# A comparison of a commercial hydrodynamics TCAD solver and Fermi kinetics
transport convergence for GaN HEMTs

Ashwin Tunga<EMAIL_ADDRESS>Holonyak Micro and Nanotechnology
Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA
Kexin Li  Holonyak Micro and Nanotechnology Laboratory, University of Illinois
at Urbana-Champaign, Urbana, IL, 61801, USA  Nicholas C. Miller  Air Force
Research Laboratory Sensors Directorate, 2241 Avionics Cir., Wright-Patterson
AFB, OH, 45433, USA  Matt Grupen  Air Force Research Laboratory Sensors
Directorate, 2241 Avionics Cir., Wright-Patterson AFB, OH, 45433, USA  John D.
Albrecht  Department of Electrical and Computer Engineering, Michigan State
University, 428 S. Shaw Lane, East Lansing, MI, 48824, USA  Shaloo Rakheja
Holonyak Micro and Nanotechnology Laboratory, University of Illinois at
Urbana-Champaign, Urbana, IL, 61801, USA<EMAIL_ADDRESS>
###### Abstract

Various simulations of a GaN HEMT are used to study the behaviors of two
different energy-transport models: the Fermi kinetics transport model and a
hydrodynamics transport model as it is implemented in the device simulator
Sentaurus from Synopsys. The electron transport and heat flow equations of the
respective solvers are described in detail. The differences in the description
of electron flux and the discretization methods are highlighted. Next, the
transport models are applied to the same simulated device structure using
identical meshes, boundary conditions, and material parameters. Static
simulations show the numerical convergence of Fermi kinetics to be
consistently quadratic or faster, whereas the hydrodynamic model is often sub-
quadratic. Further comparisons of large signal transient simulations reveal
the hydrodynamic model produces certain anomalous electron ensemble behaviors
within the transistor structure. The fundamentally different electron dynamics
produced by the two models suggest an underlying cause for their different
numerical convergence characteristics.

††preprint: JAP/Sentaurus-FKT

##  I  Introduction

Recent decades have seen increased interest in GaN devices for power and radio
frequency (RF) electronics.Hoo Teo  _et al._ (2021); Baliga (2013) Numerical
simulation is a necessary step for the design and analysis of these devices.
Robust simulation models rely on producing predictive results. Accurate
simulation of these devices must capture the important physical processes
involved and the carrier transport mechanisms. While it is possible to obtain
reasonable results using ad hoc solutions despite incorrectly modeling the
physics, these results are not predictive and do not aid in device design.
Stettler, Alam, and Lundstrom (1993); Bude (2000); Grasser, Kosina, and
Selberherr (2001) Furthermore, these numerical methods may be problematic for
large-signal transient simulations, which are particularly important for RF
power applications. This paper seeks to explore some numerical behaviors of
established charge transport solvers for RF GaN HEMTs.

Charge transport in a semiconductor is described by the Boltzmann transport
equation (BTE). The BTE is an integro-differential equation that describes the
evolution of the non-equilibrium distribution function of electrons under the
influence of an external electric field. The BTE is given as Hess (1988)

$\begin{split}&\frac{\partial f(\bm{k},\bm{r},t)}{\partial
t}+\bm{v}\cdot\nabla_{r}f(\bm{k},\bm{r},t)+\frac{\bm{F}}{\hbar}\cdot\nabla_{k}f(\bm{k},\bm{r},t)\\\
&=\left(\frac{\partial f(\bm{k},\bm{r},t)}{\partial t}\right)_{\text{coll
}},\end{split}$ (1)

where $f$ is the distribution function, $\bm{F}$ is the external force,
$\bm{v}$ is the group velocity, $\bm{k}$ is the wavevector in the reciprocal
space, $\bm{r}$ is position vector in the real space, $t$ is the time, and the
term on the right-hand side represents the rate of change of $f$ due to the
collisions between the carriers.

Monte Carlo solvers are a useful statistical method to solve the BTE. The
Monte Carlo technique generally employed is the ensemble Monte Carlo (EMC)
method. Fischetti and Laux (1988) The accurate results produced by this method
come at the cost of immense computational load. To reduce the computational
load encountered by the EMC, the hybrid cellular automaton/Monte Carlo (CA/MC)
algorithm was developed. The CA/MC scheme attempts to speed-up the simulation
by pre-calculating the scattering rates and storing them in a table. It was
shown that the CA/MC method required about 0.2-0.5 CPU-sec per iteration, up
to 25 times faster than the traditional EMC. Saraniti and Goodnick (2000)
However, this speed improvement offered by the hybrid scheme is still not
comparable to the shorter simulation time taken by deterministic solvers.

Deterministic solvers are an alternative approach that does not rely on random
sampling to solve the BTE. While the deterministic methods are generally much
faster than the Monte Carlo methods, they rely on physical approximations to
the BTE to obtain the solutions. Selberherr (1984) Deterministic solvers
convert the BTE into an equivalent infinite series of differential equations
in phase space by applying the method of moments over the reciprocal space.
Truncating the series after a finite number of moments typically results in
one solution variable more than the available equations. Therefore, a closure
relation is required to make the system of equations tractable. Grasser  _et
al._ (2003)

Scharfetter and Gummel Scharfetter and Gummel (1969) developed one of the
first deterministic Boltzmann solvers. They proposed a robust discretization
of the drift-diffusion (DD) model with physical approximations. However, as
the device dimensions were scaled, the DD model failed to account for the
large electric fields and the hot-carrier effects that arise, e.g., in FET
channels. These effects are typically accounted for in the DD model by
empirically modifying the mobility. Penzin  _et al._ (2017)

The DD model was extended by Stratton Stratton (1962a) and Blotekjaer
Blotekjaer (1970) to include a balance equation for the average energy of the
carriers. These models are generally referred to as energy transport models,
and the hydrodynamic model is a prime example. Grasser  _et al._ (2003)
Hydrodynamic models typically use the parabolic band approximation to describe
the kinetic energy vs momentum dispersion relation. This approximation fails
to hold when the electric fields in the device reach large magnitudes. To
account for this, the model relies on empirical field-dependent and
temperature-dependent mobility models. The commonly used closure relation in
hydrodynamic models is the description of heat flow using Fourier’s law and an
approximate thermal conductivity.

Synopsys Sentaurus is an industry-leading suite of semiconductor device and
process simulation tools. Sentaurus has been widely used to simulate devices
for various applications including power electronics, Strauss  _et al._
(2014); Kraus and Castellazzi (2016) RF electronics, Ajayan  _et al._ (2018)
and CMOS technology. Zheng  _et al._ (2015); Wu and Jhan (2017) Sentaurus
provides a wide range of simulation packages to handle the vast application
space and support a streamlined simulation workflow. The range of packages
available includes Sentaurus Process for process simulation, Sentaurus Device
Monte Carlo, Sentaurus Device Electromagnetic Wave Solver, and, the package
under study in this work, Sentaurus Device (SDevice). Sen (2020) The history
of SDevice can be traced back to the development of DESSIS in 1992. DESSIS was
developed by merging HFIELDS from the University of Bologna, Baccarani (1985)
SIMUL from ETH Zurich, Litsios, Müller, and Fichtner (1993) BONSIM from Bosch,
and in collaboration with ST Microelectronics. Baccarani  _et al._ (1996) In
1993, Integrated Systems Engineering (ISE) was founded and assumed
responsibility for the simulator. ISE merged with Synopsys in 2004, and DESSIS
became the basis of SDevice. Sarkar (2018) SDevice supports several carrier
transport models that can be used depending on the device under investigation.
The hydrodynamic model, implemented in SDevice, will be investigated in this
work.

FKT is also an energy transport based BTE solver that employs the method of
moments but uses an alternative treatment of the electronic heat flow. It
substitutes the idea of thermal conductivity with the heat capacity of an
ideal Fermi gas to be used as the closure relation. Grupen (2009) It has been
demonstrated that FKT, unlike the hydrodynamic model, could incorporate the
electronic band structure Grupen (2011) and full-wave EM. Grupen (2014) FKT
has also demonstrated large-signal RF simulations Miller, Albrecht, and Grupen
(2016a) and defect dynamics simulationsGrupen (2019). More details on the
solver are reported in the literature. Miller, Albrecht, and Grupen (2016b);
Miller (2017); Miller  _et al._ (2018)

This paper examines the physical applicability of both the FKT and the
hydrodynamic model employed by Sentaurus, hereafter referred to as commercial
hydrodynamic transport (CHT) solver. First, the system of equations and the
discretization scheme employed by the solvers are presented in Section II.
Section III presents the GaN HEMT structure and the simulation parameters used
in the respective simulators. Next, the thermal equilibrium simulations are
performed and the results are compared. The output characteristics and the
electron temperatures of the HEMT structure, produced by the non-equilibrium
simulations, are compared. Further, the differences in the simulation
frameworks are investigated by analyzing the numerical convergence behavior
after varying the simulation setup, to ensure similar simulation conditions
for the purpose of comparison. Finally, the effect of varying the polarization
sheet charge density at the AlGaN/GaN interface on the convergence
characteristics and the transient simulation behavior of the simulators are
investigated.

##  II Computational Background

The semiconductor device simulators considered here are both of the energy
transport variety. As such, they have a great deal in common. For example,
they both tessellate a problem domain with a suitable mesh comprised of
triangles for 2D simulations or tetrahedra for 3D simulations. This mesh
conforms to the simulated device’s geometry and is refined appropriately in
regions where solution variables change rapidly, such as material interfaces
and metallurgical junctions. Solution variables assigned to each mesh point
include an electrostatic potential to account for voltage drops and the
corresponding electric fields throughout the device. To each mesh point is
also assigned a Fermi-Dirac distribution for an electron gas occupying the
semiconductor’s conduction band to account for currents flowing through the
device under the influence of the internal fields. Integrals of the
distribution function over the density of conduction band states can provide
the particle and energy densities of mobile electrons throughout the device.
These densities can be treated as additional solution variables at each mesh
point, or they can be equivalently associated with alternative solution
variables such as the Fermi distribution’s chemical potential (Fermi level)
and temperature.

Both device simulators calculate the solution variables by solving Poisson’s
equation, the electron continuity equation, and the conservation of electron
energy at each mesh point using the box-integration method Selberherr (1984).
The electrostatic potentials are approximated as piecewise linear along the
edges connecting neighboring mesh points, and the electron fluxes are
approximated as piecewise constant between mesh points. The electron particle
and kinetic energy fluxes can be derived from the 1st and 3rd moments of the
BTE, respectively Hess (1988). Both simulators represent these fluxes in a
discrete form on the device mesh using the powerful Scharfetter-Gummel method
Selberherr (1984); Scharfetter and Gummel (1969). Because the mobile electrons
are treated as ideal Fermi gases, both simulators also account for the
electronic heat that flows between neighboring electron distributions.

Although the simulators use these common methodologies, there are some
differences in how they are implemented that lead to significant differences
in the simulation results and numerical convergence characteristics. Principal
differences include the precise forms of the electron fluxes obtained from
moments of the BTE, the way Scharfetter-Gummel discretization is applied to
these fluxes, and the way electronic heat flow is treated. The following
details these differences along with the associated simulation results and
numerical behaviors. For simplicity, effective mass and constant mobility
approximations will be used.

