Mat. Res. Bull. VoI. 8, pp. 3l-42,

the United States.

d ,-rlt'a-ar, q

LZtrl<

.v

GROWN

rBM Thonas

J,

Pergarnon Press,

Inc. Printed in

',

C*,

{ #2 L) Cru LIQUID PHASE EPITA(IAL

1973.

GROWTII

A*o K, tu'7 cs

KINETICS OF MACIIIETIC GARNET TIL}fS

BY ISOTHERMAL DIPPING WIT1I AXIAL ROTATION

R. Ghez and E. A. Giess watson Research center, yorktown

lleights,

New

york

1059g

(Received October 30, l97Z; Refereed) ABSTRACT

Epitaxial nagnetic films of Eu1.tYbt.9Fe5o12 have been gror,rn under controlled lsothernal conditions on (Lrl) Gd3Ga5o12 substrates with and without rotation. The growth kloetics aiE adequatelv described by an analytlcal model lnvolvlng diffusion (o!r.to:5 cmz/sec) through a stagnant boundary layer (6 e 100 pm) and subsequent integratign into the filsr according to a first order reaction (k = 3'10-J cm/sec), rhese magnetlc fiLms support bubbles wlth a sub-mieron di.ameter under proper bias field conditions. Introduction

Interest in magnetic bubble devices has created a need for well-controlLed fllurs and for an understanding of their growth process. Prevlous work (1) has

shor^rn

that magnetic garnet fllms

can be grown reproducibly by

liquid

epltaxy on substrates which are dipped while undergolng axial rotation whlch are posltloned

ln a horizontal plane. This technique takes

phase

end

advantage of

the supercoollng tendency of PbO-BrO, flu:red melts (2), which allows crystal growth under isothermal conditions.

This paper analyzes the filn growth data obtained with a well-controlled automated dlpping

apparatus.

We

propose a model invoLving

diffusion of

units through the fluxed nelt and their subsequent lncorporation reaction at the solid-llquid interface. We derive an expression for the growth rate as a functlon of the rotation rate whtch is also valid ln the growth

3l

Vol. 8, No.

MAGNETIC GARNET FILMS

3Z

limiting case of zero rotation.

The model adequately describes

I

the experi-

mental data and allows an estimate of both the diffusion coefficient and the

reaction constant. Experimental Results The

film growth

equipment

differed from that used in earlier studies

(1)

for pre-heating the substrate

was

only as follows: first, a separate

chaurber

located above the growth chamber; second, the substrate inunersion was

mechanism

controlled by digital timers which coul-d be preset to an accuraey of 0.1

sec.

The preheater chanber r^ras

isolated from the growth

charnber

by a Pt

shutter to prevent substrate contamination by condensing PbO vapor during heating prior to gro!ilth. The garnet composition, E t.1Ybt.9Fe5Ol2, supports magnetic bubbles of sub-micron size when the filrn is thinner than I pm. The lattlce parameter of

this garnet closely

matches

that of fr3t.5012 substrates cut from

by the Czochralski method. The substrates The rnelt composition

in

moLe 7., Er2O3

\47ere

bouLes grown

oriented ln the (11f) plane.

@.264), Yb2O3 (0.456), F.2O3 (9.0),

PbO

(84.83), B2o3 (5.45) r was similar to that enployed by Levinsteln et al . (2). The

liquidus

temperat,ure

of this melt is

^, 930'C and

the growth temperature

was 880oC.

Film thiekness \ras determined non-destructively from reflectance inlerference frlnges measured at a 10o incidence angle in the tr = 6000 to 7000 i, range, with correctlons fd? the change in index of refraction as a function

of l.

The standard deviation

filrns measuied by the

of thickness measurements

same technique

was 0.03

um.

but with another interferometer

Two

and

operator gave the same film thicknesses within 0.03 pn. However, a direct measurement obtained

by cracking a sample and observing it on edge in a

at 18r200x magnification

showed

SEM

a thickness of 0.616 + 0.0l7 irm.whereas the

optical" measurement yielded 0.74 + 0.03 um. This discrepancy is dlscussed l-ater.

