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QUASI-BOUND STATES IN AN ASYMMETRIC GaAs-AlAs SUPERLATTICE
J. Singleton, R. Nicholas, N. Pulsford, N. Couch, M. Kelly
To cite this version:
J. Singleton, R. Nicholas, N. Pulsford, N. Couch, M. Kelly. QUASI-BOUND STATES IN AN ASYM-
METRIC GaAs-AlAs SUPERLATTICE. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-435-
C5-437. �10.1051/jphyscol:1987592�. �jpa-00226796�
JOURNAL DE PHYSIQUE
Colloque C5, supplement au nOll, Tome 48, novembre 1987
QUASI-BOUND STATES I N AN ASYMMETRIC GaAs-A1As SUPERLATTICE
J. SINGLETON, R.J. NICHOLAS, N.J. PULSFORD, N.R. COUCH* and M. J. KELLY*
T h e C l a r e n d o n L a b o r a t o r y , Parks R o a d , G B - O x f o r d OX1 3 P U , G r e a t - B r i t a i n
*GEC R e s e a r c h L i m i t e d , H i r s t R e s e a r c h C e n t r e , E a s t L a n e , GB-Wembley HA9 7 P P , M i d d l e s e x , G r e a t - B r i t a i n
Interband photoconductivity measurements have been performed on a "Quasi-Graded-Gap"
Superlattice (QGGSL). A number of features, both sharp and broad, are observed in the photoresponse, corresponding to transitions between the quasi-bound states of the superlattice. These features are described by a calculation, within an envelope-function approximation, of the pseudo-stationary states of the QGGSL. It is found that the electrons chiefly tunnel through the barriers in the device via states associated with the X-conduction band minimum of AlAs. The calculation also shows that the emission lines seen in electroluminescence spectra are due to recombinations between electrons and light holes, as the light holes tunnel through the QGGSL much more readily.
The band gap of Gal-,AlXAs becomes indirect when x exceeds 0.45, and there has been much recent interest in the effect that this transition in behaviour has on the properties of GaAs-Gal -,AIxAs heterostructures. In cases where tunnelling through the Gal -xAlxAs can readily occur, it is then necessary to determine whether the electrons tunnel via states associated with the direct
r
tor
or the indirectr
to X band offset or a mixture of both. In this paper we report measurements of the photoresponse of a GaAs-AlAs quasi-graded gap superlattice (QGGSL) as a function of photon energy: the results of the study have direct bearing on this question.The QGGSL simulates the grading of the band-gap from that of GaAs tn that of Ga0.7Al0.3As by varying the well to barrier ratio in the manner shown in figure 1 : quasi-grading has in the past been proposed as a means of increasing the band gap in a ramp-like fashion without the detrimental effects associated with direct alloy grading [I]. The structure was grown by molecular beam epitaxy on an n+ (1 0 0) GaAs wafer, and the growth sequence was as follows: i) 0.5 p GaAs buffer layer n-doped to 5x1018 cm-3, ii) 0.5 p of GaAs n-doped to 1016 cm-3, iii) 3 nm of AIAs, iv) 7nm of GaAs, v) 2.4 nm of AlAs, vi) 7.6 nm of GaAs, vii) 1.8 nm of AAs, viii) 8.2 nm of GaAs, ix) 1.2 nm 'of AlAs, x) 8.8 nm of GaAs, xi) 0.6 nm of AAs, xii) 9.4 nm of GaAs (layers iii)-xii) undo ed), xiii) 0.5 pm of GaAs p-doped to 1016 cm-3 and finally xiv) 0.5 p GaAs p-doped to 5xIOP8 ~ m - ~ . Well and barrier thicknesses for the QGGSL measured by transmission electron microscopy (TEM) [2] are shown in table 1. The superlattice thus forms the intrinsic region of a p-i-n diode [3,4]. In d.c. current-voltage measurements, the structure has been shown to exhibit , .
strong negative differential resistance, which was thought to be due to the electrons tunnelling through the AlAs X conduction-band minimum in the barriers [3,4].
Fi ure 1 shows the constant-current photoresponse of the device under a small forward bias (1.10-lg A) at a temperature of 4.2 K. Several features, both sharp and broad, due to transitions between the quasi-bound states of the superlattice, were observed at energies well above the GaAs band-gap. Three lines were also observed in the electroluminescence spectrum of the device in hard forward bias a t 4.2 K [5], and, on transfering the electroluminescence peaks to figure 1 , it can be seen that they occur at very similar energies to three of the sharp features in the photoresponse spectrum.
