Single quantum level electron turnstile -...

Hervé Courtois, B. Dutta, D. van Zanten, A. Garcia-Corral, C. B. Winkelmann Institut Néel

D. M. Basko, I. M. Khaymovich

LPMMC CNRS, Université Grenoble Alpes and Grenoble INP

Single quantum level electron turnstile

J. P. Pekola Low Temperature Laboratory

Aalto University

A single electron transistor

C C

Cg Vbias/2 -Vbias/2

Vg

Well-defined charge states. At degeneracy : non-interacting conduction channel. Gate oscillation between 1 and 1 states: I = e.f obtained but limited accuracy.

Ec =e2

2CΣ>> kBT

RT >>RK =h

2πe2

2Ec

n=0 n=1

Vbias Vg Degeneracy point

ne-

A metallic island tunnel-coupled to two leads, under the influence of a gate. Charging energy large : Tunnel resistance large :

S-I-N-I-S turnstile

S SN

RFgate400nm

J. P. Pekola et al., Nature Phys. (2008), Rev. Mod. Phys. (2013)

I = e f

The sc gap makes the stability regions 0 and 1 overlap: Improved turnstile accuracy.

2Ec+∆

2∆

n=0 n=1

Vbias

Vg

A quantum-dot transistor

C C

Cg Vbias/2 -Vbias/2

Vg

A metallic quantum dot tunnel-coupled to two leads, with a gate Same Coulomb blockade physics but with a sizeable energy level difference δ Charge and heat transport ?

2Ec

n=0 n=1

Vbias Vg

Degeneracy point

2Ec+∆

2∆

n=0 n=1

Vbias

Vg

δ

Outline

•  Superconductor – Quantum dot – Superconductor electron turnstile

•  Avoided tunneling due to Landau-Zener-Stückelberg physics

•  From Shiba states to Kondo effect

•  Towards the measurement of heat transport through a quantum dot SET

A Superconductor – Quantum dot – Superconductor turnstile

Connecting a single quantum dot

H. Park et al, Appl. Phys. Lett. (1999)

F. Kuemmeth, D. C. Ralph et al., Nano Lett. (2008)

100nm

N. Roch, F. Balestro, W. Wernsdorfer et al., Nature (2008)

Electromigration (voltage bias ramp up to 1 V) of a metallic constriction. Biasing stopped immediately (< 10 µs) when conductance starts to drop: nm to few-nm gaps obtained

S-Q-S devices

D. van Zanten, F. Balestro, H. Courtois, C. B. Winkelmann, Phys. Rev. B (2015).

Large charging energy Ec > 100 meV Electron level spacing δ ≈ 1-10 meV Superconducting gap ∆ = 260 µeV Tunnel coupling γ ≈ 2 µeV Clear hierarchy of energy scales:

EC ≫ δ ≫ Δ ≫ kBT ˃ γ

dI/dV maps

Degeneracy point

n n+1

300nm

All-aluminum electromigration junctions + 5 nm Au nanoparticles

Experimental turnstile operation

H = 0 mT

Vgate [V]

V bias

[mV]

-1.49 -1.485 -1.48

-1

-0.5

0

0.5

1

2Δ

-2Δ

n n+1

D. van Zanten, D. M. Basko, I. M. Khaymovich, J. P. Pekola, H. Courtois, C. B. Winkelmann, Phys. Rev. Lett. (2016).

dI/dV map IDC while sweeping bias, plateau at e.f:

eVB / ∆

I (pA

)

202 MHz

60 MHz

0 MHz

0 0.5 1 1.5 20

10

20

30

40a

IDC = e.f

IDC = e.f


H = 0 mT

Vgate [V]

V bias

[mV]

-1.49 -1.485 -1.48

-1

-0.5

0

0.5

1

2Δ

-2Δ

IDC while increasing frequency:

f (MHz)

I (pA

)

0 50 100 150 2000

10

20

30

40

I / e

f

50 55 60 650.9811.02

I / e

f

190 195 200 2050.9811.02

b


dI/dV map

n n+1

H = 0 mT

Vgate [V]

V bias

[mV]

-1.49 -1.485 -1.48

-1

-0.5

0

0.5

1

2Δ

-2Δ

f=185MHz


IDC while sweeping DC gate:


dI/dV map

IDC = e.f n n+1

Turnstile phase space

ejection

pick-up

Aδ

At degeneracy point VG = VG0:

Turnstile operation threshold

Aδ > Δ - eV/2 Backtunneling threshold

Aδ < Δ + eV/2

non-synchronized current

non-synchronized current

I = 0

I = ef

I = -ef

|I| < ef Aδ/Δ

eVbias Δ

f = 56 MHz, square wave dI/dV

Aδ/Δ

Vbi

as (m

V)

I = 0

I = ef

I = - ef

|I| < ef

Vbias

t

Above-gap states ε

Δε(t)

Missed event probability:

tin tout

Missed tunnel events at high frequencies

γ ≈ 2 µeV (experiment) → P = 0.8 % at 200 MHz

P = exp −γ (tout − tin) / !⎡⎣ ⎤⎦

Nak

amur

a et

al.

