www.sciencemag.org/cgi/content/full/science.aab2277/DC1

Supplementary Materials for

Evidence for two-dimensional Ising superconductivity in gated MoS2 J. M. Lu, O. Zheliuk, I. Leermakers, N. F. Q. Yuan, U. Zeitler, K. T. Law, J. T. Ye*

*Corresponding author. E-mail: j.ye@rug.nl

Published 12 November 2015 on Science Express

DOI: 10.1126/science.aab2277

This PDF file includes:

Materials and Methods Figs. S1 to S5 Tables S1 and S2 Full Reference List

2

Materials and Methods

1. Device fabrication and transport measurement

Electric double layer transistor (EDLT) devices for transport measurements were

all fabricated on thin flakes of MoS2 exfoliated onto highly doped silicon wafer with 285

nm SiO2. Flakes of thickness ranging from sub-10 to 50 nm were chosen by their color

contrasting under optical microscopy (53). Electrodes composed by a bilayer of Ti/Au

(5nm/65nm) in Hall-bar geometry were deposited onto the flakes using e-beam

evaporation in high vacuum (10-7

mbar) after patterning 300 nm resist of PMMA by

standard e-beam lithography. To obtain a clean interface between the metal electrodes

and the samples, in situ argon ion sputtering was used to remove the resist residues before

metal evaporation without breaking the vacuum.

After lift-off in hot acetone, the samples were all immersed into a small droplet of

dehydrated ion liquid: N,N-diethyl-N-(2-methoxyethyl)-N-methylammonium bis-

(trifluoromethylsulfonyl)-imide (DEME-TFSI) and transferred to the vacuum chamber of

cryostat with minimized air exposure. Typical transfer characteristics of the transistors

are shown in Fig. S1 for T = 220 K, a temperature optimized in many previous

investigation (17,20) thereby enjoying both a sufficiently high speed of ionic movement

and a reduced chance of chemical reaction at high gate voltage. Clear ambipolar transfer

curves are obtained when the liquid gate shifts the Fermi level to access both the valence

and conduction bands. For DEME-TFSI used in this study, we restricted the chemical

window to a conservative voltage range within ± 5V, which proved to be free from

chemical reaction as shown in previous studies (17). Keeping the liquid gate voltage

constant, the samples were subsequently cooled from 220 K down to 180 K at 3 K/min to

3

freeze the ion movement, after which the top gate could be released to zero without

losing the gating effect. To approach the base temperature, cooling speed was then

lowered to about 1 K/min.

Transport measurement was performed using three lock-in amplifiers (Stanford

Research SR830). One supplied an AC voltage excitation as source-drain voltage Vds at f

= 13 Hz and measured simultaneously the AC current through the sample via a current

amplifier. The other two lock-ins measured the longitudinal (Vxx) and the Hall (Vxy)

voltage at the locked frequency, respectively.

2. Temperature dependence of Bc2 of sample D24 (Tc = 7.38 K)

Taking the standard in determining the superconducting transition using half of

the normal state resistance RN, the upper critical field Bc2 was extracted both for an out-

of-plane and an in-plane magnetic field as shown in Fig. 2A and B, respectively. To

analyse the temperature dependence of Bc2, as the first attempt, we applied two-

dimensional (2D) Ginzburg-Laudau model:

0

2GL

c2 2 01B t t

and

0

GL Tinkham

1 212//

c2 2 01

dB t t

, which is a widely used analysis method for 2D

superconductors (30). Here, ct T T denotes normalized temperature,

2

0 2 45.47 nm Th e is the flux quantum.

As shown in Fig. S2, we obtained c2 0 8.6 TB and / /

c2 0 48 TB for

superconducting state at Tc = 7.38 K. Thus we could estimate Tinkham 3.8 nmd and

GL 0 6.2 nm for the thickness of superconducting layer and Ginzburg-Landau

coherent length, respectively. Since GL is about twice of the thickness Tinkhamd , the

4

superconducting state induced in Device D24 can be regarded as two-dimensional (2D)

but seems to be in the transition between 2D and 3D. By including only the orbital pair-

breaking mechanism in the above model, the thickness obtained is larger than the doping

profile inferred by ab initio calculation (25). As suggested theoretically, the above picture

is certainly limited by neglecting both spin-orbit scattering and Pauli paramagnetism,

which turn out to be important factors in the EDLT device. In similar systems of interface

superconductivity, especially in the LaAlO3/SrTiO3 interfaces (54) and -doped SrTiO3

(55), the calculated Tinkhamd is regarded as an upper bound of the estimation of the

thickness of superconducting layers.

