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Confinamiento periódico 173Tunneling in Chains of Quantum Dots J. Phys. Chem. B, Vol. 108, No. 35, 2004 13289Figure 1. Time-dependent overlap |c p(t)| 2 , eq 16, vs time betweenω G′(r,0) and ω G(r,t), with G - G′ ) 5, for a parabolic band modeldefined by a bandwidth ∆E ) 0.1 au.Figure 3. Tunneling time τ G required by an electron to reach the Gthcell vs G (defined as the time required by |c(t)| 2 , eq 13, to attain itsfirst maximum in the Gth cell) for different band models. τ G estimatedby eq 22 is also included.t ) 0 as linear combinations of the abovementioned eigenvectors:However, they do not remain localized with the passage of time.For example, explicitly writing the time in Ψ L we haveand thereforeΨ L ) 12 (Ψ + + Ψ - ) (18)Ψ R ) 12 (Ψ + - Ψ - ) (19)Ψ L ) 12 (Ψ + e-iE+t/p + Ψ - e -iE-t/p ) (20)Figure 2. Plots of electron density distributions of a time-dependentWannier state at different times (au).TABLE 1: Tunneling Time τ 5 Required by a ParabolicBand Electron To Reach the Fifth Cell, for DifferentBandwidths ∆E∆E (au)τ 5 (au)0.1 125.01 12.510 1.2525 0.50out from the fact that different energy dispersion models withthe same bandwidth may yield different tunneling times.Finally, we show in Figure 2 how a Wannier function issmeared out with time, and in Figure 3 we have plotted thetime required by an electron to reach consecutive cells, i.e., thetime τ G required by |c(t)| 2 to reach its first maximum in theGth cell vs G, for different band models. This figure revealsthat the time-dependent Wannier function smears out almostlinearly vs time, like the half-width of a Gaussian packet.To end this section, we study the influence exerted byperiodicity on tunneling time τ. To do so an isolated doublewellis considered. If the central barrier is high and wide, wewill find an almost two-degenerate ground energy state, whichassociated eigenvectors, Ψ + and Ψ -, show even/odd symmetry,respectively. As we reduce the barrier, the even state stabilizeswhile the odd one becomes more unstable so that an energysplitting ∆E ) E - - E + is produced. We may write functionsthat are maximally localized in a given well at an arbitrary time|Ψ L | 2 ) 1 2(| Ψ +| 2 + | Ψ -| 2 + 2Ψ + Ψ - cos ∆Etp ) (21)At t ) 0 this density has a maximum in the well on the left,|Ψ L(x,0)| 2 ) 1 / 2|Ψ + + Ψ -| 2 , but at a timeτ) πp∆E(22)the density |Ψ L(x,τ)| 2 ) 1 / 2|Ψ +-Ψ -| 2 ; i.e., it has a maximumin the well on the right. We may then consider τ as the tunnelingtime.In the case of an isolated triple well, we can still obtain asimple formula for τ. We consider a symmetric splitting of theground energy into E +, E 0, and E - as the barriers decrease.Without loss of generality, we may set E 0 ) 0. The statelocalized in the left well is, except for a time-independent factorΨ L ) Ψ + e -iEt/p + aΨ 0 + Ψ - e iEt/p (23)where a is the constant coefficient required to localize theelectron in the left well at t ) 0. If t ) τ ) πp/2E, then |Ψ L| 2shows a maximum in the central well. We may rewrite τ )πp/∆E, where ∆E is the abovementioned splitting. Again τ hasthe meaning of tunneling time.If the number of wells increases simple formulas for τ cannotbe obtained, but we can assume eq 22 to be (approximately)valid regardless of the number of wells. At the limit of infinitewells, ∆E represents the bandwidth. Therefore, this assumptiongoes along the same lines as the reciprocal relationship betweentunneling time and bandwidth pointed out at the beginning ofthis section and shown in Table 1.