PHYSICAL REVIEW D 97, 015004 (2018)
Detection of sub-MeV dark matter with three-dimensional Dirac materials
Yonit Hochberg,1,2 Yonatan Kahn,3 Mariangela Lisanti,3 Kathryn M. Zurek,4,5 Adolfo G. Grushin,6,7 Roni Ilan,8 Sin´ead M. Griffin,6,9 Zhen-Fei Liu,6,9 Sophie F. Weber,6,9 and Jeffrey B. Neaton6,9,10,11 1Department of Physics, LEPP, Cornell University, Ithaca, New York 14853, USA 2Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel 3Department of Physics, Princeton University, Princeton, New Jersey 08544, USA 4Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720, USA 5Berkeley Center for Theoretical Physics, University of California, Berkeley, California 94720, USA 6Department of Physics, University of California, Berkeley, California 94720, USA 7Institut N´eel, CNRS and Universit´e Grenoble Alpes, F-38042 Grenoble, France 8Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel 9Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 10Kavli Energy NanoScience Institute at Berkeley, Berkeley, California 94720, USA 11Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
(Received 10 September 2017; published 8 January 2018)
We propose the use of three-dimensional Dirac materials as targets for direct detection of sub-MeV dark matter. Dirac materials are characterized by a linear dispersion for low-energy electronic excitations, with a small band gap of OðmeVÞ if lattice symmetries are broken. Dark matter at the keV scale carrying kinetic energy as small as a few meV can scatter and excite an electron across the gap. Alternatively, bosonic dark matter as light as a few meV can be absorbed by the electrons in the target. We develop the formalism for dark matter scattering and absorption in Dirac materials and calculate the experimental reach of these target materials. We find that Dirac materials can play a crucial role in detecting dark matter in the keV to MeV mass range that scatters with electrons via a kinetically mixed dark photon, as the dark photon does not develop an in-medium effective mass. The same target materials provide excellent sensitivity to absorption of light bosonic dark matter in the meV to hundreds of meV mass range, superior to all other existing proposals when the dark matter is a kinetically mixed dark photon.
DOI: 10.1103/PhysRevD.97.015004
I. INTRODUCTION Directly detecting DM relies on observing the effects of The search for sub-GeV dark matter (DM) is a growing its interactions with an experimental target, either through frontier in direct detection experiments. This program is scattering or absorption in the material. In both cases, driven by a theoretical revolution revealing a wide and sufficient energy must be deposited to observe the inter- growing range of models for light DM. In these scenarios, action; this becomes increasingly challenging as the DM the DM typically resides in a hidden sector with either mass is reduced. The current suite of direct detection strongly or weakly interacting dynamics [1–28]. There are experiments focuses on the weakly interacting massive many ways to fix the observed DM abundance in these particle (WIMP), where the DM mass is typically above theories, including asymmetric DM [29–31], freeze-in ∼10 GeV. These experiments search for nuclei that recoil [32,33],strongdynamics[34–36], kinematic thresholds after a collision with a DM particle. Since the energy – 2 [37], and various nonstandard thermal histories [38 43], deposited in an elastic scattering process is q =2mT, where to name a few. The breadth of possible scenarios has q is the momentum transfer and mT is the mass of the stimulated a rethinking of the ideal experimental targets target, it often becomes more effective to search for energy for discovery. deposition on electron targets when DM is less massive than a nucleus. Condensed-matter systems are sensitive to Published by the American Physical Society under the terms of scattering events where the DM carries comparable kinetic the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to energy to the electron excitation energy. For many such the author(s) and the published article’s title, journal citation, systems, including semiconductors [44–46], graphene [47], and DOI. Funded by SCOAP3. scintillators [48], molecules [49], and crystal lattices [50],
2470-0010=2018=97(1)=015004(24) 015004-1 Published by the American Physical Society YONIT HOCHBERG et al. PHYS. REV. D 97, 015004 (2018) these energies are at the eV scale. This is optimal for detecting DM χ with mass mχ ≳ MeV, where the kinetic 2 −3 energy is mχvχ=2 with vχ ∼ 10 , the virial velocity of DM in the Galaxy.1 If instead χ is a boson with mass ≳eV, it can be detected via absorption on an electron in these same systems [51,52]. Extending experimental sensitivity to scattering or absorption of even lower mass DM carries many challenges. For example, fermionic DM is consistent with all astro- physical observations when its mass is greater than a few keV, but to reach these mass scales, one must find a material where the few meV of energy it deposits in scattering can FIG. 1. Cartoon of the two dark-matter-initiated processes in lead to observable signatures. Superconducting targets offer Dirac materials that we consider in this paper: interband (valence – to conduction) scattering (left) and absorption by valence-band one promising option [53 55]. These ultrapure materials, electrons (right). with a small (∼meV) gap and a large Fermi velocity, are – sensitive to DM scatters in the keV MeV mass range or to of nonrelativistic DM. A cartoon of these two processes is – meV eV DM absorption. Superfluid helium has also been illustrated in Fig. 1. As we will show, the dynamics of the shown to be sensitive to sub-MeV DM, when the DM photon interacting with Dirac fermions mimics those of collision can produce multiple phonons [56,57]. Neither ordinary relativistic QED: the Ward identity keeps the superconductors nor superfluid helium, however, has opti- photon massless in a Dirac material, leading to excellent mal sensitivity to dark photons [58,59], which can serve detection reach in models of DM involving dark photons. either as the mediator for DM-electron scattering processes When Δ ¼ 0, the low-energy degrees of freedom in a or as the DM itself which is absorbed. In the case of Dirac material correspond to two Weyl fermions of oppo- superconductors, the dark photon takes on a large effective site chiralities. Materials with this feature are classified as mass in the medium, suppressing the DM interaction rate. either Dirac or Weyl semimetals and are regarded as the 3D For helium, the leading interaction is through the polar- analogues of graphene. In Dirac semimetals, both Weyl izability of the atom, which is small. fermions occur at the same point in momentum space, but In this paper, we propose Dirac materials as a new class are decoupled due to an additional crystalline symmetry of electron targets for DM scattering or absorption. We which imposes Δ ¼ 0. Examples of Dirac semimetals define Dirac materials as three-dimensional (3D) bulk include Na3Bi [63,64] and Cd3Ar2 [65–67]. Allowing substances whose low-energy electronic excitations are – the two Weyl fermions to couple, for example by applying characterized by a Dirac Hamiltonian [60 62], strain to a Dirac semimetal or tuning a topological insulator close to the semimetal critical point [68], can lead to a finite 0 vFℓ · σ − iΔ Hℓ ¼ ; Δ ≠ 0 that is typically small, 2Δ ∼ meV. Such a gap can v ℓ · σ þ iΔ 0 qFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi suppress thermal interband transitions, which is crucial for E ¼ v2 ℓ 2 þ Δ2: ð1:1Þ making detection of meV-scale DM-induced excitations ℓ F feasible.3 While our analysis is completely general, we Here, ℓ is a lattice momentum measured from the location of propose ZrTe5 as a realistic target Dirac material. ZrTe5 has the point of the Dirac cone (e.g., the Dirac point) in been synthesized experimentally, and in this work we reciprocal space, Δ is analogous to the mass term in the compute its band structure from first principles, finding Dirac equation giving rise to a band gap 2Δ, the Fermi in particular that its small Fermi velocities and tunable Fermi level, which can be located inside the gap, make it velocity vF plays the role of the speed of light c, and the positive and negative dispersion relations correspond to the especially suitable for a dark matter search. conduction and valence bands, respectively.2 The desired This paper is organized as follows. Section II presents signal is a DM-induced interband transition from the valence the benchmark dark photon model, and then introduces the to the conduction band, where for DM scattering the formalism for describing in-medium effects in Dirac momentum transfer jqj is typically much larger than the materials. This formalism is used in Secs. III and IV to energy deposit ω, with the opposite being true for absorption calculate the DM scattering rate mediated by a dark photon and the dark photon absorption rate in Dirac materials,
1Throughout this paper, we use natural units with ℏ ¼ c ¼ 1; all velocities are expressed in units of c and all distances in units 3In Weyl semimetals, the two Weyl fermions are generically of momentum. located at different points in momentum space and thus are 2Real materials typically have anisotropic Fermi velocities, but decoupled at low energies [69–72], making it difficult to open a this complication does not affect the thrust of our arguments; we gap. As the gap is necessary to control thermal noise in our treat this case in Appendixes A and B. proposal, we do not consider these materials further in this paper.
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ϵT ¼ p1ffiffi ð0 1 0Þ respectively. For both cases, we present sensitivity projec- and 1;2 2 ; ; i; . As a result of Eq. (2.2), dark tions for couplings to electrons, comparing them to other photon interactions inside a medium depend on the electro- proposals for sub-MeV dark matter detection. We conclude magnetic response of the medium, parametrized by ΠT;L in Sec. V with a brief discussion of experimental consid- (see detailed discussion in Ref. [54]). In this section, we erations. The four appendixes describe the derivation of the describe the behavior of an ordinary photon in an optically transition form factor for a generic Dirac material, the responsive medium. We review the optical properties of generalization of the scattering rate to anisotropic semi- Dirac materials in Sec. II A and compare the results to that of metals, the scattering and absorption reach for models metals in Sec. II B. We will use these results to model dark other than the light kinetically mixed dark photon, and the photon scattering and absorption processes in later sections density functional theory (DFT) calculations used to derive of the paper. the band structure of ZrTe5.
