Motion of a ferromagnetic domain wall under influence of an electromagnetic field.

Transcription

1 Motion of a ferromagnetic domain wall under influence of an electromagnetic field. Rob van den Berg University Utrecht July 0, 01 Abstract The aim of this thesis is to gain an understanding of the motion of a domain wall under the influence of an electromagnetic field. To gain this understanding, we numerically solve the equation of motion, in which we have added a potential well to simulate impurities and extra theorized torques. We find that the pinning potential indeed pins the domain wall for field strengths below a critical value, the added torques change the order of the graphs and the field strength at which Walker Breakdown happens. Our results may be of relevance to the development of race-track memory as well as a better fundamental understanding of the motion of a domain wall. 1

2 Contents 1 Introduction Race-track memory Static domain walls 5 3 Moving domain walls Magnetic field Electric current Extra Torques λ r and β Fields in other directions Pinning Frequency of the well De-pinning De-pinning with an electric curent Large oscillatory fields Conclusion 6 A Table of constants 8

3 1 Introduction A common way to store data is the usage of magnetic memory. The general concept is to magnetize small areas in a material in one of two directions. One can consider an upwards magnetization as a one and a downwards magnetization as a zero. With binary language one can store any kind of information this way. One of the benefits of this way of saving data is that, unlike for instance a system with semi-conductors, it is non-volatile. Which means that it does not take any energy for the system to stay the same e.g. to maintain the data. A downside is that the reading and saving of the data is a lot slower than when using volatile systems. The concept of magnetic memory is rather simple. Now there only needs to be a practical way to write the information on a medium and to later retrieve it by reading it. One of the newest methods of doing this is called the race-track memory, this type of memory is being developed in the Almaden Reseach Center by a team under Stuart Parkin s leadership. How this type of memory works is discussed in the next section. 1.1 Race-track memory To understand the race-track memory one has to go back to the basis. It is widely known that electrons have a half integer spin. This spin property can be described as an angular momentum. Using this, the fact that the electron has a charge and Ampere s law, it is not hard to see that the electron has a certain magnetization, pointing in the spin direction. The reasoning above is, according to quantum mechanics, not completely true. However, that does not change the fact that an electron, in fact, does have a magnetization pointing along the spin direction. A single electron, or a polarized group of them, can thus be used as a zero or one bit. These domains, found in ferromagnetic materials below the Curie-Temperature, are called Weiss areas 1. The surface between two Weiss areas is called the domain wall. For the race-track memory one makes a long, thin wire, of such a ferromagnetic material. The different domains will represent different bits, which can be forced into a certain direction with a local magnetic field. Where in a normal harddisc the disc will have to move to bring the data to the read/write head, the race-track memory makes use of the fact that the magnetization of these areas can be transported with a current, or a magnetic field. Using this, one moves the domains (moving domains and moving domain walls will both be used to describe the moving of the magnetization vector) by changing the magnetic orientation, rather than the physical placement of any particle. Effectively bringing the data to the read/write head without moving any mass. This of course decreases the wear on the storage system. 1 Introduction to Electrodynamics, D.J. Griffiths

4 Figure 1: Schematic display of a single strand of race-track memory. In figure (1) a schematic overview of a single string of race-track memory is shown. It shows the two different ways of storage, either in 3D (a), or in a plane (b). Neither is really better than the other and the one that will be used is merely the one that can be made for the lowest price. It also gives a graphic explanation on how to read (c) and how to write (d). The writing part is rather simple, the magnetization of a Weiss area (the red and blue areas) can be changed by applying a magnetic field, this magnetic field is generated by either a positive or a negative current-pulse, called the write current in the figure. The reading head makes use of the (giant)magneto resistance effect. This effect is explained by the magnetization of the electrons. Currents of positive magnetized electrons will meet a different resistance than currents of negative magnetized electrons, when moving through a magnetic field. This effect is normally not noticed due to the fact that electron currents are non-polarized in nature and because of the random orientation of Weiss areas in a conductor. In the set up for a race-track memory these factors are removed by creating a polarized read current going through a small resistor that is placed in the magnetic field of the bit we want to read. Depending on the binary value of the bit (and this depending on its magnetic orientation), this will give either a high current or a low current. Combining a lot of these wires into a grid with multiple read/write heads finishes the race-track memory. If it is developed successfully race-track memory offers a storage density comparable to conventional disk drives and higher than flash-memory. Its data speed is quite lower than for instance conventional DRAM (which is a volatile way of saving data). Slightly slower than high performance hard drives and much faster than Flash memory. In general however, it is the best potential replacement for Flashmemory. The only downside to race-track memory is the high current density needed for the transport, which is of the order of 10 8 A/cm or higher. Another problem shows that the do- IBM Develops Ultra-Dense Magnetic Memory, Nikkei Electronics Asia, 7 (008) 4

