Microscopic imaging homogeneous and single phase superuid density in UTe 2 Yusuke Iguchi12 Huiyuan Man13 S. M. Thomas4 Filip Ronning4 Priscila F. S. Rosa4 and Kathryn A. Moler125

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Microscopic imaging homogeneous and single phase superﬂuid density in UTe2

Yusuke Iguchi1,2, Huiyuan Man1,3, S. M. Thomas4, Filip

Ronning4, Priscila F. S. Rosa4, and Kathryn A. Moler1,2,5

1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA

2Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory,

2575 Sand Hill Road, Menlo Park, California 94025, USA

3Stanford Nano Shared Facilities, Stanford University, Stanford, CA 94305, USA

4Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

5Department of Applied Physics, Stanford University, Stanford, California 94305, USA

The spin-triplet superconductor UTe2shows spontaneous time-reversal symmetry breaking and

multiple superconducting phases in some crystals, implying chiral superconductivity. Here we mi-

croscopically image the local magnetic ﬁelds and magnetic susceptibility near the surface of UTe2,

observing a homogeneous superﬂuid density nsand homogeneous pinned vortices. The temperature

dependence of nsis consistent with an anisotropic gap and shows no evidence for an additional

kink that would be expected at any second phase transition. Our ﬁndings are consistent with a

dominant B3usuperconducting order parameter in the case of a quasi-2D Fermi surface and provide

no evidence for multiple phase transitions in ns(T) in UTe2.

Strong spin-orbit coupled unconventional super-

conductors, whose superconducting (SC) state can-

not be described by electron-phonon coupling, pro-

vide a platform for experimental and theoretical

studies of emergent quantum behavior [1, 2]. Time-

reversal and parity are key symmetries to charac-

terize these materials, and striking states of mat-

ter often emerge when one (or both) of these sym-

metries are broken. For instance, odd-parity su-

perconductors have been identiﬁed as a promising

route for topological superconductivity, which hosts

edge modes or vortices with non-abelian statistics

required for topological quantum computing [3]. A

chiral superconductor further breaks time-reversal

symmetry and lowers the energy of the SC conden-

sate by removing nodes from the gap function [4].

Odd-parity chiral superconductors are remarkably

rare, but their experimental manifestation has been

observed in superﬂuid 3He and actinide supercon-

ductor UPt3[5].

UTe2is a newly-discovered candidate for odd-

parity chiral superconductivity [6, 7]. Nuclear mag-

netic resonance (NMR) Knight shift measurements

strongly suggest that UTe2is an odd-parity su-

perconductor with a dominant B3uorder parame-

ter [8–10]. Point nodes in the SC gap structure are

supported by transport measurements [6, 11, 12],

Knight shift measurements [10], and non-local su-

perﬂuid density measurements [13]. The posi-

tion of the point nodes, however, is still contro-

versial. Thermal conductivity, microwave surface

impedance, and speciﬁc heat measurements sug-

gest point nodes in the ab-plane or along a[11–13],

whereas magnetic penetration depth measurements

argue for a multicomponent SC state with multiple

point nodes near the b- and c-axes [14].

Evidence for chiral superconductivity was found

in UTe2by scanning tunneling spectroscopy on the

step edges of a (0¯

11)-plane [15] and by polar Kerr ro-

tation measurements [16]. Multiple SC phase tran-

sitions were also reported in UTe2even at ambi-

ent pressure by speciﬁc heat measurements [16, 17].

Subsequently, it was found that the observed “dou-

ble peak” in the speciﬁc heat can arise from sam-

ple inhomogeneity [18]. In addition, a single phase

transition is reported in higher quality samples with

higher SC critical temperature Tc, higher residual

resistivity ratios, lower residual resistivities, and

quantum oscillations [7, 19].

To microscopically investigate the SC state of

UTe2, here we report the temperature dependence of

the local superﬂuid response using scanning SQUID

(superconducting quantum interference device) sus-

ceptometry on a cleaved (011)-plane of UTe2. We

also image the pinned vortices induced by ﬁeld cool-

ing. Our results show no evidence for multiple phase

transitions in the temperature dependence of the su-

perﬂuid density and imply an anisotropic nodal gap

structure in UTe2.

