- Email: [email protected]
Analysis of solar cells using the IBIC technique
Nuclear Instruments and Methods in Physics Research B 158 (1999) 445±450
www.elsevier.nl/locate/nimb
Analysis of solar cells using the IBIC technique K.K. Lee *, D.N. Jamieson School of Physics, Microanalytical Research Centre, University of Melbourne, Parkville 3052, Australia
Abstract The ion beam induced charge (IBIC) technique has been applied to investigate the relative eciency in polycrystalline silicon solar cells. Relative eciency is de®ned as the charge pulse height normalised with respect to the maximum charge collected. Using IBIC, solar cells have been imaged with microbeam both normal and transverse to the collection junction to obtain the charge collection. The measured charge collection at the grain boundary (GB) of the crystals is then compared to computer simulation to obtain the diusion length of the bulk and the eective recombination veGB locity Seff at the GB. The dependence of the relative eciency on the carrier injection level is then investigated using GB both MeV alphas and molecular hydrogen. We also show that Seff depends on the excess carrier injection level. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Ion beam induced charge; Surface recombination; Diusion length; Grain boundary
1. Introduction Interest in the electronic properties of polysilicon semiconductor has grown substantially because it has found wide use in solar cells, transistors, interconnects, thin-®lm resistors, and varistors. As compared to electronic devices produced from single crystal wafer, the performance of polysilicon devices is limited due to the recombination of minority carriers at the GB. In order to improve the capability of the polysilicon devices, a better physical understanding of the in¯uence of the GB on the minority carriers is therefore required.
* Corresponding author. Tel.: +61-3-9344-5081, fax: +61-39347-4783; e-mail: [email protected]
IBIC is a powerful tool for measuring transport properties in electronic materials. A MeV ion beam, which creates a high density of electron hole pairs, is normally scanned over a sample. Any defect sites that possess trap levels in the gap is capable of trapping minority carriers and will show up as areas of diminished charge collected in an IBIC image. Hence, IBIC is an excellent technique for the investigation of the eect of GB on minority carrier transport for polysilicon. IBIC analysis and computer simulation of charge pulse height from a GB have been developed by Donolato [1]. Their analytical model is able to deterGB . mine the minority carrier diusion length and Seff GB However it assumed that Seff is a constant value. This assumption is only valid for low injection regime as pointed out by several authors [2,3] using either optical illumination or electron beam
0168-583X/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 0 3 3 0 - 4
446
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
injection. In this paper we will use numerical method to calculate the eect of GB on the minority carriers. By comparing the IBIC experimental data with the numerical solutions we shall GB also varies as a funcshow that the obtained Seff tion of excess electron density. The calculated IBIC eciency pro®les were then compared to GB is Donolato's model which assumes that Seff constant; that is the relation between charge recombination and excess minority carrier concentration at the GB is linear. 2. Numerical model The atoms at the boundaries are thought to be disordered and are also preferential sites for segregation of impurities. These introduce defect states into the forbidden gap which act as traps and recombination centers [4]. These states at the silicon GBs favour the formation of majority carrier trapping levels within the band gap [5]. The ®lled traps are also known as ionized traps, as they carry charges, creating a potential energy barrier, which impedes the movement of charge from one grain to another. In view of this, a GB is normally treated as a double depletion layer, that is two Schottky barriers back to back as shown in Fig. 1(a). Fig. 1(a) shows a band diagram of the space± charge region (SCR) region along a GB in p-type GB which silicon. The model require to compute Seff has an in¯uence on the minority carriers transport within the grain has been detailed elsewhere [6,7], we will brie¯y summarize its main points. The injection of minority carriers results in a nonequilibrium condition, as shown in Fig. 1(b). Assuming that the barrier height remains at its equilibrium value before any ion impring onto the semiconductor. Then as the excitation due to the incident ion is applied, the electron quasi-Fermi level, EFN , moves above the hole quasi-Fermi level, EFP , that is the electron density increases, which leads to an increase in the net electron capture rate of the trapping states. In the steady state, the number of electrons occupying these states is larger than that in equilibrium; the donor-like traps are ®lled more rapidly than in equilibrium. The population of donor-like traps beneath EFN will depend on the
Fig. 1. (a) Energy band diagram of GB region of a p-type semiconductor at equilibrium. Ef is the Fermi level, Ec is the energy of the bottom of the conduction band, Ev is the top of GB is the space the valence band, Eg is the band gap, and WSCR charge region of the grain boundary. The GB is at x 0. (b) Under non-equilibrium condition, EFN moves above EFP with /B as the barrier potential. The dierence between the quasiFermi levels re¯ects the level of excitation of the grain boundary.
