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Time-resolved ion beam-induced charge collection measurement of minority carrier lifetime in semiconductor power devices by using Gunn's theorem
Materials Science and Engineering B102 (2003) 193 /197 www.elsevier.com/locate/mseb
Time-resolved ion beam-induced charge collection measurement of minority carrier lifetime in semiconductor power devices by using Gunn’s theorem C. Manfredotti a,*, F. Fizzotti a, A. Lo Giudice a, M. Jaksic b, Z. Pastuovic b, C. Paolini a, P. Olivero a, E. Vittone a a
Experimental Physics Department, INFM-National Institute for the Physics of Matter, UdR University of Torino, Torino 10125, Italy b Department of Experimental Physics, Ruder Bosˇkovic´ Institute (RBI), P.O. Box 180, 10002 Zagreb, Croatia Received 13 April 2002; received in revised form 19 August 2002; accepted 21 October 2002
Abstract Ion microbeam techniques like ion beam-induced charge collection (IBICC) are very powerful methods in order to investigate and to map the transport properties in different technologically important semiconductors and in particular in materials proposed for nuclear detection. Time-resolved ion beam-induced charge collection (TRIBICC) represents a further improvement with respect to more traditional IBICC, since it can supply not only the charge collection efficiency (and through it data on mobility and trapping time of carriers in drift regions) but also the time behaviours of the charge collection. For long collection times, this means to gather information, also about diffusion lengths and lifetimes of carriers in the diffusion regions, which are always present in undepleted electronic devices, in particular power devices, and which are of paramount importance as inputs for simulation codes. By TRIBICC, in fact, some difficulties could be avoided in analysis of data collected in cases when lifetimes and shaping times of electronic chain are similar, and the sensitivity of the method is worse. In order to suitably analyse TRIBICC data, a theoretical model should be available: in general, Ramo’s theorem is used, but its validity in cases when space charge is present is questionable. A more general and powerful method is presented in this work by using Gunn’s theorem and a particular formulation of the generation function in order to solve the adjoint of the continuity equation in the time-dependent case. An application of this method to a commercial power device is presented and discussed. # 2003 Elsevier B.V. All rights reserved. Keywords: Ion beam-induced charge collection; Silicon device characterisation; Carrier lifetime PACS numbers: 73.50.Gr; 85.30.De
1. Introduction The knowledge of important parameters like minority carriers lifetime is of paramount importance not only for semiconductor materials producers, but also for computer-aided engineering CAE applications in which a first value, even approximate, of these parameters is needed in order to carry out device simulations for design implementations, being sure to remain in a
* Corresponding author. Tel.: /39-011-670-7306; fax: /39-011669-1104. E-mail address: [email protected] (C. Manfredotti). 0921-5107/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-5107(02)00656-6
realistic physical region. The methods used for minority carriers lifetime evaluation generally present some drawbacks, since either they are based on some model of the device (reverse recovery) or they can be applied only to specified samples of the material and not to the finished device (photoconductive decay). Ion beam-induced charge collection (IBICC) and time-resolved ion beaminduced charge collection (TRIBICC) in frontal version do not display any drawbacks, apart from the need to dispose of an ion accelerator, since they can be easily applied to a finished device, the depth of investigation can be pushed towards 100 mm (what is generally needed for power devices) and, particularly in the case of TRIBICC, they can be used in a wide range of lifetime
194
C. Manfredotti et al. / Materials Science and Engineering B102 (2003) 193 /197
values (a good opportunity for new materials). Power devices are in general complex, even if considered in one dimension only, display both drift and diffusion regions which can move under subsequent heat treatments and, as a consequence, they cannot be treated under simplified assumptions. Moreover, in order to analyse IBICC and TRIBICC data, the standard Ramo’s theorem cannot be a good start, since it is not valid in the presence of diffusion regions or, even worse, of space charge regions driven by an external bias. In this work, we are using a different approach based on Gunn’s theorem which has not such constraints and which can offer, by the adjoint equation method, a direct numerical solution in terms of charge collection efficiency and of its behaviour as a function of bias voltage, of the time and of the depth in the device. The treatment is onedimensional and it has been applied to a particular power diode. The results indicate that TRIBICC is a more powerful method with respect to IBICC and that it can be used in more general conditions.
