Figure 1

(a) Properties of the $p$ orbital defect model, Eq. (1). We take crystal-field parameters $V_x=0$, $V_y=400\mathrm{K}$, $V_z=2000\mathrm{K}$, and spin-orbit coupling $\lambda =-600\mathrm{K}$. (a) The six energy levels of the model. The levels do not carry definite angular momentum quantum numbers, but occur in Kramers-degenerate pairs, no matter how strong the spin-orbit coupling. The mixing of the lowest four levels when $|\lambda |$ is comparable to the crystal-field parameters $V_{y,z}$ results in a locking of the magnetic moment direction; this locking is not present if $|\lambda |\gg V_{y,z}$ or if $|\lambda |\ll V_{y,z}$. (b) The idea of locking: even if the applied field $\mathit{B}$ is at a large angle ${\theta }_B$ from the principal axis $z$ of the crystal field, the resultant magnetization vector $\mathit{M}$ lies at a small angle ${\theta }_M$ from $z$. (c) The calculated ${\theta }_M$ vs ${\theta }_B$ for $|\mathit{B}|={10}^{-4}\mathrm{T}$, 300 T, and 1000 T. For a defect with these parameters, locking is strong for any practical field; it remains strong up to near 1000 T, when the magnetic energy in Eq. (1) becomes comparable to the crystal-field and spin-orbit energies. $\mathit{M}$ unlocks as ${\theta }_B$ passes through $\frac{\pi }{2}$, rotating rapidly to the opposite direction; however, if ${\dot{\theta }}_B$ is large enough, this rapid rotation is prevented by Landau-Zener tunneling between the first and second energy levels. (d) The anticrossing of these levels near ${\theta }_B=\pi /2$; $\frac{E}{|B|}$ is in units of ${\mu }_B$. The anticrossing gap scales with $|\mathit{B}|$, so that this Landau-Zener tunneling will occur readily at low fields.Reuse & Permissions

Figure 2

Magnitude of the flux per Bohr magneton coupled to SQUID loop by a current loop moved along the line indicated. “In-plane” and “perpendicular” refer to the orientation of the magnetic moment. SQUID dimensions are $2D=52\mu \mathrm{m}$ and $2d=41.6\mu \mathrm{m}$. Inset shows configuration of SQUID loop.Reuse & Permissions

Figure 3

Computed flux noise vs loop size $D+d$ for (a) fixed loop aspect ratio $2d/W=4$ and (b) fixed width $W=20\mu \mathrm{m}$. The jagged behavior in (a) is due to the discrete mesh of FastHenry. The open triangles in (b) indicate that the accuracy of the calculations is limited. (c), (d) show the dependence of the $1/f$ noise vs the separation of the current and SQUID loops for the perpendicular and one in-plane magnetic moment orientation. Dimensions: $D=30\mu \mathrm{m}$, $d=20\mu \mathrm{m}$.Reuse & Permissions