number of excess minority carriers injected, which will in turn determine the surface recombination at GB is a function of the excess mithe GB. Thus Seff nority carrier density. As a result of the traps ®lling up, the net charge and the barrier height of the GB are reduced. The analysis is made with the assumption that a single donor trap level (either neutral or positively ionized) is located near midgap, which is consistent with experimental measurements [8]. The assumption of ¯at quasi-Fermi levels within the SCR of the GB (quasi-equilibrium condition) enables the Poisson's equation to be related to the Shockley Read Hall capture emission statistics [9] with the appropriate boundary conditions which leads to the solution of barrier height as a function of excess carrier injection. GB can be Once the barrier height is known, then Seff obtained. In this work, we have used n p polycrystalline silicon solar cells with ohmic contacts made to
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
both n-type and p-type silicon. The electron continuity equation in the p-type base is solved numerically and subject to appropriate boundary conditions. The three-dimensional excess electron density equation is o2 N o2 N o2 N N g x; y; z 2 2 ÿ 2 0; ox2 oy oz Ln Dn
1
where N N x; y; z is the excess electron density, Dn is the electron diusion coecient, Ln is the electron diusion length, and g x; y; z is the electron hole pair generation rate. In this work, we consider only vertical GB [10]. Z axis is chosen to be in the direction of the depth of the cells, y axis to be along the plane of the GB, and x axis is perpendicular to this plane. Sundaresan has explained qualitatively that the excess electron density obtained from the threedimensional equation diers from the two-dimensional one by a constant factor for all excitation conditions [7]. Therefore the reduction of the three-dimensional equation to two dimensions will not aect our interpretation as we are comparing the relative values. Then Eq. (1) reduces to o2 N o2 N N g x; z 2 ÿ 2 0; ox2 oz Ln Dn
2
where N N x; z is the excess electron density and g x; z is the two-dimensional generation rate. The boundary conditions are that the excess minority carrier at the ohmic contact vanishes, N z ! 1 0, similarly at the SCR, N z 0 0. The grain is assumed to be semiin®nite N x ! 1 0, and at the edge of the GB's SCR, x 0, Dn
oN GB N: Seff ox
3
Numerically N is determined by subdividing the grain into a rectangular grid, and Eq. (2) is approximated using a ®nite dierence method. At each grid point, the residuals of Eq. (2) are simultaneously reduced to zero by the iterative solution of a matrix equation using successive over-relaxation (SOR) with Chebyshev acceleration [11]. The generation carrier volume could be computed using the analytical method described in
447
[12]. However, for computation eciency, we have used a varying LET along the length of the path based on SRIM [13]. 3. Experimental detail Commercial polycrystalline silicon solar cells that have untextured surfaces were used in this study. In order to remove any eects due to grain size, grains much larger than the base minority carriers diusion length must be chosen. IBIC analysis on polycrystalline silicon solar cell is problematic due to the poor signal to noise ratio S=N associated with the large junction capacitance. Device bias leads to a further reduction of S=N due to the shot noise associated with leakage current. We improve the S=N by mounting the preampli®er as close as possible to the sample, thereby reducing the stray capacitance and eliminating possible ground loops. The sample target stage consists of an in-vacuum Amptek A250 charge sensitive preampli®er. Prior to IBIC analysis current±voltage characteristics and reversebias capacitance measurements were performed to determine the leakage current and the base dopant in the sample respectively. These are necessary for numerical modelling of the IBIC results. 4. Results and discussion Minority carrier diusion length of the grain can be obtained with the incident beam normal to the collection junction. Fig. 2 shows a plot of charge pulse height as a function of incident ion energy and species. The IBIC signals were extracted from regions some several hundred microns away from the GB. Using the one-dimensional expression described in [14] the proton data was ®tted with a diusion length of 43 lm. As a check for consistency the same value for the diffusion length was computed for the alpha data. In this case the result obtained are in good agreement with the experimental observation. In the calculation, the charge collected over the heavily doped n region (dopant density of the order 1019 ÿ 1020 cmÿ3 ) is taken to be negligible as compared to
448
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
Fig. 2. A plot of the charge pulse height as a function of ions energy for both alpha and proton beams. The solid line and dashed line were calculated from [14]. Solid dot denotes proton data, and open dot denotes alpha data.
