The paper analyzes the influence of the hole density, the out-of-plane or in-plane disorder, and the isotopic oxygen mass on the zero temperature energy gap (2Δ(0)) Y1-xCaxBa2Cu1-yZny3O7-δ (YCBCZO) and La1.96-xSrxHo0.04CuO4 (LSHCO) superconductors. It has been found that the energy gap is visibly correlated with the value of the pseudogap temperature (T⋆). On the other hand, no correlation between 2Δ(0) and the critical temperature (TC) has been found. The above results mean that the value of the dimensionless ratio 2Δ0/kBTC can vary very strongly together with the chemical composition, while the parameter 2Δ(0)/kBT⋆ does not change significantly. In the paper, the analytical formula which binds the zero temperature energy gap and the pseudogap temperature has been also presented.
The superconductivity in the compounds of copper oxides (cuprates) was discovered in 1986 by Bednorz and Müller [1]. It is now known that in the family of cuprates the compounds of the highest critical temperatures (TC) exist. For example, in the HgBa2Ca2Cu3O8+δ (HBCCO) superconductor under the pressure at 31 GPa, the critical temperature equals about 164 K [2]. However, Takeshita et al. have reported recently that the correct maximum value of the critical temperature for HBCCO is a little bit lower, and it appears at much lower pressure (TC=153K at 15 GPa) [3].
The thermodynamics of the high-temperature superconducting state in cuprates differs significantly from the thermodynamics predicted in the framework of the classical BCS theory [4–7]. In addition to the too high value of the critical temperature, the most important difference seems to be in the existence of the second characteristic temperature, which is called the pseudogap temperature (T⋆).
Currently, it is believed that the critical temperature in cuprates sets the maximum value of T, at which disappears the coherence of the superconducting state, while T⋆ determines the temperature, in which the energy gap (2Δ) ceases to exist at the Fermi level [8].
It should be noted that both temperatures are equal in the classical BCS theory, wherein the theory predicts the universal relationship between the value of the zero temperature energy gap and the critical temperature: 2Δ0/kBTC=3.53, where kB is the Boltzmann constant [9].
In the presented paper, we have examined the impact of the various factors (the hole density (p), the disorder, and the oxygen isotopic mass) on the energy gap in the cuprates Y1-xCaxBa2Cu1-yZny3O7-δ (YCBCZO) and La1.96-xSrxHo0.04CuO4 (LSHCO) [10–12].
The primary objective of the study was to determine the relationship between the zero temperature energy gap and TC or T⋆. The obtained results allowed then the determination of the values of the dimensionless ratios: RΔ≡2Δ(0)/kBTC and RΔ⋆≡2Δ(0)/kBT⋆.
In the last step, we have derived the analytical formula which binds the energy gap and the pseudogap temperature.
All calculations have been performed in the framework of the theory, which assumes that the pairing mechanism in cuprates is induced by the electron-phonon interaction and the electron-electron correlations renormalized by the phonons. Additionally, through the appropriate selection of the electron band energy, the influence of the quasi-two-dimensionality of the electron system (the cooper-oxygen plane) on the physical properties of the studied compounds has been taken into account.
The reader may find in [13] the detailed description of the considered theory, together with the corresponding analysis leading to the fundamental thermodynamic equation. Additional information is also contained in the following works: [14] presents analysis of the ARPES method, [15, 16] present thermodynamics and ARPES for (Hg1-xSnx)Ba2Ca2Cu3O8+δ and Bi2Sr2-xLaxCuO6+δ, and [17] presents thermodynamics of the high-temperature superconductors with the maximum TC.
In the cases considered in the presented paper, the fundamental thermodynamic equation that determines the properties of the high-temperature superconducting state of d-wave symmetry has the following form [13]:(1)1=Vη+Uη6Δ¯η21N0∑kω0η2k2EkηtanhβEkη2,where the pairing potentials for the electron-phonon and electron-electron-phonon interaction have been denoted by Vη and Uη, respectively. The quantity Δ¯η is the amplitude of the order parameter for d-wave symmetry: ηk≡2coskx-cosky.
