Superlattices and Microstructures, VoL 5, No. 1, 1989
59
THEORY OF EXC1TONS IN GaAsGal_xAlxAS QUANTUM WELLS INCLUDING VALENCE BAND MIXING Lucio Claudio Andreani and Alfredo Pasquarello, Institut de Physique Thdorique, EPFL, CH1015 Lausanne, Switzerland and Scuola Normale Superiore, 156100 Pisa, Italy (Received 8 August, 1988)
A theory of excitons in GaAsGal_xAlxAS quantum wells is presented, which includes valence band mixing via the calculated subband structure. Binding energies and oscillator strengths are calculated for various excitons in their ground and excited states. This theory implies that parity forbidden excitons have zero oscillator strength, even when valence band mixing is considered, and that some observed transitions cannot be ground state excitons. Examples are given for transitions to the second conduction subband.
The properties of quantum well (QW) excitons are of
into account, as well as the appropriate boundary conditions
central interest in the field of quasitwodimensional
on the envelope function, 3 by an exact method of solving
heterostructures. Recently, experiments performed on high
the effective mass equation for the subband structure. 4 In
quality samples have probed fine details of the absorption
the layer planes, the basis consists of nonorthogonal
spectrum, like excited states of the excitons, transitions
twodimensional hydrogenic wavefunctions with different
which become allowed due to valence band mixing, and
radii; some Gaussians in kspace are included in the basis
their properties under external perturbations. This calls for
set, in order to represent states of the ¢xciton continuum.
theories of QW excitons which fully include the effects of valence band mixing.
The only strict quantum number which characterizes exciton states is the z projection of the total angular
Here we present some results of a theory of excitons
momentum, which we denote by m. Excitons belonging to
in GaAsGaAIAs QWs in the effective mass approximation.
different subbands, but with the same value of m, are
Binding energies and oscillator strengths are calculated for
coupled by the Coulomb interaction. The exciton envelope
ground and excited states of excitons with different
function is a fourcomponent spinor function, whose
symmetries. Valence band mixing is described by the 4x4
components
Luttinger kinetic matrix, l The dispersion of the conduction
s=3/2,1/2,1/2,3/2
band (CB) inc]udes nonparabolicity as calculated by
momentum is given by l=ms, and is different for each spin
R6ssler. 2 Warping of the subbands in the layer planes is
component.
neglected. The exciton wavefunction is expanded in a
approximately recovered when one of the four spin
variational basis. In the growth direction (which we take
components, say s o , is much larger than the others: an
along the z axis), this consists of QW wavefunctions,
sstate has zero orbital angular momentum for that
which, for the valence band, already include the mixing of
component and has m=s O, pstates have m=so._+l, and so
heavy holes (HH) and light holes (LH) at finite values of
on. Note that there are two 2p states, and that they are in general not degenerate, because they correspond to different values of m.
the inplane Bloch vector k. The difference in band structure parameters between GaAs and GaA1As is taken
07496036/89/010059+05 $02.00/0
are
labelled
by
the
spin
index
of the hole: the orbital angular
The usual quantum
numbers
can be
© 1989 Academic Press Limited
Superlattices and Microstructures, Vol. 5, No. 1, 1989
60
(o)
1
,
.....
(b)
>2.s E
~_
LqqCB'
~
(2s)
',
!
q
I"
;H ~
1 . ' % ~  ' 50
".
CB1
(29)
i :_~~ , 10 0 15 0 ~ve], ¢ S d t h (/,'~
~ ,
] ,
2 D'3,
~ o "
SO
~/, ~ ..~ ,
i O0 we'i
.... ,.'~:dt

~ 50 (A}
:~S
200
Fig. 1. (a) Binding energies and (b) oscillator strengths per
excitons in GaAsGa0.6A10.4As quantum wells of various
unit area for x polarization of the HH1CB1 (m=3/2),
widths. Dashed lines: twoband approximation. Solid lines:
LH1CB1 (m=1/2) and HH2CB1 (m=1/2) excited state
including coupling between valence subbands.
