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Strain and minibands in InGaAsGaAs superlattices
Superlattices and Microstructures, Vol. 9, No. 4, 1991
521
STRAIN AND MINIBANDS IN INGAAS-GAA$ SUPERLATTICES
N.J. Puisford, R.J. Nicholas, R.J. Warburton and M.J. Lawless Clarendon Laboratory, Parks Rd., Oxford, OX1 3PU, U.K. G. Duggan, K.J. Moore, K. Woodbridge and C. Roberts Philips Research Laboratories, Redhill, Surrey, RH1 5HA, U.K. (Received 17 August 1990)
Interband transmission and reflectivity measurements are performed on InGaAS-GaAs superlattices in magnetic fields up to 16 Tesla. The in-plane dispersion shows the effect of strain decoupling between the heavy and light hole subbands as their energy separation is increased. Significant interwell coupling is demonstrated by a reduction in the exciton binding energy and the presence of a finite miniband dispersion in the growth direction. The effects of both electron and holes saturating at the top of the miniband is studied.
Electronic coupling between the confined states in adjacent well layers leads to the formation of a 3 dimensional miniband structure in 'true' semiconductor superlattices. Extended miniband states have been demonstrated in GaAs--(3al_xAlxAs superlattices by both subpicosecond transport I and coherent cyclotron motion through the superlattice layers2-4. However, these studies have yet to be extended to other material systems, where the different line-up in the band edge potentials influences the miniband structure formed by the superlattice periodicity. This paper describes the optical properties of a set of four strained layer In0.1Ga0.9As--GaAs superlattices in which the interwell coupling is controlled by varying the thickness of the InGaAs layers. The superlattice band structure is probed by applying a magnetic field both perpendicular (B J') and parallel (B II) to the superlattice layers so that the cyclotron motion samples both the in-plane and miniband dispersions. The superlattice samples were grown by molecular beam epitaxy on semi-insulating (001) orientated GaAS substrates with the following growth sequence: (i) l/~m GaAs buffer, (ii) 20 periods of InGaAS wells and 100A GaAs barriers, (iii) capped with 200A GaAS. The InGaAs wells have thicknesses of either 25, 50, 100 or 200A with an indium content of about 10% measured by X-ray analysis. AS the superlattices are grown on a GaAs buffer, the "-1% mismatch between the GaAs and lnGaAs lattices is accommodated by an elastic tetragonal distortion in the InGaAS layers. The compressive strain both shifts and splits the degenerate valence band edge 5 in the InGaAs leaving the heavy hole band edge the highest. Taking a 0.67 strained electron to heavy hole offset ratio 6, a strained layer calculation 7 predicts that the electrons and heavy holes are confined in the InGaAs by 64 meV and 32 meV wells respectively, whereas the light holes are confined in the GaAs layers by a harrier of only 7 meV, making them Type 11 with
0749-6036/91 •040521 +05 $02.00/0
respect to the electrons6. This configuration considerably simplifies the optical spectra, with the Type II character of the light holes strongly reducing their absorption for wider wells, and the additional strain splitting removing any valence hand mixing effects up to higher energies. Figure 1 shows an envelope function calculation of the miniband widths for the lowest electron, heavy hole and light hole suhbands as a function of ]nGaAs thickness. The energy origin is at the electron and heavy hole strained InGaAs band edge for the conduction and . valence subbands respectively. The arrows along the x-axis indicate the lnGaAs widths in the four samples. For narrow InGaAs layers, the shallow barrier heights allow strong coupling between the electron states even though the GaAs barrier layers are 100A wide. As the InGaAs width is increased, the states become more confined and the coupling is reduced. Compared to the more commonly studied GaAs--GaAIAs system, the small potential barriers and hence thicker layers not only avoid the problems associated with characterising and modelling structures a few monolayers thick, but they also lead to short superlattice Brillouin zones and hence measurable miniband masses for minibands only 10 meV wide. Optical interband transmission and refiectivity measurements are performed at 1.8K in magnetic fields up to 16 Tesla. The in-plane dispersion is studied first by orientating the magnetic field perpendicular (B J') to the superlattice layers. Figure 2 shows transmission spectra for the 200A lnGaAs sample. The two strong zero field transitions (minima in the transmission) at 1.415 eV and 1.448 eV correspond to the E1-HHI and E2-HH2 excitons respectively. Circular polarised photoluminescence excitation analysis shows that no light hole transitions are visible in this weakly coupled sample which is consistent with a Type II character. However, the E I - L H I transition is visible in the 25, 50 and 100A
© 1991 Academic Press Limited
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Figure 1. Calculated miniband edges for the n=l m i n i b a n d s in the c o n d u c t i o n a n d valence bands as a function of I n G a A s thickness. T h e energy origins a r e the strained electron a n d h e a v y hole I n G a A s b a n d edges.
