# Disorder induced transition from type-I to type-II superconductivity in the
Dirac semimetal PdTe2

M. V. Salis<EMAIL_ADDRESS>J. P. Lorenz  Y. K. Huang  A. de Visser
<EMAIL_ADDRESS>Van der Waals - Zeeman Institute, University of Amsterdam,
Science Park 904, 1098 XH Amsterdam, The Netherlands

###### Abstract

We report a doping study directed to intentionally induce disorder in PdTe2 by
the isoelectronic substitution of Pt. Two single-crystalline batches
Pd1-xPtxTe2 have been prepared with nominal doping concentrations $x=0.05$ and
$x=0.10$. Sample characterization by energy dispersive x-ray spectroscopy
(EDX) revealed Pt did not dissolve homogeneously in the crystals. For the
nominal value $x=0.10$ small single crystals cut from the batch appeared to
have $x=0.09$, as well as the non stoichiometric composition
Pd0.97Pt<0.004Te2.03. Magnetic and heat capacity measurements demonstrate a
transition from type-I to type-II superconducting behavior upon increasing
disorder. From transport measurements we calculate a residual resistivity
$\rho_{0}=1.4~{}\mu\Omega$cm suffices to turn PdTe2 into a superconductor of
the second kind.

##  I introduction

Recently, interest in transition metal dichalcogenides has increased
significantly due to their extraordinary electronic properties. Notably, the
opportunity to realize novel quantum states arising from the topologically
non-trivial band structure, as found by density functional theory Soluyanov
_et al._ (2015); Huang  _et al._ (2016); Yan  _et al._ (2017); Bahramy  _et
al._ (2018), attracts much attention. The formation of both type-I and type-II
bulk Dirac cones has been predicted Bahramy  _et al._ (2018). Of special
interest in this family is the semimetal PdTe2 since it undergoes a
superconducting transition at $T_{c}\sim 1.7$ K Guggenheim  _et al._ (1961).
Furthermore, PdTe2 is classified as a type-II Dirac semimetal, as uncovered by
angle-resolved photoemission spectroscopy and ab initio electronic structure
calculations Bahramy  _et al._ (2018); Yan  _et al._ (2015); Fei  _et al._
(2017); Noh  _et al._ (2017); Clark  _et al._ (2018). A type-II Dirac
semimetal is characterized by a Dirac cone with a tilt parameter $k>1$ leading
to broken Lorentz invariance Soluyanov  _et al._ (2015). It is predicted that
for Dirac semimetals with $k\approx 1$, meaning close to the topological
transition at $k=1$, superconductivity is generally of the second type (type-
II) Rosenstein  _et al._ (2018). For $k>1$, superconductivity becomes of the
first kind (type-I). Interestingly, PdTe2 Leng  _et al._ (2017); Salis  _et
al._ (2021) is a type-I superconductor and based on its $T_{c}$ Shapiro et al.
Shapiro  _et al._ (2018) estimated $k\approx 2$. In view of the effect
topology has on superconductivity in these systems, it is of interest to
investigate whether the superconductivity type can be altered by, for
instance, doping.

Superconductivity in PdTe2 has been explored in great detail. Type-I
superconductivity was uncovered with help of magnetic and transport
measurements on single crystals Leng  _et al._ (2017). The intermediate state,
a hallmark of type-I behavior, was observed through the dc magnetization
curves and the differential paramagnetic effect in the ac susceptibility data.
Here, a bulk critical field $B_{c}(0)=13.6$ mT was determined in conjunction
with a surface critical field $B_{c}^{S}(0)=34.9$ mT. Moreover, the
temperature dependence of the surface superconductivity did not follow the
Saint-James $-$ de Gennes model Saint-James and de Gennes (1963). Peculiarly,
from resistance measurements a critical field $B_{c}^{R}(0)=0.32$ T was
deduced. Weak-coupling conventional superconductivity in PdTe2 was
demonstrated via measurements of the heat capacity Amit and Singh (2018);
Salis  _et al._ (2021), penetration depth Teknowijoyo  _et al._ (2018); Salis
_et al._ (2018), scanning tunneling microscopy and spectroscopy (STM/STS)
Clark  _et al._ (2018); Das  _et al._ (2018); Sirohi  _et al._ (2019), and
side junction tunneling spectroscopy Voerman  _et al._ (2019).
Superconductivity is partly attributed to a van Hove singularity situated at
$\sim 30$ meV above the Fermi level Kim  _et al._ (2018); van Heumen  _et al._
(2019).

