Superconductivity under pressure in the Dirac semimetal PdTe2

TL;DR

This study investigates how applying pressure up to 2.5 GPa affects the superconducting properties of the Dirac semimetal PdTe2, revealing non-monotonous changes in critical temperature and robust surface superconductivity linked to topological states.

Contribution

It provides the first detailed high-pressure phase diagram of PdTe2's superconductivity, highlighting the enhancement of surface superconductivity and potential topological effects under pressure.

Findings

Superconducting transition temperature peaks at 0.91 GPa.

Surface superconductivity persists beyond bulk critical fields.

Surface Tc exceeds bulk Tc at pressures above 1.41 GPa.

Abstract

The Dirac semimetal PdTe$_2$ was recently reported to be a type-I superconductor ($T_c = $1.64 K, $\mu_0 H_c (0) = 13.6$ mT) with unusual superconductivity of the surface sheath. We here report a high-pressure study, $p \leq 2.5$ GPa, of the superconducting phase diagram extracted from ac-susceptibility and transport measurements on single crystalline samples. $T_c (p)$ shows a pronounced non-monotonous variation with a maximum $T_c = $1.91 K around 0.91 GPa, followed by a gradual decrease to 1.27 K at 2.5 GPa. The critical field of bulk superconductivity in the limit $T \rightarrow 0$, $H_c(0,p)$, follows a similar trend and consequently the $H_c(T,p)$-curves under pressure collapse on a single curve: $H_c(T,p)=H_c(0,p)[1-(T/T_c(p))^2]$. Surface superconductivity is robust under pressure as demonstrated by the large superconducting screening signal that persists for applied dc-fields $H_a > H_c$. Surprisingly, for $p \geq 1.41$ GPa the superconducting transition temperature at the surface $T_c^S$ is larger than $T_c$ of the bulk. Therefore surface superconductivity may possibly have a non-trivial nature and is connected to the topological surface states detected by ARPES. We compare the measured pressure variation of $T_c$ with recent results from band structure calculations and discuss the importance of a Van Hove singularity.

Full text

I Introduction

The family of layered transition metal dichalcogenides attracts much attention, because of the wide diversity of fascinating electronic properties. One of the present-day research interests is the possibility to realize novel quantum states as a result of the topological non-trivial nature of the electronic band structure Soluyanov et al. (2015); Huang et al. (2016); Yan et al. (2017); Bahramy et al. (2018). Especially, it has been proposed that these materials host a generic coexistence of type-I and type-II three dimensional Dirac fermion states Bahramy et al. (2018). An interesting example in this respect is PdTe2 that has been classified as a type-II Dirac semimetal following a concerted examination of ab-initio electronic structure calculations and angle-resolved photoemission spectroscopy (ARPES) experiments Yan et al. (2015a); Fei et al. (2017); Noh et al. (2017); Bahramy et al. (2018); Clark et al. (2018). In a type-II Dirac semimetal the Hamiltonian breaks Lorentz invariance because the energy dispersion relations, i.e. the Dirac cone, are tilted Soluyanov et al. (2015). The Dirac point is then the touching point between the electron and hole pockets and a nearly flat band may form near the Fermi level. Moreover, PdTe2 is a superconductor below 1.6 K Guggenheim et al. (1961); Leng et al. (2017), which solicits the intriguing question whether superconductivity is promoted by the nearly flat band and consequently has a topological nature Fei et al. (2017). Topological non-trivial superconductors attract much interest since it is predicted these may host protected Majorana zero modes at the surface (for recent reviews see Refs. Sato and Fujimoto, 2016; Sato and Ando, 2017). This in turn offers a unique design route to make devices for topological quantum computation.

