Injectance and a paradox
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Injectance and a paradox
Urbashi Satpathi and P Singha Deo
Unit for Nano Science and Technology, S. N. Bose National Centre for Basic Sciences,Block JD, Sector III, Salt Lake, Kolkata 700 098, India. ∗ (Dated: November 13, 2018)Quantum mechanics manifests in experimental observations in several ways. Hauge et al. (1987)and Leavens et al. (1989) had pointed out that interference effects dominate a physical quantitycalled injectance. We show that, very paradoxically, the interference related term vanish in aquantum regime making semi-classical formula for injectance exact in this regime. This can haveuseful implications to experimentalists as semi-classical formulas are much more simple. There areother puzzling facts in this regime like an ensemble of particles can be transmitted without any timedelay or negative time delays, whereas the reflected particles are associated with a time delay. A series of experiments has recently confirmed thatscattering phase shifts in quantum mechanics can bemeasured . Non-locality in quantum mechanics doesnot allow us to determine a particular path in whichthe electron wave propagates. This is unlike classicalwaves. This problem was overcome by using additionalprobes and controlled decoherence . Ref. [5] confirmsthat Hilbert transform of the measured conductance datagives the measured phase data which confirms that thescattering phase shift was correctly measured. Thus thescattering phase shift as well as the scattering cross sec-tion of an arbitrary quantum system (say, its impurityconfiguration and confinement potential is not known)can be measured. While measuring scattering cross sec-tion is an old story, measuring scattering phase shifts isnovel and new. So one can ask the question that fromthe measured scattering phase shift what can we learnabout the quantum system.Friedel sum rule (FSR) relates scattering phase shiftto density of states (DOS) in the system and WignerSmith delay time (WSDT) relates scattering phase shiftto a time scale at the resonances. The two are basi-cally the same as both density of states and life timeof a resonance are given by the imaginary part of theretarded Greens function. Although, it is to be notedthat different works have interpreted the WSDT in dif-ferent ways which will be discussed later. They (FSR andWSDT) are semi-classical formulas that are not a pri-ori applicable to quantum systems in mesoscopic regime.The quantum versions of these formulas are not com-pletely in terms of the experimental data. A number ofworks has studied FSR and WSDT in the single channelcase . Ref. [6] has shown that in the single prop-agating channel regime, for any arbitrary potential thathas symmetry in x -direction i.e., V ( x, y ) = V ( − x, y ),FSR and WSDT are exact at the resonances of the sys-tem, essentially due to the fact that the resonances areFano resonances. Another physical quantity of interestis injectivity or injectance . In this work we calculateinjectivity and injectance in the single channel as well astwo channel regime, as this also reveals the paradoxicalnature of the scattering phase shift at Fano resonance.Let us consider plane wave incident from left hand side on a three dimensional scatterer. The scattered orasymptotic wave function in spherical polar coordinates(r, θ , φ ) is given by ,1 r sin ( kr − lπ θ t ) P l cos ( φ )Here θ t is the scattering phase shift. Assume a largesphere of radius R and let us count the number of nodesinside the sphere. Number of states is one more than thenumber of nodes (number of nodes +1). To count thenumber of nodes inside the sphere we can set the wavefunction to zero on the boundary of the sphere, i.e., kR − lπ θ t = nπ or, dθ t dk + R = dndk π (1)In absence of scatterer, kR = n π or, R = dn dk π (2)As, E = ~ k m e , we get from (1) and (2), dθ t dE = π [ ρ ( E ) − ρ ( E )] where, ρ ( E ) = dndE is the density of states in thepresence of scatterer and ρ ( E ) = dn dE is the density ofstates in the absence of scatterer. Instead one can put θ t + kR ≡ θ t that is scattering phase shift is definedwith respect to phase shift kR in absence of scatterer. Inwhich case FSR becomes , dθ t dE = πρ ( E ) (3)In the rest of the paper we will use this