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Why d-wave superconductivity?
Kazumi Maki, Hyekyung Won
To cite this version:
Kazumi Maki, Hyekyung Won. Why d-wave superconductivity?. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.2317-2326. �10.1051/jp1:1996220�. �jpa-00247315�
J. Phys. I France 6 (1996) 231î-2326 DECEMBER 1996, PAGE 2317
Why d-wave supercoJJductivity?
1(azumi Maki (~>*) and Hyekyung Won (~)
(~) Department of Physics and Astronomy, University of Southem Califomia. Los Angeles, CA90089-0484, USA
(~) Department of Physics, and TRC, Hallym [Îniversity, Chunchon 200-702. South Korea
(Received 8 august 19§6, accepted 19 August 1996)
PACS.74.20.Mn Nonconventional mechanisms (spm fluctuations, polarons and bipolarons, resonating valence bond mortel. anion mechamsm, marginal Fermi hqmd, Luttinger Iiqmd, etc.)
PACS.74.25.Bt Thermodynamic properties
PACS.74.60.Ec Mixed state, critical fields, and surface sheath
Abstract. In recent years we have seen remarkable success of d-wave superconductivity m
descnbing a variety of features of high T~ superconducting cuprates. After brief introduction
on d-wave superconductivity we shall describe some fascinating properties of the vortex state in
high T~ cuprates.
1. Introduction
10 years after the epoch making discovery of high-T~ superconducting cuprates (HTSC) by
Bednorz and Müller iii, we are now experiencing a conceptual revolution about physical origin
and nature of high-T~ superconducting cuprates [2, 3]. In earlier days rather unusual properties of both normal and superconducting state of these compounds were quite disturbing, which leads many people to abandon cherished theoretical frameworks established for ordinary super- conducting metals; Landau's Fermi liquid theory [4j and BCS theory of superconductivity [5j.
The recent progress owes in part availability of well characterized HTSC monocrystals and
in part theoretical understanding of 2D and 3D Fermi liquid. In particular through an ele- gant renormalization group analysis the stabiiity of 2D and 3D Fermi iiquid is established [6].
Also the development of bosonization technique m 2D and 3D Fermi hquid leads to the same conclusion [7-9]. What becomes clear in the above analysis is the importance of the nesting channel, which drives the Ferini liquid system to charge density wave or spm density wave depending on whether the interaction potential between two electrons are attractive or repul-
sive. Indeed HTSC'S are layered compounds and considered as quasi-two-dimensional. Further
the fundamental Cu02 plane forms two-dimensional square lattice where Cu occupies the lat- tice points while O on the bond between two neighboring lattice points. In such a situation the nesting channel with q
= Q e (~,~) plays the crucial rote. Indeed the importance of the nesting channel m HTSC is first pomted out by Virosztek and Ruvalds [10, iii. The in-
teraction between two electrons (or two holes) in metals m general is due to the repulsive
(*) Author for correspondence (e-mail: kmaki@use.eau)
© Les Éditions de Physique 1996
Coulomb interaction and the phonon exchange. Then when the phonon exchange dominâtes the Coulomb interaction we expect charge density wave (CDW) or s-wave superconducting ground state. A simple example is provided by NbSe3 where the destruction of CDW under
pressure is followed by s-wave superconducting state [12]. In contrast the appearance of spin density wave [SDW) or antiferromagnetic state indicates clearly the dominance of the Coulomb
interaction. Indeed already in 1987 Anderson [13] proposed 2-D il band) Hubbard mortel or its descendant, t-J mortel in order to describe HTSC from their phase diagram [14]. The t-J mortel for hole-doped HTSC is now further evolved [15-19]. We believe that 2D t-J or ex- tended t-J mortel should be the simplest mortel for HTSC. When the antiferromagnetic order
is destroyed by hole-doping, there ~vill be still strong antiferromagnetic spin fluctuation (or antiparamagnon) centered around q
= Q as is seen by inelastic neutron scattering experiment from YBCO [14] and from LSCO [20] monocrystals and shown theoretically as well [21-23].-
In general the exchange of vector bosons like antiparamagnon between 2 electrons result in a
repiùsive interaction. Here the magic of d-wave superconductivity enters. As we shall see later, /h[k + Q)
= -/h(k) for the d-wave superconductor; the repulsive interaction switches to the attractive interaction due to the minus sign. This is the antiparamagnon scenano [24-26] of
d-wave superconductivity. We may think this also another realization of the Kohn-Luttinger mechanism [27]
Here we shall not enumerate the success of d-wave mortel, since they are described elsewhere [2, 3]. Perhaps we have to stress a puzzling possibility that the superconductivity of electron
doped Ndi-~Ce~Cu04 may be of s-wave [28].
