Microscopic structure and properties of superconductivity on the density wave background

1
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 Hc2z(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.
2
CDW / SDW band structure
7
E
Electron Hamiltonian in the mean field
approximation
?
2?
The order parameter is a number for CDW,
and a spin operator for SDW
kx
Energy band diagrams
Empty states
E
Energy spectrum in the CDW /SDW state
2?
?
Perfect nesting condition
The energy gap in DW state prevents from SC
ky
3
CDW superconductors
3
Review paper A.M. Gabovich, A.I. Voitenko, J.F.
Annett and M. Ausloos, Supercond. Sci. Technol.
14, R1-R27 (2001)
4
SDW superconductors
3a
Review paper A.M. Gabovich, A.I. Voitenko, J.F.
Annett and M. Ausloos, Supercond. Sci. Technol.
14, R1-R27 (2001)
5
Coexistence of CDW and superconductivity in NbSe3
4b
Fermi surface
Phase diagram of NbSe3
Phys. Rev. B 64, 235119 (2001)
S. Yasuzuka et al., J. Phys. Soc. Jpn. 74, 1782
(1982)
6
Coexistence of CDW and superconductivity in sulfur
4a
Phase diagram of sulfur
Fermi surface
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.
7
Experimental phase diagrams in organic metals
4
(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)
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
8
Quasi-1D metals and Peierls instability
4
Electron dispersion in quasi-1D metals
(tight-binding approximation)
ky
antinesting term
Nesting condition
kx
Fermi surface
External pressure increases the antinesting term
ty and damps the DW.
Nesting vector QN
What is the structure of coexisting SC and DW?
(TMTSF)2PF6
9
Macroscopic coexistence of superconductivity or
normal metal with DW
29b
insulator
SC
This model explains the anomalous increase of Hc2
and its upward curvature only if the domain size
dS ltlt?SC. The nonuniform DW structure costs
energy ?0gtgt ?SC , and the soliton structure is
more favorable, where the energy loss ?0 is
compensated by the gain tb of the kinetic
energy in the soliton band.
I. J. Lee et al, PRL 88, 207002 (2002)
dS
E
?
soliton band
2?
ky
10
Two mechanisms of microscopic coexistence of
superconductivity or normal metal with DW
29
L.P. Gor'kov, P.D. Grigoriev, Phys. Rev. B 75,
020507 (2007)
1. Ungapped pockets of FS.
Empty band
E
The antinesting dispersion
2?
?
ungapped pockets
ky
Pockets appear when
2. Soliton phase (non-uniform).
E
The SDW order parameter depends on the coordinate
along the 1D chains
?
soliton band
2?
or
ky
11
Procedure of the theoretical analysis
P1
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.
This procedure allows to investigate the
superconducting properties on the DW
background and to explain many experimental
observations !
12
DoS in the open-pocket scenario (DW-SC
separation in the momentum space)
D2
1
The density of states (DoS) in the density wave
(DW) state with open pockets remains large in DW
Empty band
E
2?
?
?(?)
?0
ungapped pockets of size ?
?
ky
?0
0
?
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 TcSC even higher than
without DW
P.D. Grigoriev, Properties of
superconductivity coexisting with a density wave
with small ungapped FS parts, Phys. Rev. B 77,
224508 (2008).
13
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)
2
Knight shift does not change as temperature
decreases
Critical magnetic field Hc exceeds 5 times the
paramagnetic limit
absorption
I.J. Lee, P. M. Chaikin, M. J. Naughton, PRB 65,
18050(R) (2002)
I.J. Lee et al., PRB 68, 092510 (2003)
! 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.
14
Equations for SC instability in SDW phase
13
If we introduce the diagonal and non-diagonal
Cooper bubbles
L
L
R
L
R
R
L
f LR
f RL
f LR
R
R
L
R
L
R
L
R
R
L
L
R
L
R
f RL
f LR
f RL
L
R
L
L
R
L
R
the self-consistency equations for
superconductivity rewrite
SDW spin structure
The spin-singlet superconducting order parameter
anticommutes with SDW order parameter
which results in the SC equation
and Tc is exponentially smaller than without SDW.
15
Triplet superconductivity in SDW or CDW.
15
The self-consistency equations for
superconductivity
The triplet superconducting order parameter is
Using the commutation identity
for triplet pairing with we
obtain the SC equation on SDW background
Infrared singularities cancel each other as for
singlet SC on SDW.
one obtains
For
Infrared singularities do not cancel.
while for
one has
16
Why the spin structure of SDW background
suppresses the spin-singlet superconductivity
(illustration)
Spin-dependent scattering the sign of the
scattered electron wave function depends on its
spin orientation.
spin-triplet SC pair
-QN
singlet SC pair after scattering by SDW
Direct SC singlet pairing
The two-electron wave function acquires ? sign
after scattering by SDW if the electron spins in
this pair look in opposite directions.
QN
Fermi surface
Nesting vector QN
This affects only the infrared divergence in the
Cooper logarithm. The ultraviolet divergence
remains unchanged.
16
17
Electron dispersion in the ungapped FS pockets on
the DW background is strongly changed
27a
3
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 dispersion in these small
pockets is
where
18
Result for Hc2z on uniform DW background
13
3
P.D. Grigoriev, Phys. Rev. B 77, 224508 (2008).

