# Finding the Right Bricks for Molecular Lego:
A Data Mining Approach to Organic Semiconductor Design

Christian Kunkel  Christoph Schober  Johannes T. Margraf  Karsten Reuter
Harald Oberhofer<EMAIL_ADDRESS>Chair for Theoretical Chemistry and
Catalysis Research Center, Technische Universität München, Lichtenbergstraße
4, D-85747 Garching, Germany

###### Abstract

Improving charge carrier mobilities in organic semiconductors is a challenging
task that has hitherto primarily been tackled by empirical structural tuning
of promising core compounds. Knowledge-based methods can greatly accelerate
such local exploration, while a systematic analysis of large chemical
databases can point towards promising design strategies. Here, we demonstrate
such data mining by clustering an in-house database of $>64.000$ organic
molecular crystals for which two charge-transport descriptors, the electronic
coupling and the reorganization energy, have been calculated from first
principles. The clustering is performed according to the Bemis-Murcko
scaffolds of the constituting molecules and according to the sidegroups with
which these molecular backbones are functionalized. In both cases, we obtain
statistically significant structure-property relationships with certain
scaffolds (sidegroups) consistently leading to favorable charge-transport
properties. Functionalizing promising scaffolds with favorable sidegroups
results in engineered molecular crystals for which we indeed compute improved
charge-transport properties.

##  1 Introduction

In the last few years, organic semiconductors have received considerable
attention due to their comparatively small ecological and economical
footprint, as well as due to their great versatility promising a wide spectrum
of novel material properties.1, 2, 3, 4, 5 Their potential uses include
organic field effect transistors (OFETs)6, photovoltaics (OPVs),7 light
emitting diodes (OLEDs)8 or even sensors9. A major limitation to the
commercial application of this promising class of materials has been the
failure to reproducibly yield high charge-carrier mobilities as a prerequisite
for the electronic performance.10, 11 Thanks to intense research efforts,12,
13, 14, 15 a number of materials now exist with mobilities exceeding even that
of amorphous silicon.16 Yet, the sheer endless materials spaces constituted by
assembling building blocks of functionalized organic molecules raise the
suspicion that these few materials are nothing but the top of an iceberg.

So far, experimental discovery of new materials commonly follows a cycle of
synthesis, device manufacture, performance evaluation and subsequent molecular
tuning, based on chemical intuition, experience and—to a certain degree—trial
and error.6, 17, 5 Theoretical investigation is then often used to understand
and exploit the key charge-transport mechanisms 18, 19, 20, 21, 22. Although
highly successful, such efforts are limited by the pace of iterative
improvement, while only locally exploring the chemical space around a compound
family. As a more recent development, a number of groups have therefore
started to conduct in silico materials discovery calculations,23, 24, 25, 26,
27, 28 to guide synthetic studies to promising targets. In these screening
approaches, readily available or easily calculable properties of the system—so
called descriptors—which are correlated with critical observables of the
system (such as the carrier mobility) are determined for a larger set of
candidate materials. These then yield a ranking of the screened candidate
materials based on their suitability as an organic semiconductor.

In other fields of science, such as heterogenous catalysis,29, 30, 31
superhard materials,32 or drug discovery,33, such screening studies are
already followed up with modern large-scale data mining or even machine
learning approaches to extract more general design criteria.34 Corresponding
approaches undertaken for organic semiconductors (OSCs) have so far been
limited to a much smaller scale.35, 36 In this work, we now take this step and
combine the statistical tools of modern data mining approaches with the
molecular analysis methods known as cheminformatics to uncover guiding
principles for the design of organic semiconducting materials.

###  1.1 Descriptor-Based Data Mining of OSCs

Two of the most prominent charge-transport descriptors for organic
semiconductors are the network of electronic couplings $|H_{\text{ab}}|$
between microscopic sites involved in the charge transport and the
reorganization energy $\lambda$, which measures the cost of accommodating a
new charge state after the carrier has moved to the next site. While
$|H_{\text{ab}}|$ and $\lambda$ are most closely associated with the small
polaron hopping mechanism of carrier transport known from Marcus theory,37, 38
both also significantly influence the carrier mobility in other transport
regimes.39 In evaluating $\lambda$, usually only the dominant local, intra-
molecular reorganization energy is considered.40 In contrast to
$|H_{\text{ab}}|$, the reorganization energy can then be estimated cheaply as
a single site property, while at the same time decisively—in Marcus-like
models even exponentially—determining the carrier mobility.41, 18

A number of screening studies thus initially focused on the optimization of
$\lambda$ through modification of molecular sidegroups in, for example,
dinaphtho-thienothiophene (DNTT),42 triarylamine-based donor molecules for
solar cells,43 or high-performance polymer semiconductors.44 Taken together,
the studies impressively demonstrated how in silico exploration of material
spaces can efficiently identify high-performance organic semiconductors. While
similar approaches are now also starting to surface in experimental studies,45
most recent in silico design studies start to optimize $\lambda$ in newly
generated derivates. The design problem is thereby mainly tackled by
construction and computational screening of highly focused molecular libraries
based on a number of different strategies. These range from the generation of
polymer semiconductors from preselected building blocks46, 41 to the
optimization of $\lambda$ through heteroatom replacement47, 48, introduction
of sidegroups41, 49, 48, 50, 51, 52, 53, 54, 55 or ring fusion56, 57.

Modern data-driven approaches instead pursue a complementary strategy. Rather
than screening a focused library for potential candidate materials, in
particular data mining techniques unveil correlations in large data sets and
thereby provide leads for a rational optimization beyond the originally
considered design space58, 59, 60. Size and especially diversity of the
underlying data source are thus key for the generality and transferability of
the identified correlations. A suitable representation of the data critically
determines the efficiency with which these correlations can be extracted,
whether by machine learning or more traditional fitting techniques. In several
fields, development and application of such knowledge-based approaches has
already led to a greatly accelerated materials design61, 62, 63.

Here, we show, that a data-mining rooted in the chemical understanding of the
structure—often termed chemical intuition—can indeed yield general design
principles for organic semiconductors. To this end, we concentrate on the
class of crystalline molecular organics and analyze an in-house dataset
containing $>64.000$ experimentally characterized mono-molecular crystals.
This dataset was originally assembled to computationally screen for hitherto
unknown organic crystals with favorable charge-transport properties.25 As
further described below, it contains an unprecedented molecular diversity and
a wealth of promising semiconductors as identified on the basis of computed
high electronic couplings $|H_{ab}|$ and low reorganization energies
$\lambda$. In this work, we now perform a two-step data mining procedure on
this dataset. Using the concept of Bemis-Murcko scaffolds64 that has already
been successfully used in pharmaceutical research, we establish a useful data
representation that uncovers statistically significant performance differences
between the individual molecular building blocks of the crystals. Clustering
the data first according to their Bemis-Murcko scaffolds, i.e. the molecular
backbones, we find clear structure-property relationships, with a range of
scaffolds consistently leading to favorable charge-transport properties,
despite the positional disorder brought about by differing sidegroups and
anchor points on them. In a second step, we then cluster the data according to
the sidegroups that are attached to the molecular scaffolds. Again, we obtain
clear relationships with a range of sidegroups generally lowering the
reorganization energy, regardless of the scaffold they are attached to. These
findings suggest the combination of certain, high-coupling scaffolds with
reorganization energy-reducing sidegroups as a promising design criterion for
optimized charge transport in organic molecular crystals. We validate the
transferability and accuracy of this criterion by a corresponding sidegroup-
engineering of molecular testsets and indeed find significant improvements in
the charge-transport properties.

