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Time dependence of transmission in semiconductor superlattices C. N. Veenstraa,b, W. van Dijka,b and D. W. L. Sprungb aRedeemer University College, Ancaster ON L9K 1J4, Canada bDepartment of Physics and Astronomy, McMaster University, Hamilton ON L8S 4M1 6 J. Martorell 0 Departament d’Estructura i Constituents de la Materia, Facultat F´ısica, University of Barcelona 0 Barcelona 08028, Spain 2 (February 2, 2008) n a Time delay in electron propagation through a finite periodic system such as a semiconductor J superlatticeisstudiedbydirectnumericalsolutionofthetime-dependentSchr¨odingerequation. We 2 compare systems with and without addition of an anti-reflection coating (ARC). With an ARC, 1 the time delay is consistent with propagation at the Bloch velocity of the periodic system, which significantly reduces thetime delay, in addition to increasing thetransmissivity. ] l l 03.65.Xp, 73.63.-b, 05.60.Cg a h I. INTRODUCTION involvesjusttworealparametersatgivenenergy. Oneof - s these is the Bloch phase, and the second, the impedance e Electron transport in a layered semiconductor super- parameter µ, is approximately constant over the mid- m lattice (SL) [1] can in many cases be treated as a one- dle of an allowed band, but diverges at the band edges. t. dimensional finite periodic system. With as few as N = Adoptingthe Kardparameterizationallowsonethe ben- a five cells, a well-developed band structure ensues [2]. efitofapowerfulanalogytotheprecessionofanelectron m Within each allowed band, where the Bloch phase φ in- spin in a magnetic field. The magnetic field direction - creases by π, the transmission shows narrow peaks de- has polar angle θ, where tanθ/2 = tanhµ/2. The angle d n termined by Nφ(E)=mπ, m=1,2,···N −1. Between of precession is twice the Bloch phase: 2φ. An electron o these peaks the transmission touches an envelope of the moving to the right corresponds to spin-up along OZ, c minima which also has a simple description. In the for- and left-moving waves to spin-down. PassingthroughN [ bidden bands, the Bloch phase φ acquires an imaginary identical cells therefore amounts to precession by angle 3 part; as a result the transmission goes rapidly to zero. 2Nφ; when this is an integer m multiple of 2π, the final v Pacher et al. [3] have used this property to design an electron state will be the same as the initial plane wave, 8 electron band-pass filter. Further, by adding a quarter- (except for a phase,) which gives perfect transmission. 1 wave cell at each end of the periodic array [4,5], they A Bloch state is one where the wave function on ei- 1 wereabletoincreasetheaveragetransmissionwithinthe ther side of the potential cell differs only by the Bloch 1 band from about 25% to about 75%. phase φ. This is analogous to an electron whose spin is 1 4 In this work we will study the time dependence of an aligned along the direction of the magnetic field. The 0 electronwavepacket passingthroughasuperlattice,com- action of an ARC can therefore be understood as taking / paring the situation with and without an anti-reflection an initial spin-up state and rotating it to lie along the t a coating(ARC).Inaseriesofpapers[6,7,8,9],someofthe field direction. That is, it converts a right-moving wave m presentauthorshaveshownhowtodesignanARCwhich into a Blochstate ofthe periodic array. Passingthrough - gives optimal transmission within a given miniband, by any number of cells only adds an overall phase Nφ, and d adding suitably configured potential cells on each end of the downstreamARC reversesthe alignmentbackto the n aperiodicarray. Thedesigndependsonthe shapeofthe spin-up state when the resonant condition holds. o potentialcellsmakinguptheperiodicarray,butnottheir Without an ARC, electrons which are transmitted do c : number. An r-cell ARC consists of r distinct potential so via narrowresonancesas mentioned above, so one ex- v cells on each end [8]. In the simplest case the periodic pects a significant time delay. Resonant states and their i X arrayconsistsofreflectionsymmetriccells; the ARC can characteristic time dependence [11,12] are key parts of r also be made reflection symmetric by using symmetric nuclear physics. Perhaps the definitive discussion of the a cells and reversing their order at the opposite end. timeevolutionofwavepacketsinthevicinityofresonant In the transfer matrix formalism, the electron wave states is that of Rosenfeld [13]. When the ratio Γ/Er of function at fixed position x is represented by two am- resonance width to energy is small, and the wave packet plitudes, which can be treated as a spinor. The upper isnarrowinenergycomparedtoΓ,thereisabroadrange componentcorrespondstoright-movingandthelowerto of intermediate times overwhich the system exhibits ex- left-movingwaves. Passingthoughanarbitrarypotential ponentialtimedependence,andthereforesignificanttime cellisdescribedbytheactionofatransfermatrixonthis delay compared to free propagation. The work of More spinor. For reflection symmetric cells, the transfer ma- andGerjuoy[14,15]shouldalsobementioned. Whilethe trix can be represented in the Kard form [8,9,10], which 3D system differs in important ways from 1D, the same 1 conclusion holds. The SL is an interesting system to the walls. In our work, transparentboundary conditions study precisely because placing N −1 resonances within remove such restrictions. The work closest to ours is by anallowedbandproducesresonanceswithwidthstheor- Pacher and Gornik [4,5], who considered the effect of an der of a few meV, satisfying one of Rosenfeld’s criteria. ARC on transmission through a superlattice, following WithanARC,propagationviaaBlochstateshouldbe Pereyra for the time evolution. much faster. To quantify this effect we have performed The numericalmethod we use was pioneered by Gold- numerical solutions of the time-dependent Schr¨odinger berg et al. [24], and greatly improvedby Moyer[25] who equation (TDSE). The subject of time delay in passing implementedtheNumerovalgorithmforthespatialvari- though a barrier is a controversial one (see the reviews ation and added transparent boundary conditions. The by Hauge and Støvneng [16], Leavens and Aers [17],and time-dependence is handled by the lowest-order Crank- de Carvalho and Nussenzveig [18] for example) but it Nicolson (implicit) method. The wave equation was fol- is not our purpose to debate the merits or demerits of lowed for typically 15 ps, on a region (the “system”) the various definitions that have been used (phase time, x <x<x of width 3 to 8µm. Transparent boundary L R dwell time, Larmor time, etc.) This controversy con- conditionswereespeciallyimportantinobtainingourre- tinues in the context of wanting to define a single time sults; they were applied at both ends of the system, so to characterize the process. Our approach is to solve thatasthewavefunctionreachesthoselimits,itwillpro- the time dependent Schr¨odinger equation directly. This ceedoutwardswithoutreflection. Byintegratingtheout- provides a direct means of comparing the two situations going flux at those boundaries, the reflection and trans- (with/without an ARC), while providing much more in- mission probabilities are accumulated. formation than a single number. The initial state is a gaussian wave packet sufficiently wide in real space to correspond to a small uncertainty σ in energy. For σ /E ∼ 3% the root mean square E E (rms) width σ is 140 nm; for σ /E ∼ 0.4% it is 1µm. X E 300 Themainrequirementonthesystemwidthx −x isto R L accommodate the initial wave packet, without overlap- 250 ping the potential array. Narrower (in energy) packets would require a wider system and more computer time 200 x) for the simulations. V( 150 100 II. THE CALCULATIONS 50 A. Wave packet transmission 0 -40 -30 -20 -10 0 10 20 30 40 x (nm) The potential array is located on a < x < b near the FIG.1. Five cell array with a single-cell ARC, comparing origin,veryclosetox . Forthis workweadaptedacase R theoriginal squarebarrier cells togaussian barrier cells. previously studied. It corresponds to a GaAs/AlGaAs superlatticeofPacheretal.,butwithfivebarriersrather There have been several papers which discuss prop- than the six of their device. Their modelling gave the agation in a superlattice without an ARC. Stamp and barriers a height of approximately 290 meV, and width McIntosh [19] made a careful study of the two-barrier 2.54 nm, separated by wells of width 6.50 nm. For con- case, which exhibits narrow quasi-bound state (QB) res- ∗ venience we used a constant effective mass m = 0.071 onances. Theiremphasiswasonthe roleofthewidth(in inalllayers. Further,to simplify the numericalwork,we energy)oftheincidentwavepacketonexcitingthequasi- replaced Pacher’s square barriers by equivalent gaussian bound states. They expanded the initial wave packet in shaped barriers. stationary states of the scattering problem and propa- gated the solution in time by quadratures. This method V(x)=V e−x2/(2w2), −d/2<x<d/2. (1) 0 is feasible for square barrier/well type potentials where the solutions are analytic. Pereyra [21] used a similar The gaussian barrier has height V = 270.084 meV and 0 approach for the superlattice problem. Bouchard and a width parameter w = 1.02 nm. The full cell width Luban[22]consideredelectronstrappedinaninfiniteSL, was set at d = 9.4 nm, only 4% wider than the square in presence of an external electric field. Their empha- barrier cell. These parameters were fitted to make the sis was on the Wannier-Stark ladder of states localized single-cell transfer matrices equivalent within the lowest by the applied field, and on finding Bloch oscillations. allowed band, which runs from 50 < E < 74 meV. The Their numerical method [23,24] is similar to ours in us- originaland the gaussianpotentials are shownin Fig. 1. ing the implicit method to propagate the state forward By equivalent, we mean that their scattering proper- intime. However,they confinedthe systeminabox and ties are accurately the same, across the allowed band of had to limit the total time so as to avoid reflection from interest. Since the transfer matrix for a symmetric cell 2 depends on only two Kard parameters, we need only fit tion(TDSE)wechosegaussianwavepackets. Theinitial these two, and the results are shown in Fig. 2. The state cosφ(E) are virtually identical for the original and the gaussian potential cell, as seen in Fig. 2(a). The third ψ(x,t=0)= 1 eik(x−x0)e−(x−x0)2/(4σX2) (2) (2πσ2 )1/4 (dotted)lineisthecosφ oftheoptimalsingle-cellARC. X A That gaussian has height VA = 135.64 meV and width is normalized to one particle. For a potential array sit- parameter wA = 0.98 nm, with a total ARC cell width ting near the origin, the initial position x0 has to be of 7.62 nm. Also µ(E) was very close for both cells: see takensufficientlynegativesothatthewavepacketiswell Fig. 2(b). (The lower line is µA for the ARC layer. For away from the periodic potential at time zero. For the a single-cell ARC the prescription is µA = 0.5µ at the more time consuming runs that followed the transmis- centre of the allowedband.) Having gaussianshape cells sion across the entire band, a width σ = 1000 nm was X allowed us to use the Numerov method without having used. When producing videos of the scattering process, to worry about points of discontinuity of the potential. a less demanding σ =140 nm was chosen. Most of the X work was done with x = −7500 nm to the left of the L potential array, and a gaussian wave packet centered at 2 themid-pointofthatrange. Forthewidestwavepackets employed, the amplitude of the wave at x and 0 was 1.5 L therefore 0.0297 for σ = 1µm, and was truncated to X 1 zero. The truncation introduces some high momentum 0.5 components which have a small effect on time develop- cos o / -0.05 mpcaaetpnioetrn,.wwHhoauidclhdwihesadvnoeoutbbelveeidnsibtahltee8dini·s1ta0an−ny7ceionsftatohm1ep5lgiµrtmaupd,hetshaenofdtrtunhnoist- at all discernible. The fractional standard deviation in -1 energy can be written as -1.5 4.24nm 60 meV -240 45 50 55 60 65 70 75 80 σE/E = σ r E . (3) a) E ( meV) X 5 1 4 0.8 3 n o 0.6 /u ssi mi 2 s an 0.4 Tr 1 0.2 0 40 45 50 55 60 65 70 75 80 0 b) E (meV) 45 50 55 60 65 70 75 80 Energy (meV) FIG.2. Comparisonof(a)Blochphasesofasquarebarrier FIG. 3. Transmission for 5-cell array, bare and with sin- cell (dash line) and our gaussian cell (solid line); also shown gle-cell and double-cell ARC(time-independentcalculation). is cosφA of a single-cell ARC (dotted line). (b) same for impedance parameters µ, µA. In Fig. 4 we show transmission for the gaussian ar- raycomputed usingthe TDSE.The right-movingflux at InFig. 3weshowthetransmissionprofilesofthe5-cell xR =x is SLwithnoARC(dottedline),asingle-cellARC(dashed ∗ ¯h dψ(x,t) dψ (x,t) ∗ line) and a two-cell ARC (solid line), for the gaussian j(x,t)= ψ (x,t) − ψ(x,t) (4) 2im m∗ dx dx cells array (computed using the usual time-independent e (cid:2) (cid:3) methods). Withthetwo-cellARC,thehightransmission and the integrated transmission is region runs from 53 to 68 meV, and the satellite peaks t are pushed closer to the band edges. This agrees well T(E ,t)= j(x,t)dt with p with the square barrier calculations in [6]. Z 0 For solution of the time-dependent Schr¨odinger equa- T (E )≡ lim T(E ,t) . (5) as p t→∞ p 3 The label E is mean energy of the wave packet. long square wave packet normalized to one particle, and p Corresponding expressions for the reflection probabil- we find similar oscillatory behaviour in the build-up of ity R(E ,t) hold with the left-moving flux monitored both |ψ(x,t)|2 and the flux, which is largely absent in p at x = x. For packets narrow in energy T the time-integrated flux. L as shouldapproachthetransmissionprobabilityofthetime- Fig. 5 shows similar results for the same array plus a independent solutions, and this is seen to occur. The single-cellARC. Againthe asymptotic transmissionis in main difference with respect to Fig. 3 is that the peaks goodagreementwithatime-independentcalculation,but are smeared by the finite width of the wave packet (ap- smearedbythefinitewidthofthewavepacket. Also,the proximately 0.5 meV). Even so, they are so narrow that build-up of the transmission profile proceeds smoothly the tops are ragged due to the finite steps in E of the with the above-noted velocity bias. p wave packets used to generate the figure. (With finer stepsandnarrower(inenergy)wavepackets,convergence 1 has been verified.) 0.9 0.8 1 0.7 0.9 n o 0.6 si 0.8 s mi 0.5 on 00..67 Trans 0.4 si 0.3 s mi 0.5 0.2 s Tran 0.4 0.1 0.3 0 45 50 55 60 65 70 75 80 0.2 Energy (meV) 0.1 FIG. 5. Transmission for gaussian array plus single-cell 0 ARC,showing development overthesame time intervals. 45 50 55 60 65 70 75 80 E (meV) FIG. 4. Transmission for gaussian array without ARC showing time development. Contours are for t = 2(1.5)14 140 ps. initial wave transmitted 120 reflected Also in Fig. 4, lines below the peaks show their build- sum R + T upovertime T(E,t); this proceedsuniformly,with some 100 bias to faster development at higher energies. This bias y 80 can be understoodas a simple velocity effect: the higher sit n energywavepacketstravelfasterandreachtherightwall nte 60 I x soonerthanthe slowermovingwavepacketsnearthe R 40 lower band edge. We note an apparentdiscrepancy with Fig. 4 of Romo [26]: at 0.6 ps he shows transmission in 20 the troughs exceeding the asymptotic (t ≥ 60 ps) trans- mission. However, as pointed out by an astuute reader, 0 47 48 49 50 51 52 53 54 55 Romo’s calculation is based on a very different initial E (meV) state than ours [27,28]. At time t = 0 his initial state FIG.6. Energycontentofthetransmitted,reflected,initial is a uniform standing wave confined by a mirror to the and final wave packetsnear the51 meV resonance. left of the array at x < a. The mirror is removed at t=0, and after an infinite time a new equilibrium state Near each band edge there is a satellite peak which is achieved which is the stationary scattering state for a resembles the narrow resonances of the periodic array. wave incident from the left. As the wave front advances To study the character of these lines we scanned over to the right, the leading edge becomes smeared out, and them using spatially narrow wave packets. We enlarged oscillations develop behind, a process which Moshinsky the system to the right, so that there was roomfor both [27] called diffraction in time. Ultimately the limit as the reflected and transmitted wave packets to become t → ∞ of |ψ(b,t)|2 should approach the time- indepen- wellseparatedfromthe array. We then Fourieranalysed denttransmissionprobabilityT(E). Thecurvesatinter- the reflected and transmitted wave packets individually. mediate times, at a fixed position, show the passing of TheFouriertransforms(modulisquared)R(E)andT(E) the wave front and the characteristic oscillations behind are plotted against energy in Fig. 6 and the results are it. They do not represent the accumulation of T(E ,t) p quite revealing. The T packet is very narrow compared as in our calculation. Indeed, we can calculate using a 4 to the incident wave, and centered just below 51 meV. positionwouldbeslightlydisplaced,correspondingtothe The R packet in contrast has a node at this energy, and shift of the outer peaks in Fig. 3. corresponds to the energies on either side. This shows An electron inside the array sees four potential wells that the transmission does proceed primarily through a between the five stronger barriers. The QB states are narrow resonance. The transform (mod squared) of the builtonthe groundstateineachwell. Throughcoupling complete packet after scattering, denoted “sum R+T” acrossthebarriersfourQBstatesarecreated,associated sits right on top of the curve for the initial wave packet, with the four transmission peaks seen in Fig. 4 for ex- which is a testament to the unitarity of the numerical ample. Fig. 8 shows that even with the ARC, the first work. of these states persists (as does the last). Rosenfeld [13] The time-development of the reflection and transmis- discussed in detail the conditions under which the expo- sion probabilities is shown in Fig. 7 for a narrow (in nentialdecayof suchstateswouldbe observed;the most energy) wave packet centered at E = 51±0.47 meV. important is that the ratio Γ /E << 1, which is well p n n T(t) and R(t) (see eq. 5) are accumulated as the waves satisfied for our superlattice. Stamp and McIntosh [19] exit from the right/left boundaries x ,x . T(t) begins emphasized another criterion, the “interaction time” of R L toriseat2ps;thiscorrespondstothetimetakenforpart the wave with the potential. This is basically the dwell oftheinitialpackettobetransmittedandreachx . The time inside the potential. If it is too long compared to R longertimescaleforR(t)isduetothewiderspaceonthe the lifetime of the QB state, the state will be decaying left which accommodated the initial wave packet. One continuously as it is being fed, and the probability will sees two steps in the R(t) curve, one corresponding to not be built up to a significant extent. For the state of direct reflection from the leading edge of the array, and Fig. 8, τ is of the same order as the time delay shown n the second one from entrapment before reflection. The in Fig. 10, below. sum R(t)+T(t) = 1 −P(t) is the complement of the probability P(t) remaining within the system (x ,x ): L R 1.4 Potential array t = 7.331 ps xR t = 7.998 ps P(t)= |ψ(x,t)|2dx . (6) 1.2 tt == 89..636351 ppss Z t = 9.998 ps xL 1.0 (A 60 meV electron with effective mass 0.071 travels at 0.8 545 nm/ps; the time offsets are consistent with this.) 0.6 1 P(t) 0.4 0.9 T(t) R(t) 0.8 0.2 0.7 0 0.6 -30 -20 -10 0 10 20 30 y x (nm) bilit 0.5 ba 0.4 FIG.8. Decay oftheQBstateat 50.8 meV.Thepotential o Pr 0.3 arraywithsingle-cellARC(solidline)isshownfororientation. 