Iron-Based Superconductors

Abstract

Superconductivity is the phenomenon of vanishing an electrical resistivity of materials below a certain low temperature and superconductors are the materials that show this property. Critical temperature is the temperature below which superconducting state occurs. Based on temperature superconductors can be grouped into high-temperature superconductors and low-temperature superconductors. Based on the mechanism, they can be grouped into conventional and unconventional superconductors. Based on magnetism superconducting materials can also be separated into two groups: type-I and type-II superconductors. In this chapter, we will discuss superconductivity, the Meissner effect, type-I and type-II superconductors, convectional and unconvectional superconductors, heavy fermions, cuprates, iron-based superconductors, and high entropy alloy superconductors. High-entropy alloys (heas) are defined as alloys containing at least five elements with concentrations between 5 and 35 atom%. The atoms randomly distribute on simple crystallographic lattices, where the high entropy of mixing can stabilize disordered solid-solution phases with simple structures. The superconducting behavior of heas is distinct from copper oxide superconductors, iron-based superconductors, conventional alloy superconductors, and amorphous superconductors, suggesting that they can be considered as a new class of superconducting materials.

1. Introduction

Superconductivity is the set of physical properties observed in certain materials at low temperatures, characterized by the complete absence of electrical resistance or the resistance of the material to the electric current flow is zero [1]. It is a phenomenon in which the resistance of a material to electric current flow is zero. Any material exhibiting these properties or which have no resistance to the flow of electricity is known as a superconductor [2]. It was discovered by Dutch physicist Heike Kamerlingh onnes of Leiden University in1911, who was studying the resistance of solid mercury at extremely low temperature using the recently discovered liquid helium as a refrigerant. At the temperature of about $42K$$\left(-452°\mathrm{F},-252°\mathrm{F}\right),$ when he cooled to the temperature of liquid helium, He observed that the resistance abruptly or suddenly disappeared. The current was flowing through the mercury wire and nothing was stopping it, the resistance was zero (see Figure 1 which shows a graph of resistance versus temperature of mercury wire which Onnes produced) [3].

Figure 1.

Resistance in ohms of a specimen of mercury versus absolute temperature. This plot by Kamerlingh Onnes marked the discovery of superconductivity [1].

2. Characteristics of superconductivity

2.1 Temperature effect

According to Onnes, “Mercury has passed into a new state; he named this new state, called superconductivity” [4]. The temperature at which superconducting state starts is called the critical temperature$\left(T_c\right)$. In one of Onnes experiments, he started a current flowing through a loop of lead wire cooled at a liquid helium range, a year later the current was still flowing without significant current loss. He found that the new material exhibited what he called persistent currents. He discovered superconductivity and was awarded the Nobel Prize in 1913. Based on the temperature effect the superconductivity grouped under low-temperature superconductor with the temperature below 30K such as He, Al, Cd, Sn, Hg, U, Nb, Nb₃Ge, etc. and high-temperature superconductors with transition temperature above 30K such as Cuprates and Iron base superconductors [5].

2.2 Meissner effect

In addition to zero resistance, a new scientific discovery is made in 1933 by W. Meissner and R. Ochsenfeld that superconductors, which have an interesting magnetic property of excluding a magnetic field, are more than a perfect conductor of electricity. Because of Faraday’s law, the magnetic field inside a superconductor cannot change; there is no electromotive force due to the lack of electric resistance [6]. The flow of current-induced magnetic field on the Fermi-surface of superconductor cancels out the external field. A superconductor, when it is cooled below the critical temperature Tc, expels the magnetic field and does not allow the magnetic field to penetrate inside it (see Figure 2(a)). This phenomenon in superconductors is called the Meissner effect [3]. The most spectacular demonstration of the Meissner effect is the levitation effect [7]. That is if a small bar magnet rests on a superconducting dish; the magnet will levitate above the superconducting dish when the temperature is lowered below T_c.

Figure 2.

Expulsion of applied magnetics flux field [7].

The Meissner effect will occur only if the magnetic field is relatively small. If the magnetic field becomes too great, it penetrates the interior of the metal and the metal loses its superconductivity [8]. The certain value of magnetic field beyond which superconductor returns back to its ordinary state is called critical magnetic field. The Meissner effect is so strong that a magnet can actually be levitated over a super-conductive material. Following the discovery of the Meissner effect, in 1935 two London brothers Fritz and Heinz proposed the first phenomenological theory known as the London equation which explains the Meissner effect, where in a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold [9]. By using the London equation, one can obtain the dependence of the magnetic field inside the superconductor on the distance to the surface.

