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Graphene Dirac plasmons, which are collective oscillations of charge carriers behaving as massless Dirac fermions, have emerged as a transformative platform for nanophotonics due to their exceptional capability for deep subwavelength light confinement in the infrared-to-terahertz spectral region and their unique dynamic tunability. Although external controls such as electrostatic doping, mechanical strain, and substrate engineering are empirically known to be able to modulate plasmonic responses, a comprehensive and quantitative theoretical framework from first principles is essential to reveal the distinct efficiency and fundamental mechanisms of each tuning strategy. To address this issue, we conduct a systematic first-principles study of three primary modulation pathways—carrier density, biaxial strain, and substrate integration—by using linear-response time-dependent density functional theory in the random-phase approximation (LR-TDDFT-RPA) as implemented in the computational code ABACUS. A truncated Coulomb potential is adopted in order to accurately model the isolated two-dimensional system, while structural and electronic properties are computed using the PBE functional with SG15 norm-conserving pseudopotentials and van der Waals corrections for heterostructures. Our research results indicate that modulating carrier concentration can cause the plasmon dispersion to follow the characteristic $\omega \propto n^{1/4}$ scaling law, thereby tuning within a wide range from 0.45 eV to 1.38 eV at the Landau damping threshold—a 207% change for the carrier density varying from 0.005 electrons per unit cell to 0.1 electrons, although efficiency decreases at higher concentrations due to the sublinear nature of the scaling law. Biaxial strain linearly changes the plasmon energy by modifying the Fermi velocity ($v_{\mathrm{F}}$) near the Dirac point, yielding a 30.4% tuning range (0.78–1.12 eV) under $\pm 10{\text{%}}$ strain. Introducing an hBN substrate induces a small band gap ($\sim 43$ meV) and causes a general redshift in plasmon energy due to band renormalization, while remarkably preserving the linear strain-tuning capability in a $30.1{\text{%}}$ energy range (0.72–1.03 eV) in the heterostructure, demonstrating robust compatibility between strain engineering and substrate integration. These results quantitatively elucidate the different physical mechanisms—Fermi level shifting, Fermi velocity modification, and substrate-induced symmetry breaking and hybridization—underpinning each strategy, thereby providing a solid theoretical foundation for designing dynamically tunable optoelectronic devices based on graphene and its van der Waals heterostructures.
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Keywords:
- first-principles calculations, graphene /
- graphene /
- linear-response time-dependent density functional theory /
- Dirac plasmons
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图 1 石墨烯的电子结构及等离激元分布 (a) 石墨烯的倒空间结构及高对称点位置与高对称路径方向, 红线和蓝线分别表示从K点出发的$ \varGamma-K $方向和$ \varGamma-M $方向; (b) 石墨烯能带图, 其中黑色曲线表示占据态, 红色曲线表示非占据态; (c) 石墨烯态密度图, 黑色线表示总态密度, 红色线为π能带态密度分布, 蓝色线为σ能带态密度分布; (d) 石墨烯在0.03电子/原胞掺杂浓度时沿$ \varGamma-M $方向不同q值对应的电子能量损失谱, 其中q的取值范围为0.029/Å 到0.294/Å
Figure 1. Graphene’s electronic structure and plasmon distributions (a) the reciprocal space structure of graphene along with the positions of high-symmetry points and the directions of high-symmetry paths, where the red and blue lines represent the $ \varGamma-K $ and $ \varGamma-M $ directions from the K point, respectively; (b) graphene's band structure, where the black curve denotes the occupied states and the red curve represents the unoccupied states; (c) graphene's density of states (DOS), with the black line indicating the total DOS, the red line showing the projected DOS of the π-band, and the blue line displaying the projected DOS of the σ-band; (d) electron energy loss spectroscopy (EELS) of graphene along the $ \varGamma-M $ direction for different q values at 0.03 electrons/unit cell concentration, where q ranges from 0.029/Å to 0.294/Å.
图 2 石墨烯狄拉克等离激元分布及色散图 (a) 沿$ \varGamma-M $方向的分布; (b) 沿$ \varGamma-K $方向的分布; (c) 两种不同方向二维狄拉克等离激元色散关系的对比图, 其中蓝色竖虚线代表费米波矢$ k_F $的位置
Figure 2. The distribution and dispersion behaviors of graphene Dirac plasmons. (a) distributions along the $ \varGamma-M $ direction; (b) distributions along the $ \varGamma-K $ direction; (c) A comparative of the dispersion relationships for two-dimensional Dirac plasmons along $ \varGamma-M $ and $ \varGamma-K $ directions, where the blue vertical dashed line represents the position of the Fermi wave vector $ k_F $.
