Critical thickness during two-dimensional and three-dimensional epitaxial growth in semiconductor heterostructures

Materials Science and Engineering, B8 ( 19 91 ) 10 7-12 4

10 7

Critical thickness during two-dimensional and three-dimensional epitaxial growth in semiconductor heterostructures K. Jagannadham and J. Narayan Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7916 (U.S.A.)

(Received May 30, 1990; accepted in revised form October 15, 1990)

Abstract The homogeneous nucleation of misfit dislocations in two-dimensional and three-dimensional epitaxial structures on rigid substrates was analyzed. It is shown that nucleation and growth of a dislocation in the epilayer involves an activation energy barrier. We propose that this activation energy barrier can be overcome from the residual coherent strain energy in the film. A critical thickness of the epilayer is defined at which a dislocation is nucleated. The activation energy and the coherent strain energy were determined for several different configurations of 60 ° glide and 90 ° climb dislocations. The discrepancies that are associated with different formulations are pointed out. Specific numerical calculations were performed for dislocation nucleation in the GaAs/Si system. The results are given in terms of the various energy contributions responsible for nucleation of misfit dislocations, and the critical thickness of the epilayer is evaluated. The critical thicknesses of the epilayer under the two-dimensional and three-dimensional growth conditions are compared and the results described in terms of the mechanisms of dislocation nucleation.

I. Introduction The growth of coherent epilayers on crystalline substrates is classified into three categories [1]: (a) two-dimensional (planar) growth, (b) three-dimensional (island) or Volmer-Weber growth, and (c) planar growth during the initial stages followed by island growth or StranskiKrastanov growth. A layer-by-layer, two-dimensional growth covering the substrate during deposition is favored at low temperatures either in heterostructures when the mismatch is small or in homoepitaxial systems [2]. The interfacial energy in these homostructures and heterostructures is small compared with the surface energy so that the epilayer spreads uniformly during deposition. The second category of growth consists of several nuclei growing simultaneously in the form of islands. The islands with identical epitaxial relationships can merge perfectly without creating defects and/or grain boundaries in the interface region. This type of growth is observed [3, 4] in heterostructures with large mismatch. The interfacial energy is usually large compared with the strain energy arising from the 0921-5107/91/$3.50

coherent strains. The region of contact along the interface is reduced by the finite size of the islands, as is observed experimentally in GaAs/Si and Ge/Si systems. The third category consists of a combination of planar and island growth with the former in the initial stages. This type of growth has been observed in refractory metal systems deposited onto semiconductor substrates

[4]. The origin of lattice misfit dislocations in epilayers can be traced to homogeneous and heterogeneous nucleation [5-12]. In the former mechanism, nucleation of 60 ° glide dislocations at the surface and their motion towards the interface has been established. Further, two 60 ° glide dislocations can react to form 90 ° misfit dislocations [8]. These observations in boron-doped silicon on silicon illustrate that in the absence of any other sources capable of generating dislocations, a free surface will be responsible for the nucleation. However, both homogeneous and heterogeneous nucleation have been observed in III-V compounds such as GaAs grown on silicon. Dislocation bending to form a segment along the interface from the threading dislocations has © Elsevier Sequoia/Printed in The Netherlands

108

been verified experimentally and analysis of such configurations has been presented [13-17]. More importantly, dislocation generation from precipitates and impurities present at the interface between the epitaxial films and the substrate has been established [9]. The precipitates introduce lattice misfit dislocations as a result of interruption in the epitaxial growth of the films. These form the threading dislocations which can multiply in the presence of coherency stresses. Thus, this mechanism of formation of misfit dislocations may not be limited by the activation energy barrier as long as an interfacial segment is not formed and the additional interracial energy is not generated. As pointed out earlier, high interfacial energy is responsible for the epitaxial growth of islands. In this paper, we derive the activation energy associated with homogeneous nucleation of lattice misfit dislocations from the surface and show that the thermal energy is not always sufficient to overcome the activation energy. The formation of lattice misfit dislocations by heterogeneous nucleation will be considered separately. Growth by the above three mechanisms can occur pseudomorphically on substrates where the lattice parameter of the epilayer is different from that associated with its equilibrium structure. The epilayer is strained elastically along the interface to match the substrate with additional strain perpendicular to the interface. The strain energy in the epilayer builds up with increasing thickness or island size. Experimental observations [3-10] have shown that lattice misfit dislocations are formed above a critical thickness at which the epilayer tries to obtain the equilibrium structure. These dislocations provide traps and recombination centers and thus limit successful operation of certain electronic and optoelectronic devices. The present methods of producing thin film heterostructures, such as GaAs on silicon substrates, result in dislocation densities of around 10 6 c m - 2 compared with the desired values of less than 10 3 c m - 2 . Therefore, the mechanisms of formation and the reduction of lattice misfit dislocations in epilayers have received considerable attention [10-15]. The dislocations are formed in the epilayers primarily from glide processes and occasionally by climb. Also, both the free surface and the interface can act as sources for dislocations. In situ electron microscopy observations have illustrated [5, 8] that glide dislocations originate from

the free surface. In addition, threading dislocations which grew from the interface are also observed. The formation of dislocations in the epilayers is considered [9-17] to be a nucleation and growth process with an activation energy barrier. The activation energy should be overcome before a stable dislocation is formed. In this work, we consider the nucleation of 60 ° glide dislocations from the free surface and their motion to the interface. The possibility of nucleation of 90 ° climb dislocations is also considered. In order to compare the ease of nucleation of dislocations from the surface and from the threading dislocations, the activation energy barriers are determined and the results for the GaAs/Si system are discussed. The _planar interface in the (001) plane with the (111) slip plane for the ½1011] glide dislocation in the [110] direction is considered. The 90 ° dislocation consists of the ½1110](001) Burgers vector with the sense vector in the [li0] direction. Dislocation loops with the same Burgers vectors are also considered to form in three-dimensional growth of epilayers. In addition, the possibility of a glide dislocation splitting into partials in the form ½[011](11 i ) ~

~[112](11i)+~[i21](11i)

(1)

and thereby lowering the energy of formation of a dislocation is considered. Similarly, the formation of the lattice misfit dislocations in the presence of a stress field associated with a step on the surface is included in our analysis to determine the reduction in the activation energy barrier. The present analysis differs from others [11-17] in the following manner. Firstly, we believe that there is an activation energy associated with the dislocation nucleation from the surface. Further, this activation energy can only be overcome as a result of coherent elastic energy stored in the epilayer and thermal activation energy at the growth temperature. The presence of an activation energy will be shown by considering an idealized configuration for the sake of clarity and simplicity. That the activation energy can be overcome from the elastic energy stored follows from comparison with recovery and recrystallization processes. The nucleation of strain-free grains in a cold worked sample of crystalline material is known to be induced by the stored energy [18, 19]. Several experimental results have shown that the stored energy is released as heat [19] on raising the temperature during recovery. The recrystallization tempera-

