Diffusive release of gas from a solid during tempering

Diffusive release tempering received 21 February 1966 ;

of gas from a solid during

accepted 8 April 1966

G Farrell, W A Grant, K Erents and G Carter, University of Liverpool

Fick’s Diffusion equation is solved for the case of an initially plane distribution of migrating species, located at a specific depth below the surface of a semi-infinite medium, whilst the medium is subjected to a heating schedule of the form (temperature)-! increasing linearly with time. It is shown that the rate at which migrating species cross the surface boundary increases to a maximum at a specific temperature and then decreases with further heating. The temperature at peak rate is found to be related to the activation energy for diffusion, the initial depth of the migrating species and the heating rate. The relevance of this diffusion behaviour to outgassing of vacuum components is considered.

Introduction

An important physical phenomenon in vacuum science is the diffusive release of gas trapped in materials exposed to the vacuum, when these materials are heated. Apart from gas atoms naturally dissolved in these materials, one may also inject atoms into a surface for specific purposes, as in ion pumps or in the ionic implantation of semiconductors, whilst the action of sublimation and getter pumps relies upon the continuous burial of gas atoms within a solid. In many cases there will be a distinct possibility of diffusion of the trapped atoms within the solid, and this will be particularly true during baking cycles where gas can be re-emitted from the surface exposed to the vacuum. One particular diffusion geometry, which is of relevance in ion injection situations, is where a concentration of atoms is initially built up at a plane some fixed distance below the surface of a target, and then the target is heated. Diffusion of trapped atoms occurs and gas is evolved at the surface and it is important to know the dependence of the gas re-emission rate during heating upon such parameters as time, activation energy for diffusion, initial depth of trapping of the migrating atoms, temperature and heating rate. This is the situation with which we are concerned in the present analysis, and in which we consider, for simplicity, a 1 1 specific form for the tempering schedule, ie- = - - bt. This T To reciprocal relation between temperature (T) and time (t) can often be realised experimentally and is used here because of the facilitation of mathematical analysis it allows. Theoretical considerations The process with which we are concerned

is the migration of atoms in a solid lattice in a random fashion, where the migrating atoms essentially hop between adjacent lattice planes of separation il. We assume, initially, that a concentration of Co atoms/sq cm is located at a plane, distance pl units from a surface at x = 0 exposed to vacuum, and further allow migration normal to this plane (and the surface) only, in the x direction. The solid extends to infinity on the direction of positive x only. At any later time t, the atoms will have diffused from the VacuumIvolume16/number6.

initial position and we are interested in the instantaneous concentration of atoms at time t and at a depth x, ie C(XJ). One normally applies Fick’s Diffusion Law to this situation which gives the equation a~

a%

- = Daxa

(1) . _

at

where C(x,t) = C and D is the diffusion coefficient at the temperature of the solid, T. D is related to T via the relation D = Do exp (-Q/RT) - (2) where Q is the activation energy for the migration process, R is the Universal Gas Constant per Mole and Do is defined through koAs. (3) Do= 2 Here ko is essentially the vibration frequency of the diffusing atoms in the interplanar positions and is generally assigned a value of 10ls set-l. Kelly and Brown1 suggest that lP/sec may be a more realistic value, however, basing their belief on measurements of Do in many solid state diffusion situations. Kelly2 has, in fact, questioned the legitimacy of using equation (1) for solid state diffusion where gradients are essentially discontinuous (ie C (x) exists only at specific values of d) and suggests that one should preferably employ a finite difference form of (1). Kelly has shown, however, that after only a very short time the solutions to equation (1) and the finite difference form converge and we therefore use the differential equation in the following. Solutions of equation (1) are to be found in many text@ on diffusion, subjected to many initial and boundary conditions, but provided that the temperature T (and hence D) are maintained constant. Thus, using the method of images, Kelly* has given the solution to (1) with the initiaI conditions C(x, 0) = 0 for x # pl and boundary conditions C(O,r) = 0; C(c0, t) = 0 which illustrates that all gas crossing the surface escapes from the solid into the vacuum, ie desorbs. This solution is at constant temperature To co (exp[-(=I-exp[-(*]). (4) ’ =2(7cDt)+ As far as we are aware no solution to this problem has been offered when T (and thus D) vary with time, except in certain

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295

G Farrell WA Grant, K Erenfsand G Carter: Diffusive release of gas from a solid during tempering titid w where D varies linearly or as a power of time. A realistic variation of temperature with time could be of the form 1 -= +o-bt (3 T and the variation of D with time thus follows as D = Do .@lRT.. eQbrlR. (6)

and we require solution of _QbtlR

1

a% -=

Do

aXa

e-Q/R”,,’

f&&p

ac

*

is very large. One then

1

(1 lc)

Equation (1 lc) has been evaluated for e as a function of T, assuming k. = 1015 set-1 an activation energy for migration, Q = 30 K Cal/mole, a value of p = 10 lattice planes, and b = lO-‘pK set and this function is displayed in Figure (1).

ar’

Shewmon4 have suggested that if solution

diffusion equation

obtains the result

Q

RT.

