Two-dimensional ferromagnetism in CrTe flakes down to atomically thin layers

Mingshan Wang a, Lixing Kang b, Jianwei Su c, Luman Zhang a, Hongwei Dai a, Hui Cheng a, Xiaotao Han a, Tianyou Zhai c, Zheng Liu b and Junbo Han *a
aWuhan National High Magnetic Field Center and Department of Physics, Huazhong University of Science and Technology (HUST), Wuhan 430074, China. E-mail: junbo.han@mail.hust.edu.cn
bSchool of Materials Science & Engineering, Nanyang Technological University, Singapore 639798, Singapore
cState Key Laboratory of Material Processing and Die and Mould Technology, School of Materials Science and Engineering, Huazhong University of Science and Technology (HUST), Wuhan 430074, China

Received 29th May 2020 , Accepted 16th July 2020

First published on 17th July 2020


Abstract

Two-dimensional (2D) ferromagnetism has attracted intense attention as it provides a platform for the investigation of fundamental physics and the emerged devices. Recently, the discovery of intrinsic 2D ferromagnet has enabled researchers to fabricate ultrathin devices, which can be controlled by external fields. Nevertheless, 2D ferromagnetic materials are mostly obtained by mechanical exfoliation methods with uncontrollable size and thickness, which make the device fabrication processes time-consuming and difficult to expand in industries. Therefore, the development of a controllable fabrication process for the synthesis of 2D intrinsic magnetic materials is necessary. In this study, a new 2D ferromagnet, chromium tellurium (CrTe), was successfully synthesized by the chemical vapor deposition (CVD) method, and the magnetism was studied by the magneto-optical Kerr effect (MOKE) technique. The results demonstrated that CrTe flakes exhibit hard magnetism with strong perpendicular anisotropy. As the thickness varies from 45 nm to 11 nm, the hard magnetism sustains quite well, with the Curie temperature TC decreasing from 205 K to 140 K. Our study presents a new ultrathin hard magnetic material, which has the potential to be fabricated and applied in spintronic devices massively.


Introduction

Two-dimensional (2D) materials have attracted considerable attention owing to their excellent electrical,1–3 optical,4–6 magnetic properties,7,8 and potential corresponding applications. Recently, intrinsic 2D ferromagnetic materials have aroused particular interest for their important roles in exploring fundamental physics and emerging spintronic devices. To date, numerous atomically thin magnetic materials and devices have been achieved.9,10 For example, ferromagnetism has been observed in monolayer CrI3, and anti-ferromagnetism has been observed in bilayer CrI3[thin space (1/6-em)]7 in which ferromagnetism–anti-ferromagnetism transitions could be controlled via electrical doping,11 laser irradiation,12 and static pressure.13 Moreover, giant tunneling magnetoresistance has been realized in CrI3-based magnetic tunnel junctions.14 In addition, Cr2Ge2Te6 and Fe3GeTe2 were also explored,15,16 where gate-induced room-temperature ferromagnetic ordering has been achieved in four-layer Fe3GeTe2 flakes.17

It is worth noting that most of the reported 2D ferromagnetic materials, such as the devices listed above, are obtained via mechanical exfoliation and stacking techniques,7,15,16 which are suitable for constructing complex structures. However, the size and thickness of 2D materials obtained by exfoliation are uncontrollable, which makes difficult to apply them in the actual production. To solve this problem, new synthesis methods have been employed to grow intrinsic ferromagnets with atomic thickness. By using the molecular beam epitaxy (MBE) method, VSe2 and MnSex monolayers with ferromagnetism were successfully synthesized.18,19 Nevertheless, the complex processes and rigorous conditions limited its applications. The CVD method is regarded as a facile and effective method for synthesizing 2D materials, such as graphene, transition metal dichalcogenides, and hexagonal boron nitride, with high crystal quality.20–22 In addition, the CVD method can also be used to synthesize 2D non-van der Waals (vdW) materials with atomic thickness, whose material library is much extensive than the vdW library. It is exciting that some non-vdW magnetic materials, such as Cr2S3 and CrSe with controllable thickness and large area, have already been fabricated successfully.23–25

