CND-03

BCS critical temperature

T_c = 1.14 ħω_D exp(−1/N(0)V). Weak-coupling BCS snapshot.

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SuperconductivityBCS

Governing equation

T_c=1.14\,\hbar\omega_D\,e^{-1/N(0)V}

$N(0)V \in \left[0.08, 0.6\right]$

where

\theta_D: Debye temperature (K)
N(0)V: Coupling (—)
T_c: Critical temperature (K)

The figure, part by part

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①

Governing law — red

T_c=1.14\,\hbar\omega_D\,e^{-1/N(0)V}

$N(0)V \in \left[0.08, 0.6\right]$

$N(0)V = 0.08$ $T_c = 0.001$

Along the curve

②

Domain of the variable — blue

N(0)V_{\min} = 0.08,\quad N(0)V_{\max} = 0.6

$N(0)V \in \left[0.08, 0.6\right]$

$N(0)V = 0.08$ $N(0)V = 0.08$

Along the curve

③

Response curve — green

T_c=1.14\,\hbar\omega_D\,e^{-1/N(0)V}

$N(0)V \in \left[0.08, 0.6\right]$

$N(0)V = 0.08$ $T_c = 0.001$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}T_c}{\mathrm{d}N(0)V}

$N(0)V \in \left[0.08, 0.6\right]$

$N(0)V = 0.083$ $T_c^{\prime} = 0.306$

Along the curve

Lecture brief

Historical brief

Drude electrons, Bloch waves, BCS pairing (1957) and Wiedemann–Franz heat are the first solids-and-metals laws. The lab is conductivity, gap and phonon heat in closed form. This sheet (CND-03 — BCS critical temperature) is the form associated with BCS. Working symbols: $\theta_D$ , $N(0)V$ $\rightarrow$ $T_c$ . N(0)V is the dimensionless pairing coupling. ħω_D is the Debye cutoff.

Purpose

Purpose: compute

T_c

from

\theta_D

N(0)V

in Condensed matter via

T_c=1.14\,\hbar\omega_D\,e^{-1/N(0)V}

T_c = 1.14 ħω_D exp(−1/N(0)V). Weak-coupling BCS snapshot. Use it when a real condensed matter question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\theta_D = 300.000\,\mathrm{K}

N(0)V = 0.300\,\mathrm{—}

, the governing relation

T_c=1.14\,\hbar\omega_D\,e^{-1/N(0)V}

yields

T_c = 12.20\,\mathrm{K}

. A gap opening below T_c. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Debye temperature \theta_D300 KCoupling N(0)V0.3 —

Outputs

Critical temperature T_c12.20 K

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Narration of this film

A gap opening below T_c.

N(0)V is the dimensionless pairing coupling. ħω_D is the Debye cutoff.

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