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Coverage of foreign atoms on surfaces as a function of adsorption, sputtering, and diffusion rates
Surface Science 83 (1979) 296-300 0 North-Holland Publishing Company
COVERAGE OF FOREIGN ATOMS ON SURFACES AS A FUNCTION OF ADSORPTION, SPUTTERING, AND DIFFUSION RATES
Received 4 July 1978; manuscript received in final form 17 November 1978
During sputtering of a surface the coverage of foreign atoms may be changed by diffusion ‘from the bulk to the surface and by adsorption from the gas phase. Assuming sufficiently high negative surface segregation enthalpy i.e. high ratios of surface to bulk concentration in equilibrium, simple expressions of the coverage time function for finite and semi-irtfiiite samples are derived. The equations were applied to experimental results of oxygen coverage on a niobium surface where the three processes mentioned above were active. From the comparison with these results the diffusion coefficients of 0 in Nb D = 2 X 10-t3 m’/s at 520°C and D = 9 x IO-r3 m2/s at 600°C were evahuated.
In this study the coverage of foreign atoms on a surface is calculated as a function of time if the following processes are active: adsorption of these atoms from the gas phase and/or desorption by ion sputtering and/or diffusion from the bulk material to the surface. The generality of the derived equations is restricted by the assumption that the enrichment factor for the foeign atoms should be very high. The results of this study are compared with the experimental findings of a previous work [l] in which the decrease of oxygen coverage B during sputtering of a thin niobium foil (SO pm thick) doped with 0.6 at% oxygen was measured. In this experiment the niobium surface acts as a sink and oxygen supplied to the sink by diffusion and adsorption is removed by sputtering. The difference between supply and removal changed the oxygen coverage of the surface which was measured by Auger electron spectroscopy. This is expressed by the following balance equation used also in ref. [I] : d@/dt = .Js -
Js
f Jd,
(1)
where Js, Js, and Jd are the current densities of oxygen atoms at the surface (x = 0) as caused by sputtering, adsorption from the residual gas, and diffusion from the bulk mater&i to the surface. With the ion current density Jp and the sputtering yield S (probability of oxygen atom desorption per Ar’ ion impact), Js is expressed 296
R. Kirchheim, S. Hofinann j Coverage of foreign atoms on surfaces
291
as usual
where Be is the maximum coverage corresponding to one monolayer [ 11. The form of the solution of eq. (1) does not depend on the assumption whether Js is constant or Ja decreases linearly with 8. For a linear relation~p Js = a + b0 the second term b19may be added to J, which is proportional to 0. For this case the quantities J, and Jr,S/Oe should be replaced in the following equations by II and JpS/f3, + b. The purpose of this letter is to combine these well known relations with a solution for Jd obtained from Fick’s Second‘Law with the following boundary and initial conditions for the oxygen concentration c in the bulk c = co
for
t =0
and
O
c=o
for
x=0
and
t>O,
for
x =I
and
t>O,
&$x
= Jg/D
where I is the thickness of the infinite sample and the boundary condition there describes the oxygen sorption from the residual gas by the surface opposite to the surface under ion bombardment. Strictly speaking the boundary condition for x = 0 states that the oxygen concentration in the first niobium layers adjacent to the surface is negligible small in comparison with the initial concentration co. This is proven by the McLean equation [2] using the parameters of surface ~gregation for Nb-0 [I ,3,4] and it will be valid only for foreign atoms with high enrichment factors. With the values of ref. [I] for instance c equals 67 at ppm 0 for the highest coverage of f3/0, = 0.4 at 520°C compared to CO= 6000 at ppm. Using the known procedure the solution of the diffusion problem was found to be
c(x, t) = ($)
X’ c
cl{sin (yx)
where
” =(2i t 1)2n2D’ Using Fick’s Law Jd was calculated:
Jd
z-D!% ax
x”*
exp (-k),
298
R. Kirchheim,
S. Hofmann / Coverage of foreign atoms on surfaces
with bi =-&e/l
+ (-I)‘+’
2rs/(2i+
I)n.
