Induced Superconductivity in the Quantum Spin Hall Edge
Induced Superconductivity in the Quantum Spin Hall Edge Sean Hart, Hechen Ren
Amir Yacoby – Harvard University
• QSHE • Fraunhofer patterns to determine Current Distribution • Crossover from bulk to edge transport as approach into the topological regime
1 0.5 0 3 2 1 0 -1 -2 -3 -8
In collaboration with: L. Molenkamp and his group. Discussions: J. Alicea, K Michaeli
-6
-4
-2
0
2
4
8
6 -6
x 10
Coupling Superconductivity to HgCdTe QW’s Sean Hart, Hechen Ren
• •
Quantum Spin Hall Effect. Coupling s-superconductor to bulk and edge modes.
Fu & Kane, PRB; Markus König et al., Science
Courtesy: J. Alicea
From Dirac to Schrödinger 𝑃𝑃2 𝑃𝑃4 𝑒𝑒ℏ 𝑒𝑒ℏ2 2 𝑉𝑉 𝐻𝐻 = + 𝑉𝑉 − + 𝜎𝜎 ⋅ 𝐸𝐸 × 𝑃𝑃 + 𝛻𝛻 2𝑚𝑚 2𝑚𝑚3 𝑐𝑐 2 4𝑚𝑚2 𝑐𝑐 2 8𝑚𝑚2 𝑐𝑐 2 Schrödinger
Mass term
Spin-Orbit
Darwin
From Dirac to Schrödinger 𝑃𝑃2 𝑃𝑃4 𝑒𝑒ℏ 𝑒𝑒ℏ2 2 𝑉𝑉 𝐻𝐻 = + 𝑉𝑉 − + 𝜎𝜎 ⋅ 𝐸𝐸 × 𝑃𝑃 + 𝛻𝛻 2𝑚𝑚 2𝑚𝑚3 𝑐𝑐 2 4𝑚𝑚2 𝑐𝑐 2 8𝑚𝑚2 𝑐𝑐 2 Schrödinger
CdTe Es
Ep
Mass term
Spin-Orbit
Darwin
From Dirac to Schrödinger 𝑃𝑃2 𝑃𝑃4 𝑒𝑒ℏ 𝑒𝑒ℏ2 2 𝑉𝑉 𝐻𝐻 = + 𝑉𝑉 − + 𝜎𝜎 ⋅ 𝐸𝐸 × 𝑃𝑃 + 𝛻𝛻 2𝑚𝑚 2𝑚𝑚3 𝑐𝑐 2 4𝑚𝑚2 𝑐𝑐 2 8𝑚𝑚2 𝑐𝑐 2 Schrödinger
Mass term
Darwin
Spin-Orbit
CdTe
HgTe
Es
Es
Ep
Ep
Mass velocity: Stronger effect in Hg vs Cd
HgTe vs CdTe - Band Structure
HgTe-CdTe Quantum Well
6.3 nm
HgTe-CdTe Quantum Well
6.3 nm
HgTe-CdTe Quantum Well
6.3 nm
x
Quantum Spin Hall Effect TiAu – Ohmic Contacts
Metal Gate – SiO2 TiAu
HgTe – Quantum Well
Majorana Fermions E TiAl – Ohmic Contacts k
Gap due to Superconductivity E Metal Gate – SiO2 TiAu
HgTe – Quantum Well
k
Gap due to inplane B field
In plane B
Experimental Evidence of Edge Modes: Quantized Conductance Vxx
µ2 0
µ1
0
µ2
µ2
µ1
µ1
µ
µ I
Quantized Conductance Vxx
µ2 0
µ1
0
µ2
µ2
µ1
µ1
µ
h RQ = 2 e µ I
σ xx
2 I e2 = = =2 Vxx RQ h
Previous Experimental Observation Large samples show large
resistance at the gap. Small samples (~1X1µm) show quantized conductance at the gap, indicating transport by edge states. g: 20-50
d (nm)
L×W (μm2)
I
5.5
20.0×13.3
II
7.3
20.0×13.3
III
7.3
1.0×1.0
IV
7.3
1.0×0.5
linelastic ~ 1µm Molenkamp et al.
