- Email: [email protected]
Modeling of silicon atoms diffusion in GaAs in view of nonuniform distribution of point defects
ARTICLE IN PRESS
Materials Science in Semiconductor Processing 7 (2004) 27–33
Modeling of silicon atoms diffusion in GaAs in view of nonuniform distribution of point defects A.M. Saada, O.I. Velichkob,* b
a Al-Balqa Applied University, P.O.Box 2041, Amman 11953, Jordan Department of Physics, Belarusian State University on Informatics and Radioelectronics, 6, P. Brovki Street, Minsk 220027, Belarus
Abstract A simulation of Si diffusion in layer-doped GaAs crystals during annealing in the ambients with no excess of Ga or As and in As-rich conditions has been carried out. A model of diffusion due to ‘‘silicon atom–gallium vacancy’’ pairs taking into account the nonuniform distribution of point defects responsible for diffusion and amphoteric behavior of silicon in GaAs has been used for the calculation of dopant concentration profiles. A number of parameters, such as intrinsic diffusivity of silicon, average migration length of gallium vacancies etc, has been derived from the fitting to the experimental data. For all annealing conditions, the surface acts as a strong sink for gallium vacancies inducing vacancy diffusion to the surface. The values of the average migration length for gallium vacancies are approximately the same for all investigated cases and are equal to B0.4 mm. It is shown that in the range of a dopant concentration up to 5ni silicon diffusion is governed by the neutral gallium vacancies VGa , but, if silicon concentration is near or exceeds 10ni, the dominant defect responsible for dopant diffusion is the triply negatively charged gallium vacancies V3 Ga. r 2004 Elsevier Ltd. All rights reserved. PACS: 66.30.Jt; 66.30.Lw; 66.30.Xj; 07.05.Tp Keywords: Thermal diffusion; Gallium arsenide; Silicon; Vacancy
1. Introduction At present diverse heterostructures on the basis of GaAs are widely used for micro-, nano-, and optoelectronic devices. Unfortunately, this material is much less investigated in comparison with silicon. There is a great uncertainty in dealing with parameters related to the diffusion and chemical reactions of point defects. The goal of the paper is to derive an average migration length of gallium vacancies in crystalline GaAs and estimate a character distribution of point defects in the vicinity of the surface during different thermal processing of GaAs substrates. The characteristics of point defects are very important for *Corresponding author. Tel.: +375-239-89-13. E-mail addresses: [email protected] (A.M. Saad), [email protected] (O.I. Velichko).
defect engineering, because the diffusion of gallium vacancies can degrade the electrical parameters of the Si doped GaAs [1]. An additional information concerning the role of point defects in dopant diffusion and defect absorption near the surface of GaAs is also obtained.
2. Model It is now well known that nonuniform distributions of point defects during dopant diffusion greatly influence the form of impurity concentration profiles. Therefore, using a model of dopant diffusion that takes into account a nonuniform distribution of point defects, one can extract parameters related to point defect distribution from the dopant profile after annealing. In this paper, to obtain parameters describing the transport of
1369-8001/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mssp.2004.01.003
ARTICLE IN PRESS A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33
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Nomenclature VGa 3 VGa Si+ Ga
Si As r r Dpp rp r0 p rAs CA Ai C n ni w=n/ni CV As CV Asi
gallium vacancy in the neutral charge state triply negatively charged gallium vacancy silicon atom substitutionally dissolved in the Ga sublattice silicon atom substitutionally dissolved in the As sublattice r charge state of the vacancy VGa point defect facilitating a conversion process of Siþ Ga and SiAs species r charge state of point defect Dpp before reaction r charge state of point defect Dpp after reaction charge state of arsenic vacancy concentration of Si atoms substitutionally dissolved in the arsenic sublattice coefficient for the conversion process between Si+ Ga and SiAs concentration of silicon atoms substitutionally dissolved in the Ga sublattice electron concentration intrinsic carrier concentration electron concentration n reduced to the intrinsic carrier concentration ni actual concentration of arsenic vacancies in the neutral charge state thermal equilibrium concentration of arsenic vacancies in the neutral charge state
gallium vacancies a simulation of the experiments [2] related to Si diffusion in GaAs is carried out. In reference [2], the GaAs wafers with periodically Si doped structures were prepared by molecular beam epitaxy. After epitaxy, annealing of semiconductor substrates was carried out at the temperature of 815 C in the Ga-rich, neutral, and As-rich ambients. Thermal processing leads to a silicon redistribution that depends on the doping level and treatment ambients. Besides, it is clearly seen from the experimental data [2] that the silicon diffusion rate depends on the distance from the surface of semiconductor. Depth dependence of the diffusion rate can be used to obtain information related to the defect subsystem. For the current theory of diffusion in crystalline semiconductors, the dopant atoms occupying substitutional sites are considered as immobile. Therefore, their migration occurs only due to interaction with point defects. Consequently, the rate of dopant diffusion is proportional to the concentration of defects governing the diffusion process. Thus, some parameters concerning point defects responsible for the migration of dopant atoms can be extracted from studying the dopant diffusion. For example, one can explain the depth
CV Ga
actual concentration of gallium vacancies in the neutral charge state CV thermal equilibrium concentration of gallium Gai vacancies in the neutral charge state IAs arsenic interstitial in the neutral charge state CT total concentration of silicon atoms D(w) effective diffusivity of silicon atoms depending on w Di intrinsic diffusivity of silicon atoms b3 empirical parameter describing a relative contribution V3 Ga to the diffusion of silicon atoms in comparison with the contribution of the neutral gallium vacancies VGa V d (w) effective diffusivity of gallium vacancies depending on w dV diffusivity of neutral gallium vacancies in i intrinsic semiconductor kV(w) effective coefficient of gallium vacancies recombination t average lifetime of gallium vacancies li average migration length of neutral gallium vacancies in intrinsic GaAs tV average lifetime of neutral gallium vacancies i in intrinsic GaAs gV actual generation rate of gallium vacancies gV equilibrium value of the generation rate of i gallium vacancies eS ¼ C eð0Þ relative concentration of neutral vacancies C on the surface
