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
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
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
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
Andreev reflections
5
Andreev reflection
Cooper pair created in S hole reflected in N phase coherence
N/TI S
1
S
2
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
Gapless Andreev bound states
6
Andreev bound states
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
Erwann Bocquillon Lyon - 03/10/2016
Gapless Andreev bound states
6
Andreev bound states
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)
Erwann Bocquillon Lyon - 03/10/2016
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)
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
(Fractional) Josephson effect
11
Josephson equations
IS() = Ic sin
d
dt=
2eV
~ = 1 2
fJ =2eV
~
1
2
fJ =2eV
~⇒ Josephson frequency
Erwann Bocquillon Lyon - 03/10/2016
(Fractional) Josephson effect
11
Josephson equations
IS() = Ic sin
d
dt=
2eV
~ = 1 2
fJ =2eV
~
1
2
fJ =2eV
~⇒ Josephson frequency
Fractional Josephson effect
sin ! sin/2
fJ ! fJ/2
Erwann Bocquillon Lyon - 03/10/2016
(Fractional) Josephson effect
11
Josephson equations
IS() = Ic sin
d
dt=
2eV
~ = 1 2
fJ =2eV
~
1
2
fJ =2eV
~⇒ Josephson frequency
Fractional Josephson effect
sin ! sin/2
fJ ! fJ/2
Detection
‘listening’ to Josephson emission beatings with ac excitation (Shapiro steps)
Erwann Bocquillon Lyon - 03/10/2016
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.
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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]
Fraunhofer2π Shapiro 4π Shapiro
SQUID
4π emission2π emission
average ratio <r>
c
fJ/2fJ2fJ
Low frequency fd= 3 GHz
High frequency fd= 5.5 GHz
Erwann Bocquillon Lyon - 03/10/2016
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]
Fraunhofer2π Shapiro 4π Shapiro
SQUID
4π emission2π emission
average ratio <r>
c
fJ/2fJ2fJ
Low frequency fd= 3 GHz
High frequency fd= 5.5 GHz
QSH n regimep regime
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
4π-periodic supercurrent
doubled steps
mixture 2π/4π ?
sin ! sin/2Vn ! V2n
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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 »
Erwann Bocquillon Lyon - 03/10/2016
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]
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
RSJ frequency dependence
20
I≳Ic
High frequency
⇒ almost sinusoidal
I≫Ic
Domínguez et al., PRB 86, 146503 (2012)
Low frequency
⇒ non-sinusoidal
Erwann Bocquillon Lyon - 03/10/2016
RSJ frequency dependence
20
I≳Ic
High frequency
⇒ almost sinusoidal
I≫Ic
Domínguez et al., PRB 86, 146503 (2012)
Low frequency
⇒ non-sinusoidal
I2 = 1, I4 = 0
I2 = 1, I4 = 0.15
Erwann Bocquillon Lyon - 03/10/2016
RSJ frequency dependence
20
I≳Ic
High frequency
⇒ almost sinusoidal
I≫Ic
Domínguez et al., PRB 86, 146503 (2012)
⇒ crossover f4π yields : 1-3 modes ⇒ no Landau-Zener transitions ?
Low frequency
⇒ non-sinusoidal
I2 = 1, I4 = 0
I2 = 1, I4 = 0.15
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Fractional Josephson effect
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)
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
Open positions !
24
Post-docs
PhD students
Thank you for your attention !
Open positions!
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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, ...
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
Frequency dependence
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]
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
Other mechanisms
32
2 4 3
i
i
"()
2
Landau-Zener transitions
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
Erwann Bocquillon Lyon - 03/10/2016
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]
Back-gate voltage Vg [V]
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)
Erwann Bocquillon Lyon - 03/10/2016
Frequency dependence
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
Frequency dependence
missing n=1 step at low frequency
subharmonic steps at high frequency
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Back-gate voltage Vg [V]
NormalstateresistanceR
n[Ω]
DC
currentI
[µA]
Back-gate voltage Vg [V]
800
600
400
200
0
100
300
500
700
dV/dI [Ω]
Erwann Bocquillon Lyon - 03/10/2016
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)
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
Landau-Zener transitions
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]
Andreev bound states
Landau-Zener transition probability
"±() = ±p42 +
2i cos
2 /2
PLZ = exp
2 42
i~
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
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
Erwann Bocquillon Lyon - 03/10/2016
Emission in 3D TI
46