Ján Nyéki, Anastasia Phillis,

Jeevak Parpia*, Brian Cowan

and John Saunders

29/09/10, IPP Moscow

Torsional Oscillator Experiments on

Helium Films on Graphite;

The Search for a Two Dimensional

Supersolid

Department of Physics, Royal Holloway University of London, Egham, UK

*LASSP, Department of Physics, Clark Hall, Cornell University, Ithaca, USA

Why Graphite as Substrate ?

Joly et al, Surf. Sci. 264 (1992) 419

•Flat - minimalised disorder

•Helium film grows in layers.

•We can control the growth.

•Two solid He layers solidify

•Triangular lattice symmetry

•Cooling to submillikelvinrange

Adsorption potential

+ Periodic potentialarising from atomic structure of surface

Ground state

2D Helium on Graphite

• Strongly correlated fermions or bosons• Subject to an external triangular lattice potential•Tuning parameters: interatomic distance

binding potential

• No impurities

Experimental techniques

3He: NMR, heat capacity, neutron diffraction,

vapour pressure

4He: heat capacity, torsional oscillator,

neutron diffraction, third sound, vapour pressure

Temperature range ~ 15 µK to 15 K

QuestsSuperfluidity of a 3He monolayer

2D superfluid-insulator transition

2D supersolid

Exfoliated Graphite

Taub et al., PRB 16 (1977) 4551

Fukuyama, JPSJ 77 (2008) 111013

Grafoil ~ 20 m2/g

Grafoil – atomically flat platelets 10-20 nm long, rolled, angle spread ±15°- surface area ~ 20 m2/g - density ~ 50% of bulk- “deep sides”, edges 3-5%- possible to cool below 1mK

Layered Helium Films

Registered solid

(commensurate phasesuperlattice)

√3 ×√3 6.4 nm-2

3He/Gr

4/7 phase

Fukuyama, JPSJ 77 (2008) 111013

Sibling system: 4/7 Phase in 3He films

Elser, PRL62 (1989) 2405Takagi, J Phys Conf Ser 150 (2009) 032102

7

4

densitylayer 1st

densitylayer 2nd

1

2 ==ρρ

Existence of 4/7 solid is clearly identified

both experimentally and theoretically

•Frustrated spin ½ magnet•Mott insulator•Gapless spin-liquid ground state

Motivation I

Can we study this phenomenonin 2D helium-4 ?

(alternative system of bosons in externally imposed lattice potential)

Revived theoretical interest

Heidarian & Damle, PRL 95, 127206 (2005),

Melko et al, PRL 95, 127207 (2005),

Bonisegni & Prokof‘ev, PRL 95, 237204 (2005) + ...

Wessel and Troyer, PRL 95, 127205 (2005)

Greiner et al, Nature 415, Jan 2002

Stoof, ibid, N&V

Triangular lattice symmetry supports formation of supersolid

Searching for Supersolid

Superfluid response Broken translational symmetry+

4He/Gr 2nd Layer - Phase Diagram

Pierce& Manusakis, PRL 81, 156 (1997) and PRB 59, 3802 (1999)

1st layersolid

Commensurate4/7 phase

Total coverage on PM scale [ nm-2 ]

11 12 13 14 15 16 17 18 19 20 21 22

L CL + G IC

vacancies ?

Puddling

IC phase

4/7 Phase in 4He films

Corboz et al., PRB78 (2008) 245414

Total 4He coverage on PM scale [ nm

-2]

15 16 17 18 19 20 21 22 23

T [

mK

]

0

200

400

600

800

1000

1200

1400

1600

1800

L CL + G IC

Total coverage [ nm-2 ]

12 13 14 15 16 17 18 19 20 21 22

C [

mJ/K

]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

5mK 10mK 15mK 20mK 30mK 40mK 50mK 60mK

L C ICL + G

Ziouzia et al.: Physica B (2003) & PhD Thesis

3He tracer

Greywall, Busch: PRL 67 (1991) 3535

IC

C

Motivation IICrowell & Reppy, PRB 53, 2701 (1993)

superfluid response in 2nd layer

20 mK – 600 mK

-lnT like dependence

no dissipation peaks reported

Total 4He coverage on PM scale [ nm

-2]

