Microscopic structure and properties of superconductivity on the density wave background P. D....

Microscopic structure and properties of superconductivity on the density wave background

P. D. Grigoriev

L. D. Landau Institute for Theoretical Physics, Chernogolovka, Russia

Superconductivity and charge/spin-density wave:

1). How can these two phenomena coexist? What is the microscopic structure of such phase?2). How do the properties of SC change on the DW background?

The results obtained explain many properties in layered organic DW superconductors: high Hc2, unconventional order, high Tc, upward curvature of Hc2

z(T), triplet pairing on SDW background, etc.

Publications: 1). L.P. Gor'kov, P.D. Grigoriev, Europhys. Lett. 71, 425 (2005). 2). L.P. Gor'kov, P.D. Grigoriev, Phys. Rev. B 75, 020507 (2007). 3). P.D. Grigoriev, Phys. Rev. B 77, 224508 (2008). 4). P.D. Grigoriev, in preparation.

CDW / SDW band structure

.)()()()( 2

2

22

NN QkkQkkkE

Energy spectrum in the CDW /SDW state

Perfect nesting condition: .)()( 0 NQkk

Empty states

2

ky

E

Electron Hamiltonian in the mean field approximation:

.)()()()()(ˆ

kQ

kQ kaQkakakakH

The order parameter is a number for CDW, and a spin operator

for SDW:

'''' )'()'(

k

kaQkag

.ˆ lQ

0

2

kx

E

Energy band diagrams

7

The energy gap in DW state prevents from SC

CDW superconductors

Review paper: A.M. Gabovich, A.I. Voitenko, J.F. Annett and M. Ausloos, Supercond. Sci. Technol. 14, R1-R27 (2001)

3

3a

Review paper: A.M. Gabovich, A.I. Voitenko, J.F. Annett and M. Ausloos, Supercond. Sci. Technol. 14, R1-R27 (2001)

SDW superconductors

Coexistence of CDW and superconductivity in NbSe3

Fermi surfacePhase diagram of NbSe3

Phys. Rev. B 64, 235119 (2001)

S. Yasuzuka et al., J. Phys. Soc. Jpn. 74, 1782 (1982)

4b

Coexistence of CDW and superconductivity in sulfur

Fermi surface

Phase diagram of sulfur

O. Degtyareva et al., PRL 99, 155505 (2007)

Observed maximum atomic displacement in S-IV and S-V as a function of pressure and temperature, shown as open diamond symbols. The temperature of the superconducting transition Tc from Ref. [E. Gregoryanz et al., Phys. Rev. B 65, 064504 (2002)] is shown by yellow triangles. The temperature is given on a logarithmic scale.

4a

Experimental phase diagrams in organic metals

External pressure damps SDW, but SC appears before SDW is completely destroyed.

! There is a pressure region where SC coexists with SDW or with CDW

(TMTSF)2PF6: T.Vuletic et al., Eur.

Phys. J. B 25, 319 (2002)

-(BEDT-TTF)2KHg(SCN)4: D. Andres et al., Phys. Rev. B 72, 174513 (2005)

4

Quasi-1D metals and Peierls instability

Nestingvector QN

Fermisurface

Electron dispersion in quasi-1D metals (tight-binding approximation)

Hbktbktkkvk yyyyFF

)cos(')cos()|(|)( 222

External pressure increases the antinesting term t’y and damps the DW.

Nesting condition: .)()( 0 NQkk kx

ky

4

antinesting term

(TMTSF)2PF6What is the structure of coexisting SC and DW?

Macroscopic coexistence of superconductivity or normal metal with DW

29b

SC insulatorThis model explains the anomalous increase of Hc2 and its upward curvature only if the domain size dS <<SC. The nonuniform DW structure costs energy 0>> SC , and the soliton structure is more favorable, where the energy loss 0 is compensated by the gain ~t’b of the kinetic energy in the soliton band.

dS

soliton band

2

ky

E

I. J. Lee et al, PRL 88, 207002 (2002)

Two mechanisms of microscopic coexistence of superconductivity or normal metal with DW

1. Ungapped pockets of FS. Empty

band 2

ky

E

ungapped pockets).2cos('2

)()()(

bkt

Qkkk

yy

yyyyA

The antinesting dispersion

soliton band

2

ky

E

2. Soliton phase (non-uniform).

The SDW order parameter depends on the coordinate along the 1D chains:

./tanh)( 00 xx

);,/()( kkxsnx kk

or

29

.'2 ytPockets appear when

[ L.P. Gor'kov, P.D. Grigoriev, Phys. Rev. B 75, 020507 (2007) ]

Procedure of the theoretical analysis

Step 1: Describe the DW in the mean field approximation.a). Calculation of the quasi-particle energy spectrum and Green functions as function of pressure (imperfect nesting). b). Renormalization of the e-e coupling by the DW critical fluctuations.