###  II.1 CHT Simulator

For the sake of simplicity, we will limit ourselves to the consideration of
only electrons in steady-state conditions. Additionally, the lattice
temperature, $T_{L}$ is set to 300 K to disentangle the effects of electron
heating and lattice heating, resolve convergence issues, and enable a fair
comparison of CHT against FKT. The electron current density, modeled using
CHT, is given as Sen (2020)

$\vec{J}_{n}=\mu_{n}\left[n\nabla E_{C}+kT_{n}\nabla
n-nkT_{n}\nabla\ln\gamma_{n}+\lambda_{n}f_{n}^{\mathrm{td}}kn\nabla
T_{n}\right].$ (2)

The first term in the above equation accounts for the spatial variation of
electrostatic potential, while the remainder terms account for the gradient of
concentration and the carrier temperature gradient. $\mu_{n}$ is the electron
mobility, $n$ is the electron concentration, and $T_{n}$ denotes the
temperature of electrons. The thermal diffusion constant $f_{n}^{\mathrm{td}}$
is set equal to zero, which corresponds to the Stratton model. Stratton
(1962b) The parameters $\gamma_{n}$ and $\lambda_{n}$ account for the Fermi-
Dirac statistics and are given as

$\displaystyle\gamma_{n}=\frac{n}{N_{C}}\exp(-\eta_{n}),$ (3)
$\displaystyle\lambda_{n}=\frac{\mathcal{F}_{1/2}(\eta_{n})}{\mathcal{F}_{-1/2}(\eta_{n})}.$
(4)

$\mathcal{F}_{j}(x)$ is the Blakemore Fermi-Dirac integral of order-$j$. The
argument $\eta_{n}$ is the reduced Fermi energy and is defined as
$\eta_{n}=(F_{n}-E_{C})/kT_{n}$. The effective conduction band density of
states in the semiconductor is specified by $N_{C}$. CHT provides an option,
in lieu of $T_{n}$, to use a blended temperature in (2) according to
$T_{n}\rightarrow gT_{n}+(1-g)T_{L}$, where $T_{L}$ is the lattice temperature
and $g$ is a dimensionless parameter between zero and unity. For all CHT
simulations reported in this paper, $g$ is set equal to unity.

The energy balance equation is given as

$\frac{\partial W_{n}}{\partial
t}+\vec{\nabla}\cdot\vec{S}_{n}=\vec{J}_{n}\cdot\vec{\nabla}\frac{E_{C}}{q}+\frac{dW_{n}}{dt}|_{\mathrm{coll.}},$
(5)

where $W_{n}$ is the electron energy density, and $\vec{S}_{n}$ is the energy
flux. $W_{n}$ comprises a thermal and a convective term; however, CHT
implements $W_{n}$ without the convective term. That is,
$W_{n}=nw_{n}=n\left(3kT_{n}/2\right)$. The term
$\vec{J}_{n}\cdot\vec{\nabla}\frac{E_{C}}{q}$ represents the energy delivered
to the carriers by the field per unit time, while
$\frac{dW_{n}}{dt}|_{\mathrm{coll.}}$ is the net energy gained from the
lattice via phonon interactions. $\vec{S}_{n}$ is given as

$\vec{S}_{n}=-\frac{5r_{n}\lambda_{n}}{2}\left[\frac{kT_{n}}{q}\vec{J}_{n}+f_{n}^{\mathrm{hf}}\hat{\kappa}_{n}\nabla
T_{n}\right],$ (6)

and

$\hat{\kappa}_{n}=\frac{k^{2}}{q}n\mu_{n}T_{n}.$ (7)

Here, we use the default parameter values for $r_{n}$ and
$f_{n}^{\mathrm{hf}}$ as listed in Table 1. These parameters allow us to
change the relative contributions of the convective and diffusive terms in the
energy flux. With the default parameters, the prefactor for the diffusive term
becomes

$\kappa_{n}=\frac{3}{2}\frac{k^{2}\lambda_{n}}{q}n\mu_{n}T_{n}.$ (8)

The collision term in the energy balance is given as

$\frac{dW_{n}}{dt}_{\mathrm{coll.}}=-\zeta_{n}\frac{W_{n}-W_{n0}}{\tau_{e}}.$
(9)

$W_{n0}$ is the equilibrium electron energy density, and $\tau_{e}$ is the
_energy relaxation time_ for electrons. The parameter $\zeta_{n}$ is adjusted
to improve the stability of the numerical algorithm and is given as

$\zeta_{n}=1+\frac{n_{\mathrm{min}}}{n}\left[\frac{n_{\mathrm{0}}}{n_{\mathrm{min}}}\right]^{\mathrm{max}[0,(T_{L}-T_{n})/100\text{
K}]},$ (10)

with $n_{\mathrm{min}}$ and $n_{0}$ are adjustable density parameters to aid
convergence. Since $T_{n}\geq T_{L}$ for our device, $n_{0}$ does not impact
$\zeta_{n}$. The default value of $n_{\mathrm{min}}$ is set to $10^{3}$ cm-3,
which leads to $\zeta_{n}\approx 1$ for all test cases reported here. While
$\tau_{e}$ is energy-dependent and would lead to a modification of the
diffusive energy flux, here we treat it as an energy-independent parameter.
However, CHT provides empirical methods, such as spline approximation of
energy relaxation time over energy, to account for $\tau_{e}$’s energy
dependence. In this study, we vary $\tau_{e}$ in the fs to ps range to analyze
its impact on I-V curves and electron heating. Additionally, we elucidate that
$\tau_{e}$’s impact on I-V convergence in CHT. Detailed results are discussed
in Sec. III.

To discretize the electron fluxes, CHT uses the ingenious Scharfettel-Gummel
method Selberherr (1984); Scharfetter and Gummel (1969). In its original form,
this method assumed the electric field $\mathcal{E}=(E_{C,1}-E_{C,0})/(qL)$
and the electron flux $J_{n}$ were constant on a mesh edge of length $L$
connecting mesh points 0 and 1. With these assumptions, the phenomenological
drift-diffusion equation was treated as a first-order differential equation
for the electron density $n$. Solving this differential equation for
variations in $n$ along the mesh edge then produced a discrete form for the
electron flux, given in (11), that proved to be numerically robust.

$J_{n}=\frac{\mu_{n}}{\ell_{ij}}kT_{n}\left(n_{i}B(u_{i}-u_{j})-n_{j}B(u_{j}-u_{i})\right),$
(11)

where $i$ and $j$ are the major mesh points, $\mu_{n}$ is the electron
mobility (assumed constant between the mesh points), $\ell_{ij}$ is the length
of the mesh edge, $B(x)=x/(e^{x}-1)$ is the Bernoulli function, and
$u_{i}-u_{j}=q\mathcal{E}\ell_{ij}/kT_{n}$, and $\mathcal{E}$ is constant
between the mesh points. For high electron densities, CHT may also add some
additional terms to (11) to account for Fermi-Dirac degeneracy.

Table 1:  Parameters in CHT. Parameter | Default Value |  Meaning
---|---|---
$r_{n}$ | 0.6 |  Parameter in energy flux
$f_{n}^{hf}$ | 1 |  Parameter in energy flux
$g$ | 1 |  Parameter to select carrier temperature $(T_{n}\rightarrow gT_{n}+(1-gT_{L}))$
$f_{n}^{td}$ | 0 |  Thermal diffusion coefficient
$n_{\mathrm{min}}$ |  $10^{3}$ cm-3 |  Parameter to improve numeric stability

###  II.2 Fermi Kinetics Transport

Because FKT, like CHT, assumes an energy-dependent electron distribution
function at points in the simulation domain, electron fluxes are computed from
moments of the BTE. This method of moments is a powerful technique for
inferring information about the part of the distribution function odd in
momentum-space, such as net particle flux, from the even part of the
distribution function. For example, drift-diffusion flux, which is often
presented phenomenologically, can be derived from the first moment of the BTE
assuming a Maxwell-Boltzmann distribution function, constant electron
temperature, and constant mobility Hess (1988). In general, FKT uses electron
fluxes derived for a Fermi-Dirac distribution, non-constant temperature, non-
parabolic bands, and an energy-dependent mobility computed from electron
scattering rates Grupen (2011). To simplify comparisons with CHT, the electron
current density from the first moment assuming parabolic bands and constant
mobility is given by

$\displaystyle\vec{J}_{n}$ $\displaystyle=$
$\displaystyle\mu_{n}\frac{(2m_{n})^{3/2}}{3\pi^{2}\hbar^{3}}\left[(kT_{n})^{3/2}F_{3/2}^{\prime}(\eta_{n})\nabla
E_{C}+\right.$ (12) $\displaystyle\mbox{\hskip
54.2025pt}\left.\nabla(kT_{n})^{5/2}F_{3/2}(\eta_{n})\right],$

where $F_{j}(\eta_{n})$ is the integral over energy $E$ of the electron Fermi
function times $E^{j}$, and $F_{j}^{\prime}(\eta_{n})$ is its derivative with
respect to $\eta_{n}$. This integral is proportional to the Fermi integrals in
(4) $\mathcal{F}_{j}(\eta_{n})=F_{j}(\eta_{n})/\Gamma(j+1)$. The electron
internal/kinetic energy flux is likewise derived from the third moment of the
BTE,

$\displaystyle\vec{S}_{n}^{\rm kin}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{n}}{q}\frac{(2m_{n})^{3/2}}{3\pi^{2}\hbar^{3}}\left[(kT_{n})^{5/2}F_{5/2}^{\prime}(\eta_{n})\nabla
E_{C}+\right.$ (13) $\displaystyle\mbox{\hskip
63.23624pt}\left.\nabla(kT_{n})^{7/2}F_{7/2}(\eta_{n})\right].$

Like CHT, FKT uses the Scharfetter-Gummel discretization, but it treats (12)
as a first order differential equation for the quantity
$(kT_{n})^{5/2}F_{3/2}$. Solving this differential equation produces a
discretized electron current density on the mesh edge given by

$\displaystyle J_{n}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{n}}{L}\frac{(2m_{n})^{3/2}}{3\pi^{2}\hbar^{3}}\left[B(\xi_{n})(kT_{0})^{5/2}F_{3/2}(\eta_{n,0})-\right.$
(14) $\displaystyle\left.\mbox{\hskip
67.75311pt}B(-\xi_{n})(kT_{1})^{5/2}F_{3/2}(\eta_{n,1})\right]$
$\displaystyle=$ $\displaystyle-q\left(J_{0\rightarrow 1}-J_{1\rightarrow
0}\right)$ $\xi_{n}=\frac{\Delta(E_{C})}{\Delta(F_{n}-E_{C})}\left[\ln
F_{3/2}(\eta_{n,1})-\ln F_{3/2}(\eta_{n,0})\right],$ (15)

where the Bernoulli function $B(\xi_{n})=\xi_{n}/[\exp(\xi_{n})-1]$, and
$\Delta(x)\equiv x_{1}-x_{0}$. The same procedure is applied to (13) to obtain
the Scharfetter-Gummel discretization of the electron kinetic energy flux

$\displaystyle S_{n}^{\rm kin}$ $\displaystyle=$
$\displaystyle\frac{\mu_{n}}{qL}\frac{(2m_{n})^{3/2}}{3\pi^{2}\hbar^{3}}\left[B(\xi_{E})(kT_{0})^{7/2}F_{5/2}(\eta_{n,0})-\right.$
(16) $\displaystyle\left.\mbox{\hskip
63.23624pt}B(-\xi_{E})(kT_{1})^{7/2}F_{5/2}(\eta_{n,1})\right]$
$\displaystyle=$ $\displaystyle S_{0\rightarrow 1}^{\rm kin}-S_{1\rightarrow
0}^{\rm kin}$ $\xi_{E}=\frac{\Delta(E_{C})}{\Delta(F_{n}-E_{C})}\left[\ln
F_{5/2}(\eta_{n,1})-\ln F_{5/2}(\eta_{n,0})\right].$ (17)

