Physical Processes in Disordered Superconductors

COMENIUS UNIVERSITY IN BRATISLAVA FACULTY OF MATHEMATICS, PHYSICS AND INFORMATICS

Physical processes in disordered superconductors

2016

RNDr. Martin Zemliˇckaˇ COMENIUS UNIVERSITY IN BRATISLAVA FACULTY OF MATHEMATICS, PHYSICS AND INFORMATICS

Physical processes in disordered superconductors

Dissertation thesis

4.1.3 Condensed matter physics and acoustics Department of experimental physics

Bratislava, 2016 RNDr. Martin Zemliˇckaˇ 51249511 Comenius University in Bratislava Faculty of Mathematics, Physics and Informatics

THESIS ASSIGNMENT

Name and Surname: Mgr. Martin Žemlička Study programme: Condense Matter Physics and Acoustics (Single degree study, Ph.D. III. deg., full time form) Field of Study: Physics Of Condensed Matter And Acoustics Type of Thesis: Dissertation thesis Language of Thesis: English Secondary language: Slovak

Title: Physical processes in disordered superconductors. Literature: 1. A Tholen, Parametric amplification with weak-link nonlinearity in superconducting microresonators, arXiv:0906.2744v2 2.G. Wendin, V.S. Shumeiko,Superconducting Quantum Circuits, Qubits and Computing arXiv:cond-mat/0508729v1 3. A. M. ZAGOSKIN, QUANTUM ENGINEERING, Cambridge University Press 2011 4. L. Skrbek a kol. Fyzika nízkých teplot, matfyzpress PRAHA 2011 Aim: The aim of the thesis is the examination of physical processes in disordered superconductors and nonlinear effects in superconducting nanostructures. Such structures can be applied as very sensitive detectors usable in astronomy, experimental physics or in quantum information processing.

Tutor: doc. RNDr. Miroslav Grajcar, DrSc. Department: FMFI.KEF - Department of Experimental Physics Head of prof. Dr. Štefan Matejčík, DrSc. department: Assigned: 01.09.2012

Approved: 01.03.2012 prof. RNDr. Peter Kúš, DrSc. Guarantor of Study Programme

Student Tutor 51249511 Univerzita Komenského v Bratislave Fakulta matematiky, fyziky a informatiky

ZADANIE ZÁVEREČNEJ PRÁCE

Meno a priezvisko študenta: Mgr. Martin Žemlička Študijný program: fyzika kondenzovaných látok a akustika (Jednoodborové štúdium, doktorandské III. st., denná forma) Študijný odbor: fyzika kondenzovaných látok a akustika Typ záverečnej práce: dizertačná Jazyk záverečnej práce: anglický Sekundárny jazyk: slovenský

Názov: Physical processes in disordered superconductors. Fyzikálne procesy v neusporiadaných supravodičoch. Literatúra: 1. A Tholen, Parametric amplification with weak-link nonlinearity in superconducting microresonators, arXiv:0906.2744v2 2.G. Wendin, V.S. Shumeiko,Superconducting Quantum Circuits, Qubits and Computing arXiv:cond-mat/0508729v1 3. A. M. ZAGOSKIN, QUANTUM ENGINEERING, Cambridge University Press 2011 4. L. Skrbek a kol. Fyzika nízkých teplot, matfyzpress PRAHA 2011 Cieľ: Cieľom práce je skúmanie fyzikálnych procesov v neusporiadaných supravodičoch a nelineárnych javov v supravodivých nanoštruktúrach. Tieto môžu byť použité ako veľmi citlivé detektory, využiteľné v astronómii, experimentálnej fyzike, ako aj pri spracovaní kvantovej informácie.

Školiteľ: doc. RNDr. Miroslav Grajcar, DrSc. Katedra: FMFI.KEF - Katedra experimentálnej fyziky Vedúci katedry: prof. Dr. Štefan Matejčík, DrSc. Dátum zadania: 01.09.2012

Dátum schválenia: 01.03.2012 prof. RNDr. Peter Kúš, DrSc. garant študijného programu

študent školiteľ Abstract

Author: RNDr. Martin Zemliˇckaˇ Title: Physical processes in disordered superconductors Supervisor: Prof. RNDr. Miroslav Grajcar, DrSc. University: Comenius university in Bratislava Faculty: Faculty of mathematics, physics and informatics Department: Department of experimental physics Field of science: 4.1.3. Condensed matter physics and acoustics Year: 2016

The submitted dissertation thesis deals with the experimental analysis of physical processes and effects in highly disordered superconductors. We optimized the deposition process of MoC thin films in order to produce the samples with enhanced sheet resistance R and critical temperature Tc as highest as possible. Optimal parameters for magnetron sputtering of MoC were determined. The sputtering rate deposition was calibrated by AFM thickness measurement. Structure, stoichiometry and surface roughness of the prepared samples were analyzed by means of XRD, EDX and STM, determining that our samples have the cubic crystallographic δ-phase and have a surface with atomic flat areas. Their stoichiometry reveals decreased carbon content, caused probably due to vacancies in the crystal lattice. We measured the transport properties of samples with different thicknesses in a wide range of magnetic fields and obtained basic material parameters. We found that Tc is well correlated with R and kf l, even as sample thickness is varied. We showed that Finkelstein’s formula can not be applied in order to explain the

Tc suppression, despite the fact that it qualitatively well describes the transport properties of our samples as well as many other thin disordered films. Further analysis by STM was performed, demonstrating an increase of in-gap states with decreasing thickness. Moreover, vortex lattice was observed, proving the long range phase coherence ascribed to fermionic scenario of superconductor insulator transition (SIT). A more detailed view of the field induced SIT of 3 nm MoC thin film revealed the atypical normal state in high magnetic field in agreement with Altshuler-Aronov model of enhanced electron-electron interactions. Moreover, we showed that the interface between the thin film and the substrate plays role as possible pair-breaker.

5 The applicability of our thin films was outlined by the transport measurements of a nanobridge patterned in 10 nm MoC film, which exhibited quantum phase slip like behavior. The scattering effects were further analyzed by microwave measurements of CPW resonators. The decrease of the internal quality factor and the resonant frequency was described by modified Mattis-Bardeen theory with introduced finite quasiparticle lifetime. The results agreed with the Dynes phenomenological parameter Γ obtained from STM. The developed model was applied also in order to describe results from terahertz spectroscopy of the complex conductivity. We used the same CPW technique for granular MgB2 superconductor, which showed periodic hysteretic detuning in magnetic field, characteristic of artificially prepared SQUID structures. A reasonable model of such behavior with percolation path interrupted by weak links between the grains was designed and successfully used in the fit.

Keywords: thin ﬁlm, supercoductivity, Mattis-Bardeen theory, quasiparticle lifetime, coplanar waveguide resonator, superconductor-insulator transition, nanowire, RF

SQUID, MoC, MgB2

6 Abstrakt

Autor: RNDr. Martin Zemliˇckaˇ Názov: Fyzikálneprocesy v neusporiadaných supravodiˇcoch Skoliteˇ ˇl: Prof. RNDr. Miroslav Grajcar, DrSc. Univerzita: Univerzita Komenskéhov Bratislave Fakulta: Fakulta matematiky, fyziky a informatiky Katedra: Katedra experimentálnejfyziky StudijnýOdbor:ˇ 4.1.3. Fyzika kondenzovaných látok a akustika Rok: 2016

V predloˇzenejprácipojednávame o experimentálnejanalýzefyzikálnych procesov vo vysoko neusporiadaných supravodiˇcoch. Optimalizovali sme pr´ıpravu tenkých ˇ ˇ MoC vrstiev s cielom dosiahnut vzorky so zvýˇsenýmodporom na ˇstvorec R a s ˇco najvyˇsˇsoukritickou teplotou Tc. Urˇcilisme optimálneparametre magnetrónoveho napraˇsovania MoC vrstiev. Rýchlosˇt napraˇsovania bola kalibrovanámeran´ımhrúbky pomocou atómovéhosilovéhomikroskopu (AFM). Struktúru,stechiometriuˇ a povr- chovúdrsnosˇt pripravených vzoriek sme analyzovali pomocou XRD, EDX a STM metód. Z výsledkov sme urˇcili, ˇze naˇse vzorky majúkubickúkryˇstalografickú ˇstruktúrua atomárnehladkéoblasti. Stechiometrickáanalýzaukázalazn´ıˇzenýob- sah uhl´ıka, spôsobenýpravdepodobne kvôlivakanciámv kryˇstálovej mrieˇzke. Merali sme tieˇztransportnévlastnosti vzoriek s rôznymi hrúbkami v ˇsirokom rozsahu hodnôtexternéhomagnetickéhopoˇla, z ˇcohosme z´ıskali základnéma- teriálovévlastnosti. Zistili sme, ˇze Tc výraznekoreluje s R a kf l, a to aj pre vzorky s odliˇsnouhrúbkou. Ukázalisme, ˇze Finkelsteinov model nie je moˇzné pouˇziˇt na popis potlaˇcenia Tc, napriek tomu, ˇzekvalitat´ıvnesúhlas´ıs výsledkami transportných meran´ına naˇsich vzorkách, ako aj na vzorkách pripravených z iných neusporiadaných supravodiˇcov. Vykonali sme tieˇzanalýzupomocou STM, ktorá ukázalanárastvnútro-medzerových stavov so zniˇzujúcousa hrúbkou. Navyˇsesme pozorovali mrieˇzkuv´ırov, ˇcodokazuje pr´ıtomnosˇt fázovej koherencie prislúchajúcej fermiónovému scenáruprechodu supravodiˇc-izolant (S=I). Detailnejˇs´ı pohˇlad na poˇlom indukovanýprechod S-I v 3 nm hrubej MoC vrstve ukázalatypickýnormálny stav vo vysokých magnetických poliach v zhode s modelom Altshulera a Aronova popisujúcisilnúelektrón-elektrónovúinterakciu. Ukázalisme navyˇse,ˇzerozhranie

7 podloˇzkya tenkej vrstvy hráúlohu moˇznéhorozb´ıjaniaCooperových párov. Apliko- vateˇlnosˇt naˇsich tenkých vrstiev sme ukázalina výsledkoch transportných meran´ı nano-most´ıka, pripravenéhoelektrónovou litografiou na 10 nm tenkej MoC vrstve, ktorévykazovali pr´ıtomnosˇt kvantovéhoprek´lzavania fázy. Rozptylovéefekty boli tieˇzanalyzovanémikrovlnnýmimeraniami koplanárnych rezonátorov. Potlaˇcenievnútornejkvality a rezonanˇcnejfrekvencie sme pop´ısali modifikovanýmmodelom Mattisa a Bardeena so zavedenou koneˇcnoudobou ˇzivota kváziˇcast´ıc.Výsledkysúhlasilis hodnotami Dynesovho fenomenologickéhoparame- tra Γ, z´ıskanéhopomocou STM. Vyvinutýmodel bol tieˇzpouˇzitýna popis výsledkov komplexnej vodivosti z´ıskanej meran´ımterahertzovej spektroskopie. Rovnakámirkovlnná technika vyuˇz´ıvajúcakoplanárny rezonátorbola pouˇzitána analýzu MgB2 granulárneho supravodiˇca,ktoráukázalaperiodickéhysterézneodladenie. Takétosprávanie je charakteristicképre umelo pripravenéˇstruktúrysupravodivéhokvantovéhointerfer- ometra. Navrhli sme model uvaˇzujúciperkolaˇcnúdrahu preruˇsenúslabýmispojmi medzi jednotlivýmisupravodivýmizrnami, ktorýsme úspeˇsnepouˇzilina popis naˇsich výsledkov.

Kˇlúˇcovéslová:tenkávrstva, Mattis-Bardeenováteória,doba ˇzivotakváziˇcast´ıc,ko- planárnyrezonátor,prechod supravodiˇc-izolant,nanodrôt,RF SQUID, MoC, MgB2

8 Declaration I declare that this thesis is my own original work and all used sources have been acknowledged by means of complete references...... Martin Zemliˇckaˇ Acknowledgment Foremost I would like to express my sincere appreciation to my advisor Miroslav Grajcar for the continuous leadership of my PhD study and research. I am also thankful to my colleagues from Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Richard Hlubina, Pavol Neilinger, MariánTrgala, MatúˇsRehákand Daniel Manca and also to colleagues from Institute of Experi- mental Physics, Slovak Academy of Sciences in Koˇsicefor possibility to perform the experiments in their professionally equipped laboratory in the field of low temperature physics and many others for lot of help and useful discussions and advices. Last but not least I would like to express the deepest appreciation to my family for moral support during my entire study.

10 Contents

1 Introduction 12

2 Theory 15 2.1 Superconductivity ...... 15 2.1.1 Phenomenological theories ...... 16 2.1.2 BCS Theory ...... 17 2.1.3 Mattis-Bardeen Theory ...... 17 2.1.4 Electron tunneling ...... 19 2.1.5 Superconducting quantum interference device (SQUID) . . . . 21 2.1.6 Disordered superconductors ...... 25 2.2 Coplanar waveguide (CPW) resonator ...... 27 2.2.1 Quality factor and resonant frequency ...... 28 2.2.2 Distributed parameters per unit length ...... 30

3 Experiment 34 3.1 Sample preparation ...... 34 3.1.1 Magnetron sputtering ...... 34 3.1.2 Vapor deposition ...... 36 3.1.3 Optical lithography ...... 36 3.2 Measurement apparatus ...... 37 3.2.1 Scanning tunneling microscope (STM) ...... 37 3.2.2 Cryogenic microwave assembly ...... 40

4 Results analysis 42 4.1 Tuning the MoC deposition process ...... 42 4.1.1 Magnetron discharge analysis ...... 42 4.1.2 Carbon content influence ...... 43 4.2 Transport properties ...... 51 4.3 Tunneling measurements ...... 58 4.4 Field induced SIT in 3 nm MoC film ...... 65 4.5 Nanobridge in MoC thin film ...... 76 4.6 Microwave analysis ...... 81 4.6.1 MoC CPW resonator ...... 81

4.6.2 MgB2 CPW resonator ...... 92

5 Conclusions 97

Bibliography 99

List of publications and conferences 105

11 1 Introduction

Recently the development of nanotechnology has allowed for the preparation of new structures and artificial materials with unique properties, which don’t have an analog in nature. Artificially prepared structures, forming metamaterials with a negative refractive index allow for the construction of ’superlens’, which overcome the resolution of the classic lens, which are limited by diffraction [1]. Objects coated with such material become invisible in visible spectrum light [2]. Such a metamaterial is created from complex periodical structures of metal resonators. Such structures can be readily produced with a wide range of refractive indices, from positive to negative values, due to relatively easy implementation by lithographic methods. Real materials are built from atoms and molecules, which are quantum objects. Their artificial analogy are two-level systems, so-called qubits, which have been recently developed. They can be used as the basic cell of a quantum computer or a quantum metamaterial [3]. Their properties can be ’in-situ’ tuned by external electric and magnetic fields [4], which allows for construction of very effective detectors [5], with a sensitivity approaching the quantum limit, as well as single photon sources [6] or single photon lasers [7] in GHz frequency spectrum. Newest experiments have shown, that it is possible to construct laser spacers [8] or highly effective photodetectors with the ability to detect a single photon in the desired spectrum [9]. Their high integrability comes from by simple preparation by optical and electron lithography methods. A superconducting flux qubit was recently demonstrated by Astafiev et al [10] by employing quantum phase slip effect in a weak link in highly disordered superconducting nanowire. In general, quantum bits can be implemented by a nonlinear effect in a nonlinear element. In an optical quantum computer, it is the Kerr effect, by which optical quantum gates can be implemented [80]. However, it is difficult to achieve strong nonlinearity for visible light, because we are restricted to using natural materials. In the case of quantum artificial structures or metamaterials, strong coupling is achievable on the base of nonlinear effects in superconducting structures [81]. Such coupling can achieve several tenths of a percent of the resonant frequency of the

12 resonator. An important step during the implementation of the detector is to implement a parametric amplifier, through which it is possible to detect very weak signals with a noise suppressed below a level of standard quantum limit. As such, one of the aims of the dissertation is the precise preparation of ultra-thin superconducting films made of superconductors with high sheet resistance ( 1kΩ). Such materials are for ∼ example molybdenum carbide (MoC), indium oxide (InO), tungsten silicide (WSi), niobium nitride (NbN), etc. A parametric amplifier on the basis of weak-coupled junctions in such superconducting films allows to prepare suppressed states, which make it possible to detect very weak signals, with noise below the standard quantum limit [82]. Highly disordered superconductors are interesting from different points of view. Recent research is dealing with the most fundamental issues of condensed matter physics involving interplay of quantum correlations, quantum and thermal fluctuations, Coulomb interactions, but is also motivated by the promise of their wide applications. Their high resistance and inductance in the critical vicinity of superconductor- insulator transition [10,12] break ground for novel microwave engineering, exploring duality between phase slips at point like centers [14] or at phase slip lines [15] and Cooper pair tunneling [16–18]. While their DC properties have enrolled notable success [12,17,19] the understanding of their AC properties remains insufficient and impedes advance in their microwave applications. Recent studies of homogeneously disordered superconducting thin films [20, 21] have revealed a discrepancy between the local density of states measured by scanning tunneling microscopy and the density of states obtained from microwave experiments. This means that the model, which assumes uniform properties of the

films fails to be applied near Ioffe-Regel limit, kf l 1, which gives rise to a call for ≈ an unified model for measurements in strongly disordered superconducting films. Another type of disordered superconductors are granular superconductors, which have different quite different superconducting properties. DC experiments reveal that their transition to superconducting states becomes smeared for decreased thickness, while their critical temperature keep constant. Moreover, the tunneling of Cooper pairs can occur between individual grains, which can be analyzed by the electromagnetic response of microwave resonators.

13 Aims of the dissertation:

Tuning the MoC deposition process in order to obtain samples with enhanced • sheet resistance R and optimal critical temperature Tc

Characterization of the superconducting properties of prepared MoC thin ﬁlms • by means of transport, tunneling and high-frequency measurement techniques

Implementation of the nanostructures in prepared thin ﬁlms and characteri- • zation of their behavior

14 2 Theory

In this chapter we introduce the theoretical background of our research. We will go through basic mechanism of superconductivity, proposed by Bardeen, Cooper and Schrieffer [23] to a response of superconductor to electromagnetic field perturbation. We will explain in details the complex conductivity dependence on temperature and frequency of the external electromagnetic field, described by Mattis-Bardeen theory [24]. Then we will describe a behaviour of superconducting structures as tunnel junctions (SIS and NIN) and superconducting quantum interference devices (SQUIDs) also in interaction with EM field in superconducting coplanar waveguides (CPW) and resonators. We will show that the quality factor and the resonant frequency is determined by the complex conductivity as we will experimentally analyse in Sec. 4.6.1. Some theoretical relationships and formulations in this chapter were taken from the author’s rigorous thesis [64] and the written part of the dissertation exam [65].

2.1 Superconductivity

In 1908 H. Kamerlingh-Onnes succeeded in the liquefaction of helium, which allowed him to cool a material to the temperature of 4.2 K at standard pressure. It opened a new way for research in the field of low temperature physics. Several measurements of the temperature dependence of resistance of different materials were performed. In good conductors such as platinum, gold or copper the resistance was decreasing with decreasing temperature, but it was saturated at finite residual value. After choosing mercury as the measured material in 1911, the resistance dropped to zero below 4.2 K (later called as ”critical temperature”) This effect was called Superconductivity and it was measured in lead, tin, niobium and many other materials. For that discovery and for other research in the field of low temperature physics Kamerlingh-Onnes received widespread recognition, including the 1913 Nobel Prize in Physics.

15 2.1.1 Phenomenological theories

The experimental evidence of superconductivity attracted huge attention of theoretical physicists. The first phenomenological explanation was published in 1935 by brothers Fritz and Heinz London, shortly after the evidence that the superconductors, besides their zero resistance completely repel an applied magnetic field (below critical magnetic field, exceeding which breaks the superconducting state). This theory is called London theory [25] which describes a current in a superconductor by single equation: 1 j = 2 A, (2.1) −µ0λ where j is the surface density of superconducting current, A is the vector potential of the magnetic field and λL is the London penetration depth of the magnetic field. This equation is often written as two London equations, which can be obtained by applying time derivation and rotation:

d 1 j = E, dt µ λ2 0 (2.2) 1 j = 2 B, ∇ × −µ0λ where E is electric intensity and B magnetic induction. The first equation shows that for the dc current there is zero electrical field in the superconductor, i.e. the infinite conductivity (zero resistance) of the material. The second equation describes the penetration of the magnetic field, which is suppressed in depth λL,which means that the magnetic field is expelled by a superconductor. This phenomenon is called Meissner effect after physicist Walther Meisner, who discovered it in 1933 by measuring the magnetic field distribution outside of superconducting tin and lead samples. The generalization of the London theory was brought by V. L. Ginzburg and L. D. Landau. They phenomenologically describe superconductivity at temperatures close to the critical temperature. The theory is well-known as the Ginzburg-Landau theory (GL theory) and it is based on the Landau theory of phase transitions [26]. They introduce a complex order parameter Ψ, which is nonzero in superconducting state. Later it was shown that there is an analogy with a complex wave function described by microscopic BCS theory [23].

