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Multi-electrode LBIC method for characterization of 1D ‘hidden’ defects
Materials Science and Engineering B91– 92 (2002) 260– 263 www.elsevier.com/locate/mseb
Multi-electrode LBIC method for characterization of 1D ‘hidden’ defects V. Sirotkin, S. Zaitsev, E. Yakimov * Institute of Microelectronics Technology, Russian Academy of Sciences, Chernogolo6ka 142432, Russia
Abstract The potentialities of a three-electrode light beam induced current (LBIC) method for the characterization of individual 1D defects located under the surface are evaluated. Using the computer simulation it is demonstrated that the considered method offers advantages over the ‘standard’ electron beam induced current (EBIC) as well as over the multi-electrode EBIC methods for the defects investigated. For the computer simulation of LBIC and EBIC signals the drift-diffusion approach is applied. The mathematical model is solved by a numerical method based on a combination of adaptive composite grids and an iterative domain decomposition algorithm. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Electrical measurements; Semiconductors; Silicon; Light beam induced current; Electron beam induced current
1. Introduction In our previous works [1,2] we have demonstrated that the ‘standard’ electron beam induced current (EBIC) method (with one ‘infinite’ Schottky electrode) can be applied successfully for the characterization of individual 1D defects perpendicular or inclined to the surface. The aim of this paper is to select an appropriate method for the defect region parameters reconstruction in the case of 1D defects located under the surface. For this purpose we examine the EBIC and light beam induced current (LBIC) techniques with two schemes for the recording of the signals: the one-electrode scheme (Fig. 1) and the three-electrode scheme (Fig. 6). In the latter scheme, two ‘semi-infinite’ Schottky electrodes (A and B) form the collection volume for a narrow ‘working’ electrode (C), which is placed between the two other electrodes and in parallel to the defect [3] (in the case considered, we suppose that the ‘working’ electrode is 1 mm in width, the distance between the electrodes equals 0.1 mm and the defect lies 2 mm beneath the surface through the center of the electrode C, see Fig. 6). Here, as in Refs. [1,2], for the
characterization of the defects we use the collected current dependence on the depletion region width. In order to improve the sensitivity of the methods, the modulation technique [4] (the measurements of first derivative of collected current depending on applied bias) is implemented.
2. Physical and mathematical models As a physical model we consider a standard version of LBIC (EBIC) signal formation based on the diffusion-drift model. We consider n-type semiconductor and the case of weak generation of excess electrons and holes. Therefore, the concentrations of excess carriers do not modify the electrostatic potential and the majority carrier concentration. Using the charge collection probability the collected current J (the LBIC or EBIC signals are denoted by Jo or Je, respectively) can be described as [5] J=
&
g d6,
d
* Corresponding author. Tel.: + 7-9652-44161; fax: +7-959628047. E-mail address: [email protected] (E. Yakimov).
where g is the excess carrier generation rate, d is the computational domain. The function satisfies the diffusion-drift equation
0921-5107/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 1 ) 0 1 0 2 4 - 8
V. Sirotkin et al. / Materials Science and Engineering B91–92 (2002) 260–263
DD(P) −v9(P)·9(P) −z(P)(P) = 0, P=(x,z)d, z(P)=n(P)/{~n(P)[p(P)+ ni]+ ~p(P)[n(P) +ni]},
(1)
with the boundary conditions: = 1 on the Schottky contact C, D
#(x,0) =S(x,0) on the free surfaces, #z
= 0 on an ’infinity’ boundary and on the Schottky contacts A and B,
261
where D and v are the diffusivity and mobility of holes, respectively, S is the surface recombination velocity (usually in experiments the surface recombination velocity can be made as low as possible, therefore, in calculations we suppose that S equals 0), ni is the intrinsic carrier concentration, ~n(P) and ~p(P) are the lifetimes of electrons and holes, respectively. In the function z(P) (this function describes the generation/ recombination of the excess carriers with regard to the concentrations of electrons and holes), the electrostatic potential (P) and the concentrations of electrons n(P) and holes p(P) are determined by the solution of the standard system of the drift-diffusion equations (SDDE) (see, for example Ref. [8]) for the structure investigated in the absence of the excess carrier generation. We assume that recombination centers around 1D defect are distributed exponentially, i.e. 1/~p(x,z)= 1/~ 0p + exp[− (r/Rd)]/~ dp, r= [(x −xd)2 + (z −zd)2]1/2, (x,z)d,
Fig. 1. Set of equidistant level lines for the charge collection probability in the case of the one-electrode scheme (C is the Schottky electrode). The position of the defect is marked by ‘× ’.