Superconductivity in PdTe2 was discovered in 1961 (Ref. Guggenheim et al., 1961), but was not investigated in detail until 2017, when Leng et al. Leng et al. (2017) carried out comprehensive magnetic and transport experiments on single-crystals. Unexpectedly, dc-magnetization measurements, $M(H)$, revealed that PdTe2 is a bulk type-I superconductor, which was further embodied by the observation of the differential paramagnetic effect in the ac-susceptibility measured in applied magnetic dc-fields. The critical field $H_{c}(T)$ follows the standard quadratic temperature variation with $\mu_{0}H_{c}(0)=13.6$ mT. The possibility of type-I superconductivity in Dirac materials was recently investigated by Shapiro et al. Shapiro et al. (2018) employing a microscopic pairing theory for an arbitrary tilt parameter of the Dirac cone. For PdTe2 these authors concluded type-I superconductivity is feasible for a tilt parameter $k=2$. Another interesting aspect of PdTe2 is the observation of surface superconductivity, as evidenced by large screening currents in the ac-susceptibility for applied dc-fields $H_{a}>H_{c}$ Leng et al. (2017). The critical field for surface superconductivity $H_{c}^{S}$ does not follow the standard Saint-James - de Gennes expression $H_{c3}=2.39\times\kappa H_{c}$ Saint-James and de Gennes (1963), where $\kappa$ is the Ginzburg-Landau parameter. This led to the proposal Leng et al. (2017) that superconductivity of the surface sheath might have a topological nature and originates from topological surface states detected by ARPES Yan et al. (2015a); Noh et al. (2017). More recently, specific heat Amit and Singh (2018) and magnetic penetration depth Salis et al. (2018); Teknowijoyo et al. (2018), measurements have been conducted. These confirm conventional weak-coupling Bardeen-Cooper-Schrieffer (BCS) superconductivity, with a full gap in the bulk. At the same time zero-field scanning tunneling microscopy (STM) and spectroscopy (STS) experiments Das et al. (2018); Clark et al. (2018) lend further support for the absence of in-gap states, which seems to rule out topological superconductivity at the surface. Dominant $s$-wave superconductivity was also concluded from tunneling spectroscopy experiments on side junctions Voerman et al. (2019). Nonetheless, the uncommon type-I behavior for a binary compound, and the unexplained superconductivity of the surface sheath, justify a further in-depth examination of the superconducting properties of PdTe2.

We here report the results of a high-pressure investigation of the superconducting phase diagram of PdTe2 single crystals ($p\leq 2.5$ GPa). Combined resistivity and ac-susceptibility measurements show $T_{c}$ increases at low pressures, then passes through a maximum of 1.91 K around 0.91 GPa, and subsequently decreases at higher pressure. The critical field for $T\rightarrow 0$, $H_{c}(0,p)$, follows a similar behavior and consequently the $H_{c}(T)$-curves at different pressures collapse on a single curve. Under pressure superconductivity maintains its type-I character. Surface superconductivity is robust under pressure as demonstrated by the large superconducting screening signal that persists for applied dc-fields $H_{a}>H_{c}$. Surprisingly, for $p\geq 1.41$ GPa the superconducting transition temperature of the surface, $T_{c}^{S}$, is larger than $T_{c}$ of the bulk. Therefore surface superconductivity may possibly have a non-trivial nature and is related to the topological surface states detected by ARPESYan et al. (2015a); Noh et al. (2017); Clark et al. (2018). The initial increase of $T_{c}$ with pressure is at variance with the smooth depression predicted by recent electronic structure calculations Xiao et al. (2017).

II Experiment

The crystals used for our high pressure study were taken from the single-crystalline boule prepared by the modified Bridgman technique Lyons et al. (1976) and characterized in Ref. Leng et al., 2017. Powder X-ray diffraction confirmed the trigonal CdI2 structure (spacegroup $P\bar{3}m1$ Thomassen (1929). Scanning electron microscopy (SEM) with energy dispersive x-ray (EDX) spectroscopy showed the proper 1:2 stoichiometry within the experimental resolution of 0.5%. Laue backscattering was used to orient the crystals. Standard four-point resistance measurements were performed in a Physical Property Measurement System (PPMS, Quantum Design) at temperatures down to 2 K. The resistivity, $\rho(T)$, of our crystals shows metallic behavior. A typical trace in the temperature range 2-300 K is shown in Fig. 1. The residual resistance ratio $R$(300K)/$R$(2K) = 30.

