J. Semicond. > 2023, Volume 44 > Issue 1 > 011902
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REVIEWS

Review of phonons in moiré superlattices

Zhenyao Li1, 2, Jia-Min Lai1, 2, and Jun Zhang1, 2, 3,

+ Author Affiliations

 Corresponding author: Jia-Min Lai, laijiamin@semi.ac.cn; Jun Zhang, zhangjwill@semi.ac.cn

DOI: 10.1088/1674-4926/44/1/011902

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Abstract: Moiré patterns in physics are interference fringes produced when a periodic template is stacked on another similar one with different displacement and twist angles. The phonon in two-dimensional (2D) material affected by moiré patterns in the lattice shows various novel physical phenomena, such as frequency shift, different linewidth, and mediation to the superconductivity. This review gives a brief overview of phonons in 2D moiré superlattice. First, we introduce the theory of the moiré phonon modes based on a continuum approach using the elastic theory and discuss the effect of the moiré pattern on phonons in 2D materials such as graphene and MoS2. Then, we discuss the electron–phonon coupling (EPC) modulated by moiré patterns, which can be detected by the spectroscopy methods. Furthermore, the phonon-mediated unconventional superconductivity in 2D moiré superlattice is introduced. The theory of phonon-mediated superconductivity in moiré superlattice sets up a general framework, which promises to predict the response of superconductivity to various perturbations, such as disorder, magnetic field, and electric displacement field.

Key words: moiré pattern, moiré phonon, electron–phonon coupling, superconductivity



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Fig. 1.  (Color online) (a) Moiré pattern obtained by the overlapping of two similar fringes with different displacement and twist angles[27]. (b) Crystallographic superlattice and moiré superlattice in (5,7)-tBLM with θ = 10.99°. The green solid parallelogram represents the crystallographic superlattice unit cell and the red dashed parallelogram represents the moiré superlattice unit cell. (c) The reciprocal lattice of (5,7)-tBLM. The green and red regular hexagons correspond to the Wigner-Seitz primitive cells of the crystallographic superlattice and moiré superlattice, and the orange and blue hexagons represent the first Brillouin zone of the bottom and top MoS2, respectively. Reproduced with permission from Ref. [19]. Copyright 2018, ACS Publications.

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Fig. 2.  (Color online) Raman spectra of tBLMs and monolayer MoS2. The region of (a) is 50−365 cm−1 and (b) is 370–425 cm−1. Raman modes in various phonon branches are labeled by different colors and symbols. (c, d) The experimental and calculated frequencies of moiré phonons vary with $ |{\mathbf{g}}| $ and $ \theta $. The solid lines and the scatter symbols are theoretical and experimental results, respectively. (e, f) The peak position of $ {\text{F}}A{'_1} $ mode vary with $|{\boldsymbol{g}}|$ and $ \theta $. The peak position of $ A{'_1} $-related branch of the monolayer MoS2 (pink square) and various stacked BLMs (crossed circles) are also shown in (e). In (f), the theoretical phonon dispersion of $ A{'_1} $- related branch along the Γ–M and Γ–K directions in the monolayer MoS2 are shown as gray lines and the phonon dispersion of $ A{'_1} $- related branch of the monolayer MoS2 along ${\boldsymbol{g}}$ is shown as a dashed line. The stars are the experimental peak position of $ {\text{F}}A{'_1} $ in tBLMs. Reproduced with permission from Ref. [19]. Copyright 2018, ACS Publications.

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Fig. 3.  (Color online) (a, b) Calculated the low-energy evolution of phonon modes developed twist angle $ \theta $ at Γ. Color bars based on optical activity project the phonon eigenmodes onto the central Γ point. Optically inactive modes are shown by grey lines in (a) entirely originating at neighboring Γ points. Reproduced with permission from Ref. [21]. Copyright 2021, Springer Nature.

