Spin-Wave Resonance in (Ga,Mn)As thin films: Probing in-plane

Spin-Wave Resonance in (Ga,Mn)As thin films: Probing in-plane surface magnetic anisotropy H. Puszkarski∗ Surface Physics Division Faculty of Physics, Adam Mickiewicz University ul. Umultowska 85, 61-614 Pozna´ n, Poland

P. Tomczak Quantum Physics Division Faculty of Physics, Adam Mickiewicz University ul. Umultowska 85, 61-614 Pozna´ n, Poland (Dated: April 15, 2015) We show that recent spin-wave resonance studies of (Ga,Mn)As thin films performed by Liu et al. [Phys. Rev. B75, 195220 (2007)] reveal a substantial increase in the in-plane uniaxial anisotropy surf at the film surface with respect to the bulk value (surface uniaxial anisotropy field H2k = 305 Oe bulk against the bulk value H2k = 77 Oe). At the same time, the cubic anisotropy at the film surface surf bulk proves substantially lower than in the bulk (H4k = 39 Oe against H4k =197 Oe). To our best knowledge this is the first report of an estimate of the surface anisotropy field values in this material.

I.

INTRODUCTION

The magnetic anisotropy is one of the most interesting properties of gallium manganese arsenide (Ga,Mn)As thin films since it determines the direction of the sample magnetization, the manipulation of which is a key factor for prospective application of the material in memory devices. This is why the magnetic anisotropy of (Ga,Mn)As films is recently intensively investigated by using many experimental techniques. Among them, spin-wave resonance (SWR) is a newly emerged method for studying surface magnetic anisotropy in (Ga,Mn)As thin films. Spin-waves and spin-wave resonance have been studied particularly intensively in (Ga,Mn)As thin films in the past decade1–13 . SWR spectra were obtained in studies with a variable configuration of the static field with respect to the film surface in the out-of-plane configuration, as well as in the in-plane configuration with the external field rotating in the plane of the film. There is common agreement in the literature on (Ga,Mn)As thin films that spin-wave resonance in such samples under in-plane external magnetic field fulfills remarkably well the assumptions of the surface inhomogeneity (SI) model. The SI model assumes that the magnetic properties relevant to SWR are homogeneous across the thin film (along the direction perpendicular to its surface) and only change at its edges (surfaces). Both the broken symmetry of the neighborhood of the surface magnetic ions and the changed parameters of neighbor interaction at the film surface result in additional surface spin pinning, which plays a major role in the determination of a variety of characteristics related to the spin dynamics in the whole film sample14–17 . Surface spin pinning is usually described by the surface parameter, which enters the spin motion equations via the boundary conditions that must be fulfilled by the motion of the surface spins. A universal formula for the

surface parameter, which can be used for describing the surface properties of (Ga,Mn)As, is provided in our earlier paper18 , in which we only analyzed the pinning effects related to rotation of the external field with respect to the surface of the sample in the out-of-plane configuration, with changing polar angle ϑH between the external field and the surface normal. Our present paper complements the previous one by studying SWR in the in-plane configuration, with the external field rotating in the plane of the film, changing the azimuthal angle ϕH . The most important feature of the in-plane SWR spectra observed in (Ga,Mn)As thin films is related to the existence of the so-called double critical angle effect. The evolution of the SWR spectrum with the external field rotated in the plane of the sample is visualized in Ref. [19], Fig. 3. The multi-peak SWR spectrum is found to become a singlepeak ferromagnetic resonance at two particular angles, namely ϕH = 24◦ and ϕH = 64◦ ; these are the abovementioned critical angles. To our best knowledge, the observation of two critical angles in the in-plane rotation in the irreducible range (0, 90◦ ) had not been reported in the literature before the paper by Liu et al.19 . The observed two critical angles in the in-plane configuration makes one wonder what peculiar type of surface magnetic anisotropy in (Ga,Mn)As might result in such an effect.

II.

