supplementary information · 2015-12-18 · formula that has contribution from both ballistic or...

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NMAT4455

NATURE MATERIALS | www.nature.com/naturematerials 1

Supplementary Material for “Unconventional Superconductivity

at Mesoscopic Point-contacts on the 3-Dimensional Dirac

Semi-metal Cd3As2”

Leena Aggarwal1,∗ Abhishek Gaurav1,∗ Gohil S. Thakur2,

Zeba Haque2, Ashok K. Ganguli2,3,† and Goutam Sheet1‡

1Department of Physical Sciences, Indian Institute of

Science Education and Research Mohali, Sector 81,

S. A. S. Nagar, Manauli, PO: 140306, India

2Department of Chemistry, Indian Institute of Technology, New Delhi 110016, India and

3Institute of Nano Science & Technology, Mohali 160064, India

∗ These authors contributed equally to the work† [email protected]‡ [email protected]

1

Unconventional superconductivity at mesoscopic point contacts on the 3D Dirac semimetal Cd3As2

© 2015 Macmillan Publishers Limited. All rights reserved

http://dx.doi.org/10.1038/nmat4455

2 NATURE MATERIALS | www.nature.com/naturematerials

SUPPLEMENTARY INFORMATION DOI: 10.1038/NMAT4455

Determination of the I − V characteristics corresponding to RM

For a point-contact between two different materials, the resistance is given by Wexler’s

formula that has contribution from both ballistic or Sharvin’s resistance (RS = 2h/e2

(akF )2)

and thermal or Maxwell’s resistance (RM = ρ(T )2a

). RS is always finite and depends only

on fundamental constants namely the Planck’s constant (h) and the charge of a single

electron (e). RM is directly dependent on the resistance of the materials forming the point-

contact and therefore becomes zero in the superconducting state. The measured data on

the point-contacts have contribution of both RM and RS. As per Wexler’s formula, the

total point-contact resistance is R = RS +RM . In order to extract the I − V corresponding

to the RM component alone we have subtracted the ballistic component (RS) from the

total resistance (R). The ballistic I − V for superconducting point-contacts is dominated

by Andreev reflection. The Andreev reflection dominated ballistic I − V was calculated

as discussed below using standard BTK theory for superconducting point-contacts and the

data before and after subtraction are presented in Figure S1.

As per BTK theory, the I − V characteristics of a superconducting point-contact can be

generated by using the expression of the current through a ballistic interface given by

Iballistic= C∫ +∞−∞ [f(E − eV )− f(E)][1 + A(E)− B(E)]dE

where, A(E) is the Andreev reflection probability and B(E) is the normal reflection

probability.

A(E) and B(E) were calculated using the following formula: Since the superconducting

phase is found to be unconventional, in order to make the analysis more general, for our

simulation we have used the modified BTK formula that also accounts for finite quasiparticle

lifetime (Γ) at the interface as described by Plecenik et.al.1:

A(E) = aa∗ and B(E) = bb∗, where the coefficients a and b are given by

a = u0v0/γ, b = −(u20 − v20)(Z

2 + ιZ)/γ, where u20 and v20 are the probabilities of an

electronic state being occupied and unoccupied respectively:

u20 =

12[1 +

√(E+ιΓ )−∆2

E+ιΓ], v20 = 1− u2

0

γ2 = γγ∗, γ = u20 + (u2

0 − v20)Z2

2

Figure S1. Representative I-V characteristic measured by experimental data (red in color) and I-V

characteristic generated from BTK theory (blue in color)

Z is the dimensionless parameter used in BTK theory and it is directly proportional to

the strength of the potential barrier at the point-contact interface. For all our simulation

the value of Γ remained zero. The constant C has been determined by matching the scales

of the experimental data and the theoretical curves.

In this context it must also be noted that there are several approximations involved in

this analysis:

(a) The information that the point-contact is made of two distinct materials out of which

one never goes to the superconducting state has been ignored for simplicity.

(b) It has been assumed that there is only one critical current for a given effective point-

contact. It is believed that in general the effective electrical point-contact is usually formed

out of multiple nano-contacts. Therefore, the resultant critical current dominated I-V must

be influenced by the geometry and size of each individual nano-contact. For some of these

contacts, with weaker contribution to the total resistance, the critical current might be very

large causing a deviation of the I-V curve from a linear behavior above the measured Ic.

