On the superconducting nature of the Bi-II phase of elemental Bismuth

TL;DR

This study combines experimental muon-spin rotation measurements and theoretical Eliashberg calculations to confirm that the Bi-II phase of Bismuth is an intermediately-coupled type-I superconductor with a transition temperature around 3.92 K.

Contribution

It provides the first combined experimental and theoretical analysis confirming the superconducting nature and properties of the Bi-II phase of Bismuth.

Findings

Bi-II is a type-I superconductor with $T_c \,\approx 3.92$ K.

Experimental and theoretical results agree on the critical field and transition temperature.

Bi-II exhibits intermediate electron-phonon coupling strength.

Abstract

The superconductivity in the Bi-II phase of elemental Bismuth (transition temperature $T_{\rm c}\simeq3.92$ K at pressure $p\simeq 2.80$ GPa) was studied experimentally by means of the muon-spin rotation as well as theoretically by using the Eliashberg theory in combination with Density Functional Theory calculations. Experiments reveal that Bi-II is a type-I superconductor with a zero temperature value of the thermodynamic critical field $B_{\rm c}(0)\simeq31.97$~mT. The Eliashberg theory approach provides a good agreement with the experimental $T_{\rm c}$ and the temperature evolution of $B_{\rm c}$. The estimated value for the retardation (coupling) parameter $k_{\rm B}T_{\rm c}/\omega_{\rm ln} \approx 0.07$ ($\omega_{\rm ln}$ is the logarithmically averaged phonon frequency) suggests that Bi-II is an intermediately-coupled superconductor.

Tables

Table 1. Table 1: Experimental and calculated material parameters for various Bismuth phases. T c subscript 𝑇 c T_{\rm c} is the superconducting transition temperature, B c subscript 𝐵 c B_{\rm c} is the thermodynamical critical field, B c2 subscript 𝐵 c2 B_{\rm c2} is the upper critical field, λ el − ph subscript 𝜆 el ph \lambda_{\rm el-ph} is the electron-phonon coupling constant, γ N subscript 𝛾 N \gamma_{\rm N} is the normal state electronic specific heat coefficient, Supplemntal_part ω ln subscript 𝜔 ln \omega_{\rm ln} is the characteristic phonon frequency, and k B ​ T c / ω ln subscript 𝑘 B subscript 𝑇 c subscript 𝜔 k_{\rm B}T_{\rm c}/\omega_{\ln} is the retardation (coupling) parameter. n/a means the parameter is not available.

	Superconductivity	$T_c$	$B_c$	$B_{\mathrm{c2}}$	${\lambda }_{\mathrm{el}-\mathrm{ph}}$	${\gamma }_N$	${\omega }_{\ln }$	$k_B{T}_c/{\omega }_{\ln }$	References
		(K)	(mT)	(T)		($\mathrm{erg}{\mathrm{cm}}^{-3}{K}^{-2}$)		(meV)
Bi-I	type-I	0.00053	0.0052	–	0.236	399	n/a	n/a	Prakash_Science_2017,; Collan_PRL_1969,; Mata-Pizon_Plos_2016,
Bi-II	type-I	3.92	31.97	–	1.02	2206	4.69	0.072	This Work
Bi-III	type-II	7.05	73.6	2.6	2.75	n/a	5.51	0.110	Brown_ScAdv_2018,; Brown_Thesis_2017,; Khasanov_PRB-Bi-III_2018,
Bi-V	n/a	8.50	n/a	n/a	n/a	n/a	n/a	n/a	Lotter_EPL_1988,; Brown_Thesis_2017,

Equations

$$\text{Bi-I}\stackrel{2.55\mathrm{GPa}}{⟶}\text{Bi-II}\stackrel{2.7\mathrm{GPa}}{⟶}\text{Bi-III}\stackrel{7.7\mathrm{GPa}}{⟶}\text{Bi-V}<220\mathrm{GPa}.$$

$${\omega }_{\ln }=\exp \left(\frac{2}{{\lambda }_{\mathrm{el}-\mathrm{ph}}}{\int }_0^{\infty }\frac{\text{d}\omega }{\omega }{\alpha }^{2}F(\omega )\ln \omega \right)$$

$$\frac{\Delta C(T_{\mathrm{c}})}{{\gamma }_{\mathrm{N}}{T}_{\mathrm{c}}}=\frac{B_{\mathrm{c}}(0{)}^2}{2\pi {\gamma }_{\mathrm{N}}{T}_{\mathrm{c}}^2}{\left[{\frac{\partial D(T/T_{\mathrm{c}})}{\partial ([T/T_{\mathrm{c}}{]}^2)}|}_{(T/T_{\mathrm{c}}{)}^2=1}-1\right]}^2.$$

Full text

Bismuth is element 83 in the periodic table. It is a brittle metal with a silvery white color. Its complex and tunable electronic structure exhibits many fascinating properties that often defy the expectations of conventional theories of metals. Most notably, measurements on Bismuth provided the first evidence of quantum oscillations and the existence of the Fermi surface, thereby experimentally confirming the underlying paradigm of all modern solid state physics.Shubnikov_Nature_1930 ; deHaas_PNRAS_1930

At ambient pressure Bismuth is a compensated semimetal with an exceptionally low carrier concentration of one free charge carrier per about $10^{5}$ atoms.Yang_PRB_2000 The Fermi surface consists of tiny electron- and hole-like pockets giving rise to a highly-anisotropic effective mass, which can become as low as $\sim 10^{-3}$ that of the electron mass in some directions.Hofmann_PiSS_2006 Such properties lead to the highest Hall coefficient, the largest diamagnetism, and an exceptionally small thermal conductivity which sets Bismuth to be quite different compared to other metals.Norman_book_1998

Upon application of pressure at room temperature, Bi undergoes a series of structural transitions:Degtyareva_HPR_2005

[TABLE]

Upon cooling, all the above phases become superconducting with the transition temperature ($T_{\rm c}$) of $T_{\rm c}\simeq 0.53$ mK for Bi-I, $T_{\rm c}\simeq 3.9$ K for Bi-II, $T_{\rm c}\simeq 7$ K for Bi-III and $T_{\rm c}\simeq 8.5$ K for Bi-V, respectively.Brandt_JETP_1963 ; Buckel_PL_1965 ; Ilina_JETP_1970 ; Lotter_EPL_1988 ; Du_PRL_2005 ; Prakash_Science_2017 ; Li_PRB_2017 ; Brown_ScAdv_2018 ; Brown_Thesis_2017 ; Khasanov_PRB-Bi-III_2018 The superconductivity in Bi-I and Bi-III phases were found to be of a type-I and type-II, respectively.Prakash_Science_2017 ; Li_PRB_2017 ; Brown_ScAdv_2018 ; Brown_Thesis_2017 ; Khasanov_PRB-Bi-III_2018 Much less information is known for other Bi phases. In particular, the Bi-I to Bi-II and Bi-II to Bi-III transitions are well-established at room-temperature, while their low temperature behavior lead to contradicting results. References Bundy_PR_1958, ; Klement_PR_1963, ; Il'ina_PSS_1967, ; Compy_JAP_1970, ; Yomo_JPSJ_1972, ; Homan_JPCS_1975, suggest that the Bi-III phase forms at 2.7 GPa at room temperature, while the Bi-II to Bi-III (or possibly Bi-I to Bi-III) phase boundary occurs at pressures $p\gtrsim 3.0$ GPa at 0 K. The Bi-II phase likely extends down to 200 K only, where the Bi-I-II-III triple point may occur.Compy_JAP_1970 ; Homan_JPCS_1975 On the other hand, the superconducting Bi-III phase was observed at pressures of $\simeq$2.7 GPa by several other research groups, as well as by us.Li_PRB_2017 ; Brown_ScAdv_2018 ; Brown_Thesis_2017 ; Khasanov_PRB-Bi-III_2018 Some groups have also reported superconductivity in Bi-II phase at pressures of $\simeq 2.5$ GPa with $T_{\rm c}\simeq 4$ K.Brandt_JETP_1963 ; Homan_JPCS_1975 ; Lotter_EPL_1988 It is worth to note here, that a pure Bi-II phase has never been observed alone, but always appeared as an admixture to the Bi-I or Bi-III phases.Li_PRB_2017 ; Brown_Thesis_2017 ; Lotter_EPL_1988 It seems therefore likely, that the Bi-II phase becomes metastable at low temperatures.

This paper presents the results of an experimental and theoretical study of the Bi-II superconductor. The bulk Bi-II phase ($T_{\rm c}\simeq 3.92$ K at $p\simeq 2.80$ GPa) was stabilized by approaching it from the preformed Bi-III one ($T_{\rm c}\simeq 7.05$ K at $p\simeq 2.72$ GPa, Ref. Khasanov_PRB-Bi-III_2018, ). Muon-spin-rotation ($\mu$SR) measurement reveal that the magnetic induction ($B$) in a cylindrical Bi-II sample (with the magnetic field applied perpendicular to the cylinder axis) is separated between normal state ($B_{\rm N}=B_{\rm c}$, $B_{\rm c}$ is the thermodynamical critical field) and superconducting ($B_{\rm S}=0$) domains thus indicating that Bi-II is a superconductor of type-I. The zero temperature thermodynamic critical field was found to be $B_{\rm c}(0)\simeq 31.97$ mT. The Eliashberg theory provides a good agreement with the experimental critical temperature ($\simeq 3.95$ K), the zero temperature critical field ($\simeq 36.6$ mT), and the temperature evolution of $B_{\rm c}(T)$. The estimated value for the retardation parameter $k_{\rm B}T_{\rm c}/\omega_{\rm ln}\approx 0.07$ ($\omega_{\rm ln}$ is the logarithmically averaged phonon frequency) suggests that Bi-II is an intermediately-coupled superconductor.

The Bi sample and the pressure cell were the same as used in our previous experiments for studying Bi-III superconductivity, Ref. Khasanov_PRB-Bi-III_2018, . The transformation of the Bi sample from Bi-III to Bi-II phase was made by allowing the sample volume to increase inside the pressure cell.Supplemntal_part ac susceptibility (ACS) measurements reveal the presence of a sharp superconducting transition at $T_{\rm c}\simeq 3.92$ K at $p\simeq 2.80$ GPa. The amount of Bi-III phase admixture, obtained in the ACS experiments, does not exceed $10-15$ % (see the Supplemental Part, Ref. Supplemntal_part, ). The transverse-field (TF) $\mu$SR experiments were carried out at the $\mu$E1 beam line by using the dedicated GPD (General Purpose Decay) spectrometer (Paul Scherrer Institute, Switzerland). The details of TF-$\mu$SR experiments performed under pressure are provided in the Supplemental Part, Ref. Supplemntal_part, , and in Refs. Khasanov_PRB_2011, ; Khasanov_HPR_2016, ; Shermadini_HPR_2017, .

Due to its microscopic nature, the $\mu$SR technique allows to directly distinguish between type-I and type-II superconductors, since both superconductivity types are characterized by very different magnetic field distributions [$P(B)$’s] inside the specimen. An ordered flux-line lattice (FLL) of type-II superconductor has the field distribution and the corresponding $P(B)$ which are shown schematically in the left and right panels of Fig. 1 (a). The calculations were performed within the framework of London model with the Gaussian cutoff for a triangular FLL ($B_{\rm ex}=20$ mT, the magnetic penetration depth $\lambda=200$ nm and the coherence length $\xi=50$ nm, see the Supplemental Part, Ref. Supplemntal_part, ). The asymmetric magnetic field distribution function $P(B)$ centers in the vicinity of $B_{\rm ex}$. It is characterized by two cutoffs fields and by the peak shifted below $B_{\rm ex}$ [see the right panel at Fig. 1 (a), and, e.g., Refs. Maisuradze_JPCM_2009, ; Khasanov_PRB_2016, and references therein]. A type-I superconductor expels a magnetic field completely, apart from a layer at the surface of thickness $\lambda$. However in samples with a finite demagnetization factor $n$, a separation between superconducting domains (with $B_{\rm S}=0$) and normal state domains (with $B_{\rm N}=B_{\rm c}>B_{\rm ex}$) can occur [see the left panel at Fig. 1 (b) showing schematically the nucleation of a plate-like sample on S/N domains, and, e.g., Refs. Tinkham_75, ; Huebener_Book_1979, ; Prozorov_PRL_2007, ; Prozorov_NatPhys_2008, and references therein]. In this case, $P(B)$ consists of two, $B=0$ and $B=B_{\rm c}$, lines [right panel of Fig. 1 (b)]. Such distributions (without, however, the $B=0$ line) were reported in earlier $\mu$SR measurements on type-I superconductors Sn, Pb and In,Gladisch_HypInt_1979 ; Grebinnik_JETP_1980 ; Egorov_PhysB_2000 ; Egorov_PRB_2001 and in recent experiments on BeAu.Singh_Arxiv_2019 ; Beare_Arxiv_2019