The London equations explained not only the Meissner effect, but also provided an expression for the first characteristic length of superconductivity known as the London penetration length (${\mathit{\lambda }}_{\mathit{L}}$). The next theoretical advance came in 1950 with the theory of Ginsburg and Landau, which described superconductivity in terms of an order parameter and provided a derivation for the London equations [10]. This theory provides an expression for the penetration length similar to the London equations and also the expression for the second characteristic length known as the Ginsburg-Landau coherence length (ξ) which is a measure of the distance within which the superconducting electron concentration cannot change drastically in a spatially varying magnetic field [11]. A. A. Abrikosov used these concepts to roll up alloy for superconductors. He observed that if the electronic structure of the superconductor were such that the coherence length becomes less than the penetration depth, on would get magnetic behavior similar to type II superconductors, with two critical fields Bc₁ and Bc₂. In the same year, the quantum theory of superconductivity was predicted theoretically by H. Frohlich that the T_c would decrease as the average isotopic mass increased. This effect is called the isotope effect which is observed in experimentally the same year by Maxwell [12]. The isotope effect provided support for the electron–phonon interaction mechanism of superconductivity.

3. Type I and Type II superconductors

Based on the applied magnetic field superconductors are grouped under Type I and Type II. Superconductors those converts into a normal state abruptly at the critical field below critical temperature are known as Type I super conductors as shown in Figure 3(a) [13]. These types of superconductors are usually of low-critical temperature materials, such as metals and metal alloys (such as aluminum, lead, indium, Nb₃Ge, etc.). These superconductors have only one critical field (H_C) at which it converts into normal state. And the critical magnetic field for such superconductor is very low; it is of the order of 0.01–0.2 Wb/m². Type II superconductors are depicted in Figure 3(b) as superconductors whose diamagnetic property increases with applied magnetic field up to a certain value (H_C1), after that, decreases gradually and drops to zero at the magnetic field H_C2 and converts completely into the normal state [14].

Figure 3.

Type of superconductors based on applied magnetics [1].

4. Conventional superconductors

The first widely accepted and detailed microscopic theory was developed in 1957 by American physicists John Bardeen, Leon Cooper, and John Schrieffer. This theory is called BCS theory. According to BCS theory, superconductivity is quantum effect which result from the cloud of electron pair, which is called the Cooper pair and was first discovered by L. Cooper [15]. A Cooper pair is the electrons that are bound together. The BCS theory assumes that superconductivity arises due to Cooper pair, a state in which the attractive interaction dominates the repulsive Coulomb force [16]. A Cooper pair is an electron–electron pair mediated by electron–phonon interaction.

This attractive interaction is due to the attraction of negative charge ion by the core ion known as positive charged ion results distort of its lattice in such a way as to attract other electrons (the electron–phonon interaction) [17]. Figure 4 shows that a Cooper pair is a bound state of two electrons with opposite spin and momentum which is one in the state ($k\mathit{\uparrow })$ and the other in the state $\left(k\downarrow \right)$ [18].

Figure 4.

The bound state of Cooper pair.

The Cooper pair of the bound state becomes boson in an ordered manner therefore, the flow of electric current is able to move easily through the lattice without any electrical résistance. The BCS theory explored superconductivity at a temperature close to zero for elements and simple alloys (conventional) superconductors. However, at high temperature and with different superconductor system, the BCS theory has subsequently become inadequate to fully explain how superconductivity occurs [19].

5. Unconventional superconductor

Unconventional superconductors are materials that display superconductivity which does not conform to either the conventional BCS theory or Nikolay Bogolyubov’s theory. The first unconventional triplet superconductor, organic material (TMTSF)₂PF₆, was discovered by Denis Jerome and Klaus Bechgaard in 1979. The superconducting properties of CeCu₂Si₂, a type of heavy fermion material, were reported in 1979 by Frank Steglich [9, 20].

5.1 Heavy fermions

In the late 1970s and early 1980s, superconductivity was discovered in heavy fermions systems and in nearly magnetic systems [10]. These heavy fermions are metallic materials that hold rare earth elements, such as Yb or Ce, or actinide elements such as U with partially filled 5f shells. The name heavy fermion is given because the effective mass of electrons in these superconductors is in the order of a hundred times larger than the mass of usual electrons at low temperatures [21]. In heavy fermion compounds, the superconducting charge carriers are bound together in pairs by magnetic spin–spin interactions [22], showing that spin fluctuations (electron–electron interaction) mediate the electron pairing that leads to superconductivity in heavy fermion compounds, such as URu₂Si₂, UPd₂Al₃, and UNi₂Al₃.

5.2 Cuprates

Until 1986, physicists had believed that BCS theory forbade superconductivity at temperatures above about 30K. Then, in 1986, a truly breakthrough discovery was made in the field of superconductivity. Georg Bednorz and Alex Muller, working at IBM in Zurich Switzerland, were experimenting with a particular class of metal oxide ceramics of lanthanum-based LaBaCuO called cuprate perovskites material which had a transition temperature of 35K (Nobel Prize in Physics, (1987). Bednorz and Muller surveyed hundreds of different oxide compounds [23]. What made this discovery so remarkable was that ceramics are normally insulators. They do not conduct electricity well at all. Similar to the heavy fermion compounds, high-temperature cuprate compounds also show a delicate balance between superconductivity and magnetism. In order to understand the coexistence of superconductivity and magnetism in cuprate, it is important to know the layered perovskite-like crystal structure of these superconductors [24]. Cuprates are the second class of unconventional superconductors.

