# The Breakdown of Mott Physics at VO2 Surfaces

Matthew J. Wahila  Department of Physics, Applied Physics and Astronomy,
Binghamton University, Binghamton, New York 13902, USA  Nicholas F.
Quackenbush  Department of Physics, Applied Physics and Astronomy, Binghamton
University, Binghamton, New York 13902, USA  Jerzy T. Sadowski  Center for
Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York
11973, USA  Jon-Olaf Krisponeit  Institute of Solid State Physics, University
of Bremen, Otto-Hahn-Allee 1, 28359 Bremen, DE  Jan Ingo Flege  Institute of
Solid State Physics, University of Bremen, Otto-Hahn-Allee 1, 28359 Bremen, DE
Applied Physics and Semiconductor Spectroscopy, Brandenburg University of
Technology Cottbus-Senftenberg, Konrad-Zuse-Str. 1, 03046 Cottbus, DE  Richard
Tran  Department of NanoEngineering, University of California San Diego, 9500
Gilman Drive 0448, La Jolla, California 92093, USA  Shyue Ping Ong  Department
of NanoEngineering, University of California San Diego, 9500 Gilman Drive
0448, La Jolla, California 92093, USA  Christoph Schlueter  Diamond Light
Source Ltd., Diamond House, Harwell Science and Innovation Campus, Didcot,
Oxfordshire OX11 0DE, UK  Tien-Lin Lee  Diamond Light Source Ltd., Diamond
House, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, UK
Megan E. Holtz  School of Applied and Engineering Physics, Cornell University,
Ithaca, NY 14853, USA  David A. Muller  School of Applied and Engineering
Physics, Cornell University, Ithaca, NY 14853, USA  Kavli Institute at Cornell
for Nanoscale Science, Ithaca, New York 14853, USA  Hanjong Paik  Department
of Materials Science and Engineering, Cornell University, Ithaca, New York
14853-1501, USA  Darrell G. Schlom  Department of Materials Science and
Engineering, Cornell University, Ithaca, New York 14853-1501, USA  Kavli
Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA  Wei-
Cheng Lee  Department of Physics, Applied Physics and Astronomy, Binghamton
University, Binghamton, New York 13902, USA  Louis F. J. Piper
<EMAIL_ADDRESS>Department of Physics, Applied Physics and Astronomy,
Binghamton University, Binghamton, New York 13902, USA  Materials Science &
Engineering, Binghamton University, Binghamton, New York 13902, USA

###### Abstract

Transition metal oxides such as vanadium dioxide (VO2), niobium dioxide
(NbO2), and titanium sesquioxide (Ti2O3) are known to undergo a temperature-
dependent metal-insulator transition (MIT) in conjunction with a structural
transition within their bulk. However, it is not typically discussed how
breaking crystal symmetry via surface termination affects the complicated MIT
physics. Using synchrotron-based x-ray spectroscopy, low energy electron
diffraction (LEED), low energy electron microscopy (LEEM), transmission
electron microscopy (TEM), and several other experimental techniques, we show
that suppression of the bulk structural transition is a common feature at VO2
surfaces. Our density functional theory (DFT) calculations further suggest
that this is due to inherent reconstructions necessary to stabilize the
surface, which deviate the electronic structure away from the bulk d1
configuration. Our findings have broader ramifications not only for the
characterization of other ”Mott-like” MITs, but also for any potential device
applications of such materials.

Bulk, unstrained vanadium dioxide (VO2) undergoes a temperature-dependent
metal-insulator transition (MIT) at 65 ∘C accompanied by a crystal symmetry
change between the high-temperature rutile and low-temperature M1 phases.
After decades of investigation and debate,Kosuge (1967); Eyert (2002) VO2 is
now generally understood to be a cooperative Mott-Peierls system with aspects
of both playing an integral role in the MIT. However, the exact interplay
between the electrons (Mott) and lattice (Peierls) is still being
unraveled.Qazilbash  _et al._ (2007); Quackenbush  _et al._ (2016, 2015);
Mukherjee  _et al._ (2016); Quackenbush  _et al._ (2017); Lee  _et al._ (2019)
For example, there is currently interest in determining if the electronic and
structural phase transitions can be truly decoupled under specific
circumstances, although studies reported so far have been plagued by issues
such as poorly-defined sample quality, non-contemporaneous measurement of the
structure and electronic properties, and other issues.Lee  _et al._ (2018);
Yang  _et al._ (2016); Tao  _et al._ (2012); Laverock  _et al._ (2014, 2018)
Despite these lingering fundamental questions, the abrupt and large
discontinuity in resistivity across the MIT has generated significant interest
in VO2 for its’ potential uses in novel switching devices.Yang, Ko, and
Ramanathan (2011); Zhou and Ramanathan (2015); Chen  _et al._ (2015, 2019)

VO2 has shown promise for many applications, such as in ionic liquid gated,
field-effect transistors that undergo bulk carrier delocalization instigated
by charge accumulation at the surface.Nakano  _et al._ (2012); Jeong  _et al._
(2013) This and other VO2 devices mainly employ ultra-thin epitaxial films, as
the MIT properties of these films can be fine-tuned for specific applications
using the strain induced by epitaxial matching with the substrate.Muraoka and
Hiroi (2002); Fischer  _et al._ (2020) Unfortunately, as the dimensions of
these VO2 films decrease, the film surface/interface properties become
increasingly important for device performance, while relatively little is
known about the intrinsic nature of VO2 surfaces.Mellan and Grau-Crespo (2012)

Here, we employ low energy electron diffraction (LEED) and microscopy (LEEM)
to perform a temperature-dependent surface study on three orientations of high
quality, epitaxial VO2 films; (100), (001), and (110). Multiple x-ray
spectroscopy techniques, including x-ray absorption and x-ray photoelectron
spectroscopies (XAS & XPS), are used in conjunction to monitor the electronic
structure and composition of the films throughout the study. None of our
temperature-dependent measurements provide any evidence of a structural change
at the surface corresponding to the bulk MIT. Mild annealing and/or electron
beam exposure is found to induce some morphological changes, however, these
are explained purely by general surface stability arguments. This study
reveals and explains the existence of unavoidable surface layer
reconstructions that do not directly participate in the bulk VO2 MIT,
regardless of film orientation.

Figure 1:  (a) Oxygen K-edge XAS and (b) valence band UPS/HAXPES of a VO2/TiO2
(100) film measured above and below TMIT. (c) Approximate photoelectron escape
probabilities with depth below sample surface for the photon energies used in
this experiment ($\sim$6 keV for HAXPES and 51 eV for UPS). (d) The most
stable DFT calculated structure for a stoichiometric VO2(100) slab terminating
in vacuum.

