On the lineshapes of temperature-dependent transport measurements of superconductors under pressure

Transport measurements at high pressures are typically conducted using a four-probe geometry, sometimes referred to as a van der Pauw (VDP) configuration. It is important to note that while a VDP geometry is often stated for these experiments, this does not imply the VDP algorithm van_der_pauw_method_1958 (34) was used to determine the sheet resistivity of a sample unless specifically stated. The majority of the measurements exhibiting unusual features were not performed using the VDP method, but instead utilized techniques based on AC bridge circuitry snider_synthesis_2021 (6, 35) or a boxcar-averaged DC circuit drozdov_conventional_2015 (1, 3) (e.g., Quantum Design PPMS options noauthor_physical_nodate (36)). Other studies reported experiments using lock-in-based techniques with circuitry similar to that shown in Fig. 2 somayazulu_evidence_2019 (2, 37, 35). As all of the aforementioned techniques rely on periodic averaging, they are inherently susceptible to dynamic effects arising from sample inductance and capacitance. The relatively small size and granular nature of hydride superconductors often cause the typically negligible frequency-dependent contribution to the measured signals to approach the magnitude of the resistive contribution troyan_high-temperature_2022 (38, 39).

In a lock-in measurement, current-regulated sinusoidal signals of the form

is injected into the sample through two points (I and II) by a regulated current source (Fig. 2). A differential voltage of the form

is then measured across the other two contacts (III and IV), buffered via an active pre-amplifier, and then measured by a lock-in amplifier. Before taking any measurements, it must be ensured that excitation frequencies are within the pass band of all circuitry in the setup.

Typically, effective in-phase and quadrature resistances are then defined in terms of $\mathit{X(t)}$ and $\mathit{Y(t)}$ using Ohms law as

and

respectively. A complex impedance,

is then used to estimate $\mathit{R_{DC}}$. For ohmic materials $\mathit{R_{ip}(t,\omega)}>>\mathit{R_{qd}(t,\omega)}$, so the apparent resistance is estimated as $\mathit{R_{apparent}(t,\omega)}=|\mathit{Z(t,\omega)}|\approx\mathit{R_{DC}(t)}$ for a wide range of conditions. While the voltage across ideal ohmic materials is expected to remain in phase with the excitation current, all real materials exhibit dynamic electromagnetic effects with at least a diamagnetic effect present jackson_john_2015 (41). For non-ohmic materials, such as superconductors, the actual $I$-$V$ relation may have a significant $\mathit{\omega}$ dependence and this approximation is invalid kaiser_electromagnetic_2004 (42, 43).

The presence of a finite offset from zero resistance in the $\mathit{R_{apparent}}$ of samples reported to be superconducting measured with AC techniques has been the subject of frequent discussion (e.g., Refs. hamlin_vector_nodate (25, 30)). Here it is important to note that the use of $|\mathit{V(t)}|$, being a vector magnitude, to estimate $\mathit{R_{DC}}$ will result in a scalar voltage offset. While lock-in amplifiers can extract extremely small signals from background noise, there is still a finite noise floor in any real measurement. For extremely low voltage signals, e.g., zero voltage drop in a superconductor below $\mathit{T_{c}}$, the signal is guaranteed to be below the lock-in noise floor of the amplifiers. It is naively expected that using Ohm’s law to calculate the temperature dependence of the DC resistance ($\mathit{R_{DC}(T)}$) based on $|\mathit{V(T,\omega)}|$ will result in $\mathit{R_{apparent}(T<T_{c})}\approx 0$ with environmental and instrumental noise causing individual readings to symmetrically fluctuate around $0$. This discrepancy is addressed by recognizing that the lock-in amplifier only measures $\mathit{X(T)}$ and $\mathit{Y(T)}$, with both $|\mathit{V(T)}|$ and $\theta(\mathit{T})$ being calculated in hardware. $|\mathit{V(T,\omega)}|$, being a positive definite phasor magnitude, is then guaranteed to produce a positive resistance with a scalar offset proportional to the total SNR. For well-isolated experiments the raw $\mathit{X(T)}$ and $\mathit{Y(T)}$ signals are often observed to fluctuate around zero resistance even without accounting for any possible phase ambiguity (see Supplementary Information of Ref. salke_evidence_nodate (20)).

Modeling $\mathit{\mathcal{H}}$ of a system undergoing a complex electronic phase change, such as a granular superconductor, is achieved using passive electrical components (Fig. 3).

To capture the dynamics inherent to the AC measurement we choose to model the system using effective resistances, capacitances, and inductances representative of the normal state bulk properties. Previous models disregarded these dynamic effects and considered only resistive networks raven_filamentary_1995 (45, 46). In this model the effective inductance takes the form $\mathit{L_{eff}=L_{self}+L_{tr}}$ with $\mathit{L_{self}}$ being the classical self-inductance and $\mathit{L_{tr}}$ being a transient inductance present at magnetic phase changes. $\mathit{L_{self}}$ is typically highly nonlinear with closed-form solutions only existing for relatively simple boundary conditions (e.g., a long thin conducting wire carrying a sinusoidal current rosa_self_1908 (47)). A finite self-capacitance ($C_{s}$) can arise from sample granularity, where adjacent grains are capacitively shunted to each other sang_interfacial_2004 (48, 49, 50, 51). In general, $\mathit{R_{DC}}$, $\mathit{L_{eff}}$, and $\mathit{C_{s}}$ are functions of extrinsic variables, e.g., $\mathit{T}$, $\mathit{P}$, $\mathit{\omega}$, sample size, and sample granularity.

