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The energy spectra of the secondary ion emission for transition metals of the fourth period
Vacuum/volume 33/number 3/pages 159 to 163/1983 Printed in Great Britain
0042-207X/83/030159-05503.00/0 Pergamon Press Ltd
The energy spectra of the secondary ion emission for transition metals of the fourth period V I Veksler,
Tashkent State University, Tashkent 700095, USSR
received 3 June 1982
The energy spectra of the secondary ion emission (SlE) for transition metals of the fourth period are calculated under the assumption that SIE is caused by the joint action of electron-exchange and kinetic ion emission (KIE) mechanisms. Some improvement of the KIE theory is carried out for the cases of high (of the order of tens of keV) and low (less than 2 keV) energies E o of bombarding ions. It is found that the KIE plays an essential role in SIE, starting from the E o value of the order of tens of keV. It is shown that the KIE energy spectra (1) are broader than the sputtering energy spectra by an order of magnitude, (2) differ essentially in both cases indicated, (3) show a peak (at high Eo in accordance with an experiment) or a break (at low Eo) near to the secondary ion energy corresponding to the KIE threshold.
hi!
1. Introduction
fOs
In a number of papers dedicated to secondary ion emission investigation 1-3, it was found that energy spectra have an unusual form for transition metals of the fourth period: in them, along with the ordinary peaks at an energy W of secondary ions of about 10 eV, peaks I or irregularities2.a for W of the order hundreds of eV were also discovered (Figure 1). The position of the high energy singularity on the spectrum for copper 2 is in good agreement with the peak position on the analogous spectrum in ref 1. But in ref 2 this singularity does not turn into a peak, since here the condition for surface purity was not so good as in ref 1 according to purity criteria~6, and it is known ~ that change for the worse of those conditions leads to a strong increase in the low energy peak. In ref 1, the high energy peak formation was interpreted in the framework of the electron-exchange mechanism, but the following dependence of the secondary ion formation probability P on its velocity v was used: Pocexp(-Vo/V), which is not in accordance with experiment s . The introduction of the correct dependence s leads to the disappearance of the corresponding peak in calculated energy spectra. The opinion that the origin of the singularities indicated in the spectra is related to the action of two secondary ion emission mechanisms has been already expressed 9. The electron-exchange mechanism is the origin of the low energy peak, whilst the secondary ion energy spectrum n~'"is connected with the sputtered atom energy spectrum n~ by the following equation independent of the bombarding ion energy Eo 5.,0
n,'-~h-~2[(-~)I/2-1](MecpW)'/2 d2(1
+cWt/2)na,
(1)
V+
fO
fb'
5'ooE,,v
Figure !. The experimental energy spectra for 3d transition metal secondary ions (Eo=43 keV) ~.
where m and M are the electron and ion mass respectively, V~and d are the ionization potential and diameter of the sputtered atom, e~b is the target work function, c~0.3 (eV) -1/2 s, h is a fitting parameter, where h,~l s'6. The validity of equation (1) is confirmed by experiment s for E 0 < 2 keV. According to supposition 9, the high energy singularity may be due to kinetic ion emission (KIE). In the corresponding Joyes' theory it is assumed 11-1a that, under violent interactions of the bombarding ions with the lattice atoms and the lattice atoms between themselves, a vacancy in one of the internal shells of some atoms is formed, and the electron is transferred into the metal conduction band. The corresponding atom can escape from the metal in an excited state, if the lifetime of the vacancy 11 is sufficiently large. It can be ionized outside the metal by means of an inneratomic Auger effect if the necessary conditions are satisfied. 159
V / Veksler: Secondary ion emission for
transition metals
Unfortunately, in the above-mentioned theory a number of unjustified assumptions have been made (criticism of it is given in refs 5 and 6). This prevents direct experimental confirmation of the theory. In the present work a more correct analysis of the shape of the corresponding energy spectrum n~' of secondary ions in the framework of the KIE notion is given. Then a comparison with the experimental t-a data of the energy distributions n~ obtained by means of summation of the n* and n~* spectra is presented.
