∎

e1Corresponding Author<EMAIL_ADDRESS>
11institutetext: Department of Physics, The University of South Dakota,
Vermillion, South Dakota 57069

# Development of Planar P-type Point Contact Germanium Detectors for Low-Mass
Dark Matter Searches

W.-Z. Weiaddr1  H. Meiaddr1 K. Kooiaddr1 D.-M. Meie1,addr1  J. Liuaddr1  J.-C.
Liaddr1  R. Panthaddr1 and G.-J. Wangaddr1

(Received: date / Accepted: date)

###### Abstract

The detection of low-energy deposition in the range of sub-eV through
ionization using germanium (Ge) with a bandgap of $\sim$0.7 eV requires
internal amplification of the charge signal. This can be achieved through high
electric field which accelerates charge carriers which then interact and
generate more charge carriers. The minimum electric field required to generate
internal charge amplification is derived for different temperatures. A point
contact Ge detector provides extremely high electric field in proximity to the
point contact. We show the development of a planar point contact detector and
its performance. The field distribution is calculated for this planar point
contact detector. We predict the required electric field can be achieved with
a point contact detector.

††journal: Eur. Phys. J. C

##  1 Introduction

From an experimental point of view, the detection of low-energy deposition
from low-mass dark matter at MeV-scale remains a daunting challenge. Until
now, the devices that can meet the energy threshold of $<$ 10 eV required for
detecting light dark matter in the MeV-scale range are still in their infancy
mei ; supercdms1 ; supercdms2 ; edelweiss ; cresst ; damic ; sensei1 ; sensei2
; collar .

High-purity Ge (HPGe) detectors are well known for providing excellent energy
resolutions over a wide energy range, and thus have been widely used in
nuclear and particle physics. A low capacitance Ge detector can be realized
with a p-type point contact (PPC) geometry, which allows the detector to
achieve the capacitance on the order of 1 pF and can be operated with sub-keV
energy threshold barb . Thus, the PPC Ge detector technology is able to offer
both excellent energy resolution and relatively low energy threshold for
ultra-low background experiments such as GeV-scale dark matter and axion
interactions. This technology has been exploited by two leading 76Ge-based
neutrinoless double beta decay experiments, the Majorana Demonstrator mjd1
and GERDA gerda . However, some physical processes requiring an extreme low-
energy threshold of $<$ 10 eV such as the detection of MeV-scale dark matter
is still unreachable with existing PPC Ge detectors whose energy threshold are
hundreds of eV due to the limitation from the read-out electronic noise. Most
recently, by the introduction of ultra-low vibration mechanical cooling and
wire bonding of a complimentary medal-oxide semiconductor (CMOS) charge
sensitive preamplifier to a sub-pF PPC Ge detector, Barton et al. barton  have
demonstrated electronic noise of 39 eV-FWHM (full width at half maximum) at 43
K. Thus, PPC Ge detectors have great potential to achieve an extreme low
energy threshold if the detectors can be operated at a even lower temperature
such as liquid helium temperature ($\sim$4 K) wei2020 .

Traditionally, as shown in Fig. 1, a PPC Ge detector is made of a cylinder of
HPGe crystal in which B-implanted p+ and Li-diffused n+ electrical contacts
are fabricated on the point contact and outer surface contact, respectively.
Compared with the traditional PPC Ge detector shown in Fig. 1, a planar PPC Ge
detector with a-Ge contacts, as shown in Fig. 2, can be fabricated much easier
due to the two following reasons: (1) four wings in a planar geometry simplify
the handling process during the detector fabrication; and (2) fabrication of
a-Ge contacts is much simpler than the fabrication of traditional B-implanted
p+ and Li-diffused n+ electrical contacts amman ; wei2020 . If a planar PPC
detector shown in Fig. 2 is able to achieve a similar detector performance
(low detector capacitance and excellent energy resolution) as a conventional
PPC detector shown in Fig. 1, due to its relative simpler fabrication process,
it would be more convenient to use a planar PPC detector instead of a
cylindrical PPC to explore some interesting physics processes such as charge
trapping due to shallow level impurities and charge lost due to deep level
impurities in Ge tyl . In addition, it could also be used as a drift detector
for medical imaging cas ; kem2 ; gat .

Figure 1: Cylindrical PPC Ge detector. Left: Sketch of the full detector mjd2
; Right: Cross section. The p+ contact is shown in orange and the n+ contact
is shown in purple. Figure 2: Cross-sectional drawing of a planar PPC Ge
detector. The point and outer surface contacts are in purple and they are both
made of a-Ge contact with Al on the top.

In order to investigate the possibility of observing internal charge
amplification with a planar PPC Ge detector, we have fabricated such a
detector in our lab at University of South Dakota (USD). The geometry of the
detector is the same as shown in Fig. 2 and the detector was made with the Ge
crystal grown by us at USD. The detector has been carefully characterized to
examine its performance including leakage current, detector capacitance and
energy resolution. Based on the detector performance, we have conducted a
numeric calculation to study the electric field distribution inside the
detector to find out if the required electric field for internal charge
amplification can be achieved with this detector. In this paper, we first show
the theoretical requirements for internal charge amplification in Sect. 2. The
details about the fabrication and characterization of a custom-build planar
PPC Ge detector are presented in Sect. 3, followed by the discussion and the
expectation of detector performance for observing internal charge
amplification in Sect. 4. Finally, the conclusions are summarized in Sect. 5.

##  2 Theoretical Requirements for Internal Charge Amplification

The initial charge carriers created by external energy deposition in Ge are
through ionization and excitation of Ge atoms or impurity atoms in Ge. The
ionization and excitation of Ge atoms to create the secondary charge carriers
require a charge carrier to have a minimum kinetic energy of $\sim$0.73 eV
(the band gap of Ge mei ) at liquid nitrogen temperature (77 K). The main
remaining impurities in the Ge after zone refining and crystal growth are
boron, aluminum, gallium and phosphorus mei . They are all shallow impurities
since their ionization energies are $\sim$0.01 eV rwrw . Charge carriers or
phonons with energies larger than 0.01 eV can easily ionize or excite
impurities from the donor level or acceptor level to create charge carriers.
Therefore, the minimum energy required to generate a charge carrier and a
charged impurity through ionization and excitation of impurities in Ge is
$\sim$0.01 eV.

Under an electric field, a charge carrier will be accelerated to gain a
kinetic energy of $\Delta E=\frac{1}{2}m^{\ast}v_{d}^{2}$, where $m^{\ast}$ is
the effective mass of the charge carrier and $v_{d}$ is the drift velocity.
During the drifting process, the charge carrier will lose energy by
interacting with the crystal lattice which manifests as emitted phonons BNVT .
These phonons are so-called Neganov-Luke phonons produced by charge transport
pnlu . As a result, the emission of Neganov-Luke phonons will reduce the
kinetic energy of the charge carrier. For a given drifting length, the energy
conservation requires the work done on the charge carrier by the externally
applied electric field to be equal to the sum of the kinetic energy gained by
the charge carrier and the energy loss to the emission of phonons. This is to
say that the work done on the charge carrier by the externally applied
electric field is always balanced. This energy balance is performed through a
competing process between the energy loss to the charge carriers scattering
off the orbital electrons and the energy loss to the charge carriers
scattering off the lattice mei . The former will ionize the orbital electrons
to create additional charge carriers and the latter will cause the emission of
phonons. To obtain internal charge amplification, one prefers the energy loss
to the charge carriers scattering off the orbital electrons. This requires the
mean scattering length (the mean ionization length) for charge carrier
scattering off the orbital electrons to be smaller than the mean scattering
length for charge carriers scattering off the lattice.

Under zero electric field, the ionization is a thermal process. Therefore, the
mean scattering length depends only on temperature. For charge carriers
scattering off the orbital electrons, the mean scattering length ($L_{th}$)
can be calculated using $L_{th}$ = $v_{th}$$\times$$\tau_{th}$, where $v_{th}$
is thermal velocity of charge carriers and $\tau_{th}$ =
$m^{\ast}\frac{\mu_{th}}{e}$ is the lifetime of charge carriers, with
$\mu_{th}$ the charge mobility. For charge carriers scattering off the
lattice, the mean scattering length ($\lambda_{th}$) can be obtained using
$\lambda_{th}$ = $v_{s}\tau_{ph}$, where $v_{s}$ = 5.4$\times$105 cm/s is the
speed of sound (phonon) in Ge and $\tau_{ph}$ = $m^{\ast}\mu_{ph}/e$, where
$\mu_{ph}$ = $\frac{4.65\times 10^{5}}{m^{\ast^{5/2}}}\cdot T^{-3/2}$ is the
charge mobility due to acoustic deformation potential scattering meihao  and
$T$ is temperature. Figure 3 shows the mean scattering length as a function of
temperature for ionization and phonon emission, respectively.

Figure 3: The mean scattering length versus temperature for charge carriers
scattering off the orbital electrons and charge carriers scattering off the
lattice, respectively at zero electric field. The solid blue line shows the
mean scattering length of holes interacting with lattice for emitting phonons.
The solid red line shows the mean scattering length of electrons interacting
with lattice for emitting phonons. The dashed blue line shows the mean
scattering length of holes interacting with orbital electrons. The dashed red
lines shows the mean scattering length of electrons interacting with orbital
electrons.

