Magnetization control in S/F/S junctions on a 3D topological insulator.
Strong dependence of the Josephson energy on the magnetization orientation in Josephson junctions with ferromagnetic interlayers and spin-orbit coupling opens a way to control magnetization by Josephson current or Josephson phase. Here we investigate the perspectives of magnetization control in S/F/S Josephson junctions on the surface of a 3D topological insulator hosting Dirac quasiparticles. Due to the spin-momentum locking of these Dirac quasiparticles a strong dependence of the Josephson current-phase relation on the magnetization orientation is realized. It is demonstrated that this can lead to splitting of the ferromagnet’s easy-axis in the voltage driven regime. We show that such a splitting can lead to stabilization of a unconventional four-fold degenerate ferromagnetic state.
By now it is well-known that current-phase relation (CPR) in Josephson junctions with multi-layered ferromagnetic interlayers is strongly sensitive to the mutual orientation of the magnetizations in the layers Waintal2002 (); Barash2004 (); Braude2007 (); Grein2009 (); Liu2010 (); Kulagina2014 (); Moor2015_1 (); Moor2015_2 (); Mironov2015 (); Silaev2017 (); Bobkova2017 (); Rabinovich2018 (). CPRs of Josephson junctions with ferromagnetic interlayers in the presence of spin-orbit coupling also depends on the magnetization orientation. This occurs primarily via the appearance of the magnetization-dependent anomalous phase shift Krive2004 (); Asano2007 (); Reynoso2008 (); Buzdin2008 (); Tanaka2009 (); Zazunov2009 (); Malshukov2010 (); Alidoust2013 (); Brunetti2013 (); Yokoyama2014 (); Bergeret2015 (); Campagnano2015 (); Konschelle2015 (); Kuzmanovski2016 (). This coupling between the Josephson and magnetic subsystems leads to the supercurrent-induced magnetization dynamics Konschelle2009 (); Kulagina2014 (); Waintal2002 (); Braude2008 (); Linder2011 (); Cai2010 (); Chudnovsky2016 (); Bobkova2018 (). In particular, the reversal of the magnetic moment by the supercurrent pulse Shukrinov2017 () was predicted. A unique possibility of controlling the magnetization dynamics via external bias current and series of specific magnetization trajectories has been reported Shukrinov2019 (). In Refs. Konschelle2009, ; Shukrinov2018, it was also reported that in the presence of spin-orbit coupling the supercurrent can cause reorientation of the magnetization easy-axis. Assuming the initial position of the easy axis along -direction these works demonstrate that under the applied supercurrent stable position of the magnetization becomes between â and -axes depending on parameters of the system.
Here we investigate prospects of S/F/S Josephson junctions constructed atop a three dimensional topological insulator (3D TI) surface, which hosts Dirac quasiparticles, in the field of supercurrent-induced magnetization control. Our motivation is that these Dirac quasiparticles on the surface of the 3D TI exhibit full spin-momentum locking: an electron spin always makes a right angle with its momentum. This gives rise to a very pronounced dependence of the CPR on the magnetization direction Tanaka2009 (); Linder2010 (); Zyuzin2016 (). In particular, the anomalous ground state phase shift proportional to the in-plane magnetization component perpendicular to the supercurrent direction was reported.
The second reason to study magnetization dynamics in such a system is that at present there is a great progress in experimental realization of F/TI hybrid structures. In particular, to introduce the ferromagnetic order into the TI, random doping of transition metal elements, e.g., Cr or V, has been employed Chang2013 (); Kou2013 (); Kou2013_2 (); Chang2015 (). The second option, which has been successfully realized experimentally, is a coupling of the nonmagnetic TI to a high magnetic insulator to induce strong exchange interaction in the surface states via the proximity effectJiang2015 (); Wei2013 (); Swartz2012 (); Jiang2015_2 (); Jiang2016 ().
