berezinskii-kosterlitz-thouless transition and ... · tsc (= tvortex-bkt) tc (the superconducting...

Berezinskii-Kosterlitz-Thouless transition and superconductor-to-insulator transition

Valerii Vinokur & Tatyana Baturina*

*Institute of Semiconductor Physics, Novosibirsk

Symposium on Spin Physics and NanomagnetismChudnovsky-Fest, March 13-14, 2009

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Dear Eugene:

Warmest congratulations and love!

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Observable signature of the Berezinskii Kosterlitz Th oulesstransition in a planar lattice of Bose Einstein condensatesTrombettoni, A.; Smerzi, A.; Sodano, P.New Journal of Physics, Volume 7, Issue 1, pp. 57 (2005).

4

Gauss law: 2 4

2

rE e

eE

r

π π=

=

r

Two-dimensional Coulomb law I

5

r

Two-dimensional Coulomb law II

Energy of the two charges interaction is thus:2

0ln( / )U e r r=

6

J. Chem. Phys. 38, 2587 (1963)

In view of recent interest in the equilibrium statistical mechanics of a one-dimensional system of point charges, it seems worth pointing out that, at least so far as the equation of state is concerned, the analogous two-dimensional system is extremely simple to treat.

Two-dimensional electrolyte

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Berezinskii (1971): XY-magnetic model

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Kosterlitz and Thouless

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BKT: superconducting films

2D vortices (“pancakes”) in a superconducting film: Logarithmic interaction:

KT transition in superconducting films: vortex binding-unbinding

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Vortices in JJA: BKT

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Resistive state

Superconducting stateTSC (= Tvortex-BKT)

TC (the superconducting order parameter appears)

Metallic stateT

0

Current, log(I)

Volta

ge,lo

g(V

)

V ∝ Ι α(Τ)

T > TBKT

T < TBKT

α α α α = 1 α α α α > 3

in experiment:

2D superconducting systems: experimental observatio n of BKT

Jump in α(T) at TBKT

12

Tc versus R

activated behavior of resistance

Disordered superconducting films

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Disordered 2D systems: superconductor-insulator tra nsition

InOx

V.F. Gantmakher, M.V. Golubkov, J.G.S. Lok,and A.K. Geim (1996)

Bi

Altering various parameters of the system, such as tunnel resistance and transmittance in JJAs, conditions of deposition, chemical composition, and thickness of the films, one can drive the system directly from the superconducting to insulating state.

R.C. Dynes, J.P. Garno, and J.M. Rowell (1978)

granular Pb

D.B. Haviland, Y. Liu, and A.M. Goldman (1989)

2D JJA

L.J. Geerligs, M. Peters, L.E.M. de Groot,A. Verbruggen, and J.E. Mooij (1989).1/T (1/K)

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criticalpoint

Superconductor Insulator

Resistance

RcR < Rc R > Rc

Localized Cooper pairsCondensate of Cooper pairsA. Gold, Z. Phys. B – Condensed Matter 52, 1 (1983); Phys. Rev. A 33, 652 (1986).Matthew P.A. Fisher, G. Grinstein, S.M. Girvin , PRL 64, 587 (1990).

Disordered 2D systems: superconductor-insulator tra nsition

Altering various parameters of the system, such as tunnel resitance and transmittance in JJAs, conditions of deposition, chemical composition, and thickness of the films, one can drive the system directly from the superconducting to insulating state.

M. Strongin, R.S. Thompson, O. F. Kammerer, and J.E. Crow, Phys. Rev. B 1, 1078 (1970).

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Resistive state



Metallic stateT

0Current, log(I)

Volta

ge,lo

g(V

)

V ∝ Ι α(Τ)

T > TBKT

T < TBKT

α α α α = 1 α α α α > 3

in experiment:

2D superconducting systems

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insulator

Resistive state



Metallic stateT

0

Superconductor InsulatorRcR < Rc R > Rc

T

0

Disorder-driven superconductor-insulator transition

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Superconductor-insulator and BKT transitions

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EOn a large spatial scale the 2D picture


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EJ>ECBKT transition:Vortices – anti-vorticesunbinding EJ<ECDual charge BKT transition:charges – anti-chargeunbinding

This condition means that all the electric field is concentratedwithin the 2D array: logarithmic interaction of charges!

