Penetration Depth of General Complex Conductivity

[IDumb]

Hi all,

I'm working through chapter 2 of Michael Tinkham's Introduction to Superconductivity. On page 40, he asserts that the skin-depth for a general complex conductivity is (In Gaussian units)
$$\delta = \frac{c}{\sqrt{2\pi\omega\left(|\sigma| + \sigma_2\right)}}$$
where $$\sigma = \sigma_1 - i\sigma_2$$

I am trying to derive this skin-depth expression, but can't seem to get it. My process is as follows. I have bolded the two places where potential issues could be. Start with combining Faraday's law and Ampere's law:
$$\vec{\nabla}\times\vec{E} = -\frac{1}{c}\frac{\partial\vec{B}}{\partial t}$$
$$ \vec{\nabla}\times\vec{\nabla}\times\vec{E} = -\frac{1}{c}\frac{\partial}{\partial t}\left(\frac{4\pi}{c}\vec{J} + \frac{1}{c}\frac{\partial\vec{E}}{\partial t}\right)$$
Now use J = \sigma E and assume E = exp(i\omega t),
$$ \vec{\nabla}(\vec{\nabla}\cdot\vec{E}) - \nabla^2\vec{E} = -\frac{1}{c}\left(\frac{i4\pi\sigma\omega}{c}\vec{E} - \frac{\omega^2}{c}\vec{E}\right) $$

Now a potentially sketchy step, I assume the displacement current term is very small (I'm dealing with a superconductor here, so it makes sense), and I also assume the charge density is spatially uniform. This results in eliminating the first and fourth terms. I think this may be where I'm losing it, but I'm not sure how else to do it. The result is

$$\nabla^2\vec{E} = \frac{i4\pi\sigma\omega}{c^2}\vec{E}$$

Solving this,

$$\vec{E} = \vec{E}_0\exp{\left(-\sqrt{\frac{i4\pi\sigma\omega}{c^2}}z\right)} $$

$$= \vec{E}_0 \exp{\left(-\sqrt{\frac{4\pi\omega(\sigma_2 + i\sigma_1)}{c^2}}z\right)}$$

I'm having trouble now. I try to separate this into real and imaginary parts, but the real part does not seem to be what Tinkham has. I think the way I'm taking the squareroot of a complex number is the problem.

$$= \vec{E}_0 \exp{\left(-\sqrt{\frac{4\pi\omega|\sigma|}{c^2}}\left(\cos{\theta/2} + i\sin{\theta/2}\right)\right)}$$
Where $$\theta = \arctan{\frac{\sigma_1}{\sigma_2}}$$

The resulting skin depth is
$$\delta = \frac{c}{\sqrt{4\pi\omega |\sigma |}}\frac{1}{\cos{\theta/2}}$$
This makes sense to me, is close to the given value, and reduces to the skin depth of a real conductivity for \sigma_2 = 0, as it should. My expression does reduce to Tinkham's if I assume \sigma_1 << \sigma_2, which is a very reasonable approximation. But the assertion in the book that this is a "genera" solution is what troubles me.

Does anyone have ideas on what I'm missing? I would really appreciate your help!

 

[John Park]

I assume \sigma_1 << \sigma_2, . . . the assertion in the book that this is a "genera" solution is what troubles me

Given the title of the book, does "general" solution simply mean it's true all superconductors (or even all reasonably good conductors)?

 

[IDumb]

Given the title of the book, does "general" solution simply mean it's true all superconductors (or even all reasonably good conductors)?

I assume so. That's the justification for removing the displacement current term at least. He says a "General complex conductivity" though... Which contradicts that.

 

[John Park]

"General complex conductivity"

I still suspect it's semantic. As far as I can tell a real, frequency-dependent skin-depth implies a good conductor. How does Tinkham set up the problem?

 

[John Park]

I just looked at Tinkham's page 40 on the web; he says he's "solving the skin depth problem", as though it's an understood procedure, presumably with standard assumptions and approximations. And he's talking about good conductors the whole time; so I think "general" here simply means both real and imaginary parts of σ are included, but they're still limited to a good conductor.

 

[malindeman]

Note: that formula from Tinkham is in Gaussian units, which might confuse (sane) people who only have worked in SI units. --M. A. Lindeman

 

[malindeman]

here is a calculation by chatgpt o1 mini