- Email: [email protected]
GaAs
Materials Chemistry and Physics 130 (2011) 1341–1345
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Wet oxidation of thin AlAs in cylindrical composite of GaAs/AlAs/GaAs Sun-Chien Ko a , Sanboh Lee b,∗ , Y.T. Chou c a
Advanced Technology Research Laboratory, Telecommunication Laboratories, Chunghwa Telecom Company, Taoyuan, Taiwan Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, Taiwan c Department of Chemical Engineering and Materials Science, University of California, Irvine, CA 92697, USA b
a r t i c l e
i n f o
Article history: Received 28 March 2011 Received in revised form 16 August 2011 Accepted 10 September 2011 Keywords: Semiconductors Diffusion Thin film Oxidation
a b s t r a c t Lateral wet oxidation in a cylindrical composite, GaAs/AlAs/GaAs, with varying thickness of the AlAs layer has been investigated. The oxidation depth in AlAs was measured in the temperature range of 400–480 ◦ C. At given temperature and time, the depth increases with the increase in thickness. The thickness effect was successfully interpreted based on the kinetic model of boundary layer diffusion. The results are consistent with the findings from early studies on samples of square and rectangular cross-sections with the same activation energy of the thermal process. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Wet oxidation of AlAs is a technically important treatment used in the fabrication of vertical cavity surface emitting lasers (VCSELs) with the distributed Bragg reflector (DBR) [1–4]. The oxide provides both optical and electrical confinement, significantly lowering the threshold current [2,5–7]. Considerable research efforts have been launched for understanding the oxidation process and its kinetics [4,8–19]. According to Ashby et al. [13], the oxidation starts when the water vapor reacts with AlAs at the lateral surface, forming the stable Al2 O3 and volatile As and AsH3 , the latter two reaction products are removed from the system. Once the oxide is formed, the diffusant (the oxidant) must diffuse in the newly-formed oxide to reach the AlAs phase for reaction as well as in the un-reacted AlAs phase. As the oxidation continues, the oxide extends at the consumption of the AlAs phase. The two rate processes, reaction and diffusion, proceed alternately, with the latter at a lower rate. Thus, in the initial stage, when only a very small amount of oxide is present, the oxide growth is dominated by the reaction kinetics. As the oxide extends, the growth is controlled by diffusion [11]. In the study of wet oxidation of GaAs/AlAs/GaAs, a surprise finding is the effect of layer thickness on the oxidation rate [4,14,15]. That is, at given temperature and time, the oxide depth increases with thickness of the AlAs layer. The thickness dependence was first treated by Naone and Coldren [14], proposing an interfacial energy model as the theoretical basis. It was shown that the
∗ Corresponding author. E-mail address: [email protected] (S. Lee). 0254-0584/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2011.09.026
model was more effective for the linear rate at the initial stage of the oxidation process. It is also debatable on the use of the Gibbs–Thomson formula, a basic thermodynamic relation valid at the equilibrium state. On the other hand, Koley et al. [15] employed a diffusion-reaction model for the oxidation kinetics in a straight mesa, and explained the thickness effect led by the assumption of zero concentration of the oxidant at the Al(Ga)As–GaAs interfaces. This assumption is weak, for the oxidant which is concurrently diffusing in the outer GaAs layers [16]. Ku and Chang-Hasnain [17] later re-examined the oxidation process in a similar system (Aly Ga1−y As/Alx Ga1−x As/Aly Ga1−x As) with a circular as well as a straight mesa. Their detailed analysis is also based on the diffusionreaction model with two different assumptions: (1) the drift term has a thickness-dependent component, and (2) the diffusion flux at the (oxide/Alx Ga1−x As)–Aly Ga1−y As boundary is zero. Despite of the good agreement between theory and experiment, the analyses presented by both groups [15,17] are limited to the steady state flow (time independent). The time dependent part, which is more crucial in the early stage of the kinetic process, remained unexplored. These limits were later removed in the work of Ko et al. [18,19], in which a transient flow model based on the boundary layer diffusion was used. However, one of the assumptions imposed in the analyses of Ko et al. [18,19] is over simplified. That is, the diffusivities of the oxidant in Al2 O3 and AlAs are assumed to be equal. Upon a close examination, this assumption can be replaced with a more realistic one, as described below. In this paper, we report a new set of measurements of oxidation depth in AlAs using cylindrical samples of the GaAs/AlAs/GaAs composite. This is an extension of the early work [20] in which the boundary layer diffusion model was analyzed and applied to the wet oxidation case. Here, we will focus on
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S.-C. Ko et al. / Materials Chemistry and Physics 130 (2011) 1341–1345
Fig. 1. A sequence of top-view cross sections showing the advance of oxide front during water vapor diffusion in a cylindrical GaAs/AlAs/GaAs sandwich composite with AlAs thickness of 700 A˚ at 420 ◦ C. The oxidation periods are (a) 3 min, (b) 6 min, (c) 12 min, (d) 18 min, (e) 24 min, and (f) 30 min. Note the slight reduction in size of the newly formed oxide as the oxidation proceeds.
