Apical charge flux-modulated in-plane transport properties of cuprate superconductors

Sooran Kim, Xi Chen, William Fitzhugh, and Xin Li*

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge,

Massachusetts 02138, United States

*Corresponding author:

E-mail: lixin@seas.harvard.edu

1. Computational details

All DFT calculations were carried out by the Vienna Ab initio Simulation Package (VASP) code,

which implements the pseudopotential plane wave band method [1]. The projector augmented wave

Perdew-Burke-Ernzerhof (PAW-PBE) functional was utilized for the exchange-correlation energy. We

have performed the non-spin polarized DFT+U calculations to account for the correlated d orbitals of Cu

ion with U=8 eV and J=1.34 eV [2]. The Dudarev’ method was used for the double-counting correction.

Atomic positions were optimized from experimental structures obtained in the inorganic crystal structure

database (ICSD). Atomic positions and lattice parameters of hypothetical structures are fully relaxed. We

used a 520 eV plane-wave energy cutoff for all calculations.

PHONOPY was employed for phonon calculations [3]. The force constants were obtained by the

supercell approach with finite displacements based on the Hellmann-Feynman theorem. The 2 X 2 X 1

supercells of the conventional structures are selected in all families except for YBCO6.5 where 1 X 2 X 1

supercells were used, respectively. Forces and phonon calculations were carried out with non-spin-

polarized DFT+U method.

Polaron calculations were performed using the hybrid functional Heyd-Scuseria-Ernzerhof (HSE)06

with a=0.25 and w=0.2 Å-1. The G-type antiferromagnetic ordering in CuO2 layers were used as an initial

magnetic configuration [4]. 2 X 2 X 1, 2 X 4 X 1, 2 X 6 X 1, and 2 X 8 X 1 of Hg-1201 were used for

considering different hole concentrations.To investigate the mobility of the small polaron hole, we have

calculated the polaron activation barrier of hopping between neighboring Cu sites in CuO2 layers. The

initial and final ion positions are denoted as {qi} and {qf}, respectively. In this case, the migration of the

polaron can be expressed by the transfer of the lattice distortion from {qi} to {qf}. A set of cell

configurations {qx} is linearly interpolated between {qi} to {qf}; {qx}=(1-x){qi}+x{qf}, where 0 < x <1.

The energies at each position {qx} are calculated to find the maximum energy on the polaron migration

pathway, which corresponds to the activation barrier [5].

2. Summary of Tc,max and dA of each material

Table S1 Summary of Tc,max, dA, and abbreviations of materials. *dA of YBCO6.5 is the averaged

value below empty and filled CuO chains.

Chemical formula Tc,max(K) dA (Å) Abbreviation Family

La2CuO4 38 [6] 2.412 [7] La214 La

La2CaCu2O6 55 [8] 2.309 [9]

Pb2Sr2Cu2O6 32 [10] 2.350 [10]

Pb2Sr2YCu3O8 70 [11] 2.238 [11]

YBa2Cu3O7 93 [12] 2.296 [13] YBCO7

Y YBa2Cu3O6.5 93 2.368* [14] YBCO6.5

YBa2Cu3O6 93 2.466 [13] YBCO6

LaBa2Cu3O7 98.5 [15] 2.165 [16]

NdBa2Cu3O7 96 [17] 2.247 [16]

DyBa2Cu3O7 92 [18] 2.269 [16]

Bi2Sr2CuO6 40 [19] 2.589 [20] Bi-2201

Bi Bi2Sr2CaCu2O8 93 [21] 2.509 [22] Bi-2212

Bi2Sr2Ca2Cu3O10 110 [12] 2.528 [23] Bi-2223

Tl2Ba2CuO6 87 [24] 2.714 [20] Tl-2201

Tl Tl2Ba2CaCu2O8 110 [24] 2.700 [25] Tl-2212

Tl2Ba2Ca2Cu3O10 125 [26] 2.740 [27] Tl-2223

HgBa2CuO4 94 [28] 2.795 [28] Hg-1201

Hg HgBa2CaCu2O6 127 [29] 2.787 [30] Hg-1212

HgBa2Ca2Cu3O8 135 [30] 2.748 [30] Hg-1223

Ca2CuO2Cl2 26 [31] 2.734 [32] CCOC

Sr2CuO2Cl2 30 [33] 2.861 [34] SCOC

3. Relation between forces on apical atoms and !",$%&

FIG. S1 Apical forces by the apical anion distortion of 0.12 Å versus Tc,max (a) Force on the apical cation

(FaC), (b) forces on the apical anion (FaA), and (c) forces on the planar Cu (FpCu) with the apical anion

distortion away from the plane. Colors represent the types of the apical cations. Error bars indicate the

apical Cu in empty and filled CuO chains along the b-axis of YBCO6.5. The inset in (a) represents the

structural unit with the direction of the apical anion distortion.

