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Abstract: The ability to tune the gaps of direct bandgap materials has tremendous potential for

applications in the fields of LEDs and solar cells. However, lack of reproducibility of bandgaps

due to quantum confinement observed in experiments on reduced dimensional materials, severely

affects tunability of their bandgaps. In this article, we report broad theoretical investigations of

direct bandgap one-dimensional functionalized isomeric system using their periodic potential profile,

where bandgap tunability is demonstrated simply by modifying the potential profile by changing the

position of the functional group in a periodic supercell.We found that bandgap in one-dimensional

isomeric systems having the same functional group depends upon the width and depth of the deepest

potential well at global minimum and derived correlations are verified for known synthetic as well as

natural polymers (biological and organic), and also for other one-dimensional direct bandgap systems.

This insight would greatly help experimentalists in designing new isomeric systems with different

bandgap values for polymers and one-dimensional inorganic systems for possible applications in

LEDs and solar cells.

Keywords: density functional theory; bandgap; polymers; nanoribbons; one-dimensional systems

1. Introduction

One-dimensional materials are in focus amongst the current research areas for their remarkable

physical properties arising as a consequence of the reduced dimensionality. However, lack of control

over reproducibility of bandgap values in one-dimensional materials 1–3 is one of the challenges for

its electronic applications like LEDs and LASER diodes, as it affects their bandgap tunability. Several

theoretical studies have been reported to tune the bandgap of one-dimensional materials using various

methods like strain, functionalization at the edges, doping, etc. 4–6, however, the methods lack in

control over the bandgap values. Since, bandgap is one of the most important factors while selecting a

material for an electronic application; therefore, different direct bandgap materials have been explored

for LEDs and LASER applications, e.g., Aluminium gallium nitride (AlGaN) for ultraviolet LEDs (below

400 nm), while Aluminium gallium arsenide (AlGaAs) for infrared LEDs (above 760 nm). Therefore,

it would be of great interest if bandgap of a given material can be tuned, and this quest has been

extended to polymers. Several experimental and theoretical studies on bandgap in polymers 7–13

have been reported aimed at their applications in organic LEDs 14–16 and solar cells 9,17–22.

However, different values of bandgap noticed in experiments on isomeric polymers 23–25 and also

in theoretical studies 12,25–27 suggest that bandgaps may be tuned in one-dimensional system, if the

Condens. Matter 2018, 3, 34; doi:10.3390/condmat3040034 www.mdpi.com/journal/condensedmatter

Condens. Matter 2018, 3, 34 2 of 10

underlying physics is understood. The fact that the work so far reported on bandgaps in isomeric

polymers having the same functional group is still inconclusive 25–27, motivated us to investigate and

define the driving elements of different bandgaps seen in the isomeric systems. Once we understand

the mechanism behind this, we may be in a position to tune the bandgaps of such materials as per

our requirements. In this work, investigations are carried out on one-dimensional isomeric polymers

(synthetic and natural), and nanoribbons having the same functional groups. Since polymers are large

chain of monomers, Therefore, they are considered as one-dimensional periodic systems for band

structure calculations 28–32.

2. Results and Discussions

For investigating and defining the driving element of different bandgaps in isomeric

systems, an example of low bandgap synthetic polymer, polydithienyl naphthodithiophenes

(DThNDT) (C20H8S4)n is considered. It exists in two isomeric forms, poly(anti-DThNDT) and

poly(syn-DThNDT) 25 having periodic unit cell of length 14.681 Å, and 13.113 Å, respectively

(see Figure 1a). Cut-off energy of 500 eV is used for band structure calculations. Ground state energy

calculated per atom for poly(anti-DThNDT) and poly(syn-DThNDT) are ?7.034 eV and ?7.003 eV,

respectively, which are in close proximity of being isomers.

Figure 1. (color online) (a) Unit cells for poly(anti-DThNDT) and poly(syn-DThNDT) are represented

in the dotted boxes. Blue, yellow, and white spheres represent carbon, sulfur, and hydrogen atoms,

respectively; (b) Band structure plots corresponding to poly(anti-DThNDTP) and poly(syn-DThNDTP).

