silicon carbide detectors for sub-gev dark matter

Silicon carbide detectors for sub-GeV dark matter

Sinead M Griffin 12 Yonit Hochberg3 Katherine Inzani 12 Noah Kurinsky 45 Tongyan Lin6 and To Chin Yu 78

1Materials Sciences Division Lawrence Berkeley National Laboratory Berkeley California 94720 USA2Molecular Foundry Lawrence Berkeley National Laboratory Berkeley California 94720 USA

3Racah Institute of Physics Hebrew University of Jerusalem Jerusalem 91904 Israel4Fermi National Accelerator Laboratory Batavia Illinois 60510 USA

5Kavli Institute for Cosmological Physics University of Chicago Chicago Illinois 60637 USA6Department of Physics University of California San Diego California 92093 USA

7Department of Physics Stanford University Stanford California 94305 USA8SLAC National Accelerator Laboratory 2575 Sand Hill Road Menlo Park California 94025 USA

(Received 2 September 2020 accepted 5 January 2021 published 6 April 2021)

We propose the use of silicon carbide (SiC) for direct detection of sub-GeV dark matter SiC hasproperties similar to both silicon and diamond but has two key advantages (i) it is a polar semiconductorwhich allows sensitivity to a broader range of dark matter candidates and (ii) it exists in many stablepolymorphs with varying physical properties and hence has tunable sensitivity to various dark mattermodels We show that SiC is an excellent target to search for electron nuclear and phonon excitations fromscattering of dark matter down to 10 keV in mass as well as for absorption processes of dark matter down to10 meV in mass Combined with its widespread use as an alternative to silicon in other detectortechnologies and its availability compared to diamond our results demonstrate that SiC holds muchpromise as a novel dark matter detector

DOI 101103PhysRevD103075002

I INTRODUCTION

The identification of the particle nature of dark matter(DM) is one of the most pressing problems facing modernphysics and will be a key focus for high-energy physics andcosmology in the coming decade [12] In the absence ofevidence for darkmatter at theweak scale interest has grownin direct searches for DM with sub-GeV mass [3ndash20]The technical challenge inherent in searching for non-

relativistic sub-GeV weakly interacting particles can beseen by considering the case of a classical nuclear recoilFor DM with a mass mχ much smaller than the targetnucleus moving at the escape velocity of the galaxy themaximum energy transfer for a classical elastic scatteringnuclear-recoil event is

ΔE asymp2 meVAT

mχ

1 MeV

2

eth1THORN

with AT the atomic number of the target This motivatesusing lighter nuclei to increase the energy transfer as well

as new detector technologies sensitive to meV-scale energydepositsIn Ref [7] a subset of the authors explored the ability of

diamond (crystalline carbon C with AT frac14 12) as a detectormedium to meet these criteria The long-lived phonon stateswith meV energies coupled with the light carbon nucleimake diamond an excellent medium with which to searchfor dark matter Diamond suffers from two significantdrawbacks however it is currently difficult to producesingle crystals in bulk at masses sufficient to achieve thekg-year exposures required to probe significant DMparameter space and the nonpolar nature of diamondlimits the DM candidates to which it can be sensitiveHere we propose for the first time the use of silicon

carbide (SiC) as a DM detector as it overcomes thesedrawbacks Largewafers and therefore also large boules ofSiC can be readily obtained at prices comparable to silicon(Si) Importantly as a polar semiconductor SiC has opticalphonon modes which can be excited by sub-GeV DM withdark photon interactions [16] Furthermore as we demon-strate in this paper SiC behaves in most ways as a nearsubstitute to diamond with many relevant properties inter-mediate between crystalline diamond and silicon SiC hasalready seen widespread adoption as a target for radiationdetectors [21] microstrip detectors [22] and UV photo-diodes [23] as a drop-in replacement for Si in environmentswhere greater radiation hardness improved UV sensitivityor higher-temperature operation are required The latter two

Corresponding authorsgriffinlblgov

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 40 International licenseFurther distribution of this work must maintain attribution tothe author(s) and the published articlersquos title journal citationand DOI Funded by SCOAP3

PHYSICAL REVIEW D 103 075002 (2021)

2470-0010=2021=103(7)=075002(29) 075002-1 Published by the American Physical Society

considerations are possible due to the higher band gap ofSiC 32 eV compared to 112 eV for Si It is thus natural toobserve the parallels between the development of Sidiamond and SiC detector technologies and explore theability of future SiC detectors to search for sub-GeV DMMoreover SiC is an attractive material to explore

because of its polymorphismmdashthe large number of stablecrystal structures which can be readily synthesizedmdashandthe resulting range of properties they possess In fact SiCexhibits polytypismmdasha special type of polymorphismwhere the crystal structures are built up from a commonunit with varying connectivity between the units (seeFig 1) The variety of available polytypes results in acorresponding variety of physical properties relevant to DMdetection such as band gap and phonon mode frequenciesIn this paper we explore six of the most common polytypes(3C 2H 4H 6H 8H and 15R described in detail in Sec II)which span the range of variation in physical properties andevaluate their suitability as target materials for a detector aswell as their differences in DM reach for given detectorperformance goals In particular we show that the hex-agonal (H) polytypes are expected to exhibit stronger dailymodulation due to a higher degree of anisotropy in theircrystal structureIn this work we explore the potential of SiC-based single-

charge detectors and meV-scale microcalorimeters for DMdetection The paper is organized as follows In Sec II wediscuss the electronic and vibrational properties of the SiCpolytypes explored in this work In Sec III we explore themeasured and modeled response of SiC crystals to nuclearand electronic energy deposits over a wide energy range and

the expected performance of SiC detectors given realisticreadout schemes for charge and phonon operating modesSections IVand V summarize the DMmodels considered inthis paper compare the reach of different SiC polymorphsinto DM parameter space for nuclear recoils direct phononproduction electron recoils and absorption processes andalso compare directional detection prospects The high-energy theorist interested primarily in the DM reach ofSiC polytypes can thus proceed directly to Sec IV We findexcellent DM sensitivity comparable and complementary toother proposals which places SiC detectors in the limelightfor rapid experimental development

II ELECTRONIC AND PHONONICPROPERTIES OF SiC POLYTYPES

Silicon carbide is an indirect-gap semiconductor with aband gap (23ndash33 eV) intermediate between those ofcrystalline silicon (11 eV) and diamond (55 eV) Whilethere exists a zinc-blende form of SiC which has the samestructural form of diamond and Si there are over 200additional stable crystal polymorphs with a range of bandgap energies and physical properties These polymorphsbroadly fall into three groups based on lattice symmetrycubic (C) hexagonal (H) and rhombohedral (R) Tocompare the expected performance of these polytypes asparticle detectors we first explore how the differences inband structure between polytypes manifests in charge andphonon dynamicsIn all SiC polytypes the common unit is a sheet of corner-

sharing tetrahedra and the polytypes are distinguished byvariations in stacking sequences The polytype 3C adopts thecubic zinc-blende structure with no hexagonal close packingof the layers whereas 2H has a wurtzite structure withhexagonal close packing between all the layers The differentpolytypes can thus be characterized by their hexagonalityfraction fH with 2H (3C) having fH frac14 1 (fH frac14 0) Thissingle number correlates strongly with the materialrsquos bandgap with 3C having the smallest gap and 2H the largest gap[25] The other polytypes including those considered in thispaper consist of lattices with different sequences of hex-agonal and cubic stacking layers and can be listed in order ofincreasing hexagonal close packing 3C 8H 6H 4H 2HThe number refers the number of layers in the stackingsequence Rhombohedral structures also occur and these arecharacterized by long-range stacking order as shown inFig 1(f) Crystal structures for the polytypes considered hereare shown in Fig 1The difference in stability between cubic and hexagonal

stacking is very small which can be understood as abalance between the attractive and repulsive interactionsbetween third-nearest neighbors stemming from the spe-cific degree of charge asymmetry in the SiC bond [26] Thisresults in a difference in total energy between the polytypesof only a few meV per atom therefore many crystalstructures of SiC are experimentally accessible To limit

FIG 1 Crystal structures of the polytypes of SiC considered inthis work Si atoms are blue and C atoms are brown

SINEAD M GRIFFIN et al PHYS REV D 103 075002 (2021)

075002-2

this paper to a reasonable scope we restrict our analysis tosix of the most common forms as shown in Fig 1 and withproperties summarized in Table I Despite the relativestability of polytypes with respect to one another onlythree of these polytypes (3C 4H and 6H) are availablecommercially [25] as of this writing of these 6H is themost widely available in the large wafer and crystal sizestypically employed in semiconductor processing To cap-ture a representative range of SiC polytype behavior in ouranalysis and to observe trends in properties relevant forsub-GeV DM detection we also include 2H 8H and 15Rin our analysisCalculations of the interaction of various DM models

with SiC requires materials-specific information for eachpolymorph namely the electron and phonon spectra toestimate sensitivity to electron and phonon interactionsrespectively We calculate these quantities using state-of-the-art density functional theory (DFT) calculations asdescribed in detail in Appendix A The electronic bandstructures for the six representative polytypes are shown inFig 2 and the phonon band structures are plotted in Fig 3For reference the Brillouin zones (BZs) for the samepolytypes are shown in Fig 11

The band structure of a material is important for under-standing its charge or phonon dynamics in particular chargemobility and lifetime and phonon losses during chargepropagation As with Si Ge and diamond the indirect bandgap of all SiC polytypes ensures long charge lifetimesallowing charge to be drifted and collected with a modestelectric field At low temperature this also produces aniso-tropic propagation of electrons due to localizedminima in thefirst BZ away from the Γ point (as shown in Si and Ge at lowtemperature [3940]) which has a significant impact onchargemobility as a function of crystal orientation relative toan applied field In Si and diamond for example theseelectron valleys lie at the three X symmetry points along thecardinal directions in momentum space Depending on thecrystal orientation relative to the electric field spatiallyseparated charge clusters are observed as charges settle intoone of these conduction valleysDue to the range of stable crystal forms of SiC in

contrast to Si diamond and Ge we cannot make a generalstatement about the location in momentum or positionspace of the indirect band gap in SiC (see eg Refs [39ndash41]) but we can locate the BZ minima from the bandstructures shown in Fig 2 The 3C polytype like Si and

FIG 2 (a)ndash(f) Calculated electronic band structures of SiC polytypes with high-symmetry paths selected using SeeK-path [24]alongside the density of states Valence band maxima and conduction band minima are highlighted with blue and pink circlesrespectively To show the conduction band valleys in momentum space (g)ndash(l) are isosurfaces of the electronic energy bands at 02 eVabove the conduction band minima plotted within the first Brillouin zone boundaries of the polytypes For the positions of the high-symmetry points see Fig 11 and for details of calculations see Appendix A

SILICON CARBIDE DETECTORS FOR SUB-GEV DARK MATTER PHYS REV D 103 075002 (2021)

075002-3

diamond has X-valley minima and therefore three chargevalleys in the first BZ [42] so we can expect that the chargemobility will behave similarly to Si and diamond Thehexagonal forms as shown in Figs 2(b)ndash2(e) generallyhave minima along the L-M symmetry line while the 2Hpolytype has minima along the K points All of thesepolytypes have six charge minima in the first BZ howevercharge propagation in 2H will be maximally restricted topropagation along the horizontal plane of the BZ (the planealigned with [100] [010] Miller indices) As we go to largerunit cells charge propagation perpendicular to that planebecomes more kinematically accessible allowing for moreisotropic charge propagation The valence bands are moreconsistent between polytypes with a concentration of the

valence band near the Γ point which is also the location ofthe valence band maximum for all polytypes consideredThe dominant influence of the carbon-silicon bond

(rather than the electronic orbital overlaps) on the phononproperties leads to the phonon dynamics being similarbetween the polytypes Since Si and C have near-identicalbonding environments the Born effective charges arealmost identical in all of the compounds considered andso will have similar dipolar magnitudes and henceresponses to dark-photon-mediated interactions The pho-non band structures for these polytypes are plotted in Fig 3While we show the entire band structure the DM-phononinteractions are most sensitive to the phonon propertiesnear the Γ point In particular the similarities of the

TABLE I Bulk material properties of diamond Si and the SiC polymorphs considered in this work (measurements taken fromRefs [2127ndash31] unless otherwise stated) All gaps are indirect as discussed in the text and shown in Fig 2 ϵ0infinperp (ϵ0infink) refer to relativepermittivity perpendicular (parallel) to the crystal c axis at low and high frequency with values from Ref [25] Optical phonon energiesand high-frequency permittivity are taken from Ref [32] Eeh values denoted by dagger have been estimated as described in the text Defectcreation energies are from Refs [33ndash35] Due to the differing commercial availability and utility of different polytypes more commonlyused crystal polytypes are better characterized than less common ones and thus for the least well-studied polytypes (2H 8H 15R) manyexperimentally determined values are unavailable Quantities denoted as [calc] were calculated in this work to fill in some of the holes inthe literature

Parameter Diamond (C) Si SiC

Polymorph 3C (β) 8H 6H (α) 4H 2H 15R

Crystal structure Cubic Hexagonal Rhombohedral

ρ (g cmminus3) 351 233 sim32 [2728]N (1023 cmminus3) 176 05 096ne (1023 cmminus3) 354 1 195ℏωp (eV) 22 166 221 [36]

a (c) (Aring) 3567 5431 436 307 (2015) 308 (1512) 307 (1005) 307 (504) 307 (3780)fH 00 00 00 025 033 05 10 04Egap (eV) 547 112 239 27 302 326 333 30Egap etheVTHORNfrac12calc 224 266 292 315 317 286Eeh (eV) sim13 36ndash38 57ndash77dagger 64ndash87dagger 67 [37] 77ndash78 [2838] 78ndash105dagger 71ndash96dagger

Edefect (eV) 38ndash48 11ndash22 19 (C) 38 (Si) 22 (C) 22ndash35 [21] 17ndash30 (C)

ϵ0perp 57 117 97 967 976ϵ0k 1003 1032

ϵfrac12calc0perp 1040 1040 1039 1036 1024 1038

ϵfrac12calc0k

1080 1090 1106 1141 1096

ϵinfinperp 65 66 66 65 65ϵinfink 67 68 68 67

ϵfrac12calcinfinperp 707 710 711 710 703 711

ϵfrac12calcinfink731 736 741 740 738

ΘDebye (K) 2220 645 1430 1200 1200ℏωDebye (meV) 190 56 122 103 103ℏωTO (meV) 148 59 987 977 988 970 988 953 990 989ℏωLO (meV) 163 63 1205 1197 1203 1195 1200 1200 1207 1196cs (m=s) 13360 5880 12600 13300 13730cs ethm=sTHORNfrac12calc 13200 16300 14300 14300 15500 11900vdsat eminus (105 m=s) 27 [28] 135 2 2 2EBd (MV=cm) gt 20 03 12 24 20


075002-4

properties described above implies that the sensitivity ofSiC for DM scattering will be similar for all polytypesAnisotropies in the phonon band structure will give rise todifferences in the directional dependence of the DM signalas will be discussed later in this paperTable I summarizes the physical properties of the

polytypes shown in Figs 2 and 3 compared to Si and CIn addition some derivative properties of the phonon bandstructures are summarized in the table it can be seen that allSiC polytypes have sound speed highest optical phononenergy and permittivity which are roughly the geometricmean of the Si and C values These characteristics willinform our detector design and results for dark matterreach as we now detail

III DETECTING ENERGY DEPOSITS IN SiC

In this section we apply the detector performance modelof Ref [7] to the six representative SiC polytypes describedabove and contrast expected device performance betweenthe SiC polytypes as well as with Si Ge and diamondtargets We begin by reviewing existing measurementsand expectations for partitioning event energy into theionization (charge) and heat (phonon) systems relevant toreconstructing total event energy for different types ofparticle interactions We then discuss expected detectorperformance in charge and phonon readout modes givenavailable measurements for polytypes considered in thispaper and comment on expected performance for thosepolytypes without direct measurements based on bandstructure properties discussed above A theorist primarily

interested in the DM reach of a given SiC crystal for anassumed threshold can proceed directly to Sec IV