109 ture is lowered when the percentage cold work or stored energy is increased. From these results, one can conclude that the elastic energy stored in the epilayer will also be released when thermal fluctuations are present. A coherent epilayer with larger mismatch and elastic stored energy should enable dislocation nucleation at a smaller thickness of epilayer. The thickness of the epilayer at which lattice dislocation is nucleated spontaneously is defined as the critical thickness. Whereas the thickness is a measure of the critical size in two-dimensionally grown epilayers, the radius of the island is used for the three-dimensional growth. The various energy terms associated with the dislocation nucleation are determined after taking into account, (a) the free surface boundary conditions, and (b) the effect of a second phase of different shear modulus. The numerical calculations are performed for the GaAs/Si system for which the substrate is considered to be very rigid because of its large thickness. The growth temperature of GaAs on the silicon substrate has been taken as 650 °C. The following numerical values were used for the different parameters [20]. At 300 K Gl(300 K)= 5.86 × 1011 dyne cm -2 G2(300 K)= 3.20 x l0 ll dyne cm -2

v 1~-0.25

v2 = 0.23

(2)

y = 1725 ergcm -2

where G1, Vl and G2, v2 are the shear moduli and Poisson's ratios of silicon and Ga_As respectively, and ~, is the free surface energy of GaAs. The temperature dependence of the shear moduli of silicon and GaAs are given in the form [21] GI(T) = G1(300 K ) - 0.078( T - 300)/T M (3) G2(T) = G2(300 K ) - 0.146( T - 300)/TM where TM(Si)= 1693 K and TM(GaAs) = 1511 K are the melting points of silicon and GaAs. The lattice frictional stress in GaAs has been shown to follow [22] of = (4.4 x 104/T - 45.52)x 107 dyne cm -2 The lattice parameter of GaAs b 0 is taken to be 5.6535 A and the lattice mismatch f between GaAs and silicon increases from 0.040 to 0.044 when the temperature is raised from 3 0 0 K to 925 K.

2. Nucleation considerations According to classical nucleation theory [23-25], the rate of nucleation J is given by J = ton exp{ - A G c / k T

}

(4)

where to is the frequency with which nuclei form, n is the atom density, AG¢ is the activation energy for formation of critical nucleus, k is Boltzmann's constant and T is the temperature. The frequency factor w takes a value of 1015 S -1 with n = 1023 cm -3. In order for a nucleation event to be recorded in an epitaxial island of 1000 nm in a reasonable time, the critical nucleation rate J* becomes 1021 cm -3 s -1. The activation energy AGc should be below 4 0 k T for the critical nucleation rate to be exceeded. Thus, the activation energy takes a value of 5.0 x 10 -12 erg at a growth temperature of 925 K. The activation energy for nucleation of several dislocation configurations determined in the present analysis will be compared with the above critical value. The activation energy for several configurations has been found to be much larger than the critical value if the surface energy due to the ledge step is included. We propose that the activation energy barrier E a may be overcome if the coherent strain energy remaining in the film Est exceeds Ea. The driving force for dislocation nucleation is also higher if ( E s t - Ea) is larger. This modified condition in the form Est > E a for nucleation to occur will be used to determine the critical thickness of two-dimensional and three-dimensional epilayers.

3. Two-dimensional growth of epUayers We consider two-dimensional growth of an epilayer on a thick substrate, as shown in Fig. l(a). The coherent strain in the epilayer e = f = (bo - a0)/ao, where b0 and a 0 are the lattice parameters of silicon and GaAs respectively. This strain gives rise to the stresses in the form [9-13] Oxx = avy = 2G2f(1 + v2)/(1 - v 2)

(5)

and the associated coherent strain energy density E c per unit area becomes E c = 2G2( 1 + v 2 ) f i h / ( 1 - '1-'2)

(6)

where G 2 and v 2 are the shear modulus and the Poisson's ratio respectively, associated with the epilayer of thickness h. These results derived from continuum elasticity can be described by

1 1 (]

l

z f

Surface Dislocations

ilayer h

!

±I±

.Interface Dislocations

± ± ±~

x

Substrate T

T

(a) ~_~i~

Surface Dislocations

h

.1. t I ± ~

±~

Interface Dislocations

/

Substrate

L=nao=(b/bi)ao

1

t

b =nb i

Epilayer

"3" (c)

~d / Epilayer h 1

and

(n+l)ao=nbo

(7)

In eqn. (7), b is the component of the Burgers vector of the lattice dislocation parallel to the interface in the cube directions. The two-dimensional film is a superposition of the repeat blocks. The conservation of Burgers' vectors in the form


1

virtual interface dislocations of the Burgers vector bi = b0 - a0 distributed uniformly along the interface, as shown in Fig. l(a). However, the free surface boundary conditions requiring the tractions to vanish on the free surface are satisfied by a uniform distribution of surface dislocations of opposite sign on the free surface. The vanishing of the tractions outside the epilayer can be shown by summing the stress fields of the two arrays. A second set of dislocations perpendicular to the first shown in Fig. l(a), is also present in the epilayer, although not shown for the sake of simplicity. The length L of the repeat block shown in Fig. l(a), is determined by the Burgers vector of the lattice dislocation which forms. Specifically, it is given by

I

Interface Dislocations

±-

Substrate -r"

-.l--

(8)

is satisfied so that the stress field in the film in the presence of the lattice dislocation is short range. The relationship between the work done by the coherency stresses and the coherent strain energy released can be determined by considering the formation of imaginary lattice dislocations of Burgers' vectors [100] and [010] with the sense vectors along [010] and [100] respectively. These results are extended to the nucleation of 60 ° glide and a 90 ° climb type dislocations which are observed experimentally. The work from the coherent stress field inducing climb of the two lattice misfit dislocations through a distance d is

(d) /

Surface Dislocations

--7d

Lattice Dislocation Epilayer

i ± ± ± ± q-

(e)

Substrate

/

Interface Dislocations

Fig. 1. (a) The repeat block of a two-dimensional epilayer with virtual interface dislocations to represent the lattice mismatch, and the surface dislocations satisfying the tractionfree surface boundary conditions; (b) same as (a) except that the surface array is coalesced to form a single dislocation; (c) same as (b) except that the surface array is converted into a lattice dislocation and moved into the epilayer by a distance d; (d) same as (c) except that the lattice dislocation is allowed to spread uniformly; (e) same as (d) except that a 60 ° glide dislocation is present; the lattice dislocation with its two components spreads uniformly and the coherency stresses in the epilayer are only partially reduced; a surface array with two perpendicular components is required to satisfy the traction-free boundary conditions.

111

work, i.e.

given by E(work) = - a,~bd - orrbd

(9)

E¢(remaining) = Est = Oxxf(h - d)

which, converted into energy density, i.e. energy in an area b/f, becomes

= E¢(initial)- Ew/2

E,~ = - 2Oxxfd

(10)

The area of the interface within which the energy resides is given by the length along the interface b / f times the unit length of the misfit dislocation. The lattice dislocations can be imagined to have formed from the coalescence of dislocations in the surface array followed by climb towards the interface by a distance d. The coalescence is shown in Fig. l(b), and the climb and spreading of the dislocations within the repeat block are shown in Fig. l(c) and (d). The final configuration shown in Fig. l(d) illustrates that the region of height d from the surface within the epilayer is free from any stresses and, therefore, the strain energy within this region should be zero. Thus, the coherent strain energy released Er when the lattice dislocation climbs down by a distance d from the free surface is given by E r=

Ec(initial ) - Ec(final)

= -(Oxxfd + oyyfd)/2

(11)

This result clearly illustrates that the strain energy released E r is o n l y half of the work done by the coherent stress field when a lattice dislocation forms and annihilates the strain in the region. This result can also be proved by considering the work done on the virtual interface dislocations of Burgers' v e c t o r b i. The total work performed by the coherency stresses in order to move the virtual interface dislocations through a distance h from each other, given as energy density becomes

