Dislocation reduction in GaAs crystal grown from the Czochralski process

Journal of

Journal of Materials Processing Technology 55 (1995) 278-287

ELSEVIER

Materials Processing Technology

Dislocation reduction in GaAs crystals grown from the Czochralski process N. Subramanyam, C.T. Tsai* Department of Mechanical Engineering, Florida Atlantic' Universi(v, Boca Raton, FL 33431, USA Received 1 April 1994

Industrial summary

The dislocation density in the gallium arsenide (GaAs) crystal is generated by excessive thermal stresses during the Czochralski (CZ) growth process. A constitutive equation which couples the dislocation density with the plastic deformation is employed to simulate the dislocation density in the crystal. The temperature distribution in the crystal during the growth process is obtained by solving the quasi-steady-state (QSS) heat-transfer equation. The thermal stresses induced by the temperature distribution are calculated using the finite-element method. The crystal is assumed to be an axisymmetrical ingot. The resolved shear stress (RSS) in each slip system is obtained by stress transformation. The RSS in each slip system is no longer axisymmetric. The dislocation motion and multiplication in each slip system are simulated using the constitutive equation. The total dislocation density in the crystal is obtained by summing the dislocation densities in all of the slip systems. Since the thermal stresses are sensitive to the temperature gradients and the dislocations move faster at a higher temperature, the dislocation densities are generated most near to the solid-melt interface. The dislocation density is also found to be affected by the growth orientation, growth speed, ambient temperature and the radius of the crystal. The dislocation density in GaAs crystals grown with the different growth orientation, growth speed, and crystal radius at various ambient temperatures has been calculated so that the influence of these growth parameters on the dislocation density can be understood. Consequently, the growth parameters can be controlled to reduce the dislocation density generated in the crystal during the CZ growth process.

1. Introduction

The concept of dislocation was first proposed independently by Taylor, Orowan and Polanyi in 1934 [1]. It was only during a 10 year period following the second world war that the dislocation concept was developed extensively and applied to every aspect of the plastic deformation of metals. Over the last three decades, extensive research and experimental work has been done on observing and studying dislocation in electronic and photonic crystals. The applications of crystals such as silicon (Si), gallium arsenide (GaAs) and indium phosphide (InP) in today's electronic industry are tremendous. A majority of these crystals are being used in microelectronic and optoelectronic devices such as laser diodes, LEDs, FETs, high performance integrated circuits, etc., which form the nucleus of today's technology and science. The presence of dislocation in the crystals considerably reduces the life and performance of the devices as well as increasing their degradation rate. It is hence of paramount importance that dislocation free/low

* Corresponding author. 0924-0136/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0 9 2 4 - 0 1 3 6 ( 9 5 ) 0 2 0 1 8 - H

dislocation density crystals be made available to the electronics industry, as the latter moves into the 21st century. It was as early as 1955 that the etch pitch density of germanium (Ge) wafers obtained from pulled ingors were found to increase with the magnitude of the imposed temperature gradient [2]. The etch pits were distributed in a definite pattern and slip bands were discovered also. It was concluded then that high thermal stresses give rise to slipping and dislocation generation. It has since been proven that the primary cause of dislocation in CZ pulled GaAs single crystals is the crystallographic glide induced by the excessive thermal stresses arising during the growth process. New dislocations are generated when the thermal stress exceed a particular critical value induced from the current dislocation barrier and the crystal deforms plastically. Several researchers have already pointed out a good correlation between the distribution of the dislocation density and the critical resolved shear stress (CRSS) in the cross-sectional plane of liquid encapsulated Czochralski (LEC)-grown GaAs crystal. In ductile crystals such as metals, dislocations are propagated easily and multiplied by the thermal stress. Brittle crystals such as oxides sometimes crack under thermal stress during growth or after the growth process.

