## Abstract

Defect engineering is one of the key technologies in materials science, enriching the modern semiconductor industry and providing good test-beds for solid-state physics. While homogenous doping prevails in conventional defect engineering, various artificial defect distributions have been predicted to induce desired physical properties in host materials, especially associated with symmetry breakings. Here, we show layer-by-layer defect-gradients in two-dimensional PtSe_{2} films developed by selective plasma treatments, which break spatial inversion symmetry and give rise to the Rashba effect. Scanning transmission electron microscopy analyses reveal that Se vacancies extend down to 7 nm from the surface and Se/Pt ratio exhibits linear variation along the layers. The Rashba effect induced by broken inversion symmetry is demonstrated through the observations of nonreciprocal transport behaviors and first-principles density functional theory calculations. Our methodology paves the way for functional defect engineering that entangles spin and momentum of itinerant electrons for emerging electronic applications.

## Introduction

Controlling defects in a pristine material is an alternative method for manipulating physical properties and activating new functionalities of a material. Its outstanding applications, especially by doping, have enriched the modern semiconductor industry and developed a distinct research area, called defect engineering. Generating defects in a target material is commonly implemented through atomic or layered implantation during material growth that guarantees uniform or periodic doping to bulky materials. Beyond atomic implantation, post-treatment, e.g., plasma treatment, is an alternative way for local defect engineering notably in nanoscale materials^{1,2,3}. In particular, its inequivalent reaction from the surface of a binary material could generate selective defect sites. This simple but elaborate manipulation has brought highly improved performance in electronics and photonics with astonishing results such as phase transitions, carrier concentrations, carrier type, etc^{4,5,6,7,8,9}.

Spintronics, exploiting both spin and charge of electrons, has been recently benefited from the defect engineering in imprinting a key feature into materials. One of the pivotal goals of defect engineering in spintronic applications is breaking time-reversal symmetry by activating magnetic moments. Intensive efforts over the past decades have been made to develop dilute magnetic semiconductors by integrating random distribution of magnetic dopants^{10}. Defect-induced magnetisms in two-dimensional (2D) materials have also been proposed in a number of theoretical studies^{11,12,13}. Various forms of point defects in 2D structures stabilize local magnetic moments, as experimentally shown in such as graphene^{14} and transition metal dichalcogenides (TMDs)^{15,16}. Another key challenge in defect engineering for spintronics could be inversion symmetry breaking, which leads to the Rashba interaction. The coupling between spin and momentum of itinerant electrons offers an alternative channel for spin sources without ferromagnets. Theoretical studies showed inversion symmetry breaking in 2D materials could be triggered by inducing organized defect distribution^{17,18}. However, experimental realization of such periodic defect distribution is unattainable. Thus, it is highly desired to develop a capable approach to break inversion symmetry, which imprints the Rashba interaction in pristine materials and allows electric control of spins in device applications.

2D TMD could be considered as one of the most appropriate platforms for defect engineering due to its ideal crystallinity even in a low dimension. Their various elements and combinations allow to utilize diverse and selective reaction for constitute atoms, especially by plasma treatment^{2,9,19}. Mild plasma treatment with Ar gas can make selective vacancies as it etches chalcogen elements, and plasma with CHF_{3} and SF_{6} gas pulls out transition metal atoms accompanied by chemical reaction. Further, the energetic reaction of plasma treatment could affect the subsequent TMD layers as dispersing excessive energy. In particular, weak interlayer coupling may allow to withstand layer-by-layer compositional difference. Thus, well-refined plasma treatment with a proper TMD can generate a unique system accommodating a gradient of defects along the layers. As the resulting structural gradient indicates the broken inversion symmetry in the system, an approach to develop a defect-gradient in TMD layers could be the innovative method to activate a Rashba system.

In this study, we demonstrate the generation of a defect gradient and the Rashba effect in 1T-phase PtSe_{2} thin films through plasma treatment. A structural analysis using scanning transmission electron microscopy shows a defect-gradient along PtSe_{2} layers in atomic resolution. The Rashba effect induced by the broken inversion symmetry from the defect-gradient is demonstrated by nonreciprocal transport measurement, and first-principles calculations support comprehensive understanding for the defect-gradient and Rashba effect in PtSe_{2}. This study introduces a facile approach to instill the Rashba interaction in layered systems, which couples spin and momentum of electrons for various spin-orbitronic applications.

