# Polarization selectivity of aloof-beam electron energy-loss spectroscopy in
one-dimensional ZnO nanorods

Yao-Wen Yeh<EMAIL_ADDRESS>Department of Physics and Astronomy,
Rutgers University, Piscataway, New Jersey 08854, USA  Sobhit Singh
<EMAIL_ADDRESS>Department of Physics and Astronomy, Rutgers
University, Piscataway, New Jersey 08854, USA  David Vanderbilt  Department of
Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
Philip E. Batson  Department of Physics and Astronomy, Rutgers University,
Piscataway, New Jersey 08854, USA

###### Abstract

Orientation dependent electronic properties of wurtzite zinc oxide nanorods
are characterized by aloof beam electron energy-loss spectroscopy (EELS)
carried out in a scanning transmission electron microscope (STEM). The two key
crystal orientation differentiating transitions specific to the in-plane (13.0
eV) and out-of-plane (11.2 eV) directions with respect to the wurtzite
structure are examined by first principles density-functional theory
calculations. We note some degree of orientation dependence at the onset of
direct band gap transition near 3.4 eV. We demonstrate that good polarization
selectivity can be achieved by placing the electron probe at different
locations around the specimen with increasing impact parameter while keeping
the beam-specimen orientation fixed. The observed results are qualitatively
elucidated in terms of the perpendicular electric fields generated by the fast
electron (60 kV) used in the microscope. The fact that good polarization
selectivity can be achieved by aloof beam EELS without the requirement of
sample reorientation is an attractive aspect from the characterization method
point of view in the STEM-EELS community.

##  I Introduction

Scanning transmission electron microscope coupled electron energy-loss
spectroscopy (STEM-EELS) plays an important role in the development and
characterization of nanomaterials due to its high spatial resolution in both
imaging and spectroscopy. In the more straightforward intersecting beam-
specimen configuration where the interaction is dominated by impact
scattering, high resolution STEM-EELS is able to probe local material
properties such as interface states Batson (1993); Muller  _et al._ (1999) and
single-atom dopant electronic properties Ramasse  _et al._ (2013), and even
vibrational properties Lagos and Batson (2018); Venkatraman  _et al._ (2019);
Hage  _et al._ (2020) aided by recent development in energy monochromators
Krivanek  _et al._ (2014). Typically, these high spatial resolution STEM-EELS
experiments are carried out with a focused electron probe with a large
convergence semi-angle ($\alpha$), which allows the incident electrons to
excite the specimen with significant momentum transfer. With a common circular
collection aperture used in the on-axis arrangement, the acquired transmitted
signals lead to the so called momentum integrated energy-loss spectra
Venkatraman  _et al._ (2019); Rafferty and Brown (1998). When the specimen of
interest has complex electronic and phonon structures in the reciprocal space,
the resulting EELS spectra would also reflect the structural complexity, which
can make it difficult to fully interpret the observed spectra without prior
knowledge of the material.

Meanwhile, in the aloof beam (non-intersecting probe-specimen) configuration
where the interaction is dominated by dipole scattering, the electron beam
behaves as a near-field spectroscopy probe Cohen  _et al._ (1998) and can be
used to study long range electric field ($\bf{E}$) related phenomena such as
vibrational properties Radtke  _et al._ (2017, 2019), surface plasmon-
polaritons in metal and doped-semiconductor nanostructures Batson (1982);
Rossouw and Botton (2013); Yang  _et al._ (2020), and surface phonon-
polaritons in insulating nanostructures Lagos  _et al._ (2017); Qi  _et al._
(2019). Recently, Rez et al. Rez  _et al._ (2016) and Crozier Crozier (2017)
discussed the non-destructive nature of the aloof beam EELS and expanded its
applications in studying vibrational and electronic properties for electron
beam sensitive materials. In addition to the non-destructive characterization,
Radtke et al. Radtke  _et al._ (2019) have further applied aloof beam EELS to
study polarization dependent vibrational properties of anisotropic materials
demonstrating that good polarization selectivity can be achieved by aloof beam
EELS. We note that orientation dependent anisotropic properties have
traditionally been measured in the beam-specimen intersecting configuration
with small beam convergence ($\alpha<0.5$ mrad) in the past. Leapman and
Silcox (1979); Leapman  _et al._ (1983); Arenal  _et al._ (2007)

