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Letters

Journal of the Physical Society of Japan 91, 083704 (2022)
https://doi.org/10.7566/JPSJ.91.083704

First Observation of the de Haas–van Alphen Eﬀect and Fermi Surfaces
in the Unconventional Superconductor UTe2
Dai Aoki1+ , Hironori Sakai2 , Petr Opletal2 , Yoshifumi Tokiwa2 , Jun Ishizuka3 ,Youichi Yanase4, Hisatomo Harima5 , Ai Nakamura1 , Dexin Li1 , Yoshiya Homma1 ,Yusei Shimizu1 , Georg Knebel6 , Jacques Flouquet6 , and Yoshinori Haga2
2

1
IMR, Tohoku University, Oarai, Ibaraki 311-1313, Japan
Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan
3
Faculty of Engineering, Niigata University, Ikarashi, Niigata 950-2181, Japan
4
Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
5
Graduate School of Science, Kobe University, Kobe 657-8501, Japan6
Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, F-38000 Grenoble, France(Received June 2, 2022; accepted June 24, 2022; published online July 12, 2022)

We report the ﬁrst observation of the de Haas–van Alphen (dHvA) eﬀect in the novel spin-triplet superconductor
UTe2 using high quality single crystals with a high residual resistivity ratio (RRR) over 200. The dHvA frequencies,
named α and β, are detected for the ﬁeld directions between c- and a-axes. The frequency of branch β increases rapidly
with the ﬁeld angle tilted from c- to a-axis, while branch α splits, owing to the maximal and minimal cross-sectional
areas from the same Fermi surface. Both dHvA branches, α and β reveal two kinds of cylindrical Fermi surfaces with astrong corrugation at least for branch α. The angular dependence of the dHvA frequencies is in very good agreementwith that calculated by the generalized gradient approximation (GGA) method taking into account the on-site Coulomb
repulsion of U = 2 eV. It indicates that the main Fermi surfaces are experimentally detected. The observed cyclotron
eﬀective masses are large in the range from 32 to 57 m0. They are approximately 10–20 times lager than thecorresponding band masses, consistent with the mass enhancement obtained from the Sommerfeld coeﬃcient, γ and thecalculated density of states at the Fermi level. The local density approximation (LDA) calculations of ThTe2 assuming
U4+ with the 5 f 2 localized model are in less agreement with our experimental results, in spite of the prediction for two
cylindrical Fermi surfaces, suggesting a mixed valence states of U4+ and U3+ in UTe2.

UTe2 attracts much attention because of the unusual
superconducting properties due to the spin-triplet state.1–3)
UTe2 is a heavy fermion paramagnet with a Sommerfeld
coeﬃcient   120 mJ K−2 mol−1. It crystallizes in the bodycentered orthorhombic structure with the space group, Immm
(No. 71, D252h ), where the U atom forms a two-leg ladderstructure along the a-axis. Superconductivity occurs below
Tc ¼ 1:5{2 K with a large speciﬁc heat jump.4) A highlightis the huge upper critical ﬁeld Hc2 with a ﬁeld-reentrant
behavior for H k b-axis (hard-magnetization axis). Superconductivity survives up to the ﬁrst-order metamagnetictransition at Hm ¼ 350 kOe,5) as a bulk property.6) The valuesof Hc2 highly exceed the Pauli limit, (∼30 kOe), for all ﬁeld
directions, suggesting a spin-triplet state. A microscopic
evidence for a spin-triplet state is obtained from NMR
experiments, in which the Knight shift is unchanged or
decreases very slightly below Tc for H k a, b, and c-axes.7)
Another highlight is the appearance of the multiple superconducting phases under pressure.8,9) Applying pressure, Tc
starts decreasing, and splits at ∼0.3 GPa. The lower Tcdecreases continuously, while the higher Tc increases up to
3 K around 1 GPa and decreases rapidly. At the criticalpressure Pc  1:5 GPa, superconductivity is suppressed andmagnetic order, most likely aniferromagnetic appears above
Pc .8–11) Under magnetic ﬁeld, multiple superconducting
phases show a remarkable ﬁeld response, displaying asudden increase of Hc2 at low temperatures.9,12) These
multiple superconducting phases are the hallmarks of aspin-triplet state with diﬀerent superconducting order parameters related to the spin and orbital degree of freedom.
It was ﬁrst pointed out1) that UTe2 is located at the verge ofthe ferromagnetic order and resembles ferromagnetic super-