Here we demonstrate that the anomalous phase shift causes the magnetization dynamics analogously to the case of a spin-orbit coupled system. However, in contrast to the spin-orbit coupled systems, where the magnetization dynamics was studied before, for the system under consideration the absolute value of the critical current also depends strongly on the magnetization orientation. It only depends on the in-plane magnetization component along the current direction. We demonstrate that such dependence, in a suitably chosen voltage-driven regime, can lead to supercurrent induced splitting of the magnetic easy axis of the ferromagnet. We show that this effect may lead to stabilization of a four-fold degenerate ferromagnetic state, which is in sharp contrast to the conventional two-fold degenerate easy-axis ferromagnetic state.
The paper is organized as follows. In Sec. II we derive a CPR for the S/F/S junction atop a topological insulator surface starting from the quasiclassical Green function formalism. This is followed by a discussion of the magnetization dynamics of such systems in Sec III. Next, in Sec IV, we discuss the stabilization of the four-fold degenerate ferromagnetic state. Finally, we conclude in Sec. V.
Ii Current-phase relation in a ballistic S/F/S junction on a 3D TI
The sketch of the system under consideration is presented in Fig. 1. Two conventional s-wave superconductors and a ferromagnet are deposited on top of a 3D TI insulator to form a Josephson junction.
First of all, we consider a current-phase relation of a Josephson junction. The interlayer of the junction consists of the TI conducting surface states with a ferromagnetic layer on top of it. It is assumed that the magnetization of the ferromagnet induces an effective exchange field in the underlying conductive surface layer. The Hamiltonian that describes the TI surface states in the presence of an in-plane exchange field reads:
where , is the Fermi velocity, is a unit vector normal to the surface of TI, is the chemical potential, and is a vector of Pauli matrices in the spin space. It was shown Zyuzin2016 (); Bobkova2016 () that in the quasiclassical approximation the Green’s function has the following spin structure: , where is the unit vector perpendicular to the direction of the quasiparticle trajectory and is the spinless matrix in the particle-hole and Keldysh spaces containing normal and anomalous quasiclassical Green’s functions. The spin structure above reflects the fact that the spin and momentum of a quasiparticle at the surface of the 3D TI are strictly locked and make a right angle. Following standard proceduresEilenberger1968 (); Usadel1970 () it was demonstratedZyuzin2016 (); Bobkova2016 (); Hugdal2017 () that the spinless retarded Green’s function obeys the following transport equations in the ballistic limit:
where and . are Pauli matrices in particle-hole space with . is the matrix structure of the superconducting order parameter in the particle-hole space. We assume . The spin-momentum locking allows for including into the gauge-covariant gradient .
Eq. (3) should be supplemented by the normalization condition and the boundary conditions at . As we assume that the Josephson junction is formed at the surface of the TI, the superconducting order parameter and are effective quantities induced in the surface states of TI by proximity to the superconductors and a ferromagnet. In this case there are no reasons to assume existence of potential barriers at the interfaces and we consider these interfaces as fully transparent. In this case the boundary conditions are extremely simple and are reduced to continuity of for a given quasiparticle trajectory at the interfaces.
To obtain the simplest sinusoidal form of the current-phase relation we linearize Eq. (3) with respect to the anomalous Green’s function. In this case the retarded component of the Green’s function . The anomalous Green’s function obeys the following equation:
Equation for is obtained from Eq. (4) by , and .
The solution of Eq. (4) satisfying asymptotic conditions at takes the form [the solution is written for ]:
where the subscript corresponds to the trajectories .
The density of electric current along the -axis is
where is the angle, which the quasiparticle trajectory makes with the -axis. is the distribution function corresponding to the trajectories .
Here we consider the voltage-biased junction. In principle, in this case the electric current through the junction consists of two parts: the Josephson current and the normal current . The Josephson current is connected to presence of the nonzero anomalous Green’s functions in the interlayer and takes place even in equilibrium. Here we assume that the deviation of the distribution function from the equilibrium is weak and can be disregarded in calculation of the Josephson current. In this case . Exploiting the normalization condition one can obtain . Taking into account that we find the following final expression for the Josephson current:
where . At high temperatures the main contribution to the current comes from the lowest Matsubara frequency and Eq. (8) can be simplified further
where . Similar expression has already been obtained for Dirac materials Hugdal2017 (). The normal current is due to deviation of the distribution function from the equilibrium. However, for the system under consideration, where we assume the ferromagnet to be metallic, practically all the normal current flows through the ferromagnet because in real experimental setups the TI resistance should be much larger as compared to the resistance of the ferromagnet. As for the Josephson current, it is carried by Cooper pairs and is strongly suppressed inside the ferromagnetic layer. Therefore, it flows through the TI surface states and we can assume that it is equal to the total electric current flowing via the TI surface states.
