# Exponential correction to Friedmann equations

Özgür Ökcüa,b Ekrem Aydinera,b
aTheoretical and Computational Physics Research Laboratory, Istanbul
University,
Istanbul, 34134, Türkiye
bDepartment of Physics, Faculty of Science, Istanbul University,
Istanbul, 34134, Türkiye Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, employing the exponential corrected entropy (Chatterjee and
Ghosh in Phys Rev Lett 125:041302, 2020), we derive the modified Friedmann
equations from the first law of thermodynamics at apparent horizon and
Verlinde’s entropic gravity scenario. First, we derive the modified Friedmann
equations from the first law of thermodynamics. We investigate the validity of
generalised second law (GSL) of thermodynamics and find that it is always
satisfied for the all eras of universe. Moreover, we investigate the
deceleration parameter for the case $k=0$ in two frameworks. Finally, we
numerically study the bouncing behaviour for the modified Friedmann equations
obtained from entropic gravity. The results indicate that the bouncing
behaviour is possible for the cases $k=1$ and $k=-1$.

Keywords Thermodynamics–gravity conjecture· Entropic gravity· FRW cosmology·
Exponential corrected entropy

##  1 Introduction

Black hole thermodynamics plays a fundamental role in setting a deep
connection among general relativity, thermodynamics, and quantum mechanics [1,
2, 3, 4, 5, 6, 7, 8]. Although this deep connection is not fully known,
indeed, it is an outcome of black hole entropy $S$ and temperature $T$ being
proportional to horizon area $A_{h}$ and surface gravity $\kappa$,
respectively. Drawn inspiration from black hole thermodynamics, Jacobson
obtained the Einstein field equations from the first law of black hole
thermodynamics [9]. Combining entropy$-$area relation with the Clausius
relation, he derived the field equation from thermodynamical viewpoint.
Therefore, the field equation can be interpreted as an equation of state. The
correspondence between gravity and thermodynamics is generally known as the
thermodynamics-gravity conjecture and the research in this direction is rich
with the studies in the literature [10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 27, 24, 25, 26, 28, 29, 30, 31, 32, 41, 42, 43, 33, 34, 50,
51, 52, 53, 54, 55, 56, 35, 36, 37, 38, 39, 40, 44, 58, 59, 60, 57, 45, 46,
47, 48, 49, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
77]. For example, obtaining the Einstein field equations from the first law of
thermodynamics is an interesting topic in the thermodynamics-gravity
conjecture [10, 11, 12, 13, 14]. Another interesting topic aims to understand
the relation between the first law and Friedmann equations. In Ref. [15],
defining the first law of thermodynamics as $-dE=TdS$ at the apparent horizon
$\tilde{r_{A}}$, Cai and Kim derived the higher dimensional Friedmann
equations. Here $-dE$, $T_{h}$ and $dS_{h}$ correspond to the energy flux
crossing the horizon, temperature and change in entropy, respectively. Using
entropy-area relations of Gauss-Bonnet and Lovelock gravity theories, they
obtained the Friedmann equations [15]. Using the thermodynamical arguments of
Ref. [15], the Friedmann equations in $f(R)$ gravity and scalar-tensor gravity
theories can be found in Ref. [16]. Assuming a slow change of the apparent
horizon, the temperature is defined by [15] 111The units
$\hbar=c=G_{N}=L^{2}_{Pl}=k_{B}=1$ are used throughout the paper.

$T_{h}=\frac{1}{2\pi\tilde{r_{A}}}.$ (1)

The entropy of apparent horizon is given by [15]

$S_{h}=\frac{A_{h}}{4},$ (2)

where the horizon area is given by $A_{h}=4\pi\tilde{r_{A}}^{2}$. The
temperature in Eq. (1) is not proportional to the surface gravity at the
apparent horizon since it is an approximation for the small change of horizon.
The surface gravity is defined as follows [17, 78]:

$\kappa=-\frac{1}{\tilde{r_{A}}}\left(1-\frac{\dot{\tilde{r_{A}}}}{2H\tilde{r_{A}}}\right),$
(3)

where dot is the derivative with respect to time, and $H$ is the Hubble
parameter. From the proportionality of temperature and apparent horizon, we
have [17]

$T_{h}=\frac{\kappa}{2\pi}=-\frac{1}{2\pi\tilde{r_{A}}}\left(1-\frac{\dot{\tilde{r_{A}}}}{2H\tilde{r_{A}}}\right).$
(4)

Taking into account the above equation with the entropy-area relation (2)
requires the revising the first law of thermodynamics as follows:

$dE=T_{h}dS_{h}+WdV,$ (5)

where $E$, $W$ and $V$ are total energy inside the apparent horizon, the work
density and the volume, respectively. Recently, there has been extensive
research in the literature on Friedmann equations and apparent horizon
thermodynamics [18, 19, 20, 21, 22, 23, 27, 24, 25, 26, 28, 29, 30, 31, 32,
41, 42, 43, 33, 34, 50, 51, 52, 53, 54, 55, 56, 35, 36, 37, 38, 39, 40, 44,
58, 59, 60, 57, 45, 46, 47, 48, 49].

The connection between gravity and thermodynamics can also be handled in
Verlinde’s entropic gravity [61]. Gravity is not regarded as a fundamental
force. According to Verlinde, gravity is an entropic force based on two
principles, namely, the equipartition law of energy and the holographic
principle. When a test particle has a displacement $\Delta x$ from the
holographic screen, the entropic force is defined by $F\Delta x=T\Delta S$.
Here $\Delta S$ and $T$ correspond to entropy change and temperature of the
screen, respectively. Employing a holographic screen with Unruh temperature
yields Newton’s second law. Moreover, starting from the holographic principle
and the equipartition law of energy, Newton’s gravitational law and Einstein
field equations can be derived in entropic gravity scenario. The entropic
gravity theory has attracted a lot of attention in recent years. There are
many studies devoted to derivations of Newton’s gravitational law, Einstein
field equations, and Friedmann equations in the literature [62, 63, 64, 65,
66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77].

Motivated by numerous reasons, modifications of entropy-area relation have
been studied in the literature [79, 80, 81, 82, 83, 84, 85, 88, 86, 87, 89,
90]. Different correction scenarios may provide new insights on both black
hole thermodynamics and the evolution of universe. Taking into account quantum
gravitational effects at Planck scale yields a well-known logarithmic
correction to entropy [79, 80, 81, 82, 83, 84, 85]. For example, loop quantum
gravity [20, 21], generalized uncertainty principle (GUP) [22, 23, 24, 25],
extended uncertainty principle (EUP) [26, 27], generalized and extended
uncertainty principle (GEUP) [28] and rainbow gravity [73, 74] modifications
of Friedmann equations can be found in the literature. The modifications of
standard uncertainty principle imply a non singular beginning of universe due
to the minimal length notion [22, 23, 24, 25, 28]. Another interesting
modification of entropy-area relation is based on the generalized statistics.
For example, Kaniadakis statistics originates from the relativistic statistics
mechanics [86, 87]. Kaniadakis entropy modified Friedmann equations were
investigated in Refs. [38, 39, 40, 75]. Apart from logarithmic correction,
there is another form of correction suggesting that the entropy is a power law
function of its area [88, 89, 90]. For example, Tsallis statistics suggests
that entropy must be nonadditive [88]. Tsallis modified Friedmann equations
can be found in Refs. [41, 42, 43]. Inspired by COVID-19, Barrow proposed that
the black hole area may be deformed due to the quantum gravitational effects
[89]. The extensions of Barrow entropy to cosmological applications can be
found in Refs. [50, 51, 52, 53, 54, 55, 56]. Just like Tsallis and Barrow
entropies, fractional entropy based on fractional quantum mechanics also
suggests that entropy is a power law function of its area [90]. In Ref. [59],
we obtained the fractional entropy modified Friedmann equations from first law
and entropic gravity.

The leading order correction to entropy-area relation may be regarded as
logarithmic [79, 80, 81, 82, 83, 84, 85]. The logarithmic term becomes
important when the black hole size shrinks. For example, thermal fluctuations
around equilibrium temperature lead to a logarithmic corrected entropy-area
relation [82]. This correction can be considered a perturbative correction.
However, non-perturbative corrections to black hole entropy are inevitable
when black hole becomes even smaller due to the Hawking radiation. Recently,
authors of Ref. [91] argued that the black hole entropy has such a correction
as an exponential term. They obtained exponential correction to entropy only
employing horizon geometry without resorting to loop quantum gravity or string
theory. Counting only the microstates for quantum states residing on black
hole horizon may yield an exponential correction in black hole entropy. The
exponential corrected entropy is given by [91]

$S=\frac{A}{4}+\alpha e^{-A/4}.$ (6)

In order to track the effects of exponential correction, we add the parameter
$\alpha$. This idea can originally be traced back to Ref. [83]. In the limit
$\alpha\rightarrow 0$, the above expression reduces to semiclassical result,
i.e., Eq. (2). The exponential correction is negligible for large areas, while
it gives a remarkable contribution to black hole entropy when black hole
becomes small. The effects of exponential correction were investigated for
black hole thermodynamics in Refs. [92, 93]. However, the effects of
exponential entropy have not yet been studied for cosmology. The non-
perturbative modification as an exponential term generates more sensitive
correction to Friedmann equations at small length scales than the perturbative
correction. Therefore, in this paper, we consider the modified Friedmann
equations for exponential corrected entropy. In order to derive the modified
Friedmann equations, we use two different approaches, i.e., the first law of
thermodynamics at apparent horizon and entropic gravity case. We try to
understand the effects of exponential correction for the cosmology.

