IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 50, NO. 6, DECEMBER 2003

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Evaluating Average and Atypical Response in Radiation Effects Simulations Robert A. Weller, Senior Member, IEEE, Andrew L. Sternberg, Member, IEEE, Lloyd W. Massengill, Senior Member, IEEE, Ronald D. Schrimpf, Fellow, IEEE, and Daniel M. Fleetwood, Fellow, IEEE

Abstract—We examine the limits of performing single-event simulations using pre-averaged radiation events. Geant4 simulations show the necessity, for future devices, to supplement current methods with ensemble averaging of device-level responses to physically realistic radiation events. Initial Monte Carlo simulations have generated a significant number of extremal events in local energy deposition. These simulations strongly suggest that proton strikes of sufficient energy, even those that initiate purely electronic interactions, can initiate device response capable in principle of producing single event upset or microdose damage in highly scaled devices. Index Terms—Charge collection, extreme value statistics, Geant4, linear energy transfer, Monte Carlo simulation, protons, radiation effects, single-event effects, soft errors.

I. INTRODUCTION

A

S device dimensions approach the nano-scale, the inherent fluctuations in the parameters used to characterize radiation effects become increasingly important, and the need for consistent, physically based simulations becomes correspondingly more urgent. Many of the basic principles important to this task related to single-event mechanisms have already been discussed in the literature, including the variability of (LET) [1], the principles of microdosimetry [2], effects (particularly displacement) of nuclear reactions [3]–[6] and reaction products [7], the application of extreme-value statistics to single event analysis [8], and most recently a theoretical description of the LET spectrum [9]. In this paper we outline a synthesis of these principles in which average track models are supplemented by ensembles of physically realistic radiation events. This approach will yield deeper insight into the mechanisms of single-event effects [10], while providing a means for analyzing processes, such as multiple bit upsets [11] and secondary radiation from materials in the local environment surrounding active devices. These effects depend on the evolution and microstructure of radiation interactions, and do not lend themselves easily to other forms of analysis. II. ENERGY DEPOSITION If the value of a circuit parameter, such as the voltage at a given node, is linearly related to a random perturbation, such as Manuscript received July 20, 2003; revised August 29, 2003. The authors are with the Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN 37235 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TNS.2003.821576

the energy deposited by an energetic ion, then the distribution of voltages is simply related to the distribution of the perturbation. In this case, it is easy to show that the average response is the response to the average perturbation. However, in the general case, and particularly when significant nonlinearities are inis the volved, the distributions are related by (1), where is the distribution of perturdistribution of responses , bations , is the Dirac delta function, and the function establishes the relationship between and , . The integral is over the full range of (1) This equation may be used to show that the average value of is given by when is linear and that if the distribution of is narrow. Otherwise, and the average must be obtained by computing this integral either analytically or numerically. In more complex depends on additional random cases, as for example when variables such as the geometric coordinates of the ion strike, or when is itself a derived random variable, numerical methods are essential. The latter case is of particular interest, and it is the one we consider here. The total energy deposited by an ion in a sensitive volume is the sum of a large number of individual discrete energy transfers of random size . LET is an average of these values and is useful when the sample is thick enough for this average to be meaningful. This general framework breaks down in two important cases: (1) when the distribution of cannot be assumed to be narrow, regardless of the thickness of the sensitive volume, and (2) when the sensitive volume is sufficiently thin that the average inherent in the definition of LET is of questionable validity. values, although rare, Events with very large individual alter the distribution of in a way that undermines the validity of the averaging process. Examples are the production of discrete rays and nuclear reactions that produce multiple, charged reaction products. While an average energy deposition including nuclear events could obviously be computed to supplement electronic LET, an average event based upon this parameter would not capture the behavior of the parent distribution or predict extreme values drawn from it. Similar arguments apply to any model that uses pre-averaging of events — including radial track models — when the inherent averaging associated with long ion tracks and large collection volumes is not realized. A logical way to deal with fluctuations of this type is via Monte Carlo simulation. Individual energy loss contributions

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with appropriate probabilities can be computed using a radiation event simulator. These values can be converted to charge with the appropriate quantity, location and other relevant parameters. The corresponding electrical response may then be computed using a device or circuit simulator. Sufficient iterain tions of this procedure must be done in order to reveal adequate detail. The process is inherently parallel but extraordinarily computationally intensive nonetheless, especially when one considers that variables such as location of the ion strike, angle of the strike, and the energy of the ion are also important. In this paper we demonstrate this strategy by selecting a set of extreme events from a large set of radiation event simulations and compute the response of a MOS transistor to them. III. SIMULATION APPROACH The radiation event simulations presented here were generated using a developmental tool based on the Geant4 libraries [12]–[15]. We investigate the case of 100 MeV protons (except as noted below) normally incident on the center of a face of a cube of Si, approximating a region of a device. The total 1. These number of events generated for this analysis was 5 and analyzed by set and collecwere computed in sets of tively. In each case, the Si was assumed to be in free space, so that no particles were scattered back into the material. This resulted in an average deposited energy per unit path length of the incident particle about 20% smaller than an equivalent thickness of Si within a larger volume, but dramatically increased the speed of the simulations. Additional computations were performed assuming a full thickness Si wafer for comparisons with LET computed by standard formulae. Primary and secondary particles, except elastically recoiling nuclei, were tracked until they either stopped in the sensitive volume, as energetic rays often did, or left the volume, as was the case universally for the incident protons and nuclear reaction products. The computer code includes a large, but not exhaustive, list of physical processes potentially capable of generating charge to induce single-event effects. Omission of elastically recoiling nuclei was not a conscious choice for these simulations, but rather was a limitation of Geant4 at the revision used in this work, 4.5.0.p01. These events will be included in future work and may significantly increase ionization in some cases. Images of radiation events shown below are two-dimensional projections of fully three-dimensional (3-D) simulations. Four events were chosen for 3-D device simulation using the Taurus Process & Device simulator [16]. The target device was gate length. The width an n-channel MOSFET with a 0.18and the depth of the bulk of the simulated device was 0.25 . The device was biased in silicon in the simulations was 0.5 the off state with 0 V applied to the gate and 3.3 V to the drain. Incident trajectories were normal to the device, and struck it in the center of the gate. The ion tracks produced by Geant4 were translated into multiple line segments with a constant photogeneration rate. The charge was deposited with Gaussian temporal and spatial profiles with standard deviations of 0.7 ps and 0.7 nm respectively. An average value of 3.6 eV to produce an electron-hole pair was used for all particles [17]. The mesh for the ion strikes was refined based on the photogeneration rate. This

