Leroy C., Rancoita P.-G. Principles Of Radiation Interaction In Matter And Detection

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540 Principles of Radiation Interaction in Matter and Detection

matic investigations for low- and high-resistivity silicon.

The particle interactions in the bulk or active volume of silicon devices are

responsible for the device degradation resulting from the absorbed dose

∗

and the

so-called single event phenomena

††

As mentioned in Sects. 4.2.3.1, 4.2.3.2, the energy deposited by ionization or

collision energy-loss processes accounts for the largest fraction of the total energy

deposited in any medium (e.g., in a silicon semiconductor). Silicon based devices

are aﬀected by the total (i.e., and by far, ionizing) absorbed dose: for instance in

MOS transistors, shifts of the threshold voltage and variations of the sub-threshold

slope occur.

Other silicon based devices, like solar cells (Sect. 6.6) and bipolar transi-

stors, are mostly aﬀected by the displacement damage generated by non-ionizing

energy-loss processes (e.g., see [Srour, Marshall and Marshall (2003)] and references

therein). For instance, at large cumulative irradiations this mechanism was found to

be responsible i) for the decrease of the gain of bipolar transistors

‡‡

mostly as a result

of the decrease of the minority-carrier lifetime (e.g., see Sect. 4.3.1) in the transistor

base and ii) for the degradation of the series-noise performance of charge-sensitive-

preampliﬁers with bipolar junction transistors in the input stage mainly because of

the increase of the base spreading-resistance (e.g., see Sects. 4.3.5, 7.1.5) of these

transistors

∗∗

. Furthermore, through systematic measurements (see Sects. 7.1, 7.1.3)

it was found that the gain degradation of bipolar transistors manufactured on VLSI

technologies depends almost linearly on the amount of displacement damage gener-

ated (e.g., the amount of energy deposited by NIEL processes) independently of the

type of incoming particle [Colder et al. (2001, 2002); Codegoni et al. (2004a,b, 2006);

Consolandi, D’Angelo, Fallica, Mangoni, Modica, Pensotti and Rancoita (2006);

D’Angelo, Fallica, Galbiati, Mangoni, Modica, Pensotti and Rancoita (2006)].

A Single Event Eﬀect (SEE) is due to an individual event caused by a single

charged particle (Z ≥ 2) traversing a semiconductor or semiconductor-based device

(e.g., integrated circuit, power supply, etc.). A SEE results in a failure of the device,

as a consequence of the charge deposited along the path of the incoming particle

∗

Both the non-ionizing (e.g., see Sect. 4.2.1) and ionizing (e.g., see Sect. 4.2.3) energy-losses

contribute to the absorbed dose. The reader can see, for instance, [Vavilov and Ukhin (1977);

Srour, Long, Millward, Fitzwilson and Chadsey (1984); Ma and Dressendorfer (1989); Messenger

and Ash (1992); Claeys and Simoen (2002); Holmes-Siedle and Adams (2002); ECSS (2005)],

Sect. 7.1 and references therein.

††

One can see, e.g., [Srour, Long, Millward, Fitzwilson and Chadsey (1984); Messenger and Ash

(1992, 1997); Claeys and Simoen (2002); Holmes-Siedle and Adams (2002); ECSS (2005)], Sect. 7.2

and references therein.

‡‡

The reader can refer, e.g., to [Frank and Larin (1965); Messenger (1966); Ramsey and Vail (1970);

Messenger (1972); Srour (1973); Vavilov and Ukhin (1977); Srour, Long, Millward, Fitzwilson and

Chadsey (1984); Srour and McGarrity (1988); Srour and Hartman (1989); Messenger and Ash

(1992); Colder et al. (2001, 2002); Codegoni et al. (2004a,b); ECSS (2005); Codegoni et al. (2006);

Consolandi, D’Angelo, Fallica, Mangoni, Modica, Pensotti and Rancoita (2006)] and references

therein.