16 2.1.2 BCS Theory

Phenomenological theories are based on classical electrodynamics broadened by knowledge from thermodynamics and quantum mechanics. They give us a reasonable description of superconducting phenomena, but they don’t explain the mechanism of the superconducting state on a microscopic level. In 1957 John Bardeen, Leon Neil Cooper, and John Robert Schrieﬀer published the microscopic theory, latter called as the BCS theory. The theory introduced a ﬁnite attraction between electrons due to interaction with lattice vibrations (phonons). This leads to the pair- ing of electrons (so-called Cooper pairs) with binding energy denoted as 2∆. The pairs are the particles that carry the superconducting current. The resultant spin of the Cooper pairs is zero, meaning that they became bosons instead of fermions and the Pauli exclusion principle is not valid for them anymore. Since the energy of Cooper pair is lower than energy of two unpaired electrons, the normal state becomes unstable, which leads to condensation of all the charge carriers into a single quantum-mechanical state in the ∆ vicinity of the Fermi level. This leads to zero losses for direct current and ideal diamagnetism in a superconductor if perturbation energies are lower than energy gap 2∆. The Cooper pairs can be described by one macroscopic wave function ψ(r) = ψ(r) eiϕ(r), (2.3) | | which is analogous to the order parameter in the GL theory. It represent the amplitude of probability of the state with the Cooper pair being occupied, and its square

2 represents the concentration of Cooper pairs in the point r, thus Ψ(r) = ns. | |

2.1.3 Mattis-Bardeen Theory

In the previous sections we explained why the superconductor has zero resistance for direct current and why it completely shields external direct magnetic field. However if an external alternating electromagnetic field with frequency ω is applied to the superconductor, it causes a perturbation of the superconducting state. Photons with frequency ω > 2∆ can break the Cooper pair to single electrons. The resulting fluid of charge carriers is a mixture of normal quasiparticles and the Cooper pairs. Therefore, there are two parallel channels for the current flowing through the sample. Thus, we can define the complex conductivity σ = σ1 + iσ2, where the real

17 part σ1 represents the contribution of the quasiparticles and the imaginary part σ2 is the conductance of the Cooper pairs. The Cooper pairs can be also break by temperature ﬂuctuations. The temperature pair-breaking lead to a suppression of the energy gap parameter and to a decrease of real and imaginary part of the complex conductivity. To summarize, the complex conductivity is frequency and temperature dependent. The complex conductivity, derived by Mattis and Bardeen [24] in 1958, reads: σ1 iσ2 I(ω, T ) − = , (2.4) σN πi~ω − where σN is the normal state conductivity and I(ω, T ) is the following integral:

f() f(0) I(ω, T ) = L(ω, , 0) − dd0, (2.5) − 0 ZZ − where f is the Fermi-Dirac distribution function and L(ω, , 0) is the energy function from second-order perturbation theory, which is deﬁned in (4.22) in BCS [23] and extended for alternating ﬁeld with angular frequency ω as follows:

2 0 0 1 E + ~(ω is) + [(∆ + )/E] L(ω, , ) = (1 2f(E)) − −2 − E02 [E + ~(ω is)]2 − − E ~(ω is) + [(∆2 + 0)/E] + − − . (2.6) E02 [E + ~(ω is)]2 − − where E = √2 + ∆2 is the excitation energy, ∆ is the superconducting energy gap and the scattering parameter s = 1/τ could be taken as an inverted value of the phenomenological relaxation time. However, Mattis and Bardeen derived the ﬁnal formulas for complex conductivity in the limit s 0 in following form: →

∞ −∆ σ1 2 1 = [f(E) f(E+~ω)]g1(E)dE+ [1 2f(E+~ω)]g1(E)dE (2.7) σ ω − ω − n ~ Z∆ ~ Z∆−~ω

∆ σ2 1 = [f(E) f(E + ~ω)]g2(E)dE, (2.8) σ ω − n ~ Zmax(∆−~ω,−∆) where

∆2 g = ig = 1 + N (E)N (E + ω) (2.9) 1 2 E(E + ω) S S ~ ~

and NS is the electron density of states, deﬁned as:

18 E NS(E) = . (2.10) (E2 ∆2) − The second term in eq. 2.7 equals zerop for frequencies ~ω < 2∆, when the lower limit of the integral in eq. 2.8 is ∆ ~ω instead of ∆. Using these formulas, we can − − model temperature and frequency dependence of the complex conductivity. As we will see in the experimental part of this work, these formulas are only good enough for conventional superconductors. If the disorder in a superconductor increases, the limit s 0 is not valid anymore. →

2.1.4 Electron tunneling

Now we will explain a quantum mechanical process, when the electrons are transported through the potential barrier, despite the fact that they have not enough energy to overcome the barrier in classical sense. Since electrons are not classical particles at all, there is a ﬁnite probability of their tunneling through the barrier. According to foundations of quantum mechanics, this probability falls exponentially with the increase of the barrier energy and its density is expressed as:

2√2m x2 T = T0exp U(x) Edx , (2.11) − ~ x1 − ! Z p where T0 is constant depending on group velocity of the quasiparticles, m and E are the quasiparticle mass and energy and U(x) is the height of the barrier between x1 and x2. A good example of electron tunneling in a real physical system is an assembly of two electrodes separated by a dielectric barrier. This technique was firstly designed by Giaever [27] in 1960, who used it to confirm the temperature dependence of the density of states and the energy gap predicted by BCS. Let us assume a tunneling junction created by two electrodes with densities of states N1(E) and N2(E) with a finite voltage biased between them. The quasiparticles are tunneling from the filled states on one side of the barrier to the empty states on the other side and their distribution is defined by Fermi-Dirac function f(E). The current between the electrodes can be expressed as a subtraction between the current flowing from the first electrode to the second and the reverse one as follows:

19 ∞ 2 I = A T N1(E)N2(E + eV )[f(E) f(E + eV )]dE, (2.12) | | − Z−∞ where A is a constant of proportionality. If the electrode is in normal state (metal), we can assume its density of state as constant, since the changes in the vicinity of the Fermi energy ( 100 meV) are much smaller than the usual value ≈ of the Fermi energy itself ( 1 eV). If the electrode is in superconducting state, ≈ a direct energy dependence of the density of states has to be taken into account. Hence we can assume that the tunneling current, which is described by Volt-Ampere (VA) characteristics of the junction, directly depends on the superconducting density of states. In ﬁgure 2.1, we show the VA characteristics of three basic tunneling junctions: metal-metal (NIN), metal-superconductor (NIS) and superconductor- superconductor (SIS).

Figure 2.1: Densities of states and the corresponding VA characteristics of NIN (a), NIS (b) and SIS (c) tunnel junctions [28]

Zero current at T = 0 in the NIS and SIS junctions below the voltage V = ∆/e and V = (∆1 + ∆2)/e respectively means, that there are no empty states on the grounded electrode at this energy range. At ﬁnite temperatures, thermal excitations cause the existence of empty states, so the VA characteristics are smeared and above the superconducting critical temperature they have Ohmic behaviour as in NIN case. From experimental point of view, the most interesting case is NIS tunnel junc-

20 tion. In order to directly compare the experimental data of the superconducting density of states with theoretical prediction, it is convenient to derive the normalized diﬀerential conductance. This can be done by diﬀerentiation of the eq. 2.12, which gives us the following expression (Eq. 2.13):

∂I(V ) ∂ ∞ GNS = = GNN f(E eV )N(E)dE ∂V ∂V − Z−∞ (2.13) G ∞ NS T=→0 δ(E eV )N(E)dE = N(eV ), G − NN Z−∞ The first derivative of the Fermi function is a bell-shaped function, which in the limit T 0 becomes delta function. Therefore, the normalized differential → conductance in this limit is directly equal to the superconducting density of states. Such a measurement can be performed by a technique called scanning tunneling microscopy (STM) described later (see section 3.2.1). Finite temperatures lead to thermal smearing of width 2kT in energy (See Fig. 2.2). Using these experimental ± data and the defined temperature, one can estimate the temperature dependence of the energy gap.

Figure 2.2: Diﬀerential conductance of NIS tunnel junction for T = 0 (solid line) and T > 0 (dashed line) [14]

2.1.5 Superconducting quantum interference device (SQUID)

Quantization of ﬂuxoid

Now we will consider a closed superconducting ring in nonzero external magnetic ﬁeld. The uniqueness of the wave function in the whole superconductor implies

21 a condition, which is that an integral of the phase gradient ϕ around a closed curve ∇ is a multiple of 2π, thus ϕdl = 2kπ, where k is an integer. The gradient of the ∇ phase can be described inH terms of the canonical momentum ~ ϕ = mvs + qsA and ∇ from the condition we get:

m 2e ~ ϕ = j dl + Adl, (2.14) ∇ 2en I s~ I ~ I where Adl = 2πΦ is the total magnetic flux, j = 2ensv s is the current density, m is the massH of the charge carrier and 2e = qs is the charge of the Cooper pair. If we consider the integration curve deep in the superconductor, where the magnetic field is completely shield and j = 0, we will get the simple equation for quantization of the magnetic flux in superconducting ring:

Φ = kΦ0, (2.15)

~ −15 2 where Φ0 = 2.0678 10 Tm is the elementary quantum of magnetic ﬂux. 2e ≈ ×

Josephson eﬀect

If two superconducting electrodes are separated by a very thin dielectric barrier (thinner than Ginzburg-Landau coherence length), the wave functions of Cooper pairs in electrodes overlap (see 2.3). It means that there is a nonzero probability of the charge carriers being present, so a superconducting current can flow through the barrier, even if there is zero voltage across a tunnel junction. This effect is called dc Josephson effect [29]. The magnitude of the current, which flows through the junction is defined by the first Josephson equation:

j = jc sin (∆ϕ), (2.16)

where jc is the critical current for the particular Josephson junction and ∆ϕ is the phase difference between the superconducting wave functions of the electrodes. In the presence of an external electromagnetic potential A one should define a gauge invariant phase difference:

2π 2 δϕ = ϕ1 ϕ2 Adl, (2.17) − − Φ 0 Z1

22 At finite voltage applied on the electrodes the phase difference and therefore the current flowing through the barrier are time dependent satisfying the second Josephson equation: d∆ϕ 2π = V (2.18) dt Φ0

Figure 2.3: Josephson junction

RF-SQUID

If the superconducting ring is interrupted by the Josephson junctions , the quantum interference of the wave function can be controlled by external magnetic field which penetrates into the ring through the junctions. Such device is called superconducting quantum interference device. If the ring is interrupted by only one junction, we are talking about a single-channel interference device so-called RF-SQUID. By using the condition of uniqueness of the macroscopic wave function ϕdl = 2kπ and the definition of gauge invariant phase difference 2.17, we get ∇ Hthe following formula: ∆ϕ = 2πn φ, (2.19) − where φ = 2π Φ is the normalized magnetic flux in the ring. The current, which is Φ0 circulating in the RF-SQUID is then:

Φ Ij = Ic sin(∆ϕ) = Ic sin(2π ). (2.20) − Φ0

By using the fact that the total magnetic flux Φ in the ring is the sum of external magnetic flux Φe and the magnetic flux created by shielding currents Φi =

LIj we get the following transcendental equation:

φe = φ + β sin(φ), (2.21)

23 where β = 2π LIc is the normalized inductance also called as shielding parameter, Φ0 which deﬁnes the properties of the SQUID. If β > 1, the dependence φ(φe) becomes hysteretic as we can see in ﬁgure 2.4.

Figure 2.4: Internal magnetic flux dependence on external magnetic flux for different values of normalized inductance β

The total energy of an RF squid is deﬁned by following formula [30]:

U βi2 = cos(φe βi(φe)), (2.22) EJ 2 − −

Φ0Ic where i = I/Ic is the normalized superconducting current in the squid and Ej = 2π is Josephson energy. By minimizing U with respect to i for given φe one can calculate

U(φe) and i(φe) (See ﬁgure 2.5). For small parameter β the dependence i(φe) is sinusoidal according to ﬁrst Josephson equation (2.20).

Figure 2.5: External magnetic ﬂux dependence of the normalized potential energy in hysteretic regime (left) and normalized supercurrent in non-hysteretic (sinusoidal) regime (right)

The properties of an RF SQUID can be measured by means of the coplanar waveguide resonators described in the chapter 2.2. If the RF SQUID is coupled to the resonator, the phase shift between the driving current and the voltage on the

24 resonator can be written as follows [30]:

βi0 tan α = k2Q (2.23) 1 + βi0 where k is the coupling constant between SQUID and resonator, Q is the quality

0 0 factor of the resonator and i = Ij/Ic is the derivative of the normalized supercurrent in the SQUID with respect to the total magnetic flux. From the phase shift one can calculate the current flowing in the SQUID as a function of external and even internal magnetic flux. We will use this technique in chapter 4 by fitting the detuning of the resonant frequency measured on a magnesium diboride (MgB2) sample.

2.1.6 Disordered superconductors

In classical metals the behaviour of free electrons is described by the Drude- Sommerfeld model [31]. The electron scattering, which leads to finite resistance is caused by thermal vibrations of the atomic lattice. If disorder is included in the system, additional elastic scattering is present and the electrical conductivity is defined by the well known Drude formula: σ = ne2τ/m, where n is the density of charge carriers, m is the electron mass and τ is the time period between two collisions. The electron-electron interactions are neglected in this case. The average distance between two collisions is called mean free path and it is defined as l = vF τ, 2 1/2 where vF = ~kF /m is Fermi velocity, kF = 2π/λF = (2mEF /~ ) is Fermi wave number and EF is Fermi energy. If disorder increases, the mean free path decreases until it is comparable with lattice constant a. At this point the electrons in the system are strongly localised and the system becomes insulating. The key parameter of this transition is kF l 1, which is called the Ioffe-Regel criterion [22]. Near the ≈ transition, the system behaves like a two-dimensional one, for which the charge

2 carrier density is deﬁned as n = kF /(2π). By applying the Drude formula, we can express the minimal conductance of the disordered system as follows:

ne2τ e2 e2 1 σ2D = = kF l = = (2.24) m 2π~ h 25.82 kΩ The microscopic explanation of the electron localisation is generally recognized as Anderson localisation [32]. Electron-electron interaction in such system was ﬁrst introduced by Altshuler and Aronov [33], who theoretically showed that it leads to

25 logarithmic decrease of the electron density of states. If disorder is included in the superconducting system, the superconductor- insulator transition (SIT) can be observed. The disorder can be represented by chemical amorphous impurities, reducing the thickness of the sample to the coherence length level, applying a magnetic ﬁeld or any other reduction of charge carrier density. By increasing the disorder control parameter, the superconducting critical temperature TC decreases, while the sheet resistance increases. The critical transi- 2 tion sheet resistance is RC = h/(2e) 6.5 kΩ (four times smaller, than in the case ≈ of standard metals, because of 2e charge of Cooper pairs), above which the superconducting state is not possible anymore. For illustration, we show a SIT driven by reducing the samples thickness observed by Ketterson and Lee [34] on molybdenum carbide (MoC) thin ﬁlms (see Fig. 2.6).

Figure 2.6: SIT of MoC thin ﬁlms prepared and analysed by Ketterson and Lee [34]. (a) R(T ) characteristics for ﬁlm thicknesses (7.0-200 A),˚ (b) temperature and thickness zoom to the point of SIT

SIT phase diagram

The phase transition between two quantum phases driven by varying physical parameters (not temperature) is called quantum phase transition. Compared to classical phase transition, it can occur at absolute zero. One of such transitions is the above discussed SIT. The quantum wave functions of electrons are localised, when the material is in its insulating state, as opposed to its superconducting state. However, real experiments are performed at ﬁnite temperatures, where the thermal

26 excitations aﬀect the resultant behaviour of the sample. A detailed overview of SIT was published by Gantmakher and Dolgopolov [35]. There are several theoretically predicted mechanisms of SIT. In ﬁg. 2.7 we show three basic mechanisms described by their phase diagrams.

Figure 2.7: Phase diagrams of transition between three quantum phases: superconductor (S), metal (M) and insulator (I). Quantum transitions - dots on x-axis, thermodynamic transitions - solid lines, boundaries between regions - dashed lines, virtual boundaries - dotted lines

Fig. 2.7a shows a continuous transition from the insulating to the metallic and then to the superconducting state (so called fermionic scenario). The transition can occur directly from the superconducting to the insulating state by localisation of the Cooper pairs (bosonic scenario), which is showed in ﬁg. 2.7b for low temperatures. In the third case, which was proposed by Larkin [35,36] the transition from metallic to insulating state is impossible at the temperature range, where the sample is superconducting.

2.2 Coplanar waveguide (CPW) resonator

In order to observe the quantum properties of a nano-structure, high quality superconducting thin films should be used. Their properties can be studied by measurement of the resonant frequency and quality factor of the resonator patterned on the superconducting thin film. The structure of the quantum device can be created directly in the resonator or at a very small distance from it. One of the most common type are CPW resonators with distributed parameters. The coplanar waveguide was firstly designed by Cheng P. Wen in 1969. It is a type of transmission line formed by a microstrip center conductor and two grounded planes besides

27 the center conductor placed on a dielectric substrate (see ﬁg 2.8). It has a few ad- vantages compared to standard twin lead transmission line. Radiation of energy to the environment is strongly reduced, so they are more suitable for transmission of signals with high frequency (microwaves). They can be easily prepared by standard lithographic methods with high precision enabling to match the input and output impedance of the waveguide.

Figure 2.8: Proﬁle of coplanar waveguide without/with grounded plane (a/b): 1 - dielectric substrate, 2 - superconductor, 3 - grounded plane

The most important application of coplanar waveguide is the coplanar resonator which is created by interruption of the waveguide on the input and output with a capacitance gap. This type of electromagnetic resonator has distributed parameters L and C and suppressed parasitic resonances in comparison with the lumped element resonators (One can avoid it by reducing the dimensions of the lumped elements as we will discuss in the next chapter).

2.2.1 Quality factor and resonant frequency

As we mentioned in the previous chapter, CPW resonators can be used for very precise measurements of the superconducting properties of the material, or the quantum properties of the artiﬁcially patterned structures. Two basic measurable parameters of a resonator are the quality factor and the resonant frequency. They can be expressed by parameters of distributed RLC circuits (see Fig. (2.9 a)) using the parameters Ll, Cl and Rl representing the distributed parameters per unit length of the resonator. Impedance of such scheme of transmission line is deﬁned as [37]:

1 + i tan(βl) tanh(αl) Z0 ZTL = Z0 π , (2.25) tanh(αl) + tan(βl) ≈ αl + i (ω ωn) ω0 −

28 where Z0 = √LlCl is the characteristic impedance, β = ω0/vph is the phase constant and, ωn is the resonant frequency of n-th mode and α = Rl/(2Z0) is the attenuation constant. In the second equation of 2.25 we made an approximation for ω ωn. In ≈ the frequency area close to the resonance we can model the CPW resonator with a parallel resonant circuit with lumped elements R, L, C (see ﬁg. 2.9 b), where Ck is the capacitance of the input and output capacitive gap. Impedance of the parallel resonant circuit can be expressed as:

1 1 −1 R ZRLC = + iωC + . (2.26) iωL R ≈ 1 + 2iRC(ω ω0) − In order to model the CPW resonator with a parallel RLC circuit the impedances 2.25 and 2.26 have to be matched, from what we get the following relations:

2L l C l Z L = l ,C = l ,R = 0 . (2.27) n n2π2 2 αl

Figure 2.9: Analogical schemes for coplanar waveguide resonator: (a)impedance scheme with distributed parameters per unit length (b)parallel resonant circuit scheme (c)Norton equivalent scheme

The quality factor and the resonant frequency of the internal circuit can be now deﬁned by the distributed parameters of the CPW resonator:

C nπ 1 nπ Qint = R = ωnRC = , ωn = = . (2.28) r L 2αl √LC l√LlCl

The measured (loaded) quality and resonant frequency are shifted because of the capacitances Ck and the resistances RL of the CPW input/output. In order to express the ﬁnal formulas, the series connection of Ck and RL can be transformed into a Norton equivalent parallel connection of resistor R∗ and capacitor C∗ (see Fig. 2.9 c) [38] with:

2 2 2 ∗ 1 + ωnCk RL ∗ Ck R = 2 2 ,C = 2 2 2 . (2.29) ωnCk RL ωnCk RL

29 For the symmetric input and output coupling the loaded quality and the resonant frequency is expressed as:

∗ ∗ ∗ C + 2C ∗ C + 2C ∗ 1 QL = ω ∗ ω ∗ , ωn = ∗ ωn, (2.30) 1/R + 2/R ≈ 1/R + 2/R Ln(C + 2C ) ≈ where we used an approximation for small capacitor Ck. The loaded quality factor can be also expressed as a parallel combination of the internal (2.28) and external quality factors as: 1 1 1 = + , (2.31) QL Qint Qext where 1 Q = ω R∗C. (2.32) ext 2 n From the expression 2.30 we can see that for weakly coupled resonators (under- coupled regime), the loaded quality is primary deﬁned by Qint, which can reach the values over 106 for some materials. This kind of resonator can conserve a photon for a long period of time and can be used as ”quantum memory”. On the other hand the resonators in an over-coupled regime are suitable for quick detection of a photon.

2.2.2 Distributed parameters per unit length

In order to calculate the exact values of the quality factor and the resonant frequency we need to determine the distributed parameters of the CPW resonator, which appear in 2.25. The capacitance per unit length Cl is deﬁned by geometry of CPW as [38]: K(k0) Cl = 40ef 0 , (2.33) K(k0) where 0 is the vacuum permittivity and ef is the eﬀective relative permittivity, which reduces to the standard relative permittivity r as:

ef = 1 + q(r 1), (2.34) − where q is the ﬁlling factor given by:

0 1 K(k1) K(k0) q = 0 , (2.35) 2 K(k1) K(k0)

30 K in the above formulas is the complete elliptic integral of the ﬁrst kind with the arguments: W a 0 2 k0 = = k = 1 k , W + 2S b 0 − 0 q (2.36)

sinh(πS/4h) 0 2 k1 = k = 1 k sinh[π(S + 2W )/4h] 1 − 1 q where a, b, W and S are dimensions of the CPW as denoted on ﬁgure 2.10.

Figure 2.10: Cross-section of CPW resonator with marked dimensions

The inductance per unit length Ll is deﬁned as the sum of the geometric and kinetic inductance (Ll = Lgeom + Lkin). The geometric one is deﬁned similarly to the capacitance by the geometry of CPW as:

µ0 K(k0) Lgeom = 0 (2.37) 4 K(k0)

The kinetic inductance Lkin is an inductive contribution due to the kinetic energy of the charge carriers, which is non-negligible when the thickness of the thin ﬁlm is comparable to the London penetration depth λ. For simple superconducting wire it can be calculated by equating the total kinetic energy of the Cooper pairs with an equivalent inductive energy:

1 1 (2m v2)(n lA) = L I2, (2.38) 2 e s s 2 k

where 2me is the mass of electron, vs is the average velocity of Cooper pairs, ns is their concentration, l is the length of the wire, A is the cross-section area and

I = 2evsnsA is the total current (e is the electron charge). The kinetic inductance is then deﬁned as: m l L = . (2.39) kin 2n e2 A s For CPW resonators the calculation of the kinetic inductance is more complicated. There were several analytic expressions published [39–41] for diﬀerent approxima-

31 tions depending on CPW dimensions. For our case the most appropriate expression was [39]:

C 1.7 0.4 Lkin = µ0λ + , (2.40) 4ADK(k) (sinh(t/2λ) [(B/A)2 1][1 (B/D)2]) − − p where A, B, C, D are constants deﬁned by the dimensions of the CPW:

t 1 2t 2 S2 A = + + S2 B = −π 2 π 4A s (2.41)

t t 2 2t C = B + + W 2 D = + C. − π π π s The third distributed parameter of the CPW resonator is the resistance per unit length Rl. As we mentioned when discussing the impedance of CPW, (eq. 2.25) it is related to the attenuation constant α as:

Rl = 2αZ0, (2.42)

where Z0 = Ll/Cl is the characteristic impedance of the CPW. The attenuation constant is deﬁnedp as the sum of the dielectric losses αd and the conductor losses

αc. They can be derived by using the conformal mapping method [42] as follows. Dielectric losses: π r αd = q tan δ, (2.43) λ0 √ef where λ0 is the free space wavelenght and tan δ is the loss tangent speciﬁc for each dielectric. Conductor losses:

2 Rsmb 1 2a (b a) 1 2b (b a) αc = 2 2 2 ln − + ln − (2.44) 16Z0[K (k0)](b a ) a x b + a b x (b + a) − where x is the stopping distance and Rsm is the modiﬁed surface impedance of the strip line, with thickness t deﬁned as

cot(k t) + cos(k t) R = ωµt Imag c c , (2.45) sm k t c

32 where ω is the frequency of the applied electromagnetic field, mu is the permeability of the material and kc is the complex wave number, which is for superconductors defined by the London penetration depth λ and the skin-effect penetration depths

δse as: 1 2 1 2 k2 = + 2i . (2.46) c λ δ se The penetration depths can be calculated from the complex conductivity by the well-known formulas:

1 2 λ = and δ = . (2.47) µωσ se µωσ r 1 r 2

In order to avoid problems with the boundaries of some approximations used in the derived formulas for the distributed parameters of CPW resonator one can use a software for simulating the electromagnetic ﬁelds applied on 3D structures (such as FastHenry, FastCap, Sonnet, AWR Design Enviroment, etc.). In chapter 4, which considers the measured results analysis we used the Sonnet software to determine the quality factor and the resonant frequency and the FastHenry software for numerical calculation of the inductance and the resistance per unit length.

