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Monte-Carlo simulation of crystal growth with diffusion employing an anisotropic model
0038-1098/91 $3.00 + .00 Pergamon Press plc
Solid State Communications, Vol. 80, No. 5, pp. 311-314, 1991. Printed in Great Britain.
M O N T E - C A R L O S I M U L A T I O N OF CRYSTAL G R O W T H W I T H D I F F U S I O N E M P L O Y I N G AN ANISOTROPIC MODEL* Hua Li, Liu Jun and Nai-Ben Mingt National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, 210008, People's Republic of China (Received 30 M a y 1991; in revised f o r m 10 July 1991 by R. Barrie)
Monte-Carlo technique is used to simulate with surface diffusion on the (1 0 0 ) face employing the anisotropic variable bond surface diffusion on crystal growth kinetic T H E G R O W T H rate of a crystal depends on the ability of the vapour phase to supply material to the surface and the ability of the crystal to assimilate this material. A molecule reaching the crystal surface from the vapour phase must find a position where it is strongly bound, such as a kink site, or it is likely to return to the vapour phase. The chance of reaching a kink site depends on the average number of these sites on the surface and therefore the growth rate depends on the precise structure of the surface during the growth. Surface migration occurs in most vapour growth systems, since it generally requires less energy to translate an atom parallel to the surface than to remove it back to the vapour phase. The adatoms generally have a good chance to go to kink sites through surface diffusion before they evaporate. So surface diffusion influences the growth rate. Gilmer and Bennema [2] have discussed the problem of crystal growth with surface diffusion quantitatively and qualitatively using the constant bond (CB) model. They have drawn a conclusion that surface diffusion increases the growth rate. But they have obtained only dimensionless growth rate as a function of supersaturation. In our paper, we will use Monte-Carlo method to obtain the dimensionless growth rate and local roughness (dangling bonds per site) as functions of supersaturation employing both the constant model and the anisotropic variable bond (AVB) model [!]. We use the solid-on-solid (SOS) model of a simple cubic in the computations. Especially at the low temperatures investigated here, the SOS model is an accurate
* Project supported by National Natural Science Foundation of China. t To whom all communication regarding this paper should be addressed.
the vapour crystal growth of a simple cubic crystal model [1]. The effects of roughening are discussed.
approximation to the lattice gas because the surface remains quite smooth and the overhangs are energetically unfavourable. In the AVB model, the internal energy ~b(n) of an atom depends on the number of its nearest neighbours n[l] ~k(n) =
-
(an + bn2),
(1)
where a and b are constants obtained from the cohesive energy per atom and the energy of bulk vacancy formation (Ev)[3]. (2)
a
=
ao-
b
=
~ao/(V -
=
(Ecoh -- Ev)/E~oh
ao =
vb,
1),
E¢oh/V,
(3) (4) (5)
v is the maximum number of the nearest neighbours and ~ is a parameter to assess the influence of many body interactions. In the CB model Ev = EeoC, therefore b = 0, a = a0, ~ = 0. However, typically, Ev < Ecoh [1], then b < 0, ~ > 0. The bond energy ~jk of any neighbouring atom j and k is given by: c~jk = 2a + b(nj + nk) =
2a0 + b(nj + nk - 2v).
(6) Let P ÷ be the impingement rate of vapour atoms per site. It is independent of the nature of the sites, but varies with faces, since it is proportional to the area of site A0: P+ = JAo [4], where Jis the impingement rate per unit area and it is independent of the face. We assume that if an atom impinges on a deposition site, it deposits there. The evaporation rate of an atom is denoted by P, {Nj }, where i is the number of the nearest neighbours of the atom considered to be evaporated and N~ is the number of the nearest neighbours of the repective neighbour j of the atom considered.
