Fermi-Liquid Nonadiabatic Highly Compressed Cesium Iodide Superconductor

Abstract

The experimental discovery that compressed sulfur hydride exhibits superconducting transition temperature of $T_c=203 \mathrm{K}$ by Drozdov et al. (Nature 2015, 525, 73–76) sparked studies of compressed hydrides. This discovery was not a straightforward experimental examination of a theoretically predicted phase, but instead it was a nearly five-decade-long experimental quest for superconductivity in highly compressed matters, varying from pure elements (hydrogen, oxygen, sulfur), hydrides (SiH₄, AlH₃) to semiconductors and ionic salts. One of these salts was cesium iodide, CsI, which exhibits the transition temperature of $T_c\cong 1.5 \mathrm{K}$ at $P=206 GPa$ (Eremets et al., Science 1998, 281, 1333–1335). Detailed first principles calculations (Xu et al., Phys Rev B 2009, 79, 144110) showed that CsI should exhibit $T_c~0.03 \mathrm{K}$ ($P=180 \mathrm{GPa}$). In an attempt to understand the nature of this discrepancy between the theory and the experiment, we analyzed the temperature-dependent resistance in compressed CsI and found that this compound is a perfect Fermi liquid metal which exhibits an extremely high ratio of Debye energy to Fermi energy, $\frac{\hslash {\omega }_D}{k_B{T}_F}\cong 17$. This implies that direct use of the Migdal–Eliashberg theory of superconductivity to calculate the transition temperature in CsI is incorrect, because the theory is valid for $\frac{\hslash {\omega }_D}{k_B{T}_F}\ll 1$. We also showed that CsI falls into the unconventional superconductors band in the Uemura plot.

1. Introduction

Since superconducting transition at 203 K was observed in highly compressed sulfur hydride by Drozdov et al. [1], dozens of superconducting hydrogen-based phases have been discovered [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] (extended reviews on the current status of the topic can be found elsewhere [18,19,20]). The report by Drozdov et al. [1] was the triumphant culmination of a nearly five-decade-long journey in the terra incognita of hydrogen-rich compounds [21] and highly compressed matter [22]. On this journey, the superconductors family was significantly extended and the superconducting transition was experimentally observed in many non-superconducting (at ambient conditions) elements/compounds. At the same time, the transition was not observed in materials for which the first principles calculations (FPC) and the Eliashberg theory [23] of the electron–phonon mediated superconductivity predicted a high critical temperature, $T_c$. We may mention AlH₃ [24,25] as an outstanding case of this class of materials.

However, more often, the superconducting transition was observed, but predicted $T_c$ significantly exceeds the experimental value. The most notable case of this class of materials is compressed SiH₄ for which Feng et al. [26] calculated a Debye temperature of $T_{\theta }=3500-4000 \mathrm{K}$ and $T_c\cong 165 \mathrm{K}$, while the experiment performed by Eremets et al. [27] showed that $T_c=7-17 \mathrm{K}$.

More intriguingly, there are several highly compressed compounds in which experimentally observed $T_c$ significantly exceeds the calculated value. The most famous case of these highly pressurized compounds is sulfur hydride, for which Li et al. [28] initially predicted $T_c\cong 80 \mathrm{K}$. However, Drozdov et al. [1] reported that the experimentally observed transition temperature is significantly higher, $T_c=203 \mathrm{K}$, and the observed value is in excellent agreement with theoretical calculations reported by Duan et al. [29].

Another material from this category is highly compressed CsI, for which detailed first principles calculations performed by Xu et al. [30] predicted $T_c=0.03 \mathrm{K}$ at a pressure of $P=180 \mathrm{GPa}$, while the experimental value reported by Eremets et al. [31] is $T_c\cong 1.5 \mathrm{K}$ (at $P=206 \mathrm{GPa}$).

In this work, we answer the question, why highly compressed CsI exhibits nearly two orders of magnitude higher $T_c$ in comparison with the predicted value by FPC and the Eliashberg theory of electron–phonon mediated superconductivity. Our answer is that highly compressed CsI exhibits an enormous ratio of the Debye temperature, $T_{\theta }$, to the Fermi temperature, $T_F$, $\frac{T_{\theta }}{T_F}\cong 17$, while the Eliashberg theory [23] of the electron–phonon mediated superconductivity is valid for $\frac{T_{\theta }}{T_F}\ll 1$.

