Superradiant THz wave emission from arrays of Josephson junctions

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Superradiant THz wave

emission from arrays of

Josephson junctions

Ievgenii Borodianskyi

Ievgenii Borodianskyi Superr ad iant THz w av e emission fr om arr ays of J osephson junctions

Department of Physics

ISBN 978-91-7911-178-6 Ievgenii Borodianskyi

"Never memorize something that you can look up." - Albert Einstein

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Superradiant THz wave emission from arrays of

Josephson junctions

Ievgenii Borodianskyi

Academic dissertation for the Degree of Doctor of Philosophy in Physics at Stockholm University to be publicly defended on Wednesday 9 September 2020 at 13.00 in sal FR4, AlbaNova universitetscentrum, Roslagstullsbacken 21.

Abstract

High-power, continuous-wave, compact and tunable THz sources are needed for a large variety of applications. Development of power-efficient sources of electromagnetic radiation in the 0.1-10 THz range is a difficult technological problem, known as the “THz gap.” Josephson junctions allow creation of monochromatic THz sources with an inherently broad range of tunability. However, emission power from a single junction is too small. It can be amplified in a coherent superradiant manner by phase-locking of many junctions. In this case, the emission power should increase as a square of the number of phase-locked junctions.The aim of this thesis is to study a possibility of achieving coherent super-radiant emission with significant power and frequency tunability from Joseph-son junction arrays. Two types of devices are studied, based either on stacks (one-dimensional arrays) of intrinsic Josephson junctions naturally formed in single crystals of high-temperature cuprate superconductor Bi2Sr2CaCu2O8+x, or two-dimensional arrays of artificial low-temperature superconducting Nb/NbSi/Nb junctions. Micron-size junctions are fabricated using micro- and nanofabrication tools.The first chapter of this thesis describes the theory of Josephson junctions and how mutual coupling between Josephson junctions can lead to self-syn-chronization, facilitating the superradiant emission of electromagnetic radia-tion. The second chapter is focused on the technical aspects of this work, with detailed descriptions of sample fabrication and experimental techniques. The third chapter presents main results and discussion. It is demonstrated that de-vices based on high-Tc cuprates allow tunable emission in a very broad fre-quency range 1-11 THz. For low- Tc junction arrays synchronization of up to 9000 junctions is successfully achieved. It is argued that an unconventional traveling-waves mechanism facilitates the phase-locking of such huge arrays. The obtained results confirm a possibility of creation of high-power, continu-ous-wave, compact and tunable THz sources, based on arrays of Josephson junctions.

Keywords: Josephson junction, Superconductor, ThZ emission, high-Tc.

Stockholm 2020

http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-181234

ISBN 978-91-7911-178-6 ISBN 978-91-7911-179-3

Department of Physics

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SUPERRADIANT THZ WAVE EMISSION FROM ARRAYS OF JOSEPHSON JUNCTIONS

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Superradiant THz wave

emission from arrays of

Josephson junctions

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©Ievgenii Borodianskyi, Stockholm University 2020 ISBN print 978-91-7911-178-6

ISBN PDF 978-91-7911-179-3

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"If we knew what it was we were doing, it would not be called research, would it?” Albert Einstein

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Abstract

High-power, continuous-wave, compact and tunable THz sources are needed for a large variety of applications. Development of power-efficient sources of electromagnetic radiation in the 0.1-10 THz range is a difficult technological problem, known as the “THz gap.” Josephson junctions allow creation of monochromatic THz sources with an inherently broad range of tunability. However, emission power from a single junction is too small. It can be amplified in a coherent superradiant manner by phase-locking of many junctions. In this case, the emission power should increase as a square of the number of phase-locked junctions.

The aim of this thesis is to study a possibility of achieving coherent super-radiant emission with significant power and frequency tunability from Joseph-son junction arrays. Two types of devices are studied, based either on stacks (one-dimensional arrays) of intrinsic Josephson junctions naturally formed in single crystals of high-temperature cuprate superconductor Bi2Sr2CaCu2O8+x,

or two-dimensional arrays of artificial low-temperature superconducting Nb/NbSi/Nb junctions. Micron-size junctions are fabricated using micro- and nanofabrication tools.

The first chapter of this thesis describes the theory of Josephson junctions and how mutual coupling between Josephson junctions can lead to self-syn-chronization, facilitating the superradiant emission of electromagnetic radia-tion. The second chapter is focused on the technical aspects of this work, with detailed descriptions of sample fabrication and experimental techniques. The third chapter presents main results and discussion. It is demonstrated that de-vices based on high-Tc cuprates allow tunable emission in a very broad

fre-quency range 1-11 THz. For low- Tc junction arrays synchronization of up to

9000 junctions is successfully achieved. It is argued that an unconventional traveling-waves mechanism facilitates the phase-locking of such huge arrays. The obtained results confirm a possibility of creation of high-power, continu-ous-wave, compact and tunable THz sources, based on arrays of Josephson junctions.

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Sammanfattning

Högeffekts, kontinuerliga våg, kompakta och inställbara THz-källor behövs för en mängd olika applikationer. Utveckling av energieffektiva källor för elektromagnetisk strålning i området 0,1-10 THz är ett svårt teknologiskt problem, känt som "THz-gap". Josephson-korsningar möjliggör skapandet av monokromatiska THz-källor med ett i sig brett spektrum av inställbarhet. Emissionskraften från en enda korsning är dock för liten. Det kan förstärkas på ett sammanhängande superradiant sätt genom faslåsning av många korsningar. I detta fall bör utsläppseffekten öka som en kvadrat av antalet faslåsta korsningar.

Syftet med denna avhandling är att studera en möjlighet att uppnå koherent superradiantemission med betydande effekt och frekvensjusterbarhet från Josephson-korsningsgrupper. Två typer av anordningar studeras, baserade antingen på staplar (endimensionella matriser) av inneboende Josephson-korsningar som är naturligt bildade i enstaka kristaller av högtemperatursuprat-superledare Bi2Sr2CaCu2O8 + x, eller tvådimensionella matriser av konstgjorda låg temperatur superledande Nb / NbSi / Nb-korsningar. Korsningar i mikronstorlek tillverkas med hjälp av mikro- och nanofabriceringsverktyg.

Det första kapitlet i denna avhandling beskriver teorin om Josephson-korsningar och hur ömsesidig koppling mellan Josephson-Josephson-korsningar: er kan leda till självsynkronisering, vilket underlättar överstrålningsutsläpp av elektromagnetisk strålning. Det andra kapitlet är inriktat på de tekniska aspekterna av detta arbete, med detaljerade beskrivningar av provtillverkning och experimentella tekniker. Det tredje kapitlet presenterar huvudresultat och diskussion. Det demonstreras att enheter baserade på hög-Tc-koppar tillåter inställbar utsläpp i ett varierande brett frekvensområde 1-11 THz. För sammankopplingsmatriser med låg Tc uppnås synkronisering av upp till 9000 korsningar med framgång. Det hävdas att faslåsning av så mycket stora matriser underlättas av en okonventionell rörelsevågsmekanism. De uppnådda resultaten bekräftar möjligheten att skapa högeffekta, kontinuerliga våg, kompakta och inställbara THz-källor, baserade på matriser av Josephson-korsningar.

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List of appended papers

This thesis is based on the following papers.

I. Borodianskyi, E.A., Krasnov, V.M. “Josephson emission with

fre-quency span 1–11 THz from small Bi2Sr2CaCu2O8+δ mesa

struc-tures.” Nat.Commun 8, 1742 (2017).

