A Penetration Depth of General Complex Conductivity

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1. Mar 20, 2017

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!

2. Mar 22, 2017

John Park

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

3. Mar 22, 2017

IDumb

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.

4. Mar 22, 2017

John Park

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?

5. Mar 22, 2017

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.