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

9 Pages·0.22 MB·English

by  P. A. Shields

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, OX1 3PU, U.K.
N. Grandjean, J. Massies
CNRS, Centre de Recherche sur l’He´te´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
1
0 spectrain fieldsupto55T.Thishasbeencaused byareversalofthestatesin thestrained AlGaN
0 barriers thus giving different barrier heights for the different quantum well hole states. From k.p
2 calculations in the quasi-cubic approximation, a change in the valence-band ordering will account
for theobserved values for theg-factors.
n
Wehavealsoobservedthewell-widthdependenceofthein-planeextentoftheexcitonicwavefunction
a
J from which we infer an increase in theexciton bindingenergy with thereduction of the well width
in general agreement with theoretical calculations of Bigenwald et al (phys. stat. sol. (b) 216, 371
5
(1999)) that usesavariational approach intheenvelopefunction formalism thatincludestheeffect

1 of theelectric field in thewells.
v
9
6
0
1 I. INTRODUCTION Due to the linewidths already mentioned, the largeef-
0 fective masses and the large exciton binding energies in
1 The recent understanding of the influence of macro- thesematerials,itis necessaryto useveryhighmagnetic
0 scopic polarisation in nitride materials has inspired a fields to see any effects. This leads us to use pulsed-field
/
t closer look at the properties of AlGaN-based quantum magnets that can currently give fields of up to 55 T.
a
m wells where these effects are stronger than found with
the more commercially attractive InGaN system.
- II. EXPERIMENT
d The typical optical characterisation of a GaN/AlGaN
n quantumwellsample finds a single peak inluminescence
o at energies that are strongly dependent on the width We have performed magneto-optical experiments in
c of the wells and the aluminium content of the barriers. both steady and pulsed magnetic fields on single quan-
:
v The energiesofthese peakshavebeenquantitativelyun- tumwellsampleswithdifferentwidthsof4,8,12,and16
i derstood in terms of the quantum confined stark effect monolayers (ML). One monolayer corresponds to 2.59˚A
X
(QCSE)fromwhichtheelectricfieldcausedbythepolar- therefore giving about 10, 20, 30 and 40˚A respectively.
r
a isation can be deduced. It has been found that the field The three samples described in this paper are described
hasastronglineardependenceonthealuminiumconcen- in Table I. The structures were grown by molecular
trationreachingalmost1.5MV/cmforanAlcomposition beam epitaxy on GaN templates on sapphire substrates
of 27%2,3. with 200nmof AlGaN grownas a buffer layer before the
The optical quality, as determined from the emission quantum wells. For samples N298 and N307, the tech-
efficiency, is better in the InGaN/GaN system compared nique of lateral overgrowth on patterned GaN (ELOG)
to the GaN/AlGaN structures, whereas for the emission was used to improve the template quality5. The sample
linewidth the opposite is true. The latter is likely to widths and the aluminium composition were determined
be related to the position of the ternary alloy in the throughRHEED(reflection highenergy electrondiffrac-
structure, it being in the active region in InGaN struc- tion) spectra observed during the growth6.
tures where an inhomogeneous composition would have The results that will be discussed are low and high
a greater effect. Typically the best linewidths in In- excitationphotoluminescence(PL)withfieldsupto55 T
GaN/GaN structures are about 40 meV4, which can be at4.2Kwheretheeffectsofthefieldcanbeseenthrough
compared with about 20 meV for GaN/AlGaN wells2. a shift of the luminescence peaks.
The development of these high quality AlGaN/GaN For the high excitation density PL the source was
QW’s has now allowed measurements to be made of a pulsed (5ns) frequency-quadrupled Nd:YAG laser at
the magneto-opticalpropertiesinnitride basedquantum 266nm,whereas for the low excitation density it was the
wells. 244nm line from a CW frequency-doubled Argon laser.
1
Sample Description
N257 4 SQWsof 4, 8, 12, 16 MLs separated
by 100˚A of AlGaN at 13%. MBE template.
N307 SQW of 8ML wide with AlGaN 3000(cid:13) 0 T(cid:13)
barriers with 8% Al. ELOG template.
