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Properties in N-Component London Superconductors 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 l .................. 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 as an order parameter that through j j2 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 + αj j + j j + j( r~ − qA~) j + (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 + αlj lj + j lj + j( r~ − qlA~) lj + (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 = r~ × 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 l ¡ 3 We will first minimize 3 d rFs with respect to a variation of l: R ¢ " 3 ∗ βl 2 ∗ 0 = d r αl l δ l + j lj 2 l δ l 3 2 R ∗ # ( ~ r~ − qlA~) · ( ~ r~ δ l − qlAδ~ l) + −i l i (2.4) 2ml Next, recall that given any scalar φ and any vector T~, we have following vector identity: r~ (φT~) = r~ φ · T~ + φr~ · T~ (2.5) ~ ~ ~ ( −i ∇−qlA) ∗ Utilizing equation 2.5 in 2.4 for φ := ~ δ l and T~ := , we then i 2ml l obtain: ¢ " ~ ~ ~ ∗ ~ # 3 ∗ 2 ∗ ( −i r − qlA) l · (−qlA) 0 = d rδ l αl l + βlj lj l + + 3 2m R l ¢ " # ¢ " # ( ~ r~ − q A~) ( ~ r~ − q A~) 3 ~ ~ −i l ∗ 3 ~ ~ −i l ∗ d rr · δ l l − d r δ lr · l (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 l is 0 outside the specimen. We want to do this change because δ l is also 0 outside the specimen and we want to utilize the condition that we can choose arbitrary δ l inside the specimen to find useful equations. 2.2. Deriving the Ginzburg-Landau equations 5 Furthermore, recall the generalized divergence theorem [1, p. 93]: ¢ d3rr~ (:::) = dS~(:::) (2.7) We can now rewrite equation 2.6 as: ¢ " ~ ~ ~ 2 # 3 ∗ 2 ∗ ( −i r − qlA) ∗ 0 = d rδ l αl l + βlj lj l + l + super 2ml " # ( ~ r~ − q A~) ~ ~ −i l ∗ δ ldS · l (2.8) @super i 2ml Which yields a Ginzburg-Landau equation and a boundary condition for each component of the superconductor: ~ ~ ~ 2 ( i r − qlA) 2 l + αl l + βlj lj l = 0 (2.9) 2ml dS~ · (~r~ − q A~) = 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: ~p^ = ~r~ (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.
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