- Email: [email protected]
Diffusive release of gas from a solid in the presence of traps using a linear tempering schedule
Diffusive of traps received
22
S E Donnelly
August
and
release of gas from a solid in the presence using a linear tempering schedule 1976
D G Armour,
Department
of Electrical
Engineering,
University
of Salford,
Salford
M5
4WT,
England
The effect of permanent trapping on the surface evolution rate of gas released from a semi-infinite solid is examined for the case of a linear tempering schedule. Analytical expressions are obtained for the rate maximum, the temperature at which the maximum occurs and the temperature width of the evolution rate transient for gas initially located in a specific plane below the surface. ln addition, precise plots of the evolution rate transients are obtained for initial planar concentrations and for an extended distribution, by numerical solution of the appropriate diffusion equations.
given by J‘ e I/KL’ where f. is the mean intertrap distance in units of mean lattice spacing, A, and where K is a constant shown by Kelly and Matzkes to be between 4-6 for cubic metals. If it is assumed that the gas atoms begin one dimensional migration at time I = 0 with a diffusion coefficient D, then Fick’s second law of diffusion modified by the presence of traps* yields:-
Introduction Thermal evolution studies following gas implantation into solid substrates at energies ranging from a few eV’ up to many MeV’ have developed into a powerful tool for the investigation of defects in crystalline solids in general, and, in particular, are playing an increasingly important role in improving the understanding of the behaviour of gas atoms subsequent to energetic penetration of the crystal lattice. In these studies the gas species of interest is injected into a solid substrate and subsequently released by scheduled tempering of the target up to a temperature sufficient to ensure emission of all the implanted gas. The rate of release of gas is monitored during the tempering schedule and detected maxima in the release rate may be associated with particular trapping sites, predicted for the gas solid system in question. The tempering schedule used may be of any convenient form and several have been discussed.3 In practice, however, although the reciprocal schedule (l/T = I/T, - bt) is probably the simplest to deal with analytically, the linear schedule (T =To -in,) is the most easily realized experimentally and therefore is the one most frequently employed. The importance of the understanding of diffusion processes in these experiments has been long understood and was examined by Farrell e/ UP and Farrell and Carter’ in 1966. It is only recently, however, that the possible importance of the perturbations to desorption spectra, caused by gas retrapping during the heating schedule, has been discussed” in the literature. In a previous communication’ Carter has examined the retrapping phenomenon for the case of the reciprocal tempering schedule. The present work is concerned with the less analytically amenable case of the linear schedule.
Theoretical
27/number
D = D,
Pergamon
Press/Printed
Great
exp(
-
Q/RT) R
T= To -i-at we can centration
rewrite equation (1) to give the variation of the of diffusing atoms with depth, x, and temperature
c72C aA2 - -exp(Q/RT) (7gz - Do It has been form :-
C=O,x=
s =O,
.g
shown’
D’
(where A and conditions:-
of gas at a solid. distrif is in
(1)
where Q is the activation energy for the diffusion process and is the universal gas constant. Thus, if we allow temperature to vary with time according to the relation
C =O,
1.
