- Email: [email protected]
Metastability of commensurate and incommensurate structures in epitaxial films
Volume
115, number
PHYSICS
8
METASTABILITY OF COMMENSURATE IN EPITAXIAL FILMS
LETTERS
A
5 May 1986
AND INCOMMENSURATE
STRUCTURES
Lj. DOBROSAVLJEVIC-GRUJIC Institute Received
of Physics, P. 0. B. 57, Belgrade, Yugoslavia 23 October
1985; revised manuscript
received
25 February
1986; accepted
for publication
3 March
1986
Limits of stability for the range of metastable phases in two-dimensional epitaxial monolayers are calculated. The mean misfit between the monolayer and the substrate changes by addition or removal of solitons (or misfit dislocations) and metastability arises from the energy barrier for these processes. The behaviour is similar to that of a one-dimensional modulated chain of atoms with free ends.
The simplest model of an epitaxial monolayer is a classical harmonic chain of atoms in a periodic potential. With the “natural” periodicity b of the chain competing with the substrate periodicity a commensurate (C) and incommensurate (I) structures may be formed as stable or metastable states [l-3] . The hamiltonian of the system (at T = 0) is N-l H = ngO
[(Ka2/8n2) (un+l - u,, - 27m~)~
+ ; W(1 - cos u,)] )
(1)
where u, is the displacement of the nth atom from the nth trough of the substrate potential and m = b/a - 1 is the natural misfit. (All the lengths are measured in units of a/2n.) Frank and van der Merwe [l] solved the problem of equilibrium configurations in the continuum limit. In this model for m m,, the ground state goes over continuously into a state characterized by a finite density m of soliton-like defects (I configuration). However, when the chain has a free end one needs an activation energy for introduction or escape of solitons through the end, where the elastic stress u vanishes [l] . Thus, the chain may (at low T) remain commensurate form > m,, provided that m does not exceed an upper limit. This is a metastable C state, which could eventual0.3759601/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ly be destroyed by thermal activation. Similarly, a metastable I state may exist for m
= l/k2 - (mlJ2.
(2)
For a particular misfit m, this condition determines a class of metastable configurations, corresponding to different values of the parameter k (or of 1, Fi), such that (l/k2 - 1)lj2 < Z,m Q l/k. Physically, the above stability limits correspond to the vanishing of the energy barriers El and E, for introduction or escape of solitons, respectively. Ei represents the work done by pulling the end atom from its stable equilibrium displacement u. to the next unstable displacement ub = 2n - uo. (Note that u < 0 for u 2 u. and u > 0 for u. < u < ub .) Similarly, E, is the work done by pushing the end atom from u. 389
Volume 115, number 8
PHYSICS LETTERS A
5 May 1986
k =O.96 2 2
3 3
4 4
5 5
6
6
7
7
8
8
9
9
10
11
12
13
10
11
12
13
14
6
7
8
9
10
14
15
15
16
16
17
(aI
1
2
3
4
5
k=O,69255
Ib)
. 2
3
4
5
6
7
0
9
Fig. 1. The soliton structure near the free end of the chain for I, = 10 and m = 0.1042. (a) The end atom is at the crest of the substrate potential (k = 0.96). (b) The end atom is at the trough of the substrate potential (k = 0.6925).
