Campbell penetration depth in low carrier density superconductor YPtBi

The superconducting phase transition temperature $T_{c}$ of the Bardeen-Cooper-Schrieffer (BCS) superconductors is typically of the order of $\sim 10^{-4}T_{F}$, where $T_{F}\sim 10^{4}$ - 10⁵ K is the Fermi temperature [1, 2]. This follows from the small value of the energy gap in the density of states, $\Delta\left(0\right)\approx 1.76k_{B}T_{c}\approx 2\hbar\omega_{D}\exp\left(-1/VN_{0}\right)$. For example, for $T_{c}=10$ K, $\Delta\left(0\right)\approx 1.5$ meV$\ll E_{F}$ $\sim 1$ - 10 eV. Here $\omega_{D}$ is the Debye frequency, $V$ is the attractive pairing potential, $N_{0}$ is the density of states (DOS) at the Fermi energy $E_{F}$. $N_{0}$ has an exponential role in determining the $T_{c}$ which is often estimated from the semi-phenomenological McMillan equation [3, 4]:

where $\lambda$ is the electron–phonon coupling parameter and $\mu^{*}$ is the screened Coulomb interaction constant. The weak-coupling BCS formula can be recovered by replacing $(\lambda-\mu^{*})$ with the product $VN_{0}$ in the limit of small $\lambda\ll 1$. This relation was successfully applied for many intermetallic compounds with a typical carrier density $n\sim 10^{22}$ cm^-3 [5].

However, discovery of superconductivity in materials with a poor metallic normal state with $n\sim 10^{17}$ - $10^{19}$ cm^-3 challenged the conventional approach. Such low $n$ superconductors include elemental bismuth [6], SrTiO₃ [7], and half-Heusler compound RTBi (R=rare earth, T=Pd or Pt) [8, 9, 10, 11]. The observed $T_{c}$s (and often critical fields) in these materials are by orders of magnitude higher than the expected $T_{c}$ from the McMillan formula [12]. These low $n$ superconductors have naturally much higher values of the ratio $T_{c}/T_{F}$, pushing them closer to the Bose-Einstein condensation (BEC) regime in which the spatial range of attractive interaction in the Cooper pair, the superconducting coherence length, $\xi$, becomes comparable to the characteristic length associated with the Fermi momentum, $\xi\sim\hbar/p_{F}$. In YPtBi the $k_{F}\approx 0.4$ nm^-1 whereas in SrPd₂Ge₂ with a similar $T_{c}$ it is $k_{F}\approx 10$ nm^-1. There are materials believed to be in the BCS-BEC crossover regime, notably FeSe_1-xS_x [13].

Determination of the Cooper-pair density $n_{s}$ is required to confirm the unusual low $n$ nature of superconductivity in the material of interest. Traditionally, the normal state electronic concentration $n$ is used to estimate $n_{s}$ by using a simple relation $n_{s}=n/2$. While the relation usually holds in the normal metals, accurate measurements of $n$ in the normal state with $E_{F}\ll 1$ eV can be challenging due to strong temperature dependence of $n$ at low temperatures, anomalous Hall effect, and the presence of the surface states. Here we probe $n_{s}$ directly in the superconducting state of YPtBi by determining the theoretical critical current density $j_{s}$, the quantity directly proportional to $n_{s}$.

The half-Heusler compound YPtBi is a topological semimetal with $n\sim 10^{18}$ cm^-3 at low temperatures [14, 9]. Its superconductivity is attracting a considerable attention because $T_{c}$ is about fourfold higher than that of doped SrTiO₃ with a similar $n\sim 10^{18}$ cm^-3, and it was suggested that its superconductivity arises from the $j=3/2$ Fermi surface [15]. The possible superconducting states include unprecedented spin-quintet and septet states [16, 15]. The topological normal state is driven by strong spin-orbit coupling that inverts the $s$-orbital derived $\Gamma_{6}$ band and the $p$-orbital derived $\Gamma_{8}$ band [15]. The chemical potential lies about 35 meV below a quadratically touching point of $\Gamma_{8}$ bands [9, 15] due to naturally occurring crystal imperfection [17].

