Zebrev_SER_Computation_Methodology_DRAFT

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DRAFT Version, Revised version published in IEEE Trans. Nucl. Sci. No.6 (2010) 1

Abstract - We have proposed a test methodology based on

successive experimental determination of angular cross-section

dependence followed by averaging over full solid angle.

Equivalence between phenomenological and chord-length

distribution averaging for soft error rate computation is shown.

Role of energy-loss straggling in subthreshold error rate

enhancement has been revealed. Nuclear reaction induced error

rate computation method providing crossover between BGR and

chord-length approaches has been proposed. Possibility of inclusion of multiple bit error rate estimation in a unified

computational scheme is shown.

Index Terms— Energy Deposition, LET, Multiple Bit Error,

Nuclear Reactions, Sensitive Volume, Soft Error Rate, Straggling

I. INTRODUCTION

ith the continuous scaling of technology node and

decreasing of supply voltage, soft error issues have

emerged as a new design challenge. Some fundamental

problems of ground-based testing and soft error rate (SER)

computation arise as a result of shrinking dimensions of devices, a use of thick overlayers and high-Z materials,

multiple bit errors etc. provoking the growing public concern

over single event rate prediction. As has been noted in Ref. [1]

a need arises “to rethink test methodologies, procedures and

models in order to predict the true behavior of emerging

technologies in space”.

The traditional rate-estimation methods are based mainly

on two non-overlapping approaches, namely, the rectangular

parallelepiped (RPP) chord length distribution for directly

ionizing heavy ions [2,3] and the Burst Generation Rate

(BGR) method [4] suitable for nuclear reaction induced error

computation. Routinely the RPP methods do not include the

variability of radiation events (e.g., nuclear reactions). Insteadthey approximate the direct ionization by assuming a validity

of approximation of mean energy deposition in the sensitive

volumes (SV) through a single (often constant) value for the

mean linear energy transfer (LET).

Manuscript received September 11, 2009.

G. I. Zebrev is with the Department of Micro- and Nanoelectronics of National Research Nuclear University “MEPHI”, Kashirskoye shosse 31,

115409, Moscow, Russia, e-mail: [email protected]; phone: +7-495-

3240184; fax: +7-495-324-2111.

I. O. Ishutin is a postgraduate of NRNU MEPHI, Moscow, Russia.

R. G. Useinov is with Research Institute of Scientific Instruments,Lytkarino, Moscow region, Russia.

V. S. Anashin is with Scientific Research Institute of Space InstrumentEngineering, Moscow, Russia.

This fundamental approximation implies that equal LET

yields equal effect if track lengths being the same irrespective

to the ion type and energy. This allows to significantly

simplify consideration using LET as a “dummy” variable at

integration over space spectra and ignoring composition of ion

species and their energies. We intend to argue in this paper

that this approximation is generally wrong.

To the best of our knowledge the first extended critique of

this statement appeared in [5]. Nevertheless it was until haveappeared numerous inconsistencies between modeling and

experiments that researchers began to take serious interest in

this problem. Particularly, the applicability of LET

approximation has been questioned recently (see [6, 7] and

references within), wherein the lack of correlation between the

measured SEU cross-sections and ion LETs is noted as the key

point of the problem.

The experimental situation is commonly consisted in the

empirical facts that the greater is the ion energy the larger the

probability of single event effects even at equal mean LET,

especially for sub-threshold regions of cross-section vs LET

curves with low LETs and high ion’s energies that turned to be

significant for some technologies [8 , 9, 10, 11].A complicated Monte Carlo simulation-based approach for

SEE response description has been developed recently [12, 13]

to obtain exhaustive description of energy deposition as

applied to SER problem. Nevertheless the numerical Monte

Carlo simulations are rather computationally intensive

methods and their results seem to be often non-transparent.

We believe that the noted contradictions should be simply

explained in terms of transparent physical models. Particularly

we have shown that for these facts to explain there must be

gone beyond the mean LET approximations and

systematically taken into account fundamental fluctuation of

energy deposition in device’s sensitive volumes (SV).

Geometric dimensions of the sensitive volumes have been

shrinking from tens to tenth of microns and now it is necessary

to consider fundamental fluctuation of energy deposition such

as energy-loss straggling [14].

The general objective of this work is to describe some new

approaches to test and SER prediction methodologies based on

the original approach proposed in [15]. The rest of the paper is

organized as follows. In Sec. II and III we propose single

event cross-section concept based on general definition

accepted in nuclear physics and discuss in a unifying manner

the complementary methods of angular averaging over full

solid angle and averaging over chord length distribution under

conditions of limited experimental information. Sec. IV is

devoted to the extension of the PRIVET chord-lengthdistribution approach beyond the average LET approximation

Methodology of Soft Error Rate Computation

in Modern MicroelectronicsG. I. Zebrev, I.O. Ishutin, R.G. Useinov, V.S. Anashin

W




to include in itself the subthreshold error effects. Sec. V and

VI show how nuclear reactions and multiple bit errors can be

included in the proposed general SER computation scheme

making a bridge between BGR and chord length descriptions.