the base minority carriers diusion length. This is because Auger recombination would result in lower lifetimes in this heavily doped layer [15]. The minority carriers diusion length can also be obtained using the lateral geometry, that is with the beam parallel to the collecting junction. The solar cells sample was then mounted in this con®guration and the beam is allowed to rastered over the width of an unbiased sample. Fig. 3 shows an IBIC relative eciency pro®le obtained in the lateral geometry. An analytical solution derived using Green's function is given by [16], Charge pulse height x0 Z R oE x0 ; z0 Q x0 ; z0 dz0 ; ox 0 0
Q x0 ; z0 eÿkx ÿ
2S pDn
1 l2 k 2 2 L;
Z
1 0
4 k
l2 l 1
0
eÿlz sin kx0 dk; 5
where S is the surface recombination velocity. R is the end of range of the incident particle. L is the minority carriers diusion length. oE=ox is the
Fig. 3. IBIC pro®le in the lateral con®guration for 2 MeV proton and alpha beams. The solid line was calculated, refer to text.
stopping power, which can be obtained from SRIM [13]. Eqs. (4) and (5) were used to obtain the relative eciency by normalising to the maximum charge pulse height and a ®t is made to both the proton and alpha data in Fig. 3. L is determine to be 41 lm and S is 1:0 104 cm sÿ1 . In the modelling of the electron transport in the grain under the in¯uence of GB, the average value of the diusion length as determined by the above mentioned two techniques was incorporated. Fig. 4 shows a plot of the relative eciency as a function of spatial distance from the GB for 2 MeV alpha and molecular hydrogen beams. IBIC linescans were rastered perpendicular to the GB. Fig. 4(a) shows that the calculated relative eciency GB is reasonable when the beam is using constant Seff far away from the GB (> 10 lm). However the GB deviates from calculation based on constant Seff the experimental value near the grain boundary. The reason for this deviation is that the injected minority carriers density that arrives at the GB is lower when the beam is far away from the GB than when the beam is in the vicinity of the boundary. The same conclusion can also be made from Fig. 4(b), which is obtain with a 2 MeV He beam;
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
449
Fig. 5. A plot of eective recombination velocity at the grain boundary as a function of depth for 2 MeV H 2 with the beam hitting the sample at a distance 3 lm from the grain boundary.
Fig. 4. Relative IBIC eciency as a function of beam position from the grain boundary for (a) 2 MeV H 2 , and (b) 2 MeV He .
