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Prerequisite for superconductivity: Appropriate spin-charge correlations Tian De Cao ∗ Department of physics, Nanjing University of Information Science & Technology, Nanjing 210044, China

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a b s t r a c t

Article history: Received 30 January 2008 Received in revised form 22 April 2008 Accepted 28 April 2008 by C. Lacroix Available online 1 May 2008

This work shows that both superconductivity and antiferromagnetism can be included in the Hubbard model. It is shown that the appropriate spin-charge correlation is the key role to any superconductivity. Some unusual properties of cuprate superconductors are discussed with regard to this correlation. © 2008 Elsevier Ltd. All rights reserved.

PACS: 74.25.Jb 74.72.-h 74.62.Yb Keywords: D. Spin correlation D. Charge correlation D. Superconductivity

1. Introduction Since mother compounds of the cuprate superconductors are typical Mott insulators with antiferromagnetic order, the relation between magnetism and superconductivity has been noted in many works. Schrieffer et al. give their spin bag model in 1988 [1]. Other theories concerning spins include spin polaronic theories [2–4] and antiferromagnetic spin fluctuation based mechanisms [5–8]. Although the coexistence between magnetism and superconductivity have been investigated by other works [9,10], how spins contribute to superconductivity has been an open and important problem. In this work, by introducing spin operator and charge operator into the Hubbard model to calculate some correlation functions, we can find the roles of different correlations. It is shown that the appropriate spin-charge correlation is the key factor of the superconductivity; this explains why the materials appearing with the larger magnetic susceptibility |χm | usually show superconductivity, and why the high-temperature superconductivities occur at the region where the short-range antiferromagnetic order [11] exists in the cuprate superconductors. 2. Calculation To consider the roles of strong correlations, we discuss the Hamiltonian X X X ˆ = H Tll0 cl+σ cl0 σ + U cl+↑ cl↑ cl+↓ cl↓ + Vll0 cl+σ clσ cl+0 σ0 cl0 σ0 , (1) l,l0 ,σ

l

l,l0 ,σ,σ 0

El ) of spin σ in the where cl+σ creates as electron at site l(≡ R model, tll0 is the intersite hopping matrix element, U the effective on-site Coulomb interaction, and Vll0 the long-range interaction, particularly, the nearest-neighbor interaction. We will find that U and Vll0 dominate short- and long-range correlations, respectively. To find the effects of spins and charges, we reduce the model in the form X X X ˆ = H Tll0 cl+σ cl0 σ + U (ρˆ 2l − sˆ2l ) + 4 ρˆ l Vll0 ρˆ l0 , (2) l,l0 ,σ

using

ρˆ l =

1X 2

σ

E-mail address: [email protected] 0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.04.033

cl+σ clσ ,

sˆl =

1X 2

σ

σcl+σ clσ .

(3)

The spin operator is sˆl ≡ sˆlz , σ = ±1 represent spin up and spin down, and the relation between charge operator ρˆ l and number operator nˆ l is ρˆ l = nˆ l /2. It is found that the on-site interactions include the part of spin–spin interactions and the charge–charge interactions. Hence spin character, and then some magnetism, is included in the Hubbard model. This can be understood, because the interaction cl+↑ cl↑ cl+↓ cl↓ means that the spin of an electron is in the opposite direction of another one during the scattering process, hence the spin of electrons may be reversed in the scattering process. To discuss the possible antiferromagnetism, we divide the lattice into A-sublattice and B-sublattice, and write the model (2) as X X + + 2 2

ˆ = H

lA ,lB ,σ

∗ Tel.: +86 025 58731 031.

l,l0

l

+U

TlA lB (clA σ clB σ + clB σ clA σ ) + U

(ρˆ lA − sˆlA )

lA

X lB

(ρˆ 2lB − sˆ2lB ) + 4

X lA ,lB

ρˆ lA VlA lB ρˆ lB .