311
312
MONTE-CARLO
SIMULATION
OF CRYSTAL GROWTH
Vol. 80, No. 5
Similarly with Jin et al. [3], we get P+
= f e x p (A#/kBT),
P,{Nj}
(7) *q. O
= f e x p [(v - 2i)a/kBT .l-
+ (i--i2
+ v 2 - - 2j=,~ Nj) b/k,T,
(8)
where f is the frequency factor and A/~ equals the chemical potential difference between the v a p o u r and the solid phases, kB is the B o l t z m a n n ' s constant. The equations are in general form. Similarly with Gilmer [5], in the case of surface migration, some flexibility is allowed in the choice o f transition probabilities. A surface a t o m with i neighbours j u m p s to a site where it would have j neighbours at the rate P,j, P,.j = f e x p [ - E s d ( i , j ) + AE(i,j)/k,T).
(9)
Unlike Gilmer [5], we chose Esd(i,j) = Esd(j, i) = Esd, Esd = a (2a is the bond energy) as the activation energy for a j u m p to a site where j /> i, and this p a r a m e t e r determines the mobility o f the adatoms. The p a r a m e t e r AE(i,j) is chosen to be positive only if the n u m b e r of neighbours at the second site is less than that o f the first, and zero otherwise:
AE(i,j)
=
AE
if j ~< i,
0
j>
(lO)
i
Microscopic reversibility is satisfied by this choice since P,j/~i = exp {AE/kB T} is satisfied in all cases. Surface diffusion processes will not disturb the equilibrium structure o f the surface. In AVB model: AE
=
E(AFTER) -
E(BEFORE),
=
@J + Z ~/,(U,) -
/
J
/-I
~ ~9(U, -
i
-- [~9i +
l)
l-I
~ rn=l
i
¢(N.,)
-
~
~9(Nm -
1)],
m--I
= 2(i - - j ) a + (i --j)(i + j -- 1)b
where Nm is the n u m b e r of the nearest neighbours o f the respective neighbour m of the a t o m considered before j u m p occurs and N~ is the n u m b e r o f the nearest neighbours of the respective neighbour of the a t o m considered after the j u m p takes place. If b = 0, a = a0, then
E(i,j)
=
2(i-- j)a
(j ~ i),
=
0
( j > i).
(12)
Then the situation becomes the same as Gilmer [5].
lk
•
-CB MODEL
o
-CE MODEL+DIFFUSION
•
- A V E MODEL
+
-AVB MODEL+DIFFUSION
+
a~ ~. O r~ r~
+
o
o
+
o ~.
+
o
0
+R;
• 2
SUPERSATURATION
A~/KT
Fig. 1. The dimensionless growth rate R/P+d vs supersaturation A#/kB T, for ao/kBT = 2.0. Note that P,j depends only on i and j, and does not change when two sites are at different levels. We assume that equations (9), (10) and (11) hold for the non-equilibrium situation, although a large supersaturation m a y have an effect on the surface diffusion transition probabilities. According to equations (8), (9), (10) and (11) a surface diffusion j u m p to a site w i t h j > i occurs m o r e frequently than evaporation as would be expected in reality. Surface migration permits atoms which impinge on sites of high energy to move to low energy positions where the evaporation rate is lower. F o r example, an a t o m absorbed along the edge of a step m a y migrate to a kink site and an isolated a t o m m a y move to the edge of a step. We have simulated the v a p o u r growth on (1 00) face of a s.e.c, for ao/k BT = 0.5, b/kB T = - 0 . 0 2 5 , Esd/k~T = 0.5; ao/kBT = 2.0, b/kBT = 0.1, Esd/ kBT = 2.0 employing the CB model and the A V b model respectively with surface diffusion and without surface diffusion. We obtained the dimensionless growth rate R/P+d and the local roughness R, as functions of supersaturation. ~ = 0.25 in two cases. We define n = exp (ao/KBT) as the extent o f surface diffusion. (n ~ (xs/a) 2, xs is the main diffusion distance, following Gilmer [2]). We could expect that the effect o f surface diffusion on growth rate will be m o r e significant as T is lowered. Figures 1 and 2 show the relation between R/P+d and Al~/kBT; R~ and AI~/kBT when T is below the t h e r m o d y n a m i c roughening temperature TR. Figure 1 indicates that surface migration causes an appreciable increase in growth rate in both CB and AVB models. F o r example, when Alt/kB T = 2.5 R/P+d is a b o u t 9
Vol. 80, No. 5
M O N T E - C A R L O S I M U L A T I O N OF C R Y S T A L G R O W T H
313
i
÷ b
i ~d
t~ D o r~d
ik i 4b
I
I
2
4
SUPERSATURATION
012
o
~/KT
