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On the envelopes of ion trajectories in planar channeling
Volume 74A, number 1,2
PHYSICS LETTERS
29 October 1979
ON THE ENVELOPES OF ION TRAJECTORIES IN PLANAR CHANNELING S.G. GAONKAR’ and A.G. WAGH NuclearPhysics Division, Bhabha Atomic Research Centre, Bombay 400 085, India Received 18 June 1979
The envelopes of ion trajectories in a continuum potential are identified with the loci ofpoints in the planar channel where the ion flux is infinite. Their evolution with depth in the crystal is discussed.
It is well known that in planar channeling large deviations from statistical equilibrium occur in the first few thousand angstroms of the crystal, which is the depth region of interest in most channeling measurements. Thus, Monte Carlo simulations [1] as well as analytical [2] and numerical [3,41 calculations based on the continuum model yield periodic variations with depth in the ion flux at a fixed transverse
where y is the transverse coordinate measured from the centre of the channel at depth x, dy/dx = tan ~i sin ~,Li and V(y) is the continuum potential due to the two adjacent planes forming the planar channel. Since the assumption made in eq. (1) ignores scattering of incident ions by the nuclei and electrons in the crystal lattice, eq. (3) represents smooth trajectories periodic in x. If E1 ~ V(d/2), d being the planar
coordinate in the planar channel. Such variations at the substitutional site manifest themselves as planar oscillations [5]. In this letter, we discuss, within the continuum model, the significance of envelopes of ion trajectories and point out their importance in the study of the preequilibrium flux distribution in the planar channel. The conservation of the transverse energy, E1, of an ion incident with energy E, at an angle i~i~to the plane, viz., 2 + V(y)=E E(dy/dx) 1=Eii~ + V(y0)const., (1)
spacing, the trajectory has an amplitude A = V~(E1) and a wavelength X(A) which is obtained by integrating eq. (2) from y =0 to A. Fig. 1 shows the variation of X/4 with A for He~
yields the differential equation of motion, 2 = [E (dy/dx) 1 V(y)]/E, -
=
1
Bombay University Research Fellow,
94
(
100)
2~S 3~0 3.5
~0
____________________________
b
0, gives the trajectory ex-
yy(x;y0,~Ji0),
~‘1’~~2~C)
1~0 LS
(2)
which, when solved with the boundary conditions, = ~o andy =Yo at x pressedas
~‘
—
______________________
~O.O
(3)
1.0
20
3~0 ‘L.O
~O
E1=V(R)eV I ~10)
6~0
Fig. 1. The quarter wavelength, ?~/4(scaled to ‘~J~), and the transverse energy, E1, as functions of the amplitude, A, of helium ion trajectories in the (110) planar channel ofsilicon.
Volume 74A, number 1,2
PHYSICS LETTERS
29 October 1979
potential is nearly harmonic, the wavelength is relatively insensitive to variations in the amplitude, but exhibits a strong amplitude dependence near A = d/2. For an ion beam incident at angle &, , the ion flux, @,normalised to its value at the crystal surface can be
ions in the (110) channel of silicon and Lindhard’s planar pqtential [6] used in the calculations. The quarter wavelengths are scaled to fi so as to represent the results for all incident energies. A/4 decreases monotonically with A. Near the channel centre, where the
c-4
--d
L _+ > co f c ”
7
o*o
Od2
0.4
0.6
0.8
1.0
1.2
1,4
la6
1.8
2,O
222
2.4
GEPT’-i/~,~[ FIU/,~J @I 0
?A
2.e
! * 1 irj
3.0
3.X
3.4
3.6
3.0
2.2
3.k
3.6
!
.
