Introduction

Notably, increasing the superconducting critical temperature \({T}_{c}\) remains the principal problem of condensed matter physics since the discovery of superconductivity1. Specifically, varying the phonon spectrum, tuning the carrier number, and increasing the pressure are three important experimental approaches to enhance \({T}_{c}\) and consequently, speculate possible theories of superconductivity. However, the effects of altering these parameters on \({T}_{c}\) are quite contrasting between weak coupling (low \({T}_{c}\)) and strong coupling (high \({T}_{c}\)) superconductivity. More precisely, first, in metallic superconductors2,3,4,5,6 and nickel-based superconductors7, which correspond to the case of weak coupling, the critical temperature \({T}_{c}\) can be increased via phonon softening. In McMillan's results3, the maximum \({T}_{c}\) appeared when the coupling strength is equal to 2. In contrast, in cuprate systems, which correspond to the case of strong coupling, \({T}_{c}\) can be increased via phonon stiffening8,9. Second, the effect of pressure on \({T}_{c}\) is negative in most metallic superconductors10,11. On the other hand, positive effects or a dome-like delineation appear in cuprate systems11,12, iron-based13,14, and hydrogen-rich superconductors15,16. Third, in the phase diagrams that illustrate the variation of \({T}_{c}\) with respect to the carrier number, the doom-like delineation may be observed in metallic17,18, cuprates19,20,21, and iron-based superconductors22. Specifically, the underdoped region is strongly coupled with a positive carrier number effect and becomes weakly coupled in the overdoped region with a negative carrier number effect23. Besides, another approach to increase the carrier number is to gate thin film materials, which also demonstrates the dome-like effect24,25,26. These three phenomena have a common dome-like delineation—the positive tendency appears in strong coupling superconductivity, and becomes negative in weak coupling superconductivity.

Furthermore, the \({T}_{c}\) relation that derives from Cooper instability is a general property of superconductivity. Specifically, this relation may be written in the form \({T}_{c}\sim \mathcal{W}\mathrm{exp}\left(-{\lambda }^{-1}\right)\)2,27,28, where \(\mathcal{W}\) is the bandwidth of superconducting electrons, and \(\lambda\) is the coupling constant of the pairing. Conventional superconductors, including metallic, MgB229, and hydrogen-rich compounds, are adequately explained by phonon-mediated pairing2,3,4,16. Meanwhile, unconventional superconductors, such as cuprates and iron-based superconductors are explained by Hubbard-type theories30,31,32,33,34. Moreover, although the strong repulsion between electrons plays a major role in the Hubbard model, the electron–phonon interaction remains nonnegligible and consequently, contributes to the unconventional superconductivity21,35,36. For this reason, in this study, we discuss the effects of varying the phonon spectrum, the carrier number, and the pressure on phonon-mediated pairing.

In the phonon-mediated pairing, the bandwidth \(\mathcal{W}\) is the characteristic phonon frequency \(\Omega\), and the coupling constant \(\lambda =g\left({E}_{F}\right){V}_{eff}\) is the product of the effective interaction between electrons \({V}_{eff}\) and the density of state at Fermi level \(g\left({E}_{F}\right)\). The effective interaction \({V}_{eff}\) based on the Migdal theory37 is \({V}_{eff}\sim 1/\left(M{\Omega }^{2}\right)\), and the density of state at Fermi level38 \(g\left({\varepsilon }_{F}\right)={m}^{*}\sqrt[3]{3{\pi }^{2}Z{n}_{ion}}/{\pi }^{2}{\mathrm{\hslash }}^{2}\), where \(M\) is the ion mass, \({m}^{*}\) is the effective mass of an electron, \(Z\) is the valence number, and \({n}_{ion}\) is the ion number density. Therefore, the explicit form of \(\lambda\) and Tc are

$$\lambda \left(\Omega \right)=\frac{C \sqrt[3]{Z{n}_{ion}}}{M{\Omega }^{2}}$$
(1)

and

$${T}_{c}\left(\Omega \right)\sim \mathrm{\Omega exp}\left[-\frac{M{\Omega }^{2}}{C\sqrt[3]{Z{n}_{ion}}}\right],$$
(2)

where \(C\) is a constant. Notice that Eq. (1) is similar to McMillan3. However, we preserve the term \(\sqrt[3]{Z{n}_{ion}}\) to describe the carrier number effect and the pressure effect. First, to investigate the phonon effect on superconductivity, the derivative of Eqs. (1) and (2) with respect to \(\Omega\) are

$$\frac{d\lambda }{d\Omega }=-2\frac{\lambda }{\Omega }$$
(3)

and

$$\frac{d{T}_{c}}{d\Omega }=\frac{{T}_{c}}{\Omega }\left(1-\frac{2}{\lambda }\right).$$
(4)

The derivative \(d{T}_{c}/d\Omega\) is equal to 0 when \(\lambda\) is equal to 2. Define the critical coupling constant corresponding to the phonon frequency \({\lambda }_{c}^{\Omega }\)=2.