## Results

### Generation of a defect-gradient in PtSe_{2}

The host material, van der Waals layered PtSe_{2} is the best candidate for this study because of its confined and controllable reaction to plasma, a symmetric structure of a pristine film, and compatibility and air stability for device applications. Surface treatments to mechanically exfoliated PtSe_{2} films were performed through plasma etching with Ar and SF_{6} mixed gas producing atomically smooth surface modifications (see the Method section and Supplementary Fig. 1). The Ar gas aims for etching of Se atoms, while SF_{6} chemically reacts with Pt atoms to be vaporized (Fig. 1a). Dispersion of plasma with excessive energy can affect the subsequent TMD layers, which could generate a layer-by-layer gradient defect density, as illustrated in Fig. 1a. Then, a defect-gradient could lead to structural asymmetry, which can derive the Rashba-type spin-splitting in a plasma-treated PtSe_{2} film. Figure 1b displays the scanning electron microscope image of a studied PtSe_{2} sample (yellow shaded area) for nonreciprocal magneto-transport measurement. The Rashba spin-orbit interaction significantly affects magneto-transport properties as the effective Rashba field (*B*_{R}) points the opposite direction for forward and reverse currents (*I*_{x}), producing nonreciprocal charge transport along the PtSe_{2} film. We used mechanically exfoliated PtSe_{2} flakes, which had a long rectangular shape along the [100] crystal direction (see Supplementary Fig. 2).

### Structural analysis through atomic resolution STEM

For the atomic level investigation of a plasma effect on the structural variation in a PtSe_{2} film, we employed scanning transmission electron microscopy (STEM). A layer-dependent elemental analysis probed by energy dispersive x-ray spectroscopy (EDS) and electron energy loss spectroscopy (EELS) showed Se vacancies with a steep gradient along the layers. Figure 2a is a cross-sectional high-angle annular dark-field (HAADF) STEM image of a plasma-treated PtSe_{2} film. Compared to the deep region of a PtSe_{2} film with high crystallinity, plasma-treated PtSe_{2} layers near the surface exhibited a decomposed crystal structure. To clarify a change in local environment of Pt, we performed x-ray photoelectron spectroscopy (XPS) analysis for both a pristine film and a plasma-treated PtSe_{2} film. A pristine PtSe_{2} film exhibited a clear Pt^{4+} and Se^{2−} state (Supplementary Fig. 3), but plasma treatment altered some of Pt^{4+} elements to Pt^{2+} and Pt^{0} states. A layer-dependent compositional variation was investigated by employing EDS analysis. Figure 2b shows the observed atomic ratio between Se and Pt according to the depth of the film from the surface. A PtSe_{2} layer deeper than 7 nm showed nearly a constant ratio of Se/Pt as 2. However, the Se/Pt ratio steeply decreased down to 1.1 as the layer got close to a surface. These steep variations of the Se/Pt ratio were repeatedly confirmed by analyzing more than 7 different regions (Supplementary Fig. 4). An EDS mapping image also displayed depth-dependent deficiency of Se atoms (the inset of Fig. 2b). EELS data (in Fig. 2c) verifies an atomic gradient of a Se component as compared with Se *M*_{4,5} and Pt *N*_{6,7} states at 57 eV and 73 eV, respectively^{20}. Since the element ratio is related to the inelastic excitation intensity ratio^{21}, the decreased Se *M*_{4,5} intensity compared to the Pt *N*_{6,7} near a surface region indicates the formation of a defect-gradient from a surface to deep PtSe_{2} layers, which is consistent with the EDS result. Additionally, the depth profile of EELS spectra of the Pt *N*_{6,7} (in Fig. 2d) showed a slightly red-shift of 0.4 eV, and Pt *N*_{6} (76 eV) becomes dominant compared to Pt *N*_{7} (73 eV) as close to the surface (Fig. 2e). These behaviors can be attributed to the change in the oxidation number of Pt and defect-mediated density of state (DOS) variation because an oxidation state is related to the excitation edge shift^{22}, and the fine structure of core loss EELS is related to the conduction band DOS^{23}. In contrast, a pristine PtSe_{2} film showed homogenous and well-crystalline structure throughout all regions (Supplementary Fig. 5). Therefore, plasma treatment induces a steep gradient of Se vacancies with a finite skin depth of 7 nm in a PtSe_{2} film, establishing a strongly inversion-broken region nearby the surface.