The layered structure includes CuO₂ planes. The CuO₂ plane acts as a charge reservoir and is responsible for doping (electrons or holes) into the CuO₂ planes. The movements of holes or electrons in the CuO2 planes cause to result in superconductivity [25]. The parent compound of cuprate compound is a Mott insulator, which ought to be metal according to the band theory of electrons, but it is insulating, due to electron–electron interactions. From a common temperature versus dopant concentration, phase diagram for cuprate at very low doping concentration, antiferromagnetic (AFM) order exists in the cuprate system, and the temperature dependence of the resistivity shows an insulating behavior [26].

Increasing the doping concentration, AFM order is felled rapidly and vanishes at a certain doping level, and the superconducting order rises. As the doping level increases, T_c increases. At an optimum value of doping level, T_c reaches a maximum and the system behaves as a non-Fermi liquid. On further increasing the doping concentration, T_c decreases and finally vanishes [15]. The phase diagram suggests that AFM order does not play a conclusive role in the suppression of superconductivity, because superconductivity does not appear immediately as AFM order vanished, rather, it rises gradually with increasing doping level. Many other cuprate superconductors have since been discovered, and the theory of superconductivity in these materials is one of the major outstanding challenges of theoretical condensed matter physics [9]. Since about 1993, the highest-temperature superconductor has been a ceramic material consisting of mercury, barium, calcium, copper, and oxygen (HgBa₂Ca₂Cu₃O_8 + δ) with Tc = 133–138K [20, 21]. The latter experiment (138K) still awaits experimental confirmation [27].

6. Iron-based superconductors

The third class of unconventional superconductors is iron-based superconductors. The first report of superconductivity in an iron superconductor was F-doped LaOFeP below 5K in 2006 [22]. On February 23, 2008, a group from Tokyo Institute of technology published paper in (JACS) Journal of the American Society, in which they reported that the fluorine-doped lanthanum Oxide Fe-As superconductors at 26K [28]. Similar to cuprates and several heavy fermion superconductors, superconductivity emerges here in close proximity to AFM state and like to cuprate superconductors, high-temperature superconductivity in the iron-based superconductors (FebSC) systems is also induced from electron or hole doping of their parent compounds. Parent compounds show long-range AFM static order. AFM order is suppressed on electron/hole doping to induce superconductivity. In FebSCs magnetism arising from nesting which induced spin density wave, unlike cuprates where the parent compounds are Mott insulators, strongly repulsive electronic correlations yield an insulating and ground state despite a half-filled conduction band [8].

There are several good reasons why FebSC is so interesting. First, they show the coexistence of superconductivity and magnetism. Second, they have too much variety of compounds for research and, their multi-band electronic structure offer the hope of finally discovering the mechanism of high-temperature superconductivity and finding a way to increase T_C. Lastly, they are quite encouraging for a wide scope of applications. Having a much higher critical field (H_C) than cuprates and high isotropic critical currents, they are attractive for electrical power and magnetic applications.

Compared to Cuprates, iron-based superconductors (FebSc) have some similarities. Firstly, both of their parent compounds are antiferromagnets. Increased doping can destroy antiferromagnetism and lead to superconductivity. Secondly, superconductivity occurs in specific planes. In cuprates, it is Cu-O plane. While in FebSc, it is Fe-As plane [29]. They also have significant differences. For Cuprates, its parent compound is a special type of antiferromagnet$—$Mott insulator, which are a result of strong local interaction; While in FebSc, it is an antiferromagnetic$—$“spin-density-wave” metal. They are magnetic bad metals, originating from long-range (non-local) magnetic correlations. Moreover, the Cu3d_x²_−y² (a single electron) contributes to superconductivity; while in iron-based superconductors, all five Fe 3d orbitals contribute to superconductivity. Superconductivity can be achieved in both cases by doping, but doping directly to the superconducting layers is allowed only in FebSc, likely because they are more itinerant than cuprates. In superconducting samples, cuprates have Sc order parameter with a sign-changing d-wave symmetry, while the symmetry of the FebSc has a sign-changing s-wave symmetry. These differences make this new type of high-temperature superconductors very interesting.

One of the most prominent issues in the physics of Fe-based superconductors is the interplay between the magnetic and superconducting order parameters when charge doping, pressure, or other parameters are modified. In this chapter, we describe literature about iron-based superconductors. Electronic and crystal structures of iron-based superconductors, magnetic ordering and spin density wave, electron, and crystal structure of SrFe₂As₂, the effect of nickel substitution on SrFe₂As₂, superconductivity in SrFe_2 − xNi_xAs₂, superconductivity, and magnetism [30].

According to a general phase diagram of temperature vs. dopant concentration for cuprate, at very low doping concentrations, the cuprate system has an antiferromagnetic order and the resistivity's temperature dependency behaves as an insulator. As the doping is increased, the antiferromagnetic order quickly dissipates and disappears, while the superconducting order emerges. TC increases as doping levels rise. Tc reaches its maximum and the system behaves as a non-Fermi liquid when doping is at its optimal level. Tc decreases with increasing doping until it reaches zero. Like cuprate superconductors, FebSc systems derive their high temperature superconductivity from electron or hole doping of their parent compounds.These parent compounds exhibit long-range AFM static order, which is suppressed by electron or hole doping to induce superconductivity. In contrast to cuprates, magnetism in FebSCs results from a nesting-induced spin density wave [31].