Quantum Spin Hall Device
Topological Insulator
Ti/Au
2 μm edge length
Ti/Au
SiO2/Ti/Au Gate
Quantum Spin Hall Measurements I
Vxx
VGate
Quantum Spin Hall Measurements I
VGate
Vxx
R < h/(2e2) Vayrynen, Goldstein, Glazman, arXiv 1303.1766 Altshuler, Aleiner, Yudson, arXiv 1306.2626
Quantum Spin Hall Measurements I
VGate
Vxx
R > h/(2e2) Vayrynen, Goldstein, Glazman, arXiv 1303.1766 Altshuler, Aleiner, Yudson, arXiv 1306.2626
Quantum Spin Hall - Comparison d (nm)
L×W (μm2)
I
5.5
20.0×13.3
II
7.3
20.0×13.3
III
7.3
1.0×1.0
IV
7.3
1.0×0.5
2 μm length
Results from Molenkamp group
Edge Resistances I
I = V1 / R1 + V5 / R5 R3
V1
V4
R2
R4
R1
R5
R8
R6
R7
V4 / R4 = V5 / R5 U1 / R1 = U 4 / R4
V5
V1 U1 V4 ⇒ I = + × R1 U 4 R1
Edge Resistances I = V1 / R1 + V5 / R5 R3
U1
U4
R2
R4
R1
R5
R8
R6
R7
I
V4 / R4 = V5 / R5 U1 / R1 = U 4 / R4 V1 U1 V4 ⇒ I = + × R1 U 4 R1
Edge Resistances I = V1 / R1 + V5 / R5 R3
U1
U4
R2
R4
R1
R5
R8
R6
R7
I
V4 / R4 = V5 / R5 U1 / R1 = U 4 / R4 V1 U1 V4 ⇒ I = + × R1 U 4 R1
Edge Resistance Plots U4
U1
30 kΩ
I
Edge Resistance Plots U4
30 kΩ U1
30 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ U1
30 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
58 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
58 kΩ 31 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
58 kΩ 31 kΩ
108 kΩ I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
58 kΩ
57 kΩ
31 kΩ
108 kΩ I
Experimental Evidence of Edge Modes: Imaging
Novac et al, ‘12
Experimental Evidence of Edge Modes: Imaging
Ma et al, ‘12
Fraunhofer Patterns and Current Distribution Impose I
Current phase relation of a single junction 𝐼𝐼𝐶𝐶 = 𝐽𝐽0 sin 𝜑𝜑
Current phase relation of the entire junction 𝑤𝑤/2 𝑤𝑤/2 2𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 Measure V 𝐼𝐼𝐼𝐼 = � 𝑗𝑗 𝑥𝑥 sin − 𝜑𝜑0 𝑑𝑑𝑑𝑑 = � 𝑗𝑗 𝑥𝑥 sin 𝛽𝛽𝑥𝑥 − 𝜑𝜑0 𝑑𝑑𝑑𝑑 𝜙𝜙0 −𝑤𝑤/2 −𝑤𝑤/2
𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼 𝑒𝑒 −𝜑𝜑0 𝑔𝑔 𝛽𝛽
𝑔𝑔 𝛽𝛽 Fourier Transform of j(x) 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼 𝑒𝑒 −𝜑𝜑0 𝑔𝑔 𝛽𝛽