Fig. 2. Dependence of ~ dp on the radius Rd for 1D defects with 10% contrast. .
where ~ 0p and ~ dp are the lifetimes in the bulk and on 1D defect, respectively, Rd is the characteristic radius of this distribution, and (xd, zd) are the coordinates of the defect. For the numerical solution of SDDE and Eq. (1), a finite-difference approach combining adaptive composite grids and iterative algorithms for domain decomposition [6] is applied. Examples of the numerical solutions of Eq. (1) are illustrated in Figs. 1 and 6 for the one- and three-electrode schemes, respectively.
Fig. 3. EBIC profiles for 1D defect with 10% contrast for different values of Rd (the one-electrode scheme).
V. Sirotkin et al. / Materials Science and Engineering B91–92 (2002) 260–263
262
3. Results of simulation The following parameters are used for the numerical simulations: the donor concentration ND =4× 1014 cm − 3, the built-in potential Ums =0.8 V, the bulk lifetimes of electrons and holes ~n =2 ×10−6 s and ~p =10−6 s. For the EBIC method the generation distribution g is described as in Ref. [7] at the electron beam energy E = 25 kV. In the LBIC method we consider the unfocused beam with the light absorption coefficient h= 1.85 ×10 − 2 mm − 1. The simulations are carried out for 1D defects with the EBIC contrast (in the case of the one-electrode scheme) equal to 10%. The EBIC contrast, as usually, is calculated as C= 1 −Je/J 0e , where J 0e is the collected Fig. 6. Set of equidistant level lines for the charge collection probability in the case of the three-electrode scheme (‘semi-infinite’ Schottky electrodes A and B form a collection volume for a narrow ‘working’ electrode C). The position of the defect is marked by ‘ ×’.
Fig. 4. Dependence of the EBIC signal on applied bias U C =UCS for different Rd (the one-electrode scheme).
Fig. 7. Dependence of the first derivative of the EBIC signal on applied bias U A =U B =U C =UCS for different Rd (three-electrode scheme).
Fig. 5. First derivatives of the EBIC signals from Fig. 4.
current far from the defect. We consider three defects with Rd = 0.05, 0.1 and 0.2 mm, respectively. In the simulations the lifetime value on 1D defect ~ dp is calculated for each Rd to maintain the constant contrast value. The relation between ~ dp and Rd for C=0.1 is presented in Fig. 2. Fig. 3 shows the EBIC profiles for three defects detected by the one-electrode scheme. For the same defects, the dependence of the EBIC signal and its first derivative on applied bias UCS is illustrated in Figs. 4 and 5, respectively. The results obtained for these defects in the case of the three-electrode scheme are presented in Figs. 7 and
V. Sirotkin et al. / Materials Science and Engineering B91–92 (2002) 260–263
Fig. 8. Dependence of the first derivative of the LBIC signal on applied bias U A =U B = U C = UCS for different Rd (three-electrode scheme).
263
Fig. 10. Effect of the recombination strength variation (three-electrode scheme for the LBIC method).
4. Conclusions The numerical results presented have shown that the three-electrode LBIC method offers advantages over the ‘standard’ EBIC as well as the three-electrode EBIC methods for the characterization of individual 1D defects located under the surface. The proposed method has demonstrated sufficient sensitivity to the recombination strength and position of the defects.
Acknowledgements
Fig. 9. Effect of 1D defect shift along the z-axis (three-electrode scheme for the LBIC method). The abscissa is calculated as follows: w = z = 0, UCS is fixed.
8 (for the EBIC and LBIC methods, respectively). Figs. 9 and 10 demonstrate the results obtained for the defect with Rd =0.05 mm slightly shifted from the initial position (along the z axis) and for a non-displaced defect with a slightly changed recombination strength (C= 0.10544). Note that all presented signals are normalized to their maximum values (that is J* =J/J max, J max = max J) and that in the figures the broken lines correU CS spond to the signals calculated without any defect.
This work was partially supported by the Russian Foundation of Basic Research (Grants 00-01-00656 and 01-01-97005p2001) and by the Russian National Scientific and Technology Program ‘Physics of Solid State Nanostructures’ (Grant 98-3007).
References [1] V.V. Sirotkin, E.B. Yakimov, S.I. Zaitsev, Mater. Sci. Eng. 42 (1996) 176. [2] V.V. Sirotkin, E.B. Yakimov, Inst. Phys. Conf. Ser. 160 (1997) 79. [3] V.V. Sirotkin, S.I. Zaitsev, Mater. Sci. Eng. B24 (1994) 87. [4] E. Yakimov, Mater. Sci. Eng. B24 (1994) 23. [5] C. Donolato, J. Appl. Phys. 66 (1989) 4524. [6] V. Sirotkin, Comput. Math. Appl. 40 (2000) 645. [7] C. Donolato, Phys. Status Solidi A 65 (1981) 649. [8] S. Sze, Physics of Semiconductor Devices, Wiley, New York, 1969.
