Instability of 2DEG interacting with drifting 3DEG
Dispersion law is studied for high-frequency longitudinal plasma waves in 2DEG (the plane z = 0), separated by thin dielectric layer from half-limited 3DEG. It is shown that drift of 3DEG provokes for special conditions instability of considered plasma waves.
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| Формат: | Стаття |
| Мова: | English |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2008
|
| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Цитувати: | Instability of 2DEG interacting with drifting 3DEG / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 2. — С. 148-150. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1188582017-06-01T03:03:08Z Instability of 2DEG interacting with drifting 3DEG Boiko, I.I. Dispersion law is studied for high-frequency longitudinal plasma waves in 2DEG (the plane z = 0), separated by thin dielectric layer from half-limited 3DEG. It is shown that drift of 3DEG provokes for special conditions instability of considered plasma waves. 2008 Article Instability of 2DEG interacting with drifting 3DEG / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 2. — С. 148-150. — Бібліогр.: 6 назв. — англ. 1560-8034 PACS 52.35.-g, 71.10.Ca, 73.21.Fg http://dspace.nbuv.gov.ua/handle/123456789/118858 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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| description |
Dispersion law is studied for high-frequency longitudinal plasma waves in
2DEG (the plane z = 0), separated by thin dielectric layer from half-limited 3DEG. It is
shown that drift of 3DEG provokes for special conditions instability of considered
plasma waves. |
| format |
Article |
| author |
Boiko, I.I. |
| spellingShingle |
Boiko, I.I. Instability of 2DEG interacting with drifting 3DEG Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Boiko, I.I. |
| author_sort |
Boiko, I.I. |
| title |
Instability of 2DEG interacting with drifting 3DEG |
| title_short |
Instability of 2DEG interacting with drifting 3DEG |
| title_full |
Instability of 2DEG interacting with drifting 3DEG |
| title_fullStr |
Instability of 2DEG interacting with drifting 3DEG |
| title_full_unstemmed |
Instability of 2DEG interacting with drifting 3DEG |
| title_sort |
instability of 2deg interacting with drifting 3deg |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118858 |
| citation_txt |
Instability of 2DEG interacting with drifting 3DEG / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2008. — Т. 11, № 2. — С. 148-150. — Бібліогр.: 6 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT boikoii instabilityof2deginteractingwithdrifting3deg |
| first_indexed |
2025-07-08T14:47:32Z |
| last_indexed |
2025-07-08T14:47:32Z |
| _version_ |
1837090527611191296 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 148-150.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
148
PACS 52.35.-g, 71.10.Ca, 73.21.Fg
Instability of 2DEG interacting with drifting 3DEG
I.I. Boiko
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine
41, prospect Nauky, 03028 Kyiv, Ukraine
Phone: (044)236-5422; e-mail: igorboiko@yandex.ru
Abstract. Dispersion law is studied for high-frequency longitudinal plasma waves in
2DEG (the plane z = 0), separated by thin dielectric layer from half-limited 3DEG. It is
shown that drift of 3DEG provokes for special conditions instability of considered
plasma waves.
Keywords: instability, electron gas, plasma.
Manuscript received 28.02.08; accepted for publication 15.05.08; published online 30.06.08.
1. Introduction
The problem of generating the high-frequency
electromagnetic waves attracts attention of a great
number of investigators. Now various low-dimensional
electronic systems are presented as candidates for
designing the suitable generators. There is a lot of
distinguished variants; all of them have their own
advantages and shortages. We propose here to
investigate one variant of well-known beam instability in
plasma (see for instance [1]), where an activating beam
is not actually a beam but plays the significant role of
source of energy transferred to plasma waves.
Controlling this source gives a possibility to manage the
frequency increment of unstable waves.
2. Dielectric susceptibility
Investigated in this work is the dielectric susceptibility
for the system of charged carriers, which includes two-
dimensional electron gas located in the plane z = 0 and
separated by an insulating layer from conducting
substance occupying the half-space z ≥ l (see Figure).
The system like that was considered in [2] and [3],
similar systems − in papers [4-6].