Inserting this in eq. (1) a linear, inhomogeneous tained with the solution
differential
equation
for B is ob-
where C should be calculated from the initial condition. The evaluation of C is not necessary for the experimental results discussed in this study since t >>0,/U, and the second exponential term vanishes as well as for t > 0.1 r1 all the exponential terms in the series with the exception of the first one can be neglected. Then the equation is reduced to the simple form
Fig. 1 shows a plot of ln(e/e, - Z!..Is/SJr,)versus t where the parameter 2Js/SJn = 0.065 was obtained from a best fit of eq. (2) to the experimental data. Unfortunately no measurements were made for higher sputtering times in order to determine this steady-state value Us/SJr, more directly. From the slope of the straight line in fig. 1, D is evaluated to be (2 + 0.1) X lo-r3 m*/s, depending only weakly on the parameter 2JalSJn. With a sticking probability of 1 and e,/,SJn = 4.1 set the parameter 2Js/SrP = 0.065 corresponds to an equivalent oxygen pressure of about 2 X 10T8 mbar in
0
1
2
3
4
sputtering time, h
Fig. 1. Logarithm of relative oxygen coverage minus steady state value (Ug/Sr,) versus time. The dope of the straight line is proportional to the diffusivity of oxygen in niobium at 520°C.
R. Kirchkeim, S. Hofmann /Coverage of fore&n atoms on surfaces
299
0.1,. 0
0
I
3
2
spurrekg
time,
4
5
h
Fig. 2. Comparison of calculated coverage values with experimental data: (0) experimental eq. (3). (-)
[ 11;
good agreement with experimental conditions [S]. The diffusion coefficient D evaluated from the slope in fig. 1 was used in eq. (2) to calculate B/BO.A comparison with experimental data is shown in fig. 2. For B. the value 1.44 X 10” 0 atoms/cm2 was inserted. Because the influence of the parameter W&Y, is small on the evaluation of 8/e,,, the second adjustable parameter D obtained from the slope in fig. 1 also gives the right absolute e/e, values. In another measurement at 600°C with co = 0.7 at% 0 and t @ r1 a steady-state value of 2Js/SIP = 0.15 was obtained. Fig. 3 shows a plot of the same quantities as in fig. 1 for this experiment. For high times the coverage depends exponentially on
50
sputferingtime
, min
Fig. 3. Logarithm of relative oxygen coverage e/e0 minus steady state value (U,/Sr,) time (as in fig. 1) for 0.7 at% oxygen in niobium at 600°C.
versus
300
R. Kirchheim,
S. Hofmann / Coverage of foreign atoms on surfaces
time again and according to eq. (2) the diffusion coefficient D = 9.2 X lo-r3 m2/s with I = 60 pm was computed from the slope of the straight line in fig. 3. For t < 20 min (open circles) the fluxes Js and Jd to the surface have always been high enough to compensate for the losses J, caused by sputtering and, therefore, the coverage was kept at its maximum value. The diffusivities‘ evaluated in this study are in excellent agreement with tracer diffusion measurements of Fedorov et al. [6] made in the same temperature range (D(520°C)= 1.7 X lo-r3 m2/sandD(600”C)=8.8 X IO-r3 m2[s). The solution for 8 given in this letter for a finite sample is easily extended to semi-infinite samples, because from the solutions of Fick’s Second Law with the boundary condition c = 0 at x = O,J, is derived Jd = ce (D/r@“2 ) and eq. (1) gives the following solution for B
0 =Jgeo/sJp + Cexp(-SJ&‘0,)
f j
(D/TT)“~ co exp[-SJ,(f
- r)/e,]
dr.
0
A special case of this relation with Jg= Jp = 0 was studied by Hofmann Erlewein [7] to determine diffusion coefficients of tin in copper.
and
References [I] [2] /3] [4] [5] [6]
S. Hofmann, G. Blank and H. Schultz, Z. MetalIk. 67 (1976) 189. D. McLean, Grain Boundaries in Metals (Clarendon, Oxford, 1957) p. 116. A. Jo&i and M. Strongin, Scripta Met. 8 (1974) 413. H.H. Farell, H.S. Isaacs and M. Strongin, Surface Sci. 38 (1973) 31. S. Hofmann and G. Blank, unpublished results. G.B. Fedrov, G.V. Fetisov, N.A. Stakum, O.N. Kar’kov, A.G. Arakelov and A.E. Kissil, Fiz. Met. Metalloved. 38 (1974) 361. [7] S. Hofmann and J. Erlewein, Scripta Met. 10 (1976) 8.57.