Iii Magnetization dynamics induced by a coupling to Josephson junction
The dynamics of the ferromagnet magnetization can be described in the framework of the Landau-Lifshitz-Gilbert (LLG) equation
where is the saturation magnetization, is the gyromagnetic ratio and is the local effective field. The electric current flowing through the TI surface states causes spin-orbital torqueYokoyama2010 (); Yokoyama2011 (); Mahfouzi2012 (); Chen2014 () due to the presence of a strong coupling between a quasiparticle spin and momentum. In principle, if the ferromagnetism and spin-orbit coupling spatially coexist, this torque is determined by the total electric current flowing through the system. However, for the case under consideration only the supercurrent flows via the TI surface states, where the spin-momentum locking takes place. Therefore, only this supercurrent generates a torque acting on the magnetization. The normal current flows through the homogeneous ferromagnet, where we assume no spin-orbit coupling. Consequently, it does not contribute to the torque.
The torque caused by the supercurrent can be accounted for as an additional contribution to the effective field. In order to find this contribution we can consider the energy of the junction as a sum of the magnetic and the Josephson energies:
where with and ( is the junction area) is the Josephson energy. is the uniaxial anisotropy energy with the easy axis assumed to be along the -axis. is the volume of the ferromagnet. The effective field and takes the form:
where we have introduced the unit vector , is the dimensionless junction length, is proportional to the ratio of the Josephson and magnetic energies, , is the Josephson frequency and .
The effective field consists of two contributions: the anisotropy field, which is directed along the easy axis, is represented by the last term in Eq. (14). The other terms are generated by the supercurrent. The same approach to study magnetization dynamics in voltage biased junctions has already been applied to systems with spin-orbit coupling in the interlayer Konschelle2009 (); Shukrinov2018 (). The qualitative difference of our system based on the TI surface states from these works is that the critical current demonstrates strong dependence on the -component of magnetization in our case, while it has been considered as independent on the magnetization direction earlier. This dependence leads to nonzero at small . This means that the easy -axis can become unstable in a voltage-driven or current-driven junction, while this axis is always stable if the critical current does not depend on magnetization direction. Moreover, there is no difference for the system between -components of the magnetization. This leads to the remarkable fact that in a driven system the easy axis is not reoriented keeping two stable magnetization directions, as it has already been obtained before, but is split demonstrating four stable magnetization directions. In the following section we study this effect in detail.
Iv Magnetization easy axis splitting
It is obvious that is an equilibrium point of Eq. (11) with determined by Eqs. (13)-(15). Now we investigate stability of this point. In the linear order with respect to the effective field can be written as follows:
The LLG equation (11) in the linear order with respect to and takes the form:
One can estimate the parameter for 3D TI Josephson junctions. Taking , as has been reported for Josephson junctions Veldhorst2012 (), and the easy-axis anisotropy field , what was reported for Py Beach2005 (); Beach2006 (), we obtain . In the regime the magnetization varies slowly at , therefore we can average Eqs. (18) over a Josephson period thus obtaining the following system:
From the above system one easily obtains that the equilibrium point becomes unstable at .
It is rather difficult to make accurate estimates of the numerical value of for realistic parameters. The main problem is the absence of experimental data on the value of . However, if we take from the measurements Rusanov2004 () on permalloy with very weak anisotropy, , from Veldhorst2012 () and the permalloy volume , then we can obtain for . Therefore, we can conclude that the range of values discussed in our work should be experimentally accessible.