Fig. 2. The local roughness R, vs supersaturation A#/ka T, for ao/kBT = 2.0. times greater than that with no surface jump (CB model), while for the same AI~/kBT, R/P+d is about 4 times greater than that with no surface jump if we employ AVB model. The effect is more significant in CB model. As we know, the surface is more rough and growth is easier on it when the AVB model is employed. So the effect of surface diffusion on crystal growth in AVB model is not so evident as in CB model. Figure 1 also shows that when A#/kBT > 1.5, R/P÷d increases linearly with Al~/knT increased (with diffusion), while only when A~/k, T > 2.2, does R/P÷d increases linearly with supersaturation enhanced if we employ CB model. The reason is that surface diffusion increases 2-D nucleation rate since it increases the probability of aggregation of the adatoms on the surface. On the other hand, surface migration increases the configuration entropy and lowers the step free energy and the value of critical driving force for kinetic roughening. Thus surface diffusion promotes kinetics roughening. But on the other hand, the critical driving force does not disappear since surface migration does not change the growth mechanism but lifts the nucleation rate. Figure 2 shows that the local roughness (dangling bonds per site) is the same for the case with diffusion and with no diffusion if we use the CB and the AVB model when AI~/kBT---0. It indicates that the surface migration process does not disturb the equilibrium structure of the surface, as expected. Local roughness R~ is significantly reduced by diffusion as Fig. 2 shows because the number of adatoms on the surface is reduced since surface diffusion enables an adatom to find a position in the edge of a step or a growing cluster where its chance on annihilation is low.
I 0.4
J 0.6
SUPERSATURATION
01.8
/k~/KT
Fig. 3. The dimensionless growth rate R/pd vs supersaturation Ala/k8T, for ao/k, T = 0.5. Figures 3 and 4 show the relation between R/P +d and Ala/kBT and R~ and Al~/kBT when T is above the thermodynamic roughening temperature T R. It can be seen that R/P÷d is raised and R, is lowered with diffusion in both the CB and the AVB model. In Fig. 4 the local roughness is not changed either with surface diffusion at equilibrium state in both models. In the range of 0-6.0 of the supersaturation in Fig. 3, we can see that the growth speed R with diffusion is a little greater than that without diffusion. From this we can recognise that the effect of surface diffusion is not so remarkable as it is when T is low. The reason is when T > TR, surfaces became rough and steps can form on the surface on account of fluctuations. Growth is
r~ •
+
D O --
o
M
O
O
1~
+
+
O
O
O
O
O
0t2
01,
0.6 '
SUPERSATURATION
0.8 ' A/~IKT
Fig. 4. The local roughness R~ vs supersaturation
A#/k,T, for ao/kBT = 0.5.
314
MONTE-CARLO SIMULATION OF CRYSTAL GROWTH
very easy since there is no necessity for a two-dimension nucleation. Every molecule which reaches the surface has a good chance of reaching a kink site and the growth rate is simply proportional to the supersaturation. Surface diffusion does not change the growth mechanism either. In conclusion, surface diffusion increases the growth rate and reduces the local roughness no matter whether the surface is rough or smooth and which model is employed (CB or AVB); the effect of diffusion is more significant when the surface is smooth and the AVB model is employed; surface diffusion lowers the critical value of the driving force for kinetic roughening
Vol. 80, No. 5
and does not change the growth mechanism and equilibrium structure. REFERENCES 1. 2. 3. 4.
5.
J.S. Chen, N.B. Ming & F. Rosenberger, J. Chem. Phys. 84, 2365 0986). G.H. Gilmer & B. Bennema, J. AppL Phys. V43, 1347 (1972). Jian-Min Jin, Jun Liu, N.B. Ming & C.K. Ong, Solid. State. Commun. V66, 529 0988). G.H. Giimer & K.A. Jackson Current Topics in Material Science (Edited by E. Kaldis & H.J. Scheels), p. 79, North-Holland, Amsterdam (1977). G.H. Gilmer, J. Cryst. Growth 35, 15 (1976).