dependence of silicon diffusion rate [2] in terms of the nonuniform distribution of point defects. Now there are many papers devoted to the simulation of Si diffusion in GaAs [3–10]. For the first model [3] the amphoteric nature of Si atoms substitutionally dissolved both in the Ga sublattice (donor Si+ Ga) and in the As sublattice (acceptor Si As) was taken into account. It was supposed in [3] that a transport of dopant atoms predominantly occurred due to formation and migration of the electrically neutral nearest neighbor pairs ‘‘Si+ Ga– Si As’’. This mechanism is in contradiction with the experimentally proved assumption that Ga self-diffusion in GaAs and Ga–Al interdiffusion in GaAs/AlAs superlattices are governed by the triply negatively charged group-III element vacancies, conventionally designated as V3 Ga [4]. Besides, the mechanism of ‘‘Si+ Ga–SiAs’’ pairs is incapable of explaining the data of Si diffusion into the Sb-doped substrate [4]. The mechanism of Si atom diffusion by means of the interaction with V3 Ga was proposed in [4,6,7,10]. The last model [10] supposed that the silicon atoms in the As sublattice Si As are immobile and that the conversion rate + of the Si As species into the SiGa species is small and hence ignorable. The dopant diffusion occurs due to the
ARTICLE IN PRESS A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33 + interaction of gallium vacancies V3 Ga with SiGa. For simulation of silicon diffusion in [7,10] the diffusion– segregation equation derived in [7] was used. Unfortunately, it is not clear what vacancy mechanism of the dopant atom diffusion is described by this equation and was used for calculations of silicon profiles in [7,10]. Is it a simple vacancy diffusion mechanism by means of the lattice sites exchange between silicon atom Si+ Ga and the vacancy V3 Ga, or a mechanism of diffusion due to formation, migration and dissociation of the pairs 3 ‘‘Si+ Ga–VGa’’? The mechanism of silicon diffusion due + + to ‘‘SiGa–V Ga’’ and ‘‘(SiGa–VGa) ’’ pairs was proposed in [11,12] to explain the strong dependence of Si diffusion rate on the background doping and on the crystal Fermi level. It was showed that the pair diffusion mechanism was consistent with the experimental data concerning both Si diffusion and impurity-induced layer disordering in Alx Ga1x As–GaAs quantum well heterostructures. It is important to note that the pair diffusion mechanism is widely used in simulation of dopant diffusion in silicon (see, for example, [13]). A model of Si diffusion in GaAs due to formation, r migration, and dissociation of the pairs ‘‘Si+ Ga–VGa’’, where r is the charge state of the vacancy, was proposed in [8,9]. The following reaction was used to describe the conversion between Si+ Ga and SiAs species
r
r0
rAs p p r Siþ Ga þ Dp þ VAs þ me "SiAs þ Dp þ VGa ;
ð1Þ
r Dpp
is the point defect facilitating a conversion where 0 process of Si+ Ga and SiAs species; rp and r p are the charge states of this point defect before and after reaction; rAs and r are the charge states of arsenic and gallium vacancies, respectively. It follows from the mass action law and local charge neutrality written for reaction (1) that concentration CA of Si atoms substitutionally dissolved in the arsenic sublattice Si As is given by [8] ! eV C As C A ¼ Ai ð2Þ w2 C; C˜ eV ¼ C V =C V ; C As As Asi
ð3Þ
e ¼ C V =C V ; C Ga Gai
ð4Þ
where Ai is the coefficient for the conversion process between Si+ Ga and SiAs; C is the concentration of silicon atoms substitutionally dissolved in the Ga sublattice Si+ Ga; w=n/ni is the electron concentration n reduced to V the intrinsic carrier concentration ni; CV As and CAsi are the actual and thermal equilibrium concentrations of arsenic vacancies in the neutral charge state, respec tively; CV and CV Ga Gai are the actual and thermal equilibrium concentrations of gallium vacancies in the neutral charge state, respectively. Thus, a local equilibrium between Si+ Ga and SiAs is supposed, but there is no restriction on the conversion
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+ rate of the Si As species into the SiGa species as it was made in Ref. [10]. It is interesting to note that in the model [10] Si+ Ga and SiAs species would convert into each other via the reaction
3 Siþ Ga þ 5e "SiAs þ IAs þ VGa ;
ð5Þ
IAs
is the arsenic interstitial in the neutral charge where state. The reactions (1) and (5) yield the same dependence CA on w and C. Really, reaction (1) is r transformed into reaction (5) if Dpp is an arsenic self interstitial IAs and a site in the As sublattice is considered as a sum of IAs and VAs . Below, reaction (1) is used because a case of nonuniform distribution of gallium vacancies is investigated and expression (2) allow to take into account this peculiarity of diffusion. As in the models [9,10] it is assumed for the present analysis that Si As atoms are immobile. Then, the diffusion equation for dopant atoms can be written in the form [9] ( " #) eC qw eCÞ C qC T q qðC DðwÞ þ ; ð6Þ ¼ w qx qx qx qt CT ¼ C þ CA:
ð7Þ
T
Here C is the total concentration of dopant and D(w) is the effective diffusivity of silicon atoms. It is also assumed that a local charge neutrality is valid and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 A 2 n C C þ ðC C Þ þ 4ni w¼ ¼ : ð8Þ ni 2ni To take into consideration the contribution of V3 Ga to the dopant diffusion, the effective diffusivity of silicon atoms is represented as D ¼ Di DC ðwÞ; 3 Di ¼ D i þ Di ;
DC ðwÞ ¼
b3 ¼
1 þ b3 w3 ; 1 þ b3
D3 i ; D i
ð9Þ
ð10Þ
where Di is the intrinsic diffusivity and b3 is the empirical parameter describing a relative contribution of V3 Ga to the diffusion of silicon atoms in comparison with the contribution of the neutral gallium vacancies VGa . The equation of point defect diffusion [14] is used to calculate the gallium vacancy distribution. For 1D case the vacancy diffusion equation can be written in the form " # eg e e C q qC kVC ðwÞC VC þ 2 ¼ 0; ð11Þ d ðwÞ þ 2 qx qx li li d VC ðwÞ ¼
d V ðwÞ ; diV
ð12Þ
ARTICLE IN PRESS A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33
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kVC ðwÞ ¼ kV ðwÞtV i ;
ð13Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diV tV i ;
ð14Þ
li ¼
V eg ¼ g t ; C V gi tV i
ð15Þ
where dV(w) is the effective diffusivity of gallium vacancies; dV is the diffusivity of neutral gallium i vacancies in intrinsic semiconductor; kV(w) is the effective coefficient of gallium vacancies recombination; t is the average lifetime of gallium vacancies; li and tV i are the average migration length and the average lifetime of neutral gallium vacancies in intrinsic GaAs, respectively; gV and gV i are the generation rate of gallium vacancies and equilibrium value of the generation rate, respectively.