here, the deviation between measured data and the GB is calculated relative eciency using constant Seff even more pronounced. Our work which uses nonGB could reproduce the experimental data linear Seff GB at the accurately. This demonstrated that the Seff GB is non-linear and is highly dependent on the excess minority carriers density in the grain. We have assumed that the trapping states are monoenergetic and are located in mid-gap. The eective density of such states is estimated to be 1012 cmÿ1 . GB Fig. 5 shows the relation between Seff for 2 MeV H2 beam and depth, z within a grain. The calcu-
lation is for beam hitting the sample 3 lm from GB the GB. The calculated Seff shows a minimum at around 15 lm which can be related to the Bragg peak for 2 MeV H 2 particle. At the Bragg peak, the degree of generation of e±h pairs caused by a ionising particle were higher than elsewhere along the track. As a consequence, the excess electron density that arrives at the GB was greater and leads to a reduction of the number of the trapping states at the GB, which in turns causes a drop in GB . the Seff
5. Conclusion We have solved the two-dimensional electron transport problem using ®nite dierence method GB . With the for the general case of nonlinear Seff assumption that the trapping states were at midgap and are monoenergetic, we were able to deGB and the trapping density. termine the nonlinear Seff GB on We have also shown that the in¯uence of Seff the minority transport within the grain does depend on the injection level of the excess minority carriers. Especially when the beam is near to the grain boundary, the relative eciency deviates
450
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
signi®cantly away from the calculation based on GB . It alone cannot be used to quantify constant Seff the GB property. IBIC in either the transverse or the normal con®gurations can provided important information such as the minority carriers diusion length and surface recombination of the grain. The diusion length obtained by these means were used in the numerical analysis. In the present work we have assumed that the qualitative argument by [7] that the results obtained from the two-dimensional is similar to the three-dimensional ones. However, for small grains, this assumption is invalid. Further works will address this issue by extending into three-dimensional modelling and the possible eect GB . that it will has on the Seff
Acknowledgements K.K. Lee wishes to acknowledgement the support of the Overseas Postgraduate Research Scholarship, the Melbourne Research Scholarship and the Melbourne Abroad Postgraduate Travelling Scholarship. This work has been supported by a grant from the Faculty of Science, University of
Melbourne. The authors wish to thank J.S. Laird for helpful discussions. References [1] C. Donolato, R. Nipoti, J. Appl. Phys. 82 (2) (1997) 742. [2] P. Panayotatos, H.C. Card, IEEE Trans. Electron Lett. EDL 1 (1980) 263. [3] J.G. Fossum, F.A. Lindholm, IEEE Trans. Electron Devices ED 27 (1980) 692. [4] H.J. Queisser, Mat. Res. Soc. Symp. Proc. 14 (1983) 323. [5] J.H. Werner, Inst. Phys. Conf. Ser. 104 (1989) 63. [6] J.G. Fossum, R. Sundaresan, IEEE Trans. Electron Devices ED 29 (1982) 1185. [7] R. Sundaresan, J.G. Fossum, E.E. Burk, J. Appl. Phys. 56 (1984) 964. [8] C.H. Seager, T.G. Castner, J. Appl. Phys. 49 (1978) 3879. [9] W. Shockley, W.T. Read, Jr., Phys. Rev. 87 (1952) 835. [10] M.A. Green, J. Appl. Phys. 80 (1996) 1515. [11] D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971. [12] E.J. Kobetich, R. Katz, Phys. Rev 170 (1968) 391. [13] J.F. Ziegler, J.P. Biersack, U. Littmark, Stopping and Range of Ions in Solids, Pergamon Press, New York, 1985. [14] M. Breese, D.N. Jamieson, P.C. King, Materials Analysis Using a Nuclear Microprobe, Wiley, New York, 1996. [15] P. Jonsson, H. Bleichner, M. Isberg, E. Nordlander, J. Appl. Phys. 81 (1997) 2256. [16] C. Donolato, Solid-St. Electron. 25 (1982) 1077.
www.elsevier.nl/locate/nimb
Analysis of solar cells using the IBIC technique K.K. Lee *, D.N. Jamieson School of Physics, Microanalytical Research Centre, University of Melbourne, Parkville 3052, Australia
Abstract The ion beam induced charge (IBIC) technique has been applied to investigate the relative eciency in polycrystalline silicon solar cells. Relative eciency is de®ned as the charge pulse height normalised with respect to the maximum charge collected. Using IBIC, solar cells have been imaged with microbeam both normal and transverse to the collection junction to obtain the charge collection. The measured charge collection at the grain boundary (GB) of the crystals is then compared to computer simulation to obtain the diusion length of the bulk and the eective recombination veGB locity Seff at the GB. The dependence of the relative eciency on the carrier injection level is then investigated using GB both MeV alphas and molecular hydrogen. We also show that Seff depends on the excess carrier injection level. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Ion beam induced charge; Surface recombination; Diusion length; Grain boundary
1. Introduction Interest in the electronic properties of polysilicon semiconductor has grown substantially because it has found wide use in solar cells, transistors, interconnects, thin-®lm resistors, and varistors. As compared to electronic devices produced from single crystal wafer, the performance of polysilicon devices is limited due to the recombination of minority carriers at the GB. In order to improve the capability of the polysilicon devices, a better physical understanding of the in¯uence of the GB on the minority carriers is therefore required.