A Particle interactions

We first turn to the expected yield for an electron recoilor nuclear recoil in SiC As discussed in eg Ref [7]interactions which probe electrons or nucleons are expectedto deposit differing amounts of energy in ionization andphonon systems in semiconductor detectors This propertywas used by the previous generation of DM experiments toreject electron-recoil backgrounds in the search for primarynucleon-coupled weakly interacting massive particle DMThe resolutions in these channels required for sub-GeVDM are just now being achieved for either heat charge orscintillation light in current experiments [43ndash52] but noneof these experiments can achieve the required resolutions inmultiple channels to employ event discrimination forrecoils much below 1 keV in energy For heat readoutexperiments this partition is relatively unimportant as allenergy remains in the crystal and is eventually recovered asheat For charge and light readout experiments thispartition is necessary to reconstruct the initial event energyand contributes significant systematic uncertainty to back-ground reconstruction at energies where the energy parti-tioning is not well constrainedA convenient shorthand is to refer to the energy in the

electron system as Ee which is related to the total recoilenergy Er according to a yield model yethErTHORN as Ee frac14yethErTHORNEr As discussed in eg Refs [753] for electronrecoils one has yethErTHORN frac14 1 while for nuclear recoils the

FIG 3 First-principles calculations of phonon band structures with high-symmetry paths selected using SeeK-path [24] For detailsof calculations see Appendix A


075002-5

yield is reduced due to charge shielding effects and lossesto phonons and crystal defects referred to as nonionizingenergy losses (NIEL) [54] Additionally this yield functionis actually derived with respect to the charge yield for ahigh-energy minimum ionizing particle [55] These eventsproduce a number of charge carriers neh in linear proportionto event energy with the relation neh frac14 Er=Eeh where Eehis taken to be a fixed property of a given material and is theeffective cost to produce a single electron-hole pair If wedefine measured Ee as Ee frac14 nehEeh we thus see thatyethErTHORN frac14 1 is only true by definition for events that obeythis linear relationshipFor SiC this factor Eeh varies along with the band gap

among the different polytypes The charge yield fromminimum ionizing particles (γ β and α) in 4H SiC isexplored in Ref [56] The response of 3C 4H and 6H tolower-energy x rays is subsequently discussed in Ref [28]The results of both studies are consistent with a highlylinear yield in electron recoils down to Oeth10 keVTHORN ener-gies but the pair creation energy Eeh is only characterizedfor two of the polytypes as shown in Table IFor the polytypes in which energy per electron-hole pair

has not been characterized we can predict Eeh based onother measured properties The generic expression for Eehis [555758]

Eeh frac14 Egap thorn 2L middot ethEie thorn EihTHORN thorn Eph eth2THORN

where L is a factor which depends on the dispersion curveof the conduction and valence bands Eie and Eih are theionization thresholds for electrons and holes respectivelyand Eph are phonon losses Reference [55] shows that forEie sim Eih prop Egap we get the formula

Eeh frac14 A middot Egap thorn Eph eth3THORN

where Eph takes on values from 025 to 12 eV and A isfound to be sim22 to 29 Reference [57] finds using abroader range of materials the parameters A sim 28 andEph sim 05ndash10 eV These allow us to predict a probablerange of Eeh values for the polytypes without existingmeasurements which we summarize in Table I For thedetector models in this paper we assume the values inTable I apply linearly for all electron-recoil events down tothe band gap energyThe response of SiC detectors to neutrons is less

characterized than the electronic response A detailedreview can be found in Ref [21] which we refer thereader to for more details on existing measurements Inparticular the NIEL for different particles in SiC iscomputed and compared to measurements for differention beams in Ref [59] but this is characterized as a loss pergram and not as a fractional energy loss compared to thatlost to ionization Reference [60] explores the thermalneutron response but only a count rate is measured a linear

response with respect to fluence is measured but thereis no characterization of ionization yield on an event-by-event basisWe instead appeal to simulations calibrated to silicon

measurements in which the single tunable parameterwith largest effect is the displacement energy thresholdfor freeing a nucleus from the lattice Edefect Known andestimated values for Edefect are summarized in Table I Thelarge range is due to the difference in thresholds for the Siand C atoms comparing the threshold values to Si anddiamond it seems that a Si atom in SiC has a diamondlikedisplacement threshold while a C atom in SiC has asiliconlike displacement threshold Reference [54] calcu-lates NIEL for Si and diamond with the difference para-meterized only in terms of defect energy This suggests thatSiC with a defect energy intermediate between Si anddiamond will behave identically to Si and diamond abovesim1 keV and give a yield below Si and above diamond forlower-energy interactionsFinally we consider the subgap excitations The most

prominent features are the optical phonons with energiesof 100ndash120 meV as shown in Table I and Fig 3 Aninteresting property of the hexagonal polytypes is that theoptical phonon energy depends on the bond direction alongwhich they propagate though weakly We can expect dueto the polar nature of SiC to see strong absorption aroundthe optical phonon energies We can also expect directoptical and acoustic phonon production by nuclear recoilssourced by DM interactionsIn contrast to the large change in electron gap energy and

expected pair-creation energy between polytypes we seevery little variation in phonon properties dielectric con-stants andmdashto some degreemdashdisplacement energy Thissuggests that different polytypes will be beneficial forenhancing signal to noise for desired DM channelsNucleon-coupled and phonon excitation channels wouldprefer higher-gap polytypes with suppressed charge pro-duction while electron-coupled channels favor the smallergap materials Optimization of readout will depend on thepolytype due to differences in phonon lifetime and chargediffusion length as discussed in the next subsection as wellas differences in phonon transmission between polytypesand choice of phonon sensor Other aspects of the designsuch as capacitance and bandwidth are constant acrosspolytypes somewhat simplifying the comparison ofpolytypes

B Charge readout

The first readout mode we consider is the direct readoutof charge produced in SiC crystals by low-noise chargeamplifiers This mode is limited to energy depositsexceeding the gap energy of the relevant polytype but isof interest due to the ability to run these charge detectors athigher temperatures without requiring a dilution refriger-ator and due to the simpler readout scheme We begin by


075002-6

considering the charge collection in SiC and how the bandstructure of the polytypes will affect charge mobility Wethen contrast the expected resolution with diamond andsilicon via the resolution model of Ref [7] The resultingexpected detector performance for these devices is sum-marized in Table II

1 Charge collection

The primary questions for charge readout of SiC arewhether complete charge collection is achievable in mono-lithic insulating samples and whether charge collectionvaries across polytypes While full charge collection for the4H polytype has been demonstrated [61] detailed studiesof charge collection efficiency suggest that semi-insulatingsamples have a fairly limited charge diffusion length atroom temperature [62] In Ref [62] this is attributed toeither recombination due to impurities or the inability toseparate electron-hole pairs in the initial interaction whichcauses rapid carrier recombination More recent studies ofcharge collection efficiency (ϵq) in SiC radiation detectorssuggest that ϵq is improving with substrate quality andfabrication techniques [63] though single-crystal 4H-SiCstill has diffusion lengths closer to polycrystalline diamondthan to single-crystal diamond [64]The only studies to demonstrate near full charge collec-

tion in SiC are Refs [212865] which all study energydeposition in thin films (sim40 μm) Studies of depositions ina 10 times larger detector volume in eg Refs [2162] doshow much reduced collection efficiency for the same biasvoltage and detector readoutThese studies suggest that there remain significant bulk

dislocations in these commercial wafers which presenttrapping or recombination-inducing defect sites While it ispossible that charge collection will improve at lowertemperatures or with higher-quality substrates there isnot yet sufficient data to show this Note that most radiationdetectors to date have been constructed of 4H and 6Hpolytypes it is possible that the 3C polytype with a moresymmetric band structure could demonstrate better chargecollection A rough analogy would be comparing thecharge collection of graphite to diamond though onewould expect 4H and 6H to be much more efficient than

graphite Charge collection is also likely dependent oncrystal orientation relative to the BZ minima as discussed inSec II with more efficient charge collection occurringwhen the electric field is aligned with an electron valleyFor the resolution calculation presented later in this

section we will assume perfect collection efficiency anincomplete efficiency will not affect resolution in thesingle-charge limit but will instead reduce effective expo-sure To minimize the effect of limited charge collection ondetector performance we require the drift length (detectorthickness) to be equal to or less than the diffusion length ofthe target charge carrier at the design voltageTo make this more quantitative one can model the ϵq in

terms of a few measured parameters Given a carriermobility μ (in principle different for electrons and holes)and saturation velocity vdsat we use an ansatz for carriervelocity as a function of voltage V

vdethV dTHORN frac14

1

vdsatthorn dμV

minus1 eth4THORN

where d is the detector thickness This gives the drift lengthD frac14 vdτscat rarr vdsatτscat in the high-field limit [21] whereτscat is the carrier scattering lifetime Given this drift lengthwe can model the ϵq as [65]

ϵq frac14Dd

1 minus exp

minusdD

eth5THORN

where for long diffusion length (D ≫ d) we have ϵq sim 1For short diffusion length and in the small-field limit wefind that charge collection goes as

ϵq asympμVτscatd2

frac14 μτscatd

E ≪ 1 eth6THORN

with E the electric field in the bulk This tells us that whenthe gain is linear in voltage the inferred ϵq will be small andthe effective diffusion length is much shorter than thecrystal thicknessThe best measure of the drift constant μτscat in 4H-SiC

(the only polytype for which detailed studies are available)was found to be μτscat sim 3 times 10minus4 cm2=V and for a

TABLE II Summary of the detector designs discussion for charge readout Voltage bias for the charge designs should be high enoughto ensure full charge collection For the lower two charge readout designs improved charge lifetime is assumed allowing for lowervoltage bias and thicker crystals We note that due to the relatively high dielectric constant of SiC the optimal geometry (given currentreadout constraints) is such that cells have a thickness greater than or equal to the side length in order to minimize capacitance per unitmass

Readout Design Dimensions Mass (g) Temp (K) Vbias σq

Charge

Single cell 10 cm side length times 05 cm thick 16

42 K

4 kV 14eminus

Single cell 05 cm side length times 05 cm thick 04 4 kV 05eminus

Single cell 10 cm diameter times 15 cm thick 48 500 V 05eminus

Segmented 02 cm side length times 02 cm thick 0025 50 V 025eminus=segment


075002-7

saturation drift field of 8 kV=cm we find a maximum driftlength D sim 24 cm [65] While this does imply full chargecollection for devices up to 1 cm thick the very highvoltages required are likely to induce some measure ofcharge breakdown despite the very high dielectric strengthof SiC The devices studied in Refs [212865] are all thinfilms which did not break down at field strengths in thisregime however for low-temperature operation of thesedevices voltages of this magnitude are atypical for mono-lithic gram-scale detectors Reference [28] suggests thereis a very small difference in mobility between the 3C 4Hand 6H polytypes but it is possible that the more isolatedvalleys of 3C and different growth process may lead to alarger charge lifetime To better determine the polytype bestsuited to charge collection more studies of drift length inhigh-purity samples are needed

2 Charge resolution

Recalling the model for charge resolution from Ref [7]the minimum resolution of a charge integrating readout iscompletely determined by the noise properties of theamplifier the bias circuit and the capacitance of thedetector (Cdet) and amplifier (Cin) (see eg Ref [66])

σq geNvethCdet thorn CinTHORN

ϵqffiffiffiτ

p eth7THORN

where Nv is assumed to be a flat voltage noise spectraldensity of the amplifier inV=

ffiffiffiffiffiffiHz

pand τ is the response time

of the detector and readout For an integrator the readouttime τ is determined by the rate at which the input is drainedby some bias resistor Rb and thus τ frac14 RbethCdet thorn CinTHORNFollowing the discussion of Ref [7] the current best

cryogenic high electron mobility transistor amplifiers [67]allow for a detector resolution of

σq asymp eth28 eminushthorn pairsTHORNethCdet=eth100 pFTHORNTHORN3=4 eth8THORNwhere we have enforced the optimal design conditionCin frac14Cdet and we assume full charge collection can be achieved(this is ensured by limiting thickness to 1 cm) Note that ifthis resolution for ϵq sim 1 is subelectron it affects theeffective resolution on the input signal rather than theresolution of the readout1

We give example design parameters for a charge detectorin Table II We consider both monolithic and segmenteddetectors the latter necessary to achieve statistically sig-nificant subelectron resolution at reasonable detector mass

One benefit of SiC over eg diamond is that larger crystalsare readily available commercially allowing for designswith Oeth1 eminusTHORN resolution All designs are limited to 15 cmthicknessmdashthe largest thickness currently available and toensure full charge collection at a field of 8 kV=cm Welikewise assume significant voltage bias in our designs forthis reason and in our latter charge designs assumeimprovements can be made in drift length by improvingmean carrier lifetime through advances in crystal growthtechnology (see Ref [65] for a more detailed discussion)The size and resolution of the segmented design suggest

that development of SiC detectors with a skipper CCDreadout is likely a more straightforward development pathlarge-scale fabrication of SiC devices has been available fordecades and feature sizes required are fairly modest Suchdevelopments would be useful for employing large-areaSiC sensors as beammonitors and UV photon detectors andwould be complementary to Si substrates for dark matterdetection thanks to the reduced leakage current due to thehigher gap and dielectric strength of SiC

C SiC calorimetry

The most promising direction for application of SiC todark matter searches is direct phonon readout at cryogenictemperatures The intrinsic phonon resolution σph is theprimary metric for determining DM reach in this caseHere we take a technology-agnostic approach to computingexpected phonon resolution rather than calculating reso-lutions for a specific detector technology we will relateresolution to phonon collection efficiency intrinsic phononproperties of the material and input-referred noise equiv-alent power (NEP) of the readout For the last quantity wewill use reference values comparable to those currentlyachievable by a range of cryogenic sensing techniques Wewill compare this with currently achieved resolutions usingother crystals as well as technologies of sufficiently lownoise temperature to achieve sub-eV resolutions andsuggest form factors and noise temperature targets forthe various thresholds discussed for DM sensitivities laterin this paperFollowing this parameterization we thus calculate res-

olution as

σph frac141

ϵph

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSphτpulse

q eth9THORN

where ϵph is the energy efficiency for phonon collectionSph is the NEP for the readout in W2=Hz prop eV2=s andτpulse is the duration of the signal in seconds

2 This is similarto the detector treatment in Refs [712] and uses the same1For the case of incomplete charge collection for detectors with

single-electron resolution the resolution is not smeared due toPoisson fluctuations but the conversion from charge to energyscale requires folding in charge collection statistics For detectorswithout single-electron resolution limited charge collectioneffectively contributes an additional Poisson smearing to theGaussian noise probability distribution function

2τpulse can also be thought of as the inverse of the bandwidth(τpulse frac14 2π=ωpulse) We use τpulse rather than ωpulse for easiercomparison with sensor response time τsensor given thatτpulse frac14 τph thorn τsensor where τph is the phonon signal time


075002-8

terminology as for transition edge sensor (TES) noisemodeling [68] but is written more generally for ease ofcomparison between readout technologies

1 Phonon collection efficiency

The primary metric which determines whether a materialwill allow for efficient phonon collection is the phononlifetime τlife As discussed in Ref [7] for pure crystals atlow temperature the lifetime is limited primarily byboundary scattering This scaling for the phonon lifetimecan be inferred from thermal conductance data givenknowledge of material density sound speed and crystalsize A model for the thermal conductance and its relationto the phonon lifetime is described in Appendix BFor diamond it was found that boundary scattering is

dominant for phonons of le 10 K implying that the bulkmean free path for phonons at and below this energy(≲1 meV) is much longer than the typical crystal lengthscale (1ndash10 mm) [7] For SiC the thermal conductivity (atleast that of 6H [69]) and sound speeds are close to that ofdiamond so we can infer that SiC will similarly be limitedby boundary scattering at least for phonons near the pair-breaking energy of the superconducting phonon sensorsThe phonon band structure calculations from Sec II(calculation details in Appendix A) were used to verifythat low-energy acoustic phonons (below 2 THz) have bulklifetimes much longer than their collection timescales(gt 10 ms) For 3C the calculated average phonon lifetimewithin 0ndash2 THz is of the order 30 ms at 2 K In thehexagonal polytypes the phonon lifetimes will be smallerbecause of increased scattering from the variations instacking sequences inherent to the structures but initialcalculation results at 10 K indicate that the 2H lifetimes willbe within an order of magnitude of those for 3CAssume a detector in the form of a prism of thickness η

and area A With only one type of phonon absorber thephonon collection time constant is [712]