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Fig. 1. Simulated particle tracks resulting from 5 10 100-MeV protons entering a 1-m Si cube. Blue tracks are protons and red tracks are rays.

made it possible to re-mesh a structure dynamically without knowing a priori the trajectory of the secondary particles. Two nuclear reaction events, an energetic -ray event and a control with no high energy secondary particles where chosen for full device simulation. The control event was an ion strike depositing energy uniformly at a rate approximating the (electronic) LET of the 100 MeV incident proton, . In this event, the 100 MeV protons deof charge in the volume of the device, as posited 1.7 determined by integrating all current flows in the simulation. In the second event, a -ray was produced by a 100 MeV proton and traveled roughly parallel to the surface. This event ) deposited 1.1 fC of charge. The third event, a classic ( nuclear reaction produced by a 60 MeV proton, is included for comparison because of the large ionization—15 fC—generated by the alpha particle. The fourth event was a spallation nuclear reaction generated by a 100 MeV proton that produced several reaction products, including both protons and neutrons. 70 fC of charge were deposited in the device volume by this event. It should be noted that events such as nuclear reactions and energetic rays not only deposit more energy than an average LET event, but also can deposit that energy in localized regions closer to the sensitive regions of the device, where more of the resulting charge can be collected. IV. RESULTS Fig. 1 shows the cumulative results of 5 protons incicube of Si. Blue tracks are positive dent from the left on a 1particles, which in this case are exclusively protons, red tracks are individual electrons and green tracks, when present, may be either neutrons or discrete photons. The average energy de) of Si was 689 keV, in posited in a full wafer thickness (500 excellent agreement with, e.g., the results of the parameteriza-

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Fig. 3. Upper plot: Gaussian distribution with unit standard deviation and related extreme value and cumulative extreme value distributions for 10 points. Lower plot: The probability that the largest of 5 10 random values will exceed a given value of energy. Curves for each of the distributions shown in Fig. 2 are presented. There is only a 0.1% probability that an event drawn from a parent Weibull distribution will exceed 19.8. There were 20 such events observed (e.g., as in Fig. 4), of which 14 were electronic and 6 nuclear.

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Fig. 2. Normalized distribution of energy deposited in a 1- Si cube, not including events involving nuclear reactions. The curves are shown on both linear (above) and logarithmic scales (below) to facilitate comparisons over the full range of energy. The mean and standard deviation are 1085 eV and 1 keV respectively. Also shown are Gaussian, Log-Normal, and Weibull distributions with the same mean and standard deviation.

tions used by TRIM [18]. Similarly, when events with nuclear reactions are omitted, the standard deviation of these values, 98 keV, was found to agree well with the energy straggling results of Yang et al. [19]. (The standard deviation of the full Si wafer including nuclear reensemble of runs on the 500 actions was 200 keV.) The LET spectrum, shown as the normalized probability of cube, is shown depositing a given amount of energy in the 1in Fig. 2, along with Gaussian, Log-Normal, and Weibull distributions with the same mean and standard deviation. It is clear by examining the linear and logarithmic plots that none of the approximate distributions capture the essential characteristics of the simulated distribution very well. These graphs illustrate one of the pitfalls of characterizing a distribution simply by its first and second moments. The relatively narrow distribution seen on the linear plot in Fig. 2 would appear to substantiate the use of LET as a measure of energy deposition as implied by (1). Similarly, the general impression created by the electron tracks in

Fig. 1 seems to reinforce the concept of a structured ion track. However, the logarithmic plot in Fig. 2 reveals that the distribution of deposited energies is very wide. The implications of this high-energy tail are best characterized by extreme value statistics [8]. Modelers have used various fits to the LET spectrum as input to device physics simulation tools. In the following paragraphs we examine the statistical prediction of extreme events using three commonly applied functions. These arguments illustrate both the sensitivity of conclusions about event rates to the details of the spectrum and the problem posed when physical processes not modeled by the parent spectrum are also present. random values are chosen from a parent population If member of this ordered set is and ordered numerically, the order statistic, and it has a probability distribution called the that is simply related to the parent distribution. The largest and or , are called extreme smallest order statistics, for values statistics. This is illustrated in Fig. 3. Above is shown a Gaussian distribution, the related extreme value probability and the integral of the extreme value density for