Author’s contribution:

I fabricated the sample, performed the measurements, contributed in data analysis and writhing of the manuscript.

II. M. A. Galin, E. A. Borodianskyi, V. V. Kurin, I. A. Shereshevskiy,

N. K. Vdovicheva, V. M. Krasnov, and A. M. Klushin “Synchroni-zation of Large Josephson-Junction Arrays by Traveling Electro-magnetic Waves” Phys. Rev. Applied 9, 054032 (2018)

I have been taking active part in measurements and sample charac-terization and participated in writing the paper.

III. A. A. Kalenyuk, A. Pagliero, E. A. Borodianskyi, S. Aswartham,

S. Wurmehl, B. Büchner, D. A. Chareev, A. A. Kordyuk, and V. M. Krasnov “Unusual two-dimensional behavior of iron-based superconductors with low anisotropy” Phys. Rev. B 96, 134512 (2017)

I helped with sample fabrication and low temperature electrical measurements.

IV. A. A. Kalenyuk, A. Pagliero, E. A. Borodianskyi, A. A. Kordyuk,

and V. M. Krasnov “Phase-Sensitive Evidence for the Sign-Reversal s± Symmetry of the Order Parameter in an Iron-Pnictide Superconductor Using Nb/Ba1-xNaxFe2As2 Josephson Junctions”Phys. Rev. Lett. 120, 067001 (2018)

I helped with sample fabrication and low temperature electrical measurements

V. M.A. Galin, F. Rudau. E.A. Borodianskyi, V.V. Kurin, D. Koelle, R. Kleiner, V.M. Krasnov, A.M. Klushin, “Direct visualization of

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phase-locking of large Josephson junction arrays by surface elec-tromagenetic waves”. To be pubished…ArXiv:2004.06623

Papers not included in this thesis

VI. R. de Andrés Prada, T. Golod, O. M. Kapran, E. A. Borodianskyi,

Ch. Bernhard, and V. M. Krasnov “Memory-functionality superconductor/ferromagnet/superconductor junctions based on the high-Tc cuprate superconductors YBa2Cu3O7−x and the colossal magnetoresistive manganite ferromagnets

La2/3X1/3MnO3+δ(X=Ca,Sr)” Phys. Rev. B 99, 214510

I took part in measurements and assist with fabrication.

VII. M.A. Galin , E. A. Borodianskyi , V. V. Kurin , I. A. Shereshev-skiy, N. K. Vdovicheva , V. M. Krasnov , and A.M. Klushin,” Ev-idence of synchronization of large Josephson-junction arrays by traveling electromagnetic waves” EPJ Web of Conferences

195,02004 (2018) Author’s contribution:

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List of abbreviations

AC alternative current Bi-2212 Bi2Sr2CaCu2O8+δ

BSCCO Bi2Sr2CaCu2O8+δ

BWO Backward-wave Oscillator DC direct current

FFO flux-flow oscillator FIB focused-ion beam

FPGA field-programmable gate array GHz gigahertz

HTS high temperature superconductor IV current-voltage

ICP inductively coupled plasma JJs Josephson junctions

NIN normal metal-insulator-normal metal NIS normal metal-insulator-superconductor PCB printed circuit board

RCSJ resistively and capacitively shunted junction RF radio frequency

RIE reactive ion etching

SEM scanning electron microscope

SIS superconductor-insulator-superconductor THz terahertz

UV ultraviolet

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I Introduction

1.1 Motivation

Thirty-five years ago, two researchers from IBM, Georg Bednorz and Alex Muller, discovered a novel class of superconductors that had significantly higher critical temperatures compared with previously known low-temperature superconductors [1]. For that, they received a Nobel prize in 1987. This discovery excited great interest in the field of superconductivity, with rising expectations for the possible development of novel applications. Such unconventional superconductors are classified as a separate group with the overall name of high-temperature superconductors (HTS or High-Tc).

At present, various types of High- Tc superconductors exist. Particularly promising are copper-based superconductors that hold a record for critical transition temperature (Tc) at ambient pressure. The most well-known among them are those that contain bismuth-strontium-calcium-copper-oxide (BSCCO) and yttrium-barium-copper-oxide (YBCO) compounds. They have critical temperatures of around 100 K. However, there are HTS with even higher transitional temperatures, e.g. thallium [2] and mercury-based cuprates [3]. The latter has the highest Tc among cuprates: ~153 K under high pressure [3]. Today, however, the record Tc no longer belongs to cuprates. A recent study of H2S under

an extremely high pressure of 150 GPa indicated the record high of Tc ~203 K [4]. This suggests that hydrogen-based compounds under higher pressures may likely reach even room-temperature superconductivity.

High Tc is not the only advantage of HTS. They also have a larger superconducting gap. Furthermore, many cuprates are quasi-two-dimensional materials with high anisotropy. The layered structure of cuprates leads to the formation of natural stacks of atomic-scale “intrinsic” Josephson junctions (which will be studied in this thesis) because interlayer transport between conducting CuO2 planes occurs via tunneling [5, 6].

Conventional low-Tc superconductors have a gap in the range of 1 meV. This corresponds to microwave to far-infrared frequencies in the sub-THz range. However, HTS have tenfold higher values with regard to the superconducting gap. This allows usage of such materials in the terahertz range [7, 8]. It is advantageous for the creation of generators and detectors in the region between 0.1 and 10 THz, i.e, within the so-called “terahertz gap” [9]. There are several competing technologies (for example, quantum-cascade lasers) for the creation of emitters and detectors in this range, but they are mostly inefficient and limited for practical applications due to their size and cost. Furthermore, they are usually not tunable.

Josephson junctions allow the direct conversion of DC voltage into high-frequency electromagnetic waves [10]. Employment of intrinsic Josephson

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junctions in cuprate HTS provides an alternative way to create THz sources that can cover the frequency span of the terahertz gap. The goal of this thesis is to study the possibility of creating compact, continuous-wave and highly tunable

THz sources based on Josephson junction arrays.

1.1.1 Overview of Terahertz sources

The Terahertz radiation in the range from 1-30THz attracts tremendous growing interest in material science, imaging, and safety, defense applications [11]. This type of radiation was first observed at the beginning of the 20th century and been poorly investigated for a long time. But in the past twenty year’s situation changed towards the rapid development of various THz sources [12]. Here I will briefly cover existing THz sources and also successes in the superconducting direction.

The semiconductor oscillators or Solid-State Oscillators are a compact source in a frequency range of 100GHz to 1THz, and due to their compactness, the application range is continuously growing. The outcome power in the range of 100GHz is about 100mW while with increasing the frequency the power drops to 0.1-1mW [12, 13].

The Quantum Cascade Laser sources are one of the new inventions in the Terahertz range. Here electrons are injected to the periodic structure of a superlattice under applied bias where during the transition, a THz photon emission occurs with correspondent excitation by resonant tunneling [12, 14]. The first commercial Quantum cascade laser emitted 4.4THz and provided 2mW of power at an operating temperature of 50K [15]. But the average power from QCL is rapidly decreased by lowering the emitted frequency.

The laser-driven THz sources are based on frequency conversion from the optical range. There are two techniques used to achieve THz emission, one is from a femtosecond laser with a wide range of THz frequencies where the upper limit is settled by carrier recombination time[16,17], and another way to get THz spectrum is thought a sub-picosecond laser pulse to a crystal with a large second-order recipiency [18]. Those sources are mainly operated in the range of 0.2 to 2 THz.

