Lecture 1 - Unconventional superconductors Notes | EduRev

: Lecture 1 - Unconventional superconductors Notes | EduRev

 Page 1


Module 10 : Unconventional superconductors
Lecture 1 : Unconventional superconductors
 
Research on superconductors started with work on metals and alloys since the conventional wisdom at that time was
that a material must atleast be a good conductor to become superconducting. Over the years, a large variety of
superconductors have been discovered. In conventional superconductors, conduction electrons merge into pairs because
of their attractive interaction via phonon exchange. The total wavefunction of the Cooper pairs has an orbital part and a
spin part. In cases where their orbital angular momentum L = 0, (analogous to atomic physics) this state is called an s-
wave state. From symmetry arguments it follows that the total spin S = 0, i.e., there is a spin-singlet. There are many
superconductors which conform to the predictions of the BCS theory (and can be called conventional). However, there
are others which exhibit different and unusal features which seem to suggest the presence of different/alternative
mechanisms giving rise to superconductivity. Herein, we review superconductors which fall in this second category.
Organic superconductors
The superconductors discovered in the early years were ``inorganic'' in nature, i.e. not primarily based on the elements
C, H, O. Subsequently many superconductors have been found which are of the organic variety. While phonons are
considered to provide the route to an effective attractive interaction which gives rise to superconductivity, in the BCS
theory, it was suggested that the polarisation of the ligands in organic materials could mediate superconductivity.
Superconductivity was, in fact, discovered (under a pressure of 12 kbar) in (TMTSF) 
2
PF
6
 in 1980 with a T
c
 of 0.9 K.
Further research led to discovery of superconductivity in the family (TMTSF) 
2
X
6
 where X stands for PF6, TAF6, ReO4,
ClO4, etc. while TMTSF stands for tetramethyl-tetraseleniumafulvalene. Superconductivity was found in another family
of organic materials (BEDT-TTF) 
2
Y. The transition temperatures in these materials are low and further the mechanism
of superconductivity is not what was originally proposed. Nevertheless, there are unusual features associated with these
organic superconductors. The BEDT-TTF salts display low-dimensional (2D) behaviour where conductivity within the
planes is much greater than that perpendicular to the planes. A large anisotropy is also detected in the critical fields
and the penetration depth. The highest T
c
 (11.5K) has been found in (BEDT-TTF)
2
Cu[Ni(CN)
2
]Br. The organic
superconductors are thought to have s-wave pairing but many of their properties are found to deviate from the BCS
predictions. The possible pairing mechanism continues to be of interest in these materials.
Another carbon containing system which displays superconductivity is derived from C
60
, also called fullerene. It is not
considered in the organic molecule category and has an fcc structure with a large lattice parameter of 14.2. It is
possible to obtain superconductivity in C
60
 by doping with alkali or alkali-earth metals. The transition temperatures of
various C
60
-based superconductors ranges from . The doping concentration (or the density of conduction electrons) can
be varied by adjusting the relative amounts of alkali and alkali-earth elements in a compound, for instance, by varying
x in Na
2
Cs
x
C
60
. The T
c
 depends strongly on  and peaks at a point where the conduction band is half-filled. This is an
unusual feature and mimics the behaviour of cuprate high-T
c
 superconductors. The highest T
c 
is found in  and is
19 K. Somewhat like in cuprates, the penetration depth is of the order of 5000  and the coherence length is much
smaller and of the order of 30 . Likewise  and  are of the order of 10 mT and 50 T, respectively. The
fulleride superconductors are thought to be of the BCS type (with an s-wave order parameter) with intramolecular
vibrations playing an important role however, electron correlations are also thought to play a role. Finally, the C
60
-
based superconductors, though unusual, can perhaps be called of the conventional BCS type.
Superconductivity in magnetic systems
As mentioned earlier, the terminology s-wave, p-wave, etc. is used to classify (in analogy to atomic orbitals) the orbital
symmetry of the order parameter. The total (spin + orbital) symmetry of the order parameter/wavefunction must be
antisymmetric for Cooper pairs. In a BCS s-wave superconductor, the Cooper pairs are in a spin-singlet state and have
a zero orbital moment. In such a case, magnetic fields and magnetic moments are thought to be detrimental to
superconductivity. This is because local moments will tend to align the electrons in the Cooper pair in the same
direction (break the singlet) as also pull the pair apart due to a Lorentz force which acts in the opposite direction for
the two members (orbital effect). This is indeed observed in conventional superconductors with a strong reduction of T
c
when magnetic impurities are doped into it.
AF ordered systems
In contrast, coexistence of long-range antiferromagnetic order and superconductivity has been observed in Chevrel
phase compounds RMo S and RMo Se and as also in RRh B (here R stands for a rare-earth atom which carries
a moment). In these systems, a transition to the superconducting state takes place with decrease of temperature. With
a further decrease of temperature, transition to an AF ordered state is found at  (from specific heat, susceptibility,
and neutron scattering measurements) which coexists with the superconducting state. The critical field is found to be
depressed around  and suggests a connection between the AF ordering and the superconducting state. Note that,
though the carriers that are responsible for superconductivity and the local moments in the above systems belong to
different atoms, a strong interaction must exist between the two (local moment and the Cooper pairs) mediated by the
conduction electrons.