##  I Probing the MIT/SPT of the VO2 epilayer

The structural transition of a clean VO2 film can be investigated using oxygen
K-edge XAS, as shown in Fig. 1a. Due to the strong O 2p - V 3d hybridization
in VO2, the O K-edge provides an indirect measure of the V-V dimer formation
associated with the MIT. The presence of dimers in the insulating M1 phase is
evidenced by the so-called d∥ feature at $\sim$530.5 eV, located between the
$\pi$* ($\sim$529.5 eV) and $\sigma$* ($\sim$532 eV) states.Quackenbush  _et
al._ (2015); Paez  _et al._ (2020) In VO2(100), this d∥ feature completely
vanishes above the MIT temperature, indicating a transition from the M1 to
rutile phase. This d∥ feature can also provide insight into the relative level
of electron correlation present in films of different orientations (see
Supplementary Figure 1).

Figure 1b shows complimentary UV and hard x-ray photoelectron spectroscopies
(UPS & HAXPES) of the topmost valence states of a VO2(100) epitaxial thin
film. The top valence states (0 - 2 eV) are mainly V 3d in character and lie
just above the predominantly O 2p states that constitute the majority of the
valence band. We observe clear differences in the V 3d states above and below
the MIT temperature using HAXPES, with the density of states increasing at the
Fermi level (EF) in the rutile phase indicating an increased metallicity. In
contrast, a much broader V 3d peak with a somewhat suppressed transition is
observed using the more surface sensitive UPS.

As shown in 1c, the probability of photoelectron escape with increasing depth
inside the sample drops off much more quickly for UPS compared to HAXPES. In
fact, the probing depth of UPS is only on the order of 1-2 nm, while that of
HAXPES is closer to $\sim$15 nm. Therefore, the difference in the observed
spectral weight transfer between UPS and HAXPES implies that the MIT does not
occur as strongly (if at all) within the topmost atomic layers of the VO2(100)
film.

The DFT calculated most stable surface structure for a stoichiometric VO2(100)
slab in vacuum, shown in Figure 1d, supports the idea of a suppressed
transition within the film surface layers. The break in crystal symmetry at
the surface is predicted to result in a surface reconstruction imparting the
topmost vanadium atoms with a different oxygen coordination than those in the
bulk. This is in agreement with previous studies that report an ultra-thin V5+
surface layer on VO2 films, even after in vacuo cleaning and surface
preparation.Quackenbush  _et al._ (2015)

Furthermore, even for those vanadium atoms with the expected ”bulk”
coordination, some distortion of inter-atomic bond distances is predicted
within the near surface layers of the slab. As both doping and strain are
known to drastically alter the VO2 MIT characteristics by pushing the
d-filling closer to or farther from the Mott criterion,Lee  _et al._ (2019)
the observed alterations to the coordination and inter-atomic distances
predicted at the terminated VO2 surface should also be expected to alter the
MIT characteristics.

##  II Suppression of the transition at the surface

Figure 2:  Averaged LEED patterns for (a) VO2(001), (b) VO2(100), and (c)
VO2(110) epitaxial thin films on TiO2 single crystal substrates, along with
structural models of the expected rutile and equivalent M1 surfaces. Vanadium
and oxygen atoms are represented by large blue and small red circles,
respectively. Surface unit cells are indicated by light blue rectangles.

Shown in Figure 2a, the LEED pattern from an as-loaded VO2(001) surface
displays square (p4mm) symmetry with lattice constants comparable to that of
the corresponding rutile TiO2 ($1\times 1$) surface. This is an averaged LEED
pattern of those measured at 37, 47, 50, and 73 eV beam energy, due to the
many spectral extinctions observed on this surface. Notably, a sharp (0,0)
spectral spot is not observed until the highest beam voltage. We observed no
changes to the LEED pattern upon cooling or heating over a temperature range
of -30–130∘C.

As epitaxial VO2/TiO2(001) films undergo their MIT near room temperature, any
MIT-related structural changes should have been observed during these
temperature-dependent experiments.Lee  _et al._ (2019); Quackenbush  _et al._
(2016) Since the cR crystal axis is out-of-plane for the (001) orientation,
the V-V dimerization in the M1 phase will occur perpendicular to the film
surface, which might be expected to show no signature in the LEED. However,
the concomitant zig-zagging of vanadium atoms in the M1 phase should result in
a doubling of the periodicity along one direction at the surface if the film
is terminated along a rutile (001) equivalent plane, as depicted by the
structure diagram in Fig. 2d. This periodicity doubling should correspond to
half-order spots in the LEED pattern, however, none were observed during the
experiments.

Shown in Fig. 2b, the LEED pattern from an as-loaded VO2(100) surface displays
rectangular (p2mm) symmetry with lattice constants comparable to that of the
corresponding rutile TiO2 ($1\times 1$) surface. This is an averaged LEED
pattern measured at 30 and 60 eV beam energy. This surface shows a (0,0) spot
at most energies, and spectral extinctions occur most often in second order
diffraction spots. Also, each diffraction spot on this surface is slightly
elongated along the (0,1) direction.

In contrast to the (001) orientation, VO2/TiO2(100) is oriented such that the
cR crystal axis is in-plane with the surface. Upon entering the low
temperature phase, the lowering of symmetry due to V-V dimerization should
result in a clear doubling of the unit cell along the cR axis, as depicted by
the structure diagram in Fig. 2e, which would be evident in the LEED as half-
order spots long the (0,1) direction.Cao  _et al._ (2009) In addition, this
in-plane cR axis is elongated by the epitaxial strain, which increases the MIT
temperature and also introduces the possibility of other intermediate phases
within the epilayer that may have signatures in the LEED pattern, such as the
M2 structure.Quackenbush  _et al._ (2016); Park  _et al._ (2013) However, no
intermediate spots were observed during the experiments.

Although the VO2(100) epilayer is known to be in the M1 phase at room
temperature, the observed room temperature surface unit cell corresponds to a
($1\times 1$) reconstruction of a bulk termination of the rutile unit cell.
And this rutile-like reconstruction is found to be stable from room
temperature up to at least 150 ∘C. This pattern is notably different to the
only other VO2(100) surface reported in the literature, which indeed showed
half-order spots, although they were observed along the direction
corresponding to the rutile b axis.Goering  _et al._ (1997)

The VO2(110) surface is similar to the VO2(100) surface in that the cR axis is
completely in-plane, however, the exposed surface for this orientation is a
diagonal cut through the 3D unit cell. The LEED pattern for this as-loaded
surface is shown in Fig. 2c. The pattern is similar to the bulk terminated
($1\times 1$) rutile TiO2(110) pattern, and corresponds to a ($1\times 1$)
rutile-like surface reconstruction of VO2.

