Download Linköping University Post Print Ion streaming instability in a quantum dusty magnetoplasma

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Ion streaming instability in a quantum dusty
magnetoplasma
Nitin Shukla, P.K. Shukla, G. Brodin and Lennart Stenflo
N.B.: When citing this work, cite the original article.
Original Publication:
Nitin Shukla, P.K. Shukla, G. Brodin and Lennart Stenflo, Ion streaming instability in a
quantum dusty magnetoplasma, 2008, Physics of Plasmas, (15), 1070-664X, 044503-1044503-3.
http://dx.doi.org/10.1063/1.2909533
Copyright: American Institute of Physics
http://www.aip.org/
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-42543
PHYSICS OF PLASMAS 15, 044503 共2008兲
Ion streaming instability in a quantum dusty magnetoplasma
Nitin Shukla,a兲 P. K. Shukla,b兲 G. Brodin,c兲 and L. Stenflod兲
Institut für Theoretische Physik IV and Nonlinear Physics Centre and Centre for Plasma Science
and Astrophysics, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum,
D-44780 Bochum, Germany
共Received 4 January 2008; accepted 25 March 2008; published online 30 April 2008兲
It is shown that a relative drift between the ions and the charged dust particles in a magnetized
quantum dusty plasma can produce an oscillatory instability in a quantum dust acousticlike wave.
The threshold and growth rate of the instability are presented. The result may explain the origin of
low-frequency electrostatic fluctuations in semiconductors quantum wells. © 2008 American
Institute of Physics. 关DOI: 10.1063/1.2909533兴
Recently, there has been a great deal of interest in investigating the linear and nonlinear properties of electrostatic1–5
and electromagnetic waves6–8 in dense quantum plasmas.
The latter, which are ubiquitous in compact astrophysical
bodies9 共e.g., the interiors of white dwarf stars, magnetars
and supernovae兲 as well as in micro- and nanoscale objects10
共e.g., nanowires, ultrasmall semiconductor devices兲, have an
extremely high electron number density. Electron tunneling
caused by the quantum Bohm force11 plays then a very important role for phenomena occurring at the nanoscales.
Large amplitude electrostatic waves in an unmagnetized
quantum plasma can be excited by two-stream12 and parametric instabilities.13 Our objective here is to investigate the
stability properties of low-frequency 共in comparison with the
ion gyrofrequency兲, long wavelength 共in comparison with
the ion gyroradius兲 electrostatic quantum dust acoustic
waves in a uniform dense magnetoplasma. The latter is composed of electrons, ions, and charged dust particles. At equilibrium, we have14 Zieni0 = ene0 − qdnd0, where Zi is the ion
charge state, e is the magnitude of the electron charge, qd is
the dust charge 共qd = −eZd0 for negatively charged dust grains
and eZd0 for positively charged dust grains, where Zd0 is the
number of charges on dust grains兲, and n j0 is the unperturbed
particle number density 共j is e for the electrons, i for ions,
and d for dust grains兲. We suppose that the micrometer 共or
nanometer兲 sized dust grains are spherical, and that they can
be treated like point charges. Charged dust impurities may
a兲
Electronic mail: [email protected]. Also at GoLP/Instituto de Plasmas
e Fusao Nuclear, Instituto Superior Técnico, 1049-001 Lisboa, Portugal,
and Department of Physics, Umeå University, SE-90187 Umeå, Sweden.
b兲
Electronic mail: [email protected]. Also at School of Physics, University of
KwaZulu-Natal, Durban 4000, South Africa; Department of Physics, Umeå
University, SE-90187 Umeå, Sweden; Max-Planck Institut für Extraterrestrische Physik, D-45741 Garching, Germany; CCLRC Centre for Fundamental Physics, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon
OX11 OQX, U.K.; SUPA Department of Physics, University of Strathclyde, Glasgow G4 ONG, Scotland, U.K.; GoLP/Instituto de Plasmas e
Fusao Nuclear, Instituto Superior Técnico, 1049-001 Lisboa, Portugal.
c兲
Electronic mail: [email protected]. Also at Department of Physics, Umeå University, SE-90187 Umeå, Sweden, and CCLRC Centre for
Fundamental Physics, Rutherford Appleton Laboratory, Chilton, Didcot,
Oxon OX11 OQX, U.K.
d兲
Electronic mail: [email protected]. Also at Department of
Physics, Umeå University, SE-90187 Umeå, Sweden, and Department of
Physics, Linköping University, SE-58183 Linköping, Sweden.
1070-664X/2008/15共4兲/044503/3/$23.00
coexist with the electrons and ions in semiconductor quantum wells, as well as in astrophysical radiative environments
and in micromechanical systems.