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Fig. 4.  (Color online) (a) The moiré pattern as seen in the tBLG. The first Brillouin zones of layers 1 and 2 are shown as two large hexagons, and small hexagons are moiré Brillouin zones of the tBLG. (b) Schematic illustrating the coupling of interlayer from the initial $ {\boldsymbol{k}} $ in layer 1 (given by as $ {\boldsymbol{K}}_ + ^{(1)} $) to the three $ {\boldsymbol{k}}' $ points in layer 2 in an undistorted tBLG, where $ {\boldsymbol{k}} $ is a two-dimensional Bloch wave vector. Reproduced with permission from Ref. [30]. Copyright 2020, American Physical Society. (c) Phonon dispersion in the inner asymmetric mode ($ {{\boldsymbol{u}}^ - } $) under different twist angles $ \theta $. (d) The gap width between the third and second branches of the phonon of $ {{\boldsymbol{u}}^ - } $ mode as a function of $ \theta $. The blue dashed line indicates the linear dependence on $ \theta $. (e) Group velocities of the second and first $ {{\boldsymbol{u}}^ - } $phonon modes dependent on $ \theta $. The velocities of transverse ($ {v_t} $) and longitudinal ($ {v_l} $) phonons in monolayer graphene are represented by horizontal dashed lines. Reproduced with permission from Ref. [20]. Copyright 2018, American Physical Society.

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Fig. 5.  (Color online) Illustration of the electron–phonon interaction in moiré superlattices. The springs representing phonons and a periodic electronic potential is on the top of the picture.

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Fig. 6.  (Color online) EPC strength $ {\bar g_n} $ as a function of the phonon band index n in various twist angles. Reproduced with permission from Ref. [30]. Copyright 2020, American Physical Society.

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Fig. 7.  (Color online) (a) Electron–phonon scattering in the extended Brillouin zone. The first Brillouin zone is shown as dashed hexagon marks; the blue circles are the Fermi surface. Yellow and orange arrows represent the umklapp processes and the purple arrow represents the normal processes. Both of them contribute to scattering. (b) Temperature dependence of the power of electron-lattice cooling for two different Wannier orbitals radii. (c) System resistivity and the resistivity from the sum of contributions from different phonon branches varying with the temperature of different electron–phonon processes. (d) At twist angles $ \theta $ = 1.20° and 1.05°, shape factors are shown for diamond and disk respectively, calculated of the continuum model at K points from the Wannier function. With different Wannier function $ \xi $, the grey lines are shape factors $ {g_q} $ proportional to Gaussian functions $ \exp ( - {\xi ^2}{q^2}/4) $. (e) Shape factors at $ \theta $ = 1.05° for different electron wave numbers $ {\boldsymbol{k}} $. Reproduced with permission from Ref. [32]. Copyright 2021, ACS Publication.

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Fig. 8.  (Color online) (a) PL spectra obtained measured from 1.356 to 1.377 eV in case of near-resonant excitation. Lorentzian functions filled are shown as black lines. (b) 2D PLE intensity map. The excess energy at 24 and 48 meV is shown with sloping black dashed lines. (c) Lorentzian fitted PLE spectra as a function of the excess energy of the PL spectra. The gray areas mean excess energy at 48 and 24 meV. Reproduced with permission from Ref. [35]. Copyright 2021, ACS Publication.

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Fig. 9.  (Color online) (a) Cross-plane thermoelectricity with different twist angles. The dotted lines show the linear $ T $ dependence of thermoelectric power. The solid lines show the fit of the thermoelectric power driven by phonons. Reproduced with permission from Ref. [26]. Copyright 2020, American Physical Society. (b) The thermal conductivity depending on twist angles of different temperatures. Reproduced with permission from Ref. [25]. Copyright 2021, AIP Publishing. (c) Calculated lattice thermal conductivities of tbBPs at 300 K with various twist angles at 300 K. Reproduced with permission from Ref. [37]. Copyright 2022, Wiley Publishing.

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Fig. 10.  (Color online) (a) Schematic of the tBLG devices fabricated on SiO2/Si substrates. (b) Current–voltage curves of two devices M1 and M2 measured in graphene superlattices and at different temperatures. Resistance Rxx was measured in two devices, with $ \theta $ = 1.05° and 1.16°, respectively. Reproduced with permission from Refs. [7, 40]. Copyright 2018, Nature. (c) Real-space map of pair amplitudes $ \Delta ({\boldsymbol{r}}) $ that satisfy s-wave linearized gap equations. (d) Real-space map of pair amplitudes $ \Delta ({\boldsymbol{r}}) $ that satisfy d-wave linearized gap equations. In (c) and (d), chemical potential $ \mu $= −0.3 meV and twist angle $ \theta $= 1.05°. Reproduced with permission from Ref. [22]. Copyright 2018, American Physical Society.