Multi-electrode LBIC method for characterization of 1D ‘hidden’ defects V. Sirotkin, S. Zaitsev, E. Yakimov * Institute of Microelectronics Technology, Russian Academy of Sciences, Chernogolo6ka 142432, Russia
Abstract The potentialities of a three-electrode light beam induced current (LBIC) method for the characterization of individual 1D defects located under the surface are evaluated. Using the computer simulation it is demonstrated that the considered method offers advantages over the ‘standard’ electron beam induced current (EBIC) as well as over the multi-electrode EBIC methods for the defects investigated. For the computer simulation of LBIC and EBIC signals the drift-diffusion approach is applied. The mathematical model is solved by a numerical method based on a combination of adaptive composite grids and an iterative domain decomposition algorithm. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Electrical measurements; Semiconductors; Silicon; Light beam induced current; Electron beam induced current
1. Introduction In our previous works [1,2] we have demonstrated that the ‘standard’ electron beam induced current (EBIC) method (with one ‘infinite’ Schottky electrode) can be applied successfully for the characterization of individual 1D defects perpendicular or inclined to the surface. The aim of this paper is to select an appropriate method for the defect region parameters reconstruction in the case of 1D defects located under the surface. For this purpose we examine the EBIC and light beam induced current (LBIC) techniques with two schemes for the recording of the signals: the one-electrode scheme (Fig. 1) and the three-electrode scheme (Fig. 6). In the latter scheme, two ‘semi-infinite’ Schottky electrodes (A and B) form the collection volume for a narrow ‘working’ electrode (C), which is placed between the two other electrodes and in parallel to the defect [3] (in the case considered, we suppose that the ‘working’ electrode is 1 mm in width, the distance between the electrodes equals 0.1 mm and the defect lies 2 mm beneath the surface through the center of the electrode C, see Fig. 6). Here, as in Refs. [1,2], for the
characterization of the defects we use the collected current dependence on the depletion region width. In order to improve the sensitivity of the methods, the modulation technique [4] (the measurements of first derivative of collected current depending on applied bias) is implemented.
2. Physical and mathematical models As a physical model we consider a standard version of LBIC (EBIC) signal formation based on the diffusion-drift model. We consider n-type semiconductor and the case of weak generation of excess electrons and holes. Therefore, the concentrations of excess carriers do not modify the electrostatic potential and the majority carrier concentration. Using the charge collection probability the collected current J (the LBIC or EBIC signals are denoted by Jo or Je, respectively) can be described as [5] J=
&
g d6,
d
* Corresponding author. Tel.: + 7-9652-44161; fax: +7-959628047. E-mail address: [email protected] (E. Yakimov).
where g is the excess carrier generation rate, d is the computational domain. The function satisfies the diffusion-drift equation
0921-5107/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 1 ) 0 1 0 2 4 - 8
V. Sirotkin et al. / Materials Science and Engineering B91–92 (2002) 260–263
DD(P) −v9(P)·9(P) −z(P)(P) = 0, P=(x,z)d, z(P)=n(P)/{~n(P)[p(P)+ ni]+ ~p(P)[n(P) +ni]},
(1)
with the boundary conditions: = 1 on the Schottky contact C, D
#(x,0) =S(x,0) on the free surfaces, #z
= 0 on an ’infinity’ boundary and on the Schottky contacts A and B,
261
where D and v are the diffusivity and mobility of holes, respectively, S is the surface recombination velocity (usually in experiments the surface recombination velocity can be made as low as possible, therefore, in calculations we suppose that S equals 0), ni is the intrinsic carrier concentration, ~n(P) and ~p(P) are the lifetimes of electrons and holes, respectively. In the function z(P) (this function describes the generation/ recombination of the excess carriers with regard to the concentrations of electrons and holes), the electrostatic potential (P) and the concentrations of electrons n(P) and holes p(P) are determined by the solution of the standard system of the drift-diffusion equations (SDDE) (see, for example Ref. [8]) for the structure investigated in the absence of the excess carrier generation. We assume that recombination centers around 1D defect are distributed exponentially, i.e. 1/~p(x,z)= 1/~ 0p + exp[− (r/Rd)]/~ dp, r= [(x −xd)2 + (z −zd)2]1/2, (x,z)d,
Fig. 1. Set of equidistant level lines for the charge collection probability in the case of the one-electrode scheme (C is the Schottky electrode). The position of the defect is marked by ‘× ’.
