2. High-Temperature Superconductors

2.1 Properties of Superconducting Materials

Figure 2.1: Original measurement data by H. Kammerlingh Onnes who first discovered the superconducting properties of mercury [13].

Superconductivity refers to a special property of many materials: vanishing electrical resistance below a certain critical temperature ( $T_{\mathrm{c}}$). This phenomenon was discovered in 1911 by Heike Kammerlingh Onnes while he was experimenting on liquid helium in order to study the behaviour of metals at low temperatures. Much to everybody's surprise, the resistance of mercury dropped to an unmeasurably low value when cooled below about 4.2 K [1]. The material had passed into a new, superconducting state. The measurement results obtained by Onnes are shown in Fig. 2.1. In 1933, W. Meissner and R. Ochsenfeld discovered another fundamental property of superconducting materials: a superconductor completely expels an external magnetic flux from its interior [4]. In subsequent years, various models were developed to explain the nature of superconductivity, but it wasn't until in 1957 that J. Bardeen, L. Cooper, and J. Schrieffer developed the microscopic theory (BCS theory) to explain the physical mechanism of superconductivity [5]. Many elements, alloys, and compounds are known to be superconducting when cooled below a specific critical temperature. Most of the materials have a very low $T_{\mathrm{c}}$ and thus cooling to the temperature of liquid helium (4.2 K) is usually required. For long, such a low temperature restricted the number of potential commercial applications of superconductors. For this reason, there was an enormous interest in discovering new superconducting materials that would have a higher transition temperature. This prospect seemed remote even in the early 1980's since the highest known critical temperature was that of Nb$_3$Ge (30 K), and it had been discovered back in 1971. A significant breakthrough was made in 1986 when J. Bednorz and K. Müller reported that the transition temperature of a Ba-La-Cu-O compound was about 35 K [2,6]. With this work, Bednorz and Müller set the scene for an explosion of research interest in high-temperature superconductivity. They were awarded the Nobel prize in Physics in 1987 [7].

2.1.1 Critical Parameters

The critical temperature is one of the most important parameters characterizing superconductors. When $T<T_{\mathrm{c}}$, the electrical resistance of a superconducting material drops sharply to zero resulting in a virtually lossless flow of current. The superconducting state can be destroyed by applying a strong enough magnetic field ($H$) or current density ($J$) on the material [4]. According to the BCS theory, superconductivity results from the interaction between electrons and the surrounding ionic crystal. When moving in the crystal, an electron leaves behind itself a deformation cloud which affects the positions of the ion cores. This distortion creates an effective positive charge which attracts other electrons to form an electron pair, a Cooper pair. The Cooper pairs move without scattering in the lattice, which leads to the resistance-free nature of the superconducting state [8]. Critical temperature is also a good way to divide superconductors into different classes. Low-temperature superconducting (LTS) materials have a $T_{\mathrm{c}}$ below 30 K while the novel high-temperature superconducting (HTS) materials have $T_{\mathrm{c}}$:s ranging from 35 K to 138 K [9]. Actually, the temperature of liquid nitrogen (77 K) is a more practical lower limit for HTS materials. Although the BCS theory explains well the origin of superconductivity in LTS materials, there exists no generally accepted microscopic theory for HTS materials [10]. In particular, the microscopic pairing mechanisms in high-temperature superconductors are not fully known. Another difference between the various superconductors is their behaviour in an external magnetic field. In the case of type I superconductors, an external magnetic field higher than a certain critical field ( $H_{\mathrm{c}}$), destroys the superconducting state. The critical magnetic field is a function of temperature such that it tends to zero when the temperature is close to $T_{\mathrm{c}}$. The qualitative $H_{\mathrm{c}}$ curves for both type I and type II superconductors are illustrated in Fig. 2.2. In Fig. 2.3, the difference between a perfect conductor and a superconductor is illustrated in two different cases. One can notice that in the superconducting state, magnetic flux vanishes inside the material in all circumstances because of the Meissner effect. In fact, the field does penetrate into the superconductor but only into a depth of the order of $\lambda_{L}$ (London penetration depth) [4]. The penetration depth is generally negligible in bulk superconductors but may be of great importance in thin films.

