Microwave Photon Number Resolving Detector Using the Topological Surface State of Superconducting Cadmium Arsenide
MMicrowave Photon Number Resolving Detector Using the Topological Surface State ofSuperconducting Cadmium Arsenide
Eric Chatterjee, Wei Pan, and Daniel Soh Sandia National Laboratories, Livermore, California 94550, USA (Dated: September 7, 2020)Photon number resolving detectors play a central role in quantum optics. A key challenge inresolving the number of absorbed photons in the microwave frequency range is finding a suitablematerial that provides not only an appropriate band structure for absorbing low-energy photonsbut also a means of detecting a discrete photoelectron excitation. To this end, we propose tomeasure the temperature gain after absorbing a photon using superconducting cadmium arsenide(Cd As ) with a topological semimetallic surface state as the detector. The surface electrons absorbthe incoming photons and then transfer the excess energy via heat to the superconducting bulk’sphonon modes. The temperature gain can be determined by measuring the change in the zero-biasbulk resistivity, which does not significantly affect the lattice dynamics. Moreover, the obtainedtemperature gain scales discretely with the number of absorbed photons, enabling a photon-numberresolving function. Here, we will calculate the temperature increase as a function of the number andfrequency of photons absorbed. We will also derive the timescale for the heat transfer process fromthe surface electrons to the bulk phonons. We will specifically show that the transfer processes arefast enough to ignore heat dissipation loss. I. INTRODUCTION
Photon number resolving detectors have been exploredsignificantly over the past decades [1–3] due to the direneed for resolving the number of photons in applica-tions such as the security of quantum communications[4, 5] and the sensitivity of quantum sensing [6]. Asphoton-based quantum computing advances, precise res-olution of photon number detection is increasingly im-portant. Microwave photons are the backbone of prolifictransmon quantum computation, and therefore, detec-tion of the microwave photons is tremendously impor-tant in the current quantum computing paradigm [7].Several non-number-resolving techniques to detect mi-crowave photons have been developed, including the cir-cuit QED technique [8], dressed-state superconductingquantum circuit [9], current-biased Josephson junction[10], and the dark-state detector [11]. It is well-knownthat building a parallel detection system, first splittingthe light path using beam splitters and then using non-number-resolving detectors in each parallel path, mayprovide a probabilistic photon number resolving detec-tion, which is further limited due to the loss associatedwith parallelization. In contrast, a single photon-numberresolving detector with a deterministic photon numberresolution would provide a immense advantage particu-larly in photonic quantum computers by reducing theerror-correcting overhead. To the best of our knowl-edge, a single-device photon-number resolving detectorthat can simultaneously detect multiple incoming pho-tons at microwave frequency has never been reported sofar.Here, we propose a photon-number resolving detec-tor operating at microwave frequencies, based on thetopological surface states of cadmium arsenide (Cd As ).Semimetals such as graphene provide an ideal detecting material for microwave photons due to their zero bandgap. Recently, Dirac and Weyl semimetals with Diraccone dispersion have gained prominence due to high mo-bility [12], along with the fact that they can be synthe-sized through conventional techniques [13–15]. Particu-larly, Cd As displays proximity-induced bulk supercon-ductivity at low temperatures, and the electronic struc-tures of the bulk and the topological surface states aredecoupled. Maintaining the Cd As semimetal materialat a very low temperature is necessary for an efficientphoton-induced electron excitation to a conduction bandjust above the Fermi level due to the low photon energy.The bulk state enters a superconducting state at a suffi-ciently low temperature, opening a band gap beyond themicrowave photon energy. Fortunately, the topologicalsurface state of Cd As is not affected by the tempera-ture, continuining to provide a gapless Dirac cone. Weuse this topological surface state as a photon absorber.Once the photon is absorbed, a rapid rethermalizationin band population occurs with a new elevated temper-ature corresponding to the absorbed photon energy. Wethen utilize the fact that the redistributed electron pop-ulation transfers its energy to the bulk’s phonon modesvia a surface electron-bulk phonon coupling, thus increas-ing the bulk’s temperature. The elevated bulk tempera-ture then reduces the conductance of the superconduct-ing bulk electron state, which is measured and used toeventually indicate the number of photons absorbed.The paper is organized as follows. In Sec. II, we brieflyreview the photon absorption in the topological surfacestate of Cd As . In Sec. III, we connect the event of pho-ton absorption in the topological surface state electronsto the bulk temperature increase via a two-step process,namely, the energy gain for surface electron modes and,then, the transfer of electron energy to the bulk phononmodes. Sec. IV resolves the important issue of the timescale of the temperature increase and shows that the ab- a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p FIG. 1: Basic layout for the Cd As photon numberresolving device. At low temperatures, the bulkbecomes superconducting, while the (112) surfaceretains a graphene-like dispersion. A photon (depictedby the green arrow) is absorbed by the surfaceelectrons. The change in bulk resistivity (measured bythe electrodes at zero bias) is used to determine thetemperature increase.sorbed photon energy indeed is most likely transferred tothe increase of the bulk temperature, rather than beinglost to radiative decay of the excited electrons. Sec. Vpresents the construction of a photon-number resolvingdetector based on the results of the previous sections.Numerical examples with realistic parameters build cred-ible real use cases of the proposed scheme. In the fi-nal section, we summarize our results and suggest thepath towards building a photon-number resolving detec-tor with near-unity efficiency on a chip. II. PHOTON ABSORPTION IN TOPOLOGICALSURFACE STATE
