Prospects to attain room temperature superconductivity

TL;DR

This paper derives simple formulas to estimate superconducting transition temperatures, suggesting room temperature superconductivity requires very high Debye temperatures and identifying potential materials like a Be-Pb alloy that could superconduct at 44 K.

Contribution

It introduces new analytical expressions for estimating Tc and gap ratios, highlighting the limits of phonon-mediated superconductivity and potential candidate materials.

Findings

Maximum phonon exchange factor is 2.67

Room temperature superconductivity requires Debye temperature ~1800 K

Be-Pb alloy could superconduct at around 44 K

Abstract

With a generic model for the electron-phonon spectral density, two simple expressions are derived to estimate the transition temperature and gap-to-temperature ratio in conventional superconductors. They entail that on average the numerical value of the phonon exchange factor, $\lambda$, is limited to 2.67, so that room temperature superconductivity may be attained only with a Debye temperature of about 1800 K or higher, in materials that may or may not involve hydrogen. They also show that a Be-Pb alloy may become a superconductor at $\sim$44 K.

Equations

$${\alpha }^{2}F(\nu )=\lambda {\nu }^2/(k_B{T}_D{)}^2,\text{mbox}if\nu \le k_{B}T,$$

$$\displaystyle T_{c}=0.23\,T_{D}\Big{(}0.25\,\lambda-0.15\Big{)}^{0.66}$$

$$\displaystyle\frac{2\Delta_{0}}{k_{B}T_{c}}=4.04\,\Big{(}0.25\,\lambda-0.15\Big{)}+3.66$$

$$T_c=\frac{T_D}{1.45}\exp \left[-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.62\lambda )}\right]$$

$${\alpha }^{2}F(\nu )=\frac{1}{N}\sum _{\ell ,\mathbf{q}}\delta (\nu -\hslash {\omega }_{\ell })\delta (ϵ-{ϵ}_F){\left|g_{\ell }(\mathbf{q})\right|}^2$$

$${\alpha }_{\text{mbox}tr}^{2}F(\nu )=\frac{1}{N}\sum _{\ell ,\mathbf{q}}\delta (\nu -\hslash {\omega }_{\ell })\delta (ϵ-{ϵ}_F){\left|g_{\ell }(\mathbf{q})\right|}^2$$

$$\times \left[\frac{\mathbf{k}\cdot (\mathbf{k}+\mathbf{q})}{\mathbf{k}\cdot \mathbf{k}}-1\right]$$

$$\alpha_{\mbox{\scriptsize tr}}^{2}F(\nu)=\left\{\begin{array}[]{cc}\lambda_{\mbox{\scriptsize tr}}(\nu/k_{B}\Theta)^{4},&\nu\leq k_{B}\Theta\\ \\ 0,&\mbox{otherwise}\end{array}\right.$$

$$\frac{\rho (T)}{\rho (\Theta )}=4.2263{\left(\frac{T}{\Theta }\right)}^5{\int }_0^{\Theta /T}\frac{x^{5}dx}{(e^x-1)(1-e^{-x})}$$

$$\frac{\mathbf{k}\cdot (\mathbf{k}+\mathbf{q})}{\mathbf{k}\cdot \mathbf{k}}-1\propto {\left(\frac{\nu }{k_B\Theta }\right)}^2$$

Full text

I introduction

In 1968 Ashcroft suggested that metallic hydrogen can be a high-temperature superconductor Ashcroft . In 2004 Ashcroft suggested that hydrogen dominant metallic alloys in diamond anvil cells can become high-temperature superconductors at pressures considerably lower than may be necessary for metallic hydrogen Ashcroft2 . In 2015 Eremets and co-workers indeed observed that sulfur hydride in a diamond anvil cell becomes a superconductor at 203 kelvin at about 150 GPa Drozdov ; Capitani . Here we attempt to clarify whether the prospect of room temperature superconductivity is achievable in principle. We also demonstrate that, via virtual crystal approximation, a Be-Pb alloy may become a superconductor at $\sim$44 K.

II theory

We wish to find a general relation between the superconducting transition temperature, $T_{c}$, and other properties of the material, including the phonon exchange factor, $\lambda$, and Debye temperature, $T_{D}$, from numerical solutions to the Eliashberg equations. We apply the following generic model to evaluate the equations:

[TABLE]

otherwise $\alpha^{2}F(\nu)=0$, where $k_{B}$ is the Boltzmann constant, and $\nu$ the phonon frequency in joule or eV, see the Appendix for a justification. It is also consistent with the definition of $\lambda$ McMillan because, when we integrate $2\alpha^{2}F(\nu)/\nu$ over $\nu$, we simply recover the value of $\lambda$. In Eq. (1) the values of $\lambda$ and $T_{D}$ embody all phonon properties, leaving no ambiguities to our discussion within its premise.