Step 2: Describe superconductivity with the new quasi-particle spectrum and new e-e interaction potential.

a). Estimate the SC transition temperature with new quasi-particle energy spectrum and new e-e interaction potential. b). Consider the influence of the spin-structure of SDW on SC. c). Calculate the upper critical field Hc2 for SC on the CDW and SDW background.

P1

This procedure allows to investigate the superconducting properties on the DW background

and to explain many experimental observations !

DoS in the open-pocket scenario (DW-SC separation in the momentum space)

The density of states (DoS) in the density wave (DW) state with open pockets remains large in DW:

0 0

0

()

D2

Due to the small open pockets at the Fermi level, the DoS is the same, as in the metallic phase. Hence, the superconducting transition

temperature is not exponentially smaller in the DW state!

Renormalization of the effective e-e interaction in the Cooper channel by critical DW fluctuations can make Tc

SC even higher than without DW

Empty

band 2

ky

E

ungapped pockets of size

[ P.D. Grigoriev, “Properties of superconductivity coexisting with a density wave with small ungapped FS parts”, Phys. Rev. B 77, 224508 (2008). ]

1

Suppression of spin-singlet SC by SDW backgroundappears in both models in agreement with experiments

[ L.P. Gor'kov, P.D. Grigoriev, Phys. Rev. B 75, 020507 (2007) ]

I.J. Lee, P. M. Chaikin, M. J. Naughton,

PRB 65, 18050(R) (2002)! Critical magnetic field and the Knight shift in (TMTSF)2PF6 in the superconductivity-SDW coexistence phase confirm the triplet paring. The absence of gap nodes suggests px symmetry of order parameter.

Knight shift does not change as temperature decreases:

abso

rpti

on

I.J. Lee et al., PRB 68, 092510 (2003)

Critical magnetic field Hc exceeds ~5 times the paramagnetic limit:

2

Equations for SC instability in SDW phase

,,,

;,,

,

,

k

RLLRbn

k

LLRRbd

kgkgTg

kgkgTg

If we introduce the diagonal and non-diagonal Cooper bubbles:

the self-consistency equations for superconductivity rewrite:

.)ˆ(ˆ)ˆ(ˆˆ TLRn

RLdb

LR lflfgf

13

SDW spin structure

R

Lf LR =

R

Lf RL

+

L

Rf RL =

L

Rf LR +

R

R

L

L R

Lf LR

L

RR

R

R

L

L

L

L

Rf RL

R

L

L

R

The spin-singlet superconducting order parameter .ˆˆ LRy

LR fif

,ˆ)ˆ(ˆ)ˆ( yT

y ll

anticommutes with SDW order parameter:

.ndbg 1which results inthe SC equation:

and Tc is exponentially smaller than without SDW.

Triplet superconductivity in SDW or CDW.

The triplet superconducting order parameter is .ˆ)ˆ(ˆ LRy

LR fidf

,ˆ)ˆ)((ˆ)ˆ()ˆ(ˆ)ˆ)(ˆ( yyT

y llddldl

2Using the commutation identity

for triplet pairing with we obtain the SC equation

on SDW background:

RLLR ff

The self-consistency equations for superconductivity:


RLdb

LR lflfgf

.)]ˆ/()ˆ)(([ dlldg ndb

211

For ,ld

,ndbg 1one obtains

while for ,|| ld .ndbg 1one has

Infrared singularitiescancel each other as for singlet SC on SDW.

Infrared singularitiesdo not cancel.

15

Why the spin structure of SDW background suppresses the spin-singlet superconductivity

(illustration)

Nestingvector QN

Fermisurface

Direct SCsingletpairing

singlet SC pair after scattering by SDW

-QN

QN

The two-electron wave function acquires “” sign after scattering by SDW if the electron spins in this pair look in opposite directions.

This affects only the infrared divergence in the Cooper logarithm. The ultraviolet divergence remains unchanged.

Spin-dependent scattering: the sign of the scattered electron wave function depends on its spin orientation.

16

spin-triplet SC pair

Electron dispersion in the ungapped FS pockets on the DW background is strongly changed

Small ungapped pockets on a FS sheet, which get formed when the antinesting term in the electron dispersion exceeds CDW energy gap.

The quasi-particle dispersionin these small pockets is

where

27a

3

Result for Hc2z on uniform DW background

For some dispersion

For tight-binding dispersion with only two harmonics

where is the

size of the new FS pockets.

In all cases, since the size of new FS

Hence, Hc2 diverges as PPc1 :

,max 10 at0 cPPk

which agrees well with experiment.

13

the constant C1 depends on electron dispersion.

