We report that four properties of cuprates and their evolution with

doping are consequences of simply counting four-site plaquettes arising from

doping, (1) the universal T_c phase diagram (superconductivity between ~0.05 and

~0.27 doping per CuO_2 plane and optimal T_c at ~0.16), (2) the universal doping

dependence of the room-temperature thermopower, (3) the superconducting

neutron spin resonance peak (the “41 meV peak”), and (4) the dispersionless

scanning tunneling conductance incommensurability. Properties (1), (3), and (4)

are explained with no adjustable parameters, and (2) is explained with exactly one.

The successful quantitative interpretation of four very distinct aspects of cuprate

phenomenology by a simple counting rule provides strong evidence for four-site

plaquette percolation in these materials. This suggests that inhomogeneity, percolation,

and plaquettes play an essential role in cuprates. This geometric analysis

may provide a useful guide to search for new compositions and structures with

improved superconducting properties

Goddard, William A., III

Tahir-Kheli, Jamil

English

Caltech Authors - Main

 1 
Universal properties of cuprate superconductors: Tc phase diagram, 
room-temperature thermopower, neutron spin resonance, and STM 
incommensurability explained in terms of Chiral Plaquette Pairing  
Jamil Tahir-Kheli and William A. Goddard III 
Materials and Process Simulation Center (MC 139-74) 
California Institute of Technology, Pasadena CA 91125 
jamil@wag.caltech.edu, wag@wag.caltech.edu 
Supporting Information 
Doping restrictions and its effect on the optimal doping value 
Figure S1(a) shows the restricted plaquette doping used in this paper and figure S1(b) shows 
the more general doping. 
 
Figure S1. (a) restricted plaquette doping where the plaquettes centers are separated by an even 
number of lattice spacings. (b) general plaquette doping with no overlap. This constraint can be 
fulfilled up to 0.187 doping. 
The restricted doping in figure S1(a) can be doped all the way to x=0.25 where there are no 
remaining d
9
 sites. For random doping of plaquettes in the general form of figure S1(b), we find 
 2 
that for x>0.187, it is not possible to satisfy the constraint of no overlap of the plaquettes. We 
calculate this value by randomly doping a large ensemble of 1000 Χ 1000 lattices and computing 
the doping where the constraint cannot be satisfied. This doping is significantly greater than our 
calculated optimal doping of 0.167. 
The detailed location of the dopants in the cuprates is determined by details of the high-
temperature fabrication and the annealing/quenching profile. For dopings less than 0.187, the 
Coulomb repulsion of the impurity dopants will lead to patches of restricted plaquettes as shown 
in figure S1(a) that are joined to patches shifted by one lattice spacing. Since the calculated 
optimal doping at 0.167 is well within the regime where four-site plaquettes can be formed 
without overlap, our estimate of the optimal doping value is reliable. 
It is interesting to continue doping beyond the plaquette overlap constraint for x<0.187. 
Assuming that the Coulomb repulsion of the dopant impurities will lead to further four-site 
plaquettes at locations that cover three d
9
 spins, this constraint can be satisfied up to 0.226. 
Above x=0.226, the impurities lead to additional plaquettes that cover two d
9
 spins. This 
constraint is exhausted at x=0.271. Finally, one-spin plaquettes can be doped up to x=0.317. 
In the paper, we argued that at x=0.25 there are no longer any remaining d
9
 spins to lead to 
superconducting pairing. In the more general doping picture described here, pairing will 
disappear when only isolated d
9
 spins remain because the nearest-neighbor AF correlation (Jdd) 
that leads to pairing is lost. This occurs at x=0.271. Experiment is ≈0.27. 
For the neutron resonance, thermopower, and STM incommensurability, the surface area of 
the plaquettes is unnecessary. Only average values for the spacing between plaquettes and d
9
 
