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![Page 1: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/1.jpg)
B.Spivak
University of Washington
with S. Kivelson, and P. Oreto
Stanford University
Theory of disordered D-wave superconductors
![Page 2: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/2.jpg)
D-wave order parameter in pure superconductors
an assumption: e-e interaction in D-channel is attractiveand in S-channel is repulsive
elastic electron scattering destroys D-wave superconductivity when the electron mean free path l becomes of order of the superconducting coherence length.
0)'(
)()',( )'(
rr
kdekrr rrki
(k)
![Page 3: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/3.jpg)
++
--
s
“corner SQUID” experiment which demonstrate d-wave symmetry of the order parameter in HTC
D. Geshkenbein, A.I. LarkinJ.R. Kirtley at all
![Page 4: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/4.jpg)
“Conventional” phase diagram
normal metalD-wave
near the point of the transition the order parameter has global S-wave symmetry regardless of its symmetry in pure state
The phase diagram of disordered D-wave superconductors
T
disorder
disorder
T
D-wave S-wave?
normal metal
![Page 5: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/5.jpg)
Possible definitions of the global S-wave symmetryin bulk samples :
1. corner SQUID experiment shows global s-wave symmetry of the order parameter
2. the quantity < Fs(r)> = <F(r=r’)> is nonzero. (The brackets stand for averaging over realizations of random potential.)
3. the system has s-wave global symmetry if P+ – P- > (<)0. P+ and P- are volume fractions where F( r=r')= Fs(r) has positive or negative sign, respectively.
before averaging over realizations of disordered potential theorder parameter (r,r’) (and the anomalous Green function F(r,r’) ) do not have any symmetry.
![Page 6: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/6.jpg)
+
d-wave superconducting puddle embedded into disordered normal metal. outside the puddle s-wave component of the order parameter is generated. Only this component survives on distances larger than elastic mean free path l
+--
-
--
-
--
-
-
-
-
-
-
-
--
+++
+ ++
+ and – indicate signs of the s-components of the anomalous Green function Fs(r,r)
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
--
-
![Page 7: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/7.jpg)
metalnormaltheintcoefficiendiffusionelectrontheisD
drFrF
rirFrird
rdD
tr
ss
str
,
,sin,;0,sin,
2
2
in diffusive metal s-component of the anomalous Green function Fs(r)=F(r,r) is described by the Usadel equation
boundary conditions at D-N boundary are after Nazarov at all
-
----
+
+-
Ds
trDs
rrF
DL
L
r
rrF
1)(
;exp1
,2
![Page 8: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/8.jpg)
..)( ccejE jii
ij
dij
if puddle concentration is big the order parameter has global d-wave symmetry, while the s-component has random sample specific sign
+
+
+
+
+
+
+
++ +-
--
---
- -
- --
++ +
+ +++ -
-
-
-
Effective mean field energy
Jij(d) is the Joshepson coupling energy between D-wave
components
![Page 9: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/9.jpg)
if the concentration of superconducting puddles is smallthe order parameter has s-wave global symmetry, while the d-wave component has random sample specific sign
+
+
+ +
+ +
+ ++
+ ++
+
++
+
++
++
+
+
++
-
-
-
-
-
-
-- -
-
-- -
-- -
-
-
-
-
-
-
-
-
-
![Page 10: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/10.jpg)
ii
ii
iij js
ij
i
ji
estategroundthein
randomareccejE
1.;.)(
more realistic picture superconducting puddles embeddedinto a metal
+ +
++++
++
-
--
--
-
--
effective energy of the system is equivalent toMattic model in the spin glasses theory :
-
- -
-
---
-+ ++
+
Jij(s) is the Joshepson coupling energy between S-wave
components
-
-
![Page 11: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/11.jpg)
randomare
ccejjE
i
i
ij
dijij
sij
ji
1
..)()(
an effective energy at intermediate concentration of superconducting puddles
is there a superconducting glass phase when J(s) ~ J(d) ?
