self energy (Fig. 10). The bubble diagram has no contribution, as the
propagator connecting the two vertices carries zero momentum, and from the
form of $\Gamma$ in Eq. (56) this contribution must vanish (recall that
$Z(\mathbf{0})$ is the zero matrix). The remaining diagram gives a
contribution to the spectral weight that, after summing over Matsubara
frequencies, reduces to

$\displaystyle\langle
S^{z}_{\sigma}(\mathbf{q},\omega)S^{z}_{\tau}(-\mathbf{q},0)\rangle$
$\displaystyle=\tilde{s}^{2}\langle\Pi_{\sigma}(\mathbf{q},\omega)\Pi_{\tau}(-\mathbf{q},0)\rangle$
$\displaystyle=\frac{\epsilon_{\mathbf{p}}\epsilon_{\mathbf{q}-\mathbf{p}}}{(2\tilde{s})^{2}}\frac{1}{2}\left(\frac{g^{\prime}}{3!}\right)^{2}\sum\limits_{\mathbf{p}\in\text{BZ}}\sum_{\lambda,\xi=0}^{3}\delta(\epsilon_{\mathbf{p}}+\epsilon_{\mathbf{q}-\mathbf{p}}-i\omega)\Gamma\begin{smallmatrix}\sigma&\lambda&\xi\\\
-\mathbf{q}&\mathbf{q}-\mathbf{p}&\mathbf{p}\end{smallmatrix}\Gamma\begin{smallmatrix}\tau&\lambda&\xi\\\
\mathbf{q}&\mathbf{p}-\mathbf{q}&-\mathbf{p}\end{smallmatrix}\left[\vphantom{\sum}1+n_{B}(\epsilon_{\mathbf{p}})+n_{B}(\epsilon_{\mathbf{q}-\mathbf{p}})\right]\,.$
(57)

Hidden in the structure of the $\Gamma$ terms is a symmetry-enforced Kronecker
delta between $\mu$ and $\rho$. We plot (the trace of) this this term in Fig.
11, with an arbitrary intensity scale.


# Effective Gain and Ion Back Flow study of triple and quadruple GEM detector

Senta V. Greene  Julia Velkovska  Brandon Blankenship  Michael Z. Reynolds
Sourav Tarafdar<EMAIL_ADDRESS>
###### Abstract

Gas electron Multipliers (GEM) are a new generation of gaseous avalanche
devices in the Micro Patterned Gaseous Detector category. GEMs are widely used
in both nuclear and high energy experiments as well as in medical science.
Several parameters define the performance of these types of devices under
various experimental conditions. This article focuses on the study of
effective gain and Ion Back Flow (IBF) in both triple and quadruple GEM
detectors. Effective gain and IBF are two of the most important parameters in
determining the performance of GEM detectors.

_K_ eywords GEM $\cdot$ Effective Gain $\cdot$ IBF $\cdot$ MPGD

##  1 Introduction

In recent years, Gas Electron Multipliers (GEMs) [1] have been used
extensively in several high energy and heavy ion experiments due to their
excellent rate capability and intrinsic position resolution. Some of these
examples are the Hadron Blind Detector [2] in the PHENIX detector at the
Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory, the
ALICE detector at the Large Hadron Collider at CERN, and the Time Projection
Chamber (TPC) for tracking charged particles in the sPHENIX detector under
construction at RHIC [3, 4]. GEMs are found to be radiation hard, capable of
amplifying small signals, have excellent energy resolution, and deliver fast
signal [5]. Some of the parameters that determine the performance of GEM
detectors are the effective gain, energy resolution, spatial resolution, time
resolution and ion back flow (IBF). Important factors that determine the
effective functioning of GEM detectors are the operating voltages, gas medium
in which the detector is being operated, readout board segmentation, and pitch
of the holes on the GEM layers. This article focuses on studying effective
gain and IBF using different gas mixtures and different voltage configurations
for both triple and quadruple GEM detectors. Further, two methods for
estimating the effective gain of GEM detectors are also discussed.