,~ 1cPP

[ P.D. Grigoriev, Phys. Rev. B 77, 224508 (2008). ]3

Critical magnetic field in the coexistence phase

(TMTSF)2PF6: J. Lee, P. M. Chaikin and M. J. Naughton, PRL 88,

207002 (2002)

26

! The critical magnetic field Hc2 has very unusual temperature and pressure dependence.


CDW + superconductivity:

Two mechanisms of microscopic coexistence of superconductivity or normal metal with DW

1. Ungapped pockets of FS lead to SC with unusual properties.

Empty

band 2

ky

E

ungapped pockets0'2),2cos('2)( yyyyA tbktk


soliton band

2

ky

E



./tanh)( 00 xx

);,/()( kkxsnx kk

or

29

[ P.D. Grigoriev, PRB 77, 224508 (2008) ]

Energy of soliton phase in Q1D case

)/exp( FvnE 004

2

2

0

0

dpb

pttAt

)()(

,)( BnEtAnv

WF

SP2

20

2

where n is the soliton wall linear density,

is the soliton wall energy per chain,

is the width of center allowed band (appearing due to periodic domain walls)

and

00

2

2

1

|/|

/)( '

dpdt

btB

gives the soliton wall interaction energy.

Soliton phase linear energy:

35

[S.A. Brazovskii, L.P. Gor'kov, A.G. Lebed', Sov. Phys. JETP 56, 683 (1982)]

BoundariesE_ of the soliton level

band

2

ky

E

Schematic picture of energy bands

The soliton level band is only half-filled and the system gainsthe energy (the second term in A)which can be greater than the soliton wall energy cost .02

! Then the soliton phase is the thermodynamically stable state.

Region of soliton phase in Q1D metals for various electron dispersions

For tight- binding model with only two harmonics in the dispersion

all critical values 2t’y=0 coincide and the soliton phase has zero region.

To determine the phase diagram one has to compare the energies of uniform DW phase, soliton phase and normal metal phase.

)( pt

For step-like dispersion

)][cos(')(' bksigntkt yyy 22

the soliton phase has very large region..'. 00 540320 yt

E

ky

)cos(')cos()|(|)( bktbktkkvk yyyyFF 222

36

[ L.P. Gor'kov, P.D. Grigoriev, "Soliton phase near antiferromagnetic quantum critical point in Q1D conductors", Europhysics Letters 71, 425 (2005) ].

Energy of soliton phase (intermediate general case dispersion)

For the intermediate electron dispersion the interval of soliton phasecan be about 10% of pC in agreement with experiment in (TMTSF)2PF6 .

The SDW–SP transition at pC1

is of the second kindwhile the SP–Metal transitionat pC is of the first kind also in

agreement with experiment.

The domain phase observed in (TMTSF)2PF6 may be the soliton phase.

37

[ L.P. Gor'kov, P.D. Grigoriev, Europhysics Letters 71, 425 (2005) ]

Superconductivity in the soliton phase (suppression of spin-singlet SC by SDW background)

The Green functions in the soliton phase are 4x4 matrices:

38

Self-consistency Gor’kov equations for superconductivity in soliton phase:

R

Lf LR =

R

Lf RL

+

L

Rf RL =

L

Rf LR +

R

R

L

L R

Lf LR

L

RR

R

R

L

L

L

L

Rf RL

R

L

L

R

The sign “-” leads to the cancellation of diagonal and non-diagonal Cooper blocks in the SC equations for singlet superconductivity in the soliton band, which means the suppression of spin-singlet SC by the DW background. This cancellation doesn’t happen for singlet SC in CDW soliton, or for triplet SC in the SDW soliton phase.

Calculation of SC upper critical field on the soliton phase background

We use again the Ginzburg-Landau approximation:

Upper critical field

where

The electron dispersion :

).2cos('2

/)|(|sin)( 0

bkt

kkEk

yy

F

40

Width of soliton band in Q1D metals

,/ln/ 142 0 EtBtAE

2

2

0

0

dpb

pttAt

)()(

,)( BnEtAnv

WF

SP2

20

2

where the soliton wall linear density

,|/|

/)( '

00

2

2

1

dpdt

btB

From the soliton phase linear energy

one obtains the width of the soliton band:

and

In the tight-binding model with only two harmonics near the

transition at P = Pc1 (where 2t’b=0)

and

41

Upper critical field in SC state on soliton-phase background.

Result:close to Tc

For tight binding dispersion

where ./ 1SDWsn

The width of the soliton band

and Hc2 diverges as PPc1 :

,max 10 at0as0 cPPkE

which agrees well with experiment.

and the constant C1s depends

on the electron dispersion.

42

Upward curvature of Hc2z(T)

Solitons create a layered structure, which is described by the Lawrence-Doniach model of 1D Josephson lattice.

This model was generalize for finite width of SC layers in [G. Deutcher and O. Entin-Wohlman, Phys. Rev. B 17, 1249, (1978) ].