Figure 1 (c) shows the Fourier transform of few representative TF-$\mu$SR time spectra (the pressure cell background subtracted) measured at $B_{\rm ex}=20$ mT. Figure 1 (d) represents the contour plot of the corresponding Fourier intensities. The overall behavior shown in Figs. 1 (c) and (d) corresponds to the response of a type-I superconductor with a nonzero demagnetization factor $n$ in an applied field $B_{\rm ex}$ of $B_{\rm c}(1-n)\leq B_{\rm ex}\leq B_{\rm c}$ [see the discussion above, Fig. 1 (b) and Refs. Gladisch_HypInt_1979, ; Grebinnik_JETP_1980, ; Egorov_PhysB_2000, ; Egorov_PRB_2001, ; Singh_Arxiv_2019, ; Beare_Arxiv_2019, ]. Indeed, the $P(B)$ distributions at $T\simeq 0.5$ and 2.0 K split into two peaks with the first one at $B=0$ and the second one $\simeq 12$ and $\simeq 5$ mT higher than the applied field $B_{\rm ex}$, respectively. With increasing temperature, the intensity of the $B=0$ peak decreases until it vanishes at $T\simeq 2.3$ K, while the intensity of the $B\geq B_{\rm ex}$ peak increases by approaching $T\simeq 2.3$ K and saturates above it [Figs. 1 (c) and (d)]. The position of the $B\geq B_{\rm ex}$ peak shifts in direction of $B_{\rm ex}$ all the way up to $\simeq 2.3$ K and coincides with $B_{\rm ex}$ for higher temperatures. The intensities of the $B=0$ and $B\geq B_{\rm ex}$ peaks are proportional to the volume fractions of the superconducting ($B_{\rm S}=0$) and the normal state ($B_{\rm N}=B_{\rm c}$) domains. The disappearance of the $B=0$ peak above $2.3$ K correspond to the transition of the sample into the normal state [$T_{\rm c}(B=20\ {\rm mT})\simeq 2.3$ K]. The position of the $B>B_{\rm ex}$ peak represents the temperature evolution of the thermodynamical critical field $B_{\rm c}$ [red dashed line in Fig. 1 (d)].

Note that our $\mu$SR data exclude the possibility of type-II superconductivity in Bi-II. Additionally, the zero-temperature critical field was found to be half the value of $B_{\rm c}(0)\simeq 73$ mT reported in Ref. Li_PRB_2017, . Field scans at $T=0.25$, 2.1 and 3.0 K with 1 mT steps (from 0.3 to 35  mT) and temperature scans at $B_{\rm ex}=3$, 10, 20 and 30 and 35 mT with 0.125 K steps (from 0.25 to 8.0 K) do not show any FLL-type $\mu$SR response. No superconductivity was detected at $B_{\rm ex}=35$ mT down to the lowest temperature of the experiment ($\simeq 0.25$ K) and for all applied fields at $T\geq 4$ K. The fact that no FLL signal was observed above $4.0$ K, suggests also that the admixture of the Bi-III phase ($T_{\rm c}\simeq 7$ K as is detected in ACS experiment, see the Supplemental part, Ref. Supplemntal_part, ) is minimal in sample volume. Our results imply, therefore, that within the full range of temperatures ($0.25\leq T\leq 8.0$ K) and fields ($0.3\leq B_{\rm ex}\leq 0.35$ mT) studied, the Bi-II phase of elemental Bismuth behaves as a typical type-I superconductor.

