Bias driven coherent carrier dynamics in a two-dimensional aperiodic potential
F.A.B.F.de Moura, L.P.Viana, M.L.Lyra, V.A.Malyshev, F.Dominguez-Adame
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p Bias driven coherent carrier dynamics in atwo-dimensional aperiodic potential
F. A. B. F. de Moura
Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o AL 57072-970, Brazil
L. P. Viana
P´olo Penedo, Universidade Federal de Alagoas, Penedo AL 57200-000, Brazil
M. L. Lyra
Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o AL 57072-970, Brazil
V. A. Malyshev
Centre for Theoretical Physis and Zernike Institute for Advanced Materials,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
F. Dom´ınguez-Adame
GISC, Departamento de F´ısica de Materiales, Universidad Complutense, E-28040Madrid, Spain
Abstract
We study the dynamics of an electron wave-packet in a two-dimensional square lat-tice with an aperiodic site potential in the presence of an external uniform electricfield. The aperiodicity is described by ǫ m = V cos ( παm ν x x ) cos ( παm ν y y ) at latticesites ( m x , m y ), with πα being a rational number, and ν x and ν y tunable parame-ters, controlling the aperiodicity. Using an exact diagonalization procedure and afinite-size scaling analysis, we show that in the weakly aperiodic regime ( ν x , ν y < Key words: aperiodic potential, coherent electron dynamics, Bloch oscillations
Preprint submitted to Physics Letters A 20 November 2018
ACS:
Materials with restricted geometry, such as semiconductor quantum-well struc-tures, [1] quantum dots and wires, [2,3] organic thin films [4] as well asquasiperiodic structures, [5] are nowadays subjects of growing interest fromboth fundamental and practical points of view. An attributive peculiarity ofalmost all of them is the presence of disorder, which can be both of an intrinsicnature (imperfections of the structure itself) and originated from a randomenvironment.Whenever disorder is involved, Anderson’s ideas about localization of quasi-particles states come into play. [6] In three dimensions (3D), the states at thecenter of the quasiparticle band remain extended for a relatively weak disor-der (of magnitude smaller than the bandwidth), while the other states (in theneighborhood of the band edges) turn out to be exponentially localized. Thisimplies the existence of two mobility edges which separates the phases of ex-tended and localized states. [7] On the contrary, uncorrelated disorder of anymagnitude causes localization of all one-particle eigenstates in one dimension(1D) [8] and two dimensions (2D). [9]Since late eighties, however, it has been realized that extended states may sur-vive on 1D systems if the disorder distribution is correlated. [10,11,12,13,14,17,15,16,18]Thus, a short-range correlated disorder was found to stabilize the extendedstates at special resonance energies. In the thermodynamic limit, such ex-tended states form a set of null measure in the density of states, [10,11,12,13,14]implying the absence of mobility edges in these systems. In contrast, systemswith long-range correlations of disorder support a set of delocalized stateswithin a finite bandwidth, [15,16] giving rise to mobility edges. Theoreticalpredictions of localization suppression on 1D geometries, due to correlationsof the disorder distribution, were confirmed experimentally in semiconductorsuperlattices with intentional correlated disorder, [17] as well as in single-modewaveguides with correlated scatterers. [18]Among 1D models with extended states, aperiodic Anderson models [19] withan incommensurate site potential represent a class of particular interest. Thesemodels have been extensively investigated in the literature, [20,21,22,19,23]and the localized or extended nature of the eigenstates has been related to On leave from V. A. Fock Institute of Physics, St. Petersburg University, 198904St.