##  2 Methods

###  2.1 Dataset

Our study is based on the “64k-dataset”25, containing information on 64.725
organic crystals composed of 61.770 unique molecular structures, extracted
from the Cambridge Structural Database65 and screened for favorable charge-
transport related descriptors $\lambda$, $|H_{ab}|$. All crystals are mono-
molecular, i.e. they contain only a single type of molecule. For a
cheminformatics based analysis, $xyz$-coordinates of the molecular structures
were extracted from the crystal and converted to canonical smiles strings 66
using Openbabel67. All subsequent molecular analysis was carried out using the
RDKit.68 Marvin was used for drawing chemical structures69. Pymatgen was used
to analyze the occurrence of molecular point groups and crystal space groups.
70

###  2.2 Electronic Structure Calculations

The computational settings and methodology were described in detail in the
preceding work that established the “64k-dataset”. 25 Thus, we here only
briefly summarize them for completeness. Charge-transport related descriptors
$\lambda$ and $|H_{ab}|$ were computed using first-principles Density-
Functional Theory (DFT) at the GGA (BLYP71, 72) level of theory. All
calculations were done using the FHI-aims package 73, 74 employing its well-
established basis-sets and integration grid settings, detailed below in this
section. Our work on the “64k-dataset” showed that even semi-local DFT was
able to recover trends among different crystals for both $|H_{ab}|$75 and
$\lambda$.25 For each crystal in the dataset, the nearest-neighbor molecular
dimers were first extracted by a purpose-built python program. Electronic
couplings $|H_{ab}|$ for each dimer were then calculated using the fragment
molecular orbital approach (FO-DFT)76 in the $H^{2n-1}@D+A$ variant.75 The
electronic wave functions were expanded in an extended ”tier 1” basis set of
FHI-aims using light integration settings. The highest electronic coupling
among all dimers was extracted from this data and constitutes the $|H_{ab}|$
entering the dataset.25 Keeping in mind that the “64k-dataset” consists solely
of crystals composed of a single species of molecule each, the internal
reorganization energy was evaluated for each molecule based on the 4-point
scheme,77 using a non-periodic QM/MM embedding setup.25 In this ONIOM-based
scheme,78 a shell of nearest neighboring molecules mimics the steric hindrance
due to a realistic solid-state environment during geometry optimization of a
central molecule. The shell of nearest neighbours was thereby extracted based
on the minimal distance found between the atoms of any two molecules in the
crystal. All molecules that approach the central molecule to within this
distance times $1.5$ were included in the neighbor shell. Then, keeping the
shell of nearest neighbors fixed, the local structure optimisation of the
central molecule was performed using the Broyden-Fletcher-Goldfarb-Shanno
(BFGS) implemented in the Atomic Simulation Environment79 using a convergence
criterion $\rm{f_{max}<}$ 0.05 eV/Å. The MM parts of these calculations were
based on the LAMMPS package80 with the ”Universal Force Field” (UFF)81 and the
QEq charge equilibration scheme82 in order to determine interaction parameters
and fixed partial charges, respectively. To ensure maximum accuracy of our
main descriptor, we re-calculated all reorganization energies $\lambda$ in the
present database using default ”tight” settings and the respective ”tier 2”
basis-sets of FHI-aims.

###  2.3 Statistical Methods

We used the two-tailed Mann-Whitney-U test83 to assess whether two samples are
drawn from the same distribution (null hypothesis) or not (alternative
hypothesis). Note that the Mann-Whitney-U test is non-parametric and therefore
avoids assumptions on the underlying distributions. To account for the great
number of statistical tests we carried out, we corrected the computed
$p$-values for each descriptor using the false discovery rate correction of
Benjamini-Hochberg,84 as implemented in the statsmodels package85. These
corrected $p$-values are then named $q$-values. Results were marked
statistically significant, if the $q$-value was found to be below 0.05, i.e.
if the probability that the statistical testing wrongly rejects the null
hypothesis is smaller than 5 %. Similar methods and criteria are commonly used
in Gene Set Enrichment Analysis86 and have also been applied to
cheminformatics87, 88 for the identification of active compound series.

##  3 Results

###  3.1 Molecular Scaffolds

Figure 1: Illustration of the Bemis-Murcko partitioning scheme. For two
example molecules, the decomposition into scaffold and sidegroups is
illustrated by highlighting the scaffold in light blue on the left and by
highlighting the sidegroups on the right.

Compared to earlier, focused studies on OSCs, the dataset used in this study
covers a wide range of molecular diversity to ensure the generality of our
data mining results. This is reflected by 83 crystal space groups occuring at
least 3 times in the dataset, while the mono-molecular building blocks are
correspondingly distributed over more than 20 molecular point groups. The
molecules comprise H, B, C, N, O, F, Si, P, S, Cl, Se, Te, Br, I, and As
species and therewith cover the entire range of elements commonly found in
OSCs. As further specified below, a variety of molecular backbones is
contained in the database, featuring single to multiple (hetero)cycles. These
backbones are functionalized with 4229 different sidegroups, 828 of which
occur in more than three different crystals.

In order to extract correlations from such a diverse set of data, a suitable
representation is required that collapses its dimensionality to a tractable
size. In contrast to more abstract substructure or statistical learning based
approaches 89, we here pursue a chemically intuitive concept that partitions
the molecular building block of the organic crystal into ring systems, linkers
and sidegroup atoms. The Bemis-Murcko (BM) scaffold 64 is thereby specifically
defined as the molecular core comprised of connected ring systems and their
linkers, resulting after removal of all sidegroup atoms. Figure 1 provides two
illustrative examples of this decomposition scheme. While BM scaffolds have so
far mainly been used in medicinal chemistry to study properties of drug-like
molecules90, 91, 92, the concept is also well-suited for our purpose. It
mainly extracts $\pi$-conjugated ring systems which to date almost exclusively
form the basis of molecular electronics.93, 94 The scaffold also constitutes
the largest part of the molecule which is likely most decisive in terms of
electronic structure, shape, conformational flexibility87 and possibly
geometric arrangement in the crystal. Consequently, we disregard some
structures in the “64k-dataset” where very extensive sidegroups constitute
more than 50 % of the non-hydrogen atoms.

Using this BM representation allows us to cluster the dataset entries based on
their respective scaffolds. In order to obtain representative sample sizes, we
required each such cluster to contain at least $15$ crystals. Although this
rigorous filtering discards many clusters due to too small cluster sizes, it
leads to a well-analyzable subset of the “64k-dataset” that contains $7.569$
molecular crystals distributed over $195$ scaffold clusters. These molecular
crystals are made of $6.936$ unique molecules, since $633$ crystals appear as
polymorphic structures or had entered the Cambridge Structural Database more
than once, e.g. coming from different reported experimental sources.
Nevertheless, each scaffold cluster comprises at least $10$ unique molecules
and we provide a list of BM cluster scaffolds, the detailed distributions of
cluster sizes, as well as $|H_{ab}|$ and $\lambda$ values in this subset in
the Supporting Information (SI) Figs. S2-4. For the remainder of the study we
will focus on this data subset and will consistently refer to it as our
database.

Figure 2: Top: Scatterplot displaying the median
$|H_{ab}|_{\text{med,scaffold}}$ and $\lambda_{\text{med,scaffold}}$ for each
of the $195$ scaffold clusters. Orange crosses mark clusters with a $|H_{ab}|$
and $\lambda$ distribution significantly differing from each of the respective
background distributions, and blue crosses mark clusters with no such
significant difference. The medians of the two descriptors over the entire
dataset are marked with dashed lines. The anthracene cluster is highlighted in
red as a reference cluster containing well-known high charge-carrier mobility
crystals. Bottom: Examples of BM-scaffolds further discussed in the text. The
datapoints for these scaffolds are labeled and depicted in green color in the
top panel.

In Fig. 2 we first analyze the median over $|H_{ab}|$ and $\lambda$ for each
scaffold cluster and contrast it with the corresponding medians over the
entire database. This provides a chemically intuitive assessment of the
performance and tuning potential of each particular scaffold. Within the scope
of structures in each scaffold cluster, this averaging step accounts for
changes in electronic and crystal structure, brought about by differing anchor
points of the varying sidegroups, thus to a certain extent accounting for the
so called positional disorder95 in each scaffold.