Figure: Critical magnetic field $H_{\mathrm{c}}$ as a function of temperature for (a) type I and (b) type II superconductors.

Figure 2.3: Comparison of a perfect conductor and a superconductor in two different cases: (a) The sample is cooled in the absence of any fields and after that a magnetic field is applied, (b) the sample is cooled in an applied magnetic field.

Type II superconductors, on the other hand, have two critical fields: the lower and the upper critical fields ( $H_{\mathrm{c1}}$ and $H_{\mathrm{c2}}$, respectively). If the applied field $H<H_{\mathrm{c1}}$, the material is in the superconducting Meissner state. When the external magnetic field is between the two critical values, the material is in a mixed state and above $H_{\mathrm{c2}}$ the superconductivity of type II materials disappears completely [4]. This behaviour can also be seen in Fig. 2.2. In the mixed state, an external flux penetrates partly into the superconducting material in the form of vortices. The core of a vortex is in the normal state, whereas the other part of the material is still in the superconducting state. Each vortex carries a discrete amount of magnetic flux, namely one flux quantum ( $\Phi_0=2.07\times 10^{-15}$ Tm$^2$). The radius of the normal core is called the coherence length $\xi$. Superconductors can be classified into type I and type II materials by defining a parameter $\kappa=\xi/\lambda_L$ [4]. For type I superconductors, $\kappa<1/\sqrt{2}$, and for type II materials, $\kappa>1/\sqrt{2}$. HTS materials belong to type II superconductors, as well as some LTS alloys and compounds. The most important property of type II materials is that they have higher critical fields than type I materials, which makes them suitable for many advanced applications. In addition, their $T_{\mathrm{c}}$ is usually higher, even in the case of LTS materials [11]. Critical current density ( $J_{\mathrm{c}}$) is another key parameter of high-temperature superconductors, in particular for thin-film applications. If $J>J_{\mathrm{c}}$, the vortices start to move in the material causing resistance and thereby losses. Fortunately, there are inhomogeneities in the material which can trap vortices and thus increase the value of $J_{\mathrm{c}}$ [11].

2.1.2 Crystal Structure

HTS materials usually have complicated crystal structures. Almost all of the compounds consist of at least three different chemical elements and the materials with the highest critical temperature have up to seven elements in the crystal lattice. Most of the materials are layered cuprates, i.e., they consist of CuO$_2$ planes separated by other planes of insulating rare-earth elements or other oxides. The crystal lattice is composed of three different building blocks, namely the perovskite, the rock-salt, and the fluorite blocks [12]. Each block represents a layer extending horizontally in the so-called ab plane. The different crystal lattices are illustrated in Fig. 2.4. In addition, HTS materials may consist of other structures, such as CuO chains between the layers.

Figure 2.4: Different crystal lattices of HTS materials: (a) perovskite structure, (b) rock-salt structure, (c) fluorite structure.

The basic configuration of the perovskite block in HTS cuprates is ACuO$_3$ where A represents a rare-earth or an alkaline-earth element (e.g., yttrium or barium in YBCO, respectively) [12]. The other blocks, the fluorite and the rock-salt layers, do not contain any copper. Rock-salt blocks consist of layers of both oxygen and atom A, and the fluorite structure is composed of alternating layers of either oxygen or atom A. One unit cell of a HTS compound is a combination of these different types of building blocks. More information on the crystal structure of superconducting oxides can be found, e.g., in Ref. [14]. Owing to the layered structure, HTS materials exhibit strong anisotropy: the values of the superconducting parameters (e.g., the London penetration depth) are different in different directions. In addition, the normal-state resistance can be many times larger in the vertical $c$ direction than in the horizontal $a$ and $b$ directions, and charge transport is confined to these layers even in the superconducting state [11]. Charge carriers are either holes (p-type superconductors) or electrons (n-type superconductors) [12,15].

