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Phelim Bradley

Quantised Conductance in Self-Breaking

Nanowires

Mentor: John MacHale

Supervisor: Dr. Aidan Quinn

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• Quantised Conductance• Feedback controlled Electromigration.• MCBJ and previous research.• Analysis of self-breaking region of nanobridges.• Degree of variability in traces.• Isolation of the contribution from individual

conductance channels.• Preferred and stable conductance levels.• Favorable transitions.• Differences between Au and Pt.

Contents

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Quantised Conductance

• When the wire length is less than the Fermi Wavelength, quantised conductance can be observed. The wire behaves like an electron wave guide with each ballistic channel contributing a maximum conductance:

• However, this does not necessarily mean that the conductance will be an integer multiple of G0.

• A quantum channel with transmission T<1 contributes < G0

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Motivation

• To understand the fabrication and properties of nanoscale metallic structures.

• Vital importance in next generation of sub 10nm electronics.

• Intellectual pursuit of understanding the quantum world.

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• Electromigration is the transport of material due to the electronic wind force.[1]

• Occurs at a critical power dissipation in the neck.[2]

“Unzipping” of bridge via FCE-assisted diffusion, Strachan et al., Phys. Rev. Lett. 100, 056805 (2008)

Feedback Controlled Electromigration5.

[1.] Rous, P.J., Driving force for adatom electromigration within mixed Cu/Al overlayers on Al(111). J. Appl. Phys., 2001. 89: p. 4809.

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Self-Breaking Regime

• At room temperature is can be high enough to break the bridge entirely without even applying a bias once the conductance has fallen below a certain value

• When the conductance reaches a certain level is unstable even when the current is reduced to 0.

2. Strachan, D.R., et al., Clean electromigrated nanogaps imaged by transmission electron microscopy. Nano Letters, 2006. 6(3): p. 441-444.

3. Van der Zant, H.S.J., et al., Room-temperature stability of Pt nanogaps formed by self-breaking. Applied Physics Letters, 2009. 94(12).

• Gold (Au) nanobridges with diameters – ~5G0 can be stable on the order of

days.– ~20G0 can be stable for months. [2]

• In Platinum (Pt) the activation energy is higher so self breaking at room temperature is uncommon. [3]

• A tunnelling regime is entered once G falls below G0 accompanied by formation of a nanogap.

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Pt Au

Variation in Traces

• Data from John MacHale Tyndall.• 40 Gold traces, 91 Platinum. ~15000 data points per trace.

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• Can we give an idea of the expected degree of variability?

• Can we isolate contributions from individual conductance channels?

• Are there preferred conductance levels?

• Which levels are more stable?

• Are there favourable transitions?

• Differences between Au and Pt.

• Is there a “Typical Behaviour”?

• If so, what is it and can we describe the outliers?

What we want to know?

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Single levels

• Single stable plateau

• Usually single Gaussian histogram – normal distribution.

• Little or no fine structure in histogram

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Step Traces

• Conductance Plateuas

• Staircase like drops in conductance ~G0

• Structure in the histogram

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Multi-Level Systems

• Multi-level systems can be viewed as a double potential with an energy difference Δ between the two (or more) configurations. [4]

• A group of atoms can have a transmission between these two states either by tunnelling or at higher temperatures thermal excitation over the barrier .

A two-level system as a double well potential with an energy difference between the two positions, and a tunnelling probability T for crossing the barrier between the two metastable states. W and d denote the width of the barrier and the distance between the minima, respectively.

[4] Halbritter, A., L. Borda, and A. Zawadowski, Slow two-level systems in point contacts. Advances in Physics, 2004. 53(8): p. 939-1010.

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n-Level systems

• Both “slow and fast” n-level systems• Slow = transition rate between the two states can be of the order

of seconds or longer. Tunnelling case. • Fast = oscillations between metastable states at a rate faster or

equal to the measuring rate

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Previous Research

• Mechanically controllable break junctions (MCBJ)• Slowly stretch the wire and measure conductance

throughout.• Mostly low temperature ~4K experiments.• Frozen atomic configurations.

Halbritter, A., S. Csonka, et al. (2002). "Connective neck evolution and conductance steps in hot point contacts." Physical Review B 65(4).

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Diffusion argument

• Based on the MCBJ data it would be nice to assume each atom gives a contributions of G0 to the conductance.

• However, we can see in individual trace histograms peaks at non integer multiples of Go with structure - 0.1G0

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Orbital Contributions and Shell Effects

• Below are theoretical models for contributions of given orbitals to transmission.

