E.E. 823 & E.E 829

ELECTRICAL MATERIALS SCIENCE

PHOTODETECTORS: CALCULATION OF

SENSTIVITY OF PbO AS A DIRECT CONVERSION

MATERIAL IN X-RAY DETECTOR

Name: ADEAGBO EMMANUEL BAMISE

Student Number: 11192805

Date: APRIL 23, 2016.

Lecturer: Professor Safa Kasap

1.0 INTRODUCTION

Photodetectors convert an incident radiation to an electrical signal such as voltage or current. In many

cases such as photoconductors and photodiodes, the conversion process is typically achieved by the

creation of free electron hole pairs (EHPs) by the absorption of photons which creates electrons in

the conduction band (CB) and holes in the valence band (VB)[1]. This study will focus on the recent

method of image detection using direct conversion flat panel x-ray detection techniques (using PbO

as the photoconductor of study) as against x-ray film/screen cassettes [3] and to derive a generalized

expression for the charge carrier transport and absorption – limited sensitivity of x-ray

photoconductors by solving the continuity equation [2]. The basic principles of pn junction

photodetectors such as pn junction photodiode and pin photodiode are also examined. The direct

conversion flat panel x-ray sensors that are currently under development are based on the following

photoconductors; a-Se, PbO, PbI2, HbI2, CdZnTe and TlBr. Good x-ray images have been reported from

PbI2, HgI2,and CdTe based x-ray sensors.[3]

Detectors are characterized by their quantum detection efficiency, sensitivity, spatial resolution

properties, noise, dynamic range, and linearity of response [4] and for the purpose of this article,

the sensitivity of the photodetector will be the focal point of discussion where PbO is used as the case

study.

2.0 BASIC PRINCIPLES OF X-RAY DETECTORS

2.0.1 Pn Junction Photodiode The pn junction photodiode has a p+n type of junction which implies

that the acceptor concentration Na in the p-side is much greater than the donor concentration Nd in

the n-side. The p+ -side is generally very thin as shown in figure 2.1.

Figure 2.1 (b) shows the net space charge distribution across the p+ n junction. These charges are in

the depletion region, or in the space charge layer (SCL), and represent the exposed negatively charged

acceptors in the p+ -side and exposed positively charged donors in the n-side. The depletion region

extends almost entirely into the lightly doped n-side and, at most, it is a few microns. [1]

Because photodiodes are normally operated in reversed bias, the applied voltage Vr drops across the

highly resistive depletion layer width W and makes the voltage across the depletion rejoin to increase

from Vr to V0 + Vr where V0 is the built-in voltage. So when a photon with an energy greater than the

bandgap Eg is incident, it is absorbed and photogenerate a free EHP in the depletion region which is

separated by the field E and drift them in opposite direction until they reach the neutral regions.

Drifting carriers generate a current, called the photocurrent Iph, in the external circuit that provides

the electrical signal. The photocurrent lasts for the duration it takes for the electron and hole to cross

the SCL (width, W) and reach the neutral regions. The photocurrent Iph depends on the number of

EHP photogenerated and the drift velocity of the carriers while transiting through the depletion

region.

Figure 2.1 (a) A schematic diagram of a reverse biased pn junction photodiode. (b) Net space charge across the

diode in the depletion region. Nd and Na are the donor and acceptor concentrations in the p- and n-sides. (c) The

field in the depletion region. [1]

A high-speed p-n junction photodiode is usually constructed in such a way that most of the photons

are absorbed in the p-emitter region. The junction is placed as deep as possible so that efficient

separation of photogenerated electron–hole pairs can be achieved. This ensures that most of the

photocurrent is carried out by electrons whose speed, either by diffusion or drift, is always faster than

that of holes. The conditions for achieving excellent low-frequency response in a p-n junction

photodiode are that sWp/Dn < Wp/(Dnτn)1/2 < 1 and Wd/(vsτn) < 1, where s is the surface recombination

velocity, Wp is the width of the p region, Dn is the electron diffusion constant, τn is the electron lifetime

in the p region, Wd is the depletion layer width, and vs is the saturation velocity of electrons in the

depletion region.[5]

Figure 2.2 (a) A reverse biased pn junction. Photogeneration inside the SCL generates an electron and a hole. Both

fall in their respective energy hills (electron along Ec and hole along Ev), i.e., they drift, and cause a photocurrent Iph

in the external circuit. (b) Photogeneration occurs in the neutral region. The electron has to diffuse to the depletion

layer and then roll down the energy hill, i.e., drift across the SCL. (c) A shorted pn junction. The photogenerated

electron and hole in the SCL roll down their energy hills, i.e., drift across the SCL, and cause a current Iph in the

external circuit. (d) The pn junction in open circuit. The photogenerated electron and hole roll down their energy

hills (drift) but there is a voltage Voc across the diode that causes them to diffuse back so that the net current is zero.

Note that Ec and Ev are only shown in (a), and are self-apparent in (b)–(d).

2.0.1 Pin Photodiode

A p-i-n photodiode consists of a highly doped p+-emitter layer, a wide undoped intrinsic layer (i

region), and a highly doped n+-base layer. [4]

Figure 2.3 (a) Schematic diagram of a p-i-n photodiode, and (b) generation rate versus distance for two different

wavelengths, where λ1 denotes the short-wavelength photons, and λ2 the long-wavelength photons. [4]

Because the simple pn junction photodiode (Figure 2.1) has two major drawbacks which is that its

junction or depletion layer capacitance is not sufficiently small to allow photodetection at high

modulation frequencies. This is known as the RC time constant limitation. The depletion layer of pn

junction is narrow at most a few microns which implies that at long wavelengths where the

penetration depth is greater than the depletion layer width, the majority of photons are absorbed

outside the depletion layer where there is no field to separate and drift the EHPs. The QE is

correspondingly low at these long wavelengths. These problems are substantially reduced in the pin

(p–intrinsic–n-type) photodiode. [1]

Figure 2.4 (a). The schematic structure of an idealized pin photodiode. (b) The net space charge density across the photodiode.

