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Magneto-photoluminescence of GaNAlGaN quantum wells valence band reordering and excitonic b

a r X i v :c o n d -m a t /0101069v 1 5 J a n 2001

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

P.A.Shields,R.J.Nicholas

Department of Physics,Oxford University,

Clarendon Laboratory,Parks Rd.,Oxford,OX13PU,U.K.

N.Grandjean,J.Massies

CNRS,Centre de Recherche sur l’H ′e t ′e ro-Epitaxie et ses Applications,Valbonne,F-06560,France.

(February 1,2008)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 (g ?～3)in magneto-luminescence spectra in ?elds up to 55T.This has been caused by a reversal of the states in the strained AlGaN barriers thus giving di?erent barrier heights for the di?erent 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 e?ect of the electric ?eld in the wells.

I.INTRODUCTION

The recent understanding of the in?uence of macro-scopic polarisation in nitride materials has inspired a closer look at the properties of AlGaN-based quantum wells where these e?ects are stronger than found with the more commercially attractive InGaN system.

The typical optical characterisation of a GaN/AlGaN quantum well sample ?nds 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 un-derstood in terms of the quantum con?ned stark e?ect (QCSE)from which the electric ?eld caused by the polar-isation can be deduced.It has been found that the ?eld has a strong linear dependence on the aluminium concen-tration reaching almost 1.5MV/cm for an Al composition of 27%2,3.

The optical quality,as determined from the emission e?ciency,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 struc-tures where an inhomogeneous composition would have a greater e?ect.Typically the best linewidths in In-GaN/GaN structures are about 40meV 4,which can be compared with about 20meV 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 ef-fective masses and the large exciton binding energies in these materials,it is necessary to use very high magnetic ?elds to see any e?ects.This leads us to use pulsed-?eld magnets that can currently give ?elds of up to 55T.

II.EXPERIMENT

We have performed magneto-optical experiments in both steady and pulsed magnetic ?elds on single quan-tum well samples with di?erent widths of 4,8,12,and 16monolayers (ML).One monolayer corresponds to 2.59?A therefore giving about 10,20,30and 40?A 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 bu?er layer before the quantum wells.For samples N298and N307,the tech-nique 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 (re?ection high energy electron di?rac-tion)spectra observed during the growth 6.

The results that will be discussed are low and high excitation photoluminescence (PL)with ?elds up to 55T at 4.2K where the e?ects of the ?eld 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.

Sample Description

I(E)dE.(1)

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 the4ML well

is approximately2meV at50T.Eventually at the highest

?eld the redshift stops and appears to be overtaken by a

20 30 40 50 60

3510

3511

3512 Magnetic field (B)

3538

3539

3540 E n e r g y (m e V )3580 3581 3582

3583

3530

3531 3532 3533

Magnetic field (B)

E n e r g y (m e V )FIG.3.Magnetic ?eld dependence of the low excitation photoluminescence for the di?erent quantum wells for sample N298(top)and N307(bottom),along with the ?ts to the data as described in the text.

smaller diamagnetic term.

For the widest well,two components are observed that have di?erent dependencies on magnetic ?eld.The ob-servation of two peaks has previously been suggested to be due to ?uctuations in the well width,corresponding to 16and 17ML’s.We see an enhancement of the 16ML peak with ?eld so that it dominates the spectra at high ?eld,which is consistent with this proposal and the suppression of the in-plane transport.The excitons are then less able to ?nd the potential minima caused by the ?uctuations before recombining.

The shifts for all the wells can be described by the expression,E (B )=E 0±1

2g ?

μB B

n ↑?n ↓

n ↓,

given by a Boltzmann distribution.

Table II shows the values deduced from the experiments for the characteristic temperatures,g-factors and dia-magnetic shift coe?cient.These last two parameters will be individually discussed in the following sections.The temperatures deduced simply re?ect a heating of the car-riers by the high excitation intensities onto the sample.These were necessarily rather high in order to detect suf-?cient signal in the short time duration of a single pulse (～0.3ms).