II. CROSS-SECTION CONCEPT

A. Definition and General Relations

It is well known that single event rate prediction is based

on experimental determination of cross-section versus LET

dependencies. Single event error cross-section can be defined

in ambiguous manner [2, 16, 17, 18]. Based on general

definition accepted in nuclear physics we define differential

cross-section with respect to primary particle flux in the only

consistent way

( )( )

( )

; ,; ,

; ,

d N

do d

θ ϕ σ θ ϕ

θ ϕ

ΛΛ =

Φ Λ Λ, (1)

where cosdo d d θ ϕ = is the solid angular element,

( ); ,d N θ ϕ Λ is the error number due to particles with LET in

the range ... d Λ Λ + Λ , incident from the solid angular element

do ( ( ),θ ϕ are polar and azimuthal angle with respect normal

to IC face, ( ); ,θ ϕ Φ Λ [cm-2 sterad-1 (MeV-cm2 /mg)-1] is

differential fluence per LET unit per solid angle. Knowing

( ); ,σ θ ϕ Λ we can compute full error rate immediately from

definition (1) as a functional

( ) ( ) ( ); , ; , cos R d d d σ θ ϕ φ θ ϕ θ ϕ = Λ Λ Λ∫∫∫ , (2)

where we use flux φ instead of fluence ( d dt φ = Φ ).

In general case one can introduce LET dependent cross-

section ( )σ Λ averaged over the full solid angle 4π, which

is defined as

( ) ( ) ( )1

; , cos4

d d σ σ θ ϕ θ ϕ π

Λ = Λ∫∫ . (3)

Cosmic ray direction distributions and corresponding LET

spectra are typically assumed to be isotropic

( ) ( )4 ; ,φ π φ θ ϕ Λ ≅ Λ . Then the general relation in Eq.2 reads

( ) ( ) R d σ φ = Λ Λ Λ∫ . (4)

Notice that hitherto the consideration remains rather

general and all these relationships can be defined not for only

LET but for ion energy E as well as with a use of differential

fluence per energy unit ( ) E φ [cm-2 s-1 MeV-1] and with a

formal replacement E → Λ

. In particular the proton-inducederror number is expressed through the cross-section

dependencies as functions of energy averaged over full solid

angle ( ) E σ

( ) ( ) R E E dE σ φ = ∫ , (5)

irrespective to the exact ionization mechanisms (direct or

nuclear-reaction-induced). This stems from the fact that the

described approach is based on a very general

phenomenological ground.

In this manner one needs a procedure of averaging over all

direction of incident particles. There exist at least three

approaches to such angular averaging.

B. Phenomenological Angular Averaging Approach

There exists in principle a possibility to perform angular

averaging directly measured experimentally cross-sections for

a large number of particle flux directions. For example in case

of mono-directional beam of particle with a specified LET (or

energy) and direction ( )0 0,θ φ we have

( ) ( ) ( ) ( )0 0; , cos cosθ ϕ δ θ θ δ ϕ ϕ Φ Λ ≅ Φ Λ − − , (5)

where ( )Φ Λ is total beam fluence with a specified LET.

Using Eqs.1-2 one has obtained a basic equation for

experimental determination of single event cross-section angle

dependence

( )( )

( )0 0

exp 0 0

; ,; ,

N θ φ σ θ φ

ΛΛ =

Φ Λ, (6)

where ( )0 0, N θ φ is the error number registered at a given

angle and LET. This equation yields experimental angular

dependence of single event cross-section with respect to, for

example, a chip surface normal (this is no more than

convenient choice). Notice that no additional cosine arises in

the basic relationship Eq.6.

Dependence on orientation of the SV relative to incident

beam direction is contained entirely in the angular dependence

of cross-section. We prefer to avoid a use of an “effective

fluence” concept since fluence represents a property of

incident beam rather than its orientation relative to the

exposed object. Therefore, a use of a concept of “effective”

angular-dependent fluence leading to cross-section definition

in a form

cos

N σ

θ

=Φ

seems to be misleading in general. The latter relation is

obviously divergent for grazing angles ( cos 0θ → ) and this

divergence implying angular cutoff [19] seems to be

unjustified especially for the cuboid-like (i.e. with nearly

equal aspect sizes) sensitive volumes, which become typical

for modern microelectronics. Recall that any particle

directions are absolutely equivalent in space environments.

Measuring in some detail the cross-section angular

dependence, one can in principle perform (at least

approximately) direct averaging over full solid angle

( )

( ) ( )

exp

1 2 1

exp exp

1 0 0

1, , , ,4 d d d

π

σ

µ ϕ σ µ ϕ σ µ µ π

+ +

−

Λ =

= Λ ≅ Λ∫ ∫ ∫

(7)

where cosµ θ ≡ is a notation for cosine of polar angle.




