Extremely Large Magnetoresistance and Anisotropic Transport in Multipolar Kondo System PrTi 2Al20 Takachika Isomae1 Akito Sakai2 Mingxuan Fu12Takanori Taniguchi3 Masashi Takigawa145 and Satoru Nakatsuji12678

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Extremely Large Magnetoresistance and Anisotropic Transport in Multipolar Kondo System

PrTi2Al20

Takachika Isomae1, Akito Sakai2, Mingxuan Fu1,2Takanori Taniguchi3, Masashi Takigawa1,4,5, and Satoru Nakatsuji1,2,6,7,8∗

1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

2Department of Physics, Faculty of Science and Graduate School of Science,

The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

3Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan

4Institute of Materials Structure Science, High Energy Accelerator Research

Organization (KEK-IMSS), Oho, Tsukuba, Ibaraki 305-0801, Japan

5Toyota Physical and Chemical Research Institute, Nagakute, Aichi 480-1192, Japan

6CREST, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan

7Institute for Quantum Matter and Department of Physics and Astronomy,

Johns Hopkins University, Baltimore, MD 21218, U.S.A

8Trans-scale Quantum Science Institute, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

(Dated: August 29, 2023)

Multipolar Kondo systems offer unprecedented opportunities to design astonishing quantum phases and func-

tionalities beyond spin-only descriptions. A model material platform of this kind is the cubic heavy-fermion

system PrT r2Al20 (T r =Ti, V), which hosts a nonmagnetic crystal-electric-ﬁeld (CEF) ground state and

substantial Kondo entanglement of the local quadrupolar and octopolar moments with the conduction electron

sea. Here, we explore magnetoresistance (MR) and Hall effect of PrTi2Al20 that develops ferroquadrupolar

(FQ) order below TQ∼2K and compare its behavior with that of the non-4fanalog, LaTi2Al20. In the FQ

ordered phase, PrTi2Al20 displays extremely large magnetoresistance (XMR) of ∼103%. The unsaturated,

quasi-linear ﬁeld (B) dependence of the XMR violates Kohler’s scaling and deﬁes description based on carrier

compensation alone. By comparing the MR and the Hall effect observed in PrTi2Al20 and LaTi2Al20 , we con-

clude that the open-orbit topology on the electron-type Fermi surface (FS) sheet is key for the observed XMR.

The low-temperature MR and the Hall resistivity in PrTi2Al20 display pronounced anisotropy in the [111] and

[001] magnetic ﬁelds, which is absent in LaTi2Al20, suggesting that the transport anisotropy ties in with the

anisotropic magnetic-ﬁeld response of the quadrupolar order parameter.

The quest for material platforms exhibiting large magneto-

transport has pushed progress in both fundamental science and

technological applications. The outstanding examples, such

as giant magnetoresistance in magnetic multilayers, are typ-

ically engendered by the interplay of the spin structure with

charge transport1,2. Recent studies uncover XMR in nonmag-

netic metals and semimetals, some featuring novel topological

band structure, such as Dirac or Weyl nodes3–10. Neverthe-

less, a universal understanding of the mechanism behind the

observed XMR is lacking. Aside from the spin and charge de-

grees of freedom, electron orbitals are a critical ingredient for

creating new quantum phases and functionalities in strongly

correlated systems11–13. Since the electronic band structure

ﬁnds its root in the interplay between orbitals and the crys-

tal lattice, the ordering and ﬂuctuations of orbitals are ex-

pected to yield remarkable effects on transport properties. In

3dtransition metal compounds, however, the spin, orbital, and

charge degrees of freedom are inextricably intertwined, thus

hindering a clear understanding of how orbital ordering and

ﬂuctuations tie in with novel transport phenomena. In con-

trast, cubic 4frare-earth materials may host a nonmagnetic

crystal-electric-ﬁeld (CEF) ground state with high-rank multi-

polar moments, offering a route to materialize novel transport

phenomena of a purely orbital origin14–17.

The multipolar Kondo system PrTi2Al20 provides a suitable

stage for investigating orbital ordering and its ties to exotic

electronic transport. In this system, the cubic Tdsymmetry of

the Pr site stabilizes a non-Kramers Γ3doublet ground state

that carries quadrupolar and octupolar, but no magnetic dipo-

lar moments18. This nonmagnetic ground-state doublet is well

separated from the ﬁrst-excited magnetic triplet by a CEF gap

of ∆CEF ∼60 K, and thus governs the low-temperature prop-

erties of the system18–20. A ferroquadrupolar (FQ) order with

the order parameter O20 (3J2

x−J2,3J2

y−J2, and 3J2

z−J2)

develops below TQ∼2K at zero-magnetic ﬁeld20–22, with

a superconducting transition inside the FQ phase23. More-

over, the cage-like local structure maximizes the number of

Al ions surrounding the Pr 4fmoments, leading to substan-

tial Kondo entanglement of the multipolar moments with the

conduction (c) electrons and formation of heavy quasipar-

ticles, as experimentally conﬁrmed by various experimen-

tal probes18,24–27. Pressure tuning of PrTi2Al20 results in a

rich phase diagram featuring strongly enhanced superconduct-

ing transition temperature Tcand quasiparticle effective mass

m∗on approaching the FQ phase boundary and robust non-

Fermi-liquid (NFL) behavior over a wide parameter range28.

