LEEM image phase contrast of MnAs stripes

Ultramicroscopy 130 (2013) 7–12

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LEEM image phase contrast of MnAs stripes A.B. Pang a,b,n, A. Pavlovska c, L. Däweritz d, A. Locatelli e, E. Bauer c, M.S. Altman b a

School of Physics and Electronic Information, Huaibei Normal University, Huaibei, Anhui, 235000, PR China Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PR China Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA d Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz 5-7, D-10117 Berlin, Germany e Sincrotrone Trieste, S.C.p.a., Basovizza, Trieste 34012, Italy b c

art ic l e i nf o

a b s t r a c t

Available online 22 March 2013

Low energy electron microscopy (LEEM) imaging of strained MnAs layers epitaxially grown on GaAs(001) reveals striped contrast features that become more pronounced and vary systematically in width with increasing defocus, but that are completely absent in focus. Weaker subsidiary fringe-like features are observed along the stripe lengths, while asymmetric contrast reversal occurs between under-focus and over-focus conditions. A Fourier optics calculation is performed that demonstrates that these unusual observations can be attributed to a phase contrast mechanism between the hexagonal α phase and orthorhombic β phase regions of the MnAs film, which self-organize into a periodic stripe array with ridge-groove morphology. The unequal widths of the α and β phase regions are determined accurately from the through focus series, while the height variation in this system can also be determined in principle from the energy dependence of contrast. & 2013 Elsevier B.V. All rights reserved.

Keywords: Low energy electron microscopy Phase contrast MnAs stripes Defocus Fourier optics calculation

1. Introduction As one of the most powerful imaging tools that provide realtime and in-situ observations of surfaces, low energy electron microscopy (LEEM) has been used in a wide range of studies of surface related phenomena [1–3]. For any imaging technique, resolution is one of the most important criteria that are used to assess its performance. However, the depth of understanding we can expect to gain from imaging depends not only on the resolution, but also on the mechanisms that give rise to image contrast and our understanding of these mechanisms. Two main types of contrast that arise due to the wave nature of imaging electrons in LEEM imaging are amplitude contrast and phase contrast. Amplitude contrast arises due to spatial variations of the amplitude of the reflected electron wave. Its occurrence can frequently be attributed to structure factor differences that arise due to variations of geometric structure, but may also be observed for other reasons, for example, as a result of exchange scattering of spin polarized electrons in spin polarized LEEM [4,5]. Phase contrast is generally produced when modifications of the phase of the imaging electron wave occur upon reflection that lead to interference effects. The most prominent examples are atomic step contrast and quantum size contrast. The former is related to a n Corresponding author at: Huaibei Normal University, School of Physics and Electronic Information, Dongshan Road #100, Huaibei, Anhui 235000, China. Tel.: þ 86 15215615782. E-mail address: [email protected] (A.B. Pang).

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spatial variation of the phase of the reflected wave across a step [6], while the latter occurs in thin films due to the phase difference of the waves reflected from the film surface and its buried interface with the substrate [7]. In both cases, path length differences give rise to phase differences that determine interference conditions and contrast. Several years ago, LEEM phase contrast produced by atomic steps was modeled as the interference of Fresnel diffracted waves generated at a step [6]. This elementary approach correctly predicted several key aspects of LEEM step contrast that have been observed experimentally, including detailed energy dependence of interference fringes near a step and contrast reversal between over-focus and under-focus conditions. However, this model only included aberrations and other imaging effects in an ad hoc way, i.e. by Gaussian image convolution. Recently, two other powerful approaches were developed to model image formation in LEEM that treat imaging errors much more realistically [8,9]. In the Fourier optics (FO) method [8], image formation is calculated by the Fourier transform and inverse Fourier transform of the object wave function. The effects of spherical and chromatic aberrations, energy spread, electrons source extension, diffraction at the contrast aperture, as well as the lens voltage and current instabilities are included into the model calculation by modifying the wave function in frequency space after the first Fourier transform. The Contrast Transfer Function (CTF) approach, which is mathematically very similar to the FO approach, was subsequently adapted to LEEM and PEEM imaging [9–11]. While the CTF approach was originally developed for weak phase objects