When the oscillator strength is calculated, it turns out
been obtained in the approximation of neglecting Coulomb
that only one spin component of the exciton envelope
coupling between excitons belonging to different subbands
function can be optically active, namely the one with orbital
(twoband approximation): we emphasize that this
angular momentum l=0 (hence s=m)57: this can be traced
approximation already contains mixing of HH and LH via
back to the trasformation properties of the valence envelope
the calculated valence subband structure. Due to this fact,
function under rotations in the xy plane. If the overlap
only a few subbands (usually two or three) need to be
integral ls=fdz c(z) vS(z) between conduction and valence
included in the furl calculation in order to have convergence
envelope functions is zero for that particular spin
at the level of 0.1  0.2 meV for the energy.
component, the oscillator strength vanishes. This is the
Binding energies and oscillator strengths of the
case, e.g., of the HH2CB1 exciton with m=3/2, or the
ground state HHICB1 and LH1CB1 excitons have been
LH1CB2 exciton with m=l/2 (usually called sstate
presented in Ref. 7. For completeness, we briefly
HH2CB1 and sstate LH1CB2 respectively). 8 The fact
summarize the main results: Coulomb coupling increases
that parity forbidden excitons have zero oscillator strength,
the binding energy and the oscillator strength of both
first pointed out by Chan, 5 is independent of valence band
excitons, particularly the LH one. For both excitons, most
mixing, and was missed in some previous investigations. 9
of the coupling comes from interaction with the continuum
Parity forbidden excitons acquire a finite oscillator strength
of other excitons. In Fig. 1 we display the binding energies
in the presence of an electric field along the growth
and oscillator strengths of the lowest excited states of the
direction: it has been shown by Vi~a 10 that the oscillator
HH1CB1 (m=3/2), LH1CB1 (m=1/2) and HH2CBI
strength vanishes when the electric field goes to zero. We
(m=1/2) excitons: they are usually called HH1CB1 (2s),
also find that the oscillator strength for light polarized along
LH1CB1 (2s) and HH2CB1 (2p) respectively. The latter
z is zero for excitons with m=+_3/2, and is four times that
two have the same symmetry and therefore interact with
for xy polarization for excitons with m=.+l/2. Also these
each other. From Fig. 1, the binding energy of LH1CB1
results are independent of valence band mixing, and hold
(2s) is larger than found from simpler models. 12 It can be
regardless of the fact that the exciton in question can be
seen from Fig. 1 that Coulomb coupling increases both the binding energies and the oscillator strengths of the
referred to a HH or LH subband. to
HH1CB1 (2s) and LH1CB1 (2s) excitons: like for the
GaAsGa0,6A10.4As QWs of widths ranging from 50 to 200
corresponding ls states, the effect is stronger for the LH
A. Standard material parameters have been used,l I and we
exciton, and is mainly due to coupling with the continuum
have assumed an offset ratio 65%:35%. Results have also
of other exciton series. The oscillator strength of the
We
now present
some
results
referring
Superlattices and Microstructures, Vol. 5, No. 1, 1989
61
Table I. Comparison of the energy difference ZaEb=Eb(lS)AEb(2S) with PLE experiments (Ref. 13), for the HH1CB 1 and LH1CB 1 excitons. The aluminum concentration is x=0.22. Energies are in meV.
theory
theory
theory
Eb(lS)
Eb(2S)
AEb
AEb
HH1
11.32
1.79
9.5
10.5
LH1
13.05
2.43
10.6
11.6
HH1
11.03
1.76
9.3
10.2
LH1
12.93
2.45
10.5
11.4
HH1
10.16
1.66
8.5
9.3
LH1
12.30
2.43
9.9
10.6
HH1
8.89
1.54
7.4
8.0
LH1
10.84
2.41
8.4
9.1 8.3
L (/~)
Hole level
38 45 64 102
experiment
132
LH1
10.00
2.33
7.7
143
LH1
9.63
2.29
7.3
7.8
218
HH1
6.64
1.35
5.3
5.8
LH1
7.76
2.00
5.8
6.6
HH2CB 1 (2p) exciton is instead reduced by Coulomb
way by noting that the discrete variational states in the
coupling (contrary to what has been found in Ref. 6). In the
energy region of the resonance give a peak in the absorption
two band approximation, the HH2CB1 (2p) exciton is
lineshape. If we compute the average energy of these states
found to have a larger oscillator strength than the LH1CB1
weighted with the oscillator strength, the resulting energy
(2s) exciton for L>90/~: however, when coupling is taken into account the oscillator strength of HH2CB I (2p) turns out to be much smaller than that of LHICB1 (2s) for all well widths. The HH2CBI (2p) exciton has attracted some
position converges to the energy of the Fano resonance
theoretical interest, because it is allowed in the presence of
region of the resonance. A similar situation arises for the
when the number of trial functions is increased. The oscillator strength of the resonance is taken to be the sam of the oscillator strengths of the individual states in the energy
valence band mixing, whereas HH2CB1 (ls) is parity
HH2CB1 (2p) exciton, but in this case the dominant
forbidden. From our results it appears that HH2CB1 (2p)
contribution is found to come from the LH1CB1 (ls)
can hardly be observed in optical spectra (note that going
exciton state, hence there is no need to apply the above
beyond the twoband approximation is essential for this
procedure.