IN-PLANE SWR EXPERIMENT VS. SURFACE PINNING PARAMETER

The basis for a present study are the SWR measurements performed by Liu et al.19 in their experiment, one of the few in which a surface parameter value (calculated in the SI model) is assigned to each experimental SWR spectrum. The spectra are taken versus the polar angle ϑH and the azimuthal angle ϕH , which allows the Authors to plot the angle dependence of the surface

2 parameter in both the out-of-plane and in-plane configurations, i.e., the functions Asurf (ϑH ) and Asurf (ϕH ), respectively. In our previous paper18 we provided a picture of surface pinning mechanisms adequate to the polar angle dependence of the surface parameter found in the experimental study19 in the out-of-plane configuration. In the present paper we are going to report a similar analysis of the SWR experiment in the in-plane configuration, finding out which of the magnetic anisotropies observed in the film bulk are present also at the film surface and what is their relative contribution to the surface pinning; it will turn out that for an adequate description of the contribution of the cubic anisotropy to the surface pinning in an (Ga,Mn)As thin film it is necessary to take account of the new anisotropy term absent in the film bulk. In the in-plane configuration, on which we will henceforth focus, both the applied resonance field H and the magnetization M lie in the plane of the film (the xy plane), and their orientations are described by their azimuthal angles ϕH and ϕM , respectively, in relation to the x axis (which is the [100] crystal axis). These two angles depend on each other by fulfilling an adequate equilibrium equation resulting from the specific form of the free energy of the system (in the out-of-plane configuration there was a similar interdependence between the polar angles ϑH and ϑM , see Ref. [18]). On the basis of their SWR measurements performed in the in-plane configuration Liu et al.19 determined the dependence of the surface parameter on the field azimuthal angle ϕH (see Fig. 9b in Ref. [19]). This dependence can be easily converted to the magnetization angle dependence by referring to the equilibrium equation. The recalculated dependence, obtained by the conversion ϕH → ϕM , is shown in Fig. 1 in the present paper; it is in this form that we will henceforth analyze the in-plane angle dependence of the surface parameter. To reproduce faithfully the experimental points plotted in Fig. 1 we have performed a number of numerical attempts to test a variety of tentative functions Asurf (ϕM ) with at least partial physical grounds reported in the literature. We will begin with presenting the end result of our numerical investigations, and only then substantiate it by describing the steps that led us to this finding. The experimental dependence of the surface parameter on the azimuthal angle ϕM , plotted in Fig. 1, is reproduced with the best numerical accuracy by the formula: π Asurf (ϕM ) = 1 + aiso + auni sin2 ϕM − 4 4 +acub 3 + cos4ϕM + acub 3 + cos4ϕM . (2.1) The terms in the equation (2.1) have the following physical meaning: the first term (unity) is an effect of breaking the symmetry at the surface (i.e. elimination of a part of neighbors); the isotropic term aiso describes the contribution to the surface pinning brought by some indeterminate surface interaction independent of the orientation of the surface magnetization; the auni term describes the

contribution brought by the uniaxial anisotropy (with an easy axis in the [¯110] direction), and the acub terms – the contribution of the cubic crystallographic anisotropy (the last term describes its higher-order contribution – fourthorder one in this case). If the presence of the uniaxial anisotropy term and the lowest-order cubic anisotropy term acub 3 + cos4ϕM is not surprising (since both anisotropies are present also in the bulk), the higherorder cubic anisotropy term does come as a surprise, as nothing in the studies published to date indicates this type of anisotropy in the bulk. Considering this, now we will describe the subsequent steps which have led us to this term in the postulated formula for the surface pinning parameter Asurf (ϕM ).

III. A SEARCH FOR IN-PLANE ANGLE DEPENDENCE OF THE SURFACE PINNING PARAMETER

In what follows we will describe the seeking procedure for a function Asurf (ϕM ) which reproduces with the best numerical accuracy the experimental data shown in Fig. 1. Four of the experimental points will be given a particular statistical weight, as we expect the sought curve to pass very precisely through these points. Thus, we single out two points representing two local maximums at the respective angles ϕM = 45◦ and ϕM = 135◦ , and the points corresponding to the critical angles: ϕcM = 27◦ and ϕcM = 62◦ , for which the surface parameter is Asurf (ϕcM ) = 1. Let us note at once that the surface parameter values representing the two (local) maximums are not equal: Asurf (ϕM = 45◦ ) = 1.065 and Asurf (ϕM = 135◦) = 1.092. This asymmetry in the two maximums will turn out to be of crucial importance for our further fitting procedure, since it allows to estimate with much certainty the value of the surface uniaxial anisotropy. At the first step of the fitting procedure we postulate the following formula for the surface parameter: π 4 3 + cos4ϕM ; (3.1)

