| 摘要: | 本研究基於費米黃金法則(Fermi’s golden rule)理論探討奈米鋁薄膜中電子於聲子作用下之帶内(intraband)散射率(scattering rate)。我們以變形散射(Deformation scattering)機制描述電子與聲子之間的交互作用。首先以原子軌域線性組合(linear combination of atomic orbitals)法求取數個原子層厚之鋁薄膜的真實能帶結構與電子的實際波函數。在這項工作中，用於展開電子波函數的基底是九個原子軌域，包括一個 $3s$ 軌域、三個 $3p$ 軌域和五個 $3d$ 軌域。通過在第一布里淵區 (FBZ) 的高對稱點處將LCAO計算之特徵能量與密度泛函理論 (DFT) 之結果進行擬合，以獲得用於決定奈米鋁(001)薄膜的能帶結構所需之各能量積分的加權係數及表面位能。為了獲得所需之精確度共需54個加權係數與9個表面位能項。與 DFT 相比，本研究提出的改進型LCAO 方法消耗極少的計算資源。
以上述改進型LCAO 方法計算在具有五層原子層的鋁(001) 奈米薄膜中的費米能量(Fermi energy) $E_{F}$得$E_{F} = 10.8380$ eV，此結果較 DFT之計算數值約高0.1 eV，尚屬合理。此外，随著奈米鋁薄膜之原子層層數增加，費米能量逐步上升並接近塊材鋁之費米能量。奈米鋁薄膜中的電子-聲子 (e-p) 散射率是在固定的薄膜體積Ｖ中，以不同之聲子模態(phonon mode)$(E_s, \mathbf{q})$進行計算，其中 $E_s$ 是在聲子波向量 $\mathbf{q}$處之聲子能量，$s$ 則表示聲子之分支(branch)。
對於具有七層的奈米鋁薄膜，由聲子模態$(20 \ \mbox {meV}, \overline{\mbox{M}}$)造成初始能量$E_i$高於費米能階 0.3 eV之熱電子散射，其散射率為$1/\tau(E_i) = 1.251\times10^{16}$ s$^{-1}$。使用相同的聲子模態但對於處於較低初始能量 $E_{i} = 0.1 + E_{F}$ eV之電子，散射率為 $1/\tau(E_i) = 3.498\times10^{15}$ s$ ^{-1}$。另一方面，當納米Al薄膜的厚度增加到17層時，聲子模態$(20 \ \mbox{meV}, \overline{\mbox{M}})$造成初始能量為 $E_{i} = 0.1 + E_{F}$ eV之電子散射率是$ 1/\tau(E_i) = 2.513\times10^16$ s$^{-1}$。電子散射率隨電子能量之變化則與電子之能態密度有緊密之關係，且奈米鋁薄膜中電子之能態密度分布與塊材鋁之情形有顯著差異。;In this research, the intraband scattering rate of electrons due to phonons based on the Fermi golden rule in nano-metallic aluminum (Al) films is theoretically investigated. First, the realistic energy band structure of an Al nanofilm of a few atomic layers in thickness and the actual wave function of an electron are resolved by the linear combination of atomic orbitals (LCAO) method. In this work, the basis for describing an electron wave function is composed of nine atomic orbitals, including one $3s$ orbital, three $3p$ orbitals, and five $3d$ orbitals. Surface potentials and weighting coefficients of energy integrals used to determine the energy band structure of nanofilm Al(001) are obtained by fitting the LCAO-based eigen-energies to those from the density functional theory (DFT) at high symmetry points in the first Brillouin zone (FBZ). A total of 54 weighting coefficients and nine surface potentials are obtained to achieve the desired accuracy. Compared to the DFT, the modified LCAO method proposed in this research consumes much less computational resources.
Using the modified LCAO approximation, the computed Fermi energy $E_F$ found in Al(001) nanofilm with five atomic layers is $E_F = 10.8380$ eV, which is about 0.1 eV above that from the DFT result. Moreover, as the number of atomic layers of the nano-Al film increases, the Fermi energy gradually increases and approaches that of bulk Al. The electron-phonon (e-p) scattering rate in a nano-Al film is carried out using a phonon mode $(E_s, \mathbf{q})$ and a fixed volume $V$ of a film, where $E_s$ denotes the phonon energy at a phonon wave vector $\mathbf{q}$ and $s$ represents a phonon branch.
For a nano-Al film with seven atomic layers, the scattering rate of hot electrons having an initial energy of 0.3 eV above the Fermi level due to a phonon mode $(20 \ \mbox{meV}, \overline{\mbox{M}}$) is $1/\tau(E_i) = 1.251\times10^{16}$ s$^{-1}$. With the same phonon mode, electrons with a lower initial energy $E_i = 0.1 + E_{F}$ eV, the scattering rate is $1/\tau(E_i) = 3.498\times10^{15}$ s$ ^{-1}$. On the other hand, when the thickness of the nano-Al film increases up to 17 layers, the scattering rate of electrons at $E_{i} = 0.1$ eV above the Fermi level due to a phonon mode $ (20 \ \mbox{meV}, \overline{\mbox{M}})$ is $ 1/\tau(E_i) = 2.513\times10^16$ s$^{-1}$. The scattering rate as a function of energy is found to follow quite closely the density of states of the electron that is very different from that in bulk Al. |