Other THz sources can be organized in a category as Free Electron Based sources. Here are such examples as Travelling Wave Tubes, Backward Wave Oscillators, and Klystrons. Those sources mainly suffer from metallic wall losses and the need for high magnetic and electrical fields [12]. But BWO is the candidate that can operate by moderate power levels up to 100mW and can be tuned in the range of 30GHz to 1.2THz.

In parallel to semiconducting THz sources, a completely different technology for THz wave generation was discovered by Ozyuzer [8, 19]. Where the sub-micro-watt power at a frequency range of 0.36 to 0.86THz coherent emission was obtained from a Bi2212 single crystal mesa. Further improvement in the fabrication processes and experimenting with mesa shapes and amount of the

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intrinsic Josephson junctions by the research groups came to a 2.4THz frequency obtained from an inner branch of IV from cylindrical mesa [20]. While for rectangular mesa were observed in the frequency range of 0.3THz to 1.6THz [19, 21-23]. The most important feature for superconducting THz emission from intrinsic Josephson junction remains the output power that nowadays varies from 1 to 110μW for different types of mesa structures [23, 24]. And 620μW power at 0.51THz frequency from synchronization of an array of three conventional mesas [24]. That multi-mesa synchronization mechanism is essential for the future development of high-power and tunable High-Tc THz generators.

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1.2 Josephson Effect in BSCCO

1.2.1 Tunnel junction

In tunnel junctions, a charged carrier performs quantum-mechanical tunneling trough an insulating barrier. A tunnel current is strongly depends on an applied voltage and keeps essential information about the electronic density of states. On this principle the scanning tunneling spectroscopy is based [25]. When a voltage (eV) is applied to the tip, electrons from occupied states in the conduction band are caused to tunnel through the barrier to unoccupied states.

A normal metal-insulator-normal metal (NIN) junction IV characteristic will be linear at small voltages, as shown in Fig. 1.1, and asymptotically rises as the bias eV approaches the barrier height value [26].

Figure 1.1: Current-voltage characteristics of a NIN junction, where ϕ

~1 eV the barrier height [26].

Consider the tunnel junction made of a normal metal-insulator-superconductor (NIS); the correspondent IV will change, at bias voltages around ~1–10 mV corresponding to the energy gap of a superconductor will be visible [25].

The following equation can describe the tunnel current:

𝐼_𝑡 _{= 1 𝑒𝑅}

𝑛

⁄ ∫ |𝐸|⁄_[𝐸2_{− ∆}2_]1/2 ∞

−∞ [𝑓(𝐸) − 𝑓(𝐸 + 𝑒𝑉)]𝑑𝐸 (1.1)

where ∆ is an energy gap of the superconductor, f(E) is the Fermi-Dirac distribution function.

Fig.1.2 (below) shows the corresponding IV for a NIS junction where differences with respect to NIN case, Fig.1.1. The step in the current at eV=∆ reflects the singularity of the electronic density of the states in a superconductor.

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For a superconducting SIS tunnel junction, with two superconducting electrodes, there are two transport channels: for Cooper pairs, residing at the Fermi level and for single electrons (quasiparticles). The density of states for the latter is split into valence and the conducting bands separated by twice the superconducting energy gap. When a small voltage, eV<2∆ is applied, a quasiparticle current cannot flow through the junction , as quasiparticles below the gap on the right electrode do not have access to empty states on the left electrode. When applied voltage exceeds this limit, the quasiparticle current becomes possible and the current increases rapidly due to singularities in the density of states. At significantly large voltages the current-voltage characteristics become ohmic with a normal resistance Rn [26].

Figure 1.2: Current-voltage characteristics of the NIS junction, the

convolution of the tunneling densities of state [26].

If the superconducting tunnel junction consists of two different types of superconducting materials, each superconductor has a different energy gap that are marked as ∆₁ and ∆2 respectively. Fig. 1.3 shows band structure for such junction. An expression for tunnel current is given by [26]:

𝐼𝑠𝑠 = 1 𝑒𝑅⁄ 𝑛∫ |𝐸| [𝐸2− ∆12] 1 2 ⁄ ∞ −∞ × |𝐸 + 𝑒𝑉| [(𝐸 + 𝑒𝑉) 2_{− ∆} 2 2_]1₂ ⁄ × [𝑓(𝐸) − 𝑓(𝐸 + 𝑒𝑉)]𝑑𝐸 (1.2) Where |𝐸| > ∆₁ and |𝐸 + 𝑒𝑉| > ∆2.

Current-voltage characteristics for such junction shown in Fig. 1.4. At a finite temperature T>0, in addition to a current step at a sum-gap voltage 𝑒𝑉 = ∆₁+ ∆2, there is also a peculiarity at a difference voltage 𝑒𝑉 = |∆1− ∆2|.

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Figure 1.3: Energy diagram for a

superconductor-insulator-superconductor (SIS) junction, in this particular case for Sb/Sb-oxide/Pb [26, 27].

Figure 1.4: Current-voltage characteristics for SIS tunnel junction

[26].

1.2.2 High-Tc BSCCO crystals

Bi2Sr2Can-1CunO2n+4+δ (BSCCO) or, to give the compound its familiar name,

bismuth-strontium-calcium-copper-oxide, belongs to a group of high-Tc superconductors. When n = 2, the general formula changes to Bi2Sr2CaCu2O8+δ

which is also called Bi-2212 due to the first four indices. It was discovered in 1988 by a Japanese group of physicists and was the first high-Tc superconductor

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known at that time that did not consist of rare earth elements, in comparison with YBa2Cu3O7 [28].

As a cuprate superconductor, it has a perovskite-type structure with copper oxide layers where superconductivity occurs. Perovskites are a class of materials that have the same type of structure as calcium titanium oxide (CaTiO3), with

oxygen atoms located at the edge centers [29].

The Bi-2212 unit cell (shown in Fig. 1.5) is orthorhombic with a scale in a and b planes of ≃ 0.544 nm and c ≃ 3.090 nm. It includes 15 layers or 2[Bi2Sr2CaCu2O8] due to reasons of symmetry [30].

Figure 1.5: A schematic structure of Bi-2212 with two copper oxide

layers that form superconducting layers with thickness d = 3 Å and insulating layers BiO and SrO t = 12 Å [31].

Copper oxide layers have a thickness of d ≃ 3Å, while the thickness of the separating layer of BiO and SrO is t ≃ 12Å; this gives the superconducting layer periodicity of s=15Å. Bonding energies between different layers vary, with the weakest bonds located between the insulating BiO planes, which allows crystal splitting along them. The layered structure with a high anisotropy creates intrinsic Josephson junctions with a high-quality factor.

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In parent, Bi-2212 compound is not superconducting. Superconducting ability can be reached by oxygen doping. A δ coefficient should be around 0.1-0.23. Additional oxygen atoms require two more electrons for each of them. To fulfill this requirement, transformation of the copper 2+ ions into 3+ state takes place, and finally leading to formation of superconducting hole-doped CuO2 planes

[32].

Critical temperatures vary with δ. The best achieved Tc of 95 K was for δ = 0.16. However, it also depends on ion substitution in prepared crystals and can be increased by substitution of Pb (Bi, Pb)2Sr2CaCu2O8, which gives a maximum

achieved Tc of 102 K [33].

