# Exploring Charge Transport Dynamics in a Cryogenic P-Type Germanium Detector

P. Acharya  Department of Physics, The University of South Dakota, Vermillion,
SD 57069, USA  M. Fritts  School of Physics and Astronomy, University of
Minnesota, Minneapolis, MN, 55455, USA  D.-M. Mei  Corresponding author.
Email<EMAIL_ADDRESS>Department of Physics, The University of South
Dakota, Vermillion, SD 57069, USA  G.-J. Wang  Department of Physics, The
University of South Dakota, Vermillion, SD 57069, USA  R. Mahapatra
Department of Physics $\&$ Astronomy, Texas A $\&$ M University, College
Station, TX 77843, USA  M. Platt  Department of Physics $\&$ Astronomy, Texas
A $\&$ M University, College Station, TX 77843, USA

(September 1, 2024)

###### Abstract

This study explores the dynamics of charge transport within a cryogenic P-type
Ge particle detector, fabricated from a crystal cultivated at the University
of South Dakota (USD). By subjecting the detector to cryogenic temperatures
and an Am-241 source, we observe evolving charge dynamics and the emergence of
cluster dipole states, leading to the impact ionization process at 40 mK. Our
analysis focuses on crucial parameters: the zero-field cross-section of
cluster dipole states and the binding energy of these states. For the Ge
detector in our investigation, the zero-field cross-section of cluster dipole
states is determined to be $8.45\times 10^{-11}\pm 4.22\times
10^{-12}\leavevmode\nobreak\ cm^{2}$. Examination of the binding energy
associated with cluster dipole states, formed by charge trapping onto dipole
states during the freeze-out process, reveals a value of $0.034\pm 0.0017$
meV. These findings shed light on the intricate charge states influenced by
the interplay of temperature and electric field, with potential implications
for the sensitivity in detecting low-mass dark matter.

Suggested keywords

††preprint: APS/123-QED

##  I Introduction

Recent astronomical investigations consistently highlight the pivotal role of
dark matter (DM) in the cosmos, characterized by its non-luminous and non-
baryonic nature, constituting the majority of the universe’s material
composition kahlhoefer2017review ; porter2011dark ; brink2005beyond ;
agnes2018low . Strong evidence supports the notion that DM influences the
cosmos by creating expansive halos around galaxies, as observed in the Milky
Way agertz2016impact ; frampton2009did ; diemand2007dark ; iocco2015evidence .
The quest for direct evidence lies within laboratory research, exploring
potential interactions between DM particles and ordinary matter beyond
gravitational forces. While weakly interacting massive particles (WIMPs) have
been a focus, recent theoretical frameworks like asymmetric DM and dark
sectors prompt exploration of low-mass DM particles. Thus, the MeV-scale
particle emerges as a noteworthy candidate for low-mass DM mei2018direct ;
angloher2017results ; an2018directly , yet detecting them poses challenges.
The need for detectors with exceptional sensitivity to discern a solitary
electron-hole (e-h) pair complicates matters due to the narrow energy span
involved. Overcoming this hurdle requires leveraging modern technology and
innovative methodologies for precise and reliable capture and quantification
of these minute energy signals. The relentless pursuit of sophisticated
detectors and pioneering detection techniques continues as researchers delve
deeper into the mysteries surrounding low-mass DM barbeau2022report . These
advancements hold the promise of unraveling one of the universe’s profound
enigmas, potentially bringing us closer to understanding its intricate
tapestry.

Owing to their exceptional sensitivity, germanium (Ge) detectors have emerged
as a promising avenue for addressing the challenge of low-mass DM detection,
presenting a compelling alternative to conventional methodologies
gascon2021low . These detectors possess a distinct advantage for probing low-
mass DM due to their remarkable ability to efficiently generate electron-hole
pairs, each requiring an average energy of 3 eV. Complementing this
characteristic is Ge’s narrow band gap of $\sim$0.7 eV, further bolstering its
suitability for such investigations at millikelvin (mK) temperatures
oh2021cryogen . Leveraging the concept of doping, it becomes possible to
significantly expand the parameter space for detecting low-mass DM using Ge
detectors. By judiciously introducing impurities into the Ge matrix,
particularly shallow-level impurities boasting binding energies around 0.01
eV, a fascinating phenomenon emerges: the creation of dipole states and
cluster dipole states in conditions below 6.5 K mei2022evidence . What
distinguishes these dipole states and cluster dipole states is their binding
energy, which plunges even lower than that of the impurities themselves
raut2023development ; bhattarai2023investigating . This intriguing aspect
opens a pathway to potentially detecting low-mass DM.

Yet, despite well-explored knowledge concerning the binding energies of
impurities in Ge, a conspicuous gap remains regarding the binding energies of
these dipole states and cluster dipole states at cryogenic temperatures,
especially those hovering below 100 mK. When Ge is cooled to these frigid
temperatures, as experienced in the vicinity of liquid helium, a compelling
process unfolds: the expulsion of residual impurities from its conduction or
valence bands. These expelled impurities then find solace in localized states,
giving rise to electric dipoles (denoted as $D^{0^{*}}$ for donors and
$A^{0^{*}}$ for acceptors) or neutral states ($D^{0}$ and $A^{0}$). These
states signify excited neutral impurity configurations, tethered by binding
energies less than 10 meV. The captivating implication here is the capacity of
these dipole states to ensnare charge, thereby orchestrating the formation of
cluster dipole states ($D^{+}$ and $D^{-}$ for donors, and $A^{+}$ and $A^{-}$
for acceptors), which exhibit even lower binding energy levels. The specifics
of these states depend on the operational temperature mei2022evidence ;
acharya2023observation ; klinkigt2013cluster .

