By Maher Zakaria Ahmed Selim
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Additional resources for Study of electronic and magnetic excitations in the 2D materials represented by graphene and magnetic nano-ribbons
17). Examples will be given later. In the above text b†k bk is interpreted as the boson number operator for magnons [56, 48, 63, 45, 64]. 48) will be used through the next chapters for our studies of the effect of impurities on the spin wave excitations of 2D magnetic stripe systems. 48) for the study of magnetic materials, it is seen that both Hamiltonians contains only quadratic products of fermion or boson operators, respectively. 48), when expressed in second-quantization operators, are closelyrelated models with the nearest neighbor interactions represented by ti j and Ji j .
The dot product Si · Sj = S ix S xj + S iy S yj + S iz S zj can be rewritten in terms of Sˆ ±j operators using S ix S xj + S iy S yj = 1 ˆ+ ˆ− ˆ− ˆ+ S S + Si S j . 41) To change the above formulation to one based on the occupation number representation and the quantized form of the elementary excitations , we can make use of the HolsteinPrimakoff (HP) transformation . 42) Sˆ Zj = S − b†j b j , where b†i and b j obey the usual boson commutation relations: bi , b†j = δi j , b†j , bi = −δi j , bi , b j = b†j , b†i = 0.
The allowed spin wave modes for that stripes are the eigenvalues of the E matrix. The matrix also used to study the effects of the edge exchange, the edge uniaxial anisotropy, and impurities on the allowed spin waves of stripes. In chapter 4 a trial understanding for the physics underling the construction of E matrix describing the short range (nearest neighbor) hopping interaction effected by the lattice geometry is developed. This understanding is used to explain the dispersion relations for 2D honeycomb lattice with zigzag and armchair edges obtained for graphene nanoribbons and magnetic stripes and also explain the appearance of peculiar zigzag localized edge states.
Study of electronic and magnetic excitations in the 2D materials represented by graphene and magnetic nano-ribbons by Maher Zakaria Ahmed Selim