∗∗

A discussion can be found, e.g., in [Baschirotto et al. (1995a,b, 1996, 1997, 1999)] and references

therein.

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Displacement Damage and Particle Interactions in Silicon Devices 541

within the sensitive volume of the device. SEE’s are treated in Sects. 7.2–7.2.8.

Substantial progress has been made in the understanding of mechanisms at the

origin of SEE and predicting their rate of occurrence for various types of devices

and circuits. Monte Carlo tools have been developed which take into account the

evolution of modern technology toward faster devices with less power consumption

and denser circuits increasingly small. This reduction in size and multiplication of

active cells lead to a re-examination of the concepts of critical charge and sensitive

volume. The RPP approximation for the sensitive volume, although convenient for

rapid estimates, has to be much more reﬁned to account for eﬀects such as funneling,

bipolar eﬀects, source-to-drain currents. These eﬀects are the source of sensitivity

variations across the devices, not easily reproducible by Monte Carlo. The use of

powerful simulation to ols such as GEANT4 allows the accurate description of the

device geometry, structure, material composition, and at the same time the inclusion

of the mechanisms of energy deposition, and cell charge collection eﬃciency. SEE

cross sections for speciﬁc species of particles and ions are measured at accelera-

tors and can be convoluted with eﬀective LET spectrum calculated by GEANT4

to infer SEE rates. These predictions are strictly viable for the experimental con-

ditions under which the cross sections have been measured. The ultimate goal is

to extend these predicted SEE rates beyond these speciﬁc conditions and rely only

on GEANT4 fed with reﬁned models of energy deposition and charge collection

eﬃciency in the various regions of the device in order to calculate the charge col-

lected which also depends on the time constant of the process. This also supposes

that progress will be made in the determination of the sensitive volume size and

shape and of the critical energy which remain a major uncertainty in Monte Carlo

simulations. Precise SEE rate predictions may then have a strong impact on device

design.

7.1 Displacement Damage in Irradiated Bipolar Transistors

The bipolar transistor

∗

(contraction for transfer resistor) consists of two successive

junctions (n − p − n, or p − n − p) made by three semiconductor regions called

emitter (heavily doped

†

), base and collector, respectively. It is one of the most im-

portant semiconductor components available, nowadays, on VLSI (Very Large Scale

∗

A general treatment of this topic can be found, e.g., in Chapter 7 of [Grove (1967)], Chapter 6

of [M¨uller and Kamins (1977)], Chapter 4 of [Sze (1985)], Sections 3.9–3.14 of [Messenger and Ash

(1992)], Chapter 10 of [Neamen (2002)] and also [Sze (1981); Goodge (1983); Bar-Lev (1993); Ng

(2002); M¨uller and Kamins - with Chan (2002); Brennan (2005)].

†

In VLSI technologies, the emitter- and base-regions are strongly graded with doping concentra-

tions which may exceed 10

and 10

−3

, respectively; while the collector-region is typically an

epitaxial-layer [Sze (1988)] with dopant concentration lower than 10

−3

(e.g., see Figs. 6.16

and 6.17 at page 304 of [M¨uller and Kamins (1977)], Fig 11 at page 25 of [Sze (1985)] and Fig 3.27

at page 133 of [Messenger and Ash (1992)] 2nd Edition).

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542 Principles of Radiation Interaction in Matter and Detection

Integration) technologies. It was invented

‡

by a research team at Bell Laboratories

in 1947 (see page 133 of [Sze (1981)]). In this device, both the electrons and holes

participate in the conduction mechanism.

