The effect of anharmonicity in epitaxial interfaces

Surface

Science

145 (1984)

North-Holland.

313-328

313

Amsterdam

THE EFFECT OF ANHARMONICITY IN EPITAXIAL III. Energy, mean dislocation density and mean strain Ivan MARKOV Institute

of Physical

Received

23 March

The

influence

properties

both

model

Frank

of

characterizing greater

of

and

van

system

the deviation for negative

The misfit energy dislocation misfit

anharmonicity

undergoes

mechanisms between substrate, nucleation

split

from

the

of Sciences,

interatomic

1040

into

the results

separate

been

Sofia,

variation

is discussed.

branches

Bulgaria

the equilibrium

form of crystallites

and the mechanism

of 2D phase transitions

for positive

that such with positive

consequences

on a foreign

equilibrium

the conventional

to both strongly

1D

properties

signs

of

asymmetric

misfit (much

value). It has been shown that: (i)

misfits. misfit.

misfits.

(ii) The mean

(iii) overgrowth

with negative

The existence

strain with cluster

concerning

on

all physical

than positive

of experimentally

of growth of thin epitaxial

are also briefly

from

appears

rather

deposit

and mean strain appears

that

corresponding

misfits of the same absolute

of the residual

the

following

limit

is lower at negative

and the possibility

The

of

density

demonstrated

reference

in thin films is always greater

is suggested

forces

mean dislocation

It has

the harmonic

than for positive

of “saw-tooth”

them

Merwe.

much larger strains

high temperatures

in

and quantitatively der

of the overgrowth

density

Academy

films such as energy,

qualitatively the

MILCHEV

Bulgarian

1984

of thin epitaxial

to change

whereby

and Andrey Chemistry,

INTERFACES

size taking place at low and

observed

the adhesion substrate,

of two different

crossover

behaviour

of the overlayer

the kinetics

to the

of heterogeneous

films as well as various critical

properties

discussed.

1. Introduction In the second part [l] of our general investigation of the influence of anharmonicity in interatomic forces of the overgrowth on some fundamental features of epitaxial interfaces [1,2], we focused our study on the stability limits (critical misfits) P, and Pm,, essential for the pseudomorphic growth of thin films on a periodic substrate. In fact, we start with the well-known one-dimensional model, proposed originally by Frenkel and Kontorova [3], which gained prominence in the developments of Frank and van der Merve [4,5] for the case of misfitting overlayers. In our work the purely elastic harmonic forces between the nearest neighbour atoms in the deposit were replaced by more realistic ones with pronounced asymmetry between the repulsive and the attractive branches of the potential. The interatomic potential of Toda [6,7] proves in this sense as extremely suitable due to the following merits: (i) it 003%6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

smoothly

transforms

tion of Frank reference

from the hard sphere limit to the harmonic

and van der Merwe

basis, and (ii) one may independently

the attractive expansion

branch

of the potential

term constant.

sense

atomic

spacing of the overgrowth,

substrate,

[4,5]

Defining

tional

as the relative

a marked

asymmetry

the sign of P may be observed.

approxima-

so that the latter may serve naturally change

while

keeping

the natural difference

its leading

lattice

misfit,

h. and the lattice constant, Several

and

(harmonic)

P, in the tradi-

of the unstrained

P = (&~)/a

of the properties

as a

both the repulsive

CI. of the “rigid”

of the system with respect

of these properties

to

have been investi-

gated in ref. [l]: (i) The length of a single misfit dislocation is smaller at negative misfit as compared with the case of a system with (equally large) positive misfit. (ii) Deposit with negative misfit is easier to strain so as to match the periodicity of the substrate

than

such with positive

therefore, that more overgrowth the misfit is negative.

misfit.

From

atoms are in registry

(i) and (ii) it follows, with the substrate

when

(iii) The critical misfits for pseudomorphic growth, Ps, and for spontaneous generation of misfit dislocations, P,,,, split considerably upon “switching on” of anharmonicity. by Frank

Thus the harmonic

critical misfit, P,” = _t 8.6%, as estimated

and van der Merwe [4,5], splits into

degree of anharmonicity. ity limit PL, = + 13.6%

Under

+ 6.7% and - 12.2% at certain

the same conditions

splits into

+ 10.2% and

the harmonic

- 23.2%.

metastabil-

Evidently,

the split-

ting itself is asymmetric. It has been concluded [l] that due to anharmonicity of the atomic interactions a negative misfit appears more favourable for the epitaxial growth of thin films than a positive

misfit.