It is clear that the mean scattering length for charge carriers scattering off
the orbital electrons is smaller than that of charge carriers scattering off
the lattice at temperature less than 30 K. To effectively generate more charge
carriers with internal charge amplification, one would prefer to run the
detector at a temperature less than 30 K.

For drifting charge carriers under the electric field, the work done on a
charge carrier can be expressed as:

$\vec{F}\cdot\vec{L}=e\vec{E}\cdot\vec{L}=eEL,$ (1)

where $F$ is the electric force, $L$ is the drifting path length along the
direction of electric force, and $\vec{E}$ is the electric field. For one
interaction length, $L$ is equal to the mean scattering length for charge
carriers scattering off the orbital electrons, that is

$L=v_{d}\cdot\tau=\mu E\frac{m^{\ast}\mu}{e}=\mu^{2}E\frac{m^{\ast}}{e},$ (2)

where $\tau$ is the relaxation time of a charge carrier and $\mu$ is the drift
mobility.

For temperature less than 30 K, the mean scattering length for charge carriers
scattering off the orbital electrons is smaller than that for charge carriers
scattering off the lattice, as shown in Fig. 3. Therefore, the work done on a
charge carrier by the applied electric field will mainly accelerate the charge
carrier to gain sufficient kinetic energy and hence generate more e-h pairs
through internal charge amplification. The minimum electric field required to
trigger internal charge amplification can be roughly estimated using Eq. 1.
Since the minimum ionization energy of Ge atoms is $\sim$0.73 eV (the band gap
energy of Ge) and the average ionization length is about 0.4 $\mu$m at zero
electric field as shown in Fig. 3, the minimum electric field required is just
under 2$\times$104 V/cm for ionizing Ge atoms to initiate internal charge
amplification. Similarly, the average ionization energy for shallow impurity
atoms in Ge is about 0.01 eV. Thus, the minimum electric field required to
trigger internal charge amplification through ionization of impurity atoms in
Ge is roughly estimated to be less than 300 V/cm. It is worth mentioning that
if temperature is significantly lower than 30 K, the average ionization length
can be significantly larger than $\sim$0.4 $\mu$m under an applied electric
field, as indicated in Fig. 3. Therefore, the minimum electric field required
to trigger internal charge amplification can be significantly lower than what
is estimated for both ionization of Ge atoms and ionization of impurity atoms.
However, the detailed discussion requires sophisticate models in calculating
the average ionization length and is beyond the scope of this work.

For temperature greater than 30 K, the average ionization length is larger
than the mean scattering length for charge carriers scattering off the
lattice. Hence, it is likely there will be energy loss to the emission of
phonons before kicking off the internal charge amplification. Thus, energy
conservation requires the work done by the electric field be equal to the sum
of the gained kinetic energy and the energy of emitted phonons. Therefore, the
minimum electric field required to generate an e-h pair on Ge atoms or
impurity atoms during a drifting process with a drifting length of $L$ can be
calculated from the following relation:

$eE\times L=\Delta E+\frac{L}{\lambda_{ph}}\times\bar{\Delta E_{ph}},$ (3)

where $\lambda_{ph}$ = $v_{s}\tau$ is the mean scattering length for producing
phonons, $\bar{\Delta E_{ph}}$ is the average energy of emitted phonons, which
is estimated as $\bar{\Delta E_{ph}}$ = $\frac{hv_{s}}{a}$ = 0.04 eV, where
$h$ is the Planck constant and $a$= 0.56 $nm$ is the lattice constant of Ge.

Combining Eq. 2 and Eq. 3, we have

$eE\times\mu^{2}E\frac{m^{\ast}}{e}=\Delta E+\frac{\mu
E}{v_{s}}\cdot\bar{\Delta E_{ph}}.$ (4)

Rearranging the terms in Eq. 4, we have

$m^{\ast}\mu^{2}E^{2}-\frac{\mu}{v_{s}}\bar{\Delta E_{ph}}E-\Delta E=0.$ (5)

For this quadratic equation, we have $E=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$,
where $a=m^{\ast}\mu^{2}$, $b=-\frac{\mu}{v_{s}}\bar{\Delta E_{ph}}$, and
$c=-\Delta E$ is the energy required to generate an e-h pair in Ge. The
minimum energy required to generate an e-h pair in Ge is $\sim$0.73 eV in Ge
at liquid nitrogen temperature wei2018 .

Similarly, energy conservation during this process can be used to calculate
the minimum electric field required to generate an e-h pair on impurity atoms
in Ge using Eq. 4. However, now $\Delta E$ is the ionization energy of shallow
impurities in Ge ($\Delta E\geq$ 0.01 eV).

When the temperature is above 30 K, the charge drift mobility in Ge is
dominated by the contribution from phonon scattering mobility and neutral
impurity scattering meihao . The Hall mobility for holes is often taken as
$\mu_{h0}$ = 42,000 cm2/(V$\cdot$s) and the Hall mobility for electrons is
$\mu_{e0}$ = 36,000 cm2/(V$\cdot$s) for Ge according to IEEE standard ieee .
The relationship between the Hall mobility and the drift mobility is given by
$\mu_{h}$ = $\mu_{h0}$/1.03 for holes and $\mu_{e}$ = $\mu_{e0}$/0.83 for
electrons, respectively ieee . Utilizing Eq. 5 with $m^{\ast}$ = 0.21 m0 mei
for holes and $m^{\ast}$ = 0.12 m0 mei  for electrons, where m0 = 511,000
eV/c2 is the electron mass in vacuum, we obtain the minimum electric field as
a function of temperature from 30 K to 100 K as shown in Fig. 4.

Figure 4:  The minimum electric field required for internal charge
amplification of Ge atoms and impurity atoms in a temperature range from 30 K
to 100 K.

It is clear that the minimum field required to generate an e-h pair in Ge is
above 104 V/cm. There is a large difference in the minimum required field for
generating more e-h pairs when drifting electrons and holes across the
detector. A weak temperature dependence is also seen in Fig. 4. It shows that
the minimum field required to generate more e-h pairs decreases as temperature
decreases. This indicates that an internal charge amplification can be
achieved at a lower electric field when the detector is operated at a lower
temperature. At liquid nitrogen temperature (77 K), the required electric
field for internal charge amplification must be obtained through a special
geometry design at which high electric field is achievable in a designated
detector region. We studied this hypothesis using a customized planar PPC
detector.

##  3 Operation and Characterization of A Custom-build Planar PPC Detector

###  3.1 Detector Fabrication Process

To convert an HPGe crystal into a planar PPC detector, the fabrication process
of depositing a-Ge and Al on all the detector surfaces is the same as in
fabricating a planar detector from an HPGe crystal. Please refer to our
previous work wei2018 ; meng  for the fabrication details. After the electric
contacts (a-Ge and Al) were deposited, as shown in Fig. 5 (left), an acid-
resistant tape was used to cover the bottom, sides and a portion of the top
surface to create the outer contact of the PPC detector. To form the point
contact, an etch resistant paint (picine), as shown in Fig. 5 (left), was
applied in the center of the top surface. The exposed Al on the point contact
face was then removed using a diluted HF dip (100:1). The tape and paint were
removed and the detector fabrication was completed. Figure 5 (right) shows the
top view of the completed planar PPC detector, USD-W06, fabricated at USD. The
diameter of the point contact is around 0.6 mm and the thickness of the
detector is 9.4 mm. The width of the outer contact on the top surface of the
crystal is around 6.0 mm. Other dimensions of this detector are presented in
Table 1.

Figure 5: Left: The detector was covered with acid resistant tape and paint (picine) to form the outer contact and point contact, respectively. Right: The top view of the completed planar PPC Ge detector, USD-W06. The uneven edge of the outer contact was caused by cutting the acid resistant tape manually. Table 1: The dimensions of the planar PPC Ge detector, USD-W06, fabricated at USD in this work. |  Length (cm) |  Width (cm)
---|---|---
Crystal Top |  1.37 |  1.37
Crystal Bottom |  2.12 |  2.08

###  3.2 Detector Characterization

After the detector was fabricated, it was loaded in a variable-temperature
cryostat to conduct electrical and spectroscopy measurements, as shown in Fig.
6. This cryostat is provided by the Lawrence Berkeley National Laboratory
(LBNL). Figure 7 shows the electronics used for signal processing and
conducting electrical and spectroscopy measurements in this work.

Figure 6: Shown is the detector USD-W06 loaded in a variable-temperature
cryostat for characterization. Figure 7: Electronic circuit for detector
characterization.

####  3.2.1 Electrical Measurements

The following two electrical measurements were conducted in this study: (1)
the leakage current versus the applied voltage at 79 K; and (2) the detector
capacitance as a function of the applied voltage at 79 K. For both electrical
measurements, a positive bias voltage was applied to the bottom contact of the
detector, USD-W06, and the signals were read out from the top point contact.

