Enhancement of d-wave pairing in the extended 2D t-J model

Y. C. Chen

Tunghai University, Taiwan

C. T. Shih, Tunghai University, Taiwan

T. K. Lee, Inst. of Physics, Academia Sinica, Taiwan

R. Eder, Forschungszentrum, Karlsruhe, Germany

C. Y. Mou, National Tsinghua University, Taiwan

cond-mat/0401307

1913 Onnes

1972 BCS

1973 Josephson

1987 Bednortz and Muller

1991 P. G. de Gennes

2003 Abrikosov, Ginzburg and Leggett

                                        

       

        

       

        

       

歷史上與超導體有關的諾貝爾獎

1911 Onnes 發現汞在 4K 的溫度下失去電阻

1933 年荷蘭的邁斯納 (Meissner) 和奧森菲爾德 (Ochsenfeld) 共同發現了超導體的一個極為重要的性質。 ( 簡稱 Meissner effect)

將超導體放入磁場中，會將其內部的磁場完全排除，其內部磁通量（ magnetic flux) 保持為零。因此，若將一超導體放在一個普通的磁體上方，則會因排斥作用而懸浮在空中。

超導體有三個臨界值，

Tc: 臨界溫度

Hc : 臨界溫度

Jc : 臨界電流密度

這三個值越大時，其應用範圍越廣，不同材料有不同的值，以前低溫超導體，雖然 Jc 已能達到高度應用價值，但因 T

c 遠低於液態氮的溫度，為著要先把溫度降低，往往需要花費很大的成本，所以，它們的應用，一直難以推廣 .

第二類超導體

Cooper Pair 電荷 charge

自旋 spin ½ 角動量與磁矩費米子 (Fermion)包利不相容原則 (Pauli Exclusion principle)Fermi Surface

動量 k 與 -k

電子與晶格振動相互作用成為電子對之間的媒介

兩個費米子配對而成等效的玻色子 , 合作型的玻色子可在費米面上凝聚形成超導

比熱　　　

相變

實驗上關於 Cooper Pair 的證據

磁通量量子化 (magnetic flux ) 穿隧效應 (tunneling effect)

地球科學

地殼的厚度

磁極強度的變化

預測地震

礦產

生物磁學

遷徙

演化

奈米科技

Scanning SQUID Microscopy

生命科學與醫學

國防 -- 偵測潛水艇

400.8 km/h in 1987

Carbody dimension (m)Length x width x height

Weight (t) Number builtSeating capacity

22.0 x 3.0 x 3.7 19.0 1 12

Superconducting Coil(SC coil)Magnetomotive force (kA)x poles x rows

Max. speed(km/h)

Site of run Year completed

700 x 6 x 2unmanned:

431manned: 411

Miyazaki Maglev Test Track

1993

On December 2, 2003, this three-car train set attained a maximum speed of 581 km/h in a manned vehicle run.

MLX01

Yamanashi Maglev Test Line

超導發電機

Electric generators made with superconducting wire are far more efficient than conventional generators wound with copper wire. In fact, their efficiency is above 99% and their size about half that of conventional generators. These facts make them very lucrative ventures for power utilities.

Hypres Superconducting Microchip, incorporating 6000 Josephson Junctions.

超導馬達

5000 horsepower motor

這艘船是世界最早的超導體電磁推進船。她在 1985 年由日本造船振興財團及執日本造船牛耳的三菱重工神戶造船所共同設計製造，為多年研究之成果。全長 26 公尺，寬 10 公尺，重 250 噸。這艘流線型船之造價為 50 億日圓。這艘船的兩邊各有一個推進器，能產生強大的磁場，電流則以直角對著磁場發送電磁力，從船尾排出海水，藉以推進船身。此船推進器很強，不振動且不製造噪音。

The most ignominious military use of superconductors may come with the deployment of "E-bombs". These are devices that make use of strong, superconductor-derived magnetic fields to create a fast, high-intensity electro-magnetic pulse (EMP) to disable an enemy's electronic equipment. Such a device saw its first use in wartime in March 2003 when US Forces attacked an Iraqi broadcast facility.

High-Temperature

Superconducting Cuprates

ij

ji SSJH

nnn

kinn

ji SSJSSJH

'

Heisenberg model Frustrated Heisenberg model

jiij

jiij

ji nnSSJchCCtH4

1.).

~~( ,,

t-J model

)1(~

, iii nCCconstraint

nnnji chCCt .).

~~(' ,,

rdji chCCt

3,, .).

~~("

i

iiij

ji nnUchCCtH .).( ,,

Hubbard model t=0.43 eVU=5.4=12tt’=-0.07~ -0.2t

J=0.128 ~0.3t

k

kkkk 0|)~~(| ,, ccvuPRVB d

)( 22~

~

kkk

k

v

u

k

k

)cos(cos yxk kk

)cos(cos2 yxk kkt

D-wave

Projected BCS

S-wave

jiij

jiij

ji nnSSJchCCtH4

1.).