33 3 Experiment

In this chapter we will explain experimental methods used in our experiments. At the beginning we will describe, how the superconducting samples were prepared. After preparation we had to analyse their properties such as thin ﬁlm thickness, surface roughness, stoichiometry and the atomic structure, so we will describe the methods which were used in order to perform such analysis. In the end of this chapter we will show the particular assemblies, which gave us the experimental results analysed by stated theories. Description of some experimental methods in this chapter were taken from the author’s rigorous thesis [64] and the written part of the dissertation exam [65].

3.1 Sample preparation

The preparation technique of thin film on dielectric substrate is generally called thin-film deposition. There are two broad categories of deposition, depending on whether the process is primarily chemical or physical. We are going to focus on two types of physical deposition. The first is called magnetron sputtering process, which we used for MoC samples preparation. The second process, which we used for MgB2 sample preparation is called thermal vapor deposition.

3.1.1 Magnetron sputtering

Sputtering system (see figure 3.1) consists of a vacuum chamber with a cascade system formed of rotary and turbo-molecular pump filled with working gases, deposited material target connected to the magnetron, and sample holder, on which the substrates are placed. The sample holder heater is connected to a DC source in order to heat the samples during annealing process. The MoC thin films are deposited by reactive sputtering from the molybdenum target in the mixture of acetylene and argon. The molybdenum reacts with carbon contained in the acetylene working gas which leads to the forming of MoC on the substrates placed on sample holder.

34 Figure 3.1: Assembly of reactive magnetron sputtering process in vacuum chamber

In our case we used sapphire substrates, which were cleaned from protective resist with acetone, isopropyl alcohol and H2O respectively in an ultrasound bath, for two minutes. After that, they were placed on the sample holder in the vacuum chamber. The chamber was evacuated firstly by the rotary pump to a pressure 10−2 mbar, and then by the turbo-molecular pump (Pfeiffer TMH 261) ∼ to 10−7 mbar. After that we annealed the substrates to de-pollute them of the ∼ residual gasses and liquids. Therefore we gradually increased the temperature of the samples to 500◦C by the ceramic heating body (boralectric heater, Momentive Performance Materials Quartz GmbH) as a part of the sample holder powered by DC current 3 A and voltage 24 V and maintained the temperature for 1 hour. The actual temperature of the samples was measured by calibrated resistive thermometer PT100. After the annealing process was finished, we reduced the temperature below 100◦C and filled the chamber with working gasses - acetylene 2.6 and argon 5.0. The partial pressures of the gasses were regulated with high precision flow controllers OMEGA FMA 763 and OMEGA FMA 760 to their precise values (argon: 5 10−3 mbar, acetylene:2.2 10−4 mbar). The substrate temperature was then × × stabilized at 100◦C and the thin films were sputtered until the desired thickness was reached. The thickness of sputtered film is proportional to the sputtering time and

35 the sputtering rate was calibrated as 10nm/min.

3.1.2 Vapor deposition

Superconducting MgB2 thin ﬁlm was prepared by co-deposition of magnesium and boron from two separate sources, on a mirror-polished sapphire substrate and ex-situ annealing in vacuum chamber. The deposition chamber was evacuated to the limit vacuum 5 10−4 Pa. Resistive thermal evaporation and e-beam evaporation × were used to make a precursor of MgB2. Ex-situ annealing process was realized in a vacuum chamber evacuated to the base pressure of 1 10−3 Pa and consecutively × ﬁlled with Ar up to working pressure of 700 Pa. The annealing temperature was 800 ◦C.

The resonators were patterned on the MgB2 thin ﬁlm with thickness of 300 nm by optical lithography using a 2.5 µm thick layer of positive tone resist AZ 6624 and by reactive ion etching in Ar and SF6 plasma.

3.1.3 Optical lithography

The structures of CPW resonators are created on the deposited thin films coated with light-sensitive photoresist. A geometric pattern of the resonator is pro- jected to the thin film by UV light through an optical mask. The lighted photoresist is then removed by exposing of the film. There are two basic types of optical lithography - positive and negative. In the case of the positive lithography the lighted areas are removed as opposed to the negative lithography, where the lighted areas are left and the rest of the thin film is removed (figure 3.2). A special case of optical lithography is so-called Lift-off method, where the structure is created by the photoresist on the substrate and the thin film is then deposited on the pattern.

36 Figure 3.2: Optical lithography principle

The exposed areas of the thin film are removed by the etching process. In our case we used dry ion-etching. The samples are placed on a rotary holder in a vacuum chamber (see figure 3.3). An electromagnetic field creates plasma with high-energy ions, which vertically impact to the film surface and react on the places without photoresist. The process is finished when the etching depth reaches the dielectric surface. For precise detection of the ending time of the dry etching process one can use the detecting system designed in [43].

Figure 3.3: Ion etching assembly: left - vacuum chamber, right - sample holder

3.2 Measurement apparatus

3.2.1 Scanning tunneling microscope (STM)

STM was invented by Gerd Binning and Heinrich Roher in IBM laboratory in Zurich, who earned a Nobel Prize in Physics in 1986 for their invention. The purpose of the STM is the observation of the surface of the conductive materials

37 with resolution at atomic level. In the STM an atomic pointed tip is brought very close to the sample surface. When a small voltage is applied between the tip and sample, a tunneling current, depending exponentially on the distance between the electrodes, is detected. There are two basic regimes of using STM: the constant current and the constant height regime. In the constant height regime, the vertical position of the tip is fixed and slight changes in current are observed, while the tip is moving above the sample horizontally. This measurement is fast, but there is a risk of tip destruction, so it is appropriate to be performed only through a small horizontal range on a very flat sample surface. In the second regime, the tunneling current is keeping constant by feedback to vertical movement of the tip so that distance between tip and sample is constant and feedback signal is proportionate to the surface profile. By horizontal scanning the topography image is obtained. This regime is much slower and the precision (dependent on the feedback settings) is lower. One has to be very careful to avoid non-conductive impurities on the sample surface, because they are ”invisible” to the microscope and could destroy the tip. When the position of the tip is fixed the volt-ampere characteristic of tunnel junction can be measured. The differential conductance taken at very low temperatures reflects corrections to normal density of states cased by many-body effects. The most complex measurement which could be performed on STM is the the so called CITS (Current Imaging Tunneling Spectroscopy). It measures the volt- ampere spectroscopy at every point on a defined surface and in combination with topography, it gives us information on the material properties inhomogeneities. In the case of superconductors, it could provide for example an image of the Abricosov vortices lattice or the superconducting energy gap mapping.

Home made STM head

The head of the STM was constructed at Centre of Low Temperature Physics in Koˇsicein collaboration with UAM Madrid. The principle of operation is demonstrated in ﬁgure below 3.4.

38 (a) (b)

Figure 3.4: Head of the STM: (a) scheme, (b) photo (The pictures are presented by the courtesy of the research group from Koˇsice)

The rough movement of the golden tip and the sample holder is provided by many-layers piezoelectric crystals. The range of the tip and the sample holder movement is 3 mm and 15 mm respectively. The position of the sample holder is controlled by capacitive sensor with precision 10 µm. Usually there are three samples placed in the head before the experiment. One of the samples is always pure gold, which ensures the in-situ remodelling of the tip, if it is accidentally damaged and no longer atomically sharp. The ﬁne movement of the tip is realised by a piezo-tube, with four separated electrodes.

Cryogenic assembly for STM

The STM head is placed in commercial JANIS 3He wet refrigerator and a thermal link is provided by copper wires. Near the sample holder the 8T superconducting magnet is situated and the whole assembly is placed in a 4He cryostat (See Fig. 3.5).

Figure 3.5: Scheme of wet 3He refrigerator: 1 - sample holder, 2 - 1 K bath, 3 - 4He tank, 4 - superconducting magnet, 5 - vacuum housing, 6 - cryosorb pump, 7 - liquid level gauge, 8 - 3He tank, 5

39 When the whole assembly is ready for cooling, the cryostat is evacuated by rotary pump and ﬁlled with liquid helium 4He (volume is usually 60 l) with temperature 4.2K. The the cryosorb pump has to by warmed up to 42 K in order ∼ to release all 3He from the active carbon. Simultaneously the 1 K bath is cooled down by evacuating the 4He vapours by rotary pump trough the capillary. 3He starts to condensate into 3He chamber. After the condensation is complete, the cryosorb pump is cooled down to 4 K (by means of the thermal contact with the 4He capillary). At this temperature the active carbon in cryosorb pump starts to absorb the 3He vapours, which rapidly decreases the pressure of 3He and ensures to reach the temperature 300 mK. In ideal case, this temperature can be maintained for up to three days.

3.2.2 Cryogenic microwave assembly

The microwave measurements of CPW resonators have been performed in the 3He refrigerator HelioxAC-V from the company Oxford. The cooling below 3 K is performed by a double-stage pulse pump [44, 45], so it is not necessary to use liquid nitrogen or helium to maintain this temperature. The refrigerator has base temperature 300 mK and cooling power 100µW [46]. Temperature of the ﬁrst and ∼ second stage are 40 K and 3 K respectively. After cooling down to the second 3 K stage, the lowest temperature is achieved by the evacuation of the 3He vapors by a cryosorb pump from the stage with condensed 3He.

Figure 3.6: Cryogenic measurement assembly of network analyser and current source connected to a sample placed in refrigerator

The sample placed in the refrigerator is connected to a network analyser (Agi- lent Technologies E3062A, 300kHz - 3GHz, ENA Series), which drives the resonator with microwave signal over a wide frequency range (see ﬁgure 3.6). In order to achieve low eﬀective temperature of the sample it is necessary to isolate the sig-

40 nals from thermal and electromagnetic noises. Input lines are therefore made of rigid coaxial cables with outer and center conductors made of stainless steel. Due to their low thermal conductivity and relatively high attenuation they decrease the thermal noises coming from parts of the assembly placed at room temperature [47]. In addition the thermalisation of the signals can be performed by implementation of microwave powder ﬁlters [43] and a set of attenuators anchored to the individual stages of refrigerator. The isolation of the sample on the base stage from the noises coming out of the ampliﬁers is performed by connecting the microwave isolator [37]. The output of our measurement performed by the network analyser is the transmission characteristics described by a transmission matrix (eq. 3.1):

M11 M12 1 Zin t11 t12 1 Zout = , (3.1)         M21 M22 0 1 t21 t22 0 1         where Zin/out = 1/iωCk is input/output impedance of the system and tij are the transmission coeﬃcients deﬁned by:

t11 = cosh γl; t12 = Z0 sinh γl; t21 = (1/Z0) sinh γl; t22 = cosh γl, (3.2) where γ = α + iβ is the wave propagation coeﬃcient in the transmission line α is the attenuation constant, and β = ω0/vph is the phase constant deﬁned by the environment properties. The transmission of the signal can also be described by the

S21 parameter, which is deﬁned as a ratio of the input and output voltage and its expression through the transmission matrix coeﬃcients is [38]:

Vin 2 S21 = = , (3.3) Vout M11 + M12/RL + M21RL + M22 where RL is the real part of the load impedance, accounting for outer circuit components. By ﬁtting the measured transmission characteristics with eq. 3.3 we can get the measured values of the resonant frequency fn of n-th mode and the quality factor QL. The measurement were performed in the temperature range from the base temperature of the refrigerator 300mK to the critical temperature of the su- ∼ perconductor. Final output of our measurement are temperature dependences of QL and fn, from which we can obtain the temperature dependence of the quasiparticle lifetime as shown in the next chapter 4.

41 4 Results analysis

In this chapter we analyse the results obtained in the experiments described in the previous chapter. Firstly we describe the optimisation of MoC thin film deposition, which produced films with high sheet resistance and critical temperature. We study transport properties of thin films with different thicknesses, which give us a better understanding of the behavior of our samples near the SIT. We show that the transition has fermionic scenario and corroborate this statement by STM measurements showing that the ratio between superconducting energy gap and critical temperature does not depend on the film thickness, while nonelastic scattering rapidly increases as thicknesses decreases below 10 nm. We focus on 3 nm MoC thin films on silicon and sapphire substrates, which are still in superconducting state, but very close to the SIT. Transport measurements in magnetic field imply, that there is significant electron-electron interaction described by Altshuler-Aronov theory. In the end, we will show the results of terahertz spectroscopy of our samples and analyze them with the same model as in the microwave experiment. We discuss the possible nonequilibrium superconducting effects, which should be reflected in our experiments as well as in the experiments performed by other groups on the thin films of similar materials.

4.1 Tuning the MoC deposition process

As mentioned earlier, the appropriate samples for our purposes are thin ﬁlms close to the superconducting-insulator transition. In practice, they are ones which are still in superconducting state, but with a sheet resistance as high as possible. By tuning and analyzing the magnetron deposition process, we calibrate the procedure in order to obtain reproducible samples with well deﬁned appropriate parameters.

4.1.1 Magnetron discharge analysis

In section 3.1.1 we generally described the reactive magnetron sputtering process. Our ﬁrst inspiration came from [34], where the SIT on MoC thin ﬁlms was

42 observed at thickness of 13 A.˚ For the best deposition conditions we need to be in stable power regime. The sputtering power is defined by the current and voltage of the magnetron plasma discharge at specific pressure in the vacuum chamber. In order to define best input parameters we measured the IV characteristics of the discharge at fixed argon pressure and different acetylene pressures (see 4.1).

Figure 4.1: IV characteristics of the magnetron plasma discharge for diﬀerent content of acetylene - the ﬂow-controller opened to 0, 10, 20, 30, 40 and 50% from bottom to top

The stable regime occurs, when the IV characteristic reaches its local minimum (The changes of the discharge power are negligible for small current or voltage ﬂuctuations). We decided to set the magnetron input current to 200mA, which is the most stable when the acetylene ﬂow-controller is opened to 40% and corresponds to the voltage of 340 V.

4.1.2 Carbon content inﬂuence

The carbon content is tuned by the acetylene partial pressure. It defines the stoichiometry and the atomic structure of the resulting MoC thin film. In order to find the best deposition conditions for desired films we needed to prepare several series of films for different proportions of acetylene in the working gas keeping film thickness constant. Film thickness was estimated from the sputtering rate calibration. In order to define it, we sputtered a testing thin film on silicon substrate with a lift-off lithography structure (see Fig. 4.2a) for exactly 3 minutes. After chemical etching the structure was measured by atomic force microscope (AFM) to get the information about the film thickness. Fig. 4.2b shows that the resulting film thickness was 30 nm, which implies a sputtering rate 10 nm / min.

43 Figure 4.2: a) A silicon substrate for the lift-oﬀ lithography, b) AFM measurement of height proﬁle, c) AFM topography

Besides information on the film thickness, AFM provides us with the topography of the film surface, showing the roughness to be below 1.5 nm (see 4.2c). These results are promising towards the use of the thin films for patterning of the CPW resonator with embedded nanostructures. The roughness measurement was confirmed by X-ray reflectivity (XRR) for different film thicknesses (See the section below 4.2). Since the sputtering rate was calibrated, we prepared several 10 nm thin films at different acetylene partial pressure and measured their RT-characteristics. The sheet resistance R and critical temperature Tc were evaluated as a function of the acetylene pressure. The results can be seen in Fig. 4.3.

(a) (b)

Figure 4.3: Acetylene partial pressure (carbon content) dependence of the sheet resistance (a) and critical temperature (b)

−4 In order to maximize Tc, the optimal acetylene pressure is 3.0 10 mbar, × which corresponds to the acetylene flow-controller opened at 40%. Thin films produced above of 5.4 10−4 mbar were not superconducting, limiting the sheet resis- × tance for 10 nm superconducting film to 420 Ω. Under the same conditions, we prepared a 30 nm thick sample set for X-ray

44 diﬀraction measurement, from which we determined the atomic and crystallographic structure. The results are shown in Fig. 4.4.

Figure 4.4: XRD measurements of samples prepared with diﬀerent acetylene content - ﬂow controller opened to 0, 10, 20, 30, 40, 50 and 60% from bottom to top

The bottom curve corresponds to a pure molybdenum film deposited in argon atmosphere without acetylene. From bottom to top, the content of acetylene increases and the top curve represents a film prepared with the acetylene flow controller opened to 60 %. According to comparison with the XRD image database the relative positions of the peaks are associated with cubic crystallographic structure (δ-phase). The peaks with the largest amount of carbon are significantly smeared, which implies that the major part of the film surface is amorphous. The optimal

Tc was achieved for acetylene content corresponding to 40%-opened ﬂow controller (brown curve in Fig. 4.4). From absolute values of the peak positions, we determined the atomic lattice constant a = 4.2 A.˚ According to the database, the lattice constant for MoC with pure stoichiometry 1:1 is expected to be a = 4.273 A.˚ In detailed study in Ref. [66] it is explained that the change in carbon concentration from MoC to MoC0.75 leads to a decrease of volume of the elementary atomic unit by 5%, corresponding to 1.7% decrease of the lattice constant. This completely agrees with our deviation of 0.07 nm. We used VESTA 3D visualization program for structural models to obtain image of the MoC cubic structure. In Fig. 4.5, we compare the simulated cubic structure with a surface image obtained by STM topography measurement. The results clearly agree with the view of the structure in plane 111.

45 Figure 4.5: a),b) cubic structure obtained by simulation in VESTA software, c) STM topography in atomary level demonstrating the cubic phase on 111 plane

Since the STM is not precisely calibrated to define the exact values of length, the lattice constant couldn’t be precisely measured. However, the qualitative surface image of the atomic structure in conjunction with the XRD results gives us sufficient evidence that the crystallographic phase of our sample is of cubic (NaCl like) type. The stoichiometry was also analyzed by energy dispersive X-ray (EDX) method. This method requires a much thicker sample, so we prepared one 1000 nm film. The results of the measurement are shown in Fig. 4.6.

Figure 4.6: EDX measurement and corresponding table showing the content of individual elements in 1 µm MoC thin ﬁlm

The atomic ratio of carbon and molybdenum is approximately 60:40 in this case. The increased proportion of carbon is possibly caused by amorphous carbon islands among crystalline grains of MoC. More details about MoC thin ﬁlm deposition and the optimization of deposition parameters are described in the publication attached below.

Applied Surface Science 312 (2014) 216–219

Contents lists available at ScienceDirect

Applied Surface Science

journal homepage: www.elsevier.com/locate/apsusc

Superconducting MoC thin ﬁlms with enhanced sheet resistance

a,∗ a a a b c a

M. Trgala , M. Zemliˇ ckaˇ , P. Neilinger , M. Rehák , M. Leporis , S.ˇ Gaziˇ , J. Gregusˇ ,

a a c a

T. Plecenik , T. Roch , E. Dobrockaˇ , M. Grajcar

Department of Experimental Physics, Comenius University, SK-84248 Bratislava, Slovakia

Department of Nanotechnology, Biont, a.s., SK-84229 Bratislava, Slovakia

Institute of Electrical Engineering, Slovak Academy of Sciences, SK-84104 Bratislava, Slovakia

r t a b

i s

c l t r

e i n f o a c t

Article history: In this paper we describe a process of MoC superconducting thin ﬁlms preparation by reactive magnetron

Received 1 February 2014

sputtering in argon–acetylene atmosphere. The deposition process was optimized to achieve very smooth

Received in revised form 5 May 2014

superconducting thin ﬁlms with high sheet resistance. We present the results of four-point resistance

Accepted 27 May 2014

measurements in cryostat cooled down to 300 mK. The roughness was measured by the atomic force

Available online 4 June 2014

microscope (AFM). The results are so far promising and enable us to pattern superconducting coplanar

waveguide resonator with embedded nanostructures.

Keywords:

Critical temperature

Sheet resistance

1. Introduction clear whether the decoherence will be suppressed as expected

by authors in [4]. By measurement of the quality factor of the

Recent experiments have shown that a superconducting high CPW resonator made from the same material as phase slip qubit,

quality coplanar waveguide (CPW) resonator can be coherently one can study dissipative processes which inﬂuence decoherence

coupled to a superconducting quantum two-level system [1]. The time. Therefore we have designed and simulated CPW resonator

resonator can be used as (i) a quantum bus for superconducting build on MoC thin layer with resonance frequency of 2.5 GHz and

qubits, (ii) a dispersive qubit read-out and (iii) a coupling element estimated quality 34,000. With decreasing thickness there is a

for individual qubits. CPW resonators have several advantageous superconductor–insulator transition of MoC thin ﬁlm due to Ander-

properties. Their distributed elements construction allows better son localization [5]. Measurement of the losses in superconducting

microwave properties than lumped element resonators which suf- materials nearby insulator state can provide valuable information

fer from uncontrolled stray inductances and capacitances. They can whether such materials are suitable for quantum devices.

be strongly coupled to superconducting qubits with coupling ener-

gies which cannot be achieved for atoms in optical resonators. On

the other hand, intrinsic quality factor is limited to 10 which is 2. Fabrication of MoC thin ﬁlms

much lower than that for optical resonators. There is experimen-

tal evidence that the quality factor is limited by two level systems Samples with MoC ﬁlms were prepared by reactive magnetron

located in the surface layer of the CPW [2,3]. It was shown empiri- sputtering. Apparatus consists of deposition chamber, magnetron

cally that residual surface resistance of superconducting resonators and Mo target powered by stabilized dc source, vacuum system

scales as square root of speciﬁc resistivity in normal state. Recently, cascade with rotary vacuum pump and turbo-molecular vacuum

a new type of superconducting qubit, so-called phase slip qubit, pump. The Mo target with 2.00 diameter and 0.125 thickness

has been proposed [4]. Since the quantum phase slip can only be with purity 99.95% was used. The rotary pump produced pressure

−3

observed in materials close to the superconductor–insulator tran- at about 7 10 mbar. Turbo-molecular vacuum pump of Pfeif-

sition, which exhibits relatively high speciﬁc resistivity, it is not fer TMH 261 type was used to achieve high vacuum parameters.