The emitter-base junction is forward-biased and, consequently, allows a large

injection into the base-region of minority (for the base) carriers, which are majority

carriers inside the emitter-region. The base-collector junction is reverse-biased and

suﬃciently close to the emitter-base junction so that most of these carriers can

reach the collector-region, where they are carriers of the majority type. Thus, in

the bipolar transistor operation, a large current ﬂows in the reverse-biased base-

collector junction due to the large injection of carriers coming from the forward-

biased emitter-base junction in its vicinity. Since not all the injected carriers from

the emitter will reach the collector-region, two quantities are of great importance

in describing the d.c. transistor characteristics

∗

a) the so-called base-transport factor given by

≡

C,m

E,m

, (7.1)

where I

C,m

is the current due to the minority (in the base-region, but majority in

the emitter-region from which they are injected) carriers reaching the collector and

E,m

is that due to the minority carriers initially injected from the emitter into the

base;

b) the current gain given by

≡

, (7.2)

where I

and I

are the collector and base current, respectively. The collector cur-

rent accounts for the carriers, which are capable to reach the collector after passing

through the base region. The base current accounts for recombination processes in

the base, recombination in the emitter-base region and diﬀusion of carriers from the

base into the emitter region. The minority-carrier ﬂow [see Eq. (7.1)] provides the

basic mechanism of the transistor operation and, as a consequence, of its degrada-

tion under irradiation.

In circuit applications, the most often conﬁguration of transistors is the so-

called common-emitter conﬁguration under active mode, that is, the emitter lead

is common to the input and output circuits, the emitter-base junction is forward-

biased and the base-collector junction is reverse-biased. The collector current can

be written as {see Equation (46) at page 128 of [Sze (1985)]}:

= β I

+ I

C,o

, (7.3)

‡

In 1948 J. Bardeen and W. Brattain announced the development of the point contact transi-

stor [Bardeen and Brattain (1948)], while in 1949 W. Shockley published his paper on junction

diodes and transistors [Shockley (1949)]: all of whom subsequently won the Nobel Prize in Physics

in 1956.

This is the reason why is called bipolar.

∗

For a general treatment of the bipolar transistor properties, one may refer, for instance, to

Chapter 7 of [Grove (1967)], Chapter 6 of [M¨uller and Kamins (1977)], Chapter 4 of [Sze (1985)]

and Chapter 10 of [Neamen (2002)].

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Displacement Damage and Particle Interactions in Silicon Devices 543

where I

C,o

is the collector-emitter leakage current with open base and β is the

so-called common-emitter current gain

. Under the assumption that the recom-

bination current in the emitter-base depletion region is negligible, β is given by

{e.g., see Equations (8) at page 115 and (44) at page 127 of [Sze (1985)]}:

β '

γ α

1 − γ α

, (7.4)

where γ = I

E,m

E,d

is the so-called emitter eﬃciency

and I

E,d

accounts for the

electron- and hole-current diﬀusing across the emitter-base junction, e.g., I

E,m

and

the diﬀusion current due to carriers injected from the base into the emitter region. γ

measures the injected current due to minority carriers with respect to the overall

current due to diﬀusion from the emitter and base regions. For silicon transistors

at normal temperature of operation, I

C,o

is negligible (e.g, see page 250 in [Bar-Lev

(1993)]) and, thus, from Eqs. (7.2, 7.3), we have β

≈ β. Furthermore, for good

transistors we have γ ≈ 1 and α

≈ 1 (e.g., see page 156 of [Lutz (2001)]); as a

consequence, since γ α

. 1, β

is usually large.

In active mode and for low-level injection

∗∗

, under the assumptions that the

transistor has i) uniform doping in each region, ii) no generation-recombination

currents in the depletion regions and iii) neglecting any series resistance in the

device, we obtain {e.g., see Equation (36) at page 121 of [Sze (1985)]}:

≈ 1 −

2 L

for

¿ 1, (7.5)

where W

is the depth of the quasi-neutral base-region

††

and L

is the diﬀusion

length of the minority-carrier in the base. For the usually well-made (and so-called)

narrow-base transistors, the condition W

¿ 1 is satisﬁed. L

is given by the

minority-carrier life-time (τ

) multiplied by the diﬀusion constant (D

) of these

carriers [see Eqs. (6.63, 6.64)]. If the emitter eﬃciency is close to unity (as it is

usual the case), the common-emitter current gain [Eq. (7.4)] can be re-expressed

using Eq. (7.5) as:

β '

1 − α

= 2

. (7.6)

This parameter is indicated as h

, when representing the transistor performance within the

framework of small-signal models for low and high frequencies (e.g., see Section 2.2.26 of [Goodge

(1983)] and Chapter 15 of [Bar-Lev (1993)]); the static common-emitter current gain is also in-

dicated as h

in the h-parameter representation (e.g., see Section 3.2.2 of [Sze (1981)] and Sec-

tion 2.2.17 of [Goodge (1983)]). Furthermore, it is indicated as β

, i.e., is the so-called forward

common-emitter current gain within the framework of the Ebers–Moll model (e.g., see the Sec-

tion 14.3(d) of [Bar-Lev (1993)]).

The term γ α

is the so-called d.c. common-base current gain {e.g., see Equation (4.36) at

page 86 of [Brennan (2005)]}.

∗∗

The condition for low-level injection is satisﬁed when the excess carrier concentration is small

in comparison with the doping concentration.

††

It extends from the edge of the depletion region of the emitter-base junction to that of the

base-collector junction.

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544 Principles of Radiation Interaction in Matter and Detection

For graded-base transistors, Eq. (7.5) is replaced by the Gover–Grinberg–Seidman

expression {Equation (9) of [Gover, Grinberg and Seidman (1972)], see also [Mes-

senger (1973)]}, which is valid for the case of a general impurity distribution:

≈ 1 − u

+ u

− . . . , (7.7)

where the coeﬃcients u

depend on the base doping-proﬁle

‡

For W

¿ 1, the right side of Eq. (7.7) converges rapidly and the terms

beyond the ﬁrst two can be neglected {e.g., see Equation (31) of [Gover, Grinberg

and Seidman (1972)]}. Thus, for γ ≈ 1 the common-emitter current gain can be

rewritten using of Eqs. (7.4, 7.7) as:

β '

1 − α

= u

. (7.8)

7.1.1 Gain Degradation of Bipolar Transistors and Messenger–

Spratt Equation

Since mid-ﬁfties, the eﬀect of fast-neutron irradiations on bipolar transistor was ex-

tensively investigated by many authors (e.g., see [Webster (1954); Loferski (1958);

Messenger and Spratt (1958); Frank and Larin (1965); Messenger (1967a); Ramsey

and Vail (1970); Messenger (1973); Vavilov and Ukhin (1977); Messenger and Ash

(1992); Codegoni et al. (2004b); Consolandi, D’Angelo, Fallica, Mangoni, Modica,

Pensotti and Rancoita (2006)]). It was shown that the knowledge of the degradation

of the minority carrier lifetime allows one to predict changes in the parameters of

transistors as a function of the fast-neutron ﬂuence: for instance, Webster, Loferski,

Messenger and Spratt could determine that the variation of the reciprocal of the

common-emitter current gain (β) linearly depends on the fast-neutron ﬂuence [Web-

ster (1954); Loferski (1958); Messenger and Spratt (1958)]).

As teated in Sect. 7.1, β

†

[Eq. (7.3)] is related to the lifetime of the minority

carriers in the base as shown by the approximate expression (7.17): the decrease of

their lifetime with increasing ﬂuence (see Sect. 4.3.1) is the fundamental mechanism

which determines the degradation of the transistor gain.

‡

For instance (see Appendices VII and VIII of [Gover, Grinberg and Seidman (1972)]), for a

homogeneous doping-proﬁle in the base we have

;

while, for an exponential doping-proﬁle of the type N = N

exp(−ηx/W

) we get

η + exp(−η) − 1

†

As discussed at page 543, in practice the common-emitter current gain does not diﬀer from the

current gain β

[Eq. (7.2)], thus the two quantities are used with no distinction in Sects. 7.1.1–7.1.5.