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = max 𝐼𝐼𝐼𝐼 𝑒𝑒 −𝜑𝜑0 𝑔𝑔 𝛽𝛽 𝜑𝜑0
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽
Fraunhofer Patterns and Current Distribution Icmax
B Icmax
B J(x)
Icmax
x
B
Dynes and Fulton ‘71
Measured Icmax
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽
∞
𝑔𝑔 𝛽𝛽 = � 𝑒𝑒 𝑖𝑖𝛽𝛽𝑥𝑥 𝑗𝑗 𝑥𝑥 𝑑𝑑𝑑𝑑 = 𝑒𝑒 𝑖𝑖𝜃𝜃 −∞
B
𝛽𝛽
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚
Hilbert Transform
𝛽𝛽 ∞ 𝑙𝑙𝑙𝑙 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏 − 𝑙𝑙𝑙𝑙 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 𝛽𝛽 𝜃𝜃 𝛽𝛽 = � 𝛽𝛽 2 − 𝑏𝑏2 2𝜋𝜋 −∞
𝑑𝑑𝑑𝑑
Dynes and Fulton ‘71 Measured Icmax
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽 ∞
𝑔𝑔 𝛽𝛽 = � 𝑒𝑒 𝑖𝑖𝛽𝛽𝑥𝑥 𝑗𝑗 𝑥𝑥 𝑑𝑑𝑑𝑑 = 𝑆𝑆 𝛽𝛽 + 𝑖𝑖𝑖𝑖 𝛽𝛽 −∞
B ∞
𝑆𝑆 𝛽𝛽 = � 𝑐𝑐𝑐𝑐𝑐𝑐 𝛽𝛽𝛽𝛽 𝑗𝑗𝑒𝑒 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞
∞
𝐴𝐴 𝛽𝛽 = � 𝑠𝑠𝑠𝑠𝑠𝑠 𝛽𝛽𝛽𝛽 𝑗𝑗𝑜𝑜 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞
𝑗𝑗 𝑥𝑥 = 𝑗𝑗𝑒𝑒 𝑥𝑥 + 𝑗𝑗𝑜𝑜 𝑥𝑥
Dynes and Fulton ‘71 Measured Icmax ∞
𝑗𝑗 𝑥𝑥 = 𝑗𝑗𝑒𝑒 𝑥𝑥
𝑔𝑔 𝛽𝛽 = � 𝑒𝑒 𝑖𝑖𝛽𝛽𝑥𝑥 𝑗𝑗 𝑥𝑥 𝑑𝑑𝑑𝑑 = 𝑆𝑆 𝛽𝛽 −∞
B ∞
𝑆𝑆 𝛽𝛽 = � 𝑐𝑐𝑐𝑐𝑐𝑐 𝛽𝛽𝛽𝛽 𝑗𝑗𝑒𝑒 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞
B
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽
Real
Dynes and Fulton ‘71 Measured Icmax
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽 ∞
𝑔𝑔 𝛽𝛽 = � 𝑒𝑒 𝑖𝑖𝛽𝛽𝑥𝑥 𝑗𝑗 𝑥𝑥 𝑑𝑑𝑑𝑑 = 𝑆𝑆 𝛽𝛽 + 𝑖𝑖𝑖𝑖 𝛽𝛽 −∞
B
𝑗𝑗 𝑥𝑥 = 𝑗𝑗𝑒𝑒 𝑥𝑥 + 𝑗𝑗𝑜𝑜 𝑥𝑥 ∞
Only A contributes
𝑆𝑆 𝛽𝛽 = � 𝑐𝑐𝑐𝑐𝑐𝑐 𝛽𝛽𝛽𝛽 𝑗𝑗𝑒𝑒 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞ ∞
B
𝐴𝐴 𝛽𝛽 = � 𝑠𝑠𝑠𝑠𝑠𝑠 𝛽𝛽𝛽𝛽 𝑗𝑗𝑜𝑜 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞
Sample structure Contact bulk and edge directly X is Cadmium 92 nm
Material Properties • Device
Ti/Au Top gate with Al2O3 dielectric Ti/Au contacts
Ion mill defined mesa of HgTe 2DEG
18um
Bulk Properties
Bulk Density
400nm Junction Ungated 400 nm between SC contacts
Ti/Al SC contacts with etching close to the 2DEG
Ion mill defined mesa of HgTe 2DEG 4um wide
400nm Junction Ungated
400nm Junction Ungated
400nm Junction Ungated 400 nm Junction 200 700 180
600
160
140 500
I (nA)
120 400 100
300
80
60 200 40 100 20
0
-5
-4
-3
-2
-1
0 B (mT)
1
2
3
4
