x_opt and x_c in Cuprate Superconductors
Jürgen Röhler

Anderson localization induced by complex potential
    R. Wang, K. L. Zhang, and Z. Song
    December 30, 2023
==================================================





§ INTRODUCTION


Two distinct  
hole concentrations, x_ opt≃ 0.16 and x_ c∼ 0.19, 
characterize the transition from the under- to the overdoped regime 
of cuprate superconductors. 
At x_ opt the critical temperature T_ c(x) reaches its maximum value.  
Fundamental properties of the superconducting phase, such as the zero-temperature superfluid density, the condensation energy, the zero temperature critical Silsbee field, however, are maximal only at 
x_ c where the critical temperature is 
10% below its maximum value (“slightly overdoped”). x_ c marks also the closure of a pseudogap which arises in the normal phase at elevated temperatures right up to the thermal  decomposition of the doped crystals. Thermodynamic and magnetic data indicate that the high temperature density of states near the chemical potential μ = E_ F is partially depleted <cit.>. Lowering T the number of 
available states is significantly reduced at T^*(x) > T_0(x) > 
T_ c(x), a characteristic temperature used to define the energy scale of the pseudogap, corroborated by the uniform magnetic susceptibility χ(T, x), 
the spin susceptibility χ_ s(T, x) (NMR Knight shift) and many   
spectroscopic data.   The onset temperature 
of nonlinear diamagnetic susceptibility from superconducting phase 
fluctuations is marked by T_0(x), and the dome-shaped onset of superconducting amplitude 
fluctuations by T_ c(x).

T^*(x) decreases linearily with increasing doping 
and falls to zero at x_ c.  The pseudogap scale Δ/k_B = J(1-x/x_c) with J≃ 1000 K  arises from both, spin and charge degrees of freedom, and is 
comparable to the antiferromagnetic superexchange energy.    
Notably this strongly entropy-depleting pseudogap is completely absent 
beyond x_ c throughout the strongly overdoped regime up to x≃ 0.25 
where superconductivity vanishes.   

The apparent disparity of the superconducting properties at  x_ opt and x_ c still stirs lively debates, particularily on a possible quantum critical point at x_ c. In this short note 
we attempt to outline, almost phenomenologically in the framework of the resonating valence bond (RVB) theory<cit.>, that both characteristic hole concentrations are intimately interrelated through the spatial structure of a non-local d-wave RVB pair state constrained by strong off-site hole – hole repulsion.  We show that geometrically limited number of connections among such d-wave RVB pair states is most likely at the origin of the pseudogap. Weak overdoping of columnar closest packed but partially unconnected RVB pair states is demonstrated to induce a displacive transition into a  fully connected zigzag structure.



















This short note attempts to outline in a 
phenomenologically, the effect of   
of elaborate that the optimal T_c atis 
characterized by a homogeneous RVB (non Fermi) liquid .



Pseudogap: where does it end? Enters the Tc line in the OD regime?
Relationship between sc gap below Tc and the pseudogap above Tc.
While they are the same the T* is a crossover , if different a phase 
transition.

 


The crossover from overdoping to underdoping was found in FRG studies 


PG opens a critical doping p* (also denoted pcrit and pc) within the 
sc phase. 











§ D-WAVE RVB STATES




 §.§ Electron-hole asymmetry

The physical properties of the hole doped cuprate superconductors 
are in essence governed by relaxation of the frustrated 
kinetic energy causing antiferromagnetic “kinetic” exchange  

in an insulating parent compound where spin-half electrons are 
kept apart by large on-site (Mott - Hubbard) Coulomb repulsion 
U_d<cit.>. The electronic excitations and the pairing interaction 
in such a liquid of short range spin singlet electron pairs
have been elaborated with Anderson's theory of resonating 
valence bonds (RVB)<cit.>, mostly under the implicit assumption of electronic homogeneity.

The strong desire of lattice electrons 
to delocalize themselves favores homogeneous spatial distributions. 
In general metals prefer most densely packed lattice structures optimizing the gain of (negative) kinetic energy and opposing segregational interactions. The normal electron liquid in doped cuprate superconductors however appears to be intrinsically strained, even in most carefully annealed stoichiometric compounds with relatively simple crystal lattices. Notorious chemical metastability and electronic symmetry breaking seem to be constituitive properties of superconducting cuprates.  One  important reason is the intrinsic electron – hole asymmetry<cit.> arising from the on-site electron – electron repulsion U_d, another off-site hole – hole repulsion rendering electron – electron and hole – hole pairs spatially asymmetric. 

In the narrow doping range between x_opt and x_c however 
the strain in the electron liquid seems to be gone, with them also doping induced lattice 
effects<cit.>.  Here the superconducting RVB d-wave function 
is found to get entirely stiff.  





 


We suppose  
the reason is in the fundamental asymmetry between the electron–electron and 
hole–hole interactions in a hole doped liquid of RVB spin singlet 
electron pairs.    