5 main walls have a tendency to get stuck at imperfections inside the material, requiring long pulses to get them moving again, removing these impurities will reduce the pulse time significantly. In this thesis I will analyse the equation of motion for domain walls to find the dependence of the domain wall s speed on the field or current strength. I will also simulate the earlier mentioned imperfections in materials to show their effects and investigate a way to efficiently de-pin the domain walls from these impurities. Static domain walls As described above, we consider a very long, thin wire pointing in the x direction. The width of the wire is equal everywhere along it. The wire is divided in a lot of little Weiss areas with domain walls between them. At any position {x,y,z} along the wire, the domain wall will have an orientation. This orientation can of course be given by a normalized spherical vector: Ω={sin θ cos φ, sin θ sin φ, cos θ}, where φ is the polar angle and θ the azimuthal one. The energy functional of this system is given by: E[θ, φ] = { Js [ ( θ) + sin (θ) ( φ) ] + K sin (θ) sin (φ) K } z dx cos (θ) a 3,(1) where a is a characteristic length scale of the wire, J s is the exchange stiffness and both K and K z are positive anisotropic constants. This functional is valid for static domain walls. It is easy to see that the lowest energy is obtained when θ is either π or π and when the angle φ is either 0 or π. Because these are the angles with the lowest energy, the domain will interpolate between these magnetization directions. Using the variation method on the energy functional we find that the angle θ(x) obeys the differential equation. d θ(x) dx = K z d cos (θ). J s dθ Multiplying with dθ(x) dx on both sides and then taking the integral gives ( ) dθ(x) + K z cos (θ(x)) = C, dx J s with C an integration constant, which can be determined with boundary conditions, dθ dx 0 and θ(x) 0, π if x ±. The distance to the domain wall is so large at the ends, that the angle does not depend on x any more. With these boundary conditions the magnetization will stand (anti)parallel to the magnetic field, therefore C = Kz J s. The differential equation will be of the next form: ( ) dθ(x) = K z sin (θ(x)). dx J s All constants are positive and θ is, per definition, an element of [0, π). Therefore it is allowed to take the root. With separation of variables we get the next result: [ ( )] θ Kz log tan = ± (x r dw ). J s 5

6 Inverting this equation gives the result for θ(x) with the domain wall at position r dw, ( θ(x) = arctan exp Q(x r ) dw) () λ with λ = K z J s and Q = ±1 the charge, this can change the magnetization from ẑ to ẑ, (Q = 1) or vice versa (Q = 1). 3 Moving domain walls The equation of motion for the magnetization is given by the extended Landau- Lifshitz-Gilbert equation 3 : ( ) ( ) d dt + v s Ω( x, t) = Ω( x, t) 1 h δe[ω] δω( x, t) + g B h ( α G Ω( d x, t) dt + β ) vs Ω( x, t), (3) α G where v s is the spin-velocity which is proportional to the current density, h the Planckconstant, g stands for the Bohr Magneton, α G is the dimensionlss Gilbert damping constant and β is the dimensionless parameter of the dissipative spin transfer torque. For Ω we have an expression in φ en θ. It is also known how we expect θ to behave, we calculated that in the previous section. We assume that φ is independent of the place and only dependent on the time. This will give [ θ( x, t) = arctan exp Q(x r ] dw) ; λ φ( x, t) = φ 0 (t). We now fill these assumption into the equation of motion. The equation of motion can now be simplified by the next trick. Take the inner product at both sides with the derivative dω( x,t) dr dw. Integrate this over the whole interval of x and we get dr dw (t) λdt + α G dφ 0 (t) dt = K h sin(φ 0(t)) + v x λ. (4) Note that due to the nature of this process, only a spin-velocity in the x direction has remained. In other words, v s = {v x, 0, 0}. We can do the whole process again, but now with the derivative dω( x,t) dφ, integrate once more and the result will 0(t) be dφ 0(t) dt + α G dr dw (t) λdt = gb z h 3 S. Zhang and Z. Li, Phys. Rev. Lett. 93, 1704 (004) + βv x λ, (5) 6

