# Spin-orbit torque-driven domain wall motion in the absence of Dzyaloshinskii-Moriya interactions

André Thiaville, Jacques Miltat

► To cite this version:

André Thiaville, Jacques Miltat. Spin-orbit torque-driven domain wall motion in the absence of Dzyaloshinskii-Moriya interactions. EPL - Europhysics Letters, 2024, 146 (6), pp.66001. 10.1209/0295-5075/ad4e62. hal-04748816

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# Spin-orbit torque-driven domain wall motion in the absence of Dzyaloshinskii-Moriya interactions

ANDRÉ THIAVILLE and JACQUES MILTAT

*Laboratoire de Physique des Solides, Université Paris-Saclay, CNRS UMR 8502, 91405 Orsay, France*

**Abstract** – The fine structure and dynamics of magnetic domain walls in ultrathin films with perpendicular magnetization, in presence of a secondary anisotropy, is analysed owing to micromagnetics. Two cases are considered, a cubic anisotropy typical for (111) oriented garnet epitaxial films, and an orthorhombic anisotropy as found in e.g. Co/W(110) films. The statics is solved first, showing that, in general, domain walls are not of the pure Bloch type. The dynamics under the spin Hall effect induced by a current flowing in an adjacent layer is then monitored. Finite and non-negligible domain wall velocities are predicted in both cases, in the absence of Dzyaloshinskii-Moriya interactions, with distinct behaviours regarding the current density and its orientation with respect to the secondary anisotropy axes. The relevance of these results to recent reports of current driven domain wall dynamics in insulating ultrathin garnet films, capped with platinum, is discussed.

**Introduction.** – A series of recent studies of ultra-thin (insulating) garnet films with perpendicular magnetization, capped by platinum along which current is fed, have reported magnetization switching and domain wall (DW) motion [1–4]. Interfacial Dzyaloshinskii-Moriya interaction (DMI) has been invoked to explain these results, although [5] (i) there is at present no mechanism to account for an interfacial DMI at the surface or interface of an insulating garnet (ii) conflicting experiments attribute this interfacial DMI to the garnet film / garnet substrate interface, or to the platinum / garnet interface (iii) the observed DMI values are quite weak, the in-plane fields sufficient to suppress their effect being of only a few milli-Tesla in thulium iron garnet films.

Being epitaxial, garnet films are characterized by the existence of a cubic magnetocrystalline anisotropy [6], on top of the dominant uniaxial perpendicular anisotropy. It is therefore of interest to scrutinize in detail the role of this secondary anisotropy on the structure of the DWs, and on their motion under spin-orbit torque (SOT). This is the object of this paper.

Before turning to the cubic anisotropy case which reveals a complex behaviour, we address the orthorhombic anisotropy case that, with its fewer secondary easy axes, proves analytically understandable. Prominent examples for this case are Co/W(110), where the two in-plane directions along W[110] and W[001] have different energies [7], or (110) oriented garnet epilayers [8].

**Definitions, parameters, and background phenomenology.** – The parameters chosen for the numerical and analytical calculations are those of a typical thulium iron garnet film at room temperature, namely $M_s = 1.1 \times 10^5$ A/m, $A = 2$ pJ/m, $K_u = 9$ kJ/m$^3$, $\gamma = 1.43 \times 10^{11}$ T$^{-1}$s$^{-1}$ (g factor 1.63) and $K_1 = 500$ J/m$^3$, standing for the spontaneous magnetization, micromagnetic exchange constant, uniaxial anisotropy (easy axis along the film normal, $z$), gyromagnetic ratio and cubic anisotropy (or orthorhombic anisotropy, when applicable), respectively. The film thickness is $d = 10$ nm, and the Gilbert damping constant was varied between $\alpha = 0.5$ and 0.005, in order to evidence the different dynamical regimes. With these values, the effective anisotropy (ultrathin film limit) is $K_0 \equiv K_u - \mu_0 M_s^2 / 2 = 1400$ J/m$^3$, from which the expected Bloch wall width parameter is $\Delta_0 = \sqrt{A/K_0} \approx 38$ nm, and the Bloch DW surface energy $\sigma_0 \equiv 4\sqrt{A K_0} \approx 0.21$ mJ/m$^2$. This DW width is large compared to the film thickness $d$, so that we can use the ultrathin limit for the demagnetizing factor of a Néel wall $N_x = d \ln 2 / (\pi\Delta)$ to estimate the Bloch wall - Néel wall shape anisotropy $K_{BN} \approx 440$ J/m$^3$. This value is slightly smaller than the secondary anisotropy, cubic or orthorhombic. Axes are defined as follows: $z$ is the film normal, $X$ and $Y$ are crystallographic axes linked to the sample, so that $(X, Y, z)$ is an orthonormal frame. On this sample, we consider that a nanostrip has been patterned along an arbitrary direction $x$ which makes an angle $\xi$
with the X axis, the frame (x, y, z) being also orthonormal. The direction x is therefore both the DW normal, and the direction of the electrical current.