definition forscattering phase shift. This is a semi-classical result be-cause it is valid only when, dn ≪ n . This assumptionis related to non-dispersive wave packets or stationaryphase approximation as shown below. In this case werestrict to one dimension as Eq. (3) can be shown in onedimension too where the large sphere of radius R becomejust two points at a large distance of x = ± R .Consider a wave packet incident from left on a one di-mensional scatterer, i.e., ψ in ( x, t ) = R + ∞−∞ a k e ikx − iωt dk .After scattering the transmitted wave packet at ( x +∆ x, t + ∆ t ) is, ψ sc ( x + ∆ x, t + ∆ t ) = FIG. 1: The figure shows a wave packet is incident on a barrierof height V in one dimension. a)The energy of incidence E ofthe centroid of the wave packet is much larger than the barrierpotential V, i.e E ≫ V , and, b) The energy of incidence E ismuch smaller than the barrier potential V, i.e E ≪ V . Theselimits correspond to semi-classical regimes. Z + ∞−∞ | t ( k ) | a k e ik ( x +∆ x ) − iω ( t +∆ t )+ iθ t dk (4)where, t ( k ) is the transmission amplitude and θ t is itsphase. A classical particle is either transmitted com-pletely or reflected completely without any distortion.Assuming there is no reflected part, along with the as-sumption of no dispersion of wave packet (known as sta-tionary phase approximation), we can write, kx + k ∆ x − ωt − ω ∆ t + θ t = K (5)where K is a constant. This implies that the phase of theamplitude component a k remain stationary or that thewave packet remains undispersed. Therefore from (5), dθ t dω = ∆ t − ∆ xv g or, ~ dθ t dE = ∆ t − ∆ t (6)where v g = dωdk is the group velocity. Semi-classicalbehavior is pictorially depicted in figures 1a) and 1b).When E ≫ V then a wave packet can get com-pletely transmitted with Eq. (5) approximately satis-fied (it is never exactly satisfied as electron dispersionis quadratic). It can also happen when E ≪ V then awave packet can get completely reflected with very littledispersion. In this case, θ t is to be replaced by θ r , thereflected phase shift. Thus we find that when a wavepacket remains undispersed, then the time spent by theparticle in the scattered region (∆ t − ∆ t ), is given byenergy derivative of scattering phase shift. This life time(WSDT) is also related to DOS as both are given by theimaginary part of retarded Green’s function [ref. [11]page 155-156]. Hence, as we have seen in Eq. (3), the DOS can also be found from the energy derivative of scat-tering phase shift, which is FSR. These formulas are obvi-ously not valid in the quantum regime i.e., E ∼ V , wherethere will be transmission as well as reflection. Also inthis regime the scattering phase shifts will be so stronglyenergy dependent that stationary phase approximationcannot remain valid and a wave packet will always un-dergo dispersion. Refs. [12,13] have looked into the onedimensional problem when both transmission and reflec-tion is present. They have computed the correction termand explained the significance of it. They have shownthat the correction term arises because of quantum inter-ference effect. Note that both dispersion and quantuminterference arises due to the superposition principle inquantum mechanics. Their result will be discussed latter.The quantum regime can be treated exactly and alsoyields the following formula for time known as Larmorprecession time (LPT) , τ ( α, r, β ) = − ~ πi " s † αβ ∆ s αβ e ∆ U ( r ) − ∆ s † αβ e ∆ U ( r ) s αβ (7)Connection between LPT and WSDT or FSR is ex-plained below. τ ( α, r, β ) is the time spent by one particleincident along channel β and scattered to channel α at r.LPT is exact in the sense that when summed over α and β and divided by ~ , it gives the exact DOS as calculatedfrom the internal wave function. ∆ stands for functionalderivative. Note that the functional derivative is with re-spect to the local potential implying that exact DOS can-not be expressed using asymptotic wave function alone.Dividing Eq. (7) by ~ we get a partial local density ofstates (PLDOS) for such a process (a particle incidentalong β and scattered to α ). That is τ ( α,r,β ) ~ = ν ( α, r, β ),where ν ( α, r, β ) is partial local density of states. Such aprocess requires to specify both the incoming and out-going channel which is impossible in quantum