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Simulation of the diffusion of point defects in structures with local elastic stresses
Applied Mathematical Modelling 35 (2011) 1134–1141
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Simulation of the diffusion of point defects in structures with local elastic stresses Oleg Velichko ⇑ Belarusian State University of Informatics and Radioelectronics, Department of Physics, 6, P. Brovki Street, 220013 Minsk, Belarus
a r t i c l e
i n f o
Article history: Received 20 July 2008 Received in revised form 29 July 2010 Accepted 2 August 2010 Available online 19 August 2010 Keywords: Semiconductors Diffusion modelling Stress-mediated diffusion Point defects Defect engineering
a b s t r a c t The stress-mediated diffusion of nonequilibrium point defects into the bulk of a semiconductor is investigated by computer simulation. It is assumed that the point defects are generated on the surface of a semiconductor and that in the course of diffusion they pass through the local region of elastic stresses because the average length of defect migration is greater than the thickness and depth of the strained layer. Within the strained layer, point defect segregation or heavy defect depletion occurs if defect drift under stresses is directed in or out of the layer, respectively. The calculations also show that, in contrast to the case of local defect sink, the local region of elastic stresses practically does not change the distribution of defects beyond this region if there is no generation/absorption of point defects within the strained layer. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction In recent years decreasing dimensions of integrated components [1,2], the use of various multilayer structures [3], such as Si/SiGe [4–6], and different nano-particles embedded in crystalline [7,8] and amorphous [9,10] matrixes have been the main trends in advanced device technologies. For this reason, much attention is given to studying the defect formation kinetics and stress evolution during semiconductor processing, since defect and stress engineering can significantly improve the device performance [1,2,11]. For example, many efforts are directed at studying the stress-mediated diffusion of impurity atoms and point defects [12–16], including the influence of elastic stresses on drift of point defects near a surface or at interfaces [17–19], and near different inhomogeneities of semiconductor crystals [12,20]. Very interesting experimental data describing hydrogen diffusion in Si/Si1xGex/Si heterostructures were presented by Lin Shao with coauthors [21,22]. It was established that during diffusion the hydrogen atoms were segregated within the Si1xGex layer forming a peak of hydrogen concentration. To explain this phenomenon, it was supposed in [21] that a preferential vacancy aggregation within the strained layer occurred resulting in subsequent trapping of hydrogen atoms. Also, many studies deal with changes in the defect subsystem of ion-implanted layers, since these changes are responsible for transient enhanced diffusion of dopant atoms [1,2,23,24]. For example, to explain experimental data, it is assumed [18] that elastic stresses arising in the region of a SiGe layer buried in silicon and compressively strained after its growth cause a drift of vacancies into this layer. Thus, accumulation of vacancies within the layer and their transformation into nanovoids occur. In Ref. [23], a transient enhanced diffusion of dopant atoms in hyperfine boron-doped layers created by molecular-beam epitaxy is investigated experimentally. It is assumed that the formation of clusters of boron atoms with silicon self-interstitials occurs in these doped regions during thermal treatment. This process and a number of similar processes proceeding during thermal treatment are characterized by local absorption of self-interstitials. Thus, investigations concerned with the influence of local strained regions and ⇑ Tel.: +375 296998078. E-mail addresses: [email protected], [email protected] 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.08.002
O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
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local sinks on diffusion of nonequilibrium point defects are of great importance for the next-generation fabrication processes. The goal of the paper is to propose a model of stress-mediated diffusion of intrinsic point defects and investigate the essential features of the transport processes when stresses or sink are localized in a thin layer in comparison with the thickness of the diffusion zone. To emphasize the generality of the results obtained no specific values of strain, temperature, and diffusivity are used and the conclusions made are valid for all cases of diffusion when the average migration length of point defects is greater than the thickness of a strained layer or a layer with a sink. 2. Model of stress-mediated diffusion of point defects In order to develop a model of diffusion of point defects in the field of elastic stresses, let us assume that there are no changes in processing conditions during thermal treatment or these changes are slow enough in comparison with the average lifetime of point defects. Then, due to the high mobility of point defects in comparison with the mobility of impurity atoms, the time derivative of the defect concentration is close to zero and the distributions of point defects are quasi-stationary with respect to the distributions of dopant atoms, clusters, extended defects, and also with respect to the changes occurring under the processing conditions. In such a situation, to calculate the concentration distributions of nonequilibrium point defects in the field of elastic stresses, the stationary diffusion equation established in Ref. [17] can be used. It is assumed in [17] that the diffusion of point defects occurs due to the concentration gradient and via a drift in the built-in electric field and the field of elastic stresses. Then, the diffusion equation for point defects Dr (vacancies or self-interstitials in the charge state r) can be written as
r dr rC r þ
r r zr ed C r d Cr ru þ rU r þ SPr ; kB T kB T
GPr Sr þ GTr þ GRr ¼ 0;
ð1Þ
r ¼ 3; 2; ; ; þ; 2þ; 3þ; where Cr, dr, and zr are the concentration, diffusivity, and the charge of a defect in the charge state r, respectively; e is the elementary charge; u is the potential of the built-in electric field; Ur is the potential energy of a defect Dr in the field of internal elastic stresses; SPr and GPrare the rates of dissociation and generation of the pairs (ADr) per unit volume of the semiconductor, respectively; Sr is the rate of absorption of point defects in the charge state r by internal sinks; GTr is the rate of thermal generation of defects Dr per unit volume, while GRr is the rate due to ion implantation. It is assumed in [17] that due to the high mobility of electrons (holes) a local equilibrium holds between the species with different charge states. Then, the concentration of charged defects can be derived from the mass action law:
Cr C
r
vzzr
¼h;
ð2Þ
