Ginzburg-Landau Theory: Model and Characteristic Lengths Discussion
Ginzburg-Landau theory is discussed in three parts, focusing on the model derivation of characteristic lengths such as penetration length and coherence length, calculation of surface energy for Type I and Type II superconductivity, and exploration of current-carrying states and phase coherence. The lecture series delves into the non-local modifications by Pippard and London, addressing the relationship between impurity concentration and penetration depth. Insightful visuals and theoretical explanations contribute to a comprehensive understanding of the topic.
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Today Lecture 4: Ginzburg-Landau Theory --- model and characteristic lengths Discussion the Ginzburg-Landau theory in three parts: 1. Presentation of the model and derivation of the penetration length and coherence length 2. Calculation of the surface energy and categorization of Type I and Type II superconductivity 3. Current-carrying states and phase coherence Lecture 5: Ginzburg-Landau theory --- surface energy and vortices Next Tuesday
Pippard Non-local Modification to ? ? Brian Pippard (Cambridge, 1953) ? ? ? depends on a weighted-average of ? over a range ? = coherence length ? 1 ? ? ? ?? ? = LONDON ? = ? ? ? ? ? ? ? ?4 1 ? 3 ?? ?3? ? ?? ? = PIPPARD 4??0 ? ? How did he get this form? CHAMBERS expression for non-local resistivity (replacing ?? ? = ?? ? ) 4? ? ? ? ? 3 ? ?3? ?? ? = ? ?4 Here, range of influence is due to memory over time between scattering events
Range of non-locality for SC: Values of ?0 ? = Pippard coherence length ?? ?? ?? ?? ???? ??3?? ???? ?,? ???? ? 1600 ?? 230 ?? 83 ?? 38 ?? 20 ?? 4 ?? 1.5 ?? 0.4 ?? DIRTY SC ?0 CLEAN SC ( ?0) ?0 ?0+ ? = ?0 ? = ~ 1 ?=1 +1 ?0 ? ~ ?0 CLEAN DIRTY
Non-locality: ? vs. ? NON-LOCAL (Pippard) LOCAL (London) ? >> ? ? ? ? ? Averages in current from higher ? effective increase in ? by ~ ? ? ? ? ? ? ? ? ? = 1 ? ~ 1 ? ? ? ? ? ? 4?? ? = 1 ? ? ? 4?? ? = 1 ? ? = ? = ? ? ? ? 1 ?2= 1 ? ? ?2? = 4? ? ? 1 ? ? 1 ?2? ?2? = 4? 1 ? = ? = ?2 ? = ?L2? ?L2 ?2 ?L2 13 13> ?P>?L ? ?L 13= ?L ? ?L ? = ?L2? ? = ?L ?= ?L 13 ?0 ?L 13= ?L ?? = ?L2?0 Pippard penetration length:
Non-locality over the coherence length modifies the penetration depth depending on the impurity concentration: In CLEAN LIMIT ? = ?0 ??> ?0 ? = ?? LONDON 13 ?0 ?? 13 PIPPARD SC ??< ?0 ? = ??= ?? > ?? PIPPARD ?0 ?? ?0> ?? ??= ?? ? LONDON SC ?0< ?? ?? ?0 ~ In moderately DIRTY limit 13 CLEAN ?? DIRTY ?0 ? = ?? ?? ?~ < ??,?0 In very DIRTY limit Pippard coherence length and non-locality was introduced to explain the impurity-dependence of the penetration depth Always in London (local) limit ? ? ? Pippard expression: 12 ?~ 1 ?0 ?0 We will see a similar but different coherence length emerge from the Ginzburg-Landau theory ? ? = ?? > ??> ?? ?