Page 2


Module 10 : Unconventional superconductors
Lecture 1 : Unconventional superconductors
 
Research on superconductors started with work on metals and alloys since the conventional wisdom at that time was
that a material must atleast be a good conductor to become superconducting. Over the years, a large variety of
superconductors have been discovered. In conventional superconductors, conduction electrons merge into pairs because
of their attractive interaction via phonon exchange. The total wavefunction of the Cooper pairs has an orbital part and a
spin part. In cases where their orbital angular momentum L = 0, (analogous to atomic physics) this state is called an s-
wave state. From symmetry arguments it follows that the total spin S = 0, i.e., there is a spin-singlet. There are many
superconductors which conform to the predictions of the BCS theory (and can be called conventional). However, there
are others which exhibit different and unusal features which seem to suggest the presence of different/alternative
mechanisms giving rise to superconductivity. Herein, we review superconductors which fall in this second category.
Organic superconductors
The superconductors discovered in the early years were ``inorganic'' in nature, i.e. not primarily based on the elements
C, H, O. Subsequently many superconductors have been found which are of the organic variety. While phonons are
considered to provide the route to an effective attractive interaction which gives rise to superconductivity, in the BCS
theory, it was suggested that the polarisation of the ligands in organic materials could mediate superconductivity.
Superconductivity was, in fact, discovered (under a pressure of 12 kbar) in (TMTSF) 
2
PF
6
 in 1980 with a T
c
 of 0.9 K.
Further research led to discovery of superconductivity in the family (TMTSF) 
2
X
6
 where X stands for PF6, TAF6, ReO4,
ClO4, etc. while TMTSF stands for tetramethyl-tetraseleniumafulvalene. Superconductivity was found in another family
of organic materials (BEDT-TTF) 
2
Y. The transition temperatures in these materials are low and further the mechanism
of superconductivity is not what was originally proposed. Nevertheless, there are unusual features associated with these
organic superconductors. The BEDT-TTF salts display low-dimensional (2D) behaviour where conductivity within the
planes is much greater than that perpendicular to the planes. A large anisotropy is also detected in the critical fields
and the penetration depth. The highest T
c
 (11.5K) has been found in (BEDT-TTF)
2
Cu[Ni(CN)
2
]Br. The organic
superconductors are thought to have s-wave pairing but many of their properties are found to deviate from the BCS
predictions. The possible pairing mechanism continues to be of interest in these materials.
Another carbon containing system which displays superconductivity is derived from C
60
, also called fullerene. It is not
considered in the organic molecule category and has an fcc structure with a large lattice parameter of 14.2. It is
possible to obtain superconductivity in C
60
 by doping with alkali or alkali-earth metals. The transition temperatures of
various C
60
-based superconductors ranges from . The doping concentration (or the density of conduction electrons) can
be varied by adjusting the relative amounts of alkali and alkali-earth elements in a compound, for instance, by varying
x in Na
2
Cs
x
C
60
. The T
c
 depends strongly on  and peaks at a point where the conduction band is half-filled. This is an
unusual feature and mimics the behaviour of cuprate high-T
c
 superconductors. The highest T
c 
is found in  and is
19 K. Somewhat like in cuprates, the penetration depth is of the order of 5000  and the coherence length is much
smaller and of the order of 30 . Likewise  and  are of the order of 10 mT and 50 T, respectively. The
fulleride superconductors are thought to be of the BCS type (with an s-wave order parameter) with intramolecular
vibrations playing an important role however, electron correlations are also thought to play a role. Finally, the C
60
-
based superconductors, though unusual, can perhaps be called of the conventional BCS type.
Superconductivity in magnetic systems
As mentioned earlier, the terminology s-wave, p-wave, etc. is used to classify (in analogy to atomic orbitals) the orbital
symmetry of the order parameter. The total (spin + orbital) symmetry of the order parameter/wavefunction must be
antisymmetric for Cooper pairs. In a BCS s-wave superconductor, the Cooper pairs are in a spin-singlet state and have
a zero orbital moment. In such a case, magnetic fields and magnetic moments are thought to be detrimental to
superconductivity. This is because local moments will tend to align the electrons in the Cooper pair in the same
direction (break the singlet) as also pull the pair apart due to a Lorentz force which acts in the opposite direction for
the two members (orbital effect). This is indeed observed in conventional superconductors with a strong reduction of T
c
when magnetic impurities are doped into it.
AF ordered systems
In contrast, coexistence of long-range antiferromagnetic order and superconductivity has been observed in Chevrel
phase compounds RMo S and RMo Se and as also in RRh B (here R stands for a rare-earth atom which carries
a moment). In these systems, a transition to the superconducting state takes place with decrease of temperature. With
a further decrease of temperature, transition to an AF ordered state is found at  (from specific heat, susceptibility,
and neutron scattering measurements) which coexists with the superconducting state. The critical field is found to be
depressed around  and suggests a connection between the AF ordering and the superconducting state. Note that,
though the carriers that are responsible for superconductivity and the local moments in the above systems belong to
different atoms, a strong interaction must exist between the two (local moment and the Cooper pairs) mediated by the
conduction electrons.