The divergence of upper critical field is

cut off by Hc2 in a superconducting slab:

where ds=s is the interlayer distance.

SC insulator

sUpper critical field in this Josephson lattice is

Upper critical field Hc2z in -(BEDT-TTF)2KHg(SCN)4


CDW + superconductivity:Tc

SC<TcDW 100 times, and the

energy of SC state is 4 orders less than DW energy.

Hence, no strong influence of SC on DW is possible(as adjusting of the size of DW domains with magnetic field), an the macroscopic domains cannot explain this Hc2

z behavior

Origin of hysteresis. 44

The observed hysteresis in resistance at temperature change can be explained in both scenarios.

For open-pocket scenario of DW1 hysteresys is due the shift of the

DW wave vector at P>Pc1

Phase diagram

In the soliton scenario of DW1 the hysteresys is due the sliding of soliton walls.

Conclusions

I. There are, at least, 2 possible structures of a DW1 state, where superconductivity coexists microscopically with density wave.

II. The SC properties of such state are investigated for both structures:

1). The DoS on the Fermi level in DW1 is rather high, giving possibility of SC. 2). The SDW background suppressed the spin-singlet SC coupling, leaving the triplet SC transition temperature almost without change. 3). The upper critical field increases at critical pressure Pc1, where SC first

appears, and shows unusual temperature (upward curvature) and pressure dependence.

III. The results agree with experiment in organic metals (TMTSF)2PF6 and -(BEDT-TTF)2KHg(SCN)4, explaining many unusual properties.

Publications: 1). L.P. Gor'kov, P.D. Grigoriev, "Soliton phase near antiferromagnetic quantum critical point in Q1D conductors", Europhys. Lett. 71, 425 (2005). 2). L.P. Gor'kov, P.D. Grigoriev, " Nature of superconducting state in the new phase in (TMTSF)2PF6 under pressure", Phys. Rev. B 75, 020507 (2007). 3). P.D. Grigoriev, “Properties of superconductivity coexisting with a density wave with small ungapped FS parts”, Phys. Rev. B 77, 224508 (2008). 4). P.D. Grigoriev, “Superconductivity on the density wave background with soliton-wall structure”, in preparation.

Thank you for the attention !

1. We developed the theory, describing superconductivity on SDW or CDW background when Tc

DW>>TcSC in quasi-1D compounds with one conducting band.

2. There are two possible microscopic structures of DW1 phase, where SC may coexist microscopically with DW: (1) uniform structure with ungapped states in momentum space (open pockets); (2) non-uniform soliton phase.

3. The DoS at the Fermi level in DW1 state in both scenarios is rather high, which makes TC

SC on DW background comparable with TCSC in pure SC state. The enhancement of

the e-e interaction by critical fluctuations may increase TcSC even to the value higher

than without DW.

4. The upper critical field is calculated in both scenarios and shown to considerably exceed the usual Hc2. It diverges at critical pressure Pc1, where SC first appear, and shows unusual temperature (upward curvature) and pressure dependence.

5. The SDW background strongly damps singlet SC. The SC, appearing on SDW background in metals with single conducting band, should be triplet.

6. The hysteresis of R(T) may appear in both scenarios (for different reasons).

7. The results obtained are in good agreement with experimental observations in organic metals (TMTSF)2PF6 and -(BEDT-TTF)2KHg(SCN)4 .

Conclusions

Publications: 1). L.P. Gor'kov, P.D. Grigoriev, "Soliton phase near antiferromagnetic quantum critical point in Q1D conductors", Europhys. Lett. 71, 425 (2005). 2). L.P. Gor'kov, P.D. Grigoriev, " Nature of superconducting state in the new phase in (TMTSF)2PF6 under pressure", Phys. Rev. B 75, 020507 (2007). 3). P.D. Grigoriev, “Properties of superconductivity coexisting with a density wave with small ungapped FS parts”, Phys. Rev. B 77, 224508 (2008). 4). P.D. Grigoriev, “Superconductivity on the density wave background with soliton-wall structure”, in preparation.

Lawrence-Doniach model

[ Lawrence, W. E., and Doniach, S., in Proceedings of the 16th International Conference on Low Temperature Physics, ed. E. Kanda, Kyoto: Academic Press of Japan, p. 361 (1971). ]

Here

SC insulator

s

Lawrence-Doniach model (2).

Introducing

The lowest eigenvalue of this equation gives upper critical field:

Which of the two proposed microscopic structures appears in the experiment?

44

The observed hysteresis in resistance for increasing and decreasing magnetic field suggests the soliton phase(spatial inhomogeneity in the form of microscopic domains).

The high upper critical field Hc2 suggests the domain size is much less than the SC coherence length, because for a SC slab

This means, that superconducting domains must be microscopically narrow, supporting that the soliton scenario takes place.