The paper is organised as follows: In Section 2, we derive the modified
Friedmann equations from the first law of thermodynamics at apparent horizon.
In order to grasp the effects of exponential correction, we compute the
deceleration parameter. Then, we investigate the validity of GSL. In Section
3, we obtain the modified Friedmann equations from entropic gravity scenario.
Similarly, we compute the deceleration parameter. Then, we investigate the
bouncing behaviour of modified Friedmann equations in entropic gravity case.
Finally, we close this paper in Section 4.

##  2 Friedmann equations from the first law of thermodynamics

First, we begin to review the basic elements of Friedmann-Robertson-Walker
(FRW) universe. In the compact form, the line element is given by

$ds^{2}=h_{ab}dx^{a}dx^{b}+\tilde{r}^{2}d\Omega^{2},$ (7)

where $\tilde{r}\,=\,a(t)r$, the coordinates $x^{a}\,=\,(t,r)$ and two-
dimensional metric $h_{ab}=\\\ diag\left(-1,a^{2}/(1-kr^{2})\right)$. Here the
spatial curvature constant $k=$ $-1$, $0$ and $1$ correspond to open, flat and
closed universe, respectively. The apparent horizon $\tilde{r_{A}}$ is defined
by

$\tilde{r_{A}}=ar=\frac{1}{\sqrt{H^{2}+k/a^{2}}},$ (8)

where $H=\dot{a}/a$. We assume the energy and matter of universe as a perfect
fluid. So we can write energy-momentum tensor as follows:

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}.$ (9)

Here $\rho$, $p$ and $u^{\mu}$ correspond to energy density, pressure and
four-velocity of fluid, respectively. From the conservation of energy-momentum
tensor, i.e., $\nabla_{\mu}T^{\mu\nu}=0$, one can obtain the continuity
equation as follows:

$\dot{\rho}+3H(\rho+p)=0.$ (10)

The work density $W$ is given by [78]

$W=-\frac{1}{2}T^{ab}h_{ab}=\frac{1}{2}(\rho-p).$ (11)

Here, work density $W$ is the the work by the volume change of the universe.
The volume is given by [17]

$V=\frac{4}{3}\pi\tilde{r_{A}}^{3}.$ (12)

Let us begin to calculate the terms of first law in Eq. (5). Using the Eqs.
(10) and (12), $dE$ is given by

$dE=\rho
dV+Vd\rho=4\pi\rho\tilde{r_{A}}^{2}d\tilde{r_{A}}-4\pi(\rho+p)\tilde{r_{A}}^{3}Hdt.$
(13)

From Eqs. (11) and (12), the term $WdV$ is found

$WdV=2\pi(\rho-p)\tilde{r_{A}}^{2}d\tilde{r_{A}}.$ (14)

The general expression of entropy is given by [22]

$S_{h}=\frac{f(A_{h})}{4},$ (15)

and its differential is expressed

$\frac{dS_{h}}{dA_{h}}=\frac{f^{\prime}(A_{h})}{4},$ (16)

where prime is the derivative with respect to area. From Eq. (6) and the above
expression, one can find

$f^{\prime}(A_{h})=1-\alpha e^{-\pi\tilde{r_{A}}^{2}}.$ (17)

At last, one can obtain the term $T_{h}dS_{h}$ as

$T_{h}dS_{h}=-\left(1-\frac{\dot{\tilde{r_{A}}}}{2H\tilde{r_{A}}}\right)f^{\prime}(A_{h})d\tilde{r_{A}}.$
(18)

Combining the terms $dE$,$WdV$, $T_{h}dS_{h}$ in the first law of
thermodynamics at apparent horizon and employing the differential of apparent
horizon

$d\tilde{r_{A}}=-H\tilde{r_{A}}^{3}\left(\dot{H}-\frac{k}{a^{2}}\right)dt,$
(19)

we obtain

$4\pi(\rho+p)\tilde{r_{A}}^{3}Hdt=f^{\prime}(A_{h})d\tilde{r_{A}}.$ (20)

Employing the continuity equation (10) in Eq. (20), one can obtain the
differential form of Friedmann equation as follows:

$\frac{f^{\prime}(A_{h})}{\tilde{r_{A}}^{3}}d\tilde{r_{A}}=-\frac{4\pi}{3}d\rho.$
(21)

Using Eq. (17) in Eq. (21) and integrating the above expression, we obtain the
first Friedmann equation

$-\frac{1}{2\tilde{r_{A}}^{2}}+\frac{\alpha}{2\tilde{r_{A}}^{2}}\left[e^{-\pi\tilde{r_{A}}^{2}}+\pi\tilde{r_{A}}^{2}Ei\left(-\pi\tilde{r_{A}}^{2}\right)\right]=-\frac{4\pi}{3}\rho,$
(22)

where we set the integration constant to zero. Here $Ei$ is the exponential
integral function. The correction term gives a remarkable contribution to
Friedmann equation in small length scale. However, the above equation does not
imply a non-zero minimum apparent horizon. Instead, the apparent horizon is
still allowed to go to zero. Thus, the modified Friedmann equation does not
remove the initial singularity. From Eqs. (17), (19) and (20), the second
Friedmann equation is given by

$-4\pi(\rho+p)=\left(\dot{H}-\frac{k}{a^{2}}\right)(1-\alpha
e^{-\pi\tilde{r_{A}}^{2}}).$ (23)

In the limit $\alpha\rightarrow 0$, these modified Friedmann equations reduce
to standard Friedmann equations. Using Eq. (8), the modified Friedmann
equations can be written in term of Hubble parameter 222The exponential
integral correction term to first Friedmann equation was also reported in Ref.
[60]. Recently, using modified gravitational potentials in Verlinde’s entropic
gravity scenario, various entropies were constructed in Ref. [60]. The author
obtained the following entropy
$S=\frac{1}{4}\left[A-2\beta\lambda^{2}e^{-\sqrt{\frac{A}{4\pi\lambda^{2}}}}\left(12\pi+6\sqrt{\frac{\pi
A}{\lambda^{2}}}+\frac{A}{\lambda^{2}}\right)\right]$ (24) from Yukawa-type of
potential (See Eq. (6) in Ref. [60]). Here, $\beta$ and $\lambda$ correspond
to strength and range, respectively. Employing the the above entropy in the
first law of thermodynamics at apparent horizon, the author obtained the
following Friedmann equation as follows:
$\left(H^{2}+\frac{k}{a^{2}}\right)-\frac{\beta}{\lambda^{2}}\left[\lambda\left(H^{2}+\frac{k}{a^{2}}\right)e^{\frac{1}{\lambda\left(\sqrt{H^{2}+k/a^{2}}\right)}}\left(-\lambda+\frac{\left(1-2\sqrt{\pi}\right)}{\sqrt{H^{2}+k/a^{2}}}\right)+\left(1-2\sqrt{\pi}\right)Ei\left(\frac{-1}{\lambda\sqrt{H^{2}+k/a^{2}}}\right)\right]=\frac{8\pi\rho}{3}.$
(25)

$\displaystyle\left(H^{2}+\frac{k}{a^{2}}\right)\left[1-\alpha\left(e^{-\pi\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}+\pi\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}Ei\left(-\pi\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}\right)\right)\right]=\frac{8\pi\rho}{3},$
$\displaystyle\left(\dot{H}-\frac{k}{a^{2}}\right)\left(1-\alpha
e^{-\pi\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}\right)=-4\pi(\rho+p).$ (26)

Now, in order to see the effects of exponential correction, we consider the
deceleration parameter. The deceleration parameter $q$ is defined by

$q=1-\frac{\dot{H}}{H^{2}},$ (27)

where the case $q>0$ corresponds to decelerated expansion phase of universe,
while the case $q<0$ corresponds to accelerated expansion phase of universe.
Using the modified Friedmann equations (2) and choosing equation of state as
$p=\omega\rho$, one can obtain the expression for the declaration parameter as

$q=-1+\frac{3}{2}\frac{(1+\omega)}{1-\alpha
e^{-\pi/H^{2}}}\left[1-\alpha\left(e^{-\pi/H^{2}}+\frac{\pi}{H^{2}}Ei\left(-\frac{\pi}{H^{2}}\right)\right)\right].$
(28)

Here we consider the flat case $k=0$ in agreement with cosmological
observations [94]. Since quantum gravity effects are sufficiently tiny for the
radiation and matter dominated eras, one can expand the deceleration parameter

$q=-1+\frac{3(1+\omega)}{2}-\frac{3\pi\alpha(1+\omega)}{2H^{2}}Ei\left(-\frac{\pi}{H^{2}}\right)+...\,.$
(29)

The deceleration parameter is given by

$q_{m}=\frac{1}{2}-\frac{3\pi\alpha}{2H^{2}}Ei\left(-\frac{\pi}{H^{2}}\right),\qquad\qquad\qquad\qquad
q_{r}=1-\frac{2\pi\alpha}{H^{2}}Ei\left(-\frac{\pi}{H^{2}}\right)$ (30)

for the matter ($\omega=0$) and radiation ($\omega=1/3$) dominated eras,
respectively. For $H>0$, the correction terms give positive contribution to
deceleration parameter. Therefore, the universe is more decelerated for the
matter and radiation dominated eras.