Although FKT particle and kinetic energy fluxes as well as the flux
discretization schemes differ somewhat from the corresponding quantities in
CHT, perhaps the most significant departure is the treatment of electronic
heat flow. It is evident from (6) that CHT treats electronic heat flow with
Fourier’s law using an electronic thermal conductivity related to the
electrical conductivity. FKT, on the other hand, exploits the kinetics of
ideal Fermi gases Grupen (2009). Because electron ensembles at points inside
the simulation are assumed to have an energy-dependent distribution, i.e.,
independent of the direction of degenerate electron momentum vectors $\bm{k}$,
the electron ensembles are symmetric in $\bm{k}$-space. This is the defining
feature of an ideal gas. When simulated electron fluxes move along a mesh edge
from an initial mesh point with Fermi level $(F_{n}-E_{C})_{0}$ and
temperature $T_{n,0}$ to a final mesh point with Fermi level
$(F_{n}-E_{C})_{1}$ and temperature $T_{n,1}$, the electrons must thermalize
in order to change their temperature from $T_{n,0}$ to $T_{n,1}$. This
thermalization process is achieved through electron-electron scattering, and
it is the source of electronic heat flow between the electron ensembles/gases.
The amount of heat $H$ needed to change the temperature of an ideal Fermi gas
from initial temperature $T_{i}$ to final temperature $T_{f}$ is precisely
defined by the thermodynamic identity Kittel and Kroemer (1980)

$H=W_{n}(T_{f})-W_{n}(T_{i})-(F_{n}-E_{C})[n(T_{f})-n(T_{i})].$ (18)

Determining the net electronic heat flux on the mesh edge requires the rate
electrons change from $T_{n,0}$ to $T_{n,1}$ and the rate electrons change
from $T_{n,1}$ to $T_{n,0}$. These rates are simply $J_{0\rightarrow 1}$ and
$J_{1\rightarrow 0}$ from (14), respectively. Therefore, the electronic heat
flux flowing from point 0 to point 1 is given by

$\displaystyle S_{0\rightarrow 1}^{\rm heat}$ $\displaystyle=$ $\displaystyle
S_{0\rightarrow 1}^{\rm kin}(T_{n,0})-S_{0\rightarrow 1}^{\rm kin}(T_{n,1})-$
(19) $\displaystyle(F_{n}-E_{C})_{0}\left[J_{0\rightarrow
1}(T_{n,0})-J_{0\rightarrow 1}(T_{n,1})\right],$

the electronic heat flux flowing from point 1 to point 0 is given by

$\displaystyle S_{1\rightarrow 0}^{\rm heat}$ $\displaystyle=$ $\displaystyle
S_{1\rightarrow 0}^{\rm kin}(T_{n,1})-S_{1\rightarrow 0}^{\rm kin}(T_{n,0})-$
(20) $\displaystyle(F_{n}-E_{C})_{1}\left[J_{1\rightarrow
0}(T_{n,1})-J_{1\rightarrow 0}(T_{n,0})\right],$

and the net electronic heat flux along the edge is $S_{n}^{\rm
heat}=S_{0\rightarrow 1}^{\rm heat}-S_{1\rightarrow 0}^{\rm heat}$. Combining
this heat flow with the kinetic energy flux gives the total electron energy
flux along the edge connecting mesh points 0 and 1, $S_{n}=S_{n}^{\rm
kin}+S_{n}^{\rm heat}$.

FKT uses the same Joule heat CHT uses in (5). FKT typically computes the
collision operator that determines the rate hot electrons transfer energy to
the semiconductor lattice by integrating the phonon scattering rates over
electron energy. However, for the sake of the comparisons with CHT presented
here, FKT was modified to use (9).

##  III Comparisons of Simulation Results

The GaN HEMT structure simulated in FKT and CHT is illustrated in Fig. 1. The
n+ GaN source/drain contacts are doped with $10^{19}$ cm-3 donors and complete
dopant ionization is assumed. The GaN region is semi-insulating containing
$9\times 10^{15}$ cm-3 fully ionized acceptor atoms compensated with $10^{16}$
cm-3 donor-like traps located at an energy level 0.6 eV below $E_{C}$ and
capture cross-section of $10^{-19}$ m2. The Schottky barrier height at the
gate contact boundary condition is assumed to be 1.5 eV, while the source and
drain contacts are ohmic boundary conditions. For both simulators, AlGaN and
SiC are treated as insulators, which means that carrier transport equations
are not solved in these regions. All the simulations for both the simulators
were run on the same mesh, shown in Fig. 2.

Figure 1: Cross-section of a GaN device simulated in FKT and CHT. The 2DEG exists in GaN at the interface between AlGaN (barrier) and GaN (channel). Figure is not drawn to scale. Figure 2: Mesh structure of the GaN HEMT device, shown in Fig. 1, used for the simulations in both CHT and FKT. The mesh is refined near the 2DEG in the channel, and near the material interfaces, to obtain accurate results. Table 2:  Material parameters that are common in both CHT and FKT. Material |  Dielectric constant |  Electron Affinity
(eV) |  Bandgap
(eV) |  Electron mobility
$(\mathrm{cm}^{2}/\mathrm{V\,s})$ |  Effective mass
($\mathrm{m}_{0}$)111$\mathrm{m}_{0}$ is the free electron mass
---|---|---|---|---|---
AlGaN |  8.9 |  3.55 |  4.14 |  — |  —
GaN |  8.9 |  4.1 |  3.457 |  440 |  0.2
SiC |  9.66 |  — |  — |  — |  —

###  III.1 Thermal Equilibrium

The thermal equilibrium simulation does not depend on the transport model
used. Therefore, one would expect identical equilibrium results from the two
simulators, given the mesh, the material and simulation parameters for the GaN
HEMT are kept alike. Keeping the aforementioned conditions identical, we
calibrate the two simulators and verify that they produce similar results.
Table 2 shows the material parameters that are kept uniform. Figures 3, 4, 5
show the z-direction band diagram, z-direction electron density and the
x-direction electron density plots, respectively. As expected, excellent
agreement between the simulators is obtained.

Figure 3: Comparison of the equilibrium band diagram produced by CHT and FKT
along the z-direction (vertical) at $\mathrm{x}=0.9$ $\mu\textrm{m}$.
$\mathrm{E_{C}}$ denotes the conduction band edge and $\mathrm{E_{F}}$ refers
to the Fermi level at thermal equilibrium. Figure 4: Comparison of the
equilibrium electron density produced by CHT and FKT along the z-direction
(vertical) at $\mathrm{x}=0.9$ $\mu\textrm{m}$. Figure 5: Comparison of the
equilibrium electron density produced by CHT and FKT along the x-direction
(channel) at $6$ nm below the AlGaN/GaN interface.

###  III.2 Non-Equilibrium Simulations

Figure 6: Drain current $I_{D}$ versus drain voltage $V_{DS}$ for different
gate voltages $V_{GS}$ simulated by CHT and FKT with no electron heating, i.e.
drift-diffusion.

First, a drift-diffusion simulation was run using both FKT and CHT. Once
again, the material parameters used for the simulations and shown in Table 2
are kept identical. The voltage step size is fixed at 0.1 V for both the
simulators. As seen in Fig. 6, both the simulators give nearly identical drain
current. The minor differences in the obtained result can be attributed to the
difference in the representation of the electron flux in CHT, (Eq. 2), and the
differences in the Scharfetter-Gummel discretization in the respective
simulators.

Figure 7: Drain current $\mathrm{I_{D}}$ versus drain voltage
$\mathrm{V_{DS}}$ for different gate voltages $\mathrm{V_{GS}}$ simulated by
CHT and FKT including hot electrons, $\tau_{e}=0.2$ ps.

Next, an energy transport simulation was run using the two simulators. The
parabolic band approximation was used to model the dispersion relation with an
electron effective mass of $0.2{m_{0}}$ ($m_{0}$ is the free electron mass).
The energy relaxation time ($\tau_{e}$) is $0.2$ ps. Ye, Wicks, and Fauchet
(1999); Zhang (2014) The CHT-specific hydrodynamic simulation parameters used
are shown in Table 1. For the FKT simulation, the voltage step size is kept
fixed and unchanged at 0.1 V. On the other hand, it was observed that an
adaptive step size was required for the convergence of the CHT simulation. The
maximum allowed voltage step size is set to 0.1 V, and the minimum allowed
(initial) step size is lowered to 2.5 mV.

The $I_{D}-V_{DS}$ characteristic and the electron temperature profile in the
channel produced by the simulators is shown in Fig. 7 and Fig. 8,
respectively. It would seem unexpected that despite the higher electron
temperature obtained by CHT, the drain current produced is also marginally
higher than that produced by FKT. This can be explained by the fact that both
the simulators are using a constant mobility. As a result, the scattering
effects associated with high electron temperatures are not considered. Thus,
in this case, higher electron temperatures signify higher carrier velocity
with constant mobility, and consequently higher current.

As noted above, while the output characteristic obtained are quite similar,
the electron temperature profile in the channel shows a marked difference
between the two simulators. The electron temperatures produced by FKT are
considerably lower than those from CHT, when $\tau_{e}=0.2$ ps is chosen for
both simulators. This is because of the significantly higher electronic heat
flow into the ohmic contacts through the degenerately doped source/drain
regions in FKT. More physically realistic simulations should include lattice
heating, and FKT would transfer electron energy entering the ohmic mesh points
directly into the semiconductor crystal, leading to lattice temperature hot
spots at ohmic contact mesh points, particularly those on the drain contact.
However, the simplified simulations presented here neglect lattice heating in
order to highlight electron energy transport, and that electronic energy
flowing into ohmic mesh points is removed instantly from the simulation
domain.

Figure 8: Comparison of electron temperatures produced by CHT and FKT along
the x-direction (channel) at 6 nm below the AlGaN/GaN interface, for
$V_{GS}=0$ V, $V_{DS}=10$ V, and $\tau_{e}=0.2$ ps for both simulators. Figure
9: Calculated Newton solver residuals of CHT (left) and FKT (right) hot
electron simulations with $\tau_{e}=0.2$ ps at $V_{GS}=-4$ V. The residuals
are functions of the Newton iteration and the quiescent drain voltage. Figure
10: Estimated ROC of CHT (left) and FKT (right) hot electron simulations with
$\tau_{e}=0.2$ ps and $V_{GS}$ ranging from -4 V to 0 V in 1 V steps. The
black dashed line indicates the region of ROC values representative of
quadratic convergence. Figure 11: Comparison of electron temperatures produced
by CHT and FKT along the x-direction (channel) at 6 nm below the AlGaN/GaN
interface at $V_{GS}=0$ V. (i) Temperature profile produced by FKT on
increasing $V_{DS}$ to 50 V, at $\tau_{e}=0.2$ ps. (ii) Temperature profile
produced by CHT for $V_{DS}=10$ V and $\tau_{e}=0.2$ ps. (iii) Temperature
profile produced by CHT for $V_{DS}=10$ V and $\tau_{e}=5.0$ fs.

We next analyze the convergence of the nonlinear solvers to explore other
differences of the transport simulators. The outputs of the Newton solvers
were stored to disk and post processed using custom Python codes. In
particular, the Newton solver residuals were used to analyze the convergence
of the transport simulators. These residuals are a function of Newton
iteration and are computed at each quiescent gate and drain voltage. We define
the Newton solver residual computed at the $n^{th}$ Newton iteration for a
specific gate and drain quiescent bias as $e_{n}\left(V_{GS},V_{DS}\right)$.

An example of Newton solver residuals is illustrated in Fig. 9. Here, residual
values computed in the CHT device and FKT simulations are reported in the left
and right of the figure, respectively. These residuals were computed at
${V_{GS}}=-4$ V over the quiescent drain voltage sweep from ${V_{DS}}=0$ to 10
V. The FKT simulations exhibit excellent convergence characteristics for all
quiescent drain voltage biases. However, as can be seen in Fig. 9, there are
some instances of slow convergence exhibited by the CHT device simulator. For
example, there is one Newton solver instance which takes 10 iterations to
complete, and the final residual value is well above the typical value of
$<10^{-10}$.

