
==== Front
iScience
iScience
iScience
2589-0042
Elsevier

S2589-0042(24)01818-2
10.1016/j.isci.2024.110593
110593
Article
Pressure-induced superconductivity domes described with a theoretical equation based on the free volume concept
Hao Tian haotian9@gmail.com
12∗
1 15905 Tanberry Dr, Chino Hills, CA 91709, USA
∗ Corresponding author haotian9@gmail.com
2 Lead contact

30 7 2024
20 9 2024
30 7 2024
27 9 11059311 3 2024
17 7 2024
24 7 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Summary

A generic conductivity equation developed in our previous work is borrowed to explore how superconductivity transition temperature Tc changes with external high pressure. The volume-pressure relationship revealed in the literature is utilized to estimate the pressure-dependent free volume based on our free volume equation. Pressure-induced single and double superconductivity domes are predicted with the obtained equation. It is also used to fit experimental data available in the literature. A good agreement with experimental observations is obtained. Our equation can fit nonlinear and polynomial relationships as well. The findings provide a theoretical foundation for pressure-induced phenomena among many superconductors.

Graphical abstract

Highlights

• Link the superconductivity transition temperature, Tc, with the pressure

• Utilize generic volume-pressure relationship for any solids in literature

• Correctly predict single- and/or double-dome relationship between Tc and the pressure

• Debunk the mystery on the pressure-induced superconductivity

Superconductivity; Theoretical physics

Subject areas

Superconductivity
Theoretical physics
Published: July 30, 2024
==== Body
pmcIntroduction

High pressure is an important approach to manipulate the superconductivity transition temperature Tc. There is an interesting phenomenon observed in many types of superconductors such as kagome metal CsV3Sb5,1,2 KMn6Bi5,3 CaFe2As2,4 and hydrogen-rich compounds5,6: pressure-induced single and/or double dome superconductivity, which is similar to doped holes and or electrons that also induce dome-shape behaviors and have been addressed extensively for last 40 years.7 Correlating Tc with the externally applied pressure using a generic superconductivity equation, various pressure-dependent superconductivity properties including the saw tooth shape can be described theoretically.8 Saw-tooth behavior predicted by assuming that the electron travel distance has an exponential relationship with pressure is very similar to double-dome phenomena observed experimentally. Since electron travel distance directly correlates with the free volume of electrons, we will further go down this route to see if we can more directly reach the double-dome superconductivity using the approach consistent with our previous publications.

Similar to what we have done before, we will start with the generic conductivity equation developed previously,9 as pressure-induced dome-shape behaviors have been observed in many superconductors and have a generic nature. The free volume would be used as a main argument in the derivation. We assume that the free volume of the electrons from the compressed portion of the material should be excluded when we estimate that of the electrons from the uncompressed or less compressed region. This compressed and uncompressed or less compressed concept is proposed based on the fact that solids are compressed layer by layer in 2D rather than in 3D fashions from all directions. We will utilize the famous Birch–Murnaghan equation developed for correlating the pressure and volume of solids detailed in the articles.10,11 In the next section, We will derive the equation to connect Tc with the externally applied pressure, and then use the equations to fit the experimental data. A discussion and a summary will be provided at the end of the article.

Theory

The generic conductivity equation developed in 20159 is borrowed to determine the superconductivity transition temperature. This equation is shown below9,12(Equation 1) σ=AT[expBαT−exp−B(1−α)T]

where A=ekBNchEexp(−ΔGRT)λ, B=eEkBλ, λ=[(9πNv−4)Vm9πNvNc]1/3, the electron travel distance. e is the electron charge, kB is the Boltzmann constant, T is the temperature, h is the Planck constant, R is the gas constant, E is the applied electric field, Nv is the number of valence electrons per unit cell, Nc is the number of conduction electrons in the whole system, Vm is the volume of a material under study, ΔG is the standard Gibbs free energy and α 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, cn, α=1cn. The detailed derivation procedure can be found in the Hao,13 we finally obtained:(Equation 2) Tc=e2Nc2hλ2

