ISSN 1063-7826, Semiconductors, 2007, Vol. 41, No. 6, pp. 621–624. © Pleiades Publishing, Ltd., 2007. Original Russian Text © S.Yu. Davydov, A.A. Lebedev, 2007, published in Fizika i Tekhnika Poluprovodnikov, 2007, Vol. 41, No. 6, pp. 641–644.
REVIEW
Vacancy Kinetics in Heteropolytype Epitaxy of SiC S. Yu. Davydov^ and A. A. Lebedev Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia ^e-mail:
[email protected] Submitted October 26, 2006; accepted for publication November 1, 2006
Abstract—A model of the transformation of SiC polytypes in the course of growth of an epitaxial layer is suggested. The model is based on the variation with time of the concentrations of carbon and silicon vacancies in the transition layer. A relationship between the lifetimes of these vacancies is obtained in terms of the model. PACS numbers: 61.50.Nw, 61.66.Fn, 61.72.Cc, 61.72.Ji, 68.55.Jk DOI: 10.1134/S1063782607060012
1. It is known that silicon carbide polytypes have very different band gaps, despite the configurations of the two first coordination spheres of these polytypes being identical. Such a combination of the electronic and crystallographic structures makes this material highly promising for fabrication of a wide variety of heterostructures. Different SiC polytypes can be conveniently characterized by the degree of hexagonality, γ [1], defined as γ = n h / ( n h + n k ),
(1)
where nh and nk are the numbers of atoms occupying hexagonal and cubic sites, respectively. Following the authors of [2–4], we regard SiC polytypes as phases of various stoichiometric compositions, which differ in the ratio of the numbers of silicon and carbon atoms, [Si]/[C], to which correspond the different concentrations of NSi and NC vacancies in these polytypes. The characteristics of some of the most widely occurring polytypes are listed in Table 1. However, there is no commonly accepted theory that would explain the known experimental data on heteropolytype epitaxy of SiC. One of the most important issues is to determine the conditions under which the polytype of a growing layer is transformed. In [5], we made an attempt to explain some aspects of heteropolytype epitaxy in terms of the model that takes into account the kinetics of variation in the concentration of
carbon vacancies. In this study, we consider the kinetics of vacancies in both sublattices and assume that the transition from the starting polytype to the final one occurs with the concentrations of carbon and silicon vacancies simultaneously attaining the values listed in Table 1. It is noteworthy that the decrease in the X-ray density of the polytypes, ρ, observed as the vacancy concentration N = NSi + NC increases, indicates that we have Schottky defects, in which case a vacancy is formed because of the emergence of an atom from a lattice site within the crystal to a site at its surface [6]. 2. In heteropolytype epitaxy, technological conditions are provided in the growth zone for formation of an epitaxial layer having a polytype different from that of the substrate. Let us consider, for the sake of definiteness, a process in which the degree of hexagonality, γ, of the layer grown increases with respect to that of the substrate. According to the aforesaid, such a process must be accompanied by a decrease in the concentration of carbon vacancies, NC, and an increase in the concentration of silicon vacancies, NSi. The hexagonality will change during a certain interval of time, tT, rather than instantaneously. Let us assume, for simplicity, that the polytype of the substrate does not change during the time tT. In this case, a transition layer of thickness LT is formed on the substrate. In this layer, the concentration of carbon vacancies, NC, will be lower than the concen-