Phy

sica

C (2

014)

No thermal back-tunneling

S-I-N-I-S thermal qp excitations

S-Q-S one level involved δ ≫ kBT

At charge degeneracy, the thresholds for the onset ofboth forward and backtunneling can be seen as the narrowblue stripes in Fig. 3(a). Both thresholds cross at VB ¼ 0when Aϵ ¼ Δ. Whereas the frequency-dependent trans-mission of the ac gate signal to the device is not preciselyknown, this crossing is used to calibrate Aϵ. The brightcolor identifies regions of voltage-independent currentcorresponding to I ¼ 0 and I ¼ "ef, respectively.When ϵ̄ is slightly detuned from ϵ̄0 by the static gate

potential, the onset of forward tunneling is linearly shiftedtowards larger Aϵ [Fig. 1(b)]. Note that turnstile operationrequires two successive tunneling events to occur. This isvisible in Fig. 3(b), where the current is shown as a functionof gate detuning and modulation amplitude. For largeramplitudes Aϵ, an increasing tolerance of the turnstileoperation with respect to the proper tuning of ϵ̄ − ϵ̄0develops.Having evidenced electron turnstile operation, let us

now identify the hallmarks of transport through a singlequantum energy level. In SINIS turnstiles, backtunnelingcan be occasioned by electrons from the high-energy tail ofthe thermal energy distribution in N. The backtunnelingprobability increases, thus, steadily and smoothly as Aϵ iscranked up [33]. Conversely, in a SQS turnstile, back-tunneling sets in abruptly when the threshold Aϵ ¼ ΔþjVBj=2e is exceeded. This is seen in Fig. 4(a), where at highenough modulation amplitudes, the current drops suddenlyfrom ef. We numerically model the turnstile currentdependence on Aϵ, both for the SINIS and the SQSturnstile, by solving the time-dependent rate equationsusing the measured output of the ac signal generator.In the SQS case, the instantaneous tunneling rates to eachlead are found from the retarded Green’s function’s pole[24,34,35], that is, beyond Fermi’s golden rule. This is

particularly important near the singularities in the super-conducting density of states (see the Supplemental Material[30]). The calculation [continuous line in Fig. 4(a)] nicelycaptures the abrupt decrease of the current as soon as thebacktunneling threshold is met. For comparison, in a SINISdevice with parameters taken from the most precise devicespresently studied [36,37], the onset of backtunneling ismarkedly smoother (dashed line).This particularly sharp onset of backtunneling is all the

more pronounced if the rise time τ of ϵðtÞ is short, or moreprecisely, if the time available for forward tunneling only isbrief. If ϵ is raised to the backtunneling threshold withinτ ≪ γ−1S;D, the probability of backtunneling may actuallyexceed that of forward tunneling. This means that a currentinversion of magnitude up to ef might eventually beproduced with proper parameter combinations. This couldnot, however, be observed in our experiment because the

FIG. 3. (a) Color map of ∂I=∂VB as a function of bias andgate modulation amplitude (f ¼ 56 MHz, ϵ̄ ¼ ϵ̄0). Narrow blueregions corresponding to rapid increase in current separate areasof voltage-independent current (white), with values I ¼ 0 andI ¼ "ef. (b) Color map of turnstile current as a function of staticgate offset from degeneracy point and gate modulation amplitude(f ¼ 60 MHz, VB ¼ 1.5Δ=e). All data are from device A.

Aε / ∆

I / e

f

0.5 1 1.50

0.2

0.4

0.6

0.8

1

T (mK)

dI/d

VB (

nS)

0 200 4000

5

10

0 200 4000.1

1

10

(ε − eVB/2) / ∆

(∆/e

)⋅dq/

dε

1 1.2 1.4 1.6

−10

−5

0

5

10

15(a)

(b)

(c)

FIG. 4. (a) Turnstile current as a function of operation signalamplitude (device A, f ¼ 56 MHz, ϵ̄ ¼ ϵ̄0, and eVB ¼ 0.7Δ).The sharp decrease in current indicates the sudden onset ofbacktunneling. The continuous line is the numerical calculationfor the SQS with all parameters determined by the device dctransport properties (see text). The dashed line is the analogouscalculation for a SINIS device with normal state resistanceRN ¼ 300 kΩ, U ¼ 3.0Δ and assuming quasiequilibrium ofelectrons in N by electron-phonon relaxation [38]. The arrowsindicate the values of Aϵ used in (c). (b) Slope at inflection pointof IðVbÞ on the turnstile plateaus, averaged over Aϵ, as a functionof temperature (device A). The dashed line is the calculationfor the SINIS device, with parameters as in (a). (c) Calculationof the energy distribution of the delivered charge per cycle,for different gate drive amplitudes Aϵ, with parameters as in (a).The negative part of the panel displays the backtunnelingcontribution. The highest position of the quantum dot level, asdetermined by the gate modulation, is represented in the inset bythe lines of corresponding colors.