To convert current densities into a damping-like (DL) SOT, we consider $\theta_H = 0.1$ the spin Hall angle, and write the Landau-Lifshitz-Gilbert-Slonczewski equation

$$\partial \vec{m} / \partial t = \gamma \mu_0 \vec{H}_{\text{eff}} \times \vec{m} + \alpha \vec{m} \times \partial \vec{m} / \partial t + \vec{m} \times (\vec{m} \times \vec{\sigma}) / \tau, \quad (1)$$

with $1/\tau = (J_x \hbar \gamma \theta_H)/(2eM_s d)$ ($J_x$ current density in the metal layer on top, $e > 0$ value of the electron charge, $\hbar$ reduced Planck constant) and $\vec{\sigma} = \hat{y}$. As the DL-SOT acts on the whole magnetization structure, it leads to a tilt of the magnetization in the domains, which ends up with the wiping out of the DW. In the cylindrical symmetry case with no secondary anisotropy, the domain magnetizations under current become $\pm(\sin \vartheta, 0, \cos \vartheta)$ with $\sin 2\vartheta = M_s/(\gamma K_0 \tau)$, so that the critical current density at which the domains abruptly magnetize along y is $J_c = (2e/\hbar)K_0 d/\theta_H$ [9]. With the values considered here one gets $J_c = 0.42$ TA/m$^2$. The analytical calculation of $J_c$ is more complicated when adding an orthorhombic term, an even more so with a cubic term, but a lower bound can be guessed by replacing $K_0$ by $K_0 - K_1$, which leads to 0.27 TA/m$^2$. Numerical results agree with these estimations.

These numerical results have been obtained via a one-dimensional micromagnetic code, in essence similar to a previous version [10], except that, here, the x and z components of the demagnetizing field are numerically computed (since the nanostrip width is much larger than the film thickness, the y component of this field is zero). The cell size is square, equal to the film thickness, and the calculation box is moved so as to keep the DW in its center.

Let us now describe the expected impact of DL-SOT on the DW structure, in the absence of a secondary anisotropy. The DL-SOT involves an equivalent field ($H_x^{SOT} = -\chi m_z$; $H_y^{SOT} = 0$; $H_z^{SOT} = +\chi m_x$), with a prefactor $\chi \propto J_x$, as $\gamma \mu_0 \chi = 1/\tau$. For the Néel DW within an ultrathin layer with perpendicular magnetization, schematically depicted in Fig.1a, the magnetization distribution belongs to the (x, z) plane and undergoes a ccw global rotation for a positive $J_x$ current density, resulting in a positive wall velocity. The DW velocity reverses with reversal of the current density, and also of the x axis DW magnetic moment (the chirality of the Néel DW). This is the well-known situation when Néel-type walls are being enforced by interfacial DMI. [10]

A Bloch DW with magnetization distribution in the (y, z) plane reacts differently. In the $m_x$ vs. $m_z$ representation of Fig.1b, the points accumulation to the right and left of the diagram reflects the magnetization rotation within domains, in no way different from the Néel wall case. The main part of the figure offers a zoom on the modification of the wall structure under current. Due to the absence of an x magnetization component within the wall, the wall distortion under current respects the initial $m_z$ distribution within the wall. Under current, both

<figure><img src="image_2.png" /><figcaption>Fig. 1: Effect of DL-SOT on DWs, in the absence of secondary anisotropy. a) Schematic Néel wall profile distortion -and motion- under DL-SOT for a right-handed wall (cross-section view of the sample). Dashed thin arrows, magnetization distribution at rest, thick full arrows, magnetization distribution under steady positive $\chi$. b) Numerical results for the profile of a Bloch wall, in a $m_x$ vs. $m_z$ representation. Black symbols: profile with no current; red (blue) symbols: magnetization distribution across the DW for a positive (negative) current density, magnitude 0.05 TA/m$^2$.</figcaption></figure>

the $m_z$ and current induced $m_x$ remain symmetrical with respect to the wall midplane with, as a result, no net force acting on the wall. The Bloch wall structure may distort but no motion is to be expected. The wall distortion still reverses with reversing current density, but proves independent of the Bloch wall chirality.

