# Dilaton generation in propagation of magnetic dipole waves of pulsar in a
galactic magnetic field

M.O.Astashenkov, A.V.Bedda, K.V.Parfenov and P.A.Vshivtseva
Physics Department, Moscow State University, Moscow 119991 Russia

###### Abstract

This study is devoted to dilaton generation in propagation of magnetic dipole
waves of pulsar in a galactic magnetic field. It is shown that in this process
it’s necessary to take into account the presence of a reflective index of
matter distributed in the galaxy.

According to obtained estimates influence of the gravitational field of
pulsars and magnetars on radiation of magnetic dipole waves is negligible and
tends to a small change of their amplitude, near about 1 per cent. That’s why
in this study one can neglect the impact of gravitational field on process of
dilaton generation.

Exact solution of the dilaton equation is found and angular distribution of
dilaton radiation is obtained in propagation of magnetic dipole waves of
pulsar in galactic magnetic field.

## Introduction

Dilaton is hypothetical scalar field which is predicted by various theories of
gravitation with extra dimensions. For example, Kaluza-Klein theory [1] which
unify gravitational and electromagnetic interactions, and also superstring
theory [2].

Various properties of dilatons are being actively studied in the scientific
literature. [3, 4], as well as other undiscovered particles described by axial
scalar fields - axions and arions [5, 6, 7].

Following the study [8], density of the Lagrange function for the dilaton
field $\Psi$ interacting with an electromagnetic field can be expressed in the
form:

${\cal L}=a_{0}(\partial\Psi)^{2}+a_{1}e^{-2{\cal K}\Psi}F_{nm}F^{nm}\;,$ (1)

where $a_{0},\ a_{1}$ and $\cal K$ are gauge constant and $F_{nm}$ is Maxwell
tensor.

Value of the constant ${\cal K}=1$ is predicted by superstring theory. The
5-th dimentional Kaluza-Klein theory result gives the value ${\cal
K}=\sqrt{3}$. It should be noted that in theoretical physics there are other
theories which predict existence of dilaton, but they don’t have any common
with gravity [9, 10]. In this study constant ${\cal K}$ is believed to be
arbitrary.

The field equations obtained from the density of the Lagrange function (1)
have the form:

$(\Delta-{1\over c^{2}}{\partial^{2}\over\partial t^{2}})\Psi={a_{1}{\cal
K}\over a_{0}}e^{-2{\cal K}\Psi}F_{nm}F^{nm}={2a_{1}{\cal K}\over
a_{0}}e^{-2{\cal K}\Psi}\big{[}{\bf B}^{2}-{\bf E}^{2}\big{]}\;.$ (2)

In the expression (2) $\bf E$ and $\bf B$ are the electric field strength and
magnetic induction correspondingly. It is expected that dilaton field is weak
and one can put: $|{\cal K}\Psi|<<1.$ In this case the equation of dilaton
field (2) will take the form:

$(\Delta-{1\over c^{2}}{\partial^{2}\over\partial t^{2}})\Psi={2a_{1}{\cal
K}\over a_{0}}\big{[}{\bf B}^{2}-{\bf E}^{2}\big{]}\;.$ (3)

According to the equation (3) in considered approximation the source of
dilaton field can be only such electromagnetic fields, which invariant${\bf
B}^{2}-{\bf E}^{2}$ is not equal 0.

As many elementary particles have been discovered in cosmic ray, it is
interesting to consider generation of dilatons in astrophysical conditions and
study angular distribution of emitted dilaton energy. Especially interesting
to consider pulsars and magnetars as the sources of electromagnetic coherent
radiaton and correspondingly the possible sources of dilatons. In this study
the dilaton generation in propagation magnetic dipole radiation of pulsars and
magnetars in galactic and intergalactic magnetic fields is considered. One
should note that galactic and intergalactic magnetic fields can be
approximately constant for a quite long distances near about coherence length
$L_{\mathrm{coh}}$. For galactic magnetic field $L_{\mathrm{coh}}\sim 100$ pc,
for intergalactic magnetic field $L_{\mathrm{coh}}\sim 10^{3}$, $10^{4}$ pc.

##  1 Basic equations

Suppose that at the origin of Cartesian coordinate there is a rotating neutron
star (pulsar or magnetar) with magnetic dipole momentum $\bf m$. This star has
a gravitational field. Beside distribution of matter in neutron stars is close
to spherically symmetrical, one can consider Schwarzschild solution as the
metric tensor of pseudo-Riemannian space. It is convenient to write down this
solution in isotropic coordinates. In these coordinates non-zero components of
the metric tensor take the form:

$g_{00}={(4r-r_{g})^{2}\over(4r+r_{g})^{2}},\ \ \ \ \ \
g_{xx}=g_{yy}=g_{zz}=-(1+{r_{g}\over 4r})^{4},$ (4)

where $r_{g}\sim 10$ km is the Schwarzschild radius and the following notation
is introduced: $r=\sqrt{x^{2}+y^{2}+z^{2}}.$

According to the expression (4), influence of gravitational field on process
of dilaton generation can be limited to a sphere of the radius for example
near $10^{6}r_{g}$. Dilaton generation in galactic magnetic field occurs in
region of space with linear size of about 100 pc. Correspondingly, the
influence of gravitational field of pulsars and magnetars on radiation of
magnetic dipole waves tends to negligible change of amplitudes of these waves,
near about 1 per cent. For this reason in this study one can neglect of impact
of gravitational field on the process of dilaton generation.

As it’s known, the equation for magnetic vector potential ${\bf A}$ of an
electromagnetic field in Minkowski space has the form:

$\big{(}\bigtriangleup-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\big{)}\ {\bf A}=-{4\pi\over c}{\bf j}({\bf r},t),$ (5)

where ${\bf j}$ is a current density inside of neutron star and $c$ is speed
of light in vacuum.

Suppose that electromagnetic field outside star is analogous to field created
by rotating around axis $z$ magnetic dipole momentum $\bf
m=\\{m_{x},m_{y},m_{z}\\}$ with frequency $\omega$:

${\bf m}(t)=|{\bf m}|\\{\cos(\omega t)\sin\alpha,\ \sin(\omega t)\sin\alpha,\
\cos\alpha\\}\;.$ (6)

In the expression (6) $\alpha$ is an angle between magnetic dipole momentum
${\bf m}$ and axis $z$. In this case neutron star radiates electromagnetic
waves in magnetic dipole approximation. One can write magnetic vector
potential ${\bf A}$ in the following way:

$\displaystyle{\bf A}_{w}({\bf r},\tau)=\frac{(\dot{{\bf m}}(\tau)\times{\bf
r})}{cr^{2}}+\frac{({\bf m}(\tau)\times{\bf r})}{r^{3}}\;.$ (7)

The dot above vector ${\bf m}$ represent the derivative on relative time
$\tau=t-r/c$. Electric strength and magnetic induction corresponding to
magnetic vector potential ${\bf A}_{w}$ (7) can be expressed as:

${\bf E}_{w}({\bf r},\tau)=-\frac{1}{c}\frac{\partial{\bf A}_{w}({\bf
r},\tau)}{\partial\tau}=\frac{({\bf r}\times\dot{\bf
m}(\tau))}{cr^{3}}+\frac{({\bf r}\times\ddot{\bf m}(\tau))}{c^{2}r^{2}}$ (8)
$\displaystyle{\bf B}_{w}({\bf r},\tau)=\mathrm{rot}{\bf A}_{w}={3({\bf
m}(\tau)\cdot{\bf r}){\bf r}-r^{2}{\bf m}(\tau)\over r^{5}}$
$\displaystyle-{{\dot{\bf m}}(\tau)\over cr^{2}}+{3({\dot{\bf
m}}(\tau)\cdot{\bf r}){\bf r}\over cr^{4}}+{(\ddot{\bf m}(\tau)\cdot{\bf
r}){\bf r}-r^{2}\ddot{\bf m}(\tau)\over c^{2}r^{3}}.$ (9)

Dilaton generation by electromagnetic fields ${\bf E}_{w}$ and ${\bf B}_{w}$
of rotating magnetic dipole momentum of pulsar or magnetar in the expressions
(8), (1) occurs at frequencies $\omega$ and $2\omega$ [8]. Interesting to
note, that the arion generation by the same electromagnetic fields has the
same spectrum [11].