As depicted by the structure diagram in Fig. 2f, terminating the M1 structure
along a rutile (110) equivalent plane would be expected to produce surface
ordering with a greatly expanded unit cell. This should produce fractional
spots along both the (0,1) and (1,0) directions in the LEED. However, similar
to the other orientations, this surface is found to show no temperature
dependence and remain rutile-like at all measured temperatures above and below
the MIT of the epilayer.

##  III Stability of different VO2 surfaces

Figure 3:  (a) LEED (43 eV), LEEM, and ADF STEM of a 7.5 nm VO2/TiO2(100) film
after annealing showing the formation of (110) facets on the film surface. (b)
Composite of LEED (25 eV) images taken before/after annealing a 10 nm
VO2/TiO2(001) film and the corresponding LEEM showing the existence of micro-
cracks after annealing. (c) Calculated Wulff shape of the rutile structure,
and surface energies for common surfaces of the VO2 rutile and monoclinic
crystal structures. Terminations were non-polar to obtain the lowest surface
energy.

Shown in Figure 3a, the VO2(100) LEED pattern begins to display a pronounced
elongation perpendicular to the cR direction following several anneal cycles
(P${}_{O_{2}}$ = $1.0\times 10^{-6}$ Torr). The corresponding LEEM image shows
the development of a highly directional, corrugated surface texture running
perpendicular to the cR direction, consistent with the striations observed in
the LEED. Cross-sectional scanning transmission electron microscopy (STEM)
annular dark field (ADF) images of the VO2 specimen viewed along the [001]
axis reveal nature of the surface texture. The anneal cycles induced the
formation of (110) facets along the surface, parallel to the [001] direction.
On this specimen, these facets have an average spacing of approximately 8.7
nm. Literature reports on TiO2(100) indicate that it also has a tendency to
reconstruct and form (110) micro-facets.Diebold (2003); Hardman  _et al._
(1993); Landree  _et al._ (1998)

Typically, changes in the observed VO2(001) LEED pattern are very small after
several anneal cycles. On average the background is reduced slightly, although
the diffraction spots can also become less sharp with indications of some
vague faceting along the rutile [110] directions.Krisponeit  _et al._ (tion)
However, on occasion it is possible to observe another LEED pattern for the
VO2(001) surface wherein the first order spectral spots completely split.
Shown in Figure 3b, this new LEED pattern is associated with heating through
the MIT, although it is completely irreversible once formed and therefore not
associated with V-V dimerization. Corresponding LEEM displays long, straight
streaks on the film surface interpreted as micro-cracks formed to relieve
mechanical stresses within the film produced by the epitaxial strain. Previous
reports indicate similar cracks tend to form along the [010] and [100]
directions during the growth of much thicker films ($\sim$30 nm),Nagashima
_et al._ (2006); Paik  _et al._ (2015) while recent investigations by
Krisponeit et al. report the appearance of micro-cracks post-growth due to
thermal cycling.Krisponeit  _et al._ (tion)

DFT calculations of surface energies, given in Figure 3c, provide some insight
into the observed behavior of these VO2 films following annealing. All rutile
VO2 surfaces are predicted to be slightly lower energy than their respective
M1 counterparts, with the rutile (110) surface being the lowest energy of all.
This suggests VO2 surfaces may reconstruct into a rutile pattern regardless of
the bulk structure. In addition, VO2 would prefer to expose the (110) surface
if possible, explaining the (110) micro-facets observed in (100) orientation
films.

##  IV Oxygen-related reconstructions of the (110) surface

Figure 4:  LEEM ($\sim$1.5 eV) and LEED (30 eV) of a VO2/TiO2(110) film
comparing a region irradiated with the electron beam for several minutes to a
non-irradiated region, both (a) below and (b) above the MIT. (c) DFT predicted
stoichiometric surface and oxygen-rich (2$\times$2) surface reconstruction of
a VO2(110) slab. (d) DFT predicted surface energies of a VO2(110) slab with
oxygen chemical potential ($\Delta\mu_{O}$). $\Delta\mu_{O}=0$ eV is set to be
the oxygen chemical potential of O2 gas. For context, we drew vertical dashed
lines for the formation energy per oxygen atom of VO2 and V2O5 ($\Delta
G^{f}_{\ce{VO2}}$ and $\Delta G^{f}_{\ce{V2O5}}$) and the enthalpy of reaction
for V2O5 relative to VO2 ($\Delta H^{f}_{\ce{V2O5}}$).

After performing anneals on VO2(110), some slight surface striations can
appear in LEEM while the LEED spots become more diffuse and some higher-order
spots become visible, similar to the (100) surface behavior. The post-anneal
texture on this surface, however, appears different to that of the (100)
surface and is more likely due to the formation of step terraces, as this is
known to occur on TiO2(110) surfaces.Diebold (2003)

Shown in Figure 4a and b, it is possible to have a ($2\times 2$) reconstructed
surface with the (110) orientation after annealing at 200 ∘C under an oxygen
partial pressure of $1\times 10^{-5}$ Torr. However, this reconstructed
surface was observed at temperatures both above and below the MIT and XPS
confirms a change in the oxygen content of the film surface after the anneal
(see Supplementary Figure 3). Furthermore, after irradiating a sample region
with the electron beam for several minutes, the fractional LEED spots get
weaker and the pattern reverts to the original ($1\times 1$) pattern. A clear
change in work function is also observed with LEEM after this extended
electron beam exposure. This behavior implies the observed ($2\times 2$)
reconstruction is related to oxygen adsorption/desorption on the film surface,
instead of the bulk structural transition. This is further supported by our
DFT calculations which demonstrate larger variation in work function as oxygen
is adsorbed/desorbed on the rutile $(110)_{R}$ surfaces as opposed to
transitioning from a rutile to monoclinic $(011)_{M}$ surface (see
Supplementary Table I).

Of note, the (100) and (001) surfaces do not have this behavior when using an
oxygen partial pressure of $1\times 10^{-5}$ Torr, indicating this
reconstruction is specific to the (110) surface. This may be due to the more
polar nature of the (110) film compared to both (100) and (001). Our DFT
calculations demonstrate that cleaving the bulk crystal structure at different
facets theoretically produces surface configurations with varying oxygen
content.

