Temperature dependent Neel wall dynamics in GaMnAs/GaAs

Extensive Kerr microscopy studies reveal a strongly temperature dependent domain wall dynamics in Hall-bars made from compressively strained GaMnAs. Depending on the temperature magnetic charging of domain walls is observed and nucleation rates depend on the Hall-geometry with respect to the crystal axes. Above a critical temperature where a biaxial-to-uniaxial anisotropy transition occurs a drastic increase of nucleation events is observed. Below this temperature, the nucleation of domains tends to be rather insensitive to temperature. This first spatially resolved study of domain wall dynamics in patterned GaMnAs at variable temperatures has important implications for potential single domain magneto-logic devices made from ferromagnetic semiconductors.

Extensive Kerr microscopy studies reveal a strongly temperature dependent domain wall dynamics in Hall-bars made from compressively strained GaMnAs. Depending on the temperature magnetic charging of domain walls is observed and nucleation rates depend on the Hall-geometry with respect to the crystal axes. Above a critical temperature where a biaxial-to-uniaxial anisotropy transition occurs a drastic increase of nucleation events is observed. Below this temperature, the nucleation of domains tends to be rather insensitive to temperature. This first spatially resolved study of domain wall dynamics in patterned GaMnAs at variable temperatures has important implications for potential single domain magneto-logic devices made from ferromagnetic semiconductors. The ferromagnetic semiconductor GaMnAs [1] has been extensively studied in the past few years not only in the viewpoint of basic science but also focusing the attention on properties that can lead to novel applications in spinbased electronics and magneto-logic devices [2,3]. For the latter, a good understanding of domain wall (DW) dynamics is needed in order to control processes such as the DW nucleation and propagation. In ferromagnetic GaMnAs with in-plane magnetization, magnetic reversal processes have been studied mostly by means of magnetotransport [4,5], however with very limited gain of local information on DW nucleation and motion. Very recently single DWs have been resolved in the static limit by means of electron holography on the scale of a few micrometers [6] with high spatial resolution. In contrast, we have shown that Kerr microscopy provides full time and spatially resolved information on the dynamics of in-plane magnetic domains during the magnetization reversal on the scale of a few hundred micrometers [7]. Due to the low Curie-temperatures T c well below room temperature of most ferromagnetic semiconductors like GaMnAs it is of technical interest to study these materials in the highest possible temperature range just below T c . In this work we present a careful characterization of the temperature dependent biaxial and uniaxial magnetic anisotropies in compressively strained GaMnAs and their influence on the evolution of the magnetic domain structure thereby identifying limits for domain wall logic devices in the high temperature regime. A preferential DW alignment is found to be linked to the change in the position of the easy axis given by the temperature dependence of the uniaxial and biaxial anisotropy contributions. An increase in the number of domain nucleation centers is observed beyond a critical temperature where a * Electronic address: j.honolka@fkf.mpg.de biaxial-to-uniaxial anisotropy transition takes place. The dependence of this behaviour on the geometry of the device is also presented. The material under study, consists of GaMnAs epilayers of 170 nm thickness grown on GaAs(001) by molecular beam epitaxy (MBE). The nominal Mn concentration is (2.3 ± 0.1)% and has been estimated on the bases of flux ratios. A more detailed description of the sample growth and material characterization has been given elsewhere [7]. The GaMnAs devices used in the Kerr microscopy experiments are Hall bars of 200µm width fabricated by standard photolithography and ion milling.

