Inhomogeneous superconductivity and the “pseudogap” state of novel superconductors
Inhomogeneous superconductivity and the “pseudogap” state of novel superconductors
Abstract and Keywords
The "pseudogap" state is strongly manifested above Tc for the underdoped compounds. Features typical of the superconducting state, such as the energy gap, anomalous diamagnetism, and the isotope effect, can be observed. These features are caused by an intrinsic inhomogeneity of the compound. Various energy scales (Tc, Tc*, T*) can be introduced. The ordered doping should lead to an increase in the value of the critical temperature
Keywords: inhomogeneity, anomalous diamagnetism, isotope effect, ordered doping
The highT_{c} oxides, as well as some other superconducting systems, display many properties in the normal state above T_{c} which are drastically different from those for conventional materials. This unusual normal (it is called “normal” because of finite resistance) state has been dubbed the “pseudogap” state. The first observation of this state was reported in 1989; that is, shortly after the discovery of highT_{c} superconductivity. NMR measurements demonstrated the presence of an energy gap for spin excitations (Alloul et al., 1989; Warren et al., 1989).
The subsequent studies reveal a number of other features. It is clear that above T_{c} the sample is in a peculiar state which is intermediate between fully superconducting and normal (see Table 11.1). Indeed, as any normal metal,
Table 11.1 “Pseudogap” state versus conventional superconducting and normal states
Superconducting state (T 〈 T_{c}) 
Normal state (T $\u3009$ T_{c}) 
“Pseudogap” state $\left({T}_{c}^{\ast}\u3009T\u3009{T}_{c}\right)$ 


Resistance 
R = 0 
R ≠ 0 
R ≠ 0 
Energy gap 
Δ ≠ 0 
Δ = 0 
Δ ≠ 0 
Anomalous diamagnetism 
Yes 
No 
Yes 
Macroscopic phase coherence 
Yes 
No 
No 
Josephson effect 
Yes 
No 
“Giant” effect 
Isotope effect 
Yes 
No 
Yes 
Impedance Z 
ReZ ≠ ImZ 
ReZ = ImZ 
ReZ ≠ ImZ 
(p.148) 11.1 “Pseudogap” state: main properties
Let us, at first, compare the behavior of novel superconductors above T _{c} with that of the usual systems. Table 11.1 contrasts the two classes.
The most fundamental feature is the anomalous diamagnetism (the Meissner effect), usually observed below the critical temperature. As for the region above T_{c}, the magnetic response of conventional metals is relatively small and almost temperature independent.
As we know, superconductors have a finite resistance above T_{c}, whereas below the critical temperature they are in the dissipationless state (R = 0). In addition, a.c. transport behaves differently above and below T_{c}; namely, above T_{c}, with high accuracy (see, for example, Landau et al., 2004), ReZ = ImZ (Z is the surface impedance). Contrary to this, below T_{c} in usual superconductors one can observe a strong inequality ReZ ≠ ImZ.
The superconducting state is also characterized by macroscopic phase coherence. For example, such a remarkable phenomenon as the Josephson effect is related directly to this feature.
As can be seen in Table 11.1, the “pseudogap” state appears to be rather peculiar. Let us describe its properties in more detail.
11.1.1 Anomalous diamagnetism above T_{c}
Usual normal metals display relatively weak response to a small external magnetic field. Indeed, the electronic gas is characterized by small Pauli paramagnetism. The magnetic susceptibility of real metals consists of several contributions (see, for example, Ashchroft and Mermin, 1976), and the resultant response might be diamagnetic, but the total susceptibility is almost temperature independent.
The situation above T_{c} in the cuprates appears to be drastically different. Unusual magnetic properties of the “pseudogap” state have been observed by using various techniques.
Scanning SQUID microscopy was used by Iguchi et al. (2001) to study the underdoped La_{2–x}Sr_{x}CuO_{4} compound. This technique allows one to create a local magnetic image of the surface: its “magnetic” map. The critical temperature of the underdoped LSCO films was T_{c} ≈ 18 K. A peculiar inhomogeneous picture has been observed: the film contains diamagnetic domains, and their presence persists up to 80 K(!). The total size of the diamagnetic regions grows as the temperature is decreased (Fig. 11.1). As a result, the diamagnetic response appears to be strongly temperature dependent—a very unusual feature of the materials.
A strong temperaturedependent diamagnetic response has also been observed by Bergemann et al. (1998) by using the torque magnetometry technique for the overdoped Tl_{2}Ba_{2}CaO_{+δ} compound above T_{c} ≈ 15 K. As in the case of LSCO, the diamagnetic moment was also strongly temperature dependent (Fig. 11.2). Torque magnetometry was also employed recently by Wang et al. (2005) to study the Bi2212 compound. Similarly, diamagnetic response was observed. It was essential that the analysis ruled out fluctuations as a key source of the observed diamagnetism.
11.1.2 Energy gap
The presence of the energy gap above T_{c} has been observed by using various experimental methods. Even the title “pseudogap state” reflects the existence of the gap structure. In connection with this it is worth noting that this title is confusing, since we are dealing not with a “pseudogap” but with a real gap; that is, with a real dip of the density states in the lowenergy region.
The energy gap ɛ is a fundamental microscopic parameter. As we know, ɛ = 0 in the normal phase and opens up at T_{c}. One should note, however, that the energy gap is indeed an important parameter, but its presence, unlike the order parameters, is not a crucial factor for superconductivity. For example, one can observe “gapless” (p.151) superconductivity (see Chapter 8) caused by the pairbreaking effect, for example, by the presence of localized magnetic moments.
Let us begin with tunneling spectroscopy, which allows one to perform the most detailed and reliable study of the gap spectrum. The data obtained by Renner et al. (1998) for an underdoped Bi2212 crystal are shown in Fig. 11.3. Scanning tunneling microscopy (STM) of the crystals cleaved in vacuum was employed. One can see directly the dip in the density of states (energy gap) which persists above T_{c} ≈ 83 K and persists up to ≈ 200 K (!). It should be stressed that the gap structure changes continuously from the superconducting region T 〈 T_{c} to the “pseudogap” state (T 〉 T_{c}); there is no noticeable change at T_{c}. This can be considered as an indication that the gap structure above T_{c} is related to superconducting pairing.
The presence of an energy gap for spin excitations has been established using NMR. Actually, as mentioned previously, the “pseudogap” state was observed initially by using this method. The gap has been observed in YBCO for different nuclei in both the Knight shift and spinrelaxation rate experiments: for ^{89}Y (Alloul et al., 1989), for ^{63}Cu (Warren et al., 1989; Walstedt and Warren, 1990), and for ^{17}O (Takigawa et al., 1989). It follows also from optical data (Homes et al., 1993; Fig. 11.4).
(p.152) Photoemission spectroscopy has also revealed the presence of an energy gap at T 〉 T_{c} (see the reviews by Shen and Dessau (1995) and by Randeria and Campuzano (1998)). For example, it has been demonstrated that the energy gap persists in an underdoped sample of Bi2212 at T 〉 T_{c} (Loeser et al., 1996). Ding et al. (1996) come to a similar conclusion.
