Ecole Normale Supérieure de Lyon stage 2007-2008

Université Claude Bernard, Lyon 1 SAVARY Lucile

Formation � Sciences de la Matière � parcours physique

3eme année : M2

STRUCTURE OF FRACTIONAL VORTICES ON TWIN BOUNDARIES INNON-CENTROSYMMETRIC SUPERCONDUCTORS

Abstract: Non-centrosymmetric superconductors were discovered experimentally four year ago by Bauer et al.. The

special symmetry properties they carry, in particular the necessary mixing of s- and p-wave superconductivity, make

them particularly interesting to study. Here, I report on a purely symmetry-based phenomenological study of bound-

aries between regions of non-centrosymmetric superconductors with opposite spin-orbit coupling �twin boundaries,

and in particular on the fractional vortices that were shown to possibly appear by Iniotakis et al.. In two di�erent

coupling limits I recover the fractional �ux, whose appearance relies on the existence of two order parameters and

that of time-reversal symmetry breaking induced by the geometry. In doing so, I show that helical terms renormalize

coe�cients of the weak-coupling theory and that the renormalization depends on the geometry of the twins at the

boundary �an observation which has not yet been reported. Some speci�c results described here include the structure

of the magnetic �eld and currents about the vortices, the lower critical magnetic �eld Hc1 and how it changes with

di�erent geometries.

Keywords: condensed matter physics � non-centrosymmetric superconductors � twin boundaries � fractional vortices

� Ginzburg-Landau theory � double sine-Gordon equation � Josephson junction.

Maître de stage :

Manfred Sigrist

sigrist@itp.phys.ethz.ch

Eidgenössische Technische Hochschule Zürich � Institut für Theoretische Physik

ETH Hönggerberg � 8093 Zürich � Suisse

http://www.itp.phys.ethz.ch

7 avril 2008 � 25 juillet 2008

Contents i

Contents

1 Homogeneous sample considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 The tunneling regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 E�ect of the helical terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Zero- and single-vortex solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Outlook into multi-vortex solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Approximate interaction energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Thermodynamic phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Multi-vortex numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.4 Magnetization below the upper critical �eld Hc2 . . . . . . . . . . . . . . . . . 142.3.5 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Strong-coupling regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 Bare twin boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Presence of a vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A Derivation of the free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

List of Tables

1 Groups and representations under which the terms involved in the free energy transform. 192 Systematic approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Shape of the possible terms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

List of Figures

1 Unit cell of CePt3Si. Cerium is represented as a yellow sphere. The blue spheres repre-

sent Platinum atoms, and light and dark blue is used to distinguish between symmetry-

unequivalent locations. The only pink sphere is that of Silicium. . . . . . . . . . . . . 32 Contour used for determining the �ux-phase relation. . . . . . . . . . . . . . . . . . . 73 The di�erent types of boundary con�gurations. . . . . . . . . . . . . . . . . . . . . . . 94 Shape of the potential V (ψ) = − cosψ + ρ−2 cos 2ψ for di�erent values of ρ. . . . . . 105 Kink size Λ as a function of ρ = Λ2/Λ1. . . . . . . . . . . . . . . . . . . . . . . . . . 116 Kink, magnetization an current for values below and above ρc and for di�erent kink types. 117 Approximation to a kink solution of the double sine-Gordon equation. . . . . . . . . . 128 Linearized interaction between two vortices. . . . . . . . . . . . . . . . . . . . . . . . . 139 Schematic diagrams relevant for phase transitions between junctions containing di�erent

number of vortices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410 Temperature phase diagram between fractional and integer vortices. . . . . . . . . . . 1611 Plot of evolution of the phase di�erence ∆φ = φs− φp across the boundary in the linear

approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

List of Figures 1

Introduction

Superconductivity is often referred to as an electronic phase macroscopically characterized by dissi-pationless current �ow. It is indeed one of the most celebrated features of superconductors, togetherwith the screening of external magnetic �elds �the Meissner e�ect. Microscopically, those e�ects aredue to attractive interactions between electrons close to the Fermi surface. Pairs of electrons can formbound states that can Bose-condense into the state with the lowest possible energy. Mathematicallysuperconductivity encompasses the gauge symmetry breaking con�gurations of the electrons in thematerial [1-5].

Beyond those universal characterizations of superconductors, the possible phases within this classare many and depend on the favorable pairing of electrons governed by the interactions at play. Severaldiverse microscopic origins for the latter have already been exhibited, from the phonon-mediatedattraction appropriate for Bardeen-Cooper-Schrie�er superconductors [6, 7, 8, 9] to the interactioninduced by spin �uctuations in some heavy-fermion superconductors [9, 11]; but theoretical models arestill missing for many materials and �nding the theory behind the so-called �high-Tc� superconductors isone of the great challenges in condensed matter physics. Nevertheless, many of the properties commonto superconductors can be accounted for by solely taking into account the existence of an attractiveinteraction between electrons. Consequently, even when the microscopic details of the interactionsare not known, theoretical exploration of superconducting states and e�ects may still be carried out.By additionally imposing that electrons that pair are close to the Fermi surface and have oppositemomentum [12], one can make further predictions, speci�c to certain classes of materials. In particular,crystal-symmetry considerations, combined with identical-fermion odd-exchange properties and thefact that a Cooper pair is a spin 0 or 1 system, can help rule out a priori conceivable pairing states[12, 15]. In view of this, superconductors are often classi�ed according to the symmetries of theorbital part of their Cooper pairs. For example, within this framework, BCS superconductors areof �s-wave� type, the high-Tc cuprates are �d-wave�, and it is now strongly believed that StrontiumRuthenate superconductors are �p-wave� [9]. Given such input, a possible approach is to set up aphenomenological theory. Ginzburg-Landau theory stands at the apex of systematic implementationof symmetry considerations for the superconducting phase transition. Symmetry considerations indeedenter yet at another level, as sums of terms invariant under the symmetries of the Hamiltonian shouldform the Ginzburg-Landau free energy functional whose minimization gives the ground state of thesystem. Results predicted by Ginzburg-Landau theory are accurate close to the superconducting phasetransition.