the thickness effect and its interpretation. As will be seen, the predictions based on the modified diffusion model agree well with the experimental data, and are consistent with the findings obtained from the early studies on samples of square and rectangular crosssections.
2. Experimental Cylindrical samples were grown by the low-pressure metal organic chemical vapor deposition (LP-MOCVD) technique with the structure of 500 A˚ GaAs cap/AlAs/(1 0 0) GaAs. The thickness of the AlAs layer was controlled at 400, 500, ˚ Each sample was masked by SiO2 and selectively etched on the 700, and 1000 A. mesa sidewall by exposing the AlAs layer and part of GaAs phase for wet oxidation. These samples were then placed in a three-zone furnace and heated to desired temperature with flowing N2 gas of flux at 0.8 L min−1 . Wet oxidation took place by passing the air bubbles through a water bath at 98 ◦ C. After oxidation, the SiO2 coating was removed. The change of refractive indices between the oxide and AlAs was clearly visible under light illumination. Thus the oxidation front in the AlAs layer can be directly examined under an optical microscope attached with a chargecoupling device camera. The oxidation depth, defined as the distance from the outer surface to the oxidation front, was measured at time of 3, 6, 12,18, 24, 30 and 36 min consecutively for a given temperature.
3. Results and discussion The oxidation depths were recorded on a sequence of micrographs to be quantitatively analyzed. Fig. 1 illustrates a typical set taken for samples with 700 A˚ AlAs oxidized at 420 ◦ C, where the oxide appears as the white ring and the un-reacted AlAs the gray core. The measured oxidation depths were plotted against time for the given temperature range, as shown in Fig. 2a–e, where the geometrical indicators represent the measured data and the lines (solid and dashed) are the calculated results. These data clearly show the thickness dependence on the oxidation rate. As proposed in the early work [18,19], the effect can be explained using a kinetic model of boundary layer diffusion. The boundary layer diffusion model was first proposed by Lee and co-workers in their study on diffusion and diffusion-induced chemical stress in a composite structure of the BAB type [18,21]. It is essentially an extension of grain boundary diffusion in which the central A layer has the same chemical composition as that of the outer B layers. When layer A is extremely thin with a much greater diffusivity of the diffusant than layer B and there is no
S.-C. Ko et al. / Materials Chemistry and Physics 130 (2011) 1341–1345
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˚ (b) 400 A, ˚ (c) 500 A, ˚ (d) 700 A, ˚ and (e) 1000 A. ˚ Solid and dashed Fig. 2. Oxidation depth as a function of diffusion time at different temperatures and thicknesses: (a) 2a = 300 A, lines are based on the boundary layer models for circular and square mesa [19], respectively. Full and open squares, circles, triangles, and diamonds attached to the solid and dashed lines denote the experimental data. Note that for a given temperature and time the depth in circular case is greater.