2500(cid:13)
N298 4 SQWsof 4, 8, 12, 16 MLs separated
by100˚A of AlGaN at 8%. ELOG template.
b.)(cid:13) 2000(cid:13)
ar
TABLEI. Details of the samples grown byMBE. e (
c
n 1500(cid:13)
e
c
s
e
n
(cid:13) mi 1000(cid:13) 52 T(cid:13)
u
L
4 ML(cid:13)
500(cid:13)
minescence(cid:13) 0(cid:13)
minescence (arb)(cid:13) 16 ML(cid:13) Region of bulk lu 8 ML(cid:13) high(cid:13) (cid:13) FIG. 2. Fi3g5u90r(cid:13)e 23:600M(cid:13) ag36n1e0(cid:13)tEince3rfi6g2ey0 l(cid:13)(dmed3V6e)3(cid:13)p0(cid:13)end3e64n0c(cid:13)e o36f50t(cid:13)he low
Lu x10(cid:13) 12 ML(cid:13) AlGaN barrier(cid:13) excitation photoluminescence for the 4 monolayer quantum
luminescence(cid:13) well in sample N257. The spectra are offset for clarity.
low(cid:13)
through the observation of PL from the widest well at
3350(cid:13) 3400(cid:13) 3450(cid:13) 3500(cid:13) 3550(cid:13) 3600(cid:13) 3650(cid:13) 3700(cid:13) 3750(cid:13)
3.42 eV, which is below the band gap energy of bulk
Energy (meV)(cid:13)
GaNat3.49eV.Unfortunately,forthis sample,the bulk
feature coincides with the PL from the 12 ML QW, so
FIG.1. Highandlowexcitationphotoluminescenceofsam- that luminescence from the bulk template superimposes
ple N257 from four separate quantum wells, of widths 4, 8,
a structure onto this peak.
12, 16 monolayers (10, 20, 30, 40 ˚A) scaled for a comparison Thelowintensityexcitationisoftheorder0.25Wcm−2
of the peak positions.
and can be considered to be in the regime where self-
screening of the electric field is negligible. The exact
excitation intensity of the pulsed laser is difficult to de-
For the latter in the pulsed fields, a masked chopper
termine presisely, but can be estimated to be greater by
wheel was used to produce a train of short laser pulses,
about five orders of magnitude. The magnetic field re-
∼0.3msduration,at∼100msintervals. Apulsefromthis
sults discussed below were therefore taken using the low
train triggeredthe magnet system so that the next laser
intensity quasi-CW excitation.
pulse coincided with the maximum of the pulsed field,
The effect of the magnetic field on the 4ML QW in
where there is little change in the field (≤1%) over a 0.3
ms timescale. A CCD cooled to −70oC attached to a sample N257 can be seen in Figure 2, and shows a shift
of the luminescence to lower energy. The form of this
quarter metre spectrometer was used to detect the PL.
shiftisdependentonthewidthofthewells,andisshown
in Figure 3. In order to determine the peak positions a
centre of mass method was used in preference to a peak
III. RESULTS
fittingroutine8. Thisdecisionresultedfromthedifficulty
of deciding the correct form of the peak as it is slightly
Figure1showsthelowandhighexcitationPLforsam-
asymmetric and thus causes problems for a Gaussianfit.
ple N257. Several PL peaks can be seen that can be
The centre of mass is also known as the first moment,
attributed to the different wells. Between the low and
and is given by,
high excitation, we have observed a shift in the PL that
is strongly dependent on the width of the wells, rang-
I(E)EdE
ing from 1meV for the narrowest up to 20meV for the M1 = . (1)
I(E)dE
widest. This is in agreement with the idea that in-built R
electric fields play an important role in this sample via Forthe widestwellsthRereis adiamagneticblueshiftof
the quantum confined Stark effect7 and that they have
the transitionenergy,but as the wellwidth decreaseswe
been screened by the excited carrier population. Previ-
observeatransitiontoaredshift,whichforthe4MLwell
ous measurements have calculated the field to be 550
isapproximately2meVat50T.Eventuallyatthehighest
kV/cm2. The most obvious consequence of this is seen
field the redshift stops and appears to be overtakenby a
2
smaller diamagnetic term.
Forthewidestwell,twocomponentsareobservedthat
have different dependencies on magnetic field. The ob-
servation 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
3583(cid:13) high field, which is consistentwith this proposaland the
suppression of the in-plane transport. The excitons are
4 ML / 10 Å(cid:13) thenlessabletofindthepotentialminimacausedbythe
3582(cid:13)
fluctuations before recombining.
