Uri Lachish, *guma *science, Rehovot, Israel

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The Sun and the Moon a Riddle in the Sky

**Abstract**

Proper circuit adjustment improves the gamma resolution of
a single pixel of a detector array to the noise limit. Steady flow of gamma
generated charge, in a semiconductor detector array, induces fast signal
build-up in a single pixel, as the charge arrives near its contact. The detector
circuit response is adjusted to a short shape time of a spectroscopy amplifier
that overlaps with the build-up period. The shape time determines a distance
range, extending from the negative contact, where the detector signal does not
depend on the position of gamma generation. The adjusted detector circuit is not
sensitive to hole trapping and the gamma energy resolution is improved to the
noise limit.

**1. Introduction **

The amount of charge generated by a gamma photon
in a semiconductor detector is proportional to its energy. Therefore, the energy
is measurable by full charge collection. Hole trapping, in deep levels within
the forbidden band, is a common problem observed with semiconductors. Partial
charge collection leads to signal dispersion and low resolution of the gamma
energy.

The operation of a single pixel in a detector array is less sensitive to holes trapping, compared to a separate single detector, according to the small pixel theory [1, 2]. The electrons' contribution to the pixel's signal is dominating over the holes' contribution, and the resolution improves with reduction of the aspect ratio of the pixel side to its thickness.

This work presents a way to further improve the detector resolution up to the noise limit. The small pixel theory is discussed and the small pixel effect is utilized together with proper adjustment of a standard detector circuit.

**2. Small pixel**

Consider a detector array of thickness *d*,
biased by a voltage *V*, with common negative contact at *x* = 0, and
segmented positive contact at *x* = *d*. Fig. 1 shows a charge
*q* within a square pixel of side *a*, at a point *x*_{0}
between the contacts. *q* induces charge distribution of radial symmetry
within the contacts.

**Figure 1:** Semiconductor detector array with common negative contact at
*x* = 0, and segmented positive contact at *x* = *d*, and
electronic circuit of a pixel. A charge *q*, located at a point
*x*_{0} within a square pixel of side *a*, induces charges on
the contacts. The circuit includes charge sensitive preamplifier, spectroscopy
amplifier, and a multi channel analyzer (MCA).

The charge *q* induces a charge distribution
*qw*(*x*_{0}, *y*, *z*) on the positive contact
calculated by the method of images [3]:

(1)

where *y* and *z* are the plane coordinates of the contact. The
weight function *w*(*x*_{0}, *y*, *z*) is the
induced charge distribution on the positive contact by a unit charge within the
detector.

The charge *h'*(*x*_{0}) induced over all the positive
contact is calculated directly [4, 5]:

*h'*(*x*_{0})* = qx*_{0}/*d*
(2)

The charge *h*(*x*_{0})on a single pixel is:

(3)

Fig-2 shows the charge *h*(*x*_{0}), 0 <
*x*_{0} < *d*, for *a*/*d* = 0.4, calculated by
equation (3).

When the charge *q* moves from a position *x*_{1} to a
position *x*_{2} within the detector, it will induce electric
charge flow in each pixel circuit given by:

*e*(*x*_{1}, *x*_{2}) =
*h*(*x*_{2}) - *h*(*x*_{1}) (4)

This flow is determined by the initial and final positions of the moving
charge and not by the specific trajectory of the movement, and it can be
calculated by applying (1), (3). Thus, the charge flow and build up in the
external circuit of a pixel (4) is related to the movement of a charge within
the detector bulk by applying the weight function *w*(*x*_{0},
*y*, *z*) (1).

**Figure 2:** The charge *h*(*x*_{0}), induced at a
pixel by a charge *q*, located at a position *x*_{0} (equation
(3)), for *a*/*d* = 0.4. The same functional dependence presents
charge flow at the pixel circuit, induced by hole and electron movement within
the detector. Shifting the coordinates' origin to *x*_{0} relates
to gamma generation of hole-electron pairs at *x*_{0}. The upward
and rightward pointing coordinates relate to flow induced by electrons, and the
leftward and downward pointing coordinates relate to the flow induced by the
holes. The hole mobility is 10% of the electron mobility. The charge unit is the
gamma generated charge *q*. The time unit *t*_{e} is the
electron transition time from contact to contact.

A gamma photon absorbed at *x*_{0} generates equal amounts of
holes and electrons. These charges will separate efficiently at
*x*_{0} if the external electric field is stronger than their own
electric fields, and move towards the opposite contacts. They move at a constant
speed, *v *= μ * _{i}V*/

The same functional dependence of Fig. 2 presents the charge flow at the
pixel circuit, induced by the hole and electron movement within the detector.
Shifting the coordinates' origin to *x*_{0} relates to gamma
generation of hole-electron pairs at *x*_{0}.