Figures 8 and 9 present the measured leakage current and the detector
capacitance as a function of the detector bias voltage at 79 K. As shown in
Fig. 9, detector USD-W06 is fully depleted at 60 V with leakage current at
around 0.5 pA from Fig. 8. Also, as shown in Fig. 9, this planar PPC detector
is able to reach a low capacitance of 0.6 pF. Since the signal to background
is anti-proportional to the detector capacitance, a lower capacitance will
enhance the signal to background ratio and hence has the potential to become a
low-threshold detector if operated at an even lower temperature such as liquid
helium temperature.

Figure 8: Measured leakage current as a function of bias voltage at 79 K for
detector USD-W06. Low leakage current ($\sim$ 0.5 pA) at full depletion
voltage (60 V) shows the good quality of a-Ge contacts. Figure 9: Measured
detector capacitance as a function of bias voltage at 79 K for detector
USD-W06. The full depletion voltage of the detector was determined to be 60 V
since the detector capacitance became a constant after the detector was fully
depleted.

####  3.2.2 Spectroscopy Measurements

A radioactive source, Am-241, was used to conduct the spectroscopy
measurements. In this study, there are two setups for the collection of the
energy spectrum of Am-241: (1) the source was placed on the top of the
cryostat, facing the top side of the detector; and (2) the source was placed
at the bottom of the cryostat, facing the bottom side of the detector. For
each setup, the energy spectra of Am-241 at 79 K were acquired with positive
bias voltages of 200 V, 300 V and 400 V applied to the detector bottom with
the detector configuration shown in Fig. 6. The signals were read out from the
point contact using a charge sensitive pre-amplifier operated at room
temperature followed by an ORTEC 671 shaping amplifier set to an optimized
shaping time of 2 $\mu$s. Figure 10 shows the energy spectra of Am-241 taken
at three voltages with the source placed at the two positions mentioned above.
The data taking times for each of the applied voltages were 1.5 hours when the
source was placed on the top of the cryostat, while the data taking times were
14.5 hours, 9 hours and 8 hours for the applied bias voltages of 200 V, 300 V
and 400 V, respectively, when the source was placed at the bottom of the
cryostat. The x-axis is the energy in keV, which is converted from the ADC
count using three photon peaks from Am-241, $\sim$20.8 keV, $\sim$26.3 keV and
59.5 keV am241 , with the applied voltage at 200 V and the source placed at
the top of the cryostat. The calibration curve is shown in Fig. 11. The linear
fit function is: $E=p0+p1\cdot A$, where $E$ is the energy in keV, A is the
ADC channel number, and $p0$ and $p1$ are the fitting parameters shown in Fig.
11. From Fig. 10 we can only see that the resolution in the region of $\sim$20
keV is worse than the 59.5 keV peak. From atomic data we know that for high-Z
atoms, we expect clusters of X-rays. In addition, the energy region of
$\sim$26.3 keV, a $\gamma$ ray from Am-241 source is contaminated by several
X-rays from the $\sim$20 keV region, which are very close in energy.
Therefore, the two peaks were not well resolved.

Figure 10: Energy spectra of a Am-241 source measured with the detector
USD-W06. The spectra were taken at three voltages, 200 V, 300 V and 400 V with
the source placed at the top and the bottom of the cryostat for each voltage.
Figure 11: Converting the ADC channel number to the energy with linear
calibration function of $E=p0+p1\cdot A$, where $E$ is the energy, $A$ is the
ADC channel number, and $p0$ and $p1$ are the fitting parameters.

It is worth mentioning that there are more materials between the source and
the detector when the source was placed at the bottom of the cryostat compared
to that of the source at the top. Consequently, several low-energy X-rays and
$\gamma$ rays ($\sim$20.8 keV and 26.3 keV) seen with the source at the top
were not seen when the source was placed at the bottom. Instead, we observed
two emitted X-rays with energy 24 keV and 27.5 keV, as shown in Fig. 10, from
indium due to the process of fluorescence when 59.5 keV X-ray interacted with
the indium foil which is set underneath of the detector on the inside of the
cryostat. To confirm the fluorescent emission of X-rays from the inner
material of indium that is between the detector and the source from the
bottom, we conducted a Geant4-based Monte Carlo simulation geant4  for the
source at the bottom with a detailed geometry of the cryostat based on our
best understanding. Figure 12 shows a comparison between the measurement and
the simulation when the source is placed at the bottom. It was clear that two
distinct X-rays observed in Fig. 10 when the source was placed at the bottom
were from the fluorescent emission of indium. These two distinct X-rays
provide a good opportunity to exam the detector energy resolution for low-
energy X-rays with energy below 59.5 keV.

Figure 12: A comparison between the Monte Carlo simulation and the measurement
when the source was placed at the bottom. The blue line is the measured energy
spectrum and the red line is the simulated energy spectrum. The simulation
confirms the fluorescent emission of X-rays (24 keV and 27.5 keV) from the
inner material of indium.

To obtain the detector energy resolution, the two low energy peaks at 24 keV
and 27.5 keV were fitted together using two standard Gaussian distributions
with a linear background distribution, $p0\cdot
e^{[-\frac{1}{2}(\frac{E-p1}{p2})^{2}]}+p3\cdot
e^{[-\frac{1}{2}(\frac{E-p4}{p5})^{2}]}+p6+p7\cdot E$, where $p0$ and $p3$ are
the normalization constants, $p1$ and $p4$ are the center values in keV for
the energy peaks at 24 keV and 27.5 keV, respectively, $p2$ and $p5$ are the
Gaussian widths (sigma) in keV for the energy peaks at 24 keV and 27.5 keV,
respectively, $p6$ is a constant, $p7$ is the slope, and $E$ is the energy in
keV. The energy peak at 59.5 keV was partially fitted using a standard
Gaussian distribution, $p0\cdot e^{[-\frac{1}{2}(\frac{E-p1}{p2})^{2}]}$, with
$p0$, $p1$, $p2$ and $E$ defined above. This is because the background on the
left side is higher than the right side and we are only interested in knowing
the energy resolution. As examples, Figs. 13 and 14 show the fitting of these
three energy peaks at 400 V when the radiation source was placed at the bottom
of the cryostat. Figure 15 is a fitted artificial peak due to the injected
pulses from the high voltage line in order to obtain the noise level. The
electronic circuit presented in Fig. 7 shows how the pulse was generated to
measure the electronic noise of the detector. The peak in Fig. 15 was fitted
using a standard Gaussian distribution and linear background distribution,
$p0\cdot e^{[-\frac{1}{2}(\frac{E-p1}{p2})^{2}]}+p3+p4\cdot E$, with $p0$,
$p1$, $p2$ and $E$ defined above, $p3$ is a constant and $p4$ is the slope.
Table 2 summarizes the full width at half maximum (FWHM) of the three energy
peaks and the pulser peak shown in Figs. 13 $-$ 15. Also shown in Table 2 are
the statistically driven energy resolutions for each peak. These are
determined by subtracting in quadrature the FWHM of the injected pulse peak
from the FWHM of the respective energy peaks.

Figure 13: The fitted energy peaks at 24 keV and $\sim$27.5 keV from fluorescent emission of X-ray with applied voltage at 400 V when the source was placed at the bottom of the cryostat. Figure 14: The fitted energy peak at 59.5 keV from Am-241 with applied voltage at 400 V when the source was placed at the bottom of the cryostat. Figure 15: The fitted artificial peak due to the injected pulses from the high voltage line with applied voltage at 400 V when the source was placed at the bottom of the cryostat. Table 2: The FWHM of three energy peaks from Am-241 and the pulser peak at 400 V when the source was placed at the bottom of the cryostat. The energies presented here are the expected values. For each energy peak, the relative statistic driven energy resolution was determined by subtracting in quadrature the FWHM at the pulser from the FWHM at the energy peak and then divided by the energy. Energy Peak (keV) |  FWHM at the energy peak (keV) |  FWHM at the pulser (keV) |  Relative statistic driven energy resolution (%)
---|---|---|---
24.0 |  1.11 |  1.06 |  1.37
27.5 |  1.17 |  1.06 |  1.80
59.5 |  1.16 |  1.06 |  0.79

##  4 Discussion and Expectation

According to the discussion in Section 2, if the internal charge amplification
occurs in the spectroscopy measurements with Am-241, the energy peaks are
expected to shift significantly as a function of applied voltage. Unlike a
fraction of charge increasing due to the improvement of the charge collection
efficiency as a function of electric field, the internal charge amplification
multiplies charge production by more than a factor of 2 or higher depending on
the applied electric field. This multiplication of charge production can
significantly shift the peak positions. As shown in Fig. 10, for both
positions of the source, no evident energy position shift has been observed
for each energy peak. This indicates that the charge internal amplification
didn’t take place in the detector with applied voltage at 400 V.

