Nanoscale three-dimensional reconstruction of electric and magnetic stray fields around nanowires

Nanoscale three-dimensional reconstruction of electric and magnetic stray fieldsaround nanowiresA. Lubk, D. Wolf, P. Simon, C. Wang, S. Sturm, and C. Felser Citation: Applied Physics Letters 105, 173110 (2014); doi: 10.1063/1.4900826 View online: http://dx.doi.org/10.1063/1.4900826 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nanoscale three-dimensional reconstruction of elastic and inelastic mean free path lengths by electronholographic tomography Appl. Phys. Lett. 105, 173101 (2014); 10.1063/1.4900406 Electron holographic tomography for mapping the three-dimensional distribution of electrostatic potential in III-Vsemiconductor nanowires Appl. Phys. Lett. 98, 264103 (2011); 10.1063/1.3604793 Surface magnetization processes in soft magnetic nanowires J. Appl. Phys. 107, 09E315 (2010); 10.1063/1.3360209 Manipulation of magnetism by electrical field in a real recording system Appl. Phys. Lett. 96, 012506 (2010); 10.1063/1.3276553 Influence of three-dimensional transition elements on magnetic and structural phase transitions of Ni-Mn-Gaalloys J. Appl. Phys. 95, 1740 (2004); 10.1063/1.1641184

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Nanoscale three-dimensional reconstruction of electric and magnetic strayfields around nanowires

A. Lubk,1 D. Wolf,1 P. Simon,2 C. Wang,2 S. Sturm,1 and C. Felser2

1Triebenberg Laboratory, Institute for Structure Physics, Technische Universit€at Dresden, 01062 Dresden,Germany2Max Planck Institut for Chemical Physics of Solids, N€othnitzer Str. 40, 01187 Dresden, Germany

(Received 14 August 2014; accepted 17 October 2014; published online 30 October 2014)

Static electromagnetic stray fields around nanowires (NWs) are characteristic for a number of

important physical effects such as field emission or magnetic force microscopy. Consequently, an

accurate characterization of these fields is of high interest and electron holographic tomography

(EHT) is unique in providing tomographic 3D reconstructions at nm spatial resolution. However,

several limitations of the experimental setup and the specimen itself are influencing EHT. Here, we

show how a deliberate restriction of the tomographic reconstruction to the exterior of the NWs can

be used to mitigate these limitations facilitating a quantitative 3D tomographic reconstruction of

static electromagnetic stray fields at the nanoscale. As an example, we reconstruct the electrostatic

stray field around a GaAs-AlGaAs core shell NW and the magnetic stray field around a Co2FeGa

Heusler compound NW. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4900826]

Static electric and magnetic stray fields around nano-

structures, such as the tip of a nanowire (NW), are crucial for

various modern nanotechnological applications such as field

emission guns,1 quantitative magnetic force microscopy,2,3

and magnetic memories.4,5 Among the multitude electromag-

netic field probes currently available, electron holography

(EH) in the transmission electron microscope (TEM) is

unique in its capability to map projected electromagnetic

potentials at nm spatial resolution across fields of view

extending over several 100 nm.6,7 This has been demonstrated

through a number of EH reconstructions of electric8–10 and

magnetic stray fields,11–15 e.g., providing valuable insight

into the field emission process. A complete understanding

requires analyzing the 3D field distributions instead of 2D

projections, which may be obtained by a combination of EH

with tomography referred to as EHT.16–21 However, the pres-

ence of dynamic scattering, electron beam induced sample

charging, and incomplete tilt ranges unfortunately impede

current reconstructions, in particular those of typically weak

stray fields in the exterior region of a nanostructure.

We will show below that restricting the tomographic

reconstruction, mathematically referred to as inverse Radon

transformation, to the region outside of the nanostructure can

circumvent some of these issues. This approach is based on

the fundamental support theorem of the Radon transforma-

tion,22 which states that the so-called outer Radon problem

(ORP), i.e., the reconstruction of a scalar function in the

exterior of a convex domain from projections along subma-

nifold passing outside that domain, has a unique solution.

Cast into the specific circumstances of EHT that means that

a phase tilt series from the exterior of an object is sufficient

to reconstruct the corresponding potential in that outer

region. The advantages gained by solving the ORP are pur-

chased by disregarding any information from the blocked

region, which is usually of high interest as it contains the

object. However, the laws of electro-(magneto)statics relate

the outside potentials to object charges and dipoles,23,24

facilitating an object characterization through its stray fields.

