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A study of the trap influence on the performance of semi-insulating GaAs detectors
Nuclear
Instruments
and Methods
in Physics Research A 395 (1997)349-354
NUCLEAR INSTRUMENTS &MElwooS IN PHVSICS “EEeY
ELSEMER
A study of the trap influence on the performance of semi-insulating GaAs detectors A. Cola”**, F. Quarantaa, L. VasanellPb, C. Canali”, A. Cavallinid, F. Navae, M.E. Fantaccif aIstituto per lo studio di Nuovi Materiali per I’Elettronica (I.M.E.). C.N.R., Via Arnesano I-73100 Lecce, Italy b Dipartimento di Scienza dei Materiali, Universita di Lecce, Via Amesano I-731 00, Lecce, di Scietue deli’lngegneria, Universita di Modena. Via Campi 213/A, I-41100 Modena, and INFN ‘INFM and Dipartimento di Fisica, Universita di Bologna, Via Irnerio 46, I-40126 Bologna, ‘Dipartimento di Fisica, Vniversita di Modena. Via Campi 213/A, I-41 100 Modena, and INFN Sezione ‘Istituto Nazionale di Fisica Nucleare. S Piero a Grado Via Vecchia Livornese 582/A Pisa,
‘Dipartimento
Italy Sezione di Bologna, Italy Italy di Bologna, Ita!v Italy
Abstract
A numerical approach has been used to describe the influence of the traps on the properties of semi-insulating GaAs detectors. Based on a realistic trap scheme, the electric field distribution and the active region thickness in GaAs detectors have been calculated and compared with experimental results. From the electric field distribution and the concentration of ionized traps, taking into account detrapping processes, the carrier mean free path has been calculated and used to evaluate the charge collection efficiency. The necessity to consider the role of the high-electric-field effects on the stationary occupation of traps clearly comes out from this analysis.
1. Introduction
Semi-insulating GaAs detectors attract considerable attention, especially as detectors for low-energy X-rays [ 1,2]. However, their performance appear strongly affected by the influence of trap centers present in such kind of material. The traps influence both the spatial distribution of the electric field and the charge collection efficiency. Therefore, a central point is to correlate the trap parameters (energy level, capture cross section, concentration) to the detector performance [3]. Unfortunately, some simplifying assumptions, as the Schottky approximation, cannot be applied [4] and a numerical approach is required to obtain a description of the detector performance. Our aim is to develop a comprehensive model, based on a realistic trap scheme, able to describe the main detector properties. Hence, we start from the knowledge of trap parameters as measured by Photo-Induced Current Transient Spectroscopy (PICTS) and Photo Deep Level Transient Spectroscopy (P-DLTS) and, by solving the coupled transport and Poisson’s equations we found the electric field distribution and active thickness. The model includes high-field effects which seem to strongly
* Corresponding
2. Model 2.1. The electric field distribution Important assumptions of the model are that the problem can be treated in one-dimension and that hole
author.
0168-9002/97/$17.00 Copyright PII SO1 68-9002(97)00604-9
affect the occupation of trap [S], so causing the electric field to be flat along the detector-active region. These results have been compared with data obtained in Optical Beam-Induced Currents (OBIC) experiments. In this way, we go behind the usual modelling of the electric field which considers only the mid-gap EL2 electron trap [S-7]: we solve the governing set of equations for the real sample and we compare the results with the OBIC data which give a direct measurement of the active thickness. The spatial distribution of the electric field and the concentration of ionized traps allows us to evaluate the local mean free path of carriers and the local chargecollection efficiency. Finally, by weighing this last quantity with the distribution of carriers generated by the ionizing radiation, the charge-collection efficiency has been calculated. Comparison with experimental efficiencies are presented for or-front, m-back, and X-ray irradiation.
0
1997 Elsevier Science B.V. All rights reserved IV. SIGNAL
FORMATION
A. Cola et al. INucl. Instr. and Meth. in Phys. Res. A 395 (1997) 349-354
350
current is neglected [S]. In order to find the electric field and the free electron distribution, we cannot decouple Poisson’s equation from the transport equation and we have to solve simultaneously the governing set of equations:
After these modifications, Eq. (3) under stationary conditions gives the concentration NT of ionized traps:
an
1+
J,, - en,u,E
a.y=
aE _=_ ax
PLnkT
&a)
’
P d
(1.b)
where E is the electric field, pL, the electron mobility, T the temperature, E the dielectric constant of GaAs, k the Boltzmann constant, e the electron charge, n the free electron density. The charge density p, is given by p=e [
p-n+CNP+ I
-T;N’-]>
(2)
where p is the hole concentration, Nf+ and Nf- represent the concentration of the fixed positive and negative charges due to the ith donor and acceptor center, respectively. The two sums are carried out on all the electron and hole traps. In order to evaluate the occupation probabilityfr of the generic electron trap of concentration Nr, we consider that the electron concentration n in the conduction band varies as [9]: (3) where uthe its thermal velocity, 0 is the carrier capture cross section of the trap, and e, its emission rate. McGregor et al. have proposed [S], for the only active trap in their model, which is the EL2 center, a fieldenhanced capture effect. It is based on the experimental data of Prints et al. [lO,ll] and modelled by altering Eq. (2) in such a way that the total charge decreases dramatically at electric fields approaching a critical field E,, which they assume to be 7 kV/cm. We have used a similar filling function to describe a(E) in Eq. (3): a(E) = ~(1 + CWcI”),
(4)
where go is the zero-field capture cross-section value, E, is the critical electric field and c1an exponent. In absence of a meaningful model able to describe the real a(E) behavior, we have considered E, as adjustable parameter, equal for all the traps, while we have assumed c1= 15, as in McGregor et al. [S]. We have also accounted for the PooleeFrenkel effect by modifying the zero-field emission rate en0 with the known exponential dependence [12]: e. =
en0
ex&hE),
where in GaAs, for a singly charged 15 kV/cm [12].