τcollect frac144η

fabsnabscs eth10THORN

where fabs is the fraction of the detector surface areacovered by phonon absorber material and nabs is thetransmission probability between the bulk and the absorbernabs is calculated in detail in Appendix C with values inTable IV As a basis for comparison the worst-casescenario that phonons are completely thermalized at thecrystal sidewalls gives a bound on the phonon lifetime ofτlife ≳ η=cs a single-phonon crossing time across thecrystal In the following we will explore the case whereboundaries are highly reflective in which case τlife ≫τcollect as well as the case where boundaries are sourcesof phonon losses in which τcollect ≳ τlifeIn all cases the phonon pulse time is determined by

combining phonon collection time with phonon lifetimeas [12]

τminus1pulse asymp τminus1ph frac14 τminus1life thorn τminus1collect eth11THORN

where we assume the sensor is much faster than thetimescale of phonon dynamics (τph ≫ τsensor) Then theoverall collection efficiency is

fcollect frac14τpulseτcollect

frac14 τlifeτlife thorn τcollect

eth12THORN

The total detector efficiency is then given as a product ofthe conversion and readout efficiencies

ϵph frac14 fcollectϵqpϵtrap eth13THORN

where ϵqp is the efficiency of generating quasiparticles inthe phonon absorber and ϵtrap is the efficiency of readingout these quasiparticles before they recombine ϵqp has ageneric limit of 60 due to thermal phonon losses backinto the substrate during the quasiparticle down-conversionprocess [70] though it rises to unity as the capturedenergy approaches 2Δ sim 7kbTc=2 the Cooper pair bindingenergy for an absorber at Tc Meanwhile ϵtrap is technologydependent For quasiparticle-trap assisted TESs ϵtrap islimited by quasiparticle diffusion and losses into thesubstrate while for superconducting resonators such askinetic inductance detectors (KIDs) ϵtrap is governed by theresponse time of the resonator compared to the recombi-nation lifetime of the quasiparticles

2 Material-limited resolution

Since different readout technologies are possible herewe instead focus on the material-limited resolution of a SiCdetector We will thus consider an idealized phonon readoutwith a response time much faster than the characteristicphonon timescale and a benchmark noise temperature nearthat currently achieved by infrared photon detectors andprototype TES calorimeters Taking a single sensor withNEP

ffiffiffiffiffiSs

psim 10minus19 W=


p[71ndash73]3 and assuming our

idealized readout is limited by the timescale of phonondynamics we find a single-sensor resolution of

σph asymp 10minus19 W=ffiffiffiffiffiffiHz

p 1

ϵph

ffiffiffiffiffiffiffiffiffiffiτpulse

p eth14THORN

asymp10 meV1

fcollectϵtrap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiτpulse100 μs

reth15THORN

asymp10 meVϵtrap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ2collect

τpulse times 100 μs

s eth16THORN

3Here we are scaling the noise power measured in the referenceto the effective volume of a single quasiparticle-trap-assistedelectrothermal-feedback transition-edge sensors as characterizedin Ref [52]


075002-9

where we have set ϵqp frac14 06 We thus see that thechallenges for excellent resolution are to achieve highinternal quantum efficiency between phonon absorber andphonon sensor (ϵtrap) and to ensure fast phonon collection(short τcollect with τlife not too small compared to τcollect)Realistically readout noise power scales with sensor

volume and we can tie the above benchmark noisetemperature to a reasonable sensor area For currenttechnology a single superconducting sensor can typicallyread out about As sim 1 mm2 of area and thus we canparameterize the above equations more accurately in termsof this sensor area and detector geometry We find that

fabs frac14 NsAs=A eth17THORN

Sph frac14 NsSs frac14fabsAAs

Ss eth18THORN

For the above reference noise temperature and assumingτlife ≫ τcollect this gives an energy resolution of

σph asymp6 meVϵtrap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV

100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN

where V is the detector volume This is the generic resultthat an ideal athermal detector has a resolution that scales asffiffiffiffiV

pfor a given readout technology and as ethnabscsTHORNminus1=2 for a

given crystal and phonon absorber couplingIn the opposite limit where τlife ≪ τcollect we find the

resolution scales as


095nabs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN

where we have again used As frac14 1 mm3 and the soundspeed in SiC In this case the detector design relies on highsurface coverage fabs to maximize phonon collection andthe resolution is more sensitive to the phonon transmissionprobability nabs For the chosen parameters this is onlyabout twice the resolution of the long-lived phonon casebut it is more sensitive to details of sensor coverage and willbe more sensitive to phonon losses both in the crystal and atthe crystal-absorber interfaceThese estimates assume that the detector in question can

be read out with submicrosecond precision (such thatτsensor ≪ τph as stated earlier) while sensors at this levelof power sensitivity are not necessary capable of being readout at this rate [7374] we comment on this more belowFinally this term does not include phonon shot noisewhich will be a significant source of additional variance inthe limit of small phonon lifetime All of this goes to saythat the ideal detector design will be highly dependent onwhether phonons are completely thermalized at the boun-daries or if there is a reasonable chance of reflection of

athermal phonons such that there is a nonzero survivalprobability for each surface interactionTable III summarizes our four reference designs with

resolutions varying from 200 meV (design A) down to500 μeV (design D) Designs A and B assume the device isread out by phonon sensors comparable to those that havecurrently been demonstrated and the resolution is in thelong phonon lifetime regime of Eq (19) The designthresholds for these devices assume that the majority ofinitial phonons lie far above the absorption gap of thephonon sensor (2Δ sim 07 meV in Al) and that down-conversion at the crystal surfaces has a small impact ontotal phonon energy absorbed by the sensors The reso-lution scaling between A and B then comes just fromrelative reduction of crystal volumeFor designs C and D we consider an initial phonon

energy small enough that only a few phonon scatteringevents can occur before phonons are absorbed This implieswe will be in the short lifetime regime and need to havelarge coverage fabs rarr 1 to avoid substantial signal loss Toattain resolutions low enough to observe single-phononproduction we also assume here an order of magnitudedecrease in noise power over currently demonstratedphonon sensors Design C obeys the scaling of Eq (20)Design D has the same detector geometry but here weassume that only five or fewer sensors need to be read out toreconstruct a signal This provides an improvement inresolution by reducing the number of sensors read out by afactor of 20 without necessarily changing the detector orsensor properties The timescale for this process is still thephonon crossing time for the crystal additional resolutionreduction could still be accomplished by reducing the sizeof the crystal though gains would be fairly modestIn addition the resolution for sensors using quasiparticle

traps to read out phonons will hit a floor at the pair-breaking energy of the phonon absorber which for Al is2Δ sim 07 meV and for AlMn (with a Tc around 100 mK[75]) is sim006 meV (see also Table III) For this reason weassume that detectors with resolutionlt 50 meV (designs Cand D) will need to transition to lower-gap materials thisensures that the phonons they intend to detect can break≳100 quasiparticles per sensor to minimize shot noisecontributions to the noise budgetIn Fig 4 we show the scaling of resolution with sensor

area along with our reference designs in comparison tocurrently achieved resolutions by an array of superconduct-ing sensors These scalings are based on a fixed sensor formfactor with the given noise performance corresponding toan areal coverage of As sim 1 mm2 and the lines assume afixed power noise per unit area (as described earlier in thissection) for a variety of sensor coverage and crystal formfactors In all cases significant enhancements in sensornoise power are required to achieve less than 100 meVresolutions even for gram-scale detectors and we note thatthe detection thresholds for these detectors will be a


075002-10

multiple of these resolutions This limitation is not specificto SiC but broadly applies to any solid-state phononcalorimeters using superconducting readout In particularwe note that only designs C and D would be expected todetect single optical phonon excitationsThe choice of different polytypes in the detector designs

in Table III lead to minor changes in expected resolutiondue to sound speed (which varies by 25 betweenpolytypes) and impedance matching between the crystaland the absorber (a difference of less than 20) The samesensor design ported to different polytypes can thereforevary by up to around a factor of 2 in resolution a nontrivialamount but small compared to the range of energiesconsidered in this paper Selection of polytype is thereforeinformed more by sample quality mass and ease of

fabrication as well as potential science reach than byultimate phonon resolution The difference in science reachbetween the polytypes is the focus of the next sections ofthis paperFinally we note that the quoted resolutions apply to

readout limited by phonon dynamics and implicitly assumethat the phonon sensors used to read out these signals havea higher bandwidth than the crystal collection time Forthese designs a sensor with a response time ofsim1 μs wouldbe able to achieve within a factor of a few of these projectedresolutions This requirement and the additional require-ment that sensors be individually read out for design Dsuggests that superconducting resonator technologies suchas KIDs or switching technologies such as superconduct-ing nanowires are more likely to be the technology of

TABLE III Reference phonon detector designs Designs A and B assume performance parameters for currently demonstratedtechnology while design C assumes an improvement by a factor of 10 in noise equivalent power per sensor The main limitationaffecting design C is that for very low thresholds effective phonon lifetime may be as short as a few times the crystal crossing time dueprimarily to phonon thermalization at crystal boundaries If significant phonon thermalization is allowed to occur the phonon resolutionwill quickly be limited by statistical fluctuations in signal collection efficiency rather than sensor input noise For this reason our thirddesign assumes an effective phonon lifetime equivalent to three crystal crossings a lower gap absorber and 95 sensor coverage Thelimited absorption probability and realistic constraints on sensor area coverage severely limit the overall efficiency (ϵph) of the designrelative to the other two reference designs The polytype selection is primarily determined by the impedance match between the substrateand phonon absorber All polytypes are fairly well matched to the chosen absorbers but the 3C polytype is a close match and maximizesphonons absorbed per surface reflection

Design

Parameter A B C D

Polytype 6H or 4H 3C 3C Any

Phonon absorber Al AlMn2Δ Pair-breaking threshold 700 μeV 60 μeV

ϵqp Efficiency to generate quasiparticle in absorber 60ϵtrap Efficiency to readout quasiparticle in absorber 75τac Acoustic phonon lifetime (crystal limited) gt 30 μs

τlife Assumed phonon lifetime (boundary limited) sim100 μs sim1 μsffiffiffiffiffiffiffiffiffiffiffiffiSs=As

pNoise power per unit sensor area (W=mm middot Hz1=2) 10minus19 10minus20

affiffiffiffiffiSs

pNoise power per sensor (meV=s1=2) 600 60

A Detector area 45 cm2 5 cm2 1 cm2

η Detector thickness 1 cm 1 cm 4 mmnabs Transmission probability to absorber 083 094 sim094 b

fabs Fractional coverage of detector surface with absorber 01 07 095Ns Number of sensors 450 350 95τcollect Time scale to collect ballistic phonons 34 μs 43 μs 13 μsτpulse Time scale of phonon pulse 25 μs 42 μs 05 μsfcollect Collection efficiency into absorber 74 95 45ϵph Total signal efficiency for detector sim30 sim40 20

Detector mass 145 g 16 g 1 g

σph Resolution on phonon signal 200 meV 50 meV 2 meV sim05 meV c

aThis noise power is the best currently achievable in any quantum sensor see for example Refs [7172]bAlMn films are primarily Al containing lt 1Mn [75] and we assume the transmission coefficient will be approximately equal to

the pure Al casecThis assumes five or fewer sensors can be used to read out the total phonon signal and that the phonon dynamics are still the

bandwidth-limiting timescale


075002-11

choice than TESs or thermal sensors The former tech-nologies have noise temperature and response time that areindependent of thermal conductance and are intrinsicallymultiplexable The development of faster low-Tc TESdetectors which are capable of frequency-domain multi-plexing would allow them to be competitive at the lowerthresholds quoted here

D Combined readout

As a final note for detectors run in charge-readout modethere is still some interest in measuring total depositedphonon energy in order to separate electronic and nuclearrecoils as discussed earlier in this section The perfor-mance described in each design does not preclude combi-nation with any of the other readout modes for example acontact-free charge readout could be employed on one sideof the detector while the other side is instrumented withphonon sensors (like the CDMS-II ZIP see eg Ref [76])The need for discrimination of this sort is highly dependenton both the DM model and particular limiting backgroundand we thus leave further study of hybrid designs to futurework once charge yields in SiC at low energy can be bettercharacterized We note that for all of the models except forelectron-recoil dark matter the majority of signals arebelow the energy gap for ionization production Theseconsiderations thus mostly affect potential electron-recoilreach and nuclear-recoil reach above 05 GeV

IV THEORETICAL FRAMEWORK

We now move to describing the DM frameworks that canbe detected via SiC detectors We consider the followingpossible signals from sub-GeV DM interactions scatteringoff nuclei elastically scattering into electron excitations formχ ≳MeV phonon excitations for keV≲mχ ≲ 10 MeVand absorption of dark matter into electronic and phononexcitations for 10 meV≲mχ ≲ 100 eV In all cases ρχ frac1403 GeV=cm3 is the DM density and fχethvTHORN is the DMvelocity distribution which we take to be the standard halomodel [77] with v0 frac14 220 km=s vEarth frac14 240 km=sand vesc frac14 500 km=s

A Elastic DM-nucleus scattering

Assuming spin-independent interactions the event ratefrom dark matter scattering off of a nucleus in a detector ofmass mdet is given by the standard expression [78]

dRdEr

frac14 mdetρχσ02mχμ

2χN

F2ethqTHORNF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth21THORN

Here q frac14 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mTEr

pis the momentum transfer mT is the

target massmχ is the DM mass μχN is the reduced mass ofthe DM-nucleus system Er is the recoil energy FethErTHORNis the nuclear form factor of DM-nucleus scattering (weadopt the Helm form factor as in Ref [78]) and the form

FIG 4 Left The dependence of energy resolution on fractional surface area covered by sensors (fabs) is shown for each of the designsfrom Table III Dots indicate the resolutions quoted in the table For designs C and D the resolution is the same in the small fabs limit orwhere there are five or fewer sensors in this limit all sensors are assumed necessary to reconstruct events Meanwhile for larger fabs theimproved scaling of design D over design C results from the fixed number of sensors read out (here taken to be five) as the detectorbandwidth is increased Also shown are devices with the current best demonstrated noise power and resolution The TES and SNSPDbenchmarks come from Refs [1473] where the shaded band corresponds to the detectors listed in Ref [73] and the lines correspond tobest DM detector performance from the respective references We also include two superconducting photon detectors optimized for highdetection efficiency for THz photons where the best demonstrated NEP roughly 10minus20 W=


pin both cases is comparable to the NEP

assumed for designs C and D The quantum capacitance detector (QCD) is from Ref [72] and the SNS junction is from Ref [71] RightThe relative change in resolution as polytype and interface transmission are changed for a range of (surface-limited) phonon lifetimescompared to the nominal impedance-matched 3CAl design at the chosen phonon lifetime The resolutions of these devices are best-case scenarios for perfect phonon detection efficiency and thus represent a lower resolution limit for the given technology


075002-12

factor F2medethqTHORN captures the form factor for mediator

interactions (ie long-range or short-range) The crosssection σ0 is normalized to a target nucleus but to comparedifferent media this cross section is reparameterized as[2078]

σ0 frac14 A2

μχNμχn

2

σn eth22THORN

where A is the number of nucleons in the nucleus σn is theDM-nucleon cross section (assumed to be the same forprotons and neutrons) and μχn is the DM-nucleonreduced massFor a sub-GeV dark matter particle we have μχN rarr mχ

σ0 rarr A2σn and FethErTHORN rarr 1 such that

dRdEr

asympmdetρχA2σn2m3

χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN

which would seem to imply that a heavier nucleus is alwaysmore sensitive to dark matter from a pure event-rateperspective Hidden in the integral however is the fact that

vmin frac14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiErethmχ thornmTTHORN

2μχNmχ

srarr

ffiffiffiffiffiffiffiffiffiffiffiErmT

2m2χ

seth24THORN

in this limit which implies scattering off of heavier targetsis kinematically suppressedFor heterogeneous targets the standard modification to

this rate formula is to weight the event rate for a given atomby its fractional mass density For a SiC crystal of massmdetand equal number density of Si and C nuclei we have thetotal rate

dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C

where the rates for Si and C are computed for the givendetector mass This is a reasonable assumption for inter-actions in which the scattered DM particle only probes asingle nucleus For sufficiently low Er comparable to thetypical phonon energy the assumption is no longer validThis can be seen from the fact that the interaction of DMwith single or multiphonons is an expansion in q2=ethmTωTHORN[7980] so that we transition to the nuclear-recoil regimewhen Er ≫ ωphonon In this paper we consider elastic nuclearrecoils down to 05 eV well above the energy at the highestoptical phonon and consider DM as acting locally on asingle nucleus from the standpoint of the initial interactionFor energy depositions between the highest optical phononenergy sim120 meV and 05 eV we expect the signal rate tobe dominated by multiphonon interactionsTo compute nuclear-recoil limits the behavior at low