To provide a reference of a well-performing scaffold, we highlight in Fig. 2
the anthracene scaffold cluster. With anthracene a well-known base material
for organic electronics96, this reference cluster is predominantly composed of
high charge-carrier mobility crystals with relatively low $\lambda$ and high
$|H_{ab}|$. Compared to this reference cluster, there is a wide spread of the
median descriptors among the 195 scaffold clusters. In particular, there are
many scaffolds with exceedingly poor median charge-transport characteristics
(very high $\lambda_{\text{med,scaffold}}$ and/or very low
$|H_{ab}|_{\text{med,scaffold}}$). On the other hand, there are also several
scaffold types that exhibit at least as favorable characteristics as the
anthracene reference cluster. Note, that in particular improvements in the
reorganization energy, i.e. a decrease, can thereby also readily compensate
for a deterioration in the electronic coupling (or vice versa). Within a
Marcus model, a $50\%$ improvement (deterioration) of $\lambda$ could for
instance tolerate a decrease (increase) of $|H_{ab}|$ by up to $18$ meV ($26$
meV) to still yield the same hopping rates, and by implication mobilities, as
anthracene. Essentialy all of the scaffolds highlighted in Fig. 2 fall in
approximately the same range of mobility. Considering the immense range of
$\lambda_{\text{med,scaffold}}$ values included in our dataset and comparing
it to the narrow range of available $|H_{ab}|_{\text{med,scaffold}}$ values,
we again find that the reorganization energy is an ideal first level target
for molecular optimization. The large spread of
$\lambda_{\text{med,scaffold}}$ seen in Fig. 2 over all scaffolds indicates
thereby the vastly different degrees of inner-sphere reorganization caused by
the different molecular cores. A spread along
$|H_{ab}|_{\text{med,scaffold}}$, on the other hand, can be caused by a number
of factors, such as varying capabilities of scaffolds to stabilize their most
favorable packing motives, as well as intrinsic differences of the involved
frontier orbital geometries.39

In order to assess the statistical significance of this observed spread in the
median descriptors we employ non-parametric statistical testing based on the
two-tailed Mann-Whitney-U test.83 As further detailed in the methods section,
this test determines non-parametrically the probability $p$ of two sets of
values to belong to the same distribution. By correcting detection bias due to
the relatively large number of tests being carried out, this yields the $q$
probability and we judge from a value $q<5$ % that the distribution of
descriptor values in a given scaffold cluster differs significantly from the
distribution in the whole database. With respect to the $\lambda$ descriptor,
$105$ clusters, i.e. more than half of all considered scaffolds, fulfill this
restrictive criterion. With respect to the $|H_{ab}|$ descriptor, this number
is a bit lower, but with $69$ still extends over more than a third of the 195
scaffolds (the full data for all scaffolds is given in Table S1 in the SI). A
total of 44 clusters exhibits statistically different distributions in both
descriptors and the corresponding data is color-coded in Fig. 2. The
statistical analysis therewith confirms our initial notion that BM scaffolds
are a suitable data representation that allows to differentiate molecular
crystals in terms of their charge-transfer properties. Favorable scaffolds
with a general tendency to high electronic couplings and low reorganization
energies can thus be identified from the available data. Smaller such
scaffolds featuring one or two aromatic rings could then be used as promising
building blocks, while larger scaffolds can serve as a starting point for
further functionalization. In Fig. 2 we specifically highlight a few such
identified smaller scaffolds that we further discuss in Section III below.

###  3.2 Molecular Sidegroups

Having identified suitable molecular scaffolds, we next turn our attention to
the sidegroups. The introduction of appropriate sidegroups has been shown to
improve carrier mobilities by influencing the stable crystal structure45, 97,
20 or by suppressing vibrationally induced disorder98, 99. Here, we focus
specifically on the influence on the reorganization energy. Previous work has
confirmed this critical influence in comparative studies attaching different
sidegroups to a given single scaffold.50, 51, 49, 48, 52, 41, 53, 54 Building
on this empirical concept, we now use our extensive database to investigate
whether there are molecular sidegroups that generally lower the reorganization
energy, independent of the scaffold they are attached to. We achieve this with
the same techniques of statistical analysis as before for the scaffolds, i.e.
we first identify all sidegroups in the database and then cluster the data.
This time, one cluster corresponds to all molecules (of any scaffold type)
that contain a given sidegroup. This does not differentiate between specific
attachment positions to the scaffold or the coexistence of multiple sidegroups
in one molecule, and we return to this point below. To ensure meaningful
statistics, we only consider sidegroup clusters comprising at least 10
different molecules, with the notable exception of the -Se-CH3 sidegroup. This
sidegroup occurs only in six molecular crystals in the database, but is
nevertheless considered because of its exceptional properties (vide infra).
Altogether this yields $70$ sidegroup clusters containing a total of $6.162$
unique molecules.

Figure 3: Top: Boxplot showing the distributions of relative reorganization
energies $\Delta\lambda$ of $34$ sidegroups, see text. They are sorted by
their median $\Delta\lambda_{\text{med,sidegroup}}$ which is marked by a solid
red line. Blue boxes extend from lower to upper quartiles of the data, while
the vertical lines (whiskers) delineate the most extreme points. Gray-dashed
vertical lines serve as separators to distinguish sidegroups that generally
improve (i.e. lower) the reorganization energy
($\Delta\lambda_{\text{med,sidegroup}}<-25$,meV, to the left in the plot) and
sidegroups that generally worsen (i.e. increase) the reorganization energy
($\Delta\lambda_{\text{med,sidegroup}}>25$,meV, to the right in the plot). Due
to space constraints, the labels for the different sidegroups use (=O/S) to
denote branches, similar as is done in canonical smiles strings 66. Bottom:
Identified 12 most favorable sidegroups appearing on the left hand side of the
boxplot.

For each molecule in a given sidegroup cluster we determine its relative
reorganization energy $\Delta\lambda$ as the difference between the
reorganization energy $\lambda$ of the molecule and the median reorganization
energy $\lambda_{\text{med,scaffold}}$ over the cluster of molecules with this
particular scaffold, i.e. the $\lambda_{\text{med,scaffold}}$ evaluated in the
last section. Thus, a negative value of $\Delta\lambda$ indicates that the
attachment of the sidegroup to the scaffold leads to compounds of lower
reorganization energy as compared to the attachment of other sidegroups to the
same scaffold. Note that due to the diversity of the database it is possible
that certain sidegroups appear multiple times on a single scaffold. In order
not to over-emphasize such cases, we divide each $\Delta\lambda$ by the number
of occurrences. Taking the median of all $\Delta\lambda$ within one sidegroup
cluster, we finally arrive at $\Delta\lambda_{\text{med,sidegroup}}$, which
evaluates how the attachment of the sidegroup generally influences the
reorganization energy over different scaffold types.

For many sidegroups, the obtained $\Delta\lambda_{\text{med,sidegroup}}$
differ significantly from zero. This indicates that these sidegroups indeed
either consistently worsen or consistently improve the reorganization energy,
regardless of the scaffold they are attached to. Mann-Whitney-U testing 83 is
again employed to verify that these results are not due to chance or a biased
sample selection. For $34$ of the $70$ sidegroup clusters this statistical
testing confirms that their relative reorganization energy distributions
differ significantly from the distribution in the entire database ($q<5$ %).
In particular for the -Se-CH3 functional group a very low $q$-value of 1 %
confirms the statistical relevance despite the smaller sample size.

Figure 3 compiles the $\Delta\lambda$ distributions of these $34$ sidegroup
clusters. Groups appearing at the left side of Fig. 3 tend to reliably
decrease $\lambda$, while those on the right tend to increase the
reorganization energy. As apparent, a $\Delta\lambda$ distribution within a
sidegroup cluster differing significantly from the distribution over the
entire database does not necessarily mean that the median
$\Delta\lambda_{\text{med,sidegroup}}$ has to differ markedly from zero. For
the corresponding sidegroups in the middle of Fig. 3, the effect of the
sidegroup seems more subtle, sometimes increasing, sometimes decreasing the
reorganization energy. We speculate that in these cases the (presently
unresolved) specific attachment position or the coexistence of other
sidegroups could play a decisive role 51. More importantly, there are,
however, $12$ sidegroups which on average decrease the reorganization energy
by more than $25$ meV, see Fig. 3. These sidegroups are obviously good
candidates for $\lambda$-tuning approaches and will be further discussed in
Section III below.

###  3.3 Sidegroup Engineering

Our data analysis to this point has identified suitable scaffolds that yield
generally high $|H_{ab}|$ and low $\lambda$ values, as well as suitable
sidegroups that consistently lower $\lambda$. This suggests an appealing
design principle to arrive at new molecular crystals with optimized charge-
transfer properties: Select a favorable scaffold with high $|H_{ab}|$ and
further optimize the reorganization energy by addition of favorable sidegroups
or by exchange of unfavorable with favorable sidegroups. Unfortunately, this
neglects the known sensitive influence of sidegroups on the crystal
structure.100, 43 In contrast to $\lambda$, which is predominantly determined
by the intramolecular structure, $|H_{ab}|$ describes the electronic coupling
between molecules in the crystal and is thus critically dependent on the
crystal structure. While sidegroup tuning could improve the reorganization
energy, it could also change the crystal structure and then potentially worsen
$|H_{ab}|$.