(5) defect, E. z
NT’ = NT(l -fT) = NT
9-l expC(& - &)WI
(1 +
CW%lVev(J-EIEo) ’ 16)
where we have used the condition en0 = gaovthe x N, exp(-(E, - E,)/kT), with ET the trap energy with respect to the conduction band, Er the Fermi quasi-level, and g the degeneracy factor associated with the level. The presence of many traps does not invalidate Eq. (6) if we assume that each trap exchanges carriers only with the corresponding band. We will come on the implications of Eq. (6) in the next session. As far as the hole traps are concerned, the equation corresponding to Eq. (6) can be easily obtained starting from the balance equation of holes in the valence band. Neglecting the hole current under reverse bias, the stationary conditions imply that generation currents are also negligible and the electron density J,, is constant along the whole device (and equal to the total one). This fact allows us to use in Eq. (1.1) the experimental values J,(V), independent of X, measured at the voltage value used to solve the system of Eqs. (1). We have taken into account also the dependence of the electron drift velocity on the electric field [13]. In order to find n and E along the entire detector thickness d, we have analyzed the case of the detector not completely depleted, which is the case up to quite high voltages. We assume the presence of two distinct regions: the former, of thickness do, toward the backcontact, is ohmic and is characterized by a free-electron concentration given by its equilibrium value n = no and a constant electric field given by E
=
0-
JAW en0PO'
where pa is the ohmic mobility. In the latter region, which is the ‘active’ one under the Schottky contact, we solve the system of Eqs. (1) in n and E using the boundary conditions that at the border x = d - do with the ohmic region it still holds n = no. The total potential drop is v,i+~~~E(X)dx=~~doE(X)dX+Egdg;
(8)
where Vbi is the built-in potential. The case of very high applied voltages, which is not considered here, corresponds to the disappearance of the ohmic region and to electric field values at the backcontacts greater than the values given by Eq. (7). The real characteristics of the contacts should be taken into account in order to model this extreme situation.
351
A. Cola et al. I Nucl. Instr. and Meth. in Phys. Res. A 395 (1997) 349-354 2.2. Charge collection efJiciency
charge collection efficiency (ccc) as
In order to evaluate the influence of trapping effects on the charge collection properties, we have calculated the mean free path of electrons and holes, given by: I = tu, where u is the drift velocity and the trapping time T is given by
ccc =
d
f=Ti
(9)
I
with the index i ranging over all the electron/hole traps. The meaning of the trapping time ri emerges from Eq. (3): for the ith trap, since NT:: = (1 -fTi)NTi, if the emissivity of the trap is sufficiently low, Eq. (3) can be re-written as
an at
dxo
s
(13)
N(xo)dxo>
is
0
where N(x,) is the electric charge distribution generated by the ionizing radiation. We have used the EGS4 code [16] evaluate the distribution N(x,) of the photo-electric charges created along the GaAs detector by 60 keV X-rays, while, for the CLparticle irradiation, we have used the well-known Bragg curve.