With many issues still left unanswered and a vast range of existing and potential applications,superconductivity is still a very active �eld of research �fty years after the development of BCS theory.In this context, a new class of superconductors, namely that of non-centrosymmetric superconductors,was discovered four years ago by Bauer et al. in the non-centrosymmetric CePt3Si [21]. Jumps in thethermodynamic and NMR quantities at Tc ≈ 0.75K as temperature was varied, combined with zeroresistivity, constituted uncontroversial evidence for superconductivity [21, 20, 22]. Superconductivityin such materials is of particular interest as it had been commonly thought that p-wave superconduc-tivity in such materials was impossible on the basis of symmetry considerations [15], and the �eld ofnon-centrosymmetric superconductors was not given much attention. In fact, non-centrosymmetric su-perconductivity can be theoretically described by taking into account non-centrosymmetricity throughspin-orbit coupling induced by the absence of inversion symmetry. Due to this coupling, pair spin-channels are mixed, and it was shown that superconductivity in CePt3Si could be accounted for by aspeci�c mixture of s- and p-wave states [17, 16, 23]. Therefore, in addition to the properties relatedto the speci�cities of the absence of inversion symmetry, superconductivity in non-centrosymmetric

2 List of Figures

materials may display all the unusual features related to multi-component superconductivity [12].In particular, time-reversal symmetry-breaking states are conceivable, and thus fractional vortices �vortices that carry only a fraction of the superconducting �ux quantum Φ0 = hc

2e� as well [25]. Thelatter may act as fences for �ux creep [27]. An unexpectedly low �ux creep can constitute a strongindication for fractional vortices. But how does non-centrosymmetricity make things di�erent?

Actually, in non-centrosymmetric superconductors, time-reversal symmetry breaking does not ap-pear spontaneously [24] as it may in other two-component superconductors [25]. However, boundarycondition constraints that exist for example in the presence of experimentally-inevitable domain wallsprovide necessary conditions for the appearance of time-reversal symmetry breaking [24, 26, 12]: thecoexistence (and coupling) of di�erent degenerate electronic states. In this context, domain walls aremost often referred to as �twin boundaries�; the samples contain two regions with their crystal symme-tries lowered in opposite ways compared to the more symmetric situation. The geometries that we willconsider are the following. In CePt3Si �symmetry group C4v [33, 32], inversion symmetry is brokenalong one axis only, denoted as the �z axis� in the rest of this report. The geometry-induced spin-orbitcoupling thus possesses a privileged axis and orientation which are thus bound to the crystal. We willconsider only situations where the z axis is either parallel or perpendicular to the boundary wall onboth sides and where the orientations are opposite to one another. I insist �as is shown in this report�that degenerate homogeneous states do not exist for a single crystal of CePt3Si. Only the combinationof two samples with opposite orientation (but identical direction) with all other parameters unchanged

reproduces a situation where degenerate states would be enforced to be present and spatially adja-cent. This possibility is enabled by the crystal symmetry of CePt3Si and that of the speci�c boundarycon�guration. Twin boundaries between non-centrosymmetric superconductors thus appear to carryintricate symmetry properties which are interesting to study, both for their experimental relevance andfor fundamental theoretical insight, in particular regarding the di�erences between di�erent con�gura-tions of the boundary. Work on such systems has already been carried out by Iniotakis et al. using amicroscopic theory. In their paper [24], they con�rm the existence of fractional vortices and predict thestable phase di�erence for two di�erent boundary con�gurations. Here, in quest of information whichmight allow for understanding of experimental data, we further investigate such twin boundaries in aGinzburg-Landau phenomenological approach.

We show �as was mentioned above� that there are no discretely degenerate states in bulk samples,and that, depending on the direction of spin-orbit coupling the energetically favorable phase di�erencebetween the s- and p-component order parameter states will be either 0 or π. Thus, if the moduliof the order parameter components are kept constant across the boundary, time-reversal symmetry isforced to occur. We then show that insight into time-reversal symmetry breaking can be gained fromtwo di�erent regimes of twin coupling �weak or strong� and that con�gurations with their spin-orbitdirection pointing inwards or outwards are subtly di�erent �an observation that had not yet beenreported. This di�erence is observable in particular in the lower critical �eld Hc1 at which the �rstvortex can penetrate in the junction if a helical term is introduced in the free energy functional. Inthe weak-coupling regime where the twins are coupled via tunneling of pairs across the boundary, weshow that vortices emerge as solutions of a double sine-Gordon equation that e�ectively describes thejunction and which can have fractional kink-solutions in a certain parameter regime. As those kinksolutions contain magnetix �ux, they are interpreted as corresponding to vortices. We draw the phasediagram for this model and discuss a possible technique to study the transition between the integer andfractional �ux regimes. We also give qualitative predictions for multi-kink solutions, which, throughthe implementation of a numerical simulation of the double sine-Gordon equation, could lead to a phasediagram in (H,L) space where H is the applied magnetic �eld and L is the length of the junction. Inthe strong-coupling regime, we show that the value of the fractional �ux is intimately related to the

3

Fig. 1: Unit cell of CePt3Si. Cerium is represented as a yellow sphere. The blue spheres represent

Platinum atoms, and light and dark blue is used to distinguish between symmetry-unequivalent

locations. The only pink sphere is that of Silicium.

existence of two distinct order parameters and in particular to that of a gauge-invariant phase as adegree of freedom.