Fig. 2. Dependence of ~ dp on the radius Rd for 1D defects with 10% contrast. .
where ~ 0p and ~ dp are the lifetimes in the bulk and on 1D defect, respectively, Rd is the characteristic radius of this distribution, and (xd, zd) are the coordinates of the defect. For the numerical solution of SDDE and Eq. (1), a finite-difference approach combining adaptive composite grids and iterative algorithms for domain decomposition [6] is applied. Examples of the numerical solutions of Eq. (1) are illustrated in Figs. 1 and 6 for the one- and three-electrode schemes, respectively.
Fig. 3. EBIC profiles for 1D defect with 10% contrast for different values of Rd (the one-electrode scheme).
V. Sirotkin et al. / Materials Science and Engineering B91–92 (2002) 260–263
262
3. Results of simulation The following parameters are used for the numerical simulations: the donor concentration ND =4× 1014 cm − 3, the built-in potential Ums =0.8 V, the bulk lifetimes of electrons and holes ~n =2 ×10−6 s and ~p =10−6 s. For the EBIC method the generation distribution g is described as in Ref. [7] at the electron beam energy E = 25 kV. In the LBIC method we consider the unfocused beam with the light absorption coefficient h= 1.85 ×10 − 2 mm − 1. The simulations are carried out for 1D defects with the EBIC contrast (in the case of the one-electrode scheme) equal to 10%. The EBIC contrast, as usually, is calculated as C= 1 −Je/J 0e , where J 0e is the collected Fig. 6. Set of equidistant level lines for the charge collection probability in the case of the three-electrode scheme (‘semi-infinite’ Schottky electrodes A and B form a collection volume for a narrow ‘working’ electrode C). The position of the defect is marked by ‘ ×’.
Fig. 4. Dependence of the EBIC signal on applied bias U C =UCS for different Rd (the one-electrode scheme).
Fig. 7. Dependence of the first derivative of the EBIC signal on applied bias U A =U B =U C =UCS for different Rd (three-electrode scheme).
Fig. 5. First derivatives of the EBIC signals from Fig. 4.
current far from the defect. We consider three defects with Rd = 0.05, 0.1 and 0.2 mm, respectively. In the simulations the lifetime value on 1D defect ~ dp is calculated for each Rd to maintain the constant contrast value. The relation between ~ dp and Rd for C=0.1 is presented in Fig. 2. Fig. 3 shows the EBIC profiles for three defects detected by the one-electrode scheme. For the same defects, the dependence of the EBIC signal and its first derivative on applied bias UCS is illustrated in Figs. 4 and 5, respectively. The results obtained for these defects in the case of the three-electrode scheme are presented in Figs. 7 and
V. Sirotkin et al. / Materials Science and Engineering B91–92 (2002) 260–263
Fig. 8. Dependence of the first derivative of the LBIC signal on applied bias U A =U B = U C = UCS for different Rd (three-electrode scheme).
263
Fig. 10. Effect of the recombination strength variation (three-electrode scheme for the LBIC method).
4. Conclusions The numerical results presented have shown that the three-electrode LBIC method offers advantages over the ‘standard’ EBIC as well as the three-electrode EBIC methods for the characterization of individual 1D defects located under the surface. The proposed method has demonstrated sufficient sensitivity to the recombination strength and position of the defects.
Acknowledgements
Fig. 9. Effect of 1D defect shift along the z-axis (three-electrode scheme for the LBIC method). The abscissa is calculated as follows: w = z = 0, UCS is fixed.
8 (for the EBIC and LBIC methods, respectively). Figs. 9 and 10 demonstrate the results obtained for the defect with Rd =0.05 mm slightly shifted from the initial position (along the z axis) and for a non-displaced defect with a slightly changed recombination strength (C= 0.10544). Note that all presented signals are normalized to their maximum values (that is J* =J/J max, J max = max J) and that in the figures the broken lines correU CS spond to the signals calculated without any defect.
This work was partially supported by the Russian Foundation of Basic Research (Grants 00-01-00656 and 01-01-97005p2001) and by the Russian National Scientific and Technology Program ‘Physics of Solid State Nanostructures’ (Grant 98-3007).
References [1] V.V. Sirotkin, E.B. Yakimov, S.I. Zaitsev, Mater. Sci. Eng. 42 (1996) 176. [2] V.V. Sirotkin, E.B. Yakimov, Inst. Phys. Conf. Ser. 160 (1997) 79. [3] V.V. Sirotkin, S.I. Zaitsev, Mater. Sci. Eng. B24 (1994) 87. [4] E. Yakimov, Mater. Sci. Eng. B24 (1994) 23. [5] C. Donolato, J. Appl. Phys. 66 (1989) 4524. [6] V. Sirotkin, Comput. Math. Appl. 40 (2000) 645. [7] C. Donolato, Phys. Status Solidi A 65 (1981) 649. [8] S. Sze, Physics of Semiconductor Devices, Wiley, New York, 1969.