Ahmed S.N. Physics and Engineering of Radiation Detection

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406 Chapter 6. Scintillation Detectors and Photodetectors

Just like conventional photodiode detectors, the avalanche photodetectors can

also be designed and built in diﬀerent conﬁgurations. Fig.6.5.23(a) shows the con-

ceptual design of a simple avalanche photodetector. The detector is made of an

intrinsic (or lightly doped p type material) sandwiched between a heavy doped p

side and a heavy doped n side. Another p type region is also established between

the intrinsic material and the heavily doped n side. The contacts are made by metal

deposition on two sides of the detector. A high reverse bias applied at the two ends

creates an electric ﬁeld proﬁle similar to the one sketched in Fig.6.5.23(b). Based on

this electric ﬁeld proﬁle two regions can be identiﬁed: an absorption region R

abs

and

a multiplication region R

mult

. The incident radiation passing through the intrinsic

or absorption region creates electron hole pairs along its track. The charges move

in opposite directions under the inﬂuence of moderate electric ﬁeld inside R

abs

.The

electrons move towards the p region, which is characterized by very high electric

ﬁeld intensity. This is where avalanche multiplication takes place. The large num-

ber of electron hole pairs generated here, as a result of charge multiplication, move

in opposite directions and induce a voltage pulse that is measured by the readout

electrode. Alternatively the current generated by the motion of charges can also be

measured by the associated electronics.

mult

abs

(b)

(a)

Figure 6.5.23: (a) Conceptual design of

a simple avalanche photodetector. (b)

The electric ﬁeld proﬁle of the struc-

ture shown in (a). π represents either a

lightly doped p-material or intrinsic ma-

terial while the superscript (+) refers to

heavy doping.

The kind of structure shown in Fig.6.5.23 is generally known as reach-through

structure. Fig.6.5.24 shows the practical design of a reach-through APD. The guard

ring around the multiplication region is established to minimize the possibility of

electrical breakdown.

C.1 Basic Desirable Characteristics

The desirable characteristics of an avalanche photodetector are not much diﬀerent

from the conventional photodiode detectors as can be deduced from the list below.

 Small Leakage Current: An idea APD does not have any leakage current.

Real APDs have extremely small dark currents. The reason to strive for no

leakage current is to avoid unwanted avalanches, something that may induce

non-linearity in the detector response.

6.5. Photodetectors 407

Intrinsic

Guard Ring

SiO

Metal Contact

Antireflective Coating

Metal Contact

Photon

Figure 6.5.24: Typical structure of a sili-

con based reach-through type avalanche

photodetector.

 High Gain: Since the sensitivity of the detector depends on its gain therefore

the APDs are designed such that they have gains of up to 10

. However de-

pending on the application, one can opt for lower gain, which can be achieved

by applying lower electric ﬁeld.

 Good Frequency Response: Good frequency response is one of the most

desirable characteristics of APDs, specially the ones that are to be used in high

rate environments.

It should be noted that the last two characteristics are similar to the ones desired

in photomultipliers, which of course is not a surprise since both detectors have essen-

tially the same purpose. We will shortly see that modern APDs have characteristics

near to their ideal values.

C.2 Multiplication Process and Gain Fluctuations

The process of electron multiplication in an APD is very much diﬀerent from the one

that occurs in a PMT. It resembles the avalanche process in gases that we studied

in the Chapter on the gas ﬁlled detectors.

The process through which the charges multiply in an APD is known as carrier

impact ionization. Intuitively thinking, it might seem a very simple process since it

refers to the subsequent ionizations when high energy charge carriers make collisions

with molecules along their tracks. However, the reality is that it is an extremely

complicated process and is very diﬃcult to evaluate analytically. The diﬃculty lies

mainly in modeling the stochastically spread distribution of charges after impulse

ionization. A detailed account of diﬀerent existing models and the related analytical

computations is beyond and scope of this book and therefore the interested reader

is referred to other sources (see, for example (37; 24; 31) and the references therein).

The problems mentioned above lead to the diﬃculty in the determination of

the probability density function of the electrons that make up the output signal.

One might assume these electrons to have a Poisson distribution if the absorption

of the incident light could be described by the Poisson process. This, however, is

not really the case. In face, the distribution of electrons is fairly complicated and

extremely diﬃcult to evaluate numerically ((33)). Due to this diﬃculty a number of

408 Chapter 6. Scintillation Detectors and Photodetectors

approximations to this probability density function have been proposed, including

the one that assumes the shape of a Gaussian distribution, though with longer tails.

We will see later that even though the gain ﬂuctuates around a mean value,

still with proper attention to the noise sources one can build a detector based on

APD that works at the limit of quantum ﬂuctuations of the multiplication process.

