Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells
A study for the effects of size quantization and strain effects on the valence band spectra, the interband matrix elements, and the light gain spectrum in zinc-blende GaN quantum wells is presented. In the framework of the effective mass theory, the Schrödinger equation is solved for the valence...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2008
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irk-123456789-1190732017-06-04T03:02:46Z Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells Lokot, L.O. A study for the effects of size quantization and strain effects on the valence band spectra, the interband matrix elements, and the light gain spectrum in zinc-blende GaN quantum wells is presented. In the framework of the effective mass theory, the Schrödinger equation is solved for the valence bands with a 3×3 block Hamiltonian. The results are illustrated for the GaN/Al₀.₁₉Ga₀.₈₁N quantum well. It is shown, that the biaxial strain causes quite significant changes to the gain spectra in spatially confined structures. It is shown, that laser effect is suppressed with arising of the circular loop of valence band maxima in the heterostructure under the tensile strain, while under the compressive strain, the stimulated emission is pronounced. Our results show the internal strain effects are important in optical properties of GaN and associated quantum well structures. 2008 Article Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells / L.O. Lokot // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 4. — С. 364-369. — Бібліогр.: 36 назв. — англ. 1560-8034 PACS 61.50.Ah, 70, 81.05.Ea http://dspace.nbuv.gov.ua/handle/123456789/119073 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
A study for the effects of size quantization and strain effects on the valence
band spectra, the interband matrix elements, and the light gain spectrum in zinc-blende
GaN quantum wells is presented. In the framework of the effective mass theory, the
Schrödinger equation is solved for the valence bands with a 3×3 block Hamiltonian. The
results are illustrated for the GaN/Al₀.₁₉Ga₀.₈₁N quantum well. It is shown, that the biaxial
strain causes quite significant changes to the gain spectra in spatially confined structures.
It is shown, that laser effect is suppressed with arising of the circular loop of valence
band maxima in the heterostructure under the tensile strain, while under the compressive
strain, the stimulated emission is pronounced. Our results show the internal strain effects
are important in optical properties of GaN and associated quantum well structures. |
| format |
Article |
| author |
Lokot, L.O. |
| spellingShingle |
Lokot, L.O. Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Lokot, L.O. |
| author_sort |
Lokot, L.O. |
| title |
Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells |
| title_short |
Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells |
| title_full |
Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells |
| title_fullStr |
Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells |
| title_full_unstemmed |
Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells |
| title_sort |
strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende gan quantum wells |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119073 |
| citation_txt |
Strain effects on the valence band structure, optical transitions, and light gain spectrums in zinc-blende GaN quantum wells / L.O. Lokot // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 4. — С. 364-369. — Бібліогр.: 36 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT lokotlo straineffectsonthevalencebandstructureopticaltransitionsandlightgainspectrumsinzincblendeganquantumwells |
| first_indexed |
2025-07-08T15:10:57Z |
| last_indexed |
2025-07-08T15:10:57Z |
| _version_ |
1837092001665777664 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 4. P. 364-369.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
364
PACS 61.50.Ah, 70, 81.05.Ea
Strain effects on the valence band structure,
optical transitions, and light gain spectra
in zinc-blende GaN quantum wells
L.O. Lokot
V. Lashkaryov Institute for Semiconductor Physics, Department of Theoretical Physics,
41, prospect Nauky, 03028 Kyiv, Ukraine, e-mail: lyuba.lokot@gmail.com
Abstract. A study for the effects of size quantization and strain effects on the valence
band spectra, the interband matrix elements, and the light gain spectrum in zinc-blende
GaN quantum wells is presented. In the framework of the effective mass theory, the
Schrödinger equation is solved for the valence bands with a 3×3 block Hamiltonian. The
results are illustrated for the GaN/Al0.19Ga0.81N quantum well. It is shown, that the biaxial
strain causes quite significant changes to the gain spectra in spatially confined structures.
It is shown, that laser effect is suppressed with arising of the circular loop of valence
band maxima in the heterostructure under the tensile strain, while under the compressive
strain, the stimulated emission is pronounced. Our results show the internal strain effects
are important in optical properties of GaN and associated quantum well structures.
