Indian Journal of Chcmi stry Vol. 45A, Octobc: r 2006, pp. 2 17 1-2 178

On electronic properties of model Type I conducting triblock polymeric superlattices

Swali Agrawal & A ~ Bakhshi *

Dc:pa rtment o r Chemistry, Un ivcrsity or Delhi, Delhi 110007, India

Email: akbakhshi2000@yahoo.com

Receil'cil / 2 .Ill/V 2006; I'cI'iscil 25 AlIgllst 2006

Theorcti ca l studics on <luasi-onc-dimcnsional modcl supcrlalli ces (Am 13 IlCk) ., and (A mCIl 13 k) ., belonging to T ype I have bccn can-i ed out using simple NFC method in ti ght binding apPl'Oximation. For each system, the trends in electronic propcrti cs havc bccn studi cd as a functi on o f' bl ock sizes m, nand k, compositi on (m:n:k) and the arrangcmcnt of blocks. Increasing the block size is found to dccrease thc band gap whil c changing the arrangcmcnt o f blocks docs not altcr thc properti es as long as the block size o f' components rema ins thc same. The clectl'On ic properti cs are found to depcnd main ly on thc lowest band gap componcn t. Thcse rcsult s have also bcen compared w ith thc correspond ing results for peri od ic diblock copolymers.

Conjugated polymers have been intensively studied experimentally and theoreticalli -J for more than quarter of a century. One of the fundamental challenges in the field of conducting polymers is the molecular designing of novel polymers with tailor-made conduc­tion properties4-(l. The conduction properti es of a polymer are related to its elec tronic properties such as ioni zation potential (lP), electron affinity (EA) and the band gap (Eg). Eg or a polymer is a measure of its ability to show intrinsic conducti v ity, while IP and EA values of a polymer determine its ability to form con­ducting polymers through ox idati ve (p-) and reductive (n-) doping respecti vely. Organic conjugated polymers with relatively small band gap, small ionization potential and large electron affinity are expected to be better conductors of electricity, both intrinsically and extrinsically, To be success ful in the designing, one needs to have a complete understanding of the relati on­ship between the chemical structure of the polymers and its elec tronic properti es which determine its conduction properti es.

One exciting poss ibility in this direc tion is prov ided by the quasi-one-dimensional superlattices (or copolymers) which can have tailor-made pro­perti es depending on the choice of the semiconducting components, their relati ve amounts and their arrange­ment in the polymer chain. It has already been shown that copolymerization (peri odic or random)7-!) consi­derably influences the elec tronic properties which

generall y, though not always, remai n intermediate between those of the constituent homopolymers. Depending upon the band alignment of the constituent pol ymers, polymeri c su perlattices, like the inorganic

I · b I' 'd d' f 1011,[ super attlces, may e ( IVI e Into our types . ; ype I, Type II-staggered, Type II-misaligned and Type III (Fig. I ) . Type 1 applies to such systems where energy gap o f one componen t is contained entirely w ithin the band gap o f another component. Systems in which the top of the valence band of one component li es w ithin the band gap o f the other and the bottom of the conduction band of second li es in the band gap of the first are TyjJe II-staggered, whereas in Type 1[­

mi saligned superlattices, the band match up is such that the conduction band minimum of one component is below the valence band maximum of the second. In Type III polymeric superlattices, one component is semi metalli c while the olher is a normal semi ­conductor.

Theoretical studi es o f the copolymeric superl atli ces with two components have been reported recentlyI 2.1.1 . In thi s paper we are extending our studi es to three component systems, that is, tribl ock (A II1 BIlCk)x type of copolymers. A study on similar diblock copol y­mers has also been carried out.

Methodology The electronic density of the states (DOS ) of the

quasi-one-dimensional copolymer chain can be deter-

2 172 INDIAN J CHEM, SEC A, OCTOBER 2006

__

EC2 Eel

Evl Ev2

rEe2

Eel

Ev2

Ev l

IIEC2

Ev2 Eel

Ev l

Ec2

Eel ~ EVI~-' Ev2 III

Typ e !