We assume here that the dispersion laws for two-
dimensional and three-dimensional band carriers are
isotropic and parabolic:
2
22 2/)( mkk ⊥⊥ = h
r
ε ; 3
22 2/)( mkk h
r
=ε . (1)
The dispersion equation for longitudinal high-
frequency (collisionless) plasmons has the form
0),(),( =∆+ ⊥⊥ qqS
rr
ωεωε . (2)
Here (see [2])
⎥
⎦
⎤
⎢
⎣
⎡
+
+
+=
⊥⊥
⊥⊥
⊥ )tanh(),(
1)tanh(),(
2
1),( 21 lqq
lqq
qS r
r
r
ωβ
ωβ
εεωε ; (3)
∫
∞
∞−
⊥
⊥ ∆+
=
)],([
),(
33
2
2
qq
dqq
q z
r
r
ωεεπ
ε
ωβ ; (4)
;
02/)2(
)()(
),(
2
2
222
2
∫ +−+
−−
=
=∆
⊥⊥⊥
⊥⊥⊥
⊥
⊥
⊥
imqkq
qkfkf
kd
q
e
q
rr
h
rrr
r
h
r
ωπ
ωε
(5)
;
02/)2(
)()(
),(
3
2
333
22
2
3
∫ +−+
−−
=
=∆
imqkq
qkfkf
kd
q
e
q
rr
h
rrr
r
h
r
ωπ
ωε
(6)
f2 and f3 are distribution functions for 2DEG and 3DEG.
At 21 εε = and 0),( →⊥q
r
ωβ (the latter
corresponds to application of a metallic field electrode)
)]2exp(1/[),( 1 lqqS ⊥⊥ −−= εωε
r
.
At ∞→⊥lq , that is in the case of the infinitely
separated three-dimensional electrode, it follows from
(3):
2/)(),( 21 εεωε +=⊥qS
r
.
Limit our consideration here, for shortness, by non-
degenerated 2DEG and 3DEG. In this case, the
equilibrium distribution function for 2DEG has the form
)/(exp)/2()( 2
2
22
32
0
2 TT kkknkf −=⊥ π
v
;
hh //)2( 22
2/1
B22 TT vmTkmk == . (7)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 148-150.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
149
Fig. 1.
In the case of drifting 3DEG, we use the
hydrodynamic approximation for the non-equilibrium
distribution function:
)/()( 3
0
33 h
rvv
⊥−= umkfkf . (8)
Here ⊥u
r
is drift velocity in },{ yx -plane. Then
)/)/(exp[)/2()( 2
3
2
333
2/3
3 TT kumkknkf h
vrv
⊥−−π= ;
hh //)2( 33
2/1
B33 TT vmTkmk == . (9)
Substituting the expressions (7) and (9) in Eqs. (5)
and (6) and integrating the obtained expression over the
wave vectors ⊥k
r
and k
r
, we obtain dielectric functions
in the form:
;
2
1
2
1
2
),(
2
2
2
2
2
2
2
2
2
B
2
2
0
2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
γ
γ+λ
γ+λ
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
γ
γ−λ
γ−λ
×
×
π
=ωε∆
⊥
⊥
FF
qTk
ne
q
r
(10)
.
2
1
2
14
),(
3
2
3
2
33
2
3
2
3
2
B
3
2
3
⎥
⎥
⎦
⎤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
γ
γ+λ
γ+λ
−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
γ
γ−λ
×
×
⎢
⎢
⎣
⎡
γ−λ
π
=ωε∆
u
u
u
u
FF
qTk
ne
q
r
(11)
Here we have used the following designations (see
[1] and [2]):
TkB/ω=λ h ; Tkuqu B/)( ⊥−ω=λ
rr
h ;
Tkq 33 /=γ ; Tkq 22 /⊥=γ ;
td
ist
ts
sF ∫
∞
∞− −−
−
π
=
0
)(exp
)(
2
. (12)
Now turn to the classical limit. Adopting the
inequality 2
3,21 γ>>λ>> , that is
3
22
B 2/ mqTk hh >>ω>> and
2
22
B 2/ mqTk ⊥>>ω>> hh , (13)
and considering the case of high frequencies, where
1/;1/ 32 >>ω>>ω ⊥ TT vqvq , (14)
one obtains from (10) and (11):
2
2
2
2
2
2
2 )(2),(
ω
Ω
−=
ω
π
−=ωε∆ ⊥⊥
⊥
q
m
qneq
r
; (15)
2
2
3
32
3
3
2
3 )()(
4
),(
⊥⊥ −ω
Ω
ε−=
−ω
π
−=ωε∆
uquqm
ne
q rrrr
r
. (16)
Here
33
3
2
2
3
2
2
2
2
2
4
;2)(
m
ne
m
qneq
ε
π
=Ω
π
=Ω ⊥
⊥ . (17)
Substituting (16) into (4), we have
])[(/)(),( 2
3
2
3
2
2 Ω−−ωε−ωε=ωβ ⊥⊥⊥ uquqq
rrrrr
. (18)
3. Instability
Possible instability of 2DEG for the system shown in the
figure was investigated in [3]. In this paper, the role of a
field electrode was limited by creation of a node for EM
field on the boundary of the electrode. In this case, the
dispersion law for two-dimensional plasma waves does
not depend on the state of 3DEG. Now, we consider
quite another instability. It appears for a limited
conductivity of drifting 3DEG.
