고체상 에피텍시에서 실리콘-게르마늄 합금의

활성화 에너지의 조성에 대한 의존성

서갑양, 이홍희

서울대학교 화학공학과

Composition Dependence of Activation Energy in

Solid Phase Epitaxial Growth of Si1-xGex Alloys

K. Y. Suh , Hong H. Lee

Department of Chemical Engineering

Seoul National University

1.INTRODUCTION

Heterostructures of Si1-xGex have drawn considerable interest because of new capabilities that they can offer.

Typical experimental studies of the solid phase epitaxy involve depositing a film of Si1-xGex epilayer on a Si substrate, amorphizing from the free surface to a certain depth and then annealing so that the amorphized layer can regrow epitaxially on the underlying substrate.

Despite the significant advance made in the SPE, there still exists an anomaly with regard to the composition dependence of the activation energy of the growth. As the composition x in Si1-xGex increases, the activation energy goes through a maximum instead of showing a monotonic change from pure silicon to pure germanium.

In this paper, we present a theoretical model that can explain quantitatively the composition dependence in an a priori manner.

2.THEORETICAL MODEL

The activation energies of the SPE growth for both pure silicon and pure germanium are rather well established. The crystallization rate of pure silicon has been measured over a wide temperature range (500-650℃) and shown to be a thermally activated process characterized by an activation energy of 2.6-2.7eV. To the extent the experimental data are available for Ge, the activation energy shows a similar trend as for silicon but with an activation energy of 2.0-2.2eV in the temperature range of 300-450℃.

The recrystallization in SPE requires movement of atoms. The atoms near the amorphous-crystal interface diffuse into crystal vacancies. In the process, the diffusing atoms can encounter various types of atom configuration at the interface. The atom configuration may involve many atoms. However, the simplest are the ones involving two atoms shown in Fig.1(a). The two atom configurations that one diffusing Si atom encounters are the first three in the figure and those that one diffusing Ge atom encounters are the rest. Let Ei (i=1,2,...,6) be the diffusion activation energy for the six configurations numbered 1 through 6 from left to right so that E1, for instance, is the activation energy for silicon to diffuse into silicon (Si self-diffusion).

Any diffusing atom, whether it is silicon or germanium, can encounter any of the six configurations whatever the composition x may be. The probability that a diffusing atom encounters a certain configuration is given by the composition so that the probability for the first configuration in Fig.1(a) is (1-x)3, that for the second is 2x(1-x)2, and so on.

It follows then that the activation energy for the alloy Si1-xGex can be written as

[eqn: rm E_{Si_1-x G e_x}=k &[(1-x)^3~ E_1 +2(1-x)^2 x~ E_2 +(1-x)x^2~ E_3 rm#& rm+(1-x)^2 x~ E_4 +2(1-x)x^2~ E_5 +x^3~ E_6 ~]]     (1)

where k is a constant factor that accounts for the difference between bulk diffusion that is typical of diffusion experiments and the diffusion from near an interface to an interface crystal structure. According to theoretical treatments of the diamond lattice structure and experimental work, about two-thirds of the total activation energy is required for vacancy formation while remaining one third is expended as enthalpy of vacancy migration. If it is assumed that only vacancy formation is necessary for the diffusion from near the interface to the interface crystal structure, the value of k could be taken as 2/3, yielding

[eqn: rm E_{Si_1-x G e_x}= 2 over 3& [(1-x)^3~ E_1 +2(1-x)^2 x~ E_2 +(1-x)x^2~ E_3#& rm +(1-x)^2 x~ E_4 +2(1-x)x^2~ E_5 +x^3~ E_6 ~ ]]     (2)

There are a number of studies reported in the literature on the bulk diffusion of silicon in silicon(Si→Si), germanium in silicon(Ge→Si), silicon in germanium(Si→Ge), and germanium self-diffusion(Ge→Ge).

Hirvonen et al.7 used the (p,□) resonance broadening method, which allows the use of long annealing times and obtained a activation energy of 4.1eV in Si self diffusion. For silicon diffusion in germanium, the activation energy reported is 2.9eV6. The ranges of other system6 are narrow such that an average value may be used for each: E4(Ge→Si)=5.0eV and E6(Ge→Ge)=3.05eV. In the absence of detailed information for E2, it may be taken to be the same as E4 since both involve the same atom configuration. By the same reasoning, E5 may be set equal to E3. It is noted in this regard that E2 and E5 cannot be determined by the  usual experimental technique. Then, Eq (2) reduces to

[eqn: rm E_{Si_1-x G e_x}= 2 over 3 [(1-x)^3~ E_1 +3(1-x)x^2~ E_3 +3 (1-x)^2 x~ E_4 +x^3~ E_6 ~]]        (3)

where E1=4.1eV, E3=2.9eV, E4=5.0eV and E6=3.05eV.

