# Magnetic Fields with General Omnigenity

Daniel W. Dudt1  Alan G. Goodman2  Rory Conlin1  Dario Panici1  Egemen
Kolemen1,3<EMAIL_ADDRESS>1Princeton University, Princeton, New
Jersey 08544, USA 2Max-Planck-Institut für Plasmaphysik, D-17491 Greifswald,
Germany 3Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA

###### Abstract

Omnigenity is a desirable property of toroidal magnetic fields that ensures
confinement of trapped particles. Confining charged particles is a basic
requirement for any fusion power plant design, but it can be difficult to
satisfy with the non-axisymmetric magnetic fields used by the stellarator
approach. Every ideal magnetohydrodynamic equilibrium previously found to
approximate omnigenity has been either axisymmetric, quasi-symmetric, or has
poloidally closed contours of magnetic field strength $B$. However, general
omnigenous equilibria are a much larger design space than these subsets. A new
model is presented and employed in the DESC stellarator optimization suite to
represent and discover the full parameter space of omnigenous equilibria.
Although exact omnigenity aside from quasi-symmetry is impossible, these
results reveal that excellent particle confinement can be achieved in
practice. Examples far from quasi-symmetry with poloidally, helically, and
toroidally closed $B$ contours are attained with DESC and shown to have low
neoclassical collisional transport and fast particle losses.

##  1 Introduction

Controlled nuclear fusion offers the possibility of reliable power generation
without greenhouse gas emissions to satiate the growing global energy demand.
The magnetic confinement approach to fusion involves containing a high
temperature plasma for long periods of time, and requires a magnetic geometry
than can confine these energetic charged particles. The stellarator concept
(Spitzer, 1958) makes an attractive fusion reactor: its reliance on external
magnetic fields rather than driving plasma current to confine the plasma
enables it to operate at steady state and makes it less susceptible to plasma
disruptions (Helander et al., 2012). Yet historically, stellarators were not
pursued due to their relatively high levels of neoclassical transport. Unlike
its axisymmetric counterpart the tokamak, stellarators are not guaranteed to
confine trapped particles in a collisionless plasma. Careful attention must be
paid to design a stellarator with comparably good particle confinement, but
this is becoming increasingly achievable with modern optimization tools.

Trapped particles cannot be avoided altogether in a toroidal geometry. If the
radial drifts of trapped particles vanish everywhere it is called an
isodynamic magnetic field, but this is similarly unrealistic (Palumbo, 1968;
Helander, 2014). The next best theoretical option is omnigenity, the class of
magnetic fields in which the bounce-averaged radial drifts of trapped
particles vanish (Hall & McNamara, 1975; Cary & Shasharina, 1997b). Pseudo-
symmetry is a broader category of fields in which the contours of constant
magnetic field strength $B=|\mathbf{B}|$ on a flux surface are closed curves
(Isaev et al., 1999). Omnigenity implies pseudo-symmetry and also that the
$B_{\mathrm{max}}$ contour is straight in Boozer coordinates (and other
straight field line coordinate systems with a Jacobian that only depends on
$B$ and the flux surface label) (Helander, 2014). There are three classes of
omnigenous magnetic fields: the $B$ contours can close either poloidally,
helically, or toroidally. These cases will be referred to as omnigenity with
poloidal contours (OP), omnigenity with helical contours (OH), and omnigenity
with toroidal contours (OT). [The term quasi-isodynamic (QI) has often been
used in previous literature to mean omnigenity with poloidally closed $B$
contours. This terminology has been inconsistent, however, and is easily
conflated with more general omnigenity; QI is avoided in favor of OP in this
article for the sake of clarity.] Quasi-symmetry (QS) is a special subset of
omnigenity where all contours of $B$ are straight in these coordinates, not
only the $B_{\mathrm{max}}$ contour (Nührenberg & Zille, 1988; Rodriguez et
al., 2020). QS magnetic fields are further subdivided by their helicity:
quasi-poloidal symmetry (QP), quasi-helical symmetry (QH), and quasi-
axisymmetry (QA) are the quasi-symmetric subspaces of OP, OH, and OT,
respectfully. The Venn diagram in figure 1 visualizes the relationship between
these different classes of magnetic fields.

Since omnigenity is a less restrictive requirement than quasi-symmetry, it
provides a larger design space with equivalent neoclassical confinement. This
extra flexibility could help satisfy additional physics and engineering
constraints on a stellarator reactor, such as reduced turbulent transport and
simplified coil geometry. Unfortunately, Cary & Shasharina (1997a, b) proved
that the only omnigenous fields that are analytic are those that are quasi-
symmetric. Moreover, Garren & Boozer (1991) suggested from a near-axis
expansion that exact quasi-symmetry cannot be satisfied beyond a single flux
surface. This is not a significant restriction in practice, however: QA and QH
solutions have been found with good QS throughout a volume (Landreman & Paul,
2022; Zarnstorff et al., 2001; Bader et al., 2020; Nührenberg & Zille, 1988;
Henneberg et al., 2019), and many solutions have been discovered that approach
OP while being far from QS (Jorge et al., 2022; Camacho Mata et al., 2022;
Goodman et al., 2023; Sánchez et al., 2023). The performance of QH and OP have
also been experimentally verified by the HSX and W7-X stellarators,
respectively. These devices were optimized to have approximately quasi-
symmetric or omnigenous designs, and both experiments have demonstrated
reduced neoclassical transport (Canik et al., 2007; Beidler et al., 2021).

These recent results have been restricted to QS or OP magnetic fields, which
excludes most of the full omnigenity solution space. Plunk et al. (2019)
explain that OT and OH magnetic fields (that are not QS) cannot be achieved
with a first order expansion in the distance from the magnetic axis, but admit
that these solutions could exist more generally. These classes of omnigenity
have previously been neglected due to the lack of an efficient method to
parameterize and explore their optimization space. Accessing this larger
design space would unlock new possibilities for stellarators, and is
worthwhile to consider despite its increased computational complexity. This
article presents an approach to achieve this goal, which is made possible
through the DESC stellarator optimization code suite (Dudt et al., 2024; Dudt
& Kolemen, 2020; Panici et al., 2023; Conlin et al., 2023; Dudt et al., 2023).
It is similar to the original idea of Cary & Shasharina (1997a), in the sense
that a perfectly omnigenous magnetic field is constructed to have the same
form on every field line in terms of some new coordinate. This omnigenous
field is generalized to any helicity, as was also done by Landreman & Catto
(2012). The difference between the actual equilibrium magnetic field and the
constructed target field is then penalized, like the technique devised by
Goodman et al. (2023).

