Simulation of Al and phosphorus diffusion gettering in Si

Materials Science and Engineering B71 (2000) 175 – 181 www.elsevier.com/locate/mseb

Simulation of Al and phosphorus diffusion gettering in Si V. Kveder *, W. Schro¨ter, A. Sattler, M. Seibt IV. Physikalisches Institut der Uni6ersita¨t Go¨ttingen, Bunsenstr. 11 -15, D-37073 Go¨ttingen, Germany

Abstract We present a quantitative computer model (‘Gettering Simulator’) of phosphorus diffusion gettering (PDG) that allows to simulate the PDG process. The model was checked for Au as a typical substitutional metallic impurity elements and for Co as an example of the fast diffusing interstitial 3d metals in Si. Here we will only discuss the gettering of substitutional metals. The ‘Gettering Simulator’ includes a model for P diffusion for phosphorus concentrations [P] up to the solubility limit. In this model, the main contribution to phosphorus diffusion at [P] B 2× 1019 cm − 3 comes from the kick-out mechanism, while at higher P concentrations the diffusion is dominated by phosphorus vacancy complexes. The latter results in the development of the well-known ‘kink-and-tail’ P and specific self-interstitial profiles. The gettering mechanism is described by a combination of three factors: (1) the Fermi level effect; (2) the formation of phosphorus – metal pairs; (3) the high concentration of self-interstitials in the bulk together with nearly equilibrium concentration in the region of high phosphorus concentration near the surface. The third factor was found to be very important for the PDG of substitutional metals. No local equilibrium is assumed in the model. Instead. the calculations are based on the reaction rates between different species. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Phosphorus diffusion gettering; Aluminum gettering; Au

1. Introduction The efficiency of silicon solar cells depends very strongly on the lifetime of minority carriers. Since silicon is an indirect band-gap semiconductor, the minority carrier lifetime is mainly controlled by non-radiative recombination on impurities with deep electronic energy levels. It is well-known that transition elements like Mo, Nb, W, Ti, Au, Fe and Co in concentrations of only 1013 – 1015 cm − 3, decrease the efficiency of p-type Si solar cells by more than 30%. Since in crystalline silicon solar cells the whole wafer works as an active region, internal gettering, which is widely used in IC technology, is not applicable and external gettering techniques should be used. At present-day knowledge, the combination of back side Al gettering (AlG) with front side phosphorus diffusion gettering (PDG) is the most promising technique for increasing the carrier lifetime in silicon solar cells (see [1 –4]). The mechanism of Al gettering (AlG) is quite simple and based on segregation. A thick (usually 2 –10 mm) * Corresponding author. Present address: Institute of Solid State Physics, Chernogolovka, 142432 Russia.

Al layer deposited on the Si wafer surface, provides a gettering because the solubility of many metals in Al at 700–900°C is by 4–10 orders of magnitude higher than in crystalline Si. PDG is a much more complicated process. It depends on the type of gettered metallic impurity much stronger than AlG. During PDG not only metallic impurity diffusion occurs, but also phosphorus diffusion. The process of phosphorus diffusion influences strongly the profile of Si self-interstitials, which is dramatically important for the gettering of substitutional metals [3,4]. Therefore, to model the PDG we must solve simultaneously the strongly coupled equations describing the diffusion of phosphorus, metal atoms, Si vacancies and Si self-interstitials, including the reactions between the different species. In silicon solar cell technology the PDG process can be combined with the formation of the active p–n junction of the device. To optimize this process, we must, besides calculating the evolution of the metal impurity concentration in the wafer bulk (see [3,4]), precisely calculate the distribution of phosphorus and metal atoms in the vicinity of the p–n junction. This is important because for solar cell technology, one should adjust the gettering process in such a way that the metal

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impurities are collected in a thin surface layer, but not in the active region of the p – n junction. Such a reasonably precise description of PDG is the topic of this publication.

