# Direct detection of MeV-scale dark matter utilizing germanium internal amplification for the charge created by the ionization of impurities

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## Abstract

Light, MeV-scale dark matter (DM) is an exciting DM candidate that is undetectable by current experiments. A germanium (Ge) detector utilizing internal charge amplification for the charge carriers created by the ionization of impurities is a promising new technology with experimental sensitivity for detecting MeV-scale DM. We analyze the physics mechanisms of the signal formation, charge creation, charge internal amplification, and the projected sensitivity for directly detecting MeV-scale DM particles. We present a design for a novel Ge detector at helium temperature (\(\sim \) 4 K) enabling ionization of impurities from DM impacts. With large localized E-fields, the ionized excitations can be accelerated to kinetic energies larger than the Ge bandgap at which point they can create additional electron–hole pairs, producing intrinsic amplification to achieve an ultra-low energy threshold of \(\sim \) 0.1 eV for detecting low-mass DM particles in the MeV scale. Correspondingly, such a Ge detector with 1 kg-year exposure will have high sensitivity to a DM-nucleon cross section of \(\sim \) 5 \(\times \) 10\(^{-45}\) cm\(^{2}\) at a DM mass of \(\sim \) 10 MeV/c\(^{2}\) and a DM-electron cross section of \(\sim \) 5 \(\times \) 10\(^{-46}\) cm\(^{2}\) at a DM mass of \(\sim \) 1 MeV/c\(^2\).

## 1 Introduction

Observations from the 1930s [1] have led to the contemporary and shocking revelation that 96% of the matter and energy in the universe neither emits nor absorbs electromagnetic radiation [2, 3]. Weakly Interacting Massive Particles (WIMPs) [4] constitute a popular candidate for dark matter (DM). These particles, with mass thought to be comparable to heavy nuclei, have a feeble and extremely short range interaction with atomic nuclei. While WIMPs appear to interact with atomic nuclei very rarely, their collisions would cause atoms to recoil at a velocity in the order of a thousand times the speed of sound in air [5].

For over three decades, many experiments have conducted searches for WIMPs using various targets [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. These experiments are all sensitive to WIMPs with masses greater than a few GeV/c\(^{2}\). The best sensitivity for WIMPs masses above 10 GeV/c\(^{2}\), with a minimum of 7.7 \(\times \) 10\(^{-47}\) cm\(^{2}\) for 35 GeV/c\(^{2}\) at 90% confidence level, is given by the most recent results from XENON1T [23]. Despite great efforts have been made, WIMPs remain undetected. More experiments will soon come online [26, 27, 28]. The LZ experiment will push the experimental sensitivity for WIMPs with masses greater than 10 GeV/c\(^{2}\) very close to the boundary where neutrino induced backgrounds begin to constrain the experimental sensitivity [29, 30].

In the past decade, light DM in the MeV-scale [31, 32, 33, 34] has risen to become an exciting DM candidate, even though its low mass makes it unreachable by current experiments. The detection of MeV-scale DM requires new detectors with an extremely low-energy threshold (< 10 eV) since both electronic recoils and nuclear recoils induced by MeV-scale DM are in the range of sub-eV to 100 eV [31]. Using the data from the XENON10 experiment, the XENON Collaboration was able to set the first experimental limit on the MeV-scale DM detection [35]. More recently, Kadribasic et al. have proposed a method for using solid state detectors with directional sensitivity to DM interactions to detect low-mass DM [36]. CRESST has achieved a threshold of 20 eV with a small prototype sapphire detector [37]. DAMIC has claimed a sensitivity to ionization < 12 eV with Si CCDs and considered their method to be able to reach 1.2 eV [38].

*c*is the speed of light, \(m_{\chi }\) is the mass of DM, and \(m_{N}\) is the mass of a nucleus. As can be seen, this nuclear recoil energy is in the range of \(\sim \) 1 eV and well below the lowest threshold achieved in existing direct detection experiments.