7 where the magnetic field is chosen to be in the ẑ direction. So B = {0, 0, B z }. Solving this system of equations for the derivative of φ will give (1 + α G) dφ 0(t) dt = α GK sin(φ 0 (t)) h + (α G β)v x λ gb z h. (6) To reduce the amount of constants we reduce this to a simpler form: dφ 0(t) dt = A 1 + A sin(φ 0 (t)). This differential equation has an exact solution of the form: dφ(t) = dt [ ( A + (A 1 A )(A 1 + A ) tan( ] (A 1 A )(A 1 + A )t)) arctan. A 1 It is possible to put the derivative of this straight back into equation (4). But it is easier to calculate the average value of dφ0(t) dt and to plug that back in. φ 0 (t) and its derivatives are periodic, so the average value is easy to calculate. With some knowledge of the tan function this can be done without any hard algebra. dφ0 (t) 1 t1 π dφ 0 (t) 1 = dt = (φ 0 (t 1 ) φ 0 (t 0 )) = π π. (7) dt t 1 t 0 t 0 dt t 1 t 0 A 1 A Here t 0 and t 1 are exactly half a period apart, to determine this period, we need to find the value of t for which the arctan(y(t 0 )) = arctan(y(t 1 )) Because the arctan(y) is a bijective function on y it means that we look for a value of t for which y(t 0 ) = y(t 1 ). This comes down to tan( (A 1 A )(A 1 + A )t 0 ) = tan( (A 1 A )(A 1 + A )t 1 ), the period of the tan(at) function is given by π A, so in this case: π P = (A1 A )(A 1 + A ). Since we can take t 0 and t 0 randomly, it is easiest to just take one in a maximum and the other one in a minimum. (the periodicity of the function ensures us that these points lay indeed half a period apart), the arctan s maximum is of course π and its minimum is π. Of course we do have to take into account the sign of A 1, if its value is positive φ 0 (t) will have a positive derivative and as such the average will be positive. In other words the sign of A 1 is equal to the sign of the average therefore we multiply (7) with sign(a 1 ). Filling back the values for A 1 en A gives (1 + αg) dφ0 (t) dt ( (α G β)v x ( (αg β)v x = sign λ λ + gb ) z h gb z h ( αg K h We fill this back into equation (4) to get drdw (t) = gb z λdt α + βv x G h α G λ + 1 (1 + αg )α G 7 ) ). (8)

8 ( (αg β)v x sign gb ) ( z (α G β)v x + gb ) z λ h λ h ( ) αg K.(9) h At this point it is best to rescale the position and time into dimensionless variables using, x = r dw(t) λ, τ = K t. (10) h Putting this back into (9) we get: dxdw (t) = gb z + dτ α G K hβv x α G λ K h (1 + α G )α GK ( sign (α G β)v x + gb ) ( z (α G β)v x + gb ) z λ h λ h ( ) αg K.(11) h The whole equation for the speed of the domain wall is now given in dimensionless rescaled variables. In the next part we will look at the speed of the domain wall with only a magnetic field (v s 0) or just an electrical current(b 0). 3.1 Magnetic field The whole derivation of the domain wall s speed was done for both a magnetic field and an electric current at the same time. It is of course useful to look at the behaviour of the solution for just a magnetic field. Below, equation (11) is reduced to just be dependent on the magnetic field. In other words, we let v x 0, which gives: dxdw (t) dτ = gb z α G K h (1 + α G )α GK sign ( gbz h Introducing the critical magnetic field ) ( ) gbz h ( ) αg K. (1) h gb c = α GK, (13) will reduce the equation to dxdw (t) dτ = B z B c 1 (1 + α G ) (Bz ) 1. Set α G = 0.1 and we can plot the graph. Of course, only real values are shown. B c 8

9 Bc Figure : Exact solution for the domain wall s speed versus the magnetic field strength It is interesting to determine what exactly the velocity of the domain wall is. x (τ) is by definition dimensionless. However: x (τ) = dr dw dt λdt dτ = dr dw h dt k λ. Which changes to dr dw dt of the order of B c = α GK g λk h 15m/s. The critical magnetic field strength is Tesla. The graph shows clearly that for small values of Bz B c the speed will grow linear. When Bz = 1, we see that the domain wall suddenly decreases. This event is B c called Walker breakdown and to see what happens we will look at equation (8). After rescaling this with (10) and putting v x to zero we find, dφ(τ) = α (1 B z B ) c G dτ (1 + αg sign(b zg ) h ). Of course, only real values are shown Figure 3: Average speed of the angle versus the magnetic field strength 9