mechanicsbecause a single particle in quantum mechanics behaveprobabilistically when it encounters a potential and getsscattered. Only when an ensemble of particles is consid-ered, the probability of transmission and that of reflec-tion are given by Schr¨odinger equation. One can indi-rectly measure the consequences of such a PLDOS , butan experimental set up that can directly probe this PL-DOS requires us to take a sum over at least one of thechannels (i.e., α or β ). So, we can specify the incomingchannel and scattering can take the particle to any arbi-trary output channel. That is, P α ν ( α, r, β ) = I ( r, β ) isa physical quantity called injectivity. I ( r, β ) = X α − πi " s † αβ ∆ s αβ e ∆ U ( r ) − ∆ s † αβ e ∆ U ( r ) s αβ = X α − π | s αβ | ∆ θ s αβ e ∆ U ( r ) (8)where, θ s αβ = arctan Im ( s αβ ) Re ( s αβ ) . This quantity can be ex-perimentally observed and is the topic of study in thiswork. A typical situation where this quantity can be ob-served is when we bring a scanning tunneling microscope(STM) tip close to a mesoscopic sample at r connectedto one or more leads. Injectivity gives current deliveredby the tip.When injectivity is summed over β then we get DOSwhich has been studied earlier . Not summing over β reveals the true nature of the paradox. This is becausethen the incident wave packet comes along a single chan-nel β and after scattering it will either disperse or willnot disperse. If we sum over β then the incidence is alongall possible channels. So wave packets are incident along β channels and their scattering and dispersion can com-pensate each other. Also refs. [12,13] have calculated thecorrection term for this injectivity in the single channelcase and the paradox can be quantitatively explained interms of this correction term. The semi-classical limitcan be obtained from the following substitution, − Z Ω d r ∆ e ∆ U ( r ) −→ dd E (9)Hence, I ( E ) ≈ X α π | s αβ | ddE θ s αβ (10)This quantity is called injectance. | s αβ | appears be-cause when we allow scattering (Eq. (6) correspond to acase when everything is transmitted without scattering)then | s αβ | number of particles are scattered from chan-nel β to channel α and each ones contribution to time isrelated to dθ sαβ dE . In the single channel case, it can beexplicitly written as I ( E ) ≈ π (cid:20) | r | dθ r dE + | t | dθ t dE (cid:21) (11)The approximate equality can be replaced by an equalityif we add a correction term on the right hand side . I ( E ) = 12 π (cid:20) | r | dθ r dE + | t | dθ t dE + m e | r | ~ k sin( θ r ) (cid:21) (12)where k is the wave vector of the incident channel. Refs.[12,13] have stressed that the correction term can be zeroif | r | = 0 (corresponding to Fig.1a)), but sin( θ r ) cannotbe zero as it comes from quantum interference which is al-ways there in a quantum system. sin( θ r ) can be zero onlywhen the potential is infinitely high because then thereis obviously no interference effect (corresponding toFig.1b)). Or it sometimes become zero far away from res-onances that are almost classical regimes . Therefore,just as dispersion of a wave packet cannot disappear, in-terference effects cannot disappear. Both originate fromthe linear superposition principle in quantum mechanics.Only when dispersion or interference can be ignored weget the semi-classical limit, i.e, Eq. (10).We will first show that there is a situation where sin( θ r )can become zero at a resonance where | r |6 = 0, making FIG. 2: A general scattering problem in quasi one dimension.The reservoir L inject electrons to the left lead L and hence tothe scattering region with arbitrary potential V g ( x, y ), shownby the shaded region. Reservoir R absorb electrons transmit-ted through the shaded region, through right lead R. The elec-trons reflected from the shaded region are collected in reser-voir L. Here we have considered one propagating channel inthe leads and electrons are incident from the left. (10) exact in a quantum regime. This is in complete con-tradiction to what is known so far . A general prooffor a single channel scattering is given below. This canhave useful implications to experimentalists in the sensethat although injectance depends on the potential insidethe scatterer, it can be determined from asymptotic wavefunction that can be experimentally measured. Considera quasi one dimensional (Q1D) system with scatteringpotential V g ( x, y ) shown in Fig.2 by the shaded region.Lead L and lead R connect the system to electron reser-voirs L and R, respectively. They act as source and sinkfor electrons, respectively. The confinement potential inthe leads, in the y -direction (or, transverse direction) istaken to be hard wall, given by, V ( y ) = ∞ for | y |≥ W
