# Electron Transport in Carbon Nanotubes with Adsorbed Chromium Impurities
## 7 Spin-Dependent Transport of Carbon Nanotubes with Chromium Atoms  
In this section, we present the application of the above formalism to the problem of chromium impurities doped onto a carbon nanotube. In these final calculations, we neglect the vertex corrections in Equations (60), (61), and (64), and we neglect the static displacements of the atoms. We employ a \(2s,2p\)-states wave-function basis for the neutral carbon atoms and a \(3d\), \(4s\)-states wave-function basis for neutral chromium atoms. The initial ion core valence of C and Cr atoms \(Z^{\lambda_{i}}\) are 4 and 6, respectively. In addition, to reduce the total computational time, we only performed one iteration of the self-consistent iterative procedure. In Equation (16), we used \(Z^{\lambda}_{i\partial\sigma}=1\) for the occupied electronic states. The off-diagonal matrix elements of the Hamiltonian in Equation (1) were calculated by including the first three coordination spheres. The Green's function calculation in Equations (79) and (81) employed \(10^{3}\) points in the Brillouin zone. All calculations were performed at \(T=300\) K.  
The calculations start with a (3,0) chirality carbon nanotube that is doped with Cr atoms. The geometry for the crystal structure is then optimized by minimizing the free energy \(F\), defined in Equation (116). Note that the carbon nanotube (doped with Cr) has a one-dimensional crystal structure. The primitive cell contains 18 nonequivalent atomic positions. Carbon atoms are located at 12 positions on the surface of the inner cylinder. The distance between the carbon atoms is 0.142 nm. Cr atoms are located at the 6 positions on the outer surface of the cylinder opposite the center of a hexagon, the vertices of which are carbon atoms. The distance between carbon atoms and neighboring Cr atoms is 0.22 nm. a cross-sectional view of the crystal structure of the (3,0) chirality carbon nanotube with adsorbed Cr impurities is plotted in Figure 7.  
It turns out that the free energy is minimized by a random arrangement of Cr atoms on the surface of the nanotube. Figure 8 plots the dependence of the free energy on the pair correlations of Cr impurities with \(\varepsilon^{BB}=\varepsilon^{BB}_{i|0i}\) in Equation (100) and for the first coordination sphere. The symbol \(B\) denotes an atom of Cr. The dependence of the free energy on the pair correlations is shown in the region of the free energy minimum. Figure 7: Cross-sectional view of the crystal structure of a (3,0) chiral carbon nanotube with adsorbed Cr atoms. The unit cell of a nanotube is shown. Black solid circles are C atoms, while white dashed-line circles are Cr atoms. The smaller black circles denote Cr atoms located at a large distance from the tube end. As shown in Figure 8, the free energy \(F\) has its minimum at \(\varepsilon^{BB}=0\). This implies that the Cr atoms are randomly located on the surface of the carbon nanotube (for this density of impurities). The relative positions of the carbon atoms and the chromium impurities are similar to those found elsewhere for transition-metal dopants on carbon nanotubes using ultrasoft pseudopotentials [40; 41]. The value of the localized magneticmoment of a Cr impurity and the induced localized magnetic moment of a C atom in the direction of the magnetic field increases, as expected, with the size of the field. For carbon nanotubes that have five Cr atoms per primitive unit cell, the projection of the Cr magnetic moment varies in the range \(m_{Cr}=(1.02;2.24)\mu_{B}\), while the induced C magnetic moment lies in the range \(m_{C}=(0.0036;0.02)\mu_{B}\) as the external magnetic field increases from zero to \(H=200\) A/m. In this calculation, the magnetic field is oriented along the axis of the carbon nanotube. We also find that the pair correlations for the orientation of the localized magnetic moments in the first coordination sphere satisfies \(\varepsilon^{m}=0.235\) when the external field vanishes. This pair correlation nearly vanishes for the second and third coordination spheres.