Mem. Fac. Sci. Eng.

Series A

32, pp. 19-36 (1998)

Shimane Univ

Crystal Growth of 11 VI and I 111 V12

Compound Semiconductors by Chemical Transport

2. Thermodynamical Considerations

Yasutoshi NoDA

Department of Materia/ Science. Faculty of Science and Engineering~ Shimane University

Nishikawatsu 1060. Matsue City. Shimane 690-8504. Japan

Yoshitaka FURUKAWA

Departmen t of Materials Science. Faculty of Engineering. Tohoku University

Aoba-Aramaki. Sendai City. Miyagi 98C~8579. Japan

Snrgeatsu NAKAZAWA

Department of Metallurgy. Faculty of Engineering. Tohoku University

Aoba-Aramaki. Sendai City. Miyagi 980-85 79. Japan

(Received September 18, 1998)

Abstract

Thermodynamical calculations were performed for the chemical transport systems of ZnSe+12 and

AgGaS2+X2 (X = Cl, Br and I) . The vapor pressures of gas molecules were calculated at the source and

growth zones by using the thermodynamic data. The transport rates for ZnSe and AgGaS2 Were estimated

on the basis of the transport equation, which is limited by the diffusion and laminar flow of gaseous molec-

ules and modified by thermal convection at high pressures . The diffusibities of the molecules were estimated

by using Lennard-Jones parameters. The calculated transport rates almost reproduced the characteristic fea-

tures observed for both ZnSe and AgGaS2･ In case of ZnSe transport, the abrupt change in the transport

rate was attributable to the thermal convection. The transport rate calculated for AgGaS2 reproduced the

observed gradual change with an increase of halogen, but did not realize the observed order (CI < Br< I) in

the magnitude of the transport rate

The chalcogen (Se or S) vapor pressures calculated at the growth zone increased with an increase of

transport agent and exceeded that of dissociation of the stoichiometric ZnSe or AgGaS2･ The fact may have

caused the introduction of the non-radiative recombination centers in the grown crystals

1. Introduction

The chemical transport is one of the most important techniques for crystal growth of the

compound semiconductorsl) .However the exact knowledge on the closed tube crystal growth

has hardly been obtained. The transport equation has been proposed for chemical transport in

20 Yasutoshi NoDA ･ Yoshitaka FURUKAWA ' Shigeatsu NAKAZAWA

closed system considering diffusion and laminar flow as fundamental transport mechanism2)

and the calculation method has been applied to many systems3)

The ZnSe and AgGaS2 crystal growth by chemical transport has been performed in a

closed system. The large size ZnSe single crystal was obtained under the optimum growth condi-

tion in the vertical transport system by using iodine, where the growth is limited by thermal

convection4) . The transport reactions and the morphologies of the grown crystals were investi-

gated, but the transport mechanism has not been fully analyzed for the ZnSe-12 System5) . Preli-

minary works for AgGaS2 using iodine as a transport agent have not fully developed the

method because of its low growth rate6,7)

In the authors' previous study8) , the single crystals of ZnSe and AgGaS2 Were grown under

a variety of growth conditions using iodine and halides as the transport agent. The transport

rate of ZnSe showed a gradual increase, a plateau and a steep increase with an increase of io-

dine. The transport rate of AgGaS2 Was found to exhibit a maximum at some definite amount

of transport agent, but the relationship between transport rate and amount of iodine is steep or

slow depending upon whether the transport agent is iodine or halide. The maximum transport

rate mcreased m the order of Cl, Br and I in the halides as the transport agent. Thermodynami-

cal calculation will help to clarify transport mechanism

In the present study, the calculation of the transport rate was performed to analyze the

transport and crystal growth mechanism

2. Calculation for ZnSeil2 system

In the calculated closed-tube system, the reaction tube was held horizontally in a furnace

of two constant temperature zones, source and growth zones8) . The calculations were per-

formed as shown in Fig. I on the basis of the thermodynamic data and the conventional trans-

port equation2,3).

2.1 Transport reaction of ZnSe-12 system

In the ZnSe-12 system, the species (i) in the gas phase are assumed to be Znl2, Zn, I, 12 and

Se2. The system involves following three reactions

ZnSe (s) =Zn (g) + 1/2Se2 (g) , ( 1) Zn (g) + 12 (g) = Znl2 (g) , (2) 12 (g) = 21 (g) . (3)

The equilibrium constants (Kj) involved in these reactions ( j= 1-3) are expressed in terms of

partial pressures (P}) of the i-th gas component;

1 12

Kl = Pz* Ps*. , (4) K2 = Pznt./Pz. PI" (5) K3 = P12/ PI" (6)

Crystal Growth of II-VI and 1-III-V12 Compound Senuconductors by Chemrcal Transport

2. Thermodynamical Considerations 21 Start

C. Input temperatures

Tl.T2 (Tl<T2)

C. Input thennody namic data and

trans ort arameters AHb~8. S~98. Cp. 61K~ a

C. Calculate e(~uilibrium constants Ki(1). Ki(2)

C. Calculate vapor pressures at growih zone Pi(1) Ptotal(1)

C. Calculatevaporp~2r)es, sures at growih zone Ptotal(2)

C. Calculate derivative and new iodine pressure

C. Set new lodine pressure

P12(2)= P12(2)' No

Fig. 1.