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Figure 3. L a n d a u level fan d i a g r a m for the 2 0 0 A l n G a A s sample in B J'. T h e solid a n d dashed lines are modelled fits to the excitonic transitions originating f r o m the E I - H H I a n d E 2 - H H 2 excitons respectively.
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Figure 2. T r a n s m i s s i o n spectra for the 200/k l n G a A s sample in various m a g n e t i c fields applied p e r p e n d i c u l a r to the superlattice layers (Ba-).
l n G a A s samples due to the increased wavefunction overlap for thinner layers. Excitonic L a n d a u levels evolve out of the zero field features in B a" a n d their transition energies are plotted as a function of m a g n e t i c field in figure 3. T h e lines are a modelled fit using a 3 - d i m e n s i o n a l exciton model 8 which is a d a p t e d for a n anisotropic b a n d structure by treating the exciton binding e n e r g y a n d c y c l o t r o n mass as i n d e p e n d e n t p a r a m e t e r s9. T h e fan d i a g r a m a p p e a r s complicated at higher energies due to the presence of transitions crossing over e a c h o t h e r in the spectra. However, two sets of L a n d a u levels c a n be resolved, one originating f r o m the E I - H H 1 exciton (solid lines) a n d the o t h e r f r o m the E 2 - H H 2 exciton (dashed lines). T h e E 1 - H H 1 L a n d a u levels are seen down to 2 Tesla giving an exciton binding e n e r g y of 6.5 meV. This is consistent with an e n h a n c e m e n t over the bulk value due to c o n f i n e m e n t in a wide 200A well. T h e E 2 - H H 2 exciton is fitted with a slightly smaller binding e n e r g y (5.5 meV), intermediate between the values for bulk G a A s a n d the m o r e strongly confined E 1 - H H 1 exciton. This is in line with the theoretical calculations for an isolated q u a n t u m well 10 a n d might also reflect the finite bandwidth (4 meV) of the n=2 electron subband delocalising the exciton wavefunction. With an El i n - p l a n e mass 0 . 0 6 6 2 m o taken f r o m the envelope function model, the fitted HI i n - p l a n e mass for the 200A sample is 0 . 1 9 m o. This is close to the d e c o u p l e d limit 11 a n d reflects the weak valence
Superlattices and Microstructures, Vol. 9, No. 4, 1991
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Figure 5. E I - H H I exciton binding energy as a function of InGaAs well width: the circles are the measured values in the superlattice samples described here. The squares and solid line show the measured and calculated binding energy in isolated well samples.
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Figure 4. Landau level fan diagram for the 25A InGaAs sample in B "~. The low and high energy modelled fits are given by the dashed and solid lines respectively.
subband coupling for HH1 and LHI subbands 38 meV apart. The separation of the hole subbands decreases as the InGaAs width is reduced (figure i ) making valence subband mixing effects more evident in the thinner well samples. The Landau level fan diagram for the 25A InGaAs sample is shown in figure 4. The transition energies are measured using reflectivity rather than transmission allowing the superlattice band structure above the top of the GaAs barriers to be studied. The energy gap (1515-20 eV) corresponds to the region where the bulk GaAs l s exciton dominates the response; however, the optically thin GaAs capping layer (200A) ensures that any higher Landau levels from bulk GaAs are weak and only the transitions originating from the superlattice layers are plotted. The solid lines are fitted with a 4.5 meV exciton binding energy and a heavy hole i n - p l a n e mass of 0.36m o. At low energies, the Landau level fit is poor and much better agreement is obtained with the lighter hole mass 0.19m o (dashed lines). The heavier hole mass at high energies arises from the mixing between heavy and light hole states away from kll=0, leading to a very n o n - p a r a b o l i c hole i n - p l a n e dispersion 11. Similiar behaviour has been observed previously at very low magnetic fields in G a A s - A l G a A s quantum wells12, however the strain splitting in ] n G a A s - G a A s superlattices allows the light (decoupled) mass to be observed over a much wider field range. The linewidths in the 50 k and 100 A InGaAs samples are not as sharp as in the 25 A sample and it
