Cornell University, November 27, 2012

Fractional ac Josephson effect:

the signature of Majorana particles

Leonid Rokhinson Department of Physics, Department of Electrical Engineering

and Birck Nanotechnology Center

Purdue University, West Lafayette, Indiana USA

Jacek Furdyna (Notre Dame)

Xinyu Liu (Notre Dame)

Dirac vs Majorana

11/6/2012 Leonid Rokhinson, Purdue Univesity 2

(𝑖𝛾𝜇𝜕𝜇 − 𝑚𝑐)𝜓=0

𝜓 =𝜉𝜂

- 4-spinor

𝛾0 =0 𝐼𝐼 0

;

Dirac g-matrices:

𝜸 =0 −𝝈𝝈 0

Majorana 𝛾 -matrices

𝛾 0 = 𝑖0 −𝜎1

𝜎1 0; 𝛾 1 = 𝑖

0 𝐼𝐼 0

;

𝛾 2 = 𝑖𝐼 00 −𝐼

; 𝛾 3 =0 𝜎2

−𝜎2 0

Frank Wilczek, Majorana returns, Nature Physics 5, 614 (2009)

Majorana transformation

11/6/2012 Leonid Rokhinson, Purdue University 3

decoherence and dephasing

11/6/2012 Leonid Rokhinson, Purdue Univesity 4

|↓

|↑ 𝑠 = 𝛼 ↑ + β|↓

spin flip 𝜎𝑥|↑ = |↓ phase flip 𝜎𝑧(|↑ + ↓ = (|↑ − ↓

|0

|1 𝑠 = 𝛼 0 + β|1

good classical bit, but not quantum:

phase fluctuations Δ𝐻 ∝ 𝑎𝑙†𝑎𝑙

𝑎𝑙†|0 = |1 , 𝑎𝑙|1 = |0

fault-tolerant qubit

11/6/2012 Leonid Rokhinson, Purdue Univesity 5

|0

|1 𝑠 = 𝛼 0 + β|0

let’s create localized modes:

𝛾𝑙 = 𝑎𝑙† + 𝑎𝑙

𝛾𝑙2 = 1 ⇒ energy offset, no phase errors

new effective fermionic operators:

𝑏 = (𝛾𝑙 + 𝑖𝛾𝑚)

𝑏† = (𝛾𝑙 − 𝑖𝛾𝑚)

dephasing Δ𝐻 ∝ 𝑏†𝑏 ∝ 𝑖𝛾𝑙𝛾𝑚

separate l and m in space !!!

Kitaev, 2001

Majorana operator

statistics

11/6/2012 Leonid Rokhinson, Purdue University 6

k l k l

𝑛𝑜𝑛 − 𝑑𝑒𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒 − Abelian anyons

𝜓𝑓 =𝑒−𝑖𝜃𝑙𝑒−𝑖𝜃𝑘 𝜓𝑖 𝜓𝑓 =𝑒−𝑖𝜃𝑘𝑒−𝑖𝜃𝑙 𝜓𝑖

𝑑𝑒𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒 − non-Abelian anyons

𝝍𝒇 =𝑼𝒍 𝑼𝒌𝝍𝒊 𝝍𝒇 =𝑼𝒌 𝑼𝒍𝝍𝒊

Majorana particles in 2D are non-Abelian anyons

1 2 2 1

in general 𝑼𝒍 𝑼𝒌 ≠ 𝑼𝒌 𝑼𝒍

𝑒−𝑖𝜃𝑙𝑒−𝑖𝜃𝑘 = 𝑒−𝑖𝜃𝑘𝑒−𝑖𝜃𝑙

Wilczek ’82-84

Topological quantum computing

11/6/2012 Leonid Rokhinson, Purdue University 7

John Preskill, http://online.kitp.ucsb.edu/online/exotic_c04/preskill/oh/21.html

intrinsically fault tolerant quantum computing

can we engineer Majorana particles?

11/6/2012 Leonid Rokhinson, Purdue University 8

Kitaev’s toy model (2001)

g1 g2 g3 g4 gj gL gj+1

a1 a2 aL aj

g1 b1 gL b2 bj

𝐻 = −𝑡 𝑎𝑗†𝑎𝑗+1 + 𝑎𝑗+1

† 𝑎𝑗 − 𝜇 𝑎𝑗 𝑎𝑗† −

1

2+ Δ𝑎𝑗𝑎𝑗+1 + Δ∗𝑎†

𝑗𝑎†𝑗+1

𝑗

tunneling

between cites

# of particles

(Fermi level)

superconducting

coupling D = t > 0, m = 0

one fermion, does not enter Hamiltonian 𝐻 = 𝑖𝑡 𝑏𝑗†𝑏𝑗 − 1

2

𝐿−1

𝑗=1

𝑏𝑗 = 12(𝛾2𝑗 +𝑖𝛾2𝑗+1)

𝑏𝑗† = 1

2(𝛾2𝑗 − 𝑖𝛾2𝑗+1)

fermion transformation

g𝟐𝒋−𝟏

= 𝒂𝒋 + 𝒂𝒋†

g𝟐𝒋

= −𝒊(𝒂𝒋 − 𝒂𝒋†)

Majorana transformation

𝐻 = 𝑖𝑡 𝛾2𝑗𝛾2𝑗+1

𝑗

can we engineer Majorana particles?