DM mass is strongly dependent on the energy threshold

while the high-mass behavior depends on the upper limitfor accurate energy reconstruction Athermal phonon cal-orimeters can provide very low thresholds but are intrinsi-cally limited in dynamic range To account for this weassume 3 orders of magnitude in dynamic range similar towhat has been seen in detectors with OetheVTHORN thresholds[30] This means that the upper integration limit is set to103σt where the threshold σt is assumed to be 5 times theresolution

B DM-phonon scattering

The formalism to compute single-phonon excitations fromDM scattering was detailed previously in Refs [161779]The scattering rate per unit time and per unit target masscan be written generally as

R frac14 1

ρT

ρχmχ

Zd3vfχethvTHORNΓethvTHORN eth25THORN

where ρT is the total target density ΓethvTHORN is the scatteringrate per dark matter particle with velocity v given by

ΓethvTHORNequiv σχ4πμ2χn

Zd3qΩ

F2medethqTHORNSmedethqωTHORN eth26THORN

μχn is the DM-nucleon reduced mass and σχ is a fiducialcross section which we will define later for specific modelsΩ is the primitive cell volume and can also be written asethPd mdTHORN=ρT where d sums over all atoms in the cell Asabove the form factor F2

medethqTHORN captures the form factor formediator interactions (ie long-range or short-range)Finally the structure factor SmedethqωTHORN encapsulates thephonon excitation rate for a given momentum transfer qand energy deposition ω note that it depends on themediator through its couplings to the nuclei and electronsin a given targetAs specific examples we first consider a mediator that

couples to nuclei proportional to atomic number A inwhich case

SmedethqωTHORN frac14XνkG

δethω minus ωνkTHORN2ωνk

jFNνethqkTHORNj2δkminusqG eth27THORN

where ν labels phonon branch and k denotes crystalmomentum within the first Brillouin zone The G arereciprocal lattice vectors and for sub-MeV dark matter theG frac14 0 piece of the sum dominates The phonon form factorfor this mediator is

jFNνethqkTHORNj2 frac14Xd

Adq middot eνdkffiffiffiffiffiffimd

p eminusWdethqTHORNeiethqminuskTHORNmiddotr0d

2 eth28THORN

where d labels atoms in the primitive cell and r0d are theequilibrium atom positions We determine the phonon


075002-13

eigenvectors eνdk and band structure ωνk numericallyfrom first-principles calculations described later in thissection Finally WdethqTHORN is the Debye-Waller factor whichwe can approximate as WdethqTHORN asymp 0 since the rates for sub-MeV DM are dominated by low q With this phonon formfactor sub-MeV dark matter dominantly couples to longi-tudinal acoustic phononsWe next consider a mediator that couples to electric

charge such as a dark photon mediator A0 The structurefactor has the same form as in Eq (27) but with FNν

replaced by the phonon form factor

jFA0νethqkTHORNj2 frac14Xd

q middot Zd middot e

νdk

ϵinfinffiffiffiffiffiffimd


2where we have assumed diagonal high-frequency dielectricconstant ϵinfin and where Z

d is the matrix-valued Borneffective charge of atom d in the unit cell It is the nonzeroBorn effective charges in polar semiconductors that permitsensitivity to these models and it has been found that themost important mode excitation is the highest-energylongitudinal optical phonon mode

1 Daily modulation

The anisotropic crystal structures of SiC polymorphsimply a directional dependence of DM-phonon scatteringAs Earth rotates there is a corresponding modulation in therate over a sidereal day which can provide a uniquediscriminant for a DM signal in the event of a detectionThis effect can be captured by accounting for the time-dependent direction of Earthrsquos velocity with respect to thelab frame in the DM velocity distribution fχethvTHORNThis approach to calculating the directional signal was

previously taken in Ref [17] where it was computed forAl2O3 (sapphire) which has a rhombohedral lattice struc-ture The rate depends on the orientation of the crystalrelative to the DM wind or equivalently Earthrsquos velocitySimilar to Ref [17] we choose the crystal orientationsuch that the z axis is aligned with Earthrsquos velocity att frac14 0 Since Earthrsquos rotation axis is at an angle of θe asymp 42degrelative to Earthrsquos velocity at time t frac14 1=2 day the z axisof the crystal will be approximately perpendicular to theDM wind For the rhombohedral and hexagonal latticestructures the convention is that the z axis corresponds tothe primary crystal axis and so we expect that thisconfiguration should give a near-maximal modula-tion rate

C DM-electron scattering

Equations (25) and (26) are applicable to electronscattering as well with the appropriate substitutionsThe structure factor SethqωTHORN for electron recoil is givenby [679]

SethqωTHORN frac14 2Xi1i2

ZBZ

d3kd3k0

eth2πTHORN6 2πδethEi2k0 minus Ei1k minus ωTHORN

timesXG

eth2πTHORN3δethk0 minus kthorn G minus qTHORNjffrac12i1ki2k0Gj2 eth29THORN

where Eik is the energy of an electron in band i with crystalmomentum k and G are the reciprocal lattice vectors Thecrystal form factor ffrac12i1ki2k0G is given by

ffrac12i1ki2k0G frac14XG0

ui1ethk0 thorn Gthorn G0THORNui2ethkthorn G0THORN eth30THORN

where uiethkTHORN are the electron wave functions written in planewave basis and normalized such thatX

G

juiethkthorn GTHORNj2 frac14 1 eth31THORN

In our calculation of the electron-recoil limits we makethe isotropic approximation following the formalism out-lined in Ref [6] The scattering rate per unit time and perunit target mass is then simplified to

R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2

medethqTHORNSethqωTHORNηethvminethqωTHORNTHORN

eth32THORN

where μχe is the reduced mass of the DM and electron andthe integrated dark matter distribution ηethvminTHORN is given as inRef [6] The reference cross section σe is at a fixedreference momenta which will be taken as αme with αthe fine structure constant and me the electron massResults for daily modulation and thus directional detectionof electron-recoil signals in SiC will be presented in futurework [81]

D Absorption of sub-keV DM

For a number of models the bosonic DM absorption ratecan be determined in terms of the conductivity of thematerial and photon absorption rate Then the absorptionrate is given as

R frac14 1

ρT

ρχmχ

g2effσ1ethmχTHORN eth33THORN

where σ1ethmχTHORN is the real part of the optical conductivityσ of the material namely the absorption of photons withfrequency ω frac14 mχ and geff is an effective coupling con-stant appropriate per model [1782] as will be detailedbelowThe conductivity of the material can be obtained from

measurements or by calculation For mχ greater than theelectron band gap we use measurements on amorphousSiC thin films from Ref [83] These data do not capture the


075002-14

differences between polymorphs of SiC with band gapsranging from 236 to 325 eV for those considered here butwe expect the differences to be small for mχ well above theelectron band gapFor mχ below the electron band gap absorption can

occur into single optical phonons as well as multiphononsIn this case there is limited data or calculation availablefor sub-Kelvin temperatures To gain further insight wecan use an analytic approximation for the dielectricfunction [17]

ϵethωTHORN frac14 ϵinfinYν

ω2LOν minus ω2 thorn iωγLOν

ω2TOν minus ω2 thorn iωγTOν

eth34THORN

with a product over all optical branches and where γ is thephonon linewidth and TO (LO) abbreviates transverse(longitudinal) optical phonons The dielectric function isrelated to the complex conductivity σethωTHORN by ϵethωTHORN frac141thorn iσ=ωWe separately consider the conductivity parallel to the c

axis ϵkethωTHORN and perpendicular to the c axis ϵperpethωTHORN InSiC there is a strong optical phonon branch for each ofthese directions corresponding to the highest-energy opti-cal phonons (A1 in the parallel direction E1 in theperpendicular direction) [32] For these phonons the LOand TO frequencies are compiled in Ref [32] where thevalues are nearly identical across all polymorphs Becausethere are very limited low-temperature measurements of thephonon linewidths we use γLO frac14 26=cm and γTO frac1412=cm in all cases These values come from our calcu-lations of the linewidth of the optical phonons in the 3Cpolymorph and are also in agreement with experimentaldata [84] The calculation of linewidths is discussed inAppendix AHere we only consider the absorption into the strongest

phonon branch for the parallel and perpendicular direc-tions Accounting for the fact that the DM polarization israndom we will take an averaged absorption over thesephonon modes hRi frac14 2

3Rperp thorn 1

3Rk With the above approx-

imations we find that the absorption rate for the strongestmode is nearly identical across all polymorphs Howeverdepending on the polymorph there are additional lower-energy optical phonons with weaker absorption and whichcan have large mixing of transverse and longitudinalpolarizations Furthermore there is absorption into multi-phonons While these contributions are not included in ouranalytical computation we expect the qualitative behaviorof the low-mass absorption rate to be well captured by therange formed by the available measurements from Ref [83]and the above calculation

V RESULTS

In this section we present the potential reach of theSiC polytypes discussed in this paper under various

detector performance scenarios For the electron-recoillimits we assume single-charge resolution justified bythe detector designs in Table II For DM which interactswith the crystal via phonon production the thresholdsloosely follow the sensor design road map outlined inTable III though the thresholds have been modified tohighlight critical values (the 80 meV threshold forexample probes the production of single opticalphonons)

A DM with scalar nucleon interactions

For dark matter with spin-independent scalar interactionsto nucleons we consider both the massive and masslessmediator limit corresponding to different choices ofmediator form factor F2

medethqTHORN A discussion of the astro-physical and terrestrial constraints on both cases can befound in Ref [85]For the massive scalar mediator coupling to nucleons the

form factor is F2medethqTHORN frac14 1 The sensitivity of SiC to this

model is shown in the left panel of Fig 5 for the various SiCpolytypes and also a few different experimental thresholdsFor energy threshold ω gt 05 eV we show the reach fornuclear recoils in a SiC target and compare with arepresentative target containing heavy nucleiThe DM-phonon rate is determined using Eq (26)

where the fiducial cross section is σχ equiv σn and σn is theDM-nucleon scattering cross section With a thresholdω gt meV it is possible to access DM excitations intosingle acoustic phonons which provide by far the bestsensitivity While this threshold would be challenging toachieve we show it as a representative optimistic scenariowhere access to single acoustic phonons is possible Thereach here is primarily determined by the speed of sound[80] and is thus fairly similar for all crystal structuresFor comparison with additional polar crystal targets seeRef [86]When the threshold is ω≳ 20ndash30 meV the only exci-

tations available are optical phonons For DM whichcouples to mass number there is a destructive interferencein the rate to excite optical phonons resulting in signifi-cantly worse reach [1687] In Fig 5 we also show arepresentative optical phonon threshold of ω gt 80 meV asthis is just below the cluster of optical phonons of energy90ndash110 meV present in all polymorphs (see Fig 3) Notethat the reach for ω gt 30 meV is not significantly differentfrom ω gt 80 meV due to the destructive interferencementioned aboveWhile the optical phonon rate is much smaller than the

acoustic phonon rate the same destructive interferenceallows for a sizable directionality in the DM scattering rateand thus daily modulation The right panel of Fig 5 givesthe daily modulation for DM scattering into opticalphonons with threshold ω gt 80 meV We find that thelowest modulation is for the 3C polytype as expected givenits higher degree of symmetry and the largest modulation


075002-15

can be found in the 2H polytype While the other polytypesof SiC can give comparable modulation to 2H they containmany more phonon branches which can wash out thesignal We also note that the modulation could be evenlarger with a lower threshold on the optical phononswhich was the case for sapphire in Ref [17] Howeverif the threshold is reduced all the way to ω gt meV suchthat acoustic phonons are accessible the modulation ismuch smallerIn the massless mediator limit we assume dark

matter couples to nucleons through a scalar with massmϕ ≪ mχv sim 10minus3mχ For sub-MeV DM constraints onthis model are much less severe than in the heavy mediatorcase [85] Then we can approximate the DM-mediator formfactor as

F2medethqTHORN frac14

q0q

4

eth35THORN

where q0 frac14 mχv0 is a reference momentum transfer In thiscase σn is a reference cross section for DM-nucleonscattering with momentum transfer q0 The projectedsensitivity to the massless mediator model from single-phonon excitations in SiC is shown in the left panel ofFig 6 Here we also show the reach for a GaAs targetwhich has a lower sound speed and thus more limited reach

at low DMmass [17] For comparison with additional polarcrystal targets see Ref [86]The daily modulation amplitude for a massless scalar

mediator is shown in the right panel of Fig 6 Similar to themassive mediator case we only have a sizable modulationfor scattering into optical phonon modes and find that 2H(3C) tends to give the largest (smallest) amplitudeWe conclude with a brief discussion of how SiC

compares with other commonly considered target materialsfor DM with scalar nucleon interactions Because SiC has ahigh sound speed similar to that of diamond the sensitivityto acoustic phonon excitations extends to lower DM massthan in Si Ge or GaAs Furthermore depending on thepolytype of SiC the daily modulation in SiC is expected tobe much larger than Si Ge GaAs and diamond The lattermaterials have cubic crystal structures where the atoms in aunit cell have identical or very similar mass so we expectthe modulation to be similar to that of GaAs found to besubpercent level in Ref [17] In terms of both reach anddirectionality SiC is perhaps most similar to sapphireand has advantages over many other well-studied targetmaterials

B DM-electron interactions

We now present our results for DM that scatters withelectrons through exchange of a scalar or vector mediator

FIG 5 Reach and daily modulation for DM interactions mediated by a scalar coupling to nucleons assuming a massive mediatorLeft All reach curves are obtained assuming kg-year exposure and zero background and we show the 90 CL expected limits(corresponding to 24 signal events) For single-phonon excitations relevant formχ ≲ 10 MeV we show two representative thresholds of1 (solid lines) and 80 meV (dotted) for the different SiC polytypes We also show the reach for a superfluid He target [19] The dashedlines show sensitivity to nuclear recoils assuming threshold of 05 eV In the shaded region it is expected that the dominant DMscattering is via multiphonons (see discussion in Refs [7980]) Right The daily modulation of the DM-phonon scattering rate as afunction of DMmass where the quantity shown corresponds exactly to the modulation amplitude for a purely harmonic oscillation Themodulation is much smaller for scattering into acoustic phonons ω gt 1 meV so we only show scattering into optical phonons withω gt 80 meV The modulation amplitude is generally largest for 2H and smallest for 3C The inset compares the phase of the modulationamong the polymorphs for mχ frac14 80 keV


075002-16

(that is not kinetically mixed with the photon) Our resultsfor the reference cross section σe of Eq (32) are given inFig 7 for the heavy (left) and light (right) mediator caseswith form factors


1 heavy mediator

ethαmeTHORN4=q4 light mediatoreth36THORN

For comparison we also show the reach of Si and diamondThick blue curves indicate relic density targets fromRef [1] The gray shaded region show existing limits from

SENSEI [88] SuperCDMS HVeV [89] DAMIC [49]Xenon10 [90] Darkside [91] and Xenon1T [92]The results of Fig 7 show that the reach of SiC to DM-

electron scattering is similar to that of diamond at highmass for the case of a light mediators and comparable to thesilicon two-electron reach for the heavy mediator case Therelation of the reach between SiC polytypes is similar tothat found in Figs 5 and 6 in that the majority of thedifference at low mass can be attributed to the differentband gaps We do observe however that the reach of 3C athigh mass is roughly half an order of magnitude less than

FIG 6 Similar to Fig 5 but for DM interactions mediated by a massless scalar coupling to nucleons In this case we also comparewith the reach of another polar material GaAs for acoustic and optical branch thresholds [17]

FIG 7 The 90 CL expected limits into DM-electron scattering parameter space for 1 kg-year of exposure of select polytypes ofSiC for heavy (left) and light (right) mediators For comparison we also show the reach of Si and diamond assuming a threshold of oneelectron or energy sensitivity down to the direct band gap in the given material The reach of Si given a two-electron threshold is shownfor comparison for the case that charge leakage substantially limits the reach at one electron Relic density targets from Ref [1] areshown as thick blue lines for the freeze-in and freeze-out scenarios respectively The gray shaded region includes current limits fromSENSEI [88] SuperCDMS HVeV [89] DAMIC [49] Xenon10 [90] Darkside [91] and Xenon1T [93]