The possibility of multiple contacts has been ignored in this analysis as that cannot be

determined with absolute certainty but will make the analysis extremely complex.

3© 2015 Macmillan Publishers Limited. All rights reserved




Determination of the I − V characteristics corresponding to RM

For a point-contact between two different materials, the resistance is given by Wexler’s

formula that has contribution from both ballistic or Sharvin’s resistance (RS = 2h/e2

(akF )2)

and thermal or Maxwell’s resistance (RM = ρ(T )2a

). RS is always finite and depends only

on fundamental constants namely the Planck’s constant (h) and the charge of a single

electron (e). RM is directly dependent on the resistance of the materials forming the point-

contact and therefore becomes zero in the superconducting state. The measured data on

the point-contacts have contribution of both RM and RS. As per Wexler’s formula, the

total point-contact resistance is R = RS +RM . In order to extract the I − V corresponding

to the RM component alone we have subtracted the ballistic component (RS) from the

total resistance (R). The ballistic I − V for superconducting point-contacts is dominated

by Andreev reflection. The Andreev reflection dominated ballistic I − V was calculated

as discussed below using standard BTK theory for superconducting point-contacts and the

data before and after subtraction are presented in Figure S1.

As per BTK theory, the I − V characteristics of a superconducting point-contact can be

generated by using the expression of the current through a ballistic interface given by

Iballistic= C∫ +∞−∞ [f(E − eV )− f(E)][1 + A(E)− B(E)]dE

where, A(E) is the Andreev reflection probability and B(E) is the normal reflection

probability.

A(E) and B(E) were calculated using the following formula: Since the superconducting

phase is found to be unconventional, in order to make the analysis more general, for our

simulation we have used the modified BTK formula that also accounts for finite quasiparticle

lifetime (Γ) at the interface as described by Plecenik et.al.1:

A(E) = aa∗ and B(E) = bb∗, where the coefficients a and b are given by

a = u0v0/γ, b = −(u20 − v20)(Z

2 + ιZ)/γ, where u20 and v20 are the probabilities of an

electronic state being occupied and unoccupied respectively:

u20 =

12[1 +

√(E+ιΓ )−∆2

E+ιΓ], v20 = 1− u2

0

γ2 = γγ∗, γ = u20 + (u2

0 − v20)Z2

2

Figure S1. Representative I-V characteristic measured by experimental data (red in color) and I-V

characteristic generated from BTK theory (blue in color)

Z is the dimensionless parameter used in BTK theory and it is directly proportional to

the strength of the potential barrier at the point-contact interface. For all our simulation

the value of Γ remained zero. The constant C has been determined by matching the scales

of the experimental data and the theoretical curves.

In this context it must also be noted that there are several approximations involved in

this analysis:

(a) The information that the point-contact is made of two distinct materials out of which

one never goes to the superconducting state has been ignored for simplicity.

(b) It has been assumed that there is only one critical current for a given effective point-

contact. It is believed that in general the effective electrical point-contact is usually formed

out of multiple nano-contacts. Therefore, the resultant critical current dominated I-V must

be influenced by the geometry and size of each individual nano-contact. For some of these

contacts, with weaker contribution to the total resistance, the critical current might be very

large causing a deviation of the I-V curve from a linear behavior above the measured Ic.

The possibility of multiple contacts has been ignored in this analysis as that cannot be

determined with absolute certainty but will make the analysis extremely complex.





(c) It cannot be absolutely known how far away the contacts are from the ballistic or

the thermal regime and therefore, it is not known how much inelastic scattering/dissipation

happens in the contact region.

(d) The fact that part of the point-contact is formed by a non-trivial material that might

potentially show non-linear I − V in mesoscopic scale has also been ignored.

Due to such assumptions it may be seen that (sometimes) beyond Ic the red and the blue

lines in Figure S1 almost run parallel to each other. In fact, for a point-contact between

two simple metals, the difference in the two curves should scale linearly with increasing

current which is not seen to be the case here. Nevertheless, when the extraction of the

Andreev reflection dominated −V characteristics are more accurate for the point-contacts

being closer to the ballistic regime, the resultant I − V characteristics look more ideal.