N. Subramanvam, C.T. Tsai /Journal of Materials Processing Technology 55 (1995) 278-287

A simple steady-state heat-transfer model formulated by Brice [4], gave the experimental temperature distribution in Ge and ZnWO4, obtained by the thermocouple embedding technique, in a consistent manner. He represented the growing crystal as a stationary cylinder, the base held at a temperature Tf while its lateral surface and top dissipate the heat into a medium at a constant temperature by convection. However this solution was valid only for fixed ambient temperatures and excluded growth rate as a parameter. Jordan et al. [3] included the growth rate as a parameter in the solution considering a moving boundary quasi-steady-state partial differential equation instead of a steady-state partial differential equation. They then used the quasi-steady-state temperature profiles for deducing the radial, tangential and axial thermal stress components for a growing cylinder in a closed form. They also compared the thermal stress components in LEC-pulled GaAs, Si and InP crystals and concluded that dislocation-free GaAs and InP could be grown by the LEC technique, if the ambient temperature was sufficiently high [5]. It was found also that Si exhibited a decisive advantage over lnP and GaAs in view of its large CRSS and thermal conductivity, thereby having a greater resistance to thermal stress induced slipping. Duseaux [6] performed thermal stress calculations on GaAs crystals using LEC and LEK (liquid encapsulated Kyropoulos) techniques, the latter developed in their laboratory in France. He reported that the LEK method was a better technique as it obtained low dislocations compared to LEC. Aoyama et al. [7] studied the effects of melt stoichiometry and high In doping of GaAs grown by horizontal Bridgman (HB) growth. They concluded that the changes in dislocation densities were caused by the changes in melt stoichiometry and not by impurity segregation or differences in thermal history between the seed and tail portions of the crystal. They also proposed doping with electrically active impurities, as this shifts the Fermi energy and changes the charge states of point defects in the crystal, which in turn changes their ability to condense into dislocation loops and support dislocation climb. In 1984 Jordan et al. [8] proposed the following avenues for reduction/elimination of dislocation in crystals: (i) a sharp decrease in ambient temperature; and (ii) impurity hardening of the lattice in order to increase the CRSS. Both of these methods were successful in pulling GaAs and InP boules, but the limitations were that low gradients were not conducive to the preservation of a stable crystal shape. Further heavy doping was often undesirable from a device design point of view. They proposed a compromise strategy to reduce dislocations by combining a moderate temperature gradient with an intermediate doping level. Schvezov et al. [9] calculated, with the finite-element technique, the stress distribution in GaAs crystal assuming thermoelastic crystal behavior, the results obtained being in good agreement with ana-

279

lytical solutions. They observed that high dislocation densities occur in the central region and near to the outside surface of the crystal, which was consistent with the reported observations of etch pitch density distributions in GaAs. In order to calculate the RSS in crystals, Dobrocka [10] introduced an effective method based on double tensor transformation with arbitrary growth axis orientation. He investigated the influence of crystal orientation on the RSS for HB grown GaAs crystals. His results provided a theoretical basis for the evaluation of the role of the thermal stresses in the dislocation generation process. In 1991, Tsai [11] was the first to develop a finite-element formulation to calculate the level of dislocation density within the crystal. From the Haasen-Sumino material model for silicon, he formulated a visco-plastic response function for semiconductor materials, which relates the dislocation density to the plastic deformation. He simulated the dislocation dynamics using a non-linear finite-element formulation from the theory of viscoplasticity. The quest for growing low dislocation density GaAs, however, has not been fruitless. In 1985, Ozawa et al. [12] reported that a newly developed arsenic ambient controlled LEC growth system with the X-ray shadow imaging technique produced nearly dislocation-free undoped (100) oriented GaAs single crystals under a super low temperature gradient. The average dislocation density over a 2 inch (50.4 mm) diameter crystal were of the order of 100/cm 2 and the crystals were free from dislocation networks, lineage and slip bands. The crystals were also free from micro defects revealed by hydroxide etching. Similar success was reported by Khattak et al. [13] for large diameter GaAs crystals grown by the heatexchanger method. Crystals grown by this method were found to exhibit uniform properties in the radial as well as the vertical directions, except at the edges. The future of next generation electronics, particularly very high speed integrated circuits, looks bright as significant progress is being made in crystal growth techniques. However, the challenge of growing dislocation-free material with uniform and controlled electrical properties using the CZ technique has not been met. With this challenge in mind, an effort is undertaken here to investigate the effect of the growth parameters, i.e., the growth orientation, the growth speed, the crystal radius and the ambient temperature on the crystal dislocation density. It is expected that the results of this study will provide more insight in controlling the growth parameters, so as to be able to obtain low dislocation density GaAs crystal during the CZ growth process.