In the following we shall first summarize some of basic properties of d-wave superconductiv- ity [29]. Then we shall go on to describe intriguing properties of the vortex state m a magnetic
field B ) c.
2. Preliminary
Here we shall describe briefly the result of weak coupling mortel for d-wave superconductiv- ity [29]. As a mortel we take the interaction between two partiales
NOV(k, k')
= 2Àcas(2çi]cas(2p') ~t iii
where No is the electron density of states m the normal state per spin, is the dimensionless attractive interaction while ~t is the Coulomb repulsion in the s-channel [30, 31]. Also equa- tion il) is a simplified version of the potential used in studying the d+s mixture in the other
context [32]. Finally çi and çi' are the angle k and k' [both in the a b plane) makes from the
a axis. Then the standard gap equation is given by
~~ Î ~~~ ~°~~~~~~~~
/E2
~2
cos~(2çi)
~~~~ Î~ ~~~
where .) means average over à and E~ is the cut-off energy. We note that in the homogeneous
situation Iie. /h(x, k) independent of x) the Coulomb potential plays no role. Equation (2) gives in particular
T~ = 1.136 E~e~~/~ /ho(+ /h(01) " 2.14T~ (31
N°12 WHY d-WAVE SUPERCONDUCTIVITY? 2319
where the first one is identical to the one for s-wave superconductor, but /ho is abolit 20~
larger than the one for s-wave superconductor [5]. Further
1 3((3)(Tllho)~ + O(Tllho)~ for T < T~
/hiTillho
= ~~~n ~ i/2
~n 1/2 14)
/ho 21((3) T~
~°~ ~ ~ ~
and A(k)
= zlcos(2çi). Here ((3)
= 1.202... is Riemann's zeta function and the temperature dependence is quahtatively not much different from the one for s-wave except for larger /ho.
Also the temperature dependences of the specific heat and the superfluid density are given by
C~ = 18((3)No ~~
= ~'~j~~~N ~~ (5)
£lo
~
£lo
~~~~~ ~~~ ~Î
(~ô( /~ÎÎÎSÎÎÎ)
)3/2
1 2(In2) ~ + O(T~), T < T~
~o
~
~ (l ~ ), T m T~ (6)
3 Tc
respectively. In the clean hmit the superfluid density is accessible bath through the magnetic penetration depth à(T) and the Knight shift in NMR, since
PslT)
"
(j) ,
~s lin " Î PslT) I?l
and x~ and xn are the spin susceptibility in the superconducting and the normal state respec-
tively. The T-hnear dependence of àab(T) observed in an YBCO monocrystal [33] constitutes
one of the first expenments indicating a d-wave symmetry. Similarly the T~-dependence of low temperature specific heat is experimentally established for YBCO [34] and LSCO [35]
monocrystals. Also the same model predicts T~-law of the nuclear spm lattice relaxation rate
Tp~ in the low temperature limit which has been seen m 6~Cu-nuclei of YBCO [36].
It is now well established that Zn-substitution of Cu m the Cu02 plane has dramatic effects
on superconductivity; rapid suppression of superconducting transition temperature T~ and rapid appearance of the residual density of states Ii.e. the electronic density of states on the Fermi surface) [37-42]. The Zn impurity may be modeled as an impurity with the scattering
m the umtarity limit Ii.e. resonance on the Fermi surface). As an example the quasi-partiale density of states with different impurity concentration is shown in Figure 1. Iii the absence of the impurity the density of states is given by
~ xK(x) for z < 1
N(E) /No ~
= (8)
~K(z~~) for z > 1
~
where z
= Ellh and K(I) is the complete elhptic integral. In particular NIE) /No iiicrease like Ellh for small E. The energy dependence of the density of states appears to be now well
2.0
1.5
,-,
, ,
o ,
z ,
,
~ ~
~
~G/
o.5 ~'~'~
~'
o-o
o-o 0.5 1-ù 1.5 2.0
E / A
Fig. 1. The density of states for r/à
= 0 j-Î, 0.01(. ), 0.05 (- ), 0.1 (-.-.-.i, and 0.2 (---) vs.
E là (from Hotta [37]1.
established by STM spectroscopy for B12Sr2CaCu2 Os (B12212 [43,44] and YBCO [45] in spite of early controversies on the subject [46].
