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On the stress driven instability of thin solid films
MECHANICS RESEARCH COMMUNICATIONS Vol. 20(3), 223-225, 1993. 0093-6413/93 $6.00 + .00 Copyright (c) 1993
Printed in the U.S.A. Pergamon Press Ltd.
ON THE STRESS DRIVEN INSTABILITY OF THIN SOLID FILMS
Michael Grinfeld Department of Mathematics, Rutgers University, New Brunswick, NJ 08903
(Received 19 August 1992; acceptedfor print 9 October 1992)
Introduction
It was first announced in [1] and demonstrated theoretically in [2]-[4] that in the absence of surface tension a flat boundary of non-hydrostatically stressed solid of any symmetry is always unstable with respect to "mass rearrangement w. The physical mechanisms of the rearrangement can be different, as, for instance, a)melting-freezing or vaporizationsublimation processes at liquid-solid or vapor-solid phase boundaries, b)surface diffusion of particles along free or interfacial boundaries, c)adsorption-desorbtion of the atoms in epitaxial crystal growth, etc... This universal instability delivers new insights and provides new opportunities in different branches of materials science a part of which is discussed in [5]. In particular, it has already allowed to predict stress driven corrugations of thin solid films of 4He and and to explain the phenomenon of dislocation-free Stranski-Krastanov pattern of growth of epitaxial films of GaAs on Si substrates (see Ref. [6]-[8]). The key formula for the critical thickness of such films was announced in [9] and published in [4]. [5], [10]. In the above mentioned studies we relied on 2D theory of elasticity, and the 3D approach is becoming urgent necessity in view of (i)typical misfit stresses in the problems of epitaxy and (ii)wide opportunities appeared in the experiments with solid 4He films. In this short note we announce one essential preliminary result: the dispersion relation of the rate of amplification/decay of different Fourier components of the surface corrugations of a prestressed isotropic elastic film attached to a rigid substrate. This relation allows one to study morphological patterns of the unstable corrugations (islands) in prestressed solid films and those possible symmetry changes in the free surface morphology which accompany the increase/decrease of the film thickness.
Methods and Results
Let us consider a thin solid film of a thickness H which is uniformly stressed and attached to a rigid substrete. The uniform stresses can be produced a)by the misfit in the lattice 223
224
M. GRINFELD
parameters of the epitaxial film and the substrate or b)by the temperature changes in t w o phase systems containing 4He. We use the notation T ab for the in-plane stresses (since the upper boundary of the films is traction-free this tensor completely describes the stress state of the film w i t h a flat free boundary). We assume the Einstein summation convention and denote by x a the in-plane coordinate axes (the indices a,b,c.., take on the values 1,2). The uniformly stressed film accumulates an amount of energy Ereg consisting of t w o parts: the bulk (elastic) and the surface energy (the latter is assumed proportional
w i t h a surface
tension coefficient 0 to the area of the free boundary in the unstressed
(reference)
configuration). The same film w i t h corrugated free boundary accumulates another amount of energy Eirreg . In the case of small surface corrugations c A ( x ") (where C < < 1, A -- H) of the isotropic film (with the shear module /~ and the Poisson ratio ~ ) the difference of the energies E
irreg
- E
reg
(which determines the stability of the flat film w i t h respect to
spontaneous corrugations of its free surface) can be computed explicitly; Ref. [4] contains this computation for the 2D case. The 3D generalization leads to the following formula:
EirreQ - Ereg = - 2,~ 271"2l d k l d k 2 G(k,h)l kl-2x(k)X*(-k) R2
(1)
where G(k,h) is the following key function: (2)
G(k,h) == - O Ik[ 4 + ~ ( k l 3 {(rabeaeb) 2
(1 - •)[h + (3 - 4p)sh(h) ch(h)] 4 (! - v) 2 + h 2 + (3 - 4p) sh2h
+ITabq.e,"2 sh(h) }
In (1), (2) x ( k a) is the Fourier component of the surface corrugation A ( x a) w i t h the wavevector ka; w e use the notation I k l f o r the module of k a and notation ea, q, for the unit inplane vectors parallel and orthogonal k a, respectively; rab = Tab/#
