a thorough understanding 
 of the connection between the structure 
 and the magnetic properties of nanostructures requires characterization 
 techniques that permit direct measurements of magnetic ordering at 
 the submicron scale . therefore , together with the high control achieved 
 for the growth of thin films during the last decades , there has been 
 an increasing development of magnetic imaging techniques for two - dimensional 
 ferromagnetic nanostructures . along with the decrease of the computation 
 time of micromagnetic simulations , these techniques have significantly 
 contributed to deeper insights into nanoscale magnetic phenomena , 
 thereby boosting the development of novel technological applications 
 in the field of nanomagnetism and spintronics . recently , a new possible route has been proposed 
 in this area , based on the use of 3d magnetic nanostructures for their 
 exploitation in data storage and logic . moving toward spintronic architectures which go beyond 
 a planar configuration not only requires nonstandard lithography techniques 
 but also demands advanced imaging techniques that can resolve the 
 magnetism of nanostructures in 3d . among various magnetic imaging 
 techniques , such as kerr microscopy , 
 spin - polarized low - energy electron microscopy ( spleem ) , x - ray magnetic 
 circular dichroism photoelectron emission microscopy ( xmcd - peem ) , 
 magnetic force microscopy ( mfm ) , and spin - polarized scanning tunneling 
 microscopy ( sp - stm ) , off - axis electron holography ( eh ) is one of the 
 most powerful methods to obtain quantitative measurements of local 
 magnetization inside and around magnetic nanomaterials with nanometric 
 resolution . eh is a transmission electron microscopy 
 ( tem ) technique that utilizes 
 the interference of a split electron wave to measure the phase shift 
 of one of the partial waves when interacting with an electrostatic 
 and/or a magnetic field . aharonov and bohm have demonstrated that this phase shift between object wave and 
 unperturbed reference wave can be expressed by the phase grating approximation 
 ( pga ) , i.e.1where ce is an interaction constant 
 depending on the electron beam 
 energy , v(x , y , z ) is the three - dimensional electrostatic object potential , e is the elementary charge ,  is the reduced planck 
 constant , and az(x , y , z ) is the component 
 of the magnetic vector potential parallel to the electron beam direction z. accordingly , the first term can be considered as electric 
 phase shift el and the second one 
 as magnetic phase shift mag . consequently , 
 the projections of the lateral components of the magnetic induction b = ( bx , by , bz ) may be obtained by differentiating mag according to2and3 owing to these relations eh can directly visualize the projected 
 magnetic induction bx , y in and around nanostructures , which was successfully 
 exploited to investigate magnetic configurations in magnetotactic 
 bacteria , fe nanocubes , ni nanowires , and other types 
 of complex nanostructures such as magnetic tunnel junctions . however , a single eh experiment only permits 
 recovering the magnetic field components integrated along the beam 
 path through the sample . in order to 
 obtain information about the 3d magnetic structure from such a 2d 
 projection , additional input , e.g. , from micromagnetic simulations , combining eh with electron tomography to electron 
 holographic tomography 
 ( eht ) can overcome this problem . eht starts with acquiring a tilt 
 series of electron holograms from the same nanostructure , which is 
 subsequently reconstructed holographically to obtain a tilt series 
 of phase images , that is , projected electric or magnetic potentials . following up alignment , i.e. correction of image displacements with 
 respect to a common tilt axis , this series of projected potentials 
 can be used as input for a tomographic reconstruction . eht has been 
 successfully applied for the 3d mapping of electrostatic potentials 
 within nanostructures . similar approaches on the 3d reconstruction of magnetic 
 fields outlined and demonstrated its foundations for both off - axis 
 holography and in - line holography . very recently , a 3d observation of magnetic vortex cores in stacked 
 ferromagnetic discs using eht was reported . the work describes in a condensed manner how existing programs such 
 as the imod package are used to reconstruct 
 the 3d magnetic induction map . however , the 3d reconstruction of the 
 corresponding electric potential mandatory to clearly correlate the 
 material composition of the stacked sample with its magnetic configuration 
 is not presented . indeed , eht is a very challenging technique : 
 there are stringent 
 requirements for the tem specimen ; the object of interest should be 
 completely visible in every projection without being shadowed by the 
 holder during the tilt series . moreover , the sample should not provide 
 too strong diffraction contrast violating the projection law , eq 1 . furthermore , the experiment 
 requires a stable environment throughout the acquisition of the tilt 
