科研笔记, 个人笔记

纵向/横向电阻率和纵向/横向电导率的关系

电阻率(resistivity)张量[1]:

\rho=\begin{pmatrix}   \rho_{xx}  &  \rho _{xy}  \\  - \rho_{xy}  &  \rho_{xx}  \end{pmatrix}

电导率(conductiviy)张量:

\sigma=\begin{pmatrix}  \sigma_{xx}  &  \sigma_{xy}  \\  -\sigma_{xy}  & \sigma_{xx}  \end{pmatrix}

有用到关系式:\sigma_{yx}=-\sigma_{xy}\rho_{yx}=-\rho_{xy}。其中,\rho_{xx}\sigma_{xx}分别为纵向电阻率和纵向电导率;\rho_{xy}\sigma_{xy}分别为横向(霍尔)电阻率和横向(霍尔)电导率。

两者关系:\sigma \cdot \rho =1

展开后有:

\left\{  \begin{aligned} & \sigma_{xx} \rho_{xx} - \sigma_{xy}\rho_{xy}=1 \\   & \sigma_{xx} \rho_{xy} + \sigma_{xy}\rho_{xx}=0  \end{aligned}  \right.

推导得到以下公式:

\rho_{xx}=\frac{\sigma_{xx}}{\sigma_{xx}^2+\sigma_{xy}^2} \quad \quad    \rho_{xy}=-\frac{\sigma_{xy}}{\sigma_{xx}^2+\sigma_{xy}^2}

\sigma_{xx}=\frac{\rho_{xx}}{\rho_{xx}^2+\rho_{xy}^2}    \quad \quad   \sigma_{xy}=-\frac{\rho_{xy}}{\rho_{xx}^2+\rho_{xy}^2}

于是有结论[2]:

(1)当\rho_{xy}=0时,有常见的关系式:\sigma_{xx}=\frac{1}{\rho_{xx}}

(2)当 \rho_{xy}\neq 0时,如果\rho_{xx}=0,那么\sigma_{xx}=0

参考资料:

[1] Supriyo Datta, Electronic transport in mesoscopic systems, 2004, P23-P26.

[2] https://www.zybuluo.com/zhouhuibin/note/858185

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