Theory of excitons in GaAsGa1−xAlxAs quantum wells including valence band mixing

Theory of excitons in GaAsGa1−xAlxAs quantum wells including valence band mixing

Superlattices and Microstructures, VoL 5, No. 1, 1989 59 THEORY OF EXC1TONS IN GaAs-Gal_xAlxAS QUANTUM WELLS INCLUDING VALENCE BAND MIXING Lucio Cla...

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Superlattices and Microstructures, VoL 5, No. 1, 1989


THEORY OF EXC1TONS IN GaAs-Gal_xAlxAS QUANTUM WELLS INCLUDING VALENCE BAND MIXING Lucio Claudio Andreani and Alfredo Pasquarello, Institut de Physique Thdorique, EPFL, CH-1015 Lausanne, Switzerland and Scuola Normale Superiore, 1-56100 Pisa, Italy (Received 8 August, 1988)

A theory of excitons in GaAs-Gal_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 quasi-two-dimensional

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

two-dimensional hydrogenic wavefunctions with different

which become allowed due to valence band mixing, and

radii; some Gaussians in k-space 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 GaAs-GaAIAs 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 four-component spinor function, whose

symmetries. Valence band mixing is described by the 4x4


Luttinger kinetic matrix, l The dispersion of the conduction


band (CB) inc]udes nonparabolicity as calculated by

momentum is given by l=m-s, and is different for each spin

R6ssler. 2 Warping of the subbands in the layer planes is


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,

s-state has zero orbital angular momentum for that

which, for the valence band, already include the mixing of

component and has m=s O, p-states 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 in-plane Bloch vector k. The difference in band structure parameters between GaAs and GaA1As is taken

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of the hole: the orbital angular

The usual quantum


can be

© 1989 Academic Press Limited

Superlattices and Microstructures, Vol. 5, No. 1, 1989







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;H ~-

1 . ' % ~ - ' 50




i :_~--~ , 10 0 15 0 ~ve], ¢ S d t h (/,'~

~ ,

] ,

2 D'3,

~ o "--


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i O0 we'i

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Fig. 1. (a) Binding energies and (b) oscillator strengths per

excitons in GaAs-Ga0.6A10.4As quantum wells of various

unit area for x polarization of the HH1-CB1 (m=3/2),

widths. Dashed lines: two-band approximation. Solid lines:

LH1-CB1 (m=-1/2) and HH2-CB1 (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

(two-band approximation): we emphasize that this

angular momentum l=0 (hence s=m)5-7: 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 HH2-CB1 exciton with m=-3/2, or the

ground state HHI-CB1 and LH1-CB1 excitons have been

LH1-CB2 exciton with m=-l/2 (usually called s-state

presented in Ref. 7. For completeness, we briefly

HH2-CB1 and s-state LH1-CB2 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

HH1-CB1 (m=3/2), LH1-CB1 (m=-1/2) and HH2-CBI

strength vanishes when the electric field goes to zero. We

(m=-1/2) excitons: they are usually called HH1-CB1 (2s),

also find that the oscillator strength for light polarized along

LH1-CB1 (2s) and HH2-CB1 (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 LH1-CB1

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

HH1-CB1 (2s) and LH1-CB1 (2s) excitons: like for the

GaAs-Ga0,6A10.4As 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


now present




Superlattices and Microstructures, Vol. 5, No. 1, 1989


Table I. Comparison of the energy difference ZaEb=Eb(lS)-AEb(2S) with PLE experiments (Ref. 13), for the HH1-CB 1 and LH1-CB 1 excitons. The aluminum concentration is x=0.22. Energies are in meV.















































9.1 8.3

L (/~)

Hole level

38 45 64 102
























HH2-CB 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 HH2-CB1 (2p) exciton is

lineshape. If we compute the average energy of these states

found to have a larger oscillator strength than the LH1-CB1

weighted with the oscillator strength, the resulting energy

(2s) exciton for L>90/~: however, when coupling is taken into account the oscillator strength of HH2-CB I (2p) turns out to be much smaller than that of LHI-CB1 (2s) for all well widths. The HH2-CBI (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 HH2-CB1 (ls) is parity

HH2-CB1 (2p) exciton, but in this case the dominant

forbidden. From our results it appears that HH2-CB1 (2p)

contribution is found to come from the LH1-CB1 (ls)

can hardly be observed in optical spectra (note that going

exciton state, hence there is no need to apply the above

beyond the two-band approximation is essential for this


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


LH1-CB1 (2s) and HH2-CB1 (2p), and the peak in the

experiments performed on high quality samples, where the

band-to-band absorption at the edge of the LH1-CB1

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



It must be remarked that the LH I-CB 1 (2s) exciton is

theory always agree within 1 meV. It is interesting to

degenerate with the continuum of HH1-CB1 (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





. . . . . . . . . . . . . . '





q \


0 5O

1 O0 weh


150 (A}


Fig. 2. (a) Binding energies and (b) oscillator strengths per

i 50




1 i i i 1 O0 well width


I i 15O ('~,}



i 200

quantum wells of various widths. Dashed lines: two-band

unit area for x polarization of the LH1-CB2 (m=-3/2) and

approximation. Solid lines: including coupling between

HH2-CB2 (m=-3/2) excitons in GaAs-Ga0.6A10.4As

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

LH1-CB2 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

LH1-CB2 (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 LH1-CB2 and HH2-CB2 excitons

incorrect, because it attributes the low energy peak to the

with m=-3/2. They are usually called LH1-CB2 (2p) and

LH1-CB2 (is) exciton, which is parity forbidden). Once

HH2-CB2 (ls). Coupling to HHI-CB2 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 HH2-CB2 exciton is

oscillator strengths.

degenerate with the continuun~ of LH1-CB2 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 1-CB2 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 HH2-CB2 (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 I-CB2 (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 HH1-CB1 and LH1-CB1 excitons. We

63 2.

U. Rtissler, Solid State Communications 49, 943

confirm the result of Ref. 16 that most of the binding


energy enhancement comes from interaction with the


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,


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).


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 time-reversed 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 Yia-Chung 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. Landolt-Brrstein 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, Yia-Chung 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 LH1-CB2 (2p) exciton, whose oscillator strength relative to HH2-CB2 (ls) has been found in qualitative agreement with experiment.

15. D.A. Broido and L.J. Sham, Physical Review B 34, 3917 (1986).


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),