**High-***Tc* Superconductivity

Projects are listed in chronological order, so please skip down to see what's been going on recently.

*Tc*Superconductivity

**Cuprates**

This field needs no introduction or social commentary. I spent my graduate career and first post-doc studying high-temperature superconducting systems within the

*t-J*model, gauging the validity of different treatments of this model by comparison with experimental results. The slave-fermion approach is applicable [1,2] to the properties of lightly doped cuprates. The slave-boson scheme is suitable at higher dopings, including much of the superconducting regime. At the mean-field level it provides a framework in which to understand inelastic neutron scattering and NMR studies of spin excitations, Fermi surfaces measured by photoemission, and -- our unique contribution -- the fact that neutron- and Raman-scattering observations find phonon anomalies not at*Tc*but at the pseudogap temperature [3-8].**Ladders**

Despite me and all my colleagues being, and remaining, firmly convinced of the need for a strong-coupling (doped Hubbard-model) picture for the cuprates, we did venture to suggest that a spin-fluctuation exchange picture might be appropriate in the doped ladder material LaCuO2.5 [9], where the quasi-one-dimensionality leads to good nesting.

**Stripes**

Somewhat later, we investigated the microscopic origin of stripe formation in the known distortions of the ground-state lattice structure of La2-x-yNdySrxCuO4. Static stripes have only ever been observed in compounds with the LTT structure, which breaks the

*(x,y)*symmetry of the basal plane, and we used real-space Hartree-Fock calculations[10] to show that even a weak LTT lattice distortion is sufficient to tip the delicate balance of competing candidate ground states in favour of one-dimensional, charge-inhomogeneous configurations (stripes). This begs the question of why static stripes are not observed in YBa2Cu3O6+x, which has significant*(x,y)*anisotropy, and we traced this [11] to the much greater next-neighbour (*t'*) kinetic terms in non-LSCO systems. We also compared the relative stability of vertical and diagonal stripes [12], confirming that cuprates should prefer diagonal ones, whence a nearest-neighbour hopping (*t*) anisotropy is indeed the key physical feature responsible for the observed stripes.**Ring Exchange**

We used the same method to investigate the relation between namely circulating-current states (flux phases) and the cyclic, four-spin (ring-exchange) interaction on the square lattice. These are respectively real and virtual charge or spin currents around the plaquettes. At the Hartree-Fock level, we found that ring-exchange interactions suppress the charge-flux phase (a) on the square lattice but enhance the spin-flux phase (b) where this can be established [13]. By including projected hopping (using Gutzwiller factors), we confirmed i) that the charge-flux phase is a candidate ground state for cuprates, and ii) that the ring-exchange term may determine the magnetic phases close to half-filling (but not the flux phases) [14]. We also made a systematic investigation of ring-exchange interactions, of any magnitude, in one dimension by finding exact solutions in a generalised model on a spin ladder and thus drawing the full phase diagram [15] (see also here).

**Iron Superconductors**

Very recently, I have been helping my experimental colleagues at Renmin University with the interpretation of their data on the new iron superconductor AyFe2-xSe2 (alkali-intercalated iron selenide,

i) We used phonon Raman scattering to establish the symmetry of the structure and, in combination with first-principles lattice dynamics, the vacancy ordering pattern [16]. By selective doping studies [17] we showed that Fe-doping alters the higher (FeSe) phonons slowly, implying no dramatic structural changes, while A-doping causes strong changes in the very lowest-lying phonons.

ii) We used two-magnon Raman scattering to show where the optical magnon branches of the 1/5-depleted square lattice lie [18]. There is a

iii) We used NMR to show a clear phase separation, with a nonmagnetic phase which superconducts. The Knight shift and spin-lattice relaxation rate both rise rapidly with temperature above

iv) This led us to look elsewhere for microscopic coexistence of antiferromagnetic order and superconductivity in the Fe superconductors. The approach of choice is to study systems with isovalent doping, which results in very slow changes of the electronic properties across the phase diagram. We

