electrostatic control of spin polarization in a quantum ...in a quantum hall ferromagnet: a platform...

Luchon, France

May 24 - 29, 2015

Electrostatic control of spin polarization

in a quantum Hall ferromagnet:

a platform to realize high order non-

Abelian excitations

Aleksander Kazakov & Leonid Rokhinson Department of Physics, Purdue University

V. Kolkovsky, Z. Adamus, & Tomasz Wojtowicz Institute of Physics, Polish Academy of Science, Warsaw, Poland

George Simion & Yuli Lyanda-Geller Department of Physics, Purdue University

Engineering Majorana fermions

6/11/2015 Leonid Rokhinson, Purdue University 2

requirements:

1D

spinless (one mode)

superconductor

topological superconductor

k

2D

Bso

B

kk

EZ

B = 0 Bso || B

EZE

F

pairing

possible

Sau, et al ’10, Alicea, et al ‘10

SC SC

parameter space

6/11/2015 Leonid Rokhinson, Purdue Univesity 3

𝐸𝑍 > 𝛥2 + 𝐸𝐹

2

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 k

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

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

smallest dimension defines Eso:

small d ⇒ large Eso ⇒ large EF ⇒ less localization

Characteristic 4 energy-flux relation


modification of the Josephson phase

trivial superconductor 2 Cooper pairs, I sin(f)

topological superconductor 4 Majorana particles, I sin(f/2)

Kwon ’04 Lutchyn ‘10

Kitaev ‘01 𝑏† = (𝛾𝑙 − 𝑖𝛾𝑚)

ac Josephson effect


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

𝑉 =ℎ𝜔

2𝑒

-200 0 200

-24

-12

0

12

24

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

V (

V

)

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


f = 3 GHz

LR, X. Liu, J. Furdyna, Nature Physics 8, 795 (2012)

dc rf ~

V

Shapiro steps


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

I0 (nA) I

1I

2I

3I

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 (

V

)

dV

/dI

I (nA)

Δ𝐼𝑛=A|𝐽𝑛2𝑒𝑣𝑟𝑓

ℏ𝜔𝑟𝑓|

dV/dI vs B


step @ 6 V step @ 12 V

4-periodic Josephson supercurrent in HgTe-based 3D TI


arXiv:1503.05591

Wiedenmann, …M. Klapwijk, …, Seigo Tarucha, L. W. Molenkamp


Advantage of 1D wires: Majorana modes are localized easy to perform spectroscopy

Disadvantage of 1D wires: Majorana modes are localized almost impossible to perform exchange

quantum Hall

effect

magnetic

semiconductors superconductivity

new materials to support exotic non-Abelian excitations

reconfigurable 1D topological superconductors in 2D systems

Motivation and inspiration


Exotic non-Abelian anyons from conventional fractional quantum Hall states

David J. Clarke, Jason Alicea, and Kirill Shtengel

Nature Commun., 2012, 4,, 1348

Topological Quantum Computation - From Basic

Concepts to First Experiments

Ady Stern & Netanel Lindner

Science, 2013, 339, 1179


Mn [Ar]3d54s2

exchange split ~3 eV (Hunds rule), ½ filled

S=5/2

Ga [Ar]3d104s24p1 As [Ar]3d104s24p3

p-doping

large s-d exchange (ferromagnetic)

GaAs:Mn

CdTe:Mn Cd [Kr]4d105s2 Te [Kr]4d105s25p4

neutral impurity, large s-d exchange

Development of a new system CdTe:Mn QW

Development of a new system


High mobility 2D gas in CdTe/CdMgTe QW

m*=0.11, Eg=1.44 eV

no Mn ~1% Mn

add Mn into CdTe (neutral impurity with 5/2 spin)

FQHE in CdTe:Mn


Betthausen, et al, Phys. Rev. B 90, 115302 (2014)

T. Wojtowicz

Anomalous Zeeman splitting in CdTe:Mn


𝐸𝑛,↑↓ = (𝑛 +12 )ℏ𝜔𝑐 ±

12 𝑔

∗𝜇𝐵𝐵 + 𝑥𝑀𝑛𝐸𝑠𝑑𝔅𝑆𝑔𝜇𝐵𝑆𝐵

𝑘𝐵𝑇

cyclotron Zeeman

g*1.6

s-d exchange (>0)

1.3% Mn 0.13% Mn for n=1

En (

meV

)

En (

meV

)

En (

meV

) B (Tesla) B (Tesla) B (Tesla)

0 2 4 6 8 10-8

-6

-4

-2

0

2

4

6

8

Magnetoreflectivity studies


Wojtowicz, et al, PRB 59, R10437 (1999)

negatively charged exciton complex 𝑋−

(trion) to singlet 𝑋 transition under polarized 𝜎+/𝜎− light

0 2 4 6 8 10-8

-6

-4

-2

0

2

4

6

8

spin

ener

gy s

pli

ttin

g (

Kel

vin

)

magnetic field (Tesla)

