Analytical expression for the harmonic Hall voltages in evaluating spin orbit torques

Transcription

1 Analytical expression for the harmonic all voltages in evaluating spin orbit torques Masamitsu ayashi National Institute for Materials Science, Tsukuba 35-47, Japan Solid understanding of current induced torques is key to the development of current and voltage controlled magnetization dynamics in ultrathin magnetic heterostructures. A versatile technique is needed to evaluate such torques in various systems. ere we examine the adiabatic (low frequency) harmonic all voltage measurement that has been recently developed to study current induced effective field that originate from the spin orbit effects. We analytically derive a form that can be used to evaluate the harmonic all voltages and extract relevant parameters in two representative systems, i.e. out of plane and in-plane magnetized systems. Contributions from the anomalous all and planar all effects are considered. * hayashi.masamitsu@nims.go.jp

2 I. Introduction Application of current to systems with large spin orbit coupling in bulk or at interfaces may result in spin current generation via the spin all effect, and/or current induced spin polarization (the Rashba-Edelstein effect). 3, 4 The generated spin current can act on nearby magnetic moments via spin transfer torque 5, 6 or exchange coupling 7, 8. These effects are referred to as the "spin orbit torques", 8-4 which is to be distinguished from conventional spin transfer torque since the spin orbit coupling plays a critical role in generating the spin current. Spin orbit torques are attracting great interest as they can lead to magnetization switching in 5, 6 geometries which were not possible with conventional spin transfer torque, and unprecedented fast domain wall motion 7, 8. Solid understanding of how these torques arise is thus essential for developing devices utilizing spin orbit effects in ultrathin magnetic heterostructures. Recently, it has been reported that an adiabatic (low frequency) harmonic all voltage measurement scheme, originally developed by Pi et al., 9 can be used to evaluate the effective magnetic field 9- that generates the torque acting on the magnetic moments. -4 This technique has been used to evaluate the size and direction of the effective field in magnetic heterostructures. Using such technique, we have previously shown that the effective field shows a strong dependence on the layer thickness of Ta and CoFeB layers in Ta CoFeB MgO heterostructures. The difference in the sign of the spin all angle between Ta and Pt has been probed and reported recently. 4 It has also been shown that there is a strong angular dependence (the angle between the magnetization and the current flow direction) of the effective field in Pt Co AlOx. 3 Non-local effects, i.e. spin current generated in a Pt layer can propagate through a Cu spacer and exert torques on the magnetic layer, have been probed using a similar technique. 5

3 These results show that the adiabatic harmonic all voltage measurement is a powerful technique to study spin orbit torques in ultrathin magnetic heterostructures. To provide a simple way to estimate the effective field using this important technique, here we derive an analytical form that describes the harmonic all voltages generated when a sinusoidal current is applied to the system. We assume that both the anomalous all and the planar all effects are present. In Ref. [3], Garello et al. have shown the importance of the planar all effect contribution to the harmonic all voltage. ere we derive a formula that is applicable to the expression which we have previously used in evaluating the current induced effective field. In addition, we examine systems with out of plane and in-plane magnetization and compare the analytical solutions with numerical calculations based on a macrospin model. II. Analytical solutions A. Modulation amplitude of the magnetization angle The magnetic energy of the system can be expressed as E K cos K sin sin M. () EFF I where K EFF is the effective out of plane anisotropy energy and K I is the in-plane easy axis anisotropy energy. and are the polar and azimuthal angles, respectively, of the magnetization (see Fig. for the definition). K EFF and K I can be expressed as the following using the demagnetization coefficients K U : N ( i i X, Y, Z N i 4 ) and the uniaxial magnetic anisotropy energy K K N N M KI NX NYMS EFF U Z X S () 3