Equation 2 indicates that superconductivity transition temperature has a quadratic relationship with λ, the electron travel distance. Now we need to correlate Tc with the applied pressure P. Note that Equation 2 is reasonably approximated for simplicity, and the item on the right side of the equation may not render temperature unit in dimensionless analysis. Such an outcome does not deter from the focus of this current work which is on the free volume of electrons and how it may change under external pressure. In the end, the ultimate goal is to obtain an equation and fit/predict experimental data available in the literature regarding how the applied pressure impacts Tc. As we know high pressure can change the volume of solid materials and the relationship between high pressure and the volume has been experimentally and theoretically addressed,10,11,14,15 which can be simply written as:(Equation 3) Vm=V0exp(−aP)

V0 is the volume under zero pressure. The parameter “a” is a constant dependent on materials. Equation 3 works for all solids and will be used to correlate the applied pressure with the free volume in the system. As we can imagine, any solids contain the easily compressible portion like the voids or unoccupied spaces between molecules or atoms, and the hardcore portion like the basic crystal structures that are hard to compress. The former is named the “compressed” portion and the latter is called the “uncompressed” or less compressible portion in this article. The electrons in the compressed portion adjacent to the voids have a free volume directly or strongly dependent on pressure, while the ones in the uncompressed portion are indirectly dependent on pressure, the free volume of which is constrained by that of the electrons in the compressed portion. This physical picture should be plausible, as in the scale of an electron, more than 99% space in every atom is empty, and the world is full of empty unoccupied space among electrons, atoms, molecules, and larger particles.16 Due to these voids or free spaces, the internal pressure at each location point is not uniform. If we know the correlation between the free volume of the electrons in the compressed regions with the external pressure, we should be able to obtain that of the electrons in uncompressed or less compressible portions. We therefore can establish the correlation between Tc and the pressure.

First, let us see how to estimate the free volume of an individual particle in general. The problem was resolved based on the inter-particle spacing and the cell model17:(Equation 4) Vif3D=64r3[(ϕmϕ)1/3−1]3

(Equation 5) =3Vm4πNd[ϕm1/3−ϕ1/3]3

by considering that ϕVm=4πr3Nd3, where r is the radius of the particle, ϕm is the maximum packing fraction of particles, Nd is the number of particles in the compressed portion of the material, and ϕ is the particle volume fraction. As indicated earlier, when a high pressure is applied to a solid material, the compression is not uniform and there are compressed portions of the material in the void and out-layer regions, and uncompressed portions in the lattice structures and inside hard core regions. Since ϕ=4πr3Nd3Vm, using Equation 3 to replace Vm and ϕ leads to:(Equation 6) Vif3D=3V0exp(−aP)4πNd[ϕm1/3−(4πr3Ndexp(aP)3V0)1/3]3

The free length, Lif3D, can be defined as the basic scale of the free volume and thus can be expressed as:(Equation 7) Lif3D=(3V0exp(−aP)4πNd)1/3[ϕm1/3−(4πr3Ndexp(aP)3V0)1/3]

The chemistry and evolution of structures under high pressure indicate that the solids have layered structures and atoms and molecules pack more densely with pressure increase.18,19,20,21 The initial expectation is that the solids would adopt 3D closed-packed structures, but later the structure analysis tools reveal that it is not true,18,21 atoms are showing layered structures under high compression pressure due to the directional chemical bonds. Furthermore, the pressure inside the sample is not uniform but has a distribution against the distance from the compression center, i.e., there is a pressure gradient. Due to this unexpected compression nature of solids, we may reasonably assume the material is compressed layer by layer in a 2D fashion even if hydrostatic pressure is applied. In addition, many high-pressure experiments are carried out with diamond anvil cell that provides compression forces in two rather than four directions, which are most likely to create pressure-induced 2D structures. We may use Equation 7 to build 2D situations by assuming that the electrons can freely move only in the x and y directions in the compressed regions of the material. Therefore, the free area/volume of an individual electron in the 2D system of the compressed portions can be expressed as:(Equation 8) Vifr2D=(Lif3D)2