Photoemission spectroscopy (Kanigel et al., 2008) has revealed the presence of disconnected Fermi arcs, some of which correspond to the normal metal. One can also observe gapped regions which correspond to the superconducting state. The picture is consistent with pairing above T_{c}.
We have described the data which present direct spectroscopic observation of the gap structure above T_{c}. A gap in the spectrum can also be inferred from heat capacity data. One should note that measurements carried out by Loram et al. (1994) and Wade et al. (1994) were some of the first observations of the “pseudogap” state. Measurements of the Sommerfeld constant γ(T) display a loss of entropy caused by the gap structure. The data for the energy gap Δ(T) are derived from values of the electronic entropy S(T, x) for 0.73 〈 x 〈 0.97. Again, it has been observed that the energy gap persists for T 〉 T_{c}, and the effect is especially strong for the underdoped samples of ${\text{YBa}}_{2}{\text{Cu}}_{3}{\text{O}}_{6+x}$.
11.1.3 Isotope effect
Another interesting property of the “pseudogap” state is the strong isotope effect. This effect has been observed by Lanzara et al. (1999) for La_{2–x}Sr_{x}CuO_{4}, using Xray absorption nearedge spectroscopy (XANES). The effect has been also observed by Temprano et al. (2000) for the HoBa_{2}Cu_{4}O_{8} compound. The slightly underdoped HoBa_{2}Cu_{4}O_{8} sample was studied by using neutron spectroscopy. As we know (see, for example, the review by Mesot and Furrer, 1997), the opening of the gap, which could be associated with the “pseudogap,” affects the relaxation rate of crystal field excitations. The isotopic substitution ^{16}O → ^{18}O leads to a drastic change in the value of the pseudogap temperature ${T}_{c}^{\ast}({T}_{c}^{\ast}\approx 170\phantom{\rule{thinmathspace}{0ex}}\text{K}\to {T}_{c}^{\ast}\approx 220\phantom{\rule{thinmathspace}{0ex}}\text{K})$. Such a large isotope shift corresponds to a large value of the isotope coefficient. Note that, contrary to the ordinary superconductor, its value is negative.
11.1.4 “Giant” Josephson effect
The socalled “giant” Josephson proximity effect is another interesting phenomenon observed in the “pseudogap” region above T_{c} (Bozovic et al., 2004). Films of La_{0.85}Sr_{0.15}CuO_{4} (T_{c} ≈ 45 K) were used as electrodes, whereas the underdoped LaCuO compound $({{T}_{c}}^{{\text{}}^{\prime}}\approx 25\phantom{\rule{thinmathspace}{0ex}}\mathrm{K})$ formed the barrier which was prepared in the cgeometry (the coherence length ξ_{c} ≈ 4A) The measurements were performed at ${T}_{c}^{{\text{}}^{\prime}}\u3008T\u300835\text{}\mathrm{K}$, so that the barrier was in the “pseudogap” state. Since $T\u3009{T}_{c}^{{\text{}}^{\prime}}$, we are dealing with the SNS junction. As is known, for such a junction the thickness of the barrier should not exceed the coherence length, which is of order of ξ_{c}. However, the Josephson current was observed for thicknesses of the barrier up to 200A(!). Such (p.153) a “giant” effect cannot be explained by using conventional theory. We will discuss this effect in detail in Section 11.4.3.
11.1.5 Transport properties
The microwave properties and a.c. transport have been studied by Kusco et al., (2002). It was shown that above T_{c}, that is, in the normal state, ReZ ≠ ImZ, where Z is the surface impedance. This is an unusual property, since in ordinary normal metals with a high degree of precision, the real and imaginary part of the impedance are equal (see, for example, Landau et al., 2004), that is, ReZ_{n} ≅ ImZ_{n}. The observed inequality ReZ ≠ ImZ is typically observed in superconducting materials.
The measurements of normal resistivity (Darmaoui and Jung, 1998; Yan et al., 2000; Jung et al., 2000) have revealed a strong inhomogeneity of YBCO and TBCCO samples. The presence of two different phases and inhomogeneous structure of the order parameter has been demonstrated.
Interesting data on thermal conductivity for various high T_{c} compounds were described by Sun et al. (2006). The measurements performed for different doping levels reveal the absence of universal dependence for the thermal flow and indicate the importance of strong inhomogeneity in the cuprates.
(p.154) 11.2 Inhomogeneous state
As described previously, intensive experimental studies reveal a number of unusual features of the cuprates above the resistive transition T_{c}. We described anomalous diamagnetism, which depends strongly on temperature, an energy gap structure, a strong inequality ReZ ≠ ImZ (Z is the surface impedance), a “giant” Josephson proximity effect, and an isotope effect on ${T}_{c}^{\ast}$.
The key issue, which is still controversial, is related to the nature of the “pseudogap” state. The main question is whether this state is a normal metal or a superconductor. At first sight, observation of finite resistance provides the answer to this question. Nevertheless, as described previously (see Table 11.1), in reality the situation is more complicated.
The most fundamental fact is the observation of the Meissner effect above ${T}_{c}={T}_{c}^{res.}$ (${T}_{c}^{res.}$ corresponds to the transition into the dissipationless macroscopic state with zero resistance).
It is essential that the contribution of fluctuations is not sufficient to explain the data for the underdoped region (Caretta et al., 2000; Lascialfari et al., 2002); indeed, the temperature scale for ${T}_{c}^{\ast}$ is very large. Therefore, we are dealing with a serious challenge: we should explain the coexistence of finite resistance and anomalous diamagnetism.
The properties of the “pseudogap” state are caused by intrinsic inhomogeneity of the metallic phase (Ovchinnikov et al., 1999, 2001, 2002; and the reviews, Kresin et al., 2006; Kresin and Wolf, 2012).
11.2.1 Qualitative picture
Consider an inhomogeneous superconductor, so that T_{c} = T_{c}(r). The system contains a set of superconducting regions—“islands” embedded in a normal metallic matrix (Fig. 11.5). Properties of such system correspond to the “pseudogap” state. Indeed, (p.155) the normal metallic matrix provides finite resistance, whereas the existence of the superconducting “islands” leads to the diamagnetic moment and energygap structure.
As temperature decreases toward T_{c}, the size of the superconducting regions increases, as does the number of “islands.” The critical temperature T_{c} corresponds to the percolation transition; that is, to the formation of a macroscopic superconducting region (“infinite cluster” in terms of the percolation theory; see, for example, Shklovskii and Efros, 1984; Stuffer and Aharony, 1992), and to phase coherence and dissipationless superconducting phenomena.
