${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$/piezoelectric actuator hybrids: A model system for magnetoelastic magnetization manipulation

We have investigated the magnetic properties of a piezoelectric actuator/ferromagnetic semiconductor hybrid structure. Using a GaMnAs epilayer as the ferromagnetic semiconductor and applying the piezo stress along its [110] direction, we quantify the magnetic anisotropy as a function of the voltage $V_p$ applied to the piezoelectric actuator using anisotropic magnetoresistance techniques. As the magnetic anisotropy in GaMnAs substantially changes as a function of temperature $T$, the ratio of the magnetoelastic and the magnetocrystalline anistropies can be tuned from approximately 1/4 to 4. Thus, GaMnAs/piezoelectric actuator hybrids are an ideal model system for the investigation of different piezoelastic magnetization control regimes. At $T=5\text{K}$ the magnetoelastic term is a minor contribution to the magnetic anisotropy. Nevertheless, we show that the switching fields of $\rho \left({\mu }_{0}H\right)$ loops are shifted as a function of $V_p$ at this temperature. At 50 K—where the magnetoelastic term dominates the magnetic anisotropy—we are able to tune the magnetization orientation by about $70°$ solely by means of the electrical voltage $V_p$ applied. Furthermore, we derive the magnetostrictive constant ${\lambda }_{111}$ as a function of temperature and find values consistent with earlier results. We argue that the piezo voltage control of magnetization orientation is directly transferable to other ferromagnetic/piezoelectric hybrid structures, paving the way to innovative multifunctional device concepts. As an example, we demonstrate piezo voltage-induced irreversible magnetization switching at $T=40\text{K}$, which constitutes the basic principle of a nonvolatile memory element.

Figure 1

(Color online) (a) Schematic illustration of a $90°$ magnetization switching for a GaMnAs film affixed onto a piezoelectric actuator with the magnetic easy axis [100] parallel to the main expansion direction of the actuator. (b) Corresponding free energy surface. The free energy for the magnetization oriented along a specific direction is given by the distance from the center of the coordinate system to the surface. In this example the magnetization is oriented along $\left[\overline{1}00\right]$. (c) A positive voltage applied to the actuator leads to an elongation of the actuator/GaMnAs layer. (d) This strain induces a uniaxial anisotropy with a magnetic hard axis in elongation direction, which leads to a change in the relative strength of the in-plane magnetic easy axes [Fig. 2a] and thus a $90°$ magnetization switching into the global minimum at $\left[0\overline{1}0\right]$.

Figure 2

(Color online) Cuts through the free energy surface within the film plane for two different strain configurations. The black dashed curves illustrate the in-plane cubic anisotropy contribution with magnetic easy axes along $⟨100⟩$. (a) For the main elongation direction of the piezoelectric actuator aligned along a magnetic easy axis, the strain-induced uniaxial anisotropy leads to a change of the relative strengths of the magnetic easy axes (red full curve). (b) In contrast, for the main elongation direction of the piezoelectric actuator aligned along a locally hard axis of the in-plane cubic anisotropy, e.g., [110], the strain-induced uniaxial anisotropy leads to a rotation and a change of the relative orientations of the magnetic easy axes (red full curve).

Figure 3

(Color online) Illustration of the relative alignment of the Hall bar, the GaMnAs thin film, and the main expansion direction of the piezoelectric actuator. $\mathbf{h}=\mathbf{H}/H$, $\mathbf{m}=\mathbf{M}/M$, $\mathbf{j}$, and $\mathbf{t}$ denote the unit vectors along the orientation of the magnetic field $H$, the magnetization $M$, the current density $j$, and the direction transverse to the current, respectively.

Figure 4

(Color online) (a) Longitudinal (${ϵ}_{jj}$, black full symbols) and transverse (${ϵ}_{tt}$, red open symbols) lattice distortions for piezo voltage sweeps at 5 K (squares) and 50 K (circles) determined via strain gauges. The data show three full voltage cycles. To allow for comparison, we plot ${ϵ}_{ii}\left(V_p\right)-{ϵ}_{ii}\left(+200\text{V}\right)$, $i∊\left\{j,t\right\}$. (b) Temperature dependence of the piezo-induced lattice distortions $\Delta {ϵ}_{ii}={ϵ}_{ii}\left(V_p=+200\text{V}\right)-{ϵ}_{ii}\left(V_p=-200\text{V}\right)$, $i∊\left\{j,t\right\}$, determined from the third voltage cycle.

