Time-dependent quantum transport and power-law decay of the transient current in a nano-relay and nano-oscillator
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Time-dependent quantum transport and power-law decay of the transient current ina nano-relay and nano-oscillator
Eduardo C. Cuansing a) and Gengchiau Liang b) Department of Electrical and Computer Engineering, National University ofSingapore, Singapore 117576, Republic of Singapore (Dated: August 11, 2011)
Time-dependent nonequilibrium Green’s functions are used to study electron trans-port properties in a device consisting of two linear chain leads and a time-dependentinterleads coupling that is switched on non-adiabatically. We derive a numerically ex-act expression for the particle current and examine its characteristics as it evolves intime from the transient regime to the long-time steady-state regime. We find that justafter switch-on the current initially overshoots the expected long-time steady-statevalue, oscillates and decays as a power law, and eventually settles to a steady-statevalue consistent with the value calculated using the Landauer formula. The power-law parameters depend on the values of the applied bias voltage, the strength ofthe couplings, and the speed of the switch-on. In particular, the oscillating tran-sient current decays away longer for lower bias voltages. Furthermore, the power-lawdecay nature of the current suggests an equivalent series resistor-inductor-capacitorcircuit wherein all of the components have time-dependent properties. Such dynam-ical resistive, inductive, and capacitive influences are generic in nano-circuites wheredynamical switches are incorporated. We also examine the characteristics of thedynamical current in a nano-oscillator modeled by introducing a sinusoidally modu-lated interleads coupling between the two leads. We find that the current does notstrictly follow the sinusoidal form of the coupling. In particular, the maximum cur-rent does not occur during times when the leads are exactly aligned. Instead, thetimes when the maximum current occurs depend on the values of the bias potential,nearest-neighbor coupling, and the interleads coupling.PACS numbers: 73.63.-b,72.10.Bg,73.23.-b a) [email protected] b) [email protected] . INTRODUCTION The further miniaturization of electronicdevices will eventually lead to molecularelectronics wherein particles pass throughmolecular-scale devices whose constituentmolecules may have been manipulated andsynthetically assembled or created. Molec-ular transistors, in particular, are of signifi-cant practical interests and whose successfulimplementations are currently actively beingpursued. Several theoretical models of thetransistor have been proposed and exper-imental successes have also been reported. A related molecular-scale device, the molec-ular switch, has also garnered significant in-terests because of the switch’s important rolein circuit design and architecture. A theo-retical model of the switch makes use of amechanism that involves the reversible dis-placement of an atom in a molecular wirethrough the application of a gate voltage. In addition, experimental realizations of theatomic switch include mechanisms involvingthe reversible transfer of an atom betweentwo leads, the dynamical onset of single-atom contact between leads and the manipu-lation of atomic bonds using a dynamic forcemicroscope. Having a dynamical switch ina nano-circuit, however, necessitates the ap-pearance of time-dependent behavior, partic- ularly during the transient regime just afterswitch-on. The circuit transitions from beingdisconnected into connected in a short timeand the current does not instantly switchesinto the steady-state value upon connection.It is therefore informative to know the char-acteristics of the current just after switch-on and during the transient regime, anddetermine how this current approaches thesteady-state value. In this work, we intro-duce a model device representing a systemwherein the current can be dynamically tog-gled on and off. There are several theoret-ical approaches in treating time-dependentquantum transport. Among these includetime-dependent density functional theory, propagating the Kadanoff-Baym equations, Floquet theory, path-integral techniques, and the density matrix renormalization groupmethod. In this work we choose touse time-dependent nonequilibrium Green’sfunctions (TD-NEGF) because, as is shownin this paper, dynamically toggling the cou-pling between the leads is straightforward us-ing the technique, even without the assump-tion of weak coupling, and the resulting ex-pression for the time-dependent current is nu-merically exact.The device we examine