In conventional superconductors the resistive and Meissner transitions occur at the same temperature, T_{c}. The picture in the “pseudogap” state is different. The resistive and Meissner transition are split. The Meissner transition (the appearance of the diamagnetism) occurs at ${T}_{c}^{\ast}$, whereas the resistive transition—that is, the transition to the macroscopic dissipationless state—takes place at T_{c}, and ${T}_{c}\u3008{T}_{c}^{\ast}$. Note that in the papers (Ovchinnikov et al., 1999, 2001) the notations ${T}_{{c}^{\text{res}}}$ and ${T}_{{c}^{\text{Meis}}}$ are used. Here we use only the notations T_{c} and ${T}_{c}^{\ast}$, so that ${T}_{c}\equiv {T}_{{c}^{\mathrm{r}\mathrm{e}\mathrm{s}}}$, ${T}_{c}^{\ast}\equiv {T}_{{c}^{\mathrm{M}\mathrm{e}\mathrm{i}\mathrm{s}.}}$
It is important to note also that the “pseudogap” state in the region ${T}_{c}\u3008T\u3008{T}_{c}^{\ast}$ is not a phasecoherent one; each superconducting “island” has its own phase. At T = T_{c} the macroscopic superconducting region is formed, and below T_{c} we are dealing with macroscopically phasecoherent phenomena.
11.2.2 The origin of inhomogeneity
As noted previously, the presence of superconducting “islands” embedded in a normal metallic matrix implies an inhomogeneity of the compound. There are two possible scenarios for such an inhomogeneous structure:

1. Inhomogeneous distribution of pairbreakers.

2. Inhomogeneous distribution of carriers leading to spatial dependence of the coupling constant.
Both scenarios lead to inhomogeneous superconductivity. Pairbreaking can be caused by localized magnetic moments (Abrikosov and Gor’kov, 1961) (see Section 8.4). Qualitatively, the picture of pairbreaking can be visualized in the following way. A Cooper pair consists of two carriers with opposite spins (for singlet pairing; this is the case for both, s or dwave scenarios). A localized magnetic moment acts to align both spins in the same direction, and this leads to pairbreaking. It is known that for dwave pairing, nonmagnetic impurities are also pairbreakers.
(p.156) As discussed in Section 8.4, a pairbreaking effect leads to a depression in T_{c}. Therefore, a nonuniform distribution of pairbreakers makes the critical temperature spatially dependent: T_{c} ≡ T_{c}(r). Such a distribution is caused by the statistical nature of doping. The region which contains a larger number of pairbreakers is characterized by a smaller value of the local T_{c}.
Note that the pseudogap phenomenon is strongly manifested in the underdoped region. At optimum doping the distribution of pairbreakers becomes more uniform, and as a result, ${T}_{c}^{\ast}\simeq {T}_{c}$. The “pseudogap” state at T 〉 T_{c;opt} is characterized by the gap caused not by pairing, but by a CDW (see Section 11.3.1). In the underdoped state the spectrum is affected by both contributions.
As mentioned previously, an inhomogeneity can also be caused by an inhomogeneous distribution of carriers (see Ovchinnikov et al., 2001).
The picture described is directly related to the concept of phase separation. This concept was introduced by Gor’kov and Sokol (1987), shortly after the discovery of the high T_{c} oxides, and was then studied in many papers ((for example, Sigmund and Mueller, 1994). The concept implies the coexistence of metallic and insulating phases.
This picture of the “pseudogap” state is a next step in the inhomogeneous scenario. Namely, in addition to the mixture of metallic and insulating phases, the metallic phase is itself inhomogeneous; we are dealing with coexistence of normal and superconducting regions within the metallic phase.
11.2.3 Percolative transition
As mentioned previously, the transition at T ≡ T^{res} corresponds to the formation of the macroscopic superconducting phase. This transition is of percolative nature (Ovchinnikov et al., 1999; Mihailovich et al., 2002; Alvarez et al., 2005). In terms of percolation theory the transition to the dissipationless macroscopic state corresponds to the formation of an “infinite” cluster.
The percolative nature of the transition at T = T_{c} is due to the statistical nature of doping. The picture is similar to that introduced in manganites, which represents another family of doped oxides (see Chapter 12). Manganites (such as La_{0.7}Sr_{0.3}MnO_{3}) are characterized by the presence of ferromagnetic metallic regions embedded in the low conducting paramagnetic matrix above T_{c} ≡ T_{Curie}, where T_{Curie} is the Curie temperature. At T_{c} = T_{Curie} one can observe a percolative transition to the macroscopic ferromagnetic metallic state. The transition from the “pseudogap” to the macroscopic dissipationless state is also of percolative nature.
11.2.4 Inhomogeneity: experimental data
In Section 11.1.1 we described an interesting study of the Labased compound (Iguchi et al., 2001) performed using the STM technique with magnetic imaging (Fig. 11.1), which has demonstrated directly the presence of diamagnetic “islands” embedded in a normal matrix, and which shows the percolative picture as T → T_{c}. The same can be said about μSR spectroscopy (Sonier et al., 2008), also described in Section 11.1.1.
(p.157) Nanoscale inhomogeneity was also described in an interesting paper by Zelikovic et al. (2012). In addition to observing the inhomogeneous picture in Bi_{2–y}Sr_{2–y}CaCu_{2}O_{8+x}, they were able to correlate the structure of the “pseudogap” state with different types of oxygen dopants. It turns out that the apical oxygen vacancies are especially effective. This is an interesting observation, because we know that the apical oxygen is very important for the charge transfer, and such a defect, indeed, can act as a strong pairbreaker.
One key study was performed, using scanning tunneling spectroscopy (STM), by Gomes et al. (2007) and Parker et al. (2010). The measurements were performed on the Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} samples at finite temperature, so that the energy spectrum was measured above T_{c}, in the “pseudogap” region. These data have provided crucial information about local values of the gap and its evolution with temperature. The study of this evolution has led to the conclusion that the observed gap spectrum indeed corresponds to superconducting pairing. It is essential that the distribution of gaps turns out to be strongly inhomogeneous. As a whole, the presence of pairing persists for temperatures which greatly exceed those of T_{c}, especially for the underdoped samples. For example, for the sample with the doping level $x\approx 0.1\phantom{\rule{thinmathspace}{0ex}}({T}_{c}\approx 75\text{}\mathrm{K})$ the pairing persists up to ${T}_{c}^{\ast}\approx 180\text{}\mathrm{K}$. It is interesting also that from these measurements one can determine the local values of the gap Δ_{loc} and ${\mathrm{T}}_{\mathrm{p}}^{\mathrm{l}\mathrm{o}\mathrm{c}}$. One can conclude that the ratio appears $2{\mathrm{\Delta}}_{\mathrm{l}\mathrm{o}\mathrm{c}}/{\mathrm{T}}_{\mathrm{c}}^{\mathrm{l}\mathrm{o}\mathrm{c}}$ to be equal to $2{\mathrm{\Delta}}_{\mathrm{l}\mathrm{o}\mathrm{c}}/{\mathrm{T}}_{\mathrm{p}}^{\mathrm{l}\mathrm{o}\mathrm{c}}\approx 7.5$. Such a large value of this ratio greatly exceeds that for the usual superconductors (according to the BCS theory 2Δ/T_{c} ≈ 3.52), and corresponds to very strong electron–phonon coupling (see Section 3.5). The inhomogeneous structure with superconducting regions ~3 nm in size was observed by Lang et al. (2002).
