Topological insulators & topological superconductivity in ...tms17.dipc.org/lecture-notes/lectures-tms-2017-bocquillo.pdf · Erwann Bocquillon 3 Topological Matter School - 25/08/2017

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

http://www.lpa.ens.fr

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


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


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 !


I-A. Basics of HgTe

6

aHgTe = 0.646 nm

aCdTe = 0.648 nm


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


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


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


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)


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)


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)


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


(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


(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


(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)


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?


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)


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…


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


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)


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)



16

Strained HgTe



CdTe

8

6

1 eV

HgTe

6

8

0.3 eV

k k

"(k)"(k)



16

Strained HgTe



Strained HgTe

20meV

k

"(k)

CdTe

8

6

1 eV

HgTe

6

8

0.3 eV

k k

"(k)"(k)


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


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


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


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


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 ?


I-C. Compressive strain in 3D HgTe

21

HgTe

6

8

0.3 eV

k

"(k)

Tensile strain in HgTe

20meV

k

"(k)



21

HgTe

6

8

0.3 eV

k

"(k)


20meV

k

"(k)

Compressive strain in HgTek

"(k)

2 DP



21

HgTe

6

8

0.3 eV

k

"(k)


20meV

k

"(k)

Compressive strain in HgTek

"(k)

2 DP


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)


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)


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)


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)


ε = +1.4 %

(c) -0.3 %

+0.4 %


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


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


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


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

=



27



k ?"k ",k #



=

=

†

=1

2(c+ c†)



27



k ?"k ",k #



=

N

ST

=

†

=1

2(c+ c†)



27



k ?"k ",k #



=

N

ST

=

†

Sayonara fermions ?



27



k ?"k ",k #



=

=

†

Sayonara fermions ?

ei


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



28

S1 S2I

Josephson junctions



j =pnje

ij , j = 1, 2



28

S1 S2I

Josephson junctions



j =pnje

ij , j = 1, 2

| |2

n1 n2

x



28

S1 S2I

Josephson junctions



j =pnje

ij , j = 1, 2

IS() = Ic sin

d

dt=

2eV

~

Josephson equations

phase evolution

current-phase relation

| |2

n1 n2

x



28

S1 S2I

Josephson junctions



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)


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

" ! "



29

N/TI S

1"#

""

Conditions




E

EF + E ! EF E

kF + k ! kF k

" ! "



29

N/TI S

1"#

""

Conditions




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)


II-B. Andreev bound states

30

N/TI S

1

S

2


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


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


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


II-B. (Fractional) Josephson effect

33

Josephson equationsd

dt=

2eV

~ = 1 2

fJ =2eV

~

1

2I() =X

n

@En

@

1 2f(En)



33


dt=

2eV

~ = 1 2

fJ =2eV

~

1

2

IS() = Ic sin

fJ =2eV

~⇒ Josephson frequency

I() =X

n

@En

@

1 2f(En)



33


dt=

2eV

~ = 1 2

fJ =2eV

~

1

2

IS() = Ic sin

fJ =2eV


Fractional Josephson effect

sin ! sin/2

fJ ! fJ/2

I() =X

n

@En

@

1 2f(En)



33


dt=

2eV

~ = 1 2

fJ =2eV

~

1

2

IS() = Ic sin

fJ =2eV


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)


II-B. Fractional Josephson effect

34

Superconducting phase 𝜑 π 2π 3π 4π

Ener

gy

0

0

Δi

-Δi

EJ

topological state

conv. state


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


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


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


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


Interference patterns

37

Interference patterns

non-uniform phase

« interference » pattern

! + 2BL0

y

Hart et al., Nat. Phys. 10, 638 (2014)


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)


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.


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


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


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]


SQUID


average ratio <r>

c

fJ/2fJ2fJ

Low frequency fd= 3 GHz

High frequency fd= 5.5 GHz


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]


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


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)


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


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


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


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



45


Phase-locked motion



⇒ Vn = nhf2e

ddt = 2eV

~

t = 2n

1/f

I = IS() +~

2eR

U()

I@U() = IS() I



45


Phase-locked motion



⇒ Vn = nhf2e

ddt = 2eV

~

t = 2n

1/f

I = IS() +~

2eR



45


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


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)


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


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 »


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]


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


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)



50

I≳Ic

High frequency


I≫Ic

Low frequency

⇒ non-sinusoidal

I2 = 1, I4 = 0

I2 = 1, I4 = 0.15




50

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

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


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 ?


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


Open positions !

54

Post-docs

PhD students

Thank you for your attention !

Open positions!

Documents

Topological insulators & topological superconductivity in ...tms17.dipc.org/lecture-notes/lectures-tms-2017-bocquillo.pdf · Erwann Bocquillon 3 Topological Matter School - 25/08/2017