II. DARK MATTER INTERACTIONS A. Optical properties of Dirac materials WITH IN-MEDIUM EFFECTS In the Lorentz gauge, the in-medium photon propagator Our discussion of sub-MeV DM is focused on the is written as benchmark model of the kinetically mixed dark photon. μν μν μν P P Specifically, we consider a new Uð1Þ gauge boson that G ðqÞ¼ T þ L ; ð2:3Þ D med Π − 2 Π − 2 mixes with the ordinary photon: T q L q ¼ðω qÞ 2 ¼ ω2 − q2 1 μν 1 0 0μν ε 0μν μ where q ; is the 4-momentum transfer, q , L ¼ − FμνF − FμνF − FμνF þ eJ Aμ 4 4 2 EM and PL;T are longitudinal and transverse projection oper- 2 ators, respectively (see, e.g., Ref. [74] for a complete μ 0 mA0 0μ 0 þ g J Aμ þ A Aμ: ð : Þ derivation). From Eq. (2.3), we see that the photon can D DM 2 2 1 develop an effective mass in-medium if the real part of 0 2 Here, Fμν (Fμν) is the ordinary (dark) electromagnetic field ΠT;LðqÞ contains terms that do not vanish at q ¼ 0.In μ Π ð Þ strength, ε is the kinetic mixing parameter, and J ð Þ is general, T;L q may be a complicated function of q with EM DM Π the electromagnetic (dark) current, which couples to the no simple interpretation as an effective mass, but large T;L 4 will generally suppress electromagnetic interactions. The (dark) photon with strength e (gD). We assume that the 0 imaginary parts of Π determine the probability of photon new dark photon field Aμ acquires a mass mA0 either T;L through a dark Higgs or Stueckelberg mechanism. The absorption. ~ 0 The transverse and longitudinal components of the propagating dark photon Aμ in the mass basis can be in-medium polarization tensor are linked to the optical identified by diagonalizing the kinetic terms in Eq. (2.1) response of the medium through the complex permittivity and can serve as either the DM itself or as a mediator of the ϵ by interactions between the Standard Model and the DM r μ which comprises the dark current J . DM Π ¼ q2ð1 − ϵ Þ and Π ¼ ω2ð1 − ϵ Þ: ð2:4Þ Due to the induced coupling of the dark photon to the L r T r electromagnetic field strength, dark photon interactions are j 2j ∼ q2 ≫ ω2 modified in an optically responsive medium. The effects of In the regime q , which is relevant for DM Π Π the medium on the dark photon coupling can be derived by scattering, L dominates over T. Conversely, in the case 2 ∼ ω2 ≫ q2 Π ≃ Π considering the effects of the medium on an ordinary of DM absorption where q , L T. photon, where the propagator is modified via its inter- For Dirac materials with a band gap, it is simplest to ϵ actions with the medium. One finds [54,73] that the determine the complex permittivity r by borrowing the ~ 0T;L transverse and longitudinal dark photon fields A μ inter- expression for the one-loop polarization function in mas- act with the electromagnetic current with reduced coupling: sive QED in 3 þ 1 dimensions (see, e.g., Ref. [75]). In → α → α~ doing so, we substitute c vF and EM , where vF is 2 α~ L ⊃ ε q ~ 0T;L μ ð Þ the Fermi velocity and is the effective fine-structure e 2 A μ JEM: 2:2 q − ΠT;L constant in the medium: Π α~ ¼ α g ð Þ Here, T;L are the transverse and longitudinal com- EM × ; 2:5 Πμν ¼ κv ponentsP of the in-medium polarization tensor, F Π ϵTμϵT ν þ Π ϵLμϵLν ϵL ¼ p1ffiffiffiffi ðjqj ω q Þ κ α ¼ 2 4π T i¼1;2 i i L , with 2 ; jqj with the background dielectric constant, EM e = , q and g ¼ gsgC is the product of spin and Dirac cone 4In this paper, we follow high-energy physics conventions degeneracy [76]. In the MS scheme, to leading order in α~ andp useffiffiffiffiffiffiffiffiffiffiffiffiffi Heaviside-Lorentzpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi units for electromagnetism, where , the complex permittivity (at zero temperature and ¼ 4πα ≃ 4π 137 e EM = . doping) is therefore given by
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Z e2g 1 ð2v ΛÞ2 ðϵ Þ ¼ 1 þ dx xð1 − xÞ F r Dirac 4π2κ ln Δ2 − ð1 − Þðω2 − 2 q2Þ vF 0 x x vF sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2g 4Δ2 2Δ2 þ i 1 − 1 þ Θðω2 − v2 q2 − 4Δ2Þ; ð : Þ 24πκ ω2 − 2 q2 ω2 − 2 q2 F 2 6 vF vF vF where Λ is a UV cutoff, defined as the momentum distance case for Dirac materials, which are predicted to be weakly from the Dirac point at which the dispersion relation deviates interacting [86]. Because this is consistent with the current from linear.5 The spin degeneracy in Dirac materials is experimental and theoretical consensus in the field gs ¼ 2; taking gC ¼ 1 (hence g ¼ 2) corresponds to a single [76,84,85], we conservatively choose benchmark parame- massive Dirac fermion in QED. The complex permittivity of ters with α~ < 1 and assume the validity of perturbation isotropic semimetals can be recovered from Eq. (2.6) by theory at one loop. taking Δ → 0 and redefining Λ → expð−5=6ÞΛ to absorb The permittivity of a Dirac semimetal exhibits distinctive the finite q-independent piece. This yields the familiar behavior as a function of q2. As can be seen from Eq. (2.7), – formula [76 82] the imaginary part of ϵr approaches a constant at one-loop order, which is a signature of Dirac-like excitations with 2 1 4Λ2 – q2 ðϵ Þ ¼ 1 − e g −q2 linear dispersions [67,78 82]. The dependence on of the r semimetal 24π2κ q2 ln ω2 2 − q2 real part of ϵ is mild due to the log, and thus it is also vF =vF r approximately constant. The top panel of Fig. 2 shows 2 Π ¼ − iπq Θðω − vFjqjÞ ; ð2:7Þ the square root of the real and imaginary parts of L 2 q ð1 − ϵrÞ as a function of jqj for ω ¼ 1 (100) meV in the left (right) panel. As a benchmark, we take v ¼ 4 × 10−4, which can also be derived directly from the Lindhard F Λ ¼ 0.2 keV, κ ¼ 40, and g ¼ 2, which are representative formula, as demonstrated in Appendix A. Equation (2.7) of typical values for real Dirac materials. The vertical was recently confirmed at the 10% level with optical dashed line corresponds to jqj¼ω; below this point, measurements of Na3Bi [67]. absorption processes dominate, while scattering processes Because α~ is inversely proportional to v , materials with pffiffiffiffiffiffiffiffi F j 2j small Fermi velocities can have large effective couplings. dominate above it. To guide the eye, we plot q as the jqj This is the case of free-standing graphene, where κ ¼ 1, solid green line, which scales linearly with in the −3 scattering regime jqj ≫ ω and is constant in the absorption vF ¼ 3 × 10 , and α~ ≃ 2.2, yet perturbation theory still delivers the right predictions when compared to experiment regime. Importantly, the squarep rootffiffiffiffiffiffiffiffi of both the real Π j 2j [83]. Since QED flows to a free theory in the IR, and imaginary values of L track q , as expected from ϵ q2 perturbation theory remains valid near the Dirac point the fact that r is essentially constant in for Dirac and far from the cutoff Λ, so long as no strong coupling semimetals. phase transitions are crossed.6 This is believed to be the We discuss modifications to the complex permittivity for anisotropic Dirac materials (where there are independent Fermi velocities, v ;v ;v ) in Appendix B. 5Here we are effectively setting the renormalization scale μ~ at F;x F;y F;z the cutoff, μ~ ¼ 2vFΛ, which is perhaps unusual from a high- energy physics perspective. The unphysical parameter μ~ can be B. Comparison of metals and Dirac semimetals removed from physical quantities by matching to a measurement We will show in Secs. III and IV that Dirac materials are of the electric charge e. In QED, one typically thinks of the electric charge as being defined by a t-channel scattering process, more sensitive than superconductors to DM scattering via a e.g., e− þ e− → e− þ e−. However, the interband transition dark photon mediator, as well as to absorption of dark in a Dirac material is analogous to pair production, which is photons [51,54]. There are competing effects that drive this an s-channel process. DM scattering in Dirac materials can be result. On the one hand, the optical response of a metal is χ þ → χ þ þ γ γ → − þ þ described by N N followed by e h , much stronger than that of a Dirac semimetal, weakening where the lattice N provides the necessary recoil for the creation of an electron-hole pair. Therefore, we use the vertical transition its sensitivity to dark photon interactions. On the other rate with ðω; qÞ¼ð2vFΛ; 0Þ to measure the charge. At the cutoff hand, a metal has a much larger phase space of conduction Λ, deep inside the band structure and far from the Dirac point, we electrons at low energies, which should improve its reach. assume that the electrons behave as in an ordinary insulator and 2 2 We now discuss the balance of these effects, comparing the that the effective charge is e0 ≡ e ðμ~Þ¼4παEM=κ. 6 optical response and phase space availability in metals In gapless Dirac semimetals, vF is also renormalized [76,84,85]. We do not consider this subtlety for our benchmark versus Dirac semimetals. gapless Dirac materials, since in any realistic experiment, the For metals, intraband transitions dominate because the material will be gapped and this issue does not arise. Fermi energy lies within a single band and excitations
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pffiffiffiffiffiffiffiffiffi FIG. 2. The (square root of the) real and imaginary parts of the longitudinal in-medium polarization tensor jΠLj in Dirac semimetals (top) and metals (bottom), as a function of the momentum transfer jqj. The left (right) panel takes the deposited energy to be −4 −1 ω ¼ 1 ð100Þ meV. For semimetals, we take representative parameters vF ¼ 4 × 10 , Λ ¼ 0.2 keV ≃ 0.1 Å , g ¼ 2, and κ ¼ 40 to ~ give α ∼ 0.9. Note that for semimetals, interband transitions are only allowed for jqj < ω=vF. For metals, we choose aluminum as a representative example, with λTF ≃ 4 keV and pF ≃ 3.5 keV. occur just above the Fermi surface. In this case, the polarization function scales as q2 up to logarithmic correc- permittivity is given by tions and thus acts as a charge renormalization. The difference in behavior between the two materials is λ2 1 jqj ω 2 related to their differing Fermi surface geometries. In metals, ðϵ Þ ¼ 1 þ TF þ pF 1 − − r metal 2 the Fermi surface is not scale invariant; the dimensional jqj 2 4jqj 2pF jqjvF " # Fermi momentum pF sets the screening scale. In an jqj ω 2 − jqj þ 1 (undoped) Dirac semimetal, the Fermi surface is pointlike pF vF þðω → −ωÞ ð Þ ×ln jqj ω ; 2:8 and thus the Fermi momentum is zero by definition. 2 − jqj − 1 pF vF Consequently, there is no screening length for the photon. Alternatively, one can understand this fact from the vanish- λ2 ¼ 3 2 ð2 Þ where TF e ne= EF is the Thomas-Fermi screening ing of the density of states at the Fermi level in semimetals. length, ne is the electron density, pF is the Fermi momen- The Thomas-Fermi screening length is inversely propor- tum, and EF is the Fermi energy [87]. We plot the square root tional to the density of states, which is large for a metal and of the real and imaginary parts of ΠL in a metal in the bottom zero for a semimetal. For the case of gapped Dirac materials, panel of Fig. 2. By comparing the top and bottom panels, it is one can exploit the emergent Lorentz symmetry of the Dirac evident that the magnitudes of both the real and imaginary Hamiltonian, Eq. (1.1), to see that the Ward identity enforces 2 2 components of the polarization tensor are many orders of ΠLðq Þ ∼ q such that the photon stays exactly massless to magnitude smaller in Dirac materials than in metals. all orders in perturbation theory; the gap 2Δ does not provide Furthermore, the polarization for a metal is roughly constant a screening scale akin to pF in a metal. in jqj over a broad range of momenta near OðkeVÞ— As we have just seen, the pointlike Fermi surface in a therefore, we can think of the photon as having an effective semimetal suppresses its optical response, thereby enhanc- mass in this range. By contrast, the real part of the semimetal ing processes mediated by a kinetically mixed dark photon.