C. Set ogen pressure at growi zone P12( D

Calculate vapor pressures at growih zone P i(1 ). Ptotal(1 )

C. Set halogn prcssure at source zone Pt2(2)

. Calculate vaporp~2r)es, sures at growih zone Ptotal(2)

alculate derivative and new iodine pressure F=Ptotal(1 )_ Piotal(2) F =d F/d P12(2 ) PI 2(2)' = PI 2(2)-F'/F

l pl2(2)- pl2(2y I _l0-5<0 P12(2)

Yes C. Calculate amount of halogsn M

C. Output vapor pressures M. PiG), ptotalG)

C. Calculate transport parameters 17･D. Sc, Gr

C. Calculate flu~s and transport rates J... N, J;*. N*

C. Output fiu~~ andtransport ratcs M. J... N, J..~. N*

End

Thermodynamical calculation for chemical transport

22 Yasutoshi NODA ' Yoshitaka FURUKAWA ' Shigeatsu NAKAZAWA

Table I Thermodynamic data9,ro) and Lennard-Jones parametersl9) for ZnSe-12 system

Molecular A H.298 Cp [ J/ (K ' mol) I ') a 8/K noteb) S.298 weight (J/mol) [J/(K'mol) I a b'l03 c'l0-5 (10-3 kg/mol) (mm) (K)

ZeSe (s) - 1 63000

Znl2 (g) - 895CO

Zn (g) 1 30000 Se:2 (g) 1 39000

12 (g) 623CO I (g) 107000

84

3 OO

161

244

260

181

50.2 5.77 0.0 144.35 59.0 0.0 0_O 3 1 8 . 1 8

20_8 0.0 0.0 65.37 44.6 -2.66 -2.50 157.92

37.2 0.0 0.0 253.81 18.6 0.0 0.0 126L90

0.455 1 373 HgC12

0.350 1341 Kr 0.427 1 025 Br2 0.5 1 6 474･)

0.422 289 HI a) Cp=a+bT; cT-2, b) taken from the listed moleculel9) . c) ' taken from literaturel9) , although the averaged value of those from T* and Tb is 637 K

The Kj's were calculated for the equilibria (1)-(3) using thermodynamic data9,ro) Iisted in Ta-

ble I.

For ZnSe to be transported, the following condition of the stoichiometric composition

(Zn/Se= 1) is given in each zone

P~ + pznt, = 2Ps*,･ (7) In the growth zone at Tg, a tentative value of PI, was assumed and partial pressures (Pi) were

calculated from the equations (4)- (7) . In the source zone at T~, PI, was iterated to refine P:i's so

that the total pressure became equal to that at the growth zone. The total amount of iodine

charged m the growih ampoule was estimated in Appendix I . The calculation was performed at

Tg= 1073-1 173 K and A T=25-50 K.

The transport rate was evaluated in Appendix 2 by calculating the mean diffusivity of gase-

ous molecules on the basis of the Lennard-Jones parameters in Appendix 3 . Under the condi-

tion of high vapor pressure, flux and transport rate were modified by thermal convectionl2) .

2.2 Calculated results and discussion for ZnS~l2 System

Figure 2 shows the relationship between M and Pi, where solid and dashed lines represent

the Pj-values at the source (T* = 1098 K) and growih (Tg=1073 K) zones, respectively. It is

found that the partial pressures of Znl2, I, 12 and Se2 monotonously increase with an increase of

M. The Zn readily reacts with iodine to form Znl2. Thus, the vapor pressure of Zn is less than

10 Pa and does not appear in the figure. The selenium pressure (Ps=,) at the growth zone was

104-105 Pa in the transport atmosphere, which must be comparable to the dissociation pressure

of stoichiometnc ZnSe composttion to obtain good quality single crystal. The congruent subli-

mation is grven as follows;

ZnSe (s) =Zn (g) + I /2Se2 (g) . (8) The equilibrium constant for (8) is given by under condition of Pz*=Ps., as follows;

Crystal Growth of II-VI and 1-III-V12 Compound Semiconductors by Chemical Transport

2. Thermodynamical Considerations 23

Fig. 2.