Keywords: strain effect, valence band structure, optical transition, light gain spectra,
zinc-blende GaN, quantum well.
Manuscript received 04.08.08; accepted for publication 20.10.08; published online 11.11.08.
Direct wide band gap group III-nitride semiconductors
based on GaN and its alloys have received great
attention due to prospective applications in
optoelectronic devices such as light-emitting diodes and
lasers at green-blue and near-ultraviolet wavelengths,
solar-blind photodetectors [1, 2]. A number of ultraviolet
light-emitting diodes [3-8], and laser diodes [9-15]
already have been demonstrated. However, nitride
structures and devices are still in the developmental
stage.
Internal strain effects in thin films become
increasingly important in modern solid state technology.
An important problem in growing GaN on crystalline
substrates Si, SiC, GaAs, ZnO and sapphire is an internal
strain. Internal strains are related to the large lattice
mismatch and the difference in the thermal-expansion
coefficients of the epitaxial layer and a substrate. They
can cause large biaxial strains in the epitaxial layers.
Biaxial strains can be compressive or tensile depending
on the crystalline substrate material [16-18]. The internal
strain effects are studied in this paper. We present
studies of the influence of biaxial strain on the valence
band spectra of zinc-blende GaN/AlGaN quantum well
of width w perpendicular to the growth direction (001)
and located at 2/2/ wzw <<− . Under biaxial strain,
the transverse components of strain are proportional to
the difference in lattice constants 0a , also depend on Al
content x :
0029.0/)( GaN
0
GaN
0
AlN
0 <−=−== xaaaxyyxx εε ,
when as 0=== xzyzxy εεε [19]. The longitudinal
component of strain can be expressed as
xxzz CC εε ]/[2 1112−= , where C12 and C11 are the elastic
constants. Since 0<ε xx then the lattice mismatch causes
a compressive strain of the quantum well. Situations
under which, the crystalline substrate causes a tensile
strain of the quantum well are studied. To compare the
role of the compressive and the tensile biaxial strain
effects, we consider unstrained thin film GaN.
In this paper, we present a quantitative analysis of
strain and confinement effects on holes in zinc-blendes.
We consider an electron that being initially in the
conduction band emits a photon and ends in the valence
band top. To describe the emission or absorption
processes, the energies as well as wave functions of the
lowest conduction subband and the valence subbands are
calculated. The strain dependence of both matrix
elements for dipole optical interband transitions and the
light gain spectra in zinc-blende GaN quantum wells
have been derived.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 4. P. 364-369.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
365
The point group of the zinc-blende structure is
identical to the elements of the point group of a
tetrahedron which is denoted by dT . The space group of
the zinc-blende structure is symmorphic and is denoted
by 2
dT [20]. The bonding between nearest neighbors in
the würtzite crystal is also tetrahedral. The configuration
of nearest neighbors of the first coordination sphere in
the würtzite lattice coincides with that in zinc-blende
structures under a relatively small deformation in the
(111) direction [21]. For this reason, the physical
background for the cubic approximation is based on
similarity between the (0001) axis in the würtzite
structure and (111) direction in the cubic crystal [21].
The space group of the würtzite structure is 4
6vC .
The cohesive energy of the würtzite structure is very
close to that of the zinc-blende structure [20]. For that
reason, GaN can crystallize in both zinc-blende and
würtzite polytypes.
It is known [20-23], that the valence-band spectrum
at the Γ points originates from sixfold degenerate 15Γ
state. In zinc-blende structures, the 15Γ level splits by
the spin-orbit interaction, forming the fourfold
degenerate 8Γ heavy (HH) and light (LH) holes levels
and double degenerate 7Γ spin split-off (SH) holes
level. Under the action of the hexagonal crystal field and
the spin-orbit interaction in würtzite crystals, 15Γ splits
lead to the formation of three spin degenerate levels: 9Γ ,
upper 7Γ , and lower 7Γ levels which are denoted by
heavy holes, light holes and split-off holes, respectively.