**Orthorhombic case.** — The anisotropy function for this case reads $E_K = -K_u m_z^2 - K_p m_x^2$. In order to compare to numerical results, we first explore an analytical model based on the well-known 1D model [also called $(q, \Phi)$ model] of DW dynamics [11].

Let us first look at the DW statics. As $K_p$ is well below the effective anisotropy $K_0$, we assume that the DW width is not changed, only the angle of the DW magnetization at the DW center being free to adjust. Calling this angle $\Phi$, counted from the DW normal (x axis), this angle should minimize the density $E = K_{BN} \cos^2 \Phi - K_p \cos^2(\Phi + \xi)$. The extremum condition reads

$$\sin 2\Phi_{\text{eq}}(\cos 2\xi - k) + \cos 2\Phi_{\text{eq}} \sin 2\xi = 0, \quad (2)$$

where $k = K_{BN}/K_p$. This determines the relation $\Phi_{eq}(\xi)$ (see Fig. 2 below).

Exemplary static DW profiles numerically calculated are shown in Fig. 2a, for a series of angles $\xi$. The DW traces are close to belonging to vertical planes, which justifies the modelling of the DW structures by just one variable, the angle $\Phi$. The values of $\xi$ can be limited to the
<figure><img src="image_3.png" /><figcaption>Fig. 2: Domain walls statics in the orthorhombic case, as a function of the angle $\xi$ of the nanostrip axis with the secondary anisotropy easy axis, obtained by 1D micromagnetic calculations all starting from a Bloch wall with $m_y > 0$. a) Projection onto the $(x, y)$ plane of the unit sphere of the magnetizations across the DW. The angle starts at $\xi = 0$ and increases by 20° steps up to 140°. The DW energy is lowest at $\xi = 90°$, where the Bloch structure is favored by both magnetostatics and in-plane anisotropy. Mirror (along the $m_x$ axis) profiles are obtained when starting from a Bloch wall with $m_y < 0$. b) Equilibrium DW moment angle $\Phi_{eq}$ variation with $\xi$. Points come from the numerical calculations, and the curve is the fit of these data with the analytic equilibrium condition (2) for $\Phi_{eq}$, leading to the determination of the effective anisotropy ratio $k$. Inserts: axes definition, and schematic of the effect of the secondary anisotropy.</figcaption></figure>

[0, $\pi$] range, and if $\Phi$ is a solution so is $\Phi + \pi$: two profiles of opposite handedness are allowed for each angle $\xi$. Fitting the numerically found $\Phi_{eq}$ versus orientation $\xi$ to the static equilibrium condition given in the text, a value $k = 0.848$ is obtained as shown in Fig. 2b, not so far from the simple estimation (0.88). From these statics calculations, a more accurate value of the DW width parameter may also be obtained. It changes only slightly with angle $\xi$, and the average value $\Delta = 35.5$ nm was taken for all analytical curves shown below.

Next we look at the dynamics under DL-SOT. Calling $q$ the DW position, the relevant Slonczewski equations read [10]

$$d\Phi/dt + (\alpha/\Delta)dq/dt = \pi/(2\tau) \cos\Phi,$$

$$(1/\Delta)dq/dt - \alpha d\Phi/dt = (\gamma/M_s)dE/d\Phi. \quad (3)$$

These equations are valid in the limit of negligible tilt of the domains magnetization ($J \ll J_c$, the lowest order correction would consist in replacing the driving term $1/\tau$ by $\cos\vartheta/\tau$, with $\vartheta$ defined earlier). Eliminating $q$, the autonomous equation for the dynamics of $\Phi$ is

$$(1 + \alpha^2)d\Phi/dt = \pi/(2\tau) \cos\Phi - (\alpha\gamma/M_s)dE/d\Phi. \quad (4)$$