On the other hand, a mixed type-I and type-II superconducting state was
concluded from STM/STS Das  _et al._ (2018); Sirohi  _et al._ (2019) and point
contact spectroscopy (PCS) Le  _et al._ (2019) measurements. In a magnetic
field a range of critical fields was observed at the surface, which was
explained by spatially separated type-I and type-II regions. However, later
muon spin rotation measurements Leng  _et al._ (2019a) and scanning squid
magnetometry Garcia-Campos  _et al._ (2021) provide solid evidence for bulk
type-I superconductivity probed on the microscopic and macroscopic scale,
respectively. Finally, evidence for bulk type-I superconductivity was attained
through heat capacity measurements by demonstrating the presence of latent
heat Salis  _et al._ (2021). Measurements under hydrostatic pressure show that
superconductivity is still present at 5.5 GPa Furue  _et al._ (2021) and
remains of the first kind at least till 2.5 GPa Leng  _et al._ (2019b).

Substitution or doping studies using PdTe2 are scarce. Kudo et al. Kudo  _et
al._ (2016) examined Pd substitution in AuTe2 by preparing a series of
Au1-xPdxTe2 samples. Bulk superconductivity emerges at $x\approx 0.55$ with
$T_{c}\approx 4.0$ K as evidenced by heat capacity measurements. At lower
$x$-values the Te-Te dimer connections stabilize a monoclinic crystal
structure in which superconductivity is absent Kudo  _et al._ (2016). The
strong-coupled nature of superconductivity near $x\approx 0.55$ is attributed
to a large density of states (DOS) at the Fermi level. Further increasing the
Pd content results in weak coupling superconductivity with lower transition
temperatures, as expected from approaching the stoichiometric end compound
PdTe2. Ryu investigated Cu doping in PdTe2 by preparing a series of CuxPdTe2
samples Ryu (2015). Optimal doping was found near $x=0.05$ with bulk
superconductivity at $T_{c}\approx 2.6$ K Ryu (2015); Hooda and Yadav (2018).
The increase of $T_{c}$ is attributed to an increase in the DOS at the Fermi
level due to the hybridization of Te-p and Cu-d orbitals along the $c$-axis,
effectively reducing the 2D-nature of this layered material. This is in-line
with the Cu atoms being intercalated in the Van der Waals gaps. STM/STS
measurements provide evidence that Cu0.05PdTe2 is a homogeneous type-II
superconductor Vasdev  _et al._ (2019). This change, compared to the STM/STS
data on PdTe2 Das  _et al._ (2018); Sirohi  _et al._ (2019) that revealed a
mixed type-I/II behavior, is explained by Cu intercalation inducing disorder.
This effectively reduces the electron mean free path $l_{e}$ and the coherence
length $\xi$, thus increasing the Ginzburg Landau (GL) parameter
$\kappa=\frac{\lambda}{\xi}$ to larger than the $1/\sqrt{2}$ threshold for
type-I behavior.

Here we report the results of a doping study, directed to intentionally induce
disorder in PdTe2 by substituting Pd by iso-electronic Pt. We have prepared
Pd1-xPtxTe2 crystals with nominal doping concentrations $x=0.05$ and $x=0.10$.
Sample characterization by energy dispersive x-ray spectroscopy (EDX) revealed
that Pt did not dissolve homogeneously in the crystals. Notably, small
crystals cut from the nominal $x=0.10$ batch appeared to have $x=0.09$, or the
non stoichiometric composition Pd0.97Pt<0.004Te2.03. Transport, magnetic and
heat capacity measurements demonstrate a transition from type-I to type-II
superconducting behavior upon increasing disorder.