Bulk single crystals of UTe2were grown by chem-

ical vapor transport. Samples #1 and #2 used in

this paper were obtained from the same batch as

sample s2 in Ref. [19]. Heat capacity measurements

conﬁrmed a single SC transition at Tc= 1.68 K

with a width of 50 mK on a single crystal which

was subsequently cleaved into two samples used in

this study. We used a scanning SQUID susceptome-

ter to obtain the local ac susceptibility on a cleaved

(011)-plane of UTe2at temperatures from 80 mK to

2 K in a Bluefors LD dilution refrigerator. Our scan-

ning SQUID susceptometer has two pickup loop and

ﬁeld coil pairs conﬁgured with a gradiometric struc-

ture [20]. The inner radius of the pickup loop is 0.4

µm and the inner radius of the ﬁeld coil is 1.5 µm.

arXiv:2210.09562v1 [cond-mat.supr-con] 18 Oct 2022

FIG. 1. Local susceptibility is microscopically homogeneous on (011) surface of UTe2sample#1. (a) Optical image

of the scanned area, which includes the cleaved (011) surface with small bumps creating no signals in our scans and

the edge. (b-e) Temperature dependence of the susceptometry scan indicates the homogeneous superﬂuid density

on UTe2. Stripes along the scan directions are the scanning noise. (f) The temperature dependence of the local

susceptibility at the points from A to A’ in Fig. 1(e) has no kink below Tc. The susceptibilities are shifted by 0.2

Φ0/A for clarity except for the data at A.

The scan height is ∼500 nm. The pickup loop pro-

vides the local dc magnetic ﬂux Φ in units of the ﬂux

quantum Φ0=h/2e, where his the Planck constant

and eis the elementary charge. The pickup loop also

detects the ac magnetic ﬂux Φac in response to an

ac magnetic ﬁeld Heiωt, which was produced by an

ac current of |Iac|= 1 mA at a frequency ω/2π∼1

kHz through the ﬁeld coil, using an SR830 Lock-in-

Ampliﬁer. Here we report the local ac susceptibility

as χ= Φac/|Iac|in units of Φ0/A.

To obtain the homogeneity of the superﬂuid den-

sity and its local temperature dependence, we mea-

sured the local susceptibility near the edge of the

sample [Figs. 1(a)-1(e)]. The susceptibility far from

the edge has a homogeneous temperature depen-

dence on the micron-scale [Fig. 1(f)]. We note that

our results do not rule out possible inhomogeneity

either on the nanoscale or on scales larger than the

scan area (e.g sub-millimeter). Our data also can-

not rule out ﬂuctuations in time. There is no kink

in the temperature dependence of the local suscepti-

bility below Tc, wherein the temperature step of the

scans is 25 mK. The susceptibility is positive above

Tcdue to paramagnetism. The local susceptibility

was negative (diamagnetic) near the edge but posi-

tive far from the edge at 1.69 K [Figs. 1(c),1(f)].

We deﬁned the local Tcas the temperature that

satisﬁes the relation of χ(T > Tc)>-0.1 Φ0/A. The

local Tcmapping clearly shows that the local Tcis

weakly enhanced at the edge but is homogeneous

30 µm away from the edge into the sample [Fig. 2,

sample#1; Fig. S2, sample#2]. We note that the

reported local Tcnear the edge is a lower bound

relative to the actual value because the penetration

depth near Tcis longer than the pickup loop’s scale

and the susceptibility loses some signal at the edge.

If two phase transitions do exist, they must be closer

to each other than 25 mK, or the second kink below

Tcis much smaller than our experimental noise.

FIG. 2. Small enhancement of local Tcnear the edge of

sample#1. (a) The local Tcmapping is obtained from

the local susceptometry scans. (b) Cross section of the

local Tcfrom A to A’ shows the local Tcenhancement

of 30 mK at the edge. The plotted area and the cross

section from A to A’ are the same with Fig. 1.

Now we turn to the pinned vortex density, which

reﬂects the impurity density on the crystal surface

for small applied magnetic ﬁelds. The distance be-

tween vortices is on the order of microns. Our

magnetometry scan imaged the pinned vortices in-

FIG. 3. The vortex density is homogeneous over many-micron distances. The existence of vortices and antivortices

in low-ﬁeld scans may indicate a local magnetic source in the sample#1. (a,b) Local magnetometry scan after ﬁeld

cooling shows the vortices pinned parallel to the sample edge, as denoted by the dashed lines. (c) There are a few

vortices and antivortices pinned far from the edge after near zero ﬁeld cooling. (d) Magnetometry scan near zero

ﬁeld above Tcshows a small magnetic dipole at the sample’s edge, but no other indication of magnetism. The ”tail”

of the vortices and dipoles are due to the asymmetric shielding structure of the scanning SQUID [20].

duced by cooling in an applied uniform magnetic

ﬁeld from 2 K to 100 mK [Figs. 3(a),3(b), sam-

ple#1; Figs. S3(a),S3(b), sample#2]. The num-

ber of vortices corresponds to the applied ﬁeld, but

the vortices are preferentially pinned along lines in

one direction, which is parallel to the sample edge.

