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CHRITCAL DIAMETR OF SUPERCONDUCTING NANOWIRES |
Tony Bollinger
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Mon Apr 2 05:22:16 2007
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
| Yes, 4 nm is close to what we get from the diagram. There is a boundary line that corresponds to a critical cross-sectional area of 15.6 nm^2 or 4.5 nm diameter. |
| Created: Mon Apr 2 06:43:19 2007
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Alexey
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Mon Apr 2 04:58:18 2007
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
I see you say:
"To estimate critical diameter we can use the average T_c0 of 5 K from our magnetic field effect on I_c PRL. Then alpha_c = 6.1e-23 J. This means tau_B,c ~ hbar / alpha_c ~ 1.7 ps, which corresponds to about 5 nm diameter on our magnetic field effect on I_c PRL Fig. 3."
I get that for Tc=5K tau_b=0.26ps. Can you say why the disagreement occurs?
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| Created: Mon Apr 2 06:48:19 2007
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Rob
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Tue Apr 3 05:20:13 2007
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
Alexey. The work I did was not correct because it used the wrong formula in Tinkham. According to Tsu Chieh we need to use some different expansion for alpha = f(Tc). I am not sure what that is. It is a higher order expansion of that ugly digamma function than the one given in Tinkham.
also
2*alpha=hbar/tau
the paper has
tau= Ef/(2*pi*J*J*xm)*(missing hbar)
with
Ef=2eV and J=0.2eV
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| Created: Tue Apr 3 05:29:58 2007
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Rob
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
OOps I missed that there were 3 pages :)
Anyway I am confused by this 4nm thing. The "worst" insulating wires have a diameter of about 5.5nm and that was with a rho = 200 uOhm.cm. or 5.2 if we use the 180 value.
Tony how did you calculate this boundary line you are talking about? Is it the line you can get when try to separate them with a ruler and get L/Rn=L/2Rk + constant? Or something like that. |
| Created: Tue Apr 3 05:51:19 2007
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Alexey
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The most final formula so far. |
the formula we established with tony is this:
tau_b=exp(0.577)/pi/pi/kB/T_c0
(see the discussion over the above 3 pages)
Could it be still wrong for some reason? |
| Created: Tue Apr 3 06:06:22 2007
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Alexey
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
It might be not exact if this expression:
alpha=x_m*J^2/Ef
is not exact.
(we used this expression to derive the formula) |
| Created: Tue Apr 3 06:08:47 2007
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Alexey
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
| Tsu Chieh, do you agree with alpha=x_m*J^2/Ef or not? |
| Created: Tue Apr 3 06:09:35 2007
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Tony Bollinger
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Mon Apr 2 05:22:16 2007
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
Alexey, in response to our difference in calculated tau_B ...
There are two reasons for this difference:
a) I used T_c0 = 5 K (average of T_c0 for MoGe wires in Table I of Ic vs. B PRL)
b) I missed a factor of 2*pi
Correcting for the factor of 2*pi I get tau_B = 0.3 ps, which correspnds to a diameter of maybe 1 nm. We should determine what the correct T_c0 to use is. |
| Created: Tue Apr 3 07:48:16 2007
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Tony Bollinger
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
Rob, the boundary line we can draw across the diagram in the (R_N, R_N / L) coordinate plane representation is R_N / L = (2*R_Q - R_N) / 112 nm (here R_Q = R_K / 4 = 6.45 kOhms). Wires above this line are insulating, wires below it are superconducting.
Obviously this line has two intercepts: i) with the abscissa at R_N = 2*R_Q and ii) with the ordinate at R_N / L = 2 * R_Q / 112 nm. The first intercept just means all wires with R_N >= 2*R_Q are insulating. The second intercept means all wires with R_N / L >= 2 * R_Q / 112 nm are insulating.
Using the usual relation, R_N = rho_N * L / A, and rho_N = 180 microOhm-cm, the second condition above becomes R_N / L = rho_N / A >= 2 * R_Q / 112 nm or A <= 180 microOhms-cm * 112 nm / 12.9 kOhms = 15.6 nm^2. This corresponds to a diameter of 4.5 nm.
Of course there are some wires with diameters greater than this that are insulating but no wire with diameter less than 4.5 nm is superconducting. |
| Created: Tue Apr 3 08:13:08 2007
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Rob
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
OK this is just me rambling....
