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\title{Quantum transport in zigzag graphene nanoribbon decorated with a phenanthrene molecule%\thanks{Grants or other notes
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\author{Bo Jiang$^{1,2}$ \and Qing-Yuan Chen$^{1}$ \and Jin-Long Luo$^{1}$  \and Yi-Fen Zhao$^{1}$ \and Guojun Jin$^{1,3,^{\dagger}}$ \and Hai Yang$^{1,^{*}}$}



\institute{Guojun Jin$^{\dag}$, Hai Yang$^{*}$\at
\email{gjin@nju.edu.cn, kmyangh@263.net}
              \\1 School of Physical Science and Technology, Kunming University, Kunming 650214, China\\
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                2 School of Physics and Astronomy, Yunnan University, Kunming 650504, China\\
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                3 National Laboratory of Solid State Microstructures, Department of Physics, and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China}


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\begin{figure}
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\caption{(a) Band structure of 4-ZGNR. (b) Isosurface plots of $\Gamma$-point wave functions of $\pi$ and $\pi^{*}$ bands for 4-ZGNR.}
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Panel (a) is the band structure of pristine 4-ZGNR. Red branch and black branch represents the $\pi^{*}$ band and the $\pi$ band respectively. Panel (b) shows the electron wave function of the $\pi^{*}$ and $\pi$ band. About the yz middle plane, the parity of the $\pi^{*}$ band is even and the parity of the $\pi$ band is odd. According to the transmission selection rule of the wave function: if the parity of the wave function of the left and right electrodes is the same, it contributes to the transmission of electrons in the whole device, otherwise, the electron transport is forbidden. When the bias voltage is applied, the bands of the left and right electrodes move to the low-energy region and the high-energy region, respectively. Here, only the parity of the band near $0.1$~V in the bias window is the same, and the transmission peak appears. In the range of [$-\frac{V_{b}}{2}+0.1,~+\frac{V_{b}}{2}-0.1$~eV], the transmission peak is $0$ due to the different parity between the left and right electrodes.




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