Braess paradox at the mesoscopic scale
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Braess paradox at the mesoscopic scale
A. A. Sousa,
1, 2, ∗ Andrey Chaves, † G. A. Farias, and F. M. Peeters
2, 1, ‡ Departamento de F´ısica, Universidade Federal do Cear´a,Caixa Postal 6030, Campus do Pici, 60455-900 Fortaleza, Cear´a, Brazil Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium (Dated: August 16, 2018)We theoretically demonstrate that the transport inefficiency recently found experimentally forbranched-out mesoscopic networks can also be observed in a quantum ring of finite width with anattached central horizontal branch. This is done by investigating the time evolution of an electronwave packet in such a system. Our numerical results show that the conductivity of the ring does notnecessary improves if one adds an extra channel. This ensures that there exists a quantum analogueof the Braess Paradox, originating from quantum scattering and interference.
PACS numbers: 73.63.-b, 85.35.Ds, 73.63.Nm
I. INTRODUCTION
Suppose that two points A and B of a network areconnected only by two possible paths (e.g. roads in atraffic network, or wires in an electricity network). Onewould intuitively expect that adding to the network athird path connecting these two points would lead to animprovement of the flux through the pre-existing roadsand, consequently, to a transmission enhancement. How-ever, the so-called Braess paradox of games theorystates that this is not necessarily the case: under specificconditions, adding a third path to a network may leadto transport inefficiency instead. This effect has beeneven observed in traffic networks in big cities, where clos-ing roads improves the flux in traffic jams, or in elec-tricity networks, where it has been demonstrated thatadding extra power lines may lead to power outage, dueto desynchronization. A recent paper showed both experimental and theo-retical evidence of a very similar effect, but on a meso-scopic scale: they observed that branching out a meso-scopic network does not always improve the electronsconductance through the system. As they were deal-ing with a system consisting of wide transmission chan-nels, quantum interference effects are not expected to berelevant. In this paper, we demonstrate that the transport in-efficiency in branched out devices also occurs on a nanoscale, when only few sub-bands are involved, and trans-port is strongly influenced by quantum effects. For thispurpose, we investigate wave packet propagation througha circular quantum ring attached to input (left) and out-put (right) leads, in the presence of an extra chan-nel passing diametrically through the ring. Our resultsdemonstrate that increasing the extra channel width doesnot necessarily improve the overall current. The funda-mental reasons behind this effect, which are related toquantum scattering and interference, are discussed in de-tails in the following Sections. II. THEORETICAL MODEL
We consider an electron confined in a circular quantumring attached to input (left) and output (right) leads, in the presence of an extra channel passing diametricallythrough the ring, as sketched in Fig. 1(a). Both thering and the leads are assumed to have the same width W = 10 nm, whereas different values of the extra channelwidth W c are considered.As initial wave packet, we consider a plane wave withwave vector k = √ m e ǫ/ ~ , where ǫ is the energy and m e is the electron effective mass, multiplied by a Gaussianfunction in the x -direction, and by the ground state φ ( y )of the input channel in the y -direction,Ψ( x, y,
0) = exp (cid:20) ik x − ( x − x ) σ x (cid:21) φ ( y ) . (1)Several papers have reported calculations on wave packetpropagation in nanostructured systems, hence, anumber of numerical techniques for this kind of calcula-tion is available in the literature, such as the expansion ofthe time evolution operator in Chebyshev polynomials, and Crank-Nicolson based techniques. In the presentwork, the propagation of the wave packet in Eq. (1)is calculated by using the split-operator technique to perform successive applications of the time-evolutionoperator, i.e. Ψ( x, y, t + ∆ t ) = exp [ − iH ∆ t/ ~ ] Ψ( x, y, t ),where ∆ t is the time step. The Hamiltonian H is writtenwithin the effective mass approximation, describing anelectron constrained to move in the ( x, y )-plane and con-fined, by external potential barriers of height V , to moveinside the nanostructured region represented in gray inFig. 