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While this benefits detection rates, it simultaneously sup- III. SCATTERING IN DIRAC MATERIALS presses the available phase space for interactions with the The formalism for DM scattering in Dirac materials is a DM. One can use simple geometric arguments to estimate special case of the more general formalism for scattering in the phase space available for ultra-low-energy scattering in crystal lattices described in Ref. [88]. We describe the Dirac semimetals compared to metallic targets, for a given calculation of the DM scattering rate in Sec. III A and energy deposit ω. In a metal, the Fermi surface is a sphere, so highlight important issues pertaining to the kinematics in the volume of the initial-state phase space is given by a pffiffiffiffiffiffiffiffiffiffiffiffi Sec. III B, including the dependence of the scattering rate spherical shell of radius p and thickness δp ¼ 2m ω, F e on the Fermi velocity v . In Sec. III C, we discuss the where m is the electron mass. Numerically, the phase space F e projected sensitivity to DM scattering in a generic Dirac volume for pF ≃ 3.5 keV and ω ¼ 1 meV is target and for ZrTe5 in particular. ¼ 4π 2 δ ∼ 5 109 3 ð Þ VF;metal pF p × eV : 2:9 In a semimetal, the initial-state phase space volume is given A. Scattering rate formalism by the boundary of the hypercone traced out by the valence Consider a Dirac cone located at K in the Brillouin band. The maximum momentum available for the same zone (BZ) and a transition from k ¼ K þ ℓ in the valence ω ¼ ω 0 0 energy transfer is given by pmax =vF. The phase space band to k ¼ K þ ℓ in the conduction band with −4 jℓ j jℓ 0j ≪ jKj volume for vF ¼ 4 × 10 and ω ¼ 1 meV is ; . In order to present simplified analytic qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 results where possible, we assume the gapless, isotropic ¼ π 3 1 þ 2 ∼ 70 3 ð Þ VF;Dirac 3 pmax vF eV ; 2:10 dispersion relations: approximately eight orders of magnitude smaller than the E ¼ v jℓ j: ð3:1Þ corresponding phase space for metals. As shown in Fig. 2, ℓ F however, the phase space suppression in the scattering rate is more than offset by the gain from the reduced in-medium The main effect of a gap is to impose a kinematic response: the scale of the effective dark photon coupling in threshold 2Δ on the scattering event, but our conclusions metals can be 4–6 orders of magnitude larger. When are otherwise unchanged. A more complete discussion of squared, this leads to a huge suppression in the rate, which anisotropic materials with independent Fermi velocities dominates over the phase space suppression of semimetals. vF;x;vF;y;vF;z is included in Appendix B. We demonstrate this behavior explicitly in Secs. III and IV, The rate to scatter from the valence band (labeled where we derive the DM scattering and absorption rates in by “−”)atk to the conduction band (labeled by “þ”)at Dirac materials. k0 is given by [88]
Z ρ χ σ¯ e 3 1 2 2 2 0 ¼ q ηð ðjqj ω 0 ÞÞj ð Þj jF ð Þj j 0 ðqÞj ð Þ R−;k→þ;k 2 d jqj vmin ; kk FDM q med q f−;k→þ;k ; 3:2 mχ 8πμχe
3 −4 where ρχ ≃ 0.4 GeV=cm is the local DM density, μχe is Galactic-frame velocity v0 ¼ 220 km=s(7.3 × 10 in the DM-electron reduced mass, σ¯ e is a fiducial spin- natural units), average Earth velocity with respect to the – −4 averaged DM free electron scattering cross section, and Galactic frame vE ¼ 232 km=s(7.8 × 10 ), and escape −3 ωkk0 ¼ 550 1 8 10 is the energy difference between the final and initial velocity vesc km=s(. × ). For simplicity, states. The rate also depends on several form factors, which we will assume the DM velocity distribution is spherically ð Þ are defined explicitly below: FDM q parametrizes the symmetric. The minimum velocity for a DM particle momentum dependence of the DM–free electron interac- to scatter with momentum transfer q and energy deposit F ð Þ tion, med q parametrizes the momentum-dependent in- ωkk0 is medium effects, and f−;k→þ;k0 ðqÞ is the transition form 0 factor parametrizing the transition between bands. Because a 0 ωkk jqj vFðjℓ jþjℓ jÞ jqj ðjqj ω 0 Þ¼ þ ¼ þ distribution of DM velocities contributes to a scattering vmin ; kk : jqj 2mχ jqj 2mχ event with given k; k0, the rate depends on the halo integral: ð3:4Þ Z 3 ηð Þ¼ d v ð Þθð − Þ ð Þ vmin gχ v v vmin : 3:3 v This expression for vmin arises from solving a delta function for energy conservation assuming a spherically symmetric Here, gχðvÞ is the DM velocity distribution, which gχðvÞ—see Ref. [88] for more details. Here, we have we take to be the standard halo model with typical assumed the gapless isotropic dispersion relation near the
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K hjMj2i ≡ hjM ð Þj2i jF ð Þj2 ð Þ -point given in Eq. (3.1); the result generalizes straight- free q × med q : 3:9 forwardly to gapped or anisotropic dispersions. ð Þ There are three form factors that appear in Eq. (3.2),two The reference momentum q0 used to define FDM q in of which are related to the DM scattering interaction and Eq. (3.8) is arbitrary. Following the standard of comprehen- 2 ¼ðα Þ2 one of which depends on the initial and final wave sive reviews such as Ref. [89],wechooseq0 EMme . functions of the scattered electron. We begin by describing Finally, the fiducial cross section is defined as the latter. The transition form factor is defined as μ2 σ¯ ¼ χe hjM ð Þj2i ð Þ Z e 2 2 free q0 : 3:10 16πmχme 3 iq·x f−;k→þ;k0 ðqÞ ≡ d xΨþ k0 ðxÞΨ−;kðxÞe ; ð3:5Þ ; 0 ≪ With these definitions, we have for the light (mA keV) kinetically mixed dark photon, where Ψ−;kðþ;k0ÞðxÞ is the electron wave function in the 2 0 1 initial (final) state. An analytic expression for this factor A ;lightð Þ¼q0 F ð Þ¼ can be derived using the Hamiltonian in Eq. (1.1) and is FDM q 2 ; med q ; q ϵrðqÞ given by 16πμ2 ε2α α σ¯ ¼ χe EM D ð 2 ¼ðα Þ2Þ ð Þ e 4 q0 EMme ; 3:11 3 0 q0 2 1 ð2πÞ ℓ · ℓ 0 jf− k→þ k0 ðqÞj ¼ 1 − δðq − ðℓ − ℓ ÞÞ; ; ; 2 V jℓ jjℓ 0j α ¼ 2 4π F ð Þ ¼ where D gD= and med q is evaluated at q 0 ð3:6Þ ðωℓℓ0 ; qÞ for initial and final states labeled by ℓ and ℓ respectively. Because in Dirac materials ϵrðqÞ is effectively the ratio of unscreened charge e0 to running charge eðqÞ,the for gapless isotropic materials, where V is the crystal in-medium form factor ensures that the matrix element scales volume. A complete derivation of Eq. (3.6), generalized as e2ðqÞ rather than e2.InAppendixC1, we provide the for anisotropic gapped Dirac materials, is provided in 0 analogous form factor expressions and fiducial cross sections Appendix A. for DM scattering with electrons via other mediators. F ðqÞ F ðqÞ The other two form factors, DM and med , are The total scattering rate in the crystal is obtained from derived from the matrix element corresponding to a DM Eq. (3.2) by summing over initial and final states, which in particle scattering off an electron via the kinetically mixed this context means integrating over the initial and final BZ dark photon we are interested in: momenta: Z 16 2 2 2 2ε2 3k 3k0 2 memχgDe 2 d d hjMj i ≃ ¼ 0 2 2 2 2 2 Rcrystal gsV 6 R−;k→þ;k ðq − m 0 Þ j1 − Π ðqÞ=q j ð2πÞ A L BZZ 2 2 2 2 2 3ℓ 3ℓ 0 16memχg e ε 1 2 d d ¼ D ð Þ ¼ g g V R− ℓ →þ ℓ 0 : ð3:12Þ 2 2 2 2 ; 3:7 s C ð2πÞ6 ; ; ð − 0 Þ jϵ ð Þj cone q mA r q Note that there is no sum over bands because scattering only where gD is the dark photon gauge coupling and me is the takes place between the − and þ bands by assumption. If electron mass. Here, we are neglecting the contribution of there are several Dirac points K with identical linear Π Π ≪ Π i T to the matrix element, since T L in the regime dispersion, one can simply integrate over the region sur- jq2j ≫ ω2 relevant for scattering. The longitudinal polari- rounding one of the points and multiply by gC, giving an zation tensor Π (or equivalently, the permittivity ϵ ) L r overall factor of g ¼ gsgC. Because the rate only depends on describing the material can thus be incorporated into the the integral around the cone (we do not consider intercone event rate for DM scattering using this modified matrix scattering in this paper), and the absolute location of the cone element. We adopt standard conventions in the literature in the BZ is irrelevant, we will work exclusively in terms of and define FDM as the momentum dependence of the free the displacement vector ℓ instead of k from now on. matrix element, B. Scattering kinematics and spectrum 16 2 2 2 2ε2 memχg e We can exploit the analytic expressions for the transition hjM ðqÞj2i¼ D free ð 2 − 2 Þ2 form factor and the in-medium form factor to analyze the q mA0 ≡ hjM ð Þj2i j ð Þj2 ð Þ kinematics of scattering in a Dirac material. Using the free q0 × FDM q ; 3:8 analytic expression for the transition form factor given in Eq. (3.6), we obtain the rate for the case of an ultralight F while med captures the in-medium effects through kinetically mixed mediator:
015004-7 YONIT HOCHBERG et al. PHYS. REV. D 97, 015004 (2018)
FIG. 3. Left: Scaling of the dark matter scattering rate with the Fermi velocity vF of a gapless isotropic Dirac semimetal. The vertical dashed line indicates the point below which α~, the effective fine-structure constant in the medium, is greater than 1. Right: Spectrum 0 −4 0 dI=dE for vF ¼ 4 × 10 , where E is the final-state energy of the scattered electron. Note that the function IðvF; Λ;mχÞ is directly Λ ¼ 0 2 ¼ 2 κ ¼ 40 proportional to the total scattering rate, Rtot. In both cases, we have taken . keV, g , and . The results are shown for 10 keV and 100 keV dark matter in blue and orange, respectively.