The admittance of the circuit (Fig. 3) is then

which may be inverted to produce the complex transfer function (Eq. 9). {strip}   

$\Re\{\mathcal{H}\mathit{(t,\omega)}\}$ is the system in-phase steady-state response, which is proportional to $\mathit{X(t,\omega)}$ up to an ambiguous phase offset. $\Im\{\mathcal{H}\mathit{(t,\omega)}\}$ is the steady-state
quadrature response which is similarly proportional to $\mathit{Y(t,\omega)}$ up to a phase steinmetz_reactance_1894 (52). The phase ambiguity may be accounted for by recognizing that lock-in amplifiers inherently take differential measurements on two orthogonal quantities, so $\mathit{Y}(\mathit{t}=0)=0$ may be manually set through the use of a null detector to record only the phase change throughout the experiment. The signal magnitude and phase shift are then given by

and

We first consider the case of $\mathit{R_{apparent}}$ deviating from $\mathit{R_{DC}}$ for different resistive loads in this model with constant $\mathit{L}=1\ \mathrm{mH}$ and $\mathit{C}=1\ \mathrm{nF}$, values similar to those measured in DAC experiments han_electrostrictive_2021 (53) (Fig. 4a) where for $\mathit{R_{DC}}<0.5\ \mathrm{m\Omega}$ low excitation frequencies cause $\mathit{R_{apparent}}$ to accurately reproduce the true DC resistance. Although lower frequencies are less susceptible to resonant effects they require significantly larger integration times to obtain reasonable SNR so in practice higher frequencies are typically employed. As the load and/or frequency is increased electrically reactive effects become significant and $\mathit{R_{apparent}}$ begins to diverge from $\mathit{R_{DC}}$ due to a small resonance near $\mathit{\omega_{R1}}\sim 10$, and a larger resonance above $\mathit{\omega_{R2}}\sim 10000$. Corresponding phase shifts also indicate large load-dependent effects (Fig. 4b). For $\mathit{\omega_{R1}<\omega<\omega_{R2}}$, $\mathit{R_{apparent}}$ will approximate the true $\mathit{R_{DC}}$. Above $\mathit{\omega_{R2}}$ the dynamic effects may cause the measured resistance to saturate the lock-in amplifier with the signal exceeding the dynamic range of the instrument, as discussed in Ref. somayazulu_evidence_2019 (2) (e.g., Fig. 4). For many experiments, $\omega_{R2}$ is above the pass-band frequency of the lock-in and is not observed. We note that if a sample undergoes a significant change in electronic properties, such as a superconducting transition, as a function of temperature, the measurement will naturally sweep through one or more of these nonlinear resonances. During the transition, $\mathit{R_{apparent}}$ is expected to diverge from $\mathit{R_{DC}}$, particularly near $\mathit{T_{c}}$.

We now model the resistance as a function of temperature and pressure with $\mathit{R_{DC}(T,P)}$, $\mathit{L_{eff}(T,P)}$ and $\mathit{C_{s}}$ values assumed based on previously reported measurements of superconducting transitions under pressure salke_evidence_nodate (20) and ambient pressure measurements of Sn-whiskers miller_fluctuation_1973 (54), $\mathrm{2H}$-$\mathrm{NbSe_{2}}$ perconte_low-frequency_2020 (55), and $\mathrm{Bi_{2}Sr_{2}Ca_{2}Cu_{3}O_{8+\delta}}$ (Bi-2223) (see Supplemental Information and Ref. mark_notitle_nodate (56)). A $\mathit{R_{DC}}$-$\mathit{T}$ relation of the form

is assumed, where $\Gamma$ is a constant broadening factor and $\mathit{R_{DC}(T)}$ is the temperature dependence of the resistance above $\mathit{T_{c}}$.

Inductive effects are magnetic and well described by Lenz’s law in which any change in magnetic flux ($\Phi_{\mathit{B}}$) generates a proportional back electromotive force ($\mathcal{E}_{\mathit{Back}}$) determined by the sample’s effective inductance $(\mathit{L_{eff}}$) graf_modern_1999 (57). For a sinusoidal signal, Lenz’s law implies

