Doped holes and/or electrons induced superconductivity domes explained with the free volume concept

Abstract

The conductivity equation developed in our previous work without any restrictions to specific materials is employed to explore how superconductivity transition temperature $T_{c}$ changes with the doped hole or electron concentrations based on the free volume concept. The predicted relationship is used to fit experimental data available in the literature and a good agreement with observations is achieved. Our findings may provide an alternative explanation for doping-induced domes and/or double domes with a-dip phenomena observed among many superconductors.

Introduction

Dome-shaped relationships between superconductivity transition temperature $T_{c}$ and doped electron or hole concentration in cuprates were observed right after the breakthrough work of high-temperature superconductivity found in cuprates (Bednorz and Müller 1986; Tokura et al. 1988). It has been an extensive research topic for the last 40 years (Ding et al. 1996; Li et al. 2019; Honma and Hor 2006; Lin et al. 2021; Chen et al. 2019). A similar dome-like correlation was found in other types of superconductors like iron-based (Cho et al. 2016), hydrides (Mazziotti et al. 2021), and metal alloys (Wan et al. 2023), etc., in both electron and hole-doped systems. It seems to be a universal phenomenon across broad superconductor materials, and theoretical investigations are performed as well (Wang et al. 2023; Langmann et al. 2019). A dome-shaped relationship can be theoretically projected in principle; however, an exact equation governing superconductivity transition temperature and the doped concentrations has not been elucidated. An empirical equation obtained from experimental data regression is shown below, and it works for many superconductors (Presland et al. 1991; Honma and Hor 2006):

$$\begin{aligned} T_{c} = T_{max}^{c} \left[1-82.6(p-0.16)^{2}\right] \end{aligned}$$

(1)

where $T_{c}$ is the superconductivity transition temperature, $T_{max}^{c}$ is the maximum value of $T_{c}$ in dome-shaped curve, and p is the doped hole concentrations. Note that p is intrinsically defined as a two-dimensional (2D) quantity, the hole content per $CuO_{2}$ plane (Honma and Hor 2006). It can be determined by the excess oxygen content, the thermoelectric power at room temperature $S^{290}$, or the Hall coefficient. The number 0.16 in Eq. 1 is later experimentally found not to be a universal constant for all superconductors (Honma and Hor 2006; Honma et al. 2004). A modified version of Eq. 1 was proposed to replace the parameter p with a 3D quantity $p_{3D}$ (Honma and Hor 2006):

$$\begin{aligned} T_{c}=T_{max}^{c} \left[1-83.63(p_{3D}\times 10^{-22} -0.159)^{2}\right] \end{aligned}$$

(2)

Equation 2 was claimed to have a universal nature of fitting capability (Honma and Hor 2006). Mathematically, both Eqs. 1 and 2 are very similar. Theoretically, an equation similar to empirical Eqs. 1 and 2 is still missing, even though intensive studies have been performed to elucidate the dome-shaped phenomena.

In this article, we will use the generic conductivity equation developed before to target this issue. We will continue to use the free volume as a main argument and assume that the free volume of the doped holes or electrons should be excluded when we consider that of the conduction electrons. In other words, doped electrons/holes will take some free volume that originally belonged to the conduction carriers in the system. We will go with the theoretical derivation first and then compare the obtained equations with the experimental data through data fitting practice. A discussion and summary will be at the end of the article.

Theoretical derivations

We will borrow the generic conductivity equation developed in 2015 (Hao 2015) to determine the superconductivity transition temperature first. This equation can be simplified as (Hao 2015, Hao 2023b, Hao 2019)

$$\begin{aligned} \sigma = AT \left[\exp \frac{B\alpha }{T}-\exp \frac{-B(1-\alpha ) }{T}\right] \end{aligned}$$