Here, we explore the potential application of aloof beam EELS to characterize
the polarization-dependent electronic structures of uniaxial crystals in
scenarios where specimen reorientation is challenging to achieve by the
microscope goniometer. In particular, we study one dimensional wurtzite zinc
oxide (w-ZnO) nanorods in a fixed beam-to-specimen orientation, compare the
results obtained with the intersecting and aloof beam configurations, and
demonstrate that the aloof beam EELS offers good polarization selectivity when
the impact parameter, i.e., specimen-to-probe distance, is large enough
($\sim$6 nm). The prominent anisotropic features that differentiate the in-
plane ($\bot$ to $c$ axis) and out-of-plane ($\parallel$ to $c$ axis)
responses of w-ZnO nanorods are examined by first-principles density-
functional theory (DFT) calculations. Our results assert that the previously
debated transition at 11.2 eV in w-ZnO is not due to a surface plasmon Wang
_et al._ (2005, 2007), but instead corresponds to an orientation-dependent
interband transition from the occupied Zn-$3d$ states to the unoccupied
O-$2p_{z}$ states near the zone center Huang  _et al._ (2011). Finally, we
discuss the aloof beam results in terms of the electric fields associated with
the excitation source (incident electrons) rather than the dielectric function
that describes the response to an external perturbation. This approach
provides qualitative but intuitive understanding of the results. Moreover, the
fact that good polarization selectivity can be achieved by aloof beam EELS
without specimen reorientation is another attractive aspect to the technique,
as many technologically important materials can be anisotropic and have large
aspect ratios Xia  _et al._ (2003); Sajanlal  _et al._ (2011).

##  II Materials and methods

w-ZnO nanorods were synthesized via the hydrothermal route described by Cheng
and Samulski Cheng and Samulski (2004). The two starting solutions were
obtained by mixing (1) 0.1M of zinc acetate dihydrate (Zn(Ac)2 * 2 H2O) in
methanol and (2) 0.5M of sodium hydroxide (NaCl) in ethanol. 60 ml of of the
mixed solution (1:1 volume ratio) was then transferred to a Teflon-lined
stainless steel autoclave and heated at 150 ∘C for 8 hours. After the
autoclave was cooled to room temperature, the white precipitates containing
ZnO nanorods were washed with de-ionized water four times and dried at 60 ∘C
in an oven overnight. The dried powder was then annealed in air at 400 ∘C for
1 hour to improve its crystallinity. The electron microscopy sample was
prepared by first dispersing the annealed ZnO powder in ethanol via sonication
and the sample containing solution was then drop cast onto a lacey carbon
coated copper grid.

The synthesized nanorods have lengths that range from 90 to 1200 nm and widths
that range from 35 to 40 nm. A typical synthesized long nanorod is shown in
Fig. 1. As reported by Cheng and Samulski Cheng and Samulski (2004), the rods
preferentially grow along the $c$ axis. Regardless of the rod lengths, the $c$
axis of w-ZnO is always along the rod axis as confirmed by the high resolution
image shown in the inset of Fig. 1, where $d_{002}=2.6$ Å was observed.

Figure 1:  Annular dark-field STEM image of a single suspended 1100 by 40 nm
ZnO nanorod on lacey carbon support. The long axis of the nanorod corresponds
to the $c$ axis of w-ZnO. The inset shows a high resolution image of (002)
lattice fringes with the spacing of 2.6 Å.

The STEM-EELS experiments reported here were carried out in a monochromated
Nion UltraSTEM operating at 60 kV. The beam convergence and EELS collection
semi-angles were set at 30 and 16 mrad, respectively. The energy resolution
for the spectra is 87.2 meV according to the full-width-half-maximum of the
zero-loss peak.

