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Published in: PRB 93, 121202(R)(March 2016) Interband optical conductivity of the [001]-oriented Dirac semimetal Cd3As2 D. Neubauer,1 J. P. Carbotte,2,3 A. A. Nateprov,4 A. Lo¨hle,1 M. Dressel,1 and A. V. Pronin1,∗ 11. Physikalisches Institut, Universita¨t Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany 2Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada 3The Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada 4Institute of Applied Physics, Academy of Sciences of Moldova, Academiei str. 5, 2028 Chisinau, Moldova 6 (Dated: 13 January 2016) 1 0 We measured the optical reflectivity of [001]-oriented n-doped Cd3As2 in a broad frequency 2 range (50 – 22000 cm−1) for temperatures from 10 to 300 K. The optical conductivity, σ(ω) = σ1(ω) + iσ2(ω), is isotropic within the (001) plane; its real part follows a power law, n σ1(ω) ∝ ω1.65, in a large interval from 2000 to 8000 cm−1. This behavior is caused by interband u transitions between two Dirac bands, which are effectively described by a sublinear dispersion J relation, E(k) ∝ |k|0.6. The momentum-averaged Fermi velocity of the carriers in these bands is 9 energy dependent and ranges from 1.2×105 to 3×105 m/s, depending on the distance from the Dirac points. We detect a gaplike feature in σ1(ω) and associate it with the Fermi level positioned ] around 100 meV above theDirac points. l l a h DOI:10.1103/PhysRevB.93.121202 - s e INTRODUCTION The goal of this report is to provide insight into the m Dirac-band dispersion in Cd As by means of optical 3 2 . t The interest in measurements of the optical conduc- spectroscopy. Previous optical investigations of Cd3As2 a performed in the 70s and 80s can be divided into two m tivity, σ(ω)=σ1(ω)+iσ2(ω), in three-dimensional (3D) Diracmaterials[1]istriggeredbythe factthattheinter- groups. The first one [16–21] deals with the low-energy - d bandconductivityinthesesystemsisexpectedtodemon- partoftheopticalspectra(usually,belowsome250meV, n strate a peculiar behavior. Generally, the interband op- or2000cm−1)discussingmostlyphononmodesandfree- o tical response of d-dimensional Dirac electrons is sup- electron Drude-like absorption. Turner et al. [16] iden- c tify a very narrow (130 meV) optical gap in the optical [ posed to be universal: σ1(ω) should follow a power-law frequency dependence, absorption. Another groupof papers [22–26] mainly dis- 3 cusses absorption features at energies of a few electron v σ (ω)∝ω(d−2)/z, (1) volts and their relations to transitions between different 9 1 (high-energy) parabolic bands. No optical conductivity 9 2 where z is the exponent in the band dispersion relation, was derived from these measurements. 3 E(k)∝|k|z, [2, 3]. Recent recognition of the nontrivial electron-band .0 For example, σ1 is proportional to frequency in the topology of Cd3As2 calls for a fresh look into its optical 1 case of perfectly linear Dirac cones in three dimensions. properties. In this paper, we report broadband optical 0 Such linearity in σ1(ω) over a broad frequency range in investigations of Cd3As2. 6 a 3D system is often considered as a “smoking gun” for 1 : Diracphysics(eitheroftopologicalorofotherorigin,[4]). v Forexample,Timusket al. [5]haveclaimedthepresence EXPERIMENT i X of 3D Dirac fermions in a number of quasicrystals based r entirelyonthe observationofa linearσ1(ω)inthese ma- Single crystals of Cd3As2 have been grown by vapor a terials. Linear-in-frequencyσ (ω)has alsobeen foundin transport from material previously synthesized in argon 1 ZrTe [6],where 3Dlinearbands areevidencedbytrans- flow [27], see Supplemental Material for details. Resis- 5 port and angle-resolvedphotoemission experiments [7]. tivity and Hall measurements provide an electron den- In2013Wanget al. [8]predicted3DtopologicalDirac sity of n = 6 × 1017 cm−3 (roughly independent of e points in Cd As ; by now they are well confirmed by temperature), a metallic resistivity, and a mobility of 3 2 ARPES, scanning tunneling