IV.DISCUSSION A.Zeeman splitting

The g-factors for the di?erent samples are compared in Figure 4,and show a general trend towards g ?～3for the narrower wells,with it reducing to g ?～2for 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-re?ectivity of GaN epilayers,no such splitting is seen.From the re-sults in bulk GaN the lowest transition,associated with the A valence band,does not split for the same orienta-tion as the experiments described here 9,10.This is due

Sample N257T(K)g*γ2(μeV/T2) 4ML QW15(1) 3.1(1)0.99(7) 8ML QW13(1) 2.9(2) 1.3(1) 12ML QW13(1) 3.3(3) 2.1(2) 16ML QW5(5) 1.8(4) 3.0(3) N307

8ML QW17(3) 3.4(2) 1.5(1)

2 (?cr+?so)? (?cr+?so)2?82 ,

(3)

where?cr and?so

are the crystal?eld and spin-orbit

energy terms respectively12.

This is for the case of a bulk semiconductor.With the e?ects of a two-dimensional quantum con?nement, the energies are usually understood in terms of the en-velope function approximation that are then dependent on:the e?ective masses in the well and barrier,the va-lence band potential,and the width of the well.This causes the energies of the bands to become unrelated to the bulk energy terms in Equation3.Instead e?ective spin-orbit and crystal?eld 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?eld 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 Ga1?x N13,though this value is not experimen-tally known.The more important consequence for this work is that this will cause di?erent valence band o?sets for the di?erent bands and Figure5shows how this ef-fect 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 symmetri-cal quantum well,which is known to be invalid for these samples.However the e?ect of a strong electric?eld will be to bring the con?nement energy closer to the barrier o?sets thus making them more important and enhancing the reversal e?ect.

A reversal of states has a profound e?ect on the spin-ordering of the valence band that can be clearly seen

A-band B-band

Barrier

Bulk

T r a n s i t i o n e n e r g y Well width (?)

FIG.5.A schematic diagram based on a ?nite quantum well model showing how the quantum well energies can cross at a particular well width for two bands with di?erent o?sets.

HH-LH splitting (meV)

FIG.6.k.p results for GaN/AlGaN showing how the crys-tal ?eld splitting of the valence band a?ects the spin-ordering of the valence band states at 60T.Γ7states are indicated as solid lines,and Γ9states are dashed.The vertical arrows show the allowed optical transitions.

through the k.p calculations of Figure 6within the quasi-cubic model

a magnetic ?eld of 60T,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 ?eld 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 ?eld,we ?nd the usual zinc-blende band and spin ordering.The bulk wurtzite case is also indicated,corre-sponding to a ?nite strain or crystal ?eld,and this cor-rectly 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 Γ7states causes a reversal of the spin-ordering thus changing the sign of the g-factor for the lowest ?Γ7hole band.This process also lowers its energy below the Γ9A-valence band so that transitions involving this band are favoured in luminescence.Exci-tonic g-factors less than two occur for the lowest bands to the left of the anti-crossing in Figure 6as indicated,whereas they are greater than two to the right.

For a GaN/AlGaN quantum well structure the precise band ordering of the con?ned states is hard to predict since it is critically dependent on both the well and bar-rier properties due to the electric ?eld.As an extra tool,X-ray measurements can be used to examine the crys-tal 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,re-sulting in them having a tensile biaxial strain 14.There-fore 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 rela-tionship for the lattice constant,giving an in-plane strain,εxx =εyy =0.31%for an aluminium content of 13%.This contributes to the crystal ?eld splitting through ?cr +3C 33εxx ≈?0.51εxx ,and D 3～6eV to give 3

Lowest in terms of the hole energy.

?11,12meV from review in Kim et al 13,14.9±0.3meV from re?ectivity of sample G889

Therefore our results suggest that the ordering of the valence band has changed for narrow quantum wells,per-haps as a result of the lower barrier height for the crystal ?eld Γ7band caused by the strain and the aluminium content in the barriers.The reordering of these states will cause the ground state to have a Γ7character and the inverted spin splitting would account for the observed enhanced e?ective g-factor.

The increase of the value for sample N307in Figure 4with respect to

the same width

in the

other

samples can

understood in terms of the distributions of the electric ?elds.Where more than one quantum well is present,the ?nite thickness of the barrier allows a redis-tribution of the polarisation ?elds between both the well and the barrier,whereas the ?eld 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 2is determined by the size of the exciton wavefunction in the plane perpendicular to the ?eld.From this it is possible to infer the excitonic binding energy and the following section describes how these parameters have been determined for the di?erent quantum wells along with values corresponding to the bulk case.