In order to gain some understanding of these results, a simulation of the electron and hole energy states of the superlattice within a n envelope function approximation [6] was attempted. Wave functions of the following type were chosen:
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987592
C5-436 JOURNAL DE PHYSIQUE
h
= a eikz+
p e-ikz (wells) $b = eqz+
7 e-92 (barriers) and the following boundary conditions were imposed at the interfaces:GW
= $t, : (llm;)(s$dsz) = (lim:)(a$ ~ a z )The well and barrier states for the holes were derived from the r-point valence band maxima of the two materials. In addition, the electron states were initially treated as associated with the r-point conduction band minima of GaAs and AlAs. The light-hole and electron bands were represented using dispersion relationships derived from 3-band
k.p
theory [6]: as the three-band theory does not predict the observed band-edge masses precisely, the momentum-matrix element was adjusted separately for each type of carrier until the correct band-edge mass was obtained. Unfortunately, the three-band model predicts a totally flat heavy-hole band, and so heavy-hole states were modelled using a constant effective mass. Band-gaps for the two materials were taken from the literature [7]and the conduction band offset was fixed a t the recent consensus value of 67% [8,9]. The calculation followed the standard method for modelling resonant states in nuclear physics (eg [lo]): a plane travelling wave is incident on the superlattice from one side, and the energies of the quasi-bound states are indicated as peaks in the transmitted amplitude. As the superlattice states have finite lifetimes, the
enernv
& comolex, and hence so are k and q. The calculation is thus similar to the"Transfer Matrix" method [ l l ] , but with the addition of the complex energy, which helps to highlight the structure of the superlattice minibands. Results of the calculation are shown in figure 2 (the band offsets in this figure are explained below) for electrons, and it can be seen that the states occur in minibands numbered 1 , 2 etc. and split by wavevector quantisation into four states labelled a, b, c and d for convenience: in subsequent discussions, electron states are labelled C, heavy-hole states H and light-hole states L. For the electrons and light-holes, the lowest energy state (labelled a), predominantly confined by the 0.6 nm barrier, contributes a very small amplitude compared to the other three states, due to its very short lifetime. This is thought to be the reason why only three lines are seen in the electroluminescence spectrum.
Strong transitions were assumed to occur between states with a large spatial overlap. It was found to be impossible to fit the sharp features in the photoresponse successfully using the
r
tor
conduction band offset, as the model produced transition energies which were much too high, even when the maximum possible errors in the TEM-measured thicknesses and barrier compositions were taken into account [2]. It is not strictly valid within the envelope-function approach to match r-like to X-like wavefunctions: however, in order to try and gain some insight as to why the electrons were experiencing a much lower effective barrier-height than that provided by the r(GaAs) to r(AlAs) offset, the electron effective barrier height in the model was made adjustable. In addition, the electron dispersion relationship within the barriers was made parabolic with an adjustable effective mass. With these modifications, it was found to be possible to fit the observed sharp features in the experimental data & when the electron effective barrier height was very close to the r(GaAs) to X(AlAs) conduction band discontinuity. The effective masses and well and barrier widths used in this fit are shown in table 1: the fitted widths agreed closely with experimental thickness determinations using TEM [2], also shown in table 1. The calculated transition energies are indicated as arrows in figure 1 : the sharp features correspond to transitions from states in the HI and L1 minbands to the C1 miniband, omitting transitions from the briefly-lived "a" states, whilst the higher-energy broad features correspond to overlapping transitions involving higher minibands (see figure 1). The sharper transitions are fitted well by the model, but the agreement is not so good for the higher energy states, probably because of the parabolic electron dispersion relationship assumed for the barriers.
Nevertheless, all of the observed features can be accounted for by transitions to electron states determined by wavefunction matching to the AlAs X-minimum.
The photoresponse thus demonstrates that the AlAs X-minimum plays a vital r6le in determining the QGGSL miniband structure and hence the tunnelling of electrons through it, as indicated in recent transport measurements [3,4]. If the AlAs X-minimum did not contribute, and the electron states were determined solely by the r(GaAs) to r(Al.4~) offset, the model suggests that the electron tunnelling amplitude would be around 10,000 times smaller.