- -. -c y7 1. (7.r L2i2
where C = C(x.f) is the instantaneous concentration of untrapped atoms. The diffusion coefficient, D, varies with temperature, T, through the relation
considerations
The process we wish to consider is the random migration initially located with a planar concentration Co cm-’ plane p lattice units below the surface of a semi-infinite The solid is assumed to contain a depth independent bution of unsaturable traps whose planar concentration Vacuum/volume
?I
that
are
conT:
+$ the
solution
constants),
to an
satisfying
equation
the
of
the
boundary
all Z co,allZ
c=cc,s=pA,z=o f0”Cd.x = C,,, Z = 0 Britain
21
S E Donnelfy
and
Diffusive
D G Armour;
release
of gas from
a solid
in the presence
of traps
using
a linear
tempering
schedule
312
is :-
(3) This becomes an appropriate is made identical with the X/aZ made identical with:-
Hence, equation (3) appropriate boundary
solution pre-exponential
is a solution conditions
for
equation Do and
of equation if:-
(I)
satisfying
the
(4) Provided that 7 is not less than a few tens of degrees above and not greater than about Q/20R. then an approximate solution for equation (4) can be shown to be:Z =
$
Hence,
T2 exp(
-
diffusing appropriate
r,,
Q/RT)
substitution
D,, = 4
of
where atoms, yields boundary
kJ’,
C(x,T) =c,
this expression for Z and is the vibration frequency as a solution to equation (2) conditions:-
k,
?1
Qa
2nkoi2 RT2 exp( - Q/RT) (s -piJ2 Qa x exp 2k, I2 RT2 exp(QIRT [ ( (s +piJ2 Qa - exp 2k,12RT2 exp( - QIRT) (
putting of the and the
-
-$f$
exp(
This expression is identical and Carte? for the case trapping term :-
ex
koRT2 - 2 -PC2QaL
-
(5)
derived by Farrell multiplied by a
Q/W
which reduces to unity in the no traps limit L+cr_. Now, the parameter which is of primary interest in desorption experiments is the rate at which dissolved gas is evolved from the surface, i.e. the rate at which the diffusing species traverses the x = 0 plane. This is given, from Fick’s first law, by:ac p=D’ax
x=o
Substitution
of equation
p = $[exp(QlRT)lt(erp[
(6) (5)
into
this
gives:-
- $expK?/RT)])
- C’T’ exp( - Q/RT>
22
expression
I)
&[&+3]
= [$($+2)
However, this can leads to the result:-
Q/RT)
to the expression with no trapping,
This is essentially the same expression as that derived by Carter’ for the case of the reciprocal tempering schedule. but with the constant b replaced by the term a/T*. C’-0 and equation (7) In the case of no trapping. L -x. becomes identical to that derived for the diffusive release during tempering without traps.5 Inspection of equation (7) reveals that the release rate initially increases with increasing temperature until it reaches some maxtmum. pn, at a temperature r,,,. after which it decreases with further increase in temperature. The form of this release rate is similar both to that obtained for the no traps caseJ.s and to that obtained for the reciprocal schedule in the presence of traps.’ The temperature, T,,. at which this maximum occurs and the value of p,,, can be analytically determined by difierentiation of equation (7) and setting the derivative equal to zero. This leads to the somewhat complex relationship:-
- (7
+ 2C’Tvt) (9)
I
x exp
C’ = koR 2QaL’
(2) if D’ the term
Q
7 RTn,
exp
Q
RT,,
2
be solved
for
the
quantity
f-
4/12 3 L2
>I
Q/RT,,, which
(10)
This is identical to the expression derived by Carter’ for the reciprocal schedule, but again with the constant b replaced by the variable a/T’. Also, it is readily seen that equation (IO) relaxes to Farrell and Carter’s expression5 for diffusive release without retrapping, in the limit L-t-p. As in the earlier studies, this expression indicates the almost linear relationship between the temperature at maximum release rate, r,., and the activation energy, Q. This relationship is illustrated in Figure I for gas initially located at a depth of 100 lattice units, released using a sweep rate of 20K s-t. It can be readily seen from equation (IO) and Figure I that it is only for values of p2/L2 considerably greater than unity that the presence of traps significantly perturbs the temperature at which the release rate maximum occurs. For a defect probe experiment, wherep was of order 10, even with moderately high concentrations of traps this perturbation would be sufficiently small (0.03 % for L = 100) to be negligible when compared to the errors inherent in the experimental temperature measurement (perhaps 50.1 *A at best). The maximum evolution rate, P,,,, can be evaluated by substitution of the maximum rate criterion in equation (7) giving the results:(11)
(7)
S E Donnelly
and
D
G Armour:
Diffusive release of gas from a solid
in the
presence
traps using a linear tempering
of
schedule
e< I and relaxes to that derived for the no traps case when p2/L2 --f 0. Solution of this expressionis complex and has to be carried out graphically for each value of p2/L2. It reveals two roots, y, and y2, wherey1 correspondsto T,,, which occurs before the rate maximum and y2 correspondsto TE.., which occurs after the maximum. These roots are displayed in Figure 2 p2/Lz
. octlvotion
I 50 energy
I too tor‘&iffurion,
Q
J I50 ( k cal I mole1 y, -
Figure 1. Variation in the temperature, T, at which maximum release rate occurs with activation energy, Q, for different values of intertrap spacing, L.
before
Y2 - alter
maximum maximum
which relaxes to the expression obtained by Farrell and Carte? 1
3
I
I
KS’
I
I
I
tcr’
I
IO’
$1 L’
traps limit and can be approximated to
in the no
p”!