2. Results

2.1. The Electron–Phonon Coupling Constant and the Debye Temperature in CsI at p = 206 GPa

Cesium iodide is isoelectronic with the noble gas solid xenon (i.e., Cs⁺ and I⁻ ions in ionic salt have closed xenon-like electronic shells, and the short-range interaction between Cs⁺ and I⁻ ions in uncompressed salt and two Xe atoms in solid xenon is identical). The main difference between Xe and CsI is the very strong Coulomb interaction in cesium iodide in comparison with xenon [32,33]. The strength of the Coulomb interaction decreases on compression, and at $P\cong 110 \mathrm{GPa}$ CsI is metallized [31,33].

In the theory of the electron–phonon mediated superconductivity [23,34], the phonon spectrum is one of the primary properties that determine the superconducting transition temperature, $T_c$, and, thus, this is of great interest to determining the main characteristic parameter of this spectrum, i.e., the Debye temperature, $T_{\theta }$. This value can be deduced from the fit of temperature-dependent resistance, $R\left(T\right)$, to the Bloch–Grüneisen (BG) equation [35,36,37,38,39,40]:

$$R\left(T\right)=R_0+A\times {\left(\frac{T}{T_{\theta }}\right)}^5\times {{\displaystyle \int }}_0^{\frac{T_{\theta }}{T}}\frac{x^5}{\left(e^x-1\right)\left(1-e^{-x}\right)}\cdot dx,$$

(1)

where $R_0$ is the residual resistance at $T\to 0 \mathrm{K}$, and the second term describes the electron–phonon scattering, where $A$ and $T_{\theta }$ are free-fitting parameters. Equation (1) was applied to deduce the Debye temperature in many highly compressed superconductors, for instance, in black phosphorus [41], boron [41], sulphur [42,43], lithium [42], $\zeta$-phase of O₂ [42], SnS [37], GeAs [41], SiH₄ [41], H₃S [41,43], D₃S [41,43], LaH₁₀ [41,43], C2/m-SnH₁₂ [44], Th₄H₁₅ [45], ThH₉ [45], ThH₁₀ [45], YD₆ [45], metallic hydrogen phase-III [45], and $\left(La,Nd\right)H_{10}$ [46].

The fit of the $R\left(T\right)$ curve of compressed CsI ($P\cong 206 \mathrm{GPa}$) reported by Eremets et al. [31] in their Figure 3B of [31] to Equation (1) is shown in Figure 1a. The derived Debye temperature is $T_{\theta }=339\pm 1 \mathrm{K}$. From this value and the measured $T_c\cong 1.1 \mathrm{K}$ [31] (see Figure 5 of [31]), one can calculate the electron-–phonon coupling constant, ${\lambda }_{e-ph}$, as the root of an advanced McMillan equation [41]:

$$T_c=\left(\frac{1}{1.45}\right)\times T_{\theta }\times e^{-\left(\frac{1.04\left(1+{\lambda }_{e-ph}\right)}{{\lambda }_{e-ph}-{\mu }^{*}\left(1+0.62{\lambda }_{e-ph}\right)}\right)}\times f_1\times f_2^{*}$$

(2)

$$f_1={\left(1+{\left(\frac{{\lambda }_{e-ph}}{2.46\left(1+3.8{\mu }^{*}\right)}\right)}^{3/2}\right)}^{1/3}$$

(3)

$$f_2^{*}=1+\left(0.0241-0.0735\times {\mu }^{*}\right)\times {\lambda }_{e-ph}^2$$

(4)

where ${\mu }^{*}$ is the Coulomb pseudopotential; it can be assumed that ${\mu }^{*}=0.13$ [4,18,19,20]. In the result, ${\lambda }_{e-ph}=0.445$ was calculated as a root of Equations (2)–(4). It can be noted that the deduced ${\lambda }_{e-ph}=0.445$ is close to ${\lambda }_{e-ph}=0.43$ for aluminum [47]. It should also be mentioned that deduced $T_{\theta }=339\pm 1 K$ for CsI is not very different from $T_{\theta }=394$–428 K [48,49] for aluminum.