Let us consider a uniform electron-ion-dust plasma in an
external magnetic field B0ẑ, where B0 is the strength of the
magnetic field and ẑ is the unit vector along the z axis in a
Cartesian coordinate system. In the presence of the equilibrium electric field E0, the electrons and ions will have a drift
ue0,i0 = 共c / B0兲E0 ⫻ ẑ, while the dust grain velocity is ud0
= 共qd / md␯dn兲E共Ⰶue0,i0兲, if the electron 共ion兲 gyrofrequency
is much larger than the electron-neutral 共ion-neutral兲 collision frequency, and the dust gyrofrequency is much smaller
than the dust-neutral collision frequency ␯dn. Here, c is the
speed of light in vacuum and md is the dust grain mass. Let
us now investigate the stability of our equilibrium against
low-frequency 共in comparison with the ion gyrofrequency
␻ci = ZieB0 / mic, where mi is the ion mass兲, long wavelength
共in comparison with the ion thermal gyroradius兲 electrostatic
perturbations whose electric field is E = −ⵜ␾共r , t兲, where ␾ is
2
ne1 and
the electrostatic potential. For 兩共⳵t + ue0 · ⵜ兲2ne1兩 Ⰶ ␻ce
2
2
2
VFe兩⳵z ne1兩 Ⰶ 共ប2 / 4me 兲兩ⵜ4ne1兩, the electron number density
perturbation ne1 共Ⰶne0兲 is deduced from15
ⵜ2ne1 +
4mene0
␾ = 0,
ប2
共1兲
where ␻ce = eB0 / mec is the electron gyrofrequency, me is the
electron mass, VFe = 共kBTFe / me兲1/2, kB is the Boltzmann constant, TFe is the Fermi electron temperature, and ប is Planck’s
constant divided by 2␲. Equation 共1兲 represents a balance
between the electrostatic and quantum forces.
The ion number density perturbation ni1 共Ⰶni0兲 in the
electrostatic field is obtained from the ion continuity equation
冉
⳵
+ ui0 · ⵜ
⳵t
冊冉
ni1 −
冊
⳵viz
cni0 2
= 0,
ⵜ⬜␾ + ni0
B0␻ci
⳵z
共2兲
where the magnetic field aligned ion velocity perturbation viz
is obtained from the parallel 共to ẑ兲 component of the ion
momentum equation
15, 044503-1
© 2008 American Institute of Physics
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044503-2
冉
Phys. Plasmas 15, 044503 共2008兲
Shukla et al.
冊
⳵
Z ie ⳵ ␾
= 0.
+ ui0 · ⵜ viz +
⳵t
mi ⳵z
共3兲
We note that the second term in parenthesis of the right-hand
side of Eq. 共2兲 comes from the divergence of the ion flux
involving the ion polarization drift velocity vip = −共c / B0␻ci兲
⫻共⳵t + ui0 · ⵜ兲ⵜ⬜␾, and that Eq. 共2兲 is valid for ␯in Ⰶ 共⳵t
+ ui0 · ⵜ兲 Ⰶ ␻ci, where ␯in is the ion-neutral collision frequency. Eliminating viz from Eq. 共2兲 by using Eq. 共3兲, we
have
冉
⳵
+ ui0 · ⵜ
⳵t
冊冉
2
ni1 −
冊
共4兲
We assume that the dust grains are unmagnetized and
collisionless. This approximation holds for 兩⳵tnd1兩 Ⰷ ␻cdnd1,
␯dnnd1, where ␻cd = Zd0eB0 / mdc is the dust gyrofrequency
and md is the dust mass. The dust number density perturbation nd1 共Ⰶnd0兲 is then obtained from the dust continuity
equation
⳵nd1
+ nd0 ⵜ · vd = 0,
⳵t
共5兲
where the dust fluid velocity perturbation vd is determined
from
⳵vd
qd
ⵜ ␾.
=−
⳵t
md
共6兲
Eliminating vd from Eq. 共5兲 by using Eq. 共6兲, we obtain
共7兲
The equations above are closed with the Poisson equation
ⵜ2␾ = 4␲共ene1 − Zieni1 − qdnd1兲.
共8兲
Supposing that ␾ and n j1 are proportional to exp共−i␻t
+ ik · r兲, where ␻ and k are the frequency and the wave vector, respectively, we Fourier transform Eqs. 共1兲, 共4兲, 共7兲, and
共8兲, and combine the resultant equations to obtain the dispersion relation
1+
2
␻2pd
␻2pikz2
k4q ␻2pik⬜
+
−
−
= 0,
k4 ␻2cik2 共␻ − k · ui0兲2k2 ␻2
共9兲
2
1/2
where kq = 共4m2e ␻2pe / ប2兲1/4 ⬅ ␭−1
is the
q , ␻ pe = 共4␲ne0e / me兲
2 2
1/2
electron plasma frequency, ␻ pi = 共4␲ni0Zi e / mi兲 is the ion
2 2
e nd0 / md兲1/2 is the dust
plasma frequency, and ␻d = 共4␲Zd0
2
+ kz2, where k⬜
plasma frequency. We have denoted k2 = k⬜
and kz are the components of k across and along ẑ.