0. 7 0.0
0.Q
0.4
0.6
ore
1,o
1.2
1.4
1.6
1.8
2.0
2,2
2-4
DEFT!-l/fiI FIO/h%%I (d
2.6
2.6
I.10"
I
Fig. 2. Computer plots of helium ion trajectories in the (110) silicon channel. (a) J, o = 0, (b) $ e = $p/2, (c) J/,, = I) p, where JI p = (22, Z,e2NdCa/E,‘~2. 95
Volume 74A, number 1,2
PHYSICS LETTERS
written as
29 October 1979
tions, progressively approaching its value in statistical
CT(x, y, ‘/‘~)=
E
(4) where the summation extends over all y~subject to y(x;y~, 11’o) =y. Eq. (4) is arrived at by assuming a uniform flux across the planar channel at the surface (x = 0). To the experimenter, CT signifies the yield at the angle 111(3, normalised to its value at random incidence, of any close interaction between the ion and an atom occupying a sitey at the depthx. It is evident from eq. (4) that the flux is infinite, if
equilibrium. At sufficiently large depths (where this simplistic model may no longer be valid), the envelope very nearly passes through the crests and troughs of all trajectories, this occurring at smaller depths for larger ‘I’o and larger lyl, with the exception of IyI Amin. Then, for Iy I
8y(x; y0, 1J10)/ay0
branches, the nth pair very closely satisfying the re-
~0)y
0
=
0
.
= Y’o’
—
‘
(5)
Eq. (3) with condition (5) imposed on it represents the envelope of the11’s. whole family of trajectories with angle of incidence Hence at each point on the envelope, the flux cI~is infinite. In practice, factors such as impurity thermal vibrations and angular divergence of the incident beam smear out the infinities into flux maxima. Figs. 2a,b and c are computer plots of He’~ion trajectories in the (110) silicon channel with ‘P~= 0, 2Nd X~Pp/2 Ca/E)1/2 symbols having meaning. and ~ the respectively, wheretheir ill1, =usual (2Z1Z2e The depths are again scaled to V~In fig. 2a, only the trajectories with y~~ 0 are shown since the remaining ones can be obtained by reflecting the plot about y = 0. In figs. 2b and c, the trajectories with E 1> V(d/2) are omitted, for the sake of clarity, once they reach the atomic plane. These plots afford a pictorial visualisation of the flux variation in the planar channel. The envelopes of the trajectories and the tendency of ions to get focussed along the envelopes are clearly visible. The envelope in each plot comprises an infinite number of branches originating very close to the crests and troughs of the trajectories with amplitude d/2 (except in the vicinity ,
of the crystal surface, where no envelope exists). The envelope has zero slope alternately at y = A mm and Amin at depthsx~ = ~(2n V(0)). — l)X(Amin), = 1,2, that 1 (E11i~+ It may be71 noted where V_ site where flux peaking occurs under Iy I = ~Aminis=the statistical equilibrium. The phase difference between adjacent trajectories accumulates with penetration m to the crystal due to the variation of A withA (fig. ~ Hence the length of the branches increases with depth and the flux at any site ly I undergoes damped oscilla96
y~~v(x) ~y(x; 0, 11iç~), Iy I ~ (6a) whereas at larger ~ I, the envelope consists of pairs of ,
1YA)= xl(yA) + ~(n l)X(y,~) Xfl1((—lY~ +1 1 x,~ 2((—l)~ YA) = xl(yA) + ~YA) YA = I A —
~.
(6b)
mm
where x1 (YA) represents the smallest depth where the trajectory with amplitude y~reaches its crest. The median of the branches, x,~ = (x~1+ x,~ in eqs. (6b) hasnth thepair sameofshape as the X/4 versus A 2)/2, curve in fig. 1. Thus, at each site Iy I, a pair of closely spaced flux max~’na(just one maximum at y = 0) appears with a periodicity which tends, at large depths, to X(Iy 1)/2, if ly I ~ ~ and to X(Amin)/2 for smaller Iy I. As can be judged from the intensity variation along any branch of an envelope in fig. 2, the flux maximum at y I = Amin is the strongest, its strength decreasing with Iy I more rapidly on the larger ly I side. The flux distribution, F(x, y, 11bo), of helium ions in the (110) silicon channel, computed numerically [7] within the continuum model, reflects all the features of the envelope of trajectories incident at 11’o. References [1] J.H. Phys.Rad. Rev.Eff. 3(1971)1527. [2] M.A.Barrett, Kumakhov, 15 (1972) 85. [3] F. Abel, G. Amsel, M. Bruneaux, C. Cohen and A. L’Hoir, Phys. Rev. B13 (1976) 993. [41 J.A. Ellison, Phys. Rev. B12 (1975) 4771. [5] A.G. Wagh and J.S. Williams, Phys. Lett. 65A (1978) 175. [6] J. Lindhard, K. Dan. Vidensk, Selsk. Mat. Fys. Medd. 34 (1965) no. 14. [7] S.G. Gaonkar and A.G. Wagh, to be published.