4 K U is defined positive for out of plane magnetic easy axis. The direction of the magnetization M and the external magnetic field are expressed using and as: M M mˆ, mˆ sin cos,sinsin, cos, (3) S sin cos,sin sin,cos. (4) M s is the saturation magnetization, ˆm is a unit vector representing the magnetization direction and represents the magnitude of the external magnetic field. The equilibrium magnetization direction (, ) is calculated using the following two equations: E K EFF KI sin sin MS cos sin (cos cos sin sin ) sin cos, (5a) E K M I sin sin S sin sin sin( ). (5b) Equations (5a) and (5b) can be solved to obtain (, ), which will be discussed later. To simplify notations, we define K K EFF and M S A K I. M S When current is passed to the device under test, current induced effective field X, YZ,, including the Oersted field, can modify the magnetization angle from its equilibrium value (, ). The change in the angle, termed the modulation amplitudes, hereafter, can be 3, 6 calculated using the following equations:, (8a) X Y Z X Y Z. (8b) X Y Z X Y Z 4

5 ere i and (i=x, Y, Z) represent the degree of change in the angles when a field is i applied along one of the (Cartesian) axes. To calculate i and, we use the following i relations derived from Eqs. (5a) and (5b): E KEFF KI sin cos MS X sincos Y sinsin Z cos i i + K sin cos sin M cos sin cos cos M f I S X Y S i i E K I cossin i i + K sin cos M cos sin M g I S X Y S i i (9a) (9b) where coscos sin f cos sin, g cos. i i sin The coupled equations (9a) and (9b) can be solved for i and, which reads: i i fi Cgi (a) F f cos sin g sin cos ˆ m FF i A i K A i (b) F sin cos ˆ K A m C Acossin F sin cos cos sin A X Y C A sin sin X sin Y coscos F Substituting Eqs. (a) and (b) into Eqs. (8a) and (8b) gives the following expressions for the 5

6 modulation amplitudes of the magnetization angle: cos cos sin cos sin cos sin F C X C Y Z (a) ˆ A cos cos cos sin K cos m X sin FF + sin ˆ A sin K cos my cos FF + A sin cos sin FF Z (b) Equations (a) and (b) are valid for any equilibrium magnezation direction and are general for arbitrary values of each parameter (no approximation made). B. Expression for the all voltage The all voltage typically contains contributions from the anomalous all effect (AE) and the planar all effect (PE). We define R A and R P as the change in the all resistance due to the AE and PE, respectively, when the magnetization direction reverses. Assuming a current flow along the x-axis, the all resistance R XY is expressed as: RXY RA cos RP sin sin () If we substitute = +, = + and assume << and <<, Eq. () can be expanded to read RXY RA cos sin RP sin sin sin cos (3) The all voltage V XY is a product of the all resistance R XY and the current I passed along the device, i.e. V R I (4) XY XY 6

7 When a sinusoidal current (excitation amplitude I, frequency ) is applied, the current induced effective field oscillates in sync with the current. Thus X,Y,Z in Eqs. (a) and (b) needs to be replaced with,, sint. Substituting Eq. (3) into Eq. (4) gives: XYZ V V V sintv cos t XY V B BI, V AI, V B BI A RA cos RP sin sin B RA sin RP sin sin B os RP sin c (5) As evident in Eq. (5), the second harmonic all voltage V contains information of X,Y,Z through and. Note that Eq. (5) describes the harmonic all voltage in the limit of small and C. Relation between the current induced effective field and spin torque To illustrate the relationship between the current induced effective field X,Y,Z and the conventional spin torque terms, X,Y,Z can be added, as a vector, in the Landau-Lifshitz- Gilbert (LLG) equation: m ˆ ˆ mˆ E mˆ m (6a) t M t 7

8 ere is the Gilbert damping constant, is the gyromagnetic ratio, E is the effective M magnetic field that includes external, exchange, anisotropy and demagnetization fields. Equation (6a) can be compared to the LLG equation with the two spin torque terms. m ˆ ˆ mˆ E a ( ˆ ˆ) ˆ ˆ m J mp bjpm (6b) t M t ere ˆp represents the magnetization direction of the "reference layer" in spin valve nanopillars/magnetic tunnel junctions, a J and b J correspond to the Slonczweski-Berger 5, 6 (STT) and the field-like effective fields 7, respectively. Comparing Eqs. (6a) and (6b), we can decompose the current induced effective field into two terms, STT FL, where amˆ pˆ STT J and FL bpˆ direction whereas the field like term voltage measures J. The STT-term STT depends on the magnetization FL is independent of ˆm. Note that the harmonic all and not the torque ( ˆm ); thus one can identify whether the effective field is STT-like or field-like by measuring its dependence on the magnetization direction. For the numerical calculations, we use pˆ,, as this represents the spin direction of the electrons entering the CoFeB layer via the spin all effect in Ta when current is passed along the +x axis for Ta CoFeB MgO heterostructures. In the following, we consider two representative cases, systems with out of plane and in-plane magnetization. III. Approximate expressions for the harmonic all voltages 8