* Corresponding author. Tel.: +61-3-9344-5081, fax: +61-39347-4783; e-mail: [email protected]
IBIC is a powerful tool for measuring transport properties in electronic materials. A MeV ion beam, which creates a high density of electron hole pairs, is normally scanned over a sample. Any defect sites that possess trap levels in the gap is capable of trapping minority carriers and will show up as areas of diminished charge collected in an IBIC image. Hence, IBIC is an excellent technique for the investigation of the eect of GB on minority carrier transport for polysilicon. IBIC analysis and computer simulation of charge pulse height from a GB have been developed by Donolato [1]. Their analytical model is able to deterGB . mine the minority carrier diusion length and Seff GB However it assumed that Seff is a constant value. This assumption is only valid for low injection regime as pointed out by several authors [2,3] using either optical illumination or electron beam
0168-583X/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 0 3 3 0 - 4
446
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
injection. In this paper we will use numerical method to calculate the eect of GB on the minority carriers. By comparing the IBIC experimental data with the numerical solutions we shall GB also varies as a funcshow that the obtained Seff tion of excess electron density. The calculated IBIC eciency pro®les were then compared to GB is Donolato's model which assumes that Seff constant; that is the relation between charge recombination and excess minority carrier concentration at the GB is linear. 2. Numerical model The atoms at the boundaries are thought to be disordered and are also preferential sites for segregation of impurities. These introduce defect states into the forbidden gap which act as traps and recombination centers [4]. These states at the silicon GBs favour the formation of majority carrier trapping levels within the band gap [5]. The ®lled traps are also known as ionized traps, as they carry charges, creating a potential energy barrier, which impedes the movement of charge from one grain to another. In view of this, a GB is normally treated as a double depletion layer, that is two Schottky barriers back to back as shown in Fig. 1(a). Fig. 1(a) shows a band diagram of the space± charge region (SCR) region along a GB in p-type GB which silicon. The model require to compute Seff has an in¯uence on the minority carriers transport within the grain has been detailed elsewhere [6,7], we will brie¯y summarize its main points. The injection of minority carriers results in a nonequilibrium condition, as shown in Fig. 1(b). Assuming that the barrier height remains at its equilibrium value before any ion impring onto the semiconductor. Then as the excitation due to the incident ion is applied, the electron quasi-Fermi level, EFN , moves above the hole quasi-Fermi level, EFP , that is the electron density increases, which leads to an increase in the net electron capture rate of the trapping states. In the steady state, the number of electrons occupying these states is larger than that in equilibrium; the donor-like traps are ®lled more rapidly than in equilibrium. The population of donor-like traps beneath EFN will depend on the
Fig. 1. (a) Energy band diagram of GB region of a p-type semiconductor at equilibrium. Ef is the Fermi level, Ec is the energy of the bottom of the conduction band, Ev is the top of GB is the space the valence band, Eg is the band gap, and WSCR charge region of the grain boundary. The GB is at x 0. (b) Under non-equilibrium condition, EFN moves above EFP with /B as the barrier potential. The dierence between the quasiFermi levels re¯ects the level of excitation of the grain boundary.
