Discontinuous change of Hall coefficient across the field-induced phase boundaries in U(Ru1-xRhx)2Si2

Kee Hoon KimSchool of Physics, Seoul National University, Seoul 151-742, South Korea

Yoon Seok Oh Seoul National University

Neil Harrison NHMFL Los Alamos Marcelo Jaime ″Peter Sharma ″John Mydosh U. of KoelnHiroshi Amitsuka Hokkaido University

Collaborators

APCTP-KIAS Workshop on “Quantum Materials” July 19, 2007

• Backgrounds of URu2Si2 (Γ5 model)• Metamagnetic quantum criticality in URu2Si2• Multiple phase formation in U(Ru,Rh)2Si2• Hall effect studies: discontinuous FS change in U(Ru,Rh)2Si2• Implications to HO

Outline

A.A. Menovsky, single crystals grown by Czochralski “tri-arc” method, 1985.

W. Schlabitz et al., 4th Int. Conf. on Val. Fluc. Cologne, 1984. (polycrystals)

URu2Si2, Structure and Heavy Fermion StatesBody Centered Tetragonal

T. T. M. Palstra et al., PRL 55, 2727 (1985); PRB 33, 6527 (1986)

U(Ru,Rh)2Si2: Kondo lattice system, forming 5f quasiparticle bands with Ising like spin degrees of freedom (very anisotropic g-factor; gz=2.6, gx=gy=0)

[N. Harrison et al., cond-mat/0506384]

0 50 100 150 200 250 3000

100

200

300

4000

2

4

6

8

10

I//c-axis

ρ xx (

μΩ c

m )

T (K)

I//a-axis

URu2Si2

// a-axis

χ DC( 1

0-3em

u/m

ol f.

u. )

// c-axis

a= 4.124 Å , c=9.582 Å at 4.2 KTcoh~50K

J. Schoenes et al. PRB (1987)

URu2Si2, Hidden Order (HO) Phase

Palstraet al.,PR

L55, 2787 (1985); Ram

irez et al., PRL 68,2680 (1992);K

. H. K

im(unpublished).

Small AFM moment 0.03 μB can’t account for large entropy change

[Amitsuka et al. Physica B 312,390(2002)].

Numerous proposals for the HODipolar NondipolarOrbital antiferromagnetismUnconventional density waveHelical order Electric quadrupole order…

[Broholm et al. PRB 43, 12809(1991)]

0 5 10 15 20 25567890

20

40

600

200

400

600

χ 1 ( 10

-3 e

mu/

mol

T )

T (K)

URu2Si2

ρ (μ

Ω c

m)

C/T

( m

J/m

ol K

2 )

0.2R ln2

Specific Heat

Susceptibility

Resistivity

THO = 17.5 KTC

URu2Si2, Microscopic Magnetism: key facts

Antiferromagnetic Phase is MINORITY PHASE !!NMR intensity dependent on pressure while internal field (splitting) is independent; AFM Bragg peaks seen by neutrons is from minority phase

[Matsuda et al. PRL 87, 087203(2001)]

29Si NMR under P

One dominant picture (still under debate):

At P=1 atm, 1 % AF phase coexists with the paramagnetic HO phase (very strain sensitive)

HO competes with AFM with P or σ

P: hydrostatic pressureσ: Uniaxial pressure (c/a ratio)

P (σ)1bar 10bkar

U(Ru1-xRhx)2Si2 : Effects of Rh doping ( x < 0.05 )

Rh doping: increase of c/a ratio + disorder + carrier densitySensitive chemical tuning parameter of electronic structure;

important for present studies

0

5

10

15

20

0 0.01 0.02 0.03 0.04 0.05

T o(K

)

Rh concentration x

To

TM

HiddenOrder

HO+AF

HeavyFemion

M. Yokoyama et al., J. Phys. Soc. Jpn 73, 545 (2004)

Dilute-uranium-limit properties of URu2Si2

0

2

4

6

8

10

12

0.1 1 10 100 1000T (K)

R = Th

YLa

B//c R1-xUxRu2Si2open:filled:

χ c5f

( 10-2

emu

/ mol

U )

URu2Si2

x = 0.07x = 0.03

0

5

10

15

0 100 200 300 400

T (K)