We used DFT to investigate oxygen vacancy/adsorption on the VO2(110) surface
by removing V (adsorption) or O (vacancy) from VO2 slabs. In this context,
$\Gamma$ represents the excess number of O ions at each surface of the slab.
Only the (1$\times$1) rutile slab shown in Figure 4c was stable under a
stoichiometric configuration for the VO2(110) surface, while a (2$\times$2)
reconstruction was obtained for an oxygen rich surface ($\Gamma>1$). Slabs
with an intermediate oxygen excess/deficiency ($\Gamma=\pm 1$) were predicted
to have a ($\sqrt{2}\times\sqrt{2}$)-like surface symmetry.

More importantly, both the $\Gamma=1$ and (2$\times$2) reconstructed
$\Gamma=3$ oxygen rich slabs were predicted to possess vanadyl bonds on the
surface (see Supplementary Figure 4). These surface stabilizing vanadyl
terminations will effectively push the orbital occupation of the surface
vanadium away from the Mott criterion (e.g., $d^{n}\to 0$), similar to Ti
doping in the bulk,Quackenbush  _et al._ (2017) which should result in a
complete loss of Mott MIT physics.

As shown in Figure 4d, these oxidized surfaces are more likely to be stable
than the reduced surfaces for the $\Delta\mu_{O}$ range corresponding to where
bulk VO2 is thermodynamically stable with respect to V2O5 (-2.26 to -1.20
eV).Ong  _et al._ (2013) This is true for all film orientations investigated
in this study (see Supplementary Figure 5), explaining why the bulk transition
of the epitaxial VO2 films could not be observed at any of the surface
orientations investigated in this study.

##  V Conclusions

The inherent vanadyl forming behavior of VO2 surfaces must be considered when
developing future, ultra-thin, vanadium oxide devices, as it implies some
minimum film thickness greater than a mono-layer would be required for
switching in real-world devices. These results, combined with our previous
work showing switching in VO2 as thin as $\sim$1 nm,Quackenbush  _et al._
(2015) narrows the potential minimum thickness to between 1 to 3 unit cells.
Furthermore, these results also have important implications for
electrochemical applications of crystalline VO2, where oxygen redox active
surfaces may be desired.

Surface reconstructions of VO2 are driven primarily by the lowering of surface
energy through exposure of preferential planes and the formation (or removal)
of vanadyl species. As such, they can form and evolve independently of the
underlying bulk phase of the material. Even for the most correlated (Mott-
like) strain orientation investigated in this study, VO2/TiO2(110),Mukherjee
_et al._ (2016) we do not observe any evidence of the ’bulk’ structural
transition at the very surface. As a result, extreme caution must be taken
when employing surface sensitive spectromicroscopy techniques to investigate
MIT materials like VO2, as the independent behavior of the surface could
easily be misinterpreted as a decoupling or an appearance of exotic phases.

##  VI Methods

A set of high quality, epitaxial VO2 thin films were grown on rutile
TiO2(001), (100), and (110) single crystal substrates (Crystec GmbH, Germany)
by reactive MBE in a Veeco GEN10 system at Cornell University. Substrates were
first prepared by etching and annealing to have clean and well-defined step
and terrace microstructured surfaces.Yamamoto  _et al._ (2005) Vanadium and
distilled ozone were co-deposited onto the substrate held at 250 ∘C under a
distilled ozone background pressure of 1.0$\times$10-6 Torr. Following
deposition of the desired film thickness, the temperature of the sample was
rapidly ramped to 350 ∘C, then immediately cooled to below 100 ∘C under the
same background pressure of distilled ozone to achieve an improved film
smoothness and a more abrupt MIT. Sample thicknesses ranged from 1 to 10 nm
for each orientation, as determined by monitoring the oscillations in
reflection high energy electron diffraction (RHEED). Further details on the
growth methodology have been previously described in the literature.Tashman
_et al._ (2014); Paik  _et al._ (2015); Quackenbush  _et al._ (2016)

The quality and MIT switching behavior of the films were investigated after
growth using x-ray photoelectron and absorption spectroscopy before and after
LEED/LEEM surface studies in order to ensure the film epilayer was not
significantly altered by the employed surface preparation processes or by the
measurements themselves. All measurements were performed under an ultra-high
vacuum base pressure of $4\times 10^{-10}$ Torr or better. A washing procedure
was used to clean atmospheric contamination and over oxidation from the sample
surfacesSurnev, Ramsey, and Netzer (2003); Goering  _et al._ (1997);
Quackenbush  _et al._ (2017) before performing any surface sensitive
measurements (see Supplementary Figure 2 and the corresponding discussion).

Soft x-ray photoelectron spectroscopy (XPS) was performed at the Analytical
and Diagnostics Laboratory (ADL) at Binghamton University using a laboratory-
based monochromated PHI 5000 Versaprobe system. The spectra were measured
using a monochromated Al K$\alpha$ x-ray source and a pass energy of 23.5 eV,
corresponding to an instrumental resolution of 0.51 eV as determined from
analyzing the Fermi edge and Au 4f7/2 peak of Au foil references.

Hard x-ray photoelectron spectroscopy (HAXPES) and x-ray aborption
spectroscopy (XAS) were performed at the I09 beamline of the Diamond Light
Source (DLS) in Oxfordshire, UK. HAXPES spectra were measured at a 5.934 keV
photon energy using a VG Scienta EW4000 high-energy electron-energy analyzer
with a $\pm$ 30∘ acceptance angle. The HAXPES photon beam was monochromated
using a Si(111) double-crystal monochromator followed by a channel cut Si(004)
crystal, resulting in an energy resolution of $<$ 250 meV.Kelly and Alcock
(2013); Wangoh  _et al._ (2016); Quackenbush  _et al._ (2016) Binding energy
was calibrated using the Au 4f7/2 peak and Fermi edge position of clean Au
foil.

LEEM images were collected using Elmitec aberration-corrected low energy
electron microscope (AC-LEEM) equipped with an energy analyzer. LEED
measurements were performed using the LEEM optics, along with a microchannel
plate detector and 1.5 $\mu$m selected-area aperture. Unlike traditional LEED,
the positions of the diffraction spots on the detector do not depend on beam
energy using this experimental geometry. This allows for the straightforward
merging/averaging of multiple LEED images taken at different beam energies,
which circumvents structure factor-related extinctions that occur at certain
energies thus revealing the complete diffraction pattern. Corresponding TiO2
(1$\times$1) reconstructed surfaces for each orientation were prepared and
used for calibration of the VO2 LEED patterns.