The iron-based superconductors (FebSc) have many different systems that greatly enlarge the family of unconventional superconductors [32]. The first FebSc system LaFePO_1−xF_x was discovered by H. Hosono et al. in 2006 with Tc$\sim$4K, but it did not draw much attention until the same group substituted P by As and discovered a Tc of 26K in LaFeAsO_1−xF_x in 2008. Before 2013, many FebSc compounds were discovered with different crystal structure classes and compositions. Different systems are denoted by the stoichiometric ratios of the chemical elements in their parent compounds of FebSc. Therefore, the major FebSc systems are 1111, 122, 111, 11, and 245 (e.g., LaFeAsO, SrFe₂As₂, NaFeAs, FeTe, and Rb₂Fe₄Se₅). Even though they have different structures, all the FebSc systems share a building block: iron-based square-planar sheets. Like to cuprates, in which copper and oxygen form the superconducting layer, FebSc systems have iron-based layers which are crucial to their unconventional superconductivity [18].

The FebSc classes listed above can be grouped into two groups according to the elemental group of the atom that forms the superconducting layer with iron. The first group is the iron-pnictides (nitrogen family from the periodic table), that iron makes a zigzag layer with arsenic or phosphorous (Fe-As or Fe-P). The 1111, 111, and 122 classes grouped to iron pnictides [33]. Pnictides have similar magnetic structures but they have different crystal structures and symmetry groups. The second group is iron-chalcogenides (oxygen family from the periodic table), that is the iron-selenium (Fe-Se) or iron-tellurium (Fe-Te) makes the superconducting layer; this family includes the 11 and 245 classes. Both the 11 and 245 classes have special properties. The 11 class has only one system: Fe_1 + yTe_1−xSe_x. It does not have any buffer layers between the Fe-(Te,Se) layers, and the Tc is determined by Se doping level and related to the amount of extra iron in the compound. The 245 class is a relatively new discovered of iron-based superconductors [34].

Iron pnictides and iron chalcogenides share many intriguing common properties. They both have the highest Tcs. The superconducting gaps are close to being isotropic (not varying in magnitude) around Fermi surfaces, and the ratio between the gap and$T_c$, $2∆/T_c$, is much larger than the BCS ratio, 3.52, in both families. However, the electronic structures in the two families, and the Fermi surface topologies (geometrical properties), are quite different in the materials that reach high $T_c$ [35].

6.1 Crystal structure of iron-based superconductors

Figure 5 illustrates that the structural unit common to Fe-based materials is the square net of ${\mathit{Fe}}^{2+}$(formal charge) coordinated tetrahedrally by four pnictogen or chalcogen atoms. The iron containing plane is not flat; pnictogen or chalcogen atoms extend above and below the iron plane because the pnictogen and chalcogen atoms are much larger than iron atoms. They pack themselves in edge-sharing tetrahedral [13]. By contrast, the similar size difference between the copper and oxygen atoms in cuprate superconductor leads to corner-sharing octahedral packing. This structural difference is crucial.Due to their tetrahedral configuration of iron-based superconductors, the Fe atoms are close to each other than the Cu atoms in cuprate superconductors. Both Fe and Cu occupy the same row of the periodic table. Their valance electrons occupy 3d orbitals. But because of the Fe atoms’ close packing, all five Fe 3d orbitals contribute charge carriers. In the cuprate, only one Cu 3d orbital contributes. The chalcogen and pnictogen also play an important role. Their p orbitals also hybridize with the five 3d orbitals leading to a complicated electron band structure and a characteristic multicomponent Fermi surface. The local structure of the Fe-pnictide layer is affected directly by the atomic (or ionic) size of M (where M indicates a metallic element such as an alkali metal, alkaline earth, or rare earth element that lies between Fe-pnictide layers) because M elements in the blocking layer bond to Pn elements [23]. Most of the iron-pnictides become superconductors only when doped (adding an impurity) with holes or electrons. $T_c$ depends on doping concentration.

Figure 5.

Four families of iron-based superconductors, (a) the 1111 family compounds (P4/nmm), (b) the 122 family compounds (I4/mmm), (c) the 111 family compounds (P4/nmm), and (d) the 11 family compounds (P4/nmm) [24].

6.1.1 The 1111 iron-based superconductor family

Soon after the discovery of LaFePO in 2006, several others with similar crystal structure, were discovered. La can be substituted for almost any other rare-earth element and the superconductivity will still exist. That is because of the alternating layered structure of FeAs and RO sheets (R as rare-earth element). From theoretical studies, we can predict, that superconductivity mainly occurs in FeAs layer, while RO layer provides a charge reservoir. These materials go through structural phase transition around 160K—from tetragonal to orthorhombic lattice structure. If we drop temperature even lower, materials become antiferromagnetic [36].