where is the size of
the new FS pockets.
the constant C1 depends on electron dispersion.
For some dispersion
For tight-binding dispersion with only two
harmonics
In all cases, since the size of new FS Hence,
Hc2 diverges as P?Pc1
which agrees well with experiment.
19
Critical magnetic field in the coexistence phase
26
CDW superconductivity
?-(BEDT-TTF)2KHg(SCN)4 D. Andres et al., Phys.
Rev. B 72, 174513 (2005)
(TMTSF)2PF6 J. Lee, P. M. Chaikin and M. J.
Naughton, PRL 88, 207002 (2002)
! The critical magnetic field Hc2 has very
unusual temperature and pressure dependence.
20
Two mechanisms of microscopic coexistence of
superconductivity or normal metal with DW
29
1. Ungapped pockets of FS lead to SC with
unusual properties.
Empty band
E
2?
?
P.D. Grigoriev, PRB 77, 224508 (2008)
The antinesting dispersion
ungapped pockets
ky
2. Soliton phase (non-uniform).
E
The SDW order parameter depends on the coordinate
along the 1D chains
?
soliton band
2?
or
ky
21
Energy of soliton phase in Q1D case
35
Soliton phase linear energy
Schematic picture of energy bands
E
where n is the soliton wall linear density,
?
Boundaries E_ of the soliton level band
2?
is the soliton wall energy per chain,
ky
is the width of center allowed band (appearing
due to periodic domain walls) and
The soliton level band is only half-filled and
the system gains the energy (the second term in
A) which can be greater than the soliton wall
energy cost
gives the soliton wall interaction energy.
! Then the soliton phase is the
thermodynamically stable state.
S.A. Brazovskii, L.P. Gor'kov, A.G. Lebed', Sov.
Phys. JETP 56, 683 (1982)
22
Region of soliton phase in Q1D metals for
various electron dispersions
36
To determine the phase diagram one has to compare
the energies of uniform DW phase, soliton phase
and normal metal phase.
For tight- binding model with only two harmonics
in the dispersion all critical values 2ty?0
coincide and the soliton phase has zero region.
For step-like dispersion
E
the soliton phase has very large region
ky
L.P. Gor'kov, P.D. Grigoriev, "Soliton phase
near antiferromagnetic quantum critical point in
Q1D conductors", Europhysics Letters 71, 425
(2005) .
23
Energy of soliton phase (intermediate general
case dispersion)
37
For the intermediate electron dispersion the
interval of soliton phase can be about 10 of pC
in agreement with experiment in (TMTSF)2PF6 .
The SDWSP transition at pC1 is of the second
kind while the SPMetal transition at pC is of
the first kind also in agreement with experiment.
L.P. Gor'kov, P.D. Grigoriev, Europhysics
Letters 71, 425 (2005)
The domain phase observed in (TMTSF)2PF6 may be
the soliton phase.
24
Superconductivity in the soliton phase
(suppression of spin-singlet SC by SDW background)
38
The Green functions in the soliton phase are 4x4
matrices
L
L
R
L
R
R
L
f LR
f RL
f LR
R
R
L
R
L
R
L
Self-consistency Gorkov equations for
superconductivity in soliton phase
R
R
L
L
R
L
R
f RL
f LR
f RL
L
R
L
L
R
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 doesnt happen for singlet SC in CDW
soliton, or for triplet SC in the SDW soliton
phase.
25
Calculation of SC upper critical field on the
soliton phase background
40
We use again the Ginzburg-Landau approximation
Upper critical field
where
The electron dispersion
26
Width of soliton band in Q1D metals
41
From the soliton phase linear energy
where the soliton wall linear density
and
one obtains the width of the soliton band
In the tight-binding model with only two
harmonics near the transition at P Pc1 (where
2tb?0)
and
27
Upper critical field in SC state on soliton-phase
background.
42
Result close to Tc
and the constant C1s depends on the electron
dispersion.
where
For tight binding dispersion
The width of the soliton band and Hc2 diverges
as P?Pc1
which agrees well with experiment.
28
Upward curvature of Hc2z(T)
insulator
SC
Solitons create a layered structure, which is
described by the Lawrence-Doniach model of 1D
Josephson lattice.
Upper critical field in this Josephson lattice is
s
where dss is the interlayer distance.
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
29
Upper critical field Hc2z in ?-(BEDT-TTF)2KHg(SCN)
4
CDW superconductivity
TcSCltTcDW 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 Hc2z behavior
?-(BEDT-TTF)2KHg(SCN)4 D. Andres et al., Phys.
Rev. B 72, 174513 (2005)
30
Origin of hysteresis.
44
Phase diagram
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 PgtPc1
In the soliton scenario of DW1 the hysteresys is
due the sliding of soliton walls.
31
Conclusions