Electrical resistance, $R(T,H)$, and ac-susceptibility, $\chi_{ac}(T,H)$, measurements under high-pressure were performed utilizing a clamp-type piston-cylinder cell, which has a double-layer made of Cu-Be and NiCrAl alloys. The single crystal sizes for $R(T,H)$ and $\chi_{ac}(T,H)$ were $\sim 2.3\times 1.0\times 0.18$ mm3 and $\sim 2.9\times 1.0\times 0.67$ mm3, respectively. Both samples were mounted on a plug and loaded into a Teflon capsule together with coils and a pressure-transmitting medium, Daphne oil 7373, for hydrostatic compression. A schematic drawing of the plug with samples and coil is shown in Fig. 1. The generated pressure in the capsule relating to each load was estimated from the calibration data for this cell, which was obtained from the pressure variations of superconducting transition temperatures of lead and tin in previous experiments Slooten et al. (2009); Bay et al. (2012a). We carried out the compression experiments on the crystals twice, first up to a pressure of 1.24 GPa and in a second run up to 2.49 GPa.

Typical experimental conditions are as follows. The high-pressure cell was compressed at room temperature and then cooled down to about 0.3 K using a 3He refrigerator (Oxford Instruments Heliox VL). Electrical resistivity was measured by a resistance bridge (Linear Research LR-700) using a low-frequency ac method with an excitation current $I=300\mu$A. In order to investigate the field-suppression of $T_{c}$, a magnetic field was applied along the current, parallel to the $a$-axis. For ac-susceptibility measurements, a small cylinder, composed of an excitation coil and a pick-up coil in which the sample is situated, was prepared. The in-phase and out-of-phase signals were detected in the driving field $\mu_{0}H_{ac}=0.0047$ mT with a frequency of $f_{ac}=313$ Hz using a lock-in amplifier (EG&G Instruments Model 7260). Measurements were made in zero field and in applied dc-fields using a superconducting magnet. Special care was taken to reduce the remnant field of the superconducting magnet to close to zero, since our PdTe2 crystals show type-I superconductivity.

Overall the resistivity, $\rho(T)$, measured in the temperature range 2-300 K showed little variation with pressure and remained metallic. However, the absolute $\rho$-value at 300 K decreases smoothly with respect to pressure to about 80% of the ambient pressure value at the highest pressure 2.49 GPa.

III Results

III.1 Pressure-temperature phase diagram

The overall results of the two pressure runs are reported in Fig. 2. In the first run data were taken at pressures of 0.25, 0.58, 0.91 and 1.24 GPa. Here the normal state resistance $R_{N}\simeq 70~{}\mu\Omega$. For the second run new voltage contacts were made on the crystal resulting in $R_{N}\simeq 60~{}\mu\Omega$. The applied pressures are 0.75, 1.08, 1.41, 1.74, 2.07 and 2.49 GPa. We remark the zero-pressure data were measured after releasing the pressure. Also, the value of the ac-susceptibility differed somewhat between different cool downs and between the two pressure runs. For clarity all the $\chi_{ac}$ data in the lower panel of Fig. 2 are normalized to $-1$ in the superconducting state.

The resistance curves around $T_{c}$ at ambient pressure and $p=0.25$ GPa show a double structure which becomes more pronounced with increasing pressure. However, for $p\geq 1.08$ GPa the superconducting transition is sharp. We attribute the double structure in $R(T)$ at low pressures to parts of the crystal responding differently to pressure, because of an inhomogeneity, rather than to a pressure gradient. We remark that previous resistance experiments on crystals taken from the same single-crystalline boule revealed a single sharp superconducting transition at ambient pressure Leng et al. (2017). A similar behavior is observed in $\chi_{ac}(T)$ with relatively sharp, single transitions at pressures of 1.08 and 1.24 GPa. However, for $p\geq 1.41$ GPa the transition becomes structured again with an onset temperature of superconductivity larger than $T_{c}$ deduced from the resistivity curves (top panel). As we will demonstrate in the next Section, at these pressures the initial screening step is attributed to surface superconductivity Leng et al. (2017), while the ensuing second step with a full diamagnetic screening is attributed to bulk superconductivity.