2. Temperature distribution in a growing GaAs crystal The temperature profile of a growing crystal can be obtained by the QSS heat-transfer model proposed by

N. Subramanyam, C.T. Tsai /Journal of Materials Processing Technology 55 (1995) 278-2{47

280

lOOll I

already by Jordan et al. [3]. Tsai et al. [11], using the finite-element formulation, expressed the QSS partial differential equation as

z ~rowth Jill

i

[K] {T*} = {0},

(2)

where [K] is a symmetric matrix given by [K]

[ - I 101

;;; "" -~"

G:.~i

=

2rta

0N

0N

Q J

tolo~ Y

IOU}[ONlle-V° -z,k, RdRdZ LozJ)

[I001

+ ~

x

Fig. I. Coordinate system for axysymmetrical growth of the crystal Tm is the melting point along the solid-melt interface of the crystal.

+2' afIx}[Ul e-V°"'k'ndT. Tq

Jordan et al. [3]. This model is formulated based on the r, 0, z coordinate system as shown in Fig. 1. It is independent of 0 due to the assumption of an axisymmetric crystal. The r, 0,z coordinate system is embedded and moves with the solid-liquid interface. The QSS partial differential equation for heat conduction is of the form OaT

1 0T

0r 2 h-r~

O2T + 6~z2 - k

v 0T 0z '

(1)

where k is the thermal diffusivity, v is the growth speed and z is the vertical coordinate measured from the solid-liquid interface. The solution for the temperature distribution of a cylindrical ingot has been reported

Here N is the shape function of the element, t~ is the domain of the crystal and Tq a r e the crystal boundaries except for the solid-liquid interface along Z = 0. The Biot number, B = ha~k, and the Peclet number, P = va/k, control the variation of the temperature distribution in the crystal. The temperature distribution computed by this formulation for an identical 5 cm diameter and 5 cm long GaAs crystal with same growth parameters as reported by Jordan et al. [8] is shown in Fig. 2. It can be seen that the temperature profile is very similar to that obtained by Jordan et al. [8]. The temperature profiles obtained from the above finite-element formulation are used for computing the thermal stress components of the

""-------

0.0

0.5

1.0 Tmejt - T ml~ = 20 K

1.5

2.0

2.5

o.o

0.5

,.o

,.5

2.0

2.5

Tn~lt - T wtb = 200 K

Fig. 2. Temperature distribution obtained by the finite-element formulation for a 5 cm diameter and 5 cm long crystal pulled at a rate of 0.0004 cm/s at low and high temperature gradients•

N. Subramanyam, C T. Tsai /Journal of Materials Processing Technology 55 (1995) 278-287

ingot, where the Z axis is assigned as a growth axis, as shown in Fig. 1,

281

The velocity of moving dislocations within the crystal under deformation is given by V = Vo(g a -

D~)me -Q/kT

(9)

3. Modeling of dislocation multiplication in GaAs crystal

As already defined, dislocation is a defect that is responsible for nearly all aspect of plastic deformation. A model which relates the plastic strain rate and the dislocation multiplication rate to the thermal stress and the dislocation density in crystals with diamond or zincblende structure was first proposed by Haasen [14] and then later improved by Alexander and Haasen [15]. This model is derived from the Orowan equation based on experimental observations and supported by physical concepts related to the dynamics and interaction of dislocations. The dislocation multiplication behavior, induced by stresses generated within the crystal, can be described effectively by the Haasen model, which represents a significant advancement over the model proposed earlier by Jordan, as Jordan's model cannot predict the magnitude of the dislocation density generated in the crystal as a result of process parameters. Since most III-IV and lI-VI compounds have either diamond or zincblende structure, the Haasen model can be applied, except for the difference in material properties. The constitutive model for the dislocation multiplication of GaAs is based on the concept of effective stress, reef, which is needed to make a dislocation overcome the intrinsic resistance of the crystal lattice at a given rate. Mathematically this can be defined as

where ra is the applied stress and zi is the stress necessary to overcome the interaction force between mobile dislocations. The stress r~ can be represented as