Magnetic characterization of unpatterened, virgin GaMnAs epilayers
For a full characterization of the magnetic anisotropy within the GaMnAs epilayer we performed temperature dependent SQUID as well as magneto-optical Kerr effect (MOKE) measurements with magnetic fields applied in various in-plane directions. SQUID measurements were performed cooling the sample in a field of 1000 Oe and applying a field of 50Oe during the measurement. The results are shown with in Fig. 1 . Also plotted is the magnetization versus temperature M (T ) ( Fig. 1(a), inset) in a saturating field of H = 1T. From the temperature dependent magnetic response at non-saturating fields of H = 50Oe for different directions the temperature dependence of the anisotropy constants can be estimated assuming a Stoner-Wohlfahrt coherent rotation of the magnetization following the total energy density E(ϕ) = Kc 4 cos 2 (2ϕ) + K u cos 2 ϕ − M H cos(ϕ − ϕ H ), where K c and K u are the biaxial and uniaxial anisotropy constants, M is the magnetization, H the magnetic field, and ϕ and ϕ H are the angles of M and H with the [110] direction. For each temperature the measured SQUID signal M SQUID is determined simply by the equation system Here M SQUID is the measured projection of the magnetization M (T ) on the axis of the SQUID pick-up coils, which are aligned parallel to the magnetic field. While M (T ) is known from the SQUID measurement at saturating fields, K u and K c are temperature dependent parameters to be derived by fitting. Assuming a magnetization dependence of the anisotropy constants close to K u = αM 2 and K c = βM 4 [8] we can use equations (1) and (2) to fit the SQUID data as shown in the inset of Fig. 1(b). The fits shown for the three directions [110], [110] and [100] are derived using one and the same fit parameters α = 30.0 and β = 0.32 in addition to the magnetization exponents 1.8 and 4.1 for the expressions of K u and K c , respectively. In Fig. 1(b) the temperature dependence of K u and K c is plotted as a result of the fitting procedure. A clear crossover is observed from biaxial to uniaxial magnetic anisotropy at approximately 26K where K u = K c . As a consequence along the [110] direction the second derivative of the energy, ∂ 2 E/∂ 2 ϕ, changes sign at K u = K c and the number of local minima in E(ϕ) is reduced from 4 to 2 due to the disappearance of the biaxial induced energy barrier in the [110] direction (see Fig. 1(c)). As extensively shown in magneto-transport measurements by Pappert et al. [5] this crossover is directly visible in polar coercivity plots of Fig. 2  From Fe/GaAs thin film systems with an equivalent magnetic anisotropy symmetry it is known that the sense of rotation changes whenever the magnetic field direction ϕ H crosses a local minimum or a maximum in the magnetic energy landscape E(ϕ) [17]. Therefore, as shown in Fig. 3(a), at low temperatures where K u /K c < 1, the sense of the transition changes 8 times when ϕ H is swept over the full angle range [23]. Four of the eight sign changes occur when the magnetic field direction crosses the two equivalent global easy axis directions located at angles ±θ EA (T ) away from the [110] direction as shown in Fig. 3(a). At high temperatures T > 26K in agreement with Fig. 2(c) only one single transition at H c is observed and we expect DWs with angles ∼180 • . Accordingly, we expect that the sense of the transitions only change sign 4 times during a full angle sweep of ϕ H (see Fig. 3  producible and not critically dependent on the sample orientation. Therefore, applying the field along the [110] direction at different temperatures below and beyond the crossing point of K u and K c should allow for the observation of the transition between ∼120 • and ∼180 • DWs.
From an application point of view this direction is interesting since in this regime the transition was shown to be propagation dominated with a relatively small number of domains involved in the process [7].

Observation of temperature dependent domain wall dynamics in patterned GaMnAs Hall-bars
The Kerr-microscopic observation of magnetic domains was performed using the same procedure as described in Ref. [7]. Before presenting the microscopy results which focus on the temperature dependent dynamics of DWs for magnetic fields applied close to the [110] direction we would like to shortly discuss the expected change of the DW angle ∆ϕ and sense of rotation with temperature as well as with increasing deviations δϕ H from the [110] direction. From simple symmetry arguments reflected in Fig. 3 it is evident that generally small deviations of ±δϕ H to both sides of the [110] direction will trigger DW transitions of opposite sense. However, despite the opposite sense in rotation the absolute DW angles remain exactly the same. More specifically, at low temperatures T < 26K and δϕ H < θ EA CW (CCW) deviations lead to CW (CCW) transitions at H c1 and H c2 , whereas for δϕ H > θ EA CW (CCW) deviations lead to a CCW (CW) transition. The angle θ EA is shown in Fig. 3(a). For T > 26K CW (CCW) deviations always lead to a CCW (CW) transition. For a full understanding of DW dynamics at different temperatures it is therefore important to trace the temperature dependent global easy axis direction. To give an example of the influence of δϕ H on ∆ϕ, Fig. 4 shows the temperature dependent angle θ EA (T ) of the global easy axis direction with respect to [110] at and with a deviation of δϕH = 15 • is also plotted (filled symbols). The DW angle increases monotonously from ∼ 120 • and reaches 180 • at higher temperatures.
