Master Thesis
Properties in n-component London Superconductors
Sergio Ampuero Felix
Condensed Matter Physics, Department of Physics, School of Engineering Sciences Royal Institute of Technology, SE-106 91 Stockholm, Sweden
Stockholm, Sweden 2017 Typeset in LATEX
TRITA-FYS 2017:83 ISSN 0280-316X ISRN KTH/FYS/–17:83—SE
c Sergio Ampuero Felix, December 2017 Printed in Sweden by Universitetsservice US AB, Stockholm December 2017 Abstract
The purpose of this thesis is to analytically investigate n-component London super- conductor properties that have previously been investigated analytically in cases involving a lower amount of components. For vortex excitations in n-component London superconductor, we obtain ex- pressions for magnetic flux, magnetic vector potential, microscopic magnetic field and circulation of velocity. Furthermore, the critical applied field for vortex formation is found. Finally, the response of a cylindrical specimen to rotation is derived both in terms of a n-component analogue of the London field but also by finding the critical angular frequency for vortex formation.
iii iv Preface
This thesis assumes that the reader has some familiarity with quantum physics, functional derivation, the concept of minimizing appropriate free energy and the Ginzburg-Landau model for superconductivity. The result of work done from roughly Mars 2017 to November 2017 is man- ifested through this thesis. I would like to take this opportunity to express my utmost gratitude to my thesis supervisor Egor Bavaev for his patience and excel- lent guidance.
v vi Contents
Abstract ...... iii
Preface v
Contents vii
1 Introduction 1
2 General Considerations 3 2.1 The n-component free energy density Fs ...... 3 2.2 Deriving the Ginzburg-Landau equations ...... 4 2.2.1 Variation with respect to ψl ...... 4 2.2.2 Variation with respect to A~ ...... 5 2.3 London limit ...... 7 2.4 Supercurrents in the London limit ...... 7
3 Vortices 9 3.1 Definitions ...... 9 3.2 Vortex flux ...... 9 3.3 Calculating the magnetic vector potential in the London limit . . . 10 3.4 Calculating the microscopic field in the London limit ...... 12 3.5 Determining the constant C2 ...... 13 3.6 Calculating the circulation of velocity in the London limit . . . . . 15 3.6.1 2-component example ...... 16 3.6.2 The special case of exponentially decaying supercurrents . . 16 3.7 Free energy of a vortex in the London limit ...... 17 3.7.1 2-component example ...... 20 3.7.2 The special case of exponentially decaying supercurrents . . 21
4 Response to Applied Field H~ 23 4.1 Minimizing Gibb’s free energy ...... 23 4.1.1 2-component example ...... 25
vii viii Contents
5 Rotational Response 27 5.1 The London field ...... 27 5.1.1 1-component example ...... 28 5.1.2 2-component example ...... 28 5.2 Vortex formation condition ...... 28 5.2.1 Calculating the z-component of the angular momentum in the London limit ...... 28 5.2.2 Finding the critical angular frequency ...... 31
6 Summary and Conclusions 35
Bibliography 37 Chapter 1
Introduction
Multicomponent London superconductivity has attracted interest for an extended period of time. See for example [4] and [5] for discussion of flux fractionalization and neutral sector. The existence of a neutral sector motivated comparison to established results from analysis of neutral one-component superfluids, in particular Onsager’s [12, year 1949] and Feynman’s [14, year 1955] quantization of vortices and the associated ~ circulation quanta m . Notably, an investigation of the two-component equivalent was published in 2007 [9]. A distinct concern is how superconductors respond to rotation. In 1950, Lon- don provided pioneering analysis of rotating one-component superconductors [13]. Consequently, a dissection of the two-component correspondent was promulgated in 2007 [9]. In this thesis, the cynosure is an analytical examination of how n- component London superconductors respond to application of an external magnetic field and rotation respectively. Among germane work, one finds an article from 2004 [11] in which expressions for the n-component magnetic flux Φ and current J~ were partial results. Throughout this thesis, absence of interband couplings is assumed. An article [9] proposes that a physical example of a two-component version of a such system could be liquid metallic deuterium, liquid metallic hydrogen [15] and condensates in neutron stars [16] [23] [24]. Moreover, [11] advocates that higher component cognates should exist in ”metallic phases of light atoms under extreme pressure”. Due to metallic hydrogen requiring extreme pressures in order to be produced, the task of creating and studying such matter through experiments is a difficult one. This is a subject of broad current research efforts. For example: a prominent article [21] published in the current year, 2017, provides information to ”stimulate theoretical predictions of how to retain metastably hydrogenous materials made at high pressure P on release to ambient”.
1 2 Chapter 1. Introduction
A different article [22] aids by ”As such, the present study bridges the important but less well-explored intermediate regime between warm dense fluid and the solid phases at high P-T conditions.”. A third article [20] analyzes the closely related topic of semimetallic hydrogen. Most spectacular is an article describing a realized production of solid metallic hydrogen in a laboratory setting [17]. That claim was heavily criticized through arguments indicating that there existed severe experimental flaws [18]; as a reply the authors defended their claim [19]. These current experimental efforts motivate theoretical studies of multicompo- nent superconducting systems that are considered in this thesis. Chapter 2
General Considerations
2.1 The n-component free energy density Fs
We will investigate the n-component superconductor using the Ginzburg-Landau and London models. Following the well-known procedure by Ginzburg and Landau [25], by introduc- ing a macroscopic effective wave function ψ as an order parameter that through |ψ|2 corresponds to number density, the free energy density of a normal specimen in the absence of a magnetic field Fn, two temperature-dependent parameters α and β, charge q, mass m, magnetic vector potential A~ and magnetic field ~h one can express the free energy density of a 1-component superconductor as following:
~ 2 2 β 4 1 ~ 2 h Fs = Fn + α|ψ| + |ψ| + |( ∇~ − qA~)ψ| + (2.1) 2 2m i 2µ0 An important remark is that ~h refers to the microscopic magnetic field. The magnetic flux density B~ is an average of ~h over a suitable volume. Equation 2.1 is easily generalized to n components as following:
" # ~ 2 X 2 βl 4 1 ~ 2 h Fs = Fn + αl|ψl| + |ψl| + |( ∇~ − qlA~)ψl| + (2.2) 2 2ml i 2µ0 l
It is instructive to compare that equation to the one found at [11, equation 1] and the more general one at [10, p. 167]. As an example one can mention the liquid metallic hydrogen case [15] where different components correspond to condensates of protonic and electronic cooper pairs, respectively. A third and fourth component can arise through cooper pairs of tritium and Bose-Einstein condensation of deu- terium nuclei [11]. In a neutron star example [16] [23] [24], the different components correspond to condensates of neutronic and protonic cooper pairs, respectively.
3 4 Chapter 2. General Considerations
2.2 Deriving the Ginzburg-Landau equations
Recall following from your course in electrodynamics:
~h = ∇~ × A~ (2.3)
We will obtain the Ginzburg-Landau equations by varying the free energy func- tional with respect to the order parameter Ψ and the magnetic vector potential A~.