3. Results and discussion 3. I, Electric field distribution
n Ti'
(10)
where now n includes the electrons generated by the radiation, and 1 ri = ~ CT.V L thN+.’ TI
(11)
By looking at Eq. (11) it is clear that ri is a function of x, since either cc and NGi are functions of the position. We have considered the detrapping process for the ith trap efficient if enitp > 1, where t, is the shaping time, and e,i is the trap emissivity. In this case, the local 7(x) is given by the sum in Eq. (9) carried out on all the traps which do not detrap. From Ramo’s theorem [14,15], taking into account the exponential decay of the free carriers with mean free path n(x), the total signal induced by an electron/hole pair of unit charge deposited at the point x,, is q(xO) = n&O) +
Ann
qh(xO)
=
f
We report in Table 1 the trap parameters of the five electron traps (Al, A2, A3, A4, AS) and two hole traps (Bl, B2) which have been measured by PICTS and P-DLTS [17] on the same GaAs material used to prepare the present detector. We have solved the system of Eqs. (1) on the base of the model outlined in the previous section using the data of Table 1. In Fig. l(a) some electric field distributions at different voltages, obtained considering the a(E) dependence of Eq. (4) on all the traps are depicted. The active thickness w has been reported in lower figure (Fig. l(b)). In the same figure we have also reported the experimental behavior, as deduced from OBIC measurements [18]. There is a very good agreement between calculated and experimental results and the following comments come out: (1) the experimental w versus V relation is linear: this implies that the electric field is constant along the active
Table 1 PICTS data of electron traps used in the simulations Trap
Concentration (cm-3)
Energy level (eV)
Capture cross section (cm’)
Al A2 A3 A4 A5
6.8 x 3.7 x 5.0 x 1.2 x 2.0 x
10“’ 10’s 1013 10’5 10’6
0.175 0.315 0.400 0.560 0.790
4.5 x 2.9 x 3.0 x 2.0 x 5.0 x
Bl B2
3.0 x 10’5 7.0 x 10’4
0.410 0.510
1.0x 10-13 7.0x 10-l’
(12) Here q(xo) can be considered as the local efficiency, with partial contributes given by qe(xo) and qh(xO) for electrons and holes, respectively. Note that in the previous equation the carriers generated give a contribution to the induced signal even if they are generated in the ohmic region: the extremes of the external integrals are the detector contacts. This is quite different from the case of less resistive materials like silicon, where the electric fields in the ohmic region are several orders of magnitude lower and, therefore, negligible with respect to the fields in the active region. In order to compare the results of the calculation with the results of irradiation experiment, we calculate the
10-15 lo-l3 lo-t3 lo-l5 lo-r4
Note: Traps labeled A, B are for electrons and holes, respectively. Other material parameters are: the electron ohmic mobility p0 = 6900cm’V’s-r and resistivity p = 5.2 x 10’ $2cm. The equilibrium Fermi-level has been calculated to be at the position: E, - Ef % 0.618 eV. The detector thickness is 210 pm and the area of the Schottky contact is 3.15 mma.
IV. SIGNAL FORMATION
A. Cola et al. / Nucl. Ins&. and Meth. in Phys. Rex A 395 (1997) 349-354
352
14 -
---177v -106V ---66V
: 4
g ‘* : $z 10 8
__-__f.J4v
9
I I
:
6-
: : :
I
mainly sustain the electric field, that are the deepest traps, show the a(E) effect. Therefore, the electric field distribution like that reported in Fig. l(a) is only obtained if A4 and A5 traps show a(E) effect. It does not in any way influence the a(E) effect acting on all the other traps. 3.2. Charge collection e&ficiency
:
3
b)
5150-
4
loo-
B 9 -2
3
50-
t
ot”““““““l”‘.l 0
50
100
150
200
voltage (V) Fig. 1 (a) Electric field distribution inside detector for some applied voltages; (b) experimental (0) and calculated (---) active thickness as a function of the voltage.
region and at different voltages; (2) the flatness of the electric field observed in the calculation is caused by the o(E) rapid filling function: when the field increases above the critical field E,, the strong increase of the capture cross section decreases the fixed charge (see Eq. (6)). This effect fixes the field at values slightly above E,; (3) the role of the critical field E, is to determine the slope of the linear relation w versus I’; in this sense we have found E, = 8 kV/cm as a good fitting value. It should be noted that this value is close to the value given by McGregor et al. [S] (E, = 7 kV/vm). A crucial point concerns which trap should evidence a field-enhanced capture cross section. There are few experimental data published up to now [lO,l l] and, therefore, we have not discarded the possibility that a priori all traps could show this effect. From many simulations carried out it emerges that only the traps closest to the equilibrium Fermi level, which are A4 and A5, are decisive: it is important that the traps which
We proceed following the scheme of Section 2.2 using the results of the previous section: we have calculated the spatial distribution of trapping time, the drift velocity and, as a consequence, the mean free path of each carrier. These calculations can be carried out without free parameters, if the experimental trap data are assumed. The simulations have been performed using just the measured values for concentrations and activation energy, and values ranging in about one order of magnitude around the experimental ones for the capture cross section. This interval is quite realistic for experimental determination of capture cross section by PICTS and P-DLTS. In Fig. 2 the local mean free paths i of electron and holes are reported. We note that the values in the active region are quite greater than the values in the ohmic one. This result is related to the simple fact that the carriers are more subject to be captured when they cross the ohmic region, although in this region there are less ionized traps able to capture charge. As it can be seen from Fig. 2, for each carrier only two i values are able to describe the capture process, one value for each region. This result is in qualitative and quantitative agreement with the model of Kubiki et al. [6], who made the assumption that only two electron mean free paths were
-e-
electrons
activeregion
r.
’Schottky
depth (pm>
Fig. 2. Mean free path distribution of both carriers inside the detector. Applied reverse voltage is 107 V; on the vertical scale, the detector thickness d is also reported.