DFT calculations were performed using the Projector Augmented Wave (PAW)
method as implemented in the Vienna Ab initio Simulation Package (VASP) Kresse
and Furthmüller (1996a, b); Kresse and Joubert (1999). Twelve and six valence
electrons were considered in the zinc and oxygen PAW pseudopotentials,
respectively. The Perdew-Burke-Ernzerhof generalized-gradient approximation
for solids (GGA-PBEsol) was used to compute the exchange-correlation
functional Perdew  _et al._ (2008). The reciprocal space was sampled using a
$\Gamma$-centered k-mesh of size 16$\times$16$\times$10 together with a
kinetic energy cutoff of 650 eV for plane waves. The energy convergence and
force convergence criteria were set to $10^{-8}$ eV and $10^{-4}$ eV/Å,
respectively.

All the DFT calculations were carried out in a four-atom hexagonal unit cell
of w-ZnO (space group: $P6_{3}mc$). The GGA-PBEsol optimized lattice
parameters are $a=b=3.236$ Å, and $c=5.223$ Å. which are in good agreement
with the experimental data Adachi (1999); Wang (2004a). The employed DFT
predicts w-ZnO to be a direct band gap semiconductor with an energy band gap
of 0.7 eV at $\Gamma$. We apply a scissor operation of 2.7 eV to obtain the
reported experimental bandgap of 3.4 eV in w-ZnO Adachi (1999); Wang (2004a);
Franklin  _et al._ (2013); Nejatipour and Dadsetani (2015). The calculated
electronic bandstructure (in particular, the structure of the conduction bands
and valence bands near the Fermi level) matches reasonably well with the
previously reported hybrid and quasi-particle $GW$ calculations Preston  _et
al._ (2008); Schleife  _et al._ (2009a, b); Gori  _et al._ (2010); Ochi  _et
al._ (2017); Ataide  _et al._ (2017); Preston  _et al._ (2011); Stankovski
_et al._ (2011); Zhang and Schleife (2018). The frequency-dependent imaginary
dielectric function $\epsilon_{2}\,(\omega)$ was calculated using the method
described in Ref. Gajdoš  _et al._ (2006). The convergence of
$\epsilon_{2}\,(\omega)$ was ensured by increasing the number of empty
conduction bands. The calculated $\epsilon_{2}\,(\omega)$ is in good agreement
with the previous studies Fritsch  _et al._ (2005); Preston  _et al._ (2008);
Akiyama  _et al._ (2008); Fritsch  _et al._ (2008); Schmidt-Grund  _et al._
(2008); Schleife  _et al._ (2009a); Gori  _et al._ (2010) and gives a good
description of the recorded EELS spectra in this work.

##  III Results and discussion

w-ZnO is a uniaxial material of hexagonal structure and it possesses two sets
of dielectric properties - one associated to the in-plane direction ($a$ and
$b$ axes) and the other associated to the out-of-plane ($c$ axis) direction.
The oxygen $2p_{xy}$ (in-plane) and $2p_{z}$ (out-of-plane) orbitals play an
important role in governing the anisotropic electronic properties of w-ZnO.
Our DFT results, presented below, as well as previous studies Adachi (1999);
Wang (2004b); Ding and Wang (2005); Fritsch  _et al._ (2005); Zhang  _et al._
(2006); Preston  _et al._ (2008); Akiyama  _et al._ (2008); Fritsch  _et al._
(2008); Schmidt-Grund  _et al._ (2008); Schleife  _et al._ (2009a); Gori  _et
al._ (2010), suggest that there are several distinct optical (dipole-dipole)
interband transitions in w-ZnO at energies of $\sim$3.4, 9.1, 11.2, 13.0,
13.5, and 14.8 eV, as listed in Table 1. Although the transitions at
$\sim$3.4, 9.1, and 13.5 eV are well documented in the existing literature
Adachi (1999); Wang (2004b); Ding and Wang (2005); Fritsch  _et al._ (2005);
Zhang  _et al._ (2006); Preston  _et al._ (2008); Akiyama  _et al._ (2008);
Fritsch  _et al._ (2008); Schmidt-Grund  _et al._ (2008); Schleife  _et al._
(2009a); Gori  _et al._ (2010), less is known about the other listed
transitions. Notably, the transitions at 9.1, 13.5, and 14.8 eV are not very
sensitive to the polarization orientation, as these are common to both the in-
plane and out-of-plane polarization orientations of the $\bf{E}$-field. In
contrast, the transitions at 3.4, 3.5, 11.2, and 13.0 eV exhibit strong
orientation dependence due to the anisotropic nature of the O-$2p_{xy}$ and
O-$2p_{z}$ orbitals in w-ZnO.