+ Oyyfh

=2oxxfh

The activation energy for nucleation of a lattice dislocation in the epilayer can be determined by evaluating the total energy in the form Et=Es+Ei+Ef+Ey=Es+Ef+Ew/2+E

~

(14)

where E~ is the self energy of the dislocation, E i is its interaction energy with the interface array, E f is the energy spent in moving the dislocation against lattice frictional stress, Ew the work due to the coherency stresses on the lattice dislocation, and E~ is the surface energy associated with the ledge step. The total energy E t a s a function of the distance d from the free surface reaches a maximum which is the activation energy E a. The coherent stored energy Est remaining in the epilayer, given by eqn. (13), decreases continuously as a result of the strain energy released. We propose that if the coherent stored energy remaining in the film Est becomes greater than Ea, then a dislocation will be nucleated and it will move to the interface with continuously decreasing energy. 3.1. Nucleation of 60° glide dislocations

= - Oxxfd

E w = Oxxfh

(13)

(12)

However, this is equal to twice the coherent strain energy in an epilayer of thickness h. Thus, the elastic strain energy associated with the epilayer is half the work term performed by the coherency stresses on the interface dislocations. It is now clear that when a lattice dislocation forms at a distance d from the free surface in the epilayer, the coherency stresses perform work Ew, whereas the formation of such a lattice dislocation only reduces the coherent strain energy by half the

In order to compare the various energy terms associated with glide and climb type dislocations in two-dimensionally grown epilayers, the energy density, i.e. energy per unit area of the interface in the epilayer, should be considered. The length of the dislocation along the interface is chosen to be unity and that of the repeat block L is calculated to determine the energy density. A glide dislocation with Burgers' vector -~[011](11 i) and sense vector of [101] is assumed to nucleate. The dislocation glides on the ( 11 i) slip plane by a distance (3)1/2d when it is situated d from the free surface. The dislocation has two components, one parallel to the interface and the other perpendicular to it, as shown in Fig. l(f). The lattice dislocation with its components is shown in Fig. l(e) to spread out at a distance d from the free surface. The conservation of Burgers' vectors again requires that the sum of the Burgers' vectors of all the dislocations parallel to the interface should become zero. Similarly, the sum of the Burgers vectors of all the dislocations in the direction perpendicular to the interface should vanish. The energy density is calculated by con-

112 sidering the projected length of the misfit dislocation along [100] in the interface which is 1/(2) 1/2 per unit length. The component of Burgers' vector that relieves the strain along the interface in the [010] is b/(2) t/2. Therefore, the length L along the interface within which the coherent strain energy is released becomes, L = b/2f. The self energy associated with a straight lattice dislocation in the two-phase medium is given in the form E ={G, G2b2/4n(G, + Gz)(1 -v2) } ln(ad/b) (15) where d is the position of the dislocation from the free surface, b is the Burgers' vector and a = 4 gives the core cut-off radius of b/4. This choice of a is consistent with the core energy values determined from the atomic potentials [26]. The composite modulus, G z ( l + K ) / 2 where K = ( G 1 -G2)/(G ~ + G2) is used in order to take into account the image effects from the second phase. A detailed analysis including the image effects has already been presented [10]. The energy density associated with the self energy becomes E~={G2(1 + K ) b ( Z f ) / 8 ~ t ( 1 - v2)} ln(ad/b)

(16)

The work due to the coherent stress field on the glide dislocation can be calculated from the force acting on the dislocation. The glide component of the Peach-Koehler force on the dislocation is determined to be ox~b/(6) ~/2 so that the energy density due to the work becomes Ew = -2oxxf((3)'/Zd)/(6) 1/2 = -(2)1/2o~fd

(17)

The energy density associated with the lattice frictional stress is given by E,= 2(3)'/2afdf

(18)

whereas the surface energy due to ledge step is expressed as E~ = 2fy

(19)

3.2. Nucleation of 90 ° climb dislocations A 90 ° dislocation of Burgers' vector ~[110](001) with its sense vector along [110] is assumed to nucleate and climb from the free surface into the epilayer. The magnitude of the Burgers vector of this dislocation is the same as that of the glide dislocation. However, its Burgers' vector is responsible for reduction in the coherency strains both in the [100] and [010] directions. In addition, since its sense vector remains along the [1]0],

which lies in the plane of the interface (001) and (110), with equal components along both the [100] and [010] directions, its contribution to strain relief is twice that of the glide dislocation. Therefore, the length of the interface L along which the coherency strain energy is released becomes L = b/f The dislocation moves vertically by a distance d in the (110) plane so that the work due to the coherency stresses can be calculated using the climb component (a~xb) of the Peach-Koehler force [27]. The result in terms of energy density is E w= - a~(fd)

(20)

Similarly, the frictional energy term is given by E,=of(fd)

(21)

where af is now the lattice frictional stress against climb of dislocations. All other terms in eqns. (13) and (14) remain the same as those for glide dislocation. 3.3. Nucleation of a straight dislocation in the presence of a step on the surface The activation energy associated with the nucleation of a dislocation can be reduced in the presence of the stress concentration associated with a ledge or an atomic step on the surface. A step of height b 0 on the surface leads to a stress singularity with its maximum value given by [28] o~x(step) = oxx(4b,,/3Jzd) ~/3

(22)

where oxx is the component of applied stress present in the medium. This is taken to be the coherency stress in the epilayer. We neglect the angular dependence of the stress field at the step and assume that the orientation of the step and the dislocation are such as to give rise to a maximum reduction in the activation energy. Thus, the additional work term arising from the stress field of the step will be included in the total energy E t associated with the dislocation nucleation. This term for the 60 ° glide dislocation becomes Ew(step) = -(2)l/2a, x(step)(fd)

(23)

The coherency strain energy remaining in the epilayer is not altered by the step. We have assumed that nucleation of the dislocation introduces additional surface energy. It is also possible that part of the step is eliminated on nucleation of the dislocation so that the surface energy term is negative. The activation energy is reduced in the

113

presence of a step so that a loop will nucleate at a smaller size of the epilayer.

3.4. Nucleation of a 60°glide dislocation loop segment in the presence of a step The nucleation of a dislocation loop has already been considered in planar epitaxial films [13]. The presence of surface tension and stress field due to a step are included in the present analysis. The activation energy is reduced when a dislocation loop segment is nucleated instead of a straight dislocation. A segment of a dislocation loop with radius r is nucleated in the two-dimensional epilayer in the presence of a step. The angle, 2 0, that the radii of the arc of the dislocation segment subtend is given by 0 = tan- l{(G2b/2y)2 _ 1 }1/2

(24)

where y is the surface energy per unit area. In order to take into account the image forces exerted by the free surface, the energy of the segment is considered to be proportional to its length in the form E s = G2(1 + K ) ( 2 - v2)b2(2rO) x {ln(4d/~)- 2}/16zr(1 - v2)

(25)

The work term from the coherency stresses is given by

Ew = - oavAb/(6) 1/2

(26)

where A is the area moved by the segment into the epilayer A =r2{0 - 0.5 sin(20)}

(27)

Similarly, the frictional energy spent in overcoming the lattice frictional stress becomes

Ef =

ofbA

(28)

The work due to the stress field associated with the step is given by Ew(step) = - 3axx(Step)brd sin 0

(29)

where d is the depth to which the dislocation moves in the presence of the stress concentration due to the step. The parameter d is determined by the equivalent area rectangular loop. T h e stress field oxx(step) associated with the step in the form given by eqn. (22) was used to calculate Ew(step). The additional surface energy created by the segment is given by

Er = 2rby sin 0

(30)

If the surface ledge is eliminated as a result of dislocation nucleation, Ey becomes negative but the stress field due to the step vanishes. However, E r is positive if an additional ledge step is created and the stress field due to the step does not vanish. We have considered two different situations. In the first, the stress field due to a single atomic step is included with an additional surface energy created due to the dislocation. The presence of a double step is considered in the second situation but the nucleation of the dislocation eliminates the step by one Burgers' vector. Thus, the surface energy is reduced owing to the partial elimination of the step and additional work due to the stress field of the step is also present.