The rest of this report goes as follows. First, I give the most general Ginzburg-Landau energyfunctional for the given symmetries of the components of the order parameter and show the bulk-feature ingredients that possibly lead to time-reversal symmetry breaking. Then, I discuss the limitwhere pairs can tunnel between the twins, before �nally moving on to the regime where the twins arestrongly coupled. An appendix describes a procedure to �nd all the possible terms in the free energy.

1 Homogeneous sample considerations

Here we describe the ingredients leading to time-reversal symmetry breaking states across the boundarywhich will be discussed in the following sections.

The gap function for the appropriate superconducting state in non-centrosymmetric CePt3Si hastwo components with even and odd transformation rules under inversion symmetry (but which belongto the same representation apart from this). Thus [12], we implement Ginzburg-Landau theory usingtwo order parameters ηs and ηp with the transformation rules of respectively the Γ+

1 and the Γ−2representations of D4h ⊂ C4v, which contains inversion symmetry [32]. The C4v symmetry group,relevant for CePt3Si (see �gure 1) is reintroduced through coupling of the components of the orderparameter via the crystal parameter qz which belongs to the odd representation Γ−2 of D4h. Imposingto the free energy functional time-reversal symmetry breaking, gauge symmetry, and invariance underC4v group operations, we get the general Ginzburg-Landau free energy density, with the gauge invariantderivative D = ∇− i 2π

Φ0A (A is the vector potential):

f [r 7→ ηs (r) , r 7→ ηp (r) , r 7→ A (r)] = f0 + fmagn + fhom + fgrad + fhelical (1)

4 1 Homogeneous sample considerations

where

fmagn =B2

8π− B ·H

4π(2)

fhom =as|ηs|2 + ap|ηp|2 + bs|ηs|4 + bp|ηp|4 + c|ηs|2|ηp|2

+dqz(η∗sηp + ηsη

∗p

)+ e

(η∗sη∗sηpηp + ηsηsη

∗pη∗p

) (3)

fgrad =γs,‖

(|Dxηs|2 + |Dyηs|2

)+ γs,z|Dzηs|2 + γp,‖

(|Dxηp|2 + |Dyηp|2

)+ γp,z|Dzηp|2

+δzqz ((Dzηs)∗Dzηp +Dzηs(Dzηp)∗) + δ‖qz((D‖ηs)

∗ ·D‖ηp + D‖ηs · (D‖ηp)∗) (4)

fhelical =κsqz(z×B) ·

((D‖ηs)

∗ηs − (D‖ηs)η∗s

)+ κpqz(z×B) ·

((D‖ηp)

∗ηp − (D‖ηp)η∗p

)+ι1qz(z×B) ·

((D‖ηs)

∗Dzηp −D‖ηs(Dzηp)∗)

+ ι2qz(z×B) ·((D‖ηp)

∗Dzηs −D‖ηp(Dzηs)∗)

(5)

where the coe�cients as, ap, bs, bp, c, d, e, γs,‖, γs,z, γp,‖, γp,z, δz, δ‖, κs, κp, ι1, ι2 are material speci�c, pos-sibly temperature dependent phenomenological parameters that have to be determined experimentally,B is the local magnetic �eld and H is the external magnetic �eld. In the rest of the report, we takeδz,‖ = 0, ι1,2 = 0. We also assume that the moduli of the components of the order parameter are con-

stant, and by parameterizing the order parameters in terms of their moduli and phases ηs,p = |ηs,p|eiφs,p ,we get the considerably simpli�ed expressions:

f = f ′0 + f ′magn + f∆φ + f ′grad + fA−grad + f ′helical (6)

with

f ′0 = f0 + as|ηs|2 + ap|ηp|2 + bs|ηs|4 + bp|ηp|4 + c|ηs|2|ηp|2 (7)

f ′magn =B2

8π− B ·H

4π+

4π2

Φ20

((γzs + γzp

)A2z +

(γ‖s + γ‖p

) (A2x +A2

y

))(8)

f∆φ = qzα cos(∆φ) + β cos(2∆φ) (9)

f ′grad = γzs (∂zφs)2 + γzp (∂zφp)

2 + γ‖s

((∂xφs)

2 + (∂yφs)2)

+ γ‖p

((∂xφp)

2 + (∂yφp)2)

(10)

fA−grad =−2

2πΦ0γ‖s (Ax∂xφs +Ay∂yφs)− 2

2πΦ0γ‖p (Ax∂xφp +Ay∂yφp)

−22πΦ0γzsAz∂zφs − 2

2e~cγzpAz∂zφp

(11)

f ′helical = qz (B× z) · jeff‖ with jeff‖ = κ′s∇‖φs + κ′p∇‖φp −2πΦ0

A‖(κ′s + κ′p

)(12)

where

α = 2d|ηs||ηp| β = 2e|ηs|2|ηp|2

γzs = γs,z|ηs|2 γ‖s = γs,‖|ηs|2

γzp = γp,z|ηp|2 γ‖p = γp,‖|ηp|2(13)