• There are, at least, 2 possible structures of a
  DW1 state, where superconductivity coexists
  microscopically with density wave.
• 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.
32
Thank you for the attention !
33
Conclusions

• We developed the theory, describing
  superconductivity on SDW or CDW background when
  TcDWgtgtTcSC in quasi-1D compounds with one
  conducting band.
• 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.
• The DoS at the Fermi level in DW1 state in both
  scenarios is rather high, which makes TCSC 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.
• 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.
• The SDW background strongly damps singlet SC. The
  SC, appearing on SDW background in metals with
  single conducting band, should be triplet.
• 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 .

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.
34
Lawrence-Doniach model
insulator
SC
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).
s
Here
35
Lawrence-Doniach model (2).
Introducing
The lowest eigenvalue of this equation gives
upper critical field
36
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.
37
NMR experiments in (TMTSF)2PF6
45
NMR absorption line
Red normal state Blue zero width Blackwide
soliton.
Lineshapes for incommensurate SDWs, with
different soliton widths, using hyperbolic
tangent function for describing solitons.
Stuart Brown et al., UCLA, Dresden, 2005.
38
Upward curvature of Hc2(T)
CDW superconductivity
?-(BEDT-TTF)2KHg(SCN)4 D. Andres et al., Phys.
Rev. B 72, 174513 (2005)
(TMTSF)2PF6 J. Lee, P. M. Chaikin and M. J.
Naughton, PRL 88, 207002 (2002)
The upward curvature of Hc2(T) also suggests the
soliton structure
39
Model with two coupling constants in e-e
interactions for forward and backward scattering
21
Electron Hamiltonian is , where the
free-electron part And the e-e interaction has
two coupling constants for forward and backward
scattering
(keeps electrons on the same FS
sheet) (scatters electrons to the opposite FS
sheet)
where
The CDW or SDW onset is due to the interaction
with QQN 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.
40
Calculation of upper critical field when
superconductivity coexists with CDW or SDW
27
We use the Ginzburg-Landau approximation
then
where
L.P. Gor'kov and T.K. Melik-Barkhudarov, JETP
18, 1031 (1963)
41
Previous theoretical results on SCDW.
4t
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).
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).
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
42
Why the proposed approach is different?
P1
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.
43
Procedure of the theoretical analysis
P1
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.
44
Model for a quasi-1D metal
H
Dispersion relation of electrons in quasi-1D
metals in magnetic field
imperfect nesting term
Hamiltonian
where the free-electron term
and the electron-electron interaction is given by
For CDW or SDW UC and US are just the charge and
spin coupling constants (being taken at the wave
vector transfer QQN ).
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 QQN (the backward scattering is
enhanced).
45
Electron dispersion in the ungapped FS pockets on
the DW background in tight-binding approximation
27a
The important contribution to Cooper logarithm
and to SC properties comes from the ungapped
electron states on the Fermi level.
Empty band
E
Small ungapped pockets on a FS sheet get formed
when the antinesting term in the electron
dispersion exceeds DW energy gap.
2?
?
The quasi-particle dispersion in these small
pockets
ungapped pockets
ky
where
Effective mass
46
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

