# Spintronic RF-Direct on-off Keying Modulation Using a Frequency Division Multiplex
## 3 Design of the Spin Modulator
### Simulation on an OOK Modulation Rate in the STNO  
Although [11; 15; 16] reported that current clearly modulates pulse repetition frequency (PRF) up to 5 MHz, and an STNO has a fast settling time, nearly 1 ns, the usable frequency of PRF is limited by the bias delay driving current at the transmitter, and by a falling time of 180 ns and a rising time of 100 ns for detection at the receiver in the experiments reported in this work. By introducing the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation, we conducted a simulation to assess how the rising and falling time of oscillation for the MTJ pillar is delayed by the effect of the bias circuit to drive the STNO in the transmitter. This condition is that each matching circuit is perfectly matched to the antenna and isolated each other between two ports of an STNO. Magnetic dynamics of the free layers of an STNO are found from the LLGS equation [12; 13; 17; 18]. The LLGS equation used here is given as Equation (1). \[\frac{d\hat{m}}{dt}=-\gamma\hat{m}\times H_{eff}+\alpha\hat{m}\times\frac{d\hat {m}}{dt}+\gamma a_{J}\hat{m}\times(\hat{m}\times\hat{\sigma})+\gamma b_{J}\hat{ m}\times\hat{\sigma} \tag{1}\]  
where \(\hat{m}\) is the unit vector of free layer magnetization, \(\gamma\) is the gyromagnetic ratio, \(\alpha\) is the Gilbert damping parameter, and \(\hat{\sigma}\) is the unit vector of pinned layer magnetization. The effective magnetic field, \(H_{eff}\), is composed of an applied magnetic field, \(H_{app}\), a uniaxial anisotropy field, \(H_{k}\), and the demagnetization filed, \(H_{d}\). Therefore, the effective magnetic field can be written as:  
\[H_{eff}=H_{app}\hat{x}+H_{k}(\hat{m}\cdot\hat{x})\hat{x}-H_{d}(\hat{m}\cdot\hat {z})\hat{z} \tag{2}\]  
where \(\hat{x}\) and \(\hat{z}\) are the unit vectors along the x and z axes, respectively. The first part of Equation (1) is the Larmor precession, the second is Gilbert damping, the third is in-plane spin torque which is competes with the Gilbert damping, and the last is the perpendicular spin torque equation [12; 17; 19]. \(a_{J}\) and \(b_{J}\) in the third and the fourth part of Equation (1), respectively, are the current density, which is related to the scale of spin torque. \(a_{J}\) is the in-plane torque efficiency and \(b_{J}\) is the perpendicular torque efficiency. The in-plane field torque, \(a_{J}\), competes with the damping (\(\alpha\)) whereas the perpendicular torque, \(b_{J}\), acts similar to a magnetic field [20]. We assumed that the capacitance is located parallel with the STNO due to the bias circuit, and there are ideally no parasitic components except the internal resistance of the STNO having a value between 48 \(\Omega\) and 80 \(\Omega\). Figure 7 shows the simplified circuit driving the STNO for the simulation. Figure 7: Simplified circuit driving the STNO. (C is the capacitance located in parallel with the STNO, and Lp, Cp, and R(t) are the parasitic inductor, parasitic capacitor, and internal resistance of the STNO, respectively). In the MTJ structure of 100 \(\times\) 90 nm\({}^{2}\) with a 1 nm tunneling gap, the parasitic capacitance is around 0.1 femto-Farad, and the parasitic inductor is around 1 pico-Henry in the condition of 1.5 mA operation. These parasitic components can be neglected in the equivalent circuit and the numerical equations of a 3-5 GHz band circuit. We also assumed that the STNO directly modulates the on/off signal of the DC current for operation of OOK modulation. The high or low DC current signal, \(I_{\mathrm{DC}}\), has a level of 1.5 mA or ground for a logic level; however, we made this signal have a 1 ns delay time for the input of the bias circuit to drive the STNO. This reason is for obtaining settling time from the response of the on/off current signal into the real delay value, as the function generator, AFG3251C, outputs a rectangular wave with a 1 ns delay time into the on/off DC signal [21]. The current induced into the STNO, \(I_{S}\), can be expressed as Equation (3).