1(a), where the potential is set to zero. The interfacebetween the confinement region and the potential bar-rier is assumed to be abrupt. Nevertheless, consideringsmooth potential barriers would not affect the qualitativebehavior of the results to be presented here, since the ef-fect of such smooth interfaces has been demonstrated tobe mainly a shift on the eigenenergies of the system. The ( x, y )-plane is discretized in a ∆ x = ∆ y = 0 . FIG. 1: (a) Sketch of the system under investigation: aquantum ring with average radius R , attached to input (left)and output (right) channels with the same width as the ring( W = 10 nm ), and to an extra horizontal channel of width W c . (b) Contour plots of the transmission probabilities asa function of the extra channel width and ring radius. Thesolid, dashed and dotted lines indicate seven minima that arediscussed in the text. A zoom of the 4 nm < W c <
16 nmregion with the logarithm of the transmission is shown in (c). perform the derivatives coming from the kinetic energyterms of the Hamiltonian. Imaginary potentials areplaced on the edges of the input and output channels, inorder to absorb the propagated wave packet and avoidspurious reflection at the boundaries of the computa-tional box. As the wave packet propagates, we computethe probability density currents at the input and outputleads, which, when integrated in time, gives us the re-flection and transmission probabilities, respectively, fromwhich the conductance can be calculated.As the fabrication of InGaAs quantum ring structureshave already been reported in the literature, we assumethat the ring, channel and leads in our model are madeout of this material, so that the electron effective massis taken as m e = 0 . m . Nevertheless, the qualitativefeatures of the results presented in the following Sectiondoes not depend on specific material parameters. III. RESULTS AND DISCUSSION
Contour plots of the calculated transmission probabili-ties are shown in Fig. 1(b) as a function of the ring radius R and the width W c of the extra channel. Notice that theextra channel in the system is opened in the horizontaldirection, namely, parallel to the input and output leads,being practically just a continuation of these leads. Evenso, instead of improving the transmission, the existenceof such a channel surprisingly reduces the transmissionprobability for specific values of W c , leading to several minima in each curve. In what follows, we discuss theorigin of several of these minima, indicated by the solid,dashed and dotted curves in Fig. 1(b).The position of the minima labeled as 1, 2 and 3 inFigs. 1 (b,c) changes with the ring radius, which indi-cates that these minima are related to a path difference,i.e. to an interference effect. Let us provide other ar-guments to support this indication: in a very simplis-tic model, consider that part of the wave packet travelsthrough the central channel, while the other part passesthrough the ring arms. The latter runs a length ≈ πR while going from the input to output leads, whereas theformer runs through the 2 R diameter of the ring. Thecondition for destructive interference is: γ πRλ − R ¯ λ = n + 12 , (2)where λ = 2 π (cid:14)p m e ǫ/ ~ (¯ λ = 2 π (cid:14)p m e ( E − ¯ E j ) / ~ )is the wave length in the ring arms (extra channel), E i ( ¯ E j ) is the energy of the i -th ( j -th) eigenstate of the inputlead (extra channel), and E = ǫ + E i is the total energyof the wave packet. The parameter γ is close to one andaccounts for the fact that the effective arm length may beslightly different from πR [see Fig. 1(a)]. By substitutingthese expressions for λ and ¯ λ in Eq. (2), one obtains¯ E j = E − ~ π m e " γ r m e ~ ǫ − (cid:18) n + 12 (cid:19) R , (3)Hence, this equation gives the condition for the interfer-ence related minima in the transmission probability. Theextra channel eigenstates ¯ E j depend on W c - which canbe fairly well approximated by ¯ E j ≃ β/W . c for large W c (notice that the structure has finite potential barriers,therefore, the infinite square well relation ¯ E j ∝ /W c , isno longer valid). Therefore, the minima for large W c areexpected to occur for W ( n ) c = βE − ~ π m e h γ q m e ~ ǫ − (cid:0) n + (cid:1) R i / . , (4)which are shown in Fig. 1(b) by black dashed lines for n = 1 , γ = 0 . n = 0 minimum occurs outside of the investigatedrange W c . The wave packet in this case has total en-ergy E = 124 meV, with ǫ = 70 meV and E = 54meV (ground state of the W = 10 nm input lead). Forthe 26 nm < W c <