The symbol Ekη is defined by the following expression:(2)Ekη≡εk2+Vη+Uη6Δ¯η22Δ¯ηηk2,where the function εk determines the electron band energy, εk=-tγk, t denotes the hopping integral, and γk≡2coskx+cosky.
The inverted temperature (β) is given by the expression β≡1/kBT.
The normalization constant is given by N0≡1/∑kω0. The symbol ω0 represents the characteristic phonon frequency, which is of the order of Debye frequency.
Note that the sum over the momentums in (1) should be replaced with the integral in the following manner: ∑kω0≃∬-ππdkxdkyθω0-εkx,ky, where θ is the Heaviside function.
In order to simplify the numerical calculations and perform the analytical calculations in the subsequent part of the work, (1) should be converted into a more convenient form. For this reason, we have introduced the designations v2≡Vη, u2≡Uη/6, and(3)Δ≡v2+u2Δ¯η2Δ¯η.
Now, (1) can be rewritten in the following way:(4)1=v2+u2Δ¯η2IηΔ,T,where(5)IηΔ,T≡1N0∑kω0η2k2εk2+ηkΔ2×tanhβεk2+ηkΔ22.
Using expression (3), we can make the following transformation of (4):(6)1=v2+uΔv2+uΔ/v2+⋯222IηΔ,T.
It turns out that (6) can be written in the compact form:(7)1=v2+uΔ2IηΔ,T2IηΔ,T.
The equivalence of (6) and (7) can be most easily proven when determining the quantity IηΔ,T from (7) and then reinserting the resulting formula in the square brackets in (7).
The input parameters for (7) are as follows: the hopping integral, the characteristic phonon frequency, and the pairing potentials.
The same values of t and ω0 for the YCBCZO superconductor have been assumed as for the compound YBa2Cu3O7-δ (YBCO): t=250meV and ω0=75meV [18, 19]. In the case of the LSHCO superconductor, we have based on the values of t and ω0 obtained for La2-xSrxCuO4 (LSCO): t=240meV and ω0=96meV [20, 21].
The pairing potentials v and u have been chosen in such a way that the values of the critical temperature and the pseudogap temperature calculated on the basis of (7) would agree with the experimental values of TC and T⋆ determined in works [10–12]. It should be noted that this can be done in a relatively simple way, because the electron-phonon potential is the unique function of the critical temperature (v=vTC). Then, we have been able to determine the renormalized potential of the electron-electron interaction: u=uvTC,T⋆ [13]. The values of TC, T⋆ and the corresponding results have been presented in Figure 1 and in Table 1.
The critical temperature, the pseudogap temperature, and the values of the pairing potentials v and u for the YCBCZO and LSHCO superconductors. In the case of YCBCZO, the hole density (p) has been calculated based on the expression TCp/TC,max=1-pAp-pB2 [23]. Let us note that for the superconductors, which do not show the in-plane disorder, it has been obtained: pA=82.6 and pB=0.16. In the opposite case, the values of the parameters pA and pB increase together with the increasing disorder [24].
Material
Type
TC
T⋆
v
u
[K]
[K]
meV
meV
YCBCZO (x=0.2, y=0)
p=0.115
70.1
243.9
2.2371
2.9926
p=0.123
74.9
224.4
2.2749
2.8642
p=0.131
78.5
205.5
2.3005
2.7422
p=0.139
81.3
186.7
2.3215
2.6133
p=0.147
83.2
167.8
2.3354
2.4868
p=0.155
84.2
148.9
2.3433
2.3548
p=0.163
84.3
130.0
2.3442
2.2112
p=0.170
83.7
114.2
2.3393
2.0922
YCBCZO(x=0.2, y=0.04)
p=0.139
35.4
196.5
1.9159
2.9381
p=0.146
38.9
175.2
1.9503
2.8135
p=0.153
41.5
154.4
1.9811
2.6883
p=0.160
43.2
133.5
2.0009
2.5708
p=0.167
44.0
112.7
2.0101
2.4508
p=0.174
43.9
91.9
2.0097
2.3550
p=0.181
43.0
71.1
1.9993
2.2204
p=0.184
42.4
62.9
1.9929
2.1765
LSHCO (16O)
x=0.11
13.9
82.2
1.7435
3.0676
x=0.15
32.5
61.0
2.0325
2.7586
x=0.20
28.4
49.2
1.9836
2.7804
x=0.25
10.1
46.5
1.6474
3.0757
LSHCO (18O)
x=0.11
11.7
103.5
1.6924
3.1509
x=0.15
30.8
70.0
2.0100
2.8137
x=0.20
27.3
53.7
1.9700
2.8069
x=0.25
9.8
48.4
1.6331
3.0867
((a)-(b)) The critical temperature and the pseudogap temperature as a function of the hole density for YCBCZO superconductor [10, 11]. The results have been plotted for the systems with different degrees of disorder. ((c)-(d)) The dependence of the critical temperature and the pseudogap temperature on the concentration of strontium for LSHCO superconductor (the out-of-plane disorder). Additionally, two isotopes of oxygen have been taken into account [12].