As temperatures decrease, a remarkable exponential reduction in the density of
free charge carriers becomes apparent mei2023exploring . Approaching
temperatures lower than 10 K, donor (or acceptor) atoms predominantly maintain
their unionized configuration, confining fifth electrons (or vacant holes)
within these atomic entities. The extent of this encapsulation is governed by
the Onsager radius, denoted as
$R=\frac{e^{2}}{4\pi\varepsilon\varepsilon_{0}K_{B}T}$, where $\varepsilon$
represents the relative permittivity of Ge (with a value of 16.2),
$\varepsilon_{0}$ is the permittivity of free space, $K_{B}$ is the Boltzmann
constant, and $T$ is the temperature. Remarkably, at temperatures below 10 K,
$R$ can significantly exceed the dimensions of the donor or acceptor atom
sze2021physics . Consequently, the fifth electrons (or vacant holes)
associated with the donor (or acceptor) atoms may undergo thermal dissociation
from the atomic nucleus, initiating the formation of electric dipoles due to
the segregation of opposing charges mei2023exploring . This sequential
progression leads to the generation of dipole states with the capacity to trap
charge through Coulombic attraction, giving rise to the emergence of cluster
dipole states mei2022evidence . As depicted in Figure 1, this visualization
illustrates the formation of excited dipole states and cluster dipole states
in n-type Ge or p-type Ge, respectively. Within an excited dipole state,
whether associated with a positively charged donor ion or a negatively charged
acceptor ion, a state of profound confinement prevails due to the lattice
deformation potential. Consequently, the phase space available for capturing
charge carriers is inherently more constrained than that of bound electrons
(or holes), which retain the capability to traverse within the bounds set by
the Onsager radius (R) mei2022evidence ; mei2023exploring .

This intricate interplay foretells a higher likelihood of generating
$D^{+^{*}}$ states (or $A^{-^{*}}$ states) within an n-type detector (or a
p-type detector) than the likelihood of yielding $D^{-^{*}}$ states (or
$A^{+^{*}}$ states) mei2023exploring . This asymmetry implies that within an
n-type detector, holes undergo more pronounced entrapment effects compared to
electrons, whereas electrons experience greater susceptibility to entrapment
than holes within a p-type detector arnquist2022alpha ; meijer2003solution .
At the University of South Dakota (USD), our research has delved extensively
into the binding energy characteristics of cluster dipole states, as
documented in recent studies bhattarai2023investigating ; raut2023development
. This investigation unfolded at a low temperature of 5.2 K, uncovering a
binding energy threshold consistently below 10 meV, a parameter intricately
linked to the prevailing electric fields. Notably, our analysis has
spotlighted the pivotal role played by the Onsager radius in shaping the
spatial confines conducive to the emergence of cluster dipole states, with
temperature exerting a significant influence on this parameter. The intricate
interplay between temperature and the Onsager radius serves as a governing
factor in determining the binding energy of cluster dipole states. Our
research transcends the microscopic realm, carrying macroscopic implications.
It reveals that lower temperatures engender an expansion of the Onsager
region, consequently leading to a reduction in binding energies. Conversely,
higher temperatures induce a contraction of this region, resulting in
heightened binding energies. Our commitment to advancing this knowledge
extends further, as we seek to validate these findings across distinct
temperature regimes, including conditions exceeding 800 mK and those as low as
40 mK. Through this empirical exploration, we aim to offer concrete evidence
for the intricate thermal dynamics that influence the binding energy within
cluster dipole states.

Figure 1: The procedural framework delineating the progression towards the
genesis of both excited dipole states and cluster dipole states within a dual-
tiered Ge detector — encompassing an upper tier (n-type) and a lower tier
(p-type) — operational within sub-10 K temperatures, is embodied by the
subsequent stages. Herein, $\vec{p}$ and $\vec{q}$ symbolize the distinct
dipole moments affiliated with each state.

##  II Experimental Methods and Observed Physical Phenomenon

Leveraging cutting-edge infrastructure, USD boasts an avant-garde platform for
crystal growth and detector development. Central to this system is the
employment of a zone refining technique, artfully harnessed to attain
exceptional purification levels in commercial ingots wang2015crystal ;
yang2015zone . These meticulously purified ingots lay the foundation for
crystal growth through the renowned Czochralski method. With over a decade of
research and development experience, we have achieved increased consistency in
growing detector-grade crystals. It is noteworthy that we have integrated
advanced machine learning techniques into this process to further enhance the
growth of large-size detector-grade crystals, aiming to attain a level of
purity conducive to exploring low-mass DM phenomena acharya2023machine . The
crux of this study lies in a crystal cultivated in 2014 at USD, revealing a
net impurity level of $|N_{A}-N_{D}|=4\times 10^{11}\leavevmode\nobreak\
cm^{-3}$. This particular crystal assumes a p-type configuration. The
subsequent fabrication of the detector, orchestrated at Texas A $\&$ M
University, boasts a configuration encompassing four channels for charge
readout positioned atop the detector. Departing from conventional paradigms,
the detector’s grounding component integrates a uniform aluminum (Al)
electrode as opposed to a grid arrangement. This innovative design choice,
while safeguarding the electric field’s integrity, acts as a bulwark against
charge leakage, thereby sustaining an unwavering and uniformly distributed
electric field. Notably, this departure might have implications for the
detector’s susceptibility to effective neutralization through light-emitting
diodes (LEDs). The detector’s geometric layout mirrors the essence of
SuperCDMS-style detectors, as pictured in Figure 2. Following meticulous wire-
bonding, the detector seamlessly integrates into a dilution refrigerator,
embarking on a comprehensive testing regimen hosted at the K100 Detector
Testing Facility at the University of Minnesota (UMN).

