Interface roughness induced intrasubband scattering in a quantum well under an electric field
Scattering rates in the lowest subband in a quantum well are calculated for interface roughness scattering when an electric field is applied normally to the layer plane. It is found that the interface roughness scattering rate increases with increasing electric field. The electric field changes the...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2002
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| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Цитувати: | Interface roughness induced intrasubband scattering in a quantum well under an electric field / G.B. Ibragimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 1. — С. 39-41. — Бібліогр.: 17 назв. — англ. |
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irk-123456789-1195642017-06-08T03:06:38Z Interface roughness induced intrasubband scattering in a quantum well under an electric field Ibragimov, G.B. Scattering rates in the lowest subband in a quantum well are calculated for interface roughness scattering when an electric field is applied normally to the layer plane. It is found that the interface roughness scattering rate increases with increasing electric field. The electric field changes the interface roughness scattering rates drastically in thick QWs as compared with those for the zero-field case. 2002 Article Interface roughness induced intrasubband scattering in a quantum well under an electric field / G.B. Ibragimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 1. — С. 39-41. — Бібліогр.: 17 назв. — англ. 1560-8034 PACS: 68.65,73.20.D http://dspace.nbuv.gov.ua/handle/123456789/119564 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Scattering rates in the lowest subband in a quantum well are calculated for interface roughness scattering when an electric field is applied normally to the layer plane. It is found that the interface roughness scattering rate increases with increasing electric field. The electric field changes the interface roughness scattering rates drastically in thick QWs as compared with those for the zero-field case. |
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Article |
| author |
Ibragimov, G.B. |
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Ibragimov, G.B. Interface roughness induced intrasubband scattering in a quantum well under an electric field Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Ibragimov, G.B. |
| author_sort |
Ibragimov, G.B. |
| title |
Interface roughness induced intrasubband scattering in a quantum well under an electric field |
| title_short |
Interface roughness induced intrasubband scattering in a quantum well under an electric field |
| title_full |
Interface roughness induced intrasubband scattering in a quantum well under an electric field |
| title_fullStr |
Interface roughness induced intrasubband scattering in a quantum well under an electric field |
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Interface roughness induced intrasubband scattering in a quantum well under an electric field |
| title_sort |
interface roughness induced intrasubband scattering in a quantum well under an electric field |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2002 |
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http://dspace.nbuv.gov.ua/handle/123456789/119564 |
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Interface roughness induced intrasubband scattering in a quantum well under an electric field / G.B. Ibragimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 1. — С. 39-41. — Бібліогр.: 17 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT ibragimovgb interfaceroughnessinducedintrasubbandscatteringinaquantumwellunderanelectricfield |
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2025-07-08T16:10:56Z |
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2025-07-08T16:10:56Z |
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1837095775943786496 |
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39© 2002, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2002. V. 5, N 1. P. 39-41.
PACS: 68.65,73.20.D
Interface roughness induced intrasubband scattering
in a quantum well under an electric field
G.B. Ibragimov
Institute of Physics, Academy of Science of the Azerbaijan, 33 H. Javid av., 370143 Baku, Azerbaijan
Fax: +380 (99412) 39 5961; e-mail: physic@physics.ab.az
Abstract. Scattering rates in the lowest subband in a quantum well are calculated for interface roughness
scattering when an electric field is applied normally to the layer plane. It is found that the interface
roughness scattering rate increases with increasing electric field. The electric field changes the interface
roughness scattering rates drastically in thick QWs as compared with those for the zero-field case.
Keywords: interface roughness, intrasubband transition, quantum well.
Paper received 23.06.01; revised manuscript received 21.12.01; accepted for publication 05.03.02.
1. Introduction
There has been considerable interest in the study
of intersubband transitions in a quantum well (QW)
both in the presence and in the absence of an elec-
tric field applied perpendicular to the QW layer
plane [1-12]. The reason causing the interest is prac-
tical device application [3-4]. In [1] proposed is a
new type of infrared laser amplifier using the
intersubband transition and resonant tunneling. An
infrared detector in the 10 µm has been proposed
and fabricated [5] that relies on the intersubband
transition and resonant tunneling between adjacent
QWs.
Theoretical studies [6,7] on intersubband opti-
cal absorption under an electric field have pointed
out an increase in the oscillator strength. In all the
theories [6,7] the intra- and intersubband relaxa-
tion rates come into the calculation of the absorp-
tion coefficient. Various scattering events determine
the intra- and intersubband relaxation rates, of
which polar optic phonon scattering has been ma-
jor scattering mechanism over a considerable tem-
perature range. The rates for such scattering have
already been calculated [7]. In [8] the alloy-disor-
der scattering rate have been calculated for two-di-
mensional electrons in the lowest subband of a quan-
tum well in an applied electric field.
It is established that interface - roughness scat-
tering determines the low temperature mobility of
two - dimensional (2D) electrons in thin QWs
[13,14]. The mobility and relaxation rates due to
interface roughness scattering in QWs have been
formulated previously in the absence of an applied
electric field [15,16]. In the present work we report
similar calculations when an electric field is applied
perpendicular to the QW layer, and investigate how
the scattering rates are modified. As a first step, we
consider intrasubband relaxation for the lowest
subband.
2. Calculations
The electrons are assumed to be quantizied along
the Z direction and the wave function is taken to be
of the following form [2].