Introducing the expressions (3), (15) and (18) into
(2), we find the dispersion law for two-dimensional
plasma waves:
0)()( 2
2
2
1
2
3
22
2
22 =+ω−−ω−ω ⊥⊥ QQQQuq
rr
(19)
Here
)tanh( lq⊥=α ; )/( 23
2
33
2
1 ε+εαΩεα=Q ;
)]()(/[)(2 32223123
2
2
2
2 ε+εαε+ε+εαεε+εαΩ=Q ;
)]()([/)( 322231213
2
3
2
3 ε+εαε+ε+εαεε+εαεΩ=Q .
(20)
Consider the following limit: ∞→⊥lq . Then one
obtains a usual law for two-dimensional plasma waves:
)/(2)1( 21
2
2
2
2
2 ε+εΩ==α=ω Q . (21)
More interesting is the opposite limit: 0→⊥lq .
Then, as a consequence of the expressions (19) and (20)
we have the dispersion law of this form:
.0)/(
)]/(2[)(
313
2
3
2
31
2
2
22
=ε+εεΩω−
−ε+εΩ−ω−ω ⊥⊥uq
rr
(22)
One can see that for frequencies
)(/2 31
2
2
2 ε+εΩ<ω (23)
a real solution of the Eq. (22) does not exist.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2008. V. 11, N 2. P. 148-150.
© 2008, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
150
In the case
)/(2 31
2
2
2 ε+εΩ<<ω , (24)
the equation (22) takes the simple form:
0)(2 3
2
3
22
2
2 =εΩω+Ω−ω ⊥⊥uq
rr
. (25)
Solving this equation, we find
2
2
2
33
233
2/1
2/1
ΩΩε+
ΩεΩ±
=ω ⊥⊥
i
uq
rr
. (26)
Here we see an evident instable branch (Im ω > 0).
The form (26) is typical for the so-called beam
instability. Note that for the considered case metallic
three-dimensional electrode (Ω3→ ∞) used in [3] is not
suitable totally.
4. Discussion and conclusion
The condition of compatibility of the solution (26) and
accepted earlier conditions (13), (14) and (24) has the
form of the following set of inequalities:
;)/(2)(
]2)(/1[
312
1
2332
ε+εΩ<<
<<ΩεΩ+<<
⊥
−
⊥⊥⊥
q
quqvq T
1
3323 ]/2)(1[ −
⊥ εΩΩ+<< quv T . (27)
It is seen from here that the drift velocity of 3DEG
has to exceed significantly the thermal velocities of
2DEG and 3DEG.
It follows from the expression (26) that for the
given wave vector ⊥q increment of instability attends a
maximal value at the condition 332 2)( εΩ=Ω ⊥q , or
3223 / mnmnq =⊥ (28)
Frequency of waves for the given value of ⊥⊥uq
rr
increases, if the ratio 2
2
2
33 2/ ΩΩε goes down.
Consider now another area of frequencies:
⊥⊥<<ω uq
rr
. (29)
In this case, it follows from (22):
2
3331
2
22
22
)()(
)(2
Ωε−ε+ε
Ω
=ω
⊥⊥
⊥⊥
uq
uq
rr
rr
. (30)
Instability of two-dimensional plasma waves
appears at the condition
0)()( 2
3331
2 <Ωε−ε+ε⊥⊥uq
rr
. (31)
The inequality (29) can be represented in the form
02)()( 2
2
2
3331
2 <Ω−Ωε<<ε+ε⊥⊥uq
rr
. (32)
The latter inequality limits from the top both the
drift velocity and the wave vector ⊥q . Simultaneously,
the condition 1/ 3 >>ω ⊥ Tvq (see (14)) has to be
satisfied.
References
1. E.M. Lifshits and L.P. Pitaevckiy, Physical
Kinetics. Nauka, Moscow, 1979 (in Russian).
2. I.I. Boiko, Kinetics of Electron Gas Interacting
with Fluctuating Potential. Naukova dumka, Kyiv,
1993 (in Russian).
3. M.I. Dyakonov and M.S. Shur // Phys. Rev. Lett.
71, p. 2465 (1993).
4. J.R. Pierce // J. Appl. Phys. 19, p. 231 (1948).
5. M.V. Nezlin // Uspekhi Fizich. Nauk 11, p. 608
(1971) (in Russian).
6. Z.S.Gribnikov, N.Z. Vagidov and V.V. Mitin // J.
Appl. Phys. 88(11), p. 137 (2000).
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