2. Calculation of phosphorus diffusion The mechanisms of P diffusion in Si and the role of intrinsic point defects have been the object of extensive investigations (see [6,7] for a review). The experimentally observed enhancement of P diffusion during surface oxidation [8], as well as the ‘emitter – push effect’ and the dislocation – climb experiments [9] strongly support the idea that the kick-out mechanism gives the main contribution to P diffusion. Nevertheless, the concentration dependence of the P diffusivity is still not well-understood and different models were proposed (see, for example [10 – 12]). Obviously, many different processes contribute to the P diffusion in Si. For our purpose we have tried to make a simplified model of P diffusion that describes reasonably well the concentration profiles of P and Si self-interstitials using a minimal number of fitting parameters. First of all, our model includes the kick-out mechanism. We suppose that substitutional phosphorus PS is not mobile, while the interstitial phosphorus PI has a rather high diffusion coefficient Dp. Two reactions are taken into account in this mechanism: the kick-out reaction of substitutional phosphorus PS with Si interstitials I: P+ S +I + e l PI

(1)

and the generation – annihilation of Si vacancies and interstitials:

I+ V0 l 0

(2)

where e stands for the free electron, I is the neutral Si interstitial and V the Si vacancy. This gives four pairs of coupled partial differential equations describing the evolution of concentration profiles for PS, PI, I and (V0, V−). Our calculations show that this simple kick-out mechanism describes very well the experimentally observed P profiles in case of low phosphorus concentration [PS]B 3× 1019 cm − 3 but it cannot describe the ‘kink-and-tail’ profiles, experimentally observed at higher P concentrations (see Fig. 1A). We have found that ‘kink-and-tail’ profiles can be easily obtained if, in addition to the kick-out mechanism, we assume that the substitutional phosphorus is also mobile and its diffusivity D eff S is proportional to [PS]2. Indeed, experiments [13] show that at high P concentrations (but below the solubility limit) the effective diffusivity of P is proportional to [P]2. To give it some physical meaning, we assume that in addition to the kick-out mechanism, the substitutional phosphorus can diffuse by formation of some mobile complexes 2 with vacancies. To have D eff S 8 [PS] , the concentration of complexes must be proportional to [PS]2. We considered two possible mobile complexes of this kind: (P+ S − V2 − ) and (2P+ S V ). The first complex is just a negative E− center. The second complex consists of two P+ S atoms and a negative vacancy V−. Both complexes describe quite well the diffusion in the ‘kink’ region of the P profile. From a mathematical point of view, the sets of equations for both complexes are very similar, but there are several reasons why we prefer the second complex. Only some of them will be mentioned here:

Fig. 1. (A) The calculated concentrations profiles of PS, PI, (C), V− and I for phosphorus in-diffusion for 60 min at 900°C. Dashed curves show [PS] and [I] profiles calculated using only the kick-out mechanism (the P – V complexes were neglected). (B) Comparison of calculated and experimental [18] concentration [PS] profiles obtained after 60 min diffusion at 900°C. All parameters are the same for all curves except the P surface concentration.

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1. In the case of E− centers, the electric field in the p–n junction slows down the diffusion of complexes. To obtain a good fit of the experimental data we must then assume a rather high diffusion coefficient DC of complexes. This difficulty is absent in − case of the (2P+ S V ) complex. 2. To obtain a good fit in the case of E− center we must use a value of DV [V]eq.intr.,. two orders of magnitude smaller than the one obtained by Tan − and Go¨sele [14], while in the case of (2P+ S V ) DV [V]eq.intr. need only to be 3 – 5 times smaller. − 3. The binding energy of the (2P+ S V ) complex should + 2− be larger than for (PS V ). That should lead to a − higher concentration of (2P+ at high S V ) temperature. The formation and dissociation of complexes can be described by two reactions: − − 2P+ l (2P+ S +V S V )

(3)

− In addition to (3), the direct process of (2P+ S V ) generation and recombination is possible: + − 2P+ S +el (2PS V ) +I

(4)