QS iS
2.1. Derivation of the Ramo’s theorem by means of the Green’s reciprocal theorem Let us consider a closed region of volume V, bounded by two infinite electrodes spaced by d and maintained at constant potentials (F (x/0) /0, F (x /d) /V ) by an external power supply. Inside the bounded region, there exists a semiconductor medium with dielectric constant o and a volume charge density distribution r (r ,t)/ r2(r )/r3(r ,t ), where r2(r ) is a fixed charge distribution and r3(r,t) is due to the mobile carriers generated at time t/0 in a certain position r /r0. In order to evaluate the charge induced by the motion of carriers, Green’s reciprocal theorem [3] is considered. If F is the potential of the electromagnetic field due to the volume charge density r (r,t) with both the electrodes grounded and F ? is the potential due to the electrodes with their actual potentials in the absence of any volume charge distribution, then
g
r(r; t)F?(r) d3 r V
g s(r; t)F?(r) ds0;
(1)
Vg
dQS dt
V
1 V
g
@r3 (r; t) V
@t
(2) 3
F?(r) d r;
v iS q : d
(3)
Eq. (3) is the usual expression of Ramo’s theorem [1]. By using the continuity equation for the moving carriers, we can write the current iS as follows: 1 V
g
j(r; t)×E?(r) d3 r V
g j(r; t) d s; 2
(4)
S
where E?9F? is the electric field due to the actual potential at the electrodes in the absence of any volume charge distribution. The surface integral in Eq. (4) concerns the current entering the electrode, whereas the volume integral is relevant to the actual induced current.
2.2. Generalised Ramo/Gunn’s theorem Eq. (4) is the final result of the generalised Ramo’s theorem [2] and it is based on the assumption that the space charge distribution in the volume of the detector is independent of the external bias voltage, i.e. r2(r) does not depend on V . Such an assumption is no more valid in the case of a silicon p /n junction diode, where the space charge redistribution plays a key role in the determination of its rectifying behaviour. Hence, the analysis of the charge collection process should be based on the application of the generalised Ramo /Gunn’s theorem [4]. According to this theorem, the point charge q moving with the velocity v between two parallel electrodes spaced by d induce, in the external biased circuit, the following induced current iS :
S
where s is the surface charge density at the electrodes due to the presence of the volume charge density r , S the total area of the electrodes bounding V and ds the oriented element of S . Since F ? is constant at the electrodes and is given by the bias potentials and since the surface integral of s is equal to the total charge QS at the electrodes, Eq. (1) gives
r(r; t)F?(r) d3 r and
where iS is the induced current entering the electrode. A simple example is offered by the computation of the current due to the motion of a single charge q generated in r0 and moving with a constant velocity v along the x axis, i.e. r3(r,t) /d(x/(x0/vt)), where d is Dirac’s delta distribution function. Being F ?(r) /(V /d)x , it follows
iS
2. Theoretical model
1
iS q×v×
dE(r; V ) ; dV
(5)
where dE /dV is the derivative of the local electric field at the point charge with respect to the bias voltage applied at the electrodes. As a consequence, to evaluate the actual induced current at the electrodes, we have to consider the following expression:
C. Manfredotti et al. / Materials Science and Engineering B102 (2003) 193 /197
iS
g
j(r; t)× Vol
@E(r; V ) @V
d3 r;
(6)
obtained by the modification of Eq. (4) based on Ramo/Gunn’s theorem. The induced charge is then given by the following expression: t t @E(r; V ) 3 QS (r0 ; t) iS dt d r dt; (7) j(r; t)× @V V 0 0
g
g g