As stressed previously, one should distinguish the intrinsic critical temperature ${T}_{c}^{\ast}$ and the resistive T_{c} (${T}_{c}\equiv {T}_{{c}^{res}}$). According to the described approach, the pairing gap is related to ${T}_{c}^{\ast}$, since the Cooper pairing occurs at first at this temperature. That is why the changes in the value of ${T}_{c}^{\ast}$ lead to corresponding changes in the gap value. As for ${T}_{c}\equiv {T}_{{c}^{res}}$, this temperature describes the percolation transition to the macroscopic superconducting state and is not related directly to the pairing interaction. Correspondingly, the changes in T_{c} should not affect the gap value. It is interesting that precisely this picture has been observed by Lubashevsky et al. (2011) The values of ${T}_{c}^{\ast}$ and T_{c} were changed independently (for example, the value of T_{c} was modified by Zn substitution). Indeed, it has been observed that the value of the gap measured by using ARPES was sensitive to the changes in ${T}_{c}^{\ast}$, but was not affected by changes in the resistive ${T}_{c}$.
11.3 Energy scales
The real picture in the cuprates is complicated, as we are dealing with three different energy scales (Kresin et al., 2004, 2006) and, correspondingly, with three characteristic temperatures (we denote them T_{c}, ${T}_{c}^{\ast}$, and T ^{*}).
(p.158) 11.3.1 Highestenergy scale (T ^{*})
The highestenergy scale, which we have labeled T ^{*}(~5.10^{2} K) corresponds to the formation of the inhomogeneity and peculiar crystal structure of the compounds. For example, for YBCO the formation of the chains occurs at T ^{*}.
An energy gap could open in the region below T ^{*}. This gap is not related to the pairing, but, as mentioned previously, there are many other sources for the appearance of a gap. For example, the presence of a chain structure in YBCO is consistent with a charge density wave and, correspondingly, with a gap on part of the Fermi surface. Nesting of states might also lead to a CDW instability in other compounds.
Another important property of the compound below T ^{*} is its intrinsic inhomogeneity; this is due to the statistical nature of doping, and is manifested in phase separation (Gor’kov and Sokol, 1987; see also Sigmund and Mueller, 1994). This property implies the coexistence of metallic and insulating phases. The periodic stripe structure (Bianconi, 1994a,b; Tranquada et al., 1995, 1997; Zaanen, 2000) also appears below T ^{*}.
11.3.2 Diamagnetic transition $\left({T}_{c}^{\ast}\right)$
If the compound is cooled to below T ^{*}, then at some characteristic temperature we have labeled ${T}_{c}^{\ast}$ $\left({T}_{c}^{\ast}\approx {2.10}^{2}\text{}\mathrm{K}\right)$ one can observe a transition into the diamagnetic state.
The characteristic temperature ${T}_{c}^{\ast}$ corresponds to the appearance of superconducting regions embedded in a normal metallic matrix (Fig. 11.5). The presence of such superconducting clusters (“islands”) leads to a diamagnetic moment, whereas the resistance remains finite, because of the normal matrix. As for the energy gap, coexistence of pairing and a CDW determine its value below ${T}_{c}^{\ast}$. It is remarkable that the superconducting state appears at a temperature ${T}_{c}^{\ast}$ which is much higher than the resistive T_{c}. This value of ${T}_{c}^{\ast}$ corresponds to the real transition to the superconducting state (one can call it an “intrinsic critical temperature”; see Kresin et al., 1996).
Strictly speaking, the experimentally measured value of ${T}_{c}^{\ast}$ lies below the intrinsic critical temperature, because of the impact by the proximity effect. Nevertheless, ${T}_{c}^{\ast}$ is an important parameter measured experimentally. It corresponds to the appearance of diamagnetic “islands,” and reflects the impact of pairing. The superconducting phase appears, at first, as a set of isolated “islands.”
The picture of different energy scales T ^{*} and ${T}_{c}^{\ast}$ just described is in total agreement with interesting experimental data by Kudo et al. (2005a,b). The impact of an external magnetic field was studied by outofplane resistive measurements. According to the study, there are, indeed, two characteristic temperatures (Kudo et al. dubbed them as T ^{*} and T ^{**}; T ^{*} 〉 T ^{**}). The behavior of the resistivity appears to be independent of magnetic field in the region T ^{*} 〉 T 〉 T ^{**}, but strongly affected by the field at T 〈 T ^{**}. According to Kudo et al., the state formed below T ^{**} is related to superconductivity. The characteristic temperatures T ^{*} and T ^{**} correspond directly to the energy scales T ^{*} and ${T}_{c}^{\ast}$ (in our notation, ${T}^{\ast \ast}\equiv {T}_{c}^{\ast}$) introduced above.
(p.159) 11.3.3 Resistive transition (T_{c})
As the temperature is lowered below ${T}_{c}^{\ast}$, new superconducting clusters appear (Fig. 11.1) and existing clusters form larger “islands.” This is a typical percolation scenario. At some characteristic temperature T _{c} the macroscopic superconducting phase is formed (“infinite” cluster in terms of the percolation theory; see, for example, Shklovskii and Efros, 1984). The formation of a macroscopic phase at T _{c} leads to the appearance of a dissipationless state (R = 0).
It is also important to stress that in the region ${T}_{c}^{\ast}\u3009T\u3009{T}_{c}$, each “island” has its own phase, so that there is no phase coherence for the whole sample. Macroscopic phase coherence appears only below T_{c}.
Therefore, there are three different energy scales and, correspondingly, three characteristic temperatures ${T}^{\ast},{T}_{c}$, and ${T}_{c}^{\ast}$ (Fig. 11.6).
To conclude this section, let us stress again that the inhomogeneous distribution of pairbreakers (dopants), along with local depressions in the value of critical temperature, leads to a spatial dependence of T_{c}; that is, T_{c}(r). The value of ${T}_{c}^{\ast}$ is close to an “intrinsic” critical temperature.
What are manifestations of the hightemperature superconducting state? Of course, the presence of the normal matrix at T 〉 T_{c} excludes the possibility of observing a state with zero dc resistance (R = 0). However, one can observe a number of various phenomena (see Section 11.1) which will be analyzed in the next section.
11.4 Theory
In this section we will present the theoretical analysis of the main features of the pseudogap state: diamagnetism, a.c. properties, and the “giant” Josephson proximity (p.160) effect. The analysis should take into account the inhomogeneity of the structure. There are two essential factors:

1. The proximity effect, since the superconducting regions are embedded into normal metallic matrix.

2. The pairbreaking effect.
11.4.1 General equations
Inhomogeneity of the system is a key ingredient of the theory. Because of it, it is convenient to use a formalism describing the compound in real space. That is why we employed the method of integrated Green’s function developed by Eilenberger (1968) and independently by Larkin and Ovchinnikov (1969); see also the review by Larkin and Ovchinnikov (1986).
