Gapless Andreev bound states in HgTe-based topological...

TopoLyon 2016 03/10/2016

Physikalisches Institut (EP3) Universität Würzburg, Am Hubland, D-97074 Würzburg

http://www.physik.uni-wuerzburg.de/EP3/

Erwann Bocquillon

Gapless Andreev bound states in HgTe-based topological Josephson junctions

http://www.lpa.ens.fr

Erwann Bocquillon Lyon - 03/10/20163

Uni. Würzburg

PhD students : J. Wiedenmann, P. Leubner Staff : C. Brüne H. Buhmann L.W. Molenkamp Invited : T.M. Klapwijk Theory : F. Domínguez, E.M. Hankiewicz

RIKEN, Tōkyō

R.S. Deacon, K. Ishibashi, S. Tarucha

People involved


Induced superconductivity in a 2D TI

4

Cooper pair of helical Dirac fermions ⇒ helical pairing ⇒ p-type correlations

2D TI S

k ?"

k ",k #

gapless Andreev bound states ⇒ Majoranas

spin-orbit coupling ⇒ -junctions'0

Dolcini et al., PRB 92, 035428 (2015)

Fu et al., PRB 79, 161408 (2009)

"(k)

k


Andreev reflections

5

Andreev reflection

Cooper pair created in S hole reflected in N phase coherence

N/TI S

1

S

2


Andreev reflections

5

Andreev reflection

Cooper pair created in S hole reflected in N phase coherence

"n()

Andreev bound states

resonant modes of the ’’cavity’’ energy levels = 1 2

N/TI S

1

S

2


Gapless Andreev bound states

6


2π gapped states (bulk)

4π gapless topological state (edge)

I2 sin

I4 sin/2

(+harmonics)

Kwon et al., JLTP 30, 613 (2004) Fu et al., PRB 79, 161408 (2009)

fractional Josephson effect

π0 2π 3π 4π

Energy

"[

]

-1.0

-0.5

0.0

0.5

1.0

Phase di↵erence

I2 sin

I4 sin/2


Gapless Andreev bound states

6


2π gapped states (bulk)

4π gapless topological state (edge)

I2 sin

I4 sin/2

(+harmonics)

Kwon et al., JLTP 30, 613 (2004) Fu et al., PRB 79, 161408 (2009)

fractional Josephson effect

π0 2π 3π 4π

Energy

"[

]

-1.0

-0.5

0.0

0.5

1.0

Phase di↵erence

I2 sin

I4 sin/2

Difficult detection 2π bulk states ⇒ 2π/4π mixture finite lifetime ⇒ 2π-periodicity restored interactions ⇒ 8π-periodicity Landau-Zener transitions ⇒ 4π-periodicity

Pikulin et al., PRB 86, 140504 (2012) Badiane et al., CRP 14, 840 (2013) Zhang et al., PRL 113, 036401 (2014) Peng et al., ArXiv 1609.01896 (2016) Hui et al., ArXiv 1609.02909 (2016)


Outline

7

Josephson junctions in a 2D TI

Josephson emission

Response to AC excitation (Shapiro steps)

Bocquillon et al., Nat. Nano, DOI: 10.1038/NNANO.2016.159 Deacon et al., ArXiv 1603.09611 (2016)


QSH effect in HgTe

8

Bernevig et al., Science 314, 1757 (2006) König et al., Science 318, 766 (2007)

MBE growth, μ ≃ 3 105 cm2V-1s-1

QSH if d > dc ≃ 6.3 nm at B = 0 !

trivial if d < dc

6

8

E1

H1

d > dc

HgCdTe HgTe HgCdTe Band inversion in bulk HgTe

⇒ inverted QW

⇒ Quantum Spin Hall effect


Quantum spin Hall junctions

9

HgTeHg0.3Cd0.7Te

2μm

AlHfO2/Au

L ' 400 nm

W ' 4 µm

Josephson junctions

μ≃3 105 cm2V-1s-1

Al contacts (in situ)

