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Magneto-photoluminescence of GaN/AlGaN quantum wells: valence band reordering and excitonic binding energies

A re-ordered valence band in GaN/AlGaN quantum wells with respect to GaN epilayers has been found as a result of the observation of an enhanced g-factor in magneto-luminescence spectra in fields up to 55 T. This has been caused by a reversal of the states in the strained AlGaN barriers thus giving different barrier heights for the different quantum well hole states. From k.p calculations in the quasi-cubic approximation, a change in the valence-band ordering will account for the observed values for the g-factors. We have also observed the well-width dependence of the in-plane extent of the excitonic wavefunction from which we infer an increase in the exciton binding energy with the reduction of the well width in general agreement with theoretical calculations of Bigenwald et al (phys. stat. sol. (b) 216, 371 (1999)) that uses a variational approach in the envelope function formalism that includes the effect of the electric field in the wells.


I. INTRODUCTION
The recent understanding of the influence of macroscopic polarisation in nitride materials has inspired a closer look at the properties of AlGaN-based quantum wells where these effects are stronger than found with the more commercially attractive InGaN system.
The typical optical characterisation of a GaN/AlGaN quantum well sample finds a single peak in luminescence at energies that are strongly dependent on the width of the wells and the aluminium content of the barriers. The energies of these peaks have been quantitatively understood in terms of the quantum confined stark effect (QCSE) from which the electric field caused by the polarisation can be deduced. It has been found that the field has a strong linear dependence on the aluminium concentration reaching almost 1.5 MV/cm for an Al composition of 27% 2, 3 .
The optical quality, as determined from the emission efficiency, is better in the InGaN/GaN system compared to the GaN/AlGaN structures, whereas for the emission linewidth the opposite is true. The latter is likely to be related to the position of the ternary alloy in the structure, it being in the active region in InGaN structures where an inhomogeneous composition would have a greater effect. Typically the best linewidths in In-GaN/GaN structures are about 40 meV 4 , which can be compared with about 20 meV for GaN/AlGaN wells 2 . The development of these high quality AlGaN/GaN QW's has now allowed measurements to be made of the magneto-optical properties in nitride based quantum wells.
Due to the linewidths already mentioned, the large effective masses and the large exciton binding energies in these materials, it is necessary to use very high magnetic fields to see any effects. This leads us to use pulsed-field magnets that can currently give fields of up to 55 T.

II. EXPERIMENT
We have performed magneto-optical experiments in both steady and pulsed magnetic fields on single quantum well samples with different widths of 4, 8, 12, and 16 monolayers (ML). One monolayer corresponds to 2.59Å therefore giving about 10, 20, 30 and 40Å respectively. The three samples described in this paper are described in Table I. The structures were grown by molecular beam epitaxy on GaN templates on sapphire substrates with 200nm of AlGaN grown as a buffer layer before the quantum wells. For samples N298 and N307, the technique of lateral overgrowth on patterned GaN (ELOG) was used to improve the template quality 5 . The sample widths and the aluminium composition were determined through RHEED (reflection high energy electron diffraction) spectra observed during the growth 6 .
The results that will be discussed are low and high excitation photoluminescence (PL) with fields up to 55 T at 4.2 K where the effects of the field can be seen through a shift of the luminescence peaks.
For the high excitation density PL the source was a pulsed (5ns) frequency-quadrupled Nd:YAG laser at 266nm, whereas for the low excitation density it was the 244nm line from a CW frequency-doubled Argon laser. For the latter in the pulsed fields, a masked chopper wheel was used to produce a train of short laser pulses, ∼0.3ms duration, at ∼100ms intervals. A pulse from this train triggered the magnet system so that the next laser pulse coincided with the maximum of the pulsed field, where there is little change in the field (≤1%) over a 0.3 ms timescale. A CCD cooled to −70 o C attached to a quarter metre spectrometer was used to detect the PL.