Typ e II Staggered

Type II Misaligned

Type III

Fig. I- Va rious Jlossibiliti l:s o f polYllleri c superlat ticl:s for di block copoIYIllt: r.

mined by using a si mple negati ve fac tor counti ng I d l 4 15 b iD ' . . I met 10 . . ase( on ean s negat t ve elgen va ue

theorem l 6, The DOS arc de termi ned fro m the sec ular

determin ant whi ch is tr idiagonal in the case where on ly first neighbor in te racti ons are taken into account (t ight binding approx imati on),

ai-A f31 ~ 0 0

f3l 2 a}-A f3~.1 0 .. . 0 IH (A )1= 0 f321 a -A f3.14 ... 0 =0 .1

0 fJN- I. N

0 0 f3N - I. N a -A N

( I )

The secular determinant can be factorizecl as:

N

IH (A )1= n (/'i -A) ... (2)

Assuming that some other conveni ent factoriza ti on is fo und ,

N

1 H (A )1 = IT E i (A ) .. . (3) i= 1

The number of eigen values of the tri diagonal secul ar determinant , whi ch is small er th an a given

trial energy,).. equals the number of nega ti ve factors €i

()..) obtained by the re lati onship:

... (4)

where i =2,3, ....... N

E I (A) = (XI - A ... (5)

Equati ons 4 and 5 are obtained by transforming the trid iagonal determinant by appl ying successive Gauss ian eliminati on, U i and B i.j are the diagonal and a ll di agonal matri x clements res pect i ve ly, of effec ti ve one electron Hamiltonian IH()..)I and)" is its eigen va lue. The diagonal (u) and off diagona l (B) mat ri x elements of the secular determin an t are determined from the corresponding band structure resul ts of each co mponent constituting the polymer chain. Assumi ng the di spersion of the band (valence or conducti on) to be given by the simple relationship :

E i (k) = Ui + 2 Bi .. j cos (ka) ... (6)

Ui of a component for a given band l S taken to be the middle point (or weighted middle point) of the corresponding band . ~ i . .j is taken to be one-fourth of the band width 'i f the same component is repeated (i.e., co mponent i is foll owed by component i in the

AGRAWAL & I3 AK II SHI : ELECTRON IC PROPERTIES OF TYPE I TRIBLOCK COPOLYMERS 2 173

copolymer chain) . If on the othcr hand, i is foll owed by component j , then the olT-diagonal matri x element ~ i . .j , is assumed to be given by the simple relationship,

... (7)

In the present paper, the study of diblock polymeri c superlattices has been carri ed out and is thcn extendcd to the study of polymers having thrcc components, A , Band C, having block sizes m, n and k respectively . The model copolymers studied are of Type I super­latticcs. Vari ous poss ibiliti es of arrangements of these components were studied and the band structures are shown in Fig. 2.

IIJI~J

(A)x (B)x (C)x (A)x (c)x (B)x

Fig. 2- Various possibiliti cs o r arrange mcnt o r various compo­ncnts in Type I (Am f3 nC~), type copolymer used in present study.

Tablc 1- V;liues of (Xi and I), .j o r the th rcc componen ts (in eV)

Component i Val cncc band Conducti on band

(Xi ~i. i (Xi I)i.i

A -8 .25 0. 125 -6.0 0.25

13 -9.6 0.3 -4.2 0.4

C - 12.0 0.5 -2 .25 0.375

For our model studi es, the va lues of Ui and ~ i.j of three components as used arc given in Table I . In our calculations of the electronic DOS of various co­polymer chains we have consistently used a chai n length of 300 units and a grid size of 0.00 l eV .

Resu lts and Discussion The various polymcric superlatlices (AmBIl)' were

modelled by coup ling m units of A, and n units of B periodically . Similarly , two other copolymers (BmCIl), and (C,,AIlL were modelled by arranging the rcquired units of respective components period icall y. The block sizes m and n wcre varied keeping the m:n ratio constant.

