Cooper-pair-insulator-to-Superinsulator transition in thin TiN...

RESEARCH WORKSHOP OF THE ISRAEL SCIENCE FOUNDATION Transport in Interacting Disordered Systems - 14

Acre, September 5 - 8, 2011

Cooper-pair-insulator-to-Superinsulator transition in thin TiN films

Tatyana I. Baturina Aleksey Yu. Mironov

Svetlana V. Postolova Institute of Semiconductor Physics,

Novosibirsk, Russia

Mikhail R. Baklanov Alessandra Satta

IMEC, Belgium

Christoph Strunk Ante Bilušić David Kalok

University of Regensburg, Germany

Valerii M. Vinokur Argonne National Laboratory, USA

Benjamin Sacépé Claude Chapelier

Marc Sanquer CEA Grenoble, France

September 8, 2011

Superconductor Superinsulator

Joule loss: P = I V

Superconductor: V = 0 ⇒ P = 0 Superinsulator: I = 0 ⇒ P = 0

! Both states are nondissipative !

V. Vinokur, T. Baturina, M.V. Fistul, A.Yu. Mironov, M.R. Baklanov, C. Strunk, Nature 452, 613 (2008)

Superconductor

disorder Rsq=ρ/d (n, d, kFl) ldke

The first studies of superconductivity in the presence of disorder were performed by A.I. Shalnikov (Institute for Physical Problems, Russia).

A. Shalnikov, Nature 142, 74 (1938) A.I. Shalnikov, ZhETF 10, 630 (1940)

Amorphous metals: lead (Pb), tin (Sn) and thallium (Tl) films with thickness between 1 and 200 nanometers (!)

This was the first observation of suppression of Tc with decreasing thickness in thin superconducting films

Suppression of Superconductivity by Disorder

Tc versus Ro=ρ/d

Tc versus inverse thickness 1/d

Tc versus resistivity ρ

Tc versus Ro

activated behavior of resistance

Anderson´s theorem

predicts that nonmagnetic impurities have no effect on superconductivity

A.A. Abrikosov and L.P. Gorkov, Sov. Phys. JETP 8, 1090 (1958) (Phys. Rev. B 49, 12337 (1994)) P.W. Anderson, J. Phys. Chem. Solid 11, 26 (1959)

This theorem does not consider enhancement of electron-electron interaction with increasing disorder

the effect of Anderson localization

weak disorder strong disorder

( ) ( )2r4/r2r4/r

TT lnln0c

+−γ

−−γ

γγ−=

RGr 00 ⋅= ( )2200 2eG π= ( )/kTln1 0c τ=γ

The physical mechanism:

the decrease of the dynamical screening of the Coulomb repulsion between electrons because of the diffusive character of their motion in dirty systems

=> the decrease of the net attraction between electrons

=> the decrease of the transition temperature

Experiment J.M. Graybeal and M.R. Beasley, PRB 29, 4167 (1984)

Theory S. Maekawa, H. Fukuyama, J. Phys. Soc. Jpn. 51,

1380 (1982). A.M. Finkelstein, JETP Lett. 45, 46 (1987)

Mo79Ge21

weak disorder

Fermionic mechanism

Vanishing of Tc is accompanied by vanishing of the amplitude of the superconductive order

parameter Δ (!) There is no Cooper pairs at the transition.

Phase diagram

weak disorder

§  Superconductor – Metal – Fermi-Insulator transition (SMIT)

we can expect the following sequence of transitions

Theory S. Maekawa, H. Fukuyama, J. Phys. Soc. Jpn. 51,

1380 (1982). A.M. Finkelstein, JETP Lett. 45, 46 (1987)

Superconductor – Insulator transition

critical point

Superconductor Insulator

Resistance

Rc R < Rc R > Rc

Localized Cooper pairs Condensate of Cooper pairs

A. 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).

Bosonic mechanism

Cooper-pair insulator

ü  Suppression of Superconductivity by Disorder

§  Superconductor – Bose-Insulator transition (SIT)

§  Superconductor – Metal – Fermi-Insulator transition (SMIT)