This linear pinning indicates the existence of a line

anomaly, such as nanometer-scale step edges along

crystal axes. Near zero magnetic ﬁeld, there are still

a few vortices and antivortices pinned far from the

edge [Fig. 3(c), sample#1; Fig. S3(c), sample#2].

Notably, these vortices and antivortices do not dis-

appear after zero ﬁeld cooling with slower cooling

rates, which is expected to cancel the uniform back-

ground ﬁeld normal to the sample surface by the

application of an external ﬁeld. These data are in-

consistent with the argument from polar Kerr eﬀect

measurements that there are no vortices in UTe2

within the beam size area (∼11 µm radius) [21].

Further, our results indicate the existence of a local

magnetic source that induces vortices and antivor-

tices, in spite of the absence of long-range order or

strong magnetic sources on the scan plane above

Tc[6, 22, 23]. Small dipole ﬁelds are observed at

the edge of the sample, which may stem from U

impurities; however, these impurities cannot induce

pinned vortices and antivortices as they are too far

away [Fig. 3(d), sample#1]. Muon spin resonance

and NMR measurements have detected the presence

of strong and slow magnetic ﬂuctuations in UTe2at

low temperatures [24, 25]. Therefore, a sensible sce-

nario is that these ﬂuctuations are pinned by defects

and become locally static.

To estimate the local superﬂuid density, we mea-

sure the local susceptibility at diﬀerent tempera-

tures with the pickup loop position ﬁxed. The local

superﬂuid density is obtained using the numerical

expression of the susceptibility assuming a homo-

geneous penetration depth, λ, as described below.

Kirtley et al. developed the expression for the sus-

ceptibility as a function of the distance between the

susceptometer and the sample surface [26]. In this

model, wherein the sample surface is at z= 0, we

consider three regions. Above the sample (z > 0),

the pickup loop and ﬁeld coil are at zin vacuum and

µ1=µ0, where µ0is the permeability in vacuum.

In the sample (−t≤z≤0), the London penetra-

tion depth is λ=pm/4πne2, and the permeability

is µ2. Below the sample (z < −t), there is a non-

superconducting substrate with a permeability µ3.

The radius of the ﬁeld coil and the pickup loop are a

and b, respectively. By solving Maxwell’s equations

and the London equation for the three regions, the

SQUID height dependence of the susceptibility χ(z)

is expressed as

χ(z)/φs=Z∞

dxe−2x¯zxJ1(x)−(¯q+ ¯µ2x)(¯µ3¯q−¯µ2x) + e2¯q¯

t(¯q−¯µ2x)(¯µ3q+ ¯µ2x)

−(¯q−¯µ2x)(¯µ3¯q−¯µ2x) + e2¯q¯

t(¯q+ ¯µ2x)(¯µ3q+ ¯µ2x),(1)

where φs=Aµ0/2Φ0ais the self inductance be- tween the ﬁeld coil and the pickup loop, Ais the

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MicroscopicimaginghomogeneousandsinglephasesuperuiddensityinUTe2YusukeIguchi1;2,HuiyuanMan1;3,S.M.Thomas4,FilipRonning4,PriscilaF.S.Rosa4,andKathrynA.Moler1;2;51GeballeLaboratoryforAdvancedMaterials,StanfordUniversity,Stanford,California94305,USA2StanfordInstituteforMaterialsandEnergySciences,SLACNa...

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Microscopic imaging homogeneous and single phase superuid density in UTe 2 Yusuke Iguchi12 Huiyuan Man13 S. M. Thomas4 Filip Ronning4 Priscila F. S. Rosa4 and Kathryn A. Moler125.pdf

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Microscopic imaging homogeneous and single phase superuid density in UTe 2 Yusuke Iguchi12 Huiyuan Man13 S. M. Thomas4 Filip Ronning4 Priscila F. S. Rosa4 and Kathryn A. Moler125

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