inter-diffusion of matrix atoms between A and B, the system is soluble. This model was used in the early studies of thickness effect on wet oxidation in GaAs/AlAs/GaAs samples of square and rectangular cross-sections. To affect a solution, it was assumed that the diffusivities of the diffusant in Al2 O3 and AlAs are the same so that the oxidizing layer can be treated as a homogeneous phase. This equality assumption is obviously over-simplified. As mentioned above, during the wet oxidation of water vapor in GaAs/AlAs/GaAs, the central layer is composed of Al2 O3 and AlAs, separated by an interphase boundary. The diffusion flow would pass through the two regions and the boundary, which is moving inward due to the advancement of the chemical reaction. A formal analysis of boundary layer diffusion in such a system would be extremely difficult if not impossible. Fortunately, in the analysis of thickness effected by using the computer fitting technique, the Al2 O3 –AlAs layer can be replaced by an equivalent layer of a single-phase substance with yet-to-be-determined diffusivity. But the oxidation rate in the equivalent layer of diffusivity DEQ would be the same as that in the substituted layer of Al2 O3 –AlAs. This substitution is acceptable under the condition that the analysis is based on the use of the best fitting technique, by which the diffusivities of the oxidant in the equivalent layer need not be pre-determined, but are
estimated from the fitting process. In this way, the same boundary layer diffusion model, with either the equality or the equivalence assumption, can be effectively used in the study of oxidation kinetics in the GaAs/Al2 O3 –AlAs/GaAs system. However, the equivalence assumption is more realistic, and is an interesting topic for further exploration. The analysis of boundary layer diffusion in a cylindrical composite of the BAB type has recently been reported [20]. Let layer A and layer B be located in the regions (r ≤ r0 , |z| ≤ a) and (r ≤ r0 , a ≤ |z|), respectively, and the origin of the coordinate system be situated at the center of layer A (Fig. 3). The diffusivity of the diffusant, DA , in layer A is assumed to be much greater than the diffusivity, DB , in layer B. Under a constant source concentration C0 , on the lateral surface, the concentration function C in layer A is given by [20].
J0 (˛n r/r0 ) CA (r, t) = 1−2 exp C0 ˛n J1 (˛n ) ∞
n=1
2 ˛n
× erfc ı
ˇ
˛ 2 2
ı
n
ˇ
−
˛ 2
n
ˇ (1)
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S.-C. Ko et al. / Materials Chemistry and Physics 130 (2011) 1341–1345
Table 1 The estimated DEQ /DB and the associated activation energy (EEQ − EB ) used in the analysis of wet oxidation in square [19], circular and slab [18] mesa at different temperatures. Sample geometry
Square [19] Circular Slab [18]
EEQ − EB (eV)
DEQ /DB (×105 ) 480 ◦ C
460◦ C
440 ◦ C
420 ◦ C
400 ◦ C
2.20 2.17 2.17
1.67 1.73 1.67
1.39 1.37 1.36
1.09 1.062 1.12
0.834 0.815 0.806
0.53 ± 0.03 0.53 ± 0.02 0.53 ± 0.02
Fig. 3. Schematic diagram showing a thin layer A sandwiched between two outer layer B in a cylindrical coordinate system.
where J0 is the first kind Bessel function of zero-order, ␣n is the nth positive root of J0 (r), erfc stands for the complementary error function, with. ı=
a( − 1)
DB t
r
Fig. 4. Arrhenius plot of log(DEQ /DB ) versus 1/T.
(2a)
ˇ=
0
(2b)
=
DA DB
(2c)
DB t
Eq. (1) is valid only for layer A with a very small thickness so that its cross-sectional concentration function can be assumed uniformly distributed. Note that in the boundary layer diffusion model both layer A and layer B are assumed to be homogeneous. However, as mentioned above, in the oxidizing GaAs/AlAs/GaAs composite, layer A is composed of Al2 O3 and AlAs, having two separate sets of diffusivities of the diffusant. In order to obtain an analytic solution for the concentration function in layer A under the boundary layer diffusion, we have to replace the Al2 O3 –AlAs layer by an equivalent homogeneous layer with diffusivity DEQ such that its rate of oxidation is the same as that of the original layer. As a result of the replacement, the diffusivity D used for DA in the earlier work [19] is now being replaced by DEQ in the present study. The values of DEQ /DB are then determined from Eq. (1) using the best fitting technique on a computer. With the concentration function known, the connection between the theory and experiment can be made by the assignment that the observed oxidation depth is located at C = 0.5C0 , where C0 is the constant source concentration at the outer surface. The calculated oxidation depths for various oxide thicknesses were plotted in Fig. 2a–e as the solid lines to fit the experimental data represented by full squares, circles, triangles, and diamonds. As shown,