0.2 The fourth QB state also survives the addition of the 0.1 ARC, and implies a pole at 72−i0.767 meV. The draw- 0 ing would look very similar to Fig, 8, except that the -0.1 0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 wavefunctionhasnodes ineachbarrier. Thatis because Time (ps) the lowest state is nodeless, while the fourth state has FIG.7. Accumulated transmission (T), reflection (R) and alternating signs in the four wells. probability of remaining in thesystem (P) versus time, for a narrow wave packet at Ep =51±0.47 meV. III. TIME DELAY B. Quasi-Bound states A. Results from time-dependent calculations At times near 6 ps T(t) is approaching its asymptotic value T . Looking at the wave which is still trapped The primary motivation for this work was to compare as inside the array, one sees that it has a particular shape the time taken to traverse the array plus ARC, as com- (shown in Fig. 8) and is decaying with mean life τ = pared to the simple array. However, in the case with n 0.838(2)ps. Thiscorrespondstoahalf-widthΓ =0.786 ARC, the two satellite peaks appear to be narrow res- n meV.ItimpliesaquasiboundstatelocatedatE −iΓ /2 onances of the same type as for the simple periodic ar- n n in the lower half complex plane. Without the ARC, the ray. Hence it is also of interest to compare transmis- 5 sion through the satellite peaks with transmission via downward trend from left to right also agrees. Because the Bloch states of the broad central maximum. the Blochvelocity vanishes ata band edge, the curvedi- Fig. 7 shows that the build-up of the transmission vergesthere. Calculationswithtenandfifteencellsshow peakat51meVproceedslinearlyintime, overthe range that the satellite peaks move closer to the divergence at 0.2 < T(t) < 0.6. Similar linear behaviour applies the band edge. The delay at 51 meV, 1.6 ps, is about throughout the allowed band. In the absence of a po- double the lifetime of the quasibound state, but at 70 tentiala similarcurveis found, with some time displace- meV the two are about equal. Similar conclusions hold ment. By comparing the two, a reliable time delay (or whenatwo-cellARCisadded,butforbrevitywedonot advance) can be deduced which depends little on just include them here. which point is selected. For larger and smaller times the build-up is non-linear, and the converse is true. In Fig. 9 (a) we have plotted the ratio of T(t)/T B. Comparison with time-independent calculations as (solidline)asafunctionofenergyacrossthewholeband. The several lines correspond to different times. The in- Time delay can also be computed in the time- clined straight dotted lines are the same thing in the independent formalism. From the vast literature on this absence of any barriers (free propagation). In the back- subject, we base our discussionon relativelyrecent work groundasachainline,thetransmissionprobabilityT(E) of Nussenzveig [30,18]. He argues in favour of the “av- is plotted for reference. Except at the four transmission erage dwell time” as the most appropriate measure of peaks, the solid and dotted lines agree well. Also, the thetime takentopassabarrier,andlistsits advantages. rate of build-up is the same as in the absence ofthe bar- It is closely related to the “phase time” or transmission riers. Thatsaysthatinthetransmissiontroughs,almost groupdelayh¯dη˜/dE [31]originallyintroducedbyWigner nothing is being transmitted but it does so with no time and Eisenbud. Here η˜is the phase of the complex trans- delay. Under the four peaks, the build-up though still mission amplitude, closely related to the transfer matrix linear in time, is significantly retarded compared to free element by M =1/t, as we now explain. 11 propagation. The inclination of the dotted lines we in- In the transfer matrix method, our wave functions terpret as a velocity effect: the higher energy electrons are defined with a different phase than generally used: travel faster and arrive at xR sooner. namely the phase is setto zero oneachside (x=a, b)of To deduce a time delay, we take the difference be- the potential array, rather than at the origin. Adopting tweenthesolidanddottedlines,averagedoverthe range Nussenzveig’s notation for the asymptotic wave function 0.3 < T(t)/Tas < 0.7. This avoids using the non-linear to left and right, we compare our ψ(x) with the usual portion of the T(t) curve, as already discussed in con- convention ψ˜(x) as follows: nection with Fig. 7. It would make little difference had we simply takenthe time delay at ratio 0.5 . The result- ψ(x)∼[eik(x−a)+re−ik(x−a); teik(x−b)] ing time delay is plotted in Fig. 10 (a). This procedure ψ˜(x)∼[ eikx + r˜e−ikx ; t˜eikx] is consistent with the work of Dumont and Marchioro, who argue that for suitably chosen wave packets and by eikaψ(x)∼[eikx + re−ik(x−2a); teik(x−w)] , (7) measuring the flux of particles at x , one can define a R where w = b−a is the total width of the potential. It “tunneling time probability distribution”, which is ba- followsthatthe phaseη ofourtransmissionamplitude is sically the derivative of the integrated transmitted flux related to the usual one by η˜=η−kw. T(t) plotted in Fig. 7, but normalized to its asymptotic Then the phase time delay is value T . as Fig. 9 (b) shows the same ratio T(t)/T after adding as dη˜ dη dk a single-cell ARC. Except at the two satellite peaks, the τph =h¯ =h¯ −w , (8) dE dE dE solidanddottedlinesareroughlyparallel,butnottouch- h i ing, showing that there is a delay for propagation via a where η is the phase of our transmission amplitude and Bloch state. The delay is significantly greater under the w isthe widthofthe superlattice(nottheentiresystem) satellite peaks. The corresponding time delays are plot- under consideration. Since dE/dk is the velocity v of a f ted in Fig. 10 (b). Without an ARC, they go to zero free particle, the contribution w/v represents the time f (or evennegative)in the transmissionvalleys,but atthe taken to cross the superlattice assuming zero potential. peakstheyrangefrom0.75to2.6ps. Withasingle-layer Inviewofourearlierestimateforthefreevelocity,itisof ARC the delay is 0.25 ps over the middle of the band, the order of 0.1 ps. Thus, dη˜/dE gives phase time delay, jumping to 1.6 and 0.8 ps under the satellite peaks. So, while our dη/dE gives phase time. forthoseelectronsthataretransmitted,addingtheARC For the case of unilateral incidence on a symmetric cuts the time delay in half, while greatly increasing the potential, which we use, Nussenzweig arrives at his eq. average transmissivity. [Also shown in Fig. 10 (b) is a (20) for the mean dwell time delay as a spectral average simple estimate of the time delay for traversingfive cells over the phase time, plus an oscillatory term. In our attheBlochvelocity,ignoringanydelayintheARCcells. notation, his result can be written as in eq. A3 in the Itcanbeseenthatthisestimateistherightsize,andthe appendix. 6 In Fig. 10 we show the time delay extracted from our delaysdo not occur exactlyat the transmissionmaxima, time-dependent calculations, both before (a) and after they do lie close together for this type of potential cell, (b) a single cell ARC has been added. In both panels, and the difference would only be visible on a magnified the lower dashed line would be the phase time delay, drawing. The locus passes about 10% above the peaks, assuming that in the periodic arraythe wave propagates which we ascribe to two effects: (i) the finite steps in at the Bloch velocity of the infinite periodic system: meanwavepacketenergyinour calculations,whichmay miss the top,and(ii) the finite energywidth ofthe wave −dsinφ ∂E packetwhichsmearsouttheresultbasedonplanewaves, v = ≡d/τ , (9) Bl Bl ¯h ∂cosφ as in eq. 10. This is seen more clearly in Fig. 11. Inter- estingly,thetimedelaywithoutanARCoscillatesaround where d is the cell size. Within an allowedband, cosφ is the Bloch time delay in Fig. 10 (a); this is not partic- generally quite linear in E. ularly obvious from the present case of five cells, but if InFig. 10(b)the dash-dotline includes anestimate of one compares systems with ten to fifteen cells it is quite thetimedelayforpassingthroughtheARClayersaswell striking. as the five central cells (it is a 10% effect). This Bloch time-delay agrees quite well with the result of our time- dependent calculations, especially in its general trend. 3 1 ThisconfirmsourunderstandingoftheactionoftheARC Time Delay left scale layers: they convert the incident plane wave state into a right scale 2.5 Bloch state of the periodic system. The divergence at 0.75 the band edge arises because the Blochvelocity vanishes there. elay (ps)1.25 0.5mission D s 0 .19 Time 1 Tran 0.25 0.8 0.5 0.7 s 0.6 0 0 T / Ta 0.5 45 50 55 En6e0rgy (m6e5V) 70 75 80 0.4 a) 0.3 0.2 1.6 1 right s cale 0.1 left s cale 1.4 0 50 55 60 65 70 75 1.2 0.75 a) E (meV) 0 .19 elay (ps) .18 0.5mission D s 00..78 Time .6 Tran 0.4 0.25 s 0.6 Ta T / 0.5 0.2 0.4 0 0 0.3 45 50 55 60 65 70 75 80 Energy (meV) 0.2 0.1 b) 0 50 55 60 65 70 75 FIG.10. Time delay (solid line) for electrons transmitted b) E (meV) through(a)thefivebarrierperiodicsystemandalso(b)with single-layerARCadded. AlsoshownisthetransmissionT(E) FIG.9. Ratio T/Tas(E) with (solid line) and without po- (dotted line) for reference. tential (dotted line) for (a) five barrier periodic system and (b)withsingle-layerARCadded. Thecontoursareplottedat The locus of time delay at transmissionmaxima is de- intervals t = 2(1.5)14 ps. Also shown (dotted) is the trans- rived as follows: For a finite periodic system, we know mission T(E),for reference. that [2] In Fig. 10(a) the dash-dot line is the locus of phase 1 1 1 = sinNφ−sin(N −1)φ . (10) timedelayattransmissionmaxima. Whilemaximaltime t sinφ(cid:18)t (cid:19) N 7 We write t = |t|eiη and t = |t |eiηN for N cells. Thus similar results. They did not perform time-dependent N N η canbeexpressedintermsof|t|, ηandtheBlochphase calculations to establish the validity of the result. Fur- N φ. Using eq. (8) for the time delay we can similarly ex- thermore, in our reading, Pereyra’s derivation [21] as- press τ in terms of the single cell parameters. To look sumes that the reflection amplitude has a fixed modu- for maxima, we set the derivative with respect to energy lus, and only the phase is varying, which is obviously to be zero, and sinNφ = 0, because transmission max- questionable in the case of sharp resonances. After this ima occur when Nφ is an integer multiple of π (see first workwassubmitted,Pacheretal.