performed NMR experiments on the iron arsenide Ba(Fe1-xRux)2As2, focussing on an underdoped sample with x = 0.23 which shows a clear antiferromagnetic transition at 60 K and a superconducting one at 15 K [20]. Careful measurements of the Knight shift and spin-lattice relaxation rate established that essentially the entire sample (99.95%) was magnetic below TN and at least 70% of it was superconducting below Tc. The fact that the spin-lattice relaxation rate falls at Tc -- on sites which are already magnetic -- establishes directly that the superconductivity and antiferromagnetism are coexisting on the same Fe ions; this result cannot be the consequence of any type of phase separation. We found a high density of itinerant electrons in the magnetic state of the Ru-doped system, indicating the origin of the superconducting carriers; the lack of obvious competition between the two types of broken symmetry suggests that their carriers occupy different parts of the Fermi surface.

*Tc*= 32 K). The Fe superconductors should be good, metallic, intermediate-coupling, spin-fluctuation-exchange systems, but complexities due to the materials and the*d*-bands of Fe make this difficult to prove. The complexity in this material is that the Fe non-stoichiometry is close to*x = 0.4*, leading to a 1/5-depleted FeSe layer with, mostly, a strict vacancy ordering stablised by a very strong antiferromagnetic order, with a very large moment, among ferromagnetically coupled four-spin blocks.i) We used phonon Raman scattering to establish the symmetry of the structure and, in combination with first-principles lattice dynamics, the vacancy ordering pattern [16]. By selective doping studies [17] we showed that Fe-doping alters the higher (FeSe) phonons slowly, implying no dramatic structural changes, while A-doping causes strong changes in the very lowest-lying phonons.

ii) We used two-magnon Raman scattering to show where the optical magnon branches of the 1/5-depleted square lattice lie [18]. There is a

**highly unconventional drop**in the two-magnon intensity exactly at*Tc*, which is hard to explain except by a microscale cohabitation and strong competition of antiferromagnetism and superconductivity. Whether this cohabitation is a real coexistence or a nanoscale phase separation puzzled a lot of workers in the field for a while ...iii) We used NMR to show a clear phase separation, with a nonmagnetic phase which superconducts. The Knight shift and spin-lattice relaxation rate both rise rapidly with temperature above

*Tc*, tracking each other perfectly with a quadratic form [19]. This cannot be a pseudogap, but three-dimensional and very local spin fluctuations, which can be ascribed to the valence bands of Fe. Their contribution is simply additive with a Fermi-liquid contribution, which is due to the conduction electrons and vanishes below*Tc*. This scenario accounts for the NMR data in all the Fe superconductors away from the SDW transition, which differ only in dimensionality and in dominance of a particular ordering wavevector. There is no need for a complex correlated-electron pseudogap picture as in the cuprates. Later investigation by numerous techniques did confirm that the alkali-intercalated FeSe materials are very much a two-phase system, with nanoscale phase separation of the magnetic and superconducting regions.iv) This led us to look elsewhere for microscopic coexistence of antiferromagnetic order and superconductivity in the Fe superconductors. The approach of choice is to study systems with isovalent doping, which results in very slow changes of the electronic properties across the phase diagram. We

performed NMR experiments on the iron arsenide Ba(Fe1-xRux)2As2, focussing on an underdoped sample with x = 0.23 which shows a clear antiferromagnetic transition at 60 K and a superconducting one at 15 K [20]. Careful measurements of the Knight shift and spin-lattice relaxation rate established that essentially the entire sample (99.95%) was magnetic below TN and at least 70% of it was superconducting below Tc. The fact that the spin-lattice relaxation rate falls at Tc -- on sites which are already magnetic -- establishes directly that the superconductivity and antiferromagnetism are coexisting on the same Fe ions; this result cannot be the consequence of any type of phase separation. We found a high density of itinerant electrons in the magnetic state of the Ru-doped system, indicating the origin of the superconducting carriers; the lack of obvious competition between the two types of broken symmetry suggests that their carriers occupy different parts of the Fermi surface.

