Topological Matter School 25/08/2017
Physikalisches Institut (EP3) Universität Würzburg, Am Hubland, D-97074 Würzburg
http://www.physik.uni-wuerzburg.de/EP3/
Erwann Bocquillon
Topological insulators & topological superconductivity in HgTe heterostructures
Topological Matter School 25/08/2017
Physikalisches Institut (EP3) Universität Würzburg, Am Hubland, D-97074 Würzburg
http://www.physik.uni-wuerzburg.de/EP3/
Erwann Bocquillon
Topological insulators & topological superconductivity in HgTe heterostructures
Laboratoire Pierre Aigrain Ecole Normale Supérieure
24 rue Lhomond, 75231 Paris Cedex 05 France www.lpa.ens.fr
Erwann Bocquillon Topological Matter School - 25/08/20173
RIKEN, Tōkyō
R.S. Deacon, K. Ishibashi, S. Tarucha
People involved
Uni. Würzburg
Students : J. Wiedenmann, E. Liebhaber D. Mahler, A. Budewitz, K. Bendias, S. Hartinger Staff : C. Brüne H. Buhmann L.W. Molenkamp Invited : T.M. Klapwijk Theory : F. Domínguez, E.M. Hankiewicz
Erwann Bocquillon Topological Matter School - 25/08/2017
Outline
4
I. HgTe as a topological material
A. Basics of HgCdTe compounds B. HgTe quantum wells C. 3D layers & strain engineering
II. Search for topological superconductivity in HgTe
A. Foreword on topological superconductivity B. Physics of a Josephson junction C. Search for gapless ABS in topological JJs D. Induced superconductivity in S-N junctions
Erwann Bocquillon Topological Matter School - 25/08/2017
Part I - Topological phases in HgTe
5
A. Basics of HgCdTe compounds B. HgTe quantum wells C. 3D layers & strain engineering
II-VI semiconductors with ZnS structure (2 fcc with tetrahedral coordination) heavy + strong SOC ⇒ strong relativistic corrections different lattice constants (0.3%) ⇒ strain !
Erwann Bocquillon Topological Matter School - 25/08/2017
I-A. Basics of HgTe
6
aHgTe = 0.646 nm
aCdTe = 0.648 nm
Erwann Bocquillon Topological Matter School - 25/08/2017
I-A. Basics of HgTe
7
s-typep-type
1.6 eV
-0.3 eV
CdTe : normal ordering (like GaAs, Ge) HgTe : inverted band structure
Erwann Bocquillon Topological Matter School - 25/08/2017
I-A. Basics of HgTe
7
s-typep-type
1.6 eV
-0.3 eV
CdTe : normal ordering (like GaAs, Ge) HgTe : inverted band structure
Band crossing at the interface : topological states ! HgTe is a semi-metal
Erwann Bocquillon Topological Matter School - 25/08/2017
I-A. Basics of HgTe
8
Gap from quantum confinement
narrow 2D quantum wells (<15 nm)
2D topological insulator
QSH and QAH systems
2D TI (quantum spin Hall)
3D TI (surface states)Gap from strain
thick 3D layers (50 -150 nm)
3D topological insulator
Weyl/Dirac semi-metals
Erwann Bocquillon Topological Matter School - 25/08/2017
I-A. Basics of HgTe
9
HgTeCdTeH0 + HD + HMV + HSO H0 + HD + HMV + HSO
A few side remarks : ® Hg0.3Cd0.7Te often used instead of pure CdTe (gap 0.9 eV) ® important role of mass-velocity correction ! Hg much heavier than Cd !