consists of a semi-infinite linear chain of atoms, i.e., the leftlead, which is stationary and another semi-infinite linear chain of atoms, i.e., the right2 otatablestationary (a) left lead right lead vv v v v v v LR ε ε ε ε ε ε (b) disk (c) L e R e V S(t) (t)(t) (t)C e FIG. 1. (Color online) (a) An illustration of thenano-relay and nano-oscillator. The left lead isfirmly on a stationary substrate and the rightlead is on a rotatable disk. The left and rightleads align when the disk is rotated clockwiseto the dash line. (b) The device can be repre-sented by two semi-infinite leads connected bya time-varying coupling v LR ( t ) switched on at t = 0. In both leads, the on-site energy is ǫ andthe nearest-neighbor hopping parameter is v . (c)An equivalent resistor-inductor-capacitor circuitwith dynamical properties. lead, that is on a rotatable disk. An illus-tration of the device is shown in Fig. 1(a).When the disk is rotated clockwise to thedashed line, the left and right leads align andconduction occurs. We model how the twoleads couple in two ways: as an abrupt Heav-iside step function and as a hyperbolic tan-gent that gradually progresses in time. Forsuch cases, the off state has no current flow-ing across the leads. In addition, we can alsomodel an oscillator by swinging the disk backand forth across the dashed line. The ro- tatable disk can also be replaced by a gatevoltage, located on top of the right lead, thatcould rotate the right lead to its desired posi-tion. This latter configuration has previouslybeen examined and, in particular, the steady-state transport properties of its on and offstates have been studied . In this paper,however, we study the full time-dependenttransport properties of the device. Numeri-cally exact expressions for the current and theneeded nonequilibrium Green’s functions arederived. We then show that just after switch-on and during the transient regime the cur-rent initially overshoots the expected value ofthe steady-state current and then oscillatesaround the steady-state value while decay-ing as a power law. Such a power-law decay-ing transient current suggests the appearanceof dynamical resistance, inductance, and ca-pacitance in the system during the transientregime. Power-law decaying currents havealso recently been predicted to appear in asystem containing a quantum dot channel and in the anisotropic Kondo model. II. MODEL AND METHOD
We implement time-dependent nonequi-librium Green’s functions techniques to in-vestigate the dynamical transport proper-ties of particles traversing through a devicewith time-varying components. To deter-3ine the dynamical transport properties, aTD-NEGF approach can be used that uti-lizes either two-time
Green’s functions, ordouble-energy Green’s functions, or Green’sfunctions that depend on one time and oneenergy variables.
In a transistor withsource-channel-drain and top gate configura-tion, for example, the device can be mod-eled by a Hamiltonian constructed using den-sity functional theory or tight-bindingtheory and the dynamical transport prop-erties of small channels are calculated usingTD-NEGF. For devices with larger channels,a self-consistent calculation based on thePoisson equation and TD-NEGF can be doneto determine the consistent dynamical poten-tial and charge density in the channel. Inthis work, in contrast, we study a device con-sisting only of two leads, i.e., there is no chan-nel between the leads. The time-dependencecomes from non-adiabatically toggling on thecoupling between the leads. We use TD-NEGF to derive a numerically exact expres-sion for the dynamical current and investi-gate how the devices we call a nano-relayand a nano-oscillator respond to time-varyinginfluences. This approach has recently alsobeen used in the study of dynamical heattransport in a thermal switch .We model the system by the total Hamil-tonian H = H L + H R + H LR , where H L isfor the left lead on the stationary substrate, H R is for the right lead on the rotatabledisk, and H LR includes the dynamic inter-leads coupling. In the leads, particles followthe tight-binding Hamiltonian H α = X k ǫ αk c α † k c αk + X kj v αkj c α † k c αj , α = L , R , (1)where c α † k and c αk are particle creation and an-nihilation operators at the k th site in the α lead. ǫ k is the on-site energy at site k and v kj is the hopping parameter between nearest-neighbor sites k and j (see Fig. 1(b)). Thesums are over all sites in the α lead. H LR in-cludes the time-dependent component of thetotal Hamiltonian and is of the form H LR = X kj (cid:16) v LR kj c L † k c R j + v RL jk c R † j c L k (cid:17) , (2)where v LR jk ( t ) = v RL kj ( t ) is the time-dependentcoupling between the left and right leads andis switched-on at t = 0. Only the right-mostsite of the left lead, i.e., the site labeled 0 inFig. 