Solid State Communications, Vol. 80, No. 5, pp. 311-314, 1991. Printed in Great Britain.
M O N T E - C A R L O S I M U L A T I O N OF CRYSTAL G R O W T H W I T H D I F F U S I O N E M P L O Y I N G AN ANISOTROPIC MODEL* Hua Li, Liu Jun and Nai-Ben Mingt National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, 210008, People's Republic of China (Received 30 M a y 1991; in revised f o r m 10 July 1991 by R. Barrie)
Monte-Carlo technique is used to simulate with surface diffusion on the (1 0 0 ) face employing the anisotropic variable bond surface diffusion on crystal growth kinetic T H E G R O W T H rate of a crystal depends on the ability of the vapour phase to supply material to the surface and the ability of the crystal to assimilate this material. A molecule reaching the crystal surface from the vapour phase must find a position where it is strongly bound, such as a kink site, or it is likely to return to the vapour phase. The chance of reaching a kink site depends on the average number of these sites on the surface and therefore the growth rate depends on the precise structure of the surface during the growth. Surface migration occurs in most vapour growth systems, since it generally requires less energy to translate an atom parallel to the surface than to remove it back to the vapour phase. The adatoms generally have a good chance to go to kink sites through surface diffusion before they evaporate. So surface diffusion influences the growth rate. Gilmer and Bennema [2] have discussed the problem of crystal growth with surface diffusion quantitatively and qualitatively using the constant bond (CB) model. They have drawn a conclusion that surface diffusion increases the growth rate. But they have obtained only dimensionless growth rate as a function of supersaturation. In our paper, we will use Monte-Carlo method to obtain the dimensionless growth rate and local roughness (dangling bonds per site) as functions of supersaturation employing both the constant model and the anisotropic variable bond (AVB) model [!]. We use the solid-on-solid (SOS) model of a simple cubic in the computations. Especially at the low temperatures investigated here, the SOS model is an accurate
* Project supported by National Natural Science Foundation of China. t To whom all communication regarding this paper should be addressed.
the vapour crystal growth of a simple cubic crystal model [1]. The effects of roughening are discussed.
approximation to the lattice gas because the surface remains quite smooth and the overhangs are energetically unfavourable. In the AVB model, the internal energy ~b(n) of an atom depends on the number of its nearest neighbours n[l] ~k(n) =
-
(an + bn2),
(1)
where a and b are constants obtained from the cohesive energy per atom and the energy of bulk vacancy formation (Ev)[3]. (2)
a
=
ao-
b
=
~ao/(V -
=
(Ecoh -- Ev)/E~oh
ao =
vb,
1),
E¢oh/V,
(3) (4) (5)
v is the maximum number of the nearest neighbours and ~ is a parameter to assess the influence of many body interactions. In the CB model Ev = EeoC, therefore b = 0, a = a0, ~ = 0. However, typically, Ev < Ecoh [1], then b < 0, ~ > 0. The bond energy ~jk of any neighbouring atom j and k is given by: c~jk = 2a + b(nj + nk) =
2a0 + b(nj + nk - 2v).
(6) Let P ÷ be the impingement rate of vapour atoms per site. It is independent of the nature of the sites, but varies with faces, since it is proportional to the area of site A0: P+ = JAo [4], where Jis the impingement rate per unit area and it is independent of the face. We assume that if an atom impinges on a deposition site, it deposits there. The evaporation rate of an atom is denoted by P, {Nj }, where i is the number of the nearest neighbours of the atom considered to be evaporated and N~ is the number of the nearest neighbours of the repective neighbour j of the atom considered.
311
312
MONTE-CARLO
SIMULATION
OF CRYSTAL GROWTH
Vol. 80, No. 5
Similarly with Jin et al. [3], we get P+
= f e x p (A#/kBT),
P,{Nj}
(7) *q. O
= f e x p [(v - 2i)a/kBT .l-
+ (i--i2
+ v 2 - - 2j=,~ Nj) b/k,T,
(8)
where f is the frequency factor and A/~ equals the chemical potential difference between the v a p o u r and the solid phases, kB is the B o l t z m a n n ' s constant. The equations are in general form. Similarly with Gilmer [5], in the case of surface migration, some flexibility is allowed in the choice o f transition probabilities. A surface a t o m with i neighbours j u m p s to a site where it would have j neighbours at the rate P,j, P,.j = f e x p [ - E s d ( i , j ) + AE(i,j)/k,T).