### Magneto-transport analysis for defect-gradient PtSe_{2} films

The formation of defects with a finite depth through plasma treatment can be verified through the residual-resistance ratio (RRR). Figure 3a–c displays temperature-dependent longitudinal resistance (*R*_{xx}) for various thickness of PtSe_{2} films with and without plasma treatment. The sample thicknesses of a defect-gradient PtSe_{2} (i.e., w/ plasma) indicate a final thickness after plasma treatment. Resistances of the samples increased with increasing temperature regardless of plasma treatment, but the RRR, defined as *R*_{xx} (300 K)/*R*_{xx} (2 K), displayed a significant variation. The RRR of pristine PtSe_{2} films was higher than 3, whereas plasma-treated PtSe_{2} films exhibited much smaller ratio, down to 1.30 (see the values in Supplementary Table 1). Significantly reduced RRR can be attributed to plasma-induced defects in a PtSe_{2} film. The variation of RRR due to plasma treatment decreased with increasing the thickness of PtSe_{2} (Supplementary Fig. 6a), because the plasma effect had a finite skin depth. The defect-gradient formation also affected to the carrier concentration on PtSe_{2} films. The obtained carrier concentrations via Hall measurements were 3.2 × 10^{21} cm^{−3} and 3.6 × 10^{20} cm^{−3} for a pristine PtSe_{2} film of 10 nm and a plasma-treated PtSe_{2} film of 10 nm at 2 K, respectively (Supplementary Fig. 7). Evidence of induced defects in a PtSe_{2} film was also revealed by the magnetoresistance (MR), defined as (*R*_{xx}(*B*)−*R*_{xx} (*B* = 0))/*R*_{xx}(*B* = 0), upon a perpendicular magnetic field (*B*_{z}) (Fig. 3d–f). The magnitude of ordinary MR was significantly reduced in a plasma-treated film, because it is proportional to ~*μ*^{2}*B*^{2} and the carrier mobility *μ* is lower for defective samples^{24}. Here, the suppression of ordinary MR was also more drastic for a thin PtSe_{2} film (Supplementary Fig. 6b), because of the finite depth of the plasma effect.