ρ σ¯ 288π4κ2 2 4 ð2πÞ3 χ e vFq0 kinematics associated with linear dispersions. For scattering 0 ¼ R−;ℓ →þ;ℓ 2 4 2 very close to the Dirac point, the transition form factor mχ 8πμχe e g V Z Eq. (3.6) enforces momentum conservation q ¼ ℓ 0 − ℓ .7 1 1 ηð ðjqj ω 0 ÞÞ 3q vmin ; ℓℓ Using this relation, Eq. (3.4) becomes × d 2 2 2 2 4Λ02 2 jqj ðω 0 − q Þ j jþπ ℓℓ ln ω2 =v2 −q2 ℓℓ0 F ðjℓ 0jþjℓ jÞ jℓ 0 − ℓ j ðℓ ℓ 0Þ¼ þ ð Þ 0 ℓ · ðℓ þ qÞ vmin ; vF 0 : 3:15 δðq − ðℓ − ℓ ÞÞ 1 − ð Þ jℓ − ℓ j 2mχ × jℓ jjℓ þ qj : 3:13 The first term is at least v by the triangle inequality, and the Λ0 ¼ Λ ð12π2κ ð 2ÞÞ F We have defined exp vF= ge to absorb the second term is non-negative, so we have v >v .Ifv is ϵ min F F constant piece in Re r, and we have dropped the step greater than the largest possible DM velocity, function in Imϵr because all interband transitions satisfy 0 ωℓℓ0 >v jqj. Integrating over ℓ and ℓ in a region of size ¼ þ ≃ 2 6 10−3 ð Þ F vmax vE vesc . × ; 3:16 Λ near the Dirac point as in Eq. (3.12), and noting that the integrand only depends on the magnitudes of q and ℓ and then scattering is kinematically forbidden for any (small) mχ. the angle between them, we find the total rate in counts per Therefore, unsuppressed scattering can only occur if the unit time per unit detector mass: DM is moving faster than the electron target.8 This is in sharp contrast to the case of superconductors, where the 36π2 ρ σ¯ κ2 ¼ 4 χ e 2 ð Λ Þ ð Þ target velocity should exceed that of the DM for low-energy Rtot 4 q0 2 × neVuc vFI vF; ;mχ ; 3:14 e mχ μχe g scattering to occur [53,54]. Dirac materials exhibit a range of Fermi velocities from 6 10−3 3 10−5 where IðvF; Λ;mχÞ has dimensions of momentum—the full × for BLi to × or smaller for NbAs and NbP expression is provided in Appendix B. [90,91]. The kinematic arguments presented above suggest Equation (3.14) is related to Eq. (3.12) via that the materials with smallest vF are most desirable for ¼ Rtot Rcrystal=Mcrystal, where Mcrystal is the target mass, maximizing the DM scattering rate. However, the prefactor in ¼ 2 and V NucVuc with Nuc the total number of unit cells in Eq. (3.14) is suppressed by vF, which comes from the scaling ¼ ϵ the target and Vuc the volume of each unit cell. Then ne of r. Therefore, we do not want to drive vF too low. To illustrate this tension, the left panel of Fig. 3 plots Nuc=Mcrystal is the number of Dirac valence band electrons κ2 2 ð Λ Þ per unit mass of target material. In Eq. (3.14),wehave g vFI vF; ;mχ , which is proportional to the total scatter- separated the factors that depend on the DM model from ing rate, for two values of the DM mass. The results are those that depend only on the target material. shown assuming Λ ¼ 0.2 keV, g ¼ 2,andκ ¼ 40, Of particular interest is the behavior of IðvF; Λ;mχÞ as a function of vF, as it can suggest the optimal 7See Appendix A3 for a discussion of momentum conserva- material properties for maximizing detection rates. Firstly, tion versus lattice momentum conservation. 8 IðvF; Λ;mχÞ¼0 for large values of vF due to the peculiar We thank Justin Song for pointing out this phenomenon to us.
015004-8 DETECTION OF SUB-MEV DARK MATTER WITH THREE- … PHYS. REV. D 97, 015004 (2018) representative of typical values for real Dirac materials. For both masses, the rate is maximal for a particular choice of the Fermi velocity. When mχ ¼ 100 keV, this occurs at −4 vF ≃ 10 .Formχ ¼ 10 keV, the maximum is at even lower Fermi velocities. Such small values for the Fermi velocity lead to α~ > 1 for the material parameters assumed here. That said, the rate for either mass point only varies by a factor of a few between the vF that maximizes the rate −4 and vF ¼ 3.6 × 10 , above which the effective coupling is less than 1. Finally, we consider the energy spectrum dI=dE0 of the −4 excited electron, shown in Fig. 3 (right) for vF ¼ 4 × 10 and mχ ¼ 10; 100 keV. The spectrum peaks away from E0 ¼ 0 due to the vanishing phase space at the point of the Dirac cone. This shows that the majority of the rate comes FIG. 4. Projected reach of dark matter scattering in Dirac from final-state energies above 1 meV. At small E0, the materials through a light kinetically mixed dark photon mediator spectrum depends only weakly on mχ. This is because the with in-medium effects included. We show the expected back- energetically favorable events correspond to small initial- ground-free 95% C.L. sensitivity (3.0 events) that can be obtained “ ” state energies, such that jℓ 0 − ℓ j is small and v is with 1 kg-yr exposure. For the two curves labeled Dirac, we min assume an ideal gapless (Δ ¼ 0, green) or gapped (Δ ¼ 2.5 meV, approximately independent of mχ. As expected, heavier −4 purple) isotropic Dirac material with vF ¼ 4 × 10 , κ ¼ 40, DM masses yield scattering events with higher-energy 24 3 g ¼ 2, Λ ¼ 0.2 keV, n ¼ 5 × 10 =kg, and V ¼ 60 Å .We final-state electrons, giving a larger total rate. As we will e uc also the show the results for ZrTe5, a realistic target material. The show in Sec. III C below, these conclusions do not change red curve labeled “ZrTe5, th.” uses the parameters calculated in ≳ 10 for mχ keV even in the presence of a meV-scale gap. Appendix D, while the yellow curve labeled “ZrTe5, exp.” uses parameters extracted from experiment [96,97]. For comparison, C. Projected sensitivity reach we also show the reach of superconductors with a 1 meV threshold [54] (black), and the projected single-electron reach We are now ready to use the formalism we developed to for a silicon detector with a 1e− threshold [89] (blue dotted). The present the sensitivity reach projections for DM scattering orange curve labeled “Freeze-in” delineates where freeze-in in Dirac materials via a light kinetically mixed dark photon. production [32] results in the correct dark matter relic abundance. The results are shown in Fig. 4. The green and purple The gray shaded regions indicate bounds from white dwarfs, red curves show the expected 95% C.L. sensitivity (corre- giants, big bang nucleosynthesis, and supernovae, which are sponding to 3.0 signal events) with a kg-year exposure for derived from limits on millicharged particles [88,98]. The gray DM scattering in gapless and gapped Dirac materials, dashed line indicates bounds on self-interacting dark matter respectively. For concreteness, we choose Λ ¼ 0.2 keV, derived from observations of the Bullet Cluster [99,100]. ¼ 60 3 ¼ 5 1024 Vuc Å , and ne × =kg, typical of experimen- tally realized semimetals. In addition, we take v ¼ F anisotropic, with v ≪ v ;v . The crystal lattice 4 × 10−4, κ ¼ 40, and g ¼ 2 so that α~ ∼ 0.9 and perturba- F;y F;x F;z also has a highly anisotropic background dielectric tion theory is reliable. This corresponds to a typical tensor, with κyy ≪ κxx; κzz; we take the harmonic mean range of parameters for Dirac semimetals such as 3 κ~ ¼ 1 κ þ1 κ þ1 κ ¼ 25.3 for our estimates here, and Cd3As2 [66,92–95], for which perturbation theory is only = xx = yy = zz expected to break down at α~ ≃ 9.4 [76]. We note that the justify this approximation in the context of our inclusion of the correct wave function overlaps from assumption of a spherically symmetric DM distribution Eq. (3.6) suppresses the rate by about an order of in Appendix B1. Note that the combined effect of these magnitude compared to a naive approximation where the anisotropies may result in interesting directional depend- transition form factor is set to unity. ence of the signal, including daily modulations of the rate, In Fig. 4, we also show the projected sensitivity for a but this requires a dedicated analysis which is beyond benchmark realistic target material, ZrTe5, which has most the scope of this paper. The effective fine-structure constant α~ ¼ α κ~ð Þ1=3 ≃ 0 80 of the desired properties we have discussed. The band is g EM= vF;xvF;yvFz . . High-purity ZrTe5 structure was determined using density functional theory, can be synthesized in macroscopic quantities, and pressure as discussed in Appendix D. We find that ZrTe5 has a small or doping can shift the Fermi level inside the gap so that the −4 Fermi velocity vF;y ¼ 4.9 × 10 along one direction, a conduction band is empty at zero temperature. As shown in small degeneracy g ¼ 4, and a small gap 2Δ ¼ 35 meV at Refs. [53,54], a meV-scale gap with little or no occupation zero temperature. The remaining material parameters are of excited states is necessary for suppressing thermal noise. given in Appendix D. The band structure of ZrTe5 is highly Experimental measurements of the properties of ZrTe5