On the basis of Figure 1(a) and the results collected in work [17], it can be easily seen that the out-of-plane disorder induced in YBCO by calcium does not change significantly TC and T⋆. For this reason, the values of the potentials v and u obtained for the YCBCZO compound (x=0.2, y=0) are very close to the values of the pairing potentials received for the ordered YBCO compound.
In the case of the in-plane disorder, generated by zinc in YCBCZO (x=0.2, y=0.04), the characteristic temperatures TC and T⋆ strongly decrease together with the increasing concentration of Zn (Figure 1(b)). As a result, the value of the pairing potential for the electron-phonon interaction significantly decreases. On the other hand, u changes only slightly, because it must compensate for the strong decline of v, so the experimental value T⋆ can be reproduced.
Figures 1(c)-1(d) present the dependence of TC and T⋆ on the strontium concentration for two isotopic masses of oxygen in the LSHCO superconductor. In the whole range of concentration, we can see the completely different effect of increasing isotopic mass of oxygen on TC and T⋆. In particular for the critical temperature, we have obtained the positive isotope effect (decrease in the value of TC), and for the temperature of the pseudogap, the isotope effect is negative [22].
The results collected in Table 1 indicate that the increase of the oxygen isotopic mass causes a slight decrease of v and a slight increase of the potential u.
In the next step, we have determined a full dependence of the order parameter on temperature for the YCBCZO and LSHCO superconductors. We have considered the most interesting cases.
The results obtained for the maximum values of the critical temperature have been plotted in Figure 2.
((a)-(b)) The order parameter as a function of the temperature for the YCBCZO superconductor. The selected values of doping have been adopted. ((c)-(d)) The temperature dependence of the order parameter for the LSHCO superconductor. The blue vertical line in the drawings sets the value of the critical temperature.
We can see that the shape of the function ΔT differs very significantly from the predictions of the classical BCS theory in all analyzed cases [4, 5]. Firstly, it should be noted that the values of the order parameter very weakly depend on temperature in the range from 0 to TC. As a result, the order parameter does not vanish for T=TC. From the physical point of view, this fact means the existence of the pseudogap in the electronic density of states. Next, in the temperature range from TC to T⋆, the order parameter slightly decreases and vanishes at T⋆.
On the basis of the numerical results, Figure 3 and Table 2 present the values of the zero temperature energy gap (2Δ0) for all cases analyzed in the presented paper.
The values of the zero temperature energy gap at the Fermi level, the ratio RΔ, and the ratio RΔ⋆ for the YCBCZO and LSHCO superconductors.
Material
Type
2Δ(0)
RΔ
RΔ⋆
meV
YCBCZO (x=0.2, y=0)
p=0.115
52.30
8.66
2.49
p=0.123
47.83
7.41
2.47
p=0.131
43.35
6.41
2.45
p=0.139
38.86
5.55
2.42
p=0.147
34.49
4.81
2.39
p=0.155
30.34
4.18
2.36
p=0.163
25.84
3.56
2.31
p=0.170
22.60
3.13
2.30
YCBCZO (x=0.2, y=0.04)
p=0.139
44.04
14.43
2.60
p=0.146
39.21
11.69
2.60
p=0.153
34.32
9.60
2.58
p=0.160
30.04
8.08
2.61
p=0.167
25.46
6.72
2.62
p=0.174
21.98
5.80
2.77
p=0.181
16.73
4.51
2.73
p=0.184
14.63
4.00
2.70
LSHCO(16O)
x=0.11
25.00
20.87
3.53
x=0.15
17.90
6.39
3.41
x=0.20
16.54
6.76
3.90
x=0.25
19.45
22.35
4.85
LSHCO(18O)
x=0.11
29.67
29.43
3.33
x=0.15
19.94
7.51
3.31
x=0.20
17.28
7.35
3.74
x=0.25
19.99
23.67
4.79
(a) The zero temperature energy gap as a function of the hole density for the YCBCZO superconductor. (b) The value of the zero temperature energy gap in the dependence on the concentration of strontium for the LSHCO superconductor.