The topic of this article is the solution of the ORP,

including its technical implementation and benefits for the

determination of physical quantities within the framework of

EHT. We begin with a short introduction to the latter with a

focus on current issues of the technique, which are poten-

tially solvable by the ORP. Next, we show, how to imple-

ment the ORP into a tomographic reconstruction scheme.

In the last section, we analyze results of the reconstruction

of a weak electrostatic fringing field around a GaAs –

Al0.33Ga0.67As core-shell NW slightly charged in the elec-

tron beam and a magnetostatic stray field around a ferromag-

netic Co2FeGa Heusler compound NW. Both NWs represent

a particular application, the former is prospected for photo-

detection and solar cells25,26 while the latter could be used

for spin injection27 and spin valves.28 Moreover, both NWs

have been analyzed by standard tomographic reconstructions

previously20,29 facilitating a detailed comparison to the ORP

reconstruction.

EHT of electric and magnetic fields is based on invert-

ing the following two linear projection laws for the holo-

graphically reconstructed phase, largely valid for our

medium resolution investigations.30–32 One distinguishes

between the electric phase shift uel proportional to the pro-

jected electrostatic potential V (with the constant

CE¼ 6.5 mV/nm at 300 kV acceleration voltage)

uelðx; p; hÞ ¼ CE

ðlðp;hÞ

VðrÞds

!mod 2p (1)

and the spatial derivative of the magnetic phase shift propor-

tional to the magnetic field

@umag x; p; hð Þ@p

¼ � e

�h

ðl p;hð Þ

Bx rð Þds

!mod 2p : (2)

Here, r¼ (x, y, z)T is the spatial coordinate and the 2D detec-

tor with coordinates (x, p)T is aligned such that its first coor-

dinate coincides with x for each tilt angle h (Fig. 1).

0003-6951/2014/105(17)/173110/5/$30.00 VC 2014 AIP Publishing LLC105, 173110-1

APPLIED PHYSICS LETTERS 105, 173110 (2014)

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Provided the mod 2p phase wrapping can be inverted (by

phase unwrapping algorithms33), the integration in (1,2) is

then performed along lines l confined to planes perpendicular

to ex and (1,2) can be inverted slice wise by standard 2D to-

mographic techniques. That is, for each (y, z) plane (1,2) pos-

sess a well defined inverse provided certain prerequisites

specified by the support theorem are fulfilled.22 Note that

only the magnetic field component parallel to the tilt axis

can be reconstructed by inverting (2), which implies that fur-

ther tilt series are required for the other components.15 Until

now, only the standard (interior) Radon problem, i.e., the

reconstruction of mostly electrostatic potentials within a

compact domain (holographic field of view), has been

treated. Following the proof of concept by Lai et al.,16 3D

reconstructions of, e.g., built-in potentials across pn-junc-

tions,21,34 mean inner potential (MIP) of core-shell

NWs,20,29 or semiconductor interfaces,35 have been reported.

The EHT reconstruction of weak static magnetic fields, typi-

cally producing comparatively small phase shifts, is still

challenging with only a few semi-quantitative results

reported so far.17,36,37

Both, the acquisition of a holographic tilt series at the

TEM as well as the ensuing numerical processing including

holographic and tomographic reconstruction have been pro-

ven cumbersome with a list of issues affecting the quality of

the reconstruction.21 In the following, we will only list those

which potentially benefit from solving the ORP: (A) Due to

geometrical limitations in the TEM, tilt angles are often con-

fined to within approximately 670� (instead of 690�), ren-

dering spatial frequencies in the corresponding “missing

wedge” in Fourier space inaccessible. This double wedge fil-

ter can lead to serious artifacts in the reconstruction hamper-

ing quantitative interpretation and segmentation.38 (B) Phase

unwrapping algorithms cannot distinguish between unre-

solved phase gradients larger then p and phase jumps, pro-

ducing artifacts in that case.33 Such gradients can, however,

easily occur at object boundaries typical for tomographic

specimen such as NWs. (C) During tilting the proximity of a

low-index zone axis might not be avoided (in particular if

the object is polycrystalline). In that case, dynamical scatter-

ing violates the projection laws (1,2) producing artifacts in

the tomographic reconstruction. (D) In order to separate

electric and magnetic phase shifts, two tilt series of the same

object position are required, where the magnetic field is

reversed in between.39 However, reversal by flipping the

sample up-side down or remagnetizing typically creates

problems due to changing scattering conditions, magnetiza-

tion, and specimen charging.15

As stated above, the support theorem also ensures the

reconstruction of a unique potential from a set of line inte-

grals outside a convex region (Fig. 1). It is obvious that all

object-related issues such as dynamical scattering, phase

unwrapping as well as subtraction of two flipped tilt series

do not inflict such a reconstruction. Furthermore, the

“missing wedge” problem can be significantly mitigated by

exploiting the typically small azimuthal bandwidth of the

fringing fields in the tomographic reconstruction.29 Indeed, it

will turn out that solving the ORP facilitates the reconstruc-

tion of small fringing fields, inaccessible by the interior

reconstruction that prolong artifacts from the inner to the

outer region because of the non-local property of the 2D

Radon transformation.47

We will considerably shorten the discussion of the meth-

odological aspects by noting that our solution of the ORP

consists of numerically removing the respective projections ltraversing the circular domain containing the NW from the

