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
(ππΎπππ β ππ)π=0
π =ππ
- 4-spinor
πΎ0 =0 πΌπΌ 0
;
Dirac g-matrices:
πΈ =0 βππ 0
Majorana πΎ -matrices
πΎ 0 = π0 βπ1
π1 0; πΎ 1 = π
0 πΌπΌ 0
;
πΎ 2 = ππΌ 00 βπΌ
; πΎ 3 =0 π2
βπ2 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
|β
|β π = πΌ β + Ξ²|β
spin flip ππ₯|β = |β phase flip ππ§(|β + β = (|β β β
|0
|1 π = πΌ 0 + Ξ²|1
good classical bit, but not quantum:
phase fluctuations Ξπ» β ππβ ππ
ππβ |0 = |1 , ππ|1 = |0
fault-tolerant qubit
11/6/2012 Leonid Rokhinson, Purdue Univesity 5
|0
|1 π = πΌ 0 + Ξ²|0
letβs create localized modes:
πΎπ = ππβ + ππ
πΎπ2 = 1 β energy offset, no phase errors
new effective fermionic operators:
π = (πΎπ + ππΎπ)
πβ = (πΎπ β ππΎπ)
dephasing Ξπ» β πβ π β ππΎππΎπ
separate l and m in space !!!
Kitaev, 2001
Majorana operator
statistics
11/6/2012 Leonid Rokhinson, Purdue University 6
k l k l
πππ β πππππππππ‘π β Abelian anyons
ππ =πβππππβπππ ππ ππ =πβππππβπππ ππ
πππππππππ‘π β non-Abelian anyons
ππ =πΌπ πΌπππ ππ =πΌπ πΌπππ
Majorana particles in 2D are non-Abelian anyons
1 2 2 1
in general πΌπ πΌπ β πΌπ πΌπ
πβππππβπππ = πβππππβπππ
Wilczek β82-84
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
Kitaevβs toy model (2001)
g1 g2 g3 g4 gj gL gj+1
a1 a2 aL aj
g1 b1 gL b2 bj
π» = βπ‘ ππβ ππ+1 + ππ+1
β ππ β π ππ ππβ β
1
2+ Ξππππ+1 + Ξβπβ
ππβ π+1
π
tunneling
between cites
# of particles
(Fermi level)
superconducting
coupling D = t > 0, m = 0
one fermion, does not enter Hamiltonian π» = ππ‘ ππβ ππ β 1
2
πΏβ1
π=1
ππ = 12(πΎ2π +ππΎ2π+1)
ππβ = 1
2(πΎ2π β ππΎ2π+1)
fermion transformation
gππβπ
= ππ + ππβ
gππ
= βπ(ππ β ππβ )
Majorana transformation
π» = ππ‘ πΎ2ππΎ2π+1
π
can we engineer Majorana particles?
11/6/2012 Leonid Rokhinson, Purdue University 9
Kitaevβs toy model (2001)
requirements:
1D
spinless (one mode)
superconductor
topological superconductor
g1 b1 gL b2 bj π» = ππ‘ ππβ ππ β 1
2
πΏβ1
π=1
new operator: πΎ = βππΎ1πΎπΏ
two ground states |0 , |1 πΎ|0 = +|1 - even electron parity πΎ|1 = β|0 - odd electron parity
gβ1 bβ1 gβL bβ2 bβj
11/6/2012 Leonid Rokhinson, Purdue University 10
β’ superfluid He3 Salomaa & Volovik β87
β’ excitation in n=5/2 FQHE Moore & Read β91
β’ 1D organic semiconductors Senigupta, et al β01
β’ array of coupled flux qubits Levitov, Orlando, et al β01
β’ cold atoms Gurarie, Radzihovsky & Andreev β05
β’ p-wave superconductors (Sr2RuO4) Das Sarma, Nayak, Tewari β06
β’ topological insulator/superconductor Fu & Kane β08
β’ surface of semiconductor/superconductor Sau, et al β10, Alicea, et al β10
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:
π» =π2
2π + πΎ π Γ π + ππ΅π β π΅
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:
π» =π2
2π + πΎ π Γ π + ππ΅π β π΅
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
πΈπ > π₯2 + πΈπΉ2
Bso
B
single-spin condition:
]110[
[110] kx
ky d=20nm
w>200nm
πΈπ~πΈππ 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 β2011
Zero bias anomaly in mesoscopic physics
Kondo effect in 0D systems
β0.7 anomalyβ 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 β04 Lutchyn β10
Kitaev β01
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 β w>6 mm
TC2 β w=1 mm
TC3 β w=0.1 mm
TC β JJ proximity effect
π₯ = π₯π
π
π + π₯π
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 β 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 Γ
dI/
dV
V (mV)
0 1 2 3 4
1
2
RN Γ
dI/
dV
B (Tesla)
ac Josephson effect
11/6/2012 Leonid Rokhinson, Purdue Univesity 24
π1 π2
V
π(Ξπ)
ππ‘=
2ππ
β
πΌπ = πΌπ sin ππ½π‘ = πΌπ sin2ππ
βπ‘
Current oscillates with frequency V
direct inverse
π1 π2
I
πΌ = πΌ0 + πΌπsin(ππ‘)
Constant voltage steps w
π2 β π1 = ππ = πβπ
2π
inverse ac Josephson effect
11/6/2012 Leonid Rokhinson, Purdue Univesity 25
phase locking between external rf and Josephson frequency
Shapiro steps (Shapiro β63) ππ = πβπππ
π
-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)
π = 2π
-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: πΌπ β² πβ Ξπππ β 25 nA
experiment: π΄ β 150 nA
Q1
Q2
Q3
gap closing at the transition
π΄ β 0 for π΅ β 2 Tesla
Q4
2 or 4 periodicity?
11/6/2012 Leonid Rokhinson, Purdue Univesity 31
Infinite wire: πΌΒ± = Β±ππ€
2βsin π₯π 2 β
3ππ€2
16βπ‘sin π₯π
Lutchyn, Sau & das Sarma β10
Alicea, et al, β11
IM IC
For G~D IM ~IC
effect of finite size:
levels anticrossing
Jiang, et al β11
Pikulin & Nazarov β11
San-Jose, Prada & Aguado β12
DomΓnguez, Hassler & Platero β12
even and odd steps should be visible
2 or 4 periodicity?
11/6/2012 Leonid Rokhinson, Purdue Univesity 32
voltage bias current bias
DomΓnguez, Hassler, and Platero , β12
100 GHz
πΌΒ± = Β±ππ€
2βsin π₯π 2 β
3ππ€2
16βπ‘sin π₯π
Lutchyn, Sau & das Sarma β10
Alicea, et al, β11
IM IC
For G~D IM ~IC
πππ < 5πΊπ»π§
no odd steps for
current biased junction
11/6/2012 Leonid Rokhinson, Purdue Univesity 33
DomΓnguez, Hassler, and Platero , arXiv:1202.0642
no odd steps for πππ <2ππ ππΌπ
β
A =Ic+IM/ 2
for Icβ« IM no substantial change of step width
step width ΞπΌπ=A|π½π π½πππ |
πΌπ = 10 πΌπ
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 (π°π β π)
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
β’ 1D Josephson junction Nb/InSb/Nb
β’ Excess current - evidence of Andreev reflection
β’ Observe Shapiro steps with 2 periodicity
β’ 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
Top Related