Nuclear Instruments and Methods in Physics Research B 85 (1994) 584-587 North-Holland
NIOMI 6
Beam Interactions with Materials 8 Atoms
Scattering of heavy ions in crystals V.A. Muralev * Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Squre 4, 125047, Moscow, Russian Federation
A physical model is constructed to describe the recapture of low energy heavy ions into axial channels in kinetic approach (rechanneling effect). This model includes the main essential physical processes.
1. Introduction
A beam of ions incident on a crystal target is divided into two parts: random and channeled [l]. The ions forming these beams are assumed to be slowed down independently [l]. The random part penetrates through the crystalline target as if through an amorphous medium and is slowed down by nuclear collisions and electronic interactions, while the energy of the channeled part decreases mainly due to electronic stopping [l]. Therefore the channeled ions penetrate deeper into the target than the non-channeled ones [2]. This result was interpreted as an easy passage of heavy ions along open low index axes or planes in a crystal. Presently the investigation in the field of channeling is being carried out in two main directions, namely channeling of light charged particles (electrons, positrons) and channeling of heavy ions (boron, phosphorus, arsenic, alpha-particles, protons and so on), with energies ranging from a few keV to hundreds of GeV. The problem is far from being simple, and it can be approached in various ways, namely analytical [l], kinetic approximation [3-81 and computer simulation (see e.g. refs. [9,10]). The analytical method can only treat some simple models, while in order to build more realistic ones, it is necessary to use the other two methods that give the possibility of comparing the theoretical results with experimental data. The consistent general approach to this problem leads to a Focker-Plank type two-dimensional kinetic equation for the distribution function F(t, E I, El depending on depth t, transverse E I and total E energies of channeled ions. The one-dimensional ki-
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netic approach to channeling was developed in refs. [3-71. The purpose of the present paper is to investigate a consistent physical model constructed in kinetic approach to describe the effect of the recapture (rechanneling effect) of heavy ions into a channel when the angle of incidence upon a target equals or exceeds the critical angle for entering a channel. This model is based on the two-dimensional Fokker-Planck kinetic equation, which is solved for low energy axially rechanneled heavy ions. In the present paper we have solved two equations. The first one for the channeled fraction of ions and the second one for the quasichanneled one. At the boundary of these regions we have joined the two solutions. It should be also noted that the capture of relativistic electrons into a channel when the initial angle of incidence upon a target equals or exceeds the critical angle has been studied earlier [9-121. However the channeling of negatively charged particles differs considerably from the channeling of positively charged heavy ions, described in the present paper (see e.g. refs. [g-12]). Concluding this section it should be noted that the effect of channeling may be used for important purposes, namely the study of anisotropic properties of thermal vibrations of atoms forming a crystal lattice the electron density in channels and phonon spectra, to develop a new non-destructive method to control the degree of crystal defects, to apply the blocking technique to measure nuclear reaction lifetimes, location of impurities, controlling effects persisting through the bending of beams in smoothly bent crystals [12-161, channeling radiation [17,18] and so on. Channeling plays a decisive role in the investigations of ion implantation and implantation technology. Up-to-date applications of channeling are widespread. Now channeling experiments have been extended to the new GeV region of proton energies.