is not possible to see strong non-parabolicities due to the uncoupled mass at low fields. However the fitted high field hole masses of 0.25m o and 0.23m o r e s p e c t ively are in line with a progressive decoupling of the HHI and LH1 subbands as their energy separation increases. The binding energy of an exciton is very sensitive to its dimensionality and in a superlattice, any coupling into minibands leads to a decrease in the binding energy over to its isolated well value. Figure 5 shows the measured E 1 - H H 1 exciton binding energies in the four superlattice samples (filled circles) as a function of l n G a A s well width. Also included for comparison is some data for three I n G a A s - G a A s superlattices 13 (open squares) with much thicker 400 k GaAs barriers which effectively decouple the InGaAs wells. The measured exciton binding energies in all the uncoupled samples show good agreement with the calculated dependence for an isolated quantum well (solid line). The exciton binding energies measured in the 2 5 k and 5 0 k InGaAs samples lie substantially below the isolated well values and indicates the formation of extended minibands. For the 25 k InGaAs welt, the binding energy of 4.5 meV is little more than that of a bulk GaAs exciton (open circle), consistent with the exciton sensing the s u p e r lattice as a weakly anisotropic 3-dimensional solid. The binding energy then increases rapidly for the 5 0 k sample towards the isolated well value as the miniband width shrinks and the states become confined into the well layers. The peak in the exciton binding energy at 100A InGaAs layers indicates the strong effect of coupling in these structures across relatively thick layers. For the isolated well samples the confinement is stronger leading to a peak binding energy for much narrower lnGaAs thicknesses, around 5 0 k . A more definitive demonstration of interwell coupling in a superlattice is the presence of a finite miniband dispersion in the growth direction. This is measured by applying a magnetic field parallel (Bfl) to the superlattice layers so that the cyclotron orbits involve
Superlattices and Microstructures, Vol. 9, No. 4, 1991
524
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both i n - p l a n e a n d miniband motion 2. L a n d a u levels are t h e n only observed if the interwell coupling is strong e n o u g h to p r o d u c e extended Bloch states t h r o u g h the superlattice layers a n d the c o r r e s p o n d i n g L a n d a u level s e p a r a t i o n is smaller t h a n in a p e r p e n d i c u l a r field due to the inclusion of the heavier minibaod masses. In the strongly coupled 25.~ l n G a A s superlattice, the calculated electron and h e a v y hole m i n i b a n d widths a r e 20.3 m e V a n d 1.6 m e V respectively, which c o m p a r e with the a b s o r p t i o n linewidth of ~1.6 m e V - electrons at least should d e m o n s t r a t e c o h e r e n t miniband motion. Discrete L a n d a u levels are observed in B II which c o n f i r m s the presence of extended Bloch states a n d the corresponding transition energies are plotted as a function of m a g n e t i c field in figure 6. T h e top of the E I - H H 1 m i n i b a n d is calculated to lie at 1.512 eV so that any structure associated with the zone-edge E 1 - H H 1 e x c i t o n l 4 , 1 5 is p r o b a b l y hidden by the G a A s exciton which d o m i n a t e s the s p e c t r u m in this e n e r g y range. H o w e v e r , in c o n t r a s t to B " , no superlattice levels are seen above the G a b s b a n d edge which indicates a finite superlattice m i n i b a n d width less t h a n 30 m e V wide. T h e parallel field L a n d a u levels are fitted using the exciton model with a geometric average of the i n - p l a n e a n d m i n i b a n d masses for both electrons a n d holes. The i n - p l a n e masses a n d exciton binding e n e r g y are taken f r o m B"- a n d the m i n i b a n d masses f r o m the envelope function model. T h e a g r e e m e n t is good at low energies, giving a reduced mass a n i s o t r o p y # " 7 # " = 1.29 a n d indicates t h a t the h e a v y holes are indeed c o n t r i b u t i n g to the m i n i b a n d motion. H o w e v e r with the c o m b i n a t i o n of