zero magnetic field together with the expected DW angle of the first transition at H c1 for δϕ H = 0 • and for a field deviation δϕ H = 15 • with respect to [110]. The easy axis directions were obtained by tracing one of the two energy minima in E(ϕ) closest to the [110] uniaxial easy axis (see Fig. 1(c)). θ EA (T ) is determined by K u (T ) and K c (T ) given in Fig. 1. As expected at ∼26 Kelvin, the temperature of the crossing between K u and K c (see Fig. 1), the global easy axis starts to be fully aligned with the [110] direction. The calculation of the DW angle of the first transition includes the temperature dependence of the coercive field applied close to the [110] direction. Coherent rotation effects in two domains separated by the DW are thus taken into account. The DW angle ∆ϕ(T ) for δϕ H = 0 • and for the field deviation δϕ H = 15 • shows two distinct jumps caused by the sequential destabilization of the initial and the final magnetization state of the transition. In the three middle row plots of Fig. 2 Fig. 4) the sense of the transition changes to CW and the initial magnetization state rotates towards the [110] direction leading to a reduction in ∆ϕ. As the temperature is further elevated the initial and final magnetization states approach the global easy axis direction along [110]. In the following the effect of the temperature dependent change from four-fold to two-fold symmetry in E(ϕ) on the DW dynamics is studied on the basis of extensive Kerr microscopy measurements with magnetic fields applied close to the [110] direction. For all four cases two consecutive frames at times t = t 0 and t = t 0 + ∆t were extracted from a movie to picture the time evolution. Fig. 5 and Fig. 6 demonstrate that the alignment of the DWs with respect to the [110] direction is clearly temperature dependent. While at low temperatures the DWs avoid the alignment with the [110] direction along the Hall bar they prefer the parallel alignment at higher temperatures in both cases. Only the DW nucleation behavior seems to be dependent on the Hall-bar orientation. Here we observe that only in Fig. 6 nucleation happens preferentially at the long sides of the Hall-bar. We will discuss nucleation effects in detail in the next section. In most magnetic systems the alignment of DWs is correlated to the surface divergence of the magnetization at the domain boundary due to magnetization components normal to the DW [9,10,11]. In general, this creates so-called magnetic charges proportional to (M 1 − M 2 )·n at the DW boundary accompanied by a cost of stray field energy, where M 1 and M 2 are the magnetization vectors of the two domains separated by the DW andn the wall normal facing towards domain 2. Hence, in our GaMnAs samples, in order to avoid magnetic charges, DWs should be aligned along [110] for the low temperature ∼120 • DW transition with fields along [110]. At higher temperatures in the case of 180 • DW transitions where the magnetization vectors M 1 and M 2 are collinear with the global easy axis along [110] we expect the system to try and avoid head-to-head type of boundariesn [110] with maximum amounts of magnetic charges [13]. In agreement with the latter, the observed DWs at 27 K show typical zigzag patterns throughout the reversal dynamics withn pointing preferentially parallel to the   [110] direction where (M 1 − M 2 ) ·n = 0 holds. At low temperatures, however, out results are clearly not according to the above described model. As discussed, the reversal dynamics shown in Fig. 5 (a) with the field applied along the [110] direction corresponds to a ∼120 • DW [7] where the initial and final magnetization states M 1 and M 2 are aligned with two of the biaxial global easy axes which lay at ∼ ±25 • from the [110] direction (see Fig. 4). The preferential orientation of the DWs around 25 • away from [110] observed in the Kerr images, thus, points to significant amounts of magnetic charges ∼ M cos(25 • ) accumulated at the walls. Fig. 7(left) and (right) summarizes the experimentally observed DW orientations for the case of 120 • and 180 • DW transitions at low and high temperatures together with the respective easy axis directions. It should be noted at this point that the DW orientations are found to be the same in our virgin film samples and therefore are not a consequence of the Hall-bar patterning process.