015004-9 YONIT HOCHBERG et al. PHYS. REV. D 97, 015004 (2018) have led to some ambiguous results regarding the precise greatly enhanced at low momentum transfer due to the 1=q4 Δ j j2 values of the Fermi velocities and , so for comparison, we dependence of the DM form factor FDM . Since this also plot the projected sensitivity using the measurements momentum dependence is not spoiled by the in-medium F of Fermi velocities from Ref. [96], and a gap energy form factor med, Dirac materials are able to probe very 2Δ ¼ 23.5 meV, the median of the range of values found small couplings, which are unconstrained by any other in Ref. [97]. observations, cosmological or otherwise. As anticipated in For comparison, we provide the projections for a super- Sec. II, Dirac materials have superior reach in this case to conducting target with a 1 meV threshold [54] (solid black both superconductors, which suffer from an in-medium line) and semiconductor target [89] (blue dotted line). For suppression at low masses, and semiconductors, which have the latter, we show a silicon target with a single-electron eV-scale gaps. Ideal Dirac materials with small Fermivelocity −4 threshold. Both are low-threshold electron-scattering vF ∼ 4 × 10 and small gap 2Δ ¼ 5 meV, with 1 kg-yr of experimental proposals with complementary detection exposure, can probe cross sections many orders of magnitude modalities: the superconductor proposal exploits the break- smaller than the entire freeze-in region below 1 MeV. ing of Cooper pairs to produce quasiparticles and athermal Realistic materials such as ZrTe5 still give excellent reach, phonons from meVenergy deposits, and the semiconductor which can be improved by identifying materials with smaller proposal aims to detect valence-to-conduction excitation Fermi velocities and gaps. The exceptionally strong reach of (as we propose here) in a generic band structure with a large Dirac materials means that a zero-background experiment is gap of 1.11 eV. As we have discussed, the reach of Dirac not strictly necessary to probe the well-motivated parameter materials is superior to that of superconductors for the case space given by the freeze-in mechanism. of a light kinetically mixed dark photon mediator due to the In Appendix C1, we present the reach of Dirac materials reduced in-medium effects. Assuming the DM velocity for DM scattering via a heavy kinetically mixed dark distribution is given by the standard halo model, semi- photon, as well as via a light or heavy scalar mediator conductors are unable to probe DM lighter than 500 keV where no in-medium effects arise. For the former case, we due to their large band gaps. find that Dirac materials provide better sensitivity than The orange line in Fig. 4 shows the theory expectation superconductors; in the latter case, Dirac materials gen- for a benchmark model where the DM abundance is set erally fare worse than superconductors, as expected. Strong through freeze-in via a light mediator [32]. In such models, constraints from either stellar emission (light mediators) or the DM is very weakly coupled to the Standard Model such BBN (heavy mediators) apply at least for the most naive of that it never thermalizes, and the DM abundance is instead such models, such that typically either BBN or stellar gradually populated through very rare interactions at low emission bounds must be evaded for the models where DM temperatures. If these interactions are with the electron, as does not scatter via a light dark photon. Our results here is the case for DM coupling to a dark photon, freeze-in demonstrate, however, that Dirac materials are an ideal production gives a concrete theoretical target for electron target for light dark photon mediators. The reach of these scattering direct detection experiments. materials for millicharged dark matter, which can be The constraints on light dark photons can be quite directly inferred from the results of the light dark photon stringent; the excluded regions of parameter space (at least mediator, is similarly excellent. for the most naive of models) are indicated by the gray regions in Fig. 4. These are derived from bounds on IV. ABSORPTION IN DIRAC MATERIALS millicharged particles [98], which are also applicable to DM coupled to an ultralight kinetically mixed dark photon Having demonstrated that Dirac materials have compel- [54,88]. When the DM is lighter than the temperature of red ling reach for the case of DM scattering, we move on to the giants and white dwarfs, DM can be copiously produced, case of DM absorption. We begin by presenting the which would lead to excessive cooling; in Fig. 4, this formalism for calculating DM absorption rates, and then (approximate) region is shown in gray and marked we discuss the relevant kinematics and projected sensitiv- “RG þ WD.” In addition, the presence of dark photons ities for Dirac materials. affects the energetics of supernovae and big bang nucleo- 2 −17 A. Absorption rate formalism synthesis (BBN), implying that ε αD ≲ 10 ; in Fig. 4, this region is shown in gray and is marked “SN þ BBN.” The rate for DM absorption in counts per unit time per Constraints from DM self-interactions are generally unit detector mass is given by weaker; for example, the self-interacting DM bound from “ 1 ρ observations of the Bullet Cluster [99,100] (labeled SIDM ¼ χ h σ i ð Þ ” Rabs nT absvrel DM; 4:1 Bullet Cluster ) are subdominant to the other constraints. ρT mχ The light kinetically mixed mediator scenario we have considered here is particularly interesting for direct detec- where ρT is the mass density of the target, nT is the number σ tion with Dirac materials because the scattering rate is of target particles, abs is the DM absorption cross section
015004-10 DETECTION OF SUB-MEV DARK MATTER WITH THREE- … PHYS. REV. D 97, 015004 (2018) 1 A0 2 on the target, and vrel the relative velocity between the DM R ¼ ρχε ϵ : ð : Þ abs ρ effIm r 4 5 and the target. One can relate the absorption rate of certain T classes of DM particles to the measured optical properties For dark photon DM in the mass range of meV to hundreds of the target [51,52,55,101]. In particular, the absorption of meV, the energy deposited in absorption matches the rate of photons in a given (bulk) material is determined by regime of interest for Dirac materials. the polarization tensor via the optical theorem: ImΠðωÞ B. Absorption kinematics and scaling hn σ v i ¼ − ; ð4:2Þ T abs rel γ ω As shown in Fig. 2, ΠðqÞ has a nonvanishing imaginary part even at q ¼ 0 in a Dirac material. Indeed, this can be where ω is the energy of the incoming absorbed photon and – ΠðωÞ considered a distinctive property of Dirac materials [67,78 denotes the polarization tensor, in an isotropic material, 82]. The physical interpretation is that a Dirac material can jqj ≪ ω in the relevant limit of . For absorption of DM absorb an incoming particle with momentum transfer much ω particles, the deposited energy in the system is equal to the smaller than its mass, without the presence of additional q DM mass mχ, and the momentum transfer is equal to the particles (such as phonons). In other words, vertical v DM momentum mχ DM. Consequently, the momentum transitions from the valence band to the conduction band transfer is suppressed due to the virial velocity of the DM, are possible. This is in contrast to absorption in typical jqj ∼ 10−3ω ≪ ω Π ≈ Π ≡ Π . In this limit, L T . Using metals, where interband transitions can be neglected for Eq. (2.4), we can write the absorption rate for photons as ultralow energies and nonrelativistic absorption proceeds through emission of a phonon [55]—a process which is not h σ i ¼ ω ϵ ð Þ nT absvrel γ Im r: 4:3 described by the polarization tensor of Eq. (2.8). ϵ α~ ¼ α κ Furthermore, because Im r scales as EMg= vF, one The sensitivity of a material to DM absorption is therefore might expect the absorption rate to increase with small obtained by relating the absorption process to that of ordinary Fermi velocity vF, enhanced degeneracy g, and small photons through the complex permittivity. background dielectric constant κ. This is indeed the case We focus on the case of a kinetically mixed dark photon, for absorption of a light scalar or pseudoscalar, which does as described by Eq. (2.1), which can be absorbed by a Dirac not feel in-medium effects. In the case of the kinetically material. Such a dark photon can comprise all of the DM, mixed dark photon, Eq. (4.4) shows that the effective in- with its relic abundance set via a misalignment mechanism medium mixing angle between the dark photon and the [102,103] or gravitational production [104]. The effective photon involves both real and imaginary parts of Π, leading mixing angle between the dark photon and the photon for to a more complicated dependence on the Dirac material ε2 the case of absorption of nonrelativistic DM in the target is parameters. In Fig. 5 we show the combination eff × given by [73,101] ImðϵrÞ for a Dirac semimetal, which is proportional to the dark photon absorption rate, for two values of the mass, 2 4 ε m 0 0 ¼ 10 κ ε2 ¼ A ð Þ mA and 100 eV. The left panel fixes =g and varies eff 2 2 2 ; 4:4 ½ 0 − Πð 0 Þ þ½ Πð 0 Þ v , while the right panel fixes v and varies κ=g. The mA Re mA Im mA F F optimal Fermi velocity value is mass dependent and is of and so the rate of absorption is order 10−4 or smaller, while the optimal κ=g is Oð10Þ for
FIG. 5. Left: Scaling of the dark photon absorption rate with vF for fixed κ=g. Right: Scaling of the dark photon absorption rate with κ ε2 ðϵ Þ =g for fixed vF. Note that the absorption rate is proportional to eff ×Im r . In both cases we have considered a gapless isotropic Dirac semimetal with Λ ¼ 0.2 keV for two values of the dark photon mass, 10 eV (blue) and 100 eV (orange). The dashed line marks where the effective coupling becomes strong, α~ ¼ 1.