It has been found that the increase in the hole density in the YCBCZO superconductor causes a strong decrease in the value of the energy gap and does so irrespectively of the disorder degree. It is not difficult to notice that the dependence of Δ0 on p is clearly correlated with the shape of the function T⋆p. In contrast, there is no clear relationship between the course of Δ0 on p and the form of the function TCp.
It should be noted that the results described above stand in sharp contrast to the predictions of the classical BCS theory, in which an increase or decrease of the zero temperature energy gap is always accompanied by an increase or decrease in the critical temperature (2Δ0=3.53kBTC) [9].
An equally anomalous relationship exists between the increase in the in-plane disorder and the value of 2Δ0. Figure 3(a) clearly proves that the value of the energy gap slightly grows together with the growth of the in-plane disorder. This result is very surprising, if we take into account the fact that the value of the critical temperature drops by nearly half at the same time (see Table 1).
The doping with strontium (the out-of-plane disorder) induces the strong decrease and then the increase in the value of the order parameter in the case of the LSHCO superconductor (Figure 3(b)). The obtained result comes from the analogical behavior of the pairing potential for the renormalized electron-electron interaction with the simultaneous increase and further decrease in the potential v (see Table 1). Also in this case, the course of Δ0 on x is more correlated with the shape of the function T⋆x than with the function TCx.
Taking into account the influence of the oxygen isotopic mass on the value of the energy gap, it can be clearly seen that the increase in the isotopic mass of oxygen causes a significant increase in the value of the energy gap. The obtained result clearly correlates with the one obtained for the pseudogap temperature. However, it is completely inconsistent with the predictions of the BCS theory, where the isotope coefficient is positive (α=0.5) [9].
The estimation of the value of the zero temperature energy gap with given TC and T⋆ allows in the simple way the calculation of the values of the two dimensionless parameters:(8)RΔ≡2Δ0kBTC,RΔ⋆≡2Δ0kBT⋆.
The obtained results have been presented in Figure 4 and in Table 2.
((a)-(b)) The ratio RΔ for the YCBCZO and LSHCO superconductors. ((b)-(d)) The values of the ratio RΔ⋆ for the YCBCZO and LSHCO superconductors.
It can be noticed that the parameter RΔ for both superconductors changes in the very wide range of the values, wherein the range of the values of RΔ markedly broadens with the increasing in-plane disorder induced by zinc or by the isotope substitutions, where ^{16}O isotope is being replaced by ^{18}O isotope. It should be emphasized that the obtained result comes from the lack of the correlation between the value of the energy gap and the value of the critical temperature.
The situation changes when we consider the parameter RΔ⋆. Based on the presented data, it is clear that the value of the energy gap varies in a similar manner to the value of the pseudogap temperature, and as such, it causes a weak dependence of the ratio RΔ⋆ on the hole density, the disorder, and the isotopic mass of oxygen.
In the last part of the paper, let us turn the attention toward the fact that using (7) allows the derivation of the explicit relationship between the value of the zero temperature energy gap and the pseudogap temperature.
For this purpose, we should make use of the fact that, for T=T⋆, the derivative dΔ/dT is unspecified. Hence, when differentiating both sides of (7), we get(9)2u2ΔIηΔ,T3dΔdT=-v2+3uΔ2IηΔ,T2dIηΔ,TdT.