Table 1: Calculated optical transitions below 15.0 eV, from the occupied to
unoccupied states near the Fermi level, and their polarization orientation
with respect to the $c$ axis in w-ZnO
Energy (eV) | Orientation | Assignment
---|---|---
3.4 | $\bot$ | O-$2p_{xy}$ $\rightarrow$ Zn-$3s$
3.5 | $\parallel$ | O-$2p_{z}$ $\rightarrow$ Zn-$3s$
9.1 | $\bot\&\parallel$ | Zn-$3d$ $\rightarrow$ O-$2p$
11.2 | $\parallel$ | Zn-$3d$ $\rightarrow$ O-$2p_{z}$
13.0 | $\bot$ | Zn-$3d$ $\rightarrow$ O-$2p_{xy}$
13.5 | $\bot\&\parallel$ | O-$2p$ $\rightarrow$ Zn-$3s$
14.8 | $\bot\&\parallel$ | Zn-$3d$ $\rightarrow$ O-$2p$
$\bot\&\parallel$ | O-$2p$ $\rightarrow$ {Zn-$3d$, Zn-$3s$}
Figure 2:  Valence EELS spectrum acquired with the beam intersecting
configuration after Fourier-log deconvolution Egerton (2011) and background
removal. Three major peaks are the onset of the band gap around 3.4 eV,
Zn-$3d$ to O-$2p$ transition around 9.1 eV, and the bulk plasmon at 18.8 eV.
While not being immediately obvious, the orientation specific features can be
seen around 11.2 eV (out-of-plane) and 13.0 eV (in-plane) as guided by the
dotted orange and green lines, respectively. The inset shows the scattering
geometry with $\hat{c}\parallel\hat{q_{r}}$ and $\hat{a}$ and
$\hat{b}\parallel\hat{q_{z}}$

In our experiments, the ZnO nanorod is oriented such that the incident
electron beam is perpendicular to the crystal $c$ axis. Fig. 2 shows the EELS
spectrum of ZnO obtained using the specimen-intersecting beam and the inset
shows the scattering geometry. The three most apparent spectral features are
the direct band gap near 3.4 eV, Zn-$3d$ to O-$2p$ transition around 9.1 eV,
and the bulk plasmon at 18.8 eV, which are consistent with the previous
observations Wang  _et al._ (2005, 2007); Zhang  _et al._ (2006); Ding and
Wang (2005); Huang  _et al._ (2011). With the large beam convergent angle
($\alpha=30$ mrad) used in the present study, the beam is able to excite
transitions with polarization along both the parallel ($a$ and $b$ axes) and
perpendicular ($c$ axis) to the beam (crystal) directions.

In conjunction with the large incident beam convergence angle, the moderate
EELS collection angle ($\beta=16$ mrad) used in the present study leads to
momentum integrated EELS Rafferty and Brown (1998). Consequently, the
orientation-specific 11.2 ($\parallel\,c$) and 13.0 eV ($\bot\,c$) transitions
are always observed in the spectrum regardless of where the beam is placed, as
long as the beam intersects the sample. However, the poor polarization
selectivity observed in the intersecting configuration makes it impractical in
studying orientation specific features. We should note that the scattered
signals being collected can be further restricted to the beam parallel
direction and achieve decent polarization selectivity in the intersection
configuration by using a smaller aperture to reduce the collection angle
Batson (2008). However, such small aperture is not available in our current
microscope setup. In addition, restricting momentum transfer causes
unavoidable uncertainty-principle driven spread in the area that is sampled,
degrading the desired spatial resolution of the measurement using the STEM.

In addition, due to the considerable momentum transfer ($\vec{q}$ =
$\vec{q_{r}}+\vec{q_{z}}$) in the intersecting scattering configuration, the
transitions around 11.2, 13.0, 13.5, and 14.8 eV appear as minor humps which
are partly obscured by the nearby transitions and the bulk plasmon. Moreover,
the 11.2 eV hump was misinterpreted as a surface plasmon in the past Wang  _et
al._ (2005, 2007). In the present study, we focus on the anisotropic features
of w-ZnO, and for more detailed spectral feature assignments we refer the
readers to Refs. Zhang  _et al._ (2006); Huang  _et al._ (2011).

