The pressure dependence of T_c in high temperature superconductors was investigated in our group from 1987 to 1999 using diamond anvil cells for the generation of high pressure. Many experiments were done on the pressure dependence of T_c and H_c2.

The pressure dependence of T_c in high-temperature superconductors is experimentally found to be influenced not only by a pressure induced increase in charge carrier density but also by intrisic effects. The behaviour of superconductors with 3 or 4 CuO₂-layers is further complicated by the inequivalence of the inner and outer CuO₂-layers, which may lead to a competetion between the two pressure dependences for the two kinds of layers. We developed a consistent phenomenological picture describing the experimental findings.

Experimental technique

In our laboratory, to generate high pressure for the investigation of the pressure dependence of the critical temperature Tc of superconductors, we use cryogenic diamond anvil cells (DACs) [21]. Temperature is measured from a calibrated platinum resistor thermally anchored to one of the diamonds. Since at low temperatures diamond has an even higher thermal conductivity than copper there is a good thermal contact between sample and thermometer. The applied pressure is determined in situ, close to the superconducting samples, with the ruby fluorescence method. After correction for the temperature--induced shift of the ruby R₁ fluorescence line [11] the calibration of Mao et al. [Mao] is used.

The superconducting transition temperature T_c of the sample is determined resistively by four point resistometry either directly using a sensitive multimeter or (for low resistance samples) with a 0.5Hz ac technique thus avoiding the effect of thermovoltages. The electrical leads are placed on top of one of the diamonds. These leads consist of flattened gold wires with a diameter of 25 µm which are pressed onto the sample for electrical contact.

We use the onset values for T_c, defined as the intercept between the tangents in the midpoint of the resistive transition and in the linear regime above T_c.

In view of the extreme pressures used, the tiny pieces of sample and the resulting not fully hydrostatic stress conditions, one may rightly ask how reproducible such experiments are. To address this question, in fig. 1, experimental results of a number of different groups on T_c(p) for YBa₂Cu₄O₈ are shown. This compound is chosen, because its doping is always the same, thus reducing the influence of different samples.

Figure 1, Critical temperature Tc for YBa2Cu4O8, which has fixed oxygen stoiciometry. The line is a guide to the eye. Data are shown from a number of different experimental groups and both for single crystal and ceramic samples. Clearly there is good agreement between the various results.

Clearly, despite the different experimental techniques and pressure media, very nice agreement is obtained. This comparison shows that such experiments do yield reliable results, and that differences between experiments should be attributed to different properties of the samples used.

The pressure dependence of T_c in high-T_c superconductors

Immediately after the discovery of high-T_c superconductivity high pressure experiments have played an important role in improving and understanding these materials. Several review papers summarize these experiments [1]-[5]. Despite tremendous experimental (and theoretical) efforts, not only in the field of high pressure but extending to nearly all techniques known in solid state physics, the microscopic mechanism for high-T_c superconductivity is not yet clear, although progress is still made. Therefore, here we present a summary of our experimental findings together with a consistent phenomenological description of the behavior of these interesting materials under pressure.

Figure 1 Tc(p) for the compounds indicated. All these results were obtained in our laboratory.

The rich variety in the behavior of T_c under pressure shown in fig. 1 is essentially due to four ingredients. First, it is well known [6] from chemical doping experiments that the critical temperature T_c of high-T_c superconductors is a function of the doping of the CuO2 layers, which are responsible for the superconductivity in these materials. The function T_c(n_h) is similar to an inverted parabola with a maximum for n_h~ 0.2, where n_h is the number of holes per Cu-atom of the CuO₂ layers. Secondly, it is known both from experiments [7]-[10] and calculations [11]- [13] that n_h increases with pressure. Hence T_c(p) is expected to follow the T_c(n_h) curve. Such approximate parabolic T_c(p) curves are indeed observed for a number of compounds, see for example the curves for YBa₂Cu₄O₈ and Y₂Ba₄Cu₇O₁₅ below 20 GPa in fig. 1. Note that T_c(p) does not need to be strictly parabolic due to the possible non-linear relation n_h(p). Thirdly, in some high-T_c superconductors inequivalent superconducting layers are present due to the peculiarities of the crystallographic structure. In one unit cell there are n adjacent CuO₂ layers between a top and bottom 'block layer', as illustrated schematically in fig. 2. If n £ 2, then all CuO₂ layers are equivalent, but for n=3 and n=4 there are two different kinds (inner and outer, shown white and grey, respectively) due to the different topology of their surroundings. The presence of the inequivalent layers may lead to more complicated T_c(p) curves, for example consisting of parts of two parabolas. Fourthly, it is often found that the maximum T_c under pressure is higher than the maximum T_c at zero pressure (as a function of chemical doping), implying that there exists an intrinsic enhancement of T_c due to pressure.