 series , which may last several hours . the 3d reconstruction 
 of magnetic vector fields 
 adds another layer of complexity in comparison to the case of electric scalar potentials . the holographic 
 acquisition in the tem must be performed under field free conditions , 
 i.e. zero external magnetic fields , which can be achieved by switching 
 off the objective lens or placing the sample above the objective lens . a pair of phase images , only differing in a reversion of the magnetic 
 induction direction , is needed for each tilt angle in order to separate 
 electric and magnetic phase shifts in eq 1 , by computing half of their sum and difference , respectively . in addition , one tilt series only allows for 
 reconstructing a single - axis component of b according 
 to eqs 2 and 3 . therefore , at least four tilt series ( around two 
 mutually inclined axes ) are required to obtain the components of the 
 magnetic induction along two orthogonal axes . the third component 
 may be obtained by exploiting that the magnetic induction is divergence 
 free . furthermore , magnetic phase shifts are typically small , which , 
 in combination with the high - frequency noise amplification introduced 
 by the numeric derivative in eqs 2 and 3 and the tomographic reconstruction , 
 lead to reduced signal - to - noise ratios of the resulting tomograms . in the following , we demonstrate how to retrieve 
 quantitatively 
 the 3d magnetic ordering within a ferromagnetic nanostructure by means 
 of eht . our test case is a 100 nm thick and a few microns long cobalt 
 nanowire ( nw ) grown by focused electron beam induced deposition ( febid ) 
 in a focused ion beam ( fib)/scanning electron microscopy ( sem ) dual 
 beam instrument on top of a copper tip of a dedicated 360 tomography 
 sample holder . micromagnetic simulations validate our results and 
 provide the b - field containing all three components . the principle 
 of eht , illustrated in figure 1 , comprises the acquisition of a tilt series 
 of electron holograms , the reconstruction of phase images from each 
 hologram , and the separation of electric and magnetic phase shifts , 
 as well as the final tomographic 3d reconstruction of both electric 
 potential and magnetic field . in order to perform the crucial separation 
 of electric and magnetic phase shifts , we acquire a tilt series over 
 360 ( instead of 180 sufficient for the electric potential ) 
 using a dedicated tomography tem sample holder . the acquisition has 
 been performed semiautomatically with thomas , an in - house developed software package dedicated for efficient acquisition 
 of holographic tilt series . amplitude and phase images were reconstructed 
 from the electron holograms by fourier techniques incorporating empty 
 holograms for correction of imaging artifacts , such as distortions 
 of camera and projective lenses . we then 
 identified pairs consisting of a phase image from the first 180 
 interval and its 180 tilted counterpart , to separate the electric 
 and magnetic phase shift . prior to the separation , the phase images 
 were aligned by affine image registration methods ( see the methods 
 section ) in order to minimize artifacts in the magnetic phase shifts , 
 especially toward the boundaries of the nw . the average of these phase 
 image pairs represents the electric phase shift , whereas half of their 
 difference represents the magnetic one . artifacts in the magnetic 
 phase images may also occur , if long - ranging magnetic stray fields 
 perturb the reference wave during the hologram acquisition . however , 
 we have verified that the phase of the reference wave is virtually 
 flat in our case ( see the supporting information for details ) . both , electric 
 and magnetic contributions are displayed as gray values ( figure 2a ) , and the magnetic 
 part is visualized additionally as phase isolines ( figure 2b ) . as illustrated by line 
 profiles ( figure 2c , d ) , 
 the electric phase shift ( black ) is proportional to the projected 
 mean inner potential ( mip ) , whereas the magnetic 
 phase shift ( red ) shows a negative slope due to the magnetic flux 
 within the nw and a positive slope due to the returning magnetic flux 
 in vacuum ( stray field ) . in vacuum , also the electric phase shift 
 exhibits a gradient implying an electrical fringing field which is 
 caused by charging of the nw , in particular its oxide surface by the 
 electron beam . however , the constant phase gradient within the whole 
 tilt series indicates that the charging was constant at the entire 
 duration of the experiment . therefore , artifacts in magnetic phase 
 shift due to varying electrical fringing fields can be excluded . most 
 interestingly , in the magnetic phase images , there is a region at 
 the apex of the nw ( marked with a dashed circle in the profile figure 2d at 27 
 tilt angle ) , where the slope direction changes within the sample . this slope inversion is also visible in other projections , which suggests 
 a change in the magnetization direction at the apex of the nw . however , 
 the profiles also exhibit artifacts at the edges of the sample , e.g. 