HfO2 /Au gate

no overlap of edge states

ballistic / intermediateL . L l


Quantum spin Hall junctions

9

HgTeHg0.3Cd0.7Te

2μm

AlHfO2/Au

Josephson junctions

μ≃3 105 cm2V-1s-1

Al contacts (in situ)

HfO2 /Au gate

no overlap of edge states

ballistic / intermediateL . L l


First properties

10

I-V curve

weak hysteresis visible excess current ⇒ Andreev reflections

Gate dependence

3 regimes : p, n, and QSH asymmetry between n and p

-6 -4 -2 0 2 4 6

-400

-200

0

200

400

DC current I [µA]

DC

voltageV[µV

]

Gate voltage Vg [V]

n-type bulk

Resistance

Rn[Ω

]

Crit.

current

Ic[nA]

p-type bulk

-2.0 -1.5 -1.0 -0.5 0.00

2004006008001000120014001600

0

200

400

600

800

1000

1200

Blonder et al., PRB 25, 4515 (1982)

Rn

IcRn Ic


(Fractional) Josephson effect

11

Josephson equations

IS() = Ic sin

d

dt=

2eV

~ = 1 2

fJ =2eV

~

1

2

fJ =2eV

~⇒ Josephson frequency



11

Josephson equations

IS() = Ic sin

d

dt=

2eV

~ = 1 2

fJ =2eV

~

1

2

fJ =2eV


Fractional Josephson effect

sin ! sin/2

fJ ! fJ/2



11

Josephson equations

IS() = Ic sin

d

dt=

2eV

~ = 1 2

fJ =2eV

~

1

2

fJ =2eV



sin ! sin/2

fJ ! fJ/2

Detection

‘listening’ to Josephson emission beatings with ac excitation (Shapiro steps)


Detection setup

12

voltage bias shunt resistance Rs

current resistance R

T≃ 300 K

T≃ 4 K

T≃ 150 mK

T≃ 25 mK

rf exc.

spectrum analyzer

c

from bias-T

Vg to rf amp.

exc. current

VR

V

b

2 μm

I

RS

R

HEMT amp.

a

Superconducting phase ! π 2π 3π 4π

Ener

gy

0

0

Δi

-Δi

EJ

topological state

conv. state

bias-T

rf amplification setup 1 cryo amp. (+ 2 amps at RT) 0.1 fW (-130 dBm) in 8 MHz

a

Superconducting phase ! π 2π 3π 4π

Ener

gy

0

0

Δi

-Δi

EJ

topological state

conv. state

T≃ 300 K

T≃ 4 K

T≃ 150 mK

T≃ 25 mK

rf exc.

spectrum analyzer

c

from bias-T

Vg to rf amp.

bias current

RI I

V

2 μm

I

RS

RI

to voltage meas.

b

HEMT amp.


Emission spectra

13

voltage V swept

integrated power at fd=3 GHz (in 8 MHz bandwidth)

trivial QW : signal at fd=fJ

topological QW : at fd=fJ and fJ/2

Deacon et al., submitted, ArXiv 1603.09611 (2016)dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

a b c

d e f

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

fJ2fJ

fJ

fJ/2

2fJ

fJ/2

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

a b c

d e f

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

fJ2fJ

fJ

fJ/2

2fJ

fJ/2

Trivial QW Topological QWn- and QSH regime p-regime


Frequency dependence

14

Stronger fJ/2 signal at low frequencies

Relative intensities of fJ/2 and fJ depending on Vg

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

a b c

d e f

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

fJ2fJ

fJ

fJ/2

2fJ

fJ/2

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

a b c

d e f

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

dc v

olta

ge V

[μV

]

dc voltage V [μV]

frequency fd [GHz]

dc c

urre

nt I

[μA] rf am

p. A [a.u.]

