# Spin reorientation in tetragonally distorted spinel oxide NiCo2O4 epitaxial
films

Hiroki Koizumi  Department of Applied Physics, University of Tsukuba, Tsukuba,
Ibaraki 305-8573, Japan  Ikumi Suzuki  Institute for Chemical Research, Kyoto
University, Uji, Kyoto 611-0011, Japan  Daisuke Kan  Institute for Chemical
Research, Kyoto University, Uji, Kyoto 611-0011, Japan  Jun-ichiro Inoue
Department of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki
305-8573, Japan  Yusuke Wakabayashi  Department of Physics, Tohoku University,
Sendai 980-8578, Japan  Yuichi Shimakawa  Institute for Chemical Research,
Kyoto University, Uji, Kyoto 611-0011, Japan  Hideto Yanagihara  Department of
Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan
Tsukuba Research Center for Energy Materials Science (TREMS), University of
Tsukuba, Tsukuba, Ibaraki 305-8573, Japan

###### Abstract

We experimentally investigated the magnetic properties of NiCo2O4 epitaxial
films known to be conductive oxides with perpendicular magnetic anisotropy
(PMA) at room temperature. Both magneto-torque and magnetization measurements
at various temperatures provide clear experimental evidence of the spin
reorientation transition at which the MA changes from PMA to easy-cone
magnetic anisotropy (ECMA) at a certain temperature ($T_{\rm{SR}}$). ECMA was
commonly observed in films grown by pulsed laser deposition and reactive radio
frequency magnetron sputtering, although $T_{\mathrm{SR}}$ is dependent on the
growth method as well as the conditions. The cone angles measured from the
$c$-axis increased successively at $T_{\mathrm{SR}}$ and approached a maximum
of 45-50 degrees at the lowest measurement temperature of 5 K. Calculation
with the cluster model suggests that the Ni3+ ions occupying the $T_{d}$ site
could be the origin of the ECMA. Both the magnetic properties and the results
of the calculation based on the cluster model indicate that the ECMA is
attributable to the cation anti-site distribution of Ni3+, which is possibly
formed during the growth process of the thin films.

##  I Introduction

NiCo2O4(NCO) is a conductive inverse spinel oxide ($Fd\bar{3}m$) that exhibits
ferrimagnetism with a Curie temperature as high as $T_{C}\approx 400$ K Battle
_et al._ (1979); Marco  _et al._ (2000). Owing to the mixed valence of its
cations, NCO can be expressed in terms of the following chemical formula:
$(\mathrm{Co}^{2+}_{x}\mathrm{Co}^{3+}_{1-x})_{\rm
tet}[\mathrm{Co}^{3+}\mathrm{Ni}^{2+}_{1-x}\mathrm{Ni}^{3+}_{x}]_{\rm
oct}\mathrm{O}_{4}^{2-}$ ($0<x<1$), where the subscripts “tet” and “oct”
represent the tetrahedral ($T_{d}$) and octahedral ($O_{h}$) sites,
respectively. The $T_{d}$ sites are occupied by $\mathrm{Co}^{2+}$ ($d^{7}$;
$S=3/2$) and $\mathrm{Co}^{3+}$ ($d^{6}$; $S=2$). On the other hand, the
$O_{h}$ sites are occupied by $\mathrm{Ni}^{2+}$ ($d^{8}$; $S=1$),
$\mathrm{Ni}^{3+}$ ($d^{7}$; $S=1/2$), and $\mathrm{Co}^{3+}$ ($d^{6}$;
$S=0$). From the viewpoint of the number of $d$-electrons at each site, the
nominal saturation magnetization is 2 $\mu_{B}$/f.u. irrespective of the value
of $x$, as long as the $T_{d}$ sites are only occupied by Co ions.

The mixed valence properties of NCO confer unique characteristics upon this
transition metal oxide, such as relatively high electrical conductivity Silwal
_et al._ (2013); Ndione  _et al._ (2014); Bitla  _et al._ (2015); Zhen  _et
al._ (2018), and electrochemical activity Wang  _et al._ (2019). In
particular, a large tunnel magnetoresistance effect has recently been reported
in magnetic tunnel junctions fabricated using NCO electrodes, supporting the
half-metallic natureNdione  _et al._ (2014); Wang  _et al._ (2019); Shen  _et
al._ (2020a). In addition, NCO(001) films grown on MgAl2O4 (MAO) substrates
exhibited perpendicular magnetic anisotropy (PMA) Chen  _et al._ (2019); Kan
_et al._ (2020); Xue  _et al._ (2020); Koizumi  _et al._ (2021) of
$K^{\rm{eff}}_{u}\approx 0.3$ MJ/m3 at room temperature. The fact that both
the half-metallicity and PMA are realized at once indicates that the NCO films
possess good potential for application in spintronic devices. Mellinger et al.
remarked that the PMA is due to the magneto-elastic effect, which is provided
a semi-quantitative explanation for its attribution of the PMA to the
compressive strain of $\sim-0.3\%$ to which Co at the $T_{d}$ sites is
subjected as a result of epitaxial distortion Mellinger  _et al._ (2020).

In this paper, we report that NCO films have spin reorientation (SR)
transitions at low temperatures, that is, the magnetic anisotropy (MA) of the
films changes from PMA to easy-cone magnetic anisotropy (ECMA). This suggests
that the higher order term of the uniaxial MA must be considered at least in
the low-temperature region. To elucidate the origin of the ECMA, we also
performed tight-binding calculations for Co-O and Ni-O clusters of either the
divalent or trivalent states at both the $T_{d}$ and $O_{h}$ sites. The
results showed that the observed ECMA predominantly originated from the Ni3+
that occupy the $T_{d}$ sites, which is the anti-site disorder of cation
distribution.

This paper is organized as follows. Experimental details are provided in Sec.
II. Section III is devoted to the experimental results, numerical calculation,
and discussion, and the final section provides a summary of the work.

##  II Experiments

We prepared three epitaxial NCO films grown on single-crystal MAO(001)
substrates by reactive radio frequency magnetron sputtering (ES-250MB, Eiko
Engineering Co., Ltd.)Koizumi  _et al._ (2019a, b) or pulsed laser deposition
(Pascal Co., Ltd.). In the sputtering process, we used a 2-inch alloy target
with a nominal composition of $\mathrm{Ni:Co}=1:2$. The growth conditions of
the NCO thin films by sputtering were as follows: the Ar and O2 flow rates
were 10 and 2.5 sccm, respectively, and the input power was 100 W. The process
temperature and working pressure were 300∘C and 1.3 Pa, respectively. The film
thickness was determined to be 50 nm by X-ray reflectivity measurement.
Hereafter, we refer to this NCO film as Sample $S$. On the other hand, in the
case of the PLD process, we used a NiCo2Ox ceramic target, which was ablated
by a 6 Hz KrF excimer laser ($\lambda=248$ nm) with a laser spot density of
1.2 J/cm2, at a fixed substrate temperature of 315∘C, and under various oxygen
partial pressures. Detailed process conditions can be found in Refs. Shen  _et
al._ , 2020b, a; Suzuki  _et al._ , 2020. The NCO films prepared at an oxygen
pressure of $P_{\mathrm{O_{2}}}=10$ and 100 mTorr are denoted as Samples
$P_{10}$ and $P_{100}$, respectively. The film thicknesses of both Samples
$P_{10}$ and $P_{100}$ were 30 nm.

Thin film structures were characterized by X-ray diffraction (XRD). Magnetic
hysteresis (MH) loops and magnetic anisotropy constants were measured using a
vibrating sample magnetometer (VSM) and magneto-torque meter, respectively.
Both magnetic measurements were performed using a physical property
measurement system (PPMS, Quantum Design).

##  III Results and discussion

###  III.1 Structural characterization

Figure 1 (a) shows the reciprocal space map (RSM) around the MAO$(4,0,8)$
Bragg reflections for each sample. All three films are tetragonally distorted
by the MAO substrate. The in-plane lattice constant seems to be constrained by
the MAO substrate as $a=8.08$ Å, whereas the out-of-plane ($c$) constant
varies from sample to sample, namely $c=8.19$ Å ($c/a=1.013$), 8.27 Å
($c/a=1.023$), and 8.25 Å ($c/a=1.021$) for Samples $P_{100}$, $S$, and
$P_{10}$, respectively. The lattice parameter along the $c$-axis varies from
sample to sample, suggesting that the cation distribution varies according to
the growth conditions of the films Shen  _et al._ (2020b).