conclusion). The experimental situation is unclear, since
The energy difference AEb=Eb(lS)AEb(2S ) can be
there is no easy way to discriminate between the excitons
measured
LH1CB1 (2s) and HH2CB1 (2p), and the peak in the
experiments performed on high quality samples, where the
bandtoband absorption at the edge of the LH1CB1
2s exciton is observed as a well defined peak. In Table I we
transition, which occurs because of the negative effective
cornpare our theoretical predictions with the experimental results of Koteles and Chi, 13 for x=0.22. Experiment and
mass of the LH1 subband.
in photoluminescence
excitation
(PLE)
It must be remarked that the LH ICB 1 (2s) exciton is
theory always agree within 1 meV. It is interesting to
degenerate with the continuum of HH1CB1 (m=1/2) and
observe, however, th~tt theoretical values are systematically
is therefore a Fano resonance. When coupling is included,
lower than experimental ones, and that the discrepancy
its energy position can be determined in an approximate
seems to increase with decreasi,/g well width: the difference
Superlattices and Microstructures, Vol. 5, No. 1, 1989
62
~50IH~2SB2
b
(ls)
. . . . . . . . . . . . . . '
/,
HH2CB2
1
(ls)
q \
S
0 5O
1 O0 weh
width
150 (A}
200
Fig. 2. (a) Binding energies and (b) oscillator strengths per
i 50
i
i
r
1 i i i 1 O0 well width
i
I i 15O ('~,}
L
a
i 200
quantum wells of various widths. Dashed lines: twoband
unit area for x polarization of the LH1CB2 (m=3/2) and
approximation. Solid lines: including coupling between
HH2CB2 (m=3/2) excitons in GaAsGa0.6A10.4As
valence subbands.
cannot be accounted for by uncertainties in the material
a value of k of order IlL. Since the exciton state consists
parameters. A possible explanation for the discrepancy
mainly of subband states at a scale of k of order 1/aB,
consists in the fact that the peak assigned to the 2s state
where aB=100 ,~ is the exciton Bohr radius, the following
probably contains unresolved higher excited states: hence,
limiting cases can be understood: when L,~aB, valence band
the theoretically calculated energy of the 2s state should
mixing affects exciton states only little, and the transition
rather be replaced by a weighted average over the
LH1CB2 disappears. When L,~aB, however, the LH]
2s,3s,4s .... states. If this effect is taken into account, the
subband has more HH2 character at the scale of k relevant
theoretically calculated values for z1Eb are increased by
for the construction of exciton states, and only the
about 0.25 meV for the HH, and 0.6 meV for the LH
LH1CB2 (2p) exciton survives. Hence we predict that two
exciton. The correction goes in the right direction: however,
peaks of comparable oscillator strengths should be observed
it seems insufficient to account for the whole discrepancy,
for L=80/~, and only one peak for much smaller or much
particularly for the HH exciton.
larger L. This result is in agreement with the experimental
In Fig. 2 we show the binding energies and oscillator
data reported in Ref. 14 (the explanation given in Ref. 14 is
strengths of the lowest LH1CB2 and HH2CB2 excitons
incorrect, because it attributes the low energy peak to the
with m=3/2. They are usually called LH1CB2 (2p) and
LH1CB2 (is) exciton, which is parity forbidden). Once
HH2CB2 (ls). Coupling to HHICB2 should be small,
again, the inclusion of Coulomb coupling is found to be
because the HH1 subband has a small s=3/2 component,
essential in order to obtain the correct trends for the
and has been neglected. The HH2CB2 exciton is
oscillator strengths.
degenerate with the continuun~ of LH1CB2 for L<120/~,
Finally, we briefly compare the present theory with
and its energy has been determined by the procedure
those of other authors who include valence band mixing.