Two comments are in order. First, in the absence of the
DC electric field and dust grains, Eq. 共9兲 yields
␻=
␻ pikzk␭2q
共1 +
k4␭4q
2 2 4 2 1/2
+ ␻2pik⬜
k ␭q/␻ci兲
,
1 + k4␭4q +
−
2 2 4 2
k ␭q␻ pi
k⬜
␻2ci
−
冉
kz2k2␭4q␻2pi
␦2
冊
k4␭4q␻2pd
2␦
= 0.
2 1−
共k · ui0兲
k · ui0
共11兲
Equation 共11兲 reveals that for
cni0 2
Zieni0 ⳵2␾
ⵜ ⬜␾ −
= 0.
B0␻ci
mi ⳵z2
⳵2nd1 qdnd0 2
−
ⵜ ␾ = 0.
⳵t2
md
共1兲. Second, we investigate the two-stream instability16 involving the streaming of ions against charged dust grains.
For this purpose, we let ␻ = k · ui0 + ␦ in Eq. 共9兲 and assume
␦ Ⰶ 兩k · ui0兩 to obtain
共10兲
which is the frequency of our new, low-frequency, obliquely
共to ẑ兲 propagating dispersive electrostatic wave in a magnetized quantum plasma. The new feature to the wave is attributed to the non-Boltzmann electron response, given by Eq.
kui0 cos ␪ =
k2␭2q␻ pd
2 2 4 2
共1 + k4␭4q + k⬜
k ␭q␻ pi/␻2ci兲1/2
共12兲
,
we have
␦3 = 共kui0 cos ␪兲3kz2␻2pi/2k2␻2pd ,
共13兲
where ␪ is the angle between k and ui0. Solutions of Eq. 共13兲
are
␦=
1 ⫾ i冑3
kui0 cos ␪共kz/k兲2/3共␻ pi/␻ pd兲2/3 .
24/3
共14兲
The growth rate for our purposes reads
␥=
2
冑3 k2␭2q␻pd共kz/k兲2/3共Z2i ni0md/Zd0
nd0mi兲1/3
24/3
2 2 4 2
共1 + k4␭4q + k⬜
k ␭q␻ pi/␻2ci兲1/2
.
共15兲
The threshold condition and the growth rate, given by Eqs.
共12兲 and 共15兲, depend on the quantum parameter ␭q. Hence,
our new electron response, given by Eq. 共1兲, plays an important role in the ion streaming instability driving lowfrequency electrostatic fluctuations having the real part of the
frequency
␻r =
k2␭2q␻ pd
2 2 4 2
共1 + k4␭4q + k⬜
k ␭q␻ pi/␻2ci兲1/2
冋
1+
冉 冊册
1 kz␻ pi
24/3 k␻ pd
2/3
,
共16兲
which represents a quantum dust acoustic wave in a dense
magnetoplasma.
As an illustration, we apply our results to a gedanken
semiconductor quantum well experiment for verifying our
theoretical prediction. The typical parameters representative
of semiconductor quantum wells are:17 ne0 ⬃ 5 ⫻ 1016 cm−3,
ni0 ⬃ 4 ⫻ 1016 cm−3, nd0 ⬃ 1013 cm−3, Zi = 1, Zd0 ⬃ 103, md
= 1012mi, B0 ⬃ 104 G, and TFe ⬃ 85.7 K. Accordingly, we
␻ pe = 1.26⫻ 1013 s−1,
␻ pi / ␻ci = 3 ⫻ 103,
␻ pd = 4
have
6 −1
−7
⫻ 10 s , and ␭q = 2 ⫻ 10 cm. Taking k␭q ⬃ 0.1, kz / k
⬃ 0.01, we obtain, from Eqs. 共15兲 and 共16兲, ␥ ⬃ ␻r = 2
⫻ 103 s−1. Thus, within a fraction of milliseconds our quantum dust acoustic wave in a semiconductor quantum well can
be excited.
To summarize, we have shown that the free energy
stored in ion streams 共relative to the charged dust grains兲 can
be coupled to the low-frequency quantum dust acoustic wave
in a dense magnetoplasma containing degenerate electrons,
ions and dust grains. The knowledge of the threshold condition as well as the growth rate and the real part of the wave
frequency, as presented here, is essential for identifying non-
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044503-3
thermal electrostatic fluctuations that may originate in dense
dusty magnetoplasmas, such as those in semiconductors
quantum wells. The dust acoustic wave in a magnetized
semiconductor plasma can be used as a diagnostic tool for
inferring the dust charge.
This research was initiated while the authors were attending the 2007 Summer College on Plasma Physics at the
Abdus Salam ICTP, Trieste, Italy. It was partially supported
by the Swedish Research Council.
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