Given the one-dimensional nature of polymers, their band structures are plotted from G

to X point (see Figure 1b). Direct bandgaps are observed at G point for both the polymers of

poly(anti-DThNDT) and poly(syn-DThNDT) with bandgaps of 0.518 eV and 1.681 eV, respectively.

Bandgap for poly(anti-DThNDT) is smaller than that of poly(syn-DThNDT), which is in agreement with

the experimental report 25. Band structures for these isomeric polymers are significantly different,

even though they have same chemical formula, and practically the same ground state energy.

The polymers are one-dimensional systems with a repeating unit cell (monomer), Therefore,

their bandgap may be related to their one-dimensional periodic potential profile similar to that of

Kronig–Penney model. Since, potential is a scalar quantity, Therefore, average of potentials in the

periodic direction of isomeric unit cell may be considered for comparative analysis of bandgap values.

Average potential profile for poly(anti-DThNDT) and poly(syn-DThNDT) are plotted in the periodic

direction as shown in Figure 2.

Condens. Matter 2018, 3, 34 3 of 10

Figure 2. Average potential profile corresponding to the unit cell in the periodic direction of poly(anti-

DThNDT) and poly(syn-DThNDT) are denoted in solid and dotted lines, respectively.

The periodic average potential profiles of poly(anti-DThNDT) and poly(syn-DThNDT) are quite

different to each other, and even to ideal rectangular potentials of the Kronig–Penney model 33.

Since a system prefers to stay in its ground state, the deepest potential well at global minimum in

the periodic potential profiles are considered for comparative analysis of bandgap values 34 for

isomeric systems. The global minimum for poly(anti-DThNDT) is located at 9.849 Å, enclosed between

two crests (barriers) located at 9.058 Å and 11.432 Å, while global minimum for poly(syn-DThNDT)

is located at 5.517 Å, enclosed between two crests (barriers) located at 4.466 Å and 6.480 Å. From

Figure 2, it can be seen that shape of the potential wells in the potential profile looks like inverse

Gaussians, consisting of both well and barrier width. Therefore, for simplifying the calculations,

potential well is considered as square well potential of equal width for well and barrier (half of the

distance between crests of the potential well). Since depth of the deepest potential well at global

minimum ‘V0’, and its corresponding width ‘a’ are finite and non-zero, distinct from the KP model

(where a ! 0 and V0 ! ¥ for finite value of V0.a) 33. Hence, Schrödinger equation needs to be solved

for the periodic square well potential of finite width and depth at global minimum to get first order

bandgap, which may be correlated with the bandgap calculated using DFT. The energy eigenvalues

corresponding to Schrödinger wave equation for an electron of mass ‘m’ and energy ‘E’ (where E 0 (2)

and,

¯h2b2

2m

= ?E < 0 (3)

Condens. Matter 2018, 3, 34 4 of 10

For isomeric polymers of polydithienyl naphthodithiophenes (DThNDT) (C20H8S4)n, width

and depth of the deepest potential well at global minimum for poly(anti-DThNDT) are 1.187 Å and

0.212 eV, respectively, while for poly(syn-DThNDT) are 1.007 Å and 0.299 eV, respectively (Figure 2).

Using these values in Equation (1), it is found that poly(syn-DThNDT) has larger bandgap than

that of poly(anti-DThNDT), which is in agreement with bandgap values calculated using DFT, and

experimental reports 25. Therefore, it is concluded that bandgap of one-dimensional isomeric systems

may be correlated with depth and width of the potential well at global minimum in the periodic

average potential profile.

