Lecture schedule October 3 – 7, 2011

• #1 Kondo effect• #2 Spin glasses• #3 Giant magnetoresistance• #4 Magnetoelectrics and multiferroics• #5 High temperature superconductivity• #6 Applications of superconductivity• #7 Heavy fermions• #8 Hidden order in URu2Si2• #9 Modern experimental methods in correlated electron systems• #10 Quantum phase transitions

Present basic experimental phenomena of the above topics

Some Spectroscopy Studies of the HO State of URu2Si2

J. A. Mydosh

Kamerlingh Onnes Laboratory, Leiden University, The Netherlands

Introduction

Inelastic neutron scattering (spin)

Optical conductivity (charge)

Ultrasonic velocity (thermo.) [and attenuation (transp.)]

ARPES (charge)

STM/STS (charge and spin)

[But not PCS, & QO]

What is “Hidden Order” (HO)?

[See, e.g. N. Shah, P. Chandra, P. Coleman and JAM, PRB 6I, 564(2000).]

Now quite common usage of HO. Or as some theorists call it “Dark Quantum Matter” or as others call it “Novel Forms of Order” and “Novel Phases” {Reserve for high-field phases} or ‘‘Dark Order’’.

A clear, from bulk thermodynamic and transport measurements, phase transition at T0 where the order parameter (OP) and elementary excitations (EE) are unknown, i.e., cannot be determined from microscopic experiments.

Ψ is primary, unknown OP; m is antiferromagnetic, secondary OP

Key Unsolved Problems/Questions of HO in URu2Si2

• Local, dual or itinerant?• OP’s primary / secondary?• Mediator of phase transition?• INS resonance mode causing HO: Q0 or Q1 ?• How to probe OP experimental?• Relation of HO to LMAF (Adiabatic Continuity)?• Symmetry breaking in HO vs. LMAF?• Spin – charge duality?• HF Liq.(hybridization) or Kondo Liq. at coherence T*?• Kondo effect in (Th1-xUx)Ru2Si2?• Generic HO in other materials? Or is URu2Si2 unique?• Missing link experiments?(Hall effect under pressure, etc.)• Many theories/models -- which one is solution to HO?

Spin: Inelastic neutron scattering - “resonances” at Qo =(1,0,0) and Q1 =(1.4,0,0)

• Broholm et al. PRL & PRB(1987 – 1991)

• Wiebe et al. NP(2007)

• Bourdarot et al. JPSJ(2010) ?(2011)?

• Niklowitz et al. to be published(2011)

Excitation spectrum of URu2Si2 at 1.5K along (H,0,0)

Cones of excitations persist to higher T>To and E~10meV. Well-correlated itinerant-like spin excitations at Q1(incomm). Strongly coupled spin and charge degrees of freedom.

gapping

Resonance at E0 for magnetic response at Qo

Longitudinal mode at 1.5K with continuum of Q-E scattering persisting to higher energies.

Resonance at E1 for magnetic response at Q1

Longitudinal mode at 1.5K with continuum of Q-E scattering persisting to higher energies.

T-dependence of Qo resonance

Growth of intensity below To = 17.8K with Q-E continuum

T-dependence of resonance gap E0 at Qo

E0 represents a long lifetime (small decreasing half-width) collective mode rapidly reaching its final value 1.7 meV.

Integrated intensity of dynamical spin susceptibilityWhat about at Q1 incommensurate resonance?

Red line is a BCS-type gap fit giving T-dependence of HO-OP. No divergence of static spin susceptibility, i.e, HO non-magnetic.

Low energy excitations scanned through HO transition Niklowitz et al.(unpublished,2011)

Note peak at To for commensurate mode and step for incommen. mode

Pressure – temperature phase diagram

Collection of results by Niklowitz et al. PRL(2010).

KL

Pressure dependences of E0, E1 and bulk gap vaules

E0 disappears in LMAF phase, others persist. Note similar energy scales comparable to theoretical models.

LMAF Bragg peaks

HO

Charge: Optical Conductivity

• Bonn et al. PRL(1988)

• van der Marel et al. unpublished(2010 - 2011)

• Lobo et al. unpublished(2010)

• Timusk et al. cond-mat.(2011)

HO-gap in URu2Si2 measured through optical conductivity, D. A. Bonn et al. PRL (1988).