In contrast to previous approaches, this article provides a parameterization
of the full phase space of all omnigenous magnetic fields, and is not
restricted to a near-axis expansion formulation or to stellarator symmetry.
Section 2 gives a derivation of the omnigenity model and explains how it is
used as an optimization target. Numerical examples of equilibria obtained from
this method are provided in section 3 for all six classes of omnigenity: OP,
QP, OH, QH, OT, and QA. This is the first demonstration of good neoclassical
confinement throughout a volume for realistic equilibria with arbitrary
omnigenous magnetic fields. The implications of these results and directions
for future work are discussed in section 4.

Figure 1: Classification of magnetic fields. The decreasing area of the inner
subsets represents the smaller dimensionality of these subspaces. Note that OP
refers to the same set of magnetic fields that are sometimes labeled QI.

##  2 Method

Figure 2: Example of an exact omnigenous magnetic field, for an OP case with
helicity $M=0,~{}N=1$ and the parameters
$x_{0,1,-1}=-\frac{\pi}{8},~{}x_{0,1,0}=\frac{\pi}{8},~{}x_{0,1,+1}=\frac{\pi}{4}$.
Left: The well shape $B(\eta)$ given by a monotonic spline. Center: The
magnetic well along different field lines, which all have the same bounce
distances $\Delta h$. Right: The $B$ contours plotted in Boozer coordinates
with a rotational transform of $\mbox{$\,\iota\\!\\!$-}=\frac{1}{4}$.

Let the “helicity” of the omnigenous magnetic field be defined by the pair of
integers $M$ and $N$ such that each $B$ contour closes on itself after
traversing the torus $M$ times toroidally and $N$ times poloidally. In
addition to having closed contours of magnetic field strength, the “bounce
distance” $\delta=\sqrt{\Delta\theta_{B}^{2}+\Delta\zeta_{B}^{2}}$ in Boozer
coordinates $(\theta_{B},\zeta_{B})$ along a field line between consecutive
points with equal magnetic field strength must be independent of the field
line label

$\alpha=\frac{\theta_{B}-\mbox{$\,\iota\\!\\!$-}\zeta_{B}}{N-\mbox{$\,\iota\\!\\!$-}M}$
(1)

where $\,\iota\\!\\!$- is the rotational transform. This normalization has the
useful property that $\alpha$ increases by $2\pi$ as a constant-$B$ curve is
followed until it closes on itself (Landreman & Catto, 2012).

###  2.1 Magnetic Well

First, the “magnetic well shape” is parameterized. Assuming the magnetic field
strength has the same maximum and minimum values along each field line, it can
be written in the form

$B=B(\rho,\eta)$ (2)

where $\rho$ is the flux surface label and contours of constant magnetic field
strength are now contours of the new coordinate $\eta$ (Cary & Shasharina,
1997b). The domain is chosen to be $-\frac{\pi}{2}\leq\eta\leq\frac{\pi}{2}$
such that $B(0)=B_{\mathrm{min}}$ and $B(\pm\frac{\pi}{2})=B_{\mathrm{max}}$.
$B(\eta)$ should be monotonically increasing from $B_{\mathrm{min}}$ to
$B_{\mathrm{max}}$ and satisfy the boundary conditions

$\frac{\partial B}{\partial\eta}\bigg{|}_{\eta=0}=\frac{\partial
B}{\partial\eta}\bigg{|}_{\eta=\pm\frac{\pi}{2}}=0.$ (3)

$B$ must also be an even function of $\eta$, which will be justified later. A
magnetic well shape that satisfies these criteria is represented by the
parameters

$B_{ij}=B(\rho_{i},\eta_{j})$ (4)

where the knot locations $\eta_{j}$ are linearly spaced in the domain
$[0,\frac{\pi}{2}]$ and the parameters are sorted to ensure monotonicity. $B$
at other locations on the flux surface $\rho_{i}$ are evaluated by monotonic
spline interpolation at $\eta=|\eta|$ to ensure it is an even function. The
$C^{1}$ continuity of these monotonic splines is not a limitation since the
omnigenous magnetic field is generally not analytic anyways. Omnigenity with
multiple unique local minima and maxima of $B$ on each flux surface can exist,
as shown by Parra et al. (2015), and this work could be extended to
accommodate those cases.

###  2.2 Coordinate Transformation

Now all that remains is to parameterize the transformation between the Boozer
coordinates $(\theta_{B},\zeta_{B})$ and the computational coordinates
$(\eta,\alpha)$. This mapping will be defined through the transformation

$h:(\theta_{B},\zeta_{B})\leftrightarrow(\rho,\eta,\alpha).$ (5)

In Boozer coordinates, the function $h$ is related to the helicity of the
omnigenous field:

$h(\theta_{B},\zeta_{B})=\begin{cases}N\zeta_{B}&\text{for }M=0\\\
-\theta_{B}+\frac{N}{M}\zeta_{B}&\text{for }M\neq 0.\end{cases}$ (6)

[Note that the Boozer angles are an implicit function of $\rho$, since the
transformation to Boozer coordinates is unique to each flux surface.] Contours
of constant $h$ are parallel to the $B_{\mathrm{max}}$ contour (which must be
straight in Boozer coordinates), and in the limit of quasi-symmetry $B=B(h)$.
In the computational coordinates, the function is parameterized as

$h(\rho,\eta,\alpha)=2\eta+\pi+\sum_{l=0}^{L_{\rho}}\sum_{m=0}^{M_{\eta}}\sum_{n=-N_{\alpha}}^{N_{\alpha}}x_{lmn}T_{l}(2\rho-1)F_{m}(\eta)F_{nN_{\mathrm{FP}}}(\alpha)$
(7)

where $N_{\mathrm{FP}}$ is the number of field periods, $T_{k}(y)$ are
Chebyshev polynomials of the first kind defined as $T_{k}(\cos y)=\cos(ky)$,
and $F_{k}(y)$ are Fourier series defined as

$F_{k}(y)=\begin{cases}\cos(|k|y)&\text{for }k\geq 0\\\ \sin(|k|y)&\text{for
}k<0.\end{cases}$ (8)

Chebyshev polynomials are chosen as the radial basis functions to allow for an
independent well shape on each flux surface. [Any set of orthogonal
polynomials would suffice. Zernike polynomials are not used since they would
impose additional unphysical boundary conditions near the magnetic axis.] The
coefficients $x_{lmn}$ in Eq. 7 parameterize the variation of the well shape
on different field lines, and $N_{\alpha}=0$ represents the subspace of quasi-
symmetry. The boundary condition $h(\rho,\eta=-\frac{\pi}{2},\alpha)=0$ is
required to ensure $B_{\mathrm{max}}$ is a straight contour, which is chosen
to be located at $h=0$ by construction. This boundary condition is enforced by
a linear constraint on the even Fourier modes:

$\sum_{m=0,2,4,\ldots}^{M_{\eta}}(-1)^{\frac{m}{2}}x_{lmn}=0.$ (9)

This constraint also satisfies the periodicity requirement that
$h(\eta=\frac{\pi}{2})=2\pi$ is another $B_{\mathrm{max}}$ contour.