However, qualitatively, the simple analysis presented here shows how the typical super-

conducting I − V is extracted from the measured I − V of the point-contacts.

Comment on the possible order parameter symmetry:

Fitting of dV/dI spectrum in the intermediate regime using BTK theory.

From the general shape of the dV/dI spectra it is clear that the intermediate regime

data has contribution from both Andreev reflection and critical current. In order to confirm

the existence of the Andreev reflection contribution, we have differentiated the expression

of current (Iballistic) with respect to V and compared the same with the experimental data.

The representative fittings with the fitting parameters are shown in Figure S2.

The data obtained in the ballistic regime are significantly broader than the BTK predic-

tion. This is because in the ballistic regime the inelastic scattering processes are forbidden

and therefore, the features associated with unconventional component of the order parameter

are prominent.

The fact that we could fit the low-bias part of the spectrum with BTK theory also confirms

that the order parameter symmetry in the new superconducting phase is a mixed angular

momentum symmetry with a strong s-wave component. From the analysis and discussion

regarding the observed ZBCP presented in the main text there is a strong indication that a

s+ p-wave type of symmetry is possible in this new superconducting phase.2

The possible p-wave contribution is further supported by our experiments on the point-

contacts made with a spin-polarized metal (Cobalt) where the Andreev reflection related

4

Figure S2. Three Ag-Cd3As2 point-contact spectra obtained at different magnetic fields in the

intermediate regime with low-bias BTK fits.

Figure S3. (a) A normalized differential resistance spectrum obtained from a Co-Cd3As2 point-

contact. (b)Magnetic field dependence of another spectrum.

features are clearly visible (see Figure S3) indicating that the superconducting properties

are not strongly suppressed by the proximity of a metal with spin-polarized Fermi-surface.

Moreover, the superconductivity related features for the Co-Cd3As2 point-contacts survived

up to a high magnetic field of 25 kG.

Our experiments show a strong indication of an unconventional order parameter. How-

ever, further theoretical and experimental work (like planar tunneling, high-pressure mea-

surements etc.) will be required to establish the exact nature of the order parameter sym-

metry and the pairing mechanism.





Material Synthesis

Polycrystalline samples of Cd3As2 were obtained by heating the stoichiometric mixture

of the constituent elements. A mixture of Cd and As powder (∼ 1 gram) was sealed in an

evacuated quartz tube (∼ 10−5 mbar), heated at 5000C for 8 hours, then at 8500C for 24

hours with a typical ramping rate of 10C/min and furnace cooled to room temperature. The

shiny black crystalline product thus obtained was ground well, pelletized (φ = 8 mm) and

heated again in vacuum at 4000C for 6 h for homogenization. The pellet was shiny black

and hard in nature.

Characterization

X-ray diffraction: The samples were characterized by powder X-ray diffraction technique

using Cu-Kα radiation (λ = 1.5406) on a Bruker D8 Advance diffractometer. All the peaks

could be indexed on the basis of a centrosymmetric tetragonal cell in I41/acd space group

as reported by Cava et. al.3 The sample was pure with no apparent impurity phase present

in the resolution limit of X-ray diffraction analysis. Lattice parameters calculated using

Le‘Bail method were in close agreement with the literature values (Fig. S4). In fig.S4 we

show the Le‘Bail fit to the powder x-ray diffraction pattern of polycrystalline Cd3As2. The

vertical bars indicate the allowed Bragg reflections.

Figure S4. Powder x-ray diffraction pattern of polycrystalline Cd3As2 fitted with Le‘Bail. The


Energy dispersive X-ray analysis (EDAX): Compositional analysis was done using

a SEM-EDAX. The average stoichiometry found after collecting data on each sample at

6

many different regions was close to 3:2 (Cd:As). There were some regions were As was

slightly deficient ( 5%). Fig. S5 shows the presence of Cd and As. C and Si comes from

the carbon tape and detector respectively. Inset of Fig. S5 shows a typical electron image

of the polished pellet on which measurements were performed.

Figure S5. EDAX spectrum of Cd3As2. The inset shows image of polished pellet of Cd3As2.