Before we start to discuss the physics leading to the observed DW alignment behavior it is helpful to estimate the expected contributions of the stray field to the energy density. In the diluted ferromagnetic semiconductor GaMnAs the magnetization is about 2 orders of magnitude lower compared to typical 3d metal ferromagnets like Fe and therefore stray field energy contributions to the total energy density are generally reduced by a factor 10 −4 . One of the consequences is that in GaM-nAs films Néel walls are energetically preferred to Bloch walls up to relatively large film thicknesses d crit . In this regime the system avoids magnetic surface charges at the film surfaces and encounters volume charges within the Néel wall. It was shown that the critical thickness can be approximated by d crit = 13.8 A/4πM 2 , where A is the exchange coupling constant. With a typical value of A = 4 × 10 −8 erg/cm for GaMnAs and magnetization values of the order of 10 emu/cm 3 one gets a critical thickness of about 1µm (compare to Permalloy where d crit =50nm). Thus, we can assume that the magnetic dynamics in our GaMnAs films of 170nm thickness is governed by Néel-type walls in agreement with experi-ments by Sugawara et al.. [6]. The low magnetization value also reduces the stray field energy density ǫ S caused by magnetic charges situated at a DW. It is given by ǫ S = 2π(M 1 − M 2 ) ·n) 2 for an infinitely extended DW. For epitaxial Fe films of 150Å thickness grown on GaAs with a predominant cubic anisotropy K c > 0 and large values ǫ S of the order of 1×10 6 erg/cm 3 a strict preferential DW alignment according to the stray field minimization condition ǫ S = 0 has been reported by Gu et al. [14]. The authors observe the alignment of 90 • and 180 • DWs with the hard and easy axis, respectively, when the field is applied along the easy axis parallel to the cubic crystal symmetry direction. For our GaMnAs samples with low concentrations of Mn, however, stray field energy densities ǫ S are only of the order of 100 erg/cm 3 at most. Moreover, in thin films ǫ S is further reduced due to the limited lateral extension of the DW when oriented perpendicular to the film. It can be shown that stray fields produced by magnetic charges in laterally confined Néel walls decay like 1/x 2 at large distances away from the wall [15]. As a consequence for very thin films more complex Néel wall shapes occur and total wall energies have to be evaluated numerically including exchange stiffness and magnetic anisotropy, which leads to solutions including isolated charged walls with ǫ S = 0 [16]. From calculations by A. Hubert [16] with therein defined dimensionless parameters Q = K/2πM 2 and λ = 2Q A/K/d one expects charged 120 • DWs in our GaMnAs samples of thickness d = 170nm with Q ≈ 1 and λ ≈ 1 in accordance with our results of the Kerr-measurements at T = 3K. DW charging effects similar to ours are visible also in epitaxial Fe films in the ultrathin film limit grown on GaAs. Although the authors of Ref. [14] did not discuss this aspect in detail the film thickness dependent cross-over from uncharged to partly charged 90 • DWs clearly shows in their Lorentz microscopy data for d = 150Å and d = 35Å, respectively [17]. In GaMnAs epilayers Sugawara et al. found both 90 • Néel walls oriented along the [110] direction and 20 • away from the [110] (see DW (iii) in Fig. 1(b) of Ref. [6]). The latter configuration again should correspond to a charged wall although the authors did not comment on this issue. It should be noted, however, that since the Lorentz microscopy technique only permits the observation of domains close to the film edges where a non-magnetic reference signal is available, the local orientation of the DWs can also be affected by inhomogeneous morphology induced by the lithography process as well as flux closure processes. occur and the contact pads of the Hall bar devices tend to serve as nucleation centers, for [110] oriented Hall-bars nucleation events happen preferentially at the long sides of the bar and appear larger in number.