-Petersburg, Russia φ x and φ y varies as a power-law. The power-law exponent controls the degreeof aperiodicity in the site potential. We start by employing an exact diagonal-ization formalism to compute the dc conductance and participation numberin the absence of the electric field. We numerically demonstrate that their sizebehavior is consistent with the existence of extended states near the band cen-ter for weakly aperiodic potentials. In this regime, the wave-packet dynamics3ecomes ballistic. Furthermore, we focus on the wave-packet dynamics in thepresence of an external uniform electric field. We show that in the limit ofa weakly aperiodic site potential, the electric field promotes sustained Blochoscillations of the electronic wave-packet. The frequency of these oscillationscan be understood within the framework of a semi-classical approach.The outline of the paper is as follows. In the next section we present our modeland quantities of interest. In Sec. 3, the results of the numerical simulationsof the dc conductance, participation number and electronic wave-packet dy-namics are discussed. We summarize in Sec. 4. We consider a tight-binding single electron Hamiltonian on a regular 2D openlattice of spacing a with an aperiodic site potential and a uniform static electricfield [32,40] H = X m ( ǫ m + U · m ) | m ih m | + J X h mn i (cid:18) | m ih n | + | n ih m | (cid:19) , (1)where | m i is a Wannier state localized at site m = m x e x + m y e y , ǫ m is itsenergy, and U = e F a is the energetic bias given by the electric field F , − e being the electron charge. Here e x and e y are the corresponding Cartesian unitvectors. We will assume that the electric field F is applied along the diagonalof the square lattice, i.e., U = U ( e x + e y ) / √
2, were U is the bias. Transferintegrals are restricted to the nearest-neighbor interaction and are given by J .Hereafter, the energy scale is fixed by setting J = 1.The 2D aperiodic potential ǫ m is taken in the form: ǫ m = V cos ( παm ν x x ) cos ( παm ν y y ) , (2)where V , α , ν x , and ν y are variable parameters.Equation (2) is a 2D generalization of the aperiodic 1D potential introducedby Das Sarma et al . [19] In 1 D and at ν = 1, this potential represents justHarper’s model. In this case, a rational α describes a crystalline solid, whereasan irrational α yields an incommensurate potential. It was shown in Ref. [21]that for ν > pseudo-random regime . Oppositely, it was demonstratedthat the range 0 < ν < pseudo-random regime ,it is the average wavelength of the potential oscillations that diverges in thethermodynamic limit. Because of the slower divergence of the oscillation wave-length, this regime is usually named as weakly aperiodic . The uniform case isrecovered at ν = 0. Recently, the effect of this kind of aperiodicity on thephonon and magnon 1 D modes has attracted a renewed interest. [26,27]The time-dependent Schr¨odinger equation governs the dynamics of the Wan-nier amplitudes, [41] i ˙ ψ m = ( ǫ m + U · m ) ψ m + (cid:16) ψ m + e x + ψ m − e x + ψ m + e y + ψ m − e y (cid:17) , (3)where the Planck constant ¯ h is set to unity.We solve numerically Eq. (3) to study the time evolution of an initially Gaus-sian wave-packet centered at site m , ψ m ( t = 0) = A exp " − ( m − m ) , (4)where A is the normalization constant and we set ∆ = 1 hereafter. OnceEq. (3) is solved for the initial condition (4), we compute the projection ofthe mean position of the wave-packet (centroid) and the average velocity v ( t )along the field direction R ( t ) = 1 √ h m x ( t ) + m y ( t ) i ,v ( t ) = − X m ψ m h ψ ∗ m + e x + ψ ∗ m + e y i , (5)as well as the spread of the wave-function (square root of the mean-squaredisplacement) σ ( t ) = sX m h m − m ( t ) i | ψ m ( t ) | (6)where m ( t ) = P m m | ψ m ( t ) | .In addition, we use the exact diagonalization of the Hamiltonian (1) in theabsence of the electric field ( U = 0) to obtain the eigenvalues and eigenvectors,and to calculate the dc conductance G ( E ) as a function of energy E . For5oninteracting electrons and within the linear response theory at T = 0 K, G ( E ) is given by the Kubo-Greenwood formula [42] (in units of e / ¯ h ) G ( E ) = 2 π Ω X a,b |h a | xH − Hx | b i| × δ ( E − E a ) δ ( E − E b ) . (7)Here, the polarization is taken in the direction of the x -axis, Ω is the area ofthe system, x is the position operator, and the indices a and b label the eigen-states. We also compute the participation number of a normalized eigenstateas follows P ( E ) = (cid:20) X m | ψ m ( E ) | (cid:21) − . (8)For extended states, P ( E ) is expected to scale linearly with the number ofsites. More specifically, we will focus on the dc conductance and the partici-pation number near the band center, averaged over a narrow energy stripe atthe band center [ − W c / , W c / G = 1 N E X | E | 1, Θ being the Heavisidestep-function. The magnitude of the energy stripe in Eqs. (9a) and (9b) waschosen W c = 0 . Firstly, we discuss the character of the electronic states of the model understudy. Figure 1(a) displays the averaged conductance G at the band center( E = 0) as a function of ν ≡ ν x = ν y , obtained after numerical diagonalizationof the