where the position vector r0 indicates the point where the charge is generated at time t /0. In order to interpret TRIBICC measurements by means of equation (7), let us consider the generation at t/0 of Neh electron /hole pairs at x /x0. In the following, we will consider a unidimensional model; the coordinate x is orthogonal to the electrodes. Neh is equal to the ratio Sion =weh where Sion is the ion energy and weh the energy required to create an electron /hole pair (3.6 eV in silicon). Because of the short time involved in the ionisation process, the charge carriers are considered created simultaneously [6]. The excess electron (n) and hole (p ) density can be evaluated by solving the relevant transport equations: @n @ 2 n @n n De × ×mn ×E ; 2 @t @x @x tn @p @ 2 p @p p Dp × ×mp ×E ; 2 @t @x @x tp
(8)
where E is the total electric field due to the external bias voltage and to the built in potential, mn,p, Dn,p, and tn,p are the mobility, diffusivity, and mean lifetimes of holes and electrons, respectively. The initial and boundary conditions are (ohmic contacts) n(x; t0)p(x; t0)Nnp ×d(xx0 ); n(x0; t)p(x 0; t)p(x d; t)p(x d; t):
(9)
195
constructed as follows: @n @n @ @n n Dn Gn ; vn × @t @x @x @x tn @p @p @ @p p Dp vp × Gp ; @t @x @x @x tp
(11)
where the apex ‘‘/’’ indicates the adjoint electron concentration, and the adjoint term is @E @ @E Dn × ; @V @x @V @E @ @E Dp × ; Gp vp × @V @x @V Gn vn ×
(12)
and vn,p /mn,p ×/Etot is the drift velocity for electrons and holes, respectively. The boundary conditions can be found in [7]. It has been proven that the charge induced at the electrodes from the motion of electrons and holes generated at point x at time t is given by QS (x; t)q[n(x; t)p(x; t)]; :/ It is worth noticing that the expression of the adjoint terms Eq. (12) differ from those described in [7] for the presence of the diffusion term which is essential for the calculation of the time evolution of the charge collection profiles in partially depleted devices. Finally, in order to compare such a model with experimental data obtained in experiments when carriers are generated by MeV ion probes penetrating the semiconductor device through the electrode (see Fig. 1), we have to consider the contribution of the charges generated at different positions and weighted by the Bragg (ion energy loss) curve: QS (t)
g
R
QS (x; t)×G(x) dx;
(13)
0
The current density j flowing in the diode is given by @n @p ; (10) j q× n×mn ×E p×mp ×E Dn × Dp × @x @x where q is the elementary charge. Such a current density is then inserted in equation (7) to get the time evolution of the induced charge generated at x0. It is worth noticing that QS (x0,t) as defined by equation (7) is Green’s function for the continuity equations. 2.3. The ‘‘adjoint’’ method An efficient method to evaluate Green’s function of Eq. (8) has been recently presented by Prettyman [7]. Since the excess carrier continuity equation involves linear operators, an adjoint continuity equation can be
Fig. 1. Schematic view of TRIBICC system (DUT: device under test).
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C. Manfredotti et al. / Materials Science and Engineering B102 (2003) 193 /197
where G is the energy loss profile and R the ion range in the semiconductor.
3. Experimental set-up The investigated sample was a commercial Mesa rectifier diode (Fig. 2). The doping profile was measured by the spreading resistance method. IBICC measurements were performed at the Ruder Boskovic Institute in Zagreb (HR) using a 4 MeV proton beam. The proton flux was maintained at about 100 protons per second in order to avoid electronic pile-up, surface trapping of generated carriers, and to reduce the radiation damage. The energy loss profile G (x ) of the protons within the sample is shown in Fig. 3A as evaluated by the SRIM2000 simulation code [5]. The output of the charge-sensitive preamplifier (Canberra 1004) for each individual ion strike was digitised using a fast computercontrolled Lecroy WAVERUNNER LT342 digital oscilloscope (0.5 G sample per second). The time resolution was about 2 ns and the transient was stored in 2500 points. Fig. 4 shows the time-resolved ion beam-induced charge signals evaluated at three different reverse bias voltage.