2.2.1 Variation with respect to ψl ¡ 3 We will first minimize 3 d rFs with respect to a variation of ψl: R
¢ " 3 ∗ βl 2 ∗ 0 = d r αlψl δψl + |ψl| 2ψl δψl 3 2 R ∗ # ( ~ ∇~ − qlA~)ψ · ( ~ ∇~ δψl − qlAδψ~ l) + −i l i (2.4) 2ml
Next, recall that given any scalar φ and any vector T~, we have following vector identity:
∇~ (φT~) = ∇~ φ · T~ + φ∇~ · T~ (2.5)
~ ~ ~ ( −i ∇−qlA) ∗ Utilizing equation 2.5 in 2.4 for φ := ~ δψl and T~ := ψ , we then i 2ml l obtain:
¢ " ~ ~ ~ ∗ ~ # 3 ∗ 2 ∗ ( −i ∇ − qlA)ψl · (−qlA) 0 = d rδψl αlψl + βl|ψl| ψl + + 3 2m R l ¢ " # ¢ " # ( ~ ∇~ − q A~) ( ~ ∇~ − q A~) 3 ~ ~ −i l ∗ 3 ~ ~ −i l ∗ d r∇ · δψl ψl − d r δψl∇ · ψl (2.6) 3 i 2m 3 i 2m R l R l
Keep now in mind that equation 2.6 is valid as well if we change integration region from R3 to the superconducting specimen (denoted by ”super”) since the order parameter ψl is 0 outside the specimen. We want to do this change because δψl is also 0 outside the specimen and we want to utilize the condition that we can choose arbitrary δψl inside the specimen to find useful equations. 2.2. Deriving the Ginzburg-Landau equations 5
Furthermore, recall the generalized divergence theorem [1, p. 93]: ¢ d3r∇~ (...) = dS~(...) (2.7)
We can now rewrite equation 2.6 as:
¢ " ~ ~ ~ 2 # 3 ∗ 2 ∗ ( −i ∇ − qlA) ∗ 0 = d rδψl αlψl + βl|ψl| ψl + ψl + super 2ml " # ( ~ ∇~ − q A~) ~ ~ −i l ∗ δψldS · ψl (2.8) ∂super i 2ml
Which yields a Ginzburg-Landau equation and a boundary condition for each component of the superconductor:
~ ~ ~ 2 ( i ∇ − qlA) 2 ψl + αlψl + βl|ψl| ψl = 0 (2.9) 2ml
dS~ · (~∇~ − q A~)ψ = 0 (2.10) i l l Recall the expression for the canonical momentum in the case of a particle in a magnetic field [2, p. 308]:
~p = m~v + qA~ (2.11)
Also recall following expression for the momentum operator:
~pˆ = ~∇~ (2.12) i
We can use equations 2.11 and 2.12 to express 2.10 as following:
dS~ · ml ~vl = 0 (2.13)
Which means that the normal component of ~vl through the surface of the su- perconducting specimen is 0.
2.2.2 Variation with respect to A~
Next¡ we will utilize equation 2.3 to express ~h in terms of A~ and then minimize 3 ~ 3 d rFs with respect to a variation of A: R 6 Chapter 2. General Considerations
" ¢ # −q ψ δA~ · ( ~ ∇~ − q A~) ∗ ~ ~ ~ ~ X 3 l l −i l ∗ −qlψl δA · ( i ∇ − qlA) 0 = d r ψl + ψl 3 2ml 2ml l R ¢ " # 2(∇~ × A~) · (∇~ × δA~) + d3r (2.14) 3 2µ R 0
Recall now that for any C~ and D~ , the following vector identity holds:
∇~ · (C~ × D~ ) = D~ · (∇~ × C~ ) − C~ · (∇ × D~ ) (2.15)
By choosing C~ := ~h = ∇×~ A~ and D~ := δA~ and using equation 2.15 in 2.14, we µ0 µ0 obtain:
¢ " # X 3 −ql ~ ~ ~ ~ ∗ ∗ ~ ~ ~ 0 = d r δA · ψl( ∇ − qlA)ψl + ψl ( ∇ − qlA)ψl + 3 2ml −i i l R ¢ " # ∇~ × ~h (~h × δA~) d3r δA~ · ( ) − ∇~ · (2.16) 3 µ µ R 0 0
Next, we again use the general divergence theorem shown at equation 2.7, which yields:
¢ " ~ ~ # 3 ~ X −ql ~ ~ ~ ∗ ∗ ~ ~ ~ ∇ × h 0 = d rδA · ψl( ∇ − qlA)ψl + ψl ( ∇ − qlA)ψl + + 3 2ml −i i µ0 R l " # (~h × δA~) dS~ · − (2.17) 3 µ ∂R 0
~ ~ 1 Assuming that h × δA decays faster than r2 as r → ∞, the surface integral vanishes as r → ∞. The remaining integral yields a Ginzburg-Landau equation:
~ ~ " 2 # ∇ × h X ql~ ∗ ~ ~ ∗ ql 2 ~ = (ψl ∇ψl − ψl∇ψl ) − |ψl| A (2.18) µ0 2iml ml l
One should also keep in mind that when the displacement current is negligible and by defining J~ := J~free + J~bound , a Maxwell equation yields [3, p. 330]:
∇~ × ~h J~ = (2.19) µ0 2.4. Supercurrents in the London limit 7
2.3 London limit
As commonly taught in literature regarding superconductivity, the conventional parametrization of the order parameter ψ is ψ = |ψ|eiθ. In the London limit, one models |ψ| to be constant with respect to space co- ordinates throughout the superconducting region and zero in normal regions [13]. The London limit is reasonable when the magnetic penetration length (length scale over which the magnetic field is reduced to a small fraction of its maximum value) is much larger than the coherence lengths (length scales over which the order pa- rameters regain their corresponding bulk values) [27]. We can express the Ginzburg-Landau equations 2.9 and 2.18 as following in the London limit:
~ ~ 2 (~∇θl − qlA) 2 ψl + αlψl + βl|ψl| ψl = 0 (2.20) 2ml
" 2 # X ql~ 2 ql 2 J~ = |ψl| ∇~ θl − |ψl| A~ (2.21) ml ml l
Furthermore we can also simplify the expression for the free energy Fs at equa- tion 2.2 as following in the London limit:
" # ~ 2 X 2 βl 4 1 ~ ~ 2 2 h Fs = Fn + αl|ψl| + |ψl| + (~∇θl − qlA) |ψl| + (2.22) 2 2ml 2µ0 l
The part of Fs that is relevant in the context of vortex formation in the London limit can be described as following:
" # ~ 2 ˜ X 1 ~ ~ 2 2 h Fs := (~∇θl − qlA) |ψl| + (2.23) 2ml 2µ0 l 2.4 Supercurrents in the London limit
A few words of caution: what one means by ”current” is a matter of convention. Griffith [3, p. 212] refers to J~ as ”volume current density”, but we will refer to it as ”current” throughout this text. Recall equation 2.21:
" 2 # X ql~ 2 ql 2 J~ = |ψl| ∇~ θl − |ψl| A~ (2.24) ml ml l Following the procedure for two-component superconductors done at [4], we define supercurrents j~l as following: 8 Chapter 2. General Considerations
2 ql~ 2 ql 2 j~l := |ψl| ∇~ θl − |ψl| A~ (2.25) ml ml Given that definition it is easy to notice that following holds: X J~ = j~l (2.26) l
I.e. the current J~ that appears in Maxwell’s equations is in this case the sum of all supercurrents. The definition makes sense from another perspective as well. Using equations iθ 2.11, 2.12 and ψl = |ψl|e l we obtain:
~ ql ~vl = ∇~ θl − A~ (2.27) ml ml In general a charged current is obtained by multiplying a velocity with a charge 2 and a density. Choosing |ψl| as the density, ~vl as the velocity and ql as the charge one obtains: 2 2 ql~ 2 ql 2 j~l = ql ~vl|ψl| = |ψl| ∇~ θl − |ψl| A~ (2.28) ml ml Which is the same expression as in equation 2.25. Chapter 3
Vortices
3.1 Definitions
[26, p. 202] essentially provides following definitions: Vorticity is defined as the curl of the fluid velocity. A line drawn in R3 such that it is at each point parallel to the vorticity is called a ”vortex-line”. If we draw a closed curve in R3 and at each of its points draw a vortex line we have drawn a tube which is called a ”vortex-tube”. The fluid contained in a vortex-tube is called a ”vortex-filament” and is abbreviated as ”vortex”.
3.2 Vortex flux
Using the typical argument that J~ vanishes at large distances and using equation 2.24, we obtain following result for a contour sufficiently far away:
" # " 2 # ~ X ql~ 2 ~ ~ X ql 2 ~ 0 = J~ · dl = |ψl| ∇~ θl · dl − |ψl| A~ · dl (3.1) ml ml l l Recall the definition of magnetic flux and Stokes’ theorem to obtain: ¤ ¤ Φ := B~ · dS~ = (∇~ × A~) · dS~ = A~ · dl~ (3.2)
Also recall following: " # ∂ ∂ ∂ ∇~ θ~ · dl~ = dx + dy + dz θ = ∆θ = 2πN ,N ∈ (3.3) l ∂x ∂y ∂z l l l l Z
Nl is the so called ”winding number” which arises as a consequence of the single-valuedness of the order parameter ψl.