(Equation 9) =(3V0exp(−aP)4πNd)2/3[ϕm1/3−(4πr3Ndexp(aP)3V0)1/3]2

Electron-phonon coupling is a key concept in understanding superconductivity phenomena.22 Such interactions happen in the uncompressed portions where electrons travel through the crystal structures to contribute to the superconductivity. The free area/volume of electrons in these uncompressed portions should have a constraint with that of the electrons in the compressed portion. In other words, if the electrons cannot move freely in the compressed portions, they would travel more freely in the uncompressed portions, i.e., the free volume of electrons in the uncompressed portions would increase if that in the compressed portions decreases. Since the free volume of electrons should be a very small value, to avoid adding another unknown parameter, the total free volume of electrons in both compressed and uncompressed portions, we may simply express the free volume of electrons in the uncompressed portions as the unit volume subtracted by that in the compressed portions:(Equation 10) Vifrc−2D=1−(3V0exp(−aP)4πNd)2/3[ϕm1/3−(4πr3Ndexp(aP)3V0)1/3]2

We of course can use the volume of the material under zero pressure, V0, as the maximum total free volume of the electrons in both compressed and uncompressed portions, considering that the solids are most empty in the electron scale. For an individual electron, the maximum free volume could be V0Nd that can be used to replace “1” in Equation 10, which will be unnecessary, since it simply adds a term V0Nd on the right side of Equation 10 and will not change the regression quality of the final equation of Tc vs. the pressure P. The equation above provides a correlation between the 2D free area/volume of electrons with the applied pressure. As the free area/volume defined previously,17 for 2D systems the free area/volume of an individual conduction carrier can be correlated with the travel distance λ as:(Equation 11) Vifrc−2D=(2λ)2=4λ2

Combining Equations 2, 10, and 11 leads to the correlation between Tc and the applied pressure P:(Equation 12) Tc=e2Nc8h[1−(3V0exp(−aP)4πNd)2/3[ϕm1/3−(4πr3Ndexp(aP)3V0)1/3]2]

(Equation 13) =e2Nc8h[1−(3V04πNd)2/3exp(−2aP/3)[ϕm1/3−(4πr3Nd3V0)1/3exp(aP/3)]2]

Equation 13 shows the relationship between Tc and the pressure with several constants.

Results

Let’s plot Equation 13 with various parameters to gain an idea of whether the equation can predict a dome-shape relationship, which is shown in Figure 1. The regular and normalized superconductivity transition temperature, Tc and Tc/(e2Nc8h), do show a maximum against the pressure in most cases and occasionally a quasi-linear, really dependent on the parameters chosen. Figure 1A indicates that the normalized Tc only shows the dome-like shape when the parameter (3V04πNd)2/3 is large enough. When it is small, Tc almost linearly decreases with the applied pressure. A big dome is formed when (3V04πNd)2/3 is about 10. In Figure 1B, the parameter (e2Nc8h) controls the magnitude of Tc and the pressure has a little impact on it. However, a small dome is observed at high (e2Nc8h), implying that high Tc superconductors like LaH10, YH6, N-doped lutetium hydride etc. can be further optimized using the pressure, which is observed in articles.5,6,23 Figure 1C shows the normalized Tc changes with the pressure and the maximum packing fraction of ϕm. In this case, Tc decreases with the pressure increase but increases with ϕm. A small dome appears at high ϕm. The volume-pressure dependent parameter “a” has a huge impact on Tc, which is demonstrated in Figure 1D. A huge dome is observed at high “a” values, indicating that softer materials may show a pronounced dome phenomenon. An external pressure more likely has a big influence on Tc. Two domes, one with the pressure and another with the ϕm, are formed. Depending on the conditions or the material physical properties like the number of the conduction electrons, Tc could have a dome shape relationship with the pressures or a simple linear correlation.Figure 1 Illustration of superconductivity transition temperature against the pressure and other four key parameters in Equation 13

For further demonstrating how Tc is going to change with the number of conduction electrons in compressed region, we plot Tc/(e2Nc8h) against pressure at two different (4πr3Nd3V0)1/3=0.6 and (4πr3Nd3V0)1/3=0.4 in Figure 2 with Equation 13. A slight decrease in the number of conduction electrons shifts the dome to substantial high pressure, which is a big surprise. Insertion or intercalation of ions into the crystal structure, which may change the number of conduction electrons and electronic structures can have a profound impact on the maximum Tc values, which is already evidenced experimentally and theoretically.24,25Figure 2 The normalized superconductivity transition temperature Tc/(e2Nc8h) vs. the pressure with Equation 13

Next, let’s compare Equation 13 with experimental data, which is shown in Figure 3. The data points are extracted from the literature and the lines are fitted with Equation 13. For all three cases with superconductors LaH10, YH6/YH9, N-doped lutetium hydride, excellent fitting lines are obtained with the fitting quality parameter, R2= 0.70, 0.96, and 0.81, respectively. Of course, the values of R2 are dependent on the variation of original data points. A large variation usually leads to poor fitting quality, no matter how good the fitting equation is. Nonetheless, Equation 13 does an amazing job of fitting three hugely different relationships. As we demonstrated in Figure 1, it is not surprising to see the same equation can predict a dome and a nonlinear relationship, which is situation/material dependent. Equation 13 is versatile and universal, as it is derived from the conductivity equation without binding to any particular materials and Equation 3 is universal, too.Figure 3 Experimental data points extracted from the literature5 for lanthanum hydrides and Kong et al.6 for yttrium hydrides and yttrium-deuterium are regressed with Equation 13

The solid lines are best fitted with the equation.

(A) All data points of LaH10, (B) Only the data points on the dome are used to demonstrate if the equation can fit the skewed dome, (C) Only data of YD6 are used.

The maximum packing fraction of electrons could change when the temperature goes down to a very low level, leading to the double-dome superconductivity phenomena. As shown in Figure 2, the change in the number of the conduction electrons in the compressed region can result in double domes, separated far away from each other. According to the previously published articles,18,19,20 it is very hard for an external pressure to change the number of electrons during a compression process, however, the pressure can change the packing structures of atoms and molecules, transferring the solid from one crystal structure to another. This kind of structural re-arrangement doesn’t always mean a phase transition but can change the maximum packing fractions ϕm. Figure 4 shows the double domes induced by the pressure and predicted with Equation 13 at ϕm=0.5 and ϕm=0.8. The lowest point, the dip between these two domes, corresponds to the average of these two ϕm values divided by 2, i.e., ϕdip=(ϕm1+ϕm2)/2. ϕm can be associated with the crystal structures by X-ray diffraction or other structural analysis tools. The double domes are formed due to the suppression of the free volume from one structure to another, sharing the same origin as the doped holes or electrons process,26,27 where double domes are observed as well. The doping-induced dome and double domes associated with a dip are detailed in the Hao.13 In a word, any changes that may vary the maximum packing fraction will induce double domes. Such an example is shown in the article,28 where a dip was observed in antimony when there is a phase change and a structural transformation happened, which is confirmed with resistance measurement.Figure 4 The normalized superconductivity transition temperature Tc/(e2Nc8h) vs. pressure P with Equation 13 at different maximum packaging fractions, which forms double domes and creates a dip

In the articles,1,2,29,30 double domes are observed in pressure-induced superconductivity of CsV3Sb5, Pristine 1T−TiSe2, and CsTi3Bi5. Figure 5 shows the experimental Tc data points of CsTi3Bi5 from the sample 1 measurement and regressed with Equation 13. A good agreement with superconductivity transition temperature vs. pressure is achieved. The fitting quality, R2, is 0.80 and 0.91 for the first dataset at low-pressure range on the left and the second dataset at high-pressure range on the right side, respectively, indicating that the equation obtained in this article can fit both single and double-domes observed in pressure-induced superconductivity phenomena.Figure 5 Data points are extracted from the sample 1 in Nie et al.30 and fitted with Equation 13

The blue and black curves indicate that these data are regressed with two sets of parameters at low and high-pressure ranges.

Discussion

Pressure-induced single- and double-dome behaviors are addressed based on two universal equations, one is the conductivity equation and another is the volume-pressure equation. The former is developed by the author and the latter is developed by many other researchers. The core concept is that the volume of the solids will decrease with the pressure, and so will the free volume of electrons, which in turn controls the conductivity and superconductivity.

