Since the discovery of high-temperature superconductivity (HTSC) by Bednorz & Müller [1] in 1986, the underlying mechanism has remained an unsolved mystery to this day, although the physics community is certain that the mechanisms crucial for superconductivity take place in the CuO₂ levels and the CuO₂ layers serve as a charge carrier reservoir. For hole-doped HTSC it is called “hole conduction” because the high T_c is caused by O₂ doping, which provides holes. There is a simple relationship connecting the superconducting critical temperature T_c with P, the number of doped holes per Cu atom in the CuO₂ planes [2] ,

$T_c/T_{c,\max }=1-82.6{\left(P-P_{\max }\right)}^2,$(1)

where $T_{c,\max }$ is the maximum of the curve, which appears at $P=P_{\max }=0.16$.

In 2023 the editor of the publication by Wang et al. [3] about the superconductors Bi₂Sr₂Ca_n−1Cu_nO_2n+4 wrote a forward to it: “The mechanism of high-temperature superconductivity in copper oxide materials remains a mystery more than 30 years after its initial discovery. One way to shed light on this is to look for correlations between different observables in cuprate families…”

Such a correlation mentioned is, for instance, the correlation between T_c and the electron density n found in Al_1−x(SiO₂)_x cermets, shown in a separate paper,

$T_c/T_{c,\max }=1-\gamma \cdot n^2,$(2)

where $\gamma$ is a parameter, which is a constant for a superconductor. The special thing about this is that the change in n is caused by electron transfer, which compensates for potential differences in the material. These two aspects, Equation (2) and electron transfer, are important for an understanding of HTSC.

Applying a 35-year-old theory [4] [5] , the present article proposes a mechanism that can explain HTSC. It is based on the assumption that the charge carrier densities in HTSC are largely influenced by electron transfer, which compensates for potential differences. The Hall effect plays a key role in confirming this assumption: In YBa₂Cu₃O_6+x (YBCO), in addition to the hole Fermi surface, small electron pockets [6] were also detected in the underdoped region, indicating that electrons and holes are present simultaneously.

The parabolic curve $T_c/T_{c,\max }$ versus n, Equation (2), looks like the right part of Equation (1) ( $P>P_{\max }$), and with its left part ( $P<P_{\max }$) when mirrored on the vertical at P_max. However, while in Equation (2) the maximum of T_c occurs at $n=0$, in Equation (1) the maximum occurs on the P axis at P_max. And the question arizes, what is the reason for this shifting of $T_{c,\max }$ on the P-scale?

Let us assume that the 4s orbitals of the Cu atoms in the lowest layer in the unit cell ( Figure 1 ) form a narrow 4s band. This band we call band A. The 2p orbitals of the O atoms and the 3d orbitals of the Cu atoms overlapp forming a common pd band, called band B. The additional O atoms introduced by doping in this lowest layer produce holes in band B. Electron transfer now takes place from the band A into these holes. As a consequence, the electron density n in the band A decreases more and more as x increases. For Yba₂Cu₃O_6+x at $x=0.30$ (corresponding to $P=0.05$), n has reduced to such an extend that superconductivity can occur. And as x continues to increase, n decreases further and T_c increases until band A is completely free of “free” electrons. This is the case at $x=1$ (corresponding to P_max). As x increases further, T_c begins to decrease again.

To the left and right of P_max, T_c decreases with increasing distance from P_max because the carrier density increases; to the left of P_max, n increases with increasing distance to P_max, to the right of P_max, the hole density p increases with increasing distance to P_max. In other words, T_c decreases (parabolically) with increasing distance from the T_c maximum in the same sense as the carrier density increases, both to the left of P_max and to the right of P_max. This commonality of the two parabolic dependencies is expressed by Equation (1).

For the following calculations we assume a one-to-one correspondence between x and P in oxygen doped HTSC. In YBa₂Cu₃O_6+x such a one-to-one correspondence between x and P is only approximately fulfilled, which is the consequence of the so-called “ $P=1/8$” anomaly, which will be discussed later.

Let $N_{A,0}$ be the “free” number of electrons available for electron transfer per unit cell within the band A. Then the remaining number of “free” electrons per unit cell in band A after electron transfer is given by

$N_A\left(x\right)=N_{A,0}\left(1-\frac{x}{x_{\max }}\right)$(3)

for $0\le x\le x_{\max }$, but $N_A\left(x\right)=0$ for $x\ge x_{\max }$. $x_{\max }$ is the concentration where the maximum of T_c occurs. $N_A\left(x\right)=0$ at $x=x_{\max }$, which specify the complete emptying of the band A due to electron transfer.

Every O atom introduced by doping provides $N_{B,0}$ holes per unit cell. At $0\le x\le x_{\max }$, these holes are immediately filled with electrons transferred from band A, but at $x\ge x_{\max }$ there are $N_B\left(x\right)$ residual (unfilled) holes per unit cell in the band B, given by

$N_B\left(x\right)=N_{B,0}\left(\frac{x}{x_{\max }}-1\right)$ (4)

for $x>x_{\max }$, but $N_B\left(x\right)=0$ for $x\le x_{\max }$.