the theoretical predictions based on the equivalent boundary layer diffusion model are in good agreement with the experimental measurements. Fig. 2(a) for 2a = 300 A˚ was previously reported by Ko et al. (Fig. 4 in Ref. [20]). It is re-plotted here for completion. For comparison, the same set of diffusivities was used to compute the oxidation depth at C = 0.5C0 given by Eq. (7) in Ref. [19] for the square cross section. The calculated values for the square cross section were also plotted in Fig. 2 as dashed lines to fit the experimental data represented by open squares, circles, triangles, and diamonds. A similar comparison was reported [19] between the samples of square cross section [19] and slab cross section [18]. The oxidation rate was found higher in the former case. In general, the oxidation rate is highest in samples of circular cross section. It is worth noting that the use of C = 0.5C0 as the criterion for oxidation depth is not critical. When the concentration was set at 0.4, 0.6, or 0.8C0 , the agreement between the theory and experiment remained hold under the same set of diffusivities. In treating the thickness effect, we ignored the effect of stress arisen from the thermal expansion coefficients between the neighboring phases. However, as indicated by the close agreement between the theory and experiment, the effect seems to be small. Furthermore, the estimated DEQ /DB values satisfy the Arrhenius equation. A plot of log(DEQ /DB ) versus 1/T is shown in Fig. 4. The straight line suggests the relation DEQ = DB
0 DEQ
DB0
(EEQ − EB ) exp − kT
(3)
S.-C. Ko et al. / Materials Chemistry and Physics 130 (2011) 1341–1345 0 and D0 are respectively the pre-exponential factors of where DEQ B the activation process of diffusant in the equivalent layer and the outer layer GaAs, k is the Boltzmann constant, T is the temperature, and EEQ and EB are the corresponding activation energies. The slope of the straight line gives the value of (EEQ − EB ) to be 0.53 ± 0.02 eV, which is consistent with the corresponding values obtained for square [19] and slab [18] samples (see Table 1). It would be interesting to note that although the equality and the equivalence assumption are totally different, the difference does not affect the subsequent analysis and computations. This is because we use the fitting technique by which the diffusivity data need not be pre-assigned, but are determined by the fitting procedure. However, the equivalence assumption is more realistic and tractable.
4. Conclusions The lateral oxidation of the cylindrical GaAs/AlAs/GaAs sandwich composite in wet ambient has been investigated in the temperature range of 400–480 ◦ C. The rate of oxidation increases with the temperature and the thickness of the AlAs layer. The thickness dependence interpreted was based on the model of boundary layer diffusion. It is shown that the theory and experiment are in good agreement. These results are consistent with the findings of early studies on samples of square [19] and rectangular [18] cross sections with the same activation energy of the thermal process. Acknowlegment This work was supported by the National Science Council, Taiwan.