4. Results and discussion The mathematical method described in Section 2 was used to interpret the experimental data shown in Fig. 4. The electric field and the carrier velocity profiles evaluated by the PISCESII computer code [8] were used as input parameters of the adjoint Eq. (11) taking into account the dependence of the mobility from the
Fig. 3. Electronic energy loss for 4 MeV H-ions in the Mesa rectifier silicon diode (A); hole-generating function profile at different reverse bias voltage (B).
doping concentration and electric field [9]. The behaviour of the hole generating functions Gn at different bias voltages are plotted in Fig. 3B. The best-fit of the TRIBICC data of Fig. 4 were obtained by solving the adjoint Eq. (11) by means of a one-dimensional finite difference algorithm, assuming a hole bulk lifetime t0 /(59/1) ms and a dependence of the minority carrier lifetime t from the doping concentration given by the following expression [9]:
t
t0 ; 1 ((NA ND )=Nref )
(14)
Fig. 2. Scheme of the Mesa rectifier diode and the relevant doping profile; NA and ND represent the acceptor and donor concentrations, respectively.
C. Manfredotti et al. / Materials Science and Engineering B102 (2003) 193 /197
197
This means that their motion in the neutral region (E / 0) does not yield any charge signal and the long tail in the TRIBICC signal is relevant to their arrival time at the boundary of the DL.
5. Conclusions
Fig. 4. Experimental time behaviour of charge collection measurements carried out at different reverse bias voltage: (open circles) experimental data; (continuous line) fitting curves.
where Nref is equal to 7.1 /1015 cm 3 and NA,D are the acceptor and donor concentrations. Apart the lifetime measurements evaluated by means of the numerical solution of the adjoint equations, some qualitative observations can be drawn from the analysis of the TRIBICC signals. The source term G is obviously connected with the extension of the depletion layer (DL) where carriers experience a very rapid drift. The Ramo and Ramo / Gunn [10] theorems state that the charge is induced at the electrode only if the carriers move in the presence of the applied electric field, i.e. only when the carriers move within the DL layer. This means that carriers generated within the DL induce (less than a nanosecond) a very short current pulse, which corresponds to the sharp and intense increase of the charge collection signal. The increase of the reverse bias voltage yields an increase of the DL layer, which corresponds to larger amount of carriers generated within the DL and, consequently, a higher charge collected in the first nanosecond. After this short transient, the collection time is much longer. This is due to the minority carriers generating in the neutral region, which diffuse towards the DL. From the above-mentioned theorem, charge is induced at the electrodes only when these carriers move in the presence of the electric field, i.e. only when they enter the DL.
TRIBICC is proposed as a standard method in order to evaluate minority carriers lifetime in finished devices. In the present TRIBICC version, Gunn’s theorem and the adjoint equation method to solve the continuity equation is used in order to treat complex situations in which both drift and diffusion regions are present. The method introduces a generation function, which better defines the depletion or drift region and gives directly the time behaviour of charge collection efficiency. The fit on experimental data is carried out with only one parameter. Preliminary results obtained on a power Mesa device have been presented: they can be shown to offer minority carriers lifetime values in agreement with the expected one and suitable for CAE applications for the simulation of device design and behaviour.
