Magnetic friction of a nanometer-sized tip scanning a magnetic surface: Dynamics of a classical spin system with direct exchange and dipolar interactions between the spins

We theoretically study the occurrence of magnetic friction of a nanometer-sized tip scanning a magnetic surface by studying the dynamics of a model of classical spins interacting through dipolar and exchange interactions, neglecting thermal effects. We find that for small scanning velocities, the friction force linearly scales with the velocity, with a slope proportional to the phenomenological damping parameter of the Landau–Lifshitz–Gilbert equation. At higher velocities, the friction vs velocity relationship becomes rather complex with the presence of a maximum that is explained by the excitations of spin-wave resonances in the sample.

Figure 1

(Color online) (a) Behavior of magnetic friction $F_{\text{fric}}$ as a function of the tip scanning velocity $v_{\text{tip}}$ for different values of the damping constant and (b) as a function of the damping constant $\alpha$ for different values of the tip scanning velocity. The symbols are the numerical results while the lines are linear fits to the points. Quantities are reduced according to Eqs. (14, 15, 16, 17).

Figure 2

(Color online) $x$ and $z$ components, (a) and (b), respectively, of the layer magnetization as a function of the layer index $l$ and time $t$. The velocity of the tip is $v_{\text{tip}}=0.0284$ and $\alpha =0.1$. Quantities are reduced according to Eqs. (14, 15, 16, 17).

Figure 3

(Color online) Magnetic friction as a function of tip velocity for $\alpha =0.1$ (a) and $\alpha =0.5$ (b), and different values of $L_z$ as indicated in the labels. Quantities are reduced according to Eqs. (14, 15, 16, 17).

Figure 4

(Color online) Magnetic friction as a function of tip velocity for different values of $\alpha$, as indicated by the labels, in a wide range of tip velocities. Quantities are reduced according to Eqs. (14, 15, 16, 17).

Figure 5

Tip-substrate interaction as a function of time for $v_{\text{tip}}=0.0284$ and $\alpha =1$. Notice that the period is in agreement with Eq. (18). Quantities are reduced according to Eqs. (14, 15, 16, 17).