Th

Y

R = La

B//a URu2Si2

R0.97U0.03Ru2Si2χ a5f

(10-4

emu

/ mol

U )

Strong uniaxial magnetic anisotropyχc → ∞ (T → 0) in the dilute U limitHeavy electron state at low T and NFL behavior in χc

(interpreted as the two-channel Kondo effect)

Quite consistent with Γ5 CEF ground states

(non-Kramers doublets)

Amitsuka and Sakakibara (1994)

Γ5 Crystal Electric Field Doublets of U4+ (5f 2)

cubic (O) tetragonal (D4) orthorombic (D2)

J = 4Γ1

Γ4

Γ5

Γ3

Γ1

Γ5

Γ2Γ3

Γ5Γ1Γ4

Γ1

Γ3

Γ4Γ2Γ3Γ4Γ3Γ1

Γ4

Several experiments suggest that U(Ru,Rh)2Si2 has Γ5 CEF ground states of U4+ (5f2)

5 cos 3 sin 1α α⎧ ↑⎪Γ ± = ± + = ⎨

↓⎪⎩∓

5f 2 3H4 (S=1, L=5, and J=4)

Γ5 non-Kramers doublets

Jz

+ gz

- gz

gz = gJ (3cos2α - sin2α)=2.6

Jx, y Ogx = gy = 0

Γ5 explains Ising anisotropy since g-factors vanish in basal plane

5 cos 3 sin 1α α⎧ ↑⎪Γ ± = ± + = ⎨

↓⎪⎩∓

Γ5 possesses spin and quadrupolemoments for which quadrupole moments modeled using Sx,y pseudospin moment [Ohkawa 1999, Amitsuka, Sakakibara (1994)]

12Hmulti = - Σ Σ AΓ; ij OΓ; iOΓ; j

Hspin = - Σ Σ Jλ; ij Sλ; iSλ; jλ= x,y,z i,j

12

U4+ : 5f 2

Oz = Jz

Ox2-y2 = (J+2 + J-

2)12

Oxy = (J+2 - J-

2)- i2

mz

- mz

q1

q1

-iq2

iq2

∝ Sz

∝ Sx

∝ Sy

S = 12

pseudo-spin

gx = gy = 0gz = 0

5 cos 3 sin 1α α⎧ ↑⎪Γ ± = ± + = ⎨

↓⎪⎩∓

, ,

12Zeeman B i

i x y zH g H Sλ λ λ

λ

μ=

= − ∑ ∑

jmj multipole expansion of HCEF

Γ5 antiferro-quadrupolar ordering modelfor HO

Hspin =- Σ Σ Jλ; ij Sλ; iSλ; jλ= x,y,z i,j

12

XY

( |Jx| = |Jy| > |Jz| )

⟨ Sx ⟩2 + ⟨ Sy ⟩2 = 0√

Ising

( |Jx| = |Jy| < |Jz| )

⟨ Sz ⟩ = 0

quadrupoleorder

magneticorder

Γ5Sx, Sy

Δ ΔSz

detectable by NS

⟨ Sz ⟩ = 0

Tunable through pressure, magnetic fields etc

Γ5 quadrupolar ordering model for HO

AF and ferro. clustersserving as domain walls

of quadrupole order

weak dipolar internal fields

AF Quadrupole scenario for HO is consistent with the fact that HO is sensitive to pressure and strain

P: hydrostatic pressureσ: Uniaxial pressure (c/a)

P1bar 10kbar

Rh x

HFHOHO+AF

0.040.0

URu2Si2, Key Energy Scales

Key energy scales at H=0 T, P=1 atm

W~Tcoh~50 K

THO=17.5 K

TC≈1.8 K

T

0K

HO state in URu2Si2 has a gap in FS; pressure sensitive; competes with AFM; consistent with Γ5CEF levels.

High magnetic fields of up to 50 T; Zeeman splitting ΔE=gμBB~50 K

Pulsed and static magnetic fields : Specific heat, magnetization, resistivity, and Hall effect experiments.

H50T

Metamagnetic Transition above Phase III in URu2Si2

χ=dM/dB confirm the new phase (phase III) boundary observed through heat capacity study.