Table 1. Parameters of silicon carbide polytypes Polytype γ [Si]/[C] ρ, g cm–3 NC, 1020 cm–3 NSi, 1020 cm–3
4H
27R
15R
6H
8H
3C
0.50 1.001 3.2163 7.3 7.3
0.44 1.008 3.2162 10 6
0.40 1.012 3.2162 12 5
0.33 1.022 3.2160 14 4
0.25 1.029 3.2159 16.6 3
0 1.046 3.2154 23.5 2
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Table 2. τSi/τC ratios in transition from the initial SiC polytype with a lower degree of hexagonality to final polytypes with a higher degree of hexagonality Final → Initial ↓
8H
6H
15R
27R
4H
3C 8H 6H 15R 27R
0.86 0.59 0.69 1.00 1.60
0.75 0.64 0.83 1.31 –
0.73 0.73 1.08 – –
0.78 0.92 – – –
0.90 – – – –
Table 3. τSi/τC ratios in transition from the initial SiC polytype with a higher degree of hexagonality to final polytypes with a lower degree of hexagonality Final → Initial ↓
27R
15R
6H
8H
3C
4H 27R 15R 6H 8H
1.60 – – – –
1.31 1.00 – – –
1.08 0.83 0.69 – –
0.92 0.73 0.64 0.59 –
0.90 0.78 0.73 0.75 0.86
Hence we have a relationship between the vacancy lifetimes in the silicon and carbon sublattices: ln ( N C /N C ) -. τ Si /τ C = --------------------------L S ln ( N Si /N Si ) S
L
(5)
Expression (5) means that, for a transition from one polytype to another, the vacancy lifetimes should be related in a definite way. It is noteworthy that expression (5) is also valid for processes with a decrease in γ. Using the data in Table 1, we calculated the τSi/τC ratio for a growth process leading to an increase in the degree of hexagonality of the growing polytype, compared with the starting polytype, (Table 2), and for the opposite process, in which γ of the growing polytype is lower than that of the starting polytype (Table 3). 3. The situation described by Eqs. (2) corresponds to a process in which the concentrations of carbon and silicon vacancies vary independently of each other. Based on Table 1, we can, however, assume that there exists a relationship between the variation rates of the vacancies, dNC/dt and dNSi/dt. Let us consider again a process of heteropolytype epitaxy, in which the degree of hexagonality, γ, of the layer grown increases. Then we can write, instead of (2), d N C /dt = – N C /τ C – g Si N Si ,
S NC
tration corresponding to the polytype of the substrate, and, accordingly, the concentration of silicon S vacancies, NSi, will exceed N Si . Immediately after NC L
decreases to a certain concentration N C and NSi L
increases to N Si , the polytype of the growing layer will change. Let us assume that the variation in NC and NSi with time t obeys the equations d N C /dt = – N C /τ C , d N Si /dt = N Si /τ Si ,
where gSi and gC are coefficients. The first of Eqs. (6) takes into account the fact that an increase in the concentration of silicon vacancies automatically leads to a decrease in the concentration of carbon vacancies. By contrast, the second of these equations takes into account the fact that an increase in the concentration of silicon vacancies corresponds to a decrease in the concentration of carbon vacancies. It can be readily shown (see, e.g., [7]) that solutions to the system of Eqs. (6) can be represented as 1 1 S N C ( t ) = N C exp – ⎛ ----- + -----⎞ t , ⎝ τ C τ*⎠
(2)
from which it follows that N C ( t ) = N C exp ( – t/τ C ),
N Si ( t ) =
S
N Si ( t ) = N Si exp ( – t/τ Si ), S
(3)