The dome-shape relationship is induced by the free volume constraints between the electrons in the compressed regions and the uncompressed or less compressed regions. The reason for separating these two regions results from the fact that any solids contain the voids between molecules and atoms, and the physical entities like atoms and molecules. The former can be easily compressed, but the latter is hard to be compressed. Electrons in these two regions should behave differently. Due to the non-uniform nature of any solids and the design of commonly used high-pressure vessels, the solids are compressed layer by layer, even if hydrostatic pressure is applied; the layer that is closer to the diamond surface may be compressed harder than the layer that is further away from the diamond surface. In other words, the stress field is non-uniform, and 2D compression is assumed when we estimate the free volume. The molecular and atomic scale packing structure changes under the pressure could lead to double domes due to the maximum packing volume fraction shift.

The obtained equation, Equation 13, can fit experimental data very well, no matter what the correlation shape is, a perfect dome, a skewed dome, or a non-linear line. Four constant-type parameters in the equation can be used to fit the data. The versatility and universality of this equation stem from two fundamental equations: the conductivity and the volume-press relationships. In this article, we provide both single-dome and double-dome data fittings for different superconductors. For double-dome experimental data, we just need to fit twice with a different set of parameters.

Please note that Equation 2 borrowed from the author’s previous work13 was developed from the first principle but later was approximated to remove many parameters for simplicity reasons. We had a generic conductivity equation that works for a wide temperature range,9 the superconductivity transition temperature Tc can only happen under the condition that dσ/dT=0, an inflection point where something dramatically happens from the mathematic standpoint. The right-side equation may not be equal to temperature with dimensional analysis due to this approximation, but the value should be plausible, which is why perfect fittings with experimental data are achieved. There is no coincidence in science! This equation not only works for pressure-induced but also doped holes/electrons-induced dome phenomena. Please refer to the cited article for detailed information.

The pressure and doped holes or electrons share the same physical mechanism: change the free volume of electrons. This is the reason that the dome-like behaviors can be induced by external pressure or doped holes or electrons in the solids. The double domes create a dip that is observed in both cases.

Conclusion

Utilizing the generic conductivity equation developed in our previous work and the volume-pressure equation available in the literature, we have theoretically derived an equation to correlate the superconductivity transition temperature with the externally applied pressure. Single and/or double-dome-shaped correlations are correctly predicted, and the experimental data can be fitted very well, no matter whether the experimental data show a perfect dome, a skewed one, or a simple nonlinear polynomial. The equation indicates that Tc can increase or decrease with the pressure polynomially or exponentially, depending on the number of conduction electrons in the compressed and the less compressed regions, the compression-related material parameter, and the maximum packing volume fraction of electrons. The free volume of electrons plays a key role in this relationship. The pressure-induced phenomena share the same origin as the doped holes and/or electrons, which is the reason that both lead to very similar single- and/or double-dome-like behaviors. The dip created by the double domes is related to the structural/packing transformation that is correlated with the maximum packing fraction of electrons. Our equation inherits the universal nature of the two fundamental equations, the generic conductivity equation and the volume/pressure equation, thus it can be applied to any superconductor. The current work provides a theoretical explanation of dome-like superconductivity phenomena induced by external pressures, deepening the understanding of how and why pressure can move Tc up in various ways.

Limitations of the study

The equations shown in this article are derived based on a universal conductivity equation. They are not tied to a specific type of superconductor and should work quite universally.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER	
Software and algorithms	
	
GraphPad Prism	GraphPad Ver. 9.0	https://www.graphpad.com/	

Resource availability

Lead contact

Tian Hao, email: haotian@gmail.com. Further information and requests for resources and methods should be directed to and will be fulfilled by the Lead Contact, Tian Hao (haotian@gmail.com).

Materials availability

There is no new data generated in this article.

Data and code availability

The data points used in both Figures 3 and 5 are extracted from the corresponding literature with the software called WebPlotDigitizer https://automeris.io/. The regressions are performed automatically with a software called GraphPad Prism https://www.graphpad.com/features.

Acknowledgments

The author sincerely appreciates colleagues’ and reviewers’ feedback and comments for substantially improving the readability and rationality of this article.

Author contributions

T.H. contemplated and formulated the theory, derived the equations, extracted experimental data from the literature, regressed with the derived equations, and wrote the article.

Declaration of interests

The author declares no competing interests.
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