##  2 Experimental Configuration

The GEM foils used in this study are double mask standard GEMs from CERN with
140 micron pitch and 70 micron hole diameter with $10\times 10~{}{\rm cm}^{2}$
active area [6]. The triple GEM detector is composed of a drift cathode of
$10\times 10~{}{\rm cm}^{2}$ active area, three layers of GEMs, and a 512
channel X-Y strip readout board. The quadruple GEM detector has four layers of
stacked GEMs in between the drift cathode and 512 channel X-Y strip readout
board. The gaps between GEM leyers is known as transfer gap while between GEM
foil and readout board is induction gap. For triple GEM detector there are two
tansfer gaps while for quadruple GEM detector there are three transfer gaps.
Both triple and quadruple GEM detectors has one induction gap. Fig. 1(a) shows
the picture of the experimental configuration on the laboratory test bench and
Figures 1(b) and 1(c) show the schematics of the overall set up for triple and
quadruple GEM detectors, respectively, for measuring effective gain and IBF.
The measurement of effective gain for both triple and quadruple GEM was done
in Ar:CO2(70:30) and Ar:CO2(80:20) gas mixtures while the optimization of IBF
study was performed only using the Ar:CO2(70:30) gas mixture. The gas mixtures
used in the experiment were mixed with an in-house-built gas mixing system,
capable of mixing two different gases with good precision. Both Ar and CO2 gas
were of ultra high purity (99.999$\%$ pure). The effective gain scanning for
both triple and quadruple GEM detectors were done by biasing the all the
electrodes of detector using single channel of high voltage power supply via
voltage divider. For IBF optimization both the triple and quad GEM detectors
individual electrodes were biased using individual channels of a high voltage
power supply via 10 M$\Omega$ protection resistors. For measuring the
effective gain of the detector, a collimated monochromatic ${}^{55}{\rm Fe}$
radioactive source was used, whereas for measuring IBF, a mini X-ray tube with
1 mm collimation was used. The whole set up was placed in an Aluminum box for
shielding from X-rays during operation. The readout electronics for the
detector signal included a charge-sensitive Ortec preamplifier with rise time
of $<$ 525 ns [7], Ortec shaping amplifier and oscilloscope with the
capability of producing histograms of the detected signals. For estimating
IBF, the induced current in the readout board from avalanche electrons and
that from the drift cathode due to back-drifting positive ions were measured
using a highly-sensitive Picologic picoammeter [8]. Digital humidity and
temperature controllers were used for keeping the atmospheric temperature at
26∘C and relative humidity at 30$\%$.

(a)

(b)

(c)

Figure 1: Experimental set up a) Picture of GEM detector on experimental test
bench b) Schematics of triple GEM detector set up c) Schematics of quadruple
GEM detector set up

##  3 Experimental Results

###  3.1 Effective Gain

The effective gain of the GEM detector is defined as the ratio of the number
of avalanche electrons and the number of primary electrons. Experimentally,
the effective gain can be determined in two different ways.

  * •

Method 1 : Measuring induced current from the avalanche electrons at the
readout board. One can then extract the effective gain of the detector by
using the following formula

$G_{{\rm eff}}=\frac{I_{{\rm}anode}}{e\times n_{{\rm primary}}\times
R_{{\rm}x-ray}}$ (1)

where ,

    * –

Geff is the effective gain of the detector.

    * –

Ianode is the measured induced current from the readout board.

    * –

$e$ is the electron charge.

    * –

nprimary is the number of primary electrons in the drift volume of detector
due to complete absorption of 5.9 KeV photons from ${}^{55}{\rm Fe}$ source.

    * –

Rx-ray is the rate of the ${}^{55}{\rm Fe}$ source.

It is to be noted that the denominator in Eqn. 1 corresponds to the induced
current due to primary electrons at the drift cathode. Ianode was measured by
summing all the pads of the readout board and using a highly sensitive
picoammeter. A collimated ${}^{55}{\rm Fe}$ source was used so Rx-ray was
estimated by noting the count rate of the ${}^{55}{\rm Fe}$ spectra from the
detector.

