  We investigate the maximal violation of Bell inequalities for two
$d$-dimensional systems by using the method of Bell operator. The maximal
violation corresponds to the maximal eigenvalue of the Bell operator matrix.
The eigenvectors corresponding to these eigenvalues are described by asymmetric
entangled states. We estimate the maximum value of the eigenvalue for large
dimension. A family of elegant entangled states $|\Psi>_{\rm app}$ that violate
Bell inequality more strongly than the maximally entangled state but are
somewhat close to these eigenvectors is presented. These approximate states can
potentially be useful for quantum cryptography as well as many other important
fields of quantum information.