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in Transmission Electron Microscopy (TEM), its use to guide development of defocus conditions for optimal resolution of strong phase and amplitude objects that are frequently encountered in LEEM imaging but not in TEM was pioneered in Ref. [11]. The CTF formalism was also extended in Ref. [11] to include aberration coefficients up to fifth order, appropriate for aberration corrected instruments. The FO and CTF wave optical approaches to modeling image formation have two key strengths. On one hand, they provide a straightforward way of determining the theoretical resolution accurately from the calculated image for a given set of instrumental parameters including defocus [8,11]. This approach is more rigorous than, the simplistic geometric optics approach to estimate resolution by taking the square root of the quadratic sum of aberration and diffraction terms [11]. On the other hand, application of FO and CTF modeling provides opportunities to investigate quantitatively the structures themselves that give rise to contrast, which is the primary goal of microscopy. This may be done by trial-and-error comparison of experimental observations with intensity distributions in images calculated for features that may produce the contrast, or by utilizing more sophisticated inverse phase retrieval methods [10]. Prior work with the CTF and FO methods demonstrates that these approaches are readily applicable to two dimensional studies and that they are computationally straightforward and efficient [8–11]. Further practical applications of these approaches to real systems will add to our knowledge of the contrast characteristics that are produced by a broader range of surface features. Such work will also permit the development of methods for extracting quantitative information from this analysis. It is in this spirit that we apply the FO approach in this paper in order to understand the unusual contrast that is observed with LEEM for a strained MnAs film.

2. LEEM images of MnAs stripes Growth of MnAs layers on GaAs(001) substrate was performed by molecular-beam epitaxy using procedures described earlier [12]. After oxide desorption from the substrate, a GaAs buffer layer was first grown, followed by growth at 250 1C of the MnAs layers. After growth, the layers were capped with As for shielding against atmosphere during transport to the LEEM instrument. In the LEEM, the As was removed by slowly heating to 300 1C followed by slow cooling to room temperature. Epitaxial growth of MnAs layers on GaAs(001) results in an equilibrium phase coexistence between hexagonal ferromagnetic α-MnAs and orthorhombic paramagnetic β-MnAs phases, ranging from pure α-MnAs below about 10 1C to pure β-MnAs above 45 1C [12,13]. In this temperature range the elastic strain caused by the different lattice constants of the two phases is minimized by self-organizing into a periodic stripe pattern with a thickness dependent period p∼4.8t and a ridge-groove morphology with a height difference between α-phase and β-phase stripes of Δh∼0.01t [14]. The ridges consist of the α-phase while the grooves consist of β-phase. While the relationship between p and t is well established, the Δh values obtained from atomic force microscopy (AFM) studies at room temperature [14,15] fluctuate considerably, varying for example for 80 nm thick films similar to those analyzed in the present study from 1.1 nm to 2.2 nm [16]. A theoretical analysis of the strain state of the layers [13] gives Δh≈0.017t, i.e. about 1.4 nm for the 80 nm layer investigated here. LEEM images of the MnAs film are shown in Fig. 1. The images in Fig. 1 are all taken at the same electron energy E0 ¼14 eV for a microscope acceleration voltage V¼ 20 kV and image field of view of 4 μm  4 μm. Images are organized in order of increasing defocus from left to right. Deviation of objective lens current from