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distribution, giving the probability that the largest of a set of is greater than standard deviations from the mean, which for this example. The meaning of these curves is is that, for a normally distributed random variable, the most likely largest member of a set of a million random values is approximately 4.8 standard deviations above the mean. Fig. 3 also shows the cumulative probability that the largest of samples drawn from the Gaussian, Log-Normal, a set of 5 and Weibull distributions of Fig. 2 will exceed a given energy value. The Gaussian and Log-Normal distributions are obviously incapable of describing the observed frequency of extreme values, by grossly underestimating and overestimating their frequency, respectively. The Weibull cumulative extreme value probability curve has fallen to 0.1% at energy of 19.8 keV. In fact, there were 14 -ray events more energetic than this, as well as six nuclear reaction events, in the statistical ensemble studied. It is these extreme events that are of most concern. The point of Fig. 3 is to demonstrate that the results of an extreme-value analysis are only as reliable as the knowledge of the underlying parent distribution, and that relatively modest uncertainty in that distribution can lead to large error in the predicted extreme event rate. However, even if the LET spectrum in a specific volume of a device were known with great confidence, the total energy is only part of the picture. The specific geometric microstructure of the radiation event will affect charge collection [20] and therefore is also critically important in determining whether or not the event is significant to device function. Fig. 4 illustrates the concept of the microstructure of radiation events with two examples, which again are planar projections of fully 3-D events. Above is a single -ray event produced by a 100 MeV proton that deposits 15 keV, or about 15 times LET, cube. Below is a spallation reaction by a 1 GeV in the 1proton that deposits about 148 keV. In our simulations, similar reactions by 100 MeV protons have been observed to deposit up of Si. Had this latter to approximately 280 keV in the cubic event been in an oxide and the energy fully converted to charge, electron/hole the resulting microdose would have exceeded pairs per square centimeter [21], an amount that could produce very significant device-level effects. Comparing Figs. 4 and 1 it is clear that, at least for protons, the concept of an extended track is a consequence of ensemble averaging, and that each specific event differs greatly in detail. This is not intended to imply that there is no core of electronic excitation directly along the ion’s path. Rather, it is a statement about the relative quantity of energy and consequently of charge that is associated with that core, compared with the energy of events such as those shown in Fig. 4. The full predictive value of radiation event simulations like those shown in Fig. 4 can only be realized if they are coupled with device simulations that model the effect of the charge deposited by the events. Such simulations are inherently 3-D. Because of the complexity of the events, these challenge the present state of the art for 3-D device simulation. Fig. 5 shows the simulated drain current in a MOS transistor excited by the radiation events shown in the insets. The data from the radiation event simulation were used to establish the initial charge in three dimensions as described above. This charge

IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 50, NO. 6, DECEMBER 2003

Fig. 4. Interactions of protons (blue) traveling from left to right through the center of a 1- cube of Si. Above, energetic rays produced by a 100-MeV proton deposit 15 keV of energy. Below, a nuclear reaction initiated by a 1 GeV proton deposits a total energy of 148 keV through the secondary charged reaction products. LET for the primary track is approximately 1 keV=m. Blue trajectories denote positively charged particles, red electrons, and green neutrals, including both neutrons and photons.

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served as a perturbation to an already-converged device simulation and produced an excursion of the drain current also shown in Fig. 5. Arrows indicate the association between each of the radiation events and the resulting drain current excursion. Fig. 6 shows a visualization made directly by the device simulator of the initial condition of one of the events in Fig. 5, the spallation event, within the transistor. Since the event is three dimensional and the angle of view is different, the images have superficial differences. Note also that the neutron, shown in Fig. 5 as a

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Fig. 5. Simulated drain current in a MOS transistor following ion strikes with microstructure as shown in the insets. The energy of the primary particle and the nature of the interaction are shown in the legend. LET for the primary track is approximately 1 keV=m or about 1 keV total in the sample. The ray deposits 3 keV, and the (p; ) and spallation reactions each deposit about 100 keV, with different spatial distributions.

green trajectory does not produce charge and therefore makes no contribution to the charge shown in Fig. 6. The transistor shown in Fig. 6 is smaller than the 1 volume used to generate the radiation event, so the simulation contains only those portions of the event that lie within its

volume. Charge was generated on each trajectory leg by converting deposited energy from the Geant4 simulation to charge at a rate of 3.6 eV per electron-hole pair [17]. In Fig. 5, the energy deposited by the -ray event is only about three times that calculated from the LET, but because of its ori-

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Fig. 6. Three-dimensional visualization within the volume of the MOS transistor of the spallation event shown two dimensionally in Fig. 5. The radiation event generated by Geant4 was translated to the initial charge distribution shown here. The drain current as a function of time produced by this event was computed by the device simulator and is shown in the upper (brown) curve in Fig. 5. Both the placement and structure of the event are important in determining the device response.

entation normal to the primary ion track and its proximity to the channel as placed in the transistor, it produced an effect orders of magnitude larger than that of the control. This directionality and the magnitude of the observed response suggest an additional avenue of investigation complementing analyses based on spallation reactions to explain angular effects observed recently in proton-induced single-event upsets [22], [23]. The effects generated by nuclear reactions in the device were, of course, substantially larger. However, the frequency of energetic -electron production is several times larger. Moreover, such electrons may also be produced by photon bombardment, suggesting that they may mediate some upset processes. Simulations to analyze such connections, as well as to develop statistical and computational methodologies for combined radiation-event and device-response simulations are on-going in our laboratory. V. CONCLUSION We have examined the assumptions underlying the use of average radiation events and have noted two conditions – small devices, where inherent LET fluctuations are large, and high energies, where discrete rays are common and nuclear reactions occur – under which their use is of questionable validity. This limitation can be overcome by an ensemble averaging method, based upon a combination of device simulation with randomly generated but physically realistic radiation events. We have implemented a proof-of-concept of this approach using a radiation event simulation program based on Geant4 in conjunction with a commercial 3-D device simulator. While computationally intensive, the strategy is ideally suited to a cluster-computing environment, where it is now being further developed.