Interplay between magnetic and superconducting properties has also been found in the ternary borocarbides RNi B C
(R=Y, Tm, Ho, etc.). Compounds with the composition HoNi B C show very interesting magnetic/superconducting
behaviour. For , the substance becomes superconducting below 7 K but on further cooling, it becomes normal
around 5 K and finally regains superconductivity (re-entrant superconductivity) on cooling further. In this system,
superconductivity is lost when it orders in a spiral arrangement and superconductivity is recovered when it finally locks
into an AF ordered state.
Ferromagnetically ordered systems
Re-entrant behaviour is seen in ErRh B and HoMo S . The material becomes superconducting on decreasing
temperature but at a temperature lower than the superconducting , there is a transition to ferromagnetic order and
superconductivity is lost. Interestingly, in a temperature regime above the ferromagnetic Curie point, there exists a
modulated magnetic structure wherein the local magnetization varies sinusoidally in space. Since the periodicity of this
modulation is less than the superconducting coherence length, the Cooper pairs sense a net zero field on their size-
scale and superconductivity coexists with magnetism.
Magnetically mediated superconductivity
In the above examples, although there was coexistence of magnetism and superconductivity, it was through a
fortuitous situation where, on the length-scale of Cooper pairs (coherence length), no static magnetism was seen and
the pairing mechanism was still the electron-phonon mechanism. In contrast, we would like to ask, are there materials
wherein the magnetic interactions in fact mediate the pairing between electrons resulting in superconductivity?
CePd Si is a system which at ambient pressure is an antiferromagnet with K. As the pressure is increased,
the  drops and AF disappears around 26 kbar. Around this region of pressure, superconductivity appears while it
disappears for much higher or much lower pressures. The transition point at 26 kbar is thought to be a quantum critical
point (transition takes place at 0 K as a function of pressure). In the pressure region around the critical point one
expects large quantum fluctuations where regions of AF correlations appear and disappear. It has been shown
theoretically that these magnetic interactions can mediate superconductivity which can be thought of as a competing
ground state. It is thought that in the case of nearly ferromagnetic metals, the superconducting state should be a spin-
triplet with an odd orbital quantum number while for antiferromagnetic metals, it should be a spin-singlet. The
resistivity in CePd Si is found to have a  dependence on temperature. This unusual temperature dependence is
thought to be due to scattering of quasiparticles via magnetic interactions.
Superconductivity in itinerant ferromagnets
We now consider some compounds where superconductivity coexists with ferromagnetism and moreover the atoms
responsible for the magnetism are the same ones responsible for the carriers contributing to superconductivity. The
examples of such systems are UGe , ZrZn , and URhGe. There are various peculiar features relevant to these
systems. Firstly, the same band-like  or  electrons are responsible for both magnetism and superconductivity. As
temperature is decreased, the material first orders ferromagnetically. The electronic heat capacity coefficient  is large
(160 mJ/mol K for URhGe) indicating an abundance of low-energy magnetic excitations. Samples with a high purity
show a transition to superconductivity at even lower temperatures. However, the transition width is relatively large
probably since the sample experiences an inhomogeneous magnetic field due to the ferromagnetic domains. Moreover,
the diamagnetic response in an ac susceptibility experiment (in zero applied static magnetic field) is less than the ideal
Meissner reponse. This could be due to the large internal field which puts the sample in a mixed state.
In ZrZn , the ferromagnetic ordering temperature  and the superconducting transition temperature  both
decrease with increasing pressure and are both completely suppressed above 21 kbar. In this case then, proximity to
the quantum critical point does not assure superconductivity. The flux expulsion in ZrZn even in small magnetic fields
is negligible. The electrical resistivity does not go to zero and remains finite below . The specific heat does not
exhibit a discontinuity at  either. It is felt that superconductivity is inhomogeneous in ZrZn and perhaps has triplet
pairing.
UGe is ferromagnetic and non-superconducting at ambient pressures. It's  decreases with increasing pressure
and becomes zero at 15.8 kbar. In some pressure range (between 9 and 15.8 kbar), resistivity and  susceptibility
show that there is superconductivity with  <\textcompwordmark < .
Heavy-fermion superconductors
Superconductivity in a heavy-Fermion system CeCu Si was first discovered in 1979 by Steglich. Subsequently, other
heavy-Fermion superconductors have also been found. To start with, heavy-Fermion behaviour is itself fairly exotic and
in addition superconductivity in these systems is unusual.