9 10 Chapter 3. Vortices
Using equations 3.1, 3.2 and 3.3, we obtain a result which matches the one found at [11, equation 3]: P ql~ 2 |ψl| 2πNl Φ = l ml (3.4) q2 P j 2 |ψj| j mj
3.3 Calculating the magnetic vector potential in the London limit
We will proceed similar to how it is done in [5]. From [3, p. 44] we have for cylindrical parametrization (s, ϕ, z) following equation:
1 ∂A ∂A ∂A ∂A 1 ∂(sA ) ∂A ∇~ × A~ = ( z − ϕ )e ˆ + ( s − z )e ˆ + ( ϕ − s )e ˆ (3.5) s ∂ϕ ∂z s ∂z ∂s ϕ s ∂s ∂ϕ z
We will be looking at two cases in particular: one where one applies an external field in thee ˆz direction and one where one maintains a rotation in thee ˆz direction. ~ Therefore we can set h = hzeˆz. One way to achieve it is by following magnetic vector potential choice:
A~ = Aϕ(s, ϕ)e ˆϕ (3.6) Equations 2.3, 3.5 and 3.6 together yield:
1 ∂(sA ) ~h = ϕ eˆ (3.7) s ∂s z From [3, p. 44], we have:
1 ∂(sA ) 1 ∂A ∂A ∇~ · A~ = s + ϕ + z (3.8) s ∂s s ∂ϕ ∂z Using equation 3.6 in 3.8, we then obtain:
1 ∂A ∇~ · A~ = ϕ (3.9) s ∂ϕ
Choosing the gauge ∇~ · A~ = 0 then leads to that Aϕ is independent of ϕ:
A~ = Aϕ(s)e ˆϕ (3.10)
Through equation 3.7 we see that it means that ~h does not depend on ϕ. Using that together with 3.5 where we replace A~ with ~h, we obtain: 3.3. Calculating the magnetic vector potential in the London limit 11
~ ∇~ × h −1 ∂hz −1 ∂ 1 ∂(sAϕ) J~ = = eˆϕ = ( )e ˆϕ (3.11) µ0 µ0 ∂s µ0 ∂s s ∂s One can use following ansatz as hinted by [5]:
N ∇~ θ~ = l eˆ (3.12) l s ϕ
We verify that it yields correct winding number:
N ∇~ θ~ · dl~ = l eˆ · sdϕeˆ = 2πN (3.13) l s ϕ ϕ l
Using equations 2.24, 3.11 and 3.12 we obtain:
" 2 # −1 ∂ 1 ∂(sAϕ) X ql~ 2 Nl ql 2 ( )e ˆϕ = |ψl| − |ψl| Aϕ eˆϕ (3.14) µ0 ∂s s ∂s ml s ml l Recall the modified Bessel differential equation:
d2y dy x2 + x − (x2 + n2)y = 0 (3.15) dx2 dx It is well-known that it has following general solution [7, equation 9.6.1]:
y = C1In(x) + C2Kn(x) (3.16)
A relevant property is [7, equation 9.6.19] which shows that In(x) grows too quickly to be a valid solution for our purposes: ¢ π 1 x cos θ In(x) = e cos(nθ)dθ (3.17) π 0
A difficulty lies in connecting the flux Φ to the constant C2. It is done by Annett [6, p. 63] for following equation:
d2h dh s2 s2 z + s z − ( + 02)h = 0 (3.18) ds2 ds λ2 z That yielded following:
Φ s h (s) = K ( ) (3.19) z 2πλ2 0 λ Analogously, we can put equation 3.14 in a more useful form: 12 Chapter 3. Vortices
2 " 2 # 2 d Aϕ dAϕ X ql 2 2 2 X ql~ 2 s 2 +s − µ0 |ψl| s +1 Aϕ +µ0 |ψl| Nls = 0 (3.20) ds ds ml ml l l That looks similar to the modified Bessel differential equation for n = 1. We will make following ansatz [5, equation 7] where D is a constant and following choice for λ: s D A (s) = C K ( ) + (3.21) ϕ 2 1 λ s 1 λ := (3.22) q 2 P ql 2 µ0 |ψl| l ml Inserting those in equation 3.20 yields:
" # d2(C K ( s )) d(C K ( s )) s2 s s2 2 1 λ + s 2 1 λ − + 12 C K ( )+ ds2 ds λ2 2 1 λ
" 2 # 2 2D −D s 2 D X ql~ 2 s 3 + s 2 − 2 + 1 + µ0 |ψl| Nls = 0 (3.23) s s λ s ml l By realizing that the first row of equation 3.23 satisfies the modified Bessel differential equation and rearranging the 2nd row we obtain:
2 −s D X ql~ 2 2 + µ0 |ψl| Nls = 0 (3.24) λ s ml l Which yields: 2 X ql~ 2 D = λ µ0 |ψl| Nl (3.25) ml l
Inserting the result for D in the equation 3.21 for Aϕ yields:
2 s λ µ0 X ql~ 2 Aϕ(s) = C2K1( ) + |ψl| Nl (3.26) λ s ml l 3.4 Calculating the microscopic field in the London limit
The modified Bessel function of the second kind, Kn(x), has following two proper- ties [7, equation 9.6.28]: 3.5. Determining the constant C2 13
d (xnK (x)) = −xnK (x) (3.27) dx n n−1 d (x−nK (x)) = −x−nK (x) (3.28) dx n n+1 Recall equation 3.7 which is valid for our cases:
1 ∂(sA ) ~h = ϕ eˆ (3.29) s ∂s z
Using equation 3.26 for Aϕ together with 3.29 in 3.27 for n = 1, yields:
C d(sK ( s )) −s C s −C s ~h = 2 1 λ eˆ = 2 K ( )e ˆ = 2 K ( )e ˆ (3.30) s ds z λ s 0 λ z λ 0 λ z
3.5 Determining the constant C2 We will follow a similar approach to the one by de Gennes [8, p. 58-59]. We will first look at how ∇~ × (∇~ × ~h) looks like when we insert our expression 3.30 for ~h and use equation 3.5 to express derivatives in cylindrical coordinates:
∂h C ∂K ( s ) ∇~ × ~h = − z eˆ = 2 0 λ eˆ (3.31) ∂s ϕ λ ∂s ϕ Using equation 3.28 for the case n = 0 in 3.31 yields: C s ∇~ × ~h = − 2 K ( )e ˆ (3.32) λ2 1 λ ϕ Using equation 3.5 again yields:
C2 s 1 ∂(−s 2 K ( )) ∇~ × (∇~ × ~h) = λ 1 λ eˆ (3.33) s ∂s z Using equations 3.27, 3.30 and 3.33 for the case n = 1 then yields:
1 s C s C s ~h ∇~ × (∇~ × ~h) = 2 K ( )e ˆ = 2 K ( )e ˆ = − (3.34) s λ λ2 0 λ z λ3 0 λ z λ2 We recognize that as the well-known London equation. However, in the London model for vortices, there is a small cylindrical core of radius ξ << λ that is a normal region rather than a superconducting one. The symmetry axis of the core is parallel toe ˆz. From Griffiths [3, p. 274] we have:
~ || ~ || ~ habove − hbelow = µ0(K × nˆ) (3.35) It is an equation at the boundary between the core and the rest of the specimen where K~ is the surface current andn ˆ is the normal direction. Furthermore, Griffiths [3, p. 267] provides following equation for the bound surface current K~b: 14 Chapter 3. Vortices