3. Qualitative consideration We shall carry out first a qualitative consideration of the form of the expected energy spectrum which need not depend on the specific method of calculation of equation (2). For this purpose we alter the integration order in formulas (2) and (4): t
d("w,(~", ~)
n*((, E 0 ) = Q * ( ( , ) exp(-
t'~/~) +
d~"ws(~", ~)
2. Initial equations
¢
In the above-mentioned analysis the initial equation of the theory describing the excited atom energy spectrum under the surface at the 'metal-vacuum' interface 13 is used:
exp(-
;l
n.%, Eo)=-
t/./z),
(2)
where ( = E/E o and E is the energy of the atoms indicated. Here it is proposed that ¢ = EriE o =-~, where E~ is the energy threshold for the vacancy formation, instead of using ~ = ( as in ref 13. It is also taken into account here that the secondary ions can be initiated by bombarding ions slowing down in the lattice. In accordance with the above-mentioned assumptions nk((' ) are considered to be energy spectra of the particles moving in the solid (k = s is related to the primary ions scattered by a lattice, and k = d to the lattice displaced atoms), 0((', (") is a probability of the atom displacement with energy E" = ("E o by the particle with energy E ' = ('Eo, w((') is the probability of a vacancy formation in the former, w,((", () is the probability for slowing-down of excited atoms with energy E" to energy E, t~. is the mean time for such retardation s'6
t'~=2-a/4(1-2-~/2)-~M~/21[E-~/2-(E")-1/2],
(3)
where M and l are the mass and the free path length of atoms respectively. Equation (2) essentially is related to the KIE, provided that the frequently made assumption of the isotropy of the atomic collision cascade not being dependent on the distance from the surface (see, for example, 14-16) is valid. The threshold E, significantly exceeds the energy E~ required for vacancy formation (e.g., for the 3p vacancy in vanadium, E~ = 37.8 eV ~1 but E, found by the method ofrefs 6 and 18 is equal to ~ 200 eV). Therefore, the change in E' due to inelastic losses is subsequently neglected. Moreover, the character of slowing down for atoms in the excited and ground states is assumed to be identical. Under these conditions the energy spectrum of the latter before escaping from the metal is
n°(~)
(~ < ~ < ~,),
(6)
(~,<_~<_1),
(7)
d~"ws(~",~)
exp(-t~,/~)Q*(~")
n~,((, Eo)= f : d('nk((')w((') f;' d(" g((', (") w,(;", O e x p ( -
t~,/~)Q*(~")
n.°(~) = where
f:
d~"w~(~", OQ(~"),
Q*(O = Q(~)=
(8)
o(~', ;")[ns(~')+ nd(~')]w(U) d~',
(9)
0(~', ~")[ns(~')+nd(~')] d ( '
(10)
represent the number of excited atoms and atoms in the ground state, respectively, which are formed within the solid with energy E ('source'). We shall analyse the spectra shape for a fixed value of E o. (1) At ~ < ( < ~, the whole integration region of equation (6) is divided into two parts (Figure 2). In the region 1 (~ < ~"<_~, < (') the source Q* remains constant, independent of ~, while the mean value of Q decreases as ~ increases. Moreover, the mean value of the retained excitations, determined by the exponential, rises with increase in ~. Thus, the part of the spectrum considered (the first integral of equation (6)) should decrease with an increase in much slower than the spectrum n°((). It is not beyond possibility that this part of the spectrum can even increase since the function ws((", ~) is unknown a priori. In the region 2 ((< ~, < ~"< ~') the source O* again does not depend on ~ and the exponential rises
~
t
t
2
t
.f ~ d{'[n,(;')+ nd(~')]
:::::
I
I
II
I
r / / / /
f d("g(;', ;")w~(;",
;).
:////
(4)
One can obtain the energy spectrum of secondary ions escaping along the normal to the surface, if, according to ref 14 the influence of the surface potential barrier of height 2 = (aE o is taken into account
n*((--~, E o ) = (1 - (x/() End((, Eo)+n*~a(~, Eo)]. 160
(5)
j
I' i i
I I
I I tf
Figure 2. Integration region of equation (2).
V I Veksler: Secondary ion emission for transition metals
with an increase in (. Since here the integration over the (" interval does not depend on ~, it is more likely that this part of the spectrum (the second integral of equation (6)) will rise with an increase in (. So, in the case considered, the total spectrum n*(~, Eo) must fall much more slowly than n°(() when ~ increases and it can even rise. (2) At (, < ( < 1 the integration region is indivisible (Figure 2). Here the mean value of Q* will decrease more slowly than Q because of the increase in W, and the exponential will increase, with a ~ growth. Thus, for all the values of ~ the spectrum n* must fall o Moreover, at ( ~ ~, the spectra must much more slowly than n°. show a break owing to a decrease in Q*, when ~ increases, starting at ~ ~ ~,, i.e. in the region ( > (, the spectrum must decrease faster than in the ( < (, region. At ~ ~ ~, the spectrum will show a peak if, at ~ < ~,, n* rises with an increase in (. Now we shall assume that E0 is reduced. Then because of the decrease in Q*(~,) the significance of n* corresponding to integration region 1, must be strongly reduced for (~<(<~,, but the number of retained excitations remain invariable. For region 2, Q* also diminishes, but the mean number of retained excitations will be essentially increased. That is why the relative contribution of this part in the total spectrum will also be increased. Therefore, one can expect that the peak (singularity) at ( ~ ~, will become sharper, if E o decreases.
4.1. Case of a high energy. First, a calculation for the case when the bombarding ion energy is of the order of some tens of keV will be carried out. In this case, as is known s'14-16, the assumption that the cascade of atomic collisions is isotropic works rather well, and the sputtered atom spectrum can be presented in the form: n°(O = A(1 - [ f f ~ ) ( ~ - 2 _
1).
(15)
The relation .4(~ -2 - 1) =
dr"w ~r", ¢ ) Q ( ( " ) . ~..
(16)
can be used for the determination of w~((", ~), and the normalizing constant is found from the condition
S=A
( 1 - ~ f f ~ ) ( ( - 2 - 1) d~,
(17)
so that
A_~2(aS.
(18)
It is easily seen that equation (16) is satisfied at
ws(~", ~)= 2~'G(O/L~", 4. Specific calculations
where
We shall carry out specific calculations to illustrate the conclusion drawn. In the theory of refs 12 and 13, an inaccurate assumption that the slowing down of the excited atom and the vacancy annihilation processes occur independently of each other and a strongly underestimated value of E, (cf. refs 5 and 6) are used. Both these assumptions are ruled out in the present work. It is known that, for an isotropic cascade of the displaced atoms, the energy spectrum n~((') is determined 6"~4 by the expression
rid(~') _~2(,lS/((') 2,
( 11 )
where S is the sputtering coefficient. A question about the shape of the energy spectrum n~((') of primary ions scattered by a lattice is somewhat more complicated. The spectrum is described by a function monotonically increasing with E, if the interaction particle potential can be approximated by a dependence which is inversely proportional to the square of a distance ~9. In the diffusional approximation, the spectrum has a cupola form, but its maximum is placed sufficiently near to E 0 2o. Therefore, the following approximation will be used: n,(~') = 2 K ~ ' ( ( ' _ < 1 ),
(12)
since calculations show that the spectrum n~ form is weakly sensitive to the detailed shape of the spectrum n~(('). In equation (12), K, is the 'isotropic part' of the scattering coefficient and is later regarded as a parameter. The value 0 is determined by the equation g((', ~") d ( " = d ( " / ~ ' ,
(13)
The expression for w will be approximated by the formula (cf. ref 18) w((') = Wo[1 -- (£,/02], with w0 ~ 0.1.