Asurf (ϕM ) = 1 + aiso + auni sin2 ϕM − +acub

in this formula we take into account both main anisotropies existing in (Ga,Mn)As, namely uniaxial anisotropy and the cubic one. Indeed, since these two types of anisotropy are present in the bulk, they can be reasonably expected also at the surface (though presumably to a different extent). The fitting procedure, which we are going to carry out with the formula (3.1), should yield the values of the three coefficients aiso , auni and acub that figure in this formula. However, with the characteristic points that we have singled out at the beginning the set of fitted coefficients can be reduced to only one coefficient, acub . Indeed, note that the following equations, resulting from the formula (3.1), hold for the local

1.2

1.12 (a)

1.1

1.1 Asurf(φΜ. )

Surface pinning parameter Asurf(φΜ)

3

1 0.9

1.06 1.04

0.8

1.02

0.7

0.1 1 0.08 0.06 0.04 0.02 0

0.6 0.5 0.4 0

20 40 60 80 100 120 140 160 180 In-plane magnetization angle, φΜ [deg]

FIG. 1. Magnetization angle dependence of the in-plane surface pinning parameter Asurf (ϕM ) according to the experimental data obtained by Liu et al.19 in their SWR study of a (Ga,Mn)As thin film; the plot corresponds to that shown in Figure 9b in the cited paper, presenting the dependence on the magnetic field angle ϕH . The applied transformation between the angles ϕH and ϕM is based on our determination of the equilibrium direction of magnetization.

auni=0.027

1.08

(b)

aunisin2(φΜ-45o) auni=0.027

0


FIG. 2. Graphical representation of our method for the determination of the surface uniaxial anisotropy coefficient auni : (a) experimental plot around the local maximums (much enlarged part of the plot shown in Fig. 1); (b) angle dependence of the in-plane surface uniaxial pinning (by Eq. (2.1)). The juxtaposition of the plots (a) and (b) clearly indicates that the asymmetry between the two local maximums (upper part (a)) is correlated with the variation in the uniaxial anisotropy (b).

maximums: Asurf (ϕM = 45◦ ) = 1 + aiso + 2acub , Asurf (ϕM = 135◦ ) = 1 + aiso + auni + 2acub .

(3.2) (3.3)

Consequently, the uniaxial anisotropy coefficient is: auni = Asurf (ϕM = 135◦ ) − Asurf (ϕM = 45◦ ) = 0.027. (3.4) critical angles. Obviously, this can only be achieved by modifying The way in which we have arrived at this result is also presented graphically in Fig. 2. Also, from the formula (3.2) alone it follows that:

We have performed a number of additional attempts taking into account of further terms of the power series representing the contribution of the cubic anisotropy to the surface pinning. It turned out that neither secondorder nor third-order terms help achieve the goal: only the addition of the fourth-order cubic anisotropy term 4 acub 3+cos4ϕM in (3.1) met our expectations. Thus, in the end we conclude that the azimuthal angle dependence of the surface parameter fitting the experimental points must be postulated as the series (2.1). The complete set of values of the coefficients aiso , auni and acub obtained from this best numerical fitting is: aiso = 0.1058; auni = 0.027; acub = −0.0023.

(3.6)

Figure 4 shows the corresponding function Asurf (ϕM ) aiso = Asurf (ϕM = 45◦ ) − 1 − 2acub = 0.065 − 2acub . plotted against the experimental points; note that now (3.5) our theoretical curve passes through all our four characThus, we can consider acub as the only one indeteristic points. What physical information is carried by pendent variable in our fitting procedure; Figure the obtained coefficient values specified in (3.6)? We will 3 shows obtained Asurf (ϕM ) curve against the answer this question in the next Section. experimental points. As we can see the critical angles resulting from this calculation are ϕcM = 34criticalangles.Obviously, thiscanonlybeachievedbymodif ying◦ IV. DETERMINATION OF SURFACE and ϕcM = 56◦ , very far from the experimental ones: ◦ ◦ ANISOTROPY FIELDS 27 and 62 . Thus, we cannot accept the formula (3.1) as a good fit of the experimental plot, and must For a physical interpretation of the in-plane values of consider modifying it so as to obtain a theoretical curve the coefficients auni and acub obtained above (Eq. (3.6)) passing as close as possible to the experimental points we must first relate them to phenomenological quantities corresponding to the critical angles. Obviously, this can characterizing the (Ga,Mn)As surface, as we have done only be achieved by modifying in the Eq. (3.1) the term in our previous paper18 with the out-of-plane coefficient related to the cubic anisotropy.