The ratio of complex formation rates by reactions (3) and (4) is about (3)/(4) =exp((Eb −EG/2)/kT), where Eb is the binding energy of the complex and EG is the formation energy of the I – V pair in the bulk. Assuming that EG/2 is about 2.5 eV, we find that reaction (4) will dominate if Eb \2.5 eV. According to estimations [15], Eb for the E center is about 1.86 eV. Now, if we have one more P+ S atom in a second coordination site, it should add at least 0.5 eV to Eb, so that reaction (4) can be quite efficient. In summary, the proposed model includes the diffu− sion of PI, V0/ − , I and (2P+ S V ) complexes and takes into account the reactions (1 – 4). The calculated profiles are mainly controlled by the following parameters: 1. DP([PI]/[PS])eq.intr. where DP is the diffusion coefficient of interstitial phosphorus atoms PI. The index eq.intr. means in equilibrium conditions (including equilibrium concentrations of I and V and in intrinsic material). This first fitting parameter controls the P profile depth in the ‘tail’ region. 2. DC([C]/[PS]2)eq. Intr., where [C] is the concentration of complexes in equilibrium conditions in intrinsic material and DC is their diffusion coefficient. This second fitting parameter controls the depth of the ‘kink’ region of the P profile. 3. [PS]surf: Concentration of phosphorus at the surface. This third fitting parameter is the only one that is not universal because it is strongly processdependent. 4. (DI[I])eq: We used the value (DI[I])eq = 1.49× 1026 exp(− 4.95 eV/kT) cm − 1 s − 1 published in [16]. 5. (DV [V])eq.intr.: We used the expression (DV[V])eq.intr. =1022 exp( − 4.03 eV/kT) cm − 1 s − 1

177

that is close to the one published in [14] (note that our model is not very sensitive to this parameter). Other parameters like, for example, the lifetime of PI or the time-constants of the V–I reaction are not critical for the calculation since they were assumed to be much shorter than the diffusion time. These parameters may become important at very short diffusion times. Fig. 1(A and B) shows calculated concentration profiles in comparison with experimental data. Note that all parameters are the same for all curves shown in Fig. 1(B), except for the surface concentration. As one can see in Fig. 1(A), the [I] has almost equilibrium value in the ‘kink’ region, while there is a very strong supersaturation of self-interstitials [I] in the bulk. The reason for that is the following: the flux of [PI] carries not only phosphorus atoms from the surface to the bulk, but also ‘virtual’ self-interstitials I. Indeed, every PI atom produces an I when it comes to the substitutional state PS. If we had only the ‘kick-out’ mechanism, the PS concentration profile would show how many I were produced at given x coordinate by this mechanism. The generated I diffuse both to the left surface (that is supposed to be an ideal sink for I) and to the bulk, resulting in a high non-equilibrium concentration [I] in the wafer. The [PS] and [I] profiles, calculated under the assumption that P diffusion is controlled only by the ‘kick-out’ mechanism, are shown as dashed curves in Fig. 1(A). One can see that in the ‘kink’ region, the PV complexes give the main contribution to the phosphorus diffusion. They carry ‘virtual’ vacancies that recombine with I. For that reason there is nearly no I supersaturation in the ‘kink’ region. It will be shown below that this feature is very important for the gettering process. One should note here that the two mechanisms of diffusion (via PI and via PV complexes) suppress each other: the ‘kick-out’ diffusion generates I, which reduce the concentration of complexes, and vice versa. That makes the system strongly non-linear, especially at the early stages of in-diffusion, when both mechanisms give comparable contributions at the same depth in the sample. That can, in principle, result in some self-organization phenomena and a very non-uniform distribution of phosphorus at short diffusion time (in the range of a few minutes). To prove this idea, one needs 3D (or at least 2D) calculations.

3. PDG of substitutional metals Let us discuss the PDG of metal impurities that are dissolved predominantly on substitutional sites in intrinsic silicon (like Au, Zn, Pt). In this case, kis =([MI]/ [MS])eq.intr. B B 1 and [M]eq. = ([MI]+ [MS])eq. :[MS]eq.. That is, how does PDG work in case of metals like Au?

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PDG is a combination of three factors. Two of them are due to segregation phenomena. If for some reasons the free energy of a metal atom in a heavily P-doped region is by the value DG lower than in the undoped wafer bulk, this will result in a higher equilibrium concentration atom [MS]eq.P of the metal in the P-doped region by a factor S = [MS]eq.P/[MS]eq.B =exp(DG/kT). For all transition elements there are at least two reasons for the energy reduction of DG in the P-doped region.

