Lithographically and electrically controlled strain effects on anisotropic magnetoresistance in (Ga,Mn)As

It has been demonstrated that magnetocrystalline anisotropies in (Ga,Mn)As are sensitive to lattice strains as small as 10^-4 and that strain can be controlled by lattice parameter engineering during growth, through post growth lithography, and electrically by bonding the (Ga,Mn)As sample to a piezoelectric transducer. In this work we show that analogous effects are observed in crystalline components of the anisotropic magnetoresistance (AMR). Lithographically or electrically induced strain variations can produce crystalline AMR components which are larger than the crystalline AMR and a significant fraction of the total AMR of the unprocessed (Ga,Mn)As material. In these experiments we also observe new higher order terms in the phenomenological AMR expressions and find that strain variation effects can play important role in the micromagnetic and magnetotransport characteristics of (Ga,Mn)As lateral nanoconstrictions.


INTRODUCTION
GaAs doped with ∼ 1−10% of the magnetic acceptor Mn is a unique material for exploring spin-orbit coupling effects on micromagnetic and magnetotransport characteristics of ferromagnetic spintronic devices. Spin polarized valence band holes that mediate ferromagnetic coupling between Mn local moments produce large magnetic stiffness, resulting in a mean-field like magnetization and macroscopic single-domain behavior of these dilute moment ferromagnets. At the same time, magnetocrystalline anisotropies derived from spin-orbit coupling effects in the hole valence bands are large leading to the sensitivity of magnetic state to strains as small as 10 −4 [1,2,3,4,5]. Experimentally, strain effects can be controlled by lattice parameter engineering during growth [6,7], through post growth lithography [1,2,3], or electrically by bonding the (Ga,Mn)As sample to a piezoelectric transducer [4,5,8]. Easy axis rotations from in-plane to out-of-plane directions have been demonstrated in these studies in (Ga,Mn)As films grown under compressive and tensile lattice matching strains, and the orientation of the in-plane easy axis (axes) has been shown to respond to strain relaxation in lateral microstructures or controlled dynamically by piezoelectric transducers.
Strain control of magnetocrystalline effects on transport in (Ga,Mn)As, we focus on in this paper, has so far been explored less extensively. Our work in this direction is motivated by previous experimental and theoretical analyses of the crystalline terms of the anisotropic magnetoresistance (AMR). The studies have shown that these magnetotransport coefficients can be large and reflect the rich magnetocrystalline anisotropies of the studied (Ga,Mn)As materials [9,10,11,12,13,14,15,16,17]. Furthermore, the possibility of directly controlling the crystalline AMR by strain has been demonstrated in (Ga,Mn)As films grown under compressive and tensile lattice matching strains [10,12,18]. Here we report and analyze AMR measurements in strain relaxed (Ga,Mn)As micro and nanostructures and in a (Ga,Mn)As film bonded to a piezo-stressor. We show that post-growth induced lattice distortions can significantly modify crystalline AMR terms and give rise to new, previously undetected components in phenomenological AMR expansions.
The paper is organized as follows: In the 2nd section we derive a phenomenological description of the AMR relevant to the experimentally studied (Ga,Mn)As systems. 3rd and 4th sections report AMR data acquired in macroscopic and strain-relaxed microscopic Hall bars, in (Ga,Mn)As nanoconstriction devices, and in voltage controlled piezoelectric/(Ga,Mn)As hybrid structures. A brief summary of the results is given in the 5th Section.

