Geo-nanomaterials ION DISTRIBUTION FUNCTION IN THEIR OWN GAS PLASMA

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1 DOI.8454/PMI UDC ION DISTRIBUTION FUNCTION IN THEIR OWN GAS PLASMA А.S. MUSTAFAEV, V.S. SOUKHOMLINOV Saint-Petersburg Mining Universit Saint-Petersburg, Russia Saint-Petersburg State Universit Saint-Petersburg, Russia Flat one-sided probe was used for the first time to measure the first seven coefficients in the Legendre polynomial expansion of ion energy and angle distribution functions for He + in He and Ar + in Ar under the conditions when the ion velocity gained along its free run distance is comparable to the average thermal energy of atoms. Analytic solution of the Boltzmann kinetic equation is found for ions in their own gas for arbitrary tension of electric field in plasma when the dominating process is resonant charge exchange. The dependence of cross-section of resonant charge exchange on the relative velocity is accounted for. It is demonstrated that the ion velocity distribution function differs significantly from the Maxwell distribution and is defined by two parameters instead of just one. The results of computational and experimental data agree quite well, provided the spread function of measurement technique is taken into account. Key words: ion distribution function, flat probe, resonant charge exchange. How to cite this article: Mustafaev А.S., Soukhomlinov V.S. Ion distribution function in their own gas plasma. Zapiski Gornogo instituta. 6. Vol., p DOI.8454/PMI Introduction. Studying the ion velocity distribution function (IDF) is of particular interest for such modern applications as plasma nano-technologies, fine ion cleaning of surfaces, the technology of selective etching and producing reliefs by ion bombardment, nano-electronics of new generation (single electron transistors, spintronics, etc.) [, 6]. In conjunction with that experimental techniques of retrieving IDF in discharges of different type are of particular interest, specifically in the plasma of DC self-sustained discharge. We are not aware of any studies in which IDF would be experimentally measured in such discharge, so far. As for theoretical studies, despite the large number of such publications it is difficult to identify those among them in which the IDF in energy and motion directions in the plasma of selfsustained DC discharge would be calculated. Specificall ion drift was analyzed theoretically in a number of studies [, 6-9]. B.М.Smirnov [7] calculated the ion drift velocity in own gas plasma of inert gases in the strong field approximation on the assumption that the energy distribution of atoms has the form of delta-function. V.I.Perel [6] presented the results of calculating ion drift velocities in own gas plasma for inert gases, but the author did not provide any expression for IDF. Study [8] neglected all the processes except charge exchange to interpret the experimental data and calculate the time needed for an ion moving in its own gas would gain a velocity component along the electric field from v iz to v iz + dv iz. To the accuracy of a dimension factor that time describes the IDF in the projection of ion velocity on the direction of electric field. However it remains difficult to derive the IDF in total ion velocities from those results. When solving the problem of IDF in own gas, monograph [] neglected the process of generation of ions featuring the Maxwell velocity distribution due to charge exchange. In the result a Maxwell IDF was obtained for constant crosssection of charge exchange, its temperature defined by the electric field in the plasma. When developing an analytical theory to calculate IDF, the authors of [5] assumed, first, that the distribution of ions in the plane orthogonal to the electric field to follow Maxwell, and, second, that the cross-section of the resonant charge exchange is independent of the ion energ which is known to distort significantly the IDF. Studies [, ] develop a new approach to computing the matrix elements of the collision integral which is then applied to solving non-stationary Boltzmann equation by the momenta technique for ions in the conditions when the principal energy exchange process is the resonant charge exchange. Despite all that in strong fields when the ratio of thermal energy of atom to 864 Journal of Mining Institute. 6. Vol.. P