Figure 3:  (a) The high annular dark field image of a 1100 by 40 nm ZnO
nanorod (scale bar: 200 nm) where the green and orange dots indicate the
electron beam locations where the aloof beam spectra were acquired. Three
spectra were acquired at impact parameters of 2, 4, and 6 nm (b) along the
green arrow near the lateral-middle position of the rod and (c) along the
orange arrow near the rod tip. The dotted vertical line at 11.2 and 13.0 eV
indicate the key anisotropic transitions that are specific to the direction
parallel and perpendicular to the $c$ axis, respectively.

To explore the polarization selectivity of the aloof beam EELS while keeping
the rod orientation fixed, we acquired two sets of data points - one in the
vicinity of the rod tip and the other in the vicinity of the lateral-middle
position of the rod. Each set contains spectra acquired at three different
impact parameters ($d$) of 2, 4, and 6 nm along the directions indicated by
the orange and green arrows, as shown in Fig. 3(a). The acquired spectra near
the lateral-middle and the tip of the rod are shown in Figs. 3(b) and 3(c),
respectively. Consistent with the calculated transitions listed in Table 1,
the not very orientation-specific transitions around 3.4 eV and the 9.1 and
13.5 eV peaks are observed in both aloof beam data sets, and the broad peak
around 16.0 eV is the surface plasmon Huang  _et al._ (2011). We note that the
last non-orientation specific transition at 14.8 eV is overshadowed by the
broad surface plasmon peak, making it difficult to identify.

At small impact parameter (i.e., $d$ = 2 nm), both the 11.2 eV and 13.0 eV
transitions are visible in both sets. However, as the impact parameter
increases, the 11.2 eV and 13.0 eV transitions remain significant only when
the electron beam is placed near the tip and lateral-middle position of the
rod, respectively. Therefore, good polarization selectivity is achieved when
$d$ = 6 nm, despite the fact that the beam-parallel electric field ($\bot$ to
the specimen $c$ axis) is always present. Unlike in Ref. Radtke  _et al._
(2019), where the polarization-dependent spectra were acquired with two
different beam-to-specimen orientations, the aloof beam spectra presented here
were acquired with the same beam-to-specimen orientation but at different
locations and different values of impact parameter.

To understand the observed aloof beam EELS results, we turn our attention to
the electric fields generated by the incident electron, since the material
response (induced dipole field) also scales with the external perturbation
magnitude. The fast electrons (60 keV) used here can be considered as a
broadband electromagnetic source García de Abajo (2010); Maciel-Escudero and
Reyes-Coronado (2019) producing electric fields in the frequency domain that
can be broken down into two parts - one being perpendicular to the incident
electron beam direction $E_{R}$ and the other being parallel to the beam
$E_{z}$ García de Abajo (2010); Maciel-Escudero and Reyes-Coronado (2019);
Bigelow  _et al._ (2012) as follows

$E_{R}(x,y,z,\omega)=-\frac{2e\omega}{Rv^{2}\gamma}e^{i\omega
z/v}K_{1}(\frac{\omega R}{v\gamma})[\,(x-d)\,\vec{x}+y\,\vec{y}\,]~{},$ (1)
$\text{and}~{}~{}E_{z}(x,y,z,\omega)=i\frac{2e\omega}{Rv^{2}\gamma^{2}}e^{i\omega
z/v}K_{0}(\frac{\omega R}{v\gamma})\,\vec{z}~{}.$ (2)

Here ($x,y,z$) is the observation location in the lab frame, $e$ is the
elementary charge, $\omega$ is the angular frequency, $d$ is the impact
parameter, $K_{n}$ is the modified Bessel function of second kind with index
$n$, $R^{2}=(x-d)^{2}+y^{2}$, $v$ is the electron speed,
$\gamma=[1-(v/c)^{2}]^{-1/2}$ is the Lorentz factor, and $c$ is the speed of
light in vacuum. On the other hand, we would also like to point out that the
induced dipole field exerts forces on the incident electrons and wraps around
the nanorod, occupying a three dimensional volume as illustrated in Ref.
Batson (2008). The induced field components perpendicular and parallel to the
incident electrons are responsible for the electron deflection and energy
loss, respectively.

