How Does Increasing Length Affect Spin Hall Voltage in Materials?

[ubergewehr273]

Hi!

I'm trying to understand the dependence of spin hall voltage on various parameters of the material. I have been going through this paper, and it is mentioned that $$V_{SH} = 2 \pi R_s L j_x n \mu_B$$

In the equation, only $L$ and $j_x$ seem to be the variables. Does increasing $L$ keeping $j_x$ fixed, increase the spin hall voltage linearly (as seen from the equation)? If I simply increase $L$, is it correct to say that I cannot magically increase the spin Hall voltage simply because $j_x$ will decrease as $L^{-2}$?

Also, from the paper, doing a quick dimension analysis, I get the RHS as $[IL T^{-1}]$ whereas voltage has a dimension of $[ML^2 I T^{-3}]$. Where am I going wrong?

 

[eljose79]

Hello there!

Thank you for bringing up this interesting question. The equation you have mentioned is a simplified form of the spin Hall voltage equation and it only takes into account the spin Hall angle ($R_s$), the length of the material ($L$), the charge current density ($j_x$), and the spin magnetic moment ($\mu_B$). Other parameters such as the material's conductivity, mobility, and carrier density are not included in this equation.

To answer your first question, yes, increasing the length of the material while keeping the charge current density fixed will result in a linear increase in the spin Hall voltage. This is because the spin Hall voltage is directly proportional to the length of the material in this equation. However, in real materials, the spin Hall angle and other parameters may also change with the length, resulting in a non-linear relationship.

As for your second question, the dimensions of the equation are indeed not correct. The correct dimensions for the RHS should be [ML^3 T^-2 I^-1], which is the dimension of voltage. This is because the spin Hall voltage is a product of current density (I/L^2), spin Hall angle (dimensionless), length (L), and spin magnetic moment (I/T). Therefore, the final dimension should be [ML^3 T^-2 I^-1].

I hope this helps clear up your doubts. Keep exploring and asking questions!