From Section 2, the required electric field for the internal charge
amplification to occur is at least $\sim$15,000 V/cm at 79 K. To find out the
minimum applied voltage to allow our detector to achieve this required
electric field, we have investigated the electric field distribution inside
the detector using GeFiCa (Germanium Field Calculator) Li , which was created
to demonstrate analytic and numeric methods to calculate static electric
fields and potentials in HPGe detectors. A planar geometry with the same
dimensions as the real detector (except four wings where the electric field is
expected to be extremely low) has been built in GeFiCa. As shown in Figs. 16
and 17, the detector can be fully depleted at 60 V in GeFiCa and is in
agreement with the measurement as shown in Fig. 9. Given the inputs for the
detector dimensions and depletion voltage (60 V) in GeFiCa, the impurity
concentration of the crystal predicted by GeFiCa is 3.3$\times$109/cm3, which
is close to the measured impurity concentration of the two planar detectors
(USD-W04 and USD-RL01) mei2  made from the same crystal as the detector,
USD-W06, in this work. This validates the predication of the impurity
concentration by GeFiCa. Figures 18 and 19 show the distributions of the
electric field and potential inside the detector with an applied voltage of
400 V on the outside surface contact. As shown in Fig. 18, the maximum
electric field in the detector at 400 V is between 8,000 V/cm and 9,000 V/cm,
which is significantly lower than the required minimum electric field of
$\sim$15,000 V/cm for internal charge amplification to occur inside the
detector. This confirms that no internal charge amplification can be observed
in our detector with an applied voltage of 400 V. Figure 20 shows the expected
electric field distribution inside the detector with an applied voltage of 900
V predicted by GeFiCa. This indicates that, to observe the internal charge
amplification with our detector, the applied voltage needs to be at least
around 900 V to reach the required electric field of $\sim$15,000 V/cm.
Unfortunately, due to the fact that the leakage current of our detector jumped
to a very high value starting from around 420 V because of field penetration
into a-Ge, we were unable to apply the bias voltage beyond 400 V. For our
future work, we will optimize our fabrication process to improve the quality
of the a-Ge contact so that it will allow the detector to withstand the
required electric field to observe the internal charge amplification.

Figure 16: The potential field distribution in the detector when a positive
voltage of 55 V was applied to the outside surface contact. The white area
indicates that the detector is not fully depleted. Figure 17: The potential
field distribution in the detector when a positive voltage of 60 V was applied
to the outside surface contact.The absence of any white area indicates that
the detector is fully depleted. Figure 18: The electric field distribution in
the detector when a positive voltage of 400 V was applied to the outside
surface contact. Figure 19: The potential field distribution in the detector
when a positive voltage of 400 V was applied to the outside surface contact.
Figure 20: The electric field distribution in the detector when a positive
voltage of 900 V was applied to the outside surface contact.

##  5 Conclusions

We have experimentally shown that a planar PPC detector can be fabricated
successfully using USD-grown crystals with a-Ge contacts. The minimum required
electric field to achieve internal charge amplification is derived using the
work done on the charge carriers by the externally applied electric field.
This work done must be equal to the sum of the kinetic energy gained by the
charge carriers and the energy loss to the emission of photons. This relation
is required by the energy conservation during the drifting process. From the
calculation using GeFiCa, we find that the planar PPC Ge detector can be used
to achieve the high electric field needed for triggering internal charge
amplification. Such a detector, when operated at a bias voltage of higher than
900 V, will generate a field strength that is higher than 15,000 V/cm near the
point contact. This high field will then amplify charge carriers to generate
more charge carriers. We examined the energy resolution of the planar PPC
detector and found that a good energy resolution can be achieved with the
current electronics as shown in Table 2. If a low-noise electronic system is
adapted and the energy resolution is dominated by the statistical variation,
the energy resolution of 1.37% at 24 keV, 1.80% at 27.5 keV, and 0.79% at 59.5
keV can be achieved. If internal charge amplification can be realized by this
detector with an amplification factor of 1000, one can achieve a low energy
threshold of $\sim$0.33 eV (1.37%$\times$24 keV/1000). This suggests that
planar PPC detectors can be used for low-mass dark matter searches.

###### Acknowledgements.

The authors would like to thank Mark Amman for his instructions on fabricating
planar detectors, and the Nuclear Science Division at Lawrence Berkeley
National Laboratory for providing the vacuum cryostat. We would also like to
thank Christina Keller for a careful reading of this manuscript. This work was
supported in part by NSF NSF OISE 1743790, DOE grant DE-FG02-10ER46709, DE-
SC0004768, the Office of Research at the University of South Dakota and a
research center supported by the State of South Dakota.

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A Note on Coupled Dirac Operators

A Note on Coupled Dirac Operators††This paper is a contribution to the Special
Issue on Differential Geometry Inspired by Mathematical Physics in honor of
Jean-Pierre Bourguignon for his 75th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Bourguignon.html

Nigel J. HITCHIN

N.J. Hitchin

Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK
<EMAIL_ADDRESS>https://people.maths.ox.ac.uk/hitchin/

Received November 01, 2022, in final form January 10, 2023; Published online
January 13, 2023

The article considers some concrete solutions to the Dirac equation coupled to
a vector bundle with connection, arising in the study of Yang–Mills equations
and vector bundles on Riemann surfaces.

Dirac equation; spinor; Yang–Mills; holomorphic structure; index theorem

53C07; 14D20

Dedicated to Jean-Pierre Bourguignon
on the occasion of his 75th birthday

##  1 Introduction

I first got to know Jean-Pierre in Paris in the mid 1970s when I visited
Berger’s group in Riemannian geometry. At the time I was following up my
thesis topic, working on the Dirac operator in various contexts. In the years
that followed (when Jean-Pierre was not occupied with his tireless work for
the international community of mathematicians) he shared my interest and, as
evidenced by [5, 6, 7] for example, he was equally intrigued by questions
involving spinors – geometrically less accessible than vectors and tensors but
often quite powerful in applications.

My early focus was on index theory and Weitzenböck formulas, which provide
existence theorems for solutions of the Dirac equation. In this note we
consider some concrete examples of the Dirac operator coupled to a vector
bundle with connection, and encounter situations where the null space has
bigger then the expected value given by the index theorem, cases related to my
mathematical interests over the years.

The first concerns the ${\rm SU}(2)$ Yang–Mills equations on $S^{4}$. A
vanishing theorem for the Dirac equation coupled to the fundamental
representation gives one of the vector spaces which plays a fundamental role
in the ADHM construction of the self-dual solutions, and in fact coupling to
any representation the index theorem determines the dimension of the space of
solutions. For many years an outstanding question was whether all Yang–Mills
instantons were self-dual or anti-self-dual (indeed the papers [8, 9] were a
major contribution to the discussion) so it was natural to look for features
exhibited by any solution. We describe in Section 2 an observation from that
period showing that, using solutions to the twistor equation, a non-self-dual
instanton has at least a 4-dimensional space of solutions (of either
chirality) to the Dirac equation coupled to the adjoint representation. The
existence of non-self-dual solutions has since been proved [16, 17] and since
the latter example is reasonably explicit this gives a construction.

The second case relates to the Dirac operator on a Riemann surface, which is
the $\bar{\partial}$-operator acting on sections of a square root $K^{1/2}$ of
the canonical bundle. Coupling to a vector bundle $E$ with connection, the
null space can be identified with holomorphic sections of $E\otimes K^{1/2}$.
If $E$ has degree zero the index theorem (here the Riemann–Roch theorem) gives
no information about existence of solutions. When $E$ is associated to a
symplectic representation of a group $G$ then the moment map applied to a non-
zero section $\psi$ of $E\otimes K^{1/2}$, when it exists, gives a Higgs field
– a section of $\mathfrak{g}\otimes K$. These are important objects and,
following the suggestion of physicists, this issue was investigated further in
[13]. Here, in Section 3, we recall from that paper the construction of $\psi$
for an irreducible symplectic representation of ${\rm SU}(2)$, using points of
finite order in an abelian variety.

By contrast, when $E$ arises from a representation with an orthogonal
structure, and has odd rank, the mod 2 index theorem of Atiyah and Singer
guarantees the existence of such a $\psi$ depending on whether the spin
structure is of odd or even type. We investigate in Section 4 the construction
of such sections when $E$ is the adjoint bundle for the group ${\rm SU}(2)$,
using some classical descriptions of moduli spaces of stable bundles in genus
2.

##  2 The Yang–Mills equations

Let $E$ be a vector bundle on a Riemannian 4-manifold $M$. A connection $A$ on
$E$ defines an exterior covariant derivative ${\rm
d}_{A}\colon\Omega^{p}(M,E)\rightarrow\Omega^{p+1}(M,E)$ and ${\rm
d}_{A}^{2}=F_{A}\in\Omega^{2}(M,\operatorname{End}E)$ is its curvature. The
Yang–Mills equation for $A$ is ${\rm d}_{A}\ast F_{A}=0$, where $\ast$ is the
Hodge star operator on 2-forms. Since $\ast^{2}=1$ on 2-forms we have a
decomposition into eigenspaces
$\Lambda^{2}T^{*}=\Lambda^{2}_{+}\oplus\Lambda^{2}_{-}$ and accordingly
$F_{A}=F_{A}^{+}+F_{A}^{-}$ so the equation is ${\rm
d}_{A}(F_{A}^{+}-F_{A}^{-})=0$. The Bianchi identity ${\rm
d}_{A}(F_{A}^{+}+F_{A}^{-})=0$ implies that the Yang–Mills equation is
equivalent to ${\rm d}_{A}F_{A}^{+}=0={\rm d}_{A}F_{A}^{-}$ and so if
$F_{A}^{-}=0$, meaning the connection is self-dual, we have automatically a
solution.

When $M$ is a spin manifold we have a spinor bundle $S=S^{+}\oplus S^{-}$ with
connection induced from the Levi-Civita connection and the Clifford action
$T^{*}\otimes S^{+}\rightarrow S^{-}$, $T^{*}\otimes S^{-}\rightarrow S^{+}$.
The Dirac operator $\not{D}$ on $S^{+}$ is then defined as the composition
$C^{\infty}(S^{+})\stackrel{{\scriptstyle\nabla}}{{\rightarrow}}C^{\infty}(T^{*}\otimes
S^{+})\rightarrow C^{\infty}(S^{-})$. Using a connection on an auxiliary
vector bundle $E$ to define one on $S^{+}\otimes E$ we obtain the coupled
Dirac operator $\not{D}_{A}\colon C^{\infty}(S^{+}\otimes E)\rightarrow
C^{\infty}(S^{-}\otimes E)$.

The index of this operator is
$\dim\ker\not{D}_{A}-\dim\operatorname{coker}\not{D}_{A}=\mathrm{ch}(E)\hat{A}[M]$.
When the connection is self-dual, the curvature $F_{A}=F_{A}^{+}$ acts
trivially in the Weitzenböck formula for $\not{D}_{A}^{*}$ leaving just one
quarter of the scalar curvature and so if $M$ has positive scalar curvature,
like $M=S^{4}$, $\ker\not{D}_{A}^{*}=0$. Then the dimension of the null space
of $\not{D}_{A}$ is given by the index.

There is another differential operator defined on spinors, the twistor
operator $D$. This is obtained from $\nabla\colon C^{\infty}(S)\rightarrow
C^{\infty}(T^{*}\otimes S)$ by orthogonal projection onto the kernel of
$T^{*}\otimes S\rightarrow S$. In terms of a local orthonormal basis this is

$D\psi=\sum_{i=1}^{4}\big{(}e_{i}\otimes\nabla_{i}\psi+\tfrac{1}{4}e_{i}\otimes
e_{i}\cdot\not{D}\psi\big{)},$

where the dot denotes Clifford multiplication. This is an overdetermined
system but on the sphere $S^{4}$ it has a 4-dimensional space of solutions for
$\psi\in S^{+}$ and similarly for $S^{-}$. It is in fact conformally invariant
and by stereographic projection the solutions on $\mathbb{R}^{4}$ are affine
linear spinors $x\cdot\psi_{1}+\psi_{2}$, where $\psi_{1}$, $\psi_{2}$ are
constant spinors.

The Dirac operator is also conformally invariant acting on spinors with weight
3/2 [12] whereas the spinors for the twistor operator have weight $-1/2$. The
Yang–Mills equations are conformally invariant because the Hodge star is
conformally invariant in the middle dimension. In the context of conformal
invariance of operators this mean that the exterior derivative on 2-forms has
conformal weight $2$. A 2-form acts on a spinor by Clifford multiplication so
that a 2-form with weight 2 acting on a spinor with weight $-1/2$ gives a
spinor with weight $3/2$. The following proposition is thus conformally
invariant:

######  Proposition 2.1.

Let $F_{A}^{-}$ be the anti-self-dual component of a non-self-dual solution to
the Yang–Mills equations on $S^{4}$, and $\psi\in C^{\infty}(S^{-})$ a
solution to the twistor equation. Then
$F_{A}^{-}\cdot\psi\in\mathbb{C}^{\infty}(S^{-}\otimes\operatorname{End}E)$
satisfies the coupled Dirac equation.

######  Proof 2.2.

Let $e_{1},\dots,e_{4}$ be a local orthonormal basis of 1-forms with volume
form $e_{1}\wedge e_{2}\wedge e_{3}\wedge e_{4}$. Then a basis of anti-self-
dual 2-forms is given by $e_{1}\wedge e_{2}-e_{3}\wedge e_{4}$, $e_{2}\wedge
e_{3}-e_{1}\wedge e_{4}$, $e_{3}\wedge e_{1}-e_{2}\wedge e_{4}$. The Clifford
algebra has relations $e_{i}\cdot e_{j}+e_{j}\cdot e_{i}=-2\delta_{ij}$ and
the action of $F_{A}^{-}$ on $\psi$ is through the elements $e_{1}\cdot
e_{2}-e_{3}\cdot e_{4}$ etc. in the algebra. Using this identification of the
exterior forms with the Clifford algebra (as vector spaces) we have the
following identities: if $\omega$ is an anti-self-dual 2-form and $a$ a 1-form
then

$a\cdot\omega=a\wedge\omega-\ast(a\wedge\omega),\qquad\sum_{i=1}^{4}e_{i}\cdot\omega\cdot
e_{i}=0.$

Now

$\not{D}_{A}(F_{A}^{-}\cdot\psi)=\sum_{i=1}^{4}e_{i}\cdot\nabla_{i}F_{A}^{-}\cdot\psi+e_{i}\cdot
F_{A}^{-}\cdot\nabla_{i}\psi=\sum_{i=1}^{4}e_{i}\cdot\nabla_{i}F_{A}^{-}\cdot\psi-\tfrac{1}{4}e_{i}\cdot
F_{A}^{-}\cdot e_{i}\cdot\not{D}\psi,$

using the fact that $\psi$ satisfied the twistor equation
$\nabla_{i}\psi=-e_{i}\cdot\not{D}\psi/4$. But
$\sum_{i=1}^{4}e_{i}\cdot\nabla_{i}F_{A}^{-}={\rm d}_{A}F_{A}^{-}-\ast{\rm
d}_{A}F_{A}^{-}$ from the first identity and ${\rm d}_{A}F_{A}^{-}=0$ from the
Yang–Mills equations so the first term is zero. The second term vanishes using
the second identity.

######  Remark 2.3.

In [17], a non-self-dual solution was produced by a min-max argument for the
trivial bundle on $S^{4}$. The index is zero here and so there is a canonical
determinant section of the determinant line bundle which vanishes on those
connections for which there exists a solution to the coupled Dirac equation.
The proposition shows that all non-self-dual Yang–Mills connections lie on
this divisor. In fact since $\operatorname{End}E$ is a real bundle and $S^{-}$
is quaternionic the space of solutions is even dimensional so the determinant
section is the square of a section of a Pfaffian bundle.

######  Remark 2.4.

The non-self-dual solution given in [16] has a high degree of symmetry – it is
invariant by the action of ${\rm SO}(3)$ on the unit sphere in the
5-dimensional irreducible representation on trace-free symmetric $3\times 3$
matrices. The proof of existence reduces to analysis of the boundary
conditions for a system of ODEs. In principle we can write down the coupled
spinor fields constructed as above but the resulting formula is not
particularly rewarding.

######  Remark 2.5.

If the connection is anti-self-dual then the proposition still yields
solutions by applying the curvature $F_{A}=F_{A}^{-}$ to solutions of the
twistor equation. Note that the index theorem gives in this case the dimension
of the null space as $4k$, where $k$ is the Chern class [2, Section 10]. In
particular for $k=1$ we see that all four solutions are produced this way. The
connection in question is here the Levi-Civita connection on $S^{-}$ for a
standard constant curvature metric on $S^{4}$ .

##  3 Riemann surfaces – the symplectic case

The conformal invariance of the Dirac equation in two dimensions has a more
important consequence since it can be defined in terms of the holomorphic
structure and so becomes part of the algebraic geometry of a compact
projective curve $C$ of genus $g$. A spin structure is defined by the
holomorphic structure on a square root $K^{1/2}$ of the canonical bundle $K$
and the Dirac operator is just $\not{D}=\bar{\partial}\colon
C^{\infty}\big{(}K^{1/2}\big{)}\rightarrow
C^{\infty}\big{(}K^{1/2}\bar{K}\big{)}$. In this case, the index theorem,
which is the Riemann–Roch theorem, gives $\dim
H^{0}\big{(}C,K^{1/2}\big{)}-\dim H^{1}\big{(}C,K^{1/2}\big{)}=0$. When
coupled to a vector bundle $E$ with connection, the $(0,1)$ component of the
connection defines a $\bar{\partial}$-operator on $E$ and the null space of
the coupled Dirac operator is $H^{0}\big{(}C,E\otimes K^{1/2}\big{)}$ and if
$c_{1}(E)=0$ the index is again zero.

Suppose we are given a holomorphic principal $G$-bundle over $C$ and we take a
symplectic representation to yield a vector bundle $E$. The moment map for a
symplectic action of $G$ on a vector space $V$ is a quadratic map from $V$ to
$\mathfrak{g}^{*}$, the dual of the Lie algebra. Then, using a bi-invariant
metric, a holomorphic section of $E\otimes K^{1/2}$ gives a section of
$\mathfrak{g}\otimes K$. These are called Higgs fields and the moduli space of
bundles together with Higgs fields has gained importance in various fields.
The construction above of special Higgs fields was suggested by the physicist
D. Gaiotto [10] and described in [13].