   
    
    
    
  Tilings
    


The intrinsic metastability of the sc cuprates provides us with this 
prerequisite encountering transient localized bound electron-pairs 
well above Tc.



Elaboration of this intrinsic relationship between antiferromagnetism and 
superconductivity has been elaborated in the 
RVB theory of a liquid 


The pairing, as characterized by the variational pa-
rameter Δ_var(x), is a monotonically decreasing function
of hole doping x, largest at x = 0 and vanishing beyond
x_c.  In marked contrast, the SC order pa-
rameter Φ(x) shows a nonmonotonic doping dependence.



This is the common source of the intimate relationship 
between antiferromagnetism and superconductivity  pairing 
interaction arise from a common source 

In the RVB theory antiferromagnetism and superconductivity i
n the cuprates are 
intimately related. The quasiparticles as well as the pairing 
interaction arise from a common source 

This intrinsic relationship between antiferromagnetism and 
superconductivity in the cuprates has been elaborated in 
the RVB theory of a liquid of resonant valence bonds (RVB).    

a quantum mechanical superposition of localized 
singlet electron pairs which into a superconducting state by hole doping 
has been theoretically elaborated  in Anderson's<cit.> 


(t-t'-J to be extended by off-site repulsion) 









 §.§ Off-site hole – hole repulsion

Holes doped into a RVB liquid of short range spin singlets are empty states
assumed to move free.  Though, double occupancy of hole sites is
excluded by U_d as it is for electrons at electron sites.  It is notable that the 
mobile empty states in the CuO_2 lattice are created by singlet
combination of the symmetrized orbital on the four oxygens around a
Cu^2+.  Such holes are virtual Cu^3+
sites with a hard core diameter larger 
than the lattice constant a_0, set by the “cage" of the oxygen environment. 
The energy of this so-called Zhang-Rice (ZR) state is by ∼
5 eV distinctly lower than that of all other combined states, thus
such mobile hole states are stable throughout the entire relevant doping
regime, 0< x≤ 0.33.  
Coulomb repulsion between oxygen 2p_x,y states at the connecting
site of two ZR singlets, U_p≃ 6 eV, may be naively
held responsible for the supression of nearest neighbor hole pairs.  A
rough estimate for two ZR states built out of Wannier states
yields a repulsion R≃ U_p/32 ≃ 0.19 eV<cit.>.
More detailed theoretical work on this issue is elusive. 





Hole doping contracts strongly the basal plane area. A preliminary inspection of the 
nonlinearities in the doping dependence of basal lattice constants, bulging at x_ opt 
and collapsing at x_ c, lead us to suggest that the hard cores of the virtual Cu^3+
holes exclude not only onsite double occupancy but also nearest neighbor hole – hole configurations<cit.>. 
Hence empty replica of electron – electron spin singlets, spaced by a_0, have to be 
excluded from the possible hole – hole configuration manifolds of the doped electron system.  
Therefore also clusters of nearest neighbor holes, hole stripes, hole puddles etc.

Indirect experimental support for this electron – hole asymmetry, 
particularily by STM<cit.>, comes from the charge density modulations (CDM) 
evidenced  at pseudogap excitation energies in the underdoped regime.  
These CDM's are sinusodial in one spatial dimension (bidirectional in x, y) with a commensurate period 
4a_0 centered at oxygen sites. 
We view the stick-like CDM patterns (see inset in Fig.<ref>) as signatures of bond-centered hole – hole exchange over 3a_0, extending effectively to 4a_0 due to the large hole sizes.





 §.§ Bond-like RVB arrays


The frustrated kinetic energy causing antiferromagnetism in the 
parent compound is relaxed through interactions which are short range in 
space and time. Fig.<ref>a displays schematically the locality of an isolated doped d-wave RVB state at pseudogap excitation energies. Bond-centered hole – hole pairing in the liquid of electron singlets arises for spacings of 3a_0 along the Cu – O bonds (antinodal loci in k-space) from an axial  inversion center for all directional space and spin coordinates at the central intervening oxygen site (small rectangle in the center of Fig.<ref>a). Figs.<ref>b–e illustrate the kinetic exchange interaction effectively attracting the holes.  Hole – hole configurations spaced by even multiples of the lattice constant, and site centered holes spaced by a_0√(5) are unable to lock the spin and charge degrees of freedom in a pairing state. 

The green wavy lines in Fig.<ref> symbolize quantum fluctuations along the lattice diagonal 
exchanging holes with electrons at the next-nearest neighbor sites without spin flips. The remote hybridization of the Cu4s with the Cu3dO2p orbitals promotes nodal fluctuations such that the ratio of the spin conserving, and the spin flipping hopping matrix element -t'/t achieves crucial importance for the pairing mechanism.