The multipolar Kondo effect and quantum critical ﬂuctuations

originating from the orbital degrees of freedom are essential

in generating the observed exotic superconductivity and NFL

state.

On the other hand, magnetotransport phenomena in

PrTi2Al20 have not been explored. A giant anisotropic mag-

netoresistance ratio (AMR) of about 20% is recently reported

in the sister compound PrV2Al20 under a [001] magnetic

ﬁeld29, similar to that observed in the nematic order in iron-

based superconductors12. This AMR is believed to be driven

by quadrupolar (i.e. orbital) rearrangement and the accom-

panied Fermi surface (FS) change29, which opens intrigu-

arXiv:2210.12436v2 [cond-mat.str-el] 28 Aug 2023

ing prospects of exotic magnetotransport stemming from the

interplay of FS properties with the FQ order in PrTi2Al20.

However, investigations into the FS properties of PrT r2Al20

(T r =Ti, V) are particularly challenging. Density functional

theory (DFT) calculation of the FS is obscured by the large

number of atoms in one unit cell and the strong electronic

correlation. Experimentally, a recent de-Haas-van Alphen

(dHvA) study on PrTi2Al20 and LaTi2Al20 reveals a com-

plex FS comprising multiple electron and hole sheets in both

materials27. The resolved FS sheets in PrTi2Al20 shows sim-

ilar geometry as those in LaTi2Al20, while some electron FS

sheets in PrTi2Al20 have enhanced cyclotron effective mass

∼8−10m0, indicating their sensitivity to the c-fhybridiza-

tion effect27. Moreover, NMR22, magnetization22, and spe-

ciﬁc heat30 measurements indicate that an applied magnetic

ﬁeld of about 2 T along certain orientations induces a discon-

tinuous switching of the FQ order parameter, likely accompa-

nied by a change in the c-fhybridization which may cause

the reconstruction of FS. This possible ﬁeld-induced FS re-

construction due to changes in FQ ordering structure remains

an open question; quantum oscillations are observed only for

B≳2 T, thus offering no evidence for the potential low-ﬁeld

FS changes27. MR and Hall effect are effective alternatives for

probing FS properties; comparing their behavior in PrTi2Al20

and LaTi2Al20 may yield more profound insights into the in-

terplay of FS and electrical transport properties with multipo-

lar ordering and ﬂuctuations.

In this letter, we report transverse MR and Hall effect

ρHin high-quality single-crystal PrTi2Al20 and its non-4f

analog LaTi2Al20. Comparing the observed features for

these two materials reveals that open orbits on the electron-

type FS sheets are essential for inducing the three orders-of-

magnitude increase of transverse MR across the FQ transition

in PrTi2Al20. Moreover, the MR and Hall effect in PrTi2Al20

develop strong anisotropy on approaching the FQ phase, inti-

mately linked to the sharply distinct behavior of quadrupolar

order and ﬂuctuations in different magnetic ﬁeld orientations.

The details about material synthesis and experimental meth-

ods are in the Supplementary Materials.

Figure 1 shows the temperature Tdependence of the zero-

ﬁeld resistivity ρfor PrTi2Al20 and LaTi2Al20. The zero-ﬁeld

Pr-4felectron contribution to the resistivity, ρ4f(T)is ob-

tained by subtracting the resistivity curve of the isostructural,

non-4fanalog LaTi2Al20 from the raw data of PrTi2Al20. In

the high-Tregime of T≳∆CEF ∼60 K, ρ4f(T)shows

a logarithmic increase in cooling and reaches a broad peak

at T∼∆CEF. The behavior of ρ4f(T)in this T-regime is

governed by the magnetic Kondo effect arising from the ex-

cited magnetic triplets18. Below ∆CEF ∼60 K, the excited

triplet states become less relevant, leading to competing multi-

polar and magnetic Kondo effects. As a result, ρ4f(T)settles

into a Fermi-liquid (FL) regime with ∼T2behavior for 7 K

≲T≲17 K, rather than exhibiting the ∼T1/2non-Fermi

liquid behavior expected for a quadrupolar Kondo lattice31

(Fig. 1, inset). The T2coefﬁcient Ais about 200 times the

value found in LaTi2Al20, consistent with heavy fermion for-

mation reported by previous speciﬁc heat and dHvA exper-

iments23,27. The sharp exponential decay of ρ4f(T)below

0 . 1 1 1 0 1 0 0

2 0

4 0

6 0

T ( K )