K~b = M~ × nˆ (3.36) Assuming that the magnetization M~ just like ~h only has a z-component, we obtain: K~b = Mzeˆz × −eˆs = −Mzeˆϕ (3.37) Inserting that in equation 3.35 yields: ~ || ~ || ~ habove − hbelow = µ0(Kfree × −eˆs) + µ0(−Mzeˆϕ × −eˆs) (3.38) Assuming that there is no free surface current and that the magnetization is zero at the boundary to the core, we thus obtain that the z-component of ~h and thus ~h itself is continuous at the boundary. Furthermore, since the order parameter ψl is 0 in the normal region, then J~ is ~0 ~ according to equation 2.21. Next to satisfy that result for J~ through ∇~ × h = µ0J~, we will make the ansatz that ~h is constant in the normal region. Then due to the continuity of ~h and equation 3.30 following holds in the normal region of the vortex: −C ~h = 2 K ( )e ˆ (3.39) λ 0 λ z That expression can be simplified by using the low-argument limit of K0 [7, equation 9.6.8]: −C λ ~h = 2 ln( )e ˆ (3.40) λ z ~ Due to the discontinuity of ∇~ × h = µ0J~ at the core boundary, the London Equation 3.34 only holds in the superconducting region. To generalize the London equation to also hold in the normal region one can approximate the core with a point and use:
δ(s) λ2∇~ × (∇~ × ~h) + ~h = Φ eˆ (3.41) 2πs z Outside of the core we can identify that as the regular London Equation. To make sure that it is a proper expression in the context of the normal region we integrate over a surface within the superconducting region but that is at a distance from the core much larger than λ: ¤ ¤ ¤ δ(s) dS~ · λ2∇~ × (µ J~) + dS~ · ~h = dS~ · Φ eˆ (3.42) 0 2πs z Using Stokes’ theorem and using the expression for flux we obtain: ¤ δ(s) dl~ · λ2µ J~ + Φ = sdϕdsΦ (3.43) 0 2πs Just like in 3.1, the circulation of the current disappears at large distances, which then as expected yields: 3.6. Calculating the circulation of velocity in the London limit 15
0 + Φ = Φ (3.44)
Now that we have established that the suggested generalized London Equation is reasonable, we will look at 3.42 again but integrate this time in a circle just outside the core: ¤ ~ 2 ~ ~ dl · λ µ0J~ + dS · h = Φ (3.45) £ Since ξ << λ, the contribution from dS~ ·~h is negligible. Furthermore we can use 3.32 to express J~ explicitly. That yields:
−C ξ ξdϕλ2 2 K ( ) = Φ (3.46) λ2 1 λ
s λ Using that K1( λ ) = s for small arguments [7, equation 9.6.9], that yields: Φ C = − (3.47) 2 2πλ ~ Next using the equation 3.30 for h, equation 3.26 for Aϕ, equation 3.22 for ~ λ, equation 3.4 for Φ, equation 3.32 for ∇~ × h = µ0J~ we obtain following useful expressions: " # Φ Φ s A~ = − K ( ) eˆ (3.48) 2πs 2πλ 1 λ ϕ
Φ s ~h = K ( )e ˆ (3.49) 2πλ2 0 λ z
Φ s µ J~ = ∇~ × ~h = K ( )e ˆ (3.50) 0 2πλ3 1 λ ϕ
3.6 Calculating the circulation of velocity in the London limit
Recall equation 2.27:
~ ql ~vl = ∇~ θl − A~ (3.51) ml ml ~ Using equation 3.48 for A~ and 3.12 for ∇~ θl, we can calculate the circulation of the velocity within the superconducting region: 16 Chapter 3. Vortices
" # ~ 1 ~Nl qlΦ qlΦ s ~vl · dl = sdϕ − + K1( ) = ml s 2πs 2πλ λ " # " # 2π~ qlΦ 1 qlΦ s Nl − + sdϕ K1( ) (3.52) ml 2π~ ml 2πλ λ
For a large contour, the 2nd term vanishes due to the exponential decay of s K1( λ ) [7, equation 9.7.2]. For that case, using equation 3.4 for Φ, we obtain:
" P qn 2 # P qn 2 2π |ψn| Nn 2π |ψn| (Nlqn − Nnql) ~v ·dl~ = ~ N −q n mn = ~ n mn (3.53) l l l q2 q2 ml P j 2 ml P j 2 |ψj| |ψj| j mj j mj
3.6.1 2-component example To test the derived equation 3.53, we will compare to the result for the 2-component case at [9, equation 2]. Therefore, we will choose N1 = 0, N2 = 1, q1 = e and q2 = −e and a large contour, which yields:
h i e 2 −e 2 |ψ |2 |ψ1| (0 − 0) + |ψ2| (0 − e) 2 2π m1 m2 2π ~ ~ ~ m2 ~v1 · dl = 2 2 = 2 2 (3.54) e 2 e 2 |ψ1| |ψ2| m1 |ψ1| + |ψ2| m1 + m1 m2 m1 m2
h i e 2 −e 2 |ψ |2 |ψ1| (e − 0) + |ψ2| (−e + e) 1 2π m1 m2 2π ~ ~ ~ m1 ~v2 · dl = 2 2 = 2 2 (3.55) e 2 e 2 |ψ1| |ψ2| m2 |ψ1| + |ψ2| m2 + m1 m2 m1 m2 That result agrees with [9, equation 2]!
3.6.2 The special case of exponentially decaying supercurrents
Looking at equation 2.28 we see that the supercurrent ~jl is proportional to the velocity ~vl, hence such velocities being exponentially decaying within a supercon- ducting region means that corresponding supercurrents within the same region are exponentially decaying as well. Looking at equation 3.52 and 3.53, we notice that if Nlqn − Nnql = 0 for all choices of n and l, then ~vl is exponentially decaying and hence ~jl is exponentially decaying as well. Evidently such exponential decay means that for a large contour the circulation of ~vl is zero. The condition can be rephrased as: Nl = Nn for all choices of n and l. ql qn 3.7. Free energy of a vortex in the London limit 17
3.7 Free energy of a vortex in the London limit
Using the equation 2.23 for F˜s, 3.48 for A~ and 3.12 for ∇~ θl, we obtain:
2 " ! # ~ 2 ˜ X 1 Nl Φ Φ s 2 h Fs := ~ − ql − K1( ) |ψl| + = 2ml s 2πs 2πλ λ 2µ0 l " 2 2 X 1 Nl Φ qlΦ s ~ − ql + K1( ) + 2ml s 2πs 2πλ λ l # ~ 2 Nl Φ qlΦ s 2 h 2 ~ − ql K1( ) |ψl| + (3.56) s 2πs 2πλ λ 2µ0
Utilizing equation 3.4 for Φ then yields:
" 2 2 P qn 2 2 2 X |ψl| n m |ψn| (qnNl − qlNn) qlΦ s F˜ = ~ n + K ( ) + s 2 q2 1 2ml s P j 2 2πλ λ l |ψj| j mj P qn 2 # 2 2 |ψn| (qnNl − qlNn) q Φ s ~h ~ n mn l K ( ) + (3.57) q2 1 s P j 2 2πλ λ 2µ0 |ψj| j mj