Magnetization of the bulk Cd3As2 samples: The VSM (Vibrating Sample Magne-

tometer) experiment was done to investigate the possibility of hidden bulk superconducting

phase in the material. The measurement was done in a Quantum Design (QD) PPMS (Phys-

ical properties measurement system) with a VSM probe supplied by QD. The measurement

field was 100 Oe. The magnetization does not show any diamagnetic transition down to 2

K.





Material Synthesis

Polycrystalline samples of Cd3As2 were obtained by heating the stoichiometric mixture

of the constituent elements. A mixture of Cd and As powder (∼ 1 gram) was sealed in an

evacuated quartz tube (∼ 10−5 mbar), heated at 5000C for 8 hours, then at 8500C for 24

hours with a typical ramping rate of 10C/min and furnace cooled to room temperature. The

shiny black crystalline product thus obtained was ground well, pelletized (φ = 8 mm) and

heated again in vacuum at 4000C for 6 h for homogenization. The pellet was shiny black

and hard in nature.

Characterization

X-ray diffraction: The samples were characterized by powder X-ray diffraction technique

using Cu-Kα radiation (λ = 1.5406) on a Bruker D8 Advance diffractometer. All the peaks

could be indexed on the basis of a centrosymmetric tetragonal cell in I41/acd space group

as reported by Cava et. al.3 The sample was pure with no apparent impurity phase present

in the resolution limit of X-ray diffraction analysis. Lattice parameters calculated using

Le‘Bail method were in close agreement with the literature values (Fig. S4). In fig.S4 we

show the Le‘Bail fit to the powder x-ray diffraction pattern of polycrystalline Cd3As2. The


Figure S4. Powder x-ray diffraction pattern of polycrystalline Cd3As2 fitted with Le‘Bail. The


Energy dispersive X-ray analysis (EDAX): Compositional analysis was done using

a SEM-EDAX. The average stoichiometry found after collecting data on each sample at

6

many different regions was close to 3:2 (Cd:As). There were some regions were As was

slightly deficient ( 5%). Fig. S5 shows the presence of Cd and As. C and Si comes from

the carbon tape and detector respectively. Inset of Fig. S5 shows a typical electron image

of the polished pellet on which measurements were performed.

Figure S5. EDAX spectrum of Cd3As2. The inset shows image of polished pellet of Cd3As2.

Magnetization of the bulk Cd3As2 samples: The VSM (Vibrating Sample Magne-

tometer) experiment was done to investigate the possibility of hidden bulk superconducting

phase in the material. The measurement was done in a Quantum Design (QD) PPMS (Phys-

ical properties measurement system) with a VSM probe supplied by QD. The measurement

field was 100 Oe. The magnetization does not show any diamagnetic transition down to 2

K.





Figure S6. Magnetization vs. temperature of the bulk sample measured in a VSM in a Quantum

Design PPMS. The bulk sample does not show any diamagnetic transition at low temperature

down to 2 K.

Four-probe resistivity of bulk Cd3As2 samples: Resistivity of the samples that

were used for the point-contact spectroscopy measurements were measured by a four-probe

technique in a Quantum Design PPMS. The resistivity as a function of temperature of

three such samples is presented in Fig. S7. The samples show semi-metallic behaviour with

varying mobility. This further confirms that no superconducting phase is present in the bulk.

Figure S7. Raw data for four probe resistance vs. temperature of three bulk Cd3As2 samples

with varying mobility. All three of them lead to the new superconducting phase in a point-contact

geometry.

8

Low-temperature measurements

The low temperature measurements were performed in a liquid helium cryostat working

down to 1.4K. The cryostat is equipped with a dynamic variable temperature insert (VTI)

inside which there is one static VTI. The bottom part of the static VTI is made of copper

for efficient cooling. The sample goes inside the static VTI which is first evacuated and

then filled with dry helium exchange gas. The cryostat is also equipped with a three-axis

vector magnet. The vector magnet can apply a maximum magnetic field of 6T along the

vertical direction using a superconducting solenoid and 1T in the horizontal plane using four

superconducting Helmholtz coils. For the measurements presented in this paper, magnetic

field was applied in the vertical direction, perpendicular to the sample surface using the

solenoid.