Fig. 2. Profiles of electron concentration normalized to the intrinsic carrier concentration ni and concentration of gallium vacancies in the neutral charge state related to the equilibrium concentration value. These concentration profiles are calculated for the dopant distribution presented in Fig. 1 and correspond to the end of annealing.
3. Diffusion simulation The results of modeling of silicon redistribution in periodically Si doped structures for different thermal treatments investigated in [2] are presented in Figs. 1–6. More precisely, silicon diffusion during annealing in neutral and As-rich ambients is simulated. The temperature of processing is equal to 815 C, duration 10 h. As in Ref. [2], two different doping levels are investigated. The following parameters describing diffusion of Si in GaAs are used for fitting calculated profiles to the experimental data: intrinsic diffusivity Di; the constant
Fig. 3. Calculated profiles of the total silicon concentration and concentration of substitutionally dissolved silicon atoms Si+ Ga after annealing in neutral ambients for high doping level. Experimental profiles for the as-grown and after annealing of the total silicon concentration are taken from Ref. [2].
Fig. 1. Calculated profiles of the total silicon concentration and concentration of substitutionally dissolved silicon atoms Si+ Ga after annealing in neutral ambients for low doping level. Experimental profiles for the as-grown and after annealing of the total silicon concentration are taken from Ref. [2]. (Note that calculated total Si concentration is approximately equal to the concentration of Si+ Ga, except at the vicinity of the surface.)
Ai for conversion process between Si+ Ga and SiAs species; the parameter b3 of relative contribution of V3 Ga to the diffusion; the average migration length of gallium vacancies li; and the relative concentration of neutral eS ¼ C eð0Þ: In the ideal case vacancies on the surface C parameters Di, Ai, and b3 must be independent on the dopant concentration for the experiments with the same annealing ambients. Therefore, it is reasonable to choose the equal values of these parameters for the experiments with different doping levels. For the calculation of intrinsic charge carrier concentration ni the expression [15] is used, which yields ni=0.9811 105 mm3. The
ARTICLE IN PRESS A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33
Fig. 4. Calculated profiles of the total silicon concentration and concentration of substitutionally dissolved silicon atoms Si+ Ga after annealing in As-rich conditions for low doping level. Experimental profiles for the as-grown and after annealing of the total silicon concentration are taken from Ref. [2]. (Note that calculated total Si concentration is approximately equal to the concentration of Si+ Ga, except at the vicinity of the surface.)
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Fig. 6. Profiles of electron concentration normalized to the intrinsic carrier concentration ni and concentration of gallium vacancies in the neutral charge state related to the equilibrium concentration value. These concentration profiles are calculated for the dopant distribution presented in Fig. 5 and correspond to the end of annealing.
4. Discussion
Fig. 5. Calculated profiles of the total silicon concentration and concentration of substitutionally dissolved silicon atoms Si+ Ga after annealing in As-rich conditions for high doping level. Experimental profiles for the as-grown and after annealing of the total silicon concentration are taken from Ref. [2].
value of Ai=3.9 104 a.u. is also used for simulation of dopant diffusion in As-rich conditions. This value for Ai was derived in [8] from experimental data [16] and slightly modified in the present paper in view of new value of ni. A set of parameters derived from the condition of the best fitting to all the experimental data are presented in Table 1.
It is seen from Figs. 1, 3–5 that the calculated profiles of the total concentration of silicon atoms agree well with experimentally measured ones. However, some questions remain and require a further investigation. Really, one can see from the Table 1 that the values of intrinsic diffusivity Di are 2 times higher in the substrates with high silicon concentration in comparison with low doped GaAs. It is interesting to note that the maximum Si concentration in heavy doped substrates is also approximately 2 times higher. In the model proposed above an empirical coefficient b3 is responsible for the diffusion enhancement due to dopant level increasing. Nevertheless, the silicon profiles calculated for the higher values of b3 have a different form in comparison with the experimental data. Perhaps, the difference in the values of Di be explained by the action of the factor not taken into consideration in the model. More probably, a non-accuracy of the experimental data or different states of a defect subsystem in the low and high doped substrates leads to the difference in Di values. Besides, a very small diffusion of Si atoms occurs in the case of relatively low doping level. Therefore, experimental and fitting errors can also contribute to the difference of Di values. It is interesting also to note that there is a great uncertainty in the values of Di taken from different references. For example, for the temperature of 815 C the value of D under C ¼ ni is equal to 0.7 107 mm2/s for Si diffusion investigated in [3], Dðw ¼ 1Þ ¼ 5:249 106 mm2/s for arsenic pressure 1 atm [6], and Dðw ¼ 1Þ ¼ 4:47 1010 mm2/s for As-rich conditions, if expression presented in [10] is used. The values of Di extracted in the present paper from
ARTICLE IN PRESS 32
A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33
Table 1 Values of parameters used for the simulation of Si diffusion Cases
Di (mm2/s)
Ai (a.u.)