a n a r r o w hole miniband a n d light i n - p l a n e mass (0.19too), the superlattice L a n d a u levels reach the top of the heavy hole m i n i b a n d well before the top of the electron m i n i b a n d . T h e arrows in figure 6 indicate the field at which this is calculated to o c c u r a n d the excitonic L a n d a u levels extend well beyond the hole miniband ' s a t u r a t i o n ' points in e a c h case. Maan's calculations 16 show t h a t for levels both near a n d above the miniband top, the L a n d a u level e n e r g y depends on the orbit position in real space making the transitions very broad a n d weak. Although the n a r r o w hole m i n i b a n d width (1.6 meV) m a y prevent this h a p p e n i n g here, it is unclear which hole states the discrete electron L a n d a u levels are now c o m b i n i n g with. Intuitively, the hole mass gets very h e a v y n e a r the top of the miniband a n d the solid lines in figure 6 are calculated with a semiclassical quantisation of the superlattice band structure but assuming no f u r t h e r increase in e n e r g y o n c e the miniband top is reached. T h e saturation of the hole m i n i b a n d reduces the cyclotron energy increase - the low energy g r a d i e n t is due to the c o m b i n e d e l e c t r o n - h o l e reduced mass, whereas the gradient at higher energies is just due to the electron mass alone and the data agrees very well with this c h a n g e in gradient. If the holes are given a c o n s t a n t m i n i b a n d mass (dashed lines), it is not possible to fit the behaviour at both low a n d high energies. For the 50 ,~ InGaAs sample, the interwell coupling a n d hence the m i n i b a n d widths are reduced, however excitonic L a n d a u levels are still observed in B I1 (figure 7) indicating the presence of extended Bloch
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Superlattices and Microstructures, VoL 9, No. 4, 1991 states. From the ratio of the ls (n=0) diamagnetic shifts in B"- and B II, the reduced mass anisotropy is 1.56 at the miniband edge. The n=l Landau level weakens and dies out around 10 Tesla, which is consistent with a finite miniband width. However, a semiclassical quantisation of the calculated band structure (solid lines), which gives good agreement for the 25A InGaAs sample, clearly fails in this case. Even if the GaAs barriers are reduced to 90A, giving miniband widths of 13 and 0.5 meV for the electrons and heavy holes respectively, the calculation underestimates the n=l 'saturation' energy by about 6 meV. The fit for the ls (n=0) shift on the other hand is good indicating that the miniband masses and hence widths are approximately correct. The difference between the theory and experiment is thought to be due to the superlattice miniband structure breaking down in this sample. The transmission linewidth is relatively broad (5 meV) and has two distinct contributions to its lineshape, indicating inhomogeneities in the indium concentration 17. Any significant perturbation on the superlattice periodicity weakens the q quantum number conservation rule and hence allows Landau levels to be seen beyond the top of the miniband, with the superlattice then behaving more like an anisotropic alloy than a structure with a finite miniband width. The dashed line in figure 7 is calculated using a constant electron mass (i.e. ignoring the superlattice miniband gap) and the experimental points lie intermediate between the 'alloy' and 'superlattice' limits. In the two uncoupled samples with wider InGaAs layers, no superlattice Landau levels are observed in a parallel magnetic field and the excitonic transitions exhibit a parabolic diamagnetic shift. This is characteristic of an isolated quantum well 18 and demonstrates the absense of interwell coupling in these structures (figure 1). In summary, the influences of strain and interwell coupling have been demonstrated on the miniband structure of InGaAs-GaAs superlattices. The strain splits the valence subbands and allows the HHI decoupled in-plane mass to be seen over a wide field range. The interwell coupling is characterised by a decrease in the exciton binding energy over its isolated well value and the nresence of a miniband dispersion in the growth direction. The parallel field Landau levels are modelled
525 using an exact semiclassical quantisation of the miniband structure which, while giving good agreement for one superlattice, fails for another of poorer quality; this discrepancy is attributed to alloy inhomogeneities.
