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Linköping University Post Print 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 / mddn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 B0ci 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 Downloaded 25 Feb 2011 to 130.236.83.91. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions 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 / B0ci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−it + 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4ne0e / me兲 2 2 1/2 electron plasma frequency, pi = 共4ni0Zi e / mi兲 is the ion 2 2 e nd0 / md兲1/2 is the dust plasma frequency, and d = 共4Zd0 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 = pikzk2q 共1 + k44q 2 2 4 2 1/2 + 2pik⬜ k q/ci兲 , 1 + k44q + − 2 2 4 2 k q pi k⬜ 2ci − 冉 kz2k24q2pi ␦2 冊 k44q2pd 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. B0ci 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 = k22q pd 2 2 4 2 共1 + k44q + k⬜ k q pi/2ci兲1/2 共12兲 , we have ␦3 = 共kui0 cos 兲3kz22pi/2k22pd , 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 k22qpd共kz/k兲2/3共Z2i ni0md/Zd0 nd0mi兲1/3 24/3 2 2 4 2 共1 + k44q + 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 = k22q pd 2 2 4 2 共1 + k44q + 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 kq ⬃ 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- Downloaded 25 Feb 2011 to 130.236.83.91. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions 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. G. Manfredi and F. Haas, Phys. Rev. B 64, 075316 共2001兲. D. Anderson, M. Lisak, and M. Marklund, Phys. Rev. E 65, 046417 共2002兲. 3 F. Haas, L. G. Garcia, J. Goedert, and G. Manfredi, Phys. Plasmas 10, 3858 共2003兲. 4 G. Manfredi, Fields Inst. Commun. 46, 263 共2005兲. 5 P. K. Shukla and B. Eliasson, Phys. Rev. Lett. 96, 245001 共2006兲; 99, 096401 共2007兲; D. Shaikh and P. K. Shukla, ibid. 99, 125002 共2007兲; D. Jovanovic and R. 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