+ 2x to nb [l] . Since u. may vary between 0 and rr (in physical units between 0 and a/2) Ei and E, vary as well: When u. = 0 and l,m = (l/k2 - 1)1/2 (see fig. lb), Ei is the largest and E, = 0. For 1,m < (l/k2 - 1)lj2 the solitons spontaneously escape. When u. = TTand lam = l/k (see fig. la), Ei = 0 and E, has the largest value. For Iom > l/k, the solitons spontaneously enter. In two dimensions (2D), the free boundary and its displacement are easily obtained only for the striped structures, where the problem is reduced to the onedimensional (1D) one. This is the case for the model studied by Pokrovsky and Talapov [4] , which deals with a 2D isotropic crystal adsorbed on a uniaxially periodic substrate. In this case, the I phase consists of parallel solitons, regularly spaced along a given direction. The angle between this direction and the axis of the substrate modulation depends on the elastic properties of the monolayer. In the continuum model [4] , the latter are expressed via the Lame coefficients h and /.L The free boundary, at which all the components of the stress tensor i? vanish simultaneously, is a straight line perpendicular to the axis of the striped solitonic structure. The stability limits of the range of metastable C and I configurations are also obtained from the condition i? = 0. We find that the border lines in the corresponding 1E versus lm diagram are of the same shape as in the 1D case [ 1] , 390
but now with l,+l=(k
g$)li2,
where V. is the energy density of the adsorbate-substrate interaction. When the substrate potential is periodic in two directions, the free boundary can be determined only numerically. Instead, in this note we concentrate on finding some discrete points where the stress tensor vanishes. We suggest that in the vicinity of these points the dislocations may enter into the 2D structures, leading to the formation of metastable configurations in the bulk of the epitaxial film [5]. We consider an isotropic (with hexagonal natural lattice) monolayer of atoms adsorbed on a hexagonal substrate. This applies directly to the rare gas lattices on the graphoil substrate, which are among the most studied examples of the 2D systems exhibiting the C-I transition [6] . At T = 0 and in the continuum approximation the elastic energy of the monolayer has the usual form for an isotropic (hexagonal) 2D solid He = (a/2s)2JJ + 4Mll
dx, d.x, {;(A + /J) (~11 + u22)2 - u22)2 + (Q
+O+P)~~@~~ +52L
+ 421 (4)
5 May 1986
PHYSICS LETTERS A
Volume 115, number 8
except for the presence of the linear term. The latter appears because the displacements u i (i = 1,2) are calculated with respect to the commensurate (C(fi X 4)) lattice. The parameter ml = 2( 1 - a/b) = 2(b/a - 1) is related to the misfit m, between the natural lattice of the monolayer (with lattice constant b) and the C lattice (with the lattice constant
a): m=Imll=IGi-gjl,
i= 1,2,3.
Here the reciprocal lattice vectors Gi and gi of the C lattice and of the natural lattice, respectively, are taken to be colinear: G, = (032)
G,,,
= (4%
gj = (a/b)Gj
(i = 1,2,3).
l),
(Note that the moduli of Gj and gj are measured in
27r/aunits.) To obtain analytical expressions for the deformation tensor {uii} and the stress tensor {oV} (i,j = 1,2) in the I phase, we calculate the energy of the adatoms-substrate interaction
Fig. 2. Part of the solitonic-wall hexagonal superlattice. The “stress4ree” points r. are indicated by circles: l m large; 0 m small.
hexagonal superstructure of solitonic walls separating the nearly commensurate domains [7]. The mean misfitEiisgivenbyEi=2?r/GL. For this model the off-diagonal components of the stress tensor vanish, u12 = 02~ = 0, and
in the approximation used by Bak [7] , where the (more realistic) cosine potential [6] 3 ‘(xl
3x2) = ‘0 2
is replaced by the truncated V(xl ,x2) = I$q34 o
011 = 4
[1 - COS(G~*~)l
+ 24;)
O
_,rsh(flKxl) sh(KL/2)
[(~+2dch(&X1)
parabolic potential
+ (At 21.0Ch(KX2)] + 2(x + P)mr.
defined within the irreducible part of the hexagonal cell [7] (the shaded region in fig. 2). The soliton network in the hexagonal I phase, which is one of the ground state configurations for the hamiltonian H = He1 + Ha-s, is obtained [7] via the displacements
‘l - 3
slyKLI2)
(6)
We find that there is only one point r. = (xy,xi) within the irreducible cell where all the components Oji vanish; it lies on the cell boundary ~2 = fi xl. For a given L, the position of this stress-free point depends on the reduced natural misfit m&n/d),
sh(KX2) ’
“2 =s
Sh(KL/2)
'
(5)
where K * = 2Vtff/(X t 2~) and L is the period of the
x;=4x;.
(7)
The limits of the misfit variations are obtained, as in 391
PHYSICS LETTERS A
Volume 115, number 8
5 May 1986
the 1D case, from the requirement that the displacements of r. are non-negative and do not exceed their maximum, symmetry imposed values, i.e. O
OG+lrl~.
(8)
One can assume in the 2D case too, that the condition u. = 0 means that the escape of sohtons may occur near ‘0 without activation energy. Similarly, for the maximum displacement u. = (43, n/fi) no energy should be required for the generation of solitons in the vicinity of ro. Then the relations (5) (7) and (8) give the stability limits for the range of metastable C and I configurations. For a given misfit m, there is a class of metastable hexagonal superstructures with period L such that 1 _,&< sh (KL/2) < K?T/fi
-
Cth(K15/2).