075002-17

the hexagonal polytypes despite having the smallest bandgap This can be understood by noticing that the density ofstate near the conduction band minima is smaller than thatin the other polytypes thus limiting the available phasespace The reach of 15R is also significantly worse than theother polytypes because the size of its small Brillouin zoneis poorly matched to the typical momentum transfer(few keV)We learn that SiC can probe DM-electron scattering

processes in a complementary manner to silicon anddiamond In particular the expected rate scaling withpolytype provides a natural cross-check on potential DMsignals first seen in other crystal substrates The directionalsignal which could be used to verify a dark matter signalwould likely also be larger in some SiC polytypes Asmentioned earlier prospects for directional detection ofelectron-recoil signals in the various polytypes of SiC willbe described in future work [81]

C DM with dark photon interactions

We now consider a DM candidate with mass mχ thatcouples to a dark photon A0 of mass mA0 where the darkphoton has a kinetic mixing κ with the Standard Modelphoton

L sup minusκ

2FμνF0μν eth37THORN

We again take two representative limits of this modelscattering via a massive or nearly massless dark photonFor a massive dark photon the electron-scattering cross

section σe in terms of model parameters is

σe frac1416πκ2αχαμ

2χe

frac12ethαmeTHORN2 thorn ethmA0 THORN22 eth38THORN

and the DM-mediator form factor is F2medethqTHORN frac14 1 For the

parameter space below mχ asympMeV there are strong astro-physical and cosmological constraints [685] and the reachfrom exciting optical phonons is limited so we do notconsider DM-phonon scattering The electron-scatteringreach is the same as the heavy mediator limit of theprevious section shown in the left panel of Fig 7For a nearly massless dark photon we consider both

electron recoils and optical phonon excitations Opticalphonons are excited through the mediator coupling to theion (nucleus and core electrons) which is given in terms ofthe Born effective charges discussed in Sec IV B Forcomparison with the literature we will show both theelectron recoil and optical phonon reach in terms of theelectron-scattering cross section This electron-scatteringcross section is defined at a reference momentum transfergiven in terms of the dark fine structure constantαχ frac14 g2χ=eth4πTHORN


2χe

ethαmeTHORN4 eth39THORN

where α is the fine structure constant and μχe is DM-electron reduced mass As a result for phonon scatteringthe relevant cross section σχ in Eq (26) is

σχ equiv μ2χnμ2χe

σe eth40THORN

The DM-mediator form factor for both electron and phononscattering is


αme

q

4

eth41THORN

The reach for different polytypes of SiC to the lightmediator limit of this model is shown in the left panel ofFig 8 The reach for mχ gt MeV is from DM-electronscattering and is the same as the light mediator limit of theprevious section (shown in the right panel of Fig 7)Although there is an additional in-medium screening fordark photon mediators compared to Sec V B we expectthis to be a small effect for a relatively high-gap materialsuch as SiC The sensitivity of SiC for mχ lt MeV is fromexciting optical phonons and is very similar across allpolytypes This is because the DM dominantly excites thehighest-energy optical phonon [17] which has the largestdipole moment and has similar energy in all casesFurthermore the coupling of the DM to this phonon ischaracterized by an effective Froumlhlich coupling thatdepends only on the phonon energy ϵinfin and ϵ0 [16]Again it can be seen in Table I that all of these quantitiesare quite similar across the different polytypes For com-pleteness we also show existing constraints from stellaremission [9495] and Xenon10 [90] projections for othermaterials such as Dirac materials [15] superconductors[12] and polar materials [1617] and target relic DMcandidate curves [396]There are larger differences between polytypes in direc-

tional detection which depends on the details of the crystalstructure The results for DM-phonon scattering are pro-vided in the right panel of Fig 8 Similar to the case of DMwith scalar nucleon interactions we find that 3C has thesmallest modulation due to its higher symmetry while 2Hhas the largest modulationComparing with other proposed polar material targets

such as GaAs and sapphire the reach of SiC for darkphoton mediated scattering does not extend as low in DMmass because of the higher LO phonon energy Howeverthe directional signal is similar in size to that of sapphireand substantially larger than in GaAs For additionalproposed experiments or materials that can probe thisparameter space see for example Refs [8697ndash99]


075002-18

D Absorption of dark photon dark matter

Taking a dark photon with massmA0 and kinetic mixing κto be the dark matter candidate the effective coupling g2effin the absorption rate in Eq (33) must account for the in-medium kinetic mixing Thus we have g2eff frac14 κ2eff with in-medium mixing of

κ2eff frac14κ2m4

A0

frac12m2A0 minus ReΠethωTHORN2 thorn ImΠethωTHORN2 frac14

κ2

jϵethωTHORNj2 eth42THORN

where ΠethωTHORN frac14 ω2eth1 minus ϵethωTHORNTHORN is the in-medium polarizationtensor in the relevant limit of jqj ≪ ωThe projected reach for absorption of SiC into the

parameter space of kinetically mixed dark photons is shownin the left panel of Fig 9 As discussed in Sec IV D weconsider absorption into electron excitations using mea-surements of the optical conductivity of SiC from Ref [83](solid curve) as well as absorption into the strongest opticalphonon mode for low masses (dashed curve) These blackcurves indicate the 95 CL expected reach in SiC for akg-year exposure corresponding to three events Forcomparison we also show in dotted curves the projectedreach of superconducting aluminum targets [13] and WSinanowires [14] semiconductors such as silicon germa-nium [82] and diamond [7] Dirac materials [15] polar

crystals [17] and molecules [100] Stellar emission con-straints [101102] are shown in shaded orange while theterrestrial bounds from DAMIC [103] SuperCDMS [104]Xenon data [102] and a WSi superconducting nanowire[14] are shown in shaded gray As is evident SiC is arealistic target material that has prospects to probe deep intouncharted dark photon parameter space over a broad rangeof masses from Oeth10 meVTHORN to tens of eV

E Absorption of axionlike particles

Next we consider an axionlike particle (ALP) a withmass ma that couples to electrons via

L supgaee2me

ethpartμaTHORNeγμγ5e eth43THORN

The absorption rate on electrons can be related to theabsorption of photons via the axioelectric effect and theeffective coupling in Eq (44) is then given by

g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN

Because the ALP directly couples to electrons weconsider only the absorption above the electron bandgap (Relating the couplings of the subgap phonon

FIG 8 Reach and daily modulation for DM-phonon interactions mediated by a massless dark photon Left The 90 CL expectedlimits assuming kg-year exposure and zero background (corresponding to 24 signal events) The reach from single optical phononexcitations in SiC (solid lines) is similar for all the polytypes while the dotted lines in the same colors are the electron-recoil reach fromFig 7 The thick solid blue line is the predicted cross sections if all of the DM is produced by freeze-in interactions [396] and theshaded regions are constraints from stellar emission [9495] and Xenon10 [90] We also show the reach from phonon excitations in otherpolar materials GaAs and Al2O3 [1617] and from electron excitations in an aluminum superconductor [12] and in Dirac materialsshown here for the examples of ZrTe5 and a material with a gap ofΔ frac14 25 meV [15] (For clarity across all materials all electron-recoilcurves are dotted and all phonon excitation curves are solid) Right The daily modulation of the DM-phonon scattering rate as a functionof DM mass where the quantity shown corresponds exactly to the modulation amplitude for a purely harmonic oscillation Themodulation is negligible in the 3C polytype due to its high symmetry and is largest in 2H The inset compares the phase of themodulation among the polytypes for mχ frac14 80 keV


075002-19

excitations is less straightforward due to the spin depend-ence of the ALP coupling) The projected reach for a kg-yearexposure is shown in the right panel of Fig 9 by the solidblack curve For comparison we show the reach of super-conducting aluminum [13] targets as well as silicon [82]germanium [82] and diamond [7] by the dotted curvesConstraints from white dwarfs [107] Xenon100 [105] LUX[76] and PandaX-II [106] are also shown Constraints fromthe model-dependent loop-induced couplings to photons areindicated by shaded blue [108109] The QCD axion regionof interest is shown in shaded gray We learn that SiCdetectors can reach unexplored ALP parameter space com-plementary to stellar emission constraints

VI DISCUSSION

In this paper we proposed the use of SiC for directdetection of light DM With advantages over silicon anddiamondmdashincluding its polar nature and its many stablepolymorphsmdashwe have shown that SiC would serve as anexcellent detector across many different DM channels andmany mass scales DM-nuclear scattering (direct and viasingle- or multiple-phonon excitations) down toOeth10 keVTHORNmasses DM-electron scattering down to Oeth10 MeVTHORNmasses dark photon absorption down to Oeth10 meVTHORNmasses and axionlike absorption down to Oeth10 meVTHORNmasses with prospects for directional detection as well

In particular the high optical phonon energy in SiC(higher than that of sapphire) coupled with the high soundspeed of all polytypes and long intrinsic phonon lifetimemakes SiC an ideal substrate for calorimetric phononreadout There is substantial reach for dark photon coupledDM at higher energy thresholds than competing materialsand the presence of a strong bulk plasmon in SiC makes it apromising follow-up material for potential inelastic inter-actions of DM at the energy scales in the multiphononregime as described in Refs [53110]In fact since SiC exists in many stable polytypes it

allows us to compare the influence of crystal structure andhence bonding connectivity on their suitability as targetsfor various dark matter channels Broadly we see similarsensitivities and reach across the calculated polytypes asexpected for a set of materials comprised of the samestoichiometric combination of elements For DM-nucleonand DM-phonon interactions we find very similar reachgiven the similar phonon spectra of the SiC polytypes Onedifference is that polytypes with smaller unit cells will havethe advantage of higher intrinsic phonon lifetimes as thehigher unit cell complexity will increase scattering Morevariation in reach among the polytypes however is foundfor DM-electron scattering due to the variation in electronicband gaps across the SiC family This trend in band gapvariation in SiC polytypes is well discussed in the literatureand is a result of the third-nearest-neighbor effects [26]

FIG 9 Absorption of kinetically mixed dark photons (left) and axionlike particles (right) Left Projected reach at 95 CL forabsorption of kinetically mixed dark photons The expected reach for a kg-year exposure of SiC is shown by the solid and dashed blackcurves (using the data of Ref [83] and strongest phonon branch respectively) Projected reach for germanium and silicon [82] diamond[7] Diracmaterials [15] polar crystals [17]molecules [100] superconducting aluminum [13] andWSi nanowire [14] targets are indicatedby the dotted curves Constraints from stellar emission [101102] DAMIC [103] SuperCDMS [104] Xenon [102] data and a WSinanowire [14] are shown by the shaded orange green purple light blue and blue regions respectively Right Projected reach at 95CLfor absorption of axionlike particles The reach of a kg-year exposure of SiC is shown by the solid black curve where only excitationsabove the band gap are assumed The reach for semiconductors such as germanium and silicon [82] diamond [7] and superconductingaluminum [13] targets is depicted by the dotted curves Stellar constraints from Xenon100 [105] LUX [76] and PandaX-II [106] data andwhite dwarfs [107] are shown by the shaded red and orange regions Constraints arising from (model-dependent) loop-induced couplingsto photons are indicated by the shaded blue regions [108109] while the QCD axion region is given in shaded gray


075002-20

We indeed see that with increasing unit cell size thedecrease in band gap in the H polytypes correspondinglyleads to better reach as expectedMaterials-by-design routes explored for dark matter detec-

tion have focused on band gap tuning [111] and materialsmetrics for improved electron and phonon interactions[86112ndash114] A key advantage of SiC over other targetproposals is its prospect for directionality by designmdashgiventhe similar performance in reach across the polytypes we canselect a material that is optimized for directional detectionOur results indicate that as expected the highly symmetriccubic phase 3C exhibits no daily modulation whereas themaximal modulation is achieved for the 2H phase The 2Hphase has inequivalent in-plane and out-plane crystallo-graphic axes and so naturally has an anisotropic directionalresponse We further find that this effect is diminished forincreasing the number of out-of-plane hexagonal units(decreasing the regularity of the unit cell) as the directionalresponse becomes integrated out over repeated unit cellsAs discussed earlier one of the primary benefits of using

SiC over other carbon-based crystals is the availability oflarge samples of the 4H and 6H polytypes the 3C polytypeis not currently at the same level of fabrication scale andthe 2H 8H and 15R polytypes are scarce in the literatureand not made in significant quantities The charge mobilitymeasurements for existing SiC samples indicate that purityof these crystals is not at the same level as comparablediamond and silicon and there are few measurements ofintrinsic phonon properties at cryogenic temperatures Inorder to further develop SiC devices studies of chargetransport and phonon lifetime in a range of samples need tobe undertaken so that current limitations can be understoodand vendors can work to improve crystal purity Devicefabrication on the other hand is expected to be fairlystraightforward due to past experience with basic metal-lization and the similarity of SiC to both diamond and SiThe availability of large boules of SiC unlike for diamondmeans that scaling to large masses for large detectors ismuch more commercially viable and cost effectiveThe material response of the SiC polytypes should also

be better characterized In particular studies of the non-ionizing energy loss of nuclear recoils needs to be modeledand characterized photoabsorption cross sections at cryo-genic temperatures are needed both above and below gapand the quantum yield of ionizing energy deposits needs tobe better understood SiC has already been shown to bemuch more radiation hard than Si but more studies ofradiation-induced defects will benefit both the use of SiC asa detector as well as a better understanding of vacanciesused in quantum information storage More practicalstudies of breakdown voltage and electron-hole saturationvelocity will also inform detector modeling and readoutFinally a detailed study of potential backgrounds in

carbon-based detectors is needed and we leave this tofuture work 14C is the primary concern for SiC as well as

other carbon-based materials proposed for dark mattersearches such as diamond and graphene A dedicatedstudy of the abundance of 14C in these materials is beyondthe scope of this work [7] In addition low-energy back-grounds in this regime are poorly understood (see egRef [53] for a recent review of low-threshold DM searchexcesses) and thus speculating further on sources ofbackgrounds at meV energy scales is premature Initialdetector prototypes will also serve as a probe of thesebackgrounds and give us a better sense of the long-termpromise of SiC detectors for rare event searches

ACKNOWLEDGMENTS

We would like to thank Simon Knapen for early collabo-ration and Rouven Essig and Tien-Tien Yu for clarificationsregarding QEDark Wewould also like to thank Lauren Hsufor feedback on an early paper draft The work of Y H issupported by the Israel Science Foundation (GrantNo 111217) by the Binational Science Foundation(Grant No 2016155) by the I-CORE Program of thePlanning Budgeting Committee (Grant No 193712) bythe German Israel Foundation (Grant No I-2487-30372017) and by the Azrieli Foundation T L is supported byan Alfred P Sloan foundation fellowship and theDepartment of Energy (DOE) under Grant No DE-SC0019195 Parts of this document were prepared by NK using the resources of the Fermi National AcceleratorLaboratory (Fermilab) a US Department of Energy Officeof Science HEP User Facility Fermilab is managed byFermi Research Alliance LLC (FRA) acting underContract No DE-AC02-07CH11359 T C Y is supportedby the US Department of Energy under Contract No DE-AC02-76SF00515 S M G and K I were supported by theLaboratoryDirectedResearch andDevelopment ProgramofLBNLunder theDOEContractNoDE-AC02-05CH11231Computational resources were provided by the NationalEnergy Research Scientific Computing Center and theMolecular Foundry DOE Office of Science UserFacilities supported by the Office of Science of the USDepartment of Energy under Contract No DE-AC02-05CH11231 The work performed at the MolecularFoundry was supported by the Office of Science Officeof Basic Energy Sciences of the US Department of Energyunder the same contract S M G acknowledges supportfrom the US Department of Energy Office of SciencesQuantum Information Science Enabled Discovery(QuantISED) for High Energy Physics (KA2401032)

APPENDIX A FIRST-PRINCIPLESCALCULATION DETAILS OF ELECTRONIC

AND PHONONIC PROPERTIES

Full geometry optimizations were performed using DFTwith the Vienna Ab initio Simulation Package (VASP) [115ndash118] using projector augmented wave pseudopotentials