b3 (a.u.)
li (mm)
fS ða:u:Þ C
Annealing with no excess Ga or As (low Si concentration) Annealing with no excess Ga or As (high Si concentration) Annealing under As-rich conditions (low Si concentration) Annealing under As-rich conditions (high Si concentration)
6.67 109
6.5 104
0.003
0.4
0.3
1.33 108
6.5 104
0.004
0.4
0.3
1.25 108
3.9 104
0.002
0.4
0.6
3.17 108
3.9 104
0.002
0.4
0.1
experimental data [2] are lower than the values obtained in [3,6] but greater than the value obtained in [10]. In Figs. 2 and 6 the concentration profiles of gallium vacancies in the neutral charge state during annealing and profiles of electron concentration corresponding to the dopant distribution before the end of processing are presented. The profiles presented in Figs. 2 and 6 were calculated for the relatively low doped substrate annealed in the neutral ambients and high doped substrate annealed in As-rich ambients, respectively. The functions dV(w) and kV(w) in Eq. (11) are assumed to be approximately constant (dVC(w)E1 and kVC(w)E1), because the average migration length of neutral gallium vacancies is greater than the thickness of the separate eg ¼ 1; because there is no doped layers. Besides, C generation of nonequilibrium gallium vacancies in the experiment under consideration. In this case the nonuniform distribution of point defects is formed due to absorption of gallium vacancies on the surface of semiconductor. Table 1 shows that for all cases of eS is significantly less than 1, i.e. diffusion the values of C the surface acts as a strong sink for gallium vacancies. It can be seen from Figs. 2 and 6 that reduction in the concentration of gallium vacancies near the surface results in the decrease of the electron concentration whereas the Si concentration increases in the vicinity of interface. It is to be noted that reduction of electron concentration near the surface is experimentally observed in GaAs doped by Si [16]. The values of the coefficients b3 obtained from the fit to the form of the experimental Si profiles after diffusion are very small and lie in the range of 0.002–0.004. It means that in the regions with electron concentration up to 5ni silicon diffusion is governed by neutral gallium vacancies VGa , but if w is near or exceeds 10, the dominant defect responsible for Si diffusion is V3 Ga. Finally, it is interesting to note that the values of the average migration length for gallium vacancies li obtained from the fitting to the experimental data [2] are approximately the same for all cases of diffusion and are equal to B0.4 mm.
5. Conclusions A simulation of Si diffusion in layer-doped GaAs crystals during annealing at the temperature of 815 C has been carried out. The cases of thermal treatment in the ambients with no excess of Ga or As and in As-rich conditions have been investigated. A model of silicon diffusion due to ‘‘silicon atom—gallium vacancy’’ pairs taking into account the nonuniform distribution of point defects responsible for diffusion has been used for calculation of dopant concentration profiles. Silicon redistribution for two doping levels has been simulated to investigate the concentration dependence of Si diffusivity. The calculated profiles agree well with the experimental data [2]. A number of parameters describing dopant diffusion and point defect diffusion, such as intrinsic diffusivity of silicon, defect concentration on the surface, and average migration length of gallium vacancies, has been derived from the fitting to the experimental data. It is shown that for all annealing conditions considered in the present paper the surface acts as a strong sink for gallium vacancies and a nonuniform distribution of vacancies is formed due to absorption on the surface. The values of the average migration length for gallium vacancies obtained from a simulation of the experimental data [2] are approximately the same for all cases of diffusion and are equal to B0.4 mm. A relative contribution of neutral and charge vacancies to the silicon diffusion has also been investigated. It is shown that in the range of a dopant concentration up to 5ni silicon diffusion is governed by the neutral gallium vacancies VGa , but, if silicon concentration is near or exceeds 10ni, the dominant defect responsible for dopant diffusion is V3 Ga.
References [1] Ohbu I, Takahama M, Imamura Y. Diffusion of gallium vacancies from low-temperature grown GaAs. Jpn J Appl Phys Part 2 1992;31(12A):L1647–9.
ARTICLE IN PRESS A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33 [2] Deppe DG, Holonyak Jr N, Piano WE, Robbins VM, Dallesasse JM, Hsieh KC, Baker JE. Impurity diffusion and layer interdiffusion in AlxGa1x As–GaAs heterostructures. J Appl Phys 1988;64(4):1838–44. [3] Greiner ME, Gibbons JF. Diffusion of silicon in gallium arsenide using rapid thermal processing: experiment and model. Appl Phys Lett 1984;44(8):750–2. [4] Yu S, Go¨sele UM, Tan TY. A model of Si diffusion in GaAs based on the effect of the Fermi level. J Appl Phys 1989;66(7):2952–61. [5] Allen EL, Murray JJ, Deal MD, Plummer JD, Jones KS, Rubart WS. A comparison of the diffusion behavior of ion-implanted Sn, Ge, and Si in gallium arsenide. J Electrochem Soc 1991;138(11):3440–9. [6] You H-M, Go¨sele UM, Tan TY. A study of Si outdiffusion from predoped GaAs. J Appl Phys 1993; 73(11):7207–18. [7] You H-M, Go¨sele UM, Tan TY. Simulation of the transient indiffusion-segregation process of triply negatively charged Ga vacancies in GaAs and AlAs/GaAs superlattices. J Appl Phys 1993;74(4):2461–70. [8] Velichko OI, Egorov AA, Fedoruk SK. Modeling of silicon diffusion in gallium arsenide: I. Microscopic mechanisms of diffusion and a model of the transitions of silicon atoms between crystal sublattices. J Eng Phys Thermophys USA 1993;65(5):1091–6.