® surface states known since the 70s (works of Volkov-Pankratov)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-B. Quantum wells
10
Bernevig et al., Science 314, 1757 (2006) König et al., Science 318, 766 (2007)
HgCdTe HgCdTe
HgCdTe HgCdTe
HgTe
HgTe
6
8
6
8
E1
H1
E1H1
d > dc
d < dc
Topological phase transition
trivial if d < dc (normal order H1<E1)
QSH if d > dc ≃ 6.3 nm (H1>E1)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-B. Quantum wells
10
Bernevig et al., Science 314, 1757 (2006) König et al., Science 318, 766 (2007)
HgCdTe HgCdTe
HgCdTe HgCdTe
HgTe
HgTe
6
8
6
8
E1
H1
E1H1
d > dc
d < dc
Topological phase transition
trivial if d < dc (normal order H1<E1)
QSH if d > dc ≃ 6.3 nm (H1>E1)
6 4 2 2 4 66 8 10 12 14
E (eV
)
-0.2
-0.1
0
0.1
0.2
k (0.1 nm-1
) dQW (nm) k (0.1 nm-1
)
E2
E1
H1
H2 H3
H4
H5
H6
L1
dQW = 4 nm dQW = 15 nm
(a) (b) (c)
-100
-50
0
dQW
= 7 nm
dQW
= 6 nmdQW
= 4 nm
-0.4 0.0 0.4
-100
-50
0
E(m
eV
)
k (1/nm)
-0.4 0.0 0.4
dQW
= 15 nm
(d)
(b)
(c)
(a)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-B. Quantum wells
11
Figures of merit
MBE growth ⇒ huge mobility μ ≃ 3-5 105 cm2V-1s-1
low defect density ⇒ bulk insulating
small gap ≃ 20-25 meV strain engineering yields 40 meV
6 4 2 2 4 66 8 10 12 14
E (eV
)
-0.2
-0.1
0
0.1
0.2
k (0.1 nm-1
) dQW (nm) k (0.1 nm-1
)
E2
E1
H1
H2 H3
H4
H5
H6
L1
dQW = 4 nm dQW = 15 nm
(a) (b) (c)
-100
-50
0
dQW
= 7 nm
dQW
= 6 nmdQW
= 4 nm
-0.4 0.0 0.4
-100
-50
0
E(m
eV
)
k (1/nm)
-0.4 0.0 0.4
dQW
= 15 nm
(d)
(b)
(c)
(a)
6 4 2 2 4 66 8 10 12 14
E (eV
)
-0.2
-0.1
0
0.1
0.2
k (0.1 nm-1
) dQW (nm) k (0.1 nm-1
)
E2
E1
H1
H2 H3
H4
H5
H6
L1
dQW = 4 nm dQW = 15 nm
(a) (b) (c)
-100
-50
0
dQW
= 7 nm
dQW
= 6 nmdQW
= 4 nm
-0.4 0.0 0.4
-100
-50
0
E(m
eV
)
k (1/nm)
-0.4 0.0 0.4
dQW
= 15 nm
(d)
(b)
(c)
(a)
6 4 2 2 4 66 8 10 12 14
E (eV
)
-0.2
-0.1
0
0.1
0.2
k (0.1 nm-1
) dQW (nm) k (0.1 nm-1
)
E2
E1
H1
H2 H3
H4
H5
H6
L1
dQW = 4 nm dQW = 15 nm
(a) (b) (c)
-100
-50
0
dQW
= 7 nm
dQW
= 6 nmdQW
= 4 nm
-0.4 0.0 0.4
-100
-50
0
E(m
eV
)
k (1/nm)
-0.4 0.0 0.4
dQW
= 15 nm
(d)
(b)
(c)
(a)
P. Leubner et al., PRL 117, 86403 (2016)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-B. Quantum wells
12
M. König et al., Science 318, 766 (2007) A. Roth et al., Science 325, 294 (2009) C. Brüne et al., Nat. Physics 8, 485 (2012) K.C. Nowack et al., Nat. Materials 12, 787 (2013) M.K. Bendias et al., submitted (2017)
Observations
conductance quantization (on max. 10 μm)
non-local transport
spin transport
scanning SQUID imaging
topological superconductivity?
Erwann Bocquillon Topological Matter School - 25/08/2017
I-B. Quantum wells
13
Open questions - Robustness of edge states and topological properties
⇒ Why only 10 μm in transport ? Why not affected by bandgap ?
Disorder ? charge puddles ?