##  II Experimental

PdTe2 crystallizes in the trigonal CdI2 structure (space group $P\bar{3}m1$).
Two single-crystalline batches Pd1-xPtxTe2 were prepared with $x=0.05$ and
$x=0.10$ using a modified Bridgman technique Lyons  _et al._ (1976). The same
technique was previously used to prepare PdTe2 single crystals Leng  _et al._
(2017). Small flat crystals were cut from the prepared batches by a scalpel.
The crystals have an area of $2\times 3$ mm2 and a thickness of about 0.3 mm.
Scanning electron microscopy with energy dispersive x-ray spectroscopy
(SEM/EDX) was carried out with help of a Hitachi table top microscope TM3000.
For details of the SEM/EDX results we refer to the Supplemental Material file
Sup . SEM micrographs taken on cut crystals and other sample pieces revealed
the final composition can deviate from the nominal one and that Pt did not
dissolve in the same amount in all pieces. In fact for the cut crystal with a
nominal Pt content of 5 at.% no Pt was detected. This crystal has a
stoichiometric composition with a Pd:Te ratio of 1:2 (the error in these
numbers is 1%). Transport, ac susceptibility and heat capacity measurements
were carried out on this sample, which we labeled #ptnom5. For the experiments
on the 10 at.%Pt concentration we used two crystals. One sample had a
composition close to the nominal $x=0.10$ composition Pd0.91Pt0.09Te2. This
sample, labeled #ptnom10res, was used for transport experiments only. EDX on
the second sample showed a small Te excess and a very small Pt content ($<$
0.4%). Its composition is Pd0.97Pt<0.004Te2.03. This sample was used for
transport, ac susceptibility and heat capacity measurements and it is labeled
#ptnom10. We remark that the EDX determined compositions above each yield the
average over a large part of the sample surface and are thus representative
for the specific sample. The experimental results on the doped samples are
compared with previous resistance, ac susceptibility and heat capacity data
taken on a crystal with the stoichiometric 1:2 composition to within 0.5% as
determined by EDX Leng  _et al._ (2017); Salis  _et al._ (2021). In the
following this sample is labelled #pdte2.

Resistance measurements were performed using the standard four point method in
a Quantum Design Physical Property Measurement System (PPMS) down to 2.0 K.
Data at lower temperatures were collected in a 3-He refrigerator (Heliox,
Oxford Instruments) down to 0.3 K using a low frequency (16 Hz) ac-resistance
bridge (Linear Research LR700). The ac susceptibility was measured in the
Heliox with a custom-made coil set. Data were also taken with the LR700
bridge, operated at a driving field of 0.026 mT. The heat capacity was
measured using the dual slope thermal relaxation calorimetry technique Stewart
(1983), using a home-built set-up Salis  _et al._ (2021), where each data
point is the average of four dual slope measurements. The increase in
temperature $\Delta T$ in the measurement of the heat capacity is always in
between 1$\%$ and 1.6$\%$ of the bath temperature of the particular
measurement. In the ac susceptibility and specific heat experiment the dc
magnetic field was applied in the $ab$-plane. The demagnetization factor of
the crystals is $N\simeq 0.1$, which implies the intermediate state is formed
between $(1-N)H_{c}\simeq 0.9H_{c}$ and $H_{c}$ in the case of type-I
superconductivity. The resistance and ac susceptibility measurements in field
have been carried out by applying the field above $T_{c}$ and subsequently
cooling in field, while the specific heat data in field were taken after zero
field cooling and then applying the field.

##  III Results

Figure 1: Temperature dependence of the resistivity of crystals #ptnom5 (red
circles), #ptnom10 (blue circles) and #ptnom10res (green circles). The data
for #pdte2 (black circles) are taken from Ref. Leng  _et al._ (2017).

The resistivity of samples #ptnom5, #ptnom10 and #ptnom10res in the
temperature range 2-300 K is shown in figure 1, where we have also traced the
data for crystal #pdte2 reported in Ref. Leng  _et al._ (2017). The curves for
#ptnom5 and #pdte2 are very similar with a residual resistivity value,
$\rho_{0}$, taken at 2 K, of 0.75 and 0.76 $\mu\Omega$cm, respectively. This
is in agreement with both samples having the same stoichiometric 1:2
composition. The residual resistance ratio, RRR= $\rho(300$K$)/\rho_{0}$,
amounts to 40 and 30, respectively. For the non-stoichiometric sample #ptnom10
$\rho_{0}$ has increased to 3.6 $\mu\Omega$cm and RRR = 12. The
$\rho_{0}$-value of the substituted sample #ptnom10res is considerably higher
as expected, and equals 16.3 $\mu\Omega$cm. Its RRR is 3.

Figure 2: Resistance as a function of temperature around the superconducting
transition for crystal #ptnom5 (panel (b)) and #ptnom10res (panel (c)) in zero
field (black curves) and small applied fields, $\mu_{0}H_{a}$, as indicated.
The data in panel (a) for #pdte2 are taken from Ref. Leng  _et al._ (2017).
Figure 3: Ac susceptibility of crystals #ptnom5 (panel (b)) and #ptnom10
(panel (c)) measured in zero field (black curves) and small applied dc fields
as indicated. The field is applied in the $ab$-plane. The data of #pdte2 are
taken from Ref. Leng  _et al._ (2017). Note the ac driving field applied to
take the data in panels (b) and (c) is a factor 10 smaller than in panel (a).