One can make a comparison of the $T_{\theta }=339 \mathrm{K}$ and ${\lambda }_{e-ph}=0.445$ values derived from experiment ($P=206 \mathrm{GPa}$) with the values computed by first principles calculations [30]. Xu et al. [30] reported ${\lambda }_{e-ph}=0.262$ ($P=180 \mathrm{GPa}$) and ${\lambda }_{e-ph}=0.257$ ($P=216 \mathrm{GPa}$), and both these values are significantly lower than the one deduced from the experiment here. Xu et al. [30] also calculated the logarithmic phonon frequency $\frac{\hslash }{k_B}{\omega }_{log}=285 \mathrm{K}$ ($P=180 \mathrm{GPa}$), and $\frac{\hslash }{k_B}{\omega }_{log}=314 \mathrm{K}$ ($P=216 \mathrm{GPa}$). By its definition, ${\omega }_{log}$ is close, but does not exactly equal, to the Debye frequency, $T_{\theta }=\frac{\hslash }{k_B}{\omega }_D$, and this is what one can see for these values in highly compressed CsI.

By utilizing the Allen–Dynes equation [47,48] and ${\mu }^{*}=0.10$, Xu et al. [30] calculated $T_c=0.03 \mathrm{K}$ ($P=180 \mathrm{GPa}$) and $T_c=0.025 \mathrm{K}$ ($P=216 \mathrm{GPa}$). Both calculated that the $T_c$ values (and this was acknowledged by the authors of [30]) are by about two orders of magnitude lower than the experimental value. To explain this discrepancy, Xu et al. [30] hypothesized that because first principles calculations show that the CsI exhibits an anisotropic crystalline structure (under pressure), then the Allen–Dynes equation [48,50] (developed for the single-band isotropic superconductors) cannot accurately average an anisotropic case.

Our explanation for the discrepancy is based on a different idea which arose from a more advanced analysis of the temperature-dependent resistance curve described below.

2.2. Perfect Fermi Liquid Conductor CsI at p = 206 GPa

Despite the fit of the $R\left(T\right)$ curve for the compressed CsI to the BG equation (Equation (1)) having a high quality (Figure 1a), more advanced analysis is based on the approach when the power–law exponent in Equation (1) is a free-fitting parameter [51,52,53,54,55]:

$$R\left(T\right)=R_0+A\times {\left(\frac{T}{T_{\omega }}\right)}^p\times {{\displaystyle \int }}_0^{\frac{T_{\omega }}{T}}\frac{x^p}{\left(e^x-1\right)\left(1-e^{-x}\right)}\cdot dx.$$

(5)

In this approach, the $T_{\omega }$ (Equation (5)) is not any longer the Debye temperature; however, this temperature represents a characteristic energy scalar for the charge carrier interaction in the conductor. There are several integer $p$-values which associate with a particular charge-carrier interaction mechanism [56,57,58] and, in particular, $p=2$ implies that charge carriers in the conductor obey a perfect Landau’s Fermi liquid phenomenology [56,57,58].

It should be mentioned that for some materials, such as ReBe₂₂ [53,59] and (ScZrNb)_0.65[RhPd]_0.35 [54,60], the power–law exponent is indistinguishable from 5, which implies that these materials are pure electron–phonon-mediated superconductors. However, for the majority of highly compressed superconductors, including the ε-Fe phase, the power–law exponent, p, varies between $1.80\le p\le 3.3$ [53,54,55].

The fit of the $R\left(T\right)$ curve in the CsI (P = 206 GPa) to Equation (5) is shown in Figure 1b, where it can be seen that the deduced $p$ is indistinguishable from $p=2.0$. This means that the highly compressed CsI at ($P=206 \mathrm{GPa}$) is a perfect Fermi liquid metal.