= Dx/-N,

=

(5)

where G is the shear modulus, b is the magnitude of the Burgers vector, N is the magnitude of moving dislocations, fl is a parameter characterizing the interaction between dislocations and D is a parameter characterizing the relationship between N and % The shear modulus, G, is given by

(CI~ - Ctz) a -

2

1V = K r ~ f f N V = N K V o ( r a - D , ~ ) " + %

-e/kr,

(10)

where K and 2 are the material constants. In II1-V compounds semiconductors, the ratio of the plastic deformation of a crystal is controlled by the motion of screw dislocations. The V in eq. (10) is taken as the velocity of screw dislocation. The rate of plastic shear strain is given by [16] ~P| = N b V o ( z a --

Dx/-N)"e-Q/kr.

(11)

There is no plastic deformation nor dislocation provided that (ra - Dx/N) ~< 0. The total shear strain rate during deformation is + = kel + i:pt = G +/:ol,

(12)

(4)

reff = "Ca - - qSi'

"Ci G b ~

where Q is the Peierls potential, K is the Boltzman constant, Vo is a pre-exponential factor, T is the absolute temperature and m is the material constant. The Mawley bracket notation ( x ) is defined as ( x ) = 0 if x ~> 0 and ( x ) = 0 if x < 0. Since dislocations multiply during motion, the increase in the length of moving dislocations is assumed to be proportional to the area per unit volume swept by them. The rate of increment in the mobile dislocation density is given by [16]

'

(6)

where C11 and C12 are components of the elastic moduli, represented as functions of temperature T [18]: Cll = 1.216 x 1011 - 1.39x 107 T P a ,

(7)

Cl2 = 5.43 x 101° - 5.76x 106 TPa.

(8)

where k and ~el are the total and elastic resolved shear strain rate respectively and + is the resolved shear stress rate. The previous equation is the constitutive equation employed in the Haasen model. For the undoped GaAs crystal being investigated here, the various parameters being used are listed in Table 1. The above model can be used to simulate dislocation multiplication in each slip system in the crystal during the CZ growth process.

Table 1 List of the parameters used in the H a a s e n model of dislocation multiplication Tmelt m 2 b Q k Vo K D

1511 K 1.7 1.0 4.0x 10-1° m 1.5 eV 8.638 x 1 0 - S e V / ' K 1 . 8 x l 0 - a m Z , . + l N ms l 7.0x10 3m/N 3.13 N / m

N. Subramanyam, C T. Tsai /Journal of Materials Processing Technolo~, 55 (1995) 278-287

282

out separately and the resolved shear stress component for each slip system calculated. As cited in several publications [3, 5,8, 17] the resolved shear stress in only 5 of the 12 slip systems were independent.

Table 2 The 12 active slip systems in G a A s crystal Slip system

Slip plane

1 2 3 4 5 6 7 8 9 10 11 12

-1--1 --1 -1--1 1 --1 --1 1 --1 1 --1 --1 1 --1 --1 1--1 1 --1 -1 1 --1--1 1--1 --1 1 1 1 1 1 1 1 1 1

Burgers vector 1 0 1 0 1 1 --1 1 0 1 0--1 1 1 0 0 1 1 0--1 1 1 0 1 1 1 0 1 -1 0 1 0--1 0 1-1