###  2.3 Optimization Objective Function

The optimization function penalizes the difference between the equilibrium
magnetic field $B_{\mathrm{eq}}$ and the constructed omnigenous field $B$,
similar to the approach used by Goodman et al. (2023). The errors are
evaluated at a series of collocation points in the computational coordinates
as:

$f_{\mathrm{om}}(\rho_{i},\eta_{i},\alpha_{i})=\big{[}B_{\mathrm{eq}}(\rho_{i},\eta_{i},\alpha_{i})-B(\rho_{i},\eta_{i})\big{]}w(\eta_{i})$
(10)

where the weighting function
$w(\eta_{i})=\frac{3}{2}+\frac{1}{2}\cos(2\eta_{i})$ is used to weight the
minimum of the magnetic well relative to the maximum. The omnigenous target
field $B$ can be readily evaluated at these collocation points from the
magnetic well shape parameters $B_{ij}$ presented in Eq. 4. The equilibrium
field $B_{\mathrm{eq}}$, however, must be evaluated at the Boozer coordinates
that correspond to the desired computational coordinates, using the mapping
defined by Eq. 5.

Simultaneously solving Eqs. 1 and 6 gives the system of equations

$\displaystyle\begin{bmatrix}\alpha\\\
h(\rho,\alpha,\eta)\end{bmatrix}=\begin{bmatrix}\frac{1}{N}&-\frac{\mbox{$\,\iota\\!\\!$-}}{N}\\\
0&N\end{bmatrix}\begin{bmatrix}\theta_{B}\\\ \zeta_{B}\end{bmatrix}~{}$
$\displaystyle\text{for }M=0,N\neq 0$ (11a)
$\displaystyle\begin{bmatrix}\alpha\\\
h(\rho,\alpha,\eta)\end{bmatrix}=\begin{bmatrix}\frac{-1}{M\mbox{$\,\iota\\!\\!$-}}&\frac{1}{M}\\\
-1&0\end{bmatrix}\begin{bmatrix}\theta_{B}\\\ \zeta_{B}\end{bmatrix}~{}$
$\displaystyle\text{for }M\neq 0,N=0$ (11b)
$\displaystyle\begin{bmatrix}\alpha\\\
h(\rho,\alpha,\eta)\end{bmatrix}=\begin{bmatrix}\frac{1}{N-M\mbox{$\,\iota\\!\\!$-}}&\frac{-\mbox{$\,\iota\\!\\!$-}}{N-M\mbox{$\,\iota\\!\\!$-}}\\\
-1&\frac{N}{M}\end{bmatrix}\begin{bmatrix}\theta_{B}\\\
\zeta_{B}\end{bmatrix}~{}$ $\displaystyle\text{for }M\neq 0,N\neq 0$ (11c)

which can be inverted to yield

$\displaystyle\begin{bmatrix}\theta_{B}\\\
\zeta_{B}\end{bmatrix}=\begin{bmatrix}N&\frac{\mbox{$\,\iota\\!\\!$-}}{N}\\\
0&\frac{1}{N}\end{bmatrix}\begin{bmatrix}\alpha\\\
h(\rho,\alpha,\eta)\end{bmatrix}~{}$ $\displaystyle\text{for }M=0,N\neq 0$
(12a) $\displaystyle\begin{bmatrix}\theta_{B}\\\
\zeta_{B}\end{bmatrix}=\begin{bmatrix}0&-1\\\
M&\frac{-1}{\mbox{$\,\iota\\!\\!$-}}\end{bmatrix}\begin{bmatrix}\alpha\\\
h(\rho,\alpha,\eta)\end{bmatrix}~{}$ $\displaystyle\text{for }M\neq 0,N=0$
(12b) $\displaystyle\begin{bmatrix}\theta_{B}\\\
\zeta_{B}\end{bmatrix}=\begin{bmatrix}N&\frac{M\mbox{$\,\iota\\!\\!$-}}{N-M\mbox{$\,\iota\\!\\!$-}}\\\
M&\frac{M}{N-M\mbox{$\,\iota\\!\\!$-}}\end{bmatrix}\begin{bmatrix}\alpha\\\
h(\rho,\alpha,\eta)\end{bmatrix}~{}$ $\displaystyle\text{for }M\neq 0,N\neq
0.$ (12c)

The mapping parameters $x_{lmn}$ are used to compute $h(\rho,\eta,\alpha)$
through Eq. 7, and then Eq. 12 gives the corresponding Boozer coordinates such
that $B_{\mathrm{eq}}(\rho,\eta,\alpha)$ can be evaluated in Boozer
coordinates as $B_{\mathrm{eq}}(\theta_{B},\zeta_{B})$. These values of the
equilibrium magnetic field strength are then used to compute the deviation
from omnigenity in Eq. 10. Note that Eq. 12c is undefined when
$\mbox{$\,\iota\\!\\!$-}=\frac{N}{M}$, but this case is physically
uninteresting because it corresponds to field lines being parallel to the
$B_{\mathrm{max}}$ contour (for example, an axisymmetric vacuum field with no
rotational transform). If $M=0$ then it must be the case that $N\neq 0$, so
Eq. 12a is also always well defined. See figure 2 for an illustration of how
an omnigenous magnetic field is constructed and mapped to Boozer coordinates.

###  2.4 Results

It can be shown that this model is guaranteed to yield a perfectly omnigenous
target magnetic field. Since the straight field line coordinate system has the
property $\Delta\theta_{B}=\mbox{$\,\iota\\!\\!$-}\Delta\zeta_{B}$ along a
field line and $\,\iota\\!\\!$- is constant on each flux surface, and also
because $h(\theta_{B},\zeta_{B})$ is always a linear combination of the Boozer
angles, the bounce distance is directly proportional to the change in $h$:
$\delta\propto\Delta h$. The bounce points on any field line are always
$\pm\eta$ by construction, since Eq. 2 was chosen to be an even function of
$\eta$. Therefore the bounce distance can be computed as:

$\displaystyle\delta\propto\Delta h$
$\displaystyle=h(\rho,+\eta,\alpha)-h(\rho,-\eta,\alpha)$ (13a)
$\displaystyle=4\eta+\sum_{l=0}^{L_{\rho}}\sum_{m=0}^{M_{\eta}}\sum_{n=-N_{\alpha}}^{N_{\alpha}}x_{lmn}\Big{[}$
(13b) $\displaystyle
T_{l}(2\rho-1)F_{nN_{\mathrm{FP}}}(\alpha)\Big{(}F_{m}(+\eta)-F_{m}(-\eta)\Big{)}\Big{]}$
(13c) $\displaystyle=4\eta.$ (13d)

In the last step, the Fourier series in $\eta$ cancel each other out because
$\sum_{m\geq 0}F_{m}(\eta)$ is an even function of $\eta$. This ensures that
the bounce distances set by the parameters $B_{ij}$ will be preserved across
all field lines, regardless of the variation introduced by the parameters
$x_{lmn}$. However, there are bounds on the allowable amplitudes of these
coefficients to ensure Eq. 7 is a single-valued function.

