Optical spectroscopy study on the charge dynamics of URu2Si2

Nan-Lin Wang

Institute of PhysicsChinese Academy of Sciences

• Some basics about optical spectrscopyOptical constants, Kramers-Kronig transformation, inter- and intra-band transitions

• Example of application: URu2Si2

Collaborators:

W. T. Guo, Z. G. Chen (optical measurements)

G. Luke (single crystals)

Some basics about optical spectroscopy technique

D. N. Basov, Richard D. Averitt, Dirk van der Marel, Martin Dressel, Kristjan HauleRev. Mod. Phys., 2011

Absorption mechanisms associated withvarious excitations and collective modes insolids optical experiments

(i) measure both R(ω) and T(ω) for transparent materials

(ii) measure R(ω) or T(ω), then use Kramers-Kronig (KK) transformation

(iii) spectroscopic ellipsometry

(iv) THz time-domain spectroscopy…

complex optical constants

Optical spectroscopy of solids

Units： 1 eV = 8065 cm-1 = 11400 K1.24 eV=10000 cm-1

Optical constantsOptical constantsConsider an electromagnetic wave in a medium

index refractive :)n( ),c/n(/q vwhere

)(

0)/(

0)(

0

ωωω

ωωω

=≡

===−−− t

cnxitvxitqxi

y eEeEeEE

If there exists absorption,

K: attenuation factor

)(

0

tcnxi

cKx

y eeEE−−

=ω

ω

cKx

y eEEω2

20

2I−

=∝

))((

0 t

cxNi

y eEE−

=ω

ω

)()()( ωωω iKnN +≡Introducing a complex refractive index:

x

y E

Intensity

Reflectivity

If n, K are known, we can get R, θ; vice versa.

12

)1()1(|)(||/|

)1()1(

11

)(

22

22

2222

)(22

22

)(

−+=

+++−

==∝

+++−

=++−+

=

=≡

KnKtg

KnKnrEER

eKnKn

iKniKn

errEE

inref

i

i

in

ref

θ

ω

ω

ωθ

ωθ

1/ 2

1/ 2

1/ 2

11 2 cos

2 sin1 2 cos

RnR R

RkR R

θθ

θ

−=

+ −−

=+ −

Dielectric function

)()(2)()()()(

))()(()()()(

)()(

)()0,(,0,),(),(),(

2

221

221

ωωωεωωωε

ωωωεωεωε

ωωε

ωεωεεωωεω

KnKn

iKni

N

qqphotonqEqqD

⋅=−=

+=+≡⇒

=

=→=→≡

Q

0 π/a~1Å-1

Infrared q=2π/λ~10-4 Å-1

1 2

1 2

2 21

2 21

1 ( ) ( ) ( )2

1 ( ) ( ) ( )2

n

k

ε ω ε ω ε ω

ε ω ε ω ε ω

= + +

= + −or

Conductivity

ωωσπ

ωε

ωσωσσ)(i41)( amics,electrodynBy

)()( 21

+=

+=

ωωσπ

εωε)(i4)( += ∞

In a solid, considering the contribution from ions or from high energy electronic excitations

Now, we have several pairs of optical constants:

n(ω), K(ω)R(ω), θ (ω)ε1(ω), ε2(ω)σ1(ω), σ2(ω)

Usually, R(ω) can be easily measured experimentally.

High-ω extrapolations:R~ ω-p (p~0.5-1, for intermedate region)

R~ ω-n (n=4, above interband transition)

Low-ω extrapolations:Insulator: R~ constant

Metal: Hagen-Rubens

Superconductor: two-fluids model

2 20

ln ( ') ''

RP dω ωθ ω

π ω ω

∞=

−∫

For optical reflectance

( ) ( )ln ( ) (1/ 2) ln ( )

ir R er R i

θω ω

ω ω θ

=

⇒ = +

Kramers-Kronig relation-- the relation between the real and imaginary parts of a response function.