=
Figure2. Variation in temperature width parameter (y) with trapping parameterp2/L2.
--co eu
>(2+
T,2
Y
wherex
= -
2
in the case of high trap concentrations and deep lying initial distributions (p/L > 1). A further parameter of interest, which can be evaluated analytically, is the temperature ‘width’ of the release rate transient which may be defined as the difference in temperature between the points at which the evolution rate is e-l of the maximum rate. Thus if fe = e- ’ P,,,,then from equation (7):-
Pe=$[exP(+$)][ev( [exp( - C’T:ev( = e-l $[-4&-j$-)][(exP(
- ~exp($))I - $))I - ++$--J
(13)
1
Rearrangement of this expression and substitution of the maximum rate criterion leads to the following expression :-
which becomesidentical to that for the reciprocal sweepfor
for a range of values of pz/Lz. It can be readily be seenthat, as in the reciprocal sweepcase,y, is only slightly perturbed by the presenceof traps, whereasy2 is noticeably reducedeven for low trapping levels. For large values of p2/L2, yz is significantly decreasedwhile y,. though also reduced, is affected to a much smallerdegree. 2
e.g. for E- = 1
L2
y2 is reducedto 45% of its value with no trapping, whilst y, still retains 83% of its no traps value. Thus, the analysisso far showsthat, in the absenceof traps, the releaserate transient risesto a maximumP,,,at temperature T,,, and subsequentlydrops off towards zero with a fall time somewhat longer than the rise time. The presenceof traps reducesthe maximumP,,,accordingto equation (11) and moves it to lower temperatures(Figure 1). It also reducesthe temperature ‘width’ of the curve asymmetrically such that the slowly decaying ‘tail’ of the transient is affected considerably more than the faster rising portion of the curve before the maximum. In order to quantify theseconclusionsand to present the information in a form more amenableto comparison with experimental work, a numerical evaluation of equation (7) was performed for various values of trapping parameter,L2, and the diffusion starting depth, p. Thesedata are displayedin Figures 3 and 4 and illustrate, in some detail, the trends already derived analytically. In particular, comparison of
23
.S E Donnelly
and
D G Armour:
Diffusive
release
of gas from
a solid
in the presence
temperature
3. Variation
of desorption
transient
with
trap
4400 (K)
Cl=50 a ~20
tempering
schedule
420-
.---
initial
concentration
In an experimental system in general, the dill’using species will not be initially located in one discrete plane. but will be distributed over a range of values of/>. Unfortunately, even when assuming comparatively simple forms for the initial distributions. the resulting equations are analytically intractable f-01 non-constant D. However, provided that gas atoms originating from dilferent planes do not interact in any way with each other. the transient. resulting from a distributed initial concentration can be obtained by a summation of the contributions from each of the planes contained in the distribution, i.e. for a general distribution Co = Co(p) with all the parameters as defined above :-
no rctroppinq
p: @=
kccl/molr K s-’
a linear
concentration Distributed
0)
using
~~
(ooo
Figure
of traps
Slatticc
unit,
@I=40
p = i
A(I’P$/‘)
[exp($)-J
[exp(
- yesP(j$))]
1) / [exp(
4000
Figure depth.
4. Variation
400
temperature
of desorption
transient
4200 (K)
with
diffusion
starting
Figures 4a and b reveals that even for moderate trap levels retrapping effects have a significant influence on the release characteristics of gas from initially deep layers. e.g. for Lz = lo”, p = 100 (curve (7) Figures 4a and b) 95% of the diffusing gas atoms are trapped.