spectroscopy,and magneto- µ=8×104 cm2/Vs at 12 K. transport measurements [9–14]. Due to the presence of The investigated Cd As single crystal had lateral di- 3 2 inversion symmetry in Cd As , the bands are not spin mensionsof2.5mmby3 mmanda thicknessof300µm. 3 2 polarized [15]. The shape of the Dirac bands in Cd As It was cut out from a larger single crystal. The crystal- 3 2 is somewhat complicated by the presence of a Lifshitz- lographicaxesofthe sample werefoundbyx-raydiffrac- transition point [12]; see Fig. 1 for a sketch of the Dirac tion. The [001] axis was perpendicular to the sample’s bands. largest surface. This surface was polished prior the op- 2 at lower ω. For ω/2πc > 2000 cm−1 the reflectivity is Energy (meV) basically temperature independent. In order to extract the complex optical conductivity, 10 K we applied a Kramers-Kronig analysis procedure as de- 50 K scribedin the Supplemental Material. The results of the 100 K Kramers-Kronig analysis are plotted in Fig. 2 in terms 300 K of σ (ω) and the real part of the dielectric constant, 1 ′ ε(ω)=1−4πσ (ω)/ω. 2 Thefirststrikingresultofourinvestigationisapower- law behavior of σ (ω) between approximately 2000 and 1 8000 cm−1: σ (ω) ∝ ωn with n = 1.65 ± 0.05. This 1 power-law conductivity is basically independent on temperature. The straightforward application of Eq. (1) yields z ≃ 0.6, i.e., a sublinear dispersion, E(k) ∝ |k|0.6, of the Dirac bands in Cd As . This dispersion is valid only 3 2 FIG. 1. (Color online) Reflectivity of [001]-cut Cd3As2 for for the energies above 250 meV (∼ 2000 cm−1), as the selected temperatures between 10 and 300 K measured with observed power law in σ (ω) does not extend below this 1 Eω k [110], see text. Ordinate numbers are given for the frequency. Also, Eq. (1) does not take into account the measurements at 10 K, while the curvesobtained at T =50, asymmetry between the valence and conduction bands, 100, and 300 K are downshifted by −0.1, −0.2, and −0.3, respectively. The inset sketches the dispersion of the Dirac bands in Cd3As2. In general, the Fermi-level position is not necessarily at theDirac points. Energy (meV) 10 100 1000 tical measurements, which were performed for a few linear polarizations. The direct-current (dc) resistivity of 1000 Cd As 3 2 this sample wascharacterizedin-plane by standardfour- probe method (inset of Fig. 5). -1 m) (001) c The optical reflectivity was measured at 10 to 300 K -1 with light polarized along different crystallographic di- (1100 rections. The spectra in the far-infrared (50 cm−1 – EF 10 K 1000cm−1)wererecordedbyaBrukerIFS113vFourier- 50 K 1.65 transform infrared spectrometer using an in-situ gold 10 100 K 300 K overfillingtechnique for reference measurements [28]. At higher frequencies (700 cm−1 – 22000 cm−1) a Bruker Hyperion microscope attached to a Bruker Vertex 80v 0 spectrometer was used. Here, either freshly evaporated goldmirrorsor coatedsilver mirrorswere utilized as ref- 15 -200 erences. Both dc and optical measurements revealed an ’ 1’0 isotropic response within the (001)-plane. Hereafter, we -400 5 present the optical data obtained for E k [110] where ω Eω is the electric-field component of the probing light. 0 0 1x104 2x104 -600 -1 Frequency (cm ) EXPERIMENTAL RESULTS 100 1000 10000 -1 Frequency (cm ) Figure 1 shows the reflectivity R(ω) of [001]-oriented FIG. 2. (Color online) Overall optical conductivity on log- Cd As versus frequency at various temperatures as in- 3 2 log scale (upper frame) and dielectric permittivity (bottom dicated. Atlowfrequenciesthe reflectivityis ratherhigh frame) of Cd3As2 measured within the (001) plane. The correspondingto the metallic behaviorofthe dc resistiv- straight line in the upper frame represents σ1(ω) ∝ ω1.65. ity. At ωscr/2πc≈400cm−1 a temperature-independent The diagram in the upperframe sketches the proposed band pl (screened) plasma edge is observed in the reflectivity. A dispersion in Cd3As2 nearoneof theDiracpoints. Theinset number of phonon modes strongly affect the reflectivity of the bottom frame displays positive ε′ on a linear x scale. 