PRL 116, 166801 (2016) P HY S I CA L R EV I EW LE T T ER Sweek ending

22 APRIL 2016

166801-3

Backtunneling at large gate drive

A. Kemppinen et al, Appl. Phys. Lett. 2009

S-I-N-I-S

Sharper transition. Calculations with hybridized spectral functions.

S-Q-S one level involved δ ≫ kBT

Vbias = 0.7 & 0.8 ∆

Mono-chromaticity

With a perfect square wave on the gate, perfect mono-chromaticity expected.

At charge degeneracy, the thresholds for the onset ofboth forward and backtunneling can be seen as the narrowblue stripes in Fig. 3(a). Both thresholds cross at VB ¼ 0when Aϵ ¼ Δ. Whereas the frequency-dependent trans-mission of the ac gate signal to the device is not preciselyknown, this crossing is used to calibrate Aϵ. The brightcolor identifies regions of voltage-independent currentcorresponding to I ¼ 0 and I ¼ "ef, respectively.When ϵ̄ is slightly detuned from ϵ̄0 by the static gate

potential, the onset of forward tunneling is linearly shiftedtowards larger Aϵ [Fig. 1(b)]. Note that turnstile operationrequires two successive tunneling events to occur. This isvisible in Fig. 3(b), where the current is shown as a functionof gate detuning and modulation amplitude. For largeramplitudes Aϵ, an increasing tolerance of the turnstileoperation with respect to the proper tuning of ϵ̄ − ϵ̄0develops.Having evidenced electron turnstile operation, let us

now identify the hallmarks of transport through a singlequantum energy level. In SINIS turnstiles, backtunnelingcan be occasioned by electrons from the high-energy tail ofthe thermal energy distribution in N. The backtunnelingprobability increases, thus, steadily and smoothly as Aϵ iscranked up [33]. Conversely, in a SQS turnstile, back-tunneling sets in abruptly when the threshold Aϵ ¼ ΔþjVBj=2e is exceeded. This is seen in Fig. 4(a), where at highenough modulation amplitudes, the current drops suddenlyfrom ef. We numerically model the turnstile currentdependence on Aϵ, both for the SINIS and the SQSturnstile, by solving the time-dependent rate equationsusing the measured output of the ac signal generator.In the SQS case, the instantaneous tunneling rates to eachlead are found from the retarded Green’s function’s pole[24,34,35], that is, beyond Fermi’s golden rule. This is

particularly important near the singularities in the super-conducting density of states (see the Supplemental Material[30]). The calculation [continuous line in Fig. 4(a)] nicelycaptures the abrupt decrease of the current as soon as thebacktunneling threshold is met. For comparison, in a SINISdevice with parameters taken from the most precise devicespresently studied [36,37], the onset of backtunneling ismarkedly smoother (dashed line).This particularly sharp onset of backtunneling is all the

more pronounced if the rise time τ of ϵðtÞ is short, or moreprecisely, if the time available for forward tunneling only isbrief. If ϵ is raised to the backtunneling threshold withinτ ≪ γ−1S;D, the probability of backtunneling may actuallyexceed that of forward tunneling. This means that a currentinversion of magnitude up to ef might eventually beproduced with proper parameter combinations. This couldnot, however, be observed in our experiment because the

FIG. 3. (a) Color map of ∂I=∂VB as a function of bias andgate modulation amplitude (f ¼ 56 MHz, ϵ̄ ¼ ϵ̄0). Narrow blueregions corresponding to rapid increase in current separate areasof voltage-independent current (white), with values I ¼ 0 andI ¼ "ef. (b) Color map of turnstile current as a function of staticgate offset from degeneracy point and gate modulation amplitude(f ¼ 60 MHz, VB ¼ 1.5Δ=e). All data are from device A.