Next, we want to integrate F˜s over all space. However, since the order param- ~ eters ψl is 0 outside of the superconducting region and h decays exponentially for large distances, it is enough that we integrate over the superconducting region and the vortex core, i.e. over the specimen:
¢ ¢ " 2 2 P qn 2 2 X |ψl| n m |ψn| (qnNl − qlNn) d3rF˜ = d3r ~ n s 2 q2 2ml s P j 2 specimen specimen l |ψj| j mj 2 P qn 2 # q Φ s 2 |ψn| (qnNl − qlNn) q Φ s + l K ( ) + ~ n mn l K ( ) + 1 q2 1 2πλ λ s P j 2 2πλ λ j m |ψj| j ¢ ~h2 d3r (3.58) specimen 2µ0
To simplify the expression, we want to use the generalized London Equation 3.41. To do so we got to simplify our expression. Something to also keep in mind is that J~ = ~0 and ψl = 0 holds true inside the core. We will start by using equation s 3.50 for K1( λ ) and equation 3.22 for λ: 18 Chapter 3. Vortices
¢ ¢ " 2 2 P qn 2 2 X |ψl| n m |ψn| (qnNl − qlNn) d3rF˜ = d3r ~ n s 2 q2 2ml s P j 2 specimen specimen l |ψj| j mj 2 P qn 2 # 2 |ψn| (qnNl − qlNn) q Φ s + q λ2∇~ × ~h + ~ n mn l K ( ) + l q2 1 s P j 2 2πλ λ |ψj| j mj ¢ ¢ ~ 2 " 2 2 P qn 2 2# h X |ψl| n m |ψn| (qnNl − qlNn) d3r = d3r ~ n 2 q2 2µ0 2mls P j 2 specimen specimen l |ψj| j mj ¢ " 2 P qn 2 # X |ψl| n m |ψn| (qnNl − qlNn) qlΦ s + d3r ~ n K ( ) + q2 1 specimen mls P j 2 2πλ λ l j m |ψj| ¢ j ¢ λ2 ~h2 d3r (∇~ × ~h)2 + d3r (3.59) specimen 2µ0 specimen 2µ0
For the 3rd integral in equation 3.59, we can use the vector identity described by equation 2.15 for the case C~ := ~h and D~ := ∇~ × ~h as well as Gauss’ divergence theorem: ¢ λ2 d3r (∇~ × ~h)2 = 2µ specimen¢ 0 ¢ λ2 λ2 d3r ∇~ · (~h × (∇~ × ~h)) + d3r ~h · (∇~ × (∇~ × ~h)) = 2µ 2µ ¢specimen 0 ¢ specimen 0 λ2 λ2 dS~ · (~h × (∇~ × ~h)) + d3r ~h · (∇~ × (∇~ × ~h)) (3.60) ∂specimen 2µ0 specimen 2µ0
The surface integral in equation 3.60 disappears since both ~h and ∇~ × ~h decay exponentially at large distances. Combining the remaining integral in 3.60 with the 4th integral in equation 3.59 and using the generalized London Equation 3.41, we obtain:
¢ ¢ λ2 ~h2 d3r (∇~ × ~h)2 + d3r = 2µ 2µ ¢ specimen 0 specimen ¢0 3 1 ~ h 2 ~ ~ i 3 1 ~ δ(s) d r h · λ ∇~ × (∇~ × h) + h = d r h · Φ eˆz (3.61) specimen 2µ0 specimen 2µ0 2πs
Assuming cylindrical specimen with radius R and height ∆z as well as a vortex ~ core radius ξ and then using equation 3.49 for h and the low-argument limit for K0 as shown by equation 3.40, then yields: 3.7. Free energy of a vortex in the London limit 19
¢ ¢ λ2 ~h2 d3r (∇~ × ~h)2 + d3r = 2µ 2µ specimen ¢ 0 specimen 0 2 1 ~ δ(s) Φhz(ξ) Φ λ sdϕdsdz h · Φ eˆz = ∆z = ∆z 2 ln( ) (3.62) 2µ0 2πs 2µ0 4πµ0λ ξ
We recognize that result as being in the form analogous to the vortex energy for the 1-component case as shown at [10, equation 5.85]. Next we will turn our attention to the 2nd integral in equation 3.59, use equation 3.28 for the case n = 0 to express K1 as a derivative:
¢ " 2 P qn 2 # X |ψl| n m |ψn| (qnNl − qlNn) qlΦ s d3r ~ n K ( ) = q2 1 mls P j 2 2πλ λ specimen l |ψj| j mj ¢ " 2 P qn 2 # X |ψl| n m |ψn| (qnNl − qlNn) qlΦ d s sdϕdsdz ~ n − K ( ) q2 0 mls P j 2 2π ds λ l |ψj| j mj " 2 P qn 2 # X |ψl| n m |ψn| (qnNl − qlNn) qlΦ R ξ = 2π∆z ~ n − K ( )−K ( ) q2 0 0 ml P j 2 2π λ λ l |ψj| j mj (3.63)
We can simplify the result by again using the low-argument limit for K0 as shown by equation 3.40 and utilizing that K0 decays exponentially for large arguments [7, equation 9.7.2]:
" 2 P qn 2 # X |ψl| n m |ψn| (qnNl − qlNn) qlΦ R ξ 2π∆z ~ n − (K ( ) − K ( ) = q2 0 0 ml P j 2 2π λ λ l |ψj| j mj " 2 P qn 2 # λ X |ψl| ql n m |ψn| (qnNl − qlNn) ∆zΦ ln( ) n (3.64) ~ q2 ξ ml P j 2 l |ψj| j mj
Finally, we will look at the first integral of equation 3.59: 20 Chapter 3. Vortices
¢ " 2 2 P qn 2 2# X |ψl| n m |ψn| (qnNl − qlNn) d3r ~ n = 2 q2 2mls P j 2 specimen l |ψj| j mj ¢ 2 " 2 P qn 2 2# ds X |ψl| n m |ψn| (qnNl − qlNn) 2π∆z ~ n = q2 2 s ml P j 2 l |ψj| j mj " 2 P qn 2 2# R X |ψl| n m |ψn| (qnNl − qlNn) π∆z 2 ln n (3.65) ~ q2 ξ ml P j 2 l |ψj| j mj
Combining the results from equations 3.65, 3.64 and 3.62 we finally obtain:
¢ " 2 P qn 2 2# R X |ψl| n m |ψn| (qnNl − qlNn) d3rF˜ = π∆z 2 ln n + s ~ q2 ξ ml P j 2 specimen l |ψj| j mj " 2 P qn 2 # 2 λ X |ψl| ql n m |ψn| (qnNl − qlNn) Φ λ ∆zΦ ln( ) n + ∆z ln( ) ~ q2 2 ξ ml P j 2 4πµ0λ ξ l |ψj| j mj (3.66)
Something to keep in mind is that the right-hand side of equation 3.66 is positive due to the squares in the expression 2.23 for F˜s.
3.7.1 2-component example
Like in our previous examples, we will choose N1 = 0, N2 = 1, q1 = e and q2 = −e and a large radius R. Since the first term in equation 3.66 grows the fastest with R, we will investigate it first since if it is non-zero the remaining terms would be small compared for it: 3.7. Free energy of a vortex in the London limit 21
" 2 P qn 2 2# R X |ψl| n m |ψn| (qnNl − qlNn) π∆z 2 ln n = ~ q2 ξ ml P j 2 l |ψj| j mj " 2 P qn 2 2 2 P qn 2 2# R |ψ | |ψn| (0 − eNn) |ψ | |ψn| (qn + eNn) π∆z 2 ln 1 n mn + 2 n mn ~ q2 q2 ξ m1 P j 2 m2 P j 2 |ψj| |ψj| j mj j mj " 2 −e 2 2 2 e 2 −e 2 2# R |ψ | |ψ2| (−e) |ψ | |ψ1| (e) + |ψ2| (−e + e) = π∆z 2 ln 1 m2 + 2 m1 m2 ~ q2 q2 ξ m1 P j 2 m2 P j 2 |ψj| |ψj| j mj j mj 2 2 " |ψ2| |ψ1| # R |ψ |2 |ψ |2 = π∆z 2 ln 1 2 m2 + m1 = ~ |ψ |2 |ψ |2 ξ m1 m2 (P j )2 (P j )2 j mj j mj R |ψ |2 |ψ |2 1 π∆z 2 ln 1 2 (3.67) ~ |ψ |2 |ψ |2 ξ m1 m2 ( 1 + 2 ) m1 m2 That result is non-zero and using the argument above we do not need to calculate other contributions. This result is also the same as in [9]!