A. Cola et al. I Nucl. In&.
353
and Meth. in Phys. Rex A 395 (1997) 349-354
able to explain cl-front efficiency data; in their case a cact = 175 urn, and Rcohm= 50 urn, for the active and ohmic region, respectively. It is interesting to compare the calculated mean free path to the detector thickness: in the active region a,,,, < d (d = 210 urn), while Ahact> d. This means a better hole than electron collection. A complete electron collection should be achieved for A,,,, > d, a prediction which seems confirmed by experimental results on thin detectors [2,7]. Even if Fig. 2 refers to a particular value of the applied voltage (107 V), almost the same values of 1 are obtained at all voltages. This is especially true in the active region, where the electric field is almost constant: both the carrier drift velocity and the concentration of ionized traps, are constant. In the ohmic region there is a slightly increase of the electric field (and therefore of A) after the increase of the current on the voltage (Eq. (7)), but it is usually negligible. As a consequence, the main effect of the voltage is, by changing the w thickness, to shift the point where the mean free path changes. In Fig. 3 we compare the experimental and calculated charge collection efficiency for a-front and a-back irradiation. We recall that the collected signal is mainly due to the electron movement for irradiation on the Schottky contact (u-front) and to holes for irradiation on the ohmic contact (x-back). The saturation of a-front efficiency occurs when w > A,,,,: if electrons are captured mainly before they cross a given thickness, it is not relevant how w is greater of this thickness. In fact, there is a knee in the efficiencies at 1/ z 100 V, which corresponds to w = R,,,, z 110 urn. A different analysis holds for holes, which are created below the ohmic region and, therefore, in the low-field
loo1
100
150
200
voltage (V) Fig. 4. Charge collection efficiency as a function of the reverse voltage applied to the detector for 60 keV X-ray irradiation (0: calculation; 01 experiment).
region. In this case, Ahohmin the ohmic region is too short to allow the holes to reach the active region before being captured. Hence, holes do not produce a significant signal if they are generated too far from the high-field region. Taking into account that the penetration depth of the m-particles we have used is about 20um, only high voltages, corresponding to w z d - 20 pm = 190 urn, i.e. V cz 180 V, can allow detection of holes. On the other hand, when w x d, detectors presented in this work show breakdown, probably caused by hole injection from the backcontact [4]. If higher electric fields close to the backcontact were possible, higher efficiency should be achieved [17]. In Fig. 4 are reported experimental and calculated charge collection efficiency for 60 keV X-ray irradiation. The agreement is acceptable even if not very good. The ccc increases even at the higher voltages because charge is generated even near the backcontact.
4. Conclusion
Fig. 3. Charge collection efficiency as a function of the reverse voltage applied to the detector for a-particles irradiation. Circles are results for a-front irradiation, squares refer to a-back irradiation. Open symbols are calculation results, solid symbols are experimental results.
We have focused our attention on the influence of traps on electric field distribution and charge capture in semi-insulating GaAs detectors. We have proposed a procedure which starts from the knowledge of the trap parameters. By comparing the results of the calculation with experimental OBIC results, it comes out a strong limitation of the electric field. The field-enhanced capture cross section appears as an efficient mechanism but further and more fundamental studies are necessary to confirm this explanation and to furnish further details. After having considered the influence of traps on the electric field distribution, we have calculated the mean free path of the carriers. In this calculation, we have used again the trap parameters and we have also included an on/off detrapping effect. The comparison of the
IV. SIGNAL FORMATION
354
A. Cola et al. I Nucl. Instr. and Meth. in Phys. Res. A 395 (1997) 349-354
experimental and calculated charge collection efficiency is encouraging and it tells us that holes are collected better than electrons. In this sense, thinner detectors work better due to the better &,,,/d ratio.
Acknowledgements This work has been partially supported by the Italian National Institute of Nuclear Physics.
References Cl1 W. Bencivelli, E. Bertolucci, U. Bottigli, A. Cola, SD’ Auria, M.E. Fantacci, V. O’Shea, V. Rosso, K. Smith, A. Stefanini, L. Vasanelli, Nucl. Instr. and Meth. A 355 (1995) 425. PI E. Bertolucci, U. Bottigli, A. Cola, M.E. Fantacci, A. Stefanini, L. Vasanelli, IEEE Nucl. Trans. and Meth. 42 (1995) 254. c31 F. Nava, C. Canali, A. Castaldini, A. Cavallini, S. D’Auria, C. de1 Papa, C. Frigeri, L. Zanotti, A. Centronio, C. Lanzieri, A. Zichichi, Nucl. Instr. and Meth. A 349 (1994) 156. c41 A. Cola, in: Gallium Arsenide and Related Compounds P.G. Pelfer, J. Ludwig, K. Runge, H.S. Rupprecht (eds.), World Scientific, Singapore (1996), p. 217; S. Miniato (ed.), Proc. of 3rd Int. Workshop on Gallium Arsenide and Related Compounds, Tuscany, Italy, 22-25 May, 1995. c51 D.S. McGregor, R.A. Rojeski, G.F. Knoll, F.L. Terry Jr., J. East, Y. Eisen, J. Appl. Phys. 75 (1994) 7910. C61T. Kubicki, K. Luebelsmeyer, J. Ortmanns, D. Pandoulas, 0. Syben, M. Toporowsky, W.J. Xiao, Nucl. Instr. and Meth. A 345 (1994) 468.