3.5. Numerical analysis of dislocation nucleation in two-dimensional epilayers The results of evaluation of various energy terms associated with the nucleation of glide and climb type dislocations in terms of the total energy E t and the coherent strain energy remaining in the film Est are shown in Figs. 2(a), (b) and 2(c) for the two-dimensional epilayer. The activation energy E a is larger than the critical value of 40kT. Therefore, the thickness of the epilayer at which Est becomes larger than E t represents the critical value. Although the coherent strain energy remaining in the film Est is higher for a 60 ° glide dislocation than for a 90 ° climb dislocation, the activation energy is also higher so a larger critical thickness is expected. In contrast, the work due to the coherency stresses is larger for a 90 ° climb dislocation and hence the activation energy is also smaller compared with that for 60 ° glide dislocation. The activation energy for the nucleation of the 60 ° glide dislocation is further reduced in the presence of the stress owing to the step, as shown in Fig. 2(c). The critical thickness is reduced from 15 to 10 A owing to the presence of the step on the surface. The results also illustrate that the critical thickness at the growth temperatures (925 K) should be smaller. It is seen that E a for nucleation of 90 ° climb dislocation is 60 erg cm -2 whereas it is 120 erg cm- 2 for 60 ° glide dislocation in the presence of a step. Assuming a region b/f along the interface to be included in the nucleation process, the activation energy needed to nucleate a dislocation of length 1/~m, becomes 6.0 x 10 -1° erg. The total energy E t of the dislocation configuration when a segment of a loop is nucleated is illustrated in Fig. 2(d). The activation energy E a in the presence of

114

1~o]

250 ~

~

20~ ('i"= 925K)

\

----~i t

-

~

10/~(T= 300K)

--

Est

Et

200

E o

~150

o~ ~

lOO

~

'~,~ 10.~(T= 300K)

-\\

~

~.

\\

\\

IO

50

o

5

10

(a)

Distance,

ot

15

5

o

o

110

(c)

d(A)

lj5 d(,~)

Distance,

lOO - - - - Est -~"

80

x\~" \ \ 5,~ (T° = 300 K)

10 ~t (T = 300 K)

\'~--x-\ 10 ~ (T -- 925 K)

5 A ( T - 925 K)

&-.

Et

\

10.0

E U)

60

Single Step

0.0 6.0

=

LU

40

4.0

\

\

k

2.0

0

20

\

T=925K ~'~x

\\ B2~0

I ~

Double Step (300K and

925K)

-4.0 5

(b)

0

10

Distance, d(A)

(d)

2

'

4

I

6

,

8 Loop

I°

1

Radius

I

12 (,~)

I4

1

I

16

I8

1

Fig. 2. (a) The total energy E t and the coherent strata energy remaining in the film Est are shown as a function of distance d moved by the 60 ° glide dislocation in the two-dimensional epilayer of GaAs on silicon for two different temperatures, T= 300 K and 925 K. (b) Same as (a) except that the energy terms are determined for the nucleation of 90 ° climb dislocation. (c) Same as (a) except that the stress due to a step is included in the analysis. (d) The total energy of a 60 ° glide segment nucleated at the free surface in the presence of a single atomic step with additional surface energy on nucleation, and a double atomic step with part of the step eliminated on nucleation. The coherent strain energy remaining in the film Est is much higher than E t and could not be shown.

a single atomic step is found to be 6.0 × 10 - 12 erg at 925 K and 8.0 × 10-12 erg at 300 K. Thus, the activation energy remains above the critical value of 4 0 k T for the nucleation event to take place. It should be pointed out that the activation energy is greatly reduced in the presence of a step. The coherent stored energy remaining in the film Est is found to be larger than Ea so that dislocation nucleation is favorable. We have assumed that the single atomic step is not eliminated when the

segment is nucleated. The present analysis also showed that the total energy decreases if the single atomic step is eliminated on nucleation of a dislocation. However, in the second situation, i.e. when a double atomic step is present and a part of the step is eliminated, the energy of the configuration decreases with increasing radius of the segment of the loop, as shown in Fig. 2(d). Therefore, nucleation of a segment of the loop is favorable. The work term from the stress field

115

associated with the step is not a major contributing factor compared with the surface energy of the ledge which is partially eliminated. The stress field of the step is short range and, therefore, its contribution to the energy becomes smaller with increasing radius of the loop. The minimum in the energy E t for small values of the loop radius arises as a result of the step. The surface energy associated with the step contributes significantly to the total energy. We believe that a 60 ° glide loop will be nucleated from the surface if it results in a reduction in the surface energy by partial elimination of the step.

4. Three-dimensional growth of epilayers The coherent strain in three-dimensional growth of epilayers is illustrated in Fig. 3. The lattice mismatch is represented by an array of virtual interracial dislocation loops which replace the dipoles in the two-dimensional configuration. Although a single set of prismatic loops is shown, a second set is also present in the epilayer with its Burgers' vector perpendicular to that of the first. A lattice misfit dislocation which is also prismatic in nature and which annihilates the stress in the medium is shown in Fig. 3. The coherency strain energy in the epilayer was first determined by Cabrera [29] but the expressions are only valid for a hemispherical epilayer on a hemispherical substrate. The substrate is not very thick and therefore the expressions may not be strictly valid for the three-dimensional growth of islands on a planar substrate. In the present analysis, the dislocation model shown in Fig. 3 was used to calculate the strain energy in the hemispherical epilayer on a thick substrate. The analysis was performed using a finite number of dislocation loops and the calculated coherency energy is

VirtualInterracia"l ~ c a l A "

Epilayer

Dislocation Loops

MisfitDislocation Substrate

Fig. 3. Surface and interface dislocation array representation of a coherent three-dimensional epilayer in the form of a hemispherical island. Two arrays of prismatic dislocation loops perpendicular to each other represent the distortion. Only one set of prismatic dislocation loops is shown.

compared with the results of Cabrera [29]. Specifically, the coherent strain energy is expressed by Cabrera in the form E c = 2{$2e22/3 +4S1e12}R3/3

(31)

where R is the size of the epilayer and $ 2 = G 2 / ( 1 -v2)

S1 = G,/(1 -'Pl)

S = ( S , + $2)/2

(32)

e 1= 3aoS/16RS 1

e2 = 3 b o S / 4 R S 2

The above result shows that the strain field and the coherent strain energy in the form given by Cabrera [29] are not a function of the mismatch strain f in the epilayer. Although the strain field is non-uniform in the islands, that close to the interface should be equal to the mismatch. The coherent strain energy in the epilayer is expressed in terms of the average stress in the epilayer Oar in the two perpendicular directions in the form E c = 2¢rR3(traveav)/3 = 8yr2S2R3(1 + v2)2eav2/3

