supercurrent through grain boundaries in the presence of...

Supercurrent through grain boundaries in the presence

of strong correlations

Kolloquium zur gleichnamigen Masterarbeit

F. Alexander Wolf

Universitat Augsburg

12 July 2011

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Contents

1 Introduction

2 Gutzwiller approximation for strongly inhomogeneous systems

3 Results for the current through grain boundaries

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Experimental fact

Exponential reduction of critical supercurrent jc w.r.t. increasing grain

boundary misalignment angle

see e.g. review by Hilgenkamp and Mannhart, RMP (2002)

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Experimental fact

Exponential reduction of critical supercurrent jc w.r.t. increasing grain

boundary misalignment angle

see e.g. review by Hilgenkamp and Mannhart, RMP (2002)

Practical motivation

Largest application of conventional superconductors in the form ofsuperconducting wires for magnets (e.g. in magnetic resonance imaging machines)

High-temperature superconductors not usable for this purpose due to exponentialreduction of current at GBs

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Theoretical motivation

Microscopic modeling of current suppression already byGraser, Hirschfeld, Kopp, Gutser, Andersen, and Mannhart, Nat. Phys. (2010)

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Theoretical motivation

Microscopic modeling of current suppression already byGraser, Hirschfeld, Kopp, Gutser, Andersen, and Mannhart, Nat. Phys. (2010)

Reason for suppresion

charge fluctuations (not e.g. suppression of tunneling amplitude)

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Theoretical motivation

Microscopic modeling of current suppression already byGraser, Hirschfeld, Kopp, Gutser, Andersen, and Mannhart, Nat. Phys. (2010)

Reason for suppresion

charge fluctuations (not e.g. suppression of tunneling amplitude)

But still: Quantitatively wrong predictions (current one order of magnitude to large)Speculation: strong correlations responsible?

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Theoretical motivation

Microscopic modeling of current suppression already byGraser, Hirschfeld, Kopp, Gutser, Andersen, and Mannhart, Nat. Phys. (2010)

Reason for suppresion

charge fluctuations (not e.g. suppression of tunneling amplitude)

But still: Quantitatively wrong predictions (current one order of magnitude to large)Speculation: strong correlations responsible?

Main question of this thesis

Is it possible to model the suppression with a simple approach to strong correlations, theGutzwiller approximation? If yes, what are the results?

What could happen?

Qualitatively: Decay still exponential or stronger?

Quantitatively: Reduction in which order of magnitude (augmentation not likely)?

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Overview of model systemGraser, Hirschfeld, Kopp, Gutser, Andersen, and Mannhart, Nat. Phys. (2010)

Current transport pattern

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Overview of model systemGraser, Hirschfeld, Kopp, Gutser, Andersen, and Mannhart, Nat. Phys. (2010)

Results for the angle dependence of the critical current

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Contents

1 Introduction

2 Gutzwiller approximation for strongly inhomogeneous systems

3 Results for the current through grain boundaries

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RVB groundstate for cuprate superconductors

HHubbard = −∑

〈ij〉s

tij(c†iscjs + h.c.) + U

∑

i

(ni↑ −12)(ni↓ −

12)

Proposition for superconducting groundstate of cuprates Anderson, Science (1987)

|ψ〉 ≡ |RVB〉 ≡ P|ψ〉0 where |ψ0〉 ≡ |BCS〉 ≡∏

k

(uk + vkc†k↑c

†k↓)|vac〉

P ≡∏

i

(1− ni↑ni↓)

Focus on the t-J-model in this thesis

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RVB groundstate for cuprate superconductors

HHubbard = −∑

〈ij〉s

tij(c†iscjs + h.c.) + U

∑

i

(ni↑ −12)(ni↓ −

12)

Proposition for superconducting groundstate of cuprates Anderson, Science (1987)

|ψ〉 ≡ |RVB〉 ≡ P|ψ〉0 where |ψ0〉 ≡ |BCS〉 ≡∏

k

(uk + vkc†k↑c

†k↓)|vac〉

P ≡∏

i

(1− ni↑ni↓)

Focus on the t-J-model in this thesis

H = −∑

(ij)s

tij(c†iscjs + h.c.) + J

∑

〈ij〉

Si · Sj

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Gutzwiller approximation for homogeneous systems

