# Anomalous Hall conductivity and quantum friction

Pavel Středa and Karel Výborný  Institute of Physics, Academy of Sciences of
the Czech Republic, Cukrovarnická 10, 16253 Praha 6, Czech Republic

(Jun08, 2022)

###### Abstract

Anomalous Hall effect in high conductivity regions is studied using a two-
dimensional network model. We find that the off-diagonal conductivity
comprises two parts: one which reflects the bulk properties as obtained by the
Kubo formula and another which is sensitive to boundary conditions imposed on
the network. In fully coherent limit the latter scales with the width of the
conducting channel, while for real-world samples it is controlled by the
coherence length. It provides an alternative interpretation of the observed
behavior in the clean limit which is otherwise attributed to the skew
scattering. We highlight analogies to friction in viscous fluids responsible
for Couette flow.

##  I Introduction

Scattering is an essential ingredient to many transport phenomena. The
anomalous Hall conductivity $\sigma_{AH}$ of ferromagnetic systems provides a
notable exception to this rule but only in certain region: most materials with
moderate longitudinal conductivity $\sigma_{0}$ show almost constant
$\sigma_{AH}$ as scattering strength and hence $\sigma_{0}$ is varied. This
property seems to be well understood in terms of Berry curvature of occupied
electronic bands representing properties of ideal Bloch systems as we discuss
below; also called the intrinsic region of the anomalous Hall effect (AHE),
this interval of roughly $10^{4}$ to $10^{6}$ inverse $\Omega.$cm is
surrounded by regions in which $\sigma_{AH}(\sigma_{0})$ becomes scattering
dependent as reviewed by Nagaosa et al.Nagaosa  (we offer a brief summary of
these different regimes in Sec. V). While suppression of $\sigma_{AH}$ for
stronger disorder is natural its linear increase with $\sigma_{0}$ in high
conductivity regimeMiyasato  is, at least, surprising. It is generally
accepted that it is caused by skew scattering,Smit ; Luttinger ; Onose ;
Sinitsin ; Onoda  asymmetric scattering of electrons on impurities induced by
their non-zero spin. Its effect increases with decreasing impurity
concentration. A seemingly inevitable consequence of this argumentation is
that anomalous Hall conductivity in clean systems is driven by negligible
impurity concentration while in this limit intrinsic values obtained for ideal
Bloch systems could be expected. With few notable exceptions, the effect of
Berry phases (see Eq. II below) is ignored in the context of skew scattering
and even if it is not Ishizuka:2017_a , the lack of generality as offered by
the Kubo formula for off-diagonal conductivity (see Appendix) leads to the
contradiction mentioned above. The main aim of the present treatment is to
suggest another possible origin of the observed increase of the Hall
conductivity with the sample purity which does not rely on skew scattering.

Basic conditionnote1  for the observation of anomalous Hall effect in magnetic
systems is the existence of non-zero orbital momentum which is responsible for
violation of the time reversal symmetry, necessary condition for non-zero Hall
effect. Transport properties are measured on stripes, Hall bar samples, and
orbital momentum of atomic-type wave functions causes the space current
density oscillates across the stripe. It can be represented by current paths
with alternating current directions.Jonckheere  Physically acceptable paths at
edges should be of the chiral type leading current along opposite directions
and the total net current thus vanishes in the equilibrium. Voltage drop
applied between stripe edges induces changes of the electron concentrations
within current paths. It leads to the polarization of the system which is a
typical property accompanying the anomalous Hall effect. Karplus ; Adams ;
Fivaz  Coupling between current paths is generally represented by their mutual
friction. It defines momentum transfer between edges as well as Hall current
through the stripe cross section. In quantum coherent systems, such friction
is controlled by the wave interference. The main aim of our approach to AHE is
to show that quantum friction between chiral current paths can be responsible
for a linear increase of the anomalous Hall conductivity with $\sigma_{0}$ in
the high conductivity region. It is an external contribution due to the finite
sample dimensions.

To verify this idea, a two-dimensional network modelStreda_15  will be used.
It allows to apply theory of quantum graphsgraphs  ideally suited for studies
of interference effects. It contains all basic ingredients necessary for
existence of the anomalous Hall effect. Apart from the non-zero orbital
momentum and spin-orbit coupling, it is the exchange interaction approximated
by Zeeman splitting which removes band degeneracy. Coupling between atomic
orbitals is defined by $S$-matrix which is convenient for application of the
scattering matrix approximation. Landauer57and70  Detailed model description
and its basic properties are presented in Section II. The subsequent Section
is devoted to the properties of edge states. It will be shown that chiral edge
states crossing energy gaps which are responsible for the quantum Hall effect
can be created or removed by tuning the boundary conditions. Contrary to the
case of external rational magnetic fields, chirality is not determined by wave
function properties at the Brillouin zone boundaries.Thouless  The key part of
our treatment is described in Section IV where scattering matrix approach is
applied to obtain intrinsic Hall conductivity and enhanced Hall current given
by friction between chiral current paths. In Section V two band model is used
to obtain qualitative features of anomalous Hall conductivity for a large
range of the system disorder covering all three regions. The article will be
completed by summary of main results and concluding remarks.

##  II TWO-DIMENSIONAL NETWORK MODEL

In strictly two-dimensional systems spin-orbit interaction separates electrons
into two independent groups having spin $s_{z}=1/2$ and $s_{z}=-1/2$,
respectively. Atomic state of the orbital number $m$ and spin $s_{z}$ has the
same energy as that with $-m$ and $-s_{z}$. Degeneracy of corresponding bands
is removed by exchange interaction which will be approximated by an effective
Zeeman splitting. To estimate general features of the anomalous conductivity
the simple two-dimensional network model sketched in Fig. 1 has been used
Streda_15 . It allows to employ theory of graphs graphs  for single-mode
quantum structure with $\delta$-type coupling between orbitals. Exner  Such
type of model graphs (e.g. Chalker-Coddington modelChalker ) has already been
applied to describe localization effect in quantum Hall systemsMarston:1999_a
and properties of quantum spin-Hall systems. Kobayashi

Let us briefly recapitulate main ideas and basic properties of the used model
Streda_15  on which our treatment is based. Scattering matrix for individual
contacts defines the transmission probability $|t|^{2}$ representing the
overlap integral entering the standard tight-binding approach. The spin
quantum number allows to distinguish energy bands and define anomalous Hall
conductivity for each of the spin subsystems. For the sake of simplicity, the
spin parts of wave functions will not be shown explicitly in the following
treatment. We keep in mind that a typical AHE setting will entail two copies
of the network with opposite spins and counter-propagating wavefunctions, i.e.
composed of orbitals with opposite angular momentum.