Recently, CrxTey began receiving attention due to its out-of-plane ferromagnetic properties.26,27 Bulk CrTe crystals were proven to be ferromagnetic with strong perpendicular magnetic anisotropy.28,29 This implies that ultrathin CrTe may maintain its ferromagnetic properties even when the thickness decreases to the nanometre scale. In addition, full-potential density-functional calculations indicated that the MOKE signal was quite large for CrTe.30 Hence, we synthesized ultrathin CrTe flakes by the CVD method and investigated their ferromagnetic properties by the MOKE technique. The fabricated 11 nm CrTe flake shows obvious hard magnetic properties with a rectangular hysteresis loop. In addition, the magnetic properties for the CrTe flakes, particularly the shape of the hysteresis loop, were affected by the thickness. From the thickness-temperature phase diagram, the three-dimensional (3D) to two-dimensional (2D) evolutions of the ferromagnetism in CrTe were explored, which improves the understanding of CrTe and enables spintronic applications.

Results and discussion

Fig. 1 presents the structure and basic characterization of the CrTe flakes. Fig. 1(a) shows the crystal structure of the CrTe single crystal. The figure reveals that CrTe possesses a hexagonal NiAs structure and is in the P63/mmc space group. The lattice parameter a is 0.3978 nm, and c is 0.6228 nm.31Fig. 1(b) shows the typical optical images of the CrTe flakes. By controlling the growth temperature, the thickness could be carefully varied. Fig. 1(c) shows a typical atomic force microscopy (AFM) image of a thin CrTe flake, for which the thickness was approximately 2.4 nm. To investigate the crystal structure and crystallization quality of the as-prepared CrTe flakes, the aberration-corrected scanning transmission electron microscopy (STEM)–annular dark-field (ADF) imaging was applied. Fig. 1(d) shows a STEM image with a crystal lattice spacing of 0.34 nm, which corresponds to the (100) planes of hexagonal-phase CrTe. In addition, the corresponding fast Fourier transform (FFT) pattern reveals a single set of spots, which further corroborates the high crystallinity of CrTe (inset in Fig. 1(d)). To verify the elemental composition of the CrTe crystal, energy dispersive spectroscopy (EDS) analysis was carried out. As shown in Fig. 1(e), the as-synthesized crystal contains Cr and Te elements. Further, the quantitative analysis demonstrates that the atomic ratio of Cr/Te was approximately 1 (see Fig. S1 and Table S1). Therefore, the CVD grown material can be confirmed as CrTe from the results of STEM, SEM, EDS, and elemental analysis. Besides, the Raman spectra for the CrTe flakes were also obtained, as shown in Fig. 1(f) and S2. The figure shows that in the scanning range from 50 to 700 cm−1, two obvious Raman peaks located at 123 cm−1 and 141 cm−1 are observed. However, as the thickness of the flakes increases, the two peaks changed differently. Peak 1 (P1) shows a blueshift, while peak 2 (P2) shows no shift. This phenomenon may be attributed to the coulombic interactions and changes in the intralayer bonding.32 The temperature-dependent Raman spectra were also obtained and are shown in Fig. S2(c) and (d). Similar to other 2D non-vdW materials, all three Raman peaks show obvious redshifts when the temperature increases from 85 to 300 K.24
image file: d0nr04108d-f1.tif
Fig. 1 Structure and characterizations of the CrTe flakes. (a) Crystal structure diagrams of CrTe. (b) Representative optical images of the CrTe flakes at different growth temperatures. (c) The AFM image of 2.4 nm CrTe flake. (d) The high-angle annular dark-field (HAADF)-STEM image of the CrTe flakes. (e) EDS maps of the Cr and Te elements. (f) The thickness-dependent Raman spectra of 11 nm, 15 nm, and 19 nm CrTe flakes.