(9)
Unlike Gilmer [5], we chose Esd(i,j) = Esd(j, i) = Esd, Esd = a (2a is the bond energy) as the activation energy for a j u m p to a site where j /> i, and this p a r a m e t e r determines the mobility o f the adatoms. The p a r a m e t e r AE(i,j) is chosen to be positive only if the n u m b e r of neighbours at the second site is less than that o f the first, and zero otherwise:
AE(i,j)
=
AE
if j ~< i,
0
j>
(lO)
i
Microscopic reversibility is satisfied by this choice since P,j/~i = exp {AE/kB T} is satisfied in all cases. Surface diffusion processes will not disturb the equilibrium structure o f the surface. In AVB model: AE
=
E(AFTER) -
E(BEFORE),
=
@J + Z ~/,(U,) -
/
J
/-I
~ ~9(U, -
i
-- [~9i +
l)
l-I
~ rn=l
i
¢(N.,)
-
~
~9(Nm -
1)],
m--I
= 2(i - - j ) a + (i --j)(i + j -- 1)b
where Nm is the n u m b e r of the nearest neighbours o f the respective neighbour m of the a t o m considered before j u m p occurs and N~ is the n u m b e r o f the nearest neighbours of the respective neighbour of the a t o m considered after the j u m p takes place. If b = 0, a = a0, then
E(i,j)
=
2(i-- j)a
(j ~ i),
=
0
( j > i).
(12)
Then the situation becomes the same as Gilmer [5].
lk
•
-CB MODEL
o
-CE MODEL+DIFFUSION
•
- A V E MODEL
+
-AVB MODEL+DIFFUSION
+
a~ ~. O r~ r~
+
o
o
+
o ~.
+
o
0
+R;
• 2
SUPERSATURATION
A~/KT
Fig. 1. The dimensionless growth rate R/P+d vs supersaturation A#/kB T, for ao/kBT = 2.0. Note that P,j depends only on i and j, and does not change when two sites are at different levels. We assume that equations (9), (10) and (11) hold for the non-equilibrium situation, although a large supersaturation m a y have an effect on the surface diffusion transition probabilities. According to equations (8), (9), (10) and (11) a surface diffusion j u m p to a site w i t h j > i occurs m o r e frequently than evaporation as would be expected in reality. Surface migration permits atoms which impinge on sites of high energy to move to low energy positions where the evaporation rate is lower. F o r example, an a t o m absorbed along the edge of a step m a y migrate to a kink site and an isolated a t o m m a y move to the edge of a step. We have simulated the v a p o u r growth on (1 00) face of a s.e.c, for ao/k BT = 0.5, b/kB T = - 0 . 0 2 5 , Esd/k~T = 0.5; ao/kBT = 2.0, b/kBT = 0.1, Esd/ kBT = 2.0 employing the CB model and the A V b model respectively with surface diffusion and without surface diffusion. We obtained the dimensionless growth rate R/P+d and the local roughness R, as functions of supersaturation. ~ = 0.25 in two cases. We define n = exp (ao/KBT) as the extent o f surface diffusion. (n ~ (xs/a) 2, xs is the main diffusion distance, following Gilmer [2]). We could expect that the effect o f surface diffusion on growth rate will be m o r e significant as T is lowered. Figures 1 and 2 show the relation between R/P+d and Al~/kBT; R~ and AI~/kBT when T is below the t h e r m o d y n a m i c roughening temperature TR. Figure 1 indicates that surface migration causes an appreciable increase in growth rate in both CB and AVB models. F o r example, when Alt/kB T = 2.5 R/P+d is a b o u t 9
Vol. 80, No. 5
M O N T E - C A R L O S I M U L A T I O N OF C R Y S T A L G R O W T H
313
i
÷ b
i ~d
t~ D o r~d
ik i 4b
I
I
2
4
SUPERSATURATION
012
o
~/KT
