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In 1934, Eugene Wigner, who was studying at Princeton University, predicted the existence of electronic crystals. Electrons have both kinetic energy and potential energy of interaction. When the density of electronic states satisfies certain conditions, due to the repulsion between electrons, electrons will tend to arrange themselves in a regular lattice structure, forming electron crystals, which is also known as Wigner crystals. For nearly 90 years, Wigner crystals have fascinated condensed matter physicists. Physicists have designed many ingenious semiconductor heterojunctions to obtain lower electron densities and added magnetic fields to achieve larger effective mass of electron. In 1979, experiments revealed the existence of a phase transition from an electron liquid phase to an electron crystal on the surface of liquid helium, and subsequent experiments observed the characteristics of two-dimensional (2D) Wigner crystals in 2D electron gas under high magnetic fields. However, direct observation of 2D Wigner lattices in real space remains a formidable challenge. Through the graphene sensing layer of WSe2/WS2 moiré superlattice, Hongyuan Li, Feng Wang, et al. observed the real-space morphologies of Wigner crystals in their experiments. And in a recent study, researchers used high-resolution scanning tunneling microscopy to directly image magnetic field-induced Wigner crystals in bernal stacking bilayer graphene and investigated their structural properties as a function of electron density, magnetic field, and temperature. In this paper, we will introduce some interesting things about Wigner crystals through four representative researches briefly.
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Keywords:
- Wigner crystals /
- electron solids /
- quantum phase transitions /
- scanning tunneling microscopy
[1] Wigner E 1934 Phys. Rev. 46 1002
Google Scholar
[2] Tanatar B, Ceperley D M 1989 Phys. Rev. B 39 5005
Google Scholar
[3] Crandall R, Williams R 1971 Phys. Lett. A 34 404
Google Scholar
[4] Andrei E Y, Deville G, Glattli D C, Williams F I B, Paris E, Etienne B 1988 Phys. Rev. Lett. 60 2765
Google Scholar
[5] Tsui Y C, He M, Hu Y, Lake E, Wang T, Watanabe K, Taniguchi T, Zaletel M P, Yazdani A 2024 Nature 628 287
Google Scholar
[6] Grimes C C, Adams G 1979 Phys. Rev. Lett. 42 795
Google Scholar
[7] Monarkha Y P, Shikin V B 1975 J. Exp. Theor. Phys 41 710
[8] Fisher D S, Halperin B I, Platzman P M 1979 Phys. Rev. Lett. 42 798
Google Scholar
[9] Girvin S M, Macdonald A H, Platzman P M 1985 Phys. Rev. Lett. 54 581
Google Scholar
[10] Yoon J, Li C C, Shahar D, Tsui D C, Shayegan M 1999 Phys. Rev. Lett. 82 1744
Google Scholar
[11] Hubbard J 1978 Phys. Rev. B 17 494
Google Scholar
[12] Li H, Li S, Regan E C, et al. 2021 Nature 597 650
Google Scholar