We also measured the temperature dependence of the lattice parameters for
Sample $S$, as shown in Fig. 1 (b). Both the in-plane and out-of-plane lattice
constants $a$ and $c$, determined from the NCO$(4,0,12)$ Bragg reflections,
are less temperature dependent and are therefore similar to that of the $c/a$
ratio.

Figure 1: (a) RSMs around the NCO$(4,0,8)$ and MAO$(4,0,8)$ reflections. (b)
Temperature dependence of the lattice constant evaluated from the
NCO$(4,0,12)$ Bragg reflection in Sample $S$. The inset shows the temperature
dependence of the $c/a$ ratio.

###  III.2 Magnetic properties

Figure 2 (a)-(c) shows the temperature dependence of $MH$ loops in the
presence of a magnetic field applied perpendicular to the film plane. All
films exhibited square-shaped hysteresis loops at temperatures above 250 K,
whereas the coercivity varies markedly among the samples. Moreover, with
decreasing temperature, a stronger field is needed to saturate the
magnetization at low temperatures in Samples $P_{10}$ and $S$, meaning that
the squareness of the hysteresis is reduced. The squareness ratio starts to
decrease at a certain temperature, as shown in Fig. 2 (d). For example, the
squareness ratio of Sample $P_{10}$ decreases from 200 K and almost constant
below 70 K, and that of Sample $S$ starts decreasing at approximately 70 K.
For Sample $P_{100}$, however, the squareness ratio appears to be close to 1
at all measured temperatures. The squareness ratios at 30 K, the lowest
temperature at which we performed measurements, are 0.99, 0.81, and 0.73 for
$P_{100}$, $S$ and $P_{10}$, respectively (see the inset of Fig. 2 (d)). The
squareness at lower temperatures is obviously smaller for Samples $S$ and
$P_{10}$, although the $M(T)$ curves of none of the three films exhibited any
anomalies.

Figure 2 (e) shows the temperature dependence of the saturation magnetization
$M_{S}$, with higher values being observed for samples with higher Curie
temperature ($T_{C}$). Because $T_{C}$ and $M_{S}$ are simply associated with
the strength of the exchange interaction and the difference in the magnetic
moments between the $O_{h}$ and $T_{d}$ sites, respectively, the clear
positive correlation between $T_{C}$ and $M_{S}$ indicates that the
distribution of both Co and Ni at both the $T_{d}$ and $O_{h}$ sites changes
with the growth conditions. In fact, according to a recent structural analysis
of NCO films, the amount of cation distribution defects or anti-site defects
characteristic of NCO films is non-negligible Shen  _et al._ (2020b).

Figure 2 (f) shows the $MH$ loops of Sample $S$ with the magnetic field
parallel to the film plane. These results reveal that the $MH$ loops became
rounded and the area corresponding to $\int_{0}^{M_{s}}\mu_{0}H(M)dM$ tended
to be smaller as the temperature decreases. Considering the lowest order term
of MA, these results indicate that the effective MA decreases at low
temperatures. Note that this does not occur in conventional ferromagnetic or
ferrimagnetic materials; therefore, a detailed evaluation of MA must be
carried out, especially at low temperatures.

Figure 2:  $MH$ loops of NCO film with $\mu_{0}H\perp$film plane of (a) Sample
$P_{100}$, (b) Sample $S$, and (c) Sample $P_{10}$. (d) Temperature dependence
of the squareness ratio in each sample (inset: MH loops for the three samples
at 30 K). (e) Temperature dependence of $M_{S}$ determined from VSM results.
(f) $\mu_{0}H\parallel$ film plane of Sample $S$.

The magneto-torque curves of the $ac$-plane (around the $b$-axis,
$L_{\theta}$) and of the $ab$-plane (around the $c$-axis; $L_{\phi}$) from
room temperature to 5 K taken under $\mu_{0}H=9$ T are shown in Fig. 3. As
shown in Fig. 3 (a), the shape of $L_{\theta}$ at high temperatures exhibit
two-fold component crossing zero with a negative slope at $\theta=0$ and $180$
degree, indicating the preferential magnetization directions of the films
correspond to the direction normal to the film plane, which is consistent with
previous reports Kan  _et al._ (2020); Mellinger  _et al._ (2020). However,
four-fold components appear in the torque curves with decreasing temperature.
The existence of four-fold components means that the higher-order terms in
uniaxial MA are crucial for understanding the preferential magnetization
directions Chikazumi and Graham (1997).

In the $L_{\varphi}$ curves shown in Fig. 3 (b), on the other hand, the torque
signals are observed only at low temperature and are weaker than those of
$L_{\theta}$, meaning that the out-of-plane uniaxial components are dominant
in the NCO films.

Figure 3:  Magneto-torque curves of (a)$L_{\theta}$ ($ac$-plane), and
(b)$L_{\varphi}$ ($ab$-plane) for the three samples at various temperatures.

In tetragonal crystal systems, the magnetic anisotropy energy ($E_{A}$) and
magneto-torque for the $ac$-plane ($L_{\theta}$) and $ab$-plane
($L_{\varphi}$) can be written as follows Chikazumi and Graham (1997); Coey
(2010):

$\displaystyle E_{A}$ $\displaystyle=$ $\displaystyle
K_{u1}^{\mathrm{eff}}\sin^{2}\theta+K_{u2}\sin^{4}\theta+K_{3}\sin^{4}\theta\sin^{2}2\varphi,$
(1) $\displaystyle L_{\theta}$ $\displaystyle\equiv$
$\displaystyle-\partial_{\theta}E_{A}=-\left(K_{u1}^{\mathrm{eff}}+K_{u2}\right)\sin
2\theta+\frac{1}{2}K_{u2}\sin 4\theta,$ (2) $\displaystyle L_{\varphi}$
$\displaystyle\equiv$ $\displaystyle-\partial_{\varphi}E_{A}=-K_{3}\sin
4\varphi,$ (3)

where $K_{u1}^{\mathrm{eff}}$ and $K_{u2}$ are the uniaxial magnetic
anisotropy constants for the second- and fourth-order terms, respectively, and
$K_{3}$ is the cubic anisotropy constant. Further, $\theta$ and $\varphi$
represent the angles between the direction of the applied field and the $c$\-
and $a$-axes, respectively. The observed $K_{u1}^{\mathrm{eff}}$ comprises
both magnetic anisotropy contributions from magnetocrystalline anisotropy
$K_{u1}$ and shape anisotropy $\mu_{0}M_{S}^{2}/2$, i.e.,
$K_{u1}^{\mathrm{eff}}=K_{u1}-\mu_{0}M_{S}^{2}/2$. The MA constants enable us
to determine the shape of the preferential directions of the net magnetization
Chikazumi and Graham (1997), and these results are provided in Table 1.

Table 1: Relation between the respective magnetic anisotropy constants and schematics of preferential magnetisation orientation. $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K_{u1}^{\mathrm{eff}}>0\ \ \ \ \ \ \ \ \ \ \ \ $ | $K_{u1}^{\mathrm{eff}}<0$
---|---
$K_{u1}^{\mathrm{eff}}+K_{u2}>0$ | $K_{u1}^{\mathrm{eff}}+K_{u2}<0$ | $K_{u1}^{\mathrm{eff}}+2K_{u2}>0$
| $K_{u1}^{\mathrm{eff}}+2K_{u2}<0$ |
|  |
Perpendicular | In-plane | Easy-cone
magnetic | magnetic | magnetic
anisotropy | anisotropy | anisotropy
(PMA) |  | (ECMA)
$T_{\mathrm{SR}}<T$ |  | $T<T_{\mathrm{SR}}$

Figure 4 (a) shows the MA constants determined by fitting the torque curve
using Eq. (2), (3). The values of $K_{u1}^{\mathrm{eff}}$ were positive at
high temperatures and the sign changes to negative as the temperature
decreases. In contrast, the values of $K_{u2}$ were positive at all
temperatures. Moreover, the absolute values of both $K_{u1}^{\mathrm{eff}}$
and $K_{u2}$ increased suddenly and drastically when the temperature reached a
certain point and this depended on samples. On the other hand, $K_{3}$ is much
weaker than the out-of-plane terms of both $K_{u1}^{\mathrm{eff}}$ and
$K_{u2}$. Because $K_{u2}>0$ for all temperature ranges, if
$K_{u1}^{\mathrm{eff}}>0$ NCO exhibits PMA (see Table 1) Chikazumi and Graham
(1997). In contrast, when $K_{u1}^{\mathrm{eff}}$ changes its sign from
positive to negative, spin reorientation occurs, and the NCO exhibits ECMA.
Hereafter, we refer to the temperature at which $K_{u1}^{\mathrm{eff}}$
becomes $0$ as the spin-reorientation transition temperature
($T_{\mathrm{SR}}$). The values of $T_{\mathrm{SR}}$ for all samples are
listed in Table 2. It should also be noted that the temperature at which the
squareness ratio begins to decrease is close to the value of $T_{\mathrm{SR}}$
determined by the torque measurements, indicating that the change in the
squareness ratio is attributed to the change in MA. In addition, the cone
angle determined from the magneto-torque is in good agreement with that
estimated from the squareness ratio, as shown in Fig. 2 (d).