The wide variety of device responses we observed for a single type of particle (100 MeV proton) incident on a submicron MOSFET at the same location demonstrates the importance of this approach. In particular, events leading to the creation of energetic delta electrons can produce dramatically larger current responses than those predicted from an average LET approach (more than three orders of magnitude greater in Fig. 5). These -electron events are much more common than nuclear reactions and can produce responses that approach them in magnitude. Simulating device-level effects caused by individual extreme events, electronic and nuclear, will shed light on specific mechanisms that lead to upset, while analyzes of ensembles of such events will model the statistics of device response in a way that analysis of a single track with mean LET, even with extreme-value considerations taken into account, cannot. This approach should also lead to greater insight into the coupled response of multiple devices to individual heavy ion events analogous to the proton event shown above in Fig. 4. Combining simulations of energy deposition and device response will allow multiple forms of radiation to be treated in a consistent and physically realistic way, and make it possible to investigate the role of specific processes such as energetic secondary electron production in both device response and defect formation. These capabilities will be crucial for modeling the radiation response of emerging technologies (e.g., nanotechnologies and very deep submicron CMOS ICs) in space and terrestrial radiation environments. REFERENCES [1] M. A. Xapsos, “Applicability of LET to single events in microelectronic structures,” IEEE Trans. Nucl. Sci., vol. 39, pp. 1613–1621, Dec. 1992. [2] M. A. Xapsos, G. P. Summers, E. A. Burke, and C. Poivey, “Microdosimetry theory for microelectronics applications,” Nucl. Instrum. Methods Phys. Res. B, vol. 184, pp. 113–134, 2001. [3] R. A. Reed, P. J. McNulty, W. J. Beauvais, W. G. Abdel-Kader, E. G. Stassinopoulos, and J. L. Barth, “A simple algorithm for predicting proton SEU rates in space compared to the rates measured on the CRRES satellite,” IEEE Trans. Nucl. Sci., vol. 41, pp. 2389–2395, Dec. 1994. [4] C. J. Dale, L. Chen, P. J. McNulty, P. W. Marshall, and E. A. Burke, “A comparison of Monte Carlo and analytical treatments of displacement damage in Si microvolumes,” IEEE Trans. Nucl. Sci., vol. 41, pp. 1974–1983, Dec. 1994. [5] R. A. Reed, J. Kinnison, J. C. Pickel, S. Buchner, P. W. Marshall, S. Kniffin, and K. A. LaBel, IEEE Trans. Nucl. Sci., vol. 50, pp. 622–634, June 2003. [6] J.-M. Palau, F. Wrobel, K. Castellani-Coulié, M.-C. Calvet, P. E. Dodd, and F. W. Sexton, “Monte Carlo exploration of neutron-induced SEU sensitive volumes in SRAMs,” IEEE Trans. Nucl. Sci., vol. 49, pp. 3075–3081, Dec. 2002. [7] M. W. Savage, P. J. McNulty, D. R. Roth, and C. C. Foster, “Possible role for secondary particles in proton-induced single event upsets of modern devices,” IEEE Trans. Nucl. Sci., vol. 45, pp. 2745–2751, Dec. 1998. [8] P. W. Marshall, C. J. Dale, E. A. Burke, G. P. Summers, and G. E. Bender, “Displacement damage extremes in silicon depletion regions,” IEEE Trans. Nucl. Sci., vol. 36, pp. 1831–1839, Dec. 1989. [9] A. Akkerman and J. Barak, “Ion-track structure and its effects in small size volumes of silicon,” IEEE Trans. Nucl. Sci., vol. 49, pp. 3022–3031, Dec. 2002. [10] P. E. Dodd and L. W. Massengill, “Basic mechanisms and modeling of single-event upset in digital microelectronics,” IEEE Trans. Nucl. Sci., vol. 50, pp. 583–602, June 2003. [11] R. A. Reed, M. A. Carts, P. W. Marshall, C. J. Marshall, O. Musseau, P. J. McNulty, D. R. Roth, S. Buchner, J. Melinger, and T. Corbière, “Heavy ion and proton-induced single event multiple upset,” IEEE Trans. Nucl. Sci., vol. 44, pp. 2224–2229, Dec. 1997.

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[12] S. Agostinelli et al., “Geant4 – A simulation toolkit,” Nucl. Instrum. Methods Phys. Res. A, vol. 506, pp. 250–303, 2003. [13] F. Lei, P. R. Truscott, C. S. Dyer, B. Quaghebeur, D. Heynderickx, P. Nieminen, H. Evans, and E. Daly, “MULASSIS: A Geant4-based multilayered shielding simulation tool,” IEEE Trans. Nucl. Sci., vol. 49, pp. 2788–2793, Dec. 2003. [14] C. Inguimbert, S. Duzellier, and R. Ecoffet, “Contribution of GEANT4 to the determination of sensitive volumes in case of high-integrated RAMs,” IEEE Trans. Nucl. Sci., vol. 49, pp. 1480–1485, June 2002. [15] [Online]. Available: http://wwwasd.web.cern.ch/wwwasd/geant4/ geant4.html [16] Taurus Process & Device User Manual—, 2002.4 ed., Synopsys, Inc., Fremont, CA, 2003. [17] W. Shockley, “Problems related to p-n junctions in silicon,” Solid State Electron., vol. 2, no. 3, pp. 35–67, 1961. [18] E. Rauhala, “Energy loss,” in Handbook of Modern Ion Beam Analysis, J. R. Tesmer and M. Nastasi, Eds. Pittsburgh, PA: Materials Res. Soc., 1995, pp. 3–19.