Scanning transmission electron microscopy (STEM) data were recorded from
mechanically polished cross-sectional VO2/TiO2 specimens in the 100 keV NION
UltraSTEM, a 5th order aberration corrected microscope optimized for EELS
spectroscopic imaging with a probe size of $\sim$1 Åand a beam current of
$\sim$150 pA.

All slab and bulk DFT calculations for the R and M1 phases of VO2 were
performed using the Vienna Ab initio Simulation Package (VASP) within the
projector augmented wave (PAW) approach.Blöchl (1994); Kohn and Sham (1965);
Kresse and Furthmüller (1996); Blöchl (1994) The exchange-correlation effects
were modeled using the Perdew-Berke-Ernzerhof (PBE) generalized gradient
approximation (GGA) functional.Perdew, Burke, and Ernzerhof (1996) We also
applied the Hubbard U parameter with U=3.25 eV to all calculations in
accordance with Wang et al.Wang, Maxisch, and Ceder (2006); Jain  _et al._
(2011) as unlike GGA, GGA+U correctly demonstrates that the M1 phase is more
stable than the R phase.

All calculations were performed with a plane wave cutoff of 500 eV. The
pseudopotentials used are similar to those used in the Materials Project.Jain
_et al._ (2013) The energies and atomic forces of all calculations were
converged within $10^{-4}$ eV and 0.02 eV$\mbox{\AA}^{-1}$ respectively. Slabs
were generated using the method described by Sun et al.Sun and Ceder (2013)
$\Gamma$-centered k-point meshes of
$\frac{\bf{a}}{50}\times\frac{\bf{b}}{50}\times 1$, where $\bf{a}$ and
$\bf{b}$ are the the lattice parameters of the slab cell, were used for the
slab calculations with non-integer values rounded up to the nearest integer.
All input generation, surface energy analysis and Wulff shape generation was
performed with the Python Materials Genomics (pymatgen) materials analysis
library.Ong  _et al._ (2013); Tran  _et al._ (2016)

Like with previous DFT studies of VO2Mellan and Grau-Crespo (2012); Eyert
(2011); Wickramaratne, Bernstein, and Mazin (2019), we omitted spin-
polarization when comparing the surface energies (Table 3c) and work functions
(Supplementary Table I) of the monoclinic and rutile phases. However, both
ferromagnetic and non spin-polarized surfaces were considered when
investigating the non-stoichiometric surface energies of the rutile phase.
While both magnetic configurations yield the same stable terminations, the non
spin-polarized non-stoichiometric surface energies of the rutile phase yield
unphysical (negative) surface energies under a high oxygen chemical potential
(see Supplementary Figure 5). As such, the ferromagnetic surface energies are
shown in Figure 4d instead. For a further explanation of magnetic
considerations in DFT, the reader is directed to the Supplementary
Information.

The surface energies used to determine the relative stability of all slab
systems were calculated using the surface grand potential given by:

$\gamma=\frac{1}{2A}[E^{slab}-\sum_{i}N_{i}\mu_{i}]$ (1)

where $E^{slab}$ is the total energy of the slab, $i$ is a unique species,
$\mu_{i}$ is the chemical potential of species $i$ and $N_{i}$ is the total
number of $i$ atoms in the slab, A is the surface area and the factor of 2
accounts for the slab model containing two symmetrically equivalent
surfaces.Rogal and Reuter (2007) Further explanation and details into surface
thermodynamics and slab generation can be found in the Supplementary
Information.

##  VII References

## References

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##  VIII Acknowledgments

We thank Marc Sauerbrey for his support in the initial phase of the project
and Jens Falta for stimulating discussions. This work was supported by the Air
Force Office of Scientific Research Multi-Disciplinary University Research
Initiative (MURI) entitled, Cross-disciplinary Electronic-ionic Research
Enabling Biologically Realistic Autonomous Learning (CEREBRAL), under award
number FA9550-18-1-0024. We acknowledge Diamond Light Source for time on
Beamline I09 under Proposals SI25355 and SI13812 for XAS measurements. This
research used resources of the Center for Functional Nanomaterials, which is
U.S. Department of Energy (DOE) Office of Science facility at Brookhaven
National Laboratory, under Contract No. DE-SC0012704. We acknowledge support
from the National Science Foundation [Platform for the Accelerated
Realization, Analysis, and Discovery of Interface Materials (PARADIM)] under
Cooperative Agreement No. DMR-1539918. Work was performed in part at the
Cornell NanoScale Facility, a member of the National Nanotechnology
Coordinated Infrastructure (NNCI), which is supported by the National Science
Foundation (Grant No. ECCS-1542081). Work by R.T and S.P.O was supported by
Quantum Materials for Energy Efficient Neuromorphic Computing (Q-MEEN-C), an
Energy Frontier Research Center funded by DOE, Office of Science, BES under
Award # DE-SC0019273. J-OK and JIF gratefully acknowledge support by the
Deutsche Forschungsgemeinschaft (DFG) under project nos. 362536548 and
408002857.

##  IX Author contributions

The contributions of authors are as follows. LFJP and NFQ conceived of the
study. MJW and NFQ performed the bulk of the data analysis, generated the
figures, and wrote the manuscript. MJW and NFQ performed the temperature
dependent HAXPES/XAS. JTS, MJW, and NFQ performed the LEED/LEEM/UPS
measurements. J-OK and JIF aided with the structural analysis and
interpretation of the LEED results. JIF performed initial VO2 surface studies
and helped develop the VO2 surface cleaning recipe. RT and SPO performed the
surface stability calculations and corresponding DFT, as well as aided in
preparing the manuscript for submission. CS and T-LL aided with the
synchrotron experiments at the Diamond Light Source. MH and DAM performed the
TEM imaging. HP and DGS are responsible for the sample fabrication and initial
characterization of sample quality.

##  X Additional information

Supplementary information is available in the online version of the paper.
Reprints and permissions information is available online at
www.nature.com/reprints. Correspondence and requests for materials should be
addressed to Louis F J Piper<EMAIL_ADDRESS>Shyue Ping Ong,
<EMAIL_ADDRESS>
##  XI Competing financial interests

The authors declare no competing financial interests.