2= 0 for | y | < W ) is, (cid:20) − ~ m e (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + V ( y ) + V g ( x, y ) (cid:21) Ψ( x, y )= E Ψ( x, y ) (13)where m e is the mass of the electron, W is the width ofleads L and R. In the leads where there is no scatterer( V g ( x, y ) = 0), the Schr¨odinger equation can be decou-pled. The y -component is, (cid:20) − ~ m e d dy + V ( y ) (cid:21) χ m ( y ) = ε m χ m ( y ) (14)and the x -component is, − ~ m e d dx c m ( x ) = ( E − ε m ) c m ( x ) (15)with χ m ( y ) = q W sin mπW ( y + W ) and ε m = m π ~ m e W . E is the energy of incidence from reservoir L, given by, E = E m,k m = m π ~ m e W + ~ k m m e (16)It is known that for potentials that have symmetry in x -direction, i.e., V ( x, y ) = V ( − x, y ), we can write solutionsto (15) given below. c em ( x ) = ∞ X m =1 ( δ mn e − ik m x − S emn e ik m x ) 1 √ k m (17) c om ( x ) = ∞ X m =1 ( δ mn e − ik m x − S omn e ik m x ) 1 √ k m (18)where, c em ( x ) = c em ( − x ) and c om ( x ) = − c om ( − x ). Thenboth transmitted wave function (at x > a ) and reflectedwave function (at x < − a ) are given by c m ( x ) = c em ( x ) − c om ( x )2 (19) E can be so adjusted by adjusting the Fermi energy ofreservoir L, that π ~ m e W < E < π ~ m e W . Then k is realand from (17), (18) and (19) c ( x ) = e ik x + ˜ r e − ik x for x < − a = ˜ t e ik x for x > a where ˜ r = − ( S o + S e )2 (20)˜ t = ( S o − S e )2 (21)For m >
1, energy conservation in (16) is not violatedas k m can become negative. That yields evanescent solu-tions with k m → iκ m . Inclusion of the evanescent modesis very important to get the correct solutions. Due to thesame principle, i.e., any function can be written as a sumof an even function and odd function and any square ma-trix can be written as a sum of a symmetric matrix andan anti-symmetric matrix, the wave function in the scat-tering region ( − a < x < a ) can be written as a sum ofan even function and an odd function. We denote themas η em ( x, y ) and η om ( x, y ), i.e., Ψ( x, y ) = η em ( x,y )+ η om ( x,y )2 for − a < x < a . η em ( x, y ) = ∞ X n =1 d n ζ en ( x, y ) (22) η om ( x, y ) = ∞ X n =1 d n ζ on ( x, y ) (23) ζ en and ζ on are the basis states that satisfy the conditionthat they go to zero at the upper edge and lower edge ofthe shaded region in Fig.2.One can define the matrix elements, F eom,n = 2 W ( k m k n ) Z b − b χ m ( y ) (cid:18) ∂ζ eon ∂x (cid:19) x = a dy (24)Here eo stands for even or odd, i.e., e/o . One can matchthe wave function and conserve the current at x = ± a for all y to obtain a matrix equation given by, ∞ X n =1 [ F eorn − iδ rn ] e ik n a S eonm = [ F eorm + iδ rm ] e − ik m a (25)Solving for S eomn , we can find the scattering matrix ele-ments. Bound states can be determined from the singu-larities of the matrix equation (25), on setting right handside to zero. That is det [ F eocc − i
1] = 0 (26)Here ‘ cc ’ means evanescent channel (or closed channel)for which both k m and k n in (24, 25), are imaginary.Solving Eq. (25), one can get , S eo = e − ik a G eo + iG eo − i = e iarccot [ G eo ] = e iθ eo (27)where, G eo = F eo − X m =2 ,n =2 F eo n h ( F eocc − i − i nm F eom (28)and θ eo = arccot [ G eo ] (29)Here scattering phase shift θ eo is defined with respect tophase shift in the absence of scatterer. Putting (27) in(20), (21) and defining new variables, φ = θ e − θ o and θ r = θ e + θ o (30)we get transmission amplitude and reflection amplitude˜ t = i sin( φ ) e iθ r and ˜ r = cos( φ ) e iθ r (31)The correction term in (12) is m e | ˜ r | ~ k sin( θ r ).Threshold energy E for the second channel is givenby E = π ~ m e W , i.e., above this energy k becomes real.Below this energy the second channel can have boundstates. Such bound states will occur at energies givenby the solution to Eq. (26). At these energies the firstchannel can be propagating as its