1.2.3 DC Josephson effect

If a small current is sent through the Josephson junction or also called weak link, it would pass without resistance, even if the link material is non-superconducting. Similar wave functions describe superconductivity condensate on both sides of the junction 𝜓_𝑖 = √𝑛𝑖 𝑒𝑖𝜑𝑖 [26] (ni are density of the superconducting Cooper

pairs in electrodes, φi are phases). A potential energy of 𝑈2− 𝑈1= 2𝑒𝑉 that exists between electrodes allows writing of the following coupled Schrödinger equations:

𝑖ℏ

𝜕𝜓1 𝑑𝑡

= 𝑈

1

𝜓

1

+ 𝐾𝜓

2

,

𝑖ℏ

𝜕𝜓2 𝑑𝑡

= 𝑈

2

𝜓

2

+ 𝐾𝜓

1, (1.3)

where K as coupling constant. For a finite phase difference Δ𝜑 = 𝜑₂− 𝜑1, it leads to a supercurrent flow:

𝐼 = 𝐼

_𝑐

sin ∆𝜑

(1.4)

This equation was predicted in 1962 by David Josephson [34] and is also named as the 1st_{Josephson equation. It represents the DC Josephson effect, where}

𝐼𝑐 = 2𝐾(𝑛1𝑛2)1/2/ℏ is the critical current of the junction.

1.2.4 AC Josephson effect

The 2nd_{Josephson equation, also called the voltage-phase relation is:}

𝜕(Δ𝜑)

𝜕𝑡

=

2𝑒𝑉

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For the fixed applied voltage difference over the barrier, there is a continuous increase of the phase difference [26]

Δ𝜑(𝑡) = Δ𝜑(0) +

2𝑒𝑉

ℏ

𝑡

, (1.6)

The Josephson current will oscillate at a defined frequency, or so-called Josephson frequency:

𝑓 =

2𝑒𝑉

ℎ , (1.7)

Voltage-dependent frequency is 𝑓/𝑉 = 2𝑒/ℎ = 1/𝛷₀ = 483.6 GHz m𝑉−1. Here Φ₀ – flux quantum ≈ 2.07∙10-15_Tm2_{[26]. Therefore, the Josephson junction}

allows direct conversion of the DC voltage into high-frequency electric current.

1.2.5 Current-voltage characteristics, RCSJ model,

sine-Gordon equation

Josephson relations cannot entirely describe all the physics of the actual Josephson junction.

A more detailed description is given by a resistively and capacitively shunted junction (RCSJ) model, where the sum of the supercurrent Is, the quasiparticle

current Iqp and the displacement current Idis gives the total current through the

junction. In terms of phase difference, the overall equation can be written out as:

𝐼 = 𝐼

_𝑠

+

𝑉 𝑅𝑞𝑝

+ 𝐶

𝑑𝑉 𝑑𝑡

= 𝐼

𝑐

𝑠𝑖𝑛Δ𝜑 +

Φ0 2𝜋𝑅 𝜕𝜑 𝜕𝑡

+

𝐶Φ0 2𝜋 𝜕2𝜑 𝜕𝑡2 , (1.8) Here Rqp is the quasiparticle resistance, and C is the capacitance of the

junction. The IV curve shape depends on McCumbers parameter 𝛽_𝑐 = 2𝜋𝐼𝑐𝑅𝑞𝑝2 𝐶/Φ0 [35]. The equation simplifies through introducing the Josephson plasma frequency 𝜔_𝑝= √2𝜋𝐼𝑐/Φ0𝐶 which defines 𝜏 = 𝜔𝑝𝑡 and the quality factor𝑄 = 𝜔𝑝𝑅𝐶. 𝐼 𝐼𝑐

= 𝑠𝑖𝑛Δ𝜑 +

1 𝑄 𝜕𝜑 𝜕𝜏

+

𝜕2𝜑 𝜕𝜏2 , (1.9) 𝑄 related with so-called “damping factor” 𝛼 and the McCumbers parameter as following 𝑄 = √𝛽𝑐 = 1/𝛼. The quality factor influences the shape of the IVs. In other words, when 𝑄 or 𝛽𝑐 ≫ 1 the current-voltage characteristic become underdamped and, in the case of 𝑄 < 1, overdamped. Current-voltage

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characteristics in underdamped and overdamped cases are presented in Fig. 1.6 where the difference between hysteresis type IV and non-hysteresis IV can be seen[35].

Figure 1.6: Two IV characteristics (a) in the underdamped case

with the McCumbers parameter 𝛽𝑐 ≫ 1, large R and C, (b) in the overdamped case with the McCumbers parameter 𝛽_𝑐 < 1 and small R and C.

However, the RCSJ model has a limitation. It neglects possible screening of the magnetic field by supercurrent in the junction. Therefore, it is valid only for “short” junctions. The phase variation over the sample length in the x-direction is caused by the magnetic field applied in the y-direction at the moment when the current flows perpendicular to both of them through the junction in the z-direction. In that case, an additional term is added to equation (1.9). It describes the phase dynamics over the junction [36].

𝑗 𝑗_𝑐

= 𝑠𝑖𝑛Δ𝜑 +

1 𝑄 𝜕𝜑 𝜕𝜏

+

𝜕2𝜑 𝜕𝜏2

− 𝜆

𝐽 2 𝜕2𝜑 𝜕𝑥2

,

(1.10) This equation is called a sine-Gordon equation. The Josephson penetration depth here 𝜆_𝐽 describes how deep the magnetic field penetrates into the junction.

𝜆

_𝐽

= √

Φ0

2𝜋𝜇0𝑗𝑐Λ

(1.11) Λ is an effective magnetic thickness (penetration of the magnetic field into the junction and electrodes in the z-direction). An equation for it is [35]:

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Λ = 2𝜆

_𝑠

tanh

𝑑

2𝜆_𝑠

+ 𝑡

(1.12)

The electrode thickness is represented by d, which in the case of BSCCO crystals is 3Ǻ. 𝜆_𝑠 is the London penetration depth of a superconductor, while t represents the thickness of the insulating layer, which is 12Ǻ. Equation (1.12) is more general and can be simplified.

For 𝑑 ≫ 𝜆_𝑠when the electrode is thicker than the London penetration depth, it will transform to:

Λ = 2𝜆

_𝑠

+ 𝑡

(1.13)

And when 𝑑 ≪ 𝜆_𝑠 the equation transforms into an even simpler form:

𝛬 = 𝑑 + 𝑡

(1.14

)

This equation will give a value of

𝛬

– 1.5nm, which is equal to the single junction thickness in the BSCCO crystal [35,36].

1.2.6 Intrinsic Josephson junctions

As has been mentioned above, the naturally created Josephson junctions inside the BSCCO crystals are atomically perfect and settle between double copper oxide planes. The total size of the junction due to the lattice structure is 1.5 nm in the c-axis direction. Strong anisotropy of the crystals in the normal state and a critical current with a high bypass resistance create the underdamped junctions with a high-quality factor Q.

Bi-2212 has the London penetration depth

𝜆

_𝑠 of around 75-100 nm inside the copper oxide planes [37]. However, knowing that screening currents only fill d/s part of the total volume [38]. Therefore, in the ab-plane the effective London penetration depth for a field in the c-direction is:

𝜆

_𝑎𝑏

= 𝜆

_𝑠

√

𝑑

𝑠 (1.15)

while the effective penetration depth for a field in the ab-plane depends on the anisotropy 𝛾 and determined as:

𝜆

_𝑐

= 𝛾𝜆

_𝑎𝑏

(1.16)

Typical penetration depth values in Bi-2212 are much larger than the thickness of the junction in total (0.15 µm to 0.30 µm for

𝜆

_𝑎𝑏 and 15 to 180 µm for

𝜆

_𝑐

(30)

[37, 38]) and the effective magnetic thickness according to Eq. (1.14) is Λ = 𝑑 + 𝑡 = 𝑠 ~ 1.5𝑛𝑚. Its small value affects the Josephson penetration depth:

𝜆

_𝐽

= √

_4𝜋𝜇Φ0

0𝑗𝑐𝑠

(1.17) Which corresponds to a large

𝜆

𝑐

,

Eq. (1.16).