The shifts for all the wells can be described by the
3581(cid:13) expression, E(B) = E0 ± 21g∗µBB +γ2B2 so that the
overall dependence for each Zeeman-split component is
3580(cid:13) determined by the relative magnitudes of the linear Zee-
man and quadratic diamagnetic terms. The linear Zee-
3540(cid:13) manshiftcausesthetransitiontosplitintotwo,ofwhich
)(cid:13) 8 ML / 20 Å(cid:13)
V weseepredominantlythelowestenergycomponent. The
e
m S-shape observedespecially for the narrowerwells at the
3539(cid:13)
(
y lower fields can be explained as resulting from a signifi-
g
r cant thermal population in the upper component whilst
e 3538(cid:13)
n the Zeeman splitting is still small. This effect is partic-
E
ularly noticeable when the width of the luminescence is
large compared with the shift in magnetic field and kT,
3512(cid:13) as is the case in these samples.
12 ML / 30 Å(cid:13)
The magnetic field dependence of the centre of mass
canthenbeexpressedintermsofthepopulationsineach
3511(cid:13)
component, n↑ and n↓,
3510(cid:13) n↑−n↓
E =E0+ 21g∗µBB n↑+n↓ +γ2B2, (2)
(cid:18) (cid:19)
0(cid:13) 10(cid:13) 20(cid:13) 30(cid:13) 40(cid:13) 50(cid:13) 60(cid:13)
with the ratio, n↑, givenby a Boltzmanndistribution.
Magnetic field (B)(cid:13) n↓
Table II shows the values deduced from the experiments
for the characteristic temperatures, g-factors and dia-
magneticshiftcoefficient. Theselasttwoparameterswill
be individually discussed in the following sections. The
3533(cid:13)
temperaturesdeducedsimplyreflectaheatingofthecar-
8 ML / 20 Å(cid:13)
V(cid:13)) riers by the high excitation intensities onto the sample.
e 3532(cid:13) Thesewerenecessarilyratherhighinordertodetectsuf-
m
( ficient signal in the short time duration of a single pulse
gy 3531(cid:13) (∼0.3 ms).
r
e
n
E 3530(cid:13)
IV. DISCUSSION
0(cid:13) 10(cid:13) 20(cid:13) 30(cid:13) 40(cid:13) 50(cid:13) 60(cid:13)
Magnetic field (B)(cid:13)
A. Zeeman splitting
Theg-factorsforthedifferentsamplesarecomparedin
FIG. 3. Magnetic field dependence of the low excitation
Figure4,andshowageneraltrendtowardsg∗ ∼3forthe
photoluminescenceforthedifferentquantumwellsforsample
N298(top)andN307(bottom),alongwiththefitstothedata narrower wells, with it reducing to g∗ ∼ 2 for the wider
as described in thetext. wells. This is particularly noticeable in sample N298.
The observationof a Zeeman splitting through a clear
redshift is surprising,since in the magneto-reflectivityof
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 here9,10. This is due
3
to a cancellation of the electron and hole g-factors when
themagneticfieldisparalleltothec-axis. EvenfortheB
valence band there is a similar compensation leading to
an excitonic g-factor, g∗ = 1.24. So as both theoretical
andexperimentalworkonAlGaNheterojunctions11 sug-
gestthattheelectroneffectiveg-factorshouldnotchange
Sample N257 T (K) g* γ2(µeV/T2) from a value of ∼ 2 as a result of confinement, we con-
4 ML QW 15(1) 3.1 (1) 0.99 (7) cludethatinsteadtheholeg-factormusthavedrastically
8 ML QW 13(1) 2.9 (2) 1.3 (1) changed in the narrower quantum wells due possibly to
12 ML QW 13(1) 3.3 (3) 2.1 (2) a significant change in the valence band structure.
16 ML QW 5(5) 1.8 (4) 3.0 (3) The reduced symmetry of the wurtzite with respect
N298 to the cubic structure is represented in the Hamiltonian
4 ML QW 9(3) 3.1 (2) 1.1 (1) through the crystal field. If spin-orbit effects are ne-
8 ML QW 8(3) 2.5 (2) 1.0 (1)
glected, the crystal field reduces the degeneracy of the
12 ML QW 5(5) 2.0 (2) 1.4 (1)
valence bands with different z-components of orbital an-
N307
gular momentum. With spin-orbit terms included there
8 ML QW 17(3) 3.4 (2) 1.5 (1)
arethenthethreeusualbands;A,B,andCwithsymme-
G889
tries Γ9, Γ7, and Γ7 respectively, that have the following
Bulk - 0 2.04 (3)
separations,
TABLE II. The magnetic field fitting parameters for the 1
different GaN/AlGaN samples along with a bulk reference. E −E =1 (∆ +∆ )∓ (∆ +∆ )2−8∆ ∆ 2 ,
A B,C cr so cr so cr so
2 3
( (cid:20) (cid:21) )
(3)
where ∆ and ∆ are the crystalfield and spin-orbit
cr so
energy terms respectively12.