3.1. Fermi le6el effect If the substitutional metal atom has an acceptor level EC –EM1 in the band-gap, the concentration of nega0 tively charged metal atoms is [M− S ]= [MS]exp((EF − EM1)/kT), where EF is the Fermi energy. If the metal has a second acceptor level EC – EM2, then again [MS− 2]= [M− S ]exp( −(EF −EM2)/kT) and so on. In intrinsic material the Fermi level E 0F is almost in the middle of the band-gap, while in the P-doped region EC − EF = EC −E 0F −kT log(nP/ni), where ni and nP are the free electron concentration in undoped and P-doped regions at the process temperature T (note that nP = 0.5([PS]+([PS]2 + 4n 2i )1/2) and, therefore, nP :[PS] at [PS]\5× 1019 cm − 3). This gives the segregation factor:

SF =

[MS]eq.P = [MS]eq.B



 

nP n 2 +kF2 P +… ni ni 1 +kF1 +kF2 +…

1 + kF1

(5)

where kF1 =exp((EF0 −EM1)/kT), kF2 =kF1 exp((EF0 − EM1)/kT), and so on. For gold we have only one acceptor level EM1 =EC −0.55 eV [19] and SF is about 100 at [PS]=3× 1020 cm − 3 and T =900°C.

SFP =

]eq.P [Mtotal S [MS]eq.B

=

−

3.2. Formation of (P M ) pairs In a heavily P-doped region, the energy of negatively charged metal atoms can be drastically reduced by the formation of phosphorus – metal pairs according to the reaction: MI + e+ P+ S l (PSMS) +I

(6)

It gives an additional energy reduction EB in comparison with just a negatively charged metal atom M− S . EB is the pair binding energy. For a singly charged ion, EB is typically about 0.5 – 0.8 eV. This gives an addi− − tional segregation effect [P+ S MS ]/[MS ]P =(4[PS]/ [L])exp(EB/kT) where [L] is thc concentration of Si atoms ([L]=5 × 1022 cm − 3). The combination of both mechanisms gives a total segregation factor:

2

+…

(7)

where kB1 = 4 exp(EB/kT)/[L], kB2 = 4 exp(EB2/kT)/[L] + and so on. EB1 is the binding energy of M− S with PS , −2 + EB2 is the binding energy of MS with PS , and so on. Assuming EB1 = 0.85 eV for Au, we have SFP :1.6× 104 at [PS]= 3× 1020 cm − 3 and T= 900°C. The experimentally observed value [Mtotal]P/[M]B for Au [2] is more than 106. So, we need some additional gettering factor.

3.3. Non-equilibrium self-interstitial distribution As one can see in Fig. 1, the kick-out mechanism of P diffusion, dominating in the ‘tail’ concentration region [PS]B 3× 1019 cm − 3, results in a strong supersaturation of the wafer bulk with self-interstitials. At the same time, in the ‘kink’ region of the P profile, where the diffusion via phosphorus vacancy complexes dominates, the concentration of self-interstitials [I]P has almost its equilibrium value. During P diffusion at 900°C the ratio [I]B/[I]P is more than 100 (see Fig. 1). This gives a very large additional gettering effect for metals that mainly occupy substitution sites in Si (like gold). For such metals [MS]eq \ \ [MI]eq, but the diffusion coefficient of interstitial atoms MI is so much higher than that of the substitutional species MS that interstitial atoms give the main contribution to the diffusion flux. The balance between MS and MI is maintained by a kick-out reaction: M0S + I l M0I

(8) 0 I

+ S



n nP + kF2(1 +kB2[PS]) P ni ni 1 + kF1 + kF2 + …

1+ kF1(1 +kB1[PS])

0 S

This gives [M ]/[M ]= kis[I]/[I]eq.. The diffusion tries to equalize the chemical potential of the mobile atoms MI and therefore, the strongly non-uniform distribution of self-interstitials results in an additional gettering effect. This is illustrated by Fig. 2, which shows the chemical potentials of different metal species in the intrinsic bulk and a heavily doped region. The final maximum gettering effect is about: S= [Mtotal]P/[Mtotal]B = SI(SFP + kis)/(1 + kisSI)