The first important result is that superconductivity is enhanced under pressure with a maximum value $T_{c}=$ 1.91 K around 0.91 GPa and a gradual depression of $T_{c}$ at higher pressures. This is illustrated in Fig. 3, where we trace $T_{c}^{R}(p)$ extracted from the resistance data. Here we use the onset transition temperatures determined by extrapolation of the $R(T)$-curves just below $T_{c}$ to the normal state plateau values, as shown for the 2.49 GPa curve in Fig. 2. The same analysis for the $\chi_{ac}(T)$-data shows $T_{c}^{\chi}(p)$ tracks $T_{c}^{R}(p)$ closely up to 1.24 GPa, see Fig. 3. However, for $p\geq 1.41$ GPa it is the second, lower in temperature, transition in $\chi_{ac}(T)$ that is attributed to bulk superconductivity and tracks $T_{c}^{R}(p)$. The agreement between $T_{c}^{\chi}(p)$ and $T_{c}^{R}(p)$ obtained with different techniques on two different crystals is good. Lastly, the temperature of surface superconductivity, $T_{c}^{S}(p)$, is traced in Fig. 3. For $p\leq 1.41$ GPa $T_{c}^{S}(p)$ is obtained from the field-temperature phase diagrams by extrapolating $T_{c}^{S}(H)$ to zero field, as reported in Ref. Leng et al., 2017 and presented in the following Section. For $p\geq 1.41$ GPa we take $T_{c}^{S}(p)$ from the onset of the upper transition in $\chi_{ac}(T)$. This tells us the transition temperatures for the bulk and surface have a distinct pressure variation, and for $p\geq 1.41$ GPa $T_{c}^{S}>T_{c}^{bulk}$. This underpins surface superconductivity in PdTe2 is a unique, robust feature.

III.2 Field-temperature phase diagram

In order to investigate the pressure dependence of the superconducting phase diagram in the $H$-$T$ plane we have measured at each pressure the resistance and ac-susceptibility in applied dc-fields, $H_{a}$. A typical data set taken at $p=0.25$ GPa is shown in Fig. 4. In the lower panel with $\chi_{ac}$-data the zero-field curve shows $T_{c}=1.63$ K. In small applied fields a peak appears just below $T_{c}$ due to the differential paramagnetic effect (DPE). This peak signals the field induced intermediate state Leng et al. (2017). It shifts to lower temperatures with increasing field and for higher fields is progressively depressed because of an additional screening signal that precedes the DPE peak. The additional screening is attributed to superconductivity of the surface sheath Leng et al. (2017). Partial screening is still visible at 27 mT, but has nearly vanished at $\mu_{0}H_{a}=30$ mT down to 0.3 K. Consequently, in the limit $T\rightarrow 0$ $H_{c}^{S}(0)>H_{c}(0)$. In the upper panel, with $R(T)$ data, the transition is first rapidly depressed with field up to $\mu_{0}H_{a}\approx 13$ mT, but then the depression rate decreases, the transition broadens and signals of superconductivity persist up to $\mu_{0}H_{a}\approx 180$ mT. We remark this field is much larger than $H_{c}(0)$ or $H_{c}^{S}(0)$. The robustness of superconductivity in resistance measurements was also observed at ambient pressure, with a critical field, $H_{c}^{R}(0)$, equal to $\sim 0.3$ T Leng et al. (2017).

In the following paragraphs we present the $H$-$T$ phase diagrams determined from the $R(T)$- and $\chi_{ac}(T)$-data in applied fields, measured up to 2.49 GPa. The phase diagram at 0.25 GPa is extracted from Fig. 4. Additional data sets are presented in the Supplemental Material (SM) SM .

In Fig. 5 we present the critical field for bulk superconductivity $H_{c}(T)$. The data are obtained by tracing the $T_{c}^{\chi}$-values as a function of the applied field. The solid lines in Fig. 5 represent $H_{c}(T)=H_{c}(0)[1-(T/T_{c})^{2}]$ at different pressures, where $T_{c}=T_{c}^{\chi}$. The quadratic temperature variation is consistent with type-I superconductivity. In fact all the data under pressure collapse on one single curve, $h^{*}(t)$, as shown in the right panel of Fig. 5. Here the standard expression for plotting $H_{c}(T)$ in a reduced form is applied, with $h^{*}=(H_{c}(T)/T_{c})/(-dH_{c}/dT)|_{T_{c}}$ where $t=T/T_{c}$ Bay et al. (2012b). For a type-I superconductor $h^{*}(0)=0.5$. The collapsed curve $h^{*}(t)$ shows type-I superconductivity persists over the whole pressure range.