24
- C’T’exp(
- $))I
(15)
The results of such a summation, carried out numerically for the distribution C,(p) = p exp( --p/IO), whose maximum lies in plane p = 10, are illustrated in Figure 5. The summation was carried out over only the first 100 planes, but as the value of C,(p) has dropped to about 0.1 y’, of its peak value at plane p = 100, little accuracy is sacrificed by neglecting contributions from deeper lying planes. Inspection of Figure 5 reveals that the trends already discussed for the case of the planar initial concentration are also evident for the ‘x e-“’ type distribution and that all the trends are accentuated. For example, comparison with Figure 3 shows that for L2 = 100, for instance, T,, is altered by 5.5%. P,,, by 51% and the temperature ‘width’ by 50% for the distributed initial concentration, whereas for the planar concentration, these values are 1.6, 29 and 44% respectively.
Conclusions This analysis has shown how expressions relating to maxtmum, pm, the temperature of the maximum, T,,,, and
the the
S E Donnelly
Figure
and D G Armour:
5. Varia[ion
;,\. (,- \ 11) ;_
or
desorption
Diffusive
transient
release
with
of gas from
trap
a solid
cnncentration
tcmpcraturc ‘width’ of the dcsorption transient. resulting from a linear heating schedule, can be derived for gas initially located in a plane below the surface of a semi-infinite solid. The work has been cstendcd by numerical analysis so that the precise shapes of the dcsorption transients can be obtained for various values of the paramctcrsp and L and so that the effects of rctrapping on the release rate curve for an extended distribution can also be observed. As was the case for the reciprocal schedule.’ the solutions obtained always relax to the previously derived values4..5 in the limit where L-.-r. For low densities and shallow, narrowly distributed initial concentrations the spectra obtained from thermal dcsorption experiments will fairly accurately reflect the populations and activation energies for migration of the original trapping sites, However, for deep lying implants. cstcndcd distributions and high trapping level. interpretation of dcsorption spectra becomes extremely diflicult. In particular, for the type of defect study where low energy light ions arc used as a defect probe in solids previously damaged by heavy ion bonlbardnlent’O.” quantitative. or even qualitative, description of the damage state of the target may be impossible unless this retrapping phenomenon is taken into account. In fact, if the trap concentration is high and the initial light ion distribution broad, then observed peaks in the spectra may arise from gas initially released which is retrapped and subsequently rc-released at higher temperatures and may in no way reflect the immediate post-bombardment trapping state of the solid. In addition, if thermal dcsorption data are to be used to make precise measurements of activation energies for detrapping processes within the solid, this work illustrates the necessity of taking into account all the parameters relating to the desorption transient (i.e.p,,, r,, and the temperature width) before any unambiguous determinations can be made.
in the presence
for
gas
initially
of traps
in
an
using
‘.v
a linear
lypc
tempering
distribution
schedule
ccntred
on
P = 10
In the light of the importance of accurate quantitative interpretation of defect probe data for the understanding of gas cluster and bubble formation, it is vital to be able to deconvolutc experimental dcsorption spectra in terms of the numerous parameters which influence their shape. To this end, the present work is being extended and co-ordinatcd with experimental work currently in progress.
Acknowledgements We wish to thank D C Ingram of the Electrical Engineering Department, University of Salford for helpful advice on the numerical analysis. One of us (S E D) would also like to acknowledge the provision of financial support from the UKAEA (AERE Harwcll) for the duration of this work.
References ’ E V Kornelsen and D Edwards, Proc I~rrerrraf C’ou/’ 011 Applicafiott oJ‘hr Beams IO Metals, Albuqrrerque (1973). ’ D S Whitmell and R S Nelson, Rad Efl, 14, 1972, 249. 3 G Carter. S.wposhon o/r ha// Mass Spec~ome/ers, University of Salford (1970). ’ G Farrell, W A Grant, K Erents and G Carter, k’ncrtroa, 16, 1966, 295. ’ G Farrell and G Carter, Vacuum, 17, 1966, 15. 6 D Edwards and E V Kornelsen, Rad Ef, 26, 1975, 155. ’ G Carter, Vacrtum, 1976, 329, ’ R Kelly and Hj Matzke. A’& Mawr, 20, 1966, 171. 9 G Carter, D G Armour. C Braganza, S E Donnelly and G Farrell, 1976 Symposium on the Physics of Ionized Gases, Dubrovnik-to be published. ” E V Kornelsen, Rad Eff, 13, 1972, 227. ‘I S K Erents and G Carter, Vacuum, 17, 1967, 215.