Pressure was measured with commercial vacuum meters Pfeiffer

Vacuum PKK 251 and D 356. The chambers ultimate vacuum was

−7

∗ 1 × 10 mbar. Sputtering was realized in mixture of argon 5.0 and

Corresponding author. Tel.: +421 260295264.

acetylene 2.6. Regulation of gas ﬂow was achieved by high precision

E-mail addresses: [email protected],

[email protected] (M. Trgala). ﬂow controllers OMEGA FMA 763 and OMEGA FMA 760. Sample

http://dx.doi.org/10.1016/j.apsusc.2014.05.200

M. Trgala et al. / Applied Surface Science 312 (2014) 216–219 217

Fig. 1. Temperature dependence of sheet resistance for various thicknesses of MoC

−5 Fig. 2. Critical temperature of MoC thin ﬁlms prepared at acetylene pressure

thin ﬁlms prepared at acetylene pressure 7 × 10 mbar.

−5

7 × 10 mbar as a function of thickness.

holder made from ceramic heater (boralectric heater) allowed us

◦

regulation of sample temperature up to 1200 C. Calibrated type K

thermocouple was used as thermometer. The samples of MoC lay-

ers were prepared on single crystal C-cut sapphire substrate. The

substrate was cleaned before deposition in an ultrasonic washing

machine with acetone, isopropylalcohol and distilled water. After

the cleaning process the samples were placed on the sample holder

−6

and the chamber was pumped down to pressure of 1 × 10 mbar

◦

and the sample holder was heated to 500 C for 60 min to degasify

the sample surface. In next step the sample holder temperature was

decreased to room temperature and partial pressure of acetylene

−5 −4 −3

and Ar gas was set to 7 × 10 –7 × 10 mbar and 5.4 × 10 mbar,

respectively. During the sputtering process, temperature of the

◦

substrates was set to 200 C and discharge current was stabilized

to 200 mA. The speciﬁc substrate temperature was chosen because

of electrical continuity of MoC ﬁlms down to 0.4 nm deposited on

Fig. 3. RTG analysis of MoC ﬁlms using Bragg–Brentano method. Local maxima at

sapphire substrates. Deposition rate was optimized and best results ◦

39 corresponds to MoC layer.

were obtained for the rate 9 nm/min. Deposition time was chosen

to be in range 33–133 s which gives thickness of the sputtered MoC

−4

at acetylene partial pressure 3.0 × 10 mbar and the highest sheet

thin ﬁlms from 5 nm to 20 nm.

resistance 420 with superconducting phenomena still present

−4

was obtained at acetylene partial pressure 5.4 × 10 mbar (Fig. 5).

3. Results

To obtain the information about thickness, mass density and rough-

ness we have used X-ray reﬂectivity measurement. The reﬂectivity

The deposition process was optimized in order to achieve

curves were analysed using software LEPTOS 3.04 provided by

smooth thin ﬁlms with high critical temperature and high sheet

Bruker Company. Theoretical curves were calculated for a simple

resistance. Our system included needle valves but for better repro-

model comprising the substrate, thin interlayer (2 nm thick) and

ducibility and better control over roughness and Tc of the samples

we have decided to replace needle valves by high precision ﬂow

controllers [6].

We started our work for acetylene partial pressure similar to

those mentioned in [5]. We have produced samples with thick-

nesses 20 nm, 15 nm, 10 nm and 5 nm. In order to achieve similar

deposition conditions for a set of samples all parameters are the

same as given in Section 2 except acetylene partial pressure which

is taken as optimizing parameter. Four-point resistance measure-

ment was used to obtain superconducting critical temperature Tc

and sheet resistance R . The transport measurements were done in

cryostat cooled down to 300 mK. Results are shown in Figs. 1 and 2.

Highest sheet resistance 600 was obtained for the 5 nm sample,

as it is expected. We have used Bragg–Brentano method to clarify if

the material deposited by the described sputtering process is pure

MoC (Fig. 3). We decided to optimize the Tc on ﬁlms with thickness

10 nm suitable for nanostructure pattering. The deposition time for

10 nm thin ﬁlms was 66 s.

The critical temperature T changes rapidly with concentration

c Fig. 4. Critical temperature of MoC thin ﬁlm with thickness 10 nm as a function of

of carbide in MoC thin ﬁlms (Fig. 4). The highest Tc was obtained acetylene pressure in the deposition chamber.

218 M. Trgala et al. / Applied Surface Science 312 (2014) 216–219

Table 1

X-ray reﬂectivity (XRR) measurements for various thicknesses of MoC thin ﬁlm.

Sample 20 nm 15 nm 10 nm 5 nm

Thickness (nm) 20.08 13.67 11.20 4.74

Density (g/cm ) 8.35 8.02 7.71 8.83

Roughness (nm) 1.00 0.78 1.25 1.26

Fig. 5. Sheet resistance of MoC thin ﬁlm with thickness 10 nm as a function of

acetylene pressure.

Fig. 8. 1 ␮m × 1 ␮m AFM topography of MoC thin ﬁlm. The AFM measurements

were done by Solver P47 Pro microscope by NT-MDT. All topography measurements

were done in tapping AFM mode using standard silicon probes with force constant

of 1.45–15.1 N/m. Scanning speed was in the range of 2–3 ␮m/s. The RMS (root

mean square) roughness of the surface calculated from the AFM topography data

was about 0.29 nm.

More detailed information on roughness of our samples

was obtained by atomic force microscopy. The results (Fig. 8)

corresponds to the results obtained from X-ray reﬂectivity mea-

surement.

Next step was to produce coplanar waveguide resonator on

Fig. 6. Results of X-ray reflectivity measurement (XRR) for 20 nm thick MoC film. prepared MoC films. This goal was successfully reached. We have

produced and measured half wavelength CPW resonator with

designed quality factor of 34,000 and resonant frequency of 2.5 GHz

for ideal lossless metal. Design of the resonator is shown in Fig. 9.

MoC layer. Density of both layers changed linearly with depth. Fit- Results (Fig. 10) show a decrease of quality factor and resonant fre-

ting parameters were the thicknesses, the densities of both layers quency with temperature measured for 10 nm thick MoC ﬁlm as it is

as well as the roughnesses (root mean square) of the two interfaces expected. The loaded quality factor and resonant frequency of CPW

and of the surface. Genetic algorithm was used for simultaneous resonator patterned on 10 nm thick MoC ﬁlm was approximately

optimization of the reﬁnable parameters. The decrease of intensity 10,000 and 1.43 GHz, respectively. The resonant frequency is lower

oscillations (Kiessig fringes) is affected mainly by the roughness of than designed 2.5 GHz because of enhanced kinetic inductance in

the upper surface. Results are shown in Table 1 and Figs. 6 and 7. very thin superconducting ﬁlms [7].

Fig. 7. Reflectivity profile of 20 nm thick MoC film obtained from X-ray reflectivity measurement (XRR). Measured data (black) compared with computed data (red) from the

model. (For interpretation of the references to color in text, the reader is referred to the web version of the article.)

M. Trgala et al. / Applied Surface Science 312 (2014) 216–219 219

4. Conclusions

We have prepared superconducting MoC thin ﬁlms with very

low average roughness <2.0 nm and our samples show critical tem-

perature and sheet resistance in the range from 4.1 K to 7.3 K and

from 100 to 600 , respectively. Our goal was the preparation of

smooth ﬁlms with high Tc and high sheet resistance. The results are

promising and elaborate technology should enable us to produce

superconducting structures which could exhibit quantum phenom-

ena. We have successfully measured CPW resonators produced on

our MoC ﬁlms. Small roughness of the MoC thin ﬁlms enables to

form nanostructures inside the resonator (Fig. 11). Such resonator

exhibits nonlinear behavior typical for Dufﬁng oscillator. This latest

results are under consideration for a forthcoming publication.

Acknowledgements

Fig. 9. Design of half wavelength CPW resonator with dimensions: C = 10 ␮m;

␮ ␮

W = 30 m; S = 50 m; t – thickness. The research for this paper has been supported by the European

Community’s Seventh Framework Programme (FP7/2007-2013)

under Grant No. 270843 (iQIT), the Slovak Research and Develop-

ment Agency under the contract APVV-0515-10 (former projects

No. VVCE-0058-07, APVV-0432-07), DO7RP-0032-11 and LPP- 0159-09.

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Fig. 10. The loaded quality factor QL and resonant frequency f0 of CPW resonator

[5] S.J. Lee, J.B. Ketterson, Critical sheet resistance for the suppression of supercon-

produced on 10 nm thick MoC ﬁlm as a function of temperature.

ductivity in thin Mo-C ﬁlms, Phys. Rev. Lett. 64 (1990) 25.

[6] J. Du, A.D.M. Charles, K.D. Petersson, Study of the surface morphology of Nb ﬁlms

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[7] M. Tinkham, G. McKay, Introduction to Superconductivity, second ed., Dover

Publications, New York, 1996.

Fig. 11. Scanning electron microscope picture of nanostructures patterned inside

the resonator. The detail of the weak link with length 323 nm and width 81 nm is

shown as the inset. 4.2 Transport properties

After the MoC thin films deposition was optimized, we decided to charac- terize the transport properties of prepared samples. Firstly, we measured the RT characteristics for samples of different thickness. From the results, we established the correlation between superconducting properties of the sample and the level of disorder, which gave us more detailed insight into phase transition between the superconducting and insulating state. Then we performed transport measurements in perpendicular direct magnetic field, from which we determined basic parameters such as upper critical field BC2, superconducting coherence length ξ0, diffusion coefficient D, Hall coefficient RH , concentration of charge carriers n or Ioffe-Regel disorder parameter kf l.

Thickness dependence

In order to verify whether the thickness estimation used in sample deposition corresponds to reality, we prepared four samples with thicknesses 5, 10, 15 and 20 nm and measured them using X-ray reflectivity (XRR) method, to obtain a the mass density and intensity profile, from which we can extract information about film thickness, density and roughness (see table 4.1).

Figure 4.7: Density and intensity proﬁle measured by XRR

The measured film thickness is in very good agreement with our prediction from the calibrated sputtering rate. The roughness measurement is consistent with AFM data (see Fig. 4.2) and the fact it is below 1.5 nm allows us to prepare structures with measurable kinetic inductance. In transport experiments, we first measured the temperature dependence of sheet resistance for different thicknesses without an applied magnetic field. The results are shown in Fig. 4.8a. The superconducting transitions are very sharp and it can be seen that with

51 Sample 20nm 15nm 10nm 5nm thickness [nm] 20.08 13.67 11.20 4.74 density [g/cm3] 8.35 8.02 7.71 8.83 roughness [nm] 1.00 0.78 1.25 1.26

Table 4.1: Results of X-ray reﬂectivity measurement (XRR) for various thickness MoC ﬁlms.

(a) (b)

Figure 4.8: RT characteristics for samples with thicknesses a) 30, 20, 15, 10, 10, 5, 5, 3 nm (from bottom to top) in zoomed window 0-8 K, b) 15, 10, 10, 5, 5 nm (from bottom to top) with temperature ranging from room temperature to superconducting transition

decreasing thickness, R increases and Tc decreases. This behavior is characteristic for homogeneously disordered superconductors, in contrast with granular superconductors, where Tc keeps its bulk value and the transition are broadened signiﬁcantly. In total, we measured 8 samples with thicknesses ranging from 3 nm to 30 nm. There are two 10 nm and two 5 nm samples showing diﬀerent R and Tc, which is caused by small changes in argon and acetylene pressures during the deposition process.

We can therefore conclude that R correlates with superconducting transition onset better than ﬁlm thickness.

The negative derivative dR/dT presented for all samples (ﬁg. 4.8b) strongly indicates quantum corrections [33] to Drude conductivity caused by enhanced electron- electron interaction. From the measured data, we extracted the thickness dependence of R and Tc. The theoretical variation of R is predicted by classical percolation theory with the assumption that the thickness is proportional to the areal occupation probability in the percolation problem [34]. The thickness proportion of

−1.3 sheet resistance from this prediction is R (t) (t tc) , where tc is minimum ∼ − thickness for electrical continuity. From the ﬁtting procedure (see ﬁg. 4.9a) we determined the value tc = 1.3 nm.

52 (a) (b)

Figure 4.9: Thickness dependence of a) sheet resistance (dots) ﬁtted with −1.3 R (t) (t tc) (dashed line), b) critical temperature (dots) ﬁtted with ∼ − Tc(t) = Tc0(1 tc/t) (dashed line) −

The thickness dependence of Tc (see fig. 4.9b) shows a strong decrease for thicknesses below 15 nm, which is close to double value of the coherence length ξ(0) 5.5 nm determined from the upper critical field, as obtained from the mag- ≈ netotransport measurements. It determines the range of superconducting attraction so the decrease of Tc is caused by surface effects. As a theoretical approach, one can take the calculations made by Simonin [63], which considers the decrease in density of states because of the surface term in GL free energy. The thickness dependence of critical temperature is then expressed as:

Tc(t) = Tc0(1 tc/t), (4.1) −

where Tc0 is the bulk critical temperature and tc is the critical thickness where the superconductivity is destroyed. From the ﬁt (Fig. 4.9b), we determined the parameters Tc0 = 8.5 K and tc = 2.5 nm. The resulting values of minimum thickness for electrical continuity and the superconducting critical thickness found in Ref. [34] are bit smaller: 0.4 nm and 1.3 nm respectively. Enhanced electron-electron interactions imply the fermionic model of SIT, which means that the transition from superconductor to insulator state goes through the metallic state. This model, which describes the dependence of Tc on R was developed by Finkelstein [61] as:

Tc 1 1/γ + r/4 √r/2 log = γ + log − , (4.2) Tc0 √2r 1/γ + r/4 + √r/2

53 2 2 where r = Re /(2π ~) is dimensionless resistance and γ is the fitting parameter, which defines the ratio of the smaller value out of either the diffusive energy

ε/τ (τ is the relaxation time in the normal state) or Thouless energy π2~D/t2 (D is the diﬀusion coeﬃcient) to the non-renormalized condensation energy kBTc0, where

Tc0 = 8.5 K. The equation was used to ﬁt the experimental data and the results are shown in Fig. 4.10

Figure 4.10: Critical temperature dependence on sheet resistance (dots) ﬁtted with Finkelstein formula (dashed line)

The best ﬁt was achieved for γ = 8.3 and Tc0 = 8.5 K. Similar result with γ = 8.2 was obtained in the original paper by Finkelstein for MoGe thin ﬁlms with

Tc0 = 7.2 K. The RT characteristic across the whole temperature range can be used to determine of the electron transport dimensionality. Standard Bloch–Grüneisentheorem describing the resistivity temperature dependence gives the positive temperature coefficient dR/dT > 0. The sign is changed if the quantum corrections ascribed to weak localization and electron-electron interactions (Altshuler-Aronov) are significant. The correction to the conductivity depends on sample dimensionality and follows the relationship:

A√T + BT p/2 for 3D WL AA ∆G + ∆G =  (4.3) G00A1 log(kBT τ/~) for 2D

2 2 where p (3/2, 2), G00 = e /(2π /hbar) and A, B, A1 are the normalization ∈ constants. We compared these predictions with the data measured on the MoC sample with 5 nm thickness in superconducting state as well as in normal state achieved by an applied magnetic ﬁeld of 8 T (above Bc2). On the presented graph (see Fig. 4.11) we can see that the curves overlap above 20 K and follow the 3D

54 power law. Under this temperature, the superconducting fluctuations start to play a role and the curves differ. The superconducting conductance starts to increase and diverges to infinity at Tc = 2.3 K, while the normal state conductance decreases logarithmically as in the 2D case.

Figure 4.11: Temperature dependence of conductance for 5 nm MoC sample measured at 500 mK and magnetic ﬁeld B=0 T (red dots) and B=8 T (black dots)

Finkelstein’s mechanism becomes relevant, when the samples effectively behave as 2D. This condition seems to be satisfied in our case for all samples at low temperatures, but the strength of the Tc suppression is given by the value of the constant γ. In order to analyze its value obtained from the fitting procedure, we performed magnetotransport and Hall-effect measurements.

Magnetotransport and Hall measurements

From the RT characteristic measurement at different applied magnetic fields, one can estimate the temperature dependence of the upper critical magnetic field

Bc2(T ). We performed such measurements and the results are shown in Fig. 4.12. We can estimate the value of the upper critical field at zero temperature by Werthamer-Helfand-Hohenberg theory [67] and then the superconducting coherence length and diffusion coefficient can be determined by using the relations:

dB B (0) = 0.69 c2 T (4.4) c2 dT c φ ξ(0) = 0 (4.5) s2πBc2(0) πk dB −1 D = 0.407 B c2 (4.6) e dT

55 (a) (b)

Figure 4.12: a) Temperature dependence of upper critical magnetic ﬁeld for samples with thicknesses 5, 5, 10, 10 and 15 nm (from left to right), b) RT characteristics of 10 nm MoC sample measured at magnetic ﬁelds 0 -8 T (from right to left)

The results are summarized in Tab. 4.2. The calculated Thouless energy is not much bigger that the condensation energy kTc, which is consistent with the fact, that the samples only behave as 2D at low temperatures. In this regime, when

2 2 ~/τ π ~D/d , the suppression of Tc given by Finkelstein’s relation is very slight, which is inconsistent with our measurements. Since the same argument is also valid for MoGe [61, 68] and TiN [69–71] thin films, an alternative way to describe Tc suppression with increasing disorder has to be found. Additionally we performed Hall-effect measurements at 200 K in fields ranging from -8 to 8 T. From the results we determined the Hall coefficient RH and charge carrier densities n = 1/eRH for all samples, which are summarized in Tab. 4.2. It − is surprising that the carrier density is almost constant at the value 1.7 1023 cm−3, × which is in contrast with the measurements on TiN thin films [71]. Therefore, the decrease of the thickness decreases mobility of quasiparticles in our samples.

11 23 2 2 t R Tc RH × 10 n × 10 kf l Bc2 ξ D π ~D/d −1 −3 2 [nm] [Ω] [K] [ΩmT ] [cm ] kf l [T] [nm] [cm /s] [K] 30 56 7.95 ------20 95 7.6 ------15 120 7.4 3.75 1.7 4.1 10.7 5.48 0.52 17.4 10 263 6.5 3.75 1.7 2.8 9.4 5.78 0.53 40 10 344 4.9 3.13 1.9 2 9.5 5.8 0.39 29.4 5 850 2.86 3.8 1.7 1.46 5.3 7.8 0.39 118 5 1100 2.3 3.8 1.7 1.34 5.3 7.8 0.33 100 3 1227 1.3 3.9 1.7 1.3 - - - -

Table 4.2: Basic parameters of MoC samples with diﬀerent thicknesses, obtained and calculated from results determined by magneto-transport measurements

56 The level of disorder can be quantiﬁed by the Ioﬀe-Regel parameter kf l, where kf is the Fermi wave number and l is the mean free path. The quantum phase transition is expected at kf l 1. From our measurements, we can estimate kfl in ≈ the free electron model as follows:

2 2/3 1/3 ~(3π ) RH kf l = 5/3 (4.7) e "Rd#

In Fig. 4.13 we show the dependence of critical temperature Tc and sheet conductance 1/R on this parameter. From linear extrapolation, we can see that the conductance reaches zero if kf l equals unity, which is in agreement with Ioﬀe- Regel criterion for phase transition to insulating state.

(a) (b)

Figure 4.13: kf l dependence of a) sheet conductance (dots) with linear ﬁt (dashed line), b) critical temperature (dots) ﬁtted with Fiory-Hebard prediction (dashed line)

In order to explain the kf l dependence of the critical temperature, we used the relation introduced by Fiory and Hebard [72]:

2 Tc (kf l)c = 1 2 : (4.8) Tc0 − (kf l)

From the ﬁt, we obtained Tc0 = 8.2 K and (kf l)c = 1.2, which is slightly larger than unity. From these results we can conclude, that the superconductivity is destroyed earlier than the insulating state is reached by increasing disorder in MoC. Therefore, the quantum phase transition from superconducting to insulating state has an intermediate metallic state as predicted by the fermionic scenario of SIT. In the following section, we examine this scenario by analyzing further superconducting properties of our samples by means of a scanning tunneling microscope (STM).

57 4.3 Tunneling measurements

The transport measurement results presented in the previous section pointed out, that the superconductor - insulator phase transition has its intermediate metallic state. This scenario is called fermionic, because the Cooper pairs are ﬁrst broken by increasing disorder to the conductive electrons, which are subsequently localized into the insulating state. The alternative scenario is bosonic, where the Cooper pairs survive with ﬁnite local ∆, but they become localized. The ratio 2∆/kBTc increases in the bosonic scenario, as opposed to fermionic, where it keeps its constant value. Such behavior can be observed by measuring the density of states using a scanning tunneling microscope (STM). We used STM for the measurement of samples with thickness 3, 5, 10 and 30 nm and the results are summarized in the attached paper. We showed that in

MoC thin films Tc and ∆ decrease together with increasing disorder, which proves the fermionic scenario of SIT. Moreover, the scattering parameter Γ , obtained by fitting the spectra with Dynes formula, is rapidly increasing as thickness decreases. Therefore, strong disorder leads to pair-breaking effects. Nevertheless, for thin films with t 5 nm, long range phase coherence was confirmed by mapping of the ≥ Abrikosov vortex lattice. Presence of the vortex lattice means that there are no strong phase fluctuations, which become important for 3 nm thin films.

58 PHYSICAL REVIEW B 93, 014505 (2016)

Fermionic scenario for the destruction of superconductivity in ultrathin MoC ﬁlms evidenced by STM measurements

1,* 1 1 1 2 2 3 1, P. Szabo,´ T. Samuely, V. Haskovˇ a,´ J. Kacmarˇ cˇ´ık, M. Zemliˇ cka,ˇ M. Grajcar, J. G. Rodrigo, and P. Samuely † 1Centre of Ultra Low Temperature Physics, Institute of Experimental Physics, Slovak Academy of Sciences, and P. J. Safˇ arik´ University, SK-04001 Kosice,ˇ Slovakia 2Department of Experimental Physics, Comenius University, SK-84248 Bratislava, Slovakia 3Departamento de F´ısica de la Materia Condensada, Laboratorio de Bajas Temperaturas, Instituto de Ciencia de Materiales, Nicolas´ Cabrera, Condensed Matter Physics Center, Universidad Autonoma´ de Madrid, E-28049 Madrid, Spain (Received 9 October 2015; revised manuscript received 14 December 2015; published 11 January 2016)

We use sub-Kelvin scanning tunneling spectroscopy to investigate the suppression of superconductivity in homogeneously disordered ultrathin MoC films. We observe that the superconducting state remains spatially homogeneous even on the films of 3-nm thickness. The vortex imaging suggests the global phase coherence in our films. Upon decreasing thickness, when the superconducting transition drops from 8.5 to 1.2 K, the

superconducting energy gap follows Tc perfectly. All this is pointing to a two-stage fermionic scenario of the superconductor-insulator transition (SIT) via a metallic state as an alternative to the direct bosonic SIT scenario with a Cooper-pair insulating state evidenced by the past decade STM experiments.