PHYSICS LETTERS
29 October 1979
ON THE ENVELOPES OF ION TRAJECTORIES IN PLANAR CHANNELING S.G. GAONKAR’ and A.G. WAGH NuclearPhysics Division, Bhabha Atomic Research Centre, Bombay 400 085, India Received 18 June 1979
The envelopes of ion trajectories in a continuum potential are identified with the loci ofpoints in the planar channel where the ion flux is infinite. Their evolution with depth in the crystal is discussed.
It is well known that in planar channeling large deviations from statistical equilibrium occur in the first few thousand angstroms of the crystal, which is the depth region of interest in most channeling measurements. Thus, Monte Carlo simulations [1] as well as analytical [2] and numerical [3,41 calculations based on the continuum model yield periodic variations with depth in the ion flux at a fixed transverse
where y is the transverse coordinate measured from the centre of the channel at depth x, dy/dx = tan ~i sin ~,Li and V(y) is the continuum potential due to the two adjacent planes forming the planar channel. Since the assumption made in eq. (1) ignores scattering of incident ions by the nuclei and electrons in the crystal lattice, eq. (3) represents smooth trajectories periodic in x. If E1 ~ V(d/2), d being the planar
coordinate in the planar channel. Such variations at the substitutional site manifest themselves as planar oscillations [5]. In this letter, we discuss, within the continuum model, the significance of envelopes of ion trajectories and point out their importance in the study of the preequilibrium flux distribution in the planar channel. The conservation of the transverse energy, E1, of an ion incident with energy E, at an angle i~i~to the plane, viz., 2 + V(y)=E E(dy/dx) 1=Eii~ + V(y0)const., (1)
spacing, the trajectory has an amplitude A = V~(E1) and a wavelength X(A) which is obtained by integrating eq. (2) from y =0 to A. Fig. 1 shows the variation of X/4 with A for He~
yields the differential equation of motion, 2 = [E (dy/dx) 1 V(y)]/E, -
=
1
Bombay University Research Fellow,
94
(
100)
2~S 3~0 3.5
~0
____________________________
b
0, gives the trajectory ex-
yy(x;y0,~Ji0),
~‘1’~~2~C)
1~0 LS
(2)
which, when solved with the boundary conditions, = ~o andy =Yo at x pressedas
~‘
—
______________________
~O.O
(3)
1.0
20
3~0 ‘L.O
~O
E1=V(R)eV I ~10)
6~0
Fig. 1. The quarter wavelength, ?~/4(scaled to ‘~J~), and the transverse energy, E1, as functions of the amplitude, A, of helium ion trajectories in the (110) planar channel ofsilicon.
Volume 74A, number 1,2
PHYSICS LETTERS
29 October 1979
potential is nearly harmonic, the wavelength is relatively insensitive to variations in the amplitude, but exhibits a strong amplitude dependence near A = d/2. For an ion beam incident at angle &, , the ion flux, @,normalised to its value at the crystal surface can be
ions in the (110) channel of silicon and Lindhard’s planar pqtential [6] used in the calculations. The quarter wavelengths are scaled to fi so as to represent the results for all incident energies. A/4 decreases monotonically with A. Near the channel centre, where the
c-4
--d
L _+ > co f c ”
7
o*o
Od2
0.4
0.6
0.8
1.0
1.2
1,4
la6
1.8
2,O
222
2.4
GEPT’-i/~,~[ FIU/,~J @I 0
?A
2.e
! * 1 irj
3.0
3.X
3.4
3.6
3.0
2.2
3.k
3.6
!
.
0. 7 0.0
0.Q
0.4
0.6
ore
1,o
1.2
1.4
1.6
1.8
2.0
2,2
2-4
DEFT!-l/fiI FIO/h%%I (d
2.6
2.6
I.10"
I
Fig. 2. Computer plots of helium ion trajectories in the (110) silicon channel. (a) J, o = 0, (b) $ e = $p/2, (c) J/,, = I) p, where JI p = (22, Z,e2NdCa/E,‘~2. 95
Volume 74A, number 1,2
PHYSICS LETTERS
written as
29 October 1979
tions, progressively approaching its value in statistical
CT(x, y, ‘/‘~)=
E
(4) where the summation extends over all y~subject to y(x;y~, 11’o) =y. Eq. (4) is arrived at by assuming a uniform flux across the planar channel at the surface (x = 0). To the experimenter, CT signifies the yield at the angle 111(3, normalised to its value at random incidence, of any close interaction between the ion and an atom occupying a sitey at the depthx. It is evident from eq. (4) that the flux is infinite, if
equilibrium. At sufficiently large depths (where this simplistic model may no longer be valid), the envelope very nearly passes through the crests and troughs of all trajectories, this occurring at smaller depths for larger ‘I’o and larger lyl, with the exception of IyI Amin. Then, for Iy I
8y(x; y0, 1J10)/ay0
branches, the nth pair very closely satisfying the re-
~0)y
0
=
0
.