(c) The built-in field across the diode. (d) The pin photodiode reverse biased for photodetection. [1]

The pin structure in its designed is such that photon absorption occurs over the i-Si layer. The

photogenerated EHPs in the i-Si layer are then separated by the field E and drifted toward the n+ and

p+ -sides, respectively, as illustrated in Figure 2.4 (d). While the photogenerated carriers are drifting

through the i-Si layer they give rise to an external photocurrent which is detected as a voltage across

a small sampling resistor R in Figure 5.9 (d). The response time of the pin photo-diode is determined

by the transit times of the photogenerated carriers across the width W of the i-Si layer. Increasing W

allows more photons to be absorbed, which increases the QE (Quantum Efficiency) but it slows down

the speed of response as carrier transit times become longer. A charge carrier photogenerated at the

edge on the i-Si layer has a transit time or drift time tdrift across the i-Si layer given by

tdrift = 𝑊

𝑣𝑑 (1)

where 𝑣𝑑 is it drift velocity. In order to increase the speed of response which is to reduce the drift

time, 𝑣𝑑has to be increased by increasing the applied field E. At high fields 𝑣𝑑 does not fol-low

the expected μdE behavior, where μd is the drift mobility, but instead tends to saturate at vsat, which

is of the order of 105 m s-1 at fields greater than 106 V m-1 in the case of Si. The variation of the drift

velocity of electrons and holes with the field in Si is given in figure 2.5 below. [1]

Figure2.5 Drift velocity vs. electric field for holes and electrons in S

As already mentioned, a distinct advantage of the pin photodiode is that it allows a wider spectral

range to be absorbed in the SCL in which the photogeneration takes place this consequently improves

the responsivity R which makes it generally better than the simple pn junction photodiode and can

be controlled by adjusting the width of the i-layer. Both Si and InGaAs pin photodiodes are widely

available in the market, covering a range of wavelength from around 300 nm to 1700 nm. Ge pin

photodiodes are also available but have higher dark currents, and usually have to be cooled. [1]

2.0.3 Avalanche Photodiode Avalanche photodiodes (APDs) are high-speed, high sensitive

photodiode utilizing an internal gain mechanism that functions by applying a reverse voltage. In

comparison to the pin photodiode, APDs can measure lower level light and are used in a wide variety

of applications requiring high sensitivity such as long-distance optical communications and optical

distance measurement.[6]

Figure 2.6 (a) A schematic illustration of the structure of an avalanche photodiode (APD) biased for avalanche gain. (b) The

net space charge density across the photodiode. (c) The field across the diode and the identification of absorption and

multiplication regions. [1]

Figure 2.6 show the structure of the APD. The n+ -side is thin and it is the side that is illuminated

through a window. There are three p-type layers of different doping levels next to the n+ -layer to

suitably modify the field distribution across the diode. The first is a thin p-type layer and the second

is a thick, lightly p-type doped (almost intrinsic) layer, called the p-layer, and the third is a heavily

doped p+ -layer. The diode is reverse biased to increase the fields in the depletion regions. The net

space charge distribution across the diode due to exposed dopant ions is shown in Figure 2.6 (b).

Under zero bias, the depletion layer in the p-region (between n+ p) does not normally extend across

this layer. But when a sufficient reverse bias is applied, the depletion region in the p-layer widens to

reach through to the p-layer (and hence the name reach-through). The field extends from the exposed

positively charged donors in the thin depletion layer in n+ -side, all the way to the exposed negatively

charged acceptors in the thin depletion layer in p+ -side. [1]

Figure 1.7 (a) A pictorial view of impact ionization processes releasing EHPs and the resulting avalanche multi-plication. (b)

Impact of an energetic conduction electron with crystal vibrations transfers the electron’s kinetic energy to a valence electron

and thereby excites it to the conduction band. (c) Typical multiplication (gain) M vs. reverse bias characteristics for a typical

commercial Si APD, and the effect of temperature. (M measured for a photocurrent generated at 650 nm of illumination.) [1]

The absorption of photons which leads to photogeneration, takes place mainly in the long π-layer.

The nearly uniform field here separates the electron–hole pairs and drifts them at velocities near

saturation toward the n+ - and p+ -sides, respectively. As the drifting electrons reach the p-layer, they

experience even greater fields and therefore acquire sufficient kinetic energy (greater than Eg) to

impact-ionize some of the Si covalent bonds and release EHPs. The impact ionization process can be

visualized as shown in Figure 5.7 (a) where an electron entering the avalanche region (width w) gains

energy from the field as it “drifts” in the opposite direction to the field, and its energy (which is kinetic

energy) increases with respect to Ec. Eventually, the energy gained from the field is sufficient to excite

an electron across the bandgap Eg as illustrated in Figure 5.7(b). These impact-ionization-generated

carriers are called secondary carriers. These secondary EHPs themselves can also be accelerated by

the high fields in this region to sufficiently large kinetic energies to further cause impact ionization

and release more EHPs, which leads to an avalanche of impact ionization processes. Thus, from a

single electron entering the p-layer one can generate a large number of EHPs, all of which contribute

to the observed photocurrent. The photodiode possesses an internal gain mechanism in that single

photon absorption leads to a large number of EHPs being generated. The photocurrent in the APD in

the presence of avalanche multiplication, therefore, corresponds to an effective quantum efficiency

in excess of unity.