regions are used. Thus the main results of the paper are not dependent on the details of the 
plaquette doping. 
 3 
Integrated neutron spin susceptibility 
The imaginary part of the magnetic susceptibility due to AF clusters of size n is given by 
),( ωχ q′′  where1 
).(|0|)(|1|
)(
2)(
),( 2
2 n
cells
clus
B
n EqS
N
nN
g
q
−><⋅=
′′ + ωδ
π
µ
ωχ
h  
Nclus(n) is the number of clusters of size n, Ncells is the total cells, En is the cluster energy gap, 
g=2 is the g-factor, and )(e)( RSqS iqR ++ ∑=  is the Fourier transformed spin raising operator 
summed over the sites R in the finite cluster. The matrix element is between the ground state 
(S=0) and the lowest excited state (S=1). The units are µB
2
/eV/f.u. Integrating over energy 
removes the delta function in the above expression. The energy integrated susceptibility is the 
sum over all clusters 
∫ ∑∫ ′′=′′ .),(),(
n
n qdqd ωχωωχω  
To determine the matrix element in the equation, we computed the S=0 ground state and S=1 
first excited state for all AF cluster shapes and sizes up to 24 spins. The four-site plaquette 
doping as shown in figure S1(a) leads to finite clusters comprised of 4, 8, 12, 16, … d
9
 sites. The 
nearest-neighbor AF spin coupling is the same as for undoped systems, Jdd = 130 meV. There are 
no periodic boundary conditions applied to calculate the ground state to first excited state energy 
splitting. 
We computed the energy splitting for all possible AF clusters up to 24 spin sites. The ground 
state is determined by diagonalizing the S=0 spin state that has 2,704,156 terms. Lanczos 
diagonalization is used. This leads to the ground state energy. Since the neutron resonance is a 
spin-flip scattering, the lowest energy state with S=1 is the excited state that is probed by the 
 4 
neutrons. A second Lanczos diagonalization to determine the ground state in the state space of 
S=1 with 2,496,144 states is performed. The difference is the energy splitting. 
Figure S2 shows all the possible AF d
9
 clusters possible up to 20 sites and figure S3 shows 
all the 24-site clusters. For each size, the probability of each cluster shape was determined in 
addition to the S=0 ground state to S=1 energy splitting. The probabilities and energies were 
used to determine the mean energy splitting for each cluster size. 
 
Figure S2. All 4, 8, 12, 16, and 20 AF d
9
 site clusters. The red dot expands to four d
9
 spins as 
shown in the upper left corner. The ground state to S=1 excited state energy is shown in meV in 
blue and the probability of the cluster is shown in green. The undoped spin-spin coupling, Jdd = 
130 meV, is used in these calculations. There is only one 4-site and 8-site cluster. There are 2, 5, 
and 12 clusters for 12, 16, and 20 sites, respectively. 
 
 5 
 
Figure S3. The 35 possible 24-site AF d
9
 spin clusters. The energy splitting is shown in blue and 
the probability is in green. The most probable cluster is the second to last cluster on the lower 
right with energy splitting of 57.1 meV and probability 0.06. The clusters are arranged from 
highest energy splitting to lowest. 
24-site AF d
9
 clusters are too small to include all the cluster sizes that can appear for the 
superconducting range of dopings. Thus we must extrapolate our energies to larger clusters. 
Since the spin-wave dispersion is linear in momentum, ω ~ k, and the smallest k in a finite 
cluster is ~ N
−1/2
, where N is the cluster size, we expect our calculated energy splittings to be 
fitted by a power series in N
−1/2
. The mean energy as a function of N and a fit to the data with the 
first two terms in the power series in N
−1/2
 is shown in figure S4. The fit is excellent with an 
RMS error of 0.61 meV and a maximum percentage error of 1.03%. 
 6 
 
Figure S4. Fit of the mean energy for cluster sizes up to 24-sites with the first two terms in a 
power series expansion in N
−1/2
 where N is the cluster size. An N
−1/2
 series expansion is expected 
from spin-wave theory. The fit is excellent using only the first two terms in the series. The 
maximum error is 1.03 meV and the RMS error is 0.61 meV. The largest percentage error of the 
six data points is 1.03%. Note that the coefficient C2 is negative. 
From spin-wave theory, the matrix element, 
2|0|)(|1|
1
>< + qS
n
, increases to infinity with 
increasing n. We used our eigenstates for clusters up to 24 AF spins to calculate this term for 
q=(π,π). This term fits the power law Cn
α
 with an RMS error of 0.01 and maximum percentage 
error of 0.59%, where C=0.86 and α=0.32 as shown in figure S5. An interesting observation is 
that the expression CN
α
 is very close to (4/3)(N/4)
1/3
. 
C1 = +354.01 meV 
C2 = −192.03 meV 
N
C
N
C
NE 21)( +=
RMS error = 0.61 meV 
Max % error = 1.03% 
 7 
 