A quantitative investigation of the convergence properties of the transport
solvers is next presented. In particular, we analyze the rate of convergence
(ROC), $\alpha$, of the numerical solvers. The ROC is calculated by Senning
(2007)

$\alpha\left(V_{GS},V_{DS}\right)\approx\frac{\log\left|e_{n+1}\left(V_{GS},V_{DS}\right)/e_{n}\left(V_{GS},V_{DS}\right)\right|}{\log\left|e_{n}\left(V_{GS},V_{DS}\right)/e_{n-1}\left(V_{GS},V_{DS}\right)\right|}.$
(21)

The calculated ROCs processed from the CHT and FKT Newton solver residuals are
illustrated in Fig. 10. These results indicate that FKT exhibits at least
quadratic convergence ($\alpha=2$) for all the drain biases. It should be
noted that there exists some FKT ROC values which fall slightly below
$\alpha=2$. These values, on the order of $\alpha=1.8$, are the result of
numerical precision of the solver. For example, the set of Newton solver
residuals which contain a value at the 7th iteration shed light on the impact
of numerical precision of the solver. The slope of the residual curve changes
between iteration 6 and 7 due to hitting the numerical noise floor in the
solver. This change in slope affects the estimated ROC value and would not
occur if there was not a numerical noise floor in the calculations. The ROC of
CHT is also quadratic at several quiescent bias solutions. However, the ROC of
the CHT simulation is generally super-linear.

As noted above, the two simulators exhibit different convergence
characteristics, with FKT generally displaying quadratic convergence, compared
to CHT showing super-linear convergence. In addition, the simulators also
produced widely different electron temperatures in the channel. It would be
interesting to determine if this difference in the convergence behavior can be
attributed to the much higher electron temperatures obtained from the CHT
simulations. This can be verified by running the simulations to control for
the temperature difference. Therefore, we analyze the convergence
characteristics by changing the simulation setup so as to obtain similar peak
electron temperatures.

We investigate two methods to obtain similar channel electron temperatures
from the two simulators. First, we try to reduce the electron temperatures
produced by CHT, to the approximated range of temperatures yielded by the
unaltered FKT simulation. For that purpose, we reduce the $\tau_{e}$ used in
CHT to an artificially low value. Reducing the energy relaxation time allows
the electrons to rapidly lose their energy to the lattice, thereby reducing
the electron temperature. For the second method, we try to increase the drain
bias in the FKT simulation, keeping the CHT setup unchanged. The exceedingly
high fields produced should result in hot electrons with temperatures
comparable to CHT.

Accordingly, we first perform the CHT simulation by reducing the $\tau_{e}$ to
$5.0$ fs, while keeping the $\tau_{e}$ in FKT unchanged at $0.2$ ps. Fig. 11
shows that the peak electron temperature in the channel is in the same range
of magnitude as FKT. The convergence characteristic for the CHT simulation is
shown in Fig. 12 (left). We see that the ROC is generally super-linear, and
quadratic or above for some of the gate biases. Therefore, reducing the
electron temperature did not significantly change the CHT convergence rate,
and it still shows a slower ROC compared to FKT.

Figure 12: Estimated ROC of CHT (left) hot electron simulations with
$\tau_{e}=0.005$ ps and $\sigma_{PZ}=9\times 10^{12}\mathrm{cm}^{-2}$.
Estimated ROC of FKT (right) hot electron simulations with $\tau_{e}=0.2$ ps,
$\sigma_{PZ}=9\times 10^{12}\mathrm{cm}^{-2}$ and drain voltage up to 50 V.
The symbols correspond to $V_{GS}$, as indicated in the legends.

In addition to simulations that lowered electron temperature in CHT to
approximate the original FKT temperatures in Fig. 8, further FKT simulations
also increased electron temperatures to approximate the original CHT
temperatures in Fig. 8. Because $T_{n}$ was reduced in CHT simulations by
decreasing $\tau_{e}$ far below 0.2 ps, it might seem natural to increase
$\tau_{e}$ far above 0.2 ps in FKT to produce much higher $T_{n}$. However,
when $\tau_{e}$ increases excessively, the electronic heat flow to the ohmic
contacts in FKT limits further increase in channel electron temperature. A
more effective way to heat FKT electrons to temperatures comparable to the CHT
temperatures in Fig. 8 is to simply drive the FKT simulations to higher drain
voltages. Thus, the FKT simulation was run up to ${V_{DS}}=50$ V at
${V_{GS}}=0$ V. Fig. 11 shows that the peak electron temperature obtained for
this setup is comparable to that obtained from the original CHT simulation. We
observe that FKT continues to show excellent quadratic convergence, as seen in
Fig. 12 (right). Despite increasing the electron temperature by applying high
fields, FKT continues to show quadratic convergence compared to the super-
linear convergence of CHT.

Finally, changing the polarization sheet charge density
($\sigma_{\mathrm{PZ}}$) at AlGaN/GaN interface, changes the field at the
interface. The large electric fields that arise place further burden on the
solvers and affect the convergence behavior. Therefore, a series of
simulations were run by increasing $\sigma_{\mathrm{PZ}}$, with $\tau_{e}=0.2$
ps for both the simulators, to characterize the convergence behavior. As
summarized in TABLE 3, FKT displays greater than quadratic ROC on average for
all values of $\sigma_{\mathrm{PZ}}$, while CHT shows a super-linear ROC that
decreases as $\sigma_{\mathrm{PZ}}$ is increased. Furthermore, the CHT
simulations failed to converge for $\sigma_{\mathrm{PZ}}\geq 15\times 10^{12}$
$\mathrm{cm^{-2}}$, which is indicated as "N/A" in TABLE 3. For the
simulations to converge, we observed that $\tau_{e}$ had to be reduced to an
unphysically low value of $5.0$ fs.

Table 3:  Summary of CHT device and FKT ROC values. "N/A" refers to simulations that failed to converge. $\sigma_{PZ}\left(\times 10^{12}\hskip 2.84544pt\mathrm{cm}^{-2}\right)$ | mean$\\{\alpha\left(V_{GS},V_{DS}\right)\\}$
---|---
CHT | FKT
9 | 1.856 | 2.399
10 | 1.809 | 2.387
11 | 1.044 | 2.392
12 | 1.484 | 2.409
13 | 1.510 | 2.403
14 | 1.560 | 2.397
15 | N/A | 2.391
16 | N/A | 2.387
17 | N/A | 2.384
18 | N/A | 2.388

###  III.3 Time-Dependent Simulations

Because the above results suggest the different CHT and FKT models may have
led to different numerical convergence behavior, transient simulations were
performed to test for additional differences in behavior. Starting from the
thermal equilibrium initial condition, the drain voltage $V_{DS}$ was ramped
from 0 to 1 V in 1 ns, and the loss of electron energy to the semiconductor
crystal lattice was set to zero in order to highlight differences in real-
space energy transport. The electron temperatures across the device computed
by CHT and FKT are shown in animated Figs. 13 (Multimedia view) (left) and
(right), respectively.

Figure 13: Electron temperatures $T_{n}$ across the transistor for $V_{GS}=0$
when $V_{DS}$ is increased from 0 to 1 V in 1 ns as computed with the CHT
model (left) and the FKT model (right). The AlGaN barrier/GaN channel
interface is located at z = 0, and the position of the gate in the x-direction
is represented by the dotted lines. (Multimedia view)

Figure 14: Electron temperatures $T_{n}$ across the transistor computed by
with the CHT model (left) and the FKT model (right) over the time span
$0<t<0.15$ ns as the drain voltage $V_{DS}$ changed at a rate of 1 V/ns. The
AlGaN barrier/GaN channel interface is located at z = 0, and the position of
the gate in the x-direction is represented by the dotted lines. (Multimedia
view)

As expected from the previous static results, the electrons heat up to higher
temperatures in the CHT calculation. However, the peak CHT temperatures are
relatively distant from the gate, appearing underneath the drain contact. The
peak FKT temperatures, on the other hand, appear at the corner of the gate
closer to the drain. Because most of the $V_{DS}$ voltage drop occurs at this
corner of the gate, it is unclear why the peak CHT electron temperature would
not also occur in this location. In addition to locations of peak electron
temperature, the transient simulations also produced other differences that
become apparent when focusing on the shorter time scale $0<t<0.15$ ns. The
animated Figs. 14 (Multimedia view) (left) and (right) show these dynamic
electron temperatures across the device as produced by the CHT and FKT models,
respectively. As in the case of the longer time scales, FKT shows electron
temperatures increasing monotonically in time with the peak located at the
drain-side corner of the gate. The CHT model again produces a peak electron
temperature occurring away from the gate and underneath the ohmic drain
contact, but it also shows electron temperatures in the GaN buffer region
under the gate decreasing significantly below the 300 K ambient temperature.
As these electron dynamics occur at constant volume in the absence of physical
work, it is not immediately clear what physical mechanism could cause electron
temperatures inside the transistor to fall below the ambient temperature. The
fundamentally different physical behaviors revealed by these dynamically
evolving transient simulations may help explain the different numerical
convergence characteristics previously observed for the CHT and FKT static
simulations.

##  IV Conclusion

Deterministic Boltzmann solvers make use of the moments of the BTE to model
the charge transport in semiconductors. Hydrodynamic model and FKT are both
deterministic solvers that belong to this class of energy transport models.
They differ in the way they treat the electronic heat flow. While hydrodynamic
models utilize an electron thermal conductivity, FKT employs the heat capacity
of an ideal Fermi gas.

In this work, we compared the hydrodynamic model implemented in CHT with the
FKT model. A GaN HEMT structure was used to demonstrate the simulators. It was
observed that both the simulators gave reasonably similar non-equilibrium
output characteristics. However, the electron temperatures in the channel,
close to the AlGaN/GaN interface, were widely different. The rate of
convergence of the solvers were also different, with FKT showing quadratic
convergence or higher, and CHT showing super-linear convergence.

Next, we investigated if the difference in the convergence characteristics can
be attributed to the different electron temperatures obtained. We employed the
two methods to ensure that the two simulators produce similar temperatures.
First, we reduced the energy relaxation time used in CHT simulation, keeping
the FKT setup unchanged. Second, we increased the drain bias for the FKT
simulation, producing hot electrons with temperature comparable to the
unchanged CHT simulation. We then compared the convergence behavior of FKT and
CHT in each of the above mentioned cases. As before, FKT continued to show
greater than quadratic rate of convergence, and CHT showed super-linear
convergence. We can conclude that the different convergence behavior is due to
the difference in the transport models and the description of the electronic
heat flow, rather than the difference in the electron temperatures produced.
Additionally, simulations were run after changing the polarization sheet
charge density at the AlGaN/GaN interface. The results show that FKT exhibited
rapid quadratic convergence for all values chosen, while CHT failed to
converge for large values of the polarization charge.

To further explore the possible underlying causes of different numerical
convergence, additional large signal transient simulations were performed, and
the evolving electron temperatures inside the device, as produced by the CHT
and FKT models, were examined. During the transients, FKT electron
temperatures throughout the HEMT structure increased monotonically with drain
voltage, and the peak temperature consistently appeared at the drain side of
the gate where the lateral electric fields are largest and most of the drain
voltage drop occurs. CHT, on the other hand, produced peak electron
temperatures located away from the gate and in the degenerately doped drain
region. Also, CHT electron temperatures did not increase monotonically
throughout the device during the transient drain volt sweep. At early stages
of the voltage sweep, CHT produced electron temperatures in the buffer region
underneath the gate that dropped nearly 100 K below the ambient temperature.
Because these electron dynamics occurred under constant volume conditions,
there is no apparent physical mechanism that may have caused this electron
cooling. Instead, it may reveal some fundamental differences between the CHT
and FKT models, particularly their different treatments of electronic heat
flow. These results also suggest that FKT could potentially provide a robust
and effective means for simulating GaN HEMT structures for certain
applications including RF power electronics.

###### Acknowledgements.

This work was supported by AFOSR Grant No. LRIR 21RYCOR073.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Cross Ratio Geometry
Advances for Four Co-Linear Points in the
Desargues Affine Plane-Skew Field

Orgest ZAKA  Orgest ZAKA: Department of Mathematics-Informatics, Faculty of
Economy and Agribusiness, Agricultural University of Tirana, Tirana, Albania
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and  James F. Peters
James F. PETERS: Department of Electrical & Computer Engineering, University
of Manitoba, WPG, MB, R3T 5V6, Canada and Department of Mathematics, Faculty
of Arts and Sciences, Adi̇yaman University, 02040 Adi̇yaman, Turkey
<EMAIL_ADDRESS>Dedicated to Girard Desargues and Karl G. C. von
Staudt

###### Abstract.

This paper introduces advances in the geometry of the cross ratio of four co-
linear points in in the Desargues affine plane. The cross-ratio of co-linear
points of a skew field in the Desargues affine plane. The results given here
have a clean rendition, based on Desargues affine plane axiomatics, skew field
properties and the addition and multiplication of planar co-linear points.