Table 2 contains the calculated electroni c pro­perties IP (ionization potential corresponding to the top of VB), EA (electron affinity corresponding to the bottom of CB) and Eg (band gap) or the peri odic copo lymer chains of type (AmBIlL, (BlllCIl ), and (CmAIl), respecti vely obtained from their con'es­ponding DOS curves in ti ght binding approx imation.

The various polymeric superlattices (AmBIlC,L were modelled by coupling m units of A, n units of B and k units of C periodically. A periodic copolymer (AmBIlCkL has m units of A (Am), n units of B (BIl ) and k units of C (Ck) arranged periodicall y throughout the copolymer chai n. Variou s sequences of di fferent periodic and aperiodic copolymers can be generat ed by means of a computer program. I n our present work, we are studying (AmBIlCd, and (AmCIlBk), type

Tablc 2- Calc lilatcd c lec troni c propcrti cs (i n eV) o r periodic diblock copolymcrs (A",B,J" (B",Cn), and (CmA,J, respecti vely bclongi ng to the class o r Type I

Gl oc k (A ml3,,) , (BmC,,), (CmA,,).

sizcs (Ill .n) It> EA E .. b II' EA En

b IP EA Eg

( I, I ) 8. 127 6.2 10 1.9 17 9.357 4.47 4.887 8. 14X 6. 101 2.047

(2.2) 8.0X7 6.3 11 1.776 9.229 4.673 4.556 X.096 6.276 1.820

(5,5) 8.027 6.441 1.586 9.07 4.903 4. 167 8.028 6.437 1.59 1

( 1,2 ) 8. 17 6. 137 2.033 9.444 4.37 1 5.073 8.075 6.298 1.777

(2,4) 8.093 6.302 1.79 1 9.239 4.664 4.575 8.039 6.4 11 1.628

(2, 1) 8.066 6.348 1.71 8 9. 186 4.72 1 4.465 8. 19 6.056 2. 134

(4.2) 8.036 6.42 1 1.61 5 9.094 4.X66 4.228 8.099 6.274 1.825

(3.2) 8.054 6.384 1.670 9.14 4.80 1 4.339 8.099 6.274 I.X25 (6,4 ) 8.02 1 ().456 1.565 9.052 4.927 4. 125 8.04 6.4 11 1.629

(3, 1) 8.04 1 6.404 1.637 9. 116 4.826 4.290 8. 196 6.052 2. 144

(6.2) 8.02 6.457 1.563 9.051 4.928 4. 123 8.099 6.274 1.825

(2.3) 8.092 6.304 1.788 9.238 4.665 4.573 8.058 6.366 1.692 (4,6) 8.038 6.4 19 1.6 19 9.097 4.864 4.233 8.02 1 6.453 1.568

( 1.3) 8. 18 6.12 2.06 9.463 4.352 5.111 8.046 6.377 1.669 (2,6) 8.093 6.302 1.791 9.239 4.663 4.576 8.02 1 6.453 1.568

2174 INDIAN.I C HEM, SEC A, OCTOllER 2006

of systems where the block sizes 111 , nand k o f A , 13 and C components are varied keeping the rati o m:n:k constant. The study of these copolymers will revea l how the elect ronic structure and conduction properties of these copolymers change by increas ing the block sizes m, nand k at constant ratio.

Tab le 3 con tains the ca lculated elect ronic pro­perties [P (ionization potential corresponding to the top of VB), EA (elec tron affinity corresponding to the bottom of CB) and Eg (band gap) of the periodic copo lymer chains of type (A Ill I3 IlCk)x and (A,"CIl 13 k).x respecti vely obtai ned from thei r correspondi ng DOS curves in ti ght binding approximation.

Elcctronic properties of the copolYlIlcrs (A", BII )" (BII,C,.). .. aud (C",A II ) ... type dib/ock copolYllleric slIper­lattices

In the case of diblock copo lymeri c superlallices, on increas ing the block size of both the components, a decrease in IP values and an increase in EA values is obtained. This leads to a decrease in the Eg (band gap) values and thus to a polymer with better intrinsic cOl1ducti vi ty .

