Mechanism of pressure build-up in hydrogen bubbles during implantation

Scripta

METALLURGICA

Vol. 22, pp. 445-450, 1988 Printed in the U.S.A.

Pergamon Press plc All rights reserved

MECHANISM OF PRESSURE BUILD-UP IN HYDROGEN BUBBLES DURING IMPLANTATION Kohji Kamada and Akio Sagara I n s t i t u t e of Plasma Physics, Nagoya University, Nagoya 464, Japan (Received September 14, 1987) (Revised January 19, 1988) i. As we h a v e demonstrated

Introduction

in a previous paper (1),

hydrogen implantation

into AI creates a high density of bubbles and, at the same time, builds up large retention of hydrogen in a subsurface layer at room temperature. This is in sharp contrast to other metals such as Cu, Ni, Fe and W that we have examined, where the retention of hydrogen is not prominent at room temperature. The d i f f e r e n c e of the m i c r o s t r u c t u r a l changes between m a t e r i a l s caused by hydrogen i m p l a n t a t i o n seems to depend on several material p a r a m e t e r s such as the d i f f u s i o n c o n s t a n t s of vacancies and i n t e r s t i t i a l s and, at the same time, on the d i f f u s i o n constant, solubility and surface r e c o m b i n a t i o n constant of the hydrogen. From m e a s u r e m e n t s of the amount of hydrogen remaining in a subsurface layer with ERD (elastic recoil detection) (2) during the i m p l a n t a t i o n and of the total bubble volume created by the i m p l a n t a t i o n in the same specimen by TEM, we have d e t e r m i n e d the internal pressure of hydrogen bubbles to be more than 6 GPa in A1 (1,3,4,5), which is more than two times higher than expected from the d i s l o c a t i o n p u n c h i n g pressure e l a b o r a t e d by G r e e n w o o d et al. (6). However, this internal p r e s s u r e is reasonable when the lateral stress field due to local bubble swelling and the osmotic pressure due to vacancy s u p e r s a t u r a t i o n are taken into account in the m e c h a n i s m of a d i s l o c a t i o n p u n c h i n g process as d e s c r i b e d in (5). The latter m e c h a n i s m p r e d i c t s 7.8 GPa in A1 for the p r e s s u r e n e c e s s a r y for the d i s l o c a t i o n punching from a bubble of 12 nm in diameter. (The bubbles observed in A1 have diameters ranging from 5 to i00 nm with a sharp peak at 12 nm irrespective of the ion fluence, and their depth d i s t r i b u t i o n is similar to that of the implanted hydrogen.) N e v e r t h e l e s s , the value of the p r e s s u r e determined e x p e r i m e n t a l l y relies on the a s s u m p t i o n that all the hydrogen m e a s u r e d by ERD is c o n t a i n e d in the b u b b l e s o b s e r v e d with TEM. Although this a s s u m p t i o n can be j u s t i f i e d because of the very low s o l u b i l i t y of hydrogen in A1, it is n e c e s s a r y to examine the m e c h a n i s m to see how such high pressure builds up in bubbles in a steady state during h y d r o g e n i m p l a n t a t i o n . The b e h a v i o u r of implanted h y d r o g e n in materials, including its p e r m e a t i o n process, is c o n t r o l l e d either by a d i f f u s i o n process in a s u b s u r f a c e layer or by a surface r e c o m b i n a t i o n process at a free surface to form hydrogen m o l e c u l e s there, d e p e n d i n g on material p r o p e r t i e s and i m p l a n t a t i o n p a r a m e t e r s as well (7,8). The present model relies s t r o n g l y on the p e r m e a t i o n m e c h a n i s m of hydrogen, with the driving force r e p r e s e n t e d by its fugacity, which is d e s c r i b e d as a function of material and i m p l a n t a t i o n relevant p a r a m e t e r s . From the fugacity we can obtain the internal pressure of b u b b l e s under the steady state of hydrogen implantation. 2. Model

of

Implanted

Hydrogen

Behaviour

When h y d r o g e n ions are i m p l a n t e d into metals, they thermalize at a certain depth, d e p e n d i n g on the p r o j e c t i l e energy, and diffuse t h e r m a l l y towards both the front and b a c k w a r d surfaces of a specimen. During the