Fig.1. Illustrative sketch of cross-section dependence on polar angle at a given

LET (negative µ region corresponding to particles incident from backside of

the IC is assumed to be symmetric for brevity and omitted).

Neglecting for brevity possible dependence on azimuthal

angle the averaging is reduced to integration of experimental

dependence on µ . It is equivalent to a determination of a

square under the experimental curve ( )exp ,σ µ Λ in the

illustrative Fig.1. Comprehensive phenomenological approach

does not require any additional approximations although one

should know cross-section curves to be measured for extended

set of experimental angles. Similar approach was proposed

and utilized in Ref. [20]. Unfortunately such approaches are

impossible in full measure because of economic factors.

Nevertheless they allow to easily perform conservative SER

estimations using maximum experimental cross-section

values.

C. Monte Carlo Averaging Approach

Monte Carlo method is in essence full computer simulation

analogue of phenomenological approach since it rely on

repeated computational sampling followed be averaging over

ensemble of incident particles [21]. An advantage of Monte

Carlo method is it utilizes real space spectra. Limitation is it

demands detailed information about layout and composition of

the ICs and difficult in practical applications.

And so most of practical SER simulators (CREME96 [22],

SPENVIS (www.spenvis.oma.be) and PRIVET) are based ontrade-off approaches combining a use of experimental cross-

sections vs LET curves and averaging over chord length

distribution in the sensitive volumes.

III. SIMULATOR PRIVET METHODOLOGY

A. Chord-Length Distribution Averaging Approach

Additional concepts of the sensitive volume (SV) and the

critical energy E C should be involved due to lack of full

experimental information. Critical energy is closely connected

with the circuit parameter of the critical charge QC which is

defined as minimum amount of collected charge to upset a

memory cell. Both the sensitive volume (which is effective

volume of charge collection) and the critical charge are in

essence circuit parameters depending on geometry,

capacitances, transconductance etc. and can be computed in

principle by circuit or TCAD simulation.

We have introduced in Ref. [15] the memory cell

sensitivity function ( )C K s E Λ − which is an error probability

dependent on energy deposition overdrive, where s = ρ l is themass length chord ( ρ is the material mass density)). In the

sensitive volume approximation for isotropic flux the full

angular averaging is equivalent to averaging over the

differential chord length distribution. This allows to replace

the angular averaging of unknown in general

phenomenological cross-section in Eq.3 by averaging of

memory cell sensitivity function with a known for a concrete

sensitive volume shape the chord length distribution f (s)

( ) ( ) ( )

( ) ( )max

0

0

1; , cos

4

,

4

s

C

d d

SK s E f s ds

σ σ θ ϕ θ ϕ π

Λ = Λ =

= Λ −

∫∫

∫ (8)

where S0 is the full area of sensitive volume. Such replacement

is a sort of the so called ergodic hypothesis wherein averaging

on statistical ensemble of events can be replaced by averaging

over appropriate distribution function.

Inserting Eq.8 in Eq. 4 one obtains the expression

( ) ( ) ( )max

0

04

s

C

S R K s E f s ds d φ = Λ Λ − Λ∫ ∫ , (9)

which is indeed computational relationship proposed in [15]

and used in heavy ion upset rate simulator PRIVET with the

exact chord length distribution for RPP sensitive volume (see

Fig.2).

differentialintegral

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Chord Length, mm

D i s t r i b u t i o n s

Fig.2. Integral and differential chord length distribution in the rectangular

parallelepiped dimensions 1×2×3 µm3.

Notice that approximating error probability by the step

function ( ) ( )C C K s E s E θ Λ − Λ − , the Eq.8 can be

expressed through the integral chord distribution ( )F s

( ( ) ( ) f s dF s ds= − )

( ) ( )0( , ) 4C C

E S F E σ Λ = Λ , (10)

that is equivalent to the Bradford approach (see, [23]).




Fig. 3. Simulated with Eq. 10 normalized cross-section vs LET dependencies

for different E C .

Averaged cross-section depends not only on LET and full

area of the sensitive volume but on the critical energy (orcharge) as well which is a circuit parameter (see Fig.3). If the

argument of the integral chord length distribution tends to zero

we have F ( E C / Λ → 0) →1 and0 / 4Sσ → since any ion strike

would lead in this case to an error.

B. Error Probability Response Function

According to the microdosimetric approximation all the

circuit response effects are contained in a critical charge

(energy) QC ( E C ) ( ( )C ion C E Q qε = , ionε 3.6 eV in the Si)

which is a circuit parameter of the memory cell as a whole.

The error probability formally determined in this approach by

the step function with stochastic argument

( ), ( )err C C p E E E E θ ∆ = ∆ − , (11)

where ∆ E is the stochastic energy released in the sensitive

volume of the memory cell during pass of ionizing particle.

