Buchheit, M.

Fierling, G.

Gendry, M.

Lainé, J.P.

Letartre, X.

Sidoroff, F.

Viktorovitch, P.

English

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Copyright
Mechanical relaxation of strained semiconducting
stripes: Influence on optoelectronic properties
G. Fierling, M. Buchheit, X. Letartre, M. Gendry, P. Viktorovitch, F. Sidoroff,
J.P. Lainé
To cite this version:
G. Fierling, M. Buchheit, X. Letartre, M. Gendry, P. Viktorovitch, et al.. Mechanical relaxation
of strained semiconducting stripes: Influence on optoelectronic properties. Applied Physics Letters,
American Institute of Physics, 1997, 71 (11), pp.1516-1518. ￿10.1063/1.119953￿. ￿hal-02111632￿
Appl. Phys. Lett. 71, 1516 (1997); https://doi.org/10.1063/1.119953 71, 1516
© 1997 American Institute of Physics.
Mechanical relaxation of strained
semiconducting stripes: Influence on
optoelectronic properties
Cite as: Appl. Phys. Lett. 71, 1516 (1997); https://doi.org/10.1063/1.119953
Submitted: 08 May 1997 . Accepted: 16 July 1997 . Published Online: 05 August 1998
G. Fierling, M. Buchheit, X. Letartre, M. Gendry, P. Viktorovitch, F. Sidoroff, and J. P. Lainé
Mechanical relaxation of strained semiconducting stripes: Influence on
optoelectronic properties
G. Fierling,a) M. Buchheit, X. Letartre, M. Gendry, and P. Viktorovitch
LEAME, UMR CNRS 5512, Ecole Centrale de Lyon, F-69131 Ecully Ce´dex, France
F. Sidoroff and J. P. Lainé
LTDS, UMR CNRS 5516, Ecole Centrale de Lyon, F-69131 Ecully Ce´dex, France
~Received 8 May 1997; accepted for publication 16 July 1997!
The mechanical relaxation of strained semiconducting stripes is studied. The deformation tensor is
calculated using a classical approach of elasticity problem with predeformations; the electronic band
structure is then simulated using an 8 bandkp model including strain. To confirm the models
developed, compressively and tensely strained stripes were fabricated and characterized by
photoluminescence measurements. Theoretical and experimental results are in very good agreement
and show the importance of mechanical anisotropic relaxation phenomena in optoelectronic devices
like waveguide structures, modulators, or lasers. ©1997 American Institute of Physics.
@S0003-6951~97!04637-8#
The use of strained semiconductors offers new possibili-
ties in terms of the design of optoelectronic devices. On the
one hand, stress can be intentionally introduced into a semi-
conductor layer by the use of a stressing overlayer.1–4 On the
other hand, in the case of layers strained on their substrate,
parasitic elastic stresses can frequently be induced during the
technological fabrication of the devices, particularly at the
edges of the patterns~mesa, stripes, etc.!.5 In both cases,
inhomogeneous strain fields occur. These resulting stress and
strain distributions give rise to modulation of the electronic
and optical properties of the materials constituting the final
devices. In order to show the importance of nonhomoge-
neous elastic stresses field induced by the devices fabrica-
tion, this letter presents a theoretical and experimental study
of stress relaxation of compressively and tensely strained
stripes. A complete model describing the mechanical relax-
ation of the stripes and its influence on the electro-optic
properties was developed.
Two structures grown by molecular beam epitaxy on a
~001! semi-insulating InP substrate were analyzed. They
were comprised of a 2000-Å-thick pseudomorphic
InxGa12xAs layer on a 4000-Å-thick In0.52Al0.48As buffer
layer. In the first structure,x50.58 corresponds to a lattice
mismatcha5(a/a0)21 ~a being the lattice parameter of the
material anda0 the lattice parameter of the substrate! of
10.33% @compressively strained stripe~CS!#. In the second
one, x50.485 leads to a lattice mismatch of20.32%
@tensely strained stripe~TS!#. InGaAs-based stripes of 50mm
in width with 50 mm spacing between two adjacent stripes
were patterned using selective CH4/ 2 reactive ion etching.
6
Models were developed first in order to simulate the
structures. Then, to confirm these models, theoretical results
were compared to experimental photoluminescence measure-
ments. Figure 1 shows a schematic view of half the stripe
~along the lateral axis!. A mechanical analysis is performed
first. It consists of a classical elasticity problem with prede-
formations. The reference configuration, defined by reducing
in each material the lattice parametera to the valuea0 of the
substrate material, leads to a fictitious and stressed reference
condition but allows a simple formulation of the mechanical
problem. Considering Fig. 1, the boundary conditions are the
following: no motion at the edge of the substrate~OAD!, no
horizontal motion ~OBE!, and free surface elsewhere
~EFCD!. For the stripes studied here, the substrate and the
layer are supposed to have the same elastic properties, and to
be different only through their lattice parameter.
Practically, calculations are done in the area defined in