LaOFeAs and other rare earth substituted compounds have a layered crystal structure and they crystallize with ZrCuSiAs type structure belonging to the tetragonal P4/nmm space group [21], and the unit cell contains La₂O₂ and Fe₂As₂ molecules, and the chemical formula is represented by (La₂O₂) (Fe₂As₂). The Fe₂As₂ layer is sandwiched between the La₂O₂ layers and it serves as a carrier conduction path. Conduction carriers are two-dimensionally confined in the Fe₂As₂ layer, that causing strong interactions among the electrons [37].

6.1.2 The 122 iron-based superconductor family

Next type is AFe₂An₂, where (A stands for Alkaline earth metals like Ba, Sr. or Ca and Eu) and An is pnictide (As, P). Superconductivity can be achieved by introducing dopants. There are several ways to introduce dopants. These are (1) hole doping is achieved by substituting A for monovalent B⁺ (B = Cs, K, Na) atoms partially in the blocking layer and this substitution should add an extra hole into the system, for example, Ba_1 − xK_xFe₂As₂; (2) partially substitute Fe for transition metals (Co, Ni, Pd, Rh) into FeAs layers and yields electrons into the system. In this way, dopants are directly substituted into the Fe layer, which can additionally stabilize the system, for example, Co (A(Fe_1−xCo_x)₂As₂), Rh (A(Fe_2−xRh_x)As₂), Ni (A(Fe_1−xNi_x)₂As₂) and we get electron-doped pnictide that forms a rich phase diagram where the superconductivity and magnetism compete or coexist; and (3) replacing arsenic partially with phosphorus, and Phosphorus generates a chemical pressure effect that suppress SDW and emerges superconductivity at the corresponding unit-cell volume [38].

The 122 Fe-based superconductors crystallize with tetragonal ThCr₂Si₂-type crystal structure with space group I4/mmm. 122 systems, for example, SrFe₂As₂ system contain practically identical layers of edge-sharing FeAs4/4 tetrahedra, similar to LaOFeAs but they are separated by Sr. atoms instead of LaO sheets and Sr. layer act as a charge reservoir and FeAs as superconducting layer. The compound undergoes a structural phase transition around 205K from tetragonal (I4/mmm) to orthorhombic (Fmmm) [39].

6.1.3 The 111 iron-based superconductor family

This type is AFeAs type and A stands for alkali elements (Li or Na). Crystal structure of this type is known as the CeFeSi type, with a tetrahedral P4/nmm space group-FeAs 4 layers, separated with double layer of A ions. Distance between Fe-Fe atoms in different layers is significantly shorter than in 1111 or 122 structure [36]. The crystal structures of 111 type is similar to those of 1111 type superconductors with [LaO] layers substituted by Li layers. LiFeAs has $T_c=$18K without extra doping. Wang et al. ($T_c=$18K: LiFeAs) and Tapp et al. ($T_c=$18K: NaFeAs) first reported superconductivity without doping for 111-type materials [33].

6.1.4 The 11 iron-based superconductor family

The simplest form of Fe-based superconductors are ferrochalcogenides FeSe and FeTe and their ternary combination FeSe_xTe_1 − x and Fe_1 + ySe_xTe_1 − x. Crystal structure is similar to those FeAs layers mentioned above, only that this chalcogen does not have a separating layer. T_C for FeSe is around 8K. The FeSe is much easier to synthesize, since it does not include toxic arsenic [21].

6.1.5 The 245 iron-based superconductors family

The attempts to intercalate the 11 family FeSe, the simplest FebSc, resulted in discovery of a new family A_xFe_2 − ySe₂ (A stands for alkali metal like K, Rb, Cs, and Tl). The first alkali iron selenide (245) system K_yFe_1:6 + xSe₂ was discovered in late 2010 with $T_c=$33K. More superconductors were discovered with almost the same $T_c$ when K was replaced with other alkali metals (Rb, Cs) or alkali metals were partially substituted with Tl. This family is most often called 245 because of its parent compound A_0.8Fe_1.6Se₂$=$A₂Fe₄Se₅ [16]. This type of material has a unique crystal structure, and a unique magnetic structure with an unusually high structural/magnetic transition temperature; their phase diagrams and spin dynamics are also very different from those of other FebSc systems. Despite the strange crystal and magnetic structure, the 245 systems (1) have a huge moment of$\sim 30-34\mathit{\mu B}$, which is the highest moment among all FebSc systems; (2) have a Neel temperature of more than 550K, much higher than typical FebSC Neel temperatures $\left(<200K\right)$, and similar to those in cuprates; and (3) in the small samples, are insulators [18].

6.2 Electronic structure of electron-doped iron-based superconductors

According to theories the parent compounds of iron-based superconductors are semi-metallic and the density of state near Fermi surface is mainly contributed by the iron 3d electrons and all five of the 3d electrons cross Fermi surface. The shape of the electronic band structure depends on the doping level. In electron-doped materials, such as 122 Fe-based superconductor compounds, the Fermi surface contains several quasi-2D warped cylinders centered at $\mathrm{\Gamma }$ point $\left(k=0,0\right)$ and M point $\left(k=\pi ,\pi \right)$ in a 2D cross-section, and may also contain a quasi 3D pocket near $k_z=\pi$ as shown in the Figure 6 [15, 17, 31].

