The deduction of continuously distributed activation energy site populations from tempering schedules

Vacuum/volume32/number 5/pages 233 to 241/1982

0042-207X/82/050233-09503.00/0

Printed in Great Britain

Pergamon Press Ltd

The deduction of continuously distributed activation energy site populations from tempering schedules G C a r t e r , P Bailey, D G A r m o u r and R Collins*, Department of Electrical Engineering, University of

Salford, Salford M5 4WT, UK received 20 July 1981

The desorption rate-temperature behaviour of adatoms from sites of unique desorption energy E, obeying first order rate reaction kinetics, is analysed for the case where temperature T is varied according to a general power law tempering function d T / d t = rpTP. The analysis is then generalized to the case of adatoms initially populating sites of continuously distributed activation energy. It is shown, via several approximate and one more exact analysis, that the desorption ratetemperature function provides a close approximation to the initial population distribution. It is then demonstrated that use of both rapidly varying tempering functions and a variety of tempering functions, particularly the case of the exponential function p = 1, allows optimized determination of the initial population distribution.

Introduction

Although now complemented by many alternative sophisticated surface analysis techniques the study of thermally induced evolution ofadsorbed (or absorbed gases) still provides useful and fundamental information on gas-solid interactions, ofimportance in vacuum and allied science and technology. Recent publications= 4 have shown how developments in both instrumentation and data analysis have led to improvements in sensitivity and precision in the deduction ofthe kinetic parameters of the gas desorption process, particularly when adatoms are trapped in sites of discrete activation energy for desorption. In heterogeneous systems it may be anticipated that activation energy discreteness is blurred and that a more appropriate description is of a continuum or spectrum of activation energies for desorption. One such system appears to be that of inert gas atoms injected as low energy ions into glasses s'6. The technique of thermal evolution spectrometry, in which the total rate of gas evolution of a defined species is measured as a function ofelapsed time during constant temperature (isothermal) and controlled temperature increase (tempering) conditions has been analysed in detail theoretically by a number of authors 7'8"4 for conditions of discrete activation energy but much less attention has been paid to the case of distributed or continuous activation energy spectra. Early studies of such kinetic rate processes under distributed activation energy conditions for both isothermal and a specific tempering schedule were published by

Vand 9 and Primak =° whilst Erents et a/t~ undertook a further analysis for isothermal conditions, and Carter 8, Grant and Carter =z and Erents et al =3 developed the analysis for two tempering schedules. These tempering schedules were the linear function T= To+rot

and the hyperbolic function T - t = T o i + r2 t

(2)

where T is temperature after elapsed time t, TO is the initial temperature and the r 'S are tempering rate constants. The ability of the tempering method to resolve, via evolution rate measurements, differences in discrete activation energies was first discussed by Carter ~ for desorption rate processes obeying first and second order kinetics and by Farrell et al ~4 and Farrell and Carter =5 for gas evolution following diffusion processes in the bulk, whilst Carter, et a/=6 considered both evolution processes when the tempering schedule was of exponential form T = TO exp(rtt).

(3)

Subsequently Carter and Armour ~2 analysed rather more exotic tempering functions. It is readily deduced that the tempering functions of equations (1)-(3) belong to the class defined by dT d--t- = rp. T p.

* Department of Mathematics, University of Salford, Salford M5 4WT, UK.

(1)

(4)

It is the purpose of the present communication to develop a

233

G Carter et al: The deduction of continuously distributed activation energy site populations from tempering schedules

preliminary analysis of the time dependent total evolution rate from a continuous distribution of desorption activation energy states when exposed to the general tempering ~nction of equation (4). Attention will be focused at this stage upon processes undergoing first order rate reaction desorption kinetics alone although extension to higher order desorption and to diffusion limited situations is straightforward. It will be shown how initial populations (t = O) of adatoms in the activation energy continuum can be derived from evolution rate measurements and optimum tempering functions for different population distributions indicated. Attention will be drawn to the limitations imposed by experimental system parameters.

~.O

n

0,5

Theory Initially we consider desorption from sites of a single activation energy E. If at time t the number density of atoms is n, the desorption rate equation is given by dn

ni

(5)

where i is the order of the kinetics and a~ a temperature dependent factor. For first order kinetics (i= 1), the topic of this analysis, ai is normally assumed to be given by the Frenkel equation (6)

a i = z = z 0 e x p ( E / k T)

where r is the adatom residence time at temperature T, the preexponential factor z 0 is assigned a value of about 10- ~3 s and k is Boltzmann's constant. Is this description E is measured in eV and T in Kelvin. Under conditions of varying temperature (tempering), temperature is a prescribed function of time e.g. (7)

T=f(t)

for which equation (4) is a special case. The formal solution of equations (5)-(7) is readily deduced as n(T,t)=noexp

-

dt.--I exp-

E

.