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[9] Velichko OI, Egorov AA, Fedoruk SK. Modeling of silicon diffusion in gallium arsenide: 2. Diffusion equation. J Eng Phys Thermophys USA 1994;66(1):75–8. [10] Chen C-H, Go¨sele UM, Tan TY. Dopant diffusion and segregation in semiconductor heterostructures: part III, diffusion of Si into GaAs. Appl Phys 1999;A69:313–21. [11] Deppe DG, Holonyak Jr N, Kish FA, Baker JE. Background doping dependence of silicon diffusion in ptype GaAs. Appl Phys Lett 1987;50(15):998–1000. [12] Deppe DG, Holonyak Jr N. Atom diffusion and impurityinduced layer disordering in quantum well III–V semiconductor heterostructures. J Appl Phys 1988;64(12):R93–113. [13] Uematsu M. Simulation of boron, phosphorus, and arsenic diffusion in silicon based on an integrated diffusion model, and the anomalous phosphorus diffusion mechanism. J Appl Phys 1997;82(5):2228–46. [14] Velichko OI, Fedotov AK. Simulation of point defect diffusion in semiconductors. Diffusion and defect data, part B. Solid State Phenomena 1999;69–70:513–8. [15] Blakemore JS. Intrinsic density ni(T) in GaAs: deduced from band gap and effective mass parameters and derived independently from Cr acceptor capture and emission coefficient. J Appl Phys 1982;53(1):520–31. [16] Deppe DG, Holonyak Jr N, Baker JE. Sensitivity of Si diffusion in GaAs to column IV and VI donor species. Appl Phys Lett 1988;52(2):129–31.
Materials Science in Semiconductor Processing 7 (2004) 27–33
Modeling of silicon atoms diffusion in GaAs in view of nonuniform distribution of point defects A.M. Saada, O.I. Velichkob,* b
a Al-Balqa Applied University, P.O.Box 2041, Amman 11953, Jordan Department of Physics, Belarusian State University on Informatics and Radioelectronics, 6, P. Brovki Street, Minsk 220027, Belarus
Abstract A simulation of Si diffusion in layer-doped GaAs crystals during annealing in the ambients with no excess of Ga or As and in As-rich conditions has been carried out. A model of diffusion due to ‘‘silicon atom–gallium vacancy’’ pairs taking into account the nonuniform distribution of point defects responsible for diffusion and amphoteric behavior of silicon in GaAs has been used for the calculation of dopant concentration profiles. A number of parameters, such as intrinsic diffusivity of silicon, average migration length of gallium vacancies etc, has been derived from the fitting to the experimental data. For all annealing conditions, the surface acts as a strong sink for gallium vacancies inducing vacancy diffusion to the surface. The values of the average migration length for gallium vacancies are approximately the same for all investigated cases and are equal to B0.4 mm. It is shown that in the range of a dopant concentration up to 5ni silicon diffusion is governed by the neutral gallium vacancies VGa , but, if silicon concentration is near or exceeds 10ni, the dominant defect responsible for dopant diffusion is the triply negatively charged gallium vacancies V3 Ga. r 2004 Elsevier Ltd. All rights reserved. PACS: 66.30.Jt; 66.30.Lw; 66.30.Xj; 07.05.Tp Keywords: Thermal diffusion; Gallium arsenide; Silicon; Vacancy
1. Introduction At present diverse heterostructures on the basis of GaAs are widely used for micro-, nano-, and optoelectronic devices. Unfortunately, this material is much less investigated in comparison with silicon. There is a great uncertainty in dealing with parameters related to the diffusion and chemical reactions of point defects. The goal of the paper is to derive an average migration length of gallium vacancies in crystalline GaAs and estimate a character distribution of point defects in the vicinity of the surface during different thermal processing of GaAs substrates. The characteristics of point defects are very important for *Corresponding author. Tel.: +375-239-89-13. E-mail addresses: [email protected] (A.M. Saad), [email protected] (O.I. Velichko).
defect engineering, because the diffusion of gallium vacancies can degrade the electrical parameters of the Si doped GaAs [1]. An additional information concerning the role of point defects in dopant diffusion and defect absorption near the surface of GaAs is also obtained.
