Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice
We investigate theoretically the effect of nonparabolic band structure on the electron-confined LO-phonon scattering rate in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice. Using the quantum treatment, the new wave function of electron miniband conduction of superlattice and a reformulation of the slab model for...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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irk-123456789-1215582017-06-15T03:03:08Z Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice Abouelaoualim, D. We investigate theoretically the effect of nonparabolic band structure on the electron-confined LO-phonon scattering rate in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice. Using the quantum treatment, the new wave function of electron miniband conduction of superlattice and a reformulation of the slab model for the confined LO-phonon modes has been considered. An expression for the scattering rates has been obtained. Our results show that, for transitions related to the emission of confined LO-phonon, the scattering rates are significantly increased in the band nonparabolicity case. 2005 Article Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice / D. Abouelaoualim // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 4. — С. 60-64. — Бібліогр.: 42 назв. — англ. 1560-8034 PACS 71.20.-b, 73.21.Cd http://dspace.nbuv.gov.ua/handle/123456789/121558 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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We investigate theoretically the effect of nonparabolic band structure on the electron-confined LO-phonon scattering rate in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice. Using the quantum treatment, the new wave function of electron miniband conduction of superlattice and a reformulation of the slab model for the confined LO-phonon modes has been considered. An expression for the scattering rates has been obtained. Our results show that, for transitions related to the emission of confined LO-phonon, the scattering rates are significantly increased in the band nonparabolicity case. |
| format |
Article |
| author |
Abouelaoualim, D. |
| spellingShingle |
Abouelaoualim, D. Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Abouelaoualim, D. |
| author_sort |
Abouelaoualim, D. |
| title |
Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice |
| title_short |
Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice |
| title_full |
Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice |
| title_fullStr |
Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice |
| title_full_unstemmed |
Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice |
| title_sort |
nonparabolicity effects on electron-confined lo-phonon scattering rates in gaas-al₀.₄₅ga₀.₅₅as superlattice |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121558 |
| citation_txt |
Nonparabolicity effects on electron-confined LO-phonon scattering rates in GaAs-Al₀.₄₅Ga₀.₅₅As superlattice / D. Abouelaoualim // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 4. — С. 60-64. — Бібліогр.: 42 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT abouelaoualimd nonparabolicityeffectsonelectronconfinedlophononscatteringratesingaasal045ga055assuperlattice |
| first_indexed |
2025-07-08T20:07:18Z |
| last_indexed |
2025-07-08T20:07:18Z |
| _version_ |
1837110649023365120 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 4. P. 60-64.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
60
PACS 71.20.-b, 73.21.Cd
Nonparabolicity effects on electron-confined LO-phonon scattering
rates in GaAs-Al0.45Ga0.55As superlattice
D. Abouelaoualim
L.P.S.C.M, Physics Department, Faculty of Sciences - Semlalia,
BP:2390, 40000, Marrakech, Morocco
E-mail: abouelaoualim_d@hotmail.com
Abstract. We investigate theoretically the effect of nonparabolic band structure on the
electron-confined LO-phonon scattering rate in GaAs-Al0.45Ga0.55As superlattice. Using
the quantum treatment, the new wave function of electron miniband conduction of
superlattice and a reformulation of the slab model for the confined LO-phonon modes
has been considered. An expression for the scattering rates has been obtained. Our results
show that, for transitions related to the emission of confined LO-phonon, the scattering
rates are significantly increased in the band nonparabolicity case.
Keywords: scattering rate, band structure, nonparabolicity effect.
Manuscript received 23.08.05; accepted for publication 25.10.05.