[33]havealsodiscussed paragraphof this paper.) The result is phase time delay at the transmission maxima, deriving the result in eq. 11 and many others. dη Nd sinη N ¯h = =Nτ coshµ . (11) Bl dE (cid:12)max |t|vBl sinφ (cid:12) (cid:12) 4 1 whereτ is thetime takentocrossonecellatthe Bloch Time Delay Bl left scale velocity, and µ is the impedance parameter. The plot 3.5 right e n v eslcoaplee of coshµ in Fig. 2 explains the shape and height of the s) 3 0.75 curve immediately. p leaFdrionmg teoq.an1e1xpwreesssuiobntrfaocrttthheelloacsutstoerfmtimofe edqe.lay(8a)t, elay (2.25 0.5mission transmission maxima: e D ns m1.5 a Nd Ti Tr τ =Nτ coshµ− . (12) 1 0.25 loc Bl v free 0.5 Results using eq. (8) are in excellent agreement with 0 0 those of the time-dependent calculation, when allowance 45 50 55 60 65 70 75 80 Energy (meV) ismadeforaveragingoverenergyintheneighbourhoodof the sharp resonances. There the finite width (in energy) FIG.11. Time Delay (solid line) computed from phase of of our incident wavepacket mainly reduces the height of tN, and (dashed line) from the time-dependent calculation, the peaks of the time-independent result, by 5 to 10%, just visible below the peaks. The dash-dot line is the locus, acording to eq. (13). An example is shown in Fig. 11. eq. 12. Also shown for reference is the transmission (dotted In this drawing,we shouldhaveaveragedthe phase time line). over the spectral content of the initial state. We did not, but because our wave packets are narrow in energy, convolutiononlyreducestheheightsofthenarrowpeaks, IV. CONCLUSION which can be estimated from Time dependence of scattering from a finite periodic f(E) g(E)∼ . (13) potential array was studied by direct numerical solution 1+(2σ /Γ)2f(E) E ofthe Schr¨odingerequation. Itwasverifiedthatthenar- row transmission peaks are associated with quasi-bound This assumes that the unsmeared function f(E) is of statesofthearray. Theseresonancesentailtimedelaysof Breit-Wigner form with width Γ, and is wide compared order1to2ps,whileinthetransmissionminima thede- tothewidthinenergyσ ∼0.4meVofthewavepacket. E layvanishes. Uponadditionofananti-reflectioncoating, Eq. (13) agrees well with the reduction seen in Figs. 10 the broadcentraltransmissionmaximum correspondsto and 11, both for the time delay and transmission peaks. transmission via Bloch states, with a time delay of or- MostlyhiddenbelowthesolidlineinFig. 11isourtime- der 0.2 to 0.3 ps, as seen in Fig. 10 (b). The satellite delayfromFig.10(a). Onecanseethatexceptjustbelow peaks near the band edge continue to proceed through the peaks, the agreement is excellent. QB states, but even their time delay is cut roughly in Neither have we included the oscillatory term in the half, as compared to the bare periodic array. meandwelltimedelay,whicharisesfrominterferencebe- Our incident wave packets had widths in energy of or- tween the incident wave and the reflected wave [31,32]. der 0.4 meV; this was feasible due to the application of In our calculations there are indeed spectacular interfer- transparent boundary conditions at the edges of the re- ence effects within and to the left of the potential array, gion considered. A further improvement in the method asthereflectedwaveisgenerated. Butafterthereflected of solutionis possible, by goingfrom first-orderto third- waveiswellseparatedfromthepotentialandreachesthe order Crank-Nicolson integration for the time depen- counteratx ,theseoscillationshavedisappeared,anddo L dence [34,35]. This would match the truncation error not show up in the integrated flux. Neglect of this term, of the Numerov method, while greatly speeding up cal- in the present calculation, is justified in the appendix. culations. Systems under bias of an applied electric field PacherandGornik[4,5]havealsocomputedtunneling [36] can also be handled by this method. Extension to timesusingasimilarformulafollowingPereyra[20],with 8 two-dimensional systems is also under consideration. comparedto the wavepacket, and that leads to the con- clusion stated in eq. 13: the spectral average mainly reduces the height of each transmission peak, leaving its ACKNOWLEDGMENTS position and width unchanged. The second, oscillatory term, (OT), includes a factor We are grateful to NSERC-Canada for Discovery cos(2kxL −η(k)). Since our xL ∼ −7500 nm., this is Grants SAPIN-8672 (WvD), RGPIN-3198 (DWLS) and a very high frequency oscillation, given that we set the a Summer Research Award through Redeemer Univer- boundary xL far to the left of the potential. The re- sity College (CNV); and to DGES-Spain for continued flection amplitude |r(k)| varies on the same scale as the support through grants PB97-0915 and BFM2001-3710 phase shift. In doing the spectral average it is reason- (JM).WealsothankGigiWongforassistanceinredraw- able to treat the small prefactor h¯|r(k)|/E(k) ∼ 10 fs ing Figs. 10 and 11, and R. S. Dumont for illuminating asslowlyvaryingincomparisontothe cosine. Themean discussions. valueoftheOTcanbeestimatedbythemethodofsteep- est descents, leading to ¯h|r(k)| APPENDIX A: SPECTRAL AVERAGE OF THE <OT >∼− cos(2kx −η(k)) σ L X 2E(k) DWELL TIME DELAY ∞ 2 × exp[(i(q−k)(2x )−(q−k)22σ2 ]dq For our wave packet eq. 2, the Fourier transform is rπ Z−∞ L X ¯h|r(k)| ψ0(q)=[8πσX2 ]1/4e−iqx0e−(q−k)2σ2 , (A1) =−2E(k) cos(2kxL−η(k)) exp[−(xL/σX)2/2] . (A4) normalized according to By taking x sufficiently far to the left, the spectral av- L eragecanbemadeassmallasweplease. Mostlyweused ∞ |ψ (q)|2dq =1 . (A2) xL/σX =7.5,sotheexponentialfactorise−28,multiply- 0 Z−∞ 2π ing a term which is already small. The spectral weight function is therefore A(q) = |ψ (q)|2/(2π). Nussenzweig’s eq. (20) for the time de- 0 lay by a symmetric potential, with unilateral incidence, in our notation becomes ∞ <∆td >s→ymm= |ψ0(q)|2 1 A. Wacker, “Semiconductor superlattices, a model system Z−∞ for non-linear transport”, Phys.Repts. 357 (2002) 1-111. dη˜ ¯h|r(q)| dq 2 D.W.L. Sprung,HuaWu and J. Martorell, “Scattering by × ¯h − cos(2qx −η˜) . (A3) (cid:20) dE 2E(q) L (cid:21) 2π aFinitePeriodicPotential”,Am.J.Phys.61 (1993)1118- 24. The terms in square brackets are the dwell time delay 3 C.Pacher,C.Rauch,G.Strasser,E.Gornik,F.Elsholz,A. in the mono-energetic case. Razavy [32] for example de- Wacker, G. Kiesslich & E. Sch¨oll, “Anti Reflection Coat- rivedthem by following the method of Smith [31], albeit ing for miniband transport and Fabry-Perot resonances in withsometyposinhiseq. (18.19). (Onehastonotethat GaAs/AlGaAssuperlattices”, Appl.Phys.Lett.79(2001) 1486. forasymmetricpotentialhistwophaseshiftsarerelated 4 C. Pacher and E. Gornik, “Adjusting coherent transport by η =δ+π/2.) in finite periodic superlattices”, Phys. Rev. B 68 (2003) As stated earlier, we used a wave packet with a width 155319 (9 pp). σ of order 1000 nm. The width in energy is of order X 5 C.PacherandE.Gornik,“Tuningoftransmissionfunction 0.4 meV, which is small compared both to the width of and tunneling time in finite periodic potentials”, Physica the allowedband,25 meV,andevenofthe resonancesof E (Low-dimensional systems & nanostructures) (2004) 21 a finite periodic potential array, which are in the range 783-786. of 2 to 5 meV depending on the position in the band. 6 G.V.Morozov,D.W.L.SprungandJ.Martorell,“Optimal (Thenarroweststatesarethosecrowdedagainsttheband band-pass filter for electrons in semiconductor superlat- edge.) Itis reasonabletotreatA(q) asnarrowcompared tices” J. Phys. D 35 (2002) 2091-5. to the width of the peaks in transmission. 7 G. Morozov, D.W.L. Sprung and J. Martorell, “Design of The firstterm,dη/dE, variesonthe scaleofthe band- electronband-passfiltersforsemiconductorsuperlattices”, width divided by N, the number of cells, about 6 meV J. Phys. D 35 (2002) 3052-9. in our calculation. The slope of η is steepest at a reso- 8 D.W.L.Sprung,G.V.Morozov andJ.Martorell, “AntiRe- nance, where η =mπ, and is minimal at the mid-points flection coatings from the analogy between electron scat- between resonances. The phase shift η varies smoothly teringandspinprecession”,J.App.Phys.93(2003)4395- 4406. 9 9 D.W.L.Sprung,G.V.MorozovandJ.Martorell,“Geomet- Phys.Rev. B 71 (2005) 125317. (11 pp.) rical approach toscattering inonedimension”, J.Phys.A 34 I.V.Puzynin,A.V.SelinandS.I.Vinitsky,“Highaccuracy 37 (2004) 1861-80. methodfornumericalsolutionoftheTDSE”,Comp.Phys. 10 P. G.Kard, “Analytictheory of optical properties of mul- Comm. 123 (1999) 1-6. tilayer coatings”, Optikai Spektr.2(1957) 236-44. 35 I.V. Puzynin, A.V. Selin and S.I. Vinitsky, “Magnus- 11 P.L.KapurandR.E.Peierls,“Dispersiontheoryofnuclear factorizationmethodfornumericalsolutionoftheTDSE”, reactions”, Proc. Roy.Soc. A166 (1938) 273-306. Comp. Phys. Comm. 126 (2000) 158-61. 12 A.J.F.Siegert,“Dispersion formulafornuclearreactions”, 36 J.Martorell,D.W.L.SprungandG.V.Morozov,“Electron Phys. Rev.56 (1939) 750-2. band-passfiltersforbiasedfinitesuperlattices”,Phys.Rev. 13 L. Rosenfeld, “Time evolution of the scattering process”, B 69 (2004) 115309 (10 pp). Nucl. Phys.70 (1965) 1-27. 14 R. M. More, “Theory of decaying states”, Phys. Rev.A 4 (1971) 1782-90. 15 R. M. More and E. Gerjuoy, “Properties of resonant wave functions”, Phys.Rev. A 7 (1973) 1288-1303. 16 “Tunnelingtimes: acriticalreview”, E.H.HaugeandJ.A. Støvneng,Rev. Mod. Phys. 61 (1989) 917-36. 17 “Bohmtrajectoriesandthetunnelingtimeproblem”,C.R. LeavensandG.C.Aers,inScanningTunnelingmicroscopy III, R. Weisendanger and H.-J. Gu¨ntherodt, eds., second edition, Springer series in Surface Sciences no. 29, (1996); Ch. 6 pp.106-140. 18 C.A.A.deCarvalhoandH.M.Nussenzveig,“TimeDelay”, Phys. Repts.364 (2002) 83-174. 19 A.P. Stamp and G.C. McIntosh, “Time-dependent study of resonant tunneling through a double barrier”, Am. J. Phys. 64 (1996) 264-76. 20 P. Pereyra, “Closed formulae for tunneling time in super- lattices”, Phys.Rev.Lett. 84 (2000) 1772-5. 21 H.P. Simanjuntak and P.Pereyra, “Evolution and tunnel- ing time of electron wave packetsthrough a superlattice”, Phys. Rev.B 67 (2003) 045301 (7 pp). 22 A.M. Bouchard and Marshall Luban, “Bloch oscillations and other dynamical phenomena of electrons in semicon- ductorsuperlattices”, Phys. Rev.B 52 (1995) 5105-23. 23 S. E. Koonin, “Computational Physics”, Benjamin- Cummings, Menlo Park CA, (1986); Sec. 7.5 . 24 A. Goldberg, H.M. Schey and J.L. Schwartz, “Computer generated movies of 1D scattering phenomena”, Am. J. Phys. 35 (1967) 177-86. 25 C.A.Moyer,“Numerovextensionoftransparentboundary conditions for the 1D wave equation”, Am. J. Phys. 72 (2004) 351-8. 26 R. Romo, “Buildup dynamics of transmission resonances in superlattices”, Phys. Rev.B 66 (2002) 245311 (9 pp). 27 M.Moshinsky,“Diffractionintime”,Phys.Rev.88(1952) 625-31. 28 G.Garc´ıa-Calder´on, andAlbertoRubio“Transient effects and delay time in the dynamics of resonant tunneling”, Phys. Rev.A 55, (1997) 3361-3370. 29 Randall S. Dumont and T.L. Marchioro II, Phys. Rev. A 47 (1993) 85-97. 30 H.M. Nussenzveig, “Average dwell time and tunneling”, Phys. Rev.A 62 (2000) 042107 (5 pp). 31 F.T. Smith, “Lifetime matrix in collision theory”, Phys. Rev.118 (1960) 349-56; Erratum 119 2098. 32 M. Razavy,“Quantumtheory of tunneling”, World Scien- tific, Singapore, (2003); Ch. 18. 33 C. Pacher, W. Boxleitner,and E. Gornik, “Coherent reso- nanttunnelingtimeandvelocityinfiniteperiodicsystems” 10