9 A. Out of plane magnetization systems We first consider a system where the magnetization points along the film normal owing to its perpendicular magnetic anisotropy. To obtain analytical solutions for the harmonic all voltages, we make several approximations. First, to solve Eqs. (5a) and (5b) analytically, we assume that the in-plane uniaxial anisotropy is small compared to the external field, i.e. A <<. Equation (5b) then gives. Next we assume that the equilibrium magnetization direction does not deviate much from the z-axis, i.e. for M along ẑ and for M along ẑ. Keeping terms that are linear with, Eqs. (5a) and (5b) give: sin, (7) K Asin cos The ± sign corresponds to the case for M pointing along K, expressions (a) and (b) can be simplified to read: ẑ. Assuming and A <<, sin Asin IN Z IN IN (8a) K cos K cos sin (8b) Asin cos sin Z IN sin cos cos cossin sin sincos sin K A where we have defined the following fields: IN X cos Y sin sin cos IN X Y (9) Substituting Eqs. (7), (8a) and (8b) into Eq. (5) gives: 9

10 sin V RA RA RPsin I (a) K cos sin V sin cos RA RP IN RP IN I 4 (b) cos K We have neglected higher order terms with. In addition, if we consider cases where the external field is directed along one of the Cartesian coordinate axes (along x- or y-axis), terms with sin( ) will also vanish. Then Eqs. (a) and (b) can be simplified to read: sin V RA I K cos (a) sin V RAIN RPcos IN I 4 K cos sin RA X RPcos Ycos RA Y RPcos X sin I 4 K cos (b) Equation (b) shows that the planar all effect mixes the signal from different components of the current induced effective field. 3 For systems with negligible PE, X ( Y ) can be determined by measuring V as a function of the external in-plane field directed along the x- (y- ) axis. owever, if the PE becomes comparable to the size of AE, contribution from the orthogonal component appears in V via the PE. When R P is larger than half of R A, Y ( X ) becomes the dominant term in V for field sweep along the x- (y-) axis. Thus to estimate the effective field components accurately in systems with non-negligble PE, one needs to

11 measure V in two orthogonal directions and analytically calculate each component, as described below. We follow the procedure used previously to eliminate the prefactors that are functions of I and K in Eqs. (a) and (b). =/ is substituted in Eqs. (a) and (b) since the external field is swept along the film plane. The respective curvature and slope of V and V versus the external field are calculated to obtain their ratio B: V V RP RP B cos cos cos sin X Y Y X () RA RA We define B X V V and B Y V V, which correspond to B R measured for = and =/, respectively, and R P A, which is the ratio of the PE and AE resistances. Finally, we obtain: X Y B B X B 4 Y B 4 Y X (3) Equation (3) provides a simple method to obtain the effective field under circumstances where both AE and PE contribute to the all signal. When PE is negligible, = and we recover the form derived previously. B. In-plane magnetization systems