25
22
S E Donnelly
August
and
release of gas from a solid in the presence using a linear tempering schedule 1976
D G Armour,
Department
of Electrical
Engineering,
University
of Salford,
Salford
M5
4WT,
England
The effect of permanent trapping on the surface evolution rate of gas released from a semi-infinite solid is examined for the case of a linear tempering schedule. Analytical expressions are obtained for the rate maximum, the temperature at which the maximum occurs and the temperature width of the evolution rate transient for gas initially located in a specific plane below the surface. ln addition, precise plots of the evolution rate transients are obtained for initial planar concentrations and for an extended distribution, by numerical solution of the appropriate diffusion equations.
given by J‘ e I/KL’ where f. is the mean intertrap distance in units of mean lattice spacing, A, and where K is a constant shown by Kelly and Matzkes to be between 4-6 for cubic metals. If it is assumed that the gas atoms begin one dimensional migration at time I = 0 with a diffusion coefficient D, then Fick’s second law of diffusion modified by the presence of traps* yields:-
Introduction Thermal evolution studies following gas implantation into solid substrates at energies ranging from a few eV’ up to many MeV’ have developed into a powerful tool for the investigation of defects in crystalline solids in general, and, in particular, are playing an increasingly important role in improving the understanding of the behaviour of gas atoms subsequent to energetic penetration of the crystal lattice. In these studies the gas species of interest is injected into a solid substrate and subsequently released by scheduled tempering of the target up to a temperature sufficient to ensure emission of all the implanted gas. The rate of release of gas is monitored during the tempering schedule and detected maxima in the release rate may be associated with particular trapping sites, predicted for the gas solid system in question. The tempering schedule used may be of any convenient form and several have been discussed.3 In practice, however, although the reciprocal schedule (l/T = I/T, - bt) is probably the simplest to deal with analytically, the linear schedule (T =To -in,) is the most easily realized experimentally and therefore is the one most frequently employed. The importance of the understanding of diffusion processes in these experiments has been long understood and was examined by Farrell e/ UP and Farrell and Carter’ in 1966. It is only recently, however, that the possible importance of the perturbations to desorption spectra, caused by gas retrapping during the heating schedule, has been discussed” in the literature. In a previous communication’ Carter has examined the retrapping phenomenon for the case of the reciprocal tempering schedule. The present work is concerned with the less analytically amenable case of the linear schedule.
Theoretical
27/number
D = D,
Pergamon
Press/Printed
Great
exp(
-
Q/RT) R
T= To -i-at we can centration
rewrite equation (1) to give the variation of the of diffusing atoms with depth, x, and temperature
c72C aA2 - -exp(Q/RT) (7gz - Do It has been form :-
C=O,x=
s =O,
.g
shown’
D’
(where A and conditions:-
of gas at a solid. distrif is in
(1)
where Q is the activation energy for the diffusion process and is the universal gas constant. Thus, if we allow temperature to vary with time according to the relation
C =O,
1.