The resistance as a function of temperature around the superconducting
transition in zero field and applied magnetic fields of crystals #ptnom5 and
#ptnom10res is depicted in figure 2. Again, the data for #pdte2, shown in
panel (a), are taken from Ref. Leng  _et al._ (2017). The critical temperature
in zero field, $T_{c}(0)$, here defined by the onset of the transition, is
1.87 K and 1.56 K for the stoichiometric samples #ptnom5 and #pdte2,
respectively. Surprisingly, the higher $T_{c}$ and RRR for #ptnom5 indicate it
has a somewhat higher purity than sample #pdte2. For the substituted sample
the superconducting transition shows several steps and $T_{c}$ is lower. It
ranges from 1.44 to 1.12 K. In a magnetic field superconductivity is rapidly
suppressed. The data in panels (b) and (c) of figure 2 show these crystals
also have superconducting resistance paths in fields above the critical field
$B_{c}(0)$ determined by ac susceptibility and heat capacity (see below and
figure 5). The $B_{c}^{R}(0)$ values that can be deduced are however not as
large as the value $B_{c}^{R}(0)\approx$ 0.3 T for $H$ $\|~{}c$ reported for
PdTe2 (see figure S6 in the Supplemental Material file of Ref. Leng  _et al._
(2017)).

In figure 3 we show the in-phase component of the ac susceptibility,
$\chi_{ac}^{\prime}$, in arbitrary units measured on crystals #ptnom5 and
#ptnom10 in the temperature range 0.3-2.0 K. Again the data are compared with
those of #pdte2 (data in S.I. units taken from Ref. Leng  _et al._ (2017)).
The onset $T_{c}$ values are 1.64 K and 1.85 K for #pdte2 and #ptnom5 and
compare well to the values determined above from the resistivity. The onset
$T_{c}$ value for #ptnom10 is 1.91 K, but the transition is rather broad (the
width is 0.3 K) with a slow decrease below $T_{c}$. The resistance of this
sample was only measured in the PPMS down to 2.0 K. The RRR-value of 12 tells
us the disorder is enhanced, which is also reflected in the broad transition.
The $\chi_{ac}^{\prime}(T)$ data measured in applied magnetic fields for
#pdte2 and #ptnom5 show pronounced peaks below $T_{c}$ that are due to the
differential paramagnetic effect (DPE). The DPE is due to the positive $dM/dH$
($M$ is the magnetization) in the intermediate state Hein and Falge (1961).
The intermediate phase is due to the sample shape and is present when the
demagnetization factor, $N$, is finite. Observation of red a DPE that largely
exceeds the Meissner signal can therefore be used as solid proof for type-I
superconductivity. Most importantly, the DPE is absent for crystal #ptnom10,
which provides the first piece of evidence it is a type-II superconductor.

Figure 4: Electronic specific heat, $C_{el}$, of crystals #ptnom5 (panel (b))
and #ptnom10 (panel (c)) measured in zero field (black curves) and small
applied dc fields as indicated. The field is directed in the $ab$-plane. The
data of #pdte2 are taken from Ref. Salis  _et al._ (2021).

In figure 4 we show the electronic specific heat, $C_{el}$, of crystals
#ptnom5 and #ptnom10 in the temperature range 0.3-2.0 K. The
$C_{el}(T)$-curves are obtained by subtracting the phononic contribution from
the measured $C$ in the standard way, i.e. by using the relation $C=\gamma
T+\beta T^{3}$, where $\gamma$ is the Sommerfeld coefficient and $\beta$ the
phononic coefficient. The data are compared with $C_{el}$ of PdTe2 reported in
Ref. Salis  _et al._ (2021) (panel (a) of figure 4). This PdTe2 crystal was
cut from the same batch as the samples studied in Ref. Leng  _et al._ (2017)
and we also label it #pdte2. The onset $T_{c}$ values of crystals #pdte2 and
#ptnom5 are 1.62 K and 1.75 K and compare well to the values determined above.
The onset $T_{c}=1.60$ K for #ptnom10 is however lower than the value 1.91 K
determined by $\chi_{ac}^{\prime}(T)$.

The $\gamma$-values of the three crystals in panel (a), (b) and (c) of figure
4 amount to 4.4, 4.5 and 4.7 mJ/molK2 and the $\beta$-values are 0.7, 1.1 and
1.0 mJ/molK4, respectively. These $\gamma$-values are very similar, which
indicates the density of states near the Fermi level is not affected much by
doping. The $\beta$-values do show some variation, which is not correlated
with the amount of disorder, and likely related to an experimental uncertainty
because of the small temperature interval in which $\beta$ is obtained. To
examine the strength of the electron-phonon coupling, the step size $\Delta
C|_{T_{c}}$ is analysed using the BCS relation $\Delta C|_{T_{c}}/\gamma
T_{c}=1.43$, where $T_{c}$ is the superconducting transition temperature, here
taken as the onset of superconductivity. For crystal #pdte2 a ratio $\Delta
C|_{T_{c}}/\gamma T_{c}=1.42$ is found Salis  _et al._ (2021), which is close
to the textbook value of 1.43 for a weakly coupled BCS superconductor. For
crystal #ptnom5 a ratio of 1.41 is found, which presents a minute change from
the textbook value. However, for crystal #ptnom10 we determine a ratio of
1.48, suggesting that superconductivity is slightly more than weakly coupled.

Next we discuss the electronic specific heat measured in applied magnetic
fields (figure 4). Distinguishing between type-I and type-II superconductivity
via heat capacity can be achieved by observing the presence or absence of
latent heat. The extra energy necessary to facilitate a first order phase
transition is reflected in the heat capacity as an increased value of $C$ at
the transition. A type-I superconductor has a second order phase transition in
zero field, but a first order one in field. While for a type-II superconductor
the transition remains second order in an applied field. The excess $C_{el}$
above the standard BCS heat capacity in panel (a) provided solid thermodynamic
evidence PdTe2 is a type-I superconductor Salis  _et al._ (2021).
Surprisingly, for crystal #ptnom5 (panel b) the excess specific heat becomes
more pronounced as illustrated by the sharp peaks below $T_{c}(B)$. Thus the
contribution of the latent heat to $C_{el}$ is much larger, which indicates
the transition has a stronger first order character than observed for crystal
#pdte2. On the other hand, for crystal #ptnom10 the data in panel (c) show
latent heat is absent, which provides the second piece of evidence of type-II
superconductivity, in-line with the $\chi_{ac}^{\prime}$-data.

Finally, we trace the temperature variation of the critical field, $B_{c}(T)$,
extracted from the ac susceptibility (figure 3) and specific heat data (figure
4). The $B-T$ phase diagram is reported in figure 5. For crystals #pdte2 and
#ptnom5 we identify $T_{c}(B)$ by the onset in $C_{el}$ and the onset of the
DPE in $\chi_{ac}^{\prime}(T)$. $B_{c}(T)$ follows the standard BCS quadratic
temperature variation $B_{c}(T)=B_{c}(0)[1-(T/T_{c})^{2}]$, with
$B_{c}(0)=14.2$ mT and $T_{c}$ = 1.63 K for #ptpde2 Leng  _et al._ (2017);
Salis  _et al._ (2021), and $B_{c}(0)=15.9$ mT and $T_{c}$ = 1.77 K for
#ptnom5. For crystal #ptnom10 the transition in $\chi_{ac}^{\prime}(T)$ is
rather broad. Here we identify $T_{c}$ by the onset temperature in $C_{el}$,
which corresponds to the temperature where the magnetic transition is complete
in $\chi_{ac}^{\prime}(T)$. The $B-T$ phase-line provides further evidence for
type-II superconductivity. It compares well to the Werthamer-Helfand-Hohenberg
(WHH) model curve Werthamer  _et al._ (1966) for an orbital-limited weak-
coupling spin-singlet superconductor with an upper critical field
$B_{c2}(0)=21.8$ mT.

Figure 5: Critical field $B_{c}(T)$ of crystal #pdte2 and #ptnom5 and upper
critical field $B_{c2}(T)$ of crystal #ptnom10 extracted from the specific
heat (closed squares) and ac susceptibility (closed circles) data.

##  IV Discussion

From the sample preparation side our goal was to prepare Pd1-xPtxTe2 crystals
with $x=0.05$ and $x=0.10$. The SEM/EDX micrographs showed that Pt did not
dissolve as expected in these crystals and that the single-crystalline batches
are inhomogeneous. Crystals cut from the nominal $x=0.05$ batch appeared to be
undoped and have the 1:2 stoichiometry. From the nominal $x=0.10$ batch we
managed to obtain a crystal with $x=0.09$, and a non-stoichiometric crystal
Pd0.97Pt<0.004Te2.03. Specific heat and ac-susceptibility measurements on this
last crystal #ptnom10 demonstrated we could make a doping-induced transition
to type-II superconductivity.