(19)
(14)
G(() = (1 - ~2) [½ (1 - (2) _ (fl _ 1 )(2 In _ fl~.2 ( i _ ( ) ] -
i.
(20)
In this case, the spectrum n* (~, Eo) is calculated analytically: n* (~, Eo)=4Wo(a SyG(() {~l,[Ei(x).-;Ei(r,)]
+ n(x,, Z, x , ) - n ( x , Z, x t ) } e x p ( - g ) ( ~ ; . < ~ <~,),
(21)
n*a (~, Eo)=4Wo(~SyG(()[H(K, Z, •,) -n(z,
X, x , ) ] e x p ( - x ) (¢, < ~ < 1),
where
H(x, Z, Kt)=aEr- 2(x+ 1)exp(r) - 2 r 7 4R 1(r)] + Z- 4[ R 3(to) - (2x,4) - I g T ( r ) ] + (b -a)Ei(x), R,(r)=r!exp(~c) n=O Z ( - - l ) "+1 K=~( ~- 1/2 K, = ~
1
~" Kn'
1/2 ~ :
(F./Eo)I/2,
n,=l-(1 + ~,)2/2~2 + fl] (1 _ ~,)2,
a=flZ2/2,
b = f l ( l + ( 2 ) - (I - (~/2),
fl = 2K ff ( AS, e = 2 - 3 / 2 ( 1 - 2 - 1 / 2 ) - 2M/2.t.- 2'
Ei is the sum of the integral hyperbolic sine and cosine functions. In the case considered 7 = 1. The quantity I does not greatly exceed the lattice constant, the value of -c~ 10- ~4 s 11. Therefore, for the transition metals of the fourth period e ~ 4 keV. 161
V I Veksler: Secondary ion emission for transition metals
The total energy spectrum of secondary ions can be found with a formula __
ee
*
ni--nl +ni,
(22)
3 104
and the yield of secondary ions is found by means of integration
Ki =
(n~+n *) d~.
(23)
In Figure 3 the computed energy spectra for titanium, vanadium, nickel, copper and tin are given. It was assumed that in atoms of corresponding metals a 3p vacancy is formed by KIE.
t03
I
5
~0
20
50
t00
200
500
f000
F,eV 10~
Figure 4, The influence of e variation on the computed energy spectra (relative units) form for Ni + (E o =40 keV). e, keV (h): 1~4 (0.03); 2--0.4 (0.03); 3 ~ . 4 (0.08).
computed at e.=0.4 and 4 keV with a constant h are given. The spectra are seen to differ only slightly (curves 1 and 2). Better conformity of spectra in both cases can be obtained if the parameter h is varied (of. curves 1 and 3). The computed yield values (for example, for nickel at E 0 = 40 keV, K~ ~ 1.10-2) are believed to be quite reasonable. Thus, there is a real justification for the assumption that the metal secondary ion emission at a sufficiently large E 0 is caused by the joint action 9 of electron-exchange and KIE mechanisms.
t04
103
t
5
10
20
50
f00
200
500
f000
E.eV Figure 3. The computed energy spectra (relative units, Eo = 40 keV) of the secondary ions (h): I--V* (0.03); 2--Ti + (0.2); 3--Ni ÷ (0.03); 4-Cu+(0.005); 5--Zn+(0) (fl= 10). The spectrum 1' is computed for V+ at fl= 1000, h =0.03. The value of the parameter E,, influencing the spectra width, by the method of refs 6 and 18 was calculated. In this method it is assumed that, for head-on collisions of an atom, the kinetic energy of relative motion converts into the repulsion potential energy and into the energy of excitation of electrons. The latter was found by the molecular orbital method and the effective charge on the nucleus Ze, was calculated according to the semi-empirical Siater rule 21. It was found that E,,~200 eV for titanium and vanadium. Unfortunately, for the rest of metals, a major uncertainty in Zorf value arises because of the large reciprocal penetration of the electron shells of the colliding particles, which does not permit a sufficiently reliable determination of the E t value. Therefore, for them, E~ was taken to coincide with the corresponding value for tantalum (E~= 400 eVl a). A comparison ofenergy spectra found by experiment (Figure 1) and calculated by the method described (Figure 3) shows a satisfactory degree of accordance, and the value of the parameter h used does indeed satisfy the condition h.~ I. It should be noted that the other parameters do not much influence the form of the calculated spectrum. In Figure 3, spectra for vanadium computed at fl= 10 and 1000 ( K ~ 0 . 0 1 and 1, respectively) are given. The form of the spectra in both cases is seen to differ very slightly. In Figure 4, spectra for nickel 162
4.2. C a s e o f a law energy. At E o < 2 keV the assumption ofisotropy in the atomic collisions cascade is violated ~'6'22. In this case the function nk((')9((', (") × w~((", () in equation (2) must be regarded as the corresponding characteristics averaged over all angles. The accuracy of calculations is, naturally, reduced in this case. It is known 8 that the spectrum of atoms, sputtered along the normal to the surface for 3d transition metals, is better described by the formula no(() = 6(~S(1 - ~z/~) (~-
3 __
1)
(24)
for the case considered. Therefore, the expression 6~2S(~-