REFERENCES 1 B.Deveaud, J.Shah, T.C.Damen, B.Lambert and A.Regreny, Phys. Rev. Lett. 58, 2582 (1987) 2 G.BeUe, J.C.Maan and G.Weimann, Solid State Commun. 56, 65 (1985) 3 T.Duffield, R.Bhat, M.Koza, F.DeRosa, D.M.Huang, P.Grabbe and S.J.Allen, Phys. Rev. Lett 56, 2724 (1986) 4 N.J.Pulsford, R.J.Nicholas, P.Dawson, K.J.Moore, G.Duggan and C.T.Foxon, Phys. Rev. Lett. 63, 2284 (1989) 5 H.Hasegawa, Phys. Rev. 129, 1029 (1963) 6 J-M.Gerard and J-Y.Marzin, Phys. Rev. B40, 6450 (1989) 7 D.Gershoni, H.Temkin, M.B.Panish and R.A.Hamm, Phys. Rev B39, 5531 (1989) 8 P.C.Makado and N.McGilI, J. Phys. C19, 873 (1986) 9 N.J.Pulsford, R.J.Nicholas, K.J.Moore, P.Dawson and C.T.Foxon, Superlatt. and Microstruct. 6, 51 (1989) 10 J.A.Brum and G.Bastard, J. Phys. C18, I..789 (1985) 11 G.D.Sanders and Y.C.Chang, Phys. Rev. B31, 6893 (1985) 12 A.S.Plaut, J.Singleton, R.J.Nicholas, R.T.Harley, S.Andrews and C.T.Foxon, Phys. Rev. B38, 1323 (1988) 13 K.J.Moore, G.Duggan, K.Woodbridge and C.Roberts, Phys. Rev. B41, 1090 (1990) 14 B.Deveaud, A.Chomette, F.Clerot, A.Regreny, J.C.Maan, R.Romestain, G.Bastard, H.Chu and Y.--C.Chang, Superlatt. Microstruct. 6, 183 (1989) 15 K.J.Moore, G.Duggan, A.Raukema and C.Woodbridge, Phys. Rev. B42 xxx (1990) 16 J.C.Maan, Festkorperprobleme 27, edited by P.Grosse (Vieweg, Braunschweig) 137 (1987) 17 K.J.Moore, G.Duggan, K. Woodbridge and C.Roberts, Phys. Rev. B41, 1095 (1990) 18 N.J.Pulsford, J.Singleton, R.J.Nicholas and C.T.Foxon, J. de Physique (Paris) 48, C5-231 (1987)
521
STRAIN AND MINIBANDS IN INGAAS-GAA$ SUPERLATTICES
N.J. Puisford, R.J. Nicholas, R.J. Warburton and M.J. Lawless Clarendon Laboratory, Parks Rd., Oxford, OX1 3PU, U.K. G. Duggan, K.J. Moore, K. Woodbridge and C. Roberts Philips Research Laboratories, Redhill, Surrey, RH1 5HA, U.K. (Received 17 August 1990)
Interband transmission and reflectivity measurements are performed on InGaAS-GaAs superlattices in magnetic fields up to 16 Tesla. The in-plane dispersion shows the effect of strain decoupling between the heavy and light hole subbands as their energy separation is increased. Significant interwell coupling is demonstrated by a reduction in the exciton binding energy and the presence of a finite miniband dispersion in the growth direction. The effects of both electron and holes saturating at the top of the miniband is studied.
Electronic coupling between the confined states in adjacent well layers leads to the formation of a 3 dimensional miniband structure in 'true' semiconductor superlattices. Extended miniband states have been demonstrated in GaAs--(3al_xAlxAs superlattices by both subpicosecond transport I and coherent cyclotron motion through the superlattice layers2-4. However, these studies have yet to be extended to other material systems, where the different line-up in the band edge potentials influences the miniband structure formed by the superlattice periodicity. This paper describes the optical properties of a set of four strained layer In0.1Ga0.9As--GaAs superlattices in which the interwell coupling is controlled by varying the thickness of the InGaAs layers. The superlattice band structure is probed by applying a magnetic field both perpendicular (B J') and parallel (B II) to the superlattice layers so that the cyclotron motion samples both the in-plane and miniband dispersions. The superlattice samples were grown by molecular beam epitaxy on semi-insulating (001) orientated GaAS substrates with the following growth sequence: (i) l/~m GaAs buffer, (ii) 20 periods of InGaAS wells and 100A GaAs barriers, (iii) capped with 200A GaAS. The InGaAs wells have thicknesses of either 25, 50, 100 or 200A with an indium content of about 10% measured by X-ray analysis. AS the superlattices are grown on a GaAs buffer, the "-1% mismatch between the GaAs and lnGaAs lattices is accommodated by an elastic tetragonal distortion in the InGaAS layers. The compressive strain both shifts and splits the degenerate valence band edge 5 in the InGaAs leaving the heavy hole band edge the highest. Taking a 0.67 strained electron to heavy hole offset ratio 6, a strained layer calculation 7 predicts that the electrons and heavy holes are confined in the InGaAs by 64 meV and 32 meV wells respectively, whereas the light holes are confined in the GaAs layers by a harrier of only 7 meV, making them Type 11 with
0749-6036/91 •040521 +05 $02.00/0
respect to the electrons6. This configuration considerably simplifies the optical spectra, with the Type II character of the light holes strongly reducing their absorption for wider wells, and the additional strain splitting removing any valence hand mixing effects up to higher energies. Figure 1 shows an envelope function calculation of the miniband widths for the lowest electron, heavy hole and light hole suhbands as a function of ]nGaAs thickness. The energy origin is at the electron and heavy hole strained InGaAs band edge for the conduction and . valence subbands respectively. The arrows along the x-axis indicate the lnGaAs widths in the four samples. For narrow InGaAs layers, the shallow barrier heights allow strong coupling between the electron states even though the GaAs barrier layers are 100A wide. As the InGaAs width is increased, the states become more confined and the coupling is reduced. Compared to the more commonly studied GaAs--GaAIAs system, the small potential barriers and hence thicker layers not only avoid the problems associated with characterising and modelling structures a few monolayers thick, but they also lead to short superlattice Brillouin zones and hence measurable miniband masses for minibands only 10 meV wide. Optical interband transmission and refiectivity measurements are performed at 1.8K in magnetic fields up to 16 Tesla. The in-plane dispersion is studied first by orientating the magnetic field perpendicular (B J') to the superlattice layers. Figure 2 shows transmission spectra for the 200A lnGaAs sample. The two strong zero field transitions (minima in the transmission) at 1.415 eV and 1.448 eV correspond to the E1-HHI and E2-HH2 excitons respectively. Circular polarised photoluminescence excitation analysis shows that no light hole transitions are visible in this weakly coupled sample which is consistent with a Type II character. However, the E I - L H I transition is visible in the 25, 50 and 100A
© 1991 Academic Press Limited
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Figure 1. Calculated miniband edges for the n=l m i n i b a n d s in the c o n d u c t i o n a n d valence bands as a function of I n G a A s thickness. T h e energy origins a r e the strained electron a n d h e a v y hole I n G a A s b a n d edges.