J. I. Väyrynen et al., PRL 110, 216402 (2013)
Coulomb interactions ?
C
C. Wu et al., PRL 96, 106401 (2006) G. Dolcetto et al., Nuevo Cimento 39, 113 (2015)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-B. Quantum wells
13
Open questions - Robustness of edge states and topological properties
⇒ Why only 10 μm in transport ? Why not affected by bandgap ?
Disorder ? charge puddles ?
J. I. Väyrynen et al., PRL 110, 216402 (2013)
Coulomb interactions ?
C
C. Wu et al., PRL 96, 106401 (2006) G. Dolcetto et al., Nuevo Cimento 39, 113 (2015)
[1] J. Maciejko et al., PRL 102, 256803 (2009) [2] A. Ström et al., PRL 104, 256804 (2010) [3] F. Crépin et al., PRB 86, 121106 (2012) [4] T. L. Schmidt et al., PRL 108, 156402 (2012)
[5] F. Geissler et al., PRB 89, 235136 (2014) [6] J. I. Väyrynen et al., PRL 110, 216402 (2013) [7] J. I. Väyrynen et al., PRB 90, 115309 (2014) [8] S. Essert et al., PRB 92, 205306 (2015)
A long list of mechanisms that lead to backscattering…
Erwann Bocquillon Topological Matter School - 25/08/2017
I-B. QAH effect in HgTe - Theory
14
Mn-doped HgTe quantum wells
Mn2+ doping (0.5 to 4 %) ⇒ isoelectric, paramagnetic
2 spin splittings with opposite signs
spin polarization
GE , GH / hSi
hSi = 5
2B5/2
5
2
gMnµBB
kB(T + T0)
|H1,±i
|E1,±i
|E1,+i
|E1,i
|H1,i
|H1,+i
B
E
2GE
2GH
GE , GH
C.-X. Liu et al., PRL 101, 146802 (2008)
(almost) normal QH effect in n-type regime
early onset of ν=-1 plateau ≃70 mT (@ 30 mK)
from T dependence, at transitionhSi ' 0.1
Erwann Bocquillon Topological Matter School - 25/08/2017
I-B. QAH effect in HgTe - Experiment
15
0 1 2 3 4 5 6 7-30
-20
-10
0
10
20
30
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
-2.0 -1.5 -1.0 -0.5 0.00
10
20
30Rxy/k
Ω
B / T
Vg = 0.0VVg = -0.5VVg = -1.0VVg = -1.5VVg = -2.0V
Rxx/k
ΩB / T
~70 mT
-30
-25
-20
-15
-10
-5
0
Rxy/k
Ω
Rxx/k
Ω
Vg / V
Budewitz et al., arXiv 1706.05789 (2017)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Straining HgTe 3D layers
16
Strained HgTe
band inversion between Γ6-Γ8⇒ topological surface states
strain ⇒ bulk band gap between Γ8 subbands(20 meV)Fu et al., PRB 76, 045302 (2007)
CdTe
8
6
1 eV
HgTe
6
8
0.3 eV
k k
"(k)"(k)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Straining HgTe 3D layers
16
Strained HgTe
band inversion between Γ6-Γ8⇒ topological surface states
strain ⇒ bulk band gap between Γ8 subbands(20 meV)Fu et al., PRB 76, 045302 (2007)
CdTe
8
6
1 eV
HgTe
6
8
0.3 eV
k k
"(k)"(k)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Straining HgTe 3D layers
16
Strained HgTe
band inversion between Γ6-Γ8⇒ topological surface states
strain ⇒ bulk band gap between Γ8 subbands(20 meV)Fu et al., PRB 76, 045302 (2007)
Strained HgTe
20meV
k
"(k)
CdTe
8
6
1 eV
HgTe
6
8
0.3 eV
k k
"(k)"(k)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. QH effect in strained HgTe
17
HgTe 70 nm
CdTe 001 substrate
SiO2/Si3N4 ML 110 nmAu 100 nm
Layer structure
HgTe (70 to 90 nm) on CdTe substrate
Au gate on SiO2/Si3N4 multilayer
Brüne et al., PRL 106, 126803 (2011) Brüne et al., PRX 4, 041045 (2014)
Quantum Hall effect
electron density
QHE visible
n 1 10 1011 cm2
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Odd integer sequence