fJ2fJ

fJ

fJ/2

2fJ

fJ/2

Trivial QW Topological QWn- and QSH regime p-regime

fJ

fJ/2


Gate voltage dependence

15

dc v

olta

ge V

[μV

]

gate voltage Vg [V]

a

dc v

olta

ge V

[μV

]

gate voltage Vg [V]

b

crit.

cur

rent

I c [μ

A]

gate voltage Vg [V]

Fraunhofer2π Shapiro 4π Shapiro

SQUID

4π emission2π emission

average ratio <r>

c

fJ/2fJ2fJ

dc v

olta

ge V

[μV

]

gate voltage Vg [V]

a

dc v

olta

ge V

[μV

]

gate voltage Vg [V]

b

crit.

cur

rent

I c [μ

A]

gate voltage Vg [V]


SQUID


average ratio <r>

c

fJ/2fJ2fJ

Low frequency fd= 3 GHz

High frequency fd= 5.5 GHz


Gate voltage dependence

15

dc v

olta

ge V

[μV

]

gate voltage Vg [V]

a

dc v

olta

ge V

[μV

]

gate voltage Vg [V]

b

crit.

cur

rent

I c [μ

A]

gate voltage Vg [V]


SQUID


average ratio <r>

c

fJ/2fJ2fJ

dc v

olta

ge V

[μV

]

gate voltage Vg [V]

a

dc v

olta

ge V

[μV

]

gate voltage Vg [V]

b

crit.

cur

rent

I c [μ

A]

gate voltage Vg [V]


SQUID


average ratio <r>

c

fJ/2fJ2fJ

Low frequency fd= 3 GHz

High frequency fd= 5.5 GHz

QSH n regimep regime


AC response : Shapiro steps

16

Shapiro, PRL 11, 80 (1963) Russer, J. App. Phys. 43, 2008 (1972)

Phase-locked motion

phase dynamics (RSJ model)

motion locked to rf excitation

⇒ Vn = nhf2e

ddt = 2eV

~

t = 2n

1/f

I = IS() +~

2eR

U()

I@U() = IS() I



16


Phase-locked motion



⇒ Vn = nhf2e

ddt = 2eV

~

t = 2n

1/f

I = IS() +~

2eR



16


Phase-locked motion



⇒ Vn = nhf2e

ddt = 2eV

~

t = 2n

1/f

I = IS() +~

2eR

4π-periodic supercurrent

doubled steps

mixture 2π/4π ?

sin ! sin/2Vn ! V2n


Shapiro response : frequency

17

Shapiro response

>12 steps visible weak hysteresis on 1st step

-400 -300 -200 -100 0 100 200 300 400-8

-6

-4

-2

0

2

4

6

8

DC

voltageV[

hf

2e]

DC current I [nA]

f = 2GHz

f = 1GHz

f = 6.6GHz

multiple odd steps missing at low frequency f ≲ 4 GHz

Rokhinson et al., Nat. Phys. 86, 146503 (2012) Wiedenmann et al., Nat. Comms 7, 10303 (2016) Bocquillon et al., Nat. Nano, DOI: 10.1038/NNANO.2016.159


Shapiro response : frequency

17

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 30 60 90 120 0 10 20 30 40 0 5 10 15 0 5 10 150 10 20 30 40

DC

voltageV[

hf

2e]

Step amplitude [nA]

f = 6.6GHz f = 0.8GHz f = 3.5GHz f = 1.8GHz f = 1GHz

Shapiro response

>12 steps visible weak hysteresis on 1st step

multiple odd steps missing at low frequency f ≲ 4 GHz

Rokhinson et al., Nat. Phys. 86, 146503 (2012) Wiedenmann et al., Nat. Comms 7, 10303 (2016) Bocquillon et al., Nat. Nano, DOI: 10.1038/NNANO.2016.159

Our device missing n=1,3,5 « dark fringes »


Shapiro response : power

18

Simulated response at low power : steps forming at high power : oscillatory pattern

0 2 4 6-2-4-60

0.5

1.5

2

1

RFcu

rrentI r

f[I

c]