Hamiltonian (1) in the absence of the external electric field ( U = 0). To6void strong fluctuations of the conductance related to particular realizationsof the aperiodic potential, we further averaged the results over a narrow rangeof potential amplitudes (∆ V = 0 . ν x , ν y < 1) the conductance grows onincreasing the system size. This is a signature of macroscopic conductancein the thermodynamic limit. In contrast, the conductance is almost null for apseudo-random potential ( ν x , ν y > ξ , as shown in Fig. 1(b). For weakly aperiodicpotentials, ξ grows with the system size, while it becomes size independentfor strongly aperiodic (pseudo-random) systems. ν G N=60N=80N=100N=120 (a) ν ξ N=60N=80N=100N=120 (b) Fig. 1. (a) Averaged dc conductance G , Eq. (9a), and (b) averaged participationnumber ξ , Eq.(9b), versus the exponent ν = ν x = ν y calculated for different sys-tem sizes N . The average was performed considering a narrow range of potentialamplitudes (∆ V = 0 . 05) to reduce statistical fluctuations. 60 80 100 120 140 N ξ / N ν=0.5ν=0.7ν=2ν=3 60 80 100 120 14010 -3 -2 -1 G / N Fig. 2. Finite size scaling of the normalized participation number ξ/N for the statesin the vicinity of the band center calculated for various values of the exponent ν (shown in the legend). The inset presents the finite-size scaling of the normalizedconductance G/N . The different character of the finite-size scaling of both quantitieson increasing the exponent ν signals the change in the nature of electronic states inthe band center from delocalized ( ν < 1) to localized ones ( ν > The correlations found in the behavior of the averaged conductance G and theparticipation number ξ as a function of the exponent ν indicate that the modelunder study undergoes a metal-insulator transition associated with a changein the nature of the one-electron eigenstates. To give further support to this7nding, we performed a finite-size scaling analysis of these two quantities. InFig. 2 we plotted the size dependence of the normalized participation number ξ/N for various values of the exponent ν . As it can be noticed, ξ/N does notdepend of N in the regime of weak aperiodicity ( ν < ξ is proportional to the total number of sites N . This is a clearindication that the eigenstates at the band center are delocalized over a finitefraction of the lattice. On the contrary, for strong aperiodic potentials ( ν > ξ/N decreases with the system size, signaling the wave-function localization.The size dependence of the normalized conductance G/N in the center of theband, shown in the inset of Fig. 2, corroborates the conclusion above. In theslowly varying aperiodic regime G ∝ N leading to a macroscopic transportsupported by extended states. Such a behavior is expected for a regular squarelattice, where the conductance at the band center should be proportional tothe number of conducting channels, i.e., to the linear size of the lattice N . Inthe limit of strongly aperiodic lattices, the normalized conductance decreaseswith the system size, thus indicating the absence of macroscopic transportin the thermodynamic limit. This is in agreement with the standard scalingtheory of Anderson localization. [9] We also study the time evolution of a wave-packet initially localized in thecenter of the lattice, m = ( N/ , N/ U = 0). We numerically integrate the wave-equation (3) and calculate thewave-packet spread σ ( t ) until it reaches a stationary value as a result of mul-tiple reflections of the electron on the lattice boundaries. The extended statesthat appear in this model emerge in the regime at which the potential hasa diverging wavelength in the thermodynamic limit. These are typically nonscattered modes thus leading to a ballistic wave-packet spread, σ ( t ) ∝ t . Thismeans that, as long as σ ( t ) ≤ N , the data points calculated for differentsystem sizes N should collapse into a unique straight line after re-scaling σ ( t ) → σ ( t ) /N and t → t/N . In the localized regime, such a collapse is notexpected. We further calculated the time-dependent participation second mo-ment ξ ( t ) = 1 / P m | ψ m ( t ) | . In general, ξ ( t ) scales linearly with the numberof sites whenever the system supports extended states.In Fig. 3, we plot the results of our calculations. For a weakly aperiodic sitepotential ( ν x = ν y = 0 . σ ( t ) ≤ N [see Fig. 3(a)]. This allow us to con-clude that the wave-packet motion in this case has a ballistic nature, signalingthe presence of a phase of delocalized non scattered states. In the case of a8seudo-random site