where C is the concentration of defects in the neutral charge state; hr is a constant of local thermodynamic equilibrium for the reaction of transition of point defects from the neutral charge state to the charge state r; z is the charge of the substitutionally dissolved dopant atom; v is the concentration of electrons n (or holes p) normalized to the intrinsic carrier concentration ni in the case of doping by the donor (or acceptor) impurity, respectively. It is assumed in this paper that the region of defect diffusion can be doped nonuniformly with electrically active impurity atoms. Then, dopant atoms exert an influence upon the diffusion of charged point defects due to the drift of defects in the built-in electric field (see the second term in brackets in Eq. (1)) and due to the change in the concentrations of charged species (see Eq. (2)). Summing up equations (1) written for defects in different charge states and taking into account the mass action law (2), one can reduce the system (1) to one generalized equation of the diffusion of point defects. In Ref. [17] such a generalized equation was obtained for the concentration of point defects in the neutral charge state:
"
# ad rU d ~ SP G P S þ G T þ G R r dðvÞrða CÞ þ dðvÞ ¼ 0; C þ kB T C i d~
ð3Þ
where
~ ¼ C =C ; C Xi r r r d hi vzz ; dðvÞ ¼ SP ¼
X
r
SPr
GP ¼
r
S¼
X r
X
ð4Þ ð5Þ GPr ;
r
Sr
GT ¼
X r
GTr
GR ¼
ð6Þ X r
GRr :
ð7Þ
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O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
~ is the concentration of nonequilibrium neutral point defects normalHere d(v) is the effective diffusivity of point defects; C ized to the concentration of neutral defects at thermal equilibrium C i . It was assumed that the potential energy of point defects in the field of elastic stresses Ur and the constants of local therr r modynamic equilibrium hr depended weakly on the charge states of diffusing species, i.e., Ur Ud and h =hi ad for all r, r where hi are the values of the constants of local equilibrium in an intrinsic semiconductor. For a one-dimensional diffusion, Eq. (3) can be written in the form convenient for numerical solution as [25]
" # C Sp eÞ ~ ~ 1 þ g~R d dðad C dðv~ x CÞ k ðvÞk ðxÞad C C d ðvÞ þ ¼ 0; 2 2 dx dx dx li li
ð8Þ
where C
d ðvÞ ¼
dðvÞ ; di
ð9Þ Sp
C
kðv; xÞ ¼ kðvÞk ðxÞ k ðvÞ ¼
vx
d
kðvÞ ; ki
a dU C ¼ d ðvÞ ; di kB T dx R g g~R ¼ : gi
v~ x ¼
ð10Þ
d
ð11Þ ð12Þ
Here di is the intrinsic diffusivity of point defects; k(v) and kSp(x) are the multipliers representing, respectively, the concentration dependence and spatial distribution of the effective absorption coefficient for point defects k(v, x); kC(v) is the concentration dependence of the effective pffiffiffiffiffiffiffiffi absorption coefficient normalized to the value ki of the absorptivity in an intrinsic 1 semiconductor; si ¼ ki and li ¼ di si are the average lifetime and average migration length of point defects in an intrinsic semiconductor, respectively; vx is the x-coordinate projection of the drift velocity of point defects due to elastic stresses; gR is the rate of generation of nonequilibrium point defects per unit volume due to ion implantation or dissolution of extended defects; gi is the rate of thermally equilibrium generation of point defects in an intrinsic semiconductor. It is to be noted that Eq. (8) differs from the diffusion equation used in Ref. [24] for point defects since the drift of mobile species due to elastic stresses is included. On the other hand, the distributions of nonequilibrium point defects can be derived from Eq. (8), instead of distributions of equilibrium defects in the field of elastic stresses being computed by means of expressions used in Refs. [12,13]. Also, Eq. (8) is convenient for numerical solutions due to the following characteristic features: (i) This equation describes the diffusion–reaction–drift of self-interstitials (vacancies) with different charge states as e must be obtained to solve the equation. (ii) a whole, although only the normalized concentration of neutral point defects C The equation obtained takes into account the drift of all charged species due to the built-in electric field. At the same time, there is no explicit term that would describe this drift. To exclude this term, a special rearrangement and some mathematical manipulations of the terms describing diffusion of point defects due to concentration gradient and drift of charged species in the built-in electric field were performed during the summing up of equations (1). As a result, a drift of the charged species in the electric field is included in the functional dependence of the effective diffusivity of neutral point defects d = d(v). (iii) The effective coefficients dC(v) and kC(v) of Eq. (8) represent smooth and monotone polynomial functions of v [25]. It is assumed in this work that the thin strained layer is located in the bulk of a semiconductor. For example, this buried strained region can be created by local doping of silicon with a substitutionally dissolved impurity having other atomic radius compared to the host atom [26]. Let us consider the case of the diffusion of point defects through the strained layer. To ~ x . The calculation of this function describe this process, it is necessary to define the effective drift velocity of point defects v ~ x is considered as an unspecified is a rather complex problem calling for a separate investigation. Therefore, in this work v function which satisfies the following requirements: (i) it should be equal to zero in the middle and beyond the strained layer; (ii) it should have different signs on the left and on the right of the strained layer. Then, the drift of nonequilibrium ~ x can be approximately defined in the point defects is directed to or out of the middle of the strained layer. For example, v following way. Let us assume that the impurity distribution CB(x, t) in the strained layer is described by the Gaussian function
"
C B ðxÞ ¼ C Bm exp
ðx xmax Þ2 2DR2Bp
#
;
ð13Þ
where CBm is a maximum of impurity concentration; xmax is the maximum position and DRBp is the dispersion of the buried impurity distribution. It is assumed in [27] that the potential energy of a point defect in the field of elastic stresses is proportional to the impurity concentration, i.e., for the migration of nonequilibrium point defects we have
U d ðxÞ ¼ AC B ðxÞ: Then, it follows from expression (11) that
ð14Þ
O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
v~ x ðx; vÞ
@C B ðxÞ : @x
1137
ð15Þ
Taking into account expression (15), one can see that for the impurity distribution given by the Gaussian function (13) the effective drift velocity of point defects vx(x, v) has the following form:
"
v~ x ðx; vÞ ¼ Bðx xmax Þ exp
ðx xmax Þ2 2DR2Bp
#
;
ð16Þ
where the constant B > 0 indicates that segregation of nonequilibrium point defects within the strained layer is provided and B < 0 is for the case of depletion of the stress region of point defects. 3. Numerical solution The finite-difference method [28] is used to find a numerical solution for Eq. (8) in the one-dimensional (1D) domain [0, xB]. Following Ref. [28], the first term on the left-hand side of Eq. (8) is approximated by a symmetric difference operator of second order accuracy on the space variable x. At the same time, the second term is approximated with a first order accuracy by the asymmetric forward/backward-difference operator depending on the drift direction. On the other hand, if the flux of defects due to elastic stresses is comparable to (or lower than) the flux of defects caused by the concentration gradient, the second term is also approximated by a symmetric difference operator of second order accuracy. Comparison with exact analytical solutions for particular cases of diffusion of point defects and calculations on meshes with different step sizes were carried out to verify the approximate numerical solution. For example, in Fig. 1 a numerical solution of Eq. (8) in the case of constant diffusivity and constant coefficient of the absorption of defects is presented. For comparison with the analytical solution of the diffusion equation obtained in [29], the Gaussian distribution