###  2.1 Generalised second law of thermodynamics

Now, we consider the total entropy change of universe in the presence of
exponential correction. Reorganising Eq. (20), we obtain

$\dot{\tilde{r_{A}}}=\frac{4\pi(\rho+p)\tilde{r_{A}}^{3}H}{1-\alpha
e^{-\pi\tilde{r_{A}}^{2}}}$ (31)

Substituting the above equation and Eq. (17) into Eq. (18), we find

$T_{h}\dot{S_{h}}=4\pi(\rho+p)H\tilde{r_{A}}^{3}\left[1-\frac{2\pi(\rho+p)\tilde{r_{A}}^{2}}{1-\alpha
e^{-\pi\tilde{r_{A}}^{2}}}\right].$ (32)

The above expression does not guarantee the validity of second law. The second
law of thermodynamics may be violated since $(\rho+p)$ is negative during the
accelerated phase. Therefore, we need the consider matter fields inside the
horizon. Thus, we must check the validity of GSL. Let us begin to consider the
Gibbs equation given by [95]

$T_{f}dS_{f}=d(\rho V)+pdV=Vd\rho+(p+\rho)dV,$ (33)

where $S_{f}$ and $T_{f}$ are the entropy and temperature of matter fields
inside the horizon. We assume the thermal equilibrium between the apparent
horizon and matter fields, i.e., $T_{h}=T_{f}$. This assumption prevents the
energy flows between the horizon and matter fields. Thus, the FRW geometry is
preserved [95]. From Eqs. (10), (12), (31) and Gibbs equation (33), the
entropy change of matter fields inside the apparent horizon is given by

$T_{h}\dot{S_{f}}=-4\pi(\rho+p)H\tilde{r_{A}}^{3}\left(1-\frac{4\pi(\rho+p)\tilde{r_{A}}^{2}}{1-\alpha
e^{-\pi\tilde{r_{A}}^{2}}}\right).$ (34)

Finally, from Eq. (32) and the above equation, the total entropy evolution of
universe is expressed as follows:

$T_{h}(\dot{S_{h}}+\dot{S_{f}})=\frac{8\pi^{2}(\rho+p)^{2}H\tilde{r_{A}}^{5}}{1-\alpha
e^{-\pi\tilde{r_{A}}^{2}}}.$ (35)

In the limit $\alpha\rightarrow 0$, it is clear that the above expression
reduces to the standard form and GSL is always satisfied. It may seem that
$1-\alpha e^{-\pi\tilde{r_{A}}^{2}}>0$ must hold to satisfy the GSL. This
condition yields a lower bound $\tilde{r_{A}}>\sqrt{\frac{\ln\alpha}{\pi}}$.
However, this lower bound is imaginary in the range $\alpha\in(0,1)$. For
$\alpha=1$, the condition imply that $\tilde{r_{A}}>0$. The condition
$1-\alpha e^{-\pi\tilde{r_{A}}^{2}}>0$ always holds for $\tilde{r_{A}}>0$.
Thus, the right hand side of Eq. (35) never decreases with respect to time.
Therefore, the GSL is always valid during the all eras of the universe for any
spatial curvature.

##  3 Friedmann equations from the entropic gravity

Based on the arguments of Refs. [61, 62], we are going to obtain the modified
Friedmann equations. First, we consider a compact spatial region $\mathcal{V}$
with a holographic screen on the boundary $\mathcal{\partial V}$. Considering
the number of the bits on the holographic screen

$N=A_{h},$ (36)

with Eq. (2), the relation between $N$ and $S_{h}$ is given by

$N=4S_{h}.$ (37)

Another important assumption is that the total entropy of the screen is given
by the equipartition law of energy. It is defined by

$E=\frac{1}{2}NT_{U},$ (38)

where $T_{U}$ is the Unruh temperature of the screen. The Unruh temperature is
given by

$T=\frac{a_{r}}{2\pi}=-\frac{\ddot{a}r}{2\pi},$ (39)

where $a_{r}$ corresponds to the acceleration. It is defined by [62]

$a_{r}=-\frac{d^{2}\tilde{r_{A}}}{dt^{2}}=-\ddot{a}r.$ (40)

We also consider the active gravitational mass $\mathcal{M}$ instead of total
mass $M$ inside the horizon. $\mathcal{M}$ is given by [62]

$\mathcal{M}=2\int_{\mathcal{V}}dV\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)u^{\mu}u^{\nu}=\frac{4}{3}\pi(\rho+3p)\widetilde{r_{A}}^{3}.$
(41)

Using Eqs. (6) and (37), the modified number of bits is given by

$N=4\pi\tilde{r_{A}}^{2}+4\alpha e^{-\pi\tilde{r_{A}}^{2}}.$ (42)

Considering the assumption $\mathcal{M}=E$, from Eqs. (38)-(42), we can derive
the acceleration equation

$-\frac{\ddot{a}}{a}\left(1+\frac{\alpha}{\pi\tilde{r_{A}}^{2}}e^{-\pi\tilde{r_{A}}^{2}}\right)=\frac{4\pi}{3}(\rho+3p).$
(43)

Multiplying $2a\dot{a}$ on the both side of the above equation and employing
the continuity equation (10), we find

$\int\frac{d\dot{a}^{2}}{dt}dt=\frac{8\pi}{3}\int\frac{1}{\left(1+\frac{\alpha
e^{-\pi a^{2}r^{2}}}{\pi a^{2}r^{2}}\right)}\frac{d(\rho a^{2})}{dt}dt.$ (44)

Moreover, in order to solve this integral, we consider the equation of state
as $p=\omega\rho$. Using this equation of state in continuity equation (10)
yields

$\rho=\rho_{0}a^{-3(1+\omega)},$ (45)

where $\rho_{0}$ is a constant. From the above equation, we get

$d(\rho a^{2})=-\rho_{0}(1+3\omega)a^{-2-3\omega}da.$ (46)

Using Eq. (46) and equation of state $p=\omega\rho$ in Eq. (44), the integral
is given by

$\int\frac{d\dot{a}^{2}}{dt}dt=-\frac{8\pi\rho_{0}(1+\omega)}{3}\int\frac{a^{-2-3\omega}}{\left(1+\frac{\alpha
e^{-\pi a^{2}r^{2}}}{\pi a^{2}r^{2}}\right)}\frac{da}{dt}dt.$ (47)

In order to solve this integral, we consider the small exponential correction.
Then, the above equation can be written by

$\int\frac{d\dot{a}^{2}}{dt}dt=-\frac{8\pi\rho_{0}(1+\omega)}{3}\int
a^{-2-3\omega}\left(1+\frac{\alpha e^{-\pi a^{2}r^{2}}}{\pi
a^{2}r^{2}}\right)\frac{da}{dt}dt,$ (48)

where we use the assumption $(1+\beta)^{n}\approx 1+\beta n$ for $\beta\ll 1$.
Solving this integral, one can obtain the first Friedmann equation as follows:

$\left(\frac{\dot{a}}{a}\right)^{2}+\frac{k}{a^{2}}=\frac{8\pi\rho}{3}\left(1-\frac{\alpha(3\omega+1)}{2\pi\tilde{r_{A}}^{2}}E_{\frac{3\omega+5}{2}}(\pi\tilde{r_{A}}^{2})\right),$
(49)

where $E_{n}$ is the exponential integral function. Using $H=\dot{a}/a$, the
Friedmann equations are written in term of Hubble parameter

$\displaystyle
H^{2}+\frac{k}{a^{2}}=\frac{8\pi\rho}{3}\left[1-\frac{\alpha(3\omega+1)}{2\pi}\left(H^{2}+\frac{k}{a^{2}}\right)E_{\frac{3\omega+5}{2}}\left(\pi\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}\right)\right],$
$\displaystyle(\dot{H}+H^{2})\left[1+\frac{\alpha}{\pi}\left(H^{2}+\frac{k}{a^{2}}\right)e^{-\pi\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}\right]=-\frac{4\pi}{3}(\rho+3p).$
(50)

As can be seen, the Friedmann equations obtained in this section are different
from the Friedmann equations obtained in the previous section. In the previous
section, the modified Friedmann equations are exactly derived. In this
section, we approximately obtain the modified Friedmann equations. Moreover,
the Friedmann equations obtained in this section depend on a specific equation
of state, i.e., $p=\omega\rho$. Therefore, Eqs. (2) and (3) are not the same.