1.2.7 Washboard potential

A Josephson junction can be characterized by a Hamiltonian ℋ, as a function of the phase difference between two superconducting electrodes.

ℋ = −4𝐸

_𝑐 𝜕2

𝜕𝜑2

− 𝐸

𝐽

cos 𝜑

(1.18)

Where 𝐸_𝑐 and 𝐸_𝐽 are the charging and Josephson energies, respectively [39]. 𝐸𝑐 is usually much and can be neglected. In this case, the dynamics of phase in the junction can be represented as a motion of an imaginary particle locked into the potential gap of a washboard potential, shown in Fig.1.7.

𝑈(𝜑) = −𝐸

_𝐽

[cos 𝜑 + (

𝐼

𝐼𝑐

)𝜑]

(1.19)

If I<Ic the particle will be trapped in a potential well, where it can oscillate

with the Josephson plasma frequency 𝜔_𝑝= √2𝜋𝐼𝑐/Φ0𝐶 . Increasing I affects the potential by tilting and lowering the barrier in between two nearby potential minima. Upon, reaching the condition of I=Ic, the phase escapes and the voltage

appears. By decreasing the current, the particle will be trapped again in the minima, as I=Ir reaches the retrapping current value.

For junctions with a high-quality factor Q [39] with Ir<Ic the IV show a

hysteresis behavior, while in the case of a low-quality factor no hysteresis appears [35].

However, the particle can escape from the locked state not only by continuous biasing of the junction but also due to thermal fluctuations. When kBT<<EJ

thermal activated escapes have a small probability of ~𝑒(−Δ𝑈(𝐼)/𝑘𝐵𝑇) at each

attempt. A potential barrier for escaping from the washboard potential is:

Δ𝑈 ≃ (4√2𝑈

0

/3) (1 −

𝐼 𝐼_𝑐

)

3/2

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But in the presence of thermal fluctuations, the critical current is not well-defined, so only an escape rate or probability of switching from the stationary to the running state can be defined:

Γ

_𝑡

= 𝑎

_𝑡

(

𝜔𝑝

2𝜋

) 𝑒

(−Δ𝑈/𝑘_𝐵𝑇) _(1.21)

Here 𝑎_𝑡 is an order unity coefficient. And here clearly it can be seen that the escape rate changes from rare occasions for a small current when Δ𝑈~2𝐸_𝐽 ≫ 𝑘_𝐵𝑇 to a large rate 𝜔_𝑝/2𝜋 for I=I0.

Fig. 1.7 Tilted washboard potential diagram of the Josephson

junction. Arrow is showing possible escape of the particle via thermal or electromagnetic activation.

1.2.8 Switching current detection of electromagnetic

radiation

The incoming electromagnetic wave can also cause the escape from the potential as thermal fluctuations do. The high frequency signal induces alternating current in the junction, which shakes the wash-board potential and, therefore, switching current. This allows usage of a Josephson junction as a sensitive detector for radiation detection. The sensitivity of such detector can be tuned by adjusting the amplitude of the low-frequency ac bias current Iac, which

Potent

ial

(a.

u.

)

Phase

ΔU

U

Thermal escape

Electromagnetic wave

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determines the maximum value of the barrier height Eq. (1.20). The total current in the detector will be 𝐼 = 𝐼𝑎𝑐+ 𝐼𝑇𝐻𝑧. Where ITHz is the high-frequency current

induced by the incoming THz radiation. Therefore, a reduction of the switching current in the presence of radiation directly indicates the amplitude of the induced current +ITHz as shown in Fig. 1.8.

Fig.1.8: Sketch of the switch-current detector principle. With an

absence of incoming radiation the switching current is equal to the critical current of the junction, with incoming radiation switching occurring with suppressed switching current by THz signal.

1.3 Coherent superradiant emission

There are two steps to achieving coherent superradiant emission from Josephson junctions: firstly, by creating a stack of JJs using BSCCO crystals; secondly, by creating a large array of JJs.

The total radiative power from such structures is proportional to the total number of active junctions squared.

𝑃

_𝑟𝑎𝑑

~𝐸

2

𝐸

_𝑎𝑐

= ∑

𝑁_𝑖=1

𝐸

_𝑖

(1.22)

For the in-phase case the total power will be Ei = Ei+1, Eac = NEi, P~N2. This represents the supperradiant amplification of radiation. On the other hand, for the out-of-phase state: Ei= - Ei+1, Eac = 0 and the total power will be

P~ 0. This represent the coherent suppression of radiation (destructive

interference).

V

I

_THz

≠ 0

I

_THz

I

_THz

= 0

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1.3.1 Flux-flow emission from a single junction

Josephson junctions, due to their properties, can be used as electromagnetic generation sources [35]. One of possible ways to achieve generation is through motion of fluxons. A fluxon, is a vortex, circulation of a supercurrent within the junction. The supercurrent varies at a scale of a Josephson penetration depth from the center of the vortex and the total magnetic flux induced – equals to the single flux quantum Φ0. When a bias current is applied to the junction it exerts a Lorentz force on a vortex and causes its motion. This motion generates electromagnetic waves. Devices built on such a basis are called flux-flow oscillators or in short FFO [40-43]. The fluxon motion can be affected by interaction with Abrikosov vortices in junction electrodes, close to the junction. Unique designs of pinning centers for trapping Abrikosov vortices can be used for creating memory elements based on a single vortex and a single Josephson junction [44].

For a better understanding of how the flux-flow oscillator works, a simple illustration of a single JJ is given in Fig. 1.9. Here are some important notes about a junction for flux-flow generators. The long side length L of the junction should be larger than the Josephson penetration depth. An external magnetic field B should be applied perpendicular to the entire length of the junction and placed parallel to the dielectric layer [43].

Figure 1.9: Flux-flow oscillator, I – bias current, B – applied

magnetic field, VFF – fluxon velocity, electromagnetic wave radiated

when fluxon escapes the junction

The repulsive forces between fluxons leads to the formation of a fluxon chain within the junction. This chain of fluxons is pinned at the edges of the junction. A Lorentz force pushes fluxons when the bias current is applied through the junction. Above some critical value, the Lorentz force becomes larger than the pinning force and the fluxon chain starts moving. As a result of this unidirectional movement, fluxons enter into the junction from one side and exit on the other side. A balance is establishing between the Lorentz and the viscous damping forces at any bias value, so fluxons move with the constant velocity vFF. Fluxon

movement also induces a flux-flow voltage that can be defined via the AC-

B

I

L

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Josephson relation 𝑉⊥= Φ0𝑣𝐹𝐹/𝐿. A phase shift inducing by each fluxon is ∆𝜑 = 2𝜋 [43]. Each fluxon requires some time to pass over the junction; this time is equal to 𝑡 = 𝐿/𝑣_𝐹𝐹

The total number of the fluxons in the junction linearly depends on the length and magnetic field B, and can be calculated as 𝑁 = 𝐵𝑑_𝑒𝑓𝑓𝐿/Φ0. Therefore, the total voltage VFF is:

𝑉

_𝐹𝐹

= 𝑁𝑉

_⊥

= 𝐵𝑑

_𝑒𝑓𝑓

𝑣

_𝐹𝐹

(1.23)

This equation shows that the flux-flow voltage depends on the flux-flow velocity and the applied magnetic field but is utterly independent from the junction length.