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 en-
velope function approximation that are then dependent
on: the effective 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 Equation 3. Instead effective
4(cid:13)
spin-orbitand crystalfield terms can be used to account
forthe splittingsoftheA,BandC valencebands,where
or(cid:13)
act 3(cid:13) the nomenclature is now strictly in terms of the band
g-f symmetries and not the ordering.
e For the GaN/AlGaN system, it is known that in bulk
ctiv Ig*(cid:13)ex(cid:13)I>2(cid:13) AlN the crystal field splitting has the opposite sign to
e 2(cid:13)
eff Ig*(cid:13) I<2(cid:13) that of GaN thus inverting the order of the states. This
citonic 1(cid:13) N 2S(cid:13)5a7m p 1le3(cid:13)% Al(cid:13) ex(cid:13) wofoxulidnlAealxdGtoa1a−rxeNve1r3s,atlhoofutghhesthtaistevsaalutesoismneoctreixtipcearlivmaelune-
Ex N298 8% Al(cid:13) tally known. The more important consequence for this
N307 8% Al SQW(cid:13) work is that this will cause different valence band offsets
for the different bands and Figure 5 shows how this ef-
0(cid:13)
0(cid:13) 4(cid:13) 8(cid:13) 12(cid:13) 16(cid:13) 20(cid:13) fect can lead to a reversal of the quantum well states,
No. of monolayers, ML(cid:13) even when the same masses in the z-direction, m , are
k
used. (In fact m is identical for the A and B valence
k
bands.) This calculationwascarriedoutfora symmetri-
FIG.4. The experimentally determined well width depen-
cal quantum well, which is known to be invalid for these
denceof theg-factors.
samples. Howeverthe effect of a strong electric field will
be to bring the confinement energy closer to the barrier
offsetsthusmakingthemmoreimportantandenhancing
the reversaleffect.
A reversal of states has a profound effect on the spin-
ordering of the valence band that can be clearly seen
4
throughthek.pcalculationsofFigure6withinthequasi-
cubic model at a magnetic field of 60 T, using the band
parameters deduced by Stepniewski et al10. 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
vocabularycanthenbetranslatedintoaheavy-lighthole
Barrier(cid:13) splitting(HH-LH)incubiclanguage. Thereforewithzero
strain, corresponding to zero HH-LH splitting or zero
y(cid:13) A-band(cid:13)
g crystalfield,wefindthe usualzinc-blendebandandspin
er B-band(cid:13)
en ordering. The bulk wurtzite caseis alsoindicated, corre-
n
o sponding to a finite strain or crystal field, and this cor-
nsiti rectly reproduces the spin-ordering in GaN10, whereas
a
Tr AlN would correspond to the far right of the graph with
Bulk(cid:13) a large positive HH-LH splitting. The crossing, or more
correctly anti-crossing,of the Γ7 states causes a reversal
0(cid:13) 10(cid:13) 20(cid:13) 30(cid:13) 40(cid:13) 50(cid:13) 60(cid:13) ofthespin-orderingthuschangingthesignoftheg-factor
Well width (Å)(cid:13) forthelowest∗ Γ7 holeband. Thisprocessalsolowersits
energy below the Γ9 A-valence band so that transitions
FIG. 5. A schematic diagram based on a finite quantum involving this band are favoured in luminescence. Exci-
well model showing how thequantumwell energies can cross tonic g-factors less than two occur for the lowest bands
ataparticularwellwidthfortwobandswithdifferentoffsets. to the left of the anti-crossing in Figure 6 as indicated,
whereas they are greater than two to the right.