N. Harrison et al, PRL90, 06402 (2003)

32 34 36 38 40 420

5x10-3

1x10-2

0.5

1.0

1.5

H (T)

χ =

μ0dM

/dH 7 K

8

11 9

H || c

M (μ

B /

U)

8

0.5 K

7

URu2Si2

HM

New surprise:Metamagnetism above phase III, similar to those of CeRu2Si2 and Sr3Ru2O7Possibility of metamagnetic quantum criticality?

M. Jaime, K. H. Kim et al., PRL89, 287201 (2002); J. S. Kim et al., PRB 67, 014404 (2003)

Magnetization, susceptibility and transport resemble that seen in Sr3Ru2O7,[S. Grigera et al. Science 294, 329(2001)]

Maximum in γ (Sommerfeld coefficient) was also observed.

Heavy Fermion Metamagnetism : e.g. CeRu2Si2

[J. Flouquet et al. Physica B 319, 251(2002)]

Effective mass undergoes divergent behaviour[H. Aoki et al, PRL 1993]

Generic Picture of Metamagnetic QC in Heavy Fermions

[Edwards and Green, Z.Phys.B103,243(1997)]

Hybridized band model

Fermi surface changeZeeman splitting Δε > W (f-electron bandwidth): entire spin component of f-band can be depopulated, causing virtual f-electron localization

Tcoh

(1) System changes from HFL to PFL states forming a QCP?: continuous or discontinuous? A symmetry-breaking order parameter ?

(2) Fermi liquid scale vanishes as H → Hm on both sides of Hm

localized f ?itinerant f

[A. V. Alehandro et al., PRL 2005]

Motivated us to perform careful transport studies!!

Metamagnetic quantum criticality in URu2Si2?

Multiple phases formed near the putative QCP in URu2Si2

30 32 34 36 38 40 42 440

2

4

6

8

10

12

14

16

FermiLiquid (IV)

ρmax

μ0Η (T)

URu2Si2

III

T (K

)

HiddenOrder (I)

II

V

T*

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

0

10

20

30

40

50

38.9

39.5

42

44.5

37.736.6 50

0

T (K)

38.4

50

0

35.7

0

ρ (μ

Ω c

m)

URu2Si2

50

32.5T

(1) Multiple new phases (phase II & phase V), were found.(2) Highly hysteretic 1st order phase boundaries in (HO, II, III, V)

Metamagnetictransition

(3) Suggests the hidden metamagnetic QCP; but identification is not easy.

Resistivity intensity plot

[K. H. Kim et al. PRL 2002]

HO

HFL

NFL

PFL

H

III

II

V

45 T

Motivation to U(Ru1-xRhx)2Si2T

4%

CAN THE COMPLEX PHASE FORMATION CAN BE SIMPLIFIED?Is the creation of new phases linked to the metamagnetic QCP?

0

HFHO

HO+AFHO

HO+AF

HFL

Rh x

0.040.0

Tcoh

0

2

4

6

0

5

10

15

20

20 25 30 35 40 45

U(Ru1-xRhx)2Si2

B (T)

T (K

)

HO II III

BM

T*

ρmax

x = 2 %

A-1

/2 (

KμΩ

-1/2cm

-1/2)

0

5

10

15

20

T (K

)

HOII

ρmax

BM

T*

x = 2.5 %

III0

2

4

6

A-1

/2 (

KμΩ

-1/2cm

-1/2)

0

5

10

15

20

T (K

)

HO

T*ρ

max

II

BM

T*

x = 3 %

0

2

4

A-1

/2 (

KμΩ

-1/2cm

-1/2)

20 25 30 35 40 450

5

10

15

20

B (T)

T (K

)

T*

ρmax B

M

T*

II

x = 4%

0

2

4

A-1

/2 (

KμΩ

-1/2cm

-1/2)

BII Center of phase II

BQCP Putative QCP from ρmax, T*, A-1/2

BM Metamagnetic transition field

K H Kim et al PRL 93 126404 (2004)

Phase Diagram of U(Ru,Rh)2Si2

A clear example showing a link between BII (effective center of phases) and BQCP !!