Figure 4:  Electric field distributions, $E_{R}(R)$ generated by the fast
electron at 11.2 eV within the nanorod when the aloof beam is placed near (a)
the lateral-middle and (b) the tip of the rod. The black and green (orange)
circles represent the electron beam positions at the impact parameters of 2
and 6 nm, respectively. The nanorod area is delineated by the grey dashed
lines. The fields are more ’collimated’ toward the beam location as the impact
parameter increases from 2 to 6 nm. The direction of the field arrows is
reversed to avoid overlapping with the circles. (c) The directionality
parameter versus the impact parameter near the lateral-middle (green) and the
tip (orange) of the rod.

We now focus our attention on the field distributions in the plane
perpendicular to the beam direction (Eq. 1), as this component is larger than
the parallel one Rafferty and Brown (1998); Rez  _et al._ (2016); Radtke  _et
al._ (2019) and contributes more to the overall electron-specimen interactions
in the valence energy range considered here. As shown in Figs. 4(a) and 4(b),
we plot the perpendicular electric field distributions (Eq. 1) at 11.2 eV
within the nanorod area (delineated by the grey dashed line) when the beam is
placed outside the lateral-middle and the tip of the rod, respectively. The
$x$ and $y$ coordinates used here are referenced to the long ($c$ axis) and
short axes of the ZnO crystal, and we refer the electric field components
parallel and perpendicular to the $c$ axis as $E_{\parallel}$ and $E_{\bot}$,
respectively. It can be seen that the resultant field becomes more directional
toward the beam location when the beam is placed further away from the
specimen.

To visualize how the resultant field becomes more ‘collimated’ in the plane
perpendicular to the beam direction as the impact parameter increases, we
define a directionality parameter
$\xi=\Sigma(E_{\parallel})/\Sigma(E_{\bot})$, where $\Sigma(E_{\parallel})$
and $\Sigma(E_{\bot})$ are sums over a representative sampling of points
within the nanorod. Figure 4(c) shows the variation of $\xi$ as a function of
the impact parameter $d$. It is found that the electron beam generates
electric fields that are more parallel to the $c$ axis near the tip (orange
curve) than the lateral-middle (green curve) position of the rod. To put it
more generally, when the specimen is placed on one side next to the beam with
the extent no larger than its corresponding half-space, the fields that are
generated by the beam and felt by the specimen are inherently asymmetric and
directional in the aloof beam configuration. The larger the impact parameter,
the more the aloof beam EELS resembles polarized optical spectroscopy.
Therefore, in the case of w-ZnO, the 11.2 eV feature is only noticeable at
large impact parameter near the tip but not at the lateral-middle position of
the rod. A similar argument can be extended for the 13.0 eV feature.

Figure 5:  (a) Normalized aloof beam spectra ($d$ = 6 nm) acquired near the
tip (orange) and the lateral-middle position (green) of the rod. The two
anisotropic features at 11.2 eV ($E\parallel c$) and around 13.0 eV ($E\bot
c$), marked using magenta and purple arrows, are differentiated near the tip
and the lateral-middle position of the rod, respectively. The observed spectra
demonstrate good polarization selectivity of the aloof beam EELS. (b)
Polarization-orientation dependent imaginary dielectric functions,
$\epsilon_{2}(\omega)$, calculated along xx/yy ($E\bot c$) and zz ($E\parallel
c$) directions. The arrows mark the transitions of interest. (c) DFT
calculated orbital-resolved electronic band structure of w-ZnO. Magenta and
purple arrows depict the observed inter-band optical transitions at 11.2 eV
($E\parallel c$) and near 13.0 eV ($E\bot c$), whereas black arrow denotes the
direct band gap transition at 3.4 eV.