[7] A. Cola, F. Quaranta, M.A. Ciocci, M.E. Fantacci, presented at 9th Int. Workshop on Room Temperature Semiconductor X-ray and y-ray Detectors, Associated Electronics and Applications, Grenoble, September 1995, Nucl. Instr. and Meth. A 380 (1996) 66. [S] The hole current can be neglected because SI GaAs is slightly n-type, with an electron concentration at least two order of magnitude greater than hole concentration; moreover, the electron mobility is about one order of magnitude greater than the hole mobility. [9] A.G. Milnes, Deep Impurities in Semiconductors, Wiley, New York, 1973. [lo] V. Ya. Prints, B.A. Bobylev, Sov. Phys. Semicond. 14 (1981) 1097. [l 11 V. Ya. Prints, S.N. Rechkunov Phys. Status Solidi (B) 118 (1983) 159. [12] K.W. Boer, Survey of Semiconductor Physics, vol. II Van NostranddReinhold, New York, 1992. [13] C.S. Chang, H.R. Fetterman, Solid State Electron. 29 (1986) 1295. [14] S. Ramo, Proc. IRE 27 (1939) 584. [15] G. Cavelleri, E. Gatti, G. Fabbri, Nucl. Instr. and Meth. A 92 (1971) 137. [16] W.R. Nelson, H. Hirayama, D.W.O. Rogers, The EGS4 Code System, SLAC Report, 1985. [17] A. Castaldini, A. Cavallini, C. Canali, C. Chiossi, C. de1 Papa, C. Lanzieri, MRS-Fall Meeting, 1995, in: O.J. Glembocki, F.H. Pollak, S.W. Pang, F. Celli, (Eds.), Diagnostic Techniques for Semiconductor Materials Processing, vol. MRS-406, to appear. [18] A. Castaldini, A. Cavallini, C. de1 Papa, M. Alietti, C. Canali, F. Nava, C. Lanzieri, Scanning Microscopy 8 (1994) 969; M. Alietti, L. Berluti, C. Canali, A. Castaldini, A. Cavallini, A. Cetronio, S. D’Auria, C. de1 Papa, C. Lanzieri, F. Nava, M. Proia, P. Rinaldi, A. Zichichi, Nucl. Instr. and Meth. A 355 (1995) 420.
Instruments
and Methods
in Physics Research A 395 (1997)349-354
NUCLEAR INSTRUMENTS &MElwooS IN PHVSICS “EEeY
ELSEMER
A study of the trap influence on the performance of semi-insulating GaAs detectors A. Cola”**, F. Quarantaa, L. VasanellPb, C. Canali”, A. Cavallinid, F. Navae, M.E. Fantaccif aIstituto per lo studio di Nuovi Materiali per I’Elettronica (I.M.E.). C.N.R., Via Arnesano I-73100 Lecce, Italy b Dipartimento di Scienza dei Materiali, Universita di Lecce, Via Amesano I-731 00, Lecce, di Scietue deli’lngegneria, Universita di Modena. Via Campi 213/A, I-41100 Modena, and INFN ‘INFM and Dipartimento di Fisica, Universita di Bologna, Via Irnerio 46, I-40126 Bologna, ‘Dipartimento di Fisica, Vniversita di Modena. Via Campi 213/A, I-41 100 Modena, and INFN Sezione ‘Istituto Nazionale di Fisica Nucleare. S Piero a Grado Via Vecchia Livornese 582/A Pisa,
‘Dipartimento
Italy Sezione di Bologna, Italy Italy di Bologna, Ita!v Italy
Abstract
A numerical approach has been used to describe the influence of the traps on the properties of semi-insulating GaAs detectors. Based on a realistic trap scheme, the electric field distribution and the active region thickness in GaAs detectors have been calculated and compared with experimental results. From the electric field distribution and the concentration of ionized traps, taking into account detrapping processes, the carrier mean free path has been calculated and used to evaluate the charge collection efficiency. The necessity to consider the role of the high-electric-field effects on the stationary occupation of traps clearly comes out from this analysis.
1. Introduction
Semi-insulating GaAs detectors attract considerable attention, especially as detectors for low-energy X-rays [ 1,2]. However, their performance appear strongly affected by the influence of trap centers present in such kind of material. The traps influence both the spatial distribution of the electric field and the charge collection efficiency. Therefore, a central point is to correlate the trap parameters (energy level, capture cross section, concentration) to the detector performance [3]. Unfortunately, some simplifying assumptions, as the Schottky approximation, cannot be applied [4] and a numerical approach is required to obtain a description of the detector performance. Our aim is to develop a comprehensive model, based on a realistic trap scheme, able to describe the main detector properties. Hence, we start from the knowledge of trap parameters as measured by Photo-Induced Current Transient Spectroscopy (PICTS) and Photo Deep Level Transient Spectroscopy (P-DLTS) and, by solving the coupled transport and Poisson’s equations we found the electric field distribution and active thickness. The model includes high-field effects which seem to strongly
* Corresponding
2. Model 2.1. The electric field distribution Important assumptions of the model are that the problem can be treated in one-dimension and that hole
author.