Bosonic mechanism

Fermionic mechanism

The superinsulating state appears in the critical vicinity of the superconductor-to-insulator transition

Superconductor - Superinsulator Duality in two dimensions

Thermodynamic phase diagram

= the low-temperature

charge-BKT phase

= the low-temperature

vortex-BKT phase

0 ,0 →→ IV

Superconductor - Superinsulator Duality in two dimensions

Basic requirements for existence of superinsulation

ü Two-dimensionality ü The superinsulating state appears in the critical vicinity of the direct superconductor-to-insulator transition ü The proper size of the system

, , F Tl d lλ ξ< <

quasi-2D electronic spectrum is 3D

d – the thickness of the film

l – the mean free path

λF – Fermi wave length

ξ – the superconducting coherence length

lT – the thermal coherence length

The object

ü thin disordered superconducting films

TERMINOLOGY ONE CALLS:

metal Drude conductivity

+ quantum corrections

insulator thermally activated

or Mott-Efros-Shklovskii-like behavior of conductivity

While in the literature the term “insulator” and/or “the insulating state” is often used to characterize the materials that exhibit negative dR/dT, we find this terminology confusing and misleading: It ignores the very purpose of the terms “metallic” and “insulating”: To discriminate between the different physical mechanisms of conductivity.

InOx Be

MoSi Ta

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

Y. Qin, C.L. Vicente, J. Yoon, PRB 73, 100505(R) (2006). S. Okuma, T. Terashima, and N. Kokubo,

PRB 58, 2816 (1998).

V.F. Gantmakher, M.V. Golubkov, J.G.S. Lok, and A.K. Geim, JETP 82, 951 (1996).

E. Bielejec, J. Ruan, and W. Wu PRL 87, 36801 (2001).

Experiment TiN films

the thickness is 5 nm

ü TiN films were formed by atomic layer chemical vapor deposition onto a Si/SiO2 substrate at 350 0C. ü Composition: Ti N Cl 1 0.94 0.035 ξd = 9.3 nm – the superconducting coherence length

T. Baturina, D.R. Islamov, J. Bentner, C. Strunk, M.R. Baklanov, A. Satta, JETP Lett. 79, 337 (2004) T. Baturina, 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)

4.48 kΩ – S1 <3% (!)

Disorder-driven SIT in TiN films

Rmax = 29.4 kΩ

4.60 kΩ – I1 Rsq @ 300 K

Superconducting films

Insulating or

Superconducting ?

we are here

Superconducting films

Insulating or

Superconducting ?

Disorder-driven SIT in TiN films Q: Why one needs an access to ultra-low temperatures?

A: To determine precisely the position of the SIT

(the exact value of the critical parameters)

and to construct the phase diagram.

S1 4.48 kΩ

<3% (!)

I1 4.60 kΩ

Rsq @ 300 K

S1 4.48 kΩ

<3% (!)

I1 4.60 kΩ

Rsq @ 300 K

we are here

InOx Be

MoSi Ta

Y. Qin, C.L. Vicente, J. Yoon, PRB 73, 100505(R) (2006). S. Okuma, T. Terashima, and N. Kokubo,

PRB 58, 2816 (1998).

V.F. Gantmakher, M.V. Golubkov, J.G.S. Lok, and A.K. Geim, JETP 82, 951 (1996).

E. Bielejec, J. Ruan, and W. Wu PRL 87, 36801 (2001).

other materials

we are here

T. Baturina, A.Yu. Mironov, V. Vinokur, M.R. Baklanov, C. Strunk, JETP Lett. 88, 752 (2008)

4.48 kΩ - S1 4.76 kΩ - I4 T = 300 K

R-1 = G0+G00Aln(T)

Drude conductivity + quantum corrections: weak localization, e-e interaction

G00=e2/(πh)

A=2.55

As temperature decreases from room temperature...

4.48 kΩ - S1 4.76 kΩ - I4 T = 300 K

Drude conductivity + quantum corrections: weak localization, e-e interaction

G00=e2/(πh)

A=2.55

As temperature decreases from room temperature...

at moderate T the behavior of S-samples and I-samples is indistinguishable !! the drastic difference appears only at sufficiently low T !!