Here we clarify the reason why the calculated Tinkhamd overestimates the thickness

of the superconductive layer. If only the orbital depairing mechanism is considered, as

described in Tinkham’s model, / /

c2B will diverge at zero temperature for a purely 2D

superconducting film because zero width will forbid the formation of vortices. Therefore,

a finite / /

c2B without divergence can only be obtained by having a finite Tinkhamd for thin

superconductors.

When the spin paramagnetic effect is taken into account, finite / /

c2B can be obtained even

in purely 2D system without suffering from divergence. As a result, as long as the

paramagnetic effect acts as one of the limiting factors of / /

c2B , the deduced Tinkhamd will

appear thicker than the true thickness of superconducting layer.

On the other hand, having a relatively large out-of-plane c2B is the direct cause of

having a “thicker” superconducting layer in our previous estimation using Tinkham

model. This out-of-plane c2B is quite large considering that vortices are easily formed in

5

such a 2D system. It is worth noting that under an out-of-plane magnetic field, the

Rashba type effective magnetic field can give essential protection in competing with the

orbital pair breaking mechanism which might allow such a large c2B . This out-of-plane

spin protection is the main mechanism to protect spins of Cooper pairs in many heavy

Fermion systems (9).

3. V-I characteristics in a magnetic field

In addition to the evolution of V-I characteristics with increasing temperature (Fig.

2E), we also probed the same V-I measurement at T = 3 K with increasing magnetic field.

Instead of being controlled by increasing temperature, a similar behaviour was observed

exhibiting evolution from the superconducting to the normal metallic states, which is

controlled by increasing B field to up to 11 T.

Following the Berezinskii-Kosterlitz-Thouless (BKT) theory for 2D

superconductor, when temperature decreases from above Tc, the exponent from

V I would jump from 1 (indicating a linear relation) to a critical value of 3 because

the density of free vortices scales as a function of the current inside the superconductor is

proportional to 2I , which is indeed observed in zero magnetic field at TBKT = 6.3 K as

shown in Fig. 2F. At even lower temperature, T < TBKT, free vortices are bound into pairs

by the logarithmic interaction within the characteristic length 2 d , where is

magnetic penetration depth and d is thickness of superconductive layer. Hence the

resistance goes to zero below TBKT.

In a finite perpendicular B field, if the associated magnetic length

1 2

0BL aB (with dimensionless 4.8a denoting the distance between vortices) is

6

smaller than , free vortices appear again whose density is proportional to the resistance

(56). As a result, the resistive transition would shift to lower temperatures as shown in

Fig. 2A. Therefore the temperate TBKT, above which bound vortex pairs are broken into

free vortices, would decrease with the help of an external magnetic field (57). As shown

in Fig. S3, the exponent α equals to 3 when B = 2.2 T indicating TBKT = 3 K, which is

significantly reduced from that observed at zero field. A Similar BKT transition under

magnetic field was also observed in a LaAlO3/SrTiO3 interface (31).

4. Magneto resistance measurement under high magnetic field

In Fig. 3A, we plotted the resistance Rs as a function of in plane magnetic field at

various temperatures, from which the Bc2–Tc phase diagram for sample D1 in Fig. 3C is

derived. For Bc2 at a temperature close to Tc = 2.37 K, it is experimentally preferable to

scan the temperature at a fixed magnetic field because Bc2 is with the range of magnetic

field (< 37 T). Under a very high magnetic field (B > 20 T), the CernoxTM

temperature

sensor starts to display significant magnetoresistance , which causes an error in fixing a

temperature in the normal way (58). To fix a stable temperature below 4 K, we choose to

tune the pressure of pumped helium bath and the temperature was fixed by holding the

helium gas pressure constant without being regulated by heater. The temperatures were

read before applying the magnetic field and checked again for consistency after a cycle of

magnetic field scan when the field returned to zero. For sample D1 with Tc = 2.37 K, the

temperature dependence at constant magnetic field B = 0 and 3 T was also measured

which clearly shows a field dependence of the resistance indicating good sensitivity of

the measurement system.