# Electromagnetic Reaction on
an Accelerating Point Charge

Jerrold Franklin111Internet<EMAIL_ADDRESS>
Department of Physics
Temple University, Philadelphia, PA 19122

###### Abstract

We derive the rate of emission of electromagnetic energy by an accelerating
point charge by calculating the rate of increase of electromagnetic energy in
a localized volume. This gives the rate of emission in terms of the
acceleration at the present time. This contrasts with the usual derivation,
which calculates the energy radiated through the surface of a large sphere,
and gives the rate of radiated energy in terms of the acceleration at the
retarded time. Replacing the retarded time acceleration by the present time
acceleration is a significant correction, which resolves some problems
associated with using the retarded time to describe the emission of
electromagnetic energy.

##  1 Introduction

The standard formula for radiation reaction on a point charge, accelerating
parallel to its velocity, is usually written as222We are using Gaussian units
with $c=1$.(See, for instance [1, 2].)

$\displaystyle\frac{dW_{\rm rad}}{dt}$ $\displaystyle=$
$\displaystyle\frac{2}{3}q^{2}a^{\prime 2},$ (1)

where the acceleration, ${\bf a^{\prime}}=\gamma^{3}{\bf a}$, is the
acceleration of the charged particle in its instantaneous rest frame.

Derivations of this equation generally start with the Lienard-Wiechert field
equations for acceleration parallel to the velocity,

$\displaystyle{\bf E(r},t)$ $\displaystyle=$
$\displaystyle\frac{q({\bf\hat{r}}_{r}-{\bf
v}_{r})}{r_{r}^{2}\gamma_{r}^{2}(1-{\bf\hat{r}}_{r}\mbox{\boldmath$\cdot$}{\bf
v}_{r})^{3}}+\frac{q[{\bf\hat{r}}_{r}({\bf\hat{r}}_{r}\mbox{\boldmath$\cdot$}{\bf
a}_{r})-{\bf a_{r}}]}{r_{r}(1-{\bf\hat{r}}_{r}\mbox{\boldmath$\cdot$}{\bf
v}_{r})^{3}},$ (2) $\displaystyle{\bf B(r},t)$ $\displaystyle=$
$\displaystyle{\bf\hat{r}_{r}}\mbox{\boldmath$\times$}{\bf E(r},t),$ (3)

where the variables, ${\bf r_{r},v_{r}},\gamma_{r}=1/\sqrt{1-v_{r}^{2}}$, and
$\bf a_{r}$ are all evaluated at the retarded time,

$t_{r}=t-r_{r}.$ (4)

The radius vector, ${\bf r}_{r}$, is the distance from the charged particle’s
position at the retarded time to the point of observation of the
electromagnetic fields at the present time.

Although the derivations start with all the variables given at the retarded
time, Eq. (1) seems to give the result at the present time $t$. At some point
in derivations, the retarded time is changed to the present time with no
explanation. This means that Eq. (1) should really be written as

$\displaystyle\frac{dW_{\rm rad}}{dt_{r}}$ $\displaystyle=$
$\displaystyle\frac{2}{3}q^{2}a^{\prime 2}_{r},$ (5)

in terms of the retarded acceleration, $a^{\prime}_{r}$, and not the
acceleration at the present time.

It appears that there has been an implicit assumption in derivations that
$t_{r}\simeq t$ is a good approximation for small enough $r_{r}$ in Eq. (4).
However, the assumption that $r_{r}$ is ‘small enough’ is at odds with the
common assumption that $r_{r}$ is ‘large enough’ to drop the velocity term in
the Lienard-Wiechert fields.

In this paper, we resolve this situation by showing that Eq. (1) for the
acceleration at the present time is the appropriate equation, but the usual
derivation is at fault. We present a new derivation, giving ‘electromagnetic
reaction’ (instead of ‘radiation reaction’) as

$\displaystyle\frac{dW_{\rm EM}}{dt}$ $\displaystyle=$
$\displaystyle\frac{2}{3}q^{2}a^{\prime 2},$ (6)

correctly giving the emitted power in terms of the acceleration at the present
time.

What do we mean by ‘electromagnetic reaction’ rather than ‘radiation
reaction’? The rate of emission of electromagnetic energy is given by
Poynting’s theorem, equating the rate of change of the energy in matter to the
electromagnetic fields E and B. Poynting’s theorem can be written as

$\displaystyle\frac{dW_{\rm mat}}{dt}$ $\displaystyle=$
$\displaystyle\int_{V}{\bf
j\mbox{\boldmath$\cdot$}E}d^{3}r=-\left[\frac{1}{8\pi}\int_{V}\partial_{t}({\bf
E}^{2}+{\bf B}^{2})d^{3}r+\frac{1}{4\pi}\oint_{S}{\bf
dS\mbox{\boldmath$\cdot$}(E\mbox{\boldmath$\times$}B)}\right]$ (7)
$\displaystyle=$ $\displaystyle-\left[\frac{dW_{\rm EM}}{dt}+\frac{dW_{\rm
rad}}{dt}\right].$

This shows that the rate of change of the energy of the matter in a volume is
diminished by the rate of increase of the electromagnetic energy in the
volume,

$\displaystyle\frac{dW_{\rm EM}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int_{V}\partial_{t}({\bf E}^{2}+{\bf
B}^{2})d^{3}r,$ (8)

plus the rate at which electromagnetic energy leaves the volume through its
surface,

$\displaystyle\frac{dW_{\rm rad}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\oint_{S}{\bf
dS\mbox{\boldmath$\cdot$}(E\mbox{\boldmath$\times$}B)}.$ (9)

Derivations of Eq. (1) have been based on Eq. (9) for radiation through a
surface with radius large to eliminate the velocity fields. This would lead to
everything being evaluated at the retarded time, but somehow the switch
$t_{r}\rightarrow t$ is slipped into the derivation.

We show in this paper that it is the increase of energy within a volume, given
by Eq. (8), and not the radiation term of Eq. (9), that contributes to the
emission of electromagnetic energy. We show that this leads to a proper
derivation of $\frac{dW_{\rm EM}}{dt}$ in terms of the present time
acceleration. For some reason, the contribution of the electromagnetic energy
term of Eq. (8) has not been considered in previous derivations of Eq. (1).

In Section 2 of this paper, we find the electromagnetic energy emitted into a
volume by an accelerating point charge. Then, in Section 3, we take its time
derivative to calculate the rate of emission, $\frac{dW_{\rm EM}}{dt}$, of
electromagnetic energy by the accelerating point charge.