3 whichis presentinCd As [10,12]. Hence, the obtained fermions [2, 3, 5]: 3 2 dispersion is basically an effective approximation. Nev- ertheless, the sublinear dispersion at high energies is in e2NW ω σ (ω)= , (2) 1 qualitative agreement with the dispersion derived from 12h v F Landau-levelspectroscopy[12],withARPESresults[11], where N is the number of nondegenerate cones and all as well as with band-structure calculations [9, 12]. One W Dirac points are considered to be at the Fermi level. If could imagine that the Dirac bands in Cd As get more 3 2 theFermilevelisnotattheDiracpoint(E 6=0),Eq.(2) narrow when the Dirac point is approached as depicted F is replaced by [29]: bythesketchinFig.2. Eventually,thebandsbecomelin- ear,butwecannotprobethelineardispersionbyoptical- e2N ω conductivitymeasurements,because,aswillbediscussed σ (ω)= W θ{~ω−2E }, (3) 1 F 12h v below, the Fermi level in our sample is shifted with re- F spect to the Dirac point. whereθ{x}istheHeavisidestepfunctionandanycarrier Thedielectricconstant(bottompanelofFig.2)isneg- scattering is ignored. ative at low frequencies and crosses zero at 400 cm−1. In Fig. 3 we re-plot σ (ω) on a double-linear scale as 1 The crossingpointis independent ontemperature andis relevant for further considerations. From Figs. 2 and 3 setbythe screenedplasmafrequencyofthe free carriers, one can see that the low-temperature σ (ω) almost van- 1 ωscr. Very similar values of ωscr have been reported pre- ishes (∼ 5 Ω−1cm−1) at around 1300 cm−1 (160 meV). pl pl viously [18–20] indicating similar carrier concentrations The power-law conductivity starts at ∼ 2000 cm−1 (250 ′ in naturally grown samples. At higher frequencies ε(ω) meV). Following Ref. 29 and Eq. (3), we associate this takes a positive sign, reaching values up to 17 at 1500 – steplike feature in σ (ω) with the position of the Fermi 1 2000 cm−1. level. The power-law conductivity discussed above can be roughly approximated by a straight line (the best fit is achieved between 3000 and 4500 cm−1). By setting DISCUSSION N = 4 (two spin-degenerate cones) in Eq. (2), we ob- W tainv =1.7×105 m/s. Alternatively, onecanestimate F v from the derivative ofthe interbandconductivity, i.e. In the simplest case of symmetric 3D Dirac cones, the F from σ (ω) at ω/2πc>2000 cm−1: slopeofthelinearσ (ω)duetointerbandcontributionsis 1 1 directly relatedto the (isotropic)Fermivelocity ofDirac e2N dσ −1 W 1 v (ω)= , (4) F 12h (cid:18)dω (cid:19) Energy (meV) as it is shown in the inset of Fig. 3. As it can be seen 0 100 200 300 400 500 600 700 from the figure, the typical values of v (1.2× 105 to 800 F 3×105 m/s) are somewhat smaller than the published results,which rangefrom7.6×105 m/s to 1.5×106 m/s s) 3 Cd3As2 [9–14, 30]. Note however, that we have evaluated an 600 5 10 m/ 10 K (001) energy-dependent Fermi velocity: vF strongly increases -1 m) v (F2 as ω is reduced. In other words, vF might actually meet c the literature values when approachingthe Dirac points. -1 (1 400 1 2000 4000 6000 8000 10 K fTohrenrairsreoowfinvgFDatirωac→ba0ndiss.exactlythe behaviorexpected -1 50 K Frequency (cm ) 100 K In order to get a more quantitative description of the 200 300 K optical conductivity in Cd3As2, one needs to implement 1.65 amodel. Sofar,thereisnoaccuratemodelfortheoptical 5 vF = 1.7 x 10 m/s conductivity ofa 3Ddoped Diracsemimetalwith sublinear band dispersion. Building such a model goes beyond 0 0 1000 2000 3000 4000 5000 6000 the presentwork[31]. Below,we willuse a simple model Frequency (cm-1) of linear E(k) spectrum with the purpose of extracting the Fermi-level position and explaining its broadening. FIG. 3. (Color online) Optical conductivity from Fig. 2 re- plotted on double-linear scale for ω < 6000 cm−1. The This model will also effectively describe the leveling off straight green line represents the linear conductivity with the interband conductivity spectrum at low frequencies. vF = 1.7 × 105 m/s. The inset shows the Fermi velocity Themodelisbasedonconsideringself-energyeffectsin calculated using Eq.