Preliminary data in a – a plane, gapping(~45cm-1) into HO phase. Strong phonons. Missing Drude peak and correlation gap

Van der Marel et al., private communication, 2011

Reflectivity to optical conductivity along a and c

Clear but slow crossover (opening) of hybridization gap at 44K, persisting into HO gapping regime (not seen here).

Extracting of scattering rate -1 as function of T & ωvia extended Drude model

Note decrease of -1 into hyb. gap

Optical conductivity along a and c-axes

Opening of correlation gap ~15meV(125cm-1), clearer along a. Note low energy Drude peak and phonon modes.

Optical conductivity 20 – 70K in hybridization gap region extrapolated to ω 0 via Drude peak analysis

Note opening of hydridization gap below 50K

W(ω) is loss of spectra weight accumulation

Relaxation rate governing the frequency dependent scattering in hybribization gap region

As T increases scattering becomes incoherent

Lower frequency (E) optical conductivity above To

Labo et al., private communication, 2010.

0 20 40 60 80 1000

1000

2000

3000

4000

5000

1 (-1

cm

-1)

Energy (meV)

20 K 25 K 30 K 50 K 100 K

Clear onset of hybridization gapping(~15 meV) below 50K. Drude peak forming at 2 meV(15 cm-1). Note phonons.

Low T, low E optical conductivity probing HO

0 5 10 15 200

1000

2000

3000

4000

5000

6000

7000

8000

1 (

-1cm

-1)

Energy (meV)

5 K 7.5 K 10 K 12.5 K 15 K 20 K

HO gapping ~5meV with transfer of spectral wt. to just above gap and shifting of Drude peak to smaller E. Need lower E & T!

Some conclusions

T – E dependences of optical conductivity

Note lack of intensity(conductivity) above To – correlation gap. No clear sign of HO gap. Need lower T and E.

Low E, high T

0 2 4 6 8 100

2000

4000

6000

8000

10000

1 (-1

cm

-1)

Energy (meV)

20 K 25 K 30 K 50 K 100 K

Low E, low T

0 2 4 6 8 100

2000

4000

6000

8000

10000

1 (-1

cm

-1)

Energy (meV)

5 K 20 K 50 K

Thermodynamics: Ultrasonics velocity (attenuation as transport prop.) Determination of elastic constants, c ij

• Lüthi et al. JLTP (1994)

• Kuwahara et al. JPSJ (1997)

Elastic constants (c = ρv2 ): c11, c33, c44; c66

Note c11 only longitudinal mode showing softening for T < 80K, min. 30K and HO shoulder.

Analysis of elastic constant cij behavior of URu2Si2

Need new interpretation here: softening due to slow opening of hybridization gap. No CDW?

Charge: ARPES

•J. Denlinger et al. JES&RP(2001)

•A. Santander-Syro et al. NP(2009)

•R. Yoshida et al. PRB(2010)

•Kawasaki et al. PRB(2011)

•G. Dakovski et al. PRB(to be published, 2011)

•XXX et al. ??? (2012)

Among the many difficulties of ARPES: URu2Si2 is 3D thus depending upon the energy tuning one scans an arc through the BZ (or changing detector angle).

Note in bct the high symmetry directions Γ, Z; X

Denlinger et al.(2001) – pioneering work

• Synchrotron scans 14 - 230 eV with ΔE > 50 meV at T > 20K.• Good resolution and DFT comparisons of 4d (Ru); 5d (U) lower

bands. Poor agreement with “old” LDA bands near EF.

• But Fermi surface mapping.• Insufficient resolution for near FS and qp studies.• Surface states/bands difficulties!• X hole pocket observed in FS, not confirmed!!!• Local 5f2 model!• Awaiting new results at SCES-2011.

Fermi energy intensity maps off(85ev) / on(112eV)-resonance, 5f enhancement

DFT-LDA calculations bold=hole; fine=electron

X-point descrepancy: distinct hole pocket; LDA : small elec. pocket, also pts. vs large contours

Comparisons ARPES vs (old) LDA

Santander Syro et al. (2009) – T dependences

• Temperature scan into HO state• He lamp low energy (21 eV), high resolution ARPES• Surface states, poor vacuum• Two k space directions: [100] and [110]

• Band of heavy quasi-particles drops below EF upon entering the

HO state• Large restructuring of FS in HO• Many difficulties with data and analyses• Reproducible?