In order to study influences of geometry and temperature on the nucleation in more detail, Kerr microscopy was performed in small temperature steps on a Hall-bar having its longitudinal axis along the [110] axis. We observe that below temperatures of ∼ 24 Kelvin the number of domains involved in the transition remain fairly small and constant as shown in Fig. 8 (a) and (b) corresponding to temperatures of 3 and 16 Kelvin, respectively. However, beyond this temperature the number of nucleation events at the long sides of the Hall-bar edge grow dramatically and at the same time domains become increasingly elongated as illustrated in the Kerr images in Fig. 8(c) and (d) taken at 24 and 33 Kelvin, respectively. The number of domains involved in the reversal process versus temperature are plotted Fig. 10 (open symbols) in a temperature range going from 3 to 33 Kelvin and show an exponential behavior. Due to the decreasing contrast in the Kerr signal with decreasing magnetization values Kerr images could not be evaluated in the temperature range between 33 Kelvin and T c .
Comparing the nucleation dynamics for the Hall-bar oriented in the [110] and [110] directions it is evident that only ∼180 • DWs appearing at temperatures around 25 Kelvin are strongly affected by the orientation of the Hall bar with respect to the crystal axis and field vector. Generally, the Kerr images in Fig. 8   direction (see Fig. 9). In accordance to our earlier work [7] we see multiple nucleation events within the film at low temperatures characteristic for ∼ 60 • DW transitions and no preferential nucleation at the sides of the Hall-bar. At high temperatures T > 27K again the domains are elongated along the easy axis direction [110], however this time preferential nucleation at the long sides of the Hall-bar is not observed.
Anisotropic nucleation of domains in thin ferromagnetic films as observed in Fig. 8, where the observed preferred nucleation occurs at the Hall-bar sides [110], can have different origins: Lithography induced Anisotropies: During the Hall-bar lithography process differences in the edge profiles along [110] and [110] can be introduced. As an example it is known that the wet etching process of GaAs exhibits a different dynamics in the respective directions, leading to different edge profiles. However, the ion milling technique used in our case leads to direction independent processing and a homogeneous edge profile in all directions of the Hall-bar. This was verified using x-sectional scanning electron microscopy. Also lattice relaxation effects as observed at stripe edges [18,19] that lead to local changes in the magnetic anisotropy energy E(ϕ) should be equal in strength for edges [110] and [110]. We therefore claim the observed asymmetric nucleation behavior not to be a consequence of the Hall-bar patterning process.
Anisotropies through closure domains: Anisotropies in the nucleation rates can be induced by local dipolar fields, which decay like 1/x away from the edges and trigger flux closure domains. In micropatterned biaxial epitaxial Fe films on GaAs DW transitions of 90 • type are triggered preferentially at film edges where the rotation of the magnetization due to local dipolar fields has the same sense as the DW transition itself [20]. The local rotation of M can then be understood as a partial transition due to dipolar fields which facilitates the domain nucleation induced by the external field H. Indeed for the CCW 120 • DW transitions shown in Fig. 8 (H-field that induces the transition is slightly rotated CW from the global easy axis direction) we see nucleation at the edges [110] where the dipolar fields will rotate the magnetization vector M in the common sense. The opposite is true for edges [110]. In the measurement configurations shown in Fig. 5(a) and Fig. 6(a) due to small deviations ±δϕ H of the field direction from [110] the sense of the transition can be either CW or CCW. However, independent of that again the local rotation of M at the [110] edges is opposite in sense and therefore do not support nucleation in accord with our experimental results. At higher temperatures T = 24K and T = 33K (Fig. 8(c) Fig. 9 do not fit into this picture. At low temperatures for CCW 60 • DWs we would expect nucleation at the edges [110] and at high temperatures nucleation at those [110]. Instead we observe rather statistical nucleation within the entire device. We again tend to attribute this difference in nucleation dynamics to the reduced DW nucleation/propagation energy ǫ 60 • with respect to 120 • DWs [7]. Moreover, in the case of Fig. 9 at low temperatures stray fields proportional to the projection M sin(θ EA ) of M (H = 0) on [110] are significantly reduced when compared to those M cos(θ EA ) in Fig. 8.
