Majorana quasiparticles

in nanoscale systems

MENA5010 - 2015

Steinar Kummeneje Grinde

• Ettore Majorana (1906-1938?)

• Predicting the positron (1928)

• Majorana fermion (1937)

• Neutrinos and neutralinos

Background Theory Experiments Applications Summary

What are Majorana

particles?

Background Theory Experiments Applications Summary

What are Majorana

particles?

• Formally a fermion can be written in

terms of Majorana particles

• Two Majoranas make up one fermion

• Can these particles be

created/observed?

• Where to look for Majorana particles?

• The superposition of electrons and holes in a

superconductor (Bogoliubov quasiparticles)

• Equal amplitude hole and electron makes the

particle equal its anti-particle

Background Theory Experiments Applications Summary

Majorana particles

on the nanoscale

• Why superconductors?

• Electrons and holes are anti-particles

• Electron-hole symmetry

• hole + cooper pair ≈ electron

Background Theory Experiments Applications Summary

Majorana particles

on the nanoscale

At E = 0:

Background Theory Experiments Applications Summary

Majorana particles

on the nanoscale

At E = 0:

• Picture a quantum dot for single

electrons inside a superconductor

• If there was a level at E = 0 it would

allow for Majorana particles, but there

is zero point motion

Based on https://www.youtube.com/watch?v=7OUkHkmPGbY

• 1999 (Volovik): Majorana fermions can

form in a special type of superconductor,

a p-wave superconductor

• The zero point motion is canceled out

Background Theory Experiments Applications Summary

Recent theoretical developments:

Based on https://www.youtube.com/watch?v=7OUkHkmPGbY

• Traditional superconductors are s-wave (the Cooper pairs are in s-orbital

configuration)

• It has been suggested that superconductors with high critical

temperature are best described as d-wave

• Are there any p-wave superconductors? (Sr2RuO4)

• Magnetism tend to ruin superconductivity

Background Theory Experiments Applications Summary

p-wave superconductors

http://guweb2.gonzaga.edu/faculty/cronk/CHEM101pub/L07-index.cfm

• 2001 (Kitaev): Majorana particles can form at ends

of a (spin-less) p-wave superconducting nanowire

• In p-wave superconductivity the b configuration is

favoured over a

Background Theory Experiments Applications Summary

Recent theoretical developments:

Based on https://www.youtube.com/watch?v=7OUkHkmPGbY

a

b

• Since the triplet configuration can exist in spin up and down, both will

be present in the wire, and the unpaired Majoranas will be paired

• To avoid this the electrons must be without spin

• To sum up: Majoranas require a nanowire of a p-wave superconductor

with spin-less electrons

Background Theory Experiments Applications Summary

Recent theoretical developments:

Based on https://www.youtube.com/watch?v=7OUkHkmPGbY

• 2008: Majorana particles can appear at

the interface between topological

insulators and superconductors

• 2010: Majorana particles in

Semiconductor-Superconductor

Heterostructures

Background Theory Experiments Applications Summary

Recent theoretical developments:

Band gaps in a

topological insulator

http://en.wikipedia.org/wiki/Topological_insulator

Background Theory Experiments Applications Summary

Recent experimental results:

• 2012: Majorana particles found in

nanowires at TU Delft’s Kavli Institute

• 2014: Majorana particles found in

ferromagnetic atomic chains on a

superconductor at Princeton University

• An indium

antimonide

nanowire is

connected to a

normal metal as

well as a

superconductor

Background Theory Experiments Applications Summary

Superconductor-Semiconductor Nanowire Devices

InSb

nanowire

NbTiN

Au

500 nm

• p-wave superconductivity is induced

in the semiconductor nanowire by

proximity to the standard

superconductor

• An applied magnetic field lifts the spin

degeneracy

• InSb has a very large G-factor (∼50)

which means that a small magnetic

field creates a big energy difference

between spin up and spin down

Background Theory Experiments Applications Summary

Superconductor-Semiconductor Nanowire Devices

Schematic view of device

with energy gaps

• The device is probed with Tunnel

Andreev spectroscopy

• The presence of Majoranas allow for

midgap states for single electrons which

results in a peak at zero bias voltage

Background Theory Experiments Applications Summary

Superconductor-Semiconductor Nanowire Devices

Schematic view of device

with energy gaps

The expected conductance if no Majoranas present

• The figure shows plots

separated by 5 mT (and

are also offset for clarity)

• The peak at 0 V is

evidence for Majoranas

Background Theory Experiments Applications Summary

Superconductor-Semiconductor Nanowire Devices

• Fe atomic chains on

Pb(110) surface are

studied using scanning

tunnelling microscope at

T = 1.4 K

Background Theory Experiments Applications Summary

Ferromagnetic atomic chains on a superconductor

Background Theory Experiments Applications Summary

Ferromagnetic atomic chains on a superconductor

STM spectra at

the marked

points

along the Fe

chain. Offset by

100 nS for clarity.

Scale = 1 nm

• The Majorana particles are actually an example of

non-abelian anyons

• Can be used to build a topological quantum

computer

• Uses “braids” of the world lines of anyons as logic

gates

Background Theory Experiments Applications Summary

Topological Quantum

Computer

• Crossed Andreev effect occurs when

an incident electron in one lead cause a

hole to reflect in another spatially

separated lead

• The presence of Majoranas in a

nanowire has been seen to significantly

impact shot noise in the device

Background Theory Experiments Applications Summary

Crossed Andreev Effect

en.wikipedia.org/wiki/Andreev_reflection

• What started as a prediction in particle physics is

first discovered in nanoscale systems

• Several different experimental setups have yielded

positive results

• May have interesting applications in the years to

come

Background Theory Experiments Applications Summary

Summary

• Sergey Frolov’s lecture on Majoranas in superconductors (https://www.youtube.com/watch?v=7OUkHkmPGbY)

• Majorana Fermions in Semiconductor Nanowires: Fundamentals, Modeling, and Experiment (http://arxiv.org/abs/1302.5433)

• Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices

(http://www.sciencemag.org/content/336/6084/1003.full.pdf?sid=856badd8-93ca-4774-ba1f-8f4cd7d3ee24)

• http://en.wikipedia.org/wiki/Ettore_Majorana

• http://en.wikipedia.org/wiki/Majorana_fermion

• Topological insulators and superconductors (http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.83.1057)

• New directions in the pursuit of Majorana fermions in solid state systems (http://arxiv.org/abs/1202.1293)

• Search for Majorana fermions in superconductors (http://arxiv.org/abs/1112.1950)

• Majorana Fermion Induced Non-local Current Correlations in Spin-orbit Coupled Superconducting Wires

(http://arxiv.org/pdf/1212.5879.pdf

Background Theory Experiments Applications Summary

References and further reading: