Geometrical contributions to the exchange constants: Free electrons with spin-orbit interaction

Using thermal quantum field theory, we derive an expression for the exchange constant that resembles Fukuyama's formula for orbital magnetic susceptibility (OMS). Guided by this formal analogy between the exchange constant and OMS, we identify a contribution to the exchange constant that arises from the geometrical properties of the band structure in mixed phase space. We compute the exchange constants for free electrons and show that the geometrical contribution is generally important. Our formalism allows us to study the exchange constants in the presence of spin-orbit interaction. Thereby, we find sizable differences between the exchange constants of helical and cycloidal spin spirals. Furthermore, we discuss how to calculate the exchange constants based on a gauge-field approach in the case of the Rashba model with an additional exchange splitting, and we show that the exchange constants obtained from this gauge-field approach are in perfect agreement with those obtained from the quantum field theoretical method.

Figure 1

(a) Exchange constant ${\mathcal{A}}^{xx}$ and (b) DMI constant $D^{yx}$ in the one-dimensional Rashba model Eq. (69) as a function of Fermi energy for the model parameters $\mathrm{\Delta }V=1\mathrm{eV}$ and $\alpha =2\mathrm{eV}$ Å. Results obtained from the gauge-field approach (dashed lines) agree, respectively, to the exchange constant from the Fukuyama-type approach and to the DMI constant from the Berry-phase approach (solid lines). (c) Band structure of the one-dimensional Rashba model.

Figure 2

(a) Exchange constant ${\mathcal{A}}^{xx}$ and (b) ${\mathcal{A}}^{xy}$ in the one-dimensional Rashba model Eq. (69) as a function of the Rashba parameter $\alpha$. The Fermi energy is set to zero and $\mathrm{\Delta }V=1\mathrm{eV}$. Solid lines: complete exchange constants. The geometrical contributions ${\mathcal{A}}_{\mathrm{geo}1}$ (dashed), ${\mathcal{A}}_{\mathrm{geo}2}$ (dotted), and ${\mathcal{A}}_{\mathrm{geo}3}$ (dashed-dotted) as defined in Eqs. (52, 53, 54) are shown as well.

Figure 3

(a) Band energy of the one-dimensional Rashba model Eq. (69) with model parameters $\alpha =20\mathrm{eV}$ Å and $\mathrm{\Delta }V=1\mathrm{eV}$. (b) The corresponding exchange constant exhibits a plateau between $-0.3$ and 0.3 eV due to the gap of the band structure in (a).

Figure 4

(a) Exchange constant ${\mathcal{A}}^{xx}$ and (b) ${\mathcal{A}}^{xy}$ in the two-dimensional Rashba model Eq. (55) as a function of the Fermi energy. The model parameters are $\alpha =2\mathrm{eV}$ Å and $\mathrm{\Delta }V=1\mathrm{eV}$. Solid lines: complete exchange constants. The geometrical contributions ${\mathcal{A}}_{\mathrm{geo}1}$ (dashed), ${\mathcal{A}}_{\mathrm{geo}2}$ (dotted), and ${\mathcal{A}}_{\mathrm{geo}3}$ (dashed-dotted) as defined in Eqs. (52, 53, 54) are shown as well.