- -. -c y7 1. (7.r L2i2
where C = C(x.f) is the instantaneous concentration of untrapped atoms. The diffusion coefficient, D, varies with temperature, T, through the relation
considerations
The process we wish to consider is the random migration initially located with a planar concentration Co cm-’ plane p lattice units below the surface of a semi-infinite The solid is assumed to contain a depth independent bution of unsaturable traps whose planar concentration Vacuum/volume
?I
that
are
conT:
+$ the
solution
constants),
to an
satisfying
equation
the
of
the
boundary
all Z co,allZ
c=cc,s=pA,z=o f0”Cd.x = C,,, Z = 0 Britain
21
S E Donnelfy
and
Diffusive
D G Armour;
release
of gas from
a solid
in the presence
of traps
using
a linear
tempering
schedule
312
is :-
(3) This becomes an appropriate is made identical with the X/aZ made identical with:-
Hence, equation (3) appropriate boundary
solution pre-exponential
is a solution conditions
for
equation Do and
of equation if:-
(I)
satisfying
the
(4) Provided that 7 is not less than a few tens of degrees above and not greater than about Q/20R. then an approximate solution for equation (4) can be shown to be:Z =
$
Hence,
T2 exp(
-
diffusing appropriate
r,,
Q/RT)
substitution
D,, = 4
of
where atoms, yields boundary
kJ’,
C(x,T) =c,
this expression for Z and is the vibration frequency as a solution to equation (2) conditions:-
k,
?1
Qa
2nkoi2 RT2 exp( - Q/RT) (s -piJ2 Qa x exp 2k, I2 RT2 exp(QIRT [ ( (s +piJ2 Qa - exp 2k,12RT2 exp( - QIRT) (
putting of the and the
-
-$f$
exp(
This expression is identical and Carte? for the case trapping term :-
ex
koRT2 - 2 -PC2QaL
-
(5)
derived by Farrell multiplied by a
Q/W
which reduces to unity in the no traps limit L+cr_. Now, the parameter which is of primary interest in desorption experiments is the rate at which dissolved gas is evolved from the surface, i.e. the rate at which the diffusing species traverses the x = 0 plane. This is given, from Fick’s first law, by:ac p=D’ax
x=o
Substitution
of equation
p = $[exp(QlRT)lt(erp[
(6) (5)
into
this
gives:-
- $expK?/RT)])
- C’T’ exp( - Q/RT>
22
expression
I)
&[&+3]
= [$($+2)
However, this can leads to the result:-
Q/RT)
to the expression with no trapping,
This is essentially the same expression as that derived by Carter’ for the case of the reciprocal tempering schedule. but with the constant b replaced by the term a/T*. C’-0 and equation (7) In the case of no trapping. L -x. becomes identical to that derived for the diffusive release during tempering without traps.5 Inspection of equation (7) reveals that the release rate initially increases with increasing temperature until it reaches some maxtmum. pn, at a temperature r,,,. after which it decreases with further increase in temperature. The form of this release rate is similar both to that obtained for the no traps caseJ.s and to that obtained for the reciprocal schedule in the presence of traps.’ The temperature, T,,. at which this maximum occurs and the value of p,,, can be analytically determined by difierentiation of equation (7) and setting the derivative equal to zero. This leads to the somewhat complex relationship:-
- (7
+ 2C’Tvt) (9)
I
x exp
C’ = koR 2QaL’
(2) if D’ the term
Q
7 RTn,
exp
Q
RT,,
2
be solved
for
the
quantity
f-
4/12 3 L2
>I
Q/RT,,, which
(10)
This is identical to the expression derived by Carter’ for the reciprocal schedule, but again with the constant b replaced by the variable a/T’. Also, it is readily seen that equation (IO) relaxes to Farrell and Carter’s expression5 for diffusive release without retrapping, in the limit L-t-p. As in the earlier studies, this expression indicates the almost linear relationship between the temperature at maximum release rate, r,., and the activation energy, Q. This relationship is illustrated in Figure I for gas initially located at a depth of 100 lattice units, released using a sweep rate of 20K s-t. It can be readily seen from equation (IO) and Figure I that it is only for values of p2/L2 considerably greater than unity that the presence of traps significantly perturbs the temperature at which the release rate maximum occurs. For a defect probe experiment, wherep was of order 10, even with moderately high concentrations of traps this perturbation would be sufficiently small (0.03 % for L = 100) to be negligible when compared to the errors inherent in the experimental temperature measurement (perhaps 50.1 *A at best). The maximum evolution rate, P,,,, can be evaluated by substitution of the maximum rate criterion in equation (7) giving the results:(11)
(7)
S E Donnelly
and
D
G Armour:
Diffusive release of gas from a solid
in the
presence
traps using a linear tempering
of
schedule
e< I and relaxes to that derived for the no traps case when p2/L2 --f 0. Solution of this expressionis complex and has to be carried out graphically for each value of p2/L2. It reveals two roots, y, and y2, wherey1 correspondsto T,,, which occurs before the rate maximum and y2 correspondsto TE.., which occurs after the maximum. These roots are displayed in Figure 2 p2/Lz
. octlvotion
I 50 energy
I too tor‘&iffurion,
Q
J I50 ( k cal I mole1 y, -
Figure 1. Variation in the temperature, T, at which maximum release rate occurs with activation energy, Q, for different values of intertrap spacing, L.
before
Y2 - alter
maximum maximum
which relaxes to the expression obtained by Farrell and Carte? 1
3
I
I
KS’
I
I
I
tcr’
I
IO’
$1 L’
traps limit and can be approximated to
in the no
p”!