2. Model It is now well known that nonuniform distributions of point defects during dopant diffusion greatly influence the form of impurity concentration profiles. Therefore, using a model of dopant diffusion that takes into account a nonuniform distribution of point defects, one can extract parameters related to point defect distribution from the dopant profile after annealing. In this paper, to obtain parameters describing the transport of
1369-8001/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mssp.2004.01.003
ARTICLE IN PRESS A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33
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Nomenclature VGa 3 VGa Si+ Ga
Si As r r Dpp rp r0 p rAs CA Ai C n ni w=n/ni CV As CV Asi
gallium vacancy in the neutral charge state triply negatively charged gallium vacancy silicon atom substitutionally dissolved in the Ga sublattice silicon atom substitutionally dissolved in the As sublattice r charge state of the vacancy VGa point defect facilitating a conversion process of Siþ Ga and SiAs species r charge state of point defect Dpp before reaction r charge state of point defect Dpp after reaction charge state of arsenic vacancy concentration of Si atoms substitutionally dissolved in the arsenic sublattice coefficient for the conversion process between Si+ Ga and SiAs concentration of silicon atoms substitutionally dissolved in the Ga sublattice electron concentration intrinsic carrier concentration electron concentration n reduced to the intrinsic carrier concentration ni actual concentration of arsenic vacancies in the neutral charge state thermal equilibrium concentration of arsenic vacancies in the neutral charge state
gallium vacancies a simulation of the experiments [2] related to Si diffusion in GaAs is carried out. In reference [2], the GaAs wafers with periodically Si doped structures were prepared by molecular beam epitaxy. After epitaxy, annealing of semiconductor substrates was carried out at the temperature of 815 C in the Ga-rich, neutral, and As-rich ambients. Thermal processing leads to a silicon redistribution that depends on the doping level and treatment ambients. Besides, it is clearly seen from the experimental data [2] that the silicon diffusion rate depends on the distance from the surface of semiconductor. Depth dependence of the diffusion rate can be used to obtain information related to the defect subsystem. For the current theory of diffusion in crystalline semiconductors, the dopant atoms occupying substitutional sites are considered as immobile. Therefore, their migration occurs only due to interaction with point defects. Consequently, the rate of dopant diffusion is proportional to the concentration of defects governing the diffusion process. Thus, some parameters concerning point defects responsible for the migration of dopant atoms can be extracted from studying the dopant diffusion. For example, one can explain the depth
CV Ga
actual concentration of gallium vacancies in the neutral charge state CV thermal equilibrium concentration of gallium Gai vacancies in the neutral charge state IAs arsenic interstitial in the neutral charge state CT total concentration of silicon atoms D(w) effective diffusivity of silicon atoms depending on w Di intrinsic diffusivity of silicon atoms b3 empirical parameter describing a relative contribution V3 Ga to the diffusion of silicon atoms in comparison with the contribution of the neutral gallium vacancies VGa V d (w) effective diffusivity of gallium vacancies depending on w dV diffusivity of neutral gallium vacancies in i intrinsic semiconductor kV(w) effective coefficient of gallium vacancies recombination t average lifetime of gallium vacancies li average migration length of neutral gallium vacancies in intrinsic GaAs tV average lifetime of neutral gallium vacancies i in intrinsic GaAs gV actual generation rate of gallium vacancies gV equilibrium value of the generation rate of i gallium vacancies eS ¼ C eð0Þ relative concentration of neutral vacancies C on the surface
dependence of silicon diffusion rate [2] in terms of the nonuniform distribution of point defects. Now there are many papers devoted to the simulation of Si diffusion in GaAs [3–10]. For the first model [3] the amphoteric nature of Si atoms substitutionally dissolved both in the Ga sublattice (donor Si+ Ga) and in the As sublattice (acceptor Si As) was taken into account. It was supposed in [3] that a transport of dopant atoms predominantly occurred due to formation and migration of the electrically neutral nearest neighbor pairs ‘‘Si+ Ga– Si As’’. This mechanism is in contradiction with the experimentally proved assumption that Ga self-diffusion in GaAs and Ga–Al interdiffusion in GaAs/AlAs superlattices are governed by the triply negatively charged group-III element vacancies, conventionally designated as V3 Ga [4]. Besides, the mechanism of ‘‘Si+ Ga–SiAs’’ pairs is incapable of explaining the data of Si diffusion into the Sb-doped substrate [4]. The mechanism of Si atom diffusion by means of the interaction with V3 Ga was proposed in [4,6,7,10]. The last model [10] supposed that the silicon atoms in the As sublattice Si As are immobile and that the conversion rate + of the Si As species into the SiGa species is small and hence ignorable. The dopant diffusion occurs due to the
ARTICLE IN PRESS A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33 + interaction of gallium vacancies V3 Ga with SiGa. For simulation of silicon diffusion in [7,10] the diffusion– segregation equation derived in [7] was used. Unfortunately, it is not clear what vacancy mechanism of the dopant atom diffusion is described by this equation and was used for calculations of silicon profiles in [7,10]. Is it a simple vacancy diffusion mechanism by means of the lattice sites exchange between silicon atom Si+ Ga and the vacancy V3 Ga, or a mechanism of diffusion due to formation, migration and dissociation of the pairs 3 ‘‘Si+ Ga–VGa’’? The mechanism of silicon diffusion due + + to ‘‘SiGa–V Ga’’ and ‘‘(SiGa–VGa) ’’ pairs was proposed in [11,12] to explain the strong dependence of Si diffusion rate on the background doping and on the crystal Fermi level. It was showed that the pair diffusion mechanism was consistent with the experimental data concerning both Si diffusion and impurity-induced layer disordering in Alx Ga1x As–GaAs quantum well heterostructures. It is important to note that the pair diffusion mechanism is widely used in simulation of dopant diffusion in silicon (see, for example, [13]). A model of Si diffusion in GaAs due to formation, r migration, and dissociation of the pairs ‘‘Si+ Ga–VGa’’, where r is the charge state of the vacancy, was proposed in [8,9]. The following reaction was used to describe the conversion between Si+ Ga and SiAs species
r
r0
rAs p p r Siþ Ga þ Dp þ VAs þ me "SiAs þ Dp þ VGa ;
ð1Þ
r Dpp