DOI: 10.1103/PhysRevB.93.014505

Superconductor-insulator transition (SIT) can be tuned in for the fermionic mechanism was provided even sooner by different ways, e.g., by increasing the physical or chemical the tunneling experiments on planar junctions on amorphous disorder, by the change in charge-carrier density, by a magnetic Bi and PbBi/Ge films [15,16] indicating that the Cooper pair field, etc. Decreasing the superconducting film thickness amplitude vanishes at SIT. But the unambiguousness of these down to several atomic layers is yet another possibility [1]. results was challenged by the fact that the spatially averaged Superconductivity is characterized by the order parameter = tunneling spectra were gapless. This gaplessness could be eiφ(r) with the amplitude and phase φ. Two fundamental explained by a very inhomogeneous gap distribution due to approaches describe SIT as a consequence of either the quasi- phase fluctuations. Then, superconducting pair correlations particle fluctuations affecting the amplitude or the phase might also exist in insulators close to SIT, contradicting the fluctuations of . The quasiparticle fluctuations correlate fermionic scenario. to the fermionic scenario [2,3]. There, disorder-enhanced Here, we present sub-Kelvin STM experiments on ultrathin Coulomb interaction breaks Cooper pairs into fermionic states superconducting MoC films with atomic spatial resolution. We leading to superconductor-(bad-) metal transition with → 0 demonstrate the spatial homogeneity of the superconducting at Tc → 0. At even higher disorder metal-insulator transition state for film thicknesses down to 3 nm and the closing of the (MIT) follows due to Anderson localization. The bosonic superconducting energy gap/order parameter as Tc vanishes, in scenario [4,5] assumes a direct SIT. On the superconducting agreement with the fermionic scenario. Thus, we bring direct side even in a homogeneously disordered system supercon- evidence on the local behavior of the superconducting order ducting “puddles” with variant emerge. In the insulating parameter in this class of the transition. state Cooper pairs with a finite amplitude are still present, The MoC films were prepared by the magnetron reactive but their global phase coherence is lost. A comprehensive sputtering from a Mo target in an argon acetylene atmosphere review on SIT as a quantum phase transition [6] concludes onto a sapphire c-cut substrate. The details can be found that many characteristics in both scenarios are very similar, and elsewhere [17]. The preparation followed the procedure of Lee probably only a probe directly sensitive to the local variations and Ketterson [18] who manufactured continuous MoC films of the superconducting energy gap/order parameter can discern down to 0.4-nm thickness showing the separatrix between the the realized mechanism. The scanning tunneling microscope superconducting and the insulating films at 1.3-nm thickness (STM) is a unique probe with such a capability. with a sheet resistance RS of about 2.8 k. The available STM experiments on thin films of TiN, InOx , The inset of Fig. 1(a) shows the resistive transitions near and NbN [7–13] bring evidence that upon increasing disorder, Tc on 30-, 10-, 5-, and 3-nm-thick MoC films. The gradual decreases more slowly than Tc, the variation of on a shift of Tc (defined at 50% of RS above the transition) from scale of the superconducting coherence length ξ increases, 7.3 to 6.7, 3.75, and 1.2 K, respectively, is accompanied a pseudogap appears above Tc in the tunneling spectra, the by an increase in RS from several tens of ohms to almost spectral coherence peaks are suppressed, and the vortex lattice 1400 . The transitions remain sharp while shifting to is fading out. This phenomenology strongly supports the lower temperatures strongly suggesting that our films are bosonic scenario and raises the question about the universality homogeneously disordered [19]. The main part of Fig. 1(a) of the bosonic SIT [14]. On the other hand, potential evidence presents the overall temperature dependence of RS from room temperature down to the superconducting transition showing a negative derivative dRS /dT for all the films. Such a behavior *[email protected] can be described within the framework of the quantum †[email protected] corrections [20,21] to the standard Drude conductivity for

a 15% increase in RS before the thinnest 3-nm MoC film goes superconducting. Figure 1(b) shows a dependence of Tc on RS (the latter measured at 300 K) for the films used in the STM measurements (red squares) supplemented by another series of MoC films (circles). The solid line is a fit to Finkelstein’s model [3] indicating termination of superconductivity at Rcr ≈ 2.5k close to the value determined in Ref. [18]. Magnetotransport and Hall-effect measurements allowed for the determination of the superconducting coherence length ξ which increases from 5 to 8 nm and the Ioffe-Regel product of kF l dropping from 5 to 1.3 when the thickness of the MoC FIG. 1. (a) Temperature dependence of the sheet resistance RS film is diminished from 30 to 3 nm. Notably the carrier density normalized to its value at T = 300 K for 3-, 5-, 10-, and 30-nm is very high n ≈ 1023 cm−3 and not changing [22]. MoC films. The inset: RS near Tc. (b)The critical temperature Tc as a The STM and scanning tunneling spectroscopy (STS) function of the sheet resistance RS for the measured films (squares) experiments were performed by means of a sub-Kelvin STM and other series (circles), and the solid line is Finkelstein’s fit. system developed in Kosiceˇ enabling measurements above T = 280 mK and magnetic fields up to B = 8 T. Prior to STM disordered metals (weak localization and electron-electron experiments a protective layer on the surface of the films has interactions). These corrections are rather weak bringing about been dissolved and the samples mounted within few minutes

FIG. 2. STM surface topography and locally measured tunneling spectra for 10-, 5-, and 3-nm MoC films (a)–(c), respectively. Top: STM surface topographies at 500 mK, zoom in 5-nm film shows atomic structure. Middle: 100 STS spectra along an ∼200-nm line on the surface of the respective film taken at 450 mK. Bottom: Temperature dependence of a typical tunneling spectrum at indicated temperatures. The insets of (a) and (b): Temperature dependence of the gap determined from fits to Dynes formula (points) and BCS-like (T) dependence (line).

014505-2 FERMIONIC SCENARIO FOR THE DESTRUCTION OF . . . PHYSICAL REVIEW B 93, 014505 (2016) to the 3He cryostat. Surface topography has been performed in the constant current mode. Conductance maps and vortex imaging have been studied via the current imaging tunneling spectroscopy mode [23] in a 128 × 128 grid. The surface topography measurements have been realized on all our thin films at tunneling resistances in the range of 50–100 M. The top panels of Fig. 2 show topographic images of 400 × 400 nm2 surfaces obtained on 10-, 5-, and 3-nm films (a)–(c), respectively at T = 450 mK. Our films show clean surfaces, revealing a compact polycrystalline structure with the lateral grain size of ∼15–30 nm. Given the sharp superconducting transitions the grain boundaries are very transparent. A rms roughness of the films was typically 0.6–0.7 nm in agreement with the topography obtained by atomic force microscope [22]. A zoom of a 3 × 2.5nm2 area is displayed in the topography of the 5-nm film (inset in the FIG. 3. (a) STM topography on a 5-nm MoC film showing topography image), showing atomically resolved surfaces with protruding grain as a white spot. (b) Gap map of the same area a distorted hexagonal crystal symmetry [24,25], which allow taken at T = 450 mK. (c) Profiles of topography (full curve) and gap for correlated topographic (STM) and spectroscopic (STS) map (dashed curve) along the gray lines in (a) and (b). (d) Vortex studies. It also proves that grains are single nanocrystals. image on a flat surface of the 5-nm MoC film at 450 mK in B = 1T The tunneling conductance curves G(V,x,y) = showing 21 vortices. dI(V,x,y)/dV vs V are calculated by numerical differentiation of the locally measured I-V characteristics. Since the metallic Au tip features a constant density of states at the measured bias energies, each of the background of the tunneling conductance at higher-bias differential conductance vs voltage spectra reflects the voltages is present. The latter finding indicates an appearance superconducting density of states (SDOS) of the MoC of the Altshuler-Aronov effect of suppression of density of films, smeared by ∼±2 kB T in energy at the respective states due to a disorder-enhanced Coulomb interaction [20]. temperature. Consequently, in the low-temperature limit, For the different particular film thicknesses we analyzed the differential conductance measures directly the SDOS. numerous series of spectra, such as the ones presented in Fig. 2. The three-dimensional plots shown in the middle panels of None of them show any characteristic length scale of variations Fig. 2 display three series of 100 tunneling spectra G(V,x,y) which would indicate emerging superconducting granularity, taken along 200–250-nm lines at T = 450 mK on the 10-, such as, e.g., in TiN [7]. On the contrary, the same variations 5-, and 3-nm MoC films. The spectra are normalized to are found independently on a bias voltage for spectra taken the conductance values GN at eV = 4 above the gap along lines with a step size of 2 nm (Fig. 2) or, e.g., 0.05 nm energies. In the 10-nm films, the spectra show a pronounced (not shown). In the conductance maps (recorded at zero bias superconducting structure with well-defined coherent peaks or at biases close to the gap value) on all studied MoC films, near the gap energies at V ∼±/e without any significant a variation with the same standard deviation of about 0.05 in spatial variation across the sample surface. Notably, we normalized conductance is found. As such, these variations always observed a finite conductance inside the gap region result from the noise of the apparatus rather than physical with a value at zero bias G(V = 0,x,y) of around 0.1 GN . properties of the superconducting films. This implies a high This is not compatible with a pure BCS SDOS considering the spatial homogeneity of superconductivity in MoC. thermal smearing at 450 mK. We address this phenomenon Figures 3(a)–3(c) present a surface analysis performed on below but note that the complex conductivity derived from our another 5-nm MoC film. In (a) the STM surface topography recent experimental transmission measurements on coplanar of a small area of 95 × 65 nm2 shows a protruding grain waveguide resonators made of 10-nm MoC film shows a finite (white spot). Figure 3(b) displays the gap map (bias voltage quasiparticle lifetime consistent with the existence of in-gap of coherence peaks) of the same area taken at T = 450 mK, quasiparticle states [26]. and (c) provides the topography (red curve—left scale) and The series taken on the 5-nm film reveals gaplike features the gap map (blue curve—right scale) along the gray lines in with coherent peaks located at a smaller bias (mean value (a) and (b). On the grain where the film is thicker a larger of the peak position is V = 0.88 mV with σ = 0.022 mV) as gap is found, notably in an area of size much larger than the expected for a sample with lower Tc, but the height of the peaks superconducting coherence length ξ ࣈ 8 nm. From these data is significantly smaller than in the case of the 10-nm film. it is evident that the only systematic change in the gap value in Moreover, the intensity of the in-gap states is substantially our MoC films is due to the change in their thicknesses and that increased to a value of G(V = 0,x,y) around 0.3GN .This the superconducting state remains spatially homogeneous on tendency continues on the 3-nm MoC film where the series all MoC films with constant thicknesses. This is very different of spectra features a reduced tunneling conductance around from the STS measurements on TiN, InOx , and NbN thin the zero-bias voltage and the in-gap states bring the value of films showing an emergent granularity in the superconducting G(V = 0,x,y) to around 0.7 GN and almost no signatures condensate upon increasing disorder on the scale of the of coherence peaks. Moreover, a quasilinearly increasing coherence length [7,9,12,13].

014505-3 P. SZABO´ et al. PHYSICAL REVIEW B 93, 014505 (2016)

Figure 3(d) shows a zero-bias conductance map obtained on the 5-nm film on a flat surface of 220 × 190 nm2 at T = 450 mK in a magnetic field of B = 1 T. A distorted vortex lattice is observed. At 1 T, the average intervortex distance for 1/2 the triangular Abrikosov lattice dv = 1.075(0/B) is about 50 nm, resulting in some 21 vortices for an area equivalent to the one in the figure, as we actually find in our measurement. We have observed the presence of vortices in MoC films down to 5 -nm thickness. In the case of 3 nm the low contrast between the superconducting and normal spectra prevents vortex imaging. The presence of vortices in MoC brings evidence for a global phase coherence in the samples allowing for supercurrent flow around the vortex cores. The bottom panels of Fig. 2 show the effect of temperature FIG. 4. (a) Typical tunneling conductances of the 3-, 5-, 10-, and = on the tunneling spectra normalized to the tunneling curves 30-nm thin MoC films at T 450 mK—solid lines. The symbols are fits to the thermally smeared Dynes formula. (b) Film thickness GN (V ) measured in the normal state above Tc. GN (V )of the 10- and 5-nm films were constant in the presented bias dependence of the superconducting energy gap (red solid symbols— voltage range whereas on the 3-nm film a weak V-shaped right scale) and of the critical temperature determined from tunneling experiment (open circles—left scale). (c) Film thickness dependence normal state is present fitting the background shown in of the Dynes smearing parameter normalized to the gap— / at Fig. 2(c), middle panel. The local T ’s have been established c T = 450 mK (open circles—left scale) and of the superconducting from the STS data as the temperatures where superconducting coupling strength 2/kB Tc (red solid diamonds—right scale). features disappear. Then, Tc’s are 6.7, and 3.7, and 1.2–1.25 K for the 10-, 5-, and 3-nm films, respectively, in agreement with the transport data (see Fig. 1). No indications of a pseudogap All this strongly suggests that superconductivity is suppressed featuring a reduced DOS above the bulk Tc have been observed. (Tc → 0) when → 0. We have measured five to ten temperature dependencies on all We can now compare our findings with the bosonic SIT our films taken at different locations, but no variations of the lo- phenomenology found in the STS experiments on TiN, InOx , cal Tc on a particular film have been observed. It again supports and NbN films [7–13]. In contrast to those experiments we the spatial homogeneity of the superconductivity of the MoC have found that upon increasing disorder, decreases in the films. same way as Tc, there are no spatial variations of , and no To fit our spectra with finite in-gap states the Dynes pseudogap above Tc appears. Also, the vortex lattice is present. modification of the BCS density of states N(E) = Re{E/(E − In the strongly disordered MoC films the spectral coherence )(1/2)} where the complex energy E = E + i with a peaks are suppressed but in a different way than, for example, smearing parameter has been taken into account [27]. The in InOx [10]. In MoC their suppression is correlated with temperature dependencies of have been determined from fits the increase in the in-gap states which can be quantitatively to the thermally smeared Dynes formula on the 10- and 5-nm accounted for by the increasing relative strength of the Dynes films. The large smearing effect in the 3-nm film prevented parameter normalized to the zero-temperature gap value /. reliable fits at higher temperatures. The resulting (T) Figure 4(c) shows a steep increase in this parameter from zero dependencies presented in the insets of the bottom panels of (30-nm film) to 0.9 for the 3-nm film. Fig. 2 follow the predictions of the BCS theory (solid lines). In As mentioned above, broadened SDOS have already been the fits is found to be temperature independent up to ∼0.5 Tc, observed on other films near SIT. Valles et al. found heavily whereas at higher temperatures it is increased up to ∼4 (0) broadened SDOS in planar tunnel junctions made on ultrathin at Tc. Bi film with Tc = 0.7K [15] but also on the series of The tunneling spectra at T = 450 mK (solid lines) together ultrathin PbBi/Ge films [16] where it was attributed to with the Dynes fits (open symbols) are resumed in Fig. 4(a). spatial variations of the amplitude of the order parameter. There we have also included the result for the 30-nm-thick In the PbBi films the broadening increases as Tc decreases MoC film which displays a hard spectral gap with a flat zero with maximum / ∼ 0.4forTc = 0.96 K. Our spatially tunneling conductance in a finite range of bias voltages inside homogeneous spectra clearly demonstrate that the origin of the the gap region. The fit provides (0) = 1.23 meV with = 0 broadening is not the uneven distribution of the gap amplitude. fully in line with the BCS theory. The observation of the hard Hence, although the source of the in-gap states remains unclear gap here proves that the spectral smearing on the other samples [28], our findings unequivocally corroborate the fermionic is not due to a lack of energy resolution. As sample thickness is scenario. decreased, the inferred gap/order parameter (0) is decreased The recent measurements of the Little-Park oscillations in in a sequence (0) = 1.1, 0.62, and 0.2 meV for the 10-, the magnetoresistance on uniform a-Bi films with nanohon- 5-, and 3-nm film thicknesses, respectively. As presented in eycomb arrays of holes also support the fermionic SIT [29]. Fig. 4(b) the thickness dependence of the superconducting gap Then, a question arises what physical parameter decides on the (red squares) perfectly follows the thickness dependence of insulating ground state upon the SIT, fermionic, or Cooper-pair the transition temperature Tc (circles). Figure 4(c) shows that insulating one. The MoC superconductors differ from TiN, the superconducting coupling ratio 2/kB Tc remains constant InOx , and NbN, for example, by the fact that the quantum and being equal to about 3.8 for all the studied MoC films. corrections to the resistivity are small and the charge-carrier

014505-4 FERMIONIC SCENARIO FOR THE DESTRUCTION OF . . . PHYSICAL REVIEW B 93, 014505 (2016) density is high and not changing upon increasing disorder. route of the SIT confirming that there are at least two different If these are important ingredients for fermionic scenario, it scenarios of SIT that can be realized depending on the physical remains for further studies. parameters of the systems. In conclusion, STM and STS studies on ultrathin MoC films provide evidence that, in contrast to TiN, InOx , and We gratefully acknowledge helpful conversations with NbN where the bosonic scenario of SIT is found in the M. Skvortsov, L. Ioffe, D. Roditchev, H. Suderow, and R. ultrathin MoC films the superconducting energy gap or order Hlubina. This work was supported by Projects No. APVV- parameter terminates ( → 0) as the bulk superconductivity 0605-14, No. APVV-0036-11, No. VEGA 2/0135/13, No. ceases with Tc → 0, the global superconducting coherence in VEGA 1/0409/15, No. EU ERDF-ITMS 26220120005, and MoC films is manifested by the presence of superconducting No. EU ERDF-ITMS 26220120047 and the COST action vortices, and most importantly, the superconducting state is MP1201 as well as by the U.S. Steel Kosice, s.r.o. (spolocnosˇ ´t very homogeneous for all the thicknesses down to 3 nm srucenˇ ´ım obmedzenym)´ J.G.R. also acknowledges support where the superconducting transition is suppressed from bulk from Projects No. FIS2014-54498-R (MINECO, Spain) and Tc = 8.5 K to 1.3 K and the strong disorder is characterized by No. P2013/MIT-3007 MAD2D-CM (Comunidad de Madrid, kF l close to unity. All these observations point to the fermionic Spain).

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[25] C. I. Sathish, Y. Guo, X. Wang, Y. Tsujimoto, J. Li, S. Zhang, [28] A possible source can be a subnanometer normal metallic layer Y. Matsushita, Y. Shi, H. Tian, H. Yang, J. Li, and K. Yamaura, at the substrate-MoC interface similar to the case of thin Nb ﬁlms

Superconducting and structural properties of δ-MoC0.681 cubic [C. Delacour, L. Ortega, M. Faucher, T. Crozes, T. Fournier, B. molybdenum carbide phase, J. Sol. State Chem. 196, 579 (2012). Pannetier, and V. Bouchiat, Persistence of superconductivity in [26] M. Zemliˇ cka,ˇ P. Neilinger, M. Trgala, M. Rehak,´ D. Manca, niobium ultrathin ﬁlms grown on R-plane sapphire, Phys. Rev. M. Grajcar, P. Szabo,´ P. Samuely, S.ˇ Gazi,U.Hˇ ubner,¨ V. B 83, 144504 (2011)], in MoC due to, e.g., uncompensated M. Vinokur, and E. Il’ichev, Finite quasiparticle lifetime in orbitals. disordered superconductors, Phys. Rev. B 92, 224506 (2015). [29] S. M. Hollen, G. E. Fernandes, J. M. Xu, and J. M. Valles, [27] R. C. Dynes, V.Narayanamurti, and J. P.Garno, Direct Measure- Collapse of the Cooper pair phase coherence length at a ment of Quasiparticle-Lifetime Broadening in a Strong-Coupled superconductor-to-insulator transition, Phys.Rev.B87, 054512 Superconductor, Phys. Rev. Lett. 41, 1509 (1978). (2013).

014505-6 4.4 Field induced SIT in 3 nm MoC ﬁlm

3 nm MoC thin films prepared on silicon (Si) and sapphire (Al2O3) substrate are close to SIT. As such, quantum phase transition to the normal state can also be achieved by increasing the external magnetic field above the upper critical field. We performed RT measurements at different values of applied magnetic field as well as the measurement of field dependence of the resistance at fixed temperatures. The results exhibited a rapid increase of the resistance for T 0 at B > Bc2. We showed, → that in order to explain this behavior, the superconducting fluctuations combined with Altshuler-Aronov conductivity corrections should be taken into account. Detailed measurements by STM under external magnetic field show, that the normal state reached by magnetic field has an atypical spectroscopic character in comparison with the normal state at B = 0 and T > Tc, where the constant density of states was measured. The deviation from the constant density of states is consistent with Altshuler-Aharonov logarithmic corrections caused by enhanced electron-electron interactions in disordered systems.

Magnetotransport analysis

In Fig. 4.14, the RT characteristics at different values of fixed magnetic field are shown for both samples. The sheet resistance is normalized to the value R (50 K) ≈ 1100Ω. The relative deviation in highest magnetic field B = 8 T between the resistance measured at lowest temperature and at 20 K is very small R(0.3 K)/R(20 K) ≈ 1.03, which suggests that the field induced transition is fermionic, same as the thickness induced transition.

(a) (b)

Figure 4.14: RT characteristics measured in magnetic ﬁeld 0-8 T on 3 nm MoC sample depositend on a) silicon, b) sapphire substrate

65 From the results, temperature dependence of the upper critical field was determined (See Fig. 4.15) for different levels of superconducting transition. We can see, that the fields taken just before the resistance reaches zero qualitatively behave according to Werthamer-Helfand-Hohenberg prediction [67]. However, the fields taken at the beginning of the transition has almost the opposite character, which, in extrapolation, would give its divergence at limit T 0. Such atypical → temperature dependence of Bc2 was discussed by Spivak in [73] as a consequence of mesoscopic fluctuations, when in mean field theory the superconducting solutions exist at arbitrary B.

(a) (b)

Figure 4.15: Extracted temperature dependence of upper critical ﬁeld Bc2 taken at 95%, 75%, 50%, 25%, 10% and 5% level of superconducting transition for sample on a) silicon, b) sapphire substrate

In both graphs of RT characteristics in Fig. 4.14, we can see that for higher fields, the resistance starts to rapidly increase when T 0. Moreover, on the film → deposited on silicon substrate, the strong reentrant can be seen before the resistance starts to increase. Such behavior can be explained by superconducting fluctuations in Cooper channel in 2D systems described by Galitzki and Larkin [74]. They analyzed three terms contributing to the conductivity: Aslamasov-Larkin (AL) term, which describes the conductivity of the fluctuating Cooper pairs; density of states (DOS), which decreases the conductance because of the lower number of electrons and the Maki-Thomson term, ascribed to the coherent scattering of normal electrons. They have shown, that all these three contributions have the same order, and they derived the final formula for the correction to conductivity:

4e2 r 3 δσ = log + Ψ(r) + 4(rΨ0(r) 1) , (4.9) 3πh − b − 2r − 66 where Ψ(x) is the logarithmic derivative of the Γ function, r = (1/2γ0)(b/t)

0 with γ = exp(γ) = 1.781, and t = T/Tc0 and b = (B Bc2(T ))/Bc2(0) are the − reduced temperature and the reduced magnetic ﬁeld.