Evaluation of |ψ〉 ≡ |RVB〉 ≡ P|ψ〉0 using the Gutzwiller approximation (i.e. theassumption of complete statistical independence of site populations)

Gutzwiller, Phys. Rev. (1965)

Idea of Zhang, Gros, Rice, and Shiba, Supercond. Sci. Technol. (1988)

Employ Gutzwiller approximation for the RVB state in the derivation of an effectiveone-particle hamiltonian for the t-J-model

For that use expressions obtained in the thermodynamic limit

〈c†iscjs〉Gutzw.≃ g

t(n)〈c†iscjs〉0 , gt(n) ≡

2(1− n)

2− n

〈Si · Sj 〉Gutzw.≃ g

J (n)〈Si · Sj 〉0 , gJ(n) ≡

4

(2− n)2

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Gutzwiller approximation for inhomogeneous systems

Wang, Wang, Chen, and Zhang, Phys. Rev. B (2006) extended formalism to inhomogeneous systems

P ≡∏

i

Pi where Pi ≡ ynii (1− Di) where Di ≡ ni↑ni↓

This projection operator leads to the renormalization

gtij ≡ g

ti g

tj where g

ti ≡

√2(1− ni)

(2− ni)

gJij ≡ g

Ji g

Jj where g

Ji ≡

2

2− ni

But: What to use for the description of electron doped systems?

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Gutzwiller approx. for strongly inhom. systems

Wolf, Graser, Loder, and Kopp, arXiv:1106.5759 (2011)

gti ≡

√2|1− ni |

|1− ni |+ 1

gJi ≡

2

|1− ni |+ 1

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Gutzwiller approx. for strongly inhom. systems

Wolf, Graser, Loder, and Kopp, arXiv:1106.5759 (2011)

gti ≡

√2|1− ni |

|1− ni |+ 1≡

√2(1−ni )(2−ni )

if ni ≤ 1√2(ni−1)

niif ni > 1

gJi ≡

2

|1− ni |+ 1≡

{2

2−niif ni ≤ 1

2ni

if ni > 1

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Gutzwiller approx. for strongly inhom. systems

Wolf, Graser, Loder, and Kopp, arXiv:1106.5759 (2011)

gti ≡

√2|1− ni |

|1− ni |+ 1≡

√2(1−ni )(2−ni )

if ni ≤ 1√2(ni−1)

niif ni > 1

gJi ≡

2

|1− ni |+ 1≡

{2

2−niif ni ≤ 1

2ni

if ni > 1

This corresponds to the projection operator:

P ≡∏

i

Pi where Pi ≡

{Pd

i ≡ ynii (1− Di) if ni ≤ 1

Phi ≡ y

nii (1− Ei ) if ni > 1

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Application to GB

HGB =−∑

ijs

(g tij tij + χ

∗ij )c

†iscjs

−∑

ij

(∆ijc†j↑c

†i↓ + h.c.)−

∑

i

µi ni

with ∆ij ≡ ( 34gJij +

14)Jij∆ij ,

χij ≡ ( 34gJij −

14)Jij χij ,

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Application to GB

HGB =−∑

ijs

(g tij tij + χ

∗ij )c

†iscjs

−∑

ij

(∆ijc†j↑c

†i↓ + h.c.)−

∑

i

µi ni

with ∆ij ≡ ( 34gJij +

14)Jij∆ij ,

χij ≡ ( 34gJij −

14)Jij χij ,

where ∆ij ≡12(〈ci↓cj↑〉+ 〈cj↓ci↑〉), χij ≡

12(〈c†i↑cj↑〉+ 〈c†i↓cj↓〉), µi ≡ µ− εi , and

gti ≡

√2|1− ni |

|1− ni |+ 1and g

Ji ≡

2

|1− ni |+ 1

Solved self-consistently using the Bogoliubov - de Gennes formalism.