Figure 6.

The schematic electronic structure of electron-doped iron-based superconductors. In weakly and moderately electron-doped materials, the Fermi surface consists of quasi-2D warped cylinders centered at$\left(0,0\right)$ and $\left(\pi ,\pi \right)$ in a 2D cross-section. The ones near $\left(0,0\right)$ are hole pockets (filled states are outside cylinders), and the ones near $\left(\pi ,\pi \right)$ are electron pockets (filled states are inside cylinders) [17, 31].

6.3 The phase transitions iron-based Superconductors

The phase transitions of most FeAs-based superconductors undergo structural and/or magnetic phase transitions as presented in Figure 7. These superconductors show different ground states (structural, magnetic, and superconducting) which are close to each other and sometimes compete with each other. It can be detected by using X-ray and neutron scattering techniques [13]. The FeAs-based compound exhibits a tetragonal-to-orthorhombic structural phase transition at low temperatures. A tetragonal structure has the same length of the lattice parameters “a and b” (a = b) whereas the lattice parameters a and b (≠ a) are different in an orthorhombic structure. By measuring the difference in the peak positions, we can find the respective lattice parameters and see a tetragonal-to-orthorhombic structural phase transition. A typical measure of the tetragonal-to-orthorhombic phase transition is the distortion$\left(\delta =\frac{a-b}{a+b}\right)$. The structural transition was first noticed in macroscopic measurements. Magnetic order is found in many systems below some transition temperature [18].

Figure 7.

Phase transition of some Iron base superconductors [31, 40, 41, 42].

Magnetic phase transition of the Bragg peaks depends on the periodicity of the crystal structure. The AFM ordering gives rise to a magnetic structure that has different symmetry elements (usually a subgroup of the crystallographic space group) from the crystal structure which is in the vicinity of an antiferromagnetic (AFM) phase transition [37]. Ant-ferromagnetism in the iron-based superconductors originates from conduction electrons that also form the Cooper pairs below Tc and the antiferromagnetic is believed to arise from the Fermi surface nesting driven spin-density-wave order (when parallel sheets of the Fermi surface can be translated by a nesting vector and superposed). Magnetism in iron-based superconductors may have both itinerant-electron and local-moment characters. In this point of view, many theories and models have been proposed in which itinerant electrons and localized moments coexist in iron-based superconductors and they play some roles in magnetism. A measure of intensities of AFM Bragg peaks as a function of a control parameter, such as temperature, is termed the AFM order parameter [38].

The phase transition of some Fe base compounds are: (1) the 1111 family compounds undergo a structural phase transition from a high temperature tetragonal (P4/nmm) to an orthorhombic (Cmma) at low temperature. (2) The undoped state, the 122 family compound exhibits simultaneous structural and magnetic phase transitions below 140K, changing from the high-temperature paramagnetic tetragonal phase to the low-temperature orthorhombic phase with the collinear AFM structure. (3) The NaFeAs 111 family compounds undergo a structural phase transition from a high temperature tetragonal (P4/nmm) at temperature T_s ≈ 55K to an orthorhombic (Cmma) at low temperature AFM emerges at ∼ 37K. but not in LiFeAs. (4) The 11 family compounds undergo a structural phase transition from a high-temperature tetragonal P4/nmm to an orthorhombic Cmmn at low temperature. But the structural transitions are affected by subtle differences in the stoichiometry [39].

7. High entropy superconductors

As we have discussed above, cuprates are high-temperature superconductors, which are discovered by Bednorz and Mueller in 1986 and superconductivity occurs predominantly in the CuO₂ planes. Interlayer and intra-layer interactions in layered Cuprates play an important role in the enhancement of$T_c$, whereas $T_c$ has been found to be proportional to the number of Cu–O layer in cuprate compounds. Examples include Y-Ba-Cu-O, Bi-Sr-Ca-Cu-O, Tl-Ba-Ca-Cu-O, Hg-Ba-Ca-Cu-O, bismuth-based superconductors, etc.

Entropy is the disorder experienced in material media. For one mole of Bismuth-based cuprates, the entropy is found to be $5.603\times {10}^{-24}J{K}^{-1}$ at the$T_c$of Bi₂Sr₂CuO₆$\left(20K\right)$, Bi₂Sr₂CaCu₂O₈$\left(95K\right)$, and Bi₂Sr₂Ca₂Cu₃O₁₀$\left(110K\right)$. When the temperature is lowered from a higher value $\left(T_c\right)$ to a lower value, the entropy also decreases and the Cuprate materials become more ordered and entropy decreases with an increasing number of CuO₂ planes. Entropy per mole is constant not depending on CuO₂ planes. When considered per unit mass entropy decreases with an increase in the number of CuO₂ planes [43].