3 __ 1 ) =
f'
d("ws(~", ~)Q(~"),
(25)
permitting evaluation of w~, can be considered in this case in a certain degree as correcting for the deviation of real conditions from the isotropic cascade ones. Here, w~ is again determined by formula (19), but G ( ( ) = (1 - ( 3 ) ( - 111(1 -(fl-
_
(2)
1)( 2 In (--fl(2(1 _ ( ) ] - l
(26)
The spectrum n*((, Eo)is again determined by formula (21). In the case considered in formulas (19) and (21), the substitution 7=3(a
(27)
is made. In Figure 5, the spectra n*(E, Eo) found with formulas (5) and (21) for the two parameter fl values, along with the spectra (24), are given. It is seen that: (1) the spectrum n* is much broader than
V I Veksler: Secondary ion emission for transition metals
t~ (.,,v'~+c
the spectrum n,~ its contribution being significant even at E ~ E0; (2) In accordance with the qualitative consideration in the vicinity of E ~ E , the spectrum n* in fact shows a turning point, the lower Eo the sharper it is. F r o m the given data, it is concluded that the influence of the parameter p on the spectrum shape increases with decreasing Eo and for Eo ~ E , it is rather significant. In Figure 6, the dependences of the K I E yield, found by the formula K,=
(1 - ~J~)n*(~, Eo) d~,
(28)
-f
on the bombarding ion energy are presented. Kt is seen to depend to a lesser degree on p. The dependence of Kt on 8 also turns out to be not very strong. Previously it was found s that experimental data for niobium, m o l y b d e n u m and tantalum are described by the electronexchange mechanism. To explain similar data in the ease of vanadium other mechanisms should be applied and the K I E mechanism was proposed s. Based on results now found, this interpretation cannot be recognized as correct since: (1) for vanadium, the width of energy spectra of the secondary ion emission, found by experiment s , does not much differ from the width of the sputtered atom spectra22'23; (2) in the experimental spectra the peaks or break-points inherent to K I E are not observed. The fact that K I E does not play any significant role in the vanadium secondary ion emission at Eo < 2 keV is clear since, in this case, the experimental value ofKi ~ 10- 2 (at Eo = 1.5 keV) s well exceeds the corresponding one ( < 6 . 1 0 - ") resulting from the theory discussed (Figure 6). F o r an explanation of the indicated anomaly, the application of other mechanisms of secondary ion emission, perhaps the ion formation mechanism on the developed radiation damage 5'e, are required.
-2
-3
-4
,o'
;o
,oo '
;o
='
='
, '=
E,ev Figure 5. The computed energy spectra of the sputtered ions ( 1, 2, 3, 4 ) and atoms (5, 6) of vanadium for Eo, eV (C): 1--250 (5.0866); 2--400 (5.2907); 3--800 (5.5918); 4--1562.5 (5.8825); 5--250 (0.1314); 6--1562.5 (1.2651). The prime (two prime) superscript corresponds to/3ffi2 (10).
5
4
2
I
0
OA
I
i
i
0.8
t2
~.6 F0,keV
Figure 6. The dependencesof KIE coefficients on the bombarding ion energy. ~: 1--2; 2--10.
References i A R Bayly and R J MacDonald, Radiat Eft, 34, 169 (1977). 2 Z Jurela, Int J Mass Spectrom Ion Phys, 12, 33 (1973). 3 Z Jurela and B Perovi~, Can J Phys, 46, 773 (1968). " A A Adylov, V I Veksler and A M Reznik, Zh Tekh Fiz, 41, 2005 (1971). 5 V I Veksler, Radiat Eft, 51, 129 (1980). e V I Veksler, Secondary Ion Emission of Metals. Nauka, Moscow (1978). 7 G Slodzian and J-F Hennequin, C R Acnd Sci Parts, 263, B1246 (1966). 8 B A Tsipinyuk and V I Veksler, Vacuum, 29, 155 (1979). 9 V I Veksler, Proc Soy Nat Syrup on Interaction of Atomic Particles with Solids, p 62. Fan, Tashkent (1979). to V I Veksler, lzv Akad Nauk Uzb SSR, serfiz-mat, 4, 34 (1959). ! 1 p Joyes and H-F Hennequin, J Phys, 29, 483 (1968). 12 p Joyes, J Phys, 30, 243 (1969). 13 p Joyes, J Phys, 30, 365 (1969). 14 M W Thompson, Phil Mao, 18, 377 (1968). Is G Carter, D G Armour and K J Snowdon, Radiat Eft, 35, 175 (1978). le R B Wright and D M Gruen, J Chem Phys, 72, 147 (1980). 1: j A Bearden and A F Burr, Rev Mod Phys, 39, 125 (1967). 18 V I Veksler, Fiz Tverd Tela, 22, 2620 (1980). 19 O B Firsov, E S Mashkova, V A Molchanov and V A Snisar, Nucl lnstr Met/=, 132, 695 (1976). 20 O B Firsov, E S Mashkova and V A Molchanov, Radiat Eft, 18, 257 (1973). zl j C Slater, Phys Rev, 36, 57 (1930). 22 R V Stuart and G K Wehner, J Appl Phys, 35, 1819 (1964). 23 S V Stuart, G K Wehner and G S Anderson, J Appl Phys, 40, 803 (1969).
