電容的古典與量子效應之研究

Abstract

我們研究電容的古典以及量子的效應。 關於量子的效應，我們預期多層奈米管的電容值在外加磁場下會因為阿哈羅諾夫－玻姆效應(Aharonov-Bohm effect)而產生周期性的變化。過去的研究指出，單層奈米碳管在一個量子單位的磁通量下，可以在非金屬與金屬之間變換恰好一次。我們預期多層奈米管的每一個殼層也會有類似的效應。由於多層奈米管的每一層直徑都不相同，各自對應到不同的電性變換的磁場週期，以及不同的幾何電容的大小，因而導致了電容值隨磁場的振盪。我們的模型預估電容值的最大變化量為40aF。實驗上我們量測了21.51μm長的多層硼化氮奈米管(17.14μm在電極之間,，4.37μm在電極上方)，在縱向磁場最大到8.93T的電容值。我們發現電容值變化小於6aF。我們另外也量了5.118μm長的多層奈米碳管(3.828μm在電極之間，1.29μm在電極上方)，在縱向磁場最大到8.61T的電容值。我們發現電容變化小於15aF。但由於這個多層奈米碳管在量測後發現有異端脫落，所以有可能低估了變化量。因此我們的實驗在室溫下對多層奈米管能觀察到的阿哈羅諾夫-玻姆效應給出了一個上限。 關於古典的效應，受到位移電流產生干涉而導致電容變化的理論可能性的啟發，我們利用數值模擬四種類型的環狀共振器，將之放在不均勻的介電質中來引發位移電流的干涉。如果位移電流真的發生干涉現象，共振器的總電容值會降低，導致共振頻率的藍移。這個模擬最困難的地方在於在不同模擬模式下與邊界條件下如何取得一致的結果，以及如何辨別假訊號。在克服這些困難之後，我們提出一個簡易模型來對應共振器跟它的等效電路。當介質是均勻的時候，我們的簡易模型可以很精確地預測共振器的共振頻率，頻率相差約為0.15%，因此我們排除了邊緣場的效應。然而在不均勻介質中單共振環會出現未預期的異常紅移，無法以阻尼現象描述。另一方面，雙共振環在不均勻介質中的共振頻率符合簡易模型的預測(差距小於0.3%)。但由於模擬的頻率解析度不夠高，導致無法辨別是否產生了因為位移電流的干涉而造成的共振頻率的藍移。

We investigate the quantum and classical effect of capacitance. For the quantum effect, we predict a periodic capacitance change on multi-walled nanotube due to Aharonov-Bohm (AB) effect. The AB effect will make a single-walled carbon nanotube(SWCNT) becomes metallic as a periodic function quantum flux. We expect each shell of a multi-walled nanotube (MWNT) can have a similar feature of SWCNT, changing between an insulator and a conductor under magnetic field. Correspondingly, the capacitance of a MWNT may have the interesting phenomenon and displays oscillations of capacitance constituting from different shells under magnetic fields. We estimate the maximum change of capacitance is about 40aF. Experimentally, we measured the capacitance of a boron-nitride nanotube (BNNT) with total length 21.51μm (with 17.14μm between the electrode, 4.37μm on the electrode) under a maximum longitudinal magnetic field 8.93T at room temperature. We found that the capacitance change was smaller than 6aF. We also measured the capacitance of a total length 5.118μm MWCNT (with 3.828μm between the electrode, 1.29μm on the electrode) under a maximum longitudinal magnetic field 8.61T. The capacitance change is smaller than 15aF (which is likely to be underestimated because one side of the nanotube was found to be disconnected after the measurement). The experiments put an upper limit for observing Aharonov-Bohm effect on capacitance of MWNT at room temperature. For the classical effect, we were inspired by a possible effect of capacitance due to the interference of displacement currents. We simulate the resonance frequency of four kinds of split ring resonators (SRRs & DSRRs) and apply an inhomogeneous medium to induce the interference of effective capacitor, hoping to observe a blue-shifted resonance frequency. The most difficult part of the study is to get the same result under different simulation methods and different boundary conditions, and to verify the fake signal. We propose a simple model and find that, in a homogeneous medium, the model can explain the simulated resonance frequencies within 0.15%, thus the effect of fringe field is enough small and can be ignored. However, in an inhomogeneous medium, we find an anomalous red-shift frequency of SRR which cannot be explained by the effect of damping. On the other hand, in an inhomogeneous medium the simple model can still explain the resonance frequency of DSRRs with frequency differences smaller than 0.3%. However, the frequency resolution of the simulation is too low to verify there is a blue shift due to the effect of interference.