0168-9002/97/$17.00 Copyright PII SO1 68-9002(97)00604-9
affect the occupation of trap [S], so causing the electric field to be flat along the detector-active region. These results have been compared with data obtained in Optical Beam-Induced Currents (OBIC) experiments. In this way, we go behind the usual modelling of the electric field which considers only the mid-gap EL2 electron trap [S-7]: we solve the governing set of equations for the real sample and we compare the results with the OBIC data which give a direct measurement of the active thickness. The spatial distribution of the electric field and the concentration of ionized traps allows us to evaluate the local mean free path of carriers and the local chargecollection efficiency. Finally, by weighing this last quantity with the distribution of carriers generated by the ionizing radiation, the charge-collection efficiency has been calculated. Comparison with experimental efficiencies are presented for or-front, m-back, and X-ray irradiation.
0
1997 Elsevier Science B.V. All rights reserved IV. SIGNAL
FORMATION
A. Cola et al. INucl. Instr. and Meth. in Phys. Res. A 395 (1997) 349-354
350
current is neglected [S]. In order to find the electric field and the free electron distribution, we cannot decouple Poisson’s equation from the transport equation and we have to solve simultaneously the governing set of equations:
After these modifications, Eq. (3) under stationary conditions gives the concentration NT of ionized traps:
an
1+
J,, - en,u,E
a.y=
aE _=_ ax
PLnkT
&a)
’
P d
(1.b)
where E is the electric field, pL, the electron mobility, T the temperature, E the dielectric constant of GaAs, k the Boltzmann constant, e the electron charge, n the free electron density. The charge density p, is given by p=e [
p-n+CNP+ I
-T;N’-]>
(2)
where p is the hole concentration, Nf+ and Nf- represent the concentration of the fixed positive and negative charges due to the ith donor and acceptor center, respectively. The two sums are carried out on all the electron and hole traps. In order to evaluate the occupation probabilityfr of the generic electron trap of concentration Nr, we consider that the electron concentration n in the conduction band varies as [9]: (3) where uthe its thermal velocity, 0 is the carrier capture cross section of the trap, and e, its emission rate. McGregor et al. have proposed [S], for the only active trap in their model, which is the EL2 center, a fieldenhanced capture effect. It is based on the experimental data of Prints et al. [lO,ll] and modelled by altering Eq. (2) in such a way that the total charge decreases dramatically at electric fields approaching a critical field E,, which they assume to be 7 kV/cm. We have used a similar filling function to describe a(E) in Eq. (3): a(E) = ~(1 + CWcI”),
(4)
where go is the zero-field capture cross-section value, E, is the critical electric field and c1an exponent. In absence of a meaningful model able to describe the real a(E) behavior, we have considered E, as adjustable parameter, equal for all the traps, while we have assumed c1= 15, as in McGregor et al. [S]. We have also accounted for the PooleeFrenkel effect by modifying the zero-field emission rate en0 with the known exponential dependence [12]: e. =
en0
ex&hE),
where in GaAs, for a singly charged 15 kV/cm [12].
(5) defect, E. z
NT’ = NT(l -fT) = NT
9-l expC(& - &)WI
(1 +
CW%lVev(J-EIEo) ’ 16)
where we have used the condition en0 = gaovthe x N, exp(-(E, - E,)/kT), with ET the trap energy with respect to the conduction band, Er the Fermi quasi-level, and g the degeneracy factor associated with the level. The presence of many traps does not invalidate Eq. (6) if we assume that each trap exchanges carriers only with the corresponding band. We will come on the implications of Eq. (6) in the next session. As far as the hole traps are concerned, the equation corresponding to Eq. (6) can be easily obtained starting from the balance equation of holes in the valence band. Neglecting the hole current under reverse bias, the stationary conditions imply that generation currents are also negligible and the electron density J,, is constant along the whole device (and equal to the total one). This fact allows us to use in Eq. (1.1) the experimental values J,(V), independent of X, measured at the voltage value used to solve the system of Eqs. (1). We have taken into account also the dependence of the electron drift velocity on the electric field [13]. In order to find n and E along the entire detector thickness d, we have analyzed the case of the detector not completely depleted, which is the case up to quite high voltages. We assume the presence of two distinct regions: the former, of thickness do, toward the backcontact, is ohmic and is characterized by a free-electron concentration given by its equilibrium value n = no and a constant electric field given by E
=
0-
JAW en0PO'
where pa is the ohmic mobility. In the latter region, which is the ‘active’ one under the Schottky contact, we solve the system of Eqs. (1) in n and E using the boundary conditions that at the border x = d - do with the ohmic region it still holds n = no. The total potential drop is v,i+~~~E(X)dx=~~doE(X)dX+Egdg;
(8)
where Vbi is the built-in potential. The case of very high applied voltages, which is not considered here, corresponds to the disappearance of the ohmic region and to electric field values at the backcontacts greater than the values given by Eq. (7). The real characteristics of the contacts should be taken into account in order to model this extreme situation.