7

For all superconducting states that showed higher Tc, we obtained Bc2 by

measuring the temperature dependence of Rs at a fixed magnetic field. To compensate for

the magnetoresistance of the Cernox™ temperature sensor under high magnetic field (58),

we used temperature calibration made on the identical Cernox™ temperature sensor,

which corrected each Bc2 point on the temperature axis in the RS–T relationship (Fig. 3 B).

5. Fitting by the KLB theory

The Klemm-Luther-Beasley theory (12) describes stacked two-dimensional BCS

superconductors coupled through weak interlayer Josephson tunnelling, including

contributions from both Pauli paramagnetism and spin-orbit scattering. The model

assumes that in each layer, Cooper pairs propagate in dirty limit and spin-orbit scattering

(SOS) rate is much less than the moment scattering rate, i.e. Cooper pairs scatter many

times in each layers before tunnelling into an adjacent layer. It correctly captures the

dimensional crossover from 3D to 2D in intercalated TMD compounds, which manifests

itself as a strong upward curvature in / /

c2B T and explains the remarkable enhancement

of / /

c2 0B by spin-orbit scattering (35–38). Physically, SOS randomizes spin direction,

hence, prevent the alignment of spins in Cooper pairs by external magnetic field, which

weakens Pauli paramagnetism and helps Bc2 to go beyond the conventional Pauli limit

(11,13,33,59). The Bc2-T relationship for parallel magnetic field was formulated as

follows (12):

2 2 2 2c

1 1 1ln 0

0 2 2 2 2 22 2

a a a aT

T T Ta b a b

8

2 2

SO

1, ,

3Ba a a b a b B

(S1)

where SO is SOS time, x is the digamma function, and is the lowest eigenvalue

of the equation

2

2

c

21 cos 2 ,

2 0

h dy

dy rT

where 1 2

2 2

c2 0h eBs D rT s with the interlayer distance 𝑠 , 𝐷 is the diffusion

constant, and 𝑟 denotes the dimensionless parameter of interlayer coupling. When 1r ,

c

21

0rT

, the KLB theory reduces to the 2D superconductors case.

As shown in Fig. S5, using the KLB formula above, the fits give rise to a nearly

vanishing interlayer coupling 0r and a spin orbit scattering time SO 24 fs for both

high and low Tc superconducting states. The tiny 𝑟 indicates the 2D superconductivity of

the ion-gated MoS2. Although a short SOS time in KLB theory can analytically fit our

large Bc2 observed, it is not consistent with theoretical estimation of the SOS in MoS2

(34,60) and our transport measurement with sample of high mobility ~700 cm2/Vs, which

implies that the total scattering time is even longer than the SOS time alone.

Comparison with other intercalated TMD compounds and typical thin metallic

films can be referred to the following table S1. We also listed typical physical values of

all superconducting states in ion-gated MoS2 in table S2. According to the theory of

Abrikosov and Gor’kov (11), 4

SO 137 Z . For very heavy metallic films, ion gated

and organic molecule intercalated TMD shows SOS ratio almost 2 or 3 orders smaller

than that found in Au and Sn, respectively. For MoS2 with lighter metallic elements,

9

much larger ratio is expected. Thus the derived SO is too short, indicating there should

exist additional physical mechanism besides spin-orbit scattering, which enhances Bc2.

The above contradiction indicates that a compatible interaction should protect the

spin from Pauli paramagnetism without introducing scattering. This idea has been tried

by Bulaevskii et al. (45) where an out-of-plane effective magnetic field was introduced

(e.g. due to an in-plane lattice polarization from CDW states). It protects the

superconductivity from the in-plane magnetic field without including strong SOS. In our

work, a similar role is taken by the out-of-plane Zeeman-type intrinsic spin orbit

interaction.