5

To get the homogeneous order parameter that minimizes the free energy in the absence of externalmagnetic �eld, we can work with the further simpli�ed expression of the free energy, if ∆φ = φs − φp:

f(|ηs|, |ηp|,∆φ) =as|ηs|2 + bs|ηs|4 + ap|ηp|2 + bp|ηp|4 + c|ηs|2|ηp|2

+2dqz cos(∆φ)|ηs||ηp|+ 2e cos(2∆φ)|ηs|2|ηp|2, (14)

whose extremalization with respect to the gauge invariant phase di�erence ∆φ is readily found to give,if e 6= 0 and ηs, ηp 6= 0:

|ηs| |ηp| sin ∆φ (dqz + 2e |ηs| |ηp| cos ∆φ) = 0⇔

{|ηs| |ηp| sin ∆φ = 0or cos ∆φ = −dqz

2e|ηs||ηp|. (15)

In a reasonable range of parameters, it can be shown with microscopic arguments that the secondcondition cannot be satis�ed, leaving as the only solutions to the extremalization condition ∆φ ≡ 0 [π].Whether the latter possibilities are minimization conditions or not solely depends on the sign of dqz.Indeed, noticing that for either set of solutions the term 2e cos(2∆φ)|ηs|2|ηp|2 is a constant (the same),the energy is minimized when dqz cos(∆φ) < 0. We get the minimization condition:{

if dqz < 0 ∆φ ≡ 0 [2π]if dqz > 0 ∆φ ≡ π [2π]

(16)

As the sign of qz is characteristic of one of the twins in the boundary problem, in this framework, weclearly see the �e�ectively� degenerate solutions ∆φ ≡ 0, π [2π] of the crystal system described in theintroduction.

We are now ready to turn to the detailed study of twin boundaries at which we �rst look in thetunneling regime.

2 The tunneling regime

Here, we study the tunneling regime of twin boundaries which is relevant when the twins are weaklycoupled. Electron pairs tunnel through the boundary, e�ectively coupling the two twins. Experimen-tally, this situation is relevant when samples grown independently are glued together. In any case,this approach captures all the essential physics of samples containing at least two domains of crystalsymmetry lowered in opposite ways from a more symmetric situation.

Tunneling is introduced in the mathematical description by writing the total free energy as:

F = Fup + Fdown + Fup−down (17)

where Fup,down are the free energies in each twin (denoted by F in section 1), and Fup−down is thecoupling (tunneling) term between the twins. up and down here denote the positions along the axisperpendicular (denoted by ⊥) to the boundary plane of the the twin under consideration.1 Withthese notations, depending on the con�guration, one has either (up, down) = (qz = +, qz = −) or

1 For example, if the axis perpendicular to the boundary wall is the x axis, up will refer to the part of the sample that

lies in the half-space with x > 0 while down will characterize the other one, i.e. that with x < 0.

6 2 The tunneling regime

(up, down) = (qz = −, qz = +). This distinction will become important when the normal to theboundary n is along the z axis. We write the tunneling term as:

Fup,down =

∫Sd2r

[t1s |ηs,up − ηs,down|2 + t1p |ηp,up − ηp,down|2

+t2s |ηs,up − ηs,down|4 + t2p |ηp,up − ηp,down|4

+t1sp(|ηs,up − ηp,down|2 − |ηp,up − ηs,down|2

)] (18)

where S is the surface of the boundary, and t1s, t1p, t2s, t2p, t1sp are material dependent coe�cients whichmay also depend on the con�guration of the boundary2. Again, we write this expression in terms ofthe order parameter phases, and get the very simple-looking expression, with ψ = φs,up−φs,down (fromwhich it follows that φp,up − φp,down = ψ ± π, 0) and Fup−down =

∫S d

2rfup−down:

fup−down = ε0−ε1 cosψ+ε2 cos 2ψ

ε0 = 2

(t1s |ηs|2 + t1p |ηp|2 + 2t2s |ηs|4 + 2t2p |ηp|4

)ε1 = 2

(t1s |ηs|2 − t1p |ηp|2 ± 2t1sp |ηs| |ηp|

)ε2 = 2

(t2s |ηs|4 + t2p |ηp|4

) (19)

Note that the coe�cient ε1 can be arbitrarily small and change signs depending on the strength ofthe di�erent channels. This is an important feature as it will determine the crossover between thefractional- and integer-vortex regimes. This appears very clearly in the work done by Iniotakis et al.[24]. We can now minimize the full free energy with respect to r 7→ φs,up/down(r), r 7→ φp,up/down(r).Contributions to the boundary term are:[

−22πΦ0

(γ⊥s ∂⊥φs,up + γ⊥p ∂⊥φp,up −A⊥,up

2πΦ0

(γ⊥s + γ⊥p

))]x⊥=+∞

x⊥=0+

from the bulk terms and ε1 sinψ−2ε2 sin 2ψ from the coupling term. And we get, for a vanishing termat in�nity:

−Φ0

2πj0+ε1 sinψ−2ε2 sin 2ψ = 0, j0 = −2

2πΦ0

(γ⊥s ∂⊥φs,up + γ⊥p ∂⊥φp,up −A⊥,up

2πΦ0

(γ⊥s + γ⊥p

)).

(20)

We now show that j0 is the current that goes through the boundary. This can be qualitativelyunderstood since tunneling physically involves the transport of pairs across the boundary. Since thecurrent is given by

j =1c

∂f

∂A, (21)

we get the following relation between the gauge potential, the current and φs in a non-centrosymmetricsingle-crystal:

A =Φ2

0

8π2Γ−1j +

Φ0

2π

(∇φs − ˆγp∇∆φ

)(22)

2 i.e. depend on what the normal to the boundary is, and also whether (z > 0, z < 0) ↔ (qz = +, qz = −) or

(z < 0, z > 0)↔ (qz = −, qz = +) when z is perpendicular to the boundary wall.