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Preview Time dependence of transmission in semiconductor superlattices

Time dependence of transmission in semiconductor superlattices C. N. Veenstraa,b, W. van Dijka,b and D. W. L. Sprungb aRedeemer University College, Ancaster ON L9K 1J4, Canada bDepartment of Physics and Astronomy, McMaster University, Hamilton ON L8S 4M1 6 J. Martorell 0 Departament d’Estructura i Constituents de la Materia, Facultat F´ısica, University of Barcelona 0 Barcelona 08028, Spain 2 (February 2, 2008) n a Time delay in electron propagation through a finite periodic system such as a semiconductor J superlatticeisstudiedbydirectnumericalsolutionofthetime-dependentSchr¨odingerequation. We 2 compare systems with and without addition of an anti-reflection coating (ARC). With an ARC, 1 the time delay is consistent with propagation at the Bloch velocity of the periodic system, which significantly reduces thetime delay, in addition to increasing thetransmissivity. ] l l 03.65.Xp, 73.63.-b, 05.60.Cg a h I. INTRODUCTION involvesjusttworealparametersatgivenenergy. Oneof - s these is the Bloch phase, and the second, the impedance e Electron transport in a layered semiconductor super- parameter µ, is approximately constant over the mid- m lattice (SL) [1] can in many cases be treated as a one- dle of an allowed band, but diverges at the band edges. t. dimensional finite periodic system. With as few as N = Adoptingthe Kardparameterizationallowsonethe ben- a five cells, a well-developed band structure ensues [2]. efitofapowerfulanalogytotheprecessionofanelectron m Within each allowed band, where the Bloch phase φ in- spin in a magnetic field. The magnetic field direction - creases by π, the transmission shows narrow peaks de- has polar angle θ, where tanθ/2 = tanhµ/2. The angle d n termined by Nφ(E)=mπ, m=1,2,···N −1. Between of precession is twice the Bloch phase: 2φ. An electron o these peaks the transmission touches an envelope of the moving to the right corresponds to spin-up along OZ, c minima which also has a simple description. In the for- and left-moving waves to spin-down. PassingthroughN [ bidden bands, the Bloch phase φ acquires an imaginary identical cells therefore amounts to precession by angle 3 part; as a result the transmission goes rapidly to zero. 2Nφ; when this is an integer m multiple of 2π, the final v Pacher et al. [3] have used this property to design an electron state will be the same as the initial plane wave, 8 electron band-pass filter. Further, by adding a quarter- (except for a phase,) which gives perfect transmission. 1 wave cell at each end of the periodic array [4,5], they A Bloch state is one where the wave function on ei- 1 wereabletoincreasetheaveragetransmissionwithinthe ther side of the potential cell differs only by the Bloch 1 band from about 25% to about 75%. phase φ. This is analogous to an electron whose spin is 1 4 In this work we will study the time dependence of an aligned along the direction of the magnetic field. The 0 electronwavepacket passingthroughasuperlattice,com- action of an ARC can therefore be understood as taking / paring the situation with and without an anti-reflection an initial spin-up state and rotating it to lie along the t a coating(ARC).Inaseriesofpapers[6,7,8,9],someofthe field direction. That is, it converts a right-moving wave m presentauthorshaveshownhowtodesignanARCwhich into a Blochstate ofthe periodic array. Passingthrough - gives optimal transmission within a given miniband, by any number of cells only adds an overall phase Nφ, and d adding suitably configured potential cells on each end of the downstreamARC reversesthe alignmentbackto the n aperiodicarray. Thedesigndependsonthe shapeofthe spin-up state when the resonant condition holds. o potentialcellsmakinguptheperiodicarray,butnottheir Without an ARC, electrons which are transmitted do c : number. An r-cell ARC consists of r distinct potential so via narrowresonancesas mentioned above, so one ex- v cells on each end [8]. In the simplest case the periodic pects a significant time delay. Resonant states and their i X arrayconsistsofreflectionsymmetriccells; the ARC can characteristic time dependence [11,12] are key parts of r also be made reflection symmetric by using symmetric nuclear physics. Perhaps the definitive discussion of the a cells and reversing their order at the opposite end. timeevolutionofwavepacketsinthevicinityofresonant In the transfer matrix formalism, the electron wave states is that of Rosenfeld [13]. When the ratio Γ/Er of function at fixed position x is represented by two am- resonance width to energy is small, and the wave packet plitudes, which can be treated as a spinor. The upper isnarrowinenergycomparedtoΓ,thereisabroadrange componentcorrespondstoright-movingandthelowerto of intermediate times overwhich the system exhibits ex- left-movingwaves. Passingthoughanarbitrarypotential ponentialtimedependence,andthereforesignificanttime cellisdescribedbytheactionofatransfermatrixonthis delay compared to free propagation. The work of More spinor. For reflection symmetric cells, the transfer ma- andGerjuoy[14,15]shouldalsobementioned. Whilethe trix can be represented in the Kard form [8,9,10], which 3D system differs in important ways from 1D, the same 1 conclusion holds. The SL is an interesting system to the walls. In our work, transparentboundary conditions study precisely because placing N −1 resonances within remove such restrictions. The work closest to ours is by anallowedbandproducesresonanceswithwidthstheor- Pacher and Gornik [4,5], who considered the effect of an der of a few meV, satisfying one of Rosenfeld’s criteria. ARC on transmission through a superlattice, following WithanARC,propagationviaaBlochstateshouldbe Pereyra for the time evolution. much faster. To quantify this effect we have performed The numericalmethod we use was pioneered by Gold- numerical solutions of the time-dependent Schr¨odinger berg et al. [24], and greatly improvedby Moyer[25] who equation (TDSE). The subject of time delay in passing implementedtheNumerovalgorithmforthespatialvari- though a barrier is a controversial one (see the reviews ation and added transparent boundary conditions. The by Hauge and Støvneng [16], Leavens and Aers [17],and time-dependence is handled by the lowest-order Crank- de Carvalho and Nussenzveig [18] for example) but it Nicolson (implicit) method. The wave equation was fol- is not our purpose to debate the merits or demerits of lowed for typically 15 ps, on a region (the “system”) the various definitions that have been used (phase time, x <x<x of width 3 to 8µm. Transparent boundary L R dwell time, Larmor time, etc.) This controversy con- conditionswereespeciallyimportantinobtainingourre- tinues in the context of wanting to define a single time sults; they were applied at both ends of the system, so to characterize the process. Our approach is to solve thatasthewavefunctionreachesthoselimits,itwillpro- the time dependent Schr¨odinger equation directly. This ceedoutwardswithoutreflection. Byintegratingtheout- provides a direct means of comparing the two situations going flux at those boundaries, the reflection and trans- (with/without an ARC), while providing much more in- mission probabilities are accumulated. formation than a single number. The initial state is a gaussian wave packet sufficiently wide in real space to correspond to a small uncertainty σ in energy. For σ /E ∼ 3% the root mean square E E (rms) width σ is 140 nm; for σ /E ∼ 0.4% it is 1µm. X E 300 Themainrequirementonthesystemwidthx −x isto R L accommodate the initial wave packet, without overlap- 250 ping the potential array. Narrower (in energy) packets would require a wider system and more computer time 200 x) for the simulations. V( 150 100 II. THE CALCULATIONS 50 A. Wave packet transmission 0 -40 -30 -20 -10 0 10 20 30 40 x (nm) The potential array is located on a < x < b near the FIG.1. Five cell array with a single-cell ARC, comparing origin,veryclosetox . Forthis workweadaptedacase R theoriginal squarebarrier cells togaussian barrier cells. previously studied. It corresponds to a GaAs/AlGaAs superlatticeofPacheretal.,butwithfivebarriersrather There have been several papers which discuss prop- than the six of their device. Their modelling gave the agation in a superlattice without an ARC. Stamp and barriers a height of approximately 290 meV, and width McIntosh [19] made a careful study of the two-barrier 2.54 nm, separated by wells of width 6.50 nm. For con- case, which exhibits narrow quasi-bound state (QB) res- ∗ venience we used a constant effective mass m = 0.071 onances. Theiremphasiswasonthe roleofthewidth(in inalllayers. Further,to simplify the numericalwork,we energy)oftheincidentwavepacketonexcitingthequasi- replaced Pacher’s square barriers by equivalent gaussian bound states. They expanded the initial wave packet in shaped barriers. stationary states of the scattering problem and propa- gated the solution in time by quadratures. This method V(x)=V e−x2/(2w2), −d/2<x<d/2. (1) 0 is feasible for square barrier/well type potentials where the solutions are analytic. Pereyra [21] used a similar The gaussian barrier has height V = 270.084 meV and 0 approach for the superlattice problem. Bouchard and a width parameter w = 1.02 nm. The full cell width Luban[22]consideredelectronstrappedinaninfiniteSL, was set at d = 9.4 nm, only 4% wider than the square in presence of an external electric field. Their empha- barrier cell. These parameters were fitted to make the sis was on the Wannier-Stark ladder of states localized single-cell transfer matrices equivalent within the lowest by the applied field, and on finding Bloch oscillations. allowed band, which runs from 50 < E < 74 meV. The Their numerical method [23,24] is similar to ours in us- originaland the gaussianpotentials are shownin Fig. 1. ing the implicit method to propagate the state forward By equivalent, we mean that their scattering proper- intime. However,they confinedthe systeminabox and ties are accurately the same, across the allowed band of had to limit the total time so as to avoid reflection from interest. Since the transfer matrix for a symmetric cell 2 depends on only two Kard parameters, we need only fit tion(TDSE)wechosegaussianwavepackets. Theinitial these two, and the results are shown in Fig. 2. The state cosφ(E) are virtually identical for the original and the gaussian potential cell, as seen in Fig. 