10 In graph (3), the derivative of φ is shown as a function of Bz B c. As one clearly can see, when Bz B c = 1 or higher, the derivative is no longer zero. So in essence this is what causes the Walker Breakdown: the orientation of the domain wall will no longer remain static but will in fact become dynamic. For magnetic fields this will result in a decrease in velocity. 3. Electric current Rather than a magnetic field, one can also let an electrical current run through the wire. This means that in equation (11) we let B go to zero. Equation (11) will change into dxdw (t) dτ = hβv x α G λ K [ h (1 + αg )α sign (α ] ( G β)v x (αg β)v x GK λ λ Introducing the critical current will give ) ( ) αg K. h v c = λk h, (14) dx(τ) dτ = β α G v x v c + ( αg + v x vc (α G β) ) (1 + α G ) sign [α G β]. (15) x t vc Figure 4: The exact solution for the speed of the domain wall versus the spin velocity The current needed to reach the speeds calculated in the previous section are of the order v c = λk e vs h = 15m/s. Since J = P a A/m, which is 3 very high, unfortunately. 10

11 Φ0 t Figure 5: Average speed of the angle versus the electrical current strength. Blue, red, yellow green correspond to β =(0, 0.05, 0.1, 0.15) respectively. We see the Walker Breakdown again. For the green graph the Walker Breakdown reduces the speed because β > α G. For the red and blue curve the Walker Breakdown is positive because β < α G. The yellow line has no breakdown because the Gilbert Damping constant α G is equal to β. Below is once again a graph of the derivative of φ(τ) with respect to vx v c. The colors match with the curves in the graph (4). 4 Extra Torques Of course, the previously used model was a rather simple one. In reality there are several more terms that can be added to the equation of motion. Two kinds of these extra terms are spin-orbit interactions, these are theorized to exist in materials with a perpendicular anisotropy. We add two terms at the right side of the equation of motion for Ω. The two added torques are: λ r (J c ẑ) Ω, β r λ r Ω (J c ẑ) Ω, (16) where λ r is the constant that parametrizes the strength of the torques and β r is a dimensionless constant that determines the proportion between the two torques, J is the current density. The original way of simplifying the equation of motion still works of course. Take the inner product with the two derivatives, integrate and obtain thus two new, normal, differential equations: K sin(φ 0 (t)) hα G + v sx λ + (B ygπ + hj x πλ r ) cos(φ 0 (t)) hα G + ( B xgπ + hj y πλ r ) sin(φ 0 (t)) hα G = φ 0(t) + 1 α G λ r dw(t), (17) and hv sx β + B z gλ hα G + J xπβ r λλ r α G cos(φ 0 (t)) + J yπβ r λλ r α G sin(φ 0 (t)) = 11

12 r dw(t) λ α G φ 0(t). (18) Unfortunately, these differential equations cannot be solved analytically. We will have to solve them numerically. But first we will rescale both equations to obtain dimensionless terms using (10). After that, we reintroduce the critical magnetic field (13), critical current (14) and use that J = e v s a 3 P. The result will be and v sx π a3 B cy P α G + v cx λλ r a 3 P cos(φ 0 (t)) + π a3 B cx P α G + v cy λλ r a 3 P sin(φ 0 (t))+ sin(φ 0 (t)) + x (t) + α G φ 0(t) = 0, (19) B cz α G + v cx β πv cxβ r λλ r a 3 P cos(φ 0 (t)) πv cyβ r λλ r a 3 P sin(φ 0 (t)) α G x (t) + φ 0(t) = 0. (0) Unfortunately the solution can only be calculated when all the numerical values of the constants are filled in, this includes the six variables of v s and B. The calculated solution will rather be given as a function of t. The only way to get vs and B back as variables is to calculate the solution for different values of vs and B, then take the time average and plot those numbers out against the values of v s and B used. Using this we can now generate plots for an arbitrary configuration. The graphs (6) and (7) are the numerical calculations compared to the exact solutions Bc Figure 6: Both the numerical and the exact results for the domain wall s speed versus the magnetic field. 1

13 d x dt Vsc Figure 7: Both the numerical and the exact results for the domain wall s speed versus the electrical current. V sx vx V sc is equal to v c Do note that these numerical equations are made with the additional torques in the equation still put to zero. Its only use is to show that the numerical procedure works. Obviously the numerical solution look very much like the exact solutions, it only gives an inaccurate result at the Walker Breakdown point. This is of course, because the derivative of the solution does not exist at that point, complicating the numerical solution of the differential equations. One might be able to decrease this effect by dedicating more computing power to the generation of the graphs, but since we use graphs for qualitative results rather than quantitative, this is not really useful. 4.1 λ r and β The values of λ r and β r are of course needed to properly calculate the velocity. However, the values directly depends on the type of materials that are used. It is therefore more interesting to make plots for several values of λ r and analyse those graphs. We will therefore use values of several different orders to see whether there are any interesting effects. We expect basically three different regimes. One regime where the contribution of λ r can be neglected compared to other contributions, making λ r about 0. One regime where both contributions are of the same order and one regime where λ r dominates the whole graph. The extra torques in the equation are dependent on J and thus on v s, but not on B. The graph with only a magnetic field will therefore not change and is thus not shown again. 13

