Decoherence effects in the intraband and interband ...

JOURNAL OF APPLIED PHYSICS 105, 122412 共2009兲

Decoherence effects in the intraband and interband optical transitions in InAs/GaAs quantum dots R. Ferreira,1,a兲 A. Berthelot,1 T. Grange,1 E. Zibik,2 G. Cassabois,1 and L. Wilson2 1

Laboratoire Pierre Aigrain, Ecole Normale Supérieure, 24 Rue Lhomod, F75005 Paris, France Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom

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共Received 14 October 2008; accepted 16 April 2009; published online 18 June 2009兲 We present a review of coherence properties of interband and intraband optical transitions in self assembled InAs/GaAs quantun dots. Indeed, recent experimental and theoretical investigations of the optical transitions in both spectral domains have allowed a better understanding of the different phenomena that affects the interaction of confined carriers with light. These studies point out the many different ways the electron-phonon interactions play a role on the optical response of quantum dots. They also stress the primary role of the close environment on the coherence characteristics of quantum dots. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3130926兴 I. INTRODUCTION

In the past decade, many efforts have been devoted to the study of the energetics, population relaxation, and decoherence of optical transitions in quantum dots 共QDs兲. The most attracting feature of QD energetics is its discrete sequence of states with localized orbitals.1 This characteristics of the QD spectrum is at the origin of the so-called “macroatom” picture. For many purposes, this analogy with atoms is very useful, but, as we will see below, a QD is actually a genuine condensed matter system, with many interesting properties that are all ultimately related to the fact that we have a strong confinement in a crystalline matrix. In other words, we can also say that in a QD the primary effect is a volume effect 共since dots are small objects兲, but most of the specificities of a QD comes from its solid-state nature. In particular, the latter is responsible for specific features in the QD optical characteristics. Let us quote two significant examples: 共i兲 it is well established that the presence of a small in-plane anisotropy is of primary importance for the polarization selection rules of the optical transitions: the origin of such anisotropy is to be related either to a geometrical effect2 or to effects inherent to the underlying crystalline lattice 共such as strain3 or atomistic4 effects generated by lattice mismatch兲; and 共ii兲 it has been demonstrated that the population 共or energy兲 relaxation is to be related to the presence of phonons and to the intrinsic anharmonicity of the crystal lattice vibrations 共see below兲. In this paper we discuss some recent theoretical and experimental studies pointing out the importance of decoherence processes in the optical characteristics of QDs.5–11 We will focus on two specific cases: the ground interband 共Sec. II兲 and the ground intraband 共Sec. III兲 transitions of selfassembled InAs/GaAs QDs 共SAQD兲. When interested in decoherence effects in QDs, one has to properly account for the couplings between confined carriers and different reservoirs. Here, we will concentrate on two major sources of decoherence, namely, on the role of nearby environment and of lat-

tice vibrations on the optical decoherence and consequently also on the line shape and linewidth of the optical spectra of individual dots. II. INTERBAND PL SPECTRUM

The ground interband transition of QDs is by far the most intensively studied. This stems from the quest for a system with a small number of degrees of freedom and exhibiting controllable nonlinear optical characteristics, which should play the role of elementary brick in many different quantum devices, in particular working in the coherent optical regime. From the point of view of optical processes, one may wonder whether a QD may fit to one ideal two-level system. A very large amount of optical experiments have been so far reported for many different kinds of QDs. Tremendous advances in single dot spectroscopy techniques have been achieved the past decade. Such high-resolution 共in energy and position兲 studies revealed in particular a fine structure for the photoluminescence 共PL兲 spectrum of individual QDs: its line shape is not a Lorentzian line with linewidth limited by its sole radiative lifetime, as one would naïvely expect from the atomic analogy. As we will see later, this ideal profile exists only under strict experimental conditions 共very low temperature and quasiresonant and lowintensity excitation兲. Actually, the PL spectrum of individual QDs very often displays a much richer phenomenology, displaying many features to consider:5–7 共i兲共ii兲共iii兲

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first the existence of a broad pedestal, in its central part a sharpen Lorentzian line whose spectral width can nevertheless be much larger than the radiative limit, even at very low temperature, and finally, both the width of the Lorentzian and the height of the pedestal are temperature dependent. The spectral width presents a monotonous increase with temperature, which is linear at low T and becomes roughly exponential when T increases above some tens of kelvins. The height of the pedestal increases with T and at high temperature the latter dominates the PL spectrum. © 2009 American Institute of Physics

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FIG. 1. PL spectra of a single InAs/GaAs QD in the temperature range 5 –80 K, normalized to the spectrally integrated intensity for each temperature. Curves are calculated profiles in the framework of the Huang–Rhys model. Inset: temperature dependence of the zero-phonon linewidth 共Ref. 7兲.