2(a). The third ψ(x,t=0)= 1 eik(x−x0)e−(x−x0)2/(4σX2) (2) (2πσ2 )1/4 (dotted)lineisthecosφ oftheoptimalsingle-cellARC. X A That gaussian has height VA = 135.64 meV and width is normalized to one particle. For a potential array sit- parameter wA = 0.98 nm, with a total ARC cell width ting near the origin, the initial position x0 has to be of 7.62 nm. Also µ(E) was very close for both cells: see takensufficientlynegativesothatthewavepacketiswell Fig. 2(b). (The lower line is µA for the ARC layer. For away from the periodic potential at time zero. For the a single-cell ARC the prescription is µA = 0.5µ at the more time consuming runs that followed the transmis- centre of the allowedband.) Having gaussianshape cells sion across the entire band, a width σ = 1000 nm was X allowed us to use the Numerov method without having used. When producing videos of the scattering process, to worry about points of discontinuity of the potential. a less demanding σ =140 nm was chosen. Most of the X work was done with x = −7500 nm to the left of the L potential array, and a gaussian wave packet centered at 2 themid-pointofthatrange. Forthewidestwavepackets employed, the amplitude of the wave at x and 0 was 1.5 L therefore 0.0297 for σ = 1µm, and was truncated to X 1 zero. The truncation introduces some high momentum 0.5 components which have a small effect on time develop- cos o / -0.05 mpcaaetpnioetrn,.wwHhoauidclhdwihesadvnoeoutbbelveeidnsibtahltee8dini·s1ta0an−ny7ceionsftatohm1ep5lgiµrtmaupd,hetshaenofdtrtunhnoist- at all discernible. The fractional standard deviation in -1 energy can be written as -1.5 4.24nm 60 meV -240 45 50 55 60 65 70 75 80 σE/E = σ r E . (3) a) E ( meV) X 5 1 4 0.8 3 n o 0.6 /u ssi mi 2 s an 0.4 Tr 1 0.2 0 40 45 50 55 60 65 70 75 80 0 b) E (meV) 45 50 55 60 65 70 75 80 Energy (meV) FIG.2. Comparisonof(a)Blochphasesofasquarebarrier FIG. 3. Transmission for 5-cell array, bare and with sin- cell (dash line) and our gaussian cell (solid line); also shown gle-cell and double-cell ARC(time-independentcalculation). is cosφA of a single-cell ARC (dotted line). (b) same for impedance parameters µ, µA. In Fig. 4 we show transmission for the gaussian ar- raycomputed usingthe TDSE.The right-movingflux at InFig. 3weshowthetransmissionprofilesofthe5-cell xR =x is SLwithnoARC(dottedline),asingle-cellARC(dashed ∗ ¯h dψ(x,t) dψ (x,t) ∗ line) and a two-cell ARC (solid line), for the gaussian j(x,t)= ψ (x,t) − ψ(x,t) (4) 2im m∗ dx dx cells array (computed using the usual time-independent e (cid:2) (cid:3) methods). Withthetwo-cellARC,thehightransmission and the integrated transmission is region runs from 53 to 68 meV, and the satellite peaks t are pushed closer to the band edges. This agrees well T(E ,t)= j(x,t)dt with p with the square barrier calculations in [6]. Z 0 For solution of the time-dependent Schr¨odinger equa- T (E )≡ lim T(E ,t) . (5) as p t→∞ p 3 The label E is mean energy of the wave packet. long square wave packet normalized to one particle, and p Corresponding expressions for the reflection probabil- we find similar oscillatory behaviour in the build-up of ity R(E ,t) hold with the left-moving flux monitored both |ψ(x,t)|2 and the flux, which is largely absent in p at x = x. For packets narrow in energy T the time-integrated flux. L as shouldapproachthetransmissionprobabilityofthetime- Fig. 5 shows similar results for the same array plus a independent solutions, and this is seen to occur. The single-cellARC. Againthe asymptotic transmissionis in main difference with respect to Fig. 3 is that the peaks goodagreementwithatime-independentcalculation,but are smeared by the finite width of the wave packet (ap- smearedbythefinitewidthofthewavepacket. Also,the proximately 0.5 meV). Even so, they are so narrow that build-up of the transmission profile proceeds smoothly the tops are ragged due to the finite steps in E of the with the above-noted velocity bias. p wave packets used to generate the figure. (With finer stepsandnarrower(inenergy)wavepackets,convergence 1 has been verified.) 0.9 0.8 1 0.7 0.9 n o 0.6 si 0.8 s mi 0.5 on 00..67 Trans 0.4 si 0.3 s mi 0.5 0.2 s Tran 0.4 0.1 0.3 0 45 50 55 60 65 70 75 80 0.2 Energy (meV) 0.1 FIG. 5. Transmission for gaussian array plus single-cell 0 ARC,showing development overthesame time intervals. 45 50 55 60 65 70 75 80 E (meV) FIG. 4. Transmission for gaussian array without ARC showing time development. Contours are for t = 2(1.5)14 140 ps. initial wave transmitted 120 reflected Also in Fig. 4, lines below the peaks show their build- sum R + T upovertime T(E,t); this proceedsuniformly,with some 100 bias to faster development at higher energies. This bias y 80 can be understoodas a simple velocity effect: the higher sit n energywavepacketstravelfasterandreachtherightwall nte 60 I x soonerthanthe slowermovingwavepacketsnearthe R 40 lower band edge. We note an apparentdiscrepancy with Fig. 4 of Romo [26]: at 0.6 ps he shows transmission in 20 the troughs exceeding the asymptotic (t ≥ 60 ps) trans- mission. However, as pointed out by an astuute reader, 0 47 48 49 50 51 52 53 54 55 Romo’s calculation is based on a very different initial E (meV) state than ours [27,28]. At time t = 0 his initial state FIG.6. Energycontentofthetransmitted,reflected,initial is a uniform standing wave confined by a mirror to the and final wave packetsnear the51 meV resonance. left of the array at x < a. The mirror is removed at t=0, and after an infinite time a new equilibrium state Near each band edge there is a satellite peak which is achieved which is the stationary scattering state for a resembles the narrow resonances of the periodic array. wave incident from the left. As the wave front advances To study the character of these lines we scanned over to the right, the leading edge becomes smeared out, and them using spatially narrow wave packets. We enlarged oscillations develop behind, a process which Moshinsky the system to the right, so that there was roomfor both [27] called diffraction in time. Ultimately the limit as the reflected and transmitted wave packets to become t → ∞ of |ψ(b,t)|2 should approach the time- indepen- wellseparatedfromthe array. We then Fourieranalysed denttransmissionprobabilityT(E). Thecurvesatinter- the reflected and transmitted wave packets individually. mediate times, at a fixed position, show the passing of TheFouriertransforms(modulisquared)R(E)andT(E) the wave front and the characteristic oscillations behind are plotted against energy in Fig. 6 and the results are it. They do not represent the accumulation of T(E ,t) p quite revealing. The T packet is very narrow compared as in our calculation. Indeed, we can calculate using a 4 to the incident wave, and centered just below 51 meV. positionwouldbeslightlydisplaced,correspondingtothe The R packet in contrast has a node at this energy, and shift of the outer peaks in Fig. 3. corresponds to the energies on either side. This shows An electron inside the array sees four potential wells that the transmission does proceed primarily through a between the five stronger barriers. The QB states are narrow resonance. The transform (mod squared) of the builtonthe groundstateineachwell. Throughcoupling complete packet after scattering, denoted “sum R+T” acrossthebarriersfourQBstatesarecreated,associated sits right on top of the curve for the initial wave packet, with the four transmission peaks seen in Fig. 4 for ex- which is a testament to the unitarity of the numerical ample. Fig. 8 shows that even with the ARC, the first work. of these states persists (as does the last). Rosenfeld [13] The time-development of the reflection and transmis- discussed in detail the conditions under which the expo- sion probabilities is shown in Fig. 7 for a narrow (in nentialdecayof suchstateswouldbe observed;the most energy) wave packet centered at E = 51±0.47 meV. important is that the ratio Γ /E << 1, which is well p n n T(t) and R(t) (see eq. 5) are accumulated as the waves satisfied for our superlattice. Stamp and McIntosh [19] exit from the right/left boundaries x ,x . T(t) begins emphasized another criterion, the “interaction time” of R L toriseat2ps;thiscorrespondstothetimetakenforpart the wave with the potential. This is basically the dwell oftheinitialpackettobetransmittedandreachx . The time inside the potential. If it is too long compared to R longertimescaleforR(t)isduetothewiderspaceonthe the lifetime of the QB state, the state will be decaying left which accommodated the initial wave packet. One continuously as it is being fed, and the probability will sees two steps in the R(t) curve, one corresponding to not be built up to a significant extent. For the state of direct reflection from the leading edge of the array, and Fig. 8, τ is of the same order as the time delay shown n the second one from entrapment before reflection. The in Fig. 10, below. sum R(t)+T(t) = 1 −P(t) is the complement of the probability P(t) remaining within the system (x ,x ): L R 1.4 Potential array t = 7.331 ps xR t = 7.998 ps P(t)= |ψ(x,t)|2dx . (6) 1.2 tt == 89..636351 ppss Z t = 9.998 ps xL 1.0 (A 60 meV electron with effective mass 0.071 travels at 0.8 545 nm/ps; the time offsets are consistent with this.) 0.6 1 P(t) 0.4 0.9 T(t) R(t) 0.8 0.2 0.7 0 0.6 -30 -20 -10 0 10 20 30 y x (nm) bilit 0.5 ba 0.4 FIG.8. Decay oftheQBstateat 50.8 meV.Thepotential o Pr 0.3 arraywithsingle-cellARC(solidline)isshownfororientation. 