To aid the visual comparison of the observed aloof spectra, we scale the
spectra with respect to the Bessel function and frequency parts of Eq. 1 and
normalize the intensity around the band gap as the dielectric functions around
the gap have similar values. As shown in Fig. 5(a), the normalization
facilitates visualization of the orientation-specific transitions at 11.2 eV
($E\parallel c$) and 13.0 eV ($E\bot c$) near the tip (orange) and the
lateral-middle (green) positions, respectively.

In order to pin down the nature of the observed optical transitions, we employ
DFT calculations as detailed in the methods section. The calculated
polarization-orientation dependent imaginary dielectric function
$\epsilon_{2}(\omega)$, which contains full details of the optical-transition
matrix elements and the interband absorption processes, is in remarkable
agreement with the recorded EELS spectra for both orientations, as shown in
Fig. 5(b). Magenta and purple arrows respectively mark the orientation
specific $E\parallel c$ and $E\bot c$ transitions, respectively, whereas, the
black arrows mark the transitions that are not strongly orientation sensitive.
The orbital-resolved electronic bandstructure, shown in Fig. 5(c), further
facilitates the identification of the observed interband transitions from
occupied to unoccupied states, as discussed below.

We note that the onset of the direct band gap transition near 3.4 eV, although
marked using a black arrow, is technically orientation dependent due to the
splitting of the O-$2p$ states into a subset of O-$2p_{xy}$ and O-$2p_{z}$
states at the valence band maximum (see Fig. 7(b) in Appendix). This occurs
due to the asymmetry of the wurtzite structure, which reduces the energy of
the O-$2p_{z}$ orbitals by $\sim$0.1 eV with respect to the valence band
maximum formed by O-$2p_{xy}$ orbitals. Therefore, in principle, the onset of
direct band gap transitions for $E\bot c$ (O-$2p_{xy}$ $\rightarrow$ Zn-$3s$)
and $E\parallel c$ (O-$2p_{z}$ $\rightarrow$ Zn-$3s$) should occur near 3.4
and 3.5 eV, respectively. Indeed, this band gap anisotropy has been reported
by optical measurements Adachi (1999), but the measured energy difference is
about 70 meV rather than the calculated 100 meV presented here. While good
polarization selectivity is achieved at $d=6$ nm, the aloof-beam spectra are
not fully polarized and contain mixed signals from both polarization
directions. For the band gap onset area, the spectral identification is
hindered by the signal contribution from the in-plane direction where the
onset arises earlier than from the out-of-plane direction. However, as Fig.6
shows, we do observe some hint of anisotropy around the band gap region with a
minor onset difference of 12 meV.

Figure 6: Band gap onset region of the aloof-beam spectra acquired at $d=6$ nm
near the tip (orange) and lateral-middle (green) of the rod. Due to the mixed
polarization signals presented in aloof-beam EELS, a minor onset difference of
12 meV is observed rather than the 70 meV difference observed in the fully
polarized optical measurements reported in Ref.Adachi (1999).

The key orientation-dependent transitions at 11.2 and 13.0 eV are identified
as Zn-$3d$ $\rightarrow$ O-$2p_{z}$ near $\Gamma$ and Zn-$3d$ $\rightarrow$
O-$2p_{xy}$ near the boundaries of the hexagonal Brillouin zone (BZ),
respectively. On the other hand, orientation-independent transitions at 9.1
and 13.5 eV are identified as Zn-$3d$ $\rightarrow$ O-$2p$ and O-$2p$
$\rightarrow$ Zn-$3s$, respectively. Our DFT calculations reveal the presence
of another orientation-independent transition at 14.8 eV, which could not be
distinctly observed in the recorded EELS spectra. It appears as a minor hump,
primarily due to the presence of the nearby and broad surface plasmon. This
transition is attributed to the electric dipole transition from occupied
Zn-$3d$ to unoccupied O-$2p$ and occupied O-$2p$ to unoccupied {Zn-$3d$,
Zn-$3s$} states. Some of the above-mentioned transitions occur at various
$k$-points within the hexagonal BZ and between different sets of occupied and
unoccupied bands. In Figs. 7 and 8, we made an attempt to mark some of these
transitions having maximum optical-transition matrix elements using arrows.