Aε / ∆

I / e

f

0.5 1 1.50

0.2

0.4

0.6

0.8

1

T (mK)

dI/d

VB (

nS)

0 200 4000

5

10

0 200 4000.1

1

10

(ε − eVB/2) / ∆

(∆/e

)⋅dq/

dε1 1.2 1.4 1.6

−10

−5

0

5

10

15(a)

(b)

(c)

FIG. 4. (a) Turnstile current as a function of operation signalamplitude (device A, f ¼ 56 MHz, ϵ̄ ¼ ϵ̄0, and eVB ¼ 0.7Δ).The sharp decrease in current indicates the sudden onset ofbacktunneling. The continuous line is the numerical calculationfor the SQS with all parameters determined by the device dctransport properties (see text). The dashed line is the analogouscalculation for a SINIS device with normal state resistanceRN ¼ 300 kΩ, U ¼ 3.0Δ and assuming quasiequilibrium ofelectrons in N by electron-phonon relaxation [38]. The arrowsindicate the values of Aϵ used in (c). (b) Slope at inflection pointof IðVbÞ on the turnstile plateaus, averaged over Aϵ, as a functionof temperature (device A). The dashed line is the calculationfor the SINIS device, with parameters as in (a). (c) Calculationof the energy distribution of the delivered charge per cycle,for different gate drive amplitudes Aϵ, with parameters as in (a).The negative part of the panel displays the backtunnelingcontribution. The highest position of the quantum dot level, asdetermined by the gate modulation, is represented in the inset bythe lines of corresponding colors.

PRL 116, 166801 (2016) P HY S I CA L R EV I EW LE T T ER Sweek ending

22 APRIL 2016

166801-3

f = 56 MHz, device A, Vbias = 0.7 ∆, rise time 1.6 µs

Smearing (and peak at the gap) due to non-zero rise time of the gate.

Avoided tunneling due to Landau-Zener-Stückelberg physics

energy

tunnel coupling

γ

number of states

2∆ >> γdiscrete level εd

number of states

2∆ >> γ

discrete level coupled to semi-continuum

energy

tunnel coupling

γ

Coupling to a semi-continuum

εdε

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Spectral function γ = 0.01 ∆

εA

εqp

1 real solution εA 1 complex solution εqp εd if > ∆

with

εd (gate)

ε/∆

GR ε( ) = 1ε − εd − Σ ε( )

Σ ε( ) = − γε

Δ2 −ε2

εA − Δ ≈ −Δ2

γ02

(εd − Δ)2

D. Basko, Phys. Rev. Lett. (2016)

≈

Eigenstates can be accessed as the poles of the Green function: Related with level repulsion

t

ε

Superconducting branch

|α|2 ≈ 1 γ

Tunnel coupling

t

Superconducting branch

ε

φ ≈ α A +β qpφ ≈ β qp

Adiabaticity parameter from LZ physics: P = !

(εd − Δ)2dεddt

<1

Adiabatic transitions at anti-crossing

Naive prediction Full model, including the sub-gap states

Dependence on the gate signal waveform

I = ef I = ef

Square wave: virtually infinitely large derivative, tunneling is effective

Sine wave: frequency and amplitude are the parameters

P > 1 P > 1

P < 1 P < 1

Aδ/Δ 1

Aδ/Δ 1

Vbi

as /∆

Vbi

as /∆

f = 100 MHz f = 10 MHz

Towards the measurement of heat transport through a quantum dot

Proposed experiment

Thermometer Heater Drain, thermal bath

Gate

QD

Al

Au+Ti

Heater

Thermometer

RB

ISNS

Gate

Device

As-made constriction

VH

VSNS

Electromigrated

Grafted

Heater and Thermometer = SNS junctions: interest of a low impedance

SNS junction-based electronic thermometry

IV of a SNS junction: Thermal hysteresis observed. Sharp temperature dependence of the critical current. The critical current of a SNS junction used as an electronic thermometer. Sensitivity = dIc/dT = 1.5 µA/K H. Courtois, M. Meschke, J. T. Peltonen, and J. P. Pekola, Phys. Rev. Lett. (2008).

Ic(T)

Test experiment

Heating of the as-made constriction source by a voltage source:

Good agreement with model. Power down to 100 aW detected. B. Dutta et al (2016)

Tph

Drain

P-e-ph

PJoule

Summary

Superconductor – quantum dot –superconductor turnstile

Avoided tunneling at low frequency

From Shiba to Kondo

Josephson current electronic thermometry

eVB / ∆

I (pA

)

202 MHz

60 MHz

0 MHz

0 0.5 1 1.5 20

10

20

30

40a

300nm

Acknowledgements

Grenoble, experiments @ Institut Néel

David Van Zanten, Bivas Dutta, Alvaro Garcia-Corral, Clemens Winkelmann

Grenoble, theory @ LPMMC Helsinki, Aalto University

Ivan Khaymovich, Denis Basko Jukka Pekola, Joonas Peltonen

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