Pergamon
Solid State Communications, Vol. 98, No. 4, pp. 297301, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 00381098/96 $12.00 + .00 S00381098(96)000439
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 PhysiqueElectronique 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 latticematched GalnAs/InP quantum wells using the 6x6 LuttingerKohn Hamiltonian which includes the spinorbit coupling and compared our results with those obtained with a 4x4 Hamiltonian neglecting spinorbit effect. We show that the spinorbit coupling must be taken into account for very narrow quantum wells, even in the case of strainfree structures. Indeed, a simple 4x4 Hamiltonian can lead to overestimated confinement energies for light holes and underestimated inplane 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 spinorbit splitting, a valid approach can be obtained from a twocomponent formalism neglecting the SO band. This doubleband 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 [3,59] 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 strainedlayer QWs where the strain causes additional coupling between the F, and the Fs bands
While a simple oneband model is used for
[1,4,1012].
conduction states (sometimes corrected using the firstorder k.p perturbation approach [2]), a multiband 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 nonparabolicity [3,4]. The valence band structure consists of three components : in a zincblende type crystal, the two upper ones are degenerate at the Brillouin zone center (F8 symetry) and correspond to the socalled heavy hole (HH) and light hole (LH) bands,
into account the F7 band in the ease of very narrow QWs, even if they are latticematched. Indeed, in very narrow QWs, the confinement energy becomes comparable in magnitude with the spinorbit splitting. As an example, we show the results of calculations on latticematched InCmAs/InP QWs. This system was chosen because it is attractive for optical fiber communications [13] but
respectively ; the third one (I"7 symetry) is lowered with respect to the HH and LH bands by the spinorbit coupling and corresponds to the socalled splitoff (SO) band.
working in the 1.3pro wavelength window requires very narrow (only a few monolayzrsMLs) QWs [14]. 297
298
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 LuttingerKohn Hamiltonlan in the Ij, m) bases of the Bloch wave functions
three components :
at the zone center [15]. Under the axial approximation [3], a unitary transformation [ 1,4] leads to a blockdiagonalized Hamiltonian :
In this context, the HH, LH and SO states are no longer determined independently by solving three distinct scalar effectivemass 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 inplane wavevector k~/, the discrete eigenvalues for bound states are determined using the usual boundary conditions [4] : continuity of probability and current densities which are ensured by continuity of the enveloppe function vector (Eq. 4) and
H=IHoU
8O )
(1)
in a basis set which still refers to IRI, LH and SO states, respectively. If z is the (001) growth direction, the upper block is expressed by : I P +Q+Vh(z)
ReiS:
HU=[Rp+iSp
~Rp + ~2 So]
PQ+Vh(z)
42Q+i~/~Sp[ (2)
~42Rp ~ Sp ~r~Q_i~2~ p p+ A+ Vso(Z)fl
I
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),
Fl(z)
RP=

6
k2
'(3)
h2
SP=~mo2'[3Y3k//kz
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 (inplane 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 effectivemass equation for each block, in which the envelope function is a vector with
(4)
3 _~y3k//
4~r3k//
I
(FI (z)~
IF+C,)I<5) 3
3
l~y3kll
D
~y3kll
YI'~
\F3(z)J
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 latticematched 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 [16]). 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
(i.e.
kcFO,Le.
Table I : Parameters used in the calculations Lattice parameter
Confining potentials (meV)
Luttinger parameters
no (~)
Vh(z)
V~o(Z)
~
~2
Y3
InP
5.8688(a)
370(b)
480(b'¢)
4.95 (`t)
1.65(a)
2.35 ('t)
In0.532Ga0.468As
5.8688 (a)
0
356(c)
14.06(`1'e)
5.40(d'e)
6.20(d'e)
(a) Ref. 16 ; (b) Band offsetpercentage after Ref. 17 ; (c) Re£ 17 ; (d) Ref. 4 ; (e) Linear interpolationfrom binary compounds.
Vol. 98, No. 4
VERY NARROW InGaAs/InP QUANTUM WELLS
299
50
0
40
100 t3o
i
I f/~ /
//'~
3 0
0
400
k
20 LLI
    L with spinorbit
~ LI without spinorbit
lO
A k A A A ~
I
0
10 20 30 Well width (MLs)
40
,
i
i
l
20 30 Well width (MLs)
10
,
40
Figure 1 " Fundamental (n=l) energy levels for HIIs
Figure 2 : Error on the n=l LH confinement energy when
(circles) and LHs with (full triangles) and without (open
the spinorbit coupling is neglected, in a lattice matched InGaAs/IaP QW, as a function of the well width. The lines
triangles)
spinorbit
coupling in a lattice
matched
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 threedimensional 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 [18],
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 latticematched InCraAs, as a function of the well width. As previously reported [4], the HH energies are not affected by the spinorbit coupling. This corresponds to
In the case of the largest error, 4 MLs, the effect of the spinorbit 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 nondiagonal 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
electronlike with a negative hole mass at kcrO[3] but is pushed toward higher energies by the SO states situated below. The LHSO coupling, already significant for any k//, still increases beyond 4% of the BZ
energy levels shiR upward with respect to their position without spinorbit 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 spinorbit
remain nonzero, even when k/r=0 (see Eq. 2). In other words, taking into account the spinorbit 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 spinorbit 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 spinorbit coupling reduces the distance the inplane 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 secondorder polynomial function of k//2. The results are summarized in Table II. Taking into account the spinorbit coupling increases the inplane effective mass absolute values at kct=O : only slightly concerning the HHs (in a
300
VERY NARROW InGaAs/lnP QUANTUM WELLS I
I
I
•
I
'
I
'
factor 1.3), but more significantly for the LHs (in a factor 3.3). It should be noted that the wellknown mass reversal phenomenon [1921], consisting in an inplane LH effective mass heavier (in absolute value) than the till one in a thin QW, is accounted for, only in the case including
i
50
•
HH with spin orbit
•
LH with spin orbit
looI
D o
HH without spin orbit] LH without spin orbit1
Vol. 98, No.4
]
/
the spinorbit coupling.
 150
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
ID
g200 1 II
,,5 2so
orbit coupling.
300 (~ ~b)
350 t
I
=
I
=
I
t
I
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
k,,
Figure 3 : Valence band structure of a 4MLs wide InCmAs/InP latticematched QW. Results obtained with (full symbols) and without (open symbols) spinorbit coupling are compared : squares for HHs and circles for LHs. The inplane 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 threedimensional alloy.
In conclusion, we performed valenceband structure calculations on very narrow latticematched InGaAs/InP QWs based on the envelope function approach. We emphasized the need tO take into account the spinorbit coupling even in the case of strainfree structures. The error on the n=l confinement energy of the LHs, when the spinorbit effect is neglected, ranges between 30 and 50 meV for 28 MLs well widths. The LH subband is upshifled in the same order of magnitude by the SO states situated at lower energies. Consequently, the HIILH intersubband distance is reduced and the absolute values of inplane effective masses are enhanced, mainly for LHs.
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VERY NARROW InGaAs/InP QUANTUM WELLS
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