###### Key words and phrases:

Co-Linear Points, Cross Ratio, Skew Field, Desargues Affine Plane

###### 2010 Mathematics Subject Classification:

51-XX; 51Axx; 51A30; 51E15, 51N25, 30C20, 30F40

The research has been supported by the Natural Sciences & Engineering Research
Council of Canada (NSERC) discovery grant 185986, Instituto Nazionale di Alta
Matematica (INdAM) Francesco Severi, Gruppo Nazionale per le Strutture
Algebriche, Geometriche e Loro Applicazioni grant 9 920160 000362, n.prot U
2016/000036 and Scientific and Technological Research Council of Turkey
(TÜBİTAK) Scientific Human Resources Development (BIDEB) under grant no:
2221-1059B211301223.

###### Contents

  1. 1 Introduction and Preliminaries
    1. 1.1 Desargues Affine Plane
    2. 1.2 Addition and Multiplication of points in a line of Desargues affine plane
    3. 1.3 Some algebraic properties of Skew Fields
    4. 1.4 Ratio of two and three points
  2. 2 Cross-Ratio for Fuor points in a line of Desargues affine plane

##  1\. Introduction and Preliminaries

In the advancement of our research in the connections of axiomatic geometry
and algebraic structures, we have achieved some results which we have
presented in this paper. More recently, results are given about the
association of algebraic structures in affine planes and in Desargues affine
plane, and vice versa in [21, 22, 6, 19, 20, 18, 27, 28, 26, 25, 23]. The
foundations for the study of the connections between axiomatic geometry and
algebraic structures were set forth by D. Hilbert [9]. And some classic
research results in this context are given, for example, by E. Artin [1], D.R.
Huges and F.C. Piper [10], H. S. M Coxeter [5]. Marcel Berger in [3], Robin
Hartshorne in [7].

In this paper, we advance in study regarding the cross ratio of 4-points, in a
line of the Desargues affine plane. We study and discuses the properties and
results related to the cross ratio for four points, also we see the points of
line as a elements of a skew field which constructed over this line on
Desargues affine plane.

We use skew field properties for the proof of our results, since the cross-
ratio sketch is very confusing (even with the Euclidean interpretation).

Earlier, we study the ratio of 2 and 3 points in a line on Desargues affine
plane (see [23], [24], [24]), also we have shown that on each line on
Desargues affine plane, we can construct a skew-field simply and
constructively, using simple elements of elementary geometry, and only the
basic axioms of Desargues affine plane (see [22], [6], [18], [27] ).

In this paper, we utilize a method that is naive and direct, without requiring
the concept of coordinates. We bases only in Desargues affine plane axiomatic
and in skew field properties (the points in a line on Desargues affine plane,
we think of them as elements of skew fields, which is a construct over this
line).

###  1.1. Desargues Affine Plane

Let $\mathcal{P}$ be a nonempty space, $\mathcal{L}$ a nonempty subset of
$\mathcal{P}$. The elements $p$ of $\mathcal{P}$ are points and an element
$\ell$ of $\mathcal{L}$ is a line.

######  Definition 1.

The incidence structure $\mathcal{A}=(\mathcal{P},\mathcal{L},\mathcal{I})$,
called affine plane, where satisfies the above axioms:

  1. 1o

For each points $\left\\{P,Q\right\\}\in\mathcal{P}$, there is exactly one
line $\ell\in\mathcal{L}$ such that $\left\\{P,Q\right\\}\in\ell$.

  2. 2o

For each point $P\in\mathcal{P},\ell\in\mathcal{L},P\not\in\ell$, there is
exactly one line $\ell^{\prime}\in\mathcal{L}$ such that $P\in\ell^{\prime}$
and $\ell\cap\ell^{\prime}=\emptyset$ (Playfair Parallel Axiom [14]). Put
another way, if the point $P\not\in\ell$, then there is a unique line
$\ell^{\prime}$ on $P$ missing $\ell$ [15].

  3. 3o

There is a 3-subset of points $\left\\{P,Q,R\right\\}\in\mathcal{P}$, which is
not a subset of any $\ell$ in the plane. Put another way, there exist three
non-collinear points $\mathcal{P}$ [15].

_Desargues’ Axiom, circa 1630_ [11, §3.9, pp. 60-61] [17]. Let
$A,B,C,A^{\prime},B^{\prime},C^{\prime}\in\mathcal{P}$ and let pairwise
distinct lines
$\ell^{AA_{1}},\ell^{BB^{\prime}},\ell^{CC^{\prime}},\ell^{AC},\ell^{A^{\prime}C^{\prime}}\in\mathcal{L}$
such that

$\displaystyle\ell^{AA_{1}}\parallel\ell^{BB^{\prime}}\parallel\ell^{CC^{\prime}}\
\mbox{(Fig.~{}\ref{fig:DesarguesAxiom}(a))}$ $\displaystyle\ \mbox{{or}}\
\ell^{AA_{1}}\cap\ell^{BB^{\prime}}\cap\ell^{CC^{\prime}}=P.\mbox{(Fig.~{}\ref{fig:DesarguesAxiom}(b)
)}$ $\displaystyle\mbox{and}\ \ell^{AB}\parallel\ell^{A^{\prime}B^{\prime}}\ $
$\displaystyle\ \mbox{and}\ \ell^{BC}\parallel\ell^{B^{\prime}C^{\prime}}.$
$\displaystyle
A,B\in\ell^{AB},A^{\prime}B^{\prime}\in\ell^{A^{\prime}B^{\prime}},$
$\displaystyle\ \mbox{and}\
B,C\in\ell^{BC},B^{\prime}C^{\prime}\in\ell^{B^{\prime}C^{\prime}}.$
$\displaystyle A\neq C,A^{\prime}\neq C^{\prime},$ $\displaystyle\ \mbox{and}\
\ell^{AB}\neq\ell^{A^{\prime}B^{\prime}},\ell^{BC}\neq\ell^{B^{\prime}C^{\prime}}.$

Figure 1. Desargues Axioms: (a) For parallel lines
$\ell^{AA_{1}}\parallel\ell^{BB^{\prime}}\parallel\ell^{CC^{\prime}}$; (b) For
lines which are cutting in a single point $P$,
$\ell^{AA_{1}}\cap\ell^{BB^{\prime}}\cap\ell^{CC^{\prime}}=P$.

Then $\boldsymbol{\ell^{AC}\parallel\ell^{A^{\prime}C^{\prime}}}$.
$\blacksquare$

######  Example 1.

In Euclidean plane, three vertexes $ABC$ and $A^{\prime}B^{\prime}C^{\prime}$,
are similar (in (a) are equivalent-triangle and in (b) are homothetical-
triangle) the parallel lines,
$\ell^{AC},\ell^{A^{\prime}C^{\prime}}\in\mathcal{L}$ in Desargues’ Axiom are
represented in Fig. 1. In other words, the side $AC$ of the triangle of
$\bigtriangleup ABC$ is parallel with the side $A^{\prime}C^{\prime}$ of the
triangle $\bigtriangleup A^{\prime}B^{\prime}C^{\prime}$, provided the
restrictions on the points and lines in Desargues’ Axiom are satisfied.
$\blacksquare$

A Desargues affine plane is an affine plane that satisfies Desargues’ Axiom.

######  Notation 1.

Three vertexes $ABC$ and $A^{\prime}B^{\prime}C^{\prime}$, which, fulfilling
the conditions of the Desargues Axiom, we call _’Desarguesian’_.

###  1.2. Addition and Multiplication of points in a line of Desargues affine
plane

Addition of points in a line of affine plane: In an Desargues affine plane
$\mathcal{A_{D}}=(\mathcal{P},\mathcal{L},\mathcal{I})$ we fix two different
points $O,I\in\mathcal{P},$ which, according to Axiom 1, determine a line
$\ell^{OI}\in\mathcal{L}.$ Let $A$ and $B$ be two arbitrary points of a line
$\ell^{OI}$. In plane $\mathcal{A_{D}}$ we choose a point $B_{1}$ not incident
with $\ell^{OI}$: $B_{1}\notin\ell^{OI}$ (we call the auxiliary point).
Construct line $\ell_{OI}^{B_{1}},$ which is only according to the Axiom 2.
Then construct line $\ell_{OB_{1}}^{A},$ which also is the only according to
the Axiom 2. Marking their intersection
$P_{1}=\ell_{OI}^{B_{1}}\cap\ell_{OB_{1}}^{A}.$ Finally construct line
$\ell_{BB_{1}}^{P_{1}}.$ For as much as $\ell^{BB_{1}}$ cuts the line
$\ell^{OI}$ in point $B$, then this line, parallel with $\ell^{BB_{1}}$, cuts
the line $\ell^{OI}$ in a single point $C$, this point we called the addition
of points $A$ with point $B$ (Figure 2 (a)).

Multiplication of points in a line in affine plane. Choose in the plane
$\mathcal{A_{D}}$ one point $B_{1}$ not incident with lines $\ell^{OI},$ and
construct the line $\ell^{IB_{1}}$. Construct the line $\ell_{IB_{1}}^{A},$
which is the only accoding to the Axiom 2 and cutting the line
$\ell^{OB_{1}}$. Marking their intersection with $P_{1}=\ell_{IB_{1}}^{A}\cap
OB_{1}.$ Finally, construct the line $\ell_{BB_{1}}^{P_{1}}.$ Since
$\ell^{BB_{1}}$ cuts the line $\ell^{OI}$ in a single point $B$, then this
line, parallel with $\ell^{BB_{1}}$, cuts the line $\ell^{OI}$ in one single
point $C$, this point we called the multiplication of points $A$ with point
$B$ (Figure 2 (b)).

The process of construct the points $C$ for adition and multiplication of
points in $\ell^{OI}-$line in affine plane, is presented in the tow algorithm
form

Addition Algorithm

Step.1:

$B_{1}\notin\ell^{OI}$

Step.2:

$\ell_{OI}^{B_{1}}\cap\ell_{OB_{1}}^{A}=P_{1}$

Step.3:

$\ell_{BB_{1}}^{P_{1}}\cap\ell^{OI}=C(=A+B)$

Multiplication Algorithm

Step.1:

$B_{1}\notin\ell^{OI}$

Step.2:

$\ell_{IB_{1}}^{A}\cap\ell^{OB_{1}}=P_{1}$

Step.3:

$\ell_{BB_{1}}^{P_{1}}\cap\ell^{OI}=C(=A\cdot B)$

Figure 2.  (a) Addition of points in a line in affine plane, (b)
Multiplication of points in a line in affine plane

In [18] and [6], we have prove that $(\ell^{OI},+,\cdot)$ is a skew field in
Desargues affine plane, and is field (commutative skew field) in the Papus
affine plane.

###  1.3. Some algebraic properties of Skew Fields

I n this section $K$ will denote a skew field [8] and $z[K]$ its center, where
is the set $K$ such that

$z[K]=\left\\{k\in K\quad|\quad ak=ka,\quad\forall a\in K\right\\}$

######  Proposition 1.

$z[K]$ is a commutative subfield of a skew field $K$.

Let now $p\in K$ be a fixed element of the skew field $K$. We will denote
$z_{K}(p)$ the centralizer in $K$ of the element $p$, where is the set,

$z_{K}(p)=\left\\{k\in K|pk=kp,\right\\}.$

$z_{K}(p)$ is sub skew field of K, but, in general, it is not commutative.