Since the basis functions span the full domain, this model can approximate any
omnigenous magnetic field in the limit of increasing resolution (although this
work is restricted to fields with a single magnetic well). Whether a
particular omnigenous target can be well approximated by an equilibrium
solution is an unresolved question. The total number of coefficients used to
describe an omnigenous magnetic field in Eq. 7 is
$(L_{\rho}+1)(M_{\eta}+1)(2N_{\alpha}+1)$. Quasi-symmetry corresponds to the
condition $N_{\alpha}=0$, so that the magnetic well shape is identical on all
field lines and $B=B(h(\rho,\eta))$. Assuming $L_{\rho}=M_{\eta}=N_{\alpha}$,
the dimension of the full omnigenous parameter space scales as
$\mathcal{O}(M_{\eta}^{3})$ while the quasi-symmetric subspace has a dimension
of $\mathcal{O}(M_{\eta}^{2})$. This scaling reveals that quasi-symmetry is
only a small set of the possible solutions with good particle confinement, and
motivates the discovery of stellarator equilibria in the much broader
omnigenous design space. [This analysis is independent of the dimension of the
magnetic well parameterization $B_{ij}$.] Stellarator symmetry imposes no
constraints on the parameters $x_{lmn}$ due to the nonlinear nature of the
coordinate mapping in Eq. 5. This implies that the size of the parameter space
is not reduced by stellarator symmetry, but consequently no additional
computational complexity is required if it is not assumed.

##  3 Examples

Numerical examples from each class of omnigenity are presented: the three
types of quasi-symmetry (QP, QH, and QA) and their more general omnigenous
counterparts (OP, OH, and OT, respectfully). These six solutions were obtained
in DESC using the following optimization procedure. First, an initial guess
was generated from the near-axis expansion code pyQIC (Jorge, 2024) (for the
OP and QP case) or pyQSC (Landreman & Sengupta, 2019; Landreman et al., 2019)
(for the other cases). It was evaluated at an aspect ratio of $R_{0}/a\approx
20$, since the near-axis expansions are most accurate at high aspect ratios.
This configuration was re-solved in DESC while preserving the
$\mathcal{O}(\rho)$ behavior of the near-axis expansion solution. The
equilibrium was then optimized using a multi-objective cost function of the
form

$\mathbf{f}=\begin{bmatrix}f_{\mathrm{om}}(\rho,\eta,\alpha)\\\ \sigma
f_{\mathrm{eq}}(\rho,\theta,\zeta)\end{bmatrix}$ (14)

where $f_{\mathrm{om}}(\rho,\eta,\alpha)$ are the omnigenity errors from Eq.
10, $f_{\mathrm{eq}}(\rho,\theta,\zeta)$ are the equilibrium force balance
errors as described in Dudt & Kolemen (2020), and $\sigma$ is a relative
weighting between the two objective functions. This penalty method was used
instead of the traditional optimization approach of re-solving the fixed-
boundary equilibrium at each iteration (Dudt et al., 2023) because it can
better handle the near-axis constraints, and is equivalent in the limit
$\sigma\to\infty$. Omnigenity was targeted on five flux surfaces
$\rho=0.2,~{}0.4,~{}0.6,~{}0.8,~{}1.0$ and the shape of the magnetic well
$B(\eta)$ on each surface was parameterized by a monotonic spline with eight
knots. The force balance errors were evaluated throughout the plasma volume.
The residuals in Eq. 14 were minimized with a least-squares trust region
algorithm. The total magnetic flux, equilibrium profiles, and
$\mathcal{O}(\rho)$ behavior from the near-axis expansion solution were
constrained during the optimization. This process was repeated three times,
with the numerical resolutions and $\sigma$ increasing at each iteration.
Finally, the optimal configuration underwent a fixed-boundary solve to ensure
its volume-averaged normalized force balance error was $<1\%$ (Panici et al.,
2023).

Each optimization took less than thirty minutes to run on a single NVIDIA A100
GPU. All of the examples have a major radius of $1~{}\mathrm{m}$ and an
average magnetic field strength on axis of $1~{}\mathrm{T}$ by construction.
The results are stellarator symmetric, but this was a design choice and not a
limitation of the method. Figure 3 displays the magnetic field strength in
Boozer coordinates on the $\rho=1$ surface for these equilibria. These plots
clearly reveal the desired helicity of the (straight) $B_{\mathrm{max}}$
contours for each case, and they help picture how the bounce distances are
roughly equal on every field line (satisfying the condition
$\frac{\partial\delta}{\partial\alpha}=0$). A scalar omnigenity error is
defined as the mean of the local relative errors:

$\frac{1}{K}\sum_{i=1}^{K}\frac{\big{|}B_{\mathrm{eq}}(\rho_{i},\eta_{i},\alpha_{i})-B(\rho_{i},\eta_{i})\big{|}}{B(\rho_{i},\eta_{i})}$
(15)

and the normalized quasi-symmetry error in Boozer coordinates is defined as
(Rodriguez et al., 2022):

$\frac{\sqrt{\sum_{\frac{n}{m}\neq\frac{N}{M}}B^{2}_{mn}}}{\sqrt{\sum_{m,n}B^{2}_{mn}}}.$
(16)

These errors along with the aspect ratio and maximum elongation of each
solution are provided in table 1. The aspect ratio was indirectly constrained
during the optimization by fixing the magnetic axis shape, magnetic field
strength on axis, and total magnetic flux; the elongation was not constrained.

As another measure of the levels of omnigenity achieved, the effective ripple
$\epsilon_{\mathrm{eff}}^{3/2}$ – the magnitude of the neoclassical
collisional transport in the $\frac{1}{\nu}$ regime, where $\nu$ is the
collision frequency – is computed by NEO (Nemov et al., 1999) for each
solution. As another measure of confinement, the time history of fusion-born
alpha particles lost in a reactor scale device is also computed by SIMPLE
(Albert et al., 2020). In a perfectly omnigenous magnetic field,
$\epsilon_{\mathrm{eff}}^{3/2}$ should be zero and no particles should be lost
besides those with banana orbits that are wide enough to leave the plasma
volume. Both of these confinement measures are plotted in figure 4. The
numerical examples shown here were only optimized for omnigenity and
equilibrium quality, and other factors that may be important for a fusion
reactor design were not considered. More details about each optimized solution
are provided in the following subsections.