(9)
Of particular interest is the result that the C configuration (obtained in the limit L + -) cannot exist, neither stable nor metastable beyond m/(rtn/fl) = 1. Introducing the reduced natural misfit lm and the reduced mean misfit IEi, where I = (~n/fi)-~, relation (9) is written
(10) The equalities determine two curves in the 2D ZE versus lm diagram (fig. 3) very similar to the spontaneous escape and the spontaneous dislocation curves inthe 1Dcase [l],startingform=Oatm=Oand m = l/l respectively and approaching the line E = m for large m . The above (T = 0) diagram provides the limits of absolute instability of the metastable phases. At low temperature, the displacements of adatoms close to the points r. may occur, leading to dislocations and eventually to the formation of one phase (C or I) with stress-free boundaries, heterogeneously nucleated within the other (I or C) phase. The nucleation barrier is then related to the activation energy for the escape or generation of solitons at the nucleus boundary. This mechanism could be, among others,
392
Fig. 3. Limits of stability in the reduced mean misfit Zm versus the reduced natural misfit Im diagram (full curves); in the middle, m = m (light line).
at the origin of the hysteresis and phase coexistence observed in some C-I transitions [S-10]. References
111F.C. Frank and J.H. van der Merwe, PIOC. R. Sot. A 198 (1948) 205,216.
PI J.A. Snyman and J.H. van der Merwe, Surf. Sci. 42 (1974) 190. 131 S.R. Sharma, B. Bergersen and B. Joos, Phys. Rev. 29 (1984) 6335. 141 V.L. Pokrovsky and A.L. Talapov, Phys. Rev. Lett. 42 (1979) 65; Zh. Eksp. Theor. Fiz. 78 (1980) 269 [Sov. Phys. JETP 51 (1980) 1341. iSI L. Dobrosavljevic and Z. Radovic, Physica B 127 (1984) 433. [61 J. Villain and M. Gordon, Surf. Sci. 125 (1983) 1. [71 P. Bak, in: Solitons and condensed matter physics, eds. A.B. Bishop and T. Schneider (Springer, Berlin, 1981). VI A. Glashant, M. Jaubert and M. Bienfait, Surf. Sci. 115 (1981) 219. PI M. Nielsen, J. Ah-Nielsen, J. Bohr and J.P. McTague, Phys. Rev. Lett. 47 (1981) 582. PO1 Q.M. Shang, H.K. Kimand M.H.W. Chart, Phys. Rev. B 321 (1985) 1820.
115, number
PHYSICS
8
METASTABILITY OF COMMENSURATE IN EPITAXIAL FILMS
LETTERS
A
5 May 1986
AND INCOMMENSURATE
STRUCTURES
Lj. DOBROSAVLJEVIC-GRUJIC Institute Received
of Physics, P. 0. B. 57, Belgrade, Yugoslavia 23 October
1985; revised manuscript
received
25 February
1986; accepted
for publication
3 March
1986
Limits of stability for the range of metastable phases in two-dimensional epitaxial monolayers are calculated. The mean misfit between the monolayer and the substrate changes by addition or removal of solitons (or misfit dislocations) and metastability arises from the energy barrier for these processes. The behaviour is similar to that of a one-dimensional modulated chain of atoms with free ends.
The simplest model of an epitaxial monolayer is a classical harmonic chain of atoms in a periodic potential. With the “natural” periodicity b of the chain competing with the substrate periodicity a commensurate (C) and incommensurate (I) structures may be formed as stable or metastable states [l-3] . The hamiltonian of the system (at T = 0) is N-l H = ngO
[(Ka2/8n2) (un+l - u,, - 27m~)~
+ ; W(1 - cos u,)] )
(1)
where u, is the displacement of the nth atom from the nth trough of the substrate potential and m = b/a - 1 is the natural misfit. (All the lengths are measured in units of a/2n.) Frank and van der Merwe [l] solved the problem of equilibrium configurations in the continuum limit. In this model for m
ly be destroyed by thermal activation. Similarly, a metastable I state may exist for m
= l/k2 - (mlJ2.