Q*

~ OD ~~i ~l

O ~>

107

l06

105

104

103

102

~

7t

j

53!;~*

~;o

~~

pl

~~

/

~~

c~

9~~

7

1~~~･/

p './ ~ p' ~r

1~~l

7

10-1 100 102 I03 101

AMOUNT OF IOD~NE, M / kg m~3

Vapor pressures of gaseous molecules as a function of amount of iodine, calculated at T*= 1098

and A T= 25 K in the ZnSe+12 system. Solid line stands for the source zone and dashed line for

the growth zone.

K=P P1/2=P~!2 (9) z* s*, ,-By using thermodynamic data in Table I, the Ps,, value in (9) amounted to about I Pa, which is

much less than those in the transport systems. Therefore the excess Se2 pressure may have

caused lattice defects and their associations in the crystals resulting predominant nonradiative

recombination centers8)

The calculated gas fluxes including thermal convection are shown in Fig . 3 . The transport

direction of Znl2, I and Se2 is forward (from the source to the growth zone) , while that of 12

reverse. The result supports the transport reactionl,5) depending on M as follows,

ZnSe (s) +12 (g) =Znl2 (g) + 1/2Se (g) (10)

which neglects the reaction (3)

The forms of the grown crystals8) might be related with the gas flux as follows: the fine

granular crystals at M< I .O kg/m3 might be due to the low fiux. In the plateau region of

1 .O~M~ 5 .O kg/m3, the flux seems to become large enough for the growth of the bulk single

crystals. The large fiux due to the thermal convection at M> 5.0 kg/m3 may lead to the needle-

like crystalsl3). From these results, the optimum condition of growing large size ZnSe single

24 Yasutoshi NoDA Yoshitaka FURUKAWA Shigeatsu NAKAZAWA

Fig. 3.

(o '-,

O ~ ¥ * ~) ~~ a~ <e

1 0-7

10-8

1 0-9

10-10

1~

Se2 I

10 1 - 100 101 102 103 AMOUNT OFIODINE. M /kg m~3

Fluxes of gaseous molecules as a function of amount of iodine, calculated at T* = 1098 K and

A T= 25 K with thermal convection in the ZnSe+12 System. Solid line stands for forward trans-

port from source to growth zone and dashed one for backward

Fig. 4.

1 0-7

'ca ~bo

¥ * :~ l0-8

~ ~~:o~~:Erl 10-9 ert

(O

~

1 O- I O

o

o

o o o _

10-1 100 101 103 102

AMOUNTOFIODlNE.M /kgm~3

Transport rate of ZnSe as a function of amount of iodine. The lines were calculated at T* = 1098 K

and A T= 25 K with (solid line) and without (dashed line) consideration of thermal convection

The symbol (O) stands for experimental data

crystals can be sought in the plateau region, where the growth rate of ZnSe cryials was limited

by laminar flow and diffusion in the gas phase. . It is remarkable the calculated transport rate is almost along with the observed one as

shown in Fig. 4. The solid and dashed lines stand for the transport rates calculated with and

without considering thermal convection, respectively. At M< I .O kg/m3, both of the calculated

and observed transport rates show a gradual increase with an increase of M. The calculated

Crystal Growth of II-VI and 1-III-V12 Compound Semiconductors by Chemical Transport 2. Thermodynamical Considerations 25

transport rate and the observed data form a almost plateau at M= I .O-5.0 kg/m3. At M> 5.0

kg/m3, the contribution from thermal convection was found to be significant, resulting the

steep increase in the transport rate. The transport experiment with M>40 kg/m3 was not per-

formed, since the total vapor pressure of about I .5 x 106 Pa at M=40 kg/m3 is close to the toler-

able limit for the growth quartz ampoule

Figure 5 shows plots of observed transport rate versus calculated gas flux, which forms a

linear relationship for each data set of growth cond.ition. The transport rate do not exceed 10-9

kg/s, until the contribution from the thermal convection is significant. It is found that no great

difference in the transport rate among the data sets, although the one set at T* = I 123 K wrth

A T= 50 K is separated in gas flux from the others. The fact suggests that the growth rate mrght

be limited by deposition rate as well as by diffusion and laminar fiow

3. Calculation for AgGaS2-X2 systems

3.1 Transport reaction of AgGaS2-X2 system

The calculation was performed as was the case for ZnSe by using the thermodynamic data

in the AgGaS2-X2 System where the gas phases were AgX, GaX3, GaX, X, X2 and S2 by usmg

halogen (X= I, Br and Cl) as the transport agent. The systems involve the following three reac-

tions;

AgGaS2 (s) + 2X2=AgX (g) + GaX3 (g) + S2 (g) , (1 1 )

GaX3 (g) = GaX (g) + X2 (g) , (12) X2 (g) =2X (g) . (13)

'u) 10~8 ~ eJO ~~:

~ o ~;

~:

H ~ 10~9 h~~ e*

O e* cf)

~ H -10

1 O I 0~8 GAS FLUX. J* / mol s~1

Fig. 5. Plots of observed transport rate versus calculated gas flux of Znl2 in the ZnSe+12 system. The

growth conditions: (O) ; T*=1098 K and A T=25 K, (O) ; T,= 1123 K and A T=25 K and ([]) ;

T*= 1123 K and A T=50 K.