Again, we consider the deceleration parameter for $k=0$. From Eqs. (27) and
(3), the deceleration parameter is given by

$q=\frac{3\omega+1}{2}\left[1-\frac{\alpha(3\omega+1)H^{2}}{2\pi}E_{\frac{3\omega+5}{2}}\left(\frac{\pi}{H^{2}}\right)\right]^{-1}\left(1+\frac{\alpha}{\pi}H^{2}e^{-\frac{\pi}{H^{2}}}\right)^{-1}.$
(51)

For small quantum gravity effects, the above equation can be approximated as

$q\approx\frac{3\omega+1}{2}\left[1+\frac{\alpha
e^{-\pi/H^{2}}H^{2}}{2\pi}\left((1+3\omega)e^{\pi/H^{2}}E_{\frac{3\omega+5}{2}}\left(\frac{\pi}{H^{2}}\right)-2\right)\right].$
(52)

The deceleration parameter is given by

$q_{m}=\frac{1}{2}\left[1-\frac{\alpha
e^{-\pi/H^{2}}H^{2}}{2\pi}\left(2-e^{\pi/H^{2}}E_{\frac{5}{2}}\left(\frac{\pi}{H^{2}}\right)\right)\right],\qquad
q_{r}=1-\frac{\alpha
e^{-\pi/H^{2}}H^{2}}{\pi}\left(1-e^{\pi/H^{2}}E_{3}\left(\frac{\pi}{H^{2}}\right)\right)$
(53)

for the matter ($\omega=0$) and radiation ($\omega=1/3$) dominated eras,
respectively. The modified deceleration parameter of radiation dominated era
is positive for the condition

$\frac{\pi}{e^{-\pi/H^{2}}H^{2}\left(1-e^{\pi/H^{2}}E_{3}\left(\frac{\pi}{H^{2}}\right)\right)}>\alpha.$
(54)

For the matter dominated era, deceleration parameter is positive when the
condition

$\frac{2\pi
e^{\pi/H^{2}}}{H^{2}\left(2-e^{\pi/H^{2}}E_{5/2}\left(\frac{\pi}{H^{2}}\right)\right)}>\alpha$
(55)

is satisfied.

###  3.1 Bouncing universe

In this section, we are going to investigate the bouncing behaviour for the
modified Friedmann equations obtained from Verlinde’s entropic gravity. The
most striking feature of bouncing behaviour is the absence of the initial
singularity. In the bouncing scenario, the universe begins from a contraction
phase with $\dot{a}<0$, goes through a non-vanishing and minimal scale factor
$a_{c}$ (critical point), then starts to accelerate in expansion phase. The
bouncing behaviour has been intensively studied in the literature [76, 96, 97,
98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113,
114]. Mathematically, the bouncing conditions are expressed as $a_{c}>0$,
$\ddot{a_{c}}>0$, $H|_{a=a_{c}}=0$ at the critical point $a_{c}$ [98, 111].

The correction terms in the first Friedmann equation (3) can be written by

$H^{2}+\frac{k}{a^{2}}=\frac{8\pi\rho}{3}\left[1-\frac{4\alpha(3\omega+1)\rho}{3}E_{\frac{3\omega+5}{2}}\left(\frac{3}{8\rho}\right)\right].$
(56)

At the critical point, the above equation is given by

$\frac{k}{a_{c}^{2}}=\frac{8\pi\rho_{c}}{3}\left[1-\frac{4\alpha(3\omega+1)\rho_{c}}{3}E_{\frac{3\omega+5}{2}}\left(\frac{3}{8\rho_{c}}\right)\right],$
(57)

where $\rho_{c}$ is the energy density at the critical point. Solving this
equation, we find the solution of the critical scale factor

$a_{c}=\frac{\pm
3\sqrt{k}}{2\sqrt{2\pi\rho_{c}\left[3-4\alpha\left(1+3\omega\right)\rho_{c}E_{\frac{3\omega+5}{2}}\left(\frac{3}{8\rho_{c}}\right)\right]}}.$
(58)

We consider the positive solution since it is only physically meaningful. At
the critical point, the acceleration equation (43) is represented by

$\frac{\ddot{a_{c}}}{a_{c}}=-\frac{4\pi}{3}\left(\rho_{c}+3p_{c}\right)\left[1-\frac{8\alpha\rho_{c}}{3}e^{-\frac{3}{8\rho_{c}}}\right],$
(59)

where $p_{c}$ is the pressure at critical point.

The condition $\ddot{a_{c}}>0$ is satisfied for two cases. The first case
corresponds to $\left(\rho_{c}+3p_{c}\right)>0$ and
$1-\frac{8\alpha\rho_{c}}{3}e^{-\frac{3}{8\rho_{c}}}<0$, while the second case
corresponds to $\left(\rho_{c}+3p_{c}\right)<0$ and
$1-\frac{8\alpha\rho_{c}}{3}e^{-\frac{3}{8\rho_{c}}}>0$. Here we consider the
first case. Using the equation of state $p=\omega\rho$ and Eq. (58), Eq. (59)
is expressed by

$\ddot{a_{c}}=\frac{-\sqrt{2\pi
k}\left(1+3\omega\right)\rho_{c}}{\sqrt{\rho_{c}\left[3-4\alpha\left(1+3\omega\right)\rho_{c}E_{\frac{3\omega+5}{2}}\left(\frac{3}{8\rho_{c}}\right)\right]}}\left[1-\frac{8\alpha\rho_{c}}{3}e^{-\frac{3}{8\rho_{c}}}\right].$
(60)

We consider the condition $(3\omega+1)>0$. In order to satisfy
$\ddot{a_{c}}>0$, the condition
$1-\frac{8\alpha\rho_{c}}{3}e^{-\frac{3}{8\rho_{c}}}<0$ yields

$\rho_{c}>\frac{3}{8W(\alpha)},$ (61)

where $W$ is the Lambert function.

In order to get $a_{c}>0$ and $\ddot{a_{c}}>0$, we impose the condition

$3-4\alpha\left(1+3\omega\right)\rho_{c}E_{\frac{3\omega+5}{2}}\left(\frac{3}{8\rho_{c}}\right)>0,$
(62)

from Eqs. (58) and (60) for the case $k=1$. Taking $\alpha=1$, the $a_{c}$ is
positive in the range $0<\rho_{c}<1.753$ from the above condition for the
matter dominated universe $\omega=0$. From the conditions in Eqs. (61) and
(62), we find that $\ddot{a_{c}}$ is positive in the range
$0.661<\rho_{c}<1.753$. Thus, we conclude that the bouncing behaviour is
available in the range $0.661<\rho_{c}<1.753$ since $a_{c}$ and $\ddot{a_{c}}$
are positive.

[width=8.18cm]figures/dustScaleVsDensity.eps
[width=8.18cm]figures/accelVsDensityDust.eps

Figure 1: (a) Critical scale factor vs critical energy density. (b) Critical
$\ddot{a_{c}}$ vs critical energy density. Blue solid and red dashed lines
correspond to closed ($k=1$) and open ($k=-1$) cases, respectively. The empty
circle shows the value of $\rho_{c}$ where $\ddot{a_{c}}$ vanishes, while the
black circle shows the value of $\rho_{c}$ where $a_{c}$ is minimum. We take
$\alpha=1$ and $\omega=0$.

Let us now consider the radiation dominated universe $\omega=1/3$. This case
is relevant to the early stage of the Universe since radiation dominated.
Taking $\alpha=1$, we find that $a_{c}$ is positive in the range
$0<\rho_{c}<1.25$ from Eq. (62). Using Eqs. (61) and (62), we find that
$\ddot{a_{c}}$ is positive in the range $0.661<\rho_{c}<1.25$. We see that the
bouncing behaviour is available in the range $0.661<\rho_{c}<1.25$.

Now, we consider the case $k=-1$. From Eqs. (58) and (60), we impose the
condition

$3-4\alpha\left(1+3\omega\right)\rho_{c}E_{\frac{3\omega+5}{2}}\left(\frac{3}{8\rho_{c}}\right)<0.$
(63)

For the matter and radiation dominated universes, taking $\alpha=1$, we obtain
$a_{c}>0$ and $\ddot{a_{c}}>0$ in the ranges $\rho_{c}>1.753$ and
$\rho_{c}>1.25$, respectively. Therefore, we conclude that the bouncing
behaviour is possible for the case $k=-1$.