The multiplication of carriers in the avalanche region depends on the probability of impact ionization,

which depends strongly on the field in this region and hence on the reverse bias Vr. The overall or

effective avalanche multiplication factor M, also known as the gain, of an APD is defined as

𝑀 =𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑝ℎ𝑜𝑡𝑜𝑐𝑢𝑟𝑟𝑒𝑛𝑡

𝑃𝑟𝑖𝑚𝑎𝑟𝑦 𝑢𝑛𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑝ℎ𝑜𝑡𝑜𝑐𝑢𝑟𝑟𝑒𝑛𝑡=

𝐼𝑝ℎ

𝐼𝑝ℎ𝑜 (2)

where Iph is the APD photocurrent that has been multiplied and Ipho is the primary or unmultiplied

photocurrent,

𝑀 =1

1−(𝑉𝑟

𝑉𝑏𝑟)

𝑚 (3)

where Vbr is the avalanche breakdown voltage and m is a characteristic index that provides the best

fit to the experimental data which is temperature dependent. [1]

Table 1 summarizes some typical characteristics of pn junction, pin, and APD photodiodes based on

GaAsP, Si, Ge, InGaAs, InAs, and InSb, covering the range from the ˝200 nm in the UV, based on GaP,

to ˝5 om in the infrared (InSb).

3.0 X-RAY PHOTOCONDUCTORS AND METRICS OF PERFORMANCE

The X-ray detector is the heart and the core of a digital mammography system. Its improved

characteristics of dynamic range and signal-to-noise ratio provide inherent advantages over screen-

film technology which is gradually fading out as a result of increase in the number of potential X-ray

photoconductors such as PbO, PbI2, HgI2, CdTe, etc. Indeed, good x-ray images have been reported

from PbI2, HgI2, and CdTe based x-ray sensors. Detector technologies used for digital mammography

can be distinguished by the acquisition geometry into scanning or full-field detectors, by energy

conversion mechanism into phosphor-based and non-phosphor-based detectors and by how the

detector signal is converted into an image value into signal-integrating and quantum-counting

systems.[4][3]

The detector is one of the defining features of a digital mammography system where it produces an

electronic signal that represents the spatial pattern of X-rays transmitted by the breast. The detector

is designed to overcome several of the limitations inherent in the screen-film image receptor used in

analog mammography, and in so doing, potentially provides improved diagnostic image quality and a

reduction of dose to the breast. The detector performs function in the following sequential

operations:

(a) Interaction with the X-rays transmitted by the breast and absorption of the energy carried by the

X-rays

(b) Conversion of this energy to a usable signal generally light or electronic charge

(c) Collection of this signal

(d) Conversion of light to electronic charge (in the case of phosphor-based detectors)

(e) Readout of charge, amplification, and digitization

The optimization of these operations give the detector to provide high-quality images at appropriate

dose levels. Detectors are characterized by their quantum detection efficiency, sensitivity, spatial

resolution properties in terms of modular transfer function (MTF), noise, dynamic range, and

linearity of response.[4]

However, for the purpose of this article, the sensitivity of the photodetector will be the focal point of

discussion and polycrystalline PbO is used as the case study.

Quantum detection efficiency:

This property describes the fraction of the X-rays falling on the detector, that interact with it,

producing at least some signal.[4] Because nearly all the incident x-ray radiation should be absorbed

within a practical photoconductor thickness to avoid unnecessary exposure of the patient to the

radiation. Over the energy range of interest, the linear attenuation coefficient α must be sufficiently

large to allow the incident photons to be attenuated inside the photoconductor. Put differently, the

x-ray attenuation depth δ, the reciprocal of α, must be substantially less than the photoconductor

layer thickness L. The fraction of incident photons in the beam that are attenuated by the

photoconductor depends on α of the photoconductor material and its thickness L; and is given by [7]

AQ = Attenuated fraction = [1 − exp(−αL)] (4)

where α = α (Eph, Z, ρ)is a function of photon energy Eph, atomic number Z and density ρ of the

material. AQ is called the quantum efficiency (QE) because it describes the efficiency with which the

medium attenuates photons. The attenuation depth δ is the distance into the detector where the

beam has been attenuated by 63%.[7]

Figure 3.1 [8]

Figure 3.1 shows that photoconductors with higher-Z components such as PbO, PbI2, HgI2, CdZnTe

have very good quantum efficiencies in the high energy range that covers such modalities as chest

radiography and angiography.[7]

Detective Quantum Efficiency (DQE)

This characteristic measures the ability of the detector to transfer signal relative to noise from its

input to its output. The random nature of image quanta gives rise to random fluctuations in image

signals, which contributes to image formation and hence creates random noises. The scattering of

image quanta gives rise to image blurring, which is quantified by the modulation transfer function

(MTF (f)).[8] A meaningful comparison of different photoconductive sensors must involve the

evaluation of DQE, as a function of spatial frequency, f, which is a task that is not trivial inasmuch as

we must be able to identify and quantify all noise contributions in the imaging chain from input to

the output. DQE is defined as

DQE (f) =𝑆𝑁𝑅𝑜𝑢𝑡

2 (𝑓)