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[19] Q. Yang, D. J. O’Connor, and Z. Wang, “Empirical formula for energy loss straggling of ions in matter,” Nucl. Instrum. Methods Phys. Res. B, vol. 61, pp. 149–155, 1991. [20] P. E. Dodd, “Device simulation of charge collection and single-event upset,” IEEE Trans. Nucl. Sci, vol. 43, pp. 561–575, Apr. 1996. [21] T. R. Oldham, K. W. Bennett, J. Beaucour, T. Carriere, C. Poivey, and P. Garnier, “Total dose failures in advanced electronics from single ions,” IEEE Trans. Nucl. Sci., vol. 40, pp. 1820–1830, Dec. 1993. [22] A. H. Johnston, T. Miyahira, G. M. Swift, S. M. Guertin, and L. D. Edmonds, “Angular and energy dependence of proton upsets in optocouplers,” IEEE Trans. Nucl. Sci., vol. 46, pp. 1335–1341, Dec. 1999. [23] R. A. Reed, P. W. Marshall, H. S. Kim, P. J. McNulty, B. Fodness, T. M. Jordan, R. Reedy, C. Tabbert, M. S. T. Liu, W. Heikkila, S. Buchner, R. Ladbury, and K. A. LaBel, “Evidence for angular effects in proton-induced single-event upsets,” IEEE Trans. Nucl. Sci., vol. 49, pp. 3038–3044, Dec. 2002.

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Evaluating Average and Atypical Response in Radiation Effects Simulations Robert A. Weller, Senior Member, IEEE, Andrew L. Sternberg, Member, IEEE, Lloyd W. Massengill, Senior Member, IEEE, Ronald D. Schrimpf, Fellow, IEEE, and Daniel M. Fleetwood, Fellow, IEEE

Abstract—We examine the limits of performing single-event simulations using pre-averaged radiation events. Geant4 simulations show the necessity, for future devices, to supplement current methods with ensemble averaging of device-level responses to physically realistic radiation events. Initial Monte Carlo simulations have generated a significant number of extremal events in local energy deposition. These simulations strongly suggest that proton strikes of sufficient energy, even those that initiate purely electronic interactions, can initiate device response capable in principle of producing single event upset or microdose damage in highly scaled devices. Index Terms—Charge collection, extreme value statistics, Geant4, linear energy transfer, Monte Carlo simulation, protons, radiation effects, single-event effects, soft errors.

I. INTRODUCTION

A

S device dimensions approach the nano-scale, the inherent fluctuations in the parameters used to characterize radiation effects become increasingly important, and the need for consistent, physically based simulations becomes correspondingly more urgent. Many of the basic principles important to this task related to single-event mechanisms have already been discussed in the literature, including the variability of (LET) [1], the principles of microdosimetry [2], effects (particularly displacement) of nuclear reactions [3]–[6] and reaction products [7], the application of extreme-value statistics to single event analysis [8], and most recently a theoretical description of the LET spectrum [9]. In this paper we outline a synthesis of these principles in which average track models are supplemented by ensembles of physically realistic radiation events. This approach will yield deeper insight into the mechanisms of single-event effects [10], while providing a means for analyzing processes, such as multiple bit upsets [11] and secondary radiation from materials in the local environment surrounding active devices. These effects depend on the evolution and microstructure of radiation interactions, and do not lend themselves easily to other forms of analysis. II. ENERGY DEPOSITION If the value of a circuit parameter, such as the voltage at a given node, is linearly related to a random perturbation, such as Manuscript received July 20, 2003; revised August 29, 2003. The authors are with the Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN 37235 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TNS.2003.821576

the energy deposited by an energetic ion, then the distribution of voltages is simply related to the distribution of the perturbation. In this case, it is easy to show that the average response is the response to the average perturbation. However, in the general case, and particularly when significant nonlinearities are inis the volved, the distributions are related by (1), where is the distribution of perturdistribution of responses , bations , is the Dirac delta function, and the function establishes the relationship between and , . The integral is over the full range of (1) This equation may be used to show that the average value of is given by when is linear and that if the distribution of is narrow. Otherwise, and the average must be obtained by computing this integral either analytically or numerically. In more complex depends on additional random cases, as for example when variables such as the geometric coordinates of the ion strike, or when is itself a derived random variable, numerical methods are essential. The latter case is of particular interest, and it is the one we consider here. The total energy deposited by an ion in a sensitive volume is the sum of a large number of individual discrete energy transfers of random size . LET is an average of these values and is useful when the sample is thick enough for this average to be meaningful. This general framework breaks down in two important cases: (1) when the distribution of cannot be assumed to be narrow, regardless of the thickness of the sensitive volume, and (2) when the sensitive volume is sufficiently thin that the average inherent in the definition of LET is of questionable validity. values, although rare, Events with very large individual alter the distribution of in a way that undermines the validity of the averaging process. Examples are the production of discrete rays and nuclear reactions that produce multiple, charged reaction products. While an average energy deposition including nuclear events could obviously be computed to supplement electronic LET, an average event based upon this parameter would not capture the behavior of the parent distribution or predict extreme values drawn from it. Similar arguments apply to any model that uses pre-averaging of events — including radial track models — when the inherent averaging associated with long ion tracks and large collection volumes is not realized. A logical way to deal with fluctuations of this type is via Monte Carlo simulation. Individual energy loss contributions