Figure 4:  Temperature dependence (a) of the magnetic anisotropy energy in
each sample and (b)–(d) of the cone angles for the two different $\varphi$
directions calculated from Eq. 4. The vertical broken lines in (a) roughly
indicate the temperatures where $K_{u1}^{\mathrm{eff}}$ change their signs.

By considering $K_{3}$, the cone angle $\theta_{0}$ measured from the $c$-axis
can be expressed as

$\displaystyle\theta_{0}(\varphi)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{-K_{u1}^{\mathrm{eff}}}{2(K_{u2}+K_{3}\sin^{2}2\varphi)}.}$
(4)

Figure 4 (b)-(d) show the temperature dependence of $\theta_{0}$, determined
from Eq. (4), for the three samples. As the temperature decreases,
$\theta_{0}$ continuously increases from $\theta_{0}=0$ to a finite value. We
note that $\theta_{0}(T)$ for the MA of Co2Y-hexaferrite was reported to
exhibit similar behavior Smit and Wijn (1959).

The saturation magnetization extrapolated to 0 K ($M_{0}$), $T_{C}$,
$T_{\mathrm{SR}}$, and the values of $K_{u}$ for each sample are summarized in
Table 2. The films with higher $T_{C}$ exhibit larger $M_{0}$ values and lower
$T_{\mathrm{SR}}$. Moreover, the MA constants seem to be linearly correlated
with $T_{C}$, as shown in Fig. 5. On the other hand, at approximately room
temperature, the sample with higher $T_{C}$ exhibits stronger PMA, as shown in
Fig. 4. According to a previous study Shen  _et al._ (2020b), $T_{C}$ is
higher for NCO films with less anti-site disorder. In contrast, our
experimental results suggest that samples with lower $T_{C}$ have stronger
ECMA and higher $T_{\mathrm{SR}}$. Therefore, ECMA seems to be associated with
samples with off-stoichiometric compositions or anti-site distributions Shen
_et al._ (2020b), which is consistent with the fact that no sign of a non-
collinear spin structure was apparent even at 4.2 K in a stoichiometric powder
sample of NCO by neutron diffraction Battle  _et al._ (1979). Although
$T_{\rm{SR}}$ varies from sample to sample, the ECMA emerges at low
temperatures regardless of the sample preparation technique, indicating that
ECMA generally emerges at low temperatures in tetragonally distorted NCO
films.

Table 2:  Summary of the magnetic properties: saturation magnetization extrapolated to 0 K ($M_{0}$), magnetic anisotropy constants, $T_{C}$, and $T_{\mathrm{SR}}$. Sample | $P_{100}$ | $S$ | $P_{10}$
---|---|---|---
$M_{0}$ ($\mu_{B}$/f.u.) | 2.2 | 2.0 | 1.6
$K_{u1}$ (MJ/m3) | $-0.12$ | $-0.53$ | $-0.99$
$K_{u2}$ (MJ/m3) | $0.52$ | $0.81$ | $1.08$
$T_{C}$ (K) | $\sim 410$ | $\sim 350$ | $\sim 280$
$T_{\mathrm{SR}}$ (K) | $\sim 10$ | $\sim 70$ | $\sim 200$
Figure 5:  Curie temperature for the two respective magnetic anisotropy
constants at 5 K for the three samples.

###  III.3 Electron theory

To understand the origin of the relatively large $K_{u2}$ component bearing
ECMA, we performed a numerical calculation to evaluate the MA of each ion
present in tetragonally distorted NCO within the framework of the cluster
model Inoue  _et al._ (2014a, b). Because of the presence of site-distribution
disorders, we performed the calculations for eight different clusters using
four ions, Co2+, Co3+, Ni2+, and Ni3+, which occupy either the $T_{d}$ or
$O_{h}$ sites. Then, we estimated the total MA by considering plausible site
distributions.

The electronic structure of the cluster was calculated using a tight-binding
model for the 3$d$\- and 2$p$-orbitals of the transition metal and oxygen
ions, respectively, and by including the spin-orbit interaction (SOI). We
assumed that the 3$d$-states of the Co and Ni ions are fully spin-polarized,
and that the ions have local moments. The parameters of the inter-site $p$-$d$
hopping between the 3$d$-orbitals of a transition metal ion and the
2$p$-orbitals of the oxygen ions were obtained from Harrison’s textbook
Harrison (1980); Inoue (2015). The ground state energy was calculated by
diagonalizing the Hamiltonian as a function of the magnetization direction.
The lattice constant and Wyckoff position of NCO were obtained from a previous
report Battle  _et al._ (1979). The electron configurations and SOIs for each
ion are summarized in Table 3.

Table 3: Summary of electron configurations of the ions.  Ion | Co2+ | Co3+ | Ni2+ | Ni3+ | O2-
---|---|---|---|---|---
Electron | $3d^{7}$ | $3d^{6}$ | $3d^{8}$ | $3d^{7}$ | $2p^{6}$
SOI (meV) | 20 | 30 | 30 | 30 | —

In the calculation, we assumed that the Ni3+ ions occupying the $O_{h}$ site
are in a low-spin state of $S=1/2$ Bitla  _et al._ (2015). We excluded the
calculation of Co3+ in the $O_{h}$ site because of the low spin state of
$S=0$. To satisfy the above conditions, the Coulomb potential $U$ was set to
be 0.05 eV for Ni in the $O_{h}$ site, whereas the potential for the others
was $U=6.0$ eV.

Figure 6 (a) and (b) show the angle-dependent energy of the magnetic moment
($\Delta E$) at the $T_{d}$ and $O_{h}$ sites, respectively. The preferential
spin direction of Co3+ at the $T_{d}$ site is parallel to the $c$-axis, which
is consistent with previous tight-binding calculations Mellinger  _et al._
(2020). On the other hand, those of Co2+ at the $O_{h}$ site and Ni2+ at the
$T_{d}$ site are preferentially orientated in the $ab$-plane. $\Delta
E(\theta)$ of Ni3+ at the $T_{d}$ site has a large $K_{u2}$ with a minimum of
approximately 45 and 135 degree; therefore, it could be the origin of the
ECMA.

Figure 6: Calculated dependence of the magnetic moment on the angle $\Delta
E(\theta)$ at (a) $T_{d}$, and (b) $O_{h}$ sites. (c) Calculated total
anisotropy based on the adapted cation distribution in Table 4.

Next, we attempted to deduce the total MA by considering the site
distribution. Because the positive $K_{u2}$ possibly originates from Ni3+ at
the $T_{d}$ site, we need to introduce the anti-site disorders of Ni3+ at the
$T_{d}$ sites in our calculation. In fact, according to recent experimental
research Shen  _et al._ (2020b), approximately 10-20% of the anti-site
disorders are present at the $T_{d}$ site in the NCO films regardless of the
growth conditions. To estimate the total $\Delta E(\theta)$, we assumed the
cation distributions listed in Table 4. The distribution satisfies the
assumption that the ratio of atoms is $\mathrm{Ni:Co:O=1:2:4}$, and the total
valence becomes zero.

Table 4: Assumed cation distribution of ions for total MA calculation.  Site | $T_{d}$ | $O_{h}$
---|---|---
Ion | Co2+ | Co3+ | Ni3+ | Co3+ | Ni2+ | Ni3+
$S$ | 3/2 | 2 | 3/2 | 0 | 1 | 1/2
Occupancy | 0.85 | 0.0 | 0.15 | 1.15 | 0.15 | 0.70

Figure 6 (c) shows the results of the calculation of the total MA that takes
into account the cation distribution in Table 4. By fitting the total $\Delta
E(\theta)$ using Eq. (1), we obtain $K_{u1}=-1.34$ MJ/m3 ($-0.56$ meV/f.u.),
and $K_{u2}=1.42$ MJ/m3(0.59 meV/f.u.). Because $K^{\mathrm{eff}}_{u1}<0$ and
$K_{u2}>0$, the calculated result qualitatively reproduces the ECMA. Note that
both $K_{u1}=K^{\mathrm{eff}}_{u1}+\mu_{0}M_{0}^{2}/2$ and $K_{u2}$ are in
relatively good agreement with the experimentally obtained values, and
especially closely approximate those for Sample $P_{10}$, as in Table 2.
According to the site/ion dependencies of $\Delta E$ shown in Fig. 6 (a) and
(b), the small amount of Ni3+ that occupies the $T_{d}$ site is a key
component responsible for the ECMA.