It was shown also that this conclusion

qualitative agreement with experimental of thin films of metals, semiconductors

is in good

evidence for the pseudomorphic and ferromagnetic garnets.

growth

In the present paper (part III) we study further the effect of anharmonicity on such properties of the overlayer as energy, mean dislocation density and residual strain, where the marked asymmetry misfit is expected to show itself clearly. Apparently,

with respect

to the sign of the

it turns out that our results are in good qualitative

agreement

with experimental data obtained from the studies of the 2D commensurateincommensurate phase transitions of rare gases adsorbed on graphite [g-lo]. Eventually we also elucidate the mechanism of two different types of “saw-tooth variation” of the residual strain observed in high and low temperature regimes. 2. Model The theoretical model has been discussed in part description assisting the reader will be given here.

I [2] so that

a brief

The nearest neighbours in a 1D chain of atoms are assumed means of Toda forces, the corresponding potential being [6,7]:

where

Y measures

atoms,

and (Y and /? are constants.

the distance,

h is the natural

spacing

to interact

by

of the overgrowth

At (Y + cc and p + 0 at LIP = const., eq. (1) yields the harmonic tion of Frank and van der Merwe

approxima-

V(v)=:ap(r-h)2.

(2) (Y + 0 and /3+ cc at CID= const.,

The inverse condition,

leads to the limit of

the hard sphere potential. Assuming tude

W/2

(Toda)

a simple periodic and wavelength

chain consisting

field of the substrate

a [3-51

the potential

potential energy

with an ampli-

of the anharmonic

of N atoms is given by (cf. eq. (S), part I):

N-l +F

c

(1

r,=O

(3)

-cos257z,),

where Z,, denotes

the relative displacement

the II th potential

trough of the substrate

P=

(b-a)/a

of

(4)

is the natural equations

misfit

[2] which

parameters Minimum refs. [ll-131)

[4,5].

Minimizing

may be solved

(3),

one obtains

numerically

a set of difference

for arbitrary

values

of the

involved. energy considerations

of the harmonic

have shown that generally

ated by a homogeneous e=

of the n th atom from the bottom and

the natural

case [4,5] (for a review, see misfit is partly accommod-

strain

(6-b)/%,

(5)

where b is the average atomic The remaining part

spacing

P=(b-a)/a is accommodated

of the overgrowth.

(6) by misfit dislocations.

p is usually called mean dislocation

density [ll-131, its reciprocal giving simply the distance between the dislocations in units of a. Obviously, when h = a the natural misfit is accommodated entirely by a homogeneous strain and misfit dislocations are absent as their spacing l/P tends to infinity. The chain is in registry with the substrate.

In the opposite case of b = h the misfit misfit dislocations spaced at a distance l/P.

F= (z,_, - Z,)/(N-

is accommodated completely It is easy to show [14] that:

l),

(7)

e=a(P-P)/b.

(8)

It is thus evident that p and Z play a prominent

role in our understanding

of the structure of the epitaxial interfaces. Moreover, experimentally measured (for a review, see ref. [15]). In the present

paper

mean dislocation a function

the behaviour

these quantities

of the energy (per atom)

E/N,

can be of the

density ?’ and of the residual mean strain .? is investigated

of the natural

misfit

P and of the chain length

tions the values of the parameters Thus the harmonic Details

by

dislocation

of the computational

chosen

length procedure

as

N. In all computa-

are W = 1, u = 3, (Y = 2 and p = 6.

I, = (c$u~/~W)‘/~

= 7.35 as in ref. [4].

were given in part I [2].

3. Results 3.1. Energy Fig. 1 represents the dependence of the energy (per atom) of the ground state of a chain consisting of 30 atoms on the natural misfit. The curves consist of a number segments correspond state increasing

of curvilinear segments as in the harmonic to different number of misfit dislocations

case [14]. The in the ground

from zero by one.