MOKE and reflective magnetic circular dichroism (RMCD) were used to examine the magnetic properties for atomically thin CrTe flakes, from which the Curie temperature TC and coercive field HC could be obtained. Fig. 2(a) shows a schematic of the MOKE system. To obtain the out-of-plane ferromagnetism, a normal-irradiated polarized laser was used to detect the MOKE and RMCD signals of the samples. For MOKE and RMCD measurements, the signal was recorded with the magnetic field sweeping between −0.5 T and 0.5 T. For the Kerr mapping measurement, the θK of the whole sample was recorded point-by-point, and the sample was moved by a two-axis translation stage. Since tellurides are susceptible to ambient degradation, particularly for ultrathin CrTe flakes with a usual thickness of below 10 nm, we used a PMMA film to protect CrTe during all the MOKE and RMCD measurement procedures. Fig. 2(b) shows a typical hysteresis loop for the 11 nm CrTe flake at a temperature of 10 K. The hysteresis loop was nearly rectangular, where HC was 0.4 T, and θK was almost 19 mrad. The ratio of HC to the saturation field HS, (HC/HS) was almost equal to 1, revealing that all the spins were aligned parallelly due to strong perpendicular anisotropy, and the CrTe flake was an excellent hard magnet at low temperatures. The inset shows the optical image and MOKE mapping for the 11 nm CrTe flake, and Fig. S3 shows the corresponding AFM image. During the mapping measurement, the magnetic field first rises to 0.5 T to fully magnetize the sample and then falls to 0 T for spatial scanning. The regular hexagon, which is the same as the optical image, suggests that the MOKE signal was uniform on the surface of the sample. It is worth noticing that the CrTe flakes under the protection of PMMA were very stable. The MOKE result of the 11 nm CrTe flake remeasured after two weeks was almost identical to that of the first measured one (see Fig. S4), which guaranteed the data reliability of our work during the measurements.


image file: d0nr04108d-f2.tif
Fig. 2 The MOKE measurements for the 11 nm CrTe flake. (a) The schematic of the MOKE measurement of the CrTe flake. (b) Typical hysteresis loop for the CrTe flake measured at 10 K. The insets show the optical image and the corresponding MOKE map of the CrTe flake. (c) Hysteresis loop measured at different temperatures. (d) Extracted θK as a function of the temperature. The dots are experimental data, while the solid line is a fitted curve by using the function M = M0(1 − T/TC)β. (e) Extracted coercive field HC as a function of the temperature.

Fig. 2(c) shows θK as a function of the magnetic field for the 11 nm CrTe flake at different temperatures. As the temperature increased from 10 K, the hysteresis loop measured by the MOKE technique decreased and finally disappeared at 140 K, which means that the CrTe flake transforms from a ferromagnetic state to a paramagnetic state at approximately 140 K. To obtain the exact TC value of the CrTe flake, the temperature dependency of θK was investigated. As shown in Fig. 2(d), θK decreases slowly with the increase in temperature and then decreases rapidly near TC. By using the equation M = M0(1 − T/TC)β to fit the experimental data, TC was extracted to be approximately 140 K, where M is proportional to θK and β is the power index. Fig. 2e shows the HC as a function of the temperature extracted from Fig. 2c. As the temperature increases, HC drops rapidly and reaches zero at 140 K, where Tc is located.

In contrast to the traditional characterization of magnetic properties using electric measurements that cannot recognize the contribution of electrons with different energies, magneto-optical signals, including the θK and Kerr ellipticity, were wavelength dependent and closely dependent on the photoconductivity.30 As shown in Fig. 3(a), S5, and S6, the intensities of θK and the RMCD signals changed with the wavelength varying from 500 nm to 720 nm. The hysteresis loops were almost rectangular for all the MOKE and RMCD signals, while the signs of the saturation values differed from each other. To understand the sign alternation of the magneto-optical signals, the wavelength dependency of θK and the RMCD signals are shown in Fig. 3(b). Obviously, when the RMCD signal reached a peak value, the θK was zero and vice versa. This phenomenon is normal for the optical systems, where the real part and imaginary part of the optical function are not wholly independent but are connected by a special form Hilber transforms, which are termed as Kramers–Kronig relations.33θK and RMCD are such a pair of optical parameters that are connected by the real part and imaginary part of the complex polar Kerr rotation function,

 
image file: d0nr04108d-t1.tif(1)
where θK is the Kerr rotation angle and εK is the Kerr ellipticity, corresponding to the MOKE and RMCD signal.30σxx and σxy are optical conductivity elements of CrTe. Furthermore, the wavelength-dependent HC shows that the coercivity field remains unchanged for all the wavelengths at 80 K (inset of Fig. 3(b)), which implies that TC extracted from the temperature-dependent HC curves measured at different wavelengths should be identical.


image file: d0nr04108d-f3.tif
Fig. 3 Wavelength and temperature-dependent MOKE and RMCD signals for CrTe flakes. (a) Hysteresis loop measured at wavelengths from 500 nm to 720 nm at 80 K for 11 nm CrTe flake. (b) The extracted θK and RMCD signals as a function of wavelength. Inset: Coercivity field HC as a function of wavelength. (c) θK as a function of temperature for CrTe flakes with thicknesses of 11 nm, 15 nm, 19 nm, and 45 nm. The dots are the experimental data, while the solid lines fit the data with the function M = M0(1 − T/TC)β. (d) Hysteresis loops of the four CrTe flakes measured at 80 K. (e) MOKE signal and HC as a function of thickness. (f) Extracted θK and HC from the temperature-dependent MOKE signals for CrTe flakes.