The setup for the device is depicted in Figure 1. Thesystem is based on a Cd As crystal inside a low temper-ature refrigerator with a baseline temperature below thebulk superconducting critical temperature. Recent ex-perimental findings have demonstrated that Cd As fea-tures a topological surface state on the (112) surface witha linear band crossing around the Dirac points [16, 17].This graphene-like surface state band structure at lowenergy can be attributed to the fact that sites consistingof stacked As and Cd atoms approximately form a hon-eycomb superlattice on the (112) surface [13]. As withgraphene, the dispersion relationship can be expressedas the following linear function of the wavevector k whenthe Fermi level is at the Dirac point: E c,v ( k ) = ± (cid:126) v F | k | , (1)where (cid:126) is the Planck’s constant, c and v represent theconduction and valence bands, respectively, and v F de-notes the Fermi velocity, which is approximately 10 m/s[16, 18]. At low temperatures, the surface state electronsare decoupled from the bulk state, and the bulk becomessuperconducting below 0.7 K while the surface retains FIG. 2: Band structures for the superconducting bulk(purple) and the topological surface state (blue). Notethat a gap of frequency f ∆ is opened in the bulk, whilethe surface state remains gapless, thus restricting theabsorption of photons at microwave frequency f p to thesurface state.its semimetallic property [19]. The superconducting gapfrequency f ∆ corresponding to this critical temperature T c can be determined using the standard BCS equation[20]: f ∆ = 3 . k B T c π (cid:126) = 50 GHz , (2)where k B is the Boltzmann constant. The resulting bandstructures for the superconducting bulk and the topolog-ical surface state are compared in Figure 2. Photonsof microwave frequency below 50 GHz will be absorbedsolely by the surface state, as desired.In order to derive the absorption probability, we con-sider the physical manifestation of the Dirac cone on thenature of the Bloch states. As in graphene, each elec-tronic state in the vicinity of a Dirac point can be concep-tualized as a massless Dirac fermion with a well-definedmomentum (cid:126) k , where k represents the wavevector of theelectronic state in the reciprocal space for which the Diracpoint is the origin. Therefore, the absorption coefficientfor Cd As will equal the well-known corresponding valuefor graphene [21]: A ( ω ) = e (cid:126) (cid:15) c f (cid:18) − (cid:126) ω (cid:19)(cid:18) − f (cid:18) (cid:126) ω (cid:19)(cid:19) , (3)where e is the charge of an electron, (cid:15) is the vacuumpermittivity, c the speed of light, and f ( E ) denotes theFermi-Dirac electron occupation probability at energy E .Note that the absorption coefficient (as a function of pho-ton frequency) is invariant with respect to the Fermi ve-locity. This is because the interband dipole matrix ele-ment (corresponding to the absorption probability for asingle electronic state) increases with the Fermi velocity,while the density of states decreases with it, eventuallycanceling each other’s effect. At high frequencies, the ab-sorption coefficient is approximately invariant with fre-quency, equaling a constant value of e / (4 (cid:126) (cid:15) c ) = 2 . (cid:126) ω (cid:47) k B T , the coefficient becomes attenuated, reach-ing a minimum value of one-fourth the high-frequencyabsorption as the Dirac cone is approached. III. TEMPERATURE INCREASE VS.ABSORBED PHOTON NUMBER
Having determined the probability that the topolog-ical surface state absorbs a photon from an incomingfield, our next step is to determine how the absorptionof a single photon increases the temperature of the 3Dbulk sample. When a photon excites an electron to theconduction band, a rapid rethermalization of the Fermi-Dirac distribution through electron-electron interactionensues, leading to an electron temperature above thelattice temperature. For undoped monolayer graphene,which features a band structure approximately identicalto that for the Cd As surface state, this process oc-curs in tens of picoseconds for cryogenic baseline tem-peratures [22]. The carrier rethermalization is followedby heat transfer from the collection of electrons to thelattice via electron-acoustic phonon interaction, until athermal equilibrium is reached between the electron tem-perature and the lattice temperature. Generally, the in-teraction between electrons and acoustic phonons is muchslower than the electron-electron interaction [23–25]. Af-terward, the Fermi-Dirac distribution for the electronbands and the Bose-Einstein distribution for the phononbranches will comply with the same temperature.We now determine the temperature increase due to theabsorption of a photon of frequency ω by calculating theportion of the imparted energy that is eventually con-verted to bulk lattice vibrations (i.e. the phonon modes)and to the surface electron modes, and by deriving theheat capacity of the two systems. We will make twoimportant assumptions here: firstly, that the electron-electron interaction rate dominates over the radiative lossrate of the electrons (which we will demonstrate in a latersection), and secondly, that the very low values for theinitial and final temperatures ensure that virtually all ofthe phonons are located in the low-energy linear parts ofthe acoustic branches, thus allowing for use of the Debyeapproximation [26] in determining the heat capacity. A. Energy Gain for Surface Electron Modes
We start by writing out an expression corresponding toenergy conservation in the system given the absorptionof a photon of frequency ω : (cid:126) ω = ∆ U el + ∆ U ph , (4) where ∆ U el and ∆ U ph represent the total energy gainedby the topological surface electron modes and the bulkphonon modes, respectively, at equilibrium. We focusfirst on the energy gain for the electron modes as a func-tion of electron temperature, as this will be necessaryfor calculating the initial electron temperature gain afterphoton absorption but prior to heat transfer to the lat-tice. The total electron energy with respect to the Fermisea is calculated by taking a sum of the conduction andvalence band energies weighted by the Fermi-Dirac occu-pation probabilities (multiplied by 2 to account for thefact that each spatial state contains 2 spin states): U el ( T ) = 2 (cid:88) k (cid:18) E c, k f ( E c, k , T ) − E v, k (cid:16) − f ( E v, k , T ) (cid:17)(cid:19) . (5)The first term corresponds to the energy gained in cre-ating a conduction band electron, while the second termcorresponds to the energy gained in creating a valenceband hole. Since the deviation from the Fermi sea atlow temperatures is concentrated in the vicinity of theDirac cone, we can assume that the linear isotropic dis-persion relationship in Eq. (1) holds for the entire rele-vant wavevector range. Therefore, the summation overwavevectors can be replaced by an integral over the den-sity of spatial states ( ρ c and ρ v for the conduction andvalence bands, respectively): U el ( T ) = 2 (cid:90) −∞ dE v ρ v ( E v )( − E v ) (cid:32) − e EvkBT + 1 (cid:33) + 2 (cid:90) ∞ dE c ρ c ( E c ) E c e EckBT + 1 . (6)Since the conduction and valence band energies are op-posite at each wavevector, we can express U el as a singleintegral over the energy absolute value E , where E c = E and E v = − E . Due to the equivalent magnitudes of thedispersion slope for the conduction and valence bands ateach wavevector, we can further define a general densityof states ρ ( E ) = ρ v ( E ) = ρ c ( E ): U el ( T ) = 2 (cid:90) ∞ dEρ ( E ) E (cid:32) − e − EkBT + 1 (cid:33) + 2 (cid:90) ∞ dEρ ( E ) E e EkBT + 1= 2 (cid:90) ∞ dEρ ( E ) E e − EkBT + 1cosh (cid:16) Ek B T (cid:17) + 1 . (7)The density of states at band energy E can be solved byapplying the dispersion relationship as follows: ρ ( E ) = dNdA k dA k dk dkdE = A (2 π ) (cid:18) π E (cid:126) v F (cid:19) (cid:126) v F = A π (cid:126) v F E, (8)where A is the surface state area, A k is the area in thereciprocal space associated with the value k = | k | , and N is the number of states. As expected for the graphene-like band structure, the density of states is linear in theenergy. We are thus in a position to solve for the electronenergy U el ( T ) as a function of temperature T : U el ( T ) = Aπ (cid:126) v F (cid:90) ∞ dEE e − EkBT + 1cosh (cid:16) Ek B T (cid:17) + 1= Aπ (cid:126) v F k B T ) ζ (3) , (9)where ζ represents the Riemann zeta function, with ζ (3) ≈ .