in-focus condition increases in (a) from −0.3 mA to −2.7 mA and in (b) from þ 0.3 mA to þ2.7 mA in equal increments of 0.6 mA between neighboring images. Since lower lens current induces longer focal length, Fig. 1(a) and (b) depicts under-focus and overfocus, respectively. LEEM images at in-focus condition (not shown) exhibit no contrast. However, contrast between the coexisting α-MnAs and β-MnAs stripes appears upon defocus (Fig. 1). The α-MnAs stripes appear dark in over-focus as determined by cooling to lower temperatures, at which they broaden and finally displace the β-MnAs stripes. Three major characteristics can be distinguished from these images. The first is that the bright-dark contrast between stripes becomes stronger as the absolute value of defocus increases, for both under-focus and over-focus conditions. Secondly, contrast reversal is also observed between under-focus and over-focus conditions. Interestingly, the relative widths of the bright and dark stripes do not appear to be the same at equivalent over-focus and under-focus conditions. For example, dark stripes at over-focus condition are considerably wider than (the same) bright stripes at the equivalent under-focus condition. Likewise, the bright stripes in under-focus are wider than the bright stripes in the equivalent over-focus condition. These behavior are illustrative of the asymmetric contrast reversal about the in-focus condition, which suggests that the widths of the α-MnAs and β-MnAs stripes are unequal. Closer inspection of the intensity variation transverse to the stripe direction also reveals that intensity is not uniform (Fig. 1(c) and (d)). In particular, the bright stripe exhibits a weak dark region near the middle along the entire length of the stripe. Likewise, a weak bright area is present near the middle of every dark stripe. So, the contrast can be described as bright and dark stripes with subsidiary weak dark and bright stripes within, respectively. This kind of contrast is referred to below as duplex contrast. The appearance of duplex contrast features, which are reminiscent of interference fringes, as well as the contrast reversal between under-focus and over-focus, and the absence of contrast in focus hint at a phase contrast mechanism. In order examine the duplex contrast features of MnAs stripes in finer detail, we extract the intensity profile transverse to the MnAs stripes. Fig. 2 shows the example of how we extract the intensity. The image is first rotated in order to align the stripes nominally with the vertical y-axis. A region with more or less straight parallel stripes is then chosen, e.g. the area in the rectangular box in Fig. 2. Intensity profiles along the horizontal x-axis in this area are then extracted and averaged along the y-direction to reduce the noise. After this procedure, we obtain the one-dimensional intensity profile along the x-direction, which contains about three stripe periods. This procedure is applied to each image of the through focus series at the identical location. The extracted intensity profiles are shown for under-focus and over-focus in Fig. 3(a) and (b), respectively. The duplex contrast and asymmetry of contrast features between over-focus and under-focus are seen clearly in these intensity profiles. For example, a very strong subsidiary intensity dip in the middle of the bright stripe in under-focus (Fig. 3(a)) appears much weaker or even absent in the bright stripe at over-focus (Fig. 3(b)). Subsidiary intensity maxima in the dark stripes are narrow and decrease with increasing defocus in under-focus (Fig. 3(a)). These maxima are broader and decrease less significantly with increasing defocus in over-focus (Fig. 3(b)).

3. Fourier optics modeling of MnAs stripe phase contrast Since the surface of the MnAs film is known to have a ridge-groove structure [13,14], we calculate phase contrast for a model surface morphology with periodic height modulation along the x-direction indicated in Fig. 2. The height modulation in this

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Fig. 1. LEEM images of MnAs surface with α-MnAs and β-MnAs coexistence at different defocus conditions.: (a) under-focus: −0.3 mA, −0.9 mA, −1.5 mA, −2.1 mA, −2.7 mA from left to the right. (b) over-focus: þ 0.3 mA, þ 0.9 mA, þ1.5 mA, þ2.1 mA, and þ2.7 mA from left to the right. The magnified images in (c) and (d) show the details of the contrast in the circled regions of (a) and (b) at under- and over-focus, −1.5 mA and þ 1.5 mA, respectively.

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Fig. 2. Illustration for the area where the intensity is extracted from the underfocus image (−1.5 mA) as the example.