![Page 12: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/12.jpg)
1 jiijjiij JJX a criterion of the transition:
i is the susceptibility of a puddle
Jij is the Joshepson coupling between puddles
and Jij are random quantities
near the point of disordered quantum phase transition the system can be visualized as superconducting puddles connected by Joshepson couplings. Characteristicinterpuddle distance is much bigger than their characteristic size.
![Page 13: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/13.jpg)
a model: superconducting puddles of random radiuses Ri
are embedded into a repulsive normal metal .this model demonstrates a superconductor-metal transition
s sN
riri
Ri
Rc ~ is the critical radius
22
2
exp2
)(R
RR
R
NRP
R
i
R
i
In the framework of superconductor-insulator transition this model has been considered by Caldeira, A. Legget, B. Haperin, S. Kivelson, A. Luther, S. Chakrovarty, A.Schmid, W. Zwerger, S. Girvin, M. Fisher, S. Sachdev, N. Read, V. Ambegaokar, G. Schoen, U. Escern, D. Fisher, P. Lee, ..……………
![Page 14: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/14.jpg)
to find a critical puddle concentration Nc we use a procedure similar to that introduced by N. Mott in the theoryof variable range hopping conductivity
optRRoptoptopt JX
1. in a space of Ri we introduce an interval of a width of order R which is is centered at Ropt
2.
3. let us find a maximum of Xopt as a function of Ropt.
4. one can find the value of N=Nc from a requirement
1max optX
![Page 15: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/15.jpg)
At small temperatures the superconductingsusceptibility of puddles is exponentially big,while the Joshepson coupling between the puddles decay with the inter-puddle distance slowly. Consequently, at the point of s-superconductor-metal transition the distance between the puddles is parametrically bigger than their size.
Dij
ijA
i rJAe
1;1,
![Page 16: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/16.jpg)
near the point of the transition the order parameter has global S-wave symmetry regardless of its symmetry in pure state
The generic phase diagram of disordered D-wave superconductors
disorder strength
T
D-wave S-wave? normal metal
superconducting glass?
![Page 17: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/17.jpg)
conductivity of the “metal” is enhanced
Hall coefficient is suppressed
magnetic susceptibility is enhanced
in which sense such a metal is Fermi liquid?For example, what is the size of quasi-particles ? Is electron focusing at work in such metals ?
properties of the exotic metal near the quantumsuperconductor-metal transition:
![Page 18: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/18.jpg)
Conclusion:
in between of D-wave and normal metalphases there is a superconducting phase with S-wave “global” symmetry
![Page 19: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/19.jpg)
inter puddle Joshepson energy has a long range character
2/1)()(
2
0
*
0
2)(
/);||
exp()0()(
cos
,|||;)
;.).(
,/)(||;)
0|)|ln1(
1
||
TDLL
rrJTJ
VR
DGC
eRRRb
ccJVVC
RRRRRRa
Trrrr
CJ
trTT
jisij
sij
ijitr
eff
iicc
ijjiij
cciicc
jiND
ji
sij
i
D is the dimensionality of space, Dtr is the diffusion coefficient, V~RD is the volume of the grain,Geff is the “effective” conductance of the grainN is the repulsive interaction constant in s-channel
![Page 20: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/20.jpg)
susceptibility of an individual puddle depends on value of Ri-Rc)
0
2
2
20
42
/,
)'(
|)'()(|'
4
||||
)(
VV
ddR
RRdS
actionLandauGinzburg
iii
ii
c
icii
1) |R-Rc|<< Rc
a) R-Rc<0,
tdtiii 0*
ici
ci
icii RR
R
RRR
~;
/)(||
1~)(
![Page 21: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/21.jpg)
)||(
;)(
exp
)()(;'
'
'sin
||;0/)(||
0
02
2
0
ordersametheoftooncontrubutiagaveofnsfluctuatio
R
RR
R
RRV
R
RRAdtdt
tt
ttAS
nsfluctuatiophasetheofonContributi
eRRR
c
ciii
c
cii
c
cii
ici
susceptibility increases exponentially with (Ri-Rc) !
b) R-Rc>0,
![Page 22: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/22.jpg)
2. R~Rc quantum fluctuations of the order parameter are governed by the quantum fluctuation of EMF
,2
3
''
'sin
2
2
2
De
De
dtdttt
ttGS
D
fff
G
G
eff
Geff and G2D are conductances of a cube of normal metal of size R, and 2D normal film respectively
susceptibility is an exponential function of Geff
Kosterliz
Fegelman, Larkin, Skvortsov
![Page 23: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/23.jpg)
at T=0 the distance between optimal puddles is exponentially larger than they size !