 at positions marked with dashed arrows in the profiles figure 2c , d at + 33 tilt angle . these may be attributed to misalignment errors , lens aberrations , 
 and slight contaminations at the surface , introducing phase variations 
 of neither magnetic nor electrostatic origin . consequently , the reconstructed 
 3d distribution of the magnetic field can not be interpreted in these 
 regions as discussed further below . principle of electron holographic tomography 
 for the 3d reconstruction 
 of internal electro - magnetic fields in nanostructures . a tilt series 
 of holograms covering a tilt range of 360 is recorded . from 
 the holograms , the phase image tilt series is reconstructed and separated 
 in its electric / magnetic part by calculating half of the sum / difference 
 between opposite ( 180 tilted ) projections . to obtain the 3d 
 electric potential the tilt series of electric phase images is used 
 for tomographic reconstruction . to obtain the 3d axial magnetic field 
 component the magnetic phase images need to be differentiated in direction 
 perpendicular to the tilt axis prior the tomographic reconstruction . following eq 2 , the 
 projections of the axial component by are now obtained by a directional numerical differentiation 
 of the magnetic phase images perpendicular to the tilt axis . in our 
 experiment , tilt axis and longitudinal axis of the nw are almost parallel , 
 comprising only a small angle of 3. therefore , longitudinal b component and its projection onto the tilt axis differ 
 only slightly by ca . 0.1% and can be considered identical when discussing 
 the results of the ensuing tomographic reconstruction . the latter 
 was carried out using the weighted simultaneous iterative reconstruction 
 technique ( w - sirt ) with 5 iterations . the w - sirt developed in - house uses at each sirt iteration a weighted 
 back - projection instead of a simple back - projection used in conventional 
 sirt . ( a)/(b ) electric / magnetic phase images taken at 87 , 
 27 , and + 33 tilt angle . ( c)/(d ) line profiles 
 along the arrows marked in ( a ) and ( b ) representing the electric ( black ) 
 and magnetic ( red ) phase shift . the circle in ( d ) at 27 
 indicates the change of the phase gradient due to the change of magnetization 
 direction in a small domain at the tip . the arrows in ( c , d ) at + 33 
 indicate artifacts in the magnetic phase shift at the edge of the 
 sample . , the number of iterations was determined by visually 
 inspecting the spatial resolution and noise of the reconstruction 
 for an optimal balance . the spatial resolution of the 3d reconstruction 
 is below 10 nm , as we demonstrate by determination of the width of 
 the edge - spread function in six different directions on a representative 
 cross - section of the co nw ( see the supporting 
 information ) . the 3d distributions of the electric potential 
 and the axial magnetic induction of the co nw reconstructed by eht 
 are shown in figure 3 . here , we analyze only the b - field inside the sample where the corresponding 
 electric potential is v  16 v , because the edge regions are affected by artifacts as mentioned 
 above . a tomographic reconstruction of the stray field only is shown 
 in the supporting information . selected 
 cross sections ( figure 3b ) at positions indicated by orange boxes in the volume 
 visualization ( figure 3a ) reveal that the intrinsic electric potential is virtually homogeneous 
 along the axial direction suggesting a homogeneous chemical composition . in contrast to that , the magnetic field exhibits a domain structure 
 consisting of one large domain ( blue , by = 0.9 t ) and a small domain ( red , by = 0.3 t ) at the apex 
 with the magnetization pointing in opposite direction . these 
 findings are supported by the histograms of 3d volumes and 
 1d line profiles ( figure 3c ) along the arrows marked in figure 3b . the most frequent electric potential value 
 in the histogram is found at 21.5 v , which is lower than the theoretical 
 mip value of pure co ( 29.5 v ) calculated 
 using neutral atom potentials of rez et al . electric 