(4)

T.D. Cao / Solid State Communications 147 (2008) 4–7

Following Mahan [12], these Green’s functions are defined in GAA (lA l0A σ, τ − τ 0 ) = −hTτ clA σ (τ)cl+0 σ (τ 0 )i, A

+ FAA (lA l0A σ, τ − τ 0 ) = hTτ cl+A σ (τ)cl+0 σ¯ (τ 0 )i. A

From the definition of the τ -ordered product, we derive the first derivative of the equations for the Green’s functions, such as X ∂ + 0 0 0 0

∂τ

GAA (lA lA σ, τ − τ

TlA lB hTτ clB σ (τ)cl0 σ (τ )i

) = −δ(τ − τ )δlA ,l0A +

A

lB

1

+ U [hTτ ρˆ lA clA σ cl+0 σ (τ 0 )i + hTτ clA σ ρˆ lA cl+0 σ (τ 0 )i 2

A

A

− σ hTτ Sˆ lA clA σ cl+0 σ (τ 0 )i − σ hTτ clA σ Sˆ lA cl+0 σ (τ 0 )i] A

+2

X

A

ˆ lB cl0 σ (τ )i. VlA lB hTτ clA σ ρ +

0

(5)

A

lB

our calculation we have noted Ek¯ = Ek , Vk¯ = Vk , and Pρs = Psρ . The cell number N has been taken as N = 1, otherwise we P P should take k → N1 k in calculations. In addition, to consider the chemical potential of electron systems, we should take εk → εk − µ in our discussion. If we take the transition A → B, we + will obtain the equations of functions GBB and FBB . It is found that + + ¯ iωn ) for ¯ iωn ), and FBB (kσ, iωn ) = FAA (kσ, GBB (kσ, iωn ) = GAA (kσ, SlA = −SlB . The effects of both charge–charge correlations and spin–spin correlations are described by PA , the effects of spincharge correlations by SA . In contrast with the charge–charge correlations and the spin–spin correlations, the expressions (6) and (7) show that the effects of spin-charge correlations depend on the direction of spins. The effects of zero-range correlations are determined by U the on-site interaction, while the effects of longrange correlations are affected by Vll0 the long-range interaction. Consider the case VA∆ (kσ) → 0 for T → T pair , we get

Therefore, to calculate these Green’s functions, we must derive more equations of other correlation functions, such as hTτ ρˆ l clσ cl+0 σ (τ 0 )i. Moreover, because

GAA (kσ, iωn ) =

X ∂ 1 ˆ lA clB σ − U [ρˆ lA ρˆ lA clA σ + ρˆ lA clA σ ρˆ lA [ρˆ lA clA σ ] = − TlA lB ρ ∂τ 2 lB

where (ν)

CA

− σ ρˆ lA Sˆ lA clA σ − σ ρˆ lA clA σ Sˆ lA ] −2

X

ρˆ lA clA σ VlA lB ρˆ lB −

lB

1X 2

lB σ 0

TlA lB [cl+A σ0 clB σ0 − cl+B σ0 clA σ0 ]clA σ ,

uAσ

= −1 −

iωn

−iωn − εk + =−

1 i ωn

where X

εk =

+ 1 iωn

1 i ωn

(6)

¯ iωn ), VA+∆ (kσ)GAA (k¯ σ,

Tll00 e

,

Vk =

X

l00

(7)

ik·(l00 −l)

Vll00 e

,

l00

SA =

(0)U + 4

X

VlA lB PρBAS

(ν)

(ν)

EA

(kσ) + uAσ , (kσ) − EA(−) (kσ)

1 2

q

{εk ± ε2k + 4[vl εk + PA − σ SA ]}.