015004-11 YONIT HOCHBERG et al. PHYS. REV. D 97, 015004 (2018)
silicon semiconductors, as obtained in Refs. [51,55]. Stellar emission constraints [101,105] are shown in shaded gray. Note that we do not show the projected reach for magnetic bubble chambers [106] as they cannot be directly compared without full treatment of in-medium effects in those systems. We find that as in the case of scattering, in- medium effects suppress the response of superconductors compared to Dirac materials. We learn that Dirac materials are excellent target materials for absorption of dark photon DM, with projected reach exceeding all current proposals when 2Δ 015004-12 DETECTION OF SUB-MEV DARK MATTER WITH THREE- … PHYS. REV. D 97, 015004 (2018) As with any new detection technology, many hurdles Foundation and the Cottrell Scholar Program through the must be overcome to translate sensitivity estimates into a Research Corporation for Science Advancement. K. M. Z. is feasible experimental implementation. In a forthcoming supported by the DOE under Contract No. DE-AC02- paper, we plan to consider these issues in depth and present 05CH11231. A. G. G. was supported by the Marie Curie a detailed experimental configuration for detection of Programme under EC Grant No. 653846. S. M. G., Z.-F. L., meV-scale DM-induced excitations in Dirac materials. S. F. W. and J. B. N. are supported by the Laboratory We outline the key ingredients here: Directed Research and Development Program at the (i) Signal: Detection of single-electron excitations in Lawrence Berkeley National Laboratory under Contract semiconductors is a burgeoning field, recently dem- No. DE-AC02-05CH11231. This work is also supported by onstrated convincingly for the case of silicon in the Molecular Foundry through the DOE, Office of Basic Ref. [108]. Moreover, detection of meV athermal Energy Sciences under the same contract number. S. F. W. is phonons and quasiparticles has been proposed in supported in part by a NDSEG fellowship. This research Refs. [53,54], where the detection scheme takes used resources of the National Energy Research Scientific advantage of long excitation lifetimes made possible Computing Center, which is supported by the Office of by ultrapure materials such as superconducting alu- Science of the U.S. Department of Energy. This work minum. Ultrapure semimetals with ∼mm carrier mean was performed in part at the Aspen Center for Physics, free paths have recently been synthesized [109], and which is supported by National Science Foundation Grant our candidate material ZrTe5 can, due to its quasi-two- No. PHY-1607611. dimensional nature, be fabricated as thin flakes with sub-mm thickness [110]. A detection scheme utilizing APPENDIX A: TRANSITION FORM FACTOR the philosophy of excitation concentration should thus be feasible. Alternatively, the nontrivial elec- This appendix includes the details behind the analytical tronic properties of Dirac (and Weyl) materials could form of the transition form factor Eq. (3.6), the relation be exploited for detection of low-energy deposits. between the transition form factor and the q2 dependence of (ii) Backgrounds: For a kg detector with a 30 meV gap, the complex permittivity, and the relation to previous work thermal electron/hole excitations can be suppressed [88] on electron scatterings with generic band structure. to a negligible rate at liquid helium temperatures, T ¼ 2.7 K. As with any low-threshold experiment, 1. Derivation of the transition form factor there will also be a background spectrum extending The Dirac Hamiltonian Eq. (1.1) in a block off-diagonal to very low energies from sources such as radio- form reads active impurities, neutrons, and alpha particles. However, the strong dependence of the excitation 0 ℓ~ · σ − iΔ rate on the Fermi velocity can help reduce back- Hℓ ¼ ; ðA1Þ ℓ~ σ þ Δ 0 grounds, since interband excitations from slow- · i moving neutrons or alpha particles with speeds less where an anisotropic Fermi velocity is allowed by defining than vF are always forbidden. the rescaled momentum Sub-MeV dark matter is a viable theoretical and exper- ~ imental possibility, posing interesting challenges to both ℓ ¼ðvF;xℓx;vF;yℓy;vF;zℓzÞ: ðA2Þ theory and experiment. Many semimetal candidates have been produced and are being discovered in the laboratory The normalized eigenstates can be written as [62], which increases the likelihood of finding ideal materi- 0 1 ∓ ℓ~ als for DM detection. The DM direct detection community is B − C pursuing a robust suite of approaches for sub-GeV DM [89], 1 B ð Δ þ ℓ~ Þ C ℓ ¼ pffiffiffi B i z C and we look forward to Dirac materials joining the hunt. u1;3 B C; 2Eℓ @ 0 A Eℓ ACKNOWLEDGEMENTS 0 1 ðiΔ − ℓ~ Þ We thank Jens H. Bardarson, Ilya Belopolski, Rouven B z C Essig, Snir Gazit, Zahid Hasan, Pablo Jarillo-Herrero, and 1 B ∓ ℓ~ C ℓ ¼ pffiffiffi B þ C ð Þ Zohar Ringel for useful conversations. Y. H. is supported by u2;4 B C; A3 2Eℓ @ A the U.S. National Science Foundation, Grant No. NSF-PHY- Eℓ 1419008, the LHC Theory Initiative, by the Israel Science 0 Foundation, by the Binational Science Foundation, by the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ~ ~ ~ 2 2 I-CORE Program of the Planning Budgeting Committee and where ℓ ¼ ℓx iℓy and Eℓ ¼ ℓ þ Δ . The upper by the Azrieli Foundation. M. L. is supported by the DOE (lower) signs correspond to negative (positive) energy λ under Contract No. DESC0007968, the Alfred P. Sloan solutions Eℓ ¼ λEℓ where λ ¼∓ 1. 015004-13 YONIT HOCHBERG et al. PHYS. REV. D 97, 015004 (2018) The transition form factor appears in the expression for formally defined usingP the eigenvectors Eqs. (A3), i.e. þ ¼jℓ þihℓ þj ¼ j ℓ ih ℓ j the polarization function, which can be written Pℓ ; ; i¼1;2 ui ui with an analogous definition for P−. As is standard in many-body physics Πðω qÞ ℓ ; [76,77], the polarization function results after integrating Z 0 3 X ð λ Þ − ð λ Þ over the frequency domain and taking the trace of the g d ℓ fFD Eℓ þq fFD Eℓ 2 ¼ lim 0 jfλ ℓ →λ0 ℓ þqj : ’ η→0 3 λ λ ; ; product of two Green s functions given by Eq. (A5) V ð2πÞ 0 E − E − ω − iη 0 λ;λ ℓ þq ℓ evaluated at different momenta ℓ and ℓ ¼ ℓ þ q. After ðA4Þ evaluating the frequency integral, we are left with the explicit kernel of Eq. (A4), a factor of 1=2 and the trace Here, V is the crystal volume, g ¼ 2 is the spin degeneracy, over the product of two projectors. The product of the latter and fFD is the Fermi-Dirac distribution, which is just a step two objects defines the transition form factor as function at zero temperature. The polarization function can also be defined as the product of two Green’s functions 2 1 λ λ0 1 0 0 0 0 j 0 0 j ¼ ½ ¼ ½jℓ λihℓ λjjℓ λ ihℓ λ j [76,77], and calculating the polarization function using fλ;ℓ →λ ;ℓ 2 Tr Pℓ Pℓ 0 2 Tr ; ; ; ; Green’s functions and matching to the form of Eq. (A4) 0 1 λλ 0 allows one to extract the transition form factor. In the ¼ 1 þ ðℓ~ · ℓ~ þ Δ2Þ : ðA7Þ diagonal basis, the Green’s function is given by 2 Eℓ Eℓ 0 X 1 λ Gω ℓ ¼ P ð Þ It is worth noting that the overlap factor for Dirac ; ω − λ ℓ A5 λ¼ i Eℓ systems can be related to standard completeness relations over spinors, which are perhaps more familiar in the high- in terms of the projection operator: energy theory context. 0 1 λ ~ 1 1 ðℓ · σ −iΔÞ λ B Eℓ C 2. Relation between permittivity and Pℓ ¼jℓ ;λihℓ ;λj¼ @ A; 2 λ ðℓ~ · σ þ iΔÞ 1 transition form factor Eℓ The complex permittivity of a material is given in general ð Þ A6 by the Lindhard formula [87]: 2 Z 3 X ð 0 Þ − ð Þ 1 e 1 d k fFD Ekþq;n fFD Ek;n 2 ϵ ðω qÞ¼1 − j 0 j ð Þ r ; lim 2 3 gs;n fn;k→n ;kþq : A8 η→0 V κ q ð2πÞ Ekþq 0 − Ek − ω − iη BZ n;n0 ;n ;n Here, κ is the background dielectric constant, n and n0 are We have written Eq. (A9) in a form resembling Eq. (A8) to band indices, gs;n is the spin degeneracy of band n, Ek;n is illustrate that the interband transitions yield a form factor the energy of the nth band at lattice momentum k, and the that scales as q2 (to leading order), which multiplies the integral is taken over the first BZ. The Fermi-Dirac factors Fourier transform of the Coulomb potential e2=ðκq2Þ. This in the numerator ensure that at zero temperature, the only behavior is a direct consequence of the orthogonality transitions which contribute to ϵr are from unoccupied to of the valence and conduction bands, which implies that occupied states and vice versa. Using Eq. (A7) and the transition form factor Eq. (3.5) vanishes at q ¼ 0. This performing the momentum integral in Eq. (A8) with a ~ remains true for nonzero Δ. Indeed, defining Δ ¼ Δ=vF cutoff Λ, considering a single Dirac cone with valence and and expanding Eq. (A7) in small q yields conduction bands n ¼ − and n0 ¼þ only, yields the dielectric constant for an ideal Dirac material. This is 1 q2 ðℓ qÞ2 equivalent to evaluating Eq. (A4) and using the relationship 2 · 4 jf−;ℓ →þ;ℓ þqðqÞj ¼ − þ Oðq Þ: between the polarization function and the permittivity [87]. 4 ℓ 2 þ Δ~ 2 ðℓ 2 þ Δ~ 2Þ2 Δ ¼ 0 The integral can be performed analytically for , ðA10Þ yielding Eq. (2.7), which we repeat here for convenience: 2 1 4Λ2 This derivation illustrates an alternative perspective on ðϵ Þ ¼ 1 − e g −q2 r semimetal 24π2κ q2 ln ω2 2 − q2 why the dark photon does not develop an effective mass in- vF =vF 2 medium in a Dirac material: the vanishing of jf−;ℓ →þ;ℓ þqj 2 as q → 0 in Eq. (A8) ensures that ϵ is constant as q → 0, − iπq Θðω − vFjqjÞ : ðA9Þ r or equivalently ΠðqÞ ∼ q2. 