##  IV Conclusion

We investigated the lattice distortions and magnetic properties of epitaxial
NCO films grown on MAO(001) substrates by magnetron sputtering and PLD.
Although the NCO films exhibited PMA at room temperature, the magneto-torque
experiments revealed that the NCO films exhibited ECMA below
$T_{\mathrm{SR}}$, regardless of the preparation technique. The observed spin
reorientation seemed to occur continuously at $T_{\mathrm{SR}}$, and the cone
angle increased with decreasing temperature. $T_{\mathrm{SR}}$ and $T_{C}$ are
strongly dependent on the conditions under which the film was grown,
suggesting that these conditions affect the distribution of the different
cations. The proposed electron theory based on the cluster model successfully
explains the origin of the ECMA, which is attributed to the Ni3+ that occupy
the $T_{d}$ site. The results of both the experimental and cluster model
calculations suggest that the ECMA is the result of the cation anti-site
distribution of Ni3+ associated with the growth conditions of the thin film.
Although we succeeded in reproducing the ECMA within the framework of the
cluster model calculation, the reason for the change in the sign of $K_{u1}$
at $T_{\rm SR}$ remains to be solved. The mechanism of the spin reorientation
transition could be related to the mixed-valence nature of NCO.

###### Acknowledgements.

This project is partly supported by the Japan Science and Technology Agency
(JST) under collaborative research based on industrial demand “High
Performance Magnets: Towards Innovative Development of Next Generation
magnets” (JP-MJSK1415), KAKENHI(19KK0104, 19H05816, 21H01810) , the Ibaraki
University-Tsukuba Joint Coordination Fund, and the Tanigawa Fund Promotion of
Thermal Technology. This work was performed with the approval of the “Photon
Factory Program Advisory Committee” (proposals No.2017G602 and No.2016S2-005).
This work was also supported by the Japan Society for the Promotion of Science
(JSPS) “Core-to-Core program (A)” Advanced Research Networks by a grant from
the Integrated Research Consortium on Chemical Sciences and the International
Collaborative Research Program of the Institute for Chemical Research at Kyoto
University from the Ministry of Education, Culture, Sports, Science and
Technology (MEXT)” of Japan. H. K. acknowledges support in the form of a
Grant-in-Aid for JSPS Fellows (20J10749). H. K. thanks T. Oki for his
cooperation.

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# A Method to Detect Quantum Coherent Transport in Memristive Devices.

C Huggins  S Savel’ev  A Balanov  A Zagoskin  Physics Department, Loughborough
University, Loughborough, United Kingdom

###### Abstract

While the size of functional elements in memristors becomes of the orders of
nano-meters or even smaller, the quantum effects in their dynamics can
significantly influence their transport properties, consistent with recent
experimental observations of conductance quantisation in memristors. This
requires the development of experimentally realizable procedures to detect
quantumness of memristors. Here we developed an experimental protocol allowing
us to find evidence that the memristor can be in a superposition of states
with different memristivities. We simulate the nonlinearity induced in the
quantum memristive system via periodic projective measurements, observing how
it manifests itself in the emergence of additional spectral components in the
response to the harmonic signal. Moreover, the response demonstrates a
resonant behaviour when the frequency of the projective measurements
commensurates with the frequency of the input. We demonstrate that observation
of such harmonic mixing can be used as experimental evidence of quantum
effects in memristors.

Quantum Transport, Qubit

##  I Introduction

Memristors were first proposed back in 1971 Chua (1971) as a logically
necessary complement to the fundamental lumped circuit elements (resistors,
capacitors, and inductors). These elements parameterise the relations between
the dynamical variables of the circuit: current I, charge q, voltage V and
magnetic flux $\varphi$. Memristors were the “missing link” (directly relating
$\varphi$ to q); its appearance restores the symmetry between two active
(resistor and memristor) and two reactive (capacitor and inductor) circuit
elements. The term (a portmanteau of “memory” and “resistor”) reflects that a
memristor can be considered a resistor, the resistance of which depends on the
cumulative q or $\varphi$ (i.e., the integral over time of I or V on the
device) Chua (2019).

It is well known that reactive circuit elements have their fully quantum
analogues (see, e.g., Zagoskin (2011)) and can be realised via qubit-based
structures, which demonstrate quantum superpositions of states with different
inductances (capacitances). For active circuit elements, such behaviour may
seem impossible due to the inevitable dissipation, detrimental to quantum
coherence. Nevertheless, such a conclusion would be hasty. It is well
established, for example, that in many mesoscopic structures the region which
determines the resistance is spatially separate from those regions where the
dissipation and relaxation take place Yi  _et al._ (2016); Zagoskin (2011).
This provides a time scale on which a quantum superposition of states with
different resistances can exist. The same possibility cannot be denied to
memristances.

In this paper we investigate experimentally accessible signatures of such
paradoxical behaviour, to wit, the nonlinear effects arising from periodic
measurements of such a quantum memristor. We look at the effects of regular
measurement on spectral properties of the proposed device.

##  II Classical Memristor

First, we consider the general case of a classical current-driven memristance
M at a time t. The memristance depends on the state variables x, where M and x
are described by the following: Chua (2019)

$\displaystyle V=$ $\displaystyle M(x,I,t)I$ (1) $\displaystyle\frac{dx}{dt}=$
$\displaystyle f(x,I,t)$ (2)

For our purposes, we utilise a memristive function based on the well-studied
diffusion memristors Yi  _et al._ (2016); Strukov  _et al._ (2008); Wang and
et al (2017); Prezioso  _et al._ (2016); Jiang and et al (2017); Ushakov  _et
al._ (2021); Ushakov and et al (2021); Choi and et al (2016). Though our
scheme can be generalised to any other mechanism for memristive switching,
e.g., those outlined in ref Sun and et al (2019). The state variable for our
diffusion memristor is the (normalised) width of the doped region, X(t)=W(t)/D
(see Fig 1) where W(t) is the width of the doped region, and D is the total
width of the active part of the device (W is assumed to be constant in time).
The fewer the charge carriers, the larger the width of the undoped region,
corresponding to a higher memristance. A sinusoidal input current will drive
the memristance to oscillate between minimum/maximum memristance values,
$R_{L}$ and $R_{H}$, where $\mu_{v}$ is the average ion mobilityStrukov  _et
al._ (2008):

$M(q)=R_{H}(1-\frac{\mu_{v}}{D^{2}}R_{L}q)$ (3)

Figure 1: Diagram of an example diffusive memristor implemented via a thin
film semiconductor. The width of the doped region W(t) across the active
region of the device (total width D) varies with the bias applied to the film.
The fully undoped state corresponds to the highest memrisistance,
$M(X=0)=R_{H}$, and the fully doped corresponds to the lowest, $M(X=1)=R_{L}$.

The voltage response (for a sinusoidal input current
$I=I_{0}\sin(\omega_{s}t)$, with the initial condition q(t=0)=0) is expected
to have two dominant frequencies. One at the source frequency $\omega_{s}$,
and one at the second harmonic, $2\omega_{s}$. This second order response
arises from the multiplication of q and I.

$\begin{split}V_{C}=\left(R_{H}I_{0}-\frac{\mu_{v}}{D^{2}}R_{L}R_{H}\frac{I_{0}^{2}}{\omega_{s}}\right)\sin(\omega_{s}t)+...\\\
+\frac{\mu_{v}}{2D^{2}}R_{L}R_{H}\frac{I_{0}^{2}}{\omega_{s}}\sin(2\omega_{s}t)\end{split}$
(4)

All classical, periodically driven memristors will have a response at
$\omega_{s}$. In the next sections, we look for differences in the spectra
between the classical and quantum cases.

##  III Quantum Model

We model a qubit-controlled quantum memristor. Experimentally, this could
describe a setup with a van der Waals heterostructure Lucatto and et al (2019)
or a singularly charged pair of quantum dots, where electrons in each state
get transported to different memristive filaments. Measuring the qubit gives
an eigenvalue j, determining the memristance with j=0,1, corresponding to
qubit states $\ket{0}$, $\ket{1}$.