The distortion of 0.12 Å is similar to the apical distortions of 0.13 or 0.19 Å selected in the previous

DFT calculation [35]. We also tested smaller distortion of 2.2 pm (optically induced distortion of B1u

mode [35]), larger distortion of 0.23 Å and the HSE06 hybrid functional as in Fig. S2, which all give the

same trend. Tc,max,

When the apical anion moves down toward the CuO2 plane, the apical cation undergoes a dragging

force to move down together, while the planar Cu experiences an attractive force in most cases to move

up. When the apical anion moves away from the CuO2 layer up toward the apical cation, the apical cation

feels the pushing force to move up together, while the planar Cu experiences a repulsive force to move

down in most cases. Namely, the bonding between the apical cation and apical anion is sensitive to the

atomic displacement and tends to move the two connected atoms in-phase in the static DFT simulation,

whereas the bonding between the apical anion and planar Cu is much less sensitive to the displacement

but prefers the out-of-phase movement. Furthermore, when the apical O is moved toward the CuO2 plane,

the four planar oxygens surrounding the planar Cu experience a pushing force to move away from the

central Cu. When the apical O is moved away from the plane, the four planar oxygens feel the dragging

force and tend to move close to the central Cu. The magnitude of forces here is close to the ones on the

planar Cu and is much smaller than the ones on apical oxygen and apical cation. These interactions

indicate that the vertical distortions from the apical anion can be transferred to not only the apical cation

but also the planar CuO4 unit, which justify our definition of the out of plane structural unit that governs

the in-plane and out-of-plane phonon couplings.

4. Force calculations with HSE06 hybrid functional

FIG. S2 Force calculation with non-spin polarized HSE06 functional by the apical anion distortion of

-0.12 Å in (a,b,c) and 0.12 Å in (d,e,f) versus Tc,max. (a,d) (b,e) (c,f) are for forces on apical cation, apical

anion, and planar Cu, respectively.

5. Correlation between apical oxygen phonon frequency and Tc,max

FIG. S3 Phonon frequency of the apical oxygen at G point versus the family. (a) Anti-symmetric apical

phonon mode, where two apical atoms shift in the same direction along the c axis. (b) Symmetric apical

phonon mode, where two apical atoms move in the opposite direction along the c axis.

Phonons are directly related to the strength of the chemical bonding. Figure S3 shows the phonon

frequency of two G point modes of the apical oxygen along c-axis according to the family of

superconductors. One apical phonon mode is anti-symmetric with respect to inversion, and another one is

symmetric. For example, Figure S3(a) and S3(b) insets correspond to antisymmetric B1u and symmetric Ag

of YBCO7, respectively. The results show that higher Tc,max family in general has higher apical oxygen

phonon frequency, which does not depend on the number of CuO2 layers. These results are consistent

with our force calculations and the previous Raman scattering experiments, including the family

dependence [36–40] and the layer independence [40,41] as in Table S2. It is worth noting that the

multiple phonon coupling to the electrons have been reported in the experiment of angle-resolved

photoemission spectroscopy [42,43], where both the in-plane half-breathing mode and out-of-plane apical

oxygen phonon were considered as the bosons coupling to the electrons. We also note that the frequency

of apical oxygen phonon mode near 70 meV here coincides well with the kink energy in the quasiparticle

energy versus rescaled wavevector plot from Angle-resolved photoemission spectroscopy

(ARPES), [44,45] suggesting the possible coupling of such apical oxygen oscillations with the in-plane

charge carrier motion.

Table S2 Symmetric apical phonon mode calculated in this work and measured in experiments from Refs. [36–41].

Materials Calculation (meV) Experiment (meV)

La214 49.4 53.7 [36,39]

YBCO7 57.0 61.4 [37,38]

Hg-1201 70.5 72.7 [40,41]

Hg-1212 71.6 72.5 [40,41]

Hg-1223 71.9 72.5 [40,41]

6. Charge transfer calculations using HSE06 hybrid functional

FIG. S4 Charge transfer with the apical oxygen distortion using non-spin polarized HSE06 functional (a)

The electron density difference between the distorted structure and the equilibrium structure of Tl-2212.

Left: ∆dA= -0.12 Å, Right: ∆dA= 0.12 Å. Yellow and blue lobes represent the positive and negative parts,

respectively. (b) The partial density of states of the apical cation of Tl, the apical oxygen, the planar Cu

and the planar O of Tl-2212.

7. Bader charge analysis of the apical oxygen

FIG. S5 Bader charge of the apical oxygen in each system

8. Charge density difference depending on the number of layers

FIG. S6 The charge density difference (CDD) between the distorted structures and the equilibrium

structure (a) CDD with the anti-symmetric distortion; left: Tl-2201, middle: Tl-2212, right: Tl-2223 (b)

CDD with the symmetric distortion; left: Tl-2201, middle: Tl-2212, right: Tl-2223. Yellow and blue lobes

represent the positive and negative parts, respectively.

The difference between CDD with anti-symmetric distortion Fig. S6(a) and Fig. S6(b) becomes

smaller with increasing number of CuO2 layers. It indicates the perturbation by the distortion of apical

oxygen ions on the opposite side of CuO2 layers decreases with increasing numbers of CuO2 layers.