(8)

TO

and the instantaneous rate ofadatom depopulation or desorption is

p(T, t) =--n° exp to

kf(t)

dt . - - exp to

1 - E .Zo exp k f ( t )

is of determining importance in characterizing the depopulation as a function of temperature and time and is known as the characteristic function 9'1° (CF1). This characteristic function can be determined 4"~'a exactly analytically for the hyperbolic tempering function (equation (2)) and to a close4'7-1° approximation analytically for the linear tempering function (equation (1)). The qualitative behaviour of CF1 as a function of temperature is depicted in Figure 1 for the exponential tempering schedules for a fixed value of the activation energy E. The general behaviour of CF1 is that it remains close to unity (i.e. n"-n o and little 234

435

450

465

480

495

5(0

I

525

T (K)

Figure I. The characteristic function CF 1 (n/no) as a function of increasing temperature T for the exponential tempering schedule dT/dt=rzT. Parameters: E-~ 1.25 eV; r = 2 x 10-2 s-t; Zo=10 -~3 s. depopulation) for temperatures less than T,,,(E) then declines rapidly with increasing temperature through a point of inflection at T = TIn(E) towards zero (complete depopulation) for T > T,,(E). The temperature TIn(E) at which the inflection point occurs increases with activation energy E, but is relatively independent of the form of tempering schedule employed for each activation energy. The form of CF1 however is dependent upon the tempering schedule employed. For the linear schedule it is found that CFI reclines with increasing activation energy E and thus TIn(E) whilst for the hyperbolic schedule it is found that CFI steepens with increasing activation energy E. In the case of the exponential tempering function (equation(3)) it is readily shown ~6 that CFI is independent of energy and maintains a uniform shape independent of E, T and t. Thus Figure 1 is representative of CFI for all values of E and is merely shifted on the temperature scale with increasing E. From these three examples where the tempering rate dT/dt is proportional to T-" (linear), T ~ (exponential) and T 2 (hyperbolic) respectively it is clear empirically that as the exponent p in the general power law tempering schedule (equation (4)) increases so does the C F l function increase in elevation both for a given energy E and with increasing energy E. This is in fact a quite general result and will be derived semi-quantitatively subsequently. Equation (9) enables us to define a second characteristic function (CF2) given by

(9)

where 17o is the initial population in sites of energy E. The factor ~odt

o

which is clearly the time derivative of CF1 multiplied by the constant zo. The behaviour of CF2 with increasing activation energy can thus be readily deduced from Figure 1 by time differentiation. The form of CF2 again depends upon E and the tempering schedule. For all tempering schedules CF2 rises from (near) zero for temperatures T < T,,(E) reaches a maximum at T = T,.(E) and then declines towards zero for T > TIn(E). For the linear schedule 7 1o this peak structure of CF2 becomes broader with increasing energy (i.e. the temperature width AT(E) increases with E) but the peak height decreases with increasing E. For the exponential schedule 16 both the peak width and peak height are independent of E and indeed the whole form of CF2 is independent of E. For the hyperbolic schedule 7'8 the peak width decreases but the peak height increases with increasing E.

G Carter et al: The deduction of continuously distributed activation energy site populations from tempering schedules

Qualitatively therefore it may be concluded that the product (peak height x peak width) remains constant for all values of E and all tempering schedules. This may be proved analytically as will now be demonstrated for the general power law tempering function of equation (4). If equation (4) is integrated for arbitrary p the tempering schedule becomes

T 1 -P = T~-'P+ (1 -p)rpt.

(10)

except for the (important) case of p = 1 which gives rise to the exponential function of equation (3). This general tempering function may now be substituted in equation (9) and the CF2 function determined at least to a first order analytical approximation. Before undertaking this analysis however it is useful to examine equation (9) for the condition for the maximum value of p(T, t) and hence of CF2 by requiring that dp/dt = 0 (and d2p/dt 2 is negative). This derivation has already been undertaken elsewhere .7's'~6 with the general result that p achieves a maximum value at a temperature T,, given by the identity. E

=--1 exp -

dT - ~ - Tm

(11)

TO

~ log

1 1 dT I

l ldT

-d/-T

will not dominate the logarithmic term of equation (12) and as a general, approximate, result equation (13) is a valid result for all (reasonable) values of p. Turning, now, to evaluation of equation (9), this can be rewritten as

To

(

kT

zo

o

l exp( )}

"dT/dt

(14a) and for the general power law tempering schedule of equation (4) ro

(

kT

zor~

odT'T -pexp ~

Changing the variable in the integral by the substitution y = E / k T one deduces

To

Torp

(12) fr[ d y . y p - 2 e - y } .