(7)

e

ad

chosen sufiiciently small so that

of the form g* = Af(t)z

to a

is required, sub-

PC ac ’ dt of(t) leads to a form a;8 = A a~.

stitution of the variable Z =

s Using this technique for the present function, f(t) = e -ObtlRwe find &Jbt’R

_

1

(8)

Qb/R

z=

and 1 ac Doe-QtRTo.az

av

-=

aXa

(9)

which satisfies the condition that Z = o, t = o; Z = CO,t = CO and moreover Z-t t as b + o, ie equation (9) relaxes to equation (1) at constant temperature. Since in fact equation (9) and (1) are analogous in the variables Z and t one can immediately write the solution of (9), with the boundary conditions c(x, z=o) =o ;x+pl c (PI, z= 0) = co as C = Co (QblR

I e QtRTo}*

(x+

[&/#(eQbzlR--I)]+

- exp

[

c (0,z) = 0 c(oo,z) =o

1

--(x --pA))lQb/R e QIRTo

1 exp

[

2k,,P[eQbt'R - 11

P)?)~Qb/R e QIRTo 2/c.R[eQbr’R- l] I)

(10)

which is identical to Kelly’@ solution for Z- t, ie b - o. The important parameter from our point of view is the rate at which gas atoms cross the plane X = 0, since, if diffusion is the rate limiting step, this is the gas release rate into the aC

This rate is given by e = D a~ x = o which can be

vacuum.

I

I

derived from (10) to be

Qb eQlRTo

B=P’R

2k,

or in terms of temperature

T,

(@iRT

_e

-QIRTo)UI

exp

-

@-QfRT

(

$-Q1.T.)

1

(llb)

where A1 = Ae+laRro andB1 = BeeRTo This equation can be simpli3ed since eQ(‘IT--IIzo R is generally much larger than unity, since To can generally be 296

500 Temperature,

600 T,

OK

Figure 1. Normalised release rate (P/CO) versus temperature (T) Q = 30 Kcal/Mole ; b =lO+ OK-1 set-1; p = 10 lattice planes;

Clearly the rate increases from a very low (almost zero) value at T = To, reaches a maximum at T = T, and then declines again to zero. The temperature T, at which rate maximum occurs is readily derived from time (or temperature) differentiation of (11~) which leads to the result. 1 eQIRTm = _ 2B1

=- k,R PabQ Q

/&-QfRT e=

400

or T, = - * R lo&

(W 1 k.R

1

WC)

[ PFQ This relation between T, and Q is plotted in Figure 2 for values of the parameter pBb varying from lad to lo+* and it is clear that a closely linear relation between T, and Q exists.

G Farrell, WA Grant, K Erenfs and G Carter: Diffusive

release of gas from a solid during tempering p2b=10+2

n

Tez -

Tel = AT z constant AT cz constant

x Q = 8.92(1.98 f0.641)*.10-‘Q (Ma) x T, = 8.92(1.98 f 0.641). lO+ T,,,

p2b=1

Wb)

for p2b varying from lad to IO+*.

p2b= 1O-2 p2 b= lO-4

Further,

since i = & - bt the time difference At between

the e-l rates is given by R

p2b=10+

At = 4.46E

I

I

I

0

1

20

1

60

Activation

energy, Q,

I 100

k cal/mole

Figure 2. Temperature at which maximum release rate occurs (Tm) versus activation energy for diffusion (Q) ; for values of pzb from IO-6 to 1O+2

T, increases with increasing p and with increasing b. Substitution of (12~) into (11~) then leads to the maximum rate

-em -

A

(2B)f

e-*

Co =

Qb

(2n)+ y

e-*

(13)

The independence of e,,, upon the initial depth of location of the atoms, pt, is notable, and it is clear that e,,, increases linearly with the product bQ. The temperature (and time) interval over which release occurs at a considerable rate may be estimated by determining the temperature Tel and Tea when the rates are e-l of the maximum rate. According to (11~) this occurs when

I

= e-l A* exp

_BleQIRTm+

-

Q

2RT,

Taking logarithms, using the condition (12a) for rate maximum, and making the substitutionrelation e-X=-x+3 This may be solved graphically x2 = 2.95 ieg(&-+-) ;($

-a,

andz(&-$) Thus

Te, -

1

= x, one derives the (14) to give the roots x1 = -1.51,

=x1=-1.51 =x2=2.95 =x2---x1=4.46 Tel = Tel Te, . i

(15)

. 4.46.