26 Yasutoshi　N0DA・Yoshltaka　FURU鮎wA Shigeatsu　NA鮎zAwA

The　condensed　phase　ofAgX　wa．s　not　mc1uded．m　the　react1ons，smce　the11qu1d　phase　was　not

formedandanymd1cat1onofthecrysta1growthfromme1twasnotrecogn1zed．1ntheamoun．t　of

ha1ogen1ess　thanヱkg／m3Theequ111br1um　constants（K』）1nvo1ved1nthesereact1ons（］：1－3）

are　expressed　by，

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　K1＝P。。。p。、／p。、2，　　　　　　　　　（14）

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　K。＝P。。xp。、／p。。xヨ，　　　　　　　　　（15）

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　K。＝1）。2／P。、．　　　　　　　　　　（16）

The　cond1t1ons　ofthe　constant　meta1atom　rat1o（Ag／Ga＝1）and　the　st01ch1o㎜etry　cond1t1on

（（Ag＋Ga）／s2＝1）ho1d－1neachzoneforAgGas2to　betransported，wh1ch　areexpressed　as　fo1－

1ows；

　　　　　　　　　　　　　　　　　　　　　　　　　　　　P。。x＝P。。x、十P。。x，．　　　　　　　　　　（17）

　　　　　　　　　　　　　　　　　　　　　　　　　　　　1）・。。十P・。。，十P。。x＝P。、．　　　　　　　　　（18）

Part1a1pressures（P、’s）were　ca1cu1ated　after　the　ca1cu1at1on　of　the局s　from　the　free　energ1es

fortheequ111br1a（1）一（3）usmgthermodynam1cdata9－11）11sted1nTab1e　II　Thetota1a㎜ountof

Tab1e　II Thermod〕mamic　data州）and　Lemard－Jones　parameters工9）for　AgGaS2－X2system．

∠1凪298（J／皿o1）

　　　∫。298

［J／（K・n1o1）コ

c。［J／（K・㎜o1）］・）

a　　b・103　c・10■5

Mo1ecu1ar（、。鴇。、）（・㌫）1い…b）

AgGaS2（s）

AgC1（9）

AgBr（9）

AgI（9）

GaC1（9）

GaBr（9）

GaI（9）

GaC13（9）

GaBr3（9）

GaI。（9）

C1（9）

Br（9）

I（9）

CI。（9）

Br。（9）

I。（9）

S。（9）

AggGaS6（s）・）

Ga．S（9）

一370000

　　　100000

　　　121000

　　　138000

　　－83700

　　－33500

　　　41800

－427000

－301000

－117000

　　121000

　　111000

　　107000

　　　　　　　0

　　　30500

　　　62300

　　130000

－1480000　　　20900

172

246

257

264

241

252

259

326

362

383

165

175

181

223

245

260

228

742

290

69，7

37，7

37，7

37，7

37，7

37，7

37，7

75，4

75，4

75，4

21，0

20，9

18，6

36，9

37，8

37，2

05，7

45，3

56．O

22．6

0．0

0．O

O．0

0．0

0．O

O．O

O．0

0．O

O．0

0．0

0．0

0．0

0，06

0．O

O．0

0，28

5，65

0．275

　　O．O

　　O．0

　0．O

　　O．O

　O．O

　O．O

　O．O

　O．O

　O．0

　0．O

　O．O

　O．O

　O．O

－2．85

－1，55

　0．O

－3，31

　0．O

－9．25

241．7

143．3

187．8

234．8

105．2

149．6

196．6

176．1

309．4

450，4

　　35，5

　79．9

126．90

　70．9

159．8

253，8

　64．玉

1233．0

171．5

0．418

0．424

0．544

0．496

0．418

0．424

0．424

0．612

0．563

0．361

0．367

0．422

0．422

0．430

0．516

0．452

1726

936

430

188

1726

936

936

442

696

131

237

289

316

508

474

847

　LiI

BeBr2

BBr3

CCIF3

LiF

BeBr2

BeBr2

CBr4

HgI2

HgI2

a）

C）

0ヒa＋bτ十cr‘2，b）taken　from　the11sted　mo1ecu1e19）

est1mated　on　the　bas1s　of　the11terature19）

Crystal Growth of II-VI and 1-III-V12 Compound Semiconductors by Chemical Transport