[width=8.18cm]figures/radiationScaleVsDensity.eps
[width=8.18cm]figures/accelVsDensityRad.eps

Figure 2: (a) Critical scale factor vs critical energy density. (b) Critical
$\ddot{a_{c}}$ vs critical energy density. Blue solid and red dashed lines
correspond to closed ($k=1$) and open ($k=-1$) cases, respectively. The empty
circle shows the value of $\rho_{c}$ where $\ddot{a_{c}}$ vanishes, while the
black circle shows the value of $\rho_{c}$ where $a_{c}$ is minimum. We take
$\alpha=1$ and $\omega=1/3$.

In Figs. 1 and 2, $a_{c}$ and $\ddot{a_{c}}$ are plotted with respect to
$\rho_{c}$ for matter and radiation cases, respectively. As can be seen from
the figures, $a_{c}$ and $\ddot{a_{c}}$ exhibit different behaviours for
closed ($k=1$) and open ($k=-1$) universes. For the case $k=1$ (blue line),
$a_{c}$ has a minimum value at $\rho_{c}=0.917$ and $\rho_{c}=0.661$ for
matter and radiation cases, respectively. The minimum $a_{c}$ is $0.456$ and
$0.529$ for matter and radiation cases, respectively. In Figs. 1.a and 2.a,
the minimum $a_{c}$ is indicated by a black circle. The minimum $a_{c}$
separates the curve into two branches. $a_{c}$ decreases with respect to
$\rho_{c}$ in the left branch, while it increases with respect to $\rho_{c}$
in the right branch. Although $a_{c}$ is positive in two branches,
$\ddot{a_{c}}$ is negative when $\rho_{c}$ is less than $0.661$. In Figs. 1.b
and 2.b, we show that $\ddot{a_{c}}$ with respect to $\rho_{c}$ for matter and
radiation cases, respectively. As can be seen from the figures, $\ddot{a_{c}}$
is negative when $\ddot{a_{c}}<0.661$. In Figs. 1.a and 2.a, $\ddot{a_{c}}=0$
is represented by an empty circle. For the matter-dominated universe, the
right branch and the small portion of the left branch exhibit bouncing
behavior. Interestingly, $\rho_{c}$ where $a_{c}$ is minimum and $\rho_{c}$
where $\ddot{a_{c}}=0$ are the same for the radiation dominated universe.
Thus, the left branch in Fig. 2.a corresponds to negative $\ddot{a_{c}}$ and
does not exhibit the bouncing behaviour. The bouncing behaviour is only
available in the right branch for the radiation case.

| $\alpha=0.2$ | $\alpha=0.4$ | $\alpha=0.6$ | $\alpha=0.8$
---|---|---|---|---
$a_{c}>0(k=1)$ | $\rho_{c}\in(0,6.462)$ | $\rho_{c}\in(0,3.565)$ | $\rho_{c}\in(0,2.573)$ | $\rho_{c}\in(0,2.065)$
$\ddot{a_{c}}>0(k=1)$ | $\rho_{c}\in(2.22,6.462)$ | $\rho_{c}\in(1.262,3.565)$ | $\rho_{c}\in(0.934,2.573)$ | $\rho_{c}\in(0.765,2.065)$
$a_{c},\ddot{a_{c}}>0(k=-1)$ | $\rho_{c}>6.462$ | $\rho_{c}>3.565$ | $\rho_{c}>2.573$ | $\rho_{c}>2.065$
Table 1: The conditions $a_{c}>0$ and $\ddot{a}_{c}>0$ for the different values of $\alpha$. We take $\omega=0$. | $\alpha=0.2$ | $\alpha=0.4$ | $\alpha=0.6$ | $\alpha=0.8$
---|---|---|---|---
$a_{c}>0(k=1)$ | $\rho_{c}\in(0,4.391)$ | $\rho_{c}\in(0,2.462)$ | $\rho_{c}\in(0,1.8)$ | $\rho_{c}\in(0,1.46)$
$\ddot{a_{c}}>0(k=1)$ | $\rho_{c}\in(2.22,4.391)$ | $\rho_{c}\in(1.262,2.462)$ | $\rho_{c}\in(0.934,1.8)$ | $\rho_{c}\in(0.765,1.46)$
$a_{c},\ddot{a_{c}}>0(k=-1)$ | $\rho_{c}>4.391$ | $\rho_{c}>2.462$ | $\rho_{c}>1.8$ | $\rho_{c}>1.46$
Table 2: The conditions $a_{c}>0$ and $\ddot{a}_{c}>0$ for the different
values of $\alpha$. We take $\omega=1/3$.

The case $k=-1$ clearly shows different behavior. In Figs. 1 and 2, red dashed
line shows the case $k=-1$. As can be seen from Figs. 1.a and 2.a, $a_{c}$
always decreases while $\rho_{c}$ increases. From Figs. 1.b and 2.b, we see
that $\ddot{a_{c}}$ always increases when $\rho_{c}$ increases. In contrast to
the closed case ($k=1$), $\ddot{a_{c}}$ is always positive. Therefore, the
whole branch of $a_{c}$ shows the bouncing behaviour for matter and radiation
cases.

We only present the case $\alpha=1$ in the figures since similar behaviours
can be seen for the different values of $\alpha$. However, the range of
$\rho_{c}$ for the conditions $a_{c}>0$ and $\ddot{a_{c}}>0$ is presented in
Tables 1 and 2 for the different values of $\alpha$. In Tables 3 and 4, we
also give the minimum $a_{c}$ for the different values of $\alpha$. From
Tables 3 and 4, one can see that the minimum $a_{c}$ increases and $\rho_{c}$
decreases when $\alpha$ increases.

Finally, we consider the case $k=0$. We see that $a_{c}$ vanishes and the
condition $a_{c}>0$ is not satisfied. Therefore, we conclude that the bouncing
behaviour is absent for the case $k=0$.

| $\alpha=0.2$ | $\alpha=0.4$ | $\alpha=0.6$ | $\alpha=0.8$
---|---|---|---|---
$\rho_{c}$ | $3.263$ | $1.819$ | $1.326$ | $1.073$
$a_{c}$ | $0.258$ | $0.337$ | $0.388$ | $0.426$
Table 3: The minimum $a_{c}$ for the different values of $\alpha$. We take $\omega=0$ and $k=1$. | $\alpha=0.2$ | $\alpha=0.4$ | $\alpha=0.6$ | $\alpha=0.8$
---|---|---|---|---
$\rho_{c}$ | $2.22$ | $1.262$ | $0.934$ | $0.765$
$a_{c}$ | $0.309$ | $0.399$ | $0.455$ | $0.496$
Table 4: The minimum $a_{c}$ for the different values of $\alpha$. We take
$\omega=1/3$ and $k=1$.

##  4 Conclusions

In this paper, employing the exponential corrected entropy [91], we obtained
the modified Friedmann equations from the first law of thermodynamics and
Verlinde’s entropic gravity. First, we obtained the exponential corrected
Friedmann equation from the first law of thermodynamics at the apparent
horizon [17]. We computed the deceleration parameter for the flat case $k=0$
and found that the expansion for radiation and matter dominated eras is more
decelerated since exponential correction gives the positive contribution to
$q$ parameter. Moreover, we investigated the validity of GSL and showed that
the GSL is always satisfied for the exponential corrected Friedmann equations
obtained from the first law.

Then, we derived the exponential corrected Friedmann equations from Verlinde’s
entropic gravity scenario [61, 62]. Similarly, we computed the deceleration
parameter for the exponential corrected Friedmann equations obtained from
entropic gravity. Unlike the deceleration parameter (30) from the first law,
we found that the deceleration parameter (53) from entropic gravity is
positive for radiation and matter dominated eras when the conditions (54) and
(55) are satisfied.

Finally, we numerically investigated the bouncing behaviour for the matter and
radiation cases. The bouncing behaviour circumvents the initial singularity
since the quantum gravity effects play an important role in the early stages
of universe [111, 76]. We showed that the bouncing behaviour is available for
the cases $k=1$ and $k=-1$. In Figs. 1.a and 2.a, we showed that $a_{c}$ has
two branches for the closed universe ($k=1$). For the matter dominated and
closed universe, we concluded that the right branch and small portion of left
branch in Fig. 1.a exhibit the bouncing behavior. For the radiation dominated
and closed universe, we concluded that the right branch in Fig. 2.a only has
the bouncing behaviour. For the open case ($k=-1$), we showed that $a_{c}$ has
only one branch in Figs. 1.a and 2.a. We concluded that whole branch of
$a_{c}$ has bouncing behaviour since $\ddot{a_{c}}$ is always positive. For
the flat case ($k=0$), $a_{c}$ vanishes. Thus, the flat universe does not show
a bouncing behaviour.