References [1] S. Ramo, Proc. IRE 27 (1939) 584. [2] G. Cavalleri, E. Gatti, G. Fabri, V. Svelto, Nucl. Instrum. Meth. Phys. Res. 92 (1971) 137. [3] J.D. Jackson, Classical Electrodynamics, 2nd ed., Wiley, New York, 1975. [4] J.B. Gunn, Solid State Electron. 7 (1964) 739. [5] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon Press, New Yorkhttp://www.SRIM.org/, 1985. [6] P.A. Tove, K. Falk, Nucl. Instrum. Meth. Phys. Res. 12 (1961) 278. [7] T.H. Prettyman, Nucl. Instrum. Meth. Phys. Res. A 428 (1999) 72 /80. [8] M.R. Pinto, C.S. Rafferty, H.R. Yeager, R.W. Dutton, PISCES IIB, Supplementary Report, Stanford Electronics Lab., Department of Electrical Engineering, Stanford University, 1985. [9] S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer, Vienna, 1984. [10] E. Vittone, F. Fizzotti, A. Lo Giudice, C. Paolini, C. Manfredotti, Nucl. Instrum. Meth. Phys. Res. B 161-163 (2000) 446 /451.
Time-resolved ion beam-induced charge collection measurement of minority carrier lifetime in semiconductor power devices by using Gunn’s theorem C. Manfredotti a,*, F. Fizzotti a, A. Lo Giudice a, M. Jaksic b, Z. Pastuovic b, C. Paolini a, P. Olivero a, E. Vittone a a
Experimental Physics Department, INFM-National Institute for the Physics of Matter, UdR University of Torino, Torino 10125, Italy b Department of Experimental Physics, Ruder Bosˇkovic´ Institute (RBI), P.O. Box 180, 10002 Zagreb, Croatia Received 13 April 2002; received in revised form 19 August 2002; accepted 21 October 2002
Abstract Ion microbeam techniques like ion beam-induced charge collection (IBICC) are very powerful methods in order to investigate and to map the transport properties in different technologically important semiconductors and in particular in materials proposed for nuclear detection. Time-resolved ion beam-induced charge collection (TRIBICC) represents a further improvement with respect to more traditional IBICC, since it can supply not only the charge collection efficiency (and through it data on mobility and trapping time of carriers in drift regions) but also the time behaviours of the charge collection. For long collection times, this means to gather information, also about diffusion lengths and lifetimes of carriers in the diffusion regions, which are always present in undepleted electronic devices, in particular power devices, and which are of paramount importance as inputs for simulation codes. By TRIBICC, in fact, some difficulties could be avoided in analysis of data collected in cases when lifetimes and shaping times of electronic chain are similar, and the sensitivity of the method is worse. In order to suitably analyse TRIBICC data, a theoretical model should be available: in general, Ramo’s theorem is used, but its validity in cases when space charge is present is questionable. A more general and powerful method is presented in this work by using Gunn’s theorem and a particular formulation of the generation function in order to solve the adjoint of the continuity equation in the time-dependent case. An application of this method to a commercial power device is presented and discussed. # 2003 Elsevier B.V. All rights reserved. Keywords: Ion beam-induced charge collection; Silicon device characterisation; Carrier lifetime PACS numbers: 73.50.Gr; 85.30.De
1. Introduction The knowledge of important parameters like minority carriers lifetime is of paramount importance not only for semiconductor materials producers, but also for computer-aided engineering CAE applications in which a first value, even approximate, of these parameters is needed in order to carry out device simulations for design implementations, being sure to remain in a
* Corresponding author. Tel.: /39-011-670-7306; fax: /39-011669-1104. E-mail address: [email protected] (C. Manfredotti). 0921-5107/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0921-5107(02)00656-6
realistic physical region. The methods used for minority carriers lifetime evaluation generally present some drawbacks, since either they are based on some model of the device (reverse recovery) or they can be applied only to specified samples of the material and not to the finished device (photoconductive decay). Ion beam-induced charge collection (IBICC) and time-resolved ion beaminduced charge collection (TRIBICC) in frontal version do not display any drawbacks, apart from the need to dispose of an ion accelerator, since they can be easily applied to a finished device, the depth of investigation can be pushed towards 100 mm (what is generally needed for power devices) and, particularly in the case of TRIBICC, they can be used in a wide range of lifetime
194
C. Manfredotti et al. / Materials Science and Engineering B102 (2003) 193 /197
values (a good opportunity for new materials). Power devices are in general complex, even if considered in one dimension only, display both drift and diffusion regions which can move under subsequent heat treatments and, as a consequence, they cannot be treated under simplified assumptions. Moreover, in order to analyse IBICC and TRIBICC data, the standard Ramo’s theorem cannot be a good start, since it is not valid in the presence of diffusion regions or, even worse, of space charge regions driven by an external bias. In this work, we are using a different approach based on Gunn’s theorem which has not such constraints and which can offer, by the adjoint equation method, a direct numerical solution in terms of charge collection efficiency and of its behaviour as a function of bias voltage, of the time and of the depth in the device. The treatment is onedimensional and it has been applied to a particular power diode. The results indicate that TRIBICC is a more powerful method with respect to IBICC and that it can be used in more general conditions.