Integrated photoemission spectra along <110> Note quasiparticle peak that moves below To: Dispersing band of heavy

QP, new electron pocket in HO state

Surface state Surface state

Heavy qp band hybridized with light hole conduction band along <110> at 13 K

ARPES intensity

EDC

MDCAverging of 2nd derivatives along E and k

Heavy qp band hybridized with light hole conduction band along <100> at 15 K

ARPES intensity EDC

Yoshida et al.(2010) – Laser Arpes

• Low energy (7 eV) Laser ARPES, high resolution (2 meV), good vacuum technique

• Narrow, dispersive band in HO only, few meV from FS• Yet non-FS crossing• Destroyed with Rh doping on Ru sites• Another hole-like dispersive crossing band and surface

states at ~35 meV• “Periodicity modification“: HO doubling of unit-cell, band

backfolding, predicted by Oppeneer et al. • Low energy ARPES is only sensitive to d-bands, cannot

detect 5f-U bands. Seeing broad (partially hybridized) 4d-Ru bands which appear in HO state

Laser ARPES intensity at 7K for [110] and [100]

Surface state

Hole-like dispersion

Temperature evolution of ARPES intensity integrated over different k cuts

Kawasaki et al.(2011) Soft X-ray ARPES

• Energy 760 eV with resolution 140 meV• Vary energy or detector angle to scan BZ• Spanning vast k-space, all of high symmetry BZ• Bands below 0.6 eV are Ru-4d states, agreeing with previous ARPES

• Band above 0.6 eV to EF disagree with previous ARPES, e.g.,surface band at not observed here. No hole band at X.

• All U-based 5f bands are itinerant!!! • Quasiparticle bands clearly observed at Z(large hole FS and at (large

electron FS) with some nesting• APRES bands consistent with LDA of Oppeneer et al.

BZ with orange and blue scanning planes. Spectral image comparison with LDA band structure

Measured spectral weight along hi-sym.

Calculated BS Agreement with LDA of Oppeneer

Bands 4, 5; 6 cross EF

Photoemission intensity with FS crossings and LDA comparison

Intensity around EF

Indicated band crossings

Calculated band crossings: 6, 5; 4 with C, B; A, and 4; 5 with D; E

Fermi surface images compared with LDA

Integrated intensity Estimated Fermi surfaces with nesting vectors

Band structure FS’’s

Dakovski et al.(2011) Time Resolved ARPES • Pump (1.55 eV)– Probe (29.5 eV) method. First for SCES

• Tune ARPES on URu2Si2 to focus on “hot spots” (maximium gap) in k-space, i.e., below Z in <110> plane as determined from band structure

• Excite quasiparticles via pump, probe their fs decay

• Measurements above To rapid fs decay within hybridization gap

• Measurements below To qp excited above HO gap have longer fs decay times

• Momentum (k) dependent interactions at hot spots causing HO gapping

• Energy resolution: tr-ARPES ≈100meV; ARPES ≈10meV

Femto second spectroscopy at hot spot in HO(12 K)

Comparison spectral intensity above and below To

Note q <110> = 0.56 separating two hot spots in <110>

ARPES (34eV,12K) at Z. Note flat band above EF and agree-ment with Kawasaki for lower bands.

Cartoon model for T evolution of hybrid. and HO gaps

Hot Spots

3D FS with hot spots

See Oppeneer et al. PRB(2010)

Conclusions drawn from ARPES

• Cleaving problem solved, requires ultra high vacuum• Surface states – solved?• Need better resolution at higher E-scans for FS mapping• Inconsistencies among measurements • Present data pushed too far• Yet striving towards efficacious solution of this difficult

technique (note 1990’s ARPES in HTS)• HO gapping not clearly found or hybridization gap seen• First tr-ARPES on heavy fermion material

Stop Thanks

Charge and Spin: Charge and Spin: Visualizing the HO in URuVisualizing the HO in URu22SiSi22

Aynajian, Yazadani et al. PNAS(2010)Aynajian, Yazadani et al. PNAS(2010)