2.1.3 HTS Material YBa$_2$Cu$_3$O$_{7-\delta }$

YBa$_2$Cu$_3$O$_{7-\delta }$ (YBCO) is one of the most actively studied HTS materials and it is widely utilized in various fields of research. It is the only known stable four-element compound with a $T_{\mathrm{c}}$ above 77 K and it is relatively easy to make single-phase YBCO, in contrast to other HTS materials. Furthermore, this material includes no toxic elements (e.g., Hg) or volatile compounds [15,16]. However, one clear disadvantage of the compound is that it degrades in humid environment, even in the ambient air. Typical superconducting parameters of the material are shown in Table 2.1. The subscripts refer to the different lattice directions. The critical current density for YBCO thin films is typically $J_{\mathrm{c}}\geq 1$ MA/cm$^2$, although values of almost 10 MA/cm$^2$ have been reported [17].

Table 2.1: Typical superconducting parameters of YBCO.

Critical temperature:	$T_{\mathrm{c}}=90-92$ K [12]
Critical magnetic fields:	$H_{\mathrm{c1}}=10$ mT, $H_{\mathrm{c2}}=300$ T [4]
London penetration depths:	$\lambda_{a,b}=30$ nm, $\lambda_c=200$ nm [4]
Coherence lengths:	$\xi_{a,b}=3$ nm, $\xi_c=4$ Å [4]

Figure 2.5: Structure of the ideal YBa$_2$Cu$_3$O$_{7-\delta }$ lattice.

A single unit cell of YBCO is illustrated in Fig. 2.5. The dimensions of the cell are $a=3.8227$ Å, $b=3.8872$ Å, and $c=11.6802$ Å [14]. The lattice is composed of double perovskite layers, separated by CuO chains. The term $7-\delta$ in the chemical formula appears because the CuO plane between the adjacent BaO layers is imperfect in the sense that there is a slight deficiency of oxygen [12]. One reason for that kind of behaviour is the mobility of the oxygen atoms. Mobility increases with increasing temperature, which means that $\delta$ is also a function of temperature. When $\delta=0$, the CuO-chains are perfectly ordered and the lattice is in the orthorhombic phase. When the temperature is higher, $\delta=1$ and YBCO has a tetragonal structure [15]. Only the orthorhombic structure is superconducting but, unfortunately, it is stable only at temperatures below 500$^{\circ }$C. This complicates the deposition of thin films. Since the deposition has to be performed at high temperatures, a post-annealing is required so that the high-temperature tetragonal structure undergoes a phase transition to the orthorhombic structure. The transition temperature slightly depends on how much oxygen is present during the post-annealing step: at 0.5 mbar, the temperature is 480$^{\circ }$C and at 500 mbar, it is approximately 570$^{\circ }$C [12].

2.2 Applications of HTS Materials

The most important application areas of HTS materials are microelectronics components, superconducting wires, cables, and magnets. Microelectronics applications require superconductors typically in the form of thin films, whereas other applications are mainly realized by using bulk superconductors.

2.2.1 Microelectronics Applications

Microelectronics devices make use of high-temperature superconductors for two reasons. First, it is possible to design and construct sensitive detectors, e.g., for measuring magnetic flux or mm-wave radiation. Secondly, HTS materials could be used to replace existing designs based on normal conductors. Compared to conventional devices, HTS components can be significantly reduced in size while their performance is improved. Because, in principle, there are no electrical losses, power consumption in the components is low. High-temperature superconducting microwave components including filters, antennas, resonators, and transmission lines have already been demonstrated. In addition, there are so-called hybrid devices which combine superconducting and semiconducting circuits in a single device or system [18]. A detailed discussion of various devices and components can be found in Refs. [13,19].