These features are clearly seen in Fig. 1.7 The origin of some of these effects is nowadays well understood while others are still subject of study or have obtained a satisfactory explanation only recently. We discuss them separately in the following. Let us start with the broad pedestal feature. There is nowadays no doubt in relating the existence of a pedestal in PL lines to the coupling of confined electron/hole pairs with low energy acoustical phonons. Moreover, it is also well established that this coupling is the source of an intrinsic decoherence for the optical transitions. Let us initially recall the main physical aspects of this coupling, before going to a different aspect of the electron-phonon interaction in QDs, namely, how irreversibility effects on the interband optical transition appears from the point of view of the lattice dynamics. Electrons in semiconductors interact with longitudinal acoustical modes, which correspond to harmonic oscillations of the center-of-mass of the two atoms of each unit cell around their equilibrium positions. The ensemble of the equilibrium sites form a lattice that is periodic when the QD is empty, but no longer periodic in the QD region when it hosts one electron/hole pair, because of the electron-phonon interaction. This nonperiodicity can be identified to a distortion of the lattice in the QD region when the dot is occupied. It results that one has different phonon modes for the empty and for the occupied QD. This has direct consequences on the PL spectrum of a QD. Indeed, when evaluating the matrix element for the radiative dipolar interaction, which operates only on the electronic degrees of freedom, one has a by-product term to tackle, which is the overlap of the different phonon modes for the occupied and empty QD. Also, we have to consider a thermal distribution for the phonon popu-

lation in the initial configuration. Finally, one has to take into account in the delta for energy conservation a possible variation of the lattice energy during the recombination process. Altogether, these three ingredients provide an elegant explanation to the existence a broad pedestal in the PL spectrum of an individual dot: it reflects a reaction of the lattice to the electron state variation 共in this case a luminescence process兲. This is the famous Huang–Rhys model proposed in the 50’s to interpret similar effects observed in bulk materials containing deep impurities.12,13 So far, we have considered how the coupling to lattice vibrations affects the radiative spectrum. Let us now point out the existence of a reciprocal effect, and consider how the photoexcitation of one electron/hole pair affects the lattice vibrations. In fact, the existence of a phonon-related feature in the 共time-independent兲 PL spectrum has a dynamical correspondence: Vagov et al.14 were the first to point out that the lattice reacts dynamically to one interband excitation. First of all, as mentioned previously, the lattice becomes deformed when the QD contains one electron/hole pair. Vagov et al.14 have translated this distortion in terms of phonons occupation number. The Fig. 4 of their work14 shows the phonon occupation as a function of the distance from the center of the QD for a spherical dot of a few nanometer radius. Soon after the dot becomes occupied 共at t = 0兲, one observes the rise of a distortion inside the QD region. When times goes on after the interband photoexcitation 共i.e., for t ⬎ 0兲 we observe both the establishment of a time-independent distortion inside the QD 共more clearly seen in the inset兲 and a phonon wavepacket leaving the QD region and propagating freely with the sound velocity. The latter “phonon wind” represents a dynamical irreversible reaction of the lattice to the photocreation of one bound electron/hole pair process. This reaction is the fundamental reason why coupling to acoustical phonons leads to an intrinsic irreversibility for the optical transitions in a QD. The irreversibility of the optical transition is more clearly seen in FWM experiments such as the ones performed by Borri et al.6 In the measurements, the dephasing processes that affect the optical transition lead to a decrease of the FWM signal. Borri et al.6 observed that the TIFWM signal of an ensemble of QDs presents three salient features: 共i兲共ii兲共iii兲

at short times a fast nonexponential decay; at longer times a quasiplateau, characterized by a slower 共T-dependent兲 exponential decay; and the relative ratio of the two regimes is a function of the temperature.