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BKT transition

Vortice – antivorticebinding-unbinding

Dual charge BKT transition

Charge – antichargebinding- unbinding

N. Yoshikawa, T. Akeyoshi, M. Kojima, and M. Sugahara, “Dual Conduction Characteristics Observed in Highly Resistive NbN Granular Thin Films”,Proc. LT-18, Kyoto 1987, Jpn. J. Appl. Phys. 26 (Suppl. 26-3), 949 (1987).A. Widom and S. Badjou, “Quantum displacement-charge transitions in two-dimensional granular superconductors”, Phys. Rev. B 37, 7915 (1988).R. Fazio and G. Schon, “Charge and vortex dynamics in arrays of tunnel junctions”,Phys. Rev. B 43, 5307 (1991).B.J. van Wees, “Duality between Cooper-pair and vortex dynamics in two-dimensional Josephson-junction arrays”, Phys. Rev. B 44, 7915 (1991).

Disorder-driven Superconductor- Insulator transition in 2D-SC

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the thickness is 5 nm:Films are 2D superconductors!

TiN films were formed by atomic layer chemical vapor deposition onto a Si/SiO2 substrate at 350 0C.

Composition:Ti N Cl1 0.94 0.035

Thin disordered SC films

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T. Baturina. T. B., C. Strunk, M.R. Baklanov, A. Satta, PRL 98, 127003 (2007)T. Baturina, A.Yu. Mironov, V. Vinokur, M.R. Baklanov, C. Strunk, PRL 99, 257003 (2007)

T. Baturina, A. Bilušic, A.Yu. Mironov, V. Vinokur, M.R. Baklanov, C. Strunk, Physica C 468, 316(2008)

8.74 kΩ - S1

9.16 kΩ – I3

T = 10 K

< 5% (!)

Disorder-driven SIT in TiN films (B=0)

R(T) dependences of superconducting samples are nonmonotonic. They exhibit an ‘‘insulating trend,’’ i.e., an upward turn of

the resistance preceding its eventual drop to zero at low temperatures.

Rmax = 29.4 kΩ

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R = R0exp(TI/T)

I1: TI = 0.25 KI2: TI = 0.28 KI3: TI = 0.38 KI4: TI = 0.61 KR0 = 20 kΩ

At low temperatures we observe an Arrhenius behavior of the resistanceT. Baturina, A.Yu. Mironov, V. Vinokur, M.R. Baklanov, C. Strunk, PRL 99, 257003 (2007)

T. B., A. Bilušic, A.Yu. Mironov, V. Vinokur, M.R. Baklanov, C. Strunk, Physica C 468, 316 (2008)

Insulating side of the D-SIT in TiN films

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Activated resistance: equivalent capacitance

On the insulating side the activated resistance isdetermined by the relevant Coulomb energy

2(2 )

2B I C

ek T E

C= =

logarray

CC

N=

logB I Ck T E N≈

exp( / )IR T T∝

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Insulator [R~exp(ΤΤΤΤI/T)]TI

Weak insulator

Resistive state



Metallic stateT

0


T

0

Disorder-driven SIT in and BKT in TiN films

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V vs IS4 at B = 0T

B=0

Phase diagram forSuperconductor

TiN films

TBKT

the resistive state

V ∝ Ι

Disorder-driven SIT in and BKT in TiN films: I-V cu rves

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B=0V vs IS4 at B = 0TPhase diagram for

SuperconductorTiN films


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B=0

TiN filmsV vs I

S4 at B = 0TPhase diagram forSuperconductor

the critical current Ic(T)at I<Ic(T) the resistance is

immeasurably small


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B=0

TiN filmsV vs I

S4 at B = 0TPhase diagram forSuperconductor

the critical current Ic(T)at I<Ic(T) the resistance is

immeasurably small


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Phase diagram forinsulating side I2 at B = 0.87 T

I vs V

Ι ∝ V

B=0

TiN filmsDisorder-driven SIT in and BKT in TiN films: I-V cu rves

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I2 at B = 0.87 TI vs V B=0

TiN filmsPhase diagram forinsulating side


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I2 at B = 0.87 TI vs V B=0



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I2 at B = 0.87 TI vs V

Arrhenius behavior of the resistance

B=0



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B=0

TiN filmsI2 at B = 0.87 TI vs V

the threshold voltage VTat V<VT the resistance is

immeasurably highIF Arrhenius behavior of the resistanceretained at 20 mK the I-V would have looked like that

Phase diagram forinsulating side


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I2 at B = 0.37 TI vs V

TSI is magnetic field dependent

B=0



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BSI is the critical magnetic fieldV. Vinokur, T. B., M.V. Fistul, A.Yu. Mironov, M.R. Baklanov, C. Strunk, Nature 452, 613 (2008)

Phase diagram forinsulating side


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Insulator [R~exp(∆∆∆∆C/ kBT)]

Superinsulating stateTSI (= Tcharge-BKT)

TI (= ∆∆∆∆C/kB)Weak insulator

Resistive state



Metallic stateT

0


T

0

Disorder-driven SIT in and BKT in TiN films

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Arrhenius plots of the isomagnetic temperaturedependences of the resistance.