21 2 2

0

' ( ')d '2( )'

P ω α ω ωα ω

π ω ω

+∞

=−∫

12 2 2

0

( ')d '2( )'

P α ω ωωα ω

π ω ω

∞−=

−∫

Reflectivity measurement

From R(ω) to σ1(ω)

Si-beamsplitter for 113vBruker 113, 66v, 80v, and grating spectrometers

Energy range: 17 -50000 cm-1 (2 meV~6 eV)

C. C. Homes et al.Applied Optics 32,2976(1993)

FT-IR spectrometer In situ evaporation

In-situ overcoating technique

Optical measurement under magnetic field(10 Tesla split coils from Cryomagnetic Inc.)

KxFe2-ySe2

R. H. Yuan et al., Scientific Reports 2, 221 (2012)

2∆

Tc~30 K

Δk

Δω

Xk,Ee- k’,E

k,Ek’,E’

q,ωqe-

(a) Impurity-assisted absorption

(b) boson-assisted absorption

Holstein process, if phonons are involved.

Intraband transition

Eghν

Ec(k)

Ev(k)

Interband transition

| , , ( )| , , ( )

v

c

v k E kc k E k

>>

( ) ( ) ( )c v cvE k E k E kω = − ≡h

3( )(2 ) | ( ) |k cv

V dsJ EE kπ

=∇∫

2 22

2 2 2

8( ) ( ) | ( ) |vce J p

mπ

ε ω ω ωω

=r

h h

Kubo-Greenwood formula

)(41)( 21 ωωεπ

ωσ =

Infrared light cannot be absorbeddirectly by electron-hole excitation.

4( ) ( )iπε ω ε σ ω

ω∞= +

22

21

2

22

2

1

114

/1

τωω

τω

ωπσ

ε

τωω

εε

+==

+−=⇒ ∞

p

p

R1

ωp’

ε1/ε∞

1/ τρdc

-1

σ1(ω)

(m*/m)Neff

energy

Im(-1/ ε)

'2

2 '2 2 2 2

'

/1Im ( ) ( )

/

p

p

p p

ω ω τ

ε ω ω ω ω τ

ω ω ε

−

∞

−=

− +

=

2

10

( )8

pdω

σ ω ω∞

=∫

20 1( )

1 4 1/D

i iσ ω

σ ωωτ π τ ω

= =− −

Drude model

0 10000 20000 30000 400000.0

0.2

0.4

0.6

0.8

1.0

R

ω (cm-1)

gold

0 5000 10000 15000 200000.0

0.2

0.4

0.6

0.8

1.0

Rω (cm-1)

YBCO Bi2212

Simple metalSimple metal HighHigh--TTcc cupratescuprates

Simple metal

Correlated metal

2

1*01

10

( , )

( , )

W

co

WB

co

T T dmm

T T d

σ ω ω

σ ω ω

>=

<<

∫

∫

W1 W2

or for heavy fermions

Symmetry broken stateSuperconductor vs density wave state

- for case I

+ for case II

Ba0.6K0.4Fe2As2, SC gapAFe2As2 (A=Ba, Sr), SDW gap

W.Z.Hu et al.

PRL (08)

G. Li et al. PRL (08)

URu2Si2

Palstra et al.

ØItinerant AFM

ØCDW/SDW

ØU-moment AFM

C. Broholm et al

Kasahara et al.

Semimetal

Aynajian et al, PNAS 2010

A number of experiments indicate gap opening below TH

Optical spectroscopy

Bonn et al. PRL 1988

STM

2009

WK Park et al. arXiv:1110.5541

Point contact tunneling spectroscopy

However, some other experiments indicate gap formation up to a higher T~25 – 30 K

A gap would impedes the relaxiation of excited quasiparticles.

J. Levallois et al PRB 2011

0 10000 20000 30000 40000 500000.2

0.4

0.6

0.8

1.0

R

ω (cm-1)

300 K 200 K 100 K 70 K 50 K 20 K 15 K 8 K

Reflectance

0 1500 3000 4500 6000 75000.6

0.7

0.8

0.9

1.0

R

ω (cm-1)

300 K 200 K 100 K 70 K 50 K 20 K 15 K 8 K

0 100 200 300 400 5000.8

0.9

1.0

20 K 15 K 8 K

R

ω (cm-1)

300 K 300 K 200 K 100 K 70 K 50 K

Samples from G. Luke

Optical conductivity

0 1500 3000 4500 6000 75000

3000

6000

9000

12000

0 10000 20000 300000

4000

8000

12000

16000

ω (Ω

-1 c

m-1)

ω (cm-1)

50 K

20 K 15 K 8 K

σ 1 (Ω-1cm

-1)

ω (cm-1)

300 K 200 K 100 K 70 K 50 K

ØT>50 K, σ1(ω) shows a non-metallic behavior. The Drude component is completely absent. Actually, σ1(ω) shows a decreasing tendency with decreasing ω. Clearly, there are no well defined quasiparticles above 50 K.