Figure S5. Calculated matrix element of the spin raising operator at momentum (π, π) for all 
possible clusters up to 24 undoped AF d
9
 spins. The curve is very well fit by the power law 
expression, CN
α
, where C = 0.855 and α = 0.319. The RMS error is 0.01 and the maximum 
percentage error is 0.59%. 
Finally, to compute the integrated susceptibility, we need to know how many clusters there 
are of each size in the material. We calculated the number of clusters of size n, Nclus(n), by 
randomly doping 1000 ensembles of a 2000 X 2000 lattice. The results are shown in figure S6. 
The values were determined by averaging over 1000 ensembles. The figure shows the results for 
dopings x = 0.16 and x = 0.10. Only the x = 0.16 data is used in the main text. 
RMS error = 0.01 
Max percent error = 0.59% 
Matrix element = CN
α
 
 
C = 0.855 
α = 0.319 
 8 
 
Figure S6. The number of clusters of each size for dopings of x = 0.16 and 0.10. For x = 0.16, 
there are fewer undoped d
9
 sites leading to fewer large-sized clusters than x = 0.10. The curves in 
the figure were calculated for a 2000 x 2000 lattice and averaging was done over 1000 
ensembles. The y-axis (number of clusters) is plotted on a log scale and is not normalized by the 
number of lattice sites. If the lattice size was doubled, the number of clusters would double. 
Calculation of neutron resonance width 
The S=1 state excited by the neutron decays by electron-spin scattering with the x
2
-y
2
/pσ 
band electrons. The width of the resonance peak, Γ (half-width at half-maximum, HWHM), is 
given by Γ=π|W|
2
FBCS where FBCS is the standard electron spin-flip scattering term with BCS 
coherence factors
2
. |W|
2
=|M|
2
(1–λ)/2(1–λ)
1/2
 where λ=[1–(Eres/2Jdd)
2
]
1/2
 and the matrix element 
M≈Jdd
3
. For the bilayer systems, FBCS only includes scattering between the bonding and anti-
bonding bands. 
References 
[1] P. Bourges et al, Phys. Rev. B. 56, R11439 (1997) 
x = 0.16 
x = 0.10 
 9 
[2] D. K. Morr and D. Pines, Phys. Rev. Lett. 81, 1086 (1998). 
[3] E. Manousakis, Rev. Mod. Phys. 63, 1 (1991). 


Universal Properties of Cuprate Superconductors: T_c Phase Diagram, Room-Temperature Thermopower, Neutron Spin Resonance, and STM Incommensurability Explained in Terms of Chiral Plaquette Pairing

 1 
Universal properties of cuprate superconductors: Tc phase diagram, 
room-temperature thermopower, neutron spin resonance, and STM 
incommensurability explained in terms of Chiral Plaquette Pairing  
Jamil Tahir-Kheli and William A. Goddard III 
Materials and Process Simulation Center (MC 139-74) 
California Institute of Technology, Pasadena CA 91125 
jamil@wag.caltech.edu, wag@wag.caltech.edu 
Supporting Information 
Doping restrictions and its effect on the optimal doping value 
Figure S1(a) shows the restricted plaquette doping used in this paper and figure S1(b) shows 
the more general doping. 
 