R-1 = G0+G00Aln(T)

‘last’ superconductor

‘first’ insulator

Magnetoresistance in the critical region of SIT in TiN films

PRL 99, 257003 (2007) Physica C 468, 316 (2008)

Similar behavior for insulating and superconducting films

This suggests that the electronic structure of these films is similar: all characteristic parameters are close.

4.48 kΩ - S1

4.60 kΩ - I1

T = 300 K

In all samples, including the insulating films, R(B) varies nonmonotonically with B, starting a positive magnetoresistance (PMR) at low fields, then reaching a maximum, followed first by a rapid drop and eventually saturating at higher magnetic fields

‘last’ superconductor

‘first’ insulator

Magnetoresistance in the critical region of SIT in TiN films Cooper-pairing survives in the insulating phase !

R = R0exp(TI/T)

I1: TI = 0.25 K 4.60 kΩ I2: TI = 0.28 K 4.68 kΩ I3: TI = 0.38 K 4.73 kΩ I4: TI = 0.61 K 4.77 kΩ

R0 = 20 kΩ

an Arrhenius behavior of the resistance

Insulating side of the D-SIT in TiN films At lower temperatures...

Rsq @ 300 K

R = R0exp(TI/T)

I1: TI = 0.25 K 4.60 kΩ I2: TI = 0.28 K 4.68 kΩ I3: TI = 0.38 K 4.73 kΩ I4: TI = 0.61 K 4.77 kΩ

R0 = 20 kΩ

an Arrhenius behavior of the resistance

Insulating side of the D-SIT in TiN films At lower temperatures...

Rsq @ 300 K

Superconductor

Metallic state T

Drude conductivity + quantum corrections: weak localization, e-e interaction, superconducting fluctuations

Ψ = Ψ0 exp(iϕ)

The object !!! Two-dimensional superconducting systems: 2D JJ-array, granular films, homogeneously disordered films

Resistive state

TC (the superconducting order parameter appears) Absence of the global phase

coherence: a gas of unbound vortices and antivortices

Superconducting state

TSC (= Tvortex-BKT)

Macroscopic phase coherence: vortices and antivortices are bound in pairs

Berezinskii-Kosterlitz-Thouless transition

BKT transition is a consequence of the logarithmic interaction between vortices!

!!! Two-dimensional superconducting systems: 2D JJ-array, granular films, homogeneously disordered films

Berezinskii-Kosterlitz-Thouless transition

Energy of interacting vortices: U = EJ ln ( R / r0 ) Entropy due to arranging two vortices in the plane:

S = 2kB ln ( R / r0 ) (r0 is the size of the vortex core) Free energy: F = U – TS = EJ ln ( R / r0 ) - 2kB Tln ( R / r0 ) BKT transition at T = TBKT = EJ / 2kB

2D Coulomb interaction All electric force lines are trapped within the film

Energy of two interacting charges: the logarithmic interaction between charges ! the interaction grows with the distance between charges !

The larger the distance between the charges, the smaller the interaction energy

3D world 2D world

The larger the distance between the charges, the larger the interaction energy

rqqU 1

21 ⋅−=επε dd

qqU ρεπε

21 ⋅−=

)( dd ερ <<

ddqqU ρεπε

21 ⋅−=

)( dd ερ <<

ddqqU ρεπε

21 ⋅−=

v  Charge-vortex duality: insulator-to-superinsulator transition as Berezinskii-Kosterlitz-Thouless transition in the 2D system of charges

)( dd ερ <<

ddqqU ρεπε

21 ⋅−=

Q: How one can realize the 2D Coulomb interaction in our 3D world?

A: By arranging ε >> 1.

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

V.E. Dubrov, M.E. Levinshtein, and M.S. Shur, ZhETP 70, 2014 (1976) [Sov. Phys. JETP 43, 1050 (1976)].