Figure 5

(Color online) Schematic illustration of ADMR according to Eqs. (2, 3). In all panels, the same ${\rho }_0$, ${\rho }_7>0$ and ${\rho }_1$, ${\rho }_3<0$ have been used. The black dashed curves correspond to $\left\{{\rho }_{\text{long}}\left(\alpha \right),{\rho }_{\text{trans}}\left(\alpha \right)\right\}$ curves for a large magnetic field $H$ rotated within the film plane, so that the magnetization is aligned in field direction, i.e., $\beta =\alpha$. For small magnetic fields $H$, the magnetization tends to remain oriented close to a magnetic easy axis (e.a., green straight lines) and ${\rho }_{\text{long}}$ and ${\rho }_{\text{trans}}$ therefore tend to remain constant over a broad range of external magnetic field orientations nearby a magnetic easy axis, resulting in $\left\{{\rho }_{\text{long}}\left(\alpha \right),{\rho }_{\text{trans}}\left(\alpha \right)\right\}$ given by the red full curves. (a), (b) represent the case of a cubic magnetic anisotropy in the film plane, while in (c), (d) [(e), (f)] an in-plane uniaxial anisotropy with the easy axis along $90°$ $\left[0°\right]$ is present. Note that this figure shows the relevant ADMR features, which will be observed as a function of temperature in Figs. 6, 11.

Figure 6

(Color online) (a) Longitudinal resistivity ${\rho }_{\text{long}}\left(\alpha \right)$ and (b) transverse resistivity ${\rho }_{\text{trans}}\left(\alpha \right)$ for a constant external magnetic field ${\mu }_{0}H=60\text{mT}$ and 1 T rotated within the film plane for $V_p=0\text{V}$. The black circles represent the experimental data, the yellow dashed curves the result of the corresponding simulation using the parameter values given in Fig. 7.

Figure 7

(Color online) Temperature dependence of the anisotropy parameters $B_{110}\left(V_p\right)$, $B_{010}$, and $B_{c\parallel }$ (a), (b) and the resistivity parameters ${\rho }_0$, ${\rho }_1$, ${\rho }_3$, and ${\rho }_7$ (c), (d) as derived from the iterative fitting procedure of the ADMR curves at different external magnetic field strengths. At temperatures $T\lesssim T_C/2\approx 40\text{K}$ the magnetic anisotropy is dominated by the cubic contribution, while at higher temperatures $T\gtrsim T_C/2$ the piezo voltage-induced uniaxial magnetic anisotropy contribution becomes more important. Parameter ${\rho }_0$ in (c) was derived from measurements at $V_p=+200\text{V}$ and ${\mu }_{0}H=100\text{mT}$, the parameters ${\rho }_1$, ${\rho }_3$, and ${\rho }_7$ are found to be independent of $V_p$ within error bars.

Figure 8

(Color online) Temperature dependence of the magnetostrictive constant (full black squares) compared to the values obtained by Masmanidis et al.(Ref. 27) via a nanoelectromechanical resonator (open blue squares).

Figure 9

(Color online) (a) ${\rho }_{\text{long}}\left({\mu }_{0}H\right)$ and (b) ${\rho }_{\text{trans}}\left({\mu }_{0}H\right)$ curves measured at 5 K for the external magnetic field $H$ aligned at $\alpha =350°$. An explanation of the evolution is given in the text. (c) Illustration of the evolution of magnetization orientation during the external magnetic field sweep.

Figure 10

(Color online) (a), (c) Enlarged sections of the ${\rho }_{\text{trans}}\left({\mu }_{0}H\right)$ curves measured at 5 K for the external magnetic field $H$ aligned at $\alpha =350°$ showing the piezo voltage dependence of the first (c) and second (a) switching field. (b), (d) For $\alpha =260°$ the order of switching fields depending on the piezo voltage is inverted. The arrows indicate the direction of the field sweeps.

Figure 11

(Color online) Longitudinal resistivity ${\rho }_{\text{long}}\left(\alpha \right)$ and transverse resistivity ${\rho }_{\text{trans}}\left(\alpha \right)$ at 50 K for a constant external magnetic field rotated within the film plane at different voltages $V_p$ applied to the piezoelectric actuator: (a), (b) $V_p=+200\text{V}$, (c),(d) $V_p=0\text{V}$, and (e), (f) $V_p=-200\text{V}$. The black and gray symbols correspond to the ADMR curves obtained at ${\mu }_{0}H=10\text{mT}$ and ${\mu }_{0}H=100\text{mT}$, respectively. The yellow $\left({\mu }_{0}H=10\text{mT}\right)$ and red $\left({\mu }_{0}H=100\text{mT}\right)$ dashed curves are obtained from the iterative fitting procedure of the ADMR curves at different external magnetic field strengths using the parameter values given in Fig. 7.