" g~R ðxÞ ¼ g~m exp
ðx Rpd Þ2
#
2DR2pd
ð17Þ
is used to specify the generation rate profile. Here g~m is a maximum generation rate of point defects during ion implantation normalized to the thermal generation rate gi; Rpd and DR2pd are the position of a maximum of defect generation rate and dispersion of generation rate distribution, respectively. It is assumed that the defect generation occurs due to implantation of hydrogen ions at an energy of 20 keV (Rpd = 0.198 lm and DRpd = 0.0802 lm are taken from Ref. [30]). Numerical computations are carried out on the simulation domain [0, xB], where xB and mesh point number iB are equal to 4.0 lm and 81, respece: tively. To obtain a numerical solution, the Dirichlet boundary conditions are imposed on C
e e S; Cð0Þ ¼C
e BÞ ¼ C e B; Cðx
ð18Þ
e S and C e B are the values of normalized defect concentration on the left and right boundaries of simulation domain. where C e th ðxÞ ¼ 1 is added to the analytical solution of [29] to satisfy boundary conditions (18) with C e S ¼ 1 and The function C e B ¼ 1 and to take into account the thermal generation of point defects. As can be seen from Fig. 1, the distribution of nonC equilibrium point defects obtained by numerical computations agrees with the analytical solution suggested in Ref. [29]. 4. Simulation of diffusion of point defects The main goal of this work is to investigate the characteristic features of stress-mediated diffusion of point defects in the bulk of a semiconductor, namely, the different influence exerted by local strained regions and local sinks on the diffusion of nonequilibrium point defects. Therefore, it seems reasonable to minimize or exclude other effects governing the point defect diffusion. For example, it is supposed in this paper that only one species of point defects (vacancies or self-interstitials) is
Fig. 1. Comparison of the numerical (solid line) and analytical (dots) solutions for the equation of diffusion of point defects.
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O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
participating in diffusion. It is worth to note that this case occurs quite often in the technology of silicon devices. For example, a strong supersaturation of self-interstitials in the bulk of a semiconductor can be observed during oxidation of a silicon substrate [31]. A similar situation takes place during rapid thermal annealing of ion-implanted layers. Really, it is commonly accepted that a transient enhanced diffusion of ion-implanted dopant atoms is due to the strong supersaturation of silicon with self-interstitials. Such supersaturation results in a negligible concentration of vacancies due to their recombination with nonequilibrium self-interstitials. We suppose that in the processes when the concentration of silicon self-interstitials greatly exceeds a thermally equilibrium value, the concentration of vacancies in the diffusion zone is too small. Therefore, a segregation of the vacancies in the strained layer does not significantly influence the recombination rate and migration of silicon self-interstitials. Describing the diffusion of point defects in different charge states by means of Eq. (8) (the average migration length li is the effective migration length of all point defects in different charge states), we supposed that the local field of elastic stresses is formed due to nonuniform doping with neutral impurity. Then, v = 1 and relative concentrations of charged defects e r ¼ Cr ¼ hr do not change with an increase or decrease of the concentration of point defects. To emphasize the influence C C of stresses, we also neglect the formation and dissociation of the ‘‘impurity atom-point defects” pairs in the doped layer. To take into account these processes, it is necessary to solve Eq. (8) simultaneously with the equation of impurity diffusion. These equations can be coupled via the expressions for generation and dissociation of pairs obtained in Ref. [32]. The results of simulation of stress-mediated diffusion of point defects in layers heavily doped with an electrically active impurity (v 1), with account for intense formation and dissociation of pairs, will be published elsewhere. To clearly recognize the influence of stresses in all the calculations presented here it is also assumed that the generation of nonequilibrium point defects of one species (vacancies or self-interstitials) occurs at the surface of a semiconductor due to the reaction of noninert gas molecules with the silicon substrate or with the layer covering it. For example, a generation of silicon self-interstitials can occur during oxidation of silicon substrate or nitridation of oxide layer in NH3 [31]. For simplicity, we used the Dirichlet boundary condition (18) on the surface and in the bulk of a semiconductor in all cases under consideration. It is also assumed that the effective diffusivity and effective coefficient of absorption are constant, i.e., d(v) = di and k(v, x) = ki. The average length of migration of point defects li is selected equal to 0.7 lm, i.e., greater than the strained layer thickness. In Fig. 2a the calculated distribution of nonequilibrium point defects in the structure with a local stress field that provides the drift of defects into the strained region is demonstrated. For comparison, the solution of diffusion Eq. (8) in the case of zero stresses is also shown by a dotted line. Note, that distribution of point defects in the neutral charge state is shown in Fig. 2a. To calculate the distributions of charged point defects, one can use the expressions Cr = hrC. The distribution of the normalized drift velocity of point defects in the field of elastic stresses for the case of segregation of defects within the local strained region (see Fig. 2a) is given in Fig. 2b. As can be seen from Fig. 2b, the strained layer is within the range of depths from 0.4 to 0.6 lm, i.e., the field of elastic stresses is located between the defect generation region and the bulk of a semiconductor. To describe the normalized drift velocity of point defects, expression (16) is used. The following values of parameters are selected to provide a small thickness of the strained layer: xmax = 0.5 lm and DRBp = 0.02 lm. Fig. 3a presents the calculated distribution of nonequilibrium point defects in the structure with the local region of stresses preventing the defect diffusion into the strained layer. The distribution of the normalized drift velocity of point defects in the field of elastic stresses used for this calculation is shown in Fig. 3b. As can be seen in Fig. 3b, it is assumed that the local stress field also occupies a position between the defect generation region and the bulk. As is seen from Fig. 2a and Fig. 3a, the presence of the local stresses results either in enrichment or depletion of the stress region of point defects. On the other hand, the distributions of nonequilibrium defects beyond the region of the local stresses are practically unchanged, regardless of the stress barrier that should be overcome by the point defects migrating in the bulk of the semiconductor.
