Quantum Fluctuations in Quantum Fluctuations in LowLow--Dimensional SuperconductorsDimensional Superconductors

К. Arutyunov

MIEM HSE & Kapitza Institute, Moscow

Novosibirsk, 22-26.06.2015

Acknowledgements: Eksperiment: M. Zgirski, T. Hongisto, J. Lehtinen and T. Rantala

Discussions:

D. Averin, T. Baturina, F. Hekking, T. Heikkila, L. Ioffe, Y. Nazarov, J. Pekola, V. Vinokur, A. Zaikin

H. K. Onnes,

Commun. Phys. Lab.12,120, (1911)

x(T)

√s < x(T)<< L Long 1D wire of cross section s

If the wire is infinitely long, there is always a finite probability that in some fragment(s) the magnitude of the order parameter instantly becomes zero and the phase changes by 2p

The minimum length the superconductivity can be destroyed is the coherence length x(T)

The minimum energy corresponds to destruction of superconductivity in a volume x(T) s: DF = Bc

2 x(T) s, where Bc(T) is the critical field

In the limit rare events the probability of the process P(T) ~ exp (- DF / E )

Thermal activation: E ~ kBT.

Important at TTc.

Quantum: E ~ D.

Weak temperature dependence, exist even at T0

In current state the particular manifestation of a quantum fluctuation when magnitude of the order parameter momentary nulls and phase changes by ±2π is often called “phase slip”

QPS contributionQPS contribution A. Zaikin, D. Golubev, A. van Otterlo, and G. T. Zimanyi, PRL 78, 1552 (1997)

A. Zaikin, D. Golubev, A. van Otterlo, and G. T. Zimanyi, Uspexi Fiz. Nauk 168, 244 (1998)

D. Golubev and A. Zaikin, Phys. Rev. B 64, 014504 (2001)

Full model (G-Z)

where SQPS = A·[(RQ / x) / (RN / L)], RQ = h / 4e2 = 6.45 kW, D = energy gap,

RN = normal state resistance, L is wire length, x =0.85(x0l)1/2 is coherence length

A ≈ b ≈ 1 are numerical parameters.

Essentially, there is only one fitting parameter: A ≈ 1. For a dirty limit superconductor: GQPS ~ exp (-A’sTc

1/2/rN), where s is the wire diameter and rN is the normal state resitivity.

Materials with low Tc and high resitivity rN are of advantage!

QPS

QPSS

T

LSTbTR 2exp

)(

)()(

2

0

D

x

In the limit R(T)<<RN QPSs are activated at a rate: GQPS = EQPS /h=D0[(RQL2)/(RNx2)] exp (-SQPS)

1D samples: fabrication & shape 1D samples: fabrication & shape controlcontrol

Si

CopPMMA

EBL

patterning

Si

CopPMMA Deposition:

e-gun,

sputtering

Al,Sn,Bi

Si Liftoff

Si IBE

Ar+

~100 nm

Ion beam provides polishing and gradual reduction of the cross-section

Penetration depth of 1 keV Ar+ ions inside Al or Ti matrix < 2 nm

AFM image of a typical Ti nanowire. Inset shows the surface roughness ±1 nm.

Objective: to enable measurements of the same nanowire with progressively reduced diameter

Evolution of Al nanowire after several sessions of ion milling.

TEM analysis of the ion milled nanowire cross sectionTEM analysis of the ion milled nanowire cross section

Dashed lines: thermally activated fluctuations (LAHM) at T ~Tc Solid lines: quantum fluctuations (Golubev – Zaikin) model at T << Tc

0.1 0.2 0.3 0.4 0.5

100

1000

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1000

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1000

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1000

10000

100000

0.1 0.2 0.3 0.4 0.5

100

1000

10000

100000

R

(W)

QPS

Nanowire 1

T (K)

d = 29 nm

d = 35 nm

d = 50 nm

d = 54 nm

d = 58 nm LAHM

Aluminium Titanium

M. Zgirski, K.-P. Riikonen, V. Touboltsev, and K. Arutyunov. // Nano Letters 5, 1029--1033 (2005). M. Zgirski, K.-P. Riikonen, V. Touboltsev and K. Yu. Arutyunov.// PRB 77, 054508-1 -- 054508-6 (2008). J. S. Lehtinen, T. Sajavara, K. Yu. Arutyunov, M. Yu. Presnjakov, and A. S. Vasiliev // PRB 85, 094508 (2012).