#### Acknowledgments

The authors are grateful to the anonymous reviewer for his/her helpful and
constructive comments. This work was supported by Istanbul University Post-
Doctoral Research Project: MAB-2021-38032.

#### Author Contributions

Ö.Ö. wrote the paper, performed the calculations and prepared the figures. E.
A. is the supervisor and checked the calculations. All authors reviewed the
manuscript.

#### Funding

Open access funding provided by the Scientific and Technological Research
Council of Türkiye (TÜBiTAK).

#### Data Availability

No data was used for the research described in the article.

### Declarations

#### Conflict of interest

The authors declare that they have no conflict of interest.

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# Reliable phase quantification in focused probe electron ptychography of thin
materials

Christoph Hofer  Timothy J. Pennycook<EMAIL_ADDRESS>EMAT,
University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium

###### Abstract

Electron ptychography provides highly sensitive, dose efficient phase images
which can be corrected for aberrations after the data has been acquired. This
is crucial when very precise quantification is required, such as with
sensitivity to charge transfer due to bonding. Drift can now be essentially
eliminated as a major impediment to focused probe ptychography, which benefits
from the availability of easily interpretable simultaneous Z-contrast imaging.
However challenges have remained when quantifying the ptychographic phases of
atomic sites. The phase response of a single atom has a negative halo which
can cause atoms to reduce in phase when brought closer together. When
unaccounted for, as in integrating methods of quantification, this effect can
completely obscure the effects of charge transfer. Here we provide a new
method of quantification that overcomes this challenge, at least for 2D
materials, and is robust to experimental parameters such as noise, sample
tilt.

###### keywords:

Single side band ptychography, phase retrieval, quantification

††journal: Ultramicroscopy

##  1 Introduction

Electron ptychography addresses the phase retrieval problem in transmission
electron microscopy [1], using the probe position dependence of the convergent
beam electron diffraction. Ptychography takes advantage of redundancy in the
diffraction data, either via overlapping probe positions or overlapping
diffracted beams. Before the success of aberration correction in hardware,
direct focused probe ptychography was successfully applied to resolve the
atomic structure at a resolution three times the conventional limit a scanning
transmission electron microscope (STEM) with its ability to achieve
superresolution [2]. With the rise of aberration correction in hardware,
atomic resolution became largely routine for many materials, and the limits of
electron microscopy shifted more to the robustness of the sample to the
electron beam in many cases. However ptychography is useful for much more than
superresolution, and a primary feature of the technique is its very high dose
efficiency, which can even exceed the dose-efficiency of aberration-corrected
high-resolution transmission electron microscopy (HR-TEM) [3]. This efficiency
is of great interest for beam sensitive samples, but also when the need for
extreme sensitivity and precision arises, as is the case for accurate
measurements of charge density, especially when one wishes to detect the
subtle effect of charge transfer due to bonding.

Figure 1:  Dependency of SSB ptychographic images on the distance to
neighboring atoms. a) SSB image of isolated W with a line profile in green
showing the negative halo. The dashed lines in the plot of the line profile
indicate 0 phase. b) Examining the SSB phase variation of neighboring single
atoms as a function of their distance. The plot shows the phase at the center
of a W atom vs the distance to three neighboring S atoms. The S atoms are
moved outwards from the W atom as indicated in the SSB images at the bottom,
using a green circle to indicate the W site and red circles to label the S
atoms. Because of the negative halo, the phase at the center of W atom
displays a minimum when the negative halos from the atomic sites coinside. c)
This is important to take into account SSB images such as this simulation of
defective WS2, which shows a significant variation of the W profile depending
on positions of the neighboring atoms.

To overcome the scan speed limitations imposed by slow cameras, electron
ptychography can be performed using a defocused probe setup, which allows one
to utilize far fewer probe positions for a given area [4, 5]. This can be
useful for low dose applications, as the electrons are then distributed over a
larger area [6]. Now cameras have exceeded MHz readout times allowing one to
capture truly fast 4D data sets with a focused, atomically sharp probe [7].
This allows one to avoid both the high doses and distortions due to drift
associated with slow scans with a focused probe and allows one to
simultaneously capture the higher angle scattering for annular dark field
(ADF) Z-contrast imaging, which would be out of focus with a defocused probe.
This is an extremely useful configuration as the phase images provide the
greatest sensitivity to all the atoms, and indeed electrons, while the
Z-contrast ADF image remains very useful for its ease of interpretation and
ability to distinguish heavier elements [8, 9].

Phase images are sensitive to the total potential in the material, including
the contribution from electrons in addition to the nuclear potential, enabling
charge density mapping [10, 11, 12] and even detection of charge transfer [13,
14]. However, detecting charge transfer is particularly challenging and
requires a robust and reliable method to quantify the phases of atomic sites.
Numerous publications successfully address the quantification of atomic
intensities with the ADF signal [15, 16, 17]. Phase contrast images are,
however, significantly more complicated to analyse as the contrast transfer
function is generally more complex [18, 19].

Figure 2:  Illustration of the steps in the procedure for phase
quantification. a) The initial atomic model containing the correct elements
and the approximate positions is converted to b) a point-potential based on
the model. c) The calculated CTF of SSB ptychography is converted to the PSF
by FFT as illustrated in d). e) Initial (i) and optimized (ii) convolutional
simulations. As can be seen in the difference images in f), showing the
convolutional simulations minus the target image, the difference has been
greatly reduced from the initial result (i) in the optimized image (ii). One
can also see how the optimization increases the correlation function as a
function of iteration in g).

Single side band (SSB) ptychography is a fast, direct phase retrieval method
in STEM, extracting the phase by integrating over the double disk overlaps in
probe reciprocal space [20]. Problems quantifying SSB, or other ptychographic
images, are revealed when analysing phase images of single atoms, such as
those shown in Fig. 1. Unlike ADF images, an SSB phase image displays a
negative halo around a single atom, before converging to a zero valued
background. This negative contribution makes quantification difficult, as the
negative tail can influence the phase of neighboring atoms. Fig. 1b shows the
phase at the center of a single W atom as a function of the distance to three
neighboring S atoms. At small distances, the atomic phases significantly
overlap and add up to a large integrated phase. This occurs with any imaging
technique as the distance of the atoms are smaller than the resolution limit
of the microscopic setup. Increasing the distance decreases the W phase and
separates the atoms. Importantly, however, the phase reaches a minimum and
increases again before reaching a plateau corresponding to the phase of an
isolated independent atom. The occurrence of a local minimum is a result of
the negative halo of the atomic phase.

The negative halo can cause the phase of the same type of atom to vary
significantly depending on the configuration of the atoms surrounding it,
which can hamper quantification. To demonstrate this, we simulated an SSB
image of a defective area of monolayer WS2, introducing several S vacancies to
change the local environment of certain W atoms. This is shown in Fig. 1c. We
also included a quasi isolated W atom surrounded by 3 S divacancies, which is
not physically stable but is a useful demonstration of the difficulty of phase
quantification. A line profile across the W atoms shows a clear variation of
the observed phases. The highest phase is the quasi-isolated W site while the
lowest W phase is the pristine site without vacancies. This is because the
quasi-isolated W atom is not influenced by the negative contribution of
neighboring atoms. The W phases within the pristine lattice have a similar
distance as the minimum in Fig. 1b resulting in the lower phase observed.
Therefore, the phase of each atom in a lattice is visually reduced compared to
an isolated atom. Such effects are not accounted for using simple integration
methods of quantification. Note that this is in contrast to Z-contrast ADF
imaging, where no negative halos exist, only positive definite probe tails.
This is important when, for instance, the sample tilts and the projected
interatomic distances are changed, altering the phase of the atomic sites
without meaningful change in the potential itself.

Figure 3:  Flow chart of the phase extraction algorithm.

To account for this non-linear contrast formation, we introduce a new method
which reliably and accurately extracts the phase corresponding to the actual
potential for the atoms in arbitrary networks and orientations, and
demonstrate it with defective WS2 as shown Fig. 1c. The method optimizes a
point-potential which is convolved with a kernel. Here we focus on the
contrast transfer function (CTF) for SSB, although other kernels can be used
for other imaging modes, as we demonstrate for ePIE and iCoM. The SSB kernel
has the same shape as an isolated SSB reconstructed atom such as in Fig. 1a
and can be interpreted as the point spread function (PSF). The point-potential
represents the atomic positions as well as their phase. We compare this new
“SSB-CTF based” kernel with a kernel generated by the “ADF-CTF” which is very
similar to a Gaussian kernel. The analysis with the ADF-CTF based kernel is
very similar to simple Gaussian-fitting [21] or Voronoi-integration [17] in
the absence of aberrations [22]. The new method reliably and accurately
extracts the deconvolved phases and, importantly, is robust to environmental
changes such as local atomic configuration and sample tilt. Our new method in
principle makes quantitative analysis of phase images with arbitrary contrast
transfer functions possible.