5
0
Effective area
Period corresponds to junction area + expelled flux
800nm Junction 800 nm between SC contacts Ti/Al SC contacts with etching close to the 2DEG
Ti/Au Top gate with Al2O3 dielectric Ion mill defined mesa of HgTe 2DEG 4um wide
800nm Junction Gate
QSHE
Vg [V]
Gating of Critical Current vs. B - 800nm Junction
Gating of Critical Current vs. B - 800nm Junction
Raw data
Gating of Critical Current vs. B - 800nm Junction
Raw data
Normalized data
Gating of Critical Current vs. B - 800nm Junction
Raw data
Normalized data
Gating of Critical Current vs. B - 800nm Junction
Raw data
Normalized data
Another device - 800nm Junction
Another device - 800nm Junction
Narrower Mesa – 2um
Deeper into the Insulating Regime
Absence of critical current Periodic oscillations with period h/2e
Non Topological Well – 4.5nm
Additional Considerations Fix I and measure V Edge mode quantized
What is J(x)? 𝑤𝑤/2
𝑤𝑤/2 2𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼 = � 𝑗𝑗 𝑥𝑥 sin − 𝜑𝜑0 𝑑𝑑𝑑𝑑 = � 𝑗𝑗 𝑥𝑥 sin 𝛽𝛽 − 𝜑𝜑0 𝑑𝑑𝑑𝑑 𝜙𝜙 0 −𝑤𝑤/2 −𝑤𝑤/2
Are we measuring Ψ 𝑥𝑥
2
Can this explain the measured widths
Amir Yacoby – Harvard University
• QSHE • Fraunhofer patterns to determine Current Distribution • Crossover from bulk to edge transport as approach into the topological regime
1 0.5 0 3 2 1 0 -1 -2 -3 -8
In collaboration with: L. Molenkamp and his group. Discussions: J. Alicea, K Michaeli
-6
-4
-2
0
2
4
8
6 -6
x 10
Coupling Superconductivity to HgCdTe QW’s Sean Hart, Hechen Ren
• •
Quantum Spin Hall Effect. Coupling s-superconductor to bulk and edge modes.
Fu & Kane, PRB; Markus König et al., Science
Courtesy: J. Alicea
From Dirac to Schrödinger 𝑃𝑃2 𝑃𝑃4 𝑒𝑒ℏ 𝑒𝑒ℏ2 2 𝑉𝑉 𝐻𝐻 = + 𝑉𝑉 − + 𝜎𝜎 ⋅ 𝐸𝐸 × 𝑃𝑃 + 𝛻𝛻 2𝑚𝑚 2𝑚𝑚3 𝑐𝑐 2 4𝑚𝑚2 𝑐𝑐 2 8𝑚𝑚2 𝑐𝑐 2 Schrödinger
Mass term
Spin-Orbit
Darwin
From Dirac to Schrödinger 𝑃𝑃2 𝑃𝑃4 𝑒𝑒ℏ 𝑒𝑒ℏ2 2 𝑉𝑉 𝐻𝐻 = + 𝑉𝑉 − + 𝜎𝜎 ⋅ 𝐸𝐸 × 𝑃𝑃 + 𝛻𝛻 2𝑚𝑚 2𝑚𝑚3 𝑐𝑐 2 4𝑚𝑚2 𝑐𝑐 2 8𝑚𝑚2 𝑐𝑐 2 Schrödinger
CdTe Es
Ep
Mass term
Spin-Orbit
Darwin
From Dirac to Schrödinger 𝑃𝑃2 𝑃𝑃4 𝑒𝑒ℏ 𝑒𝑒ℏ2 2 𝑉𝑉 𝐻𝐻 = + 𝑉𝑉 − + 𝜎𝜎 ⋅ 𝐸𝐸 × 𝑃𝑃 + 𝛻𝛻 2𝑚𝑚 2𝑚𝑚3 𝑐𝑐 2 4𝑚𝑚2 𝑐𝑐 2 8𝑚𝑚2 𝑐𝑐 2 Schrödinger