The concentration dependences of $N_A\left(x\right)$ and $N_B\left(x\right)$ are shown in Figure 2 . As a result of the electron transfer, the number of electrons per unit cell, $N_A\left(x\right)$, decreases with increasing x, according to Equation (3) ( $x\le x_{\max }$), whereas the number of holes per unit cell, $N_B\left(x\right)$, increases with increasing x, according to Equation (4) ( $x>x_{\max }$).

For the electron density and hole density related to the cross-sectional area of the unit cell, $n_f$ and $p_f$, it follows:

$n_f=N_A\left(x\right)/F,$(5)

$p_f=N_B\left(x\right)/F,$(6)

where $F\approx 1.49\times {10}^{-15}{\text{cm}}^2$ is the average cross-sectional area vertical to c in the unit cell of YBCO, see Figure 1 .

Because the electrostatic Coulomb interaction acts in the volume, we are interested in the electron density and hole density related to the unit cell, n and p,

$n=N_A\left(x\right)/{\upsilon }_{UC},$(7)

$p=N_B\left(x\right)/{\upsilon }_{UC},$(8)

where the average volume of the unit cell of YBCO is ${\upsilon }_{UC}\approx 1.74\times {10}^{-22}{\text{cm}}^3$. The concentration dependences of $n,p,n_f$ and $p_f$ are shown in Figure 3 .

As $n\propto N_A$ we get with Equations (2) and (3)

$T_c\left(x\right)/T_{c,\max }=1-{\gamma }^{\prime }{\left(N_{A,0}\cdot \left(1-\frac{x}{x_{\max }}\right)\right)}^2$ (9)

for the left side of the bell curve, where ${\gamma }^{\prime }\propto \gamma$.

At $x=0$, there is one “free” electron per unit cell in band A, and in band B each doped O atom contributes one hole per unit cell, i.e. $N_{A,0}=1$ and $N_{B,0}=1$. $x_{\max }=1$, because at $x=1$ the number of holes due to doping agrees with the number of electrons in the band A at $x=0$. Therefore Equation (9) can be replaced by

$T_c\left(x\right)/T_{c,\max }=1-{\gamma }^{\prime }\cdot {\left(1-x\right)}^2.$(10)

Superconductivity begins at $x=0.30\equiv x_0$. By setting $T_c\left(x\right)/T_{c,\max }=0$ in Equation (10), we obtain

${\gamma }^{\prime }=\frac{1}{{\left(1-x_0\right)}^2}$(11)

and, introduced in Equation (10),

$T_c\left(x\right)/T_{c,\max }=1-{\left(\frac{1-x}{1-x_0}\right)}^2.$(12)

In Figure 3 , the concentration dependence of $T_c\left(x\right)/T_{c,\max }$ calculated by Equation (12) is drawn and compared with the experimental data of La_2−xSr_xCuO₄, La_2−xSr_xCaCu₂O₆ and Y_1−xCa_xBaCu₃O_7−δ with $\delta \approx 0.04$ (taken from Tallon et al. [2] ). There is a relatively good agreement between the experimental data and Equation (12).

Although Equation (12) was derived using Equations (2) and (3) only for $x\le 1$, the experimental data also follow Equation (12) very well for $x>1$. How is this possible? According to Equations (3) and (4) and in consistency with Figure 2 , for $x>1$ there should only be “free” holes but no “free” electrons.

Considering the relatively good agreement between the experimental data and equation (12) also for $x>1$, we dare to hypothesize that superconductivity is possible not only by electron-Cooper pairs but also by hole-Cooper pairs consisting of two holes and that there is a complete analogy between electron-Cooper pairs and hole-Cooper pairs. This includes the hypothesis that, for symmetry reasons, there is also an analogue to Equation (2), namely

$T_c/T_{c,\max }=1-\gamma \cdot p^2.$(13)

Since $p\propto N_B$, using Equations (13) and (4) we get

$T_c\left(x\right)/T_{c,\max }=1-{\gamma }^{\prime }{\left(N_{B,0}\cdot \left(\frac{x}{x_{\max }}-1\right)\right)}^2$(14)

for the right side of the bell curve. And if $N_{B,0}=1$, $x_{\max }=1$ and Equation (11) are considered, the identic Equation (12) follows. That means, Equation (12) follows from the Equations (2) & (3) as well as from the Equations (13) & (4), independently of each other.

If $T_c\left(x\right)/T_{c,\max }=0$ is set, Equation (12) has two solutions, $x=x_0$ and $x=2-x_0$, which define the limits within which superconductivity occurs. Superconductivity exists in the range $x_0<x<2-x_0$. For $x_0<x<1$, superconductivity is realized by electron-Cooper pairs, for $1<x<2-x_0$ by hole-Cooper pairs.

While Equation (2) corresponds to the left part of the $T_c\left(x\right)/T_{c,\max }$ bell curve in Figure 3 , Equation (13) corresponds to the right part of it.

The electrostatic repulsion of like charges, electron-electron repulsion or hole-hole repulsion, is responsible for the decrease of T_c with increasing carrier density. This is reflected by the parabolic bell-shaped curve with the T_c maximum.