In order to deduce x (and thus the relationship between E and T,,) it is necessary to determine the roots of equation (12). However for the linear, exponential and hyperbolic schedules and reasonable values of the parameter p in equation (4) which may be achieved experimentally, it has been deduced that to a very good approximation x ---30, and is almost independent of T,.. Indeed for the exponential schedule, since

Ts To =rl'

E

(15)

Remembering the qualitative behaviour of CF2 described earlier that for T,~ T,,,, p ",-0, then for any temperature T > To, the lower limit Yo in the integral of equation (15) can be replaced by an infinity. This integral can now be integrated by parts exactly and even for the case p=0, 1, 2, an approximate solution is achieved 4.7.s'x2-16 and is given in general by - -1

P(T't)=n°exp{ -y-

TO

equation (12) reveals that x is totally independent of T,. Thus for these three schedules at least, one may write

TIn(E)=30 k •

. (14b)

o(T, t) = no exp - y + -

or, writing E/kT,.=x and taking logarithms x+logex=

Thus

(k-) t-p yP-2e-Y t

(16a)

Tot p

or (13)

This essentially arises because, except for very large values of

T'mm dt T,,, the extremely large value of

1

__,,, 1013 s - t TO completely dominates the logarithmic term of equation (12). Indeed the factor 30 in equation (13) results from log =10x3. Even for larger values of p in the general power law dependence such that the contribution of

1 d_~_r" may be thought to assume importance, experimentally in order to achieve tractable tempering schedules the parameter rp will be reduced.

p ( r , t ) = n O exp{ rO

E kT

d([~[_)-' k T 2

1 zO f

E,~

exp~ - k-~)j~.

(16b)

In this solution only the first term in a series expansion of the integral of equation (15) is employed since, as already demonstrated, only for values of T"- Tmis the integral large. Near T,., y~-E/kT.~-30 so that the leading term in powers of y is sufficiently acceptable ~2-14. Recalling the criterion of equation ( 11 ) defines the temperature T,. for desorption rate maximum and substituting this condition in equation (16b), it is deduced that the desorption rate maximum for energy E, is given by

pm(E)=-no e-

1 E

dT[

. kT 2 ~--.,

(17a)

This general result has previously been derived for p = 0 and 2 .7.9 and p = 1 ' 6. Furthermore, substitution of the relationship for the temperature T,.(E) for a given energy E of equation (13) results in 235

G Carter et al: The deduction of continuously distributed activation energy site populations from tempering schedules

pm(E)=n o . rp . e -1 . (30) p - 2

(17b)

Equation (17b) reflects the qualitative conclusions outlined earlier that; for p = 0 (linear schedule), P,~ decreases with increasing E; for p = 1 (exponential schedule), p,~ is independent of E, whilst for p > 1 (hyperbolic and higher power law), Or. increases with increasing E. The temperature'width o f each desorption peak can also be deduced by further time differentiation of equations (9) or (16b) and thus evaluation of the turning points of the p(T) dependence, with, for arbitrary p somewhat more complexity than the deduction of T,.. The following simple argument is equally valid however. Since the time integral of p(T, t) is merely the total initial population n 0 the product p,.(E). AT(E) also remains constant and, of order n o. Thus from equation (17b)

" A T ( E ) ~ - e . r ; 1( 3 0 ) z - P ( E ) I - P .

(18)

This result further confirms the qualitative conclusions discussed earlier that; for p = 0 , the rate function (CF2) increases in temperature width with increasing E; for p = l the temperature width is independent of E whilst for p > l the temperature width decreases with increasing E. This behaviour is also reflected in the declining or steepening behaviour of C F I for different values of E and the tempering function exponent p, As will be noted shortly this behaviour is important since it implies that the larger is p the more nearly does C F I approach a step function and CF2 approaches a delta function both located at TIn(E), Equation {18) is also important since it reveals that the temperature width of the rate (CF2) function decreases with increasing energy E for p > 1 whereas the temperature TIn(E) is approximately linearly related to E for all p. Thus the larger the p value employed the better are individual rate spectra for states of neighbouring energy E and E + AE discriminated on a temperature scale, i.e. the energy resolution of the system E/AE improves with increasing p. This result may already be intuitively inferred from earlier studies for linear, s'1'~'15 exponentiaP 6 and hyperbolicaa4.t s schedules. We turn now to consideration of distributed or continuous activation energy spectra. In particular we consider first order depopulation kinetics from non-interacting states with variable but continuously distributed activation energy E but constant time 'parameter' Zo- All of these assumptions may be modified and subsequently analysed with differing degrees of facility. State noninteraction implies that adatoms exist only in two states, either bound to a surface site with defined desorption activation energy E or in the gas phase (vacuum) free state. No activated state transfer occurs. At time t = 0 the initial state occupancy in sites with activation energies in the range E to E + AE may be prescribed as no(E) AE, where no(E) is some arbitrary function of energy E subject to a normalization condition

: dE, no(E)=N o

(19)

where No is the total initial adatom population summed over all states. 236

For adatoms in the energy range E ~ E + A E the residual population n(E, t) and the instantaneous desorption rate p (E, t) are given by equations (8) and (9) respectively. Thus the total instantaneous residual populations and total desorption rate from all states are given by integrals of these equations over energy Thus

n(T, t) =

dE. no(E)ex p -

dt . - - exp To

d E . n°(E) exp ~ TO

-

(20a)

a~d

p(T, t) =

dt - - exp TO

.