Assume that there is a constant magnetic field ${\bf B_{0}}$ in the considered
area of space:

${\bf B_{0}}=\\{B_{0x},B_{0y},B_{0z}\\}\;.$ (10)

Such a magnetic field can be galactic and intergalactic fields which can be
considered constant at the distances smaller then coherence length
$L_{\mathrm{coh}}$.

In considered configuration of electromagnetic fields the invariant
$F_{nm}F^{nm}$ is equal:

$F_{nm}F^{nm}=2\big{[}{\bf B}^{2}-{\bf E}^{2}\big{]}=2\big{[}{\bf
B}_{w}^{2}-{\bf E}_{w}^{2}\big{]}+4({\bf B}_{0}\cdot{\bf B}_{w})\;.$ (11)

The dilaton generation by electromagnetic fields ${\bf E}_{w}$ and ${\bf
B}_{w}$ is studied in detail in [8]. This work will be focused on dilaton
generation in a constant magnetic field, which is provided by term $({\bf
B}_{0}{\bf B}_{w})$ in the expression (11).

##  2 Solution of the dilaton field equation

The dilaton generation in propagation magnetic dipole radiation (8),(1) in
constant magnetic field occurs in correspondence with equation:

$(\Delta-{1\over c^{2}}{\partial^{2}\over\partial t^{2}})\Psi={4a_{1}{\cal
K}\over a_{0}}({\bf B}_{0}\cdot{\bf B}_{w})\;.$ (12)

Taking into account expression (7), one can rewrite (1) in the following form:

${\bf B_{w}}({\bf r},t)=\mathrm{rot}\;\mathrm{rot}\frac{{\bf m}(\tau)}{r}\;.$
(13)

Putting the expression (13) into equation (12), one can obtain:

$(\Delta-{1\over c^{2}}{\partial^{2}\over\partial t^{2}})\Psi=\frac{4{\cal
K}a_{1}}{a_{0}}({\bf B_{0}}\cdot\mathrm{rot}\;\mathrm{rot}\frac{{\bf
m}(\tau)}{r})\;.$ (14)

It is easily to find a solution of the equation (14) in the following form:

$\Psi=\frac{4{\cal K}a_{1}}{a_{0}}\Big{(}{\bf
B_{0}}\cdot\mathrm{rot}\;\mathrm{rot}\;{\bf F}({\bf r},t)\Big{)}\;.$ (A1)

Thus equation for vector-function ${\bf F}$ will take the form:

$(\Delta-{1\over c^{2}}{\partial^{2}\over\partial t^{2}}){\bf F}({\bf
r},t)=\frac{{\bf m}(\tau)}{r}\;.$ (A2)

Put expression (6) in the right part of the equation (A2) and rewrite it in
the complex form:

$(\Delta-{1\over c^{2}}{\partial^{2}\over\partial t^{2}}){\bf F}({\bf
r},t)=|m|\sin\alpha({\bf e_{x}}+i{\bf e_{y}})\frac{1}{r}e^{-i(\omega
t-kr)}\;,$ (A3)

where $k=\omega/c$.

One can find solution of the equation (A3) in the form:

${\bf F}({\bf r},t)=|m|\sin\alpha({\bf e_{x}}+i{\bf e_{y}})Q({\bf r})\;.$ (A4)

In this case one can obtain following equation for function $Q$:

$(\Delta-{1\over c^{2}}{\partial^{2}\over\partial t^{2}})Q({\bf
r},t)=\frac{1}{r}e^{-i(\omega t-kr)}\;.$ (A5)

Solving equation (A5), find unknown function $Q$:

$Q({\bf r},t)=\frac{1}{2ik}e^{-i(\omega t-kr)}\;.$ (A6)

Finally, exact solution of the equation (14) takes the form:

$\displaystyle\Psi=\frac{2{\cal
K}a_{1}m\sin\alpha}{a_{0}}\Big{\\{}k(B_{0x}-({\bf B_{0}\cdot
N})N_{x})\sin(kr-\omega t)$ $\displaystyle+k(B_{0y}-({\bf B_{0}\cdot
N})N_{y})\cos(kr-\omega t)$ $\displaystyle-\frac{1}{r}(B_{0x}+({\bf B_{0}\cdot
N})N_{x})\cos(kr-\omega t)$ $\displaystyle+\frac{1}{r}(B_{0y}+({\bf B_{0}\cdot
N})N_{y})\sin(kr-\omega t)\Big{\\}}\;.$ (15)

In the expression (2) it is introduced following notation: ${\bf N}=\frac{{\bf
r}}{r}=\\{\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\\}$. It should be
noted that when $r\to\infty$ dilaton field $\Psi$ is oscillating function and
doesn’t tend to 0.

##  3 Angular distribution of the dilaton generation

By defenition amount of dilaton energy $dI$ emitted by source per unit of time
through solid angle $d\Omega$ is given by formula:

$\frac{dI}{d\Omega}=\lim_{r\to\infty}r({\bf W\cdot r})\;,$

where ${\bf W}$ is the energy flux density vector. The vector ${\bf W}$ is
associated with components of stress-energy tensor $T^{ik}$ in a following
way: $W^{\alpha}=T^{0\alpha}$. Stress-energy tensor of the free dilaton field
has the form:

$T^{ik}=2a_{0}g^{in}g^{km}\big{\\{}\frac{\partial\Psi}{\partial
x^{n}}\frac{\partial\Psi}{\partial
x^{m}}-\frac{1}{2}g_{mn}g^{sq}\frac{\partial\psi}{\partial
x^{s}}\frac{\partial\psi}{\partial x^{q}}\big{\\}}\;.$

The angular distribution of dilaton generation can be calculated as the same
way as in [3]:

${dI\over d\Omega}=-2a_{0}r({\bf r\cdot\nabla}\Psi){\partial\Psi\over\partial
t}\;.$

In this case the angular distribution of the dilaton generation in propagation
spherical electromagnetic wave in a constant magnetic field averaged over the
period of electiromagnetic wave is equal:

$\overline{\frac{dI}{d\Omega}}=\frac{4{\cal
K}^{2}a_{1}^{2}m^{2}k\omega\sin^{2}\alpha}{a_{0}}\Big{\\{}k^{2}r^{2}\frac{di_{1}}{d\Omega}+4kr\frac{di_{2}}{d\Omega}+2\frac{di_{3}}{d\Omega}\Big{\\}}\;,$
(16)

where the following notations are introduced:

$\frac{di_{1}}{d\Omega}=B_{0x}^{2}+B_{0y}^{2}-2({\bf B_{0}}\cdot{\bf
N})(B_{0x}N_{x}+B_{0y}N_{y})+({\bf B_{0}}\cdot{\bf
N})^{2}(N_{x}^{2}+N_{y}^{2})\;,$ (i1) $\frac{di_{2}}{d\Omega}=({\bf
B_{0}}\cdot{\bf N})(B_{0x}N_{y}-B_{0y}N_{x})\;,$ (i2)
$\frac{di_{3}}{d\Omega}=B_{0x}^{2}+B_{0y}^{2}+({\bf B_{0}}\cdot{\bf
N})(B_{0x}N_{x}+B_{0y}N_{y})\;.$ (i3)

The expression (i1) can be considered as lead term of the angular distribution
of dilaton generation (16) at $r\to\infty$:

$\displaystyle\overline{\frac{dI}{d\Omega}}=\frac{4{\cal
K}^{2}a_{1}^{2}m^{2}k^{3}\omega
r^{2}\sin^{2}\alpha}{a_{0}}\Big{[}B_{0x}^{2}+B_{0y}^{2}-2({\bf B_{0}}\cdot{\bf
N})$ $\displaystyle\times(B_{0x}N_{x}+B_{0y}N_{y})+({\bf B_{0}}\cdot{\bf
N})^{2}(N_{x}^{2}+N_{y}^{2})\Big{]}\;.$ (17)

Now let us analyze maximal and minimal values of the function
$\overline{dI}/d\Omega$ (16) of variables $\theta$ and $\phi$. For this
purpose denote by $\theta_{B_{0}}$, $\phi_{B_{0}}$ the azimuth and polar
angles correspondingly, along which the magnetic field ${\bf B_{0}}$ (10) is
directed:

${\bf
B_{0}}=B_{0}\\{\sin\theta_{B_{0}}\cos\phi_{B_{0}},\;\;\sin\theta_{B_{0}}\sin\phi_{B_{0}},\;\;\cos\theta_{B_{0}}\\};.$

For definiteness, let $\theta_{B_{0}}\leq\frac{\pi}{2}$ and $\phi\leq\pi$.

The expression (3) becomes 0 for the following values $\theta$ and $\phi$:

$\displaystyle\theta=\theta_{B_{0}},\;\phi=\phi_{B_{0}}\;,$
$\displaystyle\theta=\pi-\theta_{B_{0}},\;\phi=\phi_{B_{0}}+\pi,$
$\displaystyle\theta=\frac{\pi}{2},\;\phi=\phi_{B_{0}}\;,$
$\displaystyle\theta=\frac{\pi}{2},\;\phi=\phi_{B_{0}}+\pi$

and has a maximum value at:

$\displaystyle\theta=\frac{\theta_{B_{0}}}{2}+\frac{3\pi}{4}\;,\phi=\phi_{B_{0}}\;,$
$\displaystyle\theta=\frac{\pi}{4}-\frac{\theta_{B_{0}}}{2}\;,\phi=\phi_{B_{0}}+\pi\;.$

The term (i2) doesn’t give impact to total intensities. The term (i3) has
maximum values in the following values of $\theta$ and $\phi$:

$\displaystyle\theta=\frac{\theta_{B_{0}}}{2}+\frac{\pi}{4}\;,\phi=\phi_{B_{0}}\;,$
$\displaystyle\theta=\frac{3\pi}{4}-\frac{\theta_{B_{0}}}{2}\;,\phi=\phi_{B_{0}}+\pi\;.$

By integrating the expression (16) over the angles $\theta$ and $\phi$, one
can obtain total intensities of dilaton radiation in all directions:

$\displaystyle\overline{I}=\frac{16\pi{\cal
K}^{2}a_{1}^{2}m^{2}k\omega\sin^{2}\alpha}{3a_{0}}\Big{\\{}\frac{1}{5}k^{2}r^{2}\big{(}9(B_{0x}^{2}+B_{0y}^{2})+2B_{0z}^{2}\big{)}$
$\displaystyle+8(B_{0x}^{2}+B_{0y}^{2})\Big{\\}}\;.$ (18)

As one can see, the intensities of dilaton radiation (3) in propagation
magnetic dipole waves in a constant magnetic field increases proportional
$r^{2}$. However, under astrophysical conditions, galactic (intergalactic)
magnetic fields can be represented in the form (10) only at distances $r$
smaller than the coherence length of the galactic (intergalactic) magnetic
field $L_{\mathrm{coh}}$. Thus, the expression (16), (3) are applicable only
when $r<L_{\mathrm{coh}}<\infty$.

The coherence lengths $L_{\mathrm{coh}}$ of the galactic magnetic field and
the intergalactic one are estimated at $100$ pc and $10^{3}$ \- $10^{4}$ pc
respectively.

One should note that there is approximately 1 atom in 1 cm3 of outer space
environment. Therefore, in this work cosmic space is considered to be filled
rarefied neutral gas with very low density distributed uniformly. In this
case, the outer space environment has a refractive index $n$, which differs
little from $1$. As we will see below, this fact will have a strong influence
on the behavior of dilaton field $\Psi$ at $r\to\infty$, and the radiation
intensity will become finite.

##  4 Dilaton generation in propagation magnetic dipole waves in constant
magnetic field in rarefied neutral gas with low density

Assume that outer space is filled with rarefied neutral gas with low density
distributed uniformly and, accordingly, having a refractive index of $n$,
little different from $1$.

In this case the equation for magnetic vector potential ${\bf\tilde{A}}_{w}$
(5) have the form:

$\big{(}\bigtriangleup-\frac{n^{2}}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\big{)}\ {\bf\tilde{A}}_{w}({\bf r},t)=-{4\pi\over c}{\bf j}({\bf
r},t)\;.$ (19)

It follows from the equation (19) that speed of electromagnetic waves is equal
$c/n$ which is less then speed of light in vacuum. Magnetic vector potential
$\tilde{{\bf A}}_{w}$ created by a rotating magnetic dipole moment ${\bf m}$
(6) in an environment with $n>1$ will look like:

$\displaystyle{\bf\tilde{A}}_{w}({\bf r},t)=\frac{n(\dot{{\bf
m}}(\tilde{\tau})\times{\bf r})}{cr^{2}}+\frac{({\bf
m}(\tilde{\tau})\times{\bf r})}{r^{3}}\;.$ (20)

In the expression (20) magnetic dipole momentum ${\bf m}$ (6) depends on
$\tilde{\tau}$

$\tilde{\tau}=t-\frac{nr}{c}\;,$

but not on $\tau$.

Thus, the equation (14) will take the form:

$(\Delta-{1\over c^{2}}{\partial^{2}\over\partial t^{2}})\Psi=\frac{4{\cal
K}a_{1}}{a_{0}}({\bf B_{0}}\cdot\mathrm{rot}\;\mathrm{rot}\frac{{\bf
m}(\tilde{\tau})}{r})\;.$ (21)

Making analogous calculations as for case environment with $n=1$(see (A1) -
(A4)) one can obtain the following equation for function $Q$:

$(\Delta-{1\over c^{2}}{\partial^{2}\over\partial t^{2}})Q({\bf
r},t)=\frac{1}{r}e^{-i(\omega t-nkr)}\;.$ (A7)

The particular solution of the equation (A7) is found to be:

$Q_{p}=-\frac{1}{(n^{2}-1)k^{2}}\frac{1}{r}e^{-i(\omega t-nkr)}\;.$ (A8)

The obtained particular solution (A8) tends to $\infty$ at $n\to 1$. But as
one can see from the expression (A6), there is no singularity at condition
$n=1$. For this reason it’s necessary to add solution of homogeneous equation
$Q_{0}$ to eliminate this singularity. Thus, the final expression for function
$Q$ is found to be:

$Q=-\frac{1}{(n^{2}-1)k^{2}}\frac{1}{r}e^{-i\omega
t}\big{[}e^{inkr}-e^{ikr}\big{]}\;.$ (A8)

One should note, that obtained solution (A8) at $n\to 1$ tends to expression
(A6).

