The kinetics of phase formation in an ultra-thin nanoscale layer

V. M. Apalkov; Y. I. Boyko; V. V. Slezov

doi:10.1515/ijmr-2003-0210

Article

The kinetics of phase formation in an ultra-thin nanoscale layer

V. M. Apalkov , Y. I. Boyko and V. V. Slezov

Published/Copyright: February 7, 2022

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From the journal International Journal of Materials Research Volume 94 Issue 10

Abstract

The peculiarities of the kinetics of phase formation in an ultra-thin (≈ 10 nm) crystalline layer, which is formed from two components on the surface of a substrate, have been studied theoretically. It has been shown that by varying the conditions during the growth of the layer (the temperature, the vapour pressure, the type of substrate) it is possible to control the phase composition of the layer.

Keywords: Supersaturation; Substrate; Nucleation; Layer; Kinetics; Phase composition

Dedicated to Professor Dr. Dr. h. c. Herbert Gleiter on the occasion of his 65th birthday

Prof. Dr. Yuriy Boyko Forschungszentrum Karlsruhe Institut für Nanotechnologie D-76021 Karlsruhe, Germany Tel.: +49 7247 82 6380

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Appendix

From simple geometrical analysis we find that for an island in the form of a spherical segment of radius R₀ (Fig. 1) with a contact angle ϑ. The radius of the circular area of the substrate surface covered by the island:

(A.1) R=R0sinϑ

The volume of the spherical segment (island):

(A.2) V0=π3R3[ 2(1−cosϑ)sin3ϑ−cotϑ ]=π3R3κ

The area of the interface island/vapour phase:

(A.3) SF=πR221+cosϑ

The area of the interface island/substrate:

(A.4) SS=πR2

We define the effective coefficient of surface tension γ₁ of the island by the expression:

(A.5) ΔψS=γ1Rω

where Δψ_S is the change of surface energy of the island when the volume of the island is increased by the volume of one structural unit x:

(A.6) ΔV0=πκR2ΔR=ω

The subsequent change of the radius of the base of the island R is the given by:

(A.7) ΔR=ωπκR2

The change of the surface energy of the island/substrate system is then:

(A.8) ΔψS=ΔSFγF+ΔSSγS−ΔSSγ0 =ΔR2πRγF21+cosϑ+ΔR2πR(γS−γ0)

where we take into account that if the area of the interface between the island/substrate is changed by ΔS_S, then the area of the interface between the substrate/vapour phase is changed by -ΔS_S;

γ_F is the coefficient of surface tension for the island/vapour interface;

γ₀ is the coefficient of surface tension for the substrate/ vapour interface;

γ_S is the coefficient of surface tension for the island/substrate interface.

From expressions (A.7) and (A.8) we derive the effective coefficient of surface tension:

(A.9) γ1=2γFsinϑ

where we have used the relation

(A.10) γFcosϑ+γS=γ0

Taking into account (A.10) we can rewrite expression (A.9) in the form:

(A.11) γ1=2γF2−(γ0−γS)2

The effective coefficient of surface tension enters into the expression for the constant of chemical reaction between adatoms on the surface of islands with base radius R:

K(R1)=Kexp(ΔψSkT)≈K+KΔψSkT=K+Kγ1ωkTR

To find the probability of the creation of an island of base radius R it is necessary to find the energy required for the creation of such an island (Fig.1). This energy consists of both volume and surface terms:

(A.12) Ψ=ΨV+ΨS =NABμ˜AB−NAμ˜A−NBμ˜B+γSSS+γFSF−γ0S0

where N_AB is the number of the structural elements in the island, μ˜AB is the chemical potential of a structural element in the island, N_A is the number of elements A in the island; μ˜A is the chemical potential of atoms of element A outside the island; N_B is the number of atoms of element B in the island; μ˜B is the chemical potential of atoms of element B outside the island.

If the island is a compound of stoichiometric composition AvABvB then:

(A.13) NA=NABvA

(A.14) NB=NABvB

The number N _AB is determined by the total volume of the island:

(A.15) NAB=V0ω

Substituting (A.2)– (A.4) and (A.13) –(A.15) into (A–12), we find:

(A.16) Ψ=−π3ωR3κkTlnη+πR2γ12κ

where η=uAvAuBvBK.

The critical radius of the island is determined from the condition that the expression (A.16) takes its minimal value for the critical radius:

(A.17) Rcr=γ1ωkTlnη

Substituting expression (A.17) into (A.16), we derive the energy of a critical island – the minimal energy needed for the creation of an island of phase AvABvB:

(A.18) Ψcr=π6κγ3ω2(kT)2ln2η

This energy determines the probability of the creation of an island with the new phase:

(A.19) W1~exp(−ΨcrkT)=exp(−π6κγ3ω2(kT)3ln2η)

Received: 2003-05-27

Published Online: 2022-02-07

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Articles in the same Issue

https://doi.org/10.1515/ijmr-2003-0210

Keywords for this article

Supersaturation; Substrate; Nucleation; Layer; Kinetics; Phase composition