Next we show how superconductivity of the surface sheath develops with pressure. Hereto we have traced $T_{c}^{S}(H)$ obtained from the $\chi_{ac}$-curves in applied fields in Fig. 6. Phase diagrams at 0.25, 1.08 and 2.07 GPa are presented. At 0.25 GPa we start to observe the (partial) diamagnetic screening due to the surface at a finite value $H_{a}\approx 5$ mT (Fig. 4, lower panel). The corresponding $T_{c}^{S}(H)$ points are traced in the left panel of Fig. 6. By extrapolating $T_{c}^{S}(H)$ to zero field we obtain $T_{c}^{S}(0)$. In the same panel we have plotted $H_{c}(T)$ for bulk superconductivity as well. We find $T_{c}^{S}(0)<T_{c}^{\chi}(0)$, just like reported previously at ambient pressure Leng et al. (2017). However, upon further increasing the pressure the phase lines $H_{c}(T)$ and $H_{c}^{S}(T)$ move apart and do no longer intersect for $p\geq 1.41$ GPa, in which case $T_{c}^{S}(0)>T_{c}^{\chi}(0)$. This is illustrated for $p=2.07$ GPa in the right panel of Fig. 6. The distinct pressure variation of $T_{c}^{S}$ and $T_{c}^{\chi}$ demonstrates once more that surface superconductivity is not of the standard Saint-James - de Gennes type Saint-James and de Gennes (1963). We discuss the robustness and nature of this phenomenon in the next Section.

Finally we show in Fig. 7 the $H$-$T$ phase diagrams determined from the transport data at pressures up to 2.49 GPa. At each pressure we investigated the depression of superconductivity by measuring $R(T)$ in fixed applied fields. The $R(T)$-data for 0.25 GPa are shown in the upper panel in Fig. 4. Additional data sets are reported in the SM SM . In all cases superconductivity is first depressed rapidly in small fields, and $H_{c}^{R}(T)$ tracks $H_{c}(T)$ for bulk superconductivity as deduced from $\chi_{ac}$ (see Fig. 5). The $H_{c}^{R}(T)$-data in Fig. 7 show this behavior is restricted to the temperature range 1.3-1.9 K. Below 1.3 K the transition in $R(T)$ broadens and traces of superconductivity are visible up to $\sim 0.2$ T. By tracing in Fig. 7 the onset temperature for superconductivity from $R(T)$ in fixed magnetic fields below 1.3 K, we observe a steady increase of $H_{c}^{R}(T)$. A comparison with the Werthamer-Helfand-Hohenberg (WHH) model Werthamer et al. (1966) indicates the data extrapolate to $H_{c}^{R}(0)\simeq 0.2$ T for $T\rightarrow 0$. We remark that for the crystal studied in Ref. Leng et al., 2017 this value is larger, $\simeq 0.3$ T. Interestingly, $H_{c}^{R}(T)$ below 1.3 K is almost pressure independent, which shows the superconducting transition in resistance for $H_{a}>H_{c}$ is not closely connected to surface superconductivity as was proposed in Ref. Leng et al., 2017.

IV Analysis and Discussion

The mechanical and electronic properties of PdTe2 under pressure have been investigated theoretically by several groups Soulard et al. (2005); Xiao et al. (2017); Lei et al. (2017). The only experimental high-pressure study carried out so far is by Soulard et al. Soulard et al. (2005) who conducted high-pressure X-ray diffraction experiments at room temperature and 300 ∘C to investigate the possiblity of a structural phase transition. They found that an abrupt change in the interatomic distances occurs above $p=15.7$ GPa at room temperature, but the volume versus pressure curve exhibits no discontinuity. Under pressure the unit cell volume decreases by 17.6% at the maximum applied pressure of 27 GPa, and the $c/a$ ratio decreases from 1.27 to 1.24 at 27 GPa. A bulk modulus, $B_{0}$, of 102 GPa was derived from the experimental data. This value is to be compared with 71.2 GPa (74.2 GPa) derived from first principle calculations by Lei et al. Lei et al. (2017) at 300 K (0 K). Xiao et al. Xiao et al. (2017) computed the optimized lattice parameters as a function of pressure, which are slightly overestimated compared to the experimental data Soulard et al. (2005). Overall, these studies indicate there is no structural transition in the modest pressure range up to 2.5 GPa in our experiments. For a layered material the change in the $c/a$-ratio is normally an important control parameter for the electronic properties. However, for PdTe2 this change is very tiny and 0.2% at most up to 2.5 GPa Soulard et al. (2005). In the following we focus on the superconducting properties.