Atomic orbitals on individual lattice sites $\vec{R}_{i,j}$ are modeled by
rings of the radius $R$ formed by one-dimensional conductors. Each electron
subsystem (spin up and spin down) is represented by a one-way conductor. Their
eigenenergies and eigenfunctions

$E_{m}\,=\,\frac{\hbar^{2}m^{2}}{2m_{0}R^{2}}\quad,\quad\psi_{m}(\phi)\,=\,\frac{1}{\sqrt{2\pi
R}}\;e^{im\phi}\;,$ (1)

where $\phi\in(0,2\pi)$ is the polar angle are labeled by the orbital quantum
number $m=0,\pm 1,\ldots$ which defines angular momentum. The assumption that
electrons can orbit within rings in one direction only leads to a non-zero
orbital momentum, and consequently removes the time reversal symmetry which is
a necessary condition for the Hall effects to emerge.

In the square lattice shown in Fig. 1, each of the rings has four contact
points with its neighbors which separate the domain of the wave function
amplitude $A(\phi)\exp(i\delta\phi)$ into four sections listed in Tab. 1.
These allows to define four complex amplitudes $a,b,c,d$ per lattice site
fully determining the wave function for given $\delta$.

parameter | for $\phi\in$
---|---
$a_{i,j}$ | $(0,\pi/2)$
$b_{i,j}$ | $(\pi/2,\pi)$
$c_{i,j}$ | $(\pi,3\pi/2)$
$d_{i,j}$ | $(3\pi/2,2\pi)$
Table 1: Definition of $a,b,c,d$ for piece-wise constant $A(\phi)$ pertaining
to the ring at $\vec{R}_{i,j}$.

In analogy to Eq. 14 in Ref. Streda_15, , we now invoke the wave function (and
its derivative) continuity to determine the values of these parameters.
Considering the positive orbital momenta of the atomic-type orbitals,
$\delta>0$, the amplitudes are controlled by the following relations:

$\displaystyle\begin{array}[]{ccc}e^{-i\delta\pi}\,a_{i,j}&=&r\,e^{i\delta\pi}\,d_{i,j}+t\,b_{i+1,j}\;,\\\
b_{i,j}&=&r\,a_{i,j}+t\,e^{i\delta\pi}\,c_{i,j+1}\;,\\\
c_{i,j}&=&r\,b_{i,j}+t\,e^{i\delta\pi}\,d_{i-1,j}\;,\\\
d_{i,j}&=&r\,c_{i,j}+t\,e^{-i\delta\pi}\,a_{i,j-1}\;,\end{array}$ (6)

where $t$ denotes transition coefficient of the wave entering adjacent orbital
while $r$ represents part of the wave continuing the orbital motion. For the
considered $\delta$-type coupling they are of the following general form Exner

$t=\frac{i\alpha}{1-i\alpha}\quad,\quad
r=\frac{1}{1-i\alpha}\quad,\quad|r|^{2}+|t|^{2}=1\;,$ (7)

where $\alpha$ is a real parameter which is supposed to be an energy
independent constant for the sake of the simplicity.

Figure 1: Two-dimensional network model of coupled orbitals with positive
orbital momenta controlled by parameter $\delta_{\vec{k}}$. Arrows are
indicating the electron motion direction. Parameters $a,b,c,d$ are defined in
Tab. 1.

For infinite periodic network the wave functions are of the Bloch form

$\displaystyle|m,\vec{k}\rangle\equiv\Psi_{m,\vec{k}}(\vec{r})\,=\,\frac{e^{i\theta_{m}(\vec{k})}}{\sqrt{N}}\sum_{i,j=1}^{N}e^{i\vec{k}\vec{R}_{i,j}}\sqrt{\frac{1}{2\pi
R}}\,\times$ $\displaystyle\times
A_{m,\vec{k}}(\phi)\,e^{i\delta_{\vec{k}}\phi}\delta(|\vec{r}-\vec{R}_{i,j}|-R)\equiv
e^{i\vec{k}\vec{r}}\,u_{m,\vec{k}}(\vec{r})\;,$ (8)

where $m$ and $\vec{k}$ are the band number and the wave vector, respectively,
$\theta_{m}(\vec{k})$ denotes a Berry phaseXiao:2010_a  and
$u_{m,\vec{k}}(\vec{r})$ stands for the periodic part of Bloch functions. Wave
function amplitudes $A_{m,\vec{k}}(\phi)$ are subject to the following Bloch
conditions

$\displaystyle\begin{array}[]{ccc}c_{i,j+1}=e^{ik_{y}}\,c_{i,j}&,&\quad
a_{i,j-1}=e^{-ik_{y}}\,a_{i,j}\;,\\\ b_{i+1,j}=e^{ik_{x}}\,b_{i,j}&,&\quad
d_{i-1,j}=e^{-ik_{x}}\,d_{i,j}\end{array}$ (11)

where the wave vector components $k_{x,y}$ range from $-\pi$ to $\pi$ pursuant
to the choice of units (lattice constant $a_{0}=1$). Zero determinant of the
resulting equations for wave function amplitudes yields the spectral condition
for dimensionless parameter $\delta_{\vec{k}}$

$\cos k_{x}+\cos
k_{y}=-2\,\cos{\delta_{\vec{k}}\pi}-\frac{1-\alpha^{2}}{\alpha}\;\sin{\delta_{\vec{k}}\pi}\;,$
(12)

which can be transformed into dispersion relation for eigenenergies using

$E_{\vec{k}}\,=\,\frac{2\hbar^{2}\delta_{\vec{k}}^{2}}{m_{0}}$ (13)

in analogy to (1). This implicit expression for energy $E_{\vec{k}}$
corresponds to the kinetic energy of wave function (II).