Page 3


Module 10 : Unconventional superconductors
Lecture 1 : Unconventional superconductors
 
Research on superconductors started with work on metals and alloys since the conventional wisdom at that time was
that a material must atleast be a good conductor to become superconducting. Over the years, a large variety of
superconductors have been discovered. In conventional superconductors, conduction electrons merge into pairs because
of their attractive interaction via phonon exchange. The total wavefunction of the Cooper pairs has an orbital part and a
spin part. In cases where their orbital angular momentum L = 0, (analogous to atomic physics) this state is called an s-
wave state. From symmetry arguments it follows that the total spin S = 0, i.e., there is a spin-singlet. There are many
superconductors which conform to the predictions of the BCS theory (and can be called conventional). However, there
are others which exhibit different and unusal features which seem to suggest the presence of different/alternative
mechanisms giving rise to superconductivity. Herein, we review superconductors which fall in this second category.
Organic superconductors
The superconductors discovered in the early years were ``inorganic'' in nature, i.e. not primarily based on the elements
C, H, O. Subsequently many superconductors have been found which are of the organic variety. While phonons are
considered to provide the route to an effective attractive interaction which gives rise to superconductivity, in the BCS
theory, it was suggested that the polarisation of the ligands in organic materials could mediate superconductivity.
Superconductivity was, in fact, discovered (under a pressure of 12 kbar) in (TMTSF) 
2
PF
6
 in 1980 with a T
c
 of 0.9 K.
Further research led to discovery of superconductivity in the family (TMTSF) 
2
X
6
 where X stands for PF6, TAF6, ReO4,
ClO4, etc. while TMTSF stands for tetramethyl-tetraseleniumafulvalene. Superconductivity was found in another family
of organic materials (BEDT-TTF) 
2
Y. The transition temperatures in these materials are low and further the mechanism
of superconductivity is not what was originally proposed. Nevertheless, there are unusual features associated with these
organic superconductors. The BEDT-TTF salts display low-dimensional (2D) behaviour where conductivity within the
planes is much greater than that perpendicular to the planes. A large anisotropy is also detected in the critical fields
and the penetration depth. The highest T
c
 (11.5K) has been found in (BEDT-TTF)
2
Cu[Ni(CN)
2
]Br. The organic
superconductors are thought to have s-wave pairing but many of their properties are found to deviate from the BCS
predictions. The possible pairing mechanism continues to be of interest in these materials.
Another carbon containing system which displays superconductivity is derived from C
60
, also called fullerene. It is not
considered in the organic molecule category and has an fcc structure with a large lattice parameter of 14.2. It is
possible to obtain superconductivity in C
60
 by doping with alkali or alkali-earth metals. The transition temperatures of
various C
60
-based superconductors ranges from . The doping concentration (or the density of conduction electrons) can
be varied by adjusting the relative amounts of alkali and alkali-earth elements in a compound, for instance, by varying
x in Na
2
Cs
x
C
60
. The T
c
 depends strongly on  and peaks at a point where the conduction band is half-filled. This is an
unusual feature and mimics the behaviour of cuprate high-T
c
 superconductors. The highest T
c 
is found in  and is
19 K. Somewhat like in cuprates, the penetration depth is of the order of 5000  and the coherence length is much
smaller and of the order of 30 . Likewise  and  are of the order of 10 mT and 50 T, respectively. The
fulleride superconductors are thought to be of the BCS type (with an s-wave order parameter) with intramolecular
vibrations playing an important role however, electron correlations are also thought to play a role. Finally, the C
60
-
based superconductors, though unusual, can perhaps be called of the conventional BCS type.
Superconductivity in magnetic systems
As mentioned earlier, the terminology s-wave, p-wave, etc. is used to classify (in analogy to atomic orbitals) the orbital
symmetry of the order parameter. The total (spin + orbital) symmetry of the order parameter/wavefunction must be
antisymmetric for Cooper pairs. In a BCS s-wave superconductor, the Cooper pairs are in a spin-singlet state and have
a zero orbital moment. In such a case, magnetic fields and magnetic moments are thought to be detrimental to
superconductivity. This is because local moments will tend to align the electrons in the Cooper pair in the same
direction (break the singlet) as also pull the pair apart due to a Lorentz force which acts in the opposite direction for
the two members (orbital effect). This is indeed observed in conventional superconductors with a strong reduction of T
c
when magnetic impurities are doped into it.
AF ordered systems
In contrast, coexistence of long-range antiferromagnetic order and superconductivity has been observed in Chevrel
phase compounds RMo S and RMo Se and as also in RRh B (here R stands for a rare-earth atom which carries
a moment). In these systems, a transition to the superconducting state takes place with decrease of temperature. With
a further decrease of temperature, transition to an AF ordered state is found at  (from specific heat, susceptibility,
and neutron scattering measurements) which coexists with the superconducting state. The critical field is found to be
depressed around  and suggests a connection between the AF ordering and the superconducting state. Note that,
though the carriers that are responsible for superconductivity and the local moments in the above systems belong to
different atoms, a strong interaction must exist between the two (local moment and the Cooper pairs) mediated by the
conduction electrons.