The phenomenon of impact ionization prompted by impurities within Ge specimens
has received extensive attention from researchers, particularly at
temperatures surpassing 4 K zylbersztejn1962theory ; pickin1978impact .
Noteworthy contributions to this knowledge pool have been made by various
scholars. Recently, Phipps et al. have achieved a significant breakthrough by
investigating the impact ionization of impurities using SuperCDMS-style
detectors at an ultracold temperature of $\sim$40 mK phipps2016observation .
Furthermore, F. Ponce et al. demonstrated a room temperature pulsed laser’s
effect on a SuperCDMS silicon HVeV detector, unveiling coherent probabilities
for charge trapping and impact ionization within the high-purity Si substrate
under specific conditions ponce2020measuring . Aligned with these pioneering
works, our endeavor delves into the realm of time-dependent impact ionization,
mirroring the ultracold environment at $\sim$40 mK. We employ a detector
constructed from a distinctive and ultra-stable Ge crystal. This detector
experiences gradual cryogenic cooling, eventually reaching an extraordinary
temperature of approximately 40 mK. The ensuing meticulous evaluations
transpire over two separate refrigeration runs: Run 67 in 2018 and Run 74 in
2021. In the context of Run 67, the arrangement involved situating four 241Am
sources directly above individual channels on the detector, an arrangement
aptly illustrated in Figure 2. These setups employ lead collimators sporting
0.2 mm apertures, strategically permitting the transmission of 59.54 keV
$\gamma$ rays while adeptly blocking alpha particles through source
encapsulation. Our findings manifest in the spectra from each channel,
displaying distinct 59.54 keV peaks, with comprehensive measurements conducted
diligently over a two-week interval. Contrastingly, Run 74 deploys a singular
241Am source, ingeniously mounted on a carriage facilitated by a
superconducting stepper motor, as graphically represented in Figure 3. This
inventive source design incorporates a 0.5 mm collimator hole, leading to a
reduced flux of incident $\gamma$ rays on the detector, constituting
approximately 75% of the sources utilized in Run 67. Notably, the two runs
unfold over extended time spans, inherently introducing fluctuations in the
detector’s operational state. These fluctuations can wield considerable
influence over the detector’s charge collection efficiency and trapping
characteristics. For an all-encompassing comprehension of the nuanced
intricacies underpinning these experiments, an in-depth analysis is available
in the research paper authored by Acharya et al acharya2023observation .

Figure 2: This illustration features the detector scrutinized in our
investigation, equipped with four read-out electrodes labeled as $Q_{1}$,
$Q_{2}$, $Q_{3}$, and $Q_{4}$. The detector boasts dimensions of 10 cm in
diameter and 3.3 cm in thickness, endowing it with a mass of approximately
1.34 kg. Figure 3: Presented here is the arrangement of the 241Am source
manipulator, ingeniously crafted utilizing superconducting mobile coil
technology. This groundbreaking design facilitates meticulous control and
positioning of the radioactive source, thereby significantly augmenting its
effectiveness across a spectrum of scenarios mast2020situ .

##  III Experimental Data Analysis and Results

Figure 4: Depicted here is the evolving charge response from Q4, discernible
across time when subjected to favorable biases during Run-67. The foundational
signal associated with the 59.54 keV gamma-ray emission from Q4 exhibited an
initial linear rise within a matter of minutes, which swiftly transitioned
into an exponential decline spanning tens of minutes. Figure 5: Provided is
the temporal progression of the charge response in Q4, witnessed under
positive biases throughout Run-74. The initial climb of the 59.54 keV gamma
ray baseline stemming from Q4 displayed a linear trajectory during the initial
moments, rapidly transitioning into an almost exponential decay pattern
spanning a maximum duration of approximately 10 minutes.

We embarked on a thorough analysis, utilizing data collected at various bias
voltages to comprehensively delineate the charge collection efficiency for
each channel, scrutinizing its interplay with bias and time. This
investigation has yielded intriguing insights into the behavior of the 59.54
keV calibration line, as depicted in Figure 4 and Figure 5. Our observations
unveiled a multitude of time-dependent phenomena: signal amplitudes from the
59.54 keV line exhibiting an initial increase post-bias application, succeeded
by a subsequent descent towards a steady-state value. This distinctive
phenomenon exclusively manifested under positive biases, owing to the
strategic positioning of 241Am sources on the biased side of the crystal,
instigating the immediate collection of electrons and the subsequent drift of
holes across the entirety of the detector’s thickness.

Remarkably, this effect displayed varying degrees of potency across channels,
with its most pronounced manifestation occurring in the central channel (Q4),
while its presence was faintest in the outer ring channel (Q1), indicative of
a radial dependence. In light of this, our study centered its focus on the
59.54 keV line within the central channel (Q4) under positive biases, as it
showcased the most promising outcomes.

Upon delving into the equilibrium state at t=0, where
$p-n=N_{D}^{+}+N_{A}^{-}=0$, representing the charge neutralization state
hagstrum1961theory ; wei1994ion , with p signifying free holes, $n$ indicating
free electrons, $N_{D}^{+}$ representing positively ionized donor atoms, and
$N_{A}^{-}$ representing negatively ionized acceptor atoms. We also introduce
expressions for the density of acceptor impurities: $p+p_{d}=N_{A}$, where
$p_{d}$ signifies the unionized holes bound to the atom; similarly,
$n+n_{d}=N_{D}$, with $n_{d}$ representing the unionized donor electrons. When
weaving these expressions together, they coalesce into a coherent depiction.