(a)
(b)
Fig. 2. Calculated distribution of the concentration of neutral point defects passing through the region of local stresses (drift velocity directed into the strained layer) (a) and spatial distribution of the normalized drift velocity of point defects used in the calculations (b). A solution for the same diffusion equation in the case of zero stresses is given for comparison.
O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
(a)
1139
(b)
Fig. 3. Calculated distribution of the concentration of neutral point defects passing through the region of local stresses (drift velocity directed out of the strained layer) (a) and spatial distribution of the normalized drift velocity of point defects used in the calculations (b). A solution for the same diffusion equation in the case of zero stresses is given for comparison.
A qualitatively different situation takes place in the case of strong local sinks of point defects, for example, in the case of the silicon structure with an epitaxially grown Si1xCx layer [24]. This can be seen in Fig. 4a showing the point defect concentration profile calculated for the position of a local sink at the same place as the position of local stresses in the previous calculations. Fig. 4b demonstrates the spatial distribution for the effective absorption coefficient of point defects kSp(x) used in the last calculation. It is assumed that the local peak in the spatial distribution of the effective absorptivity is described by the Gaussian function similar to expression (13) with xmax = 0.5 lm and DRBp = 0.02 lm. The significant decrease in the concentration of defects in the regions before and after the sink is easily predictable and agrees with the experimental data obtained in Ref. [24]. It is interesting to note that the concentrations of nonequilibrium defects beyond the region of the local stresses can be significantly changed if the generation/absorption of point defects takes place in this region. To illustrate, in our previous work [17] it is shown that the generation of silicon interstitial atoms in the region of local stresses that provide the drift of point defects from the surface results in supersaturation of self-interstitials in the bulk of a semiconductor. It is worth to compare the results obtained with the experimental data of [22]. In [22] the heterostructure Si/Si1xGex/Si was exposed to a hydrogen plasma at 250–300 °C for 1 h and then at 300–350 °C for two more hours. The samples were grown on Si (1 0 0) substrates using molecular-beam epitaxy. A 5-nm-thick Si1xGex layer was capped by a Si layer of thickness 200 nm. It is interesting to note that this experimental schedule is similar to that of diffusion which is considered in the present paper. Really, the thickness of the strained Si1xGex is small in comparison with the distance to the surface. The hydrogen atoms are introduced into the Si layer from the hydrogen plasma, i.e., through the surface of the semiconductor. It is usually supposed that hydrogen diffuses in silicon crystals due to the interstitial species [33,34]. It is seen from Fig. 1b in [22] that the diffusing hydrogen interstitials overcome the strained Si1xGex layer and migrate into the bulk of the silicon crystal. This means that the average migration length of hydrogen interstitials is significantly greater than the region of local stresses. The diffusion equation of interstitial impurity atoms that considers the influence of elastic stresses was obtained in [35]. The terms describing diffusion and drift of impurity interstitials are similar to the appropriate terms of Eq. (8). On the other hand, the stress-mediated diffusion investigated in [22] is characterized by absorption of hydrogen atoms in the
(a)
(b)
Fig. 4. Calculated distribution of the concentration of neutral point defects passing through the region of local sink (a) and spatial distribution of the effective coefficient of absorption of point defects used in the calculations (b). For comparison a solution of the diffusion equation for the thermally equilibrium uniform absorption of point defects is also presented.
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strained Si1xGex layer by vacancies [21] or platelets that are forming in the strained region [22]. Combining the results of simulation of stress-mediated diffusion and diffusion under the condition of local sink, we can hypothesize that the main features of stress-mediated diffusion remain the same in the case of absorption of mobile species in the strained region. Only the concentration of mobile species decreases substantially due to absorption. Therefore, it is not surprising that the hydrogen concentration profile presented in Fig. 1b of [22] is similar to the concentration profile of silicon self-interstitials calculated in this paper (see Fig. 2a). 5. Conclusions The influence of local elastic stresses on the formation of distributions of point defects during diffusion of nonequilibrium defects from the surface into the bulk of a semiconductor has been investigated by computer simulation. Such local regions of stresses can be formed in the multilayer heterostructures or semiconductor substrates with heavily doped layers. Thus, point defects diffusing into the bulk must pass through the region of stresses if the average length of the migration of defects is greater than the thickness of the strained layer. Within the strained layer, the segregation of point defects or heavy depletion of defects occur if the drift of defects under stresses is directed in or out of the layer, respectively. The numerical calculations show that, in contrast to the case of local sink of defects, the local region of elastic stresses practically does not change the distribution of defects beyond this region provided that there is no additional generation/absorption of point defects within the strained layer. The results obtained for the stress-mediated diffusion of intrinsic point defects are also valid for diffusion of impurity interstitials. 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[30] A.F. Burenkov, F.F. Komarov, M.A. Kumahov, M.M. Temkin, Prostranstvennie Raspredeleniya Energii, Vydelennoi v Kaskade Atomnyh Stolknovenij v Tverdyh Telah (Space Distributions of Energy Deposited in Atomic Collisions in Solids), Energoatomizdat, Moscow, 1985 (in Russian). [31] P. Pichler, Intrinsic point defects, impurities, and their diffusion in solids, in: S. Selberherr (Ed.), Computational Microelectronics, Springer, Wien, NewYork, 2004. [32] O.I. Velichko, A set of equations of radiationenhanced diffusion of ionimplanted impurities, in: I.I. Danilovich, A.G. Koval, V.A. Labunov, et al. (Eds.), Proceedings of VII International Conference ‘‘Vzaimodeistvie Atomnyh Chastits s Tverdym Telom (Interaction of Atomic Particles with Solid)”, Part 2, Minsk, Belarus, 1984, pp. 180–181 (in Russian). [33] C.G. Van de Walle, P.J.H. Denteneer, Y. Bar-Yam, S.T. Pantelides, Theory of hydrogen diffusion and reaction in crystalline silicon, Phys. Rev. B. 39 (15) (1989) 10791–10801. [34] R. Rizk, P. de Mierry, D. Ballutaud, et al, Hydrogen diffusion and passivation processes in p- and n-type crystalline silicon, Phys. Rev. B 44 (12) (1991) 6141–6151. [35] O.I. Velichko, Macroscopic description of the diffusion of interstitial impurity atoms considering the influence of elastic stress on the drift of interstitial species, Phil. Mag. 88 (10) (2008) 1477–1491.