PHENOMENOLOGICAL DESCRIPTION OF THE AMR
We consider a thin film geometry and a magnetization vector, M /| M | = (cos ψ, sin ψ), in the plane of the film with its two components defined with respect to the orthogonal crystallographic basis {[100], [010]}. The resistivity tensor, ρ = ρ 11 (cos ψ, sin ψ) ρ 12 (cos ψ, sin ψ) ρ 21 (cos ψ, sin ψ) ρ 22 (cos ψ, sin ψ) , We first derive expressions for the non-crystalline AMR components [16,19] which depend only on the angle ψ − θ between the current (Hall bar orientation) and the magnetization vector, and which account for the AMR in isotropic (polycrystalline) materials. We expand the elements ofρ in Eq. (1) in series of cos n ψ and sin n ψ, or equivalently of cos nψ and sin nψ [19]. The form of Eq. (2) implies that corresponding expansions of ρ L and ρ T in series of cos(nψ + mθ) and sin(nψ + mθ) contain only terms with m = 0, ±2. Among those, the cos 2(ψ − θ) and sin 2(ψ − θ) are the only terms depending on ψ − θ. It explains why the non-crystalline AMR components, which are obtained by truncating Eq. (1) toρ take the simple form ∆ρ L /ρ av ≡ (ρ L − ρ av )/ρ av = C I cos 2(ψ − θ) and ρ T /ρ av = C I sin 2(ψ − θ) [16,19]. Here ρ av is the average (with respect to ψ) longitudinal resistivity, and C I is the non-crystalline AMR amplitude. All terms in the expansion of Eq. (2) which depend explicitly on the orientation of the magnetization vector with respect to the crystallographic axes contribute to the crystalline AMR [16,19]. Symmetry considerations can be used to find the form ofρ(ψ) in Eq. (1) specific to a particular crystal structure. Explicit expressions forρ(ψ) in unperturbed cubic crystals, and cubic crystals with uniaxial strains along [110] and [100] axes are derived in the Appendix. Here we write the final expression for ρ L and ρ T obtained from the particular form ofρ(ψ) and from Eq. (2). For the cubic lattice, omitting terms with the periodicity in ψ smaller than 90 • , we obtain, ∆ρ L ρ av = C I cos 2(ψ − θ) + C IC cos(2ψ + 2θ) + C C cos 4ψ + . . .
For the higher order cubic terms see the Appendix. Additional components emerge in ∆ρ L /ρ av and ρ T /ρ av for the uniaxially strained lattice which we denote as ∆ uni L and ∆ uni T , respectively. Omitting terms with the periodicity in ψ smaller than 180 • we obtain, for strain along the in-plane diagonal directions (s = [110] corresponds to "+" and [110] to "-"), and for strain along the in-plane cube edges (s = [100] corresponds to "+" and [010] to "-") For higher order uniaxial terms see again the Appendix.

EXPERIMENTS IN LITHOGRAPHICALLY PATTERNED (GA,MN)AS MICRODEVICES
We now proceed with the discussion of AMR measurements in (Ga,Mn)As microdevices in which strain effects are controlled by lithographically induced lattice relaxation [1,2,3]. Optical micrograph of the first studied device is shown in Fig. 1(a). The structure consists of four 1 µm wide Hall bars and one 40 µm wide bar connected in series. The wider bar is aligned along the [010] crystallographic direction, the micro-bars are oriented along the [110], [110], [100], and [010] axes. The Hall bars are defined by 500 nm wide trenches patterned by e-beam lithography and reactive ion etching in a 25 nm thick Ga 0.95 Mn 0.05 As epilayer, which was grown along the [001] crystal axis on a GaAs substrate. The Curie temperature of the as-grown (Ga,Mn)As is 60 K. A compressive strain in the (Ga,Mn)As epilayer grown on the GaAs substrate leads to a strong magnetocrystalline anisotropy which forces the magnetization vector to align parallel with the plane of the magnetic epilayer [6,7]. The growth strain is partly relaxed in the microbars, producing an additional, in-plane uniaxial tensile strain in the transverse direction [1,2,3]. axis. Strain relaxation due to patterning leads to a reduction of the AMR magnitude of about 30%. For better clarity we plot, instead of ∆ρL/ρav defined in the Section 2, δρL/ρav ≡ (ρL − ρL,min)/ρav. Here ρL,min is the minimum (with respect to ψ) longitudinal resistivity.
Magnetoresistance traces were measured with the saturation magnetic field applied in the plane of the device, i.e., in the pure AMR geometry with zero (antisymmetric) Hall signal and with magnetization vector aligned with the external magnetic field. The sample was rotated by 360 • with 5 • steps. Longitudinal resistances of all five Hall-bars were measured simultaneously with lock-in amplifiers.