VoI. 8, No. The

MAGNETIC GARNET FILMS

I

33

opticalLy measured fifun thickness (h) vs. tine (t) data for varying

rotation rates (r) are Presented in Fig. 1' For r * 0, the growth kinetics have essentially achieved a steady state, since h is a linear function of t. However

the positive intercepts at, the origin of the ortrapolated straight

lines lndicate an initial transient regime. rate increases with r. shows

No steady

We

also note that the growth

state is apparent for r = 0' but Fig.

2

straight line correlation between h and J/2 which suggests a

diffusive process. In this

ease the

intercept at the origin is negative.

3.O

/+(t6gl

?

,/rt(loo)

2.O

*./

-t o

^

r'^r

a lrJ

zY

9 I F

n-

r.o

o.o r. o

20

40

60

80

too

GROWTH TIME (sEC)

FIG.

1

Optically measured filn thickness vs. growth time for various rotation rates (rprn).

Theory

Our model

is

based on

the assumption of a stagnant boundary layer of

thickness (6), set up by the axial rotation, through which growth units

that these units are incorporated into the lattice by a first-order reaction at the interface. The 'rgrowth diffuse. It is

assumed

can

garnet

MAGNETIC GARNET FILMS

34

VoI. 8, No. I

unitrt is the garnet constituent

t.4

in the fluxed melt which has the t.2

?

lowest diffusion coefficient and reaction constant

r.o

is

geometry

assumed

(k).

(D)

The

to be one-

3 a o.8

dimensional, with the x-axis

lrJ

pointing into the flux and normal

U'

zY

()

o.6

to the interface at x = 0.

F = o.4

The concentratlon, C(x,t),

of o.2

growth

units nust obey the

diffusion,equation (Ref.

o

ac

2468to TIME t/2

at=n-'

6"6ttZ

FIG.

*

3)

a2c

(1)

dx

and the boundary conditions

2

Optically measured fil-rn thickness vs . square root of growth time for zero rotation rate

C(6,t) = CL ,

D#

Equation (2) specifies a flxed concentration ({)

(o,t)

=

(2)

k [c(o,r) -

c"]

(3)

beyond the bor:ndary l-ayer;

the 'fradiationrrboundary condition (3) indicates that the diffusion current

at the i-nterface is

balanced by

rium concentration.

We as'bume

a first order reaction, C" being the equlllb-

that the

boundary

layer ls set up hydrodynam-

ically by the rotatlng substrate in a time whlch i-s short wlth respect to the relaxation time for dlffusion and reaction. Therefore the inltial

con-

dltlon is C(x,0) = C, The solutlon

presented

(4)

of Eqs. (1-4) ls obtained by simple modiflcatlon of the

in Ref. 4, p. 315.

We

flnd

procedure

Vol. 8, No.

MAGNETIC GARNET FILMS

I

C" - C'(x,t) c_ LC-c

-o2 ot/t?sin a_ (r - x/6) n

-r,-r LJ

1 - x/6 1+R

35

(5)

(l_+R+ n2 o2) n' sLn o

n=1

where

n=D/dk

(6)

,

and the o-n rs are the

positlve, non-zero roots of

tano*Ro=O

Q)

The dimensionless parameter

R

measures

to reaction. Equations (5) and (7)

show

the relative importance of diffusion

that fast reaction kinetlcs (R << l,

diffusion is rate lfunlting) lnply C(0,t) 3 C" and

on =

nr + 0(R), and that

sluggish kinetics (R >> 1, the interfaclal- reactlon is rate liniting)

imply

C(g,t) 3 cl and cr, = (2n - L)nlZ + O(n-l). (tre nathematical O-symbol, fO(X),

means ttplus

neglect of terms of the order X".) Using Eqs. (5) and (7)'

the growth rate (f)

f

(t) = p-l I

of the garnet filn is given by

(t -.") -lTD

sclaxf *=0

=

exp

tt+".'E ;;;;"?- l

h(t)

the

Finally, since

flln thickness (h), obtained by integrating Eq. (8)'

6(%-c.)