4 where H2k and H4k are the in-plane uniaxial and cubic anisotropy fields, respectively, at the surface or in the bulk; d is the lattice constant (the spacing between Mn ions actively participating in the magnetic interaction), and Dex is the exchange stiffness constant. The relations (4.1)) and (4.2) allow to express both anisotropy fields at the surface by the respective surface pinning coefficients:

1.2 1.1 1 0.9 0.8

34o 56o

0.7

φcM=27o

0.6

surf bulk H2k = H2k + 2auni

φcM=62o

0.5

(4.3)

Dex . d2

(4.4)

surf bulk + 16acub H4k = H4k

0.4 0


FIG. 3. Fitting of the theoretical curve with the formula (3.1) to the experimental points; the cubic pinning coefficient value obtained for this curve is acub = −0.229 (which implies aiso = 0.523). We cannot accept this fit as satisfactory, as the curve fails to pass through the points corresponding to the critical angles 27◦ and 62◦ .


Dex , d2

1.2 1.1 1 0.9 0.8 0.7

φcM=27o

0.6

Obviously, to make effective use of these formulas, apart from the values of the coefficients auni and acub we must know the estimate value of the ratio Dex /d 2 at the surface. At this point we will refer to the results of our previous paper18 , in which we have shown (see Fig. 6 in Ref. [18]) that in the in-plane configuration the surface exchange length value is λs = 3 nm; at the same time, we know that in the in-plane configuration the sample fulfills almost perfectly the assumptions of the SI model. Thus, we can assume that the lattice constant d is close to the surface exchange length λs , i.e., d ≈ 3 nm. For the exchange stiffness constant we assume the estimate value found by Liu et al. on the basis of their SWR study19 , specifically, Dex = 3.79 T·nm2 ; this implies the estimate value of the sought ratio Dex /d 2 ≈ 4220 Oe. Now, by putting this value in the formulas (4.3) and (4.4), and using the estimate values of the pinning coefficients provided in Eq. (3.6), we obtain: surf bulk = 77 Oe), H2k = 305 Oe (against the bulk value H2k (4.5)

φcM=62o

0.5 0.4 0


FIG. 4. Magnetization angle dependence (solid line) of the in-plane surface Asurf (ϕM ) pinning parameter resulting from our model of surface pinning in (Ga,Mn)As thin films (see Eq. (2.1)). The theoretical curve is found to fit very well the experimental points obtained by Liu et al.19 ; the coefficient acub obtained with this curve is acub = −0.0023.

(see Eq. (22) in18 ). Here we find the following relations of use for this purpose: auni =

d2 bulk , H surf − H2k 2Dex 2k

(4.1)

acub =

d2 surf bulk , H4k − H4k 16Dex

(4.2)

surf bulk = 197 Oe); H4k = 39 Oe (against the bulk value H4k (4.6) above both bulk values of the anisotropy fields are as reported by Liu et al.19 . This result indicates that the uniaxial anisotropy increases substantially (four times!) at the surface of the (Ga,Mn)As film sample, whereas the surface cubic anisotropy is, on the contrary, substantially reduced with respect to the bulk value. Interestingly, the above-mentioned decrease in the cubic anisotropy field at the surface should be related to the additional fourth-order cubic anisotropy term appearing in the formula (2.1) (as a result of breaking the symmetry in the distribution of magnetic neighbors at the film surface). The substantial increase in the uniaxial anisotropy at the surface only seems to confirm that the bulk and surface uniaxial anisotropies have different origins20–22 . It is believed in the literature that a key role in the inducing of magnetocrystalline anisotropy in (Ga,Mn)As is played by hole carriers. Also, experimental studies indicate that anisotropy constants depend on hole concentration20,24–27 . On the other hand, the hole concentration in thin films is known to vary across the film