= Y’o’
—
‘
(5)
Eq. (3) with condition (5) imposed on it represents the envelope of the11’s. whole family of trajectories with angle of incidence Hence at each point on the envelope, the flux cI~is infinite. In practice, factors such as impurity thermal vibrations and angular divergence of the incident beam smear out the infinities into flux maxima. Figs. 2a,b and c are computer plots of He’~ion trajectories in the (110) silicon channel with ‘P~= 0, 2Nd X~Pp/2 Ca/E)1/2 symbols having meaning. and ~ the respectively, wheretheir ill1, =usual (2Z1Z2e The depths are again scaled to V~In fig. 2a, only the trajectories with y~~ 0 are shown since the remaining ones can be obtained by reflecting the plot about y = 0. In figs. 2b and c, the trajectories with E 1> V(d/2) are omitted, for the sake of clarity, once they reach the atomic plane. These plots afford a pictorial visualisation of the flux variation in the planar channel. The envelopes of the trajectories and the tendency of ions to get focussed along the envelopes are clearly visible. The envelope in each plot comprises an infinite number of branches originating very close to the crests and troughs of the trajectories with amplitude d/2 (except in the vicinity ,
of the crystal surface, where no envelope exists). The envelope has zero slope alternately at y = A mm and Amin at depthsx~ = ~(2n V(0)). — l)X(Amin), = 1,2, that 1 (E11i~+ It may be71 noted where V_ site where flux peaking occurs under Iy I = ~Aminis=the statistical equilibrium. The phase difference between adjacent trajectories accumulates with penetration m to the crystal due to the variation of A withA (fig. ~ Hence the length of the branches increases with depth and the flux at any site ly I undergoes damped oscilla96
y~~v(x) ~y(x; 0, 11iç~), Iy I ~ (6a) whereas at larger ~ I, the envelope consists of pairs of ,
1YA)= xl(yA) + ~(n l)X(y,~) Xfl1((—lY~ +1 1 x,~ 2((—l)~ YA) = xl(yA) + ~YA) YA = I A —
~.
(6b)
mm
where x1 (YA) represents the smallest depth where the trajectory with amplitude y~reaches its crest. The median of the branches, x,~ = (x~1+ x,~ in eqs. (6b) hasnth thepair sameofshape as the X/4 versus A 2)/2, curve in fig. 1. Thus, at each site Iy I, a pair of closely spaced flux max~’na(just one maximum at y = 0) appears with a periodicity which tends, at large depths, to X(Iy 1)/2, if ly I ~ ~ and to X(Amin)/2 for smaller Iy I. As can be judged from the intensity variation along any branch of an envelope in fig. 2, the flux maximum at y I = Amin is the strongest, its strength decreasing with Iy I more rapidly on the larger ly I side. The flux distribution, F(x, y, 11bo), of helium ions in the (110) silicon channel, computed numerically [7] within the continuum model, reflects all the features of the envelope of trajectories incident at 11’o. References [1] J.H. Phys.Rad. Rev.Eff. 3(1971)1527. [2] M.A.Barrett, Kumakhov, 15 (1972) 85. [3] F. Abel, G. Amsel, M. Bruneaux, C. Cohen and A. L’Hoir, Phys. Rev. B13 (1976) 993. [41 J.A. Ellison, Phys. Rev. B12 (1975) 4771. [5] A.G. Wagh and J.S. Williams, Phys. Lett. 65A (1978) 175. [6] J. Lindhard, K. Dan. Vidensk, Selsk. Mat. Fys. Medd. 34 (1965) no. 14. [7] S.G. Gaonkar and A.G. Wagh, to be published.