Fractional ac Josephson effect: the signature of Majorana ...Cornell University, November 27, 2012...
Transcript of Fractional ac Josephson effect: the signature of Majorana ...Cornell University, November 27, 2012...
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
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decoherence and dephasing
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|↓
|↑ 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
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|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
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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
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John Preskill, http://online.kitp.ucsb.edu/online/exotic_c04/preskill/oh/21.html
intrinsically fault tolerant quantum computing
can we engineer Majorana particles?
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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?
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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
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• 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
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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
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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?
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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
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15 nm QW
105 V/cm
parameter space
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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?
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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?
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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
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In60Ga30Sb 3 nm InSb 20 nm In60Ga30Sb 3 nm
In77Al23Sb 120 nm
InxGa1-xSb graded 1280 nm
GaSb:Te substrate
Nb
fabrication
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290 nm
120 nm
10 mm
dc rf ~
V
etch ~50 nm
T-dependence of JJs
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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
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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
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-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
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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
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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
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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
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f = 3 GHz
Shapiro steps
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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
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-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
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step @ 6 mV step @ 12 mV
consistency check
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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?
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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?
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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
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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
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-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)
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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
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• 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