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@@ -38,6 +38,7 @@ Navigation | |
| .. toctree:: | ||
| :maxdepth: 3 | ||
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| theory | ||
| modules | ||
| classes | ||
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| API | ||
| === | ||
| Modules | ||
| ============ | ||
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| .. autosummary:: | ||
| :toctree: generated | ||
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| Theory | ||
| ====================== | ||
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| The *retarded Green's function* | ||
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| .. math:: | ||
| \mathbf{G}^{\mathrm{R}}=[E \mathbf{I}-\mathbf{H}-\mathbf{\Sigma}]^{-1} | ||
| is a function of energy :math:`E` multiplied by the identity matrix :math:`\mathbf{I}`. The Hamiltonian :math:`\mathbf{H}` and self-energy :math:`\mathbf{\Sigma}` matrices are to be defined by the physical model. | ||
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| Along with the *advanced Green's function* | ||
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| .. math:: | ||
| \mathbf{G}^{\mathrm{A}} = \left[ \mathbf{G}^{\mathrm{R}} \right]^\dagger | ||
| they provide the *spectral function* | ||
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| .. math:: | ||
| \mathbf{A}=i\left[\mathbf{G}^{\mathrm{R}}-\mathbf{G}^{\mathrm{A}}\right] | ||
| and are used to solve for the *"electron occupation" Green's function* | ||
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| .. math:: | ||
| \mathbf{G}^{\mathrm{n}}=\mathbf{G}^{\mathrm{R}} \mathbf{\Sigma}^{\mathrm{in}} \mathbf{G}^{\mathrm{A}} | ||
| which gives the *density matrix* | ||
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| .. math:: | ||
| \hat{\rho} = \mathbf{G}^{\mathrm{n}} / 2\pi . | ||
| The in-scattering term :math:`\mathbf{\Sigma}^{\mathrm{in}}` is also defined by the physical model. | ||
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| Both, the self-energy :math:`\mathbf{\Sigma}` and the in-scattering term :math:`\mathbf{\Sigma}^{\mathrm{in}}` are sums of the left contact :math:`\mathbf{\Sigma}_1`, right contact :math:`\mathbf{\Sigma}_2` and an intrinsic term :math:`\mathbf{\Sigma}_0`, hence | ||
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| .. math:: | ||
| \begin{align} | ||
| \mathbf{\Sigma} &= \mathbf{\Sigma}_1 + \mathbf{\Sigma}_2 + \mathbf{\Sigma}_0 , \\ | ||
| \mathbf{\Sigma}^{\mathrm{in}} &= \mathbf{\Sigma}^{\mathrm{in}}_1 + \mathbf{\Sigma}^{\mathrm{in}}_2 + \mathbf{\Sigma}^{\mathrm{in}}_0 . | ||
| \end{align} | ||
| NOTE: We use the (physically expressive) notation of S. Datta, where the self-energies and Green's functions in relation to the standard notation (on the right) are defined as | ||
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| .. math:: | ||
| \begin{align} | ||
| \mathbf{\Sigma} &\equiv \mathbf{\Sigma}^\mathrm{R} , \\ | ||
| \mathbf{G}^\mathrm{n} &\equiv -i \mathbf{G}^< , \\ | ||
| \mathbf{\Sigma}^\mathrm{in} &\equiv -i \mathbf{\Sigma}^< . | ||
| \end{align} | ||
| Linear Chain Model | ||
| ---------------------------- | ||
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| For the ``LinearChain`` model, the **Hamiltonian** | ||
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| .. math:: | ||
| H_{ij} = \begin{cases} | ||
| \epsilon_0, & \text { if } i=j \\ | ||
| t, & \text{ if } i \neq j | ||
| \end{cases} | ||
| Impurity potential :math:`U` can be added to the on-site energy as | ||
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| .. math:: | ||
| \mathbf{H}=\left[ | ||
| \begin{array}{ccccc} | ||
| \ddots & \vdots & \vdots & \vdots & \ddots \\ | ||
| \cdots & \varepsilon & t & 0 & \cdots \\ | ||