163
0042-207X/83/030159-05503.00/0 Pergamon Press Ltd
The energy spectra of the secondary ion emission for transition metals of the fourth period V I Veksler,
Tashkent State University, Tashkent 700095, USSR
received 3 June 1982
The energy spectra of the secondary ion emission (SlE) for transition metals of the fourth period are calculated under the assumption that SIE is caused by the joint action of electron-exchange and kinetic ion emission (KIE) mechanisms. Some improvement of the KIE theory is carried out for the cases of high (of the order of tens of keV) and low (less than 2 keV) energies E o of bombarding ions. It is found that the KIE plays an essential role in SIE, starting from the E o value of the order of tens of keV. It is shown that the KIE energy spectra (1) are broader than the sputtering energy spectra by an order of magnitude, (2) differ essentially in both cases indicated, (3) show a peak (at high Eo in accordance with an experiment) or a break (at low Eo) near to the secondary ion energy corresponding to the KIE threshold.
hi!
1. Introduction
fOs
In a number of papers dedicated to secondary ion emission investigation 1-3, it was found that energy spectra have an unusual form for transition metals of the fourth period: in them, along with the ordinary peaks at an energy W of secondary ions of about 10 eV, peaks I or irregularities2.a for W of the order hundreds of eV were also discovered (Figure 1). The position of the high energy singularity on the spectrum for copper 2 is in good agreement with the peak position on the analogous spectrum in ref 1. But in ref 2 this singularity does not turn into a peak, since here the condition for surface purity was not so good as in ref 1 according to purity criteria~6, and it is known ~ that change for the worse of those conditions leads to a strong increase in the low energy peak. In ref 1, the high energy peak formation was interpreted in the framework of the electron-exchange mechanism, but the following dependence of the secondary ion formation probability P on its velocity v was used: Pocexp(-Vo/V), which is not in accordance with experiment s . The introduction of the correct dependence s leads to the disappearance of the corresponding peak in calculated energy spectra. The opinion that the origin of the singularities indicated in the spectra is related to the action of two secondary ion emission mechanisms has been already expressed 9. The electron-exchange mechanism is the origin of the low energy peak, whilst the secondary ion energy spectrum n~'"is connected with the sputtered atom energy spectrum n~ by the following equation independent of the bombarding ion energy Eo 5.,0
n,'-~h-~2[(-~)I/2-1](MecpW)'/2 d2(1
+cWt/2)na,
(1)
V+
fO
fb'
5'ooE,,v
Figure !. The experimental energy spectra for 3d transition metal secondary ions (Eo=43 keV) ~.
where m and M are the electron and ion mass respectively, V~and d are the ionization potential and diameter of the sputtered atom, e~b is the target work function, c~0.3 (eV) -1/2 s, h is a fitting parameter, where h,~l s'6. The validity of equation (1) is confirmed by experiment s for E 0 < 2 keV. According to supposition 9, the high energy singularity may be due to kinetic ion emission (KIE). In the corresponding Joyes' theory it is assumed 11-1a that, under violent interactions of the bombarding ions with the lattice atoms and the lattice atoms between themselves, a vacancy in one of the internal shells of some atoms is formed, and the electron is transferred into the metal conduction band. The corresponding atom can escape from the metal in an excited state, if the lifetime of the vacancy 11 is sufficiently large. It can be ionized outside the metal by means of an inneratomic Auger effect if the necessary conditions are satisfied. 159
V / Veksler: Secondary ion emission for
transition metals
Unfortunately, in the above-mentioned theory a number of unjustified assumptions have been made (criticism of it is given in refs 5 and 6). This prevents direct experimental confirmation of the theory. In the present work a more correct analysis of the shape of the corresponding energy spectrum n~' of secondary ions in the framework of the KIE notion is given. Then a comparison with the experimental t-a data of the energy distributions n~ obtained by means of summation of the n* and n~* spectra is presented.