(3)

where $A= \frac{ek_{B}N_{c}}{hE}\exp \left(\frac{-\Delta G}{RT}\right)\lambda$, $B=\frac{e E}{k_{B}}\lambda$, $\lambda = \left[\frac{(9\pi N_{v}-4)V_{m}}{9\pi N_{v}N_{c}}\right]^{1/3}$, the electron travel distance. e is the electron charge, $k_{B}$ is the Boltzmann constant, T is the temperature, h is the Planck constant, R is the gas constant, E is the applied electric field, $N_{v}$ is the number of valence electrons per unit cell, $N_{c}$ is the number of conduction electrons in the whole system, $V_{m}$ is the volume of a material under study, $\Delta G$ is the standard Gibbs free energy, and $\alpha$ is a parameter related to the packing structures of electrons and has a simple relationship with the coordinate number of an electron in the system, $c_{n}$, $\alpha =\frac{1}{c_{n}}$. As indicated in the articles (Hao 2015 and Hao 2023a), the parameter $\lambda$ is temperature dependent, as $V_{m}$ is dependent on temperature with a generic relationship $V_{m}=a+bT^{n}$, typically $n<1.5$ (Kaptay 2015; Srivastava 2006). However, it can be reasonably assumed to be independent of temperature unless temperature approaches zero Kelvin, as demonstrated in the article (Hao 2023a). We want to find the relationship between $T_{c}$ and $\lambda$, so approximately we will assume that $\lambda$ is independent of temperature. There are several articles cited therein (Hao 2015) showing that $\lambda$ can be assumed independent of temperature if the temperature variation range is not too wide. Differentiating Eq. 3 and assuming $d\sigma /dT=0$, $e^{-x}\approx 1-x+\frac{x^{2}}{2}$, we obtain:

$$\begin{aligned} T_{c} & = \frac{\frac{e^{2}\lambda ^{2}N_{c}}{h} \left( \frac{\Delta G}{R}-\alpha \frac{eE\lambda }{k_{B}}+\frac{eE\lambda }{k_{B}} \right) }{2\left( \frac{\Delta G}{R}-\alpha \frac{eE\lambda }{k_{B}}+\frac{eE\lambda }{2k_{B}} \right) }\end{aligned}$$

(4)

$$\begin{aligned} & =\frac{e^{2}\lambda ^{2}N_{c}}{2h} \left( 1+\frac{1}{\frac{2k_{B}\Delta G}{eE\lambda R}-2\alpha +1} \right) \end{aligned}$$

(5)

Equation 5 indicates that $T_{c}$ has a complicated relationship with $\lambda$, and it also varies with the parameter $\alpha$ that has been demonstrated in our previous articles (Hao 2015, Hao 2019, Hao 2023a, Hao 2023b). However, this equation can be simplified if we assume $\frac{\Delta G}{R}-\alpha \frac{eE\lambda }{k_{B}}+\frac{eE\lambda }{k_{B}} \approx \frac{\Delta G}{R}-\alpha \frac{eE\lambda }{k_{B}}+\frac{eE\lambda }{2k_{B}}$. Hence, we can rewrite Eq. 5 as:

$$\begin{aligned} T_{c}=\frac{e^{2}N_{c}}{2h}\lambda ^{2} \end{aligned}$$

(6)

Equation 6 indicates that superconductivity transition temperature has a quadratic relationship with $\lambda$. Note that Eq. 6 is obtained under several approximations to correlate to $\lambda$. The different expressions of $T_{c}$ shown in the articles (Hao 2023a, Hao 2023b) are from the same original Eq. 3 and approximated for particular purpose. Now, we need to correlate $T_{c}$ with the free volume of doped electrons or holes via the inter-particle spacing (IPS) concept developed for particulate dispersion systems. The inter-particle spacing is derived in the article (Hao and Riman 2006). For 3D systems, we assume that the particles can freely move in x, y, and z directions for a distance two times the inter-particle spacing and obtain the free volume of an individual particle as:

$$\begin{aligned} V_{if}^{3D} & = 64r^{3} \left[ \left(\frac{\phi _{m}}{\phi }\right)^{1/3}-1\right]^{3}\end{aligned}$$

(7)

$$\begin{aligned} & = \frac{3V_{m}}{4\pi N_{d}} \left[\phi _{m}^{1/3}-\phi ^{1/3}\right]^{3} \end{aligned}$$

(8)

by considering that $\phi V_{m}=\frac{4\pi r^{3} N_{d} }{3}$, where r is the radius of the particle, $V_{m}$ is the volume of the material, $\phi _{m}$ is the maximum packing fraction of particles, $N_{d}$ is the number of holes or electrons doped in the system, and $\phi$ is the doped particle volume fraction. The free length, $L_{if}^{3D}$, can be defined as the basic scale of the free volume and thus can be expressed as:

$$\begin{aligned} L_{if}^{3D} & = 4r \left[\left(\frac{\phi _{m}}{\phi }\right)^{1/3}-1\right]\end{aligned}$$