2. Thermodynamical Considerations 27 halogen (M) charged in the reaction tube was estimated in Appendix I by using partial pres-

sures of all gaseous halides and halogens in both zones

The fiux (J*) of the i-th gas molecule and the transport rate of AgGaS2 Were calculated on

the basis of the transport equation in Appendix 2 using the Lennard-Jones parameters m Ap-

pendix 3 . Under the condition of high vapor pressure, flux and transport rate were modified by

thermal convectionl2) . The calculation was performed at Tg of 1 173-1273 K and the temperture

difference (A T= T*- Tg) of ICH50 K

3.2 Calculated results and diseussion for AgGaS2-12 system

Figure 6 shows the relationship between M and Pi, where solid and dashed lines represent

those at source (Ts= 1248 K) and growth (Tg= 1223 K) zones, respectively. It is found that the

vapor pressures of halides, halogen and sulfur molecules monotonously increase with an in-

crease of M at M> 10-1 kg/m3. Chlorine and bromine are ready to react with Ag and Ga to

~ ~ e~

~ ;~ (t~ c,~ ~e' eg: o ~ >

107

10210-2 10 -1 10 O 101 I02 AMOUNT OF CHLORINE, M / kg m~3

~ I07

~ e･ 106

~I~ 105

c') c'~

~ l04 e*

~ 103

~ <:

> I02

eEl l07

(c)

~ 106 eq_ 10S

Total _ _ = - ~ - ~I ~

104 i

~ S2 _ ceDca) I03

. " 12 Agl _ ~ - -

~e~ l02 ~ e~:oe~:1010 ~d~ ~~~-----

G_.__~ <!10 dGal3 ~~-~--_

:> 10r:1

10-2 10-1 100 I~l l02 AMOUNT OF IODINE, M / kg m~3

Fig . 6.

10-2 10-1 100 1c1 I02 AMOUNT OF BROMINE, M / kg nf3

Plots of vapor pressure versus amount of halogen, calculated at T*= 1248 and A T=25 K m the

AgGaS2+x2 Systems. Transport agent: (a) Cl, (b) Br and (c) I. Solid line stands for the source

zone and dashed line for the growth zone

28 Yasutoshi NoDA ･ Yoshitaka FURUKAWA ' Shigeatsu NAKAZAWA

form the halide so that the vapor pressures of the halides are larger than those of halogens. On

the other hand, the vapor pressures of iodines (1 and 12) are by sev~ral orders of magnitude larg-

er than those of the iodides

The sulfur pressure in Fig. 6c was found to increase with an increased amount of iodine

and amounted to about 5 x 101 Pa at M= 100 kg m~3 by using iodine. Figure 7 shows plots of

emission intensity (IEx) due to free exciton8) versus sulfur pressure evaluated from amount of io-

dine as the transport agent. The sulfur pressures at the growth zone must be close to the dissocia-

tion ptessure of the stoichiometric AgGaS2, m order to obtain good quality single crystals by

chemical transport. The following dissociation of AgGaS2 might be induced from the results of

the heat-treatment experimentsl4) where Ag2S-rich Ag9GaS6 and Ga2S3 deposited;

AgGaS2 (s) = I /9Ag9GaS6 (s) + 4/9Ga2S (g) +4/9S2 (g) , ( 19)

The equilibrium constant for the reaction (19) is given under the condition of PG~s =Ps, as fol-

lows ,

K= (PG.,s Ps,) 4/9 = P8/9 (20) s, ,

By using the thermodynamic data listed in the Table II, the Ps, value in (20) amounted to about

10-1 Pa. It is noteworthy that Ps, on the chemical transport exceeds that on the dissociation

103

at 4.2 K

~ ~ ~~:

~lF~l

>? I02

~I

~ caD ~; ~~~

E~ ~; ~I ~;o 101

~l c'D

~ ~~

o

100

102

Ps2 / Pa

Fig. 7. Plots of observed emission intensity of free exciton versus calculated sulfur pressure in the AgGaS2

+12 systems. The transport experiment using iodine at Tg= I 173-ll23 K, A T= IC~50 K

Crystal Growth of II-VI and 1-III-V12 Compound Semiconductors by Chemical Transport

2. Thermodynamical Considerations 29 Therefore the srgnificant decrease of IE* with an increased amount of iodine indicates that the

non-radiative recombination centers and the degradation of the grown crystals might have been

caused from such excess sulfur pressure on the chemical transport

Figure 8 shows the relationshrp between M and J* which were calculated by using Pi's in

Fig. 6. AgX, GaX, X and S2 are always forward transported from source to the growth zone