42 nm range in Fig. 1(b), the eigen-states of the channel, which are accessible by the electronwith this energy, are the ground state and the second ex-cited state. The first and third excited states, althoughstill having energy lower than 124 meV for this range of W c , are not accessible by the wave packet because of theeven symmetry of the initial wave packet with respect tothe x -axis, while these excited states of the channel are FIG. 2: Snapshot of the propagating wave function at t = 900fs for two values of the extra channel width: 19 nm (a) and20 nm (b). odd. Therefore, the part of the wave packet that goesthrough the central channel under these conditions pop-ulates mostly the second excited state, but has also someprojection on the ground state and none on the otherstates. The fitting of ¯ E j for the second excited state ( j = 2) has β ≈ . , which is the value usedin Eq. (4) to obtain the dashed curves in Fig. 1(b).The n = 1 , < W c <
15 nm in Fig. 1(b) can also be obtained from Eq. (4)but, since this is a lower W c range, the dependence of¯ E j on W c will have a different exponent, one needs toreplace 1.85 by 1.50 in Eq. (4). Besides, for such low W c , the wave function travels predominantly through theground state sub-band of the extra channel, so that onemust consider the j = 0 state of this channel, which has β ≈ .
99 meV nm . in this range. The results for thismodel are shown as black dotted lines in Fig. 1(b). Toshow these minima more clearly, we present in Fig. 1(c), a magnification of the logarithm of the probabilityin the low W c region. The numerically obtained minimaare well fitted by the model of Eq. (4) for γ = 0 .
925 with n = 1 , , ... W c the wave function inside the extra channel ispredominantly in its ground (second excited) state, Fig.2 shows a snapshot of the propagating wave function at -606121824 0 100 200 300 400036912 (a) W c = 0 nm W c = 2 nm W c = 5 nm W c = 7 nm W c = 10 nm J t ( - f s - ) Input lead (b) J t ( - f s - ) t (fs) Extra channel
FIG. 3: Probability density currents as a function of time,calculated (a) in the input lead and (b) in the extra channel,for different values of the extra channel width W c , for wavepacket energy ǫ = 70 meV. t = 900 fs for two values of the extra channel width: W c = 19 nm (a) and 20 nm (b). In the former case, the wavefunction inside the extra channel exhibits predominantlya single maximum peak around y = 0, which suggestsa large contribution of the ground state eigenfunctionin the wave packet within this region. Similar resultsare obtained for lower values of the channel width W c .However, the results for a slightly larger W c = 20 nmare qualitatively different, exhibiting three peaks alongthe y -direction inside the extra channel, which impliesa higher contribution of the second excited state on thewave function in this region.Differently from the other minima, the position of thefirst minimum M in Fig. 1(b) appears around W c = 5nm and does not change with the radius R . Therefore,this minimum cannot be related to the above discussedinterference effect. In order to understand the origin ofthe M minima, we show in Fig. 3 the integrated current J t in the input lead and the extra channel. Fig. 3(a)exhibits a high negative peak for W c = 2 nm and 5 nmat ≈
100 fs and ≈