where τC and τSi are the lifetimes of carbon and silicon vacancies in the transition layer. If the process occurs with a decrease in γ, the subscripts in (2) and (3) should be interchanged. For the interval of time tT, in which the transformation of the growing layer from the initial polytype (S) to that being grown (L) occurs, we have L –1
t T = τ C ln ( N C /N C ) , S
–1
t T = τ Si ln ( N Si /N Si ) . L
S
(4)
(6)
d N Si /dt = – N Si /τ Si – g C N C ,
S N Si exp
(7)
1 1 – ⎛ ------ + -----⎞ t , ⎝ τ Si τ*⎠
where g C g Si τ C τ Si 1 -. ----- = ----------------------τ C + τ Si τ*
(8)
Then we obtain, instead of relation (5), the equation ln ( N C /N C ) 1 + ( τ C /τ* ) - ---------------------------- . τ Si /τ C = --------------------------L S ln ( N Si /N Si ) 1 + ( τ Si /τ* ) S
L
(9)
It is reasonable to assume that it is the sign of the first term on the right-hand side of Eqs. (6) that determines the direction of the process, with the second term only introducing a minor correction. In other words, it SEMICONDUCTORS
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VACANCY KINETICS IN HETEROPOLYTYPE EPITAXY OF SiC
is assumed that the change in the concentration of, say, carbon vacancies is determined just by their concentration at a given instant of time, whereas the presence of silicon vacancies has a weaker effect. Then it can be assumed that gCgSiτCτSi 1, and, consequently, (τ*)–1 τ C , τ Si . On this assumption, the results listed in Tables 2 and 3 can be regarded as quite adequate estimates of the τSi/τC ratio. It should be emphasized, however, that the scatter of τSi/τC values will now be narrower than that in Tables 2 and 3. Indeed, the τSi/τC ratio will decrease somewhat, in accordance with (9), for the tabulated values corresponding to the inequality τSi/τC > 1, and will increase in the case of τSi/τC < 1 (see Appendix). 4. Let us now consider the transition layer [5], whose thickness is given by –1
–1
L T = Gt T ,
(10)
where G is the layer growth rate. The velocities of vacancy motion, νC, Si, are given by the expression ν C, Si =
D C, Si /τ C, Si ,
(11)
where DC, Si are the diffusion coefficients of carbon and silicon vacancies, respectively. It is noteworthy that, by definition of the lifetime of a vacancy in the transition layer, its diffusion length l C, Si =
D C, Si τ C, Si
(12)
should not exceed LT [5]. Based on (10), we can rewrite expressions (4) as –1
η C = Gτ C /L T = [ ln ( N C /N C ) ] , S
L
–1
η Si = Gτ Si /L T = [ ln ( N Si /N Si ) ] . L
S
(13)
Using the data of Table 1, we can readily demonstrate that, for all technological processes leading, e.g., to a structural transition from polytype 1 to polytype 2, the parameter ηC, Si should have the same value. It is noteworthy that the ratio of parameters (13), ηSi/ηC, is equal to τSi/τC, and, consequently, it is determined by expression (5). Let us rewrite (13) in the form (14)
An analysis of experimental data, made in [5] for carbon vacancies, demonstrated that ηC > 1 and, since lC < LT, G > νC. Then the parameter ηSi also exceeds unity in the case where τSi/τC > 1. We can conclude on the basis of the results presented in Tables 2 and 3 that the parameter ηSi > 1 in the transformations (6H 15R), (15R 8H, 6H), (27R 8H), (4H 27R, 15R, 6H), and (27R 15R). In other cases, ηSi may be either larger or smaller than unity. 5. The diffusion coefficient D ∝ exp(–Ed/kBT), where Ed is the activation energy of vacancy diffusion; Vol. 41
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T is temperature; kB is the Boltzmann constant; and τSi/τC = νC/νSi. Therefore, we have from (11) τ Si /τ C = exp ( – ∆E d /k B T ).