Now we turn to study the stationary points of the magnetization and their stability. Beyond the linear approximation (with respect to and ) it is convenient to parametrize the magnetization as . Then from LLG equation one obtains:
Fig. 2 shows vector fields in the plane , according to Eq. (20) at four different values of . The stationary solutions are indicated by color points. The blue points correspond to unstable stationary solutions, while the red points indicate the stable magnetization directions. The Gilbert damping constant . We have chosen such a large unrealistic value of the Gilbert constant in order to clearly show the stability/instability of the stationary points because for , which is more appropriate for a realistic situation, stability/instability of a solution is not clearly seen in the figure [compare Figs. 3(a) and (b)], although in fact the topology of the vector field is not changed. Fig. 2(a) represents the regime , when the only stable solutions are , what corresponds to upper and bottom horizontal lines in the figure. Panels (b) and (c) demonstrate the vector fields in the regime of not very large . Four stable red points are clearly seen. Upon further increase of the stable points get closer to and finally merge into two stable points at some , as it is shown in Fig. 2(d). Therefore, there exists a finite range of , where the ferromagnet has four stable magnetization directions in the voltage-biased regime considered here. It can be easily shown from Eqs. (20) that the regime with four stable solutions exists when the nonlinear equation has nonzero solutions. All the stable solutions correspond to , that is, they lie in the -plane. In addition there always exist two unstable out of plane stationary points .
Further in Fig. 4 we demonstrate the full time evolution of the magnetization obtained from the numerical solution of the LLG equation. It is seen that starting from different initial conditions it is possible to reach all four stable magnetization solutions. The results are obtained at , but the averaged values of magnetization at large times are in good agreement with the results for stable points obtained in the limit , which are demonstrated in Fig. 3(a) for the same parameters , , and . Fig. 3(b) only differs from (a) by the value of . While the topology of the vector fields presented in panels (a) and (b) is the same, the stability/instability of all the stationary points is more clearly seen for larger values of the damping constant . At finite values of the magnetization oscillates around the vector trajectory presented in Fig. 3 and the amplitude of the oscillations is suppressed at .
In order to show that the system under consideration can demonstrate spontaneous behavior we investigate the system evolution starting from one of unstable points . A small noise is introduced to the system in order to allow for leaving the unstable equilibrium point. From the vector fields represented in Fig. 3(a) it is seen that at small values of the system finally comes to one of the four stable states with practically equal probabilities. It is shown in Fig. 5, where different panels correspond to all the possible final states.
Fig. 6 demonstrates the behavior of the absolute values of averaged magnetization at depending on essential parameters of the system. The dependence on is represented in Fig. 6(a). It is seen that at tend to constant values and, in particular, , as it follows from our analysis of stationary points of Eqs. (20).
The dependence on is plotted in Fig. 6(b). is linearly proportional to . For this reason one can explicitly see in this panel the range of where four stable limiting magnetization directions exist: it corresponds to the regions, where and are nonzero simultaneously.
Panel (c) of Fig. 6 represents the dependence of on the junction length. Analogously to the previous panel, the range of existence of four stable limiting magnetization directions is also clearly seen. The dependence on is qualitatively very similar to the dependence on , therefore we do not represent it.
Some aspects of magnetization control in S/F/S Josephson junctions on 3D topological insulator are investigated. Due to the spin-momentum locking in this system not only an anomalous ground state phase appears, but also the critical Josephson current depends strongly on the magnetization component along the current flow. This means that the easy -axis (perpendicular to the current flow) can become unstable in a voltage-driven or current-driven junction, while this axis is always stable if the critical current does not depend on magnetization direction. Moreover, there is no difference for the system between -components in the magnetization. This leads to the remarkable fact that in a driven system the easy axis is split demonstrating four stable magnetization directions. From the applied point of view it is interesting to switch between these states by means of voltage or current impulses. We postpone this problem for future work.
The work of I.V.B. and A.M.B. was carried out within the state task of ISSP RAS. The reported study was partially funded by the RFBR research projects 18-02-00318 and 18-52-45011-IND. Numerical calculations have been made in the framework of the RSF project 18-71-10095. K.S. thanks DST for support through INT/RUS/RFBR/P-314.
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