𝑆𝑁𝑅𝑖𝑛2 (𝑓)

(5)

where SNRin and SNRout are the signal to noise ratio at the input and output stages of the detector,

respectively.[7]

The random nature of charge-carrier trapping which is the incomplete charge collection in the

photoconductor layer creates fluctuations in the collected charge and hence creates additional noise

in the photodetector. Thus carrier trapping de-grades signal-to-noise performance of the image and

reduces the DQE. But in order to improve the DQE performance of an X-ray image detector, high

conversion gain and high charge collection efficiency are required. Figure 3.2 (a) shows DQE (0) as a

function of X-ray exposure for a-Se, HgI2, and CZT detectors for a 60-keV X-ray beam. The X-ray

exposure (X) is varied from 0.1 µR to 10µR, which is the range of X-ray exposure for fluoroscopic

applications. It is assumed that the pixel area, A = 200 µm × 200 µm, and the effective fill factor is 1.0

for all types of photoconductors. The average E is 60 keV and the additive electronic noise is assumed

to be 2000 electrons per pixel.[8]

However, figure 3.2 (b) shows the results of the calculations and the comparison with the

experimental DQE vs spatial frequency f data. It should be noted that the symbols in the figure are

the experimental points reported by Simon and coworkers and the solid line is the calculations based

on a cascaded linear system model with further details in [9, 10].

(a) (b)

Figure 3.2 (a) DQE (0) versus X-ray exposure for a-Se, poly-HgI2, and poly-CZT detectors and for a 60-keV monoenergetic X-

ray beam. The electronic noise is 2000 e per pixel. It is assumed that F = 10 V/µmfora-Se,0.5V/µm for HgI2 and 0.25 V/µm for

CZT [8] (b) DQE(f) vs. spatial frequency f for a negatively biased PbO photoconductive x-ray sensor at three different applied

fields, F = 0.5 V µm−1, 1 V µm−1, and 2 V µm−1. The symbols are the experimental points reported by Simon and coworkers

and the solid line is the calculations based on a cascaded linear system model; further details may be found in [9, 10]

Modulation Transfer Function (MTF)

Resolution or resolving power is the ability to record separate images of small objects that are placed

very closely together. The overall resolution of a system can be expressed as a convolution of the

component resolutions. However, the spatial resolution of an imaging device or a system can also be

described in terms of the MTF, which is the relative response of the system as a function of spatial

frequency or the efficiency of an imaging system to resolve (transfer) different spatial frequencies of

information in an image.[8][7]

𝑀𝑇𝐹 (𝑓) =𝐼𝑚𝑎𝑔𝑒 𝑐𝑜𝑛𝑡𝑟𝑎𝑠𝑡 𝑎𝑡 𝑡ℎ𝑒 𝑜𝑢𝑡𝑝𝑢𝑡 𝑎𝑡 𝑠𝑝𝑎𝑡𝑖𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 (𝑓)

𝐼𝑚𝑎𝑔𝑒 𝑐𝑜𝑛𝑡𝑟𝑎𝑠𝑡 𝑎𝑡 𝑡ℎ𝑒 𝑖𝑛𝑝𝑢𝑡 𝑎𝑡 𝑠𝑝𝑎𝑡𝑖𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 (𝑓) (6)

The MTF (f) and the DQF are related by [11]

𝐷𝑄𝐸 (𝑓) = 𝑞0𝐺2 𝑀𝑇𝐹2(𝑓)

𝑁𝑃𝑆(𝑓) (7)

where NPS (f) is the noise power density spectrum of the output image, 𝑞0 is the average quanta

incident onto the detector per unit area, and G is the detector gain.

The type of carrier that is trapped has effect on the MTF and it depends on whether the carriers

moving towards the top (radiation receiving) or bottom (pixelated) electrode are trapped. Charge

trapping effects on the MTF of large area photoconductive detectors have been studied in detail by

a number of researchers with applications to a-Se, CdZnTe and PbO based detectors [10,12,13,14]

Figure 3.3. The effects of charge trapping on the resolution (MTF) depends on the type of carriers that have been trapped; whether

carriers were drifting to the top or bottom electrode. C is the central (reference) pixel and L and R are the neighboring left and

right pixels. The transient currents flowing into the pixels are integrated and eventually yield the collected charges at the pixels.

[7]

Figure 3.4 Measured presampling MTF vs. f of a polycrystalline CdZnTe detector in comparison with a calculated MTF in which

deep trapping of charge carriers is included in the model. Blurring due to charge carrier trapping in the bulk of the photoconductor

cannot be neglected. The detector thickness is 300μm and the pixel pitch is 150μm. After [14]. Data from [15].

The over-all MTF (or presampling MTF) of an image detector can be expressed as,

MTF (f) = MTFm (f) × MTFa(f) (8)

where MTFm(f) is the modulation transfer function of the detector material and MTFa(f) is the

modulation transfer function associated with the aperture function of the pixel electrodes. MTFa (f)

arises due to averaging of the signal over a pixel area. If the aperture is square with dimension a,

then, MTFa (f) will be of the form sinc (af). The aperture MTF describes how spatial frequencies are

passed through the detector elements.