NbN is a highly disordered (= highly resistive) superconductor

0.8 0.9 1.0

1E-4

1E-3

0.01

0.1

1 RN (W) Tc(K) A

Sample 211 100000 12.0 0.33

Sample 313 83181 12.0 0.30

Sample 315 39436 11.5 0.16

Sample 421 32343 12.8 0.16

Sample 425 21000 13.3 0.11

Sample 116 14200 13.0 0.10

R/R

N

T/Tc

x = 5 nm

L 5 m

We studied more than 40 samples:

4 and 8 nm thick nanowires, width

from 30 nm to 60 nm, length 5 um.

R□ varied from 15 W to 100 W .

The experimental width of the R(T) transition correlates well with the resistance in normal state RN. The shape of the R(T) transition can be nicely fit with the model of QPS. As expected, TAPS model provides much steeper transition.

Samples were fabricated in MPI-Moscow (Goltsman, Korneev, Semenov, An)

In collaboration with A. Zaikin , A. Semenov

& P. Lähteenmäki, T. Nieminen and P. Hakonen

LT Lab, Aalto University, Finland

ExperimentExperiment

Array of 100 parallel wires

Expectations

eV<<kBT (Johnson noise)

Sv

T Tc

Bulk superconductor

eV≥kBT (shot noise)

SI

I Ic

Bulk superconductor

f=Fano factor

QPS nanowire QPS nanowire

Shot noise due to phase slipsShot noise due to phase slips Current shot noise due to charge pulses:

22 2IS eI e f random pulses – white noise

Voltage shot noise due to phase slips:

2

0 02 2VS f V random pulses – white noise

Q Idt

Vdt

0 =2

h

e

Switch to flux quantum:

02

VV

SF

V

0e

ThermallyThermally activated phase slips activated phase slips

00 sinh

2 B

IV

k T

G

2 002 cosh

2V

B

IS

k T

G

0

0

coth2 2

VV

B

S IF

V k T

D.S. Golubev and A.D. Zaikin, Phys. Rev. B 78, 144502 (2008)

0

0 0

(0) 2coth

2 2

V V BV

B

S S I k TF

V k T I

0 2 BI k T

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

Experimental results: B=0

B=0: ‘Slices’

Johnson noise @ various currents

Finite noise @ T<Tc

Shot noise @ various temperatures

Finite noise @ I<Ic Jump @ I≈Ic

Experimental results: B=35 G

B=35 G: ‘Slices’

Johnson noise @ various currents

Finite noise @ T<Tc (not much difference compared to B=0)

Shot noise @ various temperatures

Finite noise @ I<Ic higher than in normal state!

(0)V VS S

Excess noise vs Excess noise vs biasbias

FanoFano factor of factor of 34 nm wire34 nm wire 10

-910

-80

5000

10000

15000

dV

/dI (W

)

I (A)

10-9

10-8

0

2

4

6

8

F

I (A) Shot noise due to phase slips observed: quantum? Shot noise of phase slips close to the Poissonian value Above the critical current, large shot noise, Fano ~ 5

ConclusionConclusion on transport on transport measurementsmeasurements

In aluminium nanowires with diameters below 15 nm in titanium – below 35 nm and in NbN below 30 nm quantum fluctuations manifest themselves by broadening the R(T) transition. In the thinnest samples zero resistivity is not reached even at T→0.

In thinnest titanium nanowires finite noise is observed at T<<Tc and I<<Ic (?).