The flow may be presented either vs. the position of the moving charges, or
vs. time, since the charges move at a constant speed. It is assumed that the
hole mobility is 10% of the electron mobility. The electron transition time from
contact to contact *t*_{e} = *d ^{2}*/μ

The upward and rightward pointing coordinates of fig. 2 relate to flow induced by electrons, and the leftward and downward pointing coordinates relate to the flow induced by the holes.

Fig. 2 shows that the electrons mainly contribute to the charge build-up when they arrive near the pixel contact. The holes' contribution is distributed over a number of pixels.

The charge build-up induced over all the positive contact, *q
*(*x*(*t*) - x_{0})/*d*, (equation (2)), is equal
to the integral flow at any cross section of the detector circuit. Since all the
positive contact is at the same potential, there will be also lateral charge
flow between adjacent pixels, via the circuit connections. This converging
radial flow, towards the arrival point of the gamma charge, is the origin of the
small pixel effect.

Assume now that the holes are stationary. The holes stay at
*x*_{0} and the electrons flow all the way to *d*. The induced
charge build-up at the pixel circuit, when the electrons arrive to the pixel,
is:

*e*(*x*_{0}*, d*) = *q* -
*h*(*x*_{0}) (5)

since *h*(*d*) = *q* by (3).

Fig-3(a) shows the dependence of this charge on the position*
x*_{0} of gamma generation. The line shape is the inverted line of
fig-2.

**Figure 3:** (a) The charge step, *e*(*x*_{0},
*d*) (equation (4)), at a pixel contact, induced by gamma charge flow from
the point of photon absorption,* x*_{0}, to the positive contact at
*d*. (b) The pulse height, of a spectroscopy amplifier output, that
corresponds to the input (a).

Repetition of gamma generation events many times, will yield spectral signal
distribution. The distribution may be expressed in terms of the charge
distribution *n*(*e*) or the absorption depth spatial distribution,
*n*(*x*_{0}).

**Figure 4:** The number of events in the distance interval Δ*x*
is equal to the number of events in the charge
interval Δ*e*.

Fig. 4 shows that the number of events in the distance interval Δ*x*
is equal to the number of events in the charge
interval Δ*e*:

*n*(*e*)Δ(*e*) =
*n*(*x*_{0})Δ*x*_{0})
(6)

For uniform gamma charge generation within the detector bulk,
*n*(*x*_{0}) = 1, the charge distribution is calculated from
(5):

*n*(*e*) = (d(*e*(*x*_{0},*
d*))/d*x*_{0})^{-1} =
-(d*h*(*x*_{0})/d*x*_{0})^{-1 }(7)

Since the gamma generated charge is proportional to the gamma energy, the
distribution *n*(*e*) is the ordinary gamma spectrum.

Fig. 5 shows the spectra for the series: *a*/*d* = 0.2, 0.4, 0.8,
1.6, according to equation (7). The spectral lines are narrow as the aspect
ratio is low and they tend towards a square shape at high ratio. The missing
charge of the stationary holes generates a low energy tail in the spectrum of
the small pixel detector.

**Figure 5:** The charge spectral distribution, *n*(*e*)
(equation (7)), for uniform gamma generation within the detector, for the series
of aspect ratios: *a*/*d* = 0.2, 0.4, 0.8, 1.6.

**3. Detector circuit.**

The detector circuit includes a charge
sensitive preamplifier, a spectroscopy amplifier, and a multi channel analyzer
(MCA) as presented in Fig. 1.

The charge sensitive preamplifier measures the charge flow and build-up on
the pixel contact, and outputs a step like signal, *e*(*t*), with
rise-up that follows equation (4) and fig-2. Equation (5) gives the step's
height.

The charge signal is inputted into the input of a spectroscopy amplifier that
outputs a moving average of the derivative of the input. The exact output
depends on the transfer function of the specific device. Here the unit will be
considered a true integrator of a moving time window of a duration
*t*_{shape}. This operation cuts the system noise for all times
except the time of charge flow through the detector.

The output is proportional to the input's finite difference,
*e*(*t*) - *e*(*t* - *t*_{shape}), and the
pulse peaks when the electrons arrive to the pixel by eq. (5) :

*S *= *q* - *h*(*x*_{shape}) (8)

where *x*_{shape} = *d* - *v t*_{shape}.