(a) (b)

Figure 4.16: a) Galitzki-Larkin (GL) calculations of normalized resistance in diﬀerent magnetic ﬁeld b = B/Bc, b) GL + AA corrections

Fig. 4.16a shows the calculated curves for different values of magnetic field. We can see that the reentrant is observed, but in higher magnetic fields the resistance saturates. This is in contrast with our measurement, where the resistance exhibits a strong increase. As was discussed in detail by Gantmakher [35], the Althusler-Aronov term accounting for enhanced electron-electron interactions needs to be taken into account for highly disordered superconductors. Subsequently, the final relation takes the form:

2 −1 e T R (B,T ) = σ0 + δσ(B,T ) α log , (4.10) − h T ∗

where σ0 is the classical conductivity, δσ is the Galitzki-Larkin correction (Eq. 4.9) and the last term is the above mentioned Altshuler-Aronov correction with T ∗ being the temperature at which this term diminishes. In Fig. 4.16b the calculated curves are shown. We can see, that the combination of GL and AA corrections to the conductivity provides the desired temperature dependence to resistance. Further improvement can be made by taking into account the atypical dependence of upper critical ﬁeld Bc2, which could provide a far greater increase of the resistance and even better agreement with experiment. From the above, we can conclude that the normal state achieved by exceeding Bc2 much diﬀers from the normal state for

T > Tc. This behavior can be examined further by measuring the density of states by STM.

67 STM analysis

We have shown in Sec. 2.1.4 that the tunneling measurements provide information about the density of states, from which the superconducting energy gap ∆ and scattering parameter Γ can be determined. We performed the measurements at B = 0 and T =500 mK, on two samples deposited on silicon and sapphire substrate. The results were ﬁtted by standard Dynes formula and they are shown in Fig. 4.17.

Figure 4.17: Comparison of tunneling differential conductance at T 500 mK of 3 nm ≈ MoC film deposited on silicon (red) and sapphire (blue) substrate fitted with Dynes formula (solid lines). Obtained parameters were: ∆ =0.66 meV, 0.49 meV; 2∆/kTc =3.85, 3.85; Γ =0.18 meV=0.27∆; 0.2 meV=0.41∆ for film on silicon and sapphire substrate respectively

The thin film deposited on silicon substrate has larger ∆ and smaller scattering Γ than the film on sapphire substrate. This is consistent with the RT characteristics, which exhibit superconducting transition at 3.97 K and 2.95 K for the silicon and sapphire substrates respectively. Since both samples have the same normal state sheet resistance R 1100Ω and the energy gap ratio 2∆/kTc = 3.85, we can assume ≈ that the substrate influences the superconducting properties, which are better for the Si substrate. The interface between the substrate and the film could lead to possible pair breaking, which is stronger for the sapphire substrate. The topography by STM was made on both samples in order to examine the influence of the substrate on surface structure. The results displayed in Fig. 4.18 show the different structure of the samples. While in the film on sapphire substrate the individual grains are well recognized, the film on silicon substrate has interesting boomerang-shaped structure. This could imply the different influence of the substrate on the epitaxial growth of the film.

68 Figure 4.18: Surface topography taken by STM on 3 nm MoC ﬁlm on a) silicon, b) sapphire substrate

Further experiments were performed on the MoC thin ﬁlm on the silicon substrate. We measured the temperature dependence of the diﬀerential conductivity and the results are shown on graphs below (see Fig. 4.19).

Figure 4.19: Temperature dependence of the normalized diﬀerential conductance of 3 nm MoC thin ﬁlm on Si substrate measured by tunneling spectroscopy

On the 3D graphs we can see the decrease of the superconducting gap, which completely diminishes at Tc 4 K. Above this temperature, the background is not ≈ changing anymore and it is approximately constant. Several differential conductance curves were fitted with the Dynes formula, from which we obtained the temperature dependence of ∆ and Γ (see Fig. 4.20). Closing of the superconducting gap with increasing temperature is in agreement with the BCS prediction. The scattering parameter Γ increases with temperature, which was also found on thicker films presented in Sec. 4.3. Since the differential conductance is normalized to its value at normal state, it can be assumed, that the superconducting features of the sample scale with the film surface, as opposed to the normal state features. We verified this assumption by

69 Figure 4.20: Temperature dependence of Dynes scattering parameter Γ (red dots) and the superconducting energy gap ∆ (black dots) ﬁtted with BCS theory (solid line) comparing the ∆-map of the picked area of the sample with the topography image of the same area (See Fig. 4.21a,b).

Figure 4.21: Comparison of a) ∆-map and b) surface topography taken by STM at the same area of 3 nm MoC film on silicon substrate. c) Differential conductance profile taken along 100 nm line

The relative fluctuations of ∆ have the same order as the surface deviations. Fig. 4.21c shows that the fluctuations are sufficiently small to make a statement that the sample is superconducting homogeneous. Conductance mapping showed that the correlation between superconducting and surface properties completely diminishes when the voltage exceeds the superconducting energy gap. We can conclude that there are not remarkable superconducting features in the normal state.

STM analysis under external magnetic ﬁeld

Next, we focused on the density of states analysis by STM measurement of differential conductance under external magnetic field. As we mentioned earlier, the corrections to the conductivity caused by enhanced electron-electron interactions are characterized by Altshuler-Aronov (AA) effect [33], which has a logarithmic dependence in 2D disordered samples. For the density of states this correction has the following limits :

70 log(Eτ),T E τ −1 δN(E,T ) (4.11) ∼  log(T τ),E T τ −1  In very large energy windows the logarithmic background saturates to the constant value log(T τ) in limit E 0. We can assume the corrected density of → states as follows:

N(E) = Nn + N0 (1 + log(Eτ)) (4.12)

In STM we primarily measure the tunneling current in dependence on biased voltage. Taking the above formula 4.12, the analytic expression of the tunneling current and corresponding tunneling conductivity can be calculated:

∞ I(V ) = C N(E)[f(E) f(E + V )] − Z−∞ V I(V ) T=→0 CV N + N log n 0 τ I(V ) = C + C log(V ) (4.13) V 1 2

This formula can be used to describe the experimental results obtained at B=8 T and T =0.4 K in appropriate limits (see Fig. 4.22).

(a) (b)

Figure 4.22: Tunneling conductance I/V measured by STM (blue circles) at temperature T 400 mK and magnetic field B =8 T, fitted with the logarithmic voltage ≈ dependence (black solid line) for E T and the finite temperature saturation (red dashed line) for T E. a) X-axis in linear scale from -30 mV to 30 mV, b) X-axis for logarithm of voltages from 0 to 30 mV

Logarithmic scale 4.22b shows that the background has the expected loga-

71 rithmic dependence. The saturation at limit E 0 is much higher than it is → expected from the ambient temperature (0.4 K). As such, we can introduce an effective temperature, that represents the scattering effect which increases the zero bias conductance. The value of this parameter can be determined spectroscopically from the intersection of two asymptotic curves (Eq. 4.11). The obtained values are in range 0.2 - 0.5 meV, which agrees with the Dynes parameter Γ obtained at B = 0. One of the explanations could be a finite quasiparticle lifetime, which does not depend on external magnetic field. We will turn back to this concept later in microwave analysis of our samples (Sec. 4.6). Small discrepancies from the logarithmic curve in Fig. 4.22 can be ascribed to the additional effects of the magnetic field. They should be more visible in spectroscopy of the differential conductance dI/dV at low temperature, which directly reflects the density of states. Altshuler-Aronov effect in magnetic field at finite temperature leads to corrections in density of states (Eq. 6.1 in Ref. [75]):

E λ1 E E + ωS E ωS δN(E,T,B) = λ0 f + f + f + f − , T 2λ0 T T T (4.14) f e where ωS = eB/me is cyclotron frequency and

1 1/T τ 1 sinh(y) f(x) = dy −2 y cosh(y) + cosh(x) Z0 , where τ is a relaxation time, which in our case satisﬁes the condition τ −1 E. −1 λ1 The equation has 4 parameters τ ,Teff , λ0, , which have the following 2λ0 impact of the resulting DOS curve: e

τ −1: increases the curve by additive constant log(τ −1). In our case, we ﬁxed the value τ −1 = 0.1 eV , which is suﬃciently high and in approximate agreement with results obtained by transport properties on similar samples (see Sec. 4.2).

Teff : smears the peak at E 0 and also the Zeeman splitting peaks at E = → ωS. It deﬁnes the value of the zero bias conductance. ±

λ0: normalizes the whole curve.

72 λ1 : ampliﬁes the logarithmic dependence as well as the deepening of the 2λ0 • zero bias conductance with increasing magnetic ﬁeld.

we numerically found that in our case it should lie in interval < • 1/3 , 0 >. − if λ1 1/3: the deepening with increasing B is strongest, but the • 2λ0 → − logarithmic dependence at B = 0 vanishes.

if λ1 0: the logarithmic dependence at B = 0 is strongest, but • 2λ0 → the deepening with increasing B vanishes (the Zeeman splitting peaks don’t appear).

if λ1 < 1/3 : the logarithmic dependence at B=0 changes its sign, • 2λ0 − which seems nonphysical.

λ1 if > 0: peaks at E = ωS change their sign (deepening is changed • 2λ0 ± to narrowing), which is opposite to our experiment, but such behavior was observed by Adams in [76]

We used the equation to ﬁt the measured data of dI/dV at T =640 mK and

B=6 T Bc2, where the superconducting DOS should not contribute. The results ≈ are shown in Fig. 4.23

Figure 4.23: Normalized differential conductance measured by STM at temperature T 400 mK and magnetic field B = 6 T (red circles) fitted with temperature dependent ≈ Altshuler-Aronov correction to DOS (black line)

The obtained effective temperature is Teff = 3.4 K = 0.3 meV , which agrees with values obtained spectroscopically with approximate limit equations. The deviations of measured data are caused by larger noise in high magnetic field amplified by numerical differentiation of the measured I(V ) curves.

73 In order to fit the experimental data at finite magnetic fields below Bc2 we have to combine the AA contribution with the superconducting DOS as follows:

N(E) = NS(E)(1 + δN(E)), (4.15)

where NS is deﬁned by Dynes formula:

E + iΓ N (E) = , (4.16) S (E + iΓ)2 ∆(B,T )2 − where ∆(B,T ) can be approximately obtained from following relations:

∆(T ) ∆(T ) T = tanh C ∆(0) ∆(0) T ∆(B) B 2 = 1 . (4.17) ∆(0) − B s C2 In Fig. 4.24 the calculated curves for superconducting DOS, AA correction and their combination respectively in magnetic ﬁelds from 0 T to 8 T are shown.

74 (a) (b)

(c)

Figure 4.24: Simulated curves of DOS in magnetic ﬁelds from 0 to 8 T according to a) Dynes formula, b) Altshuler-Aronov (AA) correction, c) combination of Dynes and AA

Finally, we can compare the measured data with the theoretical curves in the whole range of magnetic ﬁelds displayed on 3D graphs (see Fig. 4.25).

(a) (b)

Figure 4.25: 3D plot of the magnetic ﬁeld dependence of a) normalized diﬀerential conductance measured by STM, b) DOS simulated by combination of superconducting and AA contributions

We can see that the curves, calculated from Eq. 4.14 using Dynes formula, describe the measured data very well. The results qualitatively agree for a wide range

75 of magnetic fields. The energy gap is closing with increasing magnetic field, with the minimum in upper critical field Bc2, above which it is opening again because of other, non-superconducting effects. This effect could be ascribed to Zeeman splitting in paramagnetic limit, which has the same order as the superconducting energy gap ( 1 meV). The deepening of the zero bias conductance with magnetic field above ∼ Bc2 can also corresponds to a rapid increase of sheet resistance observed by transport measurements. We can conclude that the normal state achieved by magnetic field has peculiar character and considerably differs from the normal state in B = 0 at temperatures exceeding the critical temperature Tc. In both cases, the deviations can be ascribed to AA effects in a disordered system.

4.5 Nanobridge in MoC thin ﬁlm

The optimized 10 nm MoC thin ﬁlms were patterned with nanobridges by electron lithography at IPHT Jena, Germany. Four-probe transport measurements were performed in cryogenic assembly in our laboratory. The results are summarized in the paper attached below.

76 TRANSPORT PROPERTIES OF NANOBRIDGES CREATED ON MOLYBDENUM CARBIDE SUPERCONDUTING THIN FILMS

Martin Žemlička1,Pavol Neillinger1, Matúš Rehák1, Marián Trgala1,D. Manca2,Uwe Hübner3, Evgeni Il´ichev3,Miroslav Grajcar1,2 1Dept. of Exp. Physics, FMFI, Comenius University, 84248 Bratislava, Slovakia 2 Institute of Physics, Slovak Academy of Science, 845 11 Bratislava, Slovakia, 3LeibnizInstitute of Photonic Technology, D-07702 Jena, Germany E-mail: [email protected]

Received 12 May 2014; accepted 25 May 2014

1. Introduction Nowadays, highly disordered superconductors are being extensively investigated. Their large sheet resistance implies a large kinetic inductance [1], which can be used for precise detection [2] or quantum information processing [3,4]. Many interesting phenomena occur if the dimensions of the superconducting structures are smaller than the Ginzburg- Landau (GL) coherence length ξ and the London penetration depth λ. For example,the temperature dependence of the resistance (RT) of a nanowire decreases exponentially below the critical temperature Tc and saturates at a residual value. A temperature dependent resistance below Tcis associated with thermally activated phase slip (TAPS),whereas its saturation at T<

224

2. Sample preparation and measurement The investigated nanobridges were prepared on 10nm MoC thin films deposited on sapphire substrate using d. c. magnetron sputtering from Mo target (99.99%, 50mm in diameter) in Ar glow discharge andacetylene reactive gas as carbon source. All substrates were ultrasonically cleaned in acetone and isopropyl alcohol. The vacuum chamber was evacuated initially to a residual gas pressure of 4 x 10-5Pa. Subsequently, the substrates were heated to500°C for 60min to remove water vapor from the surface. The Ar+ acetylene reactive atmosphere was maintained on Ar/acetylene ratio ~1:10[11] by means of mass flow controllers during deposition. Substrates were biased negatively Us= -400V. Working temperature of deposition was 200°C. The 4 point probe measurement structures (Fig.1a) with defined geometry 10umx100um and one nanobridge at the center were patterned by electron lithography.

Fig. 1: a) Scheme of the four probe measurement connected to a sample, b) SEM image of MoC nanobridge

The measurement of the DC transport properties of the nanobridges werecarried out in an Oxford HelioxAC-V 3He refrigerator at temperatures ranging down to 340 mK. The sample was wire-bonded to a printed circuit board sample holder with 50μm aluminum wires. The transmission properties were measured by four-probe technique with Keithley 6221 current source and Picolog ADC-24 24-bit precision data logger.

3. Results and discussion At first the temperature dependence of the nanobridge resistance was measured (Fig. 2a) with 100 nA current, corresponding to current density of ~ 2104 A/cm2. The resistance of the nanobridge was found to slightly rise from room temperature value (R300K=5570Ω), saturating at 11 K (R11K=5840Ω) and at Tc=6K(see fig. 2).The sharp resistance drop corresponds to superconducting phase transition. However, the resistance of the nanobridge does not decrease to zero, as it is expected for bulk superconductor, but exponentially decays and at ~4 K reaches its minimum (R4K=160Ω) and at lower temperatures slightly rises again. The exponential decay of the resistance could be explained either by thermally activated phase-slipor phase slip centers in the nanobridge [5,6], while the residual resistance of the nanobridge at temperatures well below Tc can be explained by quantum phase-slips (QPS), according to Giordano[13]. However, neither theory of these theories explains the resistance minimum at 4 K. Similar behavior was measured for 20nm Sn whiskers [7] and for granular In wire of 41 nm in diameter [12], but the origin of this phenomena is unexplained yet.

225

Fig. 2: a) Temperature dependence of the nanowire resistance b) V-I characteristics at 3.5K with discrete voltage steps

The I-V characteristics were measured at 3.5 K(Fig. 2 b). Discrete steps in voltage were observed corresponding to discrete values of resistance (Fig. 3a). Very similar steps were also measured in Sn whiskers [14] and associated with spatially localized “weak spots” or phase slip centers (PSCs) [15]. Moreover, the zoom of dependence close to zero current shows the saturation of resistance at the finite value of 180Ωat zero current (Fig 3b) in agreement with RT measurement.

Fig. 3: a) Discrete resistance steps recalculated from IV characteristics b) detailed zoom at saturated value at zero current

Fig 2b) shows that the I-V characteristic exhibits hysteretic behavior which is theoretically predicted for PCS [16].

Fig. 4:Discrete steps of the resistance in time caused probably by local heating of the sample

Measuring the resistance of the nanobridge at 120 A fixed current reveals instability of the resistance in time (Fig 4). The resistance of the nanobridge increases is two discrete steps up to 6000 Ω. Similar effects were observed, and argued to be the effect of local heating in the sample [8].

226

Conclusion In summary, we have studied DC electrical transport properties of a 50 nm wide nanobridge patterned on 10nm MoC thin film. The nanobridge exhibits superconducting phase transition at 6 K, however, the resistance decreases exponentially with temperature and saturates at temperatures below 4 K, as possible consequence of the quantum fluctuations induced dissipation in low temperature regime. The I-V characteristics of the nanobridge exhibit discrete steps with hysteresis in the resistance. Similar effects caused by phase slip centers were observed in Sn and In nanowires.

Acknowledgement The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under Grant No. 270843 (iQIT). This work was also supported by the Slovak Research and Development Agency under the contract APVV-0515-10 (former projects No. VVCE-0058-07, APVV-0432-07), DO7RP-0032-11 and LPP-0159-09. The authors gratefully acknowledge the financial support of the EU through the ERDF OP R&D, Project CE QUTE &metaQUTE, ITMS: 24240120032 and CE SAS QUTE.

References:

[1] B. H. Eom: Nature Physics 8, 623 (2012) [2] G.Hammer, Supercond. Sci. Technol.20 408 (2007) [3] A. Wallraff, Nature 431, 162 (2004) [4] O. V. Astafiev, Nature 484, 355 (2012) [5] N. Giordano, Phys. Rev. Lett.61, 2137 (1988); Phys. Rev. B 41,6350 (1990) [6] Tinkham, M.: Introduction to Superconductivity, 2d ed.,(McGraw-Hill, Inc. 1996) [7] N. Giordano, Phys. Rev. Lett.61, 2137 (1988); Phys. Rev. B 41, 6350 (1990). [8] M. Tian, Physical Review B 71, 104521 (2005) [9] N. Giordano, Phys. Rev. B22, 5635 (1980). [10] A. G. P. Troeman, Phys. Rev. B 77, 024509 (2008) [11] M. Trgala, Applied Surface Science(2014), in print [12] U. Schulz, J. Low Temp. Phys.71, 151 (1988) [13] G. Slama and R. Tidecks, Solid State Commun.44, 425 (1982) [14] T. Werner,J. Cryst. Growth73,467 (1985). [15] L. Kramer, Phys. Rev. Lett. 40, 1041 (1978)

227

4.6 Microwave analysis

Complex conductivity characterizes the microwave response of a superconductor. Its real and imaginary parts represent the contribution of the normal electrons and Cooper pairs, respectively. The dependence on external electromagnetic field is well described by Mattis-Bardeen theory, introduced in theoretical review (see Sec. 2.1.3). The complex conductivity can be directly measured, using a coplanar waveguide (CPW) resonator patterned on the superconducting thin films. As it is shown in section 2.2, the quality factor of a CPW resonator reflects the dissipative losses in the material, so Q 1/σ1, while the resonant frequency represents the ∼ inductive reflections caused by the kinetic inductance of Cooper pairs (besides the geometrical factors), so ω0 σ2. ∼ Disorder in superconductors directly affects CPW resonator behavior. In this section, we present a microwave analysis of homogeneously disordered MoC thin films and show that scattering effects are significantly stronger than it is expected by original MB theory. Subsequently we analyze a granular MgB2 thin film, where scattering is weak, but the tunneling between individual granules causes detuning of the resonant frequency in magnetic field, which is characteristic of RF-SQUID structures.

4.6.1 MoC CPW resonator

We prepared and measured several samples of MoC thin ﬁlms (see Tab. 4.3) on which a CPW resonator was patterned by optical lithography and dry ion-etching.

The resonator was designed to have a quality factor Qext = 30 000 and a resonant frequency ω0 = 2π 2.5 GHz for films with a high thickness, where scattering is × negligible (Qint Qext). This was verified on the 200 nm thick MoC film (sample 4). If the thickness is lowered close to or even below superconducting coherence length, the kinetic inductance is increased, which decreases the resonant frequency. Moreover, the scattering process driven by interface scattering centers considerably suppresses the internal quality factor. Films with 10 nm thickness are optimal for further implementation of quantum nanostructures [10], so a more detailed examination of their behaviour in microwave electromagnetic field is needed.

81 R(Ω) t(nm) Tc(K) Q f0(GHz) Sample 1 173 10 5.8 9600 1.42 Sample 2 182 10 7.2 24000 1.51 Sample 3 169 10 5.5 8300 1.35 Sample 4 18 200 6.7 30000 2.49 Sample 5 380 5 3.8 1700 0.75

Table 4.3: Summarized properties of the MoC CPW resonator samples

We measured the temperature dependence of the resonant frequency and the loaded quality factor of our samples. The measured curves were compared with the Mattis-Bardeen theory 2.1.3. The obtained complex conductivity was then recalculated to complex impedance by using FastHenry 3.0 software, with input values

σ1 and λ = 1/µωσ2 and output of the form of Z = R + iωL. From the real and imaginary impedance,p we can directly determine Q and ω0 by using the well known relations [38].

Figure 4.26: Temperature dependence of the resonant frequency (circles - measured data, lines - theoretical model) for samples 1, 2, 3, 4 respectively

82 In Fig. 4.26, we compare the measured data of the resonant frequency for samples 1-4 with the numerical model and as we can see, the theoretical curves ﬁt the measured data very well.

Figure 4.27: Temperature dependence of the quality factor (circles - measured data, lines - theoretical model) for 10 nm MoC sample

The same model was used to describe the quality factor temperature dependence. The internal quality factor was calculated from attenuation constant, which is defined by the real part of the impedance 2.28. From the results (Fig. 4.27) we can see, that the difference is considerable in a wide temperature range. The standard MB model is therefore not appropriate to describe the dissipative losses in the disordered superconductors, because of the enhanced scattering effects. A modification of the model was made by the introduction of the finite quasiparticle lifetime. The modified model fits our data very well. The details are presented in the attached paper.

