The 63 Cu nuclear quadrupole resonance (NQR) frequency Q in the multilayered cuprates is calculated in the cluster model by the exact diagonalization method. The charge imbalance between the outer CuO 2 plane (OP) with apical oxygen O A and the inner plane (IP) without O A in three-layered Tl 2 Ba 2 Ca 2 Cu 3 O 10 is estimated by comparing our results with the experimental Q . In Tl-based cuprates with more than three layers, we predict a large enhancement of the splitting of Q between OP and IP by taking into account the reduction in bond length between Cu and O A and the resulting increase in charge transfer energy. Our results show that the NQR frequency is a useful quantity for estimating the charge imbalance in the multilayered cuprates. Many studies have been aimed at a high superconducting transition temperature, T c . Regarding high-T c cuprates, maximum T c depends on materials, e.g., T c $ 40 K for the La family and T c $ 90 K for the Y family, even if carrier densities are optimized. In particular, among hole-doped cuprates, such material dependence closely correlates with an energy difference between in-plane and apical oxygens, Á" A .1) Actually, a spatial variation in pairing gap due to a modulation of the apical oxygen O A is observed by scanning tunneling microscopy.2,3) In addition, it is known that the disorder around the apical site is much more harmful on T c than other randomness.
4)On the other hand, the T c of multilayered cuprates correlates with the number of CuO 2 planes in a unit cell, n, and shows a maximum at n ¼ 3.5,6) One of the reasons why T c is suppressed for n > 3 is charge imbalance, [7][8][9][10][11][12] i.e., the hole concentration in pyramidally coordinated outer planes (OPs) is different from that in square-coordinated inner planes (IPs). 5,[7][8][9][10][11][12][13][14] A nuclear magnetic resonance (NMR) study has revealed that the hole concentration in OP, OP , is larger than that in IP, IP .8) The magnitude of charge imbalance, Á ¼ OP À IP , increases with n and becomes 0.10 at most for n ¼ 3.8) Such charge imbalance induces some interesting phases, e.g., two kinds of superconducting gap 7) and the coexistence of superconducting and antiferromagnetic states. [9][10][11][12]15,16) To estimate the local charge density, the nuclear quadrupole resonance (NQR) frequency Q is a useful quantity, 17,18) since Q is determined by a local charge distribution in the ground state. In the NQR measurement for three-layered Tl 2 Ba 2 Ca 2 Cu 3 O 10 (Tl2223), 18) charge imbalance has been observed as two separated peaks with an approximately 6.6 MHz difference. Here, we note that the Knight shift K c is also proportional to a local charge density on a measured site. However, one needs to examine not only the ground state but also the excited states and the neighboring spinspin interactions, since K c is proportional to the susceptibility and the transferred hyperfine couplings. 19) In addition, one needs to measure not only the 63 Cu site but also the 17 O site to estimate the hole densi...