bands Quantum oscillations are capable of measuring the electronic sta - PDF document

1 x …‹‡–‹
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¤…‡’‘”–• bands. Quantum oscillations are capable of measuring the electronic states at the Fermi level, but are insufciently sensitive to detect tiny but important pockets, such as Dirac-cone quantum states. In the meanwhile, a relatively large eld is required, which may greatly change carriers’ properties, especially for eld-sensitive materials, such as Weyl semimetals. Moreover, none of the above measurements can estimate the conductivity contributions from dierent bands.Measuring electrical transport of both longitudinal and transverse directions was an important way of obtaining carriers’ type, mobility, and concentration in single band semiconductors. For materials containing one electron band and one hole band, two carrier model was wildly used. However, for multiple-band materials, clear deviation appears. erefore, new methods are required to t the measured experimental curves. A straightforward way is hypothesizing both carrier types and number of bands, then, calculating mobility and concentration via tting. In order to avoid the hypothesis, the technique of mobility spectrum (MS), which was initially developed to study carriers in semiconductors, was applied to Ba(FeAs)  and FeSe. e MS resulting from longitudinal and transverse transport under a wide range of magnetic eld B up to 50 T, shows a physically reasonable and intrinsic interpretation on the

2 electronic states in low temperature ph
electronic states in low temperature phase. More recently, it was also used to study Type-II Weyl semimetal T  -MoTe  27Note that, application of MS requires material to have relatively large magnetoresistance. PtBi  with a layered hexagonal crystal structure was reported to exhibit a large magnetoresistance2832. Both band structures and  invariant calculations suggest PtBi  as a possible candidate for bulk topological metal28. ARPES measurement found a Dirac-cone-like surface state on the boundary of the Brillouin zone, which is identied as an accidental Dirac band without topological protection. Moreover, triply degenerate point (TDP) fermions were predicated by ab-initio calculations, and veried by quantum oscillation. More recently, APRES measurement and rst principle calculation detected ve bands that could contribute to the TDP. Magnetoresistance measurement for eld up to 22T found that magnetoresistance is related to the angle between the magnetic eld and the crystalline axis. In this paper, we use maximum entropy mobility spectrum analysis (MEMSA) to study carrier properties of PtBi  . e numerical MEMSA method is capable of nicely t the experimental data (see Figs.S1–S2 of Supplementary Materials). For comparison, We also calculated band structure using rst principle calculation. Our results suggest that, MEMSA is a useful tool of detecting carrier type, mobility, and ch

3 arge concentration in recently discovere
arge concentration in recently discovered Iron-based superconductors, and topological materials.MEMSA starts from the experimentally measured Magnetoresistivity  and Hall resistivity  from which, one can calculate the conductivity tensor by:Within the MS model, the relation between conductivity tensor and mobility is given by23263034where the MS is evaluated as follows:       is the concentration of the carriers with mobility  . It is assumed that in Eqs. (2), mobilities are negative for electrons and positive for holes.Mathmatically, Eqs. (2) belong to Fredholm equations of the rst kind. MS can be achieved by an inversing method. To reach a high resolution, we use numerical iterations based on maximum entropy principle. First, we dene the reduced conductivity tensor:Calculation of the probability is performed using the Lagrangian multiplier  where                is the partition function. Assuming      , and  , it can be proved that  , where,    , the matrix  is given by:Replacing  with experimental data, one have:            (2)            (3)          (4)          (5)    

4           (6)
          (6)      (7)             y …‹‡–‹
¤…‡’‘”–• e matrices  were inverted using singular value decomposition. Eq. () gives the approximate dierence between the new value of  and the old one  . e Lagrangian multipliers can be found using the following numerical iterative procedure: First, assume a group of initial values of  , and use Eq. () to calculate the probability ; second, calculate the corresponding conductivity matrix  (Eqs. () ) and matrix (Eq. ()); third, use Eq. () to calculate the modication of Lagrangian multipliers and calculate new set of Lagrangian multipliers      , where  . e loop continues until each element of  is suciently small. e corresponding MS can be calculated from Eq. ().‡•—Ž–•ƒ†In Fig.a–c, we show MS for temperature  K, 10K, and 20K, respectively. e corresponding tting to the conductivity tenser, as well as measured experimental data are shown in Fig.. It is clear that comparing previous methods, our MEMSA method perfectly ts the experimental data of conductivity  and Hall conductivity  spontaneously. e resolution is greatly improved, resulting in well separation of peaks on the MS curve. Each of these peaks are corresponding to an electron or a hole pockets. For a specic pocket , we calculate the fol