COVERAGE OF FOREIGN ATOMS ON SURFACES AS A FUNCTION OF ADSORPTION, SPUTTERING, AND DIFFUSION RATES
Received 4 July 1978; manuscript received in final form 17 November 1978
During sputtering of a surface the coverage of foreign atoms may be changed by diffusion ‘from the bulk to the surface and by adsorption from the gas phase. Assuming sufficiently high negative surface segregation enthalpy i.e. high ratios of surface to bulk concentration in equilibrium, simple expressions of the coverage time function for finite and semi-irtfiiite samples are derived. The equations were applied to experimental results of oxygen coverage on a niobium surface where the three processes mentioned above were active. From the comparison with these results the diffusion coefficients of 0 in Nb D = 2 X 10-t3 m’/s at 520°C and D = 9 x IO-r3 m2/s at 600°C were evahuated.
In this study the coverage of foreign atoms on a surface is calculated as a function of time if the following processes are active: adsorption of these atoms from the gas phase and/or desorption by ion sputtering and/or diffusion from the bulk material to the surface. The generality of the derived equations is restricted by the assumption that the enrichment factor for the foeign atoms should be very high. The results of this study are compared with the experimental findings of a previous work [l] in which the decrease of oxygen coverage B during sputtering of a thin niobium foil (SO pm thick) doped with 0.6 at% oxygen was measured. In this experiment the niobium surface acts as a sink and oxygen supplied to the sink by diffusion and adsorption is removed by sputtering. The difference between supply and removal changed the oxygen coverage of the surface which was measured by Auger electron spectroscopy. This is expressed by the following balance equation used also in ref. [I] : d@/dt = .Js -
Js
f Jd,
(1)
where Js, Js, and Jd are the current densities of oxygen atoms at the surface (x = 0) as caused by sputtering, adsorption from the residual gas, and diffusion from the bulk mater&i to the surface. With the ion current density Jp and the sputtering yield S (probability of oxygen atom desorption per Ar’ ion impact), Js is expressed 296
R. Kirchheim, S. Hofinann j Coverage of foreign atoms on surfaces
291
as usual
where Be is the maximum coverage corresponding to one monolayer [ 11. The form of the solution of eq. (1) does not depend on the assumption whether Js is constant or Ja decreases linearly with 8. For a linear relation~p Js = a + b0 the second term b19may be added to J, which is proportional to 0. For this case the quantities J, and Jr,S/Oe should be replaced in the following equations by II and JpS/f3, + b. The purpose of this letter is to combine these well known relations with a solution for Jd obtained from Fick’s Second‘Law with the following boundary and initial conditions for the oxygen concentration c in the bulk c = co
for
t =0
and
O
c=o
for
x=0
and
t>O,
for
x =I
and
t>O,
&$x
= Jg/D
where I is the thickness of the infinite sample and the boundary condition there describes the oxygen sorption from the residual gas by the surface opposite to the surface under ion bombardment. Strictly speaking the boundary condition for x = 0 states that the oxygen concentration in the first niobium layers adjacent to the surface is negligible small in comparison with the initial concentration co. This is proven by the McLean equation [2] using the parameters of surface ~gregation for Nb-0 [I ,3,4] and it will be valid only for foreign atoms with high enrichment factors. With the values of ref. [I] for instance c equals 67 at ppm 0 for the highest coverage of f3/0, = 0.4 at 520°C compared to CO= 6000 at ppm. Using the known procedure the solution of the diffusion problem was found to be
c(x, t) = ($)
X’ c
cl{sin (yx)
where
” =(2i t 1)2n2D’ Using Fick’s Law Jd was calculated:
Jd
z-D!% ax
x”*
exp (-k),
298
R. Kirchheim,
S. Hofmann / Coverage of foreign atoms on surfaces
with bi =-&e/l
+ (-I)‘+’
2rs/(2i+
I)n.