(solid line) and while X2 backward as shown by dashed lines. The thermal convection was sig-

nificant at large amount of transport agent, where the dotted dashed lines indicate the fluxes

without thermal convection. Figure 9 shows plots of the calculated transport rate of AgGaS2 in

terms ofM. The solid and dashed lines stand for the transport rates calculated with and without

considering thermal convection, respectively. The effect of the thermal convection is apprecia-

ble as a steep increase of the transport rate at M> 101 kg/m3. At M= 10-1- 101 kg/m3, the

transport rate usmg bromine and iodine as the transport agent decreases with an increase of M,

while the rate increases with an increased amount of chlorine

~ ~o S ¥ ~~h~ ~~ 1~ ~I ~4

(a)

10 -8 //_ ~..'- -

C12_~ ~~ 10 -9 /~ ~ AgCl S2 ~ GaC13

10-10 cl -.-_. 10-11

GaCl 10-t0-2 10 -1 100 101 I02

AMOUNT OF CHLORINE, M / kg m~3

10 -7

(b)

10 '8 - Br2 ~ ~ ~ ~

GaBr32 10 -9 Br __. _ .- . Br +*

10-10

10-11 GaBr

10-7 (c)

~ 10-8 I ~ ~ ,Q 12 ~--____~]~~~_ ~ 10-9 ~+~'~:~'~ -

~ ~~~ 10-10 Agl S2 ~<:

h~ 10-11 ¥ Gal3 ~)

10'12 10-2 1(rl 100 . 101 102 AMOUNT OF IODINE, M / kg m~3

FI

-O ~ ¥ ~~ ~~ ~!I ~rf

Fig. 8.

12 10 ~ 10-2 10fl 10 O 101 I02 AMOUNT OF BROMINE, M / kg m~3

Fluxes of gaseous molecules as a function of amount of halogen, calculated at T* = 1248 K and

A T=25 K under consideration of thermal convection in the AgGaS2+X2 systems. Transport agent: (a) C1, (b) Br and (c). I. Solid line stands for forward transport from source to growth

zone and dashed one for backward.

30

'Fig. 9.

Yasutoshi NODA ' Yoshitaka FURUKAWA ' Shigeatsu NAKAZAWA

FI '(eD 10-7

eJD ~~l

¥ 10-8 ~~ ~:

^ dl Br ~~IE~ 10-7

:~ 10 -10

Oeq 10-11

ca~

~; ~Ai! 12 ~ 10 - 10-2 10-1 100 101 I02

AMOUNT OF HALOGEN9 M / kg IDl~3 Calculated transport rate of AgGaS2 as a function of amount of halogen, at T*= 1248 K and

A T= 25 K with (solid line) and without (dashed line) consideration of thermal convection

Frgure 10 shows the observed transport rate in terms of Mmeasured by using halide and io-

dine as the source material of halogen. The transport rate using iodine and iodide was by 10

times larger than those using other halogens. The calculation reproduced the observed gradual

decrease in the transport rate at M= 100- 101 kg/m3 in the cases of bromine and iodine. The

decreased transport rate was observed for Agl (1) and AgBr (CdBr2) with a decrease of Min the

region of M< l0-1 kg/m3 which might be caused from the process limited by the supply of the

gaseous molecules. As for the transport agents, the discrepancy between the observed and calcu-

lated transport rates was found to exist in the magnitude. The calculation also did not

reproduce the order of Cl < Br< I observed in the transport rate. The order corresponds to that

of the enthalpy (AH398) of Ag and Ga halides lisfed in Table II. Therefore further thermody-

namical considerations including the deposition process might be necessary to clarify the dis-

cre pancy .

4. Summary

(1) Thermodynamical calculations were performed for the chemical transport systems of

ZnSe+12 and AgGaS2+x2 (X= Cl, Br and I) . The vapor pressures and the fiuxes of the gase-

ous molecules were calculated. The transport rates of ZnSe and AgGaS2 Were estimated on the

basrs of the transport equation, where the transport is limited by the diffusion and laminar fiow

of gaseous molecules and the diffusibities of the molecules were estimated by using Lennard-

Jones parameters.

(2) The chalcogen vapor pressures (Se2 in ZnSe or S2 in AgGaS2) were found'to increase

with an mcrease of transport agent. These values at growth zones exceeded those of the dissocia-

Crystal Growth of II-VI and 1-III-V12 Compound Semiconductors by Chemical Transport 2. Thermodynamical Considerations 31

Fig. 10.

~D

~ ~;

~ ~ ~~

~ egj

O ~ ~; <!