conductors.13,14) However, no ferromagnetic ﬂuctuations areexperimentally established; instead antiferromagnetic ﬂuctuations with an incommensurate wave vector are detected ininelastic neutron scattering experiments.15,16) Furthermore,
above Pc , antiferromagnetic order is conﬁrmed directly bymagnetic susceptibility measurements.17) These results suggest that multiple ﬂuctuations, such as ferromagnetic,
antiferromagnetic ﬂuctuations, and valence instabilities,10,18)
exist in UTe2, playing important roles for the unusual
superconducting properties.
The electronic structure of UTe2 has been investigatedby angle-resolved photoemission spectroscopy (ARPES) at
20 K. The results obtained from the soft X-ray19) and thevacuum ultraviolet synchrotron radiation20) are contradicting,probably related to the diﬀerent inelastic mean free path ofthe photoelectrons. The ﬁne structure near the Fermi level
was unresolved in the soft X-ray experiments. On the other
hand, the high resolution ARPES experiments revealed twolight quasi-one dimensional bands and one heavy band at the
Fermi level, which is, however, inconsistent with the soft
X-ray ARPES experiments. Thus, no clear conclusion on theelectronic structure emerges up to now. The determination of
the Fermi surface topology at low temperatures through thedirect observation of quantum oscillations is highly desired,
which will be a key experiment to investigate the topologicalsuperconducting phenomena as well.
In order to clarify the electronic structure, we performed
de Haas–van Alphen (dHvA) experiments on new highquality single crystals. Clear dHvA oscillations weresuccessfully detected, and the angular dependence of thedHvA frequencies reveals two kinds of cylindrical Fermi
surfaces. The results are well explained by the generalized

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Letters

J. Phys. Soc. Jpn. 91, 083704 (2022)

D. Aoki et al.

200

500
(a)

300
200

10
5
0
0

100
0
0

1

2 3
T (K)

100

4

5

200

UTe2
70 mK

150

UTe2
J || a-axis

Hc2 (kOe)(µΩ cm)(µΩ cm)

400

100

50

300

-2

0.2

-1

0.3

0.04

-2

-1

Ce/T (J K mol )0.4

Ce/T (J K mol )

T (K)

0.02
0
0

0
0
c

(b)

1

60

90
a
Field Angle (deg)

0
30
60
a
Field Angle (deg)90bFig. 2. (Color online) Anisotropy of Hc2 for the ﬁeld direction from c- to
a-axis, and from a- to b-axis at 70 mK determined by the AC susceptibility
measurements in UTe2 (sample #2).

0.5
T (K)

0.1
0
0

30

2

3

Fig. 1. (Color online) (a) Temperature dependence of the resistivity forthe current along a-axis in UTe2 (sample #1). The residual resistivity ratio(RRR) is 220. The dotted line is the results of ﬁtting. (b) Temperaturedependence of the electronic speciﬁc heat in the form of Ce =T vs T in UTe2
(sample #1). The dotted line is the results of ﬁtting between 0.34 and 0.6 K
assuming Ce =T ¼ 0 þ BT 2 .

gradient approximation (GGA) calculation with the on-site
Coulomb repulsion, U ¼ 2 eV. The detected large cyclotron
eﬀective masses are consistent with the Sommerfeld
coeﬃcient.
High quality single crystals of UTe2 were grown at Oarai(sample #1) and Tokai (sample #2). The details of single
crystal growth will be published elsewhere.21) The dHvA
experiments were performed at low temperatures down to
60 mK and at high ﬁelds up to 147 kOe, as well as resistivity,
speciﬁc heat and AC susceptibility measurements.22) The
band calculations were done by the GGA+U method in
UTe223,24) and the local density approximation (LDA) methodin ThTe2 as a reference.25)
First we present the superconducting properties of our highquality single crystals. Figure 1(a) shows the temperaturedependence of the resistivity for the current along the a-axis.
The superconducting transition at Tc ¼ 2:06 K deﬁned byzero resistivity is very high and very sharp. The resistivity
follows the T 2 dependence at low temperatures below 3.5 K.
The residual resistivity 0 and the residual resistivity ratio
RRR ( 300K =0 ) are 1.7 µΩ·cm and 220, respectively.
Figure 1(b) shows the temperature dependence of theelectronic speciﬁc heat in the form of Ce =T vs T for
sample #1 after subtracting the phonon contribution. A sharp
and large single-jump at Tc ¼ 2:05 K with the width of
0.04 K and Ce =ðTc Þ ¼ 2:64 associated with a small
residual γ-value (0 ), which is only 3% of the normal stateγ-value (N ), is observed. The small 0 and high Tc insample #1 are compared to those in diﬀerent quality
samples.26) All these properties indicate the high quality ofour dHvA samples.