Let $K$ be a skew field, $p\in K$, and let us denote by $[p_{K}]$ the
conjugacy class of $p$:

$[p_{K}]=\left\\{q^{-1}pq\quad|\quad q\in K\setminus\\{0\\}\right\\}$

If, $p\in z[K]$, for all $q\in K$ we have that $q^{-1}pq=p.$

###  1.4. Ratio of two and three points

In the paper [23], we have done a detailed study, related to the ratio of two
and three points in a line of Desargues affine plane. Below we are listing
some of the results for ratio of two and three points.

######  Definition 2.

[23] Lets have two different points $A,B\in\ell^{OI}-$line, and $B\neq O$, in
Desargues affine plane. We define as ratio of this tow points, a point
$R\in\ell^{OI}$, such that,

$R=B^{-1}A,\qquad\text{ we mark this, with,}\qquad R=r(A:B)=B^{-1}A$

For a ’ratio-point’ $R\in\ell^{OI}$, and for point $B\neq O$ in line
$\ell^{OI}$, is a unique defined point, $A\in\ell^{OI}$, such that
$R=B^{-1}A=r(A:B)$.

Figure 3. Ilustrate the Ratio-Point, of 2-Points in a line of Desargues affine
plane $R=r(A:B)=B^{-1}A$.

Some results for Ratio of 2-points in Desargues affine plane (see [23]).

  * •

If have two different points $A,B\in\ell^{OI}-$line, and $B\neq O$, in
Desargues affine plane, then, $r^{-1}(A:B)=r(B:A)$.

  * •

For three collinear point $A,B,C$ and $C\neq O$, in $\ell^{OI}-$line, have,

$r(A+B:C)=r(A:C)+r(B:C).$

  * •

For three collinear point $A,B,C$ and $C\neq O$, in $\ell^{OI}-$line, have,

    1. (1)

$r(A\cdot B:C)=r(A:C)\cdot B.$

    2. (2)

$r(A:B\cdot C)=C^{-1}r(A:C).$

  * •

Let’s have the points $A,B\in\ell^{OI}-$line where $B\neq O$. Then have that,

$r(A:B)=r(B:A)\Leftrightarrow A=B.$

  * •

This ratio-map, $r_{B}:\ell^{OI}\to\ell^{OI}$ is a bijection in
$\ell^{OI}-$line in Desargues affine plane.

  * •

The ratio-maps-set $\mathcal{R}_{2}=\\{r_{B}(X)|\forall X\in\ell^{OI}\\}$, for
a fixed point $B$ in $\ell^{OI}-$line, forms a skew-field with ’addition and
multiplication’ of points. This, skew field $(\mathcal{R}_{2},+,\cdot)$ is
sub-skew field of the skew field $(\ell^{OI},+,\cdot)$.

Ratio of three points in a line on Desargues affine plane. (see [23])

######  Definition 3.

If $A,B,C$ are three points on a line $\ell^{OI}$ (collinear) in Desargues
affine plane, then we define their ratio to be a point $R\in\ell^{OI}$, such
that:

$(B-C)\cdot R=A-C,\quad\mbox{concisely}\quad R=(B-C)^{-1}(A-C),$

and we mark this with $r(A,B;C)=(B-C)^{-1}(A-C)$.

Figure 4. Ratio of 3-Points in a line of Desargues affine plane $R=r(A,B;C)$.

Some Results for Ratio of 3-points in Desargues affine plane ([23]).

  * •

For 3-points $A,B,C$ in a line $\ell^{OI}$ of Desargues affine plane, we have
that,

$r(-A,-B;-C)=r(A,B;C).$

  * •

For 3-points $A,B,C$ in a line $\ell^{OI}$ in the Desargues affine plane, have

$r^{-1}(A,B;C)=r(B,A;C).$

  * •

If $A,B,C$, are three different points, and different from point $O$, in a
line $\ell^{OI}$ on Desargues affine plane, then

$r(A^{-1},B^{-1};C^{-1})=B[r(A,B;C)]A^{-1}.$

  * •

In the Pappus affine plane, for three point different from point $O$, in
$\ell^{OI}-$line, we have $r(A^{-1},B^{-1};C^{-1})=r(A,B;C)\cdot r(B,A;O).$

  * •

This ratio-map, $r_{BC}:\ell^{OI}\to\ell^{OI}$ is a bijection in
$\ell^{OI}-$line in Desargues affine plane.

  * •

The ratio-maps-set $\mathcal{R}_{3}=\\{r_{BC}(X)|\forall X\in\ell^{OI}\\}$,
for a different fixed points $B,C$ in $\ell^{OI}-$line, forms a skew-field
with ’addition and multiplication’ of points in $\ell^{OI}-$line. This, skew
field $(\mathcal{R}_{3},+,\cdot)$ is sub-skew field of the skew field
$(\ell^{OI},+,\cdot)$.

##  2\. Cross-Ratio for Fuor points in a line of Desargues affine plane

This section culminates in a main result in this paper. We consider the cross-
ratio of co-linear points in Desargues affine planes, utilizing a method that
is naive and direct without requiring planar coordinates. We define the cross-
ratio of four co-linear points in a line on Desargues affine plane _as a point
in this line_. This work carries forward earlier results that reveal the close
connection between lines in the Desargues affine planes and corresponding skew
fields. Skew fields properties in our proofs. Mainly, we rely on our results
regarding the addition and multiplication of co-linear points in the Desargues
affine plane, and the fact that a line (set of points), with addition and
multiplication, forms a skew field (for more about this, see [18], [22], [6],
[20], [19], [21], [27], [28], [24]).

The classical definition of the cross-ratio (see [13, 9, 2, 3]) for 4-points,
is given as a product of tow ratio of lengths. So, for example, for four co-
linear points $A,B,C,D$,

$c_{r}(A,B;C,D)=\frac{AC}{BC}\cdot\frac{BD}{AD},$

where $AC,BC,BD,AD$ are the lengths of segments $[AB],[BC],[BD].[AD]$,
respectively.

Since we will not use coordinates and metrics, our definitions are rely solely
on the algebra and axiomatics for the Desargues affine plane.

Let us have the line $\ell^{OI}$ in Desargues affine plane $\mathcal{A_{D}}$,
and four points, $A,B,C,D\in\ell^{OI}$

######  Definition 4.

If $A,B,C,D$ are four points on a line $\ell^{OI}$ in Desarges affine plane
$\mathcal{A_{D}}$, no three of them equal, then we define their cross ratio to
be a point:

$c_{r}(A,B;C,D)=\left[(A-D)^{-1}(B-D)\right]\left[(B-C)^{-1}(A-C)\right]$

######  Remark 1.

Similar to ’ratio’, we can define it, the cross-ratio, also as

$c_{r}(A,B;C,D)=[(B-D)(A-D)^{-1}][(A-C)(B-C)^{-1}],$

or

$c_{r}(A,B;C,D)=[(B-D)(A-C)][(A-D)^{-1}(B-C)^{-1}],$

(or all combination of product of this 4-factors) the results would be
similar, _but the obtained point will always be different for each case_. In
$\ell^{OI}-$line, in Desargues affine planes, these are a different point from
that of our definition, since:

$[(B-D)(A-D)^{-1}][(A-C)(B-C)^{-1}]\neq\left[(A-D)^{-1}(B-D)\right]\left[(B-C)^{-1}(A-C)\right].$

and

$[(B-D)(A-C)][(A-D)^{-1}(B-C)^{-1}]\neq\left[(A-D)^{-1}(B-D)\right]\left[(B-C)^{-1}(A-C)\right].$

also for the other cases, we would have a difference for each pair, found for
the cross ratio, according to any definition we take. We are keeping our
definition.

######  Definition 5.

If the line $\ell^{OI}$ in Desargues affine plane, is a infinite line (number
of points in this line is $+\infty$), we define as follows:

$\displaystyle c_{r}(\infty,B;C,D)$ $\displaystyle=(B-D)(B-C)^{-1}$
$\displaystyle c_{r}(A,\infty;C,D)$ $\displaystyle=(A-D)^{-1}(A-C)$
$\displaystyle c_{r}(A,B;\infty,D)$ $\displaystyle=(A-D)^{-1}(B-D)$
$\displaystyle c_{r}(A,B;C,\infty)$ $\displaystyle=(B-C)^{-1}(A-C)$

From this definition and from ratio definition 3 we have that,

  * •

$c_{r}(A,B;C,D)=\left[(A-D)^{-1}(B-D)\right]\left[(B-C)^{-1}(A-C)\right]$, so

$c_{r}(A,B;C,D)=r(B,A;D)\cdot r(A,B;C).$

  * •

$c_{r}(\infty,B;C,D)=(B-D)(B-C)^{-1}=[(D-B)^{-1}(C-B)]^{-1}$, so,

$c_{r}(\infty,B;C,D)=r^{-1}(C,D;B).$

  * •

$c_{r}(A,\infty;C,D)=(A-D)^{-1}(A-C)=(D-A)^{-1}(C-A)$, so,

$c_{r}(A,\infty;C,D)=r(C,D;A).$

  * •

$c_{r}(A,B;\infty,D)=(A-D)^{-1}(B-D)$, so

$c_{r}(A,B;\infty,D)=r(A,B;D).$

  * •

$c_{r}(A,B;C,\infty)=(B-C)^{-1}(A-C)$, so,

$c_{r}(A,B;C,\infty)=r(A,B;C).$

Some simple properties of Cross-Ratios, which derive directly from the
definition, related to the position of the points $A,B,C,D$ in
$\ell^{OI}-$line in Desargues affine plane.

  * •

If $A=B$, then

$\displaystyle c_{r}(A,B;C,D)$ $\displaystyle=c_{r}(A,A;C,D)$
$\displaystyle=[(A-D)^{-1}(A-D)][(A-C)^{-1}(A-C)]$ $\displaystyle=[I][I]$
$\displaystyle=I.$

  * •

If $A=C$, then

$\displaystyle c_{r}(A,B;C,D)$ $\displaystyle=c_{r}(A,B;A,D)$
$\displaystyle=[(A-D)^{-1}(B-D)][(B-A)^{-1}(A-A)]$
$\displaystyle=[(A-D)^{-1}(B-D)][(B-A)^{-1}\cdot O]$ $\displaystyle=O.$

  * •

If $A=D$, then

$\displaystyle c_{r}(A,B;C,D)$ $\displaystyle=c_{r}(A,B;C,A)$
$\displaystyle=[(A-A)^{-1}(B-A)][(B-C)^{-1}(A-C)]$
$\displaystyle=[O^{-1}(B-A)][(B-C)^{-1}(A-C)]$ (think that
$O^{-1}=\infty$(point in infinity)) $\displaystyle=\infty.$

  * •

If $B=C$, then

$\displaystyle c_{r}(A,B;C,D)$ $\displaystyle=c_{r}(A,B;B,D)$
$\displaystyle=[(A-D)^{-1}(B-D)][(B-B)^{-1}(A-B)]$
$\displaystyle=[(A-D)^{-1}(B-D)][O^{-1}(A-B)]$ (think that
$O^{-1}=\infty$(point in infinity)) $\displaystyle=\infty.$

  * •

If $B=D$, then

$\displaystyle c_{r}(A,B;C,D)$ $\displaystyle=c_{r}(A,B;C,B)$
$\displaystyle=[(A-B)^{-1}(B-B)][(B-C)^{-1}(A-C)]$
$\displaystyle=[(A-B)^{-1}\cdot O][(B-C)^{-1}(A-C)]$ $\displaystyle=O.$

  * •

If $C=D$, then

$\displaystyle c_{r}(A,B;C,D)$ $\displaystyle=c_{r}(A,B;C,C)$
$\displaystyle=[(A-C)^{-1}(B-C)][(B-C)^{-1}(A-C)]$
$\displaystyle=(A-C)^{-1}[(B-C)(B-C)^{-1}](A-C)$
$\displaystyle=(A-C)^{-1}\cdot I\cdot(A-C)$ $\displaystyle=(A-C)^{-1}(A-C)$
$\displaystyle=I.$