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References [1] J.M. Dallesasse, N. Holonyak Jr., Appl. Phys. Lett. 58 (1991) 394. [2] F.A. Kish, S.J. Caracci, N. Holonyak Jr., J.M. Dallesasse, K.C. Hsieh, M.J. Ries, S.C. Smith, R.D. Burnham, Appl. Phys. Lett. 59 (1991) 1755. [3] S.A. Maranowski, F.A. Kish, S.J. Caracci, N. Holonyak Jr., J.M. Dallesasse, D.P. Bour, D.W. Treat, Appl. Phys. Lett. 61 (1992) 1688. [4] K.D. Choquette, K.M. Geib, C.I.H. Ashby, R.D. Twesten, O. Blum, H.Q. Hou, D.M. Follstaedt, R.E. Hammons, D. Mathes, R. Hull, IEEE Select. Top. Quantum Electron. 3 (1997) 916. [5] D.F. Huffaker, D.G. Deppe, K. Kumar, T.J. Rogers, Appl. Phys. Lett. 65 (1994) 97. [6] K.D. Choquette, R.S. Schneider Jr., K.L. Lear, K.M. Geib, Electron. Lett. 30 (1994) 2043. [7] M.H. MacDougal, R.D. Dapkus, V. Padikov, H. Zhao, G.M. Yang, IEEE Photon. Technol. Lett. 7 (1995) 229. [8] F.A. Kish, S.A. Maranowski, G.E. Hofler, N. Holonyak Jr., S.J. Caracci, J.M. Dallesasse, K.C. Hsieh, Appl. Phys. Lett. 60 (1992) 3165. [9] R.S. Burton, T.E. Schlesinger, J. Appl. Phys. 76 (1994) 5503. [10] H. Nickel, J. Appl. Phys. 78 (1995) 5201. [11] M. Ochisi, G.E. Giudice, H. Temkin, J.W. Scott, T.M. Cockerill, Appl. Phys. Lett. 68 (1996) 1757. [12] J.H. Kim, D.H. Lim, K.S. Kim, G.M. Yang, K.Y. Lim, H.J. Lee, Appl. Phys. Lett. 69 (1996) 3357. [13] C.I.H. Ashby, J.P. Sullivan, K.D. Choquette, K.M. Geib, H.Q. Hou, J. Appl. Phys. 82 (1997) 3134. [14] R.L. Naone, L.A. Coldren, J. Appl. Phys. 82 (1997) 2277. [15] B. Koley, M. Dagenais, R. Jin, G. Simonis, J. Pham, G. McLane, F. Johnson, R. Whaley Jr., J. Appl. Phys. 84 (1998) 600. [16] T.H. Oh, P.L. Huffaker, L.A. Graham, H. Deng, D.G. Deppe, Electron. Lett. 32 (1996) 2024. [17] P.C. Ku, C.J. Chang-Hasnain, IEEE J. Quantum Electron. 39 (2003) 577. [18] S.C. Ko, S. Lee, H.L. Wang, Y.T. Chou, J. Mater. Res. 18 (2003) 1027. [19] S.C. Ko, S. Lee, Y.T. Chou, J. Electron. Mater. 36 (2007) 1652. [20] S.C. Ko, S. Lee, Y.T. Chou, Mater. Chem. Phys. 115 (2009) 488. [21] H.Y. Lin, S.C. Ko, S. Lee, J. Appl. Phys. 96 (2004) 6183.
Contents lists available at SciVerse ScienceDirect
Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Wet oxidation of thin AlAs in cylindrical composite of GaAs/AlAs/GaAs Sun-Chien Ko a , Sanboh Lee b,∗ , Y.T. Chou c a
Advanced Technology Research Laboratory, Telecommunication Laboratories, Chunghwa Telecom Company, Taoyuan, Taiwan Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, Taiwan c Department of Chemical Engineering and Materials Science, University of California, Irvine, CA 92697, USA b
a r t i c l e
i n f o
Article history: Received 28 March 2011 Received in revised form 16 August 2011 Accepted 10 September 2011 Keywords: Semiconductors Diffusion Thin film Oxidation
a b s t r a c t Lateral wet oxidation in a cylindrical composite, GaAs/AlAs/GaAs, with varying thickness of the AlAs layer has been investigated. The oxidation depth in AlAs was measured in the temperature range of 400–480 ◦ C. At given temperature and time, the depth increases with the increase in thickness. The thickness effect was successfully interpreted based on the kinetic model of boundary layer diffusion. The results are consistent with the findings from early studies on samples of square and rectangular cross-sections with the same activation energy of the thermal process. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Wet oxidation of AlAs is a technically important treatment used in the fabrication of vertical cavity surface emitting lasers (VCSELs) with the distributed Bragg reflector (DBR) [1–4]. The oxide provides both optical and electrical confinement, significantly lowering the threshold current [2,5–7]. Considerable research efforts have been launched for understanding the oxidation process and its kinetics [4,8–19]. According to Ashby et al. [13], the oxidation starts when the water vapor reacts with AlAs at the lateral surface, forming the stable Al2 O3 and volatile As and AsH3 , the latter two reaction products are removed from the system. Once the oxide is formed, the diffusant (the oxidant) must diffuse in the newly-formed oxide to reach the AlAs phase for reaction as well as in the un-reacted AlAs phase. As the oxidation continues, the oxide extends at the consumption of the AlAs phase. The two rate processes, reaction and diffusion, proceed alternately, with the latter at a lower rate. Thus, in the initial stage, when only a very small amount of oxide is present, the oxide growth is dominated by the reaction kinetics. As the oxide extends, the growth is controlled by diffusion [11]. In the study of wet oxidation of GaAs/AlAs/GaAs, a surprise finding is the effect of layer thickness on the oxidation rate [4,14,15]. That is, at given temperature and time, the oxide depth increases with thickness of the AlAs layer. The thickness dependence was first treated by Naone and Coldren [14], proposing an interfacial energy model as the theoretical basis. It was shown that the