1. Introduction
Recently there has been much interest in the study of
electron-phonon interaction in III-V semiconductor
quantum wells (QWs) and superlattices (SLs) [1-3]. This
is because the phonon scattering determines the electron
transport properties at room temperature and high
electric fields as well as at low temperatures. For
instance, the cooling of photoexcited carriers, carrier
tunnelling and mobility high-speed heterostructure
devices are primarily governed by the scattering of
electrons by polar-optical-phonons. Some results in the
Raman scattering, cyclotron-resonance and magneto-
phonon-resonance measurements show the dominance of
electron interaction with LO-phonons and reveal an
important information about the vibration modes in the
layers forming SL [4-10]. The electron-LO-phonon
interaction was found to be strongly dependent on both
the geometrical shape and the parameters of the
constituent materials [11-12]. The polaron effect in
heterostructures of confined size is, however, quite
different from that in bulk materials. Several models
have been proposed to describe the electron-confined
LO-phonon interaction in superlattices. Dielectric
continuum models [13-14], microscopic lattice
dynamical models [17-19], or slab model [20-21] are
well known. Already the several theoretical studies
reported on calculations of the relaxation time related to
scattering of carriers in semiconductor heterostructures
by optical phonons treated the case of single or multiple
quantum wells [22-25].
The purpose of this paper is to present a set of
calculated results for scattering rates in superlattices, we
have considered the carrier scattered by LO-phonon. The
effect of band nonparabolicity on the calculated
scattering rates has been analyzed. The organization of
the present paper is as follows: Section II summarizes
the theoretical framework used in the calculations, while
Section III describes the discussion of numerical results
presented graphically, then a brief conclusion is given.
2. Theoretical model
A. Miniband structure and envelope wave functions
Using an effective-mass Hamiltonian and the transfer-
matrix method, the total energy of electron associated to
the first miniband and analytically the exact normalized
wave function [26] are:
( )lkE
m
k
z
w
cos
22
1*
12*
22 Δ
−+= ⊥h
ξε (1)
( ) −
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+=Ψ
))
2
l–(–(–))
2
l–(–(
2
2
l–
nziknziklinkik
w eeb
N
eez
z
β
−L+ bl blnlz << – , (2)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 4. P. 60-64.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
61
( )
( )[ ] ( )[ ]{ }×+−++×
=Ψ
bb
linkik
b
lnlzqlnlzp
N
eez
z
ρρ sinh–cosh
2
l–
(3)
× bl blnlz 〈〈 – ,
Emk w
*22 2=h , )(2 *22 EVmh bb −=ρ ,
** / bw mm=λ , L = bl + wl . (4)
x = kl w , y = ρ l b ,
)sin()sinh()cos()cosh()sin( lkyxKyx z−−= −−β . (5)
( ) ( )/2–/2
2 ee ixixbp −+= β , ( ) ( )( )/2–/2 ee ixixikq −−= β
λρ
.(6)
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
±=±
λρ
λρ k
k
k
2
1 , ( )ykb sinh2
+= , (7)
( ) −− +⎟
⎠
⎞
⎜
⎝
⎛ += ββ '22
2
' BbAN . (8)
±
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=Π Kk
λρ
2m ,
( ) ( )
( ) ( ) ,sin
2
12cosh1
cos
4
2sinh
2
–'
⎟
⎟
⎠
⎞
⎥
⎦
⎤
⎢
⎣
⎡ −
−+
⎜
⎜
⎝
⎛
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
Π+
Π
+=
−
xyk
k
x
y
yllA bw
ρλρ
(10)
[{
[
).sin(]1/
)2/)2cosh(1()cos(
)]4/)2(sinh(2/2
2
2
'
w
bw
bbb
kLk
kLkL
LLLbB
+
+−+
×Π+Π= −+
λρρ
ρρ
(11)
B. Scattering rates
The interaction electron-phonon Hamiltonian in low-
dimensional systems depends on the specific phonon
spectra of the system and differs from the Fröhlich
Hamiltonian for a bulk phonon. The macroscopic
dielectric continuum model [27-30] gives the functional
form of the interface modes, confined and half space