The main equations have the form:
Here α and β are the usual and pairing Green’s functions averaged over energy, Δ is the order parameter, $\mathrm{\Gamma}\equiv {{\mathrm{\tau}}_{s}}^{1}$ is the spinflip relaxation time, and λ is the pairing coupling constant. Because of the inhomogeneity, all of these quantities are spatially dependent. In addition, ∂ _{±} = ∂ _{r} ± 2ie A, where A is the vector potential, ${\mathrm{\partial}}_{\mathit{\text{r}}}\mathbf{=}\left(\mathrm{\partial}/\mathrm{\partial}\mathit{\text{r}}\right)$. We consider the “dirty” case, so that D is the diffusion coefficient.
These equations contain the spatially dependent functions α, β, and Δ. The method is very effective for treatment of spatially dependent properties.
11.4.2 Diamagnetism
The Cu–O layers contain superconducting “islands,” and their presence leads to an observed diamagnetic moment. Because of the dependence T_{c}(r), the size of the superconducting region occupied by the “islands” decreases as temperature is increased. As a result, one can observe strongly temperature dependent diamagnetism.
Let us describe the evaluation of the diamagnetic moment (Ovchinnikov et al., 1999). Based on eqns. (11.1)–(11.1^{′′}), one can calculate the order parameter Δ(r) and then the current j(r) Then, one can calculate the magnetic moment, since the magnetic moment for an isolated cluster is
Here, L is the effective thickness of the superconducting layer; axis z is chosen to be perpendicular to the layers, and $\mathrm{\rho}$ is perpendicular to OZ.
(p.161) Assume that the sample contains a sufficient amount of magnetic impurities so that ${\mathrm{\tau}}_{s}\text{}{T}_{c}^{\circ}\u3008\u30081$; as a result, ${T}_{c}\u3008\u3008{T}_{c}^{\circ}$, where T_{c} is the average value of the critical temperature, and ${T}_{c}^{\circ}$ corresponds to the transition temperature with no magnetic impurities. In this case, with the use of eqns. (11.1–11.1^{′′}), we obtain
The order parameter can be found in the form Δ = CΔ_{0}, where C ≡ C(T), and Δ_{0} is the solution of the equation:
Here Γ_{∞} is the value of Γ outside of the “island,” and γ is the minimum eigenvalue. As a result, one arrives at the following equation:
Here, ψ is the Euler function, and the notation (f, g) corresponds to the scalar product of the functions. The transition temperature T_{c} is determined by the equation which can be obtained from eqn. (11.5) if we insert C = 0 and γ = 0:
The current density is described by the expression (Larkin and Ovchinnikov, 1969):
As a result, we can obtain the following expression for the current density of an isolated cluster:
Here, Δ_{0} is the solution of eqn. (11.4) and the vectorpotential has been chosen as $\mathit{\text{A}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{[}\mathit{\text{Hr}}\mathbf{]}$. Note also that because the cluster size is smaller than the penetration depth, one can neglect the spatial variation of the magnetic field.
Consider the most interesting case when the variation of the amplitude $\mathrm{\delta}\mathrm{\Gamma}={\mathrm{\Gamma}}_{\mathrm{\infty}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{\Gamma}$ has the form:
(p.162) With use of eqns. (11.4–11.6) one can obtain the following expression for the magnetic moment:
Here
The value of λ _{1} (and therefore the value of $\tilde{\mathrm{\beta}}$) depends on an interplay of two terms. The first term reflects the impact of pairbreaking, and the second term describes the proximity effect. It is natural that the impact of the proximity effect increases with a decrease in the size of the inhomogeneity ρ _{0}.
One can see directly from eqn. (11.10) that it is possible to observe a noticeable diamagnetic moment. Indeed, if we assume realistic values: p _{F} = 10^{–20} gcmsec^{1}, l = 40 Å (l is a mean free path: D = v_{F} l/3), T_{c} = 10 K, Γ_{∞} = 10^{2} K, δΓ = 50 K we put k_{B} = 1, ρ _{0} = 80 Å, and n _{s} ≅ 0.1, we obtain the following values of the parameters: A ≅ 10^{–5}, B = 3, γ = 5 K. Then, for example, at T = 11 K one can observe χ _{D} = M _{z}/H = −3 × 10^{–5}. This contribution greatly exceeds the usual paramagnetic response of a normal metal, χ _{P} ≅ 10^{–6}.
A diamagnetic response can be observed in the region $(T/{T}_{c})\u3008\tilde{B}$. This is natural, since the influence of the proximity effect (see, for example, Gilabert, 1977) to depress the superconductivity grows with a decrease in the size ρ _{0} of the superconducting grain.
11.4.3 Transport properties; “giant” Josephson effect
The dc transport properties of the inhomogeneous system above T_{c} are determined by the normal phase, since only this phase can provide a continuous path. The situation with ac transport is entirely different, and the superconducting “islands” make a direct contribution to the ac conductivity and to surface impedance.
As we know, the real and imaginary parts of the surface impedance of a normal metal are almost equal (see, for example, Landau et al., 2004). Indeed, the surface impedance Z is determined by the relation:
For normal metals the difference between ReZ and ImZ is negligibly small, and is connected with the dependence: σ(ω) = σ_{0}(1 – iωτ_{tr})^{−1}; in our case, ωτ_{tr} ≪ 1. The situation in superconductors is entirely different (see, for example, Tinkham, 1996), and the same is true for the “pseudogap” state. Indeed, a metallic compound (p.163) which contains superconducting “islands” is characterized by a strong inequality: Re(Z) ≠ Im(z). This can be shown theoretically (Ovchinnikov and Kresin, 2002) and measured experimentally (Kusco et al., 2002) for HgBa_{2}Ca_{2}Cu_{3}0_{8–δ} compound at T 〉 T_{c}.
Let us discuss the situation with the Josephson current. We mentioned previously an interesting experimental study of S–N–S Josephson junctions (Bozovic et al., 2004). This phenomenon cannot be explained by the usual theory of S–N–S proximity junctions.
We focus on the especially interesting case of S–N^{′}–S junctions where the electrodes are the high T_{c} superconducting films (such as La_{0.85}Sr_{0.15}CuO_{4}, or YBa_{2}Cu_{3}O_{7}), and the barrier N ^{′} is made of the underdoped cuprate; ${T}_{c}^{{\text{}}^{\prime}}$ is the critical temperature of the underdoped barrier, and T_{c} is the critical temperature of the electrode. The generally accepted notation N ^{′} emphasizes a difference between S–N^{′}–S and a typical S–N–S junction (then ${{T}_{c}}^{N}=0\text{}\mathrm{K}$), so that ${T}_{c}^{{\text{}}^{\prime}}\u3008{T}_{c}$. Here we consider temperatures when the barrier is in the normal resistive state because $T\u3009{{T}_{c}}^{{\text{}}^{\prime}}$. The use of the underdoped cuprate as a barrier is beneficial for various device applications because the structural similarities between the electrodes S and the barrier N ^{′} eliminate many interface problems.