The state evolves via the Schrödinger equation, determining probability of
switching state at each measurement ($P_{sw}$). We describe state switching
via pulse functions $\Pi_{j}(t)$. If the state is measured j on the $k^{th}$
measurement, then over the proceeding time interval (between times
[$t_{mk}$,$t_{mk+1}$]), $\Pi_{j}=1$ and $\Pi_{1-j}=0$. This acts as an on/off
switch for each state, illustrated in fig. 2.

$V_{Q}(t)=\sum_{j=0}^{1}\Pi_{j}(t)V_{Cj}(t)$ (5)

Pulse widths in $\Pi_{j}$ are characterised by the statistics of measurement
events and $P_{sw}$. $P_{sw}$ acts as a mediator between the classical and
quantum cases. The $P_{sw}=0$ case corresponds to a classical system of
electron transport through only one of the memristive filaments (no
superpositions of electron position). $P_{sw}$ depends on the state, governed
by the Schrödinger equation. We utilise a pseudospin Hamiltonian for the
purposes of this report (equation (6)) where $\Delta$ and $\epsilon$ have been
chosen to give a high probability of switching:

$\displaystyle H_{0}=-\frac{1}{2}(\Delta\sigma_{x}+\epsilon\sigma_{z})$ (6)
$\displaystyle\frac{d}{dt}\ket{\psi}=\frac{1}{i\hbar}H_{0}\ket{\psi}$ (7)

We also consider examples where there is some uncertainty in the measurement
frequency (modelling realistically imperfect measurement protocols). This
uncertainty is normally distributed around the average measurement frequency
$\omega_{m}$, by a variance of $\sigma_{\omega}^{2}$.

We predict the output spectrum using equation (5), $\tilde{V}_{Q}(\omega)$ is
the convolution of $\tilde{\Pi}_{j}(\omega)$ and $\tilde{V}_{Cj}(\omega)$
(equation (10)). To determine $\tilde{\Pi}_{j}(\omega)$, we use the ideal
model for $\Pi_{j}(t)$, a periodic square wave of frequency $\omega_{m}/2\pi$
(halved since 4 measurements are needed for a full wave form, see fig. 2).
This model assumes a 100% switching probability, a suitable approximation,
since we selected a Hamiltonian to give a high switching rate (at our
measurement frequency).

Figure 2: Square wave switching functions $\Pi_{j}$. a) is an example
$\Pi_{0}$, with a 100% switching probability, $P_{s}w=1$. The arrows
demonstrate the period of the square wave, $2\pi/\omega_{m}$. b) is an example
with a lower $P_{sw}$=0.4.

The most significant peaks in $\tilde{\Pi}_{j}(\omega)$ are at odd multiples
of this square-wave frequency $\omega_{m}/2\pi$. To find
$\tilde{\Pi}_{j}(\omega)$ (equation (8)) we first found the Fourier
coefficients for $\Pi_{j}(t)$, before using this to calculate the one-sided
Fourier transform. Considering $V_{Q}$ we assume that $\overline{\psi_{j}}$ is
the time-average of $\Pi_{j}$ (for a high switching rate, this will be
$\approx$0.5). We also consider that the time-average of $V_{Cj}$ is
negligible. For brevity, let
$A_{2\omega_{s}}=\frac{\mu_{v}}{2D^{2}}R_{Lj}R_{Hj}\frac{I_{0}^{2}}{\omega_{s}}$:

$\displaystyle\tilde{\Pi}_{j}(\omega)\approx$
$\displaystyle\overline{\psi_{j}}\delta(\omega)+i\sqrt{\frac{\pi\omega_{m}}{2}}\sum_{n=0}^{N}\frac{\delta(\omega-\frac{(2n+1)\omega_{m}}{2})}{\omega}$
(8) $\displaystyle\tilde{V}_{Cj}(\omega)=$ $\displaystyle
i(R_{Hj}I_{0}-2A_{2\omega_{s}})\delta(\omega-\omega_{s})+...$
$\displaystyle+iA_{2\omega_{s}}\delta(\omega-2\omega_{s})$ (9)
$\displaystyle\tilde{V}_{Q}(\omega)=$
$\displaystyle\sum_{j=0}^{1}\tilde{\Pi}_{j}(\omega)*\tilde{V}_{Cj}(\omega)$
(10)

By evaluating equation (10) at the primary peak, we get equation (11). From
this, we expect resonant peaks in $\tilde{V}_{Q}(\omega_{s})$ when
$\omega_{s}=(2n+1)\omega_{m}/4\pi$. This frequency mixing is characteristic of
the application of projective measurements, we do not expect to observe this
in the classical case.

$\displaystyle\begin{split}&\tilde{V}_{Q}(\omega_{s})=\sum_{j=0}^{1}(R_{Hj}I_{0}-A_{2\omega_{s}})...\\\
&\left(i\overline{\psi_{j}}-\sqrt{\frac{\pi\omega_{m}}{2}}\sum_{n=0}^{N}\frac{1}{\omega_{s}}\delta(\omega_{s}-\frac{(2n+1)\omega_{m}}{4\pi})\right)\end{split}$
(11)

##  IV Simulations

We use powers of two for many variables, since having $2^{k}$ data points is
most efficient for calculations using the fast Fourier transform. We find
appropriate convergence with a step size of $dt=2^{-14}s$, and chose a
measurement frequency of 16Hz and a simulation time of T=32s. To look for
periodicity, we collect data up to the $10^{th}$ harmonic of $\omega_{m}$.
Each trial consists of 160 simulations of the device, 1 Hz$<\omega_{s}<160$
Hz. The maximum $\omega_{s}$ is within realistic bounds for timescales of ion
diffusion Choi and et al (2016). We use reported memristance parameters from
ref. Stewart and et al (2004): $R_{H0}=20k\Omega$, $R_{L0}=2k\Omega$,
$R_{H1}=2.1k\Omega$ $R_{L1}=250\Omega$. Ref. Stewart and et al (2004) also
motivated the choice of $I_{0}$, since their device is stable for $V>$0.5V, we
set $I_{0}=0.5/R_{H0}=25\times 10^{-6}$A.

At each time-step $1<n<Tdt$, we calculate $V_{n}$, $\varphi_{n}$, $I_{n}$,
$q_{n}$, and the wave function $\ket{\psi_{n}}$. These are calculated via the
following, where all parameters are initialised at 0 (including
$\ket{\psi}=\ket{0}$). Solutions to equations (16) and (17) were computed
using the 4th order Runge-Kutta method:

$\displaystyle t_{n}=$ $\displaystyle t_{n-1}+dt$ (12) $\displaystyle I_{n}=$
$\displaystyle I_{0}\sin(\omega_{s}t_{n})$ (13) $\displaystyle q_{n}=$
$\displaystyle\frac{I_{0}}{\omega_{s}}(1-\cos(\omega_{s}t))$ (14)
$\displaystyle V_{n}=$ $\displaystyle
I_{n}R_{Hj}(1-\frac{\mu_{v}}{D}R_{Lj}q_{n})$ (15)
$\displaystyle\frac{d\varphi_{n}}{dt}=$ $\displaystyle V_{n}$ (16)
$\displaystyle\frac{d\psi_{n}}{dt}=$
$\displaystyle\frac{1}{i\hbar}H\ket{\psi_{n-1}}$ (17)

##  V Results

First, we look for the emergence of signal mixing in $\tilde{V}_{Q}(\omega)$
as $P_{sw}$ increases. This demonstrates that $P_{sw}$ is a mediator between
the quantum and classical cases. In fig. 3, we see characteristic signal
mixing of $\omega_{m}$ and $\omega_{s}$ for mid and high values of $P_{sw}$
Savel’ev and et al (2004). The $P_{sw}=98.8\%$ data set is close to the
switching function in fig. 2a, the $P_{sw}=46.5\%$ data set has a similar to
fig. 2b. We do not illustrate the $P_{sw}=0$ example since there is no real
component to the spectra in this case (deducible from equation (4)).

Figure 3: Emergence of periodic spikes in the spectra as the probability of
switching ($P_{sw}$) increases for $\omega_{s}$=100Hz. When $P_{sw}=0$, there
is no real part to the spectrum. For low switching probabilities, the real
part mimics the imaginary part, with a reduced amplitude. We see the emergence
of the characteristic peaks/troughs at mid-range probabilities. At high
$P_{sw}$ we see all resonances, the odd set having a higher amplitude than the
even set.

Fig. 4 demonstrates how the real amplitude of the primary peak varies with the
source frequency. In 4a we compare the classical data (no peaks) with the
quantum data (peaks at odd multiples of $\omega_{m}/4\pi$, see equation (11))
seeing a clear difference in behaviour. The average probability of switching
$P_{sw}$ across this trial was 98.7$\pm$0.9%. $P_{sw}$ is high enough for our
model $\Pi_{j}$ function to be a suitable approximation. As such, we can
confidently conclude that the presence of resonance at the predicted points is
indicative of the application of regular projective measurements.