=
Figure2. Variation in temperature width parameter (y) with trapping parameterp2/L2.
--co eu
>(2+
T,2
Y
wherex
= -
2
in the case of high trap concentrations and deep lying initial distributions (p/L > 1). A further parameter of interest, which can be evaluated analytically, is the temperature ‘width’ of the release rate transient which may be defined as the difference in temperature between the points at which the evolution rate is e-l of the maximum rate. Thus if fe = e- ’ P,,,,then from equation (7):-
Pe=$[exP(+$)][ev( [exp( - C’T:ev( = e-l $[-4&-j$-)][(exP(
- ~exp($))I - $))I - ++$--J
(13)
1
Rearrangement of this expression and substitution of the maximum rate criterion leads to the following expression :-
which becomesidentical to that for the reciprocal sweepfor
for a range of values of pz/Lz. It can be readily be seenthat, as in the reciprocal sweepcase,y, is only slightly perturbed by the presenceof traps, whereasy2 is noticeably reducedeven for low trapping levels. For large values of p2/L2, yz is significantly decreasedwhile y,. though also reduced, is affected to a much smallerdegree. 2
e.g. for E- = 1
L2
y2 is reducedto 45% of its value with no trapping, whilst y, still retains 83% of its no traps value. Thus, the analysisso far showsthat, in the absenceof traps, the releaserate transient risesto a maximumP,,,at temperature T,,, and subsequentlydrops off towards zero with a fall time somewhat longer than the rise time. The presenceof traps reducesthe maximumP,,,accordingto equation (11) and moves it to lower temperatures(Figure 1). It also reducesthe temperature ‘width’ of the curve asymmetrically such that the slowly decaying ‘tail’ of the transient is affected considerably more than the faster rising portion of the curve before the maximum. In order to quantify theseconclusionsand to present the information in a form more amenableto comparison with experimental work, a numerical evaluation of equation (7) was performed for various values of trapping parameter,L2, and the diffusion starting depth, p. Thesedata are displayedin Figures 3 and 4 and illustrate, in some detail, the trends already derived analytically. In particular, comparison of
23
.S E Donnelly
and
D G Armour:
Diffusive
release
of gas from
a solid
in the presence
temperature
3. Variation
of desorption
transient
with
trap
4400 (K)
Cl=50 a ~20
tempering
schedule
420-
.---
initial
concentration
In an experimental system in general, the dill’using species will not be initially located in one discrete plane. but will be distributed over a range of values of/>. Unfortunately, even when assuming comparatively simple forms for the initial distributions. the resulting equations are analytically intractable f-01 non-constant D. However, provided that gas atoms originating from dilferent planes do not interact in any way with each other. the transient. resulting from a distributed initial concentration can be obtained by a summation of the contributions from each of the planes contained in the distribution, i.e. for a general distribution Co = Co(p) with all the parameters as defined above :-
no rctroppinq
p: @=
kccl/molr K s-’
a linear
concentration Distributed
0)
using
~~
(ooo
Figure
of traps
Slatticc
unit,
@I=40
p = i
A(I’P$/‘)
[exp($)-J
[exp(
- yesP(j$))]
1) / [exp(
4000
Figure depth.
4. Variation
400
temperature
of desorption
transient
4200 (K)
with
diffusion
starting
Figures 4a and b reveals that even for moderate trap levels retrapping effects have a significant influence on the release characteristics of gas from initially deep layers. e.g. for Lz = lo”, p = 100 (curve (7) Figures 4a and b) 95% of the diffusing gas atoms are trapped.