Figure 4 and Figure 5 visually articulate the temporal evolution of the impact
ionization phenomenon, underscored by the drift of hole carriers under
positive bias conditions. The process unfolds with hole impact ionization
exhibiting a linear surge, modulated by bias and event rate. This amplified
charge signal experiences an exponential decay, the timescale of which is
intrinsically linked to the prevailing bias. This behavior is attributed to
the attenuation of charge collection efficiency, stemming from the disruption
of the bulk electric field due to the presence of trapped charges. The linear
enhancement in the signal arising from hole impact ionization, in conjunction
with the absence of comparable electron impact ionization, suggests that the
spectrum of plausible sites for hole impact ionization is modest at its
inception, originating from the capture of drifting electrons.

This intricate phenomenon encompasses the interplay of three distinct
processes: $e^{-}+A^{0^{*}}\rightarrow A^{-^{*}}$, $h^{+}+A^{-^{*}}\rightarrow
e^{-}+2h^{+}+A^{-}$, and $h^{+}+A^{-}\rightarrow A^{0^{*}}$. The first process
entails the creation of cluster dipole states as a consequence of electron
migration within the detector, spurred by background radiation. The second
corresponds to the ionization of cluster dipole states initiated by hole
carriers, while the third involves the confinement of $h^{+}$ carriers,
culminating in the cessation of neutralization in the series’ denouement
acharya2022observation . These intricate mechanisms find graphical
representation in Figure 1.

##  IV The Theoretical Framework

To comprehensively understand the intricate nuances influencing the detector’s
performance under low-temperature conditions, it is crucial to subject various
theoretical models to rigorous testing. The insights gained from these
systematic evaluations have the potential to provide invaluable information,
shaping our understanding of the detector’s efficacy in low-mass DM detection.
We have extensively explored several theoretical frameworks, supplementing our
investigation with calculations. A detailed presentation of these
comprehensive findings and discussions are outlined below.

###  IV.1 Impact Ionization

At low cryogenic temperatures, the Ge detector showcases intriguing phenomena
related to impact ionization (Fig. 4). Our earlier analysis of the impact
ionization phenomenon employed a well-defined physical model that accounted
for the observed behavior of charge carriers. We have established a
relationship between the detected charge energy, denoted as $E(t)$, and the
input 59.54 keV $\gamma$ rays through the following equation
acharya2023observation :

$E(t)=E_{\gamma}\\{p_{0}+p_{1}exp[\frac{p_{2}}{p_{3}}(1-exp(-p_{3}t))]\\}exp(-p_{4}t),$
(1)

where $E_{\gamma}$ = 59.54 keV. Our focus lies on two parameters, $p_{0}$ and
$\&$ $p_{1}$, where $p_{0}+p_{1}$ signifies the impact ionization factor,
specifically for absolute impact ionization observation. If $p_{0}+p_{1}>1$,
it indicates a gain in charge energy due to impact ionization.

The sole instance of observed absolute impact ionization transpired at $t=0$
during Run-67. Consequently, our analysis will center on understanding the
detector’s distinctive signature for impact ionization at low temperatures,
leveraging the data obtained from Run-67. Upon applying the data from Run-67
to fit the model outlined in Equation 1, the fitted parameters for $p_{0}$ and
$p_{1}$ under various biases were determined as,

Run 67
---
Bias | $p_{0}$ | $p_{1}$
4.5 V |  1.05 |  0.015
5 V |  1.07 |  0.018
5.5 V |  1.09 |  0.019
6 V |  1.13 |  0.020
7 V |  1.22 |  0.028
8 V |  1.28 |  0.032
9 V |  1.39 |  0.038
Table 1: Summary of the fitting for parameters $p_{0}$ and $p_{1}$ under
various biases in Run-67 acharya2023observation . Figure 6: Shown are the
fitting parameters, $p_{0}$ and $p_{1}$ using equation 1 as a function of the
applied field, $E$, and fits a linear regression model, which demonstrates the
correlation of the parameters with the electric field using the data from
Run-67. The fitting functions for those parameters in Run 67 are:
$p_{0}$=0.25$E$+0.69, $p_{1}$=0.017$E$-0.0087.

Figure 6 represents the correlation between two parameters, $p_{0}$ and
$p_{1}$, and the applied electric field at 40 mK. Regression analysis
demonstrates a strong correlation, indicating these parameters vary according
to the applied bias. The sum of these parameters, $p_{0}+p_{1}$, defines an
amplification factor, $A$. If $n_{t}$ represents the charge carriers at any
time t, it can be expressed as $n_{t}=n_{0}e^{t/\tau_{imp}}$, where $n_{0}$
denotes the charge carriers at time $t=0$ mei2023exploring . Here,
$\tau_{imp}=\frac{1}{N_{a}\sigma_{imp}(E)v_{d}}$ stands for the charge
generation time through impact ionization. Here, $N_{a}$ represents the
impurity concentration, $\sigma_{imp}(E)$ is the impact ionization scattering
cross-section as a function of the electric field, and $v_{d}$ signifies the
drift velocity, given by $v_{d}=d/t$, where $d$ denotes the detector width and
$t$ is the drift time. It should be noted that the amplification factor,
$A=n_{t}/n_{0}$. After incorporating the equations
$n_{t}=n_{0}e^{t/\tau_{imp}}$ and
$\tau_{imp}=\frac{1}{N_{a}\sigma_{imp}(E)v_{d}}$ alongside the amplification
factor, the derived equation elucidating impact ionization due to the cluster
dipole at cryogenic temperatures is succinctly expressed as:

$log(A)=N_{a}\times\sigma_{imp}(E)\times d,$ (2)

###  IV.2 Zero Field Cross-Section

The zero-field impact ionization cross-section for charged carriers within a
detector denotes the detector’s effective interaction area in the absence of
an external electric field and quantifies the likelihood per unit area of
charged carriers inducing an impact ionization event within the detector
material. This parameter is intricately impacted by several factors, including
the characteristics of the particle (such as charge and energy), alongside the
specific properties inherent to the detecting material. Notably, the zero-
field cross-section for cluster dipoles unveils a captivating domain within
low-temperature physics. Understanding and harnessing this cross-section hold
considerable potential for advancing our comprehension of elusive particles
and their interaction dynamics within the context of low-temperature
environments. If $\sigma_{0}$ denotes the zero field cross-section, its
mathematical expression can be written as palmier1972impurity ;
gutierrez2000low ; phipps2016ionization ; sundqvist2012carrier :

$\sigma_{0}=\frac{\pi}{2}\times\frac{\hbar^{2}}{2m_{h}^{*}E_{BB^{\prime}}},$
(3)

Here, $m_{h}^{*}$ represents the effective mass of holes in the conduction
band, and $E_{BB^{\prime}}$ denotes the binding energy for the cluster dipole
state, which will be discussed in detail later in the section dedicated to
calculating the binding energy of cluster dipole states, as defined by
Equation 8. Considering $m_{h}^{*}=0.21m_{e}$, where $m_{e}$ represents the
mass of an electron, equivalent to $0.51\leavevmode\nobreak\ MeV/C^{2}$. We
can plug these parameters into Equation 3 to obtain
$\sigma_{0}=\frac{2.85\times 10^{-15}}{E_{BB^{\prime}}}\leavevmode\nobreak\
cm^{2}$, where $E_{BB^{\prime}}$ has the unit in eV.

###  IV.3 Impurity Freeze-Out

In a recent study, Mei et al. demonstrated a freeze-out process in both an
n-type detector with N${d}$ = 7.02$\times$1010/cm3 and a p-type detector with
N${a}$ = 6.2$\times$109/cm3 mei2022evidence ; mei2023exploring . In these
detectors, as the temperature falls below 11 K, there is a notable decrease in
relative capacitance in the Ge detectors. Simultaneously, impurities in the
detectors freeze out of the conduction or valence band, leading to a decrease
in free charges within the detector volume as it is cooled down to 6.5 K
mei2023exploring . Further cooling towards 5.2 K maintains a constant
capacitance, indicating the absence of free charges. Nearly all impurity atoms
then form electric dipole states ($D^{0^{*}}$ for donors and $A^{0^{*}}$ for
acceptors) mei2023exploring .

In the case of a p-type Ge detector, if an impurity atom is in its ground
state, it is unable to capture charges. Nevertheless, when the detector is
cooled to 5.2 K, this impurity atom undergoes a transition to a dipole state,
gaining the ability to capture charges and forming a cluster dipole state.
During this freeze-out process, impurity atoms, previously responsible for
generating free carriers above 11 K, undergo a transition into bound states.
These bound states manifest as electrically neutral, reflecting the
transformation of impurities in the Ge detector into localized states. This
phenomenon is commonly referred to as ”freeze-out” in the context of
semiconductor physics mei2023exploring .

If the same detectors continue to cool down, as the detector temperature
reaches 40 mK, it is anticipated that the dipole moments of these dipole
states will increase, influenced by the temperature-dependent size of dipoles.
These enhanced dipole states may weakly trap charge carriers, potentially
resulting in the formation of cluster dipole states ($D^{+^{*}}$, $D^{-^{*}}$
for donors, and $A^{+^{*}}$, $A^{-^{*}}$ for acceptors) mei2022evidence ;
raut2023development ; bhattarai2023investigating . Figure 7 illustrates both a
p-type dipole state and a p-type cluster dipole state mei2023exploring .

Figure 7: Shown are a p-type dipole state and a p-type cluster dipole state.

The freeze-out phenomenon significantly influences the behavior of Ge
detectors at extremely low temperatures. In the state of freeze-out, the
proportion of holes associated with acceptor levels in a p-type material with
an impurity density of $N_{a}$ is mei2023exploring :

$\frac{p_{a}}{p+p_{a}}=\frac{1}{\frac{N_{V}}{4N_{a}}exp[-\frac{E_{a}-E_{v}}{K_{B}T}]+1},$
(4)

where $p$ signifies the number density of free holes in the valence band and
$p_{a}$ denotes the number density of holes bound to acceptors, the
mathematical relationship is expressed as $N_{a}=p+p_{a}$. Here, $N_{V}$
signifies the density of states in the valence band, and $E_{a}-E_{v}\simeq
0.01\leavevmode\nobreak\ eV$ represents the ionization energy of acceptors.
The fraction of holes bound to acceptors, as defined by Equation 4, is
visually represented in Figure 8.

Figure 8: The presented graph illustrates the temperature-dependent ratio of
holes bound to the acceptors, with $N_{A}=4\times 10^{11}\leavevmode\nobreak\
cm^{-3}$ and $E_{a}-E_{v}\simeq 0.01\leavevmode\nobreak\ eV$ as the parameters
for generating this plot. Additionally, it’s important to observe that the
inset plot conveys the same information across a wider temperature range,
displayed in a logarithmic scale.

