1959-17

Workshop on Supersolid 2008

A. Kuklov

18 - 22 August 2008

CUNY, USA

Dislocations and Supersolidity in solid He-4

Dislocations and Supersolidity in 4He

Anatoly Kuklov (CUNY,CSI)

supersolid 2008, ICTP, Trieste

Massimo Boninsegni (Univ of Alberta)Lode Pollet (ETH)Nikolay Prokof’ev (UMASS )Gunes Soyler (UMASS)Boris Svistunov (UMASS)Matthias Troyer (ETH)

First principles QMC Dislocation roughening

Darya Aleinikava (CUNY,CSI)Eugene Dedits (CUNY,CSI)David Schmeltzer (CUNY,CCNY)

Thanks to: NSF and CUNY grants

• Motivation: search for superfluid network of defects • History: Studies of dislocations in He4

• Types of dislocations and their superfluid properties• Core splitting and self-pinning• Mechanical strain and vacancy gap • Approximate phase diagram

• Quantum roughening • Long-range forces between kinks • Kosterlitz-Thouless, RG and MC arguments against quantum

roughening• J. Day & J. Beamish experiment

• Summary

Outline

in metals: dislocation densities~106-1014 cm-2

Superfluid network of dislocation (?)

Shevchenko state of quasi-1D SF random netwokS.I. Shevchenko, Sov. J. Low Temp. Phys. 14, 553 (1988).

Coreless vortices

Tc in the network T*a/D ~1/DT*~1K, nd ~ (a/D)2

Tc ~1mK for nd ~109cm-2

Tc ~10mK for nd ~1011 cm-2

a

“Incompressible vortex fluid” by P.W. Anderson, cond-mat/0705.1174

Tc <T<T* , wide range of quasi-superfluid frequency dispersive response

ω τ> superfluid response at T>Tc

ω τ< normal state at T>Tc

21*

2 1*

110 10

SFKTs

T Tτ

−

− −� �−� �

� �� �

V.A. Kashurnikov, et al, PRB, 53, 13091 (1996); Yu. Kagan, et al, PRA,61, 045601 (2000).

KSF -Luttinger parameter; T* ~ 1K; K =0.205(20); Tc~0.1K

time of phase slip in a single loop:

Not much is known about dynamics in the network!

Edge and screw dislocations

ϕr

deformation vector u ~

2

bϕ

πstrain and stress /bu rσ∂ � �

b - Burgers vector

Dislocation as almost free classical string

X

Y

Y(x,t)

J.Appl.Phys. 27,583; 789(1956)

Measurements of sound absorption and internal friction in solid He4

R.Wanner, I.Iwasa, S.Wales, Solid State Commun., 18,853(1976)Y.Hiki, F. Tsuruoka, Phys.Lett. 56A, 484; 62A, 50(1977); V.L.Tsymbalenko, JETP 47,787(1978); 49, 859(1979)I.Iwasa,K.Araki, H.Suzuki, J.Phys.Soc.Jpn 46, 1119(1979)J.Beamish, J.P.Frank, PRL 47,1736(1981).....................................................

Peierls barrier (still classical)

Peierls, Frenkel-Kontorova, in the book A. M. Kosevich, “The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices”, Wiley, 2005

Quantum effects

Semiclassical tunneling of kinks: B.V. Petukhov and V.L. Pokrovsky, JETP 36, 336 (1973)

Kinks as quantum quasiparticles: A.Andreev, Soviet Uspekhi, 19, 137(1976)

Semiclassical description of dislocation dynamics in sound absorption: A. Y.Hiki, F. Tsuruoka, PRB 27, 696(1983)I.Iwasa, N.Saito, H.Suzuki, J.Phys.Soc. Japan 52, 952(1983)A.V.Markelov, JETP 61,118(1985)

Models of quantum kinks+ superfluid properties: P.G. de Gennes ,C.R. Physique 7, 561(2006)G.Biroli,J.P. Bouchaud, cond-mat 0710.3087

see review S. Balibar and F. Caupin, J. Phys.: Condens. Matter 20, 173201 (2008)