These three observations can be directly related to the different features observed in the cw PL experiments: the broad pedestal generates a fast initial decay of the FWM signal, while the slow decay at longer times is due to the thinner Lorentzian part. Moreover, the relative importance of the two regimes in both experiments reflects the increase with T of the pedestal height. In conclusion, as far as dephasing is concerned, the phonon-related pedestal leads to a fast but partial decay of the coherence, whereas the remaining contribution to decoherence comes from slower processes affecting mainly the central Lorentzian line.


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FIG. 2. 共Color online兲 Interferogram contrasts of the PL signal of a single InAs/GaAs QD at T = 10 K, on semilogarithmic plots for three different incident powers: 0.18 共a兲, 0.72 共b兲, and 2.88 共c兲 kW cm−2. Data 共squares兲, system response function 共dotted line兲, theoretical fits 共solid line兲共Ref. 9兲.

FIG. 3. 共Color online兲 Variation of Kubo’s line shape parameter 共see text兲 obtained from a fit of contrast data for a single QD showing a crossover from a Lorentzian 共l兲 to a Gaussian 共g兲 line when increasing the excitation power 共left panel兲 or the lattice temperature 共right panel兲共Ref. 9兲.

Let us now discuss the width of the PL central line, the so-called zero-phonon line 共ZPL兲. Different experiments6,7,9,10,15–18 have been performed and some theoretical studies17,19 have been realized in order to understand the behavior of the low temperature width of the central ZPL of self-assembled QDs: ⌫共T兲 ⬇ ⌫0 + A0T 共in this paper we concentrate on this linear variation兲. It is worth to recall that ⌫0 can be 10–100 times the intrinsic broadening ⌫rad⬇ few meV for an ideal QD 共see inset of Fig. 1兲. We briefly present in the following the main points of our work. We have considered a model based on the spectral diffusion effect, which provides a satisfactory interpretation for both the very low temperature width 共⌫0兲 and also 共and simultaneously兲 for the linear increase with temperature 共A0兲 measured for the linewidth of QDs excited nonresonantly. The model relies on the presence of carriers in the WL and barrier, in addition to the electron/hole pair inside the QD.9,10 Let us briefly sketch how such an ingredient unavoidably brings spectral diffusion into play. We additionally assume that outside carriers undergo random trapping/detrapping processes in defects located nearby the QD. These processes ultimately affect the energy of the confined electron/hole pair by electrostatic interaction: free and trapped carriers do not interact with the same strength with the confined pair. Then, finally, radiative recombination in a QD occurs in a spectral diffusion regime, as due to such a dynamical Stark shift of the energy of the recombining electron/hole pair. If we assume Gaussian fluctuation for the shift of the electron/hole energy, then we have only two parameters to take into account: the correlation time and the average amplitude of the fluctuations. This situation corresponds to the well-known Kubo’s20 stochastic theory of line shape, for which the line shape is exactly known. We know also very well that this theory predicts a spectrum that evolves from a Gaussian toward a Lorentzian when the electrostatic fluctuations becomes faster and faster. We show in Fig. 2 the results of contrast measurements performed by Berthelot et al.9 in single QDs under different experimental conditions 共the sample is nonresonantly excited above the GaAs gap兲. As shown in Fig. 3, we clearly observe a crossover from a Lorentzian to a Gaussian line when increasing the excitation power 共left panel兲 or the lattice temperature 共right panel兲, this transition being characterized by the product of the correlation time by the modulation amplitude 共in units of ប兲. The observation of such a crossover