QS iS
2.1. Derivation of the Ramo’s theorem by means of the Green’s reciprocal theorem Let us consider a closed region of volume V, bounded by two infinite electrodes spaced by d and maintained at constant potentials (F (x/0) /0, F (x /d) /V ) by an external power supply. Inside the bounded region, there exists a semiconductor medium with dielectric constant o and a volume charge density distribution r (r ,t)/ r2(r )/r3(r ,t ), where r2(r ) is a fixed charge distribution and r3(r,t) is due to the mobile carriers generated at time t/0 in a certain position r /r0. In order to evaluate the charge induced by the motion of carriers, Green’s reciprocal theorem [3] is considered. If F is the potential of the electromagnetic field due to the volume charge density r (r,t) with both the electrodes grounded and F ? is the potential due to the electrodes with their actual potentials in the absence of any volume charge distribution, then
g
r(r; t)F?(r) d3 r V
g s(r; t)F?(r) ds0;
(1)
Vg
dQS dt
V
1 V
g
@r3 (r; t) V
@t
(2) 3
F?(r) d r;
v iS q : d
(3)
Eq. (3) is the usual expression of Ramo’s theorem [1]. By using the continuity equation for the moving carriers, we can write the current iS as follows: 1 V
g
j(r; t)×E?(r) d3 r V
g j(r; t) d s; 2
(4)
S
where E?9F? is the electric field due to the actual potential at the electrodes in the absence of any volume charge distribution. The surface integral in Eq. (4) concerns the current entering the electrode, whereas the volume integral is relevant to the actual induced current.
2.2. Generalised Ramo/Gunn’s theorem Eq. (4) is the final result of the generalised Ramo’s theorem [2] and it is based on the assumption that the space charge distribution in the volume of the detector is independent of the external bias voltage, i.e. r2(r) does not depend on V . Such an assumption is no more valid in the case of a silicon p /n junction diode, where the space charge redistribution plays a key role in the determination of its rectifying behaviour. Hence, the analysis of the charge collection process should be based on the application of the generalised Ramo /Gunn’s theorem [4]. According to this theorem, the point charge q moving with the velocity v between two parallel electrodes spaced by d induce, in the external biased circuit, the following induced current iS :
S
where s is the surface charge density at the electrodes due to the presence of the volume charge density r , S the total area of the electrodes bounding V and ds the oriented element of S . Since F ? is constant at the electrodes and is given by the bias potentials and since the surface integral of s is equal to the total charge QS at the electrodes, Eq. (1) gives
r(r; t)F?(r) d3 r and
where iS is the induced current entering the electrode. A simple example is offered by the computation of the current due to the motion of a single charge q generated in r0 and moving with a constant velocity v along the x axis, i.e. r3(r,t) /d(x/(x0/vt)), where d is Dirac’s delta distribution function. Being F ?(r) /(V /d)x , it follows
iS
2. Theoretical model
1
iS q×v×
dE(r; V ) ; dV
(5)
where dE /dV is the derivative of the local electric field at the point charge with respect to the bias voltage applied at the electrodes. As a consequence, to evaluate the actual induced current at the electrodes, we have to consider the following expression:
C. Manfredotti et al. / Materials Science and Engineering B102 (2003) 193 /197
iS
g