12 We next consider systems with in-plane magnetization (easy axis is along the x-axis). We make the same approximation used in the previous section, A <<, K (note that K < for in-plane magnetized systems). This may not apply to systems where the shape anisotropy is large, e.g. in high aspect ratio elements such as nanowires. owever, lifting this restriction will introduce more complexity in the analysis of the harmonic voltage, in particular, because one cannot make the assumption ~. ere we limit systems with small A. To obtain each component of the effective field (the STT and the field-like terms), the external field must be swept along two directions orthogonal to the magnetization direction. From symmetry arguments, the out of plane field sweep gives information of the field-like term whereas the in-plane transverse field sweep (transverse to the magnetization direction) provides the STT term. owever, the assumption A << used here causes the magnetization to rotate along the transverse field direction as soon as is larger than A, thus hindering evaluation of the STT term. To circumvent this difficulty, we make use of the anisotropic magnetoresistance (AMR) of the magnetic material and measure the longitudinal voltage (V XX ), which in turn provides information of the STT term. We start from the all voltage measurements and move on to the longitudinal resistance measurement in the next section.. armonic all voltage and the field-like term We assume that the equilibrium magnetization direction does not deviate much from the film for M pointing along one direction within the film plane and plane, i.e. + for the case where the direction of M is reversed from the former. Note that for A (see Eq. (5b)). Keeping terms that are linear with, Eqs. (5a) and

13 (5b) give: K Asin sin cos. (4), The ± sign corresponds to the case for. Assuming and A <<, K, expressions (a) and (b) can be simplified to read: Asin cos Z IN IN Z (5a) K sin sin K sin IN cos cos sin A A cos sin sin Z cos sin cos cos sin coscos sin K A (5b) Substituting Eqs. (4), (5a) and (5b) into Eq. (5) gives: cos V RA RPsin I K sin (6) R A cos V Z IN RPcos IN I 4 K sin K sin A cos sin (7) We have neglected higher order terms with. The second term (the R P term) in Eq. (7) dominates the second harmonic signal for samples with large K. Thus for a typical in-plane magnetized samples, one can ignore the first term in Eq. (7) to obtain: X sin Y cos V RPcos I Acos sin (8) It should be noted that the sign of V at a given, which determines the sign of the effective 3

14 field, changes when the field is slightly tilted in one direction or the other (e.g. ~ or deg). When the field is directed exactly along the film normal ( =), the field dependence of V vanishes. For in-plane magnetized samples, the STT effective field is directed along the film normal (not to be confused with the STT torque that points along the film plane). That is, Z corresponds to the STT term. See the inset of Fig. 3(b) in which we show the three effective field components X,Y,Z when the magnetization and the current flow directions are along +x and the incoming electrons spin polarization is set to +y ( pˆ,, ). owever, in Eq. (8), Z does not show up in the expression, illustrating the difficulty of evaluating the STT term in this geometry. To extract the STT term, one needs to measure the harmonic longitudinal voltages, which are shown in the next section. ere we proceed to obtain the field-like term Y. Using Eqs. (6) and (8), we derive a simple formula, similar to that shown in Eq. (3), to estimate Y. If the external field is slightly tilted from the film normal, it is preferable to apply the in-plane field component along the magnetic easy axis ( = deg) to unambiguously set the equilibrium magnetization azimuthal angle. Substituting = in Eqs. (6a) and (8) gives: V cos RAI K (9a) V sin R I P Y (9b) Thus Y (the field-like term) can be obtained by: sin V V Y K (3) 4

15 Unlike the case derived in the previous section (Eq. (3)) here one needs to substitute K and to calculate the effective field. This is because the field dependence of the first harmonic voltage is primarily determined by the change in the magnetization direction along the z-axis (relevant anisotropy is K ) whereas that of the second harmonic voltage is dominated by the magnetization angular change within the film plane (relevant anisotropy is A ); therefore, taking the ratio of the two will not cancel out K (and ). Since Y scales with the tilt angle, one needs to determine the value of in good precision.. armonic longitudinal voltage and the STT term As noted above, in order to obtain the STT term, one can make use of the AMR effect, if any, of the magnetic material. The longitudinal resistance R XX is expressed as: sin cos (3) RXX R RMR where R and R MR are, respectively, the resistance independent of the magnetization direction and the change in the resistance due to the AMR effect. Current is assumed to flow along the x- axis. We substitute = +, = + into Eq. (3) and assume << and <<, which then reads: RXX R RMR sin cos sin cos sin sin ( )sin cos (3) ere, we have kept the nd order terms that scales with and to show that these terms cannot be neglected when the current induced effective field X,Y,Z (that determines the magnitude of and ) becomes larger. This is because, for a typical geometry that would be employed here (current flow and the equilibrium magnetization directions pointing along +x), 5