Layered superconductors often exhibit high superconducting transition temperatures, such as cuprate superconductors, iron-based superconductors, and nitride-based layered superconductors. High-entropy alloys (HEAs) are defined as alloys containing at least five elements with concentrations between 5 and 35 atom%. The atoms randomly distribute on simple crystallographic lattices, where the high entropy of mixing can stabilize disordered solid-solution phases with simple structures. The HEA concept can be useful to develop new superconducting materials containing an HEA site and/or HEA-type layers. ROBiS₂ (R = La + Ce + Pr + Nd + Sm) is a BiS₂-based layered superconductor that is composed of alternating stacking sequences of BiS₂ and RO layers. Superconductivity of BiS₂-based compounds can be induced by carrier doping and/or in-plane chemical pressure. The critical temperatures of ROBiS₂ single crystals were nearly 2–4$K$. The superconducting critical temperature and superconducting anisotropies of R-site mixed high-entropy samples increased with a decrease in the average ionic radius of the R-site. Moreover, a deviation in the tendency to exhibit superconducting properties was observed based on the difference in the R-site mixed entropy. R-site mixed entropy in ROBiS₂ superconductors may affect their superconducting properties.

Experimentally, ROBiS₂ (R = La + Ce + Pr + Nd + Sm) single crystals were grown using CsCl flux. The starting materials for the growth of ROBiS₂ single crystals were La₂S₃, Ce₂S₃, Pr₂S₃, Nd₂S₃, Sm₂S₃, Bi₂S₃, Bi₂O₃, and CsCl flux. Scanning electron microscopy (SEM) was conducted using a TM3030 system from Hitachi High-Technologies. The compositional ratio of the grown ROBiS2 single crystals was evaluated using energy-dispersive X-ray spectrometry. The valence states of the La, Ce, Pr, Nd, and Sm components in the obtained single crystals were estimated by X-ray absorption spectroscopy [40].

HEA superconductor displays an excellent mechanical properties and it robust superconductivity and quiet high upper critical field that occur to be favorable for potential practical applications. The flux-pinning mechanism that control the field and temperature dependence of critical current density is very important to the practical application [41].

The superconducting behavior of HEAs is distinct from copper oxide superconductors, Fe-based superconductors, conventional alloy superconductors, and amorphous superconductors, suggesting that they can be considered as a new class of superconducting material. Until now, four types of HEA superconductors have been discovered. These are: (1) type-A HEA superconductors (e.g., the Ta-Nb-Hf-Zr-Ti superconductors) crystallize on a small unit cell BCC lattice, (2) type-B HEA-superconductors (e.g., the (HfTaWIr)_1−xRe_x superconductors, x < 0.6) crystallize on a larger-unit-cell cluster-based BCC lattice, (3) type-C HEA superconductors (e.g., the Sc − Zr − Nb − Ta − Rh − Pd superconductors) crystallize on small cell CsCl-type lattice, and (4) type-D HEA superconductors (e.g., the Re_0.56Nb_0.11Ti_0.11Zr_0.11Hf_0.11 superconductor) crystallize on an HCP lattice. The HEA superconductors that crystallize on the small cell BCC or CsCl-type lattices have the highest transition temperatures.

Even if the type-A and type-B HEA superconductors have highly disordered atoms on simple lattices, the effects of elemental makeup and valence electron count on their physical properties are important. For this property, the $T_c$ values mimic the classic Mathias behavior observed for binary alloys, although not in detail, and are limited by the chemical stability. Increasing the configurational entropy by adding elements has no conclusive effect on the $T_c$ of the HEA superconductors, but can stabilize their cocktail-like crystal structures. Applying pressure, the HEA superconductors exhibit vigorous superconductivity against volume shrinkage without structural phase transitions, with the common feature that $T_c$ saturates at a constant optimal value at a critical pressure that changes from system to system.

At the present time, the HEA superconductors display type-II superconducting behavior. Therefore, the upper critical magnetic fields of the current HEA superconductors are not as high as those of NbTi or Nb₃Sn. They are employed in fabrication of the majority of commercial superconducting magnets at this time, researches expect that future superconducting HEAs may be good candidate materials for the fabrication of the superconducting magnets. The superconducting $T_c$s of the HEAs so far found are intermediate between those of amorphous alloys and simple binary alloys, at a fixed VEC following a trend of increasing $T_c$ with decreasing disorder [42] as shown in Figure 8.

Figure 8.

Valence electron count (VEC) dependence of the superconducting transition temperatures for type-A, type-B and type-C HEA superconductors compared to amorphous alloys and classic crystalline alloys [42].

8. Conclusion

The phenomenon of vanishing of electrical resistivity of materials below a particular low temperature is called superconductivity and the materials which exhibit this property are called superconductors. The superconducting state of a material is decided by three parameters such as temperature, external magnetic field, and the current density flowing through the material. These three parameters are coupled together to define the superconducting limits of a material. For the occurrence of superconductivity in a material, the temperature must be below $T_c$, the external magnetic field must be below $H_c$ and the current density flowing through the material must be below $J_c$ [20].