015004-14 DETECTION OF SUB-MEV DARK MATTER WITH THREE- … PHYS. REV. D 97, 015004 (2018) 3. Lattice momentum conservation and comparison to If m ≠ 0 we have jG0j ≳ keV. In a Dirac material, formalism for generic band structure transitions near the Dirac point satisfy k0 − k ¼ ℓ 0 − ℓ with jℓ j;jℓ 0j ≪ jG0j by assumption. Thus jqj ∼ jG0j ≳ keV. Note that Eq. (A7) is written only as a function of initial- jqj Referring to Eq. (3.4), v ≥ ≳ 10−2 ≫ v for and final-state momenta. To make contact with the for- min 2mχ DM malism of Ref. [88], we will also write it as a function of the mχ ≲ 100 keV. In other words, scattering is kinematically q 0 momentum transfer by inserting unity in the form impossible for mχ ≲ 100 keV unless G ¼ 0.Evenif 3 ð2πÞ δðq − ðℓ 0 − ℓ ÞÞ G0 ≠ 0 V , where V is the volume of the crystal: scattering is kinematically allowed for , we will be primarily concerned with form factors which scale as ð2πÞ3 2 5 2 0 F ðqÞ ∼ 1=q 1=jqj jfλ ℓ →λ0 ℓ 0 ðqÞj ¼ δðq − ðℓ − ℓ ÞÞ DM , so that the rate (A13) scales as . This ; ; V G0 ≠ 0 represents an enormous suppression when of the ~ ~ 0 2 ð Þ5 ≃ 10−15 1 0 ℓ · ℓ þ Δ order of eV=keV . Thus in our kinematic × 1 þ λλ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; G0 ≠ 0 2 ℓ~ 2 þ Δ2 ℓ~ 02 þ Δ2 regime, reciprocal vectors can be safely neglected. When G0 ¼ 0, the sum in Eq. (A15) collapses to a single ðA11Þ term, and we can identify which reduces to Eq. (3.6) in the gapless isotropic limit. ~ ~ 0 2 2 1 0 ℓ · ℓ þ Δ j 0 0 j ¼ 1 þ λλ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The delta function in Eq. (A11) enforces exact momen- f½ðn¼−Þk;ðn ¼þÞk ;0 : 2 ℓ~ 2 þ Δ2 ℓ~ 02 þ Δ2 tum conservation, while typically in problems involving condensed matter systems, momentum is only conserved ðA16Þ up to addition of a reciprocal lattice vector. We can justify the exact momentum conservation for the cases relevant to The single delta function in Eq. (A15) now enforces q ¼ ℓ 0 − ℓ DM scattering by using kinematic arguments. A convenient , establishing that in our kinematic regime, q parametrization of a general wave function in a periodic the physical momentum transfer is equal to the difference system is as a linear combination of Bloch waves, in lattice momenta between initial and final states. 1 X pffiffiffiffi iðkþGÞ·x APPENDIX B: MODIFICATIONS FOR Ψn;kðxÞ¼ unðk þ GÞe ; ðA12Þ V G ANISOTROPIC DIRAC MATERIALS In this appendix we discuss modifications to our analysis where G runs over all reciprocal lattice vectors and the un are complex numbers. The velocity- and directionally for scattering of light DM in Dirac materials for the case of ≠ ≠ averaged scattering rate for a single electron in the anisotropic materials, with vF;x vF;y vF;z. Bloch basis is [88] 1. Anisotropic permittivity ρ π2σ¯ 1 X 1 χ e For anisotropic Dirac materials, one may make a change R k→ 0 k0 ¼ ηðv ðjqj; ωkk0 ÞÞ n; n ; 2 G0 jqj min mχ μχe V of variables in the integrand of Eq. (A8) and evaluate the 2 2 permittivity at a correspondingly rescaled value of the × jF ðqÞj jf½ k 0k0 G0 j jq¼k0−kþG0 ; ðA13Þ DM n ;n ; momentum [79,111]: where the crystal form factor is q~ 2 1 ðϵ Þan: ¼ 1 − ð1 − ϵisoðq~Þj Þ ð Þ X r Dirac 2 r v ¼1 : B1 0 0 q v v v F 0 0 0 ¼ ðk þ G þ G Þ ðk þ GÞ ð Þ F;x F;y F;z f½nk;n k ;G un0 un ; A14 G ~ ~ Here, q is defined as in Eq. (A2), q ¼ðvF;xqx;vF;yqy;vF;zqzÞ, and on the right-hand side the isotropic form factor is which is related to the transition form factor as defined in evaluated for v ¼ 1 and at the rescaled momentum q~. Eq. (3.5) by F For example, in the gapless case, Eq. (2.7) is modified to 2 j 0 0 ðqÞj fn;k→n ;k 1 e2g ðϵ Þan ¼ 1 − X 3 r semimetal 2 2 ð2πÞ 0 0 2 q 24π κv v v ¼ δðq − ðk − k þ G ÞÞj 0 0 0 j ð Þ F;x F;y F;z f½nk;n k ;G : A15 G0 V 4Λ~ 2 −q~ 2 j j − πq~ 2Θðω − jq~jÞ × ln ω2 − q~ 2 i : In the Bloch wave basis, orthogonality of different Fourier components leads to lattice momentum conserva- ðB2Þ tion q ¼ k0 − k þ G0. Here, G0 is a multiple of a reciprocal 0 ~ lattice vector whose size jG j is either 0 or ∼2πm=a for The cutoff Λ must also be rescaled: we choose Λ ¼ Λ × 0 1=3 integers m> , where a is the lattice spacing. A typical maxðvF;x;vF;y;vF;zÞ rather than e.g., Λ × ðvF;xvF;yvF;zÞ lattice will have a ∼ 1–10 Å, so 2π=a ∼ keV. to recover the correct scaling when one of the vF;i is much 015004-15 YONIT HOCHBERG et al. PHYS. REV. D 97, 015004 (2018) smaller than the other two, as is typically the case with real where materials. In addition to the anisotropy of the band structure, the 3 ϵ κ~ ¼ : ð Þ crystal lattice itself may be anisotropic, in which case r is 1 κ þ 1 κ þ 1 κ B5 ðϵ Þ = xx = yy = zz more properly described by a full tensor r ij. In this situation, Eq. (A8) should be interpreted as a tensor equation. In the basis of principal components where the background Therefore, in our analysis of scattering in ZrTe5 where κ ðϵ Þ κ dielectric tensor ij is diagonal, r ij is also diagonal. In the ij is anisotropic, we compute spherically symmetric rates gapless case its diagonal components are given by using κ~. 1 2 ðϵ Þ ¼ 1 − e g 2. Scattering in anisotropic Dirac materials r ii 2 2 q 24π κiivF;xvF;yvF;z The impact of anisotropic dispersions on the DM scatter- 4Λ~ 2 ing rate, Eqs. (3.13) and (3.14), can be estimated in a −q~ 2 − iπq~ 2Θðω − jq~jÞ ; ð Þ × ln ω2 − q~ 2 B3 straightforward manner. In typical Dirac materials, the anisotropy of the Dirac cones often involves a hierarchy ≪ ≃ ≡ with straightforward modifications for the gapped case. of Fermi velocities vF;z vF;x;vF;y, where vF;x vF;y Strictly speaking, the formalism of Sec. II does not apply vF;⊥. In this limit, Eq. (B2) becomes because longitudinal and transverse modes are not decoupled in anisotropic media. However, for the case of scattering, q 2 2 4Λ2 vF≪v⊥ ⊥ e g ϵr ≈ 1 − − ln ΠL ≫ ΠT and the dominant effects are still given by ΠL. 2 2 2 2 κq v 24π ω =v ⊥ − q⊥ Assuming a spherically symmetric velocity distribution, the F;z F; leading effects of the anisotropic tensor ðϵ Þ can be captured ig r ij − Θðω − v ⊥jq⊥jÞ ; ðB6Þ 1 24π F; by its rotationally invariant component 3 TrðϵrÞ: 2 1 1 where q⊥ ¼ðq ;q ; 0Þ. Following the arguments of ðϵ Þ¼1 − e g x y 3 Tr r 2 2 Sec. III B, the total scattering rate will then be proportional q 24π κ~vF;xvF;yvF;z to v2 , the smallest of the Fermi velocities. However, v , 4Λ~ 2 F;z min −q~ 2 − πq~ 2Θðω − jq~jÞ ð Þ × ln 2 2 i ; B4 which controls the behavior of the integral as a function of the ω − q~ Fermi velocities, now takes the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðℓ þ qÞ2 þ 2 ðℓ þ Þ2 þ 2 ℓ 2 þ 2 ℓ2 v ⊥ ⊥ v z qz v ⊥ ⊥ v z jqj ðjqj ω Þ¼ F; F;z F; F;z þ vmin ; ℓ ;ℓ þq jqj 2mχ jℓ þ q jþjℓ j jqj v2 ¼ ⊥ ⊥ ⊥ þ þ O F;z ð Þ vF;⊥ jqj 2 2 : B7 mχ vF;⊥ Here, the argument of Sec. III B that DM scattering is see that we need vDM >vF;z for scattering to occur. As a allowed only when vF Z Z Z Λ 1 2 qmax jℓ j jqj ηðv ðjqj; ωℓ ℓ þqÞÞ ℓ ðℓ þ qÞ ð Λ Þ¼ jℓ j θ jqj min ; 1 − · ð Þ I vF; ;mχ d d cos ℓq d 2 2 2 2 4Λ02 2 : B8 0 −1 0 ðω − q Þ j jþπ jℓ jjℓ þ qj ℓ ;ℓ þq ln ω2 2 −q2 ℓ ;ℓ þq=vF qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ −jℓ j θ þ Λ2 − ℓ 2ð1 − 2θ Þ jℓ þ qj Λ The limit of integration qmax cos ℓq cos ℓq ensures < . We now define a generalized anisotropic rate integral 015004-16 DETECTION OF SUB-MEV DARK MATTER WITH THREE- … PHYS. REV. D 97, 015004 (2018) Z 4 2 1 1 ηð ðjqj ω ÞÞ ℓ~ ðℓ~ þ q~Þ ~ e g 3 3 vmin ; ℓ ;ℓ þq · Iðv⃗; Λ;mχÞ¼ d qd ℓ 1 − ; ðB9Þ F 2304π6 jqj ðω2 − q2Þ2 jϵanðω qÞj2 ~ ~ ℓ ;ℓ þq r ℓ ;ℓ þq; jℓ jjℓ þ q~j an where v⃗F ¼ðvF;x;vF;y;vF;zÞ; ϵr is defined in Eq. (B2); APPENDIX C: REACH FOR OTHER MODELS q~ ℓ~ and the rescaled momenta ; are defined as in Eq. (A2). In this appendix we consider the cases of scattering ~ This is related to the isotropic rate integral (B8) by I ¼ through a mediator ϕ with no in-medium interactions (such 2 κ v2 I in the isotropic case v⃗ ¼ðv ;v ;v Þ. In Fig. 7,we as a scalar), as well as the case of a heavy kinetically mixed g F F F F F 0 “ ” 0 ≫ ~ mediator A , where heavy means mA ;ϕ keV. We also plot Iðv⃗; Λ;mχÞ for mχ ¼ 10 keV, Λ ¼ 0.2 keV, and F consider the case of absorption of pseudoscalar dark v ¼ v ¼ 10−3 as a function of v . As anticipated, F;x F;y F;z matter (ALPs). the shape of the two curves is qualitatively similar for vF;z ≪ vF;x=y, with both curves scaling similarly at small 1. Scattering reach for other mediator models vF;z. However, the rate is suppressed by about an order of magnitude in the anisotropic case, showing that isotropic The form factors and fiducial cross sections for light Dirac materials are preferred for scattering. scalar mediators, heavy scalar mediators, and heavy kineti- cally mixed mediators take the following form: q2 ϕ;light∶ F ðqÞ¼ 0 ; F ðqÞ¼1; DM q2 med 16πμ2 ϵ2α α σ¯ ¼ χe EM D ð 2 ¼ðα Þ2Þ; e 4 q0 EMme q0 ðC1Þ ϕ ∶ ð Þ¼1 F ð Þ¼1 ; heavy FDM q ; med q ; 16πμ2 ϵ2α α σ¯ ¼ χe EM D ; ð Þ e 4 C2 mϕ 0 1 ¼ 10 A ; heavy∶ F ðqÞ¼1; F ðqÞ¼ ; FIG. 7. Scaling of the DM scattering rate for mχ keV DM med ϵ ðqÞ with the Fermi velocity v of a gapless Dirac material, r F;z 2 2 comparing an isotropic dispersion v ¼ v ¼ v to one with 16πμχ ϵ α α F;x F;y F;z σ¯ ¼ e EM D : ðC3Þ v ¼ v ¼ 10−3. e 4 F;x F;y mA0 FIG. 8. Projected scattering reach for a light (left) and heavy (right) mediator ϕ without in-medium effects. Such models are subject to strong constraints; see text for discussion. We show the expected background-free 95% C.L. sensitivity (3.0 events) that can be obtained with 1 kg-yr exposure. Dirac material parameters are the same as in Fig. 4. 015004-17 YONIT HOCHBERG et al. PHYS. REV. D 97, 015004 (2018) FIG. 9. Projected scattering reach for a heavy kinetically mixed FIG. 10. Projected reach for absorption of ALPs in Dirac mediator A0 including in-medium effects. Such models are materials, given in terms of the ALP-electron coupling g . subject to strong constraints; see text for discussion. We show aee We show the expected background-free 95% C.L. sensitivity (3.0 the expected background-free 95% C.L. sensitivity (3.0 events) events) that can be obtained with 1 kg-yr exposure. The green that can be obtained with 1 kg-yr exposure. Dirac material (purple) curves are gapless (gapped) isotropic Dirac materials parameters are the same as in Fig. 4. 