Carrier lifetimes and localisation in coupled GaAs- GaAlAs quantum wells in high electric fields

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1 Journal of Physics C: Solid State Physics Carrier lifetimes and localisation in coupled GaAs- GaAlAs quantum wells in high electric fields To cite this article: E J Austin and M Jaros 1986 J. Phys. C: Solid State Phys View the article online for updates and enhancements. Related content - Electric field effects on spectroscopic lineshapes in GaAs-GaAlAs quantum wells E J Austin and M Jaros - Electronic properties of semiconductor alloy systems M Jaros - Theory of quantum wells in external electric fields D P Barrio, M L Glasser, V R Velasco et al. Recent citations - Optically induced level anticrossing in undoped GaAs/AlGaAs coupled double quantum wells Y.H. Shin et al - Nonlinear optical transitions of GaAsAlGaAs asymmetric double-well structures E. H. Kim - Shallow donor impurities in different shaped double quantum wells under the electric field E. Kasapoglu and I. Sokmen This content was downloaded from IP address on 22/08/2018 at 21:13

2 J. Phys. C: Solid State Phys. 19 (1986) Printed in Great Britain Carrier lifetimes and localisation in coupled GaAs-GaAlAs quantum wells in high electric fields E J Austin? and M Jaros School of Physics, The University, Newcastle upon Tyne NE1 7RU, UK Received 10 June 1985, in final form 17 July 1985 Abstract. An exact numerical calculation for a one-dimensional effective-mass model of a double GaAs-GaAIAs quantum well structure subjected to a strong electric field is presented. Both the Stark shifts and the field-induced tunnelling rates associated with the confined resonances are obtained. In particular, in fields in the range (2-5) x lo5 V cm-i the apparent hole ground state ceases to be the longest-lived level. This stabilisation of a higherenergy state can be interpreted in terms of field-induced charge localisation. The process of broadening of the bound states into resonances depends strongly on coupling to the continuum, and confined field-induced anti-resonances appear above the top of the barrier. 1. Introduction The effects of high electric fields on the electronic structure of quantum well systems is a topic of considerable current interest. A number of approximate calculations (Miller et a1 1984, Bastard er a1 1983) have been performed in which the confined states are viewed as bound states and their field-induced broadening is ignored. Recently we have presented strong-field calculations for an isolated quantum well (Austin and Jaros 1985a, b, hereafter referred to as I, 11). This calculation is exact (within the limitations of the one-dimensional effective-mass approximation) and predicts qualitatively new phenomena not obtained from the bound-state calculations. For an isolated quantum well the Stark shifts predicted from this calculation are found (11) to be in reasonable agreement with experimental results (Alibert er a1 1985) and the width calcualtions allows the observed (Kash et a1 1985) decrease of luminescence lifetime with field to be interpreted in terms of the field-induced tunnelling of the carriers. The extension of this calculation to coupled wells is clearly of interest in view of current developments in the employment of multi-well systems in novel devices (Linh 1984). In these devices the applied fields are high and a non-perturbative treatment of the effects of the field on the electronic structure is thus required. These field effects have significant implications for the calculation of high-field transport properties (see, for example, Barker 1973, Herbert and Till 1981, Ridley 1982, Ferry 1982, Hess 1982) for such systems. A full theoretical treatment of coupled quantum well systems, taking into account the full details of the band structure of the GaAs-GaAlAs systems and the field effects,? Present address: Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 OQX, UK /86/ $ The Institute of Physics 533

3 534 E J Austin and M Jaros has not to date been attempted. Considerable theoretical difficulties are involved in modelling the effect of the applied field on the electronic structure in a self-consistent manner, whilst also taking into account the details of the zero-field band structure. In view of the problems associated with a full calculation, it is of interest to study the simple model presented here, although its validity is limited at high fields (11). The qualitative features of the results obtained are characteristic of a system comprising a small number of bound states coupled to a continuum, so it is hoped that these would be paralleled in large-scale calculations. In this paper both symmetric and asymmetric double-quantum-well systems are studied. For symmetric double wells the Stark shifts and lifetimes are obtained. The effect of the applied field on the wavefunction localisation properties of the system are also considered. A consequence of field-induced charge localisation is that in fields in the range (2-5) X 10' V cm-' the apparent hole ground state ceases to be the longestlived level; the first excited state (which has a more favourable spatial localisation) becomes more stable. In addition, field-induced anti-resonance states appear at energies above the top of the barrier; these states can be related to the zero-field virtual bound states. The field dependence of the anti-resonance states provides a simple model for the behaviour of the confined above-barrier states found in large calculations (Jaros et a1 1985). 2. Resonance calculations for double-well systems The double-well system considered is illustrated in figure 1. In the approximation employed here, this structure can be described by the one-dimensional Schrodinger equation for a particle of effective mass m*. This is h2 d2y2 2m* dx2 efxy2 = Eyy, where F is the electric field applied perpendicular to the interface plane. Changing to scaled atomic units (denoted by primes) and applying the coordinate transformation w Figure 1. (a) The double-well structure used in the calculations. (b) The effect of the field on this structure.