After 20 years of the discovery of Onnes, a major breakthrough came in 1933 when Walther Meissner and his student Robert Ochsenfeld discovered an important magnetic property of superconductors. They observed that [12] when a specimen (sample) is placed in a magnetic field and is then cooled through the transition temperature for superconductivity, the magnetic flux originally present is ejected from the specimen [35] and exhibits diamagnetic behavior [8]. The Meissner effect suggests that perfect diamagnetism is an essential property of the superconducting state [35].

BCS theory is a comprehensive theory developed in 1957 by the American physicists John Bardeen, Leon N. Cooper, and John R. Schrieffer to explain the microscopic behavior of superconducting materials [30]. The principal insight of the theory is that superconductivity results when electrons in a material form microscopic particles known as Cooper pairs. Electrons are fermions, which are subject to the Pauli Exclusion Principle; Cooper pairs are bosons, meaning they can collect in the same low energy ground state that is the superconducting state [39]. It makes a crucial assumption that is an attractive force exists between electrons. This force is due to the Coulomb attraction between the electron and the crystal lattice. An electron passes through the lattice and the positive ions are attracted to it, causing a distortion in their nominal positions and a slight increase in positive charges around it. This increase in positive charge will, in turn, attract another electron. These two electrons are Cooper pairs [31] which are discovered by Cooper [30]. These are also referred to as supperelectrons [8]. Superconductivity requires a low temperature that means the thermal vibration of the lattice must be small enough to allow the forming of Cooper pairs. In a superconductor, the current is made up of these Cooper pairs, rather than individual electrons [2].

Based on temperature superconductors can be grouped into high-temperature superconductors and low-temperature superconductors, and based on the mechanism they can be grouped into conventional and unconventional superconductors [7]. Based on magnetism superconducting materials can also be separated into two groups: type-I and type-II superconductors [9, 20]. Type-I materials, while in the superconducting state, are completely diamagnetic which is characterized by the Meissner effect [8]. They have sharp critical magnetic field, which is usually very low. There are two critical fields for type-II superconductors, the lower critical field and the upper critical field. If the external magnetic field is less than the lower field, the field is completely eject and the material act the same as a type-I superconductor [8, 35].

The discovery of high transition temperature $\left(T_c\right)$ superconductor is a landmark in the history of condensed matter physics. In 1979, the discovery of superconductivity in the heavy fermion compound, CeCu₂ Si₂ [22] came as a surprise, because the pairing of heavy fermions through electron–phonon interaction, as postulated by BCS theory [7], is highly unlikely. After this discovery, other heavy fermions were discovered. It has been suggested that in these compounds, the superconducting charge carriers are bound together in pairs by magnetic spin–spin interactions. Cuprates are the second class of high $T_c$ superconductors and was discover in 1986 by J. G. Bednorz and K. A. Müller with $\mathit{Tc}=35K$ in La_2−xBa_xCuO₄. The other class of high $T_c$ materials are iron-based superconductors (FebSc) which were discovered in 2008 by Hosono and co-workers with $T_c=$26K in LaOFeAs [7].

The first group is the iron-pnictides (nitrogen family from the periodic table [8]), in which iron forms a zigzag layer with arsenic or phosphorous (Fe-As or Fe-P). The 1111, 111, and 122 classes group to iron pnictides. Despite their different crystal structures and symmetry groups, they have similar magnetic structures. The second group is iron-chalcogenides (oxygen family from the periodic table [8]), in which iron-selenium (Fe-Se) or iron-tellurium (Fe-Te) forms the superconducting layer; this group includes the 11 and 245 classes.

The superconducting dome is asymmetric, with rather sharp onset and more gradual offset of superconductivity as a function of concentration. At the edge of the dome, the width in temperature of the superconducting transitions increases, and the diamagnetic screening fraction is substantially decreased. These characteristics may be taken as signatures of inhomogeneous superconductivity appearing at the edges of the superconducting phase region, even though the chemical doping distribution appears chemically homogeneous throughout the entire substitutional range [37].

High-entropy alloys (HEAs), newly discovered materials that were proposed in 2004, are typically composed of five or more major elements in similar concentrations, ranging from 5 to 35 atom% for each element. Until now, four types of HEA superconductors have been discovered. The type-A HEA superconductors (e.g., the Ta-Nb-Hf-Zr-Ti superconductors) consist of the early transition metals and crystalize on a small unit cell BCC lattice. Type-B HEA-superconductors (e.g., the (HfTaWIr)_1−xRe_x superconductors, x < 0.6) mainly consist of the 5d transition metals, and crystallize on a larger-unit-cell cluster-based BCC lattice. Type-C HEA superconductors (e.g., the Sc − Zr − Nb − Ta − Rh − Pd superconductors) are composed of the early transition metals and the late transition metals and crystallize on a small cell CsCl-type lattice. Type-D HEA superconductors (e.g., the Re_0.56Nb_0.11Ti_0.11Zr_0.11Hf_0.11 superconductor) crystallize on a HCP lattice [42].

Acknowledgments

We acknowledge Mrs. Mintamr Lewoyehu for typing the first draft of the chapter.

Conflict of interest

We confirm there are no conflicts of interest.

Funding

Not applicable.