The “giant” phenomenon is manifested in a finite superconducting current through the S–N^{′}–S Josephson junction with a thick barrier, so that L ≫ ξ _{N ′}, (where L is the thickness of the barrier, and ξ _{N ′} is the proximity coherence length). The configuration is such that the layers forming the barrier N ^{′} are parallel to the electrodes so that the Josephson current flows in the cdirection. Then the coherence length is very short ξ _{c} ≈ 4 Å, so that we are dealing with the “clean” limit. This type of junction using the LaSrCuO material was studied by Bozovic et al. (2004). The films of La_{0.85}Sr_{0.15}CuO_{4} (T _{c} ≈ 45 K) were used as electrodes, whereas the underdoped LaCuO compound $({T}_{c}^{{\text{}}^{\prime}}\approx 25\text{}\mathrm{K})$ formed the barrier. The atomiclayerbylayer molecular beam epitaxy technique was used for these junctions and provides atomically smooth interfaces. The barrier was prepared in the caxis geometry. As noted previously, the coherence length ${\mathrm{\xi}}_{c}\approx 4\text{}\text{\xc5}$. The measurements were performed at ${T}_{c}^{{\text{}}^{\prime}}\u3008T\u300835\text{}\mathrm{K}$. The Josephson current was observed for a thickness of L up to 200 Å(!). Such a “giant” effect cannot be explained with use of conventional theory. Indeed, as we know (see, for example, Barone and Paterno, 1982), the amplitude of the Josephson current for the “clean” limit is
The thickness of the barrier L should be comparable with the barrier coherence length ξ _{N}, and this condition is satisfied for conventional Josephson junctions. The picture described previously for the junctions with the cuprates is entirely different, since L ≫ ξ _{N ′}. The superconducting current in the cdirection occurs via an intrinsic Josephson effect between the neighboring layers (see Kleiner et al., 1992; Kleiner and Muller, 1994). If the barrier contains several homogeneous normal layers, then the Josephson current through such a barrier is practically absent.
To understand the nature of the “giant” Josephson proximity effect it is very important to stress that the barriers we are considering are formed by underdoped (p.164) cuprates. As a result, the barriers are not in the usual normal state but in the “pseudogap” state; indeed, ${T}_{c}^{{\text{}}^{\prime}}\u3008T\u3008{T}_{c}^{\ast}$. For example, see the study (Iguchi et al., 2001; see Section 11.1.1) of the compound La_{2−x}Sr_{x}CuO_{4} (x ≅ 0.1; T_{c} ≅ 18 K), in which the stoichiometry is close to that for the sample used by Bozovic et al. (2004) as the barrier. According to Iguchi et al. (2001), this compound has a value of ${T}_{c}^{\ast}\cong 80\text{}\mathrm{K}$ whereas ${{T}_{c}}^{{\text{}}^{\prime}}={T}_{c}^{res}\cong 18\phantom{\rule{thinmathspace}{0ex}}\mathrm{K}$ describes the resistive transition to the dissipationless state. Since the diamagnetic moment measured by Iguchi et al. (2002) persists up to ${T}_{c}^{\ast}$, the question of the origin of the “giant” proximity effect is related directly to the general problem of the nature of the “pseudogap” state.
This effect can be explained (Kresin et al., 2003) by the approach described in this chapter and based on the intrinsically inhomogeneous structure of the compound. According to the model, the CuO layers forming the N^{′}barrier contain superconducting “islands,” and these “islands” form the path for the Josephson tunneling current.
For typical S–N–S junctions the propagation of a Josephson current requires the overlap of the pairing functions F_{L} and F_{R} (see, for example, Kresin, 1986); F_{R} and F_{L} are pairing Gor’kov functions for left and rightside electrodes. This overlap is caused by the penetration of F _{L} and F_{R} (“proximity”) to the Nbarrier. For the system of interest here the situation is quite different. Each “island” has its own pairing function with its own phase. As a result, the Josephson current is caused by the overlap of F_{L} and F _{1}, F _{1} and F _{2}, and so on, where F _{1} corresponds to the “island” located at the layer nearest to the left electrode, and so forth. The superconducting “islands” form the network with the path for the superconducting current.
The propagation of a Josephson current through the S–N^{′}–S junction requires the formation of a channel between the electrodes. The transport of the charge in superconducting cuprates in the cdirection is provided by the interlayer Josephson tunneling (intrinsic Josephson effect). Therefore, the Josephson current through the barrier is measurable because of the superconducting state present in the layers.
The transfer of the Josephson current in the model described implies that the electrons tunnel inside the layers between the superconducting “islands” until one of them appears to be close to some “island” in the neighboring layer. Then the next step—the interlayer charge transfer via the intrinsic Josephson effect—occurs, and so on. As a result, the chain formed by the superconducting “islands” provides the Josephson tunneling between the electrodes, and the path represents a sequence of superconducting links. It is important to note that the amplitude of the total current is determined by the “weakest” link in the chain.
The density of the critical current is determined by the equation:
Assume that p is described by a Gaussian distribution; that is:
Here
We have assumed that δ 〈〈 1, m 〉〉 1.
It can be seen directly from eqns. (11.17) and (11.18) that the current amplitude depends strongly on temperature and is determined mainly by the dependence of the area occupied by the superconducting “islands” on T: n_{s} ≡ n_{s}(T). In addition, there is a weak dependence of j _{max} on the barrier thickness.
The dependence n_{s}(T) is different for various systems and is determined by the function T_{c}(r); that is, by the nature of the doping. Note that near T_{c} ^{*} the value of n_{s}(T) is very small and the current amplitude is negligibly small. However, the situation is different in the intermediate temperature region and in the region T 〈〈 T_{c} ^{*}, which is not far from T_{c}. This is true for the data by Bozovic et al. (2004): ${T}_{c}^{{\text{}}^{\prime}}=25\phantom{\rule{thinmathspace}{0ex}}\text{K}$ and ${T}_{c}^{{\text{}}^{\prime}}$ 〈 T 〈 35 K. For example, the value T =30 K is relatively close to ${T}_{c}^{{\text{}}^{\prime}}$ but is much below T ^{*} = 80–100 K. At T = 30 K there are many superconducting “islands”, so that the value of n_{s}(T) is relatively large. At temperatures close to T_{c} one can use eqn. (11.18) with η = a(t − 1)^{v}; t = T/T_{c}, and a = const (we have chosen a = 10). One can see (Fig. 11.7) that such a dependence with v = 1.3 is in good agreement with the experimental data. Note that this value for v is close to the value of the critical index for the correlation radius in the percolation theory (see, for example, Shklovskii and Efros, 1984).
Therefore, the “giant” Josephson proximity effect is also caused by intrinsic inhomogeneity of the cuprates. The “giant” scale of the phenomenon is provided by the presence of “superconducting” islands embedded in the metallic matrix and forming the chain transferring the current. The use of superconductors in the “pseudogap” state represents an interesting opportunity for “tuning” the Josephson junction on a “giant” scale.