Fig. 1.X50 is a symmetry plane for the stripe, and along the
Y direction there is a continuous translation symmetry. An
aspect ratio, width over thickness, of 50 is chosen in order to
have a fully strained region~i.e., no strain relaxation! around
X50 ~i.e., the center of the stripe!. A rectangular mesh is
used, namely, 10 elements along theZ axis and 50 elements
along theX axis in the layer. Theexx and ezz strain tensor
components and the shear deformationsexz are given in Fig.
2 for the CS stripe. Calculations are done in each element of
the mesh, but for better understanding, the results are given
in terms of variations of these deformations along theX axis
~corresponding to the stripe width! for several vertical coor-
dinates in the stripe~Z50.05,Z50.55, andZ50.95; Z51
corresponding to the top stripe surface!. Relaxation phenom-
ena clearly occur near the stripe edges, over distances of a
few stripe thicknesses. Results are thus only depicted near
the edges of the stripes. In the middle of the stripe, the de-
formations are those related to the pseudomorphic layer,
namely,exx520.0033,eyy520.0033, andezz510.0034,
shear deformationsexz being obviously equal to 0; all these
a!Electronic mail: fierling@ec-lyon.fr FIG. 1. Schematic view of the stripe.
1516 Appl. Phys. Lett. 71 (11), 15 September 1997 0003-6951/97/71(11)/1516/3/$10.00 © 1997 American Institute of Physics
components significantly change near the edges. ForZ la ger
than about half of the stripe thickness, there is a phenomenon
of overrelaxation along theX axis: the strain tensor compo-
nent,exx , which is at first negative, becomes weakly posi-
tive. This is due to the stress singularity which appears at
corner C ~Fig. 1!. These well known singularities can be
studied from an analytic point of view, leading to an
asymptotic expression for the displacementui and the stress
s i j :
ui5r
aUi~u!, s i j 5
1
r 12a
w i j ~u!, ~1!
wherer andu are the polar coordinates around C. The eigen-
shapesUi(u) andw i j (u) can be obtained through the solu-
tion of an appropriate spectrum. The characteristic exponent
a has been roughly estimated as 0.7.7
The strain tensor components corresponding to the TS
stripe are the opposite of the CS stripe ones for directionsX,
Y, andZ whereas shear deformations keep the same values
in both cases.
Once the strain tensor components are known, the elec-
tronic band structure calculation can be performed. The elec-
tronic states and the energy levels are found by solving the
Schrödinger equation. An 8 bandkp model, based on Kane’s
formalism8 including the spin–orbit interactions is used.
Strain states have been introduced by Bahder9 by repeating
the original derivation given by Bir and Pikus.10,11 At each
point of the material the strain tensor components are known
and electronic properties can thus be calculated.
Calculations were done for both structures. The results
are presented in Fig. 3 for the CS stripe. Variations of the
mean band gap@Eg mean5(Eg(HH)1Eg(LH))/2, HH, and
LH corresponding to the heavy and light holes, respectively#
are plotted along theX axis for the threeZ coordinates de-
fined previously. HH/LH splitting is also represented. Far
from the edges, for both strain cases, the energy transitions
corresponding to the pseudomorphic layer band gaps are ob-
tained. When a layer is under compressive~tensile! biaxial
strain, the uniaxial component increases the heavy~light!
hole band level. The band gaps are thusEg ~HH!50.689 eV
for the CS stripe, andEg ~LH!50.741 eV for the TS stripe.
In order to understand the change of the mean band gap and
HH/LH splitting versus the strain relaxation, approximated
analytic expressions from Ref. 12 can be used. The mean
band gap is directly related to the hydrostatic part of the
strain:
Egmean5Eg1ag~exx1eyy1ezz!, ~2!
whereEg is the band gap of the unstrained material, andg
is the hydrostatic deformation potential.
HH/LH splitting, Es5HH2LH is given by
Es5H bn22 @~exx2eyy!21~exx2ezz!21~eyy2ezz!2#
1dn
2exz
2 J 1/2, ~3!
wherebn is the valence band deformation potential associ-
ated with strain along the@100# direction, anddn is the shear
deformation potential for strain along the@111# direction.
For the CS stripe, variations of the mean band gap near
the edges of the stripe are directly related to the change of
tr (ē)5(exx1eyy1ezz). This part is clearly demonstrated by
a comparison between Fig. 2~b! and Fig. 3~a!. For Z50.05,
i.e., near the interface between the stripe and the substrate,
the mean band gap is drastically reduced close to the edge
due to the singularity at the point called C in Fig. 1, where
the tensor components become infinite. The compressive
character of the strain is intensified throughexx and tr (ē)
thus becomes strongly negative. ForZ50.55 andZ50.95,
where an overrelaxation phenomenon occurs near the edges,
the exx component becomes positive andtr (ē) is finally in-
creasing, inducing an increase of the mean band gap near the
edges.