4 GaAs- GaA 1A s q uan rum w el Is 535 y = ( 2~713~ + [2/(2~')2/3 ~(v;, + E') y = ( 2~7~1~~' + 2 ~'/(2~')~'~ allows (1) to be written in the standard form d q/dy* = -yv (3) with solutions that are linear combinations of Airy functions Ai( -y) and Bi( -y). The numerical values of the coefficients of the Airy functions are readily found by requiring q and its derivative to be continuous at each interface. As in the case of the single-well system, studied in I and 11, the Stark resonances can be obtained by applying the method of phase-shift analysis. In region 2 on the extreme right of figure l(b) the wavefunction takes the form in 1 q = c1 Ai( -y) + c2 Bi( -y) (4) and the phase shift is given by tan q = c2/c,. Since the potential term in (1) tends to --x. as x+ -x. there are no bound states in the presence of the field; the zero-field bound states of the quantum well system evolve into Stark resonances. The phase shift (5) can be used to give information about these resonances. The resonance positions and widths can be fitted to a Breit-Wigner formula; in addition the phase shift can be related to the change in density of states via Ap(E) = (2/n)dq(E)/dE. (6) As the solutions to (1) are based on Airy functions, which describe the behaviour of a free charged particle in an electric field, Ap is to be interpreted as the change in the density of states induced by adding the potential well to the free-particle system. In I it was shown that for the single-well system in high fields the Stark resonances are broad and asymmetrical, so the Breit-Wigner parametrisation is clearly only of limited validity. It seems likely that the experimental spectroscopic observations on Stark shifts relate to the position of the density-of-states maximum; in this work we therefore adopt this definition of the resonance position. (An alternative approach to this system has been described by Kelly (1984); in this work a transmission coefficient calculation based on the Airy function solutions (3) was used to obtain current-voltage curves for quantum well and related structures.) in 2 (2) (5) 3. Stark shifts and wavefunction localisation for a symmetric well As in previous calculations (I, 11) we choose parameters that are appropriate to holes in GaAs-GaAlAs quantum wells; due to the smaller binding energies of holes compared with electrons in these structures, the largest field effects are found for holes. For the symmetric double-well system the parameters used were al = a2 = 30A V, = 70 mev L=30A m* = 0.45 me. An isolated quantum well with these parameters supports a single bound state of energy mev. In the double-well structure, tunnelling occurs between the two wells and (7)

5 536 E J Austin and M Jaros the zero-field eigenstates are the symmetric and antisymmetric linear combinations of the left- and right-hand well states yz = 2-' 2(yL 5 ylr). (8) Here q+ is the ground-state wavefunction; in both states the particle has an equal probability of being found in either well. The small splitting in energy between I$+ and Y- A= E- -E+ is caused by the quantum tunnelling. (9) F (io5 v cm-') Figure 2. Stark shifts and energies for the upper (U), lower (L) and anti-resonance (A) states for the symmetric double well. A second anti-resonance (not shown) appears at 9 x lo5 V cm-'. Study of the wavefunctions confirms that the states U and A cross without interacting. In the study of field effects two peaks are found in the density-of-states plots, corresponding to the two zero-field bound states (8). Figure 2 shows the variation in the positions of the density-of-states maxima as a function of field. In the low-field region the two states undergo a linear Stark shift. This behaviour is readily predictable on the basis of a simple perturbation theory argument. In the presence of the field the coupling of the two near-degenerate bound states dominates and the effective two-level system is described by the Hamiltonian matrix where v= -(vj+iefxlv-) (11) and E is the average energy of the unperturbed states. The eigenvalues of (10) can be obtained as E = E? $(A2 + 4V2)'/' (12)