To address this aspect we use matched molecular pair (MMP) analysis 101, 102
to identify those molecules in the database, where the resulting molecule
after addition or exchange of a single sidegroup is still contained in the
database. This way, the crystal structure of both the initial and the modified
molecule is available, enabling us to explicitly analyze the influence of the
sidegroup engineering on the electronic coupling $|H_{ab}|$. Specifically, we
thereby focus on the exchange of an unfavorable sidegroup
($\Delta\lambda_{\text{med,sidegroup}}>0$ meV) with a sidegroup identified as
very favorable above ($\Delta\lambda_{\text{med,sidegroup}}<-25$ meV), cf.
Fig. 3. This generates a testset S1 containing 206 optimized molecules. As a
control series, we also generated a testset S2 containing $124$ molecules,
where a favorable sidegroup ($\Delta\lambda_{\text{med,sidegroup}}<0$ meV) is
replaced with a highly unfavorable one
($\Delta\lambda_{\text{med,sidegroup}}>25$ meV). In analogy, testset S3 with
$161$ molecules (S4 with $91$ molecules) contains molecules where a highly
favorable (unfavorable) sidegroup has been added to the molecular scaffold.
The size discrepancy between the four testsets is due to the different
availability of experimental crystal structures. A full description of the
testsets and the ensuing analysis is provided in the SI.

Figure 4: Charge-transfer parameter tuning by sidegroup exchange for the
series of molecules contained in the “optimized” testset S1 (blue data points)
and the control testset S2 (red data points), which was deliberately made
worse. Shown are changes in a) $\lambda$, b) $|H_{ab}|$ and c) Marcus type
hopping rate $k_{\rm Marcus}$ when going from a reference molecule to a new
molecule (see text).

Fully supporting our sidegroup analysis above, we obtain marked improvements
of the reorganization energy (up to changes of a factor of 5) for the
overwhelming majority of modified crystals in the two “optimized” testsets
(87% of all crystals in S1, 77% in S3), see Fig. 4. Similarly, the “worsened”
control sets S2 and S4 show predominantly a large deterioration of the
$\lambda$ (in 80% of cases in S2, 73% in S4). As expected, the sidegroup
engineering also influences the electronic coupling. However, the changes
there are mostly unsystematic. In the “optimized” testsets S1 and S3 we find a
roughly equal number of cases where $|H_{ab}|$ is improved to those where the
coupling deteriorates. Interestingly, in the “worsened” testsets S2 and S4, we
still obtain a slight improvement of $|H_{ab}|$ in 58 % and 59 % of all cases,
respectively. This is most likely attributed to selection bias, as the same
correlation of large couplings with regrettably large reorganization energies
is already present in the molecules of “64k-dataset”25. Notwithstanding, in
particular for the relevant larger coupling values, the induced unsystematic
changes are comparatively small. As the reorganization energy furthermore
enters e.g. the Marcus model exponentially, this suggests that these changes
in $|H_{ab}|$ should not break the overall trend. Indeed, when we compute the
Marcus transfer rates $k_{\rm Marcus}$ on the basis of the obtained couplings
and reorganization energies, 18 we still find increased rates in 71% (61%) of
the molecular crystals in the “optimized” testset S1 (S3), and decreased rates
in 67% (59%) of the molecular crystals in the “worsened” control testsets S2
(S4). This suggests that sidegroup engineering is (at least to a certain
extent) robust with respect to induced changes in the crystal structure and is
therefore a viable approach to optimize the charge-transport properties of
molecular crystals.

##  4 Discussion

In this work we explored the possibility of optimizing organic crystals for
charge transport using a data-driven approach. Mining a large database of two
major charge-transport descriptors, the reorganization energy $\lambda$ and
the electronic coupling $|H_{ab}|$, for close to 7.000 mono-molecular
crystals, we transferred the concept of Bemis-Murcko scaffolds from
pharmaceutical research to cluster the data according to molecular scaffolds
and the attached sidegroups. For both types of clusters we obtain
statistically confirmed structure-property relationships, i.e. we identify
scaffolds or sidegroups that are generally favorable for charge transport.

A number of promising scaffolds with generally low $\lambda$ and high
$|H_{ab}|$ regardless of their functionalization are identified in the lower
right quadrant of Fig. 2. Among them are scaffolds contained in well-known
OSCs: anthracene96 (Fig. 2 I), pyrene103 (Fig. 2 II), or carbazole21 (Fig. 2
III). Based on our strict statistical significance criterion, all three
exhibit consistently improved charge-transport parameters compared to the
database background. The only exception is $|H_{ab}|_{\text{med,scaffold}}$ of
the pyrene scaffold II, which shows only little statistical significance due
to the limited occurrence of this scaffold in our database.

Next to these more common aromatic systems, there are also a few promising
linked ring systems like scaffolds IV and V shown in Fig. 2, which one would
not immediately associate with the OSC context. They do, however, exhibit much
higher median electronic couplings and comparable reorganization energies as
the anthracene reference cluster. Multiple other scaffolds are particularly
appealing with regard to only one of the two charge-transport descriptors. In
particular, scaffolds like Fig. 2 VI exhibit extraordinarily high electronic
couplings suitable for organic electronics, but feature on average unsuitably
high reorganization energies. These scaffolds are naturally of specific
interest for a targeted optimization through functionalization with favorable
sidegroups.

Our data-driven analysis of sidegroups paired with tests of statistical
significance allows for a broader perspective on existing empirical strategies
of sidegroup tuning. Among the $34$ identified sidegroups that significantly
alter the reorganization energies of their respective scaffolds are sidegroups
like -OH, -O-CH3, -NH2, -F, -Cl, -CF3, or -C#N that had been considered in
preceding comparative studies 50, 51, 49, 48, 52, 41, 53, 54, 55, 104. In
parts, these earlier studies found trends that match the general structure-
property relationships established in this work, for example for -C#N, -F,
-CF3, and -NH2 substitutions on thiophenes, furans, and pyrroles.41.
Interestingly, with the exception of the cyano group, these often studied
sidegroups not always seem to have a favorable effect on reorganization
energies, with some (-CF3, and -NH2) even showing—on average—the opposite
trend. The here confirmed favorable influence of the cyano group has been
explained in the literature50 by the introduction of a local nonbonding
character in the frontier orbitals. In contrast, the common and much studied
class of alkyl sidegroups is, with the exception of -CH(CH3)2, not represented
among the $34$ sidegroups with significant influence. On the basis of our
present database we can thus not confirm any systematic improvement or
deterioration of the reorganization induced by this class of sidegroups.
Instead and most intriguingly, many of the here identified $12$ sidegroups
that lead to a marked improvement of $\lambda$ have to the best of our
knowledge only rarely been discussed in the OSC context. This holds for -Se-
CH3, -N(CH2-CH3)2, -SH,54 -I, -CH2-NO2, -S-CH_3104, -C#CH,54, -CH=O,54 or
-N(CH3)2 55, 104, where we added the few references to works addressing these
sidegroups we could find at all. A detailed analysis of the reason for the
consistent $\lambda$ reduction induced by these sidegroups is presented in the
SI. It confirms that each of these sidegroups yields first of all an expected
additional contribution to the reorganization energy.51 However, this
contribution is more than compensated by a reduction of the scaffold
reorganization energy. Especially the best performing -Se-CH3 sidegroup
achieves this by efficiently delocalizing the charge throughout the scaffold
and the sidegroup.