(33)

where eav is the average strain in the epilayer. We have modified Cabrera's expressions for the average strain to include the lattice mismatch f: eav= 3fS/8yrS2(1 +

'1~2)

(34)

Therefore, the average stress in the epilayer becomes Oar = 3(1 + v2)fS/2

(35)

which will be used to replace the Oxx and the Cryy components of stress in the two-dimensional epilayer. The strain in the thick silicon substrate is close to zero and all the misfit strain is accommodated in the island. The semicircular prismatic dislocation loops shown in Fig. 3 represent the coherent strain. The coherent strain energy in the epilayer E c becomes E c = 2(E~ + Ei)

(36)

where E s is the self energy and E i the interaction energy of the loops. The factor 2 in eqn. (36) arises from the presence of two sets of non-interacting prismatic loops in the epilayer. The effect of the second phase is taken into account through the stress field associated with the loop in the two-phase medium [30]. The stress field associated with a finite dislocation loop in a two-phase medium may be obtained by integrating the expressions of stress associated with an infinitesi-

116

mal loop over the area of the finite loop in the form oij(loop) =

f

~rij(infinitesimal) dAl

(37)

Area, A I

The interaction energy between two dislocation loops in the medium can be determined from t"

E~=(b2/2)

J

aij(loop 1)dA2

(38)

Area. A 2

In the above equations, the quantities A1, b 1 and /t2, b 2 are the area and the Burgers' vector associated with the loops 1 and 2, respectively. The self energy is determined to be half that of the interaction energy when the cores of the dislocations overlap [27]. The integrations are performed over the area elements using the double precision gaussian quadrature. The accuracy of the evaluation of the integrals is maintained by separating the energy terms into two parts. Specifically, the energy of the configuration in the two-phase medium, E(two-phase), is given by

the epilayer. The dislocation loops are distributed uniformly. The sum of the Burgers' vectors associated with the loops is equal to the total lattice mismatch contained within the epilayer. The coherent strain energy determined from discrete dislocation modeling and that obtained from the modified analytical expressions of Cabrera [29] are compared in Fig. 4. The coherent strain energy from both analyses agree closely, illustrating that the strain energy is partitioned. When the substrate is thick, the strain energy in the substrate is zero. However, it is included in the two phases of the spherical nucleus considered by Cabrera [29]. Nucleation of a dislocation in the epilayer will be considered for the following different configurations: (a) 60 ° glide dislocation loop, (b) 60 ° glide dislocation loop split into partials, (c) 60 ° glide segment nucleated in the presence of surface tension and a step, (d) 60 ° glide segment 105

E (two-phase) = E (single-phase) +(b/2) f o~j(image)dA

(39) Discrete Analysis

where oij(image) are the image stress field terms obtained by subtracting the stress in the single phase from the stress field in the two-phase medium. The singular part of the stress field gives rise to the energy in the single phase whereas the non-singular image terms are included in the second term. The close form expressions for self and interaction energy terms in the single phase are used so that numerical integration of the singular terms is avoided. The self energy of a dislocation loop in a single phase medium is given in the form

E~ = G2b2r{in( 4r/~ ) - 1}/2( 1 - v2)

104

103

(40)

where ~ = b/e(1 - v2) and e is the naperian base with a numerical value of 2.7 and r is the dislocation loop radius. The detailed expressions for the interaction energy associated with two dislocations loops will not be given here since these are very lengthy [31, 32]. The self and interaction energy terms associated with two semicircular loops are taken to be half that of the energy associated with circular loops. We used 32 prismatic dislocation loops to represent the strain in

102 0

I 20

I 40

[

60

I 80

I 100

Epilayer Radius, R(A) Fig. 4. A comparison of coherent strain energy E c in a hemispherical epilayer of GaAs on silicon determined from expressions given by Cabrera and that obtained using discrete dislocation analysis.

117

nucleated in the presence of image forces, (e) 90 ° dislocation loop, (f) 90 ° dislocation segment nucleated in the presence of surface tension, (g) 90 ° dislocation segment nucleated in the presence of image forces and (h) 90 ° dislocation segment nucleated at the interface. The numerical analysis of the activation energy and the coherent strain energy remaining in the film is presented to compare the various configurations.

4.1. Nucleation of 60° glide dislocation loop at the surface The total energy of the dislocation configuration when a 60 ° glide loop is nucleated at the surface can be written in the form E t = Es+Ef+Er+Ew/2

(41)

where the self energy of the glide loop is given by E s = G2(1 + K)b2(2 - v2)r{ln(4r/~)- 2}/8(1 - v2)

(42) In the above equation, we have used the composite shear modulus as before. The work due to the coherency stresses is obtained using the Peach-Koehler force on a glide dislocation of Burgers' vector [011 ] on the (111) plane. Thus

Ew = - ~tr20avb/(6) 1/2

(43)

with the frictional energy Ev = 7tofbr 2

Ewl = - (2)1/27tr 2bp Oar/3

and

(44)

(47)

Ew2 = - ( 2)1/2~r2 bpoav/24

where r is the radius of the outer loop and bp the Burgers' vector of the partial dislocation. The frictional energy terms from the two partials are Efl = 7trZbptXf

and

Ef2 = jtr2bpof/4

(48)

The energy of the fault separating the two partials becomes Esf = 3 ytr 2ysf/4

and the surface energy associated with the ledge step given by

Er = 2~bRfl

partial dislocation loops is responsible for their glide into the epilayer. The loop radii bear a ratio of the glide force acting on the partials which is given by Oav(2)1/2/3 and Oav(2)1/2/6 for the outer and inner loops respectively. Therefore, the radius of the outer loop is twice that of the inner loop. The total energy of the dislocation configuration is again given by eqn. (41). The stacking fault energy associated with the loops is an additional term. In addition, the energy should include terms due to both the loops. The self energy of the dislocation loops is given by eqn. (42) and the interaction energy Epi between the partial dislocation loops is determined from standard expressions [31]. The work term from the coherency stresses is determined from

(49)

where Vsfis the stacking fault energy per unit area which is assumed to be 40 erg cm- 2 in GaAs [20]. The surface energy of the ledge step is the same as in eqn. (45 ).

where fl = tan- l{r/(R2 - r2) 1/2}

(45) In the above equation, the angle 2fl subtended by the dislocation loop is taken to be that made by its diameter at the center of the epilayer. The net coherent strain energy remaining in the film becomes Est = Ec(initial)+ Ew/2

(46)

As before, we propose that if the activation energy is not below the critical value of 40kT, the nucleation of a dislocation loop will take place if E s t > E a and this gives the critical size of the epilayer.