is the point defect facilitating a conversion where 0 process of Si+ Ga and SiAs species; rp and r p are the charge states of this point defect before and after reaction; rAs and r are the charge states of arsenic and gallium vacancies, respectively. It follows from the mass action law and local charge neutrality written for reaction (1) that concentration CA of Si atoms substitutionally dissolved in the arsenic sublattice Si As is given by [8] ! eV C As C A ¼ Ai ð2Þ w2 C; C˜ eV ¼ C V =C V ; C As As Asi
ð3Þ
e ¼ C V =C V ; C Ga Gai
ð4Þ
where Ai is the coefficient for the conversion process between Si+ Ga and SiAs; C is the concentration of silicon atoms substitutionally dissolved in the Ga sublattice Si+ Ga; w=n/ni is the electron concentration n reduced to V the intrinsic carrier concentration ni; CV As and CAsi are the actual and thermal equilibrium concentrations of arsenic vacancies in the neutral charge state, respec tively; CV and CV Ga Gai are the actual and thermal equilibrium concentrations of gallium vacancies in the neutral charge state, respectively. Thus, a local equilibrium between Si+ Ga and SiAs is supposed, but there is no restriction on the conversion
29
+ rate of the Si As species into the SiGa species as it was made in Ref. [10]. It is interesting to note that in the model [10] Si+ Ga and SiAs species would convert into each other via the reaction
3 Siþ Ga þ 5e "SiAs þ IAs þ VGa ;
ð5Þ
IAs
is the arsenic interstitial in the neutral charge where state. The reactions (1) and (5) yield the same dependence CA on w and C. Really, reaction (1) is r transformed into reaction (5) if Dpp is an arsenic self interstitial IAs and a site in the As sublattice is considered as a sum of IAs and VAs . Below, reaction (1) is used because a case of nonuniform distribution of gallium vacancies is investigated and expression (2) allow to take into account this peculiarity of diffusion. As in the models [9,10] it is assumed for the present analysis that Si As atoms are immobile. Then, the diffusion equation for dopant atoms can be written in the form [9] ( " #) eC qw eCÞ C qC T q qðC DðwÞ þ ; ð6Þ ¼ w qx qx qx qt CT ¼ C þ CA:
ð7Þ
T
Here C is the total concentration of dopant and D(w) is the effective diffusivity of silicon atoms. It is also assumed that a local charge neutrality is valid and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 A 2 n C C þ ðC C Þ þ 4ni w¼ ¼ : ð8Þ ni 2ni To take into consideration the contribution of V3 Ga to the dopant diffusion, the effective diffusivity of silicon atoms is represented as D ¼ Di DC ðwÞ; 3 Di ¼ D i þ Di ;
DC ðwÞ ¼
b3 ¼
1 þ b3 w3 ; 1 þ b3
D3 i ; D i
ð9Þ
ð10Þ
where Di is the intrinsic diffusivity and b3 is the empirical parameter describing a relative contribution of V3 Ga to the diffusion of silicon atoms in comparison with the contribution of the neutral gallium vacancies VGa . The equation of point defect diffusion [14] is used to calculate the gallium vacancy distribution. For 1D case the vacancy diffusion equation can be written in the form " # eg e e C q qC kVC ðwÞC VC þ 2 ¼ 0; ð11Þ d ðwÞ þ 2 qx qx li li d VC ðwÞ ¼
d V ðwÞ ; diV
ð12Þ
ARTICLE IN PRESS A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33
30
kVC ðwÞ ¼ kV ðwÞtV i ;
ð13Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diV tV i ;
ð14Þ
li ¼
V eg ¼ g t ; C V gi tV i
ð15Þ
where dV(w) is the effective diffusivity of gallium vacancies; dV is the diffusivity of neutral gallium i vacancies in intrinsic semiconductor; kV(w) is the effective coefficient of gallium vacancies recombination; t is the average lifetime of gallium vacancies; li and tV i are the average migration length and the average lifetime of neutral gallium vacancies in intrinsic GaAs, respectively; gV and gV i are the generation rate of gallium vacancies and equilibrium value of the generation rate, respectively.
Fig. 2. Profiles of electron concentration normalized to the intrinsic carrier concentration ni and concentration of gallium vacancies in the neutral charge state related to the equilibrium concentration value. These concentration profiles are calculated for the dopant distribution presented in Fig. 1 and correspond to the end of annealing.
3. Diffusion simulation The results of modeling of silicon redistribution in periodically Si doped structures for different thermal treatments investigated in [2] are presented in Figs. 1–6. More precisely, silicon diffusion during annealing in neutral and As-rich ambients is simulated. The temperature of processing is equal to 815 C, duration 10 h. As in Ref. [2], two different doping levels are investigated. The following parameters describing diffusion of Si in GaAs are used for fitting calculated profiles to the experimental data: intrinsic diffusivity Di; the constant
Fig. 3. Calculated profiles of the total silicon concentration and concentration of substitutionally dissolved silicon atoms Si+ Ga after annealing in neutral ambients for high doping level. Experimental profiles for the as-grown and after annealing of the total silicon concentration are taken from Ref. [2].
Fig. 1. Calculated profiles of the total silicon concentration and concentration of substitutionally dissolved silicon atoms Si+ Ga after annealing in neutral ambients for low doping level. Experimental profiles for the as-grown and after annealing of the total silicon concentration are taken from Ref. [2]. (Note that calculated total Si concentration is approximately equal to the concentration of Si+ Ga, except at the vicinity of the surface.)
Ai for conversion process between Si+ Ga and SiAs species; the parameter b3 of relative contribution of V3 Ga to the diffusion; the average migration length of gallium vacancies li; and the relative concentration of neutral eS ¼ C eð0Þ: In the ideal case vacancies on the surface C parameters Di, Ai, and b3 must be independent on the dopant concentration for the experiments with the same annealing ambients. Therefore, it is reasonable to choose the equal values of these parameters for the experiments with different doping levels. For the calculation of intrinsic charge carrier concentration ni the expression [15] is used, which yields ni=0.9811 105 mm3. The
ARTICLE IN PRESS A.M. Saad, O.I. Velichko / Materials Science in Semiconductor Processing 7 (2004) 27–33
Fig. 4. Calculated profiles of the total silicon concentration and concentration of substitutionally dissolved silicon atoms Si+ Ga after annealing in As-rich conditions for low doping level. Experimental profiles for the as-grown and after annealing of the total silicon concentration are taken from Ref. [2]. (Note that calculated total Si concentration is approximately equal to the concentration of Si+ Ga, except at the vicinity of the surface.)
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Fig. 6. Profiles of electron concentration normalized to the intrinsic carrier concentration ni and concentration of gallium vacancies in the neutral charge state related to the equilibrium concentration value. These concentration profiles are calculated for the dopant distribution presented in Fig. 5 and correspond to the end of annealing.