DC voltage V [hf/2e]RFpowerPrf[dBm]

T ' 25mKf = 1GHzVg = 1.1V

DC voltage V [hf/2e]


Non-topological HgTe quantum well

19

DC voltage V [

hf2e ]

RFpowerPrf[dBm]

-2.0 -1.5 -1.0 -0.5 0.0

0

50

100

150

200

250

300

350

400

-6 -4 -2 0 2 4 6-46

-44

-42

-40

-38

-36

-34

-32

-30

100

150

200

250

300

350

400

Vg = 1V

f = 0.6GHz

T ' 25mK

Vg = 1V

f = 0.6GHz

T ' 25mK

Vg = 1V

f = 0.6GHz

T ' 25mK

Non-topological QW

narrow well (5 nm) no band inversion similar mobility 1.5 105 cm2V-1s-1

Shapiro steps

no missing steps n-, p- regimes and gap verified down to f= 0.6 GHz


RSJ frequency dependence

20

I≳Ic

High frequency

⇒ almost sinusoidal

I≫Ic

Domínguez et al., PRB 86, 146503 (2012)

Low frequency

⇒ non-sinusoidal



20

I≳Ic

High frequency


I≫Ic


Low frequency

⇒ non-sinusoidal

I2 = 1, I4 = 0

I2 = 1, I4 = 0.15



20

I≳Ic

High frequency


I≫Ic


⇒ crossover f4π yields : 1-3 modes ⇒ no Landau-Zener transitions ?

Low frequency

⇒ non-sinusoidal

I2 = 1, I4 = 0

I2 = 1, I4 = 0.15


RSJ simulations

21

Theory : F. Domínguez & E. M. Hankiewicz

a b

dc v

olta

ge V

[μV

]dc voltage V [μV] frequency fd [GHz]

dc c

urre

nt I

[μA]

fJ/2

fJ


Summary

22

Gate voltage Vg [V]

FraunhoferSQUID

4π Shapiro2π Shapiro

n-type bulk

Resistance

Rn[Ω]

Crit.

current

Ic[nA]

p-type bulk

F.

-2.0 -1.5 -1.0 -0.5 0.00

2004006008001000120014001600

0

200

400

600

800

1000

1200

fJ/2 emissionfJ emission


even sequence of Shapiro steps emission at fJ/2 Landau-Zener transitions unlikely edge currents (SQUID pattern)

contribution: 1-3 modes coexistence with conduction band ? discrepancy with Rn ?

Dai et al., PRB 77, 125319 (2008) Hart et al., Nat. Phys. 10, 638 (2014)


Outlook

23

Pillet et al., Nature Phys. 6, 965 (2010) Bretheau et al., Nature 499, 312 (2013) Astafiev et al., Science 327 840 (2010) Peng et al., arXiv 1604.04287 (2016)

Spectroscopy of Andreev bound states

tunneling DOS absorption spectroscopy SN junctions

Other HgTe systems

HgTe nanowires QSH, QH, QAH, Weyl

⇒ towards Majorana qu-bits


Open positions !

24

Post-docs

PhD students

Thank you for your attention !

Open positions!


Shapiro response : RSJ model

25

4π-supercurrent

≃ 20 nA at Vg=-1.1 V 1-2 modes

Dominguez et al., PRB 86, 146503 (2012) Wiedenmann et al., Nat. Comms 7, 10303 (2016)

RSJ simulations

anharmonic motion doubled step due to 4π transition at f f4 = 2eRnI4

h

Landau-Zener transitions

stronger at high frequency

I2 = 1, I4 = 0

I2 = 1, I4 = 0.15

n = 0,±1,±2, ...


Shapiro steps

26

High frequency

>12 steps visible all steps seen:

Low frequency

first step missing: 4π supercurrent?

n = 0,±2,±3...