number of excess minority carriers injected, which will in turn determine the surface recombination at GB is a function of the excess mithe GB. Thus Seff nority carrier density. As a result of the traps ®lling up, the net charge and the barrier height of the GB are reduced. The analysis is made with the assumption that a single donor trap level (either neutral or positively ionized) is located near midgap, which is consistent with experimental measurements [8]. The assumption of ¯at quasi-Fermi levels within the SCR of the GB (quasi-equilibrium condition) enables the Poisson's equation to be related to the Shockley Read Hall capture emission statistics [9] with the appropriate boundary conditions which leads to the solution of barrier height as a function of excess carrier injection. GB can be Once the barrier height is known, then Seff obtained. In this work, we have used n p polycrystalline silicon solar cells with ohmic contacts made to
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
both n-type and p-type silicon. The electron continuity equation in the p-type base is solved numerically and subject to appropriate boundary conditions. The three-dimensional excess electron density equation is o2 N o2 N o2 N N g x; y; z 2 2 ÿ 2 0; ox2 oy oz Ln Dn
1
where N N x; y; z is the excess electron density, Dn is the electron diusion coecient, Ln is the electron diusion length, and g x; y; z is the electron hole pair generation rate. In this work, we consider only vertical GB [10]. Z axis is chosen to be in the direction of the depth of the cells, y axis to be along the plane of the GB, and x axis is perpendicular to this plane. Sundaresan has explained qualitatively that the excess electron density obtained from the threedimensional equation diers from the two-dimensional one by a constant factor for all excitation conditions [7]. Therefore the reduction of the three-dimensional equation to two dimensions will not aect our interpretation as we are comparing the relative values. Then Eq. (1) reduces to o2 N o2 N N g x; z 2 ÿ 2 0; ox2 oz Ln Dn
2
where N N x; z is the excess electron density and g x; z is the two-dimensional generation rate. The boundary conditions are that the excess minority carrier at the ohmic contact vanishes, N z ! 1 0, similarly at the SCR, N z 0 0. The grain is assumed to be semiin®nite N x ! 1 0, and at the edge of the GB's SCR, x 0, Dn
oN GB N: Seff ox
3
Numerically N is determined by subdividing the grain into a rectangular grid, and Eq. (2) is approximated using a ®nite dierence method. At each grid point, the residuals of Eq. (2) are simultaneously reduced to zero by the iterative solution of a matrix equation using successive over-relaxation (SOR) with Chebyshev acceleration [11]. The generation carrier volume could be computed using the analytical method described in
447
[12]. However, for computation eciency, we have used a varying LET along the length of the path based on SRIM [13]. 3. Experimental detail Commercial polycrystalline silicon solar cells that have untextured surfaces were used in this study. In order to remove any eects due to grain size, grains much larger than the base minority carriers diusion length must be chosen. IBIC analysis on polycrystalline silicon solar cell is problematic due to the poor signal to noise ratio S=N associated with the large junction capacitance. Device bias leads to a further reduction of S=N due to the shot noise associated with leakage current. We improve the S=N by mounting the preampli®er as close as possible to the sample, thereby reducing the stray capacitance and eliminating possible ground loops. The sample target stage consists of an in-vacuum Amptek A250 charge sensitive preampli®er. Prior to IBIC analysis current±voltage characteristics and reversebias capacitance measurements were performed to determine the leakage current and the base dopant in the sample respectively. These are necessary for numerical modelling of the IBIC results. 4. Results and discussion Minority carrier diusion length of the grain can be obtained with the incident beam normal to the collection junction. Fig. 2 shows a plot of charge pulse height as a function of incident ion energy and species. The IBIC signals were extracted from regions some several hundred microns away from the GB. Using the one-dimensional expression described in [14] the proton data was ®tted with a diusion length of 43 lm. As a check for consistency the same value for the diffusion length was computed for the alpha data. In this case the result obtained are in good agreement with the experimental observation. In the calculation, the charge collected over the heavily doped n region (dopant density of the order 1019 ÿ 1020 cmÿ3 ) is taken to be negligible as compared to
448
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
Fig. 2. A plot of the charge pulse height as a function of ions energy for both alpha and proton beams. The solid line and dashed line were calculated from [14]. Solid dot denotes proton data, and open dot denotes alpha data.
the base minority carriers diusion length. This is because Auger recombination would result in lower lifetimes in this heavily doped layer [15]. The minority carriers diusion length can also be obtained using the lateral geometry, that is with the beam parallel to the collecting junction. The solar cells sample was then mounted in this con®guration and the beam is allowed to rastered over the width of an unbiased sample. Fig. 3 shows an IBIC relative eciency pro®le obtained in the lateral geometry. An analytical solution derived using Green's function is given by [16], Charge pulse height x0 Z R oE x0 ; z0 Q x0 ; z0 dz0 ; ox 0 0
Q x0 ; z0 eÿkx ÿ
2S pDn
1 l2 k 2 2 L;
Z
1 0
4 k
l2 l 1
0
eÿlz sin kx0 dk; 5
where S is the surface recombination velocity. R is the end of range of the incident particle. L is the minority carriers diusion length. oE=ox is the
Fig. 3. IBIC pro®le in the lateral con®guration for 2 MeV proton and alpha beams. The solid line was calculated, refer to text.
