Abrikosov vortex lattice solution

Abrikosov vortex lattice solution A brief exploration O. Ogunnaike Final Presentation Ogunnaike Abrikosov vortex lattice solution Physics 295b 1 / 31

Table of Contents 1 Background 2 Quantization 3 Abrikosov vorticies 4 Interesting properties Ogunnaike Abrikosov vortex lattice solution Physics 295b 2 / 31

Table of Contents 1 Background 2 Quantization 3 Abrikosov vorticies 4 Interesting properties Ogunnaike Abrikosov vortex lattice solution Physics 295b 3 / 31

History In the 1950 s Landau and Ginzburg had published their famous paper on superconductivity. But only for small values of the dimensionless material constant apple, producing the familiar first order phase transition out of the superconducting state Dr. Alexei Abrikosov a Investigated apple>1/ p 2 two second order phase transitions Type II superconductors a A. A. Abrikosov, Doklady Akademii Nauk SSSR 86, 489 (1952). Ogunnaike Abrikosov vortex lattice solution Physics 295b 4 / 31

AnewOrderparameter Recall a Bose gas is characterized by a local phase bulk: (r) sothatinthe 0(r) = 0 (r) e i (r) We will focus on solutions where 0 (r) Const. Interesting properties arise from the phase: (r), which will actually become our order parameter. Ogunnaike Abrikosov vortex lattice solution Physics 295b 5 / 31

Landau-Ginzburg formalism Z F { (r)} = d d ~r " ~ 2 2m (r # ie ~c A) 2 2 + 4 2 (1) From which we pull out the section of the hamiltonian a ected by the presence of ~ A H e = s 2 Z d d ~r r e 2 A ~c ~ (2) In Coulumb guage (r ~A = 0), we can find the current: ~ J = c H ~A ~J = e ~ 2m i ( r r ) = e m 2 (~r e c ~ A) e 2 m c 2 ~ A (3) Ogunnaike Abrikosov vortex lattice solution Physics 295b 6 / 31

Type I and Type II superconductors Without going into detailed calculations, we cite penetration depth and and correlation lengths: 1 = c r m 2e / 1 p (Tc T ), = ~ p 2m / 1 p (Tc T ) (4) From this we get the unit-less parameter, the london penetration depth: apple =, which allows us to define Type I and Type II superconductors: Examples Type I: apple< 1 p 2 Type II: apple> 1 p 2 1 Tinkman, 118 Ogunnaike Abrikosov vortex lattice solution Physics 295b 7 / 31

What distinguishes the two? 2 Surface energy For apple<<1, the surface energy is positive because there is a region ( )wherethe magnetic field is excluded from the bulk, contributing to the positive diamagnetic energy. For apple>>1, the surface energy is negative because there is a region ( )wherethe magnetic field penetrates the bulk, preventing the contribution of the full condensation energy when saturates. 2 Image from: http://inspirehep.net/record/1389160/plots Ogunnaike Abrikosov vortex lattice solution Physics 295b 8 / 31

Three regions Vorticies in Type II super conductors: 1 H < H c1 : Meissner e ect 2 H c1 < H < H c2 :Vorticiesform, creating patches of normal state 3 H > H c12 :Normalstate Ogunnaike Abrikosov vortex lattice solution Physics 295b 9 / 31

Phase Diagram Ogunnaike Abrikosov vortex lattice solution Physics 295b 10 / 31

Table of Contents 1 Background 2 Quantization 3 Abrikosov vorticies 4 Interesting properties Ogunnaike Abrikosov vortex lattice solution Physics 295b 11 / 31

Normal Regions Consider a normal region enclosed within the superconductor bulk. If we circle around the region, the phase must return I r d~s =2 n And if we minimize the kinetic energy, we see: I I Z r d~s = e ~A d~s = e r AdS ~c ~c ~ = e ~c B =2 n (5) Remark The magnetic flux is quantized in units 0 = hc e Ogunnaike Abrikosov vortex lattice solution Physics 295b 12 / 31