3.7.2 The special case of exponentially decaying supercurrents
Just like in chapter 3.6.2, we will look at the special case Nlqn − Nnql = 0 for all choices of l and n at large distances R. Looking at the equation 3.66 for the free energy of a vortex, we notice that the two first terms are zero for this case while the third remains: ¢ 2 3 ˜ Φ λ d rFs = ∆z 2 ln( ) (3.68) specimen 4πµ0λ ξ This scales as O(R0) rather than O(ln(R)) which would have been the case if the first term of equation 3.66 did not vanish. 22 Chapter 4
Response to Applied Field H~
4.1 Minimizing Gibb’s free energy
In this section we will investigate the critical applied field at which a specimen which was previously composed of a purely superconducting region forms a vortex and hence creates a small normal region as a response to increasing an external applied field H~ . An external constant field H~ = Hzeˆz is used. This can for instance be realized by using a solenoid surrounding the superconducting specimen. Furthermore, ~h points in the same direction as H~ . Hence, to the free energy we will add following term to form the relevant Gibb’s free energy:
¢ ¢ 3 ~ 3 ~ − d r H~ · h = −Hz d r eˆz · h = specimen specimen ¢ ~ ~ − Hz dz dS · h = −Hz∆zΦ (4.1) specimen Together with equation 3.66 for the vortex energy we receive following condition for vortex formation:
" 2 P qn 2 2# R X |ψl| n m |ψn| (qnNl − qlNn) π∆z 2 ln n + ~ q2 ξ ml P j 2 l |ψj| j mj " 2 P qn 2 # λ X |ψl| ql n m |ψn| (qnNl − qlNn) ∆zΦ ln( ) n + ~ q2 ξ ml P j 2 l |ψj| j mj Φ2 λ ∆z 2 ln( ) − Hz∆zΦ ≤ 0 (4.2) 4πµ0λ ξ
23 24 Chapter 4. Response to Applied Field H~
That can be re-expressed as a condition for |H~ |:
2 " 2 P qn 2 2# π R X |ψl| n m |ψn| (qnNl − qlNn) |H~ | ≥ ~ ln n + q2 Φ ξ ml P j 2 l |ψj| j mj " 2 P qn 2 # λ X |ψl| ql n m |ψn| (qnNl − qlNn) Φ λ ln( ) n + ln( ) (4.3) ~ q2 2 ξ ml P j 2 4πµ0λ ξ l |ψj| j mj
To find which vortex type appear first as you increase |H~ |, we will notice that for the case of Nlqn − Nnql = 0 for all choices of l and n, i.e. exponentially decaying supercurrents (see chapter 3.6.2), the right-hand side of inequality 4.3 grows as O(R0) rather than O(ln(R)). For that case, following inequality holds:
~ Φ λ |H| ≥ 2 ln( ) (4.4) 4πµ0λ ξ We thus see that the first vortex type to appear is the one that corresponds to the lowest |Φ| under the condition of Nlqn −Nnql = 0 for all choices of l and n. Looking P ql 2 at the equation 3.4 for Φ, we see that we will have to minimize | |ψl| Nl|. We l ml notice that we can express that in terms of an arbitrary winding number Nj:
X ql 2 X ql 2 Njql X ql 2 ql |ψl| Nl = |ψl| = |Nj| |ψl| (4.5) ml ml qj ml qj l l l
Hence we need to minimize |Nj| given the condition Nl ∈ Z for all l and the condition Nlqn − Nnql = 0 for all choices of l and n. First we will make the observation that the first condition ensures that if one winding number is zero then so are all other; therefore, the sought vortex has to have non-zero winding in all its components. We can therefore rewrite the first condition in following more illuminating form:
N q l = l (4.6) Nn qn
Assuming that the charges are always integer multiples of the electron charge qe and by introducing gcd({ ql }) as the greatest common divisor of the set of all charges qe divided by the electron charge, we notice that the minimal |Nj| is given by:
qj qe |Nj| = (4.7) gcd({ ql }) qe
The sign of Φ is found by noticing that inequality 4.2 can only hold if HzΦ is positive, hence Φ has to have the same sign as Hz. Therefore, with the aid of 4.1. Minimizing Gibb’s free energy 25 equation 3.4 for Φ and looking at the middle term in equation 4.5 we see that the sign of Φ is the same sign as following:
2 X ql 2 Njql Nj X ql 2 |ψl| = |ψl| (4.8) ml qj qj ml l l Hence Φ has the same sign as Nj , and since we have shown that Φ has the same qj sign as Hz it means that Nj has the same sign as Hzqj. By also using equation 4.7 as well as observing that qe is negative and that gcd is always positive, we obtain:
qj qe Nj = − sgn(Hz) (4.9) gcd({ ql }) qe Using equation 3.4 for Φ and equation 4.9, we obtain:
2 P ql ~ 2 1 |ψl| 2π 2π Φ = − sgn(H ) l ml = − ~ sgn(H ) (4.10) ql z q2 ql z qe gcd({ }) P j 2 qe gcd({ }) qe |ψj| qe j mj We notice that equation 4.10 shows that the flux Φ is only equal to the flux in the 1-component electron cooper pair case when gcd({ ql }) = 2. qe Next, we can use that equation to re-express inequality 4.4 as: 2π λ |H~ | ≥ ~ ln( ) (4.11) 2 ql 4πµ0λ qe gcd({ }) ξ qe
Hence the critical applied field |H~ crit| for vortex formation is given by: 2π λ |H~ | = ~ ln( ) (4.12) crit 2 ql 4πµ0λ qe gcd({ }) ξ qe
4.1.1 2-component example
As usual we will look at the case q1 = e and q2 = −e where e < 0 and is the charge of an electron cooper pair. We will start by a more time-consuming approach. We e 2 −e 2 have to minimize |ψ1| N1 + |ψ2| N2 under the condition −N1e − N2e = 0. m1 m2 That condition yields N2 = −N1, which means that we will have to minimize 2 2 |ψ1| |ψ2| eN1 + 2 . That leads to that we have to choose between (N1 = 1,N2 = m1 m −1) and (N1 = −1,N2 = 1). The correct choice is obtained from the condition that HzΦ has to be positive. Therefore, for positive Hz, we obtain positive Φ for (N1 = −1,N2 = 1). Which is consistent with the result in [9]. A faster method is to just see that the gcd of q1 and q2 is 2 in this case qe qe qj and therefore by using equation 4.9, we obtain Nj = − sgn(Hz), which yields 2qe (N1 = −sgn(Hz),N2 = sgn(Hz)) just like we obtained through the more tedious method. 26 Chapter 5
Rotational Response
5.1 The London field
We will follow the approach outlined by London in the 1-component case at [13] and by Babaev and Ashcroft in the 2-component case at [9]. Therefore we will con- sider a cylindrical specimen composed of a pure superconducting region rotating at a constant angular velocity Ω~ = Ωzeˆz where the symmetry axis of the speci- men coincides with the direction of Ω.~ The central idea is that the neutralizing background, like for instance ions, is rotating rigidly and that that the sum of the normal current and the total supercurrent is zero in the bulk:
J~n + J~s = ~0 (5.1)
Using definition 2.25 for individual supercurrents ~jl and equation 2.26 for the total supercurrent J~s, we then obtain:
" 2 # X ql~ 2 ql 2 J~s = |ψl| ∇~ θl − |ψl| A~ (5.2) ml ml l " # X 2 J~n = − ql|ψl| sΩzeˆϕ (5.3) l Then by inserting equation 5.2 and 5.3 in 5.1, using the equation 3.5 for curl in cylindrical coordinates and applying the curl to 5.1 as well as utilizing the fact that for no vortices the curl of the gradient of the phases in a simply connected region is ~0, we obtain:
" 2 # " # X ql 2~ X 2 − |ψl| h + − 2ql|ψl| Ωzeˆz = ~0 (5.4) ml l l Which yields following equation for the magnetic field in the bulk:
27 28 Chapter 5. Rotational Response
" # P 2 − 2ql|ψl| Ωzeˆz l P 2 ~ ~ l ql|ψl| h = 2 = −2Ω 2 (5.5) P qn 2 P qn 2 |ψn| |ψn| n mn n mn [13] and [9] argues that the induced magnetic field is created by surface effects.
5.1.1 1-component example
Choosing q1 = e in equation 5.5, where e < 0, we obtain:
e|ψ |2 m ~h = −2Ω~ 1 = −2Ω~ 1 (5.6) e2 2 |ψ1| e m1 Which agrees with the result in [9].
5.1.2 2-component example
Making the usual choices of q1 = e and q2 = −e, yields:
2 2 ~ 2 2 ~ ~ e|ψ1| + (−e)|ψ2| 2Ω (|ψ1| − |ψ2| ) h = −2Ω 2 = − 2 2 (5.7) e2 2 (−e) 2 |ψ1| |ψ2| |ψ1| + |ψ2| e + m1 m2 m1 m2 Which agrees with the result in [9].
5.2 Vortex formation condition
We will be following the procedure described for the 2-component case in [9]. This means that we will look at a rotating cylindrical specimen as in previous section but we will switch frame of reference to the rotating frame. By doing so, an extra term to the free energy is added: −L~ · Ω~ = −LzΩz.