Fig. 2. The local roughness R, vs supersaturation A#/ka T, for ao/kBT = 2.0. times greater than that with no surface jump (CB model), while for the same AI~/kBT, R/P+d is about 4 times greater than that with no surface jump if we employ AVB model. The effect is more significant in CB model. As we know, the surface is more rough and growth is easier on it when the AVB model is employed. So the effect of surface diffusion on crystal growth in AVB model is not so evident as in CB model. Figure 1 also shows that when A#/kBT > 1.5, R/P÷d increases linearly with Al~/knT increased (with diffusion), while only when A~/k, T > 2.2, does R/P÷d increases linearly with supersaturation enhanced if we employ CB model. The reason is that surface diffusion increases 2-D nucleation rate since it increases the probability of aggregation of the adatoms on the surface. On the other hand, surface migration increases the configuration entropy and lowers the step free energy and the value of critical driving force for kinetic roughening. Thus surface diffusion promotes kinetics roughening. But on the other hand, the critical driving force does not disappear since surface migration does not change the growth mechanism but lifts the nucleation rate. Figure 2 shows that the local roughness (dangling bonds per site) is the same for the case with diffusion and with no diffusion if we use the CB and the AVB model when AI~/kBT---0. It indicates that the surface migration process does not disturb the equilibrium structure of the surface, as expected. Local roughness R~ is significantly reduced by diffusion as Fig. 2 shows because the number of adatoms on the surface is reduced since surface diffusion enables an adatom to find a position in the edge of a step or a growing cluster where its chance on annihilation is low.
I 0.4
J 0.6
SUPERSATURATION
01.8
/k~/KT
Fig. 3. The dimensionless growth rate R/pd vs supersaturation Ala/k8T, for ao/k, T = 0.5. Figures 3 and 4 show the relation between R/P +d and Ala/kBT and R~ and Al~/kBT when T is above the thermodynamic roughening temperature T R. It can be seen that R/P÷d is raised and R, is lowered with diffusion in both the CB and the AVB model. In Fig. 4 the local roughness is not changed either with surface diffusion at equilibrium state in both models. In the range of 0-6.0 of the supersaturation in Fig. 3, we can see that the growth speed R with diffusion is a little greater than that without diffusion. From this we can recognise that the effect of surface diffusion is not so remarkable as it is when T is low. The reason is when T > TR, surfaces became rough and steps can form on the surface on account of fluctuations. Growth is
r~ •
+
D O --
o
M
O
O
1~
+
+
O
O
O
O
O
0t2
01,
0.6 '
SUPERSATURATION
0.8 ' A/~IKT
Fig. 4. The local roughness R~ vs supersaturation
A#/k,T, for ao/kBT = 0.5.
314
MONTE-CARLO SIMULATION OF CRYSTAL GROWTH
very easy since there is no necessity for a two-dimension nucleation. Every molecule which reaches the surface has a good chance of reaching a kink site and the growth rate is simply proportional to the supersaturation. Surface diffusion does not change the growth mechanism either. In conclusion, surface diffusion increases the growth rate and reduces the local roughness no matter whether the surface is rough or smooth and which model is employed (CB or AVB); the effect of diffusion is more significant when the surface is smooth and the AVB model is employed; surface diffusion lowers the critical value of the driving force for kinetic roughening
Vol. 80, No. 5
and does not change the growth mechanism and equilibrium structure. REFERENCES 1. 2. 3. 4.
5.
J.S. Chen, N.B. Ming & F. Rosenberger, J. Chem. Phys. 84, 2365 0986). G.H. Gilmer & B. Bennema, J. AppL Phys. V43, 1347 (1972). Jian-Min Jin, Jun Liu, N.B. Ming & C.K. Ong, Solid. State. Commun. V66, 529 0988). G.H. Giimer & K.A. Jackson Current Topics in Material Science (Edited by E. Kaldis & H.J. Scheels), p. 79, North-Holland, Amsterdam (1977). G.H. Gilmer, J. Cryst. Growth 35, 15 (1976).