Generalization of Faraday’s Law to Include Nonconservative Spin Forces

The usual Faraday’s Law $\mathcal{E}=-d\Phi /dt$ determines an electromotive force $\mathcal{E}$ which accounts only for forces resulting from the charge of electrons. In ferromagnetic materials, in general, there exist nonconservative spin forces which also contribute to $\mathcal{E}$. These might be included in Faraday’s Law if the magnetic flux $\Phi$ is replaced by $[\hslash /(-e)]\gamma$, where $\gamma$ is a Berry phase suitably averaged over the electron spin direction. These contributions to $\mathcal{E}$ represent the requirements of energy conservation in itinerant ferromagnets with time dependent order parameters.

Figure 1

The wire is aligned along the $z$ axis, as is the longitudinal field $\overrightarrow{B}=B_0\widehat{\mathbf{z}}$. The wire contains a single domain wall near its center. The unit vector $\widehat{\mathbf{n}}$ defines the direction of the ferromagnetic order parameter, and the internal exchange field $JM\widehat{\mathbf{n}}$, within this wall. This vector makes an angle $\theta (z)=2{cot}^{-1}{e}^{-[(z-⟨z_0⟩)/w]}$ to the $z$ axis. The emf developed along the wire is measured by an ideal voltmeter connected as shown. In this illustration the domain wall lies $x-z$ plane. In fact, the wall processes so that it makes a time dependent angle $\varphi$ with this plane, with $d\varphi /dt=2\mu B_0/\hslash$.

Figure 2

The spin Berry phase ${\gamma }_s^{\pm }$ is $\mp \Omega /2$, where $\Omega$ is the solid angle subtended by the path in spin space, as illustrated in (a). The electron in (c) is at position $\overrightarrow{r}$ and sees a field with a direction defined by the Euler angles $\theta$, $\varphi$, while, as shown in (b), in spin space, the vector $\overrightarrow{R}$, defined by the same angles, passes from the South to North pole. The solid angle $\Omega$ for such a path is gauge dependent; however, the solid angle $\Delta \Omega$ sweep out in a time $\Delta t$ is well defined, as illustrated in red in (d). Reflecting the precession of the wall, the path has an angular speed $\omega =\dot{\varphi }=2\mu B_0/\hslash$ and the derivative ${\dot{\gamma }}_s^{\pm }=\mp (1/2)\dot{\Omega }=\mp \dot{\varphi }$. While this is a “motional motive force” in spin space it is not so in real space since in (c) the electron has no transverse velocity.

Figure 3

The balance of forces. In (a), shown in blue, is ${\phi }_s$ the spin potential energy for the majority spins. The difference in potential energy at the ends is $2\mu B_0$. The net force ${\overrightarrow{f}}_c+{\overrightarrow{f}}_{\mathrm{nc}}=0$ corresponding to equilibrium. (b) Compare this with the forces which occur when there is the usual motional emf. Charge accumulates at the ends of the bar producing an electrical potential energy ${\phi }_e$ the equivalent of the ${\phi }_s$. The resulting conservative force $(-e)E$ is equal and opposite to the nonconservative force $(-e)vB$, to give a null net force. (c) If the wall is fixed so that $\Delta z_0^{\pm }=0$, the initial potential is again that due to the Zeeman energy shown in blue. However, since now ${\overrightarrow{f}}_{\mathrm{nc}}=0$ there is a force ${\overrightarrow{f}}_c$ on the electrons which leads to an accumulation of charge such that the total force ${\overrightarrow{f}}_c+{\overrightarrow{f}}_e=0$, corresponding again to equilibrium. In purple is the electrical potential, the mirror image of the Zeeman energy, and the flat sum, shown in green.