Marcelo Z. Maialle. Liceu Vivere, Rua Duque de Caxias Norte 550, 13635-000 Pirassununga-SP, Brazil. (Received 17 February 2004; published 23 July 2004).
PHYSICAL REVIEW B 70, 033309 (2004)
Dynamic localization in finite quantum-dot superlattices: A pure ac field effect Justino R. Madureira and Peter A. Schulz Instituto de Física, Universidade Estadual de Campinas, CP 6165, 13083-970, Campinas-SP, Brazil
Marcelo Z. Maialle Liceu Vivere, Rua Duque de Caxias Norte 550, 13635-000 Pirassununga-SP, Brazil (Received 17 February 2004; published 23 July 2004) We propose that dynamic localization can be unambiguously observed through the linear optical absorption coefficient of a finite quantum dot superlattice. It is shown that the miniband collapse is almost exact in a pure ac field driven system. In the presence of Coulomb interaction, the dynamic localization is partially preserved, showing clear fingerprints in the modulation of the excitonic spectra. The spectra are obtained within an extended semiconductor Bloch equation framework, taking into account the center-of-mass motion for a finite system. DOI: 10.1103/PhysRevB.70.033309
PACS number(s): 78.67.Hc, 73.21.Cd, 73.63.2b, 78.66.2w
The dynamic properties of electrons and holes in low dimensional systems, driven by ac fields, reveal exciting emergent phenomena in the time span around the turn of the century. Such a rich scenario has been established by the concurrent development of powerful theoretical analysis tools, design and realization of high quality nanostructured devices, as well as of tunable microwave and THz ac field sources. These striking developments made possible the exploration of the interaction of THz fields with condensed matter, leading even to biological tissue imaging. Therefore a microscopic understanding of the THz field effects on designed nanostructures constitutes an important framework for further developments. A very interesting example in this context is the prediction of dynamic localization,1 which has been a subject of intense research in the past few years, from both theoretical and experimental point of views. The initial prediction states that, within a single band tight-binding approximation, an initially localized particle will return to its initial state following the periodical evolution of a driving pure sinusoidal field. This phenomenon can be simply visualized by the related collapse of the quasienergy minibands, i.e., the localization of electronic states of a periodic unidimensional structure in real space driven by a field periodic in time.2 Such collapses occur whenever the field intensity/frequency ratio, eaF / "v, is a root of the zero-order Bessel function of the first kind. The quest for experimental signatures of dynamic localization is an involved task, since a variety of perturbations to an ideal situation is always present in real systems. The question that has to be answered is how the dynamic localization, related to the quasienergy miniband collapses, may be identified in a context where concurring effects also tend to modify the quasienergy spectra. For semiconductor superlattices, dynamic localization has been suggested as the mechanism for an absolute negative conductance observed in photon assisted tunneling effects.3 A definitive evidence, however, is still lacking. A far more difficult problem concerns the dynamic localization in the presence of Coulomb interactions, unavoidable in optically excited semiconductor superlattices. The exis0163-1829/2004/70(3)/033309(4)/$22.50
tence of dynamic localization in the presence of Coulomb interaction has been suggested for semiconductor quantum well superlattices, within the framework of the semiconductor Bloch equations (SBE).4 It has been done in k-space, considering the same energy dispersion relations for electron and hole, for a unidimensional and infinite superlattice, and neglecting the carrier center-of-mass motion. Recently Zhang et al.5 have suggested that dynamic localization may not be observed in a pure ac field. Their results have been obtained using an excitonic basis, in k-space with zero center-of-mass momentum, to treat the dynamics. They have said, however, that for the pure ac field case, the unbound excitons (not included in their excitonic basis) play a much larger role and may dominate the dynamics. The aim of this paper is to show that a signature of dynamic localization can be seen unambiguously in the linear optical absorption spectra in an applied ac field, even in the presence of Coulomb interactions. We propose that such effect would be observable in a novel, realistic, and promising nanostructure, namely semiconductor finite quantum dot superlattice in a nanowire, as sketched in Fig. 1. Important steps towards the realization of such unidimensional heterostructures have been the focus of recent experimental investigations.6 It is worth noting that the system we are dealing with is very close to a heuristic 1D chain of atomic sites, in which excitonic spectrum reflects the miniband structure. It also has the essences of large, but finite, molecu-
FIG. 1. Quantum rod 3.5 mm long of InP and a superlattice of 20 In0.53Ga0.47As dots (InP barriers) in its central part.
