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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
RF
sq 2
2
2
3π=
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
Suppression of Superconductivity by Disorder
Tc versus Ro=ρ/d
Tc versus inverse thickness 1/d
Tc versus resistivity ρ
!!!
Suppression of Superconductivity by Disorder
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
Suppression of Superconductivity by Disorder
( ) ( )2r4/r2r4/r
211
TT lnln0c
c
+−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
Suppression of Superconductivity by Disorder
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
Suppression of Superconductivity by 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.
T
S
M
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
Cooper-pair insulator
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.
Bi
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
Disorder-driven SIT in TiN films
Superconducting films
Insulating or
Superconducting ?
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)
we are here
Disorder-driven SIT in TiN films
Superconducting films
Insulating or
Superconducting ?
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)
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)
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
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)
Disorder-driven SIT in TiN films
S1 4.48 kΩ
<3% (!)
I1 4.60 kΩ
Rsq @ 300 K
we are here
Bi
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).
other materials
other materials
Bi
D.B. Haviland, Y. Liu, and A.M. Goldman (1989)
Disorder-driven SIT in TiN films
we are here
Disorder-driven SIT in TiN films
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...
Disorder-driven SIT in TiN films
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
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)
B=0
‘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
B=0
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 !
Cooper-pair insulator
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...
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)
T. Baturina, A.Yu. Mironov, V. Vinokur, M.R. Baklanov, C. Strunk, JETP Lett. 88, 752 (2008)
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...
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)
T. Baturina, A.Yu. Mironov, V. Vinokur, M.R. Baklanov, C. Strunk, JETP Lett. 88, 752 (2008)
Rsq @ 300 K
o o
o
o
Superconductor
Metallic state T
0
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
4 0
21 ⋅−=επε dd
qqU ρεπε
ln2 0
21 ⋅−=
)( dd ερ <<
ddqqU ρεπε
ln2 0
21 ⋅−=
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 !
)( dd ερ <<
ddqqU ρεπε
ln2 0
21 ⋅−=
BKT transition is a consequence of the logarithmic interaction between vortices!
v Charge-vortex duality: insulator-to-superinsulator transition as Berezinskii-Kosterlitz-Thouless transition in the 2D system of charges
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 !
)( dd ερ <<
ddqqU ρεπε
ln2 0
21 ⋅−=
BKT transition is a consequence of the logarithmic interaction between vortices!
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
0
Ψ = Ψ0 exp(iϕ)
The object !!! Two-dimensional superconducting systems: 2D JJ-array, granular films, homogeneously disordered films
Current, log(I) Vo
ltag
e, lo
g(V
)
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
0
Current-Voltage Characteristics
in experiment:
Superinsulator
Voltage, log(V)
Curr
ent,
log(
I)
I ∝ V α(Τ)
T > Tc-BKT
T < Tc-BKT
α = 1 α > 3
Charge BKT transition
linear response regime current - voltage characteristics
D. Kalok, A. Bilušic, T. Baturina, V. Vinokur, C. Strunk,
arXiv:1004.5153
Charge BKT transition
linear response regime current - voltage characteristics
D. Kalok, A. Bilušic, T. Baturina, V. Vinokur, C. Strunk,
arXiv:1004.5153
Charge BKT transition
linear response regime current - voltage characteristics
D. Kalok, A. Bilušic, T. Baturina, V. Vinokur, C. Strunk,
arXiv:1004.5153
Cooper-pair insulator: linear (Ohmic) response
R = R0exp(TI/T)
Charge BKT transition
linear response regime current - voltage characteristics
D. Kalok, A. Bilušic, T. Baturina, V. Vinokur, C. Strunk,
arXiv:1004.5153
Charge BKT transition
linear response regime current - voltage characteristics
D. Kalok, A. Bilušic, T. Baturina, V. Vinokur, C. Strunk,
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).
Superconductor Insulator
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).
Superconductor Insulator
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
TiN2
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
TiN2
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
B. Sacépé, C. Chapelier, T. Baturina., V. Vinokur, M.R. Baklanov, M. Sanquer, PRL 101, 157006 (2008)
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 .
Δ
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 ü
ü