24
- C’T’exp(
- $))I
(15)
The results of such a summation, carried out numerically for the distribution C,(p) = p exp( --p/IO), whose maximum lies in plane p = 10, are illustrated in Figure 5. The summation was carried out over only the first 100 planes, but as the value of C,(p) has dropped to about 0.1 y’, of its peak value at plane p = 100, little accuracy is sacrificed by neglecting contributions from deeper lying planes. Inspection of Figure 5 reveals that the trends already discussed for the case of the planar initial concentration are also evident for the ‘x e-“’ type distribution and that all the trends are accentuated. For example, comparison with Figure 3 shows that for L2 = 100, for instance, T,, is altered by 5.5%. P,,, by 51% and the temperature ‘width’ by 50% for the distributed initial concentration, whereas for the planar concentration, these values are 1.6, 29 and 44% respectively.
Conclusions This analysis has shown how expressions relating to maxtmum, pm, the temperature of the maximum, T,,,, and
the the
S E Donnelly
Figure
and D G Armour:
5. Varia[ion
;,\. (,- \ 11) ;_
or
desorption
Diffusive
transient
release
with
of gas from
trap
a solid
cnncentration
tcmpcraturc ‘width’ of the dcsorption transient. resulting from a linear heating schedule, can be derived for gas initially located in a plane below the surface of a semi-infinite solid. The work has been cstendcd by numerical analysis so that the precise shapes of the dcsorption transients can be obtained for various values of the paramctcrsp and L and so that the effects of rctrapping on the release rate curve for an extended distribution can also be observed. As was the case for the reciprocal schedule.’ the solutions obtained always relax to the previously derived values4..5 in the limit where L-.-r. For low densities and shallow, narrowly distributed initial concentrations the spectra obtained from thermal dcsorption experiments will fairly accurately reflect the populations and activation energies for migration of the original trapping sites, However, for deep lying implants. cstcndcd distributions and high trapping level. interpretation of dcsorption spectra becomes extremely diflicult. In particular, for the type of defect study where low energy light ions arc used as a defect probe in solids previously damaged by heavy ion bonlbardnlent’O.” quantitative. or even qualitative, description of the damage state of the target may be impossible unless this retrapping phenomenon is taken into account. In fact, if the trap concentration is high and the initial light ion distribution broad, then observed peaks in the spectra may arise from gas initially released which is retrapped and subsequently rc-released at higher temperatures and may in no way reflect the immediate post-bombardment trapping state of the solid. In addition, if thermal dcsorption data are to be used to make precise measurements of activation energies for detrapping processes within the solid, this work illustrates the necessity of taking into account all the parameters relating to the desorption transient (i.e.p,,, r,, and the temperature width) before any unambiguous determinations can be made.
in the presence
for
gas
initially
of traps
in
an
using
‘.v
a linear
lypc
tempering
distribution
schedule
ccntred
on
P = 10
In the light of the importance of accurate quantitative interpretation of defect probe data for the understanding of gas cluster and bubble formation, it is vital to be able to deconvolutc experimental dcsorption spectra in terms of the numerous parameters which influence their shape. To this end, the present work is being extended and co-ordinatcd with experimental work currently in progress.
Acknowledgements We wish to thank D C Ingram of the Electrical Engineering Department, University of Salford for helpful advice on the numerical analysis. One of us (S E D) would also like to acknowledge the provision of financial support from the UKAEA (AERE Harwcll) for the duration of this work.
References ’ E V Kornelsen and D Edwards, Proc I~rrerrraf C’ou/’ 011 Applicafiott oJ‘hr Beams IO Metals, Albuqrrerque (1973). ’ D S Whitmell and R S Nelson, Rad Efl, 14, 1972, 249. 3 G Carter. S.wposhon o/r ha// Mass Spec~ome/ers, University of Salford (1970). ’ G Farrell, W A Grant, K Erents and G Carter, k’ncrtroa, 16, 1966, 295. ’ G Farrell and G Carter, Vacuum, 17, 1966, 15. 6 D Edwards and E V Kornelsen, Rad Ef, 26, 1975, 155. ’ G Carter, Vacrtum, 1976, 329, ’ R Kelly and Hj Matzke. A’& Mawr, 20, 1966, 171. 9 G Carter, D G Armour. C Braganza, S E Donnelly and G Farrell, 1976 Symposium on the Physics of Ionized Gases, Dubrovnik-to be published. ” E V Kornelsen, Rad Eff, 13, 1972, 227. ‘I S K Erents and G Carter, Vacuum, 17, 1967, 215.
25