§7 Gas-Liquid Transition of He Monolayers

A few monolayers of 3He and 4Hephysisorbed on atomically flat graphite surface

4He1 2

34 5

M. Roger et al., JLTP 112, 45 (1998)

Ideal, simple and highly tunable 2D quantum systems of Bosons and Fermions

3He 4He HD

3He/4He/gr3He/3He/gr3He/gr 3He/HD/HD/gr

• wide range of particle correlation (4He: 4 ≤ r ≤ 12 nm-2, 3He: 0.6 ≤ r ≤ 11 nm-2)

• a variety of depth and corrugation of confinement potential depending on underlayers

Fabrication of 2D He systems

Two kinds of graphite substrates to check size effects

STM characterization: Y. Niimi et al., Rev. Sci. Instrum. 74, 4448 (2003); PRB. 73, 085421 (2006)

Ø mother material: HOPGØ surface area: 2 m2/gØ platelet size: 100 - 300 nm

ZYX (longer coherence length)

Ø mother material: graphite foamØ surface area: 20 m2/gØ platelet size: 10-20 nm

Grafoil (shorter coherence length)

atomically flat

Exfoliated graphite fabricated by graphite intercalation technique

Liquefaction of bulk 3HeTheoretical predictions (1948)

3He would not liquefy due to too large zero-point energyfrom theoretical viewpoint.

Fritz London Laszlo Tisza

3He:20 cm3 STP

P (c

m H

g)

V (cm3)

Liquefaction of 3He was experimentally found.

S.G. Sydoriak, E.R. Grilly and E.F. Hammel, Phys. Rev. 75, 303 (1949)

Experimental finding (1949)

Tc = 3.3 K,Pc = 1.2 atom Tboil = 3.2 K

half-life = 12.4 y,ne: electron neutrino

6Li + 1n → 3H + 4He, 3H → 3He+ + e- + ne

Do 3He atoms liquefy (self-bound) in 2D?

2D Fermi liquid

gas

2D Fermi gas

substrate

?OR

• smaller coordination numbers• larger fluctuations

Lowering spatial dimension:a clean method to increase quantum parameter (h) effectively

s, e : L-J parameters

Gas-Liquid transition in 3D and 2D 3He-4He3He-4He bulk liquid (3D)

4He

phase separation

Wallance-Meyer (1972)

3He

Tc =3.3 K

Liquid 3He provided us many physics and applications.

3He + 4He(X = 6.6 %)

pure 3He

R. Ramos and O. Vilches, JLTP 34, 55 (2004)R. Ramos, Ph.D thesis (1999)

3He-4He monolayer (2D)

4He 3He

Tc =0−0.1 K ?

unexplored

?phase separation

(U. Tokyo)

on HD/HD/gron 4He/gr

Tc

Liquefaction of 3He in 2D is marginal…

Liquefaction of 4He in 2D

12

14

16

18

20

22

24

0 0.4 0.8 1.2 1.6

ρ C

over

age:

(n

m-2)

Temperature: T (K)

12

14

16

18

20

22

24

0 0.4 0.8 1.2 1.6

ρ C

over

age:

(n

m-2)

Temperature: T (K)

Phase diagram of 4He in 2nd layer

2nd layerpromotion

IC2 (incommen-surate solid)

C2 (commensurate solid)C2 + IC2

L2 + C2

L2 (liquid)

G2 + L2

F2(fluid)

Self-binding of 4He in 2D Tc = 0.75 Krc = 4.3 nm-2

M.C. Gordillo and D.M. CeperleyPRB 58, 6447 (1998)

PIMC calculation for 4He in strict 2D (without potential corrugation)

1.510.500

2

4

6

8

T (K)

G + L

L(superfluid)

IC

r(n

m-2

)

L + IC

Very good agreement with PIMC calculations

HC data with ZYX graphite

S. Nakamura et al., PRB 94, 180501 (2016)

S. Nakamura et al., to appear

Logarithmic divergence expected from 2D-XY model would be smeared by substrate corrugation potential.

