Possible permanent Dirac- to Weyl-semimetal phase transition by ion implantation

Three-dimensional (3D) topological semimetals (TSMs) are a new class of Dirac materials that can be viewed as 3D graphene and are referred to as Dirac semimetals (DSMs) or Weyl semimetals (WSMs) depending on whether time reversal symmetry and/or inversion symmetry are protected, respectively. Despite some interesting results on Dirac- to Weyl-semimetal phase transitions under conditions of low temperature or strong magnetic field (B), all of them are reversible phenomena. Here, we report for the first time a possible permanent transition in a single TSM by ion implantation. A Dirac- to Weyl-semimetal phase transition in a Bi0.96Sb0.04 DSM results from inversion-symmetry breaking induced by implantation with nonmagnetic Au ions for implant fluences (ϕG) ≥ 3.2 × 1016 Au cm−2. This phenomenon is evidenced by the ϕG-dependent behavior of the Raman spectra and quantum-oscillation parameters extracted from magnetoresistance (MR) measurements, which show abrupt changes at ϕG ≥ 3.2 × 1016 Au cm−2. The verification of the transition is further supported by observations of negative MR in the longitudinal B // electric field orientation, indicating the existence of a chiral anomaly in Weyl fermions induced by implantation with nonmagnetic Au ions. In contrast, implantation with magnetic Mn ions exhibits no such particular behavior. Our findings demonstrate the first realization of a possible permanent DSM-to-WSM phase transition in a single material by the simple approach of implantation using nonmagnetic elements. (Left, above) Schematic diagram describing the process of the Au implantation into the Dirac semimetal (DSM) Bi0.96Sb0.04. (Left, below) Positive longitudinal magnetic resistance (LMR) is observed in the crystal without ion implantation, but negative LMR behavior becomes apparent for ϕG ≥ 3.2 × 1016 Au cm−2 (≡ ϕC, critical implant fluence), and reaches a maximum for ϕG = 8.0 × 1016 Au cm−2. (Middle) Quantum oscillation parameters for the β Fermi pockets showing abrupt changes near ϕC, typical of phase transition. No such behavior is observed for the α Fermi pockets. (Right) A drastic change in the Raman spectra is observed for ϕG ≥ ϕC. In particular, a new peak appears at 85.7 cm−1, between the well-known Raman modes, suggesting that the inversion symmetry breaking in the crystal occurred for ϕG ≥ ϕC, resulting in the transition of the DSM to a Weyl semimetal.


Introduction
Studies on topological semimetals (TSMs), highly attractive as a novel class of topological matter, are at the frontiers of materials science and technology, with many theoretical and experimental breakthroughs being made on several TSM phases, such as Dirac semimetals (DSMs), Weyl semimetals (WSMs), nodal-line semimetals, and triple-point semimetals 1,2 . A broad range of novel transport properties, such as extremely large magnetoresistances, chiral anomalies, and the intrinsic anomalous Hall effect, have been experimentally demonstrated and attributed to relativistic fermions that represent lowenergy excitations around the Dirac/Weyl nodal points or nodal line. DSM fermions are the three-dimensional (3D) counterpart of the two-dimensional (2D) Dirac fermions in graphene 3 . Unlike the Dirac cone in graphene, the dispersion surfaces in DSMs have linear energymomentum relations along all three directions 4,5 . A crystal needs both time reversal symmetry and inversion symmetry to protect the degeneracy of a Dirac node from splitting into two Weyl nodes 6,7 . At the critical point during the transition from a topological insulator to a normal insulator, the conduction and valence band contact points are 3D Dirac or Weyl points, depending on whether inversion symmetry exists or not, respectively. Berry curvature, a quantity that can be used to characterize the topological entanglement between the conduction and valence bands, becomes singular at Weyl points that act as monopoles in momentum space with a fixed chirality [8][9][10] .
The crystal structure of pure Bi, pure Sb, and their crystalline alloys (Bi 1−x Sb x ) that can be formed over a wide range of x is rhombohedral, and all have R3m symmetry with two atoms in each unit cell 11 . In Bi, pockets of electrons are located at the L points of the Brillouin zone (BZ), while those of holes are located at the T points. The valence band (VB) and conduction band (CB) at L originate from antisymmetric (L a ) and symmetric (L s ) p-type orbitals, respectively 12 . In Sb, pockets of electrons are also located at the L points of the BZ, while those of holes are located at the H points. The substitution of Sb atoms for Bi atoms in the Bi lattice has a significant effect on the band structure of Bi, including the overlap between the L and T bands, the magnitude of the direct band gap E g , and the maximum energy of the H bands. For Sb concentrations (x) between 0 ≤ x < 0.04, the overlap and band gap decrease. At x = 0.04, the CB and VB edges exchange their symmetries at the L points, thereby closing the L-point energy gap between L a and L s , resulting in the formation of a DSM 13,14 .