“square- wave” model, Δh, is assumed to be in the range from 1.1 nm to 2.2 nm, which is in line with the range of values measured with AFM [16] and predicted earlier [13] for 80 nm thick films studied here. The period and the widths of the ridge and groove regions are determined by measuring the stripe widths of the bright and dark stripes in a through focus series of LEEM images. The widths are measured at the average image intensity level, as indicated for example by the dashed line in Fig. 4. We tentatively identify the bright and dark stripes at under-focus in Fig. 4 as α and β regions. Due to contrast reversal upon going through focus, bright and dark stripes in under-focus images become dark and bright stripes in over-focus images. A plot of width as a function of defocus is shown in Fig. 5. The widths of the α (β) stripes clearly decrease (increase) linearly with

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Position (nm) Fig. 3. Intensity profiles from the area shown in Fig. 2 for (a) under-focus and (b) over-focus conditions.

decreasing over-focus. The linear trend continues with increasing defocus at under-focus condition. Linear fits to all the α and β stripe widths are shown in Fig. 5. The intercepts of the fitted lines with the vertical axis at the in-focus condition are taken to be the global average intrinsic widths of the α-MnAs and β-MnAs stripes, 229 nm and 172 nm, respectively. The greatest contributions to the uncertainty of stripe widths come from averaging the non-uniform stripe

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Fig. 6. Height model, h(x), of the MnAs surface is a periodic square wave function with height magnitude Δh. The vertical axis is expanded by a factor of ∼200  . Fig. 4. Intensity profile transverse to stripes in the selected section of the image at under-focus of −1.5 mA shown in Fig. 2. The widths of the stripes tentatively identified as α and β phase are measured at the average image intensity level indicated by the horizontal dashed line.

Fig. 5. Plots of widths versus defocus for the α and β stripes identified in Fig. 4. The linear fits to all of the stripes (black lines) intercept the y-axis at in-focus condition at 229 nm and 172 nm. These intercepts represent the global average intrinsic widths of the α and β phase stripes correspondingly.

width over the length of the stripes being analyzed (Fig. 2) and statistical variations of the stripe widths among the ensemble that are manifested in different y-intercepts in Fig. 5. The uncertainty of the stripe widths considering these two dominant contributions are 715 nm. A simple one-dimensional “square-wave” model of the height variation of the MnAs films, h(x), is shown in Fig. 6. The spatial period of 401 nm is comprised of ridges (α) and grooves (β) with widths that were determined respectively for the α-MnAs and β-MnAs stripes from the experimental through-focus series (Fig. 5). The electron waves reflected from the surface at the two height levels of the α and β phase stripes shown in Fig. 6 do so with a path length difference of 2Δh [6,8–11]. This path length difference is indicated in the wave function of the reflected electrons by a phase shift of ϕ¼4πΔh/λ0, where λ0 is the incident electron wavelength at 14 eV. The object wave function can be expressed as ψðxÞ ¼ ψ o expð−i4πhðxÞ=λ0 Þ, where h(x) is the height profile (e.g. Fig. 6) and the wave amplitude ψ o is assumed to be the same for α-MnAs and β-MnAs phases. This assumption is justified by the absence of contrast at the in-focus condition. Amplitude contrast would be observed in-focus if the amplitude would not be uniform. Thus, contrast has its origin purely in the phase of the imaging wave. For Δh in the range from 1.1 nm to 2.2 nm, the phase shift difference between electrons reflected from the top of the ridge and the bottom of the groove correspondingly ranges from 13.43π to 26.85π for 14 eV electrons. The calculated contrast changes periodically in phase with 2π period. For Δh¼ 1.1 nm, for example, a phase change of ϕ ¼ 72π is effected