445 0036-9748/88 $3.00 + .00 C o p y r i g h t ( c ) 1988 P e r g a m o n P r e s s

plc

446

HYDROGEN

BUBBLES

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diffusion, hydrogen atoms may be trapped by some lattice defects and then released from them. In case of AI, hydrogen i m p l a n t a t i o n creates a large number of bubbles in a subsurface layer. Since the s o l u b i l i t y in the matrix is very low, a c o n s i d e r a b l e amount of the hydrogens is presumed to be in the bubbles (4). We assume that implanted hydrogen in a r e p r e s e n t a t i v e spherical region with a radius L o flows into a bubble with a radius r o situated at the center of the r e p r e s e n t a t i v e region, a~d is contained in the bubble as H 2 molecules. On the other hand, the H2 M o l e c u l e s thus contained in the bubble dissolve again into the matrix and dlffuse toward a free surface as hydrogen atoms and finally are released into vacuum after the formation of Hn m o l e c u l e s again a t the free surface. In the steady state of hydrogen flow, t~e flux into the bubble J = -D 8c/Sr at r_, with D and c being, respectively, the V d i f f u s i o n constant and c o n c e n t r a t l o n of hydrogen, must be b a l a n c e d with the release rate R from the bubble, which is identified with the p e r m e a t i o n rate through the subsurface layer; that is, (1)

J = R

This model of trapping and detrapping of hydrogen from a bubble on the basis of a previous experiment (3), in which the exchange and deuterium through a thermal diffusion process of the former demonstrated. Solving the diffusion equation with the spherical coordinate representative spherical region, we obtain

3 r----y

is justifiable of hydrogen was clearly in

the

(2) 0

where n ff is the effective implantation rate per cm 3 diminished by the loss of hydrogen atoms which diffuse far into the bulk of a specimen. A similar treatment has been given by Wiedersich (9). In the steady state, the release rate of hydrogen out of the bubble must be equal to the permeation rate of hydrogen through the subsurface layer. The situation is illustrated in Fig. i. Hydrogen in a bubble, situated at x o beneath the surface, dissolves into the matrix giving the surface concentration c* which is given by Sievert's law c * = Ks

(f

I/2

,

(3)

with the solubility Ks . Since very high internal pressure is expected, f the fugacity of hydrogen is used instead of the pressure (8). In considering the permeation of hydrogen, we s h o u l d d i s c r i m i n a t e between the diffusion limited mechanism and the surface recombination mechanism. In the former case, the hydrogen concentration at the bubble surface c a should be larger than that at the free surface Co, a n d a s h a r p c o n c e n t r a t i o n gradient builds up in the subsurface layer. In this case, as i l l u s t r a t e d in Fig. i,

Rd =

D(c d -Co) x O

=

D cd x o

(4)

'

since c ( cH . E q u a t i n g R d to J in eq.(1) and using o b t a i n ~he fdgacity of hydrogen in the bubble

neffLo.2 Xo.2 fd = tD-~s) [ 3 ro

Lo

(~

2 r°)] 2

~

.

eqs.(2)

and

(3),

we

(5)

o O

In the surface r e c o m b i n a t i o n limited case, on the other hand, h y d r o g e n c o n c e n t r a t i o n s at the bubble and free surfaces must be nearly equal since the r e c o m b i n a t i o n process at the free surface is s l o w e r than the diffusion through the subsurface layer. Therefore, the permeation rate Rs, which is r e p r e s e n t e d

4

Vol.

by of

22,

No.