III. RESULTS Figure 1 shows the low and high excitation PL for sample N257. Several PL peaks can be seen that can be attributed to the different wells. Between the low and high excitation, we have observed a shift in the PL that is strongly dependent on the width of the wells, ranging from 1meV for the narrowest up to 20meV for the widest. This is in agreement with the idea that in-built electric fields play an important role in this sample via the quantum confined Stark effect 7 and that they have been screened by the excited carrier population. Previous measurements have calculated the field to be 550 kV/cm 2 . The most obvious consequence of this is seen through the observation of PL from the widest well at 3.42 eV, which is below the band gap energy of bulk GaN at 3.49 eV. Unfortunately, for this sample, the bulk feature coincides with the PL from the 12 ML QW, so that luminescence from the bulk template superimposes a structure onto this peak.
The low intensity excitation is of the order 0.25W cm −2 and can be considered to be in the regime where selfscreening of the electric field is negligible. The exact excitation intensity of the pulsed laser is difficult to determine presisely, but can be estimated to be greater by about five orders of magnitude. The magnetic field results discussed below were therefore taken using the low intensity quasi-CW excitation.
The effect of the magnetic field on the 4ML QW in sample N257 can be seen in Figure 2, and shows a shift of the luminescence to lower energy. The form of this shift is dependent on the width of the wells, and is shown in Figure 3. In order to determine the peak positions a centre of mass method was used in preference to a peak fitting routine 8 . This decision resulted from the difficulty of deciding the correct form of the peak as it is slightly asymmetric and thus causes problems for a Gaussian fit. The centre of mass is also known as the first moment, and is given by, For the widest wells there is a diamagnetic blueshift of the transition energy, but as the well width decreases we observe a transition to a redshift, which for the 4ML well is approximately 2meV at 50T. Eventually at the highest field the redshift stops and appears to be overtaken by a smaller diamagnetic term.
For the widest well, two components are observed that have different dependencies on magnetic field. The observation of two peaks has previously been suggested to be due to fluctuations in the well width, corresponding to 16 and 17 ML's. We see an enhancement of the 16 ML peak with field so that it dominates the spectra at high field, which is consistent with this proposal and the suppression of the in-plane transport. The excitons are then less able to find the potential minima caused by the fluctuations before recombining.
The shifts for all the wells can be described by the expression, E(B) = E 0 ± 1 2 g * µ B B + γ 2 B 2 so that the overall dependence for each Zeeman-split component is determined by the relative magnitudes of the linear Zeeman and quadratic diamagnetic terms. The linear Zeeman shift causes the transition to split into two, of which we see predominantly the lowest energy component. The S-shape observed especially for the narrower wells at the lower fields can be explained as resulting from a significant thermal population in the upper component whilst the Zeeman splitting is still small. This effect is particularly noticeable when the width of the luminescence is large compared with the shift in magnetic field and kT , as is the case in these samples.
The magnetic field dependence of the centre of mass can then be expressed in terms of the populations in each component, n ↑ and n ↓, with the ratio, n↑ n↓ , given by a Boltzmann distribution. Table II shows the values deduced from the experiments for the characteristic temperatures, g-factors and diamagnetic shift coefficient. These last two parameters will be individually discussed in the following sections. The temperatures deduced simply reflect a heating of the carriers by the high excitation intensities onto the sample. These were necessarily rather high in order to detect sufficient signal in the short time duration of a single pulse (∼0.3 ms).

A. Zeeman splitting
The g-factors for the different samples are compared in Figure 4, and show a general trend towards g * ∼ 3 for the narrower wells, with it reducing to g * ∼ 2 for the wider wells. This is particularly noticeable in sample N298.
The observation of a Zeeman splitting through a clear redshift is surprising, since in the magneto-reflectivity of GaN epilayers, no such splitting is seen. From the results in bulk GaN the lowest transition, associated with the A valence band, does not split for the same orientation as the experiments described here 9,10 . This is due  to a cancellation of the electron and hole g-factors when the magnetic field is parallel to the c-axis. Even for the B valence band there is a similar compensation leading to an excitonic g-factor, g * = 1.24. So as both theoretical and experimental work on AlGaN heterojunctions 11 suggest that the electron effective g-factor should not change from a value of ∼ 2 as a result of confinement, we conclude that instead the hole g-factor must have drastically changed in the narrower quantum wells due possibly to a significant change in the valence band structure.