It will be noted that the fitted effective mass for the X-point electrons is only 0.11 me. The X-point conduction band minimum is very anisotropic, leading to two possible values of the effective mass: for momentum parallel to the (1 0 0) direction, the electron motion will be determined by the longitudinal mass, whereas perpendicular to this direction the motion will be controlled by the transverse mass. In &As, the longitudinal and transverse masses are thought to be 1.9 me and 0.19 me respectively 1121, although no reliable experimental determination of the latter has been made. A large effective mass inhibits tunnelling, and so if some mechanism is available to overcome the difference in momentum between the GaAs r-minimum and the transverse AlAs X-minimum, the lighter mass would be expected to dominate. As the AlAs barriers are very thin, there is a possibility of them acting rather like an (Ga,Al)As alloy: alloy disorder would provide a mechanism for breaking
the momentum conservation, and allowing states involving the lighter mass. Recent more sophisticated calculations [13] have indicated that the barrier states in the AlAs are in fact a mixture of X- and r-like states, and this would also act to lower the barrier mass. Another possible reason for the discrepancy with the supposed bulk value is that a smaller mass can be played off against larger barrier thicknesses: as a 1.2 nm AlAs layer contains only two planes of AlAs atoms, its representation as a 1.2 nm thick rectangular barrier is certainly an oversimplification. All of the factors mentioned here will modify the character of the barrier states, which is possibly why the envelope function approach, although not strictly valid, gives a reasonable representation of the data.
1 he llnes observed in the electroluminescence spectrum correspond to predicted energies for the L l b C l b , L l c C l c and L l d C l d transitions. As the transit times of the carriers through the superlattice are very short compared to typical recombination times 141, tpe electroluminescence will be dominated by the carriers which can tunnel into the device most readily. The calculated transmitted amplitudes for the heavy holes are approximately six' orders of magnitude lower than those for the light holes, and so recombination between the light hole and electron states will dominate the electroluminescence, as observed.
In summary, by observing the photoresponse of a GaAs-AlAs QGGSL, and by describing the quasi-bound states of the structure using a simple model, we have been able to demonstrate that the electronic properties of the device are primarily determined by the AlAs X-minimum. These results have important consequences for the performance of devices containing Gal-xAlxAs, with x>0.45.
This work was supported in part by ESPRIT under programme number 514. We would like to thank T.M.Kerr for growing the device.
l a y e r l a y e r w i d t h l a y e r w i d t h f i t t e d / m m e a s u r e d / m AlAs
GaAs AlAs Ga As AlAs GaAs
T a b l e 1 E f f e c t i v e masses and l a y e r w i d t h s u s e d i n f i t t i n g .
AlAs The measured w i d t h s were o b t a i n e d
u s i n g TEM [ 2 ] .
h he
band-edge e f f e c t i v e masses were a l l f i x e d a t v a l u e s t a k e n from t h e l i t e r a t u r e [ 7 ] e x c e p t f o r t h e X-minimum mass, which was a n a d j u s t a b l e p a r a m e t e r .Band e d g e e f f e c t i v e masses /me
11 Capasso, F., Cox, H.M., Hutchinson, A.L., Olsson, N.A. and Hummel, S.G, Appl.Phys.Lett
45
(1986) 1193
r - e l e c t r o n X - e l e c t r o n l i g h t - h o l e heavy-ho 1 e
21 Britton, E.G., Alexander, K.B., Stobbs, W.M., Kelly, M.J and Ken, T.M., GEC Journal of Research 5_ (1987) 31
31 Couch, N.R., Kelly, M.J., Kerr, T.M., Britton. E.G. and Stobbs. W.M., Semicond. Sci.
Technol.
2
(1987) 24441 Couch, N.R., Parker, D.G., Kelly, M.J. and Kerr, T.M., Electron.Lett. 22 (1986) 637 51 Phillips, R. et al, to be published.
61 Bastard, G., Acta Electronica
2
(1983) 14771 Harbeke, G., Madelung, 0. and Rossler, U., Landholt-Bornstein New Series III
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(1982) 164ff, 218ff81 Wolford, D.J.. Keuch, T.F. Bradley, J.A.. Gell, M.A., Ninno, D and Jaros M., J.Vac. Sci.
Technol. @ (1986) 1043
91 Dawson, P., Moore, K.J. and Foxon, C.T.B., Proc. SPIE Ze2, (1987). in press
IO] Blatt, J.M. and Weisskopf, V.F., "Theoretical Nuclear Physicsn, pp 379-399, Springer-Verlag, New York (1979)
111 Kane, E.O. in "Tunnelling phenomena in solids" E. Burstein and S. Lundqvist, eds. Plenum, New York (1969)
121 kheinlander, B., Neumann, H., Fischer, P. and Kuhn, G., Phys. Stat. Solidi
B49
(1972) K167 131 Cade, N. et al, in press.AlAs 0 . 1 5 0.11*
0 . 2 6 0 . 4 5
GaAs 0 . 0 6 6
---
0 . 0 8 2 0 . 5 0