5 lowing three values: I. the ratio of con
lowing three values: I. the ratio of conductivity contribution at zero eld  ; II. the location of the peak, i.e., average mobility  ; and III. carrier concentration . Note that, small peaks whose contribution  (  ) are neglected, since they might be generated by white noises in the experiment. For  K, two electron and three hole pockets are identied (see Fig.a). e number of bands is agree nicely with ARPES measurements and rst principle calculations. e carriers’ concentration are  for electron band   for electron band   for hole band  , and  for hole band  Electron band  is of one order smaller (  ), but its mobility is high, leading to a relatively large conductivity contribution (  ). In turn, it is very likely corresponding to a Dirac-cone pocket that was previously reported. e summation of concentration for electrons  , whereas, for holes,        . ose values are similar to Pyrite PtBi  , and many other reported large magnetoresistance topological materials (e.g. WTe  and LaSb), but is higher than Cr  As  . Note that,  , which is similar to Figure1.e MS of carriers in PtBi  for temperature  ), and 20), respectively. z …‹‡–‹
¤…‡’‘”–• Pyrite PtBi  35 and WTe  , indicating that, the conductivity c

6 ontribution of electrons and holes are c
ontribution of electrons and holes are comparable when their mobility dierence is small. For  , all the peaks move toward low mobility direction (see Fig.b), causing a decrease of distance between peaks. As a result, very closely neighbored peaks start to merge (see  and  in Fig.b), leading to a decrease of peak number. is rule persists for higher temperatures (see Fig.c for  ). For suciently high temperature, the number of peaks decreases to just one or two. As a result, the commonly used two carrier model will also become suitable. As show in Ref., the mismatch between two carrier model tting and experimental result indeed gradually disappears as temperature increases.An advantage of MSA is that it can analyse carrier’s temperature dependence properties. Note that, the accuracy of MS is sensitive to mobility as higher mobility carriers are more impressionable to external eld. erefore, here, we limit our analysis to high mobility carriers only, i.e., neglecting  and  . In Fig., we plot ratio of conductivity contribution  , mobility  , and carrier concentration as a function of temperature for  ,  ,  , respectively. e ratio of conductivity contribution  and carrier concentration are robust to temperature change (see Fig.a,c), whereas, the mobility of each peak decreases with increasing temperature (see Fig.b), especially for the Dirac-

7 like pocket  . e information of all
like pocket  . e information of all the peaks for dierent temperature are summarized in Tablee MS allows us to explain many physics phenomena. For example, the sign change phenomenon of Hall conductivity  : At low eld  , conductivity is the summation of all carriers’ contribution. Aer increasing eld, high mobility carriers with  are gradually localized, i.e., only low mobility carriers are responding for conducting. erefore, the winner of the competing between electron and hole conductivity contribution may swap, which leads to change of Hall conductivity’s sign. For PtBi  , as was discussed previously, the conductivity contribution for electrons and holes is comparable (  for electrons and  for holes) at zero eld. Contribution of holes are slightly larger, so the Hall conductivity sign is positive. As eld increases, the Dirac-like hole pocket  , which has the highest mobility, will rst be localized. erefore, holes’ contributions decreases faster than that of electron. Once the electrons win the competition, the sign of Hall conductivity changes. Figure2.e experimentally measured conductivity (le panel, dots), Hall conductivity (right panel, dots), and their MEMSA tting (solid lines). Figure3.e ratio of () conductivity contribution  , () mobility  , and () carrier concentration as a function of temperature for  (black squares),  (red c

8 ircles),  (blue triangles), respecti
ircles),  (blue triangles), respectively. { …‹‡–‹
¤…‡’‘”–• We further compare our MEMSA results with electron band structure obtained by rst principle calculation. e band structure of PtBi  is calculated using the full-potential linearized augmented plane wave (FPLAPW) method implemented in the WIEN2K code. In Fig.a, we show the fat band of PtBi  with orbital characters. ere are ve bands crossing Fermi level  , hybridized from Pt and Bi orbitals. A distinct Dirac cone near  locates at the point. Further band analysis demonstrates that, there are three hole-like bands which construct hole pockets locate around the Brillouin zone corner (band 1, 2 and 3), and two electron-like bands which construct bowl-shaped electron pockets centered around point (band 4 and band 5), as shown in Fig.b–f.We further calculate the density of state (DOS) at  as a function of  (  and  are free-electron mass and quasiparticle eective mass, respectively). As shown in Fig.g, the DOS also has peak structures: including Figure4.) e calculated electronic band structure of PtBi  . e green and red lines indicates the Bi- and orbital contributions, respectively. () ree-dimensional Fermi surfaces of PtBi  for each band in the rst Brillouin zone. () e DOS as a function of eective mass. Five bands including a Dirac-cone-like band (black line) is identied