11/6/2012 Leonid Rokhinson, Purdue University 9

Kitaev’s toy model (2001)

requirements:

1D

spinless (one mode)

superconductor

topological superconductor

g1 b1 gL b2 bj 𝐻 = 𝑖𝑡 𝑏𝑗†𝑏𝑗 − 1

2

𝐿−1

𝑗=1

new operator: 𝐾 = −𝑖𝛾1𝛾𝐿

two ground states |0 , |1 𝐾|0 = +|1 - even electron parity 𝐾|1 = −|0 - odd electron parity

g’1 b’1 g’L b’2 b’j

11/6/2012 Leonid Rokhinson, Purdue University 10

• superfluid He3 Salomaa & Volovik ‘87

• excitation in n=5/2 FQHE Moore & Read ’91

• 1D organic semiconductors Senigupta, et al ’01

• array of coupled flux qubits Levitov, Orlando, et al ‘01

• cold atoms Gurarie, Radzihovsky & Andreev ‘05

• p-wave superconductors (Sr2RuO4) Das Sarma, Nayak, Tewari ’06

• topological insulator/superconductor Fu & Kane ’08

• surface of semiconductor/superconductor Sau, et al ’10, Alicea, et al ‘10

low dimensionality

spinless quasiparticles

superconducting interactions

can we engineer Majorana particles?

+

Semiconductor / s-wave superconductor

11/6/2012 Leonid Rokhinson, Purdue University 11

s-wave superconductor quasiparticles:

semiconductor with spin-orbit interaction:

𝐻 =𝑝2

2𝑚 + 𝛾 𝜎 × 𝑝 + 𝜇𝐵𝜎 ∙ 𝐵

kCooper pairs k k

k

2g

Semiconductor / s-wave superconductor

11/6/2012 Leonid Rokhinson, Purdue University 12

k

2gD

Bso

B

kk

EZ

B = 0 Bso

|| B

EZE

F

s-wave superconductor quasiparticles:

semiconductor with spin-orbit interaction:

𝐻 =𝑝2

2𝑚 + 𝛾 𝜎 × 𝑝 + 𝜇𝐵𝜎 ∙ 𝐵

p-wave pairng

possible

kCooper pairs k k

-1.0 -0.5 0.0 0.5 1.0

-2

-1

0

1

2

3

4

5

ma

gn

eto

resis

tan

ce

(k

)

B (Tesla)

magnetic

focusing

GCGinj

Gdet

1

2

4

3

Can we see k-splitting?

11/6/2012 Leonid Rokhinson, Purdue University 13

magnetic focusing

V I

R2D gas

eB

kRkkE F

cFFF

& : @

Rokhinson, Larkina, Lyanda-Geller, Pfeiffer & K.W. West

"Spin separation in cyclotron motion", PRL 93, 146601 (2004)

p

E

EF

g g

4 1/ 4 5 10 cmBeLg D

choice of material

11/6/2012 Leonid Rokhinson, Purdue University 14

15 nm QW

105 V/cm

parameter space

11/6/2012 Leonid Rokhinson, Purdue Univesity 15

𝐸𝑍 > 𝛥2 + 𝐸𝐹2

Bso

B

single-spin condition:

]110[

[110] kx

ky d=20nm

w>200nm

𝐸𝑍~𝐸𝑆𝑂 to protect superconductivity:

2 22 2 ( / )SO D z DE k k d kg g

6 12.6 [meV], [10 cm ]SOE k k

d=100nm 6 10.1 [meV], [10 cm ]SOE k k

What are we looking for?

11/6/2012 Leonid Rokhinson, Purdue Univesity 16

a. States at zero energy: enhanced tunneling at zero bias

density of states

trivial superconductor

topological superconductor

simulated tunneling conductance

as a function of a tuning parameter

Stanescu, Lutchyn & Das Sarma ’2011

Zero bias anomaly in mesoscopic physics

Kondo effect in 0D systems

“0.7 anomaly” in 1D wires

etc.

What are we looking for?