14 x Τ Vc Figure 8: Speed of the domain wall versus the spin velocity for the values λ r = 10 4, β r = 0.78 and four different values of β. The Walker Breakdown points in figure (8) are almost at the same place as they were originally, indicating that the terms with λ r are much smaller than the original ones, but not yet enough to completely dismiss them. Note that there seem to be two breakdowns in the green curve. x Τ Vc Figure 9: Speed of the domain wall versus the spin velocity for the values λ r = 10, β r = 0.78 and four different values of β. In figure (9), λ r is around the same order as the original terms, which gives Walker Breakdown at several points, note that the ordering of graphs has shifted. 14

15 x Τ Vc Figure 10: Speed of the domain wall versus the spin velocity for the values λ r = 10 1, β r = 0.78 and four different values of β. In figure (10) there are merely straight lines, because the λ r term dominates the equation. Do note that the original ordering has returned. It is easy to see that while λ r has some effect on the graphs, the order of magnitude of the speed remains the same, regardless of the order of λ r. Instead it changes when or if the Walker Breakdown happens. 4. Fields in other directions Due to the extra torques we also find that the motion of the domain wall depends on fields in other directions. For example v y is a new variable. However, plotting these showed that their contributions are small compared to the B z and v x variables. Therefore, we have decided not to analyse this new behaviour. It might be an interesting subject for another thesis. 5 Pinning As mentioned at the start, we will simulate imperfections in the structure. This will be done with the help of a pinning potential V pin = V 0 exp( r rdw (t) ξ ). Its derivative can be added to the differential equation. This changes the differential equations into v sx π a3 B cy P α G + v cx λλ r a 3 P sin(φ 0 (t)) + x (t) + α G φ 0(t) = 0 cos(φ 0 (t)) + π a3 B cx P α G + v cy λλ r a 3 P sin(φ 0 (t))+ 15

16 and x(t) exp( x (t) χ ) B cz α G 4gV 0 hχ + v cx β πv cxβ r λλ r λ a 3 cos(φ 0 (t)) πv cyβ r λλ r P a 3 sin(φ 0 (t)) P α G x (t) + φ 0(t) = 0, in dimensionless variables. V 0 is the strength of the well, χ = λξ is the dimensionless width of the well. Using some tuned values for these variables is needed, if the well is too strong or too small in width (which will give a steeper derivative and thus a stronger force) the domain wall cannot escape. In reality of course these parameters are given by the defect in the material and as a result can require different values than we look at Bc Figure 11: The speed of the domain wall versus the magnetic field, with pinning potential variables V 0 = K, χ = 0.8 x Τ Vc Figure 1: The speed of the domain wall versus the spin velocity, with pinning potential variables V 0 = K, χ = 0.8 Both plots show exactly what was desired; if the field is too weak the domain wall cannot escape out of the well, therefore its average speed will stay zero. 16

17 If the field is strong enough then it will kick the domain wall out of the well, quickly returning to the velocity it would have without the well. It is now also easy to see why some phenomenon are really hard to witness in nature, for a small impurity the whole Walker Breakdown is invisible if one would only look at the magnetic field. 5.1 Frequency of the well Every well has an eigenfrequency, if one were to drive the field or current with an oscillator with the same frequency, the domain wall would need merely some time to escape the well. If one has a material with periodic impurities, it would be beneficial to find this eigenfrequency to drive the field or current. The eigenfrequency can be found with the following procedure. Both equations are of the form A i x (τ) + B i φ(τ) = f i (x, φ). Where i = 1,. Setting f i (x, φ) = 0 and solving it for x and φ will give an answer of the form y = y 0. The equations we need to solve are f 1 (x, φ) = 4 hv 0x(τ) K χ λ exp( x(τ) χ ) = 0; f (x, φ) = sin(φ 0 (τ)) = 0. The solutions are x 0 = 0 and φ 0 = k π where k is an integer. We now let x(τ) = x 0 + δx(τ) = δx(τ) and φ = k π + δφ(τ). Plug this this back into the original equations gives α G δx (τ) + δφ (τ) = 4 hv 0δx(τ) K χ λ δx (τ) + α G δφ (τ) = sin(k + δφ(τ)). exp( δx(τ) χ ); Expanding the right side of the equations until the first order will give a linear 4 hv set of equations: 0δx(τ) K χ λ exp( δx(τ) χ ) 4 hv0 K χ λ (1 + δx(τ)) and sin(πk + δφ(τ)) (sin(πk) + cos(πk)δφ(τ). This will give a set of linear equations. α G δx (τ) + δφ (τ) = hv 0 K χ λ δx(τ); δx (τ) + α G δφ (τ) = cos(πk)δφ(τ). The trick now, is to let δy = C y exp( iωτ) for y = x, φ. This will let δy = iωc y exp( iωτ) = iωδy. One can put the result into a matrix, the first column corresponds with the coefficient of δx and the second with δφ. The first and second row correspond with the first and second equation. Requiring the determinant of this matrix to be zero will give us a quadratic equation for ω The matrix will look like ( iωαg hv0 K χ λ iω iω iωα G + 4 cos(kπ) The ( determinant of ) this matrix is given by 0 = ω (1 + α G ) + hv 0 K χ λ + iα G cos(kπ) iω 4 hv0 K χ λ cos(kπ). The solution is given by ). 17