We expect the nodal fluctuations to generate effectively axial hole – hole  
attraction by kinetic exchange in an RVB array with at least 
3× 4=12 copper sites, see Fig.<ref> and caption. The size of the resulting pairing gap Δ_ p will be almost independent on the average hole concentration, and larger than the energy of the superconducting amplitude fluctuations k_ BT_c.  
The spatial extension of  the bond-like RVB array sets the full filling of the CuO_2 lattice with paired holes to  x=1/6≃ 0.16=x_ opt, see Fig.<ref> (right). Hence “overdoping” may only accomodate additional holes (still constrained by off-site repulsion) in ferromagnetic site-centered hole – hole configurations, which will destroy the antiferromagnetic pairing symmetry of the kinetic exchange, see Fig.<ref> (right). Thus three holes, still constrained by off-site repulsion even in the overdoped regime, will depair a bond-like RVB array. Complete destruction of the pairing gap Δ_ p will occur at x=3/12 = 0.25. The unusual electronic properties of the overdoped regime, e.g the paramagnetic susceptibility, may have its origin in a two-component fluid of paired bond-like RVB arrays and free electrons. It is then reasonable to expect that the superconducting stiffness decreases linearily also in the overdoped regime<cit.>. 






  §.§.§ Nano-sized columns


These bond-like RVB arrays  break the rotational symmetry of 
the lattice and give rise to uniaxial (“nematic") strain in the electronic lattice spreading macroscopically 
with increasing hole concentrations. Well known macroscopic deformable bodies favor 
relaxation of uniaxial microscopic strain  by formation of meso-scale unidirectional areas, and orientating them most appropriately. Large field STM maps<cit.> of underdoped samples evidence nanometer-scale unidirectional clusters in  x and y direction with no preference of one over the other. 
Apparently doping in the C_4 symmetric  RVB liquid proceeds as 
correlation of doped holes in nano-sized clusters with columns of  bond-like RVB arrays. 
The interconnection of bond-like RVB arrays of neighbored columns underlies the 
off-site occupation constraint as each single bond-like RVB array. Fig.<ref> (left)
displays schematically an adiabatic snapshot of an underdoped, i.e. only partially occupied columnar cluster of  bond-like RVB arrays. The STM pattern exhibits clearly the bond-like structure of interchanging holes and the intercolumnar repulsion with periodicity 4a_0.  
Naturally it does not yield information on the spins and their dynamics and not on the dynamics of the hole–hole exchange; its integrated intensity may be taken as a measure of the average number of holes<cit.>.  










  §.§.§ Zigzag packing

The organization of the bond-like RVB arrays on mesoscopic and macroscopic scales encounters  a  connectivity problem.  Fig.<ref> schematizes the nodal electron – hole quantum fluctuations (green wavy lines) in different most closely packed configurations: columnar (left), and `tweedy" zigzag (middle) patterns. Inter-columnar electron – hole fluctuations connecting nearest neighbored bond-like RVB arrays are obviously geometrically constrained by "ferro" equal spin hole – hole paths. Intra-columnar  electron – hole fluctuations however are able to connect via common electron spin singlet "antiferro" paths. 



The fully packed columnar configuration may be displacively transformed into the energetically more favorable zigzag configuration, see Fig.<ref> (left). For that columnar organized holes have to be collectively displaced along the lattice diagonals to next nearest neighbor sites, while reversing the displacive shifts from striation to striation. Note the arrows from the hole sites in the columnar configuration (open rectangles). This transformation of  a “martensitic" type creates a most densely packed zigzag configuration in which all bond-like RVB arrays are able to connect through common electron spin singlet states. Clearly this is the energetically superior configuration of the two enhancing electron delocalization through unconstrained long range phase correlation. Nevertheless the columnar configuration may still survive at x_ opt with most densely packed bond-like RVB states, because uniaxial microscopic strain inhibits long range collective displacements. Thus the number of pair states is maximal, but the superconducting stiffness stays below its optimal value. 









§ CLOSURE OF THE PSEUDOGAP AT X_ C > X_ OPT


The columnar organization of  x_ opt at x_ opt  is apparently metastable. A tiny amount of overdoping excess holes however may transform the nano-sized colums into the fully connected zigzag configuration, however with the consequence of partial pair breaking by "ferro" excess holes in some bond-like RVB arrays.  This situation is illustrated in Fig.<ref> (right). The "ferro" holes are marked by open and closed diamonds for spin up and spin down electron site, and local depairing by the grey rectangles around the broken hole pairs (omitted orange bar). Certainly local depairing will affect the superconducting amplitude fluctuations decreasing T_ c, however at the same time establishes unconstrained long range phase coherence thus maximal superconducting stiffness. Hence superconductivity at x_ c turns out to be more robust than at x_ opt, though at a diminuished value of T_ c.

Hence x_ opt at x_ opt are intimately connected with each other through the geometrical constraints governing the spatial organization of fully coherent assembly of nano-sized bond-like RVB arrays.













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