*Gauge Theory and Superconductivity in the t'-J Model*

B. Normand, P. A. Lee and N. Nagaosa,

Physica C**185-189**, 1479 (1991).*Dynamic Susceptibility and Photoemission in the t-t'-J Model*

B. Normand and P. A. Lee,

Phys. Rev. B**51**, 15519 (1995).*Properties of a Mixed-Symmetry Superconductor*

B. Normand, H. Kohno and H. Fukuyama,

Physica C**235-240**, 2275 (1994).*Phonons and Spin Excitations in the Extended t-J Model*

B. Normand, H. Kohno and H. Fukuyama,

J. Low Temp. Phys.**99**, 531 (1995).*Superconductive Phonon Anomalies in High-Tc Cuprates*

B. Normand, H. Kohno and H. Fukuyama,

J. Phys. Chem. Solids**56**, 1739 (1995).*Spin-Phonon Coupling in the Single-Layer Extended t-J Model*

B. Normand, H. Kohno and H. Fukuyama,

Phys. Rev. B**53**, 856 (1996).*Dynamic Susceptibility and Phonon Anomalies in the Bilayer t-J Model*

B. Normand, H. Kohno and H. Fukuyama,

J. Phys. Soc. Jpn.**64**, 3903 (1995).*\pi-Excitation and Resonant Neutron Scattering: a Mean-Field Study*

H. Kohno, B. Normand and H. Fukuyama,

Physica C**282-287**, 1685 (1997).*Possible Superconductivity in the doped Ladder Compound La1-xSrxCuO2.5*

B. Normand, D. F. Agterberg and T. M. Rice,

Phys. Rev. Lett**82**, 4296 (1999).*Microscopic Origin of Static Stripes in Cuprates*

B. Normand and A. P. Kampf,

Phys. Rev. B**64**, 024521 (2001).*Suppression of Static Stripe Formation by Next-Neighbor Hopping*

B. Normand and A. P. Kampf,

Phys. Rev. B**65**, 020509 (2002).*Vertical and Diagonal Stripes in the Extended Hubbard Model*

M. Raczkowski, B. Normand and A. M. Oleś,

Phys. Stat. Sol. (b)**236**, 376 (2003).*Circulating-Current States and Ring-Exchange Interactions in Cuprates*

B. Normand and A. M. Oleś,

Physica C**408-410**, 252 (2004).*Circulating-Current States and Ring-Exchange Interactions in Cuprates*

B. Normand and A. M. Oleś,

Phys. Rev. B**70**, 134407 (2004).*Phase Diagram of the Heisenberg Spin Ladder with Ring Exchange*

V. Gritsev, B. Normand and D. Baeriswyl,

Phys. Rev. B**69**, 094431 (2004).*Vacancy Ordering and Phonon Spectrum in the Fe Superconductor K0.8Fe1.6Se2*

A. M. Zhang, K. Liu, J. H. Xiao, J. B. He, D. M. Wang, G. F. Chen, B. Normand and Q. M. Zhang,

Phys. Rev. B**85**, 024518 (2012).*Effect of Iron Content and Potassium Substitution in A0.8Fe1.6Se2 (A = K, Rb, Tl) Superconductors: a Raman-Scattering Investigation*

A. M. Zhang, K. Liu, J. H. Xiao, J. B. He, D. M. Wang, G. F. Chen, B. Normand and Q. M. Zhang,

Phys. Rev. B 86, 134502 (2012).*Two-magnon Raman-scattering**in A0.8Fe1.6Se2 systems*(A = K, Cs, Rb, and Tl):*competition between superconductivity and antiferromagnetic order*

A. M. Zhang, J. H. Xiao, Y. S. Li, J. B. He, D. M. Wang, G. F. Chen, B. Normand, Q. M. Zhang and T. Xiang,

Phys. Rev. B 85, 214508 (2012).*Local spin fluctuations in iron-based superconductors: 77Se and 87Rb NMR measurements on Tl0.47Rb0.34Fe1.63Se2*

L. Ma, G. F. Ji, J. Dai, J. B. He, D. M. Wang, G. F. Chen, B. Normand and W. Q. Yu,

Phys. Rev. B**84**, 220505(R) (2012).- Microscopic Coexistence of Superconductivity and Antiferromagnetic Order in Underdoped Ba(Fe1-xRux)2As2, L. Ma, G. F. Ji, J. Dai, X. R. Lu, M. J. Eom, J. S. Kim, B. Normand and W. Q. Yu, Phys. Rev. Lett. 109, 197002 (2012).