In Fig. 1(b) we show AMR data from magnetization rotation experiments in the 40 µm and 1 µm wide bars aligned along the [010] direction. Both curves have a minimum for magnetization oriented parallel to the Hall bar axis and a maximum when magnetization is rotated by 90 • . Although this is a typical characteristic of the non-crystalline AMR term in (Ga,Mn)As the large difference between the AMR magnitudes in the two devices points to a strong contribution of the crystalline AMR coefficient C in Eq. (7), originating from the strain induced by transverse lattice relaxation in the microbar. We find that the magnitude of the coefficient, C   Fig. 2(a) and in each of the two traces in Fig. 2(b). While in the former case this behavior can be captured by Eq.
the shape of the AMR curves in Fig. 2 coefficient which is independent of the microbar orientation, i.e., assuming that its origin is distinct from transverse strains induced by the micropatterning. From the difference between the two AMR curves in Fig. 2(a) and from the [110]-bar AMR in Fig. 2(b) we obtained that this contribution is 0.3%, and from the [110]-bar AMR we obtained 0.1%. A bar independent contribution to C [100] U therefore explains only part of the observed [100]-[010] broken symmetry effects; we attribute the remaining part to possible material inhomogeneities or non-uniformities and misalignments in the micropatterning. Importantly, the above experimental uncertainties have no effect on the main conclusion of our experiments that the lattice relaxation induced uniaxial AMR coefficient is larger than the cubic crystalline component and a significant fraction of the total AMR of the unpatterned material. By normalizing the value of the transverse strain-induced C [010] U coefficient and the C C coefficient to the respective values at 4 K, we can also compare their temperature dependencies within the measured range of temperatures of 4 to 70 K. Clearly the C C coefficient decreases more rapidly with increasing temperature. This recalls the behavior of the magnetocrystalline anisotropy terms in magnetization, where the uniaxial term decreases in a less pronounced way than the cubic one, since the former scales roughly with M 2 , while the latter with M 4 . As a result of this, the transverse strain-induced term becomes more dominant at higher temperatures, changing from 31% of the total AMR at 4 K, to 38% at 70 K (not shown). (Note that non-zero AMR is still observable at 70 K which is above the T c of the as-grown material. This is presumably because of the increase in the Curie temperature due to partial annealing during the device fabrication processes.) A detail analysis of the longitudinal resistance measurements in the microbars allows us to identify higher order cubic terms (see Eq. (12) in the Appendix). By subtracting the 2nd and 4th order terms from AMR data measured on the [010] microbar we find a clear signature of an 8th order (45 • -periodic) cubic term with an amplitude of 0.04%, as shown in Figure 2(d). In Sections 4 we give another example of the unusual high order AMR terms (and explain in more detail how these are extracted from the data) which emerge from post-growth induced lattice distortion experiments.
Measurements in the Hall bars discussed above demonstrate that (sub)micrometer lithography of (Ga,Mn)As materials grown under lattice matching strains inevitably produces strain relaxation which may be large enough to significantly modify magnetotransport characteristics of the structure. Lateral micro and nanoconstrictions, utilized in magnetotransport studies of non-uniformly magnetized systems or as pinning centers for domain wall dynamics studies, are an important class of devices for which these effects are highly relevant. In Fig. 3 we show data measured in devices consisting of two 4 µm wide bars patterned from the same (Ga,Mn)As wafer as above along the [110]    We again identify the uniaxial crystalline AMR term in the constriction due to microfabrication which is of the same sign and similar magnitude as observed in the micro Hall bars. Consistency is also found when comparing the character of the AMR curves for the micro Hall-bars and for the constriction devices patterned along different crystallographic directions (see Fig. 2

EXPERIMENTS IN (GA,MN)AS/PIEZO-TRANSDUCER HYBRID STRUCTURES
Lithographic patterning of micro and nanostructures in (Ga,Mn)As provides powerful means for engineering the crystalline AMR components. In this section we show that further, dynamical control of these effects is achieved in hybrid piezoelectric/(Ga,Mn)As structures. A 25 nm thick Ga 0.94 Mn 0.06 As epilayer utilized in the study was grown by low-temperature molecular-beam-epitaxy on GaAs substrate and buffer layers [4]. A macroscopic Hall bar, fabricated in the (Ga,Mn)As wafer by optical lithography, and orientated along the [110] direction, was bonded to the PZT piezo-transducer using a two-component epoxy after thinning the substrate to 150 ± 10 µm by chemical etching. The stressor was slightly misaligned so that a positive/negative voltage produces a uniaxial tensile/compressive strain at ≈ −10 • to the [110] direction. The induced strain was measured by strain gauges, aligned along the [110] and [110] directions, mounted on a second piece of 150 ± 10 µm thick wafer bonded to the piezo-stressor. Differential thermal contraction of GaAs and PZT on cooling to 50 K produces a measured biaxial, in-plane tensile strain at zero bias of 10 −3 and a uniaxial strain estimated to be of the order of ∼ 10 −4 [20]. At 50 K, the magnitude of the additional strain for a piezo voltage of ±150 V is approximately 2 × 10 −4 . [010] [110] [100] (c) Fourth order components of the transverse AMR at piezo voltages ±150V and 0V (2nd order components were subtracted as described in the text). In all cases T=50 K and the field of 1 T was rotated in the plane of the (Ga,Mn)As layer. As in Fig.s 1-3 we plot better clarity δρL/ρav ≡ (ρL−ρL,min)/ρav and δρT /ρav ≡ (ρT −ρT,min)/ρav. Here ρ L(T ),min is the minimum (with respect to ψ) longitudinal(transverse) resistivity.