Consider a case where the shape time, *t*_{shape}, is shorter
than the electron transition time from contact to contact
(*t*_{e}). *x*_{shape} is then the position where it
takes a gamma generated charge excactly *t*_{shape} to arrive to
the pixel.

If charge is generated at a position *x*_{0} shorter of
*x*_{shape} , then the flow time will be longer than
*t*_{shape}. However, the output peak (8) takes into account input
values only in the time window *t*_{shape} before arrival to the
pixel. Therefore, the output signal (8) will be the same for all events where
*x*_{0} < *x*_{shape}.

On the other hand, for each *x*_{0} >
*x*_{shape}, the amplifier's output will be proportional to its
input. Fig. 3(b) shows the the output of the spectroscopy amplifier vs. the
position *x*_{0} of gamma generation, corresponding to the input
3(a).

Fig. 3 indicates that shape time reduction below the electron transition time leads to only small reduction of the output signal. This property is unique to the small pixel detector. In single detectors the output signal is proportional to the shape time.

**4. Optimal Resolution **

The gamma generated electrons, flow at a
steady speed within the detector, but induce nonlinear response of charge
build-up at the pixel contact. The build-up rate is slow as long as the
electrons are far from the contact, and it speeds up when they arrive near it
(equation (4), fig. 2). This unique property of the small pixel enables further
improvement of its spectral resolution.

Fig-6 shows the time dependent signal build-up at the pixel contact, as
electrons are flowing towards it. Lines (a) and (b) show the output signal of
the charge sensitive amplifier for gamma charge generation at two points,
*x*_{0}' and *x*_{0}''. *x*_{0}' is
shorter than *x*_{shape} and *x*_{0}'' is longer than
it. The figure is read either vs. the time *t*, or, the position
*x*(*t*) of the moving electrons. The position
*x*_{shape} is marked on the figure.

**Figure 6:** The signal build-up at the pixel contact,
*e*(*x*_{0}, *x*(*t*)), as electrons are flowing
towards it, observed by the charge sensitive preamplifier. The figure is read
vs. either the time *t*, or the position *x*(*t*). The position
*x*_{shape} is where it takes the electrons exactly
*t*_{shape} to arrive to the pixel. The pulse height
*S*_{1}, of the spectroscopy amplifier's output, is the same for
all positions *x*_{0}' < *x*_{shape }. The height
*S*_{2} is equal to the step height of the input charge for each
position *x*_{0}'' > *x*_{shape}.

The pulse heights *S* of the spectroscopy amplifier, for
*x*_{0}' and *x*_{0}'' are determined by eq. (8).

The signals of many gamma events are collected in a multi channel analyzer (MCA), and their height distribution yields the signal spectrum. For uniform gamma generation, the spectral distribution of the spectroscopy amplifier's signal will be (similarly to equation (7)):

*n*(*S*(*x*_{0}) =
(d*S*/d*x*_{0})^{-1} (9)

The flat part of* S*(*x*_{0}), equation(8), fig-3(b),
introduces singularity to equation (9), so that the spectrum includes a δ function.
A fraction *x*_{shape}/*d* of
all the events of gamma generation, have the same value and appear as a single
line. The rest of the events appear as a tail of lower values.

The single line response in the detector spectrum is broadened by the noise in the system. The main sources of noise are the detector's leakage current and the preamplifier's electronics noise.

In summary, the shape time of a spectroscopy amplifier is adjusted below the electron's transition time through a detector, to the fast signal build-up at the pixel contact. The reduced time somewhat decreases the amplifier's signal height, but also introduces a δ function contribution to its spectral response. The spectral line is broadened by the noise in the system.

The electrons flow within a single pixel, and induce fast charge build-up at its contact when they arrive near it. This unique property allows shape time reduction and adjustment to the build-up with little loss of signal height.

Fig-6 shows the calculated spectra of a single pixel detector, with
*x*_{shape}/*d* = 0.5, for the series *a*/*d* = 0.2,
0.4, 0.8, 1.6 (equation (9)). The line for *a*/*d* = 0.2 is
arbitrarily noise broadened to 1%, then the noise is doubled for each
consecutive line. The figure corresponds to an array where the noise is
proportional to the square root of the pixel's area.