The clear structure-property relationships for both scaffolds and sidegroups
suggest the possibility of a targeted molecular design by combining favorable
scaffolds with reorganization energy-lowering sidegroups. This effect is
expected to lead to improved charge transport rates, with the reorganization
energy entering e.g. a Marcus-like38 model exponentially, while $|H_{ab}|$
only enters quadratically.18 Our data analysis indicates that such a design
strategy is robust (at least to a certain extent) to a change in the crystal
structure that might be induced by such functionalization. Accordingly, we
would presently predict the addition for instance of most favorable -Se-CH3 or
-N(CH2-CH3)2 sidegroups to any of the favorable scaffolds highlighted in Fig.
2 to yield highly promising OSC candidates for experimental synthesis. Much of
the uncertainty in these predictions could be removed by explicitly validating
that the electronic couplings remain indeed favorable in the crystal structure
adopted by the corresponding compound. Unfortunately, computational crystal
structure prediction is still a highly challenging task.105, 106 In fact, the
limitations in rapidly and reliably determining the crystal structure for a
larger number of promising candidates represent in our view currently the
major bottleneck to a large-scale data-driven in silico discovery of improved
OSCs. This prevents a targeted extension of the database through additional
first-principles calculations, and correspondingly restricts the data mining
to compounds for which experimental crystal structures are available in the
Cambridge Structural Database 65. The concomitant relative scarcity of data
also prohibits refined data clustering that notably would distinguish
different anchor points of sidegroups to a scaffold or the coexistence of
multiple sidegroups. Efficient crystal structure prediction and the concerted
buildup of experimental and computational structural databases are therefore
key to unleash the full potential of modern data mining or machine learning
approaches for organic semiconductors, and thereby navigate the vast and
largely unexplored design spaces of this intriguing class of molecular
materials.

Additional detailed information on the BM scaffold dataset: Distributions of
$|H_{ab}|$, $\lambda$ over the dataset, as well as a distribution of the
number of molecules in the scaffolds and a list of all scaffolds. In addition,
an extended boxplot for all sidegroup clusters as well as a depiction of all
identified sidegroups is provided. A series of -H, -S-CH3 and -Se-CH3
exchanges on different molecules further illustrates the favorable impact of
the latter two groups on gas-phase reorganization energies, while a detailed
discussion of the underlying factors is provided for one example.

We want to thank Lynne Stecher for very helpful discussions on the statistical
methods. We further acknowledge support from the Solar Technologies Go Hybrid
initiative of the State of Bavaria and the Leibniz Supercomputing Centre for
high-performance computing time at the SuperMUC facility.

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11institutetext:  Department of Theoretical Physics, Faculty of Science,
University of Mazandaran,
P. O. Box 47416-95447, Babolsar, IRAN
ICRANet-Mazandaran, University of Mazandaran,
P. O. Box 47416-95447, Babolsar, IRAN

# Observational Viability of the Intermediate DBI Inflation in the Presence of
a Minimal Length

N<EMAIL_ADDRESS>(Corresponding Author) 1122 M.
<EMAIL_ADDRESS>1122 K<EMAIL_ADDRESS>11221122

###### Abstract

We consider an intermediate Dirac-Born-Infeld (DBI) inflationary model in the
presence of a minimal measurable length in the theory. We show that, the
presence of a minimal measurable length modifies the definitions of the scalar
and tensor spectral indices and also other inflation observables. This is due
to modification of the momentum and corresponding wave number of the
perturbations in the presence of a minimal length. By using the deformed
definition of the scalar and tensor spectral indices, we perform numerical
analysis on the intermediate DBI inflation model to find some constraints on
the deformation parameter. In this regard, we compare our numerical results
with both Planck2018 TT, TE, EE +lowE +lensing +BAO+ BK14 and Planck2018 TT,
TE,EE +lowE+lensing+BK14 +BAO+LIGO $\&$ Virgo2016 data at the $68\%$ CL and
$95\%$ CL. Our numerical study shows that the intermediate DBI inflation model
in the presence of a minimal measurable length is observationally viable if
the upper bound on the deformation parameter to be considered of the order of
$10^{48}$ at $68\%$ CL and $10^{49}$ at $95\%$ CL. This is consistent with the
results of other approaches to constrain such a quantity.
Key Words: DBI Inflation, Observational Viability, Natural Cutoffs

##  1 Introduction

Given that the standard model of cosmology suffers from some problems,
cosmologists have tried to solve these problems by introducing the inflation
paradigm. The inflation, in its simplest form, is caused by a so-called
“inflaton” field which is a canonical single scalar field. By slow-rolling of
this field on its almost flat potential, the early universe experienced an
exponential phase of expansion. This inflationary expansion, which is supposed
to have been occurred in the very early universe, can address some of the
problems of the standard model. Although the simple single field model
predicts scale-invariant, adiabatic and Gaussian dominant mode of the
primordial perturbations, there are many inflation models which predict the
dominant modes of the perturbations to be non-Gaussian distributed [1, 2, 3,
4, 5, 6, 7, 8, 9]. One of these models is the DBI (Dirac-Born-Infeld)
inflation [10, 11, 12, 13, 14, 15, 16, 17, 18]. A DBI field in the string
theory is defined as the radial position of a D3 brane. Cosmologist have shown
that in the inflationary models based on the DBI field, the non-Gaussianity of
distribution of perturbations is possible.

On the other hand, since in the early universe we deal with high energy
regime, incorporating the quantum effects seems logical and in fact
inevitable. In this regard, authors of Refs. [19, 20, 21, 22] have studied the
inflationary phase of the universe in the presence of quantum effects.
However, in this work, we include the quantum effects in a different way. In
fact, we consider the inflationary phase in the presence of a natural cutoff.
In this respect, natural cutoffs are constraints that have appeared by the
merging of the two respected theories; quantum mechanics and general
relativity, as a unified theory. Therefore, considering natural cutoffs as a
necessity in the very nature of spacetime leads to fundamental modifications
in the structure of the standard quantum mechanics. It has been designated
that gravity in very small length scales makes a consequential modification in
the fabric of spacetime. These modifications induce minimal uncertainty in the
positions of the particles in atomic and subatomic levels [23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34]. Actually, there exists minimal uncertainty in the
measurement of the position of a test particle in a quantum mechanical system.
The essence of this intrinsic cutoff demands deformity of the standard
Heisenberg Uncertainty Principle to the Generalized Uncertainty Principle
(GUP) (see, for more more details [35, 36, 37, 38, 39, 40]).

In this paper we investigate the effects of the existence of a minimal length
as an inherent natural cutoff in the fabric of spacetime, on a specific
inflation model within a distinctive approach. Given that, in the presence of
the minimal length, the momentum is modified as $p^{i}\rightarrow
p^{i}(1+\alpha p^{2})$, the corresponding wave number also is modified as
$k^{i}\rightarrow k^{i}(1+\alpha k^{2})$. This modification in wave number
shows its effects in the definition of the scalar and tensor spectral indices
as well. In this way, it seems that the presence of the minimal length can
affect the observational viability of the inflation models or at least changes
the viable ranges of the model’s parameter space. To check the observational
viability of our model, we use Planck2018 data sets. For instance, the
Planck2018 TT, TE, EE+lowE+lensing+BAO+BK14 data, based on the $\Lambda
CDM+r+\frac{dn_{s}}{d\ln k}$ model, implies a constraint on the value of the
scalar spectral index as $n_{s}=0.9658\pm 0.0038$ [41, 42]. The constraint on
the tensor-to-scalar ratio, obtained from the same data set is $r<0.072$ [41,
42]. There is also a constraint on the tensor spectral index, obtained from
Planck2018 TT, TE, EE +lowE+lensing+BK14+BAO+LIGO and Virgo2016 data as
$-0.62<n_{T}<0.53$ [41, 42]. By comparing the numerical results of our
analysis in this model with the mentioned observational data, we find some
constraints on the model parameter’s space, specially the deformation
parameter.

In this regard, this paper is organized as follows. In section 2, we review
the machinery of the DBI inflation briefly. In this section, we present the
background equations, slow-roll parameters, and some important perturbation
parameters. In section 3, we reconstruct the model and obtain the slow-roll
parameters in terms of the Hubble parameter. In section 4, the effects of the
presence of a minimal length are studied. In section 5, we consider an
intermediate DBI model and study the model in the presence of a minimal
measurable length. In this section, we also perform numerical analysis on the
model and compare the result with both Planck2018 TT, TE, EE +lowE +lensing
+BAO+ BK14 data and Planck2018 TT, TE, EE +lowE+lensing+BK14 +BAO+LIGO $\&$
Virgo2016 data at $68\%$ CL and $95\%$ CL. In section 6, we present a summary
of our work.

