content_image/379129.jpg
ous contradiction is due to the neglect of a Coriolis field of ⃗ B = − ⃗ ω γ along wire axis in the rotation frame. Thus, a conduction electron virtually moves across the DW will experience an static potential gain of E s = 2 ⃗ µ$_{·}$⃗ ω γ = ℏ ω e , exactly what is predicted by Niu’s theory[22] in the lab- oratory frame. 

A realistic magnetic wire will not be completely sym- metric around wire axis. In order to show the robust- ness of the physics discussed here, we consider a biaxial anisotropy field ⃗ h$_{an}$ = 1 µ 0 M s ( K 1 m$_{z}$ ˆ z − K 2 m$_{x}$ ˆ x ) , K 1 and K 2 describe the anisotropies along the easy- and hard- axis, respectively. Generally speaking, the synchroniza- tion would not be perfect in the presence of K 2 because local spin needs to climb over an extra energy barrier when it follows the motion of a CPMF. Fig. 3 shows the numerical results of velocity-frequency dependence of DW with biaxial anisotropies, where the general fea- tures are very similar to those in Fig. 2. It compares well with Eqs. (8) and (9). The magnitude of the peak velocity can be estimated by using materials parameters of Co: M$_{s}$ = 1 . 4 × 10 $^{6}$A/m, A = 4 × 10 − $^{11}$J/m, K 1 = 5 . 2 × 10 $^{5}$J/m$^{3}$, and α = 0 . 1 . We find the peak velocity of υ = 21 . 5 m/s at the critical frequency ω$_{c}$ = 0 . 17 GHz for h 0 = 100 Oe. 

<img src='content_image/379129.jpg'>

In conclusion, CPMF at GHz frequency is an efficient control parameter for DW motion. Two propagation modes are identified. Under a low frequency ( ω < ω$_{c}$ ), a DW propagates like a rigid body at a constant velocity that increases linearly with the CPMF frequency. At a high frequency ( ω > ω$_{c}$ ), DW propagation speed is os- cillatory whose time-averaged value decreases with the frequency. For a uniaxial wire, a DW under a CPMF 

can be mapped to the DW under STT due to uniform spin-polarized current. In the map, the CPMF frequency plays the role of the spin polarized current. 

This work is supported by Hong Kong UGC/CERG grants (#603007, 603508 and SBI07/08.SC09). We thank J. Lu for helping numerical simulations. 

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