For any energy-independent value of the parameter $\alpha$, the spectrum
comprises a series of non-overlapping bands linked to states having orbital
number $m$. Dispersions of the dimensionless parameter $\delta\sim\sqrt{E}$
are periodic with the period 2. Change of $\delta$ by one gives the same
dispersion but shifted in $\vec{k}$-space by $\Delta\vec{k}=(\pi,\pi)$. All
these features can be seen in spectrum obtained for a stripe samples shown in
Fig. 2. Energy gaps become closed for $|t|^{2}$ approaching the value
$|t|^{2}=0.5$ ($\alpha=\pm 1$) at which the band width $w$ equals to that
defined by $\Delta\delta=1$. Fixed-energy contours (see Fig. 2 in Ref.
Streda_15, ) are identical to the case of cosine band dispersion produced by
the well-known square-lattice tight-binding model but energy scaling (13) is
different.

In most real-world systems, the coupling between adjacent atomic orbitals is
weaker than that to the atomic core and crystal formation lowers energy of
electron states. For these reasons $|t|^{2}<0.5$ and $\alpha>0$ satisfying the
above requirements will be preferred in the following treatment. For
$|t|^{2}>0.5$, electrons ’orbit the stars’ rather than the circular orbitals
around $\vec{R}_{i,j}$, i.e. they are pushed into the interstitial positions.
In these cases the coupling gives rise to energy of electron states. For
$|t|^{2}=1-|r|^{2}=0,1$ the limit of isolated orbitals is achieved whereupon
the dispersions reduce to flat bands.

##  III EDGE STATES

Quantization of the anomalous Hall conductivity has also been observed on
systems endowed with non-zero orbital momentum.Berne ; Konig ; Chang:2013_a
Generally it is attributed to the existence of chiral edge states within gap
regions, i.e. states having opposite velocity at opposite sample edges.
Existence of such states has been first predicted for two dimensional systems
subjected to a strong external magnetic field. In this case there are two
scaling areas, the area per unit magnetic flux $A_{\phi}$ and the unit cell
area $A_{0}$. For rational values of $A_{0}/A_{\phi}$ eigenfuctions are of the
Bloch form but the corresponding translation symmetry differs from that at
zero magnetic field. As it has been shown by Thouless et al. Thouless  number
of chiral edge states, Chern number, is fully determined by eigenfunction
properties at the Brillouin zone boundary. External magnetic field induces
orbital momentum of atomic type states leading to an increase of the system
energy. Chiral edge states are induced to minimize it. For this reason they
are insensitive to the boundary conditions at the sample edges. Streda_1994
These general arguments are not applicable in the zero field limit. Using two
dimensional network model the decisive role of boundary conditions for
existence of chiral edge states will be shown.

A stripe open along the $\hat{x}$ direction parallel to main crystallographic
axis will be considered. Bloch conditions in the $\hat{x}$ direction,
$b_{i+1,j}=e^{ik_{x}}\,b_{i,j}$ and $d_{i-1,j}=e^{-ik_{x}}\,d_{i,j}$, inserted
into the basic equation set (6) give

$\displaystyle\begin{array}[]{ccc}-e^{-i\delta\pi}\,a_{i,j}+t\,e^{ik_{x}}\,b_{i,j}+r\,e^{i\delta\pi}\,d_{i,j}&=&0\;,\\\
r\,e^{-i\delta\pi}\,a_{i,j}-e^{-i\delta\pi}\,b_{i,j}+t\,c_{i,j+1}&=&0\;,\\\
r\,e^{-i\delta\pi}\,b_{i,j}-e^{-i\delta\pi}\,c_{i,j}+t\,e^{-ik_{x}}\,d_{i,j}&=&0\;,\\\
t\,a_{i,j-1}+r\,e^{i\delta\pi}\,c_{i,j}-e^{i\delta\pi}\,d_{i,j}&=&0\;.\end{array}$
(18)

For a given $k_{x}$ the eigenvalue problem reduces to the problem for a single
column of orbitals. It is independent of its position defined by the index
$i$. Two types of boundary conditions in the $y$ direction will be considered.
(i) Hard walls leaving circular orbitals unattached and (ii) those which cut
orbitals in half as shown in Fig. 1 by dashed lines. Electrons are thus
skimming or skipping along stripe walls.

Branch dispersions representing the case (i) for the column composed of $N=20$
circular orbitals controlled by the boundary conditions $b_{i,N}=a_{i,N}$ and
$d_{i,1}=c_{i,1}$ are shown in Fig. 2a. At any band energy the electron path
at the upper edge ($\cdots\rightarrow a_{i,N}\rightarrow b_{i,N}\rightarrow
a_{i-1,N}\rightarrow\cdots$) and that at the lower edge ($\cdots\rightarrow
c_{i,1}\rightarrow d_{i,1}\rightarrow c_{i+1,1}\rightarrow\cdots$) carry
electrons in opposite directions.

As for the case (ii), edge states crossing the energy gaps appear if hard
walls cut the orbitals in half on both sides of the sample. Branch dispersions
controlled by boundary conditions
$d_{i,N+1}=-c_{i+1,N+1}=-\exp(ik_{x})\,c_{i,N+1}$ and
$b_{i,1}=-a_{i-1,1}=-\exp(-ik_{x})\,a_{i,1}$ are shown in Fig. 2c. In real
space, electrons at edge current paths are skipping along stripe edges. They
are of chiral character (see also the inset of Fig. 5) and electrons flow in
opposite directions compared to the previous case of skimming electrons.

Figure 2: Energy branches for a stripe subjected to different boundary
conditions, $|t|^{2}=0.3$ and $\alpha>0$. Thick full and dashed lines
represent edge states at lower and upper strip edges, respectively.

There exists a peculiar possibility of imposing mixed boundary conditions
giving energy dispersions shown in Fig. 2b. In this case, the symmetry leading
to the presence of chiral edge states is lost and for chemical potential
within the gap region the current flow is allowed only along one direction. It
represents an ideal diode. Comparison with cases a) and c) suggests that edge
states are exclusively determined by boundary conditions. This conclusion is
supported by close connection of the anomalous Hall effect with the
polarization which is known to be affected by boundary conditions. Opposite to
the case of external magnetic fields (corresponding to rational values of
$A_{0}/A_{\phi}$), the appearance of chiral edge states is not determined by
the Chern number which in our case obeys zero value. Necessary conditions for
their appearance are relevant boundary conditions giving rise the chiral
symmetry of the current distribution across the stripe.

Note that for transition probability $|t|^{2}>0.5$ edge states crossing energy
gaps appear at edges for which hard wall leaves circular orbitals untouched
only. Nevertheless general conclusions remain unchanged.