(2)
For a particular misfit m, this condition determines a class of metastable configurations, corresponding to different values of the parameter k (or of 1, Fi), such that (l/k2 - 1)lj2 < Z,m Q l/k. Physically, the above stability limits correspond to the vanishing of the energy barriers El and E, for introduction or escape of solitons, respectively. Ei represents the work done by pulling the end atom from its stable equilibrium displacement u. to the next unstable displacement ub = 2n - uo. (Note that u < 0 for u 2 u. and u > 0 for u. < u < ub .) Similarly, E, is the work done by pushing the end atom from u. 389
Volume 115, number 8
PHYSICS LETTERS A
5 May 1986
k =O.96 2 2
3 3
4 4
5 5
6
6
7
7
8
8
9
9
10
11
12
13
10
11
12
13
14
6
7
8
9
10
14
15
15
16
16
17
(aI
1
2
3
4
5
k=O,69255
Ib)
. 2
3
4
5
6
7
0
9
Fig. 1. The soliton structure near the free end of the chain for I, = 10 and m = 0.1042. (a) The end atom is at the crest of the substrate potential (k = 0.96). (b) The end atom is at the trough of the substrate potential (k = 0.6925).
+ 2x to nb [l] . Since u. may vary between 0 and rr (in physical units between 0 and a/2) Ei and E, vary as well: When u. = 0 and l,m = (l/k2 - 1)1/2 (see fig. lb), Ei is the largest and E, = 0. For 1,m < (l/k2 - 1)lj2 the solitons spontaneously escape. When u. = TTand lam = l/k (see fig. la), Ei = 0 and E, has the largest value. For Iom > l/k, the solitons spontaneously enter. In two dimensions (2D), the free boundary and its displacement are easily obtained only for the striped structures, where the problem is reduced to the onedimensional (1D) one. This is the case for the model studied by Pokrovsky and Talapov [4] , which deals with a 2D isotropic crystal adsorbed on a uniaxially periodic substrate. In this case, the I phase consists of parallel solitons, regularly spaced along a given direction. The angle between this direction and the axis of the substrate modulation depends on the elastic properties of the monolayer. In the continuum model [4] , the latter are expressed via the Lame coefficients h and /.L The free boundary, at which all the components of the stress tensor i? vanish simultaneously, is a straight line perpendicular to the axis of the striped solitonic structure. The stability limits of the range of metastable C and I configurations are also obtained from the condition i? = 0. We find that the border lines in the corresponding 1E versus lm diagram are of the same shape as in the 1D case [ 1] , 390
but now with l,+l=(k
g$)li2,
where V. is the energy density of the adsorbate-substrate interaction. When the substrate potential is periodic in two directions, the free boundary can be determined only numerically. Instead, in this note we concentrate on finding some discrete points where the stress tensor vanishes. We suggest that in the vicinity of these points the dislocations may enter into the 2D structures, leading to the formation of metastable configurations in the bulk of the epitaxial film [5]. We consider an isotropic (with hexagonal natural lattice) monolayer of atoms adsorbed on a hexagonal substrate. This applies directly to the rare gas lattices on the graphoil substrate, which are among the most studied examples of the 2D systems exhibiting the C-I transition [6] . At T = 0 and in the continuum approximation the elastic energy of the monolayer has the usual form for an isotropic (hexagonal) 2D solid He = (a/2s)2JJ + 4Mll
dx, d.x, {;(A + /J) (~11 + u22)2 - u22)2 + (Q
+O+P)~~@~~ +52L
+ 421 (4)
5 May 1986
PHYSICS LETTERS A
Volume 115, number 8
except for the presence of the linear term. The latter appears because the displacements u i (i = 1,2) are calculated with respect to the commensurate (C(fi X 4)) lattice. The parameter ml = 2( 1 - a/b) = 2(b/a - 1) is related to the misfit m, between the natural lattice of the monolayer (with lattice constant b) and the C lattice (with the lattice constant
a): m=Imll=IGi-gjl,
i= 1,2,3.
Here the reciprocal lattice vectors Gi and gi of the C lattice and of the natural lattice, respectively, are taken to be colinear: G, = (032)
G,,,
= (4%
gj = (a/b)Gj
(i = 1,2,3).
l),
(Note that the moduli of Gj and gj are measured in
27r/aunits.) To obtain analytical expressions for the deformation tensor {uii} and the stress tensor {oV} (i,j = 1,2) in the I phase, we calculate the energy of the adatoms-substrate interaction
Fig. 2. Part of the solitonic-wall hexagonal superlattice. The “stress4ree” points r. are indicated by circles: l m large; 0 m small.