Nexus among BM , BII , and BQCP

U(Ru1-xRhx)2Si2 K. H. Kim et al. PRL 93, 126404 (2004)

20 25 30 35 40 45

0.1

1

10

100

A-1/2

A (μ

Ωcm

K-2

)

μ0H (T)

5

10

15

20 x = 4 %

T (K

)

T*

ρmax HM

T*

II0

2

4

A-1

/2 (K

μΩ-1

/2cm

-1/2)

U(Ru1-xRhx)2Si2

ρ=ρ0+AT2

K. H. Kim et al. PRL 93, 126404 (2004)

Avoid effective mass divergence by forming a phase

Quasiparticle instability (i.e. large density of states, heavy electron mass) is avoided inside phase II.

PFLHFL

T* crossover temperature fromT 2 (FL) to NFL (dρ/dT|max)

Collapse of T*, A-1/2 with the same scaling |H-HQCP|α1

ρmax scales with |H-HQCP|α2

T * ∝ A-1/2 (∝ 1/D(εF) ∝ εF ∝ 1/m*)

U(Ru0.96Rh0.04)2Si2 : A new example of a single phase formation at the metamagnetic QCP

Analagous to phase formation (i.e. superconductivity) as a function of pressure or doping, where new phase conceals QCP.

Extreme complexity results when other phases (such as HO) come within close proximity

Two Fermi liquids: one paramagnetic and one with polarized f-electrons

New phase forms so as to avoid quasiparticle instability (i.e. large density of states, mass divergence, fluctuations)

Phase II

Motivation to Hall Effect Studies

FS change discontinuously[H. Aoki et al, PRL 1993]

longitudinal Hall

FS change continuously[R. Daou et al., PRL 2006]VS.

CeRu2Si2

YbRh2Si2

A jump in at the QCP

from AFM to heavy Fermi liquid ground state.

Consistent with the local criticality picture.

xyddBρ

[S. Paschen et al, Nature 432, 881 (2006) ]

AFM HFL

0 5 10 15 20 25 30 35 40 45

0.00000

H//c, I//ab

18

2.54

55.566.578

91011

12

14

20 K

0000000

URu2Si2

ρ xy (Ω

cm

)

0

μ0H (T)

00

00

0

μ0H (T)

Shape of ρxy at high temperatures is similar to ρxx

Hall Resistivity: U(Ru1-xRhx)2Si2 , Rh 0 %[Y. S. Oh et al. to be published]

0 5 10 15 20 25 30 35 40 45

0

0.5

URu2Si2

360

ρ xx (μ

Ωcm

) 360

1.62.545

67891011121418

I//ab, H//c

20 K

Fig. 1. Kim, Harrison, and Mydosh

0

μ0H (T)

ρxy suggestive of large gap in the Fermi surface inside HO

Hall Resistivity: U(Ru1-xRhx)2Si2 , 4 %[Y. S. Oh et al. submitted]

0 10 20 30 400

20

0

20

B (T)

1.5

5

7

8.5

9.5

11

ρ xy( μ

Ω c

m )

13

16.2

21

T=30K

U(Ru0.96Rh0.04)2Si2

Shape of ρxy at high temperatures is similar to ρxx.

0 10 20 30 40

ρ xy (A

.U.)

B (T)

T=30.0 24.0 21.0 18.0 16.2 14.0 13.0 12.0 11.0 10.0 9.5 9.0 8.5 8.0 7.0 6.0 5.0 4.0 3.0 2.22 1.5 0.64

ρxy suggestive of large gap in the Fermi surface inside phase II

Large FS gap inside HO & phase IIThree different linear slopes of ρxy strongly suggest the Fermi surface with different volumes in each phase region.

Key Features of Hall Resistivity: Rh 0 & 4 %

[Y. S. Oh et al. to be published]

/

1

H xyR B

ne

ρ=

∝

Rh 0 % Rh 4 %

Information on Fermi Surface for Rh 0 %

Rh 0 %

Experiments observe mostly light hole bands with m*=13 (α) 25 (β) 8 2m (γ)

H. Ohkuni et al., Phil. Mag. B 79, 1045 (1999)

Key Features of Hall Resistivity: Rh 0 & 4 %

[Y. S. Oh et al. to be published]Rh 0 %Hall effect is dominated by the most mobile α band.

*

, , and 120,25, and 35

is t T fo

he dHvA frr the ,

equen

cy of each band.

( 2.6, experimentally determined

, and .