(4) from σ1(ω)at 10 K. Reddotted line thespiritofRef.32andisgivenintheSupplementalMa- is a guide to theeye(vF saturates at some point as ω→0). terial. Therealpartsoftheconductionandvalence-band 4 selfenergiesareapproximatedbytheconstantReΣ =∆ 120 cm−1, and v = 2.4×105 m/s. The value of ω , c F g and ReΣ = −∆, respectively. We hence obtain for the whichisfoundtobesmallerthantheFermienergy,hasto v interband conductivity be consideredas a fit parameteronly. The fit is not per- fect,asthemodeldoesn’tincludethedeviationsfromthe e2N (ω−ω )2 σ (ω)= W g θ{~ω−2max[E ,∆]}, (5) band linearity discussed above. Nevertheless, the model 1 F 12hvF ω graspsthemainfeaturesoftheinterbandconductivityin Cd As . where ~ω = 2∆ [33]. The conduction-band energies 3 2 g The obtained position of the Fermi level (100 meV) have been pushed up by ∆, while the valence-band en- seems to be quite reasonable for our sample, taking into ergies are lowered by the same amount. Alternatively, account its carrier concentration (6 × 1017 cm−3) and Eq. (5) is the interband conductivity of a simple band keeping in mind that E = 200 meV was reported for a structure with E(k) = ±[~v k + ∆], which could be F F samplewithn =2×1018 cm−3 [12]andE =286meV thought of as a first rough approximation to the case e F for n =1.67×1018 cm−3 [30]. of the Dirac cones that get narrow as energy approaches e Although the scope of this work is the interband con- the Dirac point. The Fermi energy is measured from ductivity in Cd As , let us briefly discuss the experi- the Dirac point without self-energycorrectionsincluded, 3 2 mental results at the lowest frequencies measured. The and so E must be larger than ∆ for finite doping away F intrabandconductivityisrepresentedbyanarrowDrude from charge neutrality. If impurity scattering cannot be component (best seen in Fig. 5) and an absorptionband neglected, the Heaviside function can be replaced by ofpeculiarshape at300−1300cm−1 (seeFigs.3and4). 1 1 ω−2max[E ,∆]/~ The nature of the band might be related to localization F + arctan . (6) and/or correlation effects. In any case, the presence of 2 π γ thisbandmakesitimpossibletofittheintrabandconduc- where γ represents a frequency-independent impurity tivityforω/2πc<1300cm−1 withasimple free-electron scattering rate. Drude term. One can see, however, that the narrow A combination of Eqs. (5) and (6) is now employed Drude peak is getting somewhat narrower as T →0 due to model the experimental data; the best fit is plotted to a modest decrease of scattering. Let us note that the in Fig. 4. Note, that this model does not include the spectral weight related to the low-frequency absorbtion intraband conductivity. The best description of the ex- of delocalized carriers remains temperature independent perimental curve was achieved with EF/hc = 800 cm−1 because the screened plasma frequency, discussed above (EF ≈ 100 meV), ωg/2πc = 450 cm−1, γ/2πc = in connection to the dielectric constant, is independent of temperature. In addition to electronic contributions, the low- Energy (meV) frequencyconductivityrendersalargenumberofphonon 0 100 200 300 400 modesmarkedbyarrowsinFig.5. Wecandistinguish14 400 infrared-activephononmodes inthe frequenciesbetween approximately100and250cm−1. As the temperatureis Cd As reduced, the phonons become sharper. More details on 3 2 300 the low-frequency conductivity in Cd As will be given (001) 3 2 -1 m) in a separate paper. c -1 200 experiment 10 K (1 interband-conductivity fit CONCLUSIONS We found the dc resistivity and the optical conduc- 100 tivity of [001]-oriented n-doped (n = 6 × 1017 cm−3) e Cd As to be isotropic within the (001) plane. The real 3 2 part of the frequency-dependent conductivity follows a 0 power law, σ (ω) ∝ ω1.65, in a broad frequency range, 0 1000 2000 3000 1 -1 2000 to 8000 cm−1. We interpret this behavior as the Frequency (cm ) manifestationofinterbandtransitionsbetweentwoDirac FIG. 4. (Color online) Optical conductivity of Cd3As2 mea- bandswithasublineardispersionrelation,E(k)∝|k|0.6. sured within the (001) plane at T = 10 K, plotted together The Fermi velocity falls in the range between 1.2×105 with a description of the interband portion of σ1(ω) given and 3 × 105 m/s, depending on the distance from the by Eqs. (5) and (6). Deviations between the experiment and theory at higher frequencies are due to neglecting the devia- Dirac points. tionsfromlinearityoftheDiracbandsinEqs.