Figure 12

(Color online) Free energy normalized to the saturation magnetization $F/M\left(\beta \right)$ as a function of magnetization orientation $\beta$ for an external magnetic field ${\mu }_{0}H=10\text{mT}$ applied along $\alpha =-25°$ taking into account the magnetic anisotropy parameters derived from the ADMR at different magnetic field strengths at 50 K depicted in Fig. 11. The magnetization orientation is expected to change by $\Delta \beta =76°$ when the piezo voltage $V_p$ is varied between $+200\text{V}$ and $-200\text{V}$.

Figure 13

(Color online) Evolution of (a) ${\rho }_{\text{long}}\left(V_p\right)$ and (b) ${\rho }_{\text{trans}}\left(V_p\right)$ as a function of the variation of the piezo voltage $V_p$ between $+200\text{V}$ and $-200\text{V}$ for ${\mu }_{0}H=10\text{mT}$ applied along $\alpha =-25°$ at 50 K. The solid green line in (a) displays the elongation dependence of the resistivity parameter ${\rho }_0$ in ${\rho }_{\text{long}}$ [compare Eq. (2)] as derived from the corresponding ADMR curve, which is subtracted from ${\rho }_{\text{long}}$ for the magnetization orientation determination. (c),(d) Corresponding measurements for ${\mu }_{0}H=100\text{mT}$. In all panels the orange open circles at $V_p=-200\text{V}$, 0 V, and $+200\text{V}$ display the resistivity values obtained from the ADMR measurements at $\alpha =-25°$ (cf. Fig. 11), demonstrating the good reproducibility.

Figure 14

(Color online) The magnetization orientation $\beta$ as a function of piezo voltage $V_p$ at 50 K for different external magnetic field strengths ${\mu }_{0}H$ applied along $\alpha =-25°$ calculated from ${\rho }_{\text{long}}$ via Eq. (2) (black open squares) and ${\rho }_{\text{trans}}$ via Eq. (3) (red full circles). The good agreement between the $\beta$ values calculated from ${\rho }_{\text{long}}$ and ${\rho }_{\text{trans}}$ corroborates our analysis and demonstrates that for ${\mu }_{0}H=10\text{mT}$, the magnetization orientation can be continuously and reversibly rotated by about $\Delta \beta \approx 70°$ solely by varying the piezo voltage between $+200\text{V}$ and $-200\text{V}$, as expected from the free energy contours (Fig. 12). For increasing ${\mu }_{0}H$ the maximum angle of rotation is decreased to $\Delta \beta \approx 15°$ for ${\mu }_{0}H=100\text{mT}$.

Figure 15

(Color online) (a) Transverse resistivity loop ${\rho }_{\text{trans}}\left({\mu }_{0}H\right)$ at 40 K with $V_p=+200\text{V}$ and the external magnetic field oriented along $\alpha =335°$. The open blue circles describe the evolution of ${\rho }_{\text{trans}}\left({\mu }_{0}H\right)$ during the measurements sequence $A$ to $G$ described in the text. The fact that the symmetry axis of the full ${\rho }_{\text{trans}}\left({\mu }_{0}H\right)$ loop is not at zero magnetic field, we attribute to trapped flux in the superconducting magnet. (b) Piezo voltage-induced change in free energy normalized on saturation magnetization $F/M\left(\beta \right)$ as a function of magnetization orientation $\beta$ for ${\mu }_{0}H=-5\text{mT}$. (c) Evolution of ${\rho }_{\text{trans}}$ during the piezo voltage sweep from $V_p=+200\text{V}$ $\left(B_1\right)\to 0\text{V}$ $\left(B_2\right)\to -200\text{V}$ $\left(B_3\right)\to 0\text{V}$ $\left(B_4\right)\to +200\text{V}$ $\left(B_5\right)$.

Figure 16

(Color online) Sketch of a spin valve structure based on strain modulation. The magnetization of the central part of the Hall bar is manipulated by the actuator positioned above a ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$ layer prestrained by a self-supporting ${\text{Ga}}_{1-y}{\text{In}}_y\text{As}$ virtual substrate. In this particular realization, the ${\text{Ga}}_{1-y}{\text{In}}_y\text{As}$ buffer with a lattice constant greater than that of the GaMnAs layer leads to an out-of-plane orientation of the magnetization, which is flipped in-plane by a biaxial compressive strain induced by the actuator.