Recent experimental results support unconventional superconductivity in YPtBi. $T_{c}$ can be enhanced by physical pressure with an initial linear rate of 0.044 K/GPa [18]. The upper critical field at zero temperature is $\mu_{0}H_{c2}(0)=1.5$ T [9] that is higher than the Pauli limiting field 1.4 T [9] for a weak-coupling spin-1/2 singlet superconductor. The temperature dependence of $H_{c2}(T)$ is practically linear over almost entire superconducting temperature-range, quite different from conventional parabolic behavior [9]. A muon spin rotation study determined $\lambda_{L}(0)=1.6$ $\mu$m [19] which is an order of magnitude greater than that of strong type II superconductor CeCoIn₅ where $\lambda(0)\approx 0.26$ $\mu$m [20]. Coherence length at zero temperature is $\xi(0)=\sqrt{\phi_{0}/2\pi H_{c2}\left(0\right)}\approx 15\>\mathrm{nm}$. The Ginzburg-Landau parameter is $\kappa=\lambda_{L}(0)/\xi(0)\approx 10^{2}\gg 1/\sqrt{2}$, placing it in the strong type-II regime of superconductivity.

In the mixed state of a type II superconductor, the small-amplitude AC magnetic penetration depth is governed by the elastic properties of the Abrikosov vortex lattice in the linear response regime. This means that the amplitude of the AC field excitation, $H_{ac}$, is not large enough to displace the vortex out of the pinning potential well, and it only perturbs the vortex position within the validity of Hooke’s law. In this case, the penetration depth is described by the Campbell penetration depth $\lambda_{C}$ that determines the attenuation range of the AC perturbation from the sample surface to the interior, $B_{ac}\left(x\right)\propto\mu H_{ac}e^{-x/\lambda_{C}}$ in a semi-infinite superconductor [21, 22, 23, 24, 25, 26, 27]. Here $\mu$ is magnetic permeability, and $x$ is the distance from the surface. Since $\lambda_{C}$ is not commonly measured due to amplitude/sensitivity limitations of the conventional AC techniques, we provide a simple derivation of $\lambda_{C}$ in the Appendix for completeness. The important advantage of employing $\lambda_{C}$ is that it gives access to the shielding current density via a relation $\lambda_{C}^{2}=H_{0}r_{p}/j_{c}$. Here $r_{p}$ is the radius of the pinning potential, and $H_{0}$ is the applied external DC magnetic field (see Appendix for details). Importantly, the critical current density is estimated at the frequency of the measurement, and the rf regime gives access to almost unrelaxed values. The initial vortex relaxation is exponential, and hence the conventional techniques estimate relaxed values far from the true $j_{c}$ [28, 29, 30].

While analysis of the relaxed shielding current is complicated because of inclusion of the magnetic relaxation parameters, the unrelaxed critical current, $j=H_{0}r_{p}/\lambda_{C}^{2}$ offers direct access to the superfluid density $n_{s}\propto j_{c}$ [31, 32, 2]. In this work, we use a tunnel diode oscillator (TDO) technique to measure $\lambda_{C}(T,H)$ and determine $j_{c}(T,H)$ in YPtBi. The determined $j_{c}$ is orders of magnitude smaller than that of well-known superconductors with the typical carrier density, and its rapid suppression by the magnetic field provides valuable insight into the fascinating nature of superconductivity at the low carrier density regime exemplified by YPtBi.

YPtBi single crystals were grown out of molten Bi with starting composition Y:Pt:Bi = 1:1:20 (atomic ratio).[33, 34, 9, 15] The starting materials Y ingot (99.5%), Pt powder (99.95%), and Bi chunk (99.999%) were put into an alumina crucible, and the crucible was sealed inside an evacuated quartz ampule. The ampule was heated slowly to 1150°C, kept for 10 hours, and then cooled down to 500°C at a 3°C/hour rate, where the excess of molten Bi was decanted by centrifugation.