The graph vividly illustrates a significant rise in the percentage of holes
bound to acceptors within the temperature range of 20 K to 8 K. It’s important
to note that this freeze-out range in temperature slightly deviates from the
observations in Mei et al.mei2022evidence ; mei2023exploring  due to
variations in detector impurity levels. This dependence on impurity levels is
also evident from Equation4. By rearranging the terms, Equation 4 can be
reformulated as:

$p=\frac{p_{a}N_{V}}{4N_{a}}exp[-\frac{E_{a}-E_{v}}{K_{B}T}],$ (5)

While residual impurity atoms remain ionized until approximately 20 K, a
distinctive phenomenon emerges within the temperature range of 20 K to 8 K.
During this period, these impurity atoms undergo a freeze-out process,
transitioning into bound states that manifest as effectively charge-neutral
entities. This abrupt transformation is intricately linked to the band
structures illustrated in Figure 9, where $E_{c}-E_{d}\simeq 0.01$ eV and
$E_{a}-E_{v}\simeq 0.01\leavevmode\nobreak\ eV$ mei2020impact .

Figure 9: Displayed are the energy levels of donors and acceptors within
semiconductor materials ferry2013semiconductors ; berger2020semiconductor .

The abrupt transition depicted in Figure 8 highlights the intricate
relationship between temperature fluctuations and the corresponding band
structures. It reveals a significant shift in the behavior of impurity atoms
within this particular temperature range (20 K to 8 K). Under such
circumstances, the expression for the number density of free holes ($p$)
existing within the valence band can also be delineated as mei2018direct ;
van1950theory :

$p=[\frac{N_{a}N_{V}}{4}]^{1/2}exp[-\frac{E_{a}-E_{v}}{2K_{B}T}],$ (6)

where $N_{V}$ is the effective density of states in the valence band for Ge
materials, which is given by $N_{V}=9.6\times
10^{14}T^{3/2}\leavevmode\nobreak\ cm^{-3}$.

In situations where the temperature drops below 8 K for the p-type detector
under investigation, the number density of free holes is significantly
smaller, by several orders of magnitude, than the number density of holes
bound to acceptors. Consequently, it is reasonable to assume that $p_{a}$
equals $N_{a}$ and $E_{a}-E_{v}=E_{B}$, where $E_{B}$ is the binding energy of
the bound states. Therefore, combining Equations 5 and 6 facilitates the
determination of the binding energy for the bound states - dipole states:

$E_{BB}=K_{B}T\leavevmode\nobreak\ log(\frac{N_{V}}{4p_{a}}),$ (7)

Please note that the binding energy of dipole states, represented by $E_{BB}$,
differs from the ionization energy of acceptors, as illustrated in Figure 9.
This distinction arises due to the formation of dipole states within a
particular range of the Onsager radius when holes bound to acceptors.
Consequently, the binding energy of these dipole states, influenced by the
size of the dipole, may differ from the ionization energy of impurity atoms.

Utilizing Equation 7, we can assess the binding energy for various bound
states, including dipole states and cluster dipole states, as a function of
energy. As an illustrative example, Equation 7 is employed to calculate the
binding energy of dipole states. Figure 10 portrays the binding energy of a
dipole state with a dipole density of $N_{a}=4\times 10^{11}\text{cm}^{-3}$
across different temperatures. Two specific temperature points are highlighted
in the plot to emphasize the dipole state’s binding energy. At a temperature
of 5.2 K, the binding energy of the dipole states is calculated at 3.98 meV.
However, as the temperature drops to 40 mK, the binding energy decreases
substantially to 0.005 meV, reaching a level comparable to thermal energy.

Figure 10: The plot illustrates the binding energy of the dipole state, while
the inset plot extends the representation across a broader temperature range.
Additionally, the plot furnishes details on the binding energy of the dipole
state at two distinct temperatures, namely 5.2 K and 40 mK.

The binding energy of dipole states and cluster dipole states was measured at
5.2 K using both an n-type detector and a p-type detector in our previous
study bhattarai2023investigating ; raut2023development . The obtained values
were found to be similar, as demonstrated in our research at 5.2 K
bhattarai2023investigating ; raut2023development . Generally, the binding
energy of cluster dipole states is expected to differ from that of dipole
states at the same temperature. This expectation arises because the binding
energy of dipole states is confined by the Coulomb potential associated with
the dipole size, whereas the binding energy of cluster dipole states is
influenced by both the Coulomb potential determined by the trapping distance
between two charge carriers with opposite charges and the dipole size. This
was demonstrated by the previous measurements at 5.2 K
bhattarai2023investigating ; raut2023development , where the dipole size and
the trapping distance between two mobile charge carriers with opposite charges
are slightly different. However, this scenario may change when the detector is
operated at 40 mK, where the dipole size becomes significantly larger than the
trapping distance of two opposite charge carriers. In this scenario, the
binding energy of dipole states is anticipated to be smaller than that of
cluster dipole states. To assess the binding energy of cluster dipole states
at 40 mK, we utilize the data obtained from Run-67.

###  IV.4 Binding Energy of Cluster Dipole States

The presence of dipole states and cluster dipole states within the detector is
influenced by both the detector’s configuration and its operational mode. For
this p-type detector, during the dilution refrigeration process, as the
temperature approaches 40 mK, 100% of impurity atoms precipitate out from the
conduction band, leading to the formation of electric dipole states identified
as $A^{0^{*}}$. Subsequently, electrons generated by the background radiation
from the environment become trapped by the excited dipole state ($A^{0^{*}}$),
resulting in the emergence of the cluster dipole state labeled as $A^{-^{*}}$.
This phenomenon was observed during Run-67 when the detector was exposed to a
241Am source in two operational modes. The creation of electron-hole pairs
near the surface of the detector, primarily induced by 59.54 keV gamma rays,
was a key aspect. In the first mode, a negative bias was applied to the
detector, prompting electrons to drift across its surface. Conversely, in the
second mode, a positive bias was employed, leading to the drift of holes
across the detector. For illustration, Figure 11 depicts the detector
operating at both -4 V and +4 V. The figure clearly indicates that electrons
become trapped, whereas holes do not exhibit the same behavior.