( )Z0ψwhere is the unperturbed ground-state wave func-
tion. The normalization factor of the electron wave func-
tion are
(2)
L is the thickness of the well, and k and r are, respectively
the 2D wave vector and position vector of the electron in
the plane of free motion (X-Y plane). In Eq. (1) b is vari-
ational parameter related to the electric field F as in [2]
( ) ( )8223
3
2
−+
=
πβπ
πβN
(1)
( ) ( ) ( ) ( ) ( ) ( ),exp1exp 0 ikrz
L
z
NikrZR Ψ
+=Ψ=Φ ββ
2/2/ LzL <<−
40 SQO, 5(1), 2002
G.B. Ibragimov: Interface roughness induced intrasubband scattering...
2
0
22
0
2
2
,
*2
Lzz
LFme
<<><><= ββ
h
where
The interface roughness is characterized by the height
∆ and the lateral correlation length ∧ of the Gaussian
fluctuation. For the perturbing potential due to interface
roughness we use the standard model [10,17], for which
the random potential is assumed to have the Gaussian
distribution and expressed as the autocorrelation func-
tion
( ) ( )>=′< RVRV
( ) ( ) ( )
Λ
′−−−′−∆=
2
2
22
0 exp2/2/
rr
LZLZV δδ (4)
>=<>< 0
2
0
2 ψψ zz
〉〈......where means an ensemble average, V0 is the
barrier height.
Using Eq. (4) we got for the square of the matrix
element for roughness scattering from the k state to
the k� state
( ) =′ 2,KKM
Λ−
∆Λ=
4
exp
2
22
42
0
22 qL
V ψπ (5)
qwhere q=k�-k is the 2D scattering wave vector and q = .
Also, as the scattering is elastic
The transition probability from a state k to all other states
k` is then given by
( ) ( )∑
′
′ −′=
K
KK EEKKM δπ 2
,
2
h
( )
3
42
0
22 2/*
h
LVm ψπ ∆Λ
= ( )KG ,Λ (6)
,` kk = 2`/cos kkk=θ
where
( ) ( ) θθ
π
π
dKKG ]2/cos1exp[
2
1
, 22
2
0
−Λ−=Λ ∫
( ) ( ) 2/
2/1
0
2
|
*2
2/ LZdZ
d
EVm
L =
−
= ψψ h
( )
( ) ( )KGN
EV
V
Lm
WF ,
2
1
*
4
4
2
0
2
0
6
225
Λ⋅
+
−
⋅∆Λ= ββπ h
( )
( ) ( )
4
4
0 2
1
+== ββγ N
KW
KWF
( )2/42
0 LV ψ
The interface roughness scattering rate is proportional
to , and using [9]
We obtain
∞→0V
As shown in Eq. (8), the scattering rate is proportional
to L-6, this means that interface roughness scattering rate
is much more important for narrow wells and the inter-
face roughness scattering limited mobility in quantum
wells is proportional to L6 [15].
For the interface roughness calculation we have not
bothered to take screening into account. It seems evident
that screening can only reduce an effect that is already
very small.
In order to facilitate comparison, a similar expres-
sion for the transition probability or scattering rate with-
out an applied field is needed. Thus, if we take the limit
and in Eq.(8), we may write for the
scattering rate without a field, W0(K) as [15]
0=β
( )KG
Lm
W ,
* 6
225
0 Λ∆Λ= hπ
∞→0VIn the limit from Eqs (8) - (9) the ratio is expressed
as
Fig. 1. Variation of scattering rate, WF(k), with applied electric
field for a GaAs QW of 40, 50 or 60A0. The rate WF is normal-
ized by the rate W0(k) in the absence of field, and the ratio is
denoted by g.
3. Results
We have calculated the scattering rates in QWs by using
the parameters characteristic of GaAs. The values of the
ratio γ calculated from Eq. (10) plotted against the electric
field F in Fig.1. It is found from the figure that the scattering
rate increases with an increase in the electric field. How-
ever, as seen from the figure, the electric field changes the
interface roughness scattering rates drastically in thick QWs
γ
4
3
2
1
100 200 300 400
F ield (kV /cm )
60
50
40
0
(3)
(7)
(8)
(9)
(10)
( ),cos12 22 θ−= kq
G.B. Ibragimov: Interface roughness induced intrasubband scattering...
41SQO, 5(1), 2002
as compared with those for zero-field case. These results
predict qualitatively the trends observed [2]. Smaller elec-
tric field dependences can be explained by the fact that
the wave function change due to the applied electric field
of thin QWs is smaller than that thick QWs. We find that
γ is also a strong function of field and this ratio becomes
as small as 1.1 to 1.5 below 100 kV/cm. Another observa-
tion is that although the value for γ is 1.5÷4.5 field above
400 kv/cm. In the case of polar optic phonon scattering
γ≈1.2 (F=200 kV/cm) [7] and for alloy-disorder scatter-
ing γ≈4.5 (F=400 kV/cm) [8].
4. Conclusion
We have demonstrated that there is an increase of the
scattering rate of electrons in the first subband of a QW,
with increase in the perpendicular electric field, when
the scattering is due to interface roughness. We find that
WF/W0 have a stronger dependence on the well width.
Acknowledgments
The author would like to thank Prof. M.I. Aliev and
Prof. F.M. Gashimzade for helpful discussions.
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