18
0 2 4 6 8 10 12 14 160
10
20
30
40
50
60
70
Rxx
/ kΩ
B / T
0
5
10
15
20
25
i=3
i=5
Rxy
/ kΩ
i=1
Rxx
[k] R
xy
[k]
B [T]
Degenerate Dirac cones
⇒ odd integers only
= 2
n+
1
2
, n 2 Z
only if densities equal ⇒ 1 value of Vg
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Odd/even sequences
19
0 2 4 6 8 10 12 14 160
5
10
15
20
25
Rxx
[kΩ"
B [T]
0
5
10
15
20
25
53
2
Rxy
[kΩ
"
i=1
Rxx
[k] R
xy
[k]
B [T]0 2 4 6 8 10 12 14 16
0
5
10
15
20
25
30
Rxx
[kΩ
]
B [T]
0
5
10
15
20
5
3
i=2
Rxy
[kΩ
]
Rxx
[k] R
xy
[k]
B [T]
Different densities = n1 + n2 + 1, n1, n2 2 Z ⇒ ν=2 reappears
Identification of 2 surfaces : - broadening of SdH oscillations- gate influence on peak positions
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Tracing Landau levels
20
0 2 4 6 8 10 12 14 16-3-2-1012345
7 3
5
6
5
4
3
2
V g [V]
B [T]
i=1
Magnetic field B [T]
GatevoltageVg[V]
odd integer sequence
max. of
2 subsets of LL top surface bottom surface
@Rxy
@B
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Tracing Landau levels
20
0 2 4 6 8 10 12 14 16-3-2-1012345
7 3
5
6
5
4
3
2
V g [V]
B [T]
i=1
Magnetic field B [T]
GatevoltageVg[V]
odd integer sequence
max. of
2 subsets of LL top surface bottom surface
@Rxy
@B
how do surfaces/bulk exchange charge ? screening of the bulk by TSS ? (gating?) connect to one surface only ?
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Compressive strain in 3D HgTe
21
HgTe
6
8
0.3 eV
k
"(k)
Tensile strain in HgTe
20meV
k
"(k)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Compressive strain in 3D HgTe
21
HgTe
6
8
0.3 eV
k
"(k)
Tensile strain in HgTe
20meV
k
"(k)
Compressive strain in HgTek
"(k)
2 DP
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Compressive strain in 3D HgTe
21
HgTe
6
8
0.3 eV
k
"(k)
Tensile strain in HgTe
20meV
k
"(k)
Compressive strain in HgTek
"(k)
2 DP
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Dirac/Weyl points in HgTe
22
DFT calculations : D. Di Sante, G. Sangiovanni, R. Thomale and E.M. Hankiewicz (Würzburg)
2 Dirac/Weyl points (degeneracy lifted by BIA)
anticrossing of TSS states
® tiny energy scales! J. Ruan et al., Nature Comm. 7, 11136 (2016)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Strain engineering
23
50 55 60 65
-6 -5-4
-3-2
-1
54
32
1
0
Inte
nsity
(arb
. uni
ts)
2θ (°)
(a)
S
meassim
0.0 0.5 1.06.36
6.40
6.44
6.48
a SLS (
Å)
tCT x φTe (10-4 s x Torr)
ε = +1.4 %
(c) -0.3 %
+0.4 %
Virtuel substrates
superlattice with: - 1 layer of Cd0.5Zn0.5Te - x layers of CdTe
effective lattice parameter
ae↵ = aCdTe
1 +
f
1 +mr
m
r
f lattice mismatch stiffness ratio thickness ratio
aHgTe = 0.646 nm
aCdTe = 0.648 nm
aZnTe = 0.610 nmP. Leubner et al., PRL 117, 86403 (2016)
Erwann Bocquillon Topological Matter School - 25/08/2017
I-C. Strain engineering
23
50 55 60 65
-6 -5-4
-3-2
-1
54
32
1
0
Inte
nsity
(arb
. uni
ts)
2θ (°)
(a)
S
meassim
0.0 0.5 1.06.36
6.40
6.44
6.48
a SLS (
Å)
tCT x φTe (10-4 s x Torr)
ε = +1.4 %
(c) -0.3 %
+0.4 %
Virtuel substrates
superlattice with: - 1 layer of Cd0.5Zn0.5Te - x layers of CdTe
effective lattice parameter
ae↵ = aCdTe
1 +
f
1 +mr
m
r
f lattice mismatch stiffness ratio thickness ratio