2. This implies that the electron temperaturevaries with the total electron energy as U / el , and therelationship between the gain in electron energy and thetemperature change from T i to T f takes the followingform: ∆ U el = 3 . Ak B π (cid:126) v F (cid:16) T f − T i (cid:17) . (10)For detecting a moderate number of microwave photons,we are primarily interested in the limit ∆ T ( ≡ T f − T i ) (cid:28) T i where T i > . Q for the electron modes is related to the rate of change ofthe temperature by taking the derivative of ∆ U el withrespect to time, yielding the following function of thetemperature T ≈ T i : Q = − ddt (cid:18) . Ak B π (cid:126) v F T (cid:19) = − . Ak B π (cid:126) v F T dTdt . (11) B. Energy Gain for Bulk Phonon Modes
Next, we look to determine how the energy gainedby the lattice vibrations relates to the lattice temper-ature. In general, the total phonon energy is determinedas a function of temperature by summing over the modescorresponding to various phonon branches µ and phononwavevectors q , weighted by the occupation number (cid:104) n µ, q (cid:105) for each phonon mode: U ph ( T ) = (cid:88) µ (cid:88) q (cid:104) n µ, q ( T ) (cid:105) (cid:126) ω µ, q , (12)where ω µ, q is the frequency of the phonon mode of branch µ and wavevector q , and the occupation number at a given temperature T is calculated from the Bose-Einsteindistribution: (cid:104) n µ, q ( T ) (cid:105) = 1 e (cid:126) ωµ, q kBT − . (13)As previously mentioned, the fact that the sample is inthe low temperature regime (below 0.5 K) indicates thatthe Debye model, with its assumption of a linear phonondispersion, is approximately valid for the phonon modeswith non-negligible occupation numbers. Therefore, wecan restrict the summation over branches to just the 3acoustic branches (corresponding to the 3 polarizations).In general, the slope of each of these branches is slightlyanisotropic in reciprocal space due to the varying an-gle between the polarization and propagation directions.However, per the treatment in Kittel [27], we can ap-proximate the composite effect of the 3 branches on thesummation in Eq. (12) as equivalent to a summation over3 isotropic branches, each featuring a slope of v s (i.e. thespeed of sound in the material), such that: U ph ( T ) = 3 (cid:88) q (cid:126) ω q e (cid:126) ωqkBT − , (14)where ω q = v s q , and the speed of sound v s for Cd As isestimated as 2 . × m/s [25].We now replace the summation over wavevectors withan integral over the density of states for each branch interms of frequency, D ( ω ). For a 3-dimensional latticewith a speed of sound v s , this density is determined asfollows: D ( ω ) = dNdV q dV q dq dqdω = V (2 π ) (cid:32) π (cid:18) ωv s (cid:19) (cid:33) v s = V ω π v s , (15)where V is the bulk volume of the lattice and V q is thevolume in the reciprocal space associated with q = | q | .We therefore solve for the total phonon energy as a func-tion of temperature through the following integral: U ph ( T ) = 3 (cid:90) ∞ dωD ( ω ) (cid:126) ωe (cid:126) ωkBT −
1= 3 V (cid:126) π v s (cid:90) ∞ dω ω e (cid:126) ωkBT − . (16)Note that the expression differs from that in Kittel inthat we use ∞ instead of a specific Debye cutoff for theupper bound of the frequency range. As in the case ofthe integral over energy for the electronic modes in theprevious section, this is justified by the rapid convergenceof the integrand to 0 at very low temperatures [28], cor-responding to the fact that only the linear regime of theacoustic branches are non-negligibly occupied. Then, wefind the following result: U ph ( T ) = 3 V k B T π v s (cid:126) (cid:90) ∞ d (cid:18) (cid:126) ωk B T (cid:19) (cid:16) (cid:126) ωk B T (cid:17) e (cid:126) ωkBT − (cid:18) V k B T π v s (cid:126) (cid:19)(cid:18) π (cid:19) (17)Unlike the total surface electron energy, which scales as T , the total bulk phonon energy scales as T . This dif-ference can be attributed to the fact that the surface is 2Dwhereas the bulk is 3D, having more degrees of freedom,thus implying that all else being equal, a given change inenergy would have a weaker effect on bulk temperaturethan on surface temperature.The change in the total phonon energy can thereforebe related to the initial and final temperatures T i and T f as follows: ∆ U ph = π V k B (cid:126) v s (cid:16) T f − T i (cid:17) . (18)Note that, for a prism-shaped sample (such as a thin film)for which the surface state forms one of the two bases forthe prism, the bulk volume is proportional to the surfacearea as V = Ad , where d represents the sample thickness. C. Comparison of Specific Heat
Having derived the energy gain for the surface elec-tron and the bulk phonon modes for given initial andfinal temperatures, we now seek to compare the specificheat values for the two mode types in order to glean anunderstanding of how the excess thermal energy is dis-tributed between the modes. From the result in Eq. (10),the specific heat for the collection of surface electrons isdetermined as follows: C el ( T ) = dU el dT = 10 . Ak B π (cid:126) v F T . (19)The specific heat for the bulk lattice is calculated in ananalogous manner from the result in Eq. (18): C ph ( T ) = dU ph dT = 2 π V k B (cid:126) v s T . (20)Using the relationship V = Ad , where d denotes the bulkdepth of the lattice, we divide Eq. (20) by (19) in orderto determine the ratio of the specific heat values for agiven temperature: C ph ( T ) C el ( T ) = π k B v F (cid:126) v s T d = (cid:16) . × K − m − (cid:17) T d. (21)Since the baseline refrigerator temperature is at least 0.25K, this sets the minimum value for T . The lattice depth FIG. 3: Plots of temperature gain (in nanoKelvin) vs.density of absorbed photons (per 10 − m ) for baselinetemperatures T i = 0 . , . , and 0 .
45 K given aphoton frequency f = 5 GHz. d must be at least multiple times longer than the latticeconstant, which is 3 − As [29]. Therefore,the phonon specific heat is far higher than the electronspecific heat (by at least 3 orders of magnitude), indi-cating that nearly all of the thermal energy gained fromthe photon absorption is eventually stored in the latticevibrational modes. As such, if N photons of frequency ω are absorbed, then the relationship between the finalequilibrium temperature T f and the initial temperature T i can be determined from Eq. (18): T f = (cid:32) T i + 10 (cid:126) v s π V k B N (cid:126) ω (cid:33) / = (cid:32) T i + (cid:16) . × − m K s (cid:17) N ωV (cid:33) / (22)As this expression shows, the determinitive factor in thetemperature increase is the total photon energy absorbedper unit volume of the lattice, which is proportional to N ω/V . We plot the temperature gain as a function ofphotoelectron density
N/V for 3 different baseline tem-peratures 0.35 K, 0.40 K, and 0.45 K, for an input photonfrequency of 5 GHz (corresponding to ω = π × s − )in Figure 3. Note that for low photon densities such thatthe temperature gain is small compared to the baselinetemperature, the relationship between temperature gainand photon density is approximately linear, as expected.The imbalance between the electron and lattice specificheat values also has significant implications for the heattransfer between the electron and phonon distributionsthat ultimately yields the equilibrium state. In partic-ular, when comparing the quasiequilibrium electron andlattice temperatures to the final equilibrium temperaturefor the whole system, the equilibrium temperature willbe much closer to the quasiequilbrium lattice tempera-ture than to the electron temperature. We consider thetimescale for the electron-lattice heat transfer as a func-tion of temperature in the next section. IV. ELECTRON-PHONON INTERACTIONTIMESCALE
Having derived the equilibrium temperature for thebulk lattice upon photon absorption, we now aim toestimate the timescale over which that equilibrium isreached. As previously mentioned, this energy trans-fer takes place in two steps: a rapid electron-electronrethermalization, followed by heat transfer from the elec-trons to the acoustic vibrations of the lattice (which ismuch slower than the electron-electron interaction [23–25]). Here, we will focus on the latter process, sinceit serves as the limiting factor in setting the minimumtimescale for reaching equilibrium. We will characterizethe available phase space area for bulk phonon emissionby the surface electrons, derive the matrix element forthe electron-phonon interaction, and finally calculate therate for the electron-phonon heat transfer.