(9)

$$\begin{aligned} & = \left(\frac{3V_{m}}{4\pi N_{d}}\right)^{1/3} \left[\phi _{m}^{1/3}-\phi ^{1/3}\right] \end{aligned}$$

(10)

We then may use Eq. 10 to build 2D situations by assuming that the particles can freely move only in x and y directions. Therefore, the free area/volume of an individual particle in a 2D system can be expressed as:

$$\begin{aligned} V_{if-r}^{2D} & = 16r^{2} \left[\left(\frac{\phi _{m}}{\phi }\right)^{1/3}-1\right]^{2}\end{aligned}$$

(11)

$$\begin{aligned} & =\frac{16V_{m}\phi }{\pi N_{d}}\left[\left(\frac{\phi _{m}}{\phi }\right)^{1/3}-1\right]^{2}\end{aligned}$$

(12)

$$\begin{aligned} & =\frac{16V_{m}\phi ^{1/3}}{\pi N_{d}}\left[\phi _{m}^{1/3}-\phi ^{1/3}\right]^{2} \end{aligned}$$

(13)

if we use the relationship with the particle radius in 3D and replace $r^{2}$ with $V_{m}\phi /\pi N_{d}$ in 2D systems. Alternatively, we may use the relationship with the volume fraction in 3D to obtain the free area/volume:

$$\begin{aligned} V_{if-\phi }^{2D} = \left(\frac{3V_{m}}{4\pi N_{d}}\right)^{2/3} \left[\phi _{m}^{1/3}-\phi ^{1/3}\right]^2 \end{aligned}$$

(14)

Note that Eqs. 13 and 14 are slightly different, and we will show later that they are very similar in terms of data fitting. They are just derived from a different route. Equations 13 and 14 can be used to describe the free area/volume of a doped individual hole or electron in 2D systems. The free area/volume of conduction carriers that participate in the superconductivity must be excluded from the free area/volume of doped holes or electrons, i.e., these two items should be constrained by each other. The free volume of carriers in a conduction system is dependent on the temperature, packing structures of the carriers, the atomic lattice structures of the material, etc. In other words, on the condition that the material is under study, the free volume has a certain value. Once extra holes/electrons are introduced into the system, they will take some free volume from the original “majority” of the conduction carriers, as the doped holes/electrons will participate in conduction behaviors. This is the reason that they must be excluded from each other. The free volume of an individual conduction carrier may be expressed as a subtraction of the unit volume from that of a doped electron/hole:

$$\begin{aligned} V_{ifr}^{c-2D}=1-\frac{16V_{m}\phi ^{1/3}}{\pi N_{d}}\left[\phi _{m}^{1/3}-{\phi }^{1/3}\right]^{2} \end{aligned}$$

(15)

if we start with Eq. 13. If we start with Eq.14, we may obtain another equation:

$$\begin{aligned} V_{if\phi }^{c-2D}=1- \left(\frac{3V_{m}}{4\pi N_{d}}\right)^{2/3} \left[\phi _{m}^{1/3}-\phi ^{1/3}\right]^{2} \end{aligned}$$

(16)

As the free area/volume defined previously (Hao 2015), for 2D systems the free area/volume of an individual conduction carrier can be correlated with the travel distance $\lambda$ as:

$$\begin{aligned} V_{ifr}^{c-2D}\; or\; V_{if\phi }^{c-2D}=(2\lambda )^{2}=4\lambda ^{2} \end{aligned}$$

(17)

Combining Eqs. 6, 15, 16, and 17 leads to the correlation between $T_{c}$ and the doped hole or electron concentration for 2D system

$$\begin{aligned} T_{c} = \frac{e^{2}N_{c}}{8h} \left[1-\frac{16V_{m}\phi ^{1/3}}{\pi N_{d}}\left(\phi ^{1/3}-\phi _{m} ^{1/3}\right)^{2}\right] \end{aligned}$$