When a small amount of impurity is added, it will produce a small peak at E = 0. As the
impurity concentration increases the peak broadens very rapidly. This feature is clearly seen by low temperature specific heat [15] and NMR Knight shift [47].
Of course trie sign change of /h(k) as you go around in the k-plane from the k~-direction to
the ky-direction is most clearly seen by the phase sensitive Josephson interference expenments.
However since there are now a few excellent reviews [48, 49] on the subject, we will not touch
on this subject.
3. Vortex state in the vicinity of B
= H~2
Here we limit ourselves to the vortex state m a magnetic field parallel to the c axis. First the
upper critical field is obtained by solving the linearized gap equation à la Luk'yanchuk and
Mineev [50]
~ = cosj2~jji + cjatj4j + àjat12iio1 191
where ut
= (2/1)~~(-i ~ + ~ +2ieBx) and )0) is the reference Abrikosov state [51]
ôz ôy
as in s-wave superconductor [52]. As we shall see the coefficient C plays the crucial role
m the stability of the square vortex lattice while coefficient b describes the admixture of s-
wave component through Coulomb potential. The upper critical field H~2(t) thus determined
is shown in Figure 2 which is rather similar to the one for s~wave superconductor. More remarkable is the Abrikosov parameter @_4 = (1/h(x))~)/()/h(x))~)~ which is calculated for a
N°12 WHY d-WAVE SUPERCONDUCTIVITY? 2321
120
ioo
~ go
%~ 60 Z$
40
20
0
0 0.2 0.4 0.6 0.8
t=T/T~
Fig. 2. The Upper critical field Hc2 is shown as function of the reduced temperature t
= T/Tc for
Aie À~~ + ~L~~)
# oc (sobd Iine), A
= 10 (dashed Iine), and A
= 7 (dashed-dot Iine). Here we took
ôH~2(t) /ôT)T=r~
= -1.9 T/K.
1.22
1.2 "....,_
_~
l.18
cill.16
1.14
1.12
1.1
0 0.2 0.4 0.6 0.8
t=T/ T~
Fig. 3. The Abrikosov parameters, PA's, of a square Iattice with à
= ~/4 and a hexagonal Iattice
are shown as functions of the reduced temperature t = T/T~ for A = oc (sohd Iine), A
= 10 (dashed hne), and A
= 7 (dotted Iine). The PA's of a square Iattice with à
= 1/4 are smaller than the ones for
a hexagonal Iattice for T < 0.8Tc, implying the square lattice is more stable at Iow temperatures.
hexagonal lattice aiid for a square lattice tilted by ~/4 from the a axis. As is readily seen from Figure 3. the hexagonal lattice is more stable iii the Ginzburg Landau regime (fit < Pi for
T cf T~ as m s-wave superconductor. As the temperature decreases Pi(~ /4) decreases rapidly
and pi(~ /4) < pi for T < 0.8T~ independent of the strength of ~t, Coulomb potential. This
means for T < 0.8T~ the square vortex lattice becomes more stable. It is noteworthy that
Pi(9) depends strongly on the orientation of the square lattice 9 from the a axis and takes the minimum value for 9
= ~/4, while Pi is mdependent of the lattice orientation. We believe that the stability of this particular square lattice owed to the electron exchange between two
neighboring vortices which is available only when they align parallel to il, 1, 0j or il, -1, 0j
directions. Indeed such square lattices are observed recently by small angle neutron scattenng expenment [53] and STM imaging one [45] both on YBCO monocrystals at low temperatures, though the lozenge is elongated in the b direction. However, we beheve this distortion is due to the orthorhombicity of YBCO monocrystals. The STM image of a vortex core shows
a black ellipse ii-e- no zero energy excitation) [45]. If we mterpret that its semi axis are
proportional to fa and (b. we will have (b/(a CÎ 1.5 which is fully consistent with the above distortion. We note there are alternative interpretations of rhombic lattice based on the d + s
admixtures [54, 55]. We have shown earlier that de Haas van Alphen effect should be visible in the vortex state at least for 1 > > 1/4, where b
= B/H~2(T) [52,56]. For this purpose Green's function introduced by Brandt, Pesch and Tewordt [57,58] for a clean type II superconductor
is extremely useful. We predict that de Haas van Alphen effect exists also in the vortex state.