Radon transformations (1,2). That means acquisition, holo-

graphic reconstruction, phase unwrapping (with less care on

the inner region though), alignment and possible further

steps are the same as in previously reported EHT experi-

ments.29 Nevertheless, we find that hardware realizations of

the ORP by shaping the illumination in such a way to

shadow certain parts of the object (“hollow cone

illumination”40) might be an interesting approach to follow

in order to reduce electron beam induced specimen modifica-

tions (charging, radiation damage).

Our numerical solution to the ORP is a slight modifica-

tion of the typical Algebraic Reconstruction Techniques.41

First, we discretize the Radon transformation to a set of lin-

ear equations, which project a grid representation (circular in

our case) of the underlying potential to pixel-wise projected

potential data reconstructed holographically. The prefactors

in this linear system (¼projection weights) are computed by

means of a dedicated digital difference analyzer.42 Next, we

remove that particular subset of equations, which contain

any entries from the blocked circular region (Fig. 1). In a

final step, we invert the reduced linear system by means of a

conjugate gradient LSQR algorithm.43 The support theorem

noted previously ensures the unique inversion of that

reduced linear equation system. Thanks to the smoothness of

the fringing fields (containing no discontinuities like object

boundaries), the exterior field typically possess a small azi-

muthal band width, which can be favorably exploited in our

algebraic reconstruction by increasing the azimuthal width

of the reconstructed circular pixels.

In order to discuss the benefits of the ORP, we analyze

two examples: We start with the reconstruction of a weak

charging field around a core-shell GaAs–Al0.33Ga0.67As

NW and proceed with the axial component of the magnetic

stray field around a Co2FeGa Heusler compound NW.

In both cases, we compare the solution of the ORP to the

standard interior case as well as matched finite element

FIG. 1. Schematics of tomographic reconstruction. The 3D domain is sliced

perpendicular to the tilt axis x, and the slices are independently recon-

structed. In each slice, the convex support of the NW is excluded from the

projections. The support theorem ensures the reconstruction of the outer field

(green color).

173110-2 Lubk et al. Appl. Phys. Lett. 105, 173110 (2014)

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electro-/magnetostatic models. The GaAs–Al0.33Ga0.67As

NW has been grown by metal-organic vapor phase epitaxy

(MOVPE) using colloidal Au nanoparticles as metal catalysts

(see Refs. 20 and 29 for further details). The amount of beam

induced charging of the specimen, present in the TEM, is

determined by the conductivity and the secondary electron

yield of the sample. It typically represents an unintended

artifact, affecting in particular all sorts of holographic

investigations including EHT.29 In case of the

GaAs–Al0.33Ga0.67As NW, only a weak, nearly radially sym-

metric, projected charging potential could be observed out-

side the NW (Fig. 2(a)), which is furthermore artificially

modulated during the tilt series acquisition by various effects.

The non-continuous variations in the sinogram (Fig. 2(a))

indicate a varying amount of charging as well as varying

phase wedges. The latter effect could be caused by perturbed

reference waves tilted by the weak stray fields or the numeri-

cal procedure determining the holographic sideband center.44

The standard (interior) reconstruction of the stray field

largely fails (Fig. 2(b)), mainly due to non-local artifacts

from the missing wedge: One observes the typical anisotropic

loss of resolution in wedge direction (artifact A in Fig. 2(b))

as well as sharp streaks (artifact B in Fig. 2(b)) originating

from the missing wedge convolution kernel interacting with

sharp object boundaries. For the reconstruction of the poten-

tial inside the NW we refer to Ref. 29. As discussed above,

the ORP solution does not suffer from object features leaking

erroneously into vacuum. However, the missing wedge still

introduces artificial modulations in azimuthal direction, sur-

mounting those of the almost radially symmetric charging

field. Consequently, we restrict the ORP reconstruction to the

radial symmetric part in the following (Fig. 2(c)). A compari-

son to a finite element electrostatic simulation (Fig. 2(d), see

Ref. 45 for details) shows good agreement with the stray field

of a homogeneous positively charged “nanobottle” with an

increased charge at the Au tip (qNW¼ 3.8 � 1015 cm�3,

qAu¼ 1.3 � 1017 cm�3). The latter can be explained by the

higher yield of expulsed secondary electrons at the Au tip.