potential ( ν x = ν y = 1 . σ ( t ) ≪ N [see Fig. 3(b)]: the wave-packet motion is ballistic whenever σ ( t ) issmaller than the largest localization length. For long times, a collapse is absent[see Fig. 3(b)], indicating the wave-packet localization. In Fig. 3(c), we showresults for the time-dependent participation second moment for ν x = ν y = 0 . ν x = ν y = 1 . ν < 1, the participationdynamics show a ballistic behavior until the wave-packet reaches the latticeboundaries. For ν > ν < D aperiodic model, whichwe claimed in the previous section on the basis of the finite size scaling of theconductance G and participation number ξ for the states in the vicinity of theband center. -4 -2 t/N -4 -3 -2 -1 σ ( t ) / N N=200N=400N=800 ν =0.5(a) -4 -2 t/N -4 -3 -2 -1 σ ( t ) / N N=200N=400N=800 ν=1.5 (b) -2 t ξ ( t ) ν=0.5ν=1.5 t (c) Fig. 3. The normalized wave-packet spread σ ( t ) /N , Eq. (6), as a function of there-scaled time t/N computed for lattices of N × N = 200 × 200 to 800 × U = 0). (a) A slowly varying (weakly aperiodic) site poten-tial ( ν x = ν y = ν = 0 . σ ( t ) ≤ N indicates the presence of non scattered extended states. (b) A pseu-do-random potential ( ν x = ν y = ν = 1 . ξ ( t ) number versus time t computedfor N × N = 1600 × ν x = ν y = ν = 0 . ν x = ν y = ν = 1 . ξ ( t ) ∝ t ) for ν < Now, we turn to the wave-packet dynamics of an electron subjected to a uni-form electric field ( U = 0). It is well known that in disorder-free systems,9 t R ( t ) ω R ( ω ) Fig. 4. Bias driven dynamics of the centroid R ( t ), Eq. (5), in a pseudo-randompotential ( ν x = ν y = 1 . 5) with N × N = 200 × 200 sites and bias magnitude U = 0 . 5. No signature of coherent Bloch oscillations is seen in this case. The Fouriertransform ˜ R ( ω ) of the centroid R ( t ) is broad, supporting the absence of a typicaloscillation frequency in R ( t ). R ( t ) t U=0.5U=0.75 (n ,m )=(N/2,N/2)(n ,m )=(N/2,N/2+d )(a)(b)(c) (n ,m )=(N/2,N/2) U=0.5 ω R ( ω ) (d)(e)(f) Fig. 5. Bias driven dynamics of the centroid R ( t ) in a slowly varying (weakly ape-riodic) potential ( ν x = ν y = 0 . 5) for two magnitudes of the bias: (a) U = 0 . U = 0 . 75. The initial location of the wave-packet is ( n , m ) = ( N/ , N/ R ( ω ) of the centroids depicted on panels (a) and(b), respectively. Note that ˜ R ( ω ) is peaked at a frequency ω ≃ U . (c) Same as inpanel (a), but for the initial condition ( n , m ) = ( N/ , N/ d ) with d = 10.(f) Fourier transform of the centroid depicted on panel (c). Note that here ˜ R ( ω )exhibits a doublet structure. a uniform electric field causes dynamic localization of the electron and givesrise to an oscillatory motion of the wave-packet, the so-called Bloch oscilla-tions. [28,29] The size of the segment over which the electron oscillates andthe period of the oscillations are estimated from semiclassical arguments tobe L F = aW/U and τ B = 2 π/U , respectively, [43] where W is the width ofthe Bloch band.Firstly, we compute the wave-packet centroid R ( t ) in a pseudo-random po-tential ( ν x = ν y = 1 . 5) of size N × N = 200 × 200 with the bias magnitude U = 0 . 5. As deduced from Fig. 4, there is no signature of Bloch oscillations inthis case. Coherent oscillations, which are present immediately after the initialwave-packet is released, are quickly destroyed (not shown). The asymptoticbehavior of the centroid resembles a stochastic motion around some mean po-10 40 142 144 146 148 150 152-4-2024 v v 148 150 152 154 156 158 160 R -4-2024 v ν x = ν y =1.5 ν x = ν y =0.5 ν x = ν y =0.5d =0d =10 Fig. 6. Phase diagrams velocity-vs-position computed for: (a) ν x = ν y = 1 . U = 0 . n , m ) = ( N/ , N/ ν x = ν y = 0 . U = 0 . n , m ) = ( N/ , N/ n , m ) = ( N/ , N/ d ) with d = 10. For ν > ν < 1, the Bloch-like oscillationspromotes a coherent dynamics through the phase space. In panel (c), the ampli-tude of the oscillations varies faster than in panel (b) due to the doublet structuregenerated by the anisotropic gradient of the local potential. sition. The Fourier spectrum ˜ R ( ω ) of the centroid, plotted in the right panel ofFig. (4), confirms this claim. The spectrum is rather broad, with no character-istic frequencies, indicating that R ( t ) is similar to a white noise signal. Thus,for ν x , ν y > ν x = ν y = 0 . R ( ω ) displays a well-defined narrow peak, the locationof which is slightly deviated from the expected value ω = U (panels d ande); this small shift is caused by the local contribution to the external biasproduced by the modulated potential. Indeed, the bias at position ( n , m ) isgiven by U eff = ( δε n,m /δn + U, δε n,m /δm + U ). The local contribution shall berelevant whenever the potential gradient (in appropriate units) is of the orderof the external bias U . Therefore, the frequency obtained in not exactly ω = U [Figs. 5(a) and 5(b)]. The two peak structure depicted in Fig. 5(c) has its originin the distinct contributions given by the anisotropic gradient of the localpotential to the effective local bias. Once the initial wave-packet is located ata non-symmetrical position of the lattice ( n , m ) = ( N/ , N/ d ) with d =10, the potential derivatives along the orthogonal lattice directions give raiseto distinct frequency shifts when the electron is forced by a diagonal externalfield. We stress that such frequency splitting is not found in 1 D aperiodic11 R ( t ) R ( ω ) t ω ν x =0.5 ν y =1.5 ν x =0.5 ν y =0.7 U=0.5U=0.5 a ) b) Fig. 7. The dynamics of the centroid R ( t ) and its Fourier transform (left and rightpanel respectively) for ν x = 0 . ν y = 1 . ν y = 0 . 7. For both ν x and ν y below 1, Bloch-oscillation with a frequency close to the one predicted by thesemi-classical approach was obtained. However, in the case of mixed pseudo-ran-dom and weakly aperiodic components, the pseudo-random character for ν y > potentials, being thus a specific feature of higher dimensional systems.In Fig. 6, we present the phase space plots, namely velocity versus position .All calculations were performed using N × N = 200 × ν x = ν y = 1 . U = 0 . n , m ) = ( N/ , N/ ν x = ν y = 0 . U = 0 . n , m ) = ( N/ , N/ n , m ) = ( N/ , N/ d ) with d = 10. For all cases, the integration time used was t = 1000. We can see in(a) that the velocity versus position diagram exhibits orbits with no coherency,as it was observed in the centroid dynamics (see Fig. 4). In the other cases,the diagrams show a coherent dynamics through the phase space. In the caseof bimodal Bloch oscillations [see Fig. 6(c)], the wave-packet orbits in phasespace show a pronounced breathing pattern. This breathing reflects the factthat both centroid and velocity display amplitude-modulated envelopes. Thefrequency of this amplitude-modulated envelope depends on the differencebetween the two frequency peaks appearing the Bloch oscillations spectrum[Fig. 5(c)].Before concluding, some words concerning the electronic dynamics in systemswith distinct degrees of aperiodicity along the orthogonal lattice directions ν x = ν y . In Fig.7, we show the centroid R ( t ) and its Fourier transform (leftand right panel respectively) for an initial Gaussian wave-packet located atthe center of the lattice with aperiodicity exponents ν x = 0 . ν y = 1 . ν y = 0 . 7. For both ν x and ν y below 1, we can see oscillations witha frequency close to the Bloch-oscillation frequency predicted by the semi-12lassical approach. However, in the case of mixed pseudo-random and weaklyaperiodic components, the pseudo-random character for ν y > In this work, we studied numerically the dynamics of a single electron wave-packet moving in a 2D square lattice with an aperiodic site potential ǫ m = V cos ( παm ν x x ) cos ( παm ν y y ) in the presence of an external uniform electric field.For fixed parameters V and πα , the exponents ν x and ν y allow to control thedegree of aperiodicity.We have discussed the character of the electronic states of the model in theabsence of an external electric field. Diagonalizing numerically the Hamilto-nian at zero electric field, we computed the dc conductance and the partici-pation number for various systems sizes. After a finite-size scaling analysis ofdata close to the band center, we found that, for a weakly aperiodic potential( ν x , ν y < N while the conductance scales proportionalto N . Such size dependencies are consistent with the presence of extendedstates and macroscopic transport in the thermodynamic limit. In contrast,both participation number and conductance reach a plateau upon increasingthe size for a pseudo-random potential ( ν x , ν y > ν x , ν y < ν x , ν y > D aperiodic model under study undergoes ametal-insulator transition.We have also investigated the interplay between the delocalization effect, pre-served by the weakly aperiodic structure, and the dynamic localization, causedby an electric field acting on the system. We demonstrated that in weaklyaperiodic potentials ( ν x , ν y < D , the frequencyspectrum of the centroid time-evolution can exhibit a two-mode structuredepending on the initial position of the wave-packet. 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