by changing the energy by about 74 eV, i.e. to 10.14 eV or 18.48 eV. For Δh¼2.2 nm, the corresponding energy interval is about 72 eV. The Fourier optics calculation was performed using the spherical aberration coefficient CS ¼ 0.209 m, and the chromatic aberration coefficient CC ¼−0.0913 m appropriate for the Elmitec instrument at 14 eV and microscope potential 20 kV. An energy spread of ΔE¼0.5 eV is assumed, which is nominal for LEEM imaging. Other important quantities are the illumination divergence angle, αill ¼0.125 mrad, which characterizes the source extension, and the acceptance angle of the contrast aperture, 3.04 mrad. The voltage and current instabilities are ignored here because they have been shown to have a negligible effect [8]. This value of the aperture angle is expected to optimize image resolution, based on scaling of the optimum aperture angle determined at 10 eV in Ref. [8]. The source extension that is chosen actually gives the best agreement between experimental observations and calculated image contrast here. This value is consistent with the value reported in Ref. [8]. The relationship between defocus current and focal length is determined empirically to be 5.2370.08 μm/mA. This calibration is made by moving the sample toward and away from the objective lens by several accurately measured distances over a 0.4 mm range and refocusing. All of the parameters mentioned here refer to the virtual object with magnification 1, except for the defocus length/current, which refers to the real object. The relation between real and virtual object parameters is described in the appendix of Ref. [8]. The smearing effect of the electron detector placed at the image plane is considered by convoluting the calculated intensity with Gaussian function with FWHM of 60 nm. Calculated contrast is shown in Fig. 7 for phase shifts of ϕ¼ (2nþ0.5)π, (2nþ1)π and (2nþ1.5)π, where n is integer, at large overfocus and under-focus, 72.7 mA. No contrast reversal between overfocus and under-focus is seen for the out-of-phase condition, ϕ¼ (2nþ1)π. While asymmetric contrast reversal is observed for the other two cases, the calculated contrast for ϕ¼(2nþ1.5)π more closely resembles the experimental observations (Fig. 3). We find by trial-and-error the greatest similarity between calculated and experimentally observed image contrast for ϕ¼(2nþ 1.75)π, corresponding to possible values of Δh¼1.13, 1.29, 1.45, 1.62, 1.78, 1.95, and 2.11 nm within the estimated range, corresponding to integer values 6≤n≤12. Phase retrieval methods could be used to reach a similar conclusion directly [10]. The results of the calculation shown in Fig. 8 for this value of ϕ faithfully reproduce many salient features of the duplex contrast, particularly the aspects of asymmetric contrast reversal between under-focus and over-focus conditions noted in Section 2, contrast enhancement and evolution of duplex contrast features with increasing defocus, and variation of stripe width with varying defocus. The evolution of duplex fringe-like features with increasing defocus can be understood to have its origin in the phase nature of contrast and the difference between α-MnAs and β-MnAs stripe widths. In real instruments, phase contrast at atomic steps typically appears as one dark fringe bounded symmetrically by two weaker bright fringes in the ideal out-of-phase conditions,

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position (nm) Fig. 7. Intensity profiles determined from the Fourier optics calculation for phase shift (2nþ 0.5)π, (2nþ 1)π, (2nþ 1.5)π, n¼ integer, are shown at (a) −2.7 mA underfocus, and (b) þ 2.7 mA over-focus.

ϕ¼ (2n þ1)π, or by one dark and one bright fringe at intermediate phase conditions that differ sufficiently from the out of phase condition, e.g. ϕ ¼(2n þ0.5)π and (2n þ1.5)π [6,8,9,11]. Analogous fringes are produced by the multiple height steps in the square wave model considered here (Fig. 6). These fringes appear to be broader and more intense than for single atomic steps. They are either sufficiently separated laterally for the wider α-MnAs stripes to be resolved, or they tend to overlap, particularly in overfocus, for the narrower β-MnAs stripes. In profiles calculated for both under- and over-focus conditions (Fig. 8), the fringe overlap for α-MnAs stripes is minimal, and the intensity in the middle of the stripe approaches the average intensity. On the contrary, the fringe overlap for β-MnAs tends to suppress the intensity at the middle of the stripe for large under-focus and practically smears out the fringes produced at the two edges of the stripe at large over-focus. Very similar effects are observed experimentally (Fig. 3). The asymmetry between under- and over-focus is believed to be related to the fact that the amplitudes of the bright and dark fringes produced by an atomic step for intermediate phase condition are not equal [8,11]. The stripe widths in simulated images change with defocus in a way that is qualitatively similar to the experimental observations. However, the variation of stripe width with defocus in simulated images, 9.7 nm/mA, is somewhat weaker than the variation that is determined experimentally, 13.7 nm/mA. The origin of this difference is not fully understood, but may be due to a combination of experimental uncertainties and inaccuracy of the idealized “square-wave” model of morphology considered here (Fig. 6). Further refinement of the height modulation model may bring even better agreement between simulated and experimental images. Scanning probe microscopy measurements made using bare tapered optical fiber tips of unknown radius indicated flat topped α-stripes separated by sloping edges to cusp-like, not flat, β-stripe regions [14]. A model of height modulation proposed in Ref. 14 based on elastic considerations [13] indicated cusp-like β-stripe regions and α-stripe regions with rounded caps. We performed image simulations for this model and found that it produced phase contrast that looked nothing like the duplex contrast observed experimentally. The observation that duplex