**Figure 7:** The pulse height spectrum of the spectroscopy amplifier's
output (equation (9)), for uniform gamma generation within the detector, for the
series: *a*/*d* = 0.2, 0.4, 0.8, 1.6.
*t*_{shape}/*t*_{e} = 0.5 (therefore
*x*_{shape}/*d* = 0.5). The line widths are adjusted to 1, 2,
4, 8% respectively, corresponding to noise level proportional to the square root
of the pixel's area. Reduction of *t*_{shape} below
*t*_{e}, the electron transition time from contact to contact,
leads to improvement of the spectral resolution, as compared to fig. 5.

The figure shows significant spectral improvement, compared to fig. 5 where
the shape time is equal to the electron's transition time (and therefore,
*x*_{shape} = 0). Good resolution is observed for higher aspect
ratio of pixel side to detector thickness, so the "small" pixel needs not be
really small.

**5. In practice**

Bias voltage and shape time adjustment optimizes the
detector circuit operation, so that the single pixel resolution approaches the
noise limit with little loss of signal. The voltage is adjusted high enough to
avoid electron trapping, by reducing the electron's transition time through the
detector below its lifetime. The shape time is then adjusted to the fast signal
build-up period at the pixel's contact.

The optimal shape time is significantly shorter than the electron transition
time, as observed in fig-3(b). In some systems the test equipment limits the
short shape time value, or, the value is fixed in an integrated circuit.
However, the same ratio of short shape time to the longer transition time is
achieved by slowing down the electrons' flow by lower bias voltage. The
resolution improves in spite of some electron trapping, as observed by Shor
*et al*. in [6] fig. 1.

**6. Conclusion**

The shape time is adjusted to the fast build-up of
induced charge, at a pixel contact, and improves the energy resolution of the
detector circuit towards the noise limit. The small pixel acts as a spatial and
temporal filter with proper parameters' adjustment.

The arguments presented in this paper are valid also for a single detector. The difference is that reduction of the shape time, below the charge transition time through the detector, reduces also the output signal. Such adjustment is justified in a single detector with very low noise level.

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on the net: 17, March 2000. Revised June 2001.

See poster, presented at the IEEE Nuclear Science Symposium, Lyon France, October 15 - 20, 2000.

by the author:

- "Improving CdTe and CdZnTe Crystals by Agitation during the Crystal Growth", https://urila.tripod.com/Agitate.pdf, Jun (2017).
- "Driving Spectral Resolution to the Noise Limit in Semiconductor Gamma Detector Arrays ", IEEE Trans. Nucl. Sci., Vol 48(3), pp 520 - 523, June (2001).
- "Semiconductor Crystal Optimization of Gamma Detection", J. crystal growth, Vol 225(2 - 4), pp. 114 - 117, May (2001).
- "Electron lifetime determination in semiconductor gamma detector arrays ", https://urila.tripod.com/hecht.htm, March (2000).
- "Driving Spectral Resolution to Noise Limit in CdZnTe Gamma Detector Arrays", https://urila.tripod.com/pixel.htm , March (2000).
- "Driving Spectral Resolution to Noise Limit in CdZnTe Gamma Detector Arrays", https://urila.tripod.com/pixel.pdf , March (2000).
- "CdTe and CdZnTe semiconductor gamma detectors equipped with ohmic
contacts", Nucl. Instr. and Methods
**A436**, 146 - 149 (1999). - "Ohmic Contact Gamma Radiation Detectors", in R.B. James and R.C. Schirato
Eds. "Hard X-Ray, Gamma Ray, and Neutron Detector Physics", SPIE Proc.
**3768**, Denver Colorado, July (1999). - "The role of semiconductors in digital x-ray medical imaging", https://urila.tripod.com/xray.htm , April (1999).
- "The role of semiconductors in digital x-ray medical imaging", https://urila.tripod.com/xray.pdf , April (1999).
- "CdTe and CdZnTe Gamma Detectors - Model of an Ohmic Contact", https://urila.tripod.com/ohmic.htm, July (1998).
- "CdTe and CdZnTe Crystal Growth and Fabrication of Gamma Radiation Detectors", https://urila.tripod.com/crystal.htm, March (1998).
- "CdTe Semiconductor Gamma Radiation Detectors equipped with ohmic contacts", https://urila.tripod.com/cdte.htm, February (1998).
- "The role of contacts in semiconductor gamma radiation detectors", Nucl.
Instr. and Methods
**A403**, 417 - 424 (1998)

Links:

- Thermodynamics Research Laboratory, http://www.uic.edu/~mansoori/Thermodynamics.Educational.Sites_html
- Thermodynamik - Warmelehre, http://www.schulphysik.de/thermodyn.html
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- Urgent-Meeting, https://urila.tripod.com/Urgent-Meeting.pdf