Second, to investigate the dependency of \(\lambda\) and \({T}_{c}\) on carrier number \(Z\), Eqs. (1) and (2) can be rewritten as a function of the carrier number \(Z\). Here, the characteristic phonon frequency \(\Omega\) uses the jellium phonon frequency39 \(\Omega =\sqrt{{\mathrm{Z}}^{2}{e}^{2}{n}_{ion}/{\epsilon }_{0}M}\), where \({\epsilon }_{0}\) is the permittivity. The explicit form of \(\lambda\) and \({T}_{c}\) as a function of \(Z\) are

$$\lambda \left(Z\right)=\frac{C{\epsilon }_{0}}{{e}^{2}{n}_{ion}^{2/3}}{Z}^{-\frac{5}{3}}$$
(5)

and

$${T}_{c}\left(Z\right)\sim {\left(\frac{{\mathrm{Z}}^{2}{e}^{2}{n}_{ion}}{{\epsilon }_{0}M}\right)}^\frac{1}{2}\mathrm{exp}\left(-\frac{C{\epsilon }_{0}}{{e}^{2}{n}_{ion}^{2/3}}{Z}^{-\frac{5}{3}}\right).$$
(6)

The derivative of Eqs. (5) and (6) with respect to \(Z\) are

$$\frac{d\lambda }{dZ}=-\frac{5}{3}\frac{\lambda }{Z}$$
(7)

and

$$\frac{d{T}_{c}}{dZ}=\frac{{T}_{c}}{Z}\left[1-\frac{\left(\frac{5}{3}\right)}{\lambda }\right].$$
(8)

Similarly, the derivative \(d{T}_{c}/dZ\) is equal to 0 when \(\lambda\) is equal to 5/3, denoted as \({\lambda }_{c}^{Z}\).

Third, to investigate the dependency of \({T}_{c}\) on pressure \(P\), the compressibility \(\beta =-\left(1/V\right)*\left(dV/dP\right)\) can be adopted to relate the pressure \(P\) and the volume \(V\). Using \({n}_{ion}={N}_{ion}/V\), where \({N}_{ion}\) is the number of ions; Eqs. (5) and (6) can thus be rewritten into a function of \(V\):

$$\lambda \left(V\right)=\frac{C{\epsilon }_{0}}{{e}^{2}{Z}^\frac{5}{3}{N}_{ion}^\frac{2}{3}}{V}^\frac{2}{3}$$
(9)

and

$${T}_{c}\left(V\right)\sim {\left(\frac{{Z}^{2}{e}^{2}{N}_{ion}}{{\epsilon }_{0}MV}\right)}^\frac{1}{2}\mathrm{exp}\left(-\frac{{e}^{2}{Z}^\frac{5}{3}{N}_{ion}^\frac{2}{3}}{C{\epsilon }_{0}}{V}^{-\frac{2}{3}}\right).$$
(10)

The derivative of Eqs. (9) and (10) with respect to the pressure \(P\) are:

$$\frac{d\lambda }{dP}=\frac{dV}{dP}*\frac{d\lambda }{dV}=-\beta V*\frac{2}{3}\frac{\lambda }{V}$$
(11)

and

$$\frac{d{T}_{c}}{dP}=\frac{dV}{dP}*\frac{d{T}_{c}}{dV}=\beta V*\frac{1}{2}\frac{{T}_{c}}{V}\left[1-\frac{\left(\frac{4}{3}\right)}{\lambda }\right].$$
(12)

The critical coupling constant corresponding to the pressure is \({\lambda }_{c}^{P}\)=4/3. The schematic diagram based on the result from Eqs. (1) to (12), which demonstrates the effects of tuning \(\Omega\), \(Z\), or \(P\) on \({T}_{c}\) is plotted in Fig. 1.