7

��

����

������

�

��

��

Fig. 2: Contour used for determining the �ux-phase relation.

with Γ =

γ‖s + γ

‖p

γ‖s + γ

‖p

γzs + γzp

x,y,z

ˆγp =

γ‖p

γ‖p

γzp

x,y,z

Γ−1 (23)

over which we may make a contour integral to �nd a relation for the magnetic �ux Φ instead of oneinvolving the gauge potential A:

Φ =∮

A · dl =∮ [

Φ20

8π2Γ−1j +

Φ0

2π

(∇φs − ˆγp∇∆φ

)]· dl. (24)

Now it is important to pay attention to signs. We use a rectangular contour such that the distancebetween the lines perpendicular to the boundary is in�nitely small. Moreover, this contour is oriented

positively about the x/⊥2axis where

(x/⊥1

, x/⊥2, x⊥

)forms a positively oriented triad, see �gure 2. We

get:

2πΦΦ0

= ∂/⊥1ψ. (25)

Now, assuming that the magnetic �eld in the x/⊥2direction can be summed in the ⊥ direction to

2λBx/⊥2= Φ =

Φ0

2π∂/⊥1

ψ (26)

where λ is the characteristic length of decay of the magnetic �eld in the normal to the boundary, usingMaxwell's equation on the current curlB = 4π

c j, and assuming invariance of the /⊥1 component of themagnetic �eld along the /⊥2 axis, we �nd that the tunneling current across the boundary is related tothe curvature of the phase jump along the boundary via the relation:

∂2/⊥1 /⊥1

ψ =16π2λ

Φ0cj⊥. (27)

8 2 The tunneling regime

Since j⊥ = j0 from Eq. (20), using the latter, the phase jump ψ = φs,up − φs,down appears to bespatially governed by a double sine-Gordon equation [38, 39]:

∂2/⊥1 /⊥1

ψ − Λ−21 sinψ + 2Λ−2

2 sin 2ψ = 0 where Λ1,2 =Φ0√c

4π√

2πλε1,2(28)

which can be seen as derived from the Lagrangian density:

L =12

(∂/⊥1

ψ)2− Λ−2

1 cosψ + Λ−22 cos 2ψ, (29)

and, by adjusting the constant K, is associated with the energy density of the junction:

E =K

2

(∂/⊥1

ψ)2−KΛ−2

1 cosψ +KΛ−22 cos 2ψ. (30)

In particular, K should contain the coe�cient associated with the magnetic energy density contributionB2

8π . Also note that within this approach, where a contour is taken o� of the edges, the energy densityterm −B·H

4π is not included. However, this is not a problem, since due to its form (one derivative), itacts as a coupling to a magnetic �ux reservoir, �xing how many vortices should be inside the junction.We implicitely use it in this way until we consider phase transitions for which it is responsible. There,we insert the corresponding energy density term on the junction −CH/⊥2

∂/⊥1explicitely, where C is a

constant that should also be adjusted.

Before computing solutions to this equation, we investigate the e�ects of terms that we have omitteduntil now.

2.1 E�ect of the helical terms

Recall in Eq. (6) the helical terms that couple the magnetic �eld to a current-like term fhelical. Thetotal energy density contribution from this term is, with a continuous magnetic �eld at the boundary:

f tothelical = (B× z) ·(jeff‖+ − jeff‖−

)= (B× z) ·

(κ′s + κ′p

)∇‖α, α = φs+ − φs−. (31)

We now have to specialize to the di�erent boundary con�gurations. The latter are depicted in�gure 3.

Cases where n ‖ z We then have(x/⊥1

= x, x/⊥2= y, x⊥ = z

)and so

f tothelical =(κ′s + κ′p

)By∂xα

⇒Eq. (26) f tothelical =

(κ′s + κ′p

)Φ0

4πλ∂xα∂xψ (32)

whose sign depends on the con�guration of the twins:

f tothelical = ±(κ′s + κ′p

)Φ0

4πλ(∂xψ)2

{+ if (up, down) = (+,−)⇔ ψ = α

− if (up, down) = (−,+)⇔ ψ = −α(33)

So reinserting this energy density term in the free energy density Eq. (30), we �nd renormalizedparameters which depend on the twin con�guration, namely whether spin-orbit couplings point inwardsor outwards:

E =K ′

2(∂xψ)2 −KΛ−2

1 cosψ +KΛ−22 cos 2ψ (34)

2.2 Zero- and single-vortex solutions 9

� �

���

����

�� ��

���� ����

� �

�

� �

�

Fig. 3: The di�erent types of boundary con�gurations.

or in the Lagrangian density:

L =12

(∂xψ)2 − Λ′−21 cosψ + Λ′−2

2 cos 2ψ (35)

with K ′ = K ± Σ and Λ′1,2 =

√K ± ΣK

Λ1,2 where Σ =

(κ′s + κ′p

)Φ0

4πλ(36)

Case where n ⊥ z In this case, the axes can be chosen such that x⊥ = y. Then B = Bxx + Bzz,which gives:

f tothelical = −(κ′s + κ′p

)By∂yα = 0 since By = 0, (37)

so the helical terms give no contribution to the free energy in this case..

2.2 Zero- and single-vortex solutions

We now investigate solutions of Eq. (28). It is useful to see L (Eq. (29)) as the Lagragian of a pointparticle in a potential [38, 39] whose shape changes as ρ = Λ2/Λ1 goes through ρc =

√2, as shown on

�gure 4.