ØWith T from 300 K to 50 K, the spectral weight is transferred from the low-ωregime (below 2000 cm-1) to higher ω region (centered at about 4000 cm-1).

0 1500 3000 4500 6000 75000

3000

6000

9000

12000

σ 1 (Ω-1cm

-1)

ω (cm-1)

300 K 50 K 8 K

0 1500 3000 4500 6000 75000

3000

6000

9000

12000

0 10000 20000 300000

4000

8000

12000

16000

ω (Ω

-1 c

m-1)

ω (cm-1)

50 K

20 K 15 K 8 K

σ 1 (Ω-1cm

-1)

ω (cm-1)

300 K 200 K 100 K 70 K 50 K

0 100 200 300 400 5000

3000

6000

9000

12000

20 K 15 K 8 K

σ 1 (Ω

-1cm

-1)

ω (cm-1)

300 K 200 K 100 K 70 K 50 K

Ø T<< Tco, roughly below 20 K, there appears an abrupt spectral weight suppression below 400 cm-1. formation of hybridization gap. Ø A small part of the suppressed spectral weight was transferred to the low-ωside, leading to a narrow Drude component, while the majority of the suppressed spectral weight was still transferred to the high ω side centered near 4000 cm-1.

Thes energy scale of the spectral weight transfer is high compared to some 4f electron-based heavy fermion systems, e.g. CeCoIn5.

CeCoIn5

Singley, et al. PRB 2002

0 3000 6000 9000 120000

20

40

60

spec

tral w

eigh

t (x1

06 Ω-1 c

m-2)

ω (cm-1)

70 K 20 K 8 K

10 100 1000 100000

1

2

3

Spe

ctra

l wei

gh ra

tio

ω (cm-1)

20 K/70 K 8 K/70 K

Spectral weight transfer

21

0

( )W

p dω σ ω ω∝ ∫W=2000 cm-1 for T=50 K and 300 K;

W=135 cm-1 for T=20 K

W=42 cm-1 for T=8 K

3 14.28 10p cmω −= ×

4 11.95 10p cmω −= ×

3 12.094 10p cmω −= ×

0 1500 3000 4500 6000 75000

3000

6000

9000

12000

0 10000 20000 300000

4000

8000

12000

16000

ω (Ω

-1 c

m-1)

ω (cm-1)

50 K

20 K 15 K 8 K

σ 1 (Ω-1cm

-1)

ω (cm-1)

300 K 200 K 100 K 70 K 50 K

0 100 200 300 400 5000

3000

6000

9000

12000

20 K 15 K 8 K

σ 1 (Ω

-1cm

-1)

ω (cm-1)

300 K 200 K 100 K 70 K 50 K

ØThe formation of hybridization gap is associated with renormalization of heavy quasiparticle. The mass enhancement is roughly m*/mB~21.

ØBelow the TH, we observe clearly the opening of a density-wave type energy gap: a large fraction of the coherent Drude spectral weight was removed and piled up just above the energy gap of 60 cm-1 (2∆/kBTc~5.7). n8K/n20K~0.24.

“Text book”-like density wave type energy gap below TH.

Conclusions:

Ø different from the point contact tunneling spectroscopy measurement,

the Hybridization gap (~15 meV) is completely different

from the density-wave type gap (2∆~8 emV) in the hidden order state;

Ø No other pseudogap is detected;

Ø The formation of the hybridization gap is a crossover phenomenon, being associated with the formation of a narrow Drude component (or well-defined heavy quasiparticles);

Ø The opening of the density wave gap below TH results in the removal of a large fraction of Fermi surfaces (about 75%) and a rapid reduction of the scatterings.

Hubbard U physics:

DFMT

V2O3