For materials research, ion implantation offers two main advantages: (1) it is a room temperature (RT) process, enabling doping and annealing processes to be studied independently; and (2) it enables selective doping of devices and structures with different species. Here, we compare the magnetoresistance (MR) response of Bi 0.96 Sb 0.04 DSMs ion-implanted with Au and Mn ions and demonstrate a possible permanent Dirac-to Weylsemimetal phase transition in samples implanted with Au. No such behavior is observed in samples implanted with Mn, suggesting that only nonmagnetic elements are effective for the transition.

Materials and methods
Bi 0.96 Sb 0.04 bulk single crystals with 99.99% purity were grown in a high-temperature furnace using a method previously reported 12 . Stoichiometric mixtures of highpurity Bi and Sb elements were sealed in a vacuum quartz tube to avoid oxidation. The mixtures were heated to 650°C and cooled to 270°C over a period of five days. Then, they were annealed for seven days at 270°C. X-ray diffraction measurements demonstrated that the samples were single phase and had a rhombohedral crystal structure. Bi 0.96 Sb 0.04 crystals cleaved along the (001) plane from the bulk crystals were implanted with 1 MeV Au + ions to nominal fluences of (0.8, 3.2, 8.0, 10.4, and 12.8) × 10 16 Au cm −2 or with 300 keV Mn + ions to nominal fluences of (4.0 and 8.0) × 10 16 Mn cm −2 at RT. The peak Au or Mn concentrations for these implantations were calculated from dynamic range simulations using the TriDyn code 15 to be in the range of 1.7-5.6 at% (0.48-1.6 × 10 21 Au cm −3 ) or 5.8-10.3 at% (1.6-2.9 × 10 21 Mn cm −3 ), respectively. The implanted crystals were subsequently annealed at 230°C for 1 h in a furnace under Ar flow to remove the implantation-induced damage.
Raman spectroscopy was used to characterize the optical properties of the Bi 0.96 Sb 0.04 crystals cleaved along the (001) plane from the bulk crystals in a confocal microscope system (HEDA, NOST) with an excitation wavelength of 532 nm at RT. The laser was coupled to a ×100 objective lens (numerical aperture = 0.90) and focused to a submicrometer spot on the sample surface. The Raman signal collected with the same objective was dispersed through a monochromator with a diffraction grating of 2400 grooves/mm and detected by a chargecoupled device (Andor Technology). The Raman spectra were measured from the shiny silver-colored surface of the crystals for various ϕ G in the frequency range from 50 to 180 cm −1 . The penetration depth of the X-rays is much deeper than the implantation depth, so a technique of glancing incidence was employed for the XRD analysis. Magnetotransport experiments were carried out on the Bi 0.96 Sb 0.04 crystals in a cryogen-free magnet system under B ranging from −9 T to +9 T along the directions parallel and perpendicular (B // trigonal axis for transverse MR (TMR) and B // binary axis for longitudinal MR (LMR), respectively) to the applied current I (// binary axis) using the six-probe method. Homogenous contact configurations were employed to suppress the current jetting effect of the MR, as described in our previous report 16 . All electrical contacts were prepared at RT with silver paste. The TMR and Hall resistance were simultaneously measured when B was perpendicular to I. The electrical resistivity of the crystals under no B field was also measured as a function of temperature from 1.7 to 300 K.