6 which reduces to GaAs-GaAlAs quantum wells 537 E=EkV for A e V i.e. provided that the effect of the applied field dominates the zero-field tunnelling splitting. By considering the eigenfunctions of (10) it can also be predicted that the application of the field will induce wavefunction localisation; the lower state is predicted to localise in the right-hand (low-energy) well and the upper state in the left-hand (highenergy) well of figure l(b). Inspection of the resonance wavefunctions confirms that this localisation occurs. Similar localisation effects have been found (Lang and Nishi 1984) in calculations on asymmetric double-quantum-well structures, where the asymmetry takes the form of a small difference in the parameters (width or depth) of the two wells. The simple analysis given above allows qualitative prediction of the variation in the Stark shifts and splittings and in the degree of wavefunction localisation for different choices of the parameters in (7). For structures with large tunnelling splittings (very narrow wells and/or barriers) the linear Stark shifts found here would be replaced by quadratic Stark shifts. In the linear Stark regime the magnitude of the shift is proportional to the matrix element V and thus increases with a; the wavefunction localisation effects depend on the dominance of the interaction with the field over the zero-field tunnelling and will thus be most significant for systems with a small value of the tunnelling probability exp{ -[2m*( 1 v0 1 - I E / ) 1 2~/fi]}. (14) The occurrence of the large Stark shifts illustrated in figure 2 at experimentally attainable fields shows that field effects are significant in this type of structure and can lift the neardegeneracy of the zero-field levels. By analogy the mini-bands present in multi-well systems at zero field would be expected to undergo significant Stark splitting. 4. High-field effects and Stark widths The analysis of the preceding section was essentially perturbative but various features of the results obtained illustrate that, as for the single-well case (I), perturbation theory breakdown occurs at experimentally attainable fields. The deviation from linearity of the energy shift plots in figure 2 at high fields (particularly for the upper state) provides one example of this effect. The conversion of the bound states into resonances with finite lifetimes, discussed in more detail below, itself indicates a strong coupling to the continuum and breakdown of the two-level approximation. In addition, as in the singlewell case (I), the coupling to the continuum is shown explicitly by the appearance of anti-resonance states. An anti-resonance state corresponds to a decrease of n in the phase Q, and is thus (from (6)) associated with a negative contribution to Ap; these states can be identified with virtual bound states. In the single-well problem it was found (I) that an anti-resonance state is drawn from the continuum into the well to annihilate formally each resonance at high fields. For the case of a single well supporting one bound state, study of the anti-resonance wavefunction shows that this state has a node at the well centre and is similar in appearance to the first odd-parity virtual bound state of the zero-field system. The existence and field dependence of these localised above-barrier states is of interest, since such states are known to play a role in carrier capture and in optical transitions (Bastard er a1 1984). In the double-well system described here antiresonance states also appear. The behaviour of the lower anti-resonance state as a

7 538 E J A ustin and A4 Jaros function of field is illustrated in figure 2 and a plot of the low-field wavefunction in figure 3. This wavefunction has the apgearance of a field-distorted version of the first evenparity virtual bound state of the zero-field system. A significant degree of localisation in the barrier is apparent; this state thus has the character of the above-barrier states found in large-scale (zero-field) calculations (Jaros et af 1985). An effect of the double-well structure is that the field dependence of the anti-resonance energy is qualitatively different from that of the anti-resonance observed for the single-well structure (I). At low fields the single-well anti-resonance shows an increase in energy as a function of field before being drawn into the well. In the double-well structure the anti-resonance energy plot has a consistent negative slope, and wavefunction localisation in the barrier is apparent at the lowest fields studied. As mentioned previously, the single-well calculations have demonstrated that a Breit-Wigner parametrisation is of limited validity in calculating the resonance widths (its use leads to an overestimate of the width and thus an underestimate of the tunnelling lifetime. Here an alternative approach, based on the time-delay formalism, is adopted, in which the width is calculated from the maximum rate of change of the phase shift with energy: Using this definition (which coincides with the Breit-Wigner result for sharp resonances) of the resonance width, the field dependence of the tunnelling widths and lifetime for the two states can be obtained; the tunnelling lifetimes are illustrated in figure 4. The most interesting feature of these results is that there is a range of field values for which the upper state is the more long-lived. This result can be interpreted in terms of the wavefunction localisation described previously; the higher-energy state is localised in the left-hand well in figure l(b) and thus has a lower tunnelling probability. This stabilisation of a higher-energy state due to wavefunction localisation has also been observed in atomic systems (see, for example, Damburg and Kolosov 1976). The mixing of various momentum wavefunctions, which is implied by the strong involvement of states lying above the confining barriers, is likely to affect transport +A 0.b LO -30 Figure3. A plot of the anti-resonance wavefunction at F = 2 x lo4 V cm-i. The positions of the two wells and the barrier are indicated.