######  Theorem 1.

Let $R\in\ell^{OI}$, such that $R\neq O$ and $R\neq I$. If $A,B,C\in\ell^{OI}$
are three different points, then exist a single point $D\in\ell^{OI}$, such
that $c_{r}(A,B;C,D)=R.$

###### Proof.

Suppose that exist tow different points $D$ an $D^{\prime}$ in
$\ell^{OI}-$line, such that

$c_{r}(A,B;C,D)=c_{r}(A,B;C,D^{\prime})$

We rewrite them, cross ratios, as products of ’ratios’, and we have,

$c_{r}(A,B;C,D)=\left[(A-D)^{-1}(B-D)\right]\left[(B-C)^{-1}(A-C)\right]=r(B,A;D)\cdot
r(A,B;C)$

and

$c_{r}(A,B;C,D^{\prime})=\left[(A-D^{\prime})^{-1}(B-D^{\prime})\right]\left[(B-C)^{-1}(A-C)\right]=r(B,A;D^{\prime})\cdot
r(A,B;C)$

So, have,

$r(B,A;D)\cdot r(A,B;C)=r(B,A;D^{\prime})\cdot r(A,B;C)$

we mark $r(B,A;D)=R_{1};r(A,B;C)=R_{2},r(B,A;D^{\prime})=R_{3}$, remember that
these are points of the line $\ell^{OI}$, so they are elements of the skew-
fields $(\ell^{OI},+,\cdot)$, and have

$R=R_{1}\cdot R_{2}\quad\text{and}\quad R=R_{3}\cdot R_{2}$

Thus, for it, we have

$R_{1}\cdot R_{2}=R_{3}\cdot R_{2}\Rightarrow R_{1}\cdot R_{2}-R_{3}\cdot
R_{2}=O\Rightarrow(R_{1}-R_{3})\cdot R_{2}=O$

But the points, $R_{1},R_{2},R_{3}$, are points of $\ell^{OI}-$line in
Desargues affine plane, therefore, they are elements of skew-fields
$K=(\ell^{OI},+,\cdot)$. We also know the fact that ’a skew field does not
have a divisor of zero’ (more on skew fields, see [4], [8], [16], [12])

$R_{1}-R_{3}=O\quad\text{or}\quad R_{2}=O,\quad\text{but}\quad R_{2}\neq
O\Rightarrow R_{1}-R_{3}=O$

so,

$R_{1}=R_{3}\Rightarrow r(B,A;D)=r(B,A;D^{\prime})$

and from the uniqueness of the definition for ’ratio’, we have,

$D=D^{\prime}$

∎

######  Theorem 2.

If $A,B,C,D$ are distinct points in a $\ell^{OI}-$line, in Desargues affine
plane, then

$c_{r}(-A,-B;-C,-D)=c_{r}(A,B;D,C)$

###### Proof.

From cross-ratio definition 4, we have

$\displaystyle c_{r}(-A,-B;-C,-D)$
$\displaystyle=[(-A-(-C))^{-1}(-B-(-C))][(-B-(-D))^{-1}(-A-(-D))]$
$\displaystyle=[(-A+C)^{-1}(-B+C)][(-B+D)^{-1}(-A+D)]$
$\displaystyle=[([-I](A-C))^{-1}[-I](B-C)][([-I](B-D))^{-1}[-I](A-D)]$
$\displaystyle=[(A-C)^{-1}[-I]^{-1}[-I](B-C)][(B-D)^{-1}[-I]^{-1}[-I](A-D)]$
$\displaystyle=[(A-C)^{-1}[-I][-I](B-C)][(B-D)^{-1}[-I][-I](A-D)]$
$\displaystyle=[(A-C)^{-1}(B-C)][(B-D)^{-1}(A-D)]$
$\displaystyle=c_{r}(A,B;D,C)$

From skew fields properties we have that $(ab)^{-1}=b^{-1}a^{-1}$and $ab\neq
ba$, $[-I]^{-1}=-I$, and $[-I][-I]=I.$ ∎

######  Theorem 3.

If $A,B,C,D$ are distinct points in a line, in Desargues affine plane, then

$c_{r}^{-1}(A,B;C,D)=c_{r}(A,B;D,C)$

###### Proof.

From cross-ratio Definition 4, have

$\displaystyle c_{r}^{-1}(A,B;C,D)$
$\displaystyle=\left\\{\left[(A-D)^{-1}(B-D)\right]\left[(B-C)^{-1}(A-C)\right]\right\\}^{-1}$
$\displaystyle=\left[(B-C)^{-1}(A-C)\right]^{-1}\left[(A-D)^{-1}(B-D)\right]^{-1}$
$\displaystyle=[(A-C)^{-1}(B-C)][(B-D)^{-1}(A-D)]$
$\displaystyle=c_{r}(A,B;D,C)$

∎

######  Theorem 4.

For 4 co-linear points $A,B,C,D$ in a line $\ell^{OI}$ in the Desargues affine
plane, the cross-ratio satisfies the equation,

$c_{r}(A,B;C,D)=\left[(A-B)^{-1}-(A-D)^{-1}\right]\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}$

###### Proof.

From the definition 4 we have
$c_{r}(A,B;C,D)=\left[(A-D)^{-1}(B-D)\right]\left[(B-C)^{-1}(A-C)\right]$, and
the points of this line forms a skew-field, therefore, we have association
property:

$\displaystyle R$ $\displaystyle=c_{r}(A,B;C,D)$
$\displaystyle=[(A-D)^{-1}(B-D)][(B-C)^{-1}(A-C)]$ (since this factor are
elements of skew field, so have the associative property)
$\displaystyle=\left[(A-D)^{-1}(B-D)(B-C)^{-1}\right](A-C)$

So, the point $R$, is,

$R=\left[(A-D)^{-1}(B-D)(B-C)^{-1}\right](A-C)$

multiply in the right side by side with $(A-C)^{-1}$, and have

$R\cdot(A-C)^{-1}=(A-D)^{-1}(B-D)(B-C)^{-1}$

now multiply in the right side by side with $(B-C)$, and have

$(A-D)^{-1}(B-D)=\left(R\cdot(A-C)^{-1}\right)(B-C)$

multiply side by side with $(A-B)^{-1}$, and have,

$\left[(A-D)^{-1}(B-D)\right](A-B)^{-1}=\left[R\cdot(A-C)^{-1}(B-C)\right](A-B)^{-1}$

Transform the left side of this equation as

$\displaystyle\left[(A-D)^{-1}(B-D)\right](A-B)^{-1}$
$\displaystyle=\left[(A-D)^{-1}(B+A-A-D)\right](A-B)^{-1}$
$\displaystyle=\left[(A-D)^{-1}([A-D]-[A-B])\right](A-B)^{-1}$

rewrite it, and have

$\displaystyle\left[(A-D)^{-1}(B-D)\right](A-B)^{-1}$
$\displaystyle=[(A-D)^{-1}([A-D]-[A-B])](A-B)^{-1}$
$\displaystyle=[(A-D)^{-1}[A-D]-(A-D)^{-1}[A-B]](A-B)^{-1}$
$\displaystyle=[I-(A-D)^{-1}[A-B]](A-B)^{-1}$
$\displaystyle=(A-B)^{-1}-(A-D)^{-1}[A-B](A-B)^{-1}$
$\displaystyle=(A-B)^{-1}-(A-D)^{-1}$

So, have

$(A-B)^{-1}-(A-D)^{-1}=\left[R(A-C)^{-1}(B-C)\right](A-B)^{-1}$

In the same way as above, (always bearing in mind that the points of a line of
Desargues affine planes form a skew-field related to the addition and
multiplication of the points, on this line, and the properties that satisfy a
skew-field) we do the following transformations.

First we have the associative property for the multiplication of points on a
line,

$\left[R(A-C)^{-1}(B-C)\right](A-B)^{-1}=R\left[(A-C)^{-1}(B-C)(A-B)^{-1}\right]$

Now we transform the expression

$\displaystyle\left[(A-C)^{-1}(B-C)(A-B)^{-1}\right]$
$\displaystyle=\left[(A-C)^{-1}(B+A-A-C)(A-B)^{-1}\right]$
$\displaystyle=\left[(A-C)^{-1}([A-C]-[A-B])(A-B)^{-1}\right]$
$\displaystyle=[I-(A-C)^{-1}[A-B]](A-B)^{-1}$
$\displaystyle=(A-B)^{-1}-(A-C)^{-1}[A-B](A-B)^{-1}$
$\displaystyle=(A-B)^{-1}-(A-C)^{-1}$

So, have

$(A-B)^{-1}-(A-D)^{-1}=R\left[(A-B)^{-1}-(A-C)^{-1}\right]$

Hence

$R=\left[(A-B)^{-1}-(A-D)^{-1}\right]\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}$

so,

$c_{r}(A,B;C,D)=\left[(A-B)^{-1}-(A-D)^{-1}\right]\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}.$

∎

######  Theorem 5.

If $A,B,C,D$ are distinct points in a line, in Desargues affine plane and $I$
is unital point for multiplications of points in same line, then

$I-c_{r}(A,B;C,D)=c_{r}(A,C;B,D)$

###### Proof.

Let’s start the calculations, using the result of the theorem

$I-\left[(A-B)^{-1}-(A-D)^{-1}\right]\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}=$
$=\left\\{(A-C)^{-1}-(A-D)^{-1}\right\\}\left[\left\\{(A-C)^{-1}-(A-B)^{-1}\right\\}\right]^{-1}$

write,

$I=\left[(A-B)^{-1}-(A-C)^{-1}\right]\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}$

so,

$\displaystyle I-c_{r}(A,B;C,D)$
$\displaystyle=I-\left[(A-B)^{-1}-(A-D)^{-1}\right]\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}$
$\displaystyle=\left[(A-B)^{-1}-(A-C)^{-1}\right]\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}$
$\displaystyle-\left[(A-B)^{-1}-(A-D)^{-1}\right]\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}$
$\displaystyle=\left\\{\left[(A-B)^{-1}-(A-C)^{-1}\right]-\left[(A-B)^{-1}-(A-D)^{-1}\right]\right\\}$
$\displaystyle\cdot\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}$
$\displaystyle=\left\\{-(A-C)^{-1}+(A-D)^{-1}\right\\}\left[(A-B)^{-1}-(A-C)^{-1}\right]^{-1}$
$\displaystyle=(-I)\left\\{(A-C)^{-1}-(A-D)^{-1}\right\\}\left[(-I)\left\\{(A-C)^{-1}-(A-B)^{-1}\right\\}\right]^{-1}$
$\displaystyle=(-I)\left\\{(A-C)^{-1}-(A-D)^{-1}\right\\}\left[\left\\{(A-C)^{-1}-(A-B)^{-1}\right\\}\right]^{-1}(-I)^{-1}$
$\displaystyle=\left\\{(A-C)^{-1}-(A-D)^{-1}\right\\}\left[\left\\{(A-C)^{-1}-(A-B)^{-1}\right\\}\right]^{-1}$
$\displaystyle=c_{r}(A,C;B,D)$

from skew-field properties, we have $(-I)^{-1}=-I$ and $(-I)(-I)=I$ ∎

######  Theorem 6.

If $A,B,C,D$ are distinct points in a line, in Desargues affine plane and $I$
is unitary point for multiplications of points in same line, then,

(a):

$c_{r}(A,D;B,C)=I-c_{r}^{-1}(A,B;C,D)$

(b):

$c_{r}(A,C;D,B)=[I-c_{r}(A,B;C,D)]^{-1}$

(c):

$c_{r}(A,D;C,B)=[c_{r}(A,B;C,D)-I]^{-1}c_{r}(A,B;C,D)$

###### Proof.

(a) In theorem 3 we have prove that $c_{r}^{-1}(A,B;C,D)=c_{r}(A,B;D,C)$, and
from theorem 5, have that $I-c_{r}(A,B;D,C)=c_{r}(A,D;B,C)$. So, we have prove
that

$I-c_{r}^{-1}(A,B;C,D)=I-c_{r}(A,B;D,C)=c_{r}(A,D;B,C).$

(b) From theorem 5, we have that, $I-c_{r}(A,B;C,D)=c_{r}(A,C;B,D)$, and from
theorem 3 have that $[c_{r}(A,C;B,D)]^{-1}=c_{r}(A,C;D,B)$, so have that

$c_{r}(A,C;D,B)=[c_{r}(A,C;B,D)]^{-1}=[I-c_{r}(A,B;C,D)]^{-1}.$

(c) At this point we will prove that:
$c_{r}(A,D;C,B)=[c_{r}(A,B;C,D)-I]^{-1}c_{r}(A,B;C,D)$.