Results and discussion
Bi 0.96 Sb 0.04 crystals cleaved along the (001) plane from bulk crystals with 99.99% purity were implanted with 1 MeV Au + ions to nominal fluences (ϕ G ) of (0.8, 3.2, 8.0, 10.4, and 12.8) × 10 16 Au cm −2 at RT. The Au range distributions were calculated using the TriDyn code 15 , which includes sputtering effects. This showed that the peak Au concentration saturated at~5.6 at% for ion fluences abovẽ 6 × 10 16 Au cm −2 . (Supplementary Fig. S1). X-ray diffraction analysis of the implanted crystals was performed for various ϕ G in the angular range of 2θ min = 20°and 2θ max = 80°( Supplementary Fig. S2). All the samples exhibited sharp (00l) reflections that were well indexed along the c-axis, indicating a single-crystalline, c-axisoriented rhombohedral structure of the space group R3 m [11][12][13] . An additional peak appeared at 2θ ≈ 49.5°along with the characteristic peaks of the (00l) plane for all the samples and was identified as the (202) peak, originating from the misaligned plane of Bi 0.96 Sb 0.04 17 . The XRD patterns did not show any particular dependence on ϕ G , even though a technique of glancing incidence was employed considering that the penetration depth of the X-rays was much deeper than the implanted layer. This is possible because the XRD patterns do not reflect the (111) plane (L point of the BZ) directly related to the inversion symmetry of Bi 0.96 Sb 0.04 DSMs 12,18 . Figure 1a shows the ϕ G -dependent Raman spectrum of Bi 0.96 Sb 0.04 . At ϕ G = 0 (without implantation), two peaks are clearly observed at 72.3 and 97.5 cm −1 , referred to as E g (Bi) and A 1g (Bi), respectively, while one other peak is weakly visible at 119 cm −1 , referred to as E g (Sb) 17,19,20 . It is well known that the Bi 1−x Sb x system commonly exhibits four major Raman peaks, two of which are observed at~72-75 and~97-100 cm −1 and correspond to the Bi-Bi vibrations of the E g (Bi) and A 1g (Bi) modes, respectively, typical of single crystals with rhombohedral R3m symmetry 17,21 , and the other two peaks are observed at~118-120 and~138-141 cm −1 and correspond to the Sb-Sb vibrations of the E g (Sb) and A 1g (Sb) modes 17,19,20 . The A 1g mode is singly degenerate, while the E g mode is doubly degenerate. In our Bi 0.96 Sb 0.04 system, the A 1g (Sb) mode is absent before implantation, possibly due to local deviation from the stoichiometric condition, as the A 1g (Sb) mode is very sensitive to the atomic compositions of the crystal 19,20 .  16 Au cm −2 , but a drastic change is observed for fluences above ϕ G = 3.2 × 10 16 Au cm −2 . In particular, a new peak appears at 85.7 cm −1 between the E g (Bi) and A 1g (Bi) modes, named U(Bi). It is not clear whether the U(Bi) mode is split from the E g (Bi) mode or the A 1g (Bi) mode. Moreover, the A 1g (Sb) peak, absent in the pristine Bi 0.96 Sb 0.04 crystal, appears at 149.7 cm −1 , which is a higher wavenumber than that conventionally reported (~138-141 cm −1 ), indicating that the A 1g (Sb) peak is blueshifted by implantation to ϕ G = 3.2 × 10 16 Au cm −2 . The overall shape of the Raman spectrum for ϕ G = 3.2 × 10 16 Au cm −2 is almost unchanged as the fluence is increased to 12.8 × 10 16 Au cm −2 except for a gradual increase in the peak wavenumber. Some of the Raman spectra were resolved into the E g (Bi), A 1g (Bi), and U(Bi) bands to find their peak positions more accurately based on the fitting program ( Supplementary Fig. S3). Figure 1b summarizes the shifts of the Raman modes as functions of ϕ G . As ϕ G increases from 0 to 12.8 × 10 16 Au cm −2 , the four E g (Bi), U (Bi), A 1g (Bi), and A 1g (Sb) modes tend to be gradually blueshifted.
All the Raman peaks of MoTe 2 are known to stem from two kinds of vibrations: (1) those along the zigzag Mo atomic chain (z-modes) and (2) those in the mirror plane (m-modes) perpendicular to the zigzag chain 22 . Some Raman inactive modes in the centrosymmetric monoclinic phase appear in the cooling-driven transition to the orthorhombic phase 22,23 , resulting from inversionsymmetry breaking, which is evidence for the formation of a type II WSM. On the other hand, the appearance of a Raman doublet at a higher composition of W in monoclinic Mo 1−x W x Te 2 alloys was attributed to the breakdown of inversion symmetry, which originates from the random substitution of Mo atoms with W atoms 24 . These results suggest that whether inversion symmetry in a crystalline structure is broken can be determined by analyzing the Raman scattering behaviors.