18 ω = iα G( hv 0 K χ λ cos(kπ)) K χ λ (1 + α G ) ± 16 hk χ λ (1 + α G ) cos(kπ) 4α g( hv 0 + K χ λ cos(kπ)) K χ λ (1 + α G ). Filling in all the numbers 4 gives four solutions (two for each value of k), only 1 of these is a physical solution ω 0 = i. (1) The sign of the complex part is negative, this corresponds with a damped harmonic. This is expected behaviour, because after a while, the orientation will start processing around its original axis, decreasing the energy available for translational motion. 5. De-pinning Now that there is an equation for the eigenfrequency of the well we can test if it works. With this in mind we change the constant fields of before, to frequency dependent fields. From now on F = F 0 +F 1 sin(ωτ). Herein F can be either the spin velocity or the magnetic field, both in some direction. The idea is to start with the original cut-off graph, identify the critical value for which the domain wall s velocity will grow larger than zero (F crit ). We choose F 0 a bit smaller than this critical value, pick a value for ω and plot the domain wall s velocity as a function of F 1. It is important to realise that ω 0 was calculated without any external fields, it is therefore altogether possible that the best frequency is not ω 0, but in fact some other value. If F 1 is large enough, then for just about any frequency the domain wall will de-pin. The idea is of course to minimize the Energy needed for de-pinning. The energy density of a magnetic field in vacuum is given by u = 1 µ 0 B. With the structure used above it will translate to u = 1 µ 0 (B1 + B0 sin(ωτ) + B 1 B 0 sin(ωτ)). However, we are working with average values, so it is logical to take the average value of the energy density as well. The average value of sin(ωτ) is zero, while the average value of sin(ωτ) is a half. So this would result in a energy density of u B = 1 µ 0 (B 0 + B 1 ) for a magnetic field and likewise for an electric field: u E = ɛ 0 (E 0 + E 1 ). Of course, we will need the non-dimensionless values for the E and B field. For the electric field we have that E = J σ = evs a 3 P σ. For the energy density this means u E = ɛ0 v ( s evc ) vc σa 3 P in general, for our specific field this will give ( ) u E = v 0 v + v c 1 v J/m. c ( ) For the magnetic field we then get u B = B c B 0 µ B + B c 1 B J/m. In the following c 4 For the values of the constants, see Table (1) in Appendix A. 18

19 sessions we will merely compare the dimensionless energies (We note that u F F with some prefactors, so with dimensionless energy we mean the number gained where F is a dimensionless variable), it is therefore not needed to get all the dimensions right, they are given here as an indicator of how small the densities really are. It is of importance to note that these energy densities only give the energy generated by the field in vacuum, it does not resemble the energy density inside the conductor. Figure (13) is the graph for two different frequencies. The red curve has ω = 0.9, whereas the blue line fragment has ω = ω 0. As stated above B 0 = 5 and B 1 is variated. Unfortunately it is immediately clear that the eigenfrequency is not the optimal value for de-pinning. In further graphs I will however show the results for that frequency as well, merely for comparison. 0.5 x t Figure 13: The speed of the domain wall versus the field B = 5 + B 1 sin(ωτ) for two different frequencies. One can easily see that the de-pinning will only happen until B This means that the energy associated with the field is E B 6.13 as opposed to E B 36 which is the energy of the field at the original de-pinning strength. It is however possible to generate an even better result by increasing B 1 and keeping B 0 rather small. This will make it easier to de-pin the domain wall, while the constant term will keep it out once it is. Below is the graph for B = B 1 sin(ωτ) + B1 10, the frequencies are changed to the optimal value and still ω 0. 19