The reduced symmetry of the wurtzite with respect to the cubic structure is represented in the Hamiltonian through the crystal field. If spin-orbit effects are neglected, the crystal field reduces the degeneracy of the valence bands with different z-components of orbital angular momentum. With spin-orbit terms included there are then the three usual bands; A, B, and C with symmetries Γ 9 , Γ 7 , and Γ 7 respectively, that have the following separations, where ∆ cr and ∆ so are the crystal field and spin-orbit energy terms respectively 12 .
This is for the case of a bulk semiconductor. With the effects of a two-dimensional quantum confinement, the energies are usually understood in terms of the envelope function approximation that are then dependent on: the effective masses in the well and barrier, the valence band potential, and the width of the well. This causes the energies of the bands to become unrelated to the bulk energy terms in Equation 3. Instead effective spin-orbit and crystal field terms can be used to account for the splittings of the A, B and C valence bands, where the nomenclature is now strictly in terms of the band symmetries and not the ordering.
For the GaN/AlGaN system, it is known that in bulk AlN the crystal field splitting has the opposite sign to that of GaN thus inverting the order of the states. This would lead to a reversal of the states at some critical value of x in Al x Ga 1−x N 13 , though this value is not experimentally known. The more important consequence for this work is that this will cause different valence band offsets for the different bands and Figure 5 shows how this effect can lead to a reversal of the quantum well states, even when the same masses in the z-direction, m , are used. (In fact m is identical for the A and B valence bands.) This calculation was carried out for a symmetrical quantum well, which is known to be invalid for these samples. However the effect of a strong electric field will be to bring the confinement energy closer to the barrier offsets thus making them more important and enhancing the reversal effect.
A reversal of states has a profound effect on the spinordering of the valence band that can be clearly seen  through the k.p calculations of Figure 6 within the quasicubic model at a magnetic field of 60 T, using the band parameters deduced by Stepniewski et al 10 . With this model, the wurtzite Hamiltonian can be approximated as a cubic Hamiltonian that is uniaxially strained along the [111] direction. The crystal field from the wurtzite vocabulary can then be translated into a heavy-light hole splitting (HH-LH) in cubic language. Therefore with zero strain, corresponding to zero HH-LH splitting or zero crystal field, we find the usual zinc-blende band and spin ordering. The bulk wurtzite case is also indicated, corresponding to a finite strain or crystal field, and this correctly reproduces the spin-ordering in GaN 10 , whereas AlN would correspond to the far right of the graph with a large positive HH-LH splitting. The crossing, or more correctly anti-crossing, of the Γ 7 states causes a reversal of the spin-ordering thus changing the sign of the g-factor for the lowest * Γ 7 hole band. This process also lowers its energy below the Γ 9 A-valence band so that transitions involving this band are favoured in luminescence. Excitonic g-factors less than two occur for the lowest bands to the left of the anti-crossing in Figure 6 as indicated, whereas they are greater than two to the right.
For a GaN/AlGaN quantum well structure the precise band ordering of the confined states is hard to predict since it is critically dependent on both the well and barrier properties due to the electric field. As an extra tool, X-ray measurements can be used to examine the crystal properties and give information on the strain that is present. In these samples, it has been shown that the Al-GaN barriers have the same lattice constant as the GaN template, it being larger than if they were unstrained, resulting in them having a tensile biaxial strain 14 . Therefore the well material is unstrained and has a normal wurtzite GaN valence band structure, whilst the barrier has the opposite ordering due to the AlN contribution in the alloy and the strain present.