5. Dislocation multiplication in each slip system During QSS growth, the crystal is growing in the positive z direction at a constant rate v, as shown in Fig. 1. During growth process, the dislocations are assumed to multiply from the solid-liquid interface to the far end of the crystal, along the growth direction. Therefore the rate of dislocation multiplication within crystals is given by ON,. 0z ON,. 0z 0 t - 0z v,

4. Thermoelastic stresses in the slip systems

N,.-

As mentioned earlier, the deformation rate in GaAs is controlled by the motion of screw dislocations, and in the present model the dislocations are assumed to have screw character. A total of 12 slip systems are possible in GaAs crystal. The 12 permissible glide systems are presented in Table 2. The thermal stress components obtained from the axisymmetric finite-element formulation are transformed as follows: (i) The stress components aoo, a,,, azz and 7f,zaxisymmetric are first transformed into the global Cartesian coordinate system (x, y, z) taking the x, y, z axes parallel to the (I00), (010), (001) directions, respectively. The transformation equation is given by

where t is the time. The stress rate due to plastic deformation on each slip system is given by

[~'3 = [T]TEa] [ T ] ,

(13)

where [a], [or], and the transformation matrix given by

7fro "frz 1

IO'xx

LTf ' 7fzo , z=j cos0

[T]=

i

sin0 0

-sin0

LTfzx

0

0

1

where ~ is the rate of resolved shear stress. The resolved shear stress relaxation due to plastic deformation is obtained by solving Eq. (16). By integrating Eq. (15) along the growth direction, the dislocation densities within the crystal for each slip system are obtained. Since dislocation multiplication is very sensitive and in order to obtain an accurate result, numerical integration was carried out considering very small incremental lengths along the growth direction.

6. Numerical results and discussion

,;z j

0

cos0

07f

[T] are

7fxy 7fxz 1 7f ,

(15)

(14)

(ii) The stress tensor in the global Cartesian coordinate system is then transformed into the stress tensor in the local Cartesian coordinate system (x', y', z') for each of the 12 slip systems. The Burgers vector b is along x', y' is normal to the slip plane and z' is parallel to the vector b × n, where n is the unit vector normal to the slip plane. Jordan et al. [8] have presented the resolved shear stress 7fxy for each of the 12 distinct slip systems in both the (100) and (111) pulling directions explicitly. However, in the present case tensorial transformations were carried

It has been stated already that the dislocation density in an undoped GaAs crystal grown by CZ technique is affected by the growth orientation, the growth speed, the crystal radius and the ambient temperature. In order to determine explicitly the influence of each of these parameters on the dislocation density, the dislocation generation model described in the previous sections was simulated, as below: (i) The pulling speed was kept constant at 0.0005 cm/s. (ii) Three different crystal lengths of 5, 10, and 15 cm were considered. (iii) For each crystal length, the crystal radius was increased from 1 cm to 2 cm and then to 3 cm. (iv) A low temperature gradient of 20 K (Tmelt -- Tarnb ~ 20) as well as a high temperature gradient of 200 K (Tmelt - - T a m b = 200) were considered. (v) Simulations were carried out in both the ( 0 0 1 ) and (111) pulling directions.

N, Subramanyam, C.T. Tsai /Journal of Materials Processing Technology 55 (1995) 278-287

283

Table 3 Selected physical parameters of GaAs Crystal

Tmelt (K

Tmelt - Tamb (K)

h .... (cm- 1)

hrad (cm- t)

htot = h + h (cm - l)

K (W/cm 2 K)

k (cma/s)