An analytical expression for the MTF as a result of distributed carrier trapping in the bulk of the

photoconductor have been developed by Kabir and Kasap [16] and the effect of the charge-carrier

trapping on the resolution of direct-conversion active matrix flat panel imagers (AMFPI) has also been

examined. The effect of trapping on MTF increases with decreasing normalized carrier lifetime which

is the normalized schubweg. Trapping of the carriers that move towards the pixel electrodes degrades

the MTF performance, whereas trapping of the other type of carriers, which move away from the

pixels, improves the sharpness of the X-ray image. [8]

A case study involving a polycrystalline CdZnTe detector where the MTF has been measured [15] as

shown in figure 3.4. The photoconductor thickness was 300 μm and the pixel size was 150 μm; the

Nyquist frequency is 3.3 lp/mm. The operating electric field is 0.25 V/μm with the receiving electrode

biased positively. The hole and electron ranges were adjusted until the model matched the MTF

measurements. The best fit μτ products of electrons and holes are μeτe = 2.4 × 10−4 cm2/V and μhτh =

3.2 10−6 cm2/V, which are not too different than typical μτ values reported in the literature [17,18].

In fact, Mainprize et al. [17] reported a value of μeτe = 2.4 ×10−4 cm2/V for polycrystalline CZT by

modeling the charge collection efficiency (not MTF), which is in remarkable agreement with the value

for μeτe from MTF modeling even though the two samples are different.

Sensitivity

Sensitivity is defined as the collected charge per unit area per unit exposure of radiation. If Q is the

amount of charge collected when the incident radiation exposure is X and A is the radiation receiving

area, then the sensitivity S is defined as [3]

𝑆 =𝑄

𝐴𝑋 (9)

This is considered an important performance measure for a superior image. High S permits the use

of low detector radiation-exposure levels which also increases the dynamic range of the active matrix

flat panel imager (AMFPI). The selection of the X-ray photoconductor is highly influenced by the value

of S which can be incident radiation that is useful for the generation of electron–hole pairs (EHPs),

which is characterized by the quantum efficiency AQ of the detector and depends on the value of α

of the photo-conductor and L through AQ = 1 − e−αL, where the value of α is X-ray photon-energy

dependent. Also, the generation of EHPs by X-ray interactions, which is characterized by the value of

W± of the photoconductor and the average absorbed energy (Eab) per attenuated X-ray photon of

energy E, where W± depends on the material properties of the photoconductor, and Eab depends on

the incident X-ray photon energy [22] and the material properties. The number of X-ray generated

charge is actually collected in the external circuit and is characterized by the charge-carrier drift

mobilities (µ) and lifetimes (τ), the applied F and L. [8]

Considering a high resistivity semiconductor as a model. A photoconductor that has been

sandwiched between parallel plate electrodes and biased with a voltage V across the terminals to

establish an electric field F as shown in Figure 3.5 with the assumption that the thermal equilibrium

concentration of charge carriers is negligibly small. [3]

Figure 3.5 Electron and hole concentration profiles, n’(x’,t’) and p’(x’,t’), respectively, due to bulk photogeneration

and subsequent drift of injected carriers.

Considering a monoenergetic beam. The radiation energy passing through the photoconductor

decreases exponentially as x-ray photons are absorbed and scattered in the photoconductor material

as shown in figure 3.6. The radiation energy through the semiconductor decreases exponentially as

E’(x) = E exp (−αx) (10)

where E is the incident energy at x = 0 and α is the x-ray attenuation coefficient that depends on the

photon energy. [20]

Figure 3.6. The radiation energy E’ decreases exponentially throughout the sample and causes electron–hole pairs to be

generated in the same exponential manner. The radiation absorbed within δx generates an amount of charge δq. Consider

an electron and a hole generated at x and drifting under the influence of the field F. [20]

In order to derive the general expression for the sensitivity from the solution of the continuity

equation, the following assumptions were made:

i. The field remains relatively uniform, which is a valid assumption for small signal operation.

ii. The diffusion of carriers is negligible compared with their drift

iii. We assign a constant drift mobility μ and a single deep trapping time τ’ to each species of

carriers.

iv. The loss of carriers by deep trapping is more significant than bulk recombination. [3]

v. The photoconductor is exposed to a monoenergetic pulse of x-ray radiation that has a very

short duration compare to the charge carrier transit times across the sample thickness.

vi. Small pixel effects are neglected so that the collected charge corresponds to the integration

of the total photocurrent.[2]

The continuity equation for electrons with these assumptions considering figure 3.5 is given as

𝜕𝑛(𝑥,𝑡)

𝜕𝑡= −𝜇𝑒𝐹

𝜕𝑛(𝑥,𝑡)

𝜕𝑥−

𝑛(𝑥,𝑡)

𝜏𝑒 (11)

where 𝑛(𝑥, 𝑡) is the concentration of electrons at location x at time t, and F is the applied field V/L. If

E is the x-ray photon energy and N0 is number of x-ray photons per unit area incident on the

photoconductor area A, then, αenN0Eexp(-αx)/W± is the initial hole or electron concentration at

location x. The solution of Eq. 11 is a drifting electron distribution that is given by

𝑛(𝑥, 𝑡) = B exp[−𝛼(𝑥)−𝜇𝑒𝐹𝑡] exp(−𝑡/𝜏𝑒); −𝜇𝑒𝐹𝑡˂𝑥˂𝐿

𝑛(𝑥, 𝑡) = 0; 𝑥˂ μe𝐹𝑡 and 𝑥˃𝐿, (12)

where 𝐵 =𝛼𝑒𝑛𝐸𝑁0

𝑊± (13)

The collected charge in the external circuit due to electron drift is

𝑄𝑒 =𝑒𝐴𝜇𝑒𝐹

𝐿∫ ∫ 𝑛(𝑥, 𝑡)𝑑𝑥𝑑𝑡,

𝐿

𝜇𝑒𝐹𝑡

𝐿 𝜇𝑒𝐹⁄

𝑡=0 (14)

Solving equation (14) gives

𝑄𝑒 =𝑒𝐴𝐵𝜇𝑒𝐹𝜏𝑒

𝐿 [(1 − 𝑒−𝛼𝐿) +

1

(1 𝛼𝜇𝑒𝜏𝑒𝐹)⁄(𝑒−𝐿 𝜇𝑒𝜏𝑒𝐹⁄ − 𝑒−𝛼𝐿)], (15)

where e is the elementary charge. Similarly, we can calculate the collected charge Qh due to hole drift.