LO-modes. The electron-confined LO-phonon
interaction Hamiltonian as derived from the Fröhlich
interaction is given by [31, 32]
[ ],)()(
)()()(e
,,
,,
,
⊥
+
⊥
⊥
±=
−+×
×= ∑∑
⊥
⊥
qaqa
zuqtzHH
nn
nn
nq
riq
e-p
αα
αα
α
λ
(12)
where a(q) and a+(q) are the creation and annihilation
operators for a bulk phonon in the mode q , the even (−)
and odd (+) confined phonon modes and n is the
miniband index, while the coupling
2λ = μiC / qV , (13)
where V is the volume. From [33] C can be written
explicitly as
,
)0(
1
)(
1
2
2
1
0
LO
2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
∞
=
εεε
ωhe
C (14)
where ħωLO is the energy of LO-phonons in the n-th
miniband, )(∞ε and )0(ε are the optical static dielectric
constants, respectively, Ω is the volume and e is the
electronic charge. For the slab model [27, 34] un,α(z) is
defined as
) / cos()z( wn Lznu π=+ n = 1, 3, 5, … (15)
) / sin()( wn Lznzu π=− n = 2, 4, 6, … (16)
α,nt is given by
α,nt =
( )[ ] 2122 /
1
wLnq π+⊥
n = 1, 2, 3, … (17)
Finally
H(z) =
⎪
⎩
⎪
⎨
⎧ ≤≤
otherwise 0,
– if , 1 wzw
. (18)
The scattering rate fiw → is obtained from the Fermi
Golden Rule
22)( ie-p
f
ffi Hkw ξξπ ∑=→
h
. (19)
With the Hamiltonian given by (14), we obtain
.),,()(
)0(
1
)(
1)
2
1
2
1(
2
LO
2
LO
f
f
z
i
z
fi
dNqkkIU
q
e
N
V
w
⊥
±
±
→
×
×⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
∞
±+= ∫
δ
εε
ω
π
π h
h
In this expression, the integration is over the number
of final states Nf
, where
In ( ),, ⊥qkk f
z
i
z =
2
,
2
,
, )()( ⊥
→∑∑
⊥
qt,kkG n
n
f
z
i
z
fi
n
q
α
α
α . (21)
A δ-function represents the energy conservation
quantity
=± )(Uδ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
±−+= ±⊥⊥ )()–(
2
LO
22
*
2
qEEkk
m
i
z
f
z kk
fi ωδ h
h
,
± denote the absorption and emission processes. For
optical phonon scattering
(20)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 4. P. 60-64.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
62
cte)–()cos(k2–– 2222 =+= ⊥⊥⊥⊥± Gkkkkkq f
z
i
z
fifi mθ .
(22)
G is the reciprocal lattice vector of the SL. LON is the
LO phonon occupation number defined:
1
B
LO
LO )1(exp −−=
Tk
N ωh . (23)
)(,
f
z
i
z
fi
n ,kkG →
α is the overlap integral of the electron
wave function and the z-dependent of the electron-
confined-phonon Hamiltonian
)(,
f
z
i
z
fi
n ,kkG →
α = dzzuz *
inf )()(
l/2
l/2-
,
* ψψ α∫ , (24)
where iψ , fψ are the electron envelope miniband
wave function in the initial and final states, respectively
[31]. L is the period of SL: L = Lw+Lb. At 0=±U , fk⊥
and ik⊥ terms must be equal.
C. Nonparabolicity effect
According to the Kane model [35-37], the eigenfunc-
tions of the Hamiltonian in the direction of the super-
lattice (with kx = ky = 0) associated with the conduction
band electron [38, 39] with an energy 0 < E < Vb, are
solutions of the Schrödinger equation [40]:
(
2
2
2
2
z
a
∂
∂
h + 4
4
4
4
za ∂
∂
h ) y(z) + (E − V(z))y(z) = 0 , (25)
a2 =
( )zm∗2
1 , a4 =
gE
1
0
* 2
1
2
1
mm
− . (26)
The corrective term reflects the nonparabolicity
effect (via a4). The integration of Eq. (25) over the
interface of a small arbitrary thickness provides the new
boundary conditions:
.2
,4,2
2
,4,2
bbbb
wwww
aa
aa
ψψ
ψψ
′′′+′=
=′′′+′
h
h
(27)
This expression that ensures the continuity of the
local current density generalizes that of Refs [41, 43]
where a4 = 0. In case of nonparabolicity, the wave
functions corresponding to the new condition (27)
generalize those where the continuity of ( )
dz
zd
m
Ψ
∗
1 is
used. As the latter Hamiltonian does not take
parabolicity into account, the wave functions are given
at the n-th well and barrier by Eqs (2) and (3). Due to the
new conditions (27) on the derivative of the wave
function, the analysis of the preceding sections can be
used with λ replaced by μ, which we define as follows.