Figure 1
figure 1

A schematic diagram of the critical temperature \({T}_{c}\) and the critical coupling constants \({\lambda }_{c}^{\mathcal{X}}\) corresponding to the phonon frequency \(\Omega\), the carrier number \(Z\), and the pressure \(P\). There is a sign change of derivative \(d\lambda /d\mathcal{X}\) at \({\lambda }_{c}^{\mathcal{X}}\). If \(\lambda\) is larger than \({\lambda }_{c}^{\mathcal{X}}\), the effect of varying \(\mathcal{X}\) on superconductivity is positive with \({T}_{c}\) increasing corresponding to strong coupling superconductivity. Meanwhile, if \(\lambda\) is smaller than \({\lambda }_{c}^{\mathcal{X}}\), then the effect of varying \(\mathcal{X}\) on superconductivity is negative with \({T}_{c}\) decreasing corresponding to the weak coupling superconductivity.

Full size image

Results and discussion

The origin of the dome

The results of our derivations reveal two common features: (1) \(d{T}_{c}/d\mathcal{X}\propto 1-\left({\lambda }_{c}^{\mathcal{X}}/\lambda \right)\), and (2) \(d\lambda /d\mathcal{X}<0\), where \(\mathcal{X}\) represents \(\Omega\), \(Z\), or \(P\). Specifically, feature (1) indicates that there is a sign change of derivative \(d\lambda /d\mathcal{X}\) at \({\lambda }_{c}^{\mathcal{X}}\). If \(\lambda\) is larger than \({\lambda }_{c}^{\mathcal{X}}\), the effect of varying \(\mathcal{X}\) on superconductivity is positive with \({T}_{c}\) increasing, which corresponds to strong coupling superconductivity. On the other hand, if \(\lambda\) is smaller than \({\lambda }_{c}^{\mathcal{X}}\), then the effect of varying \(\mathcal{X}\) on superconductivity is negative with \({T}_{c}\) decreasing, which corresponds to weak coupling superconductivity. These observations explain the contrasting behavior between weak and strong coupling superconductors. Feature (2) is the reason for the dome-like delineation observed in many strong coupling superconductors. Specifically, for strong coupling superconductors, the coupling constant is expected to be larger than \({\lambda }_{c}^{\mathcal{X}}\), to ensure the effect of varying \(\mathcal{X}\) on superconductivity is positive. In addition, since the derivative of \(\lambda\) is negative, \(\lambda\) decreases when the parameter \(\mathcal{X}\) increases. In particular, once \(\lambda\) becomes smaller than \({\lambda }_{c}^{\mathcal{X}}\), the effect of varying \(\mathcal{X}\) on superconductivity becomes negative. This transition from positive to negative demonstrates the dome-like delineation and can be observed when tuning the carrier number or varying the pressure in the experiments.

Critical temperature \({{\varvec{T}}}_{{\varvec{c}}}\) as a function of phonon frequency \({\varvec{\Omega}}\)

Critical temperature \({T}_{c}\) is influenced when the phonon frequency \(\Omega\) is changing. Specifically, the maximum \({T}_{c}\) appears at \({\lambda }_{c}^{\Omega }\) = 2 by varying the phonon frequency Ω, which is consistent with McMillan3. More precisely, in the \(\lambda >{\lambda }_{c}^{\Omega }\) region, the sign of \(d{T}_{c}/d\Omega\) is positive, and \({T}_{c}\) increases when increasing \(\Omega\). This region may correspond to cuprate superconductors. Particularly, LaBaCuO, YBaCuO, BiSrCaCuO, HgBaCaCuO, and TlBaCaCuO demonstrate phonon stiffening effect8,9. In Table 1, the \(\lambda\) value of phonon stiffening effect superconductors are listed. For example, the value of the coupling constant \(\lambda\) at optimal \({T}_{c}\) of Bi2Sr2Ca0Cu1Ox, Bi2Sr2Ca1Cu2Ox, and Bi2Sr2Ca2Cu3Ox are 2.95, 2.15, and 2.18, respectively21. These \(\lambda\) values are larger than \({\lambda }_{c}^{\Omega }\) indicating that \({T}_{c}\) can be enhanced by the phonon stiffening effect.