=

1/m

new platform for non-Abelian excitations


B*

Ohmic SC contact

0 2 4 6 8 10-8

-6

-4

-2

0

2

4

6

8

sp

in e

ne

rgy s

plit

tin

g (

Ke

lvin

)

magnetic field (Tesla)

=

1/m

new platform for non-Abelian excitations


𝜈 =1

𝑚, 𝐸𝑍↑ > 0 𝜈 =

1

𝑚, 𝐸𝑍↑ < 0

parafermions

𝐸𝑍

𝑥 𝐸𝐹 B* B*

braiding sequence

Ohmic SC contact

SC

S

C

0 5 10-20

0

20

40

60

80

7 8 9 10-0.2

0.0

0.2

0.4

(a) 0 3 6 9 12

-1.0

-0.5

0.0

0.5

1.0

𝜈 𝐸𝑠↑↓

(K)

magnetic field (T)

𝐵𝜈

𝜈 =1

𝑚 |↓

parafermions

𝐸𝑠↑↓

𝑥 𝐸𝐹

𝜈 =1

𝑚 |↑

magnetic field (T)

magnetic field (T)

𝑛+1 2ℏ𝜔𝑐+𝐸𝑠↑↓

(meV

) 𝐸𝑝𝐶𝐹+𝐸𝑠↑↓ (m

eV)

(b)

(c)

(d)

|0, ↓

|1, ↓

|2, ↓

|3, ↓

|0, ↑

|1, ↑

|2, ↑

|3, ↑

|1, ↓

|2, ↓ |3, ↓ |4, ↓

|1, ↑

|2, ↑ |3, ↑ |4, ↑

|𝑝, 𝑠

|𝑛, 𝑠

𝜈 = 2

1

3,5

3

𝜈 =2

3,4

3,2

5,8

5

Crossing of neighboring LLs

Quantum Hall ferromagnet & level crossing


Jaroszynski, et al, PRL 89, 266802 (2002)

Jaroszynski, et al, AIP conference proceedings (2005)

uniformly Mn-doped quantum well

Gate control of exchange


0 20 40 60 80 100 120 140 160 180 200

0.0

0.1

0.2

0.3

0.4

ba

nd

ed

ge

(m

eV

)

distance (nm)

𝐸𝑠𝑑 ∝ 𝜓𝑒 𝑥 𝜒𝑀𝑛(𝑥) 𝑑𝑥

100 120 140

0.0

0.2

0.4

VG1

VG2

bandedge [

meV

]

depth from surface [nm]

(x

)

100 120 140

0.0

0.2

0.4

b

an

de

dg

e [

me

V]


0.0

0.1

0.2

0.3

wa

ve

fu

nction

dencity

front gate

exchange

100 120 140

0.0

0.2

0.4

ba

nd

edg

e [

me

V]


0.0

0.1

0.2

0.3

wa

ve

fu

nction

de

ncity

back gate

overlap

exch

an

ge

VFG

VBG

Structures with asymmetric doping


0 10 20 30 40 50 60 70 80 90

7x(5x1)

Layer number

011414A

0 10 20 30 40 50 60 70 80 90

7x(2x1)

Layer number

011514A

0 10 20 30 40 50 60 70 80 90

8x(3x1)

Layer number

011614A

0 10 20 30 40 50 60 70 80 90

6x(1x1)

Layer number

011714A

0.00

0.03

0.06

0.00

0.03

0.06

0.00

0.03

0.06

75 100 125 150 1750.00

0.03

0.06

b

an

de

dg

e [

eV

]

wafer #011414A0.0

0.3

wafer #011514A0.0

0.3

wafer #011614A0.0

0.3

x [nm]

wafer #011714A0.0

0.3

Gate control of s-d exchange


0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Exc

= const

E xc

simulation: 5-12

simulation: 21-23

data: wafer #011414

data: B*(V

g)/B

*(-200) vs (V

g)/(-200)

log

no

ralis

ed

ove

rla

p

log normalized dencity

change of the overlap with density

Exc

1/

4 5 6 7 8 9

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

1 2

Rxx (

k

)

B (T)

#011414T=400 mKB @ 18° d023

VBG

3

5 6 7 8 9

-200

-150

-100

-50

0

50

100

150

2

2

1

B (Tesla)

VB

G (

Vo

lts)

0.0

1.2

2.4

3.6

4.8

6.0

0

5

10

VBG

= 0

Rxx

(k)

Rxx (

k

)

0.0

0.5

1.0

#011414

Rxy (

h/e

2)

B @ 18° T=400 mK (1 ↓)

(0 ↑)

(0 ↓)

crossing 1 and 1.3% Mn

Gate control of the crossing


4 6 80

50

100

150

1

2

>

B [T]

Vb

ack g

ate [

V]

= 2

|1,>

4 6 80,0

0,1

0,2

0,3

B [T]

Vto

p g

ate [V

]