stopping power, which can be obtained from SRIM [13]. Eqs. (4) and (5) were used to obtain the relative eciency by normalising to the maximum charge pulse height and a ®t is made to both the proton and alpha data in Fig. 3. L is determine to be 41 lm and S is 1:0 104 cm sÿ1 . In the modelling of the electron transport in the grain under the in¯uence of GB, the average value of the diusion length as determined by the above mentioned two techniques was incorporated. Fig. 4 shows a plot of the relative eciency as a function of spatial distance from the GB for 2 MeV alpha and molecular hydrogen beams. IBIC linescans were rastered perpendicular to the GB. Fig. 4(a) shows that the calculated relative eciency GB is reasonable when the beam is using constant Seff far away from the GB (> 10 lm). However the GB deviates from calculation based on constant Seff the experimental value near the grain boundary. The reason for this deviation is that the injected minority carriers density that arrives at the GB is lower when the beam is far away from the GB than when the beam is in the vicinity of the boundary. The same conclusion can also be made from Fig. 4(b), which is obtain with a 2 MeV He beam;
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
449
Fig. 5. A plot of eective recombination velocity at the grain boundary as a function of depth for 2 MeV H 2 with the beam hitting the sample at a distance 3 lm from the grain boundary.
Fig. 4. Relative IBIC eciency as a function of beam position from the grain boundary for (a) 2 MeV H 2 , and (b) 2 MeV He .
here, the deviation between measured data and the GB is calculated relative eciency using constant Seff even more pronounced. Our work which uses nonGB could reproduce the experimental data linear Seff GB at the accurately. This demonstrated that the Seff GB is non-linear and is highly dependent on the excess minority carriers density in the grain. We have assumed that the trapping states are monoenergetic and are located in mid-gap. The eective density of such states is estimated to be 1012 cmÿ1 . GB Fig. 5 shows the relation between Seff for 2 MeV H2 beam and depth, z within a grain. The calcu-
lation is for beam hitting the sample 3 lm from GB the GB. The calculated Seff shows a minimum at around 15 lm which can be related to the Bragg peak for 2 MeV H 2 particle. At the Bragg peak, the degree of generation of e±h pairs caused by a ionising particle were higher than elsewhere along the track. As a consequence, the excess electron density that arrives at the GB was greater and leads to a reduction of the number of the trapping states at the GB, which in turns causes a drop in GB . the Seff
5. Conclusion We have solved the two-dimensional electron transport problem using ®nite dierence method GB . With the for the general case of nonlinear Seff assumption that the trapping states were at midgap and are monoenergetic, we were able to deGB and the trapping density. termine the nonlinear Seff GB on We have also shown that the in¯uence of Seff the minority transport within the grain does depend on the injection level of the excess minority carriers. Especially when the beam is near to the grain boundary, the relative eciency deviates
450
K.K. Lee, D.N. Jamieson / Nucl. Instr. and Meth. in Phys. Res. B 158 (1999) 445±450
signi®cantly away from the calculation based on GB . It alone cannot be used to quantify constant Seff the GB property. IBIC in either the transverse or the normal con®gurations can provided important information such as the minority carriers diusion length and surface recombination of the grain. The diusion length obtained by these means were used in the numerical analysis. In the present work we have assumed that the qualitative argument by [7] that the results obtained from the two-dimensional is similar to the three-dimensional ones. However, for small grains, this assumption is invalid. Further works will address this issue by extending into three-dimensional modelling and the possible eect GB . that it will has on the Seff
Acknowledgements K.K. Lee wishes to acknowledgement the support of the Overseas Postgraduate Research Scholarship, the Melbourne Research Scholarship and the Melbourne Abroad Postgraduate Travelling Scholarship. This work has been supported by a grant from the Faculty of Science, University of
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