Mass term
Darwin
Spin-Orbit
CdTe
HgTe
Es
Es
Ep
Ep
Mass velocity: Stronger effect in Hg vs Cd
HgTe vs CdTe - Band Structure
HgTe-CdTe Quantum Well
6.3 nm
HgTe-CdTe Quantum Well
6.3 nm
HgTe-CdTe Quantum Well
6.3 nm
x
Quantum Spin Hall Effect TiAu – Ohmic Contacts
Metal Gate – SiO2 TiAu
HgTe – Quantum Well
Majorana Fermions E TiAl – Ohmic Contacts k
Gap due to Superconductivity E Metal Gate – SiO2 TiAu
HgTe – Quantum Well
k
Gap due to inplane B field
In plane B
Experimental Evidence of Edge Modes: Quantized Conductance Vxx
µ2 0
µ1
0
µ2
µ2
µ1
µ1
µ
µ I
Quantized Conductance Vxx
µ2 0
µ1
0
µ2
µ2
µ1
µ1
µ
h RQ = 2 e µ I
σ xx
2 I e2 = = =2 Vxx RQ h
Previous Experimental Observation Large samples show large
resistance at the gap. Small samples (~1X1µm) show quantized conductance at the gap, indicating transport by edge states. g: 20-50
d (nm)
L×W (μm2)
I
5.5
20.0×13.3
II
7.3
20.0×13.3
III
7.3
1.0×1.0
IV
7.3
1.0×0.5
linelastic ~ 1µm Molenkamp et al.
Quantum Spin Hall Device
Topological Insulator
Ti/Au
2 μm edge length
Ti/Au
SiO2/Ti/Au Gate
Quantum Spin Hall Measurements I
Vxx
VGate
Quantum Spin Hall Measurements I
VGate
Vxx
R < h/(2e2) Vayrynen, Goldstein, Glazman, arXiv 1303.1766 Altshuler, Aleiner, Yudson, arXiv 1306.2626
Quantum Spin Hall Measurements I
VGate
Vxx
R > h/(2e2) Vayrynen, Goldstein, Glazman, arXiv 1303.1766 Altshuler, Aleiner, Yudson, arXiv 1306.2626
Quantum Spin Hall - Comparison d (nm)
L×W (μm2)
I
5.5
20.0×13.3
II
7.3
20.0×13.3
III
7.3
1.0×1.0
IV
7.3
1.0×0.5
2 μm length
Results from Molenkamp group
Edge Resistances I
I = V1 / R1 + V5 / R5 R3
V1
V4
R2
R4
R1
R5
R8
R6
R7
V4 / R4 = V5 / R5 U1 / R1 = U 4 / R4
V5
V1 U1 V4 ⇒ I = + × R1 U 4 R1
Edge Resistances I = V1 / R1 + V5 / R5 R3
U1
U4
R2
R4
R1
R5
R8
R6
R7
I
V4 / R4 = V5 / R5 U1 / R1 = U 4 / R4 V1 U1 V4 ⇒ I = + × R1 U 4 R1
Edge Resistances I = V1 / R1 + V5 / R5 R3
U1
U4
R2
R4
R1
R5
R8
R6
R7
I
V4 / R4 = V5 / R5 U1 / R1 = U 4 / R4 V1 U1 V4 ⇒ I = + × R1 U 4 R1
Edge Resistance Plots U4
U1
30 kΩ
I
Edge Resistance Plots U4
30 kΩ U1