15 16 17 18 19 20 21 22 23

T [

mK

]

0

200

400

600

800

1000

1200

1400

1600

1800

L CL + G IC

3D “Supersolidity”

Sample annealing(disorder) and impurities playimportant role

Chan, Science 319, 1207 (2008)

Torsional Oscillator

E.L.Andronikashvili, Zh.Eksp.Teor.Fiz. 16 (1946) 780

Andronikashvili’s experimentBulk liquid 4He

ρn/ρ

ρs/ρBishop & Reppy, PRB 22, 2701 (1980)

Torsional Oscillator Signal

Kosterlitz-Thouless theory

Resonant frequency

12

22)(

h

mk

T

TB

c

cs

πρ

=−

I

k=ω

Bishop & Reppy, PRB 22, 2701 (1980)

AmplitudeDrive=const ~ Q

≈

∆

Q

1Dissipated Energy

Liquid 4He films

Experimental Cell: Torsional Oscillator

torsion element 1: contains exfoliated graphite sample ~ 35 m2

torsion rod

torsion element 2: coin silver cylinder, with two Mg electrodes biased at 100V

torsion rod

isolation mass + torsion rod (copper) mounted to cold-plate

Measure frequency shift and dissipationquality factor Q ~ 250000 - 550000working frequency ~ 1423 Hz; detecting changes < 10 µHz ~ 2x10-7g

2 mK - 4000 mK (weak thermal linkto the copper demagnetisation stage)

L+G Region

Temperature [ K ]

0.0001 0.01 0.1 1

Fre

qu

en

cy

- 1

42

3 H

z

[ m

Hz ]

-208

-206

-204

-202

-200

-198

-196

13.25 nm-2

15.42 nm-2

16.35 nm-2

17.31 nm-2

17.57 nm-2

17.91 nm-2

18.14 nm-2

18.29 nm-2

18.45 nm-2

18.57 nm-2

18.69 nm-2

Temperature [ K ]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10

8 x

∆∆ ∆∆(1

/Q)

0

10

20

30

40

50

60

70

80

90

Total coverage on PM scale [ nm-2 ]

10 11 12 13 14 15 16 17 18 19 20 21 22

L CL + G IC1st

layer

Dissipation maximum

at constant temperature

in L + G region;

moves to lower T

in L region

10 11 12 13 14 15 16 17 18 19

T [

mK

]

0

300

600

900

1200

L + G

1st layer

∆∆∆∆f

∆∆∆∆(1/Q)

Transition L+G ���� L

Temperature [ K ]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10

8 x

∆∆ ∆∆(1

/Q)

0

10

20

30

40

50

60

70

80

90

Total coverage on PM scale [ nm-2 ]

10 11 12 13 14 15 16 17 18 19 20 21 22

L CL + G IC1st

layer

18.14 nm-2

chosen as a reference to

evaluate additional decoupling

on top of the slip process

Temperature [ K ]

0.0001 0.01 0.1

Fre

qu

en

cy -

14

23

Hz

[ m

Hz ]

-200

-199

-198

-197

-196

13.25 nm-2

15.42 nm-2

16.35 nm-2

17.31 nm-2

17.57 nm-2

17.91 nm-2

18.14 nm-2

18.29 nm-2

18.45 nm-2

18.57 nm-2

18.69 nm-2

Extra decoupling = superfluid

For n ≥ 18.14 nm-2

0.001

∆∆∆∆(1/Q)

∆∆∆∆f

Homogeneous Fluid Region

Temperature [ K ]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10

8 x

∆∆ ∆∆(1

/Q)

0

10

20

30

40

50

60

70

80

90

Total coverage on PM scale [ nm-2 ]

10 11 12 13 14 15 16 17 18 19 20 21 22

L CL + G IC1st

layer

New peak appears

&

sharp increase of ∆∆∆∆f

for n4 > 18.85 nm-2

13 14 15 16 17 18 19 20

T [

mK

]

0

100

200

300

400

LL + G

Temperature [ K ]