It is seen that in the case of positive misfit and especially

at small misfits the

energy is higher. At larger misfits the energy curves go closer and merge eventually. This effect is illuminating. At low misfits, both positive and negative,

the first sum (the strain

energy)

in eq. (3) is dominant.

At positive

misfit, the steeper repulsive part of the interatomic potential is mainly involved and accordingly the energy is higher than in the case of negative misfit where the strain energy is determined At larger

misfits,

both

by the weaker attractive

positive

and negative,

part of the interaction.

the second

sum in eq. (3)

predominates, the influence of the anharmonicity may be neglected and the energy difference between two cases gradually vanishes. The dependence of the energy (per atom) of the ground state on the length of the chain, N, is shown in fig. 2 for three different absolute values of the misfit: (a) 8%, (b) 12.5% and (c) 16%. It is seen that in all cases the energy at positive misfit is higher than the energy of the harmonic approximation and that the latter is in turn higher than the energy at negative misfit. This difference is again more pronounced at lower absolute values of the misfit as discussed above. It is interesting to note that with increasing misfit (in absolute

terms) the positive misfit energy first merges with the harmonic negative

misfit energy remains

(fig. 2c) both the positive (the harmonic Irrespective dependence case [l&17], ground

E(N)

consist of curvilinear

misfit

energy curves

tend to coincide.

corresponding

to growing

segments as in the continuous number

of misfit

the

harmonic

dislocations

in the

state as N is increased.

density

The mean dislocation both positive (dashed Frank

and negative

curve while the

below (fig. 2b). At still larger misfits

curve is thereby omitted for clarity). of the sign of the misfit, that is, of the anharmonicity,

3.2. Mean dislocation

The

markedly

dotted

density

p as a function

line) and negative

line corresponds

of the natural

to the continuous

harmonic

and van der Merwe [5]. As in ref. [14], the non-zero

respective

critical

to.1

misfits

MISFIT

P for

in fig. 3.

representation

of

value of p below the

J’\+ and Pse is due to the finite

to.2

misfit

(full line) sign is represented

size of the chain

to.3

Fig. 1. Dependence of the energy of the ground state (per atom) on the natural misfit P: negative misfit, full line: positive misfit, dashed line. The figures denote the number of the misfit dislocations in the ground state (N = 30, W = 1, a = 3, a = 2, p = 6 and /” = 7.35).

b i 10

30

20 NUMBER

OF

10 NUMBER

ATOMS

20 OF

30 ATOMS

C

10

20 NUMBER

Fig. 2. Dependence of three different values anharmonic case with anharmonic case with and /3 = 6).

OF

30

ATOMS

the energy of the ground state (per atom) on the number of atoma, N, at of the natural misfit: (a) ix%; (b) +12.5%; (c) + 16%. Solid circles, negative misfit: half-open circles, harmonic approximation; open circles, positive misfit. In (c) the harmonic curve is omitted ( W = 1, u = 3. a = 2

(Iv = 30). From fig. 3 several important conclusions follow: (i) The negative critical misfit, I’-, below which the non-dislocated chain is energetically favoured is nearly twice greater in absolute value than the critical positive misfit, Ps+. (ii) The mean dislocation density in the case of negative misfit is always under (in absolute value) the positive misfit one. Consequently, the dislocation spacing for negative misfit is greater than that for positive misfit although this difference is gradually diminished at large natural misfits. (iii) The mean misfit attains faster the value of the natural misfit for positive rather than for negative sign of the latter. An analytical solution of the problem (which is not available at present) would demonstrate this more convincingly. (iv)The harmonic approximation is markedly nearer to the positive misfit curve,. This result appears important when theory is compared with experiment. It has been pointed out earlier [S-IO] that the observed discrepancy between experimental data on the critical parameters (pressure, temperature, expressed in terms of P,) which govern the commensurate-incommensurate phase transition in rare-gas monolayers on graphite and the respective theoretical (harmonic) estimations must be traced back to anharmonicity effects. Indeed, mean dislocation density (mean misfit) versus pressure diagrams of the kind shown in fig. 3 suggest I, values which are in reasonable agreement for Xe

MlSFIl

Fig. 3. Dependence of the mean dislocation density, p, on the natural misfit, P, for both positive (dashed line) and negative (full line) signs of the latter. The harmonic continuous approximation of Frank and van der Merwe [5] is represented for comparison by the dotted line. The two dependences are given in one and the same quadrant for easier comparison (N = 30. W = 1, a = 3, a = 2, p = 6 and I, = 7.35).