16 sin( )~ and sin( )~, and thus the linear terms (that scales with and ) can be smaller than the nd order terms when or is larger than a critical value. As a consequence, there is a limit in X,Y,Z above which we cannot neglect the nd order terms and this limit is much smaller than the other geometries described previously. For simplicity, we only consider the small limit of X,Y,Z here. for M pointing Again, for in-plane magnetized systems, we substitute + along one direction within the film plane and when M is reversed. With A <<, ~, Eq. (3) reads: RXX R RMR cos cos sin cos cos sin (33) Substituting Eqs. (5a) and (5b) into Eq. (33) results in the following expression: IN sin Z cos RXX R RMR cos cos sin cos K cos sin( ) (34) We have dropped the terms with sin, which is zero when = or. Substitution of Eq. (4) into Eq. (34) leads to: cos cos RXX R RMR cos K sin K sin K Z where we have omitted terms that are small. A sinusoidal excitation current (35) I Isint is applied to the device and the resulting voltage is expressed as: V R I R Isint (36) XX XX XX Again, assuming that the current induced effective fields are in sync with the excitation current, Eqs. (35) and (36) give: 6

17 XX XX XX VXX V V sintv cos t XX V BI XX V AI V XX BI A R R cos B RMR cos sin cos MR cos K sin K K Z (37) These expressions are similar to those of Eqs. (a) and (b). The ratio of the field derivatives of the first and second harmonic signals directly provides the current induced effective field (STT term): XX XX V V Z (38) Unlike the harmonic all voltage measurements, where both the AE and the PE contribute to the signal, here the longitudinal voltage V XX is determined solely by the AMR effect. Equations (3) and (38) show that combination of the all and longitudinal voltage measurements can provide means to evaluate both components of the effective field for in-plane magnetized samples. IV. Comparison to numerical calculations The analytical solutions derived above are compared to numerical calculations. We solve Eq. (6b) numerically to obtain the equilibrium magnetization direction and the associated harmonic voltage signals. A macrospin model 8, 9 is used to describe the system. Substituting Eq. () into Eq. (6b), the following differential equations are obtained: 7

18 h t sin h t h a cos b cossinh coscosh cossinh sin J J X Y Z h a cossinb cosh sinh cos J J X Y X X S X Y Y S Y Z h h M N h M N h h sincos sinsin M N cos Z S Z K (46) The two coupled differential equations are numerically solved to obtain the equilibrium magnetization direction when both the external and the current induced effective fields are turned on. To mimic the experimental setup, a sinusoidal current is passed along the x-axis and the resulting all and longitudinal voltages are evaluated. Contributions from the anomalous all effect (AE) and the planar all effect (PE) are considered for the all voltage and that from the anisotropic magnetoresistance (AMR) is taken into account for the longitudinal voltage. For a given time during one cycle of the sinusoidal current application, we calculate the equilibrium magnetization direction and the corresponding all and longitudinal voltages. One cycle is divided into two hundred time steps to obtain the temporal variation of the all and longitudinal voltages. The calculated voltages are fitted with Eq. (5): i.e. VXY VV sint V cos t, to obtain the first and second harmonic signals. We compare the numerical results with the analytical solutions derived in the previous sections. A. Out of plane magnetization systems Figure shows results for the out of plane magnetized samples. The equilibrium magnetization angle ( ) with respect to the film normal (a), first (b) and second (d) harmonic all voltages are plotted against an in-plane field directed along the current flow direction (i.e. along the x-axis). 8