20 x t Figure 14: The speed of the domain wall versus the field B = B B 1sin(ωτ) for two different frequencies. Because of the new structure of the driven field the energy would now be E F F F 1 = F The value beneath the second peak is about F 1 = 6, this would result in E F Which is quite a bit lower than the energy cost in (13), while the speed reached is even larger. 5.3 De-pinning with an electric curent. The whole procedure can be redone with the electrical current, rather than the magnetic field. x Τ Vc Figure 15: The speed of the domain wall versus the spin velocity, with pinningpotential variables V 0 = K, χ = 0.8 The critical value depends on β, it is either 1 or 0.8. To compare everything though, we set v 0 = 0.7 and variate v 1. This time we use the frequencies 0

21 ω = ω 0 and ω = 1.5, below are the results for varying values for λ r, the thick line is for the black frequency (ω = 1.5) whereas the blue frequency is for ω 0. x t Figure 16: Speed of the domain wall versus the spin velocity for the values λ r = 10 5, β r = 0.78, four different values of β and two different frequencies. In figure (16) the speed of the domain wall will eventually reach a constant value. This is of course, due to the fact that increasing the variable will only increase the amplitude of the oscillating part. This part will have an average effect of zero, once it is been used to get out of the well. For this situation it seems that ω = 1.5 works better than ω 0, this is probably due to the fact that although the current is rather small (in dimensionless variables), it is still to large for the eqigenfrequency to be optimal. It is noteworthy that if β = 0 there will be no de-pinning at all. This is of course due to its original behaviour. x t Figure 17: Speed of the domain wall versus the spin velocity for the values λ r = 10 4, β r = 0.78, four different values of β and two different frequencies. 1

22 There is only a small difference between graph (16) and (17), it seems that even for β = 0 there is a small de-pinning. Although it is not a great improvement. x t Figure 18: Speed of the domain wall versus the spin velocity for the values λ r = 10 3, β r = 0.78, four different values of β and two different frequencies. Compared to the graphs (16) and (17), graph (18) shows a clear increase in the critical spin velocity, and also shows the shifting of the order between the graphs beloning to different values of β x t Vc Figure 19: Speed of the domain wall versus the spin velocity for the values λ r = 10, β r = 0.78, four different values of β and two different frequencies. Further increasing the value of λ r has completely shifted the order of β backwards in graph (19). Additionally, the critical spin velocity has increased even more than before.

23 x t Vc Figure 0: Speed of the domain wall versus the spin velocity for the values λ r = 10 1, β r = 0.78, four different values of β and two different frequencies. In figure (0), the ordering of different β has shifted again, back to its original order. The critical spin velocity has increased once more, while the resulting domain wall s speed seems to be diminished. It also appears that for the value β = 0.15 the optimal frequency is suddenly the eigenfrequency of the well. Which is rather strange, because increasing the value of λ r gives an increase in the critical spin velocity, which would mean that the calculations for the eigenfrequency is more inaccurate. 5.4 Large oscillatory fields Since a large oscillating field and a smaller constant field delivered better results for the magnetic field, we will take the same approach for the electrical current. We once again use the velocity v = v 1 sin(ωτ) + v1 10. And plot these for different regimes of λ r. x t Figure 1: Speed of the domain wall versus the spin velocity for the values λ r = 10 4, β r = 0.78, four different values of β and two different frequencies. 3

24 x t Figure : Speed of the domain wall versus the spin velocity for the values λ r = 10 3, β r = 0.78, four different values of β and two different frequencies. As we expected, the spin-velocity needed to de-pin the domain wall in figure (1) is much less than original. However, the resulting velocity of the domain wall is a lot smaller as well. It is once again clear that the eigenfrequency (the thin lines) of the potential well, is not the optimal value. Note that the colorordering is shifted though the corresponding values were kept the same. In figure () the increase of λ r now actually reduces the critical spin velocity needed to de-pin the domain wall. It also seems that larger values do not repress the dependence on β as much, although the green curve is still small in comparison with the rest. The curves for ω 0 are still invisible. x t Figure 3: Speed of the domain wall versus the spin velocity for the values λ r = 10 1, β r = 0.78, four different values of β and two different frequencies. 4

25 Although the graph (3) itself is not very informative, it does show us something interesting. With this particular value for λ r, the de-pinning spin-velocity is lower than previous while the resulting domain wall s velocity is rather large. This is a clear indication that λ r together with β are useful parameters to efficiently de-pin the domain wall. 5