[13] Liu X, Farahi G, Chiu C L, Papic Z, Watanabe K, Taniguchi T, Zaletel M P, Yazdani A 2022 Science 375 321
Google Scholar
[14] Farahi G, Chiu C L, Liu X, Papic Z, Watanabe K, Taniguchi T, Zaletel M P, Yazdani A 2023 Nat. Phys. 19 1482
Google Scholar
[15] Coissard A, Wander D, Vignaud H, et al. 2022 Nature 605 51
Google Scholar
[16] Li S Y, Zhang Y, Yin L J, He L 2019 Phys. Rev. B 100 085437
Google Scholar
[17] Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559
Google Scholar
[18] Falson J, Sodemann I, Skinner B, et al. 2021 Nat. Mater. 21 311
Google Scholar
[19] Santos M B, Suen Y W, Shayegan M, Li Y P, Engel L W, Tsui D C 1992 Phys. Rev. Lett. 68 1188
Google Scholar
[20] Yang F, Zibrov A A, Bai R, Taniguchi T, Young A F 2021 Phys. Rev. Lett. 126 156802
Google Scholar
[21] Zhou Y, Sung J, Brutschea E, et al. 2021 Nature 595 48
Google Scholar
[22] Nazarov Y V, Khaetskii A V 1994 Phys. Rev. B 49 5077
Google Scholar
[23] Li H, Xiang Z, Reddy A P, et al. 2024 Science 385 86
Google Scholar
[24] Li H, Xiang Z, Wang T, et al. 2024 Nature 631 765
Google Scholar
[25] Hossain M S, Ma M K, Rosales K A V, et al. 2020 Proc. Nat. Acad. Sci. 117 32244
Google Scholar
[26] Kosterlitz. J M, Thouless. D J 1973 J. Phys. C: Solid State Phys. 6 1181
Google Scholar
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图 1 (a)随着温度的降低, 突然出现了耦合等离激元-涟波子共振, 该共振仅在 0.457 K 以下出现, 此时电子平面已结晶成三角形晶格; (b)经典二维电子平面的固液相界部分, 数据点表示在不同的电子平均密度$ {N}_{{\mathrm{s}}} $下测量到的熔化温度, 在图中的线上, 每个电子的势能与动能之比$ \varGamma = 137 $[6]
Figure 1. (a) Experimental traces displaying the sudden appearance with decreasing temperature of coupled plasmon-ripplon resonances. The resonances only appear below 0.457 K where the sheet of electrons has crystallized into a triangular lattice. (b) Portion of the solid-liquid phase boundary for a classical, two-dimensional sheet of electrons. The data points denote the melting temperatures measured at various values of the electron areal density, $ {N}_{{\mathrm{s}}} $. Along the line, the quantity Γ, which is a measure of the ratio of potential energy to kinetic energy per electron, Γ is 137[6].
图 2 密度为$ 0. 77\times {10}^{11}\;{{\mathrm{c}}{\mathrm{m}}}^{-2} $ 时在 28 T 和 60 mK 下的吸收光谱(填充因子$ \nu $ = 1/8.7), 在 $ f{\text{-}}{p}^{3/2} $ 的附图中, $ p $ 值的选择是为了实线经过原点, 虚线是对低混合模式频率的零阶先验计算[4]
Figure 2. Absorption spectra at 28 T and 60 mK for a density of $ 0. 77\times {10}^{11}\;{{\mathrm{c}}{\mathrm{m}}}^{-2} $ (filling factor ν = 1/8.7). In the accompanying plots of $ f{\text{-}}{p}^{3/2} $), the value of p is chosen so that the solid line passes through the origin; the dashed line is the zeroth-order a-priori computation of the frequency of the low-mixing mode[4].