Current increase leads to an acceleration of fluxons to the limiting velocity, or Swihart velocity c0. It is the speed of light in the junction, which behaves as the

superconducting transmission line.

Electromagnetic waves appear when the fluxon escapes from the junction. The amount of fluxons that reach the edge of the junction per second affects the final frequency where 𝑓 = 𝐵𝑑_𝑒𝑓𝑓𝑣𝐹𝐹/Φ0 [43] The same frequency can be obtained via the AC-Josephson equation (1.7) 𝑓 = 𝑉_𝐹𝐹/Φ0 . Some part of the emission is reflected back and thus excites cavity resonances and standing waves in the junction. As a result, geometrical (Fiske) resonances occur.

Oscillators based on the single Josephson junction are well developed and show budding promise as local oscillators [45]. Increasing the total number of junctions can enable a higher emission power, but coherent oscillation for such purposes is required. Junctions should be coupled with each other for that. Such a result has been achieved in Ref. [46] with an array of 2D JJs where junctions have been synchronized by an external resonator.

1.3.2 The Coupled sine-Gordon equation for inductively

coupled stacked Josephson junctions

As seen in part 1.2.6 above, superconducting layers in Bi-2212 are thinner in comparison with the penetration depth

𝜆

_𝑎𝑏. Therefore, magnetic field cannot be screened by a single layer and screening currents in one of the junctions will influence nearby junctions. This leads to all junctions being inductively coupled via a shared magnetic field.

A two-junction stack

The simplest case is that for a stack with two identical junctions [47]. In this case coupled-sine Gordon equation can be written as:

𝜆

_𝐽2 𝜕2 𝜕𝑥2

(

𝜑

₁

𝜑

₂

) = (

_−𝑆

1 −𝑆

₁

) (

𝐽

₁

𝐽

₂

)

(1.24)

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Where the coupling parameter is determined as

𝑆 =

𝜆𝑠

Λ sinh𝑑 𝜆⁄ 𝑠

(1.25) Where Λ an effective magnetic thickness and is given by Eq. (1.12) and 𝐽1,2 are currents through the two junctions according to RCSJ model.

This equation can be solved analytically under certain conditions. In the case of small Josephson current and zero bias current 𝐽𝑖

𝐽_𝑐 = 𝜔𝑝 −2 𝜕2𝜑𝑖

𝜕𝑡2 and

𝜑

𝑖 = 𝐴𝑖𝑒𝑖𝑘𝑥(𝑥−𝑢𝑡) the equation can be written as:

𝑢−2(

𝜑

_𝜑

1 2)

=

𝑐0 −2₍

1 −𝑆

−𝑆

1

) (

𝜑

₁

𝜑

₂)

(1.26) Where 𝑐₀= 𝜆𝐽𝜔𝑝= 𝑐√ 1

4𝜋Λ𝐶 is the Swihart velocity of the single junction. Calculation of eigenvalues will result in:

|𝑐0 −2_{− 𝑢}−2 _−𝑐 0−2𝑆 −𝑐₀−2_𝑆 _𝑐 0−2− 𝑢−2 | = 0

𝑢

_±

=

𝑐0 √1±𝑆

(1.27)

This equation gives two characteristic velocities in the stack of two junctions, and it is seen that one of them is higher, while another is lower than the velocity 𝑐0 for a single junction. Higher velocity corresponds to the in-phase mode 𝜑1= 𝜑2; the lower corresponds to the out-of-phase mode 𝜑1= −𝜑2.

A N-junction stack

In the stack of N junctions, the phase change in i-junction is related to i±1 junction, and the sine-Gordon equation transforms to a coupled equation or CSGE [39, 47-53]. Considering the case when all junctions are identical, the CSGE equation is:

𝜆

_𝐽2 𝜕2 𝜕𝑥2

(

𝜑

₁

⋮

𝜑

_𝑁

₎

=

(

1 𝑆

0 𝑆

1 𝑆

0 ⋮

⋱

0 ⋱ ⋱

⋮

𝑆

1 𝑆

0 𝑆

₁₎

₍

𝐽

₁

⋮

𝐽

_𝑖

⋮

𝐽

_𝑁

₎

(1.28)

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where:

𝐽

_𝑖

=

𝜕𝜑𝑖2 𝜕𝜏2

+

1 𝑄 𝜕𝜑𝑖 𝜕𝜏

+ sin 𝜑

𝑖

−

𝑗 𝑗𝑐 (1.29) A system of coupled differential equations sets the boundary conditions in the external magnetic field [39, 48, 52]

+

𝜕𝜑𝑖

𝜕𝑥

|

_𝑥=0,𝐿

=

2𝜋𝜇0

Φ0

𝐻

0

Λ(1 − 2𝑆)

(1.30) As the sine-Gordon equation has the form of a nonlinear wave equation, the Josephson junction can be represented as a chain of pendulums. For small amplitudes, plasma waves can exist in the junction with the following dispersion relation for a single junction:

𝜔(𝑘)

2

= 𝜔

_𝑝2

+ 𝑘

2

𝑐

₀2 (1.31) As we have already seen for the two-junction stack, Eq. (1.27), coupling in the multi-junction stack with the coupled Sine-Gordon equation leads to splitting of dispersion relation of electromagnetic waves into N branches with different characteristic velocities [53].

𝑐

_𝑛

=

𝑐0

√1+2𝑆 cos_𝑁+1𝑛𝜋

, 𝑛 = 1, … , 𝑁

(1.32)

Here N is the number of junctions in the stack

With a few simplifications, it can be seen that the slowest velocity is

𝑐 N≈𝑐 0/√1 + 2𝑆 and almost does not depend on N [53] as it is shown in Fig. 1.10.

For the single-junction the plasma waves 𝜔(𝑘) dispersion relation depends on the in-plane wave vector k. Meanwhile for stacked junctions, there is an additional component, the wave vector q for the c direction [53]. The amount of modes in the c direction is determined by the number of junctions, N, in the stack. Wave numbers are quantized, 𝑞_𝑛=𝑛𝜋

𝑁𝑠 n=1…, N, and characteristic velocities are given by Eq. (1.32) cn. For standing waves, in-plane, there are another standing

wave mode number l =1, 2…., 𝑘_𝑙 =𝑙𝜋

𝐿 [39, 54], where L is the in-plane length of the stack.

1.3.3 Geometrical resonances in stacked junctions

For a single Josephson junction, geometrical resonances occur when integer number of half-wave length of electromagnetic waves fit into the junction length.

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This leads to formation of standing waves in a transmission line, formed by the junction. Geometrical resonances lead to appearance of Fiske steps in current-voltage characteristics at 𝑉_𝑛 =Φ0𝐶0

2𝐿 𝑛, which correspond to the condition that Josephson frequency coincides with the geometrical resonance (cavity mode) frequency.

Fiske steps appear due to the interaction between the Josephson current and the standing electromagnetic wave. Such behavior forms step-like current levels which were firstly observed by M. Fiske in 1965 [55] and named after him.

For stacked Josephson junctions the number of cavity modes is enhanced due to splitting of the dispersion relation of electromagnetic waves. Therefore, Fiske steps may appear at:

𝑉

_𝑛,𝑙

=

Φ0

2𝐿

𝑙𝑐

𝑛 (1.33)

Different modes correspond to different configurations of electric fields in the stack. As for the two-junction stack, the slowest mode cN corresponds to the

out-of phase state, 𝐸_𝑖 = −𝐸𝑖+1, and the fastes mode c1, to the in-phase state 𝐸𝑖 = 𝐸𝑖+1. Consequently, the superradiant amplification of emission should occur only for the c1 mode.