2 DOI.8454/PMI the ion energy gained along its free run distance is less than., using such an approach to retrieve stationary IDF is difficult. Study [4] compares the analytical solution of Boltzmann equation for ions in the BGK-model [3] with numerical solution for a constant cross-section of resonant charge exchange. It is demonstrated that the analytical solution yields an erroneous asymptotic for ion drift velocity in strong fields. Study [4] derived an analytical solution of the problem of IDF in own gas in strong field. Comparing computational results with the available experimental data on drift velocities in strong fields demonstrated their good agreement. Besides, the same study was the first to measure the energy IDF of ions Hg + in Hg vapor using the probe technique. Agreement was demonstrated between the experimental data and computational results. Technique for retrieving the ion distribution functions in gas discharge plasma. The current study offers a new experimental technique for retrieving the ion energy distribution function and motion directions for an arbitrary tension of electric field in plasma. The principal limitation on the applicability range for that technique is that the Debye layer immediately surrounding the probe is quite thin compared to the size of the probe itself. The technique uses the part of probe characteristic that corresponds to the positive potential of the probe with respect to the plasma potential. The largest experimental difficulty in measuring IDF with flat probe is the need to rotate it simultaneously with moving it along the axis of the discharge. To cope with that problem a flat rotating one-sided probe of tantalum foil 3 mcm thick shaped as a circle of.5 or.8 mm diameter was introduced into plasma via its side boundary. The probe had a tantalum wire conduit of. mm diameter welded to it. The probe was positioned on the axial line of gas discharge tube mounted on the tree-coordinate micrometer remote shifting system which used a bellows connection to provide the spatial positioning of the probe with an accuracy of. mm and its orientation with respect to the symmetry axis of the discharge within the range of (-8) angles at a 5 discrete step of accuracy not worse than ". n ia To measure Legendre coefficients F of IDF second derivatives were recorded of probe current I U, detected by the radio technique of twin modulation of probe potential [5]. It was for the first time that the complete ion energy distribution function was retrieved, including its first seven coefficients of expansion into the Legendre polynomials P n (cos( )), where is the angle between the vectors of ion velocity and electric field tension in plasma using the technique of flat one-sided probe [, 3] for He + in He and Ar + in Ar. The analytical theory was developed to describe IDF in own gas under the condition that the principal process involving ions in the plasma is resonant charge exchange. Note that the tension of electric field may be arbitrary while the cross-section of charge exchange is considered to depend on the relative velocity of ion and atom. IDF, f i is found in the form of analytical solution of Boltzmann equation: ee vi r ( fi ) v ( f ) Si ), () m where v i is the ion velocity; E is the tension of electric field; m is the mass of ion; Si ) na{ fa ) ( vr ) vr fin ) dvi fin ) ( vr ) vr fa ( va ) dva }; f a ( v i ) is the Maxwell function of atom velocity distribution; ( v r ) is the cross-section of charge exchange; v r is the module of relative velocity of ion and atom. The retrieved analytical solution of equation () differs from the Maxwell function and depends on two parameters, one of them defining the most probable ion velocit which is close to thermal, and the other the average ion velocity that depends on the tension of electric field in plasma: F(, ) f i (, ). () F(, ) dd 865

3 n F ia А.S.Mustafaev, V.S.Soukhomlinov ( ), эв.. n= n= n=3 n=4 n=5 n=6 Е-3.., ev DOI.8454/PMI n Calculations Experiment n F ia Fig.. The dependence on energy of the first seven coefficients in IDF n Ar + in Ar expansion into Legendre polynomials for the differentiating signal value =.5 V; Т = 45 K; Е/Р = 9 Torr; Р =. mm Hg ( ), эв.. Е-3.., ev n Calculations Experiment for where For ( ) Fig.. The dependence on energy of the first seven coefficients in IDF n Не + in He expansion into Legendre polynomials for the differentiating signal value =.5 V; Т = 6 K; Е/Р = Torr; Р =. mm Hg F (, ) exp exp x ( ) ( ) x exp y ( ) x x ( ) ( ) x exp y ( ) ( ) dy ( ) dy; x ( ) ( x) ( ) x exp y ( x) ( ) ( ) d x F (, ) exp x,5,5 y x ( ) exp y x ( ),5y x ( ) dy; y ( ) x v i ; cos; ( mna x x) ; ee,5,5 z x ( ) exp z x ( ),5z x ( ) ; ( ) dz y x ( ) V (,, ) i m x y ;. kt a 866 Journal of Mining Institute. 6. Vol.. P