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lar systems. The optical absorption coefficient is obtained from the SBE considering the electron and hole degrees of freedom. It takes into account the translational invariance breaking down of the system, which is in contrast to the usual approach of solving only the electron-hole relative motion. The SBE is extended to include the finite number of quantum dot potentials for electron Feszed and hole Fhszhd. Although we consider here all dots with the same thickness and confining potentials, our scheme can treat equally well a finite superlattice with quantum dots of different formats and arrangements in the superlattice. The system is handled as a finite quantum wire with infinity potential on the lateral and ending walls (see Fig. 1). Along the wire there is a potential modulation, due to alloy alternation, as in a usual superlattice. The system is under the action of a short optical pulse and an oscillating homogeneous electric field along the wire in terahertz frequency scale. The optical absorption coefficient that gives us the signature of the dynamic localization is obtained from the interband polarization. Its dynamic takes place on the coherent excitation regime, i.e., in femtosecond scale. Therefore the SBE is dealt within relaxation-time approximation. The space-time evolution of the interband polarization Psze , zh , td is given by
H
i"
J
d − H Psze,zh,td = − Estdd0dszh − zed, dt
s1d
where zeshd denotes the coordinate of the electron (hole) along the nanowire (see Fig. 1), d denotes the delta Dirac function, d0 the interband dipole, and Estd the optical pulse. H is the space- and time-dependent differential operator "2 ]2 "2 ]2 − − Vszh − zed + Feszed H = ˜Eg − i"G − 2me ]z2e 2mh ]z2h + Fhszhd − eszh − zedFzstd.
s2d
2D The effective band gap ˜Eg = Eg + E2D e − Eh is given by the material bang gap Eg and the lateral ground-state energies of electron and hole. The inverse of the dephasing time G accounts, within the relaxation-time approximation, for scattering processes causing dephasing of the optical polarization. The optical pulse is assumed as Estd = E0e−iv0t exph−s4 / s2dt2j. The ac field along the quantum rod is given by Fzstd = F0 cossVtd. We consider an infinite potential at the lateral wall of the thin nanowire, which is adequate considering a system based on free standing nano whiskers.6 That also implies a zeroorder Bessel function for the envelope of the lateral ground state wave function and, for the present, consider only the fundamental electron and hole subbands. The quasi-onedimensional Coulomb interaction Vszh − zed is obtained performing the average of the three-dimensional Coulomb interaction with the lateral ground state,7–9 namely
E E E E R
Vszh − zed = k
2p
R
dre
0
0
rerhJ20 3
2p
due
drh
0
duh
0
S DS D a 0r e 2 a 0r h J0 R R
Îszh − zed2 + r2e + r2h − 2rerh cossuh − ued , s3d
where k = e2 / f4p3«R4J41sa0dg, and R is the radius of the nanowire. The absorption coefficient is obtained from the total polarization Pstd = ok Pkstd = edze Psze , ze , td in the usual way calculating the complex dielectric function,7,9 namely
asvd =
« 9s v d v , c Î 1 h«8svd + Îf«8svdg2 + f«9svdg2j
s4d
2
where the real and imaginary part can be obtained from the solution of the SBE, Eqs. (1)–(3), i.e.,
H
«8svd = Reh«svdj = «bg + Re
H
«9svd = Imh«svdj = Im
J
Psvd , V«0Esvd
J
Psvd . n«0Esvd
s5d
s6d
We have investigated a superlattice of 20 In0.53Ga0.47As cylindrical quantum dots with radius R = 3.5 nm and 5.0 nm thick with InP barriers 6.0 nm wide. The superlattice lies on the central part of a InP nanowire 3.5 mm long. The electron effective mass is me = 0.042m0 and effective heavy hole mass mh = 0.429m0, with m0 being the free electron mass. The effective masses are considered to be the same for the well and barrier materials. The potential barriers, due to band-gap energy difference of the materials, are V0e = 245.0 meV for the conduction electron and V0h = 367.6 meV for the valence heavy hole. The electron (hole) potential Feshdszeshdd = 0 inside the dots and Feshdszeshdd = V0eshd in the InP barriers. The background dielectric constant is «bg = 13.9, and the interband dipole moment is d0 = 0.3 e nm. The width of the Gaussian optical pumping pulse is s = 40 fs and frequency centered at the effective material band gap, i.e., "v0 = ˜Eg = 1259 meV. The dephasing time is G−1 = 500 fs.12 We initially consider the absorption spectra in the absence of Coulomb interaction, i.e., for band-to-band absorption. The absorption spectrum reveals the underlying unidimensional-like density of states of the minibands, as it is shown by the solid line in Fig. 2 for the field-free case. A finite system with 20 quantum dots already has the main features of an infinite periodic superlattice. This absorption spectrum also indicates that the minibands are highly symmetric, a condition for an exact dynamic localization.10 Figure 2 also presents the ac field effects on the absorption spectra in the absence of Coulomb interaction. The absorption coefficient is shown for different ratios x = edF0 / "V, with fixed THz field frequency "V = 20 meV. The cusps at the edges of the absorption profile reflect the
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FIG. 2. (Color online) Optical absorption without Coulomb interaction. Solid line for free ac field; short dashed line for edF0 / "V = x = 0.5; dot-dashed line for x = 1.0; dotted line for x = 2.4, and two dots-dashed line for x = 3.5. The inset on the right shows the miniband width obtained from the optical absorption spectra (full circle) and from the formula (solid line) DE = J0sxdDE0, where DE0 is the miniband width for field-free and without Coulomb interaction. In the inset on the left we show the spectrum for the case with x = 2.4 (dotted line) and the spectrum for a single dot, without Coulomb interaction and ac field (solid line).