A promising technology for sensitivity to MeV-scale DM is a germanium (Ge) detector which utilizes internal charge amplification for the charge carriers created by the ionization of impurities. We describe the design of a novel Ge detector that develops ionization amplification technology for Ge in which very large localized E-fields are used to accelerate ionized excitations produced by particle interaction to kinetic energies larger than the Ge bandgap at which point they can create additional electron–hole (e–h) pairs, producing internal amplification. This amplified charge signal could then be readout with standard high impedance JFET or HEMT [41] based charge amplifiers. Such a system would potentially be sensitive to single ionized excitations produced by DM interactions with both nuclei and electrons. In addition, purposeful doping of the Ge could lower the ionization threshold by \(\sim \, \times \) 10 (\(\sim \) 0.1 eV), making the detector sensitive to 100 keV DM via electronic recoils.

## 2 The formation of signal

### 2.1 DM-nucleus and DM-electron elastic scattering processes

*km*s\(^{-1}\). The relative motion of the solar system (\(v=220\)

*km*s\(^{-1}\)) through the DM halo is considered in the calculation. The energy deposition spectrum arises due to kinematics of elastic scattering. In the center-of-momentum frame, a DM particle scatters off a nucleus or a bound electron through an angle \(\theta \), uniformly distributed between 0 and 180\(^{o}\) for the isotropic scattering that occurs with zero-momentum transfer. The DM\('\)s initial energy in the laboratory frame can be expressed as: \(E_{i}\) = m\(_{\chi }v^2\)/2. The nuclear recoil energy can be calculated as:

Effective nuclear charges for Ge electron configuration

Electron configuration | Z\(_{eff}\) |
---|---|

1s | 31.294 |

2s | 23.365 |

2p | 28.082 |

3s | 17.790 |

3p | 17.014 |

4s | 8.044 |

3d | 16.251 |

4p | 6.780 |

*q*to excite an electron depends on the effective atomic number, \(\overline{Z_{eff}}\), and the incident angle of a DM particle.

It is clear that the energy deposited by nuclear recoils is mainly in the range of sub-eV to \(\sim \) 100 eV. The dissipation of such a small amount of energy in Ge is largely through the emission of phonons. DM interacting with electrons can lead to visible signals of \(\sim \) 10 eV through the following channels: electron ionization and electronic excitation. Similar to DM-nucleus scattering, the energy deposited by electronic recoils is also in the range of sub-eV to \(\sim \) 100 eV. Such a small amount of energy is again largely dissipated through the emission of phonons. Therefore, the detection of those phonons is a major consideration for the design of the next generation of Ge detectors.

### 2.2 The form of the detectable signature

The bandgap energy of Ge is 0.67 eV at room temperature [44]. It increases slightly as temperature decreases [45]. The band structure of Ge is an indirect bandgap, which means that the minimum of conduction band and the maximum of valance band lie at different momentum, *k*, values. When an e–h pair is created in Ge, phonons must be involved to conserve momentum [47]. Therefore, the average energy expended per e–h pair in Ge at 77 K is \(\sim \) 3 eV, much higher than the bandgap energy of 0.73 eV [47] at the same temperature.

Since the energy dissipation, induced by DM interactions with a nucleus or an electron, is mainly released through the emission of phonons, the energy of phonons (E\(_{phonon}\)), can be estimated through E\(_{phonon}=hv_{s}/a\), where *h* is the Planck constant, \(v_{s}\) is the speed of sound in Ge, and *a* (0.565 nm) is the lattice constant of Ge. The value of the speed of sound in Ge depends on the polarization of phonons and the orientation of Ge crystal. For a [100] Ge crystal, \(v_{s}\) = 5.4 \(\times \) 10\(^{3}\) m/s [49] for longitudinal acoustic (LA) phonons and \(v_{s}\) = 3.58 \(\times \) 10\(^{3}\) m/s for transverse acoustic (TA) phonons [50]. Therefore, E\(_{phonon}\) can be 0.037 eV for LA phonons and 0.026 eV for TA phonons, respectively. These values agree with the early measurements made by Brockhouse and Iyengar [51] and Others [52]. There are also measured phonons in longitudinal optical (LO) and transverse optical (TO) bench with energies up to 0.063 eV [51]. The energies of LA, TA, LO, and TO phonons are much less than 3 eV required to generate an e–h pair at 77 K. Those phonons are not capable of generating e–h pairs through excitation of Ge atoms. Indirect detection of those phonons has been demonstrated by CDMS [6], EDELWEISS [15] and SuperCDMS [20] with threshold energy as low as \(\sim \) 50 eV.