Thus, the exact solution of the dilaton field equation (21) in the complex
form is found to be:

$\displaystyle\Psi=-\mathrm{Re}\;\;\frac{4{\cal
K}a_{1}m\sin\alpha}{a_{0}(n^{2}-1)k^{2}}e^{-i\omega
t}\Big{\\{}\frac{k^{2}}{r}\big{[}B_{0x}+iB_{0y}$ $\displaystyle-({\bf
B_{0}\cdot
N})(N_{x}+iN_{y})\big{]}\big{[}n^{2}e^{inkr}-e^{ikr}\big{]}+\frac{ik}{r^{2}}\big{[}B_{0x}+iB_{0y}$
$\displaystyle-3({\bf B_{0}\cdot
N})(N_{x}+iN_{y})\big{]}\big{[}ne^{inkr}-e^{ikr}\big{]}-\frac{1}{r^{3}}\big{[}B_{0x}+iB_{0y}$
$\displaystyle-3({\bf B_{0}\cdot
N})(N_{x}+iN_{y})\big{]}\big{[}e^{inkr}-e^{ikr}\big{]}\Big{\\}}\;.$ (22)

In contrast to (2), dilaton field $\Psi$ in the expression (4) teds to $0$ at
$r\to\infty$ in inverse proportion to $r$. In this case the dilaton field
$\Psi$ is a superposition of dilaton waves propagating with different speeds:
$c$ and $c/n$. Therefore the resulted wave can be either be strengthened or
weakened depending of phase difference.

Since $n$ differs little from $1$, one can put $n=1+\chi$, $\chi\ll 1$. Then
the expression (4) can be represented as follows:

$\displaystyle\Psi=-\mathrm{Re}\;\frac{2{\cal K}a_{1}m\sin\alpha}{a_{0}\chi
k^{2}}e^{-i(\omega t-kr)}\Big{\\{}\frac{k^{2}}{r}\big{[}B_{0x}+iB_{0y}$
$\displaystyle-({\bf B_{0}\cdot
N})(N_{x}+iN_{y})\big{]}\big{[}(1+2\chi)e^{i\chi
kr}-1\big{]}+\frac{ik}{r^{2}}\big{[}B_{0x}+iB_{0y}$ $\displaystyle-3({\bf
B_{0}\cdot N})(N_{x}+iN_{y})\big{]}\big{[}(1+\chi)e^{i\chi
kr}-1\big{]}-\frac{1}{r^{3}}\big{[}B_{0x}+iB_{0y}$ $\displaystyle-3({\bf
B_{0}\cdot N})(N_{x}+iN_{y})\big{]}\big{[}e^{i\chi kr}-1\big{]}\Big{\\}}\;.$
(23)

Let us analyze the obtained expression (4) in detail. In case $\chi kr\ll 1$
one can obtain the expression (2) by expanding in Teylor series and neglecting
terms of order $\chi$ and higher. Otherwise dilaton field $\Psi$ will have
oscillating behavior.

For observation points which $\chi kr$ is a multiple of $\pi$ the
superposition of dilaton waves propagating with speeds $c$ and $c/n$ turns out
to be in phase and there is a maximum. In case $\chi kr$ is a multiple of
$2\pi$ then it is in anti-phase and there is a minimum.

For further investigation of angular distribution of dilaton energy one can
consider dilaton field $\Psi_{w}$ in the wave zone which consists of terms
inverse proportional $r$:

$\displaystyle\Psi_{w}=-\frac{2{\cal K}a_{1}m\sin\alpha}{a_{0}\chi
r}\Big{\\{}\big{[}B_{0x}-({\bf B_{0}\cdot N})N_{x}\big{]}$
$\displaystyle\times\big{[}(1+2\chi)\cos(kr-\omega t+\chi kr)-\cos(kr-\omega
t)\big{]}$ $\displaystyle-\big{[}B_{0y}-({\bf B_{0}\cdot
N})N_{y}\big{]}\big{[}(1+2\chi)\sin(kr-\omega t+\chi kr)$
$\displaystyle-\sin(kr-\omega t)\big{]}\Big{\\}}\;.$

In a result the average over the period of magnetic dipole wave angular
distribution of the dilaton generation in propagation of magnetic dipole waves
in a constant magnetic field is equal:

$\displaystyle\overline{\frac{dI}{d\Omega}}=\frac{8{\cal
K}^{2}a_{1}^{2}k\omega
m^{2}\sin^{2}\alpha}{a_{0}\chi^{2}}\Big{[}1-\cos\big{(}\chi
kr\big{)}\Big{]}\Big{[}B_{0x}^{2}+B_{0y}^{2}$ $\displaystyle-2({\bf B_{0}\cdot
N})(B_{0x}N_{x}+B_{0y}N_{y})+({\bf B_{0}\cdot
N})^{2}(N_{x}^{2}+N_{y}^{2})\Big{]}\;.$ (24)

One should note, that at $\chi\to 0$ expression (4) tends to (3).

Total intensities of such dilaton generation averaged over the period of
magnetic dipole wave is equal:

$\displaystyle\bar{I}=\frac{32\pi{\cal K}^{2}a_{1}^{2}k\omega
m^{2}\sin^{2}\alpha}{15a_{0}\chi^{2}}\Big{[}1-\cos\big{(}\chi
kr\big{)}\Big{]}$
$\displaystyle\times\big{[}9(B_{0x}^{2}+B_{0y}^{2})+2B_{0z}^{2}\big{]}\;.$
(25)

As one can see, $\cos(\chi kr)$ in the expression (4) can be expanded in
Teylor series at $\chi kr\ll 1$ and total intensities of dilaton generation
increases proportionally $r^{2}$. But at some distance $r_{0}$ ($\chi
kr_{0}\sim 1$) total intensities will have an oscillating behavior.

## Conclusion

In this study dilaton generation in propagation of magnetic dipole radiation
of pulsars and magnetars (8), (1) in a constant magnetic field (10) is
considered.

It was shown, that total intensities of dilaton generation (3) increases
proportionally to the square of distance $r^{2}$ from the pulsar to
observation point:

$\displaystyle\overline{I}=\frac{16\pi{\cal
K}^{2}a_{1}^{2}m^{2}k\omega\sin^{2}\alpha}{3a_{0}}\Big{\\{}\frac{1}{5}k^{2}r^{2}\big{(}9(B_{0x}^{2}+B_{0y}^{2})+2B_{0z}^{2}\big{)}$
$\displaystyle+8(B_{0x}^{2}+B_{0y}^{2})\Big{\\}}\;.$

Hence intergalactic and galactic magnetic fields can be considered constant
(10) only at distances smaller then the coherence length $L_{\mathrm{coh}}$.
It means that obtained result is applicable at distances
$r<L_{\mathrm{coh}}<\infty$. For the galactic magnetic field
$L_{\mathrm{coh}}\approx 100$ pc, for the intergalactic
$L_{\mathrm{coh}}\approx 10^{3}-10^{4}$ pc.

It should be taken into account that outer space environment is filled with
particles, mainly hydrogen, which leads to a refractive index $n=1+\chi$,
$\chi\ll 1$. Taking this into account, the quadratic increase in intensity of
dilaton generation stops at distances $r\approx r_{0}=\frac{1}{k\chi}$ and is
replaced by oscillations:

$\displaystyle\bar{I}=\frac{32\pi{\cal K}^{2}a_{1}^{2}k\omega
m^{2}\sin^{2}\alpha}{15a_{0}\chi^{2}}\Big{[}1-\cos\big{(}\chi
kr\big{)}\Big{]}$
$\displaystyle\times\big{[}9(B_{0x}^{2}+B_{0y}^{2})+2B_{0z}^{2}\big{]}$

Considering outer space environment to be a rarefied neutral gas distributed
uniformly, one can estimate $\chi\approx 10^{-15}$. In this case,
$r_{0}\approx 10^{3}$ pc, which correlates with the coherence length of the
intergalactic magnetic field.

##  5 Acknowledgements

This study was conducted within the scientific program of the National Center
for Physics and Mathematics, section $\\#5$ $<<$Particle Physics and
Cosmology$>>.$ Stage 2023-2025.

## References

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Authors to whom correspondence should be addressed: ]<EMAIL_ADDRESS>(S.S.Y.) and<EMAIL_ADDRESS>(J.J.L.)