higher electron-hole joint density of states at the center and border of the Brillouin zone. Increasing x, up to x = 2.4, we can see a progressive shrinking of the optical absorption spectrum. The distance between the cusps at the edges of the absorption spectrum (signature of the conduction and valence miniband widths) reaches a minimum at x = 2.4. The optical absorption coefficient becomes single peaked at the same peak position, and with the same intensity, of the optical absorption coefficient for a single quantum dot, with the same radius R = 5 nm and 6 nm thick, in the absence of Coulomb interaction and ac field (see inset on the left). For x . 2.4, the miniband width first increases and subsequently decreases. This behavior can be clearly seen in the inset on the right of Fig. 2, where the distance between the cusps of the edges of the spectrum DE is shown as a function of x. The inset on the right shows also the analytical prediction for the evolution of dressed minibands given by DE = J0sxdDE0, where J0 denotes the zero-order Bessel function, and DE0 denotes the miniband width for the case without Coulomb interaction and that is ac field free. The striking agreement indicates that the finite 1D quantum dot superlattice is a suitable system for analyzing the effect of Coulomb interaction on the dynamic localization through optical absorption experiments. The absorption spectra as a function of the THz field intensity is drastically modified in the presence of Coulomb interaction. The oscillator strength is almost completely transfered from the minibands to the exciton and shifts (redshift) the optical spectrum (exciton binding energy). Therefore a direct evidence for photon dressed minibands could be hindered in optical measurements. Nevertheless, for a quasione-dimensional system, the excitonic dispersion relation
FIG. 3. (Color online) Optical absorption considering the Coulomb interaction. Solid line for free terahertz fields; short dashed line for x = edF0 / "V = 1.5; dotted line for x = 2.4; long dashed line for x = 3.5, and dot-dashed line for x = 5.52. The inset on the right shows the dynamical localization coefficient defined by h = sDE − DE1dd / DE1d, where DE is width of the excitonic spectrum (full width at half maximum) and DE1d the excitonic linewidth of a single dot for ac field free (inset on the left).
follows the dispersion relation of the minibands.11 In Fig. 3 we show the excitonic absorption spectra for a finite quantum dot superlattice under the influence of Coulomb interaction and of THz field. The excitonic spectra for different x = edF0 / "V with fixed THz field frequency "V = 20 meV are shown. In the absence of the THz field (solid line in Fig. 3), a characteristic asymmetric line shape for excitons in 1D periodic systems can clearly be seen. Turning on the THz field, several interesting features appear, as the changing of the linewidth of the excitonic peak (signature of the dynamic localization) and the appearance of satellite peaks on the abf sorption spectra. Those peaks, on the left at E − Eef g ef f . 75 meV and on the right at E − Eg . 95 meV (see Fig. 3), are replicas of the excitonic states dressed by terahertz photons.13 The linewidth of the excitonic peak is a function of the THz field strength. It decreases when we increase the THz field amplitude for a fixed frequency V. The linewidth reaches a minimum at x = 2.4 but its modulation cannot be related directly to J0sxd, since there is a finite minimum due to the finite lifetime of the exciton. A more reliable measurement of the dynamic localization is given by h = sDE − DE1dd / DE1d, where DE is the linewidth (full width at half maximum) of the excitonic spectrum and DE1d the excitonic spectrum of a single quantum dot in the absence of the THz field (inset on the left in Fig. 3). The quantity h gives the broadening of the excitonic peak relative to the linewidth of the excitonic spectrum of a single quantum dot. We can see the signature of the dynamic localization around x = 2.4, where DE . 1.2DE1d while for the field-free case DE . 1.8DE1d. Therefore, at the first root of the zero-order Bessel function x . 2.4, the dynamic localization is maximum, although it is not complete. In conclusion, we have investigated the dynamic localiza-
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tion in a linear and finite quantum dot superlattice with a longitudinal and homogeneous ac field in terahertz scale. We have found that the signature of the dynamic localization can be seen through the linear optical absorption spectra. The latter was obtained from the SBE, which has been extended to consider the nonrelative coordinates of the carriers in direct space, taken also into account a realistic and finite superlattive potential. Switching off the Coulomb interaction, we have found that the dynamic localization occurs and indeed follows the rule DE = J0sedF0 / "vd DE0 for the mini-
1 D.
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band collapse. Switching on the Coulomb interaction, the dynamic localization still leads to the spectrum collapses at the zeros of the zero-order Bessel function, however it occurs partially s.80%d. We propose that the observation of this effect can be realized in unidimensional superlattices that are currently being developed. We thank Fernando Iikawa and Marcos H. Degani for helpful discussions. J. R. M. and P. A. S. acknowledge financial support from the Brazilian agency FAPESP.
7 H.
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