In order to find out how bandgap of one-dimensional isomeric systems may be correlated with

their deepest potential well at global minimum in the periodic direction; a general correlation needs

to be formulated and establish its validity for other isomeric systems. Since isomeric systems usually

would have different dimensions (V0 and a) of the deepest potential well at global minimum, Therefore,

the transcendental Equation (1) needs to be solved for bound states of different ‘V0.a’ varying both

‘V0’ and ‘a’. In fact for bound states (E V2) then 4Eg1 > 4Eg2

Case III (a1 > a2,V1 = V2) then 4Eg2 > 4Eg1

Case IV (a1 > a2,V2 > V1) then 4Eg2 > 4Eg1

Case V (a1 > a2,V1 > V2)

In this case, sign of slopes for bandgap as a function of ‘V0.a’ may change on changing ‘V0’ and ‘a’

w.r.t. a reference point, Therefore, bandgap correlation can be predicted only on solving Equation (1)

for corresponding ‘V0’ and ‘a’.

Condens. Matter 2018, 3, 34 5 of 10

For isomeric polymers of polydithienyl naphthodithiophenes (DThNDT) (C20H8S4)n, the width

and depth of the deepest potential well at global minimum in the periodic potential profile

for poly(anti-DThNDT) are 1.187 Å (say ‘a1’) and 0.212 eV (say V1), respectively, while for

poly(syn-DThNDT) are 1.007 Å (say ‘a2’) and 0.299 eV (say V2), respectively (see Figure 2). Since

a1 > a2 and V2 > V1, Therefore, according to correlations of Case IV, poly(syn-DThNDT) should have

larger bandgap than poly(anti-DThNDT); which agrees with our band structure calculations using

DFT, and other experimental reports 25. The agreement of derived correlation with theoretical and

experimental results establishes its validity.

To verify it further, the investigation is extended to other isomeric synthetic polymer

polydialkylterthiophenes (C36H54S3)n (see Supplementary Material Figure S1) and natural polymers

(biopolymers and organic polymers) (see Supplementary Material Figure S2 and S3), bandgaps are

found to be correlated with the dimension of the potential well at a global minimum as per the derived

correlations. The investigations have been extended to natural polymers for extensive validity of the

correlations, even though they are insulating and of little importance to electronic applications.

On the basis of theoretical analyses, it is established that bandgaps of isomeric systems are

correlated with width and depth of the deepest potential well at global minimum in their periodic

potential profile. From derived correlations, it may be predicted that bandgap of one-dimensional

periodic system may be tuned, if width and depth of the deepest potential well in the periodic potential

profile is altered on changing the position of functional group in the periodic unit cell.

To establish the concept of bandgap tunability in one-dimensional systems, the investigation is

extended to theoretical GNRs (same molecular formula for the unit cells) having the same functional

group in the periodic unit cell but of different arrangements. Zigzag GNRs (ZGNRs) of the same

width functionalized at the edges with oxygen atoms in two typical ways (say Config. I and Config.

II as shown in Figure 4a) are considered for calculations. sp2 and sp3 hybridized carbon atoms are

considered at edges for visible distinction of isomeric change in the periodic unit cell (7.378 Å) of

zigzag GNRs. Typical edge configurations of ZGNRs (Nz = 7) are shown in Figure 4a. Periodic average

potential profiles corresponding to Config. I and Config. II are plotted in Figure 4b. Even though their

average potential profiles look different, their potential profiles superpose on each other on relative

shifting in the periodic direction. The width and depth of potential well at global minimum for Config.

I and Config. II are exactly same 0.614 Å (a1 = a2) and 0.648 eV (V1 = V2). Since a1 = a2 and V1 = V2,

Therefore, according to the derived correlations (Case I), bandgap of both the configurations of ZGNRs

should be equal.

Figure 4. (color online) (a) Unit cells corresponding to two configurations Config. I and Config. II of

7-ZGNRs, where blue and red spheres represent carbon and oxygen atoms, respectively. (b) Average

potential profile of the unit cell for Config. I and Config. II in the periodic direction of 7-ZGNRs are

denoted with solid and dotted curve, respectively. Inset shows the overlap of potential profiles on

relative shifting along the periodic direction.