##  2 Methods

###  2.1 Computational method

The method works by iteratively updating the atomic positions and their point
potential representations such that images simulated using the convolution of
the point potentials by the PSF match the input images as closely as possible.
Figure 2 illustrates the process using defective WS2 as an example, and the
complete method is summarized in a flow chart in Fig 3). The WS2 model in Fig.
2a results in the point-potential image shown in b.

In general, the 2D point potential takes the form

$V(\textbf{r})=\sum_{\textbf{r}_{Z}}\Phi_{\textbf{r}_{Z}}\cdot\delta_{\textbf{r},\textbf{r}_{Z}}$
(1)

where $\textbf{r}_{Z}$ are the exact positions of the atoms in the model $M$
and $\delta$ is the Kronecker delta function, being one if
$\textbf{r}=\textbf{r}_{Z}$ and zero otherwise. We chose $\Phi$ as the
variable for non-zero pixel values as they represent the phase cross-section
of the atoms. In practise, the potential is calculated by iterating through
all atoms and adding the corresponding phase value at the corresponding pixel
position. For spatial precision, sub-pixel precision is enabled by weighting
four neighboring pixels in the potential map, as the potential maps have the
sample pixel dimensions as the input images, which can be courser than the
positions of the atoms. In practise, the initial model is often relatively
easy to match to the experimental data manually via scaling and translations.

Once a point potential map is produced, a phase image, $\phi$, is simulated
simply by convolving the potential with the chosen PSF [23], the inverse
Fourier transform of the CTF,

$\phi^{sim}=V(\textbf{r})\ast FT^{-1}\left(CTF(Q_{p})\right).$ (2)

The error is then calculated between the convolutionally simulated image,
$\phi^{\mathrm{sim}}$, and the input phase image, $\phi^{\mathrm{target}}$,
and the positions of the atoms in the model and their point potential values
is then iteratively updated such that the difference between the image
simulated from the point potential map and the input image is minimized. In
addition the algorithm additionally optimizes the PSF width, which can be
altered by either the convergence angle or the field of view via the probe
position step size. The optimization process is described in detail below.
Using convolutional simulations is advantageous for the speed of the
computations, but by optimizing a single value of the potential for each
atomic site in this way, the point is also to provide the best possible
quantification of “the phase” of an atom or atomic column, including the
effect of screening by electrons which are distributed further away from the
nucleus[13].

####  2.1.1 CTFs

The SSB CTF is related to the geometric size of the double disk overlaps for
each spatial frequency $Q_{p}$ for a given convergence angle $\alpha$ [19] as

$\displaystyle CTF(Q_{p})=$
$\displaystyle\frac{4}{\pi}\cdot\left[cos^{-1}\left(\frac{Q_{p}}{2}\right)-\frac{Q_{p}}{2}\cdot\sqrt{1-\left(\frac{Q_{p}}{2}\right)^{2}}-\right.$
$\displaystyle\left.cos^{-1}(Q_{p})+Q_{p}\cdot\sqrt{1-\left(Q_{p}\right)^{2}}\right]$

for $0\leq Q_{p}\leq\alpha$ and

$CTF(Q_{p})=\frac{4}{\pi}\cdot\left[cos^{-1}\left(\frac{Q_{p}}{2}\right)-\frac{Q_{p}}{2}\cdot\sqrt{1-\left(\frac{Q_{p}}{2}\right)^{2}}\right]$

for $\alpha<Q_{p}\leq 2~{}\alpha$ and 0 otherwise. The 2D CTF of SSB is shown
in Fig. 2c and the real spaced transformation of this, the PSF, in panel d,
which in this paper, we refer to as the SSB CTF based kernel.

Note that the PSF (Fig. 2d), the inverse Fourier transform of the CTF, is a
result of coherent interference and contains negative values. In contrast, the
ADF-based PSF is obtained via Zernike polynomials and the result of the
inverse Fourier transform is squared as a result of the incoherent image
formation process. We further note that an alternative to an analytical
calculation of the CTF is to extract the CTF numerically from simulations of
an isolated atom via the same algorithm as used for the target image. This is
indeed possible as the exact profile is negligibly dependent on the element
[18].

Figure 4:  Comparison between phase extraction using a Gaussian kernel (based
on an ADF-CTF) and a SSB-CTF based kernel.a) Target image which is used as
benchmark for the methods. b) ADF-CTF based and SSB-CTF based kernel (top and
bottom, respectively). c) Optimized convolution simulation using the ADF-based
kernel and SSB-based kernel in top and bottom, respectively. d) Difference
image between (a) and (c). e) Extracted phases using both kernels.

####  2.1.2 Optimization

To quantify the merit of a particular configuration we use the correlation
between the input target image and the convolutional simulation

$\displaystyle
R=\sum\limits_{i=1}^{N}\frac{\left(\mu^{\mathrm{sim}}-\phi^{\mathrm{sim}}_{i}\right)\left(\mu^{\mathrm{target}}-\phi^{\mathrm{target}}_{i}\right)}{\sigma^{\mathrm{sim}}\sigma^{\mathrm{target}}\left(N-1\right)},$
(3)

with $\mu$ being the mean value and $\sigma$ being the standard deviation. The
sum iterates through all pixel values $\phi$ of the convolutional simulation
and the input target image. This is shown in Fig. 2e for the initial and
optimized simulation in top and bottom, respectively. The correlation function
is maximized by adjusting the parameters for the atomic positions, width of
the kernel and the strength of the potential. The difference images for both
cases are shown in (f). The correlation in the course of the optimization is
shown in (g). This function is maximized based on quadratic interpolation
where the gradients in each direction are calculated by the finite difference
method and the minimum of a quadratic fit is estimated to be the next
iteration value. The source size can also be optimized, and the atomic
positions collectively changed by applying translations, scalings, rotations
and importantly sample tilts. This is crucial for experimental conditions,
where a sufficient alignment between the simulation of the model and the
experimental image is the task of the optimization.

Figure 5:  Specimen tilt sensitivity of ptychographic reconstructions. a) SSB
image of WS2. b) 8 degree tilted SSB image. c) Line profile of a) and b). d)
Extracted phases using a Gaussian kernel (purple) and a SSB-CTF based kernel
(red).

##  3 Results and Discussion

To compare a conventional integration of the atomic sites with the new method,
we performed the optimization analysis with an ADF-CTF based kernel and an
SSB-CTF based kernel. The target input image is a quantitative SSB simulation
of defective WS2 (Fig. 4a), where the positions are labeled with blue (W),
yellow (S vacancies) and orange (S2). The input simulation is based on an
independent atomic model potential, in which electron redistribution is not
taken into account. The convergence angle is set to 20 mrad and the
acceleration voltage is 60 kV. Therefore, the same elements have the same
phase in the phase image which is an ideal test for the different methods. The
ADF-CTF based kernel is very similar to a Gaussian kernel and the analysis
with it is very similar to simple Gaussian-fitting [21] or Voronoi-integration
[17] in the absence of aberrations [22]. However, we will show that this
approach is problematic for quantifying SSB images as the phase of the atoms
also depends on the local environment, as discussed in the introduction. The
second kernel is obtained via the SSB-CTF, as described in the Methods
section. Crucial for an accurate analysis is the negative halo in the kernel,
which is also visible in the SSB reconstruction as shown in Fig. 1a. Both
kernels are shown in Fig. 4b.

Figure 6:  Precision of optimization method at different conditions. Analysis
of SSB phase extraction with different electron doses (a), with and without
temporal incoherence (b) and with and without the effect of finite source
sizes (c). Each panel shows the extracted phases of two conditions left and
the mean values as well as the error bars right.

When optimizing the simulations using the atomic positions and point-potential
values as parameters, both methods result in similar simulated images (cf.
Fig. 4c). The difference images between the optimized convolutional
simulations and the input target image, however, shows some residual features
in the case of the ADF-CTF based image (cf. panel d). This is because the
contrast cannot be perfectly matched in this case. The difference image using
the SSB-CTF kernel is almost perfectly flat as a result of an excellent match
between the simulation and the target image. This is reflected in the higher
correlation coefficient in the SSB-CTF kernel based optimization.