∗ Corresponding author. E-mail address: [email protected] (S. Lee). 0254-0584/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2011.09.026
model was more effective for the linear rate at the initial stage of the oxidation process. It is also debatable on the use of the Gibbs–Thomson formula, a basic thermodynamic relation valid at the equilibrium state. On the other hand, Koley et al. [15] employed a diffusion-reaction model for the oxidation kinetics in a straight mesa, and explained the thickness effect led by the assumption of zero concentration of the oxidant at the Al(Ga)As–GaAs interfaces. This assumption is weak, for the oxidant which is concurrently diffusing in the outer GaAs layers [16]. Ku and Chang-Hasnain [17] later re-examined the oxidation process in a similar system (Aly Ga1−y As/Alx Ga1−x As/Aly Ga1−x As) with a circular as well as a straight mesa. Their detailed analysis is also based on the diffusionreaction model with two different assumptions: (1) the drift term has a thickness-dependent component, and (2) the diffusion flux at the (oxide/Alx Ga1−x As)–Aly Ga1−y As boundary is zero. Despite of the good agreement between theory and experiment, the analyses presented by both groups [15,17] are limited to the steady state flow (time independent). The time dependent part, which is more crucial in the early stage of the kinetic process, remained unexplored. These limits were later removed in the work of Ko et al. [18,19], in which a transient flow model based on the boundary layer diffusion was used. However, one of the assumptions imposed in the analyses of Ko et al. [18,19] is over simplified. That is, the diffusivities of the oxidant in Al2 O3 and AlAs are assumed to be equal. Upon a close examination, this assumption can be replaced with a more realistic one, as described below. In this paper, we report a new set of measurements of oxidation depth in AlAs using cylindrical samples of the GaAs/AlAs/GaAs composite. This is an extension of the early work [20] in which the boundary layer diffusion model was analyzed and applied to the wet oxidation case. Here, we will focus on
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S.-C. Ko et al. / Materials Chemistry and Physics 130 (2011) 1341–1345
Fig. 1. A sequence of top-view cross sections showing the advance of oxide front during water vapor diffusion in a cylindrical GaAs/AlAs/GaAs sandwich composite with AlAs thickness of 700 A˚ at 420 ◦ C. The oxidation periods are (a) 3 min, (b) 6 min, (c) 12 min, (d) 18 min, (e) 24 min, and (f) 30 min. Note the slight reduction in size of the newly formed oxide as the oxidation proceeds.
the thickness effect and its interpretation. As will be seen, the predictions based on the modified diffusion model agree well with the experimental data, and are consistent with the findings obtained from the early studies on samples of square and rectangular crosssections.
2. Experimental Cylindrical samples were grown by the low-pressure metal organic chemical vapor deposition (LP-MOCVD) technique with the structure of 500 A˚ GaAs cap/AlAs/(1 0 0) GaAs. The thickness of the AlAs layer was controlled at 400, 500, ˚ Each sample was masked by SiO2 and selectively etched on the 700, and 1000 A. mesa sidewall by exposing the AlAs layer and part of GaAs phase for wet oxidation. These samples were then placed in a three-zone furnace and heated to desired temperature with flowing N2 gas of flux at 0.8 L min−1 . Wet oxidation took place by passing the air bubbles through a water bath at 98 ◦ C. After oxidation, the SiO2 coating was removed. The change of refractive indices between the oxide and AlAs was clearly visible under light illumination. Thus the oxidation front in the AlAs layer can be directly examined under an optical microscope attached with a chargecoupling device camera. The oxidation depth, defined as the distance from the outer surface to the oxidation front, was measured at time of 3, 6, 12,18, 24, 30 and 36 min consecutively for a given temperature.