strongly corroborates the assumption of a ZPL governed by spectral diffusion effects. Note however that the crossover is nonconventional, in the sense that it goes from a Lorentzian line at low power and low temperature toward a Gaussian line at high excitation power and high temperature, and not the contrary as one could naively expect. This unexpected behavior can nevertheless be understood in the framework of our simple model, which takes into account trapping and detrapping rates that are temperature and power dependent. Indeed, capture and detrapping processes are expected to be dependent both upon temperature, via the electron-phonon coupling, and excitation power, via Auger scattering involving the density of photoinjected carriers. We will not detail the calculations, which are given in Refs. 9, 10, and 21. It is however worth mentioning that this model leads to a low temperature width of the central line that is linear in temperature. Moreover, the model allows to express the two constants ⌫0 and A0 in terms of dot dependent parameters, as well as parameters characterizing the environment fluctuations 共essentially the density of defects, the rates of capture and detrapping, and mean fluctuation of the electron/hole energy兲. Thus, the fit of the measurements brings information about the QD near environment. Note however that our model cannot provide detailed information about this environment 共which is most possibly sample dependent兲. For instance, for the experiment in Fig. 2, the model leads to the existence of ⬇100 trapping sites around the QD, but their nature cannot be extracted from the analysis of the data, and further studies remain to be done aiming a better identification of the QD environment. In conclusion, significant advances have been obtained these past years, both on the experimental and theoretical sides, concerning the optical response of individual selfassembled QDs. These studies provide a solid basis for the build up of a comprehensive description of many different phenomena affecting the interband optical transitions and their coherence properties. In particular, it emerges from these works the specific role of both the external 共WL and barrier兲 and internal 共crystal lattice兲 environment of a QD on its optical characteristics. Depending on the actual experimental conditions, these can become activated and perturb the properties of dots hosting one electron/hole pairs that, otherwise, would behave as solid-state macroatoms.


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FIG. 4. 共Color online兲 cw absorption spectra of one ensemble of dots doped with one electron on the average per dot for light polarizations along the 关0 ¯ 1兴共S → Px transition兲. 1 1兴 direction 共S → Py transition兲 and 关01

III. COHERENCE IN THE FAR-INFRARED: INTRABAND TRANSITIONS IN DOPED DOTS

Let us turn now to the intraband transitions in QDs. For intraband spectroscopy the samples are doped with one electron per dot on average. Samples of SAQDs display inhomogenous features. We take advantage of the inhomogeneous distribution of sizes in the dot ensemble to selectively excite and probe dots with different sizes, more precisely, with different separation between the ground 共S兲 and first excited 共P兲 energy shells.22,23 These shells contain respectively one and two levels. Moreover, SAQDs have usually a small in-plane anisotropy 共either of geometrical,2 strain,3 or atomistic3 origin兲, which splits the two P states into a doublet labeled Px and Py in the following These states are close in energy and are excited by light with different 共and orthogonal兲 in-plane polarization, and we take advantage of this polarization selection rule to selectively excite and probe only one of the two S-to-P transitions. Depending on the dot parameters, the energy separation between the S and P shells is ⌬SP⬇ tens of meV, and the Px-to-Py anisotropy splitting is ⌬ PP⬇ few meV. In the experiments so far reported, the energy h␯ of photons probing the S-to-Px energy was in the interval ប␻LO ⬍ h␯ ⱕ 55 meV, where ប␻LO = 36 meV is the longitudinal optical 共LO兲 phonon energy, while for such samples ⌬ PP was in the 5–8 meV range. We show in Fig. 4 the cw absorption spectrum of two samples, which illustrate these different features related to the stationary states of QDs 共absorption anisotropy between two orthogonal directions, ⌬SP Ⰷ ⌬ PP, and inhomogeneous broadening for ⌬SP of ⬇5 meV兲. As compared to the interband case, much less has been done for the coherence properties of intraband transitions.8,11 It is worth mentioning the experiments performed by Bras et al.8 in the regime of strong coupling to light, which demonstrated the existence of a fast decoherence for the intraband transition. Here we discuss the more recent results by Zibik et al.11 to study the coherence of the S-to-Px intraband transition. This work presents results of pump-probe and FWM measurements in the far-infrared 共FIR兲. Figure 5共a兲 shows the FWM signal measured at different temperatures for the sample with ⌬ PP = 5 meV in Fig. 4, excited by pulses centered at h␯ = 53 meV. We observe many similarities with the

FIG. 5. 共Color online兲 Temperature dependent four wave mixing signals for the ensemble of dots: experiments 共a兲 and simulations 共b兲 from Ref. 11.

signal obtained for interband transition in the band-gap region, in particular a fast nonexponential decay at early times followed by a slower exponential evolution at longer times. Moreover, both the initial and the slower decay rates increase with increasing temperature. We present as symbols in the Fig. 6 the measured variation of the exponential decay rate with temperature. Finally, we also see a small oscillatory feature superimposed to the fast initial decay. In order to interpret these results, we have to take into account different sources of decoherence for a single QD out of the QD ensemble: 共i兲共ii兲共iii兲

population decay toward ground state 共T1兲, real and virtual transitions toward the higher in energy Py-state 共Tⴱ2兲, and acoustic phonon sidebands 共partial decay of the coherence兲. These different contributions are discussed separately in

FIG. 6. 共Color online兲 Temperature dependence of the homogeneous linewidth of the fundamental intraband transition of one ensemble of dots: experiments 共squares兲, calculation including real 共real line兲, virtual 共“virtual” line兲, and both virtual and real 共solid line兲 transitions. These three theoretical curves contain the polaron lifetime contribution 共“⌫2” curve兲.