In Eq. (1) $T_{D}$, if not available, can be replaced by the characteristic temperature, $\Theta$, in the Bloch-Grüneisen formula for electrical resistivity, which is usually close to $T_{D}$, where $\Theta=1223$ K in H3S, found from Eq. (A4) and FIG. S3 in Capitani via numerical fitting. We also let the Coulomb pseudopotential $\mu^{*}$ = 0.09, 0.13 and 0.17, a usual range encountered experimentally Allen ; Mitrovic . We will let $T=T_{c}$ once $\Delta(T)=0.1\Delta_{0}$ instead of $\Delta(T)=0$ McMillan . This amounts to a new and just as accurate definition of $T_{c}$, because $\Delta(T)$ vanishes quickly as soon as it reaches $0.1\Delta_{0}$ with increasing $T$, a fact well known from the exemplary $\Delta(T)$ curve in BCS .

III Results

We are now adequately equipped to solve the Eliashberg equations. Usually the procedure starts with a destination value of $\Delta_{0}$, giving $T_{c}$ after the procedure terminates McMillan+Rowell ; we instead start with a destination value of $T_{c}$. With given values of $T_{D}$ and $\mu^{*}$, we find $\lambda$ via an optimization program of two dimensional search. In one dimension we search $T$ to let the Eliashberg equations produce $\Delta(T)=0.1\Delta_{0}$. In the other dimension we search $\lambda$ until $|T-T_{c}|\sim 10^{-7}$ K. Meanwhile $\Delta_{0}$ keeps evolving until our procedure terminates.

Our results, shown as dots in FIG. 1, align themselves into narrow bands, largely regardless of the values of $\mu^{*}$. In particular, when $T_{D}=86$ and 400 K, the dots appear to move along the same curve, in response to the varying values of $\mu^{*}$ — we therefore have an expression for $T_{c}$ in terms of only $\lambda$ and $T_{D}$, without involving $\mu^{*}$ but still retaining reasonable accuracy. We find the following power law relationship from numerical fitting

[TABLE]

which is applicable when $\lambda\geq 0.6$. We also find the following linear relationship

[TABLE]

which is likewise applicable when $\lambda\geq 0.6$. Outcomes of Eqs. (2) and (3) are shown as curves and lines in FIGs. 1 and 2 respectively.

We always encounter an upper limit of $\lambda$ in our numerical procedure. For example, at $T_{D}=400$ K, our numerical procedure fails to converge once $\lambda>2.57$, or $T_{c}>77$ K, which we will refer to as the observed maximum value of the transition temperature (possible reasons discussed in Section VI). In FIG. 3 we plot the empirical maximum values of $T_{c}$ (filled squares) found at $86\leq T_{D}\leq 2500$ K. We also plot a line for $T_{c}$ from Eq. (2), with $\lambda=2.67$ to ensure a minimum r.m.s. difference between the output of the formula and values of the observed maximum $T_{c}$.

In TABLE 1 the values of $T_{D}$ in dense Li and H3S are replaced by the Bloch-Grüneisen characteristic temperatures Ziman we extracted from the electrical resistance data in Drozdov ; Shimizu . MgB2 is a two band superconductor, with two values of $\Delta_{0}$, and we evaluate $2\Delta_{0}/k_{B}T_{c}$ with the larger $\Delta_{0}$ Masui . The experimental $\lambda$ in MgB2 is an average between its values in the $\sigma$ and $\pi$ bands Dolgov . The theoretical $\lambda$ is calculated from Eq. (2) with experimental $T_{D}$ and $T_{c}$. The theoretical gap-to-$T_{c}$ ratios are from Eq. (3) with theoretical $\lambda$.

To the best of our knowledge experimental values of $\lambda$ and $2\Delta_{0}/k_{B}T_{c}$ in dense Li and H3S are not yet available. According to Errea and co-workers $\lambda=2.64$ or 1.84 for harmonic or anharmonic phonons respectively in H3S in theory, with $\Delta_{0}\approx 43$ meV and $T_{c}=190$ K ($\mu^{*}=0.16$, anharmonic phonons) giving $2\Delta_{0}/k_{B}T_{c}\approx 5.25$ Errea , compared with $\lambda$ between 2.07 and 2.19 in (H2S)2H2 in theory Duan , largely consistent with our results considering the many uncertainties in theoretical evaluations.

IV McMillan formula

In 1968 McMillan McMillan solved the Eliashberg equations via iteration with a number of simplifications. The equations are linearised at $T=T_{c}$, the solutions are assumed to have just two values, $\Delta_{0}$ and $\Delta_{\infty}$, defined immediately beneath the surface and deeply inside of the Fermi sphere, respectively, and $\alpha^{2}F(\nu)=\alpha^{2}\times F(\nu)$, $\alpha$ being a constant and $F(\nu)$ the phonon density of states (from Nb neutron scattering experiment for any bcc lattice, assumed to vanish below 8.6 meV). During the iteration $T_{c}$ and $\mu^{*}$ are kept constant, $\alpha$ adjusted continuously to keep $\Delta_{0}$ constant. The formula

[TABLE]

results from numerical fitting, and has proven to be highly successful in guiding both theoreticians and experimentalists. For clarity we choose $\mu^{*}=0.13$ and plot the outcome of Eq. (4) in FIG. 4 as dots, which matches the outcome of Eq. (2) closely if $\lambda<2$, but otherwise underestimates $T_{c}$, rather significantly in the parameter region of H3S.

In 1975 Allen and Dynes Allen published a slightly modified $T_{c}$ formula where $T_{D}/1.45$ in Eq. (4) is replaced by $\omega_{\mbox{\scriptsize ln}}/1.2$, $\omega_{\mbox{\scriptsize ln}}=(2/3)T_{D}$ with our generic model in Eq. (1) in place, that is $T_{c}$ from Eq. (4) will be suppressed by $\sim$20%. In 1984 Mitrović, Zarate and Carbotte Mitrovic found an approximate formula for $2\Delta_{0}/k_{B}T_{c}$ in terms of $T_{c}/\omega_{\mbox{\scriptsize ln}}$, not directly comparable with Eq. (3).

V beryllium-lead alloy

Eqs. (2) and (3) enable us to predict superconductivity quantitatively, not only in simple metals but also in alloys. For example, in Pb we have $T_{D}\simeq\Theta_{1}=86$ K (Bloch-Grüneisen characteristic temperature) and $\lambda=\lambda_{1}=1.55$, giving $T_{c}=7.66$ K via Eq. (2), compared with an empirical value of 7.19 K Mitrovic . In Tl we have $T_{D}=\Theta_{2}=78.5$ K Kittel and $\lambda=\lambda_{2}=0.795$, giving $T_{c}=2.46$ K, compared with the empirical 2.36 K Mitrovic . Letting $T_{D}=x\Theta_{1}+(1-x)\Theta_{2}$ and $\lambda=x\lambda_{1}+(1-x)\lambda_{2}$, where $x$ is the content of Pb in a Tl-Pb alloy in accordance with the well-established virtual crystal approximation Nordheim , we find $T_{c}$ in the alloys from Eq. (2) with r.m.s. deviation = 4.2% against experimental data, as is shown in FIG. 5 and TABLE 2.

Replacing Tl with Be allows us to now make a prediction for $T_{c}$ in Be-Pb alloys, by applying the same virtual crystal approximation. With $\Theta_{3}=1440$ K (Debye temperature) and $\lambda_{3}=0.6$ to let $T_{c}\sim 0$ (= 0.026 K in reality) for Be Kittel , we let $T_{D}=x\Theta_{1}+(1-x)\Theta_{3}$ and $\lambda=x\lambda_{1}+(1-x)\lambda_{3}$. This leads through Eq. (2) to give the upper curve in FIG. 5, where the value of $T_{c}$ reaches a peak of 43.8 K.

VI discussion

Eqs. (2) and (3) entail a few interesting physical consequences. First, there must be a upper limit of $\lambda$. If $\lambda\gg 1$ then we have $\Delta_{0}\sim 0.51\lambda k_{B}T_{c}$ from Eq. (3), and this leads through Eq. (2) to $\Delta_{0}\sim 0.47\lambda^{1.66}k_{B}T_{D}$. Consequently we have $\Delta_{0}/\lambda k_{B}T_{D}\sim 0.47\lambda^{0.66}\rightarrow\infty$ when $\lambda\rightarrow\infty$. This is impossible, because $\Delta_{0}$ is the edge of superconducting energy gap function, whereas $\lambda k_{B}T_{D}$ measures the strength of the electron-phonon interactions, the underlying reasons for superconductivity to arise, and the two must be compatible. This explains why in FIGs. 1, 2 and 3 the numerical values of $\lambda$ are limited to 2.67 on average.

It is remarkable that McMillan also found a value of $\lambda$ in association with maximum $T_{c}$ McMillan . He argued that in Eq. (4) $T_{D}\sim\langle\omega\rangle$ and $\lambda\sim 1/\langle\omega^{2}\rangle$, where $\omega$ stands for phonon frequency, so that $T_{c}\rightarrow 0$ when $\langle\omega\rangle$ is either too large or too small. Searching for maximum $T_{c}$ leads to $\lambda=2$, not far from our value 2.67. He estimated $T_{c}$ may reach 9.2 K for lead-based alloys and 40 K for V3Si McMillan . It appears that $T_{c}$ in the BCS theory is indeed curbed by an intrinsic limit, which cannot be lifted by strong electron-phonon interactions, reminiscent of the speed limit imposed by the Lorentz transformation.

Second, it is immediately apparent from Eq. (2) that, with $\lambda\sim 2.67$, the only realistic try for us to achieve high $T_{c}$ is in materials with high $T_{D}$. Indeed we know from TABLE I that in MgB2 we have $T_{c}=39$ K with $\lambda=0.97$ and $T_{D}=815$ K, compared with $T_{c}=7.19$ K, $\lambda=1.24$ but $T_{D}=105$ K in Pb. If somehow we were able to let $\lambda=2.67$ in both cases, then in MgB2 we would have $T_{c}=121$ K from Eq, (2), compared with $T_{c}=16$ K in Pb. Indeed, it is clear from FIG. 1 that, ascending along the $T_{c}$ curves with $T_{D}=86$ or 400 K, there is hardly any hope for us to reach $T_{c}\sim 300$ K.

Third, we might be able to push $T_{c}$ into the range of room temperatures, but just barely. We know from TABLE I that we may have $\lambda=3.04$, and this leads through Eq. (2) to $T_{c}=300$ K with $T_{D}\sim 1800$ K. From direct numerical calculation we find $T_{c}=255$ K with $\lambda=2.39$ and $T_{D}=1800$ K. Whether or not we can achieve higher $T_{c}$ with some $\alpha^{2}F(\nu)$ other than the generic model in Eq. (1) remains an open question, but we suspect the potential is rather limited, unless $T_{D}\sim 2500$ K or higher were an option at our disposal.

Finally, the close fit between the dots and the lower curve in FIG. 5 tells us that $T_{c}$, from Eq. (2), in virtual crystal approximation, is trustworthy in the case of the Tl-Pb alloys. If the higher curve in FIG. 5 is just as accurate, we may achieve $T_{c}\sim 44$ K in Be-Pb alloys, due to Fermi surface enlargement, on account of the large valence in Pb (= 4). The Be-Bi alloy may also be worth considering because Bi, though not a superconductor in bulk form, was proven to be highly effective to raise $T_{c}$ in Pb-Bi alloys Mitrovic apparently likely due to its very large valance (= 5).

VII conclusions

We conclude with suggestions for future work. We wish for the experimental values of $\lambda$ and $2\Delta_{0}/k_{B}T_{c}$ in dense lithium and H3S to be made available, to validate or dispute our theoretical predictions in TABLE I. In the periodic table $T_{D}$ can reach 1440 and 2230 K in beryllium and carbon respectively Kittel formally nearly or more than enough for $T_{c}\sim 300$ K with $\lambda\sim 3$. Indeed $T_{c}=18$ K in potassium-doped C60 Hebard . It would be interesting to see what will happen in beryllium, doped with lead or other metals to become an alloy with an enlarged Fermi surface. Experimentally this may be achieved via the vacuum-sputtering technique inam to make a uniform film of an alloy of two or more metals or, if necessary, a pancake film of multiple layers of different atoms.

APPENDIX

Here we justify Eq. (1) in the main text. By definition

[TABLE]

for a spherical Fermi surface, where $\ell$ identifies phonon polarization, $N$ is the number of atoms per unit volume, $\bf q$ and $\omega_{\ell}=\omega_{\ell}({\bf q})$ phonon momentum and frequency, $\bf k$ and $\epsilon=\epsilon({\bf k})$ electron momentum and energy, $\epsilon_{F}$ Fermi energy, and $g_{\ell}({\bf q})$ matrix element Truant . In the normal state

[TABLE]

and its model

[TABLE]

lead through relevant formulations Tomlinson to the Bloch-Grüneisen formula

[TABLE]

where $\rho(T)$ and $\rho(\Theta)$ are electrical resistivity measured at $T$ and $\Theta$, respectively, $\Theta$ being the characteristic temperature, highly accurate in most metals over a broad range of temperatures Ziman . If the sound velocity is constant anywhere within the first Brillouin zone in reciprocal space (Debye model) then

[TABLE]

for normal phonons. Assuming that Eq. (A5) also applies to umklapp phonons, that is assuming the size of the phonon sphere is always proportionally measured by the value of phonon frequency (energy), we find Eq. (1) via Eqs. (A1–A3).

ACKNOWLEDGEMENTS

The authors wish to thank Professor Nikolay Plakida for useful discussions.