The mean value of the gain is an important parameter for design optimization, noise

consideration, and calibrations. For the output signal due to electrons the expression

for the mean gain can be written as (22)

G

 =

1 − u

−αd(1−u)

− u

, (6.5.57)

where u = 1 is the ratio of the hole ionization rate to the electron ionization rate,

α is the electron ionization rate, and d is the thickness of the depletion region. The

ionization ratio u can have any value between 0 and 1 but is generally very small,

on the order of 10

−2

or 10

−3

. Let us see what happens when we substitute u =0in

the above expression. This gives

G

≈e

αd

, (6.5.58)

which, the reader might recall, looks remarkably similar to the expression that was

obtained for gas multiplication in the chapter on gas ﬁlled detectors. From this

expression one might conclude that higher gain can be obtained by simply increasing

the depletion width. Although this, in principle, is true but is far away from the

actual practice as it has the downside of increasing the ﬂuctuations in the gain. We

will return to this discussion in the next section. One thing to note is that this

simpliﬁed expression for mean gain should not be used in actual computations as

the eﬀect of u on gain is not negligible (see example below).

Example:

An APD has a mean gain of 50 at hole-to-electron ionization rate ratio u

of 0.1. Compute the relative change in its mean gain if u changes to 0.01.

Assume that all other parameters remain the same.

Solution:

Using the given values we ﬁrst compute the value of αd in equation 6.5.57.

G

 =

1 − u

−αd(1−u)

− u

⇒ αd = −



1−u

G

+ u



1 − u

= −



1−0.1

+0.1



1 − 0.1

=2.37

6.5. Photodetectors 409

Now, using this value of αd we can calculate the gain at u =0.01 as follows.

G

 =

1 − u

−αd(1−u)

− u

1 − 0.01

−2.37(1−0.01)

− 0.01

=11.5

The relative change in gain is

G

 =

50 − 11.5

×100

= 77%.

In general, the gain of an APD can be any value between 1 and 10

. However the

standard APDs are mostly operated between gains of 10 and 1000. APDs of gains

higher than this are also used, though in situations where light level is extremely

low, such as in single photon counting experiments.

The mean internal gain of an APD is a function of the applied voltage since

the energy gained by the electrons between collisions depends on the electric ﬁeld

strength. The bias-gain curve of a typical APD is shown in Fig.6.5.25. It is apparent

from this ﬁgure that an APD can, in principle, be operated in a range of gains, which

makes it a versatile device that can be tuned according to the level of the incident

light.

200

400

6000

Reverse Bias (V)

Figure 6.5.25: Variation of gain

with applied reverse bias in a

typical APD.

In most applications, APDs are used in a region of the voltage-gain curve where

the relationship is approximately linear. In such a region the variation of gain with

410 Chapter 6. Scintillation Detectors and Photodetectors

voltage can be written as

= k

G %V

−1

, (6.5.59)

where V is the applied voltage and k

is a constant that depends on the APD

material and construction. Generally its value lies between 3 and 4.

Another parameter on which the gain depends heavily is the temperature. The

gain increases with decrease in temperature according to the simple relation

= −k

G %

−1

, (6.5.60)

where T is the temperature in Celcius and k

is a constant that depends on the type

of APD. It normally assumes a value between 2 and 3. The negative sign simply

signiﬁes the inverse relationship between the gain and the temperature.

The above equations imply that operating an APD at a ﬁxed temperature and

voltage would yield a ﬁxed gain. What these equations do not imply is that there

is no uncertainty associated with the gain. As a matter of fact, the biggest prob-

lem with APD gain is actually its inherent uncertainty and not its variation with

temperature and voltage. This uncertainty stems from the noisy nature of the car-

rier impact ionization process that multiplies the charges. Any measurement from

an APD is therefore subject to these ﬂuctuations as well, which of course can be-

come undesirable for high precision measurements such as single photon counting.

The ﬂuctuations in the APD gain has thus become an active area of research and

development.

Although some researchers prefer to work directly with the gain but most prefer

the gain ﬂuctuations that are characterized by the so called excess noise factor

deﬁned as

F =

G



G

, (6.5.61)

where G is the APD gain and  represents the mean value. A number of mathemat-

ical models exist that transform the above equation into a numerically computable

form (see, for example (23)). We will not go into that discussion but it is worth men-

tioning here that the excess noise factor and hence ﬂuctuations in the gain depend

 the mean APD gain,

 the ionization coeﬃcients of the charge carriers (electrons and holes), and

 the multiplication region.

The variation of the excess noise factor with the mean gain is the most profound

one among the three. Though some APDs behave a bit diﬀerently, but generally

there exists linear relationships between these two parameters, that is

F ≈ AG + B, (6.5.62)

where A and B are constants that depend on the APD material and construction.