Role of deformations in inducing core SF :

V. M. Nabutovskii and V. Ya. Shapiro, JETP 48, 480 (1978)S. I. Shevchenko, Sov. J. Low Temp. Phys. 13, 61; 553 (1987)A. T. Dorsey,P.M. Goldbart, J. Toner PRL 96, 055301 (2006)J. Toner, PRL 100, 035302(2008)

Deformations, structure and core superfluidity)M. Boninsegni, et al., PRL 99, 035301 (2007)L. Pollet, et al, cond-mat 08053713

Mechanically gapped dislocations andJ. Day & J. Beamish experiment

Nature 450, 853 (2007)

Main focus of the talk

z-screw x-screw zy-edge xy-edge xz-edge

SF splits by I

ordertransition

SF SFstrongly

split

stronglysplit

insulator

insulator

Stable dislocations in hcp He4

83b a= 8

3b a=a

b a= b a= b a=

Core splitting (partial dislocations) in He4

Full edge dislocation

b- Burger’s vector

Z

X

two atoms in unit cell

Full edge dislocation

b- Burger’s vector

Z

X

two atoms in unit cell

2~defE b

Partial (split) edge dislocation

Z

X

fault

L

two atoms in unit cell

( )2 2

~ 2 ~2 2defb bE

PIMC simulations (columnar view): split xy-edge dislocation

11 layers 11 layers

10 layers 10 layers

bz

Z

X

L

bz /2

bz /2

A

A

A

B

B

B

C

Split ZY-edge dislocation

A BY

X

Fault

Fault: C-layerA B

½ cores

Y

X

Columnar view of fault + split core: many layers

Y

X

PIMC simulations (columnar view): zy-edge (SF) dislocation

½ cores

Strong self-pinning of zy-edge dislocation

Z

X

bz/2 bz/2

L

gliding motion is pinned by the barrier ~ L

bx/2

bx/2

gliding motion is not pinned

split zy-edge dislocation

split xy-edge dislocationY

Fault size

2

2

2

fault surface te

ln4

ln

nsi

8

, 8

on

full

split full L L

L

bE R

bE E

b

σ

μ

π

μ

π

μ

πσ

σ

=

− = − +

= −

o

o

3.7 A,

MC simulati

100 2

00b

ons

ar

: 0.1

75 150A

/K particle

L

b μ

σ

= ≈

> −

−

<

Z-Screw

M. Boninsegni, et al., PRL 99, 035301 (2007)

condensate map

X-screw

G. Soyler et al.

condensate map

z-screw zy-edge

SF splits by I

order transition

SFstronglysplit and

selfpinned

Fully SF loops

+

Splitting of z-screw dislocation makes it insulator

Z

L

bz/2

bz/2

split=insulator

Z

L

bz/2

bz/2

insulator

superfluid

(Preliminary) phase diagram for dislocation SF

superfluid screw dislocations

insulating screw dislocations

I order transition

Strain and SF gap

2in t

2

( . . . ) | | . . . .

| | ~ ( . . . )

s y m m e try o f e la s t ic m o d u

;

l i

i j k l

i

i j i j k

jk l

i j

x x y y z z

i

l

i j i j kj j k l li

F r u uT

T T r T r

T

T

T

T

u

r u u u

ψ

ψ

⊥

= − − − +

− + + +

−

= = = �

no proximity to superfuidity of ideal crystal (no perfect supersolid)

Higher order terms in strain are equally important

Density change and vacancy gap

/ 0.13 n nδ ≈

L. Pollet, et al, cond-mat 08053713

ii

xx yy zz

n u n

u u u

δ ≈ −

= =

No significant quadratic terms!!!