NMR experiments in (TMTSF)2PF6

Lineshapes for incommensurate SDWs, with different soliton widths, using hyperbolic tangent function for describing solitons.

Stuart Brown et al., UCLA, Dresden, 2005.

).()(

;)(

rSHrH

rHIrdI

SDW

B

0

03

I

Red= normal state; Blue= zero width;

Black=wide soliton.

NMR absorption line

45

Upward curvature of Hc2(T)

(TMTSF)2PF6: J. Lee, P. M. Chaikin and M. J. Naughton, PRL 88,

207002 (2002)


CDW + superconductivity:

The upward curvature of Hc2(T) also suggests the soliton structure

Model with two coupling constants in e-e interactions for forward and backward scattering

Electron Hamiltonian is ,

where the free-electron part

And the e-e interaction has two coupling constants for forward and backward scattering:

intˆˆˆ HHH 0

),()()(ˆ kakakHk

0

,)()()'()'(ˆ,'',

'''int

Qkk

kaQkakaQkaQgH

2

1

The CDW or SDW onset is due to the interaction with Q=QN only,

while the SC onset is due to the interaction with all other Q. Therefore, the same interaction constants lead to both DW and SC.

where

.,

,,

Fxb

Fxf

kQg

kQgQg

2

2 (keeps electrons on the same FS sheet)

(scatters electrons to the opposite FS sheet)

21

Calculation of upper critical field when superconductivity coexists with CDW or SDW

We use the Ginzburg-Landau approximation:

then

where

27

[ L.P. Gor'kov and T.K. Melik-Barkhudarov, JETP 18, 1031 (1963) ]

Previous theoretical results on SC+DW. 4t

Model with initially imperfect nesting or with several conducting bands. ( CDW leaves some electron states on the Fermi level and does not affect the dispersion of the unnested parts of Fermi surface. )

[ General properties: K. Machida, J. Phys. Soc. Jpn. 50, 2195 (1981); Hc2 : A. M. Gabovich and A. S. Shpigel, Phys. Rev. B 38, 297 (1988). ]

DW reduces the SC transition temperature since it creates an energy gap on the part or on the whole Fermi surface.[ K. Levin, D. L. Mills, and S. L. Cunningham, Phys. Rev. B 10, 3821 (1974); C. A. Balseiro and L. M. Falicov, Phys. Rev. B 20, 4457 (1979). ]

.exp

F

metC g

T

1

3). Proximity to the Peierls (DW) instability increases the effective e-e interaction g(Q) with the wave vector Q QN:

The RPA result gives

.,,

NQQg

QQg

QgQg

1

1 00

0

Why the proposed approach is different?

In fact, the DW may considerably change the quasi-particle dispersion even on the ungapped parts of Fermi surface !

New properties in DW superconductors appear:

1). SC transition temperature Tc is higher than expected (not exponentially smaller than Tc without DW). With renormalization of the coupling constant g(Q) by critical fluctuations it may be even higher than without DW.

2). The upper critical field Hc2 may be strongly enhanced as compared to SC without DW.

P1

Procedure of the theoretical analysis

Step 1: Describe the DW in the mean field approximation.a). Calculation of the quasi-particle energy spectrum and Green functions as function of pressure (imperfect nesting). b). Renormalization of the e-e coupling by the DW critical fluctuations.

Step 2: Describe SC with the new quasi-particle spectrum and new e-e interaction potential.a). Estimate the SC transition temperature with new quasi-particle energy spectrum and new e-e interaction potential. b). Consider the influence of the spin-structure of SDW on SC. c). Calculate the upper critical field Hc2 for SC on the CDW and SDW background.

P1

Model for a quasi-1D metal

Hbktbktkkvk yyyyFF )cos(')cos()|(|)( 222

,ˆˆˆintHHH 0

),()()(ˆ,

0 kakakHk

Dispersion relation of electrons in quasi-1D metals in magnetic field

Hamiltonian

).'()'()()('ˆ''

','int kaQkakaQkaQUQUH

Qkksc

2

1

where the free-electron term

For CDW or SDW UC and US are just the charge and spin coupling

constants (being taken at the wave vector transfer Q=QN ).

imperfect nesting term

H

and the electron-electron interaction is given by

For SC the functional dependence of UC (Q) and US (Q) is important (it

determines the type of pairing). The couplings have maximum at the

wave vector transfer Q=QN (the backward scattering is enhanced).

Electron dispersion in the ungapped FS pockets on the DW background in tight-binding approximation

Small ungapped pockets on a FS sheet get formed when the antinesting term in the electron dispersion exceeds DW energy gap.

The quasi-particle dispersionin these small pockets

where

The important contribution to Cooper logarithm and to SC properties comes from the ungapped electron states on the Fermi level.