"

(20b) As already discussed, and evaluated in equation (16) the exponential (CF1 and CF2) functions of equations (20a and 20b) are, for fixed energy, functions of temperature T. Thus these characteristic functions are for fixed temperature T, defined functions of energy E (CF3 and CF4). Thus if we write these characteristic functions as hi(E, T) and h,_(E, T) respectively equations (20a) and (20b) reduce to

n(T, t) = Jl ~ dE. no(E)hl(E, T)

(21a)

and

p(r, t)=

dE. n°(E) hz(E, r). To

(21b)

The physical meaning of the characteristic functions in the context of distributed activation energy conditions now becomes clarified in that they convolute, transform or map the initial adatom spectrum in the energy domain into a residual total instantaneous population or total depopulation (desorption) rate in the temperature (and thus time) domains. The essential questions are: what is the nature ofthis mapping? and, if this can be deduced, can total desorption rate piT), generally the most readily observable experimental quantity 4"7's, measured as a function of temperature be employed to determine the initial population distribution no(E)? These questions can be answered affirmatively and have been approximately analysed for the linear tempering schedule (p=0) 8a°'12'z3 and the hyperbolic tempering schedule (p = 2)a.12.~3. In the succeeding discussion similar analysis will be developed for the general power law tempering function of equation (4). Before detailing this analysis however, a useful and indeed rather accurate zero order approximation, introduced by Vand 9 and Primak ~° will be outlined and further developed. It has already been demonstrated analytically that, for all p, the CF1 function declines rapidly from unity to zero centred upon a temperature T,,(E) and of width AT(E) whilst the CF2 function is very sharply peaked about Tm(E). Vand 9 and Primak ~° suggested that as a zero order approximation CFI may be regarded as a step function at TmlE) and consequently CF2 as a delta function at T,,(E). This was in fact proposed for the linear tempering schedule (p---0) but as we have demonstrated earlier the proposition is, in fact, even more acceptable for p > 1 and particularly as energy E is increased. A somewhat more precise first order approximation to CFI would be a linear ramp function centred upon T,,(E) and of

G Carter et al: The deduction of continuously distributed activation energy site populations from tempering schedules

width AT(E) and for CF2 a triangular function centred upon T~,(E). In fact the more detailed subsequent analysis will reveal that, for all p, even the zero order approximation is quite acceptable. In this zero order approximation equations (21a) and (21b) become respectively

Finally, therefore, dT

p(t) = As(30k) s+ ~T s d t

and for the general power law tempering function of equation (4)

p(t) = rpA~(3Ok) s+ 1 T~+p. n(T, t ) =

fo•

=

d E . no(E). H ( E - E

l)

d E , no(E)

(22a)

!

and

o( T, t) = n°(E) . 8(E - E ~) "ro

(22b)

where H ( E - E I ) is the step (up) function with unit step height at a temperature T corresponding to T=-E~/30k and f ( E - E ~) is the delta function at a temperature T corresponding to T-E~/30k. The physical meaning of these equations can be clarified as follows. At any temperature T, all adatoms with energy E < 30k T have been depopulated from the surface and all adatoms with energy.

(24a)

(24b)

For the linear (p = 0) and hyperbolic (p = 2) tempering functions it is thus deduced that p(t)oc T" and p(t)oc 7~.2, results which have already been obtained 12'1z by a more complete numerical analysis of equation (20b). It may also be observed that for p = 1 (the exponential tempering function) p(t)oc T ~* i, i.e. p(t)ocE ~+ I. Thus for the linear tempering function the measured desorption rate function reveals directly the (closely) approximate nature of the initial population distribution. This analysis can be extended to more general initial population distribution functions which may be approximated by power law series expansions ~2'13 i.e.

no(E ) = ~ A~E s $

with the result

p(t) = ~ rpA~(3Ok) s+ 1 . T z +p

(24c)

$

E>3OkT(i.e. f3~krdE.no(E) )

or

remain adsorbed. Thus the tempering ramp acts as an opening gate which continuously admits increasing desorption energy atoms to the gas phase. At any temperature all atoms with desorption energy less than a value linearly related to that temperature will have been admitted to the gas phase. In terms of desorption rate, the tempering ramp acts as a window which selectively admits adatoms of unique desorption energy to the gas phase, this unique energy being linearly related to instantaneous temperature. The more precise interpretation, in view of the peaked behaviour of CF2, is that the window is of variable 'height' (sensitivity) and 'width' (resolution) according to the activation energy explored and the tempering function employed. Thus the window tends to blur the precise no(E) distribution. Pursuing the zero order approximation a little further it may be noted from equation (22a) that, as temperature increases by an increment fiT so do sites in an energy band fE = 30k f T become depopulated. Thus the atoms admitted to the gas phase increase in number by no(E). 30k fT. Thus the 'temperature' rate of admission to the gas phase dn

-dT = n°(E)" 30k and the time rate of admission, the desorption rate p, is given by dn

dn

dT

p ( t ) = d--[ = d-T" d---i-=

30k

n E) d T

• o(

(23)

. ~-~- .