1

=

Expressions

-

exp 1-ol n.loz>

(t o2 nn + (5),

(8)

, and (9)

(8)

n

where p is the density of the growing garnet filn.

f = dh/dt,

(-cit1 nt/6-)

enbody two

R

l_s

(e)

+ n2 o2)

approximations; (i) rnaterial

at the interface actually requires (3) that f = (p-C)-l D 3C/3xlx=0, so that Eq. (8) results when p tt CL > C(O,t), a condition which ls experimentally verlfied since in our case I = 6.8 g/cn" and a, = 0. L95 elen3;

balance

(ii) strictly speaking our solution ls

onLy

valid for low growth rates since

MAGNETIC GARNET FILMS

36

VoI. B, No.

the full Stephan problem (3,4) of a moving phase boundary has been neglected. However, Burton, Prim, and Sliehter (5) have shown that, aL least in the case of fast ki.netics, such a procedure is legitimate vrhenever f6/D << 1,

a conditlon which is met, as will be seen later. We

now

point out several general features of the solution. First,

we

consider Coehrants (6) analysis of the boundary layer as discussed by Burton

et al. (5)

who show that

6=

1

.6

DL/3

uLlo ,-t/2 = 6o

(10)

E-L/2

where v is the kinematic viscosity of the fluxed me1t.

The

validtty of

stagnant layer approximations has been recently discussed by Wilcox (7).

We

then appreeiate that expressions (8) and (9) depend on rotation rate as well as on time. Secondly, the growth rate (8) consists of a steady state expression

r = p-l o (t -€.)/(6

+ D/k)

(11 )

,

plus exponentially decaying terms. Equation (11) has been reviewed long

(8) and has been analyzed in detail by Brice (9).

We

ago

also note that Eq. (9)

yields a positive intercept at the origin

h, in

=

-l zp-L

o

(t

.")

In=1

orr2

{r + n + n2 ol)-1 ,

accordance wLth the experimental data

model

in Fig. 1. Thirdlyr

(12)

we expect our

to be stlll valid in ful1y transient liniting case, r = 0, for

6 is lnfinite according to Eq. (10). This

case

which

is obtained either by solving

Eqs. (1), (3), and (4), and replaeing Eq. (2) by C(-,t) = C, (Ref.4, P.305)

letting 6 + * in the general solution Eqs. (5-9).

or

(more tediously) by

We

then obtain the analogues of Eqs. (5)' (8), and (9):

I

Vol. B, No.

MAGNETIC GARNET FILMS

I

C" - C(x,t) c_ l-e-c

erfc

x

2 6it

-

(exp

o-1 (t x + t2t) ) erfc

(

37

x

2ffi

*

LF) (13)

f

(r) =p -'l

k

1 (c-LE -c) (exp (D-' k /nt)') erfc D-1' k r'Dt -', -

h(r) = (pk)-t o(cl - ce)

[2D-1 u

StIi - I *

(o-1

(exp

(L4)

,

t /Dr)\ errc D-1 k 4itl (1s)

For small values of t

Eq. (15) can be expanded (10) and yields

h(r) = o-t u (t

ce)r +

o(rt/')

In fact, this initial linear growth law

(16)

can be deduced from

the general

expression Eq. (9). It is independent of 6, since lnitial- concentration

gradlents extend over regions much smaller than the boundary layer.

The

initial driving force is thus the product of the supersaturation (or undercooling) and the int.erfacial reaction constant. For large values of t an asymptotic expansion (L0) of Eq. (15) yields

h(t) r

(ok1-1

o (% - .")

[2D-1

k ffi7n

- 1 + 01.-112;1

(L7)

Equation (17) shows that, in this case, h should increase linearly with

in

J/2

accordance

with the data on Fig. 2. Furthermore, the negative

i-ntercept ho

=

-(pk)-l D (q, -

(18)

c")

indicates a rate limitation at the filrn-liquid interface since h, k + *.

Another argument

+ 0 when

in favor of a finite reaction rate will be presented

in the next section. To suurnarize, we have presented a simple diffusion model all-owing for

an additional surface rate limitation.

The model

is valid for any rotation

MAGNETIC GARNET FILMS

38

Vol. 8, No. I

rate since the qualitative features of the data on both Fig. 1 and Fig.

2

have been orplained. Comparison

If

with

Experlnent

we anaLyze the optlcaLly determined thickness data contained

Flgs. L and 2 aceording to the procedure which follows, we find

in

comparabJ.e

values for the diffuslon coefflcient ln the o
of k are tu 8'LO-3 cm/sec for the r = 0 cn/sec for the r # 0 ease. This discrepancy can easily be case and n,3.10-3 understood if the optlcal measurements, although reproducible to * 0.03 um, rotation rate.

Ilowever the values

overestlmate the thickness by a constant amount of che order of 0.1 un. It

is well

known

that flux

grown

crystals'can dissolve Pb (1,1-1) which can sig-

niflcantl-y influence the optical constants. surement by

SEM

mentloned

in "Experimental

The lndependent thlckness mea-

Resul-tstr i-ndlcates

that opticaL

deterninations overestlmate the thlckness by 0.L2 prn. The analysls whlch

follows

assumes

that all- thicknesses ln Flgs. 1 and 2 have been reduced by

this amount. The (steady state)

grohrth

rate is obviously unaffect,ed by this

procedure, but the intercepts are reduced. We analyze

squares

the corrected data according to Eqs. (11) and (17). Least-

fitting yields a correlation coefflclent better than 0.999 for all

straight l"ines. case are given

The measured

ln Table

slopes f and intercept" he

1.

TABLE

]-

Growth Parameter Data Calcul-ated from ResuLts

r

(rpm)

for the r *

f

(un/sec)

h.".

(un)

in Fig. 6

(um)

36

0.023

0.108

L57.9

100

0.031

0.079

94.7

169

0 .036

0.077

72.9

1.

0

Vol. B, No. I According

MAGNETIC GARNET FILMS

to Briee (9), Eq. (11) should yleld a straighr li-ne when tt-L/2 i,

plotted against f

(assuuring

a first order reaction).

with correlation coefflcient -0.993 slope [-

O/0o kl

and intercept

= -0.260

tD (q,

Using the estlnates Cl

"*2/"."

39

(sec

and such

lrad)L/z

We

get sueh a line

that its

,

(19)

- c")/6ool = 0.0179 un/(sec t^a)L/2

- ce

= 15.10-3

I

e/"rfr g = 5.8 g/crn3, v=

_t 9.10 '

(L2), we obtain the values D

= 2.49.10-q- cm2,/sec

,

k

= 3.1.10-?" cm/sec

,

(20)

= 0.0306 cn (r^d,/t"")Ll2 as

well as the values of 6

We

note that the values of the diurensionless parameter R, recorded in Table 2,

computed from

Eq. (10) and recorded ln Table 1.

indicate a signlficant rate llnitation at the lnterface. In fact, if very small, Eqs. (10) and (1-1)

show

R

were

that a plot of f u". tLl2 should yield

a straight line Lhrough the origin; this ls at variance with the data.

TABLE

2

Parameters Supporting the Assunptions

r (rpn)

0-

t

of the

(sec)

ModeL.

f6/D

0 .504

2.285

L.92

L.47.L0 -3

L00

0.84

2.089

0

.83

1.18.10 -3

L69

r_

2.0

0.53

36

Slnllari.y,

.09

when compared

dara ln Fig.

2

for the r

L

.06.10 -3

to Eqs. (17) and (18), the stralght Llne fit of the =

0

case yieLds

a

Vol. 8, No.

MAGNETIC GARNET tr'ILMS

40

slope lzp-L

- ce) {'-o/nl = 0.148 ,^/t""L/2 * l-(pk)-1 D (t - cu)l = -0.217 pm

(CL

and intercePt

(2L)

from which 2, D = 3.5.10 -5 cm /sec

:l

k = 3. 6.10 -3 cm/sec

(22)

In vlew of the uncertainties in the values of the supersaturatlon vlseoslty, Eqs. (20) and (22) are in We

are now able to support

and the

good agreement.

of the fundamental

some

model. The duration of the transient

regime when

assumptlons

r * 0

of

our

ean be estimated

by writing the largest ocponential in Eqs. (8) and (9) as .*p {-o2, ot/021

(-t/t).

e:cp

, = and

transient relaxation time is

The d2/D

thus

2 ort

(23)

is recorded in Table 2 as a function of r.-

much

We see

that t is

less than the first recorded observatlon time, which

tially a steady state growLh

=

means

always

that

essen-

has been achieved. Next, we calculate the dirnensionless

rate f6/D which

appears

Thls quantlty, recorded ln

TabJ.e

in Burton, Prirn and Slichterrs theory (5). 2, is

always very small

indicating that

neglect of the interface motion is a legitimate assumption. Finally, it is easy

to

see

that an upper bound for h.,

occurs when R-)

0. Equation

(12)

becomes then

h

{l)

=

zp-L

6 (q.