2.2.1.1 SQUIDs

Superconducting QUantum Interference Devices, SQUIDs, are sensitive detectors of magnetic flux. SQUIDs are used, in particular, in medical applications for mapping the magnetic fields of the human body. The key parts of a SQUID are weak junctions called Josephson junctions which connect two superconducting regions. When a DC bias current is driven through the SQUID, an AC voltage is generated across the loop. This voltage is a periodic function of the magnetic flux through the loop. The SQUID is the leading commercial application of LTS materials. The next step would be a high-temperature SQUID but so far they have been inferior to the conventional low-temperature SQUIDs. The most severe limitation is the large 1/$f$ noise, which results in the poor sensitivity of the devices. The Josephson junctions for high-temperature SQUIDs can be fabricated, e.g., by etching a steep step edge on the substrate prior to the deposition, or by using a bicrystal substrate which consists of two parts fused together such that their crystal axes are rotated by a certain amount. The processing of Josephson junctions and the refining of a YBCO film into a SQUID are discussed in Refs. [20,21,22,23] and the SQUIDs are discussed in general in Refs. [13,24].

2.2.1.2 Bolometers

Bolometers are resistance thermometers which are used as radiation detectors. Radiation heats the active part of the bolometer (absorber) whose temperature increases. The temperature change is determined by measuring the resistivity of the absorber. Superconducting bolometers make use of the large resistivity change at the superconducting transition: even a tiny change in temperature (small power) will produce a measurable signal. In principle, bolometers are suitable for any spectral region but, in practice, they become attractive only when other detectors (e.g., PIN diodes) cannot be used. One such region is from the far-infrared to submillimeter waves. For these wavelengths, the most severe problem of conventional detectors based on semiconductors is the lack of suitable materials: the bandgap becomes so small that thermal excitations disturb the measurements at room temperature. Essential properties of a HTS bolometer are small size of the absorber (small heat capacity) and good thermal isolation from the substrate. One approach to fabricate such a component is to produce a suspended microbridge by using isotropic micromechanical etching [28]. The etching recipes are best known for silicon and, therefore, silicon is an attractive substrate material. The realization of a YBCO bolometer is discussed, e.g., in Refs. [25,26,27]. Further information and an overview of bolometers can be found in Ref. [29]. A microscope image of a bolometer realized in this project is shown in Fig. 2.6.

Figure 2.6: Microscope image of a HTS bolometer fabricated in a YBCO thin film.

2.2.1.3 Superconducting Filters

Superconducting filters consist of coupled resonators, just like conventional filters. However, at the microwave frequencies it becomes difficult to realize the component using lumped elements (inductors and capacitors). Instead, they are usually converted into distributed elements (transmission lines). The design of such a bandpass filter is described in Ref. [11]. The most noticeable advantage of a superconducting microwave filter is the higher $Q$-values of resonator cavities. Therefore, the component has a large capacity to store energy and, furthermore, the edges of the frequency band are steep. These aspects have a great importance in telecommunications applications [11]. A disadvantage of superconducting components is, however, that they suffer high residual losses at high frequencies due to a phenomenon known as surface resistance [30].

2.2.2 Other Applications

In electric-power networks, several components, such as transformers, current limiters, and motors, could be replaced by the corresponding HTS components [31]. A typical example is a Fault Current Limiter (FCL). The FCL is inductively coupled to the power supply of an application. If current peaks occur in the circuit, the superconductor is driven into the normal state and the excess power is consumed in the FCL - not in the sensitive component [32,33]. Another example is a Superconductor Magnetic Energy Storage (SMES), which is able to store and discharge energy faster than conventional batteries. In transport applications, the main research effort is currently focused on vehicles which are able to levitate on magnetic field. These so-called maglev vehicles levitate on a track by magnetic repulsion. HTS electromagnets would reduce power losses and would generate magnetic fields strong enough for levitation [12,31]. Superconducting technology can also be applied for magnetohydrodynamic propulsion of, e.g., ships. The technology is based on producing a thrust by passing current through a conducting fluid in the presence of a magnetic field [31]. In addition to vehicles, superconducting magnets are nowadays used almost everywhere where there is a need of high magnetic fields. Typical examples include magnetic resonance imaging (MRI), high-energy particle accelerators, chemical processing, and thermonuclear fusion research. Most of these applications still rely on LTS materials because of the cost of suitable HTS materials but in certain fields, HTS test components have already been demonstrated. Advances in different large-scale applications of superconductors are discussed in Ref. [31,34].