4.2. Nucleation of 60° glide loop split into partials A glide loop can split into partials by the reaction given in eqn. (1). The stress acting on the

4.3. Nucleation of 60° glide loop in the presence of surface tension and a step The total energy of the dislocation configuration in the presence of a step has an additional work term arising from its stress field. This term is determined from eqn. (43) with the replacement of aav by the stress field of the step, given by eqn. (22). As before, we determine the maximum reduction in the activation energy, assuming that the orientation of the step is favorable, and thus the angular term in the stress field is taken to be unity. A 60 ° glide segment nucleated on the surface in the presence of line tension and the surface tension associated with the ledge step is illustrated in Fig. 5. The dislocation in the form of a segment glides towards the interface. The angle 0 that the radii of the arc of the dislocation segment

118 Ledge Step

~

/'N.20

r

/-- GaAs Hemispherical

Ex~ndG~lDislocation ~Plan e -~f~-.~-.7 -../.. _

Si, Substrate

\

/]

V

Fig. 5. The geometryof the dislocationconfiguration when a dislocation segment is nucleated at the surface of a hemispherical epilayer. subtend is given by 0 = tan- '{(G2b/2),) 2 - 1 }1/2

(50)

whereas the angle made by the dislocation segment at the center of the hemispherical epilayer is fl = tan-1{ d2/(g 2

_

_

d2)}1/2

(51)

where d = r sin(0). The self energy of the glide segment E Sis taken to be proportional to its length in the form E s = G2(1 +K)(2 - v 2 ) b 2 ( 2 r O ) x {ln(4r/~)- 2}/16x(1 - Y2)

(52)

with the work term from the coherency stress given by E w = - aavAb/(6) 1/2

(53)

where A is the area moved by the segment into the epilayer A = r 2 { O - 0 . 5 sin(ZO)}+RZ{fl-0.5 sin(Zfl)} (54) Similarly, the frictional energy spent on the segment is Ef =

afbA

(55)

part of the step is eliminated on nucleation of a dislocation is considered. The stress field associated with the step helps nucleation of the glide segment and in addition, the surface energy is reduced by the same magnitude as given in eqn. (49). These two situations illustrate that the surface energy associated with a ledge is responsible for the activation energy barrier. In the above formulations, the composite shear modulus G2(1 + K)/2 has been used in place of the shear modulus of GaAs. The free surface effects are taken into account by considering the segment to be part of a circular loop. 4. 4. Nucleation of 60 °glide segment in the presence of image forces The self energy terms of a glide and a prismatic dislocation loop in a two-phase medium are known in close form after inclusion of the image forces [33, 34]. However, the plane of the loop is considered parallel to the interface in these analyses so that the results are not directly applicable to the present configurations. The glide loop in the present analysis is nucleated on the (111) slip plane which is inclined at 54 ° to the interface. The silicon substrate has a higher shear modulus, about twice as large as that of the GaAs epilayer. This image force is repulsive, thus opposing the dislocation glide towards the interface. The image forces increase inversely with the distance from the interface so that major effects only arise when the dislocation is close to the interface. The nucleation of the dislocation is assumed to take place at the surface and hence the image forces become small and may not be important. We have determined the increase in the activation energy in the presence of the repulsive image forces. The self energy of a dislocation loop in a two-phase medium is expressed in the form [33]

with the surface energy of the ledge step given by

Es= G2b2(2rO)(El + E 2 -E3)/8(k 2 + 1)

E r = 2ybRfl

where

(56)

Here again we consider two situations. In the first, a single step on the flee surface which is not eliminated on nucleation of the dislocation is present. The work term due to the stress field associated with the step becomes Ew(step) = - a(step)Ab

(57)

The surface energy contribution is positive since the step is not eliminated. In addition, a second situation where a double step is present and a

(58)

E, =(k 2 + 5){ln(8r/~)- 2}/~t

E z = {2(2 - k 2)K1 - 2E1}(A + B + 0.5H( k 2 + 1 ))/~rk E3 = 8 A ¢ 2 k [ { 1 / ( 1 - k 2) +3 + 2k2}E1 - ( 4 + k2)K1]

A = ( F - 1)/(Fk2 +1) B =(Fk2 -k,)/(r + kl) F=G,/G2 H = ( F - 1)/(r+ 1) k 1 =3 - 4 v 1 k = l / ( l +~2) ~/2

k2 =3 -4v 2

~=(rcosO+Rcosfl)/r

119

Fig. 6(a) shows that the activation energy cannot be overcome in an epilayer of 6 0 / k when growth takes place at 300 K. However, it becomes possible to nucleate a glide loop at 925 K in an epilayer of 60 A. A higher lattice mismatch f = 0.044 exists at T = 925 K as a result of the differences in the thermal expansion coefficients of the two phases. The activation energy, both at 300 and 925 K, is reduced as a result of splitting of a dislocation loop into partials, as shown in Fig. 6(b). A comparison of Figs. 6(a) and 6(b) shows that a 50% reduction in energy occurs at both temperatures. This reduction in energy reduces the critical size of the epilayer to well below 60/k using the condition E s t > E a . Figure 6(c) shows the total energy E t a s s o c i a t e d with the nucleation of a 60 ° glide loop segment in the presence of a single or a double atomic step. The energy increases with the size of the loop and reaches a maximum for different values of the size of the epilayer R. This increase

In addition, K~ and E 1 are the complete elliptic integrals of the first and second kind respectively. The remaining parameters have already been defined. The total energy of the configurations E t is determined using eqn. (41).

5. Numerical analysis of 60° glide dislocations in three-dimensionally grown epilayers The numerical analysis of the various energy terms associated with nucleation of a glide loop was carried out and is shown in Fig. 6(a) and 6(b). The size of the epilayer R and the temperature of growth T are given in the figures. The small decrease in the total energy associated with a loop E t with the increase in size of the epilayer arises from the decrease in the length of the ledge formed on the surface. Specifically, the angle is smaller for a larger size of epilayer. The activation energy is much larger than the critical value of 40kT. Use of the criterion in the form Est > E~ in 1.0

---- Est

0.10 [

60 ~ (T = 300 K)

-- E~

6o~:r=3ooK).....A ..~ . . . . . . .

\

~o~:r=3ooK) I 6OI(T=,0OK)

\

R = 50~\

0.08

1O0A(T=300K)K~ [ /6O~t(T=9=6K)

0.8

\\~

R : aOA

--

Et

----

E,t

0.06 0.04

O.6

/',',,

0.02

o. ........ ', ! ~

oI

O.4

w" -0.02 f -0.64

100 .~ (T = 925 K)

-0.O6

0.2

R=lO~

R~R'~ ~,6OA Double Step

-0.66 0

I 10

l 20

I 40

30

-0.10 5Q

60

(c)

o Loop Size (A)

(a)

I 2

I 4

I 6

I 8

;0

,'2

,'

,'6

,;

2o

Loop Radius (A)

0.5 ~,f X.

4O,(T=300K)

\.~ 0.3

-- -- Est

60 ~. (T = 925 K)

~

!~6O~,(T=30OK)--Et

40~T= 026K) ~--"~,

40~ ~T=6OOK~

\


0

(b)

,'

20

~

'0

100' (T. 925 K)

" o

Loop S i z e (A)

"

60

Fig. 6. The total energy E t and the coherent strain energy remaining in the film Est shown as a function of loop radius r for a 60* glide dislocation. The results are shown for different sizes of the epilayer at two different temperatures, T = 300 K and 925 K..(b) Same as (a) except that the dislocation splits into partial dislocations bound by a stacking fault. (c) The total energy E t associated with the nucleation of a 60* glide segment in a hemispherical epilayer of different sizes at 925 K in the presence of a single atomic step with additional surface energy upon nucleation, and a double atomic step with part of the step eliminated upon nucleation. The stored coherent strain energy remaining in the epilayer E=t is also shown.

