Module 2C: Micromagnetics

Debanjan Bhowmik Department of Electrical Engineering Indian Institute of Technology Delhi

Abstract In this part of the second module (2C) we will show how the Stoner Wolfarth/ single domain mode of ferromagnetism often fails to match with experimental data. We will introduce domain walls in that context and explain the basic framework of micromagnetics, which can be used to model domain walls. Then we will discuss how an anisotropic exchange interaction, known as Dzyaloshinskii Moriya interaction, can lead to chirality of the domain walls. Then we will introduce another non-uniform magnetic structure known as skyrmion and discuss its stability, based on topological arguments.

1 1 Brown’s paradox in ferromagnetic thin films exhibiting perpendicular magnetic anisotropy

We study ferromagnetic thin films exhibiting Perpendicular Magnetic Anisotropy (PMA) to demonstrate the failure of the previously discussed Stoner Wolfarth/ single domain model to explain experimentally observed magnetic switching curves. PMA is a heavily sought after property in magnetic materials for memory and logic applications (we will talk about that in details in the next module). This makes our analysis in this section even more relevant.

The ferromagnetic layer in the Ta/CoFeB/MgO stack, grown by room temperature sput- tering, exhibits perpendicular magnetic anisotropy with an anisotropy field Hk of around 2kG needed to align the magnetic moment in-plane (Fig. 1a). Thus if the ferromagnetic layer is considered as a giant macro-spin in the Stoner Wolfarth model an energy barrier equivalent to ∼2kG exists between the up (+z) and down state (-z) (Fig. 1b). Yet mea- surement shows that the magnet can be switched by a field, called the coercive field, as small as ∼50 G, which is 2 orders of magnitude smaller than the anisotropy field Hk, as observed in the Vibrating Sample Magnetometry measurement on the stack (Fig. 1a). This significant deviation from the Stoner Wolfarth model is known as Brown’s paradox in magnetism literature [1]. Within the Stoner Wolfarth model the magnet needs to cross the Hk 0 in-plane (x-y plane) energy barrier 2 to switch by 180 from up (+z) to down (-z) and as a result a switching field close to Hk will be necessary (Fig. 1b). However if a domain wall is introduced in the system the magnet can switch through domain wall motion at a switching field much smaller than Hk. This is because across the width of the domain wall the magnetic moment changes gradually from up to down. For the wall to move, each moment inside the wall needs to turn only by a small angle, which needs much lower energy than Hk (Fig. 1c). Starting from the magnet saturated in the up (+z) direction such a domain wall can be introduced by applying a magnetic field in the negative direction much smaller in magnitude than Hk. The ferromagnetic layer has several defects where the anisotropy is much lower than rest of the magnet. So reverse domains nucleate at these defects with domain walls surrounding them. Theoretically if the applied magnetic field is infinitesimally small but negative the domain wall can move such that the reverse polarized domains expand and the entire magnet switches from up (+z) to down (-z). However in reality the domain wall gets pinned at defects where the domain wall sits at a local energy minimum and an external field is needed to ”depin” the domain wall. This field is called the depinning field. When the externally applied reverse magnetic field exceeds the depinning field in magnitude, the domain wall moves entirely to switch the magnet over (Fig. 1d). Thus under the domain wall depinning based switching mechanism, the depinning field determines the coercivity of the magnet. If the reverse magnetic field is applied at an angle θ with respect to the film normal, the magnet switches when the component of the applied

2 magnetic field along the normal exceeds the depinning field. Thus the coercivity of the 1 1 magnet varies as cos(θ) . Such cos(θ) dependence of coercivity has been observed in anoma- lous Hall effect measurements we performed on Hall bars made from the Ta/CoFeB/MgO stack (Fig. 1e), confirming that the magnetization of ferromagnetic layer in these stacks indeed switch under a magnetic field by nucleation of reverse domain followed by motion of depinned domain walls [2].

2 Micromagnetics

As seen in the previous section, domain wall is a non uniform magnetic structure. Across the thickness of the wall, the magnetization gradually turns from a vertically upward direction to a vertically downward direction. Such non-uniformity can of course not be modeled by single domain model. Hence we need to develop a formalism to model a large number of magnetic moments interacting with each other through different energy terms we have discussed in the previous module- exchange, anisotropy, dipole interaction, etc. This formalism is known as micromagnetics.