Fluctuations of the order parameter D=D0exp(i)

T. Rantala, MSc thesis, University of Jyvaskyla, 2013

Direct Direct determinationdetermination of the of the fluctuatingfluctuating energyenergy gapgap

Superconducting gap can be associated with E/M radiation absorption threshold

dD / D ~ (SQPS ) -1

Characteristic scale D(Al)~80 to100 GHz, D(Ti)~15 to 25 GHz.

dD /D can reach 30% in sufficently thin nanowires

BULK QPS

TunnelingTunneling in NIS and SIS in NIS and SIS systemssystems M. Tinkham, Introduction to supercondcutivity, Mc. Graw-Hill, 1996

Quantum fluctuations of the gap D should lead to extra (size dependent) broadening of the gap edge

NIS SIS

ExperimentExperiment

0 100 200 300 400 5001000

10000

100000

Wire#2 DM31: deff

= 31±3 nm

Wire#4 DM31: deff

= 36±3 nm

Wire#2 DM30: deff

= 37±3 nm

Wire#4 DM30: deff

= 40±3 nm

Wire#5 DM30: deff

= 45±3 nm

R(O

hm

s)

T(mK)

R(T) of each nanowire and I(V) chartacteristic between the nanowire and the aluminium electrode can be measured independently.

R(T) dependencies at these diameters are broad associated with QPS.

Schematic of the experiment

SIS I(V) at various diameters of nanowiresSIS I(V) at various diameters of nanowires

-0.2 -0.1 0.0 0.1 0.2

5.0x10-5

1.0x10-4

1.5x10-4

wire#2, deff

=31 nm

wire#4, deff

=36 nm

dI/dV (S)

V(mV)

T=20 mK

Sample DM31

-0.2 -0.1 0.0 0.1 0.20.0

2.0x10-6

4.0x10-6

6.0x10-6

8.0x10-6

1.0x10-5

1.2x10-5

wire#2, deff

=31 nm

wire#4, deff

=36 nm

dI/dV (S)

V(mV)

T=20 mK

Sample DM31

D1D2 features and Coulomb blockade and/or Josephson current (?)

0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27

0.0

5.0x10-5

1.0x10-4

1.5x10-4

wire#2, deff

=31 nm

wire#4, deff

=36 nm

dI/dV (S)

V(mV)

T=20 mK

Sample DM31

D1+D2 features

The smaller the nanowire diameter (1) the smaller the average value <D> and (2) the larger the gap smearing dD.

SimulationsSimulations We simulate I(V) dependencies assuming Gaussian distribution of the

gap fluctuations centered at <D> with standard deviation dD.

0

20

40

60

80

100

35 40 45 50 55 60

N

<D>=48 V dD3 V

Limitations & postulates: • Gaussian distribution of quantum fluctuation amplitude

• Cut-off Dbulk /2 < D < Dbulk

• Finite magnitude of voltage modulation and finite lock-in integration time

0.0E+00

4.0E-05

8.0E-05

1.2E-04

1.6E-04

2.0E-04

2.0E-04 2.2E-04 2.4E-04 2.6E-04

dI/dV (S)

V (V)

DM31, Wire#4 d = 36 ±3 nm

Thick (conventional)

superconducting wire

Thin wire and low-W environment

Very thin wire and high-W environment:

Coulomb blockade

Ti nanowire in Ti nanowire in lowlow--OhmicOhmic environmentenvironment

Differential resistance dV/dI, normalized by the normal state resistance RN, as function of the bias current I.

In ‘thick’ samples one observes the zero resistance state below the critical current. With reduction of the wire diameter the ‘residual’ critical current disappears and the finite resistance is observed .

D. V. Averin and A. A. Odintsov, Phys. Lett. A 140 (1989) 251 J. E. Mooij and Yu. V. Nazarov, Nature Physics 2 (2006) 169

A. M. Hriscu and Yu. V. Nazarov. Phys. Rev. B 83, 174511 (2011)

Hamiltonian of a supercondcuting nanowire in the regime of quantum fluctuations:

is dual to the corresponding Hamiltonian of a Josephson junction:

with the accuracy of substitution:

The extensively developed physics for Josephson systems can be ‘mapped’ on

the superconducting nanowires in the regime of quantum fluctuations.

eqEEEE QPSJLC 2/, , p

EL , EC , EJ and EQPS are the inductive, charging, Josephson coupling and

QPS energies, is phase and q is quasicharge.