Earlier works on 2D 3He floated on superfluid 4He thin films (1)

B.K. Bhattacharyya and F.M. Gasparini, PRL 49, 919 (1982);PRB 31, 2719 (1985)

Nuclepore substrate(200 nm dia. pore)

3-4 layers of superfluid 4He 2D 3He:Andreev state

1.3 nm-2

uniformliquid

(32.8 nm-2)(37.8 nm-2)

substrate effects?

G-L transition ?

monolayer of 3He = 6.4 nm-2

G + L ?

3He

4He thickness

4He: 24.8 nm-2

Earlier works on 2D 3He floated on superfluid 4He thin films (2)

H. Akimoto, J.D. Cummings andR.B. Hallock, PRB 73, 012507 (2006)

Later measurements with Nuclepore by other workers did not observe finite-T HC anomalies.

G. A. Csáthy, E. Kim, and M. H.W. Chan, PRL 88, 045301 (2002)

Monolayer 3He-4He shows reentrant superfluidity as a function of 3He density on porous Au (60 nm dia.) preplated with H2.

Ts: superfluid onset temperature

cf. 1 µmol/m2 = 0.602 nm-2

4Hethickness(µmolm2)11.5610.529.939.799.378.697.977.607.227.066.736.39

3 nm-2

Theories predict quantum gas ground-state

Strictly 2D:VMC (gas) A. D. Novaco and C. E. Campbell,

Phys. Rev. B 11, 2525 (1975).

VMC (gas) M. D. Miller and L. H. Nosanow, J. Low Temp. Phys. 32, 145 (1978).

VMC (gas) B. Krishnamachari and G. V. Chester, Phys. Rev. B 59, 8852 (1999).

DMC (gas) V. Grau, J. Boronat and J. Casulleras, PRL,89, 045301 (2002)

Quasi-2D:VMC (liquid) B. Brami, F. Joly, and C. Lhuillier,

JLTP 94, 63 (1994) ••• out-of-plane motions

substrate

0

0.5

1

-0.5

-1

Ene

rgy

per a

tom

(K

)

1 3 5 7Areal density (nm-2)

2D 4He(Boson)

2D 3He(Boson)

2D 3He(Fermion)

2D 3He should be a unique material which stays quantum gaseven at the ground state. (cf. ↑H)

experiment

DMC (gas) M. Ruggeri, S. Moroni, and M. Boninsegni, PRL 111, 045303 (2013)

Heat capacities of low-density 3He of first layer

nm-2

D. Sato et al., PRL 109, 235306 (2012)

• Total HC consists of two contributions, CFL and Camor, with quite different T-dependencies.

C = gT – aT2 + bCamor(T)

nuclear spin HC of amorphous3He trapped in substrate heterogeneities

HC of degenerate Fermi liquid

Degenerate 2D Fermi liquid:

CFL(T) = gT + aT2 + •••

g = (pkB2/3ℏ2)Am*

A : surface aream* : QP effective massa T2 : 2D spin fluctuations

D. Sato et al., PRL 109, 235306 (2012)

Substrate heterogeneity effect can safely subtracted

• Initial increase of b at 0 ≤ r ≤ 0.6 nm-2 corresponds to filling of heterogeneous sites (5-10%) of substrate with 3He.

• Subsequent r-linear increase of g at r ≥ 0.6 nm-2

corresponds to area expansion of 3He puddles on uniform region (90-95%) of substrate.