Table 1: Comparison between the six omnigenous solutions of geometric parameters and magnetic field errors at the $\rho=1$ surface. The aspect ratios are calculated using the same formula as the VMEC equilibrium code (Hirshman & Whitson, 1983). The omnigenity errors were computed with different collocation points than those used in the optimization. | Aspect | Maximum | Quasi-symmetry | Omnigenity
---|---|---|---|---
| Ratio | Elongation | Error | Error
OP | 19.35 | 6.70 | 3.73E-2 | 8.96E-4
QP | 17.91 | 10.10 | 1.78E-2 | 1.02E-2
OH | 18.27 | 5.74 | 2.24E-2 | 3.15E-3
QH | 18.24 | 2.48 | 1.62E-3 | 8.64E-4
OT | 19.56 | 3.37 | 1.41E-2 | 2.74E-3
QA | 19.44 | 1.95 | 2.81E-4 | 1.78E-4
Figure 3: Contours of constant magnetic field magnitude in Boozer coordinates
at the boundary surface $\rho=1$ for a single field period. The magnetic field
line at $\alpha=0$ is shown as a black dashed line in each plot. The insets
show a three-dimensional view of the field strength on each surface. Figure 4:
Measures of confinement. The quasi-symmetric solutions are plotted as dotted
lines, while the more general omnigenous solutions are plotted as solid lines.
Solutions with the same type of helicity are plotted in the same color. The
Wendelstein 7-X configuration with $\beta=4\%$ is plotted in a black dashed
line for comparison (Landreman, 2021). Top: Neoclassical collisional transport
magnitude, computed by NEO. Bottom: Collisionless losses of fusion-born alpha
particles initialized at $\rho=0.5$, computed by SIMPLE. All configurations
were scaled to the minor radius and magnetic field strength of ARIES-CS.

###  3.1 Poloidal Contours

Omnigenity with poloidally closed $B$ contours is characterized by a helicity
of $M=0$. These examples of OP and QP are vacuum equilibria
($\langle\beta\rangle=0$) that were based on initial guesses generated by
pyQIC. The OP case is a two field period configuration with a mirror ratio of
$\Delta=0.13$ and the QP case has a single field period with $\Delta=0.17$,
where the mirror ratio is defined as

$\Delta=\frac{B_{\mathrm{max}}-B_{\mathrm{min}}}{B_{\mathrm{max}}+B_{\mathrm{min}}}$
(17)

on the magnetic axis. In addition to the flux surface geometry, the omnigenity
parameters $B_{ij}$ and $x_{lmn}$ were also considered free variables during
the optimization. A resolution of $L_{\rho}=M_{\eta}=N_{\alpha}=4$ was used
for the OP case, while the QP case used a resolution of $L_{\rho}=M_{\eta}=4$,
$N_{\alpha}=0$ to enforce quasi-symmetry. The OP solution has worse QS error
than the QP example, as expected, but achieves an order of magnitude lower
omnigenity error and effective ripple, and two orders of magnitude lower fast
particle losses (see table 1 and figure 4).

###  3.2 Helical Contours

The typical case of omnigenity with helically closed $B$ contours is $M=1$ and
$N=N_{\mathrm{FP}}$. These examples of OH and QH are five field period vacuum
equilibria ($\langle\beta\rangle=0$) using the same initial guess from pyQSC.
As in the previous poloidal examples, the parameters $B_{ij}$ describing the
magnetic well shapes on each surface were considered free optimization
variables. However, in these runs the coordinate mapping of Eq. 7 was
prescribed: $x_{lmn}=0~{}\forall~{}l,m,n$ in the QH case, and
$x_{0,1,+1}=-\frac{\pi}{6}$ was the only non-zero parameter in the OH case.
This parameter value for the OH example was chosen such that the resulting
field, shown in the middle left panel of figure 3, qualitatively resembles the
$M=1$, $N=4$ omnigenous field example constructed by Landreman & Catto (2012).
In both cases, all of the $x_{lmn}$ variables were held fixed during the
optimizations to ensure that they converged to a quasi-symmetric and general
omnigenous magnetic field, respectively. The QS error of the OH solution is an
order of magnitude higher than that of the QH result, verifying that it is far
from quasi-symmetry (see table 1).

###  3.3 Toroidal Contours

$N=0$ describes omnigenity with toroidally closed $B$ contours. As explained
by Landreman & Catto (2012), “it is difficult to construct $N=0$ omnigenous
fields that depart strongly from quasi-symmetry …because the $B$ contours in a
$N=0$ quasi-symmetric field are already nearly parallel to the field lines
when $\mbox{$\,\iota\\!\\!$-}/N_{\mathrm{FP}}<1$”. These examples of OT and QA
are single field period equilibria based on the same initial guess from pyQSC,
with a finite toroidal current profile and a volume averaged normalized plasma
pressure of $\langle\beta\rangle=1.2\%$. The plasma current provides
additional rotational transform, which allows for an OT case that is far from
quasi-symmetry (as seen in the top right panel of figure 3 and by the
relatively high QS error in table 1). These toroidal solutions were found
using the same optimization process as the previous helical examples, except
that the coordinate mapping for the target OT field had
$x_{0,1,-1}=\frac{\pi}{6}$ as the only non-zero parameter. As in the helical
examples, all of the $x_{lmn}$ variables were constrained to their target
values while all of the $B_{ij}$ were free variables during the optimization.

##  4 Discussion

This is the first demonstration of equilibria solutions that closely
approximate every class of omnigenous magnetic fields. All of the $B$ contour
plots in figure 3 have their expected helicities, and it can be seen that the
distances between equal contours are roughly equivalent along any field line.
The average relative omnigenity errors listed in table 1 are all $\lessapprox
1\%$, indicating the optimized equilibria closely resemble their omnigenous
targets. The good confinement properties of these omnigenous solutions at
moderate aspect ratio are verified by the neoclassical collisional transport
magnitudes $\epsilon_{\mathrm{eff}}^{3/2}$ and the loss fractions displayed in
figure 4. These relatively low values of both confinement metrics are all
comparable in magnitude to previous optimized configurations (Landreman &
Paul, 2022). In particular, the OT solution has very low effective ripple and
the OP and OH solutions have almost no alpha particle losses – on par with the
QS results. This indicates the existence of stellarator equilibria other than
OP that can approach omnigenity without being quasi-symmetric.