图 3 载流子浓度对石墨烯狄拉克等离激元的调控 (a) 不同载流子浓度下沿$ \varGamma-M $方向的等离激元色散关系, 正值表示电子掺杂, 负值表示空穴掺杂, 子图为同浓度下电子/空穴掺杂的色散行为对比; (b) 狄拉克等离激元阈值能量随载流子浓度的演化规律
Figure 3. Modulation of graphene Dirac plasmons by carrier concentrations. (a) Plasmon dispersion along the $ \varGamma-M $ direction under varied carrier concentrations. Positive/negative values indicate electron/hole doping. Inset: comparative dispersion behavior of electron- and hole-doped systems at identical carrier concentration; (b) Evolution of Dirac plasmon threshold energy as a function of carrier concentration.
图 4 双轴应变对石墨烯狄拉克等离激元的调控 (a) 10%压缩应变下石墨烯的能带结构, 其中红线表示非占据态, 黑线表示占据态; (b) 掺杂浓度为0.03电子/原胞时, 不同程度双轴应变下石墨烯狄拉克等离激元沿$ \varGamma-M $方向的色散关系; (c) 狄拉克等离激元阈值能量随双轴应变大小的演化规律
Figure 4. Tuning graphene Dirac plasmons via biaxial strain. (a) Band structure of graphene under 10%! compressive strain, the red lines represent the unoccupied states and the black lines represent the occupied states; (b) Dispersion relation of Dirac plasmons under varying biaxial strains along the $ \varGamma-M $ direction at 0.03 electrons per unit cell; (c) Evolution of Dirac plasmon threshold energy with biaxial strain.
图 5 基底引入对石墨烯狄拉克等离激元的调控 (a) 石墨烯/六角氮化硼异质结在费米能级附近的能带结构, 其中黑线表示石墨烯的贡献, 红线表示六方氮化硼的贡献; (b) 掺杂浓度为0.03电子/原胞时, 纯石墨烯与石墨烯/六角氮化硼沿$ \varGamma-M $方向狄拉克等离激元色散关系对比图; (c) 掺杂浓度为0.03电子/原胞时, 不同程度双轴应变下石墨烯/六角氮化硼狄拉克等离激元沿$ \varGamma-M $方向的色散关系; (d) 石墨烯/六角氮化硼狄拉克等离激元阈值能量随双轴应变大小的演化规律
Figure 5. Tuning graphene Dirac plasmons via substrate integration. (a) Band structure of graphene/hexagonal boron nitride(hBN) heterostructure near the Fermi level, where the black line indicates the contribution from graphene and the red line represents the contribution from hBN; (b) Dirac plasmon dispersions along $ \varGamma-M $ direction at 0.03 electrons per unit cell; (c) Strain-dependent Dirac plasmon dispersions(graphene/hBN) under biaxial strains (–10% to 10%); (d) Evolution of plasmon threshold energy with biaxial strain in graphene/hBN heterostructure.
图 6 石墨烯等离激元激发能与载流子浓度($ n^{1/4} $)在固定波矢处(q = 0.029/Å)的标度关系
Figure 6. Scaling relation between plasmon excitation energy and carrier concentration($ n^{1/4} $) in graphene at fixed wave vector (q = 0.029/Å).
图 7 双轴应变与基底引入调控石墨烯等离激元机制 (a) 狄拉克锥斜率对等离激元激发的影响示意图; (b) 施加不同应变(–10%压缩应变到10%拉伸应变)对石墨烯K点附近能带结构的影响; (c) 基底效应诱变能带形变对等离激元激发的影响示意图; (d) 纯净石墨烯与石墨烯/六角氮化硼异质结K点附近能带结构对比
Figure 7. Mechanisms for tuning graphene plasmons via biaxial strain and substrate effects. (a) The impact of Dirac cone slope on plasmon excitations; (b) Strain-dependent band structure near K-point under varying biaxial strains (–10% compressive to 10% tensile); (c) Substrate-induced band deformation modulating plasmon excitation; (d) Band structure comparison near K-point: pristine graphene vs. graphene/hBN heterostructure.
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