120

in energy is only found when a single atomic step is present and nucleation of a dislocation is accompanied by creation of additional surface energy. Figure 6(c) also shows that the energy decreases continuously if a double atomic step is present initially and a part of the step is eliminated. The total energy E t is also found to decrease in the presence of a single step that is eliminated on nucleation of a dislocation. The stored coherent strain energy remaining in the epilayer Est is also shown in Fig. 6(c) when a single atomic step is present. The activation energy for nucleation of a dislocation in the presence of a single step is above the critical value of 4 0 k T and hence the condition Est>E t is employed to determine the critical thickness to be 30 A. Similar to the two-dimensional epilayer, it is found that the stress field due to the step does not contribute significantly to the reduction in activation energy. In contrast, the surface energy due to the ledge step is an important term responsible for the activation energy barrier. The effect of the image forces on the nucleation of the dislocation is also analyzed numerically. The use of the self energy in the form given by eqn. (58) which includes the image terms increases the activation energy: firstly, because the logarithmic term is higher by a factor of two, and secondly, because the shear modulus of silicon is higher than that of GaAs. Also, the dislocation is assumed to create an addtional surface energy. Therefore, the critical size of the epilayer is also increased. Table 1 shows the results of a numerical analysis of different 60 ° glide dislocation configurations. It is clear that the activation energy is sufficiently high compared with the thermal energy. Thus, there is a barrier to dislocation nucleation and coherent strain energy should be released to overcome the activation barrier.

TABLE 1

5.1. Nucleation of 90° dislocation loop at the surface

aThe activation energy is given in erg c m - 2 for two-dimensional epilayers and in 10 -1° erg for three-dimensional epilayers.

A 90 ° dislocation loop with Burgers' vector a/2[110] is nucleated at the surface and climbs on the (110) plane towards the interface. The self energy of the prismatic loop takes the form Es= G2(1 +K)b2r{ln(4r/~) - 1}/2(1 - v2)

(59)

with the work term from the coherency stresses given by E w = - oavztrZb

(60)

The remaining terms in the energy contributions are the same as in the nucleation of a glide loop.

The activation energy associated with the formation of a stable dislocation and the critical size shown for different configurations in two-dimensional and three-dimensional epilayers Configuration and temperature (K)

Distance d radius r at maximum (A)

Activation energy (ergs cm- 2 or 10-10 erg),

Critical thickness or radius (A)

Two-dimensional 60°glide 300 7 200 925 6 200 60°glide withstressduetosuffacestep 300 4 122 925 3 121 90°climb 3O0 2 65 925 2 65 60°glideloopsegmentwithledgesuffaceenergyand stress dueto asuffacestep 300 12 0.06 aq0 15 0.08 Thre~dimensional 60°glideloop 300 40 925 30 60°glide,~ultedloop 300 30 925 25

15 20 8 10 6 6 12 15

0.85 0.50

80 60

).38 0.27

60 45

60°~ideloopsegment withledgesuffaceenergyand stressdueto asuffacestepepilayersZre30 A 3O0 10 0.07 30 925 10 0.05 30 60°~idesegmentwi~ image forces, epilayersize80 A 3O0 6O 1 ~2 100 925 50 1.39 80 90°climbloop, epilayersize40 A 300 18 0.42 60 925 15 0.29 50 90°climbsegment, epilayersize 40 A 3OO 18 O.30 5O 925 15 0.25 50 90°climbsegmentwithimageforces, epilayersize60 A 3O0 28 0.65 6O 925 25 0.50 60 90°climbloop from theintefface 300 15 0.70 80 925 12 0.50 60

5.2. Nucleation of 90 ° dislocation segment at the surface in the presence of line tension

The formulation of various energy terms is similar to that of a 60 ° glide dislocation segment. The nucleation of the dislocation is assumed to create an additional surface energy. The self energy of a prismatic dislocation segment follows from eqn. (59) and it is proportional to the length of the segment. The work term from coherency

121

stresses is given by the climb force oavb acting on the dislocation with the area covered by the dislocation in the climb motion, given by eqn. (54).

5.3. Nucleation of 90 ° dislocation segment in the presence of image forces The close form expressions for self energy of a prismatic loop situated on a plane parallel to the interface in a two-phase medium is given in the form [34]

Es = G2b2(2rO)(E1 +E2 +E3)/(k2 +1)

(61)

where E~ ={ln(8r/~)- 2}/at E2 =(C +D2){(2 - k2)K1 -2E1}/(1-D2)k

E a = k( C-O)~[{3/( 1 - k 2) + 1 - 2ka}E1 - ( 4 - k2)K1]/2(1 + d) k 2 = 1 / ( 1 + ~ 2)

~=(rcosO+Rcosfl)/r

C =[r(k2 +1)-(k~ + 1)]/[r(k2 + 1 ) + ( k , +1)] O =[r(k2 - 1 ) - ( k l - 1)]/[r(k2 +1)+(kx +1)] The remaining energy terms are the same as in Section 4 where nucleation of a dislocation segment is considered. The numerical analysis of the energy terms associated with the nucleation of a 90 ° dislocation loop was carried out and the results are shown in Table 1. Similar to the results for glide dislocation loops, the surface energy associated with the ledge step decreases with increasing epilayer size so that the activation energy also decreases. The activation energy for nucleation of a prismatic loop is about 50% smaller than that for a glide loop and it is above the critical value of 40kT. This result arises from the larger work due to the coherency stresses on the prismatic loop. The critical size of the epilayer is reduced below 60 A at room temperature and to 50 A at the growth temperature. The activation energy for nucleation of a dislocation segment is lower than that for a loop due to the larger work term. The result of including the image effects on the glide and climb dislocation segments on the activation energy can be obtained by comparison in Table 1. A note of caution is present in this comparison since the self energy in the form of eqn. (61) is very much different from that given by eqn. (59). Specifically, the first term in eqn. (61) is different from that in eqn. (59) since these represent the single phase. This discrepancy accounts for the major differ-

ence in the activation energy obtained for the two situations. The activation energy is increased in the presence of the image effects so that the critical size of the epilayer is also higher. A comparison of the activation energy between nucleation of the glide and the prismatic dislocation segments in Table 1 for the various situations illustrates that nucleation of the 90 ° dislocation should be more favorable.

5.4. Nucleation of 90° dislocation loop from the interface The nucleation of ½[110] dislocation loop on the (110) plane from the interface was also considered to examine the possibility that the interface acts as a source of misfit dislocations. Figure 3 illustrates the dislocation model. The sense vector of the dislocations along the interface is [110] which lies both in the (110) and (001) planes. The elastic interface dislocations can also be transformed to [110] and [1 i0] directions with the Burgers vectors perpendicular to the sense vectors. An elastic lattice mismatch strain in two perpendicular cube directions (exx and eyy) can also be represented by an equivalent set of elastic strains in the [110] directions (e=' and eyy').Since the coordinate axes are rotated by 45 ° about the [001] axis, the strain tensor is transformed to obtain equal strains in the direction of the new axes. The coherent strain energy calculated using the dislocation model shown in Fig. 2 and illustrated in Fig. 4 was used to evaluate the critical size of the epilayer. The total energy of the misfit dislocation nucleated at the interface is given by eqn. (41). Specifically, the self energy of the dislocation loops is calculated using the stress field expressions valid for the two-phase medium [30]. The stress field expressions associated with an infinitesimal loop situated in a plane perpendicular to the interface are known in the twophase medium. This stress field integrated over the area of the finite loop gives the resultant stress. The singular terms in the stress field are integrated using close form expressions and only the image terms are integrated numerically. The work term from the coherency stresses Ew on the misfit dislocation is determined from the interaction energy with the virtual interface dislocations. The surface energy associated with the ledge steps at the interface is an important term given by

