電漿子共振行為: 從一維原子鏈到三維原子長方體

Plasmonic Resonances : From 1D Atomic Chains to 3D Atomic Cuboids

黃郁暉(Yu-Hui Huang)

指導教授 ： 梁贊全

國立中正大學/理學院/物理學系/博士(2016年)

摘要

我們利用時域密度泛函理論(TDDFT)研究一維原子鏈之電漿子共振行為。在一維原子鏈中存在電場極化方向沿著鏈長方向的縱向模態(L mode)以及電場極化方向垂直鏈長的橫向模態(T mode)。我們將探討在碳原子鏈中的電漿子共振行為。我們使用時域密度泛函理論研究在不同種類的碳原子鏈中之電子集體震盪現象以及觀察當參與振盪的電子增加時，單電子躍遷所引起的電漿子共振行為。我們考慮不同長度的累積烯 CnH4, 多炔基 CnH2 和碳氫鏈 CnH2n+2。我們使用以時域密度泛函理論來做計算的套裝軟體 Turbomole以及考慮TZVPP 之基底來計算以上所考慮之系統的激發能量和偶極振盪強度。我們的計算結果發現在累積烯和多炔基中皆存在縱向模態，但並沒有看到橫向模態的存在。我們也使用以局部密度近似方法(以套裝軟體 VASP)計算之無窮長原子鏈的能帶結構來解釋我們的結果，再把單電子躍遷與集體電漿子共振現象做一個連結。我們由單粒子之量子井圖像，分析出原子鏈中之縱向模態主要是由Δq=1之同能帶間躍遷所引起，其中q為量子井態的量子數。這種Δq=1的同帶間躍遷可以在金屬原子鏈(如鈉原子鏈)以及碳原子鏈(如累積烯和多炔基)中發現。此外，鈉原子鏈中的橫向模態主要是來自於Δq為偶數(主要為0)之不同能帶間的躍遷。這種橫向的集體躍遷行為只有當兩條能帶互相平行時才能夠形成，而此種橫向模態可以在簡單金屬中發現，但於碳原子鏈中卻沒有觀察到橫向模態。
為了瞭解電漿子的行為如何從一維的鈉原子鏈結構過渡到二維原子平面結構，我們考慮 [8x1]-[8x1], [8x2]-[8x2]和[8x4]-[8x4]耦合鈉原子鏈，以時域密度泛函套裝軟體Octopus進行計算，觀察電漿子模態如何隨著原子鏈間的間距改變而演化。在耦合原子鏈中，X 模態和Z模態隨著原子鏈間之間距減少而藍移，而Y模態則是隨著間距減少而紅移。這些模態的能量移動行為我們可以用一個簡單的偶極交互作用模型來解釋。此外，我們也針對不同長寬的鈉原子長方形來計算並分析其電漿子共振行為。在固定長度、改變寬度的情況下，當長方形的長度小於寬度時，X1模態之能量隨著長方形的寬度增加而緩慢的藍移，接著會逐漸收斂到一極限值。相對的，在固定寬度的情況下，X1模態會隨著長度增加而快速的紅移，最終X1模態之能量會隨著長度增加到無窮大時趨近到零。當長方形的寬度比長度大很多時，X模態可視為原子鏈中的橫向模態。為了更進一步了解電漿子行為如何從二維結構過渡到三維結構，我們考慮了 [8x8x1]-[8x8x1], [8x8x2]-[8x8x2], 及 [8x8x4]-[8x8x4] 耦合鈉長方體，改變耦合長方體之間的間距來計算並探討其電漿子共振的特徵。隨著間距減少，X(Y) 模態藍移而Z模態紅移，此現象同樣可以用類似的偶極交互作用模型來解釋。接下來我們考慮SC 及 BCC 立方體結構。隨著立方體的尺寸變大，在SC 立方體結構中有三個主要的共振峰出現，而在BCC立方體結構中則有四個主要的共振峰出現。在SC和BCC 立方體中的第一個主峰能量基本上不隨著立方體的尺寸改變而變化，這是由於兩種主要的效應，第一種是因為L1 模態隨著原子鏈鏈長增加的紅移現象，第二種是因為L模態隨著原子鏈逐漸靠近形成原子平面，進而形成立方體的藍移現象。最後我們考慮了固定底面積，改變高度的BCC 長方體結構來分析其電漿子模態如何隨著高度改變而變化。X(Y)模態能量一開始隨著高度增加而藍移，而最後慢慢收斂到一極限值而不再隨著長方體之高度變化而改變。Z模態隨著高度增加則是逐漸紅移。