Moreover, we obtain + FAA (kσ, iωn ) = VA+∆ (kσ)

VlA lB ρlB ,

gkσ =

(i )

Ekσ + uAσ¯

(i )

BA (kσ) =

4 Q

(1) Ekσ

=

(2) Ekσ

=

(3) Ekσ

=

(4) Ekσ

=

1 2 1 2 1 2 1 2

[Ek(iσ) − Ek(jσ) ]

,

q

{−εk + ε2k + 4(PA − σ SA + vl εk + gkσ )}, q

{−εk − ε2k + 4(PA − σ SA + vl εk + gkσ )}, ) {εk + ε2k + 4(PA + σ SA + vl εk )} = Ek(+ σ¯ ,

q

(9)

) {εk − ε2k + 4(PA + σ SA + vl εk )} = Ek(− σ¯ .

q

+ FAA (kσ, 0) = VA+∆ (kσ)

(lB − lA )U,

uAσ = U (ρlA − σ SlA ) + 2

(i )

4 X (i ) (i ) BA (kσ)nF (ξkσ ), i=1

X

[U ξk0 + Vk ξk0 − U ξk − Vk0 ξk ]

k0

X

Vl0B lA ρl0B ,

[U εk + Vk0 εk − U εk0 − Vk εk0 ]GAA (k0 σ, 0),

k0

X

,

where

VA+∆ (kσ) = −

l0B

VA∆ (kσ) = −

(i )

iωn − Ekσ

or

lB

X

(i ) 4 X BA (kσ) i=1

l0B ,lB

vl = 2

(8)

EA (+)

lB

X

,

(i )

lB

2UPρAAS

(kσ)

iωn − EA (kσ)

be found. Take the transition εk → εk − µ, and Ekσ → ξkσ , we get

ElA − RElB ) = hρˆ lA Sˆ lB i, PρABS (lA − lB ) ≡ PρABS (R X AA BB 0 AA PA = UPρρ (0)U + UPSS (0)U + 4 VlA l0B Pρρ (lB − lB )VlB lA BA VlA lB Pρρ (lB − lA )U,

(ν)

CA

(i )

AB AB E Pρρ (lA − lB ) ≡ Pρρ (RlA − RElB ) = hρˆ lA ρˆ lB i,

X

ν=±

ν

We can predict that an exact result should be Ekσ = 21 {±εk ± q ε2k + 4(PA ± σ SA + γkσ )}, where γkσ are the number that have to

and we introduced

+4

(kσ) =

X

j=1(6=i)

+ ¯ ¯ iωn ), VA∆ (kσ)FAA (kσ,

+ (kσ, iωn ) (PA − σSA + vl εk + gkσ ) · FAA

ik·(l00 −l)

(kσ) =

(±)

EA

to close the relations between these correlated functions, we approximate ρˆ l ρˆ l clσ = hρˆ l ρˆ l iclσ , and so forth. In the right side of the Eq. (6), lB 6= l0A ; if we consider the weak antiferromagnetic states, SlA = −SlB → 0, we can approximate GBA (ll0 σ, τ − τ 0 ) → GAA (ll0 σ, τ − τ 0 ) for l 6= l0 in the right side of Eq. (6). After exact calculations, we obtain PA − σ SA + vl εk −iωn + εk + GA (kσ, iωn ) iωn

5

[U εk0 + Vk εk0 − U εk − Vk0 εk ]FAA (k0 σ, 0).

× VA+∆ (k0 σ)

(10)

i=1

If we take Vll0 = V1 and tll0 = t1 for the nearest-neighbor approximation, we get V1 X VA+∆ (kσ) = − U + 2µ [εk0 − εk ]VA+∆ (k0 σ) t1

k0

In these expressions, we denote k ≡ Ek, sl ≡ hˆsl i, and ρl ≡ hρˆ l i. Sl and ρl represent spin and charge at each site respectively. In

4 X (i ) (i ) BA (k0 σ)nF (ξk0 σ ).

×

4 X i=1

(i )

k0

(i )

BA (k0 σ)nF (ξk0 σ ),

(11)

T.D. Cao / Solid State Communications 147 (2008) 4–7

6

where µ is the chemical potential of the electron systems. We can write It+∆ (kσ) of the identity (11) as VA+∆ (kσ) = VA+∆ (0σ, T ) + γσ (T )εk .