(15)
∆E d = E d ( V C ) – E d ( V Si )
(16)
Here,
is the difference of the activation energies of diffusion of carbon Ed(VC) and silicon Ed(VSi) vacancies in the initial polytype. According to the data obtained in [8] for the 3C- and 4H-SiC polytypes, ∆Ed > 0 and is equal to several tenths of an electronvolt for each of the polytypes, for both neutral and charged vacancies. The same conclusions follow from the results of [9] for vacancy migration in 3C-SiC, according to which the difference ∆Ed is also positive. Thus, the τSi/τC ratio must be smaller than unity. This condition is satisfied by the transitions (3C 8H, 6H, 15R, 27R, 4H) and (4H 8H, 3C), and, possibly, with account of the mutual influence of the diffusion of silicon and carbon vacancies, also by the transition (4H 6H). One more remark should be made. In accordance with the results of the calculations (see Tables 2 and 3), the τSi/τC ratio is on the order of unity. Then it follows from relation (15) that |∆Ed| ~ kBT. This condition is only satisfied by the result obtained in [9] for neutral vacancies in a cubic polytype: ∆Ed = 0.1 eV (which corresponds to T = 1160 K). In other cases, ∆Ed exceeds 0.5 eV in [8, 9]. It should, however, be emphasized that the values of Ed(VC) and Ed(VSi), reported in [8, 9], refer directly to the migration of the corresponding vacancies. At the same time, the lifetimes τC and τSi, introduced into Eqs. (2), phenomenologically describe all the processes leading to a change in the concentrations of carbon and silicon vacancies in the transition layer. It is not improbable that the existence of several alternative ways in which vacancies can leave the transition layer (or, by contrast, enter this layer), which does take place (see [8, 9]), can be taken into account by introducing effective barriers E d* , such that |∆ E d* | ~ kBT. APPENDIX
G l C, Si -. η C, Si = ----------- -------ν C, Si L T
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Introducing the designations ln ( N C /N C ) -, χ 0 = --------------------------L S ln ( N Si /N Si ) S
χ = τ Si /τ C ,
L
β = g C g Si τ C , (A.1) 2
we can rewrite Eq. (9) in the form βχ + χ + [ 1 – χ 0 ( 1 + β ) ]χ – χ 0 = 0. 3
2
(A.2)
Assuming that β 1, we find the first correction in the parameter β to the solution χ0 to Eq. (5), by setting χ = χ 0 + bβ.
(A.3)
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Then,
REFERENCES 1–χ b = χ 0 -------------0-. 1 + χ0
(A.4)
At χ0 < 1, the coefficient b > 0 and χ > χ0; at χ0 > 1 we have b < 0 and χ < χ0, which is exactly noted in the text. Let us now consider the opposite case of β 1 taking into account the fact that χ0 ~ 1. It can be readily shown that, in this case, χ ≈ χ 0 . Then, we again obtain from (A.2) that χ > χ0 for χ0 < 1 and χ < χ0 for χ0 > 1. Assuming that χ0 ~ 1 and β ~ 1 in (A.2), we obtain χ ~ 1. Thus, taking the correction into account results, in the limiting cases, in the situation where the τSi/τC ratio tends to unity. The study was supported by the Russian Foundation for Basic Research, project nos. 03-02-16054 and 04-02-16632.
1. Silicon Carbide-1968, Ed. by H. K. Henisch and R. Roy (Pergamon, Oxford, 1968; Mir, Moscow, 1972). 2. Yu. A. Vodakov, G. A. Lomakina, and E. N. Mokhov, Fiz. Tverd. Tela (Leningrad) 24, 1377 (1982) [Sov. Phys. Solid State 24, 780 (1982)]. 3. N. D. Sorokin, Yu. M. Tairov, V. F. Tsvetkov, and M. A. Chernov, Kristallografiya 28, 910 (1983) [Sov. Phys. Crystallogr. 28, 539 (1983)]. 4. A. A. Lebedev, Fiz. Tekh. Poluprovodn. (St. Petersburg) 33, 769 (1999) [Semiconductors 33, 707 (1999)]. 5. A. A. Lebedev and S. Yu. Davydov, Fiz. Tekh. Poluprovodn. (St. Petersburg) 39, 296 (2005) [Semiconductors 39, 277 (2005)]. 6. C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976; Nauka, Moscow, 1978). 7. É. Kamke, Gewöhnliche Differentialgleichungen (Akademie, Leipzig, 1959; Nauka, Moscow, 1971). 8. E. Rauls, Th. Frauenheim, A. Gali, and P. Deak, Phys. Rev. B 68, 155208 (2003). 9. M. Bockstedte, A. Mattaush, and O. Pankratov, Phys. Rev. B 68, 205201 (2003).
Translated by M. Tagirdzhanov
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