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Fig. 11.  (Color online) (a) Moiré surface state DOS with sharp higher-order Van hove singularities-like peaks at C6 potential with various potential $ U $[23]. (b) Transition temperature ${T_{\rm c}}$ at $ {v_F}/L $ = 435 K and ${\varepsilon _{\rm D}}$ = 80 K for different potential $ U $. The blue axis on the right and red axis on the left correspond to the ${T_{\rm c}}$, which shows broad peaks corresponding to higher-order Van Hove singularities around potentials. Reproduced with permission from Ref. [23]. Copyright 2021, American Physical Society.

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Table 1.   Electron–phonon coupling strength λi, mode-resolved. Reproduced with permission from Ref. [24]. Copyright 2021, American Physical Society.

Conditionλi (tBLG)λi (tTLG)λi (tDBLG)λi (tMBLG)
ωi = 10 meV0.2970.2330.0640.037
ωi = 167 meV0.9140.7430.0260.045
ωi = 197 meV0.6480.5320.0180.030
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Table 2.   Tc and D calculated for tBLG , tTLG , tDBLG, and tMBLG. Reproduced with permission from Ref. [24]. Copyright 2021, American Physical Society.

ParametertBLGtTLGtDBLGtMBLG
D (meV)0.530.673.46.7
Tc (μ = 0.15) (K)3.333.550.00.0
Tc (μ = 0.05) (K)3.453.6710–710–6
DownLoad: CSV
[1]
Encyclopedia Britannica. Chicago: British Encyclopedia Publishing Company, 2022
[2]
Kobayashi K. Moiré pattern in scanning tunneling microscopy: Mechanism in observation of subsurface nanostructures. Phys Rev B, 1996, 53, 11091 doi: 10.1103/PhysRevB.53.11091
[3]
N'Diaye A T, Bleikamp S, Feibelman P J, et al. Two-dimensional Ir cluster lattice on a graphene moiré on Ir(111). Phys Rev Lett, 2006, 97, 215501 doi: 10.1103/PhysRevLett.97.215501
[4]
Liu Y, Weiss N O, Duan X D, et al. Van der Waals heterostructures and devices. Nat Rev Mater, 2016, 1, 16042 doi: 10.1038/natrevmats.2016.42
[5]
Lopes Dos Santos J M B, Peres N M R, Castro Neto A H. Graphene bilayer with a twist: Electronic structure. Phys Rev Lett, 2007, 99, 256802 doi: 10.1103/PhysRevLett.99.256802
[6]
Bistritzer R, MacDonald A H. Moire bands in twisted double-layer graphene. Proc Natl Acad Sci USA, 2011, 108, 12233 doi: 10.1073/pnas.1108174108
[7]
Cao Y, Fatemi V, Fang S A, et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature, 2018, 556, 43 doi: 10.1038/nature26160
[8]
Huang D, Choi J, Shih C K, et al. Excitons in semiconductor moiré superlattices. Nat Nanotechnol, 2022, 17, 227 doi: 10.1038/s41565-021-01068-y
[9]
Schmitt D, Bange J P, Bennecke W, et al. Formation of moiré interlayer excitons in space and time. Nature, 2022, 608, 499 doi: 10.1038/s41586-022-04977-7
[10]
Tran K, Moody G, Wu F C, et al. Evidence for moiré excitons in van der waals heterostructures. Nature, 2019, 567, 71 doi: 10.1038/s41586-019-0975-z
[11]
Jin C H, Regan E C, Yan A M, et al. Observation of moiré excitons in WSe2/WS2 heterostructure superlattices. Nature, 2019, 567, 76 doi: 10.1038/s41586-019-0976-y
[12]
Yan J A, Ruan W Y, Chou M Y. Phonon dispersions and vibrational properties of monolayer, bilayer, and trilayer graphene: Density-functional perturbation theory. Phys Rev B, 2008, 77, 125401 doi: 10.1103/PhysRevB.77.125401
[13]
Nika D L, Balandin A A. Phonons and thermal transport in graphene and graphene-based materials. Rep Prog Phys Phys Soc G B, 2017, 80, 036502 doi: 10.1088/1361-6633/80/3/036502
[14]
Ribeiro-Palau R, Zhang C J, Watanabe K, et al. Twistable electronics with dynamically rotatable heterostructures. Science, 2018, 361, 690 doi: 10.1126/science.aat6981
[15]
Choi Y W, Choi H J. Strong electron-phonon coupling, electron-hole asymmetry, and nonadiabaticity in magic-angle twisted bilayer graphene. Phys Rev B, 2018, 98, 241412 doi: 10.1103/PhysRevB.98.241412
[16]
Jiang J W, Wang B S, Rabczuk T. Acoustic and breathing phonon modes in bilayer graphene with Moiré patterns. Appl Phys Lett, 2012, 101, 023113 doi: 10.1063/1.4735246
[17]
Cocemasov A I, Nika D L, Balandin A A. Phonons in twisted bilayer graphene. Phys Rev B, 2013, 88, 12 doi: 10.1103/PhysRevB.88.035428
[18]