IV.1 Bulk superconductivity

A major result is the non-monotonous variation of $T_{c}$ with pressure reported in Fig. 3. $T_{c}$ first increases to 1.91 K at 0.91 GPa and then is gradually depressed. We first compare the experimental results with theoretical calculations. The evolution of superconductivity with pressure was investigated theoretically by Xiao et al. Xiao et al. (2017). The authors used the Allen-Dynes-modified McMillan equation to calculate $T_{c}$, with the characteristic phonon frequency $\omega_{log}$, the electron-phonon coupling constant $\lambda$ and the Coulomb pseudopotential $\mu^{*}\simeq 0.1$ as input parameters. Combined electronic structure and phonon-density of states calculations show a gradual decrease of $\lambda$ and an increase of $\omega_{log}$ (blue shift), but overall the calculated $T_{c}$ decreases from 2.0 K at ambient pressure to 0.6 K at 10 GPa. Note the calculated $T_{c}$ at $p=0$ is larger than our experimental value of 1.6 K. While a decrease to 0.6 K at 10 GPa is within bounds of the extrapolation of $T_{c}(p)$ in Fig. 3, the calculations by Xiao et al. Xiao et al. (2017) clearly do not capture the initial increase of $T_{c}$ and its maximum value at 0.91 GPa. The superconducting properties of PdTe2 were also investigated by Kim et al. Kim et al. (2018) employing the same McMillan formalism. Their phonon band structure calculations show the electron-phonon interaction is dominated by the optical $O_{1,2}$ and $O_{3}$ phonon modes. Furthermore, they emphasize the importance of a saddle-point van Hove singularity (vHs) close to the Fermi energy. The computed $T_{c}$ is 1.79 K at ambient pressure. The importance of a vHs is further illustrated by the case of PtTe2, which is isoelectronic with PdTe2 but does not show superconductivity. Here the vHs-band has a broad dispersion along $k_{z}$ leading to a lower density of states at the Fermi level and absence of superconductivity Kim et al. (2018). Calculations for PdTe2 with a 15% volume contraction, which corresponds to a pressure of $\sim$20 GPa, indicate the vHs band moves close to the Fermi level Kim et al. (2018), which would produce a higher $T_{c}$. However, this is at variance with the experimental data presented in Fig. 3.

Another way to tune $T_{c}$ besides pressure is via doping or substitution. Recently, it was demonstrated that Cu intercalation enhances $T_{c}$ to a maximum value of 2.6 K in CuxPdTe2 Yan et al. (2015b); Ryu (2015); Hooda, M. K. and Yadav, C. S. (2018) for $x=0.06$. Upon intercalation the volume contracts, but changes are minute: $\Delta V/V=-$0.07% for $x=0.04$ Hooda, M. K. and Yadav, C. S. (2018), which corresponds to an applied pressure of 0.07 GPa. This shows Cu intercalation cannot be equated to chemical pressure in tuning superconductivity. The same holds for the substitution series (AuxPd1-x)Te2 Kudo et al. (2016). Upon alloying with Au, $T_{c}$ increases up to 4.65 K for $x=0.40$. Simultaneously, the volume increases by 2.5%, which corresponds to a negative pressure of $\sim$2.5 GPa. The experimental and calculated variation of $T_{c}$ with pressure and doping are summarized in Fig. 8. Here we trace the relative change of $T_{c}$ as a function of the relative volume change, $(V-V_{0})/V_{0}$, where a bulk modulus of 102 GPa is used Soulard et al. (2005). Although $T_{c}$ generally decreases with a smaller volume, the experimentally observed positive $dT_{c}/dp$ for PdTe2 up to 0.91 GPa is at odds with this trend.