1(b), can couple to the left-most site ofthe right lead, i.e., the site labeled 1 in thefigure.The current can be determined by not-ing how the number operator, N α = P k c α † k c αk , changes with time, i.e., I R ( t ) = − q (cid:10) dN R /dt (cid:11) , where q is the electron charge.Defining the two-time lesser Green’s functionas G RL , III. RESULTS AND DISCUSSION Firstly, we examine the impact of theswitch-on speed to current characteristics.The functional form of the interleads cou-pling, v LR ( t ), describes how the device isswitched on. We examine two types ofswitch-ons: an abrupt Heaviside step func-tion switch-on and a gradually progressinghyperbolic tangent switch-on of the form v LR ( t ) = v LR tanh( ω d t ), where ω d is the driv-ing frequency. The step function switch-on isactually the limit when ω d → ∞ of the hy-perbolic tangent switch-on. Furthermore, weset the on-site energy ǫ = 0. The left andright leads have temperature T L = T R =300 K and chemical potential µ L = ǫ F and µ R = ǫ F − qV b , where we set the Fermi en-ergy ǫ F = 0. The bias potential V b is appliedto the right lead and the bias potential energyis written as U b = qV b .Figure 2 shows the current flowing out ofthe left lead for the step function and hy-perbolic tangent switch-on with driving fre-quencies ω d = 0 . 25 [1 / t] and 0 . / t], where[1 / t] = 10 rad / s. The steady-state val-ues of the current calculated separately us-ing the Landauer formula for a linear chain6 c u rr e n t [ µ A ] time [fs] (a)(b)(c) FIG. 2. (Color online) The current as a functionof time when the interleads coupling v LR ( t ) isswitched on as a step function at t = 0 (blacklines), gradually as a hyperbolic tangent withdriving frequency ω d = 0 . / t] (red lines), and ω d = 0 . 25 [1 / t] (green lines), where [1 / t] =10 rad / s. The bias potentials are (a) U b =0 . U b = 0 . U b = 0 . v = − . with unit transmission are also shown inFig. 2 as dashed lines. During the timesjust after the switch-on, the current rapidlyincreases and overshoots the expected long-time steady-state value. It then oscillatesand decays in time, eventually settling to thesteady-state value. It can be seen that as thedriving frequency is decreased the amplitudeof the oscillations also decreases. In addition,the peaks are displaced to later times becauseof the more gradual progress of the interleadscoupling v LR ( t ). In Fig. 2 we also see the de- pendence of the decay time of the transientcurrent to the applied bias potential and thespeed of the switch-on. The higher bias re-sults in a faster decay time for the oscillatingtransient current.The transient current oscillates and de-cays in time until it settles to a steady-state value. During the transient regimethe strength of the interleads coupling dy-namically changes resulting in particles tem-porarily accumulating at the left and rightsides of that coupling. Although we do notexplicitly consider Coulomb interactions be-tween charges, the temporary accumulationof charges at the sides of the interleads cou-pling, together with the distance between theaccumulated charges and the applied biasvoltage across the interleads coupling, canbe regarded to generate a temporary dy-namical capacitance. Similarly, a dynami-cal inductance may arise because the tran-sient current is varying in time. By consider-ing possible equivalent circuit combinationsand performing least-squares fitting to theenvelope of the decaying transient current wefind that the decay closely follows a powerlaw, indicating an equivalent series resistor-inductor-capacitor (RLC) circuit whose com-ponents have time-dependent properties. Ina previous study using semiclassical Boltz-mann transport theory on quantum wires, itis found that the wire can be modeled by an7quivalent series RLC circuit. A series RLCcircuit consisting of components with con-stant resistance, inductance, and capacitanceresults in a transient current whose envelopedecays as an exponential function. However,when the resistance, inductance, and capac- itance vary in time, the resulting transientcurrent can oscillate and could decay as apower law. Making therefore such an anal-ogy to the quantum device we are examining(see Fig. 1(c)), applying Kirchhoff’s law to aseries RLC circuit with time-varying compo-nents leads to the equation d Idt + (cid:18) RL + 1 L dLdt (cid:19) dIdt + (cid:18) L dRdt + 1 LC (cid:19) I = QLC dCdt , (10)where I is the time-dependent currentthrough the circuit, R ≡ R ( t ) is the resis-tance, L ≡ L ( t ) is the inductance, C ≡ C ( t )is the capacitance, and Q = R t I dt is thetime-dependent