Angeli M, Tosatti E, Fabrizio M. Valley jahn-teller effect in twisted bilayer graphene. Phys Rev X, 2019, 9, 041010 doi: 10.1103/PhysRevX.9.041010
[19]
Lin M L, Tan Q H, Wu J B, et al. Moiré phonons in twisted bilayer MoS2. ACS Nano, 2018, 12, 8770 doi: 10.1021/acsnano.8b05006
[20]
Koshino M, Son Y W. Moiré phonons in twisted bilayer graphene. Phys Rev B, 2019, 100, 075416 doi: 10.1103/PhysRevB.100.075416
[21]
Quan J M, Linhart L, Lin M L, et al. Phonon renormalization in reconstructed MoS2 moiré superlattices. Nat Mater, 2021, 20, 1100 doi: 10.1038/s41563-021-00960-1
[22]
Wu F C, MacDonald A H, Martin I. Theory of phonon-mediated superconductivity in twisted bilayer graphene. Phys Rev Lett, 2018, 121, 257001 doi: 10.1103/PhysRevLett.121.257001
[23]
Wang T G, Yuan N F Q, Fu L. Moiré surface states and enhanced superconductivity in topological insulators. Phys Rev X, 2021, 11, 021024 doi: 10.1103/PhysRevX.11.021024
[24]
Choi Y W, Choi H J. Dichotomy of electron-phonon coupling in graphene moiré flat bands. Phys Rev Lett, 2021, 127, 167001 doi: 10.1103/PhysRevLett.127.167001
[25]
Han S, Nie X H, Gu S Z, et al. Twist-angle-dependent thermal conduction in single-crystalline bilayer graphene. Appl Phys Lett, 2021, 118, 193104 doi: 10.1063/5.0045386
[26]
Mahapatra P S, Ghawri B, Garg M, et al. Misorientation-controlled cross-plane thermoelectricity in twisted bilayer graphene. Phys Rev Lett, 2020, 125, 226802 doi: 10.1103/PhysRevLett.125.226802
[27]
Wiki. Moiré pattern, 2022, https://en.wikipedia.org/wiki/Moir%C3%A9_pattern#Materials_science_and_condensed_matter_physics
[28]
Jung J, Raoux A, Qiao Z H, et al. Ab initiotheory of moiré superlattice bands in layered two-dimensional materials. Phys Rev B, 2014, 89, 205414 doi: 10.1103/PhysRevB.89.205414
[29]
San-Jose P, Gutiérrez-Rubio A, Sturla M, et al. Spontaneous strains and gap in graphene on boron nitride. Phys Rev B, 2014, 90, 075428 doi: 10.1103/PhysRevB.90.075428
[30]
Koshino M, Nam N N T. Effective continuum model for relaxed twisted bilayer graphene and moiré electron-phonon interaction. Phys Rev B, 2020, 101, 195425 doi: 10.1103/PhysRevB.101.195425
[31]
Lai J M, Farooq M U, Sun Y J, et al. Multiphonon process in Mn-doped ZnO nanowires. Nano Lett, 2022, 22, 5385 doi: 10.1021/acs.nanolett.2c01428
[32]
Ishizuka H, Fahimniya A, Guinea F, et al. Purcell-like enhancement of electron-phonon interactions in long-period superlattices: Linear-temperature resistivity and cooling power. Nano Lett, 2021, 21, 7465 doi: 10.1021/acs.nanolett.1c00565
[33]
Sun Z X, Hu Y H. How magical is magic-angle graphene. Matter, 2020, 2, 1106 doi: 10.1016/j.matt.2020.03.010
[34]
Tan P H. Signatures of moiré excitons. J Semicond, 2019, 40, 040202 doi: 10.1088/1674-4926/40/4/040202
[35]
Shinokita K, Miyauchi Y, Watanabe K, et al. Resonant coupling of a moiré exciton to a phonon in a WSe2/MoSe2 heterobilayer. Nano Lett, 2021, 21, 5938 doi: 10.1021/acs.nanolett.1c00733
[36]
Mahapatra P S, Sarkar K, Krishnamurthy H R, et al. Seebeck coefficient of a single van der waals junction in twisted bilayer graphene. Nano Lett, 2017, 17, 6822 doi: 10.1021/acs.nanolett.7b03097
[37]
Duan S, Cui Y, Yi W, et al. Enhanced thermoelectric performance in black phosphorene via tunable interlayer twist. Small, 2022, 2204197 doi: 10.1002/smll.202204197
[38]
Peltonen T J, Ojajärvi R, Heikkilä T T. Mean-field theory for superconductivity in twisted bilayer graphene. Phys Rev B, 2018, 98, 220504 doi: 10.1103/PhysRevB.98.220504
[39]
Lian B, Wang Z J, Bernevig B A. Twisted bilayer graphene: A phonon-driven superconductor. Phys Rev Lett, 2019, 122, 257002 doi: 10.1103/PhysRevLett.122.257002
[40]
Cao Y, Fatemi V, Demir A, et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature, 2018, 556, 80 doi: 10.1038/nature26154
[41]
Xu C K, Balents L. Topological superconductivity in twisted multilayer graphene. Phys Rev Lett, 2018, 121, 087001 doi: 10.1103/PhysRevLett.121.087001
[42]
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    Received: 15 October 2022 Revised: 23 November 2022 Online: Uncorrected proof: 07 December 2022Published: 14 January 2023