Previous measurements [4] of the device identified large changes in the magnetic easy axis orientation induced by the piezoelectric stressor. Here we focus on the effects of the stressor on the magnetotransport coefficients. The AMR measured at 50 K for ±150 V on the transducer is shown in Fig. 4(a). The modification of the AMR induced by the strain can be extracted by subtracting curves at ±150 V (see Fig. 4(b)). It is expected that only the crystalline terms are modified; indeed the modification in the longitudinal resistivity ρ L is due to the second and fourth order crystalline AMR terms. This is consistent with our previous analysis on unstressed Hall bars where we found that there were second and fourth order crystalline terms representing approximately 10% of the total AMR. There is also a modification of ρ T of similar magnitude. This is predominantly due to the fourth order term.
To extract the absolute value of the fourth order term in ρ T at each voltage we have performed the following analysis: Starting with the raw ρ T data we subtract any offset due to mixing of ρ L into the ρ T signal which may occur as a consequence of small inaccuracies in the Hall bar geometry or small inhomogeneity in the wafer. This is a correction of approx 0.4% of the ρ L signal which should have no significant effect on the subsequent analysis of fourth order terms. (The fourth order components in ρ L are typically 0.1%, so the effect on ρ T would be 0.4%×0.1% = 0.0004%, i.e., negligibly small.) We then remove any unintentional antisymmetric (Hall) component from ρ T by shifting the data by 180 • and averaging. The second order terms are subsequently removed from ρ T by shifting the data by 90 • and averaging. The result of this procedure is plotted in Fig. 4(c).
At 0 V the fourth order component is approximately 0.03% (peak to trough). At +150V it is further enhanced to approximately 0.1% while at -150V the magnitude is reduced to approximately 0.01% which is a value similar to the fourth order term observed after carefully reexamining a (Ga,Mn)As wafer without the piezo-stressor attached to it [16]. For the present device, measurements of the magnetic anisotropy indicate that the application of -150V to the piezo transducer counteracts the uniaxial strain induced by differential thermal contraction on cooling to return the device close to the unstrained state [4]. The presence of a fourth order term in the transverse AMR is allowed under a uniaxial distortion, see Eq. (16), but is not expected if only cubic symmetry is present. The data presented in Fig. 4(c) clearly demonstrates that the uniaxial strain produced by the piezo transducer induces a significant fourth order term in the transverse AMR, which is usually considered to be of insignificant magnitude in the unstrained wafer. The analysis demonstrates that by applying voltage on the piezoelectric transducer one can significantly enhance crystalline AMR components, as compared to the bare (Ga,Mn)As wafer, as well as efficiently compensate additional strain effects induced by, e.g., different thermal expansion coefficients in hybrid multilayer structures.
Eqs. (9)-(11) together with Eq. (2) yield, after transforming all products of goniometric functions and recollecting them into sines and cosines of sums of angles, the following structure of the longitudinal and transverse AMR expressions: ∆ρ L ρ av = C C cos 4ψ + C C8 cos 8ψ + . . .
The terms which contain at most 2ψ reproduce Eq. (6).