075002-21

[119120] and the Perdew-Becke-Ernzerhof exchange-cor-relation functional revised for solids (PBEsol) [121] Thisgave lattice constants within 05 of experimental valuesThe high-frequency dielectric constants and Born effectivecharges were calculated using the density functionalperturbation routines implemented in VASP Force constantsfor generating the phonon dispersion spectra were calcu-lated with the finite displacement method using VASP andPHONOPY [122]The pseudopotentials used in the DFT calculations

included s and p electrons as valence A plane wave cutoffenergy of 800 eV was used with a Γ-centered k-point gridwith k-point spacing lt 028 Aringminus1 This is equal to a 9 times9 times 9 grid for the 3C unit cell and the equivalent k-pointspacing was used for the other polytypes and supercellsThe cutoff energy and k-point spacing were chosen toensure convergence of the total energy to within 1 meV performula unit and dielectric functions Born effectivecharges and elastic moduli to within 1 The self-consistent field energy and force convergence criteria were1 times 10minus8 eV and 1 times 10minus5 eVAringminus1 respectively For cal-culation of force constants within the finite displacementmethod the following supercell sizes were used 5 times 5 times 5for 3C (250 atoms) 4 times 4 times 4 for 2H (256 atoms) 3 times 3 times3 for 4H (216 atoms) 3 times 3 times 3 for 6H (324 atoms) 3 times3 times 3 for 8H (432 atoms) and 3 times 3 times 3 for 15R(270 atoms)For phonon lifetimes and linewidths the PHONO3PY code

[123] was used for calculation of phonon-phonon inter-actions within the supercell approach Third-order forceconstants were obtained from 4 times 4 times 4 supercells for 3Cand 3 times 3 times 3 supercells for 2H Lifetimes were computedon grids up to 90 times 90 times 90 The acoustic lifetimes wereaveraged close to Γ in the 0ndash2 THz frequency range At 2 Kthe acoustic lifetimes are converged to an order of magni-tude with respect to the sampling grid The optical lifetimeswere averaged over all frequencies and these converged tothree significant figures The corresponding optical line-widths were sampled close to Γ for use in Eq (34)The electronic structures and wave function coefficients

were also calculated by DFT however the Heyd-Scuseria-Ernzerhof (HSE06) screened hybrid functional [124124]was used on PBEsol lattice parameters which gave excel-lent agreement with experimental band gaps (Table I)Band structures and isosurfaces were calculated with VASP

using an increased k-point density to ensure convergenceof the band gap (12 times 12 times 12 grid for the 3C unit celland equivalent for the other polytypes) For calculationof the electron wave function coefficients the Quantum

ESPRESSOcode [125126] was used to enable the use ofnorm-conserving Vanderbilt-type pseudopotentials [127]All other calculation choices between VASP and Quantum

ESPRESSOwere kept consistent The calculated detectionrate was checked with respect to the k-point densityand plane wave energy cutoff of the underlying DFT

calculations and was found to be converged within 10The following k grids were used 8 times 8 times 4 for 2H 8 times8 times 8 for 3C 8 times 8 times 2 for 4H 8 times 8 times 2 for 6H 4 times 4 times 1for 8H and 4 times 4 times 4 for 15R Bands up to 50 eV aboveand below the Fermi energy were used to evaluate theelectron-DM matrix elements

APPENDIX B THERMAL CONDUCTANCEAND PHONON LIFETIME

Here we show how thermal conductance measurementsinform our estimations of phonon lifetime which in turnare used in our discussion of phonon collection efficiencyin Sec IV B This model was used previously in Ref [7] toestimate phonon lifetimes in diamond but was not givenexplicitly in that paperReference [128] defines the phonon relaxation time in

the limit that T rarr 0 as4

τminus1r frac14 Aω4 thorn csLthorn B1T3ω2 ethB1THORN

frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN

frac14 τminus1b frac121thorn ethω=ωpTHORN4 thorn ethω=ωBTHORN2 ethB3THORN

where L is the characteristic length scale of the crystal5 ωis the angular phonon frequency A describes the strength ofisotopic scattering and B1 is the strength of the three-phonon scattering interactions All of these interactionsrepresent an inelastic scattering event which can contributeto phonon thermalization and therefore signal loss for aphonon calorimeterWe have defined a benchmark time constant for boun-

dary scattering (τb frac14 Lcs) and the critical frequency for

phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN

below which boundary scattering dominates phonon relax-ation time and above which isotopic scattering is the moreimportant process We also defined the three-phononscattering frequency

ωB frac14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

τbB1T3

sfrac14

ffiffiffiffiffiffiffiffiffiffiffiffiffiffics

LB1T3

r ethB5THORN

4The assumption of zero temperature allows us to ignoreumklapp processes significantly simplifying this analysis

5For rough boundaries L is simply the geometric size of thecrystal For highly reflective conservative boundaries it may bemany times larger than the crystalrsquos size Using the crystal sizeshould therefore be a lower bound on thermal conductivity for asufficiently pure sample


075002-22

which is explicitly temperature dependent and is assumedto be larger than ωp hereIn a perfect crystal of finite size at low temperature

A rarr 0 and thermal conductivity is determined entirely bycrystal geometry and temperature In particular thermalconductivity is determined by losses at the surface (thismodel implicitly assumes reflections are diffusive andnonconservative not specular) In most real crystals A willbe nonzero and is thus included in the calculation as aperturbation term In this limit the thermal conductivity ofa sample obeys the equation [128]

κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN

where ωT is the mean phonon frequency at temperature Tdefined as

ωT frac14 kBTℏ

ethB7THORN

This is the mean phonon frequency at the given temper-ature but to good approximation we can use this frequencyto bound the phonon energies that are expected to belimited by boundary scattering or bulk processes based ondeviations from the leading-order conductivity in the aboveequationThermal conductivity measurements at low temperature

allow us to determine the purity and phonon interactionlength scales for high-quality crystals given that the volu-metric specific heat is an intrinsic quantity while the thermalconductivity depends on the extrinsic length scale of thecrystal Fitting the thermal conductivity to a model with threefree parameters L ωp and ωB we can infer the parametersL A and B1 assuming the sound speed and volumetric heatcapacity are known for a given materialHigh-purity samples of Ge Si SiC and diamond all

demonstrate T3 dependence and scaling with crystal sizefor temperatures below 10 K [69] and millimeter-scalecrystals This allows us to assert that phonons with energybelow 10 K have a lifetime limited by boundary scatteringin 1 mm size crystals for all four substrates The scaling lawhere also implies that high-purity SiC crystals should havesimilar κ0 to diamond which is indeed shown to be the casein Ref [69]To understand this scaling we can rewrite Eq (B6) in

terms of the specific heat of the crystal We use thevolumetric Debye specific heat

CV frac14 12π4nkB5

TTD

3

ethB8THORN

where kBTD equiv πℏcseth6ρ=πTHORN1=3 is the Debye temperature(values for ℏωDebye frac14 kBTD are given in Table I) Thisgives us the modified equation

κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN

with κ0 frac14 CVcsL=3 We thus find that the temperaturedependence of the leading-order term is only due to theincrease in the thermal phonon population and thusdividing a measurement of thermal conductivity by heatcapacity gives a temperature-dependent measure of effec-tive mean phonon lifetime ( κ

CVfrac14 1

3c2sτb)

The ultimate limit to ballistic phonon propagation for acrystal of given purity can be taken in the limit T rarr 0 inwhich phonons below ωp can be considered ballistic andthose above ωp are unlikely to propagate to sensors at thecrystal surface Because ωp depends on crystal purity andcrystal size we can use this property to inform our designrules Suppose we have a superconducting phonon absorberwith minimum gap frequency

ωg sim7kbTc

2ℏ ethB11THORN

In order to achieve a high collection efficiency we need alarge crystal sufficiently pure enough to ensure that

ωg ≪ ωp ethB12THORN

7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN

This matches the general intuition that higher sound speedimplies that the mean phonons have a higher energy andhigher-gap absorbers are acceptable This condition allowsfor a quantitative crystal size optimization given knowncrystal purity and allows us to compare crystals using low-temperature thermal conductance data (which can be usedto extract A)

APPENDIX C PHONON TRANSMISSIONPROBABILITIES

The phonon transmission probabilities across materialinterfaces are estimated using the acoustic mismatch modelin [129] The calculation is completely analogous to that ofelectromagnetic wave propagation across a boundaryexcept for phonons we also have an additional longitudinalmodeAn exemplary situation is illustrated by Fig 10 An

longitudinal wave is propagating in the x-z plane with theinterface between medium 1 and medium 2 situated along


075002-23

the x axis The incoming wave can be reflected andrefracted into both longitudinal and coplanar transversemode (but not the transverse mode parallel to the y axis)The various angles are related via laws of geometric opticsFor example we have

sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN

where c denotes the speed of sound with subscriptsdenoting the polarization and the medium We assumeisotropy and do not include any angular dependence in thespeed of sounds

In order to calculate the transmission coefficient weassume all our waves are plane waves with variousamplitudes For example the incident wave can bewritten as

vinc frac14 B expethikinc middot rTHORN frac14 B expethminusiβzthorn iσxTHORN ethC2THORN

where vinc denotes the particle velocity due to the incidentacoustic wave B is the amplitude of the incident waveβ frac14 ω=cl1 cos θ1 and σ frac14 ω=cl1 sin θ1We can relate the various amplitudes using four boun-

dary conditions(1) The sum of normal (tangential) components of

the particle velocity at the boundary should becontinuous

v1perp frac14 v2perp v1k frac14 v2k ethC3THORN

(2) The sum of normal (tangential) components ofthe mechanical stress at the boundary should becontinuous

ρ1c21partv1partz frac14 ρ2c22

partv2partz etc ethC4THORN

Writing these boundary conditions and combining withEqs (C1) and (C2) would produce a system of linearequations for the various amplitudes which can be solvedto obtain the transmission coefficient η as a function ofthe incident angle The phonon transmission probability n

FIG 10 An incident longitudinal (L) acoustic wave is bothreflected and refracted into longitudinal and transverse (SV)modes in the two media The various angles here satisfy geo-metric-optical relations such as law of reflection and Snellrsquos lawReproduced from [129]

TABLE IV Phonon transmission probabilities for materials relevant to the detector designs discussed in this paper The probability nl(nt) is the probability for a longitudinal (transverse) phonon incident on the interface from the substrate to get into the film n is theprobability averaged by the density of states of the two modes

Substrate Si Diamoncd 3C-SiC 4H6H-SiC

Material nl nt n nl nt n nl nt n nl nt n

Al 098 089 091 072 063 065 085 095 094 086 082 083Al2O3 076 061 064 096 088 090 090 032 035 095 097 096Diamond 030 011 015 100 100 100 042 006 008 061 029 034Ga 097 089 090 067 058 061 084 090 089 084 079 080Ge 090 083 084 090 082 084 087 090 089 094 091 091In 097 089 090 068 059 061 088 090 090 086 079 080Ir 047 036 038 084 076 078 063 037 038 070 055 058Nb 083 074 075 093 085 087 092 086 086 097 088 089Si 100 100 100 079 071 073 086 048 051 089 085 0853C-SiC 077 096 093 084 075 078 100 100 100 097 080 0834H6H-SiC 058 048 050 095 086 089 071 024 027 100 100 100Sn 094 085 086 084 076 078 090 087 087 094 089 089Ta 065 054 056 095 087 089 080 058 059 088 076 078Ti 095 090 091 086 077 079 091 095 095 095 089 090W 051 041 043 088 080 082 067 042 044 075 061 064Zn 091 081 082 090 081 083 089 087 087 095 090 090


075002-24

across the boundary is then defined as the angular averageof the transmission coefficient

n frac14Z

π=2

0

ηethθ1THORN sineth2θTHORNdθ

frac14Z

θc

0


where θc is the critical angle For detailed derivation werefer reader to the Appendix of Ref [129]Table IV contains a collection of transmission proba-

bilities calculated in this manner

APPENDIX D BRILLOUIN ZONES

Figure 11 shows the Brillouin zones for each of thelattice symmetries considered in this paper Of particularimportance are the X valleys in the face-centered cubic typeand the L-M symmetry line in the hexagonal type as shownin Fig 2 These points have a basic rotational symmetryabout one of the cardinal axes The rhombohedral type ismore complex integrating a twisted crystal structure whichresults in asymmetric valleys The F symmetry point is thelocation of the indirect gap

[1] M Battaglieri et al US cosmic visions New ideas in darkmatter 2017 Community report arXiv170704591

[2] R Kolb H Weerts N Toro R Van de Water R EssigD McKinsey K Zurek A Chou P Graham J Estrada JIncandela and T Tait Basic research needs for dark mattersmall projects new initiatives Department of Energytechnical report 2018

[3] R Essig J Mardon and T Volansky Direct detection ofsub-GeV dark matter Phys Rev D 85 076007 (2012)

[4] R Essig A Manalaysay J Mardon P Sorensen and TVolansky First Direct Detection Limits on Sub-GeV DarkMatter from XENON10 Phys Rev Lett 109 021301(2012)

[5] P W Graham D E Kaplan S Rajendran and M TWalters Semiconductor probes of light dark matter PhysDark Universe 1 32 (2012)

[6] R Essig M Fernandez-Serra J Mardon A Soto TVolansky and T-T Yu Direct detection of sub-GeV darkmatter with semiconductor targets J High Energy Phys05 (2016) 046

[7] N Kurinsky T C Yu Y Hochberg and B CabreraDiamond detectors for direct detection of sub-GeV darkmatter Phys Rev D 99 123005 (2019)

[8] R Budnik O Chesnovsky O Slone and T VolanskyDirect detection of light dark matter and solar neutrinos viacolor center production in crystals Phys Lett B 782 242(2018)

[9] Y Hochberg Y Kahn M Lisanti C G Tully and K MZurek Directional detection of dark matter with two-dimensional targets Phys Lett B 772 239 (2017)

[10] G Cavoto F Luchetta and A D Polosa Sub-GeV darkmatter detection with electron recoils in carbon nanotubesPhys Lett B 776 338 (2018)

[11] Y Hochberg Y Zhao and KM Zurek SuperconductingDetectors for Superlight Dark Matter Phys Rev Lett 116011301 (2016)

[12] Y Hochberg M Pyle Y Zhao and KM ZurekDetecting superlight dark matter with fermi-degeneratematerials J High Energy Phys 08 (2016) 057

[13] Y Hochberg T Lin and K M Zurek Detecting ultralightbosonic dark matter via sbsorption in superconductorsPhys Rev D 94 015019 (2016)

[14] Y Hochberg I Charaev S-W Nam V Verma MColangelo and K K Berggren Detecting Sub-GeV DarkMatter with Superconducting Nanowires Phys Rev Lett123 151802 (2019)

[15] Y Hochberg Y Kahn M Lisanti K M Zurek A GGrushin R Ilan S M Griffin Z-F Liu S F Weber andJ B Neaton Detection of sub-MeV dark matter withthree-dimensional dirac materials Phys Rev D 97015004 (2018)

[16] S Knapen T Lin M Pyle and K M Zurek Detection oflight dark matter with optical phonons in polar materialsPhys Lett B 785 386 (2018)

L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)

FIG 11 Brillouin zones and high-symmetry points for thepolytypes of SiC (a) Face-centered cubic type for the 3Cpolytype (b) Primitive hexagonal type for the 2H 4H 6H and8H polytypes (c) Rhombohedral hexagonal type for the 15Rpolytype


075002-25

[17] S Griffin S Knapen T Lin and K M Zurek Directionaldetection of light dark matter with polar materials PhysRev D 98 115034 (2018)

[18] K Schutz and KM Zurek Detectability of Light DarkMatter with Superfluid Helium Phys Rev Lett 117121302 (2016)

[19] S Knapen T Lin and KM Zurek Light dark matter insuperfluid helium Detection with multi-excitation produc-tion Phys Rev D 95 056019 (2017)

[20] S A Hertel A Biekert J Lin V Velan and D NMcKinsey A path to the direct detection of sub-GeV darkmatter using calorimetric readout of a superfluid 4Hetarget Phys Rev D 100 092007 (2019)

[21] F Nava G Bertuccio A Cavallini and E Vittone Siliconcarbide and its use as a radiation detector material MeasSci Technol 19 102001 (2008)

[22] D Puglisi and G Bertuccio Silicon carbide microstripradiation detectors Micromachines 10 835 (2019)

[23] A Laine A Mezzasalma G Mondio P Parisi GCubiotti and Y Kucherenko Optical properties of cubicsilicon carbide J Electron Spectrosc Relat Phenom 93251 (1998)