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with appropriate probabilities can be computed using a radiation event simulator. These values can be converted to charge with the appropriate quantity, location and other relevant parameters. The corresponding electrical response may then be computed using a device or circuit simulator. Sufficient iterain tions of this procedure must be done in order to reveal adequate detail. The process is inherently parallel but extraordinarily computationally intensive nonetheless, especially when one considers that variables such as location of the ion strike, angle of the strike, and the energy of the ion are also important. In this paper we demonstrate this strategy by selecting a set of extreme events from a large set of radiation event simulations and compute the response of a MOS transistor to them. III. SIMULATION APPROACH The radiation event simulations presented here were generated using a developmental tool based on the Geant4 libraries [12]–[15]. We investigate the case of 100 MeV protons (except as noted below) normally incident on the center of a face of a cube of Si, approximating a region of a device. The total 1. These number of events generated for this analysis was 5 and analyzed by set and collecwere computed in sets of tively. In each case, the Si was assumed to be in free space, so that no particles were scattered back into the material. This resulted in an average deposited energy per unit path length of the incident particle about 20% smaller than an equivalent thickness of Si within a larger volume, but dramatically increased the speed of the simulations. Additional computations were performed assuming a full thickness Si wafer for comparisons with LET computed by standard formulae. Primary and secondary particles, except elastically recoiling nuclei, were tracked until they either stopped in the sensitive volume, as energetic rays often did, or left the volume, as was the case universally for the incident protons and nuclear reaction products. The computer code includes a large, but not exhaustive, list of physical processes potentially capable of generating charge to induce single-event effects. Omission of elastically recoiling nuclei was not a conscious choice for these simulations, but rather was a limitation of Geant4 at the revision used in this work, 4.5.0.p01. These events will be included in future work and may significantly increase ionization in some cases. Images of radiation events shown below are two-dimensional projections of fully three-dimensional (3-D) simulations. Four events were chosen for 3-D device simulation using the Taurus Process & Device simulator [16]. The target device was gate length. The width an n-channel MOSFET with a 0.18and the depth of the bulk of the simulated device was 0.25 . The device was biased in silicon in the simulations was 0.5 the off state with 0 V applied to the gate and 3.3 V to the drain. Incident trajectories were normal to the device, and struck it in the center of the gate. The ion tracks produced by Geant4 were translated into multiple line segments with a constant photogeneration rate. The charge was deposited with Gaussian temporal and spatial profiles with standard deviations of 0.7 ps and 0.7 nm respectively. An average value of 3.6 eV to produce an electron-hole pair was used for all particles [17]. The mesh for the ion strikes was refined based on the photogeneration rate. This

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Fig. 1. Simulated particle tracks resulting from 5 10 100-MeV protons entering a 1-m Si cube. Blue tracks are protons and red tracks are rays.

made it possible to re-mesh a structure dynamically without knowing a priori the trajectory of the secondary particles. Two nuclear reaction events, an energetic -ray event and a control with no high energy secondary particles where chosen for full device simulation. The control event was an ion strike depositing energy uniformly at a rate approximating the (electronic) LET of the 100 MeV incident proton, . In this event, the 100 MeV protons deof charge in the volume of the device, as posited 1.7 determined by integrating all current flows in the simulation. In the second event, a -ray was produced by a 100 MeV proton and traveled roughly parallel to the surface. This event ) deposited 1.1 fC of charge. The third event, a classic ( nuclear reaction produced by a 60 MeV proton, is included for comparison because of the large ionization—15 fC—generated by the alpha particle. The fourth event was a spallation nuclear reaction generated by a 100 MeV proton that produced several reaction products, including both protons and neutrons. 70 fC of charge were deposited in the device volume by this event. It should be noted that events such as nuclear reactions and energetic rays not only deposit more energy than an average LET event, but also can deposit that energy in localized regions closer to the sensitive regions of the device, where more of the resulting charge can be collected. IV. RESULTS Fig. 1 shows the cumulative results of 5 protons incicube of Si. Blue tracks are positive dent from the left on a 1particles, which in this case are exclusively protons, red tracks are individual electrons and green tracks, when present, may be either neutrons or discrete photons. The average energy de) of Si was 689 keV, in posited in a full wafer thickness (500 excellent agreement with, e.g., the results of the parameteriza-

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Fig. 3. Upper plot: Gaussian distribution with unit standard deviation and related extreme value and cumulative extreme value distributions for 10 points. Lower plot: The probability that the largest of 5 10 random values will exceed a given value of energy. Curves for each of the distributions shown in Fig. 2 are presented. There is only a 0.1% probability that an event drawn from a parent Weibull distribution will exceed 19.8. There were 20 such events observed (e.g., as in Fig. 4), of which 14 were electronic and 6 nuclear.

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Fig. 2. Normalized distribution of energy deposited in a 1- Si cube, not including events involving nuclear reactions. The curves are shown on both linear (above) and logarithmic scales (below) to facilitate comparisons over the full range of energy. The mean and standard deviation are 1085 eV and 1 keV respectively. Also shown are Gaussian, Log-Normal, and Weibull distributions with the same mean and standard deviation.

tions used by TRIM [18]. Similarly, when events with nuclear reactions are omitted, the standard deviation of these values, 98 keV, was found to agree well with the energy straggling results of Yang et al. [19]. (The standard deviation of the full Si wafer including nuclear reensemble of runs on the 500 actions was 200 keV.) The LET spectrum, shown as the normalized probability of cube, is shown depositing a given amount of energy in the 1in Fig. 2, along with Gaussian, Log-Normal, and Weibull distributions with the same mean and standard deviation. It is clear by examining the linear and logarithmic plots that none of the approximate distributions capture the essential characteristics of the simulated distribution very well. These graphs illustrate one of the pitfalls of characterizing a distribution simply by its first and second moments. The relatively narrow distribution seen on the linear plot in Fig. 2 would appear to substantiate the use of LET as a measure of energy deposition as implied by (1). Similarly, the general impression created by the electron tracks in

Fig. 1 seems to reinforce the concept of a structured ion track. However, the logarithmic plot in Fig. 2 reveals that the distribution of deposited energies is very wide. The implications of this high-energy tail are best characterized by extreme value statistics [8]. Modelers have used various fits to the LET spectrum as input to device physics simulation tools. In the following paragraphs we examine the statistical prediction of extreme events using three commonly applied functions. These arguments illustrate both the sensitivity of conclusions about event rates to the details of the spectrum and the problem posed when physical processes not modeled by the parent spectrum are also present. random values are chosen from a parent population If member of this ordered set is and ordered numerically, the order statistic, and it has a probability distribution called the that is simply related to the parent distribution. The largest and or , are called extreme smallest order statistics, for values statistics. This is illustrated in Fig. 3. Above is shown a Gaussian distribution, the related extreme value probability and the integral of the extreme value density for