0.0001 0.01 0.1 1

∆∆ ∆∆f

[ m

Hz ]

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

18.14 nm-2

18.29 nm-2

18.45 nm-2

18.57 nm-2

18.69 nm-2

18.85 nm-2

19.01 nm-2

19.09 nm-2

19.18 nm-2

0.001

∆∆∆∆(1/Q)

∆∆∆∆f

18.14 nm-2

Super- response around C phase

Total coverage on PM scale [ nm-2 ]

10 11 12 13 14 15 16 17 18 19 20 21 22

L CL + G IC1st

layer

Temperature [ K ]

0.0001 0.01 0.1 1

∆∆ ∆∆f

[ m

Hz ]

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

19.18 nm-2

19.26 nm-2

19.34 nm-2

19.42 nm-2

19.51 nm-2

19.60 nm-2

19.69 nm-2

19.78 nm-2

19.87 nm-2

19.96 nm-2

20.05 nm-2

20.16 nm-2

20.28 nm-2

20.40 nm-2

20.52 nm-2

20.64 nm-2

20.74 nm-2

Temperature [ K ]

0.0001 0.01 0.1 1

10

8 x

∆∆ ∆∆(1

/Q)

-10

0

10

20

30

40

50

60

70

80

90

0.001 0.001

∆∆∆∆(1/Q)∆∆∆∆f

Temperature [ K ]

0.00 0.05 0.10 0.15 0.20 0.25 0.30

∆∆ ∆∆f

[ m

Hz ]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Super- response around C phase

Total coverage on PM scale [ nm-2 ]

10 11 12 13 14 15 16 17 18 19 20 21 22

L CL + G IC1st

layer

19.18 nm-2

19.26 nm-2

19.34 nm-2

19.42 nm-2

19.51 nm-2

19.60 nm-2

19.69 nm-2

19.78 nm-2

19.87 nm-2

19.96 nm-2

20.05 nm-2

20.16 nm-2

20.28 nm-2

20.40 nm-2

20.52 nm-2

20.64 nm-2

20.74 nm-2

after full composite background subtraction

∆f ~ ρs( *)( *) 1

(0) (0) 2

s

s

Tf T

f

ρρ

∆= =

∆

Super- Response Overwiev

Total coverage on PM scale [ nm-2

]

15 16 17 18 19 20 21 22

∆∆ ∆∆f(

0)

[ m

Hz ]

0

1

2

3

T

[ m

K ]

0

400

800

1200

1600

L CL + G IC

Not a simple structureless superfluid

Total coverage on PM scale [ nm-2

]

19.0 19.5 20.0 20.5 21.0

∆∆ ∆∆f(

0)

[ m

Hz ]

0

1

2

3

4

Superlattice density is a “pivotal” point in

coverage dependence of super- response

Scaling

C

T/T*

0 1 2 3 4 5 6 7 8 9 10

∆∆ ∆∆f(

T)/∆∆ ∆∆

f(0)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

T/T*

0 1 2 3 4 5 6 7 8 9 10

∆∆ ∆∆f(

T)/∆∆ ∆∆

f(0)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ρρρρs(T)/ρρρρs(0)vs

T/T*

ρρρρs(T)/ρρρρs(0)vs

T/T*( *)( *) 1

(0) (0) 2

s

s

Tf T

f

ρρ

∆= =

∆

T/T*

0.1 1 10

∆∆ ∆∆f(

T)/∆∆ ∆∆

f(0

)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

T/T*

0.1 1 10

∆∆ ∆∆f(

T)/∆∆ ∆∆

f(0

)

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Scaling

Jason Ho, private communication, 2010

α

ρρ

)/(1

)0()(

0TTT ss +

=

n4 on PM scale [ nm

-2 ]

19.0 19.5 20.0 20.5 21.0

T*

[ m

K ]

0

10

20

30

40

50

60

*TT

α~1.0

≤

*4TT 0.4T

α~1.0

≤≤*

Super- Fraction

Temperature [ K ]

0.0001 0.01 0.1 1

∆∆ ∆∆f

[ m

Hz ]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

19.18 nm-2

19.26 nm-2

19.34 nm-2

19.42 nm-2

19.51 nm-2

19.60 nm-2

19.69 nm-2

19.78 nm-2

3rd layer

What is the value of super- fraction?