In an attempt to shed further light on the pressure variation of $T_{c}$, we have conducted Hall effect measurements on two PdTe2 crystals under pressure up to 2.07 GPa SM . At the lowest pressure of 0.25 GPa the carrier concentration, $n$, amounts to 1.5-1.7$\times 10^{22}$ cm*-3* at 2 K. It varies quasi-linearly with pressure resulting in an increase of $\sim$20% at 2.07 GPa. No anomalous behavior is observed around 0.9 GPa. In the most simple model the increase of $n$ is expected to result in an increase of the density of states at the Fermi level and a monotonous enhancement of $T_{c}$.

The non-monotonous variation of $T_{c}$ indicates the density of states and the electron phonon-coupling constant are affected in an intricate manner by doping and/or pressure. Possibly this is a result from band structure subtleties that have not been probed in the coarse-grained calculations carried out so far Soulard et al. (2005); Xiao et al. (2017); Lei et al. (2017). In order to access the electronic band structure under pressure, a quantum oscillations study is highly desirable. The feasibility to observe the Shubnikov - de Haas effect and the de Haas - van Alphen effect at ambient pressure has been demonstrated in Refs. Dunsworth, 1975; Fei et al., 2017; Zheng et al., 2018. In the same context, small structural modifications that might influence $T_{c}$, such as changes in the $z$-coordinate of Te atoms in the unit cell that would affect the $O_{1,2}$ and $O_{3}$ phonon modes, cannot be excluded based on the X-ray diffraction experiment with a first pressure point at 2.2 GPa Soulard et al. (2005). This calls for high-precision low-pressure ($p\leq 2.5$ GPa) single-crystal X-ray diffraction measurements.

IV.2 Surface superconductivity

The distinct pressure variation of the superconducting transition temperature of the surface sheath, $T_{c}^{S}$, and of the bulk, $T_{c}^{\chi}$, reported in Figs. 3 and  6, is an extraordinary result. We recall this feature is derived from the ac-susceptibility curves measured in fixed magnetic fields at eleven different pressures. Selected data sets at 0.25 GPa are shown in Fig. 4 and at 1.08 and 2.07 GPa in the SM SM . The data show how type-I superconductivity in the bulk, probed by the DPE-peaks in small applied dc-fields, is progressively depressed with field, while surface superconductivity is observed for $H_{a}>H_{c}$ (see also Ref. Leng et al., 2017). Upon increasing the pressure, the DPE peak is more rapidly depressed compared to surface screening. At 2.07 GPa the DPE effect is - already in the lowest applied fields - almost completely screened by the surface SM . Hence for $p\geq 1.41$ GPa $T_{c}^{S}>T_{c}^{\chi}$. This is further underpinned by the observation that $H_{c}(T)$, defined by $T_{c}^{\chi}(H)$, follows the quadratic temperature variation at all pressures, characteristic for bulk type-I superconductivity (Fig. 5). Note that $T_{c}^{S}$ is defined as the onset temperature for the diamagnetic signal due to surface superconductivity, while the transition itself may become very broad. $H_{c}^{S}(p)$ has a maximum near 0.9 GPa, similar to $H_{c}(p)$, as reported in the SM SM . When the $H_{c}^{S}(T,p)$ data is traced in the reduced form $h^{*}(t)$ the data do not collapse on a single curve as, see SM SM . Instead the trend is that the values $h^{*}(t)$ increase with respect to pressure, which indicates the superconducting pairing interaction changes in a non-trivial way. The distinct $H_{c}(T)$- and $H_{c}^{S}$-curves and their dissimilar pressure dependence strongly suggest surface and bulk superconductivity are independent phenomena and not tightly connected, in contrast to the familiar Saint James - de Gennes surface superconductivity Saint-James and de Gennes (1963). It remains tempting to relate surface superconductivity in PdTe2 to topological surface states detected by ARPESYan et al. (2015a); Noh et al. (2017); Clark et al. (2018). These surface states could possibly be investigated by STM experiments in small applied fields ($H_{a}>H_{c}$). The STM experiments performed so far were predominantly directed to probe bulk superconductivity Das et al. (2018); Clark et al. (2018). Moreover, for the spectra taken in a magnetic field the intermediate state that occurs below $H_{c}$ for a finite demagnetization factor was not taken into account.