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the following. As we show below, these contributions to decoherence differ radically from those for interband transitions, either qualitatively or quantitatively. A. Population decay

The mechanism responsible for energy relaxation in QDs has been for long time a subject of discussion. It is nevertheless nowadays well established that intradot relaxation of excited carriers follows from the consideration of two main ingredients: 共i兲 carriers confined in a dot strongly couple to LO phonons and form polaron states,24,25 and 共ii兲 LO phonons entering in the formation of such polarons have actually a finite lifetime, because of the intrinsic anharmonicity of the crystal lattice.26 Polarons would be stationnary states of the charged dots, but anharmonicity render them unstable and trigger the energy relaxation in SAQDs.27–30 Actually, anharmonicity acts as a small perturbation, which couple different polaron states and brings into play reservoirs of two-phonon states. Thus polaron energies acquire a small imaginary part, which has been evaluated using the Fermi golden rule in Refs. 28–30. As regards the coherence properties, population decay leads to a diagonal contribution to the total dephasing rate. Pump-probe experiments,22,23 corroborated by model calculations, have shown that T1⬇ few tens of picoseconds in SAQDs. This polaron lifetime is almost two orders of magnitude shorter than the electron/hole recombination time 共usually in the nanosecond time scale兲. This unavoidable contribution imposes a very stringent limit to the coherence of the FIR response of doped dots. The calculated temperature dependence of the polaron lifetime contribution to decoherence is shown in Fig. 6 共⌫1 line兲 and compared to the measured long time decay rate.29,30 As we see, polaron lifetime limits the coherence of the S-Px dipole at low temperatures, but cannot explain the important temperature dependence observed experimentally. Note finally that a radiative relaxation channel also exists in the intraband case, but this contribution is negligible 共its characteristic time is much larger than the interband one, despite the large S-to-P dipole, because of the unfavorable ratio of the optical/ FIR frequencies兲. B. Real and virtual transitions toward Py

The coherence of the S-Px dipole has another important contribution coming from the proximity of the Py state. The latter is actually only blueshifted a few meVs from Px. This energy is accessible to acoustical phonons, and thus a direct population transfer is possible by phonon absorption. The corresponding contribution of such absorption processes to the decoherence rate is shown in Fig. 6 as a function of temperature 共altogether with the previously obtained polaron lifetime contribution; “real” line兲. Phonon absorption gives an important contribution to decoherence, which is strongly dependent with temperature for typically T ⬎ 10 K 共contrarily to polaron lifetime, which is only weakly dependent on T兲. However, as we see in the Fig. 6, this contribution is not enough to account for the measured decoherence time. Actually, the proximity of Py brings about another important contribution to decoherence, namely, the existence of virtual

FIG. 7. 共Color online兲 Calculated InAs single dot intraband cw absorption spectrum at different temperatures. The energy origin is taken at the zerophonon-line center.

transitions that affect the phonon states but let unchanged the electronic 共more correctly the polaron兲 state. The latter can be related to scattering processes that become available for acoustical phonons after populating the Px state by the FIR excitation. Indeed, if one considers the situation from the point-of-view of the lattice, the excited dot acts as a scattering center, and one incoming acoustical phonon not having the right energy to be absorbed can nevertheless be scattered, a process that becomes enhanced by the proximity of the Py state. This second order process 共its amplitude involves one creation and one annihilation兲 recalls the quasiresonant scattering of photons involving virtual intermediate atomic states.31 As a consequence, both real and virtual transitions should be considered. The role of virtual processes on the decoherence was initially studied by Takagahara32 共who considered large “interface-related” QDs兲 and by Muljarov and Zimmermann19 共for SAQDs兲. After, Muljarov et al.33 proposed a nonperturbative theory to tackle simultaneously both real and virtual processes in QD molecules. These works were done for optical interband transitions. More recently, we developed a different nonperturbative treatment that allows working out the two contributions at the same footing from the very beginning and applied it to the study of the intraband coherence.29,30 The outcome of such calculations is presented as a solid line in Fig. 6 共⌫2 line兲, and compared to experimental results 共symbols兲 obtained in a large temperature range. We see that none of the three contributions 共polaron lifetime, real and virtual transitions兲 alone provide a satisfactory explanation to the whole set of experimental data: polaron lifetime introduces a lower bound to the decoherence rate that applies at low temperature, while thermally activated real and virtual channels are equally important and altogether dominate the high temperature behavior of the dephasing. C. Acoustic phonon sidebands and partial decay of the coherence