3. Results and discussion The oxidation depths were recorded on a sequence of micrographs to be quantitatively analyzed. Fig. 1 illustrates a typical set taken for samples with 700 A˚ AlAs oxidized at 420 ◦ C, where the oxide appears as the white ring and the un-reacted AlAs the gray core. The measured oxidation depths were plotted against time for the given temperature range, as shown in Fig. 2a–e, where the geometrical indicators represent the measured data and the lines (solid and dashed) are the calculated results. These data clearly show the thickness dependence on the oxidation rate. As proposed in the early work [18,19], the effect can be explained using a kinetic model of boundary layer diffusion. The boundary layer diffusion model was first proposed by Lee and co-workers in their study on diffusion and diffusion-induced chemical stress in a composite structure of the BAB type [18,21]. It is essentially an extension of grain boundary diffusion in which the central A layer has the same chemical composition as that of the outer B layers. When layer A is extremely thin with a much greater diffusivity of the diffusant than layer B and there is no
S.-C. Ko et al. / Materials Chemistry and Physics 130 (2011) 1341–1345
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˚ (b) 400 A, ˚ (c) 500 A, ˚ (d) 700 A, ˚ and (e) 1000 A. ˚ Solid and dashed Fig. 2. Oxidation depth as a function of diffusion time at different temperatures and thicknesses: (a) 2a = 300 A, lines are based on the boundary layer models for circular and square mesa [19], respectively. Full and open squares, circles, triangles, and diamonds attached to the solid and dashed lines denote the experimental data. Note that for a given temperature and time the depth in circular case is greater.
inter-diffusion of matrix atoms between A and B, the system is soluble. This model was used in the early studies of thickness effect on wet oxidation in GaAs/AlAs/GaAs samples of square and rectangular cross-sections. To affect a solution, it was assumed that the diffusivities of the diffusant in Al2 O3 and AlAs are the same so that the oxidizing layer can be treated as a homogeneous phase. This equality assumption is obviously over-simplified. As mentioned above, during the wet oxidation of water vapor in GaAs/AlAs/GaAs, the central layer is composed of Al2 O3 and AlAs, separated by an interphase boundary. The diffusion flow would pass through the two regions and the boundary, which is moving inward due to the advancement of the chemical reaction. A formal analysis of boundary layer diffusion in such a system would be extremely difficult if not impossible. Fortunately, in the analysis of thickness effected by using the computer fitting technique, the Al2 O3 –AlAs layer can be replaced by an equivalent layer of a single-phase substance with yet-to-be-determined diffusivity. But the oxidation rate in the equivalent layer of diffusivity DEQ would be the same as that in the substituted layer of Al2 O3 –AlAs. This substitution is acceptable under the condition that the analysis is based on the use of the best fitting technique, by which the diffusivities of the oxidant in the equivalent layer need not be pre-determined, but are
estimated from the fitting process. In this way, the same boundary layer diffusion model, with either the equality or the equivalence assumption, can be effectively used in the study of oxidation kinetics in the GaAs/Al2 O3 –AlAs/GaAs system. However, the equivalence assumption is more realistic, and is an interesting topic for further exploration. The analysis of boundary layer diffusion in a cylindrical composite of the BAB type has recently been reported [20]. Let layer A and layer B be located in the regions (r ≤ r0 , |z| ≤ a) and (r ≤ r0 , a ≤ |z|), respectively, and the origin of the coordinate system be situated at the center of layer A (Fig. 3). The diffusivity of the diffusant, DA , in layer A is assumed to be much greater than the diffusivity, DB , in layer B. Under a constant source concentration C0 , on the lateral surface, the concentration function C in layer A is given by [20].
J0 (˛n r/r0 ) CA (r, t) = 1−2 exp C0 ˛n J1 (˛n ) ∞
n=1
2 ˛n
× erfc ı
ˇ
˛ 2 2
ı
n
ˇ
−
˛ 2
n
ˇ (1)
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S.-C. Ko et al. / Materials Chemistry and Physics 130 (2011) 1341–1345
Table 1 The estimated DEQ /DB and the associated activation energy (EEQ − EB ) used in the analysis of wet oxidation in square [19], circular and slab [18] mesa at different temperatures. Sample geometry
Square [19] Circular Slab [18]
EEQ − EB (eV)
DEQ /DB (×105 ) 480 ◦ C
460◦ C
440 ◦ C
420 ◦ C
400 ◦ C
2.20 2.17 2.17
1.67 1.73 1.67
1.39 1.37 1.36
1.09 1.062 1.12
0.834 0.815 0.806
0.53 ± 0.03 0.53 ± 0.02 0.53 ± 0.02
Fig. 3. Schematic diagram showing a thin layer A sandwiched between two outer layer B in a cylindrical coordinate system.
where J0 is the first kind Bessel function of zero-order, ␣n is the nth positive root of J0 (r), erfc stands for the complementary error function, with. ı=
a( − 1)
DB t
r
Fig. 4. Arrhenius plot of log(DEQ /DB ) versus 1/T.