j(r; t)× Vol
@E(r; V ) @V
d3 r;
(6)
obtained by the modification of Eq. (4) based on Ramo/Gunn’s theorem. The induced charge is then given by the following expression: t t @E(r; V ) 3 QS (r0 ; t) iS dt d r dt; (7) j(r; t)× @V V 0 0
g
g g
where the position vector r0 indicates the point where the charge is generated at time t /0. In order to interpret TRIBICC measurements by means of equation (7), let us consider the generation at t/0 of Neh electron /hole pairs at x /x0. In the following, we will consider a unidimensional model; the coordinate x is orthogonal to the electrodes. Neh is equal to the ratio Sion =weh where Sion is the ion energy and weh the energy required to create an electron /hole pair (3.6 eV in silicon). Because of the short time involved in the ionisation process, the charge carriers are considered created simultaneously [6]. The excess electron (n) and hole (p ) density can be evaluated by solving the relevant transport equations: @n @ 2 n @n n De × ×mn ×E ; 2 @t @x @x tn @p @ 2 p @p p Dp × ×mp ×E ; 2 @t @x @x tp
(8)
where E is the total electric field due to the external bias voltage and to the built in potential, mn,p, Dn,p, and tn,p are the mobility, diffusivity, and mean lifetimes of holes and electrons, respectively. The initial and boundary conditions are (ohmic contacts) n(x; t0)p(x; t0)Nnp ×d(xx0 ); n(x0; t)p(x 0; t)p(x d; t)p(x d; t):
(9)
195
constructed as follows: @n @n @ @n n Dn Gn ; vn × @t @x @x @x tn @p @p @ @p p Dp vp × Gp ; @t @x @x @x tp
(11)
where the apex ‘‘/’’ indicates the adjoint electron concentration, and the adjoint term is @E @ @E Dn × ; @V @x @V @E @ @E Dp × ; Gp vp × @V @x @V Gn vn ×
(12)
and vn,p /mn,p ×/Etot is the drift velocity for electrons and holes, respectively. The boundary conditions can be found in [7]. It has been proven that the charge induced at the electrodes from the motion of electrons and holes generated at point x at time t is given by QS (x; t)q[n(x; t)p(x; t)]; :/ It is worth noticing that the expression of the adjoint terms Eq. (12) differ from those described in [7] for the presence of the diffusion term which is essential for the calculation of the time evolution of the charge collection profiles in partially depleted devices. Finally, in order to compare such a model with experimental data obtained in experiments when carriers are generated by MeV ion probes penetrating the semiconductor device through the electrode (see Fig. 1), we have to consider the contribution of the charges generated at different positions and weighted by the Bragg (ion energy loss) curve: QS (t)
g
R
QS (x; t)×G(x) dx;
(13)
0
The current density j flowing in the diode is given by @n @p ; (10) j q× n×mn ×E p×mp ×E Dn × Dp × @x @x where q is the elementary charge. Such a current density is then inserted in equation (7) to get the time evolution of the induced charge generated at x0. It is worth noticing that QS (x0,t) as defined by equation (7) is Green’s function for the continuity equations. 2.3. The ‘‘adjoint’’ method An efficient method to evaluate Green’s function of Eq. (8) has been recently presented by Prettyman [7]. Since the excess carrier continuity equation involves linear operators, an adjoint continuity equation can be
Fig. 1. Schematic view of TRIBICC system (DUT: device under test).
196
C. Manfredotti et al. / Materials Science and Engineering B102 (2003) 193 /197
where G is the energy loss profile and R the ion range in the semiconductor.