For a GaN/AlGaNquantum wellstructure the precise
band ordering of the confined states is hard to predict
since it is critically dependent on both the well and bar-
rier propertiesdue to the electric field. As an extratool,
X-ray measurements can be used to examine the crys-
tal properties and give information on the strain that is
present. Inthesesamples,ithasbeenshownthattheAl-
GaN barriers have the same lattice constant as the GaN
template,itbeinglargerthaniftheywereunstrained,re-
sulting in them having a tensile biaxial strain14. There-
electrons (g* = -2)(cid:13) fore the well material is unstrained and has a normal
B = 60T(cid:13) +1/2(cid:13) (cid:13)
-1/2(cid:13) (cid:13) wurtzite GaN valence band structure, whilst the barrier
hasthe opposite orderingdue tothe AlN contributionin
140(cid:13)
+1/2(cid:13) (cid:13) the alloy and the strain present.
120(cid:13) |g(cid:13)ex(cid:13)*| < 2(cid:13) |g(cid:13)ex(cid:13)*| > 2(cid:13) -1/2(cid:13) (cid:13) The latter can be estimated from using a linear rela-
eV)(cid:13) 100(cid:13) tionshipforthelatticeconstant,givinganin-planestrain,
m
ε = ε = 0.31% for an aluminium content of 13%.
Energy ( 6800(cid:13)(cid:13) ++--3131////2222(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13) T∆xhcxris+co23nyDtyr3iεbzuztewsittho εtzhze =cry−st2aCCl3133fiεexldx ≈spl−itt0i.n5g1εtxhxr,oaungdh
40(cid:13) D3 ∼6eV to give 32D3εzz = 14meV13. As experimental
20(cid:13) Wurzite(cid:13) Zinc(cid:13) values of ∆cr for GaN are very close to this value (11-15
Bulk GaN(cid:13) Blende(cid:13) meV†),oncethecontributionoftheAlNisalsoincluded,
0(cid:13)
-100(cid:13) -80(cid:13) -60(cid:13) -40(cid:13) -20(cid:13) 0(cid:13) 20(cid:13) 40(cid:13) 60(cid:13) the barriersareverymuch inthe regimefor a reversalof
HH-LH splitting (meV)(cid:13) states to occur. Figure 7 of Kim et al13 shows a calcula-
tionforasimilartensilebiaxialstraininGaNthatclearly
FIG.6. k.p results forGaN/AlGaN showing howthecrys- shows a reversalof the states with the anti-crossing also
talfieldsplittingofthevalencebandaffectsthespin-ordering givingthelowestvalencebandaheaviermassthanwould
of the valence band states at 60 T. Γ7 states are indicated be otherwise expected.
as solid lines, and Γ9 states are dashed. The vertical arrows
show the allowed optical transitions.
∗Lowest in terms of thehole energy.
†11,12meV from review in Kim et al13,14.9±0.3meV from
reflectivity of sample G889
5
Therefore our results suggest that the ordering of the
Sample Width,(˚A) In-planeextent Exciton binding
valencebandhaschangedfornarrowquantumwells,per-
N257 (approx) of exciton, (˚A) energy,(meV)
hapsasaresultofthelowerbarrierheightforthecrystal
4 ML 10 28.5 (1.0) 36.5 (1.3)
field Γ7 band caused by the strain and the aluminium 8 ML 20 33.0 (1.2) 31.5 (1.2)
content in the barriers. The reordering of these states
12 ML 30 41.7 (2.0) 24.9 (1.2)
will cause the ground state to have a Γ7 character and 16 ML 40 49.4 (2.5) 21.0 (1.1)
theinvertedspinsplittingwouldaccountfortheobserved N298
enhanced effective g-factor. 4 ML 10 30.0 (1.4) 34.6 (1.6)
The increase of the value for sample N307 in Figure 8 ML 20 28.3 (1.7) 36.6 (2.2)
4 with respect to the same width in the other samples 12 ML 30 33.9 (1.2) 30.7 (1.1)
can be understood in terms of the distributions of the N307
electric fields. Where more than one quantum well is 8 ML 20 35.0 (1.2) 39.6 (1.0)
present, the finite thickness of the barrier allows a redis- G889
tribution of the polarisationfields between both the well Bulk - 40.9 (5) 25.4 (2)
andthebarrier,whereasthefieldissolelyinthewellsfor
the SQW. This will push the hole states further into the TABLE III. The in-plane extent and exciton binding en-
barrier material. ergy for the different quantum well widths deduced from the
diamagnetic shift of the luminescence.
