Physics Help Forum The tight-binding hamiltonian for a two-dimensional vanadium dioxide
 Apr 12th 2015, 10:46 AM #1 Junior Member   Join Date: Dec 2014 Posts: 2 The tight-binding hamiltonian for a two-dimensional vanadium dioxide Code:  In his book, Quantum Field Theory for the Gifted Amateur, author Stephen Blundell explains that in the tight-binding model, one considers a lattice of fixed atoms with electrons moving between them (as shown in Fig. 4.6 above). These electrons can lower their kinetic energies by hopping from lattice site to lattice site. To deal with the discrete lattice in the model, we need to work in a basis where the fermion creation operator $\hat{c}_i^\dagger$ creates a particle at a particular lattice site labelled by $i$. The kinetic energy saving for a particle to hop between points $j$ and $i$ is called $t_{ij}$ . Clearly $t_{ij}$ will have some fundamental dependence on the overlap of atomic contains a sum over all processes wave functions. The Hamiltonian $H$ in which an electron hops between sites, and so is a sum over pairs of sites: $\hat{H}_{hopping}=\sum_{ij}(-t_{ij})\hat{c}_i^\dagger\hat{c}_j^\dagger$ Ok. After reading that book, I am trying to work on the kinetic term of the Hamiltonian for a $VO_2$ system using tight-binding approximation, but I'm not quite sure whether my work is right or wrong. Would you all be so kind as to check my work? In my case, $VO_2$ is modelled by taking the (110) plane from its unit cell as shown in second figure above. The model unit cell has two basis oxygen atoms ($O_A$ and $O_B$), vertically located $a \approx 2.87$ angstrom apart, each of which contributing two p orbitals ($p_x$ and $p_y$) and two vanadium atoms ($V_A$ and $V_B$), vertically located $a \approx 2.87$ angstrom apart, each of which contributing two d orbitals ($d_{x^2-y^2}$ and $d_{xy}$). Here, we choose 8 basis orbitals to construct our hilbert space, which we order as follows: $\left| O_A-p_x \right\rangle$, $\left| O_A-p_y \right\rangle$, $\left| O_B-p_x \right\rangle$, $\left| O_B-p_y \right\rangle$, $\left| V_A-d_{x^2-y^2} \right\rangle$, $\left| V_A-d_{xy} \right\rangle$, $\left| V_B-d_{x^2-y^2} \right\rangle$, and $\left| V_B-d_{xy} \right\rangle$. Using this set of bases and tight-binding approximation, the kinetic part of the hamiltonian can be written in $k$ space as $\hspace{1cm}$ $H=H_{on-site}+H_{hopping}$ $\hspace{1cm}$ where $\hspace{1cm}$ $H_{on-site}=\sum_{\bar{k}}\left( \epsilon_{d_{A}} d_{A_{1\bar{k}}}^{\dagger} d_{A_{1\bar{k}}} + \epsilon_{d_{A}} d_{A_{2\bar{k}}}^{\dagger} d_{A_{2\bar{k}}} + \epsilon_{d_{B}} d_{B_{1\bar{k}}}^{\dagger} d_{B_{1\bar{k}}}+\epsilon_{d_{B}} d_{B_{2\bar{k}}}^{\dagger} d_{B_{2\bar{k}}}+\epsilon_{p_{A}} p_{A_{x\bar{k}}}^{\dagger} p_{A_{x\bar{k}}}+\epsilon_{p_{A}} p_{A_{y\bar{k}}}^{\dagger} p_{A_{y\bar{k}}}+\epsilon_{p_{B}} p_{B_{x\bar{k}}}^{\dagger} p_{B_{x\bar{k}}}+\epsilon_{p_{B}} p_{B_{y\bar{k}}}^{\dagger} p_{B_{y\bar{k}}}\right)$ $\hspace{1cm}$ and $\hspace{1cm}$ $H_{hopping}=-t_{V_A-V_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot a \hat{y}}d_{A_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot (-a \hat{y})}d_{A_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}\right)-t_{V_A-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y}\right)}d_{A_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}} +\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{B_{2\bar{k}}}\right)-t_{V_A-O_A}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y}\right)}d_{A_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}} +\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{A_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}\right)-t_{V_B-O_A}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y}\right)}d_{B_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{A_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}+\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{A_{2\bar{k}}}\right)-t_{V_B-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot\left(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y}\right)}d_{B_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{B_{1\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(-\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{B_{2\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot(\frac{1}{2}c\hat{x}-\frac{1}{2}a\hat{y})}d_{B_{2\bar{k}}}^{\dagger}p_{B_{2\bar{k}}}\right)-t_{O_A-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot a \hat{y}}p_{A_{y\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot (-a \hat{y})}p_{A_{y\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}\right)-t_{O_A-O_A}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot c \hat{x}}p_{A_{x\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot (-c \hat{x})}p_{A_{x\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}\right)-t_{O_B-O_B}\left(\sum_{\bar{k}}e^{-i\bar{k}\cdot c \hat{x}}p_{B_{x\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}+\sum_{\bar{k}}e^{-i\bar{k}\cdot (-c \hat{x})}p_{B_{x\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}\right)$ $\hspace{1cm}$ Thus, $\hspace{1cm}$ $H =\sum_{\bar{k}}\left(\epsilon_{d_{A}}d_{A_{1\bar{k}}}^{\dagger} d_{A_{1\bar{k}}}+\epsilon_{d_{A}}d_{A_{2\bar{k}}}^{\dagger}d_{A_{2\bar{k}}}+\epsilon_{d_{B}}d_{B_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}+\epsilon_{d_{B}} d_{B_{2\bar{k}}}^{\dagger} d_{B_{2\bar{k}}}+\epsilon_{p_{A}} p_{A_{x\bar{k}}}^{\dagger} p_{A_{x\bar{k}}}+\epsilon_{p_{A}} p_{A_{y\bar{k}}}^{\dagger} p_{A_{y\bar{k}}}+\epsilon_{p_{B}} p_{B_{x\bar{k}}}^{\dagger} p_{B_{x\bar{k}}}+\epsilon_{p_{B}} p_{B_{y\bar{k}}}^{\dagger} p_{B_{y\bar{k}}}-2t_{V_{A}V_{B}} d_{A_{1\bar{k}}}^{\dagger}d_{B_{1\bar{k}}}\cos{\left( k_{y}a \right)}-2t_{V_AO_B}\left(d_{A_{2\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{A_{2\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{V_AO_A}\left(d_{A_{2\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{A_{2\bar{k}}}^{\dagger}p_{A_{y\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{V_BO_A}\left(d_{B_{2\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{B_{2\bar{k}}}^{\dagger}p_{A_{y\bar{k}}}e^{-i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{V_BO_B}\left(d_{B_{2\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+d_{B_{2\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}e^{i\frac{1}{2}k_ya}\cos{\left(\frac{k_{x}c}{2}\right)}+h.c\right)-2t_{O_{A}O_{B}} p_{A_{y\bar{k}}}^{\dagger}p_{B_{y\bar{k}}}\cos{\left( k_{y}a \right)}-2t_{O_{A}O_{A}}p_{A_{x\bar{k}}}^{\dagger}p_{A_{x\bar{k}}}\cos{\left( k_{x}c \right)}-2t_{O_{B}O_{B}}p_{B_{x\bar{k}}}^{\dagger}p_{B_{x\bar{k}}}\cos{\left( k_{x}c \right)}\right)$ $\hspace{1cm}$ Note: $h.c$ = hermitian conjugate $\hspace{1cm}$ In equations above, $d_{A_{1\bar{k}}}^{\dagger} (d_{A_{1\bar{k}}})$,$d_{A_{2\bar{k}}}^{\dagger} (d_{A_{2\bar{k}}})$, $d_{B_{1\bar{k}}}^{\dagger} (d_{B_{1\bar{k}}})$, $d_{B_{2\bar{k}}}^{\dagger} (d_{B_{2\bar{k}}})$, $p_{A_{x\bar{k}}}^{\dagger} (p_{A_{x\bar{k}}})$, $p_{A_{y\bar{k}}}^{\dagger} (p_{A_{y\bar{k}}})$, $p_{B_{x\bar{k}}}^{\dagger} (p_{B_{x\bar{k}}})$, and $p_{B_{y\bar{k}}}^{\dagger} (p_{B_{y\bar{k}}})$ create (annihilate) an electron at $V_A$($d_{x^2-y^2}$), $V_A$($d_{xy}$), $V_B$($d_{x^2-y^2}$), $V_B$($d_{xy}$), $O_A$($p_x$), $O_A$($p_y$), $O_B$($p_x$), and $O_B$($p_y$) orbitals, respectively, with momentum $\bar{k}$. $\epsilon_{d_A}$, $\epsilon_{d_B}$, $\epsilon_{p_A}$, and $\epsilon_{p_B}$ indicate on-site energy of $V_A$, $V_B$, $O_A$, and $O_B$, respectively. Furthermore, $t_{V_AV_B}$, $t_{V_AO_B}$, $t_{V_AO_A}$, $t_{V_BO_A}$, $t_{V_BO_B}$, $t_{O_AO_B}$, $t_{O_AO_A}$, and $t_{O_BO_B}$ being hopping parameter of $V_A$-$V_B$, $V_A$-$O_B$, $V_A$-$O_A$, $V_B$-$O_A$, $V_B$-$O_B$, $O_A$-$O_B$, $O_A$-$O_A$, and $O_B$-$O_B$ respectively. Then, a and c being the lattice constant. Attached Thumbnails     Last edited by gerryliyana; Apr 12th 2015 at 10:52 AM.