hexagonal superstructure of solitonic walls separating the nearly commensurate domains [7]. The mean misfitEiisgivenbyEi=2?r/GL. For this model the off-diagonal components of the stress tensor vanish, u12 = 02~ = 0, and
in the approximation used by Bak [7] , where the (more realistic) cosine potential [6] 3 ‘(xl
3x2) = ‘0 2
is replaced by the truncated V(xl ,x2) = I$q34 o
011 = 4
[1 - COS(G~*~)l
+ 24;)
O
_,rsh(flKxl) sh(KL/2)
[(~+2dch(&X1)
parabolic potential
+ (At 21.0Ch(KX2)] + 2(x + P)mr.
defined within the irreducible part of the hexagonal cell [7] (the shaded region in fig. 2). The soliton network in the hexagonal I phase, which is one of the ground state configurations for the hamiltonian H = He1 + Ha-s, is obtained [7] via the displacements
‘l - 3
slyKLI2)
(6)
We find that there is only one point r. = (xy,xi) within the irreducible cell where all the components Oji vanish; it lies on the cell boundary ~2 = fi xl. For a given L, the position of this stress-free point depends on the reduced natural misfit m&n/d),
sh(KX2) ’
“2 =s
Sh(KL/2)
'
(5)
where K * = 2Vtff/(X t 2~) and L is the period of the
x;=4x;.
(7)
The limits of the misfit variations are obtained, as in 391
PHYSICS LETTERS A
Volume 115, number 8
5 May 1986
the 1D case, from the requirement that the displacements of r. are non-negative and do not exceed their maximum, symmetry imposed values, i.e. O
OG+lrl~.
(8)
One can assume in the 2D case too, that the condition u. = 0 means that the escape of sohtons may occur near ‘0 without activation energy. Similarly, for the maximum displacement u. = (43, n/fi) no energy should be required for the generation of solitons in the vicinity of ro. Then the relations (5) (7) and (8) give the stability limits for the range of metastable C and I configurations. For a given misfit m, there is a class of metastable hexagonal superstructures with period L such that 1 _,&< sh (KL/2) < K?T/fi
-
Cth(K15/2).
(9)
Of particular interest is the result that the C configuration (obtained in the limit L + -) cannot exist, neither stable nor metastable beyond m/(rtn/fl) = 1. Introducing the reduced natural misfit lm and the reduced mean misfit IEi, where I = (~n/fi)-~, relation (9) is written
(10) The equalities determine two curves in the 2D ZE versus lm diagram (fig. 3) very similar to the spontaneous escape and the spontaneous dislocation curves inthe 1Dcase [l],startingform=Oatm=Oand m = l/l respectively and approaching the line E = m for large m . The above (T = 0) diagram provides the limits of absolute instability of the metastable phases. At low temperature, the displacements of adatoms close to the points r. may occur, leading to dislocations and eventually to the formation of one phase (C or I) with stress-free boundaries, heterogeneously nucleated within the other (I or C) phase. The nucleation barrier is then related to the activation energy for the escape or generation of solitons at the nucleus boundary. This mechanism could be, among others,
392
Fig. 3. Limits of stability in the reduced mean misfit Zm versus the reduced natural misfit Im diagram (full curves); in the middle, m = m (light line).
at the origin of the hysteresis and phase coexistence observed in some C-I transitions [S-10]. References
111F.C. Frank and J.H. van der Merwe, PIOC. R. Sot. A 198 (1948) 205,216.
PI J.A. Snyman and J.H. van der Merwe, Surf. Sci. 42 (1974) 190. 131 S.R. Sharma, B. Bergersen and B. Joos, Phys. Rev. 29 (1984) 6335. 141 V.L. Pokrovsky and A.L. Talapov, Phys. Rev. Lett. 42 (1979) 65; Zh. Eksp. Theor. Fiz. 78 (1980) 269 [Sov. Phys. JETP 51 (1980) 1341. iSI L. Dobrosavljevic and Z. Radovic, Physica B 127 (1984) 433. [61 J. Villain and M. Gordon, Surf. Sci. 125 (1983) 1. [71 P. Bak, in: Solitons and condensed matter physics, eds. A.B. Bishop and T. Schneider (Springer, Berlin, 1981). VI A. Glashant, M. Jaubert and M. Bienfait, Surf. Sci. 115 (1981) 219. PI M. Nielsen, J. Ah-Nielsen, J. Bohr and J.P. McTague, Phys. Rev. Lett. 47 (1981) 582. PO1 Q.M. Shang, H.K. Kimand M.H.W. Chart, Phys. Rev. B 321 (1985) 1820.