The OH and OT results do have higher omnigenity errors and worse confinement
than their respective quasi-symmetric counterparts, but the target parameters
for these examples were chosen arbitrarily and other omnigenous fields with
better confinement properties might exist. Omnigenous fields that are not
quasi-symmetric are impossible to achieve exactly, and more examples are
needed to understand if they can generally be approximated well. The OT result
also has surprisingly high alpha particle losses considering its very low
effective ripple, exemplifying the complicated relationship between these two
confinement metrics. This could be explained by wide banana orbits that extend
beyond the plasma volume, but more investigation is required to determine the
cause. The outlier of all the results is the QP solution, which cannot exist
near the magnetic axis (Plunk et al., 2019) and is known to be difficult to
achieve far from the axis without highly elongated surfaces. This QP example
may have unrealistic geometry for a reactor, but it is still encouraging to
find a quasi-poloidally symmetric configuration with similar levels of
confinement as the Wendelstein 7-X experiment.

The purpose of the examples in Section 3 are to establish solutions that are
representative of each type of omnigenity that can exist. The model presented
in Section 2 and the optimization capabilities of DESC provide flexibility for
a variety of applications. The parameters $B_{ij}$ and $x_{lmn}$ can either be
constrained to target a specific location in the omnigenity design space (as
in the OH, OT, QH, and QA examples), or they can be free optimization
variables (as in the OP and QP examples). Providing more degrees of freedom
would likely result in the discovery of configurations with even better
confinement properties or that satisfy additional optimization objectives.
Eliminating the assumption of stellarator symmetry is a logical step in this
direction, as it doubles the size of the omnigenity solution space that can be
explored. As another example, the boundary condition of Eq. 9 could be relaxed
and only satisfied approximately; this would violate the already impossible
goal of exact omnigenity, but might improve the solution quality overall.
Alternatively, constraining most of the parameters would enable a systematic
scan of the full omnigenity design space. Mapping this space, like the surveys
of quasi-symmetry made in Rodríguez et al. (2023); Landreman (2022), is an
exciting direction for future research. Other ideas for future work include
investigating how the larger omnigenity phase space (in contrast to QS) could
simplify stellarator coil geometry and yield more promising candidates for
fusion power plants.

The code and data that support the findings of this study are openly available
in the DESC repository at https://github.com/PlasmaControl/DESC. José Luis
Velasco and collaborators have presented a similar method to parameterize a
subset of omnigenous magnetic fields with poloidal contours, and they are
thanked for their contributions. This work was supported by the U.S.
Department of Energy under contract numbers DE-AC02-09CH11466, DE-SC0022005,
and Field Work Proposal No. 1019. The United States Government retains a non-
exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this manuscript, or allow others to do so, for United
States Government purposes. The authors report no conflicts of interest.

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# Enhancing electrical conductivity of room temperature deposited Sn-doped
In2O3 thin films by hematite seed layers

Christian Lohaus  Céline Steinert  Getnet Deyu  Joachim Brötz  Wolfram
Jaegermann  Andreas Klein  Technische Universität Darmstadt, Institute of
Materials Science, 64287 Darmstadt, Germany

(published online 12 April 2018)

###### Abstract

Hematite Fe2O3 seed layers are shown to constitute a pathway to prepare highly
conductive transparent tin-doped indium oxide (ITO) thin films by room
temperature magnetron sputtering. Conductivities of up to $\sigma=3300\,{\rm
S/cm}$ are observed. The improved conductivity is not restricted to the
interface but related to an enhanced crystallization of the films, which
proceeds in the rhombohedral phase.
This article has been published in Applied Physics Letters, Volume 12, Article
Number 152105, 2018; The published version is accessible at doi:
10.1063/1.5022683

Transparent conductive oxide (TCO) thin films are employed in a variety of
applications such as displays, solar cells and LEDs.Ginley, Hosono, and Paine
(2010) For most TCOs, high electrical conductivity requires high temperature
deposition, which limits their application with temperature sensitive
substrates. Sn-doped In2O3 (ITO), for example, can be prepared with
conductivities $\geq 10^{4}\,{\rm S/cm}$ at temperatures of $400\,^{\circ}{\rm
C}$,Ohta  _et al._ (2000); Tahar  _et al._ (1998) while conductivities for
films deposited at room temperature is typically $<10^{3}\,{\rm S/cm}$.Song
_et al._ (1999); de Carvalho  _et al._ (2002); Kim  _et al._ (2000); Hao  _et
al._ (2008); Gassmann, Brötz, and Klein (2012); Kim  _et al._ (2016) The lower
conductivity of films deposited at room temperature is mostly caused by lower
carrier concentrations. This is related to the amorphous nature of such films
and the fact that electron generation in amorphous TCOs is not caused by the
heterovalent substituent but rather by the off-stoichiometry, which changes
the coordination of the indium atoms.Khanal  _et al._ (2015); Medvedeva,
Buchholz, and Chang (2017)

In this work it is shown that the conductivity of ITO thin films, deposited at
room temperature by magnetron sputtering without further annealing, can be
significantly increased by the insertion of a thin hematite Fe2O3 seed layer.
The achieved electrical conductivities are as good as the best values obtained
with doped polycrystalline ZnO and SnO2 layers grown at elevated substrate
temperature. Ellmer (2012) The hematite layer, even with only a few nanometer
thickness where it doesn’t affect the optical properties, enhances the
crystallinity of the samples. It also stabilizes the metastable rhombohedral
phase of In2O3. Seed layers have been used before to enhance electrical
properties of transparent conductive oxides,Nakao  _et al._ (2010) but not for
lowering substrate temperatures. Samples were prepared by RF-magnetron
sputtering in the DArmstadt Integrated SYstem for MATerials Research (DAISY-
MAT), which combines a multitechnique surface analysis system with multiple
thin film deposition chambers.Klein (2013) In a first step ultrathin hematite
films with $\leq 10\,{\rm nm}$ thickness were deposited from a metallic iron
target in an Ar-O2 mix with $8\,\%$ oxygen at a pressure of $0.5\,{\rm Pa}$.
Amorphous Quartz and c-cut Sapphire (S0001) were used as substrates. If not
noted otherwise, the deposition temperature was $400\,{\rm{}^{\circ}C}$. With
these parameters phase pure, crystalline hematite ($\alpha$-Fe2O3) thin films
are formed.Lohaus  _et al._ (2017) After cooling to room temperature the
samples were transferred in ultra-high vacuum (UHV) to a second deposition
chamber for ITO deposition. This was performed using a ceramic ITO target with
$10\,{\rm wt.\%}$ SnO2 doping. The depositions were performed at room
temperature in pure Argon atmosphere at a pressure of $0.5\,{\rm Pa}$ and a
power density of $1.23\,{\rm W/cm^{2}}$.