 potential and axial magnetic b - field component of a co 
 nanowire . ( a ) volume rendering , i.e. the colors correspond to the 
 potential / b - field values . ( b ) 15 nm thick 2d slices through the 3d 
 data as indicated by the orange boxes in ( a ) . ( c ) histograms of 3d 
 volumes and 1d line profiles along the arrows marked in ( b ) . the peak 
 in the histogram of the electric potential at 21.5 v can be interpreted 
 as the mean inner potential of this co nw . the most frequent value 
 in the histogram of the magnetic induction is at 0.9 t. the observed difference is mainly 
 caused by both the mass density 
 and the co purity of the nanocrystalline wire produced by febid being 
 lower compared to bulk material , hence reducing the mean inner potential . the co purity was determined by stem - edx ( see the supporting information for details ) and a grain size of 36 
 nm in diameter was revealed by tem analysis . furthermore , the theoretical crystal potential may be overestimated 
 when using independent neutral atom potentials without taking into 
 account chemical bonding . similarly , the average magnetic induction 
 along the wire axis of 
 0.9 t is lower than the value expected for pure bulk cobalt ( 1.76 
 t ) . this reduction of magnetization has 
 been observed in electron holography experiments for similar fe nanowires 
 grown by febid and is mainly caused by 
 low purity of the material forming the nw , as well as demagnetization 
 fields associated with the specimen geometry reducing the magnetization 
 inside the sample . another possible source of error may be the fact 
 that we determine the average magnetic induction only from the axial 
 b - field component instead of from the magnitude of the entire vector 
 field |b| . however , the simulation results discussed below suggest 
 that the contributions of the other two b - field components to the 
 histogram are indeed small ( figure 5e ) , i.e. the peak positions of the by and -|b| histogram 
 differ only by 10 mt . furthermore , the electric potential decreases 
 from the center ( red ) to the surface ( green ) in radial direction , 
 whereas the magnetic induction remains constant . the reduced potential 
 can be explained by oxidation of the surface as confirmed by stem - edx 
 ( see the supporting information ) and the 
 reduced mass density as confirmed by tomographic reconstruction of 
 mass thickness ( see the supporting information ) . according to the line profile in figure 3c , there is a drop in the electric potential 
 at the position where the magnetic induction changes its direction . a lower electric potential may indicate a structural modulation of 
 material resulting in a local reduction of the magnetization at that 
 area . a more detailed explanation of such an electromagnetic coupling is 
 however beyond the scope of this article , and we restrict ourselves 
 to verify our results by micromagnetic simulations in the following . g - l ) b - field in axial ( y- ) direction . in the volume 
 rendering of experimental ( a ) and simulated ( g ) magnetic induction 
 maps , the z - positions of the slices are indicated as ( b - f ) and ( h - l ) , 
 respectively . given the reconstructed 
 3d distribution of the axial component 
 of the magnetic induction we may now elucidate the 3d distribution 
 of all three components and the origin of the inverse apex domain 
 with the help of micromagnetic simulations ( see the methods section 
 for details ) . in figure 4 representative slices through the co nw and corresponding simulation 
 are displayed . most of the spins are aligned along the y - direction , forming a monodomain state ( blue region in figure 4 ) along the wire axis , as observed 
 experimentally . furthermore , a closure domain is present at the tip 
 of the wire representing a c state with a 180 domain wall which 
 is in excellent agreement with the experimental result . this remanent 
 magnetization state is common to soft magnetic nanostructures , where the strong demagnetizing field twists 
 the magnetization at their end to avoid the generation of magnetic 
 charges at the surface . the simulations shown here were performed 