Pegor Aynajian, Eduardo H. da Silva Neto, Colin V. ParkerDepartment of Physics, Princeton University

Yingkai Huangvan der Walls-Zeeman Institute, University of Amsterdam

Abhay PasupathyDepartment of Physics, Columbia University, New York

John MydoshKamerlingh Onnes Laboratory, Leiden University

Ali YazdaniDepartment of Physics, Princeton University

Supported by

Kondo-Fano resonance in URuKondo-Fano resonance in URu22SiSi22

Reminiscent of Fano lineshape in single Kondo impurities

2

2

)/)((1

)/)(()(

o

o

EV

qEVVG

Fano Lineshape

TK=120±10K

q : Ratio of tunneling probability to the descrete level and the continuum.

0 20 40 60 80 100

16

20

24

28

32

36

[m

eV]

Temperature [K]

2 (kBT )2 2(kBTK )2

q=1.3±0.3 ; Eo=5±2meV

D(V) = (V – V0 –iγ) / [(V –V0 –iγ)2 – Δ2]1/2 with γ ~ 1.5 mV

V0

Decomposition

Some ConclusionsSome Conclusions

Recent theoretical work:K. Haule and G. Kotliar, Nature Phys. (2009)M. Maltseva, M. Dzero, and P. Coleman, PRL (2009)Y-f. Yang, PRB(RC) (2009)J. Figgins and D. Morr, PRL (2010)

- Kondo resonance with Fano lineshape.

- Mean field-like T dependence of HO.

- HO asymmetric around EF.

- HO strongest between the surface atoms where the Kondo resonance is enhanced.

-60 -40 -20 0 20 40 60

0.4

0.6

0.8

1

Voltage [mV]

Con

duct

ance

The Hidden Order in URuThe Hidden Order in URu22SiSi22

Palstra et. al, PRL (1985)

Palstra et. al, PRL (1986)

Interplay of the U’s f electrons with the spd electrons and with each other, results in a rich variety of electronic phases.

Variable Temperature STM

Operates between 6K – 180K

Gomes et al. Nature (2007), Pasupathy et al. Science (2008), Pushp et al. Science (2009), Parker et al. PRL (2010)

STM topography on URuSTM topography on URu22SiSi22

Atomically ordered lattice:a~4.2Å corresponding to U or Si

200Å

100Å

0.6

-0.6

STM spectroscopy on URuSTM spectroscopy on URu22SiSi22

Averaged electronic density of states:

Above THO=17.5K Below THO=17.5K

-100 -50 0 50 1000

0.5

1

1.5

2

2.5

Voltage [mV]

Con

duct

ance

[nS

]

-100 -50 0 50 1000

0.5

1

1.5

2

Voltage [mV]

Con

duct

ance

[nS

]

120K

100K

85K

70K

60K

50K

40K

30K

20K

18K

15K

13K

11.7K

10.2K

8.4K

6.6K

-20 -10 0 10 201

2

3

4

5

6

7

8

9

10

Voltage [mV]

Con

duct

ance

[pS

]

6.6K

4K

2K

Entering the hidden order in URuEntering the hidden order in URu22SiSi22

-20 -10 0 10 200

0.5

1

1.5

2

2.5

3

3.5

4

Voltage [mV]

Con

duct

ance

18K

15K

13K

11.7K

10.2K

8.4K

6.6K

4K

2K

A gap in the DOS develops below THO

Entering the hidden order in URuEntering the hidden order in URu22SiSi22

HO turns on with a mean field-like temperature dependece.

Asymmetric gap around EF. Palstra et al. PRL (1985)Maple et al. PRL (1986)Bonn et al. PRL (1998)Wiebe et al. Nature Physics (2009)

0 4 8 12 16 200

1

2

3

4

Normalized by 18KFit to Fano x BCS

[m

eV]

Temperature [K]

Kondo lattice in URuKondo lattice in URu22SiSi2 2 ?

Topography Conductance at 6mV q map

1.3

1.2

1.1

1.6

1.4

1.2

nS

- Atomic scale modulations.

- q anti-correlated with topography.

- In single Kondo impurity limit, large q indicates higher tunneling probability to the Kondo resonance.

T=18K