##  2 DBI Inflation

The action for the DBI model is as follows

$S=\int
d^{4}x\sqrt{-g}\Bigg{[}\frac{1}{2\kappa^{2}}R-f^{-1}(\phi)\,\sqrt{1-2f(\phi)X}-V(\phi)\Bigg{]},$
(1)

where, $R$ is the Ricci scalar,
$X=-\frac{1}{2}\partial_{\nu}\phi\,\partial^{\nu}\phi$, and the potential of
the field is denoted by $V(\phi)$. Also, $f^{-1}(\phi)$ is the inverse of the
brane tension. This parameter in the DBI model is related to the geometry of
the throat of the compact manifold which is passed by the D3-brane.

When we vary the action (1) with respect to the metric, we obtain Einstein’s
field equations as

$\displaystyle
G_{\mu\nu}=\kappa^{2}\Bigg{[}-g_{\mu\nu}f^{-1}\sqrt{1+f\,g^{\mu\nu}\,\partial_{\mu}\phi\,\partial_{\nu}\phi}-g_{\mu\nu}V$
$\displaystyle+\partial_{\mu}\phi\,\partial_{\nu}\phi\Big{(}1+f\,g^{\mu\nu}\,\partial_{\mu}\phi\,\partial_{\nu}\phi\Big{)}^{-\frac{1}{2}}\Bigg{]}\,.$
(2)

In a spatially flat FRW background, characterized by the following metric

$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}\,,$ (3)

Einstein’s field equations give the following Friedmann equations

$\displaystyle
3H^{2}=\kappa^{2}\Bigg{[}\frac{f^{-1}}{\sqrt{1-f\dot{\phi}^{2}}}+V\Bigg{]}\,,$
(4)

$\displaystyle
2\dot{H}+3H^{2}=\kappa^{2}\Bigg{[}f^{-1}\,\sqrt{1-f\dot{\phi}^{2}}+V\Bigg{]}\,.$
(5)

The equation of motion of the scalar field in the DBI model is given by

$\frac{\ddot{\phi}}{(1-f\dot{\phi}^{2})^{\frac{3}{2}}}+\frac{3H\dot{\phi}}{(1-f\dot{\phi}^{2})^{\frac{1}{2}}}+V^{\prime}=-\frac{f^{\prime}}{f^{2}}\Bigg{[}\frac{3f\dot{\phi}^{2}-2}{2(1-f\dot{\phi}^{2})^{\frac{1}{2}}}\Bigg{]}\,,$
(6)

which is obtained by variation of the action (1) with respect to $\phi$.

To study the inflationary phase derived by the slow-rolling of the field on
its potential, the following slow-roll parameters are defined

$\epsilon\equiv-\frac{\dot{H}}{H^{2}}\,,\quad\eta=\frac{1}{H}\frac{d\ln\epsilon}{dt}\,,\quad
s=\frac{1}{H}\frac{d\ln c_{s}}{dt}\,,$ (7)

where $c_{s}$ is the sound speed of the perturbations and is given by

$\displaystyle c_{s}^{2}=1-f\dot{\phi}^{2}\,.$ (8)

The inflationary expansion is obtained by considering $f\dot{\phi}^{2}\ll 1$
and $\ddot{\phi}\ll 3H\dot{\phi}$ (corresponding to $|\epsilon,\,\eta,\,s|\ll
1$). Now, the slow-roll parameters in the DBI model take the following forms

$\epsilon\simeq{\frac{{f}^{2}{{\it
V^{\prime}}}^{2}}{2{\kappa}^{2}\left(Vf+1\right)^{2}}}-{\frac{{\it
V^{\prime}}\,{\it
f^{\prime}}}{{\kappa}^{2}\left(Vf+1\right)^{2}}}+{\frac{{{\it
f^{\prime}}}^{2}}{2{f}^{2}{\kappa}^{2}\left(Vf+1\right)^{2}}}\,,$ (9)
$\eta\simeq-{\kappa}^{-2}\left(\frac{\left(2\,{\it
V^{\prime\prime}}-2\,{\frac{{\it
f^{\prime\prime}}}{{f}^{2}}}\right)}{\left(V+{f}^{-1}\right)}-\frac{\left({\it
V^{\prime}}-{\frac{{\it
f^{\prime}}}{{f}^{2}}}\right)^{2}}{\left(V+{f}^{-1}\right)}\right)\,.$ (10)

and

$s\simeq-\frac{f^{\prime}\left(\frac{f^{\prime}}{f^{2}}-V^{\prime}\right)^{2}}{18\left(\frac{\kappa^{2}}{3}(f^{-1}+V)\right)^{\frac{3}{2}}}$
(11)

The e-folds number during inflation, which is defined as

$N=\int_{t_{hc}}^{t_{end}}H\,dt\,,$ (12)

is another important parameter in inflation model. Note that the subscript
$``hc"$ shows the time of the _horizon crossing_ of the physical scales and
$``end"$ refers to the _end of inflation_.

The important issue in studying an inflation model is its observational
viability. The Planck collaboration, in order to compare the inflationary
parameters with observational data, has expanded the scalar and tensor power
spectra in a model-independent forms as follows [42, 43]

${\cal{A}}_{s}(k)=A_{s}\left(\frac{k}{k_{*}}\right)^{n_{s}-1+\frac{1}{2}\frac{dn_{s}}{d\ln
k}\ln\big{(}\frac{k}{k_{*}}\big{)}+\frac{1}{6}\frac{d^{2}n_{s}}{d\ln
k^{2}}\ln\big{(}\frac{k}{k_{*}}\big{)}^{2}+...}\,,$ (13)
$\displaystyle{\cal{A}}_{T}(k)=A_{T}\left(\frac{k}{k_{*}}\right)^{n_{T}+\frac{1}{2}\frac{dn_{T}}{d\ln
k}\ln\big{(}\frac{k}{k_{*}}\big{)}+...}\,.$ (14)

In the above equations, the amplitude of the scalar and also tensor
perturbations have been characterized by $A_{s}$ and $A_{T}$, respectively.
Also, $\frac{dn_{s}}{d\ln k}$ and $\frac{dn_{T}}{d\ln k}$ are respectively the
running of the scalar and tensor spectral index. The term
$\frac{d^{2}n_{s}}{d\ln k^{2}}$ is the running of the running of the scalar
spectral index. Another interesting and important parameter is the tensor-to-
scalar ratio which in terms of the power spectra is given by

$\displaystyle
r=\frac{{\cal{A}}_{T}(k_{*})}{{\cal{A}}_{s}(k_{*})}=16\epsilon\,.$ (15)

In this setup one can defined the scalar and tensor spectral indices as

$n_{s}=1+\frac{d\ln{\cal{A}}_{s}}{d\ln
k}\Bigg{|}_{c_{s}k=aH}=1+(-2\epsilon-\eta-s)\,,$ (16)
$n_{T}=\frac{d\ln{\cal{A}}_{T}}{d\ln k}\Bigg{|}_{c_{s}k=aH}=-2\epsilon\,,$
(17)

respectively. These are important parameters in studying the inflation models
and checking their observational viability. To find an efficient way to use
these parameters in inflation analysis, one can express these quantities in
terms of the e-folds number via the Hubble parameter and its derivatives as
follows.