图 3 (a) n = 1 莫特绝缘体的dI/dV 图($ {V}_{{\mathrm{b}}{\mathrm{i}}{\mathrm{a}}{\mathrm{s}}} $ = 160 mV, $ {V}_{{\mathrm{B}}{\mathrm{G}}} $ = 30 V和$ {V}_{{\mathrm{T}}{\mathrm{G}}} $ = 0.53 V); (b) n = 2/3广义维格纳晶体态的dI/dV 图($ {V}_{{\mathrm{b}}{\mathrm{i}}{\mathrm{a}}{\mathrm{s}}} $ = 160 mV, $ {V}_{{\mathrm{B}}{\mathrm{G}}} $ = 21.8 V和$ {V}_{{\mathrm{T}}{\mathrm{G}}} $ = 0.458 V); (c) n = 1/3广义维格纳晶体态的dI/dV图($ {V}_{{\mathrm{b}}{\mathrm{i}}{\mathrm{a}}{\mathrm{s}}} $ = 130 mV, $ {V}_{{\mathrm{B}}{\mathrm{G}}} $ = 14.9 V和$ {V}_{{\mathrm{T}}{\mathrm{G}}} $ = 0.458 V); (d) n = 1/2广义维格纳晶体态的dI/dV 图($ {V}_{{\mathrm{b}}{\mathrm{i}}{\mathrm{a}}{\mathrm{s}}} $ = 125 mV, $ {V}_{{\mathrm{B}}{\mathrm{G}}} $ = 18.7 V和$ {V}_{{\mathrm{T}}{\mathrm{G}}} $ = 0.458 V); (e) 样品结构示意图[12]
Figure 3. (a) dI/dV diagrams for n = 1 Mott insulator ($ {V}_{{\mathrm{b}}{\mathrm{i}}{\mathrm{a}}{\mathrm{s}}} $ = 160 mV, $ {V}_{{\mathrm{B}}{\mathrm{G}}} $ = 30 V and $ {V}_{{\mathrm{T}}{\mathrm{G}}} $ = 0.53 V); (b) dI/dV diagrams for n = 2/3 generalized Wigner crystal states ($ {V}_{{\mathrm{b}}{\mathrm{i}}{\mathrm{a}}{\mathrm{s}}} $ = 160 mV, $ {V}_{{\mathrm{B}}{\mathrm{G}}} $ = 21.8 V and $ {V}_{{\mathrm{T}}{\mathrm{G}}} $ = 0.458 V); (c) dI/dV diagrams for n = 1/3 generalized Wigner crystal states ($ {V}_{{\mathrm{b}}{\mathrm{i}}{\mathrm{a}}{\mathrm{s}}} $ = 130 mV, $ {V}_{{\mathrm{B}}{\mathrm{G}}} $ = 14.9 V and $ {V}_{{\mathrm{T}}{\mathrm{G}}} $ = 0.458 V); (d) dI/dV diagrams for n = 1/2 generalized Wigner crystal states ($ {V}_{{\mathrm{b}}{\mathrm{i}}{\mathrm{a}}{\mathrm{s}}} $ = 125 mV, $ {V}_{{\mathrm{B}}{\mathrm{G}}} $ = 18.7 V and $ {V}_{{\mathrm{T}}{\mathrm{G}}} $ = 0.458 V); (e) schematic diagram of sample structure[12].
图 4 (a) 200 nm×200 nm区域内的空间分辨隧穿电流调制$ {\text{δ}} {I}_{{\mathrm{d}}{\mathrm{c}}} $, 测量采用$ {V}_{{\mathrm{B}}} $ = 4.6 mV, 填充因子ν = 0.317, 图中刻度长度为50 nm; (b) 图4(a)的FFT图像[5]
Figure 4. (a) Spatially resolved tunneling current modulation $ {\text{δ}} {I}_{{\mathrm{d}}{\mathrm{c}}} $ in the 200 nm × 200 nm region, measured with $ {V}_{{\mathrm{B}}} $ = 4.6 mV and fill factor ν = 0.317, scale length in the panel is 50 nm. (b) FFT of the tunnelling current modulation $ {\text{δ}} {I}_{{\mathrm{d}}{\mathrm{c}}} $ in panel (a)[5] .
图 5 (a)—(h)在一系列不同的填充因子ν下测量的同一区域的空间分辨隧穿电流调制$ {\text{δ}} {I}_{{\mathrm{d}}{\mathrm{c}}} $, $ {V}_{{\mathrm{B}}} $分别为5.2, 5.2, 4.6, 4.4, 4.4, 7.2, 8.0和8.8 mV, 磁场B = 13.95 T, 图中刻度长度为100 nm; (i)—(p)相应的图(a)—(h)中隧穿电流调制$ {\text{δ}} {I}_{{\mathrm{d}}{\mathrm{c}}} $的结构因子S(q), 图中刻度长度为$ 0.2\;{{\mathrm{n}}{\mathrm{m}}}^{-1} $[5]
Figure 5. (a)–(h) Spatially resolved tunneling current modulation $ {\text{δ}} {I}_{{\mathrm{d}}{\mathrm{c}}} $ in the same region measured at a series of different filling factors ν, with $ {V}_{{\mathrm{B}}} $ of 5.2, 5.2, 4.6, 4.4, 4.4, 7.2, 8.0, and 8.8 mV, and a magnetic field B = 13.95 T, the scale lengths in the plots is 100 nm; (i)–(p) corresponding to the tunneling current modulation $ {\text{δ}} {I}_{{\mathrm{d}}{\mathrm{c}}} $ in panel (a)–(h) of the structure factor S(q), scale length in the plots is $ 0.2\;{{\mathrm{n}}{\mathrm{m}}}^{-1} $ [5].