Er = 2rby/f

(62)

122 where 2r is the length of the ledge at the interface and bff is the region along which coherency is lost. The surface energy associated with the ledge is assumed to be equal to that of the free surface. A misfit dislocation formed at the interface is annihilated with the virtual interface dislocations in a region of length b/f Thus, the interface becomes non-coherent or partially coherent in that region. We determined the surface energy term assuming that a ledge of one Burgers' vector is created and found that the activation energy is sufficiently high at the growth temperature. The numerical analysis using 32 interface dislocations showed that the activation energy associated with the nucleation of the prismatic dislocation loop at the interface is comparable with that for nucleation at the free surface. The critical size of the epilayer is also comparable, as shown in Table 1. 6. Discussion We have analyzed the nucleation of lattice misfit dislocations in two-dimensional and threedimensional epilayers. The activation energy associated with different dislocation configurations has been determined. A dislocation is expected to nucleate when the activation energy is below the critical value given by nucleation theory. When the activation energy is above the critical value, we propose that it can be overcome by the coherent strain energy remaining in the film. The coherent strain energy is expected to be released in much the same manner as the stored energy of cold work during recovery and recrystallization processes. The activation energy for several dislocation configurations was found to be above the critical value of 40kT. Surface sources are considered responsible for nucleation of dislocations in the epilayers. Specific numerical evaluations have been performed for the GaAs/Si system. The activation energy for different configurations and the critical size of the epilayer at which the stored coherent energy overcomes the energy barrier are given in Table 1. The results illustrate that the two-dimensional or planar films become non-coherent, i.e. a misfit dislocation will be generated at a smaller thickness compared with the three-dimensional or island epilayers. There are subtle differences between the analysis in two-dimensional and three-dimensional growth conditions for the following reasons. In the case of a two-dimensional film, we use a repeat block

and consider that the energy terms are confined within it. All the energy terms except the self energy are proportional to the lattice mismatch f. The conservation of Burgers' vectors is satisfied such that the sum of the Burgers' vectors of the virtual interface dislocations equals that of the component of the misfit dislocation parallel to the interface. However, such a conservation rule is not always followed in the analysis of finite threedimensional epilayers. A single misfit dislocation loop is envisaged to form in an epilayer of different sizes in three-dimensional analysis. If an equivalent size of epilayer were to be chosen, similar to the two-dimensional configuration, there could only be one size with the sum of the Burgers' vectors of interfacial loops equal to that of the misfit dislocation. Results presented in Table 1 clearly illustrate that dislocation nucleation at growth temperatures (925 K) is much easier. The mismatch parameter is higher (0.044) at 925 K compared with its value (0.040) at room temperature. Thus, the stored elastic energy is also higher at 925 K. In addition, the lattice frictional stress is reduced at the growth temperature. The decrease in shear moduli at higher temperatures is not very large and hence it should not be a major factor since all the energy terms except the frictional energy, the surface energy and the stacking fault energy are linearly dependent on the shear moduli. The results of the present analysis illustrate that the activation energy for the nucleation of a 90 ° dislocation is lower than that for a 60 ° glide dislocation. However, the critical size of the epilayer is not very much different in these two situations because the work term is present in both E t and Est. In contrast, the effect of a step on the surface or splitting into the partial dislocations is different. Specifically, the work term from the stresses associated with the step are included as an additional work term without decreasing the coherent energy stored in the epilayer. The net effect is to decrease the total energy E t without decreasing the coherent strain energy Est and hence a reduction in the critical size of the epilayer is obtained. The influence of frictional stress is similar. A higher frictional stress will increase the activation energy without increasing the coherent strain energy. Thus, alloying only the surface regions with electrically active elements such as dopants increases the lattice frictional stress. The critical size of the epilayer can be increased very effec-

123

tively by increasing the lattice frictional stress in the surface layers. The activation energy for nucleation of a dislocation segment with line tension is lower than that of a loop. This result is found both for glide and prismatic segments without increasing the critical size of the epilayer. The segment spreads through the epilayer, thus increasing the ledge length and the surface energy term. However, this is not the major contribution. Instead, the work term from coherency stresses is higher since the dislocation moves through a larger area in the epilayer as a result of line tension. The calculations were performed using a composite shear modulus in place of the shear modulus of GaAs. This is an approximation to take into account the image effects due to the second phase. A detailed calculation using the stress field in a two-phase medium showed that the image effects are not very important when the critical size of the epilayer is large compared with the loop size at which the maximum in the energy is reached. The activation energy is higher in the presence of image forces and correspondingly a higher critical size is reached. If the discrepancy in the logarithmic. term is removed from the self energy of a dislocation loop, the critical size of the epilayer should not be very different from that obtained using the composite shear modulus. The activation energy barrier vanishes and the total energy of the dislocation configuration in the epilayer becomes negative when a ledge step is eliminated. Thus, surface steps which can be eliminated as a result of nucleation are sources of dislocations. The nucleation of a prismatic dislocation loop at the interface seems to be energetically unfavorable since the activation energy is higher. The surface energy associated with the ledge at the interface is always positive and assumed to be same as the free surface energy which may not be strictly true. Three-dimensional epilayer growth is mostly observed in systems where the interfacial energy is higher than the free surface energy and, therefore, the ledge energy at the interface will be higher. The peripheral region where the epilayer bonds with the substrate is a favorable site for nucleation of a glide loop, as shown in our previous analysis [35]. The presence of a sharp corner introduces stress concentration which helps to reduce the activation energy, similar to that of a step on the free surface. Experimental observations in metals during early stages of island growth suggest nucleation along the inter-

face [11]. However, the high interfacial energy in the island growth may not be favorable for dislocation nucleation along the interface in the GaAs/Si system. The analysis of activation energy associated with the nucleation of lattice misfit dislocations from the threading dislocations [11, 13, 17, 25] has not been presented in this work. Previous analyses [11, 13, 17, 25] have not included the interracial energy and the frictional energy in the calculation of the activation energy associated with the formation of the misfit dislocation. As mentioned earlier, coherency is disturbed over a length b/f along the interface when a misfit dislocation is formed, and this leads to the creation of interfacial energy. Our analysis showed that the activation energy reaches sufficiently large values such that even thermal energy at the growth temperatures is insufficient to overcome the barrier when the interracial energy of the partially coherent interface is included. Thus, we believe that the coherent strain energy in the film is required to overcome the barrier. If the lattice frictional stress and the surface energy due to the ledge step are included in the analysis, there should be an activation energy for nucleation of a misfit dislocation from the threading dislocations as well. It is possible for the threading dislocation to bend parallel to the interface and not form an interfacial segment. In this situation, no additional interfacial energy is generated and hence the total energy will decrease. This is not expected, however. We have explained [10] the greater critical thickness in island growth on the basis that the coherent strain energy is much smaller for threedimensional growth compared with two-dimensional growth. Thus, large island sizes are required before the coherent energy overcomes the activation energy barrier. This result is clearly seen in Table 1 with the critical size of the threedimensional epilayer varying from 30 to 60/k, whereas for two-dimensional films it remains between 5 and 15/k. The present analysis also illustrated that surface steps which are reduced as a result of the nucleation of a dislocation segment are favorable sites for nucleation.

Acknowledgments This work was partially supported by the National Science Foundation, Engineering Centers Program through NCSU Center for

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Advanced Electronic (Grant CDR-8721505).

Materials

Processing

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