Calculations that are not difficult give the result:(10)dIηΔ,TdT=1kBTJAηΔ,T-JBηΔ,TΔdΔdT-14kBT2JCηΔ,T,where(11)JAηΔ,T≡1N0∑kω0η4k4εk2+ηkΔ2·sech2βεk2+ηkΔ22,JBηΔ,T≡1N0∑kω0η4k2εk2+ηkΔ23/2·tanhβεk2+ηkΔ22,JCηΔ,T≡1N0∑kω0η2k·sech2βεk2+ηkΔ22.
Substituting (10) into (9), we have obtained the explicit expression for the derivative dΔ/dT:(12)dΔdT=1/4kBT2v2+3uΔ2IηΔ,T2JCηΔ,T2u2IηΔ,T3+v2+3uΔ2IηΔ,T21/kBTJAηΔ,T-JBηΔ,TΔ.
The equation for Δ0 and T⋆ can be obtained when we assume that the denominator in (12) is equal to zero. Thus,(13)kBT⋆=JAησΔ0,T⋆JBησΔ0,T⋆-2u2IησΔ0,T⋆3/v2+3u2σΔ02IησΔ0,T⋆2,where the parameter σ equals approximately 0.76.
Let us note that the value of σ has been chosen in such a way that it can, in the most precise way, allow the reproduction of the numerical results (see Table 3).
The results of the calculations for the values of the parameter σ=0.76. The symbol ΔT⋆ has been defined with the help of the formula ΔT⋆≡100%∗T⋆n-T⋆Eq./T⋆n, where T⋆n and T⋆Eq. denote the values of the pseudogap temperature obtained numerically or by using (13), respectively.
Material
Type
T⋆n [K]
T⋆Eq. [K]
|ΔT⋆| %
YCBCZO (x=0.2, y=0)
p=0.115
243.9
279.0
14.4
p=0.123
224.4
258.1
15.0
p=0.131
205.5
233.3
13.5
p=0.139
186.7
211.6
13.3
p=0.147
167.8
191.3
14.0
p=0.155
148.9
169.0
13.5
p=0.163
130.0
147.8
13.7
p=0.170
114.2
136.4
19.4
YCBCZO(x=0.2, y=0.04)
p=0.139
196.5
214.5
9.2
p=0.146
175.2
189.2
8.0
p=0.153
154.4
163.8
6.1
p=0.160
133.5
140.7
5.4
p=0.167
112.7
117.5
4.3
p=0.174
91.9
101.1
10.1
p=0.181
71.1
77.1
8.4
p=0.184
62.9
68.3
8.6
LSHCO(16O)
x=0.11
82.2
69.1
15.9
x=0.15
61.0
64.6
5.9
x=0.20
49.2
54.0
9.8
x=0.25
46.5
—
—
LSHCO(18O)
x=0.11
103.5
95.9
7.3
x=0.15
70.0
69.9
0.1
x=0.20
53.7
56.6
5.4
x=0.25
48.4
—
—
Additionally, it can be seen that, in the boundary of v/u→0, (13) takes the particularly simple form:(14)kBT⋆=JAησΔ0,T⋆JBησΔ0,T⋆-2/31/σΔ02IησΔ0,T⋆.
In conclusion, the study has examined the effect of the hole density, the out-of-plane or in-plane disorder, and the isotopic mass of oxygen on the value of the zero temperature energy gap in the YCBCZO and LSHCO superconductors.
It has been found that, regardless of the type of the studied material, the zero temperature energy gap is closely correlated with the pseudogap temperature. In contrast, there was no correlation between 2Δ0 and the critical temperature.
The obtained results indicate that the value of the ratio RΔ can vary and can widely depend on the deviations from the initial chemical composition. On the other hand, changes in the value of the parameter RΔ⋆ will be rather small.
In the presented paper, we have explicitly included all important numerical results, and for that reason we strongly encourage all readers to verify them quantitatively by means of the available experimental methods.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank Professor K. Dziliński for creating excellent working conditions. R. Szczȩśniak would like to express his gratitude to Sisi and Okta for the exhaustive scientific pieces of advice related to the topic tackled by the presented work. Additionally, the authors are grateful to Częstochowa University of Technology/MSK CzestMAN for granting access to the computing infrastructure built in Project no. POIG.02.03.00-00-028/08 “PLATON-Science Services Platform.”
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