1

2

>

|1>

= 2100 120 140

0.0

0.2

0.4

b

an

de

dg

e [

me

V]


0.0

0.1

0.2

0.3w

ave

fu

nction

dencity

front gate

exchange

100 120 140

0,0

0,2

0,4

b

an

de

dg

e [

me

V]


0,0

0,1

0,2

0,3

wa

ve

fu

nction

de

ncity

back gate

overlap

exch

an

ge

VFG

VBG

𝜈 = 2

low Mn concentration


beating in SdH

Node position:

𝐸𝑍 = 𝑁 +1

2ℏ𝜔𝐶 = 𝑔

∗𝜇𝐵𝐵 + 𝛼𝑥𝑒𝑓𝑓Sℬ𝑆𝑔𝜇𝐵𝑆𝐵

𝑘𝐵𝑇0

0.0

0.3

0.6

Rxx [

k

]

#081214

425 mK

0.00 0.25 0.50 0.750.0

0.8

1.6

E [m

eV

]

B [T]

𝑛 +1

2ℏ𝜔𝑐

0.0 0.2 0.4 0.6 0.80.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Rxx [

k

]

B [T]

#081214

625 mK

33 mK

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

xeff

= 0.34%

T0 = 84 mK

no

de p

ositio

n [T

]

T [K]

# 081214

node 1

node 2

node 3

node 4

node 5

node 6

node 7

node 8

𝑁 +1

2ℏ𝜔𝐶 = 𝑔

∗𝜇𝐵𝐵 + 𝛼𝑥𝑒𝑓𝑓Sℬ𝑆𝑔𝜇𝐵𝑆𝐵

𝑘𝐵(𝑇𝐴𝐹 + 𝑇)

SdH beating, xeff and TFM

temperature dependence

Gate control of SdH beating

0,1 0,2 0,3 0,4 0,50,0

0,2

0,4

0,6

0,8

1,0

Rxx [k

]

B [T]

#090514A200 V

-200 V

-200 -100 0 100 2000.85

0.90

0.95

1.00

1.05

1.10#090514

node 2

node 3

node 4

node 5

node p

ositio

n (

Vbg)

/ n

od

e p

ositio

n (

0V

)

Vbg

[V]

back gate dependence

Comparison of high and low Mn concentrations

Anticrossing between 1st and 2nd LLs


200 300 400 600 800

0.1

1

resis

tance

T, mK

1.12exp

KR

T

0 1an tic ro ss in g g ap b e tw een an d E E

EAC

= 1.12K = 96 eV

6.5 7.0 7.5 8.00

2

4

T =200mK

resis

tan

ce

B (tesla)

T =800mK

The role of SO interactions

𝐻𝑆𝑂 = 𝛾𝐷𝝈 ⋅ 𝜿 + 𝛾𝑅 𝝈 × 𝒌 ⋅ 𝓔

𝜅 = ( 𝑘𝑥, 𝑘𝑦2 − 𝑘𝑧

2 , 𝑘𝑦, 𝑘𝑧2 − 𝑘𝑥

2 , 𝑘𝑧, 𝑘𝑥2 − 𝑘𝑦

2 )

Only Rashba coupling contributes to N=1 and N+2 anticrossing

Transport across a gate


0

5

10

15

20

25

30

R [

k

]

Vfg = 50 mV

Rg-u

xx

Rgated

xy

Rungated

xy

3 4 5 6 7 8 90

50

100

150

200

250

300

B [T]

Vfg [

mV

]

0.000

2.000

4.000

6.000

8.000

10.00

12.00

14.00

Gg-u

xx [e

2/h]V

bg = 100 V

n, m

Landauer-Buttiker:

𝐺 = 𝐺0𝑛(𝑛 +𝑚)

𝑚

𝜈 = 2 𝑎𝑛𝑑 𝜈 = 3 𝑛 = 2,𝑚 = 1 G = 6G0

Transport across QHFm domain wall


4 6 8 10

0

6

12

18

24

R [k

]

B [T]

Vbg

= 60 V

Vfg = 0.3 V

T = 100 mK

ungated area

gated area

across the gate

6.6 6.8 7.0 7.2 7.4 7.60

1

2

4 6 8 10

0

6

12

18

24

R [

k

]

B [T]

Vbg

= 60 V

Vfg = 0.3 V

T = 100 mK

ungated area

gated area

across the gate

6 7 8 9

0

6

R [

k

]

B [T]

Vbg

= 60 V

Vfg = 0.3 V

T = 100 mK

ungated area

gated area

across the gate

people involved


Aleksander Kazakov, Purdue University

Tomasz Wojtowicz

Institute of Physics, Polish Academy of Sciences

V. Kolkovsky & Z. Adamus, Polish Academy of Science

Yuli Lyanda-Geller, Purdue University
http://www.ncn.gov.pl/?language=en

electrostatic control of spin polarization in a quantum ...in a quantum hall ferromagnet: a platform...

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