charge accumulating at thecapacitor. Furthermore, the power law fitsimply that the current is of the form I ( t ) = I t − α sin( ωt + φ ) + I , (11)where α is the power-law exponent de-termined from the fits, ω is the time-independent frequency of oscillation of thetransient current, φ is the phase determinedfrom initial conditions, and I is the time-independent steady-state current. Taking thetime derivative of Eq. (11) twice, we find d Idt + 2 αt dIdt + ω t I = ω t I , (12)where ω t = ω + α ( α − t . Comparing Eqs. (10) and (12) we get the coupled equations RL + 1 L dLdt = 2 αt , L dRdt + 1 LC = ω t ,QLC dCdt = ω t I , (13)which can be solved to determine how R ( t ), L ( t ), and C ( t ) vary in time for specific valuesof α and ω .The power-law exponent α determineshow fast the transient current decays until itreaches the steady-state value. In Fig. 3(a),it can be observed that by increasing thebias potential the value of α also increases,thereby speeding up the decay of the tran-sient current. α and the power-law coeffi-cient I actually also follow power-law fitswhen the bias potential is varied, as can beseen in Fig. 3. It suggests α = α U βb and I = I U γb , where α and I are indepen-dent of U b . The power-law exponents β and γ determine how fast α and I , respectively,8 bias potential [eV] e xpon e n t α bias potential [eV] c o e ff i c i e n t I [ µ A ] (a) (b) FIG. 3. (Color online) (a) The power-law ex-ponent α and (b) coefficient I as functions ofthe bias potential. The (green) dots are for theHeaviside step function interleads coupling. The(red) triangles are for the hyperbolic tangent in-terleads coupling with ω d = 0 . / t]. The (blue)squares are for ω d = 0 . 25 [1 / t]. The dashedlines are power-law fits to the correspondingdata points. The amplitude of the couplings are v = v LR = − . change when the bias potential is varied. InTable I we show how the values of the power-law fitting parameters change when U b is var-ied.From Table I, we see that the values of I and γ are independent of the speed ofthe switch-on. In addition, the exponent γ isabout one. These imply that I increases lin-early with the bias potential and is consistentwith the identification that I is the steady-state current. For the exponent α , we findthat as the speed of the switch-on is increased TABLE I. Values from the power-law fits to α and I when the interleads coupling is in theform of a hyperbolic tangent and a step func-tion. The couplings are v = − . v LR = − . α is (1 / eV) β and I is µ A / (eV) γ . ω d [1/t] α β I γ the coefficient α also increases but the ex-ponent β decreases. The increasing α sug-gests that for a given bias potential, the fasterswitch-on results in a faster decay of the tran-sient current. Since the slightly decreasing β is still positive, increasing the bias potentialstill speeds up the decay of the transient cur-rent. As a result, when the device is operatedunder low bias its current suffers oscillationsand overshootings longer than when it is op-erated under higher bias. Furthermore, it canbe observed that the power-law parameters α and β control the speed of decay of thetransient current. The values of these twoparameters vary depending on the speed ofthe switch-on. If we want the system to havea fast decaying transient current, then our re-sults indicate that we need a switch-on thatis as fast as possible.Next, we investigate the effects of vary-9 c u rr e n t [ µ A ] time [fs] (a)(b) FIG. 4. (Color online) The current as a functionof time for a step function switch-on. (a) Thecouplings have values v = v LR = − . v = v LR = − . v = v LR = − . v = − . v LR = − . v LR = − . v LR = − . U b = 0 . ing the values of the hopping parameter v and the interleads coupling v LR . The val-ues of these tight-binding parameters dependon the material used and varying them effec-tively means that we change the material weuse for the device. The parameters can bevaried separately or they can be varied whilemaintaining v = v LR . Firstly, we consider thelatter case. In order to determine only theeffects of varying the couplings, and not theeffects of the speed of the switch-on, we em- ploy the step function switch-on. The currentas a function of time is shown in Fig. 4(a).Since v = v LR , the long-time steady-state val-ues of the current can be calculated from theLandauer formula with a transmission coef-ficient T = 1 (no scattering involved). Thissteady-state value is shown in Fig. 4(a) asa dashed line. We examine coupling values v = v LR = − . , − . − . U b = 0 . v and v LR separately. When v LR isdifferent from v , the interleads distance is dif-ferent from the nearest-neighbor distance be-tween sites in the leads. This results