The temperature dependence of the thermodynamical critical field $B_{\rm c}$, as determined from the measured field value in the normal-state domain [$B_{\rm c}=B_{\rm N}$, see Figs. 1 (b), (c), and (d)], is shown in Fig. 1 (e). The points are obtained with several applied fields ($B_{\rm ex}=3$, 10, 20, and 30 mT) and they overlap within certain temperature and field regions. The reason for such overlapping is caused by the intermediate state formation condition: $B_{\rm c}(T)\;(1-n)\leq B_{\rm ex}\leq B_{\rm c}(T)$, showing that similar $B_{\rm c}(T)$ can be obtained for different $B_{\rm ex}$’s.Gladisch_HypInt_1979 ; Grebinnik_JETP_1980 ; Egorov_PhysB_2000 ; Egorov_PRB_2001

The obtained experimental data were compared with quantitative predictions based on ab-initio Eliashberg calculations using Density Functional Theory (DFT). The details of calculations are given in the Supplemental part, Ref. Supplemntal_part, . The experimental and calculated material parameters for Bi-II phase are summarized in Table 1.

Figures 2 (a) and (b) display the phonon density of states (DOS), the Eliashberg electron-phonon spectral function [$\alpha^{2}F(\omega)$] and the integrated electron-phonon coupling constant: $\lambda_{\rm el-ph}(\omega)=2\int_{0}^{\omega}\frac{\text{d}\omega^{\prime}}{\omega^{\prime}}\alpha^{2}F(\omega^{\prime})$. In the high frequency limit $\lambda_{\rm el-ph}$ was estimated to be $\lambda_{\rm el-ph}(\omega\rightarrow\infty)\simeq 1.02$. The logarithmically averaged phonon frequency $\omega_{\rm ln}$, representing a characteristic phonon energy mediating the pairing,al.dy.75 was calculated via:

[TABLE]

and found to be $\omega_{\rm ln}=4.69\,$meV.

The dashed line in Fig. 1 (e) represents the temperature evolution of $B_{\rm c,T}(T)$ computed from the free energy difference between the normal and superconducting states ($\Delta F$) via $B_{\rm c}(T)=\sqrt{-8\pi\Delta F}$ (here after the index ’T’ accounts for the parameter obtained from the theory). $\Delta F$ was calculated within the strong-coupling Eliashberg theory following the approach developed by Bardeen and Stephen.ba.st.64 The transition temperature $T_{\rm c,T}=3.95$ K and the zero temperature value of the thermodynamical field $B_{\rm c,T}(0)=36.6$ mT are found. Scaling the $B_{\rm c}(T)$ curve further allows direct comparison with the experimental data. The adjusted curve with $T_{\rm c}\simeq 3.922$ K and $B_{\rm c}(0)\simeq 31.97$ mT is shown by the solid line in Fig. 1 (e).

In order to better visualize the difference between the theory and the experiment, the deviation function $D(T/T_{\rm c})=B_{\rm c}(T)/B_{\rm c}(0)-(1-[T/T_{\rm c}]^{2})$ is plotted in Fig. 2 (c). For comparison, the weak coupling BCS results are also shown. Obviously, the BCS theory underestimates the experimental $D(T/T_{\rm c})$ and a significant improvement is obtained using the Eliashberg theory. Although some quantitative discrepancies remain, the main features are captured.

Many thermodynamic quantities, like the condensation energy or the specific heat jump $\Delta C(T_{\rm c})/\gamma_{\rm N}T_{\rm c}=C_{\rm eS}(T_{\rm c})/\gamma_{\rm N}T_{\rm c}-1$, can be expressed directly by using the derivative of $D(T/T_{\rm c})$ as follows:Padamsee_JLTP_1973

[TABLE]

Here $\gamma_{\rm N}$ is the electronic specific heat coefficient in the normal state (see the Supplemental part, Ref. Supplemntal_part, , for $\gamma_{\rm N}$ estimate) and $C_{\rm eS}(T)/\gamma_{\rm N}T_{\rm c}$ is the electronic specific heat in the superconducting state. We proceed with the direct numerical calculation of $C_{\rm eS}(T)/\gamma_{\rm N}T_{\rm c}$ within the Eliashberg theory (see the Supplemental part, Ref. Supplemntal_part, ). The heat capacity jump ${\Delta C(T_{\rm c})}/{\gamma_{\rm N}T_{\rm c}}\simeq 2.40$ was found, which is large in comparison with the universal BCS value of $1.43$. Such a large jump in the specific heat for Bi-II is certainly accessible for calorimetric measurements.