Figure 11: The depicted figure showcases the dynamic charge response of Q4
over time, influenced by two opposing biases: +4 V and -4 V. These biases
symbolize the migration of holes under positive bias and electrons under
negative bias across the entire detector during Run-67. The data unveils that
the detection of electron-hole pairs originating from Q4, induced by 59.54 keV
gamma rays, is fully realized under a +4 V bias. Conversely, under a -4 V
bias, the energy detection is notably lower than 59.54 keV throughout a
30-minute period. This observation suggests a substantial entrapment of
electrons within the p-type Ge detector.

In the UMN K100 laboratory, we contend with the environmental background of
gamma-rays inducing electron-hole pairs within the detector. Additionally,
cosmic-ray muons interact with the detector, resulting in the generation of
electron-hole pairs. These background electrons become ensnared in a dipole
state, leading to the formation of cluster dipole states, while holes
experience less entrapment in a p-type detector subjected to the same bias
voltage. This discrepancy arises from the limited space available for
immobilized negative ions, responsible for trapping charge carriers, compared
to the ample space for mobile bound holes.

The mobility of holes is dictated by the Onsager radius, denoted as $R$ as
defined earlier. Since the trapping probability is proportional to the
available space where charges can be trapped, the likelihood of trapping for
immobilized negative ions is smaller than that of mobile holes. Consequently,
the probability of creating $A^{-^{*}}$ states is greater than that of forming
$A^{+^{*}}$ states in a p-type detector. Note that $A^{+^{*}}$ and $A^{-^{*}}$
represent excited states that differ from the ground states of $A^{+}$ and
$A^{-}$. This elucidates why electrons are more profoundly trapped than holes
in a p-type detector, as clearly articulated in the recent publication
mei2023exploring .

Under a sufficiently electric field, holes traverse the detector undergoing
impact ionization of a dipole state or a cluster dipole state at low
temperatures. At a temperature of 40 mK, despite the binding energy of dipole
states being comparable to thermal energy, the impact ionization of holes
takes place in conjunction with a cluster dipole state. This phenomenon arises
due to Coulomb repulsion among holes, which inhibits the occurrence of this
process at the operational voltage employed within the investigation, limited
to a few volts.

Upon applying a positive bias voltage from the upper region of the detector
and exposing it to an 241Am source positioned above, hole carriers traverse
the detector. Subsequently, these drifting holes engage in impact ionization
interactions with the cluster dipole states ($A^{-^{*}}$). As the bias voltage
increases, the holes, serving as liberated charge carriers, accumulate greater
kinetic energy, leading to the emission of electrons from the traps. This
phenomenon results in a decrease in the count of cluster dipole states
($A^{-^{*}}$), inducing the transition of $A^{-}$ states, as depicted in
Figure 7. The amplification of a single hole to two holes per reaction
($h^{+}+A^{-^{*}}\rightarrow e^{-}+2h^{+}+A^{-}$) tangibly demonstrates the
occurrence of hole impact ionization within the detector.

Our investigation has revealed the presence of a cluster dipole state at the
low cryogenic temperature of 40 mK. While the exact density of the cluster
dipole state requires determination, we can utilize Equation 7 to calculate
the binding energy associated with this state. To ascertain the density of the
cluster dipole state, we will employ the relationship provided by Equation 2.
Consequently, the updated equation for calculating the binding energy of the
cluster dipole state is as follows:

$E_{BB^{\prime}}=K_{B}T\leavevmode\nobreak\ log(\frac{N_{V}}{4p_{cd}}),$ (8)

where $E_{BB^{\prime}}$ represents the binding energy for the cluster dipole
state, and $p_{cd}$ denotes the density of the cluster dipole state.
Considering the model in Equation 2 to determine the density of the cluster
dipole state at a cryogenic temperature of 40 mK, when the detector response
is in an impact ionization mode with sufficient bias voltage and some cluster
dipole states ($A^{-^{*}}$) are ionized through impact ionization, we can
express Equation 2 as $\log(A)=p_{cd}\times\sigma_{0}E^{[p_{1}]}\times d$.
Here, $\sigma_{imp}(E)=\sigma_{0}E^{[p_{1}]}$, with $[p_{1}]$ as a fitting
parameter. To simplify further, let’s assume $[p_{0}]=p_{cd}\times\sigma_{0}$.
Now, the simplified expression for calculating the binding energy for fitting
the observed data is:

$log(A)=[p_{0}]\times\leavevmode\nobreak\ E^{[p_{1}]}\times d,$ (9)

Figure 12 displays the fitting line of Equation 9. From this, we obtained the
values of two fitting parameters: $[p_{0}]=8.9\times 10^{-3}\pm 4.7\times
10^{-4}$ and $[p_{1}]=2.5\pm 7.7\times 10^{-2}$. We know that $[p_{0}]$ equals
$p_{cd}\times\sigma_{0}$. By substituting the values of $\sigma_{0}$ and
$[p_{0}]$, we find $p_{cd}=3.12\times 10^{12}\times
E_{BB^{\prime}}\leavevmode\nobreak\ cm^{-3}$. Thus, according to Equation 8,
the binding energy for cluster dipole states ($A^{-^{*}}$) at a cryogenic
temperature can be expressed as:

$E_{BB^{\prime}}=K_{B}T\leavevmode\nobreak\ log(\frac{N_{V}}{4\times
3.12\times 10^{12}\leavevmode\nobreak\ E_{BB^{\prime}}}),$ (10)

Plugging in all the constant values and solving Equation 10 numerically at a
cryogenic temperature of 40 mK, we obtain $E_{BB^{\prime}}=0.034\pm 0.0017$
meV. This ultra-low binding energy of cluster dipole states ($A^{-^{*}}$) at
cryogenic temperatures suggests a delicate balance between the forces holding
these clusters together. At such low temperatures, the energy states of these
clusters exhibit minimal thermal agitation, revealing a fundamental
characteristic of their stability.