351
A. Cola et al. I Nucl. Instr. and Meth. in Phys. Res. A 395 (1997) 349-354 2.2. Charge collection efJiciency
charge collection efficiency (ccc) as
In order to evaluate the influence of trapping effects on the charge collection properties, we have calculated the mean free path of electrons and holes, given by: I = tu, where u is the drift velocity and the trapping time T is given by
ccc =
d
f=Ti
(9)
I
with the index i ranging over all the electron/hole traps. The meaning of the trapping time ri emerges from Eq. (3): for the ith trap, since NT:: = (1 -fTi)NTi, if the emissivity of the trap is sufficiently low, Eq. (3) can be re-written as
an at
dxo
s
(13)
N(xo)dxo>
is
0
where N(x,) is the electric charge distribution generated by the ionizing radiation. We have used the EGS4 code [16] evaluate the distribution N(x,) of the photo-electric charges created along the GaAs detector by 60 keV X-rays, while, for the CLparticle irradiation, we have used the well-known Bragg curve.
3. Results and discussion 3. I, Electric field distribution
n Ti'
(10)
where now n includes the electrons generated by the radiation, and 1 ri = ~ CT.V L thN+.’ TI
(11)
By looking at Eq. (11) it is clear that ri is a function of x, since either cc and NGi are functions of the position. We have considered the detrapping process for the ith trap efficient if enitp > 1, where t, is the shaping time, and e,i is the trap emissivity. In this case, the local 7(x) is given by the sum in Eq. (9) carried out on all the traps which do not detrap. From Ramo’s theorem [14,15], taking into account the exponential decay of the free carriers with mean free path n(x), the total signal induced by an electron/hole pair of unit charge deposited at the point x,, is q(xO) = n&O) +
Ann
qh(xO)
=
f
We report in Table 1 the trap parameters of the five electron traps (Al, A2, A3, A4, AS) and two hole traps (Bl, B2) which have been measured by PICTS and P-DLTS [17] on the same GaAs material used to prepare the present detector. We have solved the system of Eqs. (1) on the base of the model outlined in the previous section using the data of Table 1. In Fig. l(a) some electric field distributions at different voltages, obtained considering the a(E) dependence of Eq. (4) on all the traps are depicted. The active thickness w has been reported in lower figure (Fig. l(b)). In the same figure we have also reported the experimental behavior, as deduced from OBIC measurements [18]. There is a very good agreement between calculated and experimental results and the following comments come out: (1) the experimental w versus V relation is linear: this implies that the electric field is constant along the active
Table 1 PICTS data of electron traps used in the simulations Trap
Concentration (cm-3)
Energy level (eV)
Capture cross section (cm’)
Al A2 A3 A4 A5
6.8 x 3.7 x 5.0 x 1.2 x 2.0 x
10“’ 10’s 1013 10’5 10’6
0.175 0.315 0.400 0.560 0.790
4.5 x 2.9 x 3.0 x 2.0 x 5.0 x
Bl B2
3.0 x 10’5 7.0 x 10’4
0.410 0.510
1.0x 10-13 7.0x 10-l’
(12) Here q(xo) can be considered as the local efficiency, with partial contributes given by qe(xo) and qh(xO) for electrons and holes, respectively. Note that in the previous equation the carriers generated give a contribution to the induced signal even if they are generated in the ohmic region: the extremes of the external integrals are the detector contacts. This is quite different from the case of less resistive materials like silicon, where the electric fields in the ohmic region are several orders of magnitude lower and, therefore, negligible with respect to the fields in the active region. In order to compare the results of the calculation with the results of irradiation experiment, we calculate the
10-15 lo-l3 lo-t3 lo-l5 lo-r4
Note: Traps labeled A, B are for electrons and holes, respectively. Other material parameters are: the electron ohmic mobility p0 = 6900cm’V’s-r and resistivity p = 5.2 x 10’ $2cm. The equilibrium Fermi-level has been calculated to be at the position: E, - Ef % 0.618 eV. The detector thickness is 210 pm and the area of the Schottky contact is 3.15 mma.
IV. SIGNAL FORMATION
A. Cola et al. / Nucl. Ins&. and Meth. in Phys. Rex A 395 (1997) 349-354
352
14 -
---177v -106V ---66V
: 4
g ‘* : $z 10 8
__-__f.J4v
9
I I
:
6-
: : :
I
mainly sustain the electric field, that are the deepest traps, show the a(E) effect. Therefore, the electric field distribution like that reported in Fig. l(a) is only obtained if A4 and A5 traps show a(E) effect. It does not in any way influence the a(E) effect acting on all the other traps. 3.2. Charge collection e&ficiency
:
3
b)
5150-
4
loo-
B 9 -2
3
50-
t
ot”““““““l”‘.l 0
50
100
150
200
voltage (V) Fig. 1 (a) Electric field distribution inside detector for some applied voltages; (b) experimental (0) and calculated (---) active thickness as a function of the voltage.