Figure S1. (a) restricted plaquette doping where the plaquettes centers are separated by an even 
number of lattice spacings. (b) general plaquette doping with no overlap. This constraint can be 
fulfilled up to 0.187 doping. 
The restricted doping in figure S1(a) can be doped all the way to x=0.25 where there are no 
remaining d9 sites. For random doping of plaquettes in the general form of figure S1(b), we find 
 2 
that for x>0.187, it is not possible to satisfy the constraint of no overlap of the plaquettes. We 
calculate this value by randomly doping a large ensemble of 1000 Χ 1000 lattices and computing 
the doping where the constraint cannot be satisfied. This doping is significantly greater than our 
calculated optimal doping of 0.167. 
The detailed location of the dopants in the cuprates is determined by details of the high-
temperature fabrication and the annealing/quenching profile. For dopings less than 0.187, the 
Coulomb repulsion of the impurity dopants will lead to patches of restricted plaquettes as shown 
in figure S1(a) that are joined to patches shifted by one lattice spacing. Since the calculated 
optimal doping at 0.167 is well within the regime where four-site plaquettes can be formed 
without overlap, our estimate of the optimal doping value is reliable. 
It is interesting to continue doping beyond the plaquette overlap constraint for x<0.187. 
Assuming that the Coulomb repulsion of the dopant impurities will lead to further four-site 
plaquettes at locations that cover three d9 spins, this constraint can be satisfied up to 0.226. 
Above x=0.226, the impurities lead to additional plaquettes that cover two d9 spins. This 
constraint is exhausted at x=0.271. Finally, one-spin plaquettes can be doped up to x=0.317. 
In the paper, we argued that at x=0.25 there are no longer any remaining d9 spins to lead to 
superconducting pairing. In the more general doping picture described here, pairing will 
disappear when only isolated d9 spins remain because the nearest-neighbor AF correlation (Jdd) 
that leads to pairing is lost. This occurs at x=0.271. Experiment is ≈0.27. 
For the neutron resonance, thermopower, and STM incommensurability, the surface area of 
the plaquettes is unnecessary. Only average values for the spacing between plaquettes and d9 
regions are used. Thus the main results of the paper are not dependent on the details of the 
plaquette doping. 
 3 
Integrated neutron spin susceptibility 
The imaginary part of the magnetic susceptibility due to AF clusters of size n is given by 
),( ωχ q′′  where1 
).(|0|)(|1|)(
2)(
),( 2
2 n
cells
clus
B
n EqS
N
nN
g
q
−><⋅=
′′ + ωδ
π
µ
ωχ
h
 
Nclus(n) is the number of clusters of size n, Ncells is the total cells, En is the cluster energy gap, 
g=2 is the g-factor, and )(e)( RSqS iqR ++ ∑=  is the Fourier transformed spin raising operator 
summed over the sites R in the finite cluster. The matrix element is between the ground state 
(S=0) and the lowest excited state (S=1). The units are µB2/eV/f.u. Integrating over energy 
removes the delta function in the above expression. The energy integrated susceptibility is the 
sum over all clusters 
∫ ∑∫ ′′=′′ .),(),(
n
n qdqd ωχωωχω
 
To determine the matrix element in the equation, we computed the S=0 ground state and S=1 
first excited state for all AF cluster shapes and sizes up to 24 spins. The four-site plaquette 
doping as shown in figure S1(a) leads to finite clusters comprised of 4, 8, 12, 16, … d9 sites. The 
nearest-neighbor AF spin coupling is the same as for undoped systems, Jdd = 130 meV. There are 
no periodic boundary conditions applied to calculate the ground state to first excited state energy 
splitting. 
We computed the energy splitting for all possible AF clusters up to 24 spin sites. The ground 
state is determined by diagonalizing the S=0 spin state that has 2,704,156 terms. Lanczos 
diagonalization is used. This leads to the ground state energy. Since the neutron resonance is a 
spin-flip scattering, the lowest energy state with S=1 is the excited state that is probed by the 
 4 
neutrons. A second Lanczos diagonalization to determine the ground state in the state space of 
S=1 with 2,496,144 states is performed. The difference is the energy splitting. 
Figure S2 shows all the possible AF d9 clusters possible up to 20 sites and figure S3 shows 
all the 24-site clusters. For each size, the probability of each cluster shape was determined in 
addition to the S=0 ground state to S=1 energy splitting. The probabilities and energies were 
used to determine the mean energy splitting for each cluster size. 
 
Figure S2. All 4, 8, 12, 16, and 20 AF d9 site clusters. The red dot expands to four d9 spins as 
shown in the upper left corner. The ground state to S=1 excited state energy is shown in meV in 
blue and the probability of the cluster is shown in green. The undoped spin-spin coupling, Jdd = 
130 meV, is used in these calculations. There is only one 4-site and 8-site cluster. There are 2, 5, 
and 12 clusters for 12, 16, and 20 sites, respectively. 
 