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Contents

1 Introduction

2 Gutzwiller approximation for strongly inhomogeneous systems

3 Results for the current through grain boundaries

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System parametersWolf, Graser, Loder, and Kopp, arXiv:1106.5759 (2011)

00.10.20.3

|∆ij|

(i) without correl.(ii) with correlations

0.51

1.52

n i

-30

-15

0

ε i

-50 -40 -30 -20 -10 0 10 20 30 40 50x [Å]

00.5

1

t ij

0

10

20

y [Å

]

t

t’

(a)

(b)

(c)

(d)

(e)

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Supercurrent through grain boundariesWolf, Graser, Loder, and Kopp, arXiv:1106.5759 (2011)

10-7

10-6

10-5

10-4

10-3

j c

(i) without correl.(ii) with correl.(iii) experiment

15 20 25 30 35 40 45GB angle [°]

10-7

10-6

10-5

10-4

10-3

j c

10-7

10-6

10-5

10-4

10-3

j c

(a)

(b)

(c)

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Density distribution

hole-doped

-50 -40 -30 -20 -10 0 10 20 30 4050x [Å]

0,5

1

1,5

2

n

(510)

electron-doped

-50 -40 -30 -20 -10 0 10 20 30 4050x [Å]

0,5

1

1,5

2

n

(510)

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Mechanism for current transportWolf, Graser, Loder, and Kopp, arXiv:1106.5759 (2011)

10-6

10-5

10-4

10-3

10-2

10-1

j ij(i) without correl.(ii) with correl.

10-9

10-8

10-7

10-6

10-5

10-4

10-3

-1-0,5

0

g ijt t ij

-0,200,2

0 2 4 6 8 10 12 1416 18channel number

-12-8-40

ε i

0 2 4 6 8-30-20-100

(a) (b)

(c) (d)

(e)(f)

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Summary

First Part – Theoretical methods

Review of Gutzwiller approximation as employed within the BdG formalism

Presentation of a particle-hole symmetric form of Gutzwiller factors

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Summary

First Part – Theoretical methods

Review of Gutzwiller approximation as employed within the BdG formalism

Presentation of a particle-hole symmetric form of Gutzwiller factors

Second Part – Application to modeling of the GB problem

Reduction of critical current of one order of magnitude as compared to standardHartree-Fock calculation

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Summary

First Part – Theoretical methods

Review of Gutzwiller approximation as employed within the BdG formalism

Presentation of a particle-hole symmetric form of Gutzwiller factors

Second Part – Application to modeling of the GB problem

Reduction of critical current of one order of magnitude as compared to standardHartree-Fock calculation

Quantitative agreement of critical current with experiment

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Summary

First Part – Theoretical methods

Review of Gutzwiller approximation as employed within the BdG formalism

Presentation of a particle-hole symmetric form of Gutzwiller factors

Second Part – Application to modeling of the GB problem

Reduction of critical current of one order of magnitude as compared to standardHartree-Fock calculation

Quantitative agreement of critical current with experiment

Current transport mechanism: Mott insulator driven reduction in areas in theneighborhood of the GB – regions around half filling

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Summary

First Part – Theoretical methods

Review of Gutzwiller approximation as employed within the BdG formalism

Presentation of a particle-hole symmetric form of Gutzwiller factors

Second Part – Application to modeling of the GB problem

Reduction of critical current of one order of magnitude as compared to standardHartree-Fock calculation

Quantitative agreement of critical current with experiment

Current transport mechanism: Mott insulator driven reduction in areas in theneighborhood of the GB – regions around half filling

Conclusion: enhance current carrying properties by further hole-doping

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Summary

First Part – Theoretical methods

Review of Gutzwiller approximation as employed within the BdG formalism

Presentation of a particle-hole symmetric form of Gutzwiller factors

Second Part – Application to modeling of the GB problem

Reduction of critical current of one order of magnitude as compared to standardHartree-Fock calculation

Quantitative agreement of critical current with experiment

Current transport mechanism: Mott insulator driven reduction in areas in theneighborhood of the GB – regions around half filling

Conclusion: enhance current carrying properties by further hole-doping

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Summary

First Part – Theoretical methods

Review of Gutzwiller approximation as employed within the BdG formalism

Presentation of a particle-hole symmetric form of Gutzwiller factors

Second Part – Application to modeling of the GB problem

Reduction of critical current of one order of magnitude as compared to standardHartree-Fock calculation

Quantitative agreement of critical current with experiment

Current transport mechanism: Mott insulator driven reduction in areas in theneighborhood of the GB – regions around half filling

Conclusion: enhance current carrying properties by further hole-doping

Thank you for your attention!

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