aHgTe = 0.646 nm
aCdTe = 0.648 nm
aZnTe = 0.610 nmP. Leubner et al., PRL 117, 86403 (2016)
50 55 60 65
-6 -5-4
-3-2
-1
54
32
1
0
Inte
nsity
(arb
. uni
ts)
2θ (°)
(a)
S
meassim
0.0 0.5 1.06.36
6.40
6.44
6.48
a SLS (
Å)
tCT x φTe (10-4 s x Torr)
ε = +1.4 %
(c) -0.3 %
+0.4 %
Erwann Bocquillon Topological Matter School - 25/08/2017
Summary of Part I
24
Versatility of HgTe platform
several topological systems possible: 2D/3D TIs, Weyl semi-metal, QAH/QH/QSH + core-shell NW
high quality samples: mobility μ ≃ 3-5 105 cm2V-1s-1 insulating bulk
Erwann Bocquillon Topological Matter School - 25/08/2017
Summary of Part I
24
Versatility of HgTe platform
several topological systems possible: 2D/3D TIs, Weyl semi-metal, QAH/QH/QSH + core-shell NW
high quality samples: mobility μ ≃ 3-5 105 cm2V-1s-1 insulating bulk
Ready for topological quantum computing ?
reproducible results & easily scalable (large MBE-grown layers)
robust topological states topological protection (QSH) ? band engineering?
complex systems with coexisting topological phases : Weyl semi-metal + TSS
Erwann Bocquillon Topological Matter School - 25/08/2017
Part II - Superconductivity in HgTe
25
A. Foreword on topological superconductivity B. Physics of a Josephson junction C. Search for gapless ABS in topological JJs D. Induced superconductivity in S-N junctions
Erwann Bocquillon Topological Matter School - 25/08/2017
II-A. Topological superconductivity
26
Cooper pair of helical Dirac fermions ⇒ helical pairing ⇒ p-type correlations
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 Gothenburg - 11/08/2017
II-A. Majoranas and superconductivity
27
J. Alicea, Rep. Prog. Phys. 75, 076501 (2012) C.W.J. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 (2013)
V. Mourik et al., Science 336, 1003 (2012) … S. M. Albrecht et al., Nature 531, 206 (2016)
k ?"k ",k #
+Cooper pair of helical Dirac fermions
⇒ helical pairing ⇒ p-type correlations
=
Erwann Bocquillon Gothenburg - 11/08/2017
II-A. Majoranas and superconductivity
27
J. Alicea, Rep. Prog. Phys. 75, 076501 (2012) C.W.J. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 (2013)
V. Mourik et al., Science 336, 1003 (2012) … S. M. Albrecht et al., Nature 531, 206 (2016)
k ?"k ",k #
+Cooper pair of helical Dirac fermions
⇒ helical pairing ⇒ p-type correlations
=
=
†
=1
2(c+ c†)
Erwann Bocquillon Gothenburg - 11/08/2017
II-A. Majoranas and superconductivity
27
J. Alicea, Rep. Prog. Phys. 75, 076501 (2012) C.W.J. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 (2013)
V. Mourik et al., Science 336, 1003 (2012) … S. M. Albrecht et al., Nature 531, 206 (2016)
k ?"k ",k #
+Cooper pair of helical Dirac fermions
⇒ helical pairing ⇒ p-type correlations
=
N
ST
=
†
=1
2(c+ c†)
Erwann Bocquillon Gothenburg - 11/08/2017
II-A. Majoranas and superconductivity
27
J. Alicea, Rep. Prog. Phys. 75, 076501 (2012) C.W.J. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 (2013)
V. Mourik et al., Science 336, 1003 (2012) … S. M. Albrecht et al., Nature 531, 206 (2016)
k ?"k ",k #
+Cooper pair of helical Dirac fermions
⇒ helical pairing ⇒ p-type correlations
=
N
ST
=
†
Sayonara fermions ?