A. Phase Space
We start by examining the available phase space for theinteraction between 2D surface state electrons and thebulk phonon modes. Unlike the spherical equal-energymanifolds for the 3D Dirac cone carrier modes, the 2DDirac cone electron modes take a cylindrical equal-energymanifolds, with a degree of freedom in the k z -direction(corresponding to the axis perpendicular to the surface).We therefore use cylindrical coordinates, expanding thefinal electron wavevector p as ( p cos θ p , p sin θ p , p z ) andthe initial wavevector k as ( k, , q can be ex-panded as follows: q = k − p = ( p cos θ p − k, p sin θ p , p z ) . (23)Note that the bulk phonon’s equal-energy manifolds re-tain a 3D spherical shape defined by ω = v s q . We mapthis onto the electron’s manifolds using energy conserva-tion: v F k − v F p = v s (cid:113) ( p cos θ p − k ) + p sin θ p + p z . (24)Squaring both sides and solving for p , we find that p varies with both θ p and p z : p = k (cid:32) θ p ) − (cid:115)(cid:16) θ p ) (cid:17) − (cid:18) v s v F − v s (cid:19) p z k (cid:33) , (25)where ∆( θ p ) is defined as follows:∆( θ p ) = v F − v s cos θ p v F − v s − . (26) For our material, v s (cid:28) v F , thus yielding ∆( θ p ) (cid:28) θ p . Therefore, p can be approximately expressed assolely a linear function of p z : p ≈ k − v s v F | p z | . (27)Geometrically, the available phase space can be envi-sioned as pair of cones aligned along the p z -axis, withthe bases overlapping at p z = 0, as depicted in Figure 4.The radius attains is maximal value of k at p z = 0 andtapers off as the magnitude of p z increases. The verticesare reached at the following values of p z : | p z | max ≈ v F v s k. (28)Finally, we determine the phonon wavevector q from thecalculated value of p as a function of k . As shown inEq. (23), the amplitudes of p z and q z will equal eachother, since the electron and phonon dispersion centerslie on the same xy -plane. We label q xy as the componentof q perpendicular to the q z -axis. For a given p z and θ p pair, the amplitudes q xy , k , and p form the 3 legs ofa triangle for which θ p represents the angle between thesides of lengths k and p . Therefore, q xy can be calculatedas follows: q xy ≈ (cid:112) k + K − kK cos θ k , (29)and since q z = − p z , the frequency ω q of the emittedphonon is straightforwardly calculated from the speed ofsound: ω q = v s (cid:113) q xy + q z ≈ v s (cid:113) p + k − pk cos θ p + p z . (30)Since v F (cid:29) v s (by a factor of 400), the length of eachcone is far longer than the diameter, implying that theapproximation q ≈ | q z | = | p z | ≈ p will be valid for nearlyall of the available phase space. B. Heat Transfer Rate
Having determined the phase space for the electron-phonon interaction, we are now in a position to calculatethe heat transfer rate between the two modes from thecomposite interaction. Labeling the electron energy for ageneric wavevector k (cid:48) as E k (cid:48) , the matrix element corre-sponding to the electronic transition from k to p throughthe emission of a phonon in branch µ and wavevector q as M µ, qk , p , the Bose-Einstein phonon occupation number forthe mode frequency ω at temperature T as n T ( ω ), andthe Fermi-Dirac distribution value at T as f ( T ), the rate Q is determined through the following summation overinitial carrier wavevectors k , final carrier wavevectors p , (a) 2D cross-section of the phase space along the p x p y -plane. (b) Phase space double cone along the p z -axis. FIG. 4: Depiction of the phase space area of final electronic states for a given initial state k = ( k, , k on the p x p y -plane, and tapering off in the + p z and − p z directionswith a length of v F k/v s along each. Note that the base vs. height ratio for the double-cone in (b) is not to scale.and phonon branches and wavevectors ( µ, q ) [30, 31]: Q = 2 π (cid:126) (cid:88) k (cid:88) p (cid:88) µ, q (cid:16) E k − E p (cid:17)(cid:12)(cid:12)(cid:12) M µ, qk , p (cid:12)(cid:12)(cid:12) × (cid:16) f ( E k ) − f ( E p ) (cid:17)(cid:16) n T L ( ω µ, q ) − n T e ( ω µ, q ) (cid:17) δ k , p + q × δ (cid:16) E k − E p − (cid:126) ω µ, q (cid:17) . (31)As previously discussed, the low temperature restrictsthe occupied phonon modes to the long-wavelengthacoustic regime. The interaction between electrons andlong-wavelength acoustic phonons is dominated by thedeformation potential [32, 33], as recently applied tothe interaction between bulk electrons and phonons inCd As [25, 34]. As discussed in Appendix A, the equiv-alent matrix elements apply for the interaction betweensurface electrons and bulk phonons. Therefore, the ma-trix element amplitude-squared reduces to a functionvarying solely with and linear in the phonon amplitude q : (cid:88) µ (cid:12)(cid:12)(cid:12) M µ, qk , p (cid:12)(cid:12)(cid:12) = CV q, (32) where V represents the lattice volume and C is a constantthat varies with the square of the deformation potential.Substituting this, along with the electron and phonondispersion relationships into the expression for Q , we findthat it takes the following form: Q = 2 π (cid:126) (cid:88) k , p , q (cid:126) v F (cid:16) k − p xy (cid:17) CV q (cid:16) f ( (cid:126) v F k ) − f ( (cid:126) v F p xy ) (cid:17) × (cid:16) n T L ( v s q ) − n T e ( v s q ) (cid:17) × δ k , p + q δ (cid:16) (cid:126) v F k − (cid:126) v F p xy − (cid:126) v s q (cid:17) . (33)The summation is simplified in Appendix B in the limit∆ T (cid:28) T , where T ≈ T e ≈ T L and ∆ T is defined as T e − T L . We find that the Dirac and Kronecker deltafunctions combine to reduce the integral over the phasespace volume to the double-cone phase space area de-rived previously, as expected. This yields the followingexpression for the carrier-phonon heat transfer rate dueto intraband (valence-valence or conduction-conduction)transitions: Q ≈ AC π (cid:126) v s v F (cid:18) (cid:126) v s k B T (cid:19) ∆ TT (cid:18) k B T (cid:126) v F (cid:19) (cid:18) k B T (cid:126) v s (cid:19) (cid:90) ∞ dxx (cid:90) x dy ( x − y ) y e y ( e y − (cid:32) e x + 1 − e x − y + 1 (cid:33) . (34)Solving the integral numerically, we obtain a value of −
32. Therefore, Q is further reduced to the following: Q ≈ − ACk B π (cid:126) v F v s T ∆ T. (35) Next, we solve for the heat transfer rate due to inter-band transitions. Based again on Appendix B, we usethe following expression: Q inter ≈ AC π (cid:126) v s v F (cid:18) (cid:126) v s k B T (cid:19) ∆ TT (cid:18) k B T (cid:126) v F (cid:19) (cid:18) k B T (cid:126) v s (cid:19) (cid:90) ∞ dxx (cid:90) ∞ x dy ( y − x ) y e y ( e y − (cid:32) e x + 1 − e x − y + 1 (cid:33) . (36)Note that the constants in front of the integral are iden-tical to that for the intraband case. Solving this integralnumerically yields a value of − Q total from the surface carriers to the lattice vi-brations is determined by multipyling the intraband rate Q by 2 (to account for both bands) and then summingwith the interband rate Q inter : Q total = 2 Q + Q inter ≈ − ACk B π (cid:126) v F v s T ∆ T. (37)In order to determine the heat transfer timescale, we sub-stitute the previously derived relationship between theelectron cooling rate and the rate of change of electrontemperature from Eq. (11) into the left-hand-side of theabove expression: − . Ak B π (cid:126) v F T d (∆ T ) dt ≈ − ACk B π (cid:126) v F v s T ∆ T,d (∆ T ) dt ≈ − . Ck B T π (cid:126) v F v s ∆ T. (38)As the result shows, the electron temperature decays ex-ponentially toward the lattice temperature, with the ratevarying as T .The remaining task is to determine the value of theconstant C , which derives from the electron-phonon ma-trix