Interplay between magnetic and superconducting properties has also been found in the ternary borocarbides RNi B C
(R=Y, Tm, Ho, etc.). Compounds with the composition HoNi B C show very interesting magnetic/superconducting
behaviour. For , the substance becomes superconducting below 7 K but on further cooling, it becomes normal
around 5 K and finally regains superconductivity (re-entrant superconductivity) on cooling further. In this system,
superconductivity is lost when it orders in a spiral arrangement and superconductivity is recovered when it finally locks
into an AF ordered state.
Ferromagnetically ordered systems
Re-entrant behaviour is seen in ErRh B and HoMo S . The material becomes superconducting on decreasing
temperature but at a temperature lower than the superconducting , there is a transition to ferromagnetic order and
superconductivity is lost. Interestingly, in a temperature regime above the ferromagnetic Curie point, there exists a
modulated magnetic structure wherein the local magnetization varies sinusoidally in space. Since the periodicity of this
modulation is less than the superconducting coherence length, the Cooper pairs sense a net zero field on their size-
scale and superconductivity coexists with magnetism.
Magnetically mediated superconductivity
In the above examples, although there was coexistence of magnetism and superconductivity, it was through a
fortuitous situation where, on the length-scale of Cooper pairs (coherence length), no static magnetism was seen and
the pairing mechanism was still the electron-phonon mechanism. In contrast, we would like to ask, are there materials
wherein the magnetic interactions in fact mediate the pairing between electrons resulting in superconductivity?
CePd Si is a system which at ambient pressure is an antiferromagnet with K. As the pressure is increased,
the  drops and AF disappears around 26 kbar. Around this region of pressure, superconductivity appears while it
disappears for much higher or much lower pressures. The transition point at 26 kbar is thought to be a quantum critical
point (transition takes place at 0 K as a function of pressure). In the pressure region around the critical point one
expects large quantum fluctuations where regions of AF correlations appear and disappear. It has been shown
theoretically that these magnetic interactions can mediate superconductivity which can be thought of as a competing
ground state. It is thought that in the case of nearly ferromagnetic metals, the superconducting state should be a spin-
triplet with an odd orbital quantum number while for antiferromagnetic metals, it should be a spin-singlet. The
resistivity in CePd Si is found to have a  dependence on temperature. This unusual temperature dependence is
thought to be due to scattering of quasiparticles via magnetic interactions.
Superconductivity in itinerant ferromagnets
We now consider some compounds where superconductivity coexists with ferromagnetism and moreover the atoms
responsible for the magnetism are the same ones responsible for the carriers contributing to superconductivity. The
examples of such systems are UGe , ZrZn , and URhGe. There are various peculiar features relevant to these
systems. Firstly, the same band-like  or  electrons are responsible for both magnetism and superconductivity. As
temperature is decreased, the material first orders ferromagnetically. The electronic heat capacity coefficient  is large
(160 mJ/mol K for URhGe) indicating an abundance of low-energy magnetic excitations. Samples with a high purity
show a transition to superconductivity at even lower temperatures. However, the transition width is relatively large
probably since the sample experiences an inhomogeneous magnetic field due to the ferromagnetic domains. Moreover,
the diamagnetic response in an ac susceptibility experiment (in zero applied static magnetic field) is less than the ideal
Meissner reponse. This could be due to the large internal field which puts the sample in a mixed state.
In ZrZn , the ferromagnetic ordering temperature  and the superconducting transition temperature  both
decrease with increasing pressure and are both completely suppressed above 21 kbar. In this case then, proximity to
the quantum critical point does not assure superconductivity. The flux expulsion in ZrZn even in small magnetic fields
is negligible. The electrical resistivity does not go to zero and remains finite below . The specific heat does not
exhibit a discontinuity at  either. It is felt that superconductivity is inhomogeneous in ZrZn and perhaps has triplet
pairing.
UGe is ferromagnetic and non-superconducting at ambient pressures. It's  decreases with increasing pressure
and becomes zero at 15.8 kbar. In some pressure range (between 9 and 15.8 kbar), resistivity and  susceptibility
show that there is superconductivity with  <\textcompwordmark < .
Heavy-fermion superconductors
Superconductivity in a heavy-Fermion system CeCu Si was first discovered in 1979 by Steglich. Subsequently, other
heavy-Fermion superconductors have also been found. To start with, heavy-Fermion behaviour is itself fairly exotic and
in addition superconductivity in these systems is unusual.
In UPt , there are three superconducting phases with different gap symmetries. The phases exist in different regions of
the  phase diagram. The symmetry of the gap is thought to be spin-triplet -wave. In UPd Al ,  = 14.3 K
and  = 2 K. The ratio  which is much higher than the BCS value. It show both local moment and
heavy-Fermion behaviour. It is thought that magnetic excitons mediate the pairing between heavy quasiparticles and
cause superconductivity.
High- T
c
 cuprate superconductors
While superconductivity at room temperature still remains a dream, a major advancement in this direction took place
with the discovery of cuprate superconductors. In the Hg-Ba-Ca-Cu-O system, a of about 155 K has been achieved
under pressure. Many properties of the cuprates are unconventional. One striking feature in all the cuprates is their
layered structure. They have CuO planes which are separated by other atoms. An example is shown in the
accompanying figure which pertains to the schematic crystal crystal structure of YBa Cu O . This system has bilayers
of CuO with an Y atom sandwiched between them. This bilayer unit is repeated in the crystallographic -direction.
