Changes of Resistivity and Hall

Coefficient in Copper-Silica Compounds:

A New Model

M.Sc. Thesis by Shaked Zychlinski

Under the supervision of Prof. Alexander Gerber

Tel Aviv University

February 2016

Introduction

Background: Percolation Theory

Let’s assume we replace the parts of the metal with parts of an insulator:

At some point, the substance will no longer be conductive.

That point is known as the percolation threshold, or the critical concentration: 𝑥𝑐.

metal

insulator

Background: The Hall Effect

The Hall Effect occurs when an out-of-layer magnetic field is applied to a current carrying material.

Perpendicular resistance:

𝑉 =𝐵

𝑛𝑒𝑡𝐼 ⇒ 𝑅𝐻𝑎𝑙𝑙 =

𝐵

𝑛𝑒𝑡

Hall Coefficient : 𝑹𝟎 =𝟏

𝒏𝒆

The Giant Hall Effect

In 1995, Pakhomov, Yan and Zhao, found that around percolation threshold of metal-insulator composites, Hall coefficient was up to 103 − 104 larger than that of the bulk-metal value of the metal alone.

When using percolation theory to explain this, changes are up to a 100 times larger than what theory expect.

Taken from Zhang et al., “Giant Hall Effect in Nonmagnetic Granular Metal Films”, Phys. Rev. Let., 2001

The Giant Hall Effect

Several attempts to explain this phenomenon were made, none succeeded to find a complete model explaining it.

We have noticed a repeating concept in all attempts: dependency of inner-structure.

This has initiated this study, set to determine if this could really be the cause.

Brief Theory

Theory of Juretschke, Landauer and Swanson

One of the earliest attempts of discussing the changes of Hall coefficient in non-pure (1956).

They have shown the following changes of the conductivity and Hall coefficient:

𝝈 = 𝝈𝟎 ⋅𝟏−𝝐

𝟏+𝝐

𝟐

; 𝑹 = 𝑹𝟎 ⋅𝟏−

𝝐

𝟒

𝟏−𝝐

𝜖 = 1 − 𝑥 is the fraction of insulator

Later theorems, such as the Effective Medium Theory and Kirkpartick’s approach also yield similar behavior to JLS.

Percolative Hall Theory

Bergman and Stroud found for metal-insulator compound:

𝑹𝟎 ∝ 𝒙 − 𝒙𝒄−𝒈

Based on this finding, researchers Zhang et al. and Liu et al. determined that the Hall coefficient should only increase by one order of magnitude.

Rescission of Power-Law Scaling

Li and Östling discussed the changes of behavior close and far from percolation threshold:

𝑹 ∝ 𝑹𝟎 𝒙 − 𝒙𝒄

−𝒈 ; 𝒙 → 𝒙𝒄+

𝑹 ∝ 𝑹𝟎 𝒙 − 𝒙𝟎−𝒔 ; 𝒙 ≫ 𝒙𝟎

They distinguish 𝑔 from 𝑠, defining two different behaviors, which only look alike.

So what really happens inside?

Our New Model

𝑹𝟎(𝒙)

• Measurements have shown: 𝑅0~ 𝑥 − 𝑥𝑐−1

• From which we understand: 𝑛~𝑥−𝑥𝑐

1−𝑥𝑐

Meaning: clusters of insulator in the metal reduce the charge carriers’ density.

𝑳 𝑺

Building the Model

• The fraction of insulator: 𝜖 = 1 − 𝑥 = 𝑁𝑆3

• The insulator cross-section: 𝜍 = 𝑁1

3𝑆2

• From here we find: 𝑙 ∝ 1 − 𝑥 −1

3 ⋅ 𝑆−1

• Drude’s Model: 𝜎 ∝ 𝑛𝑙

We therefore have:

𝜎𝑖 𝑥 ∝𝑥 − 𝑥𝑐

𝑆 ⋅ 1 − 𝑥13 ⋅ 1 − 𝑥𝑐

Building the Model

• The overall resistivity is that of the accumulative-insulator and of the pure-metal: 𝜌 = 𝜌𝑖 𝑥 + 𝜌𝑚

Therefore:

𝜌 𝑥 = 𝐴 ⋅𝑆 ⋅ 1 − 𝑥

13 ⋅ 1 − 𝑥𝑐

𝑥 − 𝑥𝑐+ 𝜌𝑚

𝐴 is a proportion-constant

The Clusters’ Size

What is the size of the insulator cluster 𝑆?