Table 1 Examples of phonon effect \(\Omega\) on superconductivity and \(\lambda\) value.
Full size table

Meanwhile, in the \(\lambda <{\lambda }_{c}^{\Omega }\) region, the coupling constant \(\lambda\) is smaller than \({\lambda }_{c}^{\Omega }\). Specifically, the sign of \(d{T}_{c}/d\Omega\) is negative and \({T}_{c}\) decreases when \(\Omega\) increases. More precisely, this region may correspond to traditional metallic superconductors. Particularly, VCr, ZrRh, NbMo, MoRe, WRe, PbTl, PbBi, Nb3Al, and Nb3Ge demonstrate the phonon softening effect3,4,5,6. In Table 1, the \(\lambda\) value of phonon softening effect superconductors are listed. For example, when the Debye temperature \({\Theta }_{\mathrm{D}}\) of the VCr alloy increases from 370 to 470 K, \({T}_{c}\) decreases from 3.21 to 0.10 K, and \(\lambda\) decreases from 0.53 to 0.33. Taking another example, when the Debye temperature \({\Theta }_{\mathrm{D}}\) of the ZrRh alloy increases from 192 to 244 K, \({T}_{c}\) decreases from 5.95 to 3.10 K, and \(\lambda\) decreases from 0.80 to 0.59. The experiment data show that the coupling constant \(\lambda\) decreases when the phonon frequency increases, which is consistent with our result \(d\lambda /d\Omega <0\) from Eq. (3). A similar effect also appears in nickel-based superconductors7, which validates that the superconductivity is enhanced by giant phonon softening.

Critical temperature \({{\varvec{T}}}_{{\varvec{c}}}\) as a function of carrier number \({\varvec{Z}}\)

The effect of carrier number on superconductivity through varying the alloy composition, doping concentration, or gating voltage has been widely studied in material physics. Particularly, the dome-like delineation was observed in metallic materials17, cuprates19,20,21, iron-based systems22, and gating thin film materials24. More precisely, the maximum \({T}_{c}\) appears at \({\lambda }_{c}^{\mathrm{Z}}\) = 5/3 by varying the carrier number \(Z\).

For conventional superconductors, we take Nb3Al1−xGex as an example43, which is showing in Fig. 2a. Specifically, when the Ge component rises from x = 0 to x = 0.29, the electrons per atom ratio (e/a) rises from 4.50 to 4.57 and \({T}_{c}\) increases from 18 to 21 K. The coupling constant of Nb3Al1 is \(\lambda\) > 1.841, which is greater than \({\lambda }_{c}^{Z}\) and is consistent with the positive carrier effect. Moreover, when the Ge component rises from x = 0.29 to x = 1, the e/a ratio rises from 4.57 to 4.75 and \({T}_{c}\) decreases from 21 to 7 K. The coupling constant of Nb3Ge1 is \(\lambda\) < 16 which is smaller than \({\lambda }_{c}^{Z}\) and is consistent with the negative carrier effect.

Figure 2
figure 2

Examples of carrier number effect \(Z\) on superconductivity. (a) The conventional superconductors Nb3Al1−xGex demonstrates carrier effect5,6,41,43. Specifically, the electrons per atom ratio (e/a) rises from 4.50 to 4.75 and \({T}_{c}\) shows dome-like effect. (b) The hole-doped h (holes/Cu) experiment of the unconventional superconductors Bi2Sr2Ca1Cu2Ox demonstrates dome-like effect21. The critical coupling constant corresponding to \(Z\) is \({\lambda }_{c}^{Z}\) = 5/3. Both cases show that \({T}_{c}\) is increasing when \(\lambda >{\lambda }_{c}^{Z}\), and \({T}_{c}\) is decreasing when \(\lambda <{\lambda }_{c}^{Z}\).

Full size image

For unconventional superconductors, we take the hole-doped experiment of Bi2Sr2Ca1Cu2Ox as an example21, which is showing in Fig. 2b. Specifically, when the hole doping h (holes/Cu) rises from h = 0.11 to h = 0.21, the coupling constant monotonically decreases from \(\lambda\) = 2.66 to \(\lambda\) = 0.97. Additionally, the critical temperature first rises from 66 K (\(\lambda\) = 2.66 > \({\lambda }_{c}^{Z}\) at h = 0.11) to 88 K (\(\lambda\) = 2.15 > \({\lambda }_{c}^{Z}\) at h = 0.16), then drops to 77 K (\(\lambda\) = 1.5 < \({\lambda }_{c}^{Z}\) at h = 0.20) and further to 67 K (\(\lambda\) = 0.97 < \({\lambda }_{c}^{Z}\) at h = 0.21). This dome-like delineation showed that the superconductivity is increasing in the underdoped region with coupling \(\lambda\) stronger than \({\lambda }_{c}^{Z}\), and decreasing in the overdoped region with coupling \(\lambda\) weaker than \({\lambda }_{c}^{Z}\).