  * •

Method 2 : By extracting a monochromatic source spectra from the bottom of the
last GEM or readout board and extracting the number of avalanche electrons
embedded inside the spectra. The ratio of the number of avalanche electrons
and the number of primary electrons from ionization of the gas inside the
drift region because of the passage of primary ionizing particles will be the
effective gain of the detector. In this method, effective gain can be
formulated as follows

$G_{{\rm eff}}=\frac{N_{{\rm avalanche}}^{{\rm spectra}}}{n_{{\rm primary}}}$
(2)

where

    * –

Geff is the effective gain of the detector.

    * –

N${}_{{\rm avalanche}}^{{\rm spectra}}$ is the number of avalanche electrons
in the monochromatic source spectra from GEM detector.

    * –

nprimary is the number of primary electrons in the drift volume of detector
due to complete absorption of 5.9 KeV photons from ${}^{55}{\rm Fe}$ source.

The total number of avalanche electrons was estimated by first measuring the
correlation between the known amount of input charge via high precission 1 pF
capacitor to the readout electronics and the corresponding signal from the
readout electronics. One can correlate this 1 pF capacitor to GEM detector.
The schematic for such a measurement is shown in Fig. 2(a). The resulting
calibration curve is shown in Fig. 2(b). It is clear from the calibration
curve that the electronics in the experimental set up reaches saturation
region when the signal goes above 1.6 V, which is fairly common in
electronics. So, care must be taken while operating the detector to ensure the
electronics don’t reaches the saturation region in order to avoid an incorrect
measurement of the effective gain.

It is expected that the measurements of effective gain from the methods
described above will provide different results because of the mismatch in the
number of avalanche electrons that reach the bottom of the last GEM due to the
bending of electric field lines and the avalanche electrons reaching the
readout board which causes variation in electron transparency [5]. The
difference in the number of avalanche electrons reaching the readout board and
accumulating at the bottom of the last GEM is largely dependent on the
electric field between the last GEM and the readout board (induction gap
field).

(a)

(b)

Figure 2: Calibration curve measurement a) Schematic for estimating
calibration curve b) Calibration curve

This can be shown by effective gain measurements using both methods for
different fields across the induction gap. This measurement was done by using
voltage divider circuit to bias each electrode of the triple GEM detector,
which uses a single channel of the high voltage power supply. Varying the
output voltage from the power supply provided different fields across the
induction gap and also potential differences across each GEM layer. The
resistors in the voltage divider circuit were chosen so that the variation of
potential difference across each GEM during the variation of the output
voltage from the power supply was the same. Table 1 shows the output voltage
from the power supply and the corresponding potential difference across each
GEM and the induction field.

Input voltage to | Potential difference across | TG1 [kV/cm] | TG2 [kV/cm] | Induction
---|---|---|---|---
voltage divider [V] | each GEM [Volt] |  |  | field [kV/cm]
-3550 | 355 | 2.23 | 2.23 | 1.7
-3700 | 370 | 2.32 | 2.32 | 1.9
-3750 | 375 | 2.36 | 2.36 | 2.0
Table 1: Table for input voltage to voltage divider and corresponding
potential difference across GEMs and induction field.

Fig. 3(b) shows ${}^{55}{\rm Fe}$ spectra in the triple GEM detector using the
Ar:CO2 (70:30) gas mixture for voltage of -3700 V to the detector via
potential divider while Fig. 3(c) shows the effective gain calculation from
both the methods described earlier. The systematic error on effective gain is
estimated to be 5$\%$ by taking into account of uncertainty of 2$\%$ in gas
mixing unit, 4$\%$ tolerance in capactor of calibration set up and 2$\%$
uncertainty in output voltage from voltage divider. One can clearly see that,
as the induction field becomes larger, more avalanche electrons reach the
readout board. which in turn increases the effective gain with respect to that
estimated by using the spectra from the bottom of the last GEM. Also at higher
applied voltage the effective gain measured from anode current increases at
faster rate as compared to the calculation where signal from bottom of last
GEM foil was used. This is because effective gain is an exponential function
of voltage across the GEM foil [1] and is also a function of electron
extraction efficiency in each of the transfer gaps and induction gap. At large
input voltage to the GEM detector via voltage divider , the gap fields
increases and also potential difference acorss each GEM foil. Both these
factors causes exponential increase in avalanche electrons reaching the anode
and hence larger difference in effective gain at higher input voltage from
voltage divider to the triple GEM detector.