Point-contact Spectroscopy: Point-contact spectroscopy experiments were performed

using a home-built low-temperature probe. The probe consists of a long stainless steel tube

at the end of which the probe-head is mounted. The probe head is equipped with a 100

threads per inch (t.p.i.) differential screw that is rotated by a shaft running to the top of

the cryostat. The screw drives a tip-holder up and down with respect to the sample. The

sample-holder is made of a 1” dia. copper disk. A cernox thermometer was mounted on

the copper disc for the measurement of the temperature. The temperature of the disc was

varied by a heater mounted on the same copper disc. The tips were fabricated by cutting a

0.25 mm dia. metal wire at an angle. The tip was mounted on the tip holder and two gold

contact leads were made on the tip with silver epoxy. The samples were mounted on the

sample holder and two silver-epoxy contact leads were mounted on the sample as well. These

four leads were used to measure the differential resistance (dV/dI) across the point-contacts.

The leads 3 and 4 were used to carry out the two-probe resitivity measurements.

The point-contact spectra were captured by ac-modulation technique using a lock-in-

amplifier (Model: SR830 DSP) (as shown in the schematic diagram). A voltage to current

converter was fabricated to which a dc input coupled with a very small ac input was fed.

The output current had a dc and a small ac component. This current passed through the

point-contact. The dc output voltage across the point-contact, V was measured by a digital

multimeter (model: Keithley 2000 ) and the ac output voltage was measured by a lock-

in amplifier working at 670 Hz. The first harmonic response of the lockin could be taken





Figure S8. Schematic diagram describing the point-contact spectroscopy measurements.

to be proportional to the differential change in the voltage dV . The current was passed

through a standard 3.3 kΩ resistor in series with the point contact. The drop across this

resistor was measured by another lock in amplifier. This gave the estimate of the ac current

passing through the point contact which is proportional to dI. dI/dV is plotted against V

to generate the point-contact spectrum. We have normalized the spectra that are presented

in this paper. The software for data acquisition was developed in house using lab-view.

How did we determine the critical temperature (Tc)?

For all the point-contacts reported here we have measured the temperature (T ) depen-

dence of the point-contact resistance (R) with V = 0. The R − T data show a broad

transition to the superconducting state. We have drawn the slope of the R − T curves

above and below the onset of transitions. The temperature at which the two slopes for a

given R− T curve meet has been taken as the Tc for the corresponding point-contacts. It is

important to note that for the ballistic point-contacts we cannot measure the Tc as for such

point-contacts the contact-resistance depends only on fundamental constants and remain

10

temperature independent. However, from the thermal limit point-contacts we learn that the

Tc does not have a strong dependence on contact size for a given sample and therefore it is

rational to conclude that for the ballistic point-contacts Tc remains close to 6 K. However,

for the samples with higher mobility, the highest critical temperature that we have measured

is slightly higher (8 K).

Measurements with Pt and Au tips: In order to investigate whether the new super-

conducting phase emerges only in a point-contact with silver (Ag) tip, we have performed

measurements with platinum (Pt) and gold (Au) tips as well. It is observed that for all

metallic tips the superconducting phase emerges. The data obtained between Cd3As2 and

Pt and Au tips are shown in Fig. S9.

Figure S9. (a) and (b) showing representative spectra with superconducting dips that were obtained

from two different point-contacts on Cd3As2 with a Pt tip. (c) A spectrum obtained with Au tip.

This spectrum also shows the signature of the gap structure. (d) Resistance vs. temperature of

the point-contact shown in (a). The superconducting transition is clearly seen at 5.3 K.





On the size of the point-contacts: The point-contact spectroscopy measurements

were done at different points on two samples grown in two different batches. The contact

size was estimated from the normal state resistance (at high V ) following Wexler’s formula

given by:

RPC =2h/e2

(akF )2+ Γ(l/a)

ρ(T )

2a

where, Γ(l/a) is a numerical factor close to unity. a is the contact diameter and 2h/e2

is quantum resistance i.e. 50kΩ. kF is the magnitude of the Fermi wave vector which is

0.04A for Cd3As2 and ρ(T ) is resistivity at temperature T(in K). ρ(1.5) = 28µΩ − cm for

Cd3As2 (measured by conventional four-probe method).

Table S1. A representative list of the normal state resistance and size of the point-contacts.

12

Raw data for the magnetic field dependence R-T

Figure S10. The raw data corresponding to the Figure 1(f) in the main manuscript.