Finally, we note that while localized surface plasmons (LSPs) are commonly
observed in metallic nanorods Rossouw and Botton (2013); Martin  _et al._
(2014), such distinct LSPs are not observed here due to the
$\epsilon_{1}(\omega)$ behaviors of ZnO. Considering the vacuum/ZnO interface
configuration studied here, surface plamons (SPs) are expected when
$\epsilon_{1}(\omega)\leq-1$. However, as Fig. 9 in the Appendix shows, unlike
in a metal, neither $\epsilon_{1,\bot}$ nor $\epsilon_{1,\parallel}$ of ZnO
goes much below $-1$, and the energy range that supports SPs in ZnO is rather
narrow, roughly from 15 to 17 eV. While there must be signals contributed from
LSPs related to the nanorod geometry, the LSPs in the ZnO nanorod are
suppressed due to its dielectric function, and the energy range where the LSPs
are expected to reside does not affect the interband transition discussion
presented here. While not shown here, we also performed plasmonic simulations
of a 1100 by 40 nm ZnO nanorod using MNPBEM Hohenester and Trügler (2012)
without observing the distinct LSPs that are typically observed in a metal
nanorod.

##  IV Summary

We have characterized the electronic structures of single crystal w-ZnO
nanorods by both intersecting and aloof beam configurations with a focused
electron beam ($\alpha$ = 30 mrad) and moderate collection angle ($\beta$ = 16
mrad). The key anisotropic features of w-ZnO that are specific to the in-plane
and out-of-plane directions were identified by DFT calculations. It was
demonstrated experimentally that when keeping the beam-to-specimen orientation
fixed, good polarization selectivity can only be achieved by the aloof beam
configuration, not the intersecting one. We found that the good polarization
selectivity brought by the aloof beam can be explained by analyzing the
perpendicular component of the electric fields (Eq. 1) generated by the fast
electron used in the STEM-EELS experiments. The fact that the field
directionality can be tuned by varying the impact parameter can be very useful
in reducing spectral complexity and characterizing anisotropic nanomaterials
such as the w-ZnO nanorods presented here.

## Acknowledgements

P.E.B. acknowledges the financial supports of U.S. Department of Energy,
Office of Science, Basic Energy Sciences under award no. DE-SC0005132. S.S.
acknowledges the support from the Office of Naval Research (ONR) grant
N00014-21-1-2107. D.V. acknowledges the support from National Science
Foundation grant DMR-1954856. We also thank N. Dellby and M. Hoffman for
discussions regarding the optical configuration of the microscope. First-
principles calculations were performed using the computational resources
provided by the Rutgers University Parallel Computing clusters.

##  V Appendix

Here we provide orbital-resolved electronic band structures marked with
orientation-dependent (Fig. 7) and orientation-independent (Fig. 8) interband
transitions. Only selected transitions having large optical-transition matrix
elements are marked using arrows. There are numerous other transitions with
relatively smaller transition probability which are not marked for the sake of
clarity. Zn-$3s$ and Zn-$3p$ orbitals are depicted in orange and green colors,
whereas O-$2p_{xy}$ and O-$2p_{z}$ orbitals are shown in blue and red colors,
respectively, in all orbital-resolved band structures.

Figure 7: (a) Orientation-dependent interband transitions at 3.4, 11.2, and
13.0 eV marked using arrows in the orbital-resolved electronic band structure
of w-ZnO. (b) An enlarged of bands near the Fermi level at zone center
$\Gamma$. Inset shows a further amplification of valence band maximum near
$\Gamma$. Crystal-field induced splitting of O-$2p_{xy}$ and O-$2p_{z}$
orbitals can be noticed.  Figure 8: Orientation-independent interband
transitions at 9.1, 13.5, and 14.8 eV marked using arrows in the orbital-
resolved electronic band structure of w-ZnO. Only selected transitions having
large optical-transition matrix elements are marked.  Figure 9: Replotted ZnO
dielectric function using the data reported in Ref. Adachi (1999). The real
(solid curves) and imaginary (dotted curves) parts of the dielectric function
along the in-plane and out-of-plane directions are shown in green and orange,
respectively. Note that the energy range where $\epsilon_{1}<-1$ is between 15
and 17 eV.

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