[24] Y Hinuma G Pizzi Y Kumagai F Oba and I TanakaBand structure diagram paths based on crystallographyComput Mater Sci 128 140 (2017)

[25] Physical properties of silicon carbide in Fundamentalsof Silicon Carbide Technology edited by T Kimoto(John Wiley amp Sons Ltd New York 2014) Chap 2pp 11ndash38

[26] C H Park B-H Cheong K-H Lee and K J ChangStructural and electronic properties of cubic 2H 4H and6H SiC Phys Rev B 49 4485 (1994)

[27] G L Harris Properties of Silicon Carbide (E M I SDatareviews Series) (INSPEC London 1995)

[28] G Bertuccio and R Casiraghi Study of silicon carbide forx-ray detection and spectroscopy IEEE Trans Nucl Sci50 175 (2003)

[29] C Jacoboni and L Reggiani The Monte Carlo method forthe solution of charge transport in semiconductors withapplications to covalent materials Rev Mod Phys 55 645(1983)

[30] N Kurinsky P Brink B Cabrera R Partridge and MPyle (SuperCDMS Collaboration) SuperCDMS SNOLABlow-mass detectors Ultra-sensitive phonon calorimetersfor a sub-GeV dark matter search in Proceedings of the38th International Conference on High Energy Physics(ICHEP2016) Chicago (2016) p 1116

[31] D Pines Collective energy losses in solids Rev ModPhys 28 184 (1956)

[32] H Mutschke A C Andersen D Clement T Henningand G Peiter Infrared properties of SiC particles AstronAstrophys 345 187 (1999)

[33] J Koike D M Parkin and T E Mitchell Displacementthreshold energy for type iia diamond Appl Phys Lett60 1450 (1992)

[34] G Lucas and L Pizzagalli Ab initio molecular dynamicscalculations of threshold displacement energies in siliconcarbide Phys Rev B 72 161202 (2005)

[35] A L Barry B Lehmann D Fritsch and D BraunigEnergy dependence of electron damage and displacement

threshold energy in 6H silicon carbide IEEE Trans NuclSci 38 1111 (1991)

[36] W G Aulbur L Joumlnsson and J W WilkinsQuasiparticleCalculations in Solids (Academic Press New York 2000)pp 1ndash218

[37] A A Lebedev N S Savkina A M Ivanov N B Strokanand D V Davydov 6H-SiC epilayers as nuclear particledetectors Semiconductors 34 243 (2000)

[38] AM Ivanov E V Kalinina G Kholuyanov N B StrokanG Onushkin A O Konstantinov A Hallen and A YKuznetsov High energy resolution detectors based on4h-sic in Silicon Carbide and Related Materials 2004Materials Science Forum Vol 483 (Trans Tech PublicationsLtd 2005) pp 1029ndash1032

[39] R A Moffatt N A Kurinsky C Stanford J Allen P LBrink B Cabrera M Cherry F Insulla F Ponce KSundqvist S Yellin J J Yen and B A Young Spatialimaging of charge transport in silicon at low temperatureAppl Phys Lett 114 032104 (2019)

[40] C Stanford R A Moffatt N A Kurinsky P L Brink BCabrera M Cherry F Insulla M Kelsey F Ponce KSundqvist S Yellin and B A Young High-field spatialimaging of charge transport in silicon at low temperatureAIP Adv 10 025316 (2020)

[41] R Moffatt Two-dimensional spatial imaging of chargetransport in Germanium crystals at cryogenic temperaturesPh D thesis Stanford University 2016

[42] M Shur S Rumyantsev and M Levinshtein SiC Materi-als and Devices Selected Topics in Electronics andSystems Vol 1 (World Scientific Singapore 2006)

[43] CRESST Collaboration et al First results on low-massdark matter from the CRESST-III experiment J PhysConf Ser 1342 012076 (2020)

[44] Q Arnaud et al (EDELWEISS Collaboration) FirstGermanium-Based Constraints on Sub-MeV Dark Matterwith the EDELWEISS Experiment Phys Rev Lett 125141301 (2020)

[45] E Armengaud C Augier A Benot A Benoit L BergJ Billard A Broniatowski P Camus A Cazes MChapellier et al Searching for low-mass dark matterparticles with a massive Ge bolometer operated aboveground Phys Rev D 99 082003 (2019)

[46] S Watkins Performance of a large area photon detectorand applications Proceedings of the 18th InternationalWorkshop on Low Temperature Detectors (LTD-18)(2019) httpsagendainfnitevent15448contributions95785

[47] O Abramoff L Barak I M Bloch L Chaplinsky MCrisler Dawa A Drlica-Wagner R Essig J Estrada EEtzion et al SENSEI Direct-detection constraints on sub-GeV dark matter from a shallow underground run using aprototype skipper CCD Phys Rev Lett 122 (2019)

[48] A Aguilar-Arevalo et al First Direct-Detection Con-straints on eV-Scale Hidden-Photon Dark Matter withDAMIC at SNOLAB Phys Rev Lett 118 141803(2017)

[49] A Aguilar-Arevalo et al (DAMIC Collaboration) Con-straints on Light Dark Matter Particles Interacting withElectrons from DAMIC at SNOLAB Phys Rev Lett 123181802 (2019)


075002-26

[50] G Angloher et al Results on MeV-scale dark matter froma gram-scale cryogenic calorimeter operated above groundEur Phys J C 77 637 (2017)

[51] R Strauss J Rothe G Angloher A Bento A Guumltlein DHauff H Kluck M Mancuso L Oberauer F PetriccaF Proumlbst J Schieck S Schoumlnert W Seidel and LStodolsky Gram-scale cryogenic calorimeters for rare-event searches Phys Rev D 96 022009 (2017)

[52] Z Hong R Ren N Kurinsky E Figueroa-Feliciano LWills S Ganjam R Mahapatra N Mirabolfathi BNebolsky H D Pinckney et al Single electronhole pairsensitive silicon detector with surface event discriminationNucl Instrum Methods Phys Res Sect A 963 163757(2020)

[53] N Kurinsky D Baxter Y Kahn and G Krnjaic A darkmatter interpretation of excesses in multiple direct detec-tion experiments Phys Rev D 102 015017 (2020)

[54] W de Boer J Bol A Furgeri S Muumlller C Sander EBerdermann M Pomorski and M Huhtinen Radiationhardness of diamond and silicon sensors compared PhysStatus Solidi Appl Res 204 3004 (2007)

[55] C Canali M Martini G Ottaviani and A A QuarantaMeasurements of the average energy per electron-hole pairgeneration in silicon between 5-320k IEEE Trans NuclSci 19 9 (1972)

[56] F Nava P Vanni M Bruzzi S Lagomarsino S SciortinoG Wagner and C Lanzieri Minimum ionizing and alphaparticles detectors based on epitaxial semiconductor sili-con carbide IEEE Trans Nucl Sci 51 238 (2004)

[57] C A Klein Bandgap dependence and related features ofradiation ionization energies in semiconductors J ApplPhys 39 2029 (1968)

[58] A Rothwarf Plasmon theory of electronhole pair produc-tion Efficiency of cathode ray phosphors J Appl Phys44 752 (1973)

[59] K Lee T Ohshima A Saint T Kamiya D Jamieson andH Itoh A comparative study of the radiation hardness ofsilicon carbide using light ions Nucl Instrum MethodsPhys Res Sect B 210 489 (2003)

[60] A Dulloo F Ruddy J Seidel J Adams J Nico and DGilliam The thermal neutron response of miniature siliconcarbide semiconductor detectors Nucl Instrum MethodsPhys Res Sect A 498 415 (2003)

[61] E Vittone N Skukan Ž Pastuovi P Olivero and M JakiCharge collection efficiency mapping of interdigitated4HSiC detectors Nucl Instrum Methods Phys ResSect A 267 2197 (2009)

[62] F H Ruddy J G Seidel R W Flammang R Singh andJ Schroeder Development of radiation detectors based onsemi-insulating silicon carbide in Proceedings of the 2008IEEE Nuclear Science Symposium Conference Record(IEEE New York 2008) pp 449ndash455

[63] K C Mandal R M Krishna P G Muzykov S Das andT S Sudarshan Characterization of semi-insulating 4Hsilicon carbide for radiation detectors IEEE Trans NuclSci 58 1992 (2011)

[64] M Hodgson A Lohstroh P Sellin and D ThomasNeutron detection performance of silicon carbide anddiamond detectors with incomplete charge collection

properties Nucl Instrum Methods Phys Res Sect A847 1 (2017)

[65] P Bryant A Lohstroh and P Sellin Electrical character-istics and fast neutron response of semi-insulating bulksilicon carbide IEEE Trans Nucl Sci 60 1432 (2013)

[66] T A Shutt A dark matter detector based on the simulta-neous measurement of phonons and ionization at 20 mKPhD thesis UC Berkeley 1993

[67] A Phipps A Juillard B Sadoulet B Serfass and Y JinA HEMT-based cryogenic charge amplifier with sub-100 eVee ionization resolution for massive semiconductordark matter detectors Nucl Instrum Methods Phys ResSect A 940 181 (2019)

[68] K Irwin and G Hilton Transition-edge sensors inCryogenic Particle Detection edited by C Enss (SpringerBerlin Heidelberg 2005) pp 63ndash150

[69] G Slack Nonmetallic crystals with high thermal conduc-tivity J Phys Chem Solids 34 321 (1973)

[70] T Guruswamy D J Goldie and S Withington Quasi-particle generation efficiency in superconducting thinfilms Supercond Sci Technol 27 055012 (2014)

[71] R Kokkoniemi J Govenius V Vesterinen R E LakeA M Gunyhoacute K Y Tan S Simbierowicz L GroumlnbergJ Lehtinen M Prunnila J Hassel A Lamminen O-PSaira and M Moumlttoumlnen Nanobolometer with ultralownoise equivalent power Commun Phys 2 124 (2019)

[72] P M Echternach B J Pepper T Reck and C MBradford Single photon detection of 15 THz radiationwith the quantum capacitance detector Nat Astron 2 90(2018)

[73] C W Fink et al Characterizing tes power noise for futuresingle optical-phonon and infrared-photon detectors AIPAdv 10 085221 (2020)

[74] N A Kurinsky The low-mass limit Dark matter detectorswith eV-scale energy resolution Ph D thesis StanfordUniversity 2018

[75] S W Deiker W Doriese G C Hilton K D Irwin W HRippard J N Ullom L R Vale S T Ruggiero AWilliams and B A Young Superconducting transitionedge sensor using dilute almn alloys Appl Phys Lett 852137 (2004)

[76] D Akerib et al (LUX Collaboration) First Searches forAxions and Axionlike Particles with the LUX ExperimentPhys Rev Lett 118 261301 (2017)

[77] A K Drukier K Freese and D N Spergel Detectingcold dark-matter candidates Phys Rev D 33 3495(1986)

[78] J Lewin and P Smith Review of mathematics numericalfactors and corrections for dark matter experiments basedon elastic nuclear recoil Astropart Phys 6 87 (1996)

[79] T Trickle Z Zhang K M Zurek K Inzani and S MGriffin Multi-channel direct detection of light dark matterTheoretical framework J High Energy Phys 03 (2020)036

[80] B Campbell-Deem P Cox S Knapen T Lin and TMelia Multiphonon excitations from dark matter scatter-ing in crystals Phys Rev D 101 036006 (2020)

[81] S Griffin Y Hochberg K Inzani N Kurinsky T Lin andT C Yu (to be published)


075002-27

[82] Y Hochberg T Lin and K M Zurek Absorption of lightdark matter in semiconductors Phys Rev D 95 023013(2017)

[83] J I Larruquert A P Perez-Mariacuten S Garciacutea-CortesL R de Marcos J A Aznaacuterez and J A Mendez Self-consistent optical constants of SiC thin films J Opt SocAm A 28 2340 (2011)

[84] A Debernardi C Ulrich K Syassen and M CardonaRaman linewidths of optical phonons in 3C-SiC underpressure First-principles calculations and experimentalresults Phys Rev B 59 6774 (1999)

[85] S Knapen T Lin and K M Zurek Light dark matterModels and constraints Phys Rev D 96 115021 (2017)

[86] S M Griffin K Inzani T Trickle Z Zhang and K MZurek Multichannel direct detection of light dark matterTarget comparison Phys Rev D 101 055004 (2020)

[87] P Cox T Melia and S Rajendran Dark matter phononcoupling Phys Rev D 100 055011 (2019)

[88] L Barak et al SENSEI Direct-Detection Results on sub-GeV Dark Matter from a New Skipper-CCD Phys RevLett 125 171802 (2020)

[89] SuperCDMS Collaboration et al Constraints on low-mass relic dark matter candidates from a surface-operatedSuperCDMS single-charge sensitive detector Phys Rev D102 091101 (2020)

[90] R Essig T Volansky and T-T Yu New constraints andprospects for sub-GeV dark matter scattering off electronsin xenon Phys Rev D 96 043017 (2017)

[91] P Agnes et al (DarkSide Collaboration) Constraints onSub-GeV Dark-MatterndashElectron Scattering from the Dark-Side-50 Experiment Phys Rev Lett 121 111303 (2018)

[92] E Aprile et al Dark Matter Search Results from a OneTon-Year Exposure of XENON1T Phys Rev Lett 121111302 (2018)

[93] E Aprile et al (XENON Collaboration) Light DarkMatter Search with Ionization Signals in XENON1TPhys Rev Lett 123 251801 (2019)

[94] H Vogel and J Redondo Dark Radiation constraints onminicharged particles in models with a hidden photonJ Cosmol Astropart Phys 02 (2014) 029

[95] J H Chang R Essig and S D McDermott Supernova1987A constraints on sub-GeV dark sectors millichargedparticles the QCD axion and an axion-like particleJ High Energy Phys 09 (2018) 051

[96] C Dvorkin T Lin and K Schutz Making dark matter outof light Freeze-in from plasma effects Phys Rev D 99115009 (2019)

[97] A Berlin R T DrsquoAgnolo S A Ellis P Schuster and NToro Directly Deflecting Particle Dark Matter Phys RevLett 124 011801 (2020)

[98] A Coskuner A Mitridate A Olivares and K M ZurekDirectional dark matter detection in anisotropic diracmaterials Phys Rev D 103 016006 (2021)

[99] R M Geilhufe F Kahlhoefer and MW Winkler Diracmaterials for sub-MeV dark matter detection New targetsand improved formalism Phys Rev D 101 055005(2020)

[100] A Arvanitaki S Dimopoulos and K Van TilburgResonant Absorption of Bosonic Dark Matter in Mole-cules Phys Rev X 8 041001 (2018)

[101] H An M Pospelov and J Pradler Dark Matter Detectorsas Dark Photon Helioscopes Phys Rev Lett 111 041302(2013)

[102] H An M Pospelov J Pradler and A Ritz Directdetection constraints on dark photon dark matter PhysLett B 747 331 (2015)

[103] A Aguilar-Arevalo et al (DAMIC Collaboration) FirstDirect-Detection Constraints on eV-Scale Hidden-PhotonDark Matter with DAMIC at SNOLAB Phys Rev Lett118 141803 (2017)

[104] R Agnese et al (SuperCDMS Collaboration) FirstDark Matter Constraints from a SuperCDMS Single-Charge Sensitive Detector Phys Rev Lett 121 051301(2018)

[105] E Aprile et al (XENON100 Collaboration) First axionresults from the XENON100 experiment Phys Rev D 90062009 (2014) Erratum Phys Rev D 95 029904 (2017)

[106] C Fu et al (PandaX Collaboration) Limits on AxionCouplings from the First 80 Days of Data of the PandaX-IIExperiment Phys Rev Lett 119 181806 (2017)

[107] G G Raffelt Astrophysical axion bounds Lect NotesPhys 741 51 (2008)

[108] D Grin G Covone J-P Kneib M Kamionkowski ABlain and E Jullo A telescope search for decaying relicaxions Phys Rev D 75 105018 (2007)

[109] P Arias D Cadamuro M Goodsell J Jaeckel JRedondo and A Ringwald WISPy cold dark matterJ Cosmol Astropart Phys 06 (2012) 013

[110] J Kozaczuk and T Lin Plasmon production from darkmatter scattering Phys Rev D 101 123012 (2020)

[111] K Inzani A Faghaninia and S M Griffin Prediction oftunable spin-orbit gapped materials for dark matter de-tection arXiv200805062