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      Article Navigation >  Journal of Semiconductors  > 2023  >  44(1): 011902
      Zhenyao Li, Jia-Min Lai, Jun Zhang. Review of phonons in moiré superlattices[J]. Journal of Semiconductors, 2023, 44(1): 011902. doi: 10.1088/1674-4926/44/1/011902 ****Z Y Li, J M Lai, J Zhang. Review of phonons in moiré superlattices[J]. J. Semicond, 2023, 44(1): 011902. doi: 10.1088/1674-4926/44/1/011902
      Citation:
      Zhenyao Li, Jia-Min Lai, Jun Zhang. Review of phonons in moiré superlattices[J]. Journal of Semiconductors, 2023, 44(1): 011902. doi: 10.1088/1674-4926/44/1/011902 ****
      Z Y Li, J M Lai, J Zhang. Review of phonons in moiré superlattices[J]. J. Semicond, 2023, 44(1): 011902. doi: 10.1088/1674-4926/44/1/011902
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      Review of phonons in moiré superlattices

      DOI: 10.1088/1674-4926/44/1/011902
      • 1.

        State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China

      • 2.

        Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China

      • 3.

        CAS Center of Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100049, China

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      • Zhenyao Li：is now a M.S. student supervised by Prof. Jun Zhang in the State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences. He received his bachelor's degree from Nankai University in China. His current research interest focuses on quantum optomechanics
      • Jia-Min Lai：is now a Ph.D. student supervised by Prof. Jun Zhang in the State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences. She received her bachelor's degree from Northeastern University in China. Her current research interest focuses on electron–phonon coupling in semiconductors
      • Jun Zhang：received a bachelor's degree from Inner Mongolia University in China in 2004, and a Ph.D. from the Institute of Semiconductors, Chinese Academy of Sciences in 2010. Then he worked as a postdoctoral fellow at Nanyang Technological University in Singapore from 2010 to 2015 and joined the State Key laboratory of Superlattice for Semiconductors (CAS) as a professor in 2015. His current researches focus on light-matter interactions in semiconductor materials including Raman and Brillouin scattering, and laser cooling in semiconductors
      • Corresponding author: laijiamin@semi.ac.cn; zhangjwill@semi.ac.cn
      • Received Date: 2022-10-15
      • Revised Date: 2022-11-23
        • Available Online: 2022-12-07

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