27a

Empty

band 2

ky

E

ungapped pockets

Effective mass ;//~/ '''*

bbybFby ttmtmbtm 0022 42

Enhancement of the e-e coupling by the proximity to DW transition (critical fluctuations)

+ + ..=

In RPA the renormalized e-e interaction is given by the sum of diagrams:

This gives ,QQg

QgQg

00

0

1 where g0(Q)<<1 is

the bare interaction,

and the susceptibility may diverge at some (nesting) wave vector,

so that Then the new coupling also diverges at some Q.

Q0

.100 QQg

The original coupling g0(Q) may be more complicated (include spin).

Then the renormalized coupling includes all components of g0(Q).

The new coupling g(Q) is strongly Q-dependent, being considerably changed only in the vicinity of the DW wave-vector. Therefore, the SC coupling doesn’t change for almost the whole FS except “hot spots”.

The enhancement of e-e coupling depends very strongly on the bare e-e interaction (example)

Consider the Hubbard model with two coupling functions U and V(Q)

Y. Tanaka and K. Kuroki, PRB 70, 060502(R) (2004)

Then the RPA gives the following renormalization of the couplings in the superconducting singlet and triplet channels:

where the spin and charge susceptibilities

and

The renormalized SC couplings depend very

strongly on the bare interaction U and V(Q)

The density of states at the Fermi level (1)

Without DW the DoS in Q1D metal is

In the presence of DW

or

where .)()()()(

,2

2

21 22

NN QkkQkkkE

and for small FS pockets

D1

Result1: Comparison of singlet Tc on metallic, CDW and SDW states without change of e-e interaction

,ln

CFb Tg

1

,lnln

Tg

CDW

CDWFb

1

.lnln

Tg

SDW

SDWSDWFb

1

1. Normal metal background:

and .exp

Fb

metC g

T

1

2. CDW background:

.expCDW

metC

FbCDW

CDWC T

gT

1and

3. SDW background:

.

1

SDWSDW

SDW

metCCDW

CSDW

metC

SDWSDW

C

TT

TT

which gives very low Tc:

Not toosmall.

17

is the size of the ungapped parts of FS

Why the spin structure of SDW background suppresses the spin-singlet superconductivity

(illustration)

Nestingvector QN

Fermisurface

Direct SCsingletpairing

singlet SC pair after scattering by SDW

-QN

QN




16

spin-triplet SC pair

Nature of superconductivity in (TMTSF)2PF6

I.J. Lee, P. M. Chaikin, M. J. Naughton,

PRB 65, 18050(R) (2002)

! Critical magnetic field and the Knight shift in (TMTSF)2PF6 in the superconductivity-SDW coexistence phase confirm the triplet paring:

Knight shift does not change as temperature decreases:

abso

rpti

on

I.J. Lee et al., PRB 68, 092510 (2003)

Critical magnetic field Hc exceeds ~5 times the paramagnetic limit:

19

Result2: Comparison of triplet Tc in normal metal, CDW and SDW background for

,ln

CFb Tg

1

,lnln

Tg

SDW

SDWFb

1

.lnln

Tg

SDW

SDWFb

1


and .exp

Fb

metC g

T

1

2. CDW background:

.expCDW

metC

FbCDW

CDWC T

gT

1and

3. SDW backgroundat :

which gives:

Not toosmall.

ld||

ld||

.exp CDWC

SDW

metC

FbSDW

SDWC TT

gT

1

18

The enhancement of e-e coupling depends very strongly on the bare e-e interaction (example)

Consider the Hubbard model with two coupling functions U and V(Q)

Y. Tanaka and K. Kuroki, PRB 70, 060502(R) (2004)

Then the RPA gives the following renormalization of the couplings in the superconducting singlet and triplet channels:

where the spin and charge susceptibilities

and

The renormalized SC couplings depend very

strongly on the bare interaction U and V(Q)

Enhancement of the e-e coupling helps to SC

(TMTSF)2PF6: T.Vuletic et al., Eur.

Phys. J. B 25, 319 (2002)


3

SC transition temperature considerably increase as the DW instability is approached. This increase is attributed to the critical fluctuation.

Two mechanisms of microscopic coexistence of superconductivity or normal metal with SDW

1. Ungapped pockets of FS. Empty

band 2

ky

E

ungapped pockets).cos('

)()()(

bkt

Qkkk

yy

yyyyA

22


soliton band

2

ky

E



./tanh)( 00 xx

);,/()( kkxsnx kk

or

29

Solitons in CDW or SDW.31

Soliton phase.


./tanh)( 00 xx

BoundariesE_ of the soliton level band

2

ky

E

Schematic picture of energy bandsin the soliton phase in Q1D case.

The soliton level band is only half-filled and the system gains the energy due to the dispersion along ky, which can be greater than the soliton wall energy cost .02

./02Each soliton costs energy

Upward curvature of Hc2(T)

Solitons create a layered structure, which is described by the Lawrence-Doniach model of 1D Josephson lattice.