3 with ρT ¼ 10 g=cm and all other parameters as in Fig. 4. We cut off the plot at mA0 ¼ 2ΛvF ¼ 160 meV, the largest energy deposit consistent with the linear dispersion relation with Λ ¼ 0 2 Note that since ϵrðqÞ is roughly constant from Eq. (2.7), the momentum cutoff . keV. We also show the reach of in-medium form factors for the two heavy mediators are superconductors with a 1 meV threshold [55] (black) as well as roughly proportional, with in-medium effects providing an constraints from Xenon100 [115] (shaded gray) and white dwarfs order-1 suppression. [116] (shaded blue), and the QCD axion region (shaded red). As shown in Fig. 8, Dirac materials have inferior reach to superconductors for mediators which are not kinetically temperatures below an MeV [112].10 In the massless mixed. Since in these models, the mediator does not acquire mediator case, stellar constraints on the emission of light a large in-medium mass in superconductors, the larger mediators imply the couplings to electrons are generally too phase space of superconductors dominates, especially for small to be detectable [101]. (The exception is a light vector the light scalar where smaller momentum transfers are particle whose mass is given by a Stueckelberg mechanism; favored. The reach of Dirac materials compared to super- this is the benchmark model utilized in Fig. 4.) These conductors is slightly better for a heavy mediator than for a constraints are reviewed in Ref. [54]. light mediator; the reason is that the phasep spaceffiffiffiffi volume for ω3 ω a semimetal grows as compared to for a metal, 2. Absorption reach for axionlike particles allowing much of the phase space suppression to be made up at larger energy transfers. The weakening reach of Dirac An ALP of mass ma which comprises DM can couple to materials at masses mχ ≳ 200 keV for the heavy mediator electrons via the operator is due to the phase space cutoff at Λ ¼ 0.2 keV. On the L ⊃ gaee ð∂ Þ¯γμγ5 ð Þ other hand, as shown in Fig. 9, Dirac materials have μa e e: C4 2me superior reach for the heavy kinetically mixed mediator, because the part of the in-medium polarization which scales The absorption of an ALP on electrons through this as q2 still suppresses the effective dark photon coupling operator is related to the photon absorption rate and is significantly in metals. given by (see, e.g., Refs. [51,52,55]) While the DM masses and cross sections are too small to 1 3 2 2 a ma gaee be constrained by current direct detection experiments, R ¼ ρχ Imϵ : ðC5Þ abs ρ 4 2 2 r these models are in strong tension with astrophysical and T me e cosmological constraints which must be evaded, at least for The projected reach of Dirac materials for ALP absorption is the most naive of models. In the massive mediator case, shown in Fig. 10, assuming the ALP comprises all of the DM, detectable cross sections imply thermalization of the DM sector, including both the mediator and the DM; this, 10For sufficiently large cross sections, multiple scattering in the however, is in tension with big bang nucleosynthesis, Earth may either prevent the DM from reaching the detector [113] or which requires at the 2σ level that only one real scalar, cause excessive heating in the Earth [114]. However, the relevant in addition to the Standard Model, can be thermalized at cross sections are much larger than those depicted in our plots. 015004-18 DETECTION OF SUB-MEV DARK MATTER WITH THREE- … PHYS. REV. D 97, 015004 (2018) 1 (a) (b) No spin-orbit Spin-orbit 0.5 0 (eV) F E-E -0.5 Te Zr -1 -1.5 Y ZTS R FIG. 11. (a) ZrTe5 in the Cmcm space group. (b) Calculated electronic band structure for ZrTe5 with and without spin-orbit coupling. The Fermi level is set to 0 eV and marked by the dashed line. and for the same parameters as Fig. 4. The reach of super- eightfold-coordinated by Te atoms, which occupy three conductors is shown for comparison [55], along with the inequivalent lattice sites. The precise nature of the topo- parameter space for the QCD axion (shaded red). Constraints logical character of the ZrTe5 electronic structure has been from Xenon100 data [115] (shaded gray) and white dwarf controversial, with several conflicting experiments conclud- cooling [116] (shaded blue) apply, and rule out the parameter ing it to be a Dirac semimetal [96,110,123–126], a topo- space that can be probed even by an ideal gapless Dirac logical insulator [127–132], and a normal semiconductor material in the mass range of interest. As expected, we learn [133]. Our first-principles PBE calculations of the electronic that for absorption of axionlike particles, superconductors band structure show a Dirac cone near Γ without spin-orbit have superior reach due to the absence of in-medium effects and larger phase space density of target electrons. TABLE I. Material parameters for ZrTe5. vF;i (i ¼ 1, 2, 3) are Fermi velocities; 2Δ is the gap, Λ is the linear dispersion cutoff; g ¼ gsgC is the product of spin and Dirac cone degeneracies; κii APPENDIX D: BAND STRUCTURE (i ¼ 1, 2, 3) are principal components of the background CALCULATIONS FOR ZrTe5 dielectric tensor; ρT is the density; ne is the mass density of Dirac Among already-synthesized Dirac materials appropriate valence-band electrons, and Vuc is the unit cell volume. Where no experimental value is listed, we use the theoretical value. The for detector targets, we identified ZrTe5 as a strong candi- date, having a linear dispersion near the Fermi level while theoretical values of the Fermi velocities were calculated along the high-symmetry directions, while the experimental values are mid- being slightly gapped by the spin-orbit interaction. plane velocities. For the experimental value of 2Δ, we take the First-principles calculations based on density functional median of the range of values presented in [97]. Λ was taken to be theory (DFT) are performed using the projector augmented the distance between the Γ and Z points in the BZ; see Figs. 11 wave (PAW) method in the Vienna ab initio Simulation and 12. The static ion-clamped dielectric tensor κij was calculated Package (VASP) [117,118] code. Zr(4s, 4p, 5s, 4d); Te(5s, using density functional perturbation theory. The unit cell is 5p); Se(4s, 4p); Nb(4p, 5s, 4d); and Ta(5p, 6s, 5d) electrons defined as containing four formula units; see Fig. 11(a). were treated as valence electrons, and the wave functions of the system were expanded in plane waves to an energy Parameter Value (th) Value (exp) −3 −3 cutoff of 600 eV. Monkhorst-Pack [119] k-point grids of vF;1 2.9 × 10 cðvF;xÞ 1.3 × 10 cðvF;xyÞ [96] −4 −4 14 × 14 × 4 were used for BZ sampling. We performed vF;2 5.0 × 10 cðvF;yÞ 6.5 × 10 cðvF;yzÞ [96] −3 −3 calculations with the generalized gradient approximation vF;3 2.1 × 10 cðvF;zÞ 1.6 × 10 cðvF;xzÞ [96] (GGA) using the Perdew-Burke-Ernzerhof (PBE) func- 2Δ (meV) 35 23.5 [97] tional [120]. Spin-orbit (SO) interactions are included self- Λ (keV) 0.14 consistently in all calculations. Our calculations on ZrTe5 g 4 κ were performed using experimentally determined lattice xx 187.5 κ parameters and internal coordinates [121]; our structural yy 9.8 κzz 90.9 relaxations of ZrSe5 was performed including DFT-D3 van 3 ρT (g=cm ) 6.1 der Waals corrections [122]. − 23 ne (e =kg) 8.3 × 10 ZrTe5 crystallizes in the Cmcm structure (Space V 3 795 Group No. 63) as shown in Fig. 11(a). Each Zr ion is uc (Å ) 015004-19 YONIT HOCHBERG et al. PHYS. REV. D 97, 015004 (2018) 1 1 1 (a) (b)Nb d-states (c) 0.5 0.5 0.5 0 0 0 (eV) (eV) (eV) F F F E-E E-E -0.5 -0.5 E-E -0.5 -1 -1 -1 -1.5 -1.5 -1.5 Y ZTS R Y ZT S R Y ZTS R FIG. 12. Calculated band structure for (a) stoichiometric ZrTe5 with 99% the lattice volume of the experimental lattice parameters, (b) ZrTe5 with 12.5% Nb substitution on the Zr site and (c) ZrSe5. In (b), the Nb d-states near the Fermi level are indicated by the weighted line. In each plot the Fermi level is set to 0 eV and marked by the dashed line. coupling which is then slightly gapped (to 35 meV) with the and Ta. We calculate the band structure of substitution of inclusion of the spin-orbit interaction, consistent with one Nb/Ta for eight formula units, resulting in electron previous DFT calculations [134,135].Wenotethatalthough doping of 0.25 electrons per formula unit as shown in DFT-GGA-SO is not expected to be qualitative for the band Fig. 12(b) for the Nb case. While the Fermi level shifts as gap, our calculations are very consistent with previous expected, Nb contributes d-states near the Fermi level, experimental findings [97,128,129]. Table I lists the making the material a metal. The same also occurs for the material parameters we use to calculate DM scattering case of Ta substitution. Substitution of Te with Br alters the rates, with the theoretical values derived from the DFT band structure near the Fermi level as well. calculations, and the experimental values used from the In light of this, and with the additional motivation of references given. If no experimental value is listed, we use reducing the band gap, we consider replacing Te with Se in the theoretical value. Our estimate of Λ was derived from the hypothetical new compound ZrSe5 in the same Cmcm the distance between the Γ and Z points in the BZ. structure as shown in Fig. 12(c). This chemical substitution While the band structure shows the gapping of the Dirac has three effects on the electronic properties of the material. cone near Γ, the Fermi level cuts the top of the band to form Firstly, the smaller ionic radius of Se reduces the total a holelike pocket. To engineer a semiconducting band volume of the compound which results in a Fermi level in structure, with the Fermi level in the gap, we recompute the the gap without any external pressure; however, this also band structure of electron-doped ZrTe5 by adding a small has the undesired effect of increasing the band gap. fraction of electrons per unit cell and compensating this Independent of the volume change, the lower spin-orbit additional electron density with a uniform positive back- coupling in Se reduces the spin-orbit splitting of the bands ground. We find that electron doping by 0.2 electrons per to 2Δ ≃ 15 meV. Therefore, our DFT estimates suggest unit cell shifts the Fermi level into the gap. 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