We refer to the expectation value of the error probability

which is yielded by averaging over all stochastic factors

(denoted by ... AV

) as the response function of the memory

cell [15]

( , ) ( )C C AV K E E E E θ ∆ = ∆ − . (12)

We use at present in the PRIVET simulator the standard

mean LET approximation where the energy deposition across

the chord is represented by its mean value( ) E s E s∆ ≅ ∆ = Λ .

There are at least four contributions to the chargedeposition variation in a given sensitive volume:

(i) variance of track length for different hits;

(ii) variance of mean linear energy transfer of particles fordifferent particles;

(iii) energy-loss straggling for fixed track length and ion’s

energy and Z;

(iv) variance in charge collection for fixed track lengthand ion’s energy.

Factors (i) and (ii) are explicitly taken into account by

averaging over the chord length distribution and convolution

with the LET spectrum in Eq.9 and they are irrelevant to a

structure of error probability K (Λs, E C ). Unfortunately we are

not capable to compute the factor (iv) in general case. A part

of deposition energy variation connected with straggling canbe in principle computed on physical level (see Sec. IV

below). Both contributions are important and cannot be

generally ignored in formal computation.In practice due to a lack of full information about factors

(iii) and (iv) we would be forced to compensate knowledge

shortage by a phenomenological information. More exactly, in

view of inevitable variations in straggling, charge collection,

and circuit response processes the step function of errorprobability is expected to smear into a more smooth

phenomenological function ( )C w E E ∆ − with a finite energy

scale of the rise W E (MeV)

( )( )

,

C C AV AV

C

E

E E s E

s aw

W

θ θ

ρ

∆ − = Λ − ≅

⎛ ⎞Λ − Λ≅ ⎜ ⎟

⎝ ⎠

(13)

where the critical (threshold) energy is assumed to be

C C E a ρ ≡ Λ by a definition. This function should be

associated with the experimental data routinely obtained as

Weibull functions of LET rather than transferred energy E ∆ .

We are faced at this point with ambiguity of converting

between energy and LET variables.

C. PRIVET Choice

The following form of phenomenological parameterization

of experimental curves is chosen in [15]

( )( , ) 1 exp C C C

s E K s s E

W s

α

θ Λ

⎡ ⎤⎛ ⎞⎛ ⎞Λ −⎢ ⎥⎜ ⎟Λ Λ = − − Λ −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

, (14)

where a is the sensitive volume thickness, α is a

dimensionless fitting constant, and W Λ (MeV-cm2 /mg)

corresponds to an empirical constant of the Weibull

distribution. This choice assumes that energy ( E W ) and LET

(W ) dispersions are connected through the relation

E W W s= .

The response function of a memory cell is assumed in the

PRIVET simulator to be proportional to a normalized shape of

cross-section versus effective LET dependence under

normally incident ion (with s = a ρ )

( )exp C C

SAT

K W

σ

σ Λ

Λ − Λ⎛ ⎞Λ − Λ=⎜ ⎟⎝ ⎠

, (15)

where the empiric values of the saturation cross-sectionSAT σ

and the critical LETC Λ should be determined

experimentally. At present the straggling effects are not taken

into account in the PRIVET computational scheme.

Knowing from the experiments the sensitivity function

( )C K Λ − Λ one can perform numerical integration over two-

dimensional “LET-chord length” domain with two-variable

integrand according to Eq.9. Typical view of the integrand is

visualized in Figs.4-5. As can be seen from Fig.4 and 5 (the

latter is the level curve map version of Fig.4) there exists the

pronounced “optimal” region of the “LET-chord length”




domain with maximum contribution in soft error rate. Phase

curve (hyperbola) defined by equation E C = Λ s divide a

domain in Fig.2 on two regions where conditional error

probability ≅1 (upper-right region, Λ s > E C ) and where error

probability ≅ 0 (lower-left region, Λ s < E C ). Partial SER is

determined both by chord length distribution and LETspectrum.

Fig. 4. Partial contribution to SER as 2D function of chord length and LET.

Fig. 5. Density plot of SER (light color corresponds to maximum error rate).

SER simulator PRIVET is one of computation modules of

the program complex OSOT for SER prediction in space

environments which has been developed under the aegis of the

“ROSCOSMOS” (Russian Federal Space Agency) for the

establishing accelerator based test center [24].

Fig. 6. Comparison of OSOT simulation results with CREME96 and flight

data adapted from Ref. [25]. Input circuit data for PRIVET computation are

taken from Ref. [25]. Results computed with CREME96 “CREME96 choice”

with CREME96 spectra are practically coincided with the online CREME96

results given in Ref. [25]. Results for two version of PRIVET with OSOT

spectra are located closely in this case.

An example of simulation with the OSOT in comparison

with CREME96 is shown in Fig.6.