(a)

(b)

(c)

Figure 3: Signal from triple GEM detector a) Voltage divider conifguration for
this measurement b) ${}^{55}{\rm Fe}$ spectra from triple GEM detector in
Ar:CO2 (70:30) gas mixture c) Effective gain measurement from the two methods
described in section 3.1.Systematic error of 5$\%$ is assigned to red data
points which is based on calibration curve and spectra measurement as per
description in the text. A systematic error of 3$\%$ is assigned to blue data
point which is based on anode current measurement. The 4$\%$ uncertainty
involves 2$\%$ uncertainty in anode current due to noise , 2$\%$ uncertainty
in gas mixing unit and 2$\%$ uncertianty in voltage divider circuit. The
statistical errors are rather small and are of the order of size of markers.

Effective gain scanning for both quadruple and triple GEM detectors were
performed for various argon-based gas mixtures and is shown in fig. 4. The
study was done using a voltage divider scheme for both triple and quadruple
GEM detectors. For a higher argon content in the gas mixture , one can attain
a greater effective gain at the same operating voltage. This is because, since
argon is the primary ionising gas component, there will be larger number of
primaries with the higher argon fraction. Also, the quadruple GEM detector
will attain the same effective gain at a lower operating voltage as compared
to the triple GEM detector using the same gas mixture because of the
additional avalanche layer.

Figure 4: Effective gain from triple and quadruple GEM detectors for various
composition of Ar:CO2 gas mixtures. The systematic error of 5$\%$ is shown by
the gray band while statistical errors are of the order of size of markers.

####  3.1.1 Ion Back Flow

Ion Back Flow (IBF) is the fraction of positive ions drifting towards to the
drift region of GEM detector. Experimentally it is determined by taking the
ratio of measured current from cathode and from anode [9].

$IBF=\frac{I_{{\rm cathode}}}{I_{{\rm anode}}}$ (3)

For the measurement and minimization of IBF, both triple and quadruple GEM
detector electrodes were biased by individual channels of a multichannel power
supply as shown in Fig. 1(b) and in Fig. 1(c). Only Ar:CO2 70:30 gas was used
for detailed study of IBF. The detector drift area was irradiated with a Ag
target X-ray tube fitted with a collimator of 1mm diameter. The X-ray tube was
operated at 20 kV peak voltage and 18 uA current. During IBF estimation for
both triple and quadruple GEM detector, all the layers of the GEM were
operated at the same potential difference. Fig. 5(a) shows the current
measured from the anode and cathode of triple GEM detector after turning on
the X-ray tube for different potential differences across the GEM foil. The
measurements were done by keeping the drift field at 1 kV/cm while the
transfer gap and induction fields were at 2 kV/cm. Using equation 3 and
results from Fig. 5(a) , IBF was estimated for different potential differences
across the GEM as shown in Fig. 5(b). The IBF is assigned a systematic
uncertainty of 3$\%$ by taking into account of 2$\%$ systematic uncertainty on
each of the anode and cathode current measurement due to noise and 1$\%$
uncertainty in the fitting procedure to anode and cathode current data.
Further the noise in anode and cathode current was estimated by measuring them
for long time before biasing the GEM detector and after turning off the X-Ray
tube. At the same time, effective gain for different voltages across the GEM
foils was measured and the results are shown in Fig. 5(b).