We shall consider the irreducible symplectic representations of ${\rm
SL}(2,\mathbb{C})$. These are the symmetric powers $S^{m}\mathbb{C}^{2}$ for
odd $m$ of the basic representation on $\mathbb{C}^{2}$ or equivalently the
polynomials in one variable of odd degree $m$ under the action
$p(z)\mapsto(cz+d)^{m}p((az+b)/(cz+d)).$ Set $m=2k-1$ and write
$p(z)=a_{0}z^{m}+a_{1}z^{m-1}+\dots+a_{m}$, then the invariant symplectic form
on the space $S^{m}\mathbb{C}^{2}$ is up to a constant

$\omega=\sum_{\ell=0}^{k-1}(-1)^{\ell}\ell!(m-\ell)!{\rm d}a_{i}\wedge{\rm
d}a_{m-i}.$ (1)

The even symmetric powers $S^{m}\mathbb{C}^{2}$ are the orthogonal irreducible
representations and $m=2$ is the adjoint representation. The moment map can be
viewed as $\mu\colon S^{m}\mathbb{C}^{2}\otimes S^{m}\mathbb{C}^{2}\rightarrow
S^{2}\mathbb{C}^{2}$ given by contraction using the symplectic form.

First consider $m=1$. We start with a rank 2 holomorphic vector bundle $V$
with $\Lambda^{2}V$ trivial. Multiples of a holomorphic section $\psi$ of
$V\otimes K^{1/2}$ generate a subbundle $L^{*}\subset V$ with quotient $L$.
Then $\psi$ is a section of the line bundle $L^{*}K^{1/2}$. This has degree
$g-1-\deg L$. So to construct such a $\psi$ we take an effective divisor $D$
of degree $<g-1$ and define $L$ to be $K^{1/2}(-D)$ and the vector bundle to
be given by an extension class in
$H^{1}\big{(}C,L^{-2}\big{)}=H^{1}(C,K(-2D))$.

The moment map in this case gives an everywhere nilpotent Higgs field
$\Phi=\psi\otimes\psi$, but for $m>1$ this is not necessarily so and we can
make use of the relation to Higgs bundles and adopt the spectral curve
approach. We start with a holomorphic section $q$ of $K^{2}$ with simple zeros
and define the curve $S$ in the total space $|K|$ of $K$ (i.e., the cotangent
bundle of $C$) by the equation $x^{2}=\pi^{*}q$. Here $\pi\colon|K|\rightarrow
C$ is the projection and $x$ is the tautological section of $\pi^{*}K$ on the
total space $|K|$. Then $S$ is a smooth curve of genus $4g-3$ and has a
covering involution $\sigma$ defined by $x\mapsto-x$. The direct image of a
line bundle $L$ on $S$ is a rank 2 bundle $V$ on $C$ and if $L\cong
U\pi^{*}K^{1/2}$ and $\sigma^{*}U\cong U^{*}$ then $\Lambda^{2}V$ is trivial
(see for example [4]). The condition on $U$ can be interpreted as saying it
lies in the Prym variety of the map $\pi\colon S\rightarrow C$, an abelian
variety of dimension $4g-3-g=3g-3$.

Proposition 3 of [13] shows how to find a solution to the Dirac operator
coupled to $S^{m}V$ from a line bundle of order $m$ on $S$ – a point of finite
order in the Prym variety. For simplicity we consider the value $m=3$. For our
purposes we are not interested in the Higgs field, only its spectral curve.

######  Proposition 3.1.

Let $U$ be a non-trivial line bundle on $S$ such that $U^{3}$ is trivial. Then
the direct image $V$ of $U\pi^{*}K^{1/2}$ admits a holomorphic section of
$S^{3}V\otimes K^{1/2}$.

######  Proof 3.2.

The definition of direct image $V=\pi_{*}L$ is that for each open set
$W\subset C$, $H^{0}(W,\pi_{*}L)=H^{0}\big{(}\pi^{-1}(W),L\big{)}$ so that
evaluation of a local section of $L$ expresses the pull-back $\pi^{*}V$ as an
extension

$0\rightarrow L^{*}\rightarrow\pi^{*}V\rightarrow L\rightarrow 0.$ (2)

and this induces a filtration of $\pi^{*}S^{3}V$ into subbundles $V_{0}\subset
V_{1}\subset V_{2}\subset V_{3}$ with $V_{0}\cong L^{-3}$.

Consider the subbundle $V_{1}$. Comparing with the filtration by degree of
polynomials $a_{0}z^{3}+a_{1}z^{2}+a_{2}z+a_{3}$ it is the subspace
$a_{0}=a_{1}=0$, the polynomials which vanish to order $2$ at $\infty$. It is
maximally isotropic with respect to the symplectic form (1) with $m=3$. It
follows that $\pi^{*}S^{3}V$ is an extension

$0\rightarrow V_{1}\rightarrow\pi^{*}S^{3}V\rightarrow V_{1}^{*}\rightarrow 0$
(3)

Now $\pi^{*}V$ is invariant by $\sigma$ so from (2) we have another extension

$0\rightarrow\sigma^{*}L^{*}\rightarrow\pi^{*}V\rightarrow\sigma^{*}L\rightarrow
0$

and this splits (2) outside the divisor given by $x=0$. The extension class is
then supported on $x=0$, the fixed point set of $\sigma$, where
$L\cong\sigma^{*}L$. In Dolbeault terms we have a section of
$L\sigma^{*}L^{*}$ on $x=0$, extend it holomorphically to a neighbourhood and
then everywhere as a $C^{\infty}$ section $s$ supported in a neighbourhood of
the fixed point set. The extension class in $H^{1}\big{(}S,L^{-2}\big{)}$ is
then represented by $\bar{\partial}s/x\in\Omega^{0,1}\big{(}S,L^{-2}\big{)}$.

The Dolbeault representative for the holomorphic structure of $V$ as an
extension gives a $C^{\infty}$-splitting $\pi^{*}V=L\oplus L^{*}$ and a
$\bar{\partial}$-operator $\bar{\partial}+\alpha$, where
$\alpha\in\Omega^{0,1}\big{(}S,L^{-2}\big{)}$. Then $\pi^{*}S^{3}V=L^{3}\oplus
L\oplus L^{-1}\oplus L^{-3}$ and a holomorphic section of
$\pi^{*}\big{(}S^{3}V\otimes K^{1/2}\big{)}$ can be written as
$a_{0}z^{3}+a_{1}z^{2}+a_{1}z+a_{0}$, where

$\bar{\partial}a_{0}=0,\qquad\bar{\partial}a_{1}+3\alpha
a_{0}=0,\qquad\bar{\partial}a_{2}+2\alpha
a_{1}=0,\qquad\bar{\partial}a_{3}+\alpha a_{2}=0$

and $a_{0}\in\Omega^{0}\big{(}S,L^{3}\pi^{*}K^{1/2}\big{)}$,
$a_{1}\in\Omega^{0}\big{(}S,L\pi^{*}K^{1/2}\big{)}$, etc.

The line bundle $U^{3}$ is trivial. Let $u$ be a trivialization such that
$\sigma^{*}u=u^{-1}$ and set $a_{0}=ux^{2}$. Then $3\alpha
a_{0}=3u(\bar{\partial}s)x=\bar{\partial}(3usx)$ so we can take $a_{1}=-3usx$
and extend $a_{0}$ to get a holomorphic section $v$ of
$V_{1}^{*}\otimes\pi^{*}K^{1/2}$. This vanishes on $x=0$ because $a_{0}$ and
$a_{1}$ do. Any two choices of $a_{1}$ differ by an element of
$H^{0}\big{(}S,L\pi^{*}K^{1/2}\big{)}$ and if they are divisible by $x$ then
this is equivalent to sections of $L\pi^{*}K^{-1/2}=U$. Thus if $U$ is non-
trivial there is a unique extension divisible by $x$.

To extend $v$ to a section of $\pi^{*}\big{(}S^{3}V\otimes K^{1/2}\big{)}$ we
observe that the vector bundle extension (3) is defined by a class in
$H^{1}\big{(}S,\operatorname{Hom}(V_{1}^{*},V_{1})\big{)}$ also of the form
$\bar{\partial}a/x$ and since $v$ as constructed is divisible by $x$, $v$
extends. In this case any two extensions differ by $v^{\prime}\in
H^{0}\big{(}S,V_{1}\otimes\pi^{*}K^{1/2}\big{)}$.

But we have

$0\rightarrow L^{-3}=V_{0}\rightarrow V_{1}\rightarrow L^{-1}\rightarrow 0$

and $L^{-3}\cong\pi^{*}K^{-3/2}$, so $V_{0}\otimes\pi^{*}K^{1/2}$ is a line
bundle of negative degree and so has no sections. Moreover
$L^{-1}\pi^{*}K^{1/2}\cong U^{-1}$ has degree zero and is non-trivial so this
has no holomorphic sections either. It follows from the exact cohomology
sequence that $H^{0}\big{(}S,V_{1}\otimes\pi^{*}K^{1/2}\big{)}=0$ and there is
a unique extension.