6. Theory including Zeeman- and Rashba-type spin-orbit coupling

For a monolayer MoS2, first-principle calculations show that near the K and K’

points, the lowest two conduction bands are dominated by the spinful 2zd -orbitals. As a

good approximation, we describe the lowest two conduction bands by the following

normal state Hamiltonian

SO0 z RH k

k K g σ b σ (S2)

where 2

2m k

k denotes the kinetic energy with chemical potential , and effective

mass 𝑚 , , ,x y z σ are the Pauli matrices, , ,0y xk k g denotes the Rashba

vector (always lying in-plane), R and SO are the strength of Rashba and intrinsic SOC,

respectively. The 1 is the valley index which equals to 1 at the K valley and –1 at

the K’ valley and Bb B is the external Zeeman field.

10

We introduce a phenomenological interaction Hamiltonian to induce

superconductivity. Suppose the superconducting state is singlet-pairing with the critical

temperature Tc (0) at zero magnetic field. Although the SOC can mix the spin-singlet with

spin-triplet pairings, it is reasonable for us to ignore the triplet pairing component as the

ratio between triplet and singlet pairings is proportional to that between the difference

and the sum of the densities of states of two Fermi surfaces split by the SOC, which is

usually very small. Furthermore, we assume the phase transition is always second order.

Then in the clean limit the relation between in-plane upper critical field 𝐵𝑐2 and critical

temperature Tc is given by the linearized gap equation:

SO

SO SO

2 2

c

c

ln 00

F

F F

T

T

g b

g b βg b

β

β, (S3)

where the arguments are defined as

c

SO SO

2

F F

T

g b g bβ β,

with the parameters

c2 SO SO,0,0 , , ,0 , 0,0,B F R F R FB k k βb g

SO

2 2

SO

2

c2F B R F R FB k k g bβ

and the function is related to the digamma function as

1 1 1

Re2 2 2

i

11

7. High doping states in gate-induced and bulk intercalated superconducting MoS2

In both gate-induced and alkali (earth) metal-intercalated MoS2, doping induced

carriers raise the Fermi level and occupy the low lying bands at Q and/or K valleys. In

bulk crystal, electrons firstly fill the bands at Q valleys, contributing to superconductivity

when carrier density reaches a critical value. Due to upshifting of Q points in monolayer

band structure, bands near K/K’ valleys take over the minima. Charge carriers initially fill

the K/K’ valleys instead, making them the contributing pockets to superconductivity as

long as charge carrier density surpasses the quantum critical point (QCP in Fig.1A).

With increasing carrier density and enhanced Tc after QCP, the gradually elevating

Fermi level EF may touch the second band minima at Q valleys near the middle of K-

lines as well. The filling of Q pockets are also influenced by lowering of band minima at

Q points as a result of doping-induced modification of band structure (25,29). Therefore,

superconducting state induced close to the QCP complies mostly with the model of Ising

superconductivity having Zeeman type protection at K/K’ valleys. On the other hand, Q

valleys have an even larger SO 80 meV (61), hence our physical picture remains valid

that effective Zeeman field strongly pins the spins of Cooper pairs in out-of-plane

direction protecting superconductivity against external in-plane magnetic field.

In intercalated bulk TMD, superconductivity can appear in both conduction and

valence bands, which are contributed by multi-band carriers. For instance, in intercalated

TaS2 compounds, both and K valleys of valence bands actually cross Fermi surfaces,

similar to those in pristine NbSe2. While G valley is spin-degenerate, K/K’ valleys

exhibit spin splitting if in-plane inversion symmetry is lowered by intercalating organic

molecules. In table I of Ref. (37), organic intercalation compounds can have very large

12

interlayer distances ranging from 9.7, 12.0 to 18.1 Å by doping collidine, pyridine and

aniline, respectively. Expanding the lattice distance can effectively weaken the interlayer

coupling and lower the symmetry along out of plane direction. Hence, effective Zeeman

field in individual layer–originally cancelled out due to interlayer coupling–can be

partially resumed, which effectively protects the superconducting state against an

external in-plane magnetic field. In alkaline (earth) metal intercalated MoS2 (41), similar

effect was observed when lattices are expanded by different dopants. Comparing different

kinds of alkali (earth) doping, Cs intercalation shows the highest Bc2 far above other

smaller dopants, in consistent with stronger symmetry lowering by expanding lattices

using an element of large radius.

13

Fig. S1

Typical bipolar transfer characteristics of EDLT devices. With VG = 0 V, MoS2 flakes are

usually slightly electron doped. The insulating regime spans around 2 V corresponding to

the direct bandgap of MoS2 monolayer. Forward (red) and backward (black) scanning of

top gate lead to an observable hysteresis due to the slow movement of the large ions in

ion liquid. Nevertheless, repeatability was found in most devices between different runs,

indicating the chemical stability of ion liquid within the chemical window.

14

Fig. S2

Phase diagram of the EDLT device D24 (Tc = 7.38K). Bc2-Tc curves in a perpendicular

(black square) and a parallel (red circle) magnetic field are fitted by the 2D GL theory

with c2 0 8.6 TB and / /

c2 0 48 TB . The derived coherence length GL 0 6.2 nm is

more or less twice of Tinkham 3.8 nmd ， indicating the two dimensionality the of

superconductivity. Taking into account the absence of paramagnetism in the GL theory,

the estimated thickness is an upper bound for the actual thickness of the superconducting

layer. Dashed green line denotes the slope of / /

c2B close to critical temperature around 500

T/K. We can estimate that

c

0

/ /

/ / c c2

0 0.014 nm0 2

TT dB dT

and in KLB theory

coupling strength between layers 04

0.002/ 2

rs

. Note that the estimation from

WHH theory c

/ / / /

c2 c c20T

B T dB dT is not valid anymore, where 0 is

underestimated and interlayer coupling strength r should also be larger than 10-3

.

15

Fig. S3

Evolution of current-voltage characteristics from superconductor (red) to normal metallic

(purple) under perpendicular magnetic fields at T = 3 K. The exponent derived from

V I at B = 2.2 T has a critical value of 3 denoted by the black straight line, which is

similar to that in temperature dependent BKT transition without magnetic field. However,

TBKT = 3 K is much smaller than TBKT = 6.3 K in zero field, showing that perpendicular

magnetic field effectively decreases the transition temperature.

16

Fig. S4

Temperature dependent resistance at zero magnetic field (black) and in a parallel

magnetic field B = 3 T (red) for sample D1 (Tc = 2.37 K).

17

Fig. S5

Fitting Bc2/Bp–T/Tc phase diagram of sample D1 with Tc = 2.37K with KLB’s theory

without interlayer coupling. Evolution of the spin-orbit scattering time from 1 to 1000 fs

is plotted with the best parameters 𝜏𝑆𝑂 equal to 24 fs.

18

Table S1

In ionic gated MoS2, SO is obtained by KLB fitting of Bc2(T) and tr is from normal state

resistance which agrees well with 0.065 ~ 0.233 ps extracted from Reference (62). For

intercalated TaS2 compound (36), SO is obtained from fitted parameter SO c0T and tr is

from Reference (63). Parameters for Sn and Au films are from Ref. (64) and (65),

respectively.

Materials Heaviest element

(Z)

SO

(ps)

tr

(ps)

Ionic gated MoS2 Mo (42) 0.024 0.185

TaS2(Py)0.5 Ta(73) 0.05 0.004

TaS2(MeA)0.5 Ta(73) 0.04 0.004

Sn film Sn(50) 30 – 60 0.05

Au film Au(79) 1 – 10 0.03

19

Table S2

Physical parameters of D1 and D24 including superconducting transition temperature Tc,

2D carrier density 2Dn , energy of the superconducting gap , coherence length ,

normal state resistivity RN, the total scattering time tr , Drude mobility Drude and

diffusion constant D and spin-orbit scattering time SO fitted by KLB theory are

summarized for different five measurements shown in Fig. 1E.

Tc (50%)

(K)

n

(1014

cm-2

)

(meV)

(nm)

RN

(Ω)

tr

(ps)

SO

(ps)

Drude

(cm2V

-1s

-1)

D

(m2s

-1)

8.17 0.87 1.24 9.01 331 0.062 0.027 218 0.0058

7.64 0.85 1.16 - 760 0.028 0.021 98 0.0026

7.38 0.83 1.12 6.19 1266 0.017 0.042 60 0.0016

5.5 0.7 0.84 8.12 1723 0.015 0.034 53 0.0014

2.37 0.61 0.36 - 158 0.185 0.024 650 0.0173

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