Ionization energies of shallow impurities in Ge

Impurity | Boron | Aluminum | Gallium | Phosphorus |
---|---|---|---|---|

Ionization energy (eV) | 0.0104 | 0.0102 | 0.0108 | 0.012 |

As can be seen from Table 2, the ionization energies of impurities are all in the range of \(\sim \) 0.01 eV, phonons with energies of 0.037 and 0.026 eV can certainly ionize or excite impurities to produce charge carriers. However, since the deposited energy from nuclear motion or electronic motion is in the range of sub-eV, the ionization or excitation of impurities can only produce a few charge carriers per interaction induced by DM particles. Such a small amount of charge needs to be amplified internally in order to overcome the electronic noise in the digitization. We describe a Ge detector utilizing internal amplification of charge carriers created by the ionization of impurities below.

## 3 Development of a Detector with Internal Amplification of Charge Created by the Ionization of Impurities

### 3.1 Zone refining

*x*is the length of the Ge ingot, \(C_{i}\)(x) is the initial purity,

*i*runs from 1 to 4 for B, Al, Ga, and P in the Ge ingot respectively, \(k_{i}\) is the effective segregation coefficient in Ge,

*L*is the width of the melting zone, and

*n*is the number of passes. The value of \(k_{i}\), depending on the zone speed, zone width, and the number of passes, is determined individually by the experimental data. The zone-refined Ge ingots usually appear to be p-type after 10 passes and can be directly used to grow a p-type Ge crystal or intentionally doped to grow an n-type crystal.

### 3.2 High-purity Ge crystal growth

*i*representing B, Al, Ga, and P, \(k_{i}\) the effective segregation coefficient for B, Al, Ga, and P respectively, and

*g*being the fraction of crystal. A comparison between the Hall effect measurements (data points in black) and the fitted function (solid line in red), using Eq. 10, is shown in Fig. 4, where a high quality crystal with diameter up to 10 cm was achieved. The measured mobility is greater than 45,000 cm\(^2\)/Vs along both the radial and axial directions. This indicates that the distribution of impurities inside crystal is uniform, which is critical to the proposed detector technology in this paper.

### 3.3 Internal charge amplification

*h*is the length of avalanche region and

*l*is the free electron path of inelastic scattering. The value of

*l*in Ge at \(\sim \) 4 K is about 0.5 \(\mu \)m and

*h*can be 5 \(\mu \)m for a planar detector of 3 cm thick, as shown in Fig. 5. Thus, it is possible to achieve a value of

*K*= 10\(^{3}\) with a threshold as low as 0.1 eV to guarantee a low-energy threshold.

A Ge detector with internal avalanche amplification can be fabricated at a mass of about 1.0 kg, as shown in Fig. 5. The detector can be made of a high-purity (an impurity level of (1–3) \(\times \) 10\(^{10}\)/cm\(^{3}\)) multistrip planar Ge detector and has a dimension of 9 \(\times \) 7 \(\times \) 3 cm\(^{3}\). There can be 15 anode strips fabricated using the photomask method, at a width of 20 \(\mu \)m and a length of 7 cm. The expected cathode area will be 9 \(\times \) 7 cm\(^{2}\) and the fiducial volume will be approximately 190 cm\(^{3}\). Guard electrodes in the anode and cathode planes can also be designed. To fabricate a Ge detector with internal amplification, one must: (1) use a Ge crystal that guarantees a uniform distribution of impurities to provide a homogeneous electric field near the anode; (2) create a wide shallow junction layer under the strips so that the electric field near the strips is defined by junction dimensions; and (3) guarantee reliable cooling of the crystal, since the critical electric field and amplification factor depend on the free path of charge carriers, which in turn depend on the temperature.

