2.6.2.2 Discussion about the function $b(\textbf {{m}})$

Actually, the value of beta is generally even smaller due to the contribution of the function $b(\textbf {{m}})$. Then, let us now analyze this function and its order of magnitude. We denote with $b(\chi)$ the following function

$$b=b(\chi)=\left[ -4 +(1+P)^3 \frac{(3+ \cos(\chi) )}{4P^{3/2}} \right]^{-1} .$$ (2.104)

where $\chi$ is the angle between $\textbf{{m}}$ and $\textbf{p}$. It is evident that $b(\chi)$ is periodic function of $\chi$ and it assumes its minimum at $\chi=0$ and its maximum at $\chi=\pm \pi$. If we take as value of $P$ the one indicated for Cobalt in the paper of Slonczewski, we have $P=0.35$. Plots of the function $b(\chi)$ versus $\chi$ are reported for three different value of $P$ in Fig. 2.21 .

Figure: Plot of $b(\chi)$ versus $\chi$ for different values of $P$.

In the case $P=0.35$ the function $b$ assume a minimum value $b(0)\approx 0.13$ (parallel configuration of fixed and free layers) and a maximum value $b(\pi) \approx 0.52$ (antiparallel configuration of fixed and free layers). With this further information we can estimate the value of $\beta$ which according to the treatment above is in the order of $10^{-3} \div 10^{-2}$.

The second parameter in the normalized equation (2.94) is $\alpha$ which is generally considered also in the range $10^{-3} \div 10^{-2}$.