Critical temperature \({{\varvec{T}}}_{{\varvec{c}}}\) as a function of pressure effect \({\varvec{P}}\)

The effect of pressure on superconductivity is an important topic in the field of condensed matter physics. In particular, for strong coupling superconductors, the pressure effect is positive and may lead to a dome-like delineation at higher pressure. These characteristics were observed in cuprate superconductors12,44, iron-based superconductors13,14,45, and hydrogen-rich superconductors15,16. In Table 2, the \(\lambda\) value of positive or dome-like pressure effect superconductors are listed. Specifically, the coupling constants of cuprates are greater than \({\lambda }_{c}^{P}\) as mentioned in the previous section. Here, we note the couple constant of iron-based superconductors; for instance, LaFeAsO1−xFx and Ba(Fe1−xCox)2As2 are \(\lambda\) = 2.38 and 2.83, respectively46. These strong coupling superconductors have coupling strength \(\lambda\) larger than \({\lambda }_{c}^{P}\), thus the positive or dome-like delineation of the pressure effect can be observed.

Table 2 Examples of pressure effect \(P\) on superconductivity and \(\lambda\) value.
Full size table

Meanwhile, most metallic superconductors are weak-coupling superconductors. Particularly, Al, Cd, Sn, In, and Pb are examples with negative pressure effect10,11. Specifically, the coupling constant \(\lambda\) of Al, Cd, Sn, In, and Pb are less than \({\lambda }_{c}^{P}\) (see Table 2). Clearly, these superconductors have \(\lambda\) less than \({\lambda }_{c}^{P}\). Additionally, the negative pressure effect is also observed in the covalent compound MgB211, with \(\lambda\) equal to 0.749, which is smaller than \({\lambda }_{c}^{P}\).

The strategies of \({{\varvec{T}}}_{{\varvec{c}}}\) enhancement

According to these three critical coupling constants: \({\lambda }_{c}^{\Omega }\), \({\lambda }_{c}^{Z}\), and \({\lambda }_{c}^{P}\), superconductors can be classified by their coupling strength: weak coupling (\(\lambda <{\lambda }_{c}^{P}\)), intermediate coupling (\({\lambda }_{c}^{P}<\lambda <{\lambda }_{c}^{\Omega }\)), and strong coupling (\(\lambda >{\lambda }_{c}^{\Omega }\)). A schematic diagram can be seen in Fig. 3.

Figure 3
figure 3

Classification and enhancement strategy for superconductivity. Using three critical coupling constants: \({\lambda }_{c}^{\Omega }\), \({\lambda }_{c}^{Z}\), and \({\lambda }_{c}^{P}\), superconductors can be classified by their coupling strength: weak coupling (\(\lambda <{\lambda }_{c}^{P}\)), intermediate coupling (\({\lambda }_{c}^{P}<\lambda <{\lambda }_{c}^{\Omega }\)), and strong coupling (\(\lambda >{\lambda }_{c}^{\Omega }\)).

Full size image

In the weak coupling region (\(\lambda <{\lambda }_{c}^{P}\)), the coupling constant \(\lambda\) is less than \({\lambda }_{c}^{P}\). Particularly, all derivatives of critical temperature \(d{T}_{c}/d\Omega\), \(d{T}_{c}/dZ\), and \(d{T}_{c}/dP\) are negative, as shown in Fig. 3. Hence, by decreasing \(\Omega\), \(Z\) or \(P\), the superconductivity can be enhanced. These procedures increase the coupling constant \(\lambda\), because the derivatives of the coupling constant \(d\lambda /d\Omega\), \(d\lambda /dZ\), and \(d\lambda /dP\) are negative. We can accurately infer that the superconductivity of weak coupling superconductors is enhanced by increasing the coupling strength. In addition, once the coupling constant \(\lambda\) becomes greater than \({\lambda }_{c}^{P}\), the superconductivity enters into the intermediate region, and the tendencies of \({T}_{c}\) become complicated.

Meanwhile, in the strong coupling region (\(\lambda >{\lambda }_{c}^{\Omega }\)), the coupling constant \(\lambda\) is larger than \({\lambda }_{c}^{\Omega }.\) Particularly, all derivatives of the critical temperature \(d{T}_{c}/d\Omega\), \(d{T}_{c}/dZ\), and \(d{T}_{c}/dP\) are positive in Fig. 3, thus the superconductivity can be increased by increasing \(\Omega\), \(Z\) or \(P\). These procedures decrease the coupling constant \(\lambda\), such that we can adequately infer that the superconductivity of strong coupling superconductors is enhanced by decreasing the coupling strength. Once the coupling constant \(\lambda\) is smaller than \({\lambda }_{c}^{\Omega }\), the superconductivity enters into the intermediate coupling region and the tendencies of \({T}_{c}\) become complicated. Besides, comparing the strong coupling region with the weak coupling region, three tendencies of \({T}_{c}\) are contrasting between the two regions.