Conduction-band states in the vicinity of Brillouin zone
center are spin degenerate and characterized by a single
effective mass for cubic symmetry and two effective
mass parameters for hexagonal symmetry.
The Hamiltonian for the würtzite valence band
which accounts for the interaction of 7Γ , 7Γ , and 7Γ
levels has been derived within the kp method [22]. Later,
a derivation of the würtzite Hamiltonian based on the
method of invariants with including the effects of strain
on the hole spectra has been proposed [21, 24]. A
transformation of the Hamiltonian written in the basis
sm ,2/1,1 to the basis of angular momenta 3/2 and
1/2 with the spin-orbit split-off band included has been
performed in papers [23, 25]. The basis of angular
momenta 3/2 and 1/2 is frequently used for the 6×6
Luttinger-Kohn Hamiltonian for zinc-blende structures.
Using a unitary transformation, a more convenient
block-diagonal form with two 3×3 blocks in the
framework of the envelope function formalism has been
proposed [23, 25, 26]. Latter approachs have been used
in this work. Here we use the effective-mass parameters,
spin-orbit splitting energy, and deformation potential
parameters as in the papers [23, 27, 28].
We consider the pseudomorphically strained zinc-
blende GaN/Al0.19Ga0.81N quantum well of width 5.2 nm.
We assume a rectangular form of potential for the
quantum well.
The results of numerical calculations of the valence
band spectra and the k-dependence of the matrix
elements as a function of the wave vector },{ yxt kkk = ,
which lies in the plain of the quantum well, are
presented in Figs 1 to 3. Here for all the structures, the
two highest hole bands are the heavy hole band and light
hole one.
1a
1b
1c
Fig. 1. Unstrained thin film GaN: (a) the valence subband
structure; the momentum matrix elements for (b) x (or y)
polarization and (c) z polarization.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 4. P. 364-369.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
366
To clarify the role of compressive and tensile
biaxial strain effects, we consider the unstrained thin
film GaN. In Fig. 1, we show the valence-band structure
and the k-dependence of the matrix elements. Fig. 1a
shows that each band consists of a mixture of heavy
hole, light hole and spin-orbit split hole states.
It is known that the p-like sixfold degenerated
valence band at the Γ point is separated into a fourfold
degenerated 8Γ state and a twofold degenerated 7Γ
state by the spin-orbit coupling. The value of this
separation is the spin-orbit splitting energy, which is
defined from the matrix element of the spin-orbit
interaction Hamiltonian between the atomic orbitals and
is usually referred to as the spin-orbit splitting width.
The degeneracy is governed by the cubic symmetry. In
the case when the quantum well is grown along the (001)
direction, the crystal symmetry is changed to the
tetragonal [29]. This implies removing the degeneracy of
8Γ state. It is shown in Fig. 1a.
In Fig. 2, we show the valence-band structure and
the k-dependence of matrix elements of the quantum
well under the compressive biaxial strain. Components
of strain are equal: 0054.0−=ε xx , 0063.0=ε zz . This
strain is consistent with the aluminum content 19 %. In
Fig. 2a, one can see that the compressive strain causes a
downward shift of the valence bands. Such behavior
agrees with the calculations of strain effects on the
valence band structure in würtzite GaN quantum wells
for nitride-based devices, which are fabricated on (0001)
sapphire substrates [30].
We consider optical transitions between the initial
and final states such as: with angular momentum
2/3=J and the magnetic quantum numbers
2/3±=jm , 2/1±=jm as well as with 2/1=J ,
2/1±=jm of the valence band and with 2/1=J ,
2/1±=jm of the conduction band. Transitions from the
valence band states 2/1±=jm obey selection rules
such as 0=∆m and 1±=∆m , therefore they have both
the x (or y) and z light polarizations. Transitions from the
valence band states 2/3±=jm obey selection rules
1±=∆m , therefore they have only the x (or y)
polarization [31].
Under the compressive strain, an increase in the
aluminum content is accompanied with both an increase
of the splitting width between the heavy hole and light
hole bands and a decrease of valence-band mixing
effects. Consider transitions from the heavy hole band.