To conclude, the superconductivity in the Bi-II phase of elemental Bismuth was studied experimentally by means of muon-spin rotation, as well as theoretically using the Eliashberg theory in combination with Density Functional Theory calculations. Experiments reveal that the magnetic induction in the cylindrical Bi-II sample is separated into normal state and superconducting domains thus suggesting that Bi-II is a superconductor of type-I. The transition temperature and the zero temperature thermodynamic critical field were found to be $T_{\rm c}\simeq 3.92$ K and $B_{\rm c}(0)\simeq 31.97$ mT, respectively. The electronic, and the superconducting properties of Bi-II were computed from first principles. Following the phenomenological approach of Carbotte, Ref. Carbotte_RMP_1990, , the strong coupling corrections were embodied via the retardation parameter $k_{B}T_{\rm c}/\omega_{\rm ln}$. Including retardation effects, the Eliashberg theory provides better agreement with the experimental data than the weak coupling BCS approach. The theory values for the critical temperature ($T_{\rm c,T}\simeq 3.95$ K) and the zero temperature critical field $B_{\rm c,T}(0)=36.6$ mT, as well as the temperature evolution of $B_{\rm c}(T)$ are in agreement with the experiment. The specific heat jump, as estimated from the deviation function $D(T/T_{\rm c})$, was found to be ${\Delta C(T_{\rm c})}/{\gamma_{\rm N}T_{\rm c}}=2.40$, which is large in comparison with the universal BCS value of $1.43$. The ab-initio calculations result in the value of the retardation parameter $k_{B}T_{\rm c}/\omega_{\rm ln}\approx 0.07$ and put Bi-II in the category of intermediate coupling superconductors, being away from the very strong coupling limit $k_{B}T_{\rm c}/\omega_{\rm ln}\approx 0.25$. Finally, our analysis reveals that the Cooper pairing in Bi-II is a consequence of balance between the electron-phonon attraction and a significant direct Coulomb repulsion. Compared to our previous study of Bi-III,Khasanov_PRB-Bi-III_2018 , the retardation effects in Bi-II were found to be less efficient than in Bi-III. While Bi-III is a type-II strong-coupled superconductor,Brown_ScAdv_2018 ; Brown_Thesis_2017 ; Khasanov_PRB-Bi-III_2018 the Bi-II and Bi-I are a type-I superconductors with the intermediate (present study) and weak-coupling (Ref. Mata-Pizon_Plos_2016, ) strength, respectively (see also the Table 1 summarizing experimental and calculated material parameters for various Bismuth phases). In this respect the high pressure $\mu$SR experiments, as those presented here and in Ref. Khasanov_PRB-Bi-III_2018, on elemental Bi, are essential tools to elucidate the nature of the interplay between structural and superconducting phases in conventional superconductors.

This work was performed at the Swiss Muon Source (S$\mu$S), Paul Scherrer Institute (PSI, Switzerland). The work of GS is supported by the Swiss National Science Foundation, grants $200021\_149486$ and $200021\_175935$. M.M.R. acknowledges the support from the Ministry of Education, Science, and Technological Development of the Republic of Serbia under project ON171017. Numerical simulations were run on the PARADOX supercomputing facility at the Scientific Computing Laboratory of the Institute of Physics Belgrade. L.C. and A.Ö acknowledges DFG for the financial support through TRR80/F6 project. S.S. acknowledges financial support by the Swedish Research Council and computational facilities provided by the Swedish National Infrastructure for Computing at the National Supercomputer Centers in Linköping and Umeå.