X-Ray and ultraviolet photoelectron spectroscopy (XPS, UPS) were performed
without breaking vacuum using a Physical Electronics PHI 5700 spectrometer.
Core level and XP valence band spectra were recorded using monochromatized Al
Kα radiation. UP valence band spectra were acquired by using He I radiation
from a Helium discharge lamp. Calibration of the binding energies was achieved
by measuring the Ag 3d5/2 core level and the Fermi edge of a sputter cleaned
Ag foil at the same day of the measurements. X-ray diffraction was carried out
in $\theta-2\theta$ geometry using a Seifert XRD 30003 PTS-S with Cu Kα
radiation. Conductivity and Hall-measurements were performed on a home made
setup in van-der-Pauw geometry.Hohmann, Wachau, and Klein (2014) The sample
current was $1\,{\rm mA}$ in all measurements. Optical measurements were
performed using a Agilent Cary 7000 Universal Measurement Spectrophotometer.

Electrical as well as surface chemical and electronic properties of ITO thin
films with and without Fe2O3 seed layer are presented in Figure 1. The
conductivity of the films with seed layer are about an order of magnitude
higher compared to those deposited on the bare quartz and sapphire substrates.
A higher carrier concentration is the origin of the increased conductivity.
The carrier mobility is slightly reduced from $40-46\,{\rm cm^{2}/Vs}$
$29-40\,{\rm cm^{2}/Vs}$. The conductivity of ITO films grown on Fe2O3 seed
layers is rather independent on film thickness as evident from Fig. 1(b). The
enhanced conductivity is therefore clearly a bulk phenomena and not related to
the region next to the Fe2O3 interface.

Figure 1: Electrical measurements (a, b), In 3d5/2 core levels (c), XP valence
bands (d), and magnified UP Fermi edge spectra (e) of room temperature
deposited ITO thin films with and without Fe2O3 seed layers. In (a) and (b)
films grown on sapphire (0001) are indicated by circles and films grown on
quartz by triangles. The colored symbols in (a) and (b) and the spectra in
(c)-(e) correspond to the same samples as identified in the table giving the
Sn:In ratio determined from XP core level intensities.

XP spectra, in which the effect has actually been identified originally,
confirm the high conductivity. Films grown on Fe2O3 seed layers show
consistently higher core level and valence band maximum binding energies
compared to their counterparts without Fe2O3 seed layer. The higher
conductivity of the spectra is also evident from the stronger asymmetry of the
core level lines towards higher binding energy, which relates to a higher
plasmon energy and therefore carrier concentration.Körber  _et al._ (2010)
Finally, occupied conduction band states are also observed by UPS as a Fermi
edge emission shown in Fig. 1(e). The sample grown on sapphire (0001) with
Fe2O3 layer shows the highest intensity of the Fermi edge, corresponding well
with the highest charge carrier concentration.

Optical transmission spectra of different films are compared in Fig. 2.
Relative to the ITO film grown at room temperature on sapphire without seed
layer, an even improved transmission is observed for a seed layer thickness of
$2\,{\rm nm}$. Increasing the seed layer thickness to $10\,{\rm nm}$ reduces
the transmission for wavelengths between $370-630\,{\rm nm}$. In this range,
the transmission follows the one recorded for a $10\,{\rm nm}$ thick Fe2O3
layer, indicating that the transmission of the ITO layer is still high and
that an optimized Fe2O3/ITO stack can fully satisfy the requirements of a
transparent electrode.

Figure 2: Optical transmission spectra of $150\,{\rm nm}$ thick ITO films
grown at room temperature on sapphire with $0$, $2$ and $10\,{\rm nm}$ thick
Fe2O3 seed layers. Transmission spectra recorded for a $10\,{\rm nm}$ thick
Fe2O3 film on sapphire and of a $150\,{\rm nm}$ thick ITO film grown at
$400\,{\rm{}^{\circ}C}$ on quartz without seed layer are added for comparison.

The Sn/In ratio determined from XPS for films with and without Fe2O3 seed
layer on quartz and sapphire are all within $10-11\,{\rm at.\%}$ as indicated
in the table included in Fig. 1. This corresponds well with the nominal target
composition. In contrast, films with high conductivity grown at higher
substrate temperatures, exhibit a significantly enhanced Sn/In ratio,
indicating a segregation of Sn to the surface.Gassenbauer  _et al._ (2006);
Frischbier  _et al._ (2016) Such a segregation is evidently not observed when
the films are grown at room temperature. This is reasonable given the fact
that Sn-donors are mobile only at $T\gtrsim 300\,{\rm{}^{\circ}C}$.Gassenbauer
_et al._ (2006); Hohmann, Wachau, and Klein (2014)

The XP valence band spectra of samples grown with and without Fe2O3 seed layer
exhibit a different shape. In particular, three distinct features are observed
for films grown on Fe2O3, which are less pronounced in the spectra of samples
without Fe2O3 layer. The differences might be related to different
crystallinity, orientation or phase composition. In order to reveal the
influence of the hematite layer on the structural properties of the ITO, X-ray
diffraction has been performed on $150\,{\rm nm}$ thick films. The
corresponding diffraction patterns are shown in Figure 3.

Figure 3: XRD pattern from $\theta-2\theta$-measurements on samples with a ITO
thickness of $150\,{\rm nm}$. From bottom to top the samples are the
following: quartz/ITO (green), quartz/Fe2O3/ITO (black), S0001/ITO (magenta),
and S0001/Fe2O3/ITO (blue). All Fe2O3 layers had a thickness of $10\,{\rm
nm}$. All reflections have been identified and labeled with the help of pdf-
card reference data (pdf card numbers rh-Al2O3: 11-661, rh-Fe2O3: 84-311, rh-
In2O3: 73-1809, and cu-In2O3: 6-416 (strained)).

The diffraction pattern for the ITO film on quartz without the iron oxide
layer does not show any reflections, indicating an amorphous structure. In
contrast to this, a number of reflections appear after insertion of the Fe2O3
layer. These can be assigned to cubic bixbyite In2O3 with $222$ being the most
intense one. There are, however, also some reflections, which do not belong to
this phase. These additional signals can be assigned to the rhombohedral In2O3
phase.

In contrast to the ITO film deposited on quartz, the sample on S0001 without
Fe2O3 layer shows some reflections. Besides a very small reflection from the
Sapphire substrate, only signals from the bixbyite In2O3 phase are observed.
Again, $222$ from this phase is the most intense reflection. The addition of a
Fe2O3 layer changes the diffraction pattern considerably. The intensity of all
reflections is increased by a factor of 20. Most intense is now the (006)
reflection of the rhombohedral ITO phase. In addition, the rhombohedral $006$
reflections of Fe2O3 and Al2O3 can be identified. This indicates epitaxial
growth of rhombohedral In2O3 and Fe2O3 on sapphire (0001), which all share the
same space group ($\mbox{R}\overline{3}\mbox{c}$).De Boer  _et al._ (2016);
Kim, Gao, and Chambers (1997) Among the reflections of the ITO bixbyite phase,
the $332$ is the most intense one.