This model was generalize for finite width of SC layers in [G. Deutcher and O. Entin-Wohlman, Phys. Rev. B 17, 1249, (1978) ].

The divergence of upper critical field is

cut off by Hc2 in a superconducting slab:

where s is the interlayer distance.

SC insulator

sUpper critical field in this Josephson lattice is

Lawrence-Doniach model

[ Lawrence, W. E., and Doniach, S., in Proceedings of the 16th International Conference on Low Temperature Physics, ed. E. Kanda, Kyoto: Academic Press of Japan, p. 361 (1971). ]

Here

SC insulator

s

Lawrence-Doniach model (2).

Introducing

The lowest eigenvalue of this equation gives upper critical field:

Gor’kov equations with forward and backward scattering

R

Lf LR =

R

Lf RL

+

L

Rf RL =

L

Rf LR +

R

R

L

L R

Lf LR +

L

RR

R

R

L

L

L

L

Rf RL +

R

L

L

R

R

Lf RL

+L

R R

Lf LR ;

R

LR

RL

L

L

Rf LR +

L

R

R

L

L

Rf RL

L

R

L

R

backward scattering forward scattering

Self-consistency equations for superconductivity order parameter:

.ˆ)ˆ(ˆ)ˆ()ˆ(ˆ)ˆ(ˆˆ LRd

TRLnf

TLRn

RLdb

LR flflglflfgf

In analytical form this rewrites:

.ˆ)ˆ(ˆ)ˆ()ˆ(ˆ)ˆ(ˆˆ RLd

TLRnf

TRLn

LRdb

RL flflglflfgf

22

Equations on Tc with forward and backward scattering

1. Normal metal or CDW background.

LRLRndbf

LRLR ffggff

LRLRndbf

LRLR ffggff

Singlet SC equation

Triplet SC equation

2. Superconductivity on SDW background.

LRLRndbf

LRLR ffggff

LRLRndbf

LRLR ffdlldggff )]ˆ/()ˆ)(([

21

Singlet SC equation

Triplet SC equation

23

Discussion1). Usually, the coupling constants, gf , gb , have the same sign, and Hence, in the normal-metal state SC is usually singlet.

On CDW background the triplet order is even less favorable.

.0 nd

2). On SDW background the spin structures of SC and SDW order parameters interfere, which leads to different self-consistency equations:

The non-diagonal block of the Cooper bubble enters with the oppositesign and cancels the infrared singularity from the diagonal block. This leads to the strong reduction of Tc for singlet SC in SDW.This cancellation happens for singlet SC but may not happen for triplet.

LRLRndbf

LRLR ffggff

LRLRndbf

LRLR ffggff

LRLRndbf

LRLR ffggff

LRLRndbf

LRLR ffdlldggff )]ˆ/()ˆ)(([

21

24

Outlook

The proposed study opens a new field in the investigation of density-wave superconductors rather than closes this problem.

1. There are many other DW superconductors.2. Most results obtained qualitatively and require further

elaboration. 3. The results depend on a particular electron dispersion.4. Many other properties are left for investigation.5. More complicated models can be studied (with more

complex e-e interaction and impurity scattering, etc.)

The Green functions in the uniform SDW state.

.',,'ˆ)ˆ(,,'ˆ kkQkkglkkgki SDW

The equations for the Green functions in the SDW state

In the matrix form these equations rewrite:

,ˆ)ˆ(

)ˆ(*

IGQkil

lki

nSDW

SDWn

where the matrix Green function

.,,)ˆ(,,

)ˆ(,,,,ˆ

n

LLn

RLn

LRn

RR

QkQkglQkkg

lkQkgkkgG

,)]([)(,22

21 kkkE

Diagonalization of the 2x2 matrix Hamiltonian gives the new energy spectrum:

where .)()(

)(2

Qkkk

10

Expressions for the electron Green functions in the SDW state

n

LL

nn

nn

RR kkgkEikEi

kikkg

,,,,

21

The diagonal elements of the Green function matrix:

,,,kEikEi

kQkgnn

SDWn

LR

21

The non-diagonal elements of the Green function matrix:

.,,*

kEikEiQkkg

nn

SDWn

RL

21

11

Equations for superconducting instability

R

Lf LR =

R

Lf RL

+

L

Rf RL =

L

Rf LR +

R

R

L

L R

Lf LR

L

RR

R

R

L

L

L

L

Rf RL

R

L

L

R

The Gor’kov functionsat t1=t2+0 :

.)()()(

;)()()(

rrrf

rrrf

LRRL

RLLR

In the presence of SDW or CDW the SC equations contain two additional terms, coming from non-diagonal elements in the Green functions:

In the normal metal state (without SDW or CDW)the SC self-consistency equation in diagram form

R

Lf LR =

R

Lf RL

L

Rf RL =

L

Rf LR

R

R

L

L

R

R

L

L

.)( LRLR frf In the uniform phase

12

Equations for SC instability in SDW phase

,

,)ˆ(ˆ,)ˆ(,ˆ,ˆk

RLTLRLRLLRLRRb

LR kglfkglkgfkgTgf

With backward scattering only the SC equation are

due to SDW spin structure

,,,

;,,

,

,

k

RLLRbn

k

LLRRbd

kgkgTg

kgkgTgIf we introduce the diagonal and non-diagonal Cooper bubbles:

the self-consistency equations for superconductivity rewrite:


RLdb

LR lflfgf

13

Singlet superconductivity in SDW or CDW.