In the vortex state both the Dingle temperature and the quasi-partiale mass are changed as
1 ~2
~~ ~ ~~~ ~
/ f ~~~~
~~~~ 1+1(il/f)2
~
~~~~
where e = uF(2eBl~/~ and T(~ is the Dingle Temperature due to the impurity. Also at low temperatures /hIf is approximately given by (1h/fi~ cf Il/3.91(b~~ -1) [52]. In the vortex state the Dingle temperature increases while the effective mass decreases with decreasing B, When the magnetic field is tilted from the c axis, it is also possible to explore the fourfold symmetry of /h(k) as (F(/h(k)))orb where (...)orb is taken over the quasi-particle orbit perpendicular to
the magnetic field [59].
4. Vortex state when B < H~2
A quasi-dassical analysis of isolated vortex shows the quasi-partiale spectrum with fourfold symmetry [60]. On the contrary a recent STM spectroscopy and imagmg of the vortex of
an YBCO monocrystal [45] suggests the vortex has cylindrical symmetry. We believe that it
is due to the breakdown of the quasi-classical approximation m iBCO in particular and in
high T~ cuprates in general. Perhaps a big surprise is that instead of peaks in the zero biased density of states at the vortex sites as seen in the vortex state of s-wave superconductor in NbSe2 Î61], the vortex core m YBCO is empty of the density of states [4.5]. The peaks in the
density of states in the vortex core of s-wave superconductor is well accounted for in terms of the detailed solution of Bogoliubov-de Gennes equation [62]. According to Caroli, de Gennes and Matricon [63] a series of bound states are formed with energy
Em = ~ (m + )/h(pF()~~ with m = 0.1.. (12)
around a vortex where pF is the Fermi momentum and ( is the coherence length (r~ i,Fllh).
In ordinary (or classic) superconductors pF(
r~ (10~ rw 10~) and the series of bound states are
well approximated by a continuum resulting m the concept of the normal core. Also in such
a vortex the energy dissipation is mostly due to the normal core as envisioned by Bardeen
and Stephen [64]. Also this gives rise to the low temperature specific heat proportional to
(B /©DÎT. On the other hand in YBCO it is found Eo " 64 K ci il/41/h [45] where Eo is the energy of the lowest bound state. This implies pF( ~- 1 or /h
rw BF1 the vortex state in YBCO
is in the quantum limit. Though this conclusion may appear some~v.hat surprising, we have
N°12 WHY d-WAVE SUPERCONDUCTIVITY? 2323
2
1.6
~i.2
~
o-g
0.4
0
0.4 0.5 0.6 0.7 0.8 0.9
x
Fig. 4. The ratio Eo là is shown as function of dopage x, At O we used measured value Eg [14j,
shown earher [21] from the analysis of inelastic neutron scattering experiments from YBCO moiiocrystals by Rossat-Mignod et ai. [14j that the spm gap Eg at T = 0 K is given m terms of the chemical potential ( measured from the half-filling Eg = 2)(). Then making use of the
data of Eg for YBa2Cu306+~, we have deduced [65]
( = -345(z 0.45) K (13)
with ( = -190 K for an optimally doped YBCO. Indeed shghtly modifying the quasi-classical limit result (12), we may predict
Eollho
=
0.51~ (14)
1(1
where we assumed rlo = 2.80T~ for YBCO. We show equation (14) in Figure 4 as function off the oxygen dopage. We expect that the ratio Eollho mcrease in the underdoped region.
We note also this small chemical potential is very consistent with the temperature dependence of the chemical potential in YBCO, which is determmed through the change of the work function [66,67j. Of course in this latter experiment the chemical potential (rw 10~ K) is
measured from the bottom of the band.
Within the quasi-classical approximation Volovik [68] has predicted the residual density of
states in the vortex state in d-wave superconductor mcreases as /Ù ~vith the magnetic field,
which is seen by specific heat measurement of YBCO monocrystal [34]. We have reached the
same conclusion [52] with different method, though unfortunately the experimental situation
is still controversial. We believe that this /Ù term should be accessible by NMR
as well and
provides the crucial dissipation mechamsm m the vortex dynamics of d-wave superconductor [69j. Also the vortex with empty core may be difficult to be pmned resulting m the extended
reversible magnetization region m the B T phase diagram. It appears that the vortex
dynamics m the quantum limit holds a lot of surprises.
5. Concluding Remarks
We have seen a rhombic vortex lattice is one of characteristics of d-wave superconductors.
Another charactenstic of high T~ cuprates is that they are in the extreme quantum hmit. This