Authors to whom correspondence should be addressed: ]<EMAIL_ADDRESS>(S.S.Y.) and<EMAIL_ADDRESS>(J.J.L.)

# Quantum-interference origin and magnitude of 1/$f$ noise in Dirac nodal line
IrO2 nanowires at low temperatures

Po-Yu Chien  Department of Electrophysics, National Yang Ming Chiao Tung
University, Hsinchu 30010, Taiwan  Chih-Yuan Wu  Department of Physics, Fu Jen
Catholic University, Taipei 24205, Taiwan  Ruey-Tay Wang  Department of
Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010,
Taiwan  Shao-Pin Chiu  Department of Electrophysics, National Yang Ming Chiao
Tung University, Hsinchu 30010, Taiwan  Stefan Kirchner  Department of
Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010,
Taiwan  Center for Emergent Functional Matter Science, National Yang Ming
Chiao Tung University, Hsinchu 30010, Taiwan  Sheng-Shiuan Yeh  [  Center for
Emergent Functional Matter Science, National Yang Ming Chiao Tung University,
Hsinchu 30010, Taiwan  International College of Semiconductor Technology,
National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan  Juhn-Jong Lin
[  Department of Electrophysics, National Yang Ming Chiao Tung University,
Hsinchu 30010, Taiwan

###### Abstract

We present 1/$f$ noise measurements of IrO2 nanowires from 1.7 to 350 K.
Results reveal that the noise magnitude (represented by Hooge parameter
$\gamma$) increases at low temperatures, indicating low-frequency resistance
noise from universal conductance fluctuations. The cause of this noise is
determined to be due to oxygen vacancies in the rutile structure of IrO2.
Additionally, the number density of these mobile defects can be calculated
from the $\sqrt{T}$ resistance rise caused by the orbital two-channel Kondo
effect in the Dirac nodal line metal IrO2.

††preprint: AIP/123-QED

The metal iridium dioxide (IrO2) has drawn significant attention as a
promising material for spintronic applications due to its giant spin Hall
resistivity.Fujiwara  _et al._ (2013) IrO2 is also widely used in glucose and
pH sensors.Dong  _et al._ (2018) Recently, the elusive orbital two-channel
Kondo (2CK) effect was observed in IrO2 nanowires (NWs).Yeh  _et al._ (2020)
It results in a square-root-in-temperature increase of the resistivity, i.e.,
$\rho\propto-\sqrt{T}$ (where $\rho$ is resistivity, and $T$ is temperature)
below a characteristic temperature due to oxygen vacancies (VO’s).Yeh  _et
al._ (2020) These observations make IrO2 NWs an attractive platform for
exploring strongly correlated electron systems.Kirchner (2020) Despite its
potential, there is limited information on the low-$T$ low-frequency $1/f$
noise and its origin in IrO2. The $1/f$ noise is a key factor which will
ultimately limit the performance of every nanoelectronic device.Balandin
(2013)

The standard theory of 1/$f$ noise in metals assumes the existence of mobile
defects, often modeled as a set of independent two-level systems (TLSs) with a
wide distribution of relaxation times. It can be shown that the resulting
resistance fluctuations power spectrum density (PSD) possesses a
$1/f^{\alpha}$ dependence, with $\alpha\simeq 1$.Dutta and Horn (1981);
Weissman (1988); Fleetwood (2015) In this study, we examine the relationship
between the number density of mobile defects ($n_{m}$) in IrO2 NWs and the
$1/f$ noise measured within the $T$ range of 1.7 to 350 K. We find unexpected
results at low $T$, where the noise magnitude increases with decreasing $T$.
We explain that this low-$T$ $1/f$ noise originates from the universal
conductance fluctuations (UCF), which stem from quantum interference of
conduction electrons scattered off mobile defects.Giordano (1991); Feng
(1991); Gupta (1994) We identify these mobile defects are associate with the
VO’s in the IrO2 rutile structure. While 1/$f$ noise has been measured in a
wide variety of conductors, both the origin and the amount of the underlying
mobile defects in a given material are usually difficult to identify and
quantize. In the present case, we demonstrate that combined with the
measurements of the 2CK effect, the $n_{m}$ value in every NW can be inferred.

Our IrO2 NWs were grown via the metal-organic chemical vapor deposition
method.Chen  _et al._ (2004a); Lin  _et al._ (2008) The single-crystalline
rutile structure was characterized by selected-area electron diffraction
patternsChen  _et al._ (2004a) and X-ray diffraction (XRD) patterns.Chen  _et
al._ (2004b) To generate VO’s, the NWs were thermally annealed at 300∘C in
vacuum. The annealing time was 3 h for NWs 1 to 3, and 4 h for NW 4. To
facilitate electrical measurements, submicron Cr/Au (10/120 nm) electrodes
attaching a NW device were fabricated by the electron-beam lithographic
technique. The inset of Fig. 1 shows a scanning electron microscopy (SEM)
image of a typical NW device. An ac method for 1/$f$ noise measurement was
applied.Scofield (1987) The background voltage noise PSD in our setup was
$S_{V}^{0}\approx 1.6\times 10^{-17}$ V2/Hz, which was limited by the voltage
preamplifier (Stanford Research model SR560). The relevant parameters of the
four IrO2 NWs studied in the present work are listed in Table 1.

Figure 1: Voltage power spectrum density as a function of frequency for NW 4 at (a) 280 K and (b) 1.7 K. The dashed line is drawn proportional to $1/f$ and is a guide to the eye. Insets: (a) an SEM image of a typical IrO2 NW device, and (b) $\langle fS_{V}\rangle$ versus $V^{2}$ at four $T$ values.  Table 1:  Relevant parameters for IrO2 NWs. $L$ is the NW length between the voltage probes in a 4-probe configuration, $d$ is the NW diameter, $n_{m}^{\rm 2CK}$ is number density of mobile defects determined from the 2CK effect, $n_{i}$ is number density of total (static and mobile) defects, $L_{e}$ is electron elastic mean free path, and $L_{\varphi}$ is electron dephasing length. $(L_{e}/L_{\varphi})^{2}$ is calculated with $L_{\varphi}\simeq 100$ nm.  Nanowire |  $L$ ($\mu$m) |  $d$ (nm) |  $n_{m}^{\rm 2CK}$ (m-3) | $n_{m}^{\rm 2CK}/n_{i}$ | $(L_{e}/L_{\varphi})^{2}$
---|---|---|---|---|---
NW 1 | 1.9 | 143 | $7.5\times 10^{25}$ | 0.074 | 0.0015
NW 2 | 1.0 | 123 | $1.8\times 10^{25}$ | 0.020 | 0.0019
NW 3 | 0.75 | 125 | $3.4\times 10^{25}$ | 0.037 | 0.0018
NW 4 | 0.90 | 147 | $7.8\times 10^{25}$ | 0.065 | 0.0010

Power spectrum density and Hooge parameter $\gamma$. For an ohmic conductor
under a small bias current $I$, the measured voltage noise is usually
expressed by the empirical formHooge (1969)

$S_{V}=\frac{\gamma}{N_{c}f^{\alpha}}V^{2}+S_{V}^{0}\,,$ (1)

where $\gamma$ is the Hooge parameter which characterizes the 1/$f$ noise
magnitude, $N_{c}$ is the total carrier number in the sample, $V$ is the
voltage dropped on the sample.