The variation of the radio-frequency (rf) magnetic penetration depth $\Delta\lambda_{m}$ was measured in a dilution refrigerator by using a tunnel diode oscillator (TDO) technique [35] (for review, see Ref. [36, 37]).

The sample with dimensions (0.29$\times$0.69$\times$0.24) mm³ positioned with the shortest direction along $H_{ac}$ was mounted on a sapphire rod and inserted into a 2 mm inner diameter copper coil that (when empty) produces rf excitation field with amplitude $H_{ac}\approx 20\>\mathrm{mOe}$ and frequency of $f_{0}\approx 22\>\mathrm{MHz}$. The shift of the resonant frequency (in cgs units), $\Delta f(T)=-G4\pi\chi(T)$, where $\chi(T)$ is the differential magnetic susceptibility, $G\approx f_{0}V_{s}/2V_{c}(1-N)$ is a constant, $N$ is the effective demagnetization factor, $V_{s}$ is the sample volume and $V_{c}$ is the coil volume [36]. The constant $G$ was determined from the full frequency change by physically pulling the sample out of the coil. With the characteristic sample size, $R$, $4\pi\chi=(\lambda_{m}/R)\tanh(R/\lambda_{m})-1$, from which $\Delta\lambda_{m}$ can be obtained [36].

Figure 1(a) shows temperature variation of the rf magnetic penetration depth $\lambda_{m}(T)$ in a single crystal of YPtBi in various applied DC magnetic fields $H_{dc}$ from 0 to 30 kOe (bottom to top). For $H_{dc}=0$, measured $\Delta\lambda_{m}(T)$ is the zero field limiting London penetration depth $\Delta\lambda_{L}(T)$ which exhibits a sharp superconducting phase transition at $T\approx 0.8$ K. We found $\Delta\lambda_{L}(T)=AT^{\alpha}$ where $A=1.98\>\mathrm{\mu m}/\mathrm{K}^{\alpha}$ and $\alpha=1.2$ [15]. The observed exponent $\alpha$ is consistent with the presence of line-nodes in the superconducting gap and moderate impurity scattering. The large pre-factor, $A\propto\lambda_{L}(0)/\Delta(0)$ is compatible with a low carrier density superconductor within London theory $\lambda_{L}(0)=(mc^{2}/4\pi ne^{2})^{1/2}$. For comparison, $A=4$ - 15 Å/K is observed in $d$-wave line-nodal high-temperature cuprate superconductors [38] and 190 - 370 Å/K in CeCoIn₅ [39, 20, 40].

Furthermore, YPtBi exhibits very pronounced field dependence of $\lambda_{m}$. It is notable even at the lowest $H_{dc}=100$ Oe which is 0.007$H_{c2}(0)$. Here we use $H_{c2}(0)=15$ kOe taken from Ref. [9]. By measuring $\lambda_{m}(H,T)$ as a function of temperature in different applied fields we constructed the magnetic field-temperature $H$-$T$ phase diagram of YPtBi. Due to the broadness of the superconducting transition, we used three different criteria for the determination of $T_{c}$, as illustrated in the inset of Fig. 1(b). $T_{1}$ was determined at the sharp maximum of $d\lambda_{m}/dT$ (black squares). $T_{2}$ was determined at the intersection of the lines through the data in the superconducting state and the normal states (blue circles). $T_{3}$ was determined at the onset of $\Delta\lambda_{m}(T)$ deviation from the normal state behavior (red up triangles). The phase diagram from rf magnetic penetration depth data is shown in the main panel of Fig. 1(b). For reference, we show the diagram as determined from resistivity measurements by Butch et al. [9], using zero resistivity (black solid line), crossing point of linear extrapolations (blue dashes) and onset of deviation (red dash-dot) criteria. The phase diagram as determined from maximum of $d\lambda_{m}/dT$ line is closely following the phase diagram as determined from resistivity measurements using $\rho=0$ [9]. However, we detected apparent diamagnetic response in YPtBi at notably higher fields than in resistivity measurements, suggesting persistence of superconductivity in some non-bulk form [41]. A discrepancy between $H_{c2}$ as determined from bulk thermal conductivity and non-bulk resistivity measurements is a known problem [42] and it is usually assigned to the superconducting layer surviving at the surface. Perhaps, it is related to the third critical field in superconductors predicted theoretically by Saint-James and Gennes [43] when a thin superconducting layer is formed on the flat surface parallel to the field. Recently, a surface-sensitive tunneling experiment on YPtBi detected energy gap spectra at higher temperatures than $T_{c}\approx 0.8$ K [44]. A similar signature of the superconducting phase was reported in another half-Heusler compound LuPtBi, which was attributed to the presence of van Hove singularity near $E_{F}$ [45] and surface pairing states [46]. The shape of tunneling spectra in the superconducting state is inconsistent with an isotropic $s$-wave in both YPtBi and LuPtBi [44, 45].