83 PHYSICAL REVIEW B 92, 224506 (2015)

Finite quasiparticle lifetime in disordered superconductors

M. Zemliˇ cka,ˇ P. Neilinger, M. Trgala, M. Rehak,´ D. Manca, and M. Grajcar* Department of Experimental Physics, Comenius University, SK-84248 Bratislava, and Institute of Physics, Slovak Academy of Sciences, Dubravsk´ a´ cesta, Bratislava, Slovakia

P. Szabo´ and P. Samuely Centre of Low Temperature Physics, Institute of Experimental Physics, Slovak Academy of Sciences & Institute of Physics, P. J. Safˇ arik´ University, SK-04001 Kosice,ˇ Slovakia

S.ˇ Gaziˇ Institute of Electrical Engineering, Slovak Academy of Sciences, Dubravsk´ a´ cesta, SK-84104 Bratislava, Slovakia

U. Hubner¨ Leibniz Institute of Photonic Technology, P.O. Box 100239, D-07702 Jena, Germany

V. M. Vinokur Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA

E. Il’ichev Leibniz Institute of Photonic Technology, P.O. Box 100239, D-07702 Jena, Germany, and Novosibirsk State Technical University, 20 Karl Marx Avenue, 630092 Novosibirsk, Russia (Received 19 August 2014; revised manuscript received 27 October 2015; published 8 December 2015)

We investigate the complex conductivity of a highly disordered MoC superconducting film with kF l ≈ 1, where kF is the Fermi wave number and l is the mean free path, derived from experimental transmission characteristics of coplanar waveguide resonators in a wide temperature range below the superconducting transition temperature Tc. We find that the original Mattis-Bardeen model with a finite quasiparticle lifetime, τ, offers a perfect description of the experimentally observed complex conductivity. We show that τ is appreciably reduced by scattering effects. Characteristics of the scattering centers are independently found by scanning tunneling spectroscopy and agree with those determined from the complex conductivity.

DOI: 10.1103/PhysRevB.92.224506 PACS number(s): 74.81.Bd, 74.25.nn, 74.55.+v, 74.78.−w

I. INTRODUCTION tunneling spectroscopy and the density of states, which has been assumed to describe the microwave response. This Disordered superconductors are a subject of intense current implies that a model assuming uniform properties of the film attention. This interest is motivated not only by the appeal of fails to describe films near the Ioffe-Regel limit, k l ≈ 1. And dealing with the most fundamental issues of condensed matter F although the authors succeeded in explaining the behavior of physics involving interplay of quantum correlations, disorder, the imaginary part of the complex conductivity σ = σ − iσ quantum and thermal fluctuations, and Coulomb interac- s 1 2 in a narrow temperature range, the understanding of the real tions [1–4], but also by the high promise for applications. The part σ , which is mostly influenced by disorder, is far from existence of states with giant capacitance and inductance in the 1 being complete. Hence a call arises for a simple unified critical vicinity of the superconductor-insulator transition [2,5] model that can explain both the microwave and the tunneling breaks ground for novel microwave engineering exploring conductance measurements in strongly disordered supercon- duality between phase slips at point-like centers [6] or at phase ducting films. In this paper, we discuss a model that meets this slip lines [7] and Cooper pair tunneling [8–10]. The feasibility challenge and experimentally demonstrate its validity. of building a superconducting flux qubit by employing quantum phase slips in a weak link created by highly disordered II. MOC THIN FILM AND CPW RESONATOR superconducting wire was demonstrated by Astafiev et al. [5]. Yet while there has been notable recent success in describing One of the ways to measure the microwave complex the dc properties of disordered superconductors [2,9,11], the conductivity is to use a coplanar waveguide (CPW) resonator understanding of their ac response remains insufficient and patterned on a thin film of the desired superconductor. The impedes advancement in their microwave applications. resonator is characterized by two main quantities, the resonant Recent studies of the electromagnetic response of strongly frequency and the quality factor, which can be directly disordered superconducting films [12,13] revealed a discrep- calculated from the complex conductivity. The capacitance ancy between the local density of states measured by scanning of the CPW is explicitly defined by its geometry [14]. The imaginary part of the impedance is mostly represented by the inductance of the CPW, and therefore it determines CPW resonator resonant frequency. The real part of the impedance *[email protected] is determined by the resistive losses in the CPW and therefore

400 350 105 300 250 )

Ω 200 4 ( 10 R

150

100 factor Quality FIG. 1. Scheme of the coplanar waveguide resonator with di- 50 mensions W = 50 μm, S = 30 μm, C = 10 μm, t = 10 nm, and = 0 103 h 430 μm. 110100 t (nm) influences the internal quality factor [14]. Taking into account the external quality factor Qext due to input/output coupling FIG. 2. Thickness dependence of sheet resistances (squares) at capacitances, one can calculate the required loaded quality temperatures just above Tc and internal quality factors (triangles) of factor. The structure of the CPW resonator (see Fig. 1)was MoC coplanar resonators at temperatures T Tc. Internal quality patterned by optical lithography and argon ion etching of the factors, limited by dielectric losses, calculated from the Mattis- deposited superconducting thin films. We focused on the study Bardeen model for the corresponding R and t (circles) exhibit the of the properties of disordered 10-nm thin MoC films with opposite trend. Solid lines are guides for the eye. ≈ sheet resistance R 180 [15]. The chosen thickness of 10 nm is optimal for further patterning of superconducting and Q completely agreed with the design. To compare our nanostructures which are expected to exhibit quantum phase experimental data with theory, we calculated the complex slips [5]. The films were fabricated by magnetron reactive impedance of the CPW resonator with known geometry sputtering, where particles of molybdenum were sputtered (see Fig. 1) using the complex conductivity given by the from a Mo target onto a sapphire c-cut substrate in an argon- Mattis-Bardeen (MB) theory [18]. acetylene atmosphere. The partial pressure of acetylene and We measured several MoC samples of different thicknesses Ar gas was set to 3 × 10−4 and 5.4 × 10−3 mbar, respectively. (and thus sheet resistances); their parameters are presented in The film thickness was controlled by tuning the sputtering 6 time according to the sputtering rate of 10 nm/min. The root (a) 10 mean square (RMS) roughness of the surface, 1 μm × 1 μm, 5 calculated from the AFM topography data was about 0.3 nm. 10 The preparation details are given in Ref. [16]. 4 The transport properties of the MoC thin films were ob- 10 tained by four-probe measurements. The critical temperatures of the superconducting transition Tc, are very sharp, showing 3 10 a shift from 8 K for a film thickness t 30 nm down to = Quality factor 1.3 K for t 3 nm accompanied by an increase in the sheet 2 resistance from several tens of ohms to 1300 , respectively. 10 The transport measurements in magnetic fields as well as the 1 10 Hall effect measurements allowed us to determine the charge 0 1 2 3 4 5 6 carrier density, the upper critical field, the diffusion coefficient, T (K) k l the coherence length, and the Ioffe-Regel product F in the (b) 1.5 prepared films. The carrier concentration n ≈ 1.7 × 1023 cm−3 does not depend on the thickness of the thin film for t = 15, 10, 1.4 and 5 nm, while te sheet resistance R changes considerably, 1.3 = = from 110 for t 15 nm, to 200 for t 10 nm, to 1100 1.2 for t = 5 nm. For thickness t ≈ 10 nm, kF l ≈ 2, indicating 1.1 that the film is in a highly disordered limit. The details of the (GHz)

0 1.0 transport data analysis will be published elsewhere [17]. f 0.9 III. MICROWAVE MEASUREMENT 0.8

Transmission measurements of the CPW resonators yielded 0.7 0 1 2 3 4 5 6 temperature dependencies of the resonant angular frequency T (K) ω0 and the quality factor Q, both depending on its complex conductivity. The CPW resonators were designed to have FIG. 3. Temperature dependence of the quality factor Q (a) and = × ≈ × 4 ω0 2π 2.5 GHz and Qext 4 10 for a conventional the resonant frequency f0 (b) of the 10-nm thin CPW resonator. superconductor with a large thickness. The design of the Circles are measured data and lines are data calculated from the resonator was veriﬁed by a test resonator fabricated out of original Mattis-Bardeen relations (s → 0) for parameters Tc = 5.8K, = = = a thick MoC ﬁlm (200 nm, Tc 6.7 K). The measured ω0 R 180 ,and0 1.83kTc.

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Fig. 2. The most striking feature of our data is that, while the As evidenced by the 0.43 K curve, the measured SDOS dif- MB model predicts that with an increase in the sheet resistance fers from the BCS SDOS: it reveals a significant quasiparticle the quality factor will increase as well, the experiment density of states at the Fermi level and broadened coherence reveals the opposite trend: a decrease in the measured quality peaks at the gap edges. In our case the best agreement with factors with an increase in the sheet resistance. Furthermore, the experimental data is obtained for the empiric Dynes for- the measured quality factor noticeably differs from those mula, (3), with parameters = 0.120,0 = 1.83kTc, and predicted by the MB model in a wide temperature range [see Tc = 5.8 K. Here we want to emphasize that the experimentally Fig. 3(a)]. At the same time, the measured resonant frequency obtained SDOS is broadened but spatially uniform in lateral falls below the theoretical one [Fig. 3(b)] only slightly, but directions. We have taken scanning tunneling spectroscopy systematically. This deviation was studied in Ref. [12]fora spectra along the 200-nm line and the coefficient of variation narrow temperature range. of the normalized tunneling conductance at zero voltage and It is worth noting that including mesoscopic fluctuations, the superconducting energy gap is about 0.05 and 0.025, re- which leads to the broadened superconducting density of states spectively. There is no characteristic length scale of variations (SDOS) [19], does not noticeably improve the agreement and they can be attributed to the noise and instabilities of between theory and experiment. It is clear from Fig. 3(a) that the measuring system during long scanning measurements. the loss of a 10-nm CPW resonator at low temperatures is much This is very different from the situation found, for example, higher than predicted by the MB model, explained below. in TiN [3], where variations of the superconducting gap are detected by scanning tunneling microscopy (STM) for highly disordered samples. A uniform or “granular” character of IV. TUNNELING SPECTROSCOPY the gap distribution in strongly disordered systems could High losses in disordered superconductors imply a finite be connected with the fermionic or bosonic scenario of the density of states at Fermi energy. Previous studies of disor- superconductor-insulator transition and, in MoC films, is the dered superconductors have shown that both microwave mea- subject of our further research. surements and tunneling spectroscopy indicate a broadened SDOS [13]. Therefore, we have carried out scanning tunneling V. MODIFICATION OF THE MATTIS-BARDEEN THEORY spectroscopy measurements, making use of a low-temperature Building on the extended BCS theory [20], MB derived scanning tunneling microscope. Figure 4 shows the normalized the frequency-dependent complex conductivity [18]. To do so, tunneling conductance spectra obtained between the Au tip they formally introduced an infinitesimal scattering parameter, and the MoC sample as measured at different temperatures s = 2π/τ, which was set to 0 at the end of calculations. ranging from 0.43 to 5.8 K. Each curve was normalized to the We may conjecture that in disordered superconductors the spectrum measured at 5.8 K with the sample in the normal state finite value of s may acquire the physical meaning of in order to exclude the influence of the applied voltage on the the inverse quasiparticle lifetime and use the corresponding tunneling barrier and normal density of states of electrodes. expressions derived in Ref. [21], where both phonon and Therefore each of these normalized differential conductance Coulomb contributions to the quasiparticle lifetime are taken versus voltage spectra reflects the SDOS of the MoC sample, into account. Keeping a finite value of s, as a phenomeno- smeared by 2k T in energy at the respective temperature. B logical inverse inelastic quasiparticle lifetime, one can derive Consequently, at the low-temperature limit (k T ), the B the modified formulas for the ratio of the superconducting differential conductance measures the SDOS directly. complex conductivity σs to the normal conductivity σn as σ 1 ∞ s = 1 − [1 − 2f (E)]g(E,ω)dE, (1) σn ω 1.5 where f (E) is the Fermi-Dirac distribution function and the propagator g is defined as 1.0 NN g(E,ω) = g+(E,ω) − csgn(E − (ω − is))g−(E,ω), /G E E ± (ω − is) NS g±(E,ω) = (2) G 2 0.5 (E2 − 2) (E ± (ω − is)) − 2 + . 2 − 2 2 2 0.0 (E ) (E ± (ω − is)) − -2 -1 0 1 2 V (mV) Here E is the quasiparticle energy, is the superconducting energy gap, and csgn(E − (ω − is)) is the complex ≡ − FIG. 4. Temperature dependence of differential tunneling con- signum function, defined as csgn(z) 1, 1, and sgn( (z)) ductance of the 10-nm MoC thin film measured with a scanning for (z) > 0, (z) < 0, and (z) = 0, respectively. Inspect- tunneling microscope (solid lines). From bottom to top (at zero ing Eq. (2), one sees that the first term of g±(E,ω) is a product voltage), data correspond to temperatures T = 0.43, 1, 2, 3, 4, 5, of the standard BCS quasiparticle density of states and a similar 5.4, and 5.6 K. Open circles represent the broadened density of states factor, but with broadened energy states. The broadening can given by the Dynes formula for = 0.120,0 = 1.83kTc,and be viewed as the result of Coulomb and/or phonon interactions. Tc = 5.8K. It should be noted, however, that the derived propagators

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0 10 broadened density of states in the Nam model [24] elaborated for superconductors containing magnetic impurities. Nam generalized the MB model for arbitrary complex functions −1 10 n(E) and p(E), whose real parts correspond to the densities of

10 states and Cooper pairs, respectively. For the Dynes broadened

N density of states, the complex functions n(E) and p(E) can be −2 10 10 /σ deﬁned as 1

σ 10 /G + ≡ E i −3 G n(E) sgn(E) , 10 10 (E + i)2 − 2 10 (4) 0 1 eV/Δ¯hω 2 −4 p(E) ≡ sgn(E) , 10 2 2 0 1 2 3 4 5 (E + i) − ¯hω/Δ and the normalized superconducting complex conductivity FIG. 5. Frequency dependence of the real part of the supercon- reads ∞ ducting complex conductivity σ1 for the finite inelastic scattering σ1 = 1 − + + parameters. Circles represent the original Mattis-Bardeen (MB; [f (E) f (E ω)][ (n(E)) (n(E ω)) σn ω −∞ scattering parameter s = 0). Solid and dashed lines are theoretical +(p(E))(p(E + ω))]dE, curves corresponding, from bottom to top, to the finite values s/ and / = 0.001, 0.01, 0.05, 0.1, 0.2, 0.5, and 1 for the Nam and ∞ (5) σ2 1 MB models with finite scattering, respectively. Inset: Normalized = [1 − 2f (E + ω)][(n(E))(n(E + ω)) tunneling conductance of the normal metal-insulator-superconductor σn ω −∞ = tunnel junction for the same values of s. +(p(E))(p(E + ω))]dE. The square roots are taken to mean the principal square root g±(E,ω) include the same broadened density of states as with real part 0. described by the empiric Dynes formula used in tunneling The real parts of the complex conductivity calculated for the and point contact spectroscopy [22,23], Nam model and the modified MB model with finite scattering are compared with those for the standard MB model in Fig. 5. E + i N(E) = sgn(E) , (3) Both models result in an increase in the real part of the complex (E + i)2 − 2 conductivity in comparison to the standard MB model. At frequencies ω>2 this increase is almost indistinguishable, with the inelastic scattering parameter = s. The supercon- whereas at frequencies ω<2 the difference is significant. ducting complex conductivity and the tunneling conductance = for finite inelastic scattering parameters are shown in Figs. 5 The difference is the most noticeable at ω , where the Nam and 6. Although the MB model with finite scattering provides model exhibits a peculiarity. The resonant frequency of our resonator is much lower than the energy gap ω , therefore results consistent with microwave measurements, the results ≈ contain a product of the BCS and broadened SDOS, which we cannot test the peculiarity at ω directly. The terahertz seems to be nonphysical. Nevertheless, one can include the (THz) spectroscopy performed recently on NbN samples [25] could detect this peculiarity but measurement temperatures are too high to resolve it. 0.2 In order to avoid problems with normalization and geometrical factor inaccuracy it is convenient to compare theoretical and experimental results via the ratio σ2/σ1. Since the 0.1 geometrical factors of the resonator are canceled out, this ratio MB

2 can be expressed as a function of the resonant frequency and

/σ 0.0 the quality factor of the CPW resonator: ) 2

MB σ2 ω0 2 = Q 1 − . (6) σ −0.1 σ1 ωg − 2

σ Here ωg is the resonant frequency of the resonator in the normal ( −0.2 state of a lossless metal. In Fig. 7 we compare the experimental data with the σ2/σ1 temperature dependence calculated for −0.3 different values of the parameters s and corresponding to 0 1 2 3 4 5 ¯hω/Δ the MB and Nam models, respectively. At high temperatures, T>Tc/2, small values of the FIG. 6. Relative deviation of the imaginary part of the supercon- scattering parameter provide a fair agreement between the ducting complex conductivity σ2 from the original Mattis-Bardeen experimental data and the prediction by the standard MB theory (circles) for the same values of parameter s as in Fig. 5. theory; i.e., this theory works well. However, at very low Increasing the parameter s decreases the magnitude of the σ2 values temperatures a larger scattering parameter is required to ﬁt the for frequencies below 2. measured data, and it is not possible to ﬁnd any intermediate

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6 4 (a) 10 (a) 10

two−channel modified MB model 5 10 1

1 3 4 10 /σ

10 2 /σ 2 σ σ

3 10

2 10 2 0 1 2 3 4 5 10 0 1 2 3 4 5 T (K) T (K) 4 (b) 10 6 (b) 10 two−channel Nam model

5 10 1 3 10 /σ 2 1 4 10 σ /σ 2 σ

3 10 2 10 0 1 2 3 4 5

2 T (K) 10 0 1 2 3 4 5 T (K) FIG. 8. Temperature dependence of the ratio σ2/σ1 of the 10-nm thin CPW resonator. Circles are measured data fitted by the two- FIG. 7. Measured ratio σ2/σ1 (open circles) for the 10-nm thin channel model with (a) modified Mattis-Bardeen relations (solid line) MoC CPW resonator compared with the ratio calculated (a) from = = = = for parameters κ 0.34,0 1.83kTc,Tc 5.7K,ss 0.240, the Mattis-Bardeen relations with finite inelastic scattering and (b) and sb = 0 and (b) Nam relations (solid line) for parameters κ = from the Nam model for various values of the parameters s and , 0.10,0 = 1.83kTc,Tc = 5.7K,and = 0.250. respectively. At low temperatures (below 1 K) the lines, from top to = bottom, correspond to s/0 and /0 0.01, 0.1, 0.2, and 0.5. similar two-channel model was used by Sherman et al. in Ref. [25], where the authors should have included an excess value of s or to obtain a reasonable agreement with conductivity in order to fit their experimental data measured at both tunneling and microwave experiments for the complete a much higher range of frequencies than in the current work. temperature range. The same procedure was applied to the Nam model and the results are shown in Fig. 8(b). The results are satisfactory for temperatures above T /2, but the qualitative agreement VI. TWO-CHANNEL MODELS c between theory and experiment for low temperatures is much The MB model with finite scattering parameter s provides worse than that for the modified MB model. a hint about how to solve the problem. In order to obtain a Remarkably, if we adopt the argument of Sherman et al. [26] self-consistent picture, one should include also a “channel” that the upper metallic electrode used for tunneling spec- corresponding to the photon scattering from the BCS to the troscopy measurements can screen Coulomb interactions, BCS density of states which corresponds to the standard which can, in turn, increase the measured energy gap, it is not MB model with s → 0. Hence let us adopt a two-channel necessary to fit the complex conductivity for the same values model in which the total complex conductivity is the weighted of the energy gap as measured by tunneling spectroscopy. sum of two contributions σ = (1 − κ)σb + κσs , where the σs Hence, the 0 can be taken as a fitting parameter, and a value corresponds to the channel with enhanced scattering ss taken smaller than the one obtained with tunneling spectroscopy as the fitting parameter, while the chanel without scattering is expected. The best fit of the complex conductivity was (bulk channel) sb is taken to be 0 and the parameter κ ∈ 0,1 obtained for κ → 0, which is in fact a single-channel model. is the filling factor. Surprisingly, the modified MB model with The suppressed gap obtained is 0 ≈ 1.5kTc, which is much a weighted sum of two contributions gives excellent agreement smaller than the 0 ≈ 1.83kTc measured by STM, whereas with experimental results [see Fig. 8(a)]. Here we can conclude the scattering parameter, ≈ 0.130, is very close to the that in order to fit the complex conductivity and tunneling value obtained by STM [see Fig. 9(a)]. Interestingly, for the conductivity of disordered superconductors for a similar set Nam theory the same approach, with a single channel and of parameters, one should apply the two-channel model. A suppressed 0, does not change the results in comparison with

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4 (a) 10 Nam model Modified MB model 0.6 0.6 single−channel modified MB model (a) (b) 0.5 0.5 N N

/σ 0.4 /σ 0.4 1 3 ) ) 10 /σ 2 MB MB

1 0.3 1 0.3 σ σ σ

− 0.2 − 0.2 1 1 σ σ ( ( 0.1 0.1 2 10 0 1 2 3 4 5 0 0 T (K) 0 1 2 3 4 5 6 0 1 2 3 4 5 6 4 ¯hω/Δ ¯hω/Δ (b) 10 Nam model Modified MB model single−channel Nam model 0.6 0.6 (c) (d) 0.5 0.5 N N 1 3 10 /σ 0.4 /σ 0.4 /σ ) ) 2 σ MB MB

1 0.3 1 0.3 σ σ

− 0.2 − 0.2 1 1 σ σ 2 ( ( 10 0.1 0.1 0 1 2 3 4 5 T (K) 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 FIG. 9. Same experimental data as in Fig. 8 (circles) fitted ¯hω/Δ ¯hω/Δ by the single-channel model with (a) modified Mattis-Bardeen relations (solid line) for parameters 0 = 1.5kTc,Tc = 5.7K,and FIG. 10. Normalized deviation of the real part of the complex = s 0.130 and (b) Nam relations (solid line) for parameters conductivity σ1 to the standard Mattis-Bardeen (MB) counterpart 0 = 1.7kTc,Tc = 5.7K,and = 0.230. (s = 0) for the Nam (a, c) and the modified MB (b, d) models for temperatures T = 0 (a, b) and T = 0.4Tc (c, d). From bottom to top, the curves in each graph correspond to inelastic scattering parameters the two-channel model [see Fig. 9(b)]. For both single-channel = /0 and s/0 0.05, 0.1, 0.2, and 0.5. models the ratio /kTc is below the BCS universal value, 1.76. Nevertheless, in both cases, single and two channel, considerable inelastic scattering is required, which fully justifies the concept of a finite quasiparticle lifetime in disordered into account the broadened density of states in the MB model superconductors. Surprisingly, the Nam models do not fit and more exotic models should be compared to the modified the experimental data as well as the modified MB models. MB model. Therefore, it would be interesting to check both models Our two-channel model is phenomenological and does not experimentally at ω ≈ /, where they exhibit the most address the problem of the microscopic origin of the second remarkable difference. THz spectroscopy can provide such channel with enhanced inelastic scattering. Nevertheless, the experimental data. Dynes empiric formula for the broadened density of states, In Ref. [25], THz spectroscopy reveals a deviation of the introduced in 1984 for disordered superconductors [22], is measured real part of the complex conductivity σ1 from the present in the MB model, and MB obtained it already in standard MB counterpart. The difference, which the authors 1958. Such a broadened density of states was observed in ascribe to a contribution of broken symmetry in disordered many superconducting systems such as disordered supercon- superconductors, is compared with the Higgs model [27]. ductors [22], high-Tc superconductors [23], MgB2 [28], and However the deviations presented in Fig. 3 in Ref. [25] can be iron-based superconductors [29]. It seems that, for some paths, even better described by models with a broadened density of the scattering parameter s is not renormalized to 0 and the MB states. For example, the position of the peaks clearly changes model with finite scattering rate s is a good first approximation. with the energy gap of the samples and there is no cutoff The scattering is probably caused by two-level systems located frequency at which the deviation of the measured real part very homogeneously at the MoC-sapphire interface, which of the complex conductivity from the standard MB model are present even if special precautions are made [30]. This σ1 − σ1MB saturates to 0. Both features are present in Fig. 10, hypothesis is corroborated by our STM/scanning tunneling where the deviation of the theoretical curves σ1(ω)fromthe spectroscopy, which reveal the same inelastic scattering standard MB model (s = 0) are shown for various inelastic parameter even for atomically flat surfaces with no adsorbed scattering parameters. These results show that one should take impurity.