(5)and(6),as At 1300 cm−1 (160 meV), we found a diminishing discussed in thetext. conductivity, consistent with observations of an “opti- 5 SUPPLEMENTAL MATERIAL Energy (meV) 10 15 20 25 30 600 Sample growth 3 Cd3As2 m) c2 Single crystals of Cd3As2 have been grown by vapor (001) -3 10 transport from material previously synthesized in argon -1 m) 400 1500 KK (1 flpoerwatwuirtehstihnetflhoewevraapteoroaftiaonfewancdmg3r/omwitnh[2z7o]n.eTs haerete5m20- -1 (c1 T 130000 KK 00 100 200 300 ◦aCt raonodm4t8e0m◦pCer,arteusrpeecthtieveellye.ctBroynsmubosbeiqliutyenits aennnheaanlcinedg T (K) 200 while at the same time the electron concentration de- creases [34]. Well-defined crystals with facets measured afew millimetersineachdirectionareharvestedafter24 hours of the growth process. The lattice parameters have been obtained by x-ray 0 50 100 150 200 250 diffraction: a = 1.267 nm and c = 2.548 nm, in good -1 Frequency (cm ) agrementwith recentmeasurementsreporting a tetrago- nalunitcellwithonlylightdeviationsfromacubicstruc- FIG.5. (Color online)Low-frequencypartofopticalconduc- tivity in Cd3As2. The vertical arrows indicate the positions ture [15]. of the phonons. The inset shows dc-resistivity as a function of temperature measured within the(001) plane. Kramers-Kronig analysis For Kramers-Kronig analysis, extrapolations to zero frequencyhavebeenmade usingthe Hagen-Rubensrela- cal gap”, made in the 1960s [16]. However, we hesitate tion in accordance with the temperature-dependent dc tofollowthetraditionalinterpretationofthisfeatureand resistivity measurements (Fig. 5 of the article). For to straightforwardly relate it to a gap in the density of the temperature-independent extrapolation to ω → ∞, states. Applying recent models for the optical response we utilized publishedreflectivity data,obtainedin ultra- of Dirac/Weyl semimetals, we instead relate this feature highvacuum for upto 170000cm−1 (21 eV) [24], apply- to the Fermi level, which is positioned around 100 meV ingtheprocedureproposedbyTanner[35],whichutilizes above the Dirac points, and which is consistent with the x-ray atomic scattering functions for calculating the op- carrier concentration. tical response for frequencies from 21 to 30000 eV and the free-electron R ∝ ω−4 extrapolation for frequencies above 30 keV. InordertochecktherobustnessofourKramers-Kronig analysis, we replaced the reflectivity data of Ref. [24] by three other data sets available in the literature for the frequency range from approximately 1 to 5 eV. Namely, ACKNOWLEDGEMENTS we used: (i) the polycrystalline data from Ref. [25]; (ii) the single-crystaldata from the same reference; and (iii) the most recent single-crystal data from Kozlov et al. Ruud Hendrikx at the Department of Materials Sci- [26]. All these data sets, as well as the data of Ref. [24], ence and Engineering of the Delft University of Tech- show rather similar overall reflectivity with two strong nology is acknowledged for the x-ray analysis. We are peaksatapproximately1.8and4 eV;the peak at1.8eV grateful to Gabriele Untereiner for sample preparation shows up in our data as well. We have found that at and other technical assistance. Sample-characterization frequenciesbelow1.2eV(10000cm−1)allthesedifferent work of Hadj M. Bania is highly appreciated. We thank extrapolationsaffecttheoutcomeoftheKramers-Kronig Hongbin Zhang, Sergue¨ı Tchoumakov, Marcello Civelli, analysis only marginally. 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