### Nonreciprocal transport in defect-gradient PtSe_{2} films

The Rashba effect induced by a defect-gradient in a low dimensional system can be probed through transport characteristics. One striking electrical manifestation of the inversion symmetry breaking is the nonreciprocal charge transport as observed in various polar materials^{25,26,27,28,29,30}. Applying a magnetic field in a Rashba system can break the degeneracy between a left and a right mover with opposite spins resulting in directional charge transport. This nonreciprocal response is empirically described as \(R={R}_{0}(1+\gamma (B\times z)\cdot I)\) where *R*_{0} is the resistance in a zero magnetic field, *γ* is the coefficient tensor, *z* is the polarization, and *I* is the current^{25}. Here, Δ*R* = *R*(*I*)–*R*(–*I*) shows maximum magnitude when both the direction of a magnetic field and a current are normal to the electric field due to structural asymmetry. The observation of nonreciprocal charge transport in a single nonmagnetic film reflects the presence of the Rashba spin-orbit interaction, as shown in several noncentrosymmetric systems^{25,26,28,30}. Figure 4a displays a schematic illustration for the characterizations of nonreciprocal charge transport. A current was applied in the *x*-axis and a longitudinal voltage was measured while applying a magnetic field. Figure 4b shows the nonreciprocal MR in a plasma-treated PtSe_{2} film (10 nm) in response to the direction of a DC current during a magnetic field sweep in the *y*-axis. In the presence of large *B*_{y}, the significant difference of resistance can be observed for a forward and a reverse current flow. For a positive *B*_{y}, resistance for the forward current flow was higher, while the resistance for the reverse current flow became higher for a negative *B*_{y}. This reversed MR feature clearly represents directional charge transport following, \(\triangle R/{R}_{0} \sim \gamma \,(B\times z)\cdot I\). The MR curves did not display any difference according to the direction of magnetic field sweeps (Supplementary Fig. 8), confirming the observed effects did not originate from sample degradation during the measurement. A small MR feature near *B* = 0 T represents an anisotropic magnetoresistance (AMR) from Pt-defect-induced magnetism in a thin PtSe_{2} film as observed in previous reports^{15,16}(see further explanation in Supplementary Fig. 9). An inset in Fig. 4b shows the observed nonreciprocal MR, defined as ΔMR = MR(+*I*)–MR(−*I*), is linear to the magnitude of an applied magnetic field. The magnitude of *γ* is estimated to be 5 × 10^{−2} A^{−1} T^{−1} that is comparable to those observed in Bi helix^{31}, chiral organic conductor^{27}, and α-GeTe^{32}. On the other hand, a pristine PtSe_{2} film without plasma treatment did not exhibit any asymmetry in the same MR measurements (Supplementary Fig. 10). In addition to the control MR measurement, we confirmed that both pristine and plasma-treated PtSe_{2} films formed the Ohmic contact to Au electrodes (Supplementary Fig. 11), which allowed us to exclude a contact effect on our nonreciprocal results.

The observed nonreciprocal MR could be simply understood as a result of the current-induced effective Rashba field, which is normal to a current flow in a polar system, as illustrated in Fig. 1b. Estimation of this effective Rashba field (*B*_{R}) can be done by measuring angle-dependent MR (ADMR) in the *yx* plane. Figure 4c shows the measured ADMR, defined as (*R*_{xx}(*θ*)−*R*_{xx}(*θ* = 0))/*R*_{xx}(*θ* = 0), while rotating a device in the *yx* plane. When a positive current was applied, ADMR at +*y* (*θ* = 0°) was higher than ADMR at −*y* (*θ* = 180°), but with a negative current, ADMR at +*y* was lower than ADMR at −*y*. This asymmetric behavior is equivalent to the nonreciprocal MR shown in Fig. 4b. The magnitude and direction of the effective magnetic field (*B*_{eff}) under the rotation of an applied magnetic field (*B*_{a}) are given as \({B}_{{{{{{\rm{eff}}}}}}}^{2}={B}_{{{{{{\rm{a}}}}}}}^{2}+{B}_{{{{{{\rm{R}}}}}}}^{2}+2{B}_{{{{{{\rm{a}}}}}}}{B}_{{{{{{\rm{R}}}}}}}{{\cos }}\theta\) and \(\alpha ={{{{{\rm{artan}}}}}}\left[{B}_{{{{{{\rm{a}}}}}}}{{\cos }}\theta /({B}_{{{{{{\rm{R}}}}}}}+{B}_{{{{{{\rm{a}}}}}}}{{\cos }}\theta )\right]\), respectively. Then, ADMR is proportional to \(c{B}_{{{{{{\rm{eff}}}}}}}{{{\cos }}}^{2}\alpha\), where *c* is constant^{33,34}. Here, an estimated Rashba field was 4.81 T for *I*_{x} = +500 μA at *B* = +8 T. Further measurements with varying currents showed that the current-induced effective Rashba field was linearly proportional to applied currents (Supplementary Fig. 12). The observed nonreciprocal MR even persisted up to room-temperature (Fig. 4d and Supplementary Fig. 13). The directional transport behaviors were also studied for various thicknesses of plasma-treated PtSe_{2} thin films (Supplementary Figs. 14–17). The magnitude of nonreciprocal MR tended to decrease with increasing the thickness of the PtSe_{2} thin film from 10 nm to 15 nm. This tendency could be attributed to decreasing the effective degree of the Rashba effect in a whole film due to the finite depth of a plasma effect. When the thickness of PtSe_{2} decreased below 8 nm, the material evolved into a semiconducting phase^{35,36} (Supplementary Fig. 18). In this regime, the observed nonreciprocal MR decreases with decreasing thickness (Supplementary Figs. 14 and 15).