To an adequate approximation Tel Tez z Tms, and using the equation (12~) or the slopes of curves in Figure 2 one derives, since T, M constant x Q,

Equations (16a) and (16b) reveal that the temperature width is a function of p and b, increasing with p, but (16c) shows that the time width is independent of p. One should expect this latter result, since according to (13~) the rate maximum e,,, is independent of p and, since the integrated rate/time curve is just Co, the time width of the curve must also be independent of p. One can deduce the temperatures (and times) at which the rates fall to any fraction (een) of the maximum in an identical manner to (15) and one requires solution to an analogous equation e-x = -x + (2n + 1). This shows that the temperature and time widths at any rate diminution obey exactly similar relations to (16) but with different multiplying constants. Discussion

Considering first equations (12~) and (13) for the case p = 1, (ie the atoms are located just below the surface) we see that, C. Qb ; em = -ee-f. (2n)+ R These relations are identical to the values of T, and e,, deduced by CarteP for single step desorption from surface adsorbed atoms and may appear surprising in that desorption occurs in one direction only whilst diffusion proceeds into the solid also. Undoubtedly the reason for this is the approximation made in (1 Ic) but exact evaluation from (11 b) does not change the results significantly. Secondly, equation (11~) indicates that, for any diffusion energy Q the temperature at rate maximum increases with increasing depth pl at which the atoms are initially located. However, Figure 2 reveals that the change in T, as p is increased over many orders of magnitude is not very large. Similarly, from (16b) it is evident that the temperature width also increases with increasing pfl. The same remarks apply to the variation of T, and AT with the heating rate b. It is clear, however, from Figure 2 that considerable variation in T,,, (and thus in AT) must be expected as the energy of activation for diffusion is increased. In fact, it can be seen that only a slight variation in Q exerts an equivalent effect on T, and AT as large variations in p and b. Thus in applications where it is anticipated that only a small distribution in p exists, any broadening in an observed desorption rate/time curve must be attributed to a distribution of Q values. Indeed since the release does not appear to depend critically uponp the single step desorption theory may be often used as an adequate approximation, as has been done by Carter and Lecke for the release of low energy ions trapped in glass. 297

G Farrell, WA Grant, K Erents and G Carter: Diffusive release of gas from a solid during tempering

Although the results derived above were obtained using the 1 1 specific tempering function - = 7 - bt, one should expect, T as in single step desorption theory6ihat the value of T,,, would not depend critically upon the actual time dependence of T, although the value of e,, and AT may. Consequently it is probably only necessary, in many applications, to measure T,,, to determine the Q value to an adequate accuracy. If precise information is required, however, then the tempering function must be accurately specified and some knowledge of the initial location of the atoms is necessary. However, if only a small initial distribution of p values is expected, peak broadening will be only moderate. It is evident that an experiment could be readily designed in principle to measure Q in a particular solid material by adsorbing a gas upon the solid and then evaporating known thicknesses of the same solid so that pil is accurately specified. Subsequent tempering and measurement of e,,,, T, and AT will allow evaluation of Q. Finally, in order to examplify the magnitudes of the quantities involved we choose a specific case, the diffusion of HBO in borosilicate glass, for which an activation energy of Q = 41 K Cal/mole has been reported’. If we assume that b = 10-S/“K set, then from Figure 2 and equation (15b) we deduce T, = 485”C, AT = 125°C for an initial location of p = 100 lattice planes. If we assume a high atomic concentration of lOl3 atoms/sq. cm. then we deduce em = 5 x 10” atoms/cm2/sec = 1.54 x 10” litre millitorr/sec. If, in fact, this atomic concentration is maintained deep into the glass then the rate will be increased and will persist for long periods. References

’ R Kelly and F Brown, Acta Met, 13, 169, 1965.

8 R Kelly, Can J Chem, 39, 1961, 2411. of Diffusion. Oxford University Press (London), 1956. ’ P G Shewmon, Diffusion in Solids, McGraw Hill Book Co, New York, 1963. s G Carter, Vacuum, 12 (5). St-pt-Ott 1962, 245. * G Carter and J H Leek. Proc Roy Sot, A261, 1961. 303. ’ B J Todd, J Appl Phys, 26, 1955. 1238.

aJ Crank, The Mathematics

298