以數值重整化群研究石墨烯在近藤效應下的傳輸性質

Abstract

受近期近藤效應在石墨烯中傳輸性質實驗測量的啟發，我們研究了石墨烯在不同化學能(μ)下的近藤電阻率和退相率 。石墨烯中由於碳原子空缺或者磁性吸附原子所形成的雜質都可以用單軌道的不對稱安德森來描述並且以精確的數值重整化群方法求解。我們發現雖然這裡討論的安德森雜質模型是一個混合價電子系統，但它卻可以藉由柵極電壓被驅動到近藤[μ>μC（臨界值）] ，混合價電子（μ=μC），或空軌道（μ<μC）區間，且電阻率和退相率在每個區間皆有不同特徵。具體來說，在μ<μC的情況下，電阻率（退相率）曲線形狀幾乎是相同的。然而，隨著溫度的降低，卻在比較低的T/TK開始增加，且增加的比正常金屬更為迅速[以(T/TK)^(-3/2)增加]〔在這裡，T（TK)表示（近藤）溫度] 。隨著溫度進一步降低，退相率達到最大值後，下降得比正常金屬更迅速，以(T/TK)^3下降而非(T/TK)^2。此外，電阻率在TK附近有一個明顯大於飽和電阻率的峰值。與此相反，在μ>μC的情況下，電阻率曲線有一個額外的寬肩在10TK附近而退相率則呈現出有趣的肩峰形狀。在狹窄的邊界區域（μ=μC），電阻率以及退相率曲線都和一般金屬極為相似。這解釋了最近實驗中測量到在有缺陷的石墨烯中符合傳統近藤效應電阻率曲線的結果，雖然其他情況下（μ>μC或μ<μC）的電阻率特徵尚未被觀察到。這些在電阻率和退相率中有趣的特點，我們以能譜函數，自洽能量以及重整化能階進行了分析。

Motivated by the recent observation of the Kondo effect in graphene in transport experiments, we investigate the resistivity and dephasing rate in the Kondo regime due to magnetic impurities in graphene with different chemical potentials (μ). The Kondo effect due to either carbon vacancies or magnetic adatoms in graphene is described by the single-orbital pseudo-gap asymmetric Anderson impurity model which is solved by the accurate numerical renormalization group method. We find that although the Anderson impurity model considered here is a mixed valence system, it can be driven into either the Kondo [μ>μC(critical value)], mixed-valency (μ=μC), or empty-orbital (μ<μC) regime by a gate voltage, giving rise to characteristic features in resistivity and dephasing rate in each regime. Specifically, in the case of μ<μC, the shapes of the resistivity (dephasing rate) curves for different μ are nearly identical. However, as temperature decreases, they start to increase to their maxima at a lower T/TK, but more rapidly [as (T/TK)^(-3/2)] than in normal metals [here, T (TK) denotes the (Kondo) temperature]. As T further decreases, after reaching the maximum, the dephasing rate drops more quickly than in normal metals, behaving as (T/TK)^3 instead of (T/TK)^2. Furthermore, the resistivity has a distinct peak above the saturation value near TK. In the case of μ>μC, in contrast, the resistivity curve has an additional broad shoulder above 10TK and the dephasing rate exhibits an interesting shoulder-peak shape. In the narrow boundary region (μ=μC), both the resistivity and dephasing rate curves are similar to the corresponding ones in normal metals. This explains the conventional Kondo like resistivity from recent experiments on graphene with defects, although the distinct features in the resistivity in the other cases (μ>μC or μ<μC) were not seen in the experiments. The interesting features in the resistivity and dephasing rate are analyzed in terms of the calculated T-dependent spectral function, correlation self-energy, and renormalized impurity level.