HH/LH splitting includes two terms, the first related to
the normal strains~exx , eyy , and ezz!, and the second that
depends on shear deformationexz . In our case, the first term
is found to be much smaller than the second one. Variations
of HH/LH splitting are thus close to variations of shear de-
formation exz . This is in agreement with the results pre-
sented in Fig. 4. Around the middle of the stripe, the splitting
is independent ofZ and corresponds to the strain state of the
pseudomorphic layer. ForZ50.05, near the singularity C,
the splitting is greatly enhanced as a result of the increase in
theexz stress tensor component. On the other hand, for larger
Z, near the edges, the splitting varies weakly according to
the small change ofexz .
FIG. 2. Variations of theexx ~a!, ezz ~b!, andexz ~c! strain tensor compo-
nents, and variations of the hydrostatic part (exx1eyy1ezz) ~d! for the com-
pressively strained stripe structure.
FIG. 3. Variations of the mean band gap~a! and variations of HH/LH
splitting ~b! for the compressively strained stripe structure.
1517Appl. Phys. Lett., Vol. 71, No. 11, 15 September 1997 Fierling et al.
As expected with regard to the TS stripe, the tensor com-
ponents have the same magnitude as that of the CS stripe; the
only difference lies in the signs of the normal components,
which are opposite. Variations of the mean band gap are thus
opposite to those of the previous case, but the same splitting
is obtained, the splitting being solely related to the magni-
tude of the tensor components.
To confirm the theoretical models, the samples were in-
vestigated using spectral scanning photoluminescence~SR-
SPL! carried out at room temperature, using as the light ex-
citation, a focused He–Ne laser spot. The lateral resolution
of measurement is 1mm. Since the absorption depth of the
light is on the order of the thickness of the stripe, it is con-
sidered that the entire depth of the stripe contributes equally
to the photoluminescence spectra. Calculations show that the
energy band gap variation along theZ axis is less than kT.
Therefore, it can be assumed that the photoluminescence
measurement is related to the average (Eav) over the stripe
thickness of the simulated energy band gaps. According to
Ref. 13, the measured energy variations at half-width peak
~Ehwp! at the low energy side correspond to the energy band
gap variations. The average over the stripe thickness of the
measured energies at half width peak at low energy side
~Ehwp!, in comparison withEav, is shown in Fig. 4 for both
samples and shows very good agreement between theory and
experiments. For the CS stripe, the measurements of the
band gaps were shifted by 2 meV. This value corresponds to
the slight difference between the 58% theoretically consid-
ered composition and the real composition which is 57.4% as
derived from DDX investigation. In the middle of the stripe,
the calculations and measurements give the same results.
Theoretical and experimental variations are also in good
agreement near the edges: the calculated shift between the
center and the edges of the stripe, about 15 meV, is con-
firmed by these measurements. For the TS stripe, the agree-
ment is quite good too although a small discrepancy appears
between theory and experiment.
A model describing the mechanical relaxation phenom-
ena and their effect on electro-optic properties of strained
stripes was developed. This model was confirmed by photo-
luminescence measurements. The observed band gap varia-
tions between the center and the edges of the stripes confirm
that it is necessary to take into account the mechanical stress
relaxation phenomena in the device simulation and it is es-
pecially necessary for the design of optoelectronic devices.
For the two structures studied in this letter, the initial lattice
mismatch was weak, about 0.3%. Fluctuations of the elec-
tronic and optical properties are increased when higher lat-
tice mismatches are used in quantum wells, for example, for
infrared wave length applications. Work in this direction,
including quantum confinement calculation, is currently in
progress.
The authors wish to thank J. L. Leclercq and J. F. Dam-
lencourt from LEAME for stripe fabrication. This work was
partially supported by CNET Contract No. 96 IB002.
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FIG. 4. Variations of the average over the stripe thickness of the theoretical
and measured band gap energies for the compressively and the tensely
strained stripe structures; the solid line~circle! corresponds to the calculated
~measured! values.
1518 Appl. Phys. Lett., Vol. 71, No. 11, 15 September 1997 Fierling et al.


Mechanical relaxation of strained semiconducting stripes: Influence on optoelectronic properties

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Mechanical relaxation of strained semiconducting stripes: Influence on optoelectronic properties

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