element. One method for doing so is by using thedeformation potential of 20 eV measured by Jay-Gerin et al. [35]. This yields the following value for C , using amaterial density ρ = 7 × kg/m [25]: C = (cid:126) D ρv s = 1 . × − J m . (39)This leads to the following heat transfer time constant γ : γ ≈ . Ck B T π (cid:126) v F v s ≈ (cid:16) . × K − s − (cid:17) T . (40)An alternative method for finding the deformation poten-tial is by merging the experimental results from Weber et al. [36] with the theory provided by Lundgren andFiete [25]. Specifically, Weber et al. used a bulk Cd As sample intrinsically doped to a baseline electron densityof 6 × m − , which corresponds to a Fermi energyof 170 meV and a Fermi temperature of 1130 K. Underthese conditions, they observed a timescale of 3.1 ps forelectron cooling by low-energy acoustic phonon emissionat lattice temperatures of 80 K and 300 K. This scenario is addressed by Lundgren and Fiete’s Equation (8), whichmodels the heat transfer rate for k B T (cid:28) E f (where E f is the Fermi energy): γ = D E f k B (cid:126) v F ρT . (41)We now substitute a temperature and rate data pointfrom Weber et al. into this expression to calculate thedeformation potential D . Since the limit k B T (cid:28) E f ismuch more valid for T = 80 K than for 300 K, we usethe former as the temperature corresponding to the rate γ = (3 . − for the purposes of application to Eq. (41).This yields the following value for D : D = (cid:18) k B (cid:126) v F ρE f T γ (cid:19) = 250 eV . (42)This leads to the following value for the coefficient C : C = (cid:126) D ρv s = 2 . × − J m , (43)which yields the following heat transfer time for ourmodel: γ ≈ . Ck B T π (cid:126) v F v s ≈ (cid:16) . × K − s − (cid:17) T . (44)As will be discussed in the next section, the lower boundfor the baseline temperature T (which will also set theminimum value for the heat transfer rate) will be about0.35 K. For this temperature, the above two methodsyield a lattice heating timescale approximately rangingfrom 93 ns to 15 µ s. V. PHOTON-NUMBER RESOLVINGDETECTION
We now describe the photon-number resolving detectorscheme based on our theoretical findings. First, we ad-dress the question whether the timescale for lattice tem-perature equilibration is much faster than the dissipationtime through thermal conduction or radiative decay. Re-garding the thermal conduction heat loss, we note thatthe contacts used for cooling the sample can be removedafter the material reaches the refrigerator temperature.As a result, the heat dissipation time through thermalconduction will range on the order of several hours andcan thus be ignored. Instead, we focus on the radiativeloss. Based on the results calculated for graphene, theelectron-hole interband dipole moment for a 2D Diraccone band structure is given as a function of photon ra-dial frequency ω as follows [37]: d c,v = ev F ω . (45)Substituting this into the well-known radiative decay rateexpression based on the Einstein coefficients [38], we findthat the radiative rate varies linearly with ω :Γ rad ( ω ) = ω π(cid:15) (cid:126) c (cid:12)(cid:12)(cid:12) d c,v ( ω ) (cid:12)(cid:12)(cid:12) = e v F π(cid:15) (cid:126) c ω = (cid:16) . × − (cid:17) ω (46)For frequencies up to 10 GHz, the radiative decay timeis therefore 150 µ s or greater. This is significantly longerthan the electron-phonon heat transfer time calculatedabove, which is 15 µ s or less, which in turn is much longerthan the previously discussed electron-electron rether-malization time of tens of picoseconds [22]. Therefore,a rapid rethermalization of the electron population inthe bands occurs before any radiative loss of the pho-toelectrons occurs. We thus conclude that the energyof the absorbed photons is safely transferred to, firstly,the rethermalization of the carrier band populations, andthen, to the bulk phonon modes to elevate the bulk tem-perature.Next, we discuss how the bulk temperature is mea-sured. Since the elevated bulk temperature will increasethe bulk resistance of the superconducting bulk statesas shown in Figure 5, we measure the zero-bias resis-tivity across the bulk (using a lock-in amplifier) as aproxy for the temperature. This is advantageous rela-tive to infrared-based bolometry since it does not per-turb the electronic structure of the bulk, as well as dueto the fact that electrical signals can be measured in ul-trafast picosecond-range intervals [39]. We manufactureda Cd As device to measure the superconducting bulkresistivity as a function of sample temperature. To thisend, it is important to note the lower bounds for the di-mensions of each Cd As crystal. The goal of the deviceis to measure photons in the transmon frequency range,i.e. 5 − v F , a photon of frequency f is resonantwith the band gap at the following band wavevector: k = πfv F . (47)Therefore, in order for resonance to exist at photonfrequencies as low as 5 GHz, the maximum length ofeach Bloch state in reciprocal space must be ∆ k ≈ . × m − , thus implying that the minimum lengthof the Cd As surface along each dimension is 2 π/ ∆ k =0 . As .20 nm, since this is the minimum thickness that hasbeen achieved with an MBE technique [14]. For a pho-ton frequency of 5 GHz and crystal dimensions of 0.4mm by 0.4 mm by 20 nm, the single-photon tempera-ture gain is calculated by substituting the values N = 1, ω = π × s − , and V = 3 . × − m into Eq. (22)and linearizing:∆ T = 14 T (cid:16) . × − m K s (cid:17) N ωV = 1 . × − K T . (48)For temperatures above our minimum refrigerator tem-perature of 0.25 K, the temperature gain due to the ab-sorption of a single photon is below 6.5 nK, which con-firms our previous assumption that ∆ T << T .Finally, we use the single-photon temperature gain todetermine the corresponding increase in bulk resistance.Figure 5 depicts the experimental values for zero-bias re-sistivity as a function of temperature in bulk Cd As in the superconducting regime. For temperatures above0.35 K, the resistivity steadily increases with tempera-ture. We will therefore use 0.35 K to 0.45 K as the rangeof baseline temperatures for which we will determine thesingle-photon bulk resistance gain. For a square latticesurface, the bulk resistance scales linearly with resistivityas 1 /d , where d denotes the lattice depth. Therefore, thesingle-photon resistance gain relates to the slope of theresistivity with respect to temperature ( dρ/dT ) and thesingle-photon temperature gain (∆ T ) as follows:∆ R = 1 d dρdT ∆ T (49)For the aforementioned sample dimensions, d = 20 nm.Substituting the expression for ∆ T from Eq. (48), we findthat the single-photon resistance gain ∆ R solely becomesa function of the baseline temperature T :∆ R = 5 . × − m − K T dρdT (50)0FIG. 6: Plots of bulk resistance gain (in microohms)due to absorption of a single photon vs. baselinetemperature for photon frequencies f = 5 ,
10 GHz givensample dimensions 0.4 mm by 0.4 mm by 20 nm.Figure 6 depicts the resistance gain due to the absorp-tion of a single photon for baseline temperatures rangingfrom 0.35 K to 0.45 K for the selected photon frequenciesof 5 GHz and 10 GHz. For temperatures of 0.39 K andabove, the single-photon resistance gain will be greaterthan 1 µ Ω for photon frequencies as low as 5 GHz, anincrease which is certainly measurable using a commer-cially available micro-ohm meter (such as the Keysight34420A NanoVolt/Micro-Ohm Meter by Keysight Tech-nologies) or with a Corbino geometry sample which caneven measure sub-micro-ohm resistance [42]. This prop-erty can therefore be exploited in order to precisely de-termine the number of absorbed photons for a knownfrequency.