To observe type-II superconductivity the disorder should be large enough such
that the threshold $\kappa=1/\sqrt{2}$ can be overcome. The effect of
controlled non-magnetic disorder on the normal and superconducting properties
of PdTe2 was recently studied by electron irradiation by Timmons et al.
Timmons  _et al._ (2020). The residual resistivity was found to increase from
a pristine crystal value of 0.6 $\mu\Omega$cm to 2.4 $\mu\Omega$cm for a
irradiation dose of 2.4 C/cm2, while at the same time $T_{c}$ decreased from
1.76 K to 1.65 K as identified by reaching the zero resistance state $R=0$.
Assuming a linear relation between $\rho_{0}$ and $T_{c}$, $T_{c}$ decreases
at a rate of 0.046 K/$\mu\Omega$cm. With this rate we estimate for crystal
#ptnom10res ($\Delta\rho_{0}=15.5~{}\mu\Omega$cm) $T_{c}=0.9$ K, which
compares favorably to the measured $T_{c}=1.1$ K ($R=0$), given the crude
approximation. In this electron irradiation work no discussion was made
whether disorder is strong enough to induce type-II behavior.

For crystal #ptnom10 the coherence length $\xi$ can be calculated from the
relation $B_{c2}(0)=\Phi_{0}/2\pi\xi^{2}$, where $\Phi_{0}$ is the flux
quantum. From figure 5 we determine $B_{c2}(0)=21.8$ mT and obtain $\xi=123$
nm. The coherence length can be related to the electron mean free path,
$l_{e}$, via Pippard’s relation $1/\xi=1/\xi_{0}+1/l_{e}$, where $\xi_{0}$ is
the intrinsic coherence length given by the BCS value Tinkham (1996). With
$\xi_{0}=1.8~{}\mu$m Salis  _et al._ (2018) and $\xi=~{}123$ nm we obtain
$\l_{e}=132$ nm. As expected, this value is reduced compared to $\l_{e}=531$
nm calculated from the residual resistivity value
$\rho_{0}=0.76~{}\mu\Omega$cm Salis  _et al._ (2018) of nominally pure PdTe2.
Reversely, using the experimental value $\rho_{0}=3.6~{}\mu\Omega$cm (figure
1) we calculate $\l_{e}=112$ nm for crystal #ptnom10, which is close to the
value $\l_{e}=132$ nm derived from Pippard’s relation.

Next we calculate $\kappa=\lambda/\xi$ of crystal #ptnom10. In their
controlled disorder study Timmons et al. Timmons  _et al._ (2020) measured the
penetration depth and found that upon increasing the disorder $\lambda$ stays
nearly constant Timmons  _et al._ (2020) at a value of 220 nm. This is in-line
with the minute change in the $\gamma$-value reported above. With $\xi=123$ nm
we calculate $\kappa=1.8$, which is in agreement with superconductivity being
of the second kind. For crystals #pdte2 and #ptnom5 we calculate
$\kappa\simeq$ 0.5-0.6 Leng  _et al._ (2017). Here $\xi\simeq$ 440-370 nm is
estimated from the GL relation $\xi=\Phi_{0}/(2\sqrt{2}\pi B_{c}\lambda_{L})$
Tinkham (1996), where $\lambda_{L}\propto(m_{e}/n_{s})^{1/2}$ is the London
penetration depth with $m_{e}$ the effective electron mass and $n_{s}$ the
superfluid density.

Another way to provide an estimate of $\kappa$ of crystal #ptnom10 is from the
GL relation $\kappa=B_{c2}/\sqrt{2}B_{c}$. The thermodynamic critical field,
$B_{c}$, can be determined from the specific heat by the relation $\Delta
C|_{T_{c}}=4B_{c}(0)^{2}/\mu_{0}T_{c}$ Poole  _et al._ (2007), where $C$ is in
units of J/m3. From $\Delta C|_{T_{c}}$ in figure 4 (panel (c)) we calculate
$B_{c}(0)=11.1$ mT. We remark this value is close to the calculated value
$B_{c}(0)=12.6$ mT reported for PdTe2 Leng  _et al._ (2017). Using
$B_{c}(0)=11.1$ mT and $B_{c2}=21.8$ mT in the expression above, we calculate
$\kappa=1.4$, which is similar to the value of 1.8 directly estimated from the
ratio $\lambda/\xi$.