Thus the tempering ramp directly maps the initial desorption activation energy population spectrum into the desorption rate spectrum. It is interesting to deduce the appropriate results for initial power law activation energy distributions 12"13, e.g. no(E) = A~Es. It has already been shown that energy E is equivalent to 30kTand thus n o(E) =-A~(30k T)~.

p(t) = ~ As(30k)S + 1T ~. dT/dt

(24d)

Thus fitting of the instantaneous measured total desorption rate: tempering rate ratio to a power series expansion in temperature enables directly evaluation' 2.13 of the constants/I s in the initial population distribution. An interesting case is that of an initially uniform population distribution (s = 0) for which p(t)= rp(30k)Tp. For a linear tempering function (p=0), p(t) is independent of temperature, for the exponential tempering function p(t) is a linear function of temperature and for larger values of the tempering rate exponent, p, the rate increases with time proportional to TP. It is also notable that for this uniform initial distribution that the approximate value of the constant of multiplication, 30k. rp in equation (24b) has been shown12.13 to be rather accurate for p = 0 and two by full analysis of equation (20b). Although the zero order analysis will be shown, shortly, to be rather good, it is also clear that it will be best for large values ofthe exponent p since the CFI and CF2 functions become more step and delta function like respectively, particularly as energy E increases. For uniform initial adatom populations the linear tempering function should be quite adequate since, although the CFI function reclines with increasing energy (i.e. the energy window of depopulation widens), adatoms with energy slightly less than 30k T which are not depopulated at T are approximately compensated by equal depopulation of sites with energy slightly greater than 30kT. Thus the CFI function may be regarded as approximately a linear ramp of temperature, time (and energy) varying slope which sweeps through the initial uniform population distribution and depopulates the area beneath the no(E)/E distribution at a constant rate. For initially non-uniform population distributions which increase with increasing E however this reclining linear ramp CFI will be less appropriate since, at a given temperature (and equivalent E) there will be a small depopulation of the larger number of initial sites with energy greater than E to 237

G Carter et al: The deduction of continuously distributed activation energy site populations from tempering schedules

compensate for non-depopulation of lower energy sites, Qualitatively, therefore, one could suggest that for energy increasing mmal population distributions, the functmn ofCF 1 (or its complement to unity which is the depopulation ramp function) should either maintain constant gradient or preferably steepen with increasing temperature. For the general tempering function of equation (4) the temperature peak width of CF2 is proportional, as shown in equation (18), to T t -P or E t -P. Thus the gradient of CFI is proportional to E ~- ~. The energy gradient of a power law initial site energy distribution is proportional to E ~-~. It may now be inferred that an optimized fidelity of transformation of the population distribution into a rate distribution could be achieved by matching the gradients of the ramp depopulation function and the initial situation energy site population distribution, thus s=p. This qualitative argument concurs with the observations for a uniform initial population for whil:h s = 0 and for which the linear tempering function (p = 0) was shown to map directly from the energy to the temperature domain. Thus for a site population, linearly increasing with energy ~tn exponential tempering schedule would be optimum; for a square law distribution, a hyperbolic function would be optimum etc. This identification would also indicate that for population distributions which increase less rapidly than with energy (e.g. fractional power functions of the type E~/s)less rapidly increasing tempering functions of the form dT/dtoc T~/s would be optimal. Although these suggestions have not been proved formally it is clear, from the earlier analysis and the qualitative discussion, that total fidelity of an initial population distribution will not be presented in the desorption rate domain. It is therefore suggested that for the same initial population distributions repeated experiments should be performed employing differing tempering functions of the form of equation (4), i.e. varying p and r~ and comparing the results with the predictions of equation (24d). In this way it should be possible to deduce an optimum approximation to the initial site population distribution. Turning next to a more detailed analysis of the more precise form of the desorption rate function given in equations (20b and 21b). From the analysis for unique energy conditions h2(E, T) is given by equation (16b). Thus combining equations (16b) and (21b).

p(T, t) =--1 I ~ d E . no(E)ex p To do kT

z o ~,-dt-]

~exp~

.