The latter can be estimated from using a linear relationship for the lattice constant, giving an in-plane strain, ε xx = ε yy = 0.31% for an aluminium content of 13%. This contributes to the crystal field splitting through ∆ cr + 3 2 D 3 ε zz with ε zz = − 2C13 C33 ε xx ≈ −0.51ε xx , and D 3 ∼ 6eV to give 3 2 D 3 ε zz = 14meV 13 . As experimental values of ∆ cr for GaN are very close to this value (11-15 meV † ), once the contribution of the AlN is also included, the barriers are very much in the regime for a reversal of states to occur. Figure 7 of Kim et al 13 shows a calculation for a similar tensile biaxial strain in GaN that clearly shows a reversal of the states with the anti-crossing also giving the lowest valence band a heavier mass than would be otherwise expected. * Lowest in terms of the hole energy. † 11,12meV from review in Kim et al 13 , 14.9 ± 0.3meV from reflectivity of sample G889 Therefore our results suggest that the ordering of the valence band has changed for narrow quantum wells, perhaps as a result of the lower barrier height for the crystal field Γ 7 band caused by the strain and the aluminium content in the barriers. The reordering of these states will cause the ground state to have a Γ 7 character and the inverted spin splitting would account for the observed enhanced effective g-factor.
The increase of the value for sample N307 in Figure  4 with respect to the same width in the other samples can be understood in terms of the distributions of the electric fields. Where more than one quantum well is present, the finite thickness of the barrier allows a redistribution of the polarisation fields between both the well and the barrier, whereas the field is solely in the wells for the SQW. This will push the hole states further into the barrier material.

V. DIAMAGNETIC SHIFT
The magnitude of the diamagnetic shift in equation 2 is determined by the size of the exciton wavefunction in the plane perpendicular to the field. From this it is possible to infer the excitonic binding energy and the following section describes how these parameters have been determined for the different quantum wells along with values corresponding to the bulk case.
The effect of a magnetic field can be considered as an extra confining potential on the already localised excitonic system in both 2D and 3D cases. The bulk data can be fitted very well to a full numerical calculation for a 3-dimensional hydrogen atom in a magnetic field 15 that uses the excitonic binding energy and the reduced mass as fitting parameters. The binding energy for our particular sample was determined by Neu et al 16 as 24.1 meV by identifying the 2s excited state, and this enables the fitting to give a reduced mass of 0.180(2) m e as the only free parameter. This numerical calculation reduces to a B 2 dependence in the low field limit, so that by fitting the low field data we obtain a coefficient, γ 2 = 2.04(3)µeV /T 2 that we can use in a direct comparison between the bulk and QW data.
The diamagnetic coefficient for a quantum well is given by 17 , where µ is the reduced mass of the exciton and ρ is the separation of the electron and the hole in the plane of the quantum well. Therefore the diamagnetic coefficient contains information about the zero-field properties of the exciton and can give a value for the in-plane extent of the wavefunction.
From the 1s orbital in the hydrogen atom, the expectation value of r 2 is, r 2 = 3a * 0 2 , where a * 0 is the effective Bohr radius. r 2 can then be converted to an in-plane size  through ρ 2 = 2 3 r 2 . If the exciton is assumed to remain essentially three dimensional, the effective binding energy, R * , can then be deduced through scaling with respect to the hydrogen atom, where a H 0 is the Bohr radius and R H is the Rydberg constant. This gives the following expression, The results from this calculation have been summarised in Table III (ε r = 9.8 ‡18 , µ = 0.18m 0 § ). The similarity of the binding energy for the bulk deduced in this way, 25.3 meV compared to 24.1 meV from the 1s-2s separation, illustrates the accuracy of this technique. Table III shows that as the well width is reduced, the exciton size is also reduced resulting in a strong enhancement of the binding energy for the narrow wells, in agreement with Bigenwald et al 1 and Grandjean et al 2 . We also observe a reduction of the exciton binding energy for the widest well due to the separation of the electron and hole to the separate interfaces of the QW. Our results can be compared with the calculations from Bigenwald et al 1 shown in Figure 7 and Figure 8 of the in-plane extent and binding energy of excitons in GaN/AlGaN quantum wells for different compositions and well widths. ‡ 3D average; ε = (ε ⊥ ) 2 3 (ε ) 1 3 § This value is taken from the bulk calculations and is used as a first approximation. The actual value will depend on the precise band ordering and mixing effects as described in the section entitled Zeeman splitting. These variational calculations of the excitonic wavefunctions were used to investigate the ground state of heavy-hole, free excitons under different levels of approximation. Two different forms of the variational ground state wavefunction were considered in the calculations. Both included a single variational parameter to govern the lateral extent of the wavefunction, however for one an additional variational parameter was included in the determination of the separation of the electron and hole along the confinement axis. In the single parameter trial function this separation is solely determined by the electric fields present in the structure, whereas an additional contribution from the Coulomb attraction is allowed for in the two-parameter trial function.