(a)

(b)

Figure 5: Triple GEM detector IBF and effective gain measurement in Ar:CO2
(70:30) gas mixture a) Current measured from anode and cathode during
operation of X-ray tube. Both statistical and systematic errors are small and
are of the order of the size of markers. b) Effective gain and IBF measurement
for different potential differences across the GEM layers. The systematic
errors are shown by rectangular box around the data points. The statistical
errors are small and are of the order of the size of markers.

During long-term operation of GEM-based detectors in particle physics
experiments like TPC in ALICE [3, 10] or HBD in PHENIX [2], the GEMs are
operated at the same effective gain. In GEM-based tracking detectors like TPC
or recently proposed Transition Radiation Detector (TRD) [11] where
maintaining uniform electric field is important for determining accurate
trajectories of particles, minimizing IBF is a major priority [3, 10, 12].
Large IBF tends to cause a distortion of the uniform electric field because of
the slow drift velocity of heavier positive ions. Over the past deccade
several research groups has worked significantly to reduce the IBF both
without the effect of magnetic field [13, 14, 12, 9] and also under the effect
of magenetic field [15]. One way to minimize IBF for a specific effective gain
of the GEM detector is to operate the detector such that the electric fields
in different gaps are not the same [15, 14, 12, 13]. In this study for
minimizing IBF in triple GEM detector, an effective gain of 4500 was chosen
and the corresponding IBF as in Fig 5(b) was 36$\%$. To see the effect of
different gap fields on IBF for the same effective gain, the triple GEM
detector was operated by varying a specific gap field while keeping other gap
fields constant and maintaining the same effective gain by changing the
potential difference across the GEMs. All the GEMs were kept at the same
potential difference to avoid the effect of ion blocking because of having a
different gain of each GEM. It has been shown explicitely that operating each
GEM foil at different gain compared to the other foil in the same GEM detector
will cause IBF suppression. [9] . Operating the triple GEM detector with
symmetric powering of GEM foils and only changing the gap fields will allow us
to understand only the effect of gap field on IBF suppression. Fig. 6(a) shows
the estimated IBF as a function of a specific gap field while keeping the
other gap fields constant. The study of asymmetric variation of electric field
in different gap fields on IBF was done in three steps as below .

  * •

Varying trasfer gap 1 (TG1) field while keeping transfer gap 2 (TG2) and
induction field constant at $\sim$ 2 kV/cm.

  * •

Varying TG2 field while keeping TG1 and induction field constant at $\sim$ 2
kV/cm.

  * •

Varying induction field while keeping TG1 and TG2 fields constant at $\sim$ 2
kV/cm.

The effective gain was kept constant as shown in Fig. 6(a). It can be seen
that if transfer gap 2 and transfer gap 1 fields are kept low then one can
reduce IBF. However if transfer gap 2 is kept at a lower electric field than
transfer gap 1 field, then IBF suppresion is greater. Also, keeping the
induction field at a higher value suppresses IBF. Keeping these parameters in
mind, the triple GEM detector was operated so that transfer gap 1 and
induction gap fields were large, while the transfer gap 2 field was kept low.
The measured anode and cathode currents are shown in Fig. 6(b) and using
equation 3 the estimated IBF for this voltage configuration is about 12.4$\%$,
which is significantly lower than the value obtained with effective gain of
4500.

(a)

(b)

Figure 6: Triple GEM detector IBF measurement in Ar:CO2 (70:30) gas mixture a)
IBF estimated by varying a specific gap field while keeping other gap fields
constant b) Measured anode and cathode currents while operating the triple GEM
detector with gap fields selected to reduce IBF.

A similar study was done with the quadruple GEM detector. First, the quadruple
GEM detector was scanned for different effective gain by changing the
potential difference across the GEMs, which is shown in Fig. 7(b) and, at the
same time, IBF was estimated for that specific effective gain by using the
X-ray tube as the ionizing source and measuring the anode and cathode current
from the quadruple GEM detector as shown in Fig. 7(a).