 with a saturation magnetization corresponding to the experimentally 
 obtained magnetic induction . for this value , an exchange constant 
 smaller than the one normally used for co bulk was needed in order 
 to get a closure domain at remanence for the given dimensions , whereas 
 simulations using higher exchange values resulted in single monodomain 
 states ( see methods ) . the low exchange value needed to reproduce the 
 experimental results may be due to the low purity of cobalt grown 
 by febid , but it could also be caused by effects not considered in 
 the simulations , such as roughness , presence of defects , local change 
 of composition along the wire , or influence of local magnetocrystalline 
 anisotropy . these can also explain small 
 differences in the structure of the domain between simulation and 
 experiments : for example , in the simulation the inversion domain is 
 formed in the thicker region , whereas in the experiment it is formed 
 in the thinner region of the  egg - shaped  cross - section 
 and is more extended along the edge ( figure 4 d , e and j , k ) . in cylindrical nanowires as 
 the one studied here , curling of the magnetization along the surface 
 is expected in order to minimize the dipolar energy . this is confirmed by the micromagnetic simulation , as observed 
 in figure 5 , where the three b - field components are shown . in particular , figure 5a depicts how the 
 azimuthal direction of the nw is inverted toward the apex of the nw . however , the separate visualization of the three b - field components 
 in figure 5b - d and 
 the histogram ( figure 5e ) show that the overall curling effect is small , with the component 
 along the wire axis by mostly dominant 
 compared with the other two . in summary , we have demonstrated 
 that electron holographic tomography 
 can be used to quantitatively reconstruct the 3d magnetic field configuration 
 within ferromagnetic nanostructures . as an example , we observed in 
 the reconstructed axial b - field component of a co nw a small closure 
 domain at the tip with a spatial resolution below 10 nm and a signal 
 resolution of 0.1 t. in a next step eht has to be extended to reconstruct 
 all components of the magnetic field at the above - noted figure of 
 merits . we envisage the technique to open new pathways to the characterization 
 of nanoscale ferromagnetic elements , such as spin valves , read - write 
 magnetic heads , magnetic sensors , and magnetic logic devices . ( a ) iso - surface with longitudinal slice ( 1 ) representing by and cross sections ( 2 ) and ( 3 ) visualized by arrow 
 plots . in slice ( 1 ) , the out - of - plane components are displayed as 
 vectors . in the cross sections ( 2 ) and ( 3 ) , the twist of magnetic 
 induction is illustrated by the opposite azimuthal direction of the 
 vectors . ( b - d ) volume rendering of the three b - field components with 
 field lines in the x - y plane . ( e ) histograms of b - field components 
 and negative of the total magnitude . the growth was 
 performed in an fei helios 600 dual beam system equipped with a 30 
 kv feg and a ga+ liquid metal source in the same chamber . in this 
 work , the substrate was a cu rod of 1 mm in diameter ending in a nanotip , 
 on which the cobalt pillar was grown . the end of the tip was cut and 
 sharpened using focused ion milling ( figure s1 ) to ensure that the pillar rested on the salient point of the substrate . the parameters used during the growth and more information on the 
 growth process can be found in the supporting 
 information and in refs ( 38 and 39 ) . the cu rod with 
 the co nw was transferred to a dedicated electron tomography holder 
 ( model 2050 , fischione instruments ) , and plasma was cleaned to remove 
 contamination of the nw surface . a tilt series consisting of object 
 and object - free ( empty ) holograms was recorded using the qu - ant - em , 
 a fei titan3 tem operated at 300 kv in non cs - corrected lorentz mode 
 ( objective lens off ) . a mllenstedt biprism operating at 200 
 v was introduced in the sa aperture plane to create the holograms . 