In relatively small magnetic fields the resistance curves shoot upwardsfrom Arrhenius activation behavior below some magnetic field dependent temperature This hyperactivated behavior maintains till fields not exceeding 2 T. At fields larger than 2 T the resistance curves bend downwards showing the tendencyto saturation.

Hyperactivated behavior of the resistance in the low -T charge BKT phase

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We define the characteristic temperature of the transition to the hyperactivated regime by the relation R(THA)=2R0exp(THA/TI), i.e., as the temperature at which the actually measured resistance exceeded two times the resistance obtained by extrapolation from theactivated temperature interval.

R(THA)=2R0exp(THA/TI)

Activated and hyperactivated behavior of the resistance

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Conclusions 1

Joule loss: P = I V=0

Low-temperature BKT phaseon the insulating side of the superconductor-insulator transition:zero conductivity state!!

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Conclusions 2

Questions:

How one can realize 2D Coulomb behavior in our 3D world?

What is the nature of transport in the low-temperature BKT phase?

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Trap all the field lines within the film.

How one can realize 2D Coulombbehavior in our 3D worl d?

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1ε Field lines remain trapped overdistances < εd

Trap all the field lines within the film.

How one can realize 2D Coulombbehavior in our 3D worl d?

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Why the films near the SIT realize 2D universe

Near the SIT the dielectric constant of the film ε → ∞

Superconducting film near superconductor-insulator transition

Percolation transition

The capacitance between the two adjacent clusters is proportional to the length of the insulating layer separating them. Upon approaching the transition from the insulating side of the SIT, the length of this layer diverges infinitely. It results in the divergent growth of the effective capacitance of the system, implying the divergence of the dielectric constant ε → ∞.C ∝ ε l

V.E. Dubrov, M.E. Levinshtein, and M.S. Shur, Sov. Phys. JETP 43, 1050 (1976).

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macroscopic Coulomb blockade energy ∆C (2D)EC – charging energy of a single island(local charging energy)

∆C ≈ EC ln(L/r0)

r0

LL – the linear size of the system

r0 – characteristic size of a single island

Important: macroscopic Coulomb blockade energy ∆Cwell exceeds the local charging energy EC !!

Why the films near the SIT realize 2D universe

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This is a particular case of critical divergence of physical quantities near the phase transition

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D. Kowal and Z. Ovadyahu, Physica C 468, 322 (2008).

Inx O films

L

leads

The distance between the contacts, L,varied, while preserving the width of thesample

More of experiment…

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LL

leads


Inx O films

The distance between the contacts, L,varied, while preserving the width of thesample


53

LL

leads

T0

The activation energyT0 appears to depend on thesample size: the larger the sample, the largerT0, contrary to our experience in semiconductors.



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Scale dependent activation energy

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STM measurements of LDOSinhomogeneous superconducting state (!)

B. Sacépé, C. Chapelier, T. B., V. Vinokur, M.R. Baklanov, M. Sanquer, PRL 101, 157006 (2008)

TiN1 – ∆ = 260 µeV, Teff = 0.25 KTiN2 – ∆ = 225 µeV, Teff = 0.32 KTiN3 – ∆ = 154 µeV, Teff = 0.35 K

BCS fit

in TiN3 the magnitudes of ∆ scattered in the interval from 125 µeV to 215 µeV

56

Tc and ∆∆∆∆mean : the resistance as control parameter

B. Sacépé, C. Chapelier, T. B., V. Vinokur, M.R. Baklanov, M. Sanquer, PRL 101, 157006 (2008)the only fitting parameter:

A.M. Finkel‘stein, JETP Lett. 45, 46 (1987).

The observed trend of the faster decay in Tc than in ∆∆∆∆mean,

together with inhomogeneous state, offers a strong support to

hypothesis that homogeneously disordered superconducting

films near SIT can be viewed as granular-like superconducting

structure.4.48 kΩ - S14.60 kΩ - I1

T = 300 K

< 3% (!)

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“Island” structure of disordered superconductor

Fluctuations in disorder lead to appearance of regions with enhanced Tc, i.e. to formation of

superconducting “islands” above the ‘bulk’ Tc .