A stable stationary regime is obtained when this is zero, and in addition

$$\pi/(2\tau) \sin\Phi + (\alpha\gamma/M_s)d^2E/d\Phi^2 \ge 0. \quad (5)$$

The larger the degree of $E(\Phi)$, the larger the number of potential solutions, at low enough $J$. We call these solutions $\Phi_i(J)$ branches, with $i = 1, 2$ in the orthorhombic case. For each stable steady state regime, the DW velocity is

$$V \equiv dq/dt = (\gamma\Delta/M_s)dE/d\Phi = \pi\Delta/(2\alpha\tau) \cos\Phi, \quad (6)$$

the terms on the r.h.s being evaluated at the dynamic equilibrium value $\Phi_i(J)$ (which differs from the static value $\Phi_{eq}$ due to the DL-SOT). We therefore see that, in general, a non-zero DW velocity is obtained. As the DW structure dynamics only depends on the angle $\Phi$, it follows that, everything else being equal, two successive DW driven by DL-SOT exhibit opposite velocities: race-track operation is not supported by this driving mechanism with no DMI, but magnetization switching is.

Still neglecting the tilt of the domains magnetization, let us consider the infinite current limit. From Eq. (4), $\cos\Phi$ has to go to zero, so that $\Phi(J_x = \pm\infty) = \pm\pi/2$ [if $\theta_H > 0$, from Eq. (5), otherwise the reverse has to be taken]. This shows that the maximum velocity is, quite generally,

$$V_{\max} = (\gamma\Delta/M_s) \left. \frac{dE}{d\Phi} \right|_{\pm\pi/2}. \quad (7)$$

Like the maximum velocity of a Bloch domain wall in the Walker solution, it is not dependent on the damping $\alpha$. In the orthorhombic case, the full calculation gives $V_{\max} = (\gamma\Delta K_p/M_s) \sin 2\xi$, only fixed by the secondary anisotropy. From Eq. (4), it is also clear that the current densities at which branches, other from those stable at infinite current, terminate, decrease with damping $\alpha$. To compute the predictions of the analytical model, we take $\Phi$ as a parameter, compute the associated value of $J_x$ (through $\tau$) using Eq. (4) checking that the stability is fulfilled by Eq. (5), and obtain the velocity through Eq. (6).

Some velocity versus current curves are shown in Fig. 3, for a Gilbert damping constant $\alpha = 0.5$ and $0.05$. At $\xi = 90°$, no motion is observed, the DW being of the Bloch type. At $\xi = 0°$, the DW is initially Néel (both chiralities possible). It moves linearly with current but its velocity
<figure><img src="image_4.png" /><figcaption>Fig. 3: Domain wall velocity (stationary regime) in the orthorhombic case, for two angles $\xi = 0^{\circ}$ and $\xi = 45^{\circ}$. For $\xi = 45^{\circ}$, the outer (inner) hysteresis corresponds to $\alpha = 0.5$ ($\alpha = 0.05$). Symbols are the 1D micromagnetic dynamic calculation results (red: initial $m_y > 0$, blue: initial $m_y < 0$), and the black continuous curves are the predictions of the analytical model. Vertical arrows signal inter-branch transitions. Note the different scales of the two plots.</figcaption></figure>

soon drops to zero as the DW converts to Bloch. Otherwise, two regimes are observed, corresponding to the two static profiles, each of them becoming unstable at some value of current, depending on its polarity. The velocities can become large on the unstable branch, at the cost of an increasingly large equilibration time. On the stable branch, a limiting velocity clearly appears at large currents. A notable feature of the orthorhombic case is that the two current polarities lead to velocities of the same sign. The reversal of the velocity sign is obtained by taking the angle $\pi - \xi$, as expected from the phenomenology described at the beginning.