..

where g0(Q)ltlt1 is the bare interaction,
This gives
and the susceptibility may diverge at
some (nesting) wave vector, so that Then
the new coupling also diverges at some Q.
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
doesnt change for almost the whole FS except
hot spots.
47
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)
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)
Y. Tanaka and K. Kuroki, PRB 70, 060502(R) (2004)
48
The density of states at the Fermi level (1)
D1
Without DW the DoS in Q1D metal is
In the presence of DW
or
where
and for small FS pockets
49
Result1 Comparison of singlet Tc on metallic,
CDW and SDW states without change of e-e
interaction
17
1. Normal metal background
and
? is the size of the ungapped parts of FS
2. CDW background
and
Not too small.
3. SDW background
which gives very low Tc
50
Why the spin structure of SDW background
suppresses the spin-singlet superconductivity
(illustration)
Spin-dependent scattering the sign of the
scattered electron wave function depends on its
spin orientation.
spin-triplet SC pair
-QN
singlet SC pair after scattering by SDW
Direct SC singlet pairing
The two-electron wave function acquires ? sign
after scattering by SDW if the electron spins in
this pair look in opposite directions.
QN
Fermi surface
Nesting vector QN
This affects only the infrared divergence in the
Cooper logarithm. The ultraviolet divergence
remains unchanged.
16
51
Nature of superconductivity in (TMTSF)2PF6
19
! 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
Critical magnetic field Hc exceeds 5 times the
paramagnetic limit
absorption
I.J. Lee, P. M. Chaikin, M. J. Naughton, PRB 65,
18050(R) (2002)
I.J. Lee et al., PRB 68, 092510 (2003)
52
Result2 Comparison of triplet Tc in normal
metal, CDW and SDW background for
18
1. Normal metal background
and
2. CDW background
and
Not too small.
3. SDW background at
which gives
53
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)
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)
Y. Tanaka and K. Kuroki, PRB 70, 060502(R) (2004)
54
Enhancement of the e-e coupling helps to SC
3
(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)
SC transition temperature considerably increase
as the DW instability is approached. This
increase is attributed to the critical
fluctuation.
55
Two mechanisms of microscopic coexistence of
superconductivity or normal metal with SDW
29
1. Ungapped pockets of FS.
Empty band
E
2?
?
The antinesting dispersion
ungapped pockets
ky
2. Soliton phase (non-uniform).
E
The SDW order parameter depends on the coordinate
along the 1D chains
?
soliton band
2?
or
ky
56
Solitons in CDW or SDW.
31
Soliton phase.
The SDW order parameter depends on the coordinate
along the 1D chains
Each soliton costs energy
E
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
?
Boundaries E_ of the soliton level band
2?
ky
Schematic picture of energy bands in the soliton
phase in Q1D case.
57
Upward curvature of Hc2(T)
insulator
SC
Solitons create a layered structure, which is
described by the Lawrence-Doniach model of 1D
Josephson lattice.
Upper critical field in this Josephson lattice is
s
where s is the interlayer distance.
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
58
Lawrence-Doniach model
insulator
SC
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).
s
Here
59
Lawrence-Doniach model (2).
Introducing
The lowest eigenvalue of this equation gives
upper critical field
60
Gorkov equations with forward and backward
scattering
22
Self-consistency equations for superconductivity
order parameter
L
L
R
L
R
R
L
L
L
L
L
R
L
f LR
f RL
f LR
f RL
f LR
R
R
L
R
L
R
L
R
L
R
R
R
R
R
R
L
L
R
L
R
R
R
L
R
R
R
f RL
f LR
f RL
f LR
f RL
L
R
L
R
L
L
R
L
R
L
L
L
L
backward scattering
forward scattering
In analytical form this rewrites
61
Equations on Tc with forward and backward
scattering
23
1. Normal metal or CDW background.
Singlet SC equation
Triplet SC equation
2. Superconductivity on SDW background.
Singlet SC equation
Triplet SC equation
62
Discussion
24
1). 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.
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 opposite sign 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.
63
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.)