Erwann Bocquillon Gothenburg - 11/08/2017
II-A. Majoranas and superconductivity
27
J. Alicea, Rep. Prog. Phys. 75, 076501 (2012) C.W.J. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 (2013)
V. Mourik et al., Science 336, 1003 (2012) … S. M. Albrecht et al., Nature 531, 206 (2016)
k ?"k ",k #
+Cooper pair of helical Dirac fermions
⇒ helical pairing ⇒ p-type correlations
=
=
†
Sayonara fermions ?
ei
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Josephson junctions
28
S1 S2N/I/F/TI
Josephson junctions
2 superconductors Sj
coupling through weak link : N/I/F/TI
j =pnje
ij , j = 1, 2
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Josephson junctions
28
S1 S2I
Josephson junctions
2 superconductors Sj
coupling through weak link : N/I/F/TI
j =pnje
ij , j = 1, 2
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Josephson junctions
28
S1 S2I
Josephson junctions
2 superconductors Sj
coupling through weak link : N/I/F/TI
j =pnje
ij , j = 1, 2
| |2
n1 n2
x
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Josephson junctions
28
S1 S2I
Josephson junctions
2 superconductors Sj
coupling through weak link : N/I/F/TI
j =pnje
ij , j = 1, 2
IS() = Ic sin
d
dt=
2eV
~
Josephson equations
phase evolution
current-phase relation
| |2
n1 n2
x
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Josephson junctions
28
S1 S2I
Josephson junctions
2 superconductors Sj
coupling through weak link : N/I/F/TI
j =pnje
ij , j = 1, 2
IS() = Ic sin
d
dt=
2eV
~
Josephson equations
phase evolution
current-phase relation
| |2
n1 n2
x
⇒ supercurrent (I≠0 for V=0) for I<Ic with constant 𝜙
finite voltage for I>Ic and 𝜙 time dependent (Josephson frequency)
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Andreev reflections
29
N/TI S
1
Conditions
two-electron process energy/momentum conservation ⇒ retro-reflection (exact at EF )
phase coherence ⇒ phase shift at interface
spin conservation = + arccos
E
EF + E ! EF E
kF + k ! kF k
" ! "
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Andreev reflections
29
N/TI S
1"#
""
Conditions
two-electron process energy/momentum conservation ⇒ retro-reflection (exact at EF )
phase coherence ⇒ phase shift at interface
spin conservation = + arccos
E
EF + E ! EF E
kF + k ! kF k
" ! "
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Andreev reflections
29
N/TI S
1"#
""
Conditions
two-electron process energy/momentum conservation ⇒ retro-reflection (exact at EF )
phase coherence ⇒ phase shift at interface
spin conservation = + arccos
E
EF + E ! EF E
kF + k ! kF k
" ! "
® normal reflections if imperfect interface ® perfect Andreev reflections in QSH edge states ® specular reflections possible in Dirac materials if EF≪Δ C. W. J. Beenakker, PRL 97 (2006)
P. Adroguer et al., PRB 82, 81303 (2010)
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Andreev bound states
30
N/TI S
1
S
2
Left reservoir
Right reservoir
Scatterer
ShN
SeN
ABS spectrum
resonant cavity
scattering formalism
resonance condition
SA
SN
Andreev reflection scattering in N region
det(I r(2)A ShNr(1)A Se
N ) = 0
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Conventional Andreev Bound States
31
Conventional ABS
1 pair of gapless states spin degeneracy connected to continuum avoided crossing at E=0 for D≠1
Titre
4
−1
−0,5
0
0,5
1
4
−1
−0,5
0
0,5
1
1
0 0,5 1 1,5 2
10 0,5 1 1,5 2
D = 10.990.90.5
Phase []
Energy
E[
]
E() =
r1D sin2
2
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Gapless Andreev Bound States
32