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V.A. Muralev /Nucl. Instr. and Meth. in Phys. Res. B 85 (1994) 584-587
2. Physical model
Let us consider the basic principles of the model used. Since the channeled ions do not come close to atomic nuclei, the cross sections for various physical reactions change significantly in the channeling regime. It follows that at low transverse ion energies (E,) the processes involving collisions with small impact parameters p are suppressed (nuclear reactions, backward scattering and so on>. The transverse energy of channeled particles in a real crystal is not conserved. As it is known, the motion of channelling ions is unstable (dechanneling effect). There are several sources of dechanneling, even in an ideal crystal, leading to an increase of the transverse energy of a particle, namely, (a) scattering by electrons; (b) scattering by thermally vibrating lattice atoms. Multiple scattering in this case leads to the diffusion of particles in the transverse momentum space. Let 7A@ be the mean squared scattering angle, then the mean and the mean squared gains of transverse energy for particles moving at a definite distance from the string have been obtained using the general ideas of refs. [1,3-51. We have AE:=2E(E,
-U(r))@,
AE, =Em,
(gs=a;(-g),
1 AE; -(-) 2 At
= ($$[E1
-O(r)]),
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As the transverse energy growth 6E, is small we use the Focker-Plank kinetic equation to study the behaviour of channeled ions during multiple scattering in the channeling regime, and thermal atomic vibrations. So, the main effect in the kinetic approach is the evaluation of the particle distribution function F(E, , E, t) over transverse and kinetic energies with the penetration depth t. The evolution of this distribution function can be determined from the following Fokker-Planck kinetic equation.
+ A(
Q(ENEd;
- (g)S(EdF))
(2) where NE,), Q(E) are the kinetic coefficients; S(E,) is the accessible region of a channel for a particle with transverse energy E, ; the factor ( AE,/At )iO,, is determined by the variations of the particle transverse energy due to electronic stopping from Eq. (1). If in Eq. (2) one neglects the last two terms, a one-dimensional kinetic equation is obtained. Its solutions for some cases were considered in refs. [3-71. The factor ( AE,/A t)loss is missing in ref. [6]. The kinetic coefficients are given by
(3)
(1)
where the mean square gains have been obtained using the general relations (1). The initial distribution function is of the form dU -’
2Tri,, Fo(E,E,,O)=CS
0
m
is the electron mass, u is the ion velocity, wp is the plasma oscillation frequency, h Plank’s constant divided by 2~; brackets ( . . . > mean averaging over the accessible region in a channel [l]; N is the atomic density, (YI 1 is the close collision part of the energy loss. The model suggested in the present paper includes the physical processes as follows: the exchange of particles between channeled and random fractions of a beam, scattering of particles by electrons and by thermally vibrating atoms, and density effects. The model also contains the dependence of the distribution function on angle, kind of target, crystallographic direction in the target and the total energy change of particles. The general approach to the problem is based on the derivation of the distribution function F( t, E I , E) from a two-dimensional kinetic equation.
I
dr I I=r,n
x~~[(E-E,+AE)(E,-E)], (4) where E, is the initial energy of particles So = ari, 6 is the theta function, C is the normalizing constant, AE is the beam energy straggling; rin is the entering point, U is the transverse potential; ri = l/PNd, N is the atomic density, d is the atomic spacing of a string. The ion is influenced by several types of interactions leading to a change in its transverse energy, when it is travelling along the channel. It should be noted that the scattering by electrons (AE ./At >, and by thermally vibrating atoms (AE, /A t&, leads to an increase of the transverse energy of channeled particles. On the contrary the electronic stopping IX. IBA THEORY
VA. Muralev / Nucl. Instr. and Meth. in Phys. Res. B 85 (1994) 584-587
586
(A E I /A t)loss is focusing, and therefore it leads to the decrease of the transverse energy. Thus the total change of the ion transverse energy in this model is determined by the formula
(%)=(%),o,, +(-%), +(%),; (9 The first term in Eq. (5) is determined in Eq. (l), the Lindhard approximation [l] was used for the second term and the Ohtsuki one [19] for the third. When calculating variations in the total energy and its fluctuations, a uniform distribution of valence electrons was assumed, while the distribution of inner electrons was estimated according to the Thomas-Fermi atomic model. For the case E I > E I,, of the quasi-channeling particles the kinetic equation has the diffusion form
regime g(t)=i-K(~)=~~~F(E,E~,~)~E~E~,
(10) where K(t) is the dechanneling function, and $,, is the critical angle. Planning of experiments and experimental data processing require knowledge of the dechanneling length K~,~, and the dechanneling function K(t) or the function 4(t) = 1 - K(t), which is the fraction of channeled particles. By this sort of approach the flux of particles in the center of a channel takes the form @(t,
r,)
E m F(E,E,,t) = // o ucr,) S(E,)
dE dE,,
(11)
where r I is the transverse radius.
