Solid State Communications, Vol. 98, No. 4, pp. 297-301, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/96 $12.00 + .00 S0038-1098(96)00043-9
VALENCE BAND STRUCTURE OF VERY NARROW InGaAs/InP QUANTUM WELLS F. Dujardin, N. Marreaud and J.P. Laurenti Laboratoire d'Optodectronique et de Microdectronique, Institut de Physique-Electronique et Chimie, Universite de Metz, 1 Bd Arago 57078 Metz Cedex 3, France
(Received 10 November 1995; accepted 10 January 1996 by G. Bastard) We performed valence band structure calculations on very narrow lattice-matched GalnAs/InP quantum wells using the 6x6 Luttinger-Kohn Hamiltonian which includes the spin-orbit coupling and compared our results with those obtained with a 4x4 Hamiltonian neglecting spin-orbit effect. We show that the spin-orbit coupling must be taken into account for very narrow quantum wells, even in the case of strain-free structures. Indeed, a simple 4x4 Hamiltonian can lead to overestimated confinement energies for light holes and underestimated in-plane effective masses for both light and heavy holes Keywords : A semiconductors, A. quantum wells, D. electronic band structure, D. spinorbit effects, D. optical properties.
Provided the confinement energies for holes are small in
Band structure of semiconductor quantum wells (QWs) has received great attention over the last decade
comparison with the spin-orbit splitting, a valid approach can be obtained from a two-component formalism neglecting the SO band. This double-band model has been
(see for instance Ref.1 and Refs. therein). Indeed, its knowledge is essential for applications since it governs the
widely used in valence band structure calculations up to now reported [3,5-9] on most QWs systems. However, the need to take into account the 1-'7 band has been made
main characteristics of optodectronic devices via e.g. transition energies, effective masses, state densities and dipole momentum.
evident in the case of strained-layer QWs where the strain causes additional coupling between the F, and the Fs bands
While a simple one-band model is used for
conduction states (sometimes corrected using the firstorder k.p perturbation approach ), a multi-band model is needed for valence band states, due to the coupling
In this paper, we emphasize the need to take also
between the components of the valence band and the resulting strong non-parabolicity [3,4]. The valence band structure consists of three components : in a zinc-blende type crystal, the two upper ones are degenerate at the Brillouin zone center (F8 symetry) and correspond to the so-called heavy hole (HH) and light hole (LH) bands,
into account the F7 band in the ease of very narrow QWs, even if they are lattice-matched. Indeed, in very narrow QWs, the confinement energy becomes comparable in magnitude with the spin-orbit splitting. As an example, we show the results of calculations on lattice-matched InCmAs/InP QWs. This system was chosen because it is attractive for optical fiber communications  but
respectively ; the third one (I"7 symetry) is lowered with respect to the HH and LH bands by the spin-orbit coupling and corresponds to the so-called split-off (SO) band.
working in the 1.3pro wavelength window requires very narrow (only a few monolayzrs-MLs) QWs . 297
VERY NARROW InGaAs/InP QUANTUM WELLS
Vol. 98, No.4
We use the envelope function approximation to calculate the valence band structure of QWs grown on (001) substrates. We start from the 6x6 Luttinger-Kohn Hamiltonlan in the Ij, m) bases of the Bloch wave functions
three components :
at the zone center . Under the axial approximation , a unitary transformation [ 1,4] leads to a block-diagonalized Hamiltonian :
In this context, the HH, LH and SO states are no longer determined independently by solving three distinct scalar effective-mass equations, as was done in the trivial parabolic band approximation. A set of eigenstates is found, each of them being, strictly speaking, a mixing of the zone center HH, LH and SO states. Nevertheless, according to litterature [1,4,12], we will hereafter refer to these mixed states by their name at the zone center and therefore use the assignment of HH or LH subbands. Finally, for each value of the in-plane wavevector k~/, the discrete eigenvalues for bound states are determined using the usual boundary conditions  : continuity of probability and current densities which are ensured by continuity of the enveloppe function vector (Eq. 4) and
in a basis set which still refers to I-R-I, LH and SO states, respectively. If z is the (001) growth direction, the upper block is expressed by : I P +Q+Vh(z)
-~Rp + ~-2 So]
~-42Rp- ~ Sp ~r~Q_i~2~ p p+ A+ Vso(Z)fl
F2(z)]. F 3 (Z))
I ( r l - 2 r 2 ) ~D
where : 2 2 P=2m°Yl(kll+kz)'
h2 2 2 Q = ~--~Y2(k//- 2kz),
Here, h is the reduced Planck constant, mo is the free electron mass, Yi are the Luttinger parameters ; kz and k+ are the wave vector components along the growth direction and in the well plane (in-plane wave vector), respectively. To take into account the band offsets at the interfaces, the confining potentials Vh(z) for F8 and Vso(Z) for I'7 are added to the relevant diagonal elements in Eq. 2. The lower block H L is the Hermitian conjugate of the upper one Hu. Thus we deal with an effective-mass equation for each block, in which the envelope function is a vector with
accross the interfaces. This way, a series of E(k#) subbands is found for a given QW. Since we deal with symetric confinement potentials, both 3x3 Hamiltonian blocks lead to identical sets of E(k#) dispersion curves corresponding to spin degeneracy. We performed valence band structure calculations on lattice-matched Ino.ss2Gao.esAs/InP QWs with various well thicknesses taken as integer numbers of MLs from 1 to 43 from 3 to 125 A with 1 ML ~ 2.93 A ). For the parameters we used the values listed in Table I after gels. 4, 16 and 17. First, we will focus on the subband energies at the quantum confinement energies, which
Table I : Parameters used in the calculations Lattice parameter
Confining potentials (meV)
(a) Ref. 16 ; (b) Band offsetpercentage after Ref. 17 ; (c) Re£ 17 ; (d) Ref. 4 ; (e) Linear interpolationfrom binary compounds.