(25)

If the factor ( d T / d t ) - L T were absent in this expression the exponential term would behave, with respect to variations in E, in an identical manner as with respect to T. Thus the mapping function h2(E, T) at a fixed temperature T would behave in an identical manner to CF2 with respect to temperature at fixed E. In the case of the exponential tempering function (p=l), T(d T/dO- ~= r~ ~and the h2(E, T) function is in fact identical in E to the CF2 function in T. It also turns out that even for other tempering functions the two functions ha(E, T) and CF2 are almost identical with respect to their prime variables. That this should be so may be illustrated by differentiation of h2(E, T) with respect to energy E and demanding the condition for a maximum with respect to E at fixed 7". This leads to the criterion

E,, d T

1

{-EmXx f k T

]

k T 2 dt = x--oe x p ~ , - - - ~ J l~---£ + 1 . ; 238

(26)

Recalling that, in general, processes occurring at energies E~,~ 30kT will be dominant, it is clear that equation (26) is very closely similar in the variable E to equation (11) in the variable T. This close similarity in h2(E, T) is further confirmed by numerical evaluation of h~(E, T) for fixed temperature, variable energy and specific forms of the tempering function. As an example the results of such a calculation, taken from ref 13, for energies in the range 1 eV < E < 2.5 eV at different selected temperatures during a linear tempering schedule are shown in Figure 2. Quite clearly the peak structure in E is evident with peak height declining and peak 35

30

2s

:y. 2o

%

._~ ~. to

5

o 0.88

1.20 4.52 1.84 desorption energy, E (eV.atom-I)

Figure 2. Theoretical desorption rate vs desorption energy transients for different temperatures. Tempering rate constant re= 1 Ks-1.

width increasing as temperature is increased in common with the temperature behaviour of CF2 for fixed energy and the linear tempering schedule. Indeed the rate maximum criterion of equation(26) with kT/E=~l, when substituted into CF4 (h2(E, T)) reveals that the maximum rate at temperature T from sites of desorption activation energy E is approximately.

Pro(T) .,.no(E,~)e-t • kETm2 d T dt

(27) E.,

which is the energy equivalent of equation (17a). In view of the linear correspondence of E and T, equation (27) indicates that pro(T) is proportional to T~-I as a function of activation energy E. Thus, as shown in Figure 2, pro(T) decreases with increasing temperature for the linear tempering schedule (p = 0), remains constant for the exponential schedule (p = l) and increases with temperature for all other values ofp > 1. The energy widths of the pro(T) functions thus behave in a reciprocal manner to the behaviour of the peak height behaviour. This analysis confirms the qualitative behaviour of the tempering function acting as a variable aperture window in depopulating the initial activation energy site distribution. Although the overall behaviour of h2(E, T) may be described

G Carter et

al: The deduction of continuously distributed activation energy site populations from tempering schedules

qualitatively, the integral of equation (25) cannot ~2'1s be evaluated analytically for arbitrary distribution no(E) and numerical analysis is necessary for specific assumed forms of no(E). If, as considered earlier, a power law initial distribution no(E)= A.,Es is assumed, then by substitution of E/kT=y equation (25) may be rewritten, for T > To as

3.C S=lO

2,.=

p(T, t ) = _( - _l ) s + ' A~(kT)~+ x f o dy(log y)~ "CO

Io

/~o\dt/

I¢

.T.~.

This result corresponds identically for the specific cases p = 0 and 2 evaluated previously ~2'13. The integral in equation (23) must be evaluated numerically for all assumed values of s, T and (dT/dt)-'. It is interesting to note, however, for the exponential tempering function (p = 1), where T dT/dt-l=r~ ~ that the value of the integral must be a constant depending only upon s, rv and ro and not upon T. For this specific tempering function therefore

p(T, t ) = c o n s t a n t . 7 ÷ 1

(29)

a result which is identical in form to the more approximate analysis leading to equation (24bl for p = 1. In general the parameter

1 (dT) -1 3o dt

.T

for typical values of s and v. For the general tempering function of equation 14), l r I_

~.

This integral was previously calculated by Erents et al ~3 and has now been re-evaluated with somewhat more precision. Values of s in the range 0-10, v in the range 109-1019, with r o = 10- ~3 were selected with the results shown in Figure 3 where the ratio 1 (log v)~÷ 1.

I,O

I

5

I0

15

ao

Ioglo

Figure 3. Plot of the ratio IJle= 1/v(Iog ols+ l :

as a function of log lo v for values ofs in the range 0-I0 for the exponential tempering function dT/dt =r, T, v = 1/'¢orI, zo = 10- t~s, r~ =10-6-10J's -l.

increasing in significance with increasing s and decreasing v. In this approximation equation (28) may be rewritten

p(T, t ) = ( -

I) s+l As(kT)S+lrvTP-1(log zorpTp-l) s+l (30a)

If the tempering rate parameter rpTP- 1 is not too large then this equation may be rewritten

TO

0

O -,-....

f~ dy(Io9 y)" exp'(log 'yy)l

in equation (28) may be regarded as an arbitrary variable so that it is necessary to evaluate

v = r;

