Majorana modes in Atomic Chains on the Surface of a Superconductor

Stevan Nadj-PergeTU DELFTCALTECH

Prof. Ali YazdaniSangjun Jeon

Prof. B. A. Bernevig

Jungpil SeoIlya Drozdov

Acknowledgments Experiment

Theory UT Austin

Prof. Allan MacDonaldHua ChenJian Li

Topology & band structure

Momentum

Ener

gy

SPIN ORBIT INTERACTION (SOI)

Momentum

Ener

gy

SOI+Magnetism

Momentum

Ener

gy

SOI + Magnetism

Momentum

Ener

gy

SOI + Magnetism

Momentum

Ener

gy

SOI + Magnetism

B < B critical B = B critical B > B critical

EF

SOI + Magnetism

B < B critical B = B critical B > B critical

EF

TOPOLOGICALLY DISTINCTNumber of crossing different in parity

SOI + Magnetic field + superconductivity

Y. Oreg et al. PRL(2010); R. M. Lutchyn et al PRL (2010);

Ener

gy

Momentum

B < B critical B = B critical B > B critical

MajoranaEdge States

TRIVIAL TRIVIALTOPOLOGICAL

Inverted band structureNon-Inverted band structure Non-Inverted band structure

Topology & bandstructure in 1D

Y. Oreg et al. PRL(2010); R. M. Lutchyn et al PRL (2010);

Majorana wires

Rokhinson et al. Nat. Phys. (2012) Mourik et al., Science (2012)Das et al., Nat. Phys. (2012)(Also: Deng et al. Nano Lett. (2012), van Harlinegen

group and C. Marcus groups (2013), and others )

Zero bias peak in tunneling spectroscopy

Majorana wires

Rokhinson et al. Nat. Phys. (2012) Mourik et al., Science (2012)Das et al., Nat. Phys. (2012)(Also: Deng et al. Nano Lett. (2012), van Harlinegen

group and C. Marcus groups (2013), and others )

Zero bias peak in tunneling spectroscopy

𝐻𝐻 = 𝐻𝐻𝐵𝐵𝐵𝐵𝐵𝐵 + 𝐵𝐵 𝑆𝑆 � 𝑠𝑠

Dens

ity o

f sta

tes

Energy-E +E

Classical spin in a superconductor

Yu-Shiba-Rusinov states for a single atom

Mn, Gd on Nb Yazdani et al., Science (1997)Cr, Mn on Pb, Ji et al., PRL (2008)

0

2ΔYu APS (1965), Shiba PTP (1968) , Rusinov JETP (1969), Balatsky et al RMP (2006)

TOPOGRAPHY

dI/dV conductance (at 1.07mV for the gap 1.35meV)

18 Å

s-wave superconductor

s-wave superconductor

STM tip

Chain of magnetic atoms on the superconductor

Magnetic atom

• Suitable for STM• Can be disorder free

Magnetism + Superconductivity Majorana proposals:Choy et. al (PRB 2011)Flensberg et. al (PRB 2011)Martin and Morpurgo (PRB 2012)

Spin-orbit coupling

s-wave superconductor

STM tip

Chain of magnetic atoms on the superconductor

Magnetic atom

• Suitable for STM• Can be disorder free

Magnetism + Superconductivity Majorana proposals:Choy et. al (PRB 2011)Flensberg et. al (PRB 2011)Martin and Morpurgo (PRB 2012)

Majorana fermions and magnetic texture

Kjaergaard, Wӧlms, Flensberg (PRB 2012)Nadj-Perge, Drozdov, Bernevig, Yazdani (PRB 2013)

See also: Martin and Morpurgo (PRB 2012); Glazman (PRB 2013), and others ..

Nakosai, Tanaka, Nagaosa (PRB 2013)Klinovaja, Stano, Loss (PRL 2012)

More than one band –Tight binding model

Momentum (π/a)

0 1-1

Ener

gy (e

V)

-4

-3

-2

-1

0

1

J

More than one band –Tight binding model

Momentum (π/a)

0 1-1

Ener

gy (e

V)

-4

-3

-2

-1

0

1

J (eV)0 1 2 4

μ(e

V)

-3

-2

-1

0

1

2

3

JFe

J

New Generation of High-Resolution STM at 1.4 Kintegrated STM/Growth/vector-field

in situ growth capabilitySTM head: dual sample holderMisra et al. Rev. Sci. Inst. (2013)

Assembled by:

Ilya Drozdov

Jungpil Seo

Experimental requirements

1) Atomically flat superconducting substrate

2) Magnetic atoms

3) Spin-orbit coupling (or spin texture)

Superconducting Pb (110) substrate

Topographic image – Pb(110)

Point spectrum

a=4.95Å

Tc=7.2K, ξ=830A, Hc=80mT, λL=370A -10 -8 -6 -4 -2 0 2 4 6 8 100

1

2

3

Diffe

rent

ial c

ondu

ctan

ce (a

.u)

Energy (meV)

Δ=1.35 meVT=1.46K

BCS fit

Bias V

Pb(110)

VacuumW Tip

100Å

Fe atomic chain growth on Pb substrate

Topographic image

200 Å

0

20

Heig

ht (Å

)

Total chain length up to 800 Å (~ 380 atoms)“Clean” segments up to 150 Å (~ 70 atoms)