There are other cuprate families with either a single layer (La CuO -based) or three or more layers per unit cell.
FIG.1 STRUCTURE OF YBCO
The parent system in all the cuprate families is an aniferromagnetic insulator and on doping with carriers (by
heterovalent substitution or oxygen addition) the system transforms to a metal which superconducts. The  varies
with the doping concentration and has a dome-like shape with a maximum at an ``optimal'' concentration.
Compositions with less than optimal doping are called underdoped and those with more than optimal doping are called
overdoped. Many of the properties (resistivity, thermal transport, coherence length, penetration depth, critical field,
etc.) are strongly anisotropic. The symmetry of the superconducting order parameter is not the s-wave variety but
rather d-wave ( , ). Another unusal feature in the normal state of the cuprate superconductors is the
existence of a pseudogap especially in the underdoped cuprates. The evidence of the pseudogap is seen in experiments
such as NMR, inelastic neutron scattering, susceptibiliy etc. A decrease in the susceptibility with a decrease in
temperature is seen well above . Many speculations have been made concerning the relevance of the pseudogap to
the mechanism of superconductivity though no clear answer is available at this point.
Iron Pnictide superconductors
In 2008, Japanese researchers discovered superconductivity in an iron based material 
LaFeAs (O F ) with a superconducting transition temperature ,K. Improvement in the transition
temperature was obtained with further work and the  was increased to 43,K with applied pressure. Currently, the
highest  in the iron-pnictide series is 56,K for Sm[O F ] FeAs. Finding superconductivity in Fe-based materials
was a big surprise since one normally associates magnetism with iron while magnetism and superconductivity are
Page 4


Module 10 : Unconventional superconductors
Lecture 1 : Unconventional superconductors
 
Research on superconductors started with work on metals and alloys since the conventional wisdom at that time was
that a material must atleast be a good conductor to become superconducting. Over the years, a large variety of
superconductors have been discovered. In conventional superconductors, conduction electrons merge into pairs because
of their attractive interaction via phonon exchange. The total wavefunction of the Cooper pairs has an orbital part and a
spin part. In cases where their orbital angular momentum L = 0, (analogous to atomic physics) this state is called an s-
wave state. From symmetry arguments it follows that the total spin S = 0, i.e., there is a spin-singlet. There are many
superconductors which conform to the predictions of the BCS theory (and can be called conventional). However, there
are others which exhibit different and unusal features which seem to suggest the presence of different/alternative
mechanisms giving rise to superconductivity. Herein, we review superconductors which fall in this second category.
Organic superconductors
The superconductors discovered in the early years were ``inorganic'' in nature, i.e. not primarily based on the elements
C, H, O. Subsequently many superconductors have been found which are of the organic variety. While phonons are
considered to provide the route to an effective attractive interaction which gives rise to superconductivity, in the BCS
theory, it was suggested that the polarisation of the ligands in organic materials could mediate superconductivity.
Superconductivity was, in fact, discovered (under a pressure of 12 kbar) in (TMTSF) 
2
PF
6
 in 1980 with a T
c
 of 0.9 K.
Further research led to discovery of superconductivity in the family (TMTSF) 
2
X
6
 where X stands for PF6, TAF6, ReO4,
ClO4, etc. while TMTSF stands for tetramethyl-tetraseleniumafulvalene. Superconductivity was found in another family
of organic materials (BEDT-TTF) 
2
Y. The transition temperatures in these materials are low and further the mechanism
of superconductivity is not what was originally proposed. Nevertheless, there are unusual features associated with these
organic superconductors. The BEDT-TTF salts display low-dimensional (2D) behaviour where conductivity within the
planes is much greater than that perpendicular to the planes. A large anisotropy is also detected in the critical fields
and the penetration depth. The highest T
c
 (11.5K) has been found in (BEDT-TTF)
2
Cu[Ni(CN)
2
]Br. The organic
superconductors are thought to have s-wave pairing but many of their properties are found to deviate from the BCS
predictions. The possible pairing mechanism continues to be of interest in these materials.
Another carbon containing system which displays superconductivity is derived from C
60
, also called fullerene. It is not
considered in the organic molecule category and has an fcc structure with a large lattice parameter of 14.2. It is
possible to obtain superconductivity in C
60
 by doping with alkali or alkali-earth metals. The transition temperatures of
various C
60
-based superconductors ranges from . The doping concentration (or the density of conduction electrons) can
be varied by adjusting the relative amounts of alkali and alkali-earth elements in a compound, for instance, by varying
x in Na
2
Cs
x
C
60
. The T
c
 depends strongly on  and peaks at a point where the conduction band is half-filled. This is an
unusual feature and mimics the behaviour of cuprate high-T
c
 superconductors. The highest T
c 
is found in  and is
19 K. Somewhat like in cuprates, the penetration depth is of the order of 5000  and the coherence length is much
smaller and of the order of 30 . Likewise  and  are of the order of 10 mT and 50 T, respectively. The
fulleride superconductors are thought to be of the BCS type (with an s-wave order parameter) with intramolecular
vibrations playing an important role however, electron correlations are also thought to play a role. Finally, the C
60
-
based superconductors, though unusual, can perhaps be called of the conventional BCS type.