2 2 4 0zzzz xxyy xxxx xxzzT T T T+ + + ≈

V. M. Nabutovskii and V. Ya. Shapiro, JETP 48, 480 (1978)S. I. Shevchenko, Sov. J. Low Temp. Phys. 13, 61; 553 (1987)J. Toner, PRL 100, 035302(2008)

ZY-edge dislocation: marginally SF

8, / 0., 2632ii

bu r a b n na

rδ

π≈≈ = = →

Splitting: / 2 / 0 13 .b n nb δ ≈→ →

XY-edge dislocation: Insulating

, , / 0.2

1 6 ii

bu r a b a

rn nδ

π≈≈ = = →

Splitting: / 2 / 0.08 0.13n nb b δ ≈ <→ →

Anistropic compression

0xx yy zzu u u+ + =

L. Pollet, et al, cond-mat 08053713

No linear term!

r r⊥=�

( )2

2 22

| | ~ ( / 2 / 2 2 ) 0.1

~ 1( 2)

zzzzzz xxyy xxxx xxzz zz

ur T T T T uψ

� �− + + + − −� �

� �

Anistropic compression

split ZY-edge is marginally SF

0.26

0.2

32 1/

zzz

zz

bu

au

π= ≈

≈

2 2 4 0zzzz xxyy xxxx xxzzT T T T+ + + ≈

22 2

2| | ~ ( 4 ) ~

0.151 ,

( )zx

xzxz xz

ur T uψ

� �− − − −� �

� �

Lx

uz~x

Z

X

Shear strain

screw dislocation:2 2

22 22

2

2

0.22

0.22

0

8, ,

4 33

( )| | ~ 1 1 0,

( )

1 0

.15 0.15

0.11

0.

1 0

5

zzx zy z

zx zy

b au u r b a

r

u u

π

ψ

ψ

+ = ≈ = =

� �+ � �− − = − + >� � � �� � � �� �

� �− + < → =� �

� �insulating if split:

superfluid if unsplit:

Quantum (non-)roughening and J.Day & J.Beamish experiment

J. Day & J. Beamish,Nature 450, 853 (2007)

Quantum dislocation

X

Y

Y(x,t)

ratio K of phonon and kink energies becomes ~1

in He4:

Quantum rough dislocation Peierls barrier up is effectively zero at T=0Zero point wandering

K – Luttinger parameter

Long-range interaction between kinks due to 3d elastic matrix

J.P.Hirth, J.Lothe, “ Theory of Dislocations”, McGraw-Hill, 1968A. M. Kosevich, “The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices”, Wiley, 2005

L

dislocation loop

Long-range space-time interaction suppresses quantum roughening

2 2( , )LR

gU x t

x t≈

+

Coulomb gas mapping

Kostrelitz-Thouless transition is destroyed for arbitrary small g

Pairs are ionized into plasma state – Peierls barrier is relevant at T=0!!!

RG

up grows at large L for any g(0) and K(0)

MC simulations (of dual model) at T=0

KT

Monte Carlo Worm algorithm (Prokof’ev&Svistunov (1998)):http://montecarlo.csi.cuny.edu/umass/index.html

Consequences

�No quantum roughening = no quantum macroscopic wandering

�Gapped spectrum rather than sound-like (gap << kink energy)

�Intrinsic hardening of shear modulus at T< gap (at pinning distances smaller than inverse concentration of thermal kinks)

2 2qω = + Δ

If gap were zero: Thermal activation of He3

linear density of He3 in units a=1:

Lp – pinning length

fit:nd~ 3.1011cm -2

nHe3~ 7.p.p.mactualnHe3~ 1.p.p.b

Single Arrhenius: poor fit

Finite gap (and Lp-1 >> soliton density)

( )0

0

2( ) exp

1 g ln p

KT T

L

π� �� �Δ ≈ Δ −� �+ Δ� �

data: courtesy of John Beamish,J. Day & J. Beamish,Nature 450, 853 (2007)

For 1p.p.b of He3, Lp=103: 2 from different curv6 s5 e DW KW D T → ≈= ≈

Modulus hardening

self-consistent!

Summary

� Selfpinned SF loops (due to core splitting)

� Splitting as I order transition for Z-screw - hysteresis and metastability

�Strain criterion for SF and non-SF dislocations

� Preliminary phase diagram

� Self - pinning in the Peierls potential at low T: no quantum macroscopic wandering

� Bimodality in the modulus hardening due to Peierls gap