26 6 Conclusion In this this thesis we have determined the dependence of the domain wall s velocity, which would in turn influence the data-speed in a memory device, on either a magnetic field, or an electric current. At first we used a simple model with exact solutions, later we introduced more terms in the equation of motion to resemble the effect of spin-orbit interaction. Afterwards we added a potential well, to simulate an impurity in the material, and changed the static field to a frequency dependent one, to see if a certain eigenfrequency would make the de-pinning easier. We managed to get the exact solutions, telling us how the speed of the domain wall behaves with regards to different field strength. These results are conform with what has been found before. Afterwards we added the the two extra torques, coming from spin-orbital interactions. These torques are theorized to exist in materials with perpendicular anisotropic attributes. The new results were plotted for several values of λ r, which is the constant associated that parametrizes the strength of these torques. One of the first things to notice is that the different values of λ r do not really change the graphs, they still grow into the same direction, they end up at the same point. There are however two things of interest, the first is of the Walker Breakdown point, this point will appear, disappear or even shift in some graphs. The second point is that the graphs with different β shift order if λ r is increased. And upon further increase the order shifts back again. Adding a potential well did exactly what was desired, it neatly cut off the domain wall s movement before a certain critical field strength. This is of course what would happen if the domain wall was stuck in some impurity; the pinning potential would first need to be overcome before it would be released. Afterwards it would quickly regain its original average speed, though perhaps a bit later. Unfortunately the calculated eigenfrequency of the well did not correspond very well to the numerical results. This is probably due to the fact that the fields were considered zero upon calculation, but were non-zero, in fact large values were needed to show any depinning. Changing the constant magnetic field into a small oscillating field, with a slightly smaller than the original critical field strength, showed a clear decrease in the critical field strength to de-pin the domain wall, though the optimal frequency was not the eigenfrequency. Changing the field to into a larger oscillating field with only a small constant value gave smaller critical values needed to de-pin the domain wall. Making the same changes to the electrical field, we could see that there is a small improvement for the critical field value, thus decreasing the energy needed to de-pin the domain wall. Changing the value of λ r changes the graphs in the same ways it did with the constant field, decreasing values will flip the order of the graphs up-side down and later back again. It is interesting to note that increasing the value of λ r increases the field strength needed for de-pinning, which is rather curious since the force with λ r as a constant works in the right direction. The effect of Walker Breakdown cannot really be noticed any more due to the oscillatory nature of the field. Making the electrical field have a large oscillatory nature with only a small constant value does not show an immediate improvement, unlike the same procedure for the magnetic field. However, when λ r is increased enough, a sudden 6

27 jump in the critical field strength is noticed. It suddenly decreased to only a fraction of its previous value. As seen, adding the extra torques changes little for the original behaviour of the domain wall. It may be of use however, in using oscillatory fields, where a certain value showed a great decrease in critical fields strength needed, while keeping the resulting domain wall velocity of the same order. It might therefore be worth the effort to invest in finding materials with such values for λ r. Though this is of course only profitable if there is no way of getting rid of all the impurities in the material. In this thesis we did not analyse the behaviour of the domain wall under the influence of fields other than a magnetic field in the z direction and a current in the x direction. It could be interesting to research the behaviour of the domain wall with fields and currents in different directions as well as using combinations of fields in arbitrary directions. 7

28 References [1] D. J. Griffiths Introduction to Electrodynamics 008 [] S. S. P. Parkin Spintronics and magnetoelectronics [3] A. Mougin, M. Cormier, J.P. Adam, P.J. Metaxas, J. Ferre. Domain wall mobility, stability and Walker breakdown in magnetic nanowires [4] T. Shinjo. Nanomagnteism and Spintronics. Elsevier, 009 [5] L. Berger, J. Appl. Phys. 55, 1954 (1984) [6] G. Tatara and H. Kohno, Phys. Rev. Lett. 9, (004) [7] S. Zhang and Z. Li Phys. Rev. Lett. 93, 1704 (004) [8] P. P. Freitas and L. Berger, J. Appl. Phys. 57, 166 (1985) [9] M. Hayashi, L. Thomas, C. Rettner, R. Moriya and S. S. P. Parkin, Narure Physics 3, 1 (007) [10] M. Oishi. IBM Develops Ultra-Dense Magnetic Memory. Nikkei Electronics Asia, 7 (008) A Table of constants Constant Value g J T 1 a m h K s λ m χ β r P α G V K Table 1: The values of the used constants. Both β and λ r are not included, their specific value is mentioned at each graph. 8