Superconductivity in magnetic systems
As mentioned earlier, the terminology s-wave, p-wave, etc. is used to classify (in analogy to atomic orbitals) the orbital
symmetry of the order parameter. The total (spin + orbital) symmetry of the order parameter/wavefunction must be
antisymmetric for Cooper pairs. In a BCS s-wave superconductor, the Cooper pairs are in a spin-singlet state and have
a zero orbital moment. In such a case, magnetic fields and magnetic moments are thought to be detrimental to
superconductivity. This is because local moments will tend to align the electrons in the Cooper pair in the same
direction (break the singlet) as also pull the pair apart due to a Lorentz force which acts in the opposite direction for
the two members (orbital effect). This is indeed observed in conventional superconductors with a strong reduction of T
c
when magnetic impurities are doped into it.
AF ordered systems
In contrast, coexistence of long-range antiferromagnetic order and superconductivity has been observed in Chevrel
phase compounds RMo S and RMo Se and as also in RRh B (here R stands for a rare-earth atom which carries
a moment). In these systems, a transition to the superconducting state takes place with decrease of temperature. With
a further decrease of temperature, transition to an AF ordered state is found at  (from specific heat, susceptibility,
and neutron scattering measurements) which coexists with the superconducting state. The critical field is found to be
depressed around  and suggests a connection between the AF ordering and the superconducting state. Note that,
though the carriers that are responsible for superconductivity and the local moments in the above systems belong to
different atoms, a strong interaction must exist between the two (local moment and the Cooper pairs) mediated by the
conduction electrons.
Interplay between magnetic and superconducting properties has also been found in the ternary borocarbides RNi B C
(R=Y, Tm, Ho, etc.). Compounds with the composition HoNi B C show very interesting magnetic/superconducting
behaviour. For , the substance becomes superconducting below 7 K but on further cooling, it becomes normal
around 5 K and finally regains superconductivity (re-entrant superconductivity) on cooling further. In this system,
superconductivity is lost when it orders in a spiral arrangement and superconductivity is recovered when it finally locks
into an AF ordered state.
Ferromagnetically ordered systems
Re-entrant behaviour is seen in ErRh B and HoMo S . The material becomes superconducting on decreasing
temperature but at a temperature lower than the superconducting , there is a transition to ferromagnetic order and
superconductivity is lost. Interestingly, in a temperature regime above the ferromagnetic Curie point, there exists a
modulated magnetic structure wherein the local magnetization varies sinusoidally in space. Since the periodicity of this
modulation is less than the superconducting coherence length, the Cooper pairs sense a net zero field on their size-
scale and superconductivity coexists with magnetism.
Magnetically mediated superconductivity
In the above examples, although there was coexistence of magnetism and superconductivity, it was through a
fortuitous situation where, on the length-scale of Cooper pairs (coherence length), no static magnetism was seen and
the pairing mechanism was still the electron-phonon mechanism. In contrast, we would like to ask, are there materials
wherein the magnetic interactions in fact mediate the pairing between electrons resulting in superconductivity?
CePd Si is a system which at ambient pressure is an antiferromagnet with K. As the pressure is increased,
the  drops and AF disappears around 26 kbar. Around this region of pressure, superconductivity appears while it
disappears for much higher or much lower pressures. The transition point at 26 kbar is thought to be a quantum critical
point (transition takes place at 0 K as a function of pressure). In the pressure region around the critical point one
expects large quantum fluctuations where regions of AF correlations appear and disappear. It has been shown
theoretically that these magnetic interactions can mediate superconductivity which can be thought of as a competing
ground state. It is thought that in the case of nearly ferromagnetic metals, the superconducting state should be a spin-
triplet with an odd orbital quantum number while for antiferromagnetic metals, it should be a spin-singlet. The
resistivity in CePd Si is found to have a  dependence on temperature. This unusual temperature dependence is
thought to be due to scattering of quasiparticles via magnetic interactions.
Superconductivity in itinerant ferromagnets
We now consider some compounds where superconductivity coexists with ferromagnetism and moreover the atoms
responsible for the magnetism are the same ones responsible for the carriers contributing to superconductivity. The
examples of such systems are UGe , ZrZn , and URhGe. There are various peculiar features relevant to these
systems. Firstly, the same band-like  or  electrons are responsible for both magnetism and superconductivity. As
temperature is decreased, the material first orders ferromagnetically. The electronic heat capacity coefficient  is large
(160 mJ/mol K for URhGe) indicating an abundance of low-energy magnetic excitations. Samples with a high purity
show a transition to superconductivity at even lower temperatures. However, the transition width is relatively large
probably since the sample experiences an inhomogeneous magnetic field due to the ferromagnetic domains. Moreover,
the diamagnetic response in an ac susceptibility experiment (in zero applied static magnetic field) is less than the ideal
Meissner reponse. This could be due to the large internal field which puts the sample in a mixed state.