Furthermore, the difference between Tl-2201 (1 layer) and Tl-2212 (2 layer) is more significant than the

difference between Tl-2212 (2 layer) and Tl-2223 (3 layer).

9. Phonon-phonon interaction based on non-linear phononics

FIG. S7 Phonon-phonon interaction between out-of-plane apical and in-plane breathing modes. (a) Left:

Anti-symmetric apical phonon mode and right: anti-symmetric breathing mode (b) Left: symmetric apical

phonon mode and right: symmetric breathing mode (c) Energy potentials of anti-symmetric breathing

mode in families with a frozen displacement of the anti-symmetric apical mode of 1.5 Å u (u, atomic

mass unit), which corresponds to a change in dA of 0.18~0.24 Å depending on the system. (d) Energy

potentials of the symmetric breathing mode with a frozen displacement of the symmetric apical mode of

1.5 Å u corresponding to a change in dA of 0.18~0.26 Å

To investigate how the out-of-plane phonon affects the in-plane phonon, we calculate the phonon-

phonon interaction based on the non-linear phononics [35,46]. Two apical phonon modes and two in-

plane breathing modes at (0.5, 0.5, 0) were shown here as an example, while the coupling between apical

oxygen phonons with some typical modes in the cuprates, such as the half-breathing mode, generally

exists as checked by our DFT computation. To simplify the analysis and computation we focus on the in-

plane bond-stretching mode or breathing mode [47–49].

Two apical phonon modes and two in-plane breathing modes at (0.5, 0.5, 0) were chosen as shown in

Fig. S7(a) and S7(b) with the dominant distortions illustrated. One breathing mode is anti-symmetric to

inversion (Fig. S7(a) right), and another breathing mode is symmetric to inversion (Fig. S7(b) right). Four

combinations of apical and breathing modes are hence possible, but only the couplings between anti-

symmetric modes or symmetric modes can survive. Figure S7(c) (S7(d)) shows the energy potentials of

anti-symmetric (symmetric) modes with a frozen displacement of the anti-symmetric (symmetric) apical

mode in various families except for La214 and Hg-1201, which do not have the anti-symmetric breathing

mode due to a monolayer of CuO2. If the coupling does not exist, the energy potential follows the linear

term of '( )(*( describing harmonic oscillations with the energy minimum at the equilibrium atomic

positions of Q = 0, where ω and Q denote the frequency and normal coordinate of the mode, respectively.

However, Figure S7(c) and S7(d) clearly shows the shifts of the minimum positions away from zero Q,

indicating the coupling between two phonon modes here. Note that the anti-symmetric breathing mode

does not exist in the single CuO2 layer materials. La214 and Hg-1201 in Fig. S7(c) (S7(d)) represent the

coupling between the anti-symmetric (symmetric) apical mode and the symmetric breathing mode. The

shift of energy minimum only happens in contrast to other double layer materials. Namely, La214 and

Hg-1201 only exhibits the phonon-phonon coupling between the symmetric modes among four possible

combinations.

10. Small polaron formation

FIG. S8 Small polaron formation in the CuO2 plane by HSE hybrid functional. Polaronic distortion at hole

doping level of 0.25 in the charge density difference map between the polaron-containing cell and a

neutral cell for Hg-1212. Blue and red atoms are Cu and O, respectively. The black circle at the center

indicates the polaron formed by the center CuO4 unit. Black lines indicate the Cu-O bond with the length

less than 1.91 Å.

Figure S8 shows the polaronic distortion in the CuO2 layer after introducing a hole to the Hg-1212

cell by removing an electron and adding a compensating background charge to preserve the charge

neutrality. It clearly illustrates the hole localization around the CuO4 unit with 4 shortened Cu-O bonds

to1.855 Å from the original 1.925 Å, forming a breathing mode small polaron with the size comparable to

the lattice constants. Note that an initial small perturbation to the in-plane oxygens and the hybrid HSE

functional are needed to form such small polaron with lower energy of 106 meV/f.u. than the non-

polaronic one at 2 X 2 X 1 supercell, while DFT+U calculation cannot stabilize the polaronic structure.

The magnetic moment of planar Cu is reduced by 0.36 µB from the original 0.58 µB by the polaron

formation from our DFT simulation, indicating the localization of 0.36 hole. The magnetic moment of

each of the 4 nearest planar oxygens to the Cu ion is increased by 0.045 µB from the original 0 µB,

indicating the localization of 0.045 hole on each oxygen. In total, our DFT HSE simulation shows the

localization of 0.54 hole in the CuO4 unit at a 2 X 2 X 1 sized supercell. The hole localization in the CuO4

unit can be slightly changed in the range of 0.52 to 0.68 depending on the supercell size or hole doping

level in Hg-1201.

11. Density of states of in-plane Cu and in-plane O without/with polaron

FIG. S9 Partial density of states (PDOS) of in-plane Cu 3dx2-y2 and in-plane O 2p of Hg-1212 (a) without

polaron (b) with polaron by hybrid functional calculation.

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