In the resistance measurements (partial) superconductivity is observed up to about 0.2 T for $T\rightarrow 0$ (Fig. 7), a value that largely exceeds $H_{c}(0)$ and $H_{c}^{S}(0)$. The enhanced $H_{c}^{R}(T)$-curves below 1.3 K are quasi pressure independent. By extrapolating the data in this field range to $H_{a}\rightarrow 0$ with the WHH function a pressure independent $T_{c}=1.2$ K is found. Since $T_{c}^{S}$ has a pronounced pressure variation the resistive superconducting transitions measured in this field range are not connected to surface superconductivity. Note that for the crystal studied in Ref. Leng et al., 2017 it was concluded that the transport experiment does probe surface superconductivity, but these experiments were performed at ambient pressure only. The persistence of superconductivity in resistance measurements in field is puzzling. Normally such an effect is attributed to filamentary superconductivity. Its pressure independence indicates it might not be intrinsic to PdTe2.

V Summary and conclusions

We have carried out a high-pressure transport and ac-susceptibility study of superconductivity in the type-I superconductor PdTe2 ($T_{c}=1.64$ K). $T_{c}$ shows a pronounced variation with pressure: it increases at low pressure, then passes through a maximum of 1.91 K around 0.91 GPa, and subsequently decreases smoothly up to the highest pressure measured, $p_{max}=2.5$ GPa. The critical field, $H_{c}$, follows a similar behavior, leading the $H_{c}(T)$-curves at different pressures to collapse on a single universal curve with the characteristic quadratic in temperature depression of $H_{c}$ for type-I superconductivity. Type-I superconductivity is robust under pressure. In view of the absence of structural modifications in our pressure range and the minute change of the $c/a$-ratio  Soulard et al. (2005), the non-monotonous variation of $T_{c}$ indicates an intricate role of the dominant phonon frequency, the electron-phonon-coupling parameter and Coulomb pseudopotential used to compute $T_{c}$ with help of the McMillan formula. This effect has not been captured by band structure calculations so far Xiao et al. (2017); Kim et al. (2018), notably the electron band structure calculations predict a smooth decrease of $T_{c}$ under pressure Xiao et al. (2017). This calls for more elaborate and detailed calculations for pressures up to $p_{max}=2.5$ GPa.

The unusual surface superconductivity, first reported at ambient pressure Leng et al. (2017), persists under pressure. Surprisingly, for $p\geq 1.41$ GPa the superconducting transition temperature for the surface $T_{c}^{S}$ exceeds $T_{c}$ of the bulk. This tells us surface and bulk superconductivity are distinct phenomena. This is further confirmed by the observation that the phase lines $H_{c}(T)$ and $H_{c}^{S}(T)$ move apart under pressure and no longer intersect for $p\geq 1.41$ GPa. We propose surface superconductivity possibly has a non-trivial nature and originates from topological surface states detected by ARPESYan et al. (2015a); Noh et al. (2017); Clark et al. (2018). This calls for quantum-oscillation experiments under pressure, possibly enabling one to follow the pressure evolution of the bulk electronic structure and topological surface states.

In the same spirit it will be highly interesting to extend the experiments to higher pressures, especially because a pronounced change in the electronic properties of PdTe2 is predicted to occur in the range 4.7-6.1 GPa: the type-II Dirac points disappear at 6.1 GPa, and a new pair of type-I Dirac points emerges at 4.7 GPa Xiao et al. (2017). Thus a topological phase transition may occur in the pressure range 4.7-6.1 GPa. This in turn might have a strong effect on (surface) superconductivity, because the tilt of the Dirac cone vanishes Fei et al. (2017); Shapiro et al. (2018). We conclude further high-pressure experiments on PdTe2 provide a unique opportunity to investigate the connection between topological quantum states and superconductivy.

Acknowledgements: H.L. acknowledges the Chinese Scholarship Council for Grant No. 201604910855. This work was part of the research program on Topological Insulators funded by FOM (Dutch Foundation for Fundamental Research on Matter). It was further supported by the JSPS (Japan Society for the Promotion of Science) Program for Fostering Globally Talented Researchers, Grant Number R2903.