4 DOI.8454/PMI F i (,), ev rad , ev Exact solution of Boltzmann equation IDF into polynomials Legendre for n = 6 calculations experiment..3 8, rad Fig.3. The angle dependence of IDF for the conditions given by Fig. and different ion energy The theory describes quite reliably a large array of available experimental data on drift velocities and reduced mobilities of atomic and molecular ions in own gas plasma and the data of numerical computations using the Monte-Carlo technique for the average energy of ions. Comparison with the experiment. Comparisons were run of the measured values and those computed using the newly developed theory for the complete ion distribution function and the first seven coefficients of the expansion into Legendre polynomials for Ar + in Ar (Fig.) and He + in He (Fig.) which demonstrated their good agreement. Possibility was demonstrated of retrieving the complete IDF from the measurements of the first seven coefficients in IDF expansion into the Legendre polynomials series (Fig.3) using the designed probe technique. Note that the range of ion energies in which such retrieval is conducted at a certain accuracy level is defined by the ratio of thermal energy of atoms to the energy gained by the ion along its free run distance and grows for higher values of that ratio. Conclusion. It is for the first time that the first seven coefficients of expansion into the Legendre polynomials of ion distribution functions in both energy and angles were measured for arbitrary tension of the electric field in plasma using the technique of flat one-sided probe. The experiment was run for He + in He and Ar + in Ar under the conditions when ion velocity gained along its free run distance is close to the average thermal energy of atoms. The kinetic Boltzmann equation is solved analytically for ions in their own gas under the conditions when the dominating process is the resonant charge exchange. The dependence of cross-section of resonant charge exchange on the relative velocity is accounted for. It is demonstrated that the form of ion velocity distribution function differs significantly from the Maxwell distribution and is defined by two parameters instead of one, the first of them characterizing the most probable ion velocity which is close to thermal velocity and the second the average ion velocity which depends on the tension of electric field in plasma. The results of computational and experimental data agree quite well. REFERENCES. Golant V.E., Zhilinsky A.P., Sakharov S.A. Fundamentals of Plasma Physics. Мoscow: Atomizdat. 977, p. 5. Lapshin V.F., Mustafaev A.S. Flat-sided probe method for the diagnosis of an anisotropic plasma. Zhurnal tekhnicheskoi fiziki Vol. 59, p Mustafaev A.S. The dynamics of electron beams in plasma. Zhurnal tekhnicheskoi fiziki.. Vol. 7, p Mustafaev A.S., Sukhomlinov V.S. Ion velocity distribution function in arbitrary electric field plasma. Zapiski Gornogo instituta. 6. Vol. 7, p

5 DOI.8454/PMI Mustafaev A.S., Soukhomlinov V.S., Ainov M.A. Experimental and theoretical determination of strongly anisotropic functions in the plasma of its own gas velocity distribution at high fields. Zhurnal tekhnicheskoi fiziki. 5.Vol. 85. Iss., p Perel V.I. Calculation of the drift velocity of the ions in their own gas. Zhurnal eksperimental'noi i teoreticheskoi fiziki Vol. 3, p Smirnov B.M. The mobility of ions in their own gas. Zhurnal tekhnicheskoi fiziki Vol. 36. N, p Fok V.A. On the motion of ions in the plasma. Zhurnal eksperimental'noi i teoreticheskoi fiziki Vol. 8, p Frish S., Kagan Yu.M. Spectroscopic study of the motion of the ions in the plasma. Zhurnal eksperimental'noi i teoreticheskoi fiziki Vol. 7, p Ender A.Ya., Ender I.A. The kinetics of ion in a neutral gas with a sharp turn on the electric field. Part. SEM model. Zhurnal tekhnicheskoi fiziki.. Vol. 8. Iss., p Ender A.Ya., Ender I.A. The kinetics of ion in a neutral gas with a sharp turn on the electric field. Part. Different intereaction models. Zhurnal tekhnicheskoi fiziki.. Vol. 8. Iss., p Abe H., Yoneda M., Fujiwara N. Developments of Plasma Etching Technology for Fabricating Semiconductor Devices. Jpn. J. Appl. Phys. 8. Vol. 47, p Bhatnagar P.L., Gross E.P., Krook M. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev Vol. 94, p Else D., Kompaneets R., Vladimirov S.V. On the reliability of the Bhatnagar-Gross-Krook collision model in weakly ionized plasmas. Physics of Plasmas. 9. Vol. 6, p Lampe M., Röcker T.B., Joyce G., Zhdanov S.K., Ivlev A.V., Morfill G.E. Ion distribution function in a plasma with uniform electric field. Physics of Plasmas.. Vol. 9, p Lieberman M.A. 7 th ICPR and 63 th GEC.. Paris. France. Bull. of the APS.. Vol. 55. N 7, p Authors: A.S.Mustafaev, Doctor of Physics and Mathematics, Professor, alexmustafaev@yandex.ru (Saint-Petersburg Mining Universit Saint-Petersburg, Russia), V.S.Soukhomlinov, Doctor of Physics and Mathematics, Associate Professor, primaivs@mail.ru (Saint-Petersburg State Universit Saint-Petersburg, Russia). The paper was accepted for publication on 6 October, Journal of Mining Institute. 6. Vol.. P