Intrinsic and doping effects on T_c(p)

To separate the intrinsic and doping effects on T_c(p), the system Tl_0.5Pb_0.5Sr₂Ca_1-xY_xCu₂O₇ is investigated [18]. In this system there is only one kind of CuO₂ layer, making the results easier to interpret. The doping n_h of the CuO₂ layers can be simply changed by adjusting x, which is done by preparing various samples [19]-[21]. Thus samples are available which are at ambient pressure at different points on the T_c(n_h) curve and in each of these T_c can be studied under pressure. The experimental results are summarized in fig. 2, which is based on experimental data [18] for x=0.0, 0.1, 0.2 and 0.35.

Figure 2. Critical temperature Tc of system Tl0.5Pb0.5Sr2Ca1-xYxCu2O7. At fixed pressure p, increasing x implies decreasing the number of holes nh. However, due to pressure induced doping, lines of constant x are not lines of constant doping.

The full pressure and doping dependence shown can be expressed as T_c(x,p)=T_c max(p) [ 1-b (p) ( x-x_max(p)) ²] where T_c max(p)  describes the intrinsic pressure dependence of T_c, while b (p) gives the width of the parabola T_c(x) and x_max(p) is the doping corresponding to the highest T_c at pressure p. These can be dereived from fitting to the experimental data as shown in fig. 2, and are plotted in fig. 3.

Figure 3.  The parameters Tc max(p) , b(p)  and xmax(p) for Tl0.5Pb0.5Sr2Ca1-xYxCu2O7. The intrinsic pressure dependence of Tc is due to the pressure dependence of Tc max.

From fig. 2 it is clear that pressure can increase T_c to a value higher than obtainable at any doping under ambient pressure. Note the clear optimum T_c(7.6 GPa,x=0.204)=111.1 K.

This proves the intrinsic effect of pressure on T_c in these materials: the increase in T_c cannot be due to more optimal doping, since the whole doping range is monitored under pressure. In fact such intrinsic effect was already identified as early as 1990 in our group by Van Eenige et al. [22] after the observation that under pressure the T_c of optimally doped YBa₂Cu₃O₇ exceeded its zero pressure value (in this paper both the effect of doping and intrinsic effects are taken into account; it precedes the more detailed work on intrinsic effects by Neumeier and Zimmerman [23]). The intrinsic effect was also identified by us in an early stage in other compounds [24, 25].

Compounds with inequivalent CuO2 layers

In compounds with 3 or 4 CuO2 layers (n=3 or n=4) one can discern 'inner' and 'outer' CuO₂ layers, see fig. 2. The inner layers are sandwiched between other CuO₂ layers, while the outer layers are adjacent to one CuO₂ layer and one block layer. As a consequence of this inequivalence, the doping from the block layers may spread unevenly between the inner and outer CuO₂ layers. The charge distribution may be calculated from the model of Haines and Tallon [31]. In this model the distribution of the charge carriers between the CuO₂ layers is found by minimizing the total energy U_tot=U_band+U_Mad of the charge carriers.

Figure 4 Tc(p) for Tl2Ba2Ca2Cu3O10, Tl2Ba2Ca3Cu4O12 and Y2Ba4Cu7O15. Note that Tc(p) consists now of two parabolae, due to the presence of two inequivalent layers with different Tc(p) behaviors.

The most relevant behavior is an increase of doping n_h vs. layer spacing d in both the inner and outer layers such that n_h of the outer layers is larger and increases at a faster rate with d than the n_h of the inner layers (for a schematic picture see fig. 8 of ref. [34]). If the T_c of the sample as a whole is the maximum of the T_c 's of the layers, then T_c(p) is first a parabola due to the outer layers and at higher pressures again a parabola due to the inner layers. In fact such behavior is beautifully displayed by the n=3 and n=4 thallium based superconductors as demonstrated in fig. 4 as a function of pressure, Tc first follows one parabola but above 12 GPa (for Tl₂Ba₂Ca₂Cu₃O₁₀) or 10 GPa (for Tl₂Ba₂Ca₃Cu₄O₁₂) T_c follows another parabola.