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Simulation of the diffusion of point defects in structures with local elastic stresses Oleg Velichko ⇑ Belarusian State University of Informatics and Radioelectronics, Department of Physics, 6, P. Brovki Street, 220013 Minsk, Belarus
a r t i c l e
i n f o
Article history: Received 20 July 2008 Received in revised form 29 July 2010 Accepted 2 August 2010 Available online 19 August 2010 Keywords: Semiconductors Diffusion modelling Stress-mediated diffusion Point defects Defect engineering
a b s t r a c t The stress-mediated diffusion of nonequilibrium point defects into the bulk of a semiconductor is investigated by computer simulation. It is assumed that the point defects are generated on the surface of a semiconductor and that in the course of diffusion they pass through the local region of elastic stresses because the average length of defect migration is greater than the thickness and depth of the strained layer. Within the strained layer, point defect segregation or heavy defect depletion occurs if defect drift under stresses is directed in or out of the layer, respectively. The calculations also show that, in contrast to the case of local defect sink, the local region of elastic stresses practically does not change the distribution of defects beyond this region if there is no generation/absorption of point defects within the strained layer. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction In recent years decreasing dimensions of integrated components [1,2], the use of various multilayer structures [3], such as Si/SiGe [4–6], and different nano-particles embedded in crystalline [7,8] and amorphous [9,10] matrixes have been the main trends in advanced device technologies. For this reason, much attention is given to studying the defect formation kinetics and stress evolution during semiconductor processing, since defect and stress engineering can significantly improve the device performance [1,2,11]. For example, many efforts are directed at studying the stress-mediated diffusion of impurity atoms and point defects [12–16], including the influence of elastic stresses on drift of point defects near a surface or at interfaces [17–19], and near different inhomogeneities of semiconductor crystals [12,20]. Very interesting experimental data describing hydrogen diffusion in Si/Si1xGex/Si heterostructures were presented by Lin Shao with coauthors [21,22]. It was established that during diffusion the hydrogen atoms were segregated within the Si1xGex layer forming a peak of hydrogen concentration. To explain this phenomenon, it was supposed in [21] that a preferential vacancy aggregation within the strained layer occurred resulting in subsequent trapping of hydrogen atoms. Also, many studies deal with changes in the defect subsystem of ion-implanted layers, since these changes are responsible for transient enhanced diffusion of dopant atoms [1,2,23,24]. For example, to explain experimental data, it is assumed [18] that elastic stresses arising in the region of a SiGe layer buried in silicon and compressively strained after its growth cause a drift of vacancies into this layer. Thus, accumulation of vacancies within the layer and their transformation into nanovoids occur. In Ref. [23], a transient enhanced diffusion of dopant atoms in hyperfine boron-doped layers created by molecular-beam epitaxy is investigated experimentally. It is assumed that the formation of clusters of boron atoms with silicon self-interstitials occurs in these doped regions during thermal treatment. This process and a number of similar processes proceeding during thermal treatment are characterized by local absorption of self-interstitials. Thus, investigations concerned with the influence of local strained regions and ⇑ Tel.: +375 296998078. E-mail addresses: [email protected], [email protected] 0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.08.002
O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
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local sinks on diffusion of nonequilibrium point defects are of great importance for the next-generation fabrication processes. The goal of the paper is to propose a model of stress-mediated diffusion of intrinsic point defects and investigate the essential features of the transport processes when stresses or sink are localized in a thin layer in comparison with the thickness of the diffusion zone. To emphasize the generality of the results obtained no specific values of strain, temperature, and diffusivity are used and the conclusions made are valid for all cases of diffusion when the average migration length of point defects is greater than the thickness of a strained layer or a layer with a sink. 2. Model of stress-mediated diffusion of point defects In order to develop a model of diffusion of point defects in the field of elastic stresses, let us assume that there are no changes in processing conditions during thermal treatment or these changes are slow enough in comparison with the average lifetime of point defects. Then, due to the high mobility of point defects in comparison with the mobility of impurity atoms, the time derivative of the defect concentration is close to zero and the distributions of point defects are quasi-stationary with respect to the distributions of dopant atoms, clusters, extended defects, and also with respect to the changes occurring under the processing conditions. In such a situation, to calculate the concentration distributions of nonequilibrium point defects in the field of elastic stresses, the stationary diffusion equation established in Ref. [17] can be used. It is assumed in [17] that the diffusion of point defects occurs due to the concentration gradient and via a drift in the built-in electric field and the field of elastic stresses. Then, the diffusion equation for point defects Dr (vacancies or self-interstitials in the charge state r) can be written as
r dr rC r þ
r r zr ed C r d Cr ru þ rU r þ SPr ; kB T kB T
GPr Sr þ GTr þ GRr ¼ 0;
ð1Þ
r ¼ 3; 2; ; ; þ; 2þ; 3þ; where Cr, dr, and zr are the concentration, diffusivity, and the charge of a defect in the charge state r, respectively; e is the elementary charge; u is the potential of the built-in electric field; Ur is the potential energy of a defect Dr in the field of internal elastic stresses; SPr and GPrare the rates of dissociation and generation of the pairs (ADr) per unit volume of the semiconductor, respectively; Sr is the rate of absorption of point defects in the charge state r by internal sinks; GTr is the rate of thermal generation of defects Dr per unit volume, while GRr is the rate due to ion implantation. It is assumed in [17] that due to the high mobility of electrons (holes) a local equilibrium holds between the species with different charge states. Then, the concentration of charged defects can be derived from the mass action law:
Cr C
r
vzzr
¼h;
ð2Þ