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[1] Wigner E 1934 Phys. Rev. 46 1002
Google Scholar
[2] Tanatar B, Ceperley D M 1989 Phys. Rev. B 39 5005
Google Scholar
[3] Crandall R, Williams R 1971 Phys. Lett. A 34 404
Google Scholar
[4] Andrei E Y, Deville G, Glattli D C, Williams F I B, Paris E, Etienne B 1988 Phys. Rev. Lett. 60 2765
Google Scholar
[5] Tsui Y C, He M, Hu Y, Lake E, Wang T, Watanabe K, Taniguchi T, Zaletel M P, Yazdani A 2024 Nature 628 287
Google Scholar
[6] Grimes C C, Adams G 1979 Phys. Rev. Lett. 42 795
Google Scholar
[7] Monarkha Y P, Shikin V B 1975 J. Exp. Theor. Phys 41 710
[8] Fisher D S, Halperin B I, Platzman P M 1979 Phys. Rev. Lett. 42 798
Google Scholar
[9] Girvin S M, Macdonald A H, Platzman P M 1985 Phys. Rev. Lett. 54 581
Google Scholar
[10] Yoon J, Li C C, Shahar D, Tsui D C, Shayegan M 1999 Phys. Rev. Lett. 82 1744
Google Scholar
[11] Hubbard J 1978 Phys. Rev. B 17 494
Google Scholar
[12] Li H, Li S, Regan E C, et al. 2021 Nature 597 650
Google Scholar
[13] Liu X, Farahi G, Chiu C L, Papic Z, Watanabe K, Taniguchi T, Zaletel M P, Yazdani A 2022 Science 375 321
Google Scholar
[14] Farahi G, Chiu C L, Liu X, Papic Z, Watanabe K, Taniguchi T, Zaletel M P, Yazdani A 2023 Nat. Phys. 19 1482
Google Scholar
[15] Coissard A, Wander D, Vignaud H, et al. 2022 Nature 605 51
Google Scholar
[16] Li S Y, Zhang Y, Yin L J, He L 2019 Phys. Rev. B 100 085437
Google Scholar
[17] Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559
Google Scholar
[18] Falson J, Sodemann I, Skinner B, et al. 2021 Nat. Mater. 21 311
Google Scholar
[19] Santos M B, Suen Y W, Shayegan M, Li Y P, Engel L W, Tsui D C 1992 Phys. Rev. Lett. 68 1188
Google Scholar
[20] Yang F, Zibrov A A, Bai R, Taniguchi T, Young A F 2021 Phys. Rev. Lett. 126 156802
Google Scholar
[21] Zhou Y, Sung J, Brutschea E, et al. 2021 Nature 595 48
Google Scholar
[22] Nazarov Y V, Khaetskii A V 1994 Phys. Rev. B 49 5077
Google Scholar
[23] Li H, Xiang Z, Reddy A P, et al. 2024 Science 385 86
Google Scholar
[24] Li H, Xiang Z, Wang T, et al. 2024 Nature 631 765
Google Scholar
[25] Hossain M S, Ma M K, Rosales K A V, et al. 2020 Proc. Nat. Acad. Sci. 117 32244
Google Scholar
[26] Kosterlitz. J M, Thouless. D J 1973 J. Phys. C: Solid State Phys. 6 1181
Google Scholar
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