More importantly, significant errors appear in the resulting extracted phases
of the optimized point-potentials. When using the ADF-based kernel simulation,
the histogram of phases (Fig. 4e) shows a more scattered W phase value
distribution as the stronger phase of W close to weaker potentials (e.g.
vacancies) has to be matched by increasing the point-potential. The highest
peak corresponds to the isolated W, surrounded by double vacancies in the
structure, due to the contrast reduction for the W atoms within the pristine
lattice. Also, the S atoms in the S2 sites (orange) are slightly lower than
the single S vacancies (yellow) due to the same reason (see Fig 4e top). This
makes the phase extraction unreliable as the contrast transfer functions are
not properly taken into account. Using the SSB-CTF kernel gives, however, a
much sharper histogram as the local environment is taken into account. Again,
keep in mind that this is an independent atom model so the variation should be
essentially nil. Correctly accounting for the CTF makes the interpretation of
the phases due to actual changes in the material, such as bonding, much easier
and more reliable. Note that the width of S (orange) in the S2 site is
unavoidably larger because they are pairwise overlapping in the projection.
However, their mean value is matching the S vacancy values (yellow) as
expected for the target image. Also the W atoms show much less variation
despite the non-linear contrast formation. This is important for detecting
small differences between atomic sites, which occurs for example when
detecting charge transfer.

The strong dependency of the SSB contrast on the local environment also makes
quantification difficult when small sample tilts are present as these lead to
small projected positional shifts of the atoms. Small tilts are difficult to
measure in STEM experiments and residual sample tilts are practically
unavoidable. The contrast dependency of the sample tilt is demonstrated in
Fig. 5a and b, which show IAM-based SSB simulations of an untilted and 8
degrees tilted WS2 respectively. The contrast change is shown by the line
profile (panel c), where the S2 sites show a significantly reduced phase when
tilted. To demonstrate the robustness of the optimization method to sample
tilt, we performed the quantification analysis on both the untilted and tilted
images using both the SSB and ADF based kernels. For the untilted pristine
case, both kernels give similar S2 to W ratios. However, analysing the tilted
image gives a remarkable deviation when using the ADF-kernel (5.50%) while the
difference using the SSB-CTF based kernel is much lower (0.89%). Ideally
mistilt should be avoided experimentally, but this method can greatly reduce
the influence of mistilt, making it potentially insignificant compared to
other influences such as limited electron dose. This shows the robustness of
the new phase extraction method to sample tilt. To make this happen, the
sample tilt is included in the optimization which gives the correct value
based on the distortion of the lattice. Importantly, this requires that scan
distortion is minimized, which is now possible in 4D STEM based on truly fast
cameras that enable microsecond level dwell times, such as event-based cameras
[7].

Next we investigated the robustness of the phase extraction to other typical
experimental influences. We measured the W phase as a function of electron
dose, partial coherence and source size individually. The resulting phases are
shown in Fig. 6. For common operating conditions, by far the largest
deleterious influence on the precision is when a limited amount of signal per
diffraction pattern is available, as a result of a limited electron dose
imparted to the specimen. This is shown in Fig. 6a. However the error bar of
the mean value not only depends on the dose but also on the number of atomic
sites analysed. To evaluate if the error bar is small enough in the context of
analysing phase differences, for example due to the subtle changes occurring
with charge transfer due to bonding, the difference between the involved
phases has to be considered. We analysed the charge transfer of WS2 where the
phase difference was indeed large enough to detect the difference caused by
charge transfer at moderate doses [24]. The partial coherence is simulated via
a defocus spread ranging from 0–6 nm. Typical values are around 3 nm for a
cold FEG, greater for less coherent sources. The phase change is small because
of the robustness of ptychography to partial coherence [3]. Lastly, the source
size is simulated by using a Gaussian blur up to a FW 2.3 Å. The phase change
is insignificant for typical source sizes [25]. We therefore conclude that
dose and therefore the sample stability is the most crucial parameter to
consider if phase changes are detectable.

Phase retrieval methods can encounter the issue of phase wraps with strong
potentials. This means when increasing the strength of the potential, the
resulting atomic phase wraps to a negative value in the center of the atomic
columns as has been observed in both simulations and experiment. At this
point, the atoms deviate from their natural rotational symmetric shape and the
presented method is not reliable. However, it has been shown that these
effects can be corrected by adjusting the probe defocus post collection [9].
However, as the ptychographic contrast mechanism becomes more complex when the
sample gets thicker we consider the present method to be reliable for thin
materials such as 2D materials. The application of this method to thicker
materials is beyond the scope of this paper and will be investigated in future
work.

Figure 7:  Quantification of iterative ptychography using ePIE. a) ePIE
reconstruction of defective WS2. b) Line profile and 2D image of PSF of ePIE.
c) Extracted phases of (a) using the ePIE kernel (left) and a Gaussian kernel
(right). ePIE also exhibits a PSF with a negative halo which enhances the
phase of an isolated atom. This can be correctly accounted for using the ePIE
PSF with our method in quantification.

Our method is in principle also applicable to imaging methods with other CTFs.
The choice of CTF results in different kernel shapes representing the shape of
an individual atom in the final phase image [18]. This makes the method very
flexible to use, including for noise-normalized contrast transfer functions.
Here, we analyse the phase retrieved by iterative ptychography with the
extended ptychographic iterative engine (ePIE) as an example. We simulated the
same region as the other examples via ePIE which has a different CTF and
therefore a different PSF than SSB, as shown in Fig. 7a. Since the exact CTF
depends on the algorithm, we provided the kernel by simulating a single atom
with the same algorithm. We used 0.5 Åas the probe position step size, a 1 nm
defocus and the same 20 mrad convergence angle used earlier. Interestingly, it
also shows a negative halo, similar to the SSB, but somewhat less pronounced
(Fig. 7b). Therefore, the W within the defective network is also brighter than
the W atoms within the pristine lattice with ePIE. As with SSB data, the
kernel method again correctly accounts for this when using the ePIE-based
kernel. The extracted phases show, similar to the SSB, sharp peaks in the case
of the optimization using the ePIE-PSF. Using the Gaussian-shaped PSF from the
ADF kernel on the ePIE data, results in a much broader width of the
distribution of W phases, as expected (cf. Fig. 7c).

We also analysed integrated center-of-mass (iCoM) imaging as a further 4D STEM
imaging method. iCoM is becoming increasingly popular with the development of
fast cameras since it is a more accurate method compared to the now popular
integrated differential phase contrast (iDPC) imaging modality. However, both
images can be analysed similarly as outlined below. The only difference is the
non-rotational symmetric CTF of iDPC as a result of the segmentation of the
detector, which would need to be accounted for correctly for an accurate
analysis of iDPC images, and therefore the exact detector geometry with
respect to the diffraction patterns would have to be known. iCoM is very
simple and fast and therefore a reliable tool for live iCoM imaging [26].
However, it has been observed to be less dose efficient than ptychographic
methods [9, 27]. It also cannot correct residual aberrations and has no super-
resolution ability. Since the CTF of iCoM is starting at 1 [28], there are no
artifacts associated with the negative part of the PSF when used with the raw
data. However, high pass filters are commonly applied to iCoM and iDPC data,
and riCoM contains an intrinsic filtering based depending on the riCoM kernel
used. In all these cases the PSF for iCoM, and by extension iDPC, obtains a
negative contribution similar to the discussions above. This is demonstrated
in Fig. 8 where raw and high pass filtered iCoM images of the same defective
WS2 region are shown. Such filtering is very commonly used to sharpen iCoM or
iDPC images generally or ameliorate the effect of contamination, as well as
edge artifacts. Importantly, the isolated W atom is also brighter here in the
case of the filtered image. This is, once again, a result of the PSF which
shows a negative tail in case of the filtered image. Therefore, conventional
quantification of high-pass filtered (iCoM) images will encounter similar
problems as in the SSB image analysis in that the extracted phases highly
depend on the environmental conditions. Omitting the high pass filter however
also often leads to unreliable quantification as the low-frequency
contribution harms the image analysis [29]. Therefore, a reliable
quantification of iCoM and iDPC phases is only possible via a method such as
the optimization method we have presented in which the exact filter conditions
are reproduced in the PSF. This is usually facile since the filter parameters
are usually very accurately known.

Figure 8:  Filter effect on phase quantification of iCoM a) Raw iCoM
reconstruction of defective WS2. b) Same as (a) but with a high-pass filter.
c,d) PSF of (a) and (b), respectively. Notice that the filter causes a
negative halo in the PSF of the iCoM which again enhances the brightness of
isolated atoms compared to those with closer neighbors. This can be accounted
for with our method.

##  4 Conclusion

We have introduced a new method to reliably analyse ptychographic phase images
quantitatively. The method takes into account the specific contrast transfer
function and is therefore robust to local environments such as atomic
configuration and sample tilts. This is very important when tiny changes of
the phase images are analysed, such as with charge transfer due to bonding. We
believe this method opens a path towards a reliable quantification of phase
images as it has been achieved for ADF images.

##  5 Acknowledgement

We acknowledge funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 Research and Innovation Programme via Grant
Agreement No. 802123-HDEM and FWO Project G013122N “Advancing 4D STEM for
atomic scale structure property correlation in 2D materials” (C.H.).

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