Figs 1 and 2 clearly indicate that the matrix elements
have the stricter contribution of x (or y) light polarization
as goes from the unstrained thin film to the strained
heterostructure. Thus, in the case of the compressively
strained quantum well, the matrix elements have the
strict x (or y) light polarization. Such behavior agrees
with the calculations [21, 32, 33] of the momentum
matrix elements in crystals of würtzite symmetry and
associated quantum well structures, in which the
considered transitions are allowed for the x (or y) light
polarization, while for the z light polarization are
forbidden.
2a
2b
2c
Fig. 2. GaN/AlGaN quantum well with the compressive biaxial
strain εxx = -0.54 %: (a) the valence subband structure; the
momentum matrix elements for (b) x (or y) polarization and (c)
z polarization.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 4. P. 364-369.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
367
In Fig. 3, we show the valence band structure and
k-dependence of the matrix elements for the quantum
well under the tensile biaxial strain. The strain
components are equal: 0054.0=ε xx , 0063.0−=ε zz . In
Fig. 3a, one can see that the tensile strain causes both
strong transformation of hole effective mass and a
decrease of the splitting width due to an upward shift of
the light hole valence band to the top of the heavy hole
one. The former leads to arising of the large negative
mass at the Brillouin zone center, while the latter results
in arising of the casual twofold degeneration of double
spin degenerated heavy hole and light hole states at the
Brillouin zone center. We show that a fourfold
degeneration of the valence band states at the Brillouin
zone center may be found in zinc-blende (001) GaN
quantum well with the former deriving double spin
degeneration of heavy hole and light hole states and the
latter under the action of the tensile strain in addition to
the casual twofold degeneration. In general, arising of
the degeneracy causes an increase of the density of
states. The light hole band can be shifted upward above
of the heavy hole band with the tensile strain increase.
This implies removing the fourfold degeneration of
valence band states. Thus, at 0054.0>ε xx the highest
quantized hole subband is the light band. It is expected
that the density of states is changed by the action of
strain.
Such behavior qualitatively agrees with the
calculations of internal strain effects on the valence band
structure of xx1 GeSi − [34].
In zinc-blende GaN, the states of the light hole
band are composed of the iYX ± characters as well as
the Z character. Comparison of Figs 1 and 3 shows an
increase of presence of the Z state in the light hole
band as one moves from the unstrained thin film to the
strained heterostructure, in which 0054.0=ε xx . There is
more the Z state in the light hole band in the strained
heterostructure, than in the thin film. Therefore, in the
strained quantum well with 0054.0>ε xx , the states of
the highest-lying light hole band are almost composed of
the Z character, which yields the strong matrix
element for the z light polarization.
Although both the compressive and the tensile
biaxial strains were studied, only zinc-blende GaN
quantum well under the tensile strain exhibits a region of
the spectrum with negative effective mass and the strong
modification of the matrix elements for the z light
polarization. The density of states, carrier population
inversion, matrix elements, the light gain spectrum vary
notably with arising of the spectral region with negative
effective mass at the Brillouin zone center.
Understanding the influence of internal strain
effects under the lattice mismatch on laser gain
properties should help towards improving the laser
performance and optimal device configurations.
3a
3b
3c
Fig. 3. GaN/AlGaN quantum well with tensile biaxial strain
εxx = 0.54 %: (a) the valence subband structure; the momentum
matrix elements for (b) x (or y) polarization and (c) z
polarization.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 4. P. 364-369.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
368
In Fig. 4, we show the energy-dependence of the
light gain coefficient of quantum well with the carrier
concentration 212 cm105 −× at the temperature 4.4 K for
this light polarization, optical transitions in which are
allowed by the selection rules. As soon as we obtain the
interband population inversion, the light becomes
amplified. At high carrier densities, the stimulated
optical transitions give rise to a high optical gain. It is
shown that under the compressive strain, the shift of the
valence and conduction bands leads to a blue shift of the
gain spectra, with respect of the band-edge of unstrained
quantum well.