Figure 4: Real component of the primary peak as $\omega_{s}$ is varied. We see
the signatures of the measurement process as predicted, peaks in the spectra
at the key frequencies of $\omega_{s}=\frac{2n+1}{4\pi}\omega_{m}$. These are
in addition to the peak at the square wave frequency,
$\omega_{s}=\omega_{m}/2\pi$. 4a shows the clear differences between the
classical and quantum cases. 4b demonstrates this method can be used in noisy
systems, we still see peaks at many of the characteristic frequencies when
there is a 10% standard deviation in the measurement frequency.

Finally, we looked at cases with varying statistics of measurement events, to
test the robustness of this methodology. We simulate trials where the standard
deviations in $\omega_{m}$ is 5% and 10%. I.e. the measurement frequencies are
normally distributed around $\omega_{m}$ by 0.05$\omega_{m}$ and
0.1$\omega_{m}$. $P_{sw}$ varies between each $\sigma_{\omega}$ trial: across
the 5% trials, $P_{sw}=0.889\pm 0.014$; in the 10% trials, $P_{sw}=0.810\pm
0.017$. Fig. 4b demonstrates the differences in peak heights and widths as
expected. The higher $\sigma_{\omega}$, the larger the deviation from the
idealised case, especially at high frequencies. In the 5% trial, the first 7
peaks in the data set have similar positions to the $\sigma_{\omega}=0$ trial.
In the 10% trial, we see analogues for the first four peaks. In both cases, we
clearly see the effects of signal mixing near the predicted points, showing
our method is robust.

##  VI Conclusions

We have simulated systems to demonstrate our method to verify the presence of
quantum coherent memristance states. Our model would fit several experimental
systems, not limited to a van der Waals heterostructure Lucatto and et al
(2019) or a singularly charged pair of quantum dots, with two memristive
filaments per localised electron state.

We have demonstrated how the probability of switching state upon measurement
controls the amount of quantumness we see in the voltage spectra. The higher
the probability of switching, the more obvious the frequency mixing in
real($V_{q}(\omega))$.

We conduct frequency sweeps, tracking the real component of the amplitude of
the primary spectral peak (fig. 4a). This alternate method to find evidence of
the periodic wave function collapse would be particularly useful in noisy
systems, where the signal in the individual spectra may be shrouded. We see
clear differences in behaviour in the classical and quantum schemes.
Additionally, we carry out the same process for noisy systems, to test the
robustness of the methodology. We still find evidence of frequency mixing even
in the case where the measurement frequency varies by 10%. This method is
stable enough that the characteristic resonance patterns are still evident
across all trials we have conducted.

## References

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# Improving Tuberculosis (TB) Prediction using Synthetically Generated
Computed Tomography (CT) Images

Ashia Lewis
The University of Alabama  Evanjelin Mahmoodi
University of California, Santa Cruz
Yuyue Zhou
New York University
Megan Coffee
NYU Grossman School of Medicine
Elena Sizikova
New York University

###### Abstract

The evaluation of infectious disease processes on radiologic images is an
important and challenging task in medical image analysis. Pulmonary infections
can often be best imaged and evaluated through computed tomography (CT) scans,
which are often not available in low-resource environments and difficult to
obtain for critically ill patients. On the other hand, X-ray, a different type
of imaging procedure, is inexpensive, often available at the bedside and more
widely available, but offers a simpler, two dimensional image. We show that by
relying on a model that learns to generate CT images from X-rays
synthetically, we can improve the automatic disease classification accuracy
and provide clinicians with a different look at the pulmonary disease process.
Specifically, we investigate Tuberculosis (TB), a deadly bacterial infectious
disease that predominantly affects the lungs, but also other organ systems. We
show that relying on synthetically generated CT improves TB identification by
7.50% and distinguishes TB properties up to 12.16% better than the X-ray
baseline.

##  1 Introduction

Diagnosing and evaluating infections on medical images is an essential role of
medical imaging. Computed tomography (CT) scans provide substantial insight
into diagnosis and follow-up of patients already on treatment, uncovering
details on the patient’s particular case. However, CT scanners are expensive
devices that are often not available in low-resource, smaller or rural
communities [26], not always feasible for critically ill patients, and present
a large dose of radiation. Even if the initial diagnosis was made with CT,
follow up exams often cannot happen due to accessibility, costs, and concerns
over radiation dose. This is an important radiology challenge which precludes
millions of people worldwide from receiving an advanced medical evaluation
[23, 25]. For deadly infectious diseases such as tuberculosis (TB) that
particularly affect those in lower resource settings, clinicians are often
left without the same radiologic insights that clinicians in resource rich
settings may have for patient care. CT scans, when available, may not have as
many imaging slices or may have delays because of needed repairs or the need
for trained radiology technicians and radiologists to be involved. Improved
access to radiology with synthetic scans could also increase clinician
engagement. Tuberculosis is a major public health concern, and year after year
has been the largest infectious disease cause of death worldwide. Timely
diagnosis and treatment, with careful follow-up, is not only imperative in
saving patient lives, but also critical for preventing disease spread across
the larger community.

In this work, we leverage the power of computational modelling to estimate CT
scans from chest radiographs (X-rays), a different type of imaging procedure.
Each X-ray provides a single image from one angle, creating a 2D image. CT
scans take the same imaging technique with multiple images, generally using a
rotating X-ray tube, creating what is essentially a 3-D image with multiple
image slices available. Image quality can vary in different settings and many
X-rays, performed without quality equipment or trained technicians, may be
blurry and may be under or over-exposed. X-rays are more widely available,
cost-effective, may even be done at the bedside for some patients, and require
less radiation. We use the X-ray and CT images to learn an automatic disease
classification network, focusing on healthy and tuberculosis (TB) patients, as
well as different TB properties (see Figure 1), and show that synthetically
generated CT scans improve identification of TB by 7.50% and classification of
TB properties by 12.16%. These synthetic CT scans also give tangible imaging
for clinicians who may not have access to CT scans to use in order to diagnose
and follow patients clinically.

We generate CT scans from input X-ray images using an existing computational
model, X2CT-GAN [36]. This model is a neural network trained to predict CT
from chest radiographs (X-ray). By relying on digitally reconstructed
radiograph (DRR) technology [22], this technique obtains synthetic training
X-rays in posterior-anterior (PA) and optionally lateral views (see Section 4
for discussion) and learns a generative adversarial network (GAN) to predict
the CT image from input X-rays. The model is further refined to accept the
distribution of real X-rays by style transfer using CycleGAN [38]. Please see
[36] for additional details. In this work, we study how CT scans generated
using this model can improve the identification and classification of
tuberculosis (TB). In particular, we learn a fully automatic tuberculosis
classification model and study how different types of inputs affect disease
identification and classification performance. Our contributions can be
summarized as follows:

  * •

We show that it is possible to significantly improve automatic TB disease
identification and classify its variants by generating synthetic CT scans from
X-ray images before the analysis phase.

  * •

We conduct an extensive experimental evaluation of our approach on disease
identification and property classification, and present compelling
quantitative and qualitative results.

Figure 1: Examples of X-ray (posterior-anterior (PA) and lateral (L) ) of
healthy and diseased patients. In comparison to healthy patients (first
column), lungs of TB patients (second column) may have many different
findings. Here is shown one specific finding: infiltrates (areas gray to
white) in the upper lung fields which represent active infection. Other
examples of active TB may look very different, such as speckled pattern
throughout the lung (miliary TB named as appears as if millet thrown across
image) and effusions (fluid appearing white on imaging) at the lung base(s).
TB sequelae [20] (Sequela of tuberculosis) refers to scarring in the absence
of infection after healing from TB. This can have many different findings as
well. There may be a decrease in lung volumes, airways may narrow, and there
may be calcification (deposition of calcium) or cavitation (loss of or holes
in lung parenchyma), as described in Figure 2.  Figure 2: Examples of
different types of tuberculosis (TB) categories considered: cavitation,
granuloma, calcification. Cavitations refer to cavities (holes) in the lung
that can develop as a result of tuberculosis infection and are identified as
dark, usually round areas on an X-ray. Granulomas refer to an inflammatory
response and can often be identified by a nodule, generally white because of
calcification. Calcifications refer to any area where TB has led to the
deposition of calcium, resulting in areas white like bone, and this can
include small flecks, larger nodules (such as granulomas), and also the
pleural or pericardial (heart) lining

##  2 Related Work

### Automatic Disease Classification

The success of deep learning-based models in computer vision tasks [29]
encouraged development of many automatic disease classification networks. Such
approaches have been used to classify COVID-19 [34], tuberculosis [27], and
other infectious diseases [32, 35] from X-ray images. CT images [28, 17] have
also been widely used for automatic disease classification, in part because
analysis of 3D imaging offers additional context [40]. In particular, the
value of CT scans for tuberculosis (TB) classification using neural networks
has been recently highlighted [41, 15]. Both CT and X-ray can be used for
diagnosis and analysis of TB, but CT may offer additional visualization and
insight [24, 3, 1].