Moreover, in the intermediate coupling region (\({\lambda }_{c}^{P}<\lambda <{\lambda }_{c}^{\Omega }\)), the enhancement methods are more interesting. In particular, take that \(\lambda\) belongs in the interval \({\lambda }_{c}^{P}<\lambda <{\lambda }_{c}^{Z}\) for the following discussion without loss of generality. In this interval, superconductivity can be enhanced by increasing \(P\) or decreasing \(Z\). First, let the superconductivity be optimized by tuning \(Z\), such that \(\lambda\) is equal to \({\lambda }_{c}^{Z}\), denoted as \({\lambda }_{1}\). Second, since \({\lambda }_{1}\) is larger than \({\lambda }_{c}^{P}\), we can increase \(P\) to increase \({T}_{c}.\) The second step causes \(\lambda\) to decrease and \({T}_{c}\) is optimized when \(\lambda ={\lambda }_{c}^{P}\), denoted as \({\lambda }_{2}\). Third, now \({\lambda }_{2}\) is smaller than \({\lambda }_{c}^{Z}\), \({T}_{c}\) can be further increased by decreasing \(Z\). The third step increases \(\lambda\) and \({T}_{c}\) is optimized when \(\lambda ={\lambda }_{c}^{Z}\), denoted as \({\lambda }_{3}\). Repeat step 2 and step 3 by increasing \(P\) and decreasing \(Z\) alternately; \({T}_{c}\) can be enhanced like a zigzag mountain climbing. In this study, we propose simultaneously gating and pressurizing on thin-film superconductors to verify our discussion. Furthermore, FeSe has been observed under gating25 and pressurizing50 independently. More precisely, the negative carrier number effect and the dome-like pressure effect suggest the coupling strength \({\lambda }_{c}^{P}<{\lambda }_{FeSe}<{\lambda }_{c}^{Z}\), which agree with \({\lambda }_{FeSe}=1.6\)47. Additionally, we propose that increasing \(P\) and decreasing \(\Omega\) alternately, or increasing \(Z\) and decreasing \(\Omega\) alternately, are two other methods to enhance superconductivity in the intermediate region.

Conclusion

In this study, we proposed a phenomenological model based on phonon-mediated interaction, which explains the difference between weak and strong coupling superconductors affected by tuning phonon frequency \(\Omega\), carrier number \(Z\), and pressure \(P\). We introduced the concept of critical coupling constants and enhancement strategies for superconductivity, extending McMillan's results on coupling strength equal to 2. Specifically, the sign of the first-order derivative \(d{T}_{c}/d\mathcal{X}\) with respect to \(\mathcal{X}=\Omega\), \(Z\), or \(P\), indicates that \({T}_{c}\) is increasing or decreasing when either these three parameters change. More precisely, these three derivatives have two features in common: (1) the coupling constant \(\lambda\) beyond (or below) the critical coupling constant \({\lambda }_{c}^{\mathcal{X}}\) determined \(d{T}_{c}/d\mathcal{X}\) to be positive (or negative), and (2) the dome-like delineation observed in strong coupling superconductors because \(d\lambda /d\mathcal{X}\) is always negative. Overall, these observations explain the differences between weak and strong superconductors.

Furthermore, using three critical coupling constants \({\lambda }_{c}^{\Omega }\), \({\lambda }_{c}^{Z}\), and \({\lambda }_{c}^{P}\), superconductors can be classified by their coupling strength and consequently, correspond to different enhancement strategies. Specifically, for the weak coupling region (\(\lambda <{\lambda }_{c}^{P}\)), \({T}_{c}\) can be increased by decreasing \(\Omega\), \(Z\), and \(P\), causing \(\lambda\) to be increased, resulting in intermediate coupling. In contrast, superconductors in the strong coupling region (\(\lambda >{\lambda }_{c}^{\Omega }\)) can be enhanced by increasing \(\Omega\), \(Z\), and \(P\), causing \(\lambda\) to be decreased, resulting in intermediate coupling. Moreover, for superconductors in the intermediate coupling region (\({\lambda }_{c}^{P}<\lambda <{\lambda }_{c}^{\Omega }\)), the zigzag strategies may further enhance superconductivity.