(9)

where SI = [I]B/[I]P. For gold, kisSI B B 1 and S:SISFP Since we have SI = 100 it can give us S \ =106 in a good correlation with experimental data. In addition, the supersaturation of the wafer bulk by self-interstitials results in a strong enhancement of the effective diffusion coefficient of metals thus reducing drastically the time necessary for gettering. To simulate the Au gettering we just added one more flux equation for the diffusion of [MI] to the equations

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Fig. 2. The chemical potentials of different metal species in a bulk wafer and in a heavily P-doped layer. The supersaturation [I]/[I]eq. increases [MI], increasing the total metal concentration in the P-doped layer.

Fig. 3. Simulation of Au PDG experiment [2]. Initial Au concentration was uniform and equal to 3× 1014 cm − 3. Points show the experimental SIMS profiles for P and Au after PDG process. (A) PDG at T =980°C, 30 min, then slow cooling down to 900°C with a rate of 5 K min − 1, then fast cooling. (B) Additional annealing of sample after PDG process A at 1150°C, 15 min and quench. The PG was removed from the surface before the second annealing (no P flux on the surface).

describing the P diffusion, and the generation – annihilation terms, corresponding to reactions (6) and (8), that control concentrations [MS] and [MSPS] In addition to the parameters controlling the P diffusion, we now need a few more critical parameters controlling the gettering process: 1. DMkis, where kis =([MI]/[MS])eq.intr. and DM is the diffusion coefficient of interstitial metal atoms. For our calculations we used the value calculated in [17] from Au in-diffusion experiments in heavily dislocated Si:DAui([AuI]/[AuS])eq.intr. =6 7×10 − 3 exp(− 1.21 eV/kT) cm − 2 s − 1. 2. The acceptor energy level of MS. This value is well-known for gold: EM =EC −0.55 eV.

3. The binding energy EB of (MSPS) We have used EB = 0.87 eV. 4. The initial metallic impurity concentration in the wafer. Fig. 3 shows, as an example, the simulation of wellknown experimental results [2] concerning PDG of Au. The calculated profiles are quite close to the experimental data taken from [2]. Note, that in addition to the parameters necessary for describing the P profile, we did not use any free fitting parameter to describe the Au profiles. All the three gettering factors decrease exponentially when the temperature increases. Therefore. the lower the temperature, the lower is the residual metal concen-

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tration. But, the time necessary for gettering increases exponentially with decreasing temperature. (This holds for the AlG process as well). Therefore, the gettering process should be optimized to obtain the maximal gettering effect in minimal time (or, to optimize the (device efficiency/device cost)-ratio). The PDG process is strongly non-linear and it has some ‘memory’ because the strong I supersaturation of a wafer, obtained at high temperature, can survive quite a long time after (or during) cooling down. Therefore one should simulate the whole time – temperature regime to optimize it.

The optimum PDG regime for substitutional metals should start from annealing at high temperature until the high I supersaturation will be reached almost everywhere in the wafer. Then the sample should be cooled down with some optimum rate, slow enough to give time for metal atoms to diffuse, but fast enough to maintain the high I supersaturation. After reaching some temperature, where the PDG becomes already too slow, one can decrease the temperature quite fast. Fig. 4 shows the calculated [Au] profiles obtained in 60 min of PDG in different temperature regimes: One can see that the process with slow cooling gives a better result for the same total process time. Please note that the processes in Fig. 4 are of course not optimal. It just illustrates how important optimization may be.

4. Combined of PD and Al gettering. The program for PDG simulation can be easily used also for AlG simulation.To describe AlG it is enough to just give a boundary condition for the metal flux FM in the Si/Al boundary in the form [4,5]: FM = (D1/l)([MI]− [MAl] [MI]eq/[MAl]eq) Fig. 4. The [Au] profiles calculated for 60 min of PDG in different temperature regimes: (1) 60 min at 980°C and fast cooling. (2) 60 min at 900°C and fast cooling. (3) 38 min at 980°C then slow cooling for 22 min with a rate of 4°C min − 1 then fast cooling. The P concentration on the left surface is 3.2 × 1020 cm − 3, the initial Au concentration is 3×1014 cm − 3.