224506-6 FINITE QUASIPARTICLE LIFETIME IN DISORDERED . . . PHYSICAL REVIEW B 92, 224506 (2015)

VII. CONCLUSION even better agreement with experiment. Thus, the original MB model with finite inelastic scattering or the Nam model In conclusion, the two-channel model well describes both is able to describe microwave and tunneling experiments in microwave and tunneling conductance measurements over a disordered superconductors, while more advanced models are wide temperature range: from 300 mK up to almost T .The c failing. Our results providing a simple expression for the enhanced scattering channel dominates at low temperatures, complex conductivity call for further theoretical, as well as which is consistent with the low-temperature losses in high- experimental, research aimed at clarifying what the simple quality CPW resonators made of conventional superconduc- MB theory with finite inelastic scattering captures that is lost tors [30,31], where the quality factor is limited by the interface in more advanced machineries. scattering. However, if STM provides an overestimated value of the superconducting energy gap, the results can even be described by the single-channel model with enhanced ACKNOWLEDGMENTS scattering. In disordered superconducting films such scattering leads to the suppression of electron diffusion and, accordingly, This work was supported by the European Commu- to enhancement of the Coulomb interaction [32,33]. This is nity’s Seventh Framework Programme (FP7/2007-2013) under consistent with the rapid decrease in the quality factors for Grant No. 270843 (iQIT), by the MP-1201 COST Action, thinner superconducting films as shown in Fig. 2. by the Slovak Research and Development Agency under the The main features shown in MoC disordered superconduc- Contract Nos. DO7RP003211, APVV-0515-10, APVV-0036- tors apply also for other disordered superconductors, such as 11, APVV-0088-12, APVV-14-0605, VEGA 2/0135/13 and NbN and TiN [13,25]. For example, the deviation of the real VEGA 1/0409/15 and by the U.S. Department of Energy, Of- part of the complex conductivity σ1 from the standard MB fice of Science, Materials Sciences and Engineering Division. one (s = 0), measured in Ref. [25], can be well explained E.I. acknowledges partial support from Russian Ministry of by finite inelastic scattering as shown in Fig. 10, providing Science and Education Contract No. 8.337.2014/K.

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224506-7 As it has been mentioned above, the modified Mattis-Bardeen model is applicable in wide frequency range, so it can be used in order to fit the experimental data from THz spectroscopy. We performed such measurements at Institute of Physics, Academy of Sciences of the Czech Republic, Prague on the 20 nm MoC film. The results fitted with the model are presented in Fig. 4.28. The temperature of the experiment was too high (T = 0.8 Tc) to resolve, if there is a peculiarity at ω = ∆.

(a) (b)

Figure 4.28: Terahertz spectroscopy measurement of the real part (a) and the imaginary part (b) of the complex conductivity (red circles) ﬁtted with modiﬁed Mattis-Bardeen model (solid line) for parameters: T = 0.8 Tc, ∆ = 1.24 meV and s = 0.25 meV=0.2 ∆

However, the experimental results obtained in MoGe and NbN superconductors demonstrated in Ref. [77,78], exhibited the peculiarity even at higher temperatures. One of the explanation can be provided by the concept of the nonequlibrium processes [14], which could be present in the THz experiments due to the large power of the THz pulses. Such processes lead to redistribution of the quasiparticles, which causes the effective cooling of the sample. As it was discussed by Scalapino [79], this effect is strongest in region (∆/2, 2∆), which is qualitatively consistent with the mentioned experimental results and also with the Nam model at low temperatures. One of our future plans in this field is to perform additional THz measurements on several samples with different properties and analyze them with deeper theoretical concept considering the mentioned processes.

91 4.6.2 MgB2 CPW resonator

The same technique of the CPW resonator and MB model was also used in granular MgB2 thin film with thickness 300 nm. The temperature dependence of the complex conductivity is well described by standard Mattis-Bardeen formulas, from which we can conclude that the scattering in this material is much lower than in homogeneously disordered superconductors. In addition, we measured the dependence of the resonant frequency at the lowest temperature in our microwave cryogenic assembly ( 300 mK). Surprisingly, hysteretic periodic detuning was observed. Such ∼ behaviour is characteristic of artificially patterned RF-SQUID, where the detuning is caused by Josephson inductance, which depends on external magnetic field (see Sec. 2.1.5). We proposed the model of percolation path interrupted by the naturally created insulating barrier between grains. The more detailed view of the model as well as the measurement results are shown in paper attached below.

Applied Surface Science 312 (2014) 231–234

Contents lists available at ScienceDirect

Applied Surface Science

jou rnal homepage: www.elsevier.com/locate/apsusc

Superconducting properties of magnesium diboride thin ﬁlm

measured by using coplanar waveguide resonator

a,∗ a a a a a

M. Zemliˇ ckaˇ , P. Neilinger , M. Trgala , M. Gregor , T. Plecenik , P. Durinaˇ , a,b

M. Grajcar

Department of Experimental Physics, Comenius University, SK-84248 Bratislava, Slovakia

Institute of Physics of Slovak Academy of Science, Dúbravská cesta, Bratislava, Slovakia

a r t i c l e i n f o a b s t r a c t

Article history: In this paper we demonstrate the superconducting properties of MgB2 coplanar waveguide resonator

Received 3 February 2014

patterned from 300 nm thin ﬁlm fabricated by vapor deposition. We measured the temperature depend-

Received in revised form 15 May 2014

ence of the quality factor and the resonant frequency of the resonator. Surprisingly, we also observed

Accepted 30 May 2014

hysteretic periodic response of resonance frequency to external magnetic ﬁeld, which is characteristic of

Available online 6 June 2014

bistable systems with double-well potential, such as superconducting RF SQUID or phase-slip ﬂux qubits.

This property seems to be peculiar for granular and disordered superconductors where a superconducting

Keywords:

loop of large effective diameter with weak links can be formed.

MgB2

Thin ﬁlm

Coplanar waveguide resonator

Quality factor

RF SQUID

Phase-slip

1. Introduction be a good candidate for this purpose and we have investigated its

superconducting properties by microwaves.

The superconducting coplanar waveguide resonator (CPW)

is a device with distributed elements created on a supercon-

2. Sample fabrication

ducting thin ﬁlm. Thanks to their low parasitic capacitances and

inductances, they have been recently used with great success in

Superconducting MgB2 thin ﬁlm was prepared by co-deposition

experiments with circuit quantum electrodynamics or kinetic

of magnesium and boron from two separate sources on a mirror-

inductance detectors. One of the crucial parameters, which allows

polished sapphire substrate and ex situ annealing in vacuum

them to be used for this purpose, is their high internal quality

chamber. The deposition chamber was evacuated to the limit vac-

−

factor. The quality factor is temperature dependent and achieves its 4

uum 5 × 10 Pa. The resistive thermal evaporation and e-beam

maximum value at temperatures close to one tenth of the critical

evaporation were used to make a precursor of MgB2. Ex situ anneal-

temperature of the superconductor. Recently, superconducting

ing process was realized in vacuum chamber evacuated to the

−

ion traps with an integrated CPW resonator have been examined 3

base pressure of 1 × 10 Pa and consecutively ﬁlled with Ar up

with aim to improve their performance. It is expected that the

to working pressure of 700 Pa. The annealing temperature was

◦

quality factors will increase by several orders of magnitude while 800 C.

heat dissipation will be reduced by making use of superconducting

The resonators were patterned on the MgB2 thin ﬁlm with thick-

electrodes. The ion trap chips are cooled to temperatures ∼4 K.

ness of 300 nm by optical lithography using a 2.5 m thick layer of

Magnesium diboride (MgB ) has critical temperature of about

2 positive tone resist AZ 6624 and by reactive ion etching in Ar and

∼

40 K, so the saturated value of internal quality factor can be

SF6 plasma.

achieved by cooling with liquid helium. Therefore MgB2 seems to

3. Theoretical background

The quality factor and the resonant frequency of the CPW

∗

resonator can be recalculated from the complex conductivity

Corresponding author. Tel.: +421 914299945.

E-mail address: [email protected] (M. Zemliˇ cka).ˇ = 1 − i2 using analytic formulas [1,2]. The real part 1 arises

http://dx.doi.org/10.1016/j.apsusc.2014.05.219

232 M. Zemliˇ ckaˇ et al. / Applied Surface Science 312 (2014) 231–234

from inertial losses in the superconductor, which determine the

quality factor, while the imaginary part 2 inﬂuenced by London

penetration depth L, changes the kinetic inductance and therefore

also the resonant frequency. The complex conductivity of the super-

conductor in alternating electromagnetic ﬁeld can be calculated

from equations derived by Mattis and Bardeen [3]

∞

1 2

= [f (E) − f (E + ω)]g(E)dE N ω

−

1 Fig. 1. Experimental measurement set-up. Sample is cooled down to 0.3 K in the

+ [1 − 2f (E + ω)]g(E)dE (1) refrigerator and connected to the microwave vector network analyzer. Magnetic

− ﬁeld is created by two NbTi coils powered by DC current. ω

2 2

1 [1 − 2f (E + ω)](E + + ωE) 2 = (2)

N ω − − 2 2 2 2

ω, ( − E )[(E + ω) − ]

The transmission was measured by means of the technique

described in the previous section. Surprisingly, the resonant fre-

where f(E) is Fermi–Dirac distribution function, E is the excitation

quency exhibited hysteretic periodic detuning (Fig. 3), which is

energy, is the superconducting energy gap, ω is the angular fre-

characteristic for measurements of resonators with artiﬁcially pat-

quency of external electromagnetic ﬁeld. For a superconductor with

terned superconducting quantum interference device (SQUID) or

two gaps, such as MgB2, one has to take into account two-gap model

phase-slip ﬂux qubit [5–7]. Therefore we have analyzed the exper-

in calculations of the complex conductivity. In our case, we used the

imental results with theoretical model of hysteretic RF SQUID

method described in [4] and calculated the complex conductivity

[8,5,7]. From the ﬁtting procedure we determined the param-

for the individual gaps and we made the weighted average of the

eters of our virtual SQUID (Fig. 4) such as coupling coefﬁcient

calculated values.

k, critical current Ic and normalized inductance of the SQUID

ˇ = 2LIc/˚0.

4. Results and discussion

A superconducting quantum interference device (SQUID) is

usually implemented as a superconducting loop interrupted by

4.1. Quality factor and resonant frequency

a Josephson junction with critical current Ic. The ‘potential’

energy U of this loop with inductance L can form a double-well

The temperature dependences of the quality factor and the reso-

potential

nant frequency (Fig. 2) of the resonator were determined from the

transmission measurement by a vector network analyzer at GHz 2

U ˇi(e)

frequency range at temperatures between 40 K and 0.3 K. Sample = − cos(e − ˇi(e)) (3)

3 EJ 2

was placed in a cryogen-free He refrigerator according to scheme

shown in Fig. 1. The results were ﬁtted by Eqs. (1) and (2) taking

if parameter ˇ is larger than one. Here EJ = ˚0Ic/2 is Josephson

into account two energy gaps of MgB2 [4]. From the ﬁtting proce-

energy and ˚0 is the magnetic ﬂux quantum. The presence of RF

dure we obtained two superconducting energy gaps and the critical

SQUID in CPW resonator causes the detuning of the inductance of

temperature Tc = 38 K of the MgB2 sample (Fig. 2).

the resonator which is dependent on the external magnetic ﬂux.

The effective inductance can be expressed as [5]

4.2. Hysteretic detuning of resonant frequency

2 ˇ cos

= −

We also measured the response of the MgB2 CPW resonator Leff Lr 1 k (4)

1 + ˇ cos

to external magnetic ﬁeld, created by two NbTi coils (Fig. 1).

Fig. 2. The measured temperature dependence of the resonant frequency f0 (left, gray points) and the loaded quality factor QL (right, gray points) compare with results,

calculated by Mattis–Bardeen theory (both, solid line).

M. Zemliˇ ckaˇ et al. / Applied Surface Science 312 (2014) 231–234 233

-6 5.0x10-6 5.0x10 T=0.34K T=5K 0.0 0.0

-5.0x10-6

-5.0x10-6 -1.0x10-5 η η

-5 -1.5x10 -1.0x10-5

-2.0x10-5 -1.5x10-5

-2.5x10-5

-1 0 1 -2 -1 0 1 2 φ /φ φ /φ

e 0 e 0

Fig. 3. Hysteretic periodic response of the resonant frequency to magnetic ﬁeld for MgB2 (squares – experiment, black and gray lines - numeric ﬁtting) obtained parameters:

ˇ = 3.5, Ic = 70 ␮A, S ≈ 170 ␮m , k ≈ 0.01.

≈

where Lr 10nH is the inductance of a resonator without permeability. If we suppose that the mutual inductance between

SQUID, k is a coupling constant between the resonator and the the resonator and the ‘virtual’ RF SQUID is equal to the Joseph-

SQUID, i() = sin() is the normalized supercurrent in the loop, son inductance M = LJ = ˚0/(2Ic), we can calculate the coupling

= e − ˇi(e) and e are normalized internal and external mag- constant

netic ﬂux, respectively. Change of the effective inductance leads to a

detuning of the resonant frequency f0 1/(2 Lef C). Normalized ˚0 1 2 =

k ≈ 0.012 (6)

detuning Á can be expressed as: 2Ic Lr ˇ

−

f0 f0 max 1 2 ˇ cos ˇ which is in agreement with the result obtained from ﬁt-

Á = = k − (5)

+ +

f0 max 2 1 ˇ cos 1 ˇ ting by Eq. (5) (k = 0.01). Therefore we can conclude that our

assumption is justiﬁed and the model of the ‘virtual’ RF SQUID

where f0 is the detuned frequency and f0max is the maximal resonant weakly coupled with the resonator gives us very reasonable

frequency at zero magnetic ﬂux. results.

If we suppose that a superconducting loop is formed in MgB2 Our ﬁnding has also other implication. If the Josephson junc-

thin ﬁlm among grains, one can determine its effective area from tion in the superconducting ring is replaced with a very narrow

∼

a ﬁtting procedure using Eq. (5). The parameters of our ‘virtual’ RF ( 10 nm) but long (∼100 nm) bridge, where phase slips can occur,

SQUID, namely the normalized inductance, the effective surface, the energy of the superconducting loop has the same depend-

the critical current and the coupling constant between SQUID struc- ence on external magnetic ﬂux (Fig. 5). If the barrier between two

ture and CPW resonator are ˇ = 3.5, Ic = 70 ␮A, S ≈ 170 ␮m , k ≈ 0.01 states in crossing points is small, the degeneracy is lifted and anti-

respectively. crossing of energy levels occurs. It has been shown by Astaﬁev

The effective area was determined from the periodic response et al. [6] that this effect occurs in structures created on highly

of the SQUID with period ˚0. The magnetic ﬁeld created by NbTi disordered superconductors and it was ascribed to quantum phase-

coils was calibrated according to measurement of RF SQUID with slip. But is this effect really caused by coherent quantum phase

known geometry. The critical current Ic was determined from the slip?

value of the parameter ˇ taking into account√ that the inductance of One can equally well explain this effect by phase slip in the

RF SQUID can be estimated as L ≈ 0 S, where 0 is the vacuum Josephson junction. We want to stress here that similar results were

Fig. 4. (a) CPW resonator design – gray colored area for MgB2, (b) detailed view of the input and output capacitor with marked dimensions: W = 50 ␮m, S = 30 ␮m, C = 10 ␮m,

t = 300 nm, h = 430 ␮m, (c) model of superconducting loop formed between groups of grains inserted in central conductor of CPW resonator. Size of the individual grains is

about ∼10 nm [9].

234 M. Zemliˇ ckaˇ et al. / Applied Surface Science 312 (2014) 231–234

2.0

1.5

1.0 J 0.5 /E min

U 0.0

-0.5

-1.0 -1 0 1 φ /φ φ /φ

e 0 e 0

Fig. 5. Total energy of (a) RF SQUID with Josephson junction (b) RF SQUID with narrow bridge (phase-slip ﬂux qubit) [10].

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In the submitted thesis, we focused on the analysis of thin films of highly disordered superconductors. The preparation of MoC thin films by magnetron sputtering was tuned in order to obtain optimal sample properties. The results were samples with maximized critical temperature Tc and enhanced sheet resistance R. Material properties, such as structure, stoichiometry and surface roughness were analyzed by means of XRD, EDX and STM. The resulting thin films are superconductive homogeneous and have cubic δ-phase crystallographic structure of MoC with decreased carbon content due to lattice vacancies. Magneto-transport and Hall effect measurements determined that the critical temperature Tc is well correlated with sheet resistance R and Ioffe-Regel disorder parameter kf l, rather than with film thickness. We showed that Finkelsten’s formula can not be applied in order to explain the Tc suppression, despite the fact that it qualitatively well describes the transport properties of the disordered superconducting thin films. The results from tunneling spectroscopy demonstrated an increase of in-gap states with decreasing thickness. Moreover, a vortex lattice was observed, proving the long range phase coherence ascribed to the fermionic scenario of superconductor- insulator transition (SIT). A more detailed view of the field induced SIT of 3 nm MoC thin film revealed an atypical normal state as opposed to the normal state achieved by temperature. Results of magneto-transport and STM measurements were in agreement with the Altshuler-Aronov model of enhanced electron-electron interactions. The superconducting properties of the same MoC samples prepared on different substrates (Si and

Al 2O3) were somewhat different, proving that the interface between the substrate − and the thin film plays a role as a possible pair breaker. Homogeneous superconducting properties of our thin films near SIT predestine them for nanotechnology applications. Such prospects were outlined by the transport measurements of a nanobridge patterned on 10 nm MoC film, which exhibited quantum phase slip-like behavior. Scattering effects were also analyzed by microwave measurement of CPW res-

97 onators patterned in 10 nm and 5 nm MoC films. In order to describe the decrease of internal quality factor and resonance frequency, a modification of Mattis-Bardeen theory for complex conductivity was made. Finite quasiparticle lifetime as a possible source of scattering was introduced as a fitting parameter, and the results agreed with the Dynes phenomenological parameter Γ obtained from STM. The new model is applicable in a wide frequency range, which was proved by a terahertz spectroscopy measurements of σ1 and σ2.

The same CPW technique was used for a granular MgB2 superconductor. The results showed periodic hysteretic detuning in magnetic field, which is characteristic of artificially prepared RF SQUID nanostructures. A reasonable model of percolation path interrupted by weak links between the grains was designed and successfully applied in the fit. Future plans in the research include further detailed examination of the interface between superconducting thin film and substrate, and transport measurements in higher magnetic fields which could provide an even better understanding of superconductor-insulator transition in our samples. Since the technology of sample deposition is optimized and provides reproducible results, application in nanowires with enhanced kinetic inductance is planned as well. A new design will be proposed and the quantum properties will be examined in dilution 3He-4He refrigerator. Fi- nally, the knowledge acquired from MoC investigation will be used in preparation and application of other disordered superconductors such as WSi, W3Si5, NbN, TiN, etc.

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104 List of publications and conferences

Publications: [P1] M. Zemliˇckaˇ , P. Neilinger, M. Trgala, M. Grajcar, M. Gregor, T. Plecenik, P. Durina:ˇ Superconducting properties of magnesium diboride Thin Film Measured By Using Coplanar Waveguide Resonator, Applied Surface Science, 312, 231–234 (2014)

[P2] M. Trgala, M. Zemliˇckaˇ , P. Neilinger, M. Reh´ak,M. Leporis, S. Gaˇzi,J. Greguˇs, T. Plecenik, T. Roch, E. Dobroˇcka, M. Grajcar: Superconducting MoC thin ﬁlms with enhanced sheet resistance, Applied Surface Science, 312, 216–219 (2014)

[P3] P. Neilinger, M. Reh´ak,M. Gregor, M. Zemliˇckaˇ , T. Plecen´ık,M. Trgala, P. Durina,ˇ M. Grajcar: Periodic response of superconducting high quality MgB2 resonator to magnetic ﬁeld, APCOM Proceedings 2013

[P4] M. Zemliˇckaˇ , D. Manca, P. Neilinger, M. Grajcar: Cryogenic carbon powder ﬁlters for superconducting qubit measurement, Proceedings of ADEPT 2014

[P5] M. Zemliˇckaˇ , P. Neillinger, M. Rehák,M. Trgala, D. Manca, U. Hübner,E. Ilichev, M. Grajcar: Transport properties of nanobridges created on molybdenum carbide superconducting films, APCOM Proceedings 2014

[P6] M. Zemliˇckaˇ , P. Neilinger, M. Trgala, M. Reh´ak,D. Manca, M. Grajcar, P. Szabo, P. Samuely, S.ˇ Gaˇzi,U. H¨ubner,V. M. Vinokur, E. Ilichev: Finite quasiparticle lifetime in disordered superconductors, Physical Review B 92, 224506 (2015)

[P7] P. Szabó,T. Samuely, V. Haˇsková,J. Kaˇcmarˇc´ık,M. Zemliˇckaˇ , M. Grajcar, J. G. Rodrigo, P. Samuely: Fermionic scenario for the destruction of superconductivity in ultrathin MoC films evidenced by STM measurements, Physical Review B 93, 014505

105 Conferences: [C1] Canadian Quantum Information Students’ Conference 2013, 24-28 June 2013, Institute for Quantum Science and Technology, University of Calgary, Canada. Poster: Finite quasiparticle lifetime as possible source of decoherence in superconducting circuits

[C2] 20th conference of Slovak physicists, 2-9 September 2013, Bratislava, Slovakia. Poster: Microwave measurement and analysis of disordered superconductors with ﬁnite quasiparticle lifetime

[C3] Solid State Surfaces and Interfaces, 24-28 November 2013, Smolenice, Slovakia. Poster: Superconducting properties of magnesium Diboride Thin Film Measured By Using Coplanar Waveguide Resonator

[C4] Physics and applications of superconducting hybrid nano-engineered devices, 31 August - 4 September 2014, Santa Maria Castellabate, Italy. Poster: Finite quasiparticle lifetime in the surface layer of disordered superconductors.

[C5] Week of Doctoral Students 2015 focused on physical study branches, 2–4 June 2015, Prague, Czech republic. Lecture: Finite quasiparticle lifetime in disordered superconductors.

106