threshold is given by E = π ~ m e W and S is given by Eq. (27). But at bound FIG. 3: A similar scattering set up as in Fig.2, with gen-eral potential V g ( x, y ) replaced by a delta function poten-tial γδ ( x ) δ ( y − y j ). Here we are considering two propagatingchannels. Incidence is along only one channel which is thefundamental channel ( m = 1) from the left. state energy, G eo will diverge as it includes matrix ele-ments of [ F eocc − i − as can be seen from equations (26)and (28). That in turn implies that at resonance (as canbe seen from Eq. (29)) θ e = pπ and θ o = qπ (32)Therefore, from Eq. (32),sin( θ r ) = sin( θ e + θ o ) = 0 (33)Thus we have shown that the correction term in Eq. (12)is zero precisely because sin( θ r ) = 0 but | r |6 = 0 (30, 31).Note that if we ignore the correction term, then all termson right hand side can be determined experimentally bymeasuring asymptotic solutions. So injectance can beknown from the asymptotic solutions. So the correctionterm being zero at a resonance can have useful implica-tion to experimentalists.We cannot study the multichannel case generally. Wewill study the two channel case for a particular poten-tial, i.e., V g ( x, y ) = γδ ( x ) δ ( y − y j ), to show the para-dox there along with other puzzling facts. In Fig 3, theshaded region is a two dimensional quantum wire with adelta function potential at position (0, y j ) marked X. Thedotted lines represent the fact that the quantum wire isconnected to electron reservoirs.Injectance can be obtained from the scattering (S) ma-trix as described earlier, for the system described in Fig 3.The two channel injectance with incidence along m = 1channel from left as shown in Fig.3, in the semi-classicallimit is given by, I ( E ) ≈ π (cid:20) | r | dθ r dE + | r | dθ r dE + | t | dθ t dE + | t | dθ t dE (cid:21) (34)Here, the subscript 1 signifies the incident channel trans-verse quantum number, i.e., m = 1. We can break thisup as I ( E ) ≈ I L ( E ) + I R ( E ), where, I L ( E ) = 12 π (cid:26) | r | dθ r dE + | r | dθ r dE (cid:27) (35) and I R ( E ) = 12 π (cid:26) | t | dθ t dE + | t | dθ t dE (cid:27) (36)That is, I L ( E ) consist of reflection channels and I R ( E )consist of transmission channels. The correction termdepends on parameters of the incident channel only andgives the following identity. I ( E ) = I L ( E ) + I R ( E ) + 12 π m e | r | ~ k sin( θ r ) (37)In the similar way, injectance of channel 2 is, I ( E ) = 12 π (cid:20) | r | dθ r dE + | r | dθ r dE + | t | dθ t dE + | t | dθ t dE + m e | r | ~ k sin( θ r ) (cid:21) (38)and in the semi-classical limit, it is given by, I ( E ) ≈ π (cid:20) | r | dθ r dE + | r | dθ r dE + | t | dθ t dE + | t | dθ t dE (cid:21) (39)We have verified this numerically. The RHS can be de-termined from the S-matrix. This S-matrix approach iseasily accessible to experimentalists. We will now cal-culate this injectance from internal wave function whichallows numerical verification. I ( E ) = Z ∞−∞ dx Z W − W dy X k | Ψ( x, y, | δ ( E − E ,k )(40)where Ψ( x, y,
1) is the wave function in the scatteringregion for incidence along channel m = 1, and is of theform , Ψ( x, y,
1) = X m f m ( x, χ m ( y ) (41)Here χ m ( y ) s are solutions in the leads in the transversedirection which is a square well potential in y -direction. χ m ( y ) s form a complete set and (41) is derived from thefact that at a given point x , the wave function in thescattering region can be expanded in terms of χ m ( y ) s . f m s are generally of the form given below , f ( x,
1) = e ik x + r e − ik x for x < t e ik x for x > f ( x,
1) = r e − ik x for x < t e ik x for x > m > f m ( x,
1) = r m e κ m x for x < t m e − κ m x for x > r mn and t mn are unknowns to be determined. Thescattering problem described above can be solved usingmode matching technique . The reflection amplitudesare given by, r mn ( E ) = − i Γ mn √ k m k n P e Γ ee κ e + i P m Γ mm k m (42)Γ mn is the coupling strength between m th and n th modes,given by Γ mn = γ sin mπW ( y j + W ) sin nπW ( y j + W ). Thetransmission amplitudes are given by, t mn ( E ) = r mn ( E )for m = n and, t mm ( E ) = 1 + r mm ( E ). P e denotessum over evanescent modes and runs from 3 to ∞ , while P m denotes the same for propagating modes (i.e. m=1and m=2). θ r mn = arctan Im ( r mn ) Re ( r mn ) . We will present ourresults for the case of two propagating channels but theanalysis and results are same for any number of channels.If the delta function potential is negative ( γ <