### 3.4 Ionization of impurities

#### 3.4.1 Propagation of phonons

#### 3.4.2 Phonons interactions with impurities

## 4 Evaluation of the Detector Sensitivity

### 4.1 Signal from the ionization or excitation of impurities

*P*is the absorption probability, d is the average distance diffused before a subsequent anharmonic decay process (Eq. 11), and \(\lambda = \frac{1}{\langle \sigma \rangle \times N_{A}}\) is the mean free path of phonons with \(N_{A}\) the net impurity level in a given detector.

Note that the recombination of e–h pairs created by shallow impurities under a high electric field can be negligible, since the recombination probability of e–h pairs is in the level of less than 1% [46]. Therefore, the phonons created by recoiling particles can be detected at a level of > 50%, if one takes into account the loss of phonons due to the internal reflection at the boundary with the specific Ge detector discussed above. This can be justified using Eqs. 11 and 14. By taking into account the anharmonic decay, which increases the average distance diffused before another subsequent anharmonic decay, the detection efficiency of phonon are 55, 64 and 74% for three impurity levels, 3 \(\times \) 10\(^{10}\)/cm\(^3\), 9 \(\times \) 10\(^{10}\)/cm\(^3\), and 2 \(\times \) 10\(^{11}\)/cm\(^3\), respectively. Err on the conservative side, we choose a phonon detection efficiency of 50% for the discussion of the project sensitivity for MeV-scale DM.

It is worth mentioning that the phonons can be generated during the avalanche process in which the accelerated charge carriers have sufficient kinetic energies (greater than the Ge bandgap) to produce e–h pairs. Since Ge is an indirect bandgap, the generation of an e–h pair requires momentum conversation [47]. Consequently, there are phonons accompanying the generation of e–h pairs. The average phonon energy is estimated to be about 0.00414 eV [47], which has small probability to ionize or excite impurities, according to Eq. 16. Since there are as many as a few hundreds of phonons produced per e–h pair, the charge carriers created by these phonons could cause electric breakdown during the avalanche process. However, the avalanche process is within 100 ns and the ionization or excitation of impurities by those phonons are delayed by about 10 \(\mu \)s, as can be seen from Fig. 6. This allows us to decrease the E-field for stopping avalanche for about 1 ms right after the primary pulse generated by the avalanche process. Therefore, electric breakdown can be avoided by a strategic operation.

The avalanche quenching techniques, partitioned in active and passive methods, exist in the applications of avalanche photodiodes [48]. In active methods, the avalanche quenching can be proceeded by acting on the bias. In passive methods, the bias voltage can be self-adjusted by a ballast resistor. Recharge the bias across the detector can be re-established by a switch or through a ballast resistor. The time scale for both quenching and recharge can be obtained within 1 ms. Therefore, the dead time of the detector due to the avalanche quenching and recharging is negligible.

The event energy can be reconstructed using pulses. The e–h pairs produced by the secondary phonons will be delayed by about 10 microseconds, which allow us to distinguish the primary avalanche pulses induced by the primary phonons from the secondary pulses induced by the secondary phonons without avalanche effect. Therefore, the timing of pulses will provide a method to reconstruct the event energy using the primary pulses only, which will prevent the event energy resolution from getting worse due to the contamination of the secondary pulses.

### 4.2 Evaluation of backgrounds

In the detection region of interest (sub-eV to 100 eV), the expected background rates are essentially negligible from natural radioactivities and muon-induced processes due to the ability of detecting a single charge carrier. This can be estimated with an achieved background rate from a Ge DM experiment, such as CDEX [7]. CDEX reported a background rate of (4.09 ± 1.71) kg\(^{-1}keV^{-1}d^{-1}\), which can be translated to be (\(4.09\pm 1.71)\times 10^{-3}\) kg\(^{-1}eV^{-1}d^{-1}\). This rate is dominated by cosmogenic backgrounds such as tritium and \(^{68}\)Ge, as shown by the Majorana Demonstrator [78]. This rate can be further reduced to be about 0.04 kg\(^{-1}eV^{-1}y^{-1}\). Neutrino elastic scattering is another source of backgrounds and is estimated in the level of \(\sim \) 0.001 kg\(^{-1}eV^{-1}y^{-1}\) [31, 79].