where C is the concentration of defects in the neutral charge state; hr is a constant of local thermodynamic equilibrium for the reaction of transition of point defects from the neutral charge state to the charge state r; z is the charge of the substitutionally dissolved dopant atom; v is the concentration of electrons n (or holes p) normalized to the intrinsic carrier concentration ni in the case of doping by the donor (or acceptor) impurity, respectively. It is assumed in this paper that the region of defect diffusion can be doped nonuniformly with electrically active impurity atoms. Then, dopant atoms exert an influence upon the diffusion of charged point defects due to the drift of defects in the built-in electric field (see the second term in brackets in Eq. (1)) and due to the change in the concentrations of charged species (see Eq. (2)). Summing up equations (1) written for defects in different charge states and taking into account the mass action law (2), one can reduce the system (1) to one generalized equation of the diffusion of point defects. In Ref. [17] such a generalized equation was obtained for the concentration of point defects in the neutral charge state:
"
# ad rU d ~ SP G P S þ G T þ G R r dðvÞrða CÞ þ dðvÞ ¼ 0; C þ kB T C i d~
ð3Þ
where
~ ¼ C =C ; C Xi r r r d hi vzz ; dðvÞ ¼ SP ¼
X
r
SPr
GP ¼
r
S¼
X r
X
ð4Þ ð5Þ GPr ;
r
Sr
GT ¼
X r
GTr
GR ¼
ð6Þ X r
GRr :
ð7Þ
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O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
~ is the concentration of nonequilibrium neutral point defects normalHere d(v) is the effective diffusivity of point defects; C ized to the concentration of neutral defects at thermal equilibrium C i . It was assumed that the potential energy of point defects in the field of elastic stresses Ur and the constants of local therr r modynamic equilibrium hr depended weakly on the charge states of diffusing species, i.e., Ur Ud and h =hi ad for all r, r where hi are the values of the constants of local equilibrium in an intrinsic semiconductor. For a one-dimensional diffusion, Eq. (3) can be written in the form convenient for numerical solution as [25]
" # C Sp eÞ ~ ~ 1 þ g~R d dðad C dðv~ x CÞ k ðvÞk ðxÞad C C d ðvÞ þ ¼ 0; 2 2 dx dx dx li li
ð8Þ
where C
d ðvÞ ¼
dðvÞ ; di
ð9Þ Sp
C
kðv; xÞ ¼ kðvÞk ðxÞ k ðvÞ ¼
vx
d
kðvÞ ; ki
a dU C ¼ d ðvÞ ; di kB T dx R g g~R ¼ : gi
v~ x ¼
ð10Þ
d
ð11Þ ð12Þ
Here di is the intrinsic diffusivity of point defects; k(v) and kSp(x) are the multipliers representing, respectively, the concentration dependence and spatial distribution of the effective absorption coefficient for point defects k(v, x); kC(v) is the concentration dependence of the effective pffiffiffiffiffiffiffiffi absorption coefficient normalized to the value ki of the absorptivity in an intrinsic 1 semiconductor; si ¼ ki and li ¼ di si are the average lifetime and average migration length of point defects in an intrinsic semiconductor, respectively; vx is the x-coordinate projection of the drift velocity of point defects due to elastic stresses; gR is the rate of generation of nonequilibrium point defects per unit volume due to ion implantation or dissolution of extended defects; gi is the rate of thermally equilibrium generation of point defects in an intrinsic semiconductor. It is to be noted that Eq. (8) differs from the diffusion equation used in Ref. [24] for point defects since the drift of mobile species due to elastic stresses is included. On the other hand, the distributions of nonequilibrium point defects can be derived from Eq. (8), instead of distributions of equilibrium defects in the field of elastic stresses being computed by means of expressions used in Refs. [12,13]. Also, Eq. (8) is convenient for numerical solutions due to the following characteristic features: (i) This equation describes the diffusion–reaction–drift of self-interstitials (vacancies) with different charge states as e must be obtained to solve the equation. (ii) a whole, although only the normalized concentration of neutral point defects C The equation obtained takes into account the drift of all charged species due to the built-in electric field. At the same time, there is no explicit term that would describe this drift. To exclude this term, a special rearrangement and some mathematical manipulations of the terms describing diffusion of point defects due to concentration gradient and drift of charged species in the built-in electric field were performed during the summing up of equations (1). As a result, a drift of the charged species in the electric field is included in the functional dependence of the effective diffusivity of neutral point defects d = d(v). (iii) The effective coefficients dC(v) and kC(v) of Eq. (8) represent smooth and monotone polynomial functions of v [25]. It is assumed in this work that the thin strained layer is located in the bulk of a semiconductor. For example, this buried strained region can be created by local doping of silicon with a substitutionally dissolved impurity having other atomic radius compared to the host atom [26]. Let us consider the case of the diffusion of point defects through the strained layer. To ~ x . The calculation of this function describe this process, it is necessary to define the effective drift velocity of point defects v ~ x is considered as an unspecified is a rather complex problem calling for a separate investigation. Therefore, in this work v function which satisfies the following requirements: (i) it should be equal to zero in the middle and beyond the strained layer; (ii) it should have different signs on the left and on the right of the strained layer. Then, the drift of nonequilibrium ~ x can be approximately defined in the point defects is directed to or out of the middle of the strained layer. For example, v following way. Let us assume that the impurity distribution CB(x, t) in the strained layer is described by the Gaussian function
"
C B ðxÞ ¼ C Bm exp
ðx xmax Þ2 2DR2Bp
#
;
ð13Þ
where CBm is a maximum of impurity concentration; xmax is the maximum position and DRBp is the dispersion of the buried impurity distribution. It is assumed in [27] that the potential energy of a point defect in the field of elastic stresses is proportional to the impurity concentration, i.e., for the migration of nonequilibrium point defects we have
U d ðxÞ ¼ AC B ðxÞ: Then, it follows from expression (11) that
ð14Þ
O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
v~ x ðx; vÞ
@C B ðxÞ : @x
1137
ð15Þ
Taking into account expression (15), one can see that for the impurity distribution given by the Gaussian function (13) the effective drift velocity of point defects vx(x, v) has the following form:
"
v~ x ðx; vÞ ¼ Bðx xmax Þ exp
ðx xmax Þ2 2DR2Bp
#
;
ð16Þ
where the constant B > 0 indicates that segregation of nonequilibrium point defects within the strained layer is provided and B < 0 is for the case of depletion of the stress region of point defects. 3. Numerical solution The finite-difference method [28] is used to find a numerical solution for Eq. (8) in the one-dimensional (1D) domain [0, xB]. Following Ref. [28], the first term on the left-hand side of Eq. (8) is approximated by a symmetric difference operator of second order accuracy on the space variable x. At the same time, the second term is approximated with a first order accuracy by the asymmetric forward/backward-difference operator depending on the drift direction. On the other hand, if the flux of defects due to elastic stresses is comparable to (or lower than) the flux of defects caused by the concentration gradient, the second term is also approximated by a symmetric difference operator of second order accuracy. Comparison with exact analytical solutions for particular cases of diffusion of point defects and calculations on meshes with different step sizes were carried out to verify the approximate numerical solution. For example, in Fig. 1 a numerical solution of Eq. (8) in the case of constant diffusivity and constant coefficient of the absorption of defects is presented. For comparison with the analytical solution of the diffusion equation obtained in [29], the Gaussian distribution