On increas ing the block size of only the larger band gap component and keeping the block size o f smaller band gap component same, it is found that EA values decrease while [P va lues increase resulting in an increase in the band gap (Eg) values. This leads to a polymer with lower intrinsic conductivity, indicating that increasing the con tent of the larger band gap component leads to a copolymer which shows lesser conductivity.

On the other hand, increas ing the block size of lower band gap component lowers the band gap by decreasing [P and increas ing the EA values. Thi s i mpl ies that i ncreas i ng the lower band gap component con tent leads to a more conducting copolymer.

(A", BIICk) ... alld (A",CII/h) ... type triblock copolYllleric super­lattice.l·

[n both the copolymers, i.e., (AIl,BIlCk)x and (A Il,C ll l3k)x it is found that the band gap (Eg) values decrease with the increase in the block size of A, B and C units for a given compos ition of the copolymer. This decrease in Eg is the result of a decrease in [P and increase in EA implying hereby that an increase in block sizes of the components resu ltS in increase in the i nl ri nsic conducti vi ty of the copo lymer. A Iso, decrease in the [P values and increase in the EA values suggests that the copolymers can be doped eas ily both by ox idation as well as reducti on.

On keeping the block sizes m, nand k of the three componen ts A, 13 and C the same in the two types of copolymers, it is found that the va lues of [P and EA and conseCluently that of Eg are same. This suggests that in the ca.;e of these copolymers i t is the block size of the components and not their arrangement which is important. This implies that copo lymers w ith same block si ze m, nand k of the three components wi ll have the same electronic properties irrespective of thei r arrangement.

On keeping th e block size of the small est band gap component (i.e. component A) same, the values of [P and EA in the two types of copolymers remain almost

Tablt: 3-Calculated d.:ctro lli c properties (ill eV) o/" per iod ic (A",I3"C~), alld (A IJlC" B ~\

13 lock (i\IJlI3"C~), (AIJlCnl3~),

sizes ( 1ll .I1.k) II' Ei\ En '"

IP Ei\ Eg

( 1.1 , 1 ) S. ISO 6.093 2.087 8. 180 6.093 2.087

(2.2 ,2) 8.096 0.288 1.808 9.229 4 .673 4 .556

(5.5.5) S.028 0.4 39 I.SS9 8.028 0.439 1.589

( 1.2.3) S.190 0.084 2. 100 8. 189 6.0S4 2. 105

(2.4 .6) S.096 0288 1. 808 8.096 0.288 1. 808

(2.1,3) S.096 6.288 1.808 S.096 6.289 1. 807

(4.2 ,6) S.OY) 0.4 1 S 1.024 fUl39 0.4 1 ) 1.024

(3 .2. 1 ) 8.057 (I.">, 7.l I.( ,X·I X.()5X (d72 I J,X()

(oA.2) 8.02 1 ('. -\:'i -I !.5(,7 ~U )2 1 (,.-15-1 1.5(,7

(3. 1.2) 8.0.')8 (1 .. ,72 1.r,S() X.OS7 037.\ I. (,S-I

(6.2.4) 8.021 0.4.')-1 1.507 802 1 o.4.'i-l 1.)07

(2,3. 1) 8.096 (,. 28<) 1.807 8.()96 0.288 1.808

(4 ,6,2) 8.039 6.415 1. (,24 8.039 0.4 15 1.624

( 1.3,2) 8. 189 6.084 2 . 105 8. 190 6084 2.106

(2,6.4 ) S.096 6.288 1.808 8096 6.288 1.808

AG RAWA L & I3 A KII SHI: ELECTRONIC PROPERTI ES OF TYPE I TRIBLOCK COPOLYM ERS 2175

same. Thi s sugges ts that the overall elec troni c pro­perti es of the copolymers are being governed by the properties of the smalles t band gap component. Thi s result is important as far as des igning of the copoly­mers with small band gap is concerned. Hence additi on of a component with low band gap to a diblock copolymeri c system will res ult in decreasing the band gap of thi s sys tem and will facilitate the formati on of a low band gap copolymer.

These studi es show similariti es between diblock and tribl ock copolymeri c sys tems. The studi es on aperi odi c and Type II staggered copolymers are in progress.