2.0--

fo

dy(Iog y p exp

is plotted as a function of logto v. The function 1/v(Iog v)~÷1 was chosen as a possible result on the basis of the earlier studies is and the clear requirement that since the integrand is a function of the parameters v and s so should the value of the integral be a function of these parameters. This figure reveals that, to a.first approximation and within a factor of less than 3, for all s,

may be assumed to equal, in magnitude, 1/v(log vp ÷ i although in fact there are departures both with variations in s and in v,

p(T, t)= rvAs(3Ok)~+1T~+p

(30b)

which is identical to the zero order approximation of equation (24b). It may be noted that over any limited range of v, which, for all tempering functions except the exponential ( p = l ) , corresponds to limited variations in temperature, the value of the integral changes by only a small amount for each value ofs. For the exponential function v, and thus the integral, are independent of temperature. The preceding analysis is, in fact, only valid over a restricted range of temperature values since the first term series expansion of equation (15) is valid only for E>>kT. Thus if the initial population spectrum extends over more than an order of magnitude in activation energy so must the temperature employed in order to effect depopulation of all sites. This will require that for the higher temperatures, E/kT, will no longer be large for those sites at the lower energies of the population spectrum. Thus, provided that the initial population distribution extends over less than about a decade in activation energies, the above analysis shows that, even for rather rapidly varying energy distribution functions, both the zero order and more exact analysis agree rather well. Indeed it should be recognized that the 239

G Carter et al:

The deduction of continuously distributed activation energy site populations from tempering schedules

more detailed analysis is itself not precise because of the approximations made in deriving equation (16). Nevertheless Figure 3 does suggest that the zero order approximation is better for lower values of the exponent in the activation energy power function distribution, since the discrepancy between approximate and more exact analysis is less for such cases. This is the result indicated from the earlier qualitative arguments. In view of the quite acceptable match of zero order approximation and detailed analysis it is worthwhile exploring somewhat further the first order approximation outlined earlier, where the CF4 convolution function is replaced by a triangular function. That this is a reasonable hypothesis is confirmed by the results of Figure 2. If it is assumed that, at any temperature T, CF4 increases linearly from zero at E - 6 E / 2 , to a maximum ofp(E) or E and then decreases linearly to zero at E+fE/2, then the total desorption rate at T, from, for example, an initial population distribution of the power form employed earlier no(E ) dE = A,¢E~ dE, is given by

E 'PT(T)=p(E)AsfE_~E/2 dE'{E' (~E -

fiE~2) ,E 1,s

}~ '

+P(E)A~ff+'E/~dEI~(E+'E/2)-E1}(E1)~

(31)

Figure 3) this can be convoluted with trial functions n0(E) until optimum match with the experimentally observed p(T)/dT/dt function is obtained. It may be remarked that if an analytical form for CF4 is required for employment in the numerical analysis, a better fit to CF4, than the approximation of Figure 3 is 1

1

v

v

'~

(34) where B=0.055 and C=0.66. This identity is valid to within 5",,; for s = 0 ~ 1 0 and v~109--*10 ~9. A further beneficial feature of this tempering function may be noted from equation (33) if it is known that the initial population distribution is composed of discrete, widely energy separated states such that the desorption function from each state can be well resolved on a temperature scale and deconvoluted 8. In this case, for a given state, E/kT,, ~- - l o g r o - log rp and thus a plot of 1/% for different coefficients in the tempering function leads to evaluation of both E and log r 0 (the pre-exponential coefficient of the Frenkel equation) from the slope and intercept of this linear relationship. Better accuracy can be achieved by full use ofequation (33) by trial insertion of related values and E and r o until optimized match with the experimentally determined values of T,, are obtained for each value ofrp. Clearly good accuracy will only result for variation ofrp over several orders of magnitude.

If this integral is evaluated and it is remembered that the product

p(E) fiE is a constant (equal to unity) for all values of E and 7", the result is 12

\EJ

)"

(32)

From equation (27) it follows that the energy width BE, at energy E is proportional to E ~-P and thus the correction term of order (SE/E) 2 in equation (32) distinguishing the delta function from the triangular function approximations is proportional to E- 2p. Consequently as energy increases and the exponent p increases the first order approximation tends more closely to the zero order approximation, results already inferred from the preceding argument and analyses. Finally we note that although all the approximations lead essentially to the same conclusion, that the form of the initial population distribution can be derived with reasonable accuracy from the form of the p(T)/(dT/dt) function in the temperature domain, accurate analysis is only possible by exact evaluation of the CF2 and CF4 functions. This is only possible using numerical methods, It should be pointed out that considerable simplification results if the exponential tempering function (p = 1) is employed. Thus, from equation (12) T,, is always exactly proportional to E and is given by the solution of

Elk T,. + log E/kT,, = - log %rp.