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Displacement Damage and Particle Interactions in Silicon Devices 545

Fig. 7.1 Collector currents (I

, in A) measured at 25

◦

C as a function of the voltage (in V)

applied at the emitter-base junction (V

), before and after irradiation with 9.1 MeV electrons

for ﬂuences of 2 × 10

, 7 × 10

and 2 × 10

e/cm

(from [Leroy and Rancoita (2007)]; see

also [Consolandi, D’Angelo, Fallica, Mangoni, Modica, Pensotti and Rancoita (2006)]).

Let us consider a transistor with narrow-base (i.e., with W

¿ 1) with

a graded doping-proﬁle. As mentioned in Sect. 7.1, when this transistor has a

large gain, an emitter eﬃciency close to 1 and is operated in the common-emitter

conﬁguration under active mode, the base transport factor is approximated by

Eq. (7.7). Furthermore, it is the dominant factor compared to the common-emitter

current gain [see Eqs. (7.4, 7.8)]. Under these assumptions and before irradiation,

β can be approximated by the ratio between the minority-carrier lifetime τ

and

the transit time across the base τ

[Eq. (7.17)]. Since τ

is the largest contribution

to the overall emitter-collector delay time (τ

), i.e., τ

≈ τ

, the cutoﬀ angular

frequency is roughly given by

≈

. (7.9)

By combining Eqs. (7.17, 7.19), we have, ﬁnally,

β ≈ τ

. (7.10)

After irradiation with fast neutrons, assuming that the transistor retains a signif-

icant common-emitter gain β

irr

(e.g., β

irr

& 3 [Messenger (1973)]) and that the

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546 Principles of Radiation Interaction in Matter and Detection

Fig. 7.2 Base currents (I

, in A) measured at 25

◦

C as a function of the voltage (in V) applied to

the emitter-base junction (V

), before and after irradiation with 9.1 MeV electrons for ﬂuences

of 2 ×10

, 7 ×10

and 2 ×10

e/cm

(from [Leroy and Rancoita (2007)]; see also [Consolandi,

D’Angelo, Fallica, Mangoni, Modica, Pensotti and Rancoita (2006)]).

diﬀusion constant (D

) of the minority carriers in the base does not vary apprecia-

bly, τ

is almost constant [e.g., see Eq. (7.15)]. Thus, β

irr

can be written as

irr

≈

B,irr

≈ τ

B,irr

, (7.11)

where τ

B,irr

is the lifetime of the minority carriers after irradiation. Then, the vari-

ation of the reciprocal of the common-emitter current gain becomes

∆

≡

irr

−

≈

B,irr

−

B,irr

−

. (7.12)

As discussed in Sect. 4.3.1, the variation of the reciprocal of the lifetime of the

minority-carriers in the base is related to the fast-neutron ﬂuence and the lifetime

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Displacement Damage and Particle Interactions in Silicon Devices 547

constant damage; thus, using Eq. (4.125), Eq. (7.12) can be rewritten as:

∆

≈

∆

τ,n

, (7.13)

where K

τ,n

is the lifetime damage coeﬃcient and Φ

is the fast-neutron ﬂuence

(e.g., see Sect. 4.3.1).

Equation (7.13) is the Messenger–Spratt equation

‡

, which was initially proven

for homogeneous base transistors [Messenger and Spratt (1958)], later extended to

treat exponentially graded base transistors [Messenger (1967a)] and, ﬁnally, using

the Gover–Grinberg–Seidman expression [Eq. (7.7)] generalized to cover any base

doping-proﬁle [Messenger (1973)]. Messenger (1973) has also noted that i) Ramsey

and Vail have shown how the emitter eﬃciency contribution, resulting from the

eﬀect of recombination processes in the emitter-base ﬁeld-region, is related to the

emitter time-constant τ

[Ramsey and Vail (1970)], ii) Eq. (7.9) is strictly related to

the cutoﬀ angular frequency for the base transport, but iii) ω

can be extended by

adding the emitter time constant. Thus, by means of this latter re-expressed value

of ω

, Eq. (7.13) accounts for recombination processes in both the emitter-base

ﬁeld-region and the base (bulk) region.