For ferromagnetic materials, the sample thickness usually plays an important role in the determination of the TC and HC values. To demonstrate the evolution of the ferromagnetic properties of the CrTe flakes from the bulk to the 2D limit, the temperature dependency of θK was measured for the CrTe flakes with different thicknesses. Fig. 3(c) presents the four θKT curves for the CrTe flakes with thicknesses of 45 nm, 19 nm, 15 nm, and 11 nm. All the four θKT curves demonstrated standard ferromagnetic temperature dependencies. For each sample, θK decreased slowly with the increase in temperature and then decreased rapidly when the temperature nears TC. Besides, TC decreased as the thickness of the sample decreased. To demonstrate the differences in the CrTe films with different thicknesses, the normalized θK as a function of the magnetic field for the four samples measured at 80 K is summarized in Fig. 3(d). All the four samples demonstrated rectangular hysteresis loops, which indicate that the hard magnetic characteristics of CrTe were sustained well when the sample thickness decreased to 11 nm. When we extracted θK and HC values of the four curves from Fig. 3(d) and plotted them in Fig. 3(e), one can see that θK increases while HC decreases as the sample thickness decreases at 80 K. This leads to a significant increase in the θK/HC ratio for the thinner samples (see Fig. 3(f)), which may be useful for the application of thin CrTe crystals in magnetic memory devices with low energy consumption.

Fig. 4(a) demonstrates two hysteresis loops for the 39 nm thick CrTe flake taken at 120 K and 150 K. At 120 K, the hysteresis loop was almost rectangular. However, the hysteresis loop presents an intermediate process at 150 K. When the magnetic field sweeps upwards from −0.1 T, where the sample was completely magnetized, θK first jumps to an intermediate value at an intermediate magnetic field and then increases slowly to a saturable value. The same behavior occurs when the magnetic field is reversed, and the additional data can be found in Fig. S8. This phenomenon is popular in ferromagnetic thin samples, and the origination of the complicated magnetic states could be attributed to the formation of the labyrinthine magnetic domains inside the flakes.16,34,35


image file: d0nr04108d-f4.tif
Fig. 4 Thickness-temperature phase diagram of CrTe thin flakes. (a) Hysteresis loops of the 39 nm CrTe flake measured at 120 and 150 K. Inset: The corresponding optical image of the CrTe flake. (b) Thickness-temperature phase diagram. PM represents the paramagnetic state, FM1 represent the ferromagnetic state with single domain, and FM2 represent the ferromagnetic state with labyrinthine domains. The black square dots represent the TC for the single-domain ferromagnetic sample, the red circular dots represent the transition temperature TC1 between the single domain and labyrinthine-domain state, and the blue triangular dots represent TC2 between the ferromagnetic state and paramagnetic state.

Fig. 4(b) shows the thickness dependency of TC for CVD grown CrTe flakes. When the sample thickness is below 15.6 nm, there is only one magnetic states transition process, i.e., the ferromagnetic state to the paramagnetic state at TC. Above these thicknesses, when the temperature increased to the transition temperature TC1 (the method to determine the value of TC1 is demonstrated in Fig. S9 and the corresponding text), the intermediate process occurred and the labyrinthine domains were generated. When the temperature was above TC2, the magnetic state of the sample transformed from the ferromagnetic state to the paramagnetic state. Besides, it is clear that TC1 was close to TC2 when the intermediate process appeared first. As the thickness increased, TC1 decreased and departed from TC2, suggesting a thickness-induced formation of the magnetic labyrinthine domains.

The theory of the critical behavior reveals that the finite thickness of the flakes limits the divergence of the spin–spin correlation length at TC. The spin–spin coupling range along the out-of-plane direction can be fitted by using the formula:36

 
1 − TC(n)/TC(∞) = [(N0 + 1)/2n]λ(2)
where n is the thickness of the CrTe flake, TC (∞) is the TC for bulk sample, N0 is the spin–spin coupling range, and λ is a universal critical exponent. By using the eqn (1) and the data (the square and triangular points) shown in Fig. 4(b), the critical thickness was extracted to be 8.10 ± 1.58 nm.