(33)

and the constant of proportionality is independent of T (and E). For this reason the integrals of CF2 and CF4 are independent of T (and E) and thus, for any prescribed value of the rate coefficient rp require to be evaluated only once for all energies and temperature. Physically this means that the convolution functions are of constant shape and merely shift linearly upon the temperature scale with increasing energy or linearly upon the energy scale with increasing temperature. Thus once CF4 has been determined by precise numerical evaluation (approximations have been employed here in 240

Some experimental criteria It has been advocated in the preceding discussion, that use of as rapid a tempering function and as wide a variety of tempering functions as possible should be used to obtain optimum knowledge of initial population distributions in a continuously distributed activation energy spectrum. Such a demand places certain requirements on the experimental observation system, which will be analysed in detail at a later date but will be noted here. Essentially in order to effect the rapidity and variation of the tempering functions both the thermal time constant" of the substrate heating system and the time constant of any associated feedback control circuits should be minimized whilst the control system itself should be capable of generating the variety of tempering functions. The former requirement is best met by employing substrates of low thermal inertia (i.e. small mass targets) with a geometry which maximizes the surface area: volume ratio. In this way larger initial adatom concentrations can be employed, thus enhancing observed desorption rate and system sensitivity but minimizing thermal inertia. Control system versatility and rapid response may be best achieved by computer control systems 3. With respect to the data monitoring system it will be necessary to measure rapidly time changing desorption rates. The two system parameters which moderate this observation are the vacuum system time constant and the measuring equipment time constant. For rapidly varying tempering rates the desorption rates will also change more rapidly with time. Thus it is essential that the response time constant of detection equipment should be as small as possible and certainly smaller than any time constant associated with rapid changes in desorption rate. The order of magnitude ofdesorption rate time constant r o may be determined,

G Carter et al: The deduction of continuously distributed activation energy site populations from tempering schedules

at temperature T, from p(T)/[dp(T)/dT] and this in turn can be evaluated for anticipated initial population distributions and proposed tempering functions from equations of the form of (24b) for which ro = rp- 1T ~-~. Thus the desorption rate time constant decreases with increasing rate parameter rp and increasing temperature during tempering. Of equal importance is the vacuum time constant V/S, where V is the effective volume and S the effective speed (or conductance) of a lumped vacuum system into which desorption occurs. Generally when slowly varying desorption rates are studied the system partial pressure of adatoms is directly proportional to desorption rate in a dynamic (pumped) system where the time constant V/s is small. If however r o becomes small by comparison with V/S then even for the dynamic system it is the rate ofchange of partial pressure which will be linearly related to desorption rate. In general terms, for a dynamic system, it will be necessary to record both pressure p and rate of change of pressure dp/dt and employ the vacuum balance equation. dp

V -~ = Sp-p(t)

(35)

for precise evaluation of p(t). Again this requires care in choice of detection equipment to ensure fidelity in measurement of dp/dt. Conclusions

It has been demonstrated that general power law tempering functions may be beneficially employed to deduce initial site populations in situations of continuously distributed desorption activation energy. The role of the tempering process is to map information of the energy distribution into the (experimentally measured) desorption rate-temperature domain. Several approximate and a more exact analytical evaluation have shown

that this transformation process is of reasonable fidelity and thus that desorption rate functions can be readily employed tO obtain rather good approximations to initial population functions. It has been shown that optimized fidelity may be achieved with more rapidly varying tempering functions but that, in order to determine close approximations to the population distribution, a variety of tempering functions should be employed. If precise information of the population distribution is required then numerical evaluation of the energy-desorption rate mapping function must be employed and calculation is simplified by use of the exponential tempering function. Some comments on the implications for experimental methods are made.

References

i j T Yates, Jr, Proc 4th Symp on Fluid/Solid Surface Interactions, Washington (1978). 2 A A van Gorkum and E V Kornelsen, Vacuum, 31, 89 (1981). 3 E V Kornelsen and A A Van Gorkum, Vacuum, 31, 99 (1981). •t A A van Gorkum, J Appl Phys, 51, 67 (1980). 5 G Carter and J H Lock, Proc R Soc, A261, 303 (1961). 6 A O R Cavaleru, D G Armour and G Carter, Vacuum, 22, 321 (1972). 7 p A Redhead, Vacuum, 12, 203 (1962). s G Carter, Vacu,lm, 12, 245 (1962). 9 V Vand, Proc Phys Soc, A55, 222 (1943). to W Primak, Phys Roy, 100, 1677 (1955). 't K Erents, W A Grant and G Carter, Surface Sci, 3, 480 (1965). ~2 W A Grant and G Carter, Vacuum, 15, 281 (1965). t3 K Erents, Vacuum, 15, 529 (1965). ~4 G Farrell, W A Grant, K Erents and G Carter, Vacuum, 16, 295 (1966). 15G Farrell and G Carter, Vacuum, 17, 15 {1966). 16 G Carter, W A Grant, G Farrell and J S Colligon, Vacuum, 18, 263 (1968). 17 G Carter and D G Armour, Vactmm, 19, 459 (1969).

241