Spin-torque quantization and microwave sensitivity of a nano-sized spin diode

Rectification of microwave signal by the spin-torque diode is very promising for its practical applications in microwave imaging. This is due to a very high sensitivity of magnetic tunnel junction under the bias current, which was previously demonstrated in a number of works [1-3]. The decreasing of crosssectional area of the spin-torque diode up to the nano-sized dimensions below 10 nm allows one to reach high sensitivity without any bias current. Transverse quantization of electron states in the magnetic nanowire based on nano-sized metallic spin valves and magnetic tunnel junctions can create an additional impact not only on the magnetoresistance, but also on the spin-transfer torque in such structures. In this work we present an analysis of the quantization effect of conductance and spin-transfer torques on the microwave sensitivity of nano-sized spin-torque diodes during the reduction of its cross-sectional area. It was found that the magnetoresistance values up to 130 % can be achieved in a magnetic nanowire containing spin-valve diode with the nonmagnetic metal spacer. As a result, the maximum microwave sensitivity of spin-torque diodes based on these structures can be increased several times that opens the way for the further development of highly sensitive microwave detectors.


Introduction
Nowadays much attention in the field of spintronics is paid to the consideration of so-called spin-torque diodes based on magnetic tunnel junctions (MTJ), which have a large magnetoresistive effect and demonstrate very high sensitivity to the microwave signal. Microwave alternating current in MTJ generates dc voltage having maximum value at the resonant frequency of spin oscillations induced by the spin-transfer torque [1]. The sensitivity of the spintorque diode , where in P is the input microwave power, can exceed the sensitivity of a semiconductor Schottky diode by an order of value in the presence of the bias current [2][3][4]. High effect of the rectification of the microwave signal in MTJ makes spintorque diode attractive for practical applications in the microwave vision techniques [5]. However, the threshold current density, in the vicinity of which there is a sharp increase in the spin-torque diode sensitivity, is generally high. For that reason there are some problems of the usage of spin-torque diode shifted by bias current. In this connection, it is of interest to consider an alternative way of increasing the sensitivity of a spin-torque diode. This can be realized by reducing its lateral dimensions, since the spin-torque diode sensitivity increases inversely with the area of MTJ [6]. On the other hand, such scaling effects can crucially influence both magnetoresistance and spin-transfer torques in nano-sized region, where the effects of transverse quantization play an important role as it was shown for the conductivity of nano-point magnetic contacts [7,8]. In this work we performed an analysis of the microwave sensitivity of magnetic nanosized spin-torque diode based on magnetic tunnel structures depending on the lateral dimensions of its cross-sectional area. We also consider possible effects of quantization of the magnetoresistance and spin-transfer torques in a magnetic nanowire consisting of the metallic spin-valve structure.

Charge and spin currents
We analyze the spin-transfer torque effect in a magnetic nanowire consisting of two semi-infinite ferromagnetic (FM) layers separated by a thin nonmagnetic spacer (S) having the thickness S d and rectangular cross section b a  . Considered trilayer structure «FM-S-FM» of the magnetic nanowire is shown in Fig.1c. The magnetization unit vector P m in the reference (left) FM layer is assumed to be fixed, while the magnetization of the free (right) FM layer m is initially directed at the angle θ to the direction of magnetization in the reference layer. We use the Sommerfeld model for the description of the ballistic spin transport of free electrons. Within this model the charge and spin fluxes are determined by spinor wave functions i σ electrons with a given spin polarization according to well-known relations: in this case can be presented as the combination of plane waves We consider the potential profile   which is shown in Fig. 1a, b for the two types of «FM-S-FM» structure. In general, in the case of tunnel spacer (Fig. 1a), and in the case of metal spacer (Fig. 1b), is the Fermi level in the metal spacer, the Fermi level, spin splitting and the work function in the left (right) FM electrode, in the left (right) FM layer. The solution of the quantum-mechanical problem (2) with the potential (3) allows us to find both the charge of the proposed structures of magnetic nanowire according to formulas (1) for spin fluxes. Thus, the spin current can be represented as the difference between the thermodynamically averaged electron spin currents «from the left to the right electrode» ( ), which can be and «from the right to the left electrode» ( ), which take the following form:   [9]. In this case, it leads to the following relationships for the charge and spin current:

Quantization of the magnetoresistance and spin-transfer torque
The amplitudes of in-plane ( || τ ) and out-ofplane (  τ ) spin-transfer torques are proportional to the corresponding spin currents transmitted in the magnetic nanowire: It follows from the expressions (4) -(6) that the quantization of the energy levels for each conduction channel lead both to the conductance quantization   S G (proportional to h e / 2 ) and to the quantization / ), which should affect the microwave sensitivity of a spin-torque diode.

Microwave sensitivity of spin-torque diode based on magnetic nanowire
In [4] it was shown that the maximum microwave sensitivity MAX ε of the spin-torque diode in the absence of magnetic field for the case of mutually perpendicular magnetization geometry can be presented:

Simulation results
Numerical calculations of the conductance and spintransfer torques in the magnetic nanowire were carried out for the case of its rectangular cross section was obtained with the increase of its amplitude up to 130 % as it is shown in Fig. 2. ) is the parallel (antiparallel) magnetization alignment of FM layers. Fig. 3 shows that the quantization of the transverse energy levels in magnetic nanowire also affects the amplitudes of spin-transfer torques. It was found from Fig. 3а that the frequency and the number of quantum jumps will be growing up with increasing the bias voltage. Like in a bulk magnetic tunnel junction, the amplitudes of in-plane ( || τ ) and perpendicular (  τ ) spin-transfer torques have opposite sign in the case of positive bias voltage, but they are additionally quantized due to the confined geometry of magnetic nanowire. The dependence of maximum microwave sensitivity MAX ε of the spin-torque diode based on the spin-valve structure (Co-Au-Co) from the cross-section width a is shown in Fig. 3b. The following parameters were used: mW mV From Fig.3b it can be seen that with the decrease of the characteristic width of the spin-torque diode a the sensitivity of the diode rises and begins to oscillate, starting at approximately 3 nm. Calculations show that a decrease to lateral sizes less 10 nm makes it possible to obtain an increase in the microwave sensitivity of the spin-torque diode in the absence of a bias magnetic field by more than two order of value higher than in a nano-pillar spin-torque diode of 100 nm cross-sectional size. These results can be useful for future development of high-sensitivity microwave detectors. The considered features of the quantization of the spin-transfer torques are very important for the understanding of the physics of magnetoresistive memory elements during its scalability to technological nodes less than 10 nm [10]. Scaling less than 1 nm requires new technological and theoretical approaches related to monoatomic physics, which is beyond the scope of this work.
The work was supported by the Russian Science Foundation (project № 16-19-00181).