**Cubic case.** — We now add to the uniaxial anisotropy an ancillary cubic anisotropy, energy $E_{K_1} = K_1(\alpha_1^2\alpha_2^2 + \alpha_2^2\alpha_3^2 + \alpha_3^2\alpha_1^2)$, where $\alpha_1, \alpha_2, \alpha_3$ are the magnetization di-

<figure><img src="image_5.png" /><figcaption>Fig. 4: Stereographic projection for a cubic crystal with [111] normal. The current propagation axis $x$ makes the angle $\xi$ with the predefined crystal axis $X = [\bar{1}\bar{1}0]$. Note that all labelled crystal axes not belonging to the equatorial plane point up in the northern hemisphere. All crystal axes have, however, their negative equivalent in the southern hemisphere. The light-blue dotted line represents both the trace of the wall plane in the equatorial plane and the locus (great circle) of the magnetization orientation within a pure Bloch wall.</figcaption></figure>

rection cosines referred to the cubic axes [100], [010], and [001]. The anisotropy constant is negative, promoting $\langle 111 \rangle$ easy secondary axes as most commonly found in (cubic) garnets [6]. We first determine the wall characteristics for a non symmetrical orientation of $x$, the current propagation axis, namely $\xi = 15^{\circ}$, an orientation halfway between the $[\bar{1}\bar{1}0]$ and the $[2\bar{1}\bar{1}]$ orientations. A rapid inspection of Fig.4 shows that the distribution of the secondary $\langle 111 \rangle$ axes exhibits no symmetry with respect to the wall plane.

Let us now look at the magnetization path across the wall starting with a [111] orientation in the left domain. In the absence of cubic anisotropy, the wall would be purely Bloch. We impose here its chirality by setting the initial wall core magnetization at $m_y = +1$. Upon rotation in the wall plane, the magnetization first feels the influence of the cubic anisotropy well with $[1\bar{1}1]$ axis. The wall acquires a modest $m_x > 0$ component there. Below the equatorial plane, the magnetization becomes in turn sensitive to the $[\bar{1}1\bar{1}]$ axis anisotropy well, gaining a small $m_x < 0$ component. The wall has thus gained a (weak) mixed Bloch-Néel character with, however, an asymmetrical $m_x$ vs. $m_z$ profile (black full symbols line in Fig.5a).

The action of a spin current may, however, still be analysed in the light of the Bloch wall archetype, namely generation, for $J_x > 0$, of a $m_x > 0$ component for $m_z < 0$, and vice versa (see Fig.5a). Under the action of a $J_x > 0$ current, the wall profile (black symbols in Fig.5a) transforms (red open symbols in Fig.5a) into a rather symmetrical profile reminiscent of a pure Bloch wall under
<figure><img src="image_6.png" /><figcaption>Fig. 5: Cubic case with $\xi = 15^\circ$, for a large damping $\alpha = 0.5$. a) Wall profile (core magnetization $m_y = +1$) at equilibrium (black full symbols) and for a current density $J_x = +0.05$ TA/m$^2$ (open red symbols), b) Wall profile for, from top to bottom, $J_x = -0.05, -0.07, -0.08, -0.085, -0.09$ TA/m$^2$.</figcaption></figure>

similar conditions (red open symbols in Fig.1b). On the contrary, application of a negative current accentuates the downwards curvature of the wall profile (upper curve in Fig.5b). The current action may be opposed to by the cubic anisotropy ($J_x > 0$), or assisted by the latter ($J_x < 0$). In the first case, the wall velocity under steady current drops to almost zero, whereas, in the second, the velocity may increase with increasing current up to the Walker breakdown. In opposition to the pure Bloch wall case, however, the initial Bloch wall chirality matters, since the roles of the two cubic anisotropy wells are interchanged during magnetization rotation within the wall. All combined, numerical simulations provide the velocity hysteresis curve $V(J_x)$ specific to $\xi = 15^\circ$ in Fig.6. The symbols colors now refer to the initial Bloch chirality [Red(Blue): core magnetization $m_y = +1$ ($m_y = -1$)]. The high velocity branch velocity reversal with $J_x$ is, within the scope of this study, specific to cubic anisotropy.

Higher symmetry configurations raise specific issues addressed below. For $\xi = 30^\circ$, the highly symmetric orientation of the cubic anisotropy wells with respect to the wall plane entails a mixed wall character, similarly to the $\xi = 15^\circ$ case. The wall profile remains, however, perfectly antisymmetric with respect to the wall mid-plane, a property unaffected by the application of a current, whether

negative or positive. For $\xi = 30^\circ$, the wall is not expected to move under current whichever the chirality of the initial Bloch wall. The same property applies to the $\xi = 90^\circ$ geometry.