Fig. 5. Simulation of simultaneous PDG and AlG. Surface concentration [PS] =3 × 1020 cm − 3 at the left side, Al-layer 2 mm thick at the right side, gettering during 60 min at 980°C and fast cooling. The calculated profiles of [Autotal] and Si I are shown for AlG and PDG alone and for a combination of PDG+ AlG.

(10)

where [MI] is the concentration of mobile (usually interstitial) metal atoms in Si [MAl] is the concentration of M in the Al layer [MI]eq and [MAl]eq are the solubility of mobile metal atoms in Si and in Al at the process temperature. DI is the diffusion coefficient of mobile metal atoms and l is the atom jump distance [4]. To calculate the concentration of M in the Al layer [MAl], we integrate in time the flux of M given by Eq. (10). This gives us the amount of metal in the Al layer. The concentration [MAl] is calculated just by dividing that amount by the thickness of the Al layer (we assume that the diffusion of M in a layer of liquid Al a few microns thick is very fast so that the metal concentration is uniform). Fig. 5 shows thc results of simulation of simultaneous AlG and PDG in comparison with the left sidePDG and right side-AlG alone. The dashed curve corresponds to AlG. The AlG process is mainly limited by in-diffusion of I into the wafer. This is quite natural because only interstitial AuI atoms are mohile and therefore, to remove one substitutional AuS atom from the sample to the AI layer one I must bc absorbed in the bulk creating a large I undersaturation in the wafer. As a result, the concentration of Au at the left surface, opposite to the Al layer is much smaller than the Au concentration in the middle of wafer. This is the socalled ‘surface proximity effect’ described in [4]. AlG is quite insensitive to the individual values of [I]eq or DI, but it is very sensitive to the value of the product DI[I]eq. Fortunately, the value of DI[I]eq is quite wellknown now. Since the P-in-diffusion leads to an I

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5. Conclusions Our computer program, based on the model described above, can successfully simulate the PDG and AlG processes for substitutional metals (like Au). For substitutional metals the gettering capahility of PD is based on three effects: (1) the Fermi level effect; (2) the − formation of (P+ S M )pairs and (3) the non-equilibrium distribution of self-interstitials. For combined PD- and Al-gettering. this self-interstitial supersaturation in the bulk gives a substantially enhanced gettering effect when both methods are applied simultaneously. We show also that in the presence of a high dislocation density the efficiency of PDG is strongly reduced, while that of AlG is increased. For combined PD and Al gettering, the reduction of [I] close to [I]eq, through such a high dislocation density will destroy the synergetic effect, resulting in a gettering effect which is more or less just the sum of the single processes.

Acknowledgements

Fig. 6. The calculated profiles of [Autotal] after PDG and AlG in dislocation free (’N’) and strongly dislocated (D, ND = 107 cm − 2) wafers. At the left surface [PS] =3× 1020 cm − 3, Al-layer 2 mm thick at the right side, gettering during 50 min at 980°C, fast cooling.

supersaturation, the simultaneous use of PDG and AlG is not just a simple superposition of both, but it shows a strong synergetic effect. All calculations above were made assuming a small density of dislocations and stacking faults. Dislocations are known to be an efficient sink for I. In a sample with a high dislocation density the dislocations can strongly influence the gettering process by keeping the I concentration close to equilibrium [I]eq. Therefore, dislocations can a bit enhance AlG, but suppress PDG. This is illustrated by Fig. 6, where the profiles for [Au] and [I] are presented for a dislocation free sample and for a sample with a rather high dislocation density (about ND =107 cm − 2). By reducing [I] close to [I]eq, a high dislocation density will destroy the synergetic effect of combined PDG-AlG. rcsulting in a gettering effect which is more or less just the sum of the single processes. In addition, one should note that since PDG generates a very large I supersaturation, it will result in the growth of stacking faults in a sample if it already has dislocations or other nucleation places for stacking faults. This should be taken into account when one analyzes all consequences and details of the PDG process. .

The authors gratefully acknowledge financial support by the BMBF under contract No. 0329743C.

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