D. CREME96 Choice

As we noted above the choice of converting method from

energy to LET variables adopted in the initial version of

PRIVET is not a unique. There exists alternating approach,

which is based on the constant energy dispersion assumption

E W W a ρ Λ= (16)

This corresponds to the following form of

phenomenological parameterization of experimental curves

( )

( )( )

( , ) 1 exp

1 exp .

C C C

C

C

s E K s s E

W a

s as E

W

α

α

θ ρ

ρ θ

Λ

Λ

⎡ ⎤⎛ ⎞⎛ ⎞Λ −⎢ ⎥⎜ ⎟Λ Λ = − − Λ − =⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞Λ − Λ⎛ ⎞⎢ ⎥⎜ ⎟= − − Λ −⎜ ⎟

⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

(17)

In contrast to “PRIVET assumption” wherein E W W s= the

approach in Eqs. 16-17 assumes an explicit independence of

the sensitivity function rise’s energy scale E

W on track length.

We have introduced into the PRIVET an option allowing to

perform computations corresponding to the choice in Eqs.16-

17. Extended computational comparison between CREME96

and PRIVET yield practically equivalent results (with

differences typically no more 10%) for any input parameters

and for any orbits and shielding.

An example of comparison between two versions of




PRIVET and CREME96 is shown in Fig.7. This gives us

grounds to refer Eq.16 assumption hereinafter as to

“CREME96 choice”.

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æ

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æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

LC = 2MeV-cm2 ímg

LC = 7MeV- cm2 ímg

LC = 40MeV-cm2 ímg

0.1 0.5 1.0 5.0 10.0 50.0 100.010-12

10-10

10-8

10-6

10-4

0.01

a, mm

R ,

1 ê H b i t d a y L

Fig. 7. Comparison of PRIVET (“CREME96 choice”) and online CREME96

simulation results for GEO orbit as functions of the SV thickness for threedifferent critical LETs with CS=10×10 µm2; W =10 MeV-cm2 /mg; α=2;

shielding 1g/cm2; (simulated curves for two simulators are practically

coincided).

Note that “PRIVET-choice” version yields typically more

conservative estimation of SER although for large critical LET

and small W Λ the differences are diminished.

E. SER Scalability

Interrelations between variety of existing computational

schemes were discussed in [14]. We regard computational

approach in PRIVET as taking mathematical advantage over

traditional approaches but having the same accuracy class

since all of the approaches are based on the same physicalapproximations. Particularly, PRIVET and CREME96

simulators possess similar scalability properties.

Let us consider the scale transformation of the RPP sizes

with arbitrary non-zero positive scale factor β

's s β → , 'ds ds β → , 'a a β → ,

'b b β → , 'c c β → , 2

0 0'S S β → .

Chord length distribution obeys exactly the condition

( ) ( )' f s f s β β → .

Then the memory cell sensitivity kernel function in

PRIVET-approximation is invariant under scaling

transformation at a given critical LET (C Λ )

' '

'

C C s c s c

K K W s W s

ρ ρ ⎛ ⎞ ⎛ ⎞Λ − Λ Λ − Λ→⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠,

and the computed SER is scaled as 2' R R β → . SER in this

approximation thusly represents a homogeneous function of

the 2-nd order for the SV geometrical size parameters

( ) ( )2, , , , / , , , ,C C R s a b c E R s a b c E β β β β β β = ,

( ) ( )2, , , , , , , ,C C

R s a b c R s a b c β β β β β Λ = Λ (18)

The simulator CREME96 is empirically found to have the

same property of geometric scalability. We will show below

that this is generally the incorrect model property since such

geometric scalability should be violated under spatial scaling

of the SV due to modification of energy-loss straggling role.

IV. BEYOND AVERAGE LET APPROXIMATION

LET is defined as the mean energy lost by an ion per unit

of mass length in collisions with electrons of the material, and

represents a good quantity to characterize only mean

ionization rate averaged over large volumes. Mean LET

concept is not enough for description of ionization in rathersmall volumes. Really, we are often faced in typical

experiments with a situation when the track length can be

considered to be constant. Nevertheless the measured cross-

section curve vs LET is by far different from the step function.

The question has arisen what is a nature of spreading of the

step function characterizing error probability. This spreading

in particular means that the error is possible even for mean

deposited energies, which are less than the critical value E C .

We will consider this effect in this section as corresponding to

energy-loss straggling.