The experimental results from N257 are shown alongside the theoretical calculations of the in-plane pseudo-Bohr radius in Figure 7. Our results agree well with the theory despite the lower aluminium concentration in the barriers, 13% compared to the value of 27% used in the calculations.
There are larger differences between the binding ener- gies deduced from this parameter, and this can be understood from the simplicity of our model and the greater influence that the higher aluminium concentration is expected to have through the increased electric field. However the data favours the inclusion of the field rather than its omission in square well calculations 1 .
The single parameter trial function underestimates the binding energy for a given aluminium composition and the more accurate two-parameter function is shown only for an aluminium composition, x=0.27. From comparing results for x=0.27 for both models, it seems that the two parameter function for the appropriate composition, x=0.08, & 0.13, would accurately account for the data.
It should be pointed out the theoretical results are for the heavy-hole exciton, which we have shown is not the lowest state for these samples. This should not be significant for the purposes of comparison because the excitonic mass will be predominantly determined by the electron mass. The relevant hole mass would be an average over the top of the valence band within an exciton binding energy from the band edge, and is likely to be heavy as a result of the valence band mixing associated with the band reordering. Both theory and experiment use the heavy-hole exciton mass as a first approximation.
In converting from the diamagnetic coefficient, γ 2 , to the binding energy the three-dimensional hydrogen atom was used as a basis rather than a two-dimensional atom. This approximation remains reasonable due to both the shrinkage of the in-plane wavefunction caused by the increased binding and the considerable leakage of the bound state wavefunction out of the wells into the barriers as evidenced by both the change in the valence band ordering and the calculated maximum in the excitonic binding energy at around 4 ML. This model is not as thorough as that used in the theoretical calculations where along with the quasi-2D wavefunction, they also use a self-consistent algorithm.
The energy shifts that we have discussed are very small in comparison with the linewidth. It is worth thinking of the reasons behind this and the prospects for future experiments. In the field of nitride semiconductors, it has become automatic to blame any sample inadequacies on the lack of a suitably lattice-matched substrate. However recent work on homoepitaxially-grown quantum well samples has suggested that there is perhaps a fundamental limitation in the linewidth, similar to the linewidths present in this work, that is caused by the random positions of the group III elements in the ternary barrier combined with the large hole masses 19,20 . The strong internal fields, through causing a strong separation of the electron and hole wavefunctions to the opposite interfaces of the quantum well, thus make the optical properties very sensitive to these barrier irregularities.

VI. CONCLUSIONS
This work has presented magneto-luminescence data in fields up to 55 T for three different GaN/AlGaN quantum wells that shows a strong well width dependence of the shift of the luminescence peaks. The high pulsed fields were essential in the resolution of the different dependencies, and even when the immense amount of work on the sample growth in the future is considered with the expected improvement in quality, it is unlikely that this would allow lower continuous fields to be used due to the fundamental linewidth limitations present.
Our data shows that by observing an enhanced gfactor for the narrow wells, the valence band must have changed significantly compared with bulk GaN. We have attributed this to a reordering of the valence band states in the strained AlGaN barriers, giving different barrier heights for the different quantum well hole states. We have also observed an increase in the exciton binding energy with the reduction of the well width in agreement with calculations using a variational approach in the envelope function formalism that includes the effect of the electric field in the wells. Better agreement is obtained when the trial wavefunction includes a consideration of the three-dimensional Coulomb potential.

VII. ACKNOWLEDGEMENTS
We are grateful to the EPSRC (UK) for the support of this work and particularly the Lasers for Science Facility for the provision of the frequency-doubled Argon laser. P.A.S. acknowledges financial support from Sharp Laboratories of Europe Ltd.