6 × 4.4 nm2

amorphous 3He(contributing to b)

liquid puddle(contributing to g)

puddle uniform liq.

amor

phou

s

rc = 0.8 nm-2

graphite

WWW

heterogeneity

amorphous 3He 3He paddleD. Sato et al., PRL 109, 235306 (2012)

• r-linear increase of g (two-phase region)

• kink in g (r) near m*/m3 = 1 (completion of puddling)

Heat capacities of low-density 3He in second layerOnly CFL component is observed.

pudd

le

uniform liq.

rc = 0.6 nm-2

g0

3He4HeWWW

heterogeneity graphite

amorphous 4He 3He paddle

D. Sato et al., PRL 109, 235306 (2012)

Puddle formation in third layer of 3He

3He4He

C-IC coexistence in 2nd layer (domain wall?)

6% compression of 2nd layer + puddle in 3rd layer

D. Sato et al., PRL 109, 235306 (2012)

m*/m = 1g0

rc = 0.9nm-2

= 80 mK Tcmax

rc0

Surprisingly low density liquid !rc ≈ 0.6-0.8 nm-2 (a ≈ 1.4-1.6 nm)converted nc

3D ≈ 0.002 g/cm3

: lowest density liquid in nature

3He effective masses in the first three layers1st layer 2nd layer 3rd layer 4th layer

C2

phas

e1st layer 0.8 1.22nd layer 0.6 1.33rd layer ≤ 0.9 1.24th layer (0.7) (1.5)

rc (nm-2) m*/m

D. Sato at al., to appear

1st, 2nd, 3rd layers of 3He have puddle phaseswith surprisingly similar critical densities in spite of their quite different circumstances.

intrinsic property of 3He in strictly 2D

New measurements with different underlayer

/HD/HD/gr

/gr

/4He/gr

M. Kamada et al. (2017)

Again, we found a similar puddle transition with those observed previously with different preplating layers.

new system

Fixed-node DMC calculation using SAPT2 pair potential with Axilrod-Teller-Muto three-body potential

Recent ab initio calculations (I)

M. Ruggeri at al., PRB 93, 104102 (2016)

• For 3He/gr, effects of He-gr potential corrugation are important, which can be represented by an effective band mass (m*) enhancement by 2%.

• For 3He/4He/gr, delocalization of wave-fn. along z-direction is more important.

• Results are sensitive to the choice of He-He interaction.

N = 18 (26, 42, 58) atoms

A metastable liquid phase is emerged both in 1st and 2nd layers of 3He.

0 0.5 1 1.5 2r (nm-2)

3He/gr3He/4He/gr (ρ4He = 11.4 nm−2)

strictly 2D 3He

0 1 2 3r (nm-2)

HFDHE2Aziz / anisotropic He-gr / m*= mSAPT2+ATM / anisotropic He-gr / m*= mSAPT2+ATM / smooth He-gr / m*=1.02m0.5 1 1.5

r (nm-2)

3He/gr

Fixed-node DMC calculation using Aziz potential

Recent ab initio calculations (II)

M.C. Gordillo and J. Boronat, PRL 116, 145301 (2016); PRB 93, 104102 (2016)

N = 130 atoms

Stable liquid phase in 2nd layer of 3He

strictly 2Dsmooth first-layer potentialfixed 4He first layeractive 4He first layer

○■□

Metastable liquid phase in 1st layer of 3He

3He/gr

3211/r (nm2)

Maxwell construction

… consistent with (I) Ruggeri at al.

… inconsistent with (I) Ruggeri at al.

r (nm-2)0 0.5 1 1.5 2 2.5

3He/gr

potential corrugation effect

ρ4He = 11.2 nm−2

3He/4He/gr

r (nm-2)0 0.5 1 1.5 2 2.5

Recent theoretical calculations (III)E

(r)–

E (0

) (K

/ato

m)

r (nm-2)

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

3He/grstrictly 2D

M. Ruggeri at al., PRB 93, 104102 (2016)M.C. Gordillo and J. Boronat, PRL 116, 145301 (2016)M. Takano, T. Suzuki, and N. Sakumichi, J. Phys. Conf. Ser. 702, 012016 (2016);T. Suzuki et al., JPS Spring Meeting 18pC44-14 (2017) a)

reference

a) Variational calculation with explicit energy functional

Testing ground for state-of-the-art theoretical simulations

can reduce to a negative value?