Fig. 4. Light gain coefficient of GaN/Al0.19Ga0.81N quantum
well for: (a) transverse polarization of unstrained quantum
well; (b) x (or y) polarization with the compressive biaxial
strain; (c) z polarization with the tensile biaxial strain.
It is found that the effective mass is drastically
changed under the tensile strain. This leads to changes of
sign and polarization of the gain coefficient, as shown in
Fig. 4. It can be explained by appearance of the circular
loop with radius 1nm7.0 −≈tk in valence band structure
of zinc-blende GaN quantum well under the tensile
strain. Extremum loop effects in the band structure of
würtzite symmetry, in which the linear term in wave
vector exists, are known from papers [35]. With
appearance of the spectrum region characterized with a
negative effective mass at the Brillouin zone center,
holes are located on the loop of valence-band maxima.
As a consequence of this, the optical transitions near the
band-edge occur with high light absorption suppressing
the laser effect. Under the tensile strain, the shift of the
valence and conduction bands leads to a red shift of the
absorption spectra, with respect of the band-edge of
unstrained quantum well.
In both zinc-blende and würtzite crystal structures,
each atom is surrounded by four nearest neighbors
forming an ideal tetrahedron. The valence electrons of
this crystal structure form hybridized sp3 orbitals [20].
This sp3 hybridization is well known from the bonding
of a methane molecule. It is interesting to consider the
analogy existing between the dependence of the matrix
elements on the strain under the lattice mismatch and
strain effects on behavior of the bond angles of the
tetrahedron. It is known [36] that in the case of tensile
biaxial strain the bonding tetrahedra are compressed
along the c axis by shrinking the distance between the
Ga-N layers toward a planar structure, which changes
the bond angles. This causes dehybridization from sp3
hybrids towards sp2 and pz orbitals. The quantum
mechanical problem of dehybridization sp3 hybrids
towards sp2 and pz orbitals is consistent with a tendency
of an increase of the Z state in the light hole band
under the tensile biaxial strain, which yields the matrix
elements for the z polarization. It is shown in Fig. 3c.
We have investigated the strain effects on the
valence band structures, the interband matrix elements,
and the light gain spectrum in the pseudomorphically
strained zinc-blende nitride quantum well. With this
purpose, we use a 3×3 block Hamiltonian to calculate
the valence band spectra in the quantum well
heterostrusture. A detailed analysis is presented for the
dependence of the hole spectra, the matrix elements, and
light gain spectra on strain under the lattice mismatch in
the heterostructures. The analysis of the band structure
of quantum well under the compressive strain exhibits a
downward shift of the valence bands. For the quantum
well with the compressive strain %54.0−=εxx , the
matrix elements for transitions from the first hole band
have the strict x (or y) light polarization, while with the
tensile strain %54.0=ε xx , the considered matrix
elements have the strict z light polarization. Under the
tensile strain, both the large negative mass and the strong
modification of the matrix elements arise at the Brillouin
zone center. The casual twofold degeneration of double
spin degenerated heavy hole and light hole states arises
at the Brillouin zone center. An increase of the Z state
in the light hole band under the tensile strain is found. It
yields the strong matrix element for the z polarization.
The circular loop of valence band maxima with the finite
radius in the heterostructure under the tensile strain is
found. Under the compressive strain, holes are located
on the upper valence subbands, while under the tensile
strain holes are located on the circular loop of the
valence band maxima. It is shown that laser effect is
suppressed in GaN quantum well with the tensile strain
%54.0=ε xx , while at %54.0−=ε xx the stimulated
optical transitions give rise to a high optical gain. In the
paper, importance of the extremum loop of valence band
in zinc-blende GaN quantum well under the tensile strain
is found. Although the extremum loop effects in würtzite
crystals were studied [35], suppression of the laser
effects was not indicated in publications. It is found that
the internal strain effects play a significant role in optical
properties of the quantum well heterostructures.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 4. P. 364-369.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
369
The author acknowledges many helpful discussions
with Prof. V.A. Kochelap and Prof. V.I. Sheka.
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