(18)

and

$$\begin{aligned} T_{c} = \frac{e^{2}N_{c}}{8h} \left[1- \left(\frac{3V_{m}}{4\pi N_{d}}\right)^{2/3} \left[\phi _{m}^{1/3}-\phi ^{1/3}\right]^{2}\right] \end{aligned}$$

(19)

Equations 18 and 19 show the relationship between $T_{c}$ and the doped hole/electron concentration $\phi$ for 2D systems. Both Eqs. 18 and 19 are very similar to the empirical equation Eqs. 1 and 2, especially the Eq. 19. They render the physical meanings of several parameters/constants in Eq. 1: (1) $T_{max}^{c}$ is equal to $\frac{e^{2}N_{c}}{8h}$; (2) the number “82.6” actually is $\frac{16V_{m}\phi _{h}^{1/3}}{\pi N_{d}}$ in Eq. 18 and $\frac{3V_{m}}{4\pi N_{d}})^{2/3}$ in Eq. 19; (3) the value “0.16” actually is $\phi _{m} ^{1/3})$, depending on the maximum packing fraction; (4) it is the doped hole/electron concentration to the power of 1/3 instead of 1 that correlates with $T_{c}$; (5) $\phi$ by definition should be less than $\phi _{m}$, as $\phi _{m}$ is the maximum packing fraction of holes/electrons.

Results

First, we will plot Eqs. 18 and 19 with various parameters to gain an idea of whether these equations can predict a dome-shape relationship, which is shown in Figs. 1 and 2, respectively. The regular and normalized superconductivity transition temperature, $T_{c}$ and $T_{c}/\left(\frac{e^{2}N_{c}}{8h}\right)$, do show a maximum against the hole concentration. The heights of the maximum point increase with both $\left(\frac{e^{2}N_{c}}{8h}\right)$ and $\frac{16V_{m}}{\pi N_{d}}$, implying that both parameters control the maximum superconductivity transition temperature. The maximum packing fraction $\phi _{m}$, on the other hand, does not create a dome-like relationship between $T_{c}$ and $\phi$. The normalized superconductivity transition temperature only monotonically decreases with the doped hole/electron concentration at low $\phi _{m}$ but slightly increases at high $\phi _{m}$. Figures 1 and 2 demonstrate that Eqs. 18 and 19 seem working for the doping induced process. The dome-shape is most pronounced when both $\left(\frac{e^{2}N_{c}}{8h}\right)$ and $\frac{16V_{m}\phi ^{1/3}}{\pi N_{d}}$ are higher and diminishes when these two parameters are low. Note that $T_{c}$ and $T_{c}/\left(\frac{e^{2}N_{c}}{8h}\right)$ should be positive, and the negative axis is shown for easy viewing purpose. Comparing Figs. 1 and 2, we may easily tell that the former gives a taller dome than the latter.

Fig. 1

Illustration of superconductivity transition temperature against the hole concentration and other three key parameters in Eq. 18

Fig. 2

Illustration of superconductivity transition temperature against the hole concentration and other three key parameters in Eq. 19

Let us compare empirical equations Eqs. 1 and 2 with newly derived ones, Eqs. 18 and 19. Figure 3 shows the normalized superconductivity transition temperature vs. the hole concentrations for these three equations. For Eq. 18, $\frac{16V_{m}}{\pi N_{d}}$ is assumed to be 82.6, same as the one in Eq. 1, and $\phi _{m}$ is assumed to be 0.125. In Eq. 2, the term $p_{3D}\times 10^{-22}$ is assumed to be $\phi$. Equations 1 and 2 are overlapping together, and they peak at the same place. However, Eq. 18 peaks at different places and is unable to overlap with Eqs. 1 and 2, no matter how I change the parameters. A different shape will be obtained if they are forced to peak at the same location. Eq. 19 is very similar to the empirical equations but shows a narrower dome shape with the same parameters, $\frac{3V_{m}}{4\pi N_{d}}^{2/3}$ is assumed to be 82.6, same as the one in Eq. 1 and $\phi _{m}$ is assumed to be 0.125. At least all four equations predict a dome-like relationship.

Fig. 3

The normalized superconductivity transition temperature $T_{c}/\left(\frac{e^{2}N_{c}}{8h}\right)$ vs. the doped hole/electron concentration $p (p=\phi )$ with Eqs. 1, 2, 18 and 19.