##  IV ELECTRONIC TRANSPORT: SCATTERING MATRIX APPROACH

Corbino disc samples can be used to measure conductivity components directly,
at least in principle. In the limit of the infinite disc radius it is
equivalent to a stripe open in one direction (in our case $\hat{x}$)
coinciding with a main crystallographic axis. A voltage drop applied to the
opposite stripe edges induces current which has two components, perpendicular
and parallel with the edges representing longitudinal (i.e. along $\hat{y}$)
and Hall currents, respectively. Scattering matrix approach Landauer57and70
will be used to evaluate corresponding conductivities, $\sigma_{yy}$ and
$\sigma_{xy}$, for the already described two-dimensional network model. It
represents response to the electron concentration gradient of the fully
coherent system, i.e. no dissipation is allowed within stripe interior.
Dissipation is supposed to take place at the source and drain only where
electrons are subjected to the equilibration processes. Stripe width can thus
be identified with the equilibration length $\lambda_{e}$. Electron wave
functions are thereby also losing information about their phases and the
coherence length $\lambda_{c}$ thus coincides with with stripe width as well.

Stripe interior is composed of electron paths leading currents along positive
or negative $\hat{x}$ direction. To analyze conductivity contributions within
the stripe, it is natural to choose one of the electron paths as the source
and another as the drain, shown in Figures by red and blue lines,
respectively. Among the four possibilities sketched in Fig. 3 there are two
qualitatively distinct cases. Source and drain paths can be chosen to carry
current along the same direction (we choose to call it the Born-von Kármán
case) or their currents have opposite direction (chiral case). These two cases
will be treated separately in following Subsections and the Hall conductivity
of very clean but not fully coherent systems will be discussed in the last
Subsection.

Bloch conditions along $\hat{x}$ direction reduce problem to scattering within
the single column of orbitals for each of the wave numbers $k_{x}$. Wave
function amplitudes are defined by Eq. (18) accompanied by appropriate
boundary conditions. Averaging over $k_{x}$ gives relevant results. To get
smooth enough dependence on the parameter $\delta$ defining the energy a large
number of $k_{x}$ values has to be used. Usually $10^{4}-10^{5}$
$k_{x}$-values uniformly spread through the interval
$k_{x}\,\epsilon\,(-\pi,\pi)$ is considered. Results of the scattering matrix
approach do not depend on the column position (i.e. index $i$) and unless
necessary, the index will be skipped for brevity. For presented numerical
examples, unless explicitly stated, the model parameters $\alpha>0$ and
$|t|^{2}=0.3$ will be considered.

Figure 3: (color on line) Scheme of four possible boundary conditions applied
to a single column. Source and drain are marked in red and blue, respectively.

###  IV.1 Born-von Kármán cases

Let us first consider source and drain paths at which electron velocity along
$\hat{x}$ direction is negative, as sketched in Fig. 3 a). Electrons are
supposed to be injected into strip region via the lower path, $d_{1}=1$, while
they are absorbed by upper path and the condition $c_{N}=0$ insures zero
injection from this side. Equation set (18) together with these conditions
define uniquely all amplitudes within the column. Transition coefficient for
given $k_{x}$ is given by the amplitude $d_{N}(k_{x},\delta)$. Total
transition probability $T(\delta)$ defining current flow through a single
column along $\hat{y}$ direction reads

$T(\delta)\equiv\frac{h}{e^{2}}\,g_{0}(\delta)=\left\langle|d_{N}(k_{x},\delta)|^{2}\right\rangle_{k_{x}}\;,$
(19)

where $g_{0}(\delta)$ stands for conductance per single column. In this case
transitions between orbitals are independent on their position. Except for
fluctuations due to the size quantization it is independent on the considered
column length $(N-1/2)a_{0}$ as shown in Fig. 4. Considering an energy
independent relaxation time represented by the parameter $\gamma$, the Kubo
formula for longitudinal conductivity $\sigma_{0}(\delta)$ given by Eq. (A)
can formally be fit to approximate $g_{0}(\delta)$, as shown in Fig 4.

Figure 4: (color on line) Conductance $g_{0}(\delta)$ and anomalous Hall
conductivity $\sigma_{xy}(\delta)$ obtained by using scattering matrix
approach with Born-von Kármán type conditions. Magenta and green curves
correspond to the column lengths $N=15,\,25$ and $|t|^{2}=0.3$. Smooth thick
lines are Kubo formula results.

Current flow along $\hat{x}$ direction representing Hall current is not
uniformly spread through the sample cross-section indicating the decisive role
of wave function phases. Hall conductivity can be defined as follows

$\sigma_{AH}\equiv\sigma_{xy}(\delta)=\\!\frac{e^{2}}{h}\sum_{j=1}^{N}\\!\left\langle|d_{j}(k_{x},\delta)|^{2}\\!-|a_{j}(k_{x},\delta)|^{2}\right\rangle_{k_{x}}\\!\approx\sigma_{xy}^{\rm(int)}(\delta)\;,$
(20)

where the sum over $j$ to defines current into the right hand side column
through coupling points enhanced by the current within drain path defined by
setting $a_{N}(k_{x},\delta)=0$. Drain contribution decreases with rising
column length and currents through coupling points becomes dominant. Again,
except for the size quantization effect the obtained results are independent
on the column length and they are close to the intrinsic Hall conductivity
$\sigma_{xy}^{\rm(int)}(\delta)$ given by Eq. (A), as shown in Fig. 4. This
comparison entails no fitting procedure. Note that conductance $g_{0}(\delta)$
defines the current through the unit cell cross-section and corresponding
density is thus much larger than the Hall current density.

Another possibility is to choose source and drain paths leading electrons
along positive direction as shown in Fig. 3 b). In this case ($a_{1}=1$ and
$b_{N}=0$) the conductance $g_{0}(\delta)$ defined by probabilities
$|a_{N}(k_{x},\delta)|^{2}$ coincides with that determined in the previous
case. To obtain Hall conductivity instead of the drain path contribution the
current of the source path has to be added by setting $d_{1}(k_{x},\delta)=0$.
In this case equation (20) gives for energy gap regions quantum value
$-e^{2}/h$. This edge state effect has to be subtracted to obtain Hall
conductivity within the bulk. Resulting $\delta$-dependence then coincides
with that of the previous case.

###  IV.2 Chiral cases

Transport between source and drain electron paths which carry current in
opposite directions leads to qualitatively different results. Considering
columns of length $Na_{0}$, the relevant conditions are $d_{1}=1$ and
$b_{N}=0$ or $a_{1}=1$ and $c_{N}=0$, as sketched in Fig. 3 c) and d).