We further investigated on the nonreciprocal MR using an AC current (\(I={I}_{0}{{\sin }}\omega t\)) with phase-sensitive detection. In the same geometry displayed in Fig. 4a, a nonreciprocal voltage in response to an AC current for a magnetic field angle *θ* can be described as \({V}^{2\omega }(t)=\gamma {R}_{0}B{I}_{0}{{\sin }}\theta {{\sin }}\omega t\times {I}_{0}{{\sin }}\theta =\frac{1}{2}\gamma {R}_{0}B{I}_{0}^{2}{{\sin }}\theta \,\{1+{{\sin }}(2\omega t-\frac{\pi }{2})\}\). Here, a nonreciprocal resistance can be directly detected from the out-of-phase component, as \({R}^{2\omega }=\frac{{V}^{2\omega }}{{I}_{0}}=\frac{1}{2}\gamma {R}_{0}B{I}_{0}{{\sin }}\theta\)^{28}. Figure 4e displays angle-dependent *R*^{2ω} measured with the applied magnetic field of 8 T at *T* = 2 K. Under the rotation of a magnetic field (+8 T), a sinusoidal resistance curve appeared with the maximum amplitude at +*y* and −*y*. Applying the opposite field of ‒8 T resulted in an inverted angle-dependent *R*^{2ω} curve representing unidirectional characteristics of MR. Figure 4f shows that *R*^{2ω} is responsive only to the *y*-component of a magnetic field with linear dependence. *R*^{2ω} also exhibited linear dependence on applied currents following \(\triangle R/{R}_{0} \sim \gamma (B\times z)\cdot I\) (Supplementary Fig. 19). In short, our magneto-transport study clearly showed all the characteristics of nonreciprocal charge transport, which is the signature of the Rashba effect induced by a defect-gradient in a PtSe_{2} system.

### First-principles DFT calculation for a defect-gradient PtSe_{2}

In consistent with the nonreciprocal transport measurements, density functional theory (DFT) calculations clearly show that asymmetric distribution of the Se vacancy in PtSe_{2} layers gives rise to a large Rashba spin-splitting at the Fermi level. To describe the experimental situation, we prepared a slab geometry that was composed of 10 layers of PtSe_{2} and 20 Å of vacuum as shown in Fig. 5a. By adopting the virtual crystal approximation in our calculations (see Methods), only the topmost 5 layers contain Se vacancy and the other layers are a pristine region without any vacancy in the PtSe_{2} film. By interpolating the result of STEM analysis in Fig. 2b, the amount of Se vacancy was the largest at the first layer and linearly decreased to zero below the sixth layer in our simulations. Figure 5b, c, shows a DFT band structure without and with spin-orbit coupling (SOC), respectively. Colors in each band denote the Se/Pt ratio of the Bloch wavefunction; the darker color indicates the larger amount of Se vacancies. In the absence of spin-orbit coupling, each band is doubly degenerated, and no spin-splitting appears as shown in Fig. 5b. Once spin-orbit coupling is turned on, the bands fall into two different categories (Fig. 5c). The dark-colored bands originating from the Se-deficient region show large Rashba-type spin-splitting, whereas the bright ones from a pristine PtSe_{2} region still maintain the spin degeneracy. Electrons residing in the Se-deficient region only feel the inversion-breaking field caused by a vacancy-gradient. Estimating the Rashba coefficient of the band at the Fermi level yields 2.2 eVÅ along the ΓK line in the Brillouin zone. This value is comparable to that found in giant Rashba materials such as BiTeI (3.8 eVÅ)^{37} or (MA)PbI_{3} (1.4~1.5 eVÅ)^{38}. Pristine PtSe_{2} layers do not show spin-splitting regardless of spin-orbit coupling (Supplementary Fig. 20). Therefore, both spin-orbit coupling and the inversion-breaking field generated by a defect-gradient are essential ingredients that make Rashba-type spin-splitting.