" g~R ðxÞ ¼ g~m exp
ðx Rpd Þ2
#
2DR2pd
ð17Þ
is used to specify the generation rate profile. Here g~m is a maximum generation rate of point defects during ion implantation normalized to the thermal generation rate gi; Rpd and DR2pd are the position of a maximum of defect generation rate and dispersion of generation rate distribution, respectively. It is assumed that the defect generation occurs due to implantation of hydrogen ions at an energy of 20 keV (Rpd = 0.198 lm and DRpd = 0.0802 lm are taken from Ref. [30]). Numerical computations are carried out on the simulation domain [0, xB], where xB and mesh point number iB are equal to 4.0 lm and 81, respece: tively. To obtain a numerical solution, the Dirichlet boundary conditions are imposed on C
e e S; Cð0Þ ¼C
e BÞ ¼ C e B; Cðx
ð18Þ
e S and C e B are the values of normalized defect concentration on the left and right boundaries of simulation domain. where C e th ðxÞ ¼ 1 is added to the analytical solution of [29] to satisfy boundary conditions (18) with C e S ¼ 1 and The function C e B ¼ 1 and to take into account the thermal generation of point defects. As can be seen from Fig. 1, the distribution of nonC equilibrium point defects obtained by numerical computations agrees with the analytical solution suggested in Ref. [29]. 4. Simulation of diffusion of point defects The main goal of this work is to investigate the characteristic features of stress-mediated diffusion of point defects in the bulk of a semiconductor, namely, the different influence exerted by local strained regions and local sinks on the diffusion of nonequilibrium point defects. Therefore, it seems reasonable to minimize or exclude other effects governing the point defect diffusion. For example, it is supposed in this paper that only one species of point defects (vacancies or self-interstitials) is
Fig. 1. Comparison of the numerical (solid line) and analytical (dots) solutions for the equation of diffusion of point defects.
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O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
participating in diffusion. It is worth to note that this case occurs quite often in the technology of silicon devices. For example, a strong supersaturation of self-interstitials in the bulk of a semiconductor can be observed during oxidation of a silicon substrate [31]. A similar situation takes place during rapid thermal annealing of ion-implanted layers. Really, it is commonly accepted that a transient enhanced diffusion of ion-implanted dopant atoms is due to the strong supersaturation of silicon with self-interstitials. Such supersaturation results in a negligible concentration of vacancies due to their recombination with nonequilibrium self-interstitials. We suppose that in the processes when the concentration of silicon self-interstitials greatly exceeds a thermally equilibrium value, the concentration of vacancies in the diffusion zone is too small. Therefore, a segregation of the vacancies in the strained layer does not significantly influence the recombination rate and migration of silicon self-interstitials. Describing the diffusion of point defects in different charge states by means of Eq. (8) (the average migration length li is the effective migration length of all point defects in different charge states), we supposed that the local field of elastic stresses is formed due to nonuniform doping with neutral impurity. Then, v = 1 and relative concentrations of charged defects e r ¼ Cr ¼ hr do not change with an increase or decrease of the concentration of point defects. To emphasize the influence C C of stresses, we also neglect the formation and dissociation of the ‘‘impurity atom-point defects” pairs in the doped layer. To take into account these processes, it is necessary to solve Eq. (8) simultaneously with the equation of impurity diffusion. These equations can be coupled via the expressions for generation and dissociation of pairs obtained in Ref. [32]. The results of simulation of stress-mediated diffusion of point defects in layers heavily doped with an electrically active impurity (v 1), with account for intense formation and dissociation of pairs, will be published elsewhere. To clearly recognize the influence of stresses in all the calculations presented here it is also assumed that the generation of nonequilibrium point defects of one species (vacancies or self-interstitials) occurs at the surface of a semiconductor due to the reaction of noninert gas molecules with the silicon substrate or with the layer covering it. For example, a generation of silicon self-interstitials can occur during oxidation of silicon substrate or nitridation of oxide layer in NH3 [31]. For simplicity, we used the Dirichlet boundary condition (18) on the surface and in the bulk of a semiconductor in all cases under consideration. It is also assumed that the effective diffusivity and effective coefficient of absorption are constant, i.e., d(v) = di and k(v, x) = ki. The average length of migration of point defects li is selected equal to 0.7 lm, i.e., greater than the strained layer thickness. In Fig. 2a the calculated distribution of nonequilibrium point defects in the structure with a local stress field that provides the drift of defects into the strained region is demonstrated. For comparison, the solution of diffusion Eq. (8) in the case of zero stresses is also shown by a dotted line. Note, that distribution of point defects in the neutral charge state is shown in Fig. 2a. To calculate the distributions of charged point defects, one can use the expressions Cr = hrC. The distribution of the normalized drift velocity of point defects in the field of elastic stresses for the case of segregation of defects within the local strained region (see Fig. 2a) is given in Fig. 2b. As can be seen from Fig. 2b, the strained layer is within the range of depths from 0.4 to 0.6 lm, i.e., the field of elastic stresses is located between the defect generation region and the bulk of a semiconductor. To describe the normalized drift velocity of point defects, expression (16) is used. The following values of parameters are selected to provide a small thickness of the strained layer: xmax = 0.5 lm and DRBp = 0.02 lm. Fig. 3a presents the calculated distribution of nonequilibrium point defects in the structure with the local region of stresses preventing the defect diffusion into the strained layer. The distribution of the normalized drift velocity of point defects in the field of elastic stresses used for this calculation is shown in Fig. 3b. As can be seen in Fig. 3b, it is assumed that the local stress field also occupies a position between the defect generation region and the bulk. As is seen from Fig. 2a and Fig. 3a, the presence of the local stresses results either in enrichment or depletion of the stress region of point defects. On the other hand, the distributions of nonequilibrium defects beyond the region of the local stresses are practically unchanged, regardless of the stress barrier that should be overcome by the point defects migrating in the bulk of the semiconductor.