11/6/2012 Leonid Rokhinson, Purdue Univesity 17

b. modification of the Josephson phase

trivial superconductor charge-2e Cooper pairs, I sin(f)

topological superconductor charge-e Majorana particles, I sin(f/2)

Kwon ’04 Lutchyn ‘10

Kitaev ‘01

wafers

11/6/2012 Leonid Rokhinson, Purdue Univesity 18

In60Ga30Sb 3 nm InSb 20 nm In60Ga30Sb 3 nm

In77Al23Sb 120 nm

InxGa1-xSb graded 1280 nm

GaSb:Te substrate

Nb

fabrication

11/6/2012 Leonid Rokhinson, Purdue Univesity 19

290 nm

120 nm

10 mm

dc rf ~

V

etch ~50 nm

T-dependence of JJs

11/6/2012 Leonid Rokhinson, Purdue Univesity 20

0 1 2 3 4 5 6 7 80

1

2

L

JJ8 40 nm gap

JJ7 30 nm gap

JJ6 20 nm gap

TC2

R (

k

)

temperature (K)

TC1

TC3

TC 0 1 2

0.0

0.5

1.0

TC3

TC

TC

R (

k

)

T (K)

3He system dilution fridge

TC1 – w>6 mm

TC2 – w=1 mm

TC3 – w=0.1 mm

TC – JJ proximity effect

𝛥 = 𝛥𝑠

𝜆

𝜆 + 𝛥𝑠

Ds=1.76 kBTC3/e = 310 meV

D =1.76 kBTC/e = 180 meV

l ~ 2.6 D

junctions on i-GaAs

11/6/2012 Leonid Rokhinson, Purdue Univesity 21

0 2 4 6 8 10 120.1

1

10

100 line

junction

(40 nm gap)

R (

k

)

Tc3

Tc2

T (K)

Tc1

0 100 200 300 400

-15

-10

-5

0

5

10

15

20

WL

hei

ght

(nm

)

x (nm)

WL

devices with the gap > 20 nm are insulating

field dependence of Ic

11/6/2012 Leonid Rokhinson, Purdue Univesity 22

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0

1

2

3

4

5

6

7

I (mA)

B || I (T

esla

)

10

632

1255

1878

2500

dV/dI ()

JJ

-30 -20 -10 0 10 20 30

0

1

2

3

4

5

6

7

I (mA)

B ||

I (

Tes

la)

0.000

0.1300

0.2600

0.3250

L10

0.1 mm - wide line

Bc~2.5 Tesla

samples

11/6/2012 Leonid Rokhinson, Purdue Univesity 23

Typical V(I) characteristics excess current – Andereev reflection

sign of coherent transport

-0.9 -0.6 -0.3 0.0 0.3 0.6 0.9

-0.4

-0.2

0.0

0.2

0.4

VR

IC

V (

mV

)

I (mA)

IR

0.0 0.5 1.0 1.50

1

2

I (u

A)

V (mV)

0.0 0.4 0.8

1

2

0T

3T

RN ×

dI/

dV

V (mV)

0 1 2 3 4

1

2

RN ×

dI/

dV

B (Tesla)

ac Josephson effect

11/6/2012 Leonid Rokhinson, Purdue Univesity 24

𝜙1 𝜙2

V

𝑑(Δ𝜙)

𝑑𝑡=

2𝑒𝑉

ℏ

𝐼𝑠 = 𝐼𝑐 sin 𝜔𝐽𝑡 = 𝐼𝑐 sin2𝑒𝑉

ℏ𝑡

Current oscillates with frequency V

direct inverse

𝜙1 𝜙2

I

𝐼 = 𝐼0 + 𝐼𝜔sin(𝜔𝑡)

Constant voltage steps w

𝜙2 − 𝜙1 = 𝑉𝑛 = 𝑛ℎ𝜔

2𝑒

inverse ac Josephson effect

11/6/2012 Leonid Rokhinson, Purdue Univesity 25

phase locking between external rf and Josephson frequency

Shapiro steps (Shapiro ’63) 𝑉𝑛 = 𝑛ℎ𝜔𝑟𝑓

𝑞

-200 0 200

-24

-16

-8

0

8

16

24

f = 2 GHz

DV= 4 mV

f = 3 GHz

DV= 6 mV

V (

mV

)

I (nA)

f = 4 GHz

DV= 8 mV

-200 0 200-30

-24

-18

-12

-6

0

6

12

18

24

30

I (nA)

-200 0 200

-28

-24

-20

-16

-12

-8

-4

0

4

8

12

16

20

24

28

I (nA)

𝑞 = 2𝑒

-200 0 200

-24

-12

0

12

24

-200 0 200 -200 0 200 -200 0 200 -200 0 200

V (

mV

)

I (nA)

B=0 B=1.0 T

I (nA)

B=1.6 T

I (nA)

B=2.1 T

I (nA)

B=2.5 T

I (nA)

