phenomenon. The orbital magnetization for Bloch electrons combines two terms [10, 16], one is the conventional orbital magnetic moment to characterize the magnetic response of the single- particle energy spectrum, and the other is a Berry-phase correction. These two terms play different roles in metallic and insulating regions [17, 18]. By varying the Fermi energy, we obtain the follows: (i) Both the magnitudes of these two terms increase (decrease) with increasing (decreasing) the Fermi energy in metallic regions; (ii) In insulating regions the conventional term keeps a constant unchanged value (quantum step), while the Berry-phase linearly varies with the Fermi energy with a slope proportional to the system’s Hall conduc- tance; (iii) A general fractal structure may occur in the orbital magnetization curve (as a function of the Fermi energy) by tuning the commensurate flux. The rest of this paper is organized as follows. In Sec. II we give the Hofstadter spectrum of the kagom´ e lattice by diagonalizing the corresponding tight-binding Hamiltonian in an external magnetic field. In Sec. III the orbital magnetization and its two components are systematically discussed by varying various system parameters. A summary is given in Sec. IV. The kagom´ e lattice is a two-dimensional periodic array of corner-sharing triangles with three sites per unit cell, as illustrated in Fig. 1. Here a is the triangle edge length (we set it as the length unit in the following), and A , B , and C denote the three sites in a unit cell. Let us begin with the tight-binding Hamiltonian for the spinless electrons on the kagom´ e lattice with zero magnetic field ( B =0), which is given by where t is hopping amplitude between the nearest-neighbor link 〈 i, j 〉 and $_{c}$† i ( c$_{i}$ ) is the cre- ation (annihilation) operator of an electron on lattice i . Hamiltonian (1) can be diagonalized in the momentum space as 3 H = t ∑ k $_{Ψ}$† $_{k}$H ( k )Ψ$_{k}$ , (2) H = t ∑ 〈 i,j 〉 $_{c}$† $_{i}$c$_{j}$ , (1) II. KAGOM ´ E LATTICE IN A MAGNETIC FIELD