Typically A has a value on the order of 10

−3

, while B assumes a value of around 2.

6.5. Photodetectors 411

C.3 Quantum Eﬃciency and Responsivity

The quantum eﬃciency of a photodiode detector is used to characterize its eﬃciency

of creating charges that contribute to the output signal. It is generally deﬁned by

the ratio

ξ =

rate of electron generation

intensity of incident photons

. (6.5.63)

Note that here the rate of electron generation refers to the electrons that contribute

to the output signal. Hence, if I

is the current in amperes that generates the output

signal and is generated by the incident photons, then we can write

Rate of electron generation =

−1

, (6.5.64)

with e being the electronic charge. I

is sometimes also referred to as the pho-

tocurrent. Since the photocurrent in an APD actually gets ampliﬁed by a factor G,

therefore we can also write the above expression as

Rate of electron generation =

out

Ge

−1

, (6.5.65)

where I

out

= GI

is the current measured at the APD output and G is the

mean APD gain. The second term in the deﬁnition of the quantum eﬃciency is the

intensity of incident photons. It can be calculated from the incident power P and

the mean wavelength λ of the incident photons through the relation

Intensity of incident photons =

Pλ

−1

. (6.5.66)

Hence the expression for the quantum eﬃciency can be written as

ξ =

out

/Ge

Pλ/hc

out

P G

eλ

G

eλ

, (6.5.67)

where R = I

out

/P is known as the responsivity of the photodetector. Responsivity

is a standard parameter that is often quoted by the APD manufacturers and is used

to compare the characteristics of diﬀerent APDs. Its generally quoted units are

amperes per watt. The expression for responsivity in terms of the eﬃciency can be

deduced from the above expression as

R =

Geξλ

, (6.5.68)

or in Standard units it can be written as

R =8.07 × 10

Geξλ AW

−1

. (6.5.69)

412 Chapter 6. Scintillation Detectors and Photodetectors

Example:

An APD having a gain of 20 and responsivity of 8 A/W is subject to a

photon beam of intensity 6.5 × 10

−1

. Calculate the quantum eﬃciency of

the APD at 1.6 µm. Also compute the average photocurrent.

Solution:

The eﬃciency can be calculated from equation 6.5.67.

ξ =

G

eλ



6.63 × 10

−34



2.99 × 10



(1.602 ×10

−19

)(1.6 × 10

−6

)

=0.31

To compute the average photocurrent I

we use the basic deﬁnition of respon-

sivity, that is

R =

out

Using I

out

= GI

this can be transformed into

P R

Rφhc

Gλ

where φ = Pλ/hc is the incident photon intensity. Substituting the given

values in the above equation yields

(8)



6.5 × 10



6.63 × 10

−34



2.99 × 10



(20) (1.6 × 10

−6

)

=3.2 × 10

−10

=0.32 nA.

C.4 Modes of Operation

Just like PMTs, APDs can also be operated in analog or digital modes. However

for APDs they are generally referred to as linear and Geiger modes.

We saw in the chapter on gas ﬁlled detectors that operating a detector in Geiger

region means applying a bias voltage high enough to cause breakdown in the gas.

This is also true for APDs operating in the Geiger mode. If a high enough reverse bias

is applied to the APD, it will cause a large current pulse whenever a photon produces

charge pairs in the bulk of the material. Hence in this way the detector can be used

as a photon counting device just like a PMT. The problem, however, is the dark

current in the bulk of the material, which is signiﬁcant even at room temperatures.

The trick, therefore, is to operate the detector at very low temperatures. Silicon

APDs have been found to have very low leakage or dark currents and therefore high

6.5. Photodetectors 413

photon counting capabilities. With a pulse resolution of as low as 20ps, a gain of

as high as 10

, and a quantum eﬃciency approaching 80%, the APDs are becoming

more and more popular in single photon counting applications.

C.5 Noise Considerations

APDs are generally operated at low light levels due to their signal ampliﬁcation

characteristics. Their sensitivity is therefore limited by their inherent noise. There

are diﬀerent sources of noise in an APD, the most important of which are listed

below.