It remains to discuss the drastic increase of the number N (T ) of domains involved in the transitions above temperatures T = 23K. The problem resembles that of Fatuzzo's domain-nucleation model developed for ferroelectrics [21]. As described in the beginning of this section the increase of N is accompanied by a change in the average width w of the domains, where w is the dimension of domains measured along [110] in Fig. 8. From the time resolved dynamics in our Kerr-movies above T = 24K it is evident that after nucleation of a domain at the Hall-bar edges, DW propagation is mainly happening in the [110] direction with little change in w of the respective domain. As proposed by Fatuzzo we therefore attribute the drastic increase in N to a complex interplay of temperature dependent nucleation rates Γ(T ) at the film edges, a reduced DW mobility µ [110] along [110] and effects of coalescence of domains. If w c is the average domain width at the coercive field H c where 50% of the area of the film has switched, then the respective number of domains in a given section of the Hall-bar with a length l [110] is approximately N c = l/2w c . Here w c will be a function of the mobilities along µ [110] and µ [110] and the nucleation rates Γ. With this we can qualitatively understand the temperature dependent nucleation dynamics. The Kerr data prove that with increasing temperature and especially for T > 20K the ratio between µ [110] and µ [110] is significantly shifted towards propagation along [110], which assuming a constant Γ would reduce w c and increase N c . On the other hand we expect Γ to increase with temperature according to a thermally activated process which supports coalescence of domains at an early stage after nucleation. Generally, both a decrease in the mobilities and Γ leads to an increase in the coercive field H c at constant sweep rates of the magnetic field. Indeed, the temperature dependence of H c shown in Fig. 10 (full symbols) indicates a distinct decrease in slope at T ≈ 20K, which points towards a change in the mobilities and/or Γ (M decreases rather monotonously in this temperature range as shown in the inset of Fig. 1(a)). Sudden changes in µ [110] or Γ would not be unexpected since they occur in close vicinity to the crossing point between K u and K c (∼26 Kelvin), where the magnetic transitions change their character. Above the crossing temperature we interpret the drastic increase in N (T ) to be mainly due to a monotonous reduction of µ [110] .

Conclusions
This work presents an extensive characterization of the temperature dependent magnetic domain wall dynamics in Hall-bars made from compressively strained GaMnAs and identifies limits for single domain wall logic devices in the high temperature regime. Kerr microscopy allows to locally observe nucleation events of domains as well as the alignment and propagation behavior of domain walls. A correlation of the preferential domain wall alignment with respect to the temperature dependent magnetic easy axis direction is found. The latter is determined by the temperature dependent in-plane uniaxial and biaxial anisotropy energy contributions. At low temperatures magnetically charged domain walls with domain wall angles considerably smaller than 180 • are observed. Above the biaxial-to-uniaxial transition temperature this charging effect is lost and domain walls are oriented along the easy axis. Domain nucleation is happening almost exclusively at Hall-bar edges aligned along the [110] uniaxial hard axis direction. This behavior is attributed to small demagnetizing fields contribution at the edges of the device, that locally facilitate the magnetic transition and therefore nucleation of domains.
This effect is asymmetric and favors nucleation at edges [110]. This first extensive study of domain nucleation and propagation dynamics at variable temperatures in GaMnAs shows that multi-domain states can be avoided by a suitable device geometry. This together with our finding that the orientation of domain walls can be tuned by the ratio between uniaxial and biaxial anisotropy energy has important consequences for applications in the field of magneto-logics and in particular for single domain wall devices where domain walls are manipulated through spin-polarized currents.