Figures 1(a) and (b) show the measured $S_{V}$ of NW 4 at two representative
temperatures 280 and 1.7 K, respectively. Above several tens Hz, the measured
$S_{V}$ approaches a constant ($\approx S_{V}^{0}$). At lower frequencies,
$S_{V}$ increases with decreasing $f$, as well as with increasing $V$. A
$S_{V}\propto f^{-\alpha}$ dependence, with $\alpha\simeq 1$, is found for
$f\lesssim 1$ Hz. This dependence can be well described by Eq. (1). To extract
the $\gamma$ value, we rewrite Eq. (1) as $\left\langle
fS_{V}\right\rangle=\left\langle\gamma V^{2}/N_{c}\right\rangle+\left\langle
fS_{V}^{0}\right\rangle=\gamma V^{2}/N_{c}+\left\langle f\right\rangle
S_{V}^{0}$, where $\left\langle fS_{V}\right\rangle$ denotes the average of
the product of each discrete $f_{i}$ and $S_{Vi}$ in the data set in the
$S_{V}\propto 1/f$ regime. The inset of Fig. 1(b) shows that our data obey the
$\left\langle fS_{V}\right\rangle\propto V^{2}$ dependence at our measurement
temperatures. Thus, the $\gamma$ value can be obtained from a linear fit of
the slope $\gamma/N_{c}$, where $N_{c}=n_{c}Ld^{2}$, with $n_{c}\approx
1\times 10^{28}$ m-3 being the carrier density of IrO2,Kawasaki  _et al._
(2018) $L$ ($d$) the length (diameter) of the NW. The linear dependence
$\left\langle fS_{V}\right\rangle\propto V^{2}$ suggests that the measured
voltage noise originates from the resistance fluctuations in the NW, i.e., the
(small) applied $I$ only acts as a sensitive electrical probe, while it does
not drive the resistance fluctuations.Hooge (1969)

Figure 2: (a) Hooge parameter $\gamma$ as a function of temperature for four
IrO2 NWs between 1.7 and 350 K. Note that in NWs 1 and 4, $\gamma$(2
K)$>$$\gamma$(300 K). (b) $\gamma$ versus $T$ below 20 K. The straight solid
line is a linear fit to NW 4. The straight dashed line is a guide to the eye.
In NW 1, $\gamma$ saturates below about 3 K.

The extracted $\gamma$ values as a function of $T$ between 1.7 and 350 K for
four IrO2 NWs are plotted in Fig. 2(a). As $T$ decreases from 350 K, $\gamma$
first decreases and then saturates below around 100 K. For example, the
$\gamma$ value of NW 4 decreases from $\approx$ 0.014 at 300 K to $\approx$
0.009 at 100 K, and then remains constant until about 20 K. In this high-$T$
regime, we expect that the TLSs are thermally activated and the $1/f$ noise
essentially follows the local-interference model.Pelz and Clarke (1987) In the
rest of this Letter, we shall focus our discussion on the noise below 20 K,
which results from the UCF in the NWs.

UCF-induced 1/$f$ noise. The most interesting observation of this work is
that, below about 20 K, our extracted $\gamma$ values increase with decreasing
$T$ in all NWs. This extraordinary behavior is particularly significant in NWs
1 and 4. In NW 4, the value of $\gamma$ at 1.7 K is $\simeq 0.11$
[$\gamma(1.7\,{\rm K})\simeq 0.11$] and is one order of magnitude larger than
that at 30 K, and even larger than that at 300 K. Figure 2(b) shows that NW 1
has $\gamma(T)$ behavior similar to that of NW 4, except that the $\gamma$
value saturates below $\sim$ 3 K. The low-$T$ increase of $\gamma$ in NWs 2
and 3 is smaller than that in NWs 1 and 4.

The rapid and repeated switching of mobile defects or TLSs modifies (a large
fraction of) the electron paths traversing the NW. The resulting quantum
interference among the electron trajectories is sensitively altered. The
interference effect becomes more pronounced at lower $T$, where the electron
dephasing time ($\tau_{\varphi}$) well surpasses the electron elastic mean
free time ($\tau_{e}$), i.e., $\tau_{\varphi}\gg\tau_{e}$. This is the origin
of the temporal UCF.Feng (1991); Giordano (1991) In 1991, Feng realized that
UCF in turn could give rise to 1/$f$ noise. He showed that the
phenomenological Hooge parameter $\gamma$ was directly connected with the
variance of the conductance fluctuations, and given by,Feng (1991); Giordano
(1991)

$\gamma=\frac{N_{c}}{{\rm ln}(f_{\rm max}/f_{\rm min})}\frac{\delta
G^{2}}{G^{2}}\,,$ (2)

where $f_{\rm max}$ ($f_{\rm min}$) denotes the maximum (minimum) frequency of
the mobile defects which are relevant in the measurement, $G$ is the
conductance of the sample, and $\delta G^{2}$ is the variance of the UCF.
Because $\delta G^{2}/G^{2}$ increases as the quantum-interference effect
increases, $\gamma$ grows with decreasing $T$. An additional $T$-dependence of
$\delta G^{2}/G^{2}$ may be due to the saturation of $\tau_{\varphi}$ (i.e.,
$\tau_{\varphi}$ becomes nearly independent of $T$) as $T\rightarrow 0$.Lin
and Bird (2002)

As the electron dephasing length $L_{\varphi}=\sqrt{D\tau_{\varphi}}<L$, $d$,
where $D=v_{F}^{2}\tau_{e}/3$ is the electron diffusion constant and $v_{F}$
is the Fermi velocity, our NWs are essentially three-dimensional (3D). We
consider the UCF in the “saturated” regime, which is defined by
$n_{m}/n_{i}\gg(L_{e}/L_{\varphi})^{2}$, where
$n_{i}=1/L_{e}\langle\sigma\rangle$ is the number density of total (static and
mobile) defects, $L_{e}=v_{F}\tau_{e}$ is the electron elastic mean free path,
and $\langle\sigma\rangle$ is the averaged electron scattering cross section.
This is the regime that is pertinent to our NWs, see Table 1. The UCF theory
predicts under these conditions the normalized varianceFeng (1991)

$\frac{\delta
G^{2}}{G^{2}}=28.2\frac{1}{k_{F}^{4}L_{e}^{2}}\frac{L_{\varphi}}{Ld^{2}}\,,$
(3)

where $k_{F}$ is the Fermi wavenumber. Thus, the $T$ dependence is fully
determined by the $T$ behavior of $L_{\varphi}$. In writing down Eq. (3), the
thermal energy averaging effect has been ignored which presumably would
introduce another characteristic length scale, the thermal diffusion length
$L_{T}=\sqrt{D\hbar/k_{B}T}$, into the cutoff of the UCF ($\hbar$ is the
Planck constant divided by 2$\pi$, and $k_{B}$ is the Boltzmann constant). We
have previously found that the thermal energy averaging effect on the UCF
magnitude was negligibly smaller than theoretically predicted.Yang  _et al._
(2012) Thus, we take Eq. (3) to be sufficient for providing a good
understanding of the $\gamma(T)$ data.

Figure 2(b) shows that below about 20 K, $\gamma\propto L_{\varphi}\propto
T^{-0.8}$ in NW 4. This $T$ dependence can be readily explained in terms of an
electron dephasing rate
$1/\tau_{\varphi}(T)=1/\tau_{\varphi}^{0}+1/\tau_{e-ph}(T)$, where the first
term $1/\tau_{\varphi}^{0}$ has been widely observed in experiments and is
called the saturated electron dephasing rate as $T\rightarrow 0$.Lin and Bird
(2002) The second term $1/\tau_{e-ph}\propto T^{2}$ is the electron-phonon
scattering rate which dominates the inelastic scattering in weakly disordered
metals.Zhong  _et al._ (2010); Zhong and Lin (1998) In NW 1, $\gamma$ shows a
similar $T$ behavior until it saturates below about 3 K, indicating that
$1/\tau_{\varphi}^{0}$ is much larger in this NW than in NW 4.Trionfi, Lee,
and Natelson (2007) The magnitudes of $\gamma$ increase in NWs 2 and 3 are
small, because these two NWs have comparatively small $n_{m}$ values (see
below).