Rokhinson et al., Nat. Phys. 86, 146503 (2012)

DC Current I [µA]Bin counts [nA]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

-100 -50 0 50 100-3

-2

-1

0

1

2

3

0 100 200

-6

-4

-2

0

2

4

6

0 200 400 600

-6

-4

-2

0

2

4

6

-0.2

0.2

0

-5 50

-4 -3 -2 -1 0 1 2 3 4

-6

-4

-2

0

2

4

6sweepsweep

f=2.7 GHzf=5.3 GHzf=11.2 GHz

f=2.7 GHz f=11.2 GHz



27

f = 5.3 GHzf = 2.7 GHz f = 11.2 GHzA) B) C)

0 0.05 0.1 0.20.15 0 0.1 0.2 0.3 0 0.2 0.4 0.6 0.8

0

4321

rf exc. Irf [a.u.] rf exc. Irf [a.u.]

rf exc. Irf [a.u.]

rf exc. Irf [a.u.]

rf exc. Irf [a.u.]rf exc. Irf [a.u.]

0

2

4

6

-2

-4

-6

0

2

4

6

-2

-4

-6

0

2

4

6

-2

-4

-6

0.2 10.4 0.6 0.8 0.2 10.4 0.6 0.8 0.2 10.4 0.6 0.8

0.2 10.4 0.6 0.80.2 10.4 0.6 0.80.2 10.4 0.6 0.8

0

1

2

0

1

2

3

0

2

4

6

8

DC

voltageV

[hf/2

e]

Bin counts [µA] Bin counts [µA] Bin counts [µA]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

w4

Stepam

plitude[µA]

Stepam

plitude[µA]

Stepam

plitude[µA]


Mechanism for a missing step?

28

Additional subharmonic steps

capacitive effects non-linearities higher harmonics in the CPR

n = 1/2, 3/2, 1/3, ...

Dominguez et al., PRB 86, 146503 (2012) Badiane et al., PRL 107, 17702 (2011)

Renne et al., R. Phys. App. 9, 25 (1974) Valizadeh et al., JNMP 15, 407 (2008) Sochnikov et al., PRL 114 066801 (2014)

0 1 2 3 4 5 6

0

1

2

3

4

5

6

Norm.voltageV

[hf/2

e]

DC current I [µA]

f = 13.2GHz

Missing steps

4π supercurrent dominates at low frequency

confirmed by RSJ simulations

I4 sin/2

f f4 = 2eRnI4h


RSJ dynamics

29

U()

I

Motion of fictitious particle

@U() = IS() I

IS() = I4 sin

2+ I2 sin+ . . .

0 1 2 3 4 5 6-0.5

0

0.5

1

1.5

2

2.5

[2]

U()

Energy potentials

13% of 4π supercurrent 11% of increase of

I2 = 1, I4 = 0

I2 = 1, I4 = 0.15

Ic


RSJ simulations

30

0

2

4

6

-2

-4

-6

0

0.5

1

1.5

0 2 4 6 0 1 2 3 0 0.5 1.5 21RF current Irf [Ic] RF current Irf [Ic] RF current Irf [Ic]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

Stepam

plitude[I

c]

Stepam

plitude[I

c]

Stepam

plitude[I

c]

f = f4 = 0.15fJ

0

2

4

6

-2

-4

-6

0

2

4

6

-2

-4

-6

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 0 1 2 3 0 0.5 1.5 21RF current Irf [Ic] RF current Irf [Ic] RF current Irf [Ic]

f = 0.5fJ f = 0.05fJ

progressive disappearance of all odd steps crossover at 4π-periodicity required no effect of 2π-periodic CPR

f f4 = 2eRnI4h


Evaluation of the 4π contribution

31

Ratios

Contribution

crossover ⇒ current

f f4I4

number of 4π modes I0 = ei

~⇒ 1-3 modes

2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

150 nm400 nm600 nm

Frequency f [GHz] Frequency f [GHz]

RatioQ

12

RatioQ

34

Q12 = w1w2

! 0

Q34 = w3w4

' 1

wn max. amplitude of nth step


Other mechanisms

32

2 4 3

i

i

"()

2


enhanced at high frequency evaluated minimal transmissionD > 0.994

Bias instability/switch

instability suppressed by shunt n=1 step still missing

DC current I [µA] rf current Irf [a.u.]

dI/dV [mS]

rf current Irf [a.u.]