Tony The same line can be drawn on the usual diagram. I did that and it looked like it was just a coincidence. I am thinking the same now. I really don't think there is one line that can separate all the samples. For the short wires I think we simply have suppression of Tc to 0 due to impurities. This suppression is different for each wire. For the longer wires I am not quite sure, but again they are still more insulating than theory predicts so maybe a similar argument can hold.
Looking at the numbers I see the thinnest SC wire, 121204i has a diameter of 5.6 and the thickest I wire has a diameter of 5.2. This I wire has an Rn of 5.3KOhm. The film Tc was 4.5 or less so the wire, if it were to be SC would have a low Tc_0. Perhaps there is some way to get an estimate for Tc_0. Where did the values in the paper come from? Can we do a similar analysis on a thinner wire? I think I can actually make one, though perhaps not quickly enough. Can we simply treat each case as Tc _> 0, use xm for that diameter and back calculate what Tc_0 would have to be in order to get complete suppression of SC due to impurities? Since Tc and diameter both play a role in whether the wire goes sc or not with the concentration of impurities we have; why are we looking for a critical diameter? Doesn't it suffice to say we can explain each of the deviant wires by using the results from the impurities paper and then showing for that for that diameter one would need a Tc_0 to be a certain value and then state why that is a reasonable value? I suppose we could still argue the other way, but I think it is more correct thinking about each wire individually first. Because you know as well as I do that each sample is special in their own way :)
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| Created: Tue Apr 3 18:46:58 2007
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Alexey
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
Rob, the question is: how to construct SIT phase diagram for nanowires.
Check this one version of the diagram here:
http://www.nanogallery.info/nanogallery/?ipg=165
It is well defined. But we want to find some interpretation.
Note that Fig.3 in the preprint (also linked to that page listed above) presents another version of the SIT diagram (this might be understood as CS-like transition). The version of the diagram on Fig.4 seems to imply that the shortest wires have some sort of critical diameter. So we are trying to see how to explain this critical diameter. Surface-localized magnetic moments might be one way to do this. |
| Created: Tue Apr 3 20:56:00 2007
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TzuChiehWei
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Tue Apr 3 22:52:35 2007
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RE: CHRITCAL DIAMTER OF SUPERCONDUCTING NANOWIRES |
Hi Alexey and Tony,
First of all, from dimensional analysis,
tau_b (dimension of time) = hbar/ Energy.
J and Ef are of unit energy.
x_m is just a number (unitless).
The correct combination for tau_b is proportional to
Ef/ J^2. As to which of the four is correct, I say
tau_b = hbar Ef/ J^2 / (2pi x_m),
namely the second one is the correct answer.
Let me explain: I defined the tau_b originally in our calculations as follows
hbar/tau_b = n_i 2 pi N0 < J(q)^2>, where
J(q) is the Fourier transform of J, so is of dimension J* volume
N0 is density of state per volume, so is of dimension 1/Ef /volume
n_i is impurity concentration, so is of dimension 1/ volume.
Namely, J(q) ~ J a^3, N0 = 1/ Ef/ a^3, n_i = x_m /a^3, where a is
some lattice constant. Putting these into the expression, we get
hbar/tau_b = x_m 2pi J^2/Ef.
In connection to solving Tc, we can use Abrikosov and Gorkov's result
log[ Tc/Tc0] = Psi (1/2) - Psi (1/2 + hbar/(2pi tau_s kB Tc)),
where their tau_s is related to our tau_b by
1/tau_s = S(S+1)/ tau_b, as we did not include the magnitude of
spin angular momentum in tau_b.
According to them, the critical tau_s where superconductivity vanishes
is at tau_s_cr = 2 hbar gamma/ pi /(kB Tc0), where gamma ~ exp (0.5772).
From this we can deduce the critical x_m:
hbar Ef/ J^2 / (2pi x_m_cr) /(S(S+1)) = 2 hbar gamma /pi /(kB Tc0)
==> x_m_cr = Ef/ J^2 /(S(S+1))/4 * (kB Tc0).
Note: kB is Boltzman constant and S =1/2 for a spin-1/2 as we have
assumed in the paper.
If we assume the impurity is uniformly distributed over the surface
of a cylinder, then we can deduce the critical diameter.
Tzu-Chieh
From Alexey:
We try to figure out which of these formulas is correct:
1. tau_b=Ef/2/pi/J^2/x_m
2. tau_b=h_bar*Ef/2/pi/J^2/x_m
3. tau_b=Ef/J^2/x_m
4. tau_b=h_bar*Ef/J^2/x_m
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| Created: Tue Apr 3 23:22:59 2007
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