The diffraction angles of rhombohedral In2O3 match those of the powder
reference (pdf-card 73-1809). In contrast, the reflections from the cubic
phase are shifted to smaller angles, which corresponds to an expansion of the
lattice parameter by about $2\,\%$. This is observed for all samples showing
reflections from bixbyite In2O3.

The insertion of an ultrathin crystalline Fe2O3 film significantly enhances
the crystallinity of room temperature deposited ITO films. However, while the
cubic bixbyite structure is commonly observed when films are grown at elevated
temperature, the Fe2O3 seed layer favors the growth of rhombohedral ITO, which
is a metastable modification.Gurlo, Kroll, and Riedel (2008) The different
shape of the valence band spectra is also consistent with the presence of
rhombohedral ITO.King  _et al._ (2009)

Only few studies report on the electric properties of rhombohedral ITO thin
films. Solution processed nanoparticles fired at $600\,{\rm{}^{\circ}C}$ and
aged at $500\,{\rm{}^{\circ}C}$ show a mixture of cubic and rhombohedral
phases with a slightly smaller resistance but higher sensitivity towards
gaseous ethanol compared to purely cubic films.Huh  _et al._ (2001); Kim  _et
al._ (2003); Seo  _et al._ (2006) A transition from cubic to rhombohedral
structure with film thickness has been reported by Malik and coworkers.Malik,
Nunes, and Martins (1997). They could show that the mobility of thicker
rhombohedral films ($37\,{\rm cm^{2}/Vs}$) was higher than for thinner cubic
films ($19\,{\rm cm^{2}/Vs}$). The charge carrier concentration was slightly
lower for the rhombohedral films ($7.7\cdot 10^{20}\,{\rm cm^{-3}}$) compared
to the cubic ones ($9\cdot 10^{20}\,{\rm cm^{-3}}$). Dunlop and coworkers have
reported partially rhombohedral thin films at temperatures of
$415-500\,{\rm{}^{\circ}C}$, showing a carrier mobility of $27\,{\rm
cm^{2}/Vs}$ and a carrier concentration of $5.7\cdot 10^{20}\,{\rm cm^{-3}}$.
Dunlop, Kursumovic, and MacManus-Driscoll (2010) Both values are higher
compared to purely cubic reference samples, which was attributed to lower
internal strain. The highest conductivities reported for at least partially
rhombohedral ITO is therefore in the range of $2400-4500\,{\rm S/cm}$. These
values are comparable to those reported here. However, the previous values
have all been obtained with processing temperatures well above
$400\,{\rm{}^{\circ}C}$, while the conductivity achieved in the present work
was achieved with room temperature deposited ITO without further annealing.

Based on the studies reported here, different mechanisms might contribute to
the enhanced conductivity induced by the Fe2O3 seed layer: (i) the enhanced
crystallinity, (ii) the absence of dopant segregation, (iii) the presence of
the rhombohedral phase. The literature survey given above provides no evidence
for fundamentally superior electrical properties of the rhombohedral compared
to the cubic bixbyite In2O3 phase. It is therefore believed that the
rhombohedral structure is not responsible for the higher conductivity.

Carrier mobilities in amorphous TCOs are comparable to those of their
crystalline counterparts.Nomura  _et al._ (2004) This is caused by the low
concentration of tail states, which is a consequence of the s-orbital
character of the conduction band states.Nomura  _et al._ (2007); Kamiya and
Hosono (2010); Medvedeva, Buchholz, and Chang (2017) On the first glance, it
is therefore not surprising that the carrier mobility is not much different
for films grown with and without Fe2O3 seed layer. However, crystallization
induces grain boundaries, which act as carrier traps and scattering
centers.Seto (1975) The carrier mobilities of our films are noticeably lower
than those of single crystalline material (see solid line in Fig. 1(a) and
Ref. Preissler  _et al._ (2013)). Grain boundary scattering is therefore
active. There is evidence that the segregation of the Sn dopants to grain
boundaries reduces carrier maobility.Hohmann, Wachau, and Klein (2014);
Frischbier  _et al._ (2016) But the segregation requires elevated temperatures
to activate Sn diffusion. No Sn segregation is therefore expected, which
agrees with the absence of surface segregation of Sn. In any case, the main
improvement in electrical conductivity induced by the Fe2O3 layer is the
higher carrier concentration. Carrier mobility and the absence of segregation
should therefore also play a minor role.

Films deposited at room temperature are normally amorphous. In such films, the
free carriers are not induced by heterovalent dopants, such as Sn, but by the
off-stoichiometry (oxygen deficiency) of the samples.Khanal  _et al._ (2015);
Medvedeva, Buchholz, and Chang (2017) Crystallization will activate the
donors. It is therefore straightforward to assign the origin of the higher
carrier concentration of films grown on Fe2O3 seed layer to the enhanced
crystallinity. Given the significantly higher diffraction intensities, the
enhanced crystallinity of films grown on sapphire/Fe2O3, compared to those
grown on quartz/Fe2O3, also explains the higher carrier concentration of the
former.

Other seed layers may exits, which favor crystallization at room temperature
in the bixbyite phase. These might result in similar improvement. There are
indeed literature reports on ITO films prepared at room temperature, which
exhibit similar conductivity than the films prepared here with Fe2O3 seed
layer (see Refs. Tahar  _et al._ (1998); Wu  _et al._ (1999) and references
therein). The crystalline state of these films is not reported, however. Our
deposition conditions were optimized for deposition at elevated temperatures.
It would, therefore, be compelling to see whether there is room for additional
improvement by a seed layer in a room temperature optimized deposition.

In summary, we have shown that the use of a Fe2O3 seed layer can enhance the
electrical conductivity of ITO thin films deposited at room temperature by
about one order of magnitude reaching yet a maximum conductivity of
$3300\,{\rm S/cm}$. The hematite structure of the Fe2O3 seed layer induces a
local epitaxial growth and thereby enhances the crystallinity of the In2O3
layer. The effect is most pronounced on c-cut sapphire substrates but also
observed on amorphous quartz substrates. The Fe2O3 layer results in a
preferred formation of the metastable rhombohedral In2O3 phase, which is not
considered to affect the electrical properties. However, crystallization at
low temperature offers the advantage of suppressing dopant diffusion and
therefore segregation, which can result in higher carrier mobilities compared
to films crystallized at higher temperatures. The use of seed layers therefore
offers an approach to reach high conductivity low temperature processed TCO
films. The possibility to stabilize the rhombohedral
$\mbox{R}\overline{3}\mbox{c}$ phase of In2O3 by the Fe2O3 seed layer also
provides a simple route for the preparation of such films.

This work was supported by the European Commission within the Knowledge
Innovation Center (KIC) raw matter.

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