GINZBURG LANDAU THEORY (1950) Phenomenological allows for spatial variations of properties not included in London equations (local electrodynamics) Order parameter ? ?complex wavefunction ? ? ??? ? 2 = ?? two-fluid model Superconducting electron density ? ? Vitaly Ginzburg Lev Landau Not widely outside Russia in the early days --- thought to be too phenomenological --- until --- 1959 Gorkov (using thermal Green s functions) showed that GL BCS in limit of near ??, slowly varying fields and currents and ? ? ? ? ~ ? energy gap parameter (related to , gap in excitation spectrum) Lev Gorkov BCS gap parameter which depends on qp s Now widely accepted and applicable to many problems We will follow the presentation (by Ginzburg and Landau) --- not the microscopic derivation (by Gorkov)
GINZBURG LANDAU MODEL and free energy near phase transition in ?2 in zero field Simplest possible form: cannot expand in ? since G must be real odd terms not allowed since G is analytic at ? = 0 ? 0 = ??0 ??0 = ? ?2+1 2? ?4 ? ? ? > ? or else minimum ? at ? NORMAL STATE SUPERCONDUCTING STATE ? ? ? ?2 = 0 as before (G-C model) ? < 0 ? ? ? > 0 ? ? = ?2 ?2 ?= 1 ?2 ?= ??2 ?+1 2 2 8? ? ? Condensation energy ? ? 0 4??2 ? Mexican hat potential ? < 0 ??< ?? ??= Minimum ? at ?2= ? ?> 0 Minimum ? at ? = 0
What about ? ? What about ? ? ? = ?? ??> 0 ? > 0 ? = 0 ? = 0 ? > ?? NORMAL GL assumed ? ? = ? ??= constant ??2= 4??2 ?~ ?? ?2 ?? ? ~ ?? ? which is observed near ?? ? = ?? ? < ?? TRANSITION SC ? < 0 ? < 0 Finite fields allows currents, spatial variations: 1 ??(?) = ??0 +?2 With geometry effects (intermediate state): ? = 0 8? ? ?2 8? ??(?) = ??0 + ???? ? If field ? penetrates, energy required to expel field is smaller ?? ??0 ?? ? ? ? ? ?? ??0 ? ?? "??2 1 ? ?? "??1 ?
2 ?? 1 ? ? 2 ???= = kinetic energy 2? ? 2 2+ ?? ? 1 ? = ? ??? for 2? 2? ? ?2 = ? gradients in ? (interface N-S) kinetic energy of pairs (supercurrent) ? ? ? ?? 1 2? ??2 = ? ?? GL only strictly valid at ~?? but extensions to lower ? are useful and often give useful results All quantities obtained from 2 experiments ? vs. ? and ?? vs. ? all thermodynamics, electrodynamics, complex cases where fields and currents vary (e.g. intermediate state)
Ginzburg-Landau Theory 2 ?? ? = ?2+1 1 1 ?? ?, ? = ?? ?? ? ? 2? ?4 + 8?? ?2 + Free energy: 2? ? CONDENSATION ENERGY KINETIC ENERGY MAGNETIC FIELD EXPULSION No fields ? = ? Fields minimize total ?? of sample 2? ? ?2+ ?? 1 2? ??2 Minimize ?3? ?? ?, ? wrt ? and ? ? ? + ?? magnetic fields order parameter ? ? + ?? GL differential equations: 2 ?? ? 1 ? ? + ? ?2? + (1) ? = 0 2? ? ? 2? ? ? 2 ? ? ? ?2 ? ??= ? ?? ??? 4?? ? = (2)
Boundary conditions: ?? ? (sufficient to give no current flow) G-L ? ? ? = 0 ? ? for insulator ?? ? ? ?? ? ? ? = deGennes (necessary for current- carrying states) ? finite for normal metals, ? ? 0 for magnetic materials (no SC) ? ? (insulator) ? ? = 0 ? finite (normal metal) (magnetic material)