Nucleation Let us consider what happens in a 2D system (thin slab) when we apply a field, ~ H = Hẑ, beginning at quadratic order in 3 : Z F = da[ ~2 2m (r + i2 0 ~A) 2 2 ] (6) If we make the guage choice, A ~ =(0, Hx, 0), and recall = ~2 2m the stationary condition implies: " r 2 + 4 i 0 (Hx) @ @y + 2 H 0 2 x 2 = 1 2 (7) 3 We will later take H! H c2, wherewecanneglectthequarticterm Ogunnaike Abrikosov vortex lattice solution Physics 295b 13 / 31

Thin slab and translation invariance: (r) =e iky y f (x) From which the above can be re-written: 2 H f 00 (x)+ 0 2 (x x 0 ) 2 f (x) = 1 f (x) (8) 2 Where x 0 = ky 0 2 H. The displaced Harmonic Oscillator! 1 2 = ~(n + 1 2 ~H 2 ) 0 (9) Ogunnaike Abrikosov vortex lattice solution Physics 295b 14 / 31

Two Takeaways We can solve for f (x) and for the ground state energy to find the greatest value of H for which superconductivity can nucleate vorticies: f (x) =e (x x 0)/2 2 (x) =e iky y e (x x k)/2 2 Where x k = ky 0 2 H Might have guessed this as r vortex H c2 = 0 2 2 / (T c T ) At this field strength, the bulk should be packed with as many vorticies as possible Ogunnaike Abrikosov vortex lattice solution Physics 295b 15 / 31

Table of Contents 1 Background 2 Quantization 3 Abrikosov vorticies 4 Interesting properties Ogunnaike Abrikosov vortex lattice solution Physics 295b 16 / 31

Lattice structure We limit ourselves to a regime where H H ~ c2,andtreatthe 4 term as a perturbation. We expect some kind of crystaline array of states.: n(r) =e ikny e (x xn)/2 2 (10) k n = 2 n a x n = n 0 qh (11) y = a x = 0 ah = 0 yh (12) Remark H x y = 0 Ogunnaike Abrikosov vortex lattice solution Physics 295b 17 / 31

Lattice properties Remark H x y = 0 (holds even whenh 6= H c2 ) a = Period grows with degreasing H Flux trough one cell is constant r Holds when H 6= H c2,butmustreplaceh with its avg. over a cell hhi = 1 a 2 R da(h) H c1 when either hhi =0ora!1 0 H Ogunnaike Abrikosov vortex lattice solution Physics 295b 18 / 31

Magnetic Profile Primary assumption: The core of the vortex is so small that the vortex can be essentially taken as a line object. Take H ~ = Hẑ Equivalent to apple>>1 since >> : 4 2 r J c ~ + H ~ = X i ẑ 0 (~r ~r i ) (13) Which we can resolve with r ~ H = 4 ~ J c : r 2 ~ H ~ H 2 = 0 2 ẑ X i (~r ~r i ) (14) Which can be solved for a field profile: 0 r H(r) = 2 2 K 0 With K 0 as the zeroth order Hankel function. As r!1, K 0 (r/ ) / r e r/. Magnetic field, and thus current fall o as (15) Ogunnaike Abrikosov vortex lattice solution Physics 295b 19 / 31

With this magnetic profile for a single vortex, one solve for free energy to obtain an approximation of H c1 : H c1 0 4 2 ln apple / (T c T ) (16) Ogunnaike Abrikosov vortex lattice solution Physics 295b 20 / 31

General Lattice A more general solution would look like: (r) = X n C n n (r) = X n C n e inqy e (x xn)/2 2 (17) Ogunnaike Abrikosov vortex lattice solution Physics 295b 21 / 31

Degeneracies Unfortunately, all solutions of the form above are equally possible at H c2. To break the degeneracy, we have to bring back the 4 term. Once we restore the term, One can do numeric calculations to find that the triangular lattice (as opposed to the square one posited by Abrikosov) is the most stable Ogunnaike Abrikosov vortex lattice solution Physics 295b 22 / 31

Degeneracies Abrikosov proposed the parameter: A h 4 i h 2 i If we ignore currents and vector potentials, and write (r) =c (r), we can solve for the average free energy: In general: 2 hf i = 2 1 A 1 2 h r 2 2 i h 2 (18) i hf i = B 2 apple B 1+(2apple 2 1) A (B in units ofh c p 2) (19) Notice if Const., A 1, but if were constant over some small fraction of the volume, p, then A p 1 >> 1. The most stable lattice minimizes A. Ogunnaike Abrikosov vortex lattice solution Physics 295b 23 / 31