##  3 Reconstruction of the Model with e-folds Number

In this section, we reconstruct the main equations of our model in terms of
the Hubble parameter and its derivatives, following Refs. [44, 45]. From now
on, we assume $f^{-1}(\phi)=V(\phi)$. By introducing a new scalar field
$\varphi$, identified by the e-folds number $N$ and characterizing the DBI
field as $\phi=\phi(\varphi)$, we can find the potential $V(\varphi)$ in terms
of the Hubble parameter and its derivatives as

$\displaystyle V=\frac{3}{2}\,{\frac{\left(2\,H{\it
H^{\prime}}\,{\kappa}^{2}+3\,H^{2}\right)H^{2}}{{\kappa}^{2}\left(H{\it
H^{\prime}}\,{\kappa}^{2}+3\,H^{2}\right)}}\,,$ (18)

where, $H=H(N)$. By this potential, we can obtain the slow-roll parameters as
well. In this regard, the slow-roll parameters are given by

$\epsilon={\frac{3\,\left(4\,{{\it H^{\prime}}}^{3}{\kappa}^{4}+3\,H^{2}{\it
H^{\prime\prime}}\,{\kappa}^{2}+15\,H{{\it
H^{\prime}}}^{2}{\kappa}^{2}+18\,H^{2}{\it
H^{\prime}}\right)^{2}}{-32\,\left({\it
H^{\prime}}\,{\kappa}^{2}+3\,H\right)^{2}\left({\it
H^{\prime}}\,{\kappa}^{2}+\frac{3}{2}\,H\right)^{3}{\it
H^{\prime}}\,{\kappa}^{2}}},$ (19)

$\displaystyle\eta=\Bigg{[}\big{(}12\kappa^{6}H^{2}{H^{\prime}}^{3}+54\kappa^{4}H^{3}{H^{\prime}}^{2}+54\kappa^{2}H^{4}{H^{\prime}}\big{)}{H^{\prime\prime\prime}}$
$\displaystyle-\big{(}36\kappa^{6}H^{2}{H^{\prime}}^{2}+99\kappa^{4}H^{3}{H^{\prime}}+27\kappa^{2}H^{4}\big{)}{H^{\prime\prime}}^{2}$
$\displaystyle+\big{(}48\kappa^{6}H{H^{\prime}}^{4}+126\kappa^{4}H^{2}{H^{\prime}}^{3}+108\kappa^{2}H^{3}{H^{\prime}}^{2}$
$\displaystyle+162H^{4}{H^{\prime}}\big{)}{H^{\prime\prime}}-24{H^{\prime}}^{6}\kappa^{6}-81{H^{\prime}}^{5}H\kappa^{4}$
$\displaystyle-135{H^{\prime}}^{4}H^{2}\kappa^{2}-162{H^{\prime}}^{3}H^{3}\Bigg{]}\Bigg{[}\big{(}4{H^{\prime}}^{3}\kappa^{4}+3H^{2}{H^{\prime\prime}}\kappa^{2}$
$\displaystyle+15H{H^{\prime}}^{2}\kappa^{2}+18H^{2}{H^{\prime}}\big{)}\left({H^{\prime}}\,\kappa^{2}+3H\right)$
$\displaystyle\left(2{H^{\prime}}\,\kappa^{2}+3H\right){H^{\prime}}\Bigg{]}^{-1},$
(20)

and

$s=\frac{2\kappa^{2}\left({H^{\prime\prime}}H-{H^{\prime}}^{2}\right)}{H\left(2{H^{\prime}}\,\kappa^{2}+3H\right)}\,.$
(21)

By having the slow-roll parameters in terms of the Hubble parameter and its
derivatives, it is possible to study the model in the intermediate inflation
case. Before we do that, we formulate the effects of the presence of the
minimal length in a general inflation model.

##  4 The Effect of the Minimal Length on the Scalar Spectral Index

It is believed that quantum gravity implies a minimal measurable length that
modifies the uncertainty principle as [46]

$\Delta X_{i}\gtrsim\frac{\hbar}{\Delta
P_{i}}\left[1+\frac{\alpha\,l_{Pl}^{2}}{\hbar^{2}}\left(\Delta
P_{i}\right)^{2}\right]\,,$ (22)

where $\alpha$ is the deformation parameter of quantum mechanics and
corresponds to the ultraviolet effect of the theory. In fact, by considering
this quantum gravitational effect, the canonical commutation relation is
modified as follows

$[X_{i},P_{i}]=i\hbar(\delta_{ij}+\alpha_{ijkl}\,p^{k}\,p^{l}+...)\,.$ (23)

In the presence of a minimal measurable length, we define position and
momentum operators in phase space as follows

$\displaystyle X^{i}=x^{i}\,,\quad P^{i}=p^{i}(1+\alpha p^{2})\,,$ (24)

where $x$ and $p$ are Heisenberg’s algebraic position and momentum operators
in standard quantum mechanics. Given the relationship between wave number and
momentum ($p=\hbar k$), we have the corrected wave number as follows [47]

$\displaystyle K^{i}=k^{i}(1+\alpha k^{2})\,,$ (25)

where here and in what follows we have set $\hbar=1$. Now, due to the
correction of the wave number, the definition of the scalar spectral index is
modified as follows

$\displaystyle n_{s}=1+\frac{d\ln{\cal{A}}_{s}}{d\ln k(1+\alpha
k^{2})}\Bigg{|}_{c_{s}k=aH}$ $\displaystyle\cong 1+(1+\alpha
k^{2})\frac{d\ln{\cal{A}}_{s}}{d\ln k}\Bigg{|}_{c_{s}k=aH}$
$\displaystyle=1+(1+\alpha k^{2})(-2\epsilon-\eta-s)\,.\hskip 14.22636pt$ (26)

Also, the definition of the tensor spectral index is modified as

$\displaystyle n_{T}=\frac{d\ln{\cal{A}}_{T}}{d\ln k(1+\alpha
k^{2})}\Bigg{|}_{c_{s}k=aH}$ $\displaystyle\cong(1+\alpha
k^{2})\frac{d\ln{\cal{A}}_{T}}{d\ln k}\Bigg{|}_{c_{s}k=aH}=-(1+\alpha
k^{2})(2\epsilon)\,.$ (27)

So, if we believe in the presence of the natural cutoffs in early universe,
these cutoffs show themselves in the primordial perturbation parameters
explicitly. This can affect on the formulation and numerical results of the
inflation models. In this regard, we perform numerical analysis on the DBI
model in the presence of the quantum gravitational effect encoded as a minimal
measurable length of the order of the Planck length. For this purpose, in the
what follows, we consider an intermediate scale factor as
$a=a_{0}\exp\left(b\,t^{\beta}\right)$ and find the potential and the DBI
tension in terms of the intermediate parameter $\beta$. In this way, we get
the scalar spectral index and tensor-to-scalar ratio in terms of the model’s
parameters and analyze the model parameter space numerically.

##  5 Intermediate DBI Inflation in the Presence of a Minimal Measurable
Length and Its Observational Viability

One of the interesting scenarios in inflation is the Intermediate Inflation.
In the intermediate inflation, the scale factor evolves with time as

$\displaystyle a=a_{0}\,\exp(b\,t^{\beta})\,,$ (28)

where the constant parameter $\beta$ is in the range $0<\beta<1$. Note that in
intermediate inflation, the universe expands faster than power-law inflation
and slower than the exponential inflation. By this intermediate scale factor,
we get the following expression for the Hubble parameter

$\displaystyle H(N)=N\left(\frac{N}{b}\right)^{-\frac{1}{\beta}}\,\beta\,.$
(29)

By this definition of the Hubble parameter, we can find the slow-roll
parameters in terms of the intermediate parameter $\beta$ as follows

$\epsilon={\frac{243(1-\beta)\left(\frac{2{\kappa}^{4}\left(\beta-1\right)^{2}}{9}+\frac{5N\beta{\kappa}^{2}}{6}\left(\beta-\frac{6}{5}\right)+{N}^{2}{\beta}^{2}\right)^{2}}{{\kappa}^{2}\left(2{\kappa}^{2}\beta+3\,\beta\,N-2{\kappa}^{2}\right)^{3}\left({\kappa}^{2}\beta+3\beta\,N-{\kappa}^{2}\right)^{2}}},$
(30)

$\displaystyle\eta=\Bigg{[}\left(-24{\kappa}^{6}-81N{\kappa}^{4}-135{N}^{2}{\kappa}^{2}-162\,{N}^{3}\right){\beta}^{4}$
$\displaystyle+\left(60\,{\kappa}^{6}+171\,N{\kappa}^{4}+216{N}^{2}{\kappa}^{2}\right){\beta}^{3}+12\,\beta\,{\kappa}^{6}$
$\displaystyle-\left(48{\kappa}^{6}+90N{\kappa}^{4}\right){\beta}^{2}\Bigg{]}\Bigg{[}\Big{(}{\kappa}^{2}\beta+3\,\beta\,N-{\kappa}^{2}\Big{)}$
$\displaystyle\Big{(}2{\kappa}^{2}\beta+3\beta
N-2{\kappa}^{2}\Big{)}\Big{(}4{\beta}^{2}{\kappa}^{4}+15N{\beta}^{2}{\kappa}^{2}$
$\displaystyle-8\beta{\kappa}^{4}+18{N}^{2}{\beta}^{2}-18\beta{\kappa}^{2}N+4{\kappa}^{4}\Big{)}\Bigg{]}^{-1}\,,\hskip
11.38092pt$ (31)