DOS {~l copolYlll ers

Fi gures 3, -1- and 5 show the DOS curves of some selected (A",B"L, (B",C,\), and (C"A,,) , peri odi c copolymer chains res pec ti vely. Fi gures 6 and 7 show the DOS curves of so me se lected (A", B"Ck), and (A",C" Bk)x peri od ic copolymer chains respecti vely_

From Figs 3, 4 and 5 it can be seen that the valence and the condu cti on bands of dibl oc k peri odic copoly­mers are split into peaks whose number is equal to the sum of block sizes values m and n of the two co mponents_ For example, there are two peaks in the va lence and the conducti on bands of (ABL, (BC), and (CAL type of copolymers where the sum of m and n

Vll

10

6 N

L.---11 -1 0

E(eV) -9 -8

6

N4

l ~ -13.5 ·11.5 E(eV) ·9.5 ·7.5

:1 -13.5 ·11.5 E(eV) -9.5 ·7.5

(All),

( llC),

(CA),

valu es is equal to two . Thi s happens becau se for a homopolymer the br ill ouin zone is continuous but in a copolymer it beco mes di sco ntinuous.These discon ti ­nuit ies result in splitting up of the brillouin zone and hence separate peaks are obtained. The va lues of ioni zat ion potential (lr) and elec tron affinity (EA) of these copolymers were obtai ned usi ng the Koop man's theorem, i.e., from negative of the top of valence band and from negati ve of the bottom of conducti on band res pec ti vely.

The valence bands (the hi ghes t fi li ed bands) for the homopolymers (A), and (BL ex ist between -S.5 to -S.O eV and -10.2 to -9 .0 eV respecti ve ly, whil e fo r the peri odi c copolymer (A B), formed by coupling one unit of A with one unit of B, the band structurc fo r valence band ex ists from -9.7 to -S. l eV instead of fro m -10.2 to -S.O eV. Thi s impli es th at there is a sh ift in the band structure of (A)x and (B)x in the co­polymer (AB ), as a res ult of interacti on between A and B components to form the copolymer. Simi larl y, a shift is obtained in the case of conducti on band where for homopolymers (A), and (B)" the conducti on bands (the lowes t unfill ed bands) ex ist between -7.5 to 6.5 e V and -6.6 to -5 .0 e V res pec­ti ve ly whi le fo r the copolymer it ranges from -6.2 to -4.0 eV. A simi lar trend is observed for the ot her peri odi c dibl ock copolymers studi ed as well .

C ll

N 4

.. .) ~ ·5 -4 -3

E(eV)

'II I -5.5 -4.5 ·3.5 -2.5 ·1.5

E(eV)

10

8

6 N

4

·).5 ·5.5 E(eV)

-3.5 ·1.5

Fig. 3- DOS curves of va lcncc band (VB) and conduct ion band (e l3 ) or (A 13 ), . (BC), and (CA)x periodic copolymer chain s.

2176 L'lO IAN J C HEM, SEC A, OCTOBER 2006

VB CB

20 12

15 10 8

N 10 N6

-II -10 E(eV) -9 -8 -7 -6 E(e-V) -4

(A, B).

10

8

N6 N4 4

~ 2 I

0 --,

-1 3.5 -l l }j E(aV) -95 -7.5 -5.5 -3.5 -1.5

(B,C), E(eV)

'::lu_~ 30 25 20

N 15 10

j I 5

-13.5 -1 15 E(eV) -95 -7.5

(C,A).

-7.5 -5.5 E{eVr 3.5 -1.5

Fig_ 4- 00S curves of valence band (VB) and co nducti on band (CI3) o f (A~ I3)" (B~C), and (C~A), period ic copolymer cha ins.

VB C B

60

N:~L~ 50 40

N30 20 10

~ lid

-II -10 E(eVf

-a -7 -6 E(e\)

-4 -3

(A,B).