As discussed in Sect. 4.3.1, the lifetime damage coeﬃcient (K

τ,i

) and Φ

de-

pends on the typ e and energy of the incoming particle. Thus, for the same type of

transistor, by inspection of Eqs. (4.125, 7.13) we can see that the variation of the

reciprocal of the current gain (or any quantity proportional to it) is expected to

exhibit diﬀerent slopes

if irradiated with diﬀerent types of particle. This behavior

was experimentally determined, for instance, by investigations with fast-neutrons

for ﬂuences up to 10

n/cm

, with 2 MeV electrons up to 2 × 10

e/cm

and with

photons from a

Co source up to 4.5 × 10

photons/cm

(e.g., see Figure 42 at

page 124 of [Vavilov and Ukhin (1977)], see also other results regarding ∆(1/β) in

Figure 5.3 at page 215 of [Holmes-Siedle and Adams (2002)]). Thus, Eq. (7.13) can

be rewritten as

∆

≈

τ,i

. (7.14)

The current gain [see Eqs. (7.2, 7.3)] depends on the ratio between the collector

current (I

) and the base current (I

). Before and after irradiation with large ﬂu-

ences for which the displacement damage is expected to be dominant, the behavior

of the base and collector currents

∗

was systematically investigated as a function

‡

A complete derivation of the Messenger–Spratt equation is found, for instance, in Section 5.10

of [Messenger and Ash (1992)].

This is equivalent to exhibit a shift in the double logarithmic plot of ∆(1/β) versus the particle

ﬂuence.

∗

The devices were characterized using an HP4142B modular DC source-monitor and the IC-CAP

code (e.g., see [IC-CAP (2004)]) controlled by a workstation.

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548 Principles of Radiation Interaction in Matter and Detection

of the voltage applied to the emitter-base junction (V

). In Fig. 7.1 (Fig. 7.2),

the collector (base) currents of n − p − n transistors

with large emitter area re-

gion (50 µm ×50 µm) are shown before and after irradiation with 9.1 MeV electrons

for ﬂuences of 2 × 10

, 7 × 10

and 2 × 10

e/cm

[Leroy and Rancoita (2007)]

(see also [Consolandi, D’Angelo, Fallica, Mangoni, Modica, Pensotti and Rancoita

(2006)]). Using the electron data given in Table 4.5, these ﬂuences correspond to

NIEL

≈ 3.3, 11.7 and 33.3 Gy and D

Ion

& 0.49 ×10

, 1.71 ×10

and 4.88 ×10

Gy,

respectively. The data were measured about two years after irradiation and are

shown for 0.5 . V

. 0.72 V, i.e., for 5 × 10

−7

. I

. 10

−3

A. For all the inve-

stigated ﬂuences, the collector currents are only marginally aﬀected by irradiation

(Fig. 7.1). Larger variations are observed in the case of base currents (Fig. 7.2):

these currents, as a function of V

, are systematically larger after irradiation be-

cause, in the base, the concentration of recombination centers and the recombination

current increase with ﬂuence. These results are in agreement with those found for

irradiations with Ar-ions [Codegoni et al. (2004b); Consolandi, D’Angelo, Fallica,

Mangoni, Modica, Pensotti and Rancoita (2006)].

