ISSN 1068-3356, Bulletin of the Lebedev Physics Institute, 2016, Vol. 43, No. 1, pp. 8–11. c  Allerton Press, Inc., 2015. Original Russian Text c  G.D. Demin, A.F. Popkov, A.K. Zvezdin, A.A. Knizhnik, 2015, published in Kratkie Soobshcheniya po Fizike, 2015, Vol. 42, No. 12, pp. 56–61. Ballistic Transfer of Spin Momentum in Multiferroic Tunnel Junction G. D. Demin a , A. F. Popkov a,b , A. K. Zvezdin c , and A. A. Knizhnik d a National Research University of Electronic Technology (MIET), pl. Shokhona 1, Zelenograd, Moscow oblast, 124498 Russia; e-mail: gddemin@gmail.com b Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow Oblast, 141700 Russia; e-mail: afpopkov@inbox.ru c Prokhorov General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia; e-mail: zvezdin@gmail.com d Kintech Lab Ltd., 3-ya Khoroshevskaya ul. 12, Moscow, 123298 Russia; e-mail: knizhnik@kintechlab.com Received October 7, 2015 Abstract—The field dependence of critical voltages of switching of magnetic states of a syn- thetic multiferroic structure is studied based on a bifurcation analysis of Landau–Lifshitz–Gilbert equations with the torques caused by a tunneling spin transport, taking into account the voltage dependence of transferred spin momenta at the variations in the magnitude and direction of electric polarization. The voltage dependences of transferred spin momenta are determined based on the free electron model, taking into account exchange splitting of electron energy subbands in magnetic beaches and the effect of changing the tunnel barrier height at the variations in the polarization magnitude and state. DOI: 10.3103/S1068335616010036 Keywords: multiferroic tunnel junction, spin torque, phase diagram, threshold voltage. Magnetic tunnel structures attract researchers due to the promising development of the non-volatile memory, reprogrammable logic, and other spintronic devices. One of the attractive lines of development of non-volatile magnetic memory elements is the use of multiferroics in spin diodes, which allows an increase in the number of logic states in the unit cell from four to eight bits. As the promising direction, synthetic multiferroics representing heterostructures with two conductive magnetic layers and a dielectric interlayer ferroelectric are considered [1–5]. The resistance of such a tunnel structure can take four different values depending on its electric and magnetic state. To practically implement spintronic devices controlled by electric voltage, it is important to understand the dependence of the conditions of stability and variations of structure equilibrium states on magnetic and electric parameters. For spin diodes with multiferroic tunnel structure, this problem has not yet been adequately studied. In this study, in the approximation of almost free electrons, similarly to [6, 7], we consider ballistic transfer of spin moment (SM) in the multiferroic [4, 5] tunnel junction (MFTJ) with collinear magneti- zation orientation in semi-infinite ferromagnetic (FM) beaches with a thin ferroelectric (FE) interlayer. The example of the potential diagram of the tunnel junction in the case of a positive voltage is shown in Fig. 1. When constructing the potential barrier, we use the Thomas–Fermi model in which the effect of spin- dependent screening of the built-in FE charge at layer interfaces [3] is taken into account. Within this model, the MFTJ potential distribution can be written as U σ L σ R σ P (x)= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −σ L · Δ L ,x ≤ 0 E F + U B − eV x d + σ P · e · σ S ǫ 0  δ SR x d − δ SL  1 − x d  , 0 < x < d, −σ R · Δ R − δE F − eV, x ≥ d (1) where δE F = E FR − E FL is the difference of Fermi levels E FL and E FR in left and right FM electrodes, where subscripts i = L,B,R mean, respectively, layers of the left electrode, tunnel barrier, and right 8