Constant solutions are:

ψconst =

{0 [2π] if ρ ≥ ρc± arccos

(ρ2/2

)[2π] otherwise

(38)

Exact non-constant solutions where ψ extrapolates between neighbor minima of the potential (Eq.(38)) take di�erent forms on each side of the boundary. They, of course, also depend on the minima

10 2 The tunneling regime

����

����

����

������

�

Fig. 4: Shape of the potential V (ψ) = − cosψ + ρ−2 cos 2ψ for di�erent values of ρ.

that ψ extrapolates between:tan

ψ(x/⊥1

,x/⊥2,0)

2 = K1csch(x/⊥1

/Λ)

if ρ ≥ ρc

tanψ(x/⊥1

,x/⊥2,0)

2 =

K21 coth(x/⊥1

/Λ)

or K22 tanh(x/⊥1

/Λ) otherwise

(39)

where Λ and the Kij 's3 are functions of Λ1 and Λ2.

Λ =

{Λ1/

√1− 2ρ−2 if ρ ≥ ρc

2Λ1/√

4ρ−1 − ρ otherwise(40)

ρc separates di�erent types of solutions. As shown on �gure 5, at ρc, Λ diverges, like in criticalphenomena, where a point where the system exhibits a diverging length scale separates two regions ofdi�erent characters.

The energy of the system in the non-constant structure is higher than when the phase di�erenceis just constant, but the term fmagn coupl = −B·H

4π provides competition. Whether the system has aconstant solution of one of the type described above depends on the strength of the external magnetic�eld via the coupling fmagn coupl = −B·H

4π of the phase jump to the external magnetic �eld, which, in athermodynamical picture acts as a reservoir (note that this term also determines the sign of the jump

of ψ(x/⊥1

)along the x/⊥1

axis, i.e. the sign of ∂/⊥1ψ). The lower critical magnetic �eld Hc1 is the �eld

at which the �rst vortex penetrates, i.e. for which it becomes energetically more favorable to have anon-homogeneous solution on the junction. For

E =∫ L/2

−L/2dx/⊥1

[K

2

(∂x/⊥1

ψ)2−KΛ−2

1 cosψ +KΛ−22 cos 2ψ − CH∂x/⊥1

ψ

]. (41)

3 The indices 1, 21 and 22 are just there to label to which �ux the constants are associated with: K1 is associated

with ρ > ρc and integer �ux, K11 labels the solution for ρ < ρc and �ux Φ = Φ0π

arccos ρ2

2etc..

2.2 Zero- and single-vortex solutions 11

��

��������������

Fig. 5: Kink size Λ as a function of ρ = Λ2/Λ1.

�

�

�

�

�

�

Fig. 6: Kink, magnetization an current for values below and above ρc and for di�erent kink types.

12 2 The tunneling regime

�

��

�

�� � ��

�

Fig. 7: Approximation to a kink solution of the double sine-Gordon equation.

where L is the length of the junction and K is possibly renormalized, it is given by:

Hc1 =Φ0Ekink

2πCΦ(kink)(42)

where

Ekink =∫ L/2

−L/2dx/⊥1

[K

2

(∂x/⊥1

ψ)2−KΛ−2

1

(cosψ − cosψconst

)+KΛ−2

2

(cos 2ψ − cos 2ψconst

)]. (43)

Hc1 is seen to depend on the kink through both the energy term and the �ux enclosed, and we �nd,using the approximate linear kink depicted in �gure 7:

Hc1 =Kπ

ΛCΦΦ0

+ g (Λ1,Λ2, Φ/ Φ0) (44)

where Φ = Φ0∆ψ2π i.e.

Φ =

{Φ0 for ρ ≥ ρcΦ0π arccos ρ

2

2 or Φ0

(1− 1

π arccos ρ2

2

)otherwise

(45)

and g is a real function that depends on the �ux Φ enclosed, easily determined through

Ekink ≈ ΛK

(12

(∆ψΛ

)2

+ 2sin ∆ψ/2

∆ψ(Λ−2

1 − Λ−22 cos ∆ψ/2

)− E0

ΛK

). (46)

where E0 = ΛK(−Λ−2

1 cosψconst + Λ−22 cos 2ψconst

).

2.3 Outlook into multi-vortex solutions

ForH > Hc1, the presence of a vortex on the junction represents a gain in energy. Thus, as the magnetic�eld is increased, more and more vortices will come in, but stronger and stronger competition will beintroduced by the interaction between vortices. In view of this observation, I give results and outlookinto multiple vortex e�ects and states.

2.3 Outlook into multi-vortex solutions 13

�����

��

�������

������

��

Fig. 8: Linearized interaction between two vortices.

2.3.1 Approximate interaction energy

There are no exact multi-vortex solutions to the double sine-Gordon equation. However, vortices arewell localized as can be seen through the exponential decay of their spatial derivatives. We may thusconsider the interaction of two vortices using a perturbative approach if the density of vortices onthe junction is su�ciently small. To do so, we linearize the double sine-Gordon equation, use thelinear approximation of the kinks far from their center and compute the interaction energy obtainedby considering the superposition of two solutions (see �gure 8) as a function of the distance betweenthe center of the kinks. We �nd the gross approximation:

Eint

(∆x/⊥1

)∝ ∆x/⊥1

exp(−

∆x/⊥1

L

)(47)

where ∆x/⊥1is the distance between the centers of the kinks.