19 The transverse field (directed along the y-axis) dependence of the second harmonic all voltage is shown in Fig. (f); the corresponding magnetization angle ( ) and the first harmonic all voltage are the same with those shown in (a) and (b), respectively. The parameters used here mimic the system of Ta CoFeB MgO heterostructures (see Fig. caption for the details). The open symbols represent results from the numerical calculations (squares: magnetization pointing +z, circles: magnetization along -z) whereas the solid/dashed lines correspond to the analytical results. As evident, the analytical solutions agree well with the numerical results. The x, y, z components of the current induced effective field are shown in Fig. (c) and (e) when the in-plane field is swept along x- and y-axis, respectively. We use pˆ,,, a J =3 Oe and b J =3 Oe. The difference of X,Y,Z in (c) and (e) is due to the difference in the magnetization azimuthal angle: ~ for (c) and ~/ for (e). As shown in Fig. (c,e), at low magnetic field, one can consider X and Y as a J and b J, respectively. We test the validity of Eq. (3) by fitting the external field dependence of the first and second harmonic voltages with parabolic and linear functions, respectively, and calculate quantities corresponding to B X and B Y (Eq. ()). Substituting B X and B Y into Eq. (3), we obtain X ~3.3 Oe and Y ~.99 Oe, which match well with a J and b J used in the numerical calculations. B. In-plane magnetization systems Numerical results of in-plane magnetized systems are shown in Figs. 3. The material parameters used are relevant for in-plane magnetized systems with a small perpendicular magnetic anisotropy. 9

20 Figure 3 shows results when an out of plane external field, slightly tilted ( =5 deg) from the film normal, is applied. The in-plane component of the tilted field is directed along the magnetic easy axis, which is the x-axis here ( = deg, the in-plane anisotropy field ( A ) is ~ 4 Oe). The open symbols represent the numerical results. The equilibrium magnetization angles (, ) are plotted against the slightly tilted out of plane field in Figs. 3(a) and 3(b), respectively. The magnetization direction reverses (Fig. 3(b)) due to the in-plane component of the tilted field. Figure 3(c) and 3(d) show the first and second harmonic all voltages whereas Fig. 3(e) and 3(f) display the first and second harmonic longitudinal voltages. The solid lines represent the analytical solutions, which agree well for the all voltages (Fig. 3(c, d)) but show a small deviation at low fields for the longitudinal voltages (Fig. 3(e, f)). The deviation is due to the nonlinear (higher order) terms in R XX (Eq. (3)). To reduce contributions from the non-linear terms in R XX, we have used a J =.3 Oe and b J =.3 Oe to generate the effective field in Fig. 3 ( pˆ,, is assumed as before). For a J =3 Oe and b J =3 Oe, as used in the calculations shown in Fig., the non-linear terms dominate the harmonic longitudinal voltages: the analytical solutions do not match the numerical calculations in this external field range (the solution shows better agreement at higher fields). Note that such effect is negligible for the harmonic all voltages (Figs. 3(c) and 3(d)) with a J =3 Oe and b J =3 Oe. The resulting components of the current induced effective field ( X,Y,Z ) are shown in the inset of Fig. 3(b). One can identify that Z is the STT term, which changes its sign upon magnetization reversal, and Y is the field-like term. Since the magnetization lies along the x-axis, X is nearly zero.

21 We fit the numerically calculated harmonic all voltages vs. field with a linear function and use Eq. (3) to estimate Y (see Fig. 3(d) inset for V - vs. the external field and the corresponding linear fit). We obtain Y ~.3 Oe for both magnetization direction (pointing along +x and x). This agrees well with b J used in the numerical calculations. For the harmonic longitudinal voltages, we use Eq. (38) to obtain Z. Fitting the external field dependence of the first and second harmonic signals with parabolic and linear functions, we find Z ~.3 Oe and ~-.3 Oe for magnetization pointing along +x and x, respectively. Although these values match that of a J, it should be noted that non-linear effects start to take place when a J and b J becomes large. To overcome this difficulty, one can apply a large out of plane field to force the magnetic moments to point along the film normal, and then simultaneously apply an in-plane field to evaluate the current induced torques using Eq. (3). Another option is to use a material/system that possesses large A, such as high aspect ratio nanowires, so that a transverse in-plane magnetic field can be applied to evaluate the STT term. V. Conclusion We have derived analytical formulas that describe the adiabatic (low frequency) harmonic all and longitudinal voltages measurements when current induced spin orbit torques develop in magnetic heterostructures. We treat both out of plane and in-plane magnetized samples, taking into account the anomalous and planar all effects for the all voltage measurements and the anisotropic mangnetoresistance for the longitudinal voltage measurements. The derived forms are compared to numerical calculations using a macrospin model. The model used to describe out of plane samples agree well with the numerical calculations. For in-plane magnetized