Since three atoms are involved, the model may be called "three-body model", for which the activation energy is given by

[eqn: rm E_{Si_1-x G e_x}=(1-x)^3~ 2.73 +3(1-x)x^2~ 1.93 +3 (1-x)^2 x~ 3.33 +x^3~ 2.03]         (4)

○     ○     ●     ○     ○     ●             ○        ○        ○        ●

○ →  ○ →  ○ →  ● →  ● →  ● →          ○  ○    ●  ●    ○  ●    ●  ●

○     ●     ●     ○     ●     ●

E1     E2     E3     E4     E5     E6            4.1eV     2.9eV     5.0eV     3.05e

(a)                                              (b)

○ : silicon   ● : germanium

Fig.1 (a) Six configurations considering two neighboring atoms and one diffusing atom. E1 through E6 are activation energies for each configuration. (b) Three-body model supposing that only vacancy formation energy is needed for SPE. Activation energies are given  at the bottom.

As indicated at the outset, the simplest possible case is that shown in Fig. 1(b) involving three atoms. If four atom configurations are considered, the "four-body model" shown in Fig. 2(a) results. It follows then that the four-body model yields,

[eqn: rm E_{Si_1-x G e_x}=&(1-x)^4~ 2.73 +4(1-x)x^3~ 1.93 +6 (1-x)^2 x^2~ 2.63 rm+ 4 (1-x)^3 x~ 3.33 +x^4~ 2.03]  (5)

Perhaps the most logical choice is the five-body model shown in Fig. 2(b) for the diamond structure being considered. The five-body represents the five atoms forming the tetrahedron, which is one-eighth of the unit cell of the diamond crystal structure.    For the five-body model, the activation energy is given by

[eqn: rm E_{Si_1-x G e_x}=&(1-x)^5~ 2.73 +5(1-x)x^4~ 1.93 +10 (1-x)^3 x^2~ 2.86##& rm +10 (1-x)^2 x^3~ 2.4 + 5 (1-x)^4 x~ 3.33 +x^5~ 2.03]            (6)

○       ○       ○       ○       ●

○  ○   ○  ○   ○  ●   ●  ●   ●  ●

○       ●       ●       ●       ●

4.1eV    2..9eV   3.95eV   5.0eV    3.05eV

(a)

○        ○        ○         ○         ○         ●

○        ○        ○         ○         ●         ●

○    ○  ○    ○  ○    ●   ●    ●   ●    ●   ●    ●

○        ●        ●         ●         ●         ●

4.1eV     5.0eV     4.3eV      3.6eV      2.9eV      3.05eV

(b)

Fig.2 (a) Four-body model with activation energy at its bottom. Activation energy for the middle case is simply taken as the average of the two adjacent ones or 3.95=(2.9+5.0)/2. (b) Five-body model. Weighted average are used for activation energies for the two middle configurations or 2.86=(2×5.0+2.9)/3, 2.4=(5.0+2×2.9)/3.

3.COMPARISON WITH EXPERIMENT AND DISCUSSION

The theoretical model results can be compared with the experimental data reported in the literature.1-5 The comparison is shown in Fig.3 for both the relaxed alloys (filled symbols) and the strained alloys (unfilled symbols).

The comparison is seen to be satisfactory. As can be seen from Fig.3, the peak gets higher and the shape becomes somewhat broader as one goes from the three-body model(solid curve) to the five-body model(dotted curve).

The multi-body model can easily be extended to any other binary alloys A1-xBx in the following general form:

[eqn: rm E_{A_1-x B_x} =k sum from i f_i (x) E_i]         (7)

where fi(x) is the probability of encounter with an i type of atom configuration, and Ei is the corresponding activation energy for the bulk diffusion. The most logical choice of the number of atoms involved is that corresponding to the sub-unit of the unit cell of the alloy crystal structure.

Composition of Ge (x in Si1-xGex)

Fig.3 Comparison of multi-body model with experimental data. The data for relaxed alloys are taken from Ref.1 (filled circles), Ref.2 (filled diamonds), and Ref.3 (filled squares). The data for some strained layers are also shown (unfilled triangles from Ref.4, unfilled diamonds from Ref.5). The curves show the theoretical activation energy as determined by the multi-body model: three-body model (solid), four-body model (dashed) and five-body model (dotted).

References

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