GaAs GaAs

1511 1511

20 200

0.19 0.24

0.64 0.34

0.83 0.58

0.072 0.083

0.04 0.04

The various parameters such as the thermal conductivity (K), the thermal diffusivity (k) and the heat-transfer coefficient (h), which were used in the simulation for high and low temperature gradients are listed in Table 3. An initial dislocation density of 1/cm 2 was considered along the solid-liquid interface. The initial plastic strains along the solid-liquid interface are assumed to be zero. By summing the dislocation density in all of the 12 slip systems, the total dislocation density at the far end of the crystal for each of the various cases was obtained. The results show that increasing the length of the crystal from 5 cm to 10 cm and then to 15 cm, keeping all other parameters constant, does not have any significant impact on the total dislocation density. The magnitude and the dislocation distribution pattern were quite the same for all crystal lengths, results from the 10 cm long crystal were thus chosen for the ensuing discussion. Figs. 3 6 are typical plots for the total dislocation density distribution, at the top of the GaAs wafer. Figs. 3 and 4 pertain to low temperature gradient crystals pulled in the ( 1 1 1 ) growth direction with crystal radii of 1 cm and 2 cm respectively: here the dislocation distribution pattern exhibits a two fold symmetry. Figs. 5 and 6 are for crystals with identical growth parameters, but pulled in

0.02[

0.00

_0,02i ~ -0.02

0.00

o02

Fig. 4. As for Fig. 3, but for r = 2 cm.

o.01o

0.005

0.010 0.000 0.005 -0.005

0.000 -0.010 -0.010

-0.00'.

-0.005

0.000

0.005

0.010

Fig. 5. Dislocation density distribution (cm-a) at the top of a 10 cm long GaAs crystal pulled in the ( 0 0 l ) growth direction at a rate of 0.0005 cm/s (r = 1 cm, Tin,It -- Tamh = 20 K/

-0.01( -0.010

-0.005

0.000

0.005

0.010

Fig. 3. Dislocation density distribution (cm -2) at the top of a 10 cm long GaAs crystal pulled in the (1 1 1) growth direction at a rate of 0,0005 cm,/s (r = l cm, Tmelt - - T a m b 20 K). =

the ( 0 0 1 ) direction: they exhibit a four-fold symmetry. In all of the four cases, the magnitude of the dislocation density reaches a minimum near to mid-way along the radial direction and attains a maximum either at the

N. Subramanyam, C.T. Tsai / Journal of Materials Processing Technology 55 (1995) 278-287

284

xlO ~

0.02

4 3.5

?

3

o - middle

E

x

-

2.5

l~riphel7

I 0.00

8 i5

1

:-

0.5

Z

-0,02 -0,02

i

i

~

*

i

in cm

Fig. 8. As for Fig. 7, but for r = 2 cm.

C,02

0.0o

axis

xl04

Fig. 6. As for Fig. 5, but for r = 2 cm. 7

xl0' 1.8 * - cerll£:l"

1.6

E

o

1.4 I E

- middle

x - penplm'y

1.2

g i,i

"~

0.~

8

0.6

i

* ° ¢.~ltcr

o - middle

o.4i

x - periphery

0

0.2!

2

4

6 Z axis

0

in

8

10

em

-2

4

6

8

10

Fig. 9. As for Fig. 7, but for r = 3 cm.

Z axisin cm

Fig. 7. Dislocation density along the ( 0 01 ) growth direction (z axis) at the axis, the mid-radius and the periphery in the r direction (r = 1 cm, Tme, - Tamb = 20 K).

center or at the periphery. This type of distribution is typical and has been reported as the 'W' shape dislocation distribution pattern in many publications [3, 5, 18]. However, in order to observe the influence of the crystal radius, the growth direction and the ambient temperature on the dislocation density and compare the magnitude of dislocations, Figs. 7-18 are presented, showing the variation of dislocation density along the growth axis ( ( 0 0 1 ) and ( 1 1 1 ) ) for the various cases simulated, at three radial locations of the crystal i.e., at center, the mid-radius and the periphery. It can be noted in Figs. 7-18 that the dislocation density jumps to a high value initially, before becoming constant along the growth axis. This is because of the high temperature

xl0 ~ 7

P

"E

8

.

0

.

.

.

.

2

.

.

* -

ce.BtP.~

o -

middle

x -

periphery

z : - - ~

4

6

m

8

Z axis in cm

Fig. 10. As for Fig. 7, but for Tm~, - Tamb = 200 K.

10

N. Subramanyam, C.T. Tsai /Journal o)CMaterials Processing Technology 55 (1995) 278-287 xl07

1 2000

285

•

100(30 ~4

1.5

8ooo ~_7

. . . .