We focus now on the variation of $\lambda_{m}$ with finite $H_{dc}$ in the mixed state. The measured magnetic penetration depth satisfies the relation $\lambda_{m}^{2}=\lambda_{L}^{2}+\lambda_{C}^{2}$ in the approximation of a linear elastic response of a vortex lattice to a small amplitude AC perturbation $H_{ac}$ [21, 22, 23]. The TDO technique is a perfect probe for this measurements because of small frequency, $f_{0}=22$ MHz, and small amplitude of the perturbation, $H_{ac}=20$ mOe. Since $\lambda_{L}(T)=\lambda_{L}(0)+\Delta\lambda_{L}(T)$ where $\lambda(0)=1.6~{}\mu$m [19], we can readily calculate $\lambda_{C}(T)$ in various $H_{dc}$. However, we took a conservative approach, assuming that this approximation is valid only at temperatures below $0.5T_{1}$ because $\lambda_{m}(T)$ becomes comparable to the size of the sample as temperature increases towards $T_{c}$. The calculated $\lambda_{C}=\sqrt{\lambda_{m}^{2}-\lambda_{L}^{2}}$ at various $H_{dc}$ is shown in Fig. 2(a). In all measured $H_{dc}$, $\lambda_{C}(T)$ shows monotonic increase with temperature as the superconductor allows more penetration of the rf field with increasing temperature.

Figure 2(b) shows the field-dependent $\lambda_{C}^{2}(H)$ at several temperatures. When critical current density does not vary much with field, we expect $\lambda_{C}^{2}(H)\sim H$ and it has been observed in most cases [47, 48, 41]. However, YPtBi exhibits significantly more curved $\lambda_{C}^{2}(H)$, indicating nearly logarithmic behavior at low fields. This rapid rise of $\lambda_{m}(H)$ at low fields is unusual but may be explained considering very large values of $\lambda$ leading to strong intervortex interaction due to significant overlap already in low fields. In Figure 3, we compare this anomalous $\lambda_{C}(H)$ in YPtBi with SrPd₂Ge₂ which has a normal carrier density and exhibits $H$-linear behavior of $\lambda_{C}^{2}(H)$.

As noted above from the known $\lambda_{C}(T,H)$, one can evaluate critical current density $j_{c}$ via $j_{c}=H_{dc}r_{p}/\lambda_{C}^{2}$ (see appendix for details) where $r_{p}$ is a characteristic radius of the pinning potential, usually taken equal to the coherence length, $r_{p}(T)=\xi(T)$ [21, 22, 47, 48, 41]. Figure 4 shows the calculated $j_{c}(T)$ in YPtBi, obtained from $\lambda_{C}(T)$ measurements taken in minimum applied $H_{dc}=100$ Oe. We compare $j_{c}(T)$ in YPtBi to the $j_{c}$’s determined in similar Campbell penetration depth measurements in some representative superconductors, LiFeAs [49] and SrPd₂Ge₂ [41]. The former is known as a two-band superconductor with full gaps [50], and the latter is a single-gap BCS superconductor [41]. The compared $j_{c}(T,H)$ in these superconductors were obtained by using the same TDO technique. In particular, we used the same TDO setup for YPtBi and SrPd₂Ge₂.