140 fs, respectively, which represents astrong reflection of the wave packet at the ring - channeljunction. This is confirmed by the very low currentsobserved for these cases inside the extra channel, in Fig.3(b). On the other hand, for W c = 7 nm the reflec-tion peak in the input lead becomes very weak, while for W c = 10 nm, almost no reflection is observed. For thelatter two cases instead, large current peaks are observedinside the extra channel. This is a clear indication thatthe transmission inefficiency in the low W c case is notrelated to interference effects, but rather to scattering atthe ring-channel junction, since the wave packet barelyenters the extra channel when it is too narrow.We discuss now the possibility of having part of theincoming wave packet passing through a narrow extrachannel. Both the leads and the extra channel have dis-crete eigenstates due to the quantum well confinementin the y -direction, whose energy levels are shown in Fig.4(a) as a function of the well width. In the x -direction,parabolic sub-bands stem from these eigenstates, as illus-trated in Fig. 4(b). The incoming wave packet consid-ered in Figs. 1 - 3 has ǫ = 70 meV on top of its groundstate energy in the input lead, E = 54 meV (for W = 10nm). This energy is represented by the dotted horizontallines in Figs. 4(a) and (b). The wave packet has a Gaus-sian distribution of energies of width ∆ E = ~ /m e k ∆ k ,where ∆ k = 2 √ ln /σ x is the full width at half maxi-mum (FWHM) of the wave vector distribution, which isrepresented by the shaded area around the dotted linein Fig. 4(a). A narrow extra channel has a very highground state sub-band energy, so that no component ofthe incoming wave packet energy has enough energy topass through the channel. As the extra channel width W c increases, its sub-band energies decrease, allowing the in-coming wave packet to travel through this channel. Thesetwo situations are illustrated in the upper and lower fig-ures of Fig. 4(b), respectively. Notice that the upperboundary of the energy distribution (shaded area) in Fig.4(a) is crossed by the second excited state energy curve(blue triangles) approximately at W = 20 nm. This ex-plains the drastic difference between the wave functionswithin the extra channel with W c = 19 nm and 20 nm,observed in Fig. 2: in the latter case, the wave functionhas a significantly larger part of its energy distributionabove the second excited state energy, allowing it to havea larger projection on this state.Therefore, the counter-intuitive result observed inFigs. 1, namely, the transmission reduction as the ex-tra channel width increases for lower values of W c , is apure quantum scattering effect. For classical particles,such an extra channel with any width would allow thepassage of the particles and, consequently, improve thetransmission. However, a quantum channel has a con-finement energy (ground state) and, if the energy of theincoming particle is lower than this minimum, the par-ticle is not allowed to pass through the channel. There-fore, adding a narrow extra channel to the system, whicheffectively also adds extra scattering, does not add anextra path for the wave packet, because of the very highground state energy of the narrow channel. This mech-anism, which is illustrated by the band diagrams in Fig.4(b), leads to the strong reflections observed in Fig. 3for W c = 2 nm and 5 nm. For W c > E = 70 meV wave packet has enough energyto go through the extra channel, explaining the increas-ing transmission as W c increases above 5 nm. This alsosuggests that incoming wave packets with higher energywould need lower extra channel widths to pass, which isindeed observed, as we will discuss further on.In fact, the position of M strongly depends on the FIG. 4: (a) Eigenstates of a finite quantum well as a functionof its width. (b) Diagram representing the energy sub-bandsin the input lead and in the extra channel. The horizontaldotted line is the average energy of the wave packet usedin Figs. 1 and 3, and the shaded area in (a) illustrates theFWHM of the energy distribution of this wave packet. wave packet energy, as shown in Fig. 5(a), where thetransmission probability in the vicinity of M is plottedas a function of the channel width W c for several val-ues of the energy, ranging from 70 meV (bottom curve)to 120 meV (top curve), with 10 meV intervals. Thering radius is fixed as R = 60 nm, and each consecu-tive curve in this figure is shifted by 0.1. If the energydependence of the position of M is due to the above dis-cussed quantum effect, it should be possible to predictthe position of these minima from the following argu-ment: the highest energy components of the wave packethave energy around ≈ E + ∆ E/
2. These componentswould be allowed to pass through the extra channel, con-sequently improving the current, provided the channelwidth is wide enough to have a ground state energy aslow as their energy, i.e. if ¯ E < E + ∆ E/