UTe2
70 mK

dHvA Signal (arb.units)

T (K)H || 56.8 deg
11.8

90

100

110

120

130

140

150

H (kOe)Fig. 3. (Color online) dHvA oscillations at 70 mK at diﬀerent ﬁeld angles
with the 4.5 deg step tilted from c- to a-axis in UTe2 (sample #2). Small uparrows indicate Hc2 .

Figure 2 shows the anisotropy of Hc2 at 70 mK for the ﬁeld
directions from c- to a-axis, and from a- to b-axis using thedHvA sample (#2) with Tc ¼ 2:05 K. Because of the high Tc ,Hc2 shifts to the higher ﬁeld, compared to the previous
results.2) Hc2 for a-axis reaches 118 kOe, and Hc2 for c-axis
exceeds our highest ﬁeld 147 kOe, most likely around
160 kOe. These high Hc2 values restrict our dHvA experiments, since the dHvA oscillations appears above Hc2 asshown later. The unusual minima were found around 50 deg
tilted from c- to a-axis, and 20 deg from a- to b-axis,
associated with a sharp maximum for H k a-axis. This isprobably a mark of the ﬁeld variation of the pairing strength
along the a-axis, which should be clariﬁed in the temperaturedependence of Hc2 with angular singularities in future
experiments.
Next we show in Fig. 3 the dHvA oscillations at diﬀerent
ﬁeld angles tilted from c- to a-axis. The clear dHvA signals
were observed at the ﬁeld angles between 11.8 and 56.8 deg
above Hc2 denoted by up-arrows. Even below Hc2 , the dHvA
oscillations are observed, but the amplitudes are strongly
damped. At higher ﬁeld angles close to the a-axis, no dHvA
oscillations were detected.

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J. Phys. Soc. Jpn. 91, 083704 (2022)

D. Aoki et al.

1.5
1

FFT Amplitude (arb.unit)

dHvA Signal (arb.unit)

(a)
UTe2
H || 26 deg from c to a-axis
0.06 K

Hc2

100

110

120

130

140

1.0

1'

0.5

'
0
0

150

(b)

5

10
7

H (kOe)dHvA Frequency (x 10 Oe)

Fig. 4. (Color online) Typical dHvA oscillations and the FFT spectrum in the ﬁeld range between 110 and 147 kOe at the ﬁeld direction tilted by 26 degfrom c- to a-axis in UTe2 (sample #1).

(a) Experiment(c) Calculation

UTe2
U = 2 eV

UTe2

ThTe2
hole
electronholeelectron

7

dHvA Frequency (x 10 Oe)

10(b) Calculation

2

hole

5

, '
1'
1

#1
#2
0
0
c

30

c

electron60

Field Angle (deg)

90 0
a c

hole

c

a

a

30

60

Field Angle (deg)

90 0
a c

30

electron60

90aField Angle (deg)Fig. 5. (Color online) (a) Angular dependence of the dHvA frequency in UTe2. Two samples, #1 (square) and #2 (circle) were used for H k a ! c andH k c ! a, respectively. The lines are guides to the eyes. Panels (b) (c) show the angular dependences of the dHvA frequency calculated by the GGA+U(U ¼ 2 eV) method23) in UTe2, and the LDA method in ThTe2,25) assuming the tetravalent U atom with the localized 5f 2 conﬁguration in UTe2, respectively.
The corresponding Fermi surfaces are depicted.

Figure 4 shows the typical dHvA oscillations and thecorresponding FFT spectrum at the ﬁeld angle of 26 deg tilted
from c- to a-axis. Four dHvA frequencies, named 1 , 01 , β,and 0 were detected.
Figure 5(a) shows the angular dependence of the dHvA
frequencies from c- to a-axis. The results are obtained using
two diﬀerent samples, #1 and #2. The sample #1 was rotatedfrom H k a- to c-axis, while the sample #2 was rotated fromH k c- to a-axis, in order to clarify the dHvA signals for bothH k a- and c-axes. Both results are in good agreement withhigh reproducibility. With increasing the ﬁeld angle, branch
1 slightly decreases ﬁrst, then increases up to the ﬁeld angle
50 deg. Branch β increases continuously up to 45 deg. Branch2 also increases with the ﬁeld angle, and shows a sharp
increase around 60 deg. The highest frequency reaches morethan 1  108 Oe, indicating a large cyclotron orbit. No dHvA
signal was detected around H k a-axis, which is also
conﬁrmed in the ﬁeld directions from a- to b-axis.27)

