Supporting Information for. Persistent charge-density-wave order in single-layer TaSe 2

Transcription

1 Supporting Information for Persistent charge-density-wave order in single-layer TaSe 2 Hyejin Ryu 1,2,,*, Yi Chen 3,, Heejung Kim 4, Hsin-Zon Tsai 3, Shujie Tang 1,5, Juan Jiang 1, Franklin Liou 3, Salman Kahn 3, Caihong Jia 3, 6, Arash A. Omrani 3, Ji Hoon Shim 4,7, Zahid Hussain 1, Zhi-Xun Shen 5,8, Kyoo Kim 2,4, Byung Il Min 4, Choongyu Hwang 9, Michael F. Crommie 3, 10, 11, and Sung-Kwan Mo 1, * 1 Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 2 Max Planck POSTECH Center for Complex Phase Materials, Pohang University of Science and Technology, Pohang 37673, Korea 3 Department of Physics, University of California, Berkeley, CA 94720, USA 4 Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea 5 Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA 6 Henan Key Laboratory of Photovoltaic Materials and Laboratory of Low-dimensional Materials Science, Henan University, Kaifeng , People s Republic of China 7 Department of Chemistry and Division of Advanced Nuclear Engineering, Pohang University of Science and Technology, Pohang 37673, Korea 8 Geballe Laboratory for Advanced Materials, Departments of Physics and Applied Physics, Stanford University, Stanford, CA 94305, USA 9 Department of Physics, Pusan National University, Busan 46241, Korea

2 10 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 11 Kavli Energy Nano Sciences Institute at the University of California Berkeley and the Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA These authors contributed equally to this work. *

3 Electronic structure calculation. We have performed total energy calculations for the distorted 3x3 supercell of single-layer TaSe 2 after slightly distorting the Ta atoms according to the proposal of a previous neutron scattering experiment 1. The CDW-modulated supercell is more stable (1.9 mev/f.u.) than the normal state with this distortion of Ta, which agrees with the previous calculation results 2. We then unfolded the band structure of the CDW-modulated supercell using a band unfolding scheme 3. Figure S1 Density of states (DOS) of single-layer TaSe 2 for a 3x3 supercell in normal state (dashed blue) and CDW state (solid red). Figure S1 shows a comparison of the total density of states (DOS) of single-layer TaSe 2 in the normal state versus the CDW state. The peak of the DOS near E F in the normal state is shifted down by about 100 mev, consistent with our ARPES observation. The DOS is suppressed by 50 % at E F in the CDW state showing the partial gap-opening.

4 Figure S2 The band structure of TaSe 2 bulk by employing the mbj functional. Figure S2 shows the band structure of TaSe 2 bulk using mbj functional. The binding energy of Se p band shifts down by 0.3 ev.

5 Figure S3 STS gap determination. (a) Numerical derivative d 2 I/dV 2 (blue curve) was calculated from experimental di/dv spectrum (black curve). The mean of d 2 I/dV 2 outside of the gap region defines the left and right slope floors (Ave L and Ave R ). The boundaries of the gap region are defined as the points where d 2 I/dV 2 differs from the slope floors by one standard deviation. (b) The background (grey curve) is determined from a polynomial fit to di/dv performed outside of the gap region (i.e. to the left and to the right of the gap boundaries). The background is then subtracted from the di/dv spectrum. The STS gap width is defined as the full width at half minimum (FWHM) of the resulting gap feature (brown curve). Single-layer TaSe 2 STS gap determination. The STS gap value of 15.3 ± 3.5 mev was determined through statistical analysis of 625 di/dv spectra on a grid at the surface of monolayer TaSe 2 (Fig. 5b). Fig. S3 shows the analysis procedure for a typical

6 STS spectrum, which was performed as follows: We first calculated the numerical derivative of the di/dv spectrum (Fig. S3a, black curve) to obtain the d 2 I/dV 2 spectrum (Fig. S3a, blue curve). The gap region is typically well within ± 30 mev for most curves and so we calculated the mean value of d 2 I/dV 2 from 50 mev to 30 mev and from 30 mev to 50 mev to determine the slope floors on the left and the right sides of the di/dv spectrum (Ave L and Ave R ), as well as the corresponding standard deviation (σ L and σ R ) to determine the level of fluctuations on either side. As the gap region is entered, a sudden change of slope in di/dv occurs. The gap boundary on the left is thus defined as the energy where d 2 I/dV 2 becomes significantly smaller (by σ L ) than the average (Ave L ) in the filled state regime, and on the right where d 2 I/dV 2 becomes significantly larger (by σ R ) than the average (Ave R ) in the empty state regime. After determining the gap boundaries, we performed a 6th-order polynomial fit to the di/dv spectrum in the ungapped region (outside the boundaries). This polynomial (Fig. S3b, grey curve) was then taken as the background and subtracted from the di/dv spectrum. The resulting spectrum (Fig. S3b, brown curve) exhibits a clear gap feature emerging from a flat background. The gap width is then taken as the full width at half minimum (FWHM) of this feature. The gap width defined this way is insensitive to the precise locations of the boundaries used to determine the background.

7 Figure S4 Statistics of STS gap. (a) Low-energy STS spectra obtained along the dashed line in (b). Grey triangles mark the gap width at each location determined using technique described in text. (c) Histogram of the STS gap width. Analysis of all 625 di/dv spectra in this way leads to an average gap value of 15.3 mev with an uncertainty of 3.5 mev determined from the standard deviation of the gap width 4. A subset of the di/dv data taken along the white dashed line in Fig. S4b is plotted in Fig. S4a and a histogram of the STS gap width is shown in Fig. S4c.

8 Figure S5 Single-layer TaSe 2 CDW bandgap determination by ARPES. Fitting of the ARPES spectra intensities at position A marked on Fig. 2a at 15 K. Single-layer TaSe 2 ARPES bandgap determination. The ARPES spectra EDC (black dots) at 15 K can be well fitted by the fitting curve (red solid line) composed of three Lorentzian peaks (orange, green, and blue dashed lines) and constant background (violet dashed line). The uppermost peak marked by orange arrow located at around -0,1 ev binding energy denotes that the CDW gap size is around 100±13 mev. All other EDCs at various temperatures displayed at Fig. 4a and b are fitted by the same method mentioned above which provide reasonable fitting results. Those fitting results are marked by red vertical tick marks on Fig. 4a and b. Mean field behavior of the temperature dependent gap. The mean-field behavior of the temperature dependent gap function of single-layer TaSe 2 is observed by fitting the

9 EDCs in Fig. 4c with a phenomenological BCS gap equation based on mean-field theory 5 : 140 K. ~ tanh 1. The fitting parameters are A and T c and the best fit is obtained for A = 1.16 and T c = References (S1) Moncton, D. E.; Axe, J. D.; DiSalvo, F. J. Phys. Rev. B 1977, 16, (2), (S2) Ge, Y.; Liu, A. Y. Phys. Rev. B 2012, 86, (10), (S3) Ku, W.; Berlijn, T.; Lee, C.-C. Phys. Rev. Lett. 2010, 104, (21), (S4) Taylor, J., Introduction to Error Analysis. 2nd ed.; University Science Books: New York, (S5) Chen, P.; Chan, Y. H.; Fang, X. Y.; Zhang, Y.; Chou, M. Y.; Mo, S. K.; Hussain, Z.; Fedorov, A. V.; Chiang, T. C. Nat. Commun. 2015, 6, 8943.