4

HYDROGEN

the e v a p o r a t i o n rate of h y d r o g e n two h y d r o g e n atoms, is e x p r e s s e d

BUBBLES

447

at a free s u r f a c e due to the r e c o m b i n a t i o n in terms of the s u r f a c e c o n c e n t r a t i o n c~

KS

R s = 2 a krC °

2

= 2 ~ k r ( C ~2)-

,

(6)

w h e r e k r is the s u r f a c e r e c o m b i n a t i o n c o e f f i c i e n t , and ~ = 5 is the s u r f a c e roughness factor. E q u a t i n g R~ to J as before, and u s i n g eqs.(2) and (3) again, we have the f u g a c i t y o~ the h y d r o g e n in the b u b b l e •

= fs

2

2

i .neffLo.Lo ~ . 2[ 3----~(~-2~

for

the recombination Therefore, if we estimate the internal

Kra s

o

ro

- ~) o

]

(7)

o

limited case. can calculate fugacity as a function of pressure, pressure of bubbles from either eq.(5) or eq.(7).

3.

Calculation

of

Fugacity

of

we

can

Hydrogen

Hydrogen under a pressure of more than 6 GPa at room temperature can not be in the form of a gas phase but instead a condensed phase (i0,II). The melting pressure of solid hydrogen at room temperature can be inferred by extrapolating the empirical formula given by Liebenberg et al. (ii). It gives 5.3 GPa. The thermodynamical properties of such fluid substance can be described conveniently by use of the fugacity, a fictitious pressure, which is equal to the fugacity in the gas phase with which the fluid is in equilibrium (12). The equation of state of hydrogen at room temperature which is needed to calculate the fugacity was given both in the low pressure region below 0.3 GPa for the gas phase by Michels et al.(13), and in the high pressure region up to 2.0 GPa for fluid hydrogen by Mills et al.(10). The fugacity f can be obtained by integrating the molar volume V m as a function of the pressure P, using

dlnf

Vm = ~

dP

,

(8)

w h e r e R is the gas c o n s t a n t and T the t e m p e r a t u r e (12). of the m o l a r v o l u m e for the n-H 2 f l u i d was g i v e n in (i0)

V m = a(T)P

1

2

3 + 8(T)P

3 + y(T)p-i

,

The as

1 = __ RT

o = ~[2

m3/mol (GPa) 2/3 2.0 GPa (i0). We the fugacity from

P

f P VmdP o 2

2

1

1

g

g

g

g

13

formula

(9)

where a -- 7.8439 × 19-5~m3/mol (GPa) I/3, 8 = - 1.0950 × 10 -4 and y = 2.7907 × i0 -~ m~/mol (GPa) at room temperature below extrapolate this formula to more than 2.0 GPa, and calculate eq.(8), obtaining f in K-

empirical

~(P

- Po ) + 38(P

p

- Po ) + y l n ~ - - ] .

(I0)

0

For t h e s t a n d a r d s t a t e

we t a k e f o = 2 . 0 7 GPa a t

Po = 305 MPa, w h i c h a r e ,

448

H Y D R O G E N BUBBLES

respectively, the fugacity et al.(13). The c a l c u l a t e d pressure. In this figure, calculated by e x t r a p o l a t i n g pressure, namely 5.3 GPa.

Vol.

22, No. 4

and pressure for the gas phase measured by Michels fugacity is shown in Fig. 2 as a function of the s o l i d - d a s h e d line represents the region the equation of state, eq.(9), beyond the melting