• Calculations done at 0K• Long chain = contributions dominated by single orbital.• Short chain = contributions from many orbitals.

Pauly, F., M. Dreher, et al. (2006). "Theoretical analysis of the conductance histograms and structural properties of Ag, Pt, and Ni nanocontacts." Physical Review B 74(23): 235106.

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• Can we give an idea of the expected degree of variability?

• Can we isolate contributions from individual conductance channels?

• Are there preferred conductance levels?

• Which levels are more stable?

• Are there favourable transitions?

• Differences between Au and Pt.

• Is there a “Typical Behaviour”?

• If so, what is it and can we describe the outliers?

What we want to know?

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Histogram Analysis

• Fit multiple Gaussians to histogram.

• Isolate position and size of a quantum conductance channel.

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• Fine structure of histograms varied hugely so finding a consistent fitting regime without over constraining the fits was non-trivial.

• Some of the traces had particularly complex structure and fitting large number of Gaussians to involved minimising large search space.

Difficulties in Histogram analysis.

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• Can we give an idea of the expected degree of variability?

• Can we isolate contributions from individual conductance channels?

• Are there preferred conductance levels?

• Which levels are more stable?

• Are there favourable transitions?

• Differences between Au and Pt.

• Is there a “Typical Behaviour”?

• If so, what is it and can we describe the outliers?

What we want to know?

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Gold Platinum

Histogram Conductance Levels

• Distribution of peak conductance levels.• Shows lots of structure Pt 4-5Go and in tunnelling

regime.• Some indictation of preferred values visible.

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• Can see evidence of recurring levels.

Overview of histogram Analysis

Au

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What we want to know?

• Can we give an idea of the expected degree of variability?

• Can we isolate contributions from individual conductance channels?

• Are there preferred conductance levels?

• Which levels are more stable?

• Are there favourable transitions?

• Differences between Au and Pt.

• Is there a “Typical Behaviour”?

• If so, what is it and can we describe the outliers?

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Cumulative Histogram

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• Can we give an idea of the expected degree of variability?

• Can we isolate contributions from individual conductance channels?

• Are there preferred conductance levels?

• Which levels are more stable?

• Are there favourable transitions?

• Differences between Au and Pt.

• Is there a “Typical Behaviour”?

• If so, what is it and can we describe the outliers?

What we want to know?

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Correlation Analysis

i

j

i

j

i

j

Positive correlationPlateaus at both bin i and j

Or no plateaus at i or j

i

j

Negative correlation

A plateau at i or jbut not both

Independent

Anticorrelated

Ni and Nj

Correlated

Every bin is correlated with itself, the diagonal

Ci,i = 1

Slide adapted from presentation by Prof. Halbritter, Budapest University

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GoldPlatinum

Correlation Analysis

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• Ran n-1 correlation analysis.

Is there actually correlation?

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• Can we give an idea of the expected degree of variability?

• Can we isolate contributions from individual conductance channels?

• Are there preferred conductance levels?

• Which levels are more stable?

• Are there favourable transitions?

• Differences between Au and Pt.

• Is there a “Typical Behaviour”?

• If so, what is it and can we describe the outliers?

What we want to know?

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• Pt more stable as expected.– Pt ~1% break (1/90) Au 40% (16/40) break– Pt ~40% (37/90) Au 55% (22/40) enter tunneling

regime. – Gold tends to have more peaks in a trace.

Gold Platinum

Au vs Pt

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• Can we give an idea of the expected degree of variability?

• Can we isolate contributions from individual conductance channels?

• Are there preferred conductance levels?

• Which levels are more stable?

• Are there favourable transitions?

• Differences between Au and Pt.

• Is there a “Typical Behaviour”?

• If so, what is it and can we describe the outliers?

What we want to know?

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• Predicting the behaviour of an individual trace is extremely difficult.

• Can only really give a statistical evaluation of the life time of a state.

Typical behaviour?

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Summary and Future Research

• Evolution of conductance in self-breaking nanowires is a complex statistical process.

• Diffusion model 1G0=1atom too simple. Lots of interesting sub-structure.

• Can identify indications of preferred levels and transitions.

• Further Research• Apply some of this analysis to pre-break data.• Correlation beyond just conductance-conductance• Physical model to explain “magic numbers” – potentially

orbital contributions

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Questions?

Questions?

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• Enables to distinguish “fast” and “slow” n-level states.

• Both would show similar histograms.

• Fast will have multi “level” differential histograms.

• Slow will have close to Lorentzian differential histograms

Differential Histogram Analysis

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Conductance Level Jumps

GoldPlatinum

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Outliers