~

10 -9

10 -10

10 -11

10-12

o

cP D BD D DD DAgl::l

R Iodine

CdBr2 o o

ogo

AA

R

oo oo o 8 oooo

A AgBr A A AAgCl

10-1 10 O 10 1 10-2

AMOUNT OF HALOGEN, ,M / kg In~3

Observed transport rate of AgGaS2 plotted ag~inst amount of halogen,

A T= 25 K by using a variety of halides as the souree of transport agent

at T,=1248 K and

tion of the stoichiometric ZnSe or AgGaS2, which might have caused the poor quality by the in-

troduction of non-radiative recombination centers in the grown crystals

(3) The calculated transport rates for ZnSe was almost,along the observed data, which in-

dicates that the transport is limited by diffusion and lamin~t flow .at M~ 5 .O kg/m3, while the

steep increase at M> 5.0 kg/m3 was due to the thermal convection

(4) The calculated transport rates for AgGaS2 reproduced the observed gradual change

with an increase of halogen, but not realized their magnitudes and the order (CI < Br < I) in the

observed transport rates. Therefore further thermodynarriical considerations including the

deposition process might be necessary to clarify the discrepancy

Appendix

1. Amount of transport agent

In these calculations for ZnSe or AgGaS2, the total amount of halogen (M) in the trans-

port agent charged in the growth ampoule was estimated'by using partial pressures of all gase-

ous halides (Znl2 or Agl, Gal3, Gal) and halogens (1 and 12, or X and X2) in both zones as fol-

lows,

2~ 1 M(kg/m3) =M. x 10~3 ~ ~ mjPj(i), (A1) i j RT,

where Mx is the atomic weight of halogen atom, i is source and growth zones, R is gas constant

and mj the number of halogen (X2) in a molecule of the j-th gas component.

32 Yasutoshi NoDA Yoshitaka FURUKAWA Shrgeatsu NAKAZAWA

2. Transport equations

The transport equation was proposed on the assumption that one-dimensional diffusion

and laminar fiow limit the transport from the source to the growth zone. The flux (J*) of i-th

gas component rs grven by

Ji (mol/s) =S( -Di dCi/dx+ WCi) , (A2) DS P~i ~{ mjAP

=LRT ~AP+ mjpj ' ~; ~ (A3)

where Di and C:i are the diffusivity and the concentration of i-th gas component respectively, W

is the laminar fiow velocity, x the position parameter, and S the cross sectional area of the am-

poule. D is the mean diffusivity of gaseous molecules which was calculated by using Lennard-

Jones parameters m Appendix 3 . L is the distance between the source and the growth zone

T= (Tg+ T,) /2, (A4) P~i = (Pig + pi*) /2, (A5) APj is the pressure difference between the growth and source zones, which is regarded as a driv-

ing force for the transport. The transport rate (N) was calculated as follows,

N(kg/s) =MO X 10~3 ~ mi J, (A6) i=1

where Mo rs the molecular weight of ZnSe or AgGaS2 and mi is the number of Zn or Ag in the i-

th gas component

Under the condition of high vapor pressure and large cross-section of the growth ampoule,

the mass transfer is modified by thermal convection, which is caused from the interaction be-

tween gravrtational force and the gas density difference due to the temperature gradient. The

gas fiux and the transport rate are given by using the quantity K(12) as follows,

Ji* = Ji (K+ 1) , (A7) N* =N(K+ I ) , (A8)

where

K= [A / (ScGr) 2 +B I ~ I , (A9) where Sc and Gr are the Schmidt and the Grashof numbers, respectively, calculated in Appen-

dix 3 . The dimensionless quantities ofA = 6.95 x 107 and B= 6.56 x 10-2 were estimated for the

ampoule geometry ratio (L/d) of 12.5ra) where d is the ampoule diameter

3. Transport paralneters

The transport parameters were calculated as followsl5-19) and shown in Figs A1 and A2

Crystal Growth of II-VI and 1-III-V12 Compound Semiconductors by Chemical Transport

2. Thermodynamical Considerations 33 - 104

S ~,:~o

~ 10-5 ~ (a)

-1 100 101

lO I02 103 AMOUNT OF lODlNE, M / kg m~3

0~~)

100

10-3

Zn I

~ Se2

1 O- 1

(c)

1 04

c~'__~ 10-5

C~

l0-6

10-7

Do

(1?)

Znl2

12

l0-1 100 101 102 103 AMOUNT OF IODINE, M / kg m~3

~

106

l05

l 04

103

102

lO1

10-1 100 101 102 I03 AMOUNroFIODINE,M /kgm~3

Fig. A1.

(d)

l0-1 100 10 101 102 3 AMOUNT OF IODINE, M / kg m~3

The transport parameters of gas nuxture as a function of amount of iodine in the transport sys-

tem of ZnSe+12 at T*= 1098 K and A T=25K: (a)viscosity, (b) diffusivity, (c) Schmidt and (d)

Grashof numbers

for viscosity, diffusivity, Schmidt and Grashof numbers

(i) Lennard-Jones parameters

The values of the Lennard-Jones parameters for the gaseous molecules taken from the

literaturel9) are listed in Table I and 11 with the thermodynamic data. When the values of power

factor (8/k) are not available, we estimated by using the approximate procedures as follows,