In YPtBi, the highest $j_{c}(T)\approx 40$ A/cm² is observed at $T\approx 75$ mK, and $j_{c}(T)$ monotonically decreases with temperature. In LiFeAs, $j_{c}\approx 1\times 10^{6}$ A/cm² at the lowest temperature, and it monotonically decreases by two orders of magnitude upon warming. In SrPd₂Ge₂, $j_{c}\approx 8.3\times 10^{4}$ A/cm² at the lowest temperature with $H_{dc}=200$ Oe. However, its temperature variation is non-monotonic and exhibits a maximum at an intermediate temperature, which was attributed to a matching effect between temperature-dependent coherence length and relevant pinning length scale [41]. At a higher $H_{dc}=4$ kOe, $j_{c}(T)$ recovers a monotonic decrease with increasing temperature. Even when $T_{sc}(H)$ of SrPd₂Ge₂ was reduced to 0.86 K in $H_{dc}=$4 kOe which is close to $T_{sc}$ of YPtBi with 100 Oe, $j_{c}$ is still two orders of magnitude greater, and the difference gets even bigger at base temperature. It is also instructive to calculate the depairing current density at which Cooper pairs break apart reaching critical velocity, $4\pi c^{-1}j=\phi_{0}/\left(3\sqrt{3}\lambda_{L}^{2}\xi\right)\approx 1\times 10^{7}\>\mathrm{A/}\mathrm{cm^{2}}$, which is much larger than $j_{c}$ due to pinning, but two orders of magnitude less than in “typical” normal carrier density superconductors [28, 30].

In Table 1, we compare normal state Hall constants $R_{H}$ reported for YPtBi  [9] and SrPd₂Ge₂ [51]. In both compounds the Hall resistivity $\rho_{xy}(H)$ is field-linear, which enables $R_{H}$ definition from the slope of the curve and sample geometry. The reported $R_{H}$ values are $-1.6\times 10^{-4}$ cm³/C [51] and $+2.4$ cm³/C [9] for SrPd₂Ge₂ and YPtBi, respectively. In the single band Drude model, the carrier density satisfies a simple relation, $R_{H}=1/ne$ where $e$ is the electron charge. The calculated carrier densities are electron-like $n_{e}=3.9\times 10^{22}$ cm^-3 for SrPd₂Ge₂ and hole-like $n_{h}=2.6\times 10^{18}$ cm^-3 for YPtBi. For reference we also present $j_{c}$ in both SrPd₂Ge₂ and YPtBi in Tab. 1.

The carrier density $n$ is responsible for the observed theoretical current density because $j_{c}\propto n_{s}v_{s}$ where $v_{s}$ is the velocity of the supercurrent [32, 31, 2]. Provided $n_{s}\approx n/2$, $j_{c}$ in YPtBi is two orders of magnitude smaller than that in SrPd₂Ge₂, while the normal state $n$ in YPtBi is smaller by four orders of magnitude. Consequently, $v_{s}$ in YPtBi is two orders of magnitude greater than that in SrPd₂Ge₂. In Ginzburg-Landau theory, $j_{c}$ is associated with the depairing velocity $v_{s}=\Delta(0)/\hbar k_{F}$, and since $\Delta(0)$ values in both superconductors are of the same order, the different $v_{s}$ is accounted by different $k_{F}$ values in these two compounds. In SrPd₂Ge₂, $k_{F}\approx 10$ nm^-1 in free electron approximation, i.e., $k_{F}=(3\pi^{2}n)^{1/3}$ with $n=2.6\times 10^{22}$ cm^-3 which is about two orders of magnitude greater than that in YPtBi, $k_{F}=0.37$ nm^-1. [15]

With the confirmed low density of the Cooper pairs in the superconducting YPtBi, we can expect $E_{F}\sim n^{3/2}$ to be respectively low. In combination with the anomalously high $T_{c}$, not accounted for by the McMillan formula [12], this leads to a large value of the $T_{c}/E_{F}$ ratio and a possibility of BCS-BEC crossover in YPtBi.