The e?ect of a magnetic ?eld can be considered as an extra con?ning potential on the already localised exci-tonic system in both 2D and 3D cases.The bulk data can be ?tted very well to a full numerical calculation for a 3-dimensional hydrogen atom in a magnetic ?eld 15that uses the excitonic binding energy and the reduced mass as ?tting parameters.The binding energy for our partic-ular sample was determined by Neu et al 16as 24.1meV by identifying the 2s excited state,and this enables the ?tting to give a reduced mass of 0.180(2)m e as the only free parameter.This numerical calculation reduces to a B 2dependence in the low ?eld limit,so that by ?tting the low ?eld data we obtain a coe?cient,γ2=2.04(3)μeV/T 2that we can use in a direct comparison between the bulk and QW data.

The diamagnetic coe?cient for a quantum well is given by 17,

γ2=

e 2

N2984ML 1030.0(1.4)34.6(1.6)8ML 2028.3(1.7)36.6(2.2)12ML

3033.9(1.2)30.7(1.1)

G889Bulk

-40.9(5)25.4(2)

TABLE III.The in-plane extent and exciton binding en-ergy for the di?erent quantum well widths deduced from the diamagnetic shift of the luminescence.

through ρ2

εr

a H 0e 2

γ2

a H 0R

3D average;ε=(ε⊥)

3§

This value is taken from the bulk calculations and is used as a ?rst approximation.The actual value will depend on the precise band ordering and mixing e?ects as described in the section entitled Zeeman splitting.

I n -p l a n e B o h r r a d i u s (?)Quantum well width (ML)

https://www.wendangku.net/doc/784835326.html,parison between the experimental (points)and theoretical values (solid line)for the in-plane extent of the excitonic wavefunction for di?erent well widths.The theoret-ical values are determined from considering a two-parameter variational trial function for a series of GaN/AlGaN QW’s with Al barrier composition x=0.27.The experimental values are from sample N257and are deduced from the diamagnetic shift of the luminescence.

These variational calculations of the excitonic wave-functions were used to investigate the ground state of heavy-hole,free excitons under di?erent levels of approx-imation.Two di?erent 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 con?nement axis.In the single parameter trial function this separation is solely determined by the elec-tric ?elds 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 N257are shown along-side 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 di?erences between the binding ener-

B i n d i n g E n e r g y (m e V )Quantum well width (ML)

FIG.8.Experimental values of the binding energies shown with calculations of the well width dependence of the exci-ton binding energies for a GaN/AlGaN SQW including in-ternal electric ?elds.For a single parameter variational trial function (short dash),calculations are shown for four alu-minium compositions,x=0.27,0.17,0.13,0.08,whereas for a two-parameter function (long dash),x=0.27.

gies deduced from this parameter,and this can be under-stood from the simplicity of our model and the greater in?uence that the higher aluminium concentration is ex-pected to have through the increased electric?eld.How-ever the data favours the inclusion of the?eld rather than its omission in square well calculations1.

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=0.27for 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 signi?-cant 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?rst approximation.

In converting from the diamagnetic coe?cient,γ2,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 bar-riers as evidenced by both the change in the valence band ordering and the calculated maximum in the ex-citonic binding energy at around4ML.This model is not as thorough as that used in the theoretical calcula-tions 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?eld of nitride semiconductors,it has become automatic to blame any sample inadequacies on the lack of a suitably lattice-matched substrate.How-ever recent work on homoepitaxially-grown quantum well samples has suggested that there is perhaps a fundamen-tal limitation in the linewidth,similar to the linewidths present in this work,that is caused by the random po-sitions of the group III elements in the ternary barrier combined with the large hole masses19,20.The strong in-ternal?elds,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 ?elds up to55T for three di?erent GaN/AlGaN quantum wells that shows a strong well width dependence of the shift of the luminescence peaks.The high pulsed?elds were essential in the resolution of the di?erent depen-dencies,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?elds to be used due to the fundamental linewidth limitations present.

Our data shows that by observing an enhanced g-factor for the narrow wells,the valence band must have changed signi?cantly compared with bulk GaN.We have attributed this to a reordering of the valence band states in the strained AlGaN barriers,giving di?erent barrier heights for the di?erent quantum well hole states.We have also observed an increase in the exciton binding en-ergy with the reduction of the well width in agreement with calculations using a variational approach in the en-velope function formalism that includes the e?ect of the electric?eld 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 Fa-cility for the provision of the frequency-doubled Argon laser.P.A.S.acknowledges?nancial support from Sharp Laboratories of Europe Ltd.

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E?ects in Semiconductors(Wiley,New York,1974).

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56,7363(1997).

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