As described in the previous section, acoustical sidebands in the optical response follow from the existence of different phonon modes for the initial and final states. Intraband excitation presents the same phenomenology. We show in Fig. 7 the calculated the cw FIR absorption spectrum of an individual dot and show that it is also formed by a central


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ZPL and a broad pedestal.29,30 The latter however presents a minimum at the energy position of the ZPL, whereas it is a maximum in the interband case. The resulting “camelbacklike” shape of the sidebands leads in time domain to oscillations of the coherence, which are nevertheless strongly damped since the sidebands are spectrally broad. We show in Fig. 5 the simulations of the FWM signal at different temperatures, taking into account both the previously discussed T-dependent broadening of the central line and the effect of phonon sidebands. Phonon sidebands are thus responsible for the fast nonexponential oscillatory decay of the FWM signal at early times, whereas the exponential decay measured at long time is governed by the ZPL. IV. CONCLUSION AND PERSPECTIVES

We have discussed in this paper the coherence properties of the fundamental interband transition of one single undoped QD and of the first intraband transition of one QD hosting one electron. The decoherence time in such strongly confining objects differs in many respects from the simple T2 = 2T1 relation expected for an isolated macroatom 共where T1 is the radiative decay time兲. In the interband case, the macroatom picture is expected to hold only in the particular case of low temperature 共to eliminate phonon-related effects兲, quasiresonant and low-power light excitation 共to avoid photoexciting free carriers nearby the QD兲. In the intraband case, on the contrary, the lifetime of the excited state is not radiative, but limited at low temperatures by the polaron anharmonic disintegration. The latter is inherent to the crystal lattice and thus cannot be in principle overcome. The diverse experimental data, corroborated by modeling, point out the important role of phonon coupling on the optical response of QDs. It should be remarked that phonon effects were initially expected to be quenched in QDs, because of energy discretization of confined carriers. Phonons instead bring a very reach phenomenology to the dot states. Indeed, a full understanding of many of its electronic and optical properties cannot be achieved without accounting for a variety of couplings: 共i兲 strong coupling, such as the one leading to the formation of polarons; 共ii兲 nonperturbative effects such as the formation of particular phonon modes in the Huang–Rhys model and the existence of sidebands in the optical spectra, and 共iii兲 perturbative influence, leading to broadening and irreversible effects such as the ones related to real and virtual transitions in the intraband experiments and to the anharmonic polaron lifetime. Self-assembled InAs dots are model structures, but these effects are also expected to occur in dots made of different materials. Essentially, the ingredients are the existence of a discrete sequence of energy level for the confined carriers and coupling to phonons. Polar materials 共II-VI, GaN, and its alloys,…兲 are expected to have enhanced polaron couplings. However, material parameters 共such as LO phonon frequencies and anharmonic decay time兲 are usually not well known in such materials. Optical studies in QD could provide some important hints on these parameters. Concerning acoustical phonons, it may be also interesting to consider dots made of different well and barrier materials. Indeed, in this case pho-

non wind escaping out of the dot region would no longer be possible because of the important material mismatch.

ACKNOWLEDGMENTS

The Laboratoire Pierre Aigrain is “unité associée au Centre National de la Recherche Scientifique 共Contract No. UMR8551兲, Ecole Normale Supérieure 共ENS-Paris兲, Université Pierre et Marie Curie 共Paris 6兲 et Université Denis Diderot 共Paris 7兲.” We gratefully acknowledge Ph. Roussignol, M. Skolnick, and G. Bastard for stimulating discussions. 1

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