3. Qualitative consideration We shall carry out first a qualitative consideration of the form of the expected energy spectrum which need not depend on the specific method of calculation of equation (2). For this purpose we alter the integration order in formulas (2) and (4): t
d("w,(~", ~)
n*((, E 0 ) = Q * ( ( , ) exp(-
t'~/~) +
d~"ws(~", ~)
2. Initial equations
¢
In the above-mentioned analysis the initial equation of the theory describing the excited atom energy spectrum under the surface at the 'metal-vacuum' interface 13 is used:
exp(-
;l
n.%, Eo)=-
t/./z),
(2)
where ( = E/E o and E is the energy of the atoms indicated. Here it is proposed that ¢ = EriE o =-~, where E~ is the energy threshold for the vacancy formation, instead of using ~ = ( as in ref 13. It is also taken into account here that the secondary ions can be initiated by bombarding ions slowing down in the lattice. In accordance with the above-mentioned assumptions nk((' ) are considered to be energy spectra of the particles moving in the solid (k = s is related to the primary ions scattered by a lattice, and k = d to the lattice displaced atoms), 0((', (") is a probability of the atom displacement with energy E" = ("E o by the particle with energy E ' = ('Eo, w((') is the probability of a vacancy formation in the former, w,((", () is the probability for slowing-down of excited atoms with energy E" to energy E, t~. is the mean time for such retardation s'6
t'~=2-a/4(1-2-~/2)-~M~/21[E-~/2-(E")-1/2],
(3)
where M and l are the mass and the free path length of atoms respectively. Equation (2) essentially is related to the KIE, provided that the frequently made assumption of the isotropy of the atomic collision cascade not being dependent on the distance from the surface (see, for example, 14-16) is valid. The threshold E, significantly exceeds the energy E~ required for vacancy formation (e.g., for the 3p vacancy in vanadium, E~ = 37.8 eV ~1 but E, found by the method ofrefs 6 and 18 is equal to ~ 200 eV). Therefore, the change in E' due to inelastic losses is subsequently neglected. Moreover, the character of slowing down for atoms in the excited and ground states is assumed to be identical. Under these conditions the energy spectrum of the latter before escaping from the metal is
n°(~)
(~ < ~ < ~,),
(6)
(~,<_~<_1),
(7)
d~"ws(~",~)
exp(-t~,/~)Q*(~")
n~,((, Eo)= f : d('nk((')w((') f;' d(" g((', (") w,(;", O e x p ( -
t~,/~)Q*(~")
n.°(~) = where
f:
d~"w~(~", OQ(~"),
Q*(O = Q(~)=
(8)
o(~', ;")[ns(~')+ nd(~')]w(U) d~',
(9)
0(~', ~")[ns(~')+nd(~')] d ( '
(10)
represent the number of excited atoms and atoms in the ground state, respectively, which are formed within the solid with energy E ('source'). We shall analyse the spectra shape for a fixed value of E o. (1) At ~ < ( < ~, the whole integration region of equation (6) is divided into two parts (Figure 2). In the region 1 (~ < ~"<_~, < (') the source Q* remains constant, independent of ~, while the mean value of Q decreases as ~ increases. Moreover, the mean value of the retained excitations, determined by the exponential, rises with increase in ~. Thus, the part of the spectrum considered (the first integral of equation (6)) should decrease with an increase in much slower than the spectrum n°((). It is not beyond possibility that this part of the spectrum can even increase since the function ws((", ~) is unknown a priori. In the region 2 ((< ~, < ~"< ~') the source O* again does not depend on ~ and the exponential rises
~
t
t
2
t
.f ~ d{'[n,(;')+ nd(~')]
:::::
I
I
II
I
r / / / /
f d("g(;', ;")w~(;",
;).
:////
(4)
One can obtain the energy spectrum of secondary ions escaping along the normal to the surface, if, according to ref 14 the influence of the surface potential barrier of height 2 = (aE o is taken into account
n*((--~, E o ) = (1 - (x/() End((, Eo)+n*~a(~, Eo)]. 160
(5)
j
I' i i
I I
I I tf
Figure 2. Integration region of equation (2).
V I Veksler: Secondary ion emission for transition metals
with an increase in (. Since here the integration over the (" interval does not depend on ~, it is more likely that this part of the spectrum (the second integral of equation (6)) will rise with an increase in (. So, in the case considered, the total spectrum n*(~, Eo) must fall much more slowly than n°(() when ~ increases and it can even rise. (2) At (, < ( < 1 the integration region is indivisible (Figure 2). Here the mean value of Q* will decrease more slowly than Q because of the increase in W, and the exponential will increase, with a ~ growth. Thus, for all the values of ~ the spectrum n* must fall o Moreover, at ( ~ ~, the spectra must much more slowly than n°. show a break owing to a decrease in Q*, when ~ increases, starting at ~ ~ ~,, i.e. in the region ( > (, the spectrum must decrease faster than in the ( < (, region. At ~ ~ ~, the spectrum will show a peak if, at ~ < ~,, n* rises with an increase in (. Now we shall assume that E0 is reduced. Then because of the decrease in Q*(~,) the significance of n* corresponding to integration region 1, must be strongly reduced for (~<(<~,, but the number of retained excitations remain invariable. For region 2, Q* also diminishes, but the mean number of retained excitations will be essentially increased. That is why the relative contribution of this part in the total spectrum will also be increased. Therefore, one can expect that the peak (singularity) at ( ~ ~, will become sharper, if E o decreases.
4.1. Case of a high energy. First, a calculation for the case when the bombarding ion energy is of the order of some tens of keV will be carried out. In this case, as is known s'14-16, the assumption that the cascade of atomic collisions is isotropic works rather well, and the sputtered atom spectrum can be presented in the form: n°(O = A(1 - [ f f ~ ) ( ~ - 2 _
1).
(15)
The relation .4(~ -2 - 1) =
dr"w ~r", ¢ ) Q ( ( " ) . ~..
(16)
can be used for the determination of w~((", ~), and the normalizing constant is found from the condition
S=A
( 1 - ~ f f ~ ) ( ( - 2 - 1) d~,
(17)
so that
A_~2(aS.
(18)
It is easily seen that equation (16) is satisfied at
ws(~", ~)= 2~'G(O/L~", 4. Specific calculations
where
We shall carry out specific calculations to illustrate the conclusion drawn. In the theory of refs 12 and 13, an inaccurate assumption that the slowing down of the excited atom and the vacancy annihilation processes occur independently of each other and a strongly underestimated value of E, (cf. refs 5 and 6) are used. Both these assumptions are ruled out in the present work. It is known that, for an isotropic cascade of the displaced atoms, the energy spectrum n~((') is determined 6"~4 by the expression
rid(~') _~2(,lS/((') 2,
( 11 )
where S is the sputtering coefficient. A question about the shape of the energy spectrum n~((') of primary ions scattered by a lattice is somewhat more complicated. The spectrum is described by a function monotonically increasing with E, if the interaction particle potential can be approximated by a dependence which is inversely proportional to the square of a distance ~9. In the diffusional approximation, the spectrum has a cupola form, but its maximum is placed sufficiently near to E 0 2o. Therefore, the following approximation will be used: n,(~') = 2 K ~ ' ( ( ' _ < 1 ),
(12)
since calculations show that the spectrum n~ form is weakly sensitive to the detailed shape of the spectrum n~(('). In equation (12), K, is the 'isotropic part' of the scattering coefficient and is later regarded as a parameter. The value 0 is determined by the equation g((', ~") d ( " = d ( " / ~ ' ,
(13)
The expression for w will be approximated by the formula (cf. ref 18) w((') = Wo[1 -- (£,/02], with w0 ~ 0.1.