77..--..~777777 . . . . . . . . .

7. . . .

777 . . . . . . . . . .

T. . . . . . . . . . . . .

~6oo0

8

8 4000

* -

o middle penphe~

c~

o- middle

-

CI

x

-

x - periphery

2

0L-~

2

0

4

ce4nler

* - center

6

4

6

8

10

8 Z e.us in cm

Z axis m cm Fig. l h

Fig. 14. As for Fig. 13 b u t for r = 2 c m .

A s for Fig. 10, b u t for r = 2 c m .

xlO ~

|.g

3

2.5

xlOa

16

[

1.4

t i

"E 1.2 * -

2

cellla"

,prpR

d

x - I~.npl~y

1.5

,,.---

I

o- middle

0.8 gl

~s

0.6

!

l

¢,erlter o - middle

* -

0.4

• - periphery 0.2 t._.4 0

0 2

4

6

8

0

I0

2

4

6

8

I0

z axis in cm Z axisincm

Fig. 15. As for Fig. 13, b u t for r : 3 c m .

Fig. 12. As for Fig. 10, b u t for r = 3 c m .

xl06

6000

c~

50OO

t

7E

E

4000

-~ 3000

* - celller o - middle

2000

* - ce.qtelr

8

,~

x - p~aphery

o - huddle I000

x- p~p~.ry J

i

t

1o Z axis In

Fig. 13. As for Fig. 7 b u t for t h e (1 I 1 ) g r o w t h d i r e c t i o n .

2

~

~

s

Fig. 16. A s for Fig. 10, b u t for t h e (1 1 1 ) g r o w t h d i r e c t i o n .

~o

N. Subramanyam, C.1~ Tsai /Journal o/'Materials Processing Technology 55 (1995) 278-287

286

xlO 7 4.5 4 3.5 ~t E 3 2.5 2 8

* - ce,rllcr

1.5

o - middle x - periphery 0.:

0

2

4

6

8

10

Z ~isincm

Fig.

17. A s f o r F i g .

16, b u t

for r = 2 cm.

comparing Figs. 13-15 for the low gradient crystals where the range of peak dislocation density is between 5800 and 16000/cm: against Figs. 16-18 for the high gradient crystals where the range of peak density is between 11 x 10 6 and 4.6 x 107/cm2. In effect, the magnitude of dislocations for crystals grown with high ambient temperature (low temperature gradient) was found to be much lower than for low ambient temperature (high temperature gradient) crystals. It can also be observed in Figs. 10-12 and 16-18 that for high gradient crystals the magnitude of peripheral dislocations reaches a minimum at the periphery. This may be due primarily to the effective cooling of the crystal periphery by the low ambient temperature considered. For low gradient crystals, however, the magnitude of the peripheral dislocations are quite significant, as seen in Figs. 7-9 and 13-15.

6.2. Effect of the crystal radius on the dislocation density

xl0 ~ 4 3.5 3 2.5 * - cedlLer

2

o - middle 1.5

x - periphery

1 0.5 0 -- 0

6

8

10

Z axis in cm

Fig. 18. As for Fig. 16, but for r = 3 cm.

gradient at the solid-melt interface, 'which implies that dislocations are generated most near to the solid-melt interface. After the initial rise, the dislocation density becomes constant, as only a few dislocations are generated away from the solid-melt interface due to the decrease in stresses from the relaxation of plastic deformation. The results obtained are presented below:

6.1. Effect of the ambient temperature on dislocation density It can be observed readily in Figs. 7-9 that the peak dislocation density for low gradient crystals pulled in the (00 1) growth direction is between 15000-65000/cm 2. This is much lower compared to Figs. 10-12 for the high gradient crystals, where the peak dislocation density ranges from 2.1 × 10 7 to 5.8 × 107/cm2. In the (1 1 1) growth direction, similar results can be also be observed