0.2 The fourth QB state also survives the addition of the 0.1 ARC, and implies a pole at 72−i0.767 meV. The draw- 0 ing would look very similar to Fig, 8, except that the -0.1 0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 wavefunctionhasnodes ineachbarrier. Thatis because Time (ps) the lowest state is nodeless, while the fourth state has FIG.7. Accumulated transmission (T), reflection (R) and alternating signs in the four wells. probability of remaining in thesystem (P) versus time, for a narrow wave packet at Ep =51±0.47 meV. III. TIME DELAY B. Quasi-Bound states A. Results from time-dependent calculations At times near 6 ps T(t) is approaching its asymptotic value T . Looking at the wave which is still trapped The primary motivation for this work was to compare as inside the array, one sees that it has a particular shape the time taken to traverse the array plus ARC, as com- (shown in Fig. 8) and is decaying with mean life τ = pared to the simple array. However, in the case with n 0.838(2)ps. Thiscorrespondstoahalf-widthΓ =0.786 ARC, the two satellite peaks appear to be narrow res- n meV.ItimpliesaquasiboundstatelocatedatE −iΓ /2 onances of the same type as for the simple periodic ar- n n in the lower half complex plane. Without the ARC, the ray. Hence it is also of interest to compare transmis- 5 sion through the satellite peaks with transmission via downward trend from left to right also agrees. Because the Bloch states of the broad central maximum. the Blochvelocity vanishes ata band edge, the curvedi- Fig. 7 shows that the build-up of the transmission vergesthere. Calculationswithtenandfifteencellsshow peakat51meVproceedslinearlyintime, overthe range that the satellite peaks move closer to the divergence at 0.2 < T(t) < 0.6. Similar linear behaviour applies the band edge. The delay at 51 meV, 1.6 ps, is about throughout the allowed band. In the absence of a po- double the lifetime of the quasibound state, but at 70 tentiala similarcurveis found, with some time displace- meV the two are about equal. Similar conclusions hold ment. By comparing the two, a reliable time delay (or whenatwo-cellARCisadded,butforbrevitywedonot advance) can be deduced which depends little on just include them here. which point is selected. For larger and smaller times the build-up is non-linear, and the converse is true. In Fig. 9 (a) we have plotted the ratio of T(t)/T B. Comparison with time-independent calculations as (solidline)asafunctionofenergyacrossthewholeband. The several lines correspond to different times. The in- Time delay can also be computed in the time- clined straight dotted lines are the same thing in the independent formalism. From the vast literature on this absence of any barriers (free propagation). In the back- subject, we base our discussionon relativelyrecent work groundasachainline,thetransmissionprobabilityT(E) of Nussenzveig [30,18]. He argues in favour of the “av- is plotted for reference. Except at the four transmission erage dwell time” as the most appropriate measure of peaks, the solid and dotted lines agree well. Also, the thetime takentopassabarrier,andlistsits advantages. rate of build-up is the same as in the absence ofthe bar- It is closely related to the “phase time” or transmission riers. Thatsaysthatinthetransmissiontroughs,almost groupdelayh¯dη˜/dE [31]originallyintroducedbyWigner nothing is being transmitted but it does so with no time and Eisenbud. Here η˜is the phase of the complex trans- delay. Under the four peaks, the build-up though still mission amplitude, closely related to the transfer matrix linear in time, is significantly retarded compared to free element by M =1/t, as we now explain. 11 propagation. The inclination of the dotted lines we in- In the transfer matrix method, our wave functions terpret as a velocity effect: the higher energy electrons are defined with a different phase than generally used: travel faster and arrive at xR sooner. namely the phase is setto zero oneachside (x=a, b)of To deduce a time delay, we take the difference be- the potential array, rather than at the origin. Adopting tweenthesolidanddottedlines,averagedoverthe range Nussenzveig’s notation for the asymptotic wave function 0.3 < T(t)/Tas < 0.7. This avoids using the non-linear to left and right, we compare our ψ(x) with the usual portion of the T(t) curve, as already discussed in con- convention ψ˜(x) as follows: nection with Fig. 7. It would make little difference had we simply takenthe time delay at ratio 0.5 . The result- ψ(x)∼[eik(x−a)+re−ik(x−a); teik(x−b)] ing time delay is plotted in Fig. 10 (a). This procedure ψ˜(x)∼[ eikx + r˜e−ikx ; t˜eikx] is consistent with the work of Dumont and Marchioro, who argue that for suitably chosen wave packets and by eikaψ(x)∼[eikx + re−ik(x−2a); teik(x−w)] , (7) measuring the flux of particles at x , one can define a R where w = b−a is the total width of the potential. It “tunneling time probability distribution”, which is ba- followsthatthe phaseη ofourtransmissionamplitude is sically the derivative of the integrated transmitted flux related to the usual one by η˜=η−kw. T(t) plotted in Fig. 7, but normalized to its asymptotic Then the phase time delay is value T . as Fig. 9 (b) shows the same ratio T(t)/T after adding as dη˜ dη dk a single-cell ARC. Except at the two satellite peaks, the τph =h¯ =h¯ −w , (8) dE dE dE solidanddottedlinesareroughlyparallel,butnottouch- h i ing, showing that there is a delay for propagation via a where η is the phase of our transmission amplitude and Bloch state. The delay is significantly greater under the w isthe widthofthe superlattice(nottheentiresystem) satellite peaks. The corresponding time delays are plot- under consideration. Since dE/dk is the velocity v of a f ted in Fig. 10 (b). Without an ARC, they go to zero free particle, the contribution w/v represents the time f (or evennegative)in the transmissionvalleys,but atthe taken to cross the superlattice assuming zero potential. peakstheyrangefrom0.75to2.6ps. Withasingle-layer Inviewofourearlierestimateforthefreevelocity,itisof ARC the delay is 0.25 ps over the middle of the band, the order of 0.1 ps. Thus, dη˜/dE gives phase time delay, jumping to 1.6 and 0.8 ps under the satellite peaks. So, while our dη/dE gives phase time. forthoseelectronsthataretransmitted,addingtheARC For the case of unilateral incidence on a symmetric cuts the time delay in half, while greatly increasing the potential, which we use, Nussenzweig arrives at his eq. average transmissivity. [Also shown in Fig. 10 (b) is a (20) for the mean dwell time delay as a spectral average simple estimate of the time delay for traversingfive cells over the phase time, plus an oscillatory term. In our attheBlochvelocity,ignoringanydelayintheARCcells. notation, his result can be written as in eq. A3 in the Itcanbeseenthatthisestimateistherightsize,andthe appendix. 6 In Fig. 10 we show the time delay extracted from our delaysdo not occur exactlyat the transmissionmaxima, time-dependent calculations, both before (a) and after they do lie close together for this type of potential cell, (b) a single cell ARC has been added. In both panels, and the difference would only be visible on a magnified the lower dashed line would be the phase time delay, drawing. The locus passes about 10% above the peaks, assuming that in the periodic arraythe wave propagates which we ascribe to two effects: (i) the finite steps in at the Bloch velocity of the infinite periodic system: meanwavepacketenergyinour calculations,whichmay miss the top,and(ii) the finite energywidth ofthe wave −dsinφ ∂E packetwhichsmearsouttheresultbasedonplanewaves, v = ≡d/τ , (9) Bl Bl ¯h ∂cosφ as in eq. 10. This is seen more clearly in Fig. 11. Inter- estingly,thetimedelaywithoutanARCoscillatesaround where d is the cell size. Within an allowedband, cosφ is the Bloch time delay in Fig. 10 (a); this is not partic- generally quite linear in E. ularly obvious from the present case of five cells, but if InFig. 10(b)the dash-dotline includes anestimate of one compares systems with ten to fifteen cells it is quite thetimedelayforpassingthroughtheARClayersaswell striking. as the five central cells (it is a 10% effect). This Bloch time-delay agrees quite well with the result of our time- dependent calculations, especially in its general trend. 3 1 ThisconfirmsourunderstandingoftheactionoftheARC Time Delay left scale layers: they convert the incident plane wave state into a right scale 2.5 Bloch state of the periodic system. The divergence at 0.75 the band edge arises because the Blochvelocity vanishes there. elay (ps)1.25 0.5mission D s 0 .19 Time 1 Tran 0.25 0.8 0.5 0.7 s 0.6 0 0 T / Ta 0.5 45 50 55 En6e0rgy (m6e5V) 70 75 80 0.4 a) 0.3 0.2 1.6 1 right s cale 0.1 left s cale 1.4 0 50 55 60 65 70 75 1.2 0.75 a) E (meV) 0 .19 elay (ps) .18 0.5mission D s 00..78 Time .6 Tran 0.4 0.25 s 0.6 Ta T / 0.5 0.2 0.4 0 0 0.3 45 50 55 60 65 70 75 80 Energy (meV) 0.2 0.1 b) 0 50 55 60 65 70 75 FIG.10. Time delay (solid line) for electrons transmitted b) E (meV) through(a)thefivebarrierperiodicsystemandalso(b)with single-layerARCadded. AlsoshownisthetransmissionT(E) FIG.9. Ratio T/Tas(E) with (solid line) and without po- (dotted line) for reference. tential (dotted line) for (a) five barrier periodic system and (b)withsingle-layerARCadded. Thecontoursareplottedat The locus of time delay at transmissionmaxima is de- intervals t = 2(1.5)14 ps. Also shown (dotted) is the trans- rived as follows: For a finite periodic system, we know mission T(E),for reference. that [2] In Fig. 10(a) the dash-dot line is the locus of phase 1 1 1 = sinNφ−sin(N −1)φ . (10) timedelayattransmissionmaxima. Whilemaximaltime t sinφ(cid:18)t (cid:19) N 7 We write t = |t|eiη and t = |t |eiηN for N cells. Thus similar results. They did not perform time-dependent N N η canbeexpressedintermsof|t|, ηandtheBlochphase calculations to establish the validity of the result. Fur- N φ. Using eq. (8) for the time delay we can similarly ex- thermore, in our reading, Pereyra’s derivation [21] as- press τ in terms of the single cell parameters. To look sumes that the reflection amplitude has a fixed modu- for maxima, we set the derivative with respect to energy lus, and only the phase is varying, which is obviously to be zero, and sinNφ = 0, because transmission max- questionable in the case of sharp resonances. After this ima occur when Nφ is an integer multiple of π (see first workwassubmitted,Pacheretal.