2. For lowvalues of W c , the ground state energy of the channel iswell approximated by ¯ E = α/W . c , for α = 8 .
65 eV, asshown by the green dashed line ( f ( W ) function) in Fig.5(b). Notice it is a different power from the one used inEq. (4), which is valid only for higher W c values. Thered dotted line ( f ( W ) function) in Fig. 5(b) is an ex-ample of fitting for high values of W c , which was used inEq. (4). Figure 5(b) is in log-log scale, so that the powerlaws in f ( W ) and f ( W ) are shown as straight curves,whose slopes are the functions’ exponents. Using thisexpression for ¯ E j , one obtains the following approximateexpression for the position of the M minima W ( M ) c = 6103 (cid:16) ǫ + E + ~ q ǫ m e ∆ k (cid:17) / . , (5)which is shown by the solid curve in Fig. 5(c). Noticethe rather good agreement with the numerically obtainedpositions of the M minima, represented by the symbols.It is important to point out that the exponents 1.85,1.50 and 1.04, as well as the values of α and β , found for M FIG. 5: (a) Transmission probabilities as a function of theextra channel width in the vicinity of the minimum labeled as M in Fig. 1(a), for several values of the wave packet energy ǫ = 70 (bottom curve), 80, ... 180 meV (top curve). Thecurves were shifted 0.1 up from each other, in order to helpvisualization. (b) Energy levels (solid) as a function of thechannel width, plotted in a log scale, along with two fittingfunctions (dashed curves), for large ( f ) and small ( f ) valuesof the channel width. (c) Numerically obtained (symbols)positions of the M minima as a function of the wave packetenergy, along with the results (solid curve) of the analyticalmodel, given by Eq. (5). the fitting functions for the eigenstate energies as a func-tion of the well width and used in Eqs. (4) and (5), wereobtained for an abrupt interface between the potentialbarriers and the confining region. These values must beslightly modified in the case of smooth potential barriers.Our results, therefore, demonstrate that the M minimain Figs. 1 and 5 are a consequence of a competition be-tween two effects: (i) the quantum scattering in the ring-channel junction, which increases the reflection when anarrow extra channel is added, and (ii) the improvementin the transmission resulting from the part of the wavepacket that has enough energy to propagate through thesub-bands of the extra channel. The former suggests thatadding extra scatterers at the input lead-ring junctionleads to a larger reflection back into the input lead. Inorder to verify this, we consider two situations that mimicthe appearance of an extra “blind” channel (see insets ofFig. 6): one is the presence of an attractive Gaussianpotential V a ( x, y ) = − V G exp (cid:8) [( x − x g ) + y ] / σ G (cid:9) close to the lead-ring junction, and the other is a circularbump of radius R b in the inner boundary of the ring. Fig.6 shows the transmission probabilities for ǫ = 70 meV asa function of the Gaussian potential depth V G (bottomaxis) and the radius R b (top axis) of the circular bump.In both cases, the transmission is reduced in the presenceof the extra scatterer, which supports the idea that thetransmission reduction in the low W c range in Figs. 1 and 5 is indeed a consequence of extra scattering createdby the opening of the extra channel, which is howevereffectively blind, since the bottom of the ground statesub-band of the a narrow channel has energy higher thanthat of the incoming electron wave packet.All the results in this work were calculated for sharpconnections between the ring, the extra channel, and theinput and output leads. However, qualitatively similarresults are also obtained for smooth junctions betweenthese parts of the system. Moreover, different ring ge-ometries would shift the high W c minima, which are re-lated to quantum interference, by effectively changing theelectronic paths, while impurities in the ring could sup-press these minima, by destroying phase coherence. How-ever, neither impurities nor different ring geometries canaffect the low W c minimum ( M ), since it is related onlyto quantum scattering in the input lead-ring junction,which does not depend on these features.The original version of the Braess paradox, describedin details in Ref. 1, discusses how the travel time be-tween two points connected by only two possible roads,A and B, changes if these two roads are inter-connectedby a third road C. If one considers that the traffic at spe-cific parts of A and B depend on the number of drivers inthese roads, then, depending on the (partial) travel timethrough this new connection C, the dominant