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Figure 3. L a n d a u level fan d i a g r a m for the 2 0 0 A l n G a A s sample in B J'. T h e solid a n d dashed lines are modelled fits to the excitonic transitions originating f r o m the E I - H H I a n d E 2 - H H 2 excitons respectively.
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52
Figure 2. T r a n s m i s s i o n spectra for the 200/k l n G a A s sample in various m a g n e t i c fields applied p e r p e n d i c u l a r to the superlattice layers (Ba-).
l n G a A s samples due to the increased wavefunction overlap for thinner layers. Excitonic L a n d a u levels evolve out of the zero field features in B a" a n d their transition energies are plotted as a function of m a g n e t i c field in figure 3. T h e lines are a modelled fit using a 3 - d i m e n s i o n a l exciton model 8 which is a d a p t e d for a n anisotropic b a n d structure by treating the exciton binding e n e r g y a n d c y c l o t r o n mass as i n d e p e n d e n t p a r a m e t e r s9. T h e fan d i a g r a m a p p e a r s complicated at higher energies due to the presence of transitions crossing over e a c h o t h e r in the spectra. However, two sets of L a n d a u levels c a n be resolved, one originating f r o m the E I - H H 1 exciton (solid lines) a n d the o t h e r f r o m the E 2 - H H 2 exciton (dashed lines). T h e E 1 - H H 1 L a n d a u levels are seen down to 2 Tesla giving an exciton binding e n e r g y of 6.5 meV. This is consistent with an e n h a n c e m e n t over the bulk value due to c o n f i n e m e n t in a wide 200A well. T h e E 2 - H H 2 exciton is fitted with a slightly smaller binding e n e r g y (5.5 meV), intermediate between the values for bulk G a A s a n d the m o r e strongly confined E 1 - H H 1 exciton. This is in line with the theoretical calculations for an isolated q u a n t u m well 10 a n d might also reflect the finite bandwidth (4 meV) of the n=2 electron subband delocalising the exciton wavefunction. With an El i n - p l a n e mass 0 . 0 6 6 2 m o taken f r o m the envelope function model, the fitted HI i n - p l a n e mass for the 200A sample is 0 . 1 9 m o. This is close to the d e c o u p l e d limit 11 a n d reflects the weak valence
Superlattices and Microstructures, Vol. 9, No. 4, 1991
523
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Figure 5. E I - H H I exciton binding energy as a function of InGaAs well width: the circles are the measured values in the superlattice samples described here. The squares and solid line show the measured and calculated binding energy in isolated well samples.