We remark that for Type-I superconductivity $B_{c}(0)$ can also be obtained
from the latent heat with help of the Clausius-Clapeyron relation. We
calculate $B_{c}(0)$ = 11.2 mT and 11.1 mT for #pdte2 and #ptnom5,
respectively, in good agreement with the values obtained from $\Delta
C|_{T_{c}}$ in zero field Sup .

Our results are of relevance for the observation of a mixed type-I and type-II
superconducting state in PdTe2 probed by surface sensitive techniques Das  _et
al._ (2018); Sirohi  _et al._ (2019); Le  _et al._ (2019). Our doping study
shows that nominal pure PdTe2 crystals can already be close to the type-I/II
border. Using the value $\lambda=230$ nm Timmons  _et al._ (2020), we
calculate $\xi=310$ nm at the threshold value $\kappa=1/\sqrt{2}$. This
implies $l_{e}$ should be smaller than 375 nm for type-II superconductivity,
or $\rho_{0}>1.4~{}\mu\Omega$cm. From the resistivity graph reported in Ref.
Das  _et al._ (2018) we deduce $\rho_{0}\simeq 1~{}\mu\Omega$cm, which indeed
is not far from the type-I/II border. Thus it is plausible inhomogeneities
give rise to the mixed phase observation reported in Refs. Das  _et al._
(2018); Sirohi  _et al._ (2019); Le  _et al._ (2019).

An unsolved aspect of superconductivity in PdTe2 is the observation of surface
superconductivity detected in the screening signal $\chi_{ac}^{\prime}(T)$
measured in small applied dc fields Leng  _et al._ (2017, 2019b). The
extracted surface critical field $B_{c}^{S}(0)=34.9$ mT exceeds the value
predicted by the Saint-James $-$ de Gennes model Saint-James and de Gennes
(1963) $B_{c3}=2.39\times\kappa B_{c}=16.3$ mT. Recently, the GL model at the
superconducting-insulator boundary was revisited Samoilenka and Babaev (2021)
and it was shown that $T_{c}$ and the third critical field $B_{c3}$ can be
enhanced to exceed the Saint-James $-$ de Gennes value, which is worthy to
explore further. On the other hand, it is tempting to attribute the surface
superconductivity in PdTe2 to superconductivity of the topological surface
state detected by ARPES Bahramy  _et al._ (2018); Yan  _et al._ (2015); Fei
_et al._ (2017); Noh  _et al._ (2017); Clark  _et al._ (2018). We remark that
the $\chi_{ac}^{\prime}(T)$ data for the doped crystals, reported in figure 3
panel (b) and (c), also show superconducting screening signals above the
$B_{c}(0)$ and $B_{c2}(0)$-values reported in figure 5. Likewise, the
resistance traces in figure 2 reveal $B_{c}^{R}(0)$ is similarly enhanced.
These screening signals of enhanced superconductivity are however not as
pronounced as reported for PdTe2 in Ref. Leng  _et al._ (2017). Nonetheless,
the robustness of superconducting screening signals above $B_{c}(0)$ or
$B_{c2}(0)$ upon doping, as well as under high pressure Leng  _et al._
(2019b), calls for further experiments.

##  V Conclusion

The Dirac semimetal PdTe2 is a type-I superconductor with $T_{c}=1.7$ K. We
have carried out a doping study directed to intentionally increase the
disorder and induce type-II superconductivity. Two single-crystalline batches
Pd1-xPtxTe2 have been prepared with nominal doping concentrations $x=0.05$ and
$x=0.10$. Sample characterization by energy dispersive x-ray spectroscopy
(EDX) on small crystals cut from the batches revealed that Pt did not dissolve
homogeneously in the crystals. In fact the nominal $x=0.05$ crystal appeared
to be undoped and have the stoichiometric 1:2 composition. From the nominal
$x=0.10$ batch we obtained a small single crystal with $x=0.09$, as well as a
crystal with the non stoichiometric composition Pd0.97Pt<0.004Te2.03. The
presence of the differential paramagnetic effect in the ac susceptibility and
latent heat in the heat capacity demonstrate the nominal $x=0.05$ crystal is a
type-I superconductor, just like PdTe2. The absence of these effects for
Pd0.97Pt<0.004Te2.03 revealed it is a type-II superconductor with an upper
critical field $B_{c2}=21.8$ mT. The analysis of $B_{c2}$ and resistance data
using Pippard’s model convincingly show PdTe2 can be turned into a
superconductor of the second kind when the residual resistivity
$\rho_{0}>1.4~{}\mu\Omega$cm.