[33]havealsodiscussed paragraphof this paper.) The result is phase time delay at the transmission maxima, deriving the result in eq. 11 and many others. dη Nd sinη N ¯h = =Nτ coshµ . (11) Bl dE (cid:12)max |t|vBl sinφ (cid:12) (cid:12) 4 1 whereτ is thetime takentocrossonecellatthe Bloch Time Delay Bl left scale velocity, and µ is the impedance parameter. The plot 3.5 right e n v eslcoaplee of coshµ in Fig. 2 explains the shape and height of the s) 3 0.75 curve immediately. p leaFdrionmg teoq.an1e1xpwreesssuiobntrfaocrttthheelloacsutstoerfmtimofe edqe.lay(8a)t, elay (2.25 0.5mission transmission maxima: e D ns m1.5 a Nd Ti Tr τ =Nτ coshµ− . (12) 1 0.25 loc Bl v free 0.5 Results using eq. (8) are in excellent agreement with 0 0 those of the time-dependent calculation, when allowance 45 50 55 60 65 70 75 80 Energy (meV) ismadeforaveragingoverenergyintheneighbourhoodof the sharp resonances. There the finite width (in energy) FIG.11. Time Delay (solid line) computed from phase of of our incident wavepacket mainly reduces the height of tN, and (dashed line) from the time-dependent calculation, the peaks of the time-independent result, by 5 to 10%, just visible below the peaks. The dash-dot line is the locus, acording to eq. (13). An example is shown in Fig. 11. eq. 12. Also shown for reference is the transmission (dotted In this drawing,we shouldhaveaveragedthe phase time line). over the spectral content of the initial state. We did not, but because our wave packets are narrow in energy, convolutiononlyreducestheheightsofthenarrowpeaks, IV. CONCLUSION which can be estimated from Time dependence of scattering from a finite periodic f(E) g(E)∼ . (13) potential array was studied by direct numerical solution 1+(2σ /Γ)2f(E) E ofthe Schr¨odingerequation. Itwasverifiedthatthenar- row transmission peaks are associated with quasi-bound This assumes that the unsmeared function f(E) is of statesofthearray. Theseresonancesentailtimedelaysof Breit-Wigner form with width Γ, and is wide compared order1to2ps,whileinthetransmissionminima thede- tothewidthinenergyσ ∼0.4meVofthewavepacket. E layvanishes. Uponadditionofananti-reflectioncoating, Eq. (13) agrees well with the reduction seen in Figs. 10 the broadcentraltransmissionmaximum correspondsto and 11, both for the time delay and transmission peaks. transmission via Bloch states, with a time delay of or- MostlyhiddenbelowthesolidlineinFig. 11isourtime- der 0.2 to 0.3 ps, as seen in Fig. 10 (b). The satellite delayfromFig.10(a). Onecanseethatexceptjustbelow peaks near the band edge continue to proceed through the peaks, the agreement is excellent. QB states, but even their time delay is cut roughly in Neither have we included the oscillatory term in the half, as compared to the bare periodic array. meandwelltimedelay,whicharisesfrominterferencebe- Our incident wave packets had widths in energy of or- tween the incident wave and the reflected wave [31,32]. der 0.4 meV; this was feasible due to the application of In our calculations there are indeed spectacular interfer- transparent boundary conditions at the edges of the re- ence effects within and to the left of the potential array, gion considered. A further improvement in the method asthereflectedwaveisgenerated. Butafterthereflected of solutionis possible, by goingfrom first-orderto third- waveiswellseparatedfromthepotentialandreachesthe order Crank-Nicolson integration for the time depen- counteratx ,theseoscillationshavedisappeared,anddo L dence [34,35]. This would match the truncation error not show up in the integrated flux. Neglect of this term, of the Numerov method, while greatly speeding up cal- in the present calculation, is justified in the appendix. culations. Systems under bias of an applied electric field PacherandGornik[4,5]havealsocomputedtunneling [36] can also be handled by this method. Extension to timesusingasimilarformulafollowingPereyra[20],with 8 two-dimensional systems is also under consideration. comparedto the wavepacket, and that leads to the con- clusion stated in eq. 13: the spectral average mainly reduces the height of each transmission peak, leaving its ACKNOWLEDGMENTS position and width unchanged. The second, oscillatory term, (OT), includes a factor We are grateful to NSERC-Canada for Discovery cos(2kxL −η(k)). Since our xL ∼ −7500 nm., this is Grants SAPIN-8672 (WvD), RGPIN-3198 (DWLS) and a very high frequency oscillation, given that we set the a Summer Research Award through Redeemer Univer- boundary xL far to the left of the potential. The re- sity College (CNV); and to DGES-Spain for continued flection amplitude |r(k)| varies on the same scale as the support through grants PB97-0915 and BFM2001-3710 phase shift. In doing the spectral average it is reason- (JM).WealsothankGigiWongforassistanceinredraw- able to treat the small prefactor h¯|r(k)|/E(k) ∼ 10 fs ing Figs. 10 and 11, and R. S. Dumont for illuminating asslowlyvaryingincomparisontothe cosine. Themean discussions. valueoftheOTcanbeestimatedbythemethodofsteep- est descents, leading to ¯h|r(k)| APPENDIX A: SPECTRAL AVERAGE OF THE <OT >∼− cos(2kx −η(k)) σ L X 2E(k) DWELL TIME DELAY ∞ 2 × exp[(i(q−k)(2x )−(q−k)22σ2 ]dq For our wave packet eq. 2, the Fourier transform is rπ Z−∞ L X ¯h|r(k)| ψ0(q)=[8πσX2 ]1/4e−iqx0e−(q−k)2σ2 , (A1) =−2E(k) cos(2kxL−η(k)) exp[−(xL/σX)2/2] . (A4) normalized according to By taking x sufficiently far to the left, the spectral av- L eragecanbemadeassmallasweplease. Mostlyweused ∞ |ψ (q)|2dq =1 . (A2) xL/σX =7.5,sotheexponentialfactorise−28,multiply- 0 Z−∞ 2π ing a term which is already small. The spectral weight function is therefore A(q) = |ψ (q)|2/(2π). Nussenzweig’s eq. (20) for the time de- 0 lay by a symmetric potential, with unilateral incidence, in our notation becomes ∞ <∆td >s→ymm= |ψ0(q)|2 1 A. Wacker, “Semiconductor superlattices, a model system Z−∞ for non-linear transport”, Phys.Repts. 357 (2002) 1-111. dη˜ ¯h|r(q)| dq 2 D.W.L. Sprung,HuaWu and J. Martorell, “Scattering by × ¯h − cos(2qx −η˜) . (A3) (cid:20) dE 2E(q) L (cid:21) 2π aFinitePeriodicPotential”,Am.J.Phys.61 (1993)1118- 24. The terms in square brackets are the dwell time delay 3 C.Pacher,C.Rauch,G.Strasser,E.Gornik,F.Elsholz,A. in the mono-energetic case. Razavy [32] for example de- Wacker, G. Kiesslich & E. Sch¨oll, “Anti Reflection Coat- rivedthem by following the method of Smith [31], albeit ing for miniband transport and Fabry-Perot resonances in withsometyposinhiseq. (18.19). (Onehastonotethat GaAs/AlGaAssuperlattices”, Appl.Phys.Lett.79(2001) 1486. forasymmetricpotentialhistwophaseshiftsarerelated 4 C. Pacher and E. Gornik, “Adjusting coherent transport by η =δ+π/2.) in finite periodic superlattices”, Phys. Rev. B 68 (2003) As stated earlier, we used a wave packet with a width 155319 (9 pp). σ of order 1000 nm. The width in energy is of order X 5 C.PacherandE.Gornik,“Tuningoftransmissionfunction 0.4 meV, which is small compared both to the width of and tunneling time in finite periodic potentials”, Physica the allowedband,25 meV,andevenofthe resonancesof E (Low-dimensional systems & nanostructures) (2004) 21 a finite periodic potential array, which are in the range 783-786. of 2 to 5 meV depending on the position in the band. 6 G.V.Morozov,D.W.L.SprungandJ.Martorell,“Optimal (Thenarroweststatesarethosecrowdedagainsttheband band-pass filter for electrons in semiconductor superlat- edge.) Itis reasonabletotreatA(q) asnarrowcompared tices” J. Phys. D 35 (2002) 2091-5. to the width of the peaks in transmission. 7 G. Morozov, D.W.L. Sprung and J. Martorell, “Design of The firstterm,dη/dE, variesonthe scaleofthe band- electronband-passfiltersforsemiconductorsuperlattices”, width divided by N, the number of cells, about 6 meV J. Phys. D 35 (2002) 3052-9. in our calculation. The slope of η is steepest at a reso- 8 D.W.L.Sprung,G.V.Morozov andJ.Martorell, “AntiRe- nance, where η =mπ, and is minimal at the mid-points flection coatings from the analogy between electron scat- between resonances. The phase shift η varies smoothly teringandspinprecession”,J.App.Phys.93(2003)4395- 4406. 9 9 D.W.L.Sprung,G.V.MorozovandJ.Martorell,“Geomet- Phys.Rev. B 71 (2005) 125317. (11 pp.) rical approach toscattering inonedimension”, J.Phys.A 34 I.V.Puzynin,A.V.SelinandS.I.Vinitsky,“Highaccuracy 37 (2004) 1861-80. methodfornumericalsolutionoftheTDSE”,Comp.Phys. 10 P. G.Kard, “Analytictheory of optical properties of mul- Comm. 123 (1999) 1-6. tilayer coatings”, Optikai Spektr.2(1957) 236-44. 35 I.V. Puzynin, A.V. Selin and S.I. Vinitsky, “Magnus- 11 P.L.KapurandR.E.Peierls,“Dispersiontheoryofnuclear factorizationmethodfornumericalsolutionoftheTDSE”, reactions”, Proc. Roy.Soc. A166 (1938) 273-306. Comp. Phys. Comm. 126 (2000) 158-61. 12 A.J.F.Siegert,“Dispersion formulafornuclearreactions”, 36 J.Martorell,D.W.L.SprungandG.V.Morozov,“Electron Phys. Rev.56 (1939) 750-2. band-passfiltersforbiasedfinitesuperlattices”,Phys.Rev. 13 L. Rosenfeld, “Time evolution of the scattering process”, B 69 (2004) 115309 (10 pp). Nucl. Phys.70 (1965) 1-27. 14 R. M. More, “Theory of decaying states”, Phys. Rev.A 4 (1971) 1782-90. 15 R. M. More and E. Gerjuoy, “Properties of resonant wave functions”, Phys.Rev. A 7 (1973) 1288-1303. 16 “Tunnelingtimes: acriticalreview”, E.H.HaugeandJ.A. Støvneng,Rev. Mod. Phys. 61 (1989) 917-36. 17 “Bohmtrajectoriesandthetunnelingtimeproblem”,C.R. LeavensandG.C.Aers,inScanningTunnelingmicroscopy III, R. Weisendanger and H.-J. Gu¨ntherodt, eds., second edition, Springer series in Surface Sciences no. 29, (1996); Ch. 6 pp.106-140. 18 C.A.A.deCarvalhoandH.M.Nussenzveig,“TimeDelay”, Phys. Repts.364 (2002) 83-174. 19 A.P. Stamp and G.C. McIntosh, “Time-dependent study of resonant tunneling through a double barrier”, Am. J. Phys. 64 (1996) 264-76. 20 P. Pereyra, “Closed formulae for tunneling time in super- lattices”, Phys.Rev.Lett. 84 (2000) 1772-5. 21 H.P. Simanjuntak and P.Pereyra, “Evolution and tunnel- ing time of electron wave packetsthrough a superlattice”, Phys. Rev.B 67 (2003) 045301 (7 pp). 22 A.M. Bouchard and Marshall Luban, “Bloch oscillations and other dynamical phenomena of electrons in semicon- ductorsuperlattices”, Phys. Rev.B 52 (1995) 5105-23. 23 S. E. Koonin, “Computational Physics”, Benjamin- Cummings, Menlo Park CA, (1986); Sec. 7.5 . 24 A. Goldberg, H.M. Schey and J.L. Schwartz, “Computer generated movies of 1D scattering phenomena”, Am. J. Phys. 35 (1967) 177-86. 25 C.A.Moyer,“Numerovextensionoftransparentboundary conditions for the 1D wave equation”, Am. J. Phys. 72 (2004) 351-8. 26 R. Romo, “Buildup dynamics of transmission resonances in superlattices”, Phys. Rev.B 66 (2002) 245311 (9 pp). 27 M.Moshinsky,“Diffractionintime”,Phys.Rev.88(1952) 625-31. 28 G.Garc´ıa-Calder´on, andAlbertoRubio“Transient effects and delay time in the dynamics of resonant tunneling”, Phys. Rev.A 55, (1997) 3361-3370. 29 Randall S. Dumont and T.L. Marchioro II, Phys. Rev. A 47 (1993) 85-97. 30 H.M. Nussenzveig, “Average dwell time and tunneling”, Phys. Rev.A 62 (2000) 042107 (5 pp). 31 F.T. Smith, “Lifetime matrix in collision theory”, Phys. Rev.118 (1960) 349-56; Erratum 119 2098. 32 M. Razavy,“Quantumtheory of tunneling”, World Scien- tific, Singapore, (2003); Ch. 18. 33 C. Pacher, W. Boxleitner,and E. Gornik, “Coherent reso- nanttunnelingtimeandvelocityinfiniteperiodicsystems” 10

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