4. Discussion
Fig. 5. Calculated profiles of the total silicon concentration and concentration of substitutionally dissolved silicon atoms Si+ Ga after annealing in As-rich conditions for high doping level. Experimental profiles for the as-grown and after annealing of the total silicon concentration are taken from Ref. [2].
value of Ai=3.9 104 a.u. is also used for simulation of dopant diffusion in As-rich conditions. This value for Ai was derived in [8] from experimental data [16] and slightly modified in the present paper in view of new value of ni. A set of parameters derived from the condition of the best fitting to all the experimental data are presented in Table 1.
It is seen from Figs. 1, 3–5 that the calculated profiles of the total concentration of silicon atoms agree well with experimentally measured ones. However, some questions remain and require a further investigation. Really, one can see from the Table 1 that the values of intrinsic diffusivity Di are 2 times higher in the substrates with high silicon concentration in comparison with low doped GaAs. It is interesting to note that the maximum Si concentration in heavy doped substrates is also approximately 2 times higher. In the model proposed above an empirical coefficient b3 is responsible for the diffusion enhancement due to dopant level increasing. Nevertheless, the silicon profiles calculated for the higher values of b3 have a different form in comparison with the experimental data. Perhaps, the difference in the values of Di be explained by the action of the factor not taken into consideration in the model. More probably, a non-accuracy of the experimental data or different states of a defect subsystem in the low and high doped substrates leads to the difference in Di values. Besides, a very small diffusion of Si atoms occurs in the case of relatively low doping level. Therefore, experimental and fitting errors can also contribute to the difference of Di values. It is interesting also to note that there is a great uncertainty in the values of Di taken from different references. For example, for the temperature of 815 C the value of D under C ¼ ni is equal to 0.7 107 mm2/s for Si diffusion investigated in [3], Dðw ¼ 1Þ ¼ 5:249 106 mm2/s for arsenic pressure 1 atm [6], and Dðw ¼ 1Þ ¼ 4:47 1010 mm2/s for As-rich conditions, if expression presented in [10] is used. The values of Di extracted in the present paper from
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Table 1 Values of parameters used for the simulation of Si diffusion Cases
Di (mm2/s)
Ai (a.u.)
b3 (a.u.)
li (mm)
fS ða:u:Þ C
Annealing with no excess Ga or As (low Si concentration) Annealing with no excess Ga or As (high Si concentration) Annealing under As-rich conditions (low Si concentration) Annealing under As-rich conditions (high Si concentration)
6.67 109
6.5 104
0.003
0.4
0.3
1.33 108
6.5 104
0.004
0.4
0.3
1.25 108
3.9 104
0.002
0.4
0.6
3.17 108
3.9 104
0.002
0.4
0.1
experimental data [2] are lower than the values obtained in [3,6] but greater than the value obtained in [10]. In Figs. 2 and 6 the concentration profiles of gallium vacancies in the neutral charge state during annealing and profiles of electron concentration corresponding to the dopant distribution before the end of processing are presented. The profiles presented in Figs. 2 and 6 were calculated for the relatively low doped substrate annealed in the neutral ambients and high doped substrate annealed in As-rich ambients, respectively. The functions dV(w) and kV(w) in Eq. (11) are assumed to be approximately constant (dVC(w)E1 and kVC(w)E1), because the average migration length of neutral gallium vacancies is greater than the thickness of the separate eg ¼ 1; because there is no doped layers. Besides, C generation of nonequilibrium gallium vacancies in the experiment under consideration. In this case the nonuniform distribution of point defects is formed due to absorption of gallium vacancies on the surface of semiconductor. Table 1 shows that for all cases of eS is significantly less than 1, i.e. diffusion the values of C the surface acts as a strong sink for gallium vacancies. It can be seen from Figs. 2 and 6 that reduction in the concentration of gallium vacancies near the surface results in the decrease of the electron concentration whereas the Si concentration increases in the vicinity of interface. It is to be noted that reduction of electron concentration near the surface is experimentally observed in GaAs doped by Si [16]. The values of the coefficients b3 obtained from the fit to the form of the experimental Si profiles after diffusion are very small and lie in the range of 0.002–0.004. It means that in the regions with electron concentration up to 5ni silicon diffusion is governed by neutral gallium vacancies VGa , but if w is near or exceeds 10, the dominant defect responsible for Si diffusion is V3 Ga. Finally, it is interesting to note that the values of the average migration length for gallium vacancies li obtained from the fitting to the experimental data [2] are approximately the same for all cases of diffusion and are equal to B0.4 mm.
5. Conclusions A simulation of Si diffusion in layer-doped GaAs crystals during annealing at the temperature of 815 C has been carried out. The cases of thermal treatment in the ambients with no excess of Ga or As and in As-rich conditions have been investigated. A model of silicon diffusion due to ‘‘silicon atom—gallium vacancy’’ pairs taking into account the nonuniform distribution of point defects responsible for diffusion has been used for calculation of dopant concentration profiles. Silicon redistribution for two doping levels has been simulated to investigate the concentration dependence of Si diffusivity. The calculated profiles agree well with the experimental data [2]. A number of parameters describing dopant diffusion and point defect diffusion, such as intrinsic diffusivity of silicon, defect concentration on the surface, and average migration length of gallium vacancies, has been derived from the fitting to the experimental data. It is shown that for all annealing conditions considered in the present paper the surface acts as a strong sink for gallium vacancies and a nonuniform distribution of vacancies is formed due to absorption on the surface. The values of the average migration length for gallium vacancies obtained from a simulation of the experimental data [2] are approximately the same for all cases of diffusion and are equal to B0.4 mm. A relative contribution of neutral and charge vacancies to the silicon diffusion has also been investigated. It is shown that in the range of a dopant concentration up to 5ni silicon diffusion is governed by the neutral gallium vacancies VGa , but, if silicon concentration is near or exceeds 10ni, the dominant defect responsible for dopant diffusion is V3 Ga.
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