Conclusions

A new 2D ferromagnet CrTe with a thickness that decreases to several nanometres was synthesized by the CVD method. The MOKE measurements showed that the CrTe thin flakes were good hard ferromagnets with strong out-of-plane ferromagnetic properties. The thickness-temperature phase diagram showed that the TC of CrTe decreased from 205 K to 140 K as the thickness decreased from 45 nm to 11 nm, and the critical thickness was extracted to be approximately 8.1 nm. Our work presents a new ultrathin hard magnetic material that would find practical applications in spintronic devices.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Scientific Foundation of China (No. 11704138 and 11404124) and Hubei Provincial Natural Science Foundation of China (2019CFA002). The authors also thank the Analytical and Testing Center at Huazhong University of Science and Technology for technical support.

Notes and references

  1. S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev and A. Kis, Nat. Rev. Mater., 2017, 2, 17033 CrossRef CAS .
  2. S. Das, D. Pandey, J. Thomas and T. Roy, Adv. Mater., 2019, 31, e1802722 CrossRef PubMed .
  3. Z. Lin, Y. Liu, U. Halim, M. Ding, Y. Liu, Y. Wang, C. Jia, P. Chen, X. Duan, C. Wang, F. Song, M. Li, C. Wan, Y. Huang and X. Duan, Nature, 2018, 562, 254–258 CrossRef CAS PubMed .
  4. K. F. Mak and J. Shan, Nat. Photonics, 2016, 10, 216–226 CrossRef CAS .
  5. A. Autere, H. Jussila, Y. Dai, Y. Wang, H. Lipsanen and Z. Sun, Adv. Mater., 2018, 30, e1705963 CrossRef PubMed .
  6. X. Wang, Y. Cui, T. Li, M. Lei, J. Li and Z. Wei, Adv. Opt. Mater., 2018, 7, 1801274 CrossRef .
  7. B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero and X. Xu, Nature, 2017, 546, 270–273 CrossRef CAS PubMed .
  8. K. S. Burch, D. Mandrus and J. G. Park, Nature, 2018, 563, 47–52 CrossRef CAS PubMed .
  9. C. Gong and X. Zhang, Science, 2019, 363, 706 CrossRef PubMed .
  10. M. Gibertini, M. Koperski, A. F. Morpurgo and K. S. Novoselov, Nat. Nanotechnol., 2019, 14, 408–419 CrossRef CAS PubMed .
  11. B. Huang, G. Clark, D. R. Klein, D. MacNeill, E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. McGuire, D. H. Cobden, D. Xiao, W. Yao, P. Jarillo-Herrero and X. Xu, Nat. Nanotechnol., 2018, 13, 544–548 CrossRef CAS PubMed .
  12. K. L. Seyler, D. Zhong, B. Huang, X. Linpeng, N. P. Wilson, T. Taniguchi, K. Watanabe, W. Yao, D. Xiao, M. A. McGuire, K. C. Fu and X. Xu, Nano Lett., 2018, 18, 3823–3828 CrossRef CAS PubMed .
  13. T. Song, Z. Fei, M. Yankowitz, Z. Lin, Q. Jiang, K. Hwangbo, Q. Zhang, B. Sun, T. Taniguchi, K. Watanabe, M. A. McGuire, D. Graf, T. Cao, J. H. Chu, D. H. Cobden, C. R. Dean, D. Xiao and X. Xu, Nat. Mater., 2019, 18, 1298–1302 CrossRef CAS PubMed .
  14. T. Song, X. Cai, M. W. Tu, X. Zhang, B. Huang, N. P. Wilson, K. L. Seyler, L. Zhu, T. Taniguchi, K. Watanabe, M. A. McGuire, D. H. Cobden, D. Xiao, W. Yao and X. Xu, Science, 2018, 360, 1214–1218 CrossRef CAS PubMed .
  15. C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia and X. Zhang, Nature, 2017, 546, 265–269 CrossRef CAS PubMed .
  16. Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu, D. H. Cobden, J. H. Chu and X. Xu, Nat. Mater., 2018, 17, 778–782 CrossRef CAS PubMed .