4. Comparison with E x p e r i m e n t From eqs.(5) and (7) we can obtain the fugacity of hydrogen in a bubble, when the hydrogen p e r m e a t i o n is limited by the diffusion process and by the surface r e c o m b i n a t i o n process, respectively. The c a l c u l a t e d fugacity in turn gives the c o r r e s p o n d i n g pressure of hydrogen in the bubble through eq. (i0) or the curve in Fig. 2. This pressure is to be compared with that determined experimentally. Thus, we have calculated the fugacity in the above two cases. The numerical values of the parameters used for this calculation are llsted in Table i. The implantation relevant parameters in the table such as n ~o, x_, ro and L_ were obtained from the previous paper (i). n ~ was o~{~ine~ from the ~implantatlon flux, assuming that half of the i m ~ n t e d hydrogen atoms contribute to the b u i l d - u p of the internal pressure. This is so because the implanted depth of the hydrogen is larger than that of the bubbles formed. The latter coincides with the depth at which the implanted hydrogen loses most of its energy. Because of u n c e r t a i n t i e s involved in such p a r a m e t e r s as D, K s and kr, the calculated fugacities are shown with the arrowed ranges in Fig. 2, where the c o r r e s p o n d i n g ranges for the pressure of hydrogen are also indicated. The e x p e r i m e n t a l l y determined pressure, indicated by the horizontal line bounded by the two arrow marks, lies close to the range for the hydrogen pressure in the diffusion limited case. That is, the build-up of pressure in a hydrogen bubble is caused by the diffusion controlled mechanism. In the theory of hydrogen recycling in plasma wall interaction, the so called transport p a r a m e t e r [7] or permeation number [8] is adopted to d e t e r m i n e the relative importance of the diffusion and surface r e c o m b i n a t i o n processes, both of which have the same physical s i g n i f i c a n c e and are e x p r e s s e d in a function of almost the same material and implantation parameters. In the case of A1 u~der the present implantation conditions, both of them amount to more than i0 ~, ensuring the diffusion limited behaviour of the implanted hydrogen in q u a l i t a t i v e agreement with the present result. More specifically, each of the release rates, R d and Rs, has a different dependence on the fugacity; i.e. R d has a square roo~ dependence and Rs, on the other hand, has a linear dependence, as can be seen when one inserts eq.(3) into eqs.(4) and (6). This is s c h e m a t i c a l l y illustrated in Fig. 3, which shows that the relative magnitude of R d and R s changes in the adjoining regions of the fugacity with the critical vaIue fc at which R d = R s. Therefore, in the region of smaller fugacity, i.e~ f ( f_, R d is l~rger than Rs, that is, the surface r e c o m b i n a t i o n limits the hydrogen behaviour, and in tKe other region, i.e. f > fc' Rd becomes smaller than R~, the diffusion process limiting the phenomenon. (The permeation n u m b e r ~ m e n t i o n e d above is simply the ratio Rs/Rd). Whether the phenomenon under investigation is diffusion limited or r e c o m b i n a t i o n limited depends, consequently, on the m a g n i t u d e of the hydrogen flux into bubbles, J. When it is larger than the flux J_ at f = fc, the phenomenon is diffusion limited, and, on the other hand, ~hen it is smaller the phenomenon becomes surface r e c o m b i n a t i o n limited._ In ~he. present case, the critical value Jc ~ 8 × I0 ~ - 3 × 109 cm-Zs -I, depending on the c h o ~ e ofoth ~ numerical values listed in Table i, is far smaller than J = 3 iO cm- s- calculated by eq.(2). This ensures again the diffusion limited behaviour of the implanted hydrogen as described above. If the implantation rate neff, to which J is proportional, is kept very low so as to fulfil J < Jc, the ~ydrogen behaviour will be surface r e c o m b i n a t i o n limited even if the material relevant parameters are kept at the same values as those employed above. The "recombination limited case" inserted in Fig. 2 indicates merely the crossing point, spreading because of the choice of the parameters, between the R s curve and J, c a l c u l a t e d by eq.(2), which is larger than Jc" Therefore, it

Vol.

22, No.

4

HYDROGEN

BUBBLES

iS a trivial case which can not be realized condition. When J < Jc, on the other hand, surface recombination process becomes larger limited case which can not be realized this illustrated in Fig. 3. Table

I. Numerical

x° (nm)

r° (nm)

L° (nm)

102

6

40

values

neff 1 (cm-~s - ) 5.4xi018

449

in the present experimental the corresponding pressure than that of the diffusion time. These points are

of the parameters D (cm2s -I) 8xlO-9-2xlO

used

for hydrogen

Ks (cm-3(pa)~/~ -7

for

the

in A1

kr (cm4s -I )