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400 nm Fig. 8. Intensity profiles determined from the Fourier optics calculation for phase shift (2nþ 1.75)π, n ¼integer, are shown at different defocus conditions (same as Fig. 3) (a) under-focus, and (b) over-focus. The calculated images are shown at different defocus conditions (same as Fig. 1) (c) under-focus: −0.3 mA, −0.9 mA, −1.5 mA, −2.1 mA, and −2.7 mA from left to the right, and (d) over-focus: þ 0.3 mA, þ0.9 mA, þ 1.5 mA, þ2.1 mA, and þ 2.7 mA from left to the right.

contrast fringes with similar appearance are present on both α and β stripes in mild over- and under-focus conditions indicates some symmetry of the shape of the α and β stripe regions. The accurate predictions of these duplex contrast features that are made using the simple model in Fig. 6 strongly suggests that α and β stripes are both flat, hence contradicting the earlier predictions in Ref. 14. Viable modifications of the model that could be considered are to replace the sharp vertical steps between flat α and β stripes with sloping transitions or by slightly rounding the sharp corners at the top and bottom of the vertical steps. These modifications might have an effect on the variation of stripe width with defocus. We also speculate that they could reduce the amplitudes of the duplex fringes in simulated images, which are generally larger than observed experimentally. Another possible development of the modeling would be to consider possible field distortions in the vicinities of the sharp steps that could affect electron trajectories and phase contrast formation. Nevertheless, despite room for improvement, the current modeling approach is already a very good start on understanding contrast produced by the stripe morphology of MnAs films. Although the analysis has narrowed down the ridge-groove height modulation, Δh, to a few possible discrete values, it is still uncertain which height among this is correct. This ambiguity in

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principle can be eliminated by examining contrast at several imaging energies. Each of the possible discrete heights Δh identified here should give rise to unique energy dependence in contrast. Determination of two or three energies that phase contrast is either absent, i.e. ϕ ¼2nπ, or that contrast reversal is not observed between under-focus and over-focus, ϕ¼(2n þ 1)π, could settle this issue. On the other hand the similar appearance of all stripes under identical imaging conditions is a good indication that the height modulation is quite uniform over the image field of view.

similar ridge-groove morphologies in other systems, in particular if similar focus dependent contrast behavior is observed.

4. Conclusions

References

Fourier optics has been used to calculate LEEM image contrast for a strained MnAs film that self-organizes into hexagonal α-phase and orthorhombic β-phase stripes with ridge-groove morphology. Calculated phase contrast for a simple “square-wave” model of the surface morphology reproduces all key aspects of the experimentally observed contrast, including absence of contrast in-focus, enhancement of contrast with increasing defocus, asymmetric contrast reversal between over-focus and under-focus conditions, and contrast fine structure that appears as broad interference fringes and that we call duplex contrast. These observations resemble all of the hallmarks of LEEM atomic step phase contrast, except for the asymmetry of contrast reversal. The asymmetry stems from inequality of the widths of α and β phase regions. The intrinsic widths of the α and β phase regions are determined accurately from the through focus series. While the observations made here at one imaging energy do not allow a firm determination of the ridge-groove height, it should be possible to determine the height modulation from a broader data set including at least one more imaging energy. From a very general practical view, the experimental observations and theoretical understanding reported here can be useful for identifying the presence of

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Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 11104098), and by the Anhui Provincial Foundation of Higher Education Institutions (Nos. 2012SQRL080, KJ2013A230 and KJ2012B167).