(6) 3. Discussion
and results
where
dS(r), (7)
dS(r), where S, = rri, S(r) = Tr2. For thin targets Eqs. (2) and (6) are reduced to
(8)
(9) At the boundary of the regions (E I = E icr> the solutions of Eqs. (2) and (6) or (8) and (9) are joined. Kinetic equations (2), (61, (8) and (9) are derived by the method of Bogolyubov [20]. It should be noted that these kinetic equations can also be obtained by the standard balance method or from the ChapmanKolmogorow equation [21]. Kinetic equation (2) is a partial differential equation of a parabolic type of the second order with variable coefficients. This equation is analytically unsolvable, thus numerical methods have to be used. The numerical solutions employed the second-order difference scheme of approximation of the kinetic equation [22]. Solving Eqs. (2) and (6) or Eqs. (8) and (9) for thin targets, we obtain the function describing the depth dependence of the number of particles in channeling
According to Lindhard [l], the channeling effect takes place when the initial angle of incidence of the particle upon a string of atoms &, is smaller than some critical channeling angle +I,,. In this case a considerable part of this beam is captured into channels regime in the surface at zero target depth. It is of great interest to study the evolution of the beam, when the initial angle of incidence of ions with respect to the string is equal to or larger than the critical angle +,,. When the incident particles are moving even at a small angle to a crystal axis, the deflection in close collisions with atoms are uncorrelated due to the thermal vibrations, but the deflections in more distant collisions are highly correlated (glancing collisions). These deflections may be described as an interaction with a continuum potential [l], which is usually obtained as an average of the crystal potential over the coordinates parallel to the axis (Lindhard continuum approach). In the present paper we used the Molier [23] approximation of the potential and the Lindhard continuum approach. When the ion beam is directed
ph, 0.2 0.4
0.6 0.6
tym Fig. 1. Fraction 4(t) of boron ions captured in a (111) Si single crystal with initial energy 50 keV, target temperature 25”C, angle of incidence I& = (I~, (I&, is the critical angle), as a function of depth t (pm).
KA. Muraleu / Nucl. Instr. and Meth. in Phys. Res. B 85 (1994) 584-587
along the channel axis, the major portion of ions is localized to the central part of the channel. Solving Eqs. (2) and (6) by numerical methods [22] for F(E, E,, t) we have from Eq. (10) the function 4(t) which discribes the depth dependence of the number of particles in channels. Fig. 1 shows the depth dependence of the fraction of boron ions that are captured into channels when the incident angle equals the Lindhard critical angle $,,. Fig. 1 also shows that in this case there are no particles present at all at zero depth in a channel. It is seen from Fig. 1, that due to multiple scattering in the crystal the trapped fraction increases slowly and reaches 17% of the initial beam at a depth 0.12 km. Therefore the results of the present paper have shown that a significant fraction of positively charged heavy ions can be captured into a channel due to their multiple scattering in the crystal even in the case when the initial incidence angle of the ions is equal to Lindhard critical angle. In this case ions may be absent at zero depth in a channel and will be trapped gradually in the channel only at larger depths in the course of beam evolution in the crystal. At the moment we have no experimental verification of the theoretical results presented in this paper.
4. Conclusion In conclusion, it should be noted that the present results show that the physical model suggested in this paper for a kinetic approach to the trapping of heavy ions in axial channels is efficient enough for theoretical studing of the recapturing effect. It would be interesting to make experiments on recapture of heavy ions in to the channeling regime when their initial angles of incidence on a crystal are larger than the critical angle. In these experiments it is necessary to use a thick crystal.
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References
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IX. IBA THEORY