Except for fluctuations caused by size quantization, the transition
probability in both cases is once more independent of the column length.
Compared to the Born-von Kármán case, conductance $g_{0}$ per single column is
$\alpha$-times smaller. This ratio is only slightly $|t|$-dependent, for
example for $|t|^{2}=0.3$ and $|t|^{2}=0.2$ the ratio $\alpha$ equals to
$0.83$ and $0.87$, respectively.

Essential difference from the Born-von Kármán case is the dependence of the
Hall conductivity on the distance between source and drain $N_{s}a_{0}$ which
reads

$\sigma_{xy}^{\rm(\pm
Ch)}(\delta)=\langle\sigma_{xy}^{\rm(Ch)}(\delta)\rangle\pm
N_{s}\sigma_{xy}^{\rm(qf)}(\delta)\equiv\langle\sigma_{xy}^{\rm(Ch)}(\delta)\rangle\pm\Delta\sigma_{xy}^{\rm(Ch)}(\delta)\;,$
(21)

where

$\langle\sigma_{xy}^{\rm(Ch)}(\delta)\rangle\approx\alpha\,\sigma_{xy}^{\rm(int)}(\delta)\;,$
(22)

denotes the average value of both chiral cases, c) and d). Plus and minus sign
correspond to scattering problems with opposite chirality of current paths as
sketched in the inset of Fig. 5. It is determined by velocity sign of
electrons within the path attached to the source path. Contribution per unit
cell $\sigma_{xy}^{\rm(qf)}(\delta)$ represents average friction between the
nearest current paths. For large enough $N_{s}$ it reaches a constant value as
illustrated in Fig. 5.

The ratio $\alpha$ of the intrinsic part
$\langle\sigma_{xy}^{\rm(Ch)}(\delta)\rangle$ turns out to be the same as for
the longitudinal conductance. Its deviation of from one is the result of wave
interference modified by the change of the boundary conditions.

Hall current enhancement $\Delta\sigma_{xy}^{\rm(Ch)}(\delta)$ entering Eq.
(21) can be understood using the analogy with the classical current friction.
The largest current is flowing via the path attached to source. It is stirring
currents within adjacent stripe paths forcing them to move along the same
direction. It explains the origin of the enhanced current direction. In the
fully coherent systems the friction between current paths is determined not
only by average coupling $|t|^{2}$ but it is modified by the wave interference
which determines coupling between distant current paths.

Figure 5: (color on line) Anomalous Hall conductivity contributions
$\langle{\sigma}^{\rm(Ch)}_{xy}(\delta)\rangle/\alpha$ and
$\sigma_{xy}^{\rm(qf)}(\delta)/\alpha$ obtained by using scattering matrix
approach with chiral type boundary conditions. The two chiral scenarios are
sketched in the inset whereas the letters refer to situations shown in Fig.3.
Magenta and green curves correspond to $N=15,\,25$, scaling factor
$\alpha=0.83$ is discussed in the text and $|t|^{2}=0.3$. Smooth thick line is
the Kubo formula result.

###  IV.3 Anomalous Hall effect in high conductivity region

Let us first discuss Hall conductivity within a coherent area of infinite
systems. It has been found that the conductance $g_{0}(\delta)$ between paths
leading currents in opposite directions (chiral cases c) and d)) is smaller
than that for which they are flowing along the same direction (Born-von
K’a]rmán cases a) and b)). It would lead to different gradients of the
corresponding chemical potentials. To prevent this difference the occupation
of neighbor paths leading currents in opposite directions has to be
appropriately modified to unify both current densities. This polarization
accompanying anomalous Hall effect allows electron transfer without enforced
dissipation due to the current differences. It also insures that intrinsic
parts of the Hall conductivity is the same in both cases. Resulting Hall
conductivity is given by average value of all four contributions discussed in
previous two Subsections. They are of the same probability to appear and
consequently sum of friction contributions depending on the column length is
averaged out. Resulting anomalous Hall conductivity approaches intrinsic
values given by the Kubo formula.

Different results are obtained for stripe samples shown in Fig. 1. They are
composed of columns containing integer number of unit cells. In these cases
the edge electron paths carry current in opposite directions. Let us assume
that attached source and drain prepared from the same material are coherently
coupled to the stripe electron system which corresponds to scattering problems
shown in Fig. 3 c) and d). In the fully coherent case the Hall current
enhancement defined by Eq. (21) is proportional to the stripe width
$N_{s}a_{0}$ which for large enough $N_{s}$ dominates.

Because of equilibration processes within source and drain, electrons are
losing all information about their past. The stripe width $N_{s}a_{0}$ can be
thus identified with the equilibration length $\lambda_{e}$. Longitudinal
conductivity $\sigma_{0}\equiv\sigma_{yy}$ can be approximated by the
conductance per square area $N_{s}^{2}a_{0}^{2}$, $\sigma_{0}=N_{s}g_{0}$, and
the Hall conductivity enhancement defined by Eq. (21) increases with
$\sigma_{0}$ as observed in the high conductivity region.

Classical analogy of this effect is Couette flow observed in fluids placed
between two plates. Motion of one plate induces fluid flow along the same
direction which in the stationary case decreases linearly towards fixed one.
In our case the role of the moving plate is played by the current path
attached to the electron source.

Dissipation processes are minimizing deviation from the equilibrium. They are
thus trying to suppress enhanced Hall current by electron transitions into
paths leading current in opposite direction. For strong enough dissipation it
can be thus expected that quantum friction contributions will be averaged out
giving rise to intrinsic values of the Hall conductivity. Since current
enhancement originates in wave interference even low angle inelastic
scattering can be quite effective. It can be expected that corresponding
relaxation time $\tau_{qf}$ could be much smaller than
$\tau_{e}\propto\lambda_{e}$ which controls the longitudinal conductivity. For
the measured anomalous Hall conductivity the Eq. (21) can be used replacing
$a_{0}N_{s}$ by the length $\lambda_{qf}\propto\tau_{qf}$

$\sigma_{xy}^{\rm(\pm
Ch)}(\delta)\approx\sigma_{xy}^{\rm(int)}(\delta)\pm\sigma_{xy}^{\rm(qf)}(\delta)\,\frac{\lambda_{qf}}{a_{0}}\;,$
(23)

where plus and minus sign correspond to boundary conditions for which
electrons are skimming or skipping along strip edges, respectively. Note that
for $|t|^{2}>0.5$ the Hall conductivity has opposite sign but its general
features remain unchanged.

Estimation of the measured Hall conductivity given by Eq. (23) has to be
viewed as a rough approximation based on the assumption that the enhanced Hall
current is uniformly spread across the stripe. For quite large stripe width
$w$ it might not be true. If enhanced Hall currents become concentrated within
a slab of the width $w_{qf}$ at the edge vicinity the measured $\sigma_{xy}$
becomes proportional to $w_{qf}/w$. This problem desires more advanced
theoretical description.

##  V Two band model

As already mentioned in the introduction,Nagaosa  three regions of the AHE can
be identified across a wide range of systems: the
$\sigma_{AH}\propto\sigma_{0}^{\mu}$ scaling (with $\mu\approx 1.6$) in the
dirty-metal regime has received considerable attention already and the
intrinsic AHE calculations for intermediate $\sigma_{0}$ (see Fig. 12 in Ref.
Onoda, ) are a popular topic for ab initio studies. While there is abundance
of experimental data for systems falling into these two categories, very clean
samples are scarce and typical measurements of AHE in clean
systemsOmori:2019_a  concern conductivity up to $\sigma_{0}\approx 10^{5}\
\Omega$cm. A notable result for $\sigma_{0}$ in excess of $10^{6}\ \Omega$cm
has been publishedXu  about ultra-clean nickel thin layers.

To illustrate dependence of the measured anomalous Hall conductivity on
disorder strength covering all three regions the overlap of energy bands has
to be taken into account. For the considered two dimensional network model two
bands having opposite orbital momentum as well as spin orientation will only
be considered for simplicity. Corresponding intrinsic Hall conductivities have
opposite sign but their absolute values are supposed to be the same as shown
in the inset of the Fig. 6. Their shift due to exchange interaction is
approximated by a Zeeman splitting to obtain a nonzero Hall conductivity given
by the sum of both band contributions

$\bar{\sigma}_{xy}(\delta_{\mu})=\left\langle\sigma_{xy}^{\downarrow}(\delta_{\mu})\right\rangle_{\rm
av}+\left\langle\sigma_{xy}^{\uparrow}(\delta_{\mu})\right\rangle_{\rm av}\;.$
(24)

Effect of the disorder will be approximated by a potential energy
fluctuations. Assuming their Gauss distribution the ensemble averaging reads

$\left\langle\sigma_{xy}(\delta_{\mu})\right\rangle_{\rm
av}=\frac{1}{\gamma\sqrt{2\pi}}\int
e^{\frac{(\delta-\delta_{\mu})^{2}}{2\gamma^{2}}}\sigma_{xy}(\delta)\,d\delta\;,$
(25)

where the dimensionless parameter $\gamma\sim\hbar/\tau_{e}$ in units of the
unperturbed band width, Eq. (35), is assumed to be energy independent. Sum of
both Hall conductivities decreases with increasing band overlap caused by the
band broadening and for considered $|t|^{2}=0.3$
$\sigma_{xy}\propto\gamma^{-1.75}$ as shown in Fig. 6. Unperturbed band
separation and Fermi level position are sketched in the inset. Within
intrinsic region the effect of the band broadening vanishes.

Figure 6: Anomalous Hall conductivity for two band model as function of the
disorder strength represented by the parameter $\gamma$ ($|t|^{2}=0.3$). Width
of the unperturbed band is taken as its unit. Inset shows $\delta$-dependence
of the intrinsic $\sigma_{xy}$ of both bands.

Sum of quantum friction contributions of both bands, Eq. (23), multiplied by
$\gamma$ has been taken as a fitting parameter. Linear dependence of
$\lambda_{qf}\propto\tau_{qf}$ on a relaxation time $\tau_{e}\sim 1/\gamma$
representing longitudinal conductivity has been considered through the whole
disorder range. It has been chosen to obtain the experimentally observed range
of the intrinsic region covering approximately two orders of
$\sigma_{xx}\propto 1/\gamma$ as presented in Fig. 6. This assumption is too
simple to illustrate effect of quantum friction precisely. Like in the
classical Couette flow the friction desires some time to evolve. If it is much
larger than the relaxation time $\tau_{e}$ no effect can be expected. For this
reason the transition between intrinsic and high conductivity regions should
be sharper.

Under conditions for which electrons are skimming along strip edges the
friction contribution enhances Hall conductivity. If they are skipping the
friction contribution has opposite sign and Hall conductivity decreases as
shown in Fig. 6 by the dashed line. With further decrease of the dissipation
it even changes its sign as has already been observed for example on Co films.
Miyasato  Similar behavior can also be obtained if electrons are skimming
along edges, but one of the bands is determined by $|t|^{2}>0.5$. Possibility
that the coherence length of minority electrons within the upper band is
larger than that within lower band cannot also be excluded as origin of this
effect.

Despite of the model simplicity it gives qualitative features of scaling
relations between anomalous Hall conductivity and longitudinal one.

##  VI Summary and concluding remarks

Existence of the quantum friction in fully coherent systems is the main
message of our treatment. The basic condition of its appearance is chirality
of edge current paths within stripe samples. This effect persists even in not
fully coherent systems for which the resulting Hall current enhancement is
proportional to the quantum friction length $\lambda_{qf}$. This external
contribution to the measured anomalous Hall conductivity due to the finite
sample dimensions dominates in high conductivity regime. It represents a
quantum analogue of the classical Couette flow in fluids. It can be expected
that the observed spin Hall effect Omori:2019_a  is of the same origin.

Question arises if the enhanced Hall current is uniformly spread through the
sample cross-section. It might be possible that it is located within a
narrower slab close to a stripe edge. In this case anomalous Hall conductivity
should depend on the strip width. This possibility is partly supported by
experiments on massive Fe crystals which do not show increase in high
conductivity regime contrary to Fe-layers. Miyasato

Presented view to the origin of the Hall current enhancement suggests that all
scattering events affects all conductivity components but with different
efficiency. Electrons flowing along the voltage drop are subjected to
dissipative processes which can be characterized by a relaxation time
$\tau_{e}\propto\lambda_{e}$ determining longitudinal conductivity. On the
other side the enhanced Hall current decreases with disrupting the effective
wave interference responsible for the coupling between current paths.
Corresponding relaxation time $\tau_{qf}\propto\lambda_{qf}$ can thus be
substantially different from $\tau_{e}$. Temperature dependent inelastic
scattering of electrons by phonons and magnons is destroying phase coherence
but its effect to $\tau_{e}$ is weaker. This is consistent with existence of
the intrinsic Hall conductivity plateau. Also an increase of the diffusion
scattering of electrons at sample surfaces gives rise to the much larger
suppression of $\tau_{qf}$ than $\tau_{e}$ as observed on high conductivity Ni
samples. Xu  Ratio of the electrical and thermal conductivity components has
been studied for pure Fe and Si or Co doped samples. Shiomi  In the limiting
case of vanishing temperature where residual resistivity dominates the
validity of the Wiedemann-Franz law has been confirmed for ratios of diagonal
as well off-diagonal components. It indicates that elastic scattering affect
all components by the similar way and the much shorter $\tau_{qf}$ is
proportional to $\tau_{e}$. On the other side the more complicated relation
between both times can be expected for inelastic scattering.

Unfortunately, there are not enough experimental data for full understanding
of the electron transport in the high conductivity regions. Further detail
investigations of scattering effects in this regime is thus desired.
Especially the dependence of the anomalous Hall conductivity on the stripe
width can help to understand the evolution of the quantum friction effect in
real systems.

Direction of the Hall current enhancement is controlled by the orientation of
the current paths just touching strip edges which is determined by the
boundary conditions. At least in cases for which suppression of the anomalous
Hall effect is observed the analysis of the current distribution at the sample
edges is desirable to verify origin of this effect. It would be ideal to have
possibility to vary boundary conditions. If orbitals of magnetic impurities
periodically distributed within non-magnetic host lattice are of the radius
larger than the distance between atoms it might be possible at least in
principle. In these cases stripe edges can cut orbitals in half forcing
electrons to skip or left them untouched. Systems like Bi2Te3 family of
topological insulators with univalent 3d magnetic ions Chang:2013_a  seem to
be good candidates. In these systems skipping electrons are giving rise to
chiral edge states crossing energy gap responsible for the observed quantum
Hall effect. Creation of such systems with mixed boundary conditions might
lead to new types of spinotronic devices, diodes, allowing current flow along
one direction only. In this case periodic distribution of ions is not
necessary condition. Although this sounds as science fiction we believe that
technological progress will allow to realize such systems in the future.

###### Acknowledgements.

Authors thank V Drchal and V. Špička for valuable comments, Y. Niimi for
advice regarding experimental data, J. Kučera for technical support and head
of the department J. J. Mareš for ensuring ideal working conditions for partly
retired author P. Středa. We acknowledge financial support from 22-21974S
(granted by the Czech Science Foundation).

##  Appendix A KUBO FORMULA RESULTS

Quantum theory of the linear response to the electric field at zero
temperature gives for diagonal conductivity components the well known Kubo-
Greenwood formula Greenwood

$\sigma_{ii}(\mu)\,=\,\pi e^{2}\hbar\left\langle\,{\rm
Tr}\left\\{v_{i}\delta(\mu-H)v_{i}\delta(\mu-H)\right\\}\,\right\rangle_{\rm
av}\;,$ (26)

and for off-diagonal components the following expression derived by Bastin et
al Bastin

$\displaystyle\sigma_{ij}(\mu)=i\hbar
e^{2}\times\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$
$\displaystyle\times\\!\\!\\!\int\limits_{-\infty}^{\mu}\\!\\!\left\langle\\!{\rm
Tr}\left\\{\delta(\eta-H)\left[v_{i}\frac{dG^{+}}{d\eta}v_{j}-v_{j}\frac{dG^{-}}{d\eta}v_{i}\right]\right\\}\\!\right\rangle_{\rm
av}\\!\\!d\eta\,,\;$ (27)

where $H$ denotes a single-electron Hamiltonian, $v_{i}$ are components of the
velocity operator and delta-function operator is defined as

$\displaystyle\delta(\eta-H)$ $\displaystyle=$
$\displaystyle-\lim_{\epsilon\rightarrow
0^{+}}\,\frac{G^{+}(\eta)-G^{-}(\eta)}{2\pi i}\;\;,$ $\displaystyle
G^{\pm}(\eta)$ $\displaystyle=$ $\displaystyle\frac{1}{\eta-H\pm
i\epsilon}\,.$ (28)

For crystals with substitutional impurities the ensemble averaging
$\langle\cdots\rangle_{\rm av}$ represents averaging over impurity
configuration. Generally it is a complicated problem Levin  which can be
simplified by neglecting vertex corrections allowing to replace averaged
product of resolvents $G(z)$ by product of their averaged operators

$\langle G(z)\rangle_{\rm av}\equiv\frac{1}{z-H_{\rm eff}(z)}\;,$ (29)

where $z$ is the complex energy variable. It has the full crystal symmetry
independently on the character of the scattering events, asymmetric scattering
is not an exception. Note that this property has been ignored in description
of skew scattering effects. Effective Hamiltonian $H_{\rm eff}(z)$ is non-
Hermitian and energy dependent but it is analytic in both complex half-planes,
$H_{\rm eff}(z^{\ast})\,=\,H_{\rm eff}^{+}(z)$. Its standard form reads

$H_{\rm
eff}(z)=H_{0}+\Sigma(z)\quad,\quad\Sigma(z)\,=\,\Delta(z)-i\Gamma(z)\;,$ (30)

where $H_{0}=\langle H\rangle_{\rm av}$ represents virtual crystal and
$\Sigma(z)$ is the energy dependent self-energy determined by the coherent
potential approach CPA , as the best known theory to estimate effect of
alloying.

Inverse value of its imaginary part represents a finite electron life-time
$\tau$. Note, that matrix elements of $\Sigma(z)$ are diagonal in
representation given by eigenfunctions of the Hamiltonian $H_{0}$. Using this
representation and neglecting $\Gamma(\eta)$ entering one of the
$\delta$-operators in Eq. (26) we get

$\sigma_{ii}(\mu)=e^{2}\hbar\sum_{n,\vec{k}}\frac{\left|\langle
n,\vec{k}|v_{i}|n,\vec{k}\rangle\right|^{2}}{\Gamma_{n,\vec{k}}(\mu)}\,\delta(E^{\prime}_{n,\vec{k}}(\mu)-\mu)\;,$
(31)

where
$E^{\prime}_{n,\vec{k}}(\eta)=E^{\rm(0)}_{n,\vec{k}}+\Delta_{n,\vec{k}}(\eta)$,
$n$ and $\vec{k}$ denotes band number and wave vector, respectively. This
expression coincides with the solution of the Boltzmann equation for
longitudinal conductivity.