in apotential barrier that is different at the in-terleads coupling and thus, a particle movingfrom the left lead scatters at the interleadscoupling. Figure 4(b) shows the current asa function of time when the nearest-neighbor10opping parameter is set at v = − . v LR to values − . − . − . (cid:12)(cid:12) v LR (cid:12)(cid:12) > | v | values because that would imply a shorterinterleads distance than the natural nearest-neighbor distance, represented by v , in theleads. In contrast, decreasing (cid:12)(cid:12) v LR (cid:12)(cid:12) increasesthe potential barrier at the interleads cou-pling, and thus implying a longer interleadsdistance, and results in the reduction in theamplitude of the oscillating transient cur-rent. From Fig. 4(b), we also see that thepeaks are slightly shifted to later times. Inaddition, the long-time steady-state currentslightly decreases when (cid:12)(cid:12) v LR (cid:12)(cid:12) is decreased. Ifwe want to calculate the steady-state cur-rent using the Landauer formula, we wouldfind that the transmission coefficient is re-duced when v LR is different from v becauseof the scattering occuring at the interleadscoupling.The oscillating and decaying transient cur-rent when v and v LR are varied also follow apower law. In Figs. 5(a) and 5(b) the val-ues of v and v LR are varied together whilein Figs. 5(c) and 5(d) the values are variedseparately. Figure 5(b) shows that the val-ues of I is the same for the cases examined.Compared to Fig. 5(d), we see that the val-ues of I are slightly different. Identifying I as the steady-state current, we thus confirmthat it is the same whenever v = v LR . On the α I [ µ A ] bias potential [eV] α bias potential [eV] I [ µ A ] (a) (b)(c) (d) FIG. 5. (Color online) The exponent α and co-efficient I when v and v LR are varied. In (a)and (b), v = v LR = − . v = v LR = − . v = v LR = − . v = − . v LR to values − . − . − . other hand, the scattering that happens atthe interleads coupling when v LR is differentfrom v affects the value of the steady-statecurrent. Moreover, the plots of the exponent α and coefficient I as functions of the biaspotential can also be fitted to power laws.As shown in Table II, when v = v LR , de-creasing the value of the couplings decreasesboth the coefficient α and the exponent β ,while I and γ remain the same. The tran-sient current therefore decays slower. Thisis because decreasing the couplings decreases11he energy required for the particle to movearound. This increase in the particle’s free-dom to move increases the frequency of oscil-lation and slightly lengthens the decay of thetransient current. However, fixing the valueof v and increasing v LR decreases α and I ,but increases β . Therefore, for a given biaspotential, increasing v LR lengthens the decaybut suppresses the amplitude of oscillationof the transient current. The parameters α , β , I , and γ depend on the type of mate-rial used. The value of the interleads cou-pling v LR , in addition, depends on the dis-tance between the leads. The farther apartare the two leads, the higher is the value of v LR because of the higher interleads poten-tial barrier. Our results show that strongerinterleads scattering lengthens the decay ofthe transient current. However, the scatter-ing also suppresses the amplitude of the tran-sient current and decreases the eventual valueof the long-time steady-state current.The times when the peaks in the currentoccur can be known from the extremum ofthe power-law form of the current in Eq. (11).Taking the time derivative of Eq. (11) andthen equating the result to 0, we find the ex-tremum of the current to occur at times t p whenever the following is satisfied: ω t p α = tan ( ω t p + φ ) . (14)The left-hand side is an equation for a TABLE II. (Color online) Values from thepower-law fits to the exponent α as a function ofthe bias potential U b when the switch-on of theinterleads coupling is in the form of a Heavisidestep function. The shaded entries indicate caseswhen v = v LR . The dimension of α is (1 / eV) β and I is µ A / (eV) γ . v [eV] v LR [eV] α β I γ -2.0 -2.0 0.344 0.340 40.125 0.980-2.7 -2.7 0.307 0.332 40.214 0.983-4.0 -4.0 0.268 0.330 40.305 0.985-2.7 -2.1 0.219 0.466 37.152 0.993-2.7 -2.4 0.252 0.367 39.344 0.992 time [fs] FIG. 6. (Color online) The location of the peaksin the transient current. The (black) lines repre-sent tan( ω t + φ ), where ω = 2 [1 / t] and φ = π .The (red) dashed line represents ω t/α , where α = 0 . 2. The (blue) dashed-dot line is when α = 0 . α = 0 . 