The total collected charge, Q = Qh + Qe.

The incident x-ray photons per unit area N0 is related to x-ray exposure X by [23]

𝑁0 =5.45×1013𝑋

𝐸(𝛼𝑎𝑖𝑟 𝜌𝑎𝑖𝑟⁄ ) (16)

where αair and ρair are the energy absorption coefficient and the density of the air, respectively.

Equation (16) is achieved by the relationship between the beam energy per unit area per unit

roentgen (that is the energy fluence of the incident radiation)Ψ, and the incident energy E given

as 𝐸 = Ψ𝐴𝑋, where X is the exposure in roentgens. The energy fluence can be readily evaluated from

the definition of 1 roentgen and the mass energy absorption coefficient αair /ρair of air via (equation

(7-42) in Johns and Cunningham 1983). [20]

Ψ =0.00873 JKg−1R−1

(𝛼𝑎𝑖𝑟 𝜌𝑎𝑖𝑟⁄ )=

5.45×1013eVg−1R−1

(𝛼𝑎𝑖𝑟 𝜌𝑎𝑖𝑟⁄ ) (17)

So it follows that 𝑁0 =Ψ

𝐸=

1

𝐴𝑋 from the expression in equation (16)

Total collected charge is

𝑄 = 𝑄0 {𝜇𝑒𝐹𝜏𝑒

𝐿 [(1 − 𝑒−𝛼𝐿) +

1

(1 𝛼𝜇𝑒𝜏𝑒𝐹)−1⁄(𝑒−𝐿 𝜇𝑒𝜏𝑒𝐹⁄ − 𝑒−𝛼𝐿)] +

𝜇ℎ𝐹𝜏ℎ

𝐿 [(1 −

𝑒−𝛼𝐿) −1

(1 𝛼𝜇ℎ𝜏ℎ𝐹)+1⁄(1 − 𝑒−𝛼𝐿−𝐿 𝜇ℎ𝜏ℎ𝐹⁄ )]}, (18)

where 𝑄0 =5.45×1013×𝑒𝐴𝑋

(𝛼𝑎𝑖𝑟 𝜌𝑎𝑖𝑟⁄ )𝑊±(

𝛼𝑒𝑛

𝛼) (19)

Equation (18) is identical to the statistical derivation given by Nemirovsky.6 If W± is in eV, αair and ρair

is in cm2 g-1, X is in roentgens (R) and A in cm2, then collected charge Q is in Coulomb (C). Q0 is the

maximum collected charge that would arise if all the incident radiation were absorbed and all the

liberated carriers were collected. [2]

From equations (18) and (19) using the normalized parameters, Δ = normalized attenuation depth =

1/(αL), xe = electron schubweg per unit thickness, μeτeF/L and, xh = hole schubweg per unit thickness,

μhτhF/L. Here, μe(h) and τe(h) are the mobility and deep trapping time (lifetime) of electrons (holes),

respectively, the x-ray sensitivity of the photoconductor can be conveniently expressed as follows,

𝑆 = 𝑆0 {𝑥𝑒 [(1 − 𝑒−1/∆) +1

(∆ 𝑥𝑒)−1⁄(𝑒−1/𝑥𝑒 − 𝑒−1/∆)] + 𝑥ℎ [(1 − 𝑒−1/∆) −

1

(∆ 𝑥ℎ)+1⁄(1 − 𝑒−1/∆−1/𝑥ℎ)]}, (20)

where 𝑆0 =5.45×1013×𝑒

(𝛼𝑎𝑖𝑟 𝜌𝑎𝑖𝑟⁄ )𝑊±(

𝛼𝑒𝑛

𝛼) (21)

The x-ray sensitivity is commonly expressed in C cm-2 R-1. If the bias polarity is reversed, then xe and

xh must be interchanged in Equation (20). The term S/S0 represents the charge transport and

absorption-limited normalized sensitivity.

The general expression derived for the sensitivity is applied to estimate the sensitivity of PbO using

values of the properties of PbO derived from previous works done on the photoconductor material.

The sensitivity calculations were carried out using equation (20) for a monoenergetic beam of x-rays

at (a) 20 keV and (b) 60 keV, to very approximately represent mammographic and chest radiology

applications. The values for the mass energy absorption coefficient of air and mass energy absorption

and linear attenuation coefficients for PbO were obtained from the website of the National Institute

of Standards and Technology Physics Laboratory [24] and [25]. The density of PbO, ρ = 9.6gcm−3, was

used to calculate the actual energy absorption and linear attenuation coefficients (αen and α,

respectively) for PbO with thickness L = 200μm. All values apply at NTP (20 ◦C and 101.3 kPa).