Figure 12: Displayed is the logarithmic value of the amplification factor, A,
obtained by summing two fitting parameters, $p_{0}$ and $p_{1}$ from Table 1.
$log(A)$ is plotted against the applied electric field. The error in $log(A)$
was calculated, with the error in the electric field measurement primarily
influenced by the precision of the applied bias voltage.

Calculating the binding energy of the cluster dipole state enabled us to
determine the density of cluster dipole states, which is found to be
$p_{cd}=3.12\times 10^{12}\leavevmode\nobreak\ E_{BB^{\prime}}$. Consequently,
the density of cluster dipole states is found to be $1.05\times 10^{8}\pm
5.25\times 10^{6}\leavevmode\nobreak\ cm^{-3}$. Similarly, the zero-field
impact ionization cross-section for charged carriers within the detector is
computed using Equation 3, resulting in a value of $8.45\times 10^{-11}\pm
4.22\times 10^{-12}\leavevmode\nobreak\ cm^{2}$.

##  V Conclusion

We conducted an experimental investigation into the time-dependent impact
ionization phenomenon within a p-type Ge detector. Our findings unveil that
temporal fluctuations in charge amplification, particularly concerning the
59.54 keV emission line from an 241Am source, mainly arise from impact
ionization induced by drifting holes traversing the detector. Our study delved
into understanding the potential mechanisms driving this impact ionization,
primarily attributing it to the generation and subsequent impact ionization of
cluster dipole states.

Our study focused on a critical examination of binding energy and zero-field
impact ionization cross-section of cluster dipole states within a low-
temperature p-type Ge detector. At 40 mK, the zero-field cross-section of
cluster dipole states was determined to be $8.45\times 10^{-11}\pm 4.22\times
10^{-12}cm^{2}$, representing a two-order-of-magnitude increase compared to
the cross-section of neutral impurity in Ge reported by Phipps et
al.phipps2016observation , which stands at $5\times 10^{-13}$$\text{cm}^{2}$.
This discrepancy is attributed to the notable difference in binding energy
between cluster dipole states at 40 mK (0.034 meV) and neutral impurity states
(approximately 10 meV). We also calculated the binding energy of cluster
dipole states ($A^{-^{*}}$), resulting in $0.034\pm 0.0017$. This ultra-low
binding energy is distinctive to the detector state and plays a crucial role
in the exploration of low-mass dark matter (DM) at the MeV scale. The low-
scale binding energy, arising from impact ionization occurrences via intrinsic
amplification, has the potential to enhance the detector’s capability to
achieve an ultra-low energy threshold, reaching as low as sub-meV.

The calculated cluster dipole state ($A^{-^{*}}$) binding energy, observed
when the detector operated at cryogenic temperatures, is lower than the
typical binding energy associated with ground-state impurities in Ge, which is
$\sim$10 meV for the detector in our study. However, this binding energy is
higher than the thermal energy at 40 mK, which is 0.005 meV. This suggests
that these cluster dipole states might not significantly contribute to the
usual impurity-related effects observed in the detector at higher
temperatures. Additionally, it implies that these cluster dipole states might
remain stable and distinguishable from thermal fluctuations at this low
temperature. This stability hints at the potential for these states to
maintain a distinct identity, unaffected by thermal agitation, thus offering a
stable feature or characteristic in the detector’s behavior at cryogenic
temperatures.

At electric fields below 3 V/cm, our calculations yield a density of cluster
dipole states ($A^{-^{*}}$), denoted as $p_{cd}$, at $1.05\times 10^{8}\pm
5.25\times 10^{6}\leavevmode\nobreak\ cm^{-3}$. This value signifies the
density of states associated with cluster dipole states formed by trapping
electrons from background radiation. Our findings underscore the capacity of
energetic charge carriers to ionize these cluster dipole states, thereby
generating additional charge carriers. This phenomenon holds particular
significance in facilitating impact ionization, especially in scenarios
characterized by low electric fields.

In summary, our study offers crucial insights into the behavior of impurities
within Ge detectors. These revelations have the potential to influence the
design of innovative detectors, particularly those crafted for tasks such as
MeV-scale DM searches. Additionally, the observation of low binding energies
suggests the feasibility of utilizing appropriately doped impurities in Ge for
the creation of low-threshold detectors capable of detecting low-mass MeV
scale DM particles.

##  VI Acknowledgement

The authors extend their sincere gratitude to Vuk Mandic for his invaluable
support in facilitating this research, providing access to his dilution
refrigerator, and contributing to the data acquisition system. This research
received partial support from NSF OISE 1743790, NSF PHYS 1902577, NSF PHYS
2310027, DOE DE-SC0024519, DOE grants DE-FG02-10ER46709 and DE-SC0004768, as
well as a research center funded by the State of South Dakota.

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