Band (effective mass) effects

Low density expansion of equation of state based on termed hypernetted chain-Euler Lagrange (HNC-EL) theory

• Regardless of mass enhancement mechanism, with increasing m*/m, spinodal point (instability density) increases.

F. Gasparini, R. Holler, and E. Krotscheck, to appear

Band effects due to potential corrugationW.E. Carlos and M.W. Cole., PRB 21, 3713 (1980)L. Reatto et al., J. Phys. 25, 443001 (2013)

m*/m = 1.08 for 3He

- - - free particle

4He

m*/m = 1.2-1.3Indirect 3He-3He attractive interactions mediated by phonons or ripplons in underlayers?

3He cluster calculations

DF + mass formulaS. Weisgerber and P.-G. Reinhard, Z. Phys. D 23, 275 (1992)

VMCE. Sola, J. Casulleras, and J. Boronat, PRB 73, 092515 (2006)V.R. Pandharipande, S.C. Pieper and R.B. Wiringa, PRB 34, 4571 (1986)

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8

3D 3He (NLDF)_WRM3D 3He (VMC)_SCB3D 3He (VMC)_PPWexp (3D-3He)

E/N

(K)

N-1/3

3He in 3D

20 10 23650240 N =

3He in 3D

Nmin

esN-1/3

(surface tension)

experimental

Tiny but finite binding energies forN = 2 and 3

N. Sakumichi and H. Suno, to appear

E (3He2) = -0.02018 mKE (3He3) = -0.286 mK

Adiabatic hyperspherical coordinates calculations

E (3He2) = -0.01681 mKE (3He3) = -0.24 mK

with PCKLJS potentialwith SAPT potential

... supporting many-body binding of 3He in 2D

dimer trimer

Korona et.al., J. Chem. Phys. 106, 5109 (1997)

S. Kilic and L. Vranjes, JLTP 134, 713 (2004)cf. E (3He2) = -0.020(1) mK

3He in 2D

Przybytek,et.al., PRL 104, 183003 (2010)

3He in 2D ?

4He cluster calculations

DMCS. Kilic and L. Vranjes, JLTP 134, 713 (2004)

DMC + mass formulaA. Sarsa et al., PRB 68, 224514 (2003)DMC + mass formula

R. Guardiola et al., J. Chem. Phys. 124, 084307 (2006)

Experimental[1] R.E. Grisenti et al., PRL 85, 2284 (2000)

-8

-6

-4

-2

0

0 0.2 0.4 0.6 0.8

3D 4He (DMC)_B

M

3D 4He (DMC)_G

exp in 3D

E/N

(K)

N-1/3

4He in 3D

20 10 23650240 N =

esN-1/3

(surface tension)dimer [ref. 1]

E = – (1.1 ± 0.2) mK

4He in 3D-1

-0.5

0

0.5

0 0.2 0.4 0.6 0.8

2D 4He (DMC)_SPPN

N

2D 4He (DMC)_KVE/

N (

K)

N-1/2

20 8 234640240 N =

4He in 2D

esN-1/2

(surface tension)

dimerE = – (41 ± 1) mK

Tc = 0.72 K(gas-liquidtransition[Ref.2])

4He in 2D

Experimental[2] S. Nakamura et al., to appear

Stability of small N clusters is sufficient condition for many-body binding?

Wikimedia Commons/InvaderXan

N. Ishii, S. Aoki, and T. Hatsuda, PRL 99, 022001 (2007)

Nucleon-nucleon potentialR.A. Aziz et al., J. Chem. Phys. 70, 4330 (1979)R.A. Aziz et al., Mol. Phys. 77, 321 (1992)

He-He pair potential

Aziz potential

L-J potential

K∝ r -2

2D 3He and nuclear systems share similar interests.

V∝ r -12