0) thenthere can be bound states . In general, the quasi boundstate for channel m=s is given by,1 + ∞ X e = s Γ ee κ e = 0 (43)Only the bound state for m=1 channel is a true boundstate and it is given by the solution to the following equa-tion, 1 + P ∞ e =1 Γ ee κ e = 0. The bound state for m=2 isgiven by, 1 + P ∞ e =2 Γ ee κ e = 0. At this energy we get abound state for m=2, but at that energy m=1 channel isa propagating channel. Hence the bound state given bythis equation is a quasi bound state or a resonance.The delta function in Eq. (40) summed over k can beshown to yield a factor hv , where v = ~ k m . Using theorthonormality of χ m ( y ) s , we get, I ( E ) =1 hv "Z ∞−∞ dx X m | f m ( x, | = 1 hv (cid:20)Z ∞−∞ dx | f ( x, | + Z ∞−∞ dx | f ( x, | + Z ∞−∞ dx | f ( x, | + Z ∞−∞ dx | f ( x, | + · · · (cid:21) Substituting the values of f m ( x, s , we get, I ( E ) =1 hv (cid:20)Z −∞ dx (cid:2) | r | +2 | r | cos(2 k x + φ ) (cid:3) + Z ∞ dx | t | + Z −∞ dx | r | + Z ∞ dx | t | + | t | κ + | t | κ + · · · (cid:21)
40 50 60 70 80 90 EW I ( E ) , I ( E ) FIG. 4: The plot is of injectance versus EW for γ = −
13, and y j = . W . The solid curve shows semi-classical injectance ofchannel 1 (i.e., incidence is along m = 1 from left) calculatedusing S-matrix (Eq. (34)), the dot-dashed curve is the ex-act injectance, for the same channel calculated using internalwave function (Eq. (45)). The dotted curve is semi-classicalinjectance for channel 2 from S-matrix (Eq. (39)), the dashedcurve is the exact injectance, for channel 2 (i.e., incidence isalong m = 2 from left) calculated using internal wave func-tion (Eq. (46)). We use ~ = 1, 2 m = 1. The figure showsthat semi-classical formula becomes exact at resonance wherethere is a peak in injectance. Here, r = | r | e − iφ . Note that for m > Z ∞−∞ dx X m | f m ( x, | = | t m | (cid:20)Z −∞ e κ m x dx + Z ∞ e − κ m x dx (cid:21) = | t m | κ m Using time reversal symmetry, i.e, r = r and t = t , we get, I ( E ) = 1+ | r | + | r | hv Z −∞ dx + | t | + | t | hv Z ∞ dx + 2 | r | hv Z −∞ dx cos(2 k x + φ )+ 1 hv (cid:18) | t | κ + | t | κ + · · · (cid:19) Adding and subtracting the following terms, | t | hv Z −∞ dx , | t | hv Z −∞ dx ,
40 50 60 70 80 90 EW I ( E ) , I ( E ) FIG. 5: The plot is of injectance versus EW for γ = −
15, and y j = . W . The solid curve shows semi-classical injectance ofchannel 1 (i.e., incidence is along m = 1 from left) calculatedusing S-matrix (Eq. (34)), the dot-dashed curve is the ex-act injectance, for the same channel calculated using internalwave function (Eq. (45)). The dotted curve is semi-classicalinjectance for channel 2 from S-matrix (Eq. (39)), the dashedcurve is the exact injectance, for channel 2 (i.e., incidence isalong m = 2 from left) calculated using internal wave func-tion (Eq. (46)). We use ~ = 1, 2 m = 1. The figure showsthat semi-classical formula becomes exact at resonance wherethere is a peak in injectance. and using the fact that, | t | hv Z −∞ dx = | t | hv Z ∞ dx and, | r | + | r | + | t | + | t | = 1, we get, I ( E ) = 1 hv Z ∞−∞ dx + | r | hv Z ∞−∞ dx cos(2 k x + φ )+ 1 hv (cid:18) | t | κ + | t | κ + ·· (cid:19) (44) hv R ∞−∞ dx = I ( E ) is the injectance in the absence ofscatterer. Again if the scattering phase shift is definedwith respect to the phase shift in absence of scattererthen this term can be dropped.Now, Z ∞−∞ dx cos(2 k x + φ ) = δ (2 k ) cos( φ )As k = 0 is not a propagating state contributing totransport, hence this term of Eq. (44) reduces to zero,and we are left with, I ( E ) = 1 hv (cid:18) | t | κ + | t | κ + · · ·· (cid:19) (45)Similarly, for incidence along channel 2, one can obtainthe injectance, I ( E ) = 1 hv (cid:18) | t | κ + | t | κ + · · ·· (cid:19) (46)
40 50 60 70 80 90 EW -2 -2-1 -10 01 1 x | r | θ r x | r | ,, FIG. 6: Here we are using the same parameters as that ofFig 5. The solid curve shows 30x | r | , the dotted curve isfor θ r . The dashed curve is for 5x | r | . The figure showsthat at resonance, there is large fluctuation in | r | , θ r and | r | . Also it shows that | r |6 = 0 but sin( θ r ) = 0.