1500
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Figure 4. Landau level fan diagram for the 25A InGaAs sample in B "~. The low and high energy modelled fits are given by the dashed and solid lines respectively.
subband coupling for HH1 and LHI subbands 38 meV apart. The separation of the hole subbands decreases as the InGaAs width is reduced (figure i ) making valence subband mixing effects more evident in the thinner well samples. The Landau level fan diagram for the 25A InGaAs sample is shown in figure 4. The transition energies are measured using reflectivity rather than transmission allowing the superlattice band structure above the top of the GaAs barriers to be studied. The energy gap (1515-20 eV) corresponds to the region where the bulk GaAs l s exciton dominates the response; however, the optically thin GaAs capping layer (200A) ensures that any higher Landau levels from bulk GaAs are weak and only the transitions originating from the superlattice layers are plotted. The solid lines are fitted with a 4.5 meV exciton binding energy and a heavy hole i n - p l a n e mass of 0.36m o. At low energies, the Landau level fit is poor and much better agreement is obtained with the lighter hole mass 0.19m o (dashed lines). The heavier hole mass at high energies arises from the mixing between heavy and light hole states away from kll=0, leading to a very n o n - p a r a b o l i c hole i n - p l a n e dispersion 11. Similiar behaviour has been observed previously at very low magnetic fields in G a A s - A l G a A s quantum wells12, however the strain splitting in ] n G a A s - G a A s superlattices allows the light (decoupled) mass to be observed over a much wider field range. The linewidths in the 50 k and 100 A InGaAs samples are not as sharp as in the 25 A sample and it
is not possible to see strong non-parabolicities due to the uncoupled mass at low fields. However the fitted high field hole masses of 0.25m o and 0.23m o r e s p e c t ively are in line with a progressive decoupling of the HHI and LH1 subbands as their energy separation increases. The binding energy of an exciton is very sensitive to its dimensionality and in a superlattice, any coupling into minibands leads to a decrease in the binding energy over to its isolated well value. Figure 5 shows the measured E 1 - H H 1 exciton binding energies in the four superlattice samples (filled circles) as a function of l n G a A s well width. Also included for comparison is some data for three I n G a A s - G a A s superlattices 13 (open squares) with much thicker 400 k GaAs barriers which effectively decouple the InGaAs wells. The measured exciton binding energies in all the uncoupled samples show good agreement with the calculated dependence for an isolated quantum well (solid line). The exciton binding energies measured in the 2 5 k and 5 0 k InGaAs samples lie substantially below the isolated well values and indicates the formation of extended minibands. For the 25 k InGaAs welt, the binding energy of 4.5 meV is little more than that of a bulk GaAs exciton (open circle), consistent with the exciton sensing the s u p e r lattice as a weakly anisotropic 3-dimensional solid. The binding energy then increases rapidly for the 5 0 k sample towards the isolated well value as the miniband width shrinks and the states become confined into the well layers. The peak in the exciton binding energy at 100A InGaAs layers indicates the strong effect of coupling in these structures across relatively thick layers. For the isolated well samples the confinement is stronger leading to a peak binding energy for much narrower lnGaAs thicknesses, around 5 0 k . A more definitive demonstration of interwell coupling in a superlattice is the presence of a finite miniband dispersion in the growth direction. This is measured by applying a magnetic field parallel (Bfl) to the superlattice layers so that the cyclotron orbits involve
Superlattices and Microstructures, Vol. 9, No. 4, 1991
524
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both i n - p l a n e a n d miniband motion 2. L a n d a u levels are t h e n only observed if the interwell coupling is strong e n o u g h to p r o d u c e extended Bloch states t h r o u g h the superlattice layers a n d the c o r r e s p o n d i n g L a n d a u level s e p a r a t i o n is smaller t h a n in a p e r p e n d i c u l a r field due to the inclusion of the heavier minibaod masses. In the strongly coupled 25.~ l n G a A s superlattice, the calculated electron and h e a v y hole m i n i b a n d widths a r e 20.3 m e V a n d 1.6 m e V respectively, which c o m p a r e with the a b s o r p t i o n linewidth of ~1.6 m e V - electrons at least should d e m o n s t r a t e c o h e r e n t miniband motion. Discrete L a n d a u levels are observed in B II which c o n f i r m s the presence of extended Bloch states a n d the corresponding transition energies are plotted as a function of m a g n e t i c field in figure 6. T h e top of the E I - H H 1 m i n i b a n d is calculated to lie at 1.512 eV so that any structure associated with the zone-edge E 1 - H H 1 e x c i t o n l 4 , 1 5 is p r o b a b l y hidden by the G a A s exciton which d o m i n a t e s the s p e c t r u m in this e n e r g y range. H o w e v e r , in c o n t r a s t to B " , no superlattice levels are seen above the G a b s b a n d edge which indicates a finite superlattice m i n i b a n d width less t h a n 30 m e V wide. T h e parallel field L a n d a u levels are fitted using the exciton model with a geometric average of the i n - p l a n e a n d m i n i b a n d masses for both electrons a n d holes. The i n - p l a n e masses a n d exciton binding e n e r g y are taken f r o m B"- a n d the m i n i b a n d masses f r o m the envelope function model. T h e a g r e e m e n t is good at low energies, giving a reduced mass a n i s o t r o p y # " 7 # " = 1.29 a n d indicates t h a t the h e a v y holes are indeed c o n t r i b u t i n g to the m i n i b a n d motion. H o w e v e r with the c o m b i n a t i o n of