VI. DISCUSSIONS AND CONCLUSION
We demonstrated a microwave photon-number resolv-ing detector based on the topological surface states ofCd As material. The number of photons absorbed isproduced after measuring the increased resistivity of thesuperconducting bulk. For this, we derived in detail howmuch bulk temperature would elevate as a function of theabsorbed number of photons and the photon frequency.We showed that the energy of the absorbed photon israpidly transferred firstly to the rethermalized distribu-tion of the surface state electron band population. Then,the electron band energy is quickly transferred to thebulk phonon modes through the deformation potentialcoupling. The bulk temperature is thus elevated, andfinally, the superconducting bulk increases resistance,which is measured to resolve the absorbed number of pho-tons. To address how quickly the energy is transferredfrom the surface electron to the bulk phonon modes, wederived the deformation potential electron-phonon cou-pling rate by calculating the transition matrix elementand the phase space volume. As a result, we concluded that the coupling time constant ranged from nanosec-onds to microseconds. Therefore, it is expected that thenumber of absorbed photons would be measured withinseveral milliseconds after the absorption happens.Our proposed scheme accomplishes rapid photon de-tection based on quick (or even continuous) and accuratebulk resistance measurement. Direct measurement of theelevated temperature in bulk does not provide a feasiblepath due to the slow detection speed and the measure-ment noise in the extremely small differential tempera-ture. It is essential to understand why the use of Cd As bulk’s semimetal feature for absorbing microwave pho-tons is avoided. Recall that, if the baseline temperatureis set above the critical temperature, the bulk’s electronicbands do not open a gap f ∆ as shown in equation (2),which allows the bulk electrons to be excited by the mi-crowave photons. However, detecting the excited electronis extremely difficult for two main reasons. First, the bulkphotoelectron may easily join the resistance-measuringcurrent and be lost in the measurement process. Second,the photoelectron’s energy transfer to the bulk temper-ature is extremely inefficient due to the reduced phasespace of 3D electrons, risking the loss of photoelectronsvia radiative decay rather than energy transfer to thebulk phonon modes. In contrast, the photon absorptionfrom the surface state electrons almost surely transfersthe energy to the bulk phonon modes.Equally important is understanding the difference be-tween our proposed scheme and an alternative devicestructure of a Dirac 2D material such as graphene onthe surface of a bulk superconductor. A pure graphenelayer indeed does not possess a superconductor state [43],and thus can be used as a Dirac cone photon absorberof microwave photons even at a very low temperature.However, it is more difficult to fabricate this device thanCd As which simultaneously has both bulk supercon-ductor and surface states. In addition, the hybrid struc-ture suffers from inefficient electronic energy transfer tothe bulk phonons due to the mismatch of lattice con-stants. Instead, as previous research on graphene single-photon detectors has shown, the inefficient electronic en-ergy transfer to phonons is used for efficient capture ofthe photoelectron in the electrodes [44]. However, in thiscase, the photon-number resolving feature is lost. In com-parison, our scheme utilizes the surface state electrons ofCd As as a microwave photon absorber and the bulksuperconductor of the same material for detecting thenumber of photons absorbed. The distinct advantage ofour method is to provide a deterministic photon-numberresolving capability in microwave photon detection.We now discuss the design strategy of maximizing thephoton absorption probability of the device. Note thateach crystal surface features an absorption rate of 2.3%. Therefore, it is possible to have a near unity photondetection efficiency if one vertically stacks few hundredbulk crystal layers in a heterostructure (such that theyare in series from the point of view of the incoming pho-ton), while measuring the bulk zero-bias resistivity for1each of the crystals separately. With the advent of moreadvanced manufacturing technique, such heterostructureis increasingly becoming possible [45]. ACKNOWLEDGMENTS
Sandia National Laboratories is a multimission labo-ratory managed and operated by National Technology &Engineering Solutions of Sandia, LLC, a wholly ownedsubsidiary of Honeywell International Inc., for the U.S.Department of Energys National Nuclear Security Ad-ministration under contract DE-NA0003525.
Appendix A: Matrix Elements for InteractionBetween Surface Electrons and Bulk Phonons
In this section, we estimate the matrix elements cor-responding to interaction between the surface electronmodes and bulk phonon modes by building from the anal-ogous elements for the bulk electron-phonon interaction.Labeling the direction perpendicular to the surface as ˆ z ,we represent the surface state wavefunction as a productof the xy -plane wavefunction and a pulse-like function of z : Ψ( x, y, z ) = φ ( x, y ) ψ ( z ) , (A1)where ψ ( z ) is expressed such that its amplitude-squaredbecomes a broadened Dirac-delta function with a widthof a : ψ ( z ) = (cid:40) √ a , < z < a , otherwise . (A2) Here, a denotes the approximate width of the surfacestate.Next, we seek to express a surface mode as a superpo-sition of bulk modes by decomposing ψ ( z ) into a super-position of modes with well-defined z -wavevector k z : | ψ (cid:105) = c ( k z ) | k z (cid:105) , (A3)where | k z (cid:105) denotes the plane wave state with wavevector k z , taking the following form with respect to the latticedepth d when projected onto the position space: (cid:104) z | k z (cid:105) = 1 √ d e ik z z . (A4)From the Heisenberg uncertainty principle, we intuitivelyknow that the range of z -direction momentum is approx-imated as ∆ p z ≈ h/a , leading to a wavevector range of∆ k z ≈ π/a . We quantitatively determine the superpo-sition coefficients c ( k z ) as follows: c ( k z ) = (cid:104) k z | ψ (cid:105) = (cid:90) dz (cid:104) k z | z (cid:105) (cid:104) z | ψ (cid:105) = 1 √ ad (cid:90) a dze − ik z z = i √ ad e − ik z a − k z . (A5)For low values of k z a (i.e. k z a (cid:47) (cid:112) a/d . Since the span of each plane wavestate in reciprocal space is 2 π/d , this accords with the in-tuition that the overall recpirocal space in the z -directionspans a length of 2 π/a , subdivided into a/d wavevectorswith a roughly uniform superposition coefficient for each.The intraband matrix element corresponding to theemission of a phonon of wavevector q in branch µ by asurface electron can thus be expressed in terms of theanalogous matrix elements for bulk electrons: H emit ( k xy , q ) = (cid:104) k xy − q xy , ψ, n µ, q + 1 | (cid:126) g µ, k , q c † k − q c k b † µ, q | k xy , ψ, n µ, q (cid:105) = (cid:88) k z (cid:104) ψ | k z − q z (cid:105) (cid:104) k − q , n µ, q + 1 | (cid:126) g µ, k , q c † k − q c k b † µ, q | k , n µ, q (cid:105) (cid:104) k z | ψ (cid:105)≈ ad πa (cid:88) k z = − πa (cid:104) k − q , n µ, q + 1 | (cid:126) g µ, k , q c † k − q c k b † µ, q | k , n µ, q (cid:105) . (A6)Note that this approximation is valid specifically if themaximum amplitude of the emitted phonon wavevectoris much smaller than the maximum amplitude of k z , i.e.if q max << π/a , which holds true for long-wavelengthacoustic phonons in the linear dispersion regime. As a final step in the generic matrix element calcu-lation, we can show that the carrier-phonon matrix ele-ment is exactly invariant in the initial carrier wavevec-tor k z , since each k z corresponds to a plane-wave statewith well-defined z -momentum (cid:126) k z . Specifically, the ma-2trix element of a spatial function f ( r ) connecting an ini-tial carrier state | k (cid:105) to a final state | k − q (cid:105) (where the wavevectors are three-dimensional) is simplified as fol-lows: (cid:104) k − q | f ( r ) | k (cid:105) = (cid:104) k xy − q xy , k z − q z | f ( r ) | k xy , k z (cid:105) = (cid:90) d rφ ∗ k xy − q xy ( x, y ) 1 √ d e − i ( k z − q z ) z f ( r ) φ k xy ( x, y ) 1 √ d e ik z z = 1 d (cid:90) d re iq z z φ ∗ k xy − q xy ( x, y ) f ( r ) φ k xy ( x, y ) . (A7)As a result, the terms in the summation in Eq. (A6)are equivalent. Since the total number of valid plane-wave wavevectors in the summation is d/a (as previouslydiscussed), the matrix element H emit becomes invariantin the lattice depth d , as desired: H emit ( k xy , q ) ≈ ad da (cid:104) k − q , n µ, q + 1 | (cid:126) g µ, k , q c † k − q c k b † µ, q | k , n µ, q (cid:105)≈ (cid:104) k − q , n µ, q + 1 | (cid:126) g µ, k , q c † k − q c k b † µ, q | k , n µ, q (cid:105) . (A8)Therefore, the electron-phonon matrix element M µ, qk xy , p xy corresponding to the transition from an initial electronicwavevector k xy to a final wavevector p xy via the emissionof a phonon of wavevector q in branch µ is equivalentto the bulk matrix element M µ, qk , p , where k z = 0 and p z = − q z . Appendix B: Calculating the Surface ElectronCooling Rate
We start by representing the cooling rate of the surfaceelectrons as the following summation over initial electronwavevectors k , final electron wavevectors p , and phononwavevectors q , with the direction of k defined as the x -axis and the projection of p on the xy -plane labeled as p xy : Q = 2 π (cid:126) (cid:88) k , p , q (cid:126) v F (cid:16) k − p xy (cid:17) CV q (cid:16) f ( (cid:126) v F k ) − f ( (cid:126) v F p xy ) (cid:17) × (cid:16) n T L ( v s q ) − n T e ( v s q ) (cid:17) × δ k , p + q δ (cid:16) (cid:126) v F k − (cid:126) v F p xy − (cid:126) v s q (cid:17) . (B1)Given an overall lattice volume V and surface state area A , the discrete summation over k and p can be convertedto integrals as follows: (cid:88) k → A (2 π ) (cid:90) d k, (cid:88) p → V (2 π ) (cid:90) d p = V (2 π ) (cid:90) dp xy p xy (cid:90) dθ p (cid:90) dp z . (B2)Also, the Dirac delta can be re-written as a function of p xy : δ (cid:16) (cid:126) v F k − (cid:126) v F p xy − (cid:126) v s q (cid:17) = 1 (cid:126) v F (cid:18) p xy − k + v s v F q (cid:19) . (B3)Therefore, the Dirac delta collapses the integral over p xy to p xy = k − v s q/v F . The rate Q reduces to the followingform: Q = 2 π (cid:126) A (2 π ) V (2 π ) × (cid:88) q (cid:90) d k (cid:90) dθ p (cid:90) dp z (cid:18) k − v s v F q (cid:19) v s v F q CV q (cid:16) f ( (cid:126) v F k ) − f ( (cid:126) v F k − (cid:126) v s q ) (cid:17)(cid:16) n T L ( v s q ) − n T e ( v s q ) (cid:17) δ k , p + q . (B4)Next, we apply the Kronecker delta, which restricts thesummation over phonon wavevectors to q = k − p . Also,as discussed in the main text, we can apply the approx- imation p ≈ p z >> k for nearly all of the phase space.This argument is further strengthened by the fact thatthe integrand approaches zero for low values of q for3which this approximation is the weakest. Therefore, weset q = p z . Since all terms in the integrand are nowfunctions solely of k or p z , we have collapsed the in-tegral over p to an integral over p z multiplied by thecircumference of the circle generated by making a plane cut (along the p x p y plane) through the cone. Also, thefact that the radius of the cone at a generic value of p z equals k − ( v s /v F ) p z implies that the circumference ofthe aforementioned circle is 2 π ( k − ( v s /v F ) q z ) and theupper bound of q z (corresponding to the cone vertex) is( v F /v s ) k . Therefore, Q simplifies to the following: Q ≈ π (cid:126) A (2 π ) V (2 π ) CV v s v F × (cid:90) ∞ dk πk (cid:90) vFvs k dp z π (cid:18) k − v s v F p z (cid:19) p z (cid:16) f ( (cid:126) v F k ) − f ( (cid:126) v F k − (cid:126) v s p z ) (cid:17)(cid:16) n T L ( v s p z ) − n T e ( v s p z ) (cid:17) . (B5)Note that the factor of 2 in front derives from the factthat the available phase space is actually a double cone,extending in both the + and − directions along the p z -axis.We now examine the parenthetical term correspondingto the net phonon number. In the limit ∆ T << T e , T L ,where ∆ T = T e − T L , we solve for a generic value of T ≈ T e ≈ T L to first order in ∆ T : n T L ( v s p z ) − n T e ( v s p z )= (cid:18) e (cid:126) vspzkBTL − (cid:19) − − (cid:18) e (cid:126) vspzkBTe − (cid:19) − ≈ e (cid:126) vspzkBT (cid:16) e (cid:126) vspzkBTL − (cid:17) (cid:126) v s p z ∆ Tk B T . (B6) The exponential term will set an approximate upperbound for the emitted phonon energy. For low temper-atures (e.g. T = 0 .