From point (a), we prove that $c_{r}(A,D;C,B)=I-c_{r}^{-1}(A,B;C,D),$ and from
theorem 3 have that $c_{r}(A,D;C,B)=c_{r}^{-1}(A,D;B,C)$. So, we have that

$c_{r}(A,D;C,B)=[I-c_{r}^{-1}(A,B;C,D)]^{-1}$

Mark the cross-ratios point $R=c_{r}(A,B;C,D)$, and rewrite. So we have to
prove that the equation holds,

$[I-R^{-1}]^{-1}=[R-I]^{-1}R,$

remember that the points are points of $\ell^{OI}-$line, in Desargues affine
planes, and can also be thought of as elements of skew-fields
$K=(\ell^{OI},+,\cdot)$, therefore, we can make algebraic transformations,
allowed for skew-fields, and we have

$\displaystyle\left[I-R^{-1}\right]^{-1}$ $\displaystyle=[R-I]^{-1}R$
(multiply from the right with $R^{-1}$) $\displaystyle[I-R^{-1}]^{-1}\cdot
R^{-1}$ $\displaystyle=[R-I]^{-1}R\cdot R^{-1}$ (from skew field property have
that $p^{-1}q^{-1}=(qp)^{-1}$) $\displaystyle[R(I-R^{-1})]^{-1}$
$\displaystyle=[R-I]^{-1}[R\cdot R^{-1}]$ $\displaystyle[R\cdot I-R\cdot
R^{-1}]^{-1}$ $\displaystyle=[R-I]^{-1}\cdot I$ $\displaystyle[R-I]^{-1}$
$\displaystyle=[R-I]^{-1}$

∎

######  Theorem 7.

If $A,B,C,D$ are distinct points, and different from zero-point $O$, in a
line, in Desargues affine plane and $I$ is unitary point for multiplications
of points in same line, have,

$c_{r}(A^{-1},B^{-1};C^{-1},D^{-1})=A\cdot c_{r}(A,B;C,D)\cdot A^{-1}$

###### Proof.

From cross-ratio definition 4, we have,

$c_{r}(A^{-1},B^{-1};C^{-1},D^{-1})=[(A^{-1}-D^{-1})^{-1}(B^{-1}-D^{-1})][(B^{-1}-C^{-1})(A^{-1}-C^{-1})]$

Points $A,B,C,D$ and $A^{-1},B^{-1},C^{-1},D^{-1}$, are points of
$\ell^{OI}-$line in Desargues affine plane, so are and elements of the skew
field $K=(\ell^{OI},+,\cdot)$. First we prove that, for tow elements $X,Y$ in
a skew field $K$, we have that $X^{-1}-Y^{-1}=Y^{-1}(Y-X)X^{-1}$. Indeed

$\displaystyle Y^{-1}(Y-X)X^{-1}$ $\displaystyle=[Y^{-1}(Y-X)]X^{-1}$
$\displaystyle=(Y^{-1}Y-Y^{-1}X)X^{-1}$ $\displaystyle=(I-Y^{-1}X)X^{-1}$
$\displaystyle=IX^{-1}-Y^{-1}(XX^{-1})$ $\displaystyle=X^{-1}-Y^{-1}I$
$\displaystyle=X^{-1}-Y^{-1}.$

We use this result in the calculation of $c_{r}(A^{-1},B^{-1};C^{-1},D^{-1})$,
and have

$\displaystyle c_{r}(A^{-1},B^{-1};C^{-1},D^{-1})$
$\displaystyle=[(A^{-1}-D^{-1})^{-1}(B^{-1}-D^{-1})]$
$\displaystyle\cdot[(B^{-1}-C^{-1})(A^{-1}-C^{-1})]$
$\displaystyle=[(D^{-1}(D-A)A^{-1})^{-1}(D^{-1}(D-B)B^{-1})]$
$\displaystyle\cdot[(C^{-1}(C-B)B^{-1})(C^{-1}(C-A)A^{-1})]$
$\displaystyle=[(A(D-A)^{-1}D)(D^{-1}(D-B)B^{-1})]$
$\displaystyle\cdot[(B(C-B)^{-1}C)(C^{-1}(C-A)A^{-1})]$ (from skew field
properties $(abc)^{-1}=c^{-1}b^{-1}a^{-1}$)
$\displaystyle=[A(D-A)^{-1}(DD^{-1})(D-B)B^{-1}]$
$\displaystyle\cdot[B(C-B)^{-1}(CC^{-1})(C-A)A^{-1}]$ (from associative
properties for multiplication)
$\displaystyle=[A(D-A)^{-1}(I)(D-B)B^{-1}][B(C-B)^{-1}(I)(C-A)A^{-1}]$
$\displaystyle=[A(D-A)^{-1}(D-B)B^{-1}][B(C-B)^{-1}(C-A)A^{-1}]$
$\displaystyle=A[(D-A)^{-1}(D-B)B^{-1}][B(C-B)^{-1}(C-A)]A^{-1}$
$\displaystyle=A\left\\{[(D-A)^{-1}(D-B)B^{-1}][B(C-B)^{-1}(C-A)]\right\\}A^{-1}$
$\displaystyle=A\cdot c_{r}(A,C;B,D)\cdot A^{-1}.$

therefore, we can say that the points, $c_{r}(A,C;B,D)$ and
$c_{r}(A^{-1},B^{-1};C^{-1},D^{-1})$ are _conjugatet-points_ in a line of
Desargues affine plane. ∎

######  Corollary 1.

If the point $A\in z[K]$ (center of skew field $K=(\ell^{OI},+,\cdot)$), then,

$c_{r}(A,C;B,D)=c_{r}(A^{-1},B^{-1};C^{-1},D^{-1}).$

###### Proof.

If $A\in z[K]$ then, $AX=XA,\forall X\in K$, so $AXA^{-1}=X,\forall X\in K$.
So, for $A\in z[K]$, we have that,

$A\cdot c_{r}(A,C;B,D)\cdot A^{-1}=c_{r}(A,C;B,D).$

Hence

$c_{r}(A^{-1},B^{-1};C^{-1},D^{-1})=c_{r}(A,C;B,D)\Leftrightarrow\text{if
$A\in z[K]$}$

∎

######  Corollary 2.

In Papus affine plane, $c_{r}(A,C;B,D)=c_{r}(A^{-1},B^{-1};C^{-1},D^{-1})$.

######  Theorem 8.

If $A,B,C,D$ are distinct points in a line, in Desargues affine plane and $I$
is unitary point for multiplications of points in same line, have,

$c_{r}(A,B;C,D)\neq c_{r}(B,A;D,C)$

so, $c_{r}(A,B;C,D)$ is different point from $c_{r}(B,A;D,C)$.

###### Proof.

From Definition of Cross-Ratio we have,

$c_{r}(A,B;C,D)=\left[(A-D)^{-1}(B-D)\right]\left[(B-C)^{-1}(A-C)\right]=r(B,A;D)\cdot
r(A,B;C)$

and

$c_{r}(B,A;D,C)=\left[(B-C)^{-1}(A-C)\right]\left[(A-D)^{-1}(B-D)\right]=r(A,B;C)\cdot
r(B,A;D)$

We mark the points, like below

$R_{1}=r(A,B;C)\quad\text{and}\quad R_{2}=r(B,A;D)$

so

$c_{r}(A,B;C,D)=R_{2}\cdot R_{1}\quad\text{and}\quad c_{r}(B,A;D,C)=R_{1}\cdot
R_{2}$

This points are in $\ell^{OI}--$line in Desargues affine plane, so are
elements of the skew fields $K=(\ell^{OI},+,\cdot)$, which are constructet
over this line, so $E,F,G,H\in K$. So we have,

$R_{2}\cdot R_{1}\neq R_{1}\cdot R_{2}\Rightarrow c_{r}(A,B;C,D)\neq
c_{r}(B,A;D,C)$

∎

######  Corollary 3.

If $A,B,C,D\in\ell^{OI}$ are distinct points in a line, in Pappus affine plane
and $I$ is unital point for multiplications, then

$c_{r}(A,B;C,D)=c_{r}(B,A;D,C)$

###### Proof.

If affine plane is Pappian plane, then the skew-field $(\ell^{OI},+,\cdot)$ is
commutative, then is a Field. ∎

We marked with $K=(\ell^{OI},+,\cdot)$ the skew field over $\ell^{OI}-$line in
Desargues affine plane, we know that the center of the skew field $z[K]$, is a
sub-skew field of $K$, moreover, $z[K]$ it is also commutative.

######  Theorem 9.

If $A,B,C,D\in\ell^{OI}$ are distinct points in a line, in Desargues affine
plane and $I$ is unital point for multiplications of points in same line, then
equation

$c_{r}(A,B;C,D)=c_{r}(B,A;D,C)$

it’s true, if

(a):

points $A,B,C,D$ are in ’center of skew-field’ $z[K]$;

(b):

ratio-points $r(A,B;C)$ are in ’center of skew-field’;

(c):

ratio-point $r(B,A;D)$ are in ’center of skew-field’;

(d):

ratio-point $r(A,B;D)$ is in centaralizer of point $r(A,B;C)$, or vice versa.

###### Proof.

(a) If points $A,B,C,D\in z[K]$, we have that,

$A-D,B-D,B-C,A-C\in z[K]$

and

$(A-D)^{-1},(B-D)^{-1},(B-C)^{-1},(A-C)^{-1}\in z[K]$

also the production is commutative. Hence,

$[(A-D)^{-1}(B-D)]\cdot[(B-C)^{-1}(A-C)]=[(B-C)^{-1}(A-C)]\cdot[(A-D)^{-1}(B-D)]$

so,

$c_{r}(A,B;C,D)=c_{r}(B,A;D,C).$

(b) If ratio-points $r(A,B;C)$ are in ’center of skew-field’, we have that,

$X\cdot r(A,B;C)=r(A,B;C)\cdot X,\quad\forall X\in K\text{(so for all points
$X\in\ell^{OI}$)}$

so the equation is also true for the ratio-point $r(B,A;D)$, and have,

$\displaystyle r(A,B;C)\cdot r(B,A;D)$ $\displaystyle=r(B,A;D)\cdot r(A,B;C)$
$\displaystyle[(B-C)^{-1}(A-C)]\cdot[(A-D)^{-1}(B-D)]$
$\displaystyle=[(A-D)^{-1}(B-D)]\cdot[(B-C)^{-1}(A-C)]$ $\displaystyle
c_{r}(B,A;D,C)$ $\displaystyle=c_{r}(A,B;C,D).$

(c) in the same way, as in case (b).

(d) The Centralizer $\mathcal{C}_{K}(r(A,B;C))=\\{Y\in K|Y\cdot
r(A,B;C)=r(A,B;C)\cdot Y\\}$, and we have that,
$r(B,A;D)\in\mathcal{C}_{K}(r(A,B;C))$, so we have,

$r(B,A;D)\cdot r(A,B;C)=r(A,B;C)\cdot r(B,A;D)$

so,

$c_{r}(A,B;C,D)=c_{r}(B,A;D,C),$

in the same way, it is proved that if $r(A,B;C)\in\mathcal{C}_{K}(r(B,A;D))$,
then $c_{r}(A,B;C,D)=c_{r}(B,A;D,C).$ ∎

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