30 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ U1
30 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
58 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
58 kΩ 31 kΩ
I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
58 kΩ 31 kΩ
108 kΩ I
Edge Resistance Plots U4
39 kΩ 30 kΩ 29 kΩ U1
30 kΩ
58 kΩ
57 kΩ
31 kΩ
108 kΩ I
Experimental Evidence of Edge Modes: Imaging
Novac et al, ‘12
Experimental Evidence of Edge Modes: Imaging
Ma et al, ‘12
Fraunhofer Patterns and Current Distribution Impose I
Current phase relation of a single junction 𝐼𝐼𝐶𝐶 = 𝐽𝐽0 sin 𝜑𝜑
Current phase relation of the entire junction 𝑤𝑤/2 𝑤𝑤/2 2𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 Measure V 𝐼𝐼𝐼𝐼 = � 𝑗𝑗 𝑥𝑥 sin − 𝜑𝜑0 𝑑𝑑𝑑𝑑 = � 𝑗𝑗 𝑥𝑥 sin 𝛽𝛽𝑥𝑥 − 𝜑𝜑0 𝑑𝑑𝑑𝑑 𝜙𝜙0 −𝑤𝑤/2 −𝑤𝑤/2
𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼 𝑒𝑒 −𝜑𝜑0 𝑔𝑔 𝛽𝛽
𝑔𝑔 𝛽𝛽 Fourier Transform of j(x) 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼 𝑒𝑒 −𝜑𝜑0 𝑔𝑔 𝛽𝛽
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = max 𝐼𝐼𝐼𝐼 𝑒𝑒 −𝜑𝜑0 𝑔𝑔 𝛽𝛽 𝜑𝜑0
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽
Fraunhofer Patterns and Current Distribution Icmax
B Icmax
B J(x)
Icmax
x
B
Dynes and Fulton ‘71
Measured Icmax
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽
∞
𝑔𝑔 𝛽𝛽 = � 𝑒𝑒 𝑖𝑖𝛽𝛽𝑥𝑥 𝑗𝑗 𝑥𝑥 𝑑𝑑𝑑𝑑 = 𝑒𝑒 𝑖𝑖𝜃𝜃 −∞
B
𝛽𝛽
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚
Hilbert Transform
𝛽𝛽 ∞ 𝑙𝑙𝑙𝑙 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏 − 𝑙𝑙𝑙𝑙 𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 𝛽𝛽 𝜃𝜃 𝛽𝛽 = � 𝛽𝛽 2 − 𝑏𝑏2 2𝜋𝜋 −∞
𝑑𝑑𝑑𝑑
Dynes and Fulton ‘71 Measured Icmax
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽 ∞
𝑔𝑔 𝛽𝛽 = � 𝑒𝑒 𝑖𝑖𝛽𝛽𝑥𝑥 𝑗𝑗 𝑥𝑥 𝑑𝑑𝑑𝑑 = 𝑆𝑆 𝛽𝛽 + 𝑖𝑖𝑖𝑖 𝛽𝛽 −∞
B ∞
𝑆𝑆 𝛽𝛽 = � 𝑐𝑐𝑐𝑐𝑐𝑐 𝛽𝛽𝛽𝛽 𝑗𝑗𝑒𝑒 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞
∞
𝐴𝐴 𝛽𝛽 = � 𝑠𝑠𝑠𝑠𝑠𝑠 𝛽𝛽𝛽𝛽 𝑗𝑗𝑜𝑜 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞
𝑗𝑗 𝑥𝑥 = 𝑗𝑗𝑒𝑒 𝑥𝑥 + 𝑗𝑗𝑜𝑜 𝑥𝑥
Dynes and Fulton ‘71 Measured Icmax ∞
𝑗𝑗 𝑥𝑥 = 𝑗𝑗𝑒𝑒 𝑥𝑥
𝑔𝑔 𝛽𝛽 = � 𝑒𝑒 𝑖𝑖𝛽𝛽𝑥𝑥 𝑗𝑗 𝑥𝑥 𝑑𝑑𝑑𝑑 = 𝑆𝑆 𝛽𝛽 −∞
B ∞
𝑆𝑆 𝛽𝛽 = � 𝑐𝑐𝑐𝑐𝑐𝑐 𝛽𝛽𝛽𝛽 𝑗𝑗𝑒𝑒 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞
B
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽
Real
Dynes and Fulton ‘71 Measured Icmax
𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑔𝑔 𝛽𝛽 ∞
𝑔𝑔 𝛽𝛽 = � 𝑒𝑒 𝑖𝑖𝛽𝛽𝑥𝑥 𝑗𝑗 𝑥𝑥 𝑑𝑑𝑑𝑑 = 𝑆𝑆 𝛽𝛽 + 𝑖𝑖𝑖𝑖 𝛽𝛽 −∞
B
𝑗𝑗 𝑥𝑥 = 𝑗𝑗𝑒𝑒 𝑥𝑥 + 𝑗𝑗𝑜𝑜 𝑥𝑥 ∞
Only A contributes
𝑆𝑆 𝛽𝛽 = � 𝑐𝑐𝑐𝑐𝑐𝑐 𝛽𝛽𝛽𝛽 𝑗𝑗𝑒𝑒 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞ ∞
B
𝐴𝐴 𝛽𝛽 = � 𝑠𝑠𝑠𝑠𝑠𝑠 𝛽𝛽𝛽𝛽 𝑗𝑗𝑜𝑜 𝑥𝑥 𝑑𝑑𝑑𝑑 −∞
Sample structure Contact bulk and edge directly X is Cadmium 92 nm
Material Properties • Device
Ti/Au Top gate with Al2O3 dielectric Ti/Au contacts
Ion mill defined mesa of HgTe 2DEG
18um
Bulk Properties
Bulk Density
400nm Junction Ungated 400 nm between SC contacts
Ti/Al SC contacts with etching close to the 2DEG
Ion mill defined mesa of HgTe 2DEG 4um wide
400nm Junction Ungated
400nm Junction Ungated
400nm Junction Ungated 400 nm Junction 200 700 180
600
160
140 500
I (nA)
120 400 100
300
80
60 200 40 100 20
0
-5
-4
-3
-2
-1
0 B (mT)
1
2
3
4
5
0
Effective area
Period corresponds to junction area + expelled flux
800nm Junction 800 nm between SC contacts Ti/Al SC contacts with etching close to the 2DEG
Ti/Au Top gate with Al2O3 dielectric Ion mill defined mesa of HgTe 2DEG 4um wide
800nm Junction Gate
QSHE
Vg [V]
Gating of Critical Current vs. B - 800nm Junction
Gating of Critical Current vs. B - 800nm Junction
Raw data
Gating of Critical Current vs. B - 800nm Junction
Raw data
Normalized data
Gating of Critical Current vs. B - 800nm Junction
Raw data
Normalized data
Gating of Critical Current vs. B - 800nm Junction
Raw data
Normalized data
Another device - 800nm Junction
Another device - 800nm Junction
Narrower Mesa – 2um
Deeper into the Insulating Regime
Absence of critical current Periodic oscillations with period h/2e
Non Topological Well – 4.5nm
Additional Considerations Fix I and measure V Edge mode quantized
What is J(x)? 𝑤𝑤/2
𝑤𝑤/2 2𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 𝐼𝐼𝐼𝐼 = � 𝑗𝑗 𝑥𝑥 sin − 𝜑𝜑0 𝑑𝑑𝑑𝑑 = � 𝑗𝑗 𝑥𝑥 sin 𝛽𝛽 − 𝜑𝜑0 𝑑𝑑𝑑𝑑 𝜙𝜙 0 −𝑤𝑤/2 −𝑤𝑤/2
Are we measuring Ψ 𝑥𝑥
2
Can this explain the measured widths