45 K), this ensures that the emittedphonons fall within the long-wavelength acoustic modelimit.Next, we write out the expression for the electron oc-cupation number difference between the initial and finalstates: f ( (cid:126) v F k ) − f ( (cid:126) v F k − (cid:126) v s q z )= 1 e (cid:126) vF kkBT + 1 − e (cid:126) vF k − (cid:126) vspzkBT + 1 . (B7)We are now in a position to simplify the double in-tegral by defining the variables x = (cid:126) v F k/ ( k B T ) and y = (cid:126) v s p z / ( k B T ), yielding the following expression for Q : Q ≈ πC (cid:126) A (2 π ) v s v F (cid:126) v s ∆ Tk B T π × (cid:90) ∞ dkk (cid:90) vFvs k dp z (cid:18) k − v s v F p z (cid:19) p z e (cid:126) vspzkBT (cid:16) e (cid:126) vspzkBTL − (cid:17) (cid:32) e (cid:126) vF kkBT + 1 − e (cid:126) vF k − (cid:126) vspzkBT + 1 (cid:33) = AC π (cid:126) v s v F (cid:18) (cid:126) v s k B T (cid:19) ∆ TT (cid:18) k B T (cid:126) v F (cid:19) (cid:18) k B T (cid:126) v s (cid:19) × (cid:90) ∞ dxx (cid:90) x dy ( x − y ) y e y ( e y − (cid:32) e x + 1 − e x − y + 1 (cid:33) . (B8)Note that our analysis so far has been restricted tothe case of intraband carrier transitions through carrier-phonon interaction. Unlike the case of bulk carriers,where the band structure prohibits interband carrier-phonon scattering, the carrier dispersion for the surfacestate allows such scattering. The phase space area forthis interaction can be constructed by merging the coni- cal phase space for the electron-phonon interaction withthe corresponding cone for the hole-phonon interaction,with the pair of cones meeting at p z = v F k/v s . Justlike we restricted the range of values for p z for the in-traband scattering to p z < v F k/v s , we will restrict therange for interband scattering to p z > v F k/v s . Recallthat the amplitude-squared of the carrier-phonon matrix4element for Cd As is identical for interband and intra-band interaction except for the following proportionality[25]: (cid:88) µ (cid:12)(cid:12)(cid:12) M µ, qk , p (cid:12)(cid:12)(cid:12) ∝ s cos θ, (B9)where θ represents the angle between k and p , and s = 1 , − p z >> k for nearlyall of the phase space area for the intraband interac- tions, and since the interband interactions involve evenhigher values for p z , this is clearly true for such pro-cesses as well. Therefore, for both intraband and in-terband interactions, we can make the approximationcos θ ≈
0, causing the interband and intraband matrixelement amplitude-squared values to scale linearly withthe phonon wavevector in the same manner. In order toconstruct the equivalent of Eq. (B8) for interband carrier-phonon scattering, the only changes we make are thus therange of p z , as previously mentioned, as well as the termin the integrand corresponding to the cone radius, whichflips as k − v s p z /v F → v s p z /v F − k : Q inter ≈ πC (cid:126) A (2 π ) v s v F (cid:126) v s ∆ Tk B T π × (cid:90) ∞ dkk (cid:90) ∞ vFvs k dp z (cid:18) v s v F p z − k (cid:19) p z e (cid:126) vspzkBT (cid:16) e (cid:126) vspzkBTL − (cid:17) (cid:32) e (cid:126) vF kkBT + 1 − e (cid:126) vF k − (cid:126) vspzkBT + 1 (cid:33) = AC π (cid:126) v s v F (cid:18) (cid:126) v s k B T (cid:19) ∆ TT (cid:18) k B T (cid:126) v F (cid:19) (cid:18) k B T (cid:126) v s (cid:19) × (cid:90) ∞ dxx (cid:90) ∞ x dy ( y − x ) y e y ( e y − (cid:32) e x + 1 − e x − y + 1 (cid:33) . (B10) [1] A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni,F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Mi-naeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul,F. Lvy, and A. Fiore, Nat. Photon. , 302 (2008).[2] B. E. Kardyna, Z. L. Yuan, and A. J. Shields, Nat. Pho-ton. , 425 (2008).[3] D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam,Phys. Rev. A , 061803(R) (2005).[4] F.-G. Deng, X.-H. Li, H.-Y. Zhou, and Z.-j. Zhang, Phys.Rev. A , 044302 (2005).[5] X.-B. Chen, X.-X. Niu, X.-J. Zhou, and Y.-X. Yang,Quant. Inf. Proc. , 365 (2013).[6] I. Afek, O. Ambar, and Y. Silberberg, Science , 879(2010).[7] A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow,Jens Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio,M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys.Rev. Lett. , 080502 (2008).[8] G. Romero, J. J. Garcia-Ripoll, and E. Solano, Phys.Rev. Lett. , 173602 (2009).[9] K. Inomata, Z. Lin, K. Koshino, W. D. Oliver, J.-S.Tsai, T. Yamamoto, and Y. Nakamura, Nat. Commun. , 12303 (2016).[10] A. Poudel, R. McDermott, and M. G. Vavilov, Phys. Rev.B , 174506 (2012).[11] B. Royer, A. L. Grimsmo, A. Choquette-Poitevin, andA. Blais, Phys. Rev. Lett. , 203602 (2018).[12] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. J. Cava, andN. P. Ong, Nat. Mater. , 280 (2015). [13] M. N. Ali, Q. Gibson, S. Jeon, B. B. Zhou, A. Yazdani,and R. J. Cava, Inorg. Chem. , 4062 (2014).[14] T. Schumann, L. Galletti, D. A. Kealhofer, H. Kim, M.Goyal, and S. Stemmer, Phys. Rev. Lett , 016801(2018).[15] M. Uchida, Y. Nakazawa, S. Nishihaya, K. Akiba, M.Kriener, Y. Kozuka, A. Miyake, Y. Taguchi, M. Toku-naga, N. Nagaosa, Y. Tokura, and M. Kawasaki, Nat.Commun. , 2274 (2017).[16] H. Yi, Z. Wang, C. Chen, Y. Shi, Y. Feng, A. Liang,Z. Xie, S. He, J. He, Y. Peng, X. Liu, Y. Liu, L. Zhao,G. Liu, X. Dong, J. Zhang, M. Nakatake, M. Arita, K.Shimada, H. Namatame, M. Taniguchi, Z. Xu, C. Chen,X. Dai, Z. Fang, and X. J. Zhou, Sci. Rep. , 6106 (2015).[17] M. Goyal, H. Kim, T. Schumann, L. Galletti, A. A.Burkov, and S. Stemmer, Phys. Rev. Mater. , 064204(2019).[18] M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian,C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin,A. Bansil, F. Chou, and M. Z. Hasan, Nat Commun. ,3786 (2014).[19] W. Yu, R. Haenel, M. A. Rodriguez, S. R. Lee, F. Zhang,M. Franz, D. I. Pikulin, and W. Pan, Phys. Rev. Research , 032002(R) (2020).[20] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys.Rev. , 1175 (1957).[21] R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov,T. J. Booth, T. Stauber, N. M. R. Peres, A. K. Geim,Science , 1308 (2008). [22] M. T. Mihnev, F. Kadi, C. J. Divin, T. Winzer, S. Lee,C.-H. Liu, Z. Zhong, C. Berger, W. A. de Heer, E. Malic,A. Knorr, and T. B. Norris, Nat. Commun. , 11617(2016).[23] D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. deHeer, P. N. First, and T. B. Norris, Phys. Rev. Lett. ,157402 (2008).[24] J. M. Dawlatya, S. Shivaraman, M. Chandrashekhar, F.Rana, and M. G. Spencer, Appl. Phys. Lett. , 042116(2008).[25] R. Lundgren and G. A. Fiete, Phys. Rev. B , 125139(2015).[26] P. Debye, Annalen der Physik , 789 (1912).[27] C. Kittel, Introduction to Solid State Physics (John Wi-ley & Sons, Inc., Hoboken, NJ, 2005).[28] ”Debye Model For Specific Heat.” En-gineering, 18 May, 2020,
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