8 GaAs-GaAlAs quantum wells , F (lo5 V cm-') Figure 4. The field dependence of the lifetimes of the upper (v) and lower (L) states for the symmetric double well. parameters, e.g. scattering rates. However: in the present calculation only the simplest zero-field model was considered in order to retain analytic features even in the high-field regime. For this reason we did not attempt to estimate these rates quantitatively. 5. Asymmetric double-well systems Since realistic multiple-quantum-well structures are unlikely to be perfectly symmetric, the effect of small structural asymmetries on the results described above are of great interest. Here we consider the effect of allowing the well widths to differ; similar results to those described in this section would be expected for the case of differing well depths. In terms of figure l(b) there are two possibilities; either the left- or right-hand well can be wider. Since at zero field the ground state corresponds to localisation in the wider well, the second possibility describes a situation in which the asymmetry of the well and the field-induced asymmetry reinforce one another. Study of this case shows that the results obtained are qualitatively similar to those for the symmetric structure. The first possibility is more interesting in that the structural asymmetry and the applied field have opposite effects on the wavefunction localisation; at zero or low fields the ground-state wavefunction would be expected to be localised in the left-hand well and at high fields in the right-hand well (and conversely for the upper state). The parameters chosen for the calculation were al = 30A a2 =40A (16) with Vo, L and m* unchanged. The most interesting feature of the results is an avoided crossing between the two resonance states at low fields (the high-field behaviour is similar to that in figure 2). For the parameters chosen here the centre of the avoided crossing is at 1.2 x 104Vcm-' with a minimum separation of about 3 mev; study of the wavefunctions in this region shows that at the centre of the avoided crossing they have

9 540 E J Austin and M Jaros 1 1 Holes Figure 5. A schematic energy-level diagram and predicted optical transitions for (a) symmetric and (b) asymmetric double wells. The labels + and - refer to the zero-field symmetric and antisymmetric states: w and R refer to wavefunctions localised in the left- and right-hand wells. the symmetrical delocalised structure characteristic of the zero-field solutions for a symmetric double well. (For the symmetric system the avoided crossing and the associated wavefunction delocalisation occurs at zero field.) These results suggest the possibility of changing or maintaining the wavefunction localisation properties for the system by varying the applied field in the vicinity of the avoided crossing. A slow adiabatic variation of the field will change the localisation; under rapid-passage (sudden) conditions the wavefunction localisation will be maintained. The timescale that defines rapid or adiabatic switching is set by the separation of the two states at the avoided crossing; this can be varied by changing the system parameters. Figure 5 gives a schematic illustration of the complete energy level diagram for electrons and holes for both the symmetric and antisymmetric cases. The applied field has opposite localisation effects on electrons and holes. At the low fields in the vicinity of the avoided crossing the transition probability for electron-hole recombination is proportional to the overlap of the wavefunctions. Thus in the presence of the field the strongest transitions will occur between states with the same localisation. Figure 5(b) suggests the possibility of switching between the transitions labelled (1) and (2). Acknowledgments EJA thanks the University of Newcastle upon Tyne for the award of a research fellowship.

10 GaAs-GaAlAs quantum wells 541 References Alibert C, Gaillard S, Brum J A, Bastard G, Frijlink P and Erman M 1985 Solid State Commun Austin E J and Jaros M 1985a Phys. Rev. B b Appl. Phys. Lett Barker J R 1973 J. Phys. C: Solid State Phys Bastard G, Mendez E E, Chang L L and Esaki L 1983 Phys. Rev. B Bastard G, Ziemelis U 0, Delalande C, Voos M, Gossard A C and Wiegmann M 1984 Solid State Commun Damburg R J and Kolosov V V 1976 J. Phys. B: AI. Mol. Phys Ferry D K 1982 Adu. Elec. Electron. Phys Herbert D C and Till S J 1981 J. Physique Suppl. 42 C Hess K 1982 Adv. Elec. Electron. Phys Jaros M, Wong K B and Gel1 M A 1985 Phys. Rev. B Kash J A. Mendez E E and Morkoc H 1985 Appl. Phys. Lett Kelly M J 1984 Electron. Left Lang R and Nishi K 1984Appl. Phys. Lett Linh N T 1984 Solid State Devices 1983 (Inst. Phys. Conf. Sec. 69) pp Miller D A B, Chemla D S, Damen T C, Gossard A C, Wiegmann W, Wood T H and Burrus C A 1984 Phys. Rev. Lett Ridley B K 1982 Quantum Processes in Semiconductors (Oxford: Clarendon) C4-H