(12)

When VA+∆ (kσ) 6= 0, the substitution of (12) into (11) leads to X k0 ,k00

(εk0 εk00 − ε2k0 δk00 ,k0 )

=

t12

4 X (i ) (i ) (j) (j) BA (k0 σ)nF (ξk0 σ ) · BA (k00 σ)nF (ξk00 σ )

i,j=1

/(Ut1 + 2µV1 ) . 2

(13)

The critical pairing temperature is determined by the expression (13). 3. Analyses and results The antiferromagnetic states can be found in these expressions. The Green’s functions (8) give the electron number distributions with spin ↑ and ↓ in the forms (+)

(k ↑)nF [EA(+) (k ↑)] − CA(−) (k ↑)nF [EA(−) (k ↑)],

(+)

(k ↓)nF [EA(+) (k ↓)] − CA(−) (k ↓)nF [EA(−) (k ↓)].

nA↑ (k) = CA

and nA↓ (k) = CA

We get SA (k) = nA↑ (k) − nA↓ (k) > 0, for SlA > 0 and the large PA . The electrons with spin up in A-lattice have probability to occupy the states with spin down in the weak magnetic states. In the same way, take the translation A → B, we get nB↑ (k) − nB↓ (k) < 0 for SlB < 0, and then SlA = −SlB . The antiferromagnetic states are discovered in the Hubbard model. However, we find SlA 6= 0 require the spincharge correlation SA 6= 0, but SA 6= 0 do not mean SlA 6= 0, a large PA is necessary. That is to say, the spin-charge correlation does not require appearing magnetisms, in this case, the correlation means having short-range magnetic correlations. Of course, the antiferromagnetism usually increases with the possible increasing of SA . The superconducting states may be also shown when the spincharge correlation exists in the model. In addition, the states below and around Fermi surface also contribute to the pairing. If PA is very small, SA must be very small, all of the bands (i ) Ekσ are overlapped. The overlap states do not contribute to the superconductivity; this is because the overlap states do not contribute to the pairing functions on the basis of the Eq. (11). (i ) (i ) (j) (j) In another words, we find BA (kσ)nF (ξkσ ) = −BA (kσ)nF (ξkσ ), for P4 (i ) (j) (i ) (i ) ξkσ = ξkσ (i 6= j), and we get i=1 BA (kσ)nF (ξkσ ) = 0 for the contributions of overlap states (i, j = 1, 2, 3, 4.). That is to say, a very weak correlation can not allow superconductivity. If PA is large enough but SA = 0, superconductivity can not occur (i ) due to the overlaps between the bands ξkσ . If SA is large enough, the superconductivity may appear, it is easy to find there are the solutions of the temperature T pair > 0 in (13). Particularly, because P 4 (i ) 0 (i ) i, BA (k σ)nF (ξk0 σ )decreases with the increased temperature or the decreased SA , the critical pairing temperature or the Tc increases with the increased SA . For example, this can be easily seen if Vll0 = 0. What is the factor dominating the spin-charge correlation SA ? When both the spin correlation and the charge correlation are larger, SA should be larger, thus it depends on various interactions imposing on electrons. It is shown that both superconductivity and antiferromagnetism can be included in the Hubbard model, therefore, the coexistence between the superconductivity and the weak antiferromagnetism is possible when the temperature is low enough. However, an exception is that the antiferromagnetism is increased with the decreasing temperature, the antiferromagnetism may exclude superconductivity.