( µΩc m )

P r T i 2A l 2 0

L a T i 2A l 2 0

4 f

L a T i 2A l 2 0

4 f

~ T 2

B = 0 T

I || [110]

= 0+A T 2

0 100 200 300 400

1 0

1 5

2 0

2 5

4f ( µΩc m )

T2 ( K 2)

0 . 0

0 . 5

1 . 0

1 . 5

2 . 0

( µΩc m )

FIG. 1. Temperature Tdependence of the zero-ﬁeld resistivity ρ

for PrTi2Al20 (solid red circles) and LaTi2Al20 (solid blue circles).

The 4f-electron contribution ρ4f(solid yellow circles) is obtained by

subtracting ρof LaTi2Al20 from that of PrTi2Al20 . The resistivity of

LaTi2Al20 is measured down to 2 K. To obtain ρ4fbelow 2 K, we

estimated the resistivity of LaTi2Al20 below 2 K based on the T2

ﬁt (ρ=ρ0+AT 2) shown in the inset, with ﬁtting parameters A=

1.7×104µΩcm/K2and ρ0= 0.31 µΩcm. The downward arrow

marks the ferroquadrupolar (FQ) transition temperature TQ∼2K.

The solid black line represents the Fermi liquid behavior ρ4f∝T2.

The inset shows ρvs. T2for PrTi2Al20 (solid yellow circles) and

LaTi2Al20 (solid blue circles). The solid black lines represent the

T2ﬁts at 7 K ≤T≤17 K for PrTi2Al20 and 2 K ≤T≤20 K for

LaTi2Al20.

TQ∼2K marks the entry into the FQ ordered state with

ceased quadrupolar-ﬂuctuation scattering. We note that ρ(T)

deviates from the T2dependence and shows upward convex

curvature ρ∼Tn(n≲1) for TQ< T ≲7K. This behav-

ior can be attributed to a crossover from the FL state driven

by competing magnetic and quadrupolar Kondo effects to a

non-Fermi-liquid (NFL) state stemming from the quadrupo-

lar Kondo effect, as predicted by numerical renormalization

group calculations32,33. Previous speciﬁc heat measurements

reveal an entropy release in the same temperature range, sup-

porting this scenario18. In the close neighborhood of TQ, crit-

ical quadrupolar ﬂuctuations associated with the FQ transition

might also inﬂuence the behavior of ρ(T), while their effects

are unlikely to persist up to as high as 7K. The overall behav-

ior of ρ4f(T)is consistent with the previous report18.

The transverse MR of PrTi2Al20 measured under B∥[111]

and B∥[001] are shown in Fig. 2(a), (c) and (d). We ﬁrst

focus on the behavior observed for B∥[111]. In the high-

Tregime dominated by the magnetic Kondo effect, the MR

exhibits quadratic ﬁeld dependence MR ∝B2(Fig. 2(a),

inset and Fig. S1(a)). Once the multipolar Kondo effect kicks

in below ∆CEF ∼60 K, the MR develops a crossover from

the low-ﬁeld B2behavior to a quasi-linear ﬁeld dependence

(Fig. 2(a)); the crossover shifts to a lower ﬁeld on cooling.

Below TQ∼2K, the window of B2behavior completely

vanishes, and unsaturated quasi-linear MR persists up to 16 T.

Remarkably, the magnitude of MR undergoes three orders of

magnitude enhancement across the FQ transition, reaching ∼

103%at 0.1 K (Fig. 2(a)); this value falls in the typical range

103−108%of extremely large magnetoresistance (XMR)3.

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ExtremelyLargeMagnetoresistanceandAnisotropicTransportinMultipolarKondoSystemPrTi2Al20TakachikaIsomae1,AkitoSakai2,MingxuanFu1,2TakanoriTaniguchi3,MasashiTakigawa1,4,5,andSatoruNakatsuji1,2,6,7,8∗1InstituteforSolidStatePhysics,UniversityofTokyo,Kashiwa,Chiba277-8581,Japan2DepartmentofPhysics,Faculty...

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Extremely Large Magnetoresistance and Anisotropic Transport in Multipolar Kondo System PrTi 2Al20 Takachika Isomae1 Akito Sakai2 Mingxuan Fu12Takanori Taniguchi3 Masashi Takigawa145 and Satoru Nakatsuji12678

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