Inserting this in eq. (1) a linear, inhomogeneous tained with the solution
differential
equation
for B is ob-
where C should be calculated from the initial condition. The evaluation of C is not necessary for the experimental results discussed in this study since t >>0,/U, and the second exponential term vanishes as well as for t > 0.1 r1 all the exponential terms in the series with the exception of the first one can be neglected. Then the equation is reduced to the simple form
Fig. 1 shows a plot of ln(e/e, - Z!..Is/SJr,)versus t where the parameter 2Js/SJn = 0.065 was obtained from a best fit of eq. (2) to the experimental data. Unfortunately no measurements were made for higher sputtering times in order to determine this steady-state value Us/SJr, more directly. From the slope of the straight line in fig. 1, D is evaluated to be (2 + 0.1) X lo-r3 m*/s, depending only weakly on the parameter 2JalSJn. With a sticking probability of 1 and e,/,SJn = 4.1 set the parameter 2Js/SrP = 0.065 corresponds to an equivalent oxygen pressure of about 2 X 10T8 mbar in
0
1
2
3
4
sputtering time, h
Fig. 1. Logarithm of relative oxygen coverage minus steady state value (Ug/Sr,) versus time. The dope of the straight line is proportional to the diffusivity of oxygen in niobium at 520°C.
R. Kirchkeim, S. Hofmann /Coverage of fore&n atoms on surfaces
299
0.1,. 0
0
I
3
2
spurrekg
time,
4
5
h
Fig. 2. Comparison of calculated coverage values with experimental data: (0) experimental eq. (3). (-)
[ 11;
good agreement with experimental conditions [S]. The diffusion coefficient D evaluated from the slope in fig. 1 was used in eq. (2) to calculate B/BO.A comparison with experimental data is shown in fig. 2. For B. the value 1.44 X 10” 0 atoms/cm2 was inserted. Because the influence of the parameter W&Y, is small on the evaluation of 8/e,,, the second adjustable parameter D obtained from the slope in fig. 1 also gives the right absolute e/e, values. In another measurement at 600°C with co = 0.7 at% 0 and t @ r1 a steady-state value of 2Js/SIP = 0.15 was obtained. Fig. 3 shows a plot of the same quantities as in fig. 1 for this experiment. For high times the coverage depends exponentially on
50
sputferingtime
, min
Fig. 3. Logarithm of relative oxygen coverage e/e0 minus steady state value (U,/Sr,) time (as in fig. 1) for 0.7 at% oxygen in niobium at 600°C.
versus
300
R. Kirchheim,
S. Hofmann / Coverage of foreign atoms on surfaces
time again and according to eq. (2) the diffusion coefficient D = 9.2 X lo-r3 m2/s with I = 60 pm was computed from the slope of the straight line in fig. 3. For t < 20 min (open circles) the fluxes Js and Jd to the surface have always been high enough to compensate for the losses J, caused by sputtering and, therefore, the coverage was kept at its maximum value. The diffusivities‘ evaluated in this study are in excellent agreement with tracer diffusion measurements of Fedorov et al. [6] made in the same temperature range (D(520°C)= 1.7 X lo-r3 m2/sandD(600”C)=8.8 X IO-r3 m2[s). The solution for 8 given in this letter for a finite sample is easily extended to semi-infinite samples, because from the solutions of Fick’s Second Law with the boundary condition c = 0 at x = O,J, is derived Jd = ce (D/r@“2 ) and eq. (1) gives the following solution for B
0 =Jgeo/sJp + Cexp(-SJ&‘0,)
f j
(D/TT)“~ co exp[-SJ,(f
- r)/e,]
dr.
0
A special case of this relation with Jg= Jp = 0 was studied by Hofmann Erlewein [7] to determine diffusion coefficients of tin in copper.
and
References [I] [2] /3] [4] [5] [6]
S. Hofmann, G. Blank and H. Schultz, Z. MetalIk. 67 (1976) 189. D. McLean, Grain Boundaries in Metals (Clarendon, Oxford, 1957) p. 116. A. Jo&i and M. Strongin, Scripta Met. 8 (1974) 413. H.H. Farell, H.S. Isaacs and M. Strongin, Surface Sci. 38 (1973) 31. S. Hofmann and G. Blank, unpublished results. G.B. Fedrov, G.V. Fetisov, N.A. Stakum, O.N. Kar’kov, A.G. Arakelov and A.E. Kissil, Fiz. Met. Metalloved. 38 (1974) 361. [7] S. Hofmann and J. Erlewein, Scripta Met. 10 (1976) 8.57.