0W†W, where (W)ασ (cid:12):= tασ, equation for the reduced density matrix ρS = TrE(ρ). gives rise to the broadenings Γ := 2πν t2 and γ := The limit λ→0 thengeneratesthe markoviandynamics ασ ~ 0 ασ α 2πν t t , with a missing index indicating that it has [9]. The reduced density matrix itself is obtained by a ~ 0 α↑ α↓ 3 projection Pρ := Tr (ρ)⊗ρ , with ρ being the equi- ρ˙ =−ı[Hλ,ρ]=−ıL λρ=−ı(λ2L +L +λL )ρ reads E E E S E S–E librium distribution of the bath. With the complemen- in the interaction picture [6, 9] tary projection Q := 1−P, the von Neumann equation t t−v ρI(t)=ρI(0)−λ2 eıλ2L Sv eıλ2L SsTr L Qe−ıL λsQL ρ ds e−ıλ2L SvρI(s)dv. (5) S S E S–E S–E E S Z0 (cid:26)Z0 (cid:16) (cid:17) (cid:27) This equation incorporates the Born approximation by As it is plausible from Eq. (6), the master equation in choosing ρ(0) = ρ (0)⊗ρ as the initial condition. On thesingular-couplinglimitisformallyobtainedbysetting S E the slowmarkoviantime scale τ =λ2t, the termin curly Ω=0 in the dissipative term but retaining the non-zero brackets converges to a time-independent quantity [9]: Ω in the free-evolution Hamiltonian [3]. The splitting Ω the master equation assumes the markovian form must notappear in the Fermi functions coming fromthe trace over the bath degrees of freedom, because due to ∞ ρ˙ =−ıL ρ − Tr L e−ıL EsL ρ dsρ . (6) the limit λ→0, the temperature of the bath is too large S S S E S–E S–E E S Z0 (cid:16) (cid:17) to resolve the splitting. The Bloch–Redfield equation would, however, have such a dependence on Ω. IfH hadnotbeenrescaledbyλ2,theexponentialfactors S enclosing the curly bracketin Eq.(5) due to the interaction picture would have taken the form exp(±ıL τ/λ2) We now evaluate the master equation (6) for the S onthemarkoviantimescaleτ. Inthiscase,lettingλ→0 Anderson-modelEq.(1)andrecastitasaBlochequation wouldhavecausedfasterandfasteroscillationsforcoher- for the pseudo-spin S~ := (2Reρ ,2Imρ ,ρ − ρ ), ↑↓ ↑↓ ↑↑ ↓↓ ences belonging to non-degenerate states. In the limit, which is defined by the electronic two-level system, cou- thosetermswouldhavebeenaveragedtozero,commonly pled to an equation for the populations p , i being the i known as the secular approximation [10]. number of electrons on the device, S~˙ = f p + 1 f −(1−f ) p −(1−f )p ~n − 1 f +(1−f ) Γ S~−(B~ +Ωeˆ )×S~, (7) α 0 α2 α 1 α2 2 α α2 α α z 2 2 Xα (cid:20) (cid:16) (cid:17) (cid:21) Xα h i p −2f (1−f ) 0 p 1−f d 0 1 α α 0 1 α p = Γ 2f −f −(1−f ) 2(1−f ) p + f −(1−f ) ~n ·S~, (8) 1 α α α2 α α2 1 α2 α α dt  2    2   p2 Xα 0 fα2 −2(1−fα2) p2 Xα −fα2        1 I =Γ f p − (1−f )−f p −(1−f )p − (1−f )+f ~n ·S~. (9) α α α 0 α α2 1 α2 2 α α2 α 2 h (cid:16) (cid:17) i (cid:16) (cid:17) Eq. (9) gives the stationary current through lead α by magnetic fields describe the tunneling-induced energy I = Tr (ρIˆ ).The pseudo-magnetic fields are deter- renormalizations of the electronic levels in exactly the α S α mined by the principal-value integrals same way as any other term in the Hamiltonian of this order would have to be treated, namely in the singular- 1 1 1 B~ := P f (ε) + dε~n , (10) coupling limit. The singular-coupling limit provides a α α 2π U −ε ε α Z (cid:18) (cid:19) non-discriminating description for both, the tunneling- X inducedrenormalizationoftheon-sitelevelsastheresult with ~n :=(2γ ,0,Γ −Γ ). The Fermi functions are α α α↑ α↓ ofperturbationtheoryandanyothersplittingofthesame abbreviated by f := f(eV −µ ) and f := f(eV + α g α α2 g order, which has to be included in the model explicitly. U −µ ). For Ω = 0, Eqs. (7–9) are reminiscent of the α ones derived in [11]. Results.—Based on our theoretical considerations, we In Eqs. (7, 8), the electronic splitting term −ıL ρ ap- understand the qualitative behavior of the stationary S S pears as a contribution to the pseudo-magnetic field di- current–voltagecharacteristicsofthetwo-levelAnderson rected along eˆ . Due to this interpretation, the pseudo- model and its remarkable sensitivity to Ω. The rate- z 4 equation results of Figs. 1 (a) and (d) show the familiar Conclusions.—The omnipresence of orbital degeneracies steps in the stationary current whenever the voltage is in molecular physics and their lifting due to various largeenoughtoaddanotherelectrontothedevice. Since symmetry-breaking effects in single-molecule junctions Ω≪k T, the near-degeneracyremains unresolved. requiresthe modeling of degenerateandnear-degenerate B The Ω = 0 master equation, also yields a finite linear systems in quantum transport theory. conductance at zero bias, but below the double-charging We have shown that within the framework of master threshold, in Figs. 1 (a) and (c), the stationary current equations for sequential tunneling, such near-degenerate is strongly suppressed. Since the equation is one for the systems fall into a descriptive gap between the energy full density matrix of the