64
The Green functions in the uniform SDW state.
10
The equations for the Green functions in the SDW
state
In the matrix form these equations rewrite
where the matrix Green function
Diagonalization of the 2x2 matrix Hamiltonian
gives the new energy spectrum
where
65
Expressions for the electron Green functions in
the SDW state
11
The diagonal elements of the Green function
matrix
The non-diagonal elements of the Green function
matrix
66
Equations for superconducting instability
12
The Gorkov functions at t1t20
In the uniform phase
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
L
L
R
L
R
R
L
L
L
R
R
f LR
f RL
f LR
f LR
f RL
R
R
L
R
L
R
L
R
L
R
L
R
R
L
L
R
L
R
R
R
L
L
f RL
f LR
f RL
f RL
f LR
L
R
L
L
R
L
R
L
R
L
R
67
Equations for SC instability in SDW phase
13
With backward scattering only the SC equation are
due to SDW spin structure
If we introduce the diagonal and non-diagonal
Cooper bubbles
the self-consistency equations for
superconductivity rewrite
68
Singlet superconductivity in SDW or CDW.
14
The self-consistency equations for
superconductivity
The spin-singlet superconducting order parameter
Using the commutation identity
for spin-singlet pairing with
we obtain the SC equation on SDW background
The SC equations on the CDW background would be
69
Triplet superconductivity in SDW or CDW.
15
The self-consistency equations for
superconductivity
The triplet superconducting order parameter is
Using the commutation identity
for triplet pairing with we
obtain the SC equation on SDW background
Infrared singularities cancel each other as for
singlet SC on SDW.
one obtains
For
Infrared singularities do not cancel.
while for
one has
70
Illustration of the cancellation of different
contributions to the SC order parameter on the
SDW background
Spin-dependent scattering the sign of the
scattered electron wave function depends on its
spin orientation.
SC pairing after scattering by SDW wave vector
-QN
Direct SC pairing
The two-electron wave function acquires ? sign
after scattering by SDW if the electron spins in
this pair look in opposite directions.
QN
Fermi surface
Nesting vector QN
This affects only the infrared divergence in the
Cooper logarithm. The ultraviolet divergence
remains unchanged.
16
71
Result1 Comparison of singlet Tc in metal, CDW
and SDW states without renormalization of e-e
interaction
17
1. Normal metal background
and
? is the size of the ungapped parts of FS
2. CDW background
and
Not too small.
3. SDW background
which gives very low Tc
72
Result2 Comparison of triplet Tc in normal
metal, CDW and SDW background for
18
1. Normal metal background
and
2. CDW background
and
Not too small.
3. SDW background at
which gives
73
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).
74
Summary
46

• We developed the theory, describing
  superconductivity on SDW or CDW background when
  TcSDWgtgtTcSC in quasi-1D compounds with one
  conducting band.
• 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.
• 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 TcSC even to the value higher than
  without DW.
• 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.
• The SDW background strongly damps singlet SC. The
  SC, appearing on SDW background should be
  triplet.
• 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 .