In ZrZn , the ferromagnetic ordering temperature  and the superconducting transition temperature  both
decrease with increasing pressure and are both completely suppressed above 21 kbar. In this case then, proximity to
the quantum critical point does not assure superconductivity. The flux expulsion in ZrZn even in small magnetic fields
is negligible. The electrical resistivity does not go to zero and remains finite below . The specific heat does not
exhibit a discontinuity at  either. It is felt that superconductivity is inhomogeneous in ZrZn and perhaps has triplet
pairing.
UGe is ferromagnetic and non-superconducting at ambient pressures. It's  decreases with increasing pressure
and becomes zero at 15.8 kbar. In some pressure range (between 9 and 15.8 kbar), resistivity and  susceptibility
show that there is superconductivity with  <\textcompwordmark < .
Heavy-fermion superconductors
Superconductivity in a heavy-Fermion system CeCu Si was first discovered in 1979 by Steglich. Subsequently, other
heavy-Fermion superconductors have also been found. To start with, heavy-Fermion behaviour is itself fairly exotic and
in addition superconductivity in these systems is unusual.
In UPt , there are three superconducting phases with different gap symmetries. The phases exist in different regions of
the  phase diagram. The symmetry of the gap is thought to be spin-triplet -wave. In UPd Al ,  = 14.3 K
and  = 2 K. The ratio  which is much higher than the BCS value. It show both local moment and
heavy-Fermion behaviour. It is thought that magnetic excitons mediate the pairing between heavy quasiparticles and
cause superconductivity.
High- T
c
 cuprate superconductors
While superconductivity at room temperature still remains a dream, a major advancement in this direction took place
with the discovery of cuprate superconductors. In the Hg-Ba-Ca-Cu-O system, a of about 155 K has been achieved
under pressure. Many properties of the cuprates are unconventional. One striking feature in all the cuprates is their
layered structure. They have CuO planes which are separated by other atoms. An example is shown in the
accompanying figure which pertains to the schematic crystal crystal structure of YBa Cu O . This system has bilayers
of CuO with an Y atom sandwiched between them. This bilayer unit is repeated in the crystallographic -direction.
There are other cuprate families with either a single layer (La CuO -based) or three or more layers per unit cell.
FIG.1 STRUCTURE OF YBCO
The parent system in all the cuprate families is an aniferromagnetic insulator and on doping with carriers (by
heterovalent substitution or oxygen addition) the system transforms to a metal which superconducts. The  varies
with the doping concentration and has a dome-like shape with a maximum at an ``optimal'' concentration.
Compositions with less than optimal doping are called underdoped and those with more than optimal doping are called
overdoped. Many of the properties (resistivity, thermal transport, coherence length, penetration depth, critical field,
etc.) are strongly anisotropic. The symmetry of the superconducting order parameter is not the s-wave variety but
rather d-wave ( , ). Another unusal feature in the normal state of the cuprate superconductors is the
existence of a pseudogap especially in the underdoped cuprates. The evidence of the pseudogap is seen in experiments
such as NMR, inelastic neutron scattering, susceptibiliy etc. A decrease in the susceptibility with a decrease in
temperature is seen well above . Many speculations have been made concerning the relevance of the pseudogap to
the mechanism of superconductivity though no clear answer is available at this point.
Iron Pnictide superconductors
In 2008, Japanese researchers discovered superconductivity in an iron based material 
LaFeAs (O F ) with a superconducting transition temperature ,K. Improvement in the transition
temperature was obtained with further work and the  was increased to 43,K with applied pressure. Currently, the
highest  in the iron-pnictide series is 56,K for Sm[O F ] FeAs. Finding superconductivity in Fe-based materials
was a big surprise since one normally associates magnetism with iron while magnetism and superconductivity are
conventionally considered mutually exclusive! These iron-based materials are semimetals at room temperature.
There is at least a superficial similarity of the pnictides with the high- cuprates since there are FeAs layers similar to
the CuO planes. Their phase diagrams are also similar in the sense that superconductivity is achieved upon doping a
layered parent compound. In contrast however, the cuprate parent is an antiferromagnetic insulator while the pnictide
parent is metallic but with a spin density wave ground state.
These materials broadly belong to four groups based on their crystal structure. The quaternary 1111 compounds with
the chemical formula RFeAsO where R is a rare earth element, ternary arsenides 
(122) AFe As with A = Ba, Sr, Ca and Eu, 1 : 1 compounds (Li/Na)FeAs and the 11 binary chalcogenides FeSe .
The figure below gives the schematic crystal structures of the various families of pnicitides.
FIG 2 PNICTIDE CRYSTAL STRUCTURES
Superconductivity can be induced in the above compounds by electron/hole/isovalent doping as also by applying
external pressure. The order parameter of the pnictides is not s-wave but is thought to be of the extended s-wave
type, also called  with a sign reversal of the order parameter between the different sheets of the Fermi surface.
Read More
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!