3xlO 5

10-14-10

-13

References I. K. Kamada, A. Sagara and H. Kinoshita and H. Takahashi, Rad. Effects 103 119 (1987). 2. S. Nagata, S. Yamaguchi, Y. Fujino, Y. Hori, N. Sugiyama and K. Kamada, Nucl. Inst. Meth. BG, 533 (1985). 3. K. Kamada, A. Sagara, N. Sugiyama and S. Yamaguchi, J. Nucl. Mater. 128/129, 664 (1984). 4. K. Kamada, A. Sagara and S. Yamaguchi, Rad. Effects Lett. 85 255 (1985). 5. K. Kamada, A. Sagara, H. Kinoshita and H. Takahashi, submitted to Rad. Effects. 6. G.W. Greenwood, A.J.E. Foreman and D.E. Rimmer, J. Nucl. Mater. 4, 305 (1959). 7. B.L. Doyle and D.K. Brice, Rad. Effects 89, 21 (1985). 8. F. Waelbroeck, J~lich Report J~i-1966. 9. H. Wiedersich, Rad. Effects 12 iii (1972). i0. R.L. Mills, D.H. Liebenberg, J.C. Bronson and L.C. Schmidt, J. Chem. Phys. 66, 3076 (1977). ii. D.H. Liebenberg, R.L. Mills and J.C. Bronson, Phys. Rev. BI8 4526 (1978). 12. E.A. Guggenheim, Thermodynamics, North-Holland (Amsterdam-Oxford-NewYorkTokyo) (1986). 13. A. Michels, W. DeGraaff, T. Wassenaar, J.M.H. Levelt and P. Louwerse. Phyica 25 25 (1959). Surface

:~iiii::i::i::iiiiiiiiiiiii!iiiii!iiiiiiiii!iiiiiiiiii!iii Vacuum

C~ d

~

>

R,

.::%ii:i:ii!iiiiiiiiiiiiiiiiiiiii!iii!iiiiiiiiiiiiiiiiiiiii ..: i :iiiii:i:i:i:!:i:i:!:!:!:i:i:i:i:i:i:i:i:i:!:i:!:i:!:i:

.::::::::.%1::)~b.4. e!ii)i::)i!ii:!:!:):i:i:i -

C ~ i:i:i:i:i:!:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:iiii!iiiiiiiiii?!i!i!iiii?

co

,

R

/

,

!iiiiiiiiiilili!i!iiiiiiiiiiiiiiiiiiiiiiiii!ii!ii I i i i ili,,i,i,,i,i,,i,i,,iiiii!i,iiiii i, iiiiiiiiiiii

)

iiiiii?!iii!i!i?!iii?i!?!!~ii~i~i?!i!ii!!!!::?::!?::!::!!!!

FIG.

1

Model

of

hydrogen

i!;iiiiiiiiiiiiiiiii!iiiiiiiii iii!iiiiiiii!iiii!ii!iiiiiii!i!!i!i!!iiiiii permeation

from

a bubble

to

a surface.

450

HYDROGEN BUBBLES

ct" T'"' Hydrogen, 300°K

16

Vol.

22, No.

/

/ /~

Diffusion _ 15L limited case -

/

•'

"~ 1 S.

.D 1 m 1 o --J

~

Recombination

FIG. 2 Fagacity vs. p r e s s u r e for h y d r o g e n at 300°K. The dashed line i n d i c a t e s the range e x t r a p o l a t e d up b e y o n d the m e l t i n g p r e s s u r e of 5.3 GPa. The arrows r e f l e c t the u n c e r t a i n t i e s in p a r a m e t e r s used and that of the

limited case

experiment.

I

Experiment

I • .

9

|-

8

10

~

20

I

I

I

I

I

30

40

50

60

70

80

P (kbar)

R

Rs

J FIG. 3 S c h e m a t i c p r e s e n t a t i o n of the r e l a t i v e i m p o r t a n c e b e t w e e n the d i f f u s i o n limited and s u r f a c e r e c o m b i n a t i o n limited p r o c e s s e s , s h o w i n g d e p e n d e n c e on the h y d r o g e n flux J.

Ic v

J

fd

fs

fc

fs

fd

f

4