 5 
 
Figure S3. The 35 possible 24-site AF d9 spin clusters. The energy splitting is shown in blue and 
the probability is in green. The most probable cluster is the second to last cluster on the lower 
right with energy splitting of 57.1 meV and probability 0.06. The clusters are arranged from 
highest energy splitting to lowest. 
24-site AF d9 clusters are too small to include all the cluster sizes that can appear for the 
superconducting range of dopings. Thus we must extrapolate our energies to larger clusters. 
Since the spin-wave dispersion is linear in momentum, ω ~ k, and the smallest k in a finite 
cluster is ~ N−1/2, where N is the cluster size, we expect our calculated energy splittings to be 
fitted by a power series in N−1/2. The mean energy as a function of N and a fit to the data with the 
first two terms in the power series in N−1/2 is shown in figure S4. The fit is excellent with an 
RMS error of 0.61 meV and a maximum percentage error of 1.03%. 
 6 
 
Figure S4. Fit of the mean energy for cluster sizes up to 24-sites with the first two terms in a 
power series expansion in N−1/2 where N is the cluster size. An N−1/2 series expansion is expected 
from spin-wave theory. The fit is excellent using only the first two terms in the series. The 
maximum error is 1.03 meV and the RMS error is 0.61 meV. The largest percentage error of the 
six data points is 1.03%. Note that the coefficient C2 is negative. 
From spin-wave theory, the matrix element, 2|0|)(|1|1 >< + qS
n
, increases to infinity with 
increasing n. We used our eigenstates for clusters up to 24 AF spins to calculate this term for 
q=(π,π). This term fits the power law Cnα with an RMS error of 0.01 and maximum percentage 
error of 0.59%, where C=0.86 and α=0.32 as shown in figure S5. An interesting observation is 
that the expression CNα is very close to (4/3)(N/4)1/3. 
C1 = +354.01 meV 
C2 = −192.03 meV 
N
C
N
CNE 21)( +=
RMS error = 0.61 meV 
Max % error = 1.03% 
 7 
 
Figure S5. Calculated matrix element of the spin raising operator at momentum (π, π) for all 
possible clusters up to 24 undoped AF d9 spins. The curve is very well fit by the power law 
expression, CNα, where C = 0.855 and α = 0.319. The RMS error is 0.01 and the maximum 
percentage error is 0.59%. 
Finally, to compute the integrated susceptibility, we need to know how many clusters there 
are of each size in the material. We calculated the number of clusters of size n, Nclus(n), by 
randomly doping 1000 ensembles of a 2000 X 2000 lattice. The results are shown in figure S6. 
The values were determined by averaging over 1000 ensembles. The figure shows the results for 
dopings x = 0.16 and x = 0.10. Only the x = 0.16 data is used in the main text. 
RMS error = 0.01 
Max percent error = 0.59% 
Matrix element = CNα 
 
C = 0.855 
α = 0.319 
 8 
 
Figure S6. The number of clusters of each size for dopings of x = 0.16 and 0.10. For x = 0.16, 
there are fewer undoped d9 sites leading to fewer large-sized clusters than x = 0.10. The curves in 
the figure were calculated for a 2000 x 2000 lattice and averaging was done over 1000 
ensembles. The y-axis (number of clusters) is plotted on a log scale and is not normalized by the 
number of lattice sites. If the lattice size was doubled, the number of clusters would double. 
Calculation of neutron resonance width 
The S=1 state excited by the neutron decays by electron-spin scattering with the x2-y2/pσ 
band electrons. The width of the resonance peak, Γ (half-width at half-maximum, HWHM), is 
given by Γ=π|W|2FBCS where FBCS is the standard electron spin-flip scattering term with BCS 
coherence factors2. |W|2=|M|2(1–λ)/2(1–λ)1/2 where λ=[1–(Eres/2Jdd)2]1/2 and the matrix element 
M≈Jdd3. For the bilayer systems, FBCS only includes scattering between the bonding and anti-
bonding bands. 
References 
[1] P. Bourges et al, Phys. Rev. B. 56, R11439 (1997) 
x = 0.16 
x = 0.10 
 9 
[2] D. K. Morr and D. Pines, Phys. Rev. Lett. 81, 1086 (1998). 
[3] E. Manousakis, Rev. Mod. Phys. 63, 1 (1991). 


Generic Superconducting Phase Behavior in High-Tc Cuprates: Tc Variation with Hole Concentration in YBa2Cu3O7-δ.

https://authors.library.caltech.edu/18490/2/jz100265k_si_001.pdf

Universal Properties of Cuprate Superconductors: T_c Phase Diagram, Room-Temperature Thermopower, Neutron Spin Resonance, and STM Incommensurability Explained in Terms of Chiral Plaquette Pairing

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