On the other hand, multi-modal models, for instance, computed tomography (CT)
and electronic health records (EHR) [13] or magnetic resonance imaging (MRI)
and positron emission tomography (PET) [19], have been shown to successfully
fuse information across multiple modalities. In a similar fashion, recent
studies [11, 39] identified that the addition of different views improves
chest X-ray prediction accuracy. Fusion of X-ray and CT in particular has been
shown to help automatically diagnose infectious diseases such as the recent
coronavirus (COVID-19) [21].

### Synthetic Data in Medical Imaging

Development of realistic generative models, such as generative adversarial
networks (GAN) [9], made possible the generation of very realistic synthetic
medical data such as patient records [5], skin [8] and bone [10] lesions,
mammograms [16] and others [14]. Synthetically generated medical data has been
used to address limitations on real data availability such as class imbalance
[18] or limited availability [6]. For example, in combination with real data,
synthetic images are effective in improving localization of important
anatomical landmarks [33]. In another study [30], simulated chest X-rays with
pathologies helped improve chest pathology classification. Synthetically
generated CT images have been shown to improve a diverse set of medical data
analysis tasks such as classification [6], segmentation [31], super-resolution
[37]. In our work, we rely on a recent GAN-based approach [36] that generates
CT images from X-rays to improve. To the best of our knowledge, we are the
first to use synthetically-generated CT images for the analysis of TB.

##  3 Tuberculosis Background

TB imaging can have many findings. In our analysis, imaging with TB was
classified as having granulomas, other calcifications, and cavitations (see
Figure 2 for examples), though there are many other findings associated with
TB diagnosis. Imaging identified as having TB was also classified as either
active TB or TB sequelae. Note that patients may have multiple
classifications.

The different specific findings include granulomas and other calcifications.
Granulomas refer to an inflammatory response, which encapsulates bacteria
responsible for causing TB but also potentially in response to other triggers.
Formally, granulomas are identified by pathological analysis, but on radiology
these may be identified when there is a nodule, often calcified and hence
white, on TB imaging. Other calcifications, where tuberculosis infection has
led to the deposition of calcium, are also identified. This can include
calcification of the pleural space, which is found surrounding the outer lung.
Cavitations refer to holes in the lung which can develop in tuberculosis
infection and persist indefinitely after. These cavities appear as darker,
often rounded, areas on imaging.

TB sequelae [20] (Sequelae of tuberculosis) refers to the state of various
complications after healing from TB, and is characterized by scarring of the
lungs in the absence of signs of active infection. These sequelae can include:
cavitations and various calcifications, as well as apical fibrosis (scarring
or stiffening of the upper lung tips), cicatrization atelectasis (scarring of
lung reducing the expansion or size of the lungs), tracheobronchial stenosis
(narrowing of airways), or bronchopulmonary fistula (a passageway between
airways and the pleural space that surrounds the lungs). Patients, without
extensive medical records may be treated and retreated for findings of TB
sequelae that are misdiagnosed as active TB. Such patients may have continued
difficulty breathing or other symptoms but this will not respond to treatment,
which can be harmful especially as there is no benefit as there is no active
TB infection. In our analysis, we study how automatic classification of TB
findings and identification of TB can be obtained from X-rays and improved
using synthetically generated CT images.

##  4 Approach

We leverage a generative model that predicts CT scans from X-rays in order to
evaluate its effectiveness in improving TB prediction accuracy. We divide our
analysis into two tasks: _disease identification_ , where the goal is to
distinguish healthy and TB patients, and _disease classification_ , where the
goal is to classify different types of TB or to distinguish active from past
TB (TB sequelae).

Specifically, let $x_{i}=(x^{PA}_{i},x^{L}_{i},d_{i})$ be the X-ray images (in
posteroanterior (PA) and lateral (L) views) of a given patient, where
$d_{i}=(l_{i},s_{i},t_{i})$ is the disease label. Binary variables
$l_{i},s_{i},t_{i}$ are such that: $l_{i}$ denotes whether the patient has TB,
$s_{i}$ denotes whether the patient has TB sequelae, $t_{i}$ denotes whether
the type of TB (granuloma, cavitation, calcification) that the patient has.
Let $X=\\{x_{1},x_{2},\ldots x_{n}\\}$ be a collection of data available. We
use computational models $f_{1},f_{2}$ to predict $c_{1,i}=f_{1}(x^{PA}_{i})$
and $c_{2,i}=f_{2}(x^{PA}_{i},x^{L}_{i})$, the synthetic CT image
corresponding to example $i$.

We then train separate models $g$ such that: $\hat{q}_{i}=g(r_{i})$, where
$q_{i}=l_{i},s_{i}$ or $t_{i}$, and $r_{i}$ is a subset of
$\\{x^{PA}_{i},x^{L}_{i},c_{1,i},c_{2,i}\\}$. The loss function used for
training $g$ is the standard binary cross entropy (BCE) with logits loss:

$\mathcal{L}(q_{i},\hat{q}_{i})=-[(\hat{q}_{i})\cdot\log\sigma(q_{i})+(1-\hat{q}_{i})\cdot(1-\sigma(q_{i}))]$

where $q_{i}$ is the ground truth label and $g(r_{i})$ is the predicted label.
Note that $g(r_{i})$ and $q_{i}$ are represented as binary (multi-dimensional)
variables and above loss is applied element-wise with mean reduction in the
end.

All functions $f_{1},f_{2}$ and $g$ were implemented using neural networks and
described below.

##  5 Experimental Details

We now describe the dataset and implementation details of our approach. We
tested the proposed methodology on a large public benchmark [4]. In practice,
$f$ and $g$ are implemented as neural networks and are trained as described
below.

###  5.1 Dataset

To experimentally analyze the proposed approach, we used the PAthology
Detection in Chest radiographs (PadChest) [4] dataset, consisting of 160,868
X-ray images and medical reports from 67,000 patients. The dataset was
collected and interpreted by 18 radiologists at the Hospital Universitario de
San Juan, Alicante (Spain) from January 2009 to December 2017. About a quarter
of the reports were manually annotated by radiologists and the rest were
labelled using a supervised recurrent neural network model (achieving a 0.93
Micro-F1 score on an independent test). The dataset had 50.3% female patients,
49.7% male patients. The patient’s ages ranged from 0 to 105 years, with a
mean age of 58.5. For each experiment, we filtered the original datasets into
smaller sets with required annotations. Specifically, for the TB
identification experiment (healthy vs TB), we used 274 training, 91
validation, and 91 testing images. For TB classification into either TB or TB
sequelae, we used 291 97 testing images. For TB classification into granuloma,
cavitation, and calcification, we used 453 training, 123 validation, and 181
testing images.

The X-ray images in this dataset contain at least one PA view, L view, or both
(mean number of images per patient is 2.37). For multi-view (PA + L)
experiments, we selected a corresponding pair of views. For the single-view
(PA or L) experiments, we randomly selected a single image per patient.

##  6 Network Architecture and Training

###  6.1 CT Generation Network

We use the X2CT-GAN model [36] to generate both biplanar (B) and single-view
(SV) synthetic CT scans for input X-ray images. We relied on the publicly
available pre-trained model for all of our experiments, which was obtained
from the paper code repository. This model was trained on the LIDC-IDRI
dataset [2] that consists of 1,018 chest CT images, augmented with synthetic
data (please see [36] for additional details). Both the posteroanterior (PA)
and lateral (L) radiographic views were used to generate the biplanar
synthetic CT images, while the single-view CT scans were generated using only
the PA view.

###  6.2 TB Analysis Network

For X-ray experiments, we resized images to resolution 364 by 364 with random
resized crop of 320 and flip, and normalized them. We used the HeMIS model
[12] with DenseNet-PA architecture based on implementation in [11] to perform
classification. We use Adam as optimizer, BCE as loss function, set learning
rate as 0.0001 and batch size of 1. The model was trained until convergence.