11.4.4 Isotope effect
As mentioned previously, according to an interesting experiment (Temprano et al., 2000), the “pseudogap” state is characterized by a strong isotope effect; that is, by a large shift in ${T}_{c}^{\ast}$ caused by isotope substitution. They studied the HoBa_{2}Cu_{4}O_{8} compound with a value of ${T}_{c}^{\ast}\approx 170\text{}\mathrm{K}$. The isotopic substitution ^{16}O → ^{18}O leads to a drastic change in the value of ${T}_{c}^{\ast}$. One can observe the following change: ${T}_{c}^{\ast}$ ≈ 170 K → ${T}_{c}^{\ast}$ ≈ 220 K(!). We are dealing with a giant isotope effect; in addition, its value is negative.
Such an isotopic dependence of ${T}_{c}^{\ast}$ is also a consequence of the presence of the superconducting regions, and reflects the fact that the superconducting pairing persists above the resistive transition. It is interesting that the isotope coefficient has a negative sign. This unusual feature is consistent with the dynamic polaron model of the isotope (p.167) effect (see Section 9.7 and eqn. (9.12)). Indeed, a strong nonadiabaticity (axial oxygen in YBCO is in such state) results in a peculiar polaronic isotope effect:
Based on eqn. (11.19) one can explain why the isotope coefficient has a negative value. Indeed, $\mathrm{\alpha}\propto \mathrm{\partial}{T}_{c}^{\ast}/\mathrm{\partial}n$ and $\left(\mathrm{\partial}{T}_{c}^{\ast}/\mathrm{\partial}n\right)\u30080$: an increase in doping in the underdoped region leads to a decrease in the value of ${T}_{c}^{\ast}$. (At optimum doping ${T}_{c}^{\ast}\cong {T}_{c}$, and this is due to an increase in a number of dopants—that is, the pairbreakers—so that the distribution of pairbreakers becomes more uniform.)
11.5 Other systems
Intrinsic inhomogeneity is an essential feature of the highT_{c} oxides, and this feature is manifested in a peculiar “pseudogap” behavior. However, the scenario is a more general one, and the “pseudogap” state can also be observed in other inhomogeneous systems. Let us describe some of these systems.
11.5.1 Borocarbides
Borocarbides represent an interesting family of novel superconductors, because they allow us to study an interesting interplay between superconductivity and magnetism (see, for example, Canfield et al., 1998; Schmiedeshoff et al., 2001). According to data by Lascialfari et al. (2003), the borocarbides YNi_{2}B_{2}C display precursor diamagnetism above T_{c} (T_{c} = 15.25 K). Analysis of magnetization data taken with a highresolution SQUID magnetometer led to the conclusion that the unusual results were caused by an inhomogeneity of the compound and by the presence of superconducting droplets with a local value of the critical temperature higher than the usual T_{c} corresponding to macroscopic dissipationless current. The presence of the superconducting isolated droplets is due to an inhomogeneity similar to that observed in the cuprates and described in this chapter. Indeed, an unconventional temperature dependence of the critical fields H _{c1} and H _{c2} was observed (Suh et al., 1996; Schmiedeshoff et al., 2001). This dependence can be explained by the presence of pairbreakers (Ovchinnikov and Kresin, 2000; see Section 10.4.2). A Statistical distribution of pairbreakers leads to spatial dependence of the critical temperature: T_{c} ≡ T_{c}(r) and to the inhomogeneous “pseudogap” picture (Fig. 11.5).
11.5.2 Granular superconductors; Pb+Ag system
Granular superconducting films were studied intensively before the discovery of highT_{c} superconductivity (see, for example, Dynes and Garno, 1981; Simon et al., 1987). These films also represent inhomogeneous superconducting systems. Such inhomogeneous films could display diamagnetic moment above T_{c}. It would be interesting to carry out a study of their magnetic properties.
The results of the Pb/Ag artificial inhomogeneous superconductor model the behavior of the cuprates, which are doped substitutionally and inhomogeneously. At some concentration of doping there are regions with a sufficiently high concentration of carriers to superconduct locally and therefore reduce the lowenergy density of states. The evolution of these islands into a percolating dissipationless state would resemble the percolating proximity coupling described previously. Therefore, it is not surprising that the phase diagrams would be nearly identical.
11.6 Ordering of dopants and potential for roomtemperature superconductivity
The “pseudogap” scenario and its manifestation, which is especially strong in the underdoped region of the cuprates, implies that the transition into a superconducting state is a twostep process.
The superconducting regions appear below some characteristic temperature ${T}_{c}^{\ast}$ (we call it an “intrinsic” critical temperature); its value greatly exceeds the usual resistive ${T}_{c}\equiv {{T}_{c}}_{;\phantom{\rule{thinmathspace}{0ex}}res}$. For example, for the underdoped LaSrCuO compound (T_{c} ≈ 25 K) the value of ${T}_{c}^{\ast}\approx 80\text{}\mathrm{K}$. For the YBCO compound the value of ${T}_{c}^{\ast}\approx 250\text{}\mathrm{K}$; that is, it is close to room temperature. Decreasing the temperature toward ${T}_{c}\equiv {{T}_{c}}_{;\phantom{\rule{thinmathspace}{0ex}}res}$ leads to an increase in the number of “islands” and to an increase in their size. The second step is the transition at T_{c} which is of percolative nature and corresponds to the formation of a macroscopic superconducting phase (“infinite cluster” in terminology of the percolation theory) capable of carrying a macroscopic supercurrent. It is obvious that the set of superconducting clusters embedded in a normal matrix structure is unable to carry a superconducting current. Such a structure reflects the intrinsic inhomogeneity of the sample caused by the statistical nature of doping.
(p.169) It would be attractive to “order” the superconductive regions, and instead of statistically distributed superconducting “islands” to have a continuous superconducting phase; then one can expect the supercurrent to flow at high temperatures. In other words, it would mean an effective increase in the value of resistive T_{c}. In the following we describe the concept of “ordering of dopants,” which appears promising in achieving this goal.
The delocalized carriers whose presence is responsible for the metallic and, correspondingly, the superconducting state are created by doping. At the same time, the superconducting pairing could be depressed by the pairbreaking effect (see Sections 8.4 and 11.2.2). The dopants play a double role: they provide delocalized carriers, and are also responsible for the pairbreaking. The statistical nature of doping leads to a random distribution of dopants, and at relatively low doping the spatial distribution is rather broad. As a result, the regions with a smaller number of dopants have a larger value of the critical temperature. These regions form the superconducting “islands” inside the normal matrix.
An increase in T_{c} can be achieved by a procedure that allows for a special ordering of defects (Wolf and Kresin, 2012). To clarify the concept, let us consider a specific example: the YBCO compound. The doping of the parent YBa_{2}Cu_{3} O_{6} sample is provided by adding oxygen to the chain layer. The mixedvalence state of the inplane Cu leads to the planechain charge transfer and the appearance of a hole, initially on the Cu site. Because of diffusion, the hole enters the system of delocalized carriers responsible for the metallic and, correspondingly, superconducting behavior: superconductivity is caused by pairing of such holes. The added oxygen ion and corresponding inplane Cu form the defect mentioned previously, with its pairbreaking impact.