)

F z B P

p

z

he g

F

g

B

F Bm

ε

αβ

σμ

β γα γ

≈

≈

= =

A sudden depopulation of a carrier pocket, will cause the chemical potential to re-equilibriateitself amongst the remaining pockets, leading to changes in their sizes and net observable changes in RH.

β

γ

Rapid increase in carriers in the highest mobility α pocket is the primary cause in the destruction of the HO state.

Rh 4 %Rh 0 %

Hall Coefficients: U(Ru1-xRhx)2Si2 , Rh 0 and 4 %

Abrupt RH change suggests a discontinuous Fermi volume change. A jump in carrier number (Δn ~1 el/U) across phase II or metamagnetismsupports the possibility of FS reconstruction.

Hall Coefficients: U(Ru1-xRhx)2Si2 , Rh 0 and 4 %[Y. S. Oh et al. to be submitted]

nHall~0.15/U in Phase II in both compoundsnHall~1.0 el/U in PFL of both compoundsIn URu2Si2, nH increases with polarization

Implications from the Hall Effect Studies

3. In URu2Si2, the order parameter becomes increasingly less efficient at gapping FS as it becomes more polarized at each transition, finally yielding ~1 hole per U beyond 39 T at ~1.5μB per U.

1. URu2Si2 is a compensate semi-metal ; nHall~0.20/U at T>THO. It became more extreme by formation of HO;, nHall~0.03/U at T<THO

2. RH has well defined plaeaux in each phase region, and sharply jumps, establishing a link among the carrier density, degree of polarization and order parameters.

HO forms a gap in the α pocket, possibly through hybridization with the local XY ordering of Γ5 CEF levels (density wave like?).

Clues to Phase II & Γ5 CEF Doublets

Groundstate of UPd2Si2very well described in terms of Γ5 CEF doublets, albeit without Kondo effect

cubic (O) tetragonal (D4) orthorombic (D2)

J = 4Γ1

Γ4

Γ5

Γ3

Γ1

Γ5

Γ2Γ3

Γ5Γ1Γ4

Γ1

Γ3

Γ4Γ2Γ3Γ4Γ3Γ1

Γ4

U(Ru1-xRhx)2Si202.5%4%

Ms/3

Ms

[Honm

aet al.,J.Phys.Soc.Japan

67,1017(1998)][K. H. Kim et al. SCES05, preprint]

Opening of FS partial gap results in the increase phonon thermal conductivity in the Rh 0 % and PrFe4P12

[Pourret et al. PRL 2006]

Very similar to the case of URu2Si2:

1. Semimetal due to heavy el channel and light hole channel

2. Large Nernst effect

3. Itinerant carriers (FS) are reduced with the AFQ order (no magnetic order).

4. Increased phonon thermal conductivity

[Sharma et al. PRL 2006]

Proposal for Multiple Phases

Purely local moment picture assumed

Tem

pera

ture

Magnetic Field0

HOII

III

c-axis

AFQ

FerriM 1/3 (Msat)

RHO ~1/2 (Msat)

FerroM 1 (Msat)

HMItinerant picture requires one to consider quasiparticles with e, Jz and Jx,yquantum numbers

1/3 state perhaps like that seen in UPd2Si2 with neutrons [Honma et al. JPSJ 67 (1998) 1017]

U(Ru1-xRhx)2Si2

x = 0x = 0.025x = 0.04

ΔMII

Ms

1) U(Ru,Rh)2Si2 shows the nexus between phase formation and the field-induced quantum criticality:i) Quantum criticality similar to other itinerant metamagnetsii) Unambiguous evidence for phase formationiii) Nexus between phase formation and QCP, via avoiding quasiparticle

instabilityiv) Discontinuous Hall constant jump comparable to Δn ~1 electron/U.

strongly supports the significant FS reconstruction across phase boundaries .

2) Further thoughts:

ii) Quantum criticality with FS reconstruction?.iii) Theory of field-induced transition from heavy FL to spin-polarized FL;

A new order parameter?

i) Our data support the Γ5 doublet ground states (5f2) in U(Ru,Rh)2Si2[i.e. 1/3rd Msat in phase II, large hysteresis in phase II, highly anisotropic g value]

HO can be related to the itinerant version of antiferro-quadrupolarordering of Γ5 doublets.

Summary