40 50 60 70 80 90 EW -1.5 -1.5-1 -1-0.5 -0.50 00.5 0.51 1 . | t | , . θ t | t | , θ t FIG. 7: Here we are using the same parameters as that ofFig 5. The solid curve shows (3 . | t | − . . θ t . The dashed curve is for 5x | t | , andthe dot-dashed curve is for θ t − .
6. The figure shows thatat resonance, there is large fluctuation in | t | , θ t , θ t and | t | . Note that contribution to injectance come from evanes-cent channels only. This is a tunneling problem becausethe lateral confinement makes the effective potential verylarge and extended in the propagating direction . In fig-ures 4 and 5 we consider two propagating channels andshow that semi-classical formula for injectance becomesexact at a resonance. Injectance for both the channels isshown. I ( E ) is the injectance for incidence along chan-nel 1 and I ( E ) is the same for incidence along channel2. The resonance condition is given by Eq. (43) and thisoccur at EW = 85 .
62 in Fig.4 and at EW = 84 .
40 50 60 70 80 90 EW -0.2 -0.20 00.2 0.20.4 0.40.6 0.6 I R ( E ) x I R ( E ) FIG. 8: Here we are using the same parameters as in Fig.4.The solid curve shows 10x I R ( E ), the dashed line is for I R ( E ).Both of them get exactly zero at the resonance i.e., at EW =85 . classical I ( E ) obtained from Eq.(34)) coincides withdot-dashed curve (exact I ( E ) obtained from Eq.(45)),and, dotted curve (semi-classical I ( E ) obtained fromEq.(39)) coincides with dashed curve (exact I ( E ) ob-tained from Eq.(46)). Exactness of the semi-classicalformula is counter-intuitive because at this point scat-tering probability and scattering phase shift show strongenergy dependence, so the stationary phase approxima-tion needed to get (34), (39) cannot be valid. Also at thisenergy, | r |6 = 0 and | r |6 = 0, rather, sin( θ r ) = 0 andsin( θ r ) = 0 implying interference effects disappear andso correction term in (34, 38) disappear making semi-classical formula exact. All this along with the energydependence of the scattering matrix elements can be seenin Fig.6 and Fig.7 for the parameters used in Fig.5. Thenegative slope in the scattering phase shift θ t (see dot-ted curve in Fig.6) is due to Fano resonance and an-other puzzling feature, explained below.It can be proved that I R , ( E ) = 0 at Fano resonance,from (36) and (42). It is shown in figures 8 and 9 forthe parameters used in figures 4 and 5 respectively. Asexplained before that I R , ( E ) (within a factor of ~ ) is atime scale associated with particles that escape to theright lead. There has been a great amount of contro-versy regarding, what negative or zero times mean. Notethat I R ( E ) (dashed curve in Fig.8 and Fig.9) becomesnegative for 40 < EW <
50. So far researchers havemainly studied this low energy quantum regime [20–25]for observing and understanding negative time scales. Inthis regime the correction term is non-zero. I ( E ) asdefined in Eq. (46) (within a factor of ~ ) is known asdwell time and it is definitely positive. Each of the fourtimes on the RHS of Eq. (38) are Wigner Smith delaytime (again within a factor of ~ ). The controversies arisedue to the lack of equality (between dashed and dottedcurves in figures 4 and 5 ) between (38) and (46), whichleaves room for different interpretations of the different
40 50 60 70 80 90 EW -0.2 -0.2-0.1 -0.10 00.1 0.10.2 0.20.3 0.30.4 0.40.5 0.5 I R ( E ) x I R ( E ) FIG. 9: Here we are using the same parameters as in Fig.5.The solid curve shows 10x I R ( E ), the dashed line is for I R ( E ).Both of them get exactly zero at the resonance i.e., at EW =84 .
40 50 60 70 80 90 EW | t | FIG. 10: Here the energy dependence of | t | is shown,using the same variables as in Fig.5. times. So, negative times are also subject to interpreta-tions. Many of these times are experimentally accessible[26]. Only at EW = 39 .
