# Carrier-induced Antisymmetric-symmetric Tendencies of Spin Stiffness in Zigzag Graphene Nanoribbons

## TL;DR

This paper investigates how doping influences the spin stiffness in zigzag graphene nanoribbons, revealing antisymmetric-symmetric tendencies and their dependence on ribbon width, using the generalized Bloch theorem and spin spiral formations.

## Contribution

It introduces a method to analyze doping effects on spin stiffness in nanoribbons using spin spiral formations and the generalized Bloch theorem, highlighting antisymmetric-symmetric tendencies.

## Key findings

- High spin stiffness confirmed by previous supercell calculations.
- Doping induces antisymmetric-symmetric tendencies in spin stiffness.
- Ribbon width affects the spin stiffness dependence on doping.

## Abstract

The generalized Bloch theorem was applied to calculate the spin stiffness and to consider its tendencies when introducing the doping in zigzag graphene nanoribbons. To reach the intentions, two different flat spin spiral formations were constructed by fixing the ferromagnetic and antiferromagnetic spin arrangements at the two different edges by applying a constraint scheme method. A spin stiffness was then calculated by means of a quadratic order function, which maps the total energy difference in the self-consistent calculations onto the Heisenberg Hamiltonian. We found a very high spin stiffness, as predicted previously by the supercell calculation. We also showed that the antisymmetric-symmetric tendencies of spin stiffness are induced by the hole-electron doping. The dependence of ribbon widths of zigzag graphene nanoribbon on the spin stiffness was also provided with similar tendencies when the doping is taken into account.