~~( ,,

Quantum Critical Phenomenon

Comments on RVB theory from Varma

Green: LoramBlue: ARPESRed: STM

8x8 J/t=0.4

0.0 0.1 0.2 0.3 0.4

Pa

v

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Pairing correlation determined by the lowest variational energy

x0.0 0.1 0.2 0.3 0.4 0.5

Pav

0.000

0.005

0.010

0.015

0.020

0.025

0.030

L0L1L2stable

8x8

Pairing is reduced from the variational result

2D t-J, J=0.4 Power_Lanczos method

“GS”

E. Pavarini, O.K. Andersen and co-workers, PRL 87, 047003 (2001)

-

Tc max correlates strongly with the value of t’/t ( t’’/t’=-0.5)

A13.02 Wei Ku, et al., “Correlation Between Tc and t’/t: Perspective from First-Principle Wannier Functions”

jiij

jiij

ji nnSSJchCCtH4

1.).

~~( ,,

t-t’-t”-J model

)1(~

, iii nCCconstraint

nnnji chCCt .).

~~(' ,,

rdji chCCt

3,, .).

~~("

For cuprates, J=0.3t, t’=-0.3t, t”=-t’/2

Previous calculations:

White and Scalapino (DMRG) It is stripe instead of SC in the pure t-J model t’ suppresses the pairing correlation

Tohyama et al., (ED, DMRG, PL) Similar conclusion as WS

Martin et al., ED of 2 holes in 20-site cluster. t’ suppresses hole binding. A13.09 For pure t-J model Pryadko, Kivelson and Zachar (High-Temperature Series expansion) Pairing susceptibilities are small and decreasing with decreasing temperature.

t’ strongly enhances pairing in the overdoped region.

Variational Monte Carlo calculation forthe extended t-J model, RVB state

0)(P ,,d'

k

kkkktCCvuRVB

four variational parameters, tv’ tv’’ ∆, and μ

Pairing of 2D extended t-J on 12x12 lattice

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Pd

ave

0.00

0.01

0.02

0.03

0.04

0.05 t'=t"=0t'=-0.1, t"=0.05t'=-0.3, t"=0.15

Pairing of 2D extended t-J on 12x12 lattice

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Pd

ave

0.00

0.01

0.02

0.03

0.04

0.05 t'=t"=0t'=-0.1, t"=0.05t'=-0.3, t"=0.15t'=-0.5, t"=0.25t'=-0.7, t"=0.35

Pairing of 2D extended t-J on 12x12 lattice

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Pd

ave

0.00

0.01

0.02

0.03

0.04

0.05 t'=t"=0t'=-0.3, t"=0.15t'=-0.3, t"=0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Pd

ave

0.00

0.01

0.02

0.03

0.04

0.05

t'=t"=0t'=-0.1, t"=0.05t'=-0.3, t"=0.15t'=-0.5, t"=0.25t'=-0.7, t"=0.35t'=-0.3, t"=0

At underdoping, t’ suppresses pairing slightly, agree with White and Scalapino, Martins et al.

But for overdoping, t’ enhances pairing strongly.

t’

Exact diagonalization of 20-site cluster

circle: t’=t”=0square: t’=-0.3, t”=0diamond: t’=-0.3, t”=0.15

Using slave-boson mean-field theory, we obtain similar enhancement.

t’’ expands superconducting region

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.1 0.2 0.3 0.4 0.5 0.6

t' v

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

J=0.3t’=-0.3t”=0

12x12

Variational parameters

Pairing is not driven by J.

Fermi surface topology is important for pairing in the overdoped region.

In addition to the effect of van Hove singularity,d-wave order parameter, ∆k , is largest near (π, 0) or (0,π) . Hence occupation of electrons in these regions contributes significantly to pairing.

132/144, δ=0.08, t’=-0.3

δ=0.08, t’=0

δ=0.31, t’=0

100/144,δ=0.31, t’=-0.3

n(k)

Summary and Conclusions

• For the hole-doped system, t’/t(<0) strongly enhances pairing in the overdoped region. In addition to the effect due to van Hove density of states, the occupation of (π,0) region by electrons is important for d-wave pairing.

• t’ only slightly suppresses pairing in the underdoped region where Fermi surface changes very little.

• Underdoped and overdoped regions dominated by different physics.

• Strong correlation of maximum pairing amplitude with (–t’/t) – consistent with the result of band-structure calculations and experimental analysis that Tc max correlates with (–t’/t) .

Thanks for your attention.

Pairing of 2D extended t-J on 12x12 lattice (J/t=0.3)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Pd

ave

0.00

0.01

0.02

0.03

0.04

0.05

t'=t"=0t'=-0.1, t"=0.05t'=-0.3, t"=0.15

At underdoping, t’ suppresses pairing slightly, agree with White and Scalapino, Martins et al.But for overdoping, t’ enhances pairing strongly.

100/144, δ= 0.31 t’’=-t’/2

Enhancement of d-wave Superconductivity by t’

in the Two-Dimensional Extended t-J Model

Y. C. ChenTunghai Univ., Taiwan

Collaborators:

T. K. Lee (Academia Sinica/NCTS)

C. T. Shih (Tunghai)

C. Y. Mou (NTHU)

R. Eder (Forschungszentrum Karlsruhe)

cond-mat/0401307

Conclusions:

1.

Using slave-boson mean-field theory, we obtain similar enhancement.

t’’ expands superconducting region

"Enhancement of d-wave Superconductivity > by $t'$ in the Two-Dimensional Extended $t-J$ Model,">> has been scheduled for session N13, (HTS Theory) which will begin at > 08:00 on Wednesday morning in 511B.