50Å

50Å

50Å

20 Å

Single atom wide chains on Pb(110) substrate

Corrugation: fitting Fe atoms onto Pb

Atomic cains on Pb(110) substrate – DFT

1) Chains form zig-zag structure2) Corrugation periodicity reproduced3) Expected height matches experiment

Cut across the chain Cut along the middle of the chain

Heig

ht (Å

)

Distance (Å)

0

5

10

15

0 10 20 0 10 20Distance (Å)

Heig

ht (Å

)

0

5

10

15

Charge density (10-4e/aB3)0 1.5

Point spectrum Experiment vs DFT modeling

Spin-polarized measurements

B B

Antiferromagnetic Cr tip

Antiferromagnetic Cr tip

B = -1 T B = +1 T

Background subtracted

B (T)-1 -0.5 0.5 10

0.5

1.0

1.5

2.0

2.5

dI/ d

Von

cha

in –

dI/ d

Von

sub

stra

te (n

S)

Fe chains on Pb(110): Spin-polarized measurements

Most likely tip switching (at ±0.25T)

Average conductance on the chain

3.1

3.2

-1 -0.5 0 +0.5 +1B (T)

dI/d

V(+3

0meV

) (a

.u.)

Spin-polarized tip on the Pb(110): Hints of spin-orbit coupling

Consistent with strong SOI at the Pb surface

B B

Experimental signature:Tunneling suppressed when Tip polarized out of plane (Spins in plane)

Pb –Heavy element (Z= 82, A=204-208) Strong SOI expectedSymmetry broken at the surface

Similar experiment in 2DEG GaAs:Moser et al, Phys Rev. Lett. 99, 056601(2007)

Experimental requirements

1) Atomically flat superconducting substrate

2) Magnetic atoms (break spin degeneracy)

3) Spin-orbit coupling in substrate (to enable p-wave pairing)

HOW DO STATES INSIDE SUPERCONDCTING GAP LOOK LIKE?????

Spectroscopy of an atomic chain

515Å

Spectroscopy of an atomic chain

515Å

130Å 150Å

0

3

dI/dV (a.u.)

0

3

dI/dV (a.u.)

Spectroscopy of an atomic chain

515Å

130Å 150Å

Voltage (mV)-5 0 50

1

2

3

dI/d

V(a

rb. u

nits

))

-5 0 50

1

2

3

4

5

Voltage (mV)dI

/dV

(arb

. uni

ts))

+0.44mV-1.28mV -0.44mV +0.64mV-0.64mV +1.04mV

Spatial extent of the density of the states: another example

100Å

+0.44mV0mV-1.28mV -0.44mV +0.64mV-0.64mV +1.04mV

Spatial extent of the density of the states: another example

100Å

Zero bias peak feature localized on the scale of a 1nm

Recall Shiba states

Localization length of MQP in an atomic chain

|ψ (r)|2 ~ 1/r2ψ (r)∝kFr

2π e-r/ξ

ξ~500-1000Å λF/2=2-3Å

Realistic model of imbedded Fe chain calculations show atomic scale localization of

Majorana (not a pure 1D chain)

J. Li, H. Chen et al, Phys. Rev. B 90, 235433 (2014)

Renormalization of vF due to d-band/Shiba hybridization

Zero bias peak at the chains ends

Energy (meV)

50

100

150

-5 0 5

end 1 bulk

dI/d

V(n

S)

end 1

bulk

bulk

-5 0 525

50

75

dI/d

V(n

S)

end 2 bulk

L

HdI/dV

-5 0 5

0

50

100

Dis

tanc

e (Å

)

-5 0 5Energy (meV)

0

50

100

Dis

tanc

e (Å

)

spectroscopic linecutssubstrate

substrate

end 2

Kondo effect? ZBP due to superconductivity?

Energy (meV)

-5 0 5

0

40

80

Dis

tanc

e (Å

)

B = 0

Energy (meV)

-5 0 5

0

40

80 L

HdI/dV (a.u.)

B = 100mT > BcStates are clearly related to superconductivity!

Note: ZBP end state separate from the Shiba bands which also change at the edge

Current bound on ZERO with 90μeV resolutionFactor of 5 improvement from 1.4K

Pb at 20mKT_electrons=250mK

Predicted two gaps in Pb

End of chainMidchain avg.

90 μV

Splitting < 45μeV

Ben Feldman

Mallika Randeria

Ilya Drozdov

Imbedding the Fe Chains further in Pb1.4K measurements

Fe chain

Topo (5mV, 100pA) Conductance @ 0V

Pb SubstrateT = 1.4 K

End of chainMid-chainMid-chain (2)Pb background

Pb MQP

Fe Chain

Pb overlayers

in-gap statesYonglong Xie

Sangjun Jeon

Better energy resolution using superconducting tip

Katarina Franke’s group FU Berlin,ArXiv: 1507.03104

Also: Ernst Mayer’s group Basel,ArXiv: 1505.06078

Summary

Experimental progress:

Magnetic chains on superconductor

Spin polarized measurement:- (Modulated) Ferromagnetism- SOI present

Observed end states at zero bias:- Reproducible on many wires- Related to Magnetism and Superconductivity - Consistent with FM + SOI scenario for Majorana fermions

THANK YOU!QUESTIONS?

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