V. DIAMAGNETIC SHIFT
2 2 2
through ρ = r . If the exciton is assumed to re-
3
main essentially three dimensional, the effective binding
The magnitude of the diamagnetic shift in equation energy, R(cid:10)∗,(cid:11)can th(cid:10)en(cid:11)be deduced through scaling with
2 is determined by the size of the exciton wavefunction
respect to the hydrogen atom,
in the plane perpendicular to the field. From this it is
possible to infer the excitonic binding energy and the 1 aH
following section describes how these parameters have R∗ = 0 RH, (5)
ε a∗
been determined for the different quantum wells along r 0
with values corresponding to the bulk case.
where aH is the Bohr radius and RH is the Rydberg
0
The effect of a magnetic field can be considered as an
constant. This gives the following expression,
extra confining potential on the already localised exci-
tonic system in both 2D and 3D cases. The bulk data
e2 1 aHRH
can be fitted very well to a full numericalcalculation for R∗ = 0 . (6)
a3-dimensionalhydrogenatominamagneticfield15 that s4µγ2 εr
uses the excitonic binding energy and the reduced mass
The results from this calculation have been sum-
asfitting parameters. The binding energyforourpartic-
ular sample was determined by Neu et al16 as 24.1 meV marised in Table III (εr = 9.8 ‡18, µ = 0.18m0 §). The
similarity of the binding energy for the bulk deduced in
by identifying the 2s excited state, and this enables the
thisway,25.3meVcomparedto24.1meVfromthe1s-2s
fitting to give a reduced mass of 0.180(2)m as the only
e
separation, illustrates the accuracy of this technique.
free parameter. This numerical calculation reduces to a
B2dependenceinthelowfieldlimit,sothatbyfittingthe Table III shows that as the well width is reduced, the
lowfielddataweobtainacoefficient,γ2=2.04(3)µeV/T2 excitonsizeisalsoreducedresultinginastrongenhance-
mentofthebindingenergyforthenarrowwells,inagree-
that we can use in a directcomparisonbetween the bulk
ment with Bigenwald et al1 and Grandjean et al2. We
and QW data.
alsoobserveareductionoftheexcitonbindingenergyfor
Thediamagneticcoefficientforaquantumwellisgiven
by17, the widest well due to the separationof the electronand
holetotheseparateinterfacesoftheQW.Ourresultscan
e2 becomparedwiththecalculationsfromBigenwaldetal1
2
γ2 = ρ , (4) shown in Figure 7 and Figure 8 of the in-plane extent
8µ
andbinding energyof excitons in GaN/AlGaNquantum
(cid:10) (cid:11)
whereµisthereducedmassoftheexcitonandρisthe wells for different compositions and well widths.
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 ‡3D average; ε=(ε⊥)23(εk)31
of the wavefunction. §This value is taken from the bulk calculations and is used
Fromthe1sorbitalinthehydrogenatom,theexpecta- asafirstapproximation. Theactualvaluewilldependonthe
tion value of r2 is, r2 =3a∗02, where a∗0 is the effective precise band ordering and mixing effects as described in the
Bohrradius. r2 canthenbeconvertedtoanin-planesize section entitled Zeeman splitting.
(cid:10) (cid:11)
6
(cid:13)
70(cid:13)
60(cid:13)
)(cid:13)
Å
(
s 50(cid:13)
u
di
a
r
r
h
o (cid:13)
B 40(cid:13)
e
n
a
pl 50(cid:13) N257 13%(cid:13)
-
n
I 30(cid:13) 27%(cid:13) N298 8%(cid:13)
N307 8%(cid:13)
40(cid:13)
)(cid:13)
20(cid:13) V
e
0(cid:13) 4(cid:13) 8(cid:13) 12(cid:13) 16(cid:13) 20(cid:13) 24(cid:13) m
( 27%(cid:13)
Quantum well width (ML)(cid:13) y
g
er 30(cid:13) 17%(cid:13)
n
E
FIG. 7. Comparison between the experimental (points) ng 13%(cid:13)
andtheoreticalvalues(solidline)forthein-planeextentofthe di
n 8%(cid:13)
excitonicwavefunctionfordifferentwellwidths. Thetheoret- Bi 20(cid:13)
ical values are determined from considering a two-parameter
variational trial function for a series of GaN/AlGaN QW’s
withAlbarriercompositionx=0.27. Theexperimentalvalues
arefrom sampleN257andarededucedfrom thediamagnetic
shift of theluminescence. 10(cid:13)
0(cid:13) 4(cid:13) 8(cid:13) 12(cid:13) 16(cid:13) 20(cid:13) 24(cid:13)
Quantum well width (ML)(cid:13)
These variational calculations of the excitonic wave-
functions were used to investigate the ground state of
heavy-hole,freeexcitonsunderdifferentlevelsofapprox-
imation. Two different forms of the variational ground FIG.8. Experimentalvaluesofthebindingenergiesshown
with calculations of the well width dependence of the exci-
state wavefunction were considered in the calculations.