5 profile, being reduced at the surface. Thus, we can wonder if the results of our study, indicating that the cubic anisotropy in (Ga,Mn)As thin films diminishes at the surface, are consistent with the commonly accepted model of anisotropy in this material. To answer this question we must refer to another resonance study, performed by Furdyna et al.28 , in which the Mn-hole exchange interaction integral is shown to depend on the hole concentration, specifically, to be proportional to its one-third power: Jpd ∼ p 1 / 3 (where p denotes the hole density). Now, considering that the hole density is reduced at the surface, the above relation implies that the information on the crystal structure of the material, conveyed by hole carriers to manganese ions, is poorer at the surface than in the bulk. Consequently, localized surface Mn spins will feel the crystal axes less strongly than bulk spins. This in turn means that the surface values of the magnetic anisotropy constants will be smaller than the respective bulk anisotropy values, which is consistent with our findings. V.

FURTHER PHYSICAL IMPLICATIONS OF OUR MODEL

The results obtained in the present study shed new light on those reported in our previous paper18 , and support them at the same time. Let us recall: in Ref. [18] we analyzed the SWR measurements performed by Liu et al.19 in the out-of-plane configuration with the external field rotating from the direction normal to the sample surface to the in-plane direction. The field was rotated in the plane perpendicular to the surface defined by the [001] and [1¯10] crystal axes. Our theoretical analysis of the experimental SWR data performed in Ref. [18] led

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Corresponding author: [email protected] Y. Sasaki, X. Liu, T. Wojtowicz, and J. K. Furdyna, Spin Wave Resonance in GaMnAs, Journal of Superconductivity and Novel Magnetism 16, 143 (2003). X. Liu, Y. Sasaki, and J. K. Furdyna, Ferromagnetic resonance in Ga1−x M nx As: Effects of magnetic anisotropy, Phys. Rev. B 67, 205204 (2003). T. G. Rappoport et al., Anomalous behavior of spin-wave resonances in Ga1−x M nx As thin films, Phys. Rev. B 69, 125213 (2004). S. T. B. Goennenwein et al., Spin wave resonance in Ga1−x M nx As, Appl. Phys. Lett. 82, 730 (2003). X. Liu and J. K. Furdyna, Ferromagnetic resonance in Ga1−x M nx As dilute magnetic semiconductors, J. Phys.: Condens. Matter 18, R245 (2006). Y.-Y. Zhou et al., Ferromagnetic Resonance Study of Ultra-thin Ga1−x M nx As Films as a Function of Layer Thickness, AIP Conf. Proc. 893, 1213 (2007). C. Bihler, W. Schoch, W. Limmer, S. T. B. Goennenwein, and M. S. Brandt, Spin-wave resonances and surface spin pinning in Ga1−x M nx As thin films, Phys. Rev. B 79,

us to the conclusion (see Fig. 8 in Ref. [18]) that in the in-plane configuration the surface magnetization is larger than the bulk magnetization. In the light of our results obtained in the present study we begin to understand the underlying cause. In this study we have shown that the easiness of the uniaxial anisotropy is multiplied at the surface of the studied material, while in terms of the cubic anisotropy the [1¯10] axis becomes much less hard. Thus, we can expect that, as the rotated external field approaches the surface plane, the spins inclined to align in its direction become at the surface more numerous than in the bulk; this explains why Mksurf > Mkbulk . The model of in-plane surface anisotropy proposed in the present paper opens the way for further exploration of surface phenomena in (Ga,Mn)As. For example, SWR measurements analogous to those presented in the paper by Liu et al.19 , but at higher temperatures, would allow to verify whether the temperature dependence of the surface anisotropy in (Ga,Mn)As is identical with or different from the reported temperature dependence of the bulk anisotropy22,23 . Another possible direction of study is related to surface hydrogenation24, known to strongly affect the surface anisotropy in (Ga,Mn)As. Our model can be used for the quantitative evaluation of the effect of such hydrogenation on the surface anisotropy in (Ga,Mn)As; no quantitative data on this subject are known to us to date. This study is a part of a project financed by Narodowe Centrum Nauki (National Science Center of Poland), Grant no. DEC-2013/08/M/ST3/00967. Illuminating discussions with Professor J. Furdyna are gratefully acknowledged. The authors are also indebted to Professor A. R. Ferchmin for very useful discussions.

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