A. Energy Deposition Distribution

Energy deposition on small spatial scales is stochasticvalue which can be represented as a sum of the mean ( ) E s∆

and fluctuation parts δε

( ) ( ) E s E s sδε δε ∆ = ∆ + = Λ + . (19)

Full description of such fluctuations is yielded by

distribution function ( )( ),P E E s∆ ∆ (see, for example,

[20]). To obtain contribution of straggling to spreading of

error probability function one has to average energy

deposition over distribution of energy-loss fluctuation

( ) ( )

( ) ( ) ( )( )

( )( ) ( )

,

, ,C

C C str

C

E

K E E E E

d E E E P E E s

P E E s d E

θ

θ

+∞

∆ − = ∆ − =

= ∆ ∆ − ∆ ∆ =

= ∆ ∆ ∆

∫

∫

(20)

where ( ) E s∆ is stochastic energy loss at a fixed chord length

s. If the mean energy deposition exceeds significantly the

critical energyC

E E s<< ∆ ≅ Λ we have integration over δε

from minus to plus infinity yielding unity for error probability

due to normalization property of distribution function. Then

error cross-section for a given LET is represented as a result of

two-fold averaging over deposited energy fluctuation and

chord length instead of single integration of empirical

response function K over chord lengths

( ) ( ) ( )( ) ( )( )max

0

0

, ,4 C

s

LN E

S E Z f s P E E s d E dsσ +∞

= ∆ ∆ ∆∫ ∫ .(21)

This relation was obtained first in our early paper [26]

wherein we used the Landau distribution which is valid for

only very short tracks with very rare scatterings. In this paper

we use more pertinent and convenient Lindhard-Nielsen

approximation [27] which can be adapted in a form

( )

( )2

2200

,

1exp ,

22

LN

E E

P E E

E E E

E m v E m v E π

∆ ∆ =

⎛ ⎞∆ − ∆∆⎜ ⎟= −⎜ ⎟∆ ∆∆ ⎝ ⎠

(22)

where mean energy deposition can be approximated as

E s∆ = Λ ≅ 0.233 (Λ /(MeV-cm2 /mg)) (s /µm), (23)




m0 is the rest electron mass, c is the light velocity, v E is the

projectile ion velocity with kinetic energy E 22 2

21 N E

N in

M cv

c M c E

⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟ +⎝ ⎠ ⎝ ⎠

, (24)

M N is the rest energy of a nucleon (≅ 939 MeV/c2

), E in is thekinetic energy of the projectile ion per nucleon (or, the same,

per atomic mass unit, a.m.u).

Ar

Ar

Xe

Xe

0 1 2 3 4

0

2

4

6

8

10

DE, MeV

p r o b a b i l i t y

Fig. 8. Simulated with Eq.22 distributions of energy depositions for the ions

of Xe ( Z ion =54, E in =106 MeV/amu) and Ar ( Z ion = 18, E in = 4.5 MeV/amu)with approximately equal LET (~14 MeV cm2 /mg) for track lengths 0.8 µm

(right peaks) and 0.1 µm (left peaks). The Xe ion distribution peaks are

broader than the Ar ones.

One should emphasize that the dispersion of distribution is

specific for ion’s type since the straggling depends strongly on

Z and energy of ions. As for mean energy deposition (LET) an

increase in Z and energy of the projectile ions in a great extent

compensate each other while at the same time stragglingdramatically increases. Fig.8 shows simulated energy

deposition probability distribution for the ions with different Z

and energies but with approximately equal LETs. As can be

seen in Fig.8 the peaks of energy-loss distributions for the

high-energy and high-Z xenon are always broader than for the

low-energy and low-Z argon. This fundamental fact inevitably

leads to an appearance of more pronounced subthreshold

errors in case of high-energy ions in comparison with low-

energy ones even for equal LET.

Fig. 9. Illustrative comparison of subthreshold error probabilities for Xe

( Z ion =54, 106 MeV/amu) and Ar ( Z ion = 18, 4.5 MeV/amu) withapproximately equal LET (~ 14 MeV-cm2 /mg) with mean energy deposition

E ∆ ≅

3.3 MeV for case of the cell with the critical LETΛ

C =15 MeV-cm2 /mg corresponding to E C ≅ 3.5 MeV for the 1 µm thickness of the SV.

This is illustrated in Fig.9 where error probability is

characterized by the squares of overlap between the

distributions and the step function of error probability.

The difference of the error cross-sections between ions with

approximately equal LET but with drastically different atomic

number and energies is illustrated in Fig.10. Errorprobabilities for Ar and Xe ions in the above threshold region

C E E ∆ >> practically coincide. It is evident from Fig. 10

that the subthreshold (i.e. with the energy deposition

significantly lower than the critical energy,C Λ < Λ ) cross-

section for the Xe ion with larger energy becomes much larger

that for the low-energy Ar ion despite of approximately equal

LET.

1 mm

0.2 mm

Ion LET

10 12 14 16 18

10-8

10-6

10-4

0.01

1

LC , MeV-cm2êmg

c r o s s - s e c t i o n

r a t i o

Fig. 10. Calculated ratio of cross-sections for Ar (4.5 MeV/amu) and Xe ( Z ion

=54, 106 MeV/amu) ions with approximately equal LET (~14 MeV-cm2 /mg)as function of critical LET for a track lengths s=0.2 µm and 1 µm.

