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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
6/11/2015 Leonid Rokhinson, Purdue Univesity 4
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 5
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 6
f = 3 GHz
LR, X. Liu, J. Furdyna, Nature Physics 8, 795 (2012)
dc rf ~
V
Shapiro steps
6/11/2015 Leonid Rokhinson, Purdue Univesity 7
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 8
step @ 6 V step @ 12 V
4-periodic Josephson supercurrent in HgTe-based 3D TI
6/11/2015 Leonid Rokhinson, Purdue Univesity 9
arXiv:1503.05591
Wiedenmann, …M. Klapwijk, …, Seigo Tarucha, L. W. Molenkamp
6/11/2015 Leonid Rokhinson, Purdue Univesity 10
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 11
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
6/11/2015 Leonid Rokhinson, Purdue University 12
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
6/11/2015 Leonid Rokhinson, Purdue University 13
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 14
Betthausen, et al, Phys. Rev. B 90, 115302 (2014)
T. Wojtowicz
Anomalous Zeeman splitting in CdTe:Mn
6/11/2015 Leonid Rokhinson, Purdue University 15
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 16
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
6/11/2015 Leonid Rokhinson, Purdue University 17
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
6/11/2015 Leonid Rokhinson, Purdue University 18
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 20
Jaroszynski, et al, PRL 89, 266802 (2002)
Jaroszynski, et al, AIP conference proceedings (2005)
uniformly Mn-doped quantum well
Gate control of exchange
6/11/2015 Leonid Rokhinson, Purdue Univesity 21
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]
depth from surface [nm]
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]
depth from surface [nm]
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 22
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 23
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 24
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]
depth from surface [nm]
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]
depth from surface [nm]
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 25
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 29
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 31
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
6/11/2015 Leonid Rokhinson, Purdue Univesity 32
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
6/11/2015 Leonid Rokhinson, Purdue University 33
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