Another example of a compound with inequivalent superconducting units is the n=2 compound Y₂Ba₄Cu₇O₁₅. This perovskite is in fact a naturally formed multilayer of the YBa₂Cu₃O₇ and YBa₂Cu₄O₈ superconductors and the T_c(p) behavior of Y₂Ba₄Cu₇O₁₅ can be predicted directly from the T_c(p) behavior of YBa₂Cu₃O₇ and YBa₂Cu₄O₈, as was elegantly demonstrated by Van Eenige et al. [35]. Note also that the pressure induced change in T_c for Y₂Ba₄Cu₇O₁₅ is about 75 K, see fig. 4, which is the largest pressure induced change in T_c reported so far.

Discussion

In a homologous series (e.g. the Hg-based superconductors) the same block layers provide the doping for a different number n of CuO₂ layers. Hence, for increasing n, the (average) doping will range from overdoped to underdoped [36]. In particular the n=3 compounds are slightly underdoped and the application of pressure will raise T_c by making the doping more optimal. Since T_c also increases due to intrinsic effects, both contributions cooperate in the n=3 compounds, resulting in the occurrence of the highest T_c-values for any superconductor under pressure.

For the intrinsic pressure effect, most experimental evidence seems now in favor of a strong influence of the a-axis lattice parameter. This is also expected from e.g. Hubbard models. On the other hand, doping is closely connected to the c-axis lattice parameter. Of course, there is an interplay between these two effects and the exact nature of the block layers is also important in determining T_c, as is evident from the difference in maximum T_c values between e.g. the Bi₂Sr₂Ca₂Cu₃O₁₀ and the HgBa₂Ca₂Cu₃O₈ system.

We arrive at the following tentative scenario for T_c(p). Under pressure the block layers approach the CuO₂ layers and the hole doping increases. If the layer is originally underdoped, this enhances T_c; if it is overdoped, it decreases T_c. At the same time an intrinsic effect possibly related to in-plane (a-axis) compression enhances T_c (at least a low pressure). The resulting T_c of the compound is determined by the sum of these effects on the various CuO₂ layers.

With reference to the above, we now understand the various behavior shown in fig. 1:

For the underdoped YBa₂Cu₄O₈, pressure both increases doping and the intrinsic T_c leading to a spectacular increase of T_c under pressure by nearly 30 K. The T_c(p) curve is asymmetric and bell-shaped, which is fully consistent with a parabolic T_c(n_h) curve and a weakly increasing intrinsic T_c(p). The compound YBa_1.85La_0.15Cu₄O₈ behaves in a similar way.

By contrast, YBa₂Cu₃O₇ is close to optimal doping, the effect of pressure induced doping on T_c is hence small and the main contribution to T_c(p) is the intrinsic effect, leading to the observed weak increase of T_c(p). The observed behavior of Y₂Ba₄Cu₇O₁₅ is a proximity coupling average between YBa₂Cu₃O₇ and YBa₂Cu₄O₈. In the macroscopically tetragonal superconductor CaLaBaCu₃O₇, the chains are fragmented and charge transfer between the chains and CuO₂ layers is frustrated as found from our H_c2(p) measurements [43]. Hence, the observed Tc(p) is only due to the intrinsic T_c(p) and dT_c/dp is very modest. At ambient pressure Bi₂Sr₂CaCu₂O₈ is already overdoped and increasing n_h by pressure quickly brings down its T_c even further [44]. In the underdoped Bi₂Sr₂Ca₂Cu₃O₁₀ a positive dTc/dp is observed, as expected.

The complicated behavior which may result from the presence of inequivalent CuO₂ layers in Tl₂Ba₂Ca₂Cu₃O₁₀, Tl₂Ba₂ Ca₃Cu₄O₁₂, HgBa₂Ca₂Cu₃O₈ and HgBa₂Ca₃Cu₄O₁₀ was already discussed above.

References

The text above is a short version of the paper:

R.J. Wijngaarden, D. Tristan Jover and R. Griessen, Intrinsic and carrier density effects on the pressure dependence of Tc of high temperature superconductors, Physica B 265 (1999) 128-135

References to the text above:

References

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