5.2.1 Calculating the z-component of the angular momentum in the London limit
Since we want to calculate −L~ · Ω~ = −LzΩz, it is of interest to calculate Lz, i.e. the z-component of the angular momentum L~ . Recall from basic mechanics courses:
L~ = (seˆs + zeˆz) × m~v (5.8)
Given ~v = vϕϕˆ then yields: 5.2. Vortex formation condition 29
Lz =e ˆz · L~ =e ˆz · (seˆs + zeˆz) × mvϕeˆϕ = smvϕ (5.9)
Generalizing it to be appropriate for our need yields: ¢ 3 X 2 Lz = d r smlvϕl|ψl| (5.10) l
~ Using equation 3.51 for ~vl, equation 3.48 for A~ and 3.12 for ∇~ θl, we obtain: ¢ " # X qlΦ sqlΦ s L = sdϕdsdz N − + K ( ) |ψ |2 (5.11) z ~ l 2π 2πλ 1 λ l l Assuming cylindrical superconducting specimen of radius R and height ∆z we obtain:
¢ " # 2 X qlΦ X s qlΦ s L = ∆zπR2 N − |ψ |2 + ∆z2π ds K ( )|ψ |2 (5.12) z ~ l 2π l 2πλ 1 λ l l l
Using equation 3.27 for the case n = 2 yields:
d (x2K (x)) = −x2K (x) (5.13) dx 2 1 That allows us to rewrite the 2nd term in equation 5.12 as: ¢ ¢ 2 2 2 X s qlΦ s X −λ qlΦ d s s ∆z2π ds K ( )|ψ |2 = ∆z2π ds K ( ) |ψ |2 2πλ 1 λ l 2π ds λ2 2 λ l l l (5.14) s 2λ2 Next, using that K2( λ ) = s2 holds for small arguments [7, 9.6.9] in equation 5.14 we obtain:
¢ 2 2 2 X −λ qlΦ d s s X R R ∆z2π ds K ( ) |ψ |2 = ∆z −λ2q Φ K ( ) − 2 |ψ |2 2π ds λ2 2 λ l l λ2 2 λ l l l (5.15) s For large R, K2( λ ) decays exponentially [7, 9.7.2] and therefore what remains from equation 5.15 is:
X 2 2 ∆z 2λ qlΦ|ψl| (5.16) l Using the equation 3.4 for Φ we can rewrite the first term in 5.12 as: 30 Chapter 5. Rotational Response
" # " P qn 2 # X qlΦ X n m |ψn| Nn ∆zπR2 N − |ψ |2 = ∆zπR2 N − q n |ψ |2 = ~ l l ~ l ~ l q2 l 2π P j 2 l l |ψj| j mj "P qn 2 # X n m |ψn| (qnNl − qlNn) ∆zπR2 n |ψ |2 (5.17) ~ q2 l P j 2 l |ψj| j mj
To summarize, for large R, we can rewrite equation 5.12 through expression 5.16 and equation 5.17 as:
"P qn 2 # X n m |ψn| (qnNl − qlNn) X L = ∆zπR2 n |ψ |2 + ∆z2λ2Φ q |ψ |2 (5.18) z ~ q2 l l l P j 2 l |ψj| l j mj
2-component example Like in the circulation section of this thesis, we also want to test the derived equation 5.18; therefore, we will compare to the result for the 2-component case at [9]. For that purpose, we will choose N1 = 0, N2 = 1, q1 = e and q2 = −e and a large radius R:
" e 2 2 −e 2 2 # |ψ1| (0 − 0)|ψ1| + |ψ2| (0 − e)|ψ1| 2 m1 m2 Lz = ∆zπR + ~ e2 2 e2 2 |ψ1| + |ψ2| m1 m2 " e 2 2 −e 2 2 # |ψ1| (e − 0)|ψ2| + |ψ2| (−e + e)|ψ2| 2 m1 m2 X 2 2 ∆zπR + ∆z 2λ qlΦ|ψl| = ~ e2 2 e2 2 |ψ1| + |ψ2| m1 m2 l 2 2 " |ψ2| 2 |ψ1| 2 # |ψ1| + |ψ2| 2 m2 m1 X 2 2 ∆zπR ~ 2 2 + ∆z 2λ qlΦ|ψl| = |ψ1| + |ψ2| m1 m2 l 2 2 " # |ψ2| |ψ1| m1 + m2 X ∆zπR2 + ∆z 2λ2q Φ|ψ |2 (5.19) ~ |ψ |2 |ψ |2 l l m2 m1 1 + 2 m1 m2 l
Since R is large, the second term in equation 5.19 is negligible compared to the first term. That yields: " # |ψ |2 |ψ |2 m + m L = ∆zπR2 2 1 1 2 (5.20) z ~ |ψ |2 |ψ |2 m2 m1 1 + 2 m1 m2 Which luckily is the same result as in [9]! 5.2. Vortex formation condition 31
The special case of exponentially decaying supercurrents
As in chapter 3.6.2, we will look at the special case Nlqn − Nnql = 0 for all choices of l and n at large distances R. Looking at the equation 5.18 for Lz yields:
2 X 2 Lz = ∆z2λ Φ ql|ψl| (5.21) l
I.e. the first term in equation 5.18 vanishes and therefore the scaling of Lzis O(R0) rather than O(R2).
5.2.2 Finding the critical angular frequency
Recall that due to switching to the rotating frame we have to add following term to the free energy:
−Ω~ · L~ = −ΩzLz (5.22)
With Lz from equation 5.18 and vortex energy from equation 3.66 we obtain the condition for vortex formation:
" 2 P qn 2 2# R X |ψl| n m |ψn| (qnNl − qlNn) 0 ≥ π∆z 2 ln n + ~ q2 ξ ml P j 2 l |ψj| j mj " 2 P qn 2 # λ X |ψl| ql n m |ψn| (qnNl − qlNn) ∆zΦ ln( ) n + ~ q2 ξ ml P j 2 l |ψj| j mj Φ2 λ ∆z 2 ln( )− 4πµ0λ ξ " "P qn 2 # # X n m |ψn| (qnNl − qlNn) X Ω ∆zπR2 n |ψ |2 + ∆z2λ2Φ q |ψ |2 (5.23) z ~ q2 l l l P j 2 l |ψj| l j mj
We notice that the inequality 5.23 can only hold if ΩzLz is positive. Hence by Lz Lz using ΩzLz = Ωzsgn(Ωz) and that both Ωzsgn(Ωz) and are positive, sgn(Ωz ) sgn(Ωz ) 32 Chapter 5. Rotational Response we can rewrite the inequality as following:
" 2# 2 P qn |ψ |2(q N −q N ) 2 R P |ψl| n mn n n l l n sgn(Ωz)π~ ln ξ l m q2 l P j 2 |ψj | j mj Ωzsgn(Ωz) ≥ " # + P qn |ψ |2(q N −q N ) 2 P n mn n n l l n 2 2 P 2 πR ~ l q2 |ψl| + 2λ Φ l ql|ψl| P j 2 |ψj | j mj " # 2 P qn |ψ |2(q N −q N ) λ P |ψl| ql n mn n n l l n sgn(Ωz)Φ ln( ξ )~ l m q2 l P j 2 |ψj | j mj " # + P qn |ψ |2(q N −q N ) 2 P n mn n n l l n 2 2 P 2 πR ~ l q2 |ψl| + 2λ Φ l ql|ψl| P j 2 |ψj | j mj Φ2 λ sgn(Ωz) 2 ln( ) 4πµ0λ ξ " # (5.24) P qn |ψ |2(q N −q N ) 2 P n mn n n l l n 2 2 P 2 πR ~ l q2 |ψl| + 2λ Φ l ql|ψl| P j 2 |ψj | j mj
We notice that for the case of Nlqn − Nnql = 0 for all choices of l and n, the 0 ln(R) right-hand side of inequality 5.24 grows as O(R ) rather than O( R2 ). Hence the case where vortex formation happens first is the one where ”Nlqn − Nnql = 0 for all choices of l and n” is not true. For such case, inequality 5.24 can be simplified:
" 2# 2 P qn |ψ |2(q N −q N ) 2 R P |ψl| n mn n n l l n sgn(Ωz)π~ ln ξ l m q2 l P j 2 |ψj | j mj Ωzsgn(Ωz) ≥ " # = P qn |ψ |2(q N −q N ) 2 P n mn n n l l n 2 πR ~ l q2 |ψl| P j 2 |ψj | j mj 2 2 R P |ψl| P qn 2 sgn(Ωz) ln |ψn| (qnNl − qlNn) ~ ξ l ml n mn (5.25) q2 2 P P qn 2 2 P j 2 R |ψn| (qnNl − qlNn)|ψl| |ψj| l n mn j mj
We notice that the right-hand side of inequality 5.25 increases by factor D when we increase each winding number by factor D. Hence in principle we just need to test all winding number combinations for which its non-zero members have ”1” as their least common multiple. Equality rather than inequality yields the critical angular frequency when the right-hand side is minimized.