The ratio ( ) ( )Ar / Xeσ σ is larger for large track lengthdue to more pronounced straggling (increased energy-loss

dispersion) for longer tracks.

Critical energy increase leads to an increase of role of large

energy deposition E ∆ in subthreshold region C E E ∆ ≤ and,

consequently, to an increase in actual dispersion of energy

deposition and subthreshold errors probability. This is why the

subthreshold error effects are less pronounced for non-

hardened devices with low E C and more pronounced for rad-

hard devices and single event latchups with rather high E C .

Direct comparison of experimental dependence of upset

cross-sections versus ion’s energy at practically fixed LET [7]

with simulation according our model is exhibited in Fig. 11.




æ

æ

æ

æ

æ

æ

Dodd, 2007

simulation

1 2 5 10 2010-15

10-14

10-13

10-12

10-11

Energy per nuclon, MeVêamu

C r o s s - s e c t i o n

, c m

ê b i t

Fig. 11. Comparison experimental results (adapted from Fig.12 Ref. [7] with

simulation for ions with LET~ 4.3 MeV-cm2 /mg: C12 (~13 MeV), Si28(~450MeV), Ar40(~1200 MeV) for fitting constants E C = 3.2 MeV, CS = 2×2 µm2,

a = 1.6 µm, ΛC ~4.3 MeV-cm2 /mg.

This comparison demonstrates excellent qualitative andsatisfactory quantitative agreement between the experiments

and simulation based on taking into account the straggling

effects. We intentionally do not aim at best quantitative

agreement in Fig.11 which can be achieved by additional

fitting due to its formal character. The main point in Fig.11 is

qualitatively similar behavior of experimental and simulated

curves on three decades of cross-section magnitudes. We

believe that such dramatic dependence cannot be explained in

any different way.

B. General Approach for Direct Ionization Induced SER

Thus we have concluded that the mean LET approximation

is in general insufficient for adequate SER modeling since

LET spectra do not contain any information about energy

deposition fluctuation. To correctly compute the SER one

needs to turn to integration over energy distribution of primary

ions taking into account partial contributions from ion species

with all Z.

General expression for direct ionization induced SER in

which straggling is taken into account can be written as

( ) ( ) ( ) ( )92

1

,m m

m

m

R E E Z dE φ σ =

= ∑∫ , (25)

where ( ) ( )m E φ [cm-2 s-1 MeV-1] is energy spectrum of m-th ion

and cross-section of the m-th ion with energy E and atomic

number Zm should be computed with distribution function

specific for a given ion

( ) ( ) ( ) ( ) ( )0, , ,4

m m

m m C

S E Z K E E Z s E f s dsσ = ∆ −⎡ ⎤⎣ ⎦∫ . (26)

The contributions of ions with approximately equal LET

but different Z and energy could be significantly different for

such approach.

Note that the theoretical Eq.25 is completely consistent

with proposed phenomenological approach (cf. Eq.5) wherein

the error cross-section for specific projectile ion is supposed to

be determined from angular averaging over multiple

experimental data.

V. SECONDARY ION INDUCED ERROR RATE

In case of proton or neutron the primary particles does not

significant direct ionization along its track, producing,

nevertheless, secondary ionizing nuclei capable to upset

memory cells [28]. We are faced in this case with the situation

where the LET from the very outset is not a good measure of capability to induce error since it is technically difficult to

express energy transfer through the LET variable.

Soft error rate due to secondary ion can be estimated

through a general Eq. 5, where ( ) p E φ is flux spectrum as

function of projectile particles E p. The problem again is to

measure or compute primary particle energy dependent cross-

section ( ) p E σ . Nuclear reactions are characterized by total

cross section ( ) p E Σ , which is approximately equal in silicon

to 1-2 barns for proton (neutron) in the range from ten to

hundreds MeV (see for example, [29])). Probability of nuclear

reaction can be estimated as ( ) ( ) p at R p E N L E Σ << 1, where N at

is the atom density in material (~5 × 1022 cm-3 in silicon),

( ) R p L E is a mean range of secondary particles [30]. Then

effective flux of secondary particles can be approximated by

equation( ) ( ) ( ) ( ) ( ) ( )sec

p at R p p p E E N L E E E φ φ φ = Σ << (27)

via energy spectrum of primary particles ( ) p E φ .

One can introduce the large and small SV approximations

[30]. For the large SV typical for old technologies with

dimensions LΩ much greater 1 µm the inequality L R << LΩ is

obeyed consisting with BGR approximation. The opposite

condition of L R > LΩ allows to consider the small SV as being

positioned in field of effective flux of secondary particles (27)

for primary particles with given energy E .