For CT and CT+X-ray experiments with two classes, we relied on the 3D
convolutional architecture proposed by [41] with Adam optimizer, learning rate
0.0001 and batch size 20. The network was trained until convergence with
binary cross-entropy (BCE) loss for 2-class experiments. In this experiment,
X-ray images were normalized and resized to 128 by 128 resolution, and
Gaussian blur was applied to de-noise. The CT images were normalized and
resized to 128 by 128, and two central slices were chosen. We experimentally
found that using two center slices yielded better results than the entire CT
volume (see Table 1), possibly because the non-center slices of the generated
CT often appeared blurrier. For the joint X-ray and CT experiments, the X-ray
and CT images were concatenated before being inserted into the network. During
training, random rotation augmentations were used. For 3-class experiments, we
used the architecture proposed by [7] with the same pre-processing and
parameters as described above.

##  7 Experimental Results

In this section, we report quantitative and qualitative analyses of the
proposed methodology.

###  7.1 Disease Identification Evaluation

We start by analyzing the performance of TB identification (distinguishing
healthy and TB patients) accuracy across different types of inputs. Results
are reported in Table 1. We find that when trying to predict TB from X-rays
only, using both PA and L views yields the same prediction accuracy.
Therefore, in subsequent experiments, we used only the PA views to represent
X-ray input to the network.

We next look at the TB prediction accuracy from generated CT scans. We find
that CT images generated from both PA and L views (i.e., the biplanar (B)
images) offered 31.25% improvement over CT images generated only from a single
(PA) view. In particular, the single view CT prediction accuracy was worse
than the X-ray prediction accuracy, but the biplanar CT prediction accuracy
was better than X-ray prediction accuracy by 5.00%. This analysis also
verifies the conclusion in [36] that X2CT-GAN generates more accurate CT scans
from multi-view X-rays. An interesting observation is that the L view helps
with CT generation, but not directly with classification. Notice also that the
using a two-slice subset of the generated CT instead of the full volume offers
an improvement of 5.00%-11.11%, likely because of the better quality of the
generated images in the center.

When combined with X-rays, the combination of single view CT (generated using
PA X-ray) and X-ray yielded the same accuracy as X-ray only, suggesting that
addition of CT did not degrade performance of the X-ray based prediction when
limited information was available from CT. On the other hand, the biplanar CT
and X-ray generated the top accuracy of 0.86, which represents a 7.50%
improvement over the X-ray baselines. It appears that the CT generation
network is constrained to generate salient geometric and disease-identifying
properties that are not learned by the network that predicts disease directly
from X-rays.

Data | Accuracy
---|---
X-ray (PA) | 0.80
X-ray (PA) + (L) | 0.80
| Full CT | 2-Slice CT
CT (SV) | 0.58 | 0.64
CT (B) | 0.80 | 0.84
CT (SV) + X-ray | 0.72 | 0.80
CT (B) + X-ray | 0.80 | 0.86
Table 1: _Evaluation of disease identification (classification into healthy vs
TB)_ accuracy. Testing accuracies are reported on different types of X-ray and
generated CT images with notation: B - biplanar, SV- single view. Top results
are achieved when PA X-ray images are combined with biplanar CT images and
when 2-slice subset of the CT image is used. Best result is in bold, the
second best is underlined.

###  7.2 Disease Classification Evaluation

Next, we evaluate the models on classifying TB properties (granuloma,
cavitation, and calcification). We report the quantitative results of these
experiments in Table 2. We find that the biplanar CT images improve the
accuracy by 2.70% over the X-ray baseline, while the combination of biplanar
CT and X-rays gives the highest accuracy of 0.83, which is 12.16% better than
the X-ray baseline.

Data | Accuracy
---|---
X-ray | 0.74
CT | 0.76
CT + X-ray | 0.83
Table 2: _Evaluation of TB type classification (granuloma, cavitation,
calcification) accuracy._ Top results are achieved when X-rays are combined
with generated biplanar CT images. Best result is in bold, the second best is
underlined.

###  7.3 Distinguishing TB and TB Sequelae

Finally, in Table 3, we study the performance of the model in distinguishing
TB and TB sequelae. Similar to the previous experiment, we found that the
combination of biplanar CT images and X-rays generated the highest accuracy of
0.81, followed by the result of using only CT images. In this case, the
combination of X-ray and CT demonstrated an improvement of 5.20%, and the CT
only demonstrated an improvement of 2.60%, over X-ray only accuracies. These
experiments demonstrate the clear improvement in accuracy as a result of using
a pre-trained synthetic CT image generator in augmenting the X-ray inputs with
a CT scan.

Data | Accuracy
---|---
X-ray | 0.77
CT | 0.79
CT + X-ray | 0.81
Table 3:  _Evaluation of TB type classification (TB vs TB sequelae) accuracy._
Top results are achieved when X-rays are combined with biplanar CT images.
Best result is in bold, the second best is underlined.

Figure 3: _Sample examples of PA X-rays and corresponding generated CT scans
that were most often correctly categorized across all models._ Patient
positioning, noise, and absence of internally placed devices typically
contribute to more accurate predictions.

Figure 4: _Sample examples of PA X-rays and corresponding generated CT scans
that were most often incorrectly categorized across all models._ The first two
images are apt representations of the x-rays that were often misrepresented.
These x-rays often featured pacemakers and other devices in the image, or
featured patients whose internal anatomy was significantly injured or
anomalous.

###  7.4 Qualitative Analysis

Finally, we analyze the results qualitatively. In Figure 3, we show sample
examples of PA X-ray images and corresponding generated CT scans that are
consistently correctly categorized. We find that data points that most
consistently get categorized correctly represent patients with clear X-rays
with little noise and no internal devices. In these cases, the CT generator
also produces realistic CT images.

In contrast, we show some sample failure examples in Figure 4. Our system has
difficulty with data points from patients whose X-rays show internal devices,
such as pacemakers. Changes to internal anatomy, for instance, from injury,
also presents a particular difficulty for both our approach and for the CT
scan generator. For example, the CT generator in Example 4 produces a low-
contrast CT that is missing details and hard to interpret. This observation
highlights a known challenge of deploying a model trained on synthetic data to
real world settings [36].

##  8 Conclusion and Future Work

We present an automatic tuberculosis TB identification and classification
approach that leverages synthetic data. By learning a computational model that
predicts CT scans from X-rays, we improve disease classification in settings
when no CT images are available. Experiments on a public benchmark demonstrate
that our approach significantly improves accuracy of TB identification and
classification. We also provide synthetic CT images for clinicians to be able
to glean more insight than from Chest X-rays alone, who may not have access to
this imaging. This approach opens the possibility of allowing regular follow
up during the course of illness with synthetic chest CT imaging. Furthermore,
chest X-rays could be compared with CT images more, as the cost of repeat
Chest CT imaging is usually prohibitive.

There are a number of ways our work can be extended. First, the PadChest [4]
dataset included many images that were not manually annotated by a radiologist
or had a confirmed TB diagnosis. It would be important to test the proposed
system on data where these labels are available and additionally performance
could be compared across real and synthetic CT. We would also want to ensure a
wide diversity in the images. Tuberculosis can come in many forms, including:
miliary (scattered points throughout lung fields), effusions (fluid at base of
lung, seen as white), and bronchogenic spread (with ill-defined fluffy air-
space nodular opacities). This analysis was able to identify a wide range of
presentations as TB. It will be important to ensure the dataset is balanced
and includes rarer presentations, especially as some presentations may be more
common in certain groups (particularly age groups or in persons with
immunocompromise, such as with HIV). It will be important for the data to
include a wider range of geographic and demographic origins, especially as
presentation of TB may vary depending on diagnostic resources available and
resultant time needed for diagnosis. In areas where TB care is obtained later
in the course of illness, there may be more permanent damage and TB sequaelae
will need lifelong follow-up for care. Ensuring a wide range of TB sequelae
can help distinguish these cases from active disease and ensure more directed
treatment. Quality of plain films can also vary and will be important to
ensure films taken by a variety of devices work well.

It would also be valuable to investigate how the technique can be applied for
the study of other diseases. We hope that our explorations will encourage more
research into the use of synthetic data to improve automatic disease
diagnosis.

##  9 Acknowledgements

AL and EM were supported by the NYU Center for Data Science (CDS) Capital One
Undergraduate Research Program (CURP) in partnership with the National Society
of Black Physicists (NSBP). ES was supported by the Moore-Sloan Data Science
Environment initiative (funded by the Alfred P. Sloan Foundation and the
Gordon and Betty Moore Foundation) through the NYU Center for Data Science. We
thank anonymous reviewers for their feedback.

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