3. Experimental set-up The investigated sample was a commercial Mesa rectifier diode (Fig. 2). The doping profile was measured by the spreading resistance method. IBICC measurements were performed at the Ruder Boskovic Institute in Zagreb (HR) using a 4 MeV proton beam. The proton flux was maintained at about 100 protons per second in order to avoid electronic pile-up, surface trapping of generated carriers, and to reduce the radiation damage. The energy loss profile G (x ) of the protons within the sample is shown in Fig. 3A as evaluated by the SRIM2000 simulation code [5]. The output of the charge-sensitive preamplifier (Canberra 1004) for each individual ion strike was digitised using a fast computercontrolled Lecroy WAVERUNNER LT342 digital oscilloscope (0.5 G sample per second). The time resolution was about 2 ns and the transient was stored in 2500 points. Fig. 4 shows the time-resolved ion beam-induced charge signals evaluated at three different reverse bias voltage.
4. Results and discussion The mathematical method described in Section 2 was used to interpret the experimental data shown in Fig. 4. The electric field and the carrier velocity profiles evaluated by the PISCESII computer code [8] were used as input parameters of the adjoint Eq. (11) taking into account the dependence of the mobility from the
Fig. 3. Electronic energy loss for 4 MeV H-ions in the Mesa rectifier silicon diode (A); hole-generating function profile at different reverse bias voltage (B).
doping concentration and electric field [9]. The behaviour of the hole generating functions Gn at different bias voltages are plotted in Fig. 3B. The best-fit of the TRIBICC data of Fig. 4 were obtained by solving the adjoint Eq. (11) by means of a one-dimensional finite difference algorithm, assuming a hole bulk lifetime t0 /(59/1) ms and a dependence of the minority carrier lifetime t from the doping concentration given by the following expression [9]:
t
t0 ; 1 ((NA ND )=Nref )
(14)
Fig. 2. Scheme of the Mesa rectifier diode and the relevant doping profile; NA and ND represent the acceptor and donor concentrations, respectively.
C. Manfredotti et al. / Materials Science and Engineering B102 (2003) 193 /197
197
This means that their motion in the neutral region (E / 0) does not yield any charge signal and the long tail in the TRIBICC signal is relevant to their arrival time at the boundary of the DL.
5. Conclusions
Fig. 4. Experimental time behaviour of charge collection measurements carried out at different reverse bias voltage: (open circles) experimental data; (continuous line) fitting curves.
where Nref is equal to 7.1 /1015 cm 3 and NA,D are the acceptor and donor concentrations. Apart the lifetime measurements evaluated by means of the numerical solution of the adjoint equations, some qualitative observations can be drawn from the analysis of the TRIBICC signals. The source term G is obviously connected with the extension of the depletion layer (DL) where carriers experience a very rapid drift. The Ramo and Ramo / Gunn [10] theorems state that the charge is induced at the electrode only if the carriers move in the presence of the applied electric field, i.e. only when the carriers move within the DL layer. This means that carriers generated within the DL induce (less than a nanosecond) a very short current pulse, which corresponds to the sharp and intense increase of the charge collection signal. The increase of the reverse bias voltage yields an increase of the DL layer, which corresponds to larger amount of carriers generated within the DL and, consequently, a higher charge collected in the first nanosecond. After this short transient, the collection time is much longer. This is due to the minority carriers generating in the neutral region, which diffuse towards the DL. From the above-mentioned theorem, charge is induced at the electrodes only when these carriers move in the presence of the electric field, i.e. only when they enter the DL.
TRIBICC is proposed as a standard method in order to evaluate minority carriers lifetime in finished devices. In the present TRIBICC version, Gunn’s theorem and the adjoint equation method to solve the continuity equation is used in order to treat complex situations in which both drift and diffusion regions are present. The method introduces a generation function, which better defines the depletion or drift region and gives directly the time behaviour of charge collection efficiency. The fit on experimental data is carried out with only one parameter. Preliminary results obtained on a power Mesa device have been presented: they can be shown to offer minority carriers lifetime values in agreement with the expected one and suitable for CAE applications for the simulation of device design and behaviour.
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