The spin-singlet superconducting order parameter .ˆˆ LRy

LR fif

,ˆ)ˆ(ˆ)ˆ( yT

y ll

Using the commutation identity

for spin-singlet pairing with we obtain the SC equation

on SDW background:

RLLR ff



RLdb

LR lflfgf

.ndbg 1

The SC equations on the CDW background would be

.ndbg 1

14

Triplet superconductivity in SDW or CDW.

The triplet superconducting order parameter is .ˆ)ˆ(ˆ LRy

LR fidf

,ˆ)ˆ)((ˆ)ˆ()ˆ(ˆ)ˆ)(ˆ( yyT

y llddldl

2Using the commutation identity

for triplet pairing with we obtain the SC equation

on SDW background:

RLLR ff



RLdb

LR lflfgf

.)]ˆ/()ˆ)(([ dlldg ndb

211

For ,ld

,ndbg 1one obtains

while for ,|| ld .ndbg 1one has

Infrared singularitiescancel each other as for singlet SC on SDW.

Infrared singularitiesdo not cancel.

15

Illustration of the cancellation of different contributions to the SC order parameter on the

SDW background

Nestingvector QN

Fermisurface

Direct SCpairing

SC pairing after scattering by SDW wave vector

-QN

QN




16

Result1: Comparison of singlet Tc in metal, CDW and SDW states without renormalization of e-e interaction

,ln

CFb Tg

1

,lnln

Tg

CDW

CDWFb

1

.lnln

Tg

SDW

SDWSDWFb

1


and .exp

Fb

metC g

T

1

2. CDW background:

.expCDW

metC

FbCDW

CDWC T

gT

1and

3. SDW background:

.

1

SDWSDW

SDW

metCCDW

CSDW

metC

SDWSDW

C

TT

TT

which gives very low Tc:

Not toosmall.

17

is the size of the ungapped parts of FS

Result2: Comparison of triplet Tc in normal metal, CDW and SDW background for

,ln

CFb Tg

1

,lnln

Tg

SDW

SDWFb

1

.lnln

Tg

SDW

SDWFb

1


and .exp

Fb

metC g

T

1

2. CDW background:

.expCDW

metC

FbCDW

CDWC T

gT

1and

3. SDW backgroundat :

which gives:

Not toosmall.

ld||

ld||

.exp CDWC

SDW

metC

FbSDW

SDWC TT

gT

1

18

Publications

Publications.

1). L.P. Gor'kov, P.D. Grigoriev, "Soliton phase near antiferromagnetic quantum critical point in Q1D conductors", Europhysics Letters 71, 425 (2005).

2). L.P. Gor'kov, P.D. Grigoriev, " Nature of superconducting state in the new phase in (TMTSF)2PF6 under pressure", Phys. Rev. B 75, 020507 (2007).

3). P.D. Grigoriev, “Properties of superconductivity coexisting with a density wave with small ungapped FS parts”, Phys. Rev. B 77, 224508 (2008).

Summary

1. We developed the theory, describing superconductivity on SDW or CDW background when Tc

SDW>>TcSC in quasi-1D compounds with one conducting band.

2. There are two possible microscopic structures of superconducti-vity, coexisting with CDW or SDW in quasi-1D metals with one conducting band: (1) uniform structure with ungapped states in momentum space; (2) non-uniform soliton phase.

3. The DoS at the Fermi level in the DW phase with open pockets is the same as in the metallic state, which makes the SC transition temperature to be rather high. The enhancement of the e-e interaction by the Peirls instability may increase Tc

SC even to the value higher than without DW.

4. The upper critical field is calculated in both scenarios and shown to considerably exceed the usual Hc2, diverging at critical pressure and showing unusual temperature and pressure dependence.

5. The SDW background strongly damps singlet SC. The SC, appearing on SDW background should be triplet.

6. The proposed models and approach to study these models open new scope to investigate the coexistence of SC with DW also in many existing DW superconductors.

7. The results obtained are in good agreement with experimental observations in organic metals (TMTSF)2PF6 and -(BEDT-TTF)2KHg(SCN)4 .

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Microscopic structure and properties of superconductivity on the density wave background P. D....

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