a n a r r o w hole miniband a n d light i n - p l a n e mass (0.19too), the superlattice L a n d a u levels reach the top of the heavy hole m i n i b a n d well before the top of the electron m i n i b a n d . T h e arrows in figure 6 indicate the field at which this is calculated to o c c u r a n d the excitonic L a n d a u levels extend well beyond the hole miniband ' s a t u r a t i o n ' points in e a c h case. Maan's calculations 16 show t h a t for levels both near a n d above the miniband top, the L a n d a u level e n e r g y depends on the orbit position in real space making the transitions very broad a n d weak. Although the n a r r o w hole m i n i b a n d width (1.6 meV) m a y prevent this h a p p e n i n g here, it is unclear which hole states the discrete electron L a n d a u levels are now c o m b i n i n g with. Intuitively, the hole mass gets very h e a v y n e a r the top of the miniband a n d the solid lines in figure 6 are calculated with a semiclassical quantisation of the superlattice band structure but assuming no f u r t h e r increase in e n e r g y o n c e the miniband top is reached. T h e saturation of the hole m i n i b a n d reduces the cyclotron energy increase - the low energy g r a d i e n t is due to the c o m b i n e d e l e c t r o n - h o l e reduced mass, whereas the gradient at higher energies is just due to the electron mass alone and the data agrees very well with this c h a n g e in gradient. If the holes are given a c o n s t a n t m i n i b a n d mass (dashed lines), it is not possible to fit the behaviour at both low a n d high energies. For the 50 ,~ InGaAs sample, the interwell coupling a n d hence the m i n i b a n d widths are reduced, however excitonic L a n d a u levels are still observed in B I1 (figure 7) indicating the presence of extended Bloch
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Figure 7. T h e fan d i a g r a m for the 50A I n G a A s sample in n II. T h e solid lines include the full m i n i b a n d structure, while the dashed lines assume a c o n s t a n t electron mass a n d h e n c e ignore the effect of the miniband gap.
Superlattices and Microstructures, VoL 9, No. 4, 1991 states. From the ratio of the ls (n=0) diamagnetic shifts in B"- and B II, the reduced mass anisotropy is 1.56 at the miniband edge. The n=l Landau level weakens and dies out around 10 Tesla, which is consistent with a finite miniband width. However, a semiclassical quantisation of the calculated band structure (solid lines), which gives good agreement for the 25A InGaAs sample, clearly fails in this case. Even if the GaAs barriers are reduced to 90A, giving miniband widths of 13 and 0.5 meV for the electrons and heavy holes respectively, the calculation underestimates the n=l 'saturation' energy by about 6 meV. The fit for the ls (n=0) shift on the other hand is good indicating that the miniband masses and hence widths are approximately correct. The difference between the theory and experiment is thought to be due to the superlattice miniband structure breaking down in this sample. The transmission linewidth is relatively broad (5 meV) and has two distinct contributions to its lineshape, indicating inhomogeneities in the indium concentration 17. Any significant perturbation on the superlattice periodicity weakens the q quantum number conservation rule and hence allows Landau levels to be seen beyond the top of the miniband, with the superlattice then behaving more like an anisotropic alloy than a structure with a finite miniband width. The dashed line in figure 7 is calculated using a constant electron mass (i.e. ignoring the superlattice miniband gap) and the experimental points lie intermediate between the 'alloy' and 'superlattice' limits. In the two uncoupled samples with wider InGaAs layers, no superlattice Landau levels are observed in a parallel magnetic field and the excitonic transitions exhibit a parabolic diamagnetic shift. This is characteristic of an isolated quantum well 18 and demonstrates the absense of interwell coupling in these structures (figure 1). In summary, the influences of strain and interwell coupling have been demonstrated on the miniband structure of InGaAs-GaAs superlattices. The strain splits the valence subbands and allows the HHI decoupled in-plane mass to be seen over a wide field range. The interwell coupling is characterised by a decrease in the exciton binding energy over its isolated well value and the nresence of a miniband dispersion in the growth direction. The parallel field Landau levels are modelled
525 using an exact semiclassical quantisation of the miniband structure which, while giving good agreement for one superlattice, fails for another of poorer quality; this discrepancy is attributed to alloy inhomogeneities.
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