Condens. Matter 2018, 3, 34 6 of 10

To check the validity of the correlations, band structure calculations for 7-ZGNRs are performed

with cut-off energy of 450 eV. Band structures are plotted from G to X point for Nz = 7 (Figure 5). Direct

bandgap of 0.393 eV is observed at G point for both the configurations. It is found that bandgap values

of ZGNRs for Config. I and Config. II are same, even ground state energy per atom of the unit cell

calculated for both the configurations are same, ?8.762 eV. On further analyses of potential profiles

of ZGNRs, the same correlations are found to be held for other odd ZGNRs (Nz = 3 to 17), which are

verified with the band structure calculations using DFT.

Figure 5. Band structure plots corresponding to Config. I and Config. II for Nz = 7.

Further, we have considered even Nz-ZGNRs (Nz=8). The unit cells corresponding to Config. I

and Config. II are shown in Figure 6a for NZ = 8. The lattice parameters in periodic direction for both

the configurations are equal to 7.378 Å. The ground state energy per atom for Config. I and Config. II

are ?8.818 eV and ?8.817 eV, respectively, which are practically the same.

Average potential profiles in the periodic direction of unit cell corresponding to Config. I and

Config. II are plotted in Figure 6b. The width and depth of potential well at global minimum for Config.

I are 0.614 Å(‘a1’) and 1.091 eV (V1), respectively, while for Config. II are 0.614 Å(‘a2’) and 0.721 eV (V2),

respectively. Since a1 = a2 and V1 > V2, Therefore, from the proposed theory, Config. I should have

higher bandgap than that of Config. II (Case II). From band structure calculations, direct bandgaps

of 0.792 eV and 0.342 eV are observed at G point for Config. I and Config. II, respectively (Figure 7).

Thus, bandgap of Config. I is significantly higher than that of Config. II, which is in agreement with

derived correlation using potential profiles (Figure 6). On further analyses of potential profiles of even

ZGNRs, the same correlations are found to be hold for other even ZGNRs (Nz = 10 to 18), which are

verified with the band structure calculations using DFT. However, potential profiles of even Nz-ZGNRs

for Nz < 6 are found to fall under Case V (see Supplementary Material Figure S4). Thus, it justifies

tunability of a bandgap value in direct bandgap one-dimensional systems.

Condens. Matter 2018, 3, 34 7 of 10

Figure 6. (color online) (a) Unit cells corresponding to two different configurations Config. I and Config.

II of even ZGNRs corresponding to Nz = 8, where blue and red spheres represent carbon and oxygen

atoms, respectively; (b) their corresponding potential profiles.

Figure 7. Band structure plots corresponding to Config. I and Config. II for Nz = 8

3. Computational Details

Band structure calculations are performed using Density Functional Theory (DFT) as implemented

in Vienna ab initio simulation package (VASP) 35. Generalized gradient approximation (GGA) 36 is

used for exchange-correlation of electron-electron interactions as implemented in projected augmented

wave (PAW) formalism 37. Further, a vacuum layer of at least 15 Å is used to avoid interlayer

interactions. The system is relaxed until a force on each atom in the unit cell is less than 0.001 eV.Å?1.

k-mesh of size 25 1 1 is used in Monkhorst–Pack formalism for momentum space sampling.

4. Conclusions

On the basis of theoretical analyses of one-dimensional systems having the same functional group

in the periodic unit cell, but of different arrangements, it is observed that:

Bandgaps of one-dimensional systems are correlated to the depth and width of potential well at

global minimum in the periodic potential profile.

The correlations derived between bandgap and dimension of periodic potential well at global

minimum is verified for known isomeric systems of synthetic as well as natural polymers

Condens. Matter 2018, 3, 34 8 of 10

(biological and organic), and bandgap tunability is also established for one-dimensional

nanoribbons.

Finally, it is concluded that bandgap of one-dimensional system can be tuned by changing the

position of functional group in the periodic unit cell of the same material, which may be used for

designing materials of different bandgap values for LEDs applications and effectively harvesting

energy in solar cells; and insight may be extended to understand the different physical properties of

isomers of biopolymers such as proteins.