The best approach to test the equations would be how they can fit the experimental data. Figure 4 shows the data points extracted from the literature and fitted with Eq. 1 in Fig. 4a, with Eq. 18 in Fig. 4b, and with Eq. 19 in Fig. 4c. The fitting parameters and quality $R^{2}$ are shown in Table 1. Note that we keep using “p” to represent the doped concentration for consistency with the literature, but it is replaced with $\phi$ during the fitting process. All three equations can give a good fitting for SrD-La214 superconductor, and the fitting quality indicated with $R^{2}$ is 0.91 for Eq. 1, 0.94 for Eq. 18, and 0.91 for Eq. 19. This superconductor has a perfect dome shape. However, both CD-Bi2201 and OD-Hg1201 superconductors show skewed dome-shapes, and Eq. 1 shows a very poor fitting quality with negative $R^{2}$. However, Eqs. 18 and 19 can still fit these two materials with very good fitting quality; $R^{2}$ is 0.97 and 0.97 for OD-Hg1201 and 0.95 and 0.94 for CD-Bi2201 with Eqs. 18 and 19, respectively. Excellent fitting quality with Eqs. 18 and 19 indicates that these two equations work for dome-shaped superconductivity phenomena observed for many superconductors, though Eq. 18 shows a slight better fit than Eq. 19 does. The numeric values before the quadratic term obtained with Eq. 18 are close to an empirical constant 82.6. The average of these three values is 78, which is not far away from 82.6 and may explain why the empirical Eq. 1 sometimes works for a lot of superconductors. On the other hand, the numeric values before the quadratic term obtained with Eq. 19 are far away from 82.6.

Fig. 4

Experimental data points extracted from the literature (Honma and Hor 2006) and therein for SrD-La214 $(La_{2-x}Sr_{x}CUO_{4})$, CD-Bi2201$(Bi_{2}Sr_{2-x}La_{x}CuO_{6+\delta })$, and OD-Hg1201 $(HgBa_{2}CuO_{4+\delta })$ are regressed with Eqs. 1, 18, and 19. The solid lines are best fitted with the equations

Table 1 The fitting parameters and quality $R^{2}$ with Eqs. 1, 18, and 19

Theoretically, $\phi$ should be less than $\phi _{m}$ based on the original concept that there is a maximum packing volume fraction. However, as shown in Table 1, the obtained $\phi _{m}$ from the best fitting process is less than the doping concentration. This seems to be contradictory to our original assumptions. However, please note that the $\phi _{m}$ corresponds to the peak of $T_{c}$, implying that $\phi _{m}$ is the maximum doping level; beyond this value, $T_{c}$ is going to decrease. Over-doping is harmful but of experimental significance.

There are many examples in literature that the dome-shape is skewed rather than the perfect (Lin et al. 2021; Li et al. 2019). Equations 18 and 19 should be more powerful to describe this universal phenomenon, as it is derived from the generic conductivity equation without being bound to any specific materials. The empirical equation can only work for perfect dome-shaped superconductors.

A “dip” in the dome has been identified recently in $Nd_{1-x}Sr_{x}NiO_{2}$ type (Chow and Ariando 2022; Li et al. 2020). Such a dip happens when the Hall coefficient transfers from the negative to the positive, implying that the system experiences from the electron-doped to the hole-doped process. As we may imagine, the maximum packing fraction of holes should be larger than that of electrons, i.e., $\phi _{m}$ should increase slightly. Figure 5 demonstrates that a dip happens if we increase $\phi _{m}$ from 0.125 to 0.216. The curves are predicted with Eq. 18. The lowest point corresponds to the average of these two $\phi _{m}$ values divided by 2, i.e., $\phi _{dip}=(\phi _{m1}+\phi _{m2})/2$. How to predict $\phi _{m}$? It is dependent on the material and its electronic structure, which is out of the scope of this article. The dome is formed due to the suppression of the free volume from the doped holes or electrons. So it is not strange to see that the dome-shape superconductivity can be induced by an external pressure (Chen et al. 2021; Zhang et al. 2021). External pressure can suppress the free volume of electrons as well, as detailed in my recent article (Hao 2022). A transition from hole to electron-doped (Zeng et al. 2020), from charge density wave to superconductivity (Chen et al. 2021), or any changes that may vary the maximum packing fraction, will induce double domes.