4. Discussion Our discussion can be extended to other problems. Firstly, the strong correlations have three kinds: the strong spin–spin correlation, the strong spin-charge correlation, and the strong charge–charge correlation, while the appropriate spin-charge correlation is a key role. As discussed above, both the antiferromagnetism and the superconductivity increase with the increased SA . However, a weak antiferromagnetism is only considered in this work. In addition, Sx and Sy have not been considered in this work, hence it does not mean that the spin-charge correlation play the same role in magnetism and superconductivity. The strong spin-charge correlation can be divided into the short-range correlation and the long-range correlation. The former is dominated by the spin–spin correlation, while the latter is dominated by the charge–charge correlation. The short-range spin-charge correlation based superconductivity may be mediated by spin excitations, while the long-range spin-charge correlation based superconductivity may be mediated by charge excitations or phonons. We suggest that the underdoped cuprates are spin excitations and phonons mediated superconductors and the overdoped cuprates are charge excitations and phonons mediated superconductors for p-types of superconductors. These suggestions are consistent with their resistivity-temperature behaviors that are also found in my recent work of theory. For example, spin correlation usually lead to ρ − T −p similar behavior (‘localization of charges’), while charge correlation lead to the ρ − T +p behavior (‘excitations of charges’) in low temperatures. The spin excitations or the charge excitations should be from the nearly localized electrons. That is, there are two kinds of electron states in some materials; the nearly localized electrons provide the spin or charge excitations while the nearly free electrons provide the charge carriers. It seems to us that the appropriate spin-charge correlation is also necessary for the phonons mediated superconductivity. It is known that the materials such as Cu, Au, and Ag etc. do not show superconductivity. The BCS theory attributes the nonsuperconductivity to the extremely weak electron–phonon interaction, and hence the weak electron–phonon interaction leads to the small resistivity. We attribute the non-superconductivity to the extremely weak spin-charge correlation, and hence the weak correlation leads to the small resistivity. This is also understood because the weak spin-charge correlation means the weak scattering of charges by spins. The stronger spin-charge correlation is, the larger the resistivity is. In some aspects, our theory is similar to the BCS. For example, the BCS equations give correct but low Tc on a single energy band of normal metals. Our theory also gives the low Tc if our bands (9) form a band due to some overlap. In other aspects, our theory is different from the BCS. For example, the interactions Vk may not be negative in Fermi surface, but the spin-charge correlation is required for superconductivity. The relations between the spin-charge correlation and the attractive force between electrons are the next topics. Particularly, our theory can also explain the relation between magnetisms and superconductivity. For example, the features appearing in the larger magnetic susceptibility |χm | are beneficial to superconductivity as shown in experiments, for which the spin-charge correlation may be stronger. In addition, if the magnetic field could strengthen the spin-charge correlation, the magnetic field induced superconductivity [9] may occur, while this behavior could not be explained with BCS. Secondly, the spectral weight transfer [13] could be explained in this work. If PA is large enough, but SA = 0, the state density around the Fermi surface is zero for the half-filled model. We can understand that the spin-charge correlation is increased with increased doping in cuprates, hence the states around the Fermi surface appear with doping as shown in (8). The “increased spincharge correlation” is related to the increased charge–charge

T.D. Cao / Solid State Communications 147 (2008) 4–7

correlation. Thirdly, the gap-function is related to the VA+∆ (kσ), the s- and d-wave symmetry [14] can be found in (10) for an anisotropic model. However, VA+∆ (kσ) is the anisotropic s-wave symmetry if the nearest-neighbor approximation is taken as tll0 = t. Fourthly, the unusual isotope effects (on Tc ) in optimally doped cuprates can be understood [15], because the strong correlation may dominate the superconductivity in this region. As discussed above, a higher Tc is due to the higher SA . Fifthly, the T -linear resistivity of cuprate superconductors may be due to the appropriate charge excitations [16], this is because the higher SA may be dominated by the charge–charge correlation, and the appropriate charge correlation can lead to the charge excitations like bosons as discussed above. In addition, if weak antiferromagnetism exists, with the expression (13) we find + + + + (k ↑) in the amounts of these (k ↓) > FBB FAA (k ↑) > FAA (k ↓) and FBB functions, this is because these electrons in A-lattice have a larger probability to spin up, while these electrons in B-lattice have a larger probability to spin down. In summary, the superconducting critical temperature increases with the strengthened spin-charge correlation, and

7

many properties of materials are related to the spin-charge correlation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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