# Magnetically tunable multi-band near-field radiative heat transfer between
two graphene sheets

Lixin Ge<EMAIL_ADDRESS>School of Physics and Electronic Engineering,
Xinyang Normal University, Xinyang 464000, China  Division of Computer,
Electrical and Mathematical Sciences and Engineering, King Abdullah University
of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia  Ke Gong
School of Physics and Electronic Engineering, Xinyang Normal University,
Xinyang 464000, China  Yuping Cang  School of Physics and Electronic
Engineering, Xinyang Normal University, Xinyang 464000, China  Yongsong Luo
School of Physics and Electronic Engineering, Xinyang Normal University,
Xinyang 464000, China  Xi Shi  Department of physics, Shanghai Normal
University, Shanghai, 200234, China  Ying Wu<EMAIL_ADDRESS>Division of
Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah
University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia

###### Abstract

Near-field radiative heat transfer (NFRHT) is strongly related with many
applications such as near-field imaging, thermos-photovoltaics and thermal
circuit devices. The active control of NFRHT is of great interest since it
provides a degree of tunability by external means. In this work, a
magnetically tunable multi-band NFRHT is revealed in a system of two suspended
graphene sheets at room temperature. It is found that the single-band spectra
for B=0 split into multi-band spectra under an external magnetic field. Dual-
band spectra can be realized for a modest magnetic field (e.g., B=4 T). One
band is determined by intra-band transitions in the classical regime, which
undergoes a blue shift as the chemical potential increases. Meanwhile, the
other band is contributed by inter-Landau-level transitions in the quantum
regime, which is robust against the change of chemical potentials. For a
strong magnetic field (e.g., B=15 T), there is an additional band with the
resonant peak appearing at near-zero frequency (microwave regime), stemming
from the magneto-plasmon zero modes. The great enhancement of NFRHT at such
low frequency has not been found in any previous systems yet. This work may
pave a way for multi-band thermal information transfer based on atomically
thin graphene sheets.

##  I Introduction

The management of near-field radiative heat transfer (NFRHT) has promising
applications in energy conversion Fio:18 ; Zha:17 ; Bas:09  and information
processing Abd:17 ; Abd:16a . Thanks to the tunneling of evanescent waves
(e.g., surface plasmon polaritions (SPPs) Jou:05 ; Bor:15 , surface phonon
polaritions (SPhPs) She:09 ; Son:15 ), the radiative heat flux can exceed the
blackbody limit by several orders of magnitude for nanoscale separation Vol:07
; liu:15 . The field of NFRHT grows rapidly in recent years Cue:18 .
Particularly, the thermal analogy of the electronic devices Abd:17 ,
i.e.,thermotronics, provides a new possibility for information processing. The
fast thermal photon generated from NFRHT is an attractive information carrier
compared with its counterpart of the slow phonon (heat conduction) Li:12 . Up
to now, the proposals for fundamental circuit elements such as thermal logic
gates Abd:16a , thermal rectifiers Ote:10 , thermal memory Kub:14 ; Ito:16 ,
thermal transistors Abd:14  have been reported. Despite great progress in
thermal circuit devices, the active and fast control of NFRHT is still highly
desired for thermal information processing.

Recently, several strategies for active control of NFRHT have been proposed
Ili:18 ; Hua:14 ; Cui:13 ; Kou:18 ; Abd:16b ; Zhu:16 ; Eke:18 ; Mon:15 ;
Van:11 ; Van:12 ; Gha:18 . For instance, one can tune the NFRHT by external
electric gating in 2D materials Ili:18  and ferroelectric materials Hua:14 .
Or one can use optical pump, i.e., photo-excitation, to control NFRHT in
chiral meta-materials Cui:13  and semiconductors Kou:18 . In addition, one can
actively control NFRHT through an external magnetic field Abd:16b ; Zhu:16 ;
Eke:18 ; Mon:15 . Examples such as near-field thermal Hall effect Abd:16b ,
persistent directional heat current Zhu:16 , anisotropic thermal
magnetoresistance Eke:18 , and thermal modulation Mon:15  were reported. Other
schemes such as temperature control in phase-change materials Van:11 ; Van:12
, and mechanical strain Gha:18  are also investigated. However, previous
studies of various strategies for active control of NFRHT have been focused on
enhancing or suppressing the total heat flux. Little attention was paid on the
control of multi-band spectral bands. Inspired by the multi-band devices in
telecommunications, a controllable thermal transfer system with multi-band
spectra could be significant for the development of thermal information
processing.

Figure 1: (color online) Schematic view of near-field radiative heat transfer
between two suspended graphene sheets in free space. The upper sheet with
higher temperature ($T_{1}$) radiates electromagnetic waves to the bottom one
with lower temperature ($T_{2}$). (a) The spectrum of radiative heat flux is
single-band in the absence of an external magnetic field. (b) The spectrum
becomes multi-band under a sufficient strong magnetic field.

In order to realize multi-band NFRHT, one straight scheme is to design a
system with multiple frequency-separated surface modes. The SPPs can be
modulated under an external magnetic field (or named magneto-plasmon) Hu:15 .
The multi-band dispersions, i.e., energy gaps appear in the dispersion
relationship, can be found in magneto-plasmon of doped semiconductors Bri:72
and graphene Fer:12 ; Hu:14 . Compared with the doped semiconductors, graphene
is only one-atom thick Gei:07 . Besides, the cyclotron energy of graphene,
characterized by the quantized Landau level (LL) transitions Goe:11 , can be
much larger than those of semiconductors under the same magnetic field. The
magneto-plasmon of graphene lies in Terahertz and far infrared frequencies
Fer:12 ; Hu:14 , which guarantees a high-efficiency thermal excitation at room
temperature.

In this work, we investigate the NFRHT between two parallel graphene sheets
under a perpendicular magnetic field. It is found that the single-band
spectrum for B=0 can be split into multi-band spectra under an external
magnetic field at room temperature. The multi-band spectra are determined by
the LL transitions of graphene, which show interesting properties due to
thermal fluctuation. For a modest magnetic field (e.g., B=4 T), dual-band
spectra can be realized, contributed from intra-band and inter-band LL
transitions, respectively. Interestingly, the latter one is quite robust
against the change of chemical potential. On the other hand, triple-band
spectra can be found under a strong magnetic field (e.g., B=15 T). Remarkably,
there is an additional band with the resonant peak appearing at near-zero
frequency. The great enhancement of NFRHT at such low frequency is attributed
to the magneto-plasmon zero modes. Finally, the properties of multi-band NFRHT
due to the scattering rate, separation distance and substrate effect are
discussed at the end.

##  II Theories for magnetically tunable NFRHT

The system considered in this work is depicted in Fig. 1. Two parallel
graphene sheets are suspended in free space with a separation distance $d$.
The temperatures for the upper and bottom sheets are respectively $T_{1}$ and
$T_{2}$. The value of the perpendicular magnetic field B can be tuned though
external control. In this work, we consider the magnetic field $\text{B}<$17
T, where the Zeeman splitting is negligible Zha:06 . The magneto-optical
conductivity of graphene is given by a tensor Gus:07 ; Fer:11 :

$\left(\begin{array}[]{cc}\sigma_{xx}&\sigma_{xy}\\\
\sigma_{yx}&\sigma_{yy}\end{array}\right)=\left(\begin{array}[]{cc}\sigma_{L}&\sigma_{H}\\\
-\sigma_{H}&\sigma_{L}\end{array}\right),$ (1)

where the subscripts $L$ and $H$ represent respectively the longitudinal and
Hall conductivities, and they can be written in compact forms Fer:11 :

$\sigma_{L(H)}(\omega,B)=g_{s}g_{v}\frac{e^{2}}{4h}\underset{n\neq
m}{\overset{\infty}{\sum}}\frac{\Xi_{L(H)}^{nm}}{i\Delta_{nm}}\frac{n_{F}(E_{n})-n_{F}(E_{m})}{\hbar\omega+\Delta_{nm}+i\Gamma_{nm}},$
(2)

where $g_{s}$=$g_{v}$=2 are respectively the spin and valley degeneracy
factors, $e$ is the charge of an electron, $h$ is the plank constant,
$\Gamma_{nm}$ is the scattering rate (the broadening of LLs),
$n_{F}(E_{n})=1/[1+\exp(E_{n}-\mu_{c})/k_{B}T]$ is the Fermi-Dirac
distribution, and $\mu_{c}$ is the chemical potential, $k_{B}$ is the
Boltzmann constant, $\Delta_{nm}=E_{n}-E_{m}$ is the LL energy transition with
$n$, $m$=0, $\pm 1$, $\pm 2$… are the Landau indices, $E_{n}=$sign$(n)[\hbar
v_{F}/l_{B}]\sqrt{2|n|}$ is the energy of the $n$-th LL, $v_{F}$=$10^{6}$ m/s
is the Fermi velocity, $l_{B}\equiv\sqrt{\hbar/(eB)}$ is the magnetic length,
and the auxiliary functions are given by:

$\displaystyle\Xi_{L}^{nm}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}v_{F}^{2}}{l_{B}^{2}}(1+\delta_{m,0}+\delta_{n,0})\delta_{|m|-|n|,\pm
1},$ $\displaystyle\Xi_{H}^{nm}$ $\displaystyle=$ $\displaystyle
i\Xi_{L}^{nm}(\delta_{|m|,|n|-1}+\delta_{|m|-1,n}),$ (3)

which are determined by the selection rule of LL transitions. The radiative
heat flux between two graphene sheets under an external magnetic field can be
calculated based on the fluctuation-dissipation theory Bie:11 :

$\left\langle
S\right\rangle=\frac{1}{(2\pi)^{2}}\int_{0}^{\infty}\int_{0}^{\infty}\xi(\omega,q)q\text{d}q[\Theta(\omega,T_{1})-\Theta(\omega,T_{2})]\text{d}\omega,$
(4)

where $\Theta(\omega,T)=\hbar\omega/(\exp(\hbar\omega/k_{B}T)-1)$ is the
average energy of a Planck’s oscillator for angular frequency $\omega$ at
temperature $T$, and $q$ is the wavevector parallel to the surface planes,
$\xi(\omega,q)$ is the energy transmission coefficient, which is expressed as
follows:

$\displaystyle\xi(\omega,q)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{c}\text{Tr}[(\mathbf{I}-\mathbf{R}_{2}^{\dagger}\mathbf{R}_{2})\mathbf{D}^{12}(\mathbf{I}-\mathbf{R}_{1}^{\dagger}\mathbf{R}_{1})\mathbf{D}^{12\dagger}],\\\
\text{Tr}[(\mathbf{R}_{2}^{\dagger}-\mathbf{R}_{2})\mathbf{D}^{12}(\mathbf{R}_{1}^{\dagger}-\mathbf{R}_{1})\mathbf{D}^{12\dagger}]e^{-2|\gamma|d},\end{array}\right.\begin{array}[]{c}q<k_{0}\\\
q>k_{0}\end{array}$

where $\gamma=\sqrt{k_{0}^{2}-q^{2}}$ is the vertical wavevector with
$k_{0}=\omega/c$ being the wavevector in vacuum,
$\mathbf{D}^{12}=(\mathbf{I}-\mathbf{R}_{1}\mathbf{R}_{2}e^{2i\gamma d})^{-1}$
and $\mathbf{R}_{j}$ ($j$=1, 2) is the $2\times 2$ reflection matrix for the
$j$-th graphene sheet, having the form:

$\mathbf{R}_{j}=\left(\begin{array}[]{cc}r_{j}^{ss}&r_{j}^{sp}\\\
r_{j}^{ps}&r_{j}^{pp}\end{array}\right),$ (9)

where the superscripts $s$ and $p$ represent respectively the polarizations of
transverse electric ($\mathbf{TE}$) and transverse magnetic ($\mathbf{TM}$)
modes. The matrix element $r^{\alpha\beta}$ ($\alpha,\beta$ = $s,p$)
represents the reflection coefficient for an incoming $\alpha$-polarized plane
wave being reflected into an outgoing $\beta$-polarized wave. For a suspended
graphene sheet under a perpendicular magnetic field, the reflection
coefficients are given analytically as follows Kot:17 ; Tse:12 :

$\displaystyle r^{ss}$ $\displaystyle=$
$\displaystyle-\frac{2k_{0}^{2}\overline{\sigma}_{L}+k_{0}\gamma(\overline{\sigma}_{L}^{2}+\overline{\sigma}_{H}^{2})}{4k_{0}\gamma+(2\gamma^{2}+2k_{0}^{2})\overline{\sigma}_{L}+k_{0}\gamma(\overline{\sigma}_{L}^{2}+\overline{\sigma}_{H}^{2})},$
$\displaystyle r^{pp}$ $\displaystyle=$
$\displaystyle-\frac{2\gamma^{2}\overline{\sigma}_{L}+k_{0}\gamma(\overline{\sigma}_{L}^{2}+\overline{\sigma}_{H}^{2})}{4k_{0}\gamma+(2\gamma^{2}+2k_{0}^{2})\overline{\sigma}_{L}+k_{0}\gamma(\overline{\sigma}_{L}^{2}+\overline{\sigma}_{H}^{2})},$
(10) $\displaystyle r^{ps}$ $\displaystyle=$
$\displaystyle-r^{sp}=\frac{2k_{0}\gamma\overline{\sigma}_{H}}{4k_{0}\gamma+(2\gamma^{2}+2k_{0}^{2})\overline{\sigma}_{L}+k_{0}\gamma(\overline{\sigma}_{L}^{2}+\overline{\sigma}_{H}^{2})},$

where the elements of conductivity tensor are normalized by the free-space
impedance $\sqrt{\mu_{0}/\varepsilon_{0}}$. Note that the denominator in all
reflection coefficients is the same. By setting this denominator to be zero
and the loss term $\Gamma$=0, the dispersion of magneto-plasmon in single
graphene sheet can be obtained.

We now consider the NFRHT in the low frequency limit $\omega\rightarrow 0$,
where the normalized conductivities Im$\overline{\sigma}_{L}$,
Im$\overline{\sigma}_{H}\rightarrow 0$ and Re$\overline{\sigma}_{L}$,
Re$\overline{\sigma}_{H}\ll 1$. For very large wave-vector $q\gg k_{0}$, i.e.,
$|\gamma|\gg k_{0}$, only the reflection coefficient $r^{pp}$ plays an
important role, and it can be simplified as:

$r^{pp}=-\frac{iq\overline{\sigma}_{L}}{2k_{0}+iq\overline{\sigma}_{L}}.$ (11)

The pole of the denominator (real part) appears at $\omega$=0, which
corresponds to a magneto-plasmon zero-mode. The NFRHT is proportional to
Im$[r^{pp}]$ Ili:18 , where it has the expression
Im[$r^{pp}$]=-2$k_{0}/(q\mathrm{Re}\overline{\sigma}_{L})$. For a strong
magnetic field, the longitudinal conductivity of graphene, i.e., the term
$\mathrm{Re}\overline{\sigma}_{L}$, can be diminished with orders of magnitude
compared with the non-magnetic configuration, based on the calculations of Eq.
(2). As a result, the NFRHT at near-zero frequency can be enhanced greatly due
to the magneto-plasmon zero modes, which is an important finding in our work.
It should be pointed out that the magneto-plasmon zero modes in 2D electron
gas systems were also demonstrated in the Ref. Jin:16 .

##  III Results and discussions

Figure 2: (color online) (a) The Fermi-Dirac distribution of graphene for
different temperatures. The inset illustrates the band structure of a graphene
sheet with chemical potential $\mu_{c}$ below the first LL. The imaginary part
of longitudinal conductivity Im[$\sigma_{L}$] for (b) $\mu_{c}$=0.04 eV and
(c) $\mu_{c}$=0.08 eV. The shaded regimes in (b) and (c) indicate the peaks of
conductivity contributed by different LL transitions. The scattering rate is
set to be a constant with $\Gamma_{nm}$=$\Gamma$= 4 meV. The external magnetic
field B=4 T, and the first LL E${}_{1}\sim$0.07 eV.

Figure 3: (color online) (a) Spectral heat flux and (b) transmission
coefficients(log scale) for two parallel graphene sheets under different
values of magnetic field. The inset in (a) shows the spectrum near-zero
frequency for B=15 T. In (b), the white line illustrates the light line
q=$\omega$/c, and the gray lines represent the corresponding dispersions of
magneto-plasmon for single graphene sheet. The dashed gray lines stand for the
dispersions generated from the intra-band transition E${}_{2}\rightarrow$E3.
We set the chemical potential for two graphene sheets $\mu_{c1}=\mu_{c2}$=0.08
eV, the scattering rate $\Gamma$=4 meV, the high temperature $T_{1}$=310 K,
the low temperature $T_{2}$=$T_{1}$-10 K, and the separation distance $d$=100
nm.

To understand the magneto-plasmon of graphene at high temperature, we firstly
inspect the Fermi-Dirac distribution $n(\omega)$ in Fig. 2(a). Clearly,
$n(\omega)$ is a good quantum number at low temperature 10 K, whereas it
becomes partially occupied at high temperatures 310 K and 610 K, due to the
thermal fluctuation. The inset schematically shows the band structure of
graphene with quantized LLs and Fermi-Dirac distribution of electron at high
temperatures. The long-decay tails of $n(\omega)$ enable multiple intra-band
LL transitions take place (such as E${}_{1}\rightarrow$E2,
E${}_{2}\rightarrow$E3) even though $\mu_{c}$ is smaller than the first LL E1.

We are interest in the imaginary part of longitudinal conductivity, i.e.,
Im[$\sigma_{L}$], as the magneto-plasmon of graphene is mainly determined by
this component Fer:12 . The magneto-plasmon of graphene are confirmed strongly
when Im[$\sigma_{L}$] is a positive number, providing a capability for the
enhancement of NFRHT. The comb-like curves of Im[$\sigma_{L}$] are shown in
Figs. 2(b) and 2(c), where the magnetic field B =4 T, and the first LL
E${}_{1}\sim$0.07 eV. The values of chemical potential of graphene are
controllable though external electric gating Gri:12 . For chemical potential
$\mu_{c}$=0.04 eV$<$ E1, there is no intra-band transition for temperature 10
K. However, multiple intra-band transitions (mainly E${}_{1}\rightarrow$E2 and
E${}_{2}\rightarrow$E3) take place for 310 K and 610 K at low frequency, which
differ considerably from those of 10 K. The conductivity Im[$\sigma_{L}$] with
chemical potential located between E1 and E2 is shown in Fig. 2(c) for
$\mu_{c}$=0.08 eV. At this time, the conductivity at low temperature 10 K is
somewhat different from those of $\mu_{c}$=0.04 eV. Now there is an intra-band
transition E${}_{1}\rightarrow$E2, whereas the inter-band transition
E${}_{0}\rightarrow$E1 is absent. By contrast, both multiple intra-band
(E1$\rightarrow$ E2, E2 $\rightarrow$ E3) and inter-band (E0 $\rightarrow$ E1)
LL transitions still exist at temperatures 310 K and 610 K. However, the
dependence of Im[$\sigma_{L}$] on the temperature becomes small at high
frequency. This is because the energies of inter-band LLs transitions are much
higher than those of thermal fluctuation (e.g, $k_{B}T\sim$0.027 eV for 310
K).

Figure 4: (color online) Spectral heat flux via chemical potential
($\mu_{c1}$=$\mu_{c2}$) for (a) B=4 T. The red dashed curves represent the
corresponding spectra for B=0, where the resonant peaks have a blue shift as
the chemical potential increases. (b) B=15 T. The middle spectral band (the
orange shaded regime) changes dramatically with increasing the chemical
potential. However, the spectral bands marked by the blue shaded regimes in
(a) and (b) are almost unchanged, showing a robust feature.

Figure 3(a) shows the spectra of radiative heat flux under different magnetic
fields. In the absence of magnetic field, the spectral heat flux is single-
band as reported in Ref. Ili:12 . Compared with the non-magnetic
configuration, the spectra are modulated correspondingly after the magnetic
field is turned on. For instance, there are some small peaks for B=1 T. As the
magnetic field increases further, dual-band spectra can be found for B=4 T, 7
T and 10 T. Overall, the peaks of the two spectral bands undergo a blue shift
and the energy gap between these two peaks enlarges as the magnetic field
increases. For a strong magnetic field B=15 T, triple-band spectrum can be
seen clearly. In addition to those two spectral bands mentioned above, there
is a remarkable band close to zero frequency, and it is plotted in the inset
for clarity. The resonant peak for this low-frequency band appears at 0.8 meV
($\sim$200 GHz), which lies in the microwave regime. The great enhancement of
NFRHT in such low frequency has not been revealed in any previous systems
before. Although the amplitude of spectrum has been multiplied 4 times for
B=15 T, the radiative heat flux can still exceed the black body limit over 40
times, which is large enough for potential applications in triple-band thermal
devices.

To clarify the origin of multi-band spectra, we plot of transmission
coefficient $\xi(\omega,q)$ below the light line in Fig. 3(b). The large wave-
vector $q\gg k_{0}$ indicates that the NFRHT is mainly contributed by the
evanescent waves. Specifically, the transmission coefficient for B=0 is
continuous over the frequency regime, resulting in a single-band spectrum. By
contrast, the transmission coefficients for B=4 T, 7 T, 10 T and 15 T turn out
to be frequency separated configurations, leading to multi-band spectra.
Interestingly, there is an extraordinary band started from zero frequency,
which is generated from magneto-plasmon zero modes as mentioned before. This
band can be prominently seen, especially for strong magnetic field B=10 T and
15 T. It is worth mentioning that the transmission coefficient $\xi(\omega,q)$
due to high-order LL transitions can be seen as well for B=4 T and 7 T with
$\omega>$0.15 eV/$\hbar$. However, the contribution of high-order LL
transitions to the NFRHT is very small, due to the fast decay of the term
$\Theta(\omega,T)$ at high frequency. For comparison, the dispersions of
magneto-plasmon in single graphene sheet are given correspondingly. It is
found that the dispersions turn out to be a series of branches at the presence
of magnetic field, which agree well with transmission coefficient
$\xi(\omega,q)$. Since the transmission coefficient $\xi(\omega,q)$ generated
from intra-band transition E${}_{2}\rightarrow$E3 is very small, the
corresponding dispersions are plotted in the dashed gray lines to indicate
their insignificance in NFRHT.

The chemical potential of graphene is an important parameter, which can be
tuned greatly by electrical gating. The spectral heat flux via different
chemical potential are shown in Figs. 4(a) and 4(b). For B= 4 T, there are two
primary spectral bands. The first primary band (marked by the black arrows)
has a relative poor excitation when the chemical potential is small (e.g.,
$\mu_{c}=0.02$ eV). As the chemical potential increases, however, the
magnitudes of this band increase and the spectral peaks have a blue shift.
This evolution is attributed to the properties of intra-band transition. On
the other hand, the second band (marked by blue shaded regime) is contributed
by the inter-band LL transition (E0$\rightarrow$E1). Surprisingly, the
magnitude and line-shape of this spectral band have little change as chemical
potential increases from 0.02 eV to 0.1 eV, showing a robust feature. For
comparison, the spectral heat flux for B=0 are given by the red dashed lines.
The resonant peaks of spectra for B=0 undergo a blue shift when the chemical
potential increases, indicating a trivial feature contributed by the intra-
band transitions. For a strong magnetic field B=15 T, the first and the third
spectral bands (blue shaded regimes) are robust against the change of chemical
potential as well. Interestingly, the middle spectral band (orange shaded
regimes) contributed by intra-band transitions is suppressed strongly at low
doping 0.02 eV and 0.04 eV. This is attributed to high energy of first LL
(E1$\sim$ 0.14 eV for 15 T), resulting in low probability of intra-band
transition. As expected, the magnitude of the middle spectrum-band increases
with the chemical potential, and it has the same order with the two side bands
when the chemical potential $\mu_{c}\sim$0.08 eV. Thus, triple-band radiative
heat transfer can be realized for suitable tuning of chemical potential under
a strong magnetic field. As the chemical potential increases further to
$\mu_{c}$=0.1 eV, the middle band is dominant.

Figure 5: (color online) Contour plots of spectral heat flux for different
scattering rate. (a) B=4 T, and (b) B=15 T. The black arrows indicates the
primary bands, and the gray arrows represent secondary bands contributed from
high order LL transitions. The chemical potential, temperatures and separation
distance are kept the same as those in Fig. 3.

Figure 6: (color online) (a) The radiative heat flux as a function of magnetic
field, normalized by blackbody limit. (b)The modulation of radiative heat flux
in the presence of magnetic field, defined as $\eta=\left\langle
S(\text{B}=0)\right\rangle$/$\left\langle S(\text{B})\right\rangle$. The solid
and dash curves represent the scattering rate $\Gamma$=4 meV and 8 meV,
respectively. The chemical potential and temperatures are kept the same as
those in Fig. 3.

To study in a more general way, the scattering rate is assumed to be a
constant but within a typical range $\Gamma\in$ [1, 12]meV (see, e.g., Fer:12
; Cra:11 ). The contour plots of spectral heat flux are shown in Figs. 5(a)
and 5(b). For B=4 T, the peak frequencies of the two primary bands indicated
by the black arrows almost do not changed as $\Gamma$ increase from 1 meV to
12 meV. Although the magnitude of spectral band become smaller for large
scattering rate, the two primary bands are still dominant. The contour plot of
spectral heat flux for B=15 T is shown in Fig. 5(b). Again, three primary
bands can be seen clearly even for large scattering rate. The magnitude of the
middle band decreases monotonously as the scattering rate increases.
Interestingly, the magnitude for the low-frequency band is almost unchanged,
whereas the bandwidths become broaden with increasing scattering rate.

Figure 6(a) shows the radiative heat flux as a function of magnetic field for
scattering rate $\Gamma$=4 meV and 8 meV. It is found that the deviation of
$\left\langle S(\text{B})\right\rangle$ for these two scattering rate is small
for separation distance $d$=50 nm. For $d$=50 nm and $d$=100 nm, there exist a
maximum enhancement of spectral heat flux near B$\sim$2 T, and the values are
about 1000 times and 300 times over the blackbody limit, respectively.
Nevertheless, $\left\langle S(\text{B})\right\rangle$ decreases monotonically
as the value of magnetic field increases further. For $d$=200 nm, however,
$\left\langle S(B)\right\rangle$ decreases monotonically from B=1 T to B=16 T.
The modulation of NFRHT due to the presence of external magnetic field is
shown in Fig. 6(b), where the ratio $\eta=\left\langle
S(\text{B}=0)\right\rangle$/$\left\langle S(\text{B})\right\rangle$ is used.
This ratio is relatively small for a moderate magnetic field, whereas it can
increase dramatically as the magnetic field becomes strong. Taking B=15 T
($\Gamma$=4 meV) as an example, the ratio $\eta$ is about 4.8, 6.2 and 10
times for separation distances $d$=50 nm, 100 nm and 200 nm, respectively. The
ratio $\eta$ can be further enlarged as the scattering rate $\Gamma$=8 meV.
The modulation of NFRHT between graphene sheets here is much larger than those
of semiconductor such as InSb and doping Si under a perpendicular magnetic
field Mon:15 .

Figure 7: (color online) Spectral heat flux for graphene sheets placed on top
of substrates with permittivity $\varepsilon_{s}$, where the thickness of the
substrates is semi-infinite. The magnetic fields are (a) 4 T and (b) 15 T,
respectively. Other parameters are kept the same as those in Fig. 3.

Technically, suspended graphene sheets can be realized in the experiments
Mey:07 ; Bol:08 . However, graphene sheets are generally deposited on a
variety of different substrates Edw:13 . The substrate effect for NFRHT is
demonstrated in Figs. 7(a) and 7(b). As the permittivity of the dielectric
substrate (non-polar and non-dispersive materials) $\varepsilon_{s}$
increases, all spectral peaks undergo a red shift. Besides, the amplitudes of
the second spectral band for B=4 T and the middle spectral band for B=15 T
decrease accordingly. It is expected that the substrate effect can be
diminished as the thickness of substrates is reduced. Nevertheless, the multi-
band spectra still persist even the substrates are taken into account.
Finally, it is worth mentioning that the type of doping (electron or hole
doping) in graphene could determine the sign of Hall conductivity
$\sigma_{H}$. The sign of $\sigma_{H}$ is important for some remarkable
phenomena such as quantum Casimir effect Tse:12  and Faraday rotation Cra:11 .
However, the sign of $\sigma_{H}$ has little effect for NFRHT since the
magneto-plasmon of graphene is dominant by the longitudinal component of
conductivities.

##  IV Conclusions

In summary, magnetically tunable multi-band NFRHT is revealed in a system of
two suspended graphene sheets. The single-band spectra for B=0 can be
modulated greatly in the presence of a sufficient strong magnetic-field. Dual-
band spectra can be realized for moderate magnetic fields (e.g., B=4 T). One
band is determined by intra-band transitions in the classical regime, which
undergoes a blue shift as the chemical potential increases. The other band is
contributed by inter-band LL transition E${}_{0}\rightarrow$E1 in the quantum
regime, showing a robust feature against the change of chemical potential. On
the other hand, triple-band spectra can be found under a strong magnetic field
(e.g., B=15 T). Remarkably, there is an extremely low-frequency band with the
resonant peak appearing at microwave regime, stemming from the magneto-plasmon
zero modes. The performances of NFRHT due to the scattering rate, separation
distance and substrate effect are demonstrated as well. Although the magnetic
field considered in our work is static, it can indeed be time-dependent such
as sin-cos, sawtooth, square-wave etc. As a result, a dynamic tunable multi-
band NFRHT would be possible under a time-dependent magnetic field, providing
a possibility for multi-band information transfer.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant No. 11747100, 11704254, 11804288), and the Innovation Scientists and
Technicians Troop Construction Projects of Henan Province. The research of L.
X. Ge is further supported by Nanhu Scholars Program for Young Scholars of
XYNU. Work in Saudi Arabia was supported by King Abdullah University of
Science and Technology (KAUST) Baseline Research Fund BAS/1626-01-01.

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Here, $\varrho$ becomes infinitely large, and therefore, the lines parallel to
the symmetry axes of the movement, at a distance $\frac{1}{n}$ from them, are
asymptotes.

From Eq. (21.5) and the second equation of (21.4), we find, by eliminating
$\varrho_{1}$ and squaring

$\cot n\vartheta_{1}=nt$ (21.7)

and then from Eq. (21.5),

$\varrho_{1}^{2}=1+n^{2}t^{2}.$ (21.8)

The time is measured from the moment $\varrho_{1}=1$ and
$\vartheta_{1}=\frac{\pi}{2n}$. From here, $\varrho_{1}$ increases and
$\vartheta_{1}$ decreases.

The resulting equation for the speed is

$w_{1}^{2}=\frac{1+n^{4}t^{2}}{1+n^{2}t^{2}}=\frac{1-n^{2}+n^{2}\varrho_{1}^{2}}{\varrho_{1}^{2}}.$
(21.9)

The velocity reaches a minimum value $w_{1}=1$ at $t=0$, then increases
steadily as it approaches the limit $n$ as $t\to\infty$.

Figure 12 corresponds to the assumption $n=2$.

Figure 12: Gröbli’s Figure 12.


# Conductivity and thermoelectric coefficients of doped SrTiO3 at high
temperatures

Kh. G. Nazaryan  Moscow Institute of Physics and Technology  M. V. Feigel’man
L.D.Landau Institute for Theoretical Physics, Chernogolovka, Moscow Region,
143432, Russia  Skolkovo Institute of Science and Technology  Moscow Institute
of Physics and Technology

###### Abstract

We developed a theory of electric and thermoelectric conductivity of lightly
doped SrTiO3 in the non-degenerate region $k_{B}T\geq E_{F}$, assuming that
the major source of electron scattering is their interaction with soft
transverse optical phonons present due to proximity to ferroelectric
transition. We have used kinetic equation approach within relaxation-time
approximation and we have determined energy-dependent transport relaxation
time $\tau(E)$ by the iterative procedure. Using electron effective mass $m$
and electron-transverse phonon coupling constant $\lambda$ as two fitting
parameters, we are able to describe quantitatively a large set of the measured
temperature dependences of resistivity $R(T)$ and Seebeck coefficient
$\mathcal{S}(T)$ for a broad range of electron densities studied
experimentally in recent paper Behnia2020 . In addition, we calculated Nernst
ratio $\nu=N/B$ in the linear approximation over weak magnetic field in the
same temperature range.

Introduction. Very dilute 3D metal originating from band insulator Strontium
Titanate (STO) due to tiny doping ($10^{-6}-10^{-3}$ conduction electrons per
unit cell) demonstrates a number of rather unusual properties Spinelli2010 ;
review1 ; review2 . Their major common origin is the close proximity of
insulating STO to a ferroelectric transition, which lead to a giant low-
temperature dielectric constant $\epsilon_{0}\approx 20000$. In result,
Coulomb interaction between conduction electrons is nearly vanishing, and
standard phenomenology developed in the theory of normal metals is not
applicable. Very low attainable density $n$ of conduction electrons and its
variability allow to study transport properties of doped STO in various
temperature regimes, from strongly degenerate Fermi gas at $k_{B}T\ll E_{F}$
to highly non-degenerate one at $k_{B}T\gg E_{F}$, with $E_{F}$ being the
Fermi energy. The latter range is in the focus of recent experimental studies
Behnia2020 ; Behnia2017 ; see also similar mobility data in Ref. (Verma, ) and
thermoelectric data in Ref. Cain . It was found Behnia2017 that at high
temperatures $T\geq 300$ K conductivity drops below Mott-Ioffe-Regel limit
and, moreover, relaxation rate $1/\tau$ becomes larger than its apparent
quantum limit $k_{B}T/\hbar$. Later on, the paper Behnia2020 demonstrates that
the account for the temperature-dependent renormalization of the effective
mass $m(T)$ makes the above contradiction less drastic. However, the behavior
of $m(T)$ found in Ref. Behnia2020 via the fitting of their data for
resistance $R(T)$ and Seebeck coefficient $\mathcal{S}(T)$ is somewhat
unexpected. Namely, mass renormalization $m(T)/m_{0}$ fitted in Behnia2020
depends not only on temperature but also on the electron density $n$, which
should not be the case in the non-degenerate region $k_{B}T\gg E_{F}$ studied.
Also, initial increase on $m(T)$ with $T$ growth is replaced by its drop with
temperature at $T$ above 300 K, which does not have physical explanation.

In the present Letter we reconsider theoretically the issue of electric
conductivity and thermoelectric response in a non-degenerate electron gas,
interacting with soft transverse optical phonons. These type of phonons exist
in STO due to its proximity to ferroelectric transition YamadaShirane ;
review1 ; review2 ; Behnia2020 . Scattering of non-degenerate electrons due to
bi-quadratic coupling to transverse optical phonons was considered in somewhat
different context in Refs. Levanyuk1 ; Levanyuk2 ; see also older experiments
old1 ; old2 .

We use kinetic equation approach in the form close to the one discussed in
Ref. Theory and express both resistivity $R(T)$ and Seebeck coefficient
$\mathcal{S}(T)$ in terms of the energy-dependence transport scattering time
$\tau(E,T)$. We emphasize that the energy-dependence of $\tau(E,T)$ is not
weak, and this is the reason why simple proportionality relation Behnia2020
between Seebeck coefficient $\mathcal{S}(T)$ and thermodynamic entropy per
electron $S(T)$ is not valid actually. Indeed,
$\mathcal{S}(T)=\frac{k_{B}}{e}S(T)$ follows once $\tau(E,T)$ can be replaced
by energy-independent $\tau(T)$, as we will show below.

Real band structure of SrTiO3 is rather complicated (see for example Dirk2011
), and detailed calculation of transport coefficients is hardly possible
without use of heavy numerical procedures based upon band-structure
calculations. Some examples of the latter type of approach can be found in
Refs. numeric1 ; numeric2 . While Ref. numeric1 reported good agreement of
computed high-$T$ mobility with experimental data, the later Ref. (numeric2, )
asserts that numerically exact band-structure calculation does over-estimate
mobility at high $T$ by a factor close to 10, while providing correct
$T$-dependence. Paper (numeric2, ) hints that the discrepancy originates from
temperature-dependent polaron effect leading to the increase of effective
electron mass with temperature. Later paper (numeric3, ) presents the results
obtained by numerical implementation of this type of approach: strong polaron
effects due to electron-phonon interactions were taken into account and good
agreement with numerically obtained data on mobility was obtained. Moreover it
was found that strong incoherent effects (demonstrated by broad electron
spectral function) are developing at $T\sim 250-300^{\circ}$K.

We will accept general idea of importance of $T$-dependent mass
renormalization and develop a semi-quantitative theory based upon the simplest
model of parabolic electronic spectrum $E(p)=p^{2}/2m$ with temperature-
dependent effective mass $m=m(T)$. Our goal is to provide simple model of
electron scattering leading to reasonable agreement with the data in the high-
temperature region $T\gg E_{F}(n)$ for both conductivity $\sigma(T)$, Seebeck
coefficient $\mathcal{S}(T)$ and Nernst coefficient $\nu(T)$. We will show
that moderately strong mass renormalization, which is weakly $T$-dependent at
high temperatures, is sufficient to account for the data, even if incoherent
effects (smearing of the spectral function) are not taken into account.

Formulation of the model. We are interested in the high-temperature region
where quantum statistics of electrons is irrelevant and it is sufficient to
consider each electron individually interacting with lattice phonons. The
Hamiltonian of an electron coupled to transverse optical phonons is

$H=\int
d^{3}r\left[-\hbar^{2}\frac{\psi^{\dagger}(\mathbf{r})\nabla^{2}\psi(\mathbf{r})}{2m(T)}+g\rho_{m}\,\psi^{\dagger}(\mathbf{r})\psi(\mathbf{r})\mathbf{u}^{2}(\mathbf{r})\right]$
(1)

where $\rho_{m}=5.11$ g/cm3 is the STO mass density, $\psi(\mathbf{r})$ is the
electron annihilation operator and $\mathbf{u}(\mathbf{r})$ is the atomic
displacement field related to the optical transverse mode (TO),
$\nabla\mathbf{u}=0$. Dispersion of this mode is
$\omega(k)=\sqrt{\omega_{TO}^{2}+(sk)^{2}}$ where $s\approx 7.5\cdot 10^{5}$
cm/s is the TO mode velocity Svel and $\omega_{TO}$ is the temperature-
dependent gap which grows with temperature Behnia2020 ; YamadaShirane ; Baurle
; in particular, $\omega_{TO}$ varies between 5 meV and 18 meV when $T$ grows
from 100 K to 800 K. Bilinear coupling (1) to transverse phonons was proposed
recently as a possible mechanism for superconductivity in doped STO Dirk2019 ;
we develop this idea further in a separate publication KiselevFeigelman .
Another possible type of electron’s coupling to transverse TO phonons may
contain vector term of the type of $i(\psi^{+}\nabla\psi-h.c.)\mathbf{u}$, but
we will not consider its effects here.

Effective band mass of electrons depends Behnia2014 on their concentration at
low temperatures in the degenerate region $T\ll E_{F}$; in particular, for
zero-temperature mass $m(0)/m_{0}\approx 1.8$ for lowest concentrations $n\leq
10^{18}$ cm-3, while at higher doping $n\geq 4\cdot 10^{18}$ cm-3 the same
ratio $m(0)/m_{0}\approx 4$; here $m_{0}$ is the free electron mass. However,
at high temperatures $T\gtrsim 150$ K Hall mobility is $n$-independent, as
follows from the data present in Ref. Behnia2020 . The same is expected for
the effective mass, which may however depend on temperature, for two different
reasons. First one is due to non-parabolic dispersion with decreasing
effective curvature of $\xi(\mathbf{p})$ at large momenta $|\mathbf{p}|$, see
Fig. 3 in Ref. Dirk2011 : at high $T$ carriers with energies much above the
bottom of the band are most relevant, and effective mass seen in transport
characteristics increases; moreover, at high temperatures the major
contribution to transport is provided by the lowest band since at large
$|\mathbf{p}|$ it provides the largest density of states. The second reason
for the mass enhancement is the polaron effect due to coupling with TO phonons
numeric3 . Below we will treat $m(T)$ as a fitting function, common for all
concentrations $n$.

The coupling constant $g$ in Eq.(1) has a natural dimension
$\textrm{Length}^{3}/\textrm{Time}^{2}$. We present it in the form

$g=\lambda{a^{3}}\omega_{L}^{2}$ (2)

where $a=0.39$ nm is the STO lattice constant, $\hbar\omega_{L}\approx 100$meV
is the highest energy gap of all longitudinal optical modes and $\lambda$ is
some dimensionless constant, to be determined below from the fit of our theory
to the data.

Figure 1: a) Lines represent the results of the $3^{\text{rd}}$ iteration of
the method of successive approximations for the inverse scattering time
normalized to the inverse Planckian time $\tau_{p}/{\tau_{TO}^{(3)}}$. Here,
$p_{BZ}=\hbar\pi/a$ stands for the first Brillouin zone and
$\tau_{p}=\hbar/{k_{B}T}$. b) Full lines show the expected qualitative
temperature dependency of the effective mass. The dashed line stands for the
linear dependency chosen in the present paper. The inset shows the comparison
of the scattering time averaged as
$\overline{\tau_{TO}}=\langle\tau_{TO}\rangle/\langle 1\rangle$ (blue line)
and the Planckian time (orange line)

The kinetic equation and relaxation time approximation. We will study electric
and thermoelectric transport in an electron system using the Boltzmann kinetic
equation. To find the transport coefficients within linear response theory, we
expand the distribution function as $f_{\boldsymbol{p}}\approx n_{p}+\delta
n_{\boldsymbol{p}}$. Here,
$n_{p}=\left[\exp\left(\beta\xi_{p}\right)+1\right]^{-1}$ is the Fermi-Dirac
distribution with $\beta=1/T$, and $\xi_{p}=E(p)-\mu$ with the chemical
potential $\mu$. This helps to write the linearized Boltzmann equation in
presence of both electric field and temperature gradient:

$\displaystyle\left(-e\mathbf{E}-\xi_{\mathbf{p}}\frac{\nabla_{\mathbf{r}}T}{T}\right)\cdot
v_{\mathbf{p}}\frac{\partial
n_{p}\left(\xi_{\mathbf{p}}\right)}{\partial\xi_{\mathbf{p}}}=I_{TO}\\{\delta
n_{\boldsymbol{p}}\\}$ (3)

where
$\mathbf{v}_{\mathbf{p}}=\partial\xi_{\mathbf{p}}/\partial\mathbf{p}=\mathbf{p}/m^{*}$
is the group velocity. An exact expression for the collision integral
$I_{TO}\\{\delta n_{\boldsymbol{p}}\\}$ which describes electron scattering by
two TO phonons is of the following form (for the derivation, see Supporting
Material I):

$\displaystyle\frac{1}{A}I_{TO}\\{\delta n_{\boldsymbol{p}}\\}=$ (4)
$\displaystyle=2\int_{\boldsymbol{k_{1}k_{2}}}\frac{\delta\left(E_{p^{\prime}}+E_{k_{1}}-E_{p}-E_{k_{2}}\right)}{\omega(k_{1})\omega(k_{2})}\left(\delta
n_{\boldsymbol{p}}\left(\left(N_{k_{2}}-N_{k_{1}}\right)n_{p^{\prime}}-\left(N_{k_{1}}+1\right)N_{k_{2}}\right)+\delta
n_{\boldsymbol{p^{\prime}}}\left(n_{p}\left(N_{k_{2}}-N_{k_{1}}\right)+\left(N_{k_{2}}+1\right)N_{k_{1}}\right)\right)$
$\displaystyle+\int_{\boldsymbol{k_{1}k_{2}}}\frac{\delta\left(E_{p^{\prime}}-E_{k_{1}}-E_{p}-E_{k_{2}}\right)}{\omega(k_{1})\omega(k_{2})}\left(-\delta
n_{\boldsymbol{p}}\left(N_{k_{1}}N_{k_{2}}+\left(1+N_{k_{2}}+N_{k_{1}}\right)n_{p^{\prime}}\right)+\delta
n_{\boldsymbol{p^{\prime}}}\left(\left(N_{k_{2}}+1\right)\left(N_{k_{1}}+1\right)-n_{p}\left(1+N_{k_{2}}+N_{k_{1}}\right)\right)\right)$
$\displaystyle+\int_{\boldsymbol{k_{1}k_{2}}}\frac{\delta\left(E_{p^{\prime}}+E_{k_{1}}-E_{p}+E_{k_{2}}\right)}{\omega(k_{1})\omega(k_{2})}\left(\delta
n_{\boldsymbol{p}}\left(\left(1+N_{k_{1}}+N_{k_{2}}\right)n_{p^{\prime}}-\left(N_{k_{1}}+1\right)\left(N_{k_{2}}+1\right)\right)+\delta
n_{\boldsymbol{p^{\prime}}}\left(n_{p}\left(1+N_{k_{2}}+N_{k_{1}}\right)+N_{k_{2}}N_{k_{1}}\right)\right)$

where $N_{k}=\left[\exp\left(\beta\omega_{k}\right)-1\right]^{-1}$ is the
Bose-Einstein distribution for TO phonons, and $A=\frac{\pi}{2}{g^{2}}$. We
use the short-cut notation for the integrals
$\int_{\boldsymbol{k_{1}k_{2}}}=\int_{BZ}\frac{d^{3}\boldsymbol{k_{1}}d^{3}\boldsymbol{k_{2}}}{\left(2\pi\right)^{6}}$,
where BZ stands for the first Brillouin zone (we take it in spherical
approximation). Integration over momenta in Eq.(4) is carried out under the
condition of conservation of the total momentum (we neglect Umpklapp
processes). Conservation of the total energy is expressed in the explicit form
in Eq.(4) by the delta-functions; the forms of their arguments demonstrate
that the 1st line corresponds to the scattering of an electron by the TO
phonon, while 2nd and 3rd lines correspond to emission and absorption of two
TO phonons, correspondingly.

It appears to be possible to represent complicated collision integral (4)
within much simpler form of relaxation time approximation (RTA):

$\displaystyle I_{TO}\\{\delta n_{\boldsymbol{p}}\\}\approx-\frac{\delta
n_{\boldsymbol{p}}}{\tau_{TO}(p)}$ (5)

Effective relaxation rate $\tau^{-1}_{TO}(p;T)$ which provides the best
approximation of the exact $I_{TO}\\{\delta n_{\boldsymbol{p}}\\}$ by the RTA,
can be found by means of the successive approximation method, described in
details in the Supporting Material II. Briefly, it works as follows. First we
note that in the linear regime over weak electric field $E$, modification of
the electron distribution function can be written in the form (which follows
by comparison between Eq.(3) and Eq.(5)):

$\displaystyle\delta n_{\boldsymbol{p}}\approx
e\,\tau_{TO}(p)\cdot\left(\boldsymbol{v_{p}},\mathbf{E}\right)\frac{\partial
n_{F}\left(\xi_{p}\right)}{\partial\xi_{p}}$ (6)

The form of Eq.(6) assumes it is possible to neglect temperature gradient
$\boldsymbol{\nabla}_{r}T$ while calculating $\delta n_{\boldsymbol{p}}$ and
$\tau_{TO}(p)$, and we check it a posteriori, see the end of Supporting
Material II. The only unknown function here is
$\tau_{TO}=\tau_{TO}(|\boldsymbol{p}|)$. In the case of degenerate Fermi-gas
relevant momentum $p=p_{F}$ and we are left with just a single number $\tau$
for relaxation time. At high temperatures $T\geq E_{F}$ the situation is more
complicated. Relaxation time $\tau_{TO}(|\boldsymbol{p}|)$ appears to be a
strong function of the electron energy. To find this function we start from
the simplest trial with $\tau_{TO}(p)=\tau_{TO}$ independent of energy, and
substitute the corresponding $\delta n_{\boldsymbol{p}}^{(0)}$ into the R.H.S.
of Eq.(4). The result is then represented in the form of Eq.(6) with somewhat
different $\tau_{TO}^{(1)}(p)$ which is now energy-dependent. Then the process
is repeated upon convergence (see Supporting Material II). We have found that
the difference of the obtained $\tau_{TO}^{(2)}(p)$ and $\tau_{TO}^{(3)}(p)$
is already quite small, thus it is possible to use $\tau_{TO}^{(3)}(p)$ as our
final approximation for the relaxation rate. It is shown in Fig. 1a for a
number of different temperatures. One can clearly see in Fig. 1a that the
momentum dependence of scattering time $\tau_{TO}(p)$ is not weak. This is why
the Seebeck coefficient $\mathcal{S}(T)$ in not just proportional to the
thermodynamic entropy per electron $S(T)$ as it was supposed in the paper
(Behnia2020, ). Instead, we will employ an approach provided in Ref. Theory
where various transport coefficients were found within RTA extended for the
case of an arbitrary dependence $1/\tau(p)$.

Transport coefficients within RTA. Electric conductivity $\sigma$ and the
Seebeck coefficient $\mathcal{S}$ can be expressed via $\tau(p)$ as follows
(Theory, ):

$\displaystyle\sigma=\frac{e^{2}}{m}\mathcal{N}\left\langle\tau_{TO}(p)\right\rangle$
(7)
$\displaystyle\mathcal{S}=-\frac{1}{eT}\frac{\left\langle\xi_{\boldsymbol{p}}\tau_{TO}(p)\right\rangle}{\left\langle\tau_{TO}(p)\right\rangle}$
(8)

where $\left\langle\dots\right\rangle$ denotes the average $\left\langle
X_{\mathrm{p}}\right\rangle=-\frac{4}{3\mathcal{N}}\int_{\mathbf{p}}X_{\mathbf{p}}\left(\xi_{\mathbf{p}}+\mu\right)\frac{\partial
n_{p}\left(\xi_{\mathbf{p}}\right)}{\partial\xi_{\mathbf{p}}}$ with
$\mathcal{N}=2\int_{\boldsymbol{p}}n_{p}(\xi_{p})$. While Eq.(7) for
conductivity is evident, the result (8) for the Seebeck coefficient needs some
comments. Indeed, if one assumes $\tau(p)$ to be some constant (independent on
$p$), then Eq.(8) leads to the relation $\mathcal{S}(T)=\frac{k_{B}}{e}S(T)$
used for the analysis developed in Ref. Behnia2020 . In fact, the
$p$-dependence of relaxation rate is very strong, as shown in Fig. 1a.

The equations ((7), (8)) can be derived as follows. We allow for the presence
of both electric field $\mathbf{E}$ and temperature gradient $\nabla T$ and
express $\delta n_{\mathbf{p}}$ using Eq. (3) for the collision integral in
the RTA form (5). The electric and thermal currents are then determined as
follows:

$\displaystyle\left(\begin{array}[]{c}\mathbf{J}_{\mathbf{E}}\\\
\mathbf{J}_{T}\end{array}\right)$
$\displaystyle=2\int_{\mathbf{p}}\left(\begin{array}[]{c}-e\\\
\xi_{\mathbf{p}}\end{array}\right)\boldsymbol{v_{p}}\delta
n_{\mathbf{p}}=\left(\begin{array}[]{cc}\sigma&\alpha\\\
\beta&\kappa^{\prime}\end{array}\right)\left(\begin{array}[]{c}\mathbf{E}\\\
-\mathbf{\nabla}_{\mathbf{r}}T\end{array}\right)$ (17)

where $\sigma$ is electric conductivity, $\alpha={\beta}/{T}$ due to Onsager
reciprocity relations and Seebeck coefficient $\mathcal{S}={\alpha}/{\sigma}$.
The use of $\delta n_{\mathbf{p}}$ in the form of Eq.(3) leads then to (7),
(8). Note that now we consider kinetic equation in presence of both electric
field and temperature gradient; we checked in Supporting Material II that the
presence of $\nabla T$ does not change the function $\tau_{TO}(p)$ found
previously.

(a)  Temperature dependency of the resistivity extended to 700 K for several
Nb doped $\text{SrTiO}_{3}$ crystals.

(b) The Seebeck coefficient as a function of temperature for four Nb doped
$\text{SrTiO}_{3}$ samples from 100 K to 700 K

Figure 2: Experimental data Behnia2020 are shown by dashed lines, while full
lines are used for the theoretical results. Figure 3:  Nernst coefficient
$\nu(T)$ predicted for several Nb doped $\text{SrTiO}_{3}$ samples from 50 K
to 700 K. The lower inset shows the temperature dependency of the Nernst
coefficient in the range 300 - 700 K. The upper inset shows the same
dependency from 50 K to 300 K.

Nernst coefficient. In presence of magnetic field $\mathbf{B}$, an applied
electric field $\mathbf{E}\perp\mathbf{B}$ leads to the temperature gradient
transverse to both $\mathbf{E}$ and $\mathbf{B}$. The corresponding response
is called Nernst signal, $N=E_{y}/\nabla_{x}T$. To calculate it we start from
the general expression (See Sec.5 of the book KamranBook ) for $N$ in terms of
electric conductivity tensor $\sigma_{\alpha\beta}$ and thermoelectric tensor
$\alpha_{\alpha\beta}$:

$\displaystyle
N=\frac{E_{y}}{\nabla_{x}T}=\frac{\alpha_{xy}\sigma_{xx}-\alpha_{xx}\sigma_{xy}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}}$
(18)

We concentrate here upon the limit of weak magnetic field and thus consider
lowest-order terms in the magnitude of $B$:

$\displaystyle\sigma_{xx}\approx\sigma;\,\,\;\alpha_{xx}\approx\alpha=-\frac{e\mathcal{N}}{mT}\left\langle\xi_{p}\tau_{TO}(p)\right\rangle;$
(19)
$\displaystyle\sigma_{xy}\approx\frac{e^{3}\mathcal{N}}{m^{2}}B\left\langle\tau^{2}_{TO}(p)\right\rangle;\,\,\;\alpha_{xy}\approx-\frac{e^{2}\mathcal{N}}{m^{2}T}B\left\langle\xi_{p}\tau^{2}_{TO}(p)\right\rangle$
(20)

These equations lead us to the expression of the linear Nernst coefficient
$\nu=dN/dB|_{B=0}$:

$\displaystyle\nu=\frac{\left\langle\tau^{2}_{TO}(p)\right\rangle\left\langle\xi_{\textbf{p}}\tau_{TO}(p)\right\rangle-\left\langle\tau_{TO}(p)\right\rangle\left\langle\xi_{\textbf{p}}\tau^{2}_{TO}(p)\right\rangle}{T\left\langle\tau_{TO}(p)\right\rangle^{2}}$
(21)

Eq.(21) will be used below to compute Nernst coefficient $\nu(T)$ as function
of temperature.

Results for the electric and thermoelectric coefficients. The obtained results
for the scattering time $\tau_{TO}$ enabled calculation of the electric and
thermoelectric coefficients using (7) and (8). The results of numerical
computation for the electric resistivity and the Seebeck coefficient are
provided on Fig. 2(a) and Fig. 2(b) respectively. The mass $m(T)$ and the
coupling constant $\lambda$ were fitted to experimental data from Ref.
Behnia2020 in the way to minimize relative errors between the data and the
theory. We emphasize that the single set of $\lambda$ and $m(T)$ was used to
fit the data for five different doping concentrations (apart from $n=5.9\cdot
10^{18}cm^{-3}$), and relative error is about 7-9%. Namely, we choose
$\lambda=0.88$ (with a relative error of 5%) and effective mass $m(T)$ which
linearly interpolates between $8.7m_{0}$ for $100$ K to $10.7m_{0}$ for $700$
K, see Fig.1b. For the concentration $n_{H}=5.9\cdot 10^{18}cm^{-3}$ we found
$\lambda\approx 0.78$, which is almost 10% less than for the others.

Our predictions for the Nernst coefficient $\nu(T)$ are shown in Fig. 3. We
present our results for different electron densities but always for
temperatures $T$ above the corresponding Fermi energy $E_{F}(n)$. At these
high temperatures $\nu(T)$ is independent on density of electrons, similar to
the behaviour of mobility $\mu(T)$. While at low temperatures Nernst signal is
known to be strongly $n$-dependent Behnia2013 , we are not aware of any
measurement of this quantity in STO in the non-degenerate region $T\gg E_{F}$.
Upper inset to Fig. 3 shows $\nu(T)$ in double-logarithmic scale, where one
can infer a power-law-like behaviour $\nu(T)\propto T^{-n}$ with $n\approx
3.5$. Lower inset is concentrated in the high-$T$ region where sign change of
$\nu(T)$ is predicted at $T\approx 450$ K.

Conclusions We developed a theory of conductivity and thermoelectric effects
in doped Strontium Titanate in the non-degenerate region $T\gg E_{F}(n)$,
assuming that the major source of electron scattering is due to processes
which involve two soft optical (TO) phonons. The strength of the interaction
between electrons and these TO phonons is determined by single dimensionless
parameter $\lambda$. We have found good agreement between our theory and
experiment Behnia2020 for both conductivity and Seebeck coefficient and for
(almost) all electron concentrations, using single value of $\lambda\approx
0.88$ and an assumption of an enhanced effective mass, with the ratio
$m^{*}(T)/m_{0}$ varying between about 8.7 and 10.7 in the temperature
interval 100-700 Kelvin (independent on $n$).

The major difference between our approach and the theory part of Ref.
Behnia2020 is that we take into account energy-dependence $1/\tau_{TO}(p)$
which spoils simple proportionality between Seebeck coefficient and entropy
per particle. The same energy-dependence leads to non-zero Nernst coefficient
which we calculate in linear over magnetic field approximation.

Direct experimental measurement of the effective mass $m(T)$ at high
temperatures is difficult due to non-parabolic spectrum of STO; in particular
it complicates the use of plasma frequency measurements like the one performed
in Ref. (Mass, ), since it would provide the average of the inverse mass,
$\langle m^{-1}\rangle$, while we rather need $m_{eff}=\langle m\rangle$.
Indirect check of the proposed theory can be performed by measurements of
Nernst effect at high temperatures and comparison with our predictions, see
Fig. 3.

With the values obtained for our fit parameters $\lambda$ and $m(T)$, we find
average scattering rate $\langle 1/\tau_{TO}\rangle$ which is below, or at
most, 20$\%$ higher, than the inverse Planckian time $k_{B}T/\hbar$, see inset
to Fig. 1b. The energy-dependent relaxation time $\tau_{TO}(p)$ never becomes
less than 60$\%$ than Planckian time, and is longer than it in the major part
of the data shown in Fig. 1a. Therefore we expect that the use of standard
kinetic equation we employed is possible for the problem studied, although
corrections to it may occur to be non-negligible.

Acknowledgments We thank Kamran Behnia and Clément Collignon for many useful
discussions of experimental aspects of STO and for providing us with the raw
experimental data from (Behnia2020, ); we also thank Dirk van der Marel and
Igor Mazin for clarifications about the issue of effective electron mass in
STO. We are grateful to Anton Andreev, Denis Basko, Alexei Ioselevich and
Dmitry Kiselev for their comments on the theory side.

## References

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Supporting Material for ”Conductivity and thermoelectric coefficients of doped
SrTiO3 at high temperatures”

##  I  Collision integral

In order to treat the Hamiltonian (1) we must represent the phonon operators
in the second quantization form

$\displaystyle\hat{\boldsymbol{u}}(\boldsymbol{r})=\frac{1}{\sqrt{N}}\sum_{\boldsymbol{k},\lambda}\frac{\boldsymbol{e}^{\lambda}_{\boldsymbol{k}}}{\sqrt{2M\omega_{\boldsymbol{k}}}}\left(\hat{a}^{\lambda}_{\boldsymbol{k}}e^{-i\omega_{\boldsymbol{k}}t+i\boldsymbol{k}\boldsymbol{r}}+e.c\right)$
(22)

where $\hat{a}^{\lambda}_{\boldsymbol{k}}$ is phonon annihilation operator,
$M$ is the mass of the unit sell, $N$ is the number of cells and
$\boldsymbol{e}^{\lambda}_{\boldsymbol{k}}$ is the $\lambda$ polarization
vector. This rewrites the Hamiltonian in a form

$\displaystyle\hat{\mathcal{H}}_{e-ph}$
$\displaystyle=\frac{g}{2MN}\sum_{\boldsymbol{k_{1}},\boldsymbol{k_{2}},\lambda,\mu}\frac{\boldsymbol{e}^{\lambda}_{\boldsymbol{k_{1}}}\boldsymbol{e}^{\mu}_{\boldsymbol{k_{2}}}}{\sqrt{\omega_{\boldsymbol{k_{1}}}\omega_{\boldsymbol{k_{2}}}}}\left(\hat{\mathcal{F}}_{\boldsymbol{k_{1}k_{2}}}^{(1)\lambda\mu}+\hat{\mathcal{F}}_{\boldsymbol{k_{1}k_{2}}}^{(2)\lambda\mu}\right)$
(23)
$\displaystyle\hat{\mathcal{F}}_{\boldsymbol{k_{1}k_{2}}}^{(1)\lambda\mu}$
$\displaystyle=\left(\hat{a}^{\lambda}_{\boldsymbol{k_{1}}}\hat{a}^{\mu}_{\boldsymbol{k_{2}}}e^{i\left(\boldsymbol{k_{1}}+\boldsymbol{k_{2}}\right)\boldsymbol{r}}\right)e^{-i(\omega_{\boldsymbol{k_{1}}}+\omega_{\boldsymbol{k_{2}}})t}+e.c.$
(24)
$\displaystyle\hat{\mathcal{F}}_{\boldsymbol{k_{1}k_{2}}}^{(2)\lambda\mu}$
$\displaystyle=\left(\hat{a}_{\boldsymbol{k_{1}}}^{\lambda}\hat{a}_{\boldsymbol{k_{2}}}^{\dagger\mu}e^{i\left(\boldsymbol{k_{1}}-\boldsymbol{k_{2}}\right)\boldsymbol{r}}\right)e^{-i(\omega_{\boldsymbol{k_{1}}}-\omega_{\boldsymbol{k_{2}}})t}+e.c$
(25)

For the scattering probability we will use Fermi’s Golden rule. Here, each
term in the Hamiltonian gets a simple physical interpretation.
$\hat{\mathcal{F}}_{\boldsymbol{k_{1}k_{2}}}^{(1)}$ will stand for the
processes when two phonons are absorbed or emitted, and
$\hat{\mathcal{F}}_{\boldsymbol{k_{1}k_{2}}}^{(2)}$ for the process of
scattering on a phonon.

Now, we take into account
$\langle\langle\hat{\psi}_{\boldsymbol{p}}^{\dagger}\hat{\psi}_{\boldsymbol{p}}\rangle\rangle=f_{\boldsymbol{p}},\langle\langle\hat{\psi}_{\boldsymbol{p}}\hat{\psi}_{\boldsymbol{p}}^{\dagger}\rangle\rangle=1-f_{\boldsymbol{p}},\langle\langle\hat{a}_{\boldsymbol{k}}^{\dagger}\hat{a}_{\boldsymbol{k}}\rangle\rangle=N_{\boldsymbol{k}},\langle\langle\hat{a}_{\boldsymbol{k}}\hat{a}_{\boldsymbol{k}}^{\dagger}\rangle\rangle=N_{\boldsymbol{k}}+1$,
where $\hat{\psi}_{\boldsymbol{p}}^{\dagger},\hat{\psi}_{\boldsymbol{p}}$ are
electron creation and annihilation operators respectively, and
$f_{\boldsymbol{p}}$ and $N_{\boldsymbol{k}}$ are non-equilibrium
distributions for electrons and phonons respectively. The last step is to
represent the electron distribution function as follows
$f_{\boldsymbol{p}}\approx n_{p}+\delta n_{\boldsymbol{p}}$ and take phonon
distribution to be almost equilibrium $N_{\boldsymbol{k}}\approx N_{k}$.
Finally, the collision integral vanishes when all the distributions are taken
to be the equilibrium ones. Therefore, saving only the terms linear with
respect to $\delta n_{\boldsymbol{p}}$ we end up with (4).

##  II  Calculation of the relaxation time

###  II.1  The method of successive approximations

In order to find the relaxation time we assume that the temperature gradient
in the Boltzmann equation can be neglected. This assumption is checked in the
Supporting Material II.2. Therefore, we can rewrite the equation for $\delta
n_{\textbf{p}}$ in the following form

$\displaystyle\delta n_{\boldsymbol{p}}\approx
e\tau_{TO}(p)\cdot\left(\boldsymbol{v_{p}},\mathbf{E}\right)\frac{\partial
n_{F}\left(\xi_{p}\right)}{\partial\xi_{p}}$ (26) $\displaystyle
I_{TO}\\{\delta n_{\boldsymbol{p}}\\}\equiv-\frac{\delta
n_{\boldsymbol{p}}}{\tau_{TO}(p)}$ (27)

Now, let us introduce our modification of the method of successive
approximations for this case.
$0^{th}$ iteration: We start from the simplest trial with an energy
independent relaxation time

$\displaystyle\tau_{TO}^{(0)}=\text{const}$ (28) $\displaystyle\delta
n_{\boldsymbol{p}}^{(0)}\approx
e\tau_{TO}^{(0)}\cdot\left(\boldsymbol{v_{p}},\mathbf{E}\right)\frac{\partial
n_{F}\left(\xi_{p}\right)}{\partial\xi_{p}}$ (29)

$1^{st}$ iteration: We substitute (29) into (27):

$\displaystyle I_{TO}^{(1)}\\{\delta
n_{\boldsymbol{p}}^{(0)}\\}=\text{Integral}\\{\delta
n_{\boldsymbol{p}}^{(0)}\\}\equiv-\frac{\delta
n_{\boldsymbol{p}}^{(0)}}{\tau_{TO}^{(1)}(p)}$ (30)
$\displaystyle\longrightarrow\tau_{TO}^{(1)}(p)=-\frac{\delta
n_{\boldsymbol{p}}^{(0)}}{\text{Integral}\\{\delta
n_{\boldsymbol{p}}^{(0)}\\}}$ (31)

where the word “Integral” means that we numerically evaluate the integral in
(4) using $\delta n_{\boldsymbol{p}}^{(0)}$ defined with (29). Here, we know
$\delta n_{\boldsymbol{p}}^{(0)}$ as a function of $\boldsymbol{p}$ to within
the constant factor $\tau_{TO}^{(0)}$. The key part is that
$I_{TO}^{(1)}\\{\delta n_{\boldsymbol{p}}^{(0)}\\}$ is linear with respect to
$\delta n_{\boldsymbol{p}}^{(0)}$, which means that the value of
$\tau_{TO}^{(0)}$ cancels out from Eq.((30)). This allows us to extract
$\tau_{TO}^{(1)}$. Using it, we find modified distribution function

$\displaystyle\delta n_{\boldsymbol{p}}^{(1)}\approx
e\tau_{TO}^{(1)}(p)\cdot\left(\boldsymbol{v_{p}},\mathbf{E}\right)\frac{\partial
n_{F}\left(\xi_{p}\right)}{\partial\xi_{p}}$ (32)

$2^{nd}$ iteration: Similarly,

$\displaystyle I_{TO}^{(2)}\\{\delta
n_{\boldsymbol{p}}^{(1)}\\}=\text{Integral}\\{\delta
n_{\boldsymbol{p}}^{(1)}\\}\equiv-\frac{\delta
n_{\boldsymbol{p}}^{(1)}}{\tau_{TO}^{(2)}(p)}$ (33)

Here, we already know $\delta n_{\boldsymbol{p}}^{(1)}$, so no problems in
finding $\tau_{TO}^{(2)}$. And then

$\displaystyle\delta n_{\boldsymbol{p}}^{(2)}\approx
e\tau_{TO}^{(2)}(p)\cdot\left(\boldsymbol{v_{p}},\mathbf{E}\right)\frac{\partial
n_{F}\left(\xi_{p}\right)}{\partial\xi_{p}}$ (34)

$n^{th}$ iteration:

$\displaystyle I_{TO}^{(n)}\\{\delta
n_{\boldsymbol{p}}^{(n-1)}\\}=\text{Integral}\\{\delta
n_{\boldsymbol{p}}^{(n-1)}\\}\equiv-\frac{\delta
n_{\boldsymbol{p}}^{(n-1)}}{\tau_{TO}^{(n)}(p)}$ (35) $\displaystyle\delta
n_{\boldsymbol{p}}^{(n)}\approx
e\tau_{TO}^{(n)}(p)\cdot\left(\boldsymbol{v_{p}},\mathbf{E}\right)\frac{\partial
n_{F}\left(\xi_{p}\right)}{\partial\xi_{p}}$ (36)

(a)  The relaxation time $\frac{1}{\tau_{TO}^{(i)}}$ for $i=1,2,3$ iterations
for a fixed temperature $T=500$K

(b)  The relaxation times $\frac{1}{\tau_{TO}^{(i)}}$ for $i=1,3$ iterations
for the considered range of temperatures

Figure 4:  These plots represent the results of the implemented nethod of
successive approximations. $p_{BZ}=\hbar\pi/a$ stands for the first Brillouin
zone.

The convergence of the presented method can not be guaranteed. However, we are
working with a strictly positive function $\tau_{TO}(p)>0$; hence it has a
strictly positive average value which can be considered as the
$\tau_{TO}^{(0)}$. So, we can think of $\tau_{TO}(p)$ as
$\tau_{TO}(p)=\overline{\tau_{TO}}+\Delta\tau_{TO}(p)$ and if
$\Delta\tau_{TO}(p)$ is not much greater compared to $\overline{\tau_{TO}}$,
the method should converge.

Now, let us prove that if the numerical evaluation shows that the method
converges
$\big{(}\lim\limits_{n\to\infty}\tau_{TO}^{(n)}=\tau_{TO}^{(\infty)};$
$\quad\lim\limits_{n\to\infty}\delta n_{\boldsymbol{p}}^{(n)}=\delta
n_{\boldsymbol{p}}^{(\infty)}\big{)}$, we can guarantee that the obtained
result is the solution for (26) and (27). The reason for this is that if we
formally take a limit $n\to\infty$ in (35) and (36), we end up with:

$\displaystyle\delta n_{\boldsymbol{p}}^{(\infty)}\approx
e\tau_{TO}^{(\infty)}(p)\cdot\left(\boldsymbol{v_{p}},\mathbf{E}\right)\frac{\partial
n_{F}\left(\xi_{p}\right)}{\partial\xi_{p}}$ (37) $\displaystyle
I_{TO}\\{\delta n_{\boldsymbol{p}}^{(\infty)}\\}=\text{Integral}\\{\delta
n_{\boldsymbol{p}}^{(\infty)}\\}\equiv-\frac{\delta
n_{\boldsymbol{p}}^{(\infty)}}{\tau_{TO}^{(\infty)}}$ (38)

The above pair of equations is fully equivalent to Eqs. (26) and (27), which
concludes our proof.

###  II.2  Results for $\tau_{TO}(p)$ and the role of temperature gradient

The results of the implemented method are presented on the fig. 4. It is clear
that the method converges very decently, as the difference between the third
approximation and the second one is already quite small. Therefore, it is
sufficient to consider $\tau_{TO}^{(3)}(p)$ to be our final approximation for
the relaxation time.

As mentioned in (26), we neglected the temperature gradient in order the
method of successive approximations to work. Let us check whether this
approximation was appropriate. In order to do that we assume the absence of
the charge current $\mathbf{J}_{\mathbf{E}}=0$ which gives a correlation
between the electric field and the temperature gradient $\nabla
T=E/\mathcal{S}$. Here take the Seebeck coefficient to be $\mathcal{S}\approx
700$ $\mu$V/K. For $\xi_{p}\sim k_{B}T$ the term with the gradient in the
Boltzmann equation is already $\sim 10$ times less than the term with the
electric field. The fig. 5 shows the comparison between the relaxation times
computed with and without the temperature gradient.

Figure 5:  The results for the relaxation time with and without the
temperature gradient considered


# Adapting the Hypersphere Loss Function from Anomaly Detection to Anomaly
Segmentation

###### Abstract

We propose an incremental improvement to Fully Convolutional Data Description
(FCDD), an adaptation of the one-class classification approach from anomaly
detection to image anomaly segmentation (a.k.a. anomaly localization). We
analyze its original loss function and propose a substitute that better
resembles its predecessor, the Hypersphere Classifier (HSC). Both are compared
on the MVTec Anomaly Detection Dataset (MVTec-AD) – training images are
flawless objects/textures and the goal is to segment unseen defects – showing
that consistent improvement is achieved by better designing the pixel-wise
supervision.

Index Terms—  Anomaly Detection, Anomaly Segmentation, Loss Function, One-
class Classification

##  1 Introduction

Anomaly Detection (AD) aims to identify deviations or non-membership to a
class (a semantic concept of normality) by learning from normal instances and
none or very few examples of such irregularities. In the past few years, this
unsupervised problem has seen an increasing number of publications with novel
approaches and several literature reviews [1, 2].

Computer Vision in particular has received considerable attention thanks to
public datasets like the MVTec Anomaly Detection Dataset (MVTec-AD) [3], which
contains industrial images of objects/textures and defines an AD problem both
on the image level and on the pixel level. Image-wise AD consists of flagging
which images contain a defect, while pixel-wise AD, a.k.a. _anomaly
segmentation_ (or localization), consists of marking which pixels belong to a
defect (ground truth masks are available).

In this paper, we incrementally build on top of Fully Convolutional Data
Description (FCDD) [4], a model that outputs, by design, a heatmap (see Figure
1) from the last feature map of a convolutional neural network by applying a
Support Vector Data Description (SVDD)-inspired loss pixel-wise. We
reformulate the Hypersphere Classifier (HSC) loss function to the segmentation
task (initially designed for the image-wise setting), and Critical Difference
(CD) diagrams using the Wilcoxon signed rank test (a non-parametric version of
the paired T-test) demonstrate consistent improvement in performance with less
training epochs.

##  2 Related work

FCDD [4] is inspired by modifications of SVDD, a One-Class Classification
(OCC) model. In this section we summarize the predecessors that led to its
formulation, focusing on their incremental changes. For the sake of clarity,
some details have been omitted, such as regularization terms. An extensive
review of OCC models can be found in [5].

OCC was first formulated as an adaptation of _Support Vector Machines_ (SVMs)
that learns to separate anomalous data points from the origin with a
hyperplane [6] or a hypersphere [7] in the feature space. The latter, _SVDD_
[7], minimizes the radius of a hypersphere by incentivizing (via slack
variables) the normal/anomalous data points to fall inside/outside the
hypersphere. In other words, normal and anomalous instances are penalized in
opposite directions, therefore (respectively) pulling towards and pushing away
from the center – notice that this formulation assumes supervision.

One-Class Deep SVDD [8], a deep-learning model inspired by SVDD, applies a
neural network $\phi_{\omega}$ with parameters $\omega$ to normal instances
$x_{i}\in\mathbb{R}^{d}$, obtaining feature vectors
$\phi_{\omega}(x_{i})\in\mathbb{R}^{d_{F}}$, which are trained by minimizing

$\frac{1}{n}\sum_{i=1}^{n}\lVert\phi_{\omega}(x_{i})-a\rVert^{2}\quad,$ (1)

where $a\in\mathbb{R}^{d_{F}}$ is the center of the hypersphere in the feature
space and $n$ is the number of instances in the batch. In other words, the
mean squared Euclidean distance to the center $a$ is minimized, and this
distance is used as an anomaly score at inference time. Notice that this
version does not consider anomalous instances, which makes the training less
stable and (eventually) leads to feature collapse ($\phi_{\omega}$ converging
to a constant function).

Among several strategies proposed to prevent feature collapse, [9, 10]
proposed an extension of (1) that brings in back the possibility of using
supervision (therefore using normal and anomalous instances at training time)
by applying

$p(\cdot)=-\log(1-\exp(-\cdot))\;,$ (2)

to the anomalous instances’ scores, thus creating an incentive that pushes
their (learned) feature vectors away from the center (we call
$p:\mathbb{R}^{+}\to\mathbb{R}^{+}$ the push function). Besides the squared
distance $\lVert\cdot\rVert^{2}$ in Equation 1 was replaced by the Pseudo-
Hubert function $h(\cdot)=\sqrt{\lVert\cdot\rVert^{2}+1}-1\in\mathbb{R}^{+}$
as it is more robust to outliers, resulting in the Hypersphere Classifier
(HSC) loss function:

$\frac{1}{n}\sum_{i=1}^{n}(1-y_{i})\,h\left(\phi_{\omega}(x_{i})-a\right)+y_{i}\,p\left(h\left(\phi_{\omega}(x_{i})-a\right)\right)\,,$
(3)

where $y_{i}$ is $0$/$1$ if $x_{i}$ is normal/anomalous respectively.

Finally, FCDD, illustrated in Figure 1, reused the HSC to directly segment
anomalies by design. The first layers of a backbone convolutional neural
network (e.g. VGG) feed a feature map to an extra (learnable) 1x1 convolution
added on top of it. At inference time, a low-resolution anomaly score map, or
heatmap, is generated (from the pixel-wise vectors’ distances to the center of
the hypersphere) and upsampled to the input resolution – a more formal
formulation is presented in the next section.

Fig. 1:  Fully Convolutional Data Descriptor (FCDD). A pre-trained (and fine-
tuned) convolutional neural network without fully connected layers is combined
with an extra 1x1 convolution layer to generate locality-preserving feature
vectors. In the last (low resolution) layer, the distances to the
hypersphere’s center are upsampled to the network’s input resolution and used
as an anomaly score heatmap. Adapted from [4].

Numerous other approaches (unlike the OCC) exist and will not be covered here,
but a few major research branches are worth noting. Autoencoders
(reconstruction models more generally) have been extensively studied and
improved for anomaly detection tasks, and some variations still remain in the
state-of-the-art (SOTA) [11]. Combinations of deep learning-based feature
extraction with classic machine learning models like k-Nearest Neighbors [12,
13] and multivariate Gaussian [14, 15] also achieve effective results despite
their simplicity, and recent improved versions are also in the SOTA [16, 17].
However, nowadays most SOTA methods are based on pre-trained feature
extractors combined with normalizing flows [18].

Our work aims to contribute towards closing the performance gap between OCC
models and the current SOTA by finding a better loss function for [4],
explained in Section 3, and our results, in Section 5, demonstrate its
effectiveness.

##  3 Loss function

In this section we first present FCDD’s loss function (4), then propose our
version (5), and finally describe its rationale by comparing them.

Let $X\in\mathbb{R}^{n\times m}$ be a batch of $n$ images of $m$ pixels
(spatial dimensions confounded) and $Y\in\\{0,1\\}^{n\times m}$ its respective
batch of ground truth anomaly masks ($0$ means normal, $1$ means anomalous),
where $X_{ij}$ refers to the pixel indexed by $j\in\\{1,...,m\\}$ in the image
$X_{i}$, indexed by $i\in\\{1,...,n\\}$ ($Y_{ij}$ defined accordingly).

First, the neural network
$\phi_{\omega}:\mathbb{R}^{m}\to\mathbb{R}^{d_{F}\times m^{\prime}}$, with
parameters $\omega$, generates a feature map
$\phi_{\omega}\left(X_{i}\right)\in\mathbb{R}^{d_{F}\times m^{\prime}}$ with
low-resolution $m^{\prime}$ ($m^{\prime}<m$) and $d_{F}$ features for each
pixel. Then, a $1\times 1$ convolution with bias outputs a single value per
pixel – the following notation considers this convolution (and its parameters)
included in $\phi$, and the center $a$ has been omitted as it is implicitly
embedded in it. Finally, the Gaussian kernel-based upsampling operator outputs
the anomaly score map
$A^{\omega}_{i}=\texttt{upsample}(h(\phi_{\omega}(X_{i})))\in(\mathbb{R}^{+})^{m}$
(indices defined as for $X_{ij}$), where $h(\cdot)$ is the Pseudo-Hubert
function applied pixel-wise. [4] optimizes the parameters $\omega$ by
minimizing the loss function

$\frac{1}{n}\sum_{i=1}^{n}\left[\left(\frac{1}{m}\sum_{j=1}^{m}(1-Y_{ij})A^{\omega}_{ij}\right)+p\left(\frac{1}{m}\sum_{j=1}^{m}Y_{ij}A^{\omega}_{ij}\right)\right].$
(4)

We propose, as a substitute to Equation (4), the following loss function

$\frac{1}{nm}\sum_{i=1}^{n}\sum_{j=1}^{m}\;(1-Y_{ij})\;A^{\omega}_{ij}+\frac{|J^{0}|}{|J^{1}|}\;Y_{ij}\;p\left(A^{\omega}_{ij}\right)\;,$
(5)

where $J^{y}=\\{\,(i,j)\in\\{1,...,m\\}\times\\{1,...,n\\}\,|\,Y_{ij}=y\,\\}$
for $y\in\\{\,0,1\,\\}$ are, respectively, the sets of image and pixel indices
of all normal and anomalous pixels in a batch.

Equation (4) is the cross-image average ($\frac{1}{n}\sum_{i=1}^{n}$) of the
sum of two terms that look like per-image averages
($\frac{1}{m}\sum_{j=1}^{m}{}$) of anomaly scores on the normal pixels and on
the anomalous pixels (left and right respectively). In fact, $Y_{ij}$ zeroes
part of the iterands, so the $\frac{1}{m}\sum_{j=1}^{m}{}$ terms do not
converge to any expected value. By correcting that, one would have $p{}$
applied to the average score of the anomalous pixels (not each pixel
individually), which interferes with the backpropagation: (very) high-scored
pixels can compensate the others blocking the propagation through (very) low-
scored pixels. Therefore we propose to apply $p$ on each pixel individually.

With these modifications, one would achieve a nested average (cross-image
outside, cross-pixel inside). However, we propose to take the average directly
on all the pixels in the batch ($\frac{1}{nm}\sum_{i=1}^{n}\sum_{j=1}^{m}$)
because otherwise the nested averages would give more weight to smaller
anomalies. Finally, we scale the anomalous pixels’ scores by a factor
$\frac{|J^{0}|}{|J^{1}|}$, compensating the imbalance of normal/anomalous
pixels, which our tests have shown to be effective (details omitted for
brevity) – other (more modern and effective) techniques to compensate
imbalanced datasets exist but we deliberately do not focus on that subject.

##  4 Experiments

Fig. 2: Synthetic anomalies generated with confetti noise.

We aim to demonstrate that our proposed loss function improves from its
baseline, therefore we test it with the same procedure as in [4] (a few
differences specified below) on MVTec-AD– 10 categories of object (e.g.
transistor) and 5 categories of texture (e.g. leather), each category being
trained and tested independently.

Procedure: a VGG-11, pre-trained on ImageNet, frozen up to the third max-
pooling and cut before the forth max-pooling is used; the network is fine-
tuned using Stochastic Gradient Descent (SGD) with a batch size of 128 images,
Nesterov momentum of 0.9, weight decay of 10-4, and initial learning rate of
10-3 decreasing by a factor of 1.5% at each epoch.

One epoch is defined as 10 iterations over the training set with each image
having 50% chance to be replaced by an anomalous image (before the data
preprocessing/augmentation described below), which can be synthetic or real
depending on the experiment (never both in the same training). Synthetic
anomalies (unsupervised setting): automatically generated by superposing a
confetti noise – randomly sized, placed, and colored squares – over the normal
image, and the modified pixels are considered anomalous (Figure 2). Real
anomalies (semi-supervised): one (randomly chosen) image of each anomaly type
(between 1 and 6 types according to the MVTec-AD category) is transferred from
the test set to the training set. Notice that our focus is on the unsupervised
setting but the semi-supervised setting (a few anomalies available) is also
tested for reference.

Each experiment was repeated six times with different seeds, and the
performance is measured in terms of run-wise average of two metrics: pixel-
wise Area Under the Receiver Operator Characteristic curve (AUROC) and pixel-
wise Average Precision (AP). Differences with [4]: we run the gradient descent
for 50 epochs instead of 200, and the synthetic anomalies are not smoothed on
the borders.

The training images are pre-processed and augmented in the following sequence:
resize to 240 x 240; with 50% chance each, resize to 224 x 224 or crop to 224
x 224 in a random position; apply color jitter with all parameters set to 0.04
or all parameters set to 0.0005 with 50% chance each; apply Gaussian noise
with zero mean and standard deviation set to 10% of the image’s pixel values’
standard deviation (all channels confounded) with 50% chance on each pixel;
apply local contrast normalization (all channels confounded); normalize the
data between 0 and 1 (min-max normalization). The test images are resized to
224 x 224 then the last two last two steps from above are applied.

We ran our experiments on two cluster nodes PowerEdge C4130 (processor Xeon
E5-2680 V4, 7 cores, 2.4GHz), each with four GPUs Tesla P100-SXM2 (16 GiB),
and we used Weights & Biases to monitor our experiments. The minimum GPU
memory requirement is 5.6 GiB, and the (category-specific) average run time
spans from 33 minutes to 2 hours. We used the code publicly provided by [4]
with modifications to integrate the losses described in Section 3.

##  5 Results

(a) Pixel-wise AUROC

(b) Pixel-wise Average Precision

Fig. 3:  Critical difference diagrams from the category-specific scores on the
MVTec AD dataset using the Wilcoxon signed-rank test with the Bonferroni-Holm
correction with a confidence level of $\alpha=10\%$ at for each pair of
methods. Values on the scale are average rankings, and the methods within a
same horizontal bar (a clique) are not significantly different according to
the pair-wise statistical test. The scores of the methods marked with “*” are
from [4].

Tables 3 and 6 compare the cross-category average results obtained with the
baseline loss from [4] (Equation 4) and ours (Equation 5) in terms of,
respectively, AUROC and AP. The baseline results in terms of AUROC are both
copied from [4] (with “*”) and reproduced by us. Category-specific scores
available on GitHub:
gist.github.com/jpcbertoldo/8db759487279a5eec6fd0fc36764a4b4.

Figure 3 shows CD diagrams (with each metric) using the pairwise Wilcoxon
signed-rank test with Bonferroni-Holm correction with a (nominal) significance
level of $\alpha=10\%$ (probability of a type I error) considering the two-
sided alternative hypothesis.

The Wilcoxon signed-rank test’s null hypothesis is that two methods are
expected to rank similarly (they have the same performance) for any dataset,
and non-parametrically compares their cross-dataset performance differences –
notice that the categories in MVTec-AD can be considered as independent
datasets because they are trained and tested individually. In other words, it
provides a stronger statistical argument (than comparing average performances)
because rejecting its null hypothesis (i.e. methods that are not connected by
a horizontal red bar in Figure 3) means that a method is (in general)
significantly better than another regardless of the dataset (all other factors
fixed).

[tb] Pixel-wise Area Under the ROC curve (AUROC) scores (values in percentage)
on the MVTec AD dataset. Cross-category average scores and cross-category
average standard deviation (in parenthesis). Under the same supervision (group
of three columns), the best performance is in bold. The columns ”baseline*” is
from [4].  semi-supervised baseline* baseline ours objects 95.2* 95.7 (1.4)
95.8 (1.4) textures 97.0* 97.8 (0.4) 97.9 (0.4) all 95.8* 96.4 (1.0) 96.6
(1.0) unsupervised baseline* baseline ours objects 91.0* 92.5 (1.1) 92.5 (1.0)
textures 92.8* 92.8 (1.3) 94.1 (0.7) all 91.6* 92.6 (1.1) 93.1 (0.9)

##  6 Discussion

As shown in Tables 3 and 6 the our modifications provided, on average, an
increase in performance – categories metal nut and tile had the highest
improvements (see the link to the full per-category table in Section 5). The
average scores over object, texture, and all classes confounded show
improvement on both metrics and settings, while their average variance
decreased in the unsupervised setting (cf. standard deviations in
parenthesis).

Figure 3 confirms that indeed our loss function is ranked better than the
previous one, and, in particular, Figure 3.b shows that, in terms of AP, ours
is significantly better across different datasets.

Finally, we also observe that semi-supervised setting almost always performs
better than the unsupervised one even having very few anomalous images (1 up
to 7 seven depending on the class), so the effect of using proper anomalies in
the training set is a major factor for performance improvement.

[tb]  Pixel-wise Average Precision (AP) scores (values in percentage) on the
MVTec AD dataset. Cross-category average scores and cross-category average
standard deviation (in parenthesis). Under the same supervision (group of two
columns), the best performance is in bold.  semi-supervised unsupervised
baseline ours baseline ours objects 52.0 (5.8) 53.4 (5.8) 36.6 (4.3) 38.7
(2.3) textures 50.3 (4.5) 54.7 (4.1) 37.8 (2.8) 40.2 (2.0) all 51.4 (5.3) 53.8
(5.2) 37.0 (3.8) 39.2 (2.2)

##  7 Conclusion

This paper introduced an adaptation for the hypersphere loss functions for
anomaly segmentation on images. We have highlighted the drawbacks of the loss
function originally used in FCDD [4] (Equation 4) and proposed an alternative
loss function (Equation 5).

An evaluation of the statistical significance of the proposition is carried
out using pairwise Wilcoxon signed-rank tests, summarized in CD diagrams
(Figure 3). Our proposition reaches, on average, better performances (Tables 3
and 6) and is, in general, better ranked regardless of the image category in
MVTec-AD.

We highlight that the new loss introduced in this paper can be used with other
neural network architectures better designed for segmentation (e.g. U-Net and
its improved versions), which we expect to provide further higher
improvements.

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