 in order to increase the width of the interference area the tilt series 
 was acquired within the angular range from 0 to 360 in 
 3 steps . the electron holograms were acquired with a us1000xp 
 gatan ccd camera ( binning 1 , 5 s exposure time ) , at a pixel size of 
 0.37 nm and a fringe spacing of 2.2 nm . all image 
 processing steps were performed in gatan inc.s digital micrograph . the resolution of the reconstructed object exit wave is 4.6 nm , determined 
 by the size of the numerical mask used in the fourier reconstruction . the tilt series of phase images was divided into two parts , one below 
 and one above 180 tilt angle . then , both tilt series were prealigned 
 separately by cross - correlation to correct for coarse displacements 
 between successive projections . furthermore , the second tilt series 
 was flipped perpendicular to the tilt axis to convert the 180 
 rotation between corresponding phase images into a mirroring with 
 respect to the image plane . then , each pair of phase images was aligned 
 to each other by applying a linear affine transformation on the  flipped phase images , which considered displacements , rotation , and direction - dependent 
 magnification changes . to perform the exact fine alignment , i.e. , 
 the accurate determination of the tilt axis and correction for subpixel 
 displacement , we have used a self - implemented center - of - mass method 
 for correction of displacements perpendicular to the tilt axis in 
 the sinograms and the common line approach for the displacement correction 
 parallel to the tilt axis . the magnetic phase tilt series was aligned 
 using parameters as determined from the electric phase tilt series , 
 which has a higher signal - to - noise ratio than the magnetic one and 
 therefore yields more accurate parameters . before computation of the 
 derivatives the magnetic phase images were smoothed slightly by a 
 median filter with a 3  3 pixel kernel . a static micromagnetic simulation 
 of the remanent state of the co febid nw was performed by the gpmagnet 
 software package , which solves the landau - lifschitz - gilbert 
 equation . the 3d shape of the nw was adapted from the 
 experimental tomogram and constructed with a maximum length of 1000 
 nm by assembling cubic cells of 3.5  3.5  3.5 nm . the  egg - shaped  cross - section of the nw model was 
 varied from 90 to 110 nm , and the tip is spherical . the micromagnetic 
 calculations were performed for the nw placed inside a volume of 301.0 
  1200.5  220.5 nm . thus , the aspect ratio 
 ( length / diameter 1000/100 = 10 ) is large enough to align the 
 magnetization along the nanowire at remanence via shape anisotropy . 
 the magnetic parameters for the co nw have been chosen from the experimental 
 data : the saturation magnetization ( ms ) was set to 0.72  10 a / m , as estimated from the measured value of the b - field ; 
 the exchange constant a was suitably tuned until 
 a small magnetic domain with a similar size of that experimental one 
 was formed at the tip . we achieved such a condition using a  0.4  10 j / m , smaller 
 than the value for pure co ( 13  10 j / m ) . if we use a  0.4  10 j / m , the closure domain is not formed for the given ms . the magnetocrystalline 
 anisotropy was neglected due to the nanocrystalline morphology of 
 the nw , composed of small grains of about 5 nm size randomly oriented . for this static simulation , an initial magnetization configuration 
 was considered , with all spins oriented along the y - direction . the simulation was iterated using a second order predictor 
 corrector until minimizing the total energy of the system .
the investigation of three - dimensional 
 ( 3d ) ferromagnetic nanoscale 
 materials constitutes one of the key research areas of the current 
 magnetism roadmap and carries great potential to impact areas such 
 as data storage , sensing , and biomagnetism . 
 the properties of such 
 nanostructures are closely connected with their 3d magnetic nanostructure , 
 making their determination highly valuable . up to now , quantitative 
 3d maps providing both the internal magnetic and electric configuration 
 of the same specimen with high spatial resolution are missing . here 
 , 
 we demonstrate the quantitative 3d reconstruction of the dominant 
 axial component of the magnetic induction and electrostatic potential 
 within a cobalt nanowire ( nw ) of 100 nm in diameter with spatial resolution 
 below 10 nm by applying electron holographic tomography . 
 the tomogram 
 was obtained using a dedicated tem sample holder for acquisition , 
 in combination with advanced alignment and tomographic reconstruction 
 routines . 
 the powerful approach presented here is widely applicable 
 to a broad range of 3d magnetic nanostructures and may trigger the 
 progress of novel spintronic nonplanar nanodevices .