We first start from the Heisenberg model, we discussed in module 2B:

H = −Σi,jJex(S~i.S~j) (1) where i and j correspond to neighboring atoms. Now after normalizing the spin of individual atoms by a saturation magnetization, we have obtained reduced magnetization vectors ~mi and ~mj corresponding to two atoms i and j, separated by a displacement vector ~ri,j.

2 ( ~mi. ~mj) turns out to be cos(φi,j) which can be approximated as 1 − φi,j, given φi,j is 2 very small. Thus, the Hamiltonian and the corresponding energy depends on φi,j. Now,

|φi,j| ≈ | ~mi − ~mj| (2)

Next, a very important assumption is made which forms the very core of the micromagnetics formalism. Instead of considering the magnetization arising out of individual atoms (in this case atoms i and j) the magnetization can be assumed to be a continuous field. Hence φi,j

3 Figure 1: (a)Vibrating Sample Magnetometry (VSM) measurement on thin films of Si (substrate)/ SiO2 (100 nm)/ Ta (10 nm)/ CoFeB (1 nm)/ MgO (1 nm)/ Ta (2 nm) shows that the stack exhibits perpendicular magnetic anisotropy. As a result, a large field (∼2000 gauss here) is needed to saturate the magnet in the in-plane direction (red plot). However the out of plane hysteresis loop shows that a very small field can switch the magnet in the out of plane direction (black plot). This behavior is observed for any Co/Pt/AlOx stack or Ta/CoFeB/MgO stack that exhibits perpendicular magnetic anisotropy. (b) Energy landscape of a single domain magnet shows that an anisotropy field of ∼2000 gauss is needed to switch the magnet by 1800 in the out of plane direction. (c) The ferromagnetic domain wall moves by a distance δθ through the rotation of each moment in the domain wall by an angle δθ. (d) Switching of a magnet with perpendicular anisotropy from up/ out of the plane (blue color) to down/ into the plane (red color) is shown schematically. Starting from a saturated state (all blue), reversed domain (red circle) nucleates with a small magnetic field into the plane. However, the domain wall is pinned at defects and hence cannot move. When the field is high4 enough to depin the domain wall, it moves and the reverse domain (red circle) expands to switch the magnet to into the plane (red square). (e) Minimum magnetic field needed to switch the magnet is plotted as a function of the angle between the direction of the applied magnetic field and normal to the surface of the film [2]. can be approximated as:

|φi,j| ≈ | ~mi − ~mj| = (mi,xxˆ + mi,yyˆ + mi,zzˆ) − (mj,xxˆ + mj,yyˆ + mj,zzˆ) ∂m ∂m ∂m ∂m ∂m ∂m ≈ ( x ∆x + x ∆y + x ∆z)ˆx + ( y ∆x + y ∆y + y ∆z)ˆy ∂x ∂y ∂z ∂x ∂y ∂z ∂m ∂m ∂m + ( z ∆x + z ∆y + z ∆z)ˆz ∂x ∂y ∂z ∂ ∂ ∂ = (∆x + ∆y + ∆z )(m xˆ + m yˆ + m zˆ) ∂x ∂y ∂z x y z ∂ ∂ ∂ = ((∆xxˆ + ∆yyˆ + ∆zzˆ).( xˆ + yˆ + zˆ))(m xˆ + m yˆ + m zˆ) = ( ~r .∇~ )~m (3) ∂x ∂y ∂z x y z i,j

Energy of the system (exchange energy) can written as:

2 2 Eex = JexS Σi,j| ~ri,j.∇~m| = 2 Z JS Z ∂mx ∂mx ∂mx 2 ∂my ∂my ∂my 2 3 ((∆x + ∆y + ∆z ) + (∆x + ∆y + ∆z ) + a V ∂x ∂y ∂z ∂x ∂y ∂z ∂m ∂m ∂m (∆x z + ∆y z + ∆z z )2)dV ∂x ∂y ∂z J S2Z Z ∂m ∂m ∂m ∂m ∂m ∂m ∂m ∂m ∂m = ex (( x + x + x )2 +( y + y + y )2 +( z + z + z )2)dV a V ∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z Z 2 2 2 = A ((∇mx) + (∇my) + (∇mz) )dV (4) V where ∆x = ∆y = ∆z = a: distance between nearest neighboring atoms, Z is the nearest 2 JexS Z number of neighboring atoms for a given atom and A = a is the exchange correlation 3kB Tc constant. From module 2B, Jex is related to Curie Temperature Tc as: Jex = J(J+1)Z . So,

J S2Z 3k T S2Z 3k T A = ex = B c = B c (5) a J(J + 1)Z a a

−10 Considering Tc = 1000K and a= 2.9 Angstrom,for Fe, we calculate A = 1.38 × 10 J/m. Usually exchange parameter A used in micromagnetic simulations is in the range of 10−11 to 10−10 J/m. The energy in equation (4) is the exchange energy, which is dependent on relative alignment of spins of adjacent atoms. In addition, the anisotropy energy is given by: Z 2 2 2 Eani = (K1mx) + (K2my) + (K3mz)dV (6) V where K1,K2,K3 are the anisotropy constants.