8/k= I . 1 5Tb, (AI I ) 8/k= I .92T*, (A12)

where Tb and T* are boiling and melting temperatures in K, respectively. When Eqs. (AI 1) and

(A12) give differing values of 8/K, the values are averaged. The difference between those values

calculated by using the Eqs. (A11) and (A12) is recognized as small. Table I and 11 also lists aj:

the collision diameter of the molecule in m. In case of being unavailable, the values of aj: were

taken from those reported for the molecules having the similar molar weight

34 Yasutoshi NoDA -

10 5

10 4

10 3

10 2

10 1

10 O

10~1

10 ~2

10 ~3

10 ~4

10 ~5

Yoshitaka FURUKAWA Shigeatsu NAKAZAWA

~ ~D ~ sr >~ E~ 5~ O (, CD ~

Gr

Sc

(a)

Fig . A2.

,1

10 ~5

10~2

10 ~1 10 O 10 1

AMOUNT OF IODINE, M / kg nf3

10 6

10 5

10 4

10 3

102 ~ ~ 10 1

100 ~ ~,:

10 ~1

10 ~2

10 '3

10 ~4

10 2

10~3

I~_ 10~4

A >1 E~ ~ 10'5

OD ~ ~ N 6 ~l 10~

10 '7

s2 Agl

(lal

G･13

12

(b)

Do

I

10 1 Io 2 10 O 10 ~2 10 '1 AMOUNT OF IODINE, M / kg m~3

The transport parameters of gas nuxture as a function of amount of iodine in the transport sys-

tem of AgGaS2+12 at T,=1248 K and AT=25K: (a)viscosity(n), Schmidt (Sc) and Grashof(Gr) numbers and (b)diffusivities

(ii) Viscosity of gases

The vrscosrty of nonpolar gases at low pressures is give as follows

ni = 8.441 1 x 10~25 (Mi T) 1/2/ (ai2Qn) ' (A13)

where the collision integral of viscosity (Qn) is estimated from the table listed in terms of k~Tl

815) . M* is molar weight (kg) and ni is in kg/ (m s) .

For the multicomponent gas mixture, the following formula is given,

Crystal Growth of II-VI and 1-III-V12 Compound Semiconductors by Chemical Transport

2. Thermodynamical Considerations 35

n==~ xini (A14) i=1 ~ xi~ij

j=1

m which

1 1/2 M 1/4 2 Mj -1/2 .) M == Mi ( :) l ) [ i (

( clJ f~~ nJ

n is the number of gas species in the mixtures; xi and xj are the mole fractions of species i and j ;

ni and ~j are the viscosities of species i and j at the system temperature and pressure; and Mi and

MJ are the corresponding molar weights

(iii) Diffusivity

For the interdiffusivity D12 (in m2/s) of two unequal gases I and 2, we have the following

equation, which is strictly applicable for spherical monatomic gases

D12 = 5.9543 x I 0~24T3/2 ( I /MI + I /M2) 1/2/ (Pal22QD) , (A16)

where crl2= (al + a2) /2 collision diameter in m, QD collision integral for 1-2 mixture at dimen-

sionless temperature ( T12*) , the vales of which were taken from the table listed in terms of (81

k) 1215) .

T12* = ~T/ (8/k) 12, (A17) (8/k) 12= [ (8/k) I (8/k) 2] l/2 is the averaged intermolecular force parameter, in K; P is the sum

of average pressures for I and 2 gases,

P= P1 + p~2 in Pa. (A18) For the I , 2, ･ ･ ･ , n-component gas mixture, the diffusion coefficient of component I is ex-

pressed as follows:

D1, mi*= (1 -yl) /[ (y2/D12) + (y3/D13) + (y4/D14) + " ' + (y./Dl') I (A19)

where Dli rs the diffusion coefficient of component I in the 1-i binary system

yi;P1/~i P~i (A20) The D value was taken from the diffusivity of Se2 or S2, since the Se2 or S2 flux was considered to

determine the transport rate of ZnSe or AgGaS2 and the diffusivities of the gas species were

almost comparable as shown in Fig. Alb and A2b where Do rs diffusivity appeared in the equa-

tion

D D., (X=SeorS) (A21) D = Do ( T~/273) 3/2/ P (A22) t*tat.

3 6 Yasutoshi NODA ' Yoshitaka FURUKAWA ' Shigeatsu NAKAZAWA

(iv) Schmidt and Grashof number

The Schmidt Sc and Grashof number Gr are given as follows,

Sc = n/ ( pD ) (A23) an d

Gr = d 3g Pp2A Tln2, (A24) where P is the thermal expansion coefficient equal to I / T for ideal gas and g gravitational ac-

celeration(=9.81 m/s2) , p is gas density(kg/m3) given as follows,

p = ~ [ P~iM/ (R T) I , (A25) where R is gas constant (= 8.31451 J mol-1K-1) .

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