For a rectangular stack with in-plane sizes Lx, Ly two-dimensional cavity

modes occur. The expecting frequencies of strongly emitting in-phase geometrical resonances are:

𝑓

_𝑛,𝑙

=

𝑐1 2

√

𝑙2 𝐿2𝑥

+

𝑛2 𝐿2𝑦 (1.34)

Figure 1.10: Velocities of the fastest c1 and the slowest cN

electromagnetic wave modes in stacked Josephson junctions as a function of the number of junctions. cN almost does not depend

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From Eq. (1.32), c1 depends on N. the corresponding dependence is shown in Fig.

1.10. For Bi-2212 c1 ≈ 0.1c for 200 junctions (c – the speed of light in vacuum)

as can be seen from Fig. 1.10 [54].

Fiske steps have since been observed in stacked junctions of Bi-2212 [56-59] as shown in Fig. 1.11.

Figure 1.11: Fiske steps observed in Bi-2212 mesastructure from

Ref. [59]

One quite important notice is that the Fiske steps amplitudes depend on the applied external magnetic field. Step amplitudes oscillate with field those steps that close to velocity matching condition VFF~ c0 have the highest amplitude [35,

43]. Mode l calculated from simple equation VFF/NFF=Vn,l gives l=2Φ/Φ0. Even

l steps oscillate in-phase while odd in anti-phase with Ic(H) oscillation. And the

quality factor of geometrical resonances is Qn,l = 2πfn,lReffC, where Reff is

effective damping resistance [44, 60].

1.3.4 Synchronization of large Josephson junction arrays

High-Tc superconductors have higher values for the superconducting gap in comparison with conventional superconductors. Using intrinsic Josephson junctions, it is possible to convert DC-voltage into electromagnetic radiation in the whole THz range. However, it is most challenging to let them oscillate in a synchronized way to reach a higher power of emitted radiation. To get a sufficient power output, huge arrays with thousands of Josephson junctions are required, and such arrays may easily be larger than 1 cm and larger than the wavelength 𝜆 for sub-THz frequencies.

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One of the ways to synchronize junctions discussed above is based on a resonant electromagnetic mode by some external cavity [46] or within the junction [54, 60-62]. However, as we would like to employ very large arrays of junctions, it becomes progressively harder to achieve a synchronized superradiant emission. Junctions parameters may vary as well environmental conditions thus complicating synchronization.

The alternative proposal is to synchronize very large arrays of Josephson junctions in a non-resonant manner by travelling electromagnetic waves, similar to operation of traveling wave antenna. [63-65].

The primary feature of the traveling-wave antenna is the strong forward-backward asymmetry of the emission with a significant amount of power in the forward direction of propagation of the traveling wave [66]. In other words, coherent emission can occur due to the unidirectional propagating wave imprinting the defined phase distribution over the whole array. It has been suggested in Ref. [65, 67] that large Josephson junction arrays can work as a Josephson traveling-wave antenna with similarities to the operation of a Beverage antenna [64]. Such antenna has an asymmetric directionality diagram with a maximum in the direction of the wave propagation at the angle α = arccos ℎ/𝑘, where h is the wave number of current oscillations that occur in the antenna, and k is the wave number for the emitted wave. If the ratio between h/k

< 1, the traveling wave will be radiating to the lateral direction, and h/k > 1, when

the angle α becomes imaginary, our traveling wave turns into a surface plasmon that travels between the wafer and the electrodes, with radiation taking place from ends of the Josephson junction array.

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II Experimental part

2.1 Sample fabrication and equipment

This section describes several significant steps: the micro- and nanofabrication tools; a measurement set-up; and the sample preparation steps. The fabrication process requires the selection of a BSCCO crystal and the creation of a final stack of Josephson junctions by various micro and nano-fabrication techniques. More-over, an important part of the process is to etch a significant amount of JJs and create defined shaped mesa structures for future measurements. All samples were fabricated at the AlbaNova NanoFabLab, and electrical measurements at low-temperatures were made at the CryoLab of the EKMF group.

The key for the presented work is fabrication of high-quality samples. That fabrication process contains several steps, including: selecting a crystal; cleav-ing; e-beam evaporation; lithography processes; oxygen ashcleav-ing; Ar ion etchcleav-ing; magnetron sputtering; bonding; and finally, SEM/FIB manipulation for curing some cracks or for cutting off short circuits and shaping final mesas. The tech-nological process is similar for all devices that were produced in previous works [38, 43], but the experimental strategy entirely different.

Crystal cleaving

The first and essential task is to select and cleave the initial crystal. I use sapphire substrates with dimensions of 5×5 mm; as crystal carriers for creating a final sample with contact pads, A BSCCO crystal should be small enough, around 200×200 µm, to be placed on the center of the sapphire to simplify the creation of future contact electrodes. If the crystal is too big, it should be cut with a scalpel into smaller pieces.

The BSCCO crystal must be cleaved in two parts to create a fresh surface for future Josephson junction stacks, as a non-cleaved crystal has an oxidized sur-face, which does not allow sending current through it.

After the crystals are selected, they are fixed for the cleaving process — an epoxy glue 353ND was used for this. I use sapphire substrates with dimensions of 5×5 mm; as crystal carriers for creating a final sample with contact pads. The small drop of epoxy is placed on top of the substrate and the crystal is placed on it. Due to capillary forces, epoxy will capsulate crystal, so another substrate is put on top. It is rotated by 45 degrees respectively to the lower one. This will minimize the hardness of the cleaving process and it will be easier to manipulate with substrates. The glued crystal between two sapphire substrates is shown in Fig. 2.1. It can be seen that the epoxy reaches the bottom part of the substrate as it became more viscous during heating. It also requires time to reach its full

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strength and hardness. This can be achieved by leaving such sandwiches for 12 hours; also, it can be achieved more quickly by baking them in an oven for 4 hours at a temperature of about 110-120ºC.

Afterwards, the stack of glued sapphire substrates can be cleaved by splitting two substrates with a scalpel. The BSCCO crystal cleaves in-between BiO planes similar to the behavior of an HOPG graphite when trying to create graphene lay-ers. Bonds within isolating SrO-BiO-BiO-SrO layers are weaker than within con-ducting CuO2-Ca-CuO2 layers see Fig. 1.6 (crystal structure). Ideally, it should

be cleaved over one smooth plane, but in practice there will be steps, cracks and surface roughness that relate to different thicknesses of the crystal on its different sides and stresses applied to the crystal during the gluing process. The epoxy that has hardened can be removed with a scalpel, with care being taken not to damage the crystal.

Figure 2.1: Two sapphire substrates with 5×5 mm and 0.5 mm

thickness and BSCCO crystal (small black flake) glued in-between them.

Electron beam deposition/gold coating

After the crystal is cleaved, it cannot be exposed to the air for a long time as its surface will passivate rapidly, in roughly 10 minutes. That is why cleaved substrates are immediately put in a vacuum chamber and a 50-60 nm protective layer of gold is deposited on top. For this, the sample was placed into the vacuum chamber in the Eurovac system for E-beam evaporation.

The electron beam deposition is one of the simplest techniques that can be used for depositing of various materials. The simplicity of that method is based on the fact that a high energy electron beam from a tungsten filament heats the

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crucible with deposition material. Part of the material evaporates and atoms are ballistically transferred to the sample and are deposited. A high vacuum is needed for achieving a good purity of deposited film as any gases inside might affect it and the film properties could deteriorate. A high vacuum is also required for ma-terial particles to travel freely from the target to the substrate, so the mean free path should be significant enough (larger than the chamber size) to facilitate such deposition.