Topological gapless ABS 1 pair of gapless states no spin degeneracy disconnected from continuum for D≠1 protected crossing at E=0 « hybridized Majorana states »
4
−1
−0,5
0
0,5
1
4
−1
−0,5
0
0,5
1
1
0 0,5 1 1,5 2
10 0,5 1 1,5 2
D = 10.99
0.90.5
Phase []
Energy
E[
]
E() =pD cos
2
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. (Fractional) Josephson effect
33
Josephson equationsd
dt=
2eV
~ = 1 2
fJ =2eV
~
1
2I() =X
n
@En
@
1 2f(En)
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. (Fractional) Josephson effect
33
Josephson equationsd
dt=
2eV
~ = 1 2
fJ =2eV
~
1
2
IS() = Ic sin
fJ =2eV
~⇒ Josephson frequency
I() =X
n
@En
@
1 2f(En)
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. (Fractional) Josephson effect
33
Josephson equationsd
dt=
2eV
~ = 1 2
fJ =2eV
~
1
2
IS() = Ic sin
fJ =2eV
~⇒ Josephson frequency
Fractional Josephson effect
sin ! sin/2
fJ ! fJ/2
I() =X
n
@En
@
1 2f(En)
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. (Fractional) Josephson effect
33
Josephson equationsd
dt=
2eV
~ = 1 2
fJ =2eV
~
1
2
IS() = Ic sin
fJ =2eV
~⇒ Josephson frequency
Fractional Josephson effect
sin ! sin/2
fJ ! fJ/2
Detection
‘listening’ to Josephson emission beatings with ac excitation (Shapiro steps)
I() =X
n
@En
@
1 2f(En)
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Fractional Josephson effect
34
Superconducting phase 𝜑 π 2π 3π 4π
Ener
gy
0
0
Δi
-Δi
EJ
topological state
conv. state
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Fractional Josephson effect
34
® Many parasitic effects ! 2π bulk states ⇒ 2π/4π mixture finite lifetime/continuum ⇒ 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., PRL 117, 267001 (2016) Hui et al., PRB 95, 014505 (2017)
Superconducting phase 𝜑 π 2π 3π 4π
Ener
gy
0
0
Δi
-Δi
EJ
topological state
conv. state
Erwann Bocquillon Topological Matter School - 25/08/2017
II-B. Quantum spin Hall junctions
35
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 Topological Matter School - 25/08/2017
II-B. Quantum spin Hall junctions
35
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 Topological Matter School - 25/08/2017
II-B. First properties of HgTe JJs
36
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
Standard patterns
Fraunhofer pattern ⇔ uniform supercurrent
SQUID ⇔ edge currents
Erwann Bocquillon Topological Matter School - 25/08/2017
Interference patterns
37
Interference patterns
non-uniform phase
« interference » pattern
! + 2BL0
y
Hart et al., Nat. Phys. 10, 638 (2014)
Erwann Bocquillon Topological Matter School - 25/08/2017
Anomalous interference patterns
38
DC
currentI[nA]
Magnetic field B [mT]
Vg = 1.3V
Vg = 1.6V
Vg = 0V
Vg = 2V
Hart et al., Nat. Phys. 10, 638 (2014) Pribiag et al., Nat. Nano. 10 593 (2015)
Erwann Bocquillon Topological Matter School - 25/08/2017
Detection setup
39
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 Topological Matter School - 25/08/2017
Emission spectra
40
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 Topological Matter School - 25/08/2017
Frequency dependence
41
fJ =2eV/h
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 Topological Matter School - 25/08/2017
Gate voltage dependence
42
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 Topological Matter School - 25/08/2017
Gate voltage dependence
42
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 Topological Matter School - 25/08/2017
RSJ model : DC current bias
43
Josephson/RSJ equations
= 2eV~
IRIS
I
I = IS() +~
2eR
IS() = I4 sin
2+ I2 sin+ . . .