Neglecting vertex corrections in Eq. (A) defining Hall conductivity, using
equality $dG(\eta)/d\eta=-G^{2}(\eta)$ and since velocity matrix elements are
diagonal in $\vec{k}$ we get

$\displaystyle\sigma_{ij}(\mu)=\frac{e^{2}\hbar}{\pi}\sum_{n,n^{\prime}}^{n\neq
n^{\prime}}\sum_{\vec{k}}\int\limits_{-\infty}^{\mu}\frac{\Gamma_{n,\vec{k}}(\eta)}{[\eta-E^{\prime}_{n,\vec{k}}(\eta)]^{2}+\Gamma^{2}_{n,\vec{k}}(\eta)}\times$
$\displaystyle\times 2\,{\rm Im}\,\left\\{\frac{\langle
n,\vec{k}|v_{i}|n^{\prime},\vec{k}\rangle\langle
n^{\prime},\vec{k}|v_{j}|n,\vec{k}\rangle}{\left[\eta-E^{\prime}_{n^{\prime},\vec{k}}(\eta)+i\Gamma_{n^{\prime},\vec{k}}(\eta)\right]^{2}}\right\\}\,d\eta\;.\;\;\;\;\;$
(32)

With decreasing impurity concentration $\Gamma$ decreases as well and the
dominant contributions are those for which $\eta$-values are close to values
of $E^{\prime}_{n,\vec{k}}(\eta)$. If there is no band overlap the energy
difference $\eta-E^{\prime}_{n^{\prime},\vec{k}}(\eta)\approx
E^{\prime}_{n,\vec{k}}(\eta)-E^{\prime}_{n^{\prime},\vec{k}}(\eta)$ dominates
the denominator value and $\Gamma_{n^{\prime},\vec{k}}(\eta)$ can be neglected
if it is much smaller than the energy difference. This approach thus excludes
significant effect of the decreasing impurity concentration to the Hall
conductivity. It can be expected that vertex corrections are not able to
change this conclusion quantitatively since they are vanishing with decreasing
concentration as well. In the pure crystal limit Eq. (A) gives finite value
even in the case of the band overlap.Yao:2004_a

Evaluation of the anomalous Hall conductivity for the considered ideal network
model ($\Gamma\rightarrow 0$) is straightforward since the energy spectrum is
for given spin subsystem composed of non-overlapping bands and we have

$\displaystyle\sigma_{xy}(\mu)=e^{2}\hbar\sum_{m,m^{\prime}}^{m\neq
m^{\prime}}\sum_{\vec{k}}f_{0}\left(E_{m,\vec{k}}-\mu\right)\,\times$
$\displaystyle\times 2{\rm Im}\left\\{\frac{\langle
m,\vec{k}|v_{x}|m^{\prime},\vec{k}\rangle\langle
m^{\prime},\vec{k}|v_{y}|m,\vec{k}\rangle}{\left[E_{m,\vec{k}}-E_{m^{\prime},\vec{k}}\right]^{2}}\right\\}\;,$
(33)

where $f_{0}(E-\mu)$ denotes Fermi-Dirac distribution. Eigenenergies
$E_{m,\vec{k}}$ are functions of the dimensionless $\delta_{\vec{k}}$ defined
by Eq. (13) and velocity operator does not include spin-orbit term because of
the one dimensional character of electron orbitals,
$\vec{v}=-i\hbar\vec{\nabla}_{\vec{r}}/m_{0}$. Contributions for
$m-m^{\prime}=\pm 2$ vanish because periodicity of wave function amplitudes.
Dominant contribution originates in elements with $m-m^{\prime}=\pm 1$.

Figure 7: Intrinsic anomalous Hall conductivity as function of the
dimensionless parameter $\delta_{\mu}\sim\sqrt{\mu}$ for several values of
$|t|^{2}$ ($\alpha>0$).

Assuming anti-clockwise motion of electrons on circular orbitals ($\delta>0$)
the obtained anomalous Hall conductivities are shown in Fig. 7 for several
values of the transition probability $|t|^{2}$. Note that their dependence on
dimensionless parameter $\delta_{\mu}\sim\mu$ is the same for all bands.
Increase of $|t|^{2}$ above 0.5 changes sign of the orbital momentum since
orbiting of electrons around interstitial positions becomes dominant. Their
average radius is smaller than that for circular orbitals leading to smaller
value of the orbital momentum. Except of the sign change the lower values of
the Hall conductivity can thus be expected.

For opposite direction of the orbital motion, $\delta\rightarrow-\delta$,
representing subsystem of the opposite spin orientation the anomalous Hall
conductivity changes its sign. Resulting Hall conductivity is given by the sum
of both subsystem conductivities and its non-zero value can thus only appear
if the spin band degeneracy is removed.

To get longitudinal conductivity the simplest approach reducing effect of the
disorder to an energy dependent imaginary part $\Gamma(\mu)$ of the self
energy will be used

$\displaystyle\sigma_{0}(\mu)=e^{2}\hbar\sum_{\vec{k}}\delta(E_{\vec{k}}-\mu)\,\frac{\Gamma(\mu)\,|v_{x}(\vec{k})|^{2}}{(E_{\vec{k}}-\mu)^{2}+\Gamma^{2}(\mu)}=$
$\displaystyle=\frac{e^{2}}{h}\,\frac{1}{2\pi\gamma(\mu)}\oint\limits_{F.S.}\left|\frac{d\delta_{\vec{k}}}{dk_{x}}\right|^{2}\,\frac{dS_{\vec{k}}}{\sqrt{\left|\frac{d\delta_{\vec{k}}}{dk_{x}}\right|^{2}+\left|\frac{d\delta_{\vec{k}}}{dk_{y}}\right|^{2}}}\;,$
(34)

where dimensionless parameter $\gamma(\mu)$ relates to $\Gamma(\mu)$ as
follows

$\Gamma(\mu)\equiv\frac{4\hbar^{2}\delta_{\mu}}{m_{0}}\;\gamma(\mu)\;.$ (35)

It has good physical meaning if its value is compared with the band width
represented by the range of available $\delta$ values.

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