system’s on-site levels, the un- regimes that are discussed in the theoretical literature. derlyingmechanismisexplainedbychoosingaparticular Whereas for degenerate systems, a master equation for electronic basis. The key idea is: for general tunneling the full reduced density matrix is used, systems with en- amplitudes tασ, there always exists a unitary transfor- ergydifferenceslargerthanthetunneling-inducedbroad- mation to eliminate at least one of them and hence de- ening Γ have to be described by rate equations. In the coupleoneofthe levelsfromoneelectrode. Letthislevel interesting cross-over regime, the tunneling of electrons, be |↑i and let the electrode it is decoupled from be the the tunneling-induced splitting, and the coherenton-site drain electrode, Γd↑ =0. Then γd =0. If we neglect the dynamicsarebothoforderΓ. Byusingthenotionofthe pseudo-magneticfieldsforthetimebeing,thedegenerate singular-coupling limit, we have derived a master equa- master equation is actually only a rate equation, whose tionforthisregime. Wehavetherebyalsogivenmeaning physics is easily understood. The tunneling dynamics to the rate-equation treatment of degenerate systems as will eventually populate the decoupled state |↑i, which being the proper description when Γ≪Ω≪k T. Since B due to Γd↑ = 0 cannot be left again. Below the double- for the high-temperature regime always Γ ≪kBT is im- chargingthresholdCoulombrepulsionthenobstructsany plied,wehaveactuallypresentedamethodtodescribeall current through the device. possible energy regimes for sequential tunneling through The pseudo-magnetic fields, which describe virtual multilevel quantum nanostructures. switchingprocessesbetweenthedegeneratelevels,soften Aside from the simple model that we have used to il- thispicture. Theyinduceaprecessionofthepseudo-spin lustrateandexplainits physics,ourapproachhasawide that correspondsto movingthe electronfromthe decou- rangeofapplicationsinmolecularelectronics. Itprovides pledstate|↑iviavirtualintermediatestatesonthesource a means to generically account for symmetry-breaking electrode into the conducting state |↓i [11]. Except for mechanisms in single-molecule junctions. And it allows those voltages, for which |B~| = 0 and the above argu- the study of complex molecular models such as Jahn– mentforcompletecurrentsuppressionapplies,thedevice Telleractivesystems,pseudo-Jahn–Tellerstructures,the carries the highly voltage-dependent stationary current valleydegeneracyincarbonnanotubes,ortheinteraction shown in Figs. 1 (a) and (d). oforbitalsymmetriesandvibrationaldegreesoffreedom, In the singular-coupling limit, the splitting Ω generates whichisaleitmotifinthetheoryofmolecularelectronics. an additionalpseudo-magneticfield of the same orderas This work was supported by SPP 1243 of the Deutsche the tunneling-induced one. Its orientation along eˆ dis- z Forschungsgemeinschaft. tinguishesthisdirectioninthesystem’sHilbertspace. As Ωisbeingincreased,theprecessionfrequencyofS~ about the z-axis will dominate the virtual tunneling dynamics, whichareofthe orderofthe electronicdwelltime onthe device. In the limit Ω → ∞, the residual precession dy- [1] M.Galperin,M.A.Ratner,andA.Nitzan,J.Phys.:Con- namics is too fast compared with the average tunneling dens. Matter 19, 103201 (2007). time, such that only the projection of S~ onto the z-axis [2] M. Pustilnik and L. I. Glazman, cond-mat/0501007. is relevant for the tunneling dynamics. For very large Ω, [3] For the special case of benzene molecules, similar obser- vationstoourshavebeenmadeindependentlybyD.Da- the pseudo-spin is effectively oriented along eˆ . Because z rau, G. Begemann, A. Donarini, and M. Grifoni, cond- then S +ıS = 2ρ ≈ 0, the master equation is ren- x y ↑↓ mat/0810.2461 (2008). dered a rate equation for the electronic occupations. As [4] H. Spohn,Rev.Mod. Phys., 53, 569 (1980), Chap. V.C. still Ω ≪ kBT, the Fermi functions cannot resolve the [5] C. Timm, Phys.Rev.B, 77, 195416 (2008). physics due to Ω, and the system is effectively described [6] R. Du¨mckeand H.Spohn,Z. Physik B, 34, 419 (1979). by the Ω=0 rate equation. [7] T. V. Shahbazyan and M. E. Raikh, Phys. Rev. B, 49, The numerical results shown in Fig. 1 (b) support this 17123 (1994). [8] R.Berkovits,F.vonOppen,andJ.W.Kantelhardt,Eu- reasoning: asthesplittingΩisnumericallyincreased,the rophys. Lett., 68, 699 (2004). current-suppressioncurveis shiftedtowardpositive bias. [9] E. B. Davies, Commun. Math. Phys., 39, 91 (1974). The suppression of the stationary current is lifted and [10] H.-P. Breuer and F. Petruccione, The Theory of Open eventually lost in the flat current profile of the Ω = 0 Quantum Systems (Oxford University Press, Oxford, rate-equation result. 2002), Chap. 3.3. 5 [11] M. Braun, J. Konig, and J. Martinek, Phys. Rev. B, Phys. Rev.B, 71, 195324 (2005). 70, 195345 (2004); S. Braig and P. W. Brouwer,