(a)
(b)
Fig. 2. Calculated distribution of the concentration of neutral point defects passing through the region of local stresses (drift velocity directed into the strained layer) (a) and spatial distribution of the normalized drift velocity of point defects used in the calculations (b). A solution for the same diffusion equation in the case of zero stresses is given for comparison.
O. Velichko / Applied Mathematical Modelling 35 (2011) 1134–1141
(a)
1139
(b)
Fig. 3. Calculated distribution of the concentration of neutral point defects passing through the region of local stresses (drift velocity directed out of the strained layer) (a) and spatial distribution of the normalized drift velocity of point defects used in the calculations (b). A solution for the same diffusion equation in the case of zero stresses is given for comparison.
A qualitatively different situation takes place in the case of strong local sinks of point defects, for example, in the case of the silicon structure with an epitaxially grown Si1xCx layer [24]. This can be seen in Fig. 4a showing the point defect concentration profile calculated for the position of a local sink at the same place as the position of local stresses in the previous calculations. Fig. 4b demonstrates the spatial distribution for the effective absorption coefficient of point defects kSp(x) used in the last calculation. It is assumed that the local peak in the spatial distribution of the effective absorptivity is described by the Gaussian function similar to expression (13) with xmax = 0.5 lm and DRBp = 0.02 lm. The significant decrease in the concentration of defects in the regions before and after the sink is easily predictable and agrees with the experimental data obtained in Ref. [24]. It is interesting to note that the concentrations of nonequilibrium defects beyond the region of the local stresses can be significantly changed if the generation/absorption of point defects takes place in this region. To illustrate, in our previous work [17] it is shown that the generation of silicon interstitial atoms in the region of local stresses that provide the drift of point defects from the surface results in supersaturation of self-interstitials in the bulk of a semiconductor. It is worth to compare the results obtained with the experimental data of [22]. In [22] the heterostructure Si/Si1xGex/Si was exposed to a hydrogen plasma at 250–300 °C for 1 h and then at 300–350 °C for two more hours. The samples were grown on Si (1 0 0) substrates using molecular-beam epitaxy. A 5-nm-thick Si1xGex layer was capped by a Si layer of thickness 200 nm. It is interesting to note that this experimental schedule is similar to that of diffusion which is considered in the present paper. Really, the thickness of the strained Si1xGex is small in comparison with the distance to the surface. The hydrogen atoms are introduced into the Si layer from the hydrogen plasma, i.e., through the surface of the semiconductor. It is usually supposed that hydrogen diffuses in silicon crystals due to the interstitial species [33,34]. It is seen from Fig. 1b in [22] that the diffusing hydrogen interstitials overcome the strained Si1xGex layer and migrate into the bulk of the silicon crystal. This means that the average migration length of hydrogen interstitials is significantly greater than the region of local stresses. The diffusion equation of interstitial impurity atoms that considers the influence of elastic stresses was obtained in [35]. The terms describing diffusion and drift of impurity interstitials are similar to the appropriate terms of Eq. (8). On the other hand, the stress-mediated diffusion investigated in [22] is characterized by absorption of hydrogen atoms in the
(a)
(b)
Fig. 4. Calculated distribution of the concentration of neutral point defects passing through the region of local sink (a) and spatial distribution of the effective coefficient of absorption of point defects used in the calculations (b). For comparison a solution of the diffusion equation for the thermally equilibrium uniform absorption of point defects is also presented.
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strained Si1xGex layer by vacancies [21] or platelets that are forming in the strained region [22]. Combining the results of simulation of stress-mediated diffusion and diffusion under the condition of local sink, we can hypothesize that the main features of stress-mediated diffusion remain the same in the case of absorption of mobile species in the strained region. Only the concentration of mobile species decreases substantially due to absorption. Therefore, it is not surprising that the hydrogen concentration profile presented in Fig. 1b of [22] is similar to the concentration profile of silicon self-interstitials calculated in this paper (see Fig. 2a). 5. Conclusions The influence of local elastic stresses on the formation of distributions of point defects during diffusion of nonequilibrium defects from the surface into the bulk of a semiconductor has been investigated by computer simulation. Such local regions of stresses can be formed in the multilayer heterostructures or semiconductor substrates with heavily doped layers. Thus, point defects diffusing into the bulk must pass through the region of stresses if the average length of the migration of defects is greater than the thickness of the strained layer. Within the strained layer, the segregation of point defects or heavy depletion of defects occur if the drift of defects under stresses is directed in or out of the layer, respectively. The numerical calculations show that, in contrast to the case of local sink of defects, the local region of elastic stresses practically does not change the distribution of defects beyond this region provided that there is no additional generation/absorption of point defects within the strained layer. The results obtained for the stress-mediated diffusion of intrinsic point defects are also valid for diffusion of impurity interstitials. 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