-.") t

(nn1-2 =

6 (ct

-

c.)/30

(24)

n=1

Its

val-ues range from 0.0535 Um for

These values

for which

r = L69 rpn to 0.116 lrm for r = 36 rpm.

are eonslderably less than the intercepts on Fig' I (r 0.2

we have shown

that

R

urn)

= 1. This further supports our correction of

I

VoI. 8, No. I

MAGNETIC GARNET FILMS

4l

the optical measurements. Conclusions

unified expression for the growth rate as a function of ti-me and rotation rate fits experimental data on the growth of LPE magnetic filns. A

The growth

rate expression is derived from a model involving diffusion

through a stagnant boundary layer and a first order interfacial reaction' For a non-zero substrate rotation rate, the model predicts an initial

transient state which decays rapidly (orponentially) into a steady state during which the growth rate is constant. For zero rotation rate, the

\-.,

growth process never achieves a steady state and the growth rate scales

essentially as the square root of time. The above analysis principally differs from earlier work (1,9) in that the importance of the interfacial reaction during the initial

transient state ls

demonstrated.

Acknowledgments We

are indebted to C. F. Guerci who ably performed most of the

o
to D. C. Cronemeyer

for filrn thickness measurements.

who developed the

optical

technique

The SEM film thickness measurements by

O. C. Wells and C. G. Bremer are gratefully acknowledged. M. A. Koblenz,

\!-

L. Gulitz,

W. C. Dimarla, and

and construction

of the

A. S. Vadaszwere responsible for the

automated controls

for

LPE

design

film growth. Polished

substrates were supplied by Unig,n Carbide, Crystal Products Division,

'

Diego, California, and by D. F. OrKane, P. C. Yin, J. Karasinski' l"toldovan, and E. Mendel

San

A-

.

Referenees

1. E. A. Giess, J. D. Kuptsis, and E. A. D. I^lhite, J. Crystal Growth 1! (to be published). 2. H. J. Levinstein, S. Licht, R. I{. Landorf, and s. Blank, App.l. Phys. Letters 19, 486 (L97L). 3. J. Crank,

The Mathematics

of Diffusion.

Clarendon Press, Oxford (1956).

MACNETIC GARNET FILMS

4Z

4.

Vol. 8, No.

H. s. carslaw and J. c. Jaeger, conduction of Heat in solids, 2nd ed. Clarendon Press, Oxford (1959).

5.

J. A. Burton, R. C. Prin, (1953).

and W.

p. Slichter, J.

Chem. phys.

ZL,

LggT

phi1. Soc. 30, 365 (1934).

6.

W. G. Cochran, Proc. Camb.

7,

I^I. R. wilcox, chap. 2 ln Preparation and properties of solid state Materj-als, vol. r (R. A. Lefever, ed.). M. Dekker rnc., New york (Lg7L).

8.

L. L.

o

J. C. Brice, J. Crystal Growrh 1, 161 (L967)

10.

Bircumshaw and

A. c. Riddiford, Quarr. Reviews 6, Ls7 (Lgs2).

M. Abramowitz and r. A. stegun, Handbook of Mathematical Functions, Chapter 7. Dover Publicatlons, New york (1965).

11. L. E. Sobon, K. A. wickersheim, J. c. Robinson, and M. J. Mltche1l, J. App1. Phys. 38, 1021 (L967). L2. w. Tolksdorf, G. Bartels, G. p. Espinosa, p. Holst, D. Mateika, and F. trIelz, J. Crystal- Growth (to be published), measured the vlscosi.ty of a similar ylG-type melt.

I