strategyturns out to consist in starting in one road and changingto the other road through the connection C, and there-fore, all players (drivers) would take this path. Thisstrategy, though leading to the Nash equilibrium situ-ation of this system, represents an increase in the traveltime - lower travel times could even be reached if thedrivers agree not to use the connection C a priori , butin a scenario of selfish drivers, they would switch roadsuntil the equilibrium is reached, despite the reductionin overall performance. Therefore, the classical Braessparadox is closely related to an unsuccessful attempt tooptimize the travel time through a traffic network by thedrivers. The transport properties of the branched outmesoscopic network investigated in Refs. is reminis-cent of those of the roads network in the original Braesspaper just in the sense that it exhibits a reduced overallcurrent when an extra channel is added to the network,depending on the channel width. However, the funda-mental reason behind this phenomenon is not clear inRefs. - it cannot be an interference effect, since thisis not a coherent system, but it is not also due to anoptimization of the currents, as in the classical paradox,since the model in these papers does not involve non-linear equations or iterative calculations of the overallcurrent flow. On the other hand, for the quantum caseinvestigated here, where such a transmission reductionin the presence of an extra channel is also observed, themain reason behind this Braess-like paradoxical behavioris quite clear: for small values of the channel width, itis due to quantum scattering effects at the ring-channeljunction, whereas for larger widths, it is due to interfer-ence effects. Therefore, if one includes the transmission T r a n s m i ss i on V G (meV)R b (nm) FIG. 6: Transmission probabilities for a ǫ = 70 meV wavepacket scattered by two kinds of defects in the lead-ring junc-tion: a Gaussian attractive potential of depth V G (solid, bot-tom axis) and width σ G = 5 nm, and a circular bump of radius R b (dashed, top axis), which are schematically illustrated inthe lower and upper insets, respectively. reduction phenomena described here into the categoryof analogs of Braess paradox, one must keep in mindthat, just like most of the other analogs suggested in theliterature , although presenting results similar to thoseof the original Braess network, in the sense that morepaths leads to reduced performance, the reason behindthis reduction is not related to an attempt to optimizethe flux, but to other fundamental physical properties ofthe investigated system. IV. CONCLUSIONS
We have investigated the propagation of a wave packetthrough a quantum ring with an extra channel along itsdiameter. Surprisingly, our results demonstrate that evenwhen an extra channel is added in the horizontal direc-tion, as a continuation of the input and output leads, thetransmission through the whole system can be lower thanin the absence of such an extra current path. This is ev-idence of the ”Braess paradox analog” observed recentlyfor mesoscopic networks. Nevertheless, while the originalBraess paradox in games theory is explained in terms ofan attempt to optimize the flux, which eventually leadsto transport inefficiency in the equilibrium situation, thetransport inefficiency observed for the wave packet prop-agation in quantum systems originates from two possibleeffects: (i) the quantum scattering of the wave packetin the input channel-ring junction, along with the ab-sence of an allowed energy sub-band for propagation inthe central channel when it is too narrow, and (ii) thequantum interference between parts of the wave packetsthat passed through the central channel and those thatpropagated through the rings arms.
Acknowledgments
This work was financially supported byPRONEX/CNPq/FUNCAP and the bilateral projectCNPq-FWO. Discussions with prof. J. S. Andrade Jr.are gratefully acknowledged. A. A. Sousa has beenfinancially supported by CAPES, under the PDSEcontract BEX 7177/13-5. ∗ Electronic address: ariel@fisica.ufc.br † Electronic address: andrey@fisica.ufc.br ‡ Electronic address: [email protected] D. Braess, A. Nagurney, and T. Wakolbinger, Transporta-tion Science , 446 (2005); ibid . Unternehmensforschung , 258 (1968). J. E. Cohen and F. P. Kelly, J. Appl. Prob. , 730 (1990). H. Lin, T. Roughgarden, ´E. Tardos, and A. Walkover, SiamJ. Discrete Math. 25, 1667 (2011); Y. Xia and D. J. Hill, IEEE Trans. Circuits and Systems-II: Express Briefs , 172 (2013). V. Zverovich and E. Avineri, arXiv:1207.3251 (2012); Seealso G. Kolata,
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