Figure 3: (a) Resistivity versus temperature for four IrO2 NWs. The data
reveal typical metallic behavior above $\sim$ 25 K. At lower $T$, $\rho$
increases with decreasing $T$. (b) Main panel and inset:
$[\rho(T)-\rho(25\,{\rm K})]$ versus $\sqrt{T}$. The straight lines are linear
fits to Eq. (4).

Identifying the origin and number density of mobile defects. We will now delve
into the examination of the microscopic source of mobile defects responsible
for inducing UCF and determining the 1/$f$ noise magnitude in IrO2 NWs,
including their number density. This is based on nonmagnetic orbital 2CK
physics which was recently reported in this material. The 2CK effect in IrO2
causes a $\sqrt{T}$ increase of $\rho$ at low $T$. Figure 3(a) shows the
variation of $\rho$ with $T$ for our NWs. $\rho(T)$ reveals typical Boltzmann
transport behavior until below about 25 K where it slightly increases with
decreasing $T$. Close inspection indicates that this increase obeys a
$\sqrt{T}$ dependence down to 2 K, Fig. 3(b). Between 2 and 25 K, the
normalized increase is $\left[\rho\left(2~{}{\rm K}\right)-\rho\left(25~{}{\rm
K}\right)\right]/\rho\left(25~{}{\rm K}\right)\approx(2.2-7.7)\times 10^{-3}$
in our NWs. This amount of resistivity increase is more than one order of
magnitude larger than what would be expected from the 3D weak localization and
electron-electron interaction effects.Lee and Ramakrishnan (1985); Yeh  _et
al._ (2020) We have recently demonstrated that this $\sqrt{T}$ increase obeys
Affleck and Ludwig (1993)

$\rho_{\rm 2CK}\left(T\right)=\frac{3n_{m}^{\rm
2CK}}{4\pi\hbar\left[eN(0)v_{F}\right]^{2}}\left(1-4\sqrt{\frac{\pi T}{T_{\rm
2CK}}}\right)\,,$ (4)

where $n_{m}^{\rm 2CK}$ denotes the number density of 2CK scatterers,
$N\left(0\right)$ is the density of states at the Fermi energy per spin per
channel, and $T_{\rm 2CK}$ is a characteristic temperature scale in the 2CK
problem.

Oxygen vacancies frequently occur in IrO2 which assumes the rutile structure,
depending on the sample preparation conditions.Yeh  _et al._ (2020) Due to
charge neutrality, a VO generates two “defect electrons” in IrO2. One of the
two defect electrons, which occupies the doubly degenerate 5$d_{xz}$ or
5$d_{yz}$ iridium ion orbital, is strongly coupled to the two equivalent and
independent spin-up and spin-down conduction-electron channels. This is the
origin of the 2CK effect in IrO2.Yeh  _et al._ (2020); Kirchner (2020) With
the assumption that the density of VO’s, denoted by $n_{\rm V_{O}}$, is low
enough to safely ignore any correlations between VO’s, we can estimate $n_{\rm
V_{O}}\simeq n_{m}^{\rm 2CK}$ in each IrO2 NW.

To compare our data with Eq. (4), we take $T_{\rm 2CK}\simeq 23$ K, the
temperature below which the $\sqrt{T}$ behavior is found. Then, the
$n_{m}^{\rm 2CK}$ value in each NW can be extracted from the fitted slope
plotted in Fig. 3(b), using $N\left(0\right)\approx 1.5\times 10^{47}$ J-1 m-3
and $v_{F}\approx 5\times 10^{5}$ m/s for IrO2.Lin  _et al._ (1999); Yeh  _et
al._ (2020) These values are listed in Table 1. We see that NWs 1 and 4, which
have the largest noise magnitudes $\gamma$, also show the largest $\sqrt{T}$
resistivity increases (i.e., largest $n_{m}^{\rm 2CK}$ values), and NWs 2 and
3 have smaller noise magnitudes and simultaneously smaller resistivity
increases (i.e., smaller $n_{m}^{\rm 2CK}$ values).

In practice, the mobile defects existing in a given sample may not all
contribute to a given noise measurement, and $n_{m}$ will be given by the
number of those defects having switching frequency $f_{\rm min}\leq f\leq
f_{\rm max}$. Then, $n_{m}\lesssim n_{\rm V_{O}}\simeq n_{m}^{\rm 2CK}$. In
the regime where the resistance fluctuations are governed by the local-
interference (LI) mechanism and $n_{\rm V_{O}}\approx n_{m}^{\rm 2CK}$ holds,
we obtain an estimate for the switching TLS density $n_{m}^{\rm LI}$. This is
accomplished by applying a method developed earlier,Yeh  _et al._ (2018) where
$n_{m}^{\rm LI}$ can be estimated with the measured $\gamma$ value through
$n_{m}^{\rm LI}\approx 4\pi\gamma n_{c}\left(\rho
e^{2}/mv_{F}\left\langle\sigma\right\rangle\right)^{2}$, where $e$ ($m$) is
the electronic charge (effective mass). At 30 K, this gives in the present
case $n_{m}^{\rm LI}\approx(1.9\pm 0.1)\times 10^{25}$ m-3 for NWs 1 and 4,
and $\approx(5.5\pm 1)\times 10^{24}$ m-3 for NWs 2 and 3. These extracted
$n_{m}^{\rm LI}$ values scale with and are smaller than the $n_{\rm V_{O}}$
values within a factor of 10, indicating the fluctuators at low $T$ are
associated with the VO’s. Thus, in our NWs, we estimate $n_{m}$ $[\sim
n_{m}^{\rm LI}(30\,{\rm K})\lesssim n_{\rm V_{O}}\simeq n_{m}^{\rm 2CK}]$ has
values $\sim 10^{24}-10^{25}$ m-3, being about one order of magnitude larger
than that found in, e.g., Bi films.Birge, Golding, and Haemmerle (1989)

One possible explanation for this is the existence of an interstitial oxygen
density that is (by and large) proportional to $n_{\rm V_{O}}$. Our thermal
annealing studies suggest that this is not the reason for the proportionality
between $n_{\rm V_{O}}$ and $n_{m}^{\rm LI}$, as an increase in annealing time
in vacuum is expected to reduce the number density of oxygen interstituals
while increasing $n_{\rm V_{O}}$. This leaves us with the more interesting
alternative that the VO’s themselves are driving the 1/$f$ noise.

The characteristic feature of the one-channel (two-channel) Kondo effect is
the existence of a degenerate local level hybridizing with a band (two bands)
of electron states. This leads to a characteristic energy scale known as the
Kondo temperature ($T_{\rm K}$) or equivalently a characteristic time scale
$\tau_{\rm K}=\hbar/k_{B}T_{\rm K}$.Hewson (1993) (In the present case,
$T_{\rm K}$ = $T_{\rm 2CK}$.) On the other hand, 1/$f$ noise is associated
with a broad distribution of relaxation times of the mobile defects. To ensure
that the 2CK effect can form in IrO2, the degeneracy between 5$d_{xz}$ and
5$d_{yz}$ has to remain intact even in the presence of such a broad
distribution. Moreover, to guarantee that the flickering is not interfering
with the formation of the 2CK effect, $k_{B}T_{\rm 2CK}\gg hf$ needs to hold.
This is evidently satisfied, as can be inferred from Fig. 1 with $T_{\rm
2CK}\approx 20$ K as is appropriate for IrO2 NWs. The necessary broad
distribution of relaxation times of the fluctuators driving the 1/$f$ noise
implies a corresponding disorder distribution of the VO environment. That the
$T_{\rm K}$ value distribution in such a situation remains sharply peaked has
been addressed before.Chakravarty and Nayak (2000) A microscopic model of how
VO’s can drive an orbital Kondo effect and concomitantly 1/$f$ noise at lower
frequencies is left for future work. Intuitively, if different VO’s reside at
varying distances to grain boundaries, and grain boundaries in the NW have
differing orientations, a distribution of the relaxation times should
naturally arise. NW surfaces (and the interface with substrate) could likely
have similar effects on resulting in a distribution of relaxation times. In
any case, our recent experiments have confirmed that the 1/$f$ noise magnitude
in a series of RuO2 films is controlled by $n_{\rm V_{O}}$, which can be
reversely adjusted, i.e., increased or reduced, by annealing RuO2 in vacuum,
air, or oxygen gas.Yeh  _et al._ (2018) Since both RuO2 and IrO2 crystallize
in the rutile structure and have similar physical propertiesLin  _et al._
(2004) (except that RuO2 is antiferromagnetic which leads to an orbital one-
channel Kondo effect at low $T$, see Ref. Yeh  _et al._ , 2020), we expect
that VO’s play a similar role in producing 1/$f$ noise in IrO2.