Figure 2.2: Eurovac E-beam deposition system: 1 the main chamber,

where deposition occurs; 2 the load-lock for loading and unloading samples; 3 vacuum gauges, one for the load-lock to avoid a pressure drop in the main chamber before transporting samples and a gauge for monitoring pressure during deposition, there is a valve in-between them that separates the load-lock from the main chamber; 4 the trans-fer rod that allows samples to be transtrans-ferred to the main chamber and taken out after deposition; 5 a tilting rod for the tilting deposition, rotates by 360˚; 6 power supply; 7 a thickness monitor for controlling deposition rates and the deposited film thickness.

The Eurovac is a custom-made electron beam evaporation system. It can be used for the deposition of various materials, layer by layer, without breaking a vacuum (see Fig. 2.2). Due to the simplicity of the system and easy maintenance, it is mainly used for the deposition of materials such as gold, cobalt, titanium, copper, and calcium-fluoride as an insulating layer. The system, shown in Fig.2.2, is constructed in such a way that the main-deposition chamber 1 is kept under a vacuum of 10-7_{mbar, while a load lock 2 is separated by a safety valve}

and can be vented to atmospheric pressure for the sample mounting on a transfer rod 4. Transferring the sample can be tricky. In order not to drop it inside the

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main chamber during transference to its final position, three clamps hold the sam-ple carrier on a transfer rod, and three more are at the main chamber holder: the operator is required to catch it inside the main chamber by pressing and rotating the transfer rod. The sample holder in the main chamber can be tilted at different angles for an angular deposition 5. For example, this allows step coverage be-tween crystal and substrate edges even if the crystal is rather high. A power sup-ply 6 has a “remote” control panel that can be used to increase the applied beam power and to adjust the beam position over the target in x and y directions with a wobbling amplitude for smooth heating of the crucible. Thickness monitor 7 uses a quartz crystal with a known resonance frequency that changes when ma-terial is deposited. Calibration constants can be set with correspondence to the densities of different materials and the chamber’s geometry. As a result, a final thickness of the film can be measured and controlled in-situ, with an accuracy of up to a few Å. The deposited film thickness can be checked afterwards by a KLA Tencor P-15 surface profiler with a vertical resolution of 0.5Å and a maximum scan length of 200 mm.

Photolithography and etching

The next step after the gold deposition will mainly determine the quality of the sample at the end of fabrication. Several sequences of photolithography were conducted during device fabrication, and different masks were used. Fig. 2.3 is a sketch of the photolithography and etching steps. Etching always occurs after lithography and completes the created structure.

All lithography processes were done under a yellow room light for minimizing the chance of unwanted exposure of the resist to daylight sources.

Photolithography is a method where the pattern on the mask is transferred to the sample using a UV light source. A Canon PPC 210 projection camera was used for that purpose, as shown in Fig. 2.4. Here the UV light from a mercury lamp with well-defined intensity shine on the mask with a pattern and then passed through the optical system and focus on the sample that was spin-coated with the resist.

Before exposure, the yellow filter was used to align the sample properly to the mask pattern. Then the filter was removed, and UV light exposed the positive photoresist S1818/S1813. After the exposure, the resist was developed using a developer, in this case MF-319, a chemical reagent that removes exposed parts. In the case of negative photoresist, non-exposed parts will be removed, but an-other developer should be used.

Further use of the photoresist on top of the sample can be used as a protection layer for a subsequent etching. Alternatively, a deposited material can be re-moved together with the photoresist via the lift-off process. The photoresists used, S1818 and S1813, differ in viscosity which gives different thicknesses of 1.8 µm and 1.3 µm respectively when spin coating them at 4000 rpm for 60 sec-onds. Due to the limitations of the PPC system and its age, the best-achieved resolution is ~2 µm. One minute of baking on a hot plate under 100ºC is needed

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for the resist to harden before the actual lithography process takes place. Expo-sure takes place for 35-40 seconds, and the transmitted mask pattern should be developed for the next ~30 seconds in the developer. The sample should be checked under the optical microscope to ensure that the photolithography has taken place without any defects. If the pattern shows defects, the photoresist can be removed from the sample using acetone, then cleaned using isopropanol and distilled water, and the entire photolithography process is iterated again.

Figure 2.3: Sketch of the photolithography and etching process.

The first step is needed to create a square pattern with side length of 100 µm on the top of our crystal to prepare a working space, where at the next step a mesa line will be made. The lithography of the square is simple due to its large size. However, the second step, with the thin 5 µm mesa line, should be done with exquisite accuracy. crystal substrate Au layer 3.Development mask UV

2.UV exposure 5.O2 ashing

4.Etching

photoresist

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Figure 2.4: Canon PPC 210 system: Mercury lamp is placed in

light-house on top (not present on image due to lamp failure), then light passed across lens and retractable yellow filter (for alignment, not to expose resist before lithography), and shone on mask with pattern, transmitted through optical column to the substrate for alignment and exposure.

A KI+ water solution was used for removing the residual gold on top of the substrate and the crystal. The remaining resist preserves the gold layer from a wet gold etch.

At the second step, a mesa line 50 µm long and 5µm wide was created simi-larly, using another mask with a line pattern. After that all the remaining gold around the mesa line should be etched away.

The etching is used for creation the final shape of the mesa line with a desired number of Josephson junctions. The etching process can be divided into physical and chemical etching techniques. The physical etching occurs when the material that is to be etched is bombarded by ions and then sputtered away. Ar ions being used for this purpose. Chemical etching involves a chemical reaction with good selectivity to the different materials. For dry chemical etching volatile products that form during the chemical reactions are pumped away.

A wet chemical etching was performed with a KI+ water solution where the gold was etched with excellent selectivity to the BSCCO and the photoresist. The dry chemical etching is performed in the gas atmosphere. The chemically active gas reacts with the sample, and the reaction takes place in the inductively coupled plasma. In addition, the physical sputtering may occur if the kinetic energy of atoms is high enough.

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Figure 2.5: Oxford Instruments PlasmaLab 80 RIE-ICP used for

ox-ygen ashing of photoresist and cleanup processes (left); Oxford Plas-maLab System 100 used for argon sputtering and chemical etching

(right).

An oxygen ashing process can easily remove organic materials such as the photoresist by using the Oxford Instruments PlasmaLab 80 RIE-ICP shown in Fig. 2.5 (left) during the chemical reaction where burning of the photoresist took place, with good tolerance to the BSCCO and deposited films.

The sample was placed on a power electrode in the process chamber. First, the lithographed square can be ashed away with a soft oxygen ashing recipe (if ace-tone was not used for that), under a pressure of 100mTorr to prevent sputtering and damage of unprotected parts, with a low RF (radio frequency) generator power of 10W and an applied ICP (inducted coupled plasma) power of 50W. Increasing the ICP power will lead to an increasing etching rate and ionization, and that will affect the ashing speed, which can reach up to 70 nm/min. A hard ashing recipe is used for the hard-baked photoresist when a soft etch takes con-siderable time.

The argon sputtering process was performed on an Oxford PlasmaLab System 100, as seen in Fig. 2.5 (right). This system is more advanced, with a separate load lock and an automatic wafer transfer arm to the main process chamber. Other gases can be used here for different purposes, depending on the physical and chemical etching recipes, such as CF4, SF6, CHF3, and Cl2. The PlasmaLab 100

Superradiant THz wave emission from arrays of Josephson junctions