5 Then, the Zeeman energy (energy due to interaction of the magnetization with external magnetic field) is given by: Z EZeeman = − µ0(mxHx + myHy + mzHz)dV (7) V The magnetostatic energy/ dipole coupling energy/ energy of the demagnetizing field is given by: Z µ0 Edemag = − ~m.H~ddV (8) 2 V where H~d is the demagnetizing field. The total energy is the summation of all these energy terms, given by equations (4),(6),(7) and (8). The configuration of the magnetization, i.e. the value of the equi- librium magnetization vector as a function of position (x,y,z) is obtained by minimizing the total energy. The time evolution of the magnetization can also be obtained from the above energy terms using the dynamic equation (Landau Lifschitz Gilbert equation) which we discuss in the next part of the current module (2D). It is to be‘ noted that in this micromagnetic approach, we have moved from magnetization arising from discrete atoms, separated by the lattice constant, to a continuos magnetization distribution. In order to solve for this continuous distribution numerically (e.g. micromagnetic simulation packages like OOMMF and muMAX) a mesh can be chosen where the mesh size (distance between two adjacent grid points) can be much larger than the lattice constant. For example, a mesh size of about 1-2 nm is good to model domain walls in ferromagnetic films with perpendicular anisotropy. This mesh size is much larger than atomistic lengths which is in Angstrom. Thus micromagnetic simulations are not atomistic simulations, yet they are fine enough to model nanoscale structures like domain walls, skyrmions, etc.

3 Domain wall

We have explained in Section 1 of this module how domains can get formed in the first place during the magnetization reversal process. The reason domains expand in size is because that minimizes the magnetostatic energy of the system. Within a single domain, dipole coupling from moments of neighboring atoms oppose the moment of a particular atom. But when there are two domains with moments anti-parallely oriented then moments of one domain align in the direction of the dipole field from the moments of adjacent domain, thus minimizing the magnetostatic energy of the system. However two domains are separated by a domain wall where the magnetization gradually changes its orientation from one direction, that of the moments in the first domain, to another, that of the moments in the second domain. There is an energy cost associated with a domain wall which we will calculate next. Thus formation of domains is a balance between magnetostatic/ dipole coupling energy and energy cost of a domain wall.

6 Let us consider a magnetic material exhibiting Perpendicular Magnetic Anisotropy (PMA). Hence in one domain moment is in +z direction and in another domain moment is in -z direction with the moment in between (inside the domain wall) being: πx πx m = cos( ), m = sin( ), m = 0 (9) z W x W y where x is the direction in which the domain wall is formed, i.e. the width of the domain πx wall (W) is along x direction. mx can be 0 and my can be sin( W ) (Bloch wall vs Neel wall, to be discussed in next section) instead but it won’t change the result we derive next. Since the material exhibits PMA, only K3 in equation 6 is non-zero, we call it K. Considering only the anisotropy energy and exchange energy, energy of the domain wall=

Z W 2 ∂mz 2 ∂mx 2 E = (Kmz + A(( ) ) + A(( ) ))Btdx = 0 ∂x ∂x Z W πx π πx π πx KW Aπ2 (Kcos2( ) + A( )2sin2( ) + A( )2cos2( ))dx = Bt( + ) (10) 0 W W W W W 2 W Optimum domain wall width is such that: r ∂E 2A √ = 0 ⇒ W = π ; E = π 2KA (11) ∂W K

For the value of A = 1.38 × 10−10J/m and K = 6 × 105 J/m, domain wall width W turns out to be 67 nm. A more precise calculation can accurate more accurate value of domain wall widths. Usually for PMA materials of our focus, domain wall width is in tens of nanometers, similar to our rough calculation.

References

[1] Aharoni, A. Introduction to the Theory of Ferromagnetism, Oxford 1996

[2] Bhowmik,D. et al. Chapter: 6 (Magnetization switching and domain wall motion due to spin orbit torque) in Nanomagnetic and Spintronic Devices for Energy-Efficient Memory and Computing, Wiley (2016)

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