and h ~. H l k l a r e the
dimensionless misfit stresses and thickness, respectively. The physical meaning of the function G(k,h) is rather transparent from the viewpoint of the theory of surface diffusion in prestressed solids, and it will be discussed in detail elsewhere. Indeed, assuming the isotropy of the surface diffusion and introducing the surface diffusion coefficient D s > 0 one can find out that the product DsG(k,h) is nothing more than the rate of
exponential
growth/decay
of
the
corresponding
Fourier-component
of
surface
corrugation. Formulas (1), (2) lead, in particular, to the following value of the critical film thickness
INSTABILITY OF THIN SOLID FILMS
/~g
Hcrit = .r2max
225
(3)
where Tmax is the greater of two principal in-plane misfit stresses. Formula (3) generalizes a similar formula for critical thickness of [4], [5], [9], [10] established originally in the framework of 2D approach. Investigation of the formulas (1), (2) shows that at H
l.M.A.Grinfeld, Sov. Phys. Dokl. 31, 831 (1986). 2.M.A.Gdnfeld, Fluid Dyn. 22, 169 (1987). 3.M.A.Gdnfeld, PMM, 51,489 (1987). 4.M.A.Gdnfeld, Thermodynamic Methods in the Theory of Heterogeneous Systems, pp. 325, 364. Longman, Sussex (1991). ("Interaction of Mechanics and Mathematics Sedes"). 5.M.A.Gdnfeld, The Stress Ddven Instabilities in Crystals: Mathematical Models and Physical Manifestations. IMA Prepdnt Sedes #819, June (1991). (to appear in: J. Nonlinear Science). 6.D.J.Eaglesham and M.Cerullo, Phys. Rev. Lett., 64, 1943 (1990). 7.F.K.LeGoues, M.Copel, R.M.Tromp, Phys. Rev. B, 42, 11690 (1990). 8.R.Todi and C. Balibar, Helium Crystals Under Stress: the Grinfeld Instability. Intl. Symp. "Quantum Fluids and Solids", Penn. State, (1992) (to appear in: J. Low Temp. Phys.). 9.M.A.Gdnfeld, Equilibrium Shape and Instabilities of Deformable Elastic Crystals. Lecture given in Heirot-Watt University, Edinburgh (1989). 10.M.A.Gdnfeld, In 1991 MRS Fall Meeting Proc. 237 "Interface Dynamics and Growth"; 239, "Thin Films: Stresses and Mechanical Properties. III (to appear).
Abbreviated Paper - For further information, please, contact the author.
Printed in the U.S.A. Pergamon Press Ltd.
ON THE STRESS DRIVEN INSTABILITY OF THIN SOLID FILMS
Michael Grinfeld Department of Mathematics, Rutgers University, New Brunswick, NJ 08903
(Received 19 August 1992; acceptedfor print 9 October 1992)
Introduction
It was first announced in [1] and demonstrated theoretically in [2]-[4] that in the absence of surface tension a flat boundary of non-hydrostatically stressed solid of any symmetry is always unstable with respect to "mass rearrangement w. The physical mechanisms of the rearrangement can be different, as, for instance, a)melting-freezing or vaporizationsublimation processes at liquid-solid or vapor-solid phase boundaries, b)surface diffusion of particles along free or interfacial boundaries, c)adsorption-desorbtion of the atoms in epitaxial crystal growth, etc... This universal instability delivers new insights and provides new opportunities in different branches of materials science a part of which is discussed in [5]. In particular, it has already allowed to predict stress driven corrugations of thin solid films of 4He and and to explain the phenomenon of dislocation-free Stranski-Krastanov pattern of growth of epitaxial films of GaAs on Si substrates (see Ref. [6]-[8]). The key formula for the critical thickness of such films was announced in [9] and published in [4]. [5], [10]. In the above mentioned studies we relied on 2D theory of elasticity, and the 3D approach is becoming urgent necessity in view of (i)typical misfit stresses in the problems of epitaxy and (ii)wide opportunities appeared in the experiments with solid 4He films. In this short note we announce one essential preliminary result: the dispersion relation of the rate of amplification/decay of different Fourier components of the surface corrugations of a prestressed isotropic elastic film attached to a rigid substrate. This relation allows one to study morphological patterns of the unstable corrugations (islands) in prestressed solid films and those possible symmetry changes in the free surface morphology which accompany the increase/decrease of the film thickness.