The bulk thermal noise depends on the thermal energy, which is about 0.00033 eV at 4 K. The excitation probability with such a small thermal energy is estimated to be at a level of \(\sim \) 10\(^{-4}\). This is completely negligible. Another possible source of background is the single electron injection from electrodes. The materials with higher values of work-function will be chosen to minimize this source of background. It is worth mentioning that breakdown observed at fields of order of \(\sim \) 30 V/cm in the SuperCDMS-type detector at a base temperature of \(\sim \) 30 mK was not due to the impact ionization of the bulk of the crystals [80]. This also indicates that thermal noise will not be a significant source of background.

### 4.3 Projected sensitivity

*A*is the number of nucleons per nucleus;

*F*is the nuclear form factor; \(v_{esc}\) = 544 km/s (the escape velocity); and the minimum velocity is defined as:

*A*is the nuclear mass number. It is common that, \(f^{p} = f^{n}\) is assumed and the dependence of the cross-section with the number of nucleons

*A*takes an A\(^{2}\) form. For light DM, the momentum transfer is small and the nuclear form factor,

*F*, can be assumed to be 1. The total event rate is given by:

Figure 12 demonstrates the projected total event rate for 1 kg-day exposure, assumed a DM-nucleon cross-section of 10\(^{-42}\) cm\(^{2}\) and a DM-electron cross-section calculated using Eq. 24. As can be seen in Fig. 12, there will be a few events per day for detecting DM-electron scattering when the mass of DM is greater than 0.1 MeV/c\(^2\) and a few events per day for detecting DM-nucleus scattering when the mass of DM is a few MeV/c\(^2\) in a detector with only 1 kg of mass, if we assume the DM-nucleus cross section is 10\(^{-42}\) cm\(^{2}\). If we don’t observe any events in the region of interest, the sensitivity for such a detector is shown in Fig. 13. This is indeed a very sensitive experiment even with 1 kg of mass due to a much larger DM flux in the sub-MeV mass region comparing to that of the GeV mass region. Since the signals are single electrons in the energy region of sub-eV, the background events are negligible. For 1 kg-year exposure, as shown in Fig. 14, the event rates are calculated with a DM-nucleon cross section of 5 \(\times \) 10\(^{-45}\) cm\(^{2}\) and a DM-electron cross section calculated using Eq. 24.

## 5 Conclusions

We have demonstrated a design for a very sensitive detector in the search for light DM of MeV-scale in mass. The key features of this detector are: (1) high-purity Ge detector with a net impurity level of \(\sim \) 3 \(\times \) 10\(^{10}\) cm\(^{-3}\) uniformly distributed across the entire detector; (2) internal charge amplification with a gain factor of 1000; and (3) depleted detector at 77 K and continuously cooled down to \(\sim \) 4 K. Such a detector allows phonons, created by DM through elastic scattering off either a nucleus or electrons, to excite impurities at 0.01 eV and hence create charge carriers. Taking into account the internal amplification, the best option of such a detector is n-type, which will directly generate electrons as the charge carriers when phosphorus atoms are excited by phonons. Those electrons will then be drifted under a large field of 10\(^4\) V/cm to create avalanche amplification. The sensitivity of this detector reaches \(\sim \) 10\(^{-43}\) cm\(^2\) for DM-electron scattering and \(\sim \) 10\(^{-42}\) cm\(^2\) for DM-nucleon scattering in a day.

In comparison with the current SuperCDMS and EDELWEISS experiments, there are two main differences: (1) charge creation and (2) internal amplification. For the former, in our technology, the charge is created by ionization of impurities, which allows an experiment to access even lower energy deposition (\(\sim \) 0.1 eV) comparing to SuperCDMS or EDELWEISS at which the charge is mostly created by ionization of Ge (\(\sim \) 50 eV). In the case of the latter, we propose to internally amplify charge using avalanche while SuperCDMS and EDELWEISS internally amplify signal through emission of Luke phonons.

## Notes

### Acknowledgements

We would like to thank Christina Keller for a careful reading of this manuscript. This work was supported in part NSF OIA 1434142, DOE grant DE-FG02-10ER46709, 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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