We show in Section 3 that the actual effect due to the acceleration of a
charged particle is not the ‘radiation reaction’ of Eq. (5) at the retarded
time, but the rate of increase of electromagnetic energy (‘electromagnetic
reaction’), given by Eq. (1) at the present time.

In Section 4, we use the electromagnetic reaction derived in Section 3 to
demonstrate the modification in the differential equation for the acceleration
of a point charge by an external force. We discuss the differences in
‘electromagnetic reaction’ and ‘radiation reaction’ as seen in Eqs. (1) and
(5) in Section 5.

##  2 Electromagnetic Energy of an Accelerating Point Charge

The electromagnetic energy of the $\bf E$ and $\bf B$ fields of an
accelerating point charge is given by

$\displaystyle W_{\rm EM}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int(E^{2}+B^{2})d^{3}r.$ (10)

A problem arises because the volume integral is over the present time radius,
$\bf r$, while the Lienard-Wiechert $\bf E$ and $\bf B$ fields are given in
terms of the retarded time radius, $\bf r_{r}$. Consequently, we have to
change the variable of integration from $d^{3}r$ to $d^{3}r_{r}$.

Equation (4) for the retarded time can be put in differential form as

$dt=dt_{r}+dr_{r}=dt_{r}+{\bf{\hat{r}_{r}}\mbox{\boldmath$\cdot$}dr_{r}}=dt_{r}-{\bf{\hat{r}_{r}}\mbox{\boldmath$\cdot$}v_{r}}dt_{r}.$
(11)

The change in sign for the last term in Eq. (11) is due to the fact that the
change in the vector $\bf r_{r}$ is at the tail of the vector.

For a particle traveling in the x direction, the relation between the
differentials $dx$ and $dx_{r}$ can be seen by multiplying each term in Eq.
(11) by the velocity, $v_{r}$, resulting in

$v_{r}dt=(v_{r}/v)dx=(1-{\bf
v_{r}\mbox{\boldmath$\cdot$}{\hat{r}_{r}}})dx_{r}.$ (12)

The transverse coordinates of $\bf r$ and $\bf r_{r}$ are the same, so

$d^{3}r=\frac{v}{v_{r}}(1-{\bf
v_{r}\mbox{\boldmath$\cdot$}{\hat{r}_{r}}})d^{3}r_{r}$ (13)

gives the transformation of the variable of integration from $d^{3}r$ to
$d^{3}r_{r}$.

With this change of variables, the electromagnetic energy emitted by a point
charge accelerating parallel to its velocity is given by the integral

$\displaystyle W_{\rm EM}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int\left(\frac{v}{v_{r}}\right)(1-{\bf
v_{r}\mbox{\boldmath$\cdot$}{\hat{r}_{r}}})(E^{2}+B^{2})d^{3}r_{r}$ (14)
$\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int\left(\frac{v}{v_{r}}\right)(1-{\bf
v_{r}\mbox{\boldmath$\cdot$}{\hat{r}_{r}}})E^{2}d^{3}r_{r}.$

The electric and magnetic fields for a point charge, accelerating parallel to
its velocity, are given by the Lienard-Wiechert field equations in Eqs. (2)
and (3). We will show below that the velocity term reduces to the Lorentz
transformation of the Coulomb field to the moving frame, and should not be
considered as part of the increase in electromagnetic energy. Consequently, we
leave the velocity term out of our equations for $\bf E$. (Note that this does
not require large $r_{r}$.)

Then, the electromagnetic energy is given by

$\displaystyle W_{\rm EM}$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{4\pi}\int\frac{v[{\bf\hat{r}}_{r}({\bf\hat{r}}_{r}\mbox{\boldmath$\cdot$}{\bf
a}_{r})-{\bf
a}_{r}]^{2}d^{3}r_{r}}{v_{r}r^{2}_{r}(1-{\bf\hat{r}}_{r}\mbox{\boldmath$\cdot$}{\bf
v}_{r})^{5}}$ (15) $\displaystyle=$
$\displaystyle\frac{q^{2}}{4\pi}\int\frac{va_{r}^{2}[1-({\bf\hat{r}}_{r}\mbox{\boldmath$\cdot$}{\bf\hat{v}}_{r})^{2}]d^{3}r_{r}}{v_{r}r_{r}^{2}(1-{\bf\hat{r}}_{r}\mbox{\boldmath$\cdot$}{\bf
v}_{r})^{5}}$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{2}\int_{0}^{R}\frac{va_{r}^{2}dr_{r}}{v_{r}}\int_{0}^{\pi}\frac{\sin\theta_{r}d\theta_{r}(1-\cos^{2}\theta_{r})}{(1-v_{r}\cos\theta_{r})^{5}}$
$\displaystyle=$
$\displaystyle\frac{2q^{2}}{3}\int_{0}^{R}\frac{va_{r}^{2}\gamma_{r}^{6}dr_{r}}{v_{r}}.$

Equation (15) gives the energy inside a sphere of radius R. The radius of the
sphere can be arbitrary in calculating the rate of electromagnetic energy
emission, as long as $R<T$, the time since the acceleration started. Although
the energy inside the sphere does depend on its radius, we show below that the
rate of energy emission does not.

##  3 Electromagnetic Reaction of
an Accelerating Point Charge

The rate of emission of electromagnetic energy is given by the time derivative
of the electromagnetic energy emitted by the accelerating point charge. In
order to facilitate taking the time derivative of Eq. (15), we change the
integration variable from the retarded radius, $r_{r}$, to
$(\gamma_{r}v_{r})$, using the relation (for $\bf a$ parallel to $\bf v$)

$\displaystyle\frac{{\bf d}(\gamma{\bf v})}{dt}$ $\displaystyle=$
$\displaystyle\gamma{\bf a}+\gamma^{3}{\bf
v(v\mbox{\boldmath$\cdot$}a})=\gamma^{3}{\bf a}={\bf a^{\prime}}.$ (16)

The acceleration, ${\bf a^{\prime}}$, is the charged particle’s acceleration
in its instantaneous rest frame. (We will use the rest frame acceleration,
$\bf a^{\prime}$, in the remainder of the paper.)

From Eq. (16), we see that

$\displaystyle d(\gamma_{r}v_{r})$ $\displaystyle=$ $\displaystyle{\bf
a^{\prime}}_{r}dt_{r}=-{\bf a^{\prime}_{r}}dr_{r}.$ (17)

In the last step above, we have used the relation $dt_{r}=-dr_{r}$ from Eq.
(4), with the $t$ in Eq. (4) being fixed at the present time, $T$.