Acknowledgement: This work is part of the Projectruimte programme with project
number 680-91-109, which is financed by the Dutch Research Council (NWO).

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Supplemental Material: Disorder induced transition from type-I to type-II
superconductivity in the Dirac semimetal PdTe2

Content

1\. SEM/EDX mapping experimental and results

Fig.S1 SEM/EDX mapping of crystal #ptnom5

Fig.S2 SEM/EDX mapping of crystal #ptnom10

Fig.S3 SEM/EDX mapping of crystal #ptnom10res

2\. Latent heat and Clausius-Clapeyron relation

1\. SEM/EDX mapping: experimental
Scanning electron microscopy with energy dispersive x-ray spectroscopy
(SEM/EDX) was carried out with help of a Hitachi table top microscope TM3000.
The acceleration voltage in all measurements is 15 kV. Of each single-
crystalline boule prepared with a certain nominal Pt content small thin
crystals were isolated with typical size 2 $\times$ 3 mm2. On each of these
crystals we have investigated the composition by EDX in several areas of
typically 200 $\times$ 200 $\mu$m2. Overall the composition in the sleected
and measured crystals was found to be homogenous. The table below gives the
labels of the doped crystals, the nominal Pt content and the EDX determined
composition.

Crystal | Nominal Pt cotent | EDX compostion
---|---|---
#ptnom5 | 5 at. $\%$ | PdTe2
#ptnom10 | 10 at. $\%$ | Pd0.97Pt<0.004Te2.03
#ptnom10res | 10 at.$\%$ | Pd0.91Pt0.09Te2

Typical SEM/EDX results for crystals #ptnom5, #ptnom10 and #ptnom10res are
given in Fig. S1, Fig. S2 and Fig. S3, respectively. Shown are:

Top panel: The SEM spectrum in cps/eV (counts per second per electron-volt)
with Pd, Pt and Te peaks labelled.

Middle panel: Table with quantitative results of the composition analysis.

Lower left panel: SEM picture of the crystal with scanned area for the
composition analysis indicated.

Lower right panel: Pd, Pt and Te element distribution in the scanned area.

Figure S1 SEM/EDX mapping of crystal #ptnom5

Figure S2 SEM/EDX mapping of crystal #ptnom10

Figure S3 SEM/EDX mapping of crystal #ptnom10res

2\. Latent heat and Clausius-Clapeyron relation
As mentioned in the Discussion section in the manuscript an estimate for the
thermodynamic critical field $B_{c}(0)$ can be obtained by evaluating the
relation $\Delta C=4B_{c}(0)^{2}/\mu_{0}T_{c}$, where $\Delta C$ is the step
at $T_{c}$ in the specific heat at $B=0$. Alternatively, for a type-I
superconductor in a magnetic field $B_{c}(0)$ can be calculated from the
latent heat, $L$. Using the Clausius-Clapeyron equation the entropy change is
given by $\Delta S=L(T)/T=-2\mu_{0}B_{c}(T)\frac{dB_{c}(T)}{dT}$, with
$B_{c}(T)=B_{c}(0)(1-(T/T_{c})^{2})$. $\Delta S$ was calculated by graphically
integrating the $C/T$ data in field and subtracting the data taken at $B=0$.
This procedure is carried out for all applied field values, and from $\Delta
S(B)$ we obtain $B_{c}(0)$ as a fit parameter. For crystals #pdte2 and
#ptnom5, that exhibit type-I superconductivity, we obtain $B_{c}(0)=11.2$ mT
and 11.1 mT, respectively. We remark these values are close to the $B_{c}(0)$
values evaluated at $B=0$. On the other hand, the $B_{c}(0)$-values derived
from $\Delta C$ and $\Delta S$ are smaller than the value derived directly
from the experiment. In the table below we compare the $B_{c}(0)$ values
obtained in different ways.

Crystal |  $B_{c}(0)$ from $\Delta C$ |  $B_{c}(0)$ from $\Delta S$ |  $B_{c}(0)$ experiment
---|---|---|---
#pdte2 | 10.9 mT | 11.2 mT | 14.2 mT
#ptnom5 | 11.8 mT | 11.1 mT | 15.9 mT
#ptnom10 | 11.1 mT | - | -