(19)
(14)
G(() = (1 - ~2) [½ (1 - (2) _ (fl _ 1 )(2 In _ fl~.2 ( i _ ( ) ] -
i.
(20)
In this case, the spectrum n* (~, Eo) is calculated analytically: n* (~, Eo)=4Wo(a SyG(() {~l,[Ei(x).-;Ei(r,)]
+ n(x,, Z, x , ) - n ( x , Z, x t ) } e x p ( - g ) ( ~ ; . < ~ <~,),
(21)
n*a (~, Eo)=4Wo(~SyG(()[H(K, Z, •,) -n(z,
X, x , ) ] e x p ( - x ) (¢, < ~ < 1),
where
H(x, Z, Kt)=aEr- 2(x+ 1)exp(r) - 2 r 7 4R 1(r)] + Z- 4[ R 3(to) - (2x,4) - I g T ( r ) ] + (b -a)Ei(x), R,(r)=r!exp(~c) n=O Z ( - - l ) "+1 K=~( ~- 1/2 K, = ~
1
~" Kn'
1/2 ~ :
(F./Eo)I/2,
n,=l-(1 + ~,)2/2~2 + fl] (1 _ ~,)2,
a=flZ2/2,
b = f l ( l + ( 2 ) - (I - (~/2),
fl = 2K ff ( AS, e = 2 - 3 / 2 ( 1 - 2 - 1 / 2 ) - 2M/2.t.- 2'
Ei is the sum of the integral hyperbolic sine and cosine functions. In the case considered 7 = 1. The quantity I does not greatly exceed the lattice constant, the value of -c~ 10- ~4 s 11. Therefore, for the transition metals of the fourth period e ~ 4 keV. 161
V I Veksler: Secondary ion emission for transition metals
The total energy spectrum of secondary ions can be found with a formula __
ee
*
ni--nl +ni,
(22)
3 104
and the yield of secondary ions is found by means of integration
Ki =
(n~+n *) d~.
(23)
In Figure 3 the computed energy spectra for titanium, vanadium, nickel, copper and tin are given. It was assumed that in atoms of corresponding metals a 3p vacancy is formed by KIE.
t03
I
5
~0
20
50
t00
200
500
f000
F,eV 10~
Figure 4, The influence of e variation on the computed energy spectra (relative units) form for Ni + (E o =40 keV). e, keV (h): 1~4 (0.03); 2--0.4 (0.03); 3 ~ . 4 (0.08).
computed at e.=0.4 and 4 keV with a constant h are given. The spectra are seen to differ only slightly (curves 1 and 2). Better conformity of spectra in both cases can be obtained if the parameter h is varied (of. curves 1 and 3). The computed yield values (for example, for nickel at E 0 = 40 keV, K~ ~ 1.10-2) are believed to be quite reasonable. Thus, there is a real justification for the assumption that the metal secondary ion emission at a sufficiently large E 0 is caused by the joint action 9 of electron-exchange and KIE mechanisms.
t04
103
t
5
10
20
50
f00
200
500
f000
E.eV Figure 3. The computed energy spectra (relative units, Eo = 40 keV) of the secondary ions (h): I--V* (0.03); 2--Ti + (0.2); 3--Ni ÷ (0.03); 4-Cu+(0.005); 5--Zn+(0) (fl= 10). The spectrum 1' is computed for V+ at fl= 1000, h =0.03. The value of the parameter E,, influencing the spectra width, by the method of refs 6 and 18 was calculated. In this method it is assumed that, for head-on collisions of an atom, the kinetic energy of relative motion converts into the repulsion potential energy and into the energy of excitation of electrons. The latter was found by the molecular orbital method and the effective charge on the nucleus Ze, was calculated according to the semi-empirical Siater rule 21. It was found that E,,~200 eV for titanium and vanadium. Unfortunately, for the rest of metals, a major uncertainty in Zorf value arises because of the large reciprocal penetration of the electron shells of the colliding particles, which does not permit a sufficiently reliable determination of the E t value. Therefore, for them, E~ was taken to coincide with the corresponding value for tantalum (E~= 400 eVl a). A comparison ofenergy spectra found by experiment (Figure 1) and calculated by the method described (Figure 3) shows a satisfactory degree of accordance, and the value of the parameter h used does indeed satisfy the condition h.~ I. It should be noted that the other parameters do not much influence the form of the calculated spectrum. In Figure 3, spectra for vanadium computed at fl= 10 and 1000 ( K ~ 0 . 0 1 and 1, respectively) are given. The form of the spectra in both cases is seen to differ very slightly. In Figure 4, spectra for nickel 162
4.2. C a s e o f a law energy. At E o < 2 keV the assumption ofisotropy in the atomic collisions cascade is violated ~'6'22. In this case the function nk((')9((', (") × w~((", () in equation (2) must be regarded as the corresponding characteristics averaged over all angles. The accuracy of calculations is, naturally, reduced in this case. It is known 8 that the spectrum of atoms, sputtered along the normal to the surface for 3d transition metals, is better described by the formula no(() = 6(~S(1 - ~z/~) (~-
3 __
1)
(24)
for the case considered. Therefore, the expression 6~2S(~-