Selected properties of PbO

Eg = 1.9eV

W± = 8 – 20 eV [8]. 8.20eV was used as reported by F. Lappe [26]

μeτe = 3.5×10-7 cm2/V

μhτh ≈ 10-8 cm2/V

(a)

(b)

(c) (d)

(e)

(f) (g)

(a) X-ray sensitivity (S) versus hole schubweg per unit thickness xh and normalized attenuation depth (Δ) for 20 keV photon

(b) X-ray sensitivity (S) versus electron schubweg per unit thickness xe and normalized attenuation depth (Δ) for 20 keV photon

(c) Sensitivity at room temperature versus electric field

(d) Variation of X-ray sensitivity (S) with electric field and thickness for 20 keV photon

(e) X-ray sensitivity (S) versus detector thickness (L) and normalized attenuation depth (Δ) for 20 keV photon energy assuming

no electron and hole trapping.

(f) Sensitivity at room temperature versus electron schubweg per unit thickness xe

(g) Sensitivity at room temperature versus electron schubweg per unit thickness xh

3.0.1 Ideal Photoconductor Properties

The metric of performance of detectors has been discussed so far with much emphasis on the

sensitivity where an expression for the amount of collected charge per unit incident radiation, defined

as the x-ray sensitivity S, in terms of W±, α, αen, and the normalized parameters: normalized

attenuation depth and electron and hole schubwegs per unit thickness. [3] It is also necessary to

establish that the performance of direct-conversion X-ray detectors depends critically on the

selection and design of the photoconductor. It is therefore instructive to identify what constitutes a

nearly ideal X-ray photoconductor to guide a search for improved performance or a better material.

Therefore, an ideal photoconductor layer should possess the following material properties as

discussed in [8]:

1. High quantum efficiency (η): An ideal photoconductor must possess a high quantum efficiency

because most of the incident X-ray radiation should be absorbed within a thickness of the

photoconductor to avoid unnecessary patient exposure. This means that, over the energy range

of interest, the absorption depth δ of the X-rays must be substantially less than the device layer

thickness L.

2. High intrinsic X-ray sensitivity: The photoconductor must be able to generate as many

collectable (free) electron–hole pairs (EHPs) as possible per unit of incident radiation which

means the amount of radiation energy required, denoted by W±, to create a single free electron–

hole pair must be as low as possible. Typically, W± increases with the band gap Eg of the

photoconductor [19] and thus a low Eg is desired for maximum X-ray sensitivity.

3. Bulk recombination of electrons and holes as they drift to the collection electrodes: Bulk

recombination is proportional to the product of the concentration of holes and electrons, and

typically it is negligible for clinical exposure rates.

4. Negligible deep trapping of EHPs: Deep trapping of EHP means that for both electrons and holes,

the schubweg defined as µτF >> L where µ is the drift mobility, τ is the deep-trapping time

(lifetime), F is the electric field and L is the photoconductor layer thickness. The schubweg is the

mean distance a carrier drifts before it is trapped and unavailable for conduction. The temporal

responses of the X-ray image detector, such as lag and ghosting, depend on the rate of carrier

trapping.

5. Negligible diffusion compared to drift of the carrier: This will ensure less time for lateral carrier

diffusion leading to better spatial resolution.

6. Small dark current: The dark current serves as a source of noise in the photoconductor so it must

be kept as low as possible. This implies that the contacts to the photoconductor should be non-

injecting and the rate of thermal generation of carriers from various defects or states in the band

gap should be negligibly small.

7. Relationship between carrier transit time and image read out time: The image read out time

and the inter-frame time must be longer than the longest carrier transit time, which depends on

the smallest drift mobility.

8. Negligible X-ray fatigue and X-ray damage: This means that the properties of the

photoconductor should not change with time because of repeated exposure to X-rays. [8]

4.0 CONCLUSION

A brief study on the conversion of radiation to an electrical signal such as voltage or current typically

achieved by the creation of free electron hole pairs by the absorption photon in photodetectors such

as photodiodes and photoconductors has been carried out. The basic principles of photogeneration

in pn junction, pin and avalanche photodiode with their respective advantages and applications have

also been examined. However the metric of performance of X-ray detectors used in flat panel X-ray

imagers with their effect on the quality of X-ray images in mammography as reported by several

researchers has also been re-examined in the course of this study.

An analytical solution of the continuity equation by considering the drift of electrons and holes in the

presence of deep traps under the situation of exponentially decaying distribution of electron–hole

pair generation across the thickness of the photoconductor. The X-ray sensitivity of photoconductors

was considered in terms of the following combined effects: (i) Absorption of x-rays, controlled by the

linear attenuation coefficient α(E) and energy absorption coefficient αen(E), both x-ray photon energy

E dependent. (ii) Electric field F and x-ray photon energy dependent ionization of the medium, that

is, in terms of the electron and hole creation energy W± (E, F). (iii) The transport and trapping of

charges across the photoconductor as they drift to the collecting electrodes. (iv) The electron and

hole pairs are generated with an exponentially decaying distribution across the thickness of the

photoconductor. An expression for the x-ray sensitivity S in terms of W±, α, αen, and the normalized

parameters was obtained as in the case of Zahangir and Kasap [3]. Using the expressions obtained

and published experimental data the sensitivity of PbO has been calculated and discovered to depend

strongly on the carrier transport properties.

It was observed that if the material properties of the PbO is improved to enhance the carrier lifetime,

it would be an excellent candidate for direct conversion flat panel imaging detectors.

5.0 REFERENCES

[1] S. O. Kasap, Optoelectronics and Photonics: Principles and Practices. Pearson Education, Inc.,

2012.