2.5%) but are too low for Kr (q\._r = -4.5%) with respect to the theoretically predicted ones [18,19]: IO_, = 37.4 for Xe and 10.exp= 13 for Kr = 40 for Xe and 10,theor= 24 - 32 for Kr [18,19]. whi’e III theor Now’it is evident from fig. 3 that the introduction of anharmonicity in the model makes 9, rather sensitive to the sign of the misfit and even for equal values of I, as determined from the elastic constant and from W the respective critical misfits P,’ and Psp would move apart in opposite direction from Plh with increasing anharmonicity, the effect for negative P being much more pronounced.

( Ps.exp =

3.3. Mean strain The dependence of the residual strain F on natural misfit for a 30 atoms chain is given in fig. 4. The values of 2 for the case of negative misfit are given with the opposite sign for easier comparison with the positive misfit values. In

Fig. 4. Dependence negative

of the mean strain

(full line) values of the natural

for easier comparison

of the ground misfit,

with that of positive

state

e on both

positive

P. The sign of the negative

misfit.

(dashed

line) and

misfit strain is reversed

fact, the dependences shown in fig. 4 represent the differences between the mean dislocation density p and the natural misfit P (cf. eq. (8)). The curves consist of nearly linear parts in which the number of dislocations changes from zero onwards by one. It is seen that in the case of negative misfit (full line) the anharmonic chain undergoes considerable large strains. Also the mean strain starts changing sign at larger absolute values of P < 0. 3.4. Saw-tooth behauiour of the re.Cdualstrain Vincent [ZO]and Takayanagi et al. [Zl] observed a saw-tooth variation of the residual strain in Sn particles epitaxially grown on (001) SnTe. This system is particularly convenient for such kind of investigations because the lattice misfit is practically negligible in one direction ( + 0.58%) and large enough in absolute value ( - 8.5%) in the other direction. Thus, the 1 D theoretical model could be directly applied. However, both papers 120,211 appear to some extent controversial in their interpretation of the observed phenomenon. So Vincent [20] observes a clear saw-tooth behaviour at elevated temperatures in the range between 140 and 200°C. Takayanagi et al. [21] establish a less pronounced saw-tooth variation of the mean strain at lower temperatures arguing that the mechanism is not a dislocational one. In our opinion, two different mechanisms leading to saw-tooth variation of the mean strain and valid respectively at high and low temperatures follow from the same theoretical model. On the one hand, Jesser and van der Merwe [16,17] proposed an explanation of the phenomenon comprising a successive introduction of misfit dislocations at the free ends of the overgrowth islands when their size exceeds a critical value at which a state with one more dislocation becomes energetically favoured. Thus the first dislocation (first tooth) will be introduced when the chain length exceeds the critical value N,, that is, one goes beyond the critical limit of P, which is N-dependent at small N, so that the chain contains one dislocation in the ground state 122,233. This mechanism should be valid at high enough temperatures so that the chain can acquire sufficient thermal energy in order to overcome the activation barrier for an introduction of a new dislocation [4,5]. On the other hand, in the case studied’ by Takayanagi et al. [21], an alternative mechanism based on the spontaneous barrierless generation of misfit dislocations at the free ends should take place. Such a mechanism can occur at low temperatures but the necessary condition is that the natural misfit should be greater (in absolute value) than the metastability limit P,,,,. The first dislocation will emerge when the size of the chain exceeds the critical size N,.,.,, [22,23] which is larger than N, and beyond which the state without dislocation can no longer exist. N,, is determined by the condition Z, = & 0.5 [22,23]. Below we consider the saw-tooth variation of the mean strain with growing size

b

N-

C

Fig.