## Discussion

In summary, we successfully demonstrated the generation of a defect-gradient and a resulting Rashba effect in van der Waals 2D layered PtSe_{2}. Optimal plasma treatment with Ar and SF_{6} gases enabled a selective etching process, which induced a defect-gradient from the surface. STEM analysis revealed that the induced defect-gradient was dominated by Se vacancies and the gradient along layers was formed down to 7 nm depth. Broken spatial inversion symmetry by the defect-gradient brought out the Rashba effect in the PtSe_{2} film that was evidenced in directional charge transport. In particular, the observed nonreciprocal charge transport persisted even up to room-temperature, which can be utilized for functional two-terminal devices. Theoretical calculations clearly showed the emergence of the Rashba-type spin-splitting by disposing a gradient of Se vacancies along PtSe_{2} layers. Our study suggests that selective plasma treatment to binary elemental TMD layers could be a facile methodology to develop a defect-gradient along the layers, which could be a new platform to activate the Rashba system for various electronic and spintronic applications. This approach would be also applied for other van der Waals 2D materials standing for a novel and state-of-art technology in the defect engineering.

## Methods

### Sample preparation

PtSe_{2} thin films were prepared through mechanical exfoliation from a bulk PtSe_{2} (purchased from HQ Graphene). PtSe_{2} films exfoliated on polydimethylsiloxane (PDMS) were transferred onto substrates such as *p*-Si/SiO_{2} (300 nm), α-Al_{2}O_{3}, and TEM grid, depending on analyses. A reactive ion etching (Labstar, TTL) process with Ar and SF_{6} mixed gases was performed to the prepared PtSe_{2} film. Various amounts of the mixed gas, up to 100 sccm depending on conditions, were used with a same ratio of Ar and SF_{6}. The etching decreased the thickness of a PtSe_{2} film with the etching rate of 0.03 nm/s. Roughness and thickness of each sample were confirmed by using AFM (Veeco) measurement before and after an etching process. The initial thickness of a PtSe_{2} flake to prepare a plasma-treated PtSe_{2} was usually 20 nm to 30 nm. This thickness should be, at least, 7 nm-thicker than a target thickness after plasma treatment, to make a 7 nm defect-gradient in a PtSe_{2} film. Then, identical structures and results can be obtained for the same target thickness of PtSe_{2} films regardless of their initial thicknesses. XPS (Thermo Fisher) analysis was conducted with a K-alpha spectrometer for an 0.9 mm diameter spot in a UHV chamber (<10^{−10} Torr).

### STEM analysis

A cross-sectional STEM specimen was prepared by a focused ion beam (Helios 450HP FIB) with a carbon coating to protect the surface of the specimen during ion milling. STEM data was obtained using aberration-corrected FEI Titan Cubed G2 60-300 STEM and all measurements were performed at 200 keV. The convergence angle of the STEM incident beam was 26.6 mrad with a probe current of ~60 pA. An EDS line profile and a spectrum-imaging process were performed using a Super-X system attached to the STEM. EELS spectra were recorded by a Gatan Quantum 965 dual EELS system attached to the STEM. EELS measurements were performed with an energy resolution of 1.0 eV in a 0.1 eV/channel energy dispersion. Dual EELS measurements were used to simultaneously acquire both the zero-loss and core-loss EELS spectra at each spot to compensate for energy drift during spectrum acquisition.

### Magneto-transport measurement

PtSe_{2} flakes were mechanically exfoliated for the transport study. Long and rectangular shape of PtSe_{2} flakes less than 3 μm width was used for a transport device, because the narrow flakes were highly stable during device fabrication and measurements compared to wide PtSe_{2} flakes. Four-terminal contacts on a prepared PtSe_{2} film were patterned by using photolithography. Ti (5 nm) and Au (40 nm) were deposited on the electrode patterns in a UHV chamber (<10^{−8} Torr). The deposition chamber had a long distance (~700 mm) between a source and a sample to avoid unwanted damage to samples. All transport measurements were performed in a physical property measurement system (PPMS, Quantum Design) with a horizontal sample rotator. Keithley 2636 A sourcemeter and Keithley 2182 nanovoltmeter were used for DC measurement. For AC measurement, Keithley 6221 was used to supply an AC current with 10 Hz and the AC signal of the first- and the second-harmonic voltage was detected by SR830 DSP lock-in amplifier (Stanford Research).