with $\mathcal{E}_{Back}$ generating a voltage $90^{o}$ out of phase from the driving current. $\mathit{L_{tr}}$ then arises from the sudden repulsion of $\Phi_{\mathit{B}}$ within a superconductor at $\mathit{T_{c}}$ due to the Meissner effect. $\mathit{L_{tr}}$ is suppressed below $\mathit{T_{c}}$ due to the superconductor’s inability to sustain a voltage drop. This $\mathit{L_{tr}}$ is unrelated to the kinetic inductance that has been observed in high-frequency supercurrents meservey_measurements_2003 (58, 59) which are typically several orders of magnitude smaller at the frequencies used in these experiments and only exist well below $\mathit{T_{c}}$ vodolazov_nonlinear_2023 (60, 61, 62). Similar transient low-frequency impedances observed in 2H-$\mathrm{NbSe_{2}}$ perconte_low-frequency_2020 (55) were attributed to a coupling of the superconducting vortex state to the heat capacity. Whereas coupling to the heat capacity may cause a phase shift near $\mathit{T_{c}}$, it does not explain the observed frequency-dependent deviation of $\mathit{X(T)}$ from $\mathit{R_{DC}(T)}$ or the high-frequency behavior where $\mathit{R_{apparent}}$-$\mathit{T}$ can approach a step function in some measurements somayazulu_evidence_2019 (2). We cannot rule out variations (e.g., peaks) in $R_{DC}(T)$ near and above $T_{c}$ arising from sample or phase inhomogeneity (e.g., Ref. zhang_bosonic_2016 (63)), but effects of AC measurement techniques can introduce features in this regime as well.

In our model, the total effective inductance is described by a growing and decaying exponential surrounding $\mathit{T_{c}}$ modulated with a step function to account for the finite $\mathit{L_{self}}$ above $\mathit{T_{c}}$. Model $\mathit{R(T)}$ and $\mathit{L_{eff}(T)}$ relations that exhibit this effect are presented in Fig. 5. In inhomogeneous materials, including powders and polycrystals, a universal dielectric response causes an effective capacitance ($\mathit{C_{s}}$) between grain boundaries jonscher_universal_1977 (64, 65, 66, 67, 68). Eq. 9 implies that lock-in based techniques will produce a $Y(T,\omega)$ that scales with $C_{s}$. Similar capacitive couplings between grains have been observed in ambient pressure Josephson junctions cheng_anomalous_2016 (69, 70) and in chains of capacitively coupled granular superconductors ilin_superconducting_2020 (71). In addition to the sample’s inherent $\mathit{C_{s}}$, metal film contacts prepared similarly to those used in high-pressure experiments have been reported to exhibit very large contact capacitances that scale with pressure wang_stress-dependent_2021 (72), with some reported to exceed 1 F dervos_effect_1998 (73). In DAC experiments the collective capacitance of samples can be large ($C>1\ \mathrm{\mu}$F) and significantly impact low-frequency measurements when voltage signals are low he_situ_2007 (74, 75).

We now consider the frequency dependence of these effects using the range of frequencies employed in these measurements (17 Hz to 10 kHz salke_evidence_nodate (20, 2)). Significant effects have been reported in both low frequency perconte_low-frequency_2020 (55, 20) and high frequency somayazulu_evidence_2019 (2) experiments. These features can provide additional information about the material in both the superconducting and normal states. As $\mathit{C_{s}}$ is relatively insensitive to temperature or frequency, it is assumed to be constant for each pressure. Model signal curves for a superconducting transition using various frequencies are presented in Fig. 6. The calculated $\mathit{R_{apparent}(T)}$ in the simulated curve does not approximate $\mathit{R_{DC}}$ for all frequencies, especially for small samples where voltages from dynamic effects approach the resistive voltage drops. As $\mathit{\omega}\rightarrow\mathit{\omega_{R1}}$ a peak emerges in $\mathit{R_{apparent}}$ that distorts the transition (Fig. 6). While the signal is influenced by $\mathit{\omega_{R1}}$, $\mathit{R_{apparent}}$ overestimates $\mathit{R_{DC}}$ above $\mathit{T_{c}}$ and is seen to sharpen the transition. As $\mathit{\omega}$ is further increased, the influence of $\mathit{\omega_{R2}}$ causes the voltage signal to grow exponentially, often saturating the measurement equipment resulting in the $\mathit{R}$-$\mathit{T}$ relation approaching a step function, although this effect is not always observed due to $\mathit{\omega_{R2}}$ often being outside the range of the lock-in amplifier. Similar peaks have been reported in the resistivity of thin disordered superconductors measured using an AC bridge (e.g., Sn/In ems_resistance_1971 (77) and TiN postolova_reentrant_2017 (78)).

Additional features in $R$-$T$ curves are present if the reactive contribution from $C_{s}$ is significant (e.g. Fig. 1b somayazulu_evidence_2019 (2)). $C_{s}(P)$ is a function of pressure, with typical values ranging from 1 pF to 1 $\mathrm{\mu}$F wei_dielectric_2008 (49). This variability can cause lineshapes to be distorted under pressure (Fig. 7). For larger values of $C_{s}$ and higher $\mathit{\omega}$, the system may sweep through $\mathit{\omega_{R2}}$ during the superconducting transition, producing large $X(T,\omega)$ and $Y(T,\omega)$ signals that saturate the lock-in. While significant enough to impact the measurement, the magnitude of the large resonant effects at frequencies near $\mathit{\omega_{R2}}$ is non-physical due to significant non-linear effects not accounted for in this model.