5. Saw-tooth

the chain &24%. positive negative positive

variation

length, The

sign

ones misfit; misfit.

of the residual

N, at three of the

(W=

strain

different

strain

values

at negative

misfits

1, (I = 3, a = 2, fi = 6 and

half-open

circles,

harmonic

of the ground

of the natural is reversed I, = 7.35).

approximation;

state misfit,

C (high

for easier Solid open

temperature

P: (a) + 12.5%; circles. circles.

case) with

(b)

comparison

k 16%; (c)

with

that

at

anharmonic

case

with

anharmonic

case

with

N for both the high temperature (HT) and the low temperature (LT) mechanisms outlined above. Fig. 5 represents the variation of the mean strain with the number of atoms in the case of HT mechanism (mean strain of the lowest energy) at three different values of the natural misfit: (a) 12.5%, (b) 16% and (c) 24%. Filled circles denote the mean strain in the case of negative misfit. The sign of the strain is reversed for better comparison with the opposite case of positive misfit which is given by empty circles. The reference harmonic approximation is also presented by semifilled circles. Here, as throughout in this paper, (Y= 2, p = 6, W = 1 and a = 3. The following points can be made: (i) The mean strain splits for positive and negative misfits into two curves above and below the harmonic one. (ii) The amplitude of the strain variation is considerably larger in the case of negative misfits. (iii) The values of the mean strain at positive misfits are very near to those of the harmonic approximation. (iv) A change of sign is always observed except for the case of - 12.5% which is slightly greater in absolute value than the corresponding stability limit P;’ = - 12.2%. (v) The larger the misfit in absolute value the smaller the amplitudes of the oscillations of the mean strain irrespective of the sign of the misfit. (vi) The larger the misfit in absolute value the more symmetric the values of the mean strain around zero particularly in the case of positive misfit and in the harmonic approximation. The variation of the mean strain in the case of LT mechanism is shown in fig. 6 for the case of P = - 24% and CY= 2, /3 = 6 and P,;, = - 23.2% (filled circles). For comparison the mean strain of the ground state (the HT variation) is also given by empty circles. Points belonging to both curves are denoted by semi-filled circles . Very similar results (not shown in the paper) were obtained also in the anharmonic positive misfit case and for the harmonic approximation. The magnitude of the jumps in the mean strain versus chain size variation for positive and negative misfits as well as in the harmonic approximation is shown in fig. 7. From figs. 6 and 7 it can be concluded that: (i) The LT mechanism leads to much larger strains on the average than in the case of HT mechanism. (ii) The sign of the strain never change in the case of LT mechanism. (iii) The jumps in the variation of the mean strain in the case of HT mechanism for anharmonic positive misfit systems (fig. 7a) and in the harmonic approximation (fig. 7b) are larger than the LT jumps, while in the anharmonic negative misfit case HT and LT jumps practically coincide (fig. 7~). It is worth noting that in both experimental papers [20,21] no change of sign

LT lot Fig.

6. Saw-tooth

high

temperature

variation (HT)

of the residual

mechanism.

The

strain, figures

2: (0)

low temperature

denote

the number

(LT)

mechanism;

of the misfit

(0)

dislocations

(W=l,a=3,~~=2,/3=6,P=-24%andP;~=-23.2%).

a

-

C

b

I

2

4 NUMBER

Fig. 7. Jumps

of the mean strain resulting

(a) anharmonic

case with P = + 12.58,

I, = 7.35;

(c) anharmonic

6

10

6 OF

12

14

DISLOCATIONS

from the consecutive

introduction

a = 2 and B = 6; (b) harmonic

case with P = - 24%, OL= 2 and p = 6. (0)

of misfit dislocations:

case. with P = f 36% and

LT case, (0)

HT case.

of residual mean strain measured is observed. According to our present investigation this could suggest that a LT saw-tooth behaviour is established.