 Leakage Current: An APD can have two types of leakage currents: surface

leakage current and bulk leakage current. The bulk leakage current, which ﬂows

in the bulk of the material, is aﬀected by the high electric ﬁeld in a manner

similar to the signal current, and hence gets ampliﬁed. We will represent the

gain associated with the bulk leakage current by G

to diﬀerentiate it from

the multiplication factor for the signal. If I

and I

are the surface and bulk

leakage currents respectively then the total leakage current I

in an APD having

a mean gain G can be expressed as

= I

+ G

I

. (6.5.70)

The contribution of surface leakage current to the total leakage current is quite

small and can therefore be safely ignored for most computations. Hence we

can write

≈G

I

. (6.5.71)

The ﬂuctuations of this leakage or dark current is given by the shot noise

formula

=2eI

BG



, (6.5.72)

where e is the unit electrical charge, B is the bandwidth of the system, and F

is the excess noise factor for the leakage current. Note that here we have made

use of the deﬁnition of the excess noise factor given earlier, that is

G



G



. (6.5.73)

The excess noise factor in terms of electron ionization rate α,depletionwidth

d, ionization rate ratio u, and gain factor G

is given by (see, for example (21))

(1 + αdG

)(1+uαdG

)

G



. (6.5.74)

Determination of the noise factor and the ﬂuctuations of the dark current

requires the knowledge of the multiplication factor or gain, which can be com-

puted from the relation (21)

G

 =

1 − e

−αd(1−u)

αd



−αd(1−u)

− u



. (6.5.75)

414 Chapter 6. Scintillation Detectors and Photodetectors

It is instructive to compare equations 6.5.57 and 6.5.74, which represent the

gains of signal and dark currents respectively. It can be shown that they are

related through the relation

G

 =

αd

[G

−1] . (6.5.76)

 Excess Noise: We have already introduced the term excess noise factor to

characterize the ﬂuctuations in the APD gain. This factor actually repre-

sents the so called excess noise that is related to the statistical nature of the

underlying particle interactions that take place during the process of charge

multiplication. Due to these ﬂuctuations the distribution of amplitudes of the

output current pulses assume a ﬁnite width even if the incident light can be

described by an impulse or delta function.

We have already introduced the expression for the excess noise factor for the

dark current (see equation 6.5.74. For the signal current due to electrons the

noise factor can be written as (41)

= uG

+



2 −

G



(1 − u)



, (6.5.77)

where, as before, u represents the hole to electron ionization rate ratio and

G

 is the mean signal gain.

 Thermal Noise: Thermal noise, though important, is not actually related to

the APD itself. It refers to the noise generated in the output circuitry due

to thermal agitations of the current carriers. During the discussion on PMT

noise, we introduced the term Johnson noise to characterize the thermal noise

generated in the preampliﬁer. The same argument can be applied for the case

of APDs as well. Hence the variance of output current due to thermal noise is

given by

eqv

, (6.5.78)

where F

is the noise ﬁgure for the noise source (the output circuitry), k

is the

familiar Boltzmann’s constant, T is the absolute temperature, B is the system

bandwidth, and R

eqv

is the equivalent impedance of the output circuit.

The signal-to-noise ratio can be determined by noting that the noise sources

described above are independent of each other, since this allows us to represent the

total ﬂuctuations or variance of the output current by

out

= σ

+ σ

, (6.5.79)

where the subscripts out, c, l,andt refer to the variances corresponding to the

output signal, charge carriers, leakage current, and thermal noise respectively. If

the output signal is due to the electrons, as is usually the case, then the ﬂuctuations

in the electron current can be written as

=2eI

BG



, (6.5.80)

6.5. Photodetectors 415

where B is the bandwidth of the system, G

 is the mean electron multiplication

factor, I

is the mean electron current, and F

is the excess noise factor for the elec-

tron current. To compute the signal-to-noise ratio we also need the signal current,

which for this case is simply given by I

G

 since I

is the signal current before

multiplication. The expression for the signal-to-noise ratio can now be written as

S/N =

G



out

G





+ σ

G





2eI

BG



+2eI

BG



+4F

TBR

−1

eqv

. (6.5.81)

The signal-to-noise ration depends on the mean gain which in turn has a depen-

dence on the ratio of hole and electron ionization rates u.InfactS/N is highly

sensitive to the value of u as can be deduced from Fig.6.5.26. This implicit de-

pendence on u puts a tough constraint on the value of gain that would yield high

signal-to-noise ratio. This is specially true for higher values of u. For small u,the

curve is more or less ﬂat after a certain gain and therefore gives some ﬂexibility in

terms of choosing the gain according to the particular application.

−3

u =

−2

u =

−1

u =

S/N

(

)

0.5

1.0

100

200

Figure 6.5.26: Dependence

of signal-to-noise ratio of an

APD on the mean gain at

diﬀerent values of the ratio

of the hole ionization rate to

the electrion ionization rate.

C.6 Radiation Damage

The radiation damage to semiconductor detectors has already been discussed in

the previous Chapter. The damage mechanisms in the APDs are the same as in

conventional semiconductor detectors. The overall eﬀect of the damage is, however,

more dramatic due to the higher sensitivity of these detectors. In an APD being

operated in a hostile radiation environment, the following two distinct types of

radiation induced damages can occur.