In order to determine the cyclotron eﬀective masses, thedHvA oscillations were measured at diﬀerent temperatures.28)
The results are summarized in Table I. The detected eﬀective
masses are very large in the range from 32 to 57 m0 ,indicating a direct evidence for a heavy electronic state froma microscopic point of view.
The Dingle temperature, TD , was also derived from theﬁeld dependence of the dHvA amplitude. At the ﬁeld angle of
26 deg, TD for branch 1 is 0.16 K. From the simple relations,
F ¼ ħc=ð2eÞSF , SF ¼ kF2 , mc vF ¼ ħkF , TD ¼ ħ=ð2kB Þ,and l ¼ vF , where SF , vF , and τ are the cross-sectional areaof the Fermi surface, Fermi velocity, and scattering life time,
respectively, we obtain the mean free path l as 850 Å,
indicating the high quality of our sample.
The angular dependence of the dHvA frequencies arecompared to those obtained from the calculations.
Figures 5(b) and 5(c) are the results from the GGA+U(U ¼ 2 eV) calculation for UTe223) and from the LDA

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Letters

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Table I. Experimental dHvA frequency F, cyclotron eﬀective mass mc ,calculated dHvA frequency Fb and band mass mb on the basis of the
GGA+U (U ¼ 2 eV) at the ﬁeld angles tilted by 26, 29, 44, and 61 deg fromc- to a-axis in UTe2.
ExperimentCalculation

F(107 Oe)

mc
(m0 )

Fb
(107 Oe)

mb
(m0 )

 ¼ 26 deg
1
β

3.08
3.99

32
48

3.80
3.93

3.0
2.5

 ¼ 29 deg
1
01
β
0
2

3.15
3.38
3.94
4.09
5.05

33
40
57
34
39

3.89

3.1

3.98
4.05
5.36

2.5
2.6
2.4

 ¼ 44 deg
1
2

3.31
5.85

32
36

4.84
5.70

4.3
2.9

 ¼ 61 deg
α

9.41

55

8.95

3.6

Branchcalculation for ThTe2,25) respectively. The experimental
results are fairly in good agreement with those of the
GGA+U (U ¼ 2 eV). Therefore we can assign the detected
dHvA branches as follows. Branch β is ascribed to theelectron Fermi surface with the cylindrical shape, giving riseto nearly the 1=cos  dependence by tilting the ﬁeld angle, θfrom c- to a-axis. Branches 1 and 2 originate from thesame Fermi surface, that is a cylindrical hole Fermi surface.
Since the Fermi surface is corrugated from the cylinder
shape, the dHvA frequency splits into 1 and 2 , whichcorrespond to the minimal and maximal cross-sectional area,
respectively, at low ﬁeld angles, when the ﬁeld is titled fromc- to a-axis.
The results of the LDA calculations in ThTe2, whichcorresponds to U4+ with the localized 5f 2 conﬁgurationin UTe2, shows a less agreement with the experimental
results.29) Nevertheless, two kinds of cylindrical Fermi
surfaces are quite similar to those by GGA+U (U ¼ 2 eV)
as well as DFT with large U (U ¼ 7 eV) calculations.30) Notethat the conventional LDA calculation predicts a Kondo
semiconductor in UTe2.2,25,31) Similarly, GGA23) and DFT30)
calculations without U also shows a band gap at the Fermi
energy. These band structures are totally inconsistent with theexperimental results, indicating that the strong correlation
should be taken into account in the calculations. Small pocket
Fermi surfaces predicted by other calculations19,32) are also
inconsistent with our dHvA experiments.
Assuming the two kinds of cylindrical Fermi surfaces,
which occupy approximately 20% of volume for each in
the Brillouin zone with the carrier compensation, one canroughly calculate the γ-value derived from each Fermi
surface, from the following equation,  ¼ kB2 V=ð6ħ2 Þmc kz .33)
Here V is the molar volume and kz is the length of Brillouinzone along c-axis. If we take mc ¼ 32 m0 and 48 m0 for thehole and electron Fermi surfaces, respectively, the obtained
γ-values are 40 and 60 mJ K−2 mol−1 for each. Thus, the total
γ-value is 100 mJ K−2 mol−1, which roughly agrees with thevalue of   120 mJ K−2 mol−1 in the speciﬁc heat measurements, indicating that our dHvA experiment detects the main
Fermi surfaces of UTe2.