Because the thermal strains induced in larger diameter crystals are more difficult to be relaxed than those for smaller diameter crystals, the thermal stresses in crystals increase as the crystal diameter increase. Therefore, increasing the crystal radius consistently increases the peak dislocation density for low gradient crystals, in both pulling directions. In the (00 1) pulling direction, comparing Figs. 7-9, it can be noted that the peak dislocation density increases from 15000/cm 2 to 37000/cm 2 and 65 000/cm 2 for radii of 1 cm, 2 cm and 3 cm, respectively. At the mid-radius and periphery of the wafer the rise in dislocation density is comparatively lesser. For the (1 1 1) pulled direction, a similar result can be noted by comparing Figs. 13-15. High gradient crystals pulled in the (1 ! 1) growth direction agree to a certain extent with this phenomenon. Here the peak dislocation density increases with increase in the radius from 1 cm to 2 cm, but drops slightly as the radius is increased to 3 cm, as seen in Figs. 16-18. In the (00 1~ direction, the peak density drops nearly 50% when the radius is increased from 1 cm to 2 cm. At a radius of 3 cm the peak dislocation density increases, but the rise is relatively small. This can be observed in Figs. 10 12.

6.3. Effect of the growth direction on the dislocation density For low gradient crystals, the peak dislocation density was consistently lower in the (1 1 i ) pulling direction as compared to the (0 0 1) direction. This becomes evident when comparing Figs. 7-9 with Figs. 13-15, indicates that the resolved shear stresses in the slip planes for the (1 1 1) pulling direction are smaller than those for the (0 01) direction even though the global thermal stresses are the same for both pulling directions. For high gradient crystals with a radius of 1 cm, this phenomenon is observed in Figs. 10 and 16. The contrary was noted for

N. Subramanvam, C T. Tsai /Journal of Materials Processing Technology 55 (1995) 278~87

high gradient crystals with a radius of 2 and 3 cm from Figs. 11, 12, 17 and 18, where the peak dislocation density in the ( 0 0 1 ) direction was lower than that for the ( 1 1 1 ) growth direction, which indicates that the resolved shear stresses in the slip planes for the ( 1 1 1 ) pulling direction and greater than those for the ( 0 0 1 ) direction. Therefore, the dislocation density in the crystals pulling in a particular direction depends on the magnitude of the resolved shear stresses in the slip planes of the crystals pulling in that direction.

7. Conclusions

It has been reported already that a small crystal diameter and a low temperature gradient significantly reduce the dislocation density during the CZ growth process. The objective of the present effort, however, was to determine the influence of each of the growth parameters on the dislocation density and thereby provide some insight into controlling the growth parameters so that the dislocation density can be reduced during the CZ process. This has been achieved successfully. In the light of the results available from this study it can be stated that, by and large, the ambient temperature, the crystal radius and the growth direction can all be used effectively as tools for successfully reducing the dislocations in the CZ growth process. It has been shown that a low temperature gradient significantly reduces the magnitude of dislocations. Increasing the crystal radius has resulted in increasing the dislocation density, however the rise in magnitude in low gradient crystals is nominal. In high gradient crystals, interestingly, increasing the radius has resulted in decreasing the dislocation density in the (0 0 1) growth direction. It has also been shown that low gradient crystals grown in the (1 1 1) growth direction have much lower dislocation than those in the (1 0 0) direction. It can hence be concluded that for a pulling speed of 0.0005 cm/s, a low temperature gradient and the (1 1 1) growth direction will be most conducive for CZ growth of undoped GaAs and yield the least amount of dislocations. The peak dislocation density for a 3 cm radius crystal grown under such above recommended parameters is around 16 000/cm 2, as seen in Fig. 15. It has also been reported that dislocation density in GaAs can be significantly controlled by doping with Si or Te (S and Ge for InP) [8], which opens further avenues for dislocation reduction. The effect of the growth speed on the dislocation density will be included in the authors' next publication.

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It will also be interesting to simulate the dislocation generation model for other semiconductor materials such as CdTe and CdZn Te and compare the influence of the growth parameters on the dislocation density with the present results.

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