Very recently, inversion symmetry breaking was also demonstrated by first-principle calculations and Raman scattering of the CdTiO 3 ilmenite phase belonging to the rhombohedral R3m group 21 . Additional Raman peaks, different from the E g and A g modes expected from group theory for the ilmenite rhombohedral structure, appeared in both low-temperature and high-pressure spectra, with the Raman wavenumbers being blueshifted at higher pressures. These results were very similar to the ϕ Gdependent Raman behavior observed here for the Bi 0.96 Sb 0.04 crystal and were successfully accounted for by assuming the local loss of inversion symmetry operation, which reduces the overall space group from R3 to R3. Such arguments can be extended to the Bi 0.96 Sb 0.04 crystal with the same rhombohedral R3 symmetry, suggesting that inversion symmetry breaking in the crystal occurred for implant fluences ϕ G ≥ 3.2 × 10 16 Au.cm −2 , resulting in the transition of the DSM Bi 0.96 Sb 0.04 to a Weyl semimetal.
MR measurements under parallel (B // E) (i.e., B parallel to the binary axis for LMR and perpendicular (B ⊥ E) (i.e., B parallel to the trigonal axis for TMR) orientations in TSMs always show different behavior, as orbital quantum interference effects are sensitive to the field orientation [25][26][27] . Parallel (B // E) measurements for the present samples are shown in Fig. 2a, with perpendicular (B ⊥ E) measurements included in Supplementary Fig. S4a. It has also been observed that the LMR in some TSMs shows a sharp increase as the B field is initially increased from zero, before decreasing at intermediate fields and then further increasing at higher fields. This phenomenon is referred to as negative LMR (NMR) [27][28][29][30] . Significantly, NMR is observed at T = 1.7 K and increases with increasing T up to 100 K, but above 100 K, it decreases. c LMR at 1.7 K for various implant fluences. Positive LMR is observed in the crystal without ion implantation, but NMR behavior becomes apparent for samples implanted to a fluence of ϕ G = 3.2 × 10 16 Au cm −2 and reaches a maximum for a fluence of ϕ G = 8.0 × 10 16 Au cm −2 before decreasing as the fluence is increased further to 12.8 × 10 16 Au cm −2 .
NMR is only observed in the B // E orientation and not in the B ⊥ E orientation 30,31 , consistent with current observations (Supplementary Fig. S4b). Figure 2b shows the T-dependent LMR of the Bi 0.96 Sb 0.04 crystal at ϕ G = 3.2 × 10 16 Au cm −2 , measured from −9 to +9 T. The amplitude of the NMR is observed at T = 1.7 K and increases with increasing T up to 100 K, but above 100 K, it decreases. Similar temperature-dependent behavior has been reported in WSMs by several groups. In Cd 3 As 2 nanowires 28 , the NMR increased as T increased from 1.5 to 60 K but decreased as it increased further from 80 to 300 K. The NMR found in Cd 3 As 2 nanoplates 32 and NbP 31 increased monotonically with increasing temperature up to 150 and 300 K, respectively. In contrast, the NMR in TaAs decreased monotonically with increasing T 27,29 . Figure 2c shows the variation in the LMR at 1.7 K for various implant fluences. Positive LMR is observed in the crystal with no ion implantation, but NMR behavior becomes apparent for samples implanted to a fluence of ϕ G = 3.2 × 10 16 Au cm −2 and reaches a maximum for a fluence of ϕ G = 8.0 × 10 16 Au cm −2 before decreasing as the fluence is increased further to 12.8 × 10 16 Au cm −2 . In contrast, only positive MR is observed from Mnimplanted Bi 0.96 Sb 0.04 crystals, irrespective of the relative orientation of the B and E fields and ϕ G ( Supplementary  Fig. S5). Implantation with Mn ions is expected to break both time reversal and inversion symmetries because Mn is a magnetic element, implying that no phase transition from DSM to WSM occurred and that the crystal was potentially changed into a trivial metallic state by Mn implantation. If no magnetic exchange interaction exists, conventional metals should exhibit positive MR 33,34 , and the appearance of a negative MR in normal metals is limited to special cases 35,36 . An important aspect of NMR is that it confirms the existence of a chiral anomaly in Weyl fermions, as shown in Cd 3 As 2 28,32 , Bi 0.97 Sb 0.03 37 , TaAs 27,29 , ZrTe 5 38 , Na 3 Bi 39 , and NaP 31 . These results are similar to the ϕ B -dependent Raman behavior shown in Fig. 1 and further suggest that the implantation-induced Dirac-to Weyl-semimetal phase transition is enabled by a nonmagnetic element (Au) rather than a magnetic element (Mn). Figure 3a shows Shubnikov-de Haas (SdH) oscillations as a function of 1/B as extracted from the ϕ G -dependent LMR data in Fig. 2c after background subtraction. Fast Fourier transform (FFT) analysis for ϕ G = 0, 3.2, and 12.8 × 10 16 Au cm −2 , shown in Fig. 3b, reveals the primary and secondary frequencies (F α and F β ) of the SdH oscillations, which are plotted in Fig. 3c as a function of ϕ G . The validity of the fitting was checked by separately plotting the two components of the SdH oscillations and summing them (Supplementary Fig. S6), resulting in almost coincidence of the summed oscillatory profiles with the original experimental data. The variation in F α is almost negligible, but F β increases abruptly at a fluence of ϕ G = 3.2 × 10 16 Au cm −2 .