Fig. 5

The normalized superconductivity transition temperature $T_{c}/\left(\frac{e^{2}N_{c}}{8h}\right)$ vs. the doped hole/electron concentration with Eq. 18 at different maximum packaging fractions, which forms double domes and creates a dip

Discussion

The limitations of Eq. 18 come from the approximations of Eq. 3. (1) $N_{c}$ and $\lambda$ that contains $N_{c}$ are independent of temperature. Generally, this assumption is valid, except in extremely low temperatures; (2) $\frac{\Delta G}{R}-\alpha \frac{eE\lambda }{k_{B}}+\frac{eE\lambda }{k_{B}} \approx \frac{\Delta G}{R}-\alpha \frac{eE\lambda }{k_{B}}+\frac{eE\lambda }{2k_{B}}$, which is not exact; (3) we use $e^{-x}\approx 1-x+\frac{x^{2}}{2}$ to simplify the exponential function. They should be valid unless the temperature is extremely low close to zero Kelvin. These choices are unfortunately needed to find the relationship between $T_{c}$ and the free volume of doped holes or electrons. Equations 18 and 19 are derived for both hole and electron-doped systems. No matter which particle is, the free volume of the doped holes or electrons should be excluded from the free volume that the conduction electrons can have.

I would like to emphasize that “conduction electrons” means the electrons that participate conduction process. The term originated from the conduction equation developed for the electron conduction process. It should include any conduction carriers including both electrons and holes. Since we are dealing with doped electrons/holes in this article, the doped electrons/holes can only be distinguished from the free volume confounding effect as expressed with Eq. (16).

The free volume argument is valid for all superconductors. The impact of doped electrons and holes on $T_c$ may be very pronounced for low-$T_c$ superconductors but not too much for high-$T_c$ superconductors, which is of course material-dependent. Please take a look at Eq. (5), $T_c$ is related to the parameter $\alpha$ that is small for the low-$T_c$ superconductors and large for high-$T_c$ superconductors (Hao 2019). When $\alpha$ is large, the change of $\alpha$ may make $T_c$ lower rather than higher. The enhancement is limited for high-$T_c$ superconductors via the doping method. In overdosed regions, the doped electrons/holes would not further impact the free volume of the conduction electrons. In such a situation, we deal with a mixture of two materials rather than one compound and do not expect further suppression effect, which is the reason that there is a dome-shaped relationship.

Free volumes of electrons and holes should be quite different due to their different maximum packing fractions and packing structures of these two, which may explain the dips for any materials, including infinite-layer nickelates, during the doping process. The concepts are demonstrated in Figs. 3 and 5. Anything different in terms of the packing structures and the ultimate $\phi _m$ will shift the dome from one location to another, creating a dip.

We further demonstrate that the free volume of conduction electrons is critical. The superconductivity transition temperature is tied to the electron travel distance that in turn associated with free volume. Doped holes or electrons cannot share the free volume with the conduction electrons, which leads to dome-like behaviors. The external pressure of course can induce dome-shape correlation, as it can change the free volume of conduction electrons, the same as doped holes or electrons.

The number of conduction electrons and the maximum doping level to achieve the highest transition temperature can be obtained from Eq. 18. These parameters may be useful when we design, synthesize, and optimize superconductivity materials. Compared with empirical equations, our equation renders physical meanings for every constant.

Conclusion

Utilizing the generic conductivity equation developed in our previous article, we have correlated the superconductivity transition temperature with the doped hole or electron concentration. The basic idea is simple: the doped species, holes or electrons, will suppress the free volume of the conduction electrons, resulting in a different travel distance of electrons. The obtained equations do reproduce a dome-like relationship, show an identical prediction as the empirical equations when the dome-shape is perfect, and can accurately fit the experimental data when the dome-shape is skewed, much better than the empirical equations. Since our equation is not bound to any specific materials, it should be capable of describing various superconductors. The dome-shaped behaviors of doped systems should be a common feature across a broad of materials. Double domes with a dip can happen when the maximum packing fraction change happens, either due to the doping type or doping structure changes. Anything that can change the free volume of conduction electrons, such as external pressure, can generate a dome-shaped behavior, which will be covered in our future article. Our equations can work for both holes and electrons-doped systems.