25 15

20 10

15 NID

N

5 ~ ~ I

0 -13.5 -11 .5 E(eV) -9.5 -7.5 -5.5 -3.5 -1 .5

E(eV) (B,C),

N~:LL 50

40

30 N 20

10 I u,-0

-13.5 -1 15E(eV) -95 -7.5 -7 .5 -5.5 -35 -1.5 E(eV)

(e,A),

Fig_ 5- 00S curves of va lence band (V I3) and conduction band (C I3) o f (A, B), _ (B, C), and (C, A), peri odi c co po lymer chains.

AGRA WAL & I3AKHS HI : ELECTRON IC PROPERTIES OF TYPE I TRII3LOC K COPOLYMERS

VB

25

20

15 N 10

5

0 -13.5 -115 E(~V) -9.5

30 25 20

N15 10 5 0 -13.5 -11 .5 E(eV) -9.5

30 25

20 \

-7.5

-7.5

':: I O~~~~--~L-~~

-13.5 -1 1.5 E(eV) -9.S -7.5

CB

25

20

N 15 10

5

-7.5 -5.5 -3.5 E(eV)

-1.5

(A,B,C,)

30 25 20

H15 10 5

-7 .5 -5.5 E(eV) -3.5 -1.5

(A,B,C,),

30 1 25 20

N15 10

-7.5 -5 .5 E(eVj35 -1.5

Fig. 6- 0 0S curves of so me selec ted (AmB"Ck)x periodic copo lymer chai ns.

25

20

N15

10

VB

5 O ~LLL+~~~~~

-13.5 -1 1.5E(IV) -9 .5 -7 .5

25

20

N~~

CB

5 O+-~~~~~~~-

-7.5 -1 .5 '--_________________ ...J (A, C,B,), '-----------------"

30 25 20

N 15 10

...... ~ ll1Jt -1 15 E(eV) -9 .5 -7.5

5 O ~~~U-~~L-~

-13.5 -11 .5 -9.5 -7.5 E(IV)

30 25 20

N1 5 10 5

-7.5 -5.5 -3.5 -1 .5 E(IV) (A,C,B,), L-_________________ -'

30 25 20

N15 10 5 O ~~ ...... -L~+_~~-

-7.5 -5.5 -3.5 -1.5 E(IV)

'----- ----------------' (A,C,B.), L-________________ ...J

Fig. 7-0 0S curves o f some selec ted (A",C" i3 ,)x period ic copo lymer chain s.

2 177

2178 IND IAN J CI-IE M. SEC A. OCTOI3 ER 2006

Similarl y, 1'01' the periodic tribl ock copolymers (ArnBIICd, and (A IIlCII [3k), (Figs 6 and 7) it is found that v,dence and conduction bands spl i t into peaks equal to the sum of block sizes m, nand k of A, [3 and C components. Also, the bands (valence and conduc­ti on bands) are round to be shifted as compared to the bands or the homopo lymers. The negati ve of the top or valence band corresponds to the IP of the copoly­mer and the negati ve or the bOllom o f the conduction band corresponds to the EA of copolymcr.

The DOS curves or all the periodi c copolymers studi ed arc found to consist of narrow and well separated peaks. Thi s is becau se in the case of periodic copolymers like A [3ABA[3 A [3 . . . , i .e., (A B), or ABCABC. .. , i.e., (A BC)" the environment o f a unit (A and B for diblock copolymers or A , [3 and C 1'0 1' tribl ock copolymers) throughout the chain remains the same.

Conclusions In thi s paper, we have studied systematically the

electronic structures and conduction properti es of model periodic tribl ock copoly mcrs which represent the prototype or quasi-one-dimensional superlallices of Type I in ti ght binding approximati on. Increase in the amount or the thrce components keeping m:n:k constant improves the conducti vity of the copolymers both intrinsica ll y as we ll as ex trin sica l ly . The elec­tronic propcrti es are round to be the same and independent or the sequencc in which thcy are arrangcd ir the block size of the three components

remains the samc. The electronic properties of these copolymers depend on the properties of the sma llest band gap component which control s theil' properti es. The results obtained hence suggest that the study or diblock copolymeric supcrlat ticcs can e effec ti vely ex tendcd to tribl ock copolymers ancl these can be used for des igning copolymers w i th wilor-made pro­perties.

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