Superfluidity without Cooper pairing, i.e. BEC, can be achieved when the de Broglie wavelength of a particle becomes larger than the inter-particle distance, causing strong correlations. The possibility of BCS-BEC crossover was discussed in the low carrier density systems in which the effective size of the conduction electron is comparable to that of the Cooper pair. The mean spacing between conduction electrons, $r_{s}$, can be described with $k_{F}$ by the relation, $r_{s}\approx 7.1/k_{F}$, for the parabolic band. In YPtBi, $r_{s}\approx 19$ nm while the zero temperature coherence length is only $\xi(0)=15$ nm.

In the BCS-BEC crossover regime, the overlap between Cooper pairs is small or $\xi(0)\sim 1/k_{F}$. Many superconductors have been tested for the conditions for BEC, and recently the multigap superconductor FeSe was found to be near the regime, in which $1/k_{F}\xi(0)\approx 0.26$-$0.67$ [13]. For YPtBi, $k_{F}\approx 0.4$ nm^-1 [15] and $\xi(0)=15$ nm [9], which makes $1/k_{F}\xi(0)\approx 0.17$, and the ratio $T_{c}/T_{F}\approx 2\times 10^{-3}$ from $T_{c}\approx 0.8$ K and $k_{B}T_{F}$ = 35 meV. Several low carrier density superconductors are compared to the well-known superconductors in the summary plot of $T_{c}/E_{F}$ vs. $1/k_{F}\xi(0)$ in Fig. 5. We find that both YPtBi (red star) and SrTiO₃ are relatively close to the crossover regime whereas Bi is well in the BCS limit. Tuning chemical potential in YPtBi and SrTiO₃ by gating or charge doping, particularly towards smaller $E_{F}$, would be uppermost interesting for understanding their pairing mechanism.

There has been much effort to elucidate the unconventional superconductivity in the low carrier density superconductors including YPtBi and SrTiO₃. Recently, the unexpectedly high $T_{c}$ in YPtBi was explained by the electron-phonon pairing mechanism with polar optical phonon mode within the $j=3/2$ Luttinger-Kohn four-band model [52]. In the similar low carrier density superconductors, the plasmonic [53] and nonadiabatic [54] superconducting mechanisms were proposed in SrTiO₃.

The structure of the superconducting energy gap and the symmetry of pairing interaction are prerequisites for understanding the superconducting mechanism, but low $T_{c}$ in the low $n$ superconductors makes the experimental investigation difficult. The half-Heusler compounds RTBi (R=Y,La,Lu, T=Pt,Pd) exhibit relatively high superconducting transition temperatures $T_{c}\sim 1$ K[9, 10, 11], and a nodal superconducting gap was observed in YPtBi [15]. Subsequently, various exotic pairing symmetries were proposed including nematic $d$-wave [55, 56] and $j=3/2$ high-spin superconductivity  [16, 17, 15, 57, 46, 58]. In general, the high-spin superconductivity exhibits topological gap structures with the possibility of harboring the Majorana surface fluid [59, 46], which makes the low carrier density superconductor RTBi a promising platform for the fault-tolerant quantum devices.

We measured rf superconducting magnetic penetration depth in single-crystal YPtBi. The London penetration depth is consistent with the nodal superconductivity in YPtBi. In the finite DC magnetic fields, the measured Campbell penetration depth exhibits unusual sub-quadratic power-law behavior in the low field range. From the variation of the Campbell penetration depth, we estimated the theoretical critical current density which is orders of magnitude smaller than that of the superconductors with a typical carrier density. Therefore, we confirmed the low carrier density nature of superconductivity in YPtBi.

We would like to thank V. G. Kogan for the useful discussion. Work in Ames was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Ames Laboratory is operated for the U.S. DOE by Iowa State University under contract DE-AC02-07CH11358. Research done at the University of Maryland was supported by the U.S. Department of Energy (DOE) Award No. DE-SC-0019154 (experimental investigations), and the Gordon and Betty Moore Foundation EPiQS Initiative through Grant No. GBMF4419 (materials synthesis).