Bin counts

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

-1.5 -1 -0.5 0 0.5 1 1.5

0

2

4

6

-2

-4

-60 10.25 0.5 0.75

0

5

10

15

20

25

30

35A) B) C)

0

2

4

6

-2

-4

-6

0

2

4

6

-2

-4

-60 10.25 0.5 0.75

-60

-40

-20

20

40

60

80

100

0

-80

-100


Graphene-based junctions

33

0

1

1.5

2

-1

-1.5

-2

0.5

-0.5

-40 -20 20 400-40 -20 20 400

800

600

400

200

0

100

300

500

700

Back-gate voltage Vg [V]

NormalstateresistanceR

n[Ω]

DC

currentI

[µA]


800

600

400

200

0

100

300

500

700

dV/dI [Ω]

0

2

4

6

-2

-4

-6

8

-8

0

2

4

6

-2

-4

-6

8

-80.2 10.4 0.6 0.8 0.2 10.4 0.6 0.8

rf exc. Irf [a.u.]DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

rf exc. Irf [a.u.]

f = 7GHz f = 5.5GHz

Graphene junctions

density : −2.5 × 1012cm-2 to 2.5 × 1012 cm-2

mobility : ∼5000 cm2V-1s-1

Shapiro steps

n=1 step always visible

different densities (n, p, DP) and frequencies (4-14 GHz)



34

rf exc. Irf [a.u.]

rf exc. Irf [a.u.]

rf exc. Irf [a.u.]rf exc. Irf [a.u.]

rf exc. Irf [a.u.]

f=

3.34GHz

f=

4.12

GHz

f=

4.9GHz

f=

6.2GHz

f=

7.24GHz

f=

7.4GHz

3/2

0

2

4

6

-2

-4

-60.2 10.4 0.6 0.8 0.2 10.4 0.6 0.8

0.2 10.4 0.6 0.80.2 10.4 0.6 0.8

0.2 10.4 0.6 0.8 0.2 10.4 0.6 0.8

0

2

4

6

-2

-4

-6

0

2

4

6

-2

-4

-6

0

2

4

-2

-4

0

2

4

-2

-4

0

2

4

-2

-4

rf exc. Irf [a.u.]

10

20

30

40

10

20

30

40

10

20

30

40

10

20

30

40

10

20

30

40

10

20

30

40

dI/dV [mS]

dI/dV [mS] dI/dV [mS]

dI/dV [mS]

dI/dV [mS]dI/dV [mS]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

1/2


missing n=1 step at low frequency

subharmonic steps at high frequency


Shapiro steps : voltage bias

35

DC current I [µA] rf current Irf [a.u.]

dI/dV [mS]

rf current Irf [a.u.]

Bin counts

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

-1.5 -1 -0.5 0 0.5 1 1.5

0

2

4

6

-2

-4

-60 10.25 0.5 0.75

0

5

10

15

20

25

30

35A) B) C)

0

2

4

6

-2

-4

-6

0

2

4

6

-2

-4

-60 10.25 0.5 0.75

-60

-40

-20

20

40

60

80

100

0

-80

-100


Graphene-based junctions

36

0

1

1.5

2

-1

-1.5

-2

0.5

-0.5

-40 -20 20 400-40 -20 20 400

800

600

400

200

0

100

300

500

700


NormalstateresistanceR

n[Ω]

DC

currentI

[µA]


800

600

400

200

0

100

300

500

700

dV/dI [Ω]


Temperature dependence of Ic

37

0 0.2 0.4 0.6 0.8 1 1.21.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 1 2 3 4 50

1

2

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Temperature T [K] Temperature T [K]

CriticalcurrentI c

[µA]

CriticalcurrentI c

[µA]

150 nm 400 nm 600 nm

A) B)

fits for in 0-1K region to access induced gap

perturbation theory diverges at high temperatureTkachov et al., PRB 88, 075401 (2013)


Effect of CPR in RSJ simulations

38

0 1 2 3 4-1

-0.5

0

0.5

1

0 1 2 3 4-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.80

2

4

6

8

0 0.2 0.4 0.6 0.80

2

4

6

8

DC current I [Ic]DC current I [Ic]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

Phase di↵. Phase di↵.