The SdH quantum oscillations are usually described by the Lifshitz-Kosevich (LK) formula 8,[40][41][42][43] , and the corresponding parameters for both the α and β Fermi pockets can be obtained by fitting the FFT amplitude data based on the LK formula, as detailed previously 8,9,[40][41][42][43] . The SdH oscillation of resistance in a metal arises from the Landau quantization of electronic states under an applied B-field. Based on the Lifshitz-Onsager quantization rule WSMs possess nontrivial topological surface states that form unique Fermi arcs 1,2,46 and unusual Fermi surfaces consisting of unclosed curves that start and end at Weyl points split from a Dirac point. This explains why A F should increase in the Dirac-to Weyl-semimetal phase transition at ϕ G ≥ 3.2 × 10 16 Au cm −2 , as shown for the β Fermi pocket in Fig. 3d. In contrast, no meaningful variation in A F is visible for the α Fermi pocket, indicating that the phase transition occurs only in the β Fermi pocket. The A F of the β Fermi pocket (∼0.09 nm −2 in Fig.  3d) is extremely small, indicating linear band dispersion, as previously reported 47,48 . Combining this fact with the inversion symmetry breaking further explains why Bi 0.96 Sb 0.04 implanted to ϕ G ≥ 3.2 × 10 16 Au cm −2 should show NMR. It is well known that a π ϕ B is expected for DSMs such as the pristine Bi 0.96 Sb 0.04 crystal because their electronic band is topologically nontrivial 8,9,40,43 . As shown in Fig. 3h, the ϕ values for the α and β Fermi pockets show somewhat different dependences on ϕ G compared to the other parameters of the quantum oscillations, as shown above. They show gradually decreasing behavior with increasing fluence from 0 to 12.8 × 10 16 Au cm −2 , with the decreasing rate being much smaller for the α pocket. The ϕ values for the β Fermi pocket are within ±∼0.2. This near-zero ϕ, with zero being the widely accepted value for 3D DSMs/WSMs 8,49,50 , possibly indicates a nontrivial π ϕ B . An increase or decrease in the other parameters for the β Fermi pocket at ϕ G ≥ 3.2 × 10 16 Au cm −2 can be understood by simple physical considerations and the following related equations [40][41][42][43] : These results suggest that the α Fermi pocket remains a Dirac semimetal phase even after implantation. Figure 4a shows the ϕ G -dependent resistivity as a function of T under no B field. All the samples show an increase in resistivity with temperature, irrespective of ϕ G , typical of TSMs. Figure 4b-d shows ϕ G -dependent resistivity, carrier density, and mobility for the F β and F α Fermi pockets obtained by Hall-effect measurements. Abrupt changes at ϕ G = 3.2 × 10 16 Au cm −2 , as shown in the ϕ G -dependent behaviors of the Raman scattering and quantum oscillations, were not observed in the Hall mobility or carrier density, but both show ϕ G -dependent behavior for both Fermi pockets, consistent with what is expected from the fundamental characteristics of materials with larger impurity concentrations.

Conclusion
Here, we demonstrated a possible permanent Dirac-to Weyl-semimetal phase transition in a single DSM resulting from inversion-symmetry breaking induced by implantation with nonmagnetic Au ions. This phenomenon is evidenced by Raman analysis and the quantumoscillation parameters extracted from the LMR, which show abrupt changes at the critical fluence, and the negative LMR, indicating the existence of a chiral anomaly in Weyl fermions. The approach of implantation using nonmagnetic elements for phase transitions may be a suitable method for simple modifications of TSMs in the vertical direction as well as in the lateral direction and useful for diverse science studies and device applications.