51 when k →
0, the correctionterm disappear because Vk → ∞ (see Fig.1b)). Hence ifsomeone can do an experiment at this energy then thenegative times may be observed and experimental resultsare free of controversy. But at this point quite ironically, | t | → EW = 85 . EW = 84 .
29 in Fig.9), both the solidcurve and the dashed curve become zero, implying thereare time scales that go to zero at this energy. So thesame puzzling question can be asked again, as to howcan time scales be negative or zero? Unlike the low en-ergy regime, at this energy, the correction term is zeroimplying Wigner Smith delay times add up to give dwelltime, and also transmission is finite. We show the finite-ness of | t | , for same parameters as in Fig.5, in Fig.10.So there is no room for different interpretations and alsoone does not have to wait infinitely to observe a negativeor zero time delay. In fact | t | dθ t dE is strongly negativehere and so if one observes the transmission to the m = 2channel on the left to m = 2 channel on the right, thenone can see strongly negative delays. It should not bedifficult to separate such transmission processes from t transmission processes, as in the former case the trans-mitted wave packet move much slowly (being constructedfrom k instead of k ). Experimentalists may try to dosimilar experiments in this regime. Specially as one cansee, the transmissions at this energy is finite.Study of injectance of a quantum system coupled tofinite thickness leads has established many novel facts.First of all the correction term to injectance, obtained by [12] and [13], become zero very paradoxically in aquantum regime of a Fano resonance. For single channelleads this has been proved generally and for multichan-nel leads this has been shown for a delta function poten-tial. Thus, semi-classical formulas become exact at Fanoresonance. All the terms that appear in semi-classicalformulas can be determined experimentally from asymp-totic wave functions. Hence, although injectance stronglydepend on the scattering potential, can be determinedwithout knowing the scattering potential. Besides, thequantum interference term going to zero can unambigu-ously establish the possibility of negative time scales inquantum mechanics.The authors acknowledge useful discussions with Dr.Rajesh Parwani and DST for funding this research. ∗ Electronic address: [email protected] Yang Ji, M. Heiblum, D. Sprinzak, D. Mahalu, HadasShtrikman, Science , 779 (2000). R. Schuster et al. , Nature , 417 (1997). K. Kobayashi, H. Aikawa, S. Katsumoto and Y. Iye, Phys.Rev. B , 235304 (2003). K. Kobayashi, H. Aikawa, A. Sano, S. Katsumoto and Y.Iye, Phys. Rev. B , 035319 (2004). R. Englman and A. Yahalom, Phys. Rev. B , 2716(2000). P Singha Deo, J. Phys.Condens. Matter , 285303 (2009). H. W. Lee, Phys. Rev. Lett , 2358 (1999). M. B¨uttiker, H. Thomas, and A Pretre, Z. Phys B , 133(1994); M. B¨uttiker, Pramana Journal of Physics , 241(2002). E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, NewYork, 1997). J. M. Ziman, Principles of the Theory of Solids, 2nd ed.(Cambridge University Press, Cambridge, UK, 1979). S. Datta, Electronic transport in mesoscopic systems,(Cambridge University Press, Cambridge, UK, 1995). E. H. Hauge, J. P. Falck, and T. A. Fjeldly, Phys. Rev. B , 4203 (1987). C. R. Leavens and G. C. Aers, Phys. Rev. B , 1202(1989). Swarnali Bandopadhyay and P. Singha Deo, Phys. Rev. B ,2247 (2002). B. F. Bayman and C. J. Mehoke, Am. J. Phys. , 875(1983). P. F. Bagwell, Phys. Rev. B , 10354 (1990). H. Wu, D. W. L. Sprung and J. Martorell, Phys. Rev. B , 11960 (1992). A. L. Yeyati and M. B¨uttiker, Phys. Rev. B , 7307(2000), and references therein. T. Taniguchi and M. B¨uttiker, Phys. Rev. B 60, , 814(1999). F. T. Smith, Phys. Rev. B , 349 (1960). V. Gasparian et al. , Phys. Rev. A , 4022 (1996). H. M. Nussenzvig, Phys. Rev. A , 042107 (2000). C. Texier and M. B¨uttiker, Phys. Rev. B , 245410(2003). H. G. Winful, Phys. Rev. Lett , 260401 (2003). Time in Quantum Mechanics, edited by J. G. Muga, R.Sala Mayato, and I. L. Egusquiza (Springer, Berlin, 2002),and references therein. M. D. Stenner et al. , Nature425