ton binding energies for a GaN/AlGaN SQW including in-
Both included a single variational parameter to govern
ternal electric fields. For a single parameter variational trial
the lateral extent of the wavefunction, however for one
function (short dash), calculations are shown for four alu-
an additional variational parameter was included in the
minium compositions, x= 0.27, 0.17, 0.13, 0.08, whereas for
determination of the separation of the electron and hole
a two-parameter function (long dash), x=0.27.
alongthe confinementaxis. Inthe single parametertrial
function this separationis solely determined by the elec-
tricfields presentinthe structure,whereasanadditional
contribution from the Coulomb attraction is allowed for
in the two-parameter trial function.
The experimental results from N257 are 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 concentrationin the
barriers, 13% compared to the value of 27% used in the
calculations.
There are larger differences between the binding ener-
7
giesdeducedfromthisparameter,andthiscanbeunder- VI. CONCLUSIONS
stood from the simplicity of our model and the greater
influence that the higher aluminium concentrationis ex- Thisworkhaspresentedmagneto-luminescencedatain
pectedto havethroughthe increasedelectricfield. How- fieldsupto55TforthreedifferentGaN/AlGaNquantum
everthedatafavourstheinclusionofthefieldratherthan wells that shows a strong well width dependence of the
its omission in square well calculations1. shift of the luminescence peaks. The high pulsed fields
Thesingleparametertrialfunctionunderestimatesthe were essential in the resolution of the different depen-
binding energy for a given aluminium composition and dencies, and even when the immense amount of work on
the more accurate two-parameterfunction is shown only the sample growth in the future is considered with the
foranaluminiumcomposition,x=0.27. Fromcomparing expected improvement in quality, it is unlikely that this
results for x=0.27 for both models, it seems that the wouldallowlowercontinuousfieldstobeusedduetothe
twoparameterfunction for the appropriatecomposition, fundamental linewidth limitations present.
x=0.08, & 0.13, would accurately account for the data. Our data shows that by observing an enhanced g-
It should be pointed out the theoretical results are for factor for the narrow wells, the valence band must have
the heavy-hole exciton, which we have shown is not the changedsignificantlycomparedwithbulk GaN.We have
loweststateforthesesamples. Thisshouldnotbesignifi- attributedthis toa reorderingofthe valencebandstates
cantforthepurposesofcomparisonbecausetheexcitonic in the strained AlGaN barriers, giving different barrier
mass will be predominantly determined by the electron heights for the different quantum well hole states. We
mass. The relevant hole mass would be an average over havealsoobservedanincreaseinthe excitonbinding en-
the top of the valence band within an exciton binding ergy with the reduction of the well width in agreement
energy from the band edge, and is likely to be heavy as with calculations using a variationalapproachin the en-
a result of the valence band mixing associated with the velope function formalism that includes the effect of the
band reordering. Both theory and experiment use the electric field in the wells. Better agreement is obtained
heavy-hole exciton mass as a first approximation. when the trial wavefunction includes a consideration of
In converting from the diamagnetic coefficient, γ2, to the three-dimensional Coulomb potential.
thebindingenergythethree-dimensionalhydrogenatom
was used as a basis rather than a two-dimensionalatom.
This approximation remains reasonable due to both the VII. ACKNOWLEDGEMENTS
shrinkage of the in-plane wavefunction caused by the
increased binding and the considerable leakage of the
We are grateful to the EPSRC (UK) for the support
bound state wavefunction out of the wells into the bar-
of this work and particularly the Lasers for Science Fa-
riers as evidenced by both the change in the valence
cility for the provision of the frequency-doubled Argon
band ordering and the calculated maximum in the ex-
laser. P.A.S. acknowledges financial support from Sharp
citonic binding energy at around 4 ML. This model is
Laboratories of Europe Ltd.
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.
Theenergyshiftsthatwehavediscussedareverysmall
in comparison with the linewidth. It is worth thinking
of the reasons behind this and the prospects for future
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9