[112] R M Geilhufe B Olsthoorn A D Ferella T Koski FKahlhoefer J Conrad and A V Balatsky Materialsinformatics for dark matter detection Phys Status SolidiRRL 12 1800293 (2018)

[113] R M Geilhufe F Kahlhoefer and MW Winkler Diracmaterials for sub-mev dark matter detection New targetsand improved formalism Phys Rev D 101 055005(2020)

[114] R Catena T Emken N A Spaldin and W TarantinoAtomic responses to general dark matter-electron inter-actions Phys Rev Research 2 033195 (2020)

[115] G Kresse and J Hafner Ab initio molecular dynamics forliquid metals Phys Rev B 47 558 (1993)

[116] G Kresse and J Hafner Ab initio molecular-dynamicssimulation of the liquid-metalamorphous-semiconductortransition in germanium Phys Rev B 49 14251 (1994)

[117] G Kresse and J Furthmuumlller Efficiency of ab initio totalenergy calculations for metals and semiconductors using aplane-wave basis set Comput Mater Sci 6 15 (1996)

[118] G Kresse and J Furthmuumlller Efficient iterative schemesfor ab initio total-energy calculations using a plane-wavebasis set Phys Rev B 54 11169 (1996)

[119] P Bloumlchl Projector augmented-wave method Phys RevB 50 17953 (1994)

[120] G Kresse and D Joubert From ultrasoft pseudopotentialsto the projector augmented-wave method Phys Rev B 591758 (1999)


075002-28

[121] J P Perdew A Ruzsinszky G I Csonka O A VydrovG E Scuseria L A Constantin X Zhou and K BurkeRestoring the Density-Gradient Expansion for Exchange inSolids and Surfaces Phys Rev Lett 100 136406 (2008)

[122] A Togo and I Tanaka First principles phonon calculationsin materials science Scr Mater 108 1 (2015)

[123] A Togo L Chaput and I Tanaka Distributions of phononlifetimes in brillouin zones Phys Rev B 91 094306(2015)

[124] J Heyd G E Scuseria and M Ernzerhof Hybridfunctionals based on a screened Coulomb potentialJ Chem Phys 118 8207 (2003) Erratum J ChemPhys 124 219906 (2006)

[125] P Giannozzi et al Quantum ESPRESSO A modularand open-source software project for quantum simu-lations of materials J Phys Condens Matter 21 395502(2009)

[126] P Giannozzi et al Advanced capabilities for materialsmodelling with Quantum ESPRESSO J Phys CondensMatter 29 465901 (2017)

[127] D R Hamann Optimized norm-conserving Vanderbiltpseudopotentials Phys Rev B 88 085117 (2013)

[128] J Callaway Model for lattice thermal conductivity at lowtemperatures Phys Rev 113 1046 (1959)

[129] S Kaplan Acoustic matching of superconducting films tosubstrates J Low Temp Phys 37 343 (1979)


075002-29

considerations are possible due to the higher band gap ofSiC 32 eV compared to 112 eV for Si It is thus natural toobserve the parallels between the development of Sidiamond and SiC detector technologies and explore theability of future SiC detectors to search for sub-GeV DMMoreover SiC is an attractive material to explore

because of its polymorphismmdashthe large number of stablecrystal structures which can be readily synthesizedmdashandthe resulting range of properties they possess In fact SiCexhibits polytypismmdasha special type of polymorphismwhere the crystal structures are built up from a commonunit with varying connectivity between the units (seeFig 1) The variety of available polytypes results in acorresponding variety of physical properties relevant to DMdetection such as band gap and phonon mode frequenciesIn this paper we explore six of the most common polytypes(3C 2H 4H 6H 8H and 15R described in detail in Sec II)which span the range of variation in physical properties andevaluate their suitability as target materials for a detector aswell as their differences in DM reach for given detectorperformance goals In particular we show that the hex-agonal (H) polytypes are expected to exhibit stronger dailymodulation due to a higher degree of anisotropy in theircrystal structureIn this work we explore the potential of SiC-based single-

charge detectors and meV-scale microcalorimeters for DMdetection The paper is organized as follows In Sec II wediscuss the electronic and vibrational properties of the SiCpolytypes explored in this work In Sec III we explore themeasured and modeled response of SiC crystals to nuclearand electronic energy deposits over a wide energy range and

the expected performance of SiC detectors given realisticreadout schemes for charge and phonon operating modesSections IVand V summarize the DMmodels considered inthis paper compare the reach of different SiC polymorphsinto DM parameter space for nuclear recoils direct phononproduction electron recoils and absorption processes andalso compare directional detection prospects The high-energy theorist interested primarily in the DM reach ofSiC polytypes can thus proceed directly to Sec IV We findexcellent DM sensitivity comparable and complementary toother proposals which places SiC detectors in the limelightfor rapid experimental development

II ELECTRONIC AND PHONONICPROPERTIES OF SiC POLYTYPES

Silicon carbide is an indirect-gap semiconductor with aband gap (23ndash33 eV) intermediate between those ofcrystalline silicon (11 eV) and diamond (55 eV) Whilethere exists a zinc-blende form of SiC which has the samestructural form of diamond and Si there are over 200additional stable crystal polymorphs with a range of bandgap energies and physical properties These polymorphsbroadly fall into three groups based on lattice symmetrycubic (C) hexagonal (H) and rhombohedral (R) Tocompare the expected performance of these polytypes asparticle detectors we first explore how the differences inband structure between polytypes manifests in charge andphonon dynamicsIn all SiC polytypes the common unit is a sheet of corner-

sharing tetrahedra and the polytypes are distinguished byvariations in stacking sequences The polytype 3C adopts thecubic zinc-blende structure with no hexagonal close packingof the layers whereas 2H has a wurtzite structure withhexagonal close packing between all the layers The differentpolytypes can thus be characterized by their hexagonalityfraction fH with 2H (3C) having fH frac14 1 (fH frac14 0) Thissingle number correlates strongly with the materialrsquos bandgap with 3C having the smallest gap and 2H the largest gap[25] The other polytypes including those considered in thispaper consist of lattices with different sequences of hex-agonal and cubic stacking layers and can be listed in order ofincreasing hexagonal close packing 3C 8H 6H 4H 2HThe number refers the number of layers in the stackingsequence Rhombohedral structures also occur and these arecharacterized by long-range stacking order as shown inFig 1(f) Crystal structures for the polytypes considered hereare shown in Fig 1The difference in stability between cubic and hexagonal

stacking is very small which can be understood as abalance between the attractive and repulsive interactionsbetween third-nearest neighbors stemming from the spe-cific degree of charge asymmetry in the SiC bond [26] Thisresults in a difference in total energy between the polytypesof only a few meV per atom therefore many crystalstructures of SiC are experimentally accessible To limit

FIG 1 Crystal structures of the polytypes of SiC considered inthis work Si atoms are blue and C atoms are brown


075002-2







075002-3











ϵ0perp 57 117 97 967 976ϵ0k 1003 1032

ϵfrac12calc0perp 1040 1040 1039 1036 1024 1038

ϵfrac12calc0k

1080 1090 1106 1141 1096






075002-4











075002-5













B Charge readout



075002-6


1 Charge collection








1

vdsatthorn dμV

minus1 eth4THORN


ϵq frac14Dd

1 minus exp

minusdD

eth5THORN



frac14 μτscatd

E ≪ 1 eth6THORN





Charge


42 K

4 kV 14eminus





075002-7


2 Charge resolution



ϵqffiffiffiτ

p eth7THORN










C SiC calorimetry


olution as

σph frac141

ϵph


q eth9THORN






075002-8






τcollect frac144η








eth12THORN






ffiffiffiffiffiSs

psim 10minus19 W=





p 1

ϵph


p eth14THORN

asymp10 meV1

fcollectϵtrap


reth15THORN

asymp10 meVϵtrap



s eth16THORN



075002-9





Ss eth18THORN




100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN






095nabs


fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN









075002-10






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29







075002-3











ϵ0perp 57 117 97 967 976ϵ0k 1003 1032

ϵfrac12calc0perp 1040 1040 1039 1036 1024 1038

ϵfrac12calc0k

1080 1090 1106 1141 1096






075002-4











075002-5













B Charge readout



075002-6


1 Charge collection








1

vdsatthorn dμV

minus1 eth4THORN


ϵq frac14Dd

1 minus exp

minusdD

eth5THORN



frac14 μτscatd

E ≪ 1 eth6THORN





Charge


42 K

4 kV 14eminus





075002-7


2 Charge resolution



ϵqffiffiffiτ

p eth7THORN










C SiC calorimetry


olution as

σph frac141

ϵph


q eth9THORN






075002-8






τcollect frac144η








eth12THORN






ffiffiffiffiffiSs

psim 10minus19 W=





p 1

ϵph


p eth14THORN

asymp10 meV1

fcollectϵtrap


reth15THORN

asymp10 meVϵtrap



s eth16THORN



075002-9





Ss eth18THORN




100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN






095nabs


fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN









075002-10






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29











ϵ0perp 57 117 97 967 976ϵ0k 1003 1032

ϵfrac12calc0perp 1040 1040 1039 1036 1024 1038

ϵfrac12calc0k

1080 1090 1106 1141 1096






075002-4











075002-5













B Charge readout



075002-6


1 Charge collection








1

vdsatthorn dμV

minus1 eth4THORN


ϵq frac14Dd

1 minus exp

minusdD

eth5THORN



frac14 μτscatd

E ≪ 1 eth6THORN





Charge


42 K

4 kV 14eminus





075002-7


2 Charge resolution



ϵqffiffiffiτ

p eth7THORN










C SiC calorimetry


olution as

σph frac141

ϵph


q eth9THORN






075002-8






τcollect frac144η








eth12THORN






ffiffiffiffiffiSs

psim 10minus19 W=





p 1

ϵph


p eth14THORN

asymp10 meV1

fcollectϵtrap


reth15THORN

asymp10 meVϵtrap



s eth16THORN



075002-9





Ss eth18THORN




100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN






095nabs


fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN









075002-10






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29











075002-5













B Charge readout



075002-6


1 Charge collection








1

vdsatthorn dμV

minus1 eth4THORN


ϵq frac14Dd

1 minus exp

minusdD

eth5THORN



frac14 μτscatd

E ≪ 1 eth6THORN





Charge


42 K

4 kV 14eminus





075002-7


2 Charge resolution



ϵqffiffiffiτ

p eth7THORN










C SiC calorimetry


olution as

σph frac141

ϵph


q eth9THORN






075002-8






τcollect frac144η








eth12THORN






ffiffiffiffiffiSs

psim 10minus19 W=





p 1

ϵph


p eth14THORN

asymp10 meV1

fcollectϵtrap


reth15THORN

asymp10 meVϵtrap



s eth16THORN



075002-9





Ss eth18THORN




100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN






095nabs


fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN









075002-10






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29













B Charge readout



075002-6


1 Charge collection








1

vdsatthorn dμV

minus1 eth4THORN


ϵq frac14Dd

1 minus exp

minusdD

eth5THORN



frac14 μτscatd

E ≪ 1 eth6THORN





Charge


42 K

4 kV 14eminus





075002-7


2 Charge resolution



ϵqffiffiffiτ

p eth7THORN










C SiC calorimetry


olution as

σph frac141

ϵph


q eth9THORN






075002-8






τcollect frac144η








eth12THORN






ffiffiffiffiffiSs

psim 10minus19 W=





p 1

ϵph


p eth14THORN

asymp10 meV1

fcollectϵtrap


reth15THORN

asymp10 meVϵtrap



s eth16THORN



075002-9





Ss eth18THORN




100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN






095nabs


fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN









075002-10






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29


1 Charge collection








1

vdsatthorn dμV

minus1 eth4THORN


ϵq frac14Dd

1 minus exp

minusdD

eth5THORN



frac14 μτscatd

E ≪ 1 eth6THORN





Charge


42 K

4 kV 14eminus





075002-7


2 Charge resolution



ϵqffiffiffiτ

p eth7THORN










C SiC calorimetry


olution as

σph frac141

ϵph


q eth9THORN






075002-8






τcollect frac144η








eth12THORN






ffiffiffiffiffiSs

psim 10minus19 W=





p 1

ϵph


p eth14THORN

asymp10 meV1

fcollectϵtrap


reth15THORN

asymp10 meVϵtrap



s eth16THORN



075002-9





Ss eth18THORN




100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN






095nabs


fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN









075002-10






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29


2 Charge resolution



ϵqffiffiffiτ

p eth7THORN










C SiC calorimetry


olution as

σph frac141

ϵph


q eth9THORN






075002-8






τcollect frac144η








eth12THORN






ffiffiffiffiffiSs

psim 10minus19 W=





p 1

ϵph


p eth14THORN

asymp10 meV1

fcollectϵtrap


reth15THORN

asymp10 meVϵtrap



s eth16THORN



075002-9





Ss eth18THORN




100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN






095nabs


fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN









075002-10






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29






τcollect frac144η








eth12THORN






ffiffiffiffiffiSs

psim 10minus19 W=





p 1

ϵph


p eth14THORN

asymp10 meV1

fcollectϵtrap


reth15THORN

asymp10 meVϵtrap



s eth16THORN



075002-9





Ss eth18THORN




100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN






095nabs


fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN









075002-10






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29





Ss eth18THORN




100 mm3

1 mm2

As

095nabs

14 km=scs

s eth19THORN






095nabs


fabs

V100 mm3

ethη=csTHORNτlife

s eth20THORN









075002-10






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29






Design

Parameter A B C D






affiffiffiffiffiSs











075002-11


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29


D Combined readout






dRdEr


2χN


Zvmin

fχethvTHORNv

d3v eth21THORN









075002-12



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29



σ0 frac14 A2

μχNμχn

2

σn eth22THORN



dRdEr


χF2medethqTHORN

Zvmin

fχethvTHORNv

d3v eth23THORN



2μχNmχ

srarr


2m2χ

seth24THORN



dRdEr

SiC

frac14 1

2mSiC

mSi

dRdEr

SithornmC

dRdEr

C






R frac14 1

ρT

ρχmχ




Zd3qΩ












2 eth28THORN



075002-13






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29






νdk





1 Daily modulation






ZBZ

d3kd3k0


timesXG






G



R frac14 σe2ρTμ

2χe

ρχmχ

ZqdqdωF2


eth32THORN




R frac14 1

ρT

ρχmχ





075002-14






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29






eth34THORN




3Rperp thorn 1



V RESULTS












075002-15




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29




q0q

4

eth35THORN









075002-16



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29



1 heavy mediator








075002-17





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29





L sup minusκ





2χe






2χe




σe eth40THORN



αme

q

4

eth41THORN





075002-18



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29



κ2eff frac14κ2m4

A0


κ2







L supgaee2me



g2eff frac143m2

a

4m2e

g2aeee2

eth44THORN




075002-19


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29


VI DISCUSSION






075002-20







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29







ACKNOWLEDGMENTS






075002-21













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29













frac14 csL

1thorn AL

csω4 thorn L

csB1T3ω2

ethB2THORN




phonons

ωp frac14csAL

1=4

frac14

1

Aτb

1=4

ethB4THORN



1

τbB1T3

sfrac14


LB1T3

r ethB5THORN




075002-22



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29



κ frac14 2kBπ2

15

τbcs

ω3T

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2 ethB6THORN


ωT frac14 kBTℏ

ethB7THORN





CV frac14 12π4nkB5

TTD

3

ethB8THORN


κ frac14 1

3CVcsL

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB9THORN

frac14 κ0

1 minus

2πωT

ωp

4

minus5

7

2πωT

ωB

2

ethB10THORN


CVfrac14 1

3c2sτb)


ωg sim7kbTc

2ℏ ethB11THORN



7kbTc

2ℏ≪

csAL

1=4

ethB13THORN

Tc ≪2ℏ7kb

csAL

1=4

ethB14THORN






075002-23


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29


sin θ1cl1

frac14 sin θ2cl2

frac14 sin γ1ct1

frac14 sin γ2ct2

ethC1THORN


















075002-24


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



075002-25




































075002-26



































075002-27









































075002-28











075002-29


n frac14Z

π=2

0


frac14Z

θc

0






















L

WXK

U

b2

b3

b1

(a)

b2

b3

b1

A

M

H

K

L

(b)

b2

b3 b1

S0

H0 LS2

H2F

T

(c)



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silicon carbide detectors for sub-gev dark matter

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