### First-principles calculation

Density functional theory calculations were carried out within the local-density-approximation (LDA) with fully relativistic spin-orbit coupling using VASP code^{39,40}. A plane wave cut-off energy of 500 eV and 7 × 7 × 1 k-points was used for all calculations. Virtual crystal approximation (VCA) was used to describe the vertically-varying distribution of Se vacancy^{41}. In VCA, we constructed the thin film using virtual atoms averaged by Se and vacancy, finally producing the defect distribution shown in Fig. 2b. The VCA has been successfully applied to studies on, for example, topological phase transitions that require very precise handling of structural symmetry and SOC^{42,43,44,45}. An in-plane lattice constant was fixed at the experimental value of PtSe_{2}. Internal atomic positions were fully relaxed until the maximum force was below 5 meVÅ^{−1} while the interlayer distances were constrained to their experimental value during the relaxation for stability of the calculations. Qualitatively equivalent results were confirmed even if the interlayer distance was fully relaxed. To obtain the average value of the Se/Pt ratio of the *n*^{th} Bloch state |*ψ*_{nk}〉, we calculated the inner product of the Bloch wavefunction with each atomic orbital, \({{{{{{\mathcal{P}}}}}}}_{{{{{{\bf{k}}}}}}}^{\alpha }=\mathop{\sum}\limits_{{lm}}{\left|\left\langle {Y}_{{lm}}^{\alpha },|,{\psi }_{n{{{{{\bf{k}}}}}}}\right\rangle \right|}^{2}\), where \({Y}_{{lm}}^{\alpha }\) is the spherical harmonics centered at ion index \(\alpha\) and \(l,m\) are the angular moment and magnetic moment quantum numbers, respectively. By using the linearly-interpolated value of the Se/Pt ratio in each atom site \(\left({{{{{{\mathscr{w}}}}}}}^{\alpha }\right)\), the average value of the Se/Pt ratio of the *n*^{th} Bloch wavefunction at **k** is given by \({{{{{{\mathcal{W}}}}}}}_{n{{{{{\bf{k}}}}}}}=\mathop{\sum}\limits_{\alpha }{{{{{{\mathscr{w}}}}}}}^{\alpha }{{{{{{\mathcal{P}}}}}}}_{n{{{{{\bf{k}}}}}}}^{\alpha }\).

## Data availability

All data are available in the main text or the Supplementary Information. Additional data related to the findings of this study may be requested from the authors.

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## Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2020M3F3A2A03082444, 2021R1A2C1008431, and 2020R1F1A1070279). This work was also supported by the Ulsan National Institute of Science and Technology (No. 1.200095.01). J. Jo acknowledges the support of Basic Science Research Program through the NRF funded by the Ministry of Education (2020R1A6A3A03039086). J.H.K. and Z.L. acknowledge the support of Institute for Basic Science (IBS-R019-D1). H.J. acknowledges the support of the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2021M3H4A1A0305486411, 2019R1A2C1010498 and 2017M3D1A1040833). C.H.K. was supported by the Institute for Basic Science (IBS-R009-D1)

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J.J. and J.-W.Y. conceived and designed the project. J.J. and J.L. fabricated experimental devices. J.H.K. and Z.L. did TEM/STEM/EDS measurement and analysis. J.J. and J.L. performed magneto-transport measurement and analysis, and S.L., D.C., and I.O. assisted the experiment. C.H.K. and H.J. did density functional theory calculations. J.-W.Y. supervised this project. All the authors participated in the discussion and the preparation of a manuscript.

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Jo, J., Kim, J.H., Kim, C.H. *et al.* Defect-gradient-induced Rashba effect in van der Waals PtSe_{2} layers.
*Nat Commun* **13**, 2759 (2022). https://doi.org/10.1038/s41467-022-30414-4

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DOI: https://doi.org/10.1038/s41467-022-30414-4

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