In the $\xi = 0^\circ$ or $\xi = 60^\circ$ geometry, one $\langle 111 \rangle$ direction aside the [111] sample normal belongs to the wall plane. The remaining cubic anisotropy wells either above or below the equatorial plane exert antagonistic influences on the wall profile, so that the wall keeps a perfect Bloch character. Yet, two mobility regimes, high and low, still exist for such a wall in full analogy to the $\xi = 15^\circ$ geometry. Therefore, an asymmetry of the wall profile in static equilibrium is not a necessary condition for a fair wall mobility. A favorable alteration of the wall profile under current in the complex anisotropy landscape, together with the constraints imposed onto the wall profile by the magnetization reorientation within domains, is.

Altogether, the velocity pattern vs current orientation respects the three-fold symmetry of the $\langle 111 \rangle$ directions in [111] oriented samples. Within a $120^\circ$ sector (see Fig.6), two specific current propagation directions prove unable, due to symmetry reasons, to lead to wall propagation under SOT in the absence of DMI. Within a $120^\circ$ sector, higher order symmetries relate the initial Bloch wall chirality, current density and wall velocity. Lastly, for any velocity hysteresis cycle, and similarly to the orthorhombic anisotropy case, only the low-velocity branch is reversible; the high-velocity branch may only be accessed from the low current end.

**Discussion.** — In this work, we have concentrated on the basic physics: there are no defects limiting DW motion, and two-dimensional effects have not been considered. The calculations and the model nevertheless show that, except at special orientations of the current and DW normal, a non-zero, and sizable, DW velocity is expected, even in the absence of Dzyaloshinskii-Moriya interactions. For low currents and large damping, two regimes of motion exist. At large current however, only one DW structure is dynamically stable, the transition threshold decreasing when Gilbert damping becomes smaller. Whereas in the orthorhombic case the high mobility branch velocity does not reverse with current direction, it does in the (111) cubic case. With the thulium iron garnet parameters considered here, velocities are larger in the orthorhombic case (10-30 m/s) than in the cubic case (3-12 m/s). Referring to the analytic model developed for the orthorhombic case, velocities should scale as $\gamma\Delta K_1/M_s$, so that at angular momentum compensation (i.e. $M_s/\gamma = 0$) [12] or close to magnetization reorientation (i.e. $K_0 = 0$ so that $\Delta$ becomes large), larger velocities are anticipated.

Very few direct measurements of interfacial DMI have been performed in such samples [13–15], and they reveal low values, of the order of the detection limit. Many studies rather rely on the variation of the easy-axis coercive field under simultaneous application of current and in-plane field (normal to the DW): the coercive field shift
<figure><img src="image_7.png" /><figcaption>Fig. 6: General symmetry governing the $V(J_x)$ hysteresis curves for the three-fold symmetry 1st 120° sector, in the cubic case. Colors refer to the initial Bloch wall chirality: red: wall core magnetization $m_y = +1$; blue: $m_y = -1$. Note the velocity reversal between the $\xi = 0°$ and $\xi = 60°$ geometries as well as across the $\xi = 30°$ and $\xi = 90°$ 'no motion' current flow orientations. Higher symmetries involving wall chirality and current may be observed between orientations such as $\xi = 15°$ and $\xi = 75°$. Current values are in proportion of the high damping parameter $\alpha = 0.5$ used throughout.</figcaption></figure>
should be proportional to current, and vary linearly vs. in-plane field, with a saturation at large fields. Although a full study of this situation is beyond the scope of this paper, it is clear that this behaviour is also expected in the secondary anisotropy scenario. A recent study reporting magnetization switching by SOT in the absence of an in-plane field, in thulium iron garnet ultrathin films capped by platinum, has investigated the in-plane orientation dependence of the effect, observing a large variation (factor of 5) of the switching current [15]. Although the full angular dependence, with a fine step, has not been measured, these results appear fully consistent with what we report here.

To conclude, even if the presence of a small Dzyaloshinskii-Moriya interaction in ultrathin garnet films seems possible (symmetry allows it), we suggest that the role of the secondary anisotropy should by all means not be dismissed, although its inclusion clearly complexifies experiments.

***

This work was supported by the French National Research Agency (project DeMIuRGe, ANR-22-CE30-0014). Enriching discussions with S. Husain, J.-M. George and N. Reyren, as well as with V. Jeudy who critically read the manuscript, are gratefully acknowledged.

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