This gives us a section $v^{\prime\prime}$ of $\pi^{*}\big{(}S^{3}V\otimes
K^{1/2}\big{)}$ on $S$. But our choice of trivialization $u$ implies that
$\sigma$ takes $ux^{2}$ to $u^{-1}x^{2}$ and by uniqueness
$\sigma^{*}v^{\prime\prime}=v^{\prime\prime}$. Since $\pi^{*}V$ is pulled back
from $C$, $v^{\prime\prime}$ descends to a section $\psi$ of $S^{3}V\otimes
K^{1/2}$.

######  Remark 3.3.

Proposition 2 in [13] is the converse of this: a generic section of
$S^{3}V\otimes K^{1/2}$ is obtained this way.

######  Remark 3.4.

In the same article it is also shown that the curve $S$ in the above
construction is in fact the spectral curve of the Higgs field $\mu(\psi)$.

##  4 Riemann surfaces – the orthogonal case

###  4.1 Parity

If we attempt a construction like the one above for orthogonal representations
of ${\rm SL}(2,\mathbb{C})$ then the homomorphism for $m$ even
$S^{m}\mathbb{C}^{2}\otimes S^{m}\mathbb{C}^{2}\rightarrow
S^{2}\mathbb{C}^{2}$ is skew-symmetric. Then we need at least two solutions of
the coupled Dirac operator to combine into a Higgs field. We shall not pursue
this here but instead focus on the parity of the dimension of the null space.
The integer index theorem via Riemann–Roch gives us no information but the mod
2 index theorem [3] does:

######  Proposition 4.1.

Let $E$ be a vector bundle over $C$ with a non-degenerate symmetric bilinear
form. If $E$ has odd rank and Stiefel–Whitney class $w_{2}(E)=0$ then $\dim
H^{0}\big{(}C,E\otimes K^{1/2}\big{)}=\dim H^{0}\big{(}C,K^{1/2}\big{)}$ mod
$2$. If $w_{2}(E)\neq 0$ then the dimension is $1+\dim
H^{0}\big{(}C,K^{1/2}\big{)}$ mod $2$.

Once one has established that $\dim H^{0}\big{(}C,E\otimes K^{1/2}\big{)}$ mod
2 is a $C^{\infty}$ deformation invariant then this can be proved by deforming
$E$ continuously to be a trivial bundle $\mathbb{C}^{2m+1}$ in the
$w_{2}(E)=0$ case. Then $\dim H^{0}\big{(}C,E\otimes K^{1/2}\big{)}$ mod 2 =
$(2m+1)\dim H^{0}\big{(}C,K^{1/2}\big{)}$ mod 2 = $\dim
H^{0}\big{(}C,K^{1/2}\big{)}$ mod 2 and similarly for the other case. This is
described in [15].

In the case of $K^{1/2}$ alone (the basic Dirac operator) these square roots
are classically known as theta characteristics and of the $2^{2g}$ choices of
square root, or spin structure, there are $2^{g-1}(2^{g}-1)$ odd ones and
$2^{g-1}(2^{g}+1)$ even ones. When the parity is odd then we know that there
is a non-zero solution to the Dirac equation.

We shall restrict ourselves to the adjoint representation of ${\rm
SL}(2,\mathbb{C})$, or the rank $3$ vector bundle $\operatorname{End}_{0}V$ of
trace free endomorphisms of a rank $2$ bundle $V$ with symmetric form
$\operatorname{tr}AB$. Here $w_{2}(\operatorname{End}_{0}V)=\deg V$ mod 2.

######  Remark 4.2.

It is possible to give a spectral curve description of bundles $V$ with
sections $\psi$ of $\operatorname{End}_{0}V\otimes K^{1/2}$ by taking a
covering of $C$ branched over the zeros of a holomorphic one-form as in [15],
rather than a quadratic differential as in the previous section, but spectral
curve methods have the disadvantage of not separating the roles of bundle and
section. In the case of Higgs bundles, fixing a stable bundle and fixing a
spectral curve give two different Lagrangian submanifolds whose intersection
is in general difficult to define, but is of great interest.

In what follows we shall consider the case where $C$ has genus $2$ and try and
construct the solutions which are guaranteed by parity.

###  4.2 Genus 2 even degree

Let $C$ be a genus $2$ curve expressed as a double covering of ${\rm P}^{1}$
branched over six points $x_{1},\dots,x_{6}$. We have
$2^{g-1}\big{(}2^{g}-1\big{)}=6$ odd spin structures. In higher genus the odd
dimension of $H^{0}\big{(}C,K^{1/2}\big{)}$ can vary with the modulus of the
curve but for genus $2$ it must be one. So the line bundles $K^{1/2}$ have
degree $g-1=1$ with a single section $s$ which vanishes at one point and these
are the preimages $\tilde{x}_{i}$ of the branch points. These then define
solutions of the basic Dirac equation on $C$.

The rank 2 stable bundles $V$ with $\Lambda^{2}V$ trivial have a moduli space
which is three-dimensional and (if we include S-equivalence classes of
semistable bundles) can be identified with ${\rm P}^{3}$ [14]. The tangent
space at $[V]$ of the moduli space is naturally isomorphic to
$H^{1}(C,\operatorname{End}_{0}V)$ and the cotangent space is
$H^{0}(C,\operatorname{End}_{0}V\otimes K)$ by Serre duality. Since the rank
of $\operatorname{End}_{0}V$ is odd Proposition 4.1 implies that we have a
section $\psi$ of $\operatorname{End}_{0}V\otimes K^{1/2}$ if $K^{1/2}$ is
odd. Then $\Phi=s\psi$ is a section of $\operatorname{End}_{0}V\otimes K$
which vanishes at $\tilde{x}_{i}$. For generic bundles $E$ there is just a
one-dimensional space of sections and this in fact holds in this low genus
case. We thus have a distinguished one-dimensional subspace in each cotangent
space of ${\rm P}^{3}$, at least over the open set of stable bundles.

To find this consider $\alpha\in H^{1}\big{(}C,K^{-1}\big{)}$ and the
homomorphism

$\alpha\colon\ H^{0}(C,\operatorname{End}_{0}V\otimes K)\rightarrow
H^{1}(C,\operatorname{End}_{0}V).$

The dual space of $H^{1}\big{(}C,K^{-1}\big{)}$ is
$H^{0}\big{(}C,K^{2}\big{)}$ and evaluation of a section of $K^{2}$ at
$\tilde{x}_{i}$ defines up to a multiple a class $\alpha_{i}$ whose kernel is
the required one-dimensional space. An alternative description is to take the
Serre duality pairing
$\big{\langle}\alpha,\operatorname{tr}\Phi^{2}\big{\rangle}$ to define a conic
in each projective plane ${\rm P}\big{(}H^{0}(C,\operatorname{End}_{0}V\otimes
K)\big{)}$ and for $\alpha_{i}$ this is a pair of lines whose intersection
defines the one-dimensional space. Evaluation of this quadratic function was
achieved in a very explicit manner in [11] following basic work in [18].

A section of $K^{2}$ may be written as
$\big{(}a_{0}+a_{1}x+a_{2}x^{2}\big{)}{\rm d}x^{2}/y^{2}$, where the equation
of $C$ is $y^{2}=(x-x_{1})\cdots(x-x_{6})$. The cotangent bundle of ${\rm
P}^{3}={\rm P}(U)$ is written as the (symplectic) quotient $\\{(p,q)\in
U^{*}\times U\colon\langle p,q\rangle=0,\,q\neq 0\\}/\mathbb{C}^{*}$, where
$\lambda\in\mathbb{C}^{*}$ acts as $\big{(}\lambda^{-1}p,\lambda q\big{)}$. So
$q_{i}$ are homogeneous coordinates in ${\rm P}^{3}={\rm P}(U)$ and the
$p_{i}$ linear in the fibres of $T^{*}{\rm P}^{3}$. Then up to a factor the
quadratic map $\operatorname{tr}\Phi^{2}$ is given in [11] by