Two possible approaches:

• Far from percolation threshold: 𝑆 ≈ 𝑐𝑜𝑛𝑠𝑡

⇒ 𝜌 𝑥 = 𝐴 ⋅1 − 𝑥

13 ⋅ 1 − 𝑥𝑐

𝑥 − 𝑥𝑐+ 𝜌𝑚

• Close to percolation threshold: 𝑆 ~ 𝜉 ∝ 𝑥 − 𝑥𝑐−0.9

⇒ 𝜌 𝑥 = 𝐴 ⋅1 − 𝑥

13 ⋅ 1 − 𝑥𝑐

𝑥 − 𝑥𝑐1.9

+ 𝜌𝑚

Setting Up

Samples

Compounds of Copper and Silica were made, using E-Beam and Sputtering:

• Group I: Main group - 100nm E-Beam

• Group II: 100nm Sputtering

• Group III: 10nm E-Beam

Group I & II – Hall Bar Pattern

Group III – No Pattern

Why Using E-Beam?

Using E-Beam co-evaporation created a gradient in each series.

co-evaporation, varying gradient of metal/insulator

𝑪𝒖 𝑺𝒊𝑶𝟐

Decreasing 𝑪𝒖 gradient

Sample Examination

• TEM Examination shows decreasing copper-cluster size.

• The copper grain-size remains pretty much constant (~4nm)

69% 74% 80%

Results

Hall Resistance

Hall coefficient increase as the metallic fraction decrease

𝑹𝟎(𝒙)

We found 𝑅0 𝑥 ~ 𝑥 − 𝑥𝑐−1 for 𝑥 > 50%

𝒏(𝒙)

𝑛 𝑥 = 𝑛0

𝑥 − 𝑥𝑐

1 − 𝑥𝑐+ 𝑁𝑐

Model Adjustments

• We would have expected 𝑛 𝑛_100 = 12 when 𝑥 = 50%

• But, as seen in the TEM photos, the silica encapsulate parts of the copper fraction. Thus the sharper decrease.

𝑳 𝑺

𝝆 𝒙

𝑺 ∝ 𝒙 − 𝒙𝒄−𝟎.𝟗

𝑺 ≈ 𝒄𝒐𝒏𝒔𝒕

Effective Medium Theory

Comparison

Comparison to Previous Studies

𝑥𝑐 = 0.45

𝑥𝑐 = 0.43

Taken from Zhang et al., “Giant Hall Effect in Nonmagnetic Granular Metal Films”, Phys. Rev. Let., 2001

Comparison to Percolation Theory

𝒙 − 𝒙𝒄−𝟎.𝟒

𝒙 − 𝒙𝒄−𝟏

An Enigma

Measuring Very Low Concentrations

Group II sample with 𝑥 = 39%

• A positive magnetoresistance is clearly seen.

• A change of Hall coefficient sign is witnessed as well.

• A shift of the Hall resistance measurement is also seen.

Conclusions

Conclusions

• We have found that the charge carriers’ density 𝑛 varies linearly with the percentage of metal 𝑥.

• We have found a simple model describing the resistivity 𝝆 of an impure metal, which depends on 𝒙.

• Our model was confronted with several examinations (temperature change, annealing, oxidation) and found valid in all cases.

• Some changes exist when our data is compared to other studies.

• For 𝑥 ≫ 𝑥𝑐, our model seems to be more accurate than percolation theory.

• For 𝑥 ≪ 100%, our model seems to be more accurate than previous theorems, such as the Effective Medium Theory.

• Unexplained phenomena was witnessed for very low concnetrations.

Thank-you’s & Gratitude

Many thanks to

Professor Alexander Gerber

And many more to all the former and present crew at the High Magnetic Fields Laboratory:

Dr. Amir Segal

Dr. Gregory Kopnov

Dr. Alexander Gladkikh

Dr. Itay Kishon

Aviad Panhi

A great gratitude to

Dr. Zehava Barkay & Misha Karpovsky

Special thanks to

Professor Edwin Herbert Hall

For all his great work in this field

Presentation Bibliography

1. A. B. Pakhomov, X. Yan and B. Zhao, "Giant Hall effect in percolating ferromagnetic granular metal-insulator films," Applied Physics Letters, vol. 67, no. 23, pp. 3497-3499, 1995

2. X. X. Zhang, C. Wan, H. Liu, Z. Q. Li, P. Sheng and J. J. Lin, "Giant Hall Effect in Nonmagnetic Granular Metal Films," Physical Review Letters, vol. 86, no. 24, pp. 5562-5565, 2001

3. H. J. Juretschke, R. Landauer and J. A. Swanson, "Hall Effect and Conductivity in Porous Media," Journal of Applied Physics, vol. 27, no. 7, p. 838, 1956

4. H. Liu, R. K. Zheng, G. H. Wen and X. X. Zhang, "Giant Hall Effect in Metal/Insulator Composite Films," Vacuum, vol. 73, no. 3-4, pp. 603-610, 2004

5. D. J. Bergman and D. Stroud, "Scaling theory of the low-field Hall effect near the percolation threshold," Physical Review B, vol. 32, no. 9, pp. 9-11, 1985

6. J. Li and M. Östling, "Conductivity scaling in supercritical percolation of nanoparticles – not a power law," Nanoscale, vol. 7, no. 8, pp. 3424-3428, 2015