CurrentI s

[Ic]

CurrentI s

[Ic]

2 + 4 modes2 modes

0 1 2 3 4-1

-0.5

0

0.5

1

0 1 2 3 4-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.80

2

4

6

8

0 0.2 0.4 0.6 0.80

2

4

6

8

DC current I [Ic]DC current I [Ic]

DC

voltageV

[hf/2

e]

DC

voltageV

[hf/2

e]

Phase di↵. Phase di↵.

CurrentI s

[Ic]

CurrentI s

[Ic]

2 + 4 modes2 modes


Excess current

39

Nb/e

2Nb/e

DC

voltageV

[mV]

DC current Ic [µA] DC voltage V [mV]

N

b/e

2N

b/e

Exc.currentI e

xc

[µA

]

A) B)

-100 0 100-3

-2

-1

0

1

2

3

3

0

1

2

5

6

30 1 2 5

4

4-100 0 100-3

-2

-1

0

1

2

3

3

0

1

2

5

6

30 1 2 5

4

4

I1exc



40

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

2

4

6

8

10

0

2

4

6

8

10

Norm.voltageV(t)

[hf/2e]

Norm. time t [1/f ]

Phasedi↵.

[]

= 3

V |=3

0.7 0.8 0.9 1 1.1 1.2 1.3

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6P=1P=0.975P=0.95P=0.925P=0.9P=0.85P=0.8P=0.75P=0.70

DC

voltageV

[hf/2

e]

DC current I [Ic]


Landau-Zener transition probability

"±() = ±p42 +

2i cos

2 /2

PLZ = exp

2 42

i~


Weak suppression of third step

41

CurrentI[µA]

rf exc. Irf [a.u.]Stepam

plitude

rf exc. Irf [a.u.]

0

12

34

Reduced 3rd step


Hysteresis & Shapiro steps

42

CurrentI[µA]

rf exc. Irf [a.u.]

dV/dI []

Splitting of 1st/2nd steps

Hysteresis unknown origin common in similar systems (graphene, TIs)

Shapiro steps bistability ⇒ no phase locking Shapiro steps invisible


Frequency dependence (2)

43

CurrentI[µA]

RF exc. Irf [u.a.]

Stepam

plitude

dV/dI [] dV/dI []

CurrentI[µA]

CurrentI[µA]

RF exc. Irf [u.a.]RF exc. Irf [u.a.]

dV/dI []

Stepam

plitude

Stepam

plitude

RF exc. Irf [u.a.]RF exc. Irf [u.a.]RF exc. Irf [u.a.]

0 1 2

0

4321

f = 3.04 GHz f = 6.04 GHz f = 7.5 GHz


Band structure of quantum wells

44

-0.6 -0.3 0.0 0.3 0.6

-60

-40

-20

0

20

(110)

E (m

eV)

k (nm-1)

EG = 20 meV

(100)

dQW = 8.0 nm asub = 6.467 Angstrom (Cd0.96Zn0.04Te)

« camelback » structure ⇒ low mobility in p-regime

gap around 20 meV

edge states not calculated


Resonances at high frequencies

45

dc voltage V [μV]

trans

m.

|S21

| [dB

]δ|

S 21|

[dB]

diff.

con

d. d

V/dI

[S]

dc v

olta

ge V

[μV

]

frequency fd [GHz]frequency fd [GHz]

frequency fd [GHz]

a c

b d


Emission in 3D TI

46

Gapless Andreev bound states in HgTe-based topological...

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