N S S

S S

S S

S S S

S

S

∆

Localization in 2D Bose systems:

A. Larkin and V. Vinokur, PRL, 75, 4666 (1995)A. Lopatin and V. Vinokur, PRL, 92, 67008 (2004)

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The two-dimensional Coulomb gas: the model

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The two-dimensional Coulomb gas

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The two-dimensional Coulomb gas: KT arguments

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The two-dimensional Coulomb gas

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The two-dimensional Coulomb gas: BKT summary

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An exemplary system for study BKT: Josephson juncti on array

64

Vortices in JJA

65

Vortices in JJA

66

BKT physics

67

BKT physics

68

BKT physics: I-V characteristics (vortices)

Magnus force

friction force

69


70


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BKT physics: dynamics

( , )H

tt

ϕ δγ ξ ϕδϕ

∂ = − +∂

Overdamped dynamics

Kolmogorov, 1937: crystal growthKramers, 1940: 1D diffusionZeldovich, 1943: decay of metastable state (nuclei growth) in 1-order

phase transitionLanger, 1967: sameVentzel and Fridland, 1969

Search for an instanton solution:U

Mott VRHVortex creepBKT dynamics

exp( / )v U T∝ −

72


2a

Rmin

a

ak

73


U

74


75


76


77

Current, log(I)

Volta

ge,lo

g(V

)

V ∝ Ι α(Τ)

T > TBKT

T < TBKT

α α α α = 1 α α α α > 3

(Tatyana Baturina’s presentation)


78

BKT physics: renormalization of dielectric constant

79

BKT physics: superconducting films, finite size effe cts

80

BKT physics: charges vs. vortices

charge:

electric field:

force:

driving force: electric field

(voltage, )

response: current

e

E

eE

V

I

ur

ur

0

0

0

vortices: "charge" is

current density:

1force (Magnus):

driving force: current

1response: voltage,

(moving magnetic field

induces electric field)

V

c

j

j Lc

V j L nc

η

Φ

Φ

= Φ

81

BKT physics: finite sample size effect

Q: How will the response change in a small sample?A: The applied force is not sufficient to unbind the dipole, thus

only the pairs that have already dissociated will contribute to transport

82

BKT physics: finite sample size – penetration from th e edge

83


1E I α+∝2 , classic BKT

2 +1, wide films,

, narrow films,

w

w

βα β

β

= > Λ < Λ

84


85

InOxV.F. Gantmakher, M.V. Golubkov, J.G.S. Lok,and A.K. Geim (1996)

Bi

Altering various parameters of the system, such as tunnel resitance and transmittance in JJAs, conditions of deposition, chemical composition, and thickness of the films, one can drive the system directly from the superconducting to insulating state.

R.C. Dynes, J.P. Garno, and J.M. Rowell (1978)

granular Pb

D.B. Haviland, Y. Liu, and A.M. Goldman (1989)

2D JJA

L.J. Geerligs, M. Peters, L.E.M. de Groot,A. Verbruggen, and J.E. Mooij (1989).1/T (1/K)

Superconductor-insulator transition

86

2(2 ): charging energy

2: Josephson coupling energy

: insulator; : superconductor

c

J

c J c J

eE

CE

E E E E

=

> <

Mechanism of the transition:Coulomb blockade

An exemplary system demonstrating SIT

87

Disorder-driven Superconductor- Insulator transition in 2D-SC


88

Insulating state:R~exp(∆∆∆∆C/ kBT)

Superinsulating stateTSI (= Tcharge-BKT)

TI (= ∆∆∆∆C/kB)Weak insulator

Resistive state



Metallic stateT

0

T

0

RcSuperconductorR < Rc Insulator

R > Rc

Disorder-driven Superconductor- Insulator transition in 2D-SC:The low-temperature BKT phase in the system of loca lized Cooper pairs

89

Q: How will the response change in a small sample?A: The applied force is not sufficient to unbind the dipole, thus

only the pairs that have already dissociated will contribute to transport

22 2 /p pI en v e n V Lη= =

Linear response approximation: we have to find pn

Transport in a superinsulating state

90

Local Cooper pairs charge density :

the total probability :sI ∝


91

2 : the filling factor of a SC islandn fδ =

1, is the energy of placing

exp( / ) 1

Cooper pair at the island

cc

f EE T

=−

2 : /c cT E n f T Eδ = ≈

2 : exp( / )c cT E n f E Tδ = ≈ −


berezinskii-kosterlitz-thouless transition and ... · tsc (= tvortex-bkt) tc (the superconducting...

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