2.6.2.1 Fundamental constants and value of characteristic current density $J_p$

One of the fundamental constant of the LLG equation is the electron gyromagnetic ratio which, in SI units, is generally given as the ratio between electron charge $e$ and mass $m_e$:

$$\gamma_e=\frac{e}{m_e}=\frac{1.602 \cdot 10^{-19} \text{C}}{9.109 \cdot 10^{-31} \text{kg} }=1.7587 \cdot 10^{11} \text{s}^{-1} \text{T}^{-1}$$ (2.98)

where it has been used the fact that $(1$    C$)/(1$    kg$)=1/(1$    T$\times 1$    s$)$. The gyromagnetic ratio $\gamma$ appearing in Eq. (2.91) (and previous equations) is measured in such units that $\gamma M_s$ should have the dimension of a frequency. We will use the MKSA system and measure magnetization in A/m. To have the corresponding measure in Tesla we have to multiply magnetization by $\mu_0=4\pi \cdot 10^{-7}$ F/m, i.e. the magnetic permeability of vacuum. Therefore the value of $\gamma$ to be used in Eq. (2.91) is

$$\gamma=\gamma_e \cdot \mu_0 = 2.21 \cdot 10^5 ^{-1} ( )^{-1} .$$ (2.99)

As far as the value of $M_s$, if we assume that the free layer is constituted by cobalt, we have

$$M_s=1.42 \cdot 10^6 \quad \longrightarrow \quad \gamma M_s= 3.14 \cdot 10^{11} ^{-1} .$$ (2.100)

This means that the unit time in the normalized equation (2.91) correspond to

$$\tau=\frac{1}{\gamma M_s}=3.2 = 3.2 10^{-12} .$$ (2.101)

Another fundamental constant involved in the characteristic current density (2.90) is the Bhor magneton $\mu_B$ which, in SI units, has the following value

$$\mu_B=927.4 \cdot 10^{-26} ^2 = 9.274 \cdot 10^{-24} ( ) \cdot ^3 ,$$ (2.102)

namely, it has the physical dimension of a magnetic moment. In addition to $\mu_B$, it is necessary to specify the Landè factor $g_e$ which is a pure number very close to 2. Finally, the characteristic current $J_p$ is proportional to the thickness $d$ of the free layer. A sensible choice of this parameter should be in the range of the nanometers. Let us choose $d=3$ nm. Now, we can compute $J_p$ :

$$J_p={\gamma M_s}\frac{e \text{M}_s d}{g_e \mu_\text{B}} \approx 1.15 \cdot 10^{13} A m^{-2} = 1.15 \cdot 10^{9} \text{A cm}^{-2} .$$ (2.103)

This value of current is reference to establish if a current is small or big as far as current induced spin torque is concerned. In this respect, it is useful to mention that in most reported experiment in Co-Cu-Co pillars the largest injected current densities are in the order of $10^{7}$   A m$^{-2}$. Thus the factor $\beta$ in Eq. (2.94) is at most in the order of $10^{-2}$.