$h=\sum_{i\neq j}\frac{r_{ij}(q,p)}{(x-x_{i})(x-x_{j})}{\rm
d}x^{2}=\sum_{i\neq j}\prod_{k\neq i,j}(x-x_{k}){r_{ij}(q,p)}\frac{{\rm
d}x^{2}}{y^{2}},$

where $r_{12}(q,p)=(q_{1}p_{1}+q_{2}p_{2}-q_{3}p_{3}-q_{4}p_{4})^{2}$ and

$\displaystyle
r_{13}(q,p)=(q_{1}p_{4}-q_{2}p_{3}-q_{3}p_{2}+q_{4}p_{1})^{2},\qquad$
$\displaystyle
r_{14}(q,p)=-(q_{1}p_{4}+q_{2}p_{3}-q_{3}p_{2}-q_{4}p_{1})^{2},$
$\displaystyle
r_{15}(q,p)=-(q_{1}p_{3}-q_{2}p_{4}-q_{3}p_{1}+q_{4}p_{2})^{2},\qquad$
$\displaystyle r_{16}(q,p)=(q_{1}p_{3}+q_{2}p_{4}+q_{3}p_{1}+q_{4}p_{2})^{2},$
$\displaystyle
r_{23}(q,p)=-(q_{1}p_{4}-q_{2}p_{3}+q_{3}p_{2}-q_{4}p_{1})^{2},\qquad$
$\displaystyle r_{24}(q,p)=(q_{1}p_{4}+q_{2}p_{3}+q_{3}p_{2}+q_{4}p_{1})^{2},$
$\displaystyle
r_{25}(q,p)=(q_{1}p_{3}-q_{2}p_{4}+q_{3}p_{1}-q_{4}p_{2})^{2},\qquad$
$\displaystyle
r_{26}(q,p)=-(q_{1}p_{3}+q_{2}p_{4}-q_{3}p_{1}-q_{4}p_{2})^{2},$
$\displaystyle
r_{34}(q,p)=(q_{1}p_{1}-q_{2}p_{2}+q_{3}p_{3}-q_{4}p_{4})^{2},\qquad$
$\displaystyle r_{35}(q,p)=(q_{1}p_{2}+q_{2}p_{1}+q_{3}p_{4}+q_{4}p_{3})^{2},$
$\displaystyle
r_{36}(q,p)=-(q_{1}p_{2}-q_{2}p_{1}-q_{3}p_{4}+q_{4}p_{3})^{2},\qquad$
$\displaystyle
r_{45}(q,p)=-(q_{1}p_{2}-q_{2}p_{1}+q_{3}p_{4}-q_{4}p_{3})^{2},$
$\displaystyle
r_{46}(q,p)=(q_{1}p_{2}+q_{2}p_{1}-q_{3}p_{4}-q_{4}p_{3})^{2},\qquad$
$\displaystyle r_{56}(q,p)=(q_{1}p_{1}-q_{2}p_{2}-q_{3}p_{3}+q_{4}p_{4})^{2}.$

Evaluating at $x=x_{1}$ gives a linear combination of the terms $r_{1i}$. Each
of these is a quadratic form $\ell_{i}\otimes\ell_{i}$, where
$\ell_{i}(p_{1},p_{2},p_{3},p_{4})$ is a linear form. But on examination one
sees that $\ell_{i}(q_{2},{-}q_{1},q_{4},\allowbreak{-}q_{3})=0$ for each $i$.
Hence

######  Proposition 4.3.

At the point $[q_{1},q_{2},q_{3},q_{4}]\in{\rm P}^{3}$ representing the vector
bundle $V$, the cotangent vectors
$(p_{1},p_{2},p_{3},p_{4})=(q_{2},-q_{1},q_{4},-q_{3})$ define the one-
dimensional space of Higgs fields which vanish at $\tilde{x}_{1}$ or
equivalently the holomorphic sections of $\operatorname{End}_{0}V\otimes
K^{1/2}$ for each $V$.

######  Remark 4.4.

The evident symmetry in the formulae above arises from the action $V\mapsto
V\otimes U$ of the $2^{2g}=16$ line bundles $U$ with $U^{2}$ trivial.

###  4.3 Genus 2 odd degree

Suppose now the rank 2 bundle has odd degree. From Proposition 4.1,
$H^{0}\big{(}C,\operatorname{End}_{0}V\otimes K^{1/2}\big{)}$ has a section if
$K^{1/2}$ is one of the $10=16-6$ even spin structures. The latter have no
non-zero holomorphic sections.

We use a model of the curve $C$ using $K\otimes K^{1/2}=K^{3/2}$. By
Riemann–Roch $K^{3/2}$ has a two-dimensional space of holomorphic sections as
does $K$ and hence together they give a map $C\rightarrow{\rm P}^{1}\times{\rm
P}^{1}$ so that $C$ is a divisor of ${\mathcal{O}}(2,3)$. The line bundle
${\mathcal{O}}(1,1)$ embeds ${\rm P}^{1}\times{\rm P}^{1}\subset{\rm P}^{3}$
and so a plane in ${\rm P}^{3}$ intersects $C$ in a divisor of $K^{5/2}$. By
Riemann–Roch $\dim H^{0}\big{(}C,K^{5/2}\big{)}=4$ so all such divisors have
this form.

Now we need a description of vector bundles of odd degree. Note that
$\operatorname{End}_{0}V$ does not change if we replace $V$ by $V\otimes L$
for a line bundle $L$, which means we can make use of Atiyah’s description of
projective bundles in [1].

The construction takes a representative vector bundle as an extension
${\mathcal{O}}\rightarrow V\rightarrow L$, where $L$ has degree $1$. The
extension class lies in a two-dimensional space $H^{1}(C,L^{*})$ and up to a
scalar is determined by its annihilator under Serre duality, a section $s$ of
$LK$. This vanishes at three points $p$, $q$, $r$ in general distinct. There
are four degree zero subbundles and the same projective bundle ${\rm P}(V)$ is
defined by three other triples
$(p,\sigma(q),\sigma(r)),(\sigma(p),q,\sigma(r)),(\sigma(p),\sigma(q),r)$,
where $\sigma\colon C\rightarrow C$ is the hyperelliptic involution.

Given $\psi\in H^{0}\big{(}C,\operatorname{End}_{0}V\otimes K^{1/2}\big{)}$,
its action on the trivial subbundle followed by the projection onto the
quotient $L$ gives a holomorphic section $c$ of $LK^{1/2}$. As divisor classes
$LK\sim p+q+r$ so $LK^{1/2}\cong K^{-1/2}(p+q+r)\cong
K^{5/2}(-\sigma(p)-\sigma(q)-\sigma(r))$ since $x+\sigma(x)$ is the divisor of
a section of $K$. Thus $c$ is defined by a plane in ${\rm P}^{3}$ which passes
through the three points $\sigma(p)$, $\sigma(q)$, $\sigma(r)$.

######  Proposition 4.5.

Let $V$ be a vector bundle defined by three points $p,q,r\in C\subset{\rm
P}^{3}$. If these points are not collinear, then the plane though $\sigma(p)$,
$\sigma(q)$, $\sigma(r)$ uniquely determines a section $\psi\in
H^{0}\big{(}C,\operatorname{End}_{0}V\otimes K^{1/2}\big{)}$.

######  Proof 4.6.

We use the Dolbeault approach for the extension ${\mathcal{O}}\rightarrow
E\rightarrow L$ by a $C^{\infty}$ splitting and a form
$\alpha\in\Omega^{0,1}(C,L^{*})$ defining the extension. Then

$\psi=\begin{pmatrix}a&\hphantom{-}b\cr c&-a\end{pmatrix}$

is holomorphic if $\bar{\partial}c=0$, $\bar{\partial}a+\alpha c=0$,
$\bar{\partial}b-2\alpha a=0$, where $c$ is a holomorphic section of
$LK^{1/2}$, $a$ is a $C^{\infty}$ section of $K^{1/2}$ and $b$ is a section of
$L^{*}K^{1/2}$.

As above, the plane defines $c$, a section of
$K^{5/2}(-\sigma(p)-\sigma(q)-\sigma(r))$ and since $\sigma$ induces a
projective transformation of ${\rm P}^{3}$, $\sigma(p)$, $\sigma(q)$,
$\sigma(r)$ are not collinear by assumption. Thus there is a unique plane
passing though them, or equivalently $\dim H^{0}\big{(}C,LK^{1/2}\big{)}=1$,
generated by $c$.

The term $\alpha c$ represents a class in $H^{1}\big{(}C,K^{1/2}\big{)}$ which
by Riemann–Roch has the same dimension as $H^{0}\big{(}C,K^{1/2}\big{)}$. But
this is zero since $K^{1/2}$ is an even spin structure. Then we can solve
$\bar{\partial}a+\alpha c=0$ for $a$. Any two choices differ by a holomorphic
section of $K^{1/2}$ so $a$ is unique.

To find $b$ we need $\alpha a$ to define a trivial class in
$H^{1}\big{(}C,L^{*}K^{1/2}\big{)}$. This is dual to the space
$H^{0}\big{(}C,LK^{1/2}\big{)}$ which consists of multiples of $c$. So
$[\alpha a]=0$ if $\alpha a$ pairs to zero with $c$. But since
$\bar{\partial}a+\alpha c=0$

$\int_{C}(\alpha a)c=\int_{C}a(\alpha
c)=-\int_{C}a\big{(}\bar{\partial}a\big{)}=-\int_{C}\bar{\partial}\big{(}a^{2}/2\big{)}=0.$

Then we can solve $\bar{\partial}b-2\alpha a=0$ for $b$. Since $L^{*}K^{1/2}$
is of degree zero and non-trivial, $b$ is unique.

######  Remark 4.7.

Note that if the points are collinear, $LK^{1/2}$ has degree $2$ and a two-
dimensional space of sections and so must be isomorphic to $K$, or
equivalently $L\cong K^{1/2}$. There is then an obvious section $\psi$ given
by $a=c=0$, $b=1$.

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