  17. Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen and Y. Zhang, Nature, 2018, 563, 94–99 CrossRef CAS PubMed .
  18. M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalappattil, R. Das, T. Eggers, H. R. Gutierrez, M. H. Phan and M. Batzill, Nat. Nanotechnol., 2018, 13, 289–293 CrossRef CAS PubMed .
  19. D. J. O'Hara, T. Zhu, A. H. Trout, A. S. Ahmed, Y. K. Luo, C. H. Lee, M. R. Brenner, S. Rajan, J. A. Gupta, D. W. McComb and R. K. Kawakami, Nano Lett., 2018, 18, 3125–3131 CrossRef PubMed .
  20. X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo and R. S. Ruoff, Science, 2009, 324, 1312–1314 CrossRef CAS PubMed .
  21. K. H. Lee, H. J. Shin, J. Lee, I. Y. Lee, G. H. Kim, J. Y. Choi and S. W. Kim, Nano Lett., 2012, 12, 714–718 CrossRef CAS PubMed .
  22. Y. Zhan, Z. Liu, S. Najmaei, P. M. Ajayan and J. Lou, Small, 2012, 8, 966–971 CrossRef CAS PubMed .
  23. Y. Zhang, J. Chu, L. Yin, T. A. Shifa, Z. Cheng, R. Cheng, F. Wang, Y. Wen, X. Zhan, Z. Wang and J. He, Adv. Mater., 2019, 31, e1900056 CrossRef PubMed .
  24. S. Zhou, R. Wang, J. Han, D. Wang, H. Li, L. Gan and T. Zhai, Adv. Funct. Mater., 2019, 29, 1805880 CrossRef .
  25. J. Chu, Y. Zhang, Y. Wen, R. Qiao, C. Wu, P. He, L. Yin, R. Cheng, F. Wang, Z. Wang, J. Xiong, Y. Li and J. He, Nano Lett., 2019, 19, 2154–2161 CrossRef CAS PubMed .
  26. X. Sun, W. Li, X. Wang, Q. Sui, T. Zhang, Z. Wang, L. Liu, D. Li, S. Feng, S. Zhong, H. Wang, V. Bouchiat, M. Nunez Regueiro, N. Rougemaille, J. Coraux, Z. Wang, B. Dong, X. Wu, T. Yang, G. Yu, B. Wang, Z. Vitto Han, X. Han and Z. Zhang, 2019, arXiv:1909.09797.
  27. Y. Wen, Z. Liu, Y. Zhang, C. Xia, B. Zhai, X. Zhang, G. Zhai, C. Shen, P. He, R. Cheng, L. Yin, Y. Yao, M. Getaye Sendeku, Z. Wang, X. Ye, C. Liu, C. Jiang, C. Shan, Y. Long and J. He, Nano Lett., 2020, 20, 3130–3139 CrossRef CAS PubMed .
  28. M. G. Sreenivasan, K. L. Teo, X. Z. Cheng, M. B. A. Jalil, T. Liew, T. C. Chong, A. Y. Du, T. K. Chan and T. Osipowicz, J. Appl. Phys., 2007, 102, 053702 CrossRef .
  29. T. Hirone and S. Chiba, J. Phys. Soc. Jpn., 1960, 15, 1991–1994 CrossRef CAS .
  30. H. Weng, Y. Kawazoe and J. Dong, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 74, 085205 CrossRef .
  31. J. Dijkstra, H. H. Weitering, C. F. v. Bruggen, C. Haas and R. A. d. Groot, J. Phys.: Condens. Matter, 1989, 1, 9141–9161 CrossRef CAS .
  32. C. Lee, H. Yan, L. E. Brus, T. F. Heinz, J. Hone and S. Ryu, ACS Nano, 2010, 4, 2695–2700 CrossRef CAS PubMed .
  33. V. Lucarini, J. Saarinen, K. Peiponen and E. Vartiainen, Kramers-Kronig Relations in Optical Materials Research, Springer, Berlin-Verlag Berlin Heidelberg, 2005, ISBN: 3-540-23673-2 Search PubMed .
  34. E. A. Jagla, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 094406 CrossRef .
  35. J. M. Deutsch and T. Mai, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2005, 72, 016115 CrossRef CAS PubMed .
  36. R. Zhang and R. F. Willis, Phys. Rev. Lett., 2001, 86, 2665–2668 CrossRef CAS PubMed .

Footnotes

Electronic supplementary information (ESI) available. See DOI: 10.1039/d0nr04108d
These authors contributed equally to this work.

This journal is © The Royal Society of Chemistry 2020