A.M. Hriscu and Yu. V. Nazarov. Phys. Rev. B 83, 174511 (2011)

The mandatory requirement for the charge effects

observation is the high impedance of the environment:

current bias charge is a ‘good’ quantum number.

QPS is a dynamic equivalent of a

conventional (static) tunnel junction.

First two terms correspond to linear LC oscillator. If to shift the

charge variable by q/2e, the induced charge disappears from the

oscillator part, while the QPS amplitudes acquire the phase factors:

The charge sensitivity is purely due to the coherent

QPS term: the induced charge q/2e affects the

interference of the phase slips with two opposite

directions ±2π.

Let us consider the simplest case of a QPS Copper pair box:

HighHigh--OhmicOhmic environmentenvironment

Purely dissipative environment: high-Ohmic normal metal probes with Rprobe up to 10 MOhm

24 24 nmnm titaniumtitanium nanowirenanowire, 10 M, 10 MWW contactscontacts

All three neighboring parts of the same multiterminal structure demonstrate the same value of the Coulomb gap. The effect disappears above Tc and/or Hc.

Temperature dependenciesTemperature dependencies

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

I (1

0-1

0 A

)

V

Structure 170212

Sample#3

100 V

23 mK

100 mK

200 mK

300 mK

400 mK

0 200 400 600 800 1000

100000

1000000

1E7

1E8

S3 no sput

S3 8k sput

S2 no sput

dV

/dI(

Ohm

s)

T (K)

RN

-0.02 -0.01 0.00 0.01 0.02

1000000

2E6

3E6

SET170212

Sample#3

B=0

575 mK

500 mK

400 mK

200 mK

23 mK

dV

/dI

(Ohm

s)

Vgate(V)

Ibias

= 0 nA DC + 70 pA AC

The Coulomb gap and the gate modulation

disappear above 450 mK (the Tc of Ti?).

Magnetic field dependenciesMagnetic field dependencies

-0.02 0.00 0.02

2E8

3E8

4E8

B = 68 mT

B = 3.85 T

B = 25 mT

B = 0

dV

/dI(

Oh

ms)

Vgate(V)

SET170212

Sample#2

T=100 mK

0 1 2 3 4

0

5

10

15

20

25

30

SET170212

Sample#2

T=100 mK

dV

CB

(Ib

ias =

0),

uV

B(T)

At a given (low) temperature the Coulomb gap and the gate modulation disappear above certain magnetic field

(the Bc of Ti?).

Digest on QPS Digest on QPS transistortransistor • Quantum phase slip and Josephson tunneling are the

phenomena described by identical Hamiltonians: the quantum dynamics is indistinguishable.

• Supercondcuting nanowire (homogeneous!), in the regime of QPS, is the dynamic equivalent of a Josephson junction.

• Containing no static (in space and time) junctions, QPS nanowire can sustain much higher currents and has no undesired two-level ’fluctuators’ present in tunnel contacts.

• All phenomena, observable in Josephson systems, can be observed in QPS nanowires.

Josephson junction: critical current

QPS wire: critical voltage

Josephson junction: voltage steps

(Shapiro effect)

QPS wire: current steps

Electron transport in a Cooper pair transistor is a periodic process

corresponding to cyclic charging/discharging of the central island by 2e.

Syncronisation of the process with external drive should lead to resonance.

NotNot--tootoo--highhigh--WW environment (50 kenvironment (50 kWW))

0.00 0.34 0.68 1.02 1.36

4.1

4.2

dV

/ d

I (a

rb.

un

its)

I (nA)

TiWire290710_part34_

f(RF) =1GHz

Arrows mark the prediction I =n×2e ×fRF

0 1 2 3 4 5 6 7 8 9 10

1

2

3

TiBi260210

sample#3

T=18 mK

fRF

=350 MHz

VRF

=100 mV

dV

/ d

I (a

rb.

un

its

)

I / (2e)f

dV/dI at fRF=350 MHz and amplitude 100 mV. Current is normalized by (2e)×fRF. One can distingush steps with quantum number n≤8.

RFRF--inducedinduced stepssteps at at highhigh frequencyfrequency With the increase of the RF frequency (= larger value of the current) the width

of the steps increases, but the noise increases and (sub)harmonics appear.