4. Concluding remarks Summarizing one could claim that in general allowing for anharmonicity in the adatom interaction results in splitting of all characteristic properties of the system, such as dislocation length, energy, mean dislocation density, mean strain etc., with respect to the sign of the natural misfit. In all cases this splitting is asymmetric so that the positive misfit values are nearer to the harmonic ones. Thus the theoretical results presented in this paper seem to give ground for some general predictions which could be added to those already made in part II[l]: (i) The adhesion of the thin films to the substrates is one of the most important parameters of technological interest. It follows from the considerations in section 3.1 that negative misfit overlayer have considerably smaller misfit energy (particularly at small absolute values of misfit) than such with positive sign of the misfit. Having in mind the relation p = /?a- E [24], where &, is the specific adhesion energy between overgrowth and substrate in absence of misfit and E denotes the misfit energy (eq. (3)) it is clear that for negative misfits the specific adhesion energy j3 between overgrowth and substrate will be larger than for positive misfits under equal other conditions. As a matter of fact, this is illuminating if one recalls figs. 1 and 2 in part II [l] which show that larger fraction of atoms is in registry with the substrate and that they sit deeper in the potential throughs when the misfit is negative rather than positive. (ii) The specific adhesion energy is in turn a fundamental parameter determining the equilibrium form of crystallites on a foreign substrate according to Wulffs theorem as generalized by Kaischew [25,26]. It follows from the above considerations that negative misfit and hence stronger adhesion lead to smaller value of the height-to-width ratio the crystallites being thus flatter than in the case of positive misfit. (iii) We emphasize further the significance of the sign of the misfit for the kinetics of heterogeneous nucleation. The stronger the adhesion the smaller the work for nucleus formation and hence the higher the nucleation rate [27-291. It can be concluded that the negative misfit favours the process of nucleation under equal other conditions so that the nucleation rate and the saturation nucleus density should be greater when the lattice misfit is negative in comparison with the case of positive misfit. (iv) Concerning the mechanism of growth of thin epitaxial films, it has already been shown that better adhesion favours layer growth rather than island growth [30-321. This tendency is manifested by the higher critical temperatures upto which layer growth should be observed [31,32]. The same is valid for the

from layer growth (Frankvan der Merwe mechanism) to the Stranski-Krastanov mechanism or layer-by-layer growth followed by islands. In this case, after the completion of one or several monolayers of the deposit which are stable up to the temperature of desorption. islands are observed to form at temperatures higher than the critical one [32]. The latter is a strong function of the adhesion between the islands and the underlying strained layers of the overgrowth, being higher in case of stronger adhesion. It is thus clear that negative misfit will lead to higher critical temperature for 2D-3D transition to occur in comparison with positive misfit. (v) If epitaxial films consisting of separate islands grown at low temperature (LT) are subsequently annealed at higher temperature (HT), so that the mean strain is relieved by introduction of new dislocations (see fig. 6) an abrupt change in the moire pattern should be observed thus reflecting the attainment of the equilibrium ground state by the system. The general assertions mentioned in this section appear to be gaining additional support by a recent study of the critical temperatures, TIC, of 2D condensation in monolayers of Ar, Kr and Xe adsorbed on lamellar halides [33] whereby a broad range of misfits between substrates and deposits has been investigated. In fig. 8 experimental data of Millot et al. [33] on T2,/T,, have been replotted in units of the misfit, P, as defined in the present paper. It has transition

-20

-10

0

10

20 MISFIT

(%)

30 -

Fig. 8. Experimentally measured dependence of the critical temperature TzC for 2D condensation of rare gases Ar (solid circles). Kr (open circles) and Xe (half-open circles) on single crystal substrates of lamellar halides in units of the bulk critical temperature T,, which is replotted in terms of the natural misfit P = (h ~ u)/c( from the original drawing in the paper of Millot. Larher and Tessier [33].

been argued by the authors 1331 and also shown theoretically 134,351 that a maximum of T,,/7;, could be expected in the vicinity of the best dimensional compatibility, i.e. at vanishing misfit. While confirming this general conclusion, fig. 8 also reveals a well pronounced asymmetry of T,,/7;, with respect to the sign of the misfit. At a given absolute value of the natural misfit, a considerably higher value of T2JT3, corresponds to a negative sign of the incompatibility than to a positive one. This fact is naturally explained in the spirit of the present model when anharmonicity is accounted for. And indeed, according to our results more atoms are in registry with the substrate periodicity and on the average the atoms sit deeper in the potential troughs when the misfit is negative rather than positive and the thermal energy required for a collective phenomenon, such as the 2D condensation, to take place should be larger (T& is higher). Altogether one could claim that, as our study has demonstrated, anharmonic effects in the overlayer may be held responsible for a number of facts. experimentally observed in epitaxy and 2D phase transitions on periodic substrates. Apparently, the problems of anharmonicity effects in incompatible systems in view of their principal importance may hardly be considered as comprehended an extension of the model to related topics and into more than 111 appears imminent.

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