2-component example
Keep in mind the condition that q1N2 − q2N1 6= 0. 5.2. Vortex formation condition 33
Ωzsgn(Ωz) ≥ " 2 2# 2 2 R |ψ1| q2 2 |ψ2| q1 2 sgn(Ωz) ln |ψ2| (q2N1 − q1N2) + |ψ1| (q1N2 − q2N1) ~ ξ m1 m2 m2 m1 " # q2 2 q1 2 2 q2 2 2 P j 2 R |ψ1| (q1N2 − q2N1)|ψ2| + |ψ2| (q2N1 − q1N2)|ψ1| |ψj| m1 m2 j mj " # 2 2 2 2 2 2 R |ψ1| |ψ2| 2 q2 |ψ2| q1 |ψ1| sgn(Ωz) ln (q1N2 − q2N1) ( + ) ~ ξ m1m2 m2 m1 = " # 2 2 q2 2 |ψ1| |ψ2| P j 2 R (q1N2 − q2N1)(q1m2 − q2m1) |ψj| m1m2 j mj " # 2 2 2 2 R q2 |ψ2| q1 |ψ1| sgn(Ωz) ln (q1N2 − q2N1)( + ) ~ ξ m2 m1 = " # q2 2 P j 2 R q1m2 − q2m1 |ψj| j mj
R sgn(Ωz)~ ln ξ (q1N2 − q2N1) = " # (5.26) 2 R q1m2 − q2m1
Using the usual case of q1 = e and q2 = −e yields following two conditions :
R R sgn(Ωz)~ ln ξ (eN2 + eN1) sgn(Ωz)~ ln ξ (N2 + N1) Ωzsgn(Ωz) ≥ " # = " # (5.27) 2 2 R e m2 + e m1 R m2 + m1
N2 6= −N1 (5.28)
Recall that if we assume that Ωz is positive then Lz has to be positive as well. By also using the inequality 5.27, we see that the right-hand side of the inequality is minimized both (N1 = 1,N2 = 0) and (N1 = 0,N2 = 1) vortices. We consequently obtain following expression for the critical angular frequency:
R ~ ln Ω = ξ (5.29) z,crit 2 R m2 + m1 It agrees with the result in [9]. 34 Chapter 6
Summary and Conclusions
Due to the existence of numerous condensed matter and astrophysical systems that contain multicomponent order parameters, studying multicomponent models of superconductivity is warranted. In this thesis, the focus has been on a particular such model, namely a n-component generalization of the London model in which the order parameter |ψ| is constant with respect to space coordinates throughout the superconducting region and zero in normal regions. More specifically, a generalization in which it is assumed that there exists no term in the free energy density that is a mixture of different components, was analyzed. For this reason, emphasis was put on investigating the condition for creation of the topological defect known as ”vortex”. If the aforementioned omitted terms were allowed, it would also have been of relevance to consider other forms of topological defects, such as skyrmions [28] [29]. In particular, the ambition of this thesis has been to analytically generalize multiple of the 2-component results that were derived in an antecedent article [9]. Most conspicuously, how a superconductor responds to an application of an external magnetic field and rotation respectively. This thesis was initiated by describing a n-component generalized version of the conventional one-component free energy density and deriving corresponding Ginzburg-Landau equations. Subsequently, the London limit was explained and the notion of supercurrents was introduced. Continuing, the focus was shifted to vortices. At first, vortex as a concept was defined and afterwards expressions for utile properties such as magnetic flux, magnetic vector potential, microscopic field, circulation of velocity and the free energy of a vortex were derived. Remarkably, the special case of exponentially decaying supercurrents turned out to be significantly different from other cases. To test the derived n-component expressions, the result in the two-component version was explicitly calculated and compared with known results [9]; they were found to be congruent.
35 36 Chapter 6. Summary and Conclusions
Furthermore, the response to an applied external field was investigated in the sense of finding the value for the field at which creation of a vortex is energetically favorable. By using the mathematical concept of greatest common divisors, one could simplify the derived expressions substantially. The results were compared to those pertaining to an anterior article about the two-component correspondent [9] and were found to be consistent with each other. Finally, the response of a cylindrical specimen to rotation was examined in the leading order with respect to its radius R. Prior to vortice formation, the n-component analogue of the London magnetic field was derived. Subsequently, the expression for the critical angular frequency, at which vortices begin to be formed due to rotation, was derived. The validity of the results was appraised by comparison to corresponding results in a precursive article regarding the two- component equivalent [9]; they were found to be in accord. Bibliography
[1] Anders Ramgard, Vektoranalys, 3:e upplagan, Teknisk H¨ogskolelitteratur i Stockholm AB (THS AB), 2005.
[2] Leslie E. Ballentine, Quantum Mechanics - A Modern Development, 2nd edi- tion, World Scientific Publishing Co. Pte. Ltd., 2015.
[3] David J. Griffiths, Introduction to Electrodynamics, 3rd edition, Prentice-Hall International, Inc., 1999.
[4] Egor Babaev, Phase diagram of planar U(1) × U(1) superconductor, Conden- sation of vortices with fractional flux and a superfluid state, Nucl. Phys. B686, 397, 2004.
[5] Egor Babaev, Vortices with fractional flux in two-gap superconductors and in extended Faddeev Model, Phys. Rev. Lett. 89, 067001, 2002.
[6] James F. Annett, Superconductivity, Superfluids and Condensates, Oxford Uni- versity Press, 2004.
[7] Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, 10th printing, National Bu- reau of Standards Applied Mathematics Series 55, 1972.
[8] P. G. de Gennes, Superconductivity of Metals and Alloys, Westview Press, 1999.
[9] Egor Babaev and N. W. Ashcroft, Violation of the London Law and Onsager- Feynman quantization in multicomponent superconductors, Nature Physics 3, 530 - 533, 2007.
[10] Boris Svistunov, Egor Babaev and Nikolay Prokof’ev, Superfuild States of Mat- ter, CRC Press, Taylor & Francis Group, 2015
[11] J. Smiseth, E .Smørgrav, E. Babaev and A. Sudbø, Field- and temperature in- duced topological phase transitions in the three-dimensional N-component Lon- don superconductor, Phys. Rev. B71, 214509, 2005.
37 38 Bibliography
[12] L. Onsager, Statistical Hydrodynamics, Nuovo Cimento Series 9 Supplemento 6, 279, 1949. [13] Fritz London, Superfluids, Wiley, 1950. [14] R. P. Feynman, Application of Quantum Mechanics to Liquid Helium, Progress in Low Temperature Physics Volume 1, North-Holland, 1955. [15] N. W. Ashcroft, Hydrogen at High Density, J. Phys. A: Math. Gen. 36, 6137, 2003. [16] P. B. Jones, Type I and Two-Gap Superconductivity in Neutron Star Mag- netism, Mon. Not. Roy. Astron. Soc. 371, 1327-1333, 2006. [17] Ranga P. Dias and Isaac F. Silvera, Observation of the Wigner-Huntington Transition to Metallic Hydrogen, Science 355, issue 6326, 715-718, 2017. [18] Alexander F. Goncharov and Viktor V. Struzhkin, Comment on Observation of the Wigner-Huntington transition to metallic hydrogen, arXiv:1702.04246, 2017. [19] Ranga P. Dias and Isaac F. Silvera, Response to Critiques on Observation of the Wigner-Huntington transition to metallic hydrogen, arXiv:1703.03064, 2017. [20] M. I. Eremets, A. P. Drozdov, P. P. Kong and H. Wang, Molecular semimetallic hydrogen, arXiv:1708.05217, 2017. [21] W. J. Nellis, Metastable Ultracondensed Solid Hydrogenous Materials, arXiv:1709.04094, 2017. [22] Chang-sheng Zha, Hanyu Liu, John S. Tse and Russell J. Hemley, Melting and High P-T Transitions of Hydrogen up to 300 GPa, Phys. Rev. Lett 119, 075302, 2017. [23] Kostas Glampedakis, Nils Andersson and Lars Samuelsson, Magnetohydrody- namics of Superfluid and Superconducting Neutron Star Cores, Monthly No- tices of the Royal Astronomical Society Volume 410, Issue 2, 805-829, 2011. [24] Egor Babaev, Unconventional rotational responses of hadronic superfluids in a neutron star caused by strong entrainment and a Σ− hyperon gap, Phys. Rev. Lett. 103, 231101, 2009. [25] V. L. Ginzburg and L. D. Landau, On the Theory of Superconductivity Zh. Eksp. Teor. Fiz. 20, 1064, 1950; English version is available at Collected Papers of L. D. Landau, p. 546, Pergamon Press, 1965. [26] Sir Horace Lamb, Hydrodynamics, 6th edition, Cambridge University Press, 1932. Bibliography 39
[27] Egor Babaev, Juha J¨aykk¨aand Martin Speight, Magnetic field delocalization and flux inversion in fractional vortices in two-component superconductors, Phys. Rev. Lett. 103, 237002, 2009. [28] Julien Garaud and Egor Babaev, Properties of skyrmions and multi-quanta vortices in chiral p-wave superconductors, Sci. Rep. 5, 17540, 2015. [29] Julien Garaud and Egor Babaev, Skyrmionic state and stable half-quantum vortices in chiral p-wave superconductors, Phys. Rev. B86, 060514(R), 2012.