400 450 500 550 600 650

0.00

0.61

1.22

1.83

2.44

3/1

1/1

TiBi260210

sample#1

T=75 mK

fRF

=3.8 GHz

VRF

=45 mV

I

(nA

)

V (mV)

2/1

4/1

1/2

Red arrows correspond to the expectations for charge (2e) and the blue arrows – to single electron oscillations (e).

WidthWidth of the of the currentcurrent stepstep vs. RF vs. RF radiationradiation amplitudeamplitude

Width of the n-th step current step dVn~(-1)nJn(a)sin(q),

where q – quasicharge, a – amplitude of the RF radiation, Jn – Bessel function

[D. V. Averin and A. A. Odintsov, Fizika Nizkix Temperatur, 16, 16 (1990)]

0 50 100 150 200 250 300 350

0

10

20

30

40

50

60

70

80

90

100

110

dV

(m

V)

VRF (mV)

2/1 step

0/1 step x 0.1

(blockade width)

TiBi260210

sample#1

T=18 mK

fRF

=3.8 GHz

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1/2

3/1

1/1

4/1

TiBi260210

sample#1

T=18 mKI e

xp (

nA

)

2ef (nA)

2/1

Red symbols correspond to Bloch steps (2e), blue – single electron (e), black – 1st subharmonic of single electron oscillations.

SimulatedSimulated BlochBloch stepssteps

C. H. Webster, J. C. Fenton, T. T. Hongisto, S.P. Giblin, A. B. Zorin, and P. A. Warburton, PRB 87 144510 (2013)

Simulated I-V characteristics of a NbSi nanowire for a

continuous-wave ac signal with various amplitudes and

frequency 1 GHz at T= 0, 33, 200 and 500 mK.

where Vn(t) is noise with rms amplitude

Sharpness of the Bloch steps depends on noise. At a finite current I the Joule heating

of the resistor R increases the temperature T, and, hence, broadens the steps.

at temperature T and bandwidth df , I≡dq/dt.

HHighigh impedanceimpedance environmentenvironment M. Watanabe, D. Haviland , PRL 86, 5120 (2001); PRB 67, 094505 (2003); M. Watanabe, PRB 69, 094509 (2004).

1D chain of SQUIDs

At f/f0→p/2 both Rdyn→∞ and LK→∞.

SuperinductanceSuperinductance In more ’complicated’ Josephson systems the EJ(f) dependence can be strongy anharmonic. Within the locus of the anharmonicty the kinetic inductance can be made very large LK→∞.

M. T. Bell, at. al. PRL 109, 137003 (2012) and N. A. Masluk, et.al. PRL 109, 137002 (2012)

HybridHybrid highhigh impedanceimpedance environmentenvironment

High-Ohmic electrodes

Ti-AlOx-Ti SQUIDs

Ti - nanowires

island

High-Ohmic normal metal probes with Rprobe up to 500 KOhm and 1D array of SQUIDs.

Temperature dependence of the Coulomb blockade

The insulating state (Coulomb blockade) disappears above Tc.

Inset: temperature dependencies of the zero-bias dynamic resistance of several nanowires, normalized by the normal state resistance RN. Length L(m), effective diameter s1/2(nm) and high-bias impedance (kW) of

the environment RENV are listed in the inset.

Coulomb blockade in high impedance environment

Coulomb blockade vs. cross section s. Inset: magnetic field dependence of

the Coulomb gap.

Coulomb blockade vs. impedance of the environment Rmin

ENV. Inset: dynamic resistance dV/dI at

various biases for typical SQUID probes.

WidthWidth of the of the BlochBloch stepstep

0.0 0.5 1.0 1.5 2.0

0

2

4

dV

/dI (a

rb. units)

I (nA)

T=70 mK

fRF

=3.12 GHz

Imod

=0.5 pA rms

0.90 0.95 1.00 1.05 1.100

2

4

Imod

=0.5 pA rms

dV

/dI (a

rb. units)

I (nA)

dI=20 pA

Demonstrated accuracy is +/- 2%. Further accuracy improovement is mandatory for practical

metrology.