(a)

(b)

Figure 7: Quadruple GEM detector IBF and effective gain measurement in Ar:CO2
(70:30) gas mixture a) Current measured from anode and cathode during
operation of X-ray tube. Both of them has systematic error of 2$\%$ due to
noise. Boith statistical and systematic errors are small and are of the same
order as markers. b) Effective gain and IBF measurement for different
potential differences across the GEM layers. The systematic errors are shown
by rectangular box around the data points. The statistical errors are small
and are of the order of the size of markers.

In order to study the effect of gap fields and optimizing them for reducing
the IBF , an effective gain of 4000 and corresponding IBF of 17$\%$ as per
Fig. 7(b) were chosen. The quadruple GEM detector was then operated by varying
a specific gap field while keeping the other gap fields constant. The study of
asymmetric variation of electric field in different gap fields on IBF was done
in four steps as below .

  * •

Varying trasfer gap 1 (TG1) field while keeping transfer gap 2 (TG2), transfer
gap 3 (TG3) and induction field constant at $\sim$ 2 kV/cm.

  * •

Varying TG2 field while keeping TG1, TG3 and induction field constant at
$\sim$ 2 kV/cm.

  * •

Varying TG3 field while keeping TG1, TG2 and induction field constant at
$\sim$ 2 kV/cm.

  * •

Varying induction field while keeping TG1, TG2 and TG3 fields constant at
$\sim$ 2 kV/cm.

The effective gain throughout the variation of the gap field was kept constant
by manipulating the potential difference across each GEM foil. All the four
GEM foils were operated at the same potential difference for the same reason
as mentioned in triple gem detector studies. Fig. 8(a) shows the IBF estimated
for the quadruple GEM detector by varying a specific gap field while keeping
the other gap fields constant. From the same figure, it can be seen that the
effective gain was kept constant during the whole measurement. It is quite
clear from this estimate that operating the detector with transfer gap 1 and
induction field at larger electric field while keeping the transfer gap 2 and
transfer gap 3 field low will provide the maximum suppression of IBF. With
this configuration, the quadruple GEM detector was again operated to estimate
IBF. Fig. 8(b) shows the anode and cathode current measured from the detector
and using Equation 3 the IBF was estimated to be about 6$\%$, which is almost
3 times lower than the initial value. Comparing the results from this work
which focussed only on effect of gap fields on IBF and the results from [9] ,
which focussed on effect of asymmetric gem foil biasing , it can be seen that
gap field has larger effect on IBF suppression as compared to only asymmetric
gem foil biasing. This has been also confirmed in [16].

(a)

(b)

Figure 8: Quadruple GEM detector IBF measurement in Ar:CO2 (70:30) gas mixture
a) IBF estimated by varying a specific gap field while keeping other gap
fields constant.The systematic errors are shown by rectangular box around the
data points. The statistical errors are small and are of the order of the size
of markers. b) Measured anode and cathode currents while operating the
quadruple GEM detector with gap fields chosen to reduce IBF.

##  4 Conclusions

Studies of effective gain, IBF and reduction of IBF for both triple and
quadruple GEM detectors were performed using Argon-based gas mixtures. The
effective gain was calculated using two different methods and the variation of
results is as expected. Also, effective gain scanning was performed for
different Argon-based gas mixtures for both triple and quadruple GEM
detectors. Higher argon content increases the effective gain at the same
potential difference across the GEMs which is attributed to the fact that more
ionising component is added to the gas mixture. Also, the quadruple GEM showed
higher effective gain as compared to the triple GEM detector for the same
potential difference across the GEMs because of the additional avalanche
layer. The results for IBF shows that the IBF can be suppressed by varying the
electric field in different gap regions of the detector in such a way that the
low electric field in transfer gap 2 for both triple and quadruple GEM
detector acts as an ion blocker. In addition quadruple GEM detector shows
larger IBF suppression compared to triple GEM detector due to the presence of
additonal transfer gap where lower electric field aids in additional
suppression of IBF. The studies on IBF suppression in this technical report
quantifies IBF suppression by changing specific gap field of standard GEMs
from CERN [6] while keeping other gap fields constant for both triple and
quadruple GEM.

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