0]exp[ 22 TN Ropt
this is a generic picture of any quantum phase transition in a metal with disorder.
![Page 24: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/24.jpg)
more realistic model :
2
0
20
22
3
]2
exp[2
1
lc
l
Gll
islengthncorrelatiothe
l
lllP
pathfreemeantheoffunctionondistributi
![Page 25: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/25.jpg)
D-wave globalsymmetry
phase diagram of D-(or P) superconductors as a function of disorder (T=0)
super-conducting glass ?
S-waveglobalsymmetry
normalmetal
strength of disorder
region where quantumfluctuations of the order parameter are important
it is likely that this sequence of phases take place on over-doped site of HTC.
![Page 26: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/26.jpg)
near the point of the transition the order parameter has global S-wave symmetry regardless of its symmetry in pure state
The phase diagram of disordered D-wave superconductors
disorder
T
D-wave S-wave
superconducting glass ??
normal metal
![Page 27: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/27.jpg)
Sr2RuO4
A. P. Mackenzie at all
can we say the same thing about superconducting ruthanates which are suspects for P-wave superconductivity?
![Page 28: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/28.jpg)
conductivity of the “metal” is enhanced
Hall coefficient is suppressed
magnetic susceptibility is enhanced
in which sense such a metal is Fermi liquid?For example, what is the size of quasi-particles ? Is electron focusing at work in such metals ?
properties of the exotic metal near the quantumsuperconductor-metal transition:
![Page 29: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/29.jpg)
Conclusion:
in between of D-wave and normal metalphases there is a superconducting phase with S-wave “global” symmetry
![Page 30: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/30.jpg)
Other theoretical possibilities:
a. Quantum S-wave superconductor-metal transition in an external magnetic field
b. If the electron-electron interaction constant has random sign the system may exhibit quantum superconductor-metal transition
![Page 31: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/31.jpg)
If Ri <Rc and the variance Ris small, than
Nc~1/RD
this result can be obtained on the level of mean field
DrrrrK
arrrKrdrr
|'|
1)',(
||)'()',(')()( 2
interaction constants S >0.
![Page 32: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/32.jpg)
Examples of experimental data
![Page 33: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/33.jpg)
Rc = 4 h/e
Insulator
Superconductor
Bi layer on amorphous Ge.Disorder is varied by changing film thickness.
Experiments suggesting existence of quantumsuperconductor-insulator transition
![Page 34: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/34.jpg)
Ga layer grown on amorphous Ge
Insulator
Superconductor
Experiments suggesting existence of quantumsuperconductor-metal transition
A metal?
![Page 35: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/35.jpg)
T=0 superconductor-metal transition in a perpendicular magnetic field
There are conductors whose T=0 conductance is four order of magnitude larger than the Drude value.
N. Masson,A. Kapitulnik
![Page 36: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/36.jpg)
superconductor – glass transition in a magnetic field parallel to the film
There are long time (hours) relaxation processes reflected in the time dependence of the resistance.
H||
The mean field theory:The phase transition is of first order.
W. Wu, P. Adams,
Hc
![Page 37: B.Spivak University of Washington with S. Kivelson, and P. Oreto Stanford University Theory of disordered D-wave superconductors.](https://reader036.fdocuments.in/reader036/viewer/2022062309/56649f1c5503460f94c31c71/html5/thumbnails/37.jpg)
Is it a superconductor-glass transition in a parallel magnetic field?
W. Wo, P. Adams, 1995