[2] M. Z. Kabir and S. O. Kasap, “Charge collection and absorption-limited sensitivity of x-ray

photoconductors: Applications to a-Se and HgI2,” Appl. Phys. Lett., vol. 80, no. 9, pp. 1664–

1666, 2002.

[3] M. Zahangir Kabir and S. O. Kasap, “Sensitivity of x-ray photoconductors: Charge trapping and

absorption-limited universal sensitivity curves,” J. Vac. Sci. Technol. A, vol. 20, no. 2002, p.

1082, 2002.

[4] M. J. Yaffe, “Digital Mammography,” Technol. Cancer Res. Treat., vol. 3, no. 4, pp. 13–31,

2010.

[5] U. of F. Sheng S. Li ( Department of Electrical and Computer Engineering, Semiconductor

Physics Electronics, 2nd ed., no. 1. Gainesville: Spinger, 2014.

[6] Hamamatsu, “Characteristics and use of Si APD (Avalanche Photodiode),” p. 12, 2004.

[7] S. Kasap, J. B. Frey, G. Belev, O. Tousignant, H. Mani, J. Greenspan, L. Laperriere, O. Bubon, A.

Reznik, G. DeCrescenzo, K. S. Karim, and J. A. Rowlands, “Amorphous and polycrystalline

photoconductors for direct conversion flat panel x-ray image sensors,” Sensors, vol. 11, no. 5,

pp. 5112–5157, 2011.

[8] S. Kasap and P. Capper, “Photoconductors,” Springer Handb. Electron. Photonic Mater., pp.

1121–1137, 2006.

[9] Simon, M.; Ford, R.A.; Franklin, A.R.; Grabowski, S.P.; Mensor, B.; Much, G.; Nascetti, A.;

Overdick, M.; Powell, M.J.; Wiechert, D.U. Analysis of lead oxide (PbO) layers for direct

conversion X-ray detection. IEEE Trans. Nucl. Sci. 2005, 52, 2035-2040.

[10] Kabir, M.Z. Effects of charge carrier trapping on polycrystalline PbO x-ray imaging detectors J.

Appl. Phys. 2008, 104, 074507-074516.

[11] Shaw, R. The equivalent quantum efficiency of the photographic process. J. Photogr. Sci. 1963

11, 199-204

[12] Kabir, M.Z.; Chowdhury, L.; DeCrescenzo, G.; Tousignant, O.; Kasap, S.O.; Rowlands, J.A. Effect

of repeated x-ray exposure on the resolution of amorphous selenium based x-ray imagers.

Med. Phys. 2010, 37, 1339-1349.

[13] Kabir, M.Z.; Kasap, S.O.; Zhao, W.; Rowlands, J.A. Direct conversion x-ray sensors: Sensitivity,

DQE and MTF. IEEE Proc. G Circ. Device. Syst. 2003, 150, 258-266.

[14] Kabir, M.Z.; Kasap, S.O. Modulation transfer function of photoconductive x-ray image

detectors: effects of charge carrier trapping. J. Phys. D Appl. Phys. 2003, 36, 2352-2358.

[15] Tokuda, S.; Kishihara, H.; Adachi, S.; Sato, T.; Izumi, Y.; Teranuma, O.; Yamane, Y.; Yamada, S.

Large-area deposition of a polycrystalline CdZnTe film and its applicability to x-ray panel

detectors with superior sensitivity. Proc. SPIE 2002, 4682, 30-35.

[16] M. Z. Kabir, S. O. Kasap: J. Phys. D: Appl. Phys. 36 2352 (2003)

[17] Mainprize, J.G.; Ford, N.L.; Yin, S.; Gordon, E.E.; Hamilton, W.J.; Tümer, T.O.; Yaffe, M.J A

CdZnTe slot-scanned detector for digital mammography. Med. Phys. 2002, 29, 2767-2772.

[18] Eisen, Y.; Shor, A. CdTe and CdZnTe room-temperature X-ray and gamma ray detectors and

imaging systems. Mater. Res. Soc. Symp. P 1997, 487, 129-132.

[19] S. O. Kasap, J. A. Rowlands: Optoelectronics and Photonics: Principles and Practices

(Prentice–Hall Upper Saddle River, New Jersey 2001)

[20] S. O. Kasap, “X-ray sensitivity of photoconductors: application to stabilized a-Se,” J. Phys. D. Appl.

Phys., vol. 33, no. 21, pp. 2853–2865, 2000.

[21] O. Semeniuk, “Lead Oxide ( PbO ) for direct conversion fluoroscopic detectors,” 2012.

[22] J. M. Boone: Handbook of Medical Imaging,Vol.1, ed. by J. Beutel, H. L. Kundel, R. L. Van Metter (SPIE,

Washington 2000) Chap. 1 and references therein

[23] J. M. Boone, in Handbook of Medical Imaging, edited by J. Beutel, H. L. Kundel, and R. L. Van Metter,

(SPIE Press, Washington, 2000), Vol. 1, Chap. 1.

[24] http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html

[25] H. Buhr, L. Büermann, M. Gerlach, M. Krumrey, and H. Rabus, “Measurement of the mass energy-

absorption coefficient of air for x-rays in the range from 3 to 60 keV.,” Phys. Med. Biol., vol. 57, no. 24,

pp. 8231–47, 2012.

[26] F. Lappe, “The energy of electron-hole pair formation by X-rays in PbO,” J. Phys. Chem. Solids, vol. 20,

no. 3–4, pp. 173–176, 1961.