The picture described could be affected strongly by specific ordering of the oxygen ions. More specifically, let us consider the thin film formed by a layerbylayer deposition technique; the film is growing in the cdirection. The film should be built as a set of columns, the top layer could be imagined as a set of strips, and an additional oxygen should be placed inside specific columns. As a result, these columns would contain the chains with a composition close to YBa_{2}Cu_{3}O_{7}. The chains are parallel to the interstrip boundaries. Such columns could be called the reservoir columns, since holes are created in these areas. The holes are delocalized and diffuse into the neighboring columns, which are free of defects and represent the highT_{c} regions. Such a structure could be built, for example, using a nanoimplantation technique (see, for example, Schenkel et al., 2009; Toyli et al., 2010) with appropriate annealing or other methods for providing an ordered defect structure similar to what has been described here.
As a whole, the film would then contain alternating reservoir and highT_{c} columns. Because of the presence of the oxygen defects, the reservoir columns have value of T_{c} lower than in the neighboring columns which are free of defects. It is expected that the highT_{c} columns would have values of T_{c} close to the intrinsic critical temperature, ${T}_{c}^{\ast}$, which in fact could be close to room temperature. Moreover, such a structure would provide a continuous path for the supercurrent, which means an effective increase in the resistive T_{c}.
Note that the value of T_{c} in these highT_{c} columns could be depressed by the proximity effect with the reservoir columns, and this leads to some limitation on the width (p.170) of the highT_{c} columns, W_{H}. Indeed, we have the S_{L}–S_{H} proximity system, where L and H correspond to the reservoir and highT_{c} columns (T_{cH} 〉 T_{cL}). We know that the scale of the proximity effect is of order of the coherence length, ξ_{H}, of the highT_{c} region. To minimize the impact of the proximity effect, the width of the highT_{c} column, W_{H}, should be larger than ξ _{H}. However, the value of ξ_{H} is rather small. Indeed, ξ_{H} ≈ ν _{F}/2πT _{c;H}. If we take the values v_{F} ≈ 10^{7} sm/sec, and T_{c} ≈ (2–2.5) 10^{2} K, we obtain ξ _{H} ≈ 5–10 Å. Because of such a small value of the coherence length, the condition W_{H} ≫ ξ_{H} is perfectly realistic. Note that similar ordering could be effective for an increase in T_{c} for other cuprates: for example, LaSrCuO or Bi2212 compounds.
The proposed method conceptually is analogous to the observed increase in T_{c} for the cuprates caused by pressure (see, for example, Schilling and Klotz, 1992; Hochheimer and Etters, 1991; and the analysis by Kresin et al., 1996), by an applied field (Mannhart et al., 1991; Ann et al., 2006), or by the photoinduced effect (Yu et al., 1992; Pena et al., 2006). Indeed, the external pressure, radiation, or an applied field affects the doping without creating defects; that is, pairbreakers. The specifics of the structure proposed here is that the carriers appear in a region (highT_{c} column) which is free of defects; they are produced in a different spatially separated column. As stressed previously, we observe the impact of such separation in the existing cuprates (in the “pseudogap state”), especially in the underdoped region, where the superconducting clusters (highT_{c} regions) are embedded in a normal matrix. Due to such a separation one can observe the anomalous diamagnetism (Meissner effect) at temperatures higher than the resistive T_{c}. The proposed ordering leads to a noticeable increase of the critical temperature for the resistive transition and thus a much higher effective transition temperature.
The ordering leads to an increase in the size of superconducting clusters, and as a result, the percolative transition occurs at a higher temperature. This effective increase could be enhanced in a material with a larger dielectric constant (Mueller and Shengelaya, 2013). More specifically, they proposed to create a multilayer structure, so that the underdoped highT_{c} oxide thin film (in the range of ~1–10 nm) is sandwiched between the highdielectricconstant insulator layers; their presence reduced the Coulomb repulsion between the superconducting “islands.” Examples of such materials are SrTiO_{3} (Mueller and Burkard, 1979, with ɛ ≃ 10^{4}), or other ferroelectrics, such as Sr_{1x}Ba_{x}TiO_{3} or Pb(Zr_{x}Ti_{1–x})O_{3} (ɛ ≃ 3×10^{2}).
An additional increase in T_{c} can be achieved with the use of isotope substitution (Mueller, 2012). Indeed, as noted in Section 9.2.3, one can observe a large negative isotope effect, and as a result, the O_{16} → O_{18} substitution leads to an increase of order of 30 K(!).
The ordered doping manifested itself in experiments by Liu et al. (2006). The ordering of apical oxygen has been observed for Sr_{2}CuO_{3+δ} superconductor (T_{c} ≃ 75 K). This superconductor is characterized by a peculiar feature: the apical oxides sides are not fully occupied, and the hole doping occurs through the charge transfer between the apical oxygen and the Cu–O plane. The additional oxygen can be deposited on these sides in an ordered way, and as a result, an increase in T_{c} up to 95 K was observed.
Correlation between structure and its ordering and superconducting properties—that is, a noticeable rise in T_{c}—was described by Fratini et al. (2010) and (p.171) Pokkia et al. (2011). Xray radiation was used to create ordered superconducting regions in the La_{2}CuO_{4+y} samples, and it was accompanied by an increase in ${T}_{c}\left({T}_{c}\simeq 33.5\text{}\mathrm{K}\to {T}_{c}\simeq 41.5\text{}\mathrm{K}\right)$. The onset of highquality superconductivity was associated with 2D defect ordering.
11.7 Remarks
The approach to the “pseudogap” state described in this chapter is rather general in the sense that it is valid for any pairing force (phonons, magnons, plasmons, excitons, and so on), but, nevertheless, we are dealing (in the region T_{c} 〈 T 〈 T ^{*}) with real Cooper pairs, so that ${T}_{c}^{\ast}$ is the “intrinsic” critical temperature. Moreover, each superconducting “island” has its own phase. It is an essential feature of this picture that the superconducting regions are embedded in a normal metallic matrix which provides normal dc transport. As a result, the proximity effect between the superconducting regions and the normal metal plays an important role.
Until recently, the presence of inhomogeneities was considered as a signature of a poorquality sample (except for the “pinning” problem). However, we think that the situation is similar to that in the history of semiconductors. Indeed, initially the presence of impurities in these materials was considered as a negative factor (they were called “dirty” semiconductors). But later, when scientists developed tools allowing the precise control of the impact of various impurities (donors and acceptors), it became clear that the presence of impurity atoms is a critical ingredient; even the language has changed and sounds more “respectful” (“doped” semiconductors). The analogy between inhomogeneous novel superconductors and semiconductors is even stronger, because we are dealing with doping for both classes of materials.