The radiation eﬀect on the cutoﬀ frequency f

of (Si) bipolar transistors (see

page 549) has b een investigated after irradiation with a fast-neutron ﬂuence of

n/cm

and for collector currents 5 × 10

−2

< I

< 5 mA [Roldan, Ansley,

Cressler and Clark (1997)]. After irradiation, f

shows practically no change with

respect to the values before irradiation for I

up to ≈ 1 mA, but it becomes slightly

larger with increasing I

. These measurements indicate that the overall emitter-

to-collector delay time τ

[Eq. (7.18)] is almost not aﬀected by the fast-neutron

irradiation for Si bipolar transistor with large f

The extension of the neutral region in the base and the properties of the minority

carriers are related to the transit time of these carriers (τ

) across the base. For a

homogeneous doping-proﬁle of the base, it can be shown {e.g., see Section 7.3(a)

of [Grove (1967)]} that τ

is given by

2 D

, (7.15)

where D

is the diﬀusion constant of the minority carriers in the base. For a graded

doping-proﬁle of the base and W

¿ 1, τ

is given by {Equation (32) of [Gover,

Grinberg and Seidman (1972)]}

= u

. (7.16)

It can be remarked that, since u

= 1/2 for a homogeneous doping-proﬁle in the base

(see footnote ‡ at page 544), in such a case Eq. (7.16) reduces to Eq. (7.15). To a

ﬁrst approximation by determining W

from Eq. (7.15) [Eq. (7.16)] and introducing

These transistors are manufactured according to the HF2BiCMOS technology by STMelectro-

nics [Gola, Pessina and Rancoita (1990)].

Before irradiation, the peak value of the cutoﬀ frequency is ≈ 28.9 GHz [Roldan, Ansley, Cressler

and Clark (1997)].

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Displacement Damage and Particle Interactions in Silicon Devices 549

this parameter in Eq. (7.6) [Eq. (7.8)], we can estimate the common emitter gain

for both the homogeneous and graded base doping-proﬁles, in terms of the transit

time across the base and the lifetime of the minority carriers:

β '

. (7.17)

The transit time of minority carriers across the base may also limit the tran-

sistor operation, when the operating frequency (f = ω/2π) increases beyond a

certain critical frequency. The cutoﬀ frequency

∗

, f

, is an important ﬁgure of merit

in transistors. It is deﬁned as the frequency at which the common-emitter short-

circuit current gain (h

) is unity [Pritchard, Angell, Adler, Early and Webster

(1961)]. The frequency response of a transistor can be described in the framework

of the hybrid-π model (e.g., see Section 4.5 of [Brennan and Brown (2002)]) in which

the h-parameter, h

, is given by h

= ∂I

/∂I

. Furthermore, for ω → 0 we have

that

→ β

{e.g., see Equation (4.5.20) at page 179 of [Brennan and Brown (2002)]}, i.e., h

reduces to the d.c. common-emitter current gain. The cutoﬀ frequency is inversely

proportional to the overall emitter-to-collector delay time τ

(e.g., see Section 4.3.2

of [Sze (1985)], Sections 4.3 and 4.5 of [Brennan and Brown (2002)] and also [Ramsey

and Vail (1970)]):

2πτ

. (7.18)

is the delay resulting from i) the emitter depletion-layer charging time τ

= r

where r

is the emitter resistance and C

the electrically measurable total delay

capacitance (e.g., see [Ramsey and Vail (1970)], Section 3.3.1 of [Sze (1981)] and

Section 5.3.1 of [Sze and Ng (2007)]), ii) the collector depletion-layer transit time

and collector charging-time (e.g., see Section 3.3.1 of [Sze (1981)] and Section 5.3.1

of [Sze and Ng (2007)]) and iii) the transit time of the minority carriers across the

base region. Generally, this latter term is the most limiting parameter (e.g., see

Section 4.3.2 of [Sze (1985)] and Section 4.3 of [Brennan and Brown (2002)]) that

inﬂuences the transistor frequency-response. From Equation (7.18), we have that

the cutoﬀ angular frequency is

, (7.19)

which (as well as f

) depends on the collector current I

[i.e., ω

)].

∗

This term expresses also the so-called gain bandwidth product for the common-emitter gain

(e.g., see Section 3.10 of [Messenger and Ash (1992)]).