described above. Subband coupling always increases the
The theories of Ref. 9,15 ignore the fact that different spin
binding energy of the LH 1CB2 exciton, which is on the
components of the exciton envelope function have different
low energy side. Coupling also has a remarkable effect on
values of the orbital angular momentum I. This leads to an
the oscillator strengths. It turns out that the two excitons
incorrect treatment of Coulomb coupling and of selection
have comparable oscillator strengths for L=80 ~. For
rules for excitonic transitions. The correct selection rules
smaller L, the oscillator strength of HH2CB2 (ls) is larger
were first obtained by Chan, 5 who, however, used a limited
than that of L H I  C B 2 (2p), whereas for large L the
basis set and did not include coupling with the continuum of
oscillator strength of LH ICB2 (2p) is much larger. This is
other excitons. Our results are slightly larger than those of
due to the fact that the LH1 and HH2 subbands anticross at
Ekenberg and Altarelli 16 for the binding energies of the
Superlattices and Microstructures, VoL 5, No. 1, 1989 ground state HH1CB1 and LH1CB1 excitons. We
63 2.
U. Rtissler, Solid State Communications 49, 943
confirm the result of Ref. 16 that most of the binding
(1984).
energy enhancement comes from interaction with the
3.
exciton continuum. The present theory yields results equivalent to those of Bauer and Ando, 17 but our expansion
4. Lucio Claudio Andreani, Alfredo Pasquarello and Franco Bassani, Physical Review B 36, 5887 (1988).
set is smaller since it already includes valence band mixing,
5.
and it is more suitable in order to study excitons degenerate
6. Bangfen Zhu and Kun Huang, Physical Review B 36, 8102 (1987); Bangfen Zhu, Physical Review B 37, 4689
with the continuum. Our basis set is similar to that used by Zhu and Huang, 6 and most qualitative conclusions are the same. However, we find a much larger effect of Coulomb coupling. We attribute the difference to the fact that
M. Altarelli, Physical Review B 28, 842 (1983).
K.S. Chan, Journal of Physics C 19, L125 (1986).
(1988).
to be important for an accurate determination of binding
7. L.C. Andreani and A. Pasquarello, Europhysics Letters 6,259 (1988). 8. Kramers degenerate excitons are obtained by letting m~m and by considering timereversed valence states. Our choice for the sign of m corresponds to the convention of Ref. 4 for the valence envelope functions, and is opposite to that of Ref. 6. 9. G.D. Sanders and YiaChung Chang, Physical Review B 32, 5517 (1985); 35, 1300 (1987).
energies and oscillator strengths. Most of this coupling
10. L. Vifia, Surface Science 196, 569 (1988).
comes from interaction with the continuum of other
11. LandoltBrrstein Numerical Data and Functional Relationships in Science and Technology, Group III, vol. 17, edited by O. Madelung (Springer, Berlin, 1982). 12. Ronald L. Greene, Krishan K. Bajai and Dwight E. Phelps, Physical Review B 29, 1807 (1984). 13. Emil S. Koteles and J.Y. Chi, Physical Review B 37, 6332 (1988). 14. R.C. Miller, A.C. Gossard, G.D. Sanders, YiaChung Chang, and J.N, Schulman, Physical Review B 32, 8452 (1985).
coupling to the exciton continuum was neglected in Ref. 6. In conclusion, we have presented a comprehensive theory of excitons in quantum wells, which takes into account valence band mixing via the calculated subband structure. Coulomb coupling between excitons belonging to different subbands has been calculated and has been found
excitons: this has been taken into account, also for excitons which are degenerate with the continuum. Calculated binding energies agree within 1 meV with PLE experiments. 13 Valence band mixing does not change the fact that parity forbidden excitons have zero oscillator strength. Therefore some observed peaks must be excitons in excited states, and an example has been given for the LH1CB2 (2p) exciton, whose oscillator strength relative to HH2CB2 (ls) has been found in qualitative agreement with experiment.
15. D.A. Broido and L.J. Sham, Physical Review B 34, 3917 (1986).
References
16. U. Ekenberg and M. Altarelli, Physical Review B 35, 7585 (1987).
1. J.M. Luttinger and W. Kohn, Physical Review 97, 869 (1955).
17. Gerrit E,W. Bauer and Tsuneya Ando, Physical Review B 37, 3130 (1988),