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distribution, giving the probability that the largest of a set of is greater than standard deviations from the mean, which for this example. The meaning of these curves is is that, for a normally distributed random variable, the most likely largest member of a set of a million random values is approximately 4.8 standard deviations above the mean. Fig. 3 also shows the cumulative probability that the largest of samples drawn from the Gaussian, Log-Normal, a set of 5 and Weibull distributions of Fig. 2 will exceed a given energy value. The Gaussian and Log-Normal distributions are obviously incapable of describing the observed frequency of extreme values, by grossly underestimating and overestimating their frequency, respectively. The Weibull cumulative extreme value probability curve has fallen to 0.1% at energy of 19.8 keV. In fact, there were 14 -ray events more energetic than this, as well as six nuclear reaction events, in the statistical ensemble studied. It is these extreme events that are of most concern. The point of Fig. 3 is to demonstrate that the results of an extreme-value analysis are only as reliable as the knowledge of the underlying parent distribution, and that relatively modest uncertainty in that distribution can lead to large error in the predicted extreme event rate. However, even if the LET spectrum in a specific volume of a device were known with great confidence, the total energy is only part of the picture. The specific geometric microstructure of the radiation event will affect charge collection [20] and therefore is also critically important in determining whether or not the event is significant to device function. Fig. 4 illustrates the concept of the microstructure of radiation events with two examples, which again are planar projections of fully 3-D events. Above is a single -ray event produced by a 100 MeV proton that deposits 15 keV, or about 15 times LET, cube. Below is a spallation reaction by a 1 GeV in the 1proton that deposits about 148 keV. In our simulations, similar reactions by 100 MeV protons have been observed to deposit up of Si. Had this latter to approximately 280 keV in the cubic event been in an oxide and the energy fully converted to charge, electron/hole the resulting microdose would have exceeded pairs per square centimeter [21], an amount that could produce very significant device-level effects. Comparing Figs. 4 and 1 it is clear that, at least for protons, the concept of an extended track is a consequence of ensemble averaging, and that each specific event differs greatly in detail. This is not intended to imply that there is no core of electronic excitation directly along the ion’s path. Rather, it is a statement about the relative quantity of energy and consequently of charge that is associated with that core, compared with the energy of events such as those shown in Fig. 4. The full predictive value of radiation event simulations like those shown in Fig. 4 can only be realized if they are coupled with device simulations that model the effect of the charge deposited by the events. Such simulations are inherently 3-D. Because of the complexity of the events, these challenge the present state of the art for 3-D device simulation. Fig. 5 shows the simulated drain current in a MOS transistor excited by the radiation events shown in the insets. The data from the radiation event simulation were used to establish the initial charge in three dimensions as described above. This charge

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Fig. 4. Interactions of protons (blue) traveling from left to right through the center of a 1- cube of Si. Above, energetic rays produced by a 100-MeV proton deposit 15 keV of energy. Below, a nuclear reaction initiated by a 1 GeV proton deposits a total energy of 148 keV through the secondary charged reaction products. LET for the primary track is approximately 1 keV=m. Blue trajectories denote positively charged particles, red electrons, and green neutrals, including both neutrons and photons.

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served as a perturbation to an already-converged device simulation and produced an excursion of the drain current also shown in Fig. 5. Arrows indicate the association between each of the radiation events and the resulting drain current excursion. Fig. 6 shows a visualization made directly by the device simulator of the initial condition of one of the events in Fig. 5, the spallation event, within the transistor. Since the event is three dimensional and the angle of view is different, the images have superficial differences. Note also that the neutron, shown in Fig. 5 as a

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Fig. 5. Simulated drain current in a MOS transistor following ion strikes with microstructure as shown in the insets. The energy of the primary particle and the nature of the interaction are shown in the legend. LET for the primary track is approximately 1 keV=m or about 1 keV total in the sample. The ray deposits 3 keV, and the (p; ) and spallation reactions each deposit about 100 keV, with different spatial distributions.

green trajectory does not produce charge and therefore makes no contribution to the charge shown in Fig. 6. The transistor shown in Fig. 6 is smaller than the 1 volume used to generate the radiation event, so the simulation contains only those portions of the event that lie within its

volume. Charge was generated on each trajectory leg by converting deposited energy from the Geant4 simulation to charge at a rate of 3.6 eV per electron-hole pair [17]. In Fig. 5, the energy deposited by the -ray event is only about three times that calculated from the LET, but because of its ori-

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Fig. 6. Three-dimensional visualization within the volume of the MOS transistor of the spallation event shown two dimensionally in Fig. 5. The radiation event generated by Geant4 was translated to the initial charge distribution shown here. The drain current as a function of time produced by this event was computed by the device simulator and is shown in the upper (brown) curve in Fig. 5. Both the placement and structure of the event are important in determining the device response.

entation normal to the primary ion track and its proximity to the channel as placed in the transistor, it produced an effect orders of magnitude larger than that of the control. This directionality and the magnitude of the observed response suggest an additional avenue of investigation complementing analyses based on spallation reactions to explain angular effects observed recently in proton-induced single-event upsets [22], [23]. The effects generated by nuclear reactions in the device were, of course, substantially larger. However, the frequency of energetic -electron production is several times larger. Moreover, such electrons may also be produced by photon bombardment, suggesting that they may mediate some upset processes. Simulations to analyze such connections, as well as to develop statistical and computational methodologies for combined radiation-event and device-response simulations are on-going in our laboratory. V. CONCLUSION We have examined the assumptions underlying the use of average radiation events and have noted two conditions – small devices, where inherent LET fluctuations are large, and high energies, where discrete rays are common and nuclear reactions occur – under which their use is of questionable validity. This limitation can be overcome by an ensemble averaging method, based upon a combination of device simulation with randomly generated but physically realistic radiation events. We have implemented a proof-of-concept of this approach using a radiation event simulation program based on Geant4 in conjunction with a commercial 3-D device simulator. While computationally intensive, the strategy is ideally suited to a cluster-computing environment, where it is now being further developed.