Superconducting film near superconductor-insulator transition

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

An insulating state in which superconducting regions (shown as white areas) are separated by dielectric spacers (blue and green stripes). The green stripe highlights the “last” or “critical” insulating interface dividing two large superconducting clusters. The circle marks the critical insulating “bond.”

As soon as the “critical” insulating bond is broken, the path connecting electrodes through the superconducting clusters shown by the red dotted line appears and the film turns superconducting. The system capacitance C on the insulating side is proportional to the length of the green “critical” insulating interface growing with the linear size L of the sample as Lψ, where ψ > 1, i.e. faster than L. Thus the dielectric constant ε ~ C/L diverges upon approach to the percolation superconductor-superinsulator transition.

Superconductor

Resistive state

Superconducting state

TSC (= Tvortex-BKT)

TC (the superconducting order parameter appears)

Metallic state T

Ψ = Ψ0 exp(iϕ)

The object !!! Two-dimensional superconducting systems: 2D JJ-array, granular films, homogeneously disordered films

Current, log(I) Vo

V ∝ Ι α(Τ)

T > TBKT

T < TBKT

α = 1 α > 3

in experiment:

Vortex BKT transition

linear response regime current - voltage characteristics

V ∝ Ι α(Τ)

Insulator [R~exp(ΔC/ kBT)]

Superinsulating state

TSI (= Tcharge-BKT)

TI (= ΔC/kB)

Metallic state T

Current-Voltage Characteristics

in experiment:

Superinsulator

Voltage, log(V)

I ∝ V α(Τ)

T > Tc-BKT

T < Tc-BKT

α = 1 α > 3

Charge BKT transition

D. Kalok, A. Bilušic, T. Baturina, V. Vinokur, C. Strunk,

arXiv:1004.5153

Cooper-pair insulator: linear (Ohmic) response

R = R0exp(TI/T)

arXiv:1004.5153

Superconductor - Superinsulator Duality

Thermodynamic phase diagram Dual current - voltage characteristics

0 ,0 →→ IV

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

Vortex BKT transition Vortice – antivortice binding-unbinding

Dual charge BKT transition Charge – anticharge binding- unbinding

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).

gc g > gc g < gc

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

Vortex BKT transition Vortice – antivortice binding-unbinding

Dual charge BKT transition Charge – anticharge binding- unbinding

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).

EJ < Ec EJ = Ec EJ > Ec

Q: Whether and Why the picture of charge BKT transition developed in the context of JJA can be applied to homogeneously disordered films?

STM measurements of LDOS inhomogeneous superconducting state (!)

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

TiN1 – Δ = 260 µeV, Teff = 0.25 K TiN2 – Δ = 225 µeV, Teff = 0.32 K TiN3 – Δ = 154 µeV, Teff = 0.35 K

BCS fit

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

TiN1 2700 spectra

STM measurements of LDOS inhomogeneous superconducting state (!)

TiN1 – Δ = 260 µeV, Teff = 0.25 K TiN2 – Δ = 225 µeV, Teff = 0.32 K TiN3 – Δ = 154 µeV, Teff = 0.35 K

BCS fit

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

TiN1 2700 spectra

Tc and Δmean

The observed trend of the faster decay in Tc

than in Δmean, together with inhomogeneous

state indicates that Cooper pairs survive at

the insulating side of the SIT

BSC-predicted ratio 76.1/ =Δ cBTk

L. B. Ioffe and A. I. Larkin, Zh. Eksp. Teor. Fiz. 81, 707 (1981) [Sov. Phys. JETP 54, 378 (1981)].

“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 .

S S S S S S S S S S S S S S S S S S S S

Conclusions

In the critical region of the disorder-driven Superconductor-Insulator transition a peculiar highly inhomogeneous insulating phase with superconducting correlations, a Cooper pair insulator, forms.

It is characterized by the activated behavior of the resistance.

At ultra-low temperatures a Cooper pair insulator falls into a Superinsulating state.

Superinsulating state is the low-temperature phase of charge BKT - transition.

ü  ü 

Cooper-pair-insulator-to-Superinsulator transition in thin TiN...

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