Then Eq. (15) becomes

$\displaystyle W_{\rm EM}$ $\displaystyle=$
$\displaystyle\frac{2}{3}q^{2}\int_{(\gamma_{R}V_{R})}^{(\gamma_{V}V)}\frac{va^{\prime}_{r}d(\gamma_{r}v_{r})}{v_{r}},$
(18)

where we have changed the variable of integration from $r_{r}$ to
$(\gamma_{r}v_{r})$. The upper limit of the integral is the product of the
velocity, $V$, and $\gamma_{V}$, at the present time, $T$. The lower limit is
the product of the velocity, $V_{R}$, and $\gamma_{R}$, at the retarded time
corresponding to the radius, $R$, of the original volume integral.

The rate of emission of the electromagnetic energy is given by

$\displaystyle\frac{dW_{\rm EM}}{dT}$ $\displaystyle=$
$\displaystyle\frac{2}{3}q^{2}\frac{d}{dT}\int_{(\gamma_{0}V_{0})}^{(\gamma_{V}V)}\frac{va^{\prime}_{r}d(\gamma_{r}v_{r})}{v_{r}}.$
(19)

The time derivative in Eq. (19) acts only the upper limit, $\gamma_{V}V$, of
the integral, since $v$, $v_{r}$, and $a^{\prime}_{r}$ are variables of
integration. Then, Eq. (19) becomes

$\displaystyle\frac{dW_{\rm EM}}{dT}$ $\displaystyle=$
$\displaystyle\frac{2}{3}q^{2}a^{\prime}\left[\frac{d(\gamma_{V}V)}{dT}\right]$
(20) $\displaystyle=$ $\displaystyle\frac{2}{3}q^{2}a^{\prime 2}.$

The integration variables, $v$, and $v_{r}$, have each been evaluated at the
upper limit of the integral, where $v=v_{r}=V$, the velocity of the point
charge at the present time $T$. The radii, $r$ and $r_{r}$, are each approach
zero at the upper limit of the integral in Eq. (19), which means that the
emission of the electromagnetic energy is taking place at the present position
of the accelerating charge, which was to be expected. That is why the value of
the radius, $R$, in Eq. (15) had no effect on the emission rate.

We also note that the velocity in the Lienard-Wiechert velocity term is fixed
at the present time, so the velocity term just corresponds to the electric
field of a point charge moving at constant velocity. This justifies our
dropping it from the Lienard-Wiechert electric field at the beginning.

##  4 Application of Electromagnetic Reaction

We now want to relate $\frac{dW_{\rm EM}}{dt}$ to its effect on the motion of
an accelerating point charge. For an accelerating particle of mass $m$, the
rate of change of its kinetic energy is given (for $\bf a$ parallel to $\bf
v$) by

$\frac{dW_{\rm
mat}}{dt}=\frac{d(m\gamma-m)}{dt}=m\gamma^{3}{\bf(v\mbox{\boldmath$\cdot$}a)}=mva^{\prime}.$
(21)

If an external force acts on the charged particle, the electromagnetic power
produced by the external force, will increase the sum of the particle’s
kinetic energy and the electromagnetic energy at the rate

$\displaystyle\frac{dW_{\rm ext}}{dt}$ $\displaystyle=$
$\displaystyle\frac{dW_{\rm mat}}{dt}+\frac{dW_{\rm EM}}{dt}.$ (22)

This would reduce the particle’s velocity and acceleration by

$\displaystyle mva^{\prime}$ $\displaystyle=$ $\displaystyle
m{\bar{v}}{\bar{a}^{\prime}}-\frac{2}{3}q^{2}a^{\prime 2},$ (23)

where $\bar{v}$ and $\bar{a}^{\prime}$ would be the velocity and acceleration
of an uncharged partical.

We note that the increase in electromagnetic energy does not correspond to an
added force on the charged particle, but just electromagnetic energy produced
in space by the external force. The external force accelerates the charged
particle, and also produces electromagnetic energy. The energy going into the
electromagnetic field reduces the acceleration of the particle. The external
force acts on two separate entities, the accelerating particle and the
electromagnetic field, so the energy put into the electromagnetic field should
not be considered as a separate force on the accelerating particle.

##  5 Radiation Reaction?

What about ‘radiated reaction’, derived from the surface integral

$\displaystyle\frac{dW_{\rm rad}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\oint_{S}{\bf
dS\mbox{\boldmath$\cdot$}(E\mbox{\boldmath$\times$}B)},$ (24)

which would lead to Eq. (5):

$\displaystyle\frac{dW_{\rm rad}}{dt_{r}}$ $\displaystyle=$
$\displaystyle\frac{2}{3}q^{2}a^{\prime 2}_{r}?$ (25)

In fact, this result does not represent electromagnetic energy emitted by the
accelerating charge at the present time. It just represents the rate of
electromagnetic energy transfer from just inside the spherical surface of
radius, $R$, to just outside the surface, with no increase in the energy
produced by the accelerating charge. Therefore, it has no effect on the
emission of electromagnetic energy at the present time.

Although Eq. (25) looks somewhat like Eq. (20), the two results have
significant differences. The acceleration, $\bf a^{\prime}$, in Eq. (20) is
the acceleration at the present time, $T$, which is the appropriate time for
the rate of change of electromagnetic energy at that time. The acceleration,
${\bf a^{\prime}}_{r}$, in Eq. (25) is the acceleration at the retarded time,
$t_{r}=T-r$, due to the time delay, $\Delta t=r/c$, between the production and
observation of the electromagnetic fields. That is the appropriate time for
observing the electromagnetic fields, however it is not the appropriate time
for the rate of change of electromagnetic energy at the present time. The rate
of energy emission at the retarded time, given by Eq. (25), should not be used
in a differential equation like Eq. (23).

##  6 Conclusion

Our main conclusion is that the rate of emission of electromagnetic energy by
a point charge, accelerating parallel to its velocity, is

$\displaystyle\frac{dW_{\rm EM}}{dt}$ $\displaystyle=$
$\displaystyle\frac{2}{3}q^{2}a^{\prime 2},$ (26)

where the acceleration is evaluated at the present, and not the retarded time.

Although this result agrees with the equation that has generally been used for
radiation reaction, its correct derivation is quite different than derivations
that have been used in the past.

## References

  * [1] J. D. Jackson, Classical Electrodynamics, 3rd Ed. (John Wiley & Sons, 1999)
  * [2] D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, 1998)