$\displaystyle
s=2\,{\frac{{\kappa}^{2}\left(1-\beta\right)}{\left(2\,N{\kappa}^{2}+3\,{N}^{2}\right)\beta-2\,N{\kappa}^{2}}}\,.$
(32)

In this way, we get the scalar spectral index and tensor-to-scalar ratio
(equations (15) and (4)) in terms of the intermediate parameter $\beta$ and
deformation parameter $\alpha$. Now, we can perform numerical analysis on our
model and by comparing the results with observational data, we obtain some
constraints on the model’s parameter space. As we mentioned in the
introduction, from Planck2018 TT, TE, EE+lowE+lensing +BAO +BK14 data, based
on $\Lambda$CDM$+r+\frac{dn_{s}}{d\ln k}$ model, the constraint on the scalar
spectral index and tensor-to-scalar ratio are as $n_{s}=0.9658\pm 0.0038$ and
$r<0.072$. Also, Planck2018 TT, TE, EE +lowE+lensing+BK14+BAO+LIGO and
Virgo2016 data gives the constraint on the tensor spectral index as
$-0.62<n_{T}<0.53$. By considering these observational constraints, and using
equations (15), (4), and (4), we obtain the ranges of the parameters $\alpha$
and $\beta$ leading to the observationally viable values of the scalar
spectral index, tensor-to-scalar ratio, and the tensor spectral index. The
results are shown in figure 1. Another way to analyze the observational
viability of the model is the study of $r-n_{s}$ and $r-n_{T}$ behaviors. To
this end, we plot $r-n_{s}$ behavior in the background of the Planck2018 TT,
TE, EE +lowE +lensing +BAO+ BK14 data in figure 2. We also plot $r-n_{T}$
behavior in the background of the Planck2018 TT, TE, EE +lowE+lensing+BK14
+BAO+LIGO $\&$ Virgo2016 data in figure 3. Figures 2 and 3 have been plotted
for $0.6<\beta<1$ and $\alpha<10^{50}$. Our numerical analysis shows that the
intermediate DBI model in the presence of a minimal length is consistent with
the both Planck2018 TT, TE, EE +lowE +lensing +BAO+ BK14 data and Planck2018
TT, TE,EE +lowE+lensing+BK14 +BAO+LIGO $\&$ Virgo2016 data if
$0.93\leq\beta<1$.

Figure 1: Ranges of the parameters $\alpha$ and $\beta$ leading to the
observationally viable values of the scalar spectral index (upper-left panel),
tensor-to-scalar ratio (upper-right panel), and the tensor spectral index
(lower panel) in the intermediate DBI model in the presence of a minimal
measurable length.  Figure 2: Tensor-to-scalar ratio versus the scalar
spectral index for the intermediate DBI model in the presence of a minimal
measurable length, in the background of Planck2018 TT, TE, EE
+lowE+lensing+BK14 +BAO data. The arrows show the direction where the
parameters increase. Figure 3: Tensor-to-scalar ratio versus the tensor
spectral index for the intermediate DBI model in the presence of a minimal
measurable length, in the background of Planck2018 TT, TE, EE
+lowE+lensing+BK14 +BAO+LIGO $\&$ Virgo2016 data. The arrows show the
direction where the parameters increase.

To find some specified constraints on the model’s parameters, we have adopted
some sample values of the intermediate parameter as $\beta=0.94,\,0.96$, and
$0.98$. By these adopted values of $\beta$, we have obtained some constraints
on the parameter $\alpha$, in comparison with the Planck2018 TT, TE, EE +lowE
+lensing +BAO+ BK14 data and Planck2018 TT, TE,EE +lowE+lensing+BK14 +BAO+LIGO
$\&$ Virgo2016 data at $68\%$ CL and $95\%$ CL. The results are summarized in
Table 1. Note that, although by analyzing the tensor spectral index we find
the upper bound on the deformation parameter $\alpha$ to be of the order of
$10^{53}$, for some domains within this range the scalar spectral index is not
observationally viable. In this regard, to get the observationally viable
range of $\alpha$, we should consider all three perturbation parameters
(scalar spectral index, tensor spectral index, and tensor-to-scalar ratio). In
this way, from our analysis, we find that the observationally viable upper
bound on the deformation parameter is of the order of $10^{48}$ at $68\%$ CL
and $10^{49}$ at $95\%$ CL. This is consistent with other studies, see for
instance [40].

Table 1:  The ranges of the parameter $\alpha$ in which the tensor-to-scalar ratio and the scalar spectral index of the intermediate DBI model in the presence of a minimal measurable length are consistent with different data sets. | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE | Planck2018 TT,TE,EE+lowE
---|---|---|---|---
| +lensing+BK14+BAO | +lensing+BK14+BAO | lensing+BK14+BAO | lensing+BK14+BAO
|  |  |  +LIGO$\&$Virgo2016 |  LIGO$\&$Virgo2016
$\beta$ |  $68\%$ CL |  $95\%$ CL |  $68\%$ CL |  $95\%$ CL
$0.94$ | not consistent | $0<\alpha<1.15\times 10^{49}$ | $0<\alpha<5.18\times 10^{52}$ | $0<\alpha<1.05\times 10^{53}$
$0.96$ | $0<\alpha<5.04\times 10^{48}$ | $0<\alpha<2.67\times 10^{49}$ | $0<\alpha<7.15\times 10^{52}$ | $0<\alpha<1.80\times 10^{53}$
$0.98$ | $0<\alpha<1.84\times 10^{48}$ | $0<\alpha<4.28\times 10^{49}$ | $0<\alpha<7.85\times 10^{52}$ | $0<\alpha<4.29\times 10^{53}$

##  6 Summary and Conclusion

In this work, we have studied the intermediate DBI inflation in the presence
of a natural cutoff as a minimal measurable length. By reviewing the DBI
model, we have presented the main equations of the model and also some
important inflation observables such as the scalar spectral index, tensor
spectral index and the tensor-to-scalar ratio. After that, we reconstructed
the model with e-folds number and re-formulated the slow-roll parameters in
terms of the Hubble parameter and its derivatives. In this way, we have
prepared ourselves to consider the intermediate scale factor in the DBI
inflation model. Then, by considering that the presence of a minimal
measurable length modifies the momentum operator, we have presented the
modified form of the wave number. This is an important point in our setup. The
reason is that to obtain the scalar and tensor spectral indices, the wave
number plays a crucial role. In this regard, we have obtained these parameters
by using the modified wave number and in this way, the effects of a minimal
measurable length have been included in the perturbation parameters $n_{s}$
and $n_{T}$. Then, we considered an intermediate scale factor and obtained the
slow-roll parameters in terms of the intermediate parameter. In this way, the
perturbation parameters have been expressed in terms of the deformation and
intermediate parameters. Finally, we have performed a numerical analysis on
the model’s parameters and compared the results with Planck 2018 different
joint data sets. In this regard, by using the observational constraints on the
perturbation parameters, implied by Planck2018 TT, TE, EE +lowE +lensing +BAO+
BK14 data and Planck2018 TT, TE,EE +lowE+lensing+BK14 +BAO+LIGO $\&$ Virgo2016
data, we have obtained the ranges of the deformation parameter $\alpha$ and
intermediate parameter $\beta$, leading to observationally viable values of
$r$, $n_{s}$, and $n_{T}$. We have also explored $r-n_{s}$ behavior in the
background of Planck2018 TT, TE, EE +lowE +lensing +BAO+ BK14 data and
$r-n_{T}$ behavior in the background of Planck2018 TT, TE,EE
+lowE+lensing+BK14 +BAO+LIGO $\&$ Virgo2016 data at $68\%$ CL and $95\%$ CL.
By adopting some sample viable values of $\beta$ as $\beta=0.94,\,0.96$, and
$0.98$, we have obtained some constraints on the deformation parameter
$\alpha$. According to our numerical analysis, the intermediate DBI inflation
model in the presence of a minimal measurable length is observationally viable
if the upper bound on the deformation parameter is of the order of $10^{48}$
at $68\%$ CL and $10^{49}$ at $95\%$ CL.

Acknowledgement

We thank the referee for the insightful comments that have improved the
quality of the paper.
This research work has been supported by a research grant from the University
of Mazandaran.

Data availability statement All data generated or analyzed during this study
are included in this article.

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