D. Aoki et al.

The eﬀective masses can be compared to the band masses
from GGA+U (U ¼ 2 eV) as shown in Table I. The band
masses at the selected ﬁeld angles are in the range from 2.5
to 4:3 m0 , meaning that the mass enhancement, mc =mb areapproximately 10–20. This is also consistent with the mass
enhancement, =b obtained from the speciﬁc heat and thecalculated density of states at the Fermi level (b ¼ 8:1
mJ K−2 mol−1), where the electron correlation is taken into
account.
In the LDA calculations for ThTe2 without the electron
correlation, the band masses are much smaller. For instance,
mb ¼ 0:7 m0 is derived for the electron Fermi surface at
26 deg, while mc ¼ 48 m0 is obtained in the dHvA experiments. The mass enhancement, mc =mb is consistent with thatobtained from the Sommerfeld coeﬃcient, namely =b , inwhich b is 1.7 mJ K−2 mol−1.
A question is whether the anisotropy of the resistivity for
J k a, b, and c can be explained by these Fermi surfaces,
because the resistivity for J k c-axis, c is only twice larger
than that for J k a-axis, a , and is comparable to that forJ k b-axis, b at room temperature.34) At low temperature, theanisotropy increases, but not an order of magnitude. In c , therapid increase on cooling below 50 K with a maximum
around T   15 K is observed, suggesting that the electronic
state may change from 3D to 2D-like nature on cooling. Thisis also in agreement with the development of the low
dimensional antiferromagnetic ﬂuctuation, which only starts
developing below 60 K.16)
It should be noted that we cannot exclude the existence ofsmall pocket Fermi surfaces with heavy masses, which may
induce a Lifshitz transition under magnetic ﬁeld as proposedin thermopower experiments.35) This may also compromisewith possible topological superconductivity.
In summary, the dHvA oscillations were detected for theﬁrst time in UTe2. The angular dependence of the dHvA
frequencies from H k c- to a-axis, are in good agreementwith the results of GGA+U (U ¼ 2 eV) based on the 5f 3
itinerant model, revealing two kinds of cylindrical Fermi
surfaces from hole and electron bands. The detected holeFermi surface shows a large corrugation from the cylindricalshape. On the other hand, the dHvA results are in lessagreement with those of LDA calculations in ThTe2, whichcorresponds to U4+ with the 5f 2 -localized model. The
detected cyclotron eﬀective masses are quite large, indicating
heavy electronic states, consistent with the γ-value of thespeciﬁc heat. These suggest a mixed valence states of U4+and U3+, as proposed in the core-level spectroscopy,36) which
are sensitive to external parameters, such as pressure andﬁeld. A link between our dHvA results on small energy scalenear the Fermi level and the high energy spectroscopy19,20,36)
deserves to be clariﬁed. Our results elucidate a key part tosolve the UTe2 properties, as it happened when the Fermi
surface was determined on high Tc superconductors37) andCe-115 heavy fermion systems.38)
Acknowledgements We thank Y. Ōnuki, S.-i. Fujimori, V. Mineev, Y.
Tokunaga, M. Kimata, K. Ishida, K. Izawa, A. Miyake, J. P. Brison, D.
Braithwaite, A. Pourret, I. Sheikin, and S. Fujimoto for fruitful discussion. This
work was supported by KAKENHI (JP19H00646, JP20K20889, JP20H00130,
JP20KK0061, JP22H04933), GIMRT (20H0406), ICC-IMR, and ANR(FRESCO).

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J. Phys. Soc. Jpn. 91, 083704 (2022)

Letters+

aoki@imr.tohoku.ac.jp
1) S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz, S. R. Saha,
I.-L. Liu, M. Zic, H. Kim, J. Paglione, and N. P. Butch, Science 365,684 (2019).
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Q. Niu, S. Rousseau, A. Demuer, G. Seyfarth, G. Lapertot, D. Aoki, D.
Braithwaite, J. Flouquet, and J. P. Brison, arXiv:2205.04524.
7) G. Nakamine, K. Kinjo, S. Kitagawa, K. Ishida, Y. Tokunaga, H.
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Commun. Phys. 2, 147 (2019).
9) D. Aoki, F. Honda, G. Knebel, D. Braithwaite, A. Nakamura, D. Li, Y.
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G. Knebel, and D. Braithwaite, Commun. Phys. 4, 40 (2021).
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13) D. Aoki and J. Flouquet, J. Phys. Soc. Jpn. 81, 011003 (2012).
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16) W. Knafo, G. Knebel, P. Steﬀens, K. Kaneko, A. Rosuel, J.-P. Brison,
J. Flouquet, D. Aoki, G. Lapertot, and S. Raymond, Phys. Rev. B 104,
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17) D. Li, A. Nakamura, F. Honda, Y. J. Sato, Y. Homma, Y. Shimizu, J.
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18) A. Miyake, M. Gen, A. Ikeda, K. Miyake, Y. Shimizu, Y. J. Sato, D.

19)
20)

21)
22)
23)
24)

25)
26)
27)

28)
29)

30)
31)
32)
33)
34)

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