2.3 Deposition of Superconducting Thin Films

Fabrication of thin films is a research area that has been intensively studied since the discovery of high-temperature superconductivity. The majority of the preparation techniques can be described as deposition processes. The principles of the different methods vary a lot but they have also some common features. In most cases, the deposition takes place in a vacuum chamber in order to obtain single-crystalline films with a low number of defects and impurities. An introduction and comparison of different methods can be found in Refs. [18,35,36]. The deposition techniques can be divided into two major categories: in-situ and ex-situ film growth [18]. The films produced by in-situ methods can be made superconducting by cooling as soon as they are removed from the growth chamber whereas ex-situ films have to be annealed in an oxygen-rich environment after the deposition. In the latter growth process, an amorphous film is first deposited and the diffusion during the post-annealing leads to the crystallization of the film. Although ex-situ growth techniques require higher deposition temperatures than in-situ techniques, the deposition apparatus is usually much simpler and the oxygen pressure lower during the film growth [18]. Some of the deposition processes are based on physical phenomena, e.g., removal or evaporation of material in order to produce a particle flux. Other deposition techniques rely on chemical reactions between suitable compounds or dissociation of large molecules near the film to obtain the correct film composition. A schematic illustration of the physical and chemical deposition processes is illustrated in Fig. 2.7. In addition, there are so-called hybrid, or intermediate, processes which combine the properties of the two previously mentioned principles (for example, using a glow discharge to initiate chemical reactions).

Figure 2.7: Schematic illustration of film growth (a) by physical deposition and (b) by chemical deposition. In (a) the film constituents collect on the substrate whereas in (b) they are contained in larger molecules which dissociate near the substrate [18].

Sputter deposition makes use of energetic ions (e.g., Ar$^{+}$) that strike a target consisting of the material to be deposited. As a result of the interaction, a continuous flux of particles is produced. The emitted atoms condense on a substrate to form a film. The ions can be extracted from a plasma which is sustained by a DC voltage, RF power, or with the help of a magnetron [35]. It is also possible to use focused ion or electron beams to evaporate material. Sputtering is mainly utilized in the industry to fabricate electronics materials and protective coatings on optical components [36]. Molecular beam epitaxy (MBE) utilizes continuous molecular beams to deposit films of the desired thickness. Beams are generated by thermally evaporating elemental sources (so-called effusion cells) [35]. MBE can produce high-quality layers with good control of the thickness and the composition of the film. Atomic layer epitaxy (ALE) can be considered as a special mode of MBE. The vapour-phase reactants are supplied on the growing surface layer by layer by using shutters to turn on and off the fluxes of the reacting components. Because of the pulsed nature, epitaxial growth of monolayers is possible [36]. Chemical vapor deposition (CVD) is based on chemical reactions between two or more gaseous chemicals in the vicinity of a heated substrate. Heaters provide the necessary energy needed to sustain the reactions and also to improve the crystallization of the film. For HTS device applications, pulsed laser deposition (PLD) is currently the most straightforward film-production technique: the method is applicable to a wide pressure range and the setup is one of the simplest. Another advantage is that PLD is suitable for the deposition of all kinds of materials, in particular ceramic multicomponent materials (e.g., YBCO) [36]. In addition, PLD offers better film quality and superior structural and electrical properties compared to other vapour-deposition methods. For these reasons, we have chosen this method for our project. The PLD technique is discussed in detail in the following chapters.