關鍵字

時域密度泛函理論 ； 電漿子共振 ； 鈉原子鏈 ； 碳原子鏈 ； 鈉原子平面 ； 鈉原子 長方體 ； 吸收頻譜

並列摘要

We study plasmon-like resonances in one-dimensional (1D) atomic chain systems
using time-dependent density-functional theory (TDDFT). The induced polarization is along the atomic chain in the longitudinal mode and perpendicular to the atomic chain in the transverse mode. We choose to study carbon atomic chains where plasmonic resonances are not expected to be found. We used TDDFT to determine the emergence of collective resonances in various forms of linear carbon chains and see how plasmon-like resonance arises from single electron excitations as the number of electrons increases. We considered di erent lengths of cumulene CnH4, polyyne CnH2 and alkene CnH2n+2. The excitation energy and dipole oscillation strength of these systems were obtained by TDDFT using the Turbomole program package with TZVPP basis set. Our TDDFT results showed that there were longitudinal collective modes for cumulene and polyyne and for nite-sized chains, and there is no transverse collective mode in the these carbon chains. We also used the band structures of periodic atomic chains calculated using standard local density functional method (as implemented in the VASP package) to interpret the results and to relate the single electron excitation to the collective plasmon-like response. Within the one-particle quantum well picture, the longitudinal mode in linear atomic chain arises from intraband transition with Δq equal to 1 where q is
the quantum number of the quantum well. As such Δq=1 intraband transitions
can be found in metallic (e.g. Na) chains as well as in carbon chains (cumulene and polyyne). On the other hand, the transverse modes of the sodium chains are due to interband transitions with q being an even number (dominated by Δq = 0) and such transverse collective excitation can be formed only if the allowed Δq = 0 transitions occur between bands that are parallel to each other which can be found in simple metals but not in carbon chains.
To understand how the evolution of plasmon mode from 1D to 2D sodium
nanostructures, we study the plasmon resonances in [8x1]-[8x1], [8x2]-[8x2], and [8x4]-[8x4] sodium coupled chains with di erent interchain separation using use a real-space and real-time time-dependent density functional theory code, OCTOPUS. For coupled chains, the X-modes blue shift in energy as the interchain distance decreases. The Y-modes redshift, and the Z-modes blue shift with decreased interchain distance. The energy shift of plasmon modes in coupled chains can be explained by a simple dipole interaction model. In addition, we study the plasmon resonances for sodium atomic rectangular planes with di erent lengths and widths. The energy of X-modes increases as the width of rectangular increases when the length of the rectangular plane is xed and, as long as the width is smaller than length. The energy of X1 increases slowly as width increases when length is smaller than width, and then it will converge to a limit gradually. When the width is larger than the length, the energy of X1 increases slowly as width increases, and then it will converge to a limit gradually. In contrast, if the width of the plane is xed, the energy of X1 mode decreases rapidly as the length increases, and it will approach to zero as the length goes to in nity. When the width is much larger than length, the X-modes can be referred as T-modes. There are TE and TC modes as the length is equal to 1, and three X-modes as the length exceeds two. For understanding the plamonic properties evolve from 2D to 3D systems, we study the plasmon resonances in [8x8x1]-[8x8x1], [8x8x2]-[8x8x2], and [8x8x4]-[8x8x4] sodium coupled cuboids with di erent intercuboid separation. The X(Y)-modes blue shift, and the Z-modes redshift in energy as the intercuboid distance decreases in coupled cuboid systems, which can be explained by a similar dipole interaction model. We next consider the system of SC and BCC cubes. There are three major peaks in SC cubes and four major peak emerge in BCC cubes emerge as the size of cubes increases. The 1st peak energy in both SC and BCC cubes is independent on the cube size. It is due to two e ects which are from the red shifts of L1 mode of the chain as the length increases and the L-mode blue shifts when the chains are combined to form a cube. We study, moreover, the dependence of the energy of plsamon modes on the height of the BCC cuboids. The Z-modes red shift gradually in energy with increasing height. The energy shift behavior of L-modes for cuboid can be used to explain why the energy of the rst resonance peaks in cubes almost independent on the cube size.

並列關鍵字

time-dependent density functional theory ； plasmonic resonance ； sodium atomic chain ； carbon atomic chain ； sodium atomic plane ； sodium atomic cuboid ； absorption spectrum

參考文獻

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