22 samples, although there are some difficulties in accurately evaluating the size of the current induced effective field, partly due to the non-linear effects that takes place when current is increased, we show that one can use the all and longitudinal voltages to evaluate the two orthogonal components of the effective field (spin transfer and field-like terms). Utilizing the harmonic voltage measurements can help gaining solid understanding of the spin orbit torques, which is key to the development of ultrathin magnetic heterostructures for advanced storage class memories and logic devices. Acknowledgements The author thanks Kevin Garello and Kyung-Jin Lee for fruitful discussions which stimulated this work, and Junyeon Kim and Seiji Mitani for helpful discussions. This work was partly supported by the Grantin-Aid (5767) from MEXT and the FIRST program from JSPS.

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25 Figure Captions Fig.. Schematic illustration of the experimental setup. A all bar is patterned from a magnetic heterostructure consisting of a non-magnetic metal layer (gray), a ferromagnetic metal layer (blue) and an insulating oxide layer (red). The large gray square is the substrate with an insulating oxide surface. Definitions of the coordinate systems are illustrated together. M denotes the magnetization and represents the external field. Fig.. (a) Magnetization angle with respect to the film normal ( ), (b) first harmonic all voltage and (d,f) second harmonic all voltage plotted against in-plane external field ( =9 deg). The field is directed along the x-axis ( = deg) for (a,b,d) and along the y-axis ( =9 deg) for (f). Open symbols show numerical calculations using the macrospin model. Solid/dashed lines represent the analytical solutions: (a) Eq. (7), (b) Eq. (a), (d,f) Eq. (b). (c,e) x, y, z component of the effective field used for the numerical calculations. Left (right) panel indicates the effective field when the magnetization is pointing along +z (-z). Parameters used in the numerical calculations: K =36 Oe, A =-6 Oe, =., =7.6 Mz/Oe, a J =3 Oe, b J =3 Oe, pˆ,,, R A =, R P =., I= A. Fig. 3. (a) Polar ( ) and (b) azimuthal ( ) angles of the magnetization, (c) first and (d) second harmonic all voltages (V XY ), (e) first and (f) second harmonic longitudinal voltages (V XX ) as a function of a slightly tilted out of plane field ( =5 deg). The in-plane component of the tilted field is directed along the x-axis ( = deg). Open symbols show numerical calculations using 5

26 the macrospin model. Solid lines represent the analytical solutions: (a,b) Eq. (4), (c) Eq. (6), (d) Eq. (7), (e,f) Eq. (37). Inset of (b) shows the x, y, z component of the effective field used for the numerical calculations. Parameters used in the numerical calculations: K =-4657 Oe, A =-4 Oe, =., =7.6 Mz/Oe, a J =.3 Oe, b J =.3 Oe, pˆ,,, R A =, R P =., R MR =, R =, I= A. 6

27 Current z M y x Fig.

28 (deg) - Out of plane magnetization, field in the film plane - (a) 8 6 (b) 3 i=x,y,z i=x,y,z 9 +M Z -M Z V (mv) Field (Oe) Field (Oe) - along x-axis - (c) +Mz -Mz (d) 3 3 Y 5 Z V (V) X -5 +Mz Mz Field (Oe) Field (Oe) Field (Oe) - along y-axis - (e) +Mz -Mz (f) 3 3 Y 5 Z -3 X Field (Oe) Field (Oe) V (V) +M Z -M Z -5 +Mz -Mz Field (Oe) Fig.

29 (deg) V (mv) V (mv) - In-plane magnetization, field ~along the z-axis ( =5 deg) - (a) Field (Oe) 6 (c) Field (Oe) (e) V XY V XX Field (Oe) (deg) V (mv) (b) 36 V (V) Field (Oe) (d) Field (Oe) 4 (f) 4 V XX Fig. 3 V XY i=x,y,z.3 Y. X -.3 Z -4 4 Field (Oe) /V (mv - ) 4 +Mx -Mx Field (Oe) +Mx -4 -Mx -4 4 Field (Oe)