3 __ 1 ) =
f'
d("ws(~", ~)Q(~"),
(25)
permitting evaluation of w~, can be considered in this case in a certain degree as correcting for the deviation of real conditions from the isotropic cascade ones. Here, w~ is again determined by formula (19), but G ( ( ) = (1 - ( 3 ) ( - 111(1 -(fl-
_
(2)
1)( 2 In (--fl(2(1 _ ( ) ] - l
(26)
The spectrum n*((, Eo)is again determined by formula (21). In the case considered in formulas (19) and (21), the substitution 7=3(a
(27)
is made. In Figure 5, the spectra n*(E, Eo) found with formulas (5) and (21) for the two parameter fl values, along with the spectra (24), are given. It is seen that: (1) the spectrum n* is much broader than
V I Veksler: Secondary ion emission for transition metals
t~ (.,,v'~+c
the spectrum n,~ its contribution being significant even at E ~ E0; (2) In accordance with the qualitative consideration in the vicinity of E ~ E , the spectrum n* in fact shows a turning point, the lower Eo the sharper it is. F r o m the given data, it is concluded that the influence of the parameter p on the spectrum shape increases with decreasing Eo and for Eo ~ E , it is rather significant. In Figure 6, the dependences of the K I E yield, found by the formula K,=
(1 - ~J~)n*(~, Eo) d~,
(28)
-f
on the bombarding ion energy are presented. Kt is seen to depend to a lesser degree on p. The dependence of Kt on 8 also turns out to be not very strong. Previously it was found s that experimental data for niobium, m o l y b d e n u m and tantalum are described by the electronexchange mechanism. To explain similar data in the ease of vanadium other mechanisms should be applied and the K I E mechanism was proposed s. Based on results now found, this interpretation cannot be recognized as correct since: (1) for vanadium, the width of energy spectra of the secondary ion emission, found by experiment s , does not much differ from the width of the sputtered atom spectra22'23; (2) in the experimental spectra the peaks or break-points inherent to K I E are not observed. The fact that K I E does not play any significant role in the vanadium secondary ion emission at Eo < 2 keV is clear since, in this case, the experimental value ofKi ~ 10- 2 (at Eo = 1.5 keV) s well exceeds the corresponding one ( < 6 . 1 0 - ") resulting from the theory discussed (Figure 6). F o r an explanation of the indicated anomaly, the application of other mechanisms of secondary ion emission, perhaps the ion formation mechanism on the developed radiation damage 5'e, are required.
-2
-3
-4
,o'
;o
,oo '
;o
='
='
, '=
E,ev Figure 5. The computed energy spectra of the sputtered ions ( 1, 2, 3, 4 ) and atoms (5, 6) of vanadium for Eo, eV (C): 1--250 (5.0866); 2--400 (5.2907); 3--800 (5.5918); 4--1562.5 (5.8825); 5--250 (0.1314); 6--1562.5 (1.2651). The prime (two prime) superscript corresponds to/3ffi2 (10).
5
4
2
I
0
OA
I
i
i
0.8
t2
~.6 F0,keV
Figure 6. The dependencesof KIE coefficients on the bombarding ion energy. ~: 1--2; 2--10.
References i A R Bayly and R J MacDonald, Radiat Eft, 34, 169 (1977). 2 Z Jurela, Int J Mass Spectrom Ion Phys, 12, 33 (1973). 3 Z Jurela and B Perovi~, Can J Phys, 46, 773 (1968). " A A Adylov, V I Veksler and A M Reznik, Zh Tekh Fiz, 41, 2005 (1971). 5 V I Veksler, Radiat Eft, 51, 129 (1980). e V I Veksler, Secondary Ion Emission of Metals. Nauka, Moscow (1978). 7 G Slodzian and J-F Hennequin, C R Acnd Sci Parts, 263, B1246 (1966). 8 B A Tsipinyuk and V I Veksler, Vacuum, 29, 155 (1979). 9 V I Veksler, Proc Soy Nat Syrup on Interaction of Atomic Particles with Solids, p 62. Fan, Tashkent (1979). to V I Veksler, lzv Akad Nauk Uzb SSR, serfiz-mat, 4, 34 (1959). ! 1 p Joyes and H-F Hennequin, J Phys, 29, 483 (1968). 12 p Joyes, J Phys, 30, 243 (1969). 13 p Joyes, J Phys, 30, 365 (1969). 14 M W Thompson, Phil Mao, 18, 377 (1968). Is G Carter, D G Armour and K J Snowdon, Radiat Eft, 35, 175 (1978). le R B Wright and D M Gruen, J Chem Phys, 72, 147 (1980). 1: j A Bearden and A F Burr, Rev Mod Phys, 39, 125 (1967). 18 V I Veksler, Fiz Tverd Tela, 22, 2620 (1980). 19 O B Firsov, E S Mashkova, V A Molchanov and V A Snisar, Nucl lnstr Met/=, 132, 695 (1976). 20 O B Firsov, E S Mashkova and V A Molchanov, Radiat Eft, 18, 257 (1973). zl j C Slater, Phys Rev, 36, 57 (1930). 22 R V Stuart and G K Wehner, J Appl Phys, 35, 1819 (1964). 23 S V Stuart, G K Wehner and G S Anderson, J Appl Phys, 40, 803 (1969).
163