Why do we not see a zero-resistance here?

While a bulk superconductor shows zero resistance when measured in a four-probe geom-

etry, when a mesoscopic contact is made on a superconductor with a non-superconducting

metal, the contact-resistance cannot ever be zero. In fact, the contact resistance depends a

lot on the size of the point-contact. When the contact size is small (compared to the elastic

mean free path), the contact resistance is large and the contact is in the ballistic regime.

When the contact size is large, the contact resistance is small and the contact is said to

be in the thermal regime. In between there is an intermediate regime where the contact

resistance is given by Wexler’s formula.





Even when the bulk of the superconductor transitions to the superconducting phase,

the N-S point-contact resistance must remain finite because (I) there is always a tem-

perature independent Sharvin’s contribution, no matter how small it is and (II) the non-

superconducting electrode forming the point-contact will always show finite resistance.

Even if the normal conductor were a perfect conductor, i.e. a conductor whose resistivity

is zero but shows no Meissner effect, a contact resistance still exists. When the supercon-

ductor is normal this contact resistance is given by Rpc = 1/Gc, where the conductance

Gc = f × (e2/h) × N , with f the degeneracy (spin, valley, etc.), e2/h the quantum con-

ductance, and N the number of quantum channels = kFA (kF is the Fermi wavenumber,

and A the contact area). At low-bias with |e∆Vbias| < ∆, Andreev reflection, doubles the

conductance, or equivalently, halves the resistance.

Here we show that the same fact is true in general and show an example with the experi-

ments on Pb-Ag point-contacts. In order to show how we have obtained superconductivity-

related spectroscopic features in different regimes of mesoscopic transport on Cd3As2-Ag

point-contacts, we demonstrate a comparison between the Pb-Ag data and the Cd3As2-Ag

data. Please see Figures S11 and S12.

14

Figure S11. Transport and magneto-transport data on Pb-Ag point-contacts and Cd3As2-Ag point-

contacts. (a) Schematic of the four probe geometry for the contact resistance measurements. Cur-

rent is sent through the leads 1 and 4, voltage is measured between the leads 2 and 3. (b) Magnetic

field-dependence of R-T on one Pb-Ag point-contact with a normal-state contact resistance of 2.2

Ω. (c)Magnetic field-dependence of R-T on one Pb-Ag point-contact with a normal-state contact

resistance of 18 mΩ. (d) Magnetic field-dependence of R-T on one Cd3As2-Ag point-contact with

a normal-state contact resistance of 2.95 Ω.





Figure S12. Comparison with point-contact spectra on superconducting lead (Pb). (a) Thermal

regime data on a Pb-Ag point-contact. (b) Thermal regime data on Cd3As2-Ag point-contact.

The critical current dominated peaks in dV/dI are clearly visible in both cases. (c) Intermediate

regime data on a Pb-Ag point-contact. (d) Intermediate regime data on Cd3As2-Ag point-contact.

Peaks in dV/dI are due to critical current as in (a) and (b) and dips in dV/dI represent Andreev

reflection. (e) Ballistic regime data on Pb-Ag point-contact. (f) Ballistic regime data on Cd3As2-

Ag point-contact. In the ballistic regime we have shown dI/dV instead of dV/dI and the peaks

in dI/dV are due to Andreev reflection. The position of the gaps also give the superconducting

energy gap. Note that the resistance of the point-contacts never become zero!

[1] Plecenik, A., Grajkar, M. Benacka, S., Seidel, P. & Pfuch, A. Finite-quasiparticle-lifetime effects

in the differential conductance of Bi2Sr2CaCu2Oy/Au junctions.Physical Review B49, 10016

(1994).

[2] Kashiwaya, S. & Tanaka, Y. Tunnelling effects on surface bound states in unconventional

superconductors. Rep. Prog. Phys.63, 1641 (1999).

[3] Ali, M. N., Gibson, Q., Jeon, S., Zhou, B. B., Yazdani, A. & Cava, R. J. The crystal and

electronic structures of Cd3As2, the three-dimensional electronic analogue of graphene. Inorg.

Chem. 53, 4062 (2014).



supplementary information · 2015-12-18 · formula that has contribution from both ballistic or...

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