Stewart, APL 12, 277 (1968) McCumber, J. App. Phys. 39, 3113 (1968)
Erwann Bocquillon Topological Matter School - 25/08/2017
RSJ model : DC current bias
43
Josephson/RSJ equations
= 2eV~
IRIS
I
I = IS() +~
2eR
IS() = I4 sin
2+ I2 sin+ . . .
Stewart, APL 12, 277 (1968) McCumber, J. App. Phys. 39, 3113 (1968)
U()
I
Motion of fictitious particle
Zero-voltage state ⟷ particle trapped
Voltage state ⟷ particle falling down
@U() = IS() I
= Cst
%
I < Ic
I > Ic
Erwann Bocquillon Topological Matter School - 25/08/2017
I-V curve from RSJ model
44
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
2.5
3
3.5
DC
voltageV
=h˙ i
[RI
c]
DC current I [Ic]
V = RIp
1 (Ic/I)2I Ic Harmonic for
Anharmonic for
Frequency
I ' Ic
fJ =2eV
h
Erwann Bocquillon Topological Matter School - 25/08/2017
I-V curve from RSJ model
44
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
2.5
3
3.5
DC
voltageV
=h˙ i
[RI
c]
DC current I [Ic]
V = RIp
1 (Ic/I)2I Ic Harmonic for
Anharmonic for
Frequency
I ' Ic
fJ =2eV
h
Erwann Bocquillon Topological Matter School - 25/08/2017
AC response : Shapiro steps
45
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 Topological Matter School - 25/08/2017
AC response : Shapiro steps
45
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 Topological Matter School - 25/08/2017
AC response : Shapiro steps
45
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 Topological Matter School - 25/08/2017
AC response : Shapiro steps
45
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 Topological Matter School - 25/08/2017
Experimental realization
46
open-ended coaxial cable
frequency range
4 devices tested 2 substrates
f 0.5 12GHz
Coaxial cable
1-3 mm
CdTeHgTe
Al
CdHgTe/HgTe (mesa)
Au (gate)
Al (contacts)
Erwann Bocquillon Topological Matter School - 25/08/2017
Shapiro response : frequency
47
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 Topological Matter School - 25/08/2017
Shapiro response : frequency
47
-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 Topological Matter School - 25/08/2017
Shapiro response : power
48
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 Topological Matter School - 25/08/2017
Non-topological HgTe quantum well
49
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 Topological Matter School - 25/08/2017
RSJ frequency dependence
50
I≳Ic
High frequency
⇒ almost sinusoidal
I≫Ic
Low frequency
⇒ non-sinusoidal
Domínguez et al., PRB 86, 146503 (2012) Domínguez et al., PRB 95, 195430 (2017)
Erwann Bocquillon Topological Matter School - 25/08/2017
RSJ frequency dependence
50
I≳Ic
High frequency
⇒ almost sinusoidal
I≫Ic
Low frequency
⇒ non-sinusoidal
I2 = 1, I4 = 0
I2 = 1, I4 = 0.15
Domínguez et al., PRB 86, 146503 (2012) Domínguez et al., PRB 95, 195430 (2017)
Erwann Bocquillon Topological Matter School - 25/08/2017
RSJ frequency dependence
50
I≳Ic
High frequency
⇒ almost sinusoidal
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
Domínguez et al., PRB 86, 146503 (2012) Domínguez et al., PRB 95, 195430 (2017)
Erwann Bocquillon Topological Matter School - 25/08/2017
RSJ simulations
51
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 Topological Matter School - 25/08/2017
Summary of Part II
52
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
… of topological states ?
Landau-Zener transitions unlikely edge currents (SQUID pattern) contribution: 1-3 modes coexistence with conduction band ? discrepancy with Rn ?
Erwann Bocquillon Topological Matter School - 25/08/2017
Outlook
53
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