Methods and Results
Let us consider a thin solid film of a thickness H which is uniformly stressed and attached to a rigid substrete. The uniform stresses can be produced a)by the misfit in the lattice 223
224
M. GRINFELD
parameters of the epitaxial film and the substrate or b)by the temperature changes in t w o phase systems containing 4He. We use the notation T ab for the in-plane stresses (since the upper boundary of the films is traction-free this tensor completely describes the stress state of the film w i t h a flat free boundary). We assume the Einstein summation convention and denote by x a the in-plane coordinate axes (the indices a,b,c.., take on the values 1,2). The uniformly stressed film accumulates an amount of energy Ereg consisting of t w o parts: the bulk (elastic) and the surface energy (the latter is assumed proportional
w i t h a surface
tension coefficient 0 to the area of the free boundary in the unstressed
(reference)
configuration). The same film w i t h corrugated free boundary accumulates another amount of energy Eirreg . In the case of small surface corrugations c A ( x ") (where C < < 1, A -- H) of the isotropic film (with the shear module /~ and the Poisson ratio ~ ) the difference of the energies E
irreg
- E
reg
(which determines the stability of the flat film w i t h respect to
spontaneous corrugations of its free surface) can be computed explicitly; Ref. [4] contains this computation for the 2D case. The 3D generalization leads to the following formula:
EirreQ - Ereg = - 2,~ 271"2l d k l d k 2 G(k,h)l kl-2x(k)X*(-k) R2
(1)
where G(k,h) is the following key function: (2)
G(k,h) == - O Ik[ 4 + ~ ( k l 3 {(rabeaeb) 2
(1 - •)[h + (3 - 4p)sh(h) ch(h)] 4 (! - v) 2 + h 2 + (3 - 4p) sh2h
+ITabq.e,"2 sh(h) }
In (1), (2) x ( k a) is the Fourier component of the surface corrugation A ( x a) w i t h the wavevector ka; w e use the notation I k l f o r the module of k a and notation ea, q, for the unit inplane vectors parallel and orthogonal k a, respectively; rab = Tab/#
and h ~. H l k l a r e the
dimensionless misfit stresses and thickness, respectively. The physical meaning of the function G(k,h) is rather transparent from the viewpoint of the theory of surface diffusion in prestressed solids, and it will be discussed in detail elsewhere. Indeed, assuming the isotropy of the surface diffusion and introducing the surface diffusion coefficient D s > 0 one can find out that the product DsG(k,h) is nothing more than the rate of
exponential
growth/decay
of
the
corresponding
Fourier-component
of
surface
corrugation. Formulas (1), (2) lead, in particular, to the following value of the critical film thickness
INSTABILITY OF THIN SOLID FILMS
/~g
Hcrit = .r2max
225
(3)
where Tmax is the greater of two principal in-plane misfit stresses. Formula (3) generalizes a similar formula for critical thickness of [4], [5], [9], [10] established originally in the framework of 2D approach. Investigation of the formulas (1), (2) shows that at H
l.M.A.Grinfeld, Sov. Phys. Dokl. 31, 831 (1986). 2.M.A.Gdnfeld, Fluid Dyn. 22, 169 (1987). 3.M.A.Gdnfeld, PMM, 51,489 (1987). 4.M.A.Gdnfeld, Thermodynamic Methods in the Theory of Heterogeneous Systems, pp. 325, 364. Longman, Sussex (1991). ("Interaction of Mechanics and Mathematics Sedes"). 5.M.A.Gdnfeld, The Stress Ddven Instabilities in Crystals: Mathematical Models and Physical Manifestations. IMA Prepdnt Sedes #819, June (1991). (to appear in: J. Nonlinear Science). 6.D.J.Eaglesham and M.Cerullo, Phys. Rev. Lett., 64, 1943 (1990). 7.F.K.LeGoues, M.Copel, R.M.Tromp, Phys. Rev. B, 42, 11690 (1990). 8.R.Todi and C. Balibar, Helium Crystals Under Stress: the Grinfeld Instability. Intl. Symp. "Quantum Fluids and Solids", Penn. State, (1992) (to appear in: J. Low Temp. Phys.). 9.M.A.Gdnfeld, Equilibrium Shape and Instabilities of Deformable Elastic Crystals. Lecture given in Heirot-Watt University, Edinburgh (1989). 10.M.A.Gdnfeld, In 1991 MRS Fall Meeting Proc. 237 "Interface Dynamics and Growth"; 239, "Thin Films: Stresses and Mechanical Properties. III (to appear).
Abbreviated Paper - For further information, please, contact the author.