2.3.2 Thermodynamic phase transitions

We get insight into the multi-vortex situation by writing the energy of the junction as follows:

E = nEkink + (n− 1)Eint + 2Eedge + 2Eint(vortex−edge) + EH·B. (48)

where n is the amount of vortices on the junction, Eint is the interaction energy between two vortices,Eedge is the surface energy contribution due to the continuity of the magnetic �eld at the edges ofthe sample and Eint(vortex−edge) is the energy due to the interaction of the vortex closest to one ofthe edges with the phase variation at the edge, and EH·B is the energy due to the coupling of theinternal magnetic �eld with the external one (EH·B ≤ 0). Within this framework, competition betweeninteractions and bulk energies naturally appear and we can conjecture �rst-order phase transitionswithin thermodynamical reasoning that naturally arises. We can try and describe phase transitions ina thermodynamical phase space (and with observable parameters!). In particular, we can try and �ndtransitions between junctions with more or less vortices in (H,L) space for which e.g. the schematicdiagrams of �gure 9 can be drawn. In particular, one can try and get Ehrenfest relations on the phasetransition lines.

14 2 The tunneling regime

�

�

� ��� ���

�

�

�

���

Fig. 9: Schematic diagrams relevant for phase transitions between junctions containing di�erent number

of vortices.

By further making the following reasonable assumptions on the dependence of the above quantitieson the external parameters:

n = n (L,H)Ekink = const.

Eint = Eint

(l ∼ L

n = Ln(L,H)

)Eedge = Eedge (L,H)Eint(vortex−edge) = Eint(vortex−edge) (L,H)

, (49)

we get for example:

Ekink − 2πCHextΦΦ0

= (n− 1)(Eint

(L

n

)− Eint

(L

n+ 1

))−Eint

(L

n+ 1

)≡ ∆Eint

(L

n

)(50)

at the transition between a junction with n vortices and that with n+ 1 vortices due to sample lengthchange (if we omit the edge e�ect change).

To get quantitative results we must resort to numerical simulations of the junction to which wenow turn.

2.3.3 Multi-vortex numerical solutions

To get a complete picture, we resorted to numerical simulations of the junction, using a quasi-Newtonroutine. We should get estimates of the interaction energy, and be able to draw a phase diagram in(H,L) phase space. Unfortunately time came short and we only ran few.

2.3.4 Magnetization below the upper critical �eld Hc2

In order to investigate how the magnetization acquires a structure below Hc2, we perturb the solutionabove Hc2 �which is linear since the magnetic �eld fully penetrates� through

ψ(x/⊥1) =

4πλHext

Φ0x/⊥1

+ φ(x/⊥1) where

φ

L� 4πλHext

Φ0. (51)

We get the following equation on the perturbing �eld:

∂2/⊥1 /⊥1

φ+(

4Λ−22 cos

4πλHext

Φ0x/⊥1− Λ−2

1

)sin

4πλHext

Φ0x/⊥1

= 0 with ∂/⊥1φ∣∣∣boundary

= 0 (52)

15

which has solutions if the following quantization condition on Hext (and L) is ful�lled:

Hext ≡Φ0

4πλLarccos

Λ−21 ±

√∆

4Λ−22

[Φ0

2λL

]. (53)

where ∆ is a function of Λ1 and Λ2. Thus the magnetization changes in steps only as the magnetic�eld is lowered below Hc2.

2.3.5 Temperature dependence

Within Ginzburg-Landau theory, temperature dependence is introduced through the parameters. Here,we show that the phase diagram obtained by Iniotakis et al. [24] in a microscopic approach can berecovered at least qualitatively. Recall Eq. (14) and introduce the following temperature dependence:

as = a′s (T − Tcs)ap = a′p (T − Tcp)

, (54)

where Tcs,p are the critical temperatures associated with the s and p representations. The minimizationover |ηs| and |ηp|, yields two third-order coupled equations on |ηs| and |ηp|:

4bs,p |ηs,p|3 +(

2as,p + 2c |ηp,s|2)|ηs,p| − 2|d| |ηp,s| = 0. (55)

By plugging the solutions in the condition that separates the fractional- and integer-�ux regimesε1/ε2 = ρ2

c = 2, one gets (recall Eq. (19)):

t1s |ηs|2 − t1p |ηp|2 ± 2t1sp |ηs| |ηp|t2s |ηs|4 + t2p |ηp|4

> 2 for integer vortices. (56)

For all tunneling coe�cients, for small enough order parameters i.e. large enough temperatures, thiscondition is satis�ed. But as the temperature is decreased well below the critical temperature, thissolution is only satis�ed in some parameter domains. Furthermore, when only one order parametersurvives, only integer vortices are allowed. Accordingly, we can conjecture the schematic diagram of�gure 10. Note the resemblance to that obtained semi-analytically by Iniotakis et al..

3 Strong-coupling regime

The purpose of this section is to get further insight on the appearance of fractional �uxes as consequenceof time-reversal symmetry breaking. To do so, we exhibit the role played by the existence of two orderparameters in the limit where solutions that minimize the free energy density are continuous acrossthe boundary.

3.1 Bare twin boundary

We assume variation of the phase across the boundary only, so we drop the derivatives along thedirections parallel to the boundary. We get the following simpli�ed expression for the free energy:

f [r 7→ φs(r), r 7→ φp(r)] = f ′0 + αqz cos(∆φ) + β cos(2∆φ) + γ⊥s (∂⊥φs)2 + γ⊥p (∂⊥φp)2 (57)

16 3 Strong-coupling regime

����

����

����

����

����

��������

Fig. 10: Temperature phase diagram between fractional and integer vortices.

For this case where there are no vortices on the junction, setting the current through the boundary tozero gives:

∂⊥φs,p = (−1)0,1 γ⊥p,sγ⊥s + γ⊥p

∂⊥∆φ. (58)

Thus:

f [r 7→ φs(r), r 7→ φp(r)] = f0 + αqz cos(∆φ) + β cos(2∆φ) + γ(∂⊥∆φ)2 (59)

and

αqz sin(∆φ) + 2β sin(2∆φ) + 2γ⊥∂2⊥⊥∆φ = 0 (60)

where γ⊥ = γ⊥s γ⊥p

γ⊥s +γ⊥p.