The wide variety of device responses we observed for a single type of particle (100 MeV proton) incident on a submicron MOSFET at the same location demonstrates the importance of this approach. In particular, events leading to the creation of energetic delta electrons can produce dramatically larger current responses than those predicted from an average LET approach (more than three orders of magnitude greater in Fig. 5). These -electron events are much more common than nuclear reactions and can produce responses that approach them in magnitude. Simulating device-level effects caused by individual extreme events, electronic and nuclear, will shed light on specific mechanisms that lead to upset, while analyzes of ensembles of such events will model the statistics of device response in a way that analysis of a single track with mean LET, even with extreme-value considerations taken into account, cannot. This approach should also lead to greater insight into the coupled response of multiple devices to individual heavy ion events analogous to the proton event shown above in Fig. 4. Combining simulations of energy deposition and device response will allow multiple forms of radiation to be treated in a consistent and physically realistic way, and make it possible to investigate the role of specific processes such as energetic secondary electron production in both device response and defect formation. These capabilities will be crucial for modeling the radiation response of emerging technologies (e.g., nanotechnologies and very deep submicron CMOS ICs) in space and terrestrial radiation environments. REFERENCES [1] M. A. Xapsos, “Applicability of LET to single events in microelectronic structures,” IEEE Trans. Nucl. Sci., vol. 39, pp. 1613–1621, Dec. 1992. [2] M. A. Xapsos, G. P. Summers, E. A. Burke, and C. Poivey, “Microdosimetry theory for microelectronics applications,” Nucl. Instrum. Methods Phys. Res. B, vol. 184, pp. 113–134, 2001. [3] R. A. Reed, P. J. McNulty, W. J. Beauvais, W. G. Abdel-Kader, E. G. Stassinopoulos, and J. L. Barth, “A simple algorithm for predicting proton SEU rates in space compared to the rates measured on the CRRES satellite,” IEEE Trans. Nucl. Sci., vol. 41, pp. 2389–2395, Dec. 1994. [4] C. J. Dale, L. Chen, P. J. McNulty, P. W. Marshall, and E. A. Burke, “A comparison of Monte Carlo and analytical treatments of displacement damage in Si microvolumes,” IEEE Trans. Nucl. Sci., vol. 41, pp. 1974–1983, Dec. 1994. [5] R. A. Reed, J. Kinnison, J. C. Pickel, S. Buchner, P. W. Marshall, S. Kniffin, and K. A. LaBel, IEEE Trans. Nucl. Sci., vol. 50, pp. 622–634, June 2003. [6] J.-M. Palau, F. Wrobel, K. Castellani-Coulié, M.-C. Calvet, P. E. Dodd, and F. W. Sexton, “Monte Carlo exploration of neutron-induced SEU sensitive volumes in SRAMs,” IEEE Trans. Nucl. Sci., vol. 49, pp. 3075–3081, Dec. 2002. [7] M. W. Savage, P. J. McNulty, D. R. Roth, and C. C. Foster, “Possible role for secondary particles in proton-induced single event upsets of modern devices,” IEEE Trans. Nucl. Sci., vol. 45, pp. 2745–2751, Dec. 1998. [8] P. W. Marshall, C. J. Dale, E. A. Burke, G. P. Summers, and G. E. Bender, “Displacement damage extremes in silicon depletion regions,” IEEE Trans. Nucl. Sci., vol. 36, pp. 1831–1839, Dec. 1989. [9] A. Akkerman and J. Barak, “Ion-track structure and its effects in small size volumes of silicon,” IEEE Trans. Nucl. Sci., vol. 49, pp. 3022–3031, Dec. 2002. [10] P. E. Dodd and L. W. Massengill, “Basic mechanisms and modeling of single-event upset in digital microelectronics,” IEEE Trans. Nucl. Sci., vol. 50, pp. 583–602, June 2003. [11] R. A. Reed, M. A. Carts, P. W. Marshall, C. J. Marshall, O. Musseau, P. J. McNulty, D. R. Roth, S. Buchner, J. Melinger, and T. Corbière, “Heavy ion and proton-induced single event multiple upset,” IEEE Trans. Nucl. Sci., vol. 44, pp. 2224–2229, Dec. 1997.

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[19] Q. Yang, D. J. O’Connor, and Z. Wang, “Empirical formula for energy loss straggling of ions in matter,” Nucl. Instrum. Methods Phys. Res. B, vol. 61, pp. 149–155, 1991. [20] P. E. Dodd, “Device simulation of charge collection and single-event upset,” IEEE Trans. Nucl. Sci, vol. 43, pp. 561–575, Apr. 1996. [21] T. R. Oldham, K. W. Bennett, J. Beaucour, T. Carriere, C. Poivey, and P. Garnier, “Total dose failures in advanced electronics from single ions,” IEEE Trans. Nucl. Sci., vol. 40, pp. 1820–1830, Dec. 1993. [22] A. H. Johnston, T. Miyahira, G. M. Swift, S. M. Guertin, and L. D. Edmonds, “Angular and energy dependence of proton upsets in optocouplers,” IEEE Trans. Nucl. Sci., vol. 46, pp. 1335–1341, Dec. 1999. [23] R. A. Reed, P. W. Marshall, H. S. Kim, P. J. McNulty, B. Fodness, T. M. Jordan, R. Reedy, C. Tabbert, M. S. T. Liu, W. Heikkila, S. Buchner, R. Ladbury, and K. A. LaBel, “Evidence for angular effects in proton-induced single-event upsets,” IEEE Trans. Nucl. Sci., vol. 49, pp. 3038–3044, Dec. 2002.