From Eqs (4) and (25), expressions of k, ρ and μ are
given by:
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
=
max,
max,
22 – 1 –1 4
w
w
*
w
E
EEmkh ;
w
w
a
aE
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
4
2
2
max;
4
; (28)
,
4
2
2
max,
0–max,
0
0max,
22
4
1 1–
)–( 4
;
b
b
b
b
*
b
a
a
E
VE
EV
VEm
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧ −
++×
×=ρh
(29)
ww
bb
aka
aa
4,
22
2,
4,
22
2,
h
h
−
+
=
ρ
μ . (30)
When introducing the new expressions of wave
vectors k and ρ in Eqs (1), (2), and (3), we obtained the
new expressions for the dispersion relation and the wave
functions in the barrier and wells of SL by continuation
those of the times of relaxation and mobility. If the
effect of the nonparabolicity becomes negligible (a4 = 0),
*
*
b
w
m
m
=→λμ as defined in the parabolic case.
Expressions (28), (29) allow an explicit relationship of ρ
in relation with k. For w
b
b EVE maxmax =− (i.e.,
2
0
2
0 λρ=k ) insignificant values of ρ and k, we find the
parabolic case given by the relation (4).
3. Numerical results and discussion
For numerical computation, we have chosen
AsAlGa-GaAs xx–1 with x = 0.45 as a superlattice. The
parameters pertaining to the system are: 0067.0 mm*
w = ,
0104.0 mm*
b = , where m0 is the free electron mass. The
dielectric constant in the wells is taken equal to that in
the barrier: 8.12=dε , 9.10=∞ε , lw = 108 Å, l b = 38 Å,
V b = 495 meV, Ew,max = 2 eV, Eb,max−Vb = Ew,max, the
energy of a bulk GaAs LO-phonon LOωh = 36.8 meV,
the static and high frequency dielectric constants for
GaAs 35.12s =ε and 48.10=∞ε .
In Fig. 1, we show the calculated rates for
intraminiband transitions related to interaction electron-
confined LO-phonons as a function of the SL well width.
Note that the scattering rate does not qualitatively differ
from that to the parabolic band approximation. In that
approximation, the scattering rates related to confined
LO-phonons become larger. It may be due to the overlap
integrals given by Eq. (24), to the nonparabolic band
approximation the electron wave function becomes more
confined in the direction of SL, see Fig. 2.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 4. P. 60-64.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
63
Fig. 1. Scattering rates for intraminiband transitions in GaAs-
Al0.45Ga0.55As superlattice as a function of the well width. The
solid line is drawn with account of nonparabolicity, the dashed
line corresponds to the parabolic approximation.
Z(Å)
Fig. 2. Density of probability associated to an electron of the
first miniband to the approximation of the binding forces. Link
pace of potential is to indicate the positions of the barrier and
well of superlattice.
Fig. 3. Density of states calculated for superlattice GaAs-
Al0.45Ga0.55As. For nonparabolic (solid line) and parabolic
(dashed lines) band approximations.
Fig. 4. Ratio of the nonparabolic and parabolic scattering rates
for intraminiband as a function of the well width in super-
lattice.
Another element that influences the scattering rate is
the density of final states. In Fig. 3, we give the density
of final states to the parabolic band approximation in
comparison with that to the nonparabolic band one.
We show that the density in the case of the
nonparabolic band approximation is larger. In Fig. 4, we
display the ratio of nonparabolic and parabolic scattering
rates (wnp/wp). For the intraminiband for the narrow well
(the well width is as small as 45 Å ) all nonparabolic
scattering rates are close to those in the parabolic band
approximation. For larger quantum wells, the transition
rate with the band nonparabolicity is larger.
In conclusion with the new analytic wave function
associated to the electron in conduction minibands. We
have evaluated the expressions for the relaxation time
due to electron-confined LO-phonon, including band
nonparabolicity. It is found that for transitions from
higher energy states, the band nonparabolicity affects the
scattering rate. The enhancement of the scattering rates
with the inclusion of band nonparabolicity results from a
larger electron-phonon overlap as well as from a larger
density of final electrons states.
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