Performing simply a Taylor expansion of the potential, and imposing continuity of ∆φ, we �nd, ifqz = ±1 for x⊥ ≷ 0 and d > 0:

∆φ(x⊥) =

{π2 ex⊥/λ⊥ if x⊥ < 0

π(1− e−x⊥/λ⊥2 ) if x⊥ ≥ 0

where λ⊥ =

√2γ⊥

|α| − 4β(61)

3.2 Presence of a vortex

Now we enforce the presence of one vortex, where, by de�nition a vortex is related to the non-connectedness of the space that the values of the components of the order parameter are free tolive in. We enforce that to go from one twin to the other the order parameter spatially takes di�erentpaths in order parameter space.

3.2 Presence of a vortex 17

-10 -5 5 10

z

Λ

0.5

1.0

1.5

2.0

2.5

3.0

DΦ

Fig. 11: Plot of evolution of the phase di�erence ∆φ = φs − φp across the boundary in the linear

approximation.

To �nd the fractional �ux, we perform contour integrals. Recall Eq. (22):

Φ =∮

A · dl =∮ [

Φ20

8π2Γ−1j +

Φ0

2π

(∇φs − ˆγp∇∆φ

)]· dl (62)

We take a contour su�ciently far from the vortex core, j = 0 along it. For simplicity, we also assume

isotropy of the γ coe�cients, i.e. γ‖s,p = γzs,p = γs,p. Using the fact that the phases φs and ∆φ

live in a space homotopic to U(1) and using the continuity across the boundary associated with thestrong-coupling limit, we �nd:

∃n, n′ ∈ ZΦΦ0

= n− γpn′. (63)

Through this equation, we see that fractional vortices appear because of the existence of di�erent orderparameters (they appear with di�erent coe�cients in the free energy) and time-reversal symmetry(�ux not only appears via a gauge-dependent phase, but also through the imposed variation of agauge-dependent one, making this spatial variation a new e�ect).

Conclusion and discussion

In this work, we have investigated twin boundaries between non-centrosymmetric superconductors ina purely symmetry-based phenomenological approach. As essential input from microscopic knowledge,we used the existence of an order parameter with an odd and an even component under inversionsymmetry, and the existence of �helical� terms. In two di�erent limits we showed how fractionalvortices appeared. In both the weak and strong coupling limit, the existence of two order parametersand time-reversal symmetry breaking induced by the geometry proved to be essential. Moreover, inthe weak-coupling regime, fractional vortices appeared to be directly related to the combination of

18 3 Strong-coupling regime

tunneling processes of di�erent orders. In the strong-coupling limit, fractional �ux followed directlyfrom the fact that di�erent coe�cients were associated with the di�erent order parameters.

In doing so, we exhibited the interesting result that helical terms renormalized coe�cients in thee�ective theory and that the renormalization depended on the relative con�guration of spin-orbitcoupling on either side of the boundary. This renormalization led in particular to di�erent values ofthe lower critical �eld Hc1, even in the case of integer vortices. Time ran short to make a detailednumerical study of the interactions between vortices, but we outlined the steps of such an investigationand discussed how those results could be used to �nd other possibly experimentally-accessible data.

Unfortunately, experimental evidence for the existence of fractional vortices in non-centrosymmetricsuperconductors is still missing. Thanks to the fact that fractional vortices can only appear on bound-aries and thus act as fences for magnetic �ux, an interesting indirect probe could come from �ux creepexperiments, which this work could help account for theoretically. Of course, uncontroversial evidencefor fractional �uxes and for their structure as described in this report would come from very localmagnetic probes such as electron microcope spectroscopy. But such experiments require special instru-ments and techniques and have not yet been carried out on the recently discovered non-centrosymmetricsuperconductors.

Acknowledgements

I acknowledge Manfred Sigrist for providing me with this project and for advising me, and ChristianIniotakis for his helpful �tutorials� and discussions on superconductivity, and for carefully reading myreport. I also thank Christian, Mark Fischer and Simon Wood for creating a nice atmosphere in theo�ce. I address many thanks to Jonathan Buhmann, another Master's student for discussions onphysics and other things, and Emanuel Gull for showing me around and many conversations. Finally,I wish to thank Mario, my mother and sister for visiting, and Vincent for saving the day today bycompiling my TeX �les!

19

A Derivation of the free energy

group G representation Γ of the group G

ηs D4h Γ+1

ηp D4h Γ−2qz C4v Γ−2

Di,i = x, y D4h Γ−5Dz D4h Γ−2

Hi,i = x, y D4h Γ+5

Hz D4h Γ+2

Tab. 1: Groups and representations under which the terms involved in the free energy transform.

Second order in the order parameter:

product reprn

ηsηs Γ+1

ηpηp Γ+1

ηsηp Γ−2

product representation

DiDj Γ+1 + Γ+

2 + Γ+3 + Γ+

4

DzDz Γ+1

DiDz Γ+5

Di Γ−5Dz Γ−2

product reprn

1 Γ+1

qz Γ−2qz(z×H) Γ−5qzHz Γ−1

Tab. 2: Systematic approach.

qz (Γ−2 ) ηsηp, DiηsDjηp, DzηsDzηp, (Dzηs)ηs, (Dzηp)ηpqz(z×H) (Γ−5 ) DiηsDzηp, DiηpDzηs, (Diηs)ηs, (Diηp)ηpqzHz (Γ−1 ) DiηsDjηp

Tab. 3: Shape of the possible terms.

20 References

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