--- title: "Complete and incomplete Meissner effect" subject: "Electrical Engineering Material" chapter: 5 key: "semester_3/electrical_engineering_material:meissner-complete-incomplete" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/meissner-complete-incomplete site: BCT Engineering --- # Complete and incomplete Meissner effect Electrical Engineering Material · Chapter 5: Superconductivity - Asked in: 2069 Asar, 8 marks - Video: [Meissner Effect (Properties of Superconductor) Impurity Effect & Pressure Effect (Btech 1st year)](https://www.youtube.com/watch?v=vePYlqZTvuk) — Gautam Varde - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/meissner-complete-incomplete - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-5 ## Question What is Meissner effect? Explain how Meissner's effect is complete for type-I and incomplete for type-II superconductors. ## Short answer Meissner effect is the complete expulsion of magnetic flux from a superconductor below Tc, making it a perfect diamagnet. Type‑I superconductors expel flux entirely up to Hc, then go normal. Type‑II superconductors expel flux completely only below Hc1; between Hc1 and Hc2 flux penetrates as quantized vortices, so the Meissner effect is incomplete. ## Answer #### Meissner effect **Meissner effect** is the complete expulsion of magnetic flux from the interior of a material when it is cooled below its critical temperature $T_c$ in a weak magnetic field ($H < H_c$). The superconductor becomes a **perfect diamagnet**. Inside the superconductor: $$ \begin{aligned} B &= \mu_0 (H + M) = 0 \\ \Rightarrow M &= -H, \quad \chi_m = \frac{M}{H} = -1 \end{aligned} $$ Screening currents flow in a thin surface layer (the London penetration depth, $\lambda_L \approx 10$ to $100$ nm) and create a field that exactly cancels the applied field inside. ```text T > Tc (normal) T < Tc (superconducting) | | | | | | | \ \ / / | |+-----+| | \ \ +---+ / / | || || | cool ) )| |( ( | || || | ----> / / +---+ \ \ | |+-----+| | / / \ \ | | | | | | | flux lines bend around it ``` The effect is independent of history: whether the field is applied before or after cooling, the flux is expelled. This shows a superconductor is not just a perfect conductor (a perfect conductor would trap the flux present at cooling). #### Complete Meissner effect in type-I - In a type-I superconductor, $B = 0$ inside for **all** fields up to $H_c$. Magnetization rises linearly, $M = -H$. - At $H = H_c$, flux enters all at once and the material becomes normal; $M$ drops to zero. - There is no intermediate state in a long sample, so the expulsion is **complete** right up to $H_c$. Examples: Pb, Hg, Sn. #### Incomplete Meissner effect in type-II - Below the lower critical field $H_{c1}$, flux is fully expelled (complete Meissner effect). - Between $H_{c1}$ and $H_{c2}$ it is energetically favourable for flux to enter as thin normal cores called **vortices** (fluxons), each carrying one flux quantum $\Phi_0 = h/2e = 2.07\times10^{-15}$ Wb, surrounded by circulating supercurrents. - The material between vortices stays superconducting (zero resistance), but $B \ne 0$ inside on average. Hence the Meissner effect is **incomplete** in this **mixed (vortex) state**. - As $H$ increases, vortex density rises; at $H_{c2}$ the cores overlap and the material becomes normal. Examples: NbTi, Nb$_3$Sn, YBCO. ```text -M -M | /| | /\ | / | | / \ | / | | / \_ | / | |/ \__ |/ |____ H +----|------|--- H 0 Hc 0 Hc1 Hc2 Type-I Type-II ``` ```text Mixed state (top view of type-II slab) +-------------------+ | o o o o | o = vortex (normal | o o o | core with flux) | o o o o | rest = superconducting +-------------------+ ``` | Point | Type-I | Type-II | |---|---|---| | Meissner effect | Complete up to $H_c$ | Complete below $H_{c1}$, partial up to $H_{c2}$ | | Flux entry | All at once at $H_c$ | Gradual, as vortices | | B inside | 0 until $H_c$ | Non-zero between $H_{c1}$ and $H_{c2}$ | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the Meissner effect? A. Complete expulsion of magnetic flux from a material cooled below Tc in a weak field B. Drop of resistance to zero when a current flows C. Trapping of the existing flux when a perfect conductor is cooled D. Increase of Tc with applied field Answer: A. Below Tc and for H < Hc the interior has B = 0, so the superconductor is a perfect diamagnet. (Key point: Meissner effect definition) 2. Why is the Meissner effect complete in type-I up to Hc? A. Vortices enter gradually from the start B. B = 0 inside for all fields up to Hc, then flux enters at once with no intermediate state C. B equals the applied field inside D. It has two critical fields Answer: B. No mixed state exists, so expulsion is complete until Hc. (Key point: Why complete in type-I) 3. Why is it incomplete between Hc1 and Hc2? A. Resistance returns in the bulk B. The Cooper pairs are fully broken C. Flux enters as vortex cores, so average B is not zero, though material between vortices stays superconducting D. The applied field is below Hc1 Answer: C. In the mixed state vortices carry flux, so B is nonzero inside. (Key point: Why incomplete in type-II) 4. What flux does each vortex carry? A. h/e = 4.14e-15 Wb B. Any value depending on the field C. Zero flux D. One flux quantum, h/2e = 2.07e-15 Wb Answer: D. Flux through each vortex is quantized as Phi0 = h/2e. 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