Notes: (1) LOCAL theory ?? ? (2) EXPANSION in ? is strictly only valid near ?? but works more widely (3) SPATIAL theory allows varying fields and currents (4) PHENOMENOLOGICAL - ,? can be determined from measurements of ?? and ?? (via ) 2= ?? =1 ?2= ? 2?? ? = 0, ? uniform ? = ? 2 ?2 ? 8?= 1 2 ISOLATED SAMPLE no transport currents ? = real (choose proper gauge) 2 4??2? = ? + ?2 ??2?2? + ??3 ? 8??2 ??2 ?2 ? =4??2?? ? ?2 ? = ? using ? = 2?,? = 2? in anticipation of BCS ??2
Length Scales Variation of ? penetration depth = 12 ??2 4??2?? = ?2= ?? =1 (regain London for ? constant) 2?? (# pairs) Variation ? ? = ?? coherence length Assume no fields ? = 0 Assume no currents ? real 2 4??2? = ? + ??3 2 4???2? = ? +? ??3 ?? coherence length Define natural length scale: ? ? 12 ? = ? = ? ? Define 2 ? ? ? = 2 2???? (?) 4? ? 2 2= ? 4???2? = ? +? 2?3= ? ?3 ? ?? using ? ? = 2?2 ??2 ??2? 2? using 2 4? ??2? + ? ?3= 0 since ? < 0
Relation to Pippard coherence length: No reason why they should be related --- not based on the same physics ?? ???? ? 12 ?: ?? = 0.18 2 2???(?)?(?)~ 1 ? ??: ? = 12 1 ? 1 ? independent of ? Diverges at ?? Connection via BCS theory: 0.74 ?? ? ??0 ??? ? 0 ? ? clean limit ?0 ?~?? = ? ? = ?0 12 2 3 1 ? =0.86 ?? 1 ? dirty limit ?0?~? ? 12 3 2 ? ??= ???? 0 ? 0 Note: dependence on also different 12 ??: ? ~ ?: ? ~
Physical meaning: Pippard : range of non-local averaging Ginzburg-Landau : range of order parameter variations ?2?2? + ? ?3= 0 Let f = 1 + ? (perturbation to the bulk order parameter) ?2?2? + 1 + ? 1 + ?3= 0 ?2?2? 2? + ? ?2= 0 ? Sets range for decay of order parameter (cubic term vital) 2 ?2? ~ ?2 ? Scale over which ? can change with out large cost in free energy ? ? ~ ? ?/ ? 2 ? ? ?
? ? ? ? ? = Define ?? parameter: ratio of 2 lengths ? Type II Type II ?0 12 ? dependence: ? ~ 1 ? 12 ? ? ~ 1 ? 1 ? ? ? ? ~ constant ? Type I Type I dependence: CLEAN DIRTY 13 12 ?0 ? ?0 ~ ?, ? ~ ? (nm) 0(nm) 1600 Material Tc(K) 1.2 Al 50 0.03 12 ? ~?0 ? ~ ? ?~ ?0 ? ~ ? Sn 3.4 50 230 0.22 Pb 7.0 40 83 0.48 ~ constant ?0 Nb 9.2 85 38 2.24 PbBi 8.3 200 20 10 ?0 ?0 ?0 NbTi 9.5 300 4 75 clean limit 0.96 Nb3Sn YBCO (a,b) 18 65 3 22 ? = BCS: 95 140 1.5 ~100 dirty limit 0.72 YBCO (c) 95 700 0.3 ~1000
? > ? (Type I) ? Variation of Order Parameter near a surface ?? ? GL equations couple ? and ? so expect ? to depend on field penetration ? ? ? > ?? THICK SAMPLE Assume small neglect terms in ?? and assume ?~? to get an estimate for ? ? ???2 ?? ? ?? = GL2: ? ?? 2 ? ?2 ??2 ? = 4????2 1 ?? ? = ? ? = ???? ??2 2 4? ?2 ??2 ?2 ??2 ?2? ? + ? 3(?) = + GL1: ?? ?? = 0 2 B.C. ? ? = 0 at surface 2 ? 2? ??)2? ?? 2 2 2 ?2(? ? ? ? = {1 ? }
? 2 2 ? 2? ??)2? ?? 2 2 2 ?2(? ? ? ? = ? {1 ? } ? ?? ??= 0 INCREASING ? Double exponential in ? and ? ? ?? ? 2 H H Suppression of order parameter in field = = ( 0) x 4 2 Full self-consistent solution shows deviation of ?(x) from exponential 0 c