Disappearance of the first Shapiro step

11/6/2012 Leonid Rokhinson, Purdue Univesity 26

f = 3 GHz

Shapiro steps

11/6/2012 Leonid Rokhinson, Purdue Univesity 27

0 300 0 100 0 100 0 100 0 100-200 0 200

0

2

4

6

8

10

12

Vrf (

mV

)

I (nA)

0

5

10

dV/dIB=0, f = 3 GHz

DI0 (nA) DI

1DI

2DI

3DI

4

-40

-32

-24

-16

-8

0

8

16

24

32

40

-300 -200 -100 0 100 200 300

0

20

40

f = 4 GHz

Vrf = 14.25 mV

V (

mV

)

dV

/dI

I (nA)

more fields

11/6/2012 Leonid Rokhinson, Purdue Univesity 28

-200 0 200

I (nA)

-200 0 200

I (nA)

-200 0 200

I (nA)

-200 0 200

I (nA)

-200 0 200

I (nA)

0 T 1.0 T 1.6 T 2.1 T 2.5 T

(1)

(2) (3)

(1)

(2)

(1)

(2) (2) (2)

dV/dI vs B

11/6/2012 Leonid Rokhinson, Purdue Univesity 29

step @ 6 mV step @ 12 mV

consistency check

11/6/2012 Leonid Rokhinson, Purdue Univesity 30

2 or 4 periodicity

width of the steps

third step and higher odd steps?

f f

vs A

theory: 𝐼𝑀 ≲ 𝑒ℏ Δ𝑖𝑛𝑑 ≈ 25 nA

experiment: 𝐴 ≈ 150 nA

Q1

Q2

Q3

gap closing at the transition

𝐴 → 0 for 𝐵 ≈ 2 Tesla

Q4

2 or 4 periodicity?

11/6/2012 Leonid Rokhinson, Purdue Univesity 31

Infinite wire: 𝐼± = ±𝑒𝛤

2ℎsin 𝛥𝜑 2 −

3𝑒𝛤2

16ℎ𝑡sin 𝛥𝜑

Lutchyn, Sau & das Sarma ’10

Alicea, et al, ‘11

IM IC

For G~D IM ~IC

effect of finite size:

levels anticrossing

Jiang, et al ’11

Pikulin & Nazarov ’11

San-Jose, Prada & Aguado ’12

Domínguez, Hassler & Platero ‘12

even and odd steps should be visible

2 or 4 periodicity?

11/6/2012 Leonid Rokhinson, Purdue Univesity 32

voltage bias current bias

Domínguez, Hassler, and Platero , ‘12

100 GHz

𝐼± = ±𝑒𝛤

2ℎsin 𝛥𝜑 2 −

3𝑒𝛤2

16ℎ𝑡sin 𝛥𝜑

Lutchyn, Sau & das Sarma ’10

Alicea, et al, ‘11

IM IC

For G~D IM ~IC

𝜔𝑟𝑓 < 5𝐺𝐻𝑧

no odd steps for

current biased junction

11/6/2012 Leonid Rokhinson, Purdue Univesity 33

Domínguez, Hassler, and Platero , arXiv:1202.0642

no odd steps for 𝜔𝑟𝑓 <2𝑒𝑅𝑁𝐼𝑀

ℏ

A =Ic+IM/ 2

for Ic≫ IM no substantial change of step width

step width Δ𝐼𝑛=A|𝐽𝑛 𝛽𝑉𝑟𝑓 |

𝐼𝑐 = 10 𝐼𝑀

A1

A2

3-rd and higher odd steps

11/6/2012 Leonid Rokhinson, Purdue Univesity 34

-0.9 -0.6 -0.3 0.0 0.3 0.6 0.9

-0.4

-0.2

0.0

0.2

0.4

VRIC

V (

mV

)

I (mA)

IR

1-st step: 6 mV

3-rd step: 18 mV

vcr~20 mV

VR~60 mV

A3

no gap closing at the transition (𝑰𝒄 ≠ 𝟎)

11/6/2012 Leonid Rokhinson, Purdue Univesity 35

a. there is some reduction of the Ic at 2 Tesla: A4

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0

1

2

3

4

5

6

7

I (mA)

B || I (T

esla

)

10

632

1255

1878

2500

dV/dI ()

b. gappless superconductivity?

when density of gappless excitation

small compared to the gapped ones, Ic>0

conclusions

11/6/2012 Leonid Rokhinson, Purdue Univesity 36

• 1D Josephson junction Nb/InSb/Nb

• Excess current - evidence of Andreev reflection

• Observe Shapiro steps with 2 periodicity

• At high field first step disappears: 4 periodicity

Clear evidence of the formation of zero energy Andreev states

(Majorana particles)

arXiv: 1204.4212; Nature Physics, AOP 10.1038/nphys2429