region and at different voltages; (2) the flatness of the electric field observed in the calculation is caused by the o(E) rapid filling function: when the field increases above the critical field E,, the strong increase of the capture cross section decreases the fixed charge (see Eq. (6)). This effect fixes the field at values slightly above E,; (3) the role of the critical field E, is to determine the slope of the linear relation w versus I’; in this sense we have found E, = 8 kV/cm as a good fitting value. It should be noted that this value is close to the value given by McGregor et al. [S] (E, = 7 kV/vm). A crucial point concerns which trap should evidence a field-enhanced capture cross section. There are few experimental data published up to now [lO,l l] and, therefore, we have not discarded the possibility that a priori all traps could show this effect. From many simulations carried out it emerges that only the traps closest to the equilibrium Fermi level, which are A4 and A5, are decisive: it is important that the traps which
We proceed following the scheme of Section 2.2 using the results of the previous section: we have calculated the spatial distribution of trapping time, the drift velocity and, as a consequence, the mean free path of each carrier. These calculations can be carried out without free parameters, if the experimental trap data are assumed. The simulations have been performed using just the measured values for concentrations and activation energy, and values ranging in about one order of magnitude around the experimental ones for the capture cross section. This interval is quite realistic for experimental determination of capture cross section by PICTS and P-DLTS. In Fig. 2 the local mean free paths i of electron and holes are reported. We note that the values in the active region are quite greater than the values in the ohmic one. This result is related to the simple fact that the carriers are more subject to be captured when they cross the ohmic region, although in this region there are less ionized traps able to capture charge. As it can be seen from Fig. 2, for each carrier only two i values are able to describe the capture process, one value for each region. This result is in qualitative and quantitative agreement with the model of Kubiki et al. [6], who made the assumption that only two electron mean free paths were
-e-
electrons
activeregion
r.
’Schottky
depth (pm>
Fig. 2. Mean free path distribution of both carriers inside the detector. Applied reverse voltage is 107 V; on the vertical scale, the detector thickness d is also reported.
A. Cola et al. I Nucl. In&.
353
and Meth. in Phys. Rex A 395 (1997) 349-354
able to explain cl-front efficiency data; in their case a cact = 175 urn, and Rcohm= 50 urn, for the active and ohmic region, respectively. It is interesting to compare the calculated mean free path to the detector thickness: in the active region a,,,, < d (d = 210 urn), while Ahact> d. This means a better hole than electron collection. A complete electron collection should be achieved for A,,,, > d, a prediction which seems confirmed by experimental results on thin detectors [2,7]. Even if Fig. 2 refers to a particular value of the applied voltage (107 V), almost the same values of 1 are obtained at all voltages. This is especially true in the active region, where the electric field is almost constant: both the carrier drift velocity and the concentration of ionized traps, are constant. In the ohmic region there is a slightly increase of the electric field (and therefore of A) after the increase of the current on the voltage (Eq. (7)), but it is usually negligible. As a consequence, the main effect of the voltage is, by changing the w thickness, to shift the point where the mean free path changes. In Fig. 3 we compare the experimental and calculated charge collection efficiency for a-front and a-back irradiation. We recall that the collected signal is mainly due to the electron movement for irradiation on the Schottky contact (u-front) and to holes for irradiation on the ohmic contact (x-back). The saturation of a-front efficiency occurs when w > A,,,,: if electrons are captured mainly before they cross a given thickness, it is not relevant how w is greater of this thickness. In fact, there is a knee in the efficiencies at 1/ z 100 V, which corresponds to w = R,,,, z 110 urn. A different analysis holds for holes, which are created below the ohmic region and, therefore, in the low-field
loo1
100
150
200
voltage (V) Fig. 4. Charge collection efficiency as a function of the reverse voltage applied to the detector for 60 keV X-ray irradiation (0: calculation; 01 experiment).
region. In this case, Ahohmin the ohmic region is too short to allow the holes to reach the active region before being captured. Hence, holes do not produce a significant signal if they are generated too far from the high-field region. Taking into account that the penetration depth of the m-particles we have used is about 20um, only high voltages, corresponding to w z d - 20 pm = 190 urn, i.e. V cz 180 V, can allow detection of holes. On the other hand, when w x d, detectors presented in this work show breakdown, probably caused by hole injection from the backcontact [4]. If higher electric fields close to the backcontact were possible, higher efficiency should be achieved [17]. In Fig. 4 are reported experimental and calculated charge collection efficiency for 60 keV X-ray irradiation. The agreement is acceptable even if not very good. The ccc increases even at the higher voltages because charge is generated even near the backcontact.
4. Conclusion
Fig. 3. Charge collection efficiency as a function of the reverse voltage applied to the detector for a-particles irradiation. Circles are results for a-front irradiation, squares refer to a-back irradiation. Open symbols are calculation results, solid symbols are experimental results.
We have focused our attention on the influence of traps on electric field distribution and charge capture in semi-insulating GaAs detectors. We have proposed a procedure which starts from the knowledge of the trap parameters. By comparing the results of the calculation with experimental OBIC results, it comes out a strong limitation of the electric field. The field-enhanced capture cross section appears as an efficient mechanism but further and more fundamental studies are necessary to confirm this explanation and to furnish further details. After having considered the influence of traps on the electric field distribution, we have calculated the mean free path of the carriers. In this calculation, we have used again the trap parameters and we have also included an on/off detrapping effect. The comparison of the
IV. SIGNAL FORMATION
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A. Cola et al. I Nucl. Instr. and Meth. in Phys. Res. A 395 (1997) 349-354
experimental and calculated charge collection efficiency is encouraging and it tells us that holes are collected better than electrons. In this sense, thinner detectors work better due to the better &,,,/d ratio.
Acknowledgements This work has been partially supported by the Italian National Institute of Nuclear Physics.
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