Effective cross-section is yielded by averaging of Eq.10

over LET distribution function of secondary particles

( ); p p E Λ for primary particles with energy

p E [30, 31]

( ) ( )0, ;4

C C

S E E E d p E F σ

⎛ ⎞= Λ Λ ⎜ ⎟Λ⎝ ⎠

∫ . (28)

Using Eqs. 5 and 27 one can get soft error rate induced by

nuclear reaction

( ) ( ) ( ) ( )0 ; .4

C at R

S E R dE E N L E E d p E F φ

⎛ ⎞= Σ Λ Λ ⎜ ⎟Λ⎝ ⎠

∫ ∫ (29)

This relation can be rewritten in an equivalent form

( )0

4

C eff

S E R d F φ

⎛ ⎞= Λ Λ ⎜ ⎟Λ⎝ ⎠

∫ , (30)

where the effective LET spectrum of secondary particles can

be computed as

( ) ( ) ( ) ( ) ( );eff at RdEp E E N L E E φ φ Λ = Λ Σ∫ . (31)

If for a given mono-energy ( ( ) ( )0 p p E E E φ φ δ = − ,

[ ( )0 p E φ ] = cm2 s-1) we have a concrete nuclear reaction with

a secondary particle having energy R E ∆ assumed to be

converted to ionization and mean range R L , then LET

distribution can be roughly approximated as

( ) ( ) /

R R p E Lδ Λ Λ − ∆ . Using the property of Dirac delta-

function one reads




( ) ( ) ( ) ( ) ( )00

4

C p at p R p R p p

R

S E R E N E L E F L E E

E φ

⎛ ⎞= Σ ⎜ ⎟

∆⎝ ⎠. (32)

For large energy deposition and/or short secondary particle

ranges 2 2 2 / R C R L E E a b c∆ << + + we have ( )0 1F x → →

and reproduce the result of the known BGR approximation [4,

32]

( ) ( ) ( ) ( )

( ) ( ) ( )

00

0

4 p at p R p p

at p eff p p

S R E N E L E E

N E E E

φ

φ

≅ Σ =

= Σ Ω

(33)

with the effective sensitive volume for nuclear-induced

reactions

0

4

R

eff R

LS L

LΩ

Ω = = Ω (34)

where Ω is the sensitive volume for direct ionization,

04 / L SΩ ≡ Ω is the mean chord length of the SV. This

“quasi-BGR” form of SER with strong impact

( ( ) / R C R

E E L LΩ∆ > ) is valid for the small SV approximation

R L LΩ> while for opposite case R L LΩ< we have

“true” BGR approximation with eff Ω ≅ Ω .

VI. MULTIPLE Bit Error Rate Estimation

Modern highly scaled memory IC are susceptible to

multiple bit upset (MBE), in which more than one bit is upset

[33]. MBEs are relatively rare events in comparison to single-

bit error. In low and moderate scaled IC the SER is

proportional to the integration N 0 (i.e. density of the sensitive

volumes per unit area) ( 0 R N ∝ ). One can anticipate the totalSER per device in the high-density ICs as being represented as

a sum of single-bit ( SBE R ) and multiple-bit ( MBE R ) error rates

2

1 0 2 0tot SBE MBE R C N C N R R≅ + ≡ + . (35)

One of the main types of MBE is caused by a finite value

of track’s transversal section square ( 0a ≤ 1 um2). The

smallness parameter for such MBE characterizing MBE

probability is a dimensionless product 0 0 N a . This parameter

is assumed to be negligible for older technologies0 0

1 N a << ,

but in new technologies, due to smaller spacing between

neighboring cells (i.e. large N 0) it can no longer be ignored.

Due to a single particle can deposit sufficient charge in twoor more adjacent cells the effective LET spectrum increases

( ) ( ) ( )0 01 N aφ φ Λ → + Λ . (36)

The effective LETs in contrast are reduced due to charge

sharing between different nodes and we model this effect

roughly

( )0 01 N aΛ → Λ + . (37)

Substituting Eqs.36-37 into Eq.9 (neglecting in such way

insufficient here straggling) one reads

( ) ( ) ( )max0

0 00

0 0

14 1

,

s

tot C

SBE MBE

S s R N a ds f s d K E

N a

R R

φ ⎛ ⎞Λ

+ Λ Λ −⎜ ⎟+⎝ ⎠

≡ +

∫ ∫

(38)

where multiple-bit error rate is expressed as follows

( ) ( )max0

0 00

0 04 1

s

MBE

S s R N a ds f s d K E

N aφ

⎛ ⎞Λ= Λ Λ −⎜ ⎟

+⎝ ⎠∫ ∫

(39)

The ratio of multiple-bit to the total error rates (i.e. relativeMBE probability) in such approach is expressed as

0 0

0 01

MBU

tot

R N a

R N a=

+, (40)

tending to an unity for very high integration. For relatively

low integration we have proportionality MBE probability to

integration what is in reasonable agreement with experimental

data.

ACKNOWLEDGMENT

The authors would like to thank V.V. Emelianov, M.S.

Gorbunov, N.V. Kuznetsov and A. Bulkin for fruitful

discussions and technical support.




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