5. The times when the straight andtangent lines intersect correspond to the timeswhen the peaks in the transient current occur. straight line with a slope that depends on α . Since α varies depending on the values12f the bias potential, the couplings, and thespeed of the switch-on, changing these pa-rameters would change the slope. As a con-sequence, the location in time of the currentpeaks would also change. This can be seenby noting how the peaks in the transient cur-rent move in Fig. 2 as α is varied. t p canbe determined by the intersection points ofthe straight and tangent lines, correspondingto the left-hand side and right-hand side, re-spectively, of Eq. (14) and as shown in Fig. 6.The times when the current peaks occur arelocated whenever the two curves intersect.Since the slope of the straight line depends on α , we see that the faster decaying transientcurrent, i.e., higher values of α , correspondto earlier peak times.Finally, we investigate the transport prop-erties of the device having a regular timevariation, such as a nano-oscillator. In anano-oscillator, the rotating disk in Fig. 1(a)is rocked back and forth across the dashedline. This would result in a harmonic modu-lation of the interleads coupling and woulddynamically modulate the current throughthe device. However, compared to an al-ternating current which changes sign, themodulated current flowing out of the nano-oscillator maintains the same sign. We modelthe harmonically modulated coupling in theform v LR ( t ) = v LR / · (1 − cos ω d t ), where ω d is the driving frequency of modulation. Fig- time [fs] c u rr e n t [ µ A ] (a)(b) FIG. 7. (Color online) The current as a func-tion of time for the nano-oscillator with drivingfrequency ω d = 0 . 25 [1 / t]. (a) The bias poten-tials are U b = 0 . U b = 0 . U b = 0 . v = v LR = − . U b = 0 . v = − . v LR = − . v LR = − . ure 7 shows the current characteristics as afunction of time as the interleads couplingis swinged back and forth with driving fre-quency ω d = 0 . 25 [1 / t]. In Fig. 7(a), thecouplings are v = v LR = − . U b = 0 . v = − . v LR is varied. We findthat the current through the oscillator comes13n pulses. However, it does not exactly followthe harmonic form of the coupling. The in-terleads coupling is maximum at times whenthe left and right leads are exactly aligned.In contrast, we see that the peaks in the cur-rent do not coincide with the times when theinterleads coupling is maximum. The shapeof the curve for the current actually looks likethe truncated version of the transient currentwe examined in Fig. 2. In particular, the ini-tial overshoot of the transient current mani-fests as the first current peak in Fig. 7. Thispeak, however, does not occur when the in-terleads coupling is maximum. In addition,the times when the peak occurs depend onthe values of the bias potential and the cou-plings. This dependence of the peak loca-tion to the above physical parameters followsimilar dependence of the peak location inthe nano-relay. In the design of nano-circuitscontaining an oscillator, therefore, it shouldbe noted that the maximum current does notoccur when the leads are exactly aligned andthat the exact location of these peaks dependon the values of the applied bias potential,the nearest-neighbor coupling, and the inter-leads coupling. IV. SUMMARY ANDCONCLUSION In summary, we examine a device thatcould act as a nano-relay or a nano-oscillator.The device consists of two leads and a time-varying interleads coupling. We use NEGFto derive a nonperturbative expression for thetime-dependent current flowing from one leadto the other. In the nano-relay configuration,we model the switch-on of the interleads cou-pling in the form of either a step function ora slowly progressing hyperbolic tangent. Wefind that the current oscillates and decaysin time just after switch-on and during thetransient regime. In both the step functionand hyperbolic tangent switch-on, the de-cay of the transient current fits a power law.This leads to an equivalent RLC series circuitwhere all of the components have dynamicalproperties. We also find that the values of thecouplings v and v LR , the scattering at the in-terleads coupling, the speed of the switch-on,and the value of the bias potential affect thedecay time of the transient current. In thelong-time regime, the current approaches thesteady-state value. In the nano-oscillator, wemodel the dynamical sytem by harmonicallymodulating the interleads coupling. We findthat the current passes through the device inpulses, maintains the same sign, but does notexactly follow the functional form of the os-14illating coupling. In particular, the peaks inthe current do not occur at the times when-ever the leads are exactly aligned.The expressions for the current shown inEq. (4) and the corresponding lesser Green’sfunction shown in Eq. 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