2.3. Compressed CsI (p = 206 GPa) in the Uemura Plot

One of the widely accepted ways to classify the superconducting state in the material is to position the material in the Uemura plot, i.e., in the plot where the X-axis is the Fermi temperature, $T_F$, and the Y-axis is the transition temperature, $T_c$ [61,62]. The Fermi temperature in the superconductor can be calculated by the equation [42,63,64]:

$$T_F=\frac{{\epsilon }_F}{k_B}=\frac{{\pi }^2}{8·k_B}\times \left(1+{\lambda }_{e-ph}\right)\times {\xi }^2\left(0\right)\times {\left(\alpha \frac{k_B{T}_c}{\hslash }\right)}^2,$$

(6)

where ${\epsilon }_F$ is the Fermi energy, $k_B$ is the Boltzmann constant, $\mathsf{\alpha }=\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}$, and $\mathsf{\Delta }\left(0\right)$ is the amplitude of the ground state energy gap, $\hslash =h/2\pi$ is the reduced Planck constant, and $\mathsf{\xi }\left(0\right)$ is the ground state coherence length. Based on a very large database on electron–phonon-mediated superconductors [47], one can expect that the CsI ($P=206 \mathrm{GPa}$) which exhibits ${\lambda }_{e-ph}=0.445$ should have $\mathsf{\alpha }=\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}$ not very different from 3.53, and we used this value in our calculations. Thus, to calculate $T_F$ in the compressed CsI, one needs to estimate the ground state coherence length $\xi \left(0\right)$ (Equation (6)).

We deduce $\xi \left(0\right)$ for the compressed CsI ($P=206 \mathrm{GPa}$) from the fit of the temperature-dependent upper critical field, $B_{c2}\left(T\right)$, to the simplest equation of the Werthamer–Helfand–Hohenberg theory [65,66]:

$$B_{c2}\left(0\right)=\frac{{\varphi }_0}{2\pi {\xi }^2\left(0\right)}=-0.697\times T_c\times {\left(\frac{d{B}_{c2}\left(T\right)}{dT}\right)|}_{T~T_c}.$$

(7)

The upper critical field represents the applied magnetic field in which the superconducting state collapses in the experiment. We extracted the B_c2(T) dataset in highly compressed CsI from the magnetoresistance data reported by Eremets et al. [31] in their Figure 5. To define B_c2(T) we utilized a 50% normal state resistance criterion. The fit of the B_c2(T) data is shown in Figure 2, from which $\xi \left(0\right)=26\pm 3 \mathrm{nm}$ was estimated.

From all determined/estimated values, one can calculate $T_F=20\pm 4 \mathrm{K}$ and the ratio of $\frac{T_c}{T_F}$, which varies within a range:

$$0.04\lesssim \frac{T_c}{T_F}\lesssim 0.07$$

(8)

In the result, the CsI ($P=206 \mathrm{GPa}$) falls into the unconventional superconductors band in the Uemura plot (Figure 3).

It should be noted that the A-15 superconductor V₃Si (which exhibits the electron–phonon-mediated superconductivity with ${\lambda }_{e-ph}=0.96$ and $\mathsf{\alpha }=\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}=3.7$ [47,67]) is also located in the unconventional superconductors band in the Uemura plot (Figure 3).

Another material which can be mentioned is the magic-angle twisted bilayer graphene (MATBG) which exhibits very close $T_c=1.4\pm 0.3 \mathrm{K}$ and $T_F=23\pm 6 \mathrm{K}$ values [68,69] (Figure 3) to the compressed CsI.

Thus, our result that the highly compressed CsI (exhibited ${\lambda }_{e-ph}=0.445$) falls into the unconventional superconducting band manifests itself as an interesting, but not unique, case. The explanation of this result is based on, roughly speaking, the degree of Cooper pairs overlapping [70]. In CsI and V₃Si, the average size of the Cooper pairs at the low temperature, $\xi \left(T\to 0\right)$, is comparable with the average distance between pairs and, thus, the pairs overlapping is not significant and is close to the Bose–Einstein condensate [70]. However, in superconductors located on the BCS side of the Uemura plot (Figure 3), the distance between the Cooper pairs centers is significantly smaller than the $\xi \left(T\to 0\right)$ and the pairs’ overlapping is high. Details for the position of the superfluid ⁴He in Figure 3 can be found elsewhere [62].