Temporal resistance fluctuations. While the UCF-induced 1/$f$ noise has been
theoretically proposed in the 1980s,Feng, Lee, and Stone (1986) the
predictions have not been widely tested. The reasons are that one needs to use
microscale samples and measure the noise down to low $T$. Otherwise, the
quantum-interference effect will be negligible and the noise magnitude small
and difficult to detect. To further check that the measured 1/$f$ noise in
IrO2 below 20 K is caused by the UCF mechanism, we have extracted the
(normalized) variance of conductance fluctuations $\delta G^{2}/G^{2}$ from an
independent measurement, and compare it with that calculated from the measured
$\gamma$ value through Eq. (2).

Figure 4: (a) Voltage fluctuations as a function of time for NW 4 at 1.7 K and
under five different bias currents. (b) $\delta V_{m}^{2}$ versus $I^{2}$.
Straight solid and dashed lines are linear fits. (c) $\delta G^{2}/G^{2}$
versus $\gamma/N_{c}$. Closed symbols are calculated from $\delta V_{m}^{2}$.
Open symbols are calculated through Eq. (2). The straight line is a guide to
the eye.

In our experimental method, a current $I$ is applied to the NW device and the
total voltage fluctuations $\delta V_{m}(t)$ are measured, i.e., $\delta
V_{m}(t)=I\delta R(t)+\delta V_{0}(t)$, where $I\delta R$ is the voltage noise
of the NW, and $\delta V_{0}$ is the input noise of the preamplifier. Figure
4(a) shows the measured $\delta V_{m}(t)$ at several $I$ values for NW 4 at
1.7 K. We see that the $\delta V_{m}$ magnitude increases with increasing $I$.
The $\delta G^{2}/G^{2}$ value of a NW can be obtained by subtracting out the
background contribution $\delta V_{0}$. Because $\delta R(t)$ and $\delta
V_{0}(t)$ are uncorrelated, we write $\delta V_{m}^{2}=I^{2}\delta
R^{2}+\delta V_{0}^{2}$ for an ohmic conductor, where $\delta V_{m}^{2}$,
$\delta R^{2}$, and $\delta V_{0}^{2}$ denote the variances of $\delta
V_{m}(t)$, $\delta R(t)$, and $\delta V_{0}(t)$, respectively. Thus, the
intrinsic resistance fluctuations of the NW $\delta R^{2}$ can be obtained
from a plot of $\delta V_{m}^{2}$ versus $I^{2}$. Figure 4(b) shows the
variation of $\delta V_{m}^{2}$ with $I^{2}$ for our NWs at 1.7 K. A $\delta
V_{m}^{2}\propto I^{2}$ linear dependence is observed for all NWs, with the
fitted $\delta V_{0}^{2}\approx(5.5\pm 0.2)\times 10^{-15}$ V2. Here we note
that in our noise measurements, the sampling rate of the spectrum analyzer
(Stanford Research model SR785) was set to be 1024 Hz and the measurement time
was lasted for 30 s, corresponding to the relevant frequency range from
$f_{\rm min}\simeq 0.033$ Hz to $f_{\rm max}\simeq 512$ Hz. This $f$ range
leads to an estimated upper bound $\delta V_{0}^{2}\sim
S_{V}^{0}\times\left(f_{\rm max}-f_{\rm min}\right)\sim 8\times 10^{-15}$ V2,
being in good accord with our measurement.

Once $\delta R^{2}$ value in each NW is obtained from the slope in Fig. 4(b),
$\delta G^{2}/G^{2}=\delta R^{2}/R^{2}$ can be calculated. The calculated
$\delta G^{2}/G^{2}$ values (solid symbols) as a function of the normalized
Hooge parameter $\gamma/N_{c}$ are plotted in Fig. 4(c). Alternatively, we
have calculated $\delta G^{2}/G^{2}$ values (open symbols) from the measured
$\gamma$ (Fig. 2) through Eq. (2), by inserting ${\rm ln}(f_{\rm max}/f_{\rm
min})\approx{\rm ln}(512/0.033)\approx 10$. Figure 4(c) shows that the $\delta
G^{2}/G^{2}$ values obtained from the two independent methods agree to within
a factor of $\sim$ 2\. At $T$ = 1.7 K, $\delta G^{2}/G^{2}\approx 1.9\times
10^{-9}$ and $\approx 4.8\times 10^{-9}$ in NWs 1 and 4, respectively. We thus
obtain $L_{\varphi}\approx$ 95 (81) nm in the former (the latter) through Eq.
(3), where $k_{F}\approx 7\times 10^{9}$ m-1.Lin  _et al._ (1999) Therefore,
we have $L_{\varphi}<d$ and $n_{m}/n_{i}\gg(L_{e}/L_{\varphi})^{2}$,
justifying the application of Eq. (3).

The 1/$f$ noise has previously been studied in a wide variety of metalsDutta
and Horn (1981); Weissman (1988); Giordano (1991) and semiconductors,Fleetwood
(2015) and recently, in 2D materials.Balandin (2013); Karnatak  _et al._
(2017) UCF-induced 1/$f$ noise were observed in polycrystalline Bi
films,Birge, Golding, and Haemmerle (1989) Ag films,Meisenheimer and Giordano
(1989); McConville and Birge (1993) Au wires,Trionfi, Lee, and Natelson (2007)
RuO2 NWs,Lien  _et al._ (2011) C-Cu composites,Garfunkel  _et al._ (1988) and
epitaxial graphene.Kalmbach  _et al._ (2016) In Bi and Ag films, the UCF fell
in the “unsaturated” regime where $n_{m}/n_{i}\ll(L_{e}/L_{\varphi})^{2}$. The
origins of the mobile defects were unspecified in those samples, while the
diffusion of H atoms (the spin-glass freezing process) was identified as the
cause in amorphous Ni-Zr filmsAlers  _et al._ (1989) (narrow AuFe
wiresNeuttiens  _et al._ (2000)). In the present study, we are able to show
that VO’s are rich and generate mobile defects in IrO2.

In summary, we have measured the 1/$f$ noise of four IrO2 nanowires in the
temperature range from 1.7 to 350 K. As temperature decreases from 350 K, the
$\gamma$ value decreases down to about 100 K and then remains constant, but as
$T\lesssim 20$ K $\gamma$ increases with decreasing temperature. We explain
this low-temperature $\gamma$ increase in terms of the UCF-induced 1/$f$
noise. We identify the mobile defects to be associated with oxygen vacancies
in the IrO2 rutile structure. In combination with the orbital 2CK effect
previously observed in this Dirac nodal line metal, we have extracted the
number density of the mobile defects. The UCF origin of the 1/$f$ noise is
further confirmed by independent measurements of temporal conductance
fluctuations in the nanowires.

ACKNOWLEDGMENTS

We thank C. W. Wu for experimental help. This work was supported by National
Science and Technology Council of Taiwan through grant numbers
110-2112-M-A49-015 and 111-2119-M-007-005 (J.J.L.), and 110-2112-M-A49-033-MY3
(S.S.Y.).

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