Table of Contents 1 Background 2 Quantization 3 Abrikosov vorticies 4 Interesting properties Ogunnaike Abrikosov vortex lattice solution Physics 295b 24 / 31

Behavior near H c1 When H = H c1 the gibbs free energy is the same with or without the first vortex: G 0 noflux = F 0 = G 0 1stvortex = F 0 + 1 L = F 0 + 1 L Where 1 is free energy per unit length of the vortex (di in general). Thus a more general equation for H c1 is H c1 R hd~r 4 H c1 0 4 (20) cult to calculate H c1 = 4 1 0 (21) Ogunnaike Abrikosov vortex lattice solution Physics 295b 25 / 31

Wavefunction profile Take a more general wavefunction (~r) = 1 f (r)e i,where 1 = is the constant value in the bulk in the absence of currents and fields. Notice radial symmetry. Fixes one flux quantum: phase rotates by 2 in single circuit. GL equation becomes: f f 3 2 " 1 r With a current given by: 2 A 2 f 0 1 r d dr r df # = 0 (22) dr ~J = c dh(r) = c 4 dr 4 d 1 dr r d dr (ra) = e ~ 1 2 m 1f 2 r 2 A 0 (23) Ogunnaike Abrikosov vortex lattice solution Physics 295b 26 / 31

Solving for A(r) andf (r) requiresnumericalmethods,butweexaminethe limits of A(r). ~ The form of the wavefunction fixed vector potential: ~A = A(r) ˆ = 1 R r r 0 r 0 h(r 0 )dr 0 ˆ Near the center of the vortex: A(r) = h(0)r 2. Far from the center: A 1 = 2 r 0, (since total flux is H A ~ ds ~ =2 ra1 = 0 ) We look at r! 0, and assume solution starts as f = cr n for n 0 " 1 f f 3 2 h(0)r 2 1 d f r df # =0 r 0 r dr dr " 1 # cr r c 3 r 3n 2 h(0)r 2 cr n n 2 cr n 2 =0 r Saturates around r 2 f cr 1 0 r 2 8 2 " 1+ h(0) H c2 #! (24) Ogunnaike Abrikosov vortex lattice solution Physics 295b 27 / 31

Vortex forces, and magnetization curves It can be shown that the force on a vortex is ~f = ~ J c 0 c ẑ (25) Static equilibrium only when superfluid velocity is zero at vortex locations. Thus, symmetric lattices. B =4 M = H H c2 (2apple 2 H 1) A Ogunnaike Abrikosov vortex lattice solution Physics 295b 28 / 31

Flux Pinning Vortex lattices are like any other. Disorder and fluctuations can create vortex liquids and glasses, and with additional energy, one can have vortex excitations. One unique property manifests when the the temperature is low enough, the vorticies can be pinned Ogunnaike Abrikosov vortex lattice solution Physics 295b 29 / 31

Conclusion Type II superconductors have a mixed state with vorticies Ogunnaike Abrikosov vortex lattice solution Physics 295b 30 / 31

Conclusion Type II superconductors have a mixed state with vorticies Voriticies hold quantized magnetic flux, 0 and form a lattice structure Ogunnaike Abrikosov vortex lattice solution Physics 295b 30 / 31

Conclusion Type II superconductors have a mixed state with vorticies Voriticies hold quantized magnetic flux, structure 0 and form a lattice GL Theory produces an intuitive lattice with vorticies of radius, and supercurrents decaying as, Ogunnaike Abrikosov vortex lattice solution Physics 295b 30 / 31

Acknowledgements A. A. Abrikosov. Nobel lecture: Type-ii superconductors and the vortex lattice. Rev. Mod. Phys., 76:975?979, Dec 2004. M. Tinkham. Introduction to Superconductivity. Ogunnaike Abrikosov vortex lattice solution Physics 295b 31 / 31

Acknowledgements A. A. Abrikosov. Nobel lecture: Type-ii superconductors and the vortex lattice. Rev. Mod. Phys., 76:975?979, Dec 2004. M. Tinkham. Introduction to Superconductivity. Thank You! Ogunnaike Abrikosov vortex lattice solution Physics 295b 31 / 31