The remaining differences to that model are due a violation

of the model’s homogeneous charge density assumption as

well as the tilt series inconsistencies noted above.

The Co2FeGa Heusler compound NW was synthesized

by using of mesoporous SBA-15 silica as structure-directing

template. Methanol dispersion of iron, cobalt, and gallium

salts was added to the SBA. After removal of methanol, the

solid was annealed at 850 �C for 2 h under H2 atmosphere.

Due to the large shape anisotropy, the Heusler compound

NW possesses a large remanent magnetization close to the

saturation magnetization of Ms¼ 1016 kA/m.46 Additionally,

no significant amount of beam charging could be detected

here due to the good conductivity of the Heusler NW (see

Ref. 45 for details). Therefore, the magnetic stray field of the

remanent magnetization is determining the total phase shift

outside of the NW (Fig. 3(a)) and only one tilt series (instead

of two for the interior magnetic field) proved sufficient for a

reliable (and noise reduced) 3D reconstruction. Similar to

the electrostatic example, the standard (interior) reconstruc-

tion of the stray field fails because of non-local reconstruc-

tion artifacts (Fig. 3(b)). We furthermore note a strongly

reduced SNR typical for magnetic field reconstructions due

to the numerical derivative in the projection law (2). Again,

the reconstruction of the magnetic field inside the NW will

be considered elsewhere. The ORP reconstruction (again re-

stricted to the radial symmetric component) is largely free

from artifacts displaying a weak but continuous stray field in

the order of several tens of mT in close vicinity of the NW

tip (Fig. 3(c)). Note in particular the small part of the Bx field

being pushed out from the tapered region of the NW (Fig.

3(c)) as a consequence of the vanishing divergence of B. The

reconstructed Bx field compares well to a finite element mag-

netostatic model of a homogeneously magnetized NW

(Ms¼ 1016 kA/m) incorporating a non-magnetized layer of

10 nm at the surface deduced from the comparison of electric

and magnetic phase shifts (Fig. 3(d), see Ref. 45 for details).

The remaining differences to that model consist mainly of the

missing negative field outside of the thick part of the NW,

which may be explained by the holographic sideband center-

ing procedure removing such constant phase wedges.

We have demonstrated how a restriction of the tomo-

graphic reconstruction to a region outside of an object can be

beneficially used to reconstruct static electromagnetic stray

FIG. 2. GaAs–Al0.33Ga0.67As NW charging field characterization: (a) holo-

graphically reconstructed projected potential (artifacts indicated), (b) stand-

ard (interior) tomographic reconstruction, (c) ORP reconstruction, and (d)

electrostatic finite element model. The indicated x-interval (¼150 nm) in (a)

is used for computing the averaged sections shown in the second column.

The small insets in (b) show the position of the cross-sections.

173110-3 Lubk et al. Appl. Phys. Lett. 105, 173110 (2014)

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fields by means of EHT. The solution of the so-called outer

Radon problem is largely free from typical EHT problems

such as dynamical scattering, phase unwrapping, and limited

tilt range. That facilitates the tomographic reconstruction of

very weak stray fields, which are buried under reconstruction

artifacts emerging from the above noted issues otherwise. As

an example, we reconstructed a weak beam induced charging

potential around a GaAs – Al0.33Ga0.67As core-shell NW

(order of magnitude: 0.1 V) and a magnetic stray field

emerging from the tip of a homogeneously magnetized

Co2FeGa Heusler compound NW (order of magnitude:

10 mT). The technique can be most favorably applied to

problems where the 3D stray field distribution is the crucial

quantity, e.g., for the characterization of NWs used for field

emission as well as magnetic tips used in magnetic force mi-

croscopy and magnetic read-and-write heads.

We are grateful to P. Prete and N. Lovergine for

providing the GaAs–Al0.33Ga0.67As core-shell NW. A.L. and

D.W. acknowledge financial support from the European

Union under the Seventh Framework Programme under a

contract for an Integrated Infrastructure Initiative, Reference

312483-ESTEEM2. S.S. was funded by the European Union

(ERDF) and the Free State of Saxony via the ESF project

100087859 ENano.

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FIG. 3. Co2FeGa Heusler compound

NW axial magnetic field (Bx) charac-

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