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VERY NARROW InGaAs/InP QUANTUM WELLS
I f/~ /
- - - - L with spin-orbit
--~-- LI without spin-orbit
A k A A A ~
10 20 30 Well width (MLs)
20 30 Well width (MLs)
Figure 1 " Fundamental (n=l) energy levels for HI-Is
Figure 2 : Error on the n=l LH confinement energy when
(circles) and LHs with (full triangles) and without (open
the spin-orbit coupling is neglected, in a lattice matched InGaAs/IaP QW, as a function of the well width. The lines
coupling in a lattice
InGaAsanP QW, as a function of the well width. The
are guides to the eye.
energy is scaled with respect to the F8 valence band top of the three-dimensional alloy. The lines are guides to the eye.
with a maximum close to 50 meV for 4 MLs. Its value is close to 35 meV for 8 MLs which is the well thickness compatible with the 1.3 Bm wavelength window ,
contribute to the interband transition energies. Fig. 1 shows the first eigenvalues (n=l) of energies for HHs and LIB, scaled with respect to the F8 valence band top of three-
relative error of about 4% for interband transition energy.
dimensional lattice-matched InCraAs, as a function of the well width. As previously reported , the HH energies are not affected by the spin-orbit coupling. This corresponds to
In the case of the largest error, 4 MLs, the effect of the spin-orbit coupling on the band structure is illustrated
when interface effects are neglected. This leads to a
in Fig.3. For H/Is, this effect becomes significant beyond
the fact that, in both upper and lower 3x3 Hamiltonian
4% of the Brillouin zone (BZ). The LH subband remains
blocks, all non-diagonal elements involving HH states vanish when l~r=0 (ga = Sp = 0 in Eq. 2). On the contrary, coupling. When SO states are taken into account, the LIB
electron-like with a negative hole mass at kcr--O but is pushed toward higher energies by the SO states situated below. The LH-SO coupling, already significant for any k//, still increases beyond 4% of the BZ
energy levels shiR upward with respect to their position without spin-orbit coupling. This corresponds to the fact that the non diagonal elements involving LH and SO
between n=l HH and LH subhands. Therefore, a change in
the LIB energies are significantly affected by the spin-orbit
remain non-zero, even when k/r=0 (see Eq. 2). In other words, taking into account the spin-orbit coupling leads to LHs confinement energies weaker than those commonly obtained in the parabolic band approximation. The error on the n=l LH energy level, when the spin-orbit coupling is neglected, is plotted in Fig.2 as a function of the well width. Below 20 MLs, it ranges in several tens of meV,
The spin-orbit coupling reduces the distance the in-plane effective masses at k//=0 is expected for HH and LH respectively. To check this, we attempted a least mean square fit of every subband for 1~/< 3% of the BZ, by a second-order polynomial function of k//2. The results are summarized in Table II. Taking into account the spin-orbit coupling increases the in-plane effective mass absolute values at kct=O : only slightly concerning the HHs (in a
VERY NARROW InGaAs/lnP QUANTUM WELLS I
factor 1.3), but more significantly for the LHs (in a factor 3.3). It should be noted that the well-known mass reversal phenomenon [19-21], consisting in an in-plane LH effective mass heavier (in absolute value) than the till one in a thin QW, is accounted for, only in the case including
HH with spin orbit
LH with spin orbit
HH without spin orbit] LH without spin orbit-1
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the spin-orbit coupling.
Table H : Heavy hole (m~,/m0) and light hole (mut/m0) inplane effective masses obtained for a 8MLs wide latticematched InGaAs/InP QW with (a) and without (b) spin-
g-200 |1 II
-300 (~ ~b)
mmdmo 0.105 0.082
mm/m0 -0.248 -0.076
0.00 0.01 0.02 0.03 0.04 0.05 0.06
Figure 3 : Valence band structure of a 4MLs wide InCmAs/InP lattice-matched QW. Results obtained with (full symbols) and without (open symbols) spin-orbit coupling are compared : squares for HHs and circles for LHs. The in-plane wave vector k,¢/ is normalized by the Brillouin zone width along the (001) direction. The energy is scaled with respect to the Fs valence band top of the three-dimensional alloy.
In conclusion, we performed valence-band structure calculations on very narrow lattice-matched InGaAs/InP QWs based on the envelope function approach. We emphasized the need tO take into account the spin-orbit coupling even in the case of strain-free structures. The error on the n=l confinement energy of the LHs, when the spin-orbit effect is neglected, ranges between 30 and 50 meV for 2-8 MLs well widths. The LH subband is upshifled in the same order of magnitude by the SO states situated at lower energies. Consequently, the HI-I-LH intersubband distance is reduced and the absolute values of in-plane effective masses are enhanced, mainly for LHs.
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