| \cdots & t & \varepsilon+U & t & \cdots \\ | ||
| \cdots & 0 & t & \varepsilon & \cdots \\ | ||
| \ddots & \vdots & \vdots & \vdots & \ddots | ||
| \end{array} | ||
| \right] . | ||
| The **self-energies** | ||
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| .. math:: | ||
| \mathbf{\Sigma}_1=\left[\begin{array}{ccccc} | ||
| \mathrm{te}^{i k a} & 0 & 0 & \cdots & 0 \\ | ||
| 0 & 0 & 0 & \cdots & 0 \\ | ||
| 0 & 0 & 0 & \cdots & 0 \\ | ||
| \vdots & \vdots & \vdots & \ddots & \vdots \\ | ||
| 0 & 0 & 0 & \cdots & 0 | ||
| \end{array}\right], \quad \mathbf{\Sigma}_2=\left[\begin{array}{ccccc} | ||
| 0 & \cdots & 0 & 0 & 0 \\ | ||
| \vdots & \ddots & \vdots & \vdots & \vdots \\ | ||
| 0 & \cdots & 0 & 0 & 0 \\ | ||
| 0 & \cdots & 0 & 0 & 0 \\ | ||
| 0 & \cdots & 0 & 0 & \mathrm{te}^{i k a} | ||
| \end{array}\right] , | ||
| with the broadening functions :math:`\mathbf{\Gamma} \equiv i\left[ \mathbf{\Sigma} - \mathbf{\Sigma}^\dagger\right]` | ||
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| .. math:: | ||
| \mathbf{\Gamma}_1=\frac{\hbar v}{a}\left[\begin{array}{ccccc} | ||
| 1 & 0 & 0 & \cdots & 0 \\ | ||
| 0 & 0 & 0 & \cdots & 0 \\ | ||
| 0 & 0 & 0 & \cdots & 0 \\ | ||
| \vdots & \vdots & \vdots & \ddots & \vdots \\ | ||
| 0 & 0 & 0 & \cdots & 0 | ||
| \end{array}\right], \quad \mathbf{\Gamma}_2=\frac{\hbar v}{a}\left[\begin{array}{ccccc} | ||
| 0 & \cdots & 0 & 0 & 0 \\ | ||
| \vdots & \ddots & \vdots & \vdots & \vdots \\ | ||
| 0 & \cdots & 0 & 0 & 0 \\ | ||
| 0 & \cdots & 0 & 0 & 0 \\ | ||
| 0 & \cdots & 0 & 0 & 1 | ||
| \end{array}\right] , | ||
| where :math:`v=\mathrm{d} E /(\hbar \mathrm{d} k) = -2 a t / \hbar \sin (k a)` so that :math:`\frac{\hbar v}{a} = -2 t / \sin (k a)`. | ||
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| The in-scattering terms | ||
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| .. math:: | ||
| \mathbf{\Sigma}^\mathrm{in}_i = \mathbf{\Gamma}_i \cdot f_i(E) , | ||
| where :math:`f_i(E)` is the Fermi-Dirac distribution function for contact :math:`i \in \set{1, 2}`. | ||
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| The self-energies describing the **phase and phase-momentum relaxation** are defined in terms of the Green's functions themselves. Their strength is defined via the (scalar) coefficients :math:`D_0^\text{phase}` and :math:`D_0^\text{phase-momentum}`, creating a "mask" matrix | ||
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| .. math:: | ||
| \mathbf{D} = D_0^\text{phase} | ||
| \left[\begin{array}{ccccc} | ||
| 1 & 1 & 1 & \cdots & 1 \\ | ||
| 1 & 1 & 1 & \cdots & 1 \\ | ||
| 1 & 1 & 1 & \cdots & 1 \\ | ||
| \vdots & \vdots & \vdots & \ddots & \vdots \\ | ||
| 1 & 1 & 1 & \cdots & 1 | ||
| \end{array}\right] | ||
| + D_0^\text{phase-momentum} | ||
| \left[\begin{array}{ccccc} | ||
| 1 & 0 & 0 & \cdots & 0 \\ | ||
| 0 & 1 & 0 & \cdots & 0 \\ | ||
| 0 & 0 & 1 & \cdots & 0 \\ | ||
| \vdots & \vdots & \vdots & \ddots & \vdots \\ | ||
| 0 & 0 & 0 & \cdots & 1 | ||
| \end{array}\right] , | ||
| which is used for an element-wise multiplication :math:`\odot` of the Green's function matrices | ||
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| .. math:: | ||
| \begin{align} | ||
| \mathbf{\Sigma}_0 &= \mathbf{D} \odot \mathbf{G}^\text{R}, \\ | ||
| \mathbf{\Sigma}^\text{in}_0 &= \mathbf{D} \odot \mathbf{G}^\text{n} . | ||
| \end{align} | ||
| Since the Green's functions enter the definition of the self-energy, a self-consistent loop is performed, where :math:`\mathbf{G}^\text{R}` and :math:`\mathbf{G}^\text{n}` are initially set as zero matrices and iteratively updated, along with :math:`\mathbf{\Sigma}_0` and :math:`\mathbf{\Sigma}^\text{in}_0`. About 70 iteration steps are usually enough to reach a convergence. |
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