(2a)
ˇ=
0
(2b)
=
DA DB
(2c)
DB t
Eq. (1) is valid only for layer A with a very small thickness so that its cross-sectional concentration function can be assumed uniformly distributed. Note that in the boundary layer diffusion model both layer A and layer B are assumed to be homogeneous. However, as mentioned above, in the oxidizing GaAs/AlAs/GaAs composite, layer A is composed of Al2 O3 and AlAs, having two separate sets of diffusivities of the diffusant. In order to obtain an analytic solution for the concentration function in layer A under the boundary layer diffusion, we have to replace the Al2 O3 –AlAs layer by an equivalent homogeneous layer with diffusivity DEQ such that its rate of oxidation is the same as that of the original layer. As a result of the replacement, the diffusivity D used for DA in the earlier work [19] is now being replaced by DEQ in the present study. The values of DEQ /DB are then determined from Eq. (1) using the best fitting technique on a computer. With the concentration function known, the connection between the theory and experiment can be made by the assignment that the observed oxidation depth is located at C = 0.5C0 , where C0 is the constant source concentration at the outer surface. The calculated oxidation depths for various oxide thicknesses were plotted in Fig. 2a–e as the solid lines to fit the experimental data represented by full squares, circles, triangles, and diamonds. As shown,
the theoretical predictions based on the equivalent boundary layer diffusion model are in good agreement with the experimental measurements. Fig. 2(a) for 2a = 300 A˚ was previously reported by Ko et al. (Fig. 4 in Ref. [20]). It is re-plotted here for completion. For comparison, the same set of diffusivities was used to compute the oxidation depth at C = 0.5C0 given by Eq. (7) in Ref. [19] for the square cross section. The calculated values for the square cross section were also plotted in Fig. 2 as dashed lines to fit the experimental data represented by open squares, circles, triangles, and diamonds. A similar comparison was reported [19] between the samples of square cross section [19] and slab cross section [18]. The oxidation rate was found higher in the former case. In general, the oxidation rate is highest in samples of circular cross section. It is worth noting that the use of C = 0.5C0 as the criterion for oxidation depth is not critical. When the concentration was set at 0.4, 0.6, or 0.8C0 , the agreement between the theory and experiment remained hold under the same set of diffusivities. In treating the thickness effect, we ignored the effect of stress arisen from the thermal expansion coefficients between the neighboring phases. However, as indicated by the close agreement between the theory and experiment, the effect seems to be small. Furthermore, the estimated DEQ /DB values satisfy the Arrhenius equation. A plot of log(DEQ /DB ) versus 1/T is shown in Fig. 4. The straight line suggests the relation DEQ = DB
0 DEQ
DB0
(EEQ − EB ) exp − kT
(3)
S.-C. Ko et al. / Materials Chemistry and Physics 130 (2011) 1341–1345 0 and D0 are respectively the pre-exponential factors of where DEQ B the activation process of diffusant in the equivalent layer and the outer layer GaAs, k is the Boltzmann constant, T is the temperature, and EEQ and EB are the corresponding activation energies. The slope of the straight line gives the value of (EEQ − EB ) to be 0.53 ± 0.02 eV, which is consistent with the corresponding values obtained for square [19] and slab [18] samples (see Table 1). It would be interesting to note that although the equality and the equivalence assumption are totally different, the difference does not affect the subsequent analysis and computations. This is because we use the fitting technique by which the diffusivity data need not be pre-assigned, but are determined by the fitting procedure. However, the equivalence assumption is more realistic and tractable.
4. Conclusions The lateral oxidation of the cylindrical GaAs/AlAs/GaAs sandwich composite in wet ambient has been investigated in the temperature range of 400–480 ◦ C. The rate of oxidation increases with the temperature and the thickness of the AlAs layer. The thickness dependence interpreted was based on the model of boundary layer diffusion. It is shown that the theory and experiment are in good agreement. These results are consistent with the findings of early studies on samples of square [19] and rectangular [18] cross sections with the same activation energy of the thermal process. Acknowlegment This work was supported by the National Science Council, Taiwan.
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