Compare with measurements of a KT superfluid monolayer in the same cell (3rd layer, n3 ~ 5.5 nm

-2)

Super- fraction at T����0:

~ 39% of 2nd layer for 19.18 nm-2 sample

~ 27% of 2nd layer for 19.78 nm-2 sample

ΧΧΧΧ-factor for this cell = 0.942

Measurements at perimeter velocity 25 ÷÷÷÷ 55 µµµµm/s

checked up to ~ 500 µµµµm/s0.001

Super- response from superlattice?

These number fluctuations are intrinsic to the 4/7 superlattice structure[Watanabe, Imada, JPSJ 76 2007]. Fluctuations into the 3rd layer.

Zero-point vacancies [Andreev & Lifshitz] appear in the superlattice when ∆> 2UV .

In 2D , ∆ for 3He impuriton is much larger than in the bulk [Ziouzia, PhD Thesis, London ]

The 2nd layer superlattice phase is unstable. [Corboz et al., PRB 78 2008]

Key: since ∆N ∆φ ≥ 1, number fluctuations required to define phase.[For this reason a commensurate solid cannot be a supersolid]

g(E)

E

∆

Monte Carlo simulations

2nd

layer coverage n2 [ nm

-2 ]

3 4 5 6 7 8

∆∆ ∆∆f(

0)

[ m

Hz ]

0

1

2

3

4

G + L SF IC

Total coverage on PM scale [ nm-2

]

16 17 18 19 20 21

∆∆ ∆∆f(

0)

[ m

Hz ]

0

1

2

3

4

L CL + G IC

Corboz et al., PRB78 (2008) 245414

Pierce& Manousakis, PRL 81, 156 (1997)

and PRB 59, 3802 (1999)

Two Monte Carlo simulations

of the 2nd layer disagree

about stability of the 4/7

superlattice solid phase

stable 4/7 phase

superfluid phase only

In the 4/7 phase

a ~ 4.1 Ǻ � vc~ 3.9 m/s

Momentum deficit in Quantum Glass

Reduced rotational inertia due to presence of tunelling two-level systems.

If “bulk” velocity v(t) ≠0 the TLS has nonzero mean values of momenta

12 in both stationary states. In the ground state the projection of

1 onto v is negative.

No superflow

[Andreev, JETP Lett 85 2007, ZhETF 136 2009, Korshunov Pis’ma v ZhETF 90 2009 ]

U - characteristic height of

energy barrier

J - tunneling amplitude

∆∆∆∆ ≈ 1 mK ?Temperature [ K ]

0.0001 0.01 0.1 1

∆∆ ∆∆f

[ m

Hz ]

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Summary

Motivation: Does vacancy doped commensurate phase √7 x √7(4/7 phase) have a supersolid response?

•High precision TO measurements from 2mK to 4K. Second layer density is the tuning parameter.

•1st layer interaction with the substrate present all time ���� careful measurements can distinguish that from ‘super-’ response; sensitive to

2nd layer structure

•Unusual temperature dependence of momentum deficit; no sharp onset:∆f(T) ~ -lnT near onset while ∆f(T) ~ T as T ���� 0.

•∆f(0) is tuned by n2. Plateau-feature when n ≈ n0 (4/7 ) and vanishes near at incommensurate 2nd layer density nI. Possible QCP at nI.

•More than 25% of the 2nd layer contributes to super- signal when approaching n0.

Summary II

Motivation: Does vacancy doped commensurate phase √7 x √7(4/7 phase) have a supersolid response?

• After “pre-cursor” feature plot of ∆f(0) vs T* shows two clear regimes. Boundary correlates with qualitative change in dependence of ∆f on lnT

• For a significant coverage range there is data collapse in ∆f(T)/∆f(0) vsT/T* coordinates.

• Quantum glass model qualitatively agrees with observed ∆f(T) ~ -lnTdependence. Energy scales involved need to be addressed.

•We need experimental evidence on the structure of putative 4/7 phase in the T ���� 0 limit.