2.4. Nonadiabalic Superconductivity in CsI (P = 206 GPa)

It is important to note that one of our central findings is that our analysis shows that the compressed CsI is a remarkably prominent nonadiabatic superconductor. This finding directly follows from the ratio of the Debye temperature, $T_{\theta }$ (for which we used the value deduced from the R(T) data fit to Equation (1), to the Fermi temperature, $T_F$:

$${\frac{\hslash {\omega }_D}{k_B{T}_F}|}_{CsI, P=206 GPa}=\frac{T_{\theta }=339 K}{T_F=20 K}=17\pm 4.$$

(9)

Roughly speaking, the Eliashberg theory of electron–phonon-mediated superconductivity [23] is based on Migdal’s theorem [71], which is the many-body version of the principle of Born–Oppenheimer [72]. Namely, it is an argument over adiabaticity, since charge-carrier particles are expected to move faster than ions. Thus, instead of taking the interaction with an electron in a certain position, it is easier to consider the interaction with the full electronic cloud. This consideration is applicable if the electrons are much faster than ions, but in some materials (such as in the C₆₀ compound), the Fermi energy and the average phonon energy are just the same, and for these materials the Migdal theorem [71] and the Eliashberg theory [23] are no longer valid.

These kinds of superconductors, designated as nonadiabatic superconductors [73], were first theoretically considered by Pietronero and co-workers nearly three decades ago [73,74,75,76,77]. Pietronero and co-workers [73,74,75,76,77] considered the generalization of the many-body theory of superconductivity in the cases in which the Migdal theorem does not work, typically the systems with very low Fermi energy (or Fermi velocity, or Fermi temperature).

3. Discussion

Equation (9) shows that the compressed CsI ($P=206 \mathrm{GPa}$) exhibits relatively “very fast” phonons and relatively “very slow” charge-carriers. For instance, one can make a comparison of the ratio for the CsI (Equation (9)) with the ratio for elemental electron–phonon superconductors (data for the ratios are taken from [49]):

$${\frac{\hslash {\omega }_D}{k_B{T}_F}|}_{Pb}=\frac{T_{\theta }=88 K}{T_F=110000 K}=8\times {10}^{-4}.$$

(10)

$${\frac{\hslash {\omega }_D}{k_B{T}_F}|}_{Nb}=\frac{T_{\theta }=265 K}{T_F=61800 K}=4\times {10}^{-3}.$$

(11)

$${\frac{\hslash {\omega }_D}{k_B{T}_F}|}_{Al}=\frac{T_{\theta }=394 K}{T_F=136000 K}=3\times {10}^{-3}.$$

(12)

Equation (9) shows that the standard Migdal–Eliashberg theory [23,71] of the electron–phonon-mediated superconductivity is inapplicable for highly compressed CsI and, therefore, this is our explanation for the discrepancy between the superconducting transition temperature, $T_c$, predicted by first-principles calculations [30] (following standard Allen–Dynes methodology [48,50]) and the observed $T_c$ in experiment [31].

To the best of our knowledge, such a large ${\frac{\hslash {\omega }_D}{k_B{T}_F}|}_{CsI, P=206 GPa}=17$ ratio has not been reported for any superconductor to date.

4. Conclusions

In ambient conditions, CsI represents typical ionic salt, which transforms into a metallic state at high pressure. Xu et al. [30] performed first principle calculations for compressed CsI and established that within the Migdal–Eliashberg theory [23,69] this compound should exhibit the superconducting transition temperature T_c(p = 180 K)~0.03 K. Experiments performed by Eremets et al. [31] showed that highly compressed CsI exhibited a nearly two orders of magnitude higher transition temperature T_c(p = 206 K)~1.5 K.

In attempts to find the primary origin for the discrepancy between the theory and the experiment, here we analysed available experimental data measured in highly compressed CsI (p = 206 GPa) and found that:

• CsI is a perfect Fermi liquid metal.
• CsI exhibits an extremely high ratio of Debye energy to Fermi energy, $\frac{\hslash {\omega }_D}{k_B{T}_F}\cong 17$. Based on this, one can conclude that the superconducting state in this compound cannot be described by the Migdal–Eliashberg theory [23,69] because the theory is valid when $\frac{\hslash {\omega }_D}{k_B{T}_F}<<1$.
• CsI exhibits the ratio of $\frac{T_c=1.1 K}{T_F=20 